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A joint framework of design, control, and applications of energy generation and energy storage systems

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A joint framework of design, control, and applications of energy generation and energy storage systems

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Content A Joint Framework of Design, Control, and Applications of Energy Generation and Energy Storage Systems by Yanzhi Wang ______________________________________________________________ A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2014 Copyright 2014 Yanzhi Wang 2 Dedication To my parents, Shuyi Wang and Wenxing Zhang And to my lovely wife, Xue Lin 3 Acknowledgments First and foremost, I would like to thank my Ph.D. advisor, Prof. Massoud Pedram, for his enthusiasm, mentorship, and enouragement throughout my graduate studies. He opened my mind to a world of opportunities and taught me about research, learning, teaching, career planning, and so much more. His passion for scientific discoveries and his dedication in making technical contributions have been invaluable in shaping my academic and professional career. I still remember the first time when I entered Prof. Pedram's office and knew nothing about digital VLSI or CAD. Over my initial time of feeling frustrated and frightened, Prof. Pedram taught me how to do scientific work and kept reminding me to think of the big picture and always push an idea to its limits. Since I became a senior PhD student, Prof. Pedram gave me the freedom to investigate new topics and collaborate with people inside and outside the SPORT lab. He supported me to attend various conferences and introduced me to other professors, and gave me advice and encouragement about starting a career in academia. Without Prof. Pedram I would not been anywhere near where I am now. Next, I would like to thank other members in my thesis and qualification exam committee: Prof. Sandeep Gupta, Prof. Murali Annavaram, Prof. Viktor Prasanna, and Prof. Ming-deh Huang, for providing valuable feedback to my research and pushing me to think deeper. Special thanks to Prof. Sandeep Gupta, for his tremendous guidance on the joint work on cloud computing and hardware security. I have also tremendously enjoyed his VLSI design course who introduced me to the area of digital design. His dedication to academic excellence, clarity of presentation, and sense of humor were especially impressive to me. Also thanks Prof. Murali Annavaram for the 4 joint discussions and collaboration on the near-threshold project. His willingness to discuss technical issues freely and rigorously is very much appreciated. During my five year research and study journey at USC, I am very fortunate to have the privilege to work with top researchers inside and outside USC, including Prof. Naehyuck Chang, Prof. Shahin Nazarian, Prof. Paul Bogdan, Prof. Melvin Breuer, Prof. Mansour Rahimi, Dr. Cyrus Ashtiani, Prof. Massimo Poncino, and Dr. Louis Kerofsky and Dr. Sachin Deshpande. Special thanks go to Prof. Naehyuck Chang for the five-year-long collaboration with the outcome of 45 papers. I met Prof. Chang on the first day that I joined the SPORT lab. I have been collaborating with him ever since, and plan to continue my collaboration after graduation. His dedication and attentiveness in writing papers, and his engineering insight are impressive and have significantly affected my working style in the first two years. As a result of my collaboration with him, I have been fortunate to take part in real implementations of a number of energy generation and storage systems. I would also like to thank Prof. Shahin Nazarian with whom I took two VLSI design courses, I specially appreciate insightful discussions with him on smart grid design and current source modeling of CMOS devices, as well as his kindness in letting me mentor several directed research students. I would like to thank Prof. Paul Bogdan for his energy, rigor, and enthusiasm in conducting research, and his good sense of mathematics. He let me see what the life of a young academic scholar would be like. Sincere thanks also go to Prof. Melvin Breuer for his great insights in finding good research topics and his guidance in VLSI CAD, to Prof. Mansour Rahimi for helping me start an interesting prtoject of life-cycle assessment of advanced device technologies, to Dr. Cyrus Ashtiani for the insightful discussions about energy storage systems, to Prof. Massimo Poncino for his thoughtfulness in our joint project efforts, and to Dr. Louis 5 Kerofsky and Dr. Sachin Deshpande for their mentorship during my summer internship at Sharp Laboratory of America. During my time at USC, I enjoy myself very much as a member of the SPORT Laboratory. I am proud to say that I have collaborated with ALL of the current SPORT members, including Qing Xie, Xue Lin, Woojoo Lee, Alireza Shafaei, Mohammad Javad Dousti, Tiansong Cui, Di Zhu, Siyu Yue, Shuang Chen, and Majid Ghasemi. I sincerely appreciate the time shared with them during my Ph.D. career. I would also thank the senior members in the SPORT team, (to name a few) Hwisung Jung, Safar Hatami, Ehsan Pakbaznia, Hadi Goudarzi, Inkwon Hwang, for their guidance and instructions when I was a junior student in this group. Besides, I would thank the students in the ELPL lab of Seoul National University, for the five-year-long collaboration and friendship. They include: Younghyun Kim, Donghwa Shin, Sangyoung Park, Jaehyun Park, Jaemin Kim, and Kitae Kim. I enjoy the time when I visited their lab in the summer of 2012. My sincere thanks also go out to my collaborators, colleagues, and friends at CENG and EE, USC. In no particular order, they include (but not limited to) Yue Gao, Da Cheng, Wentao Zhu, Maoqing Yao, Hsunwei Hwiung, Jianwei Zhang, Jizhe Zhang, Ji Li, Lizhong Chen, Ruisheng Wang, Daniel Wong, Mohammad Abdul-Majeed, Yanting Wu, Wenyuan Tang, Ying Zhang, Siddharth Bhargav, and Mahboobeh Ghorbani. I would also thank the directed research students working with me (and Prof. Shahin Nazarian), including Yang Liu, Yang Li, Ji Li, and Chenxiao Guan, and my old collaborators in various course projects, including Sheng Ye, Tarun Bala Nelapatla, Siddhanth Dhodhi, Abhinav Kijan, and Wei-Lun Chao. I also sincerely thank many senior colleages at USC (who graduated several years ago), including Jianwei Chen, Xin Feng, Weirong Jiang, Qingbo Wang, for their help when I am a new student at USC. 6 Last but not least, my deepest gratitude goes to my parents for their unconditional love, support, and devotion throughout my life and especially during my education. Lastly, I would like to thank my dear and lovely wife Xue Lin for always being there through my happiness and toughness, for believing me and encouraging me to pursue my goals, and for being the best friend, listener, and research collaborator in my life. 7 Table of Contents Dedication ...................................................................................................................................... 2 Acknowledgments ......................................................................................................................... 3 List of Figures .............................................................................................................................. 11 List of Tables ............................................................................................................................... 18 Abstract ........................................................................................................................................ 22 Chapter 1. Highlights and Overview ......................................................................................... 24 Chapter 2. Photovoltaic (PV) System Modeling and Architecture ........................................ 28 2.1 PV Cell Modeling and Characterization ........................................................................... 31 2.2 Power Converter (Charger) Power Model ........................................................................ 34 2.3 Maximum Power Point Tracking (MPPT) and Maximum Power Transfer Tracking (MPTT) ..................................................................................................................................... 37 2.4 Partial Shading Effect ....................................................................................................... 38 Chapter 3. PV Reconfiguration Structure and Control to Combat Partial Shading ........... 44 3.1 PV Module Reconfiguration Architecture ........................................................................ 45 3.2 Problem Formulation ........................................................................................................ 48 3.3 PV Module Reconfiguration Control Algorithm .............................................................. 52 3.4 Experimental Results ........................................................................................................ 63 Chapter 4. Dynamic Reconfiguration of PV System in Hybrid Electric Vehicles ................ 76 4.1 PV System Modeling in the HEV Framework ................................................................. 78 4.2 Problem Formulation ........................................................................................................ 79 4.3 The Kernel Algorithm ....................................................................................................... 82 4.4 Experimental Results ........................................................................................................ 90 8 4.5 PV Array Reconfiguration Implementation and Optmization for Partial Solar Powered Vehicles .................................................................................................................................... 93 Chapter 5. Capital Cost-Aware Reconfigurable PV System Design and Optimization ..... 106 5.1 Capital Cost-Aware Reconfigurable PV Panel ............................................................... 107 5.2 Optimization Algorithm .................................................................................................. 113 5.3 Experimental Results ...................................................................................................... 120 Chapter 6. Energy Storage Elements and Energy Storage Systems: Background ............. 124 6.1 Introduction ..................................................................................................................... 124 6.2 Electrical Energy Storage Elements................................................................................ 125 6.3 EES Systems ................................................................................................................... 140 Chapter 7. Hybrid Electrical Energy Storage (HEES) Systems ........................................... 151 7.1 Motivation and Principle................................................................................................. 151 7.2 System Architecture ........................................................................................................ 155 7.3 HEES Prototype Implementation .................................................................................... 163 Chapter 8. Charge Management in HEES Systems .............................................................. 165 8.1 Basic Charge Management Schemes in HEES Systems................................................. 165 8.2 Single-Source and Single-Destination Charge Migration: Architecture, Modeling, and Efficiency Optimization .......................................................................................................... 173 8.3 Multiple-Source and Multiple-Destination Charge Migration: Architecture, Modeling, and Efficiency Optimization ................................................................................................... 196 Chapter 9. Online Fault Detection and Fault Tolerance in EES Systems ........................... 214 9.1 Structural Support ........................................................................................................... 215 9.2 Fault Detection and Fault Tolerance Algorithm ............................................................. 218 9 9.3 Experimental Results ...................................................................................................... 227 Chapter 10. State-of-Health (SoH) Aware Charge Management in HEES Systems Based on a Novel SoH Degradation Model ............................................................................................. 230 10.1 SoH Degradation Model ................................................................................................. 232 10.2 SoH-Aware Charge Management I: System Model and Problem Formulation ............. 241 10.3 SoH-Aware Charge Management II: Algorithm............................................................. 246 10.4 Experimental Results ...................................................................................................... 251 Chapter 11. Joint Control and Optimization of PV Systems and HEES Systems .............. 256 11.1 PV Power System ........................................................................................................... 257 11.2 Proposed Efficiency Enhancement Methods .................................................................. 263 11.3 Experimental Results ...................................................................................................... 276 Chapter 12. Residential/Household Applications of PV and HEES Systems ...................... 279 12.1 Optimal Control of a Grid-Connected HEES System ..................................................... 285 12.2 PV Power Generation and Load Power Consumption Profile Predictions ..................... 297 12.3 Accurate Component Model-Based Control Algorithm for Residential Photovoltaic and Energy Storage Systems Accounting for Prediction Inaccuracies .......................................... 308 Chapter 13. Applications of PV and HEES Systems in Embedded Systems ....................... 330 13.1 Energy Storage in Low-Power Embedded System Applications ................................... 330 13.2 Power Supply and Consumption Co-Optimization of Portable Embedded Systems with Hybrid Power Supply .............................................................................................................. 334 References .................................................................................................................................. 347 Conclusions ................................................................................................................................ 363 Appendix I ................................................................................................................................. 369 10 Appendix II ................................................................................................................................ 370 11 List of Figures Figure 1. Different applications of PV systems. ........................................................................... 28 Figure 2. The PV system architecture based on the string charger architecture. .......................... 30 Figure 3. PV cell (a) V-I and (b) V-P output characteristics under different solar irradiance level . The red dots denote MPPs of a PV cell, where the PV cell achieves the maximum output power under certain solar irradiance level. ................................................................................... 30 Figure 4. The equivalent circuit model (a) and the symbol (b) of a PV cell. ............................... 31 Figure 5. Matching between the measurement results and the simulation model on the V-I characteristics of a single PV cell. ................................................................................................ 34 Figure 6. The architecture of buck-boost switching converter. .................................................... 35 Figure 7. Power conversion efficiency of the Linear Technology LTM4609 buck-boost converter [24]. ............................................................................................................................................... 36 Figure 8. A PV module with one PV cell completely shaded. ........................................... 40 Figure 9. The V-I output characteristics of the partially shaded PV module and its PV groups. . 40 Figure 10. The V-P output characteristics of the partially shaded PV module and the lighted PV module........................................................................................................................................... 41 Figure 11. The reconfigurable PV module structure in [15], which requires additional PV cells (in the "adaptive bank") to perform reconfiguration. .................................................................... 43 Figure 12. The reconfigurable PV module architecture. ............................................................... 46 Figure 13. An example of PV module reconfiguration................................................................. 46 Figure 14. The architecture of the PV system with reconfigurable PV modules. ........................ 48 Figure 15. An example of optimal PV module reconfiguration according to the PV cell MPP current values at their own MPPs. ................................................................................................ 54 12 Figure 16. The near-optimal solution of the PMR problem.......................................................... 55 Figure 17. The prototype PV module with reconfiguration. ......................................................... 64 Figure 18. The computer-controlled programmable switch board. .............................................. 64 Figure 19. Different partial shading patterns. ............................................................................... 66 Figure 20. Measured V-P curves of the partially shaded PV module (from the prototype) before and after reconfiguration. .............................................................................................................. 67 Figure 21. Partial shading pattern of the single PV module in the 1st experiment....................... 69 Figure 22. V-P characteristics of the PV modules with and without reconfiguration technique. . 70 Figure 23. Partial shading pattern of the three PV modules in the 2nd experiment. .................... 73 Figure 24. Partial shading pattern of the three PV modules for maximum output power improvement. ................................................................................................................................ 74 Figure 25. PV cell (a) V-I and (b) V-P output characteristics under different temperatures. The red dots denote MPPs of a PV cell, where the PV cell achieves the maximum output power under certain temperature........................................................................................................................ 77 Figure 26. System diagram of a PV-powered HEV. ..................................................................... 78 Figure 27. Equivalent circuit of a PV system on a HEV. ............................................................. 79 Figure 28. An illustration of the PV array reconfiguration. .......................................................... 83 Figure 29. An illustration of the MCHV function. ....................................................................... 87 Figure 30. (a) Solar irradiance and (b) temperature distributions on the PV array. ..................... 91 Figure 31. The V-P characteristics of the two PV systems........................................................... 92 Figure 32. Zigbee-based solar sensor node to measure the solar irradiance. ................................ 95 Figure 33. SM5 official dimension from the vehicle manual. ...................................................... 99 13 Figure 34. Performance comparison between the proposed reconfigurable PV system with baseline systems. ......................................................................................................................... 101 Figure 35. Performance (average output power) of the proposed reconfigurable PV system with different reconfiguratin period values on two benchmark profiles. ............................................ 102 Figure 36. The capital-aware reconfigurable PV panel structure. .............................................. 108 Figure 37. The switch circuitry. .................................................................................................. 108 Figure 38. An example of PV panel reconfiguration. ................................................................. 109 Figure 39. The two indexing methods: (a) “global coordinates” and (b) “local coordinates” of PV cells in the PV panel. .................................................................................................................. 112 Figure 40. The archiecture of a capital cost-aware reconfigurable PV system. ......................... 114 Figure 41. Pareto-optimal macro-cell sizes and the corresponding PV system performance and capital cost. ................................................................................................................................. 121 Figure 42. Discharging a 350 mAh 2-cell series Li-ion GP105L35 battery at constant currents of 1C, 2C, 4C, and 4C from our experiments. ................................................................................ 127 Figure 43. Example commercial products of typical batteries and supercapacitors. .................. 133 Figure 44. Comparison between EES elements. ......................................................................... 137 Figure 45. EES element equivalent circuit model. ..................................................................... 138 Figure 46. EES system structure. ................................................................................................ 141 Figure 47. Linear Technology LTC6803-3 battery stack monitor with passive cell balancing switches [76]. .............................................................................................................................. 142 Figure 48. Reconfiguration examples of an energy storage array with four energy storage elements [111]. ............................................................................................................................ 143 14 Figure 49. Battery-supercapacitor hybrid wireless sensor nodes (a) Prometheus from Berkeley [32] and (b) AmbiMax from UC Irvine [98]. ............................................................................. 147 Figure 50. An example of battery-based EES applications, Toyota Prius PHEV with Li-ion batteries. ...................................................................................................................................... 147 Figure 51. Golden Valley Electric Association, Fairbanks, AK, a 27 MW NiCd battery EES system for the power grid. .......................................................................................................... 149 Figure 52. Comparison of computer memory hierarchy and HEES archiecture. ....................... 153 Figure 53. Conceptual diagram of a HEES system with four energy storage banks connected through a CTI. ............................................................................................................................. 155 Figure 54. HEES system architecture with three EES banks...................................................... 156 Figure 55. Supercapacitor-battery hybrid connection topologies. .............................................. 157 Figure 56. Various interconnect architectures for system-on-chip and HEES system. .............. 161 Figure 57. The architecture of a CTI router. An associated EES bank is connected via a power converter. The arrows denote the CTI links................................................................................ 162 Figure 58. HEES system prototype. ............................................................................................ 164 Figure 59. Conceptual diagram of charge allocation, replacement and migration. .................... 167 Figure 60. Comparison of charge allocation efficiency among different policies. UB: unbiased charging (equal charging current for both banks). SBF: supercapacitor bank first (charging the supercapacitor bank first). BBF: battery bank first (charging the battery bank first). ................ 170 Figure 61. Comparison of charge replacement efficiency among different policies. UB: unbiased discharging (equal discharging current for both banks). SBF: supercapacitor bank first (discharging the supercapacitor bank first). BBF: battery bank first (discharging the battery bank first). ............................................................................................................................................ 172 15 Figure 62. Single-source, single-destination charge migration structure. .................................. 174 Figure 63. Online procedure in the optimal solution of ∞ that maximizes . The lookup table is built online. ......................................................................................................... 179 Figure 64. An example of optimal time-unconstrained charge migration. ................................. 179 Figure 65. An example to illustrate the tracing back procedure. ................................................ 183 Figure 66. Traces of in the optimal solution and near-optimal solution of a time- constrained charge migration. ..................................................................................................... 196 Figure 67. MSMD charge migration system architecture. .......................................................... 200 Figure 68. Comparison of charge migration efficiencies on charge migration process from supercapacitor banks to battery banks, with different values. .............................................. 211 Figure 69. Comparison of charge migration efficiencies on charge migration process from battery banks to supercapacitor banks, with different values. .............................................. 212 Figure 70. Reconfigurable supercapacitor array structure. ......................................................... 216 Figure 71. A configuration for a supercapacitor array. .................................................... 217 Figure 72. A configuration for a supercapacitor array. .................................................... 218 Figure 73. Flow chart of the fault detection algorithm. .............................................................. 219 Figure 74. Comparison results on energy capacity. .................................................................... 228 Figure 75. Comparison results on EES system performance. ..................................................... 229 Figure 76. Li-ion battery SoH degradation versus SoC swing (at different average SoC levels) and average SoC level (at different SoC swings). ...................................................................... 235 Figure 77.Illustrative example of Motivation I. .......................................................................... 236 Figure 78.Illustrative example of Motivation II. ........................................................................ 237 Figure 79.An example SoC profile versus time of a battery element and turning points. .......... 238 16 Figure 80.Six basic cases for (charging/discharging) cycle identification. ................................ 240 Figure 81.An example of estimating the value from an arbitrary battery SoC profile. ......................................................................................................................................... 240 Figure 82.Structure of the HEES system considered in this chapter. ......................................... 242 Figure 83.Synthesized source and load power profiles. ............................................................. 252 Figure 84.SoC profiles of the Li-ion battery array of the proposed system and Baseline 1 under synthesized power profiles. ......................................................................................................... 253 Figure 85.Real source and load power profiles. ......................................................................... 254 Figure 86.SoC profiles of the lead-acid battery array of the proposed system and Baseline 1 under real power profiles. ........................................................................................................... 255 Figure 87. Architecture of the homogeneous EES-based PV system (Homogeneous EES-Based System). ....................................................................................................................................... 258 Figure 88. Architecture of the proposed HEES-based PV system (HEES-Based System). ........ 259 Figure 89. Illustration of the proposed enhancement methods. .................................................. 264 Figure 90. Illustration of the balanced reconfigurations. ............................................................ 268 Figure 91. Architecture of the proposed HEES system for a residential smart grid user with PV power generation. ........................................................................................................................ 283 Figure 92. Daily time-of-use energy pricing............................................................................... 284 Figure 93. (a) PV power generation and load power consumption profiles and (b) electricity energy price function. ................................................................................................................. 294 Figure 94. Power drawn from the Grid. ...................................................................................... 295 Figure 95. Daily profit of the proposed HEES system and two baseline systems. ..................... 296 17 Figure 96. Comparison between the peak load power consumption prediction results from initial prediction (top) and from intra-day refinement at time (bottom) and actual peak load power consumption results.................................................................................................. 306 Figure 97. Comparison between the peak PV power generation prediction results from initial prediction (top) and from intra-day refinement at time (bottom) and actual peak PV power generation results. ...................................................................................................... 307 Figure 98. Block diagram illustrating the interface between the PV module, storage system, residential load, and the Smart Grid. .......................................................................................... 309 Figure 99. Relationship between and in two types in two types of batteries. 316 Figure 100. The daily energy price component in the second type of electricity price function.324 Figure 101. Hybrid power controller board developed in [120]. ................................................ 331 Figure 102. Dual-battery powered portable system model [167]. .............................................. 332 Figure 103. Block diagram of HypoEnergy [167]. ..................................................................... 334 Figure 104. System architecture of the hierarchical reinforcement learning framework with two dedicated PMs [125]. .................................................................................................................. 334 Figure 105. Architecture of the proposed battery-supercapacitor hybrid system in embedded system applications. .................................................................................................................... 335 Figure 106. An example showing the control during system operation. .......................... 343 Figure 107. The comparison results on the total service time (in minutes) between the proposed system and two baseline systems in the first experiment. .......................................................... 345 Figure 108. The comparison results on the total service time (in minutes) between the proposed system and two baseline systems in the second experiment. ...................................................... 345 18 List of Tables Table I. Notations and definitions used in Chapter 3.................................................................... 49 Table II. MPP output power of the PV module before and after reconfiguration under all the nine partial shading patterns. ................................................................................................................ 67 Table III. Improvement of instantaneous output power of the PV system in the first experiment. ....................................................................................................................................................... 69 Table IV. Improvement of instantaneous output power of the PV system in the first experiment considering temperature effect. ..................................................................................................... 71 Table V. Improvement of instantaneous output power of the PV system in the first experiment using battery as energy storage device. ......................................................................................... 72 Table VI. Improvement of instantaneous output power of the PV system in the second experiment..................................................................................................................................... 74 Table VII. Improvement of instantaneous output power of the PV system in the third experiment to test the maximum gain. ............................................................................................................. 75 Table VIII. Output power improvement of the proposed PV system with different values. ....................................................................................................................................................... 91 Table IX. Recorded driving profiles. ............................................................................................ 96 Table X. Performance comparison between the proposed reconfigurable PV system with baseline PV systems on the six benchmark profiles. ................................................................................ 101 Table XI. Performance comparison of the reconfigurable PV system with different PV cell sizes on the six benchmark profiles. .................................................................................................... 104 Table XII. Performance comparison between the optimal reconfigurable PV system with two customized PV systems on six benchmark profiles. ................................................................... 105 19 Table XIII. Prices and characteristics of PV cells, MOSFETs and gate drivers. ....................... 120 Table XIV. Maximal performance enhancement with capital cost constraint. ........................... 122 Table XV. Minimal cost increase with performance guarantee. ................................................ 123 Table XVI. Comparison between energy storage elements. ....................................................... 136 Table XVII. Comparison of density, cost, power consumption (idle/active), 16-bit random access time (read/write/erase) of memory devices [129]. ...................................................................... 152 Table XVIII. Notations for single-source, single-destination charge migration. ....................... 175 Table XIX. Online computation complexity comparison among the three algorithms. ............. 187 Table XX. Comparison of normalized GMEs in time-unconstrained supercapacitor-to- supercapacitor charge migration. ................................................................................................ 189 Table XXI. Comparison of normalized GMEs in time-unconstrained supercapacitor-to-battery charge migration. ........................................................................................................................ 190 Table XXII. Comparison of normalized GMEs in time-unconstrained battery-to-supercapacitor charge migration. ........................................................................................................................ 190 Table XXIII. Comparison of normalized GMEs in time-unconstrained battery-to-battery charge migration. .................................................................................................................................... 190 Table XXIV. Comparison of normalized GMEs in time-constrained supercapacitor-to- supercapacitor charge migration. ................................................................................................ 193 Table XXV. Comparison of normalized GMEs in time-constrained supercapacitor-to-battery charge migration. ........................................................................................................................ 193 Table XXVI. Comparison of normalized GMEs in time-constrained battery-to-supercapacitor charge migration. ........................................................................................................................ 194 20 Table XXVII. Comparison of normalized GMEs in time-constrained battery-to-battery charge migration. .................................................................................................................................... 194 Table XXVIII. SoH degradation and cycle life comparison between the proposed system and baseline systems using synthesized power profiles. ................................................................... 253 Table XXIX. SoH degradation and cycle life comparison between the proposed system and baseline systems using real power profiles. ................................................................................ 254 Table XXX. Energy conversion efficiency results on various systems. ..................................... 277 Table XXXI. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 45Ah and Parameter Set to be 0.8. .. 326 Table XXXII. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 60Ah and Parameter Set to be 0.8. ..................................................................................................................................................... 326 Table XXXIII. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 45Ah and Parameter Set to be 1.5. ..................................................................................................................................................... 327 Table XXXIV. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 60Ah and Parameter Set to be 1.5. ..................................................................................................................................................... 328 Table XXXV. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, under the Second Type of Price Function and Parameter Set to be 0.8................................................................................................................................................ 329 21 Table XXXVI. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, under the Second Type of Price Function and Parameter Set to be 1.5................................................................................................................................................ 329 22 Abstract This thesis focuses on the design and runtime control of advanced electrical energy generation and storage systems for a wide range of applications, from the power grid to electric cars, and from households to mobile battery-operated devices. The thesis is organized in three parts, focusing on energy generation, energy storage, and their integration and applications. Due to an increasing appetite for energy and concern about environmental impacts of fossil fuels, there has been a growing demand for renewable, eco-friendly and sustainable energy resources (e.g., solar, wind, geothermal). The energy produced from these alternate energy resources must be cost-competitive with the energy produced from fossil fuels. Photovoltaic (PV) energy generation techniques have received significant attention since they utilize the abundance of solar energy and can be easily scaled up. A PV system is comprised of several (series- connected) PV modules, an energy storage, and charger connecting in between. Accurate modeling and effective control of the PV system is mandatory in order to fully exploit the solar irradiance. Moreover, PV systems are subject to the partial shading effect that can result in a significant degradation in the PV system output power. The first part of this thesis presents an accurate modeling framework of each component in the PV system and an effective control mechanism, called the maximum power transfer tracking method. In addition, a cost-effective reconfigurable PV architecture to combat the partial shading effect and a dynamic programming-based reconfiguration algorithm to maximize the overall PV system output power are described. Experimental results based on a hardware prototype of a reconfigurable PV system demonstrate the accuracy of the PV modeling and the effectiveness of 23 the PV reconfiguration algorithm. Moving towards real testbeds, this thesis next focuses on how the aforesaid reconfigurable PV system can be integrated into a hybrid electric vehicle (HEV) so as to maximize the total distance which is travelled by an HEV on a full tank of gasoline with the aid of the PV harvested energy. The issue of capital cost and economic vialibilty of the proposed reconfigurable PV system is considered next. As of today, no single type of EES (electrical energy storage) element, e.g., batteries, supercapacitors, can fulfill all desirable features of an ideal storage device. In this thesis the architecture and control mechanisms for a hybrid EES (HEES) system comprising of two or more heterogeneous EES elements are presented. The HEES system, which can realize advantages of each EES element while hiding their weaknesses, can thus exhibit superior performance compared to conventional homogeneous EES systems. Three fundamental charge management policies are defined in this thesis: charge allocation from a power source to a set of EES banks, charge replacement from EES banks to load devices, and charge migration among EES banks to improve the availability and responsiveness of the HEES system. In particular, an optimal control policy for charge migration and algorithms for fault detection and tolerance in the HEES system are presented. Finally, a state-of-health (SoH) aware joint charge management algorithm for the HEES system, which is based on an improved SoH modeling, is discussed. The last part of this thesis focuses on joint control and applications of the proposed (reconfigurable) PV and HEES systems. The target platforms range from residential units to portable embedded systems. It has been observed that both the appropriate system design and runtime control algorithm are critical in order to enhance the energy efficiency. 24 Chapter 1. Highlights and Overview Due to an increasing appetite for energy sources and environmental concerns about fossil fuels, there has been a growing demand for renewable energy resources (e.g., solar, wind, geothermal), which are clean and eco-friendly. The energy produced from renewable energy resources must be cost-competitive with the energy produced from fossil fuels. Photovoltaic (PV) energy generation techniques have received significant attention since they utilize the abundance of solar energy and can easily be scaled up. A PV system is comprised of several (series- connected) PV modules, an energy storage, and charger connecting in between. Accurate modeling and effective control of the PV system is mandatory in order to fully exploit the effectiveness. Moreover, PV systems are subject to the partial shading effect that can result in a significant degradation in the PV system output power. This thesis presents an accurate modeling framework of each component in the PV system, as well as an effective control mechanism, the maximum power point tracking (MPTT) method, in order to maximize the amount of energy accumulated in the storage system. We also present a cost-effective, reconfigurable PV module architecture with integrated switches in order to combat partial shading, as well as a dynamic programming-based algorithm to adaptively produce optimal reconfigurations of each PV module to maximize the PV system output power under any partial shading pattern. A hardware prototype of the reconfigurable PV module is implemented and demonstrates the accuracy of PV modeling and effectiveness of PV module reconfiguration. Finally we discuss about the integration of PV and optimal reconfiguration control in hybrid electric vehicles (HEVs) as well as techniques to reduce the capital cost of the PV system. 25 Moreover, in spite of extensive research it is still quite expensive to store electrical energy without converting it to a different form of energy. As of today, no single type of EES (electrical energy storage) element, e.g., batteries, supercapacitors, can fulfill all the desirable features of an ideal storage devices e.g., high-efficiency, high power/energy capacity, low-cost, and long cycle life. Furthermore, it is not likely that we will have such an ultimate EES element in the near future. In this thesis we present the architecture and control mechanism of hybrid EES system (HEES system) that consists of two or more heterogeneous EES elements, realizing the advantages of each EES element while hiding their weaknesses. HEES systems exhibit superior performance compared to conventional homogeneous EES systems when appropriate charge allocation and replacement policies are developed and used. Three fundamental charge management policies are defined in this thesis: charge allocation from a power source to a set of EES banks, charge replacement from EES banks to load devices, and charge migration among EES banks to improve the availability and responsiveness of the HEES system. We present the detailed problem formulation and efficiency optimization policy of the single-source, single- destination (SSSD) charge migration and the more general multiple-source, multiple-destination (MSMD) charge migration in this thesis. We also discuss about the fault detection and tolerance (bypassing) algorithms for the EES banks in the HEES system. Finally, we present a state-of- health (SoH) aware joint charge management algorithm for the HEES system, which is based on an improved SoH modeling capable of dealing with arbitrary charging and discharging patterns of a battery bank. The last part of this thesis is the joint control and applications of PV and HEES (or EES) systems. We discuss about their applications with different scales, including residential usage and portable embedded systems. 26 The rest of the thesis will be organized as follows. Chapter 2 - 5 focuses on PV systems design, control, and optimization. More specifically, Chapter 2 introduces the basics of PV systems, accurate modeling of PV cells and converters/chargers, as well as the MPTT control of the PV system. Chapter 3 presents the proposed PV module reconfiguration technique and control algorithm. Chapter 4 investigates the integration of PV systems in HEVs and the corresponding reconfiguration problem. Chapter 5 discusses certain techniques in order to reduce the capital cost of PV systems while maintaining the system performance/output power higher than the pre-defined level. Chapter 6 - 10 is the second part of the thesis and focuses on the motivation, introduction, design, control, and optimization of HEES systems. More specifically, Chapter 6 presents a survey of different types and their characteristics of EES elements, an accurate modeling framework of EES elements, as well as components, structures, and applications of EES systems. Chapter 7 presents the structure and control of HEES systems in analogy to that of memory hierarchy. Chapter 8 defines three fundamental charge management policies: charge allocation, charge replacement, and charge migration, and provides the optimization framework of SSSD and MSMD charge migrations. Chapter 9 presents fault detection and fault tolerance algorithms in EES banks of a HEES system. Chapter 10 discusses the state-of-health (SoH) aware joint charge management algorithm for the HEES system, which is based on an improved SoH modeling capable of dealing with arbitrary charging and discharging patterns of a battery bank. Finally, Chapter 11 - 13 is the third part of this thesis and deals with joint control and applications of PV and HEES (or EES) systems on different scales. More specifically, Chapter 11 deals with the joint control and optimization of PV and HEES systems. Chapter 12 discusses about the applications of HEES system and/or PV system for 27 residential usage, whereas Chapter 13 presents the applications of PV and HEES systems on portable embedded systems. 28 Chapter 2. Photovoltaic (PV) System Modeling and Architecture Figure 1. Different applications of PV systems. Due to an increasing appetite for energy sources and environmental concerns about fossil fuels, there has been a growing demand for renewable energy resources (e.g., solar, wind, geothermal), which are clean and eco-friendly. The energy produced from renewable energy resources must be cost-competitive with the energy produced from fossil fuels. Photovoltaic (PV) energy generation techniques are a method of generating electrical power by converting solar irradiance into direct current electricity using semiconductors that exhibit the photovoltaic effect, and have received significant attention since they utilize the abundance of solar energy and can easily be scaled up. Thanks to extensive research and development of PV energy generation technologies, various scales of PV energy generation systems (PV systems) have been deployed for many practical applications such as PV power stations, solar-powered vehicles, and solar- powered heating and lighting appliances, as shown in Figure 1. By the end of 2011, a total of 29 71.1 GW of PV systems had been installed [1], which is sufficient to generate 85 TWh/year [2]. And by the end of 2012, the 100 GW installed capacity milestone was achieved [3]. Solar photovoltaics is now, after hydro and wind power, the third most important renewable energy source in terms of globally installed capacity. The PV system output power largely depends on the solar irradiance level, which is continually changing. The solar irradiance levels in cloudy days tend to be lower than those in sunny days by an order of magnitude. Even in a sunny day, the solar irradiance level in the morning/evening is much lower than that at noon. Therefore, the output power of PV systems is quite unstable. To compensate this weakness, standalone PV systems are equipped with electrical energy storage elements (e.g., batteries, supercapacitors) [4], [5], [6]. When the solar irradiance level is high, the energy storage elements can store the excess output energy of PV systems. And when the solar irradiance level is low, the energy storage elements can complement the insufficient output energy of PV systems. Figure 2 shows a typical PV system architecture. Several PV modules are connected in series to provide a desirable output voltage level. We call such series-connected PV modules a PV (module) string. In the conventional realization of a PV system, each PV module consists of PV cells connected in series/parallel i.e., each PV module has a fixed configuration. The PV string is then fed to a charger, which is a power converter regulating its output current. The charger regulates the operation of PV modules with the help of appropriate control circuitry. This PV system architecture reduces the hardware cost due to sharing of the charger among different PV modules. We call such a PV system architecture the string charger architecture. An electrical energy storage (EES) system is connected to the charger to store the electrical energy harvested by the PV modules. 30 Charger Energy Storage PV Module 1 PV Module 2 PV Module 3 I pv I pv Figure 2. The PV system architecture based on the string charger architecture. (a) (b) 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 Voltage (V) Current (A) G = 1.0G STC G = 0.8G STC G = 0.6G STC G = 0.4G STC G = 0.2G STC MPP 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Voltage (V) Power (W) Figure 3. PV cell (a) V-I and (b) V-P output characteristics under different solar irradiance level . The red dots denote MPPs of a PV cell, where the PV cell achieves the maximum output power under certain solar irradiance level. The PV system output power level, i.e., the output power of the charger in the PV system, depends on the solar irradiance, which is changing frequently according to the time of day and weather conditions. The PV cells exhibit highly non-linear voltage-current (V-I) output characteristics (curves) that change with the solar irradiance level. Figure 3 (a) shows the PV cell V-I output characteristics under different solar irradiance levels. Figure 3 (b) shows the corresponding voltage-power (V-P) output characteristics. The red dots in Figure 3 denote the 31 maximum power points (MPPs) of a PV cell where the PV cell achieves the maximum output power for the given solar irradiance level. 2.1 PV Cell Modeling and Characterization (a) (b) pvc I pvc V S R sh I d I P R L I Figure 4. The equivalent circuit model (a) and the symbol (b) of a PV cell. Every PV module consists of multiple PV cells. Let and denote the output voltage and current of a PV cell, respectively. The PV cell equivalent circuit model is shown in Figure 4 with the V-I output characteristics given by (2.1) where (2.2) and (2.3) Parameters in (2.1) – (2.3) are defined as follows. is the solar irradiance level; is the cell temperature; is the charge of the electron; is the energy bandgap; and is the Boltzmann’s constant. STC stands for standard test condition in which the irradiance level is 1000 and 32 the cell temperature is 25 °C. The parameters listed above are either physical constants, environment-related value or configuration parameters. There are still five unknown parameters, commonly not provided by manufacturers, to be determined. These five parameters are the key that is capable of analytically describing the characteristics of a PV cell:  : the photo-generated current at standard test condition.  : dark saturation current at standard test condition.  : PV cell series resistance.  : PV cell parallel (shunt) resistance.  : the diode ideality factor. A method has been proposed in [7] to extract the above-mentioned five parameters from datasheet values at STC, which consist of the open circuit voltage , the short circuit current , the voltage and current at the MPP , and temperature coefficients. Different from [7], we extract the unknown parameters from measured PV cell V-I curves at various irradiance levels and temperatures, in which each V-I curve is measured using data acquisition equipment under one specific environmental condition . Therefore, our parameter extraction method is not confined to only the parameters at STC. Instead, it extracts and flexibly under any environmental condition from the measured data. Subsequently, the corresponding parameters at STC, and , can be determined using (2.2) and (2.3). Moreover, we know that the method in [7] extracts the five unknown parameters from only some special values at STC, and does not utilize the whole V-I characteristics of PV cell for parameter extraction. Therefore, the overall average fitting error cannot be guaranteed to be minimized although such method is quite stable and does not rely heavily on initial values for the 33 iterative procedure. Using only a small number of measurement data, instead of the whole V-I curve, does not really simplify the parameter extraction procedure because we need to measure the whole range anyway to find out the MPP unless we rely on the datasheet provided by the manufacturer. In fact, the fitting errors can be significant in some specific PV cell V-I ranges as pointed out in [37]. Hence, such method cannot fulfill the requirement of state-of-the-art researches in which the whole operating range should be accurately modeled for maximizing the energy efficiency. On the other hand, nonlinear curve fitting algorithm can be adopted here to overcome the shortcoming of the previous method in which only some specific points of the whole V-I curve have been used. The fitting parameters depend heavily on the initial values. If the initial values are not properly set, the fitting results obtained may be not optimal nor even feasible. This is because of the fact that nonlinear curve fitting is a highly non-convex optimization procedure, and it is likely to be stuck at the local optimal point. Therefore, we propose to use the parameter extraction heuristic that uses specific points of each V-I curve, accelerated by Newton-Raphson method, in the initial phase. The derived five parameters, i.e., , , , , and , serve as the "proper" initial values in the subsequent least-squares nonlinear curve fitting method based on the Levenberg-Marquardt algorithm. Furthermore, we have to also set an upper bound and a lower bound of the fitting parameters, since such bounds also play an important role in the nonlinear curve fitting for acceleration and convergence. One simple, yet effective set of bounds is given by , where is the derived PV cell parameters in the initial phase. With such properly set initial values and upper/lower bounds, nonlinear curve fitting algorithm can find the optimal PV cell parameters effectively, accounting for the whole V-I operating range. 34 We apply the proposed combined parameter extraction method on the measured PV cell V-I curves. Significant reduction (on average 8X) in root mean square (RMS) fitting error can be observed compared with the conventional method which only considers some specific points. Figure 5 demonstrates the V-I curves of the PV simulation model (after parameter extraction) as well as the V-I curves of the PV cell measurements. The V-I curves of the measured PV cell and the simulation model achieve very good match in the entire operation range at all three solar irradiance levels, demonstrating the high accuracy of the parameter extraction method as well as the PV cell simulation model. Figure 5. Matching between the measurement results and the simulation model on the V-I characteristics of a single PV cell. 2.2 Power Converter (Charger) Power Model Figure 6 shows the model of a PWM (pulse width modulation) buck-boost switching converter, which is used as the charger in the proposed PV system. The input ports of the charger are connected to the PV string, whereas the output ports are connected to the EES element. The charger regulates the operating point of the PV string by controlling the charger’s input voltage, i.e., the PV output voltage (and then the PV string output current is automatically determined by 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 Voltage (V) Current (A) Model Measurement G = 0.50 G STC G = 0.77 G STC G = 1.0 G STC 35 its V-I characteristics.) We denote the input voltage, input current, output voltage and output current of the charger by , , , and , respectively. Depending on the relationship between and , the charger operates in one of the two possible operating modes: the buck mode when and the boost mode otherwise [20], [109]. Capacitor Inductor Buck Controller Q sw,1 R sw,1 Q sw,2 R sw,2 Q sw,3 R sw,3 R L L Q sw,4 R sw,4 Boost Controller V in I in R C V out I out C Figure 6. The architecture of buck-boost switching converter. Power conversion is not free. Converting the voltage level involves non-zero amount of power loss. The overall power loss includes conduction losses by parasitic resistances of circuit components, switching losses by parasitic capacitances of switching devices, power consumption of the controller circuit, and so on. The power conversion efficiency is defined as: (2.4) where denotes the power loss in the converter/charger. This power loss is not constant, but varies depending on the input and output voltages and the amount of power that is transferred through the converter, and thus, the power conversion efficiency is also a variable. The power conversion efficiency is a critical factor because it determines the amount of harvested PV energy that can be ultimately transferred to the storage system (and to be used later.) Figure 7 shows the efficiency variation of the LTM4609 buck-boost converter from Linear Technology [24]. It shows a wide range of variation depending on input voltage, output voltage, and output current. In general, the power conversion efficiency will be maximized when (i) the input voltage 36 and output voltage are close to each other, and (ii) the output current is within a certain desirable range. Figure 7. Power conversion efficiency of the Linear Technology LTM4609 buck-boost converter [24]. We develop the converter power model based on the power model for buck switching converter provided in [20]. When the charger/converter is operating in the buck mode, its power loss is given by: (2.5) where is the PWM duty ratio and is the maximum current ripple; is the switching frequency; is the current of the micro-controller of the charger; and are the internal series resistances of the inductor and the capacitor , 37 respectively; and are the turn-on resistance and gate charge of the i th MOSFET switch shown in Figure 7, respectively. The charger power loss in the boost mode is given by: (2.6) where and . The power dissipation of the charger is minimized when (i) the input voltage and the output voltage of the charger are close to each other and (ii) the output current of the charger is within a certain range. Let denote the function that calculates based on , , and . 2.3 Maximum Power Point Tracking (MPPT) and Maximum Power Transfer Tracking (MPTT) Recall that each PV cell has a MPP under a certain solar irradiance level at which the output power can be maximized. Ideally, all PV cells in the PV system experience the same solar irradiance level and thus exhibit the same V-I and V-P output characteristics. Consequently, all the PV cells can simultaneously operate at their MPPs, and the PV string achieves the maximum output power since the output voltage of the PV string is set to be a desired value by the charger. Usually, a maximum power point tracking (MPPT) technique is implemented in the control circuitry of the PV system to find the desired output voltage for the PV string, where the PV string can achieve the maximum output power [8], [9]. The MPPT technique can be efficiently 38 implemented using the perturb & observe (P&O) algorithm. In the P&O algorithm, the controller slightly changes the duty ratio in the charger/converter (to regulate the PV output voltage and current) and observes whether the PV output power is increased. It will changes the duty ratio in the opposite way if the PV output power decreases, and finally this method will converge to the MPP of the PV string (or each PV cell.) The traditional MPPT technique only optimizes the PV string output power, without taking into account the efficiency variation of the charger. To overcome this shortcoming, we propose the maximum power transfer tracking (MPTT) method [36] that accounts for the variation in the conversion efficiency of the charger as a function of the output power of the PV string and the state-of-charge (SoC) of the EES elements. The MPTT method maximizes the output power of the charger, and thereby, maximizes the actual energy accumulation rate in the EES elements. Hence, it can be even more effective than the MPPT methods [10], [36], [38]. The MPTT method can be implemented using the similar P&O algorithm with the same implementation complexity as the conventional MPPT method. More specifically, the charger/converter regulates the PV output voltage (and current) through adjusting its duty ratio, in order to maximize the output power of the charger. 2.4 Partial Shading Effect In reality, the solar irradiance levels received by PV cells in a PV system may be different from each other when a portion of PV modules is in shadow, and such a phenomenon is known as the partial shading effect. For example, moving clouds cause partial shading for stationary applications. On the other hand, shadows from nearby objects (e.g., buildings, trees, and poles) produce partial shading for PV systems on hybrid electric vehicles, which is much more severe as vehicles are moving through shaded or lighted regions. In these cases, the partial shading 39 pattern may be quite regular (i.e., like a block), and we call this case block shading. Partial shading may also results from fallen leaves or dust on the PV modules [26], or other aging effects of PV modules [27]. In this case, the partial shading pattern may be randomized, and we call this case random shading. PV cells generally have different MPPs under the partial shading effect. Partial shading not only reduces the maximum output power of the shaded PV cells, but also makes the lighted or less-shaded PV cells that are connected in series with the shaded ones to deviate from their MPPs. In other words, the PV cells cannot simultaneously operate at their MPPs. The PV systems with the string charger architecture are extremely vulnerable to partial shading since the PV modules are connected in series. With partial shading, the maximum output power of a PV string becomes much lower than the sum of the maximum output power values of all the individual PV cells in the PV string. We demonstrate that the partial shading effect may significantly degrade the output power level of a PV module with a fixed configuration. We use a PV module with a configuration as an example. As shown in Figure 8, the PV module consists of two series- connected PV groups, and each PV group consists of two parallel-connected PV cells. The PV cell at the bottom right is completely shaded (with no solar irradiance) while the rest of PV cells receive the solar irradiance under the standard test condition i.e., . Since only one PV cell out of four is shaded, the ideal setup should exhibit the PV module output power degradation of 25% compared to the same PV module without any shading. However, the actual PV module output power degradation is much larger than 25%. 40 PV Groups Figure 8. A PV module with one PV cell completely shaded. We plot in Figure 9 the V-I characteristics of the PV module under partial shading. Curve 1 corresponds to the V-I output characteristics of the bottom PV group with the shaded PV cell, whereas Curve 2 corresponds to the V-I output characteristics of the top PV group. Curve 2 has a higher current value than Curve 1 at the same voltage value. Curve 3 is the V-I output characteristics of the PV module, which is directly derived from Curves 1 and 2 since the PV module is a series connection of the two PV groups. Note that we assume that each PV cell is integrated with a bypass diode to protect the PV cell from reverse bias operation under partial shading [21] when we derive Curve 3. Figure 9. The V-I output characteristics of the partially shaded PV module and its PV groups. 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 Voltage (V) Current (A) 1 2 3 41 Figure 10. The V-P output characteristics of the partially shaded PV module and the lighted PV module. We compare the end-to-end V-P output characteristics of the partially shaded PV module with the same PV module without shading in Figure 10. The red dots in Figure 10 show the MPPs. The maximum output power of the partially shaded PV module is about 56% of that of the same PV module without shading. As a result, one shaded PV cell degrades the PV module output power by as much as 44%, which establishes the significance of the effect of partial shading effect. In addition, partial shading may result in multiple power peaks in the V-P output characteristics of a PV string, as also can be seen in Figure 10. Therefore, the MPPT (or MPTT) techniques must be modified in order to dynamically track a global optimum operating point instead of a local optimal one [11], [12], [13], [14]. This is because the existing MPPT or MPTT techniques such as the perturb and observe method rely on the unimodality assumption about the V-P output characteristics of the PV string. The modified MPPT or MPTT techniques increase the complexity of the PV system control circuitry. 0 1 2 3 4 5 6 7 0 1 2 3 4 5 Voltage (V) Power (W) Partially shaded PV module Non-shaded PV module 42 The modified MPPT techniques may restore part of the power loss due to partial shading, but they cannot fully utilize the lighted PV cells due to the deviation from their MPPs caused by the shaded cells. On the other hand, PV module reconfiguration techniques, which have the potential of fully exploiting the MPPs of both lighted and shaded PV cells in a partially shaded PV string, can help maintain the output power level of a PV system under partial shading. Various PV reconfiguration techniques have been proposed, which are different from each other in terms of the system structure and control approach that they employ [15], [16], [17], [18], [19]. However, they suffer from one or more of the following limitations: 1) To compensate the power loss from shaded PV cells, many extra PV cells are needed for performing reconfiguration according to the shading pattern, as shown in Figure 11 from [15] as an example; 2) There is a lack of systematic and scalable structural support or effective control mechanism; 3) Variations in the conversion efficiency of the charger or inverter at different operating points are overlooked, which may result in a sizeable degradation in the overall energy conversion efficiency; 4) The PV system employs an individual charger architecture, in which each PV module has an individual charger for setting the operating point, thereby increasing the hardware cost of the PV system. A more widely used and cost-effective structure is the string charger architecture. 43 Figure 11. The reconfigurable PV module structure in [15], which requires additional PV cells (in the "adaptive bank") to perform reconfiguration. 44 Chapter 3. PV Reconfiguration Structure and Control to Combat Partial Shading In this section, we present a PV module reconfiguration approach that provides both a scalable reconfiguration architecture as well as a systematic and near-optimal control mechanism to overcome the PV system output power degradation caused by partial shading. The PV module reconfiguration controller dynamically updates the PV module configurations according to the changing partial shading pattern and conversion efficiency variation of the charger. We employ a reconfigurable PV module architecture. This reconfigurable PV module architecture can also be applied for PV systems online fault detection and tolerance [43]. We use the reconfigurable PV module architecture to realize flexible PV module configurations, where there can be an arbitrary number of PV groups connected in series. Note that a PV group consists of parallel-connected PV cells where the number of PV cells in each PV group can be different from each other. We also develop an effective reconfiguration control mechanism for PV systems with the string charger architecture. We focus on the string charger architecture since it is widely used and more cost effective than other architectures. Our reconfiguration control mechanism adaptively finds the near-optimal PV module configuration for each PV module according to the partial shading pattern and the conversion efficiency variation of the charger such that both the shaded and lighted PV cells can work at or close to their MPPs simultaneously. In this way, we improve the PV system output power level under partial shading conditions to the largest possible extent. The proposed reconfiguration control mechanism is based on a dynamic programming algorithm with polynomial time complexity, and therefore, it can be incorporated 45 into modern PV systems with negligible extra computational overhead. We implement a working prototype of reconfigurable PV module with 16 PV cells and confirm 45.2% output power level improvement. Using accurate PV cell models extracted from prototype measurement, we have demonstrated up to a factor of 2.36X output power improvement of a large-scale PV system comprised of 3 PV modules with 60 PV cells per modules. 3.1 PV Module Reconfiguration Architecture We replace the conventional PV modules (with fixed configurations) by the reconfigurable PV modules for the PV system to combat partial shading. We make the physical layout and the configuration of a PV module independent of one another. The physical layout of the PV module is an array on a panel, where there are rows and columns of PV cells. The configuration of the PV module is the actual electrical connection of PV cells in the PV module. We change the configuration of the PV module to counter partial shading. We introduce a reconfigurable PV module architecture as shown in Figure 12. Each PV cell except for the last one is integrated with three switches i.e., a top P-switch , a bottom P-switch and a S- switch . The PV module reconfiguration is realized by controlling the ON/OFF states of these switches. The two P-switches of a PV cell are always in the same state, whereas its S-switch must be in the opposite state of the P-switches. The P-switches connect PV cells in parallel to form a PV group, while the S-switches connect the PV groups in series. 46 S PT,1 S PB,1 S S,1 S PT,i S S,i S PB,i P-switch P-switch S-switch #1 #i #N Switch-Set Figure 12. The reconfigurable PV module architecture. Figure 13 is an example of PV module reconfiguration. The first four PV cells are connected in parallel to form PV Group 1; the next three PV cells form PV Group 2; and the last five PV cells form PV Group 3. These three PV groups are series-connected by the S-switches of the fourth and the seventh PV cells. S PT,1 S PB,1 S S,1 #1 S PT,2 S PB,2 S S,2 #2 S PT,3 S PB,3 S S,3 #3 S PT,4 S PB,4 S S,4 #4 S PT,5 S PB,5 S S,5 #5 S PT,6 S PB,6 S S,6 #6 S PT,7 S PB,7 S S,7 #7 S PT,10 S PB,10 S S,10 #10 S PT,11 S PB,11 S S,11 #11 #12 PV group 1 PV group 2 PV group 3 S PT,9 S PB,9 S S,9 #9 S PT,8 S PB,8 S S,8 #8 Figure 13. An example of PV module reconfiguration. 47 A reconfigurable PV module consisting of PV cells may include an arbitrary number (less than or equal to ) of PV groups. The number of parallel-connected PV cells in the j-th PV group should satisfy (3.1) where is the number of PV groups. We denote such a configuration by . This configuration can be viewed as a partition of the PV cell index set , where the elements in denote the indices of PV cells in the PV module. The partition is denoted by subsets , , , and of , which correspond to the PV groups consisting of , , , and PV cells, respectively. The subsets , , , and satisfy (3.2) and , for and (3.3) The indices of PV cells in PV group must be smaller than the indices of PV cells in PV group for any due to the structural characteristics of the reconfiguration architecture i.e., for and satisfying . A partitioning satisfying the above properties is called an alphabetical partitioning. The proposed reconfigurable PV module architecture is general and it can deal with both block shading and random shading effects, as we shall see later. If we only focus on the block shading effect that is relatively regular, we may come up with other reconfigurable module designs with a smaller switch count. For example, the basic unit of PV reconfiguration can be a 48 PV macro-cell that is essentially a series and parallel connection of PV cells (e.g., a connection of PV cells), and we can produce comparable results under only block shading. 3.2 Problem Formulation Figure 14 shows the architecture of a PV system. It contains series-connected reconfigurable PV modules, each of which has the reconfiguration architecture shown in Figure 12. The input and output ports of the charger are connected to the PV string and a supercapacitor array, respectively. The charger regulates the operation of the PV string by regulating its output voltage. The output current of the PV string is automatically determined based on its V-I characteristics. We adopt a software-based MPTT technique in the proposed PV system. It employs the perturb & observe (P&O) algorithm to maximize the charger output current through regulating the output voltage of PV string. For readers’ convenience, notation used in the rest of this section is summarized in Table I. PV group 1 PV group g 1 PV group 1 PV group g 2 PV group 1 PV group g M PV module 1 PV module 2 PV module M  pvm V 2 pvm V pvm M V pvs I pvs V Charger Supercap conv P cap I cap V Figure 14. The architecture of the PV system with reconfigurable PV modules. 49 Table I. Notations and definitions used in Chapter 3. For the i-th PV cell in the k-th PV module, the relationship between and depends on as given by Eqn. (2.1). Figure 3 illustrates the V-I curves of a PV cell under different irradiance levels. We obtain of each PV cell using on-board solar irradiance sensors. The PV cell temperature has a relatively minor effect on the V-I characteristics. We derive the V-I curve of the k-th PV module given . The output current of the k-th PV module is equal to the output current of any PV group in this PV module, i.e., (3.4) 50 The output voltage of the k-th PV module is equal to the sum of the output voltages of its PV groups, i.e., (3.5) where (3.6) Similarly, the output current and voltage of the PV string satisfy the following: (3.7) and (3.8) The charger sets the operating point of the PV string by controlling , and can be determined accordingly. Once the charger sets , the operating point of each PV cell is determined accordingly from (3.4) – (3.8). The charger power loss is determined by its input voltage, input current, output voltage, and output current, i.e., , , , and , respectively, from Eqns. (2.5) and (2.6). According to Eqn. (2.4), we have (3.9) where . Please note that the terminal voltage of supercapacitor array can be assumed constant with different values due to the negligible internal resistance. 51 We give a formal problem statement for the PV module reconfiguration (PMR) problem in the following. PMR Problem Statement: Given of each i-th PV cell in the k-th PV module and , find the optimal of each PV module and the optimal ( , such that is maximized. The objective is equivalent to maximizing the PV system output power. At the end of this section, we briefly introduce the problem formulation in two other cases, which are in general similar to the above problem formulation but have some slight differences. Battery as Energy Storage Device: In this case we still maximize the charger output current (or power) as objective function (please note that this is equivalent to maximizing the energy accumulation rate of battery.) However, we need to account for the following two differences from using supercapacitor: (i) the battery terminal voltage depends on the charger output current, and (ii) the charger output power is higher than the energy accumulation rate in the battery due to its internal resistance and rate capacity effect [75]. Considering the Effect of Load Device: In this case the storage is connected to the load device via a power converter for voltage regulation. This is the typical structure of PV energy harvesting system with load devices [32], [33]. We focus on PV reconfiguration and MPTT, and still maximize the charger output current (or power) as objective function. When supercapacitor is used for energy storage, the problem formulation is exactly the same as the above PMR problem. This is because the supercapacitor terminal voltage is almost constant due to the negligible internal resistance. In other words, the load device has negligible influence on reconfiguration and MPTT. When battery is used for energy storage, the load power will affect the terminal voltage of battery, and hence reconfiguration and MPTT, but not very significantly as we shall see in the experimental results. 52 We solve these two problems using similar methods. Details are omitted due to space limitation. We will provide experimental results about these two cases later. 3.3 PV Module Reconfiguration Control Algorithm We define the output power at MPP of a PV cell/group/module/string by MPP power. The output power of the PV string and the output power of the PV system are positively correlated but they are different from one another because the former does not account for the charger power loss. We define the MPP voltage and MPP current of a PV cell/group/module/string as the output voltage and current of that PV cell/group/module/string at its MPP, respectively. The maximum solar power harvested by a PV system is the sum of the MPP power levels of all the PV cells in this system. The reconfiguration algorithm aims at making all PV cells to simultaneously operate at or close to their MPPs. We update the optimal configurations for PV modules according to the current shading pattern to maximize the PV system output power. We make the following critical observations from the PV cell V-I and V-P characteristics shown in Figure 3. Observation I: The MPP voltage levels of a PV cell are very close to each other even under different solar irradiance levels. On the other hand, the MPP current values vary significantly under different solar irradiance levels. Observation I is mainly due to the following two facts: (i) the MPP voltage level of a PV cell is largely determined by its open-circuit voltage level. The latter is a nearly constant value corresponding to the bias of the PV cell junction [25]. On the other hand, the MPP current is largely determined by the cell's photo-generated current , which is linearly dependent on the solar irradiance according to Eqn. (2.2). Please refer to [25] for details. Observation I has 53 already been adopted in some simple MPPT methods [26] or MPPT methods to combat the partial shading effect. Observation I enables us to use a constant voltage to approximate the MPP voltage for every PV cell. We propose the ideal PV cell model based on Observation I. The V-I characteristics of an ideal PV cell is a step function such that (3.10) where is the MPP current of the PV cell, which is calculated from using the PV cell model in Section 2.1. The ideal PV cell is an efficient and accurate approximation of the real PV cell, and plays an important role in the reconfiguration algorithm design. We know that the MPP voltage of PV cell is close to at different solar irradiance levels. In the configuration shown in Figure 15, the PV cell MPP current values, which are calculated from solar irradiance levels on PV cells, are labeled beside the PV cells. The sum of PV cell MPP current values in every PV group is 0.7 A. All the PV cells simultaneously operate close to their own MPPs with the configuration as shown in Figure 15, when we set the output voltage of this PV module to . The output power of the PV module is maximized in this case. Hence, we arrive at the following observation. Observation II: The MPP current values of all PV groups should be close to each other in a PV module, while the MPP current values of all the PV modules should be close to each other in the PV string to maximize the PV string output power. 54 1 0.1A 2 0.2A 3 0.15A 4 0.25A 5 0.2A 6 0.3A 7 0.2A 8 0.1A 9 0.15A 10 0.25A 11 0.1A 12 0.1A PV group #1 PV group #2 PV group #3 Figure 15. An example of optimal PV module reconfiguration according to the PV cell MPP current values at their own MPPs. The following Observation III is also important in determining the optimal PV module configurations: Observation III: The charger power loss is minimized when the MPP voltage of the PV string and the supercapacitor terminal voltage are close to each other, according to the charger model described in Section 2.2. In this case, the output power of the whole PV system can be further optimized. We provide a brief explanation of Observation III: When the MPP voltage of the PV string (close to the input voltage of charger) and the supercapacitor terminal voltage (the output voltage of charger) are far from each other, the charger will have relatively higher imbalance between input and output currents and higher current ripples [20], [24], which result in relatively higher power loss in the charger. 3.3.1 Decomposing the Problem We simplify the original PMR problem assuming ideal PV cells and name the new problem Ideal PV Cell-based PMR (IC-PMR) problem. Let and represent the MPP voltage and current values, respectively, of the k-th ideal PV module with a 55 configuration . Based on the ideal PV cell assumption, we calculate these values using the following identities: (3.11) (3.12) Finally, the MPP voltage and current of the ideal PV string are given by and , respectively. We have the following observation on the IC-PMR problem: Observation IV: In the IC-PMR problem, the MPP of the PV string is the same PV string operating point that maximizes the output power of the PV system. The IC-PMR problem is equivalent to: find optimal of each PV module such that the charger output current, given by , is maximized. From Observation IV, we do not need to separately find the optimal PV string operating point because it is exactly the MPP of the ideal PV string. PMR Solver MPTT Control IC-PMR Solver Algorithm 1 (Kernel Algorithm): Find Pareto-Optimal Set for Each Module Algorithm 2: Find Optimal PV String Configuration Figure 16. The near-optimal solution of the PMR problem. 56 We propose a near-optimal solution of the original PMR problem in two steps: (i) PV module reconfiguration: optimally solving the IC-PMR problem and finding the corresponding optimal for , based on the solar irradiance levels and , and (ii) MPTT: finding the optimal PV string operating point ( that maximizes based on current . Figure 16 illustrates the two steps in the proposed near-optimal solution. We introduce the optimal solution of the IC-PMR problem in the following. 3.3.2 Solution to the IC-PMR Problem We propose to solve the IC-PMR problem in two steps as illustrated in Figure 16. We first solve the problem of finding the optimal of each ideal PV module for each given , such that (or equivalently, ) is maximized. The solution to this problem is named the kernel algorithm and is based on dynamic programming. Let store all the optimal configurations of the k-th PV module with different values. Next we find the optimal solution of the IC-PMR problem based on the optimization results of the kernel algorithm on every PV module. This step is based on the Pareto-optimal substructure property. We introduce these two steps one by one. Step I: Kernel Algorithm The kernel algorithm aims to maximize for given . We name the problem the reconfiguration problem to emphasize that is given. Consider a generalized problem that finds the optimal configuration for an –cell ( , corresponding to the first cells of the original cells in the k–th PV module) PV module composed of PV groups, given and . This is equivalent to finding the optimal alphabetical partitioning of the set , 57 which is optimal in the sense that is maximized. We call this problem the reconfiguration problem. When and , the reconfiguration problem becomes the original reconfiguration problem of the k–th PV module. We find the optimal substructure property of the reconfiguration problem as described below, ensuring the applicability of dynamic programming. Observation V (The optimal substructure property): Suppose that in the optimal solution of the reconfiguration problem, the last (i.e., the -th) PV group consists of PV cells. Consider the subproblem that finds the optimal configuration for the first PV cells within PV groups. This corresponds to the reconfiguration problem. The optimal solution of the reconfiguration problem thus contains within it the optimal solution of the reconfiguration problem. We have Algorithm 1 from the optimal substructure property as the kernel algorithm for solving the reconfiguration problem with a given . The time complexity of Algorithm 1 is (because we can pre-compute and store values in a matrix.) We make the size of matrices and equal to in order to solve the reconfiguration problems for all in one execution of Algorithm 1, with a total computation complexity of . 58 Step II: Optimal Solution of the IC-PMR Problem We define the -substring as the string consisting of PV modules 1, 2, …, . The whole PV string is the M-substring. The configuration of the -substring, , is a collection of the configurations of the 1 st , 2 nd , …, -th PV modules. The MPP voltage and current of the ideal -substring are and , respectively. We define the Pareto-optimality and Pareto-optimal set of the -substring configurations. Definition I (Pareto superiority): Consider two configurations and . is Pareto- superior to if and , or and . 59 Definition II (Pareto-optimal configuration and Pareto-optimal set): is a Pareto-optimal configuration of the -substring if no other configuration of the -substring is Pareto-superior to it. The Pareto-optimal (configuration) set of the -substring is represented by . We generalize the definition of Pareto-optimality and Pareto-optimal set to PV modules. The set obtained from the kernel algorithm is essentially the Pareto-optimal configuration set of the k-th PV module. Theorem I demonstrates that finding the Pareto-optimal set of the -substring, i.e., the whole PV string, will help in solving the IC-PMR problem. We rewrite the objective of the IC- PMR problem in Theorem I as finding the optimal , which is equivalent to the original objective of finding optimal for each PV module. Please see Appendix I for the proof. Theorem I (Optimal configuration and Pareto-optimal configurations): The optimal that optimizes the IC-PMR problem is an element in . Consider the problem of finding . We find the Pareto-optimal substructure property of this problem as shown in Theorem II. The proof of Theorem II is similar to that of Theorem I (by using proofs by contradiction), and hence we omit the details of the proof. Theorem II (The Pareto-optimal substructure property): Suppose that is a Pareto-optimal configuration of the -substring. Then must be a Pareto-optimal configuration of the -substring. We propose Algorithm 2 that derives from based on the Pareto-optimal substructure property. We execute Algorithm 2 iteratively and find starting from , which is obtained from kernel algorithm. 60 The MPP voltage of an ideal PV module can only take discrete values , , ..., . Therefore, contains at most different Pareto-optimal configurations while contains at most different Pareto-optimal configurations. We implement Algorithm 2 with time complexity making use of this property. The last step in the IC-PMR solution is finding the optimal from that maximizes the charger output current, . We summarize the steps of finding the optimal solution to the IC-PMR problem in Algorithm 3. 61 3.3.3 Complexity, Overhead, and Implementation Details Because the kernel algorithm dominates the computation complexity in Algorithm 3, the overall complexity is , or if there is a constraint on the maximum number of groups ( values). For a relatively large-scale PV system with and , it takes only 10 ms to calculate the optimal configuration on a 3.0 GHz desktop computer and should take less than 30 ms on a typical ARM-based embedded processor (as the reconfiguration controller) [29]. Moreover, since the switching time of MOSFET switches and voltage/current regulation time of chargers are in the order of [31], performing reconfiguration (i.e., changing the ON/OFF states of switches) and MPTT regulation are faster than computing the optimal configuration and have negligible time overhead. On the other hand, the hardware overhead of the reconfiguration architecture will be mainly the additional switches. For example, the MOSFET switch in [34], which allows 10.3 A ON- current, costs only $0.09. This is much cheaper than PV cells, which is above $ 2 - 5/W. Besides, for larger-scale PV systems, the PV cell power rating can be much larger than 1 W. The micro- 62 controller cost may be also accounted for, which is however also negligible for large-scale PV system integration. Theoretically, if we want to maximize the energy accumulation over a time period with changing shading patterns and solar irradiance, we need to maximize the PV system output current at every time in this period. In practice, we discretize the whole operating time of the PV system into a set of decision periods. At the beginning of each decision period, we obtain irradiance of each PV cell in the system using on-board solar irradiance sensors and the supercapacitor terminal voltage . We execute Algorithm 3 and find the optimal configuration of each PV module. We perform reconfiguration, i.e., changing the ON/OFF states of switches, according to the derived configurations, and keep the configurations unchanged until the beginning of next decision period. On the other hand, the MPTT control is performed much more frequently to keep tracking the optimal operating point of the PV string. The length of a decision period depends on the shading types and applications. For example, if the shading is caused by clouds or the PV system is used for vehicular applications, the decision period needs to be set small. If the shading is mainly caused by a nearby building, the decision period can be longer. Since the overhead of reconfiguration algorithm is less than 30 ms, the decision period can be set much less than one second, which may be suitable for even fast shading applications. Besides, we may use event-driven decision to trigger the reconfiguration algorithm to reduce overhead, where the trigger condition can be either significant change in shading pattern or terminal voltage change of the supercapacitor. 63 3.4 Experimental Results 3.4.1 Prototype of the PV Module with Reconfiguration We implement a prototype of PV module reconfiguration to substantiate the feasibility and effectiveness of the proposed reconfiguration structure design and control algorithm. Figure 17 shows the prototype of the reconfigurable PV module. The PV module consists of 16 PV cells. Each PV cell has a maximum output power of 1.2 W when the solar irradiance is under the standard test condition. We implement the reconfiguration network with a SPDT (single pole, double throw) switch as an S-switch and a DPDT (double pole, double throw) switch as a P-switch for each PV cell, because the two P-switches of a PV cell are always turned ON and OFF together. We mount the PV cells and toggle switches on top of an acrylic board, and route the connection wires in the back of the board. We operate the toggle switches manually in the prototype PV module. However, automatic switch control is not of high cost. We confirm the implementation of a computer-controlled programmable switch set using power MOSFETs and isolated gate drivers as shown in Figure 18. 64 Figure 17. The prototype PV module with reconfiguration. Figure 18. The computer-controlled programmable switch board. We measure the V-I characteristics and MPP values of a single PV cell in the reconfigurable PV module when the solar irradiance levels are , , and , and 65 temperature is 25 . The measured V-I characteristics are shown in Figure 5. Based on the measured V-I characteristics, we extract the unknown parameters , , , , and of the PV cell model using the method discussed in Section 2.1. The V-I characteristics of the PV simulation model are also shown in Figure 5 after parameter extraction. The V-I curves of the measured PV cell and the simulation model match with each other in the entire operation range at all three solar irradiance levels, demonstrating the accuracy of the extracted PV cell simulation model. We demonstrate using the prototype the effectiveness of PV module reconfiguration to combat partial shading. We use paperboards to shade the corresponding PV cells in the PV module to implement the case of partial shading. We use the PV module to directly drive a controllable active load and measure the whole V-I and V-P curves of the PV module before and after reconfiguration. Then we derive the improvement of MPP using reconfiguration from the measured V-P characteristics. We test the following nine partial shading patterns, which contain one to ten completely shaded PV cells, as shown in Figure 19. We aim to (i) validate the effectiveness of PV module reconfiguration and (ii) validate the PV simulation model, using hardware measurement results. 66 Figure 19. Different partial shading patterns. In the first shading pattern in Figure 19, we shade four PV cells at the bottom right corner of the PV module. Then we optimally reconfigure the PV module into a configuration to maximize the output power. We measure the output power of the PV module and confirm 36.3% output power level enhancement from reconfiguration compared with the original configuration. Figure 20 illustrates the measured V-P curves of the partially shaded PV module before and after reconfiguration, where the MPPs are marked by red dots. We can clearly see the improvement of the MPP of the partially-shaded PV module using the reconfiguration method. We also perform software simulation of the PV module and observe 36.1% output power level enhancement from reconfiguration against partial shading. This shows that the software simulation model is accurate. Pattern # 1 Pattern # 2 Pattern # 3 Pattern # 4 Pattern # 5 Pattern # 6 Pattern # 7 Pattern # 8 Pattern # 9 67 Figure 20. Measured V-P curves of the partially shaded PV module (from the prototype) before and after reconfiguration. Table II provides the MPP output power of the PV module before and after reconfiguration under all the nine partial shading patterns. It provides both measurement and simulation results. From the measurement results, we can observe that an improvement of 1.62 W to 3.90 W, or equivalent, 14.8% to 45.2%, can be achieved from the reconfiguration technique. Table II. MPP output power of the PV module before and after reconfiguration under all the nine partial shading patterns. 0 2 4 6 8 10 12 14 0 5 10 15 Voltage (V) Power (W) Without reconfiguration With reconfiguration 68 3.4.2 Large-Scale PV System Simulation We perform reconfiguration on large-scale PV modules and PV arrays using the simulation model. We ensure that the software simulation can present comparable results as those in the implementation due to the following three reasons: (i) We have derived accurate PV cell modeling (V-I and V-P characteristics) from real measurements and it matches with real measurements at different solar irradiance levels. (ii) The V-I and V-P characteristics modeling of PV module and string is also accurate because they are essentially series and parallel connection of PV cells. In Section 3.3, we have already validated the output power improvement of the simulation model using real experiments on the PV module prototype. (iii) We have utilized accurate charger power model, which have been validated using HSPICE simulation. We compare the performances of the PV system with reconfiguration and the baseline PV systems without reconfiguration. In the proposed PV system, we use reconfigurable PV modules with 60 PV cells in each module, a charger, and a 100 F supercapacitor as the energy storage. On the other hand, the PV modules in the baseline system have a fixed configuration, where 10 PV groups are series-connected with 6 PV cells per PV group. We incorporate a software- based MPTT technique [36] in both the proposed PV system and the baseline system to maximize the PV system output power. It employs the perturb & observe (P&O) algorithm to maximize the charger output current. In the baseline system, we incorporate bypass diodes for PV cells [15] in order to enhance the PV system output power and robustness under partial shading. The first experiment considers a PV system with a single 60-cell PV module with the partial shading pattern shown in Figure 21, which is essentially block shading. We use supercapacitor as energy storage device. We test the instantaneous output power level of the two PV systems. For 69 the proposed system, Figure 21 shows the physical locations of the PV cells in the PV module, instead of the actual electrical connection of the PV cells. Table III summarizes the output power improvement of the proposed PV system compared to the baseline system given the shading pattern and different values. As shown in Table III, the proposed PV system with reconfiguration achieves up to 42% output power improvement compared with the baseline system when V, thereby demonstrating the effectiveness of the reconfigurable method. Table III also shows (i) the actual PV system output power of the proposed system and baseline system, and (ii) the near-optimal PV module configuration obtained by the reconfiguration control algorithm. Figure 21. Partial shading pattern of the single PV module in the 1st experiment. Table III. Improvement of instantaneous output power of the PV system in the first experiment. 0.2 0.4 0.6 0.8 1 70 Two factors contribute to the PV system output power improvement. The first is the enhancement in the maximum output power of the PV module due to reconfiguration. Figure 22 plots the V-P curves of the PV module in the two systems with the shading pattern in Figure 21 and a value of 15 V. The proposed reconfigurable PV module achieves a peak output power much higher than that of the baseline PV module. The second factor is the ability to achieve through reconfiguration a better match between the MPP voltage of the PV module and the terminal voltage of the supercapacitor. In this way, the charger consumes the least amount of power and the output power of the PV system can be maximized. Figure 22. V-P characteristics of the PV modules with and without reconfiguration technique. In reality, temperatures of different PV cells in a system can be different due to the partial shading effect. We perform in-field measurement and confirm a maximum of 10 difference between temperature of the lighted PV cell and that of the shaded PV cell. 10 higher in temperature will result in degradation in the MPP voltage of PV cell according to the 0 10 20 30 40 0 10 20 30 40 50 Voltage (V) Power (W) Proposed system V cap = 15V Baseline system 71 model presented in Section 2.1 and reference [35]. We perform experiments using the same PV system setup as shown in Figure 21 and supercapacitor as energy storage. For simplicity, we assume two levels of temperature in the PV system: 25 (same as before) for PV cells with solar irradiance less than (0.5 ), and 35 for the other PV cells with higher solar irradiance. Table IV summarizes the actual output power of the two systems and improvement of the proposed system. We can observe that although slight output power degradation can be observed when comparing with Table III, the relative improvement remains nearly the same. This is because the output power degradation due to higher temperature affects both proposed system and baseline system. Table IV. Improvement of instantaneous output power of the PV system in the first experiment considering temperature effect. We conduct further experiments based on the PV module shading pattern shown in Figure 21, but using a 4 Ah Li-ion battery as energy storage device. Table V summarizes the actual output power of the PV system, i.e., the charger output power, and the improvement of the proposed system, at different open circuit voltages of the battery. It also provides the actual energy accumulation rate of the battery and corresponding improvement using reconfiguration. We can observe that the charger output power and the corresponding improvement are very close to those in Table III when supercapacitor is used as energy storage. The slight difference in charger 72 output power (compared to the supercapacitor case) results from the change of charger's output voltage when battery is used for storage, i.e., the change of battery terminal voltage. On the other hand, the actual energy accumulation rate inside battery is smaller than the charger output power due to internal power loss inside the battery. This effect is the most phenomenal when the battery open circuit voltage is low because of the high charging current, but not very significant in general. Moreover, the improvement in battery energy accumulation rate in Table V is comparable to the improvement in Table III, demonstrating the effectiveness of the proposed technique when battery is used as storage. Table V. Improvement of instantaneous output power of the PV system in the first experiment using battery as energy storage device. Furthermore, we conduct experiments accounting for the effect of the load device. We use the same PV and battery setup as the above experiment. The open circuit voltage of battery is set to 15 V. We consider two cases: (i) the load device will result in 1 C discharge rate of battery by itself (when PV is not considered), and (ii) it will result in 2 C discharge of battery. A discharge rate C implies that the battery can be depleted in hour(s). In the 1 C case, the charger output powers are 27.36 W and 37.91 W for the baseline and proposed systems, respectively. In the 2 C case, the charger output powers are 27.38 W and 38.19 W for the baseline and proposed 73 systems, respectively. Comparing with the results in Table V, we can see that the effect of load device on the charger output power (or MPTT) is very minor even when the load is quite high (when it can fully discharge the battery in half an hour.) The second experiment takes into account a PV system with three 60-cell PV modules with partial shading pattern shown in Figure 23. We use supercapacitor as energy storage device. We test the instantaneous output power level of the two PV systems. Table VI summarizes the output power improvement of the proposed PV system compared to the baseline system, given the shading pattern and the value. It also provides the near-optimal PV module configuration obtained by the reconfiguration control algorithm. The proposed reconfigurable PV system achieves up to 76% output power enhancement compared with the baseline system, which shows that the proposed PV module reconfiguration technique achieves more benefits for the string charger architecture. Figure 23. Partial shading pattern of the three PV modules in the 2nd experiment. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 74 Table VI. Improvement of instantaneous output power of the PV system in the second experiment. We investigate the maximum possible enhancement in the PV system output power using reconfiguration technique. We consider a PV system with three 60-cell PV modules with partial shading pattern shown in Figure 24. We use supercapacitor as the energy storage. The proposed PV module reconfiguration technique achieves up to 2.36X output power enhancement under this partial shading pattern as shown in Table VII. Figure 24. Partial shading pattern of the three PV modules for maximum output power improvement. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 75 Table VII. Improvement of instantaneous output power of the PV system in the third experiment to test the maximum gain. Finally, we test the overall efficiency of the two PV systems in a time period of 30 minutes over random shading. We use supercapacitor as the energy storage. We consider the following two test cases:  1/2 of the PV cells in the three PV modules are shaded, and the solar irradiance levels on these shaded PV cells are uniformly distributed within the range and change with time.  2/3 of all the PV cells are shaded, and the solar irradiance levels on these PV cells are uniformly distributed within the range . The proposed PV system updates its module configurations once per minute according to the current shading pattern and charger efficiency variation. We compare the electrical energy stored into the supercapacitors in the two systems during this time period. The proposed PV system achieves 53% and 88% improvements compared to the baseline system in the two test cases, respectively. 76 Chapter 4. Dynamic Reconfiguration of PV System in Hybrid Electric Vehicles With growing public consciousness about the greenhouse gas emissions and increase in the price of petroleum, hybrid electric vehicles (HEVs) are perceived as a core segment of future automotive market. HEVs employ both electric motors and internal combustion engines to improve the fuel economy. Batteries powering the electric motor in a HEV must be recharged by a generator in the HEV. On top of a regenerative braking-based battery charging scheme, a PV system mounted on the HEV can enable battery charging whenever there is solar irradiance. PV cells can be mounted on the roof panel, engine hood, trunk, and door panels of a HEV. In reality, the PV cells mounted on the HEV may receive different solar irradiances due to partial shading and variations in the solar irradiance incidence angle. In addition, PV cells may experience different temperatures. For instance, PV cells on the door panel have a different solar irradiance incidence angle from those on the roof. PV cells on top of the hood have a higher temperature than PV cells on the roof. Figure 3 and Figure 25 illustrate that a higher solar irradiance or a lower temperature lead to a higher output power at the MPP of the cell. Partial shading not only reduces the maximum output power of the shaded PV cell, but also makes non-shaded PV cells that are connected in series with the shaded one deviate from their MPPs. Similarly, a hotter PV cell makes PV cells that are connected in parallel with it deviate from their MPPs. Thus PV cells in a PV array with solar irradiance and/or temperature non-uniformity cannot operate at their MPPs simultaneously, and thus, the PV array suffers from significant power inefficiency. 77 Figure 25. PV cell (a) V-I and (b) V-P output characteristics under different temperatures. The red dots denote MPPs of a PV cell, where the PV cell achieves the maximum output power under certain temperature. In this chapter, we present a dynamic PV array reconfiguration technique, which updates the PV array configuration according to the change of irradiance and temperature distribution on the PV array during system operation in order to enhance the output power of PV systems. Our reconfiguration technique has the ability to simultaneously meet (to the extent that is possible) the MPPs for all PV cells even under non-uniform solar irradiance and temperature distributions. We use the PV reconfiguration architecture proposed in the previous chapter and provide an effective control algorithm. The control algorithm realizes near-optimal PV array reconfiguration accounting for solar irradiance and temperature distributions and charger efficiency variation. The algorithm is based on dynamic programming with polynomial time complexity so that it can be integrated into PV systems with insignificant computational overhead. Experimental results demonstrate that our proposed dynamic reconfiguration technique results in up to 6X improvement in PV system output power compared with a conventional fixed array structure PV system. 0 1 2 3 0 0.5 1 1.5 2 Voltage (V) Current (A) 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 Voltage (V) Power (W) 25 C 40 C 55 C G = 1.0 85 C 70 C 78 4.1 PV System Modeling in the HEV Framework Figure 26. System diagram of a PV-powered HEV. Figure 26 shows the system diagram of a PV-powered HEV. The PV array consists of a number of PV cells, mounted on the roof panel, engine hood, trunk, and door panels of the HEV. Conventional HEVs are equipped with battery packs as the energy storage system, connected to the traction motor via the generator charger. The PV-powered HEV also has an add-on PV system that consists of a PV array and a PV charger connecting the PV array to battery packs. The generator charger and PV charger can simultaneously charge battery packs. MPPT or MPTT techniques are integrated with the PV charger, and therefore, the PV charger can set the optimal operating point for the PV array. The PV array uses the reconfiguration structure stated in Section 3.1, and consists of a number of series-connected PV cell groups where each PV group consists of (potentially a different number of) PV cells connected in parallel. Although mounting 79 positions of PV cells are fixed on the HEV, PV array reconfiguration technique adaptively changes the electric connection of PV cells in response to changes in the environmental condition. Figure 27 provides the equivalent circuit of the PV system shown in Figure 26, where the battery bank in Figure 27 represents the battery packs on HEV. PV Charger Battery V bat I bat_pv P conv V pv I pv PV Cell Array Group #1 Group #2 Group #3 Generator Charger I bat_motor From Motor Figure 27. Equivalent circuit of a PV system on a HEV. 4.2 Problem Formulation 4.2.1 Problem Statement The solar irradiance distribution over the PV array on the HEV changes over time, due to driving direction change and shading caused by buildings, clouds, and other moving vehicles. Temperature variation is also severe because of the engine heat dissipation and convection due to airflow. We should perform the PV array reconfiguration algorithm at every decision epoch to maximize the PV system output power. For notational convenience, we drop the time index . We denote the solar irradiance and PV cell temperature of the i-th ( ) PV cell by and , respectively. We obtain and values using irradiance and temperature sensors in the system. We denote the output voltage and current of the i-th PV cell by and , respectively. Given and , the relationship between and is given by the PV 80 characteristics discussed in Section 2.1. The PV array configuration is the major control variable of the reconfiguration algorithm. We develop V-I characteristic of the PV array with a configuration that corresponds to the alphabetical partitioning , , , and as follows. We use to denote the output voltage of the j-th PV group, and use and to denote the output voltage and current of the PV array, respectively. The and values can be effectively controlled by the PV charger. From the Kirchhoff’s laws, we have: (4.1) , for and (4.2) and (4.3) We control and maintain the PV array operating point using the PV charger. The power loss of the PV charger, , is calculated from its input voltage, input current, output voltage, and output current, i.e., , , , and , respectively, according to the power modeling in Section 2.2. Due to the energy conservation law, we have: (4.4) where and are the battery’s terminal voltage and charging current from the PV system, respectively. The formal problem statement of the PV array reconfiguration (PAR) problem for HEV is described as follows: PAR Problem Statement: Given the solar irradiance and the temperature of each i-th ( ) PV cell in the PV array, and the battery terminal voltage , find the optimal PV 81 array configuration and the PV array operating point , such that the PV charger output current is maximized. 4.2.2 Problem Decomposition We propose a near-optimal solution to the PAR problem using a novel method that consists of a kernel algorithm, an MPTT control algorithm, and an outer loop finding the optimal number of PV groups in the PV array. The kernel algorithm finds the optimal PV array configuration such that all PV cells can work near their MPPs. The kernel algorithm derives the optimal PV array configuration for a given value to maximize the PV array MPP output power. The MPP output power of the PV array is the maximum achievable value under a given solar irradiance and temperature distribution on the PV array, and a given PV array configuration. The subsequent MPTT control algorithm aims to find the optimal PV array operating point taking into account the PV charger efficiency variation, such that is maximized. We use the optimal PV array configuration obtained via the kernel algorithm for the subsequent MPTT control algorithm. The outer loop enumerates all the possible values ( ) and finds the optimal one. The outline of this method is given in Algorithm 4. The next section describes the kernel algorithm in detail, which is the key part in Algorithm 4. 82 . 4.3 The Kernel Algorithm The maximum solar power that can be drawn from a PV array is the sum of MPP output power of all the PV cells in that array. Under the non-uniformity of solar irradiance and temperature, PV cells in a balanced n-by-m configuration can hardly operate at their MPPs simultaneously. Our reconfiguration technique aims at making all the PV cells operate at, or close to, their MPPs by updating the configuration of the PV array according to current solar irradiance and temperature distributions. In Figure 28, each PV cell is labeled with its MPP output voltage and current. If the PV array employs the configuration as shown in Figure 28, all PV cells work at their MPPs simultaneously with the PV array output voltage of 3.6V (1.5+0.9+1.2=3.6V) and output current of 4.5A (1.5+1.5 +1.5=4.5A). The goal of our kernel algorithm is to find a configuration for the PV array such that (i) PV cells within the same PV group have very close MPP output voltages and (ii) the sum of MPP output currents of PV cells within the same group is quite similar among all the PV groups. 83 S p,1 1 2 3 4 5 8 S p,6 6 S p,7 7 S p,2 S p,3 S p,4 9 10 11 12 S p,9 S p,10 S p,11 S s,5 S s,8 Group #1 Group #2 Group #3 1.5V 1.0A 1.5V 1.0A 1.5V 1.5A 1.5V 0.5A 1.5V 0.5A 0.9V 1.5A 0.9V 1.5A 0.9V 1.5A 1.2V 1.0A 1.2V 0.5A 1.2V 1.5A 1.2V 1.5A Figure 28. An illustration of the PV array reconfiguration. 4.3.1 Simplifying the Problem The kernel algorithm finds the optimal PV array configuration to maximize the PV array MPP output power for the specified number of PV groups in the array. We name the problem simplified PAR (SPAR) problem. We further simplify the SPAR problem assuming ideal PV cells, and name the new problem ideal PV Cell-based SPAR (IC-SPAR) problem. The V-I characteristic of an ideal PV cell is a step function such that (4.5) where and denote the MPP output voltage and current of the i-th PV cell in the PV array, which are calculated from the solar irradiance and temperature . The ideal PV cell is an effective approximation of the real PV cell. We use , , and to denote the MPP output voltage, current, and power, respectively, of the ideal PV cell array with a configuration of . These values are calculated as: (4.6) 84 (4.7) (4.8) Thus the IC-SPAR problem becomes: find the optimal array configuration (corresponding to the optimal alphabetical partitioning , , , and ) for a given value, such that the value calculated from (4.8) is maximized. 4.3.2 The Pareto-Optimal Substructure We consider a reduced-size PV array consisting of the first PV cells of the original -cell PV array. The reduced-size L-cell PV array is composed of PV groups. We denote a possible configuration of the L-cell PV array by . Finding the optimal -cell PV array configuration is equivalent to finding the optimal alphabetical partitioning of the set = . The configuration is optimal when the MPP output power is maximized. We use and to denote the MPP output voltage and current of the PV array with configuration , respectively. We name the new problem the reconfiguration problem. The more general reconfiguration problem becomes the original IC-SPAR problem when and . We find the Pareto- optimal substructure property of this reconfiguration problem as follows. Definition 1 (Pareto superiority): Consider two configurations and of the reconfiguration problem. Configuration is Pareto-superior to if 85 and , or and . Definition 2 (Pareto-optimal configuration and Pareto-optimal set): A configuration is a Pareto-optimal configuration of the reconfiguration problem if no other configuration of the reconfiguration problem is Pareto superior to . The set of all Pareto-optimal configurations of the reconfiguration problem is the Pareto-optimal (configuration) set of the problem. Next we present two theorems. Theorem 1 demonstrates that the optimal configuration of each reconfiguration problem is an element in the Pareto-optimal set of that problem, whereas Theorem 2 provides the Pareto-optimal substructure property. Proofs are omitted due to space limitation. Theorem 1 (Optimal configuration and Pareto-optimal configurations): The optimal configuration (with the maximum MPP output power ) of the reconfiguration problem is an element in the Pareto-optimal set of the reconfiguration problem. Theorem 2 (The Pareto-optimal substructure): Suppose that a configuration , corresponding to an alphabetical partitioning , is a Pareto-optimal configuration of the reconfiguration problem, and the last (the -th) PV group consists of cells. Then the alphabetical partitioning (or the corresponding configuration) must also be Pareto-optimal in the reconfiguration problem. 86 4.3.3 The MCHV Function Based on the above-described Pareto-optimal substructure property, one can find the optimal solution of the original IC-SPAR problem by storing the complete Pareto-optimal set of each reconfiguration problem with and , in a way similar to the conventional dynamic programming. However, cardinality of the Pareto-optimal set of the reconfiguration problem could be in the order of , which implies that the Pareto-optimal set cardinality could be exponential. Hence, it is impractical to store the whole Pareto-optimal set of each reconfiguration problem to solve the original IC-SPAR problem. For effectively solving the IC-SPAR problem, we introduce a new function, named maximal_current_higher_voltage (MCHV) function for each reconfiguration problem, denoted by . The MCHV function derives the maximum value achieved by any configuration of the reconfiguration problem satisfying , i.e.,: (4.9) If there exists no configuration satisfying for relatively large values, we have . The domain of the MCHV function for the reconfiguration problem is given by , where and values are the estimated lower and upper bounds of the value for any configuration , respectively. The and values are calculated by (4.10) (4.11) 87 Finding tighter upper and lower bounds is an interesting problem, but it is outside the scope of this section. An illustration of the MCHV function is shown in Figure 29. The MCHV function is a non-increasing function with a set of points of discontinuity. Each point of discontinuity in the MCHV function corresponds to a Pareto-optimal configuration of the reconfiguration problem. Point shown in Figure 29 corresponds to a Pareto-optimal configuration with and , since we have for any configuration of the reconfiguration problem satisfying . Finally, the MCHV function of each reconfiguration problem fully represents the Pareto-optimal set of that problem. ,, ,, ˆ IC MPP L l pv max V V     , , Ll a MCHV a V f V , Ll MCHV f     , , Ll b MCHV b V f V ˆ IC MPP,L,l pv, ,min V ( 1) k   k   ... ... ˆ IC MPP,L,l pv, ,min V ˆ IC MPP,L,l pv, ,min V Figure 29. An illustration of the MCHV function. 4.3.4 Implementation of the Kernel Algorithm In the MCHV function for an reconfiguration problem, the total number of points of discontinuity could also be exponential. Hence, we represent the MCHV function in discrete domain. We define a three-dimensional matrix , where 88 stores the value. In the above expression, , and is the predefined precision level in the kernel algorithm. When , the array fully represents the MCHV function , which is however not practical since the size of is infinite. When , the Pareto-optimal configurations of the reconfiguration problem can be classified into two categories: observable and unobservable Pareto-optimal configurations. Consider a Pareto-optimal configuration of the reconfiguration problem satisfying . Then is observable if , and is unobservable otherwise (i.e., .) An example of observable Pareto- optimal configuration is the one corresponding to the point in Figure 29, and an example of unobservable Pareto-optimal configuration is the one corresponding to the point The following lemma and theorem prove that we can find an approximate solution of the IC-SPAR problem by considering only the observable Pareto-optimal configurations in the kernel algorithm implementation. Lemma 1 (Bound on MPP output power of unobservable Pareto-optimal configurations): For each unobservable Pareto-optimal configuration of the reconfiguration problem satisfying , there exists a corresponding observable Pareto-optimal configuration with and . Besides, the MPP output power of the unobservable Pareto-optimal configuration is bounded as follows: (4.12) 89 Theorem 3 (Approximate solution): We use to denote the near-optimal solution (configuration) of the IC-SPAR problem obtained by considering only the observable Pareto- optimal configurations in the kernel algorithm implementation, with details shown in Algorithm 5. We use to denote the true optimal solution (configuration) of the IC-SPAR problem. Then the MPP output power of the near-optimal configuration is bounded as follows: (4.13) where is a function of the predefined parameter and converges to 0 when . Since the number of observable Pareto-optimal configurations for each reconfiguration problem is bounded by , practical implementation of the kernel algorithm is feasible. Details of the proposed dynamic programming based kernel algorithm to solve the IC-SPAR problem are provided in Algorithm 5. The solution of the IC-SPAR problem serves as the near-optimal solution of the SPAR problem, and the original PAR problem can be further solved using Algorithm 4. 90 4.4 Experimental Results We compare the performances of two PV systems: one is the proposed system with reconfiguration technique, and the other is the baseline system without reconfiguration technique. Both PV systems contain a PV array of 60 PV cells, a charger and a battery bank. The PV array in the proposed system has the reconfiguration structure shown in Figure 12 while the PV array in the baseline system has a fixed 6x10 configuration, where 6 PV cell groups are connected in 91 series and each PV group consists of 10 parallel-connected PV cells. The output power of baseline system is enhanced by the improved MPPT technique [11] and the integration of a bypass diode within each PV cell [12]. (a) (b) G = 1.00 G = 0.50 G = 0.25 G = 0.15 90 C 10 C Figure 30. (a) Solar irradiance and (b) temperature distributions on the PV array. We simulate the output power levels of the two systems under the solar irradiance and temperature distributions as shown in Figure 30. Although the mounting locations of PV cells in the proposed system are the same as those of PV cells in the baseline system, the PV array in the proposed system shows different configurations achieved by the reconfiguration technique, in response to different values. Table VIII summarizes the output power improvement of the proposed system compared to the baseline system with different battery terminal voltage values. The proposed system achieves up to 6X improvement in the output power level with the solar irradiance and temperature distributions as shown in Figure 30 when 45 V. Table VIII also provides near-optimal configurations obtained from reconfiguration algorithm. Table VIII. Output power improvement of the proposed PV system with different values. 92 0 5 10 15 20 25 0 10 20 30 40 Voltage (V) Power (W) PV System with Reconfiguration Baseline System V bat = 30 V Figure 31. The V-P characteristics of the two PV systems. In Figure 31, we plot the V-P characteristics of PV arrays in the proposed system and the baseline system with the irradiance and temperature distributions given in Figure 30 when 30 V. The proposed system achieves significantly higher peak output power than that of the baseline system. Since the operating point of the PV array is always set to be its MPP to extract maximum output power out of the PV system, the proposed system outperforms the baseline system in Figure 31. Moreover, the V-P characteristic of the proposed system is unimodal even under irradiance and temperature non-uniformity with the proposed PV reconfiguration technique, while the V-P characteristic of the baseline system has multiple peaks. Therefore, the standard MPTT technique could be incorporated into the proposed system without extra modification. However, the MPPT or MPTT technique must be modified to track a global optimal operating point instead of a local optimal point in the baseline system. A typical modification would be searching the whole operating range of the PV array output voltage and current, which results in significantly higher time overhead compared with standard MPPT or MPTT technique based on the perturb and observe approach. 93 4.5 PV Array Reconfiguration Implementation and Optmization for Partial Solar Powered Vehicles Combined usage of the PV modules on the rooftop, hood, trunk, quarter, and door panels are challenging in maintaining high performance in a string charger architecture. This section discusses the implementation and optimization of a fast PV cell array reconfiguration for the partial solar powered EV. We use the dynamic PV module reconfiguration architecture discussed previously. However, the solar irradiance levels are rapidly changing in the case of PV modules on the vehicle due to nearby shading and direction changes. We implement a high-speed, high- voltage PV reconfiguration switch network with IGBTs (insulated-gate bipolar transistors) and a controller. We derive the optimal reconfiguration period considering the on/off delay of IGBT, CAN (control-area network) delay, computation overhead, and energy overhead. We carefully decide the reconfiguration policy based on the solar irradiance/driving profiles using adaptive learning method [138]. We solve a design-time optimization problem of deriving the optimal granularity of PV reconfiguration to achieve a desirable trade-off of performance and reconfiguration complexity/overhead. This section also introduces partial PV installation. As PV modules are still costly, installation of a low-efficiency PV module is a waste. For example, the driver-side quarter and door panels do not have meaningful solar irradiance when a driver commutes to the northbound in the morning and the southbound in the afternoon. We implement an onboard PV irradiance monitoring sensor network and collect various irradiance profiles by the driving location and time, and then we customize the PV module installation according to the driving pattern. 94 We evaluate the proposed fast dynamic PV reconfiguration technique based on the actual implementation of reconfiguration network and controller. Experiments show that the fast dynamic PV array reconfiguration increases 423.0 W power from the baseline. The customized PV installation reduces 22.3% PV cell cost showing only 5.6% reduction of power generation output. 4.5.1 Solar Sensor Network and Irradiance Profile Acquirement We build a Zigbee-based solar sensor nodes network and a logger program to acquire real solar irradiance profile on each side of vehicles during driving. The actual implementation of a sensor node in the network is provided in Figure 32. Zigbee is a wireless network protocol to create personal area networks, which is commonly used for applications requiring low power, low data rate, and long battery life. The physical transceiving range of Zigbee protocol is up to 120 meters, however, we disable the boost mode to reduce power consumption. We use dual AAA-size batteries to supply power for each node without DC-DC converter to minimize power loss. A Zigbee transceiver module and an ambient sensor can operate by 2.4 V to 3.3 V supply voltage level, which has a enough operation margin for dual alkaline batteries. The lifetime of dual AAA batteries is more than 12 hours, which is enough for recording solar irradiance profile of one day driving. The Zigbee module automatically reads value from the ambient sensor with its internal ADC and sends it to a receiving node every 50 ms with 250 kbits/s data transmission speed. A specially designed logger program collects irradiance sensor data from the receiving node with vehicle speed and location information from GPS including latitude, longitude, altitude and time. We install magnets to each corner of a sensor node so that the sensor nodes can stick to vehicle easily and firmly. The purpose of this sensor node is to (i) easily attach to any 95 vehicle in a very short amount of time, and (ii) so that we can test various vehicles and locations and collect benchmark solar irradiance profiles. Figure 32. Zigbee-based solar sensor node to measure the solar irradiance. We attached five sensor nodes at hood, roof, trunk, left side and right side of a vehicle to measure benchmark profiles of , , , and , respectively. Finally we drive a vehicle along six paths to collect real benchmark vehicle drive profiles: Seoul to Incheon airport, Ontario to Riverside, west Los Angeles to Indio, west Los Angeles to Carson, Riverside, and west Los Angeles to Riverside. Details are shown in Table IX. 96 Table IX. Recorded driving profiles. 4.5.2 Reconfiguration Hardware Design and Optimization PV Reconfiguration Hardware Design In the PV reconfiguration structure proposed in Chapter 3, each switch consists of one MOSFET gate driver and one pair of N-type MOSFETs. However, the reconfigurable PV module array on moving vehicles has some unique requirements. First, the reconfiguration speed should be highly fast. For example, if there is a shade by an approaching four-meter-long vehicle in the opposite side and the speeds of both vehicles are 80 km/hour, the shade exists for a maximum of 180 ms. Moreover, rapid direction changes of vehicles will result in fast changing in solar irradiance levels at each side of the vehicles. Hence, fast reconfiguration within a few milliseconds reconfiguration time is required to fully exploit the potential benefits of the dynamic reconfiguration capability during vehicle driving. Second, high-voltage or high-current gate control is required for vehicular PV array reconfigurations. We implement an IGBT (insulated- gate bipolar transistor)-based reconfiguration network to meet both requirements. We carefully select commercial IGBTs and gate drivers for switches in the reconfiguration network. The selected IGBT IXXK200N65B4 can handle voltage and current ratings of 650 V and 370 A, respectively, which is enough rating for vehicular PV arrays. We select gate driverMC33153 that has a short propagation delay of few hundreds nanoseconds to control the 97 IGBTs. We use photo-coupler isolation between the high-voltage IGBT side and the controller logic side to prevent from damage due to power surge. We connect IGBT with 65V/5A rating power supply and apply square-wave input voltage on the gate driver to observe the step response of IGBT. The response waveform of output voltage is not distorted until the input frequency reaches tens of kHz. This shows the stability of the IGBT and gate driver selections. Then we implement a communication system based on the controller area network, as known as CAN. The CAN standard is established specially for vehicles, which require high stability. The CAN network topology is also a bus structure so that we can easily attach sensor nodes to the communication network. We carefully select ADM3053 as an isolated CAN physical layer transceiver with LM3S2965 as the control processor, which supports hardware layers of CAN communications. 1 Mbps communication speed in transmission will make the transmission delay below 1 ms. Overhead Analysis In order to derive the optimal reconfiguration control policy, we need a thorough analysis of both timing overhead and energy overhead during PV array reconfiguration. During PV array reconfiguration, the following processes are required: sensing the irradiance levels, transmitting the irradiance data from sensors to the central controller, computing the optimal configuration, changing the ON/OFF states of IGBTs, and performing MPPT control after reconfiguration. Hence, the timing overhead of PV array reconfiguration is comprised of the following components:  Sensing delay: With current sensor network setup, each sensor node senses and converts the solar irradiance value in every 50 ms, which is the sensing period. The sensing delay is less than 10 s based on the sensor ADC setup. 98  Network delay: The transmission delay is no more than 1 ms in the sensor network using CAN transmission protocol.  Computation overhead: The reconfiguration control algorithm is a polynomial-time optimal algorithm. For a moderate-scale PV array with 60 PV cells, it takes only 3 - 4 ms to calculate the optimal configuration on a 3.0 GHz desktop computer and should take less than 10 ms on a typical ARM-based embedded processor (as the reconfiguration controller) [29].  Reconfiguration delay: Our experiments show that the gate driver and IGBT can reconfigure within 10 s with only a little distortion of waveform. so 1 ms will be a safe (conservative) reconfiguration delay.  MPPT control overhead: The delay of a perturb & observe (P&O)-based MPPT control is typically less than 2.5 ms. The total timing overhead is the sum of the above-mentioned delay components, and we use 15 ms in our experiments to derive the optimal reconfiguration period. The minimum reconfiguration period is 50 ms which is limited by the sensing frequency. For the energy overhead, the vehicular PV system will have zero output power during reconfiguration (i.e., changing the ON/OFF states of IGBTs) and have sub-optimal output power during P&O-based MPPT control. We use a conservative estimate that the output power will be zero also during the MPPT control period. 99 Figure 33. SM5 official dimension from the vehicle manual. Reconfiguration Period Optimization Under rapid-changing irradiance levels on each side of the vehicle during driving, it is not possible to perform optimal reconfiguration as long as some irradiance level changes due to the timing and energy overhead associated with reconfiguration. Thus, we need to determine the optimal reconfiguration period (and policy) for the vehicular PV array based on the overhead analysis in the previous subsection. A larger reconfiguration period may not be able to capture the fast changes in solar irradiance levels. On the other hand, a smaller reconfiguration period will induce higher timing overhead and energy overhead, and may eventually degrade the PV system performance. We use the adaptive learning method to derive the optimal reconfiguration period in an online manner [138]. We maintain multiple candidate reconfiguration period values, and choose one value with the currently highest performance at the beginning of each evaluation period (say, 10 minutes.) At the end of this evaluation period, we evaluate all the reconfiguration period values 100 and update their performance using an exponential weighting function [46], and then choose the reconfiguration period value with the highest updated performance level. Experimental Results We compare the performance between the proposed reconfigurable vehicular PV system with two baseline systems. The proposed system installs reconfigurable PV array on the rooftop, hood, trunk, and left and right door panels. This should be the largest area for potential solar energy harvesting. We consider two baseline setups. Baseline B1 installs solar modules only on the rooftop, hood, and trunk panels without reconfiguration. Baseline B2 installs solar modules on the rooftop, hood, trunk, quarter, and door panels without reconfiguration. We measure a mid- size family sedan Renault-Samsung NEW-SM5 car and observe the following area parameters: hood (bonnet): 1.6 m 2 (1.024 m by 1.565 m), left door, right door: 1.7 m 2 for each (0.616 m by 2.760 m), roof: 1.99 m 2 (1.274 m by 1.565 m), trunk: 0.63 m 2 (0.400 m by 1.565 m). Note: All parameter values are measured on a real car, which are slightly smaller than official dimension in Figure 33. We assume fixed-size PV cells with 0.15 m 2 area, 20 V MPP voltage, and 2.25 A MPP current at G = 1000 W/ m 2 . We assume 200 V terminal voltage of the vehicle battery pack. We consider a realistic solar charger model with efficiency variations [109]. We first consider a fixed reconfiguration period of 0.5 s, and compare the performance, i.e., the average output power, between the proposed and baseline setups on the six benchmark solar irradiance profiles. Comparison results are illustrated in Table X. We can observe that the proposed system significantly outperforms baseline setups by a maximum of 423.0W improvement in average output power. Comparing between two baseline systems, we observe that B1 even often outperforms B2 because the solar irradiances on the left side and right side of the vehicle are often changing and have smaller magnitude compared with the rooftop, which 101 degrades the output power of the PV system. Moreover, we plot the PV array output power versus time of the proposed system and two baseline systems in Figure 34. We can observe that the proposed system consistently outperforms the two baseline systems over the whole time range. Table X. Performance comparison between the proposed reconfigurable PV system with baseline PV systems on the six benchmark profiles. Figure 34. Performance comparison between the proposed reconfigurable PV system with baseline systems. Furthermore, we consider the optimization of reconfiguration period. Figure 35 shows the average output power of the proposed reconfigurable PV system with different reconfiguration 102 period values on two benchmark profiles “Ontario to Riverside" and “Riverside". Please note that 50 ms is the lowest possible reconfiguration period because it is the sensing period. Our experiments show that the optimal reconfiguration periods for the six benchmark profiles are around 0.5 s - 1 s in general, in order to achieve a trade-off between lower timing/energy overhead and fast reconfiguration capabilities. Figure 35. Performance (average output power) of the proposed reconfigurable PV system with different reconfiguratin period values on two benchmark profiles. 4.5.3 Design-time Optimization of the Vehicular PV Array In this section, we discuss the design-time optimization of the vehicular PV array, including (i) deriving the optimal granularity of PV array reconfiguration, i.e., optimizing the size of a PV cell, and (ii) partial PV array installation. These optimizations are performed statically in the design time, and cannot be altered after installation. Optimization of the PV Cell Size The PV cell is the basic unit in the reconfiguration technique, and its size is essentially trade- off between the lower additional capital cost and reconfiguration complexity, and performance 103 enhancement. Basically, a larger PV cell size reduces the cost of the reconfigurable PV array architecture since fewer switches are required for reconfiguration and also reduces the computation overhead for the optimal array configuration. A smaller PV cell, on the other hand, achieves better flexibility and thus higher performance against partial shading. The PV cell size optimization is performed at the system design stage. We aim to find the optimal PV cell size with a capital cost limit to achieve a desirable trade-off between the lower PV system capital cost and reconfiguration overhead, and enhanced performance against partial shading. In the outer loop of algorithm, we use binary search to find the most desirable PV cell size subject to the capital cost constraint. For each given PV cell size inside the loop, we evaluate the PV system performance using the six solar irradiance benchmarks using the corresponding optimal reconfiguration policy. The timing overhead and energy overhead are taken into account in this evaluation procedure. We set the PV cell lower limit by 0.1 m 2 and compare the performance of reconfigurable PV array with different PV cell sizes on the six solar irradiance profiles. We adopt a fixed reconfiguration period of 0.5 s. Table XI shows the comparison results including four possible PV cell sizes: 0.1 m 2 , 0.15 m 2 , 0.25 m 2 , and 0.5 m 2 . We observe that a finer grained PV cell will result in higher average output power (at most 26.3% higher average output power when comparing between the 0.1 m 2 case and 0.5 m 2 case) due to the higher flexibility in reconfiguration. We would like to point out that the timing and energy overhead of all these testing cases are within the estimates discussed before. 104 Table XI. Performance comparison of the reconfigurable PV system with different PV cell sizes on the six benchmark profiles. Partial PV Array Installation In this part, we introduce partial PV array installation. As PV modules are still costly, installation of a low-efficiency PV module is a waste. For example, the driver-side quarter and door panels do not have meaningful solar irradiance when a driver commutes to the northbound in the morning and the southbound in the afternoon. Based on the PV irradiance profiles collected using the solar sensor network, we propose to customize the PV module installation according to different driving patterns. More specifically, we consider the following partial solar array installation with reconfiguration: the rooftop, hood and trunk are equipped with PV cells, and either the left side or right side of the vehicle is equipped with PV cells. We compare the performance (output power) of these two customized PV array installation cases with the optimal reconfigurable vehicular PV system. We assume fixed-size PV cells of 0.15 m 2 and adopt a fixed reconfiguration period of 0.5 s. Table XII shows the comparison results. We can observe that for some benchmark profiles such as “Ontario to River side", partial PV installation will result in significant output power degradation. However, this is opposite for some other benchmark profiles such as “Riverside" or “Incheon Airport", because the solar irradiance levels on either the left side or right side (or both) of the vehicle is relatively low in the whole benchmark. In this 105 cases, customized PV installation will be beneficial because it can significantly reduce the capital cost of the vehicular PV system. For example, if a user commutes through “Riverside" trace everyday such as a bus in a public transportation system, the customized PV installation reduces 22.3 % PV cell cost showing only 5.6 % reduction of power generation output. Table XII. Performance comparison between the optimal reconfigurable PV system with two customized PV systems on six benchmark profiles. 106 Chapter 5. Capital Cost-Aware Reconfigurable PV System Design and Optimization In the PV reconfiguration technique discussed in Chapter 3 and Chapter 4, a large number of programmable switches are used for the reconfigurable PV panel (module), which induce significant additional capital cost of the PV system. The basic unit of the PV panel reconfiguration proposed in Chapter 3 is one single PV cell, i.e., the PV panel reconfiguration changes the connection of PV cells. To reduce the additional capital cost, this chapter proposes to perform PV panel reconfiguration up to the scale of PV macro-cells. A PV macro-cell is comprised of a number of PV cells connected in series and parallel. A PV panel consists of multiple identical PV macro-cells. The PV panel reconfiguration changes the connection of the PV macro-cells, whereas the size and structure of a PV macro-cell are fixed and determined at the system design stage. That is to say, the PV macro-cell instead of a single PV cell becomes the basic unit in PV panel reconfiguration. In this way, the number of programmable switches can be significantly reduced. The size of a PV macro-cell essentially represents the trade-off between higher performance for combating partial shading and lower PV system capital cost. Basically, a larger PV macro-cell size can reduce the cost overhead of the reconfigurable PV panel since fewer switches are required, while a smaller PV macro-cell size can achieve better flexibility and thus higher performance in combating the partial shading effect. This chapter provides a design method to decide the optimal size of a PV macro-cell for the PV system design to achieve a balance between the performance and the capital cost. Experiments demonstrate 60% enhancement in PV system output power under partial shading with a negligible capital cost increase of 3.7% by selecting the proper macro-cell size. 107 5.1 Capital Cost-Aware Reconfigurable PV Panel To mitigate the significant output power loss due to partial shading, the conventional PV panel with fixed configuration is replaced with a reconfigurable PV panel in Chapter 3 and Chapter 4, where each PV cell is integrated with three programmable switches to facilitate the PV panel reconfiguration. The PV panel configuration, i.e., the connection topology of PV cells in the PV panel, can be dynamically changed according to the partial shading pattern. To reduce the additional capital cost induced by programmable switches, this chapter proposes to group a number of PV cells into a PV macro-cell and perform PV panel reconfiguration at the PV macro- cell level. The PV panel reconfiguration changes the connection of the PV macro-cells, whereas the size and structure of a PV macro-cell are fixed and determined at the system design stage. In this way, the number of programmable switches can be significantly reduced. Figure 36 shows the reconfigurable PV panel structure comprised of PV macro-cells. The connection of PV macro-cells can be dynamically changed during PV panel reconfiguration, whereas the physical locations of the PV macro-cells are fixed after the PV system installation. Please note that Figure 36 does not represent the physical locations of the PV macro-cells. Figure 36 shows an example, where the PV macro-cell has a size of . Please note that the size of the PV macro-cell should be a variable for the design optimization of the PV system. Chapter 3 and Chapter 4 simply sets the size of the PV macro-cell to be , i.e., it does not consider the effect of the PV macro-cell size on the trade-off between higher performance for combating partial shading and lower PV system capital cost. 108 S PT,1 S PB,1 S S,1 S PT,i S S,i S PB,i P-switch P-switch S-switch #1 #i #N Switch-Set PV Macro-Cell Figure 36. The capital-aware reconfigurable PV panel structure. Figure 37. The switch circuitry. Each PV macro-cell except for the -th one is integrated with three switches i.e., a top P- switch , a bottom P-switch , and a S-switch . Each switch in this structure is comprised of one MOSFET gate driver and one or multiple pairs of n-type MOSFETs. Figure 37 shows the detailed circuitry of a switch. The two n-type MOSFETs in one MOSFET pair are connected back to back. The number of MOSFET pairs is proportional to the maximum current that the switch can conduct. The minimum number of MOSFET pairs required for a switch is calculated by Gate Driver On/Off 109 (5.1) where is the number of PV cells connected in parallel in a macro-cell, and is the maximum drain current of a MOSFET. The required number of MOSFETs per PV macro-cell is . S PT,1 S PB,1 S S,1 #1 S PT,2 S PB,2 S S,2 #2 S PT,3 S PB,3 S S,3 #3 S PT,4 S PB,4 S S,4 #4 S PT,5 S PB,5 S S,5 #5 S PT,6 S PB,6 S S,6 #6 S PT,7 S PB,7 S S,7 #7 S PT,10 S PB,10 S S,10 #10 S PT,11 S PB,11 S S,11 #11 #12 PV group 1 PV group 2 PV group 3 S PT,9 S PB,9 S S,9 #9 S PT,8 S PB,8 S S,8 #8 Figure 38. An example of PV panel reconfiguration. The PV panel reconfiguration can be conducted by controlling the ON/OFF states of these switches. The two P-switches of a PV macro-cell are always in the same state, and its S-switch must be in the opposite state as the P-switches. The P-switches connect PV macro-cells in parallel forming a PV group, and the S-switches connect the PV groups in series. Figure 38 is an example of PV panel reconfiguration. Figure 38 presents the connection of PV macro-cells instead of their physical locations. The first four PV macro-cells are connected in parallel to form PV group 1; the next three PV macro-cells form PV group 2; and the last five PV macro-cells 110 form PV group 3. These three PV groups are series-connected by S-switches of the fourth and the seventh PV macro-cells. A reconfigurable PV panel comprised of PV macro-cells can include an arbitrary number (less than or equal to ) of PV groups. The number of parallel-connected PV macro-cells in the -th PV group should satisfy (5.2) where is the number of PV groups. Such a configuration is denoted by . This configuration can be viewed as a partitioning of the PV macro-cell index set , where the elements in denote the indices of PV macro-cells in the PV panel. The partitioning is denoted by subsets , , …, and of , which correspond to the PV groups consisting of , , …, and PV macro-cells, respectively. The subsets , , …, and satisfy (5.3) and , for and (5.4) The indices of PV macro-cells in PV group must be smaller than the indices of PV macro- cells in PV group for any due to the structural characteristics of the reconfigurable PV panel, i.e., for and satisfying . A partitioning satisfying the above properties is called an alphabetical partitioning. The PV cells in a PV panel are usually physically placed into an array, where there are rows and columns of PV cells. This is the physical (not electrical) layout of a PV panel. The PV macro-cell has a size of , i.e., PV cells connected in series and PV cells connected in parallel to form a PV macro-cell. and must be factors of and , respectively. Given the 111 size of the PV panel, i.e., , a larger PV macro-cell size indicates a smaller number of PV macro-cells and a smaller number of switches and therefore lower capital cost of the PV system. A smaller PV macro-cell size enables more flexibility in the PV panel reconfiguration, thereby achieving better performance against partial shading. Two indexing methods are defined to locate the PV cells in the PV panel. First, a PV cell can be indexed using its global coordinate , which means that it is in the -th row and the -th column of the PV cell array. The global coordinates represent the physical locations of the PV cells, and the solar irradiance level on a PV cell depends on its physical location. Therefore, the global coordinates are used to relate the PV cells to the solar irradiance levels on them. Second, a PV cell can be indexed using its local coordinate , which means that this PV cell belongs to the -th PV macro-cell and it is in the -th row and the -th column of the -th PV macro-cell. The local coordinates present the relative locations of PV cells in the PV macro-cells. The PV panel reconfiguration method only changes the electrical connection of the PV macro- cells, while the electrical connection of PV cells within a PV macro-cell (i.e., a series-parallel connection of PV cells with cells in series and cells in parallel) is fixed during system operation. Therefore, the local coordinates are used to represent the electrical structures of the PV macro-cells and the PV panel. 112 (1,1,1) m m = 6 n = 6 s = 2 p = 3 (1,1,2) m (1,1,3) m (1,2,1) m (1,2,2) m (1,2,3) m (2,1,1) m (2,1,2) m (2,1,3) m (2,2,1) m (2,2,2) m (2,2,3) m (4,1,1) m (4,1,2) m (4,1,3) m (4,2,1) m (4,2,2) m (4,2,3) m (3,1,1) m (3,1,2) m (3,1,3) m (3,2,1) m (3,2,2) m (3,2,3) m (5,1,1) m (5,1,2) m (5,1,3) m (5,2,1) m (5,2,2) m (5,2,3) m (6,1,1) m (6,1,2) m (6,1,3) m (6,2,1) m (6,2,2) m (6,2,3) m (1,1) o (1,2) o (1,3) o (1,4) o (1,5) o (1,6) o (2,1) o (2,2) o (2,3) o (2,4) o (2,5) o (2,6) o (3,1) o (3,2) o (3,3) o (3,4) o (3,5) o (3,6) o (4,1) o (4,2) o (4,3) o (4,4) o (4,5) o (4,6) o (5,1) o (5,2) o (5,3) o (5,4) o (5,5) o (5,6) o (6,1) o (6,2) o (6,3) o (6,4) o (6,5) o (6,6) o (a) (b) Figure 39. The two indexing methods: (a) “global coordinates” and (b) “local coordinates” of PV cells in the PV panel. Figure 39 shows the two indexing methods of PV cells, in which the PV panel has a size of , and the PV macro-cell has a size of . As shown in Figure 39 (b), the PV macro-cells 1~6 are indexed in a zigzag manner. The first PV macro-cell is on the left of the second PV macro-cell and the third PV macro-cell is on the right of the fourth PV macro-cell. This is because in the reconfigurable PV panel, the -th and the -th PV macro-cells have electrical connection between them as shown in Figure 36 and Figure 38. Thus, as shown in Figure 39 (b), by placing the third PV macro-cell right below the second PV macro-cell, the length of wires connecting the two PV macro-cells is shortened. The local coordinates reflect the electrical connection of PV cells within a PV macro-cell. A mechanism is needed to transfer the local coordinates into the global coordinates. Let denote the function transferring the local coordinates into the global coordinates. This function is derived as follows. First, the PV panel is considered as a PV macro- cell array, which consists of rows and columns of PV macro-cells. This viewpoint only 113 represents the physical locations of PV macro-cells. Suppose the -th PV macro-cell is in the -th row and the -th column of the PV macro-cell array. It has (5.5) If is odd, it has (5.6) If is even, it has (5.7) and can be obtained using Eqns. (5.5)~(5.7). The global coordinate is calculated using and as (5.8) and (5.9) The transferring function from the local coordinates to the global coordinates has been obtained. 5.2 Optimization Algorithm This section first discusses the macro-cell based PV panel reconfiguration problem targeting at maximizing the output power of the PV system against partial shading. This problem assumes that the size of the PV macro-cell is given. Secondly, the PV system design optimization problem is considered, which determines the optimal size of the PV macro-cell accounting for the trade-off between lower PV system capital cost and higher performance for combating partial shading. 114 pv I pv V Charger Supercap conv P cap I cap V PV Group 1 PV Group 2 PV Group 3 Figure 40. The archiecture of a capital cost-aware reconfigurable PV system. 5.2.1 PV Panel Reconfiguration Algorithm Figure 40 shows architecture of a reconfigurable PV system, which is comprised of a reconfigurable PV panel, a power converter, and a supercapacitor as the energy storage device. The PV cells in the PV panel are physically placed into an PV cell array (not shown in Figure 40). The PV macro-cell has a size of . Then the total number of PV macro-cells is . In Figure 40, the PV macro-cell has a size of and the PV panel configuration is . The output voltage and current of the PV panel are and , respectively. The power consumption of the power converter is . The terminal voltage of the supercapacitor is , and the charging current of the supercapacitor is . The input ports of the power converter are connected to the PV panel, and the output ports of the power converter are connected to the supercapacitor. We have 115 (5.10) The actual solar irradiance level on the PV cell with global coordinate is . However, it is assumed that the PV reconfiguration controller has no access to every for the realistic concern. Instead, the PV reconfiguration controller has only an estimate of the solar irradiance level on each -th PV macro-cell. This estimate causes certain performance degradation since different PV cells in a macro-cell may receive different solar irradiance levels. The PV panel reconfiguration problem aims at finding the optimal configuration of the PV panel, given the instantaneous solar irradiance level estimate on each PV macro-cell, in order to maximize the PV system output power under the current solar irradiance condition. The formal statement of the PV panel reconfiguration problem is: PV Panel Reconfiguration Problem Statement: Referring to Figure 40, given the solar irradiance estimate on the -th PV macro-cell and the supercapacitor terminal voltage , find the optimal PV panel configuration and the optimal PV panel operating point , such that is maximized. This work proposes the near-optimal PV panel reconfiguration algorithm comprised of a kernel algorithm and an outer loop. The kernel algorithm finds the optimal , , …, values with a given such that the PV panel MPP power is maximized. The MPP of a PV panel is the PV panel operating point where the PV panel achieves the maximum output power. The MPP power of a PV panel is dependent on the solar irradiance levels on the PV cells in the PV panel. The kernel algorithm here is based on the kernel algorithm in Chapter 3 except that the PV macro-cell is considered as the basic unit in the PV panel reconfiguration. In other words, the PV panel reconfiguration method only changes the connection of PV macro-cells whereas the size and structure of a PV macro-cell are fixed. The kernel algorithm is based on dynamic 116 programming with polynomial time complexity. On the other hand, the outer loop determines the optimal value such that is maximized when taking into account the converter efficiency variation. Similar to Chapter 3, the kernel algorithm also relies on the observation that the MPP voltages of a PV cell under different solar irradiance levels are very close to each other, but the corresponding MPP currents vary significantly. Let denote the solar irradiance- independent MPP voltage of a PV cell, and let denote its MPP current as a function of the solar irradiance . The kernel algorithm essentially treats the PV macro-cell as a super PV cell comprised of PV cells. The kernel algorithm assumes that each -th PV macro-cell receives a uniform solar irradiance level . The MPP voltage and current of each -th PV macro-cell are estimated as and , respectively. The kernel algorithm maximizes the estimated PV panel MPP power given by (5.11) Or equivalently, it maximizes . Optimal substructure property applies to this problem. Hence, this work applies dynamic programming-based kernel algorithm to solve this problem. Details of the kernel algorithm are provided in Algorithm 2. In the outer loop, Algorithm 2 is executed first to find an optimal configuration for each value. After that, the estimated under configuration is calculated using the converter function given in Section 2.2. In the calculation of the estimated value, and are estimated by and , respectively. The optimal value that maximizes the estimated , and the corresponding configuration are finally found. Details are provided in Algorithm 1. 117 During system operation, the MPTT method similar to [36] is used in order to find the optimal PV panel operating point that maximizes the actual under the optimal configuration . 5.2.2 PV System Design Optimization The PV system design optimization problem aims to find the optimal size (i.e., the optimal and ) of the PV macro-cell considering the trade-off between lower PV system capital cost and 118 higher performance. When the PV macro-cell size is smaller, the PV system capital cost becomes higher due to the increasing number of integrated switches. Consider two PV panels with the same size and the same total number of PV cells. The first PV panel has a macro-cell size of , and the second PV panel has a macro-cell size of . The first PV panel has twice the total switch cost compared with the second panel. On the other hand, the PV system achieves higher performance in combating the partial shading effect when the macro-cell size is smaller because of the following two reasons: (i) The PV panel with a smaller macro-cell size has higher flexibility in reconfiguration against partial shading. For example, the PV panel with macro-cell size can achieve all and beyond the available configurations of a PV panel with the same physical layout of PV cells but with macro-cell size . (ii) The use of a single solar irradiance level for each PV macro-cell in the reconfiguration algorithm causes certain performance degradation. The PV system design optimization is performed at system design stage. The optimization problem is formulated as: PV System Design Optimization Problem 1: Given a PV panel with fixed area and fixed physical layout of PV cells, find the optimal and , such that the PV system performance (output power) is maximized and the PV system capital cost is within a certain limit. and should be factors of and , respectively. This work effectively evaluates the performance of reconfigurable PV panels using PV panel partial shading benchmarks. The benchmarks may reflect the effect of dusts on the PV panel, the effect of clouds and nearby obstacles, the changing of solar irradiance incidence angle, and so on. The benchmarks consist of either measured or synthesized data. The benchmarks adopted in our PV system performance evaluation process are introduced in Section 5.3. 119 The proposed PV system design optimization algorithm is provided in Algorithm 3, which utilizes Observation I to reduce search space to a Pareto-optimal set of and values. Observation I: Consider two PV panels with the same area and the same physical layout of PV cells, but with different macro-cell sizes and , respectively, where and . The performance of the former PV panel is no less than that of the latter in combating partial shading effect. Similarly, PV System Design Optimization Problem 2 is formulated as minimizing the PV system capital cost with a performance degradation constraint. The performance degradation is evaluated with respect to the PV panel with macro-cell size . The optimal and values are found based on binary search algorithm, making use of the observation that a smaller macro- cell size results in a higher PV system performance against partial shading. Details are provided in the following Algorithm 4. 120 5.3 Experimental Results In this section, the PV macro-cell size is investigated as a trade-off between the lower PV system capital cost and higher performance against the partial shading effect by model-based simulation. Consider a PV panel with possible PV macro-cell sizes. Real-world examples of the PV cells, switch drivers, and switches are exploited with prices and specifications as shown in Table XIII. Commercial PV cells are carefully selected in the price range of ~$1, ~$5, ~$20. They are named as low-price, medium-price, and high-price PV cells, respectively. Also listed in Table XIII are the selected MOSFETs and gate drivers and their characteristics. Table XIII. Prices and characteristics of PV cells, MOSFETs and gate drivers. 121 Figure 41. Pareto-optimal macro-cell sizes and the corresponding PV system performance and capital cost. To effectively evaluate the performance of the reconfigurable PV panels with different PV macro-cell sizes, PV panel partial shading benchmarks are used. The proposed benchmarks are comprised of (i) extreme-case partial shading patterns similar to those in Chapter 3 and Chapter 4 for testing the robustness of the PV panel reconfiguration architecture, (ii) random shading over each PV cell to mimic the effect of dusts on the PV panel, and (iii) block-based partial shading on the PV panel to mimic the effect of clouds and nearby moving obstacles. Figure 41 shows the trade-off between the lower PV system capital cost and higher performance against partial shading of a PV panel consisting of medium-price PV cells when the PV macro-cell size changes. The x-axis of Figure 41 is the percentage of performance enhancement over the same size PV panel without any reconfiguration method. The y-axis is the percentage of capital cost decrease with respect to the same size PV panel with macro-cell size of . All the Pareto- optimal PV macro-cell sizes are marked in Figure 41. 122 Table XIV. Maximal performance enhancement with capital cost constraint. Table XIV provides experimental results on the PV system design optimization problem 1. It provides the maximal performance enhancement when the PV panel capital costs are 20%, 50%, and 100% higher than a panel without any reconfiguration method. Experimental results on the low-price, medium-price and high-price PV cell-based panels are provided. According to Table XIV, the high-price PV cell-based panel achieves more than 75% performance enhancement with a capital cost increase less than 20% compared with the PV panel without any reconfiguration method. For the low-price PV cell-based panel, about 43% performance enhancement can be achieved with a capital cost increase less than 20%. Table XV provides experimental results on the PV system design optimization problem 2. It provides the minimal amounts of capital cost increase when the PV system performances are 20%, 40%, 60%, and 80% higher than a PV system without any reconfiguration method. Experimental results on the low-price, medium-price and high-price PV cell-based panels are provided. According to Table XV, the high-price PV cell-based panel achieves more than 60% performance enhancement with a negligible capital cost increase of 3.7%, compared with the PV panel without reconfiguration ability. Hence, it is more beneficial in investing on the reconfiguration architecture than simply increasing the area of the PV panel if allowable. On the other hand, the low-price PV cell-based panel encounters 354% capital cost increase in order to enhance its output power by 80%, which is certainly not as cost-effective as increasing the area of the PV panel by 80% if allowable. However, investing on the reconfiguration architecture will 123 be cost-effective even for low-price PV cell-based panels if we set the macro-cell size relatively larger. For example, such PV panel achieves more than 40% performance gain with only 11% increase in capital cost. It is beneficial to invest on the reconfiguration architecture in this case, even though the cost of a single PV cell is cheaper than two gate drivers. Table XV. Minimal cost increase with performance guarantee. 124 Chapter 6. Energy Storage Elements and Energy Storage Systems: Background 6.1 Introduction Electrical energy is a high-quality form of energy [47], [48], [49] that can be efficiently converted into other lower quality forms of energy whereas generation of electrical energy from other forms of energy is less efficient in general. However, electricity supply and demand are commonly not balanced well with each other because (i) typical fossil fuel and nuclear power plants can hardly respond to the rapid fluctuations of the load demands, and (ii) the output power levels of most renewable power sources are not controllable and are largely dependent on the environmental factors such as solar irradiance and wind speed/direction. For utility companies, expanding power plant facilities to provide the maximum load demand results in energy inefficiency during the non-peak hours. On the other hand, storage of excessive energy during off-peak hours and compensation for the energy shortage during the peak hours is an alternative promising solution to mitigate the supply and demand mismatch. Electrical energy storage (EES) systems, which are comprised of various and potentially heterogeneous EES elements, are deployed to increase power availability, reliability, and efficiency, compensate the supply- demand mismatch, and regulate the peak-power demand. There are many practical deployments of grid-scale EES systems to mitigate the gap between the supply and demand [6], [50]. For residential consumers, integration of EES systems can store energy from the grid or residential power generation sources (i.e., photovoltaic modules) during off-peak hours and supply energy during peak hours for peak shaving, and can result in significant savings in overall electricity 125 cost [118], [119], [127]. In addition, most stand-alone renewable energy sources, such as solar energy, wind power, and hydropower, require some sort of EES system [36], [51]. The rest of this chapter is organized as follows. In Section 6.2, we will provide a detailed discussion about the distinct characteristics of various types of EES elements as well as an accurate EES element modeling framework. In Section 6.3, we will discuss about the components and classification of EES systems. 6.2 Electrical Energy Storage Elements An electrical energy storage element is an elemental device or apparatus that is capable of storing electrical energy in the potential, kinetic, chemical, or other forms of energy, and restoring the stored energy back the electrical energy on demand. There are many types of energy storage elements developed to-date. In this section, we enumerate the key metrics that are frequently used to evaluate different energy storage elements in Section 6.2.1 in detail. Next we compare some representative energy storage elements in light of the discussed metrics in Section 6.2.2. We do not discuss detailed principles behind storing (releasing) the electrical energy in (from) storage elements but focus on evaluating their performance and other characteristics. 6.2.1 Performance Metrics for EES Elements We introduce various performance metrics used to characterize energy storage elements, including cycle efficiency and internal resistance, rate capacity effect, energy density, power density, capital cost, cycle life/state-of-health, self-discharge rate, environmental impact, etc. Cycle Efficiency and Internal Resistance: Cycle efficiency of an energy storage element is defined as the round-trip energy efficiency, that is, the ratio of the amount of energy output during discharging to the energy input during 126 charging. In other words, the cycle efficiency is the product of the charging efficiency and discharging efficiency. Here, the charging efficiency is the ratio of energy stored in an energy storage element after the charging process to the total energy supplied to that element during the entire charging process, and discharging efficiency is the ratio of energy extracted from an energy storage element during discharging to the total energy stored before the discharging process begins. Power loss during the charge and discharge cycles is mostly due to internal resistance of the energy storage elements. Cycle efficiency is drastically affected by charging/discharging profiles, that is, the magnitude and shape of the charging/discharging current. For example, due to the well-known rate capacity effect, the total energy delivered by a battery goes down with the increase in load current, resulting in lower discharging efficiency. Meanwhile, the recovery effect of battery, which recovers the terminal voltage during idle periods between discharge pulses, also affects the cycle efficiency of the battery [52]. Among all types of EES elements, supercapacitor and flywheel have a very high cycle efficiency close to 100% and less affected by the charging/discharging profile. A high cycle efficiency means less energy loss during charging and discharging processes that leads to low operational cost per cycle. Therefore, it is wise to use a high-cycle efficiency EES element such as supercapacitor for frequent charging/discharging applications. Rate Capacity Effect: Rate capacity effect (also referred to rate capability or power capability) signifies the capability to provide high power without degradation of the total amount of energy. A battery's available capacity decreases as the discharging rate increases. Peukert's law describes the 127 phenomenon that the delivered capacity normalized to the rated capacity has an exponential relationship to the discharging current normalized to the rated current [53]. That is, (6.1) where is the discharge current, is the discharge time, and is the Peukert coefficient, which is typically a value between 1 and 2. Generally speaking, the Peukert constant varies according to the aging of the battery, generally increasing with age. In an ideal battery where the total delivered capacity is independent of the discharge current, . For a lead-acid battery, is between 1.1 and 1.3. In contrast, supercapacitors have a very good rate capability and can withstand very high discharge rates with virtually no loss of available capacity. Note that if is described as actual discharge current relative to 1 ampere, then will give the battery capacity at a one-ampere discharge rate. This is another common form of the Peukert's law. Figure 42. Discharging a 350 mAh 2-cell series Li-ion GP105L35 battery at constant currents of 1C, 2C, 4C, and 4C from our experiments. 128 Energy Density versus Power Density: Power density is defined as the rated output power divided by the volume (W/L) or mass (W/kg) of the EES element. Similarly, energy density is the amount of stored energy divided by the volume (Wh/L) or mass (Wh/kg). Generally speaking, power density is related to the amount of instantaneous power that an EES element can provide; on the other hand, energy density is related to the time duration that the EES element can last while supplying a certain amount of power. Supcapacitor and flywheel have a very high power density, whereas typical Li-ion batteries have a marginally high power density. High power density EES elements are suitable as temporary energy buffers to deal with some applications with short-duration high-power demand. Metal-air batteries and fuel cells have a high energy density, which is multiple orders of magnitude higher than those of typical batteries. High energy density EES elements are suitable as long-term EES means. Supercapacitor is one of the EES elements that have the lowest energy density, and thus, it is not economical to use them alone as a large-scale EES means. A HEES system can rely on high energy density EES elements for long-term energy storage and a relatively small amount of high power density EES elements for high output power rating. Battery-supercapacitor hybrid is a representative HEES system, which effectively exploits the high energy density of the battery and high power density of the supercapacitor. Capital Cost: Capital cost, which is an important consideration in the design and implementation of an EES system, is typically represented in the forms of cost per unit delivered energy ($/Wh) or per unit output power ($/W). The capital cost determines how much money should be invested in order to build an EES system with certain amount of energy capacity or power capacity. 129 Some batteries are superior to others in terms of both their energy and power densities (for example, Li-ion battery is better than lead-acid in both aspects), yet the capital cost of these batteries is much higher than lead-acid ones. In fact, the capital cost gap between energy storage elements is a key motivation for the hybrid storage approach because one may not afford to use unlimitedly large amount of good, yet expensive, EES elements to meet the system requirements. For example, we can use a very large amount of supercapacitors to meet both the high output power requirement and the high-energy storage requirement of some applications, but the cost of such an EES system will be impractically high. Instead, consider an alternative EES system that uses a small amount of supercapacitors to meet the high instantaneous power demand and a small amount of Li-ion batteries to meet the high storage requirement of the same applications. The cost will be affordable. Therefore, we should judiciously determine the portion of expensive (but of higher performance) energy storage elements and inexpensive (but of lower performance) energy storage elements for a given monetary budget. Cycle Life and State-of-Health (SoH): State-of-health (SoH) of an energy storage element is a measure of its age. It captures the general condition of the energy storage element and its ability to store and deliver energy compared to its initial state (that is, when it was a fresh device out of the manufacturing line.) During the lifetime of the energy storage element, its capacity (or SoH) gradually deteriorates due to the irreversible physical and chemical processes that take place along with usage. Cycle life is the maximum number of charging and discharging cycles that a storage element can perform before its capacity drops to a specific percentage (60% - 80% typically) of its initial capacity. It is one of the key performance parameters, which gives an indication of the expected working lifetime of the EES element. Obviously, the cycle life of an EES element is closely 130 related to the replacement period for it and thus the total cost of the element according to a life- cycle analysis. Cycle life of an EES element heavily depends on the usage pattern of the element, especially to the depth-of-discharge (DoD) that is defined as the ratio of used capacity to the initial full capacity. For lead-acid batteries and Li-ion batteries, the number of available charging/discharging cycles increases when lowering the DoD [54]. Similar conclusions hold for most electrochemical batteries. Typically, energy storage elements whose operation principles are based on electrical, mechanical or thermal technologies, such as supercapacitor, flywheel, thermal energy storage (TES), cryogenic energy storage (CES), typically have long cycle lives. In contrast, cycle lives of electrochemical batteries are not that high due to unavoidable chemical deterioration of the electrodes during their operation. Self-Discharge Rate: Self-discharge rate is a measure of how quickly a storage element loses its stored energy when there are no charging and discharging currents. It is heavily dependent on how the EES element stores the energy inside, as well as ambient conditions such as the temperature and humidity. Supercapacitor and flywheel, which have a high cycle life, have the highest self- discharge rate, that is, they typically lose all of their stored energy within a few days or even hours. One should thus not store energy in a supercapacitor if it is expected that energy will not be used for a long time. On the other hand, electrochemical batteries store energy with stable chemicals, and do not lose so much energy in self-discharging. 131 Environmental Impacts: The importance of environmental friendliness of EES elements is being emphasized from the recent past. Typically, electrochemical batteries have a negative impact on the environment due to their reliance on toxic metals such as lead and cadmium that can be quite harmful if not disposed properly. They should be recycled properly for both reducing environmental impact and conserving scarce resources [55]. The environmental impacts of an EES element are closely related to its expected cycle life. In other words, we can reduce the negative impacts of using hazardous EES elements by extending their cycle lives. Note that the supercapacitor and flywheel have very small or almost negligible impacts on the environments, not only because they do not contain harmful materials, but also because of their long cycle lives. 6.2.2 Various Types of EES Elements Lithium-ion (Li-ion) Battery: The Li-ion battery, first demonstrated in the 1970s, is a family of rechargeable batteries in which lithium ions move from a negative electrode (cathode) to a positive electrode (anode) through discharging. The ions move in opposite direction during the charging process. Li-ion battery is now the battery of choice in portable electronic devices and is growing in popularity in mobile systems ranging from EV to airplanes. The growing popularity of Li-ion battery is mainly due to the following reasons: high energy density, high efficiency, long cycle life, no memory effect, and low self-discharge rate. While commanding a 50% share of the small portable devices market [56], there are many challenges to build large-scale Li-ion battery based EES systems. The main hurdle is the high cost due to special packaging and internal overcharge protection circuitry. There are also concerns about their safety and fire risk. Manufacturers are working to reduce the manufacturing 132 cost of Li-ion batteries to capture large new energy markets, especially markets for the electrical vehicles. At the same time, many are working on making large-size Li-ion cells safer and more stable. Figure 43(a) shows a 18650 size Li-ion battery, ICR18650-22F from Samsung SDI. Multiple Li-ion batteries of this size are combined in a pack and are widely used for portable devices such as laptops. Estimating the state-of-charge (SoC), which is the remaining energy, and SoH of batteries is important for system management, but it is not an easy task [57]. Significant number of research work have been performed to accurately estimate the SoC and SoH by developing accurate models because simply measuring voltage and current does not give an good estimation. A Li- ion battery circuit model is proposed for developing a battery-supercapacitor hybrid system in [61]. A dual extended Kalman filter is used to simultaneously estimate the SoC and capacity from the open-circuit voltage [62]. A Li-ion battery model introduced in [63] employs the method of electrochemical impedance spectroscopy. 133 Figure 43. Example commercial products of typical batteries and supercapacitors. Lead-acid Battery: As one of the oldest and most developed rechargeable battery technologies, lead-acid batteries have short cycle life and low energy density due to the inherent high density of lead as a metal. Besides, they also have a poor low temperature performance and thus a thermal management system is required. In spite of these disadvantages, their ability to supply high surge currents means that such lead-acid batteries maintain a relatively high power density. These features, along with their low cost and high energy efficiency, make lead-acid batteries suitable in motor vehicles so as to provide the high current demand for automobile starter motors. Lead-acid batteries have been used in a few large-scale commercial energy management systems. Figure 134 43(b) shows an example of a valve-regulated (sealed) lead-acid battery pack, Panasonic LR- R123R4P. Lead-acid battery technology is considered quite mature now [49], but there are still efforts to find better materials for the current collectors, electrodes, electrolytes, etc [64], [65]. Nickel-Metal Hydride Batteries: Nickel-metal hydride battery, abbreviated NiMH, is a type of rechargeable battery similar to the nickel-cadmium cells (NiCd). The only difference is that the former one uses a hydrogen- absorbing alloy for the negative electrode instead of cadmium. The energy density of NiMH batteries is more than double that of lead-acid batteries and 40% higher than that of NiCd batteries. NiMH battery is relatively inexpensive. However, it suffers from the memory effect, although this is much less pronounced than that in the NiCd ones. NiMH battery also has a rather high self-discharge rate. Recently, low self-discharge (LSD) NiMH battery has been commercialized targeting portable devices. Figure 43(c) is a commercial product of an AA size LSD NiMH battery from Sanyo. Toyota Prius HEV also uses NiMH batteries [66]. The most significant feature of NiMH batteries is the high power density (250 - 1000 W/kg), which is the highest among existing battery types. Therefore, this type of batteries is widely used in high current drain consumer electronics, such as digital cameras with LCDs and flashlights. Recently, NiMH batteries have also been deployed in HEV such as the Toyota Prius, Honda Insight, Ford Escape Hybrid, Chevrolet Malibu Hybrid, and Honda Civic Hybrid. Capacitors and Supercapacitors: Electric double-layer capacitors, more commonly known as supercapacitors or ultra- capacitors, are widely exploited to mitigate load current fluctuations in the batteries. Supercapacitors have a superior cycle efficiency that reaches almost 100%, and a long cycle life [67]. Moreover, compared with batteries, supercapacitors exhibit significantly higher volumetric 135 power density (but they have much lower energy density) [68]. Therefore, they are suitable as energy storage in situations with frequent charging/discharging cycles or periodic high current pulses. In a battery-supercapacitor hybrid system, the supercapacitor stores surplus energy from the battery during low demand periods, and provides extra energy during peak load current demand period. A key disadvantage of a supercapacitor is its large self-discharge rate compared to that of ordinary batteries. A supercapacitor may lose more than 20% of its stored energy per day even if no load is connected to it. Another important concern with respect to supercapacitors is their terminal voltage variation, which in turn arises from the characteristics of any capacitor whereby its terminal voltage is linearly proportional to its SoC. The terminal voltage thus increases or decreases accordingly as the supercapacitor is charged or discharged. This terminal voltage variation is much higher than that observed in typical batteries. This effect results in a significant conversion efficiency variation in the power converters that are connected to the supercapcitors. Figure 43(d) is a commercially available supercapacitor, BCAP0310 from Maxwell Technologies. Its size is the same as that of a D size battery, and it has a 310 F capacity with the maximum voltage of 2.7 V, which results in 0.31 Wh energy capacity. In comparison, a high energy density D size Li-ion primary battery has 68.4 Wh energy capacity. Flow Batteries and Flow Capacitors: A flow battery generates electricity inside a reactor by using dissolved electroactive species stored in external tanks. This separation of reactor and tank decouples the energy density (limited by tank size) from the power density (limited by reactor size.) Flow batteries have advantages of long cycle life, environmental friendliness, quick charging by electrolyte replacement, and so on. However, they require complicated components and complex circuitry such as sensors and 136 actuators to run the system. Flow batteries are generally considered for large-scale stationary applications rather than portable applications. Vanadium redox battery [69] and zinc-bromine battery [70] are examples of flow batteries. Lead-acid flow battery is also considered as a replacement of the conventional lead-acid batteries [71], [72], [73]. A recent technology called the electrochemical flow capacitor has the advantages of both supercapacitors and flow batteries [74]. It provides rapid charging and discharging, high cycle life, and high cycle efficiency like a supercapacitor. It also provides high energy capacity like a flow battery. A flowable carbon-electrolyte mixture, called a slurry, captures or releases charged ions while flowing through a flow cell during a charging or discharging process. Other EES Elements: There are many other types of EES elements such as nickel-cadmium (NiCd) battery, sodium- sulfur (NaS) battery, metal-air battery, flywheel, cryogenically cooled thermal plates, and compressed air systems. We do not cover all the details of those EES elements here. Characteristics of some of these EES elements are summarized in Table XVI and Figure 44 together with the EES elements we discussed above. Table XVI. Comparison between energy storage elements. 137 Figure 44. Comparison between EES elements. 6.2.3 EES Element Modeling We present an accurate EES element modeling that is applicable to various types of EES elements, including Li-ion battery, lead-acid battery, and supercapacitor. The voltaic representation of the EES element SoC at time t is given by: 138 (6.2) We derive in Coulomb from the nominal capacity given in Ahr: (6.3) We interpret as the state of the EES element. EES element models specify the relationship among the SoC , the open circuit voltage (OCV) , the closed circuit voltage (CCV) , and current . Battery models for the electronic systems have been extensively studied during the past few decades [57], [58], [59], [60]. A battery model in the form of an electric circuit is suitable for our purpose. We adopt the battery model introduced in [60] as shown in Figure 45. This includes a runtime-based model on the left as well as a circuit-based model on the right for accurate capturing of the battery service life and I-V characteristics. In Figure 45, , , and are internal resistances, and , are internal capacitances, of the battery. On the other hand, supercapacitors have a very low internal resistance (< 1 m ). We introduce details of EES models in the following three aspects: OCV- SoC relationship, CCV-OCV relationship and rate capacity effect. Figure 45. EES element equivalent circuit model. 139 OCV-SoC Relationship: is a monotonically increasing function of . is modeled as a voltage- controlled voltage source controlled by of the battery as shown in Figure 45. The OCV- SoC relationship is nonlinear and is given by (6.4) where those 's are empirically determined parameters from our real pulsed charging and discharging measurements [108]. On the other hand, and for a supercapacitor satisfy a linear relationship. CCV-OCV Relationship: when for the battery due to internal resistances and capacitances. The relationship between and of the battery is given by (6.5) We have for a supercapacitor due to its negligible internal resistance. Rate Capacity Effect and Coulomb Counting: The rate capacity effect of batteries explains that the charging and discharging efficiencies decrease with the increasing of charging and discharging currents, respectively. More precisely, the Peukert’s formula [75] describes that the charging and discharging efficiencies of a battery, as functions of the charging current and discharging current , respectively, are given by (6.6) where , , , and are constants known a priori. We define the equivalent current inside the battery as the actual charge accumulating/reducing speed inside the battery, given by: 140 (6.7) In contrast, the rate capacity effect of supercapacitor is negligible, i.e., . We calculate from the initial SoC ( ) using Coulomb counting: (6.8) 6.3 EES Systems An EES system is an energy reservoir that stores electrical energy and supplies the energy when necessary. The electrical energy is stored in the EES system as potential energy, kinetic energy, chemical energy, or in some other form in the EES elements as we discussed in Section 6.2. An EES system performs many useful functions such as load leveling, contingency service, voltage stabilization, maximum power point tracking (MPPT) for renewable power sources, etc., for a wide range of applications including portable devices, household appliances, EV/HEV, and even power grid. We first discuss the general architecture and components of the EES system in Section 6.3.1. Next we provide a classification of the EES systems based on the potential applications (therefore, their size and scale) in Section 6.3.2. 6.3.1 Architecture of Electrical Energy Storage Systems Figure 46 shows a typical architecture of an EES system. Major components of an EES system include an EES array, power converter, and EES controller, which will be discussed in the following. 141 Figure 46. EES system structure. Energy Storage Array: The EES element is the most important consideration for EES system design. There are various types of EES elements as discussed before. Type and capacity of EES element are determined according to the application and requirement of the EES system with restriction on volume, weight, cost, and so forth. When a single EES element does not meet the energy or power requirement, which is usual for a large-scale EES system, multiple EES elements are connected together to form an array of EES elements. An EES array is a set of multiple identical EES elements that are connected in series and/or parallel forming a regular matrix. The dimension of the EES array is determined by power and energy capacity requirement, and the maximum voltage rating of the EES array. The regular array structure makes it possible to maintain the same SoC and SoH of all the elements in that array. Some types of EES elements require balancing between series-connected cells. Even though all the EES elements are of the same type, manufacturing variation in practice may result in imbalance of characteristics such as capacity and internal resistance that results in imbalanced 142 SoC during operation and even damage to the elements [77]. Supercapacitors and Li-ion batteries require external cell balancing circuits like Figure 47. Cell balancing is an active research area, which is critical for large-scale EES systems [78], [79], [80], [81]. A paper categorizes cell balancing methods into three categories: charging methods (steering), active methods, and passive methods (bleeding) [82]. The charging method selectively bypasses fully charged cells. The active cell balancing methods move energy from more-charged cells to less-charged cells by means of capacitors or power converters. The passive cell balancing methods simply dissipate energy from over-charged cell as heat. Figure 47. Linear Technology LTC6803-3 battery stack monitor with passive cell balancing switches [76]. Configuration of EES elements, that is, the number of series and parallel connections, may be dynamically adjusted. For example, with four supercapacitors, three configurations, 1-by-4, 2- by-2, and 4-by-1, are feasible. Supercapacitors, which have wide voltage variation according to 143 SoC, benefit the most from the dynamic reconfiguration [83]. A supercapacitor array maintains roughly constant voltage by increasing (deceasing) the number of series connections as the voltage of each energy storage element decreases (increases). We propose dynamic reconfiguration circuit that aims at maintaining marginally constant supercapacitor array voltage [111] (please see Figure 48 as a reconfiguration example.) We show that explicitly taking into account the power converter efficiency further improves the energy efficiency than simply maintaining a constant voltage [111]. Figure 48. Reconfiguration examples of an energy storage array with four energy storage elements [111]. Fault resilience is another benefit of the dynamic energy storage array reconfiguration. Energy storage elements are subject to manufacturing variations. Each energy storage element shows different aging-induced battery capacity degradation [84], [85]. Dynamic reconfiguration techniques may prolong the lifetime of the energy storage array when short-circuit faults or open-circuit faults occur in the supercapacitor array. It is more practical to operate the partly 144 degraded supercapacitor array rather than replacing the whole array when the most supercapacitors are healthy. Energy storage array reconfiguration improves dependability, efficiency, and scalability of large-scale battery arrays. The real-time controller proposed in [86], [87] makes decisions and rearranges in series or parallel while bypassing faulty batteries. Power Converters: The energy storage array is not directly connected to the power source or load because of its varying terminal voltage depending on its SoC, load current, temperature, and so on. Therefore, a power converter is placed between the energy storage array and power source or load and generates regulated voltage or current for charging or discharging the energy storage array. Batteries and supercapacitors are DC energy storages, and so storing energy from the AC power grid and supplying power from and to the AC load electronics require AC-to-DC and DC-to-AC conversions. An AC-to-DC power converter is often referred as to a rectifier, and a DC-to-AC power converter is often referred as to an inverter. Recall that we pointed out in Section 2.2 that power conversion will incur non-zero amount of power loss by parasitic resistances, parasitic capacitances, and controller circuit, and provided a power modeling of converters. This power loss is not constant, but varies depending on the input and output voltage and the amount of power that is transferred through the converter, and so the power conversion efficiency also varies. In general, the power conversion efficiency is highest when the input voltage and output voltage are close to each other, and the output (regulating) current is within a certain range. The power conversion efficiency is a critical factor, which determines the energy efficiency of the EES system together with the cycle efficiency of the energy storage elements. Note that the fact that the cycle efficiency of the supercapacitor is nearly 100% does not guarantee high-energy efficiency of the EES system composed of 145 supercapacitors only because the terminal voltage of such an EES system varies in a very wide range depending on its SoC, which makes it difficult to design an energy-efficient power converter. System Controller: Control of the EES system is a not trivial problem. Even a homogeneous EES system requires elaborate system control schemes beyond simple charging and discharging through power converters in order to maximize benefits of the EES system. Objectives of the EES control typically include enhancing the energy efficiency, cycle life, and reliability. The basic functionality of the EES control is management of the energy storage elements. The EES controller monitors the voltage, current, and temperature of each energy storage element, and prevents the energy storage elements from operating in an unsafe range. Distributed EES controllers communicate with each other or an external device for monitoring and control through a communication network. Control area network (CAN) bus communication is one of the most widely used communication networks for system control [88], [89], [90], [91], which can be used here as well. An EES system also provides application-specific system controls. EES controllers for renewable power sources perform MPPT in order to maximize power generation from the power sources [36], [92], [93]. The controller is responsible for finding and maintaining the energy- optimal operating point that dynamically varies depending on the environmental conditions. They also should determine the amounts of charge and discharge currents from the power grid considering the power generation and power demand. The optimization objectives include minimizing the daily energy cost considering time-of-use pricing and maximizing battery 146 lifetime [94], [95], [127]. EES controllers for EV/HEV applications determine the direction and amount of power flow among components such as internal combustion engine (ICE), EES system, traction motor, and alternator depending on the operation modes. Optimal system control is critical not only for traction performance, but also for energy efficiency and battery lifetime [96], [97], [130]. System control of the HEES system is much more complicated than that for a homogeneous EES system. We will discuss this in detail in Chapter 7. 6.3.2 Portable and Grid-Scale EES Systems EES systems may also be categorized by their scale, which is directly related to the total capacity, portability, and target applications. Small-scale and large-scale EES systems have distinctive performance requirements. Portable Applications Portable EES systems are used for powering portable electronics, and thus have strict constraints on their volume and weight. High energy density per unit volume or unit weight is thus the key criterion for portable EES systems. Li-ion battery is a most promising type of energy storage element for today’s portable applications. Electrical energy is increasingly becoming a major concern for today’s foot soldiers who must carry more and more advanced power-hungry electronics [68]. Some portable applications such as military radios also require high-power capability with small and light form factor. Battery-supercapacitor HEES systems may deal with the high-power demand of such applications by using supercapacitors as an energy buffer [108], [120]. Also some low-power sensor nodes like [32], [98] employ a battery- supercapacitor hybrid system as shown in Figure 49. Due to very limited capability to produce power from energy harvesting devices such as PV cells, reducing the power loss during 147 charge/discharge cycles is important. They take advantage of the high cycle efficiency of the supercapacitor while using the battery for low-leakage long-term energy storage. Figure 49. Battery-supercapacitor hybrid wireless sensor nodes (a) Prometheus from Berkeley [32] and (b) AmbiMax from UC Irvine [98]. Figure 50. An example of battery-based EES applications, Toyota Prius PHEV with Li-ion batteries. EV/HEV applications are in between portable small-scale applications and stationary large- scale applications. They can accommodate larger size and weight quota for the EES system, but 148 still require high energy density because it directly affects the cruising distance of the vehicle. Cycle life is another critical factor for EV/HEV applications because they suffer from deep DoD. Low cycle life implies high maintenance cost for battery replacement. Lead-acid batteries have been used for traditional vehicles and EV such as golf carts and forklifts for low cost, but their heavy weight is the problem. Today’s HEV such as Toyota Prius has a NiMH battery pack [99]. Li-ion batteries are being actively researched for use in EV/HEV applications to replace the NiMH batteries [100], [101]. Grid-Scale Applications: Large-scale EES systems for powering households, factories, or even towns virtually have no restriction with respect to their total volume and/or weight. Battery-based large-scale EES systems with tens of MW capacity are already deployed over the world [102]. The types of batteries are also diverse including traditional lead-acid, nickel-cadmium, and Li-ion batteries and emerging flow batteries. EES systems are already being deployed for various purposes of power supply variation compensation and peak load shaving. Reference [102] provides some examples of such homogeneous EES deployment over the world. These systems are commonly composed of lead- acid batteries, nickel-cadmium batteries, Li-ion batteries, or regenerative fuel cell. A report from the Department of Energy indicates that a battery array incorporated with a PV cell array in a house substantially enhances efficiency, and reduces overall energy cost for the owner despite the capital cost of the battery [66]. Grid-connected EES systems serve as a commodity storage for storing energy during off-peak period for use during peak usage period for arbitraging the production price [56], [102], [103]. 149 Such systems can eliminate peak load demand and thereby lower the maximum power generation capacity (peak shaving), or make the load demand uniform over time, which is better from the perspectives of generation, transmission, and distribution systems (load leveling). Another application is contingency service, which supplies power when the grid power generation plants fall off-line. A grid-scale NiCd battery-based EES system shown in Figure 51, operated by Golden Valley Electric Association (GVEA) Alaska, is designed to provide 15 minutes of community load against the power failure [103]. Grid-connected EES systems also aid the power generation by preventing unplanned transfer of power and maintaining a state of frequency equilibrium. In addition, they may start up on their own and energize the power generation system after a blackout [56]. Figure 51. Golden Valley Electric Association, Fairbanks, AK, a 27 MW NiCd battery EES system for the power grid. Stand-alone renewable power sources such as PV cells and windmill generations utilize the EES system for various benefits. More precisely, the EES system for the renewable power 150 sources mitigate mismatch between power demand and uncontrollable power generation due to environmental conditions (for example, solar irradiance, wind strength). Maximizing the power generation regardless of variations in the environmental conditions and load demand is crucial for increasing the power generation efficiency of the renewable power sources. EES systems are used to decouple the power generation and power consumption to enable MPPT [10], [36], [104], [105], [106]. 151 Chapter 7. Hybrid Electrical Energy Storage (HEES) Systems The HEES system is an emerging approach for achieving performance improvement of the EES system with current non-ideal EES element technologies relying on the optimal design and control methodology. Hybridization offers opportunities to take advantage of each EES element while hiding their shortcomings/drawbacks. Expected benefits include enhancement of energy efficiency, cycle life, power and energy capacity, and so on. The motivations for the HEES system are introduced in Section 7.1 in analogy with the hybridization and hierarchy in computer memory subsystem design. Distinctive characteristics of the HEES system in terms of the system architecture and management are discussed in Section 7.2. Finally, we present our HEES prototype in Section 7.3. 7.1 Motivation and Principle The hybrid approach for the HEES system has the same motivations as those for the designing the memory subsystem in a computer system. Table XVII shows characteristics of some representative computer memory devices. SRAM (static random access memory) has the lowest latency and highest throughout but is expensive and has low density. On the other hand, DRAM (dynamic random access memory) is inferior to the SRAM in terms of latency and throughout, but is cheap and has high density. Mass storage devices such as an HDD (hard disk drive) and NOR/NAND flash memory have even lower cost, higher capacity, non-volatility, but are subject to limited random access capability and write count. Composing the required memory space with the SRAM only is infeasible due to its high cost except for supercomputers where cost is not a 152 primary issue. On the other hand, using HDD or flash memory only cannot meet the latency and throughout requirements of the CPU core and suffers from poor random access capability and limited write count. Computer architects, therefore, have remedied this problem by building a hierarchy of different types of memory devices. A typical memory hierarchy example is illustrated in Figure 52(a). L1 (level 1) SRAM cache provides the best latency and throughout but the smallest capacity, whereas L2 (level 2) SRAM cache has a bit lower latency and throughput but is larger than the L1 cache. They both generally reside on chip to provide a fast access speed. The DRAM main memory is placed off chip, or sometimes on chip for better latency, provides a much larger capacity with a higher latency and lower throughout than the L1/L2 SRAM caches. Finally, the slowest HDD and flash memory are used to provide the largest storage space. There are many policies to utilize this memory hierarchy efficiently, but generally speaking, we use a faster memory to store frequently accessed data and/or code in order to take advantage of its high speed. We overcome the capacity limitation of fast memory by moving less frequently accessed data down to a slower memory. As a result, this memory hierarchy enables the CPU to exploit the low latency of the L1 SRAM cache and the large capacity of the HDD at the same time. Table XVII. Comparison of density, cost, power consumption (idle/active), 16-bit random access time (read/write/erase) of memory devices [129]. 153 Figure 52. Comparison of computer memory hierarchy and HEES archiecture. A HEES system aims at similar benefits by using multiple heterogeneous EES elements. Instead of relying on a single type of EES element, the HEES system exploits distinct advantages of multiple heterogeneous EES elements and hides their drawbacks. For instance, EV/HEV exhibits frequent charge and discharge cycles with a short period and a large amount of current. Conventional batteries make it difficult to maintain a high efficiency and longer cycle life in such an operational environment. Use of supercapacitors instead can be a huge upgrade in terms 154 of efficiency and cycle life. However, the current supercapacitor technologies have serious disadvantages in terms of energy density and cost, which makes it difficult to completely replace the batteries in an EV with supercapacitors (there is despite a testbed supercapacitor-only EV [128]). Use of supercapacitors in a complementary manner reinforces the drawback of the battery through high power density, long cycle life, and high efficiency [107], [130], [131], [132]. A conceptual drawing of the HEES system is shown in Figure 52(b). A HEES system is comprised of a number of EES banks, and is connected to external power sources and load devices. The HEES system in Figure 52(b) is comprised of a supercapacitor, a Li-ion battery, and a lead-acid battery. Similar to the computer memory hierarchy, the HEES system exploits different features of these three energy storages for its own benefit: the high power density and long cycle life of the supercapacitor and the relatively low cost and high energy density of Li-ion battery and lead-acid battery. The charge transfer interconnect (CTI) internally connects the energy storages, external power sources, and external load devices through appropriately- selected power converters. In spite of the similarity of the HEES system and computer memory subsystem, of course, there are differences as well. For example, there is some energy loss during energy transfer, but no data is lost during data transfer. Also, the selection of storage elements does not cause any coherence or invalidation issues that a cache memory suffers from. Employing the HEES concept comes with additional design considerations. Deployment of a HEES system does not always guarantee better performance without proper design considerations. Designers should thus carefully determine the selection of EES elements, the proportion of each EES elements, system architecture, management policy, etc., in order to maximize the benefits of the HEES system over the homogeneous EES system. 155 7.2 System Architecture We present in Figure 53 a conceptual diagram of a general HEES system. There are multiple EES banks, which are connected through a network called the CTI. Each EES bank is similar to an independent homogeneous EES system, but it performs EES array management as determined by system-level HEES management policies. We discuss the energy storage banks, CTI, and control policies for HEES systems in the following. Figure 53. Conceptual diagram of a HEES system with four energy storage banks connected through a CTI. 7.2.1 Energy Storage Bank A HEES system is a hybridization of heterogeneous EES banks, each of which is homogeneous. An EES bank is composed of a homogeneous EES array and a bi-directional power converter as shown in Figure 54, which is similar to the homogeneous EES system architecture introduced in Section 6.3. In addition to the individual control of EES array, an EES bank in the HEES system supports system-wide bank management. The EES banks communicate with a central controller, which sets the system-level management. The central 156 controller receives the current status of each bank, such as voltage, current, SoC, SoH, and so on, and sends out charge management commands to each bank. Figure 54. HEES system architecture with three EES banks. 7.2.2 Charge Transfer Interconnect Let us see how the EES elements are connected with each other starting from a simple battery-supercapacitor HEES system. Figure 55 shows three representative hybrid architectures of battery and supercapacitor. A basic hybrid approach is a passive direct connection of a battery and a supercapacitor in parallel [133] as shown in Figure 55(a). The supercapacitor handles high 157 current demands by suppressing large battery voltage variations, but this architecture cannot be generally applied for different combinations of heterogeneous EES elements due to terminal voltage mismatch. Cascaded converter architecture shown in Figure 55(b) puts a converter in between the battery and supercapacitor to control the power. A constant-current regulator-based hybrid architecture in [108], [120] improves the energy efficiency compared with the direct parallel connection. The parallel connection and cascaded converters architectures, however, are not suitable for HEES systems with three or more EES banks. Figure 55. Supercapacitor-battery hybrid connection topologies. A general hybrid architecture, which can accommodate any type of EES elements in a systematic manner, utilizes a DC bus (CTI bus) [107], [109]. In the DC bus architecture, each EES element is connected to the DC bus through a power converter (voltage regulator or current regulator). In the DC bus architecture, each EES element is connected to the DC bus through a (bi-directional) power converter (voltage regulator or current regulator). A HEES system introduced in [134] is comprised of a fuel cell, a battery, and a supercapacitor connected to a fixed-voltage DC bus as shown in Figure 55(c). The control method regulates a fast EES element with a slow EES element; it controls the DC bus voltage, supercapacitor voltage, and battery 158 voltage by controlling the supercapacitor current, battery current, and fuel cell current, respectively. Another HEES system is comprised of a photovoltaic panel, a fuel cell, and a supercapacitor, and its basic operational principle is the same [135]. This simple and intuitive control method, however, cannot achieve energy optimality because (i) the DC bus voltage is fixed, and so the conversion efficiency is not always the maximum; and (ii) the current distribution among the EES elements is determined by other EES elements, and cannot be optimized in a holistic manner. True energy optimality requires consideration of not only the dynamic response, but also other factors such as rate capacity effect, residual energy, and load demand. For example, we may want to reserve energy in the supercapacitor and use the battery instead when a high current demand is expected in a near future, and this kind of intelligent energy control is not supported in [134], [135]. In order to mitigate these shortcomings, we provide a general DC bus-based HEES control mechanism, which (i) allows to dynamically control the DC bus voltage, and (ii) is compatible with high-level charge management policies in order to achieve energy optimality [113]. The HEES system architecture is shown in Figure 54. The bidirectional converter in each EES bank in this infrastructure is typically implemented based on a switching-mode power converter and can be configured as either a voltage regulator or a current regulator:  Voltage regulating mode: The converter generates a controllable voltage output regardless of the input and output current variation. The target output voltage is set by the central controller of the HEES system. The converter compares the current output voltage with the target voltage level, and increases/decreases its switching duty ratio through a feedback control loop to match the output voltage with the target voltage level. The underlying principle is that the output voltage increases when the converter 159 increases its switching duty ratio [20]. Current technologies enable the converter to provide a precise voltage regulation. For example, the LTM4607 converter has a typical voltage regulation inaccuracy of 0.15% and a maximum inaccuracy of 0.5% [22], whereas the LTC4000 converter has a voltage regulation inaccuracy less than 0.25% [23].  Current regulating mode: The converter generates a controllable current output regardless of the input and output voltage variation. Similarly, the target output current is set by the HEES central controller, and the converter adjusts its switching duty ratio through a feedback control loop to match the output current with the target current. The current regulation accuracy is also quite high, though slightly lower than the voltage regulation accuracy. For example, the LTC4000 converter uses 12-bit resolution (4096 steps) and results in less than 1% regulation inaccuracy [23]. We can control both the CTI voltage and EES bank current simultaneously in this way. We set only one converter in the voltage regulating mode and let it control the CTI voltage (please note that the CTI voltage can be stabilized because it has a moderate capacitance.) All others operate in the current regulating mode. The output current of the voltage regulating converter is automatically determined so that the sum of input currents of the CTI is equal to the sum of output currents. The other voltages and currents in the HEES system are associate variables and are determined due to energy conservation law once the output voltage/current level of each converter is set. In summary, this control method uses the analog feedback control loop in the converters to maintain the stability of the CTI voltage and EES bank current, whereas the micro- controller executes real-time charge management tasks and properly sets the target output voltage/current level of each converter according to the charge management policies. 160 A HEES system with larger number of EES banks requires a more complicated connection than a passive parallel connection or cascaded converter architecture as the CTI bus. Connecting heterogeneous EES elements in a HEES system is more than expanding power and energy capacities, but involves complicated energy transfer among the elements. Therefore, the CTI architecture is more critical in a HEES system than in a homogeneous EES system though the former can be used for homogeneous EES systems as well. Similar to on-chip communication networks, the network topology of the CTI is one of the important design considerations for energy efficiency in a large-scale HEES system. It has a significant impact on the charge transfer efficiency, and thus, should be carefully designed in order to maximize the potential benefits of the HEES system. A system-on-chip design is subject to a similar problem of determining a suitable interconnect architecture. The interconnect architecture in a system-on-chip affects its communication latency, throughput, power consumption, and so on. For both HEES and system- on-chip designs, the interconnect architecture should be selected by considering the scale of the system. The interconnect architecture becomes more critical as the number of nodes increases. CTI architectures of the HEES system are similar to the system-on-chip interconnect architectures. Figure 56 shows four interconnect architectures of four nodes. Figure 56(a) is a shared bus interconnect. It is simple to implement, but has a limited scalability. Variances of the shared bus CTI with higher scalability include a segmented bus CTI in Figure 56(b) and multiple bus CTI in Figure 56(c). The point-to-point interconnect in Figure 56(d) provides independent paths between every pair of nodes, but its cost increases exponentially as the number of nodes increases. These architectures are well explored for the system-on-chips, and also applied for the HEES systems [116]. 161 Figure 56. Various interconnect architectures for system-on-chip and HEES system. As the number of EES banks in the HEES system further increases, the CTI architecture in Figure 56 cannot provide efficient paths for charge transfers. Higher energy efficiency may be achieved if the CTI network is able to provide more isolated paths to simultaneous charge transfers for the energy-optimal CTI voltage. We proposed a networked CTI architecture proposed in Figure 57, which is comparable to a typical network-on-chip architecture [115]. As the number of processing elements in a network-on-chip increases, the single-level on-chip bus architecture is no longer able to handle increased data exchanges between the processing elements. Similar to the network-on-chip that requires packet routing, a HEES system with a networked CTI architecture requires routing of the charge transfers. We provide optimization principles of charge transfer routing in [115]. 162 Figure 57. The architecture of a CTI router. An associated EES bank is connected via a power converter. The arrows denote the CTI links. 7.2.3 HEES System Control The HEES system requires more sophisticated management policies than a conventional EES system because of the heterogeneity of EES elements. Proper management policies are very crucial to achieving high energy efficiency in the HEES system. The HEES management policies are system-level policies for maximizing the benefits of the HEES system in energy efficiency, lifetime, state-of-health, etc., by exploiting the heterogeneity. The most important decision required in a HEES system is energy distribution among heterogeneous EES banks. We need to select particular EES banks to charge or discharge and determine the CTI voltage and amount of current that maximize the energy efficiency. Also, we may need to internally move energy from one EES bank to another in order to mitigate the self- discharge or prepare for the expected demand for energy/power capacity. These operations are called charge management, which is named after cache management in computer memory hierarchy in [129]. The charge management includes (i) charge allocation for charging EES banks [110], [122], (ii) charge replacement for discharging EES banks [112], [124], and (iii) 163 charge migration for moving energy between the EES banks [109], [113]. A SoH-aware charge management policy enhances battery lifespan by utilizing supercapacitors for high frequency power input and output [114]. We discuss optimization methods of the charge management policies in more detail in the next section. In order for effectively implementing the high-level charge management policies, we use a microcontroller as the main controller in the HEES system (consider the HEES system architecture in Figure 54) to determine the operation of the converters. At the beginning of each time slot of system operation, i.e., a decision epoch, it sets the target output voltage/current level of each converter in the HEES system according to the high-level charge management policies (i.e., charge allocation, replacement, and migration), and the target output voltage/current level of each converter will remain the same within the time slot (usually in the order of seconds.) The power consumption of a typical ARM-based embedded processor or micro-controller is 0.6 - 1.2 W [29], [30], which is much less than the incoming/output power of a typical HEES system, which is 50 - 100 W. 7.3 HEES Prototype Implementation We present our first prototype of HEES system implemented in a 19-inch rack, as shown in Figure 58. The system has three EES banks: a 165 Wh, 24 V terminal voltage lead-acid battery bank, a 115 Wh, 24 V terminal voltage Li-ion battery bank, and a 6.5 Wh supercapacitor bank. It is able to store energy from AC power grid and supply AC power to load devices. The total energy capacity is around 300 Wh, and the peak power capacity is about 350 W. The HEES system is simply placed between power grid and load devices to save energy and perform various system-level power management schemes. We implement a novel control mechanism to 164 achieve efficient and reliable operations of the HEES system. We demonstrate the charge management policies applied to the real HEES system. Figure 58. HEES system prototype. 165 Chapter 8. Charge Management in HEES Systems 8.1 Basic Charge Management Schemes in HEES Systems An operative framework for designing a HEES system should enable holistic optimizations across different HEES components and structures through unified problem formulations and efficient solutions. Simultaneously, this framework should enable multi-objective optimizations with respect to cycle efficiency, cycle life, system cost, and so on. For illustration purposes and without loss of generality, we focus on cycle efficiency enhancement. The cycle efficiency of an EES system is defined as the ratio between the energy deposited and retrieved. This ratio is always less than one because of the energy loss during charging and discharging. Mainly the EES arrays and power converters cause energy loss. IR losses of the CTI and switch elements also cause power loss, but the loss is not dominant. The type of EES array such as supercapacitors, Li-ion batteries, lead-acid batteries, and so on, largely determines the energy loss in charging and discharging operations by the internal resistance and rate capacity effect [75], [107]. However, the energy loss is also heavily dependent on the operating conditions such as the magnitude of current, SoC, and temperature. The type of converter primarily determines the power loss in the power converters such as synchronous, buck, boost, and flyback. At the same time, the operating conditions greatly affect the power conversion efficiency such as input voltage, output voltage, and current. The system efficiency is changing during the charge and discharge operation continuously. Therefore, maintaining the highest 166 possible efficiency requires continuous updating of the charging and discharging current, which is not a trivial problem. It is crucial to provide policies and methods for charge allocation to the most suitable EES banks for a given incoming power level, and for charge replacement from the most suitable EES banks for a given load demand. Hence, the HEES system runtime control comprises of policies that dynamically control the manner in which the system is used once it has been deployed. These policies include algorithmic/heuristic approaches for basic energy management operations like charge allocation, charge replacement, and charge migration [107]. These also include heuristics that predict the future energy needs [122], heuristics that predictively charge some EES banks [38], and so on. These operations are illustrated in Figure 59. Even if the optimal charge allocation and replacement policies are put in place and executed, charge migration that moves charge from one EES bank to another is often necessary to improve the overall HEES system efficiency and responsiveness. Charge migration can ensure the availability of the most suitable EES bank (in terms of its self-leakage, output power rating, etc.) to service a load demand (to the extend that is possible.) To achieve this goal, we must first invent an efficient low-cost charge migration architecture. We must also develop systematic solutions for multiple- source and multiple-destination charge migration taking into account the efficiency of chargers, rate capacity effect of the storage element, terminal voltage variation of the storage element as a function of the SoC, and so on. 167 Figure 59. Conceptual diagram of charge allocation, replacement and migration. 8.1.1 Charge Allocation Charge allocation determines (i) destination EES banks to be charged from an external power source, (ii) the amounts of charging current of destination banks, and (iii) the CTI voltage. The goal of the charge allocation problem is to maximize the system's charge allocation efficiency (also known as charging efficiency), which is given by: 168 (8.1) As HEES system comprises of multiple heterogeneous EES banks unlike conventional homogeneous EES systems, selection of the most efficient energy storage banks to be charged is the key to achieving high charge allocation efficiency [110], [122]. Charging efficiency is strongly dependent on the type of EES banks, the magnitudes of the charging currents, SoCs of the energy storage banks, voltage and current characteristics of the external power source, and so on. Stark mismatch between the input voltage level and the energy storage bank terminal voltage results in unnecessarily large power loss in the chargers. Severe mismatch between the input power level and the destination energy storage bank power capacity results in a high power loss due to IR loss and rate capacity effect. The destination energy storage banks must be compatible with the input power source in terms of the energy capacity as well. A greedy charge allocation policy may choose the destination energy storage banks only based on their instantaneous charging efficiency. For example, such a policy tries to charge mainly EES banks with a high power capacity such as a supercapacitor bank. Supercapacitor banks generally have a low energy capacity and will be fully charged soon at an early stage of the charge allocation process. During the rest of the charge allocation process only battery banks with lower power capacity can be charged, which results in a low overall charging efficiency for the whole charging process. The CTI voltage significantly affects the efficiency of the chargers, and thus, it should be carefully determined by the charge allocation control algorithm. The optimal charging currents and the CTI voltage change over time as charge allocation progresses. We continue to monitor the system status, calculate the optimal setup, and control the charge allocation process accordingly. 169 Dividing the whole charge allocation period into a fixed number of timing intervals can significantly reduce the problem complexity [110], [122]. The boundary point between two consecutive intervals is called a decision epoch. We solve an instantaneous charge allocation (ICA) optimization problem to maximize the instantaneous charge allocation efficiency by a convex programming approach at each decision epoch [136]. We avoid fully charging of high power capacity EES banks at an early stage of the charge allocation process with additional constraints on the charging currents, thereby maximizing the overall charging efficiency. A brief procedure of the near-optimal charge allocation algorithm is shown in the following algorithm. In [122], we further integrate the near-optimal charge allocation algorithm with solar power generation prediction and maximum power transfer tracking (MPTT) technique for the PV panels [36]. The EES bank reconfiguration techniques proposed in [111] can be effectively integrated with the proposed charge allocation algorithm. We find the optimal EES bank configuration (series- parallel connection) to provide a better match between the EES bank terminal voltage and the CTI voltage. We provide further improvement on the charge allocation efficiency in this way 170 through reducing power dissipation in the power converters [20]. Similar methods also apply for the charge replacement and charge migration optimizations. Figure 60 shows the instantaneous charging efficiency, battery bank charging current, and supercapacitor bank charging current. The instantaneous charge allocation efficiency of the proposed algorithm is always higher than the UB or BBF policies. Although the initial instantaneous charging efficiency of the proposed algorithm is slightly lower than the SBF baseline, the efficiency of SBF drops significantly after the supercapacitor bank becomes fully charged. The improvement of charge allocation efficiency by using the proposed near-optimal algorithm ranges from 8.6% to 35.6% compared with baseline setups. Figure 60. Comparison of charge allocation efficiency among different policies. UB: unbiased charging (equal charging current for both banks). SBF: supercapacitor bank first (charging the supercapacitor bank first). BBF: battery bank first (charging the battery bank first). 171 8.1.2 Charge Replacement Charge replacement selects one or more source EES banks and also determines the amounts of discharging current of the source EES banks for a given load demand. The goal of the charge replacement problem is to maximize the charge replacement efficiency (also known as discharging efficiency), given by: (8.2) Discharging efficiency is strongly dependent on the type of selected EES banks and the magnitudes of the discharging currents, SoC of the source EES banks, and the load voltage and current characteristics. Therefore, compatibilities in terminal voltage, power capacity and energy capacity are again key factors in selecting the source EES banks and determining their discharging currents. The best-suited set of source EES banks changes over time, and we recalculate the source bank selection, discharging current, and CTI voltage setting at each decision epoch [112]. A brief procedure of the near-optimal charge replacement algorithm is shown in the following algorithm, which is similar to the charge allocation problem. 172 As shown in Figure 61, the instantaneous charge replacement efficiency of the proposed algorithm is always higher than the UB or BBF policies. The instantaneous charge replacement efficiency of the SBF policy again drops significantly after the supercapacitor bank gets fully discharged. The improvement of charge replacement efficiency by using the proposed near- optimal algorithm ranges up to 24% compared with baseline setups. Figure 61. Comparison of charge replacement efficiency among different policies. UB: unbiased discharging (equal discharging current for both banks). SBF: supercapacitor bank first (discharging the supercapacitor bank first). BBF: battery bank first (discharging the battery bank first). 8.1.3 Charge Migration Charge migration is the internal energy transfer among EES banks. The optimal EES banks for charge allocation and for charge replacement can be different in general. Some EES banks, such as a supercapacitor bank, are not suitable for long-term energy storage due to large self- discharge rate despite their high efficiency for charge allocation and/or charge replacement [110], 173 [112]. This motivates charge migration. Charge migration maximizes the power compatibility of EES banks. For instance, even a supercapacitor bank that is compatible with an incoming power source may be no longer compatible in the near future as it becomes fully charged. However, charge migration is an expensive process due to energy loss. It is additional overhead that is not necessary in a homogeneous EES system. It should therefore be performed judiciously based on accurate prediction of power profiles of the power source and load devices. We formulate the charge migration optimization problem such that a given amount of energy from a set of source EES banks is transferred to a set of destination EES banks within a predefined deadline time. We minimize the amount of energy drawn from source EES bank, or equivalently, maximizing the charge migration efficiency or minimizing the energy loss during the migration process. A simple case of charge migration optimization problem is the single- source, single-destination (SSSD) charge migration [109], whereas a more general case will be multiple-source, multiple-destination (MSMD) charge migration problem [113]. As examples of the charge management policies, we provide the architecture, modeling, and systematic solution for efficiency optimization in the SSSD and MSMD charge migration problems in Section 8.2 and Section 8.3, respectively. 8.2 Single-Source and Single-Destination Charge Migration: Architecture, Modeling, and Efficiency Optimization 8.2.1 Problem Formulation We focus on the optimal single-source and single-destination charge migration. Figure 62 illustrates the conceptual architecture of a single-source and single-destination charge migration process. Both the discharging process of the source EES bank and the charging process of the destination EES bank are controlled by a converter. The converter in the source EES bank is 174 configured as the voltage regulating mode and properly maintains the CTI voltage. The converter in the destination EES bank is configured as the current regulating mode. There is a central HEES controller that controls the output voltage or current of the programmable converters at each decision epoch so that the HEES system can perform globally optimal charge migration. The following table summarizes the notations. Figure 62. Single-source, single-destination charge migration structure. The single-source, single-destination charge migration problem is constrained by the energy conservation law. As illustrated in Figure 62, the power flowing into the destination EES bank charges the corresponding EES element array and drives the corresponding converter, i.e., (8.3) where and are the CTI voltage and the migration current on the CTI, respectively. The destination array CCV is a strong function of its OCV and charging current . The CCV-OCV relationship of EES elements captures their relationship. The migration current on the CTI comes from the source EES array through the corresponding converter, i.e., (8.4) where the source array CCV is a strong function of its OCV and discharging current . The converter power loss values and are functions of the input voltage, output voltage, and output current of the corresponding converters as described in Section 2.2. 175 Table XVIII. Notations for single-source, single-destination charge migration. 176 Suppose that the charge migration process starts at time and ends at time . The HEES controller provides set points of the two variables and for . The HEES system monitors the source and destination EES array currents, and calculates the SoC values and using Coulomb counting: (8.5) (8.6) and depend on control variable values and for . We calculate and based on the OCV-SoC relationship of source and destination EES arrays, respectively. The rest of variables shown in Figure 62 are either given or associated variables, which are determined by the control variables and equations (8.3) and (8.4). We formally describe the time-unconstrained charge migration problem and time- constrained charge migration problem as an optimal control problem considering efficiency variations of both converters, rate capacity effect, and OCV variations of the EES element arrays.  Given The initial SoC and , the amount of charge to be migrated and the relative deadline , where in .  Find the optimal and for during the charge migration process.  Such that . In addition, the Global Migration Efficiency (GME) should be maximized, which is given by 177 (8.7) 8.2.2 Time-Unconstrained Charge Migration Instantaneous Migration Efficiency Optimization: We consider the Instantaneous Migration Efficiency (IME) in at time t. We consider the general problem where both and are optimization variables. We obtain and from Coulomb counting. We have two control variables and . We maximize the IME such that (8.8) The optimal control variable values and are given by: (8.9) and are functions of and . Maximization of the IME is in general a quasi-convex (unimodal) optimization problem over and . The IME becomes lower when due to increasing in power dissipation caused by the EES array internal resistance and rate capacity effect. The IME becomes lower when because of the converter efficiency degradation. We exploit a ternary search algorithm, which is an extension of the well-known binary search algorithm, utilizing this quasi-convex property. This makes the solution quickly converge to the global optimal or at least a near global optimal solution. A branch and bound method provides the global optimal solution of the IME optimization problem at the expense of the solution complexity. We have the following observation about : 178 Observation I: becomes relatively larger when is higher and is lower. Global Migration Efficiency Optimization: We derive the optimal solution of that maximizes in Eqn. (8.7). Charge migration makes decrease and increase as time elapses. We solve in a discrete time space. We divide the charge migration process into time slots with an equal distance . We calculate and at each decision epoch , which is the beginning of each time slot, using the Coulomb counting method given in (8.5) and (8.6). We maximize the IME and find and . We set and during the time slot. We continue this process until is migrated into the destination EES array. We avoid from and changing abruptly within two consecutive time slots by setting small enough. In this case, the proposed solution is the optimal solution of . We reduce the online computation overhead by separating the solution into offline and online phases. We build a lookup table offline. The input variables of the lookup table are and , and the values stored in the lookup table are and . The online phase only needs to index the lookup table with and at each decision epoch to find the optimal control variable values. Since there are only two input variables, the size of the lookup table does not grow much, typically smaller than 20 by 20. The experimental results also back up that the granularity level of the lookup table does not have a strong effect on the charge migration efficiency. The online procedure for solving is shown in Figure 63. 179 Figure 63. Online procedure in the optimal solution of that maximizes . The lookup table is built online. Figure 64. An example of optimal time-unconstrained charge migration. Figure 64 illustrates an example of the optimal solution of . The source and destination EES banks are both supercapacitor banks with the initial OCVs of = 8 V 180 and = 4 V, respectively. The traces of , and = are shown in Figure 64. 8.2.3 Time-Constrained Charge Migration The Optimal Solution: We derive the optimal solution of based on dynamic programming. Please refer to the Appendix II for detailed proof. Theorem I: Maximizing in Eqn. (8.7) is equivalent to minimizing the total amount of charge extracted from the source EES array, which is given by . We find the optimal substructure property of as follows. This enables us to apply dynamic programming to find the optimal solution of the problem. Property I (The optimal substructure property): Suppose we achieve the optimal solution of that maximizes . Suppose that ( ) is migrated to the destination EES array by ( < ) in the optimal solution of . This corresponds to the subproblem . The optimal solution of contains within it the optimal solution of the subproblem . Based on Property I, we introduce the optimal solution of . The optimal solution is comprised of a planning phase and a control phase. The planning phase is executed at time , i.e., the beginning of charge migration, in order to find the optimal amount of charge migrated to the destination array in each time slot, denoted by , , ..., . On the other hand, the control phase is executed along with the charge migration process to determine the optimal 181 and for and control the process. We will discuss the two phases in details as follows. 1) The Planning Phase: The optimal algorithm in the planning phase is based on dynamic programming and given in the following algorithm, which requires to take a holistic view of the whole charge migration process at the beginning of the process. For and , let denote the minimal amount of charge drawn from the source EES array during when we migrate to the destination EES array during that time period. We initialize as (8.10) for . We calculate from all the for based on Property I. We calculate in two steps as follows. We keep the control variables, i.e., and , the same within the time slot . Step I (Calculation of for ): The following equation calculates for that guarantees the destination EES array to accumulate in the time slot : (8.11) We calculate based on and , and based on and : (8.12) (8.13) 182 We find that maximizes the IME at with given , , and , using optimization methods such as ternary search or branch and bound, or the high-order curve fitting method as shall be discussed later. We set for . We calculate , which is the minimum amount of charge drawn from the source EES array during time slot to deliver to the destination EES array, using Eqns. (8.3), (8.4), and (8.5). Step II (Calculation of ): We calculate as follows: (8.14) We keep track of the optimal , i.e., (8.15) which is necessary in finding the optimal control variable values after we find . 183 After we find and , we determine the optimal amount of charge migrated to the destination array in each time slot, denoted by , , ..., , in a reverse chronological order. For example, the optimal amount of migrated charge in time slot is , and that in time slot is . This process is called tracing back in dynamic programming. We use Figure 65 as an example to illustrate the tracing back procedure with . In this example we have migrated amount of charge by the end of the third time slot. We have , which is the optimal amount of charge migrated by the end of the second time slot. Then we have . Furthermore, we have , which is the optimal amount of charge migrated by the end of the first time slot, and then . Finally, because no charge has been migrated to the destination array at the beginning of the first time slot, we have . We have finished the tracing back procedure. Figure 65. An example to illustrate the tracing back procedure. 184 2) The Control Phase: Now that we have determined the set of , , ..., values from the tracing back procedure. At each decision epoch in the actual charge migration process, we calculate and using the Coulomb counting method given in (8.5) and (8.6). We determine the value during the time slot , such that amount of charge can be migrated into the destination EES array: (8.16) We subsequently determine the optimal that maximizes the IME at with given , , and the just calculated . We find the optimal CTI voltage value through branch and bound or ternary search algorithm, or the high-order curve fitting method that will be discussed next. We set and during the time slot to control charge migration, and wait until the next time slot . High-Order Curve Fitting: The optimal solution of requires finding with given , , and in both the planning phase and the control phase. In order to reduce the online computation overhead, we use a high-order curve fitting method to find an approximation of . We use a kernel feature vector in the high- order curve fitting, which is a collection of high-order functions of , , and . The kernel feature vectors have been widely used in machine learning methods [138]. We approximate by a linear function of , i.e., (8.17) 185 where is the fitting parameter vector with elements to be determined by the offline training phase of the high-order curve fitting. The high-order curve fitting method acts better than traditional linear curve fitting in capturing the nonlinear relationship between and as pointed out in [138]. The following kernel feature vector yields the best fitting results: (8.18) We set n = 9 as the kernel feature vector has nine elements. Hence, there are ten fitting parameters in total including the constant term of . The proposed high-order curve fitting method consists of both an offline initial training phase and an online estimation phase. The initial training phase determines with an optimization method such as branch and bound or ternary search algorithm for each possible pair. We determine using a standard least square method [138]. The online estimation phase calculates for given , and from (8.17). We use separate high-order curve fitting with different set of fitting parameters for the buck mode ( > ) and the boost mode ( < ), respectively, in order to further enhance the fitting accuracy. The high-order curve fitting method is effective in providing approximation of with negligible online computation overhead. The curve fitting results in only 0.02% IME degradation in average when the system is in buck mode compared with the ideal 186 case, i.e., for given , and is given in prior. The IME degradation is 0.15% when the system is in boost mode. 8.2.4 Computation Complexity and Overhead We compare the online computation complexity among the two optimal solutions proposed in this section for and , respectively, as well as the near-optimal solution of proposed in our conference paper [109]. We show the computation complexity comparison results in Table XIX. In Table XIX, N is the number of time slots during the entire charge migration process; M is the number of discrete levels of Q used in Algorithm 1 (shown above); p is the precision level used in the branch and bound algorithm or the ternary search algorithm. The higher p indicates the higher precision. Table XIX shows that we achieve significant online computation complexity reduction through the exploitation of lookup table and high-order curve fitting. We derive the optimal control variable values in O(1) time complexity when equipped with these methods. Moreover, experiments show that N = 100 and M = 200 400 will yield good enough charge migration results that are not sensitive to further enhancing the granularity level. In this case, the online computation time for the optimal solution of is less than 200 ms on a 3.0 GHz desktop computer or less than 1 2s on a typical ARM-based embedded processor (as the micro-controller) [29]. Please note that this is the total online computation time of a whole charge migration process, instead of the computation time at a single decision epoch. In conclusion, the optimal solution of has reasonable computation complexity for online implementation in the HEES controller though its computation complexity is higher than the near-optimal solution. 187 Table XIX. Online computation complexity comparison among the three algorithms. Next, we discuss about the energy overhead in the micro-controller and peripherals in the HEES system. We compare the energy overhead with the amount of energy transferred into the destination EES array in a typical charge migration process with 20 V terminal voltage for both source and destination EES arrays, 2 A current flowing into the destination EES array, and 1000s migration time. Then the total energy transferred into the destination EES array is 40 kJ. On the other hand, the power consumption of a typical ARM-based embedded processor is 0.6 1.2 W [29]. We know from above that the total online computation time of the micro-controller is less than 1 2s for the whole charge migration process of 1000s, resulting in 0.6 2.4 J total energy overhead in the micro-controller. For the rest of time, the micro-controller can be either power gated or perform optimization for other HEES operations, and thus are not accounted for as the energy overhead of the target charge migration process. Hence, the total energy overhead in the micro-controller (0.6 2.4 J) is negligible compared with the amount of energy transferred in a charge migration process (40 kJ). This gap will become even greater for larger-scale HEES systems, e.g., for residential usages or hybrid electric vehicles. 188 8.2.5 Experimental Results We show the experimental results of the single-source, single-destination charge migration. We demonstrate the results of all four charge migration scenarios: supercapacitor-to- supercapacitor, supercapacitor-to-battery, battery-to-supercapacitor, and battery-to-battery for both the time-unconstrained and time-constrained charge migrations. We compare the results of the proposed optimal solutions with the baseline systems. We consider two types of baseline systems. Baseline systems of the first type apply constant and in the charge migration process, independent of time t, whereas the second type applies constant and optimized during charge migration. We apply Linear Technology LTM4607 converter as the converter in the HEES system. We extract the parameters required in the converter model (please refer to Section 2.2) from the datasheet [22]. We obtain characteristics of the Li-ion battery by performing measurement on a GP1051L35 Li-ion 2-cell series battery pack with 350 mAh nominal capacity [108] and extracting the parameters for the battery model shown in Figure 45. Time-Unconstrained Charge Migration: We summarize the global migration efficiency results of supercapacitor-to- supercapacitor, supercapacitor-to-battery, battery-to-supercapacitor, and battery-to-battery time- unconstrained charge migrations in Table XX, Table XXI, Table XXII, and Table XXIII, respectively. We normalize the values with respect to the results of the optimal solution in each test case. The normalized values of the proposed optimal solution are 100% as annotated with “optimal” in the tables. The actual values of the optimal solution are 84.11%, 77.88%, 80.34%, and 78.68%, respectively, in these four cases. The baseline systems of the first type apply constant and in the charge migration process. The values in the 189 baseline systems are equal to , , and , respectively. The values are equal to 0.2 A, 0.5 A, 1 A, and 2 A, respectively. The second type of baseline systems applies constant value and optimized during charge migration due to potential compliance to standards, compatibility choices, and stability issues. The results of the second type of baseline systems are listed in the rows with the title "adaptive” in the tables. The proposed system and baselines have the same initial conditions and amount of charge to be migrated to the destination array. The initial source and destination supercapacitor array OCVs are given by = 8 V and = 1 V, respectively, and the target migration charge is given by Q = 1200 C in the supercapacitor-to-supercapacitor charge migration. We set = 10 V, = 3.8 V (i.e., = 0.6), and Q = 1000 C in the supercapacitor-to-battery charge migration. We set = 8.2 V (i.e., = 0.9), = 3 V, and Q = 1000 C in the battery-to-supercapacitor charge migration. We set = 16.4 V (i.e., = 0.9), = 7.5 V (i.e., = 0.2), and Q = 2000 C in the battery-to-battery charge migration. Table XX. Comparison of normalized GMEs in time-unconstrained supercapacitor-to- supercapacitor charge migration. 190 Table XXI. Comparison of normalized GMEs in time-unconstrained supercapacitor-to-battery charge migration. Table XXII. Comparison of normalized GMEs in time-unconstrained battery-to-supercapacitor charge migration. Table XXIII. Comparison of normalized GMEs in time-unconstrained battery-to-battery charge migration. The proposed optimal charge migration control algorithm consistently outperforms the first type of baseline algorithms with constant and as illustrated from Table XX through 191 Table XXIII. Most importantly, because there exists no systematic method determining the optimal constant and in the baseline systems, it is not surprising for someone to design HEES systems with inappropriate and in charge migration processes that yield very poor . The proposed optimal solution shows up to 83.4% enhancement in over a poorly configured baseline system. Even the accidentally optimally configured baseline system is up to 4.4% less efficient than the optimal solution. This is because the optimal solution dynamically adjusts and according to the source and destination EES array SoCs in order to yield the optimal . In general, the proposed optimal algorithm achieves more significant efficiency enhancement over the first type of baseline algorithms in supercapacitor-to- supercapacitor charge migration than in battery-to-battery migration. This is because the difference in the optimal at the beginning and at the end of the charge migration process is higher in the former case due to more significant OCV variation in supercapacitor arrays during charge migration. Finally, we analyze the effect of inaccuracy in voltage and current regulation on the charge migration efficiencies. According to the state-of-the-art converter technologies [22], [23], the voltage regulation inaccuracy is less than 0.5% while the current regulation inaccuracy is less than 1%. We first study the instantaneous migration efficiency degradation of a supercapacitor- to-supercapacitor charge migration at time t with = 8 V and = 1 V. We assume a worst-case 0.5% regulation inaccuracy of and 1% regulation inaccuracy of , and this results in less than 0.02% efficiency degradation. In fact, we need a voltage regulation error of 15% or a current regulation error of 10% to have a 1% instantaneous efficiency degradation, illustrating that the instantaneous efficiency function is quite “flat” near the optimal value of and . Next, we consider the whole supercapacitor-to-supercapacitor charge 192 migration process with = 8 V, = 1 V, and Q = 1200 C. We consider that the regulation inaccuracy of is a uniform distribution between 0 and 0.5% while that of is uniformly distributed between 0 and 1%, which is closer to the realistic case. Then the degradation in migration efficiency is less than 0.01% for the whole charge migration process, which is hardly noticeable. In order to derive a general conclusion, we study the degradation of instantaneous migration efficiency in the supercapacitor-to-battery charge migration case with = 10 V and = 3.8 V (i.e., = 0.6), and assume 0.5% inaccuracy of and 1% inaccuracy of . The efficiency degradation is 0.07% in this case, which is slightly higher because of the higher power dissipation in the internal resistance of the battery array. In this case, we need a voltage regulation error of 7% or current regulation error of 10% to have a 1% efficiency degradation, which is far beyond the regulation inaccuracy of state-of-the- art converters. Based on these observations, we make the conclusion that the effect of regulation inaccuracy is negligible on the charge migration efficiency, and this conclusion also applies to the time-constrained charge migration. Time-Constrained Charge Migration: We summarize the global migration efficiency results of supercapacitor-to- supercapacitor, supercapacitor-to-battery, battery-to-supercapacitor, and battery-to-battery time- constrained charge migrations in Table XXIV, Table XXV, Table XXVI, and Table XXVII, respectively, with shown in the first column of each table. We show the normalized values of the optimal solution and the near-optimal solution proposed in our conference paper [109] in the second column and third column, respectively. These values are normalized with respect to the optimal in the time-unconstrained migration case. The , , and Q values are exactly the same as those used in the time-unconstrained experiments. We only 193 consider the first type of baseline systems with constant and due to space limitation. We use constant charging current = in the baseline systems corresponding to the given , in which is defined in (10) of [109]. Charge migration in the baseline systems finishes just before the deadline with the smallest possible constant charging current. Table XXIV. Comparison of normalized GMEs in time-constrained supercapacitor-to- supercapacitor charge migration. Table XXV. Comparison of normalized GMEs in time-constrained supercapacitor-to-battery charge migration. 194 Table XXVI. Comparison of normalized GMEs in time-constrained battery-to-supercapacitor charge migration. Table XXVII. Comparison of normalized GMEs in time-constrained battery-to-battery charge migration. The proposed optimal solution and the near-optimal solution consistently outperform the baseline algorithms under the same deadline constraint, i.e., the same , as illustrated from Table XX through Table XXIII. The main reason is the flexibility in finding the optimal and that are functions of the source and destination array SoCs. The proposed optimal algorithm outperforms the near-optimal algorithm in two aspects. First, it achieves up to 8.5% in efficiency enhancement compared with the near-optimal algorithm, when the deadline is relatively tight ( = 200). Please note that this 8.5% efficiency enhancement corresponds to around 21% reduction in energy loss during charge migration, which is significant because charge migration is a frequent operation to enhance the responsiveness and availability of the 195 HEES system. Also, the seemingly small 4.0% efficiency enhancement in supercapacitor-to- supercapacitor migration when = 300 corresponds to around 14% reduction in energy loss. There is a second benefit of the optimal solution as shown in Table XXVI. When the time constraint is relatively tight, both the near-optimal solution and baselines fail to finish the charge migration. This is because the source battery arrays cannot support the high charging power at the end of charge migration process when the SoC (and terminal voltage) of the destination supercapacitor array becomes high. In this case, only the optimal solution can finish the charge migration process. We provide reasons as follows. In order to explain the reason that the optimal algorithm outperforms the near-optimal one, Figure 66 illustrates as an example the traces of in the optimal solution and the near- optimal solution of a supercapacitor-to-supercapacitor time-constrained migration when = 400. is higher at the beginning and lower at the end of the charge migration process in the optimal solution using dynamic programming. On the other hand, is constant during the charge migration process in the near-optimal solution because of the constraint (11) in [109]. Therefore, the proposed optimal algorithm outperforms the near-optimal algorithm when the deadline is (relatively) tight due to its higher flexibility in choosing the appropriate . Please note that the charging current at the end of charge migration process in the optimal solution is lower than that in the near-optimal solution. This explains why the optimal solution does not fail in the battery-to-supercapacitor migration in Table XXVI because the source battery bank can support the relatively small charging current when the terminal voltage of the destination supercapacitor array becomes high. On the other hand, when the deadline is very loose, the results of both the optimal and the near-optimal solutions converge to the optimal in the time-unconstrained charge migration. The comparison between the optimal 196 and near-optimal solutions shed some light on when the optimal solution is more desirable to implement and when near-optimal solution is more desirable. Figure 66. Traces of in the optimal solution and near-optimal solution of a time- constrained charge migration. 8.3 Multiple-Source and Multiple-Destination Charge Migration: Architecture, Modeling, and Efficiency Optimization In the last section we discussed about single-source and single-destination charge migration and optimization. The more general multiple-source and multiple-destination (MSMD) charge migration may be often needed in a large-scale HEES system. In this section, we formally describe the MSMD charge migration problem and optimization. Compared to performing multiple SSSD charge migration operations sequentially, performing a single MSMD charge migration results in lower charge transfer time and higher migration efficiency. We formulate the MSMD charge migration optimization problem as the problem of delivering a fixed amount of energy to the destination banks and targeting at the maximization of the overall charge migration efficiency defined as the ratio of the total energy received by the destination EES element arrays to the total energy drawn from the source EES element arrays. 197 The goal of the MSMD charge migration optimization solution is to provide the optimal voltage level for the CTI, the amount of charging currents among the destination EES banks, and the amount of discharging currents among the source EES banks. The MSMD charge migration problem is formulated as a non-linear programming problem (NLP), which is generally hard to solve efficiently. The more complicating factor here, however, is the fact that the SoC of an EES array at time instance is a function of the charging or discharging currents before that time. Therefore, a one-shot solution of the NLP can result in a solution that is useless in practice. To address this latter issue, we propose solving the MSMD charge migration problem over short intervals while updating the SoC’s of various EES arrays in between consecutive intervals. More precisely, we divide the total charge migration time (also called a relative deadline) into a fixed number of timing intervals. The boundary point between two consecutive intervals is called a decision epoch. At each decision epoch, we solve a spontaneous MSMD (sMSMD) optimization problem in which the SoC and OCV levels for all EES arrays are known. Assume that the relative deadline is divided into timing intervals, i.e., . At decision epoch , the remaining amount of energy that must still be delivered to the destination banks is equally divided among the remaining time intervals. The result is called an energy quantum. The goal of the sMSMD optimization problem is to maximize the charge transfer efficiency while delivering at least this quantum of energy to the destination banks. The optimal solution to the sMSMD optimization problem will not only specify the values of the CTI voltage and charge/discharge current of various EES banks, but also determine the optimal amount of energy (which is still no less than the required quantum) that can be transferred in one timing interval. The action is taken, the SoC and OCV levels of all EES arrays are updated and the process of setting up and solving the sMSMD optimization problem at the 198 next decision epoch is repeated. The process continues until the full amount of required energy is transferred to the destination banks. The sMSMD optimization problem at each decision epoch can be solved in an iterative manner, where in each iteration we solve a quasi-convex optimization problem in polynomial time. Details are provided below. 8.3.1 Architecture and Problem Formulation Figure 67 presents the conceptual architecture for the MSMD charge migration in a HEES system. For given groups of source EES banks and destination EES banks, the proposed near- optimal MSMD charge migration algorithm shall effectively determine the CTI voltage level, discharging currents from source EES banks and charging currents for destination EES banks. The goal of the MSMD charge migration algorithm is to maximize the charge migration efficiency during the migration process, considering distinct properties of EES element arrays and efficiencies of chargers. The near-optimal CTI voltage, discharging currents and charging currents may vary over time due to the changes of SoC’s of the EES element arrays. As shown in Figure 67, we have sets of source EES banks, each consisting of EES banks, on the left of the CTI, and sets of destination EES banks, each consisting of EES banks, on the right of the CTI, where each bank consists of an EES element array, a discharging control charger and a charging control charger that connect the EES array to CTI. The charging control chargers in source EES banks and discharging control chargers in destination EES banks are turned off in the case of MSMD charge migration process and thus are removed from the schematic for simplicity. At time instance , and ( ) denote the open circuit terminal voltage (OCV) and closed circuit terminal voltage (CCV) of the k-th EES array in the i-th set of source EES banks, respectively, and and ( 199 ) denote the OCV and CCV of the l-th EES array in the j-th set of destination EES banks, respectively. These two voltage values are generally not equal to each other due to the internal resistance of the EES array. Their relation is described in Section 6.2.3. The source and destination EES arrays have self discharge with the power rate of and , respectively. The input and output currents of the discharging control charger connecting the k-th EES array in the i-th set of source EES banks to the CTI are denoted by the array discharging current, , and bank discharging current, , respectively. The power loss of the charger is denoted by and is a function of its input and output voltages and currents, as shown in Section 2.2. Similarly, we define the array charging current , bank charging current , and the power loss of the charging control charger connecting the l-th EES array in the j-th set of destination EES banks to the CTI. 200 Figure 67. MSMD charge migration system architecture. In an MSMD charge migration process, the system controller supervises the bank discharging currents of all the source EES banks, the bank charging currents of all the destination EES banks, as well as the CTI voltage at time instance . All other voltage and current values are either given (e.g., EES array OCVs) or are associated variables. The MSMD migration problem is constrained by the energy conservation law. As illustrated in Figure 67, the power flowing into each destination EES bank is used to charge the corresponding EES element array and drive the corresponding charging control charger, i.e., at time , we have: 201 (8.19) where , , , and of the charger connecting CTI and the l-th EES array in the j-th set of destination EES banks are , , , and , respectively. We further denote the charge increasing speed of the l-th EES array in the j-th set of destination EES banks by , when neglecting the self-discharge effect. Obviously, and is a concave and monotonically increasing function of , denoted by , and is given by (8.20) Moreover, each bank discharging current comes from source EES array through the corresponding discharging control charger, i.e., (8.21) where , , , and of the charger connecting the k-th EES array in the i-th set of source EES banks to CTI are , , , and , respectively. We further denote the charge decreasing speed of the k-th EES array in the i-th set of source EES banks when neglecting the self-discharge effect by . and is a convex and monotonically increasing function of , denoted by , and is given by (8.22) Furthermore, the total current flowing into the CTI equals to the total current flowing out of the CTI, i.e., 202 (8.23) The initial OCVs of all the EES element arrays can be derived based on the initial EES array SoC’s, using the OCV-SoC relation for batterys or the linear function for supercapacitors. Suppose that the migration process starts at time , we formulate the MSMD charge migration problem as transferring , , …, amounts of energy to the 1 st , 2 nd , …, N-th set of destination EES banks, respectively, within certain time limit , such that the energy loss in the 1 st , 2 nd , …, M-th set of source EES banks are no more than , , …, , respectively. We formally describe the MSMD charge migration problem as follows: Given: for and for , amounts of energy to be migrated for , energy loss constraints for , and relative deadline . Find: the optimal , ( , and ( ), for . Maximize: the charge migration efficiency , defined as: (8.24) Subject to: i) Energy Conservation: (8.19) - (8.23) are satisfied. ii) Calculating SoC: SoC values at time are given by (8.25) 203 (8.26) for and . In (8.25) and (8.26), and denote the full charge of the k-th EES array in the i-th set of source EES banks, and the l-th EES array in the j-th set of destination EES banks, respectively. The self-discharge currents and can be calculated based on the EES elemnet modeling in Section 6.2.3, and the SoC values calculated in (8.25), (8.26) should be within the range of . iii) Achieving Specified Energy Delivery to Destination Banks: the amount of energy shall be migrated into the j-th ( ) set of destination EES banks equals to , i.e., (8.27) iv) Bounding Energy Depletion of the Source Banks: the amount of energy loss in the i-th ( ) set of source EES banks is no more than , i.e., (8.28) v) Calculating OCV and CCV from SoC: the OCV-SoC relation for battery and supercapacitor (linear function), and the OCV-CCV relation, as discussed in the EES element modeling in Section 6.2.3. vi) Non-negativity of Charging/Discharging Currents: the output currents of discharging control and charging control chargers, i.e., the bank discharging currents and array charging currents, are no less than zero at any time , i.e., 204 (8.29) for each , and . 8.3.2 Optmization Method In this section we first discuss the spontaneous multiple-source, multiple-destination (sMSMD) charge migration problem targeting at maximizing the spontaneous migration efficiency, and propose an efficient solution for the sMSMD charge migration problem. After that we consider the original MSMD charge migration problem, and provide a solution in a discrete time space by performing the near-optimal sMSMD optimization at each decision epoch with lower bounds on the amount of delivered energy to destination banks and upper bound on the amount of energy depleted from the source banks. Spontaneous Charge Migration Optimization: In the sMSMD charge migration problem formulation, we omit the time index for simplicity in writing. In this problem, we have EES array OCVs for and for , derived from EES array SoC values. The system control variables are the CTI voltage , bank discharging currents ( ) and bank charging currents ( ). However, in the sMSMD charge migration optimization problem formulation we use the charge increasing speeds ( ) when neglecting self-discharge of destination EES arrays as optimization variables, instead of the bank charging currents, to make the optimization problem easier to solve, which is equivalent since the bank charging currents can be calculated and used for system control as long as the sMSMD charge 205 migration problem has been solved. The objective function to be maximized is (derived from (8.24)): (8.30) We further derive the constraints from (8.19) – (8.23) and (8.29) by omitting time index . Moreover, there exists additional constraint come from (8.25) and (8.26) that the bank charging currents (or equivalently, ) if , and the bank discharging current if . It can be observed that the sMSMD migration optimization problem is a non-linear and non-convex optimization problem, and therefore effective heuristics have to be developed to find the near-optimal solution. We first consider a quasi-convex version of sMSMD (qsMSMD) charge migration problem to find the optimal ( ) and ( ) values aiming at maximization of spontaneous charge migration efficiency , under the assumption that the CTI voltage and the source and destination EES bank CCVs are given. In the qsMSMD optimization problem, the power loss of each discharging control charger is a quadratic function of according to the modeling in Section 2.2, and therefore the array discharging current becomes a convex function of according to (8.21). In addition, due to the fact that is a convex and monotonically increasing function of , becomes a convex function of as well due to the rules of convexity of composite functions [136]. Therefore the nominator of (8.30) is a linear function of optimization variables ( ) and ( ), while the denominator of (8.30) is a convex function of those 206 optimization variables (self-discharge power terms and can be viewed as constants.) Hence the spontaneous charge migration efficiency to be maximized is a quasi-concave function of optimization variables in the qsMSMD optimization problem. Note that constraints (8.19) – (8.22) are already integrated into the objective function, and constraint (8.29) will become convex inequality constraint if we use instead of in the constraint (they are equivalent.) Moreover, constraint (8.23) will become convex inequality constraint if we modify it into the following form: (8.31) which is also intuitive. Hence, the qsMSMD charge migration optimization problem becomes a quasi-convex programming problem, and its optimal solution can be calculated in the following way: since the feasibility problem for in the qsMSMD charge migration problem setup (i.e., CTI voltage, EES bank CCVs are given) is a convex optimization problem and can be solved in polynomial time [136], we could use bisection method to effectively derive the maximum feasible value efficiency could achieve, which will be the optimal spontaneous charge migration efficiency of the qsMSMD charge migration problem. The optimal variable values can be determined accordingly. We propose the following two heuristics to determine , ( ), and ( ) values, which are assumed given in solving the qsMSMD charge migration optimization problem. Heuristic A: Assume that the optimal spontaneous migration efficiency is a quasi- concave function with respect to . We propose to solve a qsMSMD migration problem with a fixed value, and use the ternary search algorithm to effectively search feasible region of 207 to get the optimal value. Simulation results validate the assumption of quasi-concavity and prove the efficiency of ternary search for finding the best-suited value. Heuristic B: We start from the initial values and , and find the final and values iteratively. In each iteration, we solve a qsMSMD migration problem with and values assumed to be fixed and update such values based on the CCV-OCV relationship at the end of the iteration. We finally obtain the near-optimal solution to the sMSMD charge migration optimization problem by iteratively solving the qsMSMD migration problem. The complete algorithm is given in the following algorithm. 208 Global MSMD Charge Migration Optimization Algorithm: In the global MSMD charge migration optimization algorithm which may last for a few minutes or hours, we have to consider not only the charge migration efficiency, but also how to satisfy the migration energy constraint (8.27) and the energy loss constraint (8.28). Therefore, we propose to solve the global MSMD charge migration problem in a discrete time space, by solving a series of sMSMD charge migration optimization problems one at each decision epoch, with additional constraints so that it can be guaranteed that constraints (8.27) and (8.28) shall be satisfied. Note that if the migration controller makes greedy decisions so that for some specific set of destination EES banks (e.g., the j-th set, for instance), it is possible that (the target) amount of energy has been migrated into such set of banks at an early stage, while at that time the migration energy constraints of other sets of EES banks have not been satisfied. This may result in efficiency degradation since the j-th set of EES banks will be turned off without being charged any more after that time. Therefore in our proposed heuristic for global charge migration, all sets of EES banks finish charging at the same time, i.e., we know that , , …, amounts of energy shall be transferred into the 1 st , 2 nd , …, N-th set of destination EES banks, respectively, and these transfers shall finish at the same time, and we denote this finishing time by (of course and is not known until charge migration finishes.) To achieve this goal, we incorporate the following constraint to the sMSMD charge migration optimization at each decision epoch (assume at time ): (8.32) 209 for each pair and . This implies that for any set (the j- th set, for instance) of destination EES banks, the energy accumulating rate is proportional to its target migration value at any time during charge migration process. Furthermore, we shall make sure that charge migration will finish within the time limit . To achieve this goal, we use for to denote the remaining energy to be migrated into the destination EES banks at time , which can be calculated as follows: (8.33) We add the following constraint to the sMSMD charge migration optimization performed at each decision epoch so that charge migration can be guaranteed to finish before deadline : (8.34) Next we shall discuss about the fairness issue, i.e., how constraint (8.28) shall be satisfied in the global MSMD charge migration solution. For each set of source EES banks (the i-th set, for instance), we use to denote the energy drawn from that set of source EES banks from the beginning of the migration process to time , i.e., (8.35) 210 Then the following constraint on the i-th set of source EES banks ( ) can be added to the sMSMD charge migration optimization performed at each decision epoch so that the source energy loss constraint (8.28) can be satisfied: (8.36) Since constraint (8.32) is affine equality constraint, and constraints (8.34) and (8.36) are convex inequality constraints, the sMSMD charge migration optimization with addition constraints (8.32), (8.34) and (8.36) can be solved with the same order of time complexity as the original sMSMD charge migration optimization problem. 8.3.3 Experimental Results We compare our proposed near-optimal solution of the MSMD charge migration problem, with various baseline systems with constant bank discharging/charging currents. Experiments are carried out with different deadline values. In fact, the baseline systems unaware of detailed models of EES arrays and power converters cannot guarantee to meet both the deadline constraint (8.27) and the energy loss constraint (8.28). Therefore for the need of comparison we make the assumption that baseline systems are aware of such detailed models and can meet all the related constraints. The CTI voltage of the baseline systems are set to be constant throughout the charge migration process. Moreover, the overall energy migration rate, defined as the increasing speed of the amount of energy migrated into all the destination EES banks, are set to be constant (are equal to ) for the baseline setups, such that the charge migration process will finish just before the deadline. Furthermore, the bank charging currents among destination EES banks in the baseline systems are properly set such that the energy 211 accumulating speed of each set of destination EES banks (the j-th, for instance) is proportional to its target migration value , and therefore constraint (8.27) can be satisfied. Within each set of destination EES banks, the bank charging currents of different banks are set to be the same. Also, the bank discharging currents among source EES banks in the baseline systems are again properly set such that the energy decreasing speed of each set of source EES banks (the i-th, for instance) is proportional to its energy loss constraint value , and therefore constraint (8.28) can be satisfied. Finally, within each set of source EES banks, the bank discharging currents of different banks are set to be the same. Figure 68. Comparison of charge migration efficiencies on charge migration process from supercapacitor banks to battery banks, with different values. 212 Figure 69. Comparison of charge migration efficiencies on charge migration process from battery banks to supercapacitor banks, with different values. In the first test case we compare our near-optimal MSMD charge migration policy with three baseline setups with constant CTI voltage 5 V, 10 V, 15 V, respectively, on the charge migration process from one set of source EES banks, consisting of four supercapacitor banks with array OCVs 13.5 V, 13.5 V, 11.0 V, 11.0 V, respectively, to one set of destination EES banks, consisting of four battery banks with array OCVs 3.0 V, 3.0 V, 12.0 V, 12.0 V, respectively. The total energy transferred into destination banks during the charge migration process equals to 50% of the total energy stored in all the four source supercapacitor banks. The comparison results between the proposed method and baseline setups are illustrated in Figure 68. As shown in this figure, four sets of experiments are carried out with relative deadline ( ) values 300 s, 500 s, 800 s, and 1200 s, respectively. In the second test case we compare our near-optimal MSMD charge migration policy with three baseline setups with constant CTI voltage 5 V, 10 V, 15 V, respectively, on the charge migration process from one set of source EES banks, consisting of four battery banks with array OCVs 4.0 V, 4.0 V, 12.0 V, 12.0 V, respectively, to two sets of destination EES banks, both 213 consisting of two supercapacitor banks with array OCVs 2.25 V and 3.0 V, respectively. The total energy transferred into destination supercapacitor banks during the migration process equals to 30% of the total energy required to fully charge all the four destination supercapacitor banks. Besides, the 1 st and the 2 nd sets of destination supercapacitor banks are supposed to receive 70% and 30% of the total migrated energy, respectively. The comparison results are illustrated in Figure 69. As shown in this figure, four sets of experiments have been carried out with relative deadline ( ) values 600 s, 800 s, 1000 s, and 1500 s, respectively. As can be seen from Figure 68 and Figure 69, the proposed near-optimal solution for MSMD charge migration problem consistently outperforms baseline systems under the same deadline constraint (the same value), with migration efficiency enhancement compared to baseline systems ranging from 9.1 % to 35 %. In the case when the deadline is very tight, the proposed near-optimal solution, although forced to operate with overall energy migration rate restricted by the deadline value, outperforms baseline systems due to the freedom of selecting and adjusting the optimal CTI voltage, the discharging currents among various banks in each set of source EES banks, and the charging currents among various banks in each set of destination EES banks. In the case when the deadline is relatively loose, the proposed near-optimal solution has additional degree of freedom of choosing the optimal overall energy migration rate. Therefore, in the latter case the MSMD charge migration process may finish before the deadline time, and the charge migration efficiency will converge to a maximum value along with the increase of relative deadline value . 214 Chapter 9. Online Fault Detection and Fault Tolerance in EES Systems Faulty EES elements, i.e., open-circuited or short-circuited EES elements, are inevitable especially for large-scale EES systems in long-term use. Repeated deep-discharges of batteries can cause capacity loss and eventually battery faults due to electrode damage arising from cycling stress [114], [141]. Supercapacitor faults may result from long-time uneven temperature distribution across the array, imbalanced capacity, and non-uniform aging processes. A fixed EES array structure may make the entire array unusable even with a single EES element fault. Dynamic reconfiguration of battery arrays was proposed for battery fault tolerance in large- scale EES systems consisting of thousands of batteries [139], [140]. This method successfully extends lifespan of battery arrays by bypassing faulty batteries and changing the battery array configuration. The local and global battery management units in large-scale battery arrays can detect faulty batteries by monitoring individual battery behavior. Therefore, the battery array reconfiguration algorithm in [139], [140] does not include any fault detection mechanism. However, this method cannot cope with faults in supercapacitor arrays due to lack of fault detection mechanism. In this chapter, we present a framework of online fault detection and fault tolerance for supercapacitor arrays. We first identify faulty supercapacitors and then reconfigure the supercapacitor array with the faulty supercapacitors disabled. We provide both a scalable structural support and an efficient algorithm for the online fault detection and fault tolerance 215 framework. By integrating programmable switches into the supercapacitor array, we reconfigure the supercapacitor array appropriately for fault detection and isolate faulty supercapacitors for fault tolerance. Compared to the reconfiguration architectures proposed in [139], [140], the proposed reconfiguration architecture requires a smaller number of switches and achieves more flexible configurations. The proposed fault detection algorithm identifies faulty supercapacitors in a logarithmic time complexity. The proposed fault tolerance algorithm produces the optimal configuration of supercapacitor array considering the efficiency variation of power converter. Experimental results demonstrate that the proposed fault detection and tolerance technique reduces the fault-induced EES system performance degradation by up to 91%. 9.1 Structural Support Although the proposed online fault detection and fault tolerance framework is generally applicable to any type of EES element arrays, we focus on an EES system consisting of a supercapacitor array and a bidirectional converter. We propose a reconfigurable supercapacitor array structure that achieves any configuration for normal EES system operation as well as more flexible configurations to facilitate online fault detection and fault tolerance. We improve this reconfigurable structure proposed in [111] to achieve more flexibility aiming at fault detection and fault tolerance. Figure 70 shows the proposed reconfigurable supercapacitor array structure. For each supercapacitor except for the last one in the array, we integrate four switches i.e., a top P-switch , a bottom P-switch , an S-switch , and an E-switch . The last supercapacitor has only an E- switch . Although in Figure 70 the supercapacitors are arranged from left to right, the physical layout of the array does not necessarily follow this pattern. 216 #1 S BP,1 S S,1 S TP,1 S E,1 #i S BP,i S S,i S TP,i S E,i #N S E,N P-switch S-switch P-switch E-switch Figure 70. Reconfigurable supercapacitor array structure. 9.1.1 Normal Supercapacitor Array Operation The reconfiguration structure in Figure 70 achieves any configurations for an array consisting of supercapacitors. For normal operation of the array, the two P-switches i.e., and of each i-th supercapacitor must be ON or OFF together. Moreover, and are ON when is OFF, and vice versa. The P-switches connect supercapacitors in parallel to form a group, and the S-switches connect the groups in series. enables the i-th supercapacitor. All the E-switches are ON in the normal operation. Figure 71 shows how to achieve a configuration for an array consisting of 16 supercapacitors using the reconfiguration structure in Figure 70. 217 #1 S BP,1 S S,1 S TP,1 S E,1 #2 S BP,2 S S,2 S TP,2 S E,2 #3 S BP,3 S S,3 S TP,3 S E,3 #4 S BP,4 S S,4 S TP,4 S E,4 #5 S BP,5 S S,5 S TP,5 S E,5 #6 S BP,6 S S,6 S TP,6 S E,6 #7 S BP,7 S S,7 S TP,7 S E,7 #8 S BP,8 S S,8 S TP,8 S E,8 #9 S BP,9 S S,9 S TP,9 S E,9 #10 S BP,10 S S,10 S TP,10 S E,10 #11 S BP,11 S S,11 S TP,11 S E,11 #12 S BP,12 S S,12 S TP,12 S E,12 #13 S BP,13 S S,13 S TP,13 S E,13 #14 S BP,14 S S,14 S TP,14 S E,14 #15 S BP,15 S S,15 S TP,15 S E,15 #16 S E,16 Figure 71. A configuration for a supercapacitor array. 9.1.2 Fault Detection and Fault Tolerance The supercapacitor array structure in Figure 70 also achieves more flexible configurations to facilitate the online fault detection and fault tolerance. Generally speaking, for fault detection and fault tolerance purposes, we need to create a configuration using only supercapacitors out of the whole array. Let denote the supercapacitor index set, where the elements in are indices of the supercapacitors in the whole array. is any non- empty subset of . The reconfiguration structure achieves any configurations by using only the supercapacitors in , where . The E-switches for the supercapacitors in are ON meaning that these supercapacitors are enabled. The E-switches for the rest of the supercapacitors are turned off. The setting of P-switches and S-switches here is different from 218 those in Figure 70. The most important modification is that the two P-switches for a disabled supercapacitor may not be ON or OFF at the same time. Figure 72 shows how to achieve a configuration using supercapacitors in for an array consisting of 16 supercapacitors. Likewise, we disable supercapacitors to bypass faulty supercapacitors for fault tolerance. #1 S BP,1 S S,1 S TP,1 S E,1 #2 S BP,2 S S,2 S TP,2 S E,2 #3 S BP,3 S S,3 S TP,3 S E,3 #4 S BP,4 S S,4 S TP,4 S E,4 #5 S BP,5 S S,5 S TP,5 S E,5 #6 S BP,6 S S,6 S TP,6 S E,6 #7 S BP,7 S S,7 S TP,7 S E,7 #8 S BP,8 S S,8 S TP,8 S E,8 #9 S BP,9 S S,9 S TP,9 S E,9 #10 S BP,10 S S,10 S TP,10 S E,10 #11 S BP,11 S S,11 S TP,11 S E,11 #12 S BP,12 S S,12 S TP,12 S E,12 #13 S BP,13 S S,13 S TP,13 S E,13 #14 S BP,14 S S,14 S TP,14 S E,14 #15 S BP,15 S S,15 S TP,15 S E,15 #16 S E,16 Figure 72. A configuration for a supercapacitor array. 9.2 Fault Detection and Fault Tolerance Algorithm We execute the online fault detection algorithm every time interval . We assume that at most one fault occurs during each time interval. This is a reasonable assumption considering the slow rate of fault occurrences in the EES array. Therefore, we need to detect only one potential 219 newly-occurring fault at each execution of the algorithm. The fault detection algorithm identifies a fault in logarithmic time. We execute the fault tolerance algorithm only when a new fault is detected. It determines the optimal number of supercapacitors that should be enabled in the array so that the EES system reaches its maximum achievable performance. 9.2.1 Fault Detection Algorithm The fault detection algorithm detects the type of fault (short- circuit or open-circuit) as well as the location of the faulty supercapacitor or confirms the non-existence of a new fault. Let denote the set of indices of identified faulty supercapacitors. Set is empty if no faulty supercapacitor was found before. The fault detection algorithm focuses on supercapacitors in set (i.e., the set of supercapacitors that is in but not in ) in order to detect the potential fault. Figure 73 shows the flow chart of the proposed fault detection algorithm. If no open-circuited or short-circuited supercapacitor exists in , the open-circuit or short-circuit detection procedure terminates in time. Otherwise, the open circuit and short circuit detection procedures identify the faulty supercapacitor in time, where . Run <Open-Circuit Detection Procedure> Find an open-circuit? N Run <Short-Circuit Detection Procedure> Find a short-circuit? N Return 0 (no fault) or index of the faulty supercapacitor Y Y Figure 73. Flow chart of the fault detection algorithm. 220 To determine the existence of an open-circuited supercapacitor in , we make an configuration (a series connection) of supercapacitors in , where the E-switches of supercapacitors in are disconnected. We charge the array from the power converter. If no open-circuited supercapacitor exists in , the voltage of the array increases accordingly, and the open-circuit detection procedure terminates after confirming no open-circuited fault existing in . If an open-circuited supercapactior exists in , the actual charging current of the array becomes zero as the supercapacitors are in series, and the open-circuit detection procedure continues. The output of the converter is protected by its voltage feedback control mechanism when the output is open. We partition into two equal-sized subsets and to find the faulty supercapacitor. The first elements in go to and the rest of the elements go to . We apply the same method to as that to for determining whether the fault exists in . If so, we partition into two subsets for further detection. If not, the fault exists in , and we partition into two subsets for further detection. This procedure continues until the faulty supercapacitor is located. To determine the existence of a short-circuited supercapacitor in , we make a configuration (a parallel connection) of supercapacitors in , where the E-switches of supercapacitors in are disconnected. We charge the array with a regulated current by the power converter. If there is no short-circuited supercapacitor in , the voltage of the array increases accordingly, and the short-circuit detection procedure terminates, confirming no short-circuited fault in . If there is a short-circuited supercapactior in , the terminal voltage of the array remains zero, and the short-circuit detection procedure continues. The short-circuit detection procedure continues in a way analogous to the open-circuit detection, which is based on binary search. We summarize the open-circuit and short-circuit detection procedures as follows. 221 222 223 9.2.2 Fault Tolerance Algorithm We execute the fault tolerance algorithm after a new fault is identified by the fault detection algorithm. The fault tolerance algorithm finds the optimal number of fault-free supercapacitors, , that should be enabled in the array so that the EES system achieves the maximum overall performance. We enable fault-free supercapacitors after the execution of the fault tolerance algorithm for performing the EES system operation i.e., charging and discharging. We disable the rest of supercapacitors regardless whether they are faulty or fault-free by disconnecting their E-switches. The energy stored in the supercapacitor array during EES system operation is equally distributed into the supercapacitors. The rest of the fault-free supercapacitors, which are disabled, have no stored energy (of course we fully discharge these supercapacitors before we disable them.) Therefore, the fault-free supercapacitors cannot be disabled once selected, and the rest of the fault-free supercapacitors cannot be re-enabled until the next execution of the fault detection algorithm. We first introduce the optimal supercapacitor array reconfiguration during system operation with a given number of enabled fault-free supercapacitors, followed by an example showing the necessity of finding . Finally, we provide the optimization algorithm for finding . Optimal Array Reconfiguration during System Operation: The power dissipation in the converter is the major part of power losses during charging and discharging of the supercapacitor array. The converter power efficiency increases when the voltage gap between its input and output voltages, i.e., between the CTI voltage and supercapacitor array terminal voltage, becomes smaller. Therefore, it is desirable to perform optimal runtime supercapacitor array reconfiguration to provide the most suitable supercapacitor array terminal voltage for 224 maximizing the energy efficiency. Suppose there are enabled supercapacitors in the array for the EES system operation. Given the CTI voltage , the CTI current , and the energy stored in the supercapacitor array during system operation, we adaptively find the optimal configuration of the supercapacitor array using only the enabled supercapacitors, where . We provide the optimization algorithm below. When the superapacitor array has a configuration by using the enabled supercapacitors, the energy stored in each enabled supercapacitor is . The terminal voltage of each enabled supercapacitor is , where is the capacitance of each supercapacitor. The terminal voltage of the supercapacitor array is . A positive value of implies that the converter is charging the supercapacitor array. The power loss of the converter is given by: . (9.1) The charging efficiency of the EES system is . (9.2) A negative value of implies that the converter is discharging the supercapacitor array. The power loss of the converter is . (9.3) The discharging efficiency of the EES system is . (9.4) 225 Algorithm 1 optimizes the EES system charging/discharging efficiency by finding the optimal supercapacitor array configuration given , , , and . Necessity of Finding : Suppose that the supercapacitor array consists of 30 supercapacitors, and one of them is identified faulty by the fault detection algorithm. If we choose , i.e., all the 29 fault-free supercapacitors are enabled for future EES operations, we only have two available configurations and . We can hardly generate a supercapacitor terminal voltage that is compatible with the CTI voltage through array reconfiguration. If we choose , i.e., 28 out of the 29 fault-free supercapacitors are enabled, we achieve six available configurations , , , , , and . Clearly, we have more flexibility in reconfiguration by using , and thereby, we achieve higher efficiency. Therefore, the algorithm of finding should jointly consider (i) 226 the total energy capacity of the enabled fault-free supercapacitors, which is proportional to the number of enabled supercapacitors and (ii) the reconfiguration flexibility. The Fault Tolerance Algorithm: Let denote the joint probability density function of , , and . We obtain by monitoring and recording the EES system operation history. We calculate the average energy accumulation rate of the supercapacitor array given the number of enabled fault-free supercapacitors: (9.5) We calculate the EES system performance as , (9.6) which is a function of . Therefore, the fault tolerance algorithm finds the optimal value as follows , (9.7) where . We search all the values from the set and find the one with maximum value. Algorithm 2 summarizes the fault tolerance algorithm. 227 In order to reduce the effort for finding , we improve Algorithm 2 by removing some elements from . For instance, suppose and are two elements in . We remove if and the number of factors of is larger than that of . This is because means that the array with enabled supercapacitors has larger energy capacity, and the second condition means that it also has more possible configurations. 9.3 Experimental Results We compare the energy capacity and performance of the EES system with the proposed online fault detection and tolerance technique and three baseline EES systems. The supercapacitor array in the proposed EES system employs the reconfiguration architecture as shown in Figure 70, where there are 60 supercapacitors in total. The superapacitor array in the baseline systems also consists of 60 supercapactiors. The first baseline system uses a fixed array configuration and has no online fault detection and tolerance technique. The second baseline system can detect and bypass faulty supercapacitors, but it can neither perform optimal runtime array 228 reconfiguration nor find the optimal number of enabled supercapacitors in the fault tolerance mode. In other words, it enables all the fault-free supercapacitors and keeps a fixed configuration in the fault tolerance mode. The third baseline system can detect/bypass faulty supercapacitors and perform runtime array reconfiguration, but it cannot find the optimal number of enabled supercapacitors. Figure 74. Comparison results on energy capacity. Figure 74 shows the energy capacity of the supercapacitor array in the proposed system after running fault detection and fault tolerance and Baseline 1, as a function of the number of faulty supercapacitors. The energy capacity values are normalized with respect to the initial array energy capacity without faulty supercapacitors. We assume that all the supercapacitor faults are short-circuit faults. The supercapacitor array energy capacity in the proposed system is related to the number of faulty supercapacitors but unrelated to their locations. On the other hand, the supercapacitor array energy capacity in Baseline 1 is related to both the number of faulty supercapacitors and their locations. We observe from Figure 74 that the proposed system 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Number of Faulty Supercapacitors Relative Energy Capacity Proposed Baseline 1 229 consistently outperforms Baseline 1 in terms of supercapacitor array energy capacity. This demonstrates the effectiveness of the proposed fault detection and fault tolerance techniques. Figure 75. Comparison results on EES system performance. Figure 75 shows the comparison results on EES system performance of the proposed system, Baseline 2 and Baseline 3, as a function of the number of faulty supercapacitors. The EES system performance values are normalized with respect to the initial EES system performance without faulty supercapacitors. Baseline 3 outperforms Baseline 2 due to the ability of performing runtime adaptive supercapacitor array reconfiguration. The proposed system outperforms both baseline systems primarily because the proposed system is able to find the optimal number of enabled supercapacitors. For example, both Baseline 2 and Baseline 3 use all 59 fault-free supercapacitors in EES system operations when the array contains only one faulty supercapacitor. Only two possible configurations and are available – a situation that severely degrades the EES system performance. On the other hand, the proposed system selects the optimal number 56 of enabled supercapacitors in this case. It is able to reduce the fault-induced EES system performance degradation by up to 91%. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Number of Faulty Supercapacitors Relative Performance Proposed Baseline 3 Baseline 2 230 Chapter 10. State-of-Health (SoH) Aware Charge Management in HEES Systems Based on a Novel SoH Degradation Model Among all the performance metrics, the cycle life of the EES elements is one of the most important metrics that should be considered by the designers of the EES system. The cycle life is directly related to the state-of-health (SoH), which is defined as the ratio of full charge capacity of an aged EES element to its designed (nominal) capacity. This metric captures the general condition of the EES elements and their ability to store and deliver energy compared to its initial state (i.e., compared to a fresh new EES element.) The state-of-health (SoH) degradation models presented in the reference papers [141], [142] can only be applied to the case of constant-current cycled charging and discharging with same state-of-charge (SoC) swing 1 in each charging/discharging cycle. To address this shortcoming, we present a novel SoH degradation model to estimate the SoH degradation under arbitrary charging/discharging patterns of the battery elements. The proposed SoH degradation model accounts for the following important observation: both a higher SoC swing and a higher average SoC in each charging/discharging cycle will result in a higher SoH degradation rate. Unlike a single element or a homogeneous EES system, the cycle life of the EES elements in a HEES system is largely dependent on the HEES charge management policy. Some references have worked on extending the lifetime of EES elements [143], [144], [145]. However, they only 1 SoC swing is defined as the SoC change during a charging/discharging cycle. 231 focus on either a single EES element or a homogeneous EES system, which consists of only a single type of EES element array. Reference [114] proposes an SOH-aware charge management policy for the HEES systems based on the simple SoH model introduced in [141]. It uses the supercapacitor bank as a buffer to shave the spiky portion of the source or load profiles so that the battery bank can stably receive energy from the power source or provide energy to the load device. This charge management policy has the following limitations: (i) it can only be applied to a two-bank HEES architecture consisting of a Li-ion battery bank and a supercapacitor bank, and (ii) the proposed policy is based on a simply crossover filter and is far from optimality. In this chapter, based on the novel SoH model, we derive a near-optimal charge management policy focusing on extending the cycle life of battery elements in the HEES systems while simultaneously improving the overall cycle efficiency. The derivation procedure of the SoH- aware charge management policy is based on the convex optimization technique, and has the following extensions over [114]:  It is applicable to the general HEES systems consisting of multiple battery banks and multiple supercapacitor banks.  The proposed policy is no longer limited to the case of cycled charging and discharging with the same SoC swing. It can be applied to the case of HEES charge management with arbitrary power source and load profiles.  The proposed policy achieves higher performance, because the optimization of charging/discharging currents of various EES banks depends not only on the frequency components but also on the magnitudes of the power source and load profiles. On the other hand, the crossover filter-based policy only depends on the frequency components [114]. Experimental results demonstrate significant cycle life improvement up to 17.3X. 232 10.1 SoH Degradation Model First, we formally define the SoC and SoH degradation of an EES array. The SoC of an EES array is defined by (10.1) where is the amount of charge stored in the EES array, and is the amount of charge in the EES array when it is fully charged. We interpret as the state of the EES array. On the other hand, the value gradually decreases during battery aging (i.e., SoH degradation.) The amount of SoH degradation, denoted by , is defined as follows: (10.2) where is nominal value of for a fresh new battery. 10.1.1 Related Work The SoH of batteries is difficult to estimate because it is related to a capacity fading effect (i.e., SoH degradation) which is a result of long-term electrochemical reaction. The capacity fading is related to the carrier concentration loss and internal impedance growth in the batteries. These effects strongly depend on the operating condition of the battery such as charging and discharging current, number of cycles, SoC swing, average SoC, and operation temperature [146], [142], [147]. The characterization of battery cell requires time-consuming experiments. Therefore, mathematical models help us to reduce the time complexity in estimating the SoH degradation. Electrochemistry-based models [148], [149], [57] are generally accurate but not easy to implement. Hence, we will discuss in the following the SoH degradation model of Li-ion 233 batteries proposed in [141], which shows a good match with real data but can only be applied to cycled charging and discharging of the battery elements with the same SoC swing. After that we will propose a novel SoH degradation model allowing for arbitrary charging and discharging patterns of the battery elements. The SoH degradation model proposed in [141] estimates the SoH degradation for cycled charging and discharging of a Li-ion battery cell, where a (charging/discharging) cycle is defined as a charging process of the battery cell from to and a discharging process right after it from to . The SoH degradation during one cycle depends on the average SoC level and the SoC swing . We calculate and of one cycle using: (10.3) (10.4) achieves the maximum value of 1.0 (100%) for the full 100% depth of discharge cycle, i.e., the SoC ranges from 0 up to 100% and back to 0. The SoH degradation during this charging/ discharging cycle, accounting for both the average SoC level and the SoC swing, is given by 234 (10.5) where , , , and are battery specific parameters; and are the operation battery temperature and reference battery temperature, respectively; is the duration of this charging/discharging cycle; is the calendar life of the battery. We use to denote the relationship between , , and . The total SoH degradation (from a brand new battery) after charging/discharging cycles is calculated by (10.6) where denotes SoH degradation in the m th cycle. In Eqn. (10.6), the normalized SoH degradation value will increase over the battery lifetime from 0 (brand new) to 100% (no capacity left). Typically, the values of or , which indicate 80% or 70% remaining capacity, respectively, are used in the literature as a measure of the end of useful life. The relationship between the Li-ion battery SoH degradation versus the SoC swing and average SoC level is shown in Figure 76. In this experiment, we vary the duration of a cycle to achieve different average SoC levels and SoC swings. We repeat the charge and discharge cycling until the battery reaches , and record the total number of cycles (i.e., the cycle life of the battery) at that time. The results are 235 shown in Figure 76. There are two important observations from Figure 76: (i) a higher SoH degradation rate is caused by both higher SoC swing and higher average SoC level in each charging/discharging cycle, (ii) the cycle life of a Li-ion battery increases superlinearly with respect to the reduction of SoC swing and average SoC level. We will make use of these observations as well as the function in the proposed SoH degradation model and SoH-aware charge management algorithm. Figure 76. Li-ion battery SoH degradation versus SoC swing (at different average SoC levels) and average SoC level (at different SoC swings). The SoH degradation of lead-acid batteries satisfies a similar relationship with respect to the average SoC level and the SoC swing [118]. However, the SoH degradation rate of lead-acid batteries is much higher than that of Li-ion batteries. A typically lead-acid battery has a cycle life of 300 – 500 cycles when , whereas the cycle life of a Li-ion battery is around 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 x 10 4 SoC swing Life cycles SoC avg 30% SoC avg 40% SoC avg 50% SoC avg 60% SoC avg 70% 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 x 10 4 SoC avg Life cycles SoC sw ing 40% SoC sw ing 45% SoC sw ing 50% SoC sw ing 55% SoC sw ing 60% 236 1500 – 2500 cycles [107]. On the other hand, the supercapacitors have a cycle life that is orders of magnitude higher than batteries [107]. Hence, we do not consider the SoH degradation of supercapacitors in this chapter. 10.1.2 Proposed SoH Degradation Model In this section, we propose a novel SoH degradation estimation model allowing for arbitrary charging/discharging patterns of the battery elements, which extends and generalizes the SoH degradation model introduced in [141]. We use Li-ion battery as an example to illustrate the new SoH degradation model. The proposed general SoH degradation model is based on the following two motivations: Motivation I: The SoH degradation rate is a superlinear function of the SoC swing and the average SoC level as seen from the discussion before. Moreover, the SoC swing has dominant effect among these two factors. An illustrative example of Motivation I is provided in Figure 77, which shows two SoC profiles of a battery cell within the same duration of time. The SoC profile in Figure 77 (b) results in a much higher SoH degradation (about 71.6% higher when the average SoC value is the same) although it has a smaller number of charging/discharging cycles. 20% SoC 80% SoC 40% SoC 60% SoC (a) (b)   Figure 77.Illustrative example of Motivation I. 237 Motivation II (Decoupling of Cycles): Consider the SoC profile of a battery cell as shown in Figure 78(a). Although it is not possible to directly apply the model in [141] to estimate the SoH degradation, we can perceive it as a combination of two charging/discharging cycles as shown in Figure 78(b).  (a) (b) 10% SoC 30% SoC 50% SoC 70% SoC 10% SoC 70% SoC 30% SoC 50% SoC Figure 78.Illustrative example of Motivation II. Based on the two motivations, we provide the general SoH degradation model as follows. Consider the period of charge management. We assume that the duration of the period is small compared with the battery life time (300 – 500 cycles for lead-acid battery or 1500 – 2500 cycles for Li-ion battery [107]) to make any noticeable change in the value for the charge management algorithm. Let denote the total SoH degradation of the Li-ion battery over this period, which we are going to estimate. In the first step, we initialize the value of to zero. We identify a set of turning points , , ..., , at which points the battery changes from charging to discharging or from discharging to charging. Figure 79 shows an example SoC profile versus time of a battery element and the set of turning points. 238 SoC(t 0 ) SoC(t 1 ) SoC(t 2 ) SoC(t 3 ) SoC(t 4 ) SoC(t 5 ) SoC(t 6 ) SoC(t 7 ) SoC(t 8 ) Figure 79.An example SoC profile versus time of a battery element and turning points. Next, we identify four consecutive turning points from the set of turning points, satisfying one of the following six conditions: (a) , (b) , (c) , (d) , (e) , (f) , The six cases are shown in Figure 80(a) - (f). In each case, we can identify a complete charging/discharging cycle as shown by the shadowed area in Figure 80(a) - (f). We take case (a) as an example. The SoC swing and average SoC level of the identified charging/discharging cycle are given by: 239 (10.7) (10.8) Then we estimate the SoH degradation in this cycle by . We delete the period and update the value of using: (10.9) The updating procedures of in the other five cases are similar and thus not explained in detail. We continue this procedure until only one cycle, i.e., the cycle with the largest SoC swing, remains in the SoC profile of the battery. Then we will obtain an effective estimate value of . Figure 81 provides an example of estimating the value from an arbitrary battery SoC profile. It can be proved that any charging and discharging pattern of the battery can be decoupled using this procedure to a set of charging/discharging cycles with potentially different depth of charging/discharging. Therefore, we can effectively calculate using this decoupling procedure. The details of proof are omitted due to space limitation. 240 SoC(t i ) SoC(t i+1 ) SoC(t i+2 ) SoC(t i+3 ) (a) SoC(t i ) SoC(t i+1 ) SoC(t i+2 ) SoC(t i+3 ) (b) SoC(t i ) SoC(t i+1 ) SoC(t i+2 ) SoC(t i+3 ) (c) SoC(t i ) SoC(t i+1 ) SoC(t i+2 ) SoC(t i+3 ) (d) SoC(t i ) SoC(t i+1 ) SoC(t i+2 ) SoC(t i+3 ) SoC(t i ) SoC(t i+1 ) SoC(t i+2 ) SoC(t i+3 ) (e) (f) Figure 80.Six basic cases for (charging/discharging) cycle identification. SoC(t 0 ) SoC(t 1 ) SoC(t 2 ) SoC(t 3 ) SoC(t 4 ) SoC(t 5 ) SoC(t 6 ) SoC(t 7 ) SoC(t 8 ) SoC(t 0 ) SoC(t 3 ) SoC(t 4 ) SoC(t 5 ) SoC(t 6 ) SoC(t 7 ) SoC(t 8 ) SoC(t 1 ) SoC(t 2 ) SoC(t 0 ) SoC(t 3 ) SoC(t 6 ) SoC(t 7 ) SoC(t 8 ) SoC(t 1 ) SoC(t 2 ) SoC(t 4 ) SoC(t 5 ) SoC(t 0 ) SoC(t 3 ) SoC(t 6 ) SoC(t 8 ) SoC(t 1 ) SoC(t 2 ) SoC(t 4 ) SoC(t 5 ) SoC(t 7 ) SoC(t 8 ) The largest cycle remains with SoC swing = SoC(t 3 )-SoC(t 6 ) Figure 81.An example of estimating the value from an arbitrary battery SoC profile. 241 10.2 SoH-Aware Charge Management I: System Model and Problem Formulation 10.2.1 System Architecture Figure 82 presents the architecture of the HEES system, which consists of a lead-acid battery bank, a Li-ion battery bank, and a supercapacitor bank. The proposed HEES charge management algorithm is able to handle more complicated HEES systems. Generally speaking, lead-acid batteries are much cheaper than Li-ion batteries, yet they suffer from a shorter cycle life and incur higher power loss during charging and discharging due to their larger internal resistance and more severe rate capacity effect. Supercapacitors are more expensive than both types of batteries. However, supercapacitors have nearly 100% charging and discharging efficiencies and a much longer cycle life than batteries. We use a slotted time model, i.e., all the system constraints as well as decisions are provided for discrete time intervals of equal length. More specifically, the whole time period of charge management is divided into time slots, each of duration . The HEES charge management algorithm should correctly account for the distinct characteristics of different types of EES elements and power dissipation in the DC-DC conversion circuitries. 242 Lead-acid array Li-ion array Supercap array SoC LA [i] SoC LI [i] SoC S [i] Charge transfer interconnects (CTI) V LA V LI V S [i] I LA [i] I LI [i] I S [i] Converter Converter Converter P bus,LA [i] P bus,LI [i] P bus,S [i] Converter Converter P src [i] P load [i] Figure 82.Structure of the HEES system considered in this chapter. 10.2.2 System Power Model Let and (1 ) denote the power generation of the power source and the power consumption of the electric load, respectively. For the EES element arrays, let , , and denote the SoC values of the lead-acid battery array, the Li-ion battery array, and the supercapacitor array, respectively. Let and denote the terminal voltages of the lead-acid battery array and the Li-ion battery array, respectively. We neglect the dependency of the battery terminal voltages on the SoC values because the terminal voltages are nearly constant in the major SoC operation range of 20% to 80% [75]. On the other hand, let denote the terminal voltage of the supercapacitor element array at time slot i, which is a linear function of SoC. Moreover, the charging/discharging currents of the lead-acid battery array, the Li-ion battery array, and the supercapacitor array are denoted by , , and , 243 respectively. The current values are positive when charging the EES array and negative when discharging. The rate capacity effect of batteries explains that the charging and discharging efficiencies decrease with the increasing of charging and discharging currents, respectively. More precisely, the Peukert's formula [75] describes that the charging and discharging efficiencies of a battery element array, as functions of the charging current and discharging current , respectively, are given by (10.10) where , , , and are constants known a priori. We define the equivalent current inside the battery array as the actual charge accumulating/reducing speed inside the battery array. For the lead-acid battery array, the equivalent current is calculated by: (10.11) The equivalent current of the Li-ion battery array can be calculated in the similar way. Lead-acid batteries are subject to a much more severe rate capacity effect compared with Li-ion batteries. On the other hand, the supercapacitor arrays have negligible rate capacity effect, i.e., . For the lead-acid battery array, we calculate from the initial SoC using Coulomb counting: (10.12) 244 where is the full charge capacity of the lead-acid battery array. Similar notations also apply for the Li-ion battery and the supercapacitor by replacing the subscript by and , respectively. The power conversion circuitries exploited in the system accounts for a significant portion of power losses. We denote the power conversion efficiencies ( due to power losses) of various converters at the i th time slot by , , , , and . We use , , and (1 ) to denote the power flowing into the lead-acid battery bank, the Li-ion battery bank, and the supercapacitor bank from the CTI, respectively. satisfies the following equation: (10.13) and also satisfy similar relationships. Moreover, we have the following equation due to the energy conservation law: (10.14) 10.2.3 Problem Formulation The objective of the SoH-aware HEES system control algorithm is to minimize the SoH degradation while satisfying the load power requirements. However, since there are two types of batteries in the HEES system, we define a new objective function, the overall value degradation, which captures the different cycle lifes and capital cost values of the two types of batteries. We define the overall value degradation during the period of charge management as follows. Let 245 and denote the SoH degradation of the lead-acid battery array and Li-ion battery array during the period of charge management, respectively. Let denote the amount of SoH degradation indicating the end-of-life of a battery array (i.e., or ). The capital cost values of the lead-acid battery array and the Li-ion battery array are given by and , respectively. Then the overall value degradation is (10.15) The SoH-aware HEES system control problem is formally described as follows: Given: Power source and load device power profiles , , respectively, for , initial SoC’s of the supercapacitor array . Minimize: the overall value degradation given by Eqn. (10.15). Subject to: i) Load Power Requirement Constraint: Eqn. (10.14) is satisfied. ii) Capacity and Power Rating Constraints: Each EES array SoC cannot be less than zero or more than 100%, i.e., (10.16) Moreover, the charging/discharging current of each EES array cannot exceed a maximum value, i.e., (10.17) 246 (10.18) (10.19) iii) Final Stored Energy Constraints: Each EES array SoC at the end of the charge management period should be no less than the initial SoC value, i.e., (10.20) 10.3 SoH-Aware Charge Management II: Algorithm In this section, we derive a near-optimal SoH-aware charge management policy based on the convex optimization technique. We need to find the near-optimal values of the initial SoC's and , as well as the EES array current profiles , , for . The proposed optimization procedure consists of an outer loop and a kernel algorithm. The outer loop finds near-optimal values of and using the ternary search technique, in order to minimize the overall value degradation while satisfying load power requirement (10.14). The kernel algorithm finds the optimal EES array current profiles , , for . The general procedure to derive the SoH-aware charge management policy is shown in Algorithm 1. In the rest of this section, we will describe the kernel algorithm in detail. 247 10.3.1 The Kernel Algorithm The kernel algorithm consists of two steps as shown in Algorithm 1: Feasibility check and the subsequent optimization of EES array current profiles in order to minimize the overall value degradation. We will discuss these two steps as follows. 1) Feasibility Check In this step, we are given the and values from the outer loop. We perform feasibility check, i.e. checking whether it is possible to find the EES array current profiles , , for such that all the constraints (10.14), (10.16) – (10.20) are satisfied. We formulate the feasibility checking problem as a convex constraint satisfaction problem (convex CSP) and optimally solve this problem in polynomial time. First, we define the energy increasing/decreasing rates inside the lead-acid battery array, the Li-ion battery array, and the supercapacitor array by , , and , respectively, satisfying: 248 (10.21) (10.22) (10.23) In the problem formulation, we use , , and as the optimization variables instead of the EES array current profiles , , . This will transform the problem into a convex CSP as we shall see in the following. The HEES controller can easily calculate the values of control variables , , from the derived values of , , and using Eqns. (10.11), (10.21) – (10.23). We rewrite constraint (10.14) as follows to make it a convex inequality constraint: (10.24) Both the energy conservation law and the load power requirement are still satisfied in (10.24). We know that , , and are convex functions of , , and , respectively, from Eqns. (10.11) and (10.13). This proves that constraint (10.24) is a convex inequality constraint. Moreover, the other constraints (10.16) – (10.20) can be translated into linear inequality constraints of , , and . Details are omitted due to space limitation. Then the feasibility checking problem becomes a convex CSP [136] because all the constraints are convex (or linear) inequality constraints. We can set the objective function to be a 249 constant value in order to solve this feasibility checking problem using standard convex optimization tools such as CVX [136]. After we find the solution of the feasibility checking problem, we calculate the average SoC levels and of the lead-acid battery array and the Li-ion battery array, respectively, from the derived and profiles. These average SoC values are important in the subsequent optimization step. 2) Minimizing the Overall Value Degradation After feasibility checking, we perform optimization to find the optimal values of , , and in order to minimize the overall value degradation given by Eqn. (10.15). We make use of the following observation in deriving the near-optimal charge management policy: Motivation III: Notice that we can decouple the charging and discharging profile of a (lead-acid or Li-ion) battery array into a set of charging/discharging cycles. The cycle with the largest SoC swing will have the most significant contribution to the SoH degradation. Based on Motivation III, we focus on minimizing the overall value degradation induced by the charging/discharging cycle (after decoupling) of both battery arrays with the largest SoC swing. Please notice that minimizing this objective function will help in minimizing the overall value degradation induced by the other charging/discharging cycles as well. For the lead-acid battery array, the largest SoC swing in all the charging/discharging cycles is given by: 250 (10.25) is a convex function of because the pointwise maximum function of a set of convex function is still a convex function [136]. Similarly, we define and calculate the largest SoC swing for the Li-ion battery array. Moreover, let denote the SoH degradation of the lead-acid battery array in one charging/discharging cycle as a function of the SoC swing and average SoC level . Similarly, we define the function for the Li-ion battery array. We minimize the overall value degradation contributed by the charging/discharging cycles with largest SoC swing of both battery arrays, as an estimation of the original objective function (10.15). The objective is given by: + (10.26) where we use the average SoC levels obtained from the feasibility check procedure as the estimation of average SoC levels and in (10.26). Objective function (10.26) is a convex function of the variables and because: (i) and are convex and monotonically increasing functions of when is given, and (ii) and are convex functions of and , respectively, as mentioned before. 251 The constraints of this optimization problem are the same as those in the feasibility checking problem. Therefore, the overall value degradation minimization problem described in this part is a convex optimization problem because it has convex objective function and convex inequality constraints. We find the optimal solution of this problem in polynomial time complexity. After we have minimized the overall value degradation induced by the charging/discharging cycle (after decoupling) of both battery arrays with the largest SoC swing, we can continue to minimize the effect on the overall value degradation from the other charging/discharging cycles. However, detailed discussion is out of scope of this chapter. 10.4 Experimental Results We derive and implement the proposed SoH-aware charge management policy on a typical HEES system comprised of a lead-acid battery bank, a Li-ion battery bank, and a supercapacitor bank. The lead-acid battery bank has 3 Ah nominal capacity and 20 V terminal voltage. The Li- ion battery bank has 4 Ah nominal capacity and 15 V terminal voltage. The supercapacitor bank has 200 F capacitance. The energy capacity of the supercapacitor array is 10% of that of the battery array. The SoH-aware control policy minimizes the overall value degradation in the propose HEES system. We compare the cycle life of the proposed system with two baseline systems. Baseline 1 uses a HEES system comprised of a lead-acid battery bank and a Li-ion battery bank that are the same as the proposed system, but without the supercapacitor bank. Baseline 2 uses the same HEES system as the proposed system. Both baseline systems exploit the optimal HEES control policy in order to satisfy the load power requirement and improve the HEES system cycle efficiency. The load power requirement is satisfied in the proposed system and both baseline systems. 252 We perform experiments based on two sets of source and load power profiles. In the first experiment, we use synthesized source and load power profiles as shown in Figure 83. We compare the SoH degradation and cycle life of both battery arrays between the proposed system and two baseline systems, with results shown in Table XXVIII. The proposed system achieves significantly smaller SoH degradation rate, and hence, larger cycle life, compared with both baseline systems. It achieves a cycle life improvement up to 17.3X compared with Baseline 1 thanks to the contributions of both the supercapacitor bank and the SoH-aware control policy. On the other hand, the maximum cycle life improvement compared with Baseline 2 is 3.5X due to the SoH-aware control policy solely. Figure 83.Synthesized source and load power profiles. 0 4 8 12 16 20 24 0 20 40 60 80 100 Time (H) Power (W) Power Source Electric Load 253 Table XXVIII. SoH degradation and cycle life comparison between the proposed system and baseline systems using synthesized power profiles. Figure 84.SoC profiles of the Li-ion battery array of the proposed system and Baseline 1 under synthesized power profiles. We provide the SoC profile versus time for the proposed system and Baseline 1 as shown in Figure 84. The maximum SoC swing and average SoC level of the Li-ion battery array have been reduced by 35% and 30%, respectively. The former accounts for about 6X improvement in cycle life whereas the latter accounts for about 3X improvement. 0 4 8 12 16 20 24 0 0.2 0.4 0.6 0.8 1 Time (H) SoC of Li-ion Proposed Baseline 1 254 In the second experiment, we use actual source and load power profiles scaled for the proposed HEES system, as shown in Figure 85. We compare the SoH degradation and cycle life of both battery arrays between the proposed system and two baseline systems, with results shown in Table XXIX. The proposed system achieves a cycle life improvement up to 13.9X. Similarly, we provide the SoC profile of the lead-acid battery array versus time in the proposed system and Baseline 1 as shown in Figure 86. Figure 85.Real source and load power profiles. Table XXIX. SoH degradation and cycle life comparison between the proposed system and baseline systems using real power profiles. 0 4 8 12 16 20 24 0 20 40 60 80 100 Time (H) Power (W) Power Source Electric Load 255 Figure 86.SoC profiles of the lead-acid battery array of the proposed system and Baseline 1 under real power profiles. 0 4 8 12 16 20 24 0 0.2 0.4 0.6 0.8 1 Time (H) SoC of Lead-acid Proposed Baseline 1 256 Chapter 11. Joint Control and Optimization of PV Systems and HEES Systems Photovoltaic (PV) power systems have been widely applied in commercial and domestic facilities, and electrical energy storage (EES) systems are mandatory in standalone PV systems for continuous power supply. In this paper we address efficiency and robustness enhancement methods under partial shading. Partial shading due to moving clouds and shadows of nearby obstacles on a PV module array causes significant efficiency degradation, since shaded and non- shaded PV modules have large discrepancy in their maximum power points (MPPs). Use of individual charger for each PV module may mitigate the negative effect from partial shading. However, this method alone may still face severe energy efficiency degradation caused by 1) significant energy loss in the EES elements under variable incoming power from the PV modules and 2) potentially high energy loss in the chargers since partial shading may cause high imbalance between the input and output voltages of each charger. In this paper, we call a PV system robust to partial shading if the energy efficiency is consistently high under various spatiotemporal-variant shading patterns among the PV modules. We consider two major energy conversion efficiency degradation factors: 1) the energy loss due to parasitic effects in the EES elements and 2) the non-negligible energy loss in the chargers, which is a function of their input and output voltages and currents. Energy loss due to 1) is more significant when the incoming power from the PV modules is higher, while the EES elements have lower power capacity. On the other hand, energy loss due to 2) becomes more distinctive if solar irradiance strength and state of charges (SoCs) of EES elements result in greater imbalance 257 between the input and output voltages of each charger. Therefore, we propose the following three methods based on the individual charger interface topology for higher energy efficiency and robustness under partial shading: 1) incorporation of HEES into a standalone PV system and developing a near-optimal HEES control algorithm, 2) extension of the MPTT approach, and 3) a novel dynamic balanced PV module reconfiguration method to enhance the charger efficiencies. The three proposed methods can be effectively combined together, thereby yielding an energy conversion efficiency improvement ranging from 17.1% to 53.3%, compared with the baseline systems using traditional MPPT control, fixed PV module configurations, and homogeneous EES system. 11.1 PV Power System 11.1.1 PV Power System Architecture Figure 87 shows the architecture of the PV system with homogeneous EES arrays (battery arrays) for energy storage, named Homogeneous EES-Based System; while Figure 88 shows the architecture of the proposed PV system with HEES element arrays for energy storage, named HEES-Based System. Both systems consist of a power source, an energy storage (sub-) system, a single-wire CTI, and distributed chargers. The power source of both Homogeneous EES-Based System and the proposed HEES-Based System consists of a set of PV modules from a single large PV array. Certain control mechanisms, e.g., MPPT, MPTT, and/or module reconfiguration, can be integrated into each PV module for efficiency enhancement. The PV modules are connected to the CTI via distributed (individual) chargers. This “individual power interface” structure can mitigate the negative effect on the output power of the whole PV array due to partial shading. On the right-hand side of the CTI, the energy storage system in Homogeneous EES-Based System is comprised of (multiple) homogeneous EES arrays connected via charging 258 control chargers and discharging control chargers to the CTI. In contrast, the energy storage system in the HEES-Based System consists of multiple heterogeneous EES arrays with complementary characteristics connected via charging control chargers and discharging control chargers to the CTI. Each EES array is composed of multiple homogeneous EES elements as a typical single EES element has a small energy capacity. We aim to convey a simple yet important idea of introducing a HEES system for efficiency enhancement, by exploiting a two- array architecture, i.e., one battery array and one supercapacitor array. Thus the Homogeneous EES-Based System only contains a single battery array (which is the same as the battery array in the HEES-Based System) for fair comparison. In fact, the supercapacitor array capacity is much smaller than 10% of the battery array capacity. Charger Battery Array V pv,1 I pv,1 I CTI,pv,1 I CTI,B I array,B V array,B V CTI CTI Charger V pv,2 I pv,2 I CTI,pv,2 Charger V pv,M I pv,M I CTI,pv,M ( G 1 , T 1 ) Charger Charger P conv,pv,1 P conv,pv,2 P conv,pv,M P conv,B ( G 2 , T 2 ) ( G M , T M ) Figure 87. Architecture of the homogeneous EES-based PV system (Homogeneous EES-Based System). 259 Charger Battery Array V pv,1 I pv,1 I CTI,pv,1 I CTI,B I array,B V array,B SuperCap Array I CTI,C I array,C V array,C V CTI CTI Charger V pv,2 I pv,2 I CTI,pv,2 Charger V pv,M I pv,M I CTI,pv,M ( G 1 , T 1 ) Charger Charger Charger Charger P conv,pv,1 P conv,pv,2 P conv,pv,M P conv,B P conv,C ( G 2 , T 2 ) ( G M , T M ) Figure 88. Architecture of the proposed HEES-based PV system (HEES-Based System). At time instance , we use and to denote the output voltage and current of the i-th ( ) source PV module, respectively. Obviously, the relationship between and values, as given in Section 2.1, depends on the irradiance level and the module temperature on the i-th PV module at time , as well as the configuration (the number of series and parallel connections of PV cells) of the i-th PV module. Note that and also serve as the input voltage and current of the charger connecting the i-th PV module to the CTI, respectively. On the other hand, the output voltage and current of that charger are denoted by (the CTI voltage) and , respectively, and its power loss is denoted by . The value is a function of the , , and values, as shown in Section 2.2, and we have: 260 (11.1) by the energy conservation law. For the battery array in the Homogeneous EES-Based System or the HEES-Based System, we use and to denote the OCV and CCV of the battery array at time , respectively, and their relationship is given in Section 6.2. The current between the battery array and its corresponding chargers is denoted by , while the current between CTI and the corresponding chargers for the battery array is denoted by . In the HEES-Based System, we make the battery array steadily and continuously receive energy from the PV modules. Therefore the discharging control charger of the battery array is always turned off, and we have and . Obviously these two inequalities also hold for the Homogeneous EES-Based System where there is only one battery array for energy storage. Moreover, the power loss value of the corresponding charging control charger for the battery array is denoted by , which is a function of the , , , and values, as shown in Section 2.2. We have: (11.2) by the energy conservation law. Similar notations can also be applied to the supercapacitor array in the HEES-Based System, as illustrated in Figure 88. However, for the supercapacitor array, the currents and can be positive (if power flows from CTI into the supercapacitor array), negative (if power flows from the supercapacitor array into the CTI), or zero. The variable actually denotes the power loss of either the charging control charger or the discharging control charger of the supercapacitor array. Those two chargers cannot 261 be turned on at the same time since an EES array cannot be simultaneously charged and discharged. Finally, the current values flowing into and out of the CTI satisfy the Kirchhoff’s law, i.e., (11.3) where in the Homogeneous EES-Based System . 11.1.2 Problem Statement Let the system (Homogeneous EES-Based System or HEES-Based System) operation start at time . The initial SoC of the battery array, denoted by , is given, and thus its initial OCV, , can be derived using methods given in Section 6.2. On the other hand, the initial SoC (and also OCV) of the supercapacitor array in the HEES-Based System is zero (fully discharged), i.e., and , for more realistic operating scenarios. In fact, supercapacitors are not suitable for long-term energy storage due to its high self-discharge rate. The solar irradiation is available during the time period . Moreover, there exists a system operation deadline time ( ), by which time system operations must finish. The PV systems in this chapter run in an online manner, i.e., the system controller is not aware of the irradiation levels and temperatures among PV modules in the future. More specifically, at any time instance during system operation, the irradiance level and the module temperature of the i-th ( ) PV module are available (note that when ), and therefore the PV module output voltage and output 262 current satisfy the relationship given in Section 2.1 (in which and ). They also depend on the configuration, i.e., numbers of series and parallel connections of PV cells, of the i-th PV module. Moreover, at that time , the battery array and supercapacitor array (in the HEES-Based System) SoC values can be calculated via: (11.4) (11.5) where and denote the full charge of the battery array and the supercapacitor array, respectively. The OCV values of the battery array and the supercapacitor array can be calculated based on the SoC values via the OCV-SoC relation for battery (discussed in Section 6.2) and for supercapacitor (linear function.) In order for efficiency optimization, the system controller controls the PV module operation points for (through controlling the currents for ), the CTI voltage , as well as the battery and supercapacitor array currents, and , respectively, at time , based on the above-mentioned environmental parameters and EES elements state parameters (SoC, OCV, etc.). The objective of both the Homogeneous EES-Based System and the HEES-Based System is to store the maximum amount of energy inside the battery array within deadline time , i.e., we maximize 263 (11.6) This objective is equivalent to the maximization of the energy conversion efficiency within the deadline time . The energy conversion efficiency is defined as the ratio of the total energy stored into the battery array during time period to the maximum available energy generated by all the source PV modules during time period , when each PV module works at its own MPP at any time . The battery array has relatively large energy capacity and low self-discharge rate, and is therefore more suitable for long-term energy storage. 11.2 Proposed Efficiency Enhancement Methods We first specify the baseline system, as all the proposed efficiency enhancement methods can be viewed as improvements on the baseline system. The baseline system uses the Homogeneous EES-Based System architecture (with only one single battery array for energy storage, as shown in Figure 87), with a fixed (and predefined) CTI voltage, MPPT control of each PV module, and no PV module reconfiguration. Obviously, the baseline system should have the lowest energy conversion efficiency and the worst robustness under various input irradiance and temperature profiles, as revealed in experimental results. 264 Adaptive V CTI PV Reconfig. HEES MPTT MPTT+ PV Reconfig. PV Reconfig. +HEES MPTT +HEES MPTT+ PV Reconfig. +HEES HEES-Based System Homogeneous EES-Based System Baseline System (with MPPT, fixed config. and homogeneous EES system) 1 2 3 4 5 6 7 8 9 Figure 89. Illustration of the proposed enhancement methods. Figure 89 shows the evolution of the baseline PV system equipped with different (combination of) efficiency enhancement methods, in which arrows denote improvements and nodes denote corresponding systems. Although not discussed before, the most basic efficiency enhancement method is named adaptive CTI voltage. We discuss the adaptive CTI voltage enhancement on the baseline system in the following. The PV operation points for at each time are known since the PV modules with fixed configurations work on their MPPs. Therefore for each CTI voltage value , we calculate the CTI input currents for from (11.1). Furthermore, since the target system (baseline system with adaptive CTI voltage) uses the Homogeneous EES-Based System architecture which only contains battery array for energy storage, the battery array charging 265 current can be calculated using (11.2) and (11.3) in which . Therefore, the received power by the battery array, defined as , is a function of the value, denoted by . Based on the above calculation, the adaptive CTI voltage method finds the most suitable CTI voltage value at any time instance during system operation, such that the received power by the battery at that time , , can be maximized. The system stops operation when , because there is no energy input from PV modules (this claim only holds for the Homogeneous EES-Based System architecture.) Besides, maximizing the received power is equivalent to maximizing the battery array charging current , and we shall use the two objectives interchangeably. Finding the optimal value can be accelerated by the ternary search algorithm, using the quasi-concavity of the function . The ternary search converges in logarithmic time with respect to the precision. The detailed algorithm, named basic CTI voltage selection (B-VCTI) algorithm, for outputting the set of system control variables at time , is given in Algorithm 1. The and values used in Algorithm 1 denote the numbers of PV cells connected in series and in parallel in the i-th PV module, respectively. These values are fixed since we do not allow dynamic PV module reconfiguration here. The function used in Algorithm 1 finds the MPP operation point of the i-th PV module given the solar irradiation level , the temperature , and the PV module configuration . 266 As illustrated in Figure 89, we propose further energy efficiency enhancement methods, including MPTT, dynamic PV module reconfiguration, and HEES, all based on the above- described adaptive CTI voltage enhancement method. These three efficiency enhancement methods can be effectively combined with each other, thereby yielding a total of seven types of enhanced PV systems over the baseline system with adaptive CTI voltage method. The HEES- Based System architecture shown in Figure 88 is the architecture support of the “HEES” efficiency enhancement method. 11.2.1 MPTT Extension We extend our proposed MPTT method in the multiple-PV-module system setting. The motivation of MPTT is that for each PV module, the traditional MPPT approach, i.e., the PV module works at its MPP, will not guarantee maximum energy transferred into the EES elements. The MPTT idea can be effectively integrated with the above-mentioned adaptive CTI voltage method. For any given CTI voltage at time , we can apply the MPTT idea to find the 267 optimal operation point (which may not be the MPP) of each i-th PV module so that its corresponding CTI input current can be maximized. The optimal value can be obtained using the ternary search method. Finding the optimal PV operation point of each i-th PV module to optimize the value for given becomes a necessary condition of the overall efficiency maximization problem, i.e., maximizing the value at time . This is similar to the subproblem structure in a dynamic programming problem. The MPTT subproblems, one for each PV module, are independent with each other, and thus can be implemented via distributed PV module controllers. The detailed algorithm combining the adaptive method with the MPTT idea for overall efficiency maximization at each time , named the MPTT-based CTI voltage selection (M-VCTI) algorithm, is given in Algorithm 2. 268 11.2.2 Dynamic PV Module Reconfiguration We adopt the PV module reconfiguration technique proposed in Section 3.1 in order to find the “best match” between the input and output voltages of each charger in the PV system. This further mitigates the negative effect of the charger input and output voltage level imbalance. We first define a balanced configuration of a PV module consisting of identical PV cells to be the arrangement of cells in which there are sub-modules connecting in series. Each sub- module consists of a set of cells connected in parallel. Obviously we have . The - cell PV module can be organized into various balanced configurations and the number of possible configurations is equal to the number of bi-factor decompositions of the natural number . We use to denote a configuration with cells in series and cells in parallel. Figure 90 is an example of the balanced reconfiguration of a four-cell reconfigurable PV module ( ), in which three balanced reconfigurations are possible. 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 N = 4, N = 4 X 1 N = 4, N = 2 X 2 N = 4, N = 1 X 4 S-switch P-switch Figure 90. Illustration of the balanced reconfigurations. 269 The way that dynamic PV module reconfiguration can be integrated with adaptive method is similar as the M-VCTI algorithm shown in Algorithm 2, that is, for any given value at time , we search all possible balanced configurations of each i-th PV module (each PV module works at its MPP) and find its best-suited configuration such that the corresponding CTI input current can be maximized. Then the optimal value can be obtained using the ternary search method. The detailed algorithm for maximizing overall energy efficiency at time , named the reconfiguration-based CTI voltage selection (R-VCTI) algorithm, is similar to Algorithm 2 and therefore not presented in this paper due to space limitation. Integrating both the dynamic PV module reconfiguration and MPTT methods with the adaptive method will yield further enhancement. At each time , the system controller will find both the best-suited balanced configuration and the optimal operating point (based on the best-suited configuration, may not be MPP) of each i-th source PV module, as well as the optimal CTI voltage value . To avoid the time complexity arisen from determining the best-suited configuration and the optimal operation point at the same time for each PV module, we provide a near-optimal solution of first finding the near-optimal balanced configuration of each PV module by assuming that it works at the MPP, and then find the optimal MPTT operating point of that PV module with the above-determined near-optimal configuration, for each given value. Obviously, the optimal value can be determined using ternary search, similar as before. The detailed algorithm for overall efficiency maximization at time , named the MPTT and reconfiguration-based CTI voltage selection (MR-VCTI) algorithm, is given in Algorithm 3. 270 11.2.3 HEES Control Algorithm for Efficiency Enhancement We focus on the near-optimal HEES control algorithm development in this section. The algorithm determines how to control the battery and supercapacitor array currents, and , respectively, at each time , so that (11.6) can be maximized. We introduce a HEES enhancement method from the baseline system with an adaptive CTI voltage. We illustrate the HEES method based on MPPT control and fixed PV module configurations. The MPTT and dynamic PV module reconfiguration methods can also be effectively integrated with the “HEES” method and yield a higher efficiency gain. 271 The underlying idea of the proposed HEES control algorithm is to apply a crossover filter to the solar irradiation and temperature profiles on the PV modules. This allows the battery array to receive the PV modules energy steadily and continuously, and make the high-frequency components in the power source profile handled by the supercapacitor array. The proposed HEES control algorithm consists of three steps: 1) initial charging of the supercapacitor array, 2) charging the battery array using the supercapacitor array as buffer, and 3) migrating charge from the supercapacitor array to the battery array, as shall be discussed in the following. 1) Step One: Since the initial SoC of the supercapacitor array is zero, we should charge the supercapacitor array from the beginning of system operation until its SoC reaches a predefined value. This enables us to use the supercapacitor array as a charge buffer during the battery array charging process. The initial charging phase finishes at time , which is not known to the system controller in advance. We shall maximize the energy receiving rate of the supercapacitor array, defined as , at any time to achieve the optimal initial charging. We can prove that maximizing the energy receiving rate is equivalent to maximizing the supercapacitor array current . We add a new control variable to Algorithm 1 (the B-VCTI algorithm) for the supercapacitor array power optimization, and set . We set also to be the optimization objective of the modified B-VCTI algorithm. The system controller executes the modified B-VCTI algorithm at each time . Details of the algorithm modification are omitted due to space limitation. The MPTT and dynamic PV module reconfiguration methods can be incorporated 272 into the modified B-VCTI algorithm for charging supercapacitor arrays, similar to the M-VCTI, R-VCTI, and MR-VCTI algorithms. 2) Step Two: The second and most important step of the proposed HEES control algorithm takes place during the time period . In this step the supercapacitor array acts as a buffer that filters out the high-frequency components of the PV input power and provides steady battery charging operation. The motivation of this step is that the power loss in low-power-capacity EES elements (e.g., batteries) due to the parasitic effects is a superlinear function of the energy incoming rate, and therefore it is desirable to let such EES elements with low power capacity receive the steady part of the incoming power. Based on this motivation, we apply a causal crossover filter on the solar irradiation profile and temperature profile of each i-th PV module to separate the high frequency components and the low frequency components in such profiles. We use and to denote the low frequency component outputs of the crossover filter on the irradiation and temperature profiles, respectively. The and values for at each time can be calculated as the moving averages of the past solar irradiation and temperature profiles, respectively. Next we find the near-optimal target charging current for the battery array, denoted by , at each time , based on the derived and values. We first maximize the battery array charging current value through effectively finding the optimal CTI voltage , assuming that the solar irradiation and temperature on the each i-th PV module are given by and , respectively. We do not consider the supercapacitor array here. We carry out this step through performing Algorithm 1 273 (the B-VCTI algorithm) by setting and in the algorithm. Then we set to be the obtained maximal value. Similarly we find the actual available charging current for the battery array, denoted by , at each time , based on the actual and values. The actual available charging current is the actual maximum available battery array charging current, under the current solar irradiation and temperature condition. We also do not consider the supercapacitor array here. We obtain the value using Algorithm 1 (the B- VCTI algorithm) again, based on the current and values. In the next step we compare the calculated and values. Generally speaking, the proposed HEES control algorithm make the supercapacitor array act as a buffer to store the excessive energy from PV modules when , and provide energy to charge the battery array simultaneously with the PV modules when . More precisely, there are three different cases based on the comparison results between and , as follows, and we present the proposed near-optimal HEES control algorithm at time in these three cases: Case I ( ): This case is usually caused by a sudden peak of incoming PV power. In this case, we set the battery array charging current to be the target value , and we maximize the supercapacitor array current value by finding the optimal CTI voltage . To effectively achieve this goal, we add a new control variable to Algorithm 1 (the B-VCTI algorithm) for supercapacitor array power optimization, 274 and set . We set also to be the optimization objective of the modified B-VCTI algorithm. Case II ( ): This case is typically caused by a valley of incoming PV power. In this case, we set the battery array charging current to be the target value , and we minimize the absolute value of by finding the optimal CTI voltage , since in this case we have , i.e., the supercapacitor array is discharging. We adopt the modified B-VCTI algorithm used in Case I again in this optimization problem. Case III ( ): In this case the difference between and values is within a threshold value. We set the battery array charging current to be the actual maximum available value , and set the supercapacitor array current to be zero. The outline of the second step of the proposed HEES control algorithm is given in Algorithm 4. The MPTT and dynamic PV module reconfiguration methods can be effectively incorporated in this algorithm. We may replace the B-VCTI algorithm used in Algorithm 4 with the M-VCTI, R-VCTI, or MR-VCTI algorithms without difficulty. 275 3) Step Three: The third and last step takes place during time period . In this step, the charge stored in the supercapacitor array is migrated into the battery array. The motivation of this step is that our goal is to store all the solar energy in long-term storage devices, i.e., the battery array, by the deadline , but there still exists residual energy in the supercapacitor array at time , when the second step ends. Hence it is necessary to perform the charge migration during time period to maximize the energy conversion efficiency. According to Section 8.2, the charge migration problem in the third step is a time-constrained single-source single-destination charge migration problem with relative deadline time . This can be optimally solved using the method proposed in Section 8.2. Besides, we can derive the effective near-optimal solution by incorporating high-order curve fitting techniques with coefficients determined offline, with the benefit of significantly reducing the online computational efforts. 276 11.3 Experimental Results This section demonstrates the performances of the baseline system and the systems with different (combinations of) energy efficiency enhancing methods. We carry out experiments with timing parameters , s and s. The PV array consists of three PV modules, each consisting of 60 identical PV cells. The solar irradiation profile reflects partial shading in that each PV module may receive different solar irradiation. The temperature of the three PV modules is assumed to be a constant, which is the same as (the temperature at standard test condition). The battery array in both the Homogeneous EES-Based System and the HEES-Based System has a nominal capacity of 1 Ah, and a nominal voltage of 8.4 V when the battery array is fully charged. The initial SoC of the battery array at time is set to 10%. The capacitance of the supercapacitor array in the HEES-Based System is 100 F. We use Linear Technology LTM4607 converter as the charger with a modified feedback circuit, and obtain the parameters of the Li-ion battery model given in [108] by measurement. The experimental results are illustrated in Table XXX. 277 Table XXX. Energy conversion efficiency results on various systems. The System Index column in Table XXX refers to the nine PV systems including the baseline system as annotated in Figure 89. The baseline system with system index of 1 cannot adjust the value and the PV module configurations adaptively. The system with system index of 9 has all the proposed efficiency enhancement methods and yields a significant performance gain ranging from 17.1% to 53.3%, compared with the baseline system. 278 We evaluate benefits and limitations of each proposed methods. The average benefit of an enhancement method implies average energy efficiency gain from only the specific method while keeping all other conditions the same. The average benefit is an effective metric that indicates the potential of efficiency improvement from the enhancement method. For example, the adaptive CTI voltage method exhibits average benefit of 12.6%. This justifies that the system efficiency is highly dependent on the CTI voltage. The three advanced efficiency enhancement methods, i.e., HEES, MPTT, and dynamic PV module reconfiguration, exhibit average benefits of 7.95%, 0.60%, and 9.07%, respectively. The dynamic PV module reconfiguration has the highest potential in efficiency improvement. Although the MPTT shows minor performance enhancement in this experiment, it requires no additional hardware cost unlike the other two advanced methods. Moreover, the MPTT technique can be implemented just like the traditional MPPT. On the other hand, the dynamic PV module reconfiguration method requires additional hardware cost as shown in Figure 88 and also requires additional control algorithm for controlling the switches inside each PV module. The HEES enhancement method exhibits more consistent efficiency gain around 8% compared to the dynamic PV module reconfiguration. Moreover, the HEES enhancement is also capable of providing efficiency improvement when a time-variant load is connected to the PV system, which is an additional advantage compared with other methods. 279 Chapter 12. Residential/Household Applications of PV and HEES Systems The traditional (static and centrally controlled) structure of the national electricity grid (also known as the power grid) is comprised of a transmission network, which transmits electrical power generated at remote power plants through long-distance high-voltage power lines to substations, and a distribution network, which delivers electrical power from substations to local end users. In this infrastructure, the local distribution network is often statically adjusted to match the load profile from its end users. The power grid must be able to support the worst-case power demand of all the end users in order to avoid potential power delivery failure as the end user profiles often change drastically according to the day of week and time of day [150]. The smart grid infrastructure is being designed to avoid expending a large amount of capital for increasing the power generation capacity of utility companies in order to meet the expected growth of end user energy consumption at the worst case [151]. The smart grid is integrated with smart meters, which can monitor and control the power flow in the power grid to match the amount of power generation to that of power consumption, and to minimize the overall cost of electrical power delivered to the end users. Utility companies can employ dynamic electricity pricing strategies, that is, employing different electricity prices at different time periods in a day or at different locations. This policy will incentivize energy consumers to perform demand side management, also known as demand response, by adjusting their loads to match the current state of the network, that is, shifting their loads from the peak time periods to off-peak periods. There are several ways to perform demand side management, including integration of renewable 280 energy sources such as PV power or wind power at the residential level, demand shaping, household tasks scheduling, and so on [152]. Although integrating residential-level renewable energy sources into the smart grid proves useful in reducing the usage of fossil fuels, several problems need to be addressed for these benefits to be realized. First, there exists a mismatch between the peak PV power generation time (usually at noon) at the peak load power consumption time for residential users (usually in the evening.) This timing slew results in cases where the generated PV power cannot be optimally utilized for peak power shaving. Moreover, at each time instance, the PV output power is fixed depending on the solar irradiance, when performing the MPPT or MPTT control [1]. Hence, the ability of the residential user for peak shaving is also restricted by the PV output power. An effective solution of the above-mentioned problems is to incorporate EES systems, either homogeneous or hybrid, for households equipped with PV modules [153]. The proposed residential energy storage stores power from the smart grid during off peak periods of each day and (or) from the PV system, and provide power for the end users during the peak periods of that day for peak power shaving and energy cost reduction (since electrical energy tends to be the most expensive during these peak hours.) Therefore, the design of energy pricing-aware control algorithm for the residential storage system, which controls the charging and discharging of energy storage bank(s) and the magnitude of charging/discharging current, is an important task in order for the smart grid technology to deliver on its promises. Effective storage control algorithms should take into account the realistic electricity pricing function, such as [154], [155]. It consists of both an energy price component, which is a time of usage (TOU) dependent function indicating the unit energy price during each time period of the 281 billing period (a day, or a month, etc.), and a demand price component, which is an additional charge due to the peak power consumption in the billing period. The latter component is added to the price of energy consumption in order to prevent a case whereby all the customers utilize their PV power generation and energy storage systems and/or schedule their loads such that a very large amount of power is demanded from the smart grid during low-cost time slots, which can subsequently result in power delivery failure. Moreover, the size of the EES system is limited due to the relatively high cost of energy storage elements. Therefore, at each decision epoch of a billing period, it is important for the storage controller to forecast the future PV power generation and load power consumption profiles so that it can perform optimization of the total cost. References [158], [159], [160] are representative of general PV power generation and load power consumption predictions by either predicting the whole power profiles, or predicting certain statistical characteristics. Prediction techniques include (but are not limited to): machine learning-based, ant colony clustering-based, and residential activity-based methods. In [153], we propose PV power generation and load power consumption profile predictors specifically designed for the residential EES system controller, i.e., exploiting the specific form of the energy price function, and effectively avoiding underestimation of the load power consumption or overestimation of the PV power generation. Most early work on the residential EES system control and management focuses only on homogeneous systems, and performs battery management without employing systematic optimization and/or optimality consideration in spite of the significant amount of relevant work. For example, reference [161] simply limits the battery current and considers reselling the excessive energy from the PV system to the power grid. Power leveling, which controls the power drawn from the power grid [162], and peak shaving [163] can mitigate this problem, but 282 they do not provide systematic optimization of the system efficiency or the billing cost. Reference [164] provides a systematic optimization based on the Lagrangian relaxation method, but it focuses on issues from the power distribution network such as locational marginal pricing (LMP) and transmission congestion problems. Recent work provides an algorithm that determines when and how to charge and discharge the battery, but the method is ad-hoc without much reasoning about the optimality [165]. Based on the PV power generation and load power consumption prediction results, we proposed in [127] a holistic optimization framework for a homogeneous residential EES system, which can effectively mitigate the electricity demand and supply mismatch and minimize the total electricity billing cost. The proposed framework takes into account the PV module impedance, converter loss, battery rate-capacity effect, and storage capacity limit for given prediction results of solar irradiance profile, load profile, and billing policy. We proposed in [153] a near-optimal residential storage control algorithm under a realistic electricity pricing function with both the energy price component and the demand price component [154]. However, we use simpler (and less accurate) models for the EES elements in [153] and neglect the conversion power loss. Recent work [166] provides an optimal control algorithm for the residential EES system by assuming that the residential load consumption is task-based and satisfies certain stationary random process, which is known to the storage controller in prior. HEES systems provide significant benefits than homogeneous EES systems for residential storage, such as higher power capacity, higher energy capacity, faster response time, and longer cycle life, because HEES systems can exploit the strengths of each type of each storage element while hiding their weaknesses. Figure 91 illustrates the architecture of the proposed HEES systems for a residential smart grid user with PV power generation. Appropriate control 283 algorithm for residential-level HEES systems will be more complicated than the control algorithm for a homogeneous EES system because the former requires taking into account the distinct and complementary characteristics of various types of energy storage elements. Figure 91. Architecture of the proposed HEES system for a residential smart grid user with PV power generation. We justify the usage of residential-level HEES systems in [118]. We consider a HEES system comprised of a Li-ion battery bank and a lead-acid battery bank. We devise an effective control algorithm of both banks in the residential HEES system under a simple electricity pricing function [155], as shown in Figure 92, in order to minimize the daily energy cost. The electricity pricing function is different during the high season (summer) and low season (winter). Both energy storage banks will get charged during the low peak period and discharged during the high peak period under such electricity pricing function. The optimization variables are the charging/discharging currents of both energy storage banks during these two time periods. We 284 show that the optimality designed HEES system achieves an annual return on investment (ROI) of up to 60% higher than a lead-acid battery-only system or a Li-ion battery-only system, under the same amount of investment. The HEES system is also beneficial in terms of capital cost compared with the homogeneous EES system with the same performance. Figure 92. Daily time-of-use energy pricing. In the following, we formally describe the integration of a residential-level HEES system for the Smart Grid users equipped with PV power generation. We propose the optimal control algorithm for the HEES system. The objective of the optimal control algorithm is to reduce the total electricity cost over a billing period and perform peak power shaving under arbitrary energy price function set by the Smart Grid central controller. The proposed optimal HEES system control algorithm correctly accounts for the distinct characteristics of different types of EES elements, conversion efficiency variations of power converters, as well as the time-of-use (TOU) dependent energy price function. The proposed control algorithm is based on dynamic programming and therefore has polynomial time complexity. Experimental results demonstrate 285 that the proposed HEES system and control algorithm achieves 73.9% average profit enhancement over homogeneous EES systems. 12.1 Optimal Control of a Grid-Connected HEES System 12.1.1 Problem Formulation We consider the HEES system integrated for a residential Smart Grid user with PV power generation with architecture shown in Figure 91. We use a slotted time model, i.e., all the system constraints as well as decisions are provided for discrete time intervals of equal length. More specifically, each day is divided into time slots, each of duration . We consider the residential electric load with load current at the i-th time slot. We consider the HEES system consisting of a lead-acid battery bank and a Li-ion battery bank. Our HEES system control algorithm is able to handle more complicated HEES systems with relatively high computation complexity. The proposed optimal HEES system control algorithm is performed at the beginning of a day and it optimally determines the CTI voltage level and charging or discharging current for each EES array throughout the day. The other voltage or current values are either given or associate variables. The proposed algorithm should correctly account for the distinct characteristics of different types of EES elements, conversion efficiency variation of power converters, and the TOU dependent energy price function. Let and denote the charging/discharging current of the lead-acid battery array and bank at the i-th time slot, respectively. and are positive when charging the battery array and negative when discharging the battery array. We denote the equivalent current inside the lead-acid battery array by . The relationship between and are 286 given in Section 6.2.3. We denote the SoC of the lead-acid battery array by , which is calculated from its initial SoC, , using Coulomb counting: (12.1) We denote the OCV and CCV of the lead-acid battery array by and , respectively. The OCV-SoC relationship and CCV-OCV relationship are given in Section 6.2.3. Similar notations also apply for the Li-ion battery bank by replacing the subscript by . We apply the maximum power point tracking (MPPT) method for the PV panel, which ensures that the output power of the PV panel is maximized [9]. The PV panel output voltage and current are denoted by and , respectively, and they are given to the HEES system controller. The grid voltage is a constant value (say, 110 V.) We denote the current flowing from the DC-AC inverter to the Grid by . can be positive (i.e., power flows from the HEES system to the AC load) or negative (i.e., power flows from the Grid to the HEES system.) The current drawn from the Grid, denoted by , satisfies . We assume that . In other words, the Smart Grid user cannot get reimbursed from selling power back into the Grid. The power conversion circuitries exploited in the system accounts for a significant portion of power losses. We denote the power losses of various converters and the DC-AC inverter at the i- th time slot by , , , and , which are functions of their input and output voltages and currents (please refer to Section 2.2.) We have the following equations due to the energy conservation law: (12.2) 287 (12.3) (12.4) (12.5) (12.6) Equations (12.3) - (12.5) also hold when the corresponding currents are negative (i.e., power flows in the opposite direction.) The objective of the HEES system control algorithm is to maximize the total energy cost saving, or equivalently, minimizing the total energy cost, over a billing period (i.e., one day.) The electricity energy price function is pre-announced by the utility company just before the start of each billing period, and the price function does not change until the start of the next billing period. Let denote the arbitrary unit energy price at the i-th time slot. The optimal HEES system control problem is formally described as follows: Given: PV panel output profile , , residential load demand profile , , and the TOU dependent energy price for , initial SoCs of EES arrays , . Control variables: CTI voltage battery array charging/discharging current , for . The control variables not only determine when to charge or discharge the EES arrays, but also the optimal operating conditions for charging and discharging. Minimize: the total energy cost for a day, given by (12.7) 288 Subject to: i) Energy Conservation: (12.2) - (12.6) are satisfied. ii) Capacity and Power Rating Constraints: Each EES array SoC cannot be less than zero or more than 100%, i.e., (12.8) Moreover, the charging/discharging current of each EES array cannot exceed a maximum value, i.e., (12.9) (12.10) iii) Final Stored Energy Constraints: Each EES array SoC at the end of day should be no less than the initial SoC value, i.e., (12.11) 12.1.2 Optimal HEES System Control Algorithm We derive the optimal solution of the HEES system control problem based on dynamic programming. This algorithm works for the HEES system and derive the optimal solution while [127] considers the homogeneous EES system only and derive a near-optimal solution. We consider a variant version of the original HEES system control problem, named the problem. This problem minimizes the total electrical energy cost during time slots 1 to under the condition that and . The following observation describes the relationship between these two optimization problems: Observation I: Consider the optimal solutions of all the problems satisfying and (i.e., satisfying the final stored energy 289 constraints (12.11)). The optimal solution of the original HEES system control problem is the one that minimizes the total energy cost during the whole day (i.e., over time slots 1 to .) The optimal substructure property of the problem goes as follows. This enables us to apply the dynamic programming method to find the optimal solution of the problem efficiently. Observation II (The Optimal Substructure Property): Suppose that we have found the optimal solution of the problem that minimizes the total energy cost during time slots 1 to . Suppose that and in that optimal solution. This corresponds to the subproblem . The optimal solution of the problem contains within it the optimal solution of the subproblem . Based on Observations I and II, we introduce Algorithm 1 using the dynamic programming method to find the optimal solution of the original HEES system control problem. We perform Algorithm 1 at the beginning of a day to determine the control variables over all the time slots in that day in a one-shot manner. 290 For and , denotes the minimal amount of electricity energy cost during time slots 1 to under the condition that and . We initialize the matrix as: (12.12) 291 Algorithm 1 calculates from all the values for and based on Observation II. We describe the calculation procedure in two steps as follows. Step I (Calculation of battery array currents and availability check): The following equations calculate the lead acid battery array current , which guarantees that the lead-acid battery array SoC changes from to in the time slot i: (12.13) We calculate in a similar way, which guarantees that the Li-ion battery array SoC changes from to in the time slot i. However, we must make sure that and are within the range of the allowable charging/discharging current of the EES arrays. We check if constraints (12.9) or (12.10) are violated, i.e., is not in the range or is not in the range . If constraints (12.9) or (12.10) are violated, we conclude that it is not possible for the lead-acid and Li-ion battery array SoCs to change simultaneously in the time slot i from to and from to , respectively. We proceed with another pair. If constraints (12.9) and (12.10) are not violated, we go on to the next step. This procedure is named availability check in Algorithm 1. Step II (Calculation of ): First we calculate the optimal CTI voltage that minimizes the energy cost in time slot i, or equivalently, maximizes 292 , with given and . Constraints in this optimization problem are the energy conservation constraints (12.2) – (12.6). Maximization of is in general a quasi-convex (unimodal) optimization problem over the variable . We exploit a ternary search algorithm, which is an extension of the well-known binary search algorithm, utilizing this quasi-convex property. This makes the solution quickly converge to the global optimal or at least a near global optimal solution. A branch and bound method provides the global optimal solution of the optimization problem at the expense of the solution complexity. We calculate the minimal energy cost in time slot i using after we find the maximum . We keep track of this minimal energy cost using . We calculate using the following equation based on Observation II: (12.14) We keep track of the optimal and using the following two matrices: (12.15) 293 which is necessary in finding the optimal control variable values after we find . After we calculate all the values for and , we determine the battery array SoCs , at the end of day using: (12.16) This step is based on Observation I. We reflect the final stored energy constraints (12.11) in (12.16). Next, we determine the optimal amounts of SoC change in both battery arrays in each time slot in a reverse chronological order. For example, the optimal amounts of SoC change in the lead-acid battery array and Li-ion battery array in time slot are and , respectively. We subsequently determine the optimal control variables values, i.e., the optimal , and for . This process is called tracing back, and is the last step in dynamic programming. Algorithm 1 discretizes the battery array SoC into levels. This discretization approach is necessary for effectively performing dynamic programming in the way of filling up the entries of a matrix. We exploit certain branch and bound techniques to reduce the computation time of Algorithm 1. However, such techniques are out of the scope of this work. 294 12.1.3 Experimental Results We compare the proposed residential-level HEES system and its optimal control with baseline homogenous EES systems with control algorithm described in our former paper [127] in terms of the average profit. The proposed HEES system consists of a lead-acid battery array with 48 V nominal (average) terminal voltage and 30 Ahr nominal capacity, as well as a Li-ion battery array with 36 V nominal terminal voltage and 26.7 Ahr nominal capacity. The two baseline homogenous EES systems consist of either a lead-acid battery array or a Li-ion battery array. We obtain the lead-acid battery and Li-ion battery characteristics and parameters from real measurements. For fairness in comparison, we assume that the total battery array volumes in the proposed HEES system and in the two baseline systems are the same. The residential electricity load and PV power generation profiles are measured at Duffield, in the year 2007. Figure 93 (a) shows a sample daily electrical load and PV power generation profiles while Figure 93 (b) shows the daily electricity energy price function adopted in the experiments. Figure 93. (a) PV power generation and load power consumption profiles and (b) electricity energy price function. 0 4 8 12 16 20 24 0 200 400 600 800 1000 1200 1400 Hour Power (W) (a) PV Power Generation Load Power Consumption 0 4 8 12 16 20 24 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Hour Energy Price ($/kWh) (b) 295 We show the effectiveness of the proposed residential-level HEES system and its optimal control algorithm in Figure 94. Figure 94 provides the power profile drawn from the Grid without the HEES system where the load power consumption and PV power generation profiles are given in Figure 93 (a). It also provides the power profile drawn from the Grid when the proposed HEES system and the optimal HEES system control algorithm are incorporated. According to Figure 94, the HEES system could store power from the Grid and the PV system when the energy price is low, and perform peak shaving when the electricity price is high, thereby reducing the total energy cost over a billing period. Figure 94. Power drawn from the Grid. We compare the average profit enhancement between the proposed HEES system and baseline homogenous EES systems. We define the amortized cost of an EES array, or equivalently, a homogenous EES system, as the amortized daily capital cost required for purchasing and maintaining the EES array. For example, suppose that the capital cost of a Li-ion battery array is $3000 and its life time is 5 years (1825 days.) The amortized cost is $3000/1825 per day. The amortized cost of a HEES system is the sum of amortized costs of all its EES arrays. 0 4 8 12 16 20 24 -500 0 500 1000 1500 2000 Hour P grid (W) Without HEES System With HEES System 296 We define daily profit of the HEES (or baseline EES) system as the total electricity energy cost saving in a day due to the incorporation of the HEES (or baseline EES) system minus its amortized cost. We use $80/kWh and $350/kWh as the unit energy price of lead-acid battery and Li-ion battery, respectively [107]. Their lifetimes are given by 1.5 years and 5 years, respectively [107]. We show the daily profit of the proposed HEES system and the two baseline systems over a month of 30 days in Figure 95. The proposed HEES system outperforms both baseline systems in terms of the daily profit. It achieves an average daily profit enhancement of 73.9% compared with the baseline system consisting of only a lead-acid battery bank due to its relatively low energy density and charging/discharging performance. The baseline system consisting of only a Li-ion battery bank even has a negative average daily profit due to its over-high capital cost. We conclude that the HEES system is a promising candidate compared to its homogeneous counterparts for residential Smart Grid usage. Figure 95. Daily profit of the proposed HEES system and two baseline systems. 0 5 10 15 20 25 30 -0.5 0 0.5 1 Day Daily Profit ($/Day) HEES EES:Lead Acid EES:Li-on 297 12.2 PV Power Generation and Load Power Consumption Profile Predictions In this section we introduce the proposed PV power geneartion and residential load power consumption profile prediction algorithms and experimental results. For a residential smart grid user with both PV power generation and energy storage facilities, these PV power generation and load power consumption profiles will be critical for the storage controller to judiciously decide the charging/discharging profile of the storage system. We adopt a slotted time model, i.e., all system constraints as well as decisions are provided for discrete time intervals of equal and constant length. Each day is divided into T time slots, each of duration D. We use T = 96 and D = 15 minutes. As specified in [154], the electricity price function is pre-announced by the utility company just before the start of each billing period, and the price function will not change until possibly the start of the next billing period. Reference [154] also specifies five different time periods of each day, denoted by the term price periods, with (potentially) different unit energy prices and (or) demand prices. These pre-announced price periods are: the 1 st off peak (OP) period from 00:00 to 09:59, the 1 st low peak (LP) period from 10:00 to 12:59, the high peak (HP) period from 13:00 to 16:59, the 2 nd LP period from 17:00 to 19:59, and the 2 nd OP period from 20:00 to 23:59. For notation simplicity, we denote the 1 st OP, 1 st LP, HP, 2 nd LP, and 2 nd OP price periods of a day as the 1 st , 2 nd , 3 rd , 4 th , and 5 th price periods of that day. We use to denote the statement that the j th time slot belongs to the k th price period. We use and to denote the start time and end time of the k th price period in each day, respectively. We have (start time of the day), (end time of the day), and for . 298 Accurate prediction of the PV power generation and load power consumption profiles is extremely important for the development of residential storage control algorithms. Due to the fact that predicting the complete load (or PV) power profile is difficult and unnecessary (since the storage controller only needs certain characteristics of the load or PV power profiles, e.g., average values and magnitudes and time instances of peaks), we use predictors to forecast the peak and average load power consumption (or PV power generation) values for different price periods in each day, i.e., one predictor for the 1 st OP period, one for the 1 st LP period, one for the HP period, one for the 2 nd LP period, and one for the 2 nd OP period. Subsequently, we reconstruct the approximate load (or PV) power profile for each day based on the predicted average and peak values in each price period. In the following, we describe the peak power predictors. The average predictors, which tend to be more accurate, are realized by using the same algorithms. The proposed load power consumption and PV power generation prediction algorithms consist of an initial prediction phase followed by an intra-day refinement phase, as explained below. Consider the peak power (consumption or generation) prediction for the i th day of a billing period (i.e., a month.) The initial prediction of the peak power refers to the prediction performed at time 00:00 ( ) of the i th day, for the peak load power consumption (or PV power generation) in all five price periods of the i th day. The intra-day refinement of peak power (consumption or generation) prediction may be performed at the start time of the k th ( ) price periods, i.e., , of the i th day, with the goal of refining the initial prediction results in the k th , (k+1) st , …, 5 th price periods. 299 Motivation for the intra-day refinement is that at time ( ), the actual peak load power consumption (or PV power generation) values in the 1 st , 2 nd , …, (k-1) st price periods of the i th day are known to the controller. This information can thus be used to improve the accuracy of peak power prediction for the rest of price periods in the same day. The proposed intra-day refinement process is crucial since the characteristics of the load power consumption and PV power generation profiles are required to be more accurate for the 1 st LP, HP, and 2 nd LP periods compared to the 1 st OP period. This is because the energy and demand prices in the former price periods are higher than those for the 1 st OP price period. In addition, it is very unlikely for the peak power demand from the Grid to occur during the 1 st OP period. 12.2.1 Prediction for the Peak Load Power Consumption For the initial prediction phase of peak load power consumption, an adaptive regression-based algorithm is used. Consider that we are at time of the i th day. The peak load power consumption values in the k th price period of the i th day ( ) are predicted as follows: (12.17) where the feature vector captures the actual values of peak load power consumption sampled at some previous points of interest, and is the number of elements in the feature vector. In addition, denotes a dynamically-updated coefficient vector with elements. The values of elements are all initilized to . Utilizing a stochastic gradient descent method (see [138] for details), the coefficient vector is updated as follows: 300 (12.18) where is a pre-defined learning parameter. Testing on the load power consumption profile from Baltimore Gas and Electric Company [156], we have found that a value of with the feature vector defined as follows yields the best prediction results with a relatively low computational complexity (a overlarge will result in not only high computation complexity but also over-fitting [138]): (12.19) where the values denote the actual peak load power consumption values in the k th price period of the (i-1) st day, the (i-2) nd day, the (i-7) th day, the (i-14) th day, and the (i-21) st day, respectively. This is because the tested load power profile exhibits both daily dependency and weekly dependency. The daily dependency is properly captured by the feature elements of the (i- 1) st day and the (i-2) nd day, whereas the weekly dependency is captured by those of the (i-7) th day, the (i-14) th day, and the (i-21) st day. For the intra-day refinement phase of peak load power consumption prediction, consider that we are currently at time instance (the start time of the k th price period) of day i, and we intend to refine the initial prediction results of the peak load power consumption values in the th ( ) price period of that day. We denote the result of refinement as . Since at that time the actual peak load power consumption in the (k-1) st price period is already known, we calculate as follows: 301 (12.20) where the coefficient . We learn the optimal value using the same stochastic gradient descent method (see [138] for details) as the learning process of the coefficient vector . Typically, a value in the range of [0.5, 0.7] will yield the best prediction results. The intuition for this update equation (12.20) is as follows: If the actual peak load power consumption in the (k-1) st price period is higher than the predicted peak load power consumption in that period, i.e., , it is highly likely that the actual load power consumption in the th price period ( ) of the same day will also be higher than the predicted peak load power consumption in that period, and vice versa. Experimental results will demonstrate that the prediction error can be reduced to 50% of the initial prediction error by using intra-day refinement. It is important to note that underestimating peak load power consumption needs to be avoided, since it may result in larger than expected power demand from the Smart Grid, which in turn significantly increases the monthly demand price of electricity. We propose to use the following heuristic for avoiding underestimation of peak load power consumption. We modify the aforesaid intra-day refinement algorithm in that after we have derived the ( ) value using (12.20), we add the following additional correction phase: (12.21) where the correction factor . 302 The reason for this modification is as follows. Experimental results show that underestimating peak load power consumption in the th price period of the i th day is very likely to occur when . We effectively learn the optimal correction factor value using a simple learning technique [138] discussed as follows. We maintain a set of correction factors and pick one of them in each billing period. At the end of the billing period, we evaluate the cost-saving performance of the storage controller if each correction factor is chosen in this billing period, and find that results in the lowest energy cost. Then we select the correction factor in the next billing period. 12.2.2 Prediction Algorithm for the Peak PV Power Generation For the peak PV power generation prediction, an important observation is that the actual peak PV power generation over a specific price period (e.g., the k th price period) in the i th day of a billing period may be viewed as the peak PV power generation over the k th price period for a sunny day, multiplied by a decay factor, representing the effect of clouds, if that day is cloudy. Obviously, such sunny day peak PV power generation over the k th period ( ) varies with the change in seasons. This effect is however captured by a smoothing operation as described below (cf. Eqns. (12.22) and (12.23).) Therefore, we use the initial prediction, which is performed at the beginning of each day, mainly to predict the sunny day peak PV power generation over each price period of that day. Next, we rely on the intra-day refinement, which is performed at the start time of the k th ( ) price period of that day, in order to predict the decay factors (and subsequently, the actual peak PV power generation levels) in the rest of price periods. Please note that at time , the decay factor in the (k-1) st price period is already 303 known. Moreover, decay factors for different price periods of the same day are positively correlated in general. In the initial prediction phase, we adopt a variant of the well-known exponential average- based prediction method, in order for effectively predicting the sunny day peak PV power generation in each price period of day (please see the discussion below about how our version is different from the standard method.) Consider that we are at time of the i th day. We want to derive the prediction value of the sunny day peak PV power generation in the k th ( ) price period of that day, denoted by , based on the prediction result of sunny day peak PV power generation value in the k th price period of the (i-1) st day, denoted by , and the actual peak PV power generation value in the k th price period of the (i-1) st day, denoted by . Please note that we must also capture and predict the seasonal change of the sunny day peak PV power generation values, while filtering out random power decaying effects due to the presence of clouds. This is a smoothing operation. The value is calculated as follows: (12.22) In the above equation, the learning rate function is set to: (12.23) where is the basis learning rate, and is the decaying parameter for the learning rate. 304 The motivation for this smoothing operation is as follows. Since (i) we want to predict the seasonal change of sunny day peak PV power generations while filtering out the effect of clouds and (ii) << only if there are clouds, it is natural that our new predicted sunny day peak PV power generation should not be so much influenced by (which is strongly affected by the clouds.) Therefore, we adopt the exponentially decaying learning rate function (12.23), rather than the constant learning rate in the original exponential average-based prediction algorithm. The intra-day refinement phase of PV power generation prediction, which is performed at the start time of the k th ( ) price period of each day to predict the decay factors (and subsequently, the actual peak PV power generation values) in the remainder price periods, can be implemented via exactly the same algorithm (Eqn. (12.20)) as the intra-day refinement phase of load power consumption prediction. Finally, in contrast to the load power consumption prediction, overestimating peak (or average) PV power generation needs to be avoided. Therefore similarly, we modify the intra-day refinement phase of peak PV power generation prediction as follows to avoid overestimating, in that after we derive the ( ) value, we add the following conditional correction phase: (12.24) in which the correction factor . We effectively learn the optimal value using the learning technique, similar to the learning process of the optimal value. 305 The reason for this modification is as follows. Experimental results show that overestimating peak PV power generation in the th price period of the i th day is very likely to happen when . 12.2.3 Experimental Results We present the experimental results on load power consumption and PV power generation prediction algorithms proposed for the residential Smart Grid users. In the experiments, we use the electric load data from the Baltimore Gas and Electric Company measured in the year 2007 [156]. We use PV power generation profiles measured at (i) Duffield, VA, measured in the year 2007, and (ii) Los Angeles, CA, measured in the year 2012 [157]. These two profiles are representative of the PV generation profiles in the west coast and the east coast of the U.S., respectively. We show some representative experimental results on the accuracy of the peak load power consumption and PV power generation predictions. The average load power consumption and PV power generation prediction results are in general 10% - 15% more accurate than the peak predictions, and are therefore not shown due to space limitation. Figure 96 compares the peak load power consumption prediction results and the actual peak load power consumption results in the HP period of each day in a year. The peak load power consumption prediction results shown in the top subfigure come from the initial prediction performed at time 00:00 of each day, whereas the prediction results in the bottom subfigure come from the intra-day refinement performed at time . Data in the first 120 days of the year are used for initial training, and thus the prediction results over those days are not shown in Figure 96. We can observe from Figure 96 that the proposed adaptive regression-based initial prediction algorithm is effective in load power consumption prediction, resulting in an average 306 prediction error of about 8%, by making use of both the daily and weekly dependencies. The average prediction error is further reduced to less than 4%, i.e., less than 50% of the average prediction error in initial prediction, by the effective use of intra-day refinement. Figure 96. Comparison between the peak load power consumption prediction results from initial prediction (top) and from intra-day refinement at time (bottom) and actual peak load power consumption results. Figure 97 compares the peak PV power generation prediction results with the actual PV power generation results in the 1 st LP period of each day in a year, using the PV data measured at Duffield, VA. The peak PV power generation prediction results shown in the top subfigure come from the initial prediction performed at time 00:00 of each day, whereas the prediction results shown in the bottom subfigure come from the intra-day refinement procedure performed at time . Data in the first 90 days in the year are used for initial training, and thus the prediction results over those days are not shown in Figure 97. We can observe that the proposed 307 modified exponential average-based initial prediction algorithm is effective in predicting the sunny day peak (and also average) PV power generation over each day in a year, by effectively capturing the long-term seasonal change of sunny day peak power generation values throughout a year while filtering out the random effects of clouds. The proposed intra-day refinement technique also proves itself effective in predicting the decay factors due to clouds by reducing the average prediction error to about 14%. We achieve even more accurate peak PV power generation predictions on PV data measured at Los Angeles, CA, with an average error of less than 12%. Figure 97. Comparison between the peak PV power generation prediction results from initial prediction (top) and from intra-day refinement at time (bottom) and actual peak PV power generation results. 308 12.3 Accurate Component Model-Based Control Algorithm for Residential Photovoltaic and Energy Storage Systems Accounting for Prediction Inaccuracies In this section, we consider the case of a Smart Grid residential user equipped with local PV power generation and an energy storage system. We consider a realistic electricity price function comprised of both energy and demand prices, with system architecture and the storage power loss model used in the paper. Based on the PV power generation and load power consumption prediction results from Section 12.2, we present a near-optimal storage control algorithm that mitigates the inevitable prediction errors and properly accounts for the energy loss components due to power dissipation in the power conversion circuitries, as well as the rate capacity effect, which is the most significant portion of energy loss in the storage system. The proposed near- optimal storage control algorithm is effectively implemented by solving a convex optimization problem with polynomial time complexity at the beginning of each day in a billing period. Experimental results demonstrate that the proposed residential storage control algorithm achieves up to 2.62X enhancement in electricity cost reduction compared with the baseline storage control algorithm. We consider an individual Smart Grid residential user that is equipped with PV power generation and energy storage systems, as shown in Figure 98. The PV system and storage system are connected to a residential DC bus, via unidirectional and bidirectional DC-DC converters, respectively. An AC bus, which is further connected to the Smart Grid, is connected via an AC/DC interface (e.g., inverter, rectifier, and transformer circuitry) to the residential DC bus. The residential AC load (e.g. household appliances, lighting and heating equipments) is 309 connected to the AC bus. We consider the power losses in the above-mentioned power conversion circuitry for the realistic concern. Figure 98. Block diagram illustrating the interface between the PV module, storage system, residential load, and the Smart Grid. Similar to Section 12.2, we adopt a slotted time modeling approach, i.e., all system constraints as well as decisions are provided for discrete time intervals of equal and constant length. More specifically, each day is divided into time slots, each with a duration of . We use and minutes. Let set denote the set of all time slots in each day. We adopt a realistic electricity price function comprised of both the energy price component and the demand price component as discussed before, with a billing period of a month [154]. Consider a specific day i of a billing period. The residential load power consumption at the j th time slot of that day is denoted by . The output power levels of PV and storage systems at the j th time slot are denoted by and , respectively. Notice that, may be positive (discharging from the storage), negative (charging the storage), or zero. 310 We assume that the PV power generation can be accurately predicted at the beginning of the i th day based on the PV power generation characteristics. On the other hand, the residential load power consumption cannot be accurately predicted, and let denote the predicted value of in the j th time slot in the i th day. We use to denote the power required from the Smart Grid, i.e., the grid power, at the j th time slot of the i th day, where can be positive (if the Smart Grid provides power for the residential usage), negative (if the residential system sells power back into the Smart Grid), or zero. Similarly, we use to denote the predicted value of . We consider realistic power conversion circuitry (i.e., their power conversion efficiency is less than 100%) in the proposed optimization framework. Accordingly, we use , , and to denote the power conversion efficiencies of the DC-DC converter between the PV system and the DC bus, the DC-DC converter connecting between the storage system and the DC bus, and the AC/DC power conversion interface, respectively. Those power conversion efficiency values are typically in the range of 85% to 95%. There are three operating modes in the system. In the first mode, both the PV system and the storage system are providing power for the residential load (i.e., the storage system is being discharged.) For the j th time slot of the i th day, the condition that the residential system is in the first mode is given by . In this mode, the actual grid power can be calculated by: (12.25) whereas the predicted grid power is given by: 311 (12.26) In the second mode, the storage system is being charged, and the PV system is sufficient for charging the storage. For the j th time slot of the i th day, the condition that the residential system is in the second mode is given by and . In this mode, there is power flowing from the DC bus to the AC bus, and the actual grid power can be calculated by: (12.27) whereas the predicted grid power is given by: (12.28) In the third mode, the storage system is being charged, and the PV system is insufficient for charging the storage. In other words, the storage is simultaneously charged by the PV system and the Grid. For the j th time slot of the i th day, the condition that the residential system is in the third mode is given by and . In this mode, there is power flowing from the AC bus to the DC bus, and the actual grid power is given by: (12.29) whereas the predicted grid power is given by: (12.30) 312 It can be observed from the above equations (12.25) - (12.30) that (or ) is a piecewise linear (continuous) and monotonically decreasing function of , when and (or ) values are given. (or ) is also a convex function of . As specified in [154] and [155], the electricity price function is pre-announced by the utility company just before the start of each billing period, and it will not change until possibly the start of the next billing period. We use a general electricity price function as follows. We use to denote the unit energy price at the j th time slot of a day. Then the cost we actually pay in a billing period due to the energy price component is given by: (12.31) The demand price component, on the other hand, is charged for the peak power drawn from the Grid over certain time periods in a billing period. A generic definition of the demand price is given as follows. Let , , ..., be different non-empty subsets of the original set of time slots, each of which corresponds to a specific time period, named by the term price periods, in a day. A price period does not necessarily need to be continuous in time. For example, a price period can span from 10:00 to 12:59 and then from 17:00 to 19:59, as shown in [154]. Also those price periods in a day do not need to be mutually exclusive. We use to denote the statement that the j th time slot in a day belongs to the k th price period. Let denote the demand price charged over each k th price period . Then the cost we have to pay in a billing period due to the demand price component is given by 313 (12.32) Obviously, the actual total cost for the residential user in a billing period (i.e., a month) is the summation of the two aforesaid cost components. 12.3.1 Rate Capacity Effect of the Storage System The most significant cause of power losses in the storage system, which is typically made of lead-acid batteries or Li-ion batteries, is the rate capacity effect of batteries [75]. To be more specific, high discharging current of the battery will reduce the amount of available energy that can be extracted from the battery, thereby reducing the battery's service life between fully charged and fully discharged states [75]. In other words, high-peak pulsed discharging current will deplete much more of the battery's stored energy than a smooth workload with the same total energy demand. We use discharging efficiency of a battery to denote the ratio of the battery's output current to the degradation rate of its stored charge. Then the rate capacity effect specifies the fact that the discharging efficiency of a battery decreases with the increase of the battery's discharging current. The rate capacity effect also affects the energy loss in the battery during the charging process in a similar way. The rate capacity effect can be captured using the Peukert's formula, an empirical formula specifying the battery charging and discharging efficiencies as functions of the charging current and discharging current , respectively: (12.33) 314 where and are peukert's coefficients, and their values are typically in the range of 0.1 - 0.3; denotes the reference current of the battery, which is proportional to the battery's nominal capacity . Typically, is set to , indicating that it takes 20 hours to fully discharge the battery using discharging current . We name and the battery's normalized charging current and normalized discharging current, respectively. Notice that the efficiency values and in Eqn. (12.33) are greater than 100% if the magnitude of the normalized charging or discharging current is less than one, which implies that the above-mentioned Peukert's formula is not accurate in this case. We modify the Peukert's formula such that the efficiency values and become equal to 100% if the magnitude of the normalized charging/discharging current is less than one. In other words, the battery suffers from no rate capacity effect in this case. We denote the increase/degradation rate of storage energy in the j th time slot of the i th day by , which may be positive (i.e., discharging from the storage, and the amount of stored energy decreases), negative (i.e., charging the storage, and the amount of stored energy increases), or zero. Based on the modified Peukert's formula, the relationship between and the storage output power is given by: 315 (12.34) where is the storage terminal voltage and is supposed to be (near-) constant; is the reference current of the storage system, which is proportional to its nominal capacity given in Ampere-Hour (Ahr); coefficient is in the range of 0.8 - 0.9, whereas coefficient is in the range of 1.1 - 1.3. One can observe that when the storage discharging (or charging) current is the same, the discharging (or charging) efficiency becomes higher (i.e., the rate capacity effect becomes less significant) when the nominal capacity of the storage system is larger. We use the function to denote the relationship between and . An important observation is that such a function is a concave and monotonically increasing function over the input domain , as shown in Figure 99. Due to the monotonicity property, is also a monotonically increasing function of , denoted by . We can see from Figure 99 that a lead-acid battery-based storage system has more significant energy loss due to rate capacity effect than a Li-ion battery- based storage system. However, the lead-acid battery-based storage system is more often deployed in real household scenarios due to cost considerations (the capital cost of lead-acid battery is only 100 - 200 $/kWh, whereas that of Li-ion battery is > 600 $/kWh [107].) 316 Figure 99. Relationship between and in two types in two types of batteries. 12.3.2 Optimal Control Algorithm of Residential Storage System We introduce in details the proposed near-optimal residential storage control algorithm accounting for prediction inaccuracy, which could effectively utilize the combination of PV power generation and load power consumption prediction results to minimize the total electricity cost, including both the energy price and the demand price, over each billing period (i.e., a month.) The storage control optimization problem is performed at time 00:00 (i.e., at the beginning) of each day in the billing period. To be more realistic, we assume that the prediction results of PV power generation and load power consumption profiles of each i th day are not available before time 00:00 of that day. We further assume that the PV power generation profile at each day, i.e., for , can be accurately predicted from the weather forecast and prediction algorithms presented in Section 12.2. On the other hand, the load power consumption profile -2 -1 0 1 2 -2 -1 0 1 2 Pst,in (kW) Pst (kW) Li-ion battery Lead-acid battery 317 for is not perfectly accurate. We assume that follows a uniform distribution over the range of , in which the average value is the actual load power consumption and can be estimated from the previous (observed) load power consumption profiles. Hence, with given predicted value , also follows a uniform distribution over the range of . At time 00:00 of each i th day, the storage controller performs optimization to find the optimal storage system output power profile for throughout the day, which is equivalent to finding the charging/discharging current profile of the storage system. In this following, we first introduce the storage control optimization performed at the beginning of a billing period (i.e., at time 00:00 of day ), in order to achieve a balance between the expected (induced by the energy price component) and expected (induced by the demand price component) values. In this way, we can minimize the total expected energy cost. Next, we introduce the storage control optimization performed at the beginning of the other days in the billing period, properly taking into account the prediction errors. Although in reality we control the output power ( , ) of the storage system during the system operation, we use ( , ) as the control variables in the optimal storage control problem formulation because it can help transform the optimal storage control problem into a standard convex optimization problem. We observe from Eqns. (12.25), (12.27), (12.29), and (12.34) that the grid power ( , ) is a monotonically decreasing function of , denoted by , over the input domain . Furthermore, 318 is a convex function of the control variable according to the rules of convexity in function compositions [136], because of the following two reasons: (i) is a convex and monotonically decreasing function of , and (ii) is a concave function of . Similarly, the estimated grid power ( , ) is also a monotonically decreasing function of , denoted by . At any time in a billing period, let ( ) denote the peak grid power consumption value that is observed so far over the k th price period in the billing period of interest. Obviously, such values are initialized to zero at the beginning of the billing period, and are updated at the end of each day according to the actual (observed) grid power consumption profiles. Storage Control Optimization at the Beginning of a Billing Period In this part, we introduce the storage control optimization performed at the beginning of a billing period (i.e., at time 00:00 of day ), in order to achieve a desirable balance between the expected and expected values. At that time, we have for . The storage controller is only aware of the PV power generation and load power consumption predictions in the 1st day of the billing period, i.e., and for , in which is inaccurate. The storage control derives the optimal profile for . The objective of the storage controller is to minimize an estimation of the total electricity cost in the billing period. Then the Optimal Storage Control problem performed at the Beginning of a billing period (the OSC-B problem) is formally described as follows: 319 The OSC-B Problem Formulation Given the PV power generation profiles of the 1st day in the billing period, i.e., for , the predicted load power consumption profiles for , and the initial energy in the storage system at time 00:00. Find the optimal profile for . Minimize an estimation of the total electricity cost in the billing period, which is given by: (12.35) Subject to the following constraints: For each : (12.36) (12.37) (12.38) In the OSC-B problem formulation, the objective function (12.35) is an estimation of the total electricity cost in the whole billing period. In this equation, we use the predicted PV power generation and load power consumption profiles of the first day, i.e., and for 320 , as a representation for the whole billing period. This is because the storage controller can only predict the PV power generation and load power consumption profiles in the first day. Moreover, in the objective function (12.35), we use as an estimation of the expected value of . This is because of the assumption that is uniformly distributed over the range of in the OSC-B problem formulation. In the OSC-B problem, constraint (12.36) represents the restrictions on the maximum allowable amount of power flowing into and out of the storage system during charging and discharging, respectively. Constraint (12.37) ensures that the storage energy can never become less than zero or exceed a maximum value throughout the day. Finally, constraint (12.38) ensures that the remaining storage energy at the end of day, which is required for performing peak power shaving on the next day, is no less than the initial value . The OSC-B problem is a standard convex optimization problem due to the following two reasons:  The objective function (11) is a convex objective function because the pointwise maximum function of a set of convex functions is still a convex function.  The other constraints are all convex (or linear) inequality constraints of optimization variables. Although the OSC-B problem is formulated as a convex optimization problem, and therefore, it can be solved optimally with a polynomial time complexity using convex optimization algorithms [136], [137], it is difficult to directly solve the OSC-B problem by using standard convex optimization tools such as CVX [137] or the fmincon function in MATLAB. This is 321 because the function is non-differentiable at several points, and typical convex optimization tools only accept differentiable objective functions. To address this issue, we use a piecewise linear function to approximate the function , and then transform the OSC-B problem into a linear programming problem (please note that all the constraints are linear constraints), which could be optimally solved using standard optimization tools in polynomial time complexity. Details are omitted due to space limitation. Similar method will also be applied to the optimal storage control problem as shall be discussed in the following. Storage Control Optimization at the Beginning of the Other Days We introduce the storage control optimization at the beginning of the other days in the billing period (i.e., days 2, 3, and so on). Suppose that we are at the beginning of the i th day of the billing period of interest. At that time, the values may not be zero any more. The storage controller is aware of the accurate PV power predictions and inaccurate load power consumption predictions in the i th day, i.e., and for . The storage controller derives the optimal profile for . The objective of the storage controller is to minimize the expected increase of the electricity cost in the i th day of the billing period, as will be formally described as follows. The Optimal Storage Control problem performed at the beginning of the Other days in the billing period (the OSC-O problem) is formally described as follows: The OSC-O Problem Formulation Given the PV power generation profiles of the i th day in the billing period, i.e., for , the predicted load power consumption profiles for , and the initial energy in the storage system at time 00:00. 322 Find the optimal profile for . Minimize the estimated increase in the electricity cost in the i th day, which is given by (12.39) or equivalently, minimize (12.40) Subject to the following constraints: For each : (12.41) (12.42) (12.43) 323 In the OSC-O problem formulation, the objective function (12.39) is an estimation of the increase of the electricity cost in the i th day of the billing period of interest. The objective function is comprised of two parts: (i) the expected energy price-induced electricity cost in the i th day of the billing period, given by the first term of Eqn. (12.39), and (ii) the estimation of the increase in demand price-induced electricity cost in the billing period of interest, given by the second term of Eqn. (12.39). In the second term, we use as a (conservative) estimation of the expected value of when . Please note that is uniformly distributed over the range of in the problem formulation. Moreover, the constraints in the OSC-O problem are similar to the constraints in the OSC-B problem as discussed before. Again, similar to the above-mentioned OSC-B problem, the OSC-O problem has a convex objective function (12.39) or (12.40), and linear inequality constraints (12.41) - (12.43), of the optimization variables. Therefore, the OSC-O problem can also be optimally solved with polynomial time complexity using the standard convex optimization methods [136], [137]. 12.3.3 Optimal Control Algorithm of Residential Storage System In this section, we present the experimental results on the effectiveness of the proposed accurate component model-based near-optimal residential storage control algorithm accounting for prediction errors. The PV power profiles used in our experiments are measured at Duffield, VA, in the year 2007, whereas the electric load data comes from the Baltimore Gas and Electric 324 Company, also measured in the year 2007 [156]. We add some random peaks to the electric load profiles. Figure 100. The daily energy price component in the second type of electricity price function. We use two types of electricity price functions. The first type of electricity price function is a real price function similar to [154], [155], which is given as follows. The energy price component is given by: 0.01879 $/kWh during 00:00 to 09:59 and 20:00 to 23:59, 0.03952 $/kWh during 10:00 to 12:59 and 17:00 to 19:59, 0.04679 $/kWh during 13:00 to 16:59. For the monthly demand price component, there are three price periods in a day: (i) the "high peak" period from 13:00 to 16:59, with demand price of 9.00 $/kW, (ii) the "low peak" period from 10:00 to 12:59 and from 17:00 to 19:59, with demand price of 3.25 $/kW, (iii) the "overall" period from 00:00 to 23:59 (the whole day), with demand price of 5.00/kW. The second type is a synthesized electricity price function. The energy price component over a day is demonstrated in 325 Figure 100. For the monthly demand price component, we consider only one "high peak" period from 18:00 to 21:59 with demand price of 9.00 $/kW, whereas the rest is "low peak" period. We define the cost saving capability of a storage control algorithm (the proposed algorithm or the baseline algorithm) to be the average daily cost saving over a billing period due to the additional storage system, compared with the same residential Smart Grid user equipped only with the PV system. We compare the cost saving capabilities of the proposed near-optimal storage control algorithm with the baseline algorithm. The baseline algorithm charges the storage system from the Grid during the "off peak" period (00:00 to 09:59 and 20:00 to 23:59 in the first type of electricity price function or 00:00 to 17:59 and 22:00 to 23:59 in the second type of electricity price function) with constant charging power, and distributes energy stored in the storage system evenly in the "high peak" period. First we show experimental results based on the first type of electricity price function. In the experiments, we set the amount of residual energy at the end of each day to be no less than 20% of the full energy capacity of the storage system. We set the inaccuracy parameter to be 0.8. Table XXXI illustrates the comparison results on the cost saving capabilities between the proposed near-optimal storage control algorithm accounting for prediction inaccuracy and the baseline algorithm on every month throughout a year, when the capacity of the storage is 45 Ah. The improvement in cost saving capabilities using the proposed algorithm is provided in the table. Table XXXII shows the comparison results on the same testing data when the capacity of the storage system is 60 Ah. We can see from these two tables that the proposed near-optimal residential storage control algorithm consistently outperforms the baseline algorithm, with the maximum improvement of 162% (i.e., 2.62X) on the cost saving capability (on December, 45 Ah storage capacity). Furthermore, it can be observed that the proposed storage control algorithm 326 demonstrates more significant improvement on the cost saving capability over the baseline system when the storage system has a capacity of 60 Ah. It also achieves higher cost saving capability during the winter than during the summer. The reason is that the peak load power consumption generally occurs in the "high peak" price period in the summer, and therefore, the baseline algorithm achieves relatively higher performance by distributing the storage energy only in the "high peak" periods. Table XXXI. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 45Ah and Parameter Set to be 0.8. Table XXXII. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 60Ah and Parameter Set to be 0.8. 327 We set the parameter to be 1.5 and conduct the same experiments as discussed before. Table XXXIII and Table XXXIV illustrate the comparison results when the capacity of the storage is 45 Ah and 60 Ah, respectively. We can see from these two tables that the proposed near-optimal residential storage control algorithm accounting for prediction inaccuracy consistently outperforms the baseline algorithm, with the maximum improvement of 160% (i.e., 2.60X) on the cost saving capability (on December, 45 Ah storage capacity). Moreover, when comparing with Table XXXI and Table XXXII, we can observe that the improvement in cost saving capability of the proposed algorithm is slighted degraded in Table XXXIII and Table XXXIV (i.e., when parameter is 1.5). This is because the prediction accuracy is lower, thereby degrading the effectiveness of the proposed near-optimal algorithm. Table XXXIII. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 45Ah and Parameter Set to be 1.5. 328 Table XXXIV. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, When the Storage Capacity is 60Ah and Parameter Set to be 1.5. Next we show experimental results based on the second type of electricity price function. We only show experimental results on the 60 Ah storage system due to space limitation. Table XXXV illustrates the comparison results on the cost saving capabilities between the proposed near-optimal storage control algorithm accounting for prediction inaccuracy and the baseline algorithm on every month throughout a year, when the prediction inaccuracy parameter is set to 0.8. Table XXXVI illustrates the comparison results when the prediction inaccuracy parameter is set to 1.5. Once again, the proposed near-optimal residential storage control algorithm accounting for prediction inaccuracy consistently outperforms the baseline algorithm, with a maximum improvement of 106% on the cost saving capability. However, one can notice that the improvement is less significant than the results on the first type of electricity price function. This is because the energy cost due to the demand price component is less significant compared with that due to the energy price component in this case, which will degrade the improvement achieved by the proposed near-optimal solution (because the proposed solution is the most effective in reducing cost due to the demand price component.) 329 Table XXXV. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, under the Second Type of Price Function and Parameter Set to be 0.8. Table XXXVI. Improvement in Cost Saving Capability of the Proposed Algorithm Compared with the Baseline Algorithm, under the Second Type of Price Function and Parameter Set to be 1.5. 330 Chapter 13. Applications of PV and HEES Systems in Embedded Systems 13.1 Energy Storage in Low-Power Embedded System Applications Energy storage for portable electronics has unique requirements compared with EES systems for residential or HEV usages. First, it has strict constraints on the size and weight, and requirements for high-energy density per unit volume or unit weight (which is the key criteria for portable EES systems.) Hence, Lithium-ion battery is the most promising type for man portable applications nowadays due to its high-energy density. Moreover, some portable applications such as military radios and bio-sensors require small and light form factor, high-energy capacity, and high power capability for a short period of time. Finally, relatively simple structure and control policy are desirable for EES systems in portable electronics due to the limited computation capability. A battery-supercapacitor HEES system is a promising candidate to address the above- mentioned requirements. It has a simple architecture and high-energy density due to the usage of Li-ion battery, and can deal with the high-power demand by using supercapacitors as an intermittent energy buffer [108], [120]. We can enhance the total service time of the battery- supercapacitor HEES system by use of a constant-current charger circuit compared with a conventional hybrid architecture that simply connects the battery and supercapacitor in parallel. We illustrate the controller board of the HEES system in Figure 101. Our optimization method in [108] derives that a supercapacitor of 2.5 F capacity maximizes the total service time of the 331 HEES system. Experimental results of a real implementation of the constant-current charger- based HEES system show 7.7% total service time improvement. To address a similar problem, early work [167] uses dual-battery as the hybrid power source for a portable electronic system in order for exploiting both the rate-capacity characteristic and the relaxation-induced recovery effect. The block diagram of the portable electronic system is shown in Figure 102. This work maximizes the utilization of the battery capacity under a given performance constraint using continuous-time Markov decision process (CTMDP), by modeling the workload arrival times, device service times, and battery selection times as stationary stochastic processes known in prior. However, this assumption may not be realistic. Figure 101. Hybrid power controller board developed in [120]. 332 Figure 102. Dual-battery powered portable system model [167]. As an even more interesting application, battery-supercapacitor HEES system can be employed in wireless sensor nodes with energy harvesting in order to significantly extend the life time of the wireless sensors (which is typically restricted by the battery’s cycle life) to even achieve near-perpetual operation. Life time extension of the battery is achieved by relying mostly on the supercapacitor as an energy buffer and reducing the frequency in charging/discharging the battery, because the supercapacitor has a nearly infinite cycle life. This is a desirable feature of wireless sensors since frequent replacements of battery may not be an easy task after deployment. Example implementations include [32], [98] as shown in Figure 49. The analysis in [32] predicts that the sensor nodes will operate for 43 years under 1% load, 4 years under 10% load, and 1 year under 100% load, even under a simple control algorithm far from optimal. Similar potential applications of HEES systems include medical devices with energy harvesting from heat or vibration in order to significantly prolong the device’s total lifetime. Besides achieving near- perpetual operation, the HEES system can reduce the power loss during charge/discharge cycles, which is also critical due to the very limited capability and intermittent nature to produce power from energy harvesting devices such as PV cells. Effective control algorithms should be investigated to take advantage of the high cycle efficiency of the supercapacitor while using the Li-ion battery as a low-leakage long-term energy storage. 333 Recent work [168], [169], [125] focuses on joint optimization of the embedded electronic system and its hybrid power supply. Reference [168] presents HypoEnergy, a framework for extending the lifetime of the hybrid battery-supercapacitor power supply. HypoEnergy studies the hybrid supply lifetime optimization for a preemptively known workload (a given set of tasks.) The block diagram of HypoEnergy is shown in Figure 103. The optimization problem is formulated and mapped to a multiple-choice knapsack problem and solved using the dynamic programming method. The authors evaluate the efficiency and applicability of the HypoEnergy framework using iPhone load measurements. The authors further extend their work to the setup of multiple supercapacitors and workload that is not given a priori [169], and they use the machine learning technique to derive a near-optimal adaptive management policy for the hybrid power supply. Recently, we propose to use a model-free reinforcement learning (RL) technique for an adaptive dynamic power management (DPM) framework in embedded systems with bursty workloads, using a hybrid power supply comprised of Li-ion batteries and supercapacitors [125]. We propose a hierarchical power management framework with two dedicated power managers (PMs): one Supply PM for the hybrid power supply and one Device PM for the embedded device, in order to reduce the online computation overhead. The system and control architecture is shown in Figure 104. We use continuous-time Q-learning for the device PM to deal with bursty workloads and use discrete-time Q-learning for the supply PM. The supply PM makes decision at a much lower frequency than the device PM, and they exchange information with each other about embedded system states (such as busy, idle, sleep), number of waiting requests, SoC’s of the battery and supercapacitor, and so on. The proposed RL-based DPM approach enhances the power efficiency by up to 9% compared to a battery-only power supply. 334 Figure 103. Block diagram of HypoEnergy [167]. Figure 104. System architecture of the hierarchical reinforcement learning framework with two dedicated PMs [125]. 13.2 Power Supply and Consumption Co-Optimization of Portable Embedded Systems with Hybrid Power Supply In this work, we consider a portable embedded system with a hybrid power supply (a battery and a supercapacitor) and executing periodic real-time tasks. We perform system power management from both the power supply side and the power consumption side to maximize the system service time. Specifically, we use feedback control for maintaining the supercapacitor energy at a certain level by regulating the discharging current of the battery, such that the 335 supercapacitor has the capability to buffer the load current fluctuation. On the other hand, at the power consumption side we perform task scheduling to assist supercapacitor energy maintenance. Experimental results demonstrate that the proposed joint optimization framework of task scheduling and power supply control successfully prolongs the total service time by up to 57%. The system architecture diagram is shown in Figure 105. The battery is the main energy storage device for the embedded processing unit (sensor) whereas the supercapacitor serves as an energy buffer. The battery is connected to the supercapacitor through a charger, which regulates the supercapacitor charging current. The supercapacitor is connected to the embedded processing unit through a power converter, which regulates the supply voltage level of the embedded processing unit. This architecture will provide higher energy efficiency than the DC bus-based general HEES structure discussed in Section 7.2 because the supercapacitor is connected to the load device with only one converter (fewer converters/chargers are used in this architecture.) Figure 105. Architecture of the proposed battery-supercapacitor hybrid system in embedded system applications. The voltage and current notations are shown in Figure 105. The OCV and CCV (terminal voltage) of the battery are denoted by and , respectively, whereas the battery 336 discharging current is . will gradually decrease as the battery SoC decreases during the discharging process. depends on the OCV and current as shown in the CCV-OCV relationship in Section 6.2.3. The supercapacitor terminal voltage, input current, and output current are denoted by , , and , respectively. is linearly dependent on the amount of charge stored in the supercapacitor, but independent of and due to the supercapacitor's negligible internal resistance. The voltage and current levels of the embedded processing unit are denoted by (a fixed value) and , respectively, where is equal to when it is executing a task and when it is idle. Please note that this is a simple problem formulation to convey to idea of joint control of HEES system and the embedded processing unit. We can generalize to embedded sensor model to support DVFS capability or have different current levels when executing different types of tasks. The power loss values in the charger and converter depend on their input and output voltages and currents, with the power converter modeling discussed in Section 2.2. We consider the operation of the portable embedded system from the time when the battery is fully charged till the time when it is fully depleted. The processing unit executes a set of periodic tasks with a common period of . The time requirements to execute each instance of tasks are denoted by , , ..., , satisfying . The processing unit needs to execute an instance of each task in each period. 13.2.1 Joint Control and Optimization Algorithm In this work, we aim at maximizing the system service time. More specifically, at the beginning of the system operation, the battery is fully charged whereas the supercapacitor has 337 zero (or low) charge. During the system operation, the processing unit executes periodic tasks as described before, and no task dropping is allowed. We aim at maximizing the total number of finished task instances before the battery is depleted, which is equivalent to maximizing the system service time. We propose joint control and optimization of charging/discharging of the hybrid power supply with task scheduling in the embedded system based on the following two motivations: Motivation I: The battery suffers from rate capacity effect as discussed in Section 6.2, which specifies that the energy loss rate in the battery is a superlinear function of the battery's discharging current. The energy loss due to rate capacity effect will be minimized if the battery discharging current is nearly constant. This motivation has been exploited in our prior work [108]. Based on Motivation I, we are going to set the battery discharging current nearly constant. As a result, the system will operate in two modes. In Mode I, the embedded processing unit is executing a task. In this case the supercapacitor mainly provides energy for the embedded sensor and we have . In Mode II, the embedded processing unit is idle and requires little amount of power. In this case the battery will charge the supercapacitor and we have . The battery discharging currents in these two modes will be nearly the same. Motivation II: The conversion efficiency of charger/power converter is not a constant value, but variable depending on its input and output voltages and currents. In general, the power conversion efficiency will be maximized if its input and output voltages are close to each other. This leads to the motivation that a most desirable supercapacitor voltage exists, which 338 results in the highest energy transfer efficiency or equivalently, the minimum energy loss in the battery in each task scheduling period. Next, we will briefly discuss how to derive the most desirable supercapacitor voltage level . We assume that the supercapacitor voltage is constant (in order to derive .) We know that the processing unit will be busy for amount of time in each period and will be idle for amount of time. Since the supercapacitor voltage is , we can calculate the supercapacitor discharging currents when the processing unit is busy and idle, denoted by and , respectively, based on the converter power modeling provided in Section 2.2. Then we can estimate the (constant) supercapacitor charging current using the following energy balancing equation: (13.1) when the supercapacitor self-discharge is taken into account. After that we can derive the required battery discharging current based on the converter power modeling. Based on this calculation framework (from given to derive the required ), we use the ternary search algorithm, which is an extension of the well-known binary search algorithm, to find the optimal supercapacitor terminal voltage in order to minimize . The underlying assumption of the ternary search algorithm is that the battery discharging current is a quasi-convex function of the supercapacitor voltage . 339 In the following we provide the proposed joint optimization algorithm of power supply control and task scheduling, performed at the beginning of each time period, i.e., , , , , ... The proposed method is motivated by the technique of feedback control (e.g., PID control), which can provide certain level of tolerance of control and modeling inaccuracies, to effectively control the supercapacitor voltage. We need to achieve the following two goals: (i) keep the battery discharging current nearly constant and (ii) keep the supercapacitor terminal voltage near or above the most desirable value . In the feedback control method, we also take into account the following two effects: (i) the optimal supercapacitor voltage also evolves, although slowly, due to the slow degradation of during battery discharging (note that the battery OCV is a monotonically increasing function of its SoC), and (ii) the supercapacitor voltage is typically low at the beginning of system operation (because of its high self-discharge rate) and then we need to gradually increase the supercapacitor voltage towards the optimal value (maybe through multiple periods.) At the beginning of each time period (suppose is time for ), the current supercapacitor voltage level is given by , whereas the battery OCV is given by . Then we execute the following three steps to perform joint control and optimziation of the hybrid power supply and task scheduling in the time period: Step I: Derive and update the optimal supercapacitor voltage based on the current battery OCV , and set as the new target value. 340 Step II: Schedule the set of tasks such that the supercapacitor terminal voltage can maintain nearly constant during the time period. More specifically, we need to interleave task execution and idle times as shown in Figure 106. Step III: With the given task scheduling from Step II, we derive the battery discharging current value during the time period, so that the supercapacitor terminal voltage will get closer to the target value at the end of this time period based on the feedback control policy. The procedure of Step I has already been described, hence we elaborate Step II and Step III as follows: Step II (task scheduling): In this step, we have derived the optimal supercapacitor voltage derived from Step I. We assume that the supercapacitor voltage is and is a constant value to derive the task scheduling. The assumption that the supercapacitor voltage is a constant value 2 can effectively decouple the task scheduling problem with the derivation of battery discharging current . As shown in Figure 106, the best way to schedule all tasks in a time period while maintaining the supercapacitor voltage nearly constant is to interleave the task execution and idle times of each task. More specifically, we execute at the beginning of the time period, and then wait until the supercapacitor terminal voltage returns to . We subsequently execute and wait, and go on this procedure until all task instances have been executed. The ratio of task instance execution time of a task and the subsequent idle time is given by: 2 Please note that this assumption is often valid because the supercapacitor voltage is around when the system operation is stable. 341 (13.2) Of course, this procedure is based on the ideal assumption that the supercapacitor voltage is and has negligible change during task execution and idle times, which is not necessarily precise at the beginning of system operation and will inevitably result in control inaccuracy. This inaccuracy issue is effectively mitigated in Step III using the feedback control mechanism. Step III (feedback control): In this step we have the task execution schedule from Step II, and we are not going to change the task schedule in this step. On the other hand, we are going to derive the battery discharging current during this time period, such that the supercapacitor terminal voltage will get closer to the target value at the end of this time period using the feedback control policy. More specifically, the supercapacitor terminal voltage should reach at the end of this time period, where is a parameter specifying the speed to control the supercapacitor voltage. We use the binary search method to find the appropriate value in this time period to achieve this goal. There are two ways to implement the binary search algorithm. The first way uses and as the approximations of the supercapacitor terminal voltage and battery OCV, respectively. There is inevitable modeling inaccuracy by using constant values to approximate these two voltages. However, such modeling inaccuracy could be effectively mitigated by the feedback control framework because of its inaccuracy tolerance capability. On the other hand, the second way accounts for changes in supercapacitor terminal voltage and battery OCV by dividing the whole time period into a number of fine-grained time 342 slots and calculating the supercapacitor terminal voltage and battery OCV at the beginning of each time slot. The second way will result in a more accurate calculation of the value. Details of approaches are omitted due to space limitations. Recall that as described before, the proposed joint optimization algorithm needs to achieve two goals. The goal that the supercapacitor voltage is kept near the most desirable value is achieved by Step I (deriving the most desirable value) and Step III (using feedback control to maintain the voltage near the most desirable value.) Subsequently, the battery discharging current is nearly constant because (i) we keep the discharging current constant in each time period and (ii) the battery discharging current will not change significantly among various time periods since the supercapacitor voltage is kept near the most desirable value. An example of joint control of hybrid energy supply and task scheduling is shown in Figure 106. We show the system operation over multiple time periods. One can observe that the supercapacitor terminal voltage is initially low, and gradually increases thanks to the feedback control mechanism. We also see the task scheduling and supercapacitor terminal voltage change within a specific time period. We can see that although the supercapacitor terminal voltage drops during task execution and increases during idle time, the change is insignificant because task executions are interleaved and both the execution time of a task instance and the subsequent idle time are small compared to a time period or the total service time. 343 Figure 106. An example showing the control during system operation. 13.2.2 Experimental Results In this section we provide experimental results on the joint optimization framework of task scheduling and power supply control of the portable embedded system. For battery modeling, we obtain characteristics of Li-ion battery by performing measurement on a GP1051L35 Li-ion battery with 350 mAh nominal capacity [108], and extract parameters for the battery model described in Section 6.2.3. We adopt a 5 F supercapacitor in the system, which is the typical size for embedded system applications. We apply Linear Technology LTM4607 converter as the converter and charger in the embedded system. We extract the parameters required in the converter model described in Section 2.2 from the datasheet [22]. The supply voltage level of the embedded processing unit is 1.0 V. The static current of the processing unit is 0.2 A, 344 and we are going to change the active current when executing tasks in the experiments. We use one minute (60 seconds) as the time period for task scheduling. We compare the performance of the proposed joint optimization framework with two baseline systems. The first baseline system only uses battery for power supply, without incorporating the supercapacitor. In this system the battery is connected to the embedded processing unit through a power converter. The second baseline system employs the same battery-supercapacitor hybrid system for power supply as the proposed system and same configurations. However, the second baseline system does not derive the optimal supercapacitor voltage and perform feedback control accordingly. Instead, it simply keeps the supercapacitor voltage as its initial value. In the first experiment, we consider a set of 6 tasks with execution times (of each task instance) of 1 s, 1 s, 1.5 s, 1.5 s, 2 s, and 3 s, respectively. Figure 107 illustrates the comparison results on the total service time (in minutes) between the proposed system and the two baseline systems. The X-axis of Figure 107 is different active current values ’s of the processing unit, while the Y-axis is the total service time. One could observe that the proposed joint optimization framework consistently outperforms the two baseline systems, with the maximum improvements in total service time of 57% and 36%, respectively, when comparing with baseline 1 and baseline 2. Moreover, it can be observed that the improvement is more significant with higher values. This is because the more significant rate capacity effect degrades the performance of the baseline system, which could be effectively mitigated by the proposed system through maintaining the battery discharging current nearly constant. 345 Figure 107. The comparison results on the total service time (in minutes) between the proposed system and two baseline systems in the first experiment. Figure 108. The comparison results on the total service time (in minutes) between the proposed system and two baseline systems in the second experiment. 346 In the second experiment, we consider a set of 6 tasks with execution times (of each task instance) of 2 s, 2 s, 3 s, 3 s, 4 s, and 6 s, respectively. In fact, the execution time of each task instance is twice of that in the first experiment. Figure 108 illustrates the comparison results on the total service time (in minutes) between the proposed system and the two baseline systems. 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Athena Scientific, 2007. 363 Conclusions This thesis is dedicated to the design, control, and applications of photovoltaic (PV) energy generation and (hybrid) energy storage systems. More specifically, we made the following contributions to the PV energy generation systems:  For the PV energy generation system modeling, control, and architecture, we have (i) provided a PV cell modeling and parameter extraction technique, which is a two-step parameter extracion method and achieves up to 8X accracy improvement over the whole V-I operation range of the PV cell compared with reference methods, (ii) derived accurate power modeling for the (DC-DC) buck-boost power converter, which is an essential component in the PV system, and validated against real measurements and the datasheet, and (iii) an effective PV system control technique, the MPTT (maximum power transfer tracking) technique, which accounts for efficiency variations in power converters and ensures the maximum amount of power extraction from the PV system.  In order to combat the partial shading effect, we present a PV module reconfiguration approach that provides both a scalable reconfiguration architecture as well as a systematic and near-optimal control mechanism to overcome the PV system output power degradation. The PV module reconfiguration controller dynamically updates the PV module configurations according to the changing partial shading pattern and conversion efficiency variation of the charger, such that both the shaded and lighted PV cells can work at or close to their MPPs simultaneously. In this way, we improve the PV system output power level under partial shading conditions to the largest possible extent. The proposed reconfiguration control mechanism is based on a dynamic programming 364 algorithm with polynomial time complexity. We have demonstrated up to a factor of 2.36X output power improvement of a large-scale PV system comprised of 3 PV modules with 60 PV cells per modules.  We propose to mount PV cells on the roof, trunk, engine hood, and left and right sides of an HEV or EV, and they may receive different solar irradiances and have different temperature levels. Based on the PV reconfiguration architecture, we present a dynamic PV reconfiguration technique, which updates the PV configuration according to the change of irradiance and temperature distribution during system operation in order to enhance the output power of PV systems. We implement a high-speed, high-voltage PV reconfiguration switch network with IGBTs (insulated-gate bipolar transistors) and a controller. We derive the optimal reconfiguration period considering the on/off delay of IGBT, CAN (control-area network) delay, computation overhead, and energy overhead. We evaluate the proposed fast dynamic PV reconfiguration technique based on the actual implementation of reconfiguration network and controller. Experiments show that the fast dynamic PV array reconfiguration increases 423.0 W power from the baseline.  To reduce the additional capital cost of power switches in the reconfiguration architecture, we propose to perform PV reconfiguration up to the scale of PV macro-cells. A PV macro-cell is comprised of a number of PV cells connected in series and parallel, and the size and structure of a PV macro-cell are determined at the system design stage. In this way, the number of programmable switches can be significantly reduced. The size of a PV macro-cell essentially represents the trade-off between higher performance in combating partial shading and lower capital cost. We provide a design method to decide the optimal size of a PV macro-cell to achieve a balance between performance and capital 365 cost. Experiments demonstrate 60% enhancement in PV system output power with a negligible capital cost increase of 3.7% by selecting the proper macro-cell size. The HEES (hybrid electrical energy storage) system is an emerging approach for achieving performance improvement of the EES system with current non-ideal EES element technologies relying on the optimal design and control methodology. Hybridization offers opportunities to take advantage of each EES element while hiding their shortcomings/drawbacks. Expected benefits include enhancement of energy efficiency, cycle life, power and energy capacity, and so on. We made the following contributions to HEES systems:  We described the general HEES system architecture, which is comprised of energy storage element arrays, power converters, charge transfer interconnect (CTI), and a micro-controller, in analogy to the heterogeneous memory hierarchy. The micro- controller controls energy input/output of each EES bank according to high-level charge management policies. We have implemented a HEES system prototype comprised of a Li-ion battery bank, a lead-acid battery bank, and a supercapacitor bank.  We have defind three fundamental charge management policies in this thesis: charge allocation from a power source to a set of EES banks, charge replacement from EES banks to load devices, and charge migration among EES banks to improve the availability and responsiveness of the HEES system. We present the detailed problem formulation and efficiency optimization policy of the single-source, single-destination (SSSD) charge migration and the more general multiple-source, multiple-destination (MSMD) charge migration in this thesis. We have derived the optimal solution of the SSSD charge migration problem using dynamic programming, and derived the near-optimal solution of the MSMD charge migration problem using the quasi-convex optimization technique. 366  Faulty EES elements, i.e., open-circuited or short-circuited EES elements, are inevitable especially for large-scale EES systems in long-term use. In this thesis, we present a framework of online fault detection and fault tolerance for supercapacitor arrays. We first identify faulty supercapacitors and then reconfigure the supercapacitor array with the faulty supercapacitors disabled. We provide both a scalable structural support and an efficient algorithm for the online fault detection and fault tolerance framework. The proposed fault detection algorithm identifies faulty supercapacitors in a logarithmic time complexity. The proposed fault tolerance algorithm produces the optimal configuration of supercapacitor array considering the efficiency variation of power converter. Experimental results demonstrate that the proposed fault detection and tolerance technique reduces the fault-induced EES system performance degradation by up to 91%.  The cycle life of the EES elements is one of the most important metrics that should be considered by designers. The cycle life is directly related to the state-of-health (SoH), which is defined as the ratio of full charge capacity of an aged EES element to its designed (nominal) capacity. The SoH degradation models presented in reference papers can only be applied to the case of constant-current cycled charging and discharging with same SoC swing in each charging/discharging cycle. To address this shortcoming, we present a novel SoH degradation model to estimate the SoH degradation under arbitrary charging/discharging patterns of battery. Based on the novel SoH model, we derive a near-optimal charge management policy focusing on extending the cycle life of battery in HEES systems while simultaneously improving the overall cycle efficiency. The derivation procedure of the SoH-aware charge management policy is based on convex optimization, and has demonstrated significant cycle life improvement up to 17.3X. 367  Finally, we consider the joint control optimization of PV and HEES systems. We adopt the following three methods based on the individual charger interface topology for higher energy efficiency and robustness under partial shading: 1) incorporation of HEES into a standalone PV system and developing a near-optimal HEES control algorithm, 2) extension of the MPTT approach, and 3) a novel dynamic balanced PV module reconfiguration method to enhance the charger efficiencies. The three proposed methods can be effectively combined together, thereby yielding an energy conversion efficiency improvement ranging from 17.1% to 53.3%, compared with the baseline systems using traditional MPPT control, fixed PV module configurations, and homogeneous EES system. Finally, we discuss the applications of the PV and energy storage systems in two different scales, residential usage and portable embedded systems. More specifically, we have the following contributions:  At the residential usage level, we consider a residential smart grid user with equipped with both PV energy generation and (homogeneous or hybrid) energy storage systems, under dynamic energy pricing policies. We have the following three-fold contributions: (i) We develop effective prediction methods for PV power generation and residential load power consumption, taking into account the specific characteristics of dynamic energy pricing function. (ii) Based on the prediction results, we present a near-optimal storage control algorithm that mitigates the inevitable prediction errors and properly accounts for the energy loss components due to power dissipation in the power conversion circuitries as well as the rate capacity effect in the storage system. The proposed near-optimal storage control algorithm is effectively implemented by solving 368 a convex optimization problem with polynomial time complexity at the beginning of each day in a billing period. (iii) We formally describe the integration of a residential- level HEES system for the Smart Grid users equipped with PV power generation. We propose the optimal control algorithm for the HEES system. The proposed control algorithm is based on dynamic programming and therefore has polynomial time complexity. Experimental results demonstrate that the proposed HEES system and control algorithm achieves 73.9% average profit enhancement over homogeneous EES systems.  We consider a portable embedded system with a hybrid power supply (a battery and a supercapacitor) and executing periodic real-time tasks. We perform system power management from both the power supply side and the power consumption side to maximize the system service time. Specifically, we use feedback control for maintaining the supercapacitor energy at a certain level by regulating the discharging current of the battery, such that the supercapacitor has the capability to buffer the load current fluctuation. On the other hand, at the power consumption side we perform task scheduling to assist supercapacitor energy maintenance. Experimental results demonstrate that the proposed joint optimization framework of task scheduling and power supply control successfully prolongs the total service time by up to 57%. 369 Appendix I We use proofs by contradiction. Suppose that the optimal that optimizes the IC-PMR problem is NOT an element in . Then there exists a configuration of the PV string, denoted by , that is Pareto-superior to , which implies that and , or and . Then according to the property of charger as described in Section 2.2, the charger output current with PV string configuration , , is higher than the charger output current with PV string configuration , . This implies that cannot optimize the IC-PMR problem. Therefore, we have proved the theorem. 370 Appendix II We have the following two equations from (8.5) and (8.6): (A.1) (A.2) Based on the above two equations, we rewrite (8.7) in the following way: (A.3) where is the destination array OCV given SoC , and is the source array OCV given SoC . The nominator of Eqn. (A.3) is a constant since both and are constants. Hence, maximizing is equivalent to minimizing the denominator of (A.3), . It is furthermore equivalent to maximizing or minimizing the total amount of charge extracted from the source EES array since is given. We have proved the theorem by far.

Abstract (if available)

Abstract This dissertation focuses on the design and runtime control of advanced electrical energy generation and storage systems for a wide range of applications, from the power grid to electric cars, and from households to mobile battery‐operated devices. The dissertation is organized in three parts, focusing on energy generation, energy storage, and their integration and applications. ❧ Due to an increasing appetite for energy and concern about environmental impacts of fossil fuels, there has been a growing demand for renewable, eco‐friendly and sustainable energy resources (e.g., solar, wind, geothermal). The energy produced from these alternate energy resources must be cost‐competitive with the energy produced from fossil fuels. Photovoltaic (PV) energy generation techniques have received significant attention since they utilize the abundance of solar energy and can be easily scaled up. A PV system is comprised of several (series‐connected) PV modules, an energy storage, and charger connecting in between. Accurate modeling and effective control of the PV system is mandatory in order to fully exploit the solar irradiance. Moreover, PV systems are subject to the partial shading effect that can result in a significant degradation in the PV system output power. ❧ The first part of this dissertation presents an accurate modeling framework of each component in the PV system and an effective control mechanism, called the maximum power transfer tracking method. In addition, a cost‐effective reconfigurable PV architecture to combat the partial shading effect and a dynamic programming‐based reconfiguration algorithm to maximize the overall PV system output power are described. Experimental results based on a hardware prototype of a reconfigurable PV system demonstrate the accuracy of the PV modeling and the effectiveness of the PV reconfiguration algorithm. Moving towards real testbeds, this dissertation next focuses on how the aforesaid reconfigurable PV system can be integrated into a hybrid electric vehicle (HEV) so as to maximize the total distance which is travelled by an HEV on a full tank of gasoline with the aid of the PV harvested energy. The issue of capital cost and economic vialibilty of the proposed reconfigurable PV system is considered next. ❧ As of today, no single type of EES (electrical energy storage) element, e.g., batteries, supercapacitors, can fulfill all desirable features of an ideal storage device. In this dissertation the architecture and control mechanisms for a hybrid EES (HEES) system comprising of two or more heterogeneous EES elements are presented. The HEES system, which can realize advantages of each EES element while hiding their weaknesses, can thus exhibit superior performance compared to conventional homogeneous EES systems. Three fundamental charge management policies are defined in this dissertation: charge allocation from a power source to a set of EES banks, charge replacement from EES banks to load devices, and charge migration among EES banks to improve the availability and responsiveness of the HEES system. In particular, an optimal control policy for charge migration and algorithms for fault detection and tolerance in the HEES system are presented. Finally, a state‐of‐health (SoH) aware joint charge management algorithm for the HEES system, which is based on an improved SoH modeling, is discussed. ❧ The last part of this dissertation focuses on joint control and applications of the proposed (reconfigurable) PV and HEES systems. The target platforms range from residential units to portable embedded systems. It has been observed that both the appropriate system design and runtime control algorithm are critical in order to enhance the energy efficiency.

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Asset Metadata

Creator Wang, Yanzhi (author)

Core Title A joint framework of design, control, and applications of energy generation and energy storage systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Design,energy efficiency,hybrid energy storage,OAI-PMH Harvest,optimal control,photovoltaic energy generation

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Pedram, Massoud (committee chair), Gupta, Sandeep K. (committee member), Huang, Ming-Deh (committee member)

Creator Email yanzhiwa@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-465151

Unique identifier UC11287904

Identifier etd-WangYanzhi-2851.pdf (filename),usctheses-c3-465151 (legacy record id)

Legacy Identifier etd-WangYanzhi-2851.pdf

Dmrecord 465151

Document Type Dissertation

Rights Wang, Yanzhi

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA