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Integration of energy-efficient infrastructures and policies in smart grid

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Integration of energy-efficient infrastructures and policies in smart grid

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Content UNIVERSITY OF SOUTHERN CALIFORNIA Integration of Energy-Efficient Infrastructures and Policies in Smart Grid by Tiansong Cui A dissertation submitted in partial fulfillment for the degree of Doctor of Philosophy in the Prof. Shahin Nazarian and Prof. Massoud Pedram Ming Hsieh Department of Electrical Engineering May 2017 Declaration of Authorship I, Tiansong Cui, declare that this dissertation titled, ‘Integration of Energy-Efficient Infrastructures and Policies in Smart Grid’ and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this dissertation has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this dissertation is entirely my own work. I have acknowledged all main sources of help. Where the dissertation is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: i “Never tell people how to do things. Tell them what to do and they will surprise you with their ingenuity. ” General George S. Patton UNIVERSITY OF SOUTHERN CALIFORNIA Abstract Prof. Shahin Nazarian and Prof. Massoud Pedram Ming Hsieh Department of Electrical Engineering Doctor of Philosophy by Tiansong Cui The current smart grid technology is undergoing a transformation from a centralized, producer- controlled network to one that is less centralized and more consumer-interactive. To shape the power demand to reduce the peak power consumption and smoothen the variation, several energy- efficient infrastructures as well as the corresponding policies are introduced. The infrastructures include Hybrid Electrical Energy Storage (HEES) systems, Plug-in Electric Vehicles (PEVs), and energy efficient buildings. The policies include dynamic energy pricing and Regulation Service (RS) provisioning. In this dissertation, we present an integrated framework of these energy-efficient infrastructures and policies. Four works are introduced in this dissertation with different perspec- tives and focuses. They are the game-theoretic price determination algorithm for utility companies serving a community, the negotiation-based task scheduling algorithm is studied which minimizes energy user’s electricity bills under dynamic energy prices, the optimal energy co-scheduling frame- work for energy efficient smart buildings, and the optimal PEV control problem with a charging aggregator considering regulation service provisioning. Acknowledgements First and foremost, I give my deepest gratitude to my PhD advisors, Prof. Shahin Nazarian and Prof. Massoud Pedram, for being constant sources of guidance, assistance, expertise, and inspiration during the past five years. They have been and will always be tremendous mentors for me in my life. Prof. Nazarian introduced me to the world of research and helped me grow as a research scientist. At the beginning of my Ph.D., I was far from a good researcher and I was also lack of self-confidence. Prof. Nazarian was extremely patient to me even when I made mistakes. He guided me and encouraged me to try new ideas, give presentations, and write papers. With his help, my research, presentation and writing skills have been greatly improved. Apart from research, he also taught me how to be a good teaching assistant and helped me with my career plan. Prof. Pedram offered me the great opportunity to work in his group. I joined his group and it turned out to be one of the wisest decisions I have ever made. He has taught me how good research is done throughout my Ph.D. studies and consistently contributes valuable advice, feedback and encouragement to my research. Without his patient instruction, insightful criticism and expert guidance, the completion of my Ph.D. would not have been possible. His unbounded passion for research and dedication in making technical contributions will always be an inspiration to me even after I graduate. Next, I would like to thank other members of my defense and qualifying exam committee: Prof. Aiichiro Nakano, Prof. Sandeep Gupta, and Prof. Paul Bogdan, for providing valuable feedback to my research and pushing me to think deeper. Special thanks to Prof. Nakano for his excellent teaching in Scientific Computing and Visualization course, from which I have learned how to design good videos to visualize research ideas and improve presentation quality. iv My sincere thanks also go out to my collaborators. They include Yanzhi Wang, Qing Xie, Xue Lin, Shuang Chen, Woojoo Lee, Luhao Wang, Di Zhu, Mohammad Javad Dousti, Ji Li, Alireza Shaefei, Naveen Katem, Ting-Ru Lin, Hadi Goudarzi, and Safar Hatami. Without them I could not accomplish what I have done. Special thanks to Yanzhi Wang for contributing in most of my work. His intelligent ideas and unbeatable passion have been and will always be an encouragement to me. Also special thanks to Qing Xie for helping me developing the complete research skill set which is also beneficial to my job search. Another special thank goes to Shuang Chen for his extraordinary programming skills and significant contribution to many of my work. During my five-year Ph.D. study at USC, I am very proud to be a member of the SPORT family. I appreciate the time shared with all SPORT members during my PhD career, and I also appreciate the help I received from alumni members of the SPORT group, including Amir H. Salek and Hanif Fatemi, who offered me the chances to intern in their groups. I would like to thank my colleagues and friends who have in many ways contributed to the success of my academic endeavors. Special thanks to my roommate Jianwei Zhang for offering all kinds of help in my life. Also special thanks to Fangzhou Wang, Yang Zhang, Jizhe Zhang, Yuankun Xue and Xuan Zuo for being excellent cooperators in different areas. Finally, I would like to thank my family for all their support, encouragement and love. In particular, I gratefully thank my parents for always raising me up with unconditional love. No matter what obstacle I faced or how despair I was, they were always encouraging me and releasing my pressure. This last word of acknowledgment I have saved is for my dear fianc´ ee Wentao Zhang. We met each other in USC in September 2014. Out of similar outlook on life and values, our friendship evolved into a deep loving relationship and I have been happier, calmer, more confident and more content. We got engaged in 2016 and I have been even happier after that. I had never been afraid of the challenges in my long Ph.D. journey because I have Wentao by my side, to catch me when I falter, to encourage me when I lose heart, to strive for the same goal, and to share happiness and sorrow. Thank you with all my heart and soul. Contents Declaration of Authorship i Abstract iii Acknowledgements iv List of Figures x List of Tables xi 1 Introduction 1 1.1 Energy-Efficient Infrastructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Energy-Efficient Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Related Work 10 2.1 Plug-in Electrical Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 HV AC Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 HEES System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Dynamic Energy Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Regulation Service Reserve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 A Game-Theoretic Price Determination Algorithm for Utility Companies Serving a Community in Smart Grid 15 3.1 Model I: Ignoring Energy Generation Cost . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Model II: Considering Energy Generation Cost . . . . . . . . . . . . . . . . . . . 22 3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 vii 4 Negotiation-Based Task Scheduling to Minimize User’s Electricity Bills under Dynam- ic Energy Prices 28 4.1 System Model and Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Negotiation-Based Energy Cost Minimization . . . . . . . . . . . . . . . . . . . . 33 4.2.1 Negotiation-Based Routing Algorithm in FPGAs . . . . . . . . . . . . . . 34 4.2.2 Motivation to Introduce Negotiation-Based Routing Method to Task Schedul- ing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.3 Negotiation-Based Task Scheduling Algorithm . . . . . . . . . . . . . . . 37 4.2.4 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 An Optimal Energy Co-Scheduling Framework for Smart Buildings 45 5.1 System Component Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1.1 Model of Building Power Flow and Power Conversion . . . . . . . . . . . 47 5.1.2 Model of HV AC control . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1.3 Model of HEES system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.1.3.1 Battery Rate Capacity Effect . . . . . . . . . . . . . . . . . . . 51 5.1.3.2 Battery SoH Degradation . . . . . . . . . . . . . . . . . . . . . 53 5.1.3.3 Supercapacitor Self-Discharge . . . . . . . . . . . . . . . . . . 56 5.2 HV AC Control and HEES Management Co-Scheduling Algorithm . . . . . . . . . 58 5.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2.2 Adaptive Co-Scheduling Problem Formulation . . . . . . . . . . . . . . . 60 5.3 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6 Optimal Control of PEVs with a Charging Aggregator Considering Regulation Service Provisioning 72 6.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.1.1 Charging Aggregator Architecture and Power Flow . . . . . . . . . . . . . 74 6.1.2 Model of Battery Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.1.3 Model of Battery SoH Degradation . . . . . . . . . . . . . . . . . . . . . 78 6.1.4 Model of Regulation Service Provisioning Market . . . . . . . . . . . . . 81 6.2 Optimal Charging Schedule of an Individual PEV . . . . . . . . . . . . . . . . . . 84 6.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2.2 Adaptive Control Problem Formulation . . . . . . . . . . . . . . . . . . . 87 6.3 Optimal Control of the PEV Charging Aggregator . . . . . . . . . . . . . . . . . . 92 6.3.1 Charging Aggregator Cost Function, Decision Window and Billing Period . 92 6.3.2 Estimation of Future Parking PEVs . . . . . . . . . . . . . . . . . . . . . 95 6.3.3 Adaptive Solution of PEV Charging Aggregator Control . . . . . . . . . . 96 6.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.4.1 Results of Individual PEV Charging Control Optimization . . . . . . . . . 98 6.4.2 Results of PEV Charging Aggregator Control Optimization . . . . . . . . . 100 6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7 Conclusion 104 Bibliography 105 List of Figures 3.1 Initial and final price functions for each utility companies . . . . . . . . . . . . . . 26 3.2 Profit change steps for each utility companies . . . . . . . . . . . . . . . . . . . . 27 4.1 Example of a household task scheduling problem . . . . . . . . . . . . . . . . . . 31 5.1 The energy co-scheduling framework. . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 Power flow of an energy-efficient smart building including HV AC control, PV sys- tem, HEES system, building load and the external power grid. The directions of arrows represent the directions of the power flow. . . . . . . . . . . . . . . . . . . 47 5.3 Relationship betweenP bat andP bat;int considering the rate capacity effect. . . . . 53 5.4 Li-ion battery SoH degradation versus SoC swing (at different average SoC levels) and average SoC level (at different SoC swings). . . . . . . . . . . . . . . . . . . 56 5.5 An example showing self-discharge of a supercapacitor in 48 hours. . . . . . . . . 57 5.6 Relationship between daily cost and battery storage capacity for our proposed al- gorithm and four baseline schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.7 Battery daily charge/discharge schedule. . . . . . . . . . . . . . . . . . . . . . . . 69 5.8 Relationship between daily cost and peak hour energy price for our proposed algo- rithm and two baseline schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.1 Structure of the PEV charging aggregator. . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Charging Aggregator Power Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3 Relationship betweenP bat andP bat;int considering the rate capacity effect. . . . . 77 6.4 Li-ion battery SoH degradation versus SoC swing (at different average SoC levels) and average SoC level (at different SoC swings). . . . . . . . . . . . . . . . . . . 81 6.5 An example of RS tracking signalz(t). . . . . . . . . . . . . . . . . . . . . . . . 82 6.6 The billing period, decision window, and estimation of future parking PEVs in the charging aggregator optimization framework. . . . . . . . . . . . . . . . . . . . . 94 6.7 Flow of charging aggregator optimization. . . . . . . . . . . . . . . . . . . . . . . 98 6.8 Electricity Price and Bidding Decision of One Vehicle . . . . . . . . . . . . . . . . 102 x List of Tables 3.1 Profit comparison of Model I under a high initial price . . . . . . . . . . . . . . . 25 3.2 Profit comparison of Model I under a low initial price . . . . . . . . . . . . . . . . 25 3.3 Profit comparison of Model II for different utility companies . . . . . . . . . . . . 25 4.1 Performance and time complexity of the negociation-based task scheduling algo- rithm and the baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Performance and average error rate of the negociation-based task scheduling algo- rithm and the baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.1 Comparison between the Cost of Proposed Solution and Baseline Solutions for In- dividual PEV Charging Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2 Comparison between the Cost of Proposed Solution and Baseline Solutions for Charging Aggregator Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 101 xi Chapter 1 Introduction The current smart grid technology is undergoing a transformation from a centralized, producer- controlled network to one that is less centralized and more consumer-interactive [1][2]. This will lead to a change of the industry’s entire business model and its relationship with all stakeholders, involving and affecting utilities, regulators, energy service providers, technology and automation vendors and all consumers of electric power [1]. With the introduction of decentralized electrical network architecture, the usual practice of power network on matching supply to real-time demand will be increasingly challenging [3]. It is generally agreed that the amount of generation, transmis- sion and distribution capacities that utility companies need to provision depends on peak demand rather than the average [4]. Consequently, the huge difference between energy consumption lev- els at peak usage time and off-peak hours has resulted in not only cost inefficiencies and potential brownouts and blackouts, but also environmental pollution due to over provisioning of the Power Grid and the resulting energy waste [5]. For example, the US national load factor is about 55%, and 1 Chapter1 2 only 10% of generation plants and 25% of distribution facilities are used more than 400 hours per year, i.e., 5% of the time [1]. To shape the power demand to reduce the peak power consumption and smoothen the variation, several energy-efficient infrastructures as well as the corresponding policies are introduced. The infrastructures include Hybrid Electrical Energy Storage (HEES) systems, Plug-in Electric Vehicles (PEVs), and energy efficient buildings. The policies include dynamic energy pricing and Regulation Service (RS) provisioning. 1.1 Energy-Efficient Infrastructures The commercial and residential building stock is responsible for 40% of the U.S. primary energy consumption, 40% of the greenhouse gas emissions, and 70% of the electricity use [6]. To ad- dress the rising cost of energy and ensure the sustainability of the environment, reducing energy consumption in new and existing buildings has become a focus of all stakeholders including gov- ernments, building developers, owners, operators, tenants and users [7]. Energy-efficient smart buildings are thus introduced, in which flexible energy demands and heterogeneous energy sources are co-scheduled together to achieve total energy cost savings. Based on the collected information, the building management system controls the energy flow from different energy sources to a variety of building tasks. Among all the energy consumed in buildings, 50% of it is directly related to space heating, cooling, and ventilation [6]. Therefore, a great amount of energy consumption reduction will be achieved if a smart control mechanism is designed to operate the Heating, Ventilation and Air Conditioning Chapter1 3 (HV AC) system in a more efficient way. This is a challenging task because the need for HV AC changes over hours and days as does the electricity price. Level of comfort of the building occupants is, however, a primary concern, which tends to overwrite pricing. Dynamic HV AC control under a dynamic energy pricing model while meeting an acceptable level of occupants’ comfort is thus critical to achieving energy efficiency in buildings in a sustainable manner. In the literature, various HV AC control mechanisms are proposed to reduce energy cost through storage of thermal energy [8], co-design of HV AC control and embedded platform [9], minimization of total and peak energy consumption using Model Prediction Control [10], and so on. Morever, plug-in electric vehicles (PEVs), which utilize electric motors for propulsion, have shown great promise in reducing the cost of transportation as well as curbing the emission due to their battery storage systems that can be flexibly recharged in a parking garage or at home [11]. In addition to the lowered energy cost and reduced greenhouse gas emission, the development of PEVs can also improve the flexibility and reliability of the power grid. The Vehicle-to-Grid (V2G) structure has been presented [12] which uses a number of batteries of the PEVs as an equivalent large-scale energy storage in order to balance the supply and demand in the power system. Another promising methodology of improving energy efficiency is the adoption of hybrid energy supply in smart buildings, where multiple energy sources are scheduled together to reduce the peak demand and leverage renewable energy sources such as solar radiation [13]. Previous papers have proposed different approaches for efficiently scheduling multiple energy sources [14, 15]. Electri- cal energy storage (EES) is the one of the most popular hybrid energy sources in energy-efficient buildings, because the energy stored inside the EES element can be converted into electrical energy and delivered to electrical systems whenever and wherever energy is needed [16]. Through proper Chapter1 4 charging and discharging schedule, the storage ability of EES elements is exploited for frequen- cy regulation, load balancing, and peak demand shaving [11, 17, 18]. Unfortunately, none of the existing EES elements, such as Li-ion batteries, lead-acid batteries, and supercapacitors, can simul- taneously fulfill all the desirable performance metrics, e.g., long cycle life, high power and energy densities, low cost/weight per unit capacity, high cycle efficiency, and low environmental effects [19]. Hybrid Electrical Energy Storage (HEES) systems, which consist of multiple banks of inho- mogeneous EES elements with difference characteristics, are proposed to exploit the strengths of different EES elements and compensate their weaknesses for achieving a combination of superior performance metrics [20]. To fully explore the advantages of HEES systems, authors in [21–23] have studied the implementation of three essential management operations including charge alloca- tion, charge replacement, and charge migration. 1.2 Energy-Efficient Policies A promising methodology of improving power reliability is the dynamic pricing strategy in Smart Grid [24]. In the Smart Grid infrastructure, utility companies could employ real-time or time-of- use electricity pricing policies, i.e., employing different electricity prices at different time periods in a day or at different locations. This policy can incentivize consumers to perform demand side management, a.k.a. demand response, by shifting their load demands from peak hours to off-peak hours [25],[26]. Proper shaping of customer’s demands also makes it possible for utility companies to reduce their capital expenditure by not having to add new power plants to the Grid in order to meet the customers’ peak-hour demands [27]. Chapter1 5 Moreover, regulation service (RS) has been proved to be the most feasible service for grid-side use of PEVs in matching electricity supply with demand in real time while enabling PEV owners to reduce electricity cost by offering a reserved power capacity[28],[29]. In an RS contract, each PEV owner declares an average power consumption (for which he/she is charged) as well as a regulation service (for which he/she is credited). The vehicle is asked to modulate its power consumption dynamically through V2G network so as to track the RS signal provided by independent system operators (ISOs), who in turn try to match supply and demand in real time in presence of volatile and intermittent renewable energy generation [28]. Due to this fact, the payoff from RS provisioning should also be included in the PEV control mechanism. 1.3 Research Objective Although the above mechanisms are well researched, there has been little work on formulating the interactions between these aspects and addressing them together. In this research, we present an integrated framework of the above-mentioned energy-efficient infrastructures and policies. After a literature review of previous works related to this paper is presented in Chapter 2, we present the formulation and solution of four problems as follows. In Chapter 3, we present a game-theoretic price determination algorithm for utility companies serv- ing a community. Distributed power network is the major trend of future smart grid, which contains multiple non-cooperative utility companies who have incentives to maximize their own profits. The energy price competition forms an n-person game among utility companies where one’s price s- trategy will affect the payoffs of others. More interestingly, the use of dynamic energy pricing Chapter1 6 schemes incentivizes homeowners to consume electricity more prudently in order to minimize their electric bill. In this chapter, two models of price determination are introduced for utility companies under different assumptions. In the first model, a Nash equilibrium solution is presented and the uniqueness of Nash equilibrium point is proved. The second model accounts for more sophisticated factors such as the cost of energy generation and the homeowner’s reaction to the change of energy usage as a factor of energy price. Although it is no longer possible to prove the uniqueness of Nash equilibrium for the second model, we present a practical solution in which no utility compa- ny can increase its expected profit by adjusting the price function. Experimental results show the effectiveness of our two models both in reliability of solution and in runtime. In Chapter 4, a negotiation-based task scheduling algorithm is studied which minimizes energy us- er’s electricity bills under dynamic energy prices. Dynamic energy pricing is a promising technique in the smart grid that incentivizes energy consumers to consume electricity more prudently in order to minimize their electric bills meanwhile satisfying their energy requirements. This has become a particularly interesting problem with the introduction of residential photovoltaic (PV) power gen- eration facilities. This chapter addresses the problem of task scheduling of (a collection of) energy consumers with PV power generation facilities, in order to minimize the electricity bill. A general type of dynamic pricing scenario is assumed where the energy price is both time-of-use and total power consumption-dependent. A negotiation-based iterative approach has been proposed that is inspired by the state-of-the-art Field-Programmable Gate Array (FPGA) routing algorithms. More specifically, the negotiation-based algorithm is used to rip-up and re-schedule all tasks in each iter- ation, and the concept of congestion is effectively introduced to dynamically adjust the schedule of Chapter1 7 each task based on the historical scheduling results as well as the (historical) total power consump- tion in each time slot. Experimental results demonstrate that the proposed algorithm achieves up to 51.8% improvement in electric bill reduction compared with baseline methods. In Chapter 5, we focus on the optimal energy co-scheduling framework for energy efficient smart buildings. The Heating, Ventilation and Air Conditioning (HV AC) system accounts for half of the energy consumption of a typical building. Additionally, the need for HV AC changes over hours and days as does the electric energy price. Level of comfort of the building occupants is, however, a primary concern, which tends to overwrite pricing. Dynamic HV AC control under a dynamic energy pricing model while meeting an acceptable level of occupants’ comfort is thus critical to achieving energy efficiency in buildings in a sustainable manner. Finally, there is the possibility that the building is equipped with some renewable source of power such as solar panels mounted on the rooftop. The presence of Hybrid Electrical Energy Storage (HEES) system in a target building would enable peak power shaving by adopting a suitable charge and discharge schedule for each Electrical Energy Storage (EES) element, while simultaneously meeting building energy efficiency and user satisfaction. Achieving this goal requires detailed information (or predictions) about the amount of local power generation from the renewable source plus the power consumption load of the building. This chapter addresses the co-scheduling problem of HV AC control and HEES system management to achieve energy-efficient smart buildings, while also accounting for the degradation of the battery state-of-health during charging and discharging operations (which in turn determines the amortized cost of owning and utilizing a battery storage system). A time-of-use dynamic pricing scenario is assumed and various energy loss components are considered including power dissipation in the power conversion circuitry, the rate capacity effect in the batteries, and the self-discharge in Chapter1 8 the super-capacitor. A global optimization framework targeting the entire billing cycle is presented and an adaptive co-scheduling algorithm is provided to dynamically update the optimal HV AC air flow control and the HEES system management in each time slot during the billing cycle to mitigate the prediction error of unknown parameters. Experimental results show that the proposed algorithm achieves up to 15% in the total electric utility cost reduction compared with some baseline methods. The formulation and solution of the optimal PEV control problem with a charging aggregator con- sidering regulation service provisioning are presented in Chapter 6. Plug-in electric vehicles (PEVs) are considered the key to reducing the fossil fuel consumption and an important part of the smart grid. The plug-in electric vehicle-to-grid (V2G) technology in the smart grid infrastructure enables energy flow from PEV batteries to the power grid so that the grid stability is enhanced and the peak power demand is shaped. PEV owners will also benefit from V2G technology as they will be able to reduce energy cost through proper PEV charging and discharging scheduling. Moreover, power regulation service (RS) reserves have been playing an increasingly important role in modern power markets. It has been shown that by providing RS reserves, the power grid achieves a better match between energy supply and demand in presence of volatile and intermittent renewable energy gener- ation. This paper starts with the problem of PEV charging under dynamic energy pricing, properly taking into account the degradation of battery state-of-health (SoH) during V2G operations as well as RS provisioning. An overall optimization throughout the whole parking period is proposed for the PEV and an adaptive control framework is presented to dynamically update the optimal charg- ing/discharging decision at each hour to mitigate the effect of RS tracking error. As more and more PEVs are being plugged into the power grid, the control or management issue of PEV charg- ing arises, since mass unregulated charging processes of PEVs may result in degradation of power Chapter2 9 quality and damage utility equipments and customer appliances. To solve this problem, this paper also presents a SoH-aware charging aggregator design, which decides the control sequences of a group of PEVs to reduce the peak power caused by simultaneous PEV charging. An energy storage system is used in the charging aggregator to do a peak power shaving, and future arriving PEVs are properly taken care of. Experimental results show that the proposed optimal charging algorithm minimizes the combination of electricity cost and battery aging cost in the RS provisioning power market. Finally, Chapter 7 concludes the thesis. Chapter 2 Related Work 2.1 Plug-in Electrical Vehicle The increasing demands for energy resources all around the world as well as the growing public concern over the environmental effects of fossil fuels have sparked significant interests in renewable energy [30]. Plug-in electric vehicles (PEVs), which utilize electric motors for propulsion, have shown great promise in reducing the cost of transportation as well as curbing the emission due to their battery storage systems that can be flexibly recharged in a parking garage or at home [11]. In addition to the lowered energy cost and reduced greenhouse gas emission, the development of PEVs can also improve the flexibility and reliability of the power grid. The Vehicle-to-Grid (V2G) structure has been presented [12] which uses a number of batteries of the PEVs as an equivalent large-scale energy storage in order to balance the supply and demand in the power system. 10 Chapter2 11 Plug-in Electrical Vehicle (PEV) has been proposed as one of the solutions to tackle the energy crisis and global warming [30],[31]. The Plug-in option allows the battery storage systems to be flexibly recharged in a parking garage or at home [11]. As PEVs have drawn a growing attention in the past few years, the Vehicle-to-Grid (V2G) network design and PEV charging control mech- anisms are studied in previous papers with different focuses. Considering that the vehicle fleet has higher power capacity, while utility generators have longer operating life and lower operating cost- s, authors in [12] introduce several strategies and business models for using their complementary strengths in V2G network to reconcile the complementary needs of the driver and grid manager. They conclude that after the initial high-value, V2G markets saturate and production costs drop, V2G can provide storage for renewable energy generation. Authors in [18] consider PEV cus- tomers’ convenience and present the State of Charge (SoC) control method by the local control centers which enables the SoCs of the PEVs to be synchronized. An V2G algorithm is developed to optimize energy and ancillary services scheduling in [32], and an integrated framework of PEV charging and wind energy scheduling on electricity grid is presented in [33]. All the above work- s have used the idea of coordinating PEV group charging by an aggregator in order to minimize power loss in the distribution network. 2.2 HV AC Control In the literature, there are a variety of building HV AC control mechanisms proposed aiming at min- imizing energy consumption while meeting an acceptable level of building occupants’ comfort and satisfying the operational constraints. Most of these papers are based on a Model-based Predictive Control (MPC) approach. The main idea of MPC is to use the model of the plant and buildings Chapter2 12 to predict the future evolution of the system. Authors in [10] focus on the total and peak energy consumption minimization while the idea of improving energy efficiency using storage of thermal energy is presented in [8]. Considering that HV AC systems are generally complex cyber-physical systems which involve three closely-related subsystem - the control algorithm, the physical building and environment, and the embedded implementation platform, authors in [9] propose a co-design approach that analyzes the interaction between the control algorithm and the embedded platform through a set of interface variables. 2.3 HEES System Hybrid electrical energy storage (HEES) system was first introduced in [20]. The proposed HEES system builds on the concepts of computer memory system architecture and management in order to achieve the attributes of an ideal electrical energy storage (EES) system through appropriate allocation and organization of various types of EES elements. Three key problems, namely charge allocation, charge migration, and charge replacement, are studied in the design and management of HEES systems [21–23]. Charge allocation is the problem of distributing the incoming power to different destination EES banks [21]. Authors in [22] discuss the charge migration problem of transferring electrical charge from source banks to destination banks so as to improve the overall efficiency. Charge replacement is the management of discharge currents of EES banks to serve a given load demand [23]. Chapter2 13 2.4 Dynamic Energy Pricing The introduction of dynamic energy pricing gives a popular solution on shaping the demand to reduce the peak and smoothen the variation [34],[2],[3]. Dynamic pricing is an incentive-based scheme, which means both utility companies and energy users have an incentive to maximize their own profits or minimize their own costs. The research on dynamic pricing can be classified into the following two categories: on the utility company side, authors in [4] and [27] provide different models to analyze the price determination problem in order to maximize the total profit; on the en- ergy user side, authors in [5] and [26] focus on task scheduling problem for energy users to achieve lowered electricity bills. Authors in [35] combine the models of these two sides and concurrently optimize energy user’s electrical energy bill and utility company’s power generation cost. In prac- tice, dynamic energy pricing is already used in some utility companies such as Consolidated Edison Company 1 . 2.5 Regulation Service Reserve Regulation service (RS) has been proved to be the most feasible service for grid-side use of PEVs in matching electricity supply with demand in real time while enabling PEV owners to reduce electricity cost by offering a reserved power capacity[28],[29]. In an RS contract, each PEV owner declares an average power consumption (for which he/she is charged) as well as a regulation service (for which he/she is credited). The vehicle is asked to modulate its power consumption dynamically through V2G network so as to track the RS signal provided by independent system operators (ISOs), 1 http://www.coned.com/ Chapter3 14 who in turn try to match supply and demand in real time in presence of volatile and intermittent renewable energy generation [28]. Due to this fact, the payoff from RS provisioning should also be included in the PEV control mechanism. Prior works mainly focus on optimizing demand-side RS reserves in an hour-ahead power mar- ket. Authors in [36] investigate optimal dynamic pricing policies for RS bidding, while in [37] a market-based mechanism is proposed which uses the dynamic pricing policy to enable a smart grid building operator to offer RS reserves and meet the ISO requirements. In [29] an optimal aggregator design for Vehicle-to-Grid RS is provided and the corresponding control strategy for the aggregator regarding a performance measure is developed. Chapter 3 A Game-Theoretic Price Determination Algorithm for Utility Companies Serving a Community in Smart Grid In this chapter, two models of dynamic pricing are presented in this paper to solve the profit maxi- mization problem for utility companies under an oligopolistic energy market. For both models, we assume that each energy consumer has the ability to freely select any of the existing utility compa- nies without any additional cost, and it is the energy price function that determines the probability he/she selects a certain company. Based on this assumption, we propose game theoretic solutions for determining the hourly price of the electric energy in each utility company to maximize its total expected profit. In the first model, a Nash equilibrium solution is presented and the uniqueness of Nash equilibrium point is proved. The second model considers more sophisticated factors such as 15 Chapter3 16 the cost of energy generation and the homeowner’s reaction on the change of energy usage as a fac- tor of energy price. Although it is no longer possible to prove the uniqueness of Nash equilibrium, we still present a practical solution to profit maximization problem in which no utility company can increase its expected profit by adjusting the price function. 3.1 Model I: Ignoring Energy Generation Cost In this chapter, a slotted time model is assumed for all models, i.e., all system cost parameters and constraints as well as scheduling decisions are provided for discrete time intervals of constant length. The scheduling epoch is thus divided into a fixed number of equal-sized time slots (in the experiment, a day is divided into 24 time slots, each with duration of 1 hour). A unified electricity bill is used throughout the chapter. We define price function,P [c][t], as the price of one unit of energy (kWh) for each utility company c at time slott. The price is decided by the utility company and pre-announced to homeowners. In addition, for every homeownerh,con[c][h][t] is the total energy consumption at time t if he chooses companyc. It can be easily observed that the equation below calculates the total energy cost for a certain homeowner: cost h [c][h] = X t P [c][t]con[c][h][t] (3.1) As stated above, our ultimate goal is to solve profit maximization problem for each utility compa- ny in oligopolistic market. But in the classical economics problems, between sellers and buyers, Chapter3 17 economists always give suggestions to the sellers based on the reaction of the buyers or vice versa because although the government would like to maximize the total social welfare, we still need to consider sellers and buyers as non-cooperative and always making decisions based on their own best solution [38]. This is also the case for energy users and utility companies, which means utility companies need to analyze homeowner’s reactions based on a given price function. It has been proved in [39] that both Cournot Model and Bertrand Model fail in the future architecture of smart grid. First of all, demand response is a key element of the smart grid technologies, which means the usual practice of power networks is matching supply to demand instead of matching demand to supply. For this reason, the Cournot Model, which is based on competition on the amount of output each industry will produce, is not applicable. On the other hand, the Bertrand Competition Model, which assumes consumers always choose the product with the lowest price, also turns out to be oversimplified, because the introduction of dynamic prices makes it hard for the consumers to determine which utility company really offers a better price, and also the customers may never be totally free to switch from one energy supply to another [38]. Hence authors in [39] used a modification of the Bertrand Competition Model which is based on the threshold cost of each energy consumer. But it may be unrealistic to assume that homeowners will randomly choose one utility company among those who offer a lower price than their threshold cost with an equal probability. According to the fact that electricity is a product with almost zero elasticity [38], in this paper, a modified economical demand function is used to determine the probability that each homeowner h will choose a certain utility companyc (prob[c][h]) Chapter3 18 prob[c][h] = e cost h [c][h] e cost h [c][h] + P i6=c e cost h [i][h] (3.2) which reveals that even considering the information asymmetry, energy consumers will still have a preference to choose the company who offers a cheaper price. To start, we consider each utility company has no cost function, i.e., the expected profit from a certain homeowner (profit[c][h]) can be calculated as follows: profit[c][h] = cost h [c][h]e cost h [c][h] e cost h [c][h] + P i6=c e cost h [i][h] (3.3) Notice thatcost h [c][h], which is an affine function of price functionP [c][t], is the only variable in the above equation given the price information of all other companies. If we use a single variable x to representcost h [c][h] and use a constant valueA to represent the cost summation of all other companies, the previous function can be written as: f(x) = x 1 +Ae x (3.4) It can be easily proved that this is a unimodal function, which means each utility company can adjust its own price function (sayx6=x 0 is the optimal solution) to maximize the profit. But we can also observe from the equation that the optimal price function of each company is determined by the constantA, which is a function of prices of other companies. On the other hand, the price decision made by each company will in turn affect the prices of its competitors. Thus a non-cooperative game is formed. Chapter3 19 To study the Nash equilibrium solution of the non-cooperative game, we first analyze the second derivative off(x), which is: f 00 (x) = Ae x (1 +Ae x ) 3 [2Axe x (x + 2)(1 +Ae x )] (3.5) It can be determined from the above equation that there exists an inflection point, sayx =x 1 , where f 00 (x 1 ) = 0. f(x) is a strictly concave function in the area wherex x 1 . Several experiments have been made and we observed thatx 1 4x 0 under different values ofA. This means during the profit maximization process,x can rarely go beyondx1. Considering the certain issue, a new functiong(x) is built with the following properties: 1.g(x) =ax 2 +bx +c, wherea,b,c are given constant values anda< 0 2.g(x 1 ) =f(x 1 ) 3.g 0 (x 1 ) =f 0 (x 1 ) And then we modifyf(x) to a new functionk(x) as follows: k(x) = 8 > > < > > : f(x) (xx 1 ) g(x) (x>x 1 ) (3.6) Property 1:k(x) is a strictly concave function. Proof: k(x) can be determined as continuous and differentiable due to the second and the third properties ofg(x).f(x) is a strictly concave function in the area wherexx 1 , sok 0 (x) is strictly Chapter3 20 monotonically decreasing in this area. In addition,g(x) is a well-known strictly concave quadratic function, sok 0 (x) is also strictly monotonically decreasing in the areax > x 1 . We can therefore determine k 0 (x) as strictly monotonically decreasing in the whole interval of definition and thus strictly concave. Considering the fact thatx 1 is already far from the price optimization area, this kind of modification will not influence the effectiveness of the final solution. The profit maximization problem for utility companies can be defined as follows: Maximize: profit total [c] = X h k(cost h [c][h]);8c (3.7) Subjectto: P [c][t]> 0; 8c;t P [c][t]max price; 8c;t (3.8) As companies are considered as non-cooperative among each other, we are interested in the exis- tence and uniqueness of Nash equilibrium points. Property 2: Utility companies have a unique Nash equilibrium point in the profit maximization problem. Proof: Note thatk(x) has been proved to be strictly concave, whilecost h [c][h] is an affine function of price functionP [c][t]. We can determine thatk(cost h [c][h]) is also strictly concave as a function ofP [c][t], and so is its summation function toh. Therefore, the profit maximization problem is a Chapter3 21 strictly concave n-person game. In this case, the existence and uniqueness of Nash equilibrium are directly resulted from the first and third theorem in [40]. Another observation is that as we consider no energy generation cost, the objective functions for different utility companies are totally symmetric. Property 3: At the Nash equilibrium point, each utility company has exactly the same price func- tion. Proof: Assume the price function of two utility companies, namelyc 1 andc 2 , are different in a Nash equilibrium point (sayP [c 1 ][t]6=P [c 2 ][t]). As the objective functions of these two companies are totally the same, another Nash equilibrium point should exist if they exchange their price functions, which contradicts the uniqueness of Nash equilibrium in property 2. The third property makes the problem much easier to solve, as we can simply assume the price function of all utility companies are the same after the first derivative of the objective function. The detailed algorithm is presented as follows: Algorithm 1: Unique Nash equilibrium point determination. for any one of the utility companyc do for each time slott do set @profit total [c] @P [c][t] j P [i][t]=P [c][t];8i6=c = 0 end Solve the linear equation set to getP [c][t] for eacht. end For all utility companiesj6=c, set price functionP [j][t] =P [c][t] for eacht. Chapter3 22 3.2 Model II: Considering Energy Generation Cost Model I is far from complete, as utility companies should also take into consideration the cost of electricity generated by different sources, which is the cost of generating electricity at the point of connection to a load or electricity grid [1]. As the trend of the future power networks is towards distributed smart grid, the capital, maintenance and distance cost will differ significantly based on the electricity generation type (e.g., steam-power station or solar-energy-power station) as well as weather, area and seasons. In this model, we useP e [c][t] to denote the energy cost function for each utility company at a certain time slott. The energy generation cost for homeownerh (denoted by cost e [c][h]) can be calculated as: cost e [c][h] = X t P e [c][t]con[c][h][t] (3.9) As has been stated in several previous papers like [35] and [39], the introduction of dynamic pricing is to regulate homeowner’s energy usage at peak demand hours and thus achieve a global energy saving. Considering that homeowners will always have an incentive to move their tasks from high- price hours to low-price times. We therefore set a negative correlation between energy price and energy usage during a certain amount of time by usingP expect [h] to represent the expected energy price for every homeowner. The homeowner’s energy cost can be reformulated as: cost h [c][h] = X t P [c][t]con[c][h][t] [ e 2 (1 + P [c][t] P expect [h] )e P[c][t] P expect [h] (1) +]; (3.10) Chapter3 23 where is a constant value between 0 and 1 which reflects the inelastic energy consumption of each homeowner. And consequently the profit maximization problem is modified as: Maximize: profit total [c] = X h (cost h [c][h]cost e [c][h])e cost h [c][h] e cost h [c][h] + P i6=c e cost h [i][h] ;8c (3.11) Subjectto: P [c][t]> 0; 8c;t P [c][t]max price; 8c;t (3.12) Once these more sophisticated factors are taken into consideration, the properties of the first model may no longer exist. However, based on game theoretic method, this problem can be solved by iterative local maximization of each utility company. And finally we can still achieve a Nash e- quilibrium point under a given initial condition (i.e., the initial price function of utility companies). Notice that a constant valued is needed to determine the endpoint. More precisely, the optimization process stopped when no utility companies can achieve a profit increase higher thand. Obvious- ly, the value of d makes a tradeoff between accuracy of the solution and run-time. The detailed algorithm is presented as follows: 3.3 Experimental Results To demonstrate the effectiveness of the proposed solutions, cases corresponding to the aforesaid pricing models are examined. Chapter3 24 Algorithm 2: Iterative solution of the above n-person game. InitializeP [c][t] for everyc andt. whilemax profit increase>d do for each utility companyc do Calculate currentprofit total [c]. for each time slott do set @profit total [c] @P [c][t] = 0 end Solve the equation set to getP [c][t] for each t under the given price functions of other companies. Calculate optimizedprofit total [c]. end Calculate themax profit increase among all companies. end For all utility companiesj6=c, set price functionP [j][t] =P [c][t] for eacht. In these simulations, the duration of a time slot is set to one hour. For this reason, power consump- tion of the tasks is determined with a granularity of one hour. The proposed algorithms have been implemented using Matlab and tested for random cases. For both models, the energy consumption at each time slot is fixed for every homeowner, and each homeowner has a probability to choose certain utility company based on the offered price functions. For model I, we calculated the unique Nash equilibrium point for an oligopolistic energy market containing 5 utility companies serving 100 homeowners. In order to show the effectiveness of the result, we assume an initial price function is given to all companies and they applied the calculated result one after another. Table 3.1 and Table 3.2 show the comparison between different utility companies under a high initial price and a low initial price, respectively. The above tables verify that regardless of how high or low the initial price is set to, utility companies will always have incentives to adjust the price function and thus maximize their own profits. Their final price strategy turned out to be the same as the calculated result, which has been proved to be the unique Nash equilibrium solution. Note that the Nash equilibrium point does not guarantee the globally optimal solution. This can also be observed from the first table where the profit of Chapter3 25 each utility company is higher if all of them choose to retain a high energy price. But this kind of cooperation can easily be broken since each company can achieve a profit increase by lowering its price. The result also explains why energy consumers can benefit from the price competition. For model II, we assume different energy generation costs for utility companies and use the iter- ative Algorithm 2 to solve the profit maximization problem for each company. Table 3.3 is the comparison between the initial profit and final profit. TABLE 3.1: Profit comparison of Model I under a high initial price Number of utility Profit of a company Profit of a company companies that applied which applied the which did not apply the calculated price calculated price the calculated price 0 - 47.0 1 53.9 33.1 2 41.5 25.6 3 33.8 20.9 4 28.6 17.7 5 24.7 - TABLE 3.2: Profit comparison of Model I under a low initial price Number of utility Profit of a company Profit of a company companies that applied which applied the which did not apply the calculated price calculated price the calculated price 0 - 11.8 1 14.2 13.0 2 15.9 14.5 3 18.0 16.5 4 20.8 19.1 5 24.7 - TABLE 3.3: Profit comparison of Model II for different utility companies Company Initial profit Final profit 1 42.1485 27.4585 2 40.6060 24.9549 3 42.9249 28.6499 4 41.0160 26.0111 5 41.1806 26.0680 Chapter3 26 FIGURE 3.1: Initial and final price functions for each utility companies Figure 3.1 shows the change of price function for utility companies. We assume the energy price is initially set to be high, but it is finally brought down due to competition. Companies turn out to have different final price functions due to different energy generation cost. Figure 2 illustrates the change of profit for utility companies as a factor of simulation steps. It can be observed that each company achieves a profit increase when it is running the local optimization step, while suffers from a profit decrease when other companies are running their own optimization. The curves turn out to be stable in the end, which means each company can no longer pursue profit increase by adjusting its price function. In other words, a Nash equilibrium solution is achieved. Runtime of the proposed heuristic for all the 5 utility companies is about 1 second and 10 minutes for the first and second model respectively both on a machine with a dual core processor with frequency of 2.80 GHz. This run time makes it feasible to utilize our models real-time. Chapter4 27 FIGURE 3.2: Profit change steps for each utility companies 3.4 Chapter Summary Two models were presented to tackle the profit maximization problem of non-cooperative utility companies in oligopolistic market. The models are under different assumptions including its prob- lem formulation, proof of properties and optimal solution. In each model, utility companies are considered as non-cooperative, i.e., always making decisions based on their own best solution. A probability-based calculation is utilized based on consumers’ reaction to price functions from dif- ferent companies. The models were implemented and tested with some arbitrary test schemes. The results confirm that our designed algorithms lead to Nash equilibrium solutions for both problems, which means that all utility companies achieve their local optimal price functions. The real-time simulation strengthened the reliability of our proposed solution on price function with an acceptable runtime. Chapter 4 Negotiation-Based Task Scheduling to Minimize User’s Electricity Bills under Dynamic Energy Prices In this chapter, our approach to minimize users’ electricity bills under dynamic prices takes off-grid residential PV energy generation into consideration and uses an iterative approach that is inspired by the Triptych Field-Programmable Gate Array (FPGA) routing software developed in [41, 42]. The negotiation-based task scheduling algorithm in this work differs from the previous works in several aspects including the formulation of rip-up and retry as well as the introduction of off-grid PV energy generation. The concept of congestion is also introduced in the proposed algorithm which dynamically adjusts the congestion degree based on the historical scheduling results as well as the total power consumption in each time slot. Effective PV prediction algorithms can be applied 28 Chapter4 29 to predict the PV power generation profile ahead of time, however, to focus on the task scheduling instead of predicting the PV power, we assume part of the user’s energy is supplied by the off-grid PV system and we use PV power generation profiles measured at (i) Duffield, V A, measured in the year 2007, and (ii) Los Angeles, CA, measured in the year 2012 [43]. To the best of our knowledge, this is the first work that introduces off-grid PV energy and congestion cost into the demand-side task scheduling problem under dynamic pricing. 4.1 System Model and Cost Functions In this paper, we consider a single user who pays a unified electricity bill that covers the total energy consumed by a group of tenants in research laboratories, households, office spaces, fac- tories, warehouses, etc. A negotiation-based task scheduling algorithm is developed to minimize the user’s electricity bills determined by a cost function comprised of a time-dependent price, a power-dependent electricity price and an inconvenience cost. Without loss of generality, we focus on the task scheduling problem in one day in this paper and our approach can be easily extended to a longer period such as week, month, season and year. We adopt a slotted time model, i.e., all system cost parameters and constraints as well as scheduling decisions are provided for discrete time intervals of constant length. In this model, the entire task scheduling period is divided into a fixed number of equal-sized time slots and the time resolution is defined as the largest size of time slot that all timing properties of each task can slot into. For example, if every task starts at the beginning of an hour, lasts integer multiples of half hours with deadline at the end of an hour, and the price function updates every 20 minutes, the time resolution Chapter4 30 should be 10 minutes. Each day is thus divided into T time slots and we useT = 24 and = 60 minutes in our experiments. For the user of interest, we assume that there are a number of tasks that should be executed daily. These tasks are identified by indexi. The set of task indices isT =f1;:::;Ng whereN is the total number of tasks for the energy user. Each task i has an earliest start time S i , a deadline E i and a fixed operation length D i to complete the task. We denote the power consumption of task i in time slot t by p i (t) as t ranges from 1 to T and i ranges from 1 to N. We assume the tasks are non-interruptible, i.e., each task has to operate in a number of consecutive time slots. After the scheduling, task i actually starts at time i and completes at i +D i . It is preferable to start taski no earlier thanS i and complete it no later thanE i . In the previous papers such as [35], the earliest start time and deadline are hard constraints of a task. However in our model, we make the scheduling more realistic by allowing taski to be scheduled outside the preferable time window [S i ;E i ] but with an incurred inconvenience costI i which is determined by the user. The inconvenience cost represents the penalty to schedule task i outside the preferable time window [S i ;E i ]. Besides, each task consumes electricity according to a known power dissipation profile, i.e., the power consumption of taski will follow a known profile regardless of the start time i . The timing specifications, inconvenience cost and power profile for all tasks are assumed to be provided by the user before the beginning of the day and keeps unchanged during the day. Figure 4.1 gives an example of household task scheduling solution. Each bar represents a task which occupies a number of time slots. Any task that crosses the midnight can be divided into two equivalent parts, e.g. the two periods of air conditioner represents the entire air conditioner period crossing the midnight from 3pm to 6am. Chapter4 31 FIGURE 4.1: Example of a household task scheduling problem We use a combined price model (t;!(t)) comprised of a time-of-use (TOU) price that is de- termined by the time slot t, and a power-dependent price that depends on !(t) which represents instantaneous power consumption drawn from the grid in time slott, i.e., the grid power. The price function(t;!(t)) is assumed to be monotonically increasing with respect to!(t), which means the price will rise if more power is consumed in a certain time slott. Therefore, this combined price model incentivizes the user to shift loads from the peak hours to off-peak periods. However, going too far is as bad as not going far enough. If most users shift their tasks in the same way, the power plant might fail to match the load during some time periods which are originally considered as off- peak time. Considering this, we assume the utility company also sets a maximum power constraint in each time slot for the user so as to avoid potential outage. This maximum power constraint is denoted by(t) for time slott and it is considered as a soft upper bound in the proposed algorithm because it is not practical for the utility company to set a hard maximum power constraint in reality. The detailed meaning of the soft bound is that the energy price will rise dramatically when the total power consumption in a time slott exceeds(t), and the user can violate this bound but with an additional cost. By applying this soft bound, the chance to see an outage is much lower. Chapter4 32 As stated in the introduction, a part of the energy is generated by the off-grid residential PV system. The PV power generation available to the user in time slott is denoted bypv(t). We assume that the user is not equipped with energy storage equipment, and therefore, the unused PV energy cannot be stored and is wasted. The investment and maintenance of the PV system is not in the scope of this work and we assume this part of cost is covered by the user, and thus, the PV power generation can be considered to be free to the user. A wise strategy is to utilize generated PV power as much as possible. After introducing the off-grid PV power, the grid power!(t) consumed by the user in time slott is calculated by: !(t) = 8 > > > > > < > > > > > : 0; pv(t) N X i=1 p i (t) N X i=1 p i (t)pv(t); pv(t)< N X i=1 p i (t) (4.1) Using the above definitions, the energy user’s cost minimization problem can be modeled as fol- lows: Cost Minimization Problem for an Energy User. Find: the optimal start time i for 1iN. Minimize: TotalCost = T X t=1 (t;!(t))!(t) + N X i=1 I i ( i ) (4.2) where the inconvenience cost for taski is given as Chapter4 33 I i ( i ) = 8 > > < > > : 0; S i i E i D i I i ; otherwise (4.3) Subjectto(SoftBound 1 ): !(t)(t) (4.4) The available PV energy pv(t) in time slott is efficiently utilized when the following condition is satisfied: N X i=1 p i (t)pv(t) (4.5) 4.2 Negotiation-Based Energy Cost Minimization In this section, a negotiation-based approach inspired by an adaptive routing algorithm in commer- cial FPGAs [41, 42] is developed to find a suitable solution to the cost minimization problem (task scheduling problem) described in Section II. This section is organized as follows: part A provides a general description of the congestion-based routing approach in FPGAs; part B explains the moti- vation to introduce the concept of negotiation-based routing method to the task scheduling problem; part C presents the proposed negotiation-based task scheduling algorithm. 1 The scheduler is encouraged to satisfy the soft bound in all time slots. However, if there is no such solution, the scheduler can violate this soft bound and the penalty in energy cost is already captured in the objective function (4.2). Chapter4 34 4.2.1 Negotiation-Based Routing Algorithm in FPGAs Routing is an important step of the Computer Aided Design (CAD) process that is necessary to implement a circuit in FPGA. The goal of a routing algorithm is to find a feasible solution which connects input signals of digital logic elements to output signals according to the gate-level logic descriptions using the limited routing resources such as physical wires and switches. On top of that, an effective routing algorithm tries to reduce the length of the longest path (i.e., the critical path) that determines the performance of the circuit. As explained in [41], the main constraint in the routing problem is that different signals cannot share the same routing resources, e.g. two signals cannot share the same physical wire in an FPGA. In reference work [41], congestion in a routing problem indicates sharing of routing resources. As stated above, a routing solution must resolve congestion in every routing resource node. In [41], a signal router that only considers performance chooses the shortest path for each signal, which generates the fastest circuit but leads to a significant amount of congestions. Thus, a congestion cost function is introduced to each routing resource node and a global router guides the signal router to avoid those resources with high congestion cost. A solution is found when there is no congestion in the circuit. Besides the introduction of the concept of congestion, the algorithm in [41] is implemented in an iterative manner. In each iteration, every signal is ripped-up and re-routed based on the latest congestion cost that is updated at the end of the previous iteration. To avoid the situation where a resource node is always shared, a history term is introduced to the congestion cost formula of every resource node. The effect of the history term is to permanently increase the cost of using congested Chapter4 35 nodes so that routes through other nodes are attempted [41]. In some sense, the rationality behind this rip-up and re-route scheme is to guide the signal router to look at a larger scope of possible scheduling paths because a fixed signal may block a large number of potential routing solutions. The negotiation-based routing algorithm provides the best solution known so far for the FPGA routing problem. 4.2.2 Motivation to Introduce Negotiation-Based Routing Method to Task Schedul- ing Problem The cost minimization problem described in Section 4.1, also known as the task scheduling prob- lem, is an integer nonlinear programming problem subject to nonlinear constraints. In this kind of problems, no theoretically optimal results can be derived in polynomial time, nor are they likely to ever be available [44]. Using nonlinear optimizers to find the optimal solution in a task scheduling problem with a large set of unknowns is impractical because of the unacceptable run time [45]. Therefore, instead of finding the optimal task scheduling solution, we seek for a good enough solu- tion so that the implementation of the proposed algorithm is feasible with respect to computational complexity and the result is much better than the baseline approach. Let us first analyze and conclude the similarities between the FPGA routing problem and the task scheduling problem: 1. Finding the optimal solution incurs an extremely long run time, thus in both problems it is reasonable to find a suitable solution instead of the optimal one. Chapter4 36 2. Certain times of rip-up and retry are required to find a suitable solution, and hence, an iterative approach should be applied in both problems. 3. There are limited resources in both problems with a cost in each resource node. The cost is delay and congestion in the FPGA routing problem, while the cost is price (increase) in the task scheduling problem. Despite the similarities stated above, the task scheduling problem is quite different from the FP- GA routing problem. Different tasks in the task scheduling problem can share a time slot even if the power consumption drawn from the grid (i.e., the grid power) exceeds the power limit in the time slot, whereas in the routing problem resources (physical wires and switches) cannot be shared by different signals. Besides the difference in resource sharing, these two problems have different objective functions. In the task scheduling problem, the goal is to schedule all tasks in T =f1;:::;Ng so that the total energy cost function (4.2) is minimized, whereas the objective function in the routing problem is to minimize the critical path delay represented by the sum of delay values of all the routing resources in that path. Moreover, the price function(t;!(t)) in the cost minimization problem is a function of time slott and instantaneous grid power consumption !(t), and hence the price function of a time slot is more complex than the delay function of a rout- ing resource. Based on the above differences, we cannot directly map the FPGA routing problem into to the cost minimization problem. However, we apply the concept of congestion used in the routing problem to help solve the cost minimization problem. Chapter4 37 4.2.3 Negotiation-Based Task Scheduling Algorithm A Negotiation-Based Task Scheduling (NBTS) algorithm, which is inspired by the congestion- based FPGA routing algorithm in [41], is proposed to find a suitable solution with a low computa- tional complexity. At the beginning of the day, the user provides the properties of each taski, namely,S i ,E i ,D i ,I i and its power profile, and the utility company provides the price function and power constraint in each time slott. The measured PV power generationpv(t) is also given for each time slott [43]. These are the inputs to the problem. The set of scheduled start times of all tasks in is denoted by =f 1 ;:::; N g. This is the output of the proposed algorithm. In the context of our problem formulation, a suitable solution is described as a schedule in which most time slots utilize the PV energy efficiently and no slots or only a few slots exceed the power constraint set by the utility company. As mentioned in Section 4.1, an iterative approach is applied in the proposed algorithm. We setK as the maximum iteration times. At the beginning of thei-th iteration, all the tasks are ripped-up, and then we schedule task 1 to task N one by one. To introduce the iterative method, let! j i (t) denote the total grid power consumed in time slott after thej-th task has been scheduled in thei-th iteration wheret2 [1;T ],j2 [1;N] andi2 [1;K]. In the remaining subsection, we will introduce the concept of congestion during the discussion of each step in the proposed algorithm. First, we analyze the 1st iteration and introduce the concept of intra-iteration congestion (to be precisely defined later). Before we schedule the 1st task, ! 1 1 (t) is zero in all slots since no task is scheduled. We find the best possible start time 1 to schedule task 1 so that the overall energy Chapter4 38 cost P T t=1 (t;! 1 1 (t))! 1 1 (t) +I 1 ( 1 ) is minimized. Then the final grid power profile! 1 1 (t) for t2 [1;T ] is calculated based on the power profile of task 1. The time slots occupied by task 1 are more likely to see a higher energy price since the price function in each slott is monotonically increasing with respect to!(t). Therefore, it is reasonable to lower the priority of these slots in the following steps in this iteration. We define a congested time slot as the time slot that is occupied by at least one task within the iteration. To quantify the congestion degree of a time slot, we define the intra-iteration congestion termR(t) as the number of tasks that have been scheduled to occupy time slott in one iteration. Before task 1 is scheduled, R(t) is 0 in all slots and after task 1 has been scheduled,R(t) becomes 1 in those slots chosen by task 1. Before we schedule thej-th task in the 1st iteration, the total grid power in each slott is! j1 1 (t), the price in each slot is(t;! j1 1 (t)), and the value ofR(t) is in the range of 0 toj1. We will find the best start time of thej-th task to minimize the congestion cost of thej-th task. The congestion cost is defined as follows: CC = T X t=1 ((t;! j 1 (t))! j 1 (t)(t;! j1 1 (t))! j1 1 (t)) (A 0 R(t) + 1) +I j ( j ) (4.6) where P T t=1 ((t;! j 1 (t))! j 1 (t)(t; j1 1 (t))! j1 1 (t)) is the cost increase after scheduling taskj in the 1st iteration, and the termA 0 R(t) (A 0 is a positive weight coefficient) is added to guide the scheduler to avoid congested slots. The j value that minimizes (4.6) is chosen as the start time for taskj, and then the final value of! j 1 (t) for each time slott is updated based on the chosen j . Chapter4 39 At the end of the 1st iteration, we obtain a schedule that consists of j forj2 [1;N]. When the power constraint is tight, it is possible that the total energy cost is not minimized and some time slots fail to use PV energy efficiently even though the termR(t) has been introduced to avoid congestion within the 1st iteration. Thus, we need more iterations to achieve a lower total energy cost. In the subsequent iterations, it is reasonable to lower the priority of those congested slots that result in high energy cost in the 1st iteration. Meanwhile, it is also reasonable to increase the priority of the time slots in which PV energy generation has not been utilized efficiently in the 1st iteration. To guide the scheduler to achieve these two goals, we introduce two inter-iteration congestion terms. The 1st term is denoted byH 1 (t), which represents the total times in the previous iterations when the power constraint has been exceeded in time slott. The 2nd term is denoted byH 2 (t) that is the total times in the previous iterations when PV energy generation is not fully utilized in time slot t. The inter-iteration congestion terms are updated at the end of each iteration (whereas the intra- iteration term is updated within one iteration). After introducing the two inter-iteration congestion terms, we integrate the three congestion terms into C slot (t) =A 0 R(t) +A 1 H 1 (t)A 2 H 2 (t) + 1 (4.7) whereA 1 andA 2 are positive weights ofH 1 (t) andH 2 (t), respectively 2 . Thus, when the scheduler is scheduling thej-th task to start execution at time j in thei-th iteration, the congestion costCC 0 2 The proposed algorithm can be applied to a user without off-grid PV energy supply by settingA2 = 0. Chapter4 40 of thej-th task is then redefined by CC 0 = T X t=1 ((t;! j i (t))! j i (t)(t;! j1 i (t))! j1 i (t))C slot (t) +I j ( j ) (4.8) In (4.7) and (4.8), the term A 1 H 1 (t) guides the scheduler to lower the priority of the time slots which violate the power constraint in the previous iterations. Meanwhile, the termA 2 H 2 (t) slightly lowers the cost of the time slots in which PV energy generation are not used efficiently in previous iterations. The j value that minimizes the objective function (4.8) is chosen as the start time for taskj. After that, we update the final value of! j i (t) fort2 [1;T ] based on the chosen j before scheduling the next task. To solve the problem that a number of suitable solutions are blocked when a task always occupies the same time slots, we introduce the 3rd inter-iteration congestion termh(j;t) which indicates the number of times in the previous iterations when taskj occupies time slott. Initially,h(j;t) is 0 for all tasks in all time slots. This term permanently increases if this taskj occupies the time slott in one iteration, and after several iterations, the task may give up those time slots due to the increasing congestion cost. Of note, the difference betweenh(j;t) and the above-mentioned three congestion terms is that h(j;t) is introduced to each task and each time slot while the other three terms are integrated to each time slot. When the scheduler is scheduling the task j in the i-th iteration and it tries to set j as the start time, the final congestion costCC 00 is recalculated by CC 00 = T X t=1 ((t;! j i (t))! j i (t)(t;! j1 i (t))! j1 i (t)) (ah(j;t) + 1)C slot (t) +I j ( j ) (4.9) Chapter4 41 wherea is the weight ofh(j;t). The j value that minimizes (4.9) is chosen, and we update the final value of! j i (t) fort2 [1;T ] based on the chosen start time before scheduling the next task. The proposed algorithm terminates when the total energy cost is not decreased inL consecutive iter- ations or theK-th iteration is completed. The computational complexity of the proposed algorithm isO(KNT 2 ). The pseudo code for the proposed algorithm is shown as follows: Algorithm 3: The proposed negociation-based task scheduling algorithm. Initialize tasks, PV profile, price functions, inconvenience cost functions, constraints and congestion terms Set iteration counteri = 0 while Total energy cost is decreased for L iterations andi<K do i =i + 1 Rip up all tasks for each taskj do for each time slott2 [1;TD j + 1] do Set the start time j =t end Choose the j that minimizes (4.9) Update the intra-iteration congestion termR(t) end for each time slott do UpdateH 1 (t) andH 2 (t) for each taskj do Updateh(j;t) end end Calculate the overall energy cost end Return 4.2.4 Experimental Result In this section, to demonstrate the effectiveness of the proposed NBTS algorithm, cases correspond- ing to the abovementioned pricing models are examined. As mentioned above, the duration of a time slot is set to one hour, and we examine the task scheduling problem in one day. For this reason, Chapter4 42 the duration of a task is integer multiples of one hour and it cannot exceed 24 hours. Moreover, power consumptions of the tasks are determined with a granularity of one hour. We assume the input data from the user are given at the beginning of the day and remain unchanged during the day. The preferable start time, end time, duration and inconvenience cost are generated arbitrarily for each task. The power profile of each task is randomly generated based on power consumption data about real appliances. Besides, PV profile of that day is predicted using effective PV power generation prediction algorithms. We examine a household task scheduling case. The user owns a 6,000-watt (6 kW) PV system that can provide around 6 kilowatts of electricity per hour under optimum conditions. According to Phillips, the Florida Solar Energy Center has determined there are 1,489 optimum solar generating hours per year, accompanied with hours of less than optimum generation [46]. Therefore, the 6 kW PV system has around 4 hours per day on average under optimum generation conditions. We also assume the input data from the utility company are available at the start of the day and keep unchanged through the day. The utility company will set the TOU-dependent price and the power limit for each time slot. The price function presented in the task scheduling problem is assumed to be monotonically increasing. The part of the power consumption that exceeds the power limit will incur twice higher energy prices in our experiments. The baseline is implemented using the greedy algorithm that schedules each task to start at the best possible time slot based on the TOU-dependent price component of the energy pricing model and the total power consumption of all previously scheduled tasks. The baseline algorithm will check whether the power constraint is satisfied in each time slot and it will reschedule from the 1st task that has no possible start time to satisfy the power constraints in all time slots. For example, in a Chapter4 43 task scheduling problem with 5 tasks, if tasks 1, 2 and 3 have been scheduled and task 4 cannot be scheduled to make sure that the power constraint is satisfied in every time slot, in the next iteration, task 4 will be scheduled with the highest priority. Of note, our baseline is not aware of the part of PV energy, and in order to make the comparison fair, the peak PV energy cannot cover more than 6 tasks in a time slot. Table 4.1 shows the simulation results of the proposed NBTS algorithm compared to the baseline greedy optimization method when the power limit in each slot is set so that greedily scheduling each task to its best start time will violate the power constraint. The total task number N is set as a parameter of the program and we arbitrarily generate 10,000 cases for task number ranging from 5 to 50 and report the average performance in Table 4.1. In order to test the effectiveness of the rip-up policy in both algorithms, we consider a task scheduling problem that has 25 tasks and arbitrarily generate 10,000 cases for different degrees of power constraint, i.e., medium, tight and very tight, and define a figure-of-merit called error rate that is the number of cases whose schedule violate power constraint in at least one time slot over the total number of the simulated cases. The results are shown in the Table 4.2. It can be seen from Table 4.1 that the proposed algorithm can achieve 21.6%-51.8% improvement whenN is below 40 and more than 15% improvement whenN is from 40 to 50. Meanwhile, the simulation time of the NBTS is not large. Moreover, the proposed algorithm is effective when the power constraint in each time slot is tighter, and it achieves less than 6% error rate under the very tight constraint condition while the greedy baseline has 61.04% error rate. Chapter5 44 4.3 Chapter Summary The overarching goal of this chapter was to develop an effective algorithm to minimize the user’s electric bills under dynamic energy prices. The concept of congestion and off-grid PV power gener- ation was introduced to the proposed negotiation-based task scheduling algorithm, and the proposed algorithm was implemented and tested for different test schemes. The results demonstrated the a- bility of the proposed algorithm for up to 51.8% energy cost saving compared to a greedy baseline method. TABLE 4.1: Performance and time complexity of the negociation-based task scheduling algorithm and the baseline Task Average total price Average total price Average performance Average run-time number of baseline (cents) of NBTS (cents) increase factor of NBTS (s) 5 103.94 68.46 1.518 0.004 10 227.43 154.69 1.470 0.010 15 372.71 260.44 1.430 0.018 20 530.19 391.91 1.352 0.029 25 726.54 560.11 1.297 0.041 30 954.16 746.20 1.278 0.060 35 1161.51 955.03 1.216 0.086 40 1403.74 1171.31 1.198 0.115 45 1657.40 1417.57 1.169 0.108 50 1931.87 1649.42 1.171 0.153 TABLE 4.2: Performance and average error rate of the negociation-based task scheduling algorithm and the baseline Constraint Average performance Average Error Average Error Intensity increase factor Rate of Baseline Rate of NBTS Medium 1.297 11.40% 0.76% Tight 1.389 27.29% 1.59% Very Tight 1.593 61.04% 5.77% Chapter 5 An Optimal Energy Co-Scheduling Framework for Smart Buildings In this chapter, we consider the problem of co-scheduling HV AC control and HEES management for energy-efficient smart buildings as shown in Figure 5.1, aiming to minimize the total cost while maintaining the temperature within the comfort zone for building occupants. In this problem, we explicitly take into account the degradation of battery state-of-health (SoH), which is defined as the ratio of full charge capacity of an aged battery to its designed (nominal) capacity, during the charging and discharging process based on an accurate SoH modeling. We also consider various energy loss components including power dissipation in the power conversion circuitries as well as the rate capacity effect of the batteries. In addition, a time-of-use electricity pricing policy is adopted in our model. The objective function is to minimize the summation of the building’s electricity bill and the extra cost associated with the aging of the batteries used in the HEES system. 45 Chapter5 46 FIGURE 5.1: The energy co-scheduling framework. 5.1 System Component Models In this paper, we consider an energy-efficient smart building with both HV AC control and HEES systems. We use a general HEES system architecture in our paper, which consists of a battery bank and a supercapacitor bank 1 [19]. Each EES element has special performance metrics. Com- pared with batteries, supercapacitors have nearly 100% charging and discharging efficiencies and an order-of-magnitude longer cycle life, but at the same time they are more expensive and suffer from self-discharge. Considering that the photovoltaic (PV) system, which converts solar radiation into electricity, is increasingly becoming a key power source and is believed to play an important role in the process of transferring to the future low-carbon power system [47, 48], we assume that the building is also equipped with a PV power generation system as well. Our objective is to minimize the combined 1 Our proposed energy co-scheduling algorithm can also handle more complicated HEES systems. Chapter5 47 cost of electricity bill and battery SoH degradation of the building by co-scheduling the air flow of the HV AC control system and the charging/discharging process of the HEES system. 5.1.1 Model of Building Power Flow and Power Conversion FIGURE 5.2: Power flow of an energy-efficient smart building including HV AC control, PV sys- tem, HEES system, building load and the external power grid. The directions of arrows represent the directions of the power flow. The block diagram of the power flow in a typical energy-efficient smart building is shown in Figure 6.2. The PV system, providing a power of P pv , and the HEES system, in which the charging powers of the EES elements are denoted byP bat andP cap , are connected to the building DC power bus through unidirectional and bidirectional DC-DC converters, respectively. Notice thatP bat and P cap can be positive, negative or zero. A positive value means the EES element is being charged, a negative value indicates discharging from the EES element, and zero represents no charging or discharging operation. We useP hees to denote the total charging power of the HEES system which Chapter5 48 is calculated using the equation: P hees = bat P bat + cap P cap (5.1) And the conversion rate of the EES elements bat and cap are based on the direction of the charging power flow and can be calculated by: bat = 8 > > > < > > > : 1 1 ; P bat 0 1 ; P bat < 0 (5.2) cap = 8 > > > < > > > : 1 2 ; P cap 0 2 ; P cap < 0 (5.3) The DC power bus is further connected to the residential AC power bus via bidirectional AC-DC interfaces (e.g., inverters, rectifiers, etc.). The power consumption from HV AC control system (P hvac ) is the part that we need to manage in this paper. The building AC load on the AC bus (P load ) corresponds to other building tasks such as lighting, fire and security. The AC power bus is further connected to the state-level or national smart grid and the power from the gridP grid is related to the building’s electricity bill. We consider realistic power conversion circuits (i.e., the power conversion efficiency is less than 100%) in this work, and use 0 to 3 to denote the power efficiencies of the converters between the PV system and the DC power bus, the converters between the EES element and the DC power bus, and the AC-DC interface connecting the DC power bus and the AC power bus. Typical conversion efficiency values are in the range of 85% to 95% [49, 50]. Chapter5 49 There are three operation modes in the above-mentioned power flow. In the first mode, the HEES is discharging, and thus, the building loads are supplied simultaneously by the grid, the PV system as well as the battery storage. In the second mode, the HEES system is charged by the PV system only, and the surplus PV power generation is used to supply HV AC system and other building loads. In this mode, part of the power generated by the PV system flows from the DC bus to the AC bus. Both the first mode and the second mode lead to a power flow from the DC bus to the AC bus. In the last mode, the PV system is not sufficient for charging the HEES system. Thus, the HEES system is simultaneously charged by the PV system as well as the converted power from grid, resulting in a power flow from the AC bus to the DC bus. In general,P grid can be calculated using the following equation: P grid = 8 > > > < > > > : P hvac +P load 3 ( 0 P pv P hees );P hees 0 P pv P hvac +P load 1 3 ( 0 P pv P hees );P hees > 0 P pv (5.4) For the convenience of expression, the relationship in Eqn. (5.4) will be referred to asP grid = f grid (P hvac ;P load ;P pv ;P hees ) in the rest of this paper. 5.1.2 Model of HV AC control HV AC is the technology which aims at providing human thermal comfort and acceptable indoor air quality. Previous work [51] used an encoded variablesT ctrl (t) andu(t) to represent the room temperature state and air flow at a certain timet. Assuming the HV AC system keeps an air flowu(t) for a certain time period t, the room temperature updates according to the following equation: T ctrl (t + t)=A n T ctrl (t) +B n u(t) +E n D(t); (5.5) Chapter5 50 whereA n ,B n andE n are HV AC specific parameters which are functions of t,T ctrl (t) is a state (column) vector comprised of five elements, andD(t) is the disturbance from external sources (e.g. solar radiance, wind, etc.) at timet. The five elements inT ctrl (t) correspond to the temperature of the four surrounding walls of a room and the temperature of the room itself. To maintain the building temperature within a comfortable zone, at any timet,T ctrl (t) needs to satisfy the following constraint: T LB (t)C T n T ctrl (t)T UB (t); (5.6) whereC T n = [0 0 0 0 1] is used to find the temperature of the room, andT LB (t) andT UB (t) are the lower bound and upper bound of the comfort zone in terms of air temperature at a certain time. The air flow also needs to follow the constraint: U LB u(t)U UB ; (5.7) meaning that the maximal achievable value ofu(t) isU UB , while the minimal achievable value of u(t) isU LB . Given the HV AC air flow, its power consumptionP hvac can be calculated using the equation: P hvac (t) = c 1 u(t) 3 +c 2 u(t) 2 +c 3 u(t) +c 4 P AC (5.8) where c 1 , c 2 , c 3 and c 4 are HV AC power coefficients [51] and P AC represents the standard AC power in the system. Chapter5 51 5.1.3 Model of HEES system The advantage of HEES system is that it consists of multiple banks of inhomogeneous EES elements with different characteristics so that the strengths of different EES elements can be exploited and a combination of superior performance metrics can be achieved [20]. In this paper, we consider two kinds of power loss of the batteries used in the HEES system. The first one is the rate capacity effect, which affects the energy efficiency during the battery charging or discharging process; The second one is the SoH degradation, which models the degradation of full charge capacity of an aged battery to its nominal capacity. Compared with batteries, supercapacitors have nearly 100% charging and discharging efficiencies and an order-of-magnitude longer cycle life, but suffer from self-discharge. 5.1.3.1 Battery Rate Capacity Effect The most critical effect that causes power loss in the storage system of a battery is the rate capacity effect [52]. 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 define discharging efficien- cy of a battery, denoted by rate;d , as the ratio of the battery’s output current to the degradation rate of its stored charge amount. The rate capacity effect specifies that the discharging efficiency of a battery decreases with the increase of the battery’s discharging current. The charging efficiency of a battery, denoted by rate;c , is defined similarly. Chapter5 52 According to Peukert’s formulae [53], rate;d and rate;c are described as functions of the charging currentI c and the discharging currentI d , respectively, as follows: rate;c (I c ) = 1 (I c =I bat;ref ) c ; rate;d (I d ) = 1 (I d =I bat;ref ) d (5.9) where c and d are Peukert’s coefficients with typical values in the range of 0:1-0:3, andI bat;ref denotes the reference current level for charging and discharging which is typically the current that can fully deplete the battery in 20 hours. It can be observed that the efficiency values rate;c and rate;d in Eqn. (5.9) will be higher than 100% ifI c < I bat;ref orI d < I bat;ref , which contradicts common sense. To address this issue, we use a slightly modified version of Peukert’s formulae such that rate;c and rate;d will be saturated at 100% when the charging/discharging current is low. We denote the change rate of the stored energy in the battery by P bat;int , which can be positive (charging the storage), negative (discharging from the storage), or zero. The relationship between P bat;int and the input power of the battery, denoted byP bat , is given by P bat = 8 > > > > > > > > < > > > > > > > > : V bat I bat;ref P bat;int V bat I bat;ref c ; P bat;int V bat I bat;ref >1 V bat I bat;ref jP bat;int j V bat I bat;ref d ; P bat;int V bat I bat;ref <1 P bat;int ; otherwise (5.10) whereV bat is the battery terminal voltage and is assumed to be (nearly) constant, and coefficients c and d are given by c = 1 1 c ; d = 1 1 + d (5.11) Please note thatV bat andI bat;ref are always positive. Chapter5 53 For the convenience of expression, the relationship in Eqn. (5.10) will be referred to as P bat = f bat (P bat;int ) in the rest of this paper. One important observation is that this function is a convex and monotonically increasing function over the input domain1 < P bat;int <1. An example off bat () with c = 1:1 and d = 0:9 is shown in Figure 5.3, from which one can see that rate capacity effect makes the charging/discharging process less efficient. FIGURE 5.3: Relationship betweenP bat andP bat;int considering the rate capacity effect. 5.1.3.2 Battery SoH Degradation Another significant portion of power loss for batteries [19] is due to the state-of-health (SoH) degra- dation, i.e. the charge capacity of a battery will slowly degrade as the battery ages. The amount of SoH degradation, denoted byD SoH , is defined as: D SoH = C nom full C full C nom full 100% (5.12) Chapter5 54 whereC nom full is the nominal charge capacity of a new battery andC full is the charge capacity of the battery in its current state. Considering the SoH degradation, the state-of-charge (SoC) of a battery is defined as follows: SoC = C bat C full 100% (5.13) whereC bat is the remaining charge stored in the battery. The SoH degradation of a battery is d- ifficult to estimate because it is related to a set of long-term electrochemical reactions inside the battery. These effects strongly depend on the operating condition of the battery such as charging and discharging currents, number of charge-discharge cycles, SoC swing, average SoC, and oper- ating temperature [54, 55]. Although very accurate, some electrochemical models such as the one proposed in [56] are too complicated to be applied to an optimization problem like ours. In this paper, we use the SoH degradation model proposed in [57], which calculates the SoH degradation based on (partial) charging cycles and shows a good match with real data. In the applied model, a charging cycle is defined as a process of charging a battery cell from SoC low to SoC high and discharging it fromSoC high toSoC low after that. The average SoC and the SoC swing during one charging cycle, denoted bySoC avg andSoC swing , respectively, are defined as follows: SoC avg = (SoC low +SoC high )=2 (5.14) SoC swing =SoC high SoC low (5.15) Chapter5 55 The SoH degradation of one charging cycle, denoted byD SoH;cycle , is calculated as D 1 =K CO exp[(SoC swing 1) T ref K ex T B ] + 0:2 life D 2 =D 1 exp[4K SoC (SoC avg 0:5)] (1D SoH;current ) (5.16) D SoH;cycle =D 2 exp[K T (T B T ref ) T ref T B ] whereK co ,K ex ,K SoC , andK T are battery specific parameters, andT B andT ref are the battery’s operation temperature and reference temperature, respectively. is the duration of this charging/dis- charging cycle, and life is the expected lifetime of this battery.D SoH;current represents the current SoH degradation level of the battery. We useD SoH;cycle (SoC swing ;SoC avg ) to denote the relationship betweenD SoH;cycle , SoC swing , andSoC avg . The total SoH degradation (compared to a new battery) afterM charging cycles is calculated by: D SoH = M X m=1 D SoH;cycle;m (5.17) whereD SoH;cycle;m denotes the SoH degradation in them-th cycle calculated using Eqn. (6.8). One can observe in Eqn. (6.8) that the normalized SoH degradation valueD SoH accumulates over the battery lifetime. In the literature, one typically usesD SoH = 20% orD SoH = 30% as a threshold to indicate the battery’s end of life. An example of SoH degradation under differentSoC avg ’s and SoC swing ’s with a threshold of D SoH = 20% is shown in Figure 5.4. There are two important observations drawn from the figure: (i) a higher SoH degradation rate is caused by both a higher SoC swing and a higher average SoC level in each charging/discharging cycle, and (ii) the cycle life of a Li-ion battery increases superlinearly with respect to the reduction of SoC swing and average Chapter5 56 SoC. FIGURE 5.4: Li-ion battery SoH degradation versus SoC swing (at different average SoC levels) and average SoC level (at different SoC swings). 5.1.3.3 Supercapacitor Self-Discharge The self-discharge behavior of supercapacitors is an important factor when considering their suit- ability for applications in HEES system [58]. As time goes on, supercapacitors slowly lose their charge and hence their stored energy. The self-discharge process is generally modeled by repre- senting the capacitor as a capacitanceC in parallel with a constant leakage resistanceR. In this case, without external charge/discharge current, the supercapacitor terminal voltage levelV cap as a function of timet is given by: V cap (t) =V 0 e t=RC (5.18) Figure 5.5 shows the general self-discharge process of a supercapacitor in 48 hours with an initial terminal voltage of 2.7V [59]. If we consider the voltage degradation during a constant period of Chapter5 57 time, the above equation becomes: V cap (t + t) =V cap (t)e t=RC ; (5.19) FIGURE 5.5: An example showing self-discharge of a supercapacitor in 48 hours. Considering that the supercapacitor SoC levelSoC cap is proportional toCV 2 cap , the SoC degrada- tion of the supercapacitor can also be written as: SoC(t + t) = s SoC(t) (5.20) where s = e 2t=RC is the SoC degradation rate caused by self-discharge. One can see that s can be considered as a constant rate when t is fixed. In Figure 5.5, the hourly voltage degradation rate is around 99.5%, indicating a value of 99% for s if we consider t to be one hour. Chapter5 58 5.2 HV AC Control and HEES Management Co-Scheduling Algorith- m 5.2.1 Problem description In this section, we present the formulation and solution of the cost minimization problem for an energy-efficient building through the co-scheduling of HV AC control and battery management. A slotted time model is adopted for decision making in which all system constraints as well as deci- sions are provided for discrete time intervals with equal and constant length t, and we consider the optimization framework for a billing period with N time slots. The temperature variable in HV AC system, battery SoC level and supercapacitor SoC level at time sloti are denoted byT ctrl [i], SoC bat [i] andSoC cap [i] respectively where 0iN. The initial building temperature variable is given byT ctrl [0] =T ctrl;ini , the initial SoC level of the battery is given bySoC bat [0] =SoC bat;ini , and the initial SoC level of the supercapacitor is given by SoC cap [0] = SoC cap;ini . We assume the battery and the supercapacitor can be re-used in the next billing cycle, so the final SoC level SoC bat [N] andSoC cap [N] should also beSoC bat;ini andSoC cap;ini , respectively. The total cost function is comprised of two parts: the energy cost charged from the power grid, and the cost associated with battery aging 2 , given as follows: Cost total =Cost energy +Cost aging (5.21) 2 As supercapacitors have an order-of-magnitude longer cycle life, we do not consider the SoH degradation of super- capacitors. Chapter5 59 We consider a dynamic pricing function is offered by the power grid, in which the price of one unit of energy (kWh) during thei-th time slot is denoted by [i] and is pre-announced at the beginning of the billing period. The energy cost in Eqn. (5.21) is calculated as follows: Cost energy = N X i=1 [i]P grid [i] t (5.22) And the battery aging cost is given by (we assume that the battery reaches end-of-life when SoH degradation is 30%): Cost aging = D SoH 1SoH th Cost bat (5.23) whereCost bat is the cost to purchase and replace the battery,D SoH represents the amount of SoH degradation during the charging and discharging process, and SoH th is the threshold SoH level (1-30% = 70%) at which the battery should be replaced. According to Eqn. (5.4), P grid [i] is a function of battery charging, supercapacitor charging, PV power generation, building power load as well as HV AC power consumption (denoted byP bat [i], P pv [i], P load [i], andP hvac [i] respectively) at time sloti. Apart fromP hvac [i], P bat [i] andP cap [i] which can be controlled by HV AC and HEES systems, the values ofP pv [i] andP load [i] cannot be exactly determined until time sloti unfolds. In order to reflect the potential opportunity for peak power shaving through co-scheduling of HV AC control and battery management in all future time slots, we propose to estimate the values ofP pv [i] andP load [i] (denoted by ^ P pv [i] and ^ P load [i]) based on other factors such as weather report, task schedule of the building, etc. The prediction error of these parameters adds to the difficulty of accurately calculating the total cost while maintaining certain constrains in future time slots. To tackle this problem, we propose Chapter5 60 the following adaptive co-scheduling framework to handle the prediction error. At the beginning of thei-th time slot, the HV AC power consumption and battery/supercapacitor charging plan are determined for all future time slots based on the current knowledge of the system parameters (either given or estimated). While the actions ofP hvac [i] andP bat [i] take place at thei-th time slot, the values ofP hvac [i + 1];:::;P hvac [N], P bat [i + 1];:::;P bat [N] andP cap [i + 1];:::;P cap [N] will be further updated at the next decision epoch at the beginning of the (i + 1)-th time slot when our knowledge of parameters including ^ P pv [i + 1] and ^ P load [i + 1] are updated. 5.2.2 Adaptive Co-Scheduling Problem Formulation We describe the adaptive co-scheduling problem of HV AC control and battery management at the beginning of thei-th hour. At that time, the current temperature variable and the SoC level of the battery and supercapacitor are given byT ctrl [i 1],SoC bat [i 1] andSoC cap [i 1]. The amount of HV AC power consumption and battery/supercapacitor charging, i.e.P hvac [i],P bat [i] andP cap [i], are derived by jointly considering the next hour (thei-th hour) and all other hours in the future (the (i + 1)-th toN-th hour). As mentioned in Section 6.2.1, in order to consider all future time slots, the unknown parameters including the PV power generation and building power load will first be estimated. While finding these estimations, we assume that necessary information (e.g. weather report, task schedule of the building, etc.) is available. To estimate the future building temperature change under a certain level of HV AC power con- sumption, we first need to determine the air flow at the that time slot. According to Eqn. (5.8), relationship betweenP hvac [j] and the air flow at a future time slotj (denoted byu[j];ijN) Chapter5 61 is given by: P hvac [j] = (c 1 u[j] 3 +c 2 u[j] 2 +c 3 u[j] +c 4 )P AC (5.24) For the convenience of expression, the relationship in Eqn. (5.24) will be referred to asP havc [j] = f hvac (u[j]) in the rest of this paper. Notice that u[j] and P hvac [j] have a one-to-one mapping relationship. To solve the problem more effectively, we setu[j] as the optimization variable in our algorithm. And based on Eqn. (5.5), the estimated temperature variable will be updated at time slotj following the equation: ^ T ctrl [j]=A n ^ T ctrl [j 1] +B n u[j] +E n ^ D[j];ijN (5.25) where A n , B n and E n are HV AC specific parameters corresponding to t and ^ D[j] is another estimated value which should be predicted and dynamically updated based on the most recent in- formation. In addition, SoC values of the battery for future time slots, denoted bySoC bat [j], are determined by: SoC bat [j] =SoC bat [i 1] + j X k=i P bat;int [k] t V bat C full ;ijN (5.26) whereP bat;int [k] is a function ofP bat [k] and can be calculated according to Eqn. (5.10). Consider- ing that there is also a one-to-one mapping relationship betweenP bat;int [k] andP bat [k] and we set P bat;int [k] as our optimization variable in our algorithm. Chapter5 62 Similar equations can be used to calculate supercapacitor SoC values in future time slots. However, the self-discharge effect should be considered, leading to a modified equation: SoC cap [j] =SoC bat [i 1] ji+1 s + j X k=i P cap [k] t jk s ;ijN (5.27) Knowing the SoC values of future time slots, we provide as follows an estimate of the SoH degra- dation of the battery during the charging and discharging process. We approximate the combination of the process as multiple charge/discharge cycles of the battery in all the future time slots. The highest and lowest battery SoC values SoC bat;high [i] and SoC bat;low [i] in these discharge/charge cycles are: SoC bat;high [i] = max ijN SoC bat [j] (5.28) SoC bat;low [i] = min ijN SoC bat [j] (5.29) And the SoH degradation of the battery in future time slots is estimated by: D SoH [i] =N C [i]D SoH;cycle (SoC bat;swing [i];SoC bat;avg [i]) (5.30) whereD SoH;cycle (SoC bat;swing [i];SoC bat;avg [i]) is defined as in Eqn. (6.8) andN C [i] is the equiv- alent charging cycles in future time slots that can be calculated as N C [i] = N X k=i P bat;int [k] I[P bat [k]> 0] t V bat C full SoC swing (5.31) where I[-] is the indicator function. Based on the above calculations, the adaptive control problem at the beginning of time sloti (1iN) can be formulated as follows: Chapter5 63 Given: Current battery SoC levelSoC bat [i 1], current supercapacitor SoC levelSoC cap [i 1] current temperature conditionT ctrl [i 1]. Predict: ^ P pv [j], ^ P load [j] and ^ D[j] forijN. Find:u[j],P cap [j], andP bat;int [j] forijN. Minimize: Estimated objective function in Eqn.(5.21). Subject to: HV AC temperature constraint, HV AC air flow constraint, battery charging constraint, and battery SoC constraint. To solve the problem efficiently after predicting the values of the unknown parameters, we propose to use a solution framework with an outer loop and a kernel algorithm. In the outer loop, we iterate over a set of possible values of SoC high [i] and SoC low [i]. In each iteration, given the range of the SoC of the battery during the charging and discharging process, we formulate an optimization problem as follows: Find:u[j]’sP cap [j]’s andP bat;int [j]’s. Minimize: N X j=i [j]P grid [j]t + D SoH [i] 1SoH th Cost bat (5.32) Chapter5 64 Subjectto: P grid [j] =f grid (P hvac [j]; ^ P load [j]; ^ P pv [j];P bat [j]);8j (5.33) C T n ^ T ctrl [j]T LB [j] +;8j (5.34) C T n ^ T ctrl [j]T UB [j];8j (5.35) P hvac [j] =f hvac (u[j]);8j (5.36) U LB u[j]U UB ;8j (5.37) P bat [j] =f bat (P bat;int [j]);8j (5.38) P bat;max;D P bat [j]P bat;max;C ;8j (5.39) P cap;max;D P cap [j]P cap;max;C ;8j (5.40) SoC[j] =SoC[i 1] + P j k=i P bat;int [k]t V bat C full ;8j (5.41) SoC cap [j] =SoC bat [i 1] ji+1 s + P j k=i P cap [k] t jk s ;8j (5.42) SoC bat [N]SoC bat;ini ;SoC cap [N]SoC cap;ini (5.43) SoC bat;low SoC bat [j]SoC bat;high ;8j (5.44) SoC cap;min SoC cap [j]SoC cap;max ;8j (5.45) Eqn. (5.33) is the relationship in the system power flow. Eqn. (5.34)-(5.37) are HV AC related constraints. Constraints (5.34) and (5.35) ensure that the building stays at comfortable temperature zone. Notice that is a parameter that accounts for the prediction error of ^ D[j], and can be set proportional toT UB [j]T LB [j]. Constraint (5.36) captures the relationship between the HV AC air flow and HV AC power consumption and constraint (5.36) sets air flow upper bound and lower Chapter5 65 bound. Eqn. (5.38)-(5.45) are HEES system related constraints. Constraint (5.38) captures the rela- tion between the input power from the DC bus and the energy change inside the battery. Constraint (5.39) and (5.40) set the bounds for the total charging and discharging power of the battery and the supercapacitor. Constraint (5.41) and (5.42) calculate the SoC levels of the battery and supercapac- itor for each future time slot. Constraint (6.28) ensures that the SoC of EES elements reach at least the initial SOC levels at the end of the billing cycle. Constraint (5.44) ensures that the SoC of the battery will not go beyond the SoC bound set in the outer loop in future time slots. And constraint (5.45) sets the maximal and minimum SoC levels of the supercapacitor. In order to solve the kernel optimization problem, we first eliminate constraints (5.38) and (5.33) by substitutingP bat in Eqn. (5.33) using (5.38) and substituteP grid in the objective function using Eqn. (5.33). Then all the inequality constraints are convex and all the remaining equality constraints are affine. As mentioned in Section 5.1.3.1,f bat is a convex function. Noticing that usually we have c 1 < 0,c 2 > 0, andc 1 c 2 (an example can be found in [51]), it can also be proved thatP havc [j] in Eqn. (5.24) is a convex function of u[j] when u[j] c 2 3c 1 . Therefore, we can set c 2 3c 1 as an additional upper bound foru[j] to makeP havc [j] a convex function ofu[j] over its domain without significantly change the original feasible set. In addition, D SoH [i] in Eqn. (5.30) is a convex function of P bat [i]’s. Finally, the objective function in (5.32) is a convex increasing function of P hvac [j]’s,P bat [j]’s,P cap [j]’s, andD SoH [i]. According to [60], the objective function in (5.32) is a convex function of all decision variables. Based on aforementioned conclusions, we know that the kernel problem can be solved efficiently using standard convex optimization tools such as CVX [61]. Because the kernel algorithm can be solved with polynomial time complexity and the outer loop Chapter5 66 can be implemented by a search algorithm (an exhaustive search in the worst case), the overall time complexity of the algorithm is pseudo-polynomial. 5.3 Experimental Result 5.3.1 Simulation Setup In the simulations, we consider a day-ahead utility market where the time-of-use electricity prices are pre-announced at the beginning of each day, and we take one day as the billing period. The scheduling decisions of both the battery storage and the HV AC system are updated at the beginning of each hour, i.e., t is set to one hour. We adopt the electricity pricing policy from the Consolidat- ed Edison Company 3 where energy usage from 10a.m.-10p.m. is charged with a peak price, while energy usage in other hours are charged with an off-peak price. The building temperature model as in Eqn. (5.25) is extracted from a building located at 1084 Columbia Ave., Irvine, California, USA. The relationship between the HV AC power consumption and the HV AC air flow in Eqn. (5.24) from [51] is used. The peak power of the HV AC system is set to 100kW. Because of difference in the size of the building, we use a modified set of parameters listed below: c 1 =3:8784 10 5 ;c 2 = 0:0108 c 3 =0:48;c 4 = 59:2 3 http://www.coned.com/ Chapter5 67 A realistic residential PV generation system is considered, and we use PV power profiles measured at Duffield, V A, in the year 2007 [47]. The peak power generation of the PV panel is set to 20 kW. The power consumption of loads other than the HV AC (P load [i]) is assumed to follow a uniform distribution between 5kW and 15kW. The predicted values of ^ P pv [i], ^ P load [i], and external distur- bance ( ^ D n [i]) are assumed to have a maximum of20% prediction error. The prediction error also follows a uniform distribution and the average prediction error is around 10%. To model the rate capacity effect of the battery, we set factors c and d to 1.15 and 0.85, respec- tively. The parameters to calculate the SoH degradation as in Eqn. (6.8) are from [57]. The battery cost is set to $400/kWh. The power conversion efficiencies ( 0 to 3 ) are set to 90%. Data corre- sponding to supercapacitor self-discharge come from [59] and the supercapacitor SoC degradation rate s is set to 99%. 5.3.2 Simulation Results We first study the impact of energy storage capacity on the potential of building energy cost saving. Figure 5.6 shows the daily total costs (measured as the electricity bill plus the cost associated with battery aging, as described in Eqn. (5.21)) with different battery capacities (varying from 60kWh to 360kWh). The capacity of supercapacitor is set as 20% of the battery capacity. Electricity prices in June 2014 from Consolidated Edison Company are used where the electricity price is $0.3032/kWh during peak hours and $0.0116 during off-peak hours. The proposed optimization framework is compared with four baseline schemes. In the first baseline scheme, the scheduling of HV AC system is performed without any available battery energy storage (marked as “No-Bat”). In this baseline scheme, the battery aging cost is zero, and total cost is calculated directly from the electricity bill. Chapter5 68 The second and the third baselines are based on a simplified energy storage system in our previous work [? ] which contains only a battery and no supercapacitor. In the second baseline scheme, a greedy algorithm is used to schedule the HV AC air flow and the battery charging/discharging (marked as “Greedy Bat”). In the greedy algorithm, the battery will be charged to the maximal SoC level during off-peak hours, and discharged to the minimal SoC level during peak hours. The third baseline scheme comes from the solution presented in [? ]. The last baseline scheme uses the greedy algorithm but is based on the HEES system presented in this paper. FIGURE 5.6: Relationship between daily cost and battery storage capacity for our proposed algo- rithm and four baseline schemes. One can observe that the proposed co-scheduling algorithm consistently achieves lower costs com- pared with all the four baseline schemes. Please note that the greedy algorithm can achieve a reasonable performance if the rate capacity effect and the SoH degradation of the battery do not exist because it can store a maximum amount of energy and use it in peak hours. However, it can be observed from Figure 5.6 that this is not the case when a realistic battery model is considered. In the simplified energy storage system with only a battery, the result from the greedy algorithm can be Chapter5 69 even worse compared with the first baseline scheme with no battery. In addition, compared with the simplified system in [? ], our HEES system consistently performs better no matter which algorithm is used. This is due to the introduction of supercapacitor which compensates the weakness of the battery. FIGURE 5.7: Battery daily charge/discharge schedule. The benefit of the proposed algorithm becomes even more significant with the increase of battery storage capacity. With a battery storage capacity of 60kWh, the proposed algorithm achieves $4/d to $8/d cost saving, compared with the baseline schemes. When the battery energy capacity reaches 360kWh, the cost saving of the proposed algorithm cam reach $15/d to $35/d. The internal energy increasing rates of the battery (P bat;int ) with 180kWh of battery storage capacity are shown in Figure 5.7 . As can be seen from this figure, large charging/discharging power is avoided in the proposed algorithm since the realistic battery model including the rate capacity effect and the SoH degradation is properly accounted for. We also conduct a study on the impact of energy prices, and Figure 5.8 shows the equivalent daily cost under different dynamic pricing functions where the electricity price during off-peak hours Chapter5 70 stays at $0.0116/kWh while the electricity price during peak hours varies from $0.2500/kWh to $0.4000/kWh. The proposed algorithm also consistently achieves lower costs compared with both base line schemes with a cost reduction from 5% to 10%. The cost reduction rate stays the same with the change of peak hour energy price, indicating that the good performance of our proposed algorithm can be achieved under different dynamic pricing functions. FIGURE 5.8: Relationship between daily cost and peak hour energy price for our proposed algo- rithm and two baseline schemes. 5.4 Chapter Summary In this chapter, the optimal co-scheduling problem of HV AC control and HEES management was considered for energy-efficient smart buildings under a complete building power system and a dy- namic energy pricing policy. In this problem, the degradation of battery SoH during the charg- ing/discharging process was taken into consideration based on an accurate SoH modeling. The total cost function therefore became the summation of the electricity bill charged by the power grid and the extra cost associated with the aging of battery. An optimal co-scheduling algorithm was Chapter6 71 presented that adaptively adjusts its current building temperature condition and always makes the optimal building air flow control and the charging/discharging decision of each EES element in the future time slots based on the most updated information. The proposed algorithm also accurately accounted for the power loss during the charging and discharging process of each EES element, especially the rate capacity effect of batteries and the the self-discharge of the super-capacitor, and in AC-DC or DC-DC power conversion circuits, which is often neglected in the reference work. Experimental results demonstrated that the proposed optimal co-scheduling algorithm minimizes the combination of building electricity bill and battery aging cost. Chapter 6 Optimal Control of PEVs with a Charging Aggregator Considering Regulation Service Provisioning In this chapter, we consider the problem of PEV charging under dynamic pricing and RS provision- ing, with a given departure time and a given target state-of-charge (SoC) level at that time. In this problem, we explicitly take into account the degradation of battery state-of-health (SoH), which is defined as the ratio of full charge capacity of an aged battery to its designed (nominal) capacity, during V2G operations based on an accurate SoH modeling. We also consider a power market with RS provisioning, in which PEV owners are credited for the power reserves that they provide through the V2G network. The objective function to minimize therefore becomes the summation 72 Chapter6 73 of the real energy cost during PEV charging (cost from nominal power consumption minus payoff from availing their power reserves) and the extra cost associated with the aging of the PEV battery. FIGURE 6.1: Structure of the PEV charging aggregator. Another contribution of this chapter is to present an SoH-aware charging aggregator design, which decides the control sequences of a group of PEVs to reduce the peak power caused by simultaneous PEV charging. An energy storage system is also used in the charging aggregator to mitigate the impact of real-time RS signal. The system structure is shown in Fig. 6.1. The charging aggregator collects the information from each PEV and announce the total amount of power reserves that PEVs can avail to the power grid based on their current SoC levels. The benefits of the charging aggregator can be classified into two aspects: the ability to reduce peak power caused by mass unregulated PEV charging and the potential to reduce energy cost. By considering a number of PEVs as a cooperated group and with the help of the energy storage system, the charging sequences can be optimized and the peak power consumption can be shaved. Chapter6 74 6.1 System Model 6.1.1 Charging Aggregator Architecture and Power Flow In this paper, we consider the design of the charging aggregator which controls the charging se- quences of all the connected PEVs. The block diagram of the power flow in a typical charging aggregator is shown in Fig. 6.2. In this figure, we assume a total number ofM PEVs are connect- ed to a charge transfer interconnect and the charging power of each PEV is denoted byP bat;v for 1vM. A positive value ofP bat;v means the PEV is being charged, a negative value indicates discharging from the battery storage, and zero represents no charging or discharging operation. The charging aggregator is connected to the state-level or national smart grid through AC-DC interfaces (e.g., inverters, rectifiers, or transformer circuitries), and the power from the gridP grid is related to the charging aggregator’s electricity bill. An energy storage system, proving a power ofP storage , is used to mitigate the impact of real-time RS signal. Notice that bothP grid andP storage can be positive or negative, indicating bidirectional power interfaces. According to the direction of the AC-DC interfaces, we can calculate P grid using the following equation: P grid = 8 > > > > > < > > > > > : 1 C ( M X v=1 P bat;v P storage ); M X v=1 P bat;v P storage 0 D ( M X v=1 P bat;v P storage ); otherwise (6.1) where C and D are the AC-to-DC and DC-to-AC power transfer efficiencies, respectively. In this paper, we consider two kinds of power loss in the battery of each PEV or the energy storage system. The first one is the rate capacity effect, which affects the energy efficiency during the Chapter6 75 FIGURE 6.2: Charging Aggregator Power Flow. battery charging or discharging process; The second one is the SoH degradation, which models the degradation of full charge capacity of an aged battery to its nominal capacity. We also consider a power market with RS provisioning, in which PEV owners are credited for the power reserves that they provide through the V2G network. 6.1.2 Model of Battery Storage The most critical effect that causes power loss in the storage system of a PEV is the rate capacity effect [52]. 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 amount. The rate capacity effect specifies that the discharging efficiency of a battery decreases with Chapter6 76 the increase of the battery’s discharging current. The energy loss in the battery during the charging process will be affected in a similar way. Peukert’s formula [53] can be used to capture the rate capacity effect. In this empirical formula, the battery charging and discharging efficiencies are described as functions of the charging currentI c and the discharging currentI d , respectively: rate;c (I c ) = 1 (I c =I ref ) c ; rate;d (I d ) = 1 (I d =I ref ) d (6.2) where c and d are Peukert’s coefficients, and their values are typically in the range of 0:1-0:3;I ref denotes the reference current level for charging and discharging of the battery, which is proportional to the battery’s nominal capacityC full . We callI c =I ref andI d =I ref the battery’s normalized charging current and normalized discharging current, respectively. It can be observed that the efficiency values rate;c (I c ) and rate;d (I d ) in Eqn. (1) will be higher than 100% if the magnitude of the normalized charging or discharging current is less than one, which contradicts common sense. We therefore modify the Peukert’s formula such that the efficiency values rate;c (I c ) and rate;d (I d ) are saturated at 100% when the magnitude of the normalized charging/discharging current is less than one, meaning that the battery suffers from no rate capacity effect for relatively low values of the charging and discharging currents. We denote the rate of change in electric energy stored in the battery byP bat;int , a rate which may be positive (charging the storage), negative (discharging from the storage), or zero. Based on the modified Peukert’s formula, the relationship betweenP bat;int and the storage output powerP bat is Chapter6 77 given by P bat = 8 > > > > > > > > > < > > > > > > > > > : V bat I bat;ref ( P bat;int V bat I bat;ref ) 1 ; P bat;int V bat I bat;ref >1 P bat;int ; 1 P bat;int V bat I bat;ref 1 V bat I bat;ref ( jP bat;int j V bat I bat;ref ) 2 ; P bat;int V bat I bat;ref <1 (6.3) where V bat is the battery terminal voltage and is assumed to be (nearly) constant; I bat;ref is the reference current of the battery storage, which is proportional to its nominal capacityC full which is given in Ah (Ampere Hour); coefficient 1 is in the range of 1:1-1:3, and coefficient 2 is in the range of 0:8-0:9. We use the functionP bat =f bat (P bat;int ) to denote the relationship betweenP bat andP bat;int . One important observation is that this function is a convex and monotonically increasing function over the input domain1 < P bat;int <1, as shown in Fig. 6.3. Due to the monotonicity property, P bat;int is also a monotonically increasing function ofP bat , denoted byP bat;int = f 1 bat (P bat ). We can see from Fig. 6.3 that rate capacity effect makes the charging/discharging process less efficient. FIGURE 6.3: Relationship betweenP bat andP bat;int considering the rate capacity effect. Chapter6 78 6.1.3 Model of Battery SoH Degradation State-of-health (SoH) degradation is another significant portion of power loss for PEV batteries [19]. To study the SoH degradation effect, we first formally define the state-of-charge (SoC) of a battery storage bank as follows: SoC = C bat C full 100% (6.4) whereC bat is the amount of charge stored in the battery bank, andC full is the amount of charge in the battery when it is fully charged. We interpret SoC as the state of the battery bank. On the other hand, theC full value gradually decreases as a value of battery aging (i.e., SoH degradation). The amount of SoH degradation, denoted byD SoH , is defined as follows: D SoH = C nom full C full C nom full 100% (6.5) whereC nom full is nominal value ofC full for a fresh new battery. The SoH of batteries is difficult to estimate because it is related to a capacity fading effect (i.e., SoH degradation) which results from long-term electrochemical reactions inside the battery. The capacity fading is related to the carrier concentration loss and internal impedance growth in the bat- teries. These effects strongly depend on the operating condition of the battery such as charging and discharging currents, number of charge-discharge cycles, SoC swing, average SoC, and operation temperature [54],[55]. The characterization of a battery cell requires time-consuming experiments and mathematical models are used to help us reduce the time complexity in estimating the SoH degradation. Electrochemistry-based models [56] are generally accurate but not easy to implement. Chapter6 79 Hence, we apply the SoH degradation model of Li-ion batteries proposed in [57], which can be applied to cycled charging and discharging of the battery elements and shows a good match with real data. The SoH degradation model estimates the SoH degradation for cycled charging/discharging of a Li- ion battery cell, where a (charging/discharging) cycle is defined as a charging process of the battery cell fromSoC low toSoC high and a discharging process right after it fromSoC high toSoC low . The SoH degradation during one such cycle depends on the average SoC level SoC avg and the SoC swingSoC swing . We calculateSoC avg andSoC swing of one cycle using: SoC avg = (SoC low +SoC high )=2 (6.6) SoC swing =SoC high SoC low (6.7) SoC swing achieves the maximum value of 1:0 (100%) for the full 100% depth of discharge cycle, i.e., the SoC changes from 0 up to 100% and then back to 0. The SoH degradation D SoH;cycle during this charging/discharging cycle, accounting for both average SoC level and SoC swing, is: D 1 =K CO exp[(SoC swing 1) T ref KexT B ] + 0:2 life (6.8) D 2 =D 1 exp[4K SoC (SoC avg 0:5)] (1D SoH ) (6.9) D SoH;cycle =D 2 exp[K T (T B T ref ) T ref T B ] (6.10) Chapter6 80 whereK co ,K ex ,K SoC , andK T are battery specific parameters;T B andT ref are the battery’s op- eration temperature and reference temperature, respectively; is the duration of this charging/dis- charging cycle; life is the calendar life of this battery. We useD SoH;cycle (SoC swing ;SoC avg ) to denote the relationship betweenD SoH;cycle ,SoC swing , andSoC avg . The total SoH degradation (in reference to a fresh battery) afterM charging and discharging cycles is calculated by: D SoH = M X m=1 D SoH;cycle;m (6.11) whereD SoH;cycle;m denotes the SoH degradation in them th cycle. One can observe in Eqn. (6.11) that the normalized SoH degradation valueD SoH accumulates over the battery lifetime from 0 (brand new) to 100% (no capacity left). In the literature, one typically finds values of D SoH = 20% or D SoH = 30%, indicating 80% or 70% remaining capacity, re- spectively, to measure the battery’s end of life. The relationship between the Li-ion battery SoH degradation versus the SoC swing and average SoC level is shown in Fig. 6.4. In this experiment, we change 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 D SoH = 20%, and record the total number of cycles (i.e., the cycle life of the battery). The results are also shown in Fig. 6.4. There are two important observations drawn from the figure: (i) a higher SoH degradation rate is caused by both a higher SoC swing and a higher average SoC level in each charging/discharging cycle, and (ii) the cycle life of a Li-ion battery increases superlinearly with respect to the reduc- tion of SoC swing and average SoC. We make use of these observations as well as the function D SoH;cycle (SoC swing ;SoC avg ) in the rest of this paper. Chapter6 81 FIGURE 6.4: Li-ion battery SoH degradation versus SoC swing (at different average SoC levels) and average SoC level (at different SoC swings). 6.1.4 Model of Regulation Service Provisioning Market Power market RS provisions have been widely studied in recent years [28], which are adopted to match electricity supply with demand in real time. There are several power markets with different time-scales, and we focus on the hour-ahead power market in [62] because PEVs can participate in RS provisioning market in the time scale of several hours. Currently, RS reserves are mainly offered by centralized generators. However, market rules are changing to allow the demand side, especially PEVs, to the provide reserves as well. For example, PJM, one of the largest US ISOs, has allowed electricity loads to participate in reserve transactions [63]. In this power market, each energy user declares an nominal energy consumption P and an RS reserve R to the power system in advance of each hour. The market clearing prices for energy consumed and RS reserves are denoted by E and R . The energy user is charged for its nominal power consumption and credited for the RS reserves such that the participant pays a net amount Chapter6 82 of E P R R. However, the credit for the RS reserves does not come for free. As the hour unfolds, each energy user is asked to modulate its real time power consumptionP (t) dynamically so as to track the RS signalz(t) by ensuring thatP (t) = P +z(t)R where1 z(t) 1. An example of RS tracking signalz(t) is shown in Fig. 6.5. It can be observed that energy users can reduce their electricity bill by providing RS reserves, while the power grid can also achieve a better match between power supply and demand in presence of volatile and intermittent renewable energy generations. FIGURE 6.5: An example of RS tracking signalz(t). The RS provisioning market is especially promising for PEVs, because the charging policy can be expediently adjusted in order to meet the required power consumption level specified in the RS provisioning contract. When connected to the grid, each vehicle determines the optimal nominal power consumptionP and the RS reserveR of the next time slot (hour) based on its current SoC, the target SoC, dynamic energy prices, and the credit for RS reserves. OnceP andR are set, each vehicle adjusts its charging power in the next hour based on the RS signalz(t) which is dynamically broadcasted from the power grid. The RS signal z(t) is generated based on the real-time power market situation and is used to balance the supply and demand in the power market [36]. For each Chapter6 83 individual PEV owner who has very little effect on the entire power market,z(t) can be considered as a given signal. The uncertainty of thez(t) signal complicates the optimal PEV charging schedule determination because of the following reasons: First, the PEV requires a guaranteed amount of charging energy during the parking time to satisfy a target SoC, while in the worst case, the grid might require the PEV owners to always ramp down the charging power consumption; In addition, the SoC swing value has a strong dependency on the real-time charging sequence, which is directly related to the actual RS signalz(t), and hence the battery SoH degradation is also dependent on thez(t) signal; Finally, the SoC level of future time slots can not be accurately estimated due to the uncertainly of z(t). To participate in RS provisioning market, a joint optimization framework should be developed that would consider the dynamic price of the power grid, the change ofSoC swing value as well as the resultant battery SoH degradation, and the uncertainty of SoC level in future time slots. To find the optimal control of PEVs with a charging aggregator, we start with a simplified problem in Section 6.2 regarding the optimal charging scheduling of an individual PEV . In this problem, we assume each PEV is considered independently without a charging aggregator, i.e., each PEV is directly connected to the power grid. Given its departure time and target SoC level, we optimize the charging schedule of a PEV considering the energy cost (including the revenue from offering RS provisioning) and the cost associated with SoH degradation. We further extend the problem to the charging aggregator level in Section 6.3, in which a number of PEVs are considered as a cooperated group and are scheduled together to shave peak power consumption. A peak power charge from the power grid is added to the total cost function which is proportional to the maximal power Chapter6 84 consumption in a billing period. The energy storage system is included to balance the charging schedule and mitigate the impact of real-time RS signal. 6.2 Optimal Charging Schedule of an Individual PEV 6.2.1 Problem description In this section, we present the formulation and the solution of the cost minimization problem for an individual PEV in a V2G system in the context of hour-ahead power market with regulation services 1 . We assume the PEV is directly connected to the power grid, i.e.,P grid can be directly calculated from the PEV’s charging powerP bat . Assume that a PEV is scheduled to depart afterN hours of parking at home or at a public parking lot, and it can thus participate in the power market forN consecutive hours. The initial SoC level of the PEV battery is given bySoC[0] =SoC ini , and the target SoC level isSoC tar when the PEV departs. The parking timeN and target SoC level are specified by the PEV owner. We denote the dynamic energy price and the RS reserve revenue per unit of energy provided by the PEV during the i-th hour by E [i] and R [i], respectively, while the nominal power consumption and the amount of RS reserve declared by the PEV in the i-th hour are denoted by P [i] and R[i], respectively. Please note that E [i] and R [i] are announced at the beginning of thei-th hour in this hour-ahead power market, and the PEV declaresP [i] andR[i] to the grid accordingly. During thei-th hour, the 1 Please note that other types of dynamic energy prices, such as day-ahead dynamic pricing, may also be supported. Chapter6 85 real-time power drawn from the grid can be expressed as follows P grid (t) =P [i] +z(t)R[i] (6.12) wherez(t) is the regulation signal as defined in Section 6.1.4. In this problem, we don’t consider peak power charge, so the total cost function is comprised of two parts: the energy cost during PEV charging (cost from nominal power consumption minus payoff from RS reserve provisioning) and the cost associated with PEV battery aging, given as follows: Cost pev =Cost energy +Cost aging (6.13) The energy cost in Eqn. (6.13) is calculated as follows: Cost energy = N X i=1 ( E [i]P [i] R [i]R[i]) (6.14) while the battery aging cost is given by (we assume that the battery reaches end-of-life when SoH degradation is 30%): Cost aging = D SoH 1SoH th Cost bat (6.15) whereCost bat is the cost to purchase and replace the PEV battery, D SoH represents the amount of SoH degradation during the combination of driving cycle and charging process, SoH th is the threshold SoH level (typically 70%) at which the battery should be replaced. Chapter6 86 As can be seen from Eqn. (6.14) and (6.15), Cost energy andCost aging depend on the entire se- quence of E [i] and R [i], as well as the RS signalz(t) which will directly affect the battery SoC range during each hour. However, in an hour-ahead power market, one only knows the price for nominal power and credit for power regulation for the next hour. In order to reflect the potential op- portunity for power regulation in all future hours, we propose to estimate the values of E [i]’s and R [i]’s based on the pricing history. Moreover, as mentioned earlier, the uncertainty ofz(t) adds to the difficulty of accurately calculating the total cost. To tackle this problem, rather than finding an effective approach to predict the complicated dynamics of the regulation signal, we treatz(t) as a random variable that follows a specific distribution on [1; 1] which can be extracted from data in the past. Once the probability density function (PDF) ofz(t) is obtained, the expected behavior of the system can be estimated based on the statistics ofz(t). In order to find the optimal P [i]’s and R[i]’s to minimize the total cost of the PEV , we propose the following adaptive control framework. At the beginning of thei-th hour, E [i] and R [i] are given, and the HEV controller determines the nominal power consumption and the RS reserve of all future hours based on the current knowledge of the system parameters (either given or estimated). While the values of P [i] andR[i] for thei-th hour are submitted to the grid, the values of P [i + 1];:::; P [N] and R[i + 1];:::;R[N] will be further updated at the next decision epoch at the beginning of the (i+1)-th hour when our knowledge of parameters including E [i+1] and R [i+1] is updated. Chapter6 87 6.2.2 Adaptive Control Problem Formulation We describe the adaptive control problem of HEV charging at the beginning of thei-th hour. At that time E [i] and R [i] are announced from the grid, and the SoC level of the HEV battery is given bySoC[i1]. The amount of nominal power consumption and RS reserve to be submitted to the grid, i.e. P [i] andR[i], are derived by HEV controller by jointly considering the next hour (the i-th hour) and all other hours in the future (the (i + 1)-th toN-th hour). As mentioned in Section 6.2.1, in order to consider all future hours, the unknown parameters including the market prices, the SoC of the battery, and the SoH degradation of the battery will first be estimated. While finding these estimations, we assume that history information of the power market (e.g. pricing, regulation signal, etc.) are available. For the market clearing price of nominal power consumption over houri 0 (i + 1i 0 N), E [i 0 ], we make the observation that its daily fluctuation pattern is similar across different days and a high power price in a specific hour usually implicates high prices in the following hours. Therefore, the estimated value, denoted by ^ E [i 0 ], is calculated as ^ E [i 0 ] = E [i] E [i] E [i 0 ] (6.16) where E [i] and E [i 0 ] are the average market clearing prices for nominal power consumption in the i-th hour and the i 0 -th hour in the history, respectively. In this way we effectively derive the estimation ^ E [i 0 ] over thei 0 -th hour through the given value E [i]. Using the same approach, the estimated value of R [i 0 ], denoted by ^ R [i 0 ], can be calculated. Chapter6 88 To estimate the SoC change and the SoH degradation of the battery of the PEV , we first extract the empirical p.d.f. (probability density function) ofz(t), which will be denoted byf Z (z) where 1 z 1. Considering that the PEV is directly connected to the power grid, equation (6.1) becomes P [i] +z(t)R[i] = 8 > > > < > > > : 1 C P bat (t); P bat (t) 0 D P bat (t); otherwise (6.17) The relationship between P bat (t) and the internal input power of the battery P bat;int (t) is speci- fied in Eqn. (6.1). Based on these two relationships we define the function P [i] +z(t)R[i] = f tran (P bat;int (t)). Therefore, the estimated average internal input power of the battery for thei 0 -th hour, denoted by ^ P bat;int [i 0 ], can be calculated as ^ P bat;int [i 0 ] = Z 1 1 f 1 tran ( P [i 0 ] +zR[i 0 ])f Z (z)dz (6.18) And the estimated SoC values for future hours, denoted by ^ SoC[i 0 ], are determined by: ^ SoC[i 0 ] =SoC[i 1] + i 0 X j=i ^ P bat;int [j] T V bat C full ;ii 0 N (6.19) Using functionsf Z () andf tran (), the approximate SoH degradation during the charging period, denoted by ^ D SoH , can be calculated as follows ^ D SoH =N C D SoH;cycle (SoC swing ;SoC avg ) (6.20) Chapter6 89 whereD SoH;cycle (SoC swing ;SoC avg ) is defined as in Eqn. (6.8) andN C is the equivalent charging cycles that can be calculated as N C = T P i 0 R 1 1 max C ( P [i 0 ] +zR[i 0 ]); 0 f Z (z)dz 2V bat C full SoC swing (6.21) Using the Eqn. (6.20) and Eqn. (6.21), we decouple the entire charging sequence into several charging/discharging cycles so that the total SoH degradation can be calculated. Based on the above calculations, the adaptive control problem at the beginning of time sloti (1iN) can be formulated as follows: Given: Current SoC levelSoC[i 1], target SoC levelSoC tar , next-hour energy pricing function E [i], RS reserve revenue R [i] and PDF of RS tracking signalz(t). Predict: Future energy pricing function and RS reserve revenue ^ E [i 0 ] and ^ R [i 0 ] fori<i 0 N, Find:SoC high ,SoC low ,P [i 0 ], andR[i 0 ], forii 0 N. Minimize: Estimated cost function in Eqn. (6.13). Subjectto: Charging/discharging power constraint and SoC constraint. To solve the problem efficiently after predicting the values of the unknown parameters, we propose to use a solution framework with an outer loop and a kernel algorithm. In the outer loop, we iterate over a set of possible values ofSoC high andSoC low . In each iteration, given the range of the SoC of the battery during the charging period, we formulate an optimization problem as follows: Find: P [i 0 ]’s,R[i 0 ]’s, ^ P bat;int [i 0 ]’s, and ^ SoC[i 0 ]’s Chapter6 90 Minimize: E [i] P [i] R [i]R[i] + N X i 0 =i+1 ( E [i 0 ] P [i 0 ] R [i 0 ]R[i 0 ]) + ^ D SoH 1SoH th Cost bat (6.22) Subjectto: P [i 0 ] +R[i 0 ]P max;C ;8i 0 (6.23) P [i 0 ]R[i 0 ]P max;D ;8i 0 (6.24) ^ P bat;int [i 0 ]2 f 1 tran (P max;D );f 1 tran (P max;C ) ;8i 0 (6.25) ^ P bat;int [i 0 ] R 1 1 f 1 tran ( P [i 0 ] +zR[i 0 ])f Z (z)dz;8i 0 (6.26) ^ SoC[i 0 ] = ^ SoC[i 0 1] + ^ P bat;int [i 0 ] T V bat C full ;8i 0 (6.27) SoC[N]SoC tar + (6.28) SoC[i 0 1] + C P [i 0 ] +R[i 0 ] T SoC high ;8i 0 (6.29) SoC[i 0 1] +f 1 tran P [i 0 ]R[i 0 ] T SoC low ;8i 0 (6.30) R[i 0 ] 0;8i 0 (6.31) Constraints (6.23) and (6.24) set the bounds for the total charging and discharging power for the battery. Similarly, constraint (6.25) set the range for the actual energy change rate of the battery. Constraint (6.26) captures the relation between the input power from the grid and the energy change of the battery. Constraint (6.27) estimate the SoC of the battery for each future hour. Constraint (6.28) ensures that the SoC reaches the preset target value when the PEV departs, where is a parameter to account for the estimation error of the SoC, which can be set to be proportional to Chapter6 91 Algorithm 4: Adaptive Algorithm for Optimal Charging Control of an Individual PEV Input: Initial SoCSoC ini , Target SoCSoC tar , DepartureN, Historical pricing functions Output: Bidding decisionP [i] andR[i] for 1iN SoC[0] =SoC ini ; fori from 1 toN do At the beginning of thei-th hour: Get next-hour price functions E [i] and R [i] Predict all future price functions ^ E [i 0 ] and ^ R [i 0 ] fori<i 0 N Cost pev;opt =1 for each (SoC high ,SoC low ) combination do Find optimalP [i 0 ] andR[i 0 ] forii 0 N Calculate estimated cost functionCost pev using Eqn. (6.13) ifCost pev <Cost pev;opt then Cost pev;opt =Cost pev ^ E opt = ^ E [i], ^ R opt = ^ R [i] end end Apply ^ E opt and ^ R opt as the next-hour bidding decision When thei-th hour unfolds: Charge the PEV according to the real-timez(t) signal At the end of thei-th hour: UpdateSoC[i] based on the actual charging result end the averageR[i 0 ] values. Constraints (6.29) and (6.30) ensure that the SoC of the battery will not go beyond the SoC bound set in the outer loop with the input power modulated by the regulation signal. Based on [60], it can be proved thatf trans () is a convex increasing function whilef 1 trans () is a concave increasing function. Therefore, the objective function as shown above is a convex function of all decision variables, and every inequality constraints in the formulation can be trivially trans- formed into a convex form. At the same time, all equality constraints are affine. Consequently, the formulated problem that is solved in each iteration of the SoC range is a standard convex optimiza- tion problem that can be solved optimally with polynomial time complexity using algorithms such as the interior point algorithm [60]. The detailed algorithm is shown in Algorithm 4 Chapter6 92 Because the kernel algorithm can be solved with polynomial time complexity and the outer loop can be achieved by a search algorithm (an exhaustive search in the worst case), the overall time complexity of the algorithm is pseudo-polynomial. 6.3 Optimal Control of the PEV Charging Aggregator In this section, we present the formulation and the solution of the cost minimization problem for the charging aggregator shown in Fig. 6.2. We consider a total number of M T charging slots. The number of PEVs parking at the charging aggregator at i-th hour is denoted by M[i] where 0 M[i] M T ,8i, and these PEVs are labeled from 1 toM[i] ifM[i] 1. For each parking PEV labeled byv, the current SoC, the target SoC, and the departure time are denoted bySoC v [i], SoC tar;v , and D v , respectively. An energy storage system is included with a current SoC level denoted bySoC storage [i]. 6.3.1 Charging Aggregator Cost Function, Decision Window and Billing Period In this problem, considering an nominal power consumptionP [i] and an amount of RS reserveR[i] declared by the charging aggregator in thei-th hour, Eqn. (6.1) becomes P [i] +z(t)R[i] = 8 > > > > > < > > > > > : 1 C ( M X v=1 P bat;v P storage ); M X v=1 P bat;v P storage 0 D ( M X v=1 P bat;v P storage ); otherwise (6.32) Chapter6 93 As a number of PEVs are being plugged into the power grid, the control or management issue of PEV charging arises, since mass unregulated charging processes of PEVs may result in extremely high peak power consumption [64]. The charging aggregator collects the needed information (e.g., the energy price and RS revenue) and optimizes the charging sequences of all the parking PEVs. The advantage of charging the PEVs using an aggregator is to regulate the group charging process and reduce the peak power demand. To reflect this advantage, a peak power bill is considered in this paper. The cost function in a decision window thus becomes: Cost total =Cost energy;total +Cost aging;total +Cost peak (6.33) whereCost energy;total andCost aging;total are the total energy cost as well as the cost associated with battery aging for all the PEVs as well as the storage battery, which can be calculated using the equations in Section 6.2. 2 Cost peak is the peak power bill which can be calculated using: Cost peak =P peak E peak (6.34) whereP peak is the peak power consumption during a billing period, and E peak is the peak power price. Usually, the peak power bill is calculated for an entire charging aggregator based on a billing period of one month or one week [65]. We set a constant time as the decision windowN D and optimize the charging decision during this time period. If there exists a current parking PEV whose departure time is beyond our decision 2 The introduction of charging aggregator will also bring additional costs such as maintenance cost and energy storage cost. In this section, we focus on the part of cost that we can minimize. Chapter6 94 window, we extend the decision window till the PEV departs. Fig. 6.6 shows the billing period, decision window, and estimation of future parking PEVs in the charging aggregator optimization framework. In this paper, considering that a billing period is generally much longer than a decision window, we focus on the situation that the decision window is included by one billing period. FIGURE 6.6: The billing period, decision window, and estimation of future parking PEVs in the charging aggregator optimization framework. In order to consider the effect of peak power bill, we use an average method in which an increased peak power bill is considered as uniformly distributed to all the future time slots. Assume there are in total N T hours in a billing period and we are at the end of the i-th hour with a current peak power ofP peak;pre , two situations might happen in the coming decision window. If the peak power consumption during the coming decision windowP peak;i is less than or equal toP peak;pre , no additional peak power bill needs to be considered. Otherwise ifP peak;i > P peak;pre , we divide the additional peak power bill (P peak;i P peak;pre ) E peak uniformly to all the futureN T i hours and consider that each hour leads to a peak power bill increase of (P peak;i P peak;pre ) E peak N T i . In our Chapter6 95 optimization framework, Eqn. (6.34) is considered as: Cost peak = 8 > > > < > > > : (P peak;i P peak;pre ) E peak N T i N D ; P peak;i >P peak;pre 0; P peak;i P peak;pre (6.35) 6.3.2 Estimation of Future Parking PEVs One primary challenge of finding the optimal charging aggregator control is that we don’t know the information of the PEVs that arrive in the future hours. To solve this problem, we estimate the number of PEVs that are parked in the charging aggregator based on historical traffic data. We denote the estimated number of PEVs parking at the charging aggregator ati-th hour by ^ M[i] where 0 ^ M[i]M T ,8i. Knowing the information of all the current parking PEVs, there are two possible cases in any future hour i 0 . The first case is that the number of existing PEVs at hour i 0 (i.e., current parking PEVs which are still parking at houri 0 ) is greater or equal to ^ M[i 0 ]. In this case, only these PEVs need to be considered at houri 0 . On the other hand, if the number is less than ^ M[i 0 ], an extra amount of power should be taken into consideration to avoid peak power resulting from the load of future parking PEVs. We use P std [i 0 ] andR std [i 0 ] to denote the estimated nominal power and RS reserve of one future parking PEV at thei 0 -th hour, which can also be derived from historical data. The estimatedbarP [i 0 ] andR[i 0 ] can thus be calculated as: P [i 0 ] = M[i 0 ] X v=1 P v [i 0 ] + P storage [i 0 ] +M D [i 0 ] P std [i 0 ];ii 0 i +N D R[i 0 ] = M[i 0 ] X v=1 R v [i 0 ] +R storage [i 0 ] +M D [i 0 ]R std [i 0 ];ii 0 i +N D (6.36) Chapter6 96 where P v [i 0 ] andR v [i 0 ] are the bidding decision of PEV labeled byv at thei 0 -th hour, which can be calculated using the method in section 6.2. P storage [i 0 ] andR storage [i 0 ] are the corresponding decision of the storage battery. M D [i 0 ] reflects the number of future parking PEVs which can be calculated using the following equation: M D [i 0 ] = 8 > > < > > : ^ M[i 0 ]M[i]; ^ M[i 0 ]>M[i] 0; ^ M[i 0 ]M[i] (6.37) 6.3.3 Adaptive Solution of PEV Charging Aggregator Control The solution of the charging aggregator optimization problem is shown in Fig. 6.7. We use an adaptive control method to dynamically update the optimal charging/discharging decision at each hour to mitigate the effect of RS tracking error as well as PEV arriving/departing. The detailed optimization steps are as follows: 1. At the beginning of each hour, the information of each PEV as well as the storage battery are updated. We set a decision window during which the future price functions are estimated and the charging/discharing decisions of the PEVs and storage battery are calculated. 2. We set the peak power constraintP max;C of each individual PEV . 3. Under the peak power constraint, each PEV optimizes the charging/discharging decision in- dependently using the algorithm presented in Section 6.2. Notice that the peak power bill is not considered for each individual PEV . Instead, the peak power consumption of each PEV is limited byP max;C to avoid mass unregulated charging processes. Chapter6 97 4. After the optimization framework of the PEVs, the optimization of the storage battery is per- formed. In this framework, we use a total cost function in Eqn. (6.33) including the peak power bill calculated using Eqn. (6.35). Knowing the decision of all the current parking PEVs as well as the estimated decision of future parking PEVs, the storage battery finds the optimal charging/discharg- ing schedule of all the future hours in the decision window. The target SoC level of the storage battery at the end of the decision window is determined based on historical data. 5. The charging aggregator estimate the total cost under currentP max;C and update current optimal solution. 6. Repeat from Setup 2 under a different power constraintP max;C until all theP max;C values are explored. 7. After getting the decisions of all the PEVs as well as the storage battery, the charging aggregator submit the total nominal power as well as the amount of RS reserve of the next hour to the power grid. 8. When the hour unfolds, the storage battery and the PEVs adjust the charging or discharging power based on the real-time RS signal. 9. At the end of each hour, the SoH of all the batteries are updated. The charging aggregator also update the information if there are PEVs arriving or departing during this hour. 10. Repeat from Setup 1 until we reach the end of the billing cycle. Chapter6 98 FIGURE 6.7: Flow of charging aggregator optimization. 6.4 Experimental Results 6.4.1 Results of Individual PEV Charging Control Optimization We first conduct a simulation of an individual PEV charging control optimization. We consider an hour-ahead RS provisioning power market and the PEV is equipped with a 24kWh Li-ion battery. The maximum charging and discharging power, P max;C and P max;D , are set to 10kW. The The transmission/conversion efficiency is set to 90%, and the Peukert factors 1 and 2 are set to 1.05 and 0.95, respectively. Chapter6 99 All SoH related parameters in Eqn. (6.8) are from [55]. Based on the current state of the practice, we set the calendar life of the battery pack to 5 years and the price of lithium-ion battery to $500/kWh. The dynamic pricing scheme for power consumption and regulation service are extracted from the locational based marginal pricing (LBMP) and regulation pricing provided by NYISO [66] for October 12th – 13th, 2014. The pricing history used for future price estimation as in Eqn. (6.16) is also from NYISO. At the same time, we use the tracking signal from PJM [63] (sampled per 4-second interval) on May 10th, 2014 for the value ofz(t) in our simulation. 3 The proposed algorithm is compared against two baseline algorithms. Both baselines apply dynamic- pricing-aware battery control. However, Baseline 1 does not participate in the regulation service and does not account for the SoH degradation. At the same time, Baseline 2 tries to benefit from the regulation service but ignores the SoH degradation. TABLE 6.1 shows the energy cost and aging cost of the optimal solution for differentSoC ini andSoC tar values, under different total charging hoursN. The energy cost and aging cost of the two baseline solutions are also shown in this table for comparison. Notice that a the energy cost might be negative, indicating that the PEV makes some profit by offering RS reservation or buying electricity when the electricity price is low while selling back when the price is high. It can be observed from Table 6.1 that our proposed solution achieves an total cost of 10% less than the baseline solutions on average. The benefit of the proposed algorithm mainly comes from the reduction of energy cost (30% reduction on average), as PEV owners can participate in RS 3 Please note that although we have assumed these parameters as given, our adaptive control algorithm can handle the variation of these parameters, because we re-calculate the optimal solution every hour based on the most updated information. If the amount of energy storage varies, we can just predict the value and update the information every hour. The process of the algorithm is not affected. Chapter6 100 TABLE 6.1: Comparison between the Cost of Proposed Solution and Baseline Solutions for Indi- vidual PEV Charging Optimization Parameters Cost of Proposed Solution ($) Cost of Baseline Solution 1($) Cost of Baseline Solution 2($) N SoCini SoCtar Energy Aging Total Energy Aging Total Energy Aging Total 3 0.2 0.5 0.115 0.6439 0.7589 0.3229 0.6346 0.9575 0.0975 0.8542 0.9517 3 0.5 0.8 0.1236 1.9277 2.0513 0.3229 1.9048 2.2277 0.0936 2.5796 2.6732 3 0.2 0.8 0.517 1.4189 1.9359 0.6617 1.3859 2.0476 0.5153 1.4184 1.9337 4 0.2 0.5 -0.1543 0.8698 0.7155 0.2736 0.6635 0.9371 -0.0408 0.8362 0.7954 4 0.5 0.8 -0.1322 2.2687 2.1365 0.2736 1.9917 2.2653 -0.0408 2.5101 2.4693 4 0.2 0.8 0.3195 1.5951 1.9146 0.6095 1.5686 2.1781 0.5054 1.3882 1.8936 6 0.2 0.5 0.1329 0.9595 1.0924 0.1692 0.9509 1.1201 0.01 1.3305 1.3405 6 0.5 0.8 0.1435 2.8758 3.0193 0.1692 2.8543 3.0235 0.0097 3.9955 4.0052 6 0.2 0.8 0.2955 1.9679 2.2634 0.3517 1.9339 2.2856 0.2155 1.9815 2.197 provisioning power market and get credit from offering RS reservation. The adaptive solution guarantees that PEV reaches the target SoC no matter whatz(t) signal is broadcasted. Another observation from Table 6.1 is that it is better for PEVs to start charging at a relatively low SoC ini level. The reason is that under the same total charging amount, a lowerSoC ini will lead to a smallerSoC avg and thus the aging cost can be reduced. 6.4.2 Results of PEV Charging Aggregator Control Optimization In the charging aggregator control optimization problem, we consider a senario on the first day of a month. The energy and regulation pricing are the same as in the single-vehicle simulations. The peak power pricing is set to $17.092/kW according to [65]. We consider 10 PEVs which visit the aggregator within one day. The arrival time, departure time, and the amount of energy needed are extracted from [67]. Since the initial SoC of each PEV is not present in the dataset, we assume a uniform distribution between 0 and 20% for each PEV’s SoC level when arriving at the aggregator. Each PEV is equipped with a battery pack of 24kWh energy. In addition, a 24kWh storage battery pack is installed in the aggregator. Chapter6 101 TABLE 6.2: Comparison between the Cost of Proposed Solution and Baseline Solutions for Charg- ing Aggregator Optimization Total Cost ($) Cost per Vehicle Hour ($) Peak Power Energy Aging Total Energy Aging Total (W) Proposed 9.9227 27.1672 37.0899 0.1575 0.4312 0.5887 12.39 Baseline 1 12.2586 28.8547 41.1133 0.1945 0.4580 0.6525 11.55 Baseline 2 10.0361 66.4801 76.5162 0.1593 1.0552 1.2145 16.64 Baseline 3 10.4005 27.2396 37.6401 0.1651 0.4323 0.5975 13.53 TABLE 6.2 shows the energy cost and aging cost of the optimal solution and the above-mentioned two baseline solutions for the total aggregator cost as well as the average hourly cost per vehicle. It can be observed from Table 6.2 that for both total aggregator cost and average cost per vehicle hour, our proposed solution achieves an total cost reduction of 10% compared with Baseline 1, and the cost reduction ratio is even higher than 50% compared with Baseline 2. Baseline 2 tries to benefit from the regulation service, the result turns out to be even worse than Baseline 1 when ignoring the SoH degradation. Instead, our proposed method considers both RS reserve and SoH degradation and achieves reductions of both energy cost and the cost associated with energy aging. To demonstrate the effectiveness of the charging aggregator, we add a third baseline, in which each PEV optimizes its charging decision individually (consider both RS reserve and SoH degradation) using the algorithm presented in Section 6.2. The results show that the introduction of the charg- ing aggregator can bring 2% total cost saving and 10% peak power reduction. According to our experimental results, the main contribution of the charging aggregator is the reduced peak power consumption. Considering the maintenance cost and energy storage cost, the introduction of the charging aggregator might result in an increased overall cost. The decision whether to introduce and maintain the aggregator will become a tradeoff between cost and peak power reduction. Fig. 6.8 shows the detailed result of our proposed optimal solution for electricity price and bidding Chapter6 102 decision for one vehicle. One can see that in our optimal solution, the PEV offers a considerable amount of RS reserve, which result in a large amount of energy cost reduction. The RS reserve has reduced a lot in the last hour, indicating that the PEV is about to leave the charging aggregator and needs to guarantee a target SoC level when it departs. FIGURE 6.8: Electricity Price and Bidding Decision of One Vehicle 6.5 Chapter Summary In this chapter, we start with the problem of PEV charging under an hour-ahead RS reservation market with dynamic energy prices, with a given total charging time as well as a target SoC level at that time. In this problem, we explicitly take into consideration the degradation of battery SoH during V2G operations, based on an accurate SoH modeling. The total cost function therefore becomes the summation of the energy cost during PEV charging (cost from declaimed nominal energy consumption minus payoff from declaimed RS reserves) and the extra cost associated with the aging of PEV battery. We derive an optimal control algorithm of PEV that adaptively updates its current SoC level and always makes the optimal bidding decision based on an accurately estimated Chapter7 103 price function in the future hours. The proposed algorithm also accurately accounts for the power loss during the charging and discharging process of PEV batteries, especially the rate capacity effect, and in power conversion circuits, which is often neglected in the reference work. Based on the solution of the individual PEV charging problem, we present an SoH-aware charg- ing aggregator design, which decides the control sequences of a group of PEVs to reduce the peak power caused by simultaneous PEV charging. In this problem, the peak power bill and future park- ing PEVs are properly taken care of. An energy storage system is used in the charging aggregator to mitigate the impact of real-time RS signal. Experimental results demonstrate that the proposed optimal PEV charging algorithm minimizes the combination of electricity cost and battery aging cost in the RS provisioning power market, and the introduction of charging aggregator can bring significant peak power reduction. We use actual data for RS reserve and dynamic energy prices and energy storage modeling from actual experiments in simulations, and the experimental results can serve practical purpose. Chapter 7 Conclusion This thesis introduces an integrated framework of energy-efficient infrastructures and policies. Four works are presented with different perspectives and focuses. They are the game-theoretic price determination algorithm for utility companies serving a community, the negotiation-based task scheduling algorithm is studied which minimizes energy user’s electricity bills under dynamic ener- gy prices, the optimal energy co-scheduling framework for energy efficient smart buildings, and the optimal PEV control problem with a charging aggregator considering regulation service provision- ing. For each work, the system model is described and the corresponding optimization solutions are presented. 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Abstract (if available)

Abstract The current smart grid technology is undergoing a transformation from a centralized, producer-controlled network to one that is less centralized and more consumer-interactive. To shape the power demand to reduce the peak power consumption and smoothen the variation, several energy-efficient infrastructures as well as the corresponding policies are introduced. The infrastructures include Hybrid Electrical Energy Storage (HEES) systems, Plug-in Electric Vehicles (PEVs), and energy efficient buildings. The policies include dynamic energy pricing and Regulation Service (RS) provisioning. In this dissertation, we present an integrated framework of these energy-efficient infrastructures and policies. Four works are introduced in this dissertation with different perspectives and focuses. They are the game-theoretic price determination algorithm for utility companies serving a community, the negotiation-based task scheduling algorithm is studied which minimizes energy user's electricity bills under dynamic energy prices, the optimal energy co-scheduling framework for energy efficient smart buildings, and the optimal PEV control problem with a charging aggregator considering regulation service provisioning.

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Creator Cui, Tiansong (author)

Core Title Integration of energy-efficient infrastructures and policies in smart grid

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 04/27/2017

Defense Date 03/22/2017

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

Tag dynamic energy pricing,energy-efficient building,hybrid electrical energy storage (HEES) system,OAI-PMH Harvest,plug-in electric vehicles (PEV),regulation service (RS) provisioning,smart grid

Language English

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Advisor Pedram, Massoud (committee chair), Nakano, Aiichiro (committee member), Nazarian, Shahin (committee member)

Creator Email cuitiansongcts@gmail.com,tcui@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-368816

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