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The smart grid network: pricing, markets and incentives

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The smart grid network: pricing, markets and incentives

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Content THE SMART GRID NETWORK: PRICING, MARKETS AND INCENTIVES by Wenyuan Tang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2015 Copyright 2015 Wenyuan Tang In memory of my grandfather ii Acknowledgments First and the foremost, I would like to express my sincere appreciation to my advisor, Profes- sor Rahul Jain. He led me into the world of game theory, which I found a wonderful tool for understanding human interactions, designing intelligent mechanisms, and ultimately promoting the social welfare. He has put much effort in cultivating my critical thinking, and has encouraged me to explore my own research interest. His patience and support have helped me overcome many difficulties in research. His deep knowledge and broad vision has made my PhD journey smooth and rewarding. I am also grateful for his generosity for providing travel funds which have enabled me to attend various conferences, where I can practice my presentation skills and meet researchers around the world. I would like to thank Professor Ram Rajagopal, who hosted me as a visiting scholar at Stanford University during the last months of my PhD, yet our collaboration started even earlier. He has an acute awareness of state-of-the-art research. He has taught me to achieve a balance between theory and practice. His sharp insights and passion have greatly inspired my research. His guidance has broadened my perspective on this field. I would like to thank Professor Ali Zahid, whose fascinating lectures aroused my interest in networks. I am very grateful to Professor Michael Neely, who places great emphasis on math- ematical rigor. I have learned a lot from him. I thank Professor Edmond Jonckheere, Professor Urbashi Mitra and Professor Suvrajeet Sen for serving on my qualifying exam committee. I thank Professor Ashutosh Nayyar and Professor Ketan Savla for serving on my dissertation committee. I appreciate the help from Professor John Silvester, Professor Jin Ma, Professor Cauligi Raghaven- dra and Professor Michael Enright, during my study at USC. I also thank Professor Steven Low, iii Professor Adam Wierman, Professor Eilyan Bitar, Professor Kameshwar Poolla, and Professor Pravin Varaiya who have given a lot of advice on my research projects. I am grateful to the hard work of the staff at USC. Special thanks to Diane Demetras, Annie Yu and Tim Boston for processing the paperwork meticulously and efficiently along my PhD. I would like to thank my labmates Dileep Kalathil, Harsha Honnappa, Naumaan Nayyar and Abhishek Gupta. Many thanks to Baosen Zhang, Desmond Cai, Yuanzhang Xiao, Longbo Huang, Yanzhi Wang, Haojun Yu, Shuping Liu, Daoyuan Zhai, Sucha Supittayapornpong, Hongyu Wu, Yanting Wu. Special thanks to my collaborator Junjie Qin, from whom I benefit a lot through many discussions. I owe many thanks to those who guided me, helped me and brought happiness into my life, from primary school to PhD. My sincere thanks to my teachers Yiling Chen, Haigen Yin, Haixia Wang, Professor Qingqing Ding, Professor Weidong Liu and Professor Chen Shen. I thank my friends Haojie Wu, Jingyan Li, Zhiren He, Jing Xu, Liuxin Ding, Linyu Qi, Qing Fan, Chengxiao Gao, Yunwen Chen, Sien Gu, Yichen Bao, Yuzhang Ni, Yue Cheng, Mingming Yang, Yin Song, Yuan Li, Feng Xie, Xiaonan Lu, Haiwang Zhong, Miao He, Fei Wang, Xinyu Yao and Guanying Zhu. Finally, I would like to express my deepest gratitude to my dear family. I am greatly indebted to my parents. It is their constant care, support and devotion that made this achievement possible. I sincerely appreciate the company and encouragement of my fianc´ ee Yun Li. Your love is lighting the road to truth. iv Abstract The electric power system is regarded as the largest and most complex machine ever built, in which supply and demand must be balanced on a second-by-second basis. Consequently, the electricity market has to be carefully designed to prevent a recurrence of the California electricity crisis. This task is even more demanding with the popularization of the smart grid. Our work attempts to address the following question: what smart mechanisms should a smart grid be equipped with, for a smart market that can be utilized but not manipulated by smart people? Based on a two-level market model, we focus on the control, optimization and market design of the smart grid, where we incorporate three key elements: the increasing penetration of renew- able generation, the increasing participation of demand response, and the fast development and deployment of energy storage systems. On the upper level is the primary market coordinated by the independent system operator. We consider the economic dispatch problem, and develop a game-theoretic framework to investigate the market outcomes with strategic generators under locational marginal pricing. We then consider a dynamic extension, and investigate how the use of storage may affect the market structure and market outcomes. We also study a multistage energy procurement problem, and design incentiviz- ing pricing mechanisms that facilitate efficient participation of the generators. On the lower level, there are two secondary markets. The secondary market I, when it exists, involves an aggregator who buys power from a group of generators and sells it to the primary market. In particular, we study how to design market mechanisms for buying wind power. We first consider the welfare-maximizing objective, and propose the stochastic resource auction paradigm v that elicits probability distributions of wind power generation. We then consider the revenue- maximizing objective, and study how to extract the surplus given the correlation among wind power generation. The secondary market II mimics the retail electricity market. We study the distributor’s prob- lem how to utilize demand response in an adaptive manner. In a stochastic setting, the optimal pricing scheme should evolve according to the up-to-date information. We develop a distributed algorithm to compute the optimal price process that incentives the agents to choose the socially optimal decisions. Finally, we study a network resource allocation problem in a hierarchical setting. Motivated by the results on hierarchical mechanism design, we discuss the challenges of the integration of the two-level markets. This is an open problem and the direction of our future work. vi Contents Dedication ii Acknowledgments iii Abstract v List of Figures ix List of Tables xi 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Market Structure of Electric Power Systems . . . . . . . . . . . . . . . . . 1 1.1.2 Smart Grid of the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Framework of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 A Two-Level Market Model . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Economic Dispatch and Market Power 7 2.1 Economic Dispatch: Problem and Game . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Game-Theoretic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Market Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.4 Market Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.5 Economic Dispatch Game with Cournot Offers . . . . . . . . . . . . . . . 21 2.1.6 Marginal Contribution Pricing Mechanism . . . . . . . . . . . . . . . . . 24 2.2 Dynamic Economic Dispatch Game . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.2 Efficient Bids: Resolving the Issue of Storage-Unawareness . . . . . . . . 33 2.2.3 Equilibrium Analysis: the Value of Storage . . . . . . . . . . . . . . . . . 34 2.2.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 vii 3 Multistage Energy Procurement 45 3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Temporal Marginal Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.2 Vulnerability to Misreporting . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Dynamic Marginal Contribution Pricing . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Buying Random Wind Power 61 4.1 Welfare-Maximizing Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1.2 Selecting Wind Power Providers . . . . . . . . . . . . . . . . . . . . . . . 66 4.1.3 Pricing Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Revenue-Maximizing Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5 Stochastic Dynamic Pricing for Demand Response 99 5.1 User’s Problem: Stochastic Optimal Control . . . . . . . . . . . . . . . . . . . . . 101 5.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1.2 Inelastic Demand Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1.3 Elastic Demand Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 System Problem: Stochastic Dynamic Pricing . . . . . . . . . . . . . . . . . . . . 107 5.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2.2 Dual Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2.3 Offline Distributed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2.4 Investigating the Information Structure . . . . . . . . . . . . . . . . . . . 112 5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6 Toward an Integrated Smart Grid 116 6.1 Hierarchical Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.1.2 Hierarchical Auctions for Indivisible Resources . . . . . . . . . . . . . . . 121 6.1.3 Hierarchical Auctions for Divisible Resources . . . . . . . . . . . . . . . . 129 6.1.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.2 Challenges of the Integration of the Two-Level Markets . . . . . . . . . . . . . . . 139 References 141 viii List of Figures 1.1 A two-level model of the smart grid market. . . . . . . . . . . . . . . . . . . . . . 5 2.1 Comparison of offer formats used in the SFE literature and in this work. . . . . . . 13 2.2 Example of a two-bus network in which a Nash equilibrium does not exist. . . . . . 14 2.3 Example of a two-bus network in which the price of anarchy can be arbitrarily large. 16 2.4 The true cost functionc n;t (), the bid b n;t () used in practice, and the simplified bid b n;t () used in this work specified by a four-dimensional signal (r n;t ;s n;t ;r + n;t ;s + n;t ). 32 2.5 Example of a two-bus, two-generator network. . . . . . . . . . . . . . . . . . . . . 35 2.6 Example of a two-bus, four-generator network. . . . . . . . . . . . . . . . . . . . 36 2.7 The social cost versus flow limit under three scenarios, when the storage capacity is sufficiently large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.8 The social cost versus storage capacity under three scenarios, when the flow limit is sufficiently large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.9 The aggregate generation versus time under different storage capacities, when the flow limit is sufficiently large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.10 The LMP versus time under different storage capacities, when the flow limit is sufficiently large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1 Multistage energy procurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Expected profit earned by each generator. . . . . . . . . . . . . . . . . . . . . . . 59 3.3 TMP system cost with misreporting. . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4 DMCP generator profit with misreporting. . . . . . . . . . . . . . . . . . . . . . . 60 4.1 The timeline of the stochastic resource auction with independent types. . . . . . . . 66 4.2 The expected social cost versus the number of wind farms under two market archi- tectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 The share of wind power generation versus the number of wind farms under two market architectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 The expected payoff of wind farm 1 versus the number of wind farms under two market architectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 The expected payoff of wind farm 1 versus the reported mean in the proposed architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.6 The timeline of the stochastic resource auction with correlated types. . . . . . . . . 84 4.7 The expected revenue of the full information mechanism for problem (4.26) and the optimal mechanism for problem (4.23) versus the uncertainty in i ’s, whereas i ’s are fully correlated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 ix 5.1 The optimal policy (x t (y t ; t );z t (y t ; t )) of problem (5.5) as a function ofy t for a fixed t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.1 An example of a 3-tier network withN = 6. . . . . . . . . . . . . . . . . . . . . . 119 x List of Tables 2.1 Cost Functions for the IEEE 57-bus system . . . . . . . . . . . . . . . . . . . . . 40 2.2 Social Costs between SED and DED . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1 Cost Functions of Thermal Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Characteristics of Wind Farms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1 The Optimal Price Process in Example 5.1 . . . . . . . . . . . . . . . . . . . . . . 114 xi Chapter 1 Introduction 1.1 Background 1.1.1 Market Structure of Electric Power Systems The fundamental difference between electricity and other commodities is that electricity is inex- tricably linked with a physical system that functions much faster than any other market [1]. In the power system, supply and demand must be balanced on a second-by-second basis. Moreover, elec- tricity is by its nature difficult to store. This makes the design of electricity markets a tremendously complex task. The electricity industry throughout the world, which has long been dominated by vertically integrated utilities, is undergoing enormous changes [2]. The deregulation and reconstruction of the electricity industry has led to the development of open markets, in which two dominant modes of trading are bilateral trading and competitive electricity pools [3]. The former involves negotia- tion on delivery of power at some future time between two parties (a buyer and a seller). The latter is a centralized marketplace that clears the market for numerous buyers and sellers. A common pool trading structure consists of two successive ex ante markets: a day-ahead forward market and a real-time spot market [4]. A day-ahead market schedules resources at each hour of the following day. A real-time market ensures the balance of supply and demand, which is cleared 5 to 15 min before the operating interval that is on the order of 5 min. A typical deregulated electricity market involves a wholesale electricity market and a retail electricity market. In a pool-based wholesale electricity market, generators and distributors submit supply and demand curves to the independent system operator (ISO), who administers a bid-based, 1 security-constrained, economic dispatch. Distributors then resell the electricity to consumers in retail electricity markets. 1.1.2 Smart Grid of the Future The electric grid is undergoing profound transformation due to the following: increasing penetra- tion of renewable generation; increasing participation of demand response; and fast development and deployment of energy storage systems [5]. Renewable energy will increasingly constitute a greater fraction of the energy portfolio. In April 2011, the Governor of California signed legislation to require one-third of the state’s elec- tricity to come from renewable energy by 2020. There are many system challenges in integrating renewable energy into the current power grid and electricity markets. These mainly arise due to the variability and unpredictability of such energy sources. For example, wind power generation at a single wind farm can vary from 0 to 100 MW in a matter of a few hours. Solar power is equally unpredictable and highly variable, while tidal power has a cyclic nature, marked by extreme peaks (during extreme events such as storms and hurricanes). This introduces significant challenges in matching supply with demand. For this purpose, demand response solutions are being devised where consumers are exposed to time-varying prices via smart-meters over a smart-grid infras- tructure [6]. However, demand response may not be enough to compensate for the highly volatile fluctuation of the output of wind generators. The energy storage system is considered key to open- ing the possibility of high penetration of wind and solar generation [7]. Storage systems will greatly enhance the controllability of the grid as well. The smart grid is now a general term for a modernized and upgraded electricity grid, with the use of sensors, communications, computational ability and control in some form to enhance the overall functionality [8]. The smart grid is expected to employ the advanced information and communication technologies to optimize its operation and to fully accommodate new technologies of renewable generation, demand response, storage, and so on. 2 1.2 Framework of the Dissertation 1.2.1 Motivation The electric power system is regarded as the largest and most complex machine ever built. Conse- quently, the electricity market has to be carefully designed to prevent a recurrence of the California electricity crisis. This task is even more demanding with the popularization of the smart grid. Our work attempts to address the following question: What smart mechanisms should a smart grid be equipped with, for a smart market that can be utilized but not manipulated by smart people? Our general interest is control, optimization and market design of the smart grid. The problems we will study incorporate the following features: 1. Hierarchical. Since the wholesale electricity market and the retail electricity market are oper- ated on two different levels, the entire market structure is hierarchical. Thus, it is important to understand how to integrate those markets, in which there are middlemen (e.g., distribu- tors) who participate in both markets. 2. Networked. A power network is a network, with tens to thousands of nodes (or buses). Moreover, power networks are more complex than communication or transportation net- works, because power flows over the network in accordance with the Kirchoff’s circuit laws. Results based on a single bus case may not be generalized to a networked case. 3. Dynamic. In power networks, there are physical constraints such as ramp rate limits that couple problems at different periods. The use of storage also requires optimization and control in a dynamic model. 4. Stochastic. The penetration of renewable generation introduces uncertainty in power sys- tems, which affects supply and thus market prices as well. Therefore, a deterministic model may not be appropriate. We need a stochastic model to characterize the evolution of the system as well as the optimal control policy. 3 5. Strategic. The participants are economic agents each with their own objectives and some private information as well. Hence, they would be expected to behave in a way to further their own interests, or in other words, act strategically, and even give misleading information if it benefits them. To model market power and manipulation, sometimes it is necessary to formulate the problem in a game-theoretic setting. 1.2.2 A Two-Level Market Model Our work is based on a two-level model of the smart grid market, as shown in Fig. 1.1. This model is a stylized abstraction of the real world electricity market. We describe the model and the market entities in the following: 1. Primary market. This mimics the wholesale electricity market in practice. The primary mar- ket is a double-sided market consisting of aggregators as sellers and distributors as buyers. This market is coordinated by the ISO, who administers a bid-based, security-constrained, economic dispatch. 2. Secondary market I. In our model, the term aggregator refers to the one who supplies power generation. It can be a single generator who participates in the primary market directly, in which case the secondary market does not exist; or a middleman who purchases power from a group of generators and sell it in the primary market. 3. Secondary market II. This mimics the retail electricity market in practice. This market is coordinated by the distributor (or retailer, provider, utility company, load serving entity, etc., which are used interchangeably), who purchases power from the primary market and sell it to commercial and residential consumers (which we will simply refer to as users). By constructing the entire market as a primary market interacting with two secondary markets, we classify the problems in the smart grid market into three categories: the ISO’s problem, the aggregator’s problem and the distributor’s problem. The general task is modeling, analysis and 4 ISO Aggregator Distributor Wind Power Generator Coal-Fired Generator Solar Power Generator Commercial Consumer Residential Consumer Residential Consumer Primary Market Secondary Market I Secondary Market II Figure 1.1: A two-level model of the smart grid market. design for each market, where we incorporate three key elements of the smart grid: high penetra- tion of renewables, increasing participation of demand response, and widespread deployment of storage. 1.2.3 Organization We study the ISO’s problems in Chapter 2 and 3. In Chapter 2, we consider the economic dispatch problem, and develop a game-theoretic framework to investigate the market outcomes with strate- gic generators under locational marginal pricing [9, 10]. We then consider a dynamic economic dispatch problem, where the operation of the storage introduces time-coupling constraints. We investigate how the use of storage may affect the market structure and market outcomes [11, 12]. In Chapter 3, we consider a multistage energy procurement problem, and design incentivizing pricing mechanisms that facilitate efficient participation of the generators [13]. 5 In Chapter 4, we study the aggregator’s problem how to design market mechanisms for buy- ing wind power. We first consider the welfare-maximizing objective, and propose the stochastic resource auction paradigm that elicits probability distributions of wind power generation [14, 15]. We then consider the revenue-maximizing objective, and study how to extract the surplus given the correlation among wind power generation [16]. In Chapter 5, we study the distributor’s problem how to utilize demand response in an adaptive manner. In a stochastic setting, the optimal pricing scheme should evolve according to the up-to- date information. We develop a distributed algorithm to compute the optimal price process that incentives the agents to choose the socially optimal decisions [17]. To conclude, we point out an open problem in Chapter 6: how to integrate the two-level mar- kets, which evolve simultaneously and dynamically, and interact with each other. We review our earlier work on hierarchical mechanism design [18, 19]. We then discuss the challenges of the integration of the smart grid market. 6 Chapter 2 Economic Dispatch and Market Power 2.1 Economic Dispatch: Problem and Game Electric power is traded in a wholesale electricity market that involves various entities: the gen- erators who generate and sell power, the distributors who buy power and sell to consumers, and the independent system operator (ISO) who coordinates the operation of the market and the power system. The generators and the distributors submit economic signals for supply and demand to the ISO, who then determines an optimal dispatch that maximizes the social welfare while satisfy- ing the physical and operational constraints. This problem is referred to as the economic dispatch problem. While the economic dispatch is computed, the price paid (or charged) to each generator (or distributor) is determined accordingly. Locational marginal pricing (LMP) is a widely used mech- anism for pricing electricity in the wholesale electricity market [4, 1]. When the market is compet- itive, the LMP mechanism is efficient, i.e., inducing the economic dispatch. However, those market participants are strategic agents who potentially possess market power. They may not have incentives to reveal their true characteristics, or may even provide misleading information for their own interests. For example, Enron’s energy traders found ways to manipulate the congestion prices that led to the California electricity crisis of 2000–01 [20]. While market power can be measured using market power indices [21], those indices have their own limitations and may not adequately characterize the situation of market manipulation. In this work, we adopt a game-theoretic approach to study the LMP mechanism through inves- tigating the equilibrium outcomes. We focus on the strategic bidding of the supply side, assume general cost functions, and take the transmission constraints into account. Our model differs from the supply function equilibrium (SFE) models [22, 23], in that we employ piecewise constant 7 offer curves as in practice, in contrast to affine offer curves as in the SFE literature. Moreover, we explore the existence and the efficiency of Nash equilibria, while most of the existing work focuses on computing SFEs [24, 25, 26, 27]. Our main results are twofold. On the one hand, we show that a Nash equilibrium may not exist, or that the price of anarchy can be arbitrarily large. On the other hand, we provide sufficient conditions under which there exist efficient Nash equilibria. We then study the LMP mechanism with Cournot offers, in which generators submit quantity offers instead of price/quantity offer curves. Due to the tractability, Cournot models have gained popularity in electricity market analysis [28, 29, 30]. We establish the existence of Nash equilibria under certain mild conditions. Furthermore, there are other mechanisms as alternatives to the LMP mechanism. For example, the pay-as-bid pricing mechanism is compared with the LMP mechanism in [31]. We propose the marginal contribution pricing (MCP) mechanism which always induces efficient Nash equilibria, using piecewise constant offer curves. 2.1.1 Problem Statement We consider a connected power network which consists ofI buses indexed byi = 1;:::;I, and N generators indexed by n = 1;:::;N. There can be zero, one, or more generators located at each bus i, the set of which is denoted by N i . For each generator n, the cost of supplying x n is c n (x n ). The cost function c n : R + ! R + is assumed to be strictly increasing, convex and piecewise differentiable. The competitive demand at each busi is modeled by an inverse demand functionp i (y i ), which maps the quantity y i to the maximum price that the consumers are willing to pay. The inverse demand function p i : R + ! R + is assumed to be continuous and decreasing. It is sometimes convenient to work with the valuation function v i (y i ) = Z y i 0 p i (z)dz; 8 which is increasing, concave and continuously differentiable. In this work, we will also consider inelastic demand, in which case all they i ’s are fixed. For analytical and computational simplicity, we adopt a DC power flow model as a common practice [32]. In the DC flow model, a branch i-j is characterized by B ij , the negative of its susceptance, which satisfiesB ij = B ji 0. Let i be the voltage phase angle at busi. Then the active power flow over branchi-j is given by f ij =B ij ( i j ): The bus power balance equation for busi is the following: X n2N i x n y i = X j f ij : Let C ij be the flow limit of branch i-j, which satisfies C ij = C ji 0. The branch power flow constraint for branchi-j is the following: f ij C ij : Letx = (x 1 ;:::;x N ),y = (y 1 ;:::;y I ), and = ( 1 ;:::; I ). The economic dispatch problem is to determine an optimal allocation of supply and demand that maximizes the social welfare while satisfying the transmission constraints. Formally, it is a convex optimization problem: maximize x;y; X i v i (y i ) X n c n (x n ) (2.1a) subject to X n2N i x n y i = X j B ij ( i j );8i; (2.1b) B ij ( i j )C ij ;8(i;j); (2.1c) x n 0;8n; (2.1d) y i 0;8i: (2.1e) 9 The primal optimal solution to (2.1) is called the economic dispatch, denoted by (x ;y ; ). Since P i ( P n2N i x n y i ) = P i P j B ij ( i j ) = 0, the system of linear equations (2.1b) overi is underdetermined with respect to. In fact, only the phase angle differences matter. Thus, for computational purposes, we choose bus 1 as the slack bus by setting 1 = 0. The economic dispatch problem is based on the given on/off states of the generators, in com- parison with the unit commitment problem. In the proposed model, theN generators have been started up, so that the fixed cost of each generator does not affect the dispatch. Thus, we further assumec n (0) = 0 for alln without loss of generality. For a more detailed DC power flow model (e.g., where line losses are taken into account), the reader can refer to [33]. Associate the dual variables i with (2.1b) and ij with (2.1c). Let = ( 1 ;:::; I ) and = [ ij ] II . It is easily seen that the economic dispatch problem (2.1) is always feasible, and that the (refined) Slater’s condition is automatically satisfied. Therefore, strong duality holds, and the Karush-Kuhn-Tucker (KKT) conditions provide necessary and sufficient conditions for optimality [34]: (c 0 n (x n ) i )x n = 0;8n2N i ; (2.2a) c 0 n (x n ) i 0;8n2N i ; (2.2b) (v 0 i (y i ) i )y i = 0;8i; (2.2c) v 0 i (y i ) i 0;8i; (2.2d) X j B ij ( i j + ij ji ) = 0;8i; (2.2e) X n2N i x n y i X j B ij ( i j ) = 0;8i; (2.2f) ij (B ij ( i j )C ij ) = 0;8(i;j); (2.2g) B ij ( i j )C ij 0;8(i;j); (2.2h) x n 0;8n; (2.2i) y i 0;8i; (2.2j) ij 0;8(i;j): (2.2k) 10 Note that whenc n is not differentiable atx n ,c 0 n (x n ) should be interpreted as a subderivative, which always exists given the assumptions of the cost function. Denote by ( ; ) the dual optimal solution to (2.1). Then (x ;y ; ; ; ) satisfies the KKT conditions (2.2). The locational marginal price (LMP) at each busi is given by i , which is the cost of serving an additional increment of load at that bus. Thus, the payoff of generatorn located at busi is given by n = i x n c n (x n ): The LMP mechanism is efficient in a competitive environment, i.e., when generators are price takers. If the price at each bus i is fixed as i , and each generator determines its supply to maximizes its own payoff, then the resulting allocation is the economic dispatch. This can be seen from the KKT conditions, which state that the marginal cost of a generator with positive generation is exactly the LMP at that bus. 2.1.2 Game-Theoretic Formulation To solve (2.1) for the economic dispatch, the ISO needs to know the cost functions. However, strategic generators may not have incentives to reveal their true cost functions. They are often not even able to do so, since the offer is typically a low-dimensional signal that may not fully parametrize the cost function. This motivates the formulation of the economic dispatch game. We first specify the strategy space, or the offer format. An offer curvep n (x n ) maps the quantityx n to the minimum price that generatorn is willing to accept. It is supposed to approximate the marginal cost functionc 0 n (x n ), but not necessarily. The offer curvep n (x n ) corresponds to a reported cost ^ c n (x n ) = Z xn 0 p n (z)dz: 11 The offer format refers to the parametrization of the offer curve, or equivalently, the parametriza- tion of the reported cost. In the SFE literature, the offer curve often takes an affine form: p n (x n ) =a n +b n x n ; a n > 0; b n > 0; which corresponds to a quadratic reported cost ^ c n (x n ) =a n x n + (b n =2)x 2 n : In practice, however, most ISOs adopt piecewise constant (or step) offer curves. Such an offer curve corresponds to a piecewise linear reported cost. For example, the California ISO asks for 10 price/quantity pairs [35], which correspond to a 10-segment piecewise linear reported cost. In this work, we employ the realistic offer format. For simplicity, we consider a two-segment piecewise linear reported cost parametrized by a four-dimensional signal (r n ;s n ;r + n ;s + n ), where 0r n r + n ; 0s n s + n : ^ c n (x n ) = 8 > > < > > : r n x n ; x n 2 [0;s n ]; r n s n +r + n (x n s n ); x n 2 (s n ;s + n ]: Fig. 2.1 illustrates the offer format used in the SFE literature and that used in this work. We note that quadratic costs provide smooth dispatch, revenue and profit curves that facilitate calculus- based analysis, while piecewise linear costs do not produce continuously differentiable dispatch, revenue and profit curves, requiring different analysis techniques [36]. This distinguishes our work from the SFE literature. Now we specify the economic dispatch game. Let ^ c = (^ c 1 ;:::; ^ c N ) be the offer profile, i.e., the collection of reported cost. In reality, the ISO solves the following problem: maximize x;y; X i v i (y i ) X n ^ c n (x n ) (2.3a) 12 O x n p n (x n ) O x n ^ c n (x n ) (a) Affine offer curve and quadratic reported cost. O x n p n (x n ) r n r + n s n s + n O x n ^ c n (x n ) r n r + n s n s + n (b) Piecewise constant offer curve and piecewise linear reported cost. Figure 2.1: Comparison of offer formats used in the SFE literature and in this work. subject to X n2N i x n y i = X j B ij ( i j );8i; (2.3b) B ij ( i j )C ij ;8(i;j); (2.3c) x n 0;8n; (2.3d) y i 0;8i: (2.3e) Compared with (2.1), the true costs are replaced by the reported costs in (2.3). Let (x ;y ; ) be the primal optimal solution to (2.3). The LMP at each busi is given by i , the optimal dual 13 1 1 x 1 C 2 2 x 2 y Figure 2.2: Example of a two-bus network in which a Nash equilibrium does not exist. variable associated with the bus power balance equation (2.3b). Thus, the payoff of generatorn located at busi is given by n = i x n c n (x n ): We adopt the pure strategy Nash equilibrium as the solution concept. An offer profile is a Nash equilibrium if no generator can get a higher payoff by a unilateral deviation. A Nash equilibrium is efficient if it induces the economic dispatch (x ;y ; ). 2.1.3 Market Manipulation We have formulated an economic dispatch game to study the LMP mechanism with the realistic offer format. In this section, we illustrate the phenomena of market manipulation. Specifically, we present counterexamples to show that a Nash equilibrium may not exist, and that even when a Nash equilibrium exists, the price of anarchy can be arbitrarily large. The following example is adapted from [24], where SFEs are computed with affine offer curves. Our focus, however, is on the existence of Nash equilibria with piecewise linear offer curves. Example 2.1. Consider the network as shown in Fig. 2.2. There are two buses, with generator 1 at bus 1 and generator 2 at bus 2. The cost functions arec 1 (x 1 ) = 0:01x 2 1 + 10x 1 andc 2 (x 2 ) = 0:01x 2 2 + 10x 2 . There is no demand at bus 1. The inverse demand function at bus 2 is p(y) = 30 0:08y. The branch flow limit isC = 100. We prove by contradiction that a Nash equilibrium does not exist in this example. 14 Suppose that a Nash equilibrium (^ c 1 ; ^ c 2 ) exists, which induces the outcome (x 1 ;x 2 ; 1 ; 2 ). Since there is no demand at bus 1, 1 2 . If 1 < 2 , it is easily seen that generator 1 has an incentive to deviate. Thus, 1 = 2 . Sincec 0 1 (C) = 12 < p(2C) = 14, the branch must be congested; otherwise, it can be shown that at least one of the generators has an incentive to deviate. Thus, x 1 = 100; 1 = 2 12. Then (x 2 ; 2 ) must solve the following problem: maximize x 2 ; 2 2 x 2 c 2 (x 2 ) subject to 2 =p(x 1 +x 2 ); x 2 ; 2 0; which givesx 2 = 66:67; 2 = 16:67. Since a portion ofx 1 is supplied at a reported marginal cost 1 = 2 = 16:67>c 0 2 (x 2 ) = 11:33, it can be shown that generator 2 has an incentive to deviate: by decreasing 2 by an arbitrarily small number > 0 and increasingx 2 , generator 2 can get a higher payoff. Therefore, there does not exist a Nash equilibrium. Remark 2.1. In this example, we illustrate a situation of market manipulation. The result is closely related to the concept of market power. Indeed, the nonexistence of Nash equilibria is due the exercise of market power of generator 2. However, not all market power indices are relevant. For example, the nodal must-run share (NMRS) [37] is undefined here, since the demand is elastic. There are also situations in which Nash equilibria exist but some of them are undesirable in terms of efficiency. The price of anarchy is a metric that measures how the efficiency degrades due to the strategic behavior, compared with the socially optimal outcome. Consider the case of inelastic demand, in whichy i is fixed for alli. Let (x ; ) be any equilibrium dispatch. The price of anarchy is defined as the ratio between the cost of the worst equilibrium and the socially optimal cost: PoA = max P n c n (x n ) P n c n (x n ) ; 15 1 1 x 1;k 4 x 4;k y = 2C C 2 2 x 2;k 3 x 3;k Figure 2.3: Example of a two-bus network in which the price of anarchy can be arbitrarily large. where the maximum is taken over the set of equilibrium dispatch. The following example shows that the price of anarchy can be arbitrarily large. Example 2.2. Consider the network as shown in Fig. 2.3. The network has two buses, with gen- erator 1 and 4 at bus 1, and generator 2 and 3 at bus 2. The branch flow limit isC. The inelastic demand at bus 1 isy = 2C. There is no demand at bus 2. Consider a sequence of gamesf k ;k 1g. In game k , the cost functions are given by c 1;k (x 1;k ) =x 1;k ; c 2;k (x 2;k ) =kx 2;k ; c 3;k (x 3;k ) =kx 3;k ; c 4;k (x 4;k ) = 2kx 4;k : The economic dispatch is (x 1;k ;x 2;k ;x 3;k ;x 4;k ) = (2C; 0; 0; 0); with a social cost 2C. One Nash equilibrium is ^ c 1;k (x 1;k ) = 2kx 1;k ; x 1;k 2C; ^ c 2;k (x 2;k ) =kx 2;k ; x 2;k 2C; ^ c 3;k (x 3;k ) =kx 3;k ; x 3;k 2C; ^ c 4;k (x 4;k ) = 2kx 4;k ; x 4;k 2C; 16 which induces the outcome (x 1;k ;x 2;k ;x 3;k ;x 4;k ) = (C;C; 0; 0); ( 1;k ; 2;k ) = (2k;k); with a social cost C +kC. For example, generator 1 has no incentive to report ^ c 1;k (x 1;k ) = kx 1;k ;x 1;k 2C, since its new payoff, (k 1)2C, would be smaller than its current payoff, (2k 1)C. Thus, the price of anarchy is at least C +kC 2C = k + 1 2 ; which is unbounded ask!1. Remark 2.2. In this example, we illustrate another situation of market manipulation. While a Nash equilibrium always exists, it can be arbitrarily inefficient. This is due to the fact that generator 1 possesses the market power. However, the market power index NMRS is not informative here, which is zero for each generator. 2.1.4 Market Efficiency On the other hand, the LMP mechanism may work well. We propose two sufficient conditions under either of which there exists an efficient Nash equilibrium. In the game-theoretic language, the price of stability (defined as the ratio between the cost of the best equilibrium and the socially optimal cost) is equal to 1. Definition 2.1 (Congestion-Free Condition). No branch power flow constraint (2.1c) is binding at the economic dispatch (x ;y ; ). Lemma 2.1. Under the congestion-free condition, all the LMPs are equal at the economic dis- patch. 17 Proof. Under the congestion-free condition, ij = 0 for all (i;j). From the KKT conditions (2.2), we have X j B ij ( i j ) = 0;8i: LetI = arg max i i be the set of buses with the largest LMPs. For eachi2I, since i j andB ij 0 for allj, there must be j = i forj connected toi such thatB ij > 0. It follows by the connectedness of the network that all the buses belong toI. That is, i = 1 for alli. It is immediate to prove by construction the existence of efficient Nash equilibria under the congestion-free condition, which also tells how to compute such equilibria directly. In the follow- ing, letQ be a large enough constant. Theorem 2.1. Under the congestion-free condition, there exists an efficient Nash equilibrium in the economic dispatch game. Proof. By Lemma 2.1, i = 1 for alli. Let ^ c be an offer profile where ^ c n (x n ) = 1 x n ;x n Q for alln. Clearly, (x ;y ; ; ; ) is also a solution to (2.3). It remains to show that ^ c is a Nash equilibrium. Consider generatorn located at bus i. From the KKT conditions, x n 2 arg max xn 1 x n c n (x n ). Its current payoff is n = 1 x n c n (x n ). Suppose that it changes its offer to ^ c n , resulting a new dispatch (^ x; ^ y; ^ ; ^ ; ^ ) given (^ c n ; ^ c n ). In light of ^ c n , we have ^ i 1 . Its new payoff would be ^ n = ^ i ^ x n c n (^ x n ) 1 ^ x n c n (^ x n ) 1 x n c n (x n ) = n : Thus, it has no incentive to deviate. This proves that the constructed offer profile is a Nash equi- librium. 18 Remark 2.3. Since each ^ c n is linear (with a large enough capQ), it is essentially one-segment so that the value ofs n does not matter. On the other hand, if we sets n =x n for alln, this provides a suggested dispatch point for the ISO for tie-breaking purposes. To put it another way, if we further setr + n = 1 + with a small enough > 0 for alln, then we obtain an efficient-Nash equilibrium. Definition 2.2 (Monopoly-Free Condition). There are at least two, or no generators at each bus. This condition is easier to use than the congestion-free condition, since we only need to know the placement of the generators. The constructive proof is similar as before. Theorem 2.2. Under the monopoly-free condition, there exists an efficient Nash equilibrium in the economic dispatch game. Proof. Let ^ c be an offer profile where ^ c n (x n ) = i x n ;x n Q for all n 2 N i . Then (x ;y ; ; ; ) is also a solution to (2.3). It remains to show that ^ c is a Nash equilib- rium. Consider generatorn located at bus i. From the KKT conditions, x n 2 arg max xn i x n c n (x n ). Its current payoff is n = i x n c n (x n ). Suppose that it changes its offer to ^ c n , resulting a new dispatch (^ x; ^ y; ^ ; ^ ; ^ ) given (^ c n ; ^ c n ). Under the monopoly-free condition, there is at least one another generatorm located at the same busi, whose offer is ^ c m (x m ) = i x m ;x m Q. So we have ^ i i . Generatorn’s new payoff would be ^ n = ^ i ^ x n c n (^ x n ) i ^ x n c n (^ x n ) i x n c n (x n ) = n : Thus, it has no incentive to deviate. This proves that the constructed offer profile is a Nash equi- librium. 19 Example 2.3. In Example 2.2, the monopoly-free condition holds, so that an efficient Nash equi- librium exists: ^ c 1;k (x 1;k ) =x 1;k ; x 1;k 2C; ^ c 2;k (x 2;k ) =kx 2;k ; x 2;k 2C; ^ c 3;k (x 3;k ) =kx 3;k ; x 3;k 2C; ^ c 4;k (x 4;k ) =x 4;k ; x 4;k 2C; which induces the outcome (x 1;k ;x 2;k ;x 3;k ;x 4;k ) = (2C; 0; 0; 0); ( 1;k ; 2;k ) = (1; 1): Remark 2.4. The intuition of the two sufficient conditions is to limit the exercise of market power. On the other hand, neither of them is necessary for the existence of Nash equilibria, whether efficient or not. That said, they may be used together for a larger class of scenarios. As a side motivation of our work, the offer format may have a significant effect on the equi- librium outcomes. It has been shown that different parametrizations of affine offer curves lead to different outcomes [38]. We further examine the case of piecewise constant offer curves. The following example shows that while piecewise constant offer curves yield an efficient Nash equilibrium, affine offer curves may not do so. Example 2.4. Consider two identical generators located at a single bus, with cost functions c n (x n ) =x 2 n forn = 1; 2. The inverse demand function isp(y) = 2y. Since both the congestion- free and the monopoly-free conditions hold, an efficient Nash equilibrium exists with piecewise linear reported costs: ^ c 1 (x 1 ) =x 1 ; x 1 1; ^ c 2 (x 2 ) =x 2 ; x 2 1; 20 which induces the outcome (x 1 ;x 2 ;y ) = (0:5; 0:5; 1); = 1: Now consider quadratic reported costs which take the following form: ^ c n (x n ) = n x 2 n : Suppose that there exists an efficient Nash equilibrium ( 1 ; 2 ). There must be 1 = 2 = 1. Then either of them has an incentive to deviate. For example, given 2 = 1, the best response of generator 1 is 1 = 4=3. Thus, there is no efficient Nash equilibrium with quadratic reported cost. 2.1.5 Economic Dispatch Game with Cournot Offers We now study the economic dispatch game with Cournot offers. Note that the pricing mechanism is still the LMP mechanism. In this game, each generatorn submits a scalar offerx n 0 which specifies the quantity to be generated. Letx = (x 1 ;:::;x N ) be the offer profile. Then the ISO solves the following problem: maximize y; X i v i (y i ) (2.4a) subject to X n2N i x n y i = X j B ij ( i j );8i; (2.4b) B ij ( i j )C ij ;8(i;j); (2.4c) x n 0;8n; (2.4d) y i 0;8i: (2.4e) Compared with (2.1),x n ’s are the inputs of (2.4) instead of the decision variables. Lety be the primal optimal solution to (2.4). The LMP at each busi is given by i (x), the optimal dual variable 21 associated with the bus power balance equation (2.4b). Thus, the payoff of generatorn located at busi is given by n (x) = i (x)x n c n (x n ): The proposed game is different from the one in [39], in which the ISO also acts as a strategic agent who moves simultaneously with the generators. Since the action set of the ISO is constrained by the generators’ actions, the generalized Nash equilibrium is adopted as the solution concept. Our model is more realistic, in which the ISO determines the dispatch after the generators submit the offers. In the next, we establish the existence of Nash equilibria under certain mild assumptions. This result is similar to that in [39]. Assumption 2.1. For eachi, the inverse demand functionp i (y i ) is strictly decreasing on [0;D i ] withp i (D i ) = 0 for some constantD i > 0. Assumption 2.2. For anyx6= 0, the dual optimal solution (x) is unique. Forx = 0 (so that y =0), we set i = max j p j (0) for alli. Assumption 2.2 eliminates degenerate dual optimal solutions to ensure that is a function of x, and is continuous atx =0. Lemma 2.2. There exists some constantM > 0 such that in any Nash equilibriumx ,x n M for alln. Proof. LetM = P i D i . Suppose thatx n >M for somen2N i . Then i (x) = 0, so that n < 0. Thus,x cannot be a Nash equilibrium. Lemma 2.2 implies that the strategy space of each generator can be restricted to a compact set [0;M]. Lemma 2.3. In any Nash equilibrium,y i D i for alli. Proof. Suppose y i > D i for some i. Then i = 0, which implies that there must be some generatorn who can reducex n to get a higher payoff. Thus, y i > D i cannot happen in a Nash equilibrium. 22 Lemma 2.3 shows that the demand in equilibrium is bounded, so that the objective function P i v i (y i ) is strictly concave iny. To study the relationship between (x) andx, we consider the (partial) Lagrangian: L(y;;) = X i v i (y i ) + i X n2N i x i y i X j B ij ( i j ) !! ; withdomL =Y , where Y =fyj0y i D i ;8ig; =fjB ij ( i j )C ij ;8(i;j)g; =fj0 i max j p j (0);8ig: The dual functiong : !R is defined as the maximum value of the Lagrangian over (y;): g() = max y2Y;2 L(y;;) =h() + X i i X n2N i x i ; where h() = max y2Y X i (v i (y i ) i y i ) + max 2 X i i X j B ij ( i j ) ! : Lemma 2.4. h() is strictly convex and continuously differentiable on . Proof. h() (as well asg()) is convex, since it is the pointwise maximum of a family of affine functions of. By Danskin’s theorem [40], it can be shown thath() is continuously differentiable on . To prove the strict convexity, it is enough to show that for6= ^ , the corresponding maximizers (y;) and (^ y; ^ ) are not equal. Suppose (y;) = (^ y; ^ ). Then x = ^ x, which means = ^ by Assumption 2.2. 23 Now we consider the dual problem: min 2 h() + X i i X n2N i x i : The first order condition gives dh d i = X n2N i x i ;8i: Lemma 2.5. is continuous inx. Proof. Sinceh is continuously differentiable,rh is continuous in. Sinceh is strictly convex, rh is invertible. Therefore, is continuous inx. Lemma 2.6. i is decreasing inx n for alli and for alln. Proof. dh=d i is decreasing inx n forn2 N i . Sinceh is convex, dh=d j is increasing in j for allj. Thus, j is decreasing inx n for allj, whetherj equalsi or not. Theorem 2.3. Under Assumption 2.1 and 2.2, there exists a Nash equilibrium in the economic dispatch game with Cournot offers. Proof. The payoff of generatorn located at busi is given by n (x) = i (x)x n c n (x n ): By Lemma 2.5, n is continuous inx, which implies nonempty, closed-graph reaction correspon- dences. By Lemma 2.6, it can be shown that n is quasi-concave inx n , which implies the reaction correspondences are convex-valued. It follows from Kakutani’s fixed point theorem that a Nash equilibrium exists [41]. 2.1.6 Marginal Contribution Pricing Mechanism We have studied the LMP-based economic dispatch game with various offer formats. As an alter- native, we propose the MCP mechanism, in which each generator is paid a single amount of its 24 marginal contribution, instead of a unit price of the locational marginal cost as in the LMP mecha- nism. The proposed MCP mechanism is adapted from the Vickrey-Clarke-Groves (VCG) mechanism, a canonical mechanism in the mechanism design theory that implements efficient allocations in dominant strategies [42]. However, the standard VCG mechanism does not apply directly, since we require a finite-dimensional (and preferably low-dimensional) offer format while the true cost function can be infinite-dimensional. In the MCP mechanism, we still use the realistic offer format, in which the reported cost is a two-segment piecewise linear function. The dispatch rule is also given by (2.3). We only change the payment rule, so that the payoff of generatorm is m =w m c m (x m ); (2.5) wherew m is the payment made to generatorm. Let (x m ;y m ) be the dispatch when generatorm is excluded (so thatx m m = 0). Thenw m is given by w m = X i v i (y i ) X n6=m ^ c n (x n ) ! X i v i (y m i ) X n6=m ^ c n (x m n ) ! ; (2.6) which is the (positive) externality that generatorm imposes on the other generators by its partici- pation. Theorem 2.4. There exists an efficient Nash equilibrium in the MCP-based economic dispatch game. Proof. Let ^ c be a bid profile where ^ c n (x n ) = i x n ;x n Q for all n 2 N i . Then (x ;y ; ; ; ) is also a solution to problem (2.3). It remains to show that ^ c is a Nash equilibrium. Consider generatorm located at busi. Its current payoff is m = X i v i (y i ) X n6=m ^ c n (x n ) ! X i v i (y m i ) X n6=m ^ c n (x m n ) ! c m (x m ): 25 Suppose that it changes its offer to ^ c m , resulting a new dispatch (^ x; ^ y; ^ ; ^ ; ^ ) given (^ c m ; ^ c m ). Its new payoff would be ^ m = X i v i (^ y i ) X n6=m ^ c n (^ x n ) ! X i v i (^ y m i ) X n6=m ^ c n (^ x m n ) ! c m (^ x m ): So its payoff changes by ^ m m = X i v i (^ y i ) X n6=m ^ c n (^ x n ) ! c m (^ x m ) X i v i (y i ) X n6=m ^ c n (x n ) ! +c m (x m ) ^ c m (^ x m ) ^ c m (x m ) +c m (x m )c m (^ x m ) = i ^ x m i x m +c m (x m )c m (^ x m ) 0: The first equality follows from the fact that (x m ;y m ) = (^ x m ; ^ y m ). The first inequality follows since X i v i (y i ) X n ^ c n (x n ) X i v i (^ y i ) X n ^ c n (^ x n ): The last inequality follows since x m 2 arg max xm i x m c m (x m ): Thus, it has no incentive to deviate. This proves that the constructed offer profile is a Nash equi- librium. 26 2.2 Dynamic Economic Dispatch Game Increasing penetration of renewable energy sources like wind and solar introduces various new challenges due to their high variability and volatility. Thus, there is lots of effort in developing efficient energy storage systems that can scale in industrial sizes. Once large-scale storage sys- tems are widely available, they are likely to be deployed by generators (and even distributors and consumers) to manage volatility in generation and prices. From the independent system operator’s (ISO’s) perspective, how should economic dispatch be done with storage? Does the ISO need to dispatch both generation and storage? Does storage increase the room for strategic play already available to generators? This work attempts to address these questions. Consider a day-ahead market, where the ISO conducts economic dispatch over a 24-hour hori- zon. Such a problem is modeled as dynamic economic dispatch (DED), and various models have been proposed in the literature [43, 44, 45, 46]. In those models, decisions at different periods are typically coupled by ramping constraints; the fuel cost function may not be smooth due to valve-point effects [47]; there can be prohibited operating zone constraints which make the fea- sible region nonconvex. Hence, most work on DED focuses on developing efficient algorithms that solve for the optimal dispatch [48, 49, 50, 51]. The recent DED models have incorporated the uncertainty of wind power generation, and solution methods in such stochastic settings have been studied [52, 53]. We study the DED problem from a widely different point of view. First, we propose a DED model in which each generator has its own electricity storage device. The operation of storage introduces time-coupling constraints, so that the instant supply (power injected into the network) may differ from the instant generation. We note that a lot of recent work has modeled the use of storage in various contexts [6, 54, 55]. Second, we consider a game-theoretic setting in which generators are strategic agents. We investigate the equilibrium outcomes in the locational marginal pricing (LMP) mechanism [56]. We focus on how the use of storage may affect the market structure and market outcomes. To implement the optimal dispatch, there are two major challenges. First, the ISO is unaware of 27 the storage, or has no access to the control of the storage. Given the lack of information, is it possible for the ISO to conduct the optimal dispatch? Second, generators may not have incentives to reveal their true cost functions. While the use of storage may reduce the cost on the supply side, it is unclear whether it would introduce more opportunities for the generators to manipulate the market. Our contributions are the following. We first show that there exists an efficient bid profile which internalizes the operation of storage, and hence induces the optimal dispatch. The implication is that the current market structure does not have to change: the ISO makes supply decisions only, and the generators makes generation and storage decisions. Storage can be treated as generation in some periods, and as load in others. We then demonstrate that the use of storage does not expand the room for strategic play. On the contrary, the use of storage may sustain a Nash equilibrium that would otherwise not exist; it may also improve the inefficiency of the worst equilibrium. We provide sufficient conditions under which there exist efficient Nash equilibria. 2.2.1 Problem Statement Consider a connected power network consisting ofI buses andN generators. The set of generators at each busi is denoted byN i . LetT be the total number of periods in the planning horizon. The demand at each busi and each periodt is inelastic, denoted byD i;t 0. Each generatorn has an electricity storage device, with a storage capacityC n 0. Letx n;t be the supply (power injected into the network) at periodt,z n;t be the generation at periodt, andy n;t be the state of charge at the end of periodt. Then the storage dynamics is given by y n;t =y n;t1 +z n;t x n;t ;8t; withy n;0 = 0 for alln. The cost of producingz n;t isc n;t (z n;t ), where the cost functionc n;t :R + ! R + is assumed to be increasing, convex and differentiable. We adopt a DC power flow model [32] as usual in the context of economic dispatch, in which each branchi-j is characterized byB ij , the (i;j)-th element of the susceptance matrix. Let i;t be 28 the voltage phase angle at busi and periodt. Then the active power flow over branchi-j is given by f ij;t =B ij ( i;t j;t ): The bus power balance equation for busi and periodt is the following: X n2N i x n;t D i;t = X j f ij;t : Let f max ij be the flow limit of branch i-j such that f max ij = f max ji 0. The branch power flow constraint for branchi-j and periodt is the following: f ij;t f max ij : The system problem is to determine an optimal schedule of supply and generation that min- imizes the social cost while satisfying the unit and transmission constraints. Formally, it is the following convex optimization problem: min T X t=1 N X n=1 c n;t (z n;t ) (2.7a) s.t. X n2N i x n;t D i;t = X j B ij ( i;t j;t );8i;8t; (2.7b) B ij ( i;t j;t )f max ij ;8(i;j);8t; (2.7c) y n;t =y n;t1 +z n;t x n;t ;8n;8t; (2.7d) y n;t C n ;8n;8t; (2.7e) x n;t ;y n;t ;z n;t 0;8n;8t: (2.7f) For each t, since P i ( P n2N i x n;t D i;t ) = P i P j B ij ( i;t j;t ) = 0, the system of linear equations (2.7b) over i is underdetermined with respect tof i;t g. In fact, only the phase angle differences matter. Thus, for computational purposes, one may choose bus 1 as the slack bus by 29 setting 1;t = 0 for allt. Since theN generators have been started up so that the fixed cost does not affect the dispatch, we further assume thatc n;t (0) = 0 for alln and for allt without loss of generality. For future purposes, associate the dual variable i;t with (2.7b), ij;t with (2.7c), n;t with (2.7d), and n;t with (2.7e). We assume that the system problem (2.7) is always feasible. Denote the primal and dual optimal solution byfx n;t ;y n;t ;z n;t ; i;t ; i;t ; ij;t ; n;t ; n;t g, which we call the optimal dispatch. To implement the optimal dispatch, there are two major challenges: the issue of storage- unawareness, and the issue of incentive compatibility. We first formulate the storage-unaware DED problem that considers the former, and then propose the storage-unaware DED game that incorporates the latter as well. Since the ISO is unaware of the storage devices, it solves the following problem: min T X t=1 N X n=1 c n;t (x n;t ) (2.8a) s.t. X n2N i x n;t D i;t = X j B ij ( i;t j;t );8i;8t; (2.8b) B ij ( i;t j;t )f max ij ;8(i;j);8t; (2.8c) x n;t 0;8n;8t: (2.8d) In fact, problem (2.8) essentially decomposes intoT independent static economic dispatch (SED) problems. Denote the solution to problem (2.8) byfx n;t ; i;t g. Given the supply decisionsfx n;t g, each generatorn minimizes its generation cost by utilizing the storage device: minimize T X t=1 c n;t (z n;t ) (2.9a) subject to y n;t =y n;t1 +z n;t x n;t ;8t; (2.9b) y n;t C n ;8t; (2.9c) 30 y n;t ;z n;t 0;8t: (2.9d) Denote the solution to problem (2.9) byfy n;t ;z n;t g. We refer to problem (2.8)–(2.9) as the storage-unaware DED problem. It can be seen that its solution,fx n;t ;y n;t ;z n;t ; i;t g, is a feasible but suboptimal solution to the system problem (2.7). Since the generators are strategic agents, they may not have incentives to reveal their true cost functions. They are often not even able to do so, given the reported cost function constrained to be low-dimensional. Therefore, the ISO may not have the right information to conduct the optimal dispatch. To study the market outcomes with strategic generators, we reformulate the storage- unaware DED problem (2.8)–(2.9) as a game. In the storage-unaware DED game, each generatorn submits a reported cost functionb n;t (), or bid, for each periodt. In practice, the bid is typically a piecewise linear function with increasing slopes. For example, the California ISO uses 10-segment piecewise linear bids [35]. In this work, for simplicity, we consider a two-segment piecewise linear bid parametrized by a four-dimensional signal (r n;t ;s n;t ;r + n;t ;s + n;t ), where 0r n;t r + n;t ; 0s n;t s + n;t : b n;t (x n;t ) = 8 > > < > > : r n;t x n;t ; x n;t 2 [0;s n;t ]; r n;t s n;t +r + n;t (x n;t s n;t ); x n;t 2 (s n;t ;s + n;t ]: The true cost functionc n;t (), the bid b n;t () used in practice, and the simplified bidb n;t () used in this work are illustrated in Fig. 2.4. Note that we intentionally express the bid as a function of the supply, since the generation is hidden from the ISO. Given the bids for each generator and each period, the ISO solves the following problem: min T X t=1 N X n=1 b n;t (x n;t ) (2.10a) s.t. X n2N i x n;t D i;t = X j B ij ( i;t j;t );8i;8t; (2.10b) B ij ( i;t j;t )f max ij ;8(i;j);8t; (2.10c) 31 O x n;t ;z n;t c n;t (z n;t ) b n;t (x n;t ) b n;t (x n;t ) r n;t r + n;t s n;t s + n;t Figure 2.4: The true cost function c n;t (), the bid b n;t () used in practice, and the simplified bid b n;t () used in this work specified by a four-dimensional signal (r n;t ;s n;t ;r + n;t ;s + n;t ). x n;t 0;8n;8t: (2.10d) Associate the dual variable i;t with (2.10b), and ij;t with (2.10c). Denote the primal and dual optimal solution to problem (2.10) byf^ x n;t ; ^ i;t ; ^ i;t ; ^ ij;t g. Then ^ i;t gives the LMP at busi and periodt. Given the supply decisionsf^ x n;t g, each generatorn solves the same problem as (2.9): min T X t=1 c n;t (z n;t ) (2.11a) s.t. y n;t =y n;t1 +z n;t ^ x n;t ;8t; (2.11b) y n;t C n ;8t; (2.11c) y n;t ;z n;t 0;8t: (2.11d) Associate the dual variable n;t with (2.11b), and n;t with (2.11c). Denote the primal and dual optimal solution to problem (2.11) byf^ y n;t ; ^ z n;t ; ^ n;t ; ^ n;t g. The payoff of generatorn located at busi is given by n = T X t=1 ( ^ i;t ^ x n;t c n;t (^ z n;t )): 32 We adopt the pure strategy Nash equilibrium as the solution concept of the storage-unaware DED game (2.10)–(2.11). A bid profile is a Nash equilibrium if no generator can get a higher payoff by a unilateral deviation. A Nash equilibrium is efficient if it induces the optimal dispatch. 2.2.2 Efficient Bids: Resolving the Issue of Storage-Unawareness In this section, we address the first challenge. We show that there exists a bid profile which induces the optimal dispatch, despite the fact that the ISO is unaware of the storage devices. The idea is that the bids can internalize the effect of the operation of the storage, so that the optimal dispatch, as a solution to the system problem (2.7), solves the storage-unaware DED game (2.10)–(2.11). Proposition 2.1. LetM be a large enough constant. For each generatorn located at busi, let b n;t (x n;t ) = n;t x n;t ; 0x n;t M;8t: (2.12) Then the solution to the system problem (2.7) is also a solution to the storage-unaware DED game (2.10)–(2.11). Proof. It is clear that strong duality holds in problem (2.7), (2.10) and (2.11). Therefore, the Karush-Kuhn-Tucker (KKT) conditions provide necessary and sufficient conditions for optimality [34]. The solution to the system problem (2.7) satisfies the following KKT conditions: ( n;t i;t )x n;t = 0;8(n;i);8t; (2.13a) n;t i;t 0;8(n;i);8t; (2.13b) X j B ij ( i;t j;t + ij;t ji;t ) = 0;8i;8t; (2.13c) X n2N i x n;t D i;t X j B ij ( i;t j;t ) = 0;8i;8t; (2.13d) ij;t (B ij ( i;t j;t )f max ij ) = 0;8(i;j);8t; (2.13e) 33 B ij ( i;t j;t )f max ij 0;8(i;j);8t; (2.13f) x n;t 0;8n;8t; (2.13g) ij;t 0;8(i;j);8t; (2.13h) ( n;t n;t+1 + n;t )y n;t = 0;8n;8t; (2.13i) n;t n;t+1 + n;t 0;8n;8t; (2.13j) (c 0 n;t (z n;t ) n;t )z n;t = 0;8n;8t; (2.13k) c 0 n;t (z n;t ) n;t 0;8n;8t; (2.13l) y n;t y n;t1 z n;t +x n;t = 0;8n;8t; (2.13m) n;t (y n;t C n ) = 0;8n;8t; (2.13n) y n;t C n 0;8n;8t; (2.13o) y n;t ;z n;t ; n;t 0;8n;8t; (2.13p) where n;T +1 = 0 for alln. But (2.13a)–(2.13h) are exactly the KKT conditions of (2.10) given the bid profile as in (2.12). Thus,fx n;t ; i;t ; i;t ; ij;t g solves (2.10). Similarly, (2.13i)–(2.13p) are exactly the KKT conditions of (2.11) givenfx n;t g. Thus,fy n;t ;z n;t ; n;t ; n;t g solves (2.11). Hence, the solution to the system problem (2.7) is also a solution to the storage-unaware DED game (2.10)–(2.11). 2.2.3 Equilibrium Analysis: the Value of Storage We present examples to show that the use of storage may improve the equilibrium outcomes. Moreover, we provide sufficient conditions under which there exist efficient Nash equilibria. The first example shows that the use of storage may sustain a Nash equilibrium that would otherwise not exist. Example 2.5. Consider the network as shown in Fig. 2.5. The network has two buses, with gen- erator 1 at bus 1 and generator 2 at bus 2. The branch flow limit is f max =1. The planning horizon has two periods, i.e., T = 2. There is no demand at bus 1. The demand at bus 2 is 34 1 1 x 1;t f max =1 2 2 x 2;t D t Figure 2.5: Example of a two-bus, two-generator network. D 1 = 10;D 2 = 20. The cost functions arec 1;t (z 1;t ) = z 1;t ;z 1;t 16 andc 2;t (z 2;t ) = 2z 2;t , for t = 1; 2. If there is no storage, there is no Nash equilibrium. The argument is straightforward as follows. Considert = 2. Sincex 1;2 =z 1;2 16 andD 2 = 20, it is guaranteed thatx 2;2 =z 2;2 4. Then generator 2 can get an arbitrarily high payoff by bidding b 2;2 (x 2;2 ) = x 2;2 with an arbitrarily large. Thus, a Nash equilibrium does not exist. It can be shown that the claim still holds even if there is a reserve price on the demand side. On the other hand, a Nash equilibrium exists as long as generator 1 has its own storage with a sufficiently large capacity. Consider the following bid profile:b n;t =c 1;t , forn = 1; 2 andt = 1; 2, with the outcomex 1;1 = 10;x 2;1 = 20;z 1;1 = 16;z 1;2 = 14;x 2 =z 2 =0. It is easily seen thatb is a Nash equilibrium. In fact, it is also an efficient Nash equilibrium. Remark 2.5. In this example, generator 1 has an output limit 16 for each period. If there is no storage, generator 2 has enough market power to ask for an arbitrarily high price at period 2, when the demand exceeds the output limit of generator 1. On the other hand, the use of storage makes full use of the generation capacity of generator 1 whose cost is cheaper, so that a Nash equilibrium exists. The next example shows that the use of storage may improve the price of anarchy. Example 2.6. Consider the network as shown in Fig. 2.6. The network has two buses, with gen- erator 1 and 4 at bus 1, and generator 2 and 3 at bus 2. The branch flow limit isf max = f <1. The planning horizon has two periods, i.e.,T = 2. The demand at bus 1 isD 1 =D 2 = 2 f. There is no demand at bus 2. The cost functions are c 1;1 (z 1;1 ) =z 1;1 ; c 1;2 (z 1;2 ) =z 1;2 ; 35 1 1 x 1;t 2 x 2;t D t f max = f 2 3 x 3;t 4 x 4;t Figure 2.6: Example of a two-bus, four-generator network. c 2;1 (z 2;1 ) =z 2;1 ; c 2;2 (z 2;2 ) = 2kz 2;2 ; c 3;1 (z 3;1 ) =kz 3;1 ; c 3;2 (z 3;2 ) =kz 3;2 ; c 4;1 (z 4;1 ) =kz 4;1 ; c 4;2 (z 4;2 ) =kz 4;2 : If there is no storage, we can consider two independent games for the two periods. In particular, consider the game at period 2. The optimal dispatch is (x 1;2 ;x 2;2 ;x 3;2 ;x 4;2 ) = (2 f; 0; 0; 0); with a total cost 2 f. One Nash equilibrium is b 1;2 (x 1;2 ) = 2kx 1;2 ; x 1;2 2 f; b 2;2 (x 2;2 ) = 2kx 2;2 ; x 2;2 2 f; b 3;2 (x 3;2 ) =kx 3;2 ; x 3;2 2 f; b 4;2 (x 4;2 ) =kx 4;2 ; x 4;2 2 f; which induces the outcome (x 1;2 ;x 2;2 ;x 3;2 ;x 4;2 ) = ( f; 0; f; 0); ( 1;2 ; 2;2 ) = (2k;k); 36 with a total cost f +k f. For example, generator 1 has no incentive to bid ^ b 1;2 (x 1;2 ) =kx 1;2 ;x 1;2 2 f, since its new payoff, (k 1)2 f, would be smaller than its current payoff, (2k 1) f. Thus, the price of anarchy is at least f +k f 2 f = k + 1 2 ; which is unbounded ask!1. On the other hand, when there is storage, all Nash equilibria are efficient. This can be easily shown by observing: (i) generator 1 has a marginal cost 1 fort = 1; 2; and (ii) generator 2 can also effectively have a marginal cost 1 by storing enough energy at period 1. Thus, with the use of storage, the price of anarchy is 1. Remark 2.6. In this example, the cost of generator 1 is strictly the lowest at period 2. If there is no storage, it may exercise market power, which leads to an inefficient Nash equilibrium. On the other hand, the use of storage makes generator 2 essentially have the same cost as generator 1 at period 2, so that there exist only efficient Nash equilibria. Example 2.7. In Example 2.5, if the branch flow limit f max < D 2 = 20, there is still no Nash equilibrium even with the use of storage. Again, this is because generator 2 can secure an amount of supply so that it has enough market power to ask for an arbitrarily high price at period 2. In Example 2.6, if the storage capacity of generator 2 is smaller thanD 2 , it can be shown that there still exist inefficient Nash equilibria with the use of storage, due to the exercise of market power of generator 1. Remark 2.7. The use of storage may not always improve the equilibrium outcomes. The storage has a capacity, or there may exist system constraints such as transmission line capacities that limit the effect of storage. We propose two sufficient conditions under either of which not only a Nash equilibrium but also an efficient one exists in the storage-unaware DED game. The idea is based on Proposition 2.1, where an efficient bid profile is constructed. When there is enough competition, such a bid profile can be a Nash equilibrium. 37 The following assumption ensures that no generator has enough market power to ask for arbi- trarily high prices. Assumption 2.3. The system problem (2.7) is still feasible if any one of the generators is excluded. The first sufficient condition is in the following. Definition 2.3 (Congestion-Free Condition). No branch power flow constraint (2.7c) is binding in the optimal dispatch of the system problem (2.7). It is immediate to prove by construction the existence of efficient Nash equilibria under the congestion-free condition. Theorem 2.5. Under Assumption 2.3 and the congestion-free condition, there exists an efficient Nash equilibrium in the storage-unaware DED game. Proof. Consider the bid profile as in (2.12). By Proposition 2.1, the bid profile is efficient, which induces the optimal dispatchfx n;t ;y n;t ;z n;t ; i;t ; i;t ; ij;t ; n;t ; n;t g. It remains to show thatfb n;t ()g is a Nash equilibrium. Consider generatorn located at busi. By the KKT conditions (2.13), it is easy to check that fx n;t ;y n;t ;z n;t ; n;t ; n;t g solves the following problem: max T X t=1 ( i;t x n;t c n;t (z n;t )) s.t. y n;t =y n;t1 +z n;t x n;t ;8t; y n;t C n ;8t; x n;t ;y n;t ;z n;t 0;8t: Its current payoff is n = T X t=1 ( i;t x n;t c n;t (z n;t )): 38 Moreover, since the congestion-free condition holds, all the LMPs are equal at each period: i;t = 1;t for alli and for allt. Suppose it changes its bid tof ^ b n;t ()g, resulting in a new dispatch f^ x n;t ; ^ y n;t ; ^ z n;t ; ^ i;t ; ^ i;t ; ^ ij;t ; ^ n;t ; ^ n;t g: By Assumption 2.3 and in light of the others’ bidsfb m;t ();m6=ng, we have ^ i;t i;t = 1;t for alli and for allt. Its new payoff would be ^ n = T X t=1 ( ^ i;t ^ x n;t c n;t (^ z n;t )) T X t=1 ( 1;t ^ x n;t c n;t (^ z n;t )) T X t=1 ( 1;t x n;t c n;t (z n;t )) = n : Thus, it has no incentive to deviate. This proves that the constructed bid profile is a Nash equilib- rium. The second sufficient condition is the following. Definition 2.4 (Monopoly-Free Condition). There are at least two, or no generators at each bus. This condition is easier to use than the congestion-free condition, since we only need to know the placement of the generators. Moreover, Assumption 2.3 is automatically satisfied. Theorem 2.6. Under the monopoly-free condition, there exists an efficient Nash equilibrium in the storage-unaware DED game. Proof. Consider the bid profile as in (2.12). The proof is similar as before. The key is that under the monopoly-free condition and the given bid profile, the LMP at any bus and any period cannot increase by a unilateral deviation of a generator located at that bus. Consequently, no generator can get a higher payoff. This proves that the constructed bid profile is a Nash equilibrium. 39 Table 2.1: Cost Functions for the IEEE 57-bus system Generator Bus Cost ($/hr) (x: MW) 1 1 0:3879x 2 + 4x 2 2 0:0500x 2 + 8x 3 3 1:2500x 2 + 4x 4 6 0:0500x 2 + 8x 5 8 0:1111x 2 + 4x 6 9 0:0500x 2 + 8x 7 12 0:1613x 2 + 4x 2.2.4 Case Studies We present numerical results for the IEEE power system test cases [57]. Consider the IEEE 57-bus system. There are 57 buses, 7 generators and 42 loads in this power network. Consider a 24-hour planning horizon, i.e.T = 24. Since the test case only gives the load data for a single period, we take the loadD i at busi as the mean value and model the dynamic load as a negative sine wave: D i;t =D i [1 0:5 sin(t=12)]; t = 1; 2;:::; 24; which mimics the load in real-life scenarios. The cost functions for generation are listed in Table 2.1. For simplicity, we assume that the cost function of the same generator is constant over time. The flow limits of the branches are the same, i.e.,f max ij =f max for all (i;j). The storage capacities of the generators are the same, i.e.,C n =C for alln. We consider three scenarios. In the baseline scenario, there is no storage. In the second sce- nario, the ISO is unaware of the storage, while the generators bid true cost functions. This cor- responds to the storage-unaware DED problem (2.8)–(2.9). The third scenario corresponds to the system problem (2.7), in which the ISO is aware of the storage. Alternatively, we can say that it corresponds to the storage-unaware DED game (2.10)–(2.11), in which the generators submit the efficient bid profile. First, we assume that the storage capacity is sufficiently large, i.e.,C =1. We compare the social costs under three scenarios as the flow limit increases, as shown in Fig. 2.7. In all the three 40 100 110 120 130 140 150 160 170 180 190 200 7 7.5 8 8.5 9 9.5 x 10 5 Flow Limit (MW) Social Cost ($) Storage−aware DED Storage−unaware DED No storage Figure 2.7: The social cost versus flow limit under three scenarios, when the storage capacity is sufficiently large. scenarios, the social cost decreases as the flow limit increases. Moreover, the use of storage can reduce the social cost greatly. When the ISO is aware of the storage, the social cost can be even lower. In the following, we assume that the flow limit is sufficiently large, i.e.,f max =1. We focus on the optimal dispatch, i.e., the solution to the system problem (2.7). Fig. 2.8 plots the social costs under three scenarios as the storage capacity increases. Again, the use of storage can reduce the social cost, and there is an improvement when the ISO is aware of the storage. We examine the aggregate generation versus time under different storage capacities, as shown in Fig. 2.9. As the storage capacity increases, the generation curve becomes flatter, given the cost functions to be constant over time. In the case where the storage capacity is zero, the aggregate generation curve is identical to the aggregate load curve. Since there is no congestion in the power network, the LMPs are equal for any given time. Fig. 2.10 plots the LMPs versus time under different storage capacities. As the storage capacity increases, the LMP curve becomes flatter. That is, the use of storage may reduce the price volatility. Lastly, we compare the social costs between the SED approach (the dispatch in the baseline scenario) and the DED approach (the optimal dispatch) under different test cases, when the flow 41 0 100 200 300 400 500 600 6.9 7 7.1 7.2 7.3 7.4 7.5 x 10 5 Storage Capacity (MWh) Social Cost ($) Storage−aware DED Storage−unaware DED No storage Figure 2.8: The social cost versus storage capacity under three scenarios, when the flow limit is sufficiently large. 2 4 6 8 10 12 14 16 18 20 22 24 500 1000 1500 2000 Hour Aggregate Generation (MWh) 0 200 400 600 800 Storage Capacity (MWh) Figure 2.9: The aggregate generation versus time under different storage capacities, when the flow limit is sufficiently large. Table 2.2: Social Costs between SED and DED Case SED ($) DED ($) % Difference 14-bus 7:485 10 4 7:178 10 4 4.11 30-bus 1:733 10 4 1:567 10 4 9.59 57-bus 7:480 10 5 6:883 10 5 7.98 118-bus 1:379 10 6 1:310 10 6 4.97 limit and the storage capacity are sufficiently large. From the results listed in Table 2.2, we can see that overall, the DED approach reduces the social cost. 42 2 4 6 8 10 12 14 16 18 20 22 24 20 25 30 35 40 45 50 55 60 Hour LMP ($/MWh) 0 200 400 600 800 Storage Capacity (MWh) Figure 2.10: The LMP versus time under different storage capacities, when the flow limit is suffi- ciently large. 2.3 Discussion We developed a game-theoretic framework for analysis and design of market mechanisms in the context of economic dispatch. Some immediate extensions include considering demand side to be strategic as well. We focused on the LMP mechanism with piecewise constant offer curves as in practice. On the one hand, we show that a Nash equilibrium may not exist, or that even when a Nash equilibrium exists, the price of anarchy can be arbitrarily large. On the other hand, we provide the congestion- free and the monopoly-free conditions under either of which there exists an efficient Nash equilib- rium. Our findings coincide with the policy proposed in [58]: ensuring enough competition to limit the exercise of market power. The presented counterexamples also suggest that the results in the SFE literature may not apply directly to practice. Moreover, for a general transmission-constrained Cournot model in which the generators submit quantity offers, we show that there exists a Nash equilibrium under certain mild conditions. The proposed MCP mechanism always induces an efficient Nash equilibrium, at the expense of price discrimination. Specifically, the MCP mechanism assigns a total payment to each generator, while the LMP mechanism assigns a uniform price at each bus. Like the LMP mechanism, there 43 may be undesirable Nash equilibria in the MCP mechanism. Future work is needed to tackle these issues. As an extension, we formulate a DED model in which each generator has its own electricity storage device. The operation of storage introduces time-coupling constraints. To implement the optimal dispatch, there are two major challenges: the issue of storage-unawareness, and the issue of incentive compatibility. To address the first issue, we construct an efficient bid profile that internalizes the operation of storage, and hence induces the optimal dispatch. This implies that the current market structure does not have to change. As for the second issue, we demonstrate that the use of storage does not expand the room for strategic play, but may improve the equilibrium outcomes: it may sustain a Nash equilibrium that would otherwise not exist; it may also improve the price of anarchy. While the use of storage does not guarantee the existence nor the efficiency of Nash equilibria, we provide two sufficient conditions under either of which not only a Nash equilibrium but also an efficient one exists. To focus on the value of storage, and the strategic behavior of the generators, we use a simple model to provide the insights into the DED problem. One can apply the same idea to a more sophisticated model. In future work, we will investigate whether it is possible to adapt the two sufficient conditions for a larger class of scenarios, and generalize the results to the stochastic setting. 44 Chapter 3 Multistage Energy Procurement In a wholesale electricity market, the system operator (SO) clears a series of markets for the bal- ancing of power at each instant. To absorb the uncertainty of supply and demand at the real time, the current practice is allocating enough reserve capacity, and conducting load shedding if neces- sary. With the increasing penetration of renewables (e.g., considering California’s ambitious goal of 33% by 2020), that approach will no longer be cost effective, since substantial reserve capacity will be needed. Recently, a new operating paradigm called risk-limiting dispatch (RLD) has been proposed [5, 59]. RLD implicitly hedges against the risk of not meeting operating constraints, through sequential optimal energy purchases, thus eliminating the need for separate reserve markets based on deterministic factors of reliability. In particular, RLD acknowledges the fact that future deci- sions based on more accurate forecasts may correct current decisions, in contrast with the decou- pled dispatch in the current practice. More generally, stochastic programming has been employed for modeling market-clearing problems under uncertainty in electricity markets [60]. As an alternative approach, robust opti- mization can be used to minimize the total cost of system operation in the worst-case realization of the uncertainty [61]. Robust optimization can also be applied in unit commitment problems [62, 63]. In this work, we formulate a multistage stochastic control problem, in which the SO procures power sequentially from generators with varied lead times, employing the up-to-date information at each stage. As in the RLD setting, this problem can be solved by dynamic programming. Mean- while, the SO needs to set a pricing mechanism that supports the optimal dispatch policy. In other 45 words, the generators should be reimbursed properly for their costs so that they would like to par- ticipate in the market in an efficient manner. Thus, the focus of this work is to design incentivizing pricing mechanisms that facilitate efficient participation of the generators in the dynamic setting. Most wholesale electricity markets employ marginal pricing or uniform pricing as the settle- ment scheme, in which all generators are paid the same market-clearing price [64]. As an exten- sion, locational marginal pricing (LMP) reflects how this marginal price may vary at different buses when transmission congestion and losses are taken into account [56]. The LMP mechanism is used independently in each forward market, aligned with the decoupled dispatch as in the current prac- tice. Based on the two-stage stochastic programming approach, pricing mechanisms that account for uncertainty in energy-only markets have been studied in [65, 66]. We propose a generalized marginal pricing mechanism which reflects the effect of generation flexibility and information refinement as time advances, hence referred to as temporal marginal pricing (TMP). The TMP mechanism has quite a few appealing properties, but suffers from untruthful reporting of the strategic generators. In particular, a fast-start generator may pretend to be a slow-start generator so as to secure a higher profit. Generators may also change their bids from stage to stage in the dynamic setting. For example, a generator does not have to participate in the next forward market even if it claims to be flexible enough in the current market. To address the issue of incentive compatibility, we propose the dynamic marginal contribution pricing (DMCP) mechanism, based on the dynamic mechanism design framework developed recently [67]. In the DMCP mechanism, truth-telling is a best response regardless of the history and the current state of the other agents, provided that all the other agents report truthfully. 3.1 Problem Statement Consider a fixed delivery time. Let L be the load and W be the wind power generation, both of which are random variables. To meet the net demand D = LW , the SO procures power sequentially atT 1 forward markets, and make any necessary adjustments at the real-time spot market. We refer to thet-th forward market as staget (wheret = 1;:::;T1), and the spot market 46 as stageT . Denote the sensor readings at staget byy t , which provides information aboutL and W . The measurements may include current load, wind power generation, and weather forecasts. The information available at staget is thenY t =fy s ;stg. SinceL andW are realized at stage T ,Y T gives the exact information ofD. There are N generators with varied lead times, specified by the commitment deadline i 2 f1;:::;T 1g for generator i. At each stage t (where t = 1;:::; i ), generator i is scheduled a quantityx i;t 0 to be delivered at the real time, so that its total generation isx i = P i t=1 x i;t . Note thatx i;t should be based on the current informationY t , but not the future information. The cost functionc i (x i ) of generatori is assumed to be convex, non-decreasing and differentiable. At stageT whenD is realized, the SO makes a recourse decisionx 0 2R to balance the supply and demand. The recourse cost functionc 0 (x 0 ) is assumed to be convex, non-increasing onR , and non-decreasing onR + . The model of multistage energy procurement is shown in Fig. 3.1. The system problem is to find a dispatch policyf 0 (); i;t ();t i g that minimizes the social cost: minimize E " N X i=1 i X t=1 c i;t (x i;t ) +c 0 (x 0 ) # (3.1a) subject to N X i=1 i X t=1 x i;t +x 0 =D; (3.1b) x i;t 0;8(i;t) :t i ; (3.1c) x i;t = i;t (Y t );8(i;t) :t i ; (3.1d) x 0 = 0 (Y T ): (3.1e) It can be easily shown that it is always optimal not to schedulex i;t > 0 until stage i . Therefore, we obtain the following problem which is equivalent to (3.1): minimize E " N X i=0 c i (x i ) # (3.2a) subject to N X i=0 x i =D; (3.2b) 47 Gen 1 x 1;1 x 1;2 x 1; 1 x 1 = P x 1;t Gen 2 x 2;1 x 2;2 x 2; 2 x 2 = P x 2;t . . . GenN x N;1 x N;2 x N; N x N = P x N;t Forward Markets Real Time Y 1 Y 2 Y T1 Y T x 0 = D P x i Figure 3.1: Multistage energy procurement. x i 0; i = 1;:::;N; (3.2c) x i = i (Y i ); i = 1;:::;N; (3.2d) x 0 = 0 (Y T ): (3.2e) The focus of this work is to design incentivizing pricing mechanisms. Letp i be the payment made to generatori. Then the profit of generatori is given by i (x i ;p i ) =p i c i (x i ): 3.2 Temporal Marginal Pricing Problem (3.2) can be written in the following form: minimize E " N X i=0 c i (x i ) # (3.3a) subject to X i: i =1 x i =z 1 ; : 1 (3.3b) 48 z 1 + X i: i =2 x i =z 2 ; : 2 (3.3c) . . . z T2 + X i: i =T1 x i =z T1 ; : T1 (3.3d) z T1 +x 0 =D; : T (3.3e) x i 0; i = 1;:::;N; (3.3f) x i = i (Y i ); i = 1;:::;N; (3.3g) x 0 = 0 (Y T ); (3.3h) where t ’s are the dual variables associated with the constraints weighted by the corresponding probabilities. The purpose of this transform is two fold. First, problem (3.3) prompts a solution method based on dynamic programming, wherez t (the total scheduled quantity by staget) along withY t+1 can be viewed as the state at staget+1. Second, t , as a function ofY t , has an economic meaning: it is the cost of procuring an additional increment of power at staget. Hence, we define as t the temporal marginal price (TMP) at staget. Letp i = i x i , so that the profit of generatori is given by i (x i ; i ) = i x i c i (x i ): Example 3.1. LetT = 3. From stage 1’s point of view,D is a Bernoulli random variable such that Pr(D = 100) = Pr(D = 120) = 0:5. At stage 2, the forecast ofD is exact. LetN = 2, i = i, andc i (x i ) =x 2 i fori = 1; 2. Letc 0 (x 0 ) = 500 maxfx 0 ; 0g, the interpretation of which is that the excessive generation is spilled for free, and that the value of lost load is 500. The system problem is the following: minimize c 1 (x 1 ) + 0:5(c 2 (x l 2 ) +c 0 (x l 0 )) + 0:5(c 2 (x h 2 ) +c 0 (x h 0 )) subject to 0:5(x 1 +x l 2 +x l 0 ) = 0:5 100; 49 0:5(x 1 +x h 2 +x h 0 ) = 0:5 120; x 1 ;x l 2 ;x h 2 0: The solution gives the optimal dispatch and the TMPs: x 1 = 55; x l 2 = 45; x h 2 = 65; x l 0 =x h 0 = 0; 1 = 110; l 2 = l 0 = 90; h 2 = h 0 = 130: 3.2.1 Properties First, we show that given the current information, the TMPs at all future stages are the same in expectation. Proposition 3.1 (Price Equivalence). For anyr;s;t such thatrst, E[ s jY r ] =E[ t jY r ]: Proof. Consider the Karush-Kuhn-Tucker (KKT) conditions of problem (3.3). Forz T1 , we have @ @z T1 ( T1 (z T1 ) +E[ T z T1 jY T1 ]) = 0; or T1 =E[ T jY T1 ]. Similarly, we have t =E[ t+1 jY t ]; t = 1;:::;T 1: By the tower property of conditional expectation, we obtain E[ s jY r ] =E[E[ s+1 jY s ]jY r ] =E[ s+1 jY r ] = 50 =E[ t jY r ]: Second, we show that the TMP mechanism is efficient in a competitive environment, when the generators are price takers. Proposition 3.2 (Efficiency). Let (x 0 ;x 1 ;:::;x N ) be the optimal dispatch and ( 1 ;:::; T ) be the TMPs. For alli, x i 2 arg max x i i x i c i (x i ): Proof. Consider the relevant KKT conditions forx i : (c 0 i (x i ) i )x i = 0; c 0 i (x i ) i 0; x i 0; from which the result follows. Third, we are concerned about the fairness of the mechanism. In particular, we consider the following criterion: for any two generators with the same cost functions, the more flexible one always secures a weakly higher profit in expectation. Proposition 3.3 (Flexibility Subsidization). For any i;j;t such that c i () = c j () = c() and t i j , E[ i jY t ]E[ j jY t ]: 51 Proof. Define'() = max x xc(x), which is convex, sincexc(x) is linear in and pointwise maximum preserves convexity. Therefore, i ='( i ) ='(E[ j jY i ]) E['( j )jY i ] =E[ j jY i ]; where the first and last equalities follows from Proposition 3.2, the second equality follows from Proposition 3.1, and the inequality follows from Jensen’s inequality. Taking conditional expections on both sides, we obtain E[ i jY t ]E[E[ j jY i ]jY t ] =E[ j jY t ]: 3.2.2 Vulnerability to Misreporting The preceding properties are based on the underlying assumption that the characteristics of the generators are truthfully revealed. Since the generators are strategic agents, they may not have incentives to reveal such private information, as illustrated in the following example. Example 3.2. Consider Example 3.1. We show that even if the cost functions are truthfully revealed, there is still plenty of room to game the system. Suppose generator 2 claims that its lead time is the same as generator 1, so that it should also be dispatched at stage 1. Then the SO solves the following problem: minimize c 1 (x 1 ) +c 2 (x 2 ) + 0:5c 0 (x l 0 ) + 0:5c 0 (x h 0 ) subject to 0:5(x 1 +x 2 +x l 0 ) = 0:5 100; 52 0:5(x 1 +x 2 +x h 0 ) = 0:5 120; x 1 ;x 2 0: The solution gives the dispatch and the TMPs: x 1 =x 2 = 60; x l 0 =20; x h 0 = 0; 1 = 120; l 2 = l 0 = 0; h 2 = h 0 = 240: In this case, the profit of generator 2 is 3600. When it reports truthfully, the expected profit is 3125. Thus, generator 2 has an incentive to misreport its lead time. Such misreporting has serious consequences. In the short term, the resulting dispatch may not be optimal, and the social cost may increase. In the long term, it fails to provide adequate incentives for investment in fast-start generators. In the next section, we will propose an alternative mechanism that addresses the issue of incentive compatibility. 3.3 Dynamic Marginal Contribution Pricing We adapt the recent results on dynamic mechanism design for the multistage energy procurement problem, with exogenous uncertainty. The proposed DMCP mechanism pays each generator its flow marginal contribution at each stage, which induces truth-telling as a best response in the dynamic sense. 3.3.1 Setup Define a generic state vector = (t;Y;z;P;c;), wheret is the stage index,Y is the information set,z is the total scheduled power,P is the set of participating generators,c is the profile of cost 53 functions, and is the profile of commitment deadlines. Given, we define a generic optimization problem OPT(), whose optimal value is denoted byC(): minimize E " X j2P c j (x j ) +c 0 (x 0 ) Y # subject to X j2P x j +x 0 =Dz; x j 0;8j2P; x j = j (Y j );8j2P; x 0 = 0 (Y T ): Let t = (t;Y t ;z t1 ;P;c;), where z t is the total scheduled power by stage t, andP = fj : j tg. Then the social cost starting at stage t is given by C( t ). Moreover, let i t = (t;Y t ;z t1 ;Pnfig;c;). Then the social cost starting at staget when generatori is excluded is given byC( i t ). The marginal contribution of generatori starting at staget is M i ( t ) =C( i t )C( t ): The flow marginal contribution of generatori at staget is m i ( t ) =C( i t )C( t )E[C( i t+1 )C( t+1 )]: The idea of the mechanism is to make generator i secure the flow marginal contribution as the profit at each stage. Thus, the payment made to generatori at staget is given by p i;t ( t ) =C( i t )C( t )E[C( i t+1 )C( t+1 )] +1 f i =tg c i (x i ) =C( i t )E[C( i t+1 )] X j6=i: j =t c j (x j ): 54 The expected profit of generatori is then its marginal contribution starting at stage 1: E[ i ] =E " i X t=1 p i;t ( t )c i (x i ) # =M i ( 1 ): 3.3.2 Mechanism To completely specify the mechanism, we need to consider cases in which the reports are not truthful, or the generators do not follow the bids placed in the previous stages. Let (^ c i;t (); ^ i;t ) be the report of generatori at staget. Truth-telling means that ^ i;t = i , and ^ c i;t () is effectively truthful: ^ c i;t (x i ) =c i x i + t1 X s=1 x i;s ! c i t1 X s=1 x i;s ! ; wherex i;s is the scheduled power of generatori at stages, which can be positive if generatori has misreported i before. We formally specify the DMCP mechanism in the following. Algorithm 1 DMCP Mechanism z 0 0 fort 1 toT 1 do ObserveY t P fj : ^ j;t tg Receive ^ c t f^ c j;t ()g j2P ; ^ t f^ j;t g j2P ^ t (t;Y t ;z t1 ;P; ^ c t ; ^ t ) Solve OPT( ^ t ), obtainC( ^ t ), schedulefx j;t g j2P z t z t1 + P j x j;t fori2P do ^ i t (t;Y t ;z t1 ;Pnfig; ^ c t ; ^ t ) Solve OPT( ^ i t ), obtainC( ^ i t ) ^ i t+1jt (t + 1;Y t+1 jY t ;z t ;Pnfj : ^ j =tgnfig; ^ c t ; ^ t ) Solve OPT( ^ i t+1jt ), obtainC( ^ i t+1jt ) p i;t ( t ) C( ^ i t )E[C( ^ i t+1jt )] P j6=i ^ c j;t (x j;t ) end for end for x 0 Dz T1 55 Example 3.3. Consider Example 3.1. At stage 1, x 1 = 55, p 1;1 = 9075, so that the profit of generator 1 is 1 =p 1;1 c 1 (x 1 ) = 6050: Moreover,p 2;1 =16125, which means that generator 2 makes a payment to the system. At stage 2, ifD = 100,x l 2 = 45,p l 2;2 = 22500; ifD = 120,x h 2 = 65,p h 2;2 = 32500. The expected profit of generator 2 is 2 =p 2;1 + 0:5(p l 2;2 c 2 (x l 2 )) + 0:5(p h 2;2 c 2 (x h 2 )) = 8250: 3.3.3 Properties It is clear that the DMCP mechanism is efficient. Since the expected profit is the marginal contri- bution, the DMCP mechanism also satisfies the fairness criterion. Now we show that it is incentive compatible in the dynamic sense. Definition 3.1. A mechanism is periodic ex post incentive compatible, if for each agent, truth- telling is a best response regardless of the history and the current state of the other agents, provided that all the other agents report truthfully. Theorem 3.1. The DMCP mechanism is periodic ex post incentive compatible. Proof. Let t andx i;t be the state and dispatch given the truthful reports of all the agents. Let ^ t and ^ x i;t be the state and dispatch given any report of generatori, and truthful reports of the other agents. By the one-stage deviation principle [41], it suffices to prove that truth-telling is a best response for each agenti at each staget: p i;t ( t )c i (x i;t ) +E[M i ( t+1 )]p i;t ( ^ t )c i (^ x i;t ) +E[M i ( ^ t+1jt )]; 56 or C( i t )C( t )C( i t )E[C( ^ i t+1jt )] X j6=i c j (^ x j;t )c i (^ x i;t ) +E[M i ( ^ t+1jt )]; or C( t ) X j c j (^ x j;t ) +E[C( ^ t+1jt )]; which is true sinceC( t ) is the optimal value of the social cost starting att. The following result compares the expected profits between those two mechanisms. Theorem 3.2. For each generator, the expected profit in the DMCP mechanism is weakly higher than that in the TMP mechanism assuming that truthful information is revealed. Proof. Let D i be the expected profit of generatori in the DMCP mechanism, and T i be that in the TMP mechanism. We have D i =E " N X j=0 c j (x i j ) # E " N X j=0 c j (x j ) # E " X j6=i c 0 j (x j )(x i j x j )c i (x i ) # E c 0 i (x i )(x i x i i )c i (x i ) =E[c 0 i (x i )x i c i (x i )] = T i : 3.4 Case Studies Consider a setting in which there are two dispatch stages (for DA and RT markets) and a real-time adjustment stage (T = 3). The second dispatch stage is very close to the real-time adjustment stage 57 so that these two stages have the same information set. Wind and load data from BPA for year 2014 are used for this case study [68]. The annual average of hourly load and wind are 6281 MW and 1271 MW, respectively, with the mean wind penetration of 20:24%. The standard deviation for the forecast error is 214:1 MW, which accounts for 16:85% of the average wind generation and 3:41% of the average load. For numerical simplicity, we fit a discrete probability distribution with 1000 outcomes using the net forecast error data, and consider a representative problem with the DA forecast being the mean net demand (average load minus average wind). Under such setting, the information set in the first dispatch stage contains the forecast of net demand and the probability distribution of the forecast error, and the information set in the second dispatch stage and the adjustment stage contains the realization of the net forecast error (which in turn leads to the realization of the net demand). We consider 10 generators which have the cost c n (x n ) = 0:05x 2 n + c L n x n with c L n = 20; 30;:::; 100; 200, for n = 1;:::; 10, respectively. The first 5 generators are DA generators which have to be dispatched at the first stage due to their lead time requirements. The remaining generators are RT generators which can ramp up or down their generation level rapidly so that they be dispatched either in the first stage or in the second stage. We simulate both the TMP and DMCP mechanisms with the dataset. Assuming that all gen- erators report their true lead times and cost function, both mechanisms lead to efficient allocation with the optimal cost being $4:11 10 5 . The payments provided to generators by these two mech- anisms are different and so are the profits of the generators. Figure 3.2 depicts the profits of all 10 generators under TMP and DMCP mechanisms. We observe that indeed the generator profits under DMCP is higher than that under TMP. Here the payments and profits for TMP are calculated assuming that the true lead times and cost data are reported to SO. However, gaming opportunities exist for TMP as we have discussed. Figure 3.3 demonstrates the potential efficiency loss on the TMP system cost if there are RT generators misreporting their lead time, i.e., pretending that they are DA generators. As there is 58 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 x 10 4 Expected profit ($) Generator index TMP DMCP Figure 3.2: Expected profit earned by each generator. 0 1 2 3 4 5 4 4.2 4.4 4.6 4.8 5 x 10 5 Number of RT misreportors Expected system cost ($) Deterministic baseline 20% integration 40% integration 60% integration Figure 3.3: TMP system cost with misreporting. no incentive compatibility guarantee for the TMP mechanism, this is possible in view of Exam- ple 3.2. Here we have performed this calculation for various forecast error levels, corresponding to no renewable generation, 20%, 40%, and 60% penetration of variable generation resources. This is done by linearly scaling up the forecast errors with error standard deviation per renewable pen- etration fixed to be 214:1MW=20:24% as in the current BPA data. Confirming our intuition, we see from the figure that in the case without uncertainty, misreporting of the lead times does not affect the system cost; the higher the uncertainty level in the system, the more significant the effi- ciency loss for the system becomes. Since we can interpret the area between the cost curves and the deterministic baseline as the integration cost for variable resources (cf. [69]), this suggests that misreporting leads to higher integration costs. Finally, we illustrate numerically that DMCP is indeed incentive compatible. Figure 3.4 shows the profit gained by the 6th generators if it reports some cost function and lead time that may be 59 0 50 100 150 200 250 −10000 −5000 0 5000 10000 15000 Reported linear cost coefficient ($/MW) Expected profit ($) Reported as RT generator Reported as DA generator (a) Maximum profit achieved at 70$/MW and reported as RT 0 0.2 0.4 0.6 0.8 1 0 5000 10000 15000 Reported quadratic cost coefficient ($/MW 2 ) Expected profit ($) Reported as RT generator Reported as DA generator (b) Maximum profit achieved at 0:05$/MW 2 and reported as RT Figure 3.4: DMCP generator profit with misreporting. different from its true data. The profits are evaluated using DMCP under the forecast errors with 60% penetration and assuming other generators are truth-telling. Figure 3.4(a) shows that with the quadratic coefficient fixed to be its true value, the generator does not have an incentive to report a wrong lead time or a wrong linear coefficient. Similar is true for the quadratic cost coefficient if the linear coefficient is fixed (see Figure 3.4(b)). Thus truth-telling is at least the best action which the generator could take in this subset of the strategy space. 1 3.5 Discussion We have proposed two mechanisms for the multistage energy procurement problem. The TMP mechanism generalizes the uniform pricing mechanism to the stochastic dynamic setting. With a few appealing properties, the TMP mechanism reflects the effect of generation flexibility and information refinement as time advances. However, it suffers from misreporting of the strategic generators. To address the issue of incentive compatibility, the DMCP mechanism is designed such that each generator receives the flow marginal contribution as the profit. Moreover, generators would like to participate in the DMCP mechanism, where they can secure a weakly higher profit than in the TMP mechanism. The presented case study complements our theroetic analysis. 1 In fact, evaluating the profits for the generator over a grid of linear and quadratic coefficients has confirmed that truth-telling is actually the best action in the entire space. 60 Chapter 4 Buying Random Wind Power 4.1 Welfare-Maximizing Objective With the increasing penetration of wind power, challenges arise in integrating such random gen- eration into the current electricity grid and market. While wind power producers are commonly treated as negative loads and receive feed-in tariffs, this scheme would be no longer applicable when the penetration level is high, with substantial reserve margin needed. Alternatively, wind power producers can be required to participate in the competitive electricity pool through a two-settlement system. In the ex ante day-ahead (DA) forward market, the wind power producer commits to a fixed amount of generation; in the ex post real-time (RT) spot market, it pays a penalty for the shortfall if any. However, it is still unclear what are the appropriate market mechanisms. In the literature, the determination of the committed quantity can be posed as an optimization problem [61, Ch. 7], [3]. In [70], they introduce risky power contracts in addition to firm power contracts to enable flexible and efficient wind power aggregation. They focus on how optimal offer- ings and equilibria depend on exogenous price signals, and on deriving concept and expressions for critical prices from the perspective of wind power producers. To address the risk of not meet- ing operating constraints such as power balance due to the uncertainty of wind power generation, a new paradigm for power system operation called risk-limiting dispatch has been proposed [59]. That paradigm employs real-time information about supply and demand, taking into account the stochastic nature of wind power generation, and determines a risk-constrained stochastic optimal dispatch. There are two major issues of the existing approaches. First, it is assumed that wind power producers have no market power to influence the market prices when they determine the optimal 61 trading strategies. That assumption does not hold with a high penetration of wind power. Second, the optimal dispatch under uncertainty requires probabilistic models of wind power generation, which may not be revealed truthfully with strategic wind power producers. In this work, we observe that it may not be efficient for wind power producers to bid cost functions as thermal units do. Instead, bidding probability distributions of generation can poten- tially exploit the value of aggregation. This motivates the proposed paradigm referred to as the stochastic resource auction. The underlying assumption is that the current two-settlement system does not change, but aggregators are allowed to enter the market, who procure wind power through auctions and assume any risk due to the uncertainty of wind power generation. The key feature of a stochastic resource auction is that while the realized generation of the wind farm is random, the probability distribution can be learned beforehand as its private information. Auction and market design for electricity markets is a well-studied problem [1, 4, 22, 23, 71]. However, almost all of economic and auction theory deals with classical goods. The auction design problem we introduce is for stochastic goods, which has received only scant attention, if at all. Our focus is to design incentive compatible mechanisms that elicit truthful information of strategic agents who supply stochastic resource. We will explore the problem from both an aggregator’s and an independent system operator’s (ISO’s) points of view. We note that following the same motivation, one can approach the same problem in the con- tract design framework that typically considers a principal and a single agent [72, 73]. In [74], they consider an aggregator procuring both conventional and renewable power from a single producer that has multi-dimensional private information, and show that the optimal mechanism is a menu of contracts. As another context, [75] proposes a dynamic contract design problem with an appli- cation to indirect load control. The key characteristic of their setting is that the principal has no capability to monitor the agent’s control or the state of the engineered system, whereas the agent has perfect observations. 62 4.1.1 Problem Statement We consider a single-period model composed of a DA market and a RT market. There areN wind farms as strategic players (or agents), indexed by i = 1;:::;N. The wind power generation of agent i at the RT market is a random variable W i 2 [0; w], where w is a constant denoting the maximum capacity over all agents. Denote byw i the realization ofW i . We assume thatW i ’s are independent. The probability distribution ofW i is parametrized by i , referred to as agenti’s type, which is a one- or multi-dimensional parameter learned privately by agenti at the DA market. Example 4.1. Let the wind power generationW i ’s be normalized by the maximum nominal capac- ity over all agents, so that w = 1. When W i has a beta distribution, i is a two-dimensional parameter ( i ; i ), where i > 0; i > 0. At the DA market, wind farms can bid cost functions as thermal units do. While the marginal cost is negligible, the generation is random, so that agenti incurs a balancing cost as a function of the offered quantityx i . Letp> 0 be the cost of load shedding. Then the cost function of agenti is C i (x i ) =E[p(x i W i ) + ]; where we definex + = maxfx; 0g. Note that any excess generation (W i x i ) + is assumed to be spilled. The cost functionC i () is well-behaved. First, it is increasing. Moreover, it is convex, as the expectation of a convex function. In this market architecture, the agents bear the risk, since it is their own responsibility to fulfill the committed quantity. Alternatively, there can be an aggregator who aggregates wind power generation and bears the risk of buying power at the RT market to make up for any shortfall. In this market architecture, the wind farms can bid probability distributions (or equivalently, i ’s), so that they do not need to make an estimate ofp. The cost function of the aggregator is C(x) =E " p x X i W i ! + # : 63 The following result implies that bidding probability distributions is more efficient than bidding cost functions. Proposition 4.1. For any given x 0, let (x 1 ;:::;x N ) be an optimal solution to the following problem: minimize x 1 ;:::;x N X i C i (x i ) subject to X i x i =x; x i 0;8i: Then C(x) X i C i (x i ): Proof. We have C(x) =E " p X i x i X i W i ! + # X i E[p(x i W i ) + ] = X i C i (x i ); where the inequality follows from the fact that (x +y) + x + +y + for anyx andy. Therefore, bidding probability distributions can potentially exploit the value of aggregation and reduce the social cost. The surplus, P i C i (x i )C(x), provides incentives for aggregators to enter the market who bear the risk due to the uncertainty of wind power generation. Example 4.2. Letp = 1, w = 1,x = 1, andf 1 (w) = f 2 (w) = 1;w2 [0; 1], wheref i () is the probability density function ofW i . When the wind farms bid cost functions, we havex 1 = x 2 = 64 1=2, so thatC 1 (x 1 ) =C 2 (x 2 ) = 1=8. When they bid probability distributions, since the probability density function ofW 1 +W 2 is f(w) = 8 > > < > > : w; w2 [0; 1]; 2w; w2 (1; 2]; we haveC(x) = 1=6. Thus, bidding probability distributions reduces cost by 1=8 + 1=8 1=6 = 1=12: Proposition 4.1 motivates a new paradigm for wind power procurement, referred to as the stochastic resource auction, in which wind farms bid probability distributions. The timeline of the stochastic resource auction is shown in Fig. 4.1. At the DA market, each agenti learns its own type i and submits a bid ^ i , which could be different from the true type i . Based on the bid profile ^ = (^ 1 ;:::; ^ N ),M out of theN agents are selected as the wind power providers, the set of which is denoted byI. The unselected agents get zero payoffs and leave the auction. Each selected agenti makes a paymentt d;i (^ ) to the aggregator. At the RT market, upon the realization of W i , each selected agent i gets paid t r;i (^ ;w i ) from the aggregator. Note that we allow t d;i or t r;i to be negative, indicating a payment in the reverse direction. Therefore, the selection and payment schemes define a direct revelation mechanism =fI;t d;i ;t r;i g. The payoff of agenti2I is given by i (^ ;w i ) =t r;i (^ ;w i )t d;i (^ ): We will consider two settings. In the first setting, the wind farms do not directly participate in the electricity pool; instead, an aggregator selects some of them as the wind power providers who have the most desirable distributions. We refer to this as the aggregator’s problem, in which M <N. In the second setting, all the wind farms are online, and the problem is how to price wind energy for stochastic economic dispatch. We refer to this as the ISO’s problem, in whichM =N. 65 Agents Aggregator ^ = (^ 1 ;:::; ^ N ) I =fi 1 ;:::;i M g t d;i (^ );8i2I DA RT w i ;8i2I t r;i (^ ;w i );8i2I Figure 4.1: The timeline of the stochastic resource auction with independent types. In both settings, the aggregator or the ISO needs to know the probability distributions of wind power generation to find the best set of wind power providers or to make an optimal dispatch. Since the agents are strategic, they may not have incentives to reveal their private information truthfully. Our goal is to design incentive compatible mechanisms to elicit the true types of the agents. 4.1.2 Selecting Wind Power Providers We first consider a basic scenario where a single agent is selected as the provider, for technical or regulatory reasons. We will make several generalizations afterward. In the simplest scenario, the objective is to identify the agent who yields the highest expected generation. That is, the aggregators problem is maximize i E[W i ]: Since the resource is stochastic, we derive a variant of the well-known Vickrey-Clarke-Groves (VCG) mechanism [42]. We then propose an alternative mechanism which has a commitment- with-penalty payment structure [3]. 66 Denote by ^ W i the random variable parametrized by ^ i . The selection scheme,I =fig, is given by i2 arg max j E[ ^ W j ]; (4.1) where any tie-breaking rule applies. That is, the one who claims to yield the highest expected generation is selected. This makes sense provided that the incentive compatibility is achieved. Moreover, it is natural to specify t r;i (^ ;w i ) =w i ; (4.2) where> 0 is an adjustable parameter, denoting the price the aggregator pays the selected agent for the realized generation, which does not need to be the price at the DA market, nor that at the RT market. It remains to specifyt d;i (^ ) to achieve the incentive compatibility. We have E[ i (^ i ; ^ i ;W i )] =E[W i ]t d;i (^ i ; ^ i ): Fix ^ i . On the one hand, ifE[W i ]E[ ^ W j ] for allj6=i, then t d;i ( i ; ^ i )t d;i (^ i ; ^ i ); for any ^ i such thatE[ ^ W i ]E[ ^ W j ] for allj6= i. On the other hand, ifE[W i ] <E[ ^ W j ] for some j, then E[W i ]t d;i (^ i ; ^ i ) 0; for any ^ i such thatE[ ^ W i ]E[ ^ W j ] for allj6=i. Thus, a candidate DA payment scheme can be t d;i (^ ) = max j6=i E[ ^ W j ]; (4.3) 67 which also ensures the individual rationality. The proposed mechanism is referred to as the stochastic VCG (SVCG) mechanism, since it is an analogue of the standard VCG mechanism. The expected RT payment E[W i ] can be viewed as the counterpart of the valuation in the standard VCG mechanism, and the DA payment max j6=i E[ ^ W j ] as the counterpart of the usual payment (or externality). We now formally state the properties of this mechanism. Theorem 4.1. The SVCG mechanism specified by (4.1)–(4.3) is efficient, dominant strategy incen- tive compatible, and individual rational. Proof. The selection scheme (4.1) implies that the SVCG mechanism is efficient. Fix an agent i and ^ i . We show that bidding i is the best response of agent i. Note that conditioned on that agenti is selected, its expected payoff is independent of ^ i : E[ i (^ i ; ^ i ;W i )] =E[W i ] max j6=i E[ ^ W j ]: Suppose that agenti is selected by bidding i , which means E[W i ] max j6=i E[ ^ W j ]; so that its current expected payoff is nonnegative. Consider that it changes the bid to ^ i . If it is still selected, its expected payoff remains the same. If it is no longer selected, it gets a zero payoff. In either case, its expected payoff does not increase. Suppose that agenti is not selected by bidding i , which means E[W i ] max j6=i E[ ^ W j ]; so that its current payoff is zero. Consider that it changes the bid to ^ i . If it is not selected, it still gets a zero payoff. If it is selected, its expected payoff is nonpositive. In either case, its expected payoff does not increase. 68 Thus, truth-telling is a dominant strategy for each agent, which means that the SVCG mecha- nism is dominant strategy incentive compatible. Since the expected payoff for each agent is nonnegative, the SVCG mechanism is individual rational. Remark 4.1. The SVCG mechanism can be interpreted in the following way. At the DA market, the provider makes a contractual payment max j6=i E[ ^ W j ] to the aggregator, which depends on the reported expected generation of the second highest bidder. At the RT market, the aggregator makes a paymentE[W i ] to the provider for the actual generation at a price. Dominant strategy incentive compatibility means that each agent will truthfully reveal its probability distribution, regardless of what the other agents do. Consequently, truth-telling for all agents is a dominant strategy equilibrium. Since the wind farms have no intrinsic valuation for the wind power generation they produce, there exists a richer class of incentive compatible mechanisms beyond the standard mechanism design framework. We now propose an alternative mechanism with natural interpretations. Recall that W i 2 [0; w] for all i. Ideally, if W i = w with probability 1, then agent i is the desired provider. This suggests that w can serve as a reference point in a commitment-with-penalty payment structure. In the proposed mechanism, referred to as the stochastic shortfall penalty (SSP) mechanism, the selection scheme is the same as before: i2 arg max j E[ ^ W j ]: (4.4) The DA payment scheme is given by t d;i (^ ) = w: (4.5) The RT payment scheme is given by t r;i (^ ;w i ) =(^ )( ww i ); (4.6) 69 where() is a function of ^ . The interpretation is the following. At the DA market, the aggregator makes a payment to the provider for the full capacity generation w at a price. At the RT market, the provider is penalized for the shortfall ww i at a penalty price(^ ). We specify (^ ) = w w max j6=i E[ ^ W j ] : (4.7) Note that(^ ), which is necessary for the mechanism to be incentive compatible. Theorem 4.2. The SSP mechanism specified by (4.4)–(4.7) is efficient, dominant strategy incentive compatible, and individual rational. Proof. The selection scheme (4.4) implies that the SSP mechanism is efficient. Fix an agent i and ^ i . We show that bidding i is the best response of agent i. Note that conditioned on that agenti is selected, its expected payoff is independent of ^ i : E[ i (^ i ; ^ i ;W i )] = w(^ )( wE[W i ]) = w(E[W i ] max j6=i E[ ^ W j ]) w max j6=i E[ ^ W j ] : Suppose that agenti is selected by bidding i , which means E[W i ] max j6=i E[ ^ W j ]; so that its current expected payoff is nonnegative. Consider that it changes the bid to ^ i . If it is still selected, its expected payoff remains the same. If it is no longer selected, it gets a zero payoff. In either case, its expected payoff does not increase. Suppose that agenti is not selected by bidding i , which means E[W i ] max j6=i E[ ^ W j ]; 70 so that its current payoff is zero. Consider that it changes the bid to ^ i . If it is not selected, it still gets a zero payoff. If it is selected, its expected payoff is nonpositive. In either case, its expected payoff does not increase. Thus, truth-telling is a dominant strategy for each agent, which means that the SSP mechanism is dominant strategy incentive compatible. Since the expected payoff for each agent is nonnegative, the SSP mechanism is individual rational. Remark 4.2. There is a duality between those two mechanisms: At the DA market, money flows from the provider to the aggregator in SVCG (which depends on the second highest bid), while it flows from the aggregator to the provider in SSP (which is a constant). At the RT market, money flows from the aggregator to the provider in SVCG (which depends on the realization), while it flows from the provider to the aggregator in SSP (which depends on both the second highest bid and the realization). In the remaining of this work, we will focus on the SVCG mechanism. In the basic scenario, the objective is to identify the agent who yields the highest expected generation. Now we generalize the objective. The aggregator’s problem is maximize i E[g(W i )]; whereg : [0; w]!R is referred to as the objective function. We assume thatg() is continuous, which satisfies most practical purposes, so that it attains a maximum and a minimum. We can considerg() when we have a minimization problem. We give some examples of the objective function. Example 4.3. Wheng(w) =w, we recover the basic scenario. Example 4.4. Wheng(w) = minfw;Dg, the interpretation is that there is an upper boundD2 [0; w] on the demand at the RT market, beyond which the aggregator does not care about how much more power would be generated. 71 We now generalize the SVCG mechanism. The selection scheme is given by i2 arg max j E[g( ^ W j )]: (4.8) The payment schemes are given by t d;i (^ ) = max j6=i E[g( ^ W j )]; t r;i (^ ;w i ) =g(w i ): (4.9) We state the properties of this mechanism in the following theorem. The proof is similar as before. Theorem 4.3. The generalized SVCG mechanism specified by (4.8)–(4.9) is efficient, dominant strategy incentive compatible, and individual rational. We now consider the scenario in whichM (<N) agents are selected according to the criterion represented by a continuous objective functiong : [0; w] M !R. That is, the aggregator’s problem is maximize i 1 ;:::;i M E[g(W i 1 ;:::;W i M )]: We give some examples of the objective function. Example 4.5. Wheng(w i 1 ;w i 2 ) = (w i 1 w i 2 ) 2 , the interpretation is that the aggregator wants to contract with two agents who have the closest distributions in a certain sense. Example 4.6. Wheng(w i 1 ;:::;w i M ) =w i 1 + +w i M , the interpretation is that the aggregator wants to contract withM agents who yield the highest expected generation. Theorem 4.4. Let the selection scheme be (i 1 ;:::;i M )2 arg max j 1 ;:::;j M E[g( ^ W j 1 ;:::; ^ W j M )]: For allk = 1;:::;M, let the DA payment be t d;i k (^ ) = max j 1 6=i k ;:::;j M 6=i k E[g( ^ W j 1 ;:::; ^ W j M )]; 72 and the RT payment be t r;i k (^ ;w i k ) =E[g( ^ W i 1 ;:::; ^ W i M )j ^ W i k =w i k ]: Then the proposed mechanism is efficient, dominant strategy incentive compatible, and individual rational. Remark 4.3. For certain objective functions, the proposed mechanism may have an explicit form. Consider Example 4.6. Rank the agents in order of the reported expected generation: E[ ^ W i 1 ]E[ ^ W i N ]: Then agentsi 1 ;:::;i M are selected. For allk = 1;:::;M, the payment schemes are given by t d;i k (^ ) =E[ ^ W i M+1 ]; t r;i k (^ ;w i k ) =w i k : While the marginal cost is negligible, each agent i may have a fixed cost i 0 when it is selected and produces generation. In this case, agent i’s type is a pair ( i ; i ), and the reported type is denoted by (^ i ; ^ i ). Let ^ = (^ 1 ;:::; ^ N ). The payment schemes may also depend on ^ , and the payoff of a selected agenti is given by i (^ ; ^ ;w i ) =t r;i (^ ; ^ ;w i )t d;i (^ ; ^ ) i : We can employ the same idea as before to design incentive compatible mechanisms. To illustrate, consider the following aggregator’s problem: maximize i E[W i ] i : 73 Theorem 4.5. Let the selection scheme be i2 arg max j E[ ^ W j ] ^ j ; Let the payment schemes be t d;i (^ ; ^ ) = max j6=i E[ ^ W j ] ^ j ; t r;i (^ ; ^ ;w i ) =w i : Then the proposed mechanism is efficient, dominant strategy incentive compatible, and individual rational. Remark 4.4. The fixed cost i is typically the operating cost. But it may also be interpreted as the reservation utility, e.g., the expected payoff when agenti participates in the market in the conven- tional way. In that case, the proposed mechanism ensures that each agent voluntarily participates in the auction before seeking outside options. 4.1.3 Pricing Wind Energy Economic dispatch is the short-term determination of the optimal generation schedule to meet the system load, subject to the transmission constraints. In a two-settlement market system, such market clearing can be based on a two-stage stochastic programming approach [61], taking into account the uncertainty of wind power generation. We refer to this problem as the stochastic economic dispatch problem. We consider a connected power network which consists ofN buses indexed byi = 1;:::;N, with agent i located at bus i. We assume that the supply from the thermal units at each bus i is competitive, represented by a cost function c i () which is increasing, convex and differentiable. The demand at each busi is inelastic, denoted byD i 0. For analytical and computational simplicity, we adopt a DC power flow model as a common practice [32]. In the DC flow model, a branch i-j is characterized by B ij , the negative of its susceptance, which satisfiesB ij = B ji 0. Let i be the voltage phase angle at busi. Let f ij be 74 the flow limit of branchi-j, which satisfies f ij = f ji 0. Then the active power flow over branch i-j is given by f ij =B ij ( i j ) f ij : Letx i andy i be the thermal and wind power production scheduled at the DA market at busi, and d;i be the voltage phase angle. Letx = (x 1 ;:::;x N ), and similarly for other variables. At the DA market, the ISO determines an optimal dispatch (x;y; d ) that minimizes the expected social cost: minimize x;y; d X i c i (x i ) +E[Q(y; d ;W )] (4.10a) subject to x i +y i D i = X j B ij ( d;i d;j );8i; (4.10b) B ij ( d;i d;j ) f ij ;8(i;j); (4.10c) x i ;y i 0;8i; (4.10d) whereQ(y; d ;w) is the recourse cost due to the uncertainty of wind power generation. The recourse costQ(y; d ;w) is the minimum cost to balance the power throughout the network at the RT market. Similarly as before, letp i > 0 be the cost of load shedding at busi, and assume that any excess power is spilled. Lety + i andy i be the amount of power procured and disposed, respectively, at busi, and r;i be the voltage phase angle. Lety + = (y + 1 ;:::;y + N ), and similarly for other variables. At the RT market, the ISO determines an optimal dispatch (y + ;y ; r ) that minimizes the balancing cost: minimize y + ;y ;r X i p i y + i (4.11a) subject to w i +y + i y i y i = X j B ij ( r;i r;j d;i + d;j );8i; (4.11b) B ij ( r;i r;j ) f ij ;8(i;j); (4.11c) 75 y + i ;y i 0;8i: (4.11d) The optimal value of (4.11) gives the recourse costQ(y; d ;w). Thus, we have formulated a two-stage stochastic programming problem (4.10)–(4.11). More- over, it can be shown that the the first-stage problem (4.10) is a convex optimization problem. To solve the problem (4.10)–(4.11) exactly, the ISO needs to know the true probability distribu- tions of wind power generation. This can be potentially addressed through an incentivized pricing mechanism. Traditionally, locational marginal pricing (LMP) is widely used as a settlement scheme for economic dispatch. In the two-settlement market with stochastic production, an extended LMP mechanism is proposed in [61, Ch. 3]. Specifically, the LMPs at the DA (or RT) market are the dual variables associated with the bus power balance equation (4.10b) (or (4.11b)). Those mecha- nisms are efficient in a competitive environment when agents are price takers. For strategic agents, however, such LMP-based mechanisms may not incentivize them to reveal their characteristics truthfully, as illustrated in the following example. Example 4.7. There is a single bus. The cost function of thermal power is c(x) = x 2 . The probability distribution ofW is given by the probability density functionf(w) = 1;w2 [0; 1]. The demand isD = 1. The cost of load shedding isp = 2. When the true distribution is revealed, the ISO solves the following problem at the DA market: minimize x;y c(x) +p Z y 0 (yw)f(w)dw subject to x +yD = 0; x;y 0; which gives the optimal dispatchx =y = 1=2, with the DA market price d = 1. The RT market price is r =p if there is any shortfall. The expected payoff of the wind farm is = d y p Z y 0 (y w)f(w)dw = 1=4: 76 Suppose that the wind farm claims that it can produce 1=4 amount of power with probability 1. Then the optimal dispatch is ^ x = 3=4, ^ y = 1=4, with the DA market price ^ d = 3=2. The RT market price is ^ r =p when there is any shortfall. The expected payoff of the wind farm is ^ = ^ d ^ yp Z ^ y 0 (^ yw)f(w)dw = 5=16> : By misreporting the distribution, the wind farm is able to raise the DA market price so that it can be better off. Therefore, the extended LMP mechanism is not incentive compatible. To elicit the true distribution of wind power generation, we propose an alternative mechanism following the same idea as in the SVCG mechanism. Let (x ;y ; d ) be a solution to the following problem: minimize x;y; d X i c i (x i ) +E[Q(y; d ; ^ W )] subject to x i +y i D i = X j B ij ( d;i d;j );8i; B ij ( d;i d;j ) f ij ;8(i;j); x i ;y i 0;8i: Let (x k ;y k ; k d ) be a solution to the following problem: minimize x;y; d X i c i (x i ) +E[Q(y; d ; ^ W )j ^ W i = 0] subject to x i +y i D i = X j B ij ( d;i d;j );8i; B ij ( d;i d;j ) f ij ;8(i;j); x i ;y i 0;8i: 77 Table 4.1: Cost Functions of Thermal Units Unit Cost ($/hr) (x: MW) 1 0:0776x 2 + 20x 2 0:0100x 2 + 40x 3 0:2500x 2 + 20x 4 0:0100x 2 + 40x 5 0:0222x 2 + 20x 6 0:0100x 2 + 40x 7 0:0323x 2 + 20x Let the DA payment scheme be t d;k (^ ) = X i (c i (x i )c i (x k i )); (4.12) and the RT payment scheme be t r;k (^ ;w k ) =E[Q(y k ; k d ; ^ W )j ^ W k = 0]E[Q(y ; d ; ^ W )j ^ W k =w k ]: (4.13) Theorem 4.6. The mechanism specified by (4.12)–(4.13) is efficient, dominant strategy incentive compatible, and individual rational. 4.1.4 Case Studies In this section, we focus on the ISO’s problem and present a case study based on the IEEE 57-bus system [57]. There are 57 buses, 7 thermal units and 42 loads in the system. For the cost functions of the thermal units, we use the data from MATPOWER [76], which is listed in Table 4.1. Consider that there is a wind farm colocated with each thermal unit. We obtain wind power generation data of 7 wind farms from the NREL dataset [77], and calculate the mean and standard deviation (SD) for each wind farm. Assume that eachW i is a truncated normal random variable which is bounded below by zero and bounded above by its capacity. Let the cost of load shedding bep = 100 $/MWh. We use the Monte Carlo method to compute the costC i (x i ) =E[p(x i W i ) + ] 78 Table 4.2: Characteristics of Wind Farms Unit Capacity Mean SD Fitted Cost (p = 100 $/MWh) (MW) (MW) (MW) ($/hr) (x: MW) 1 101.1 37.02 33.06 0:4933x 2 + 13:82x 2 101.0 33.01 29.66 0:5156x 2 + 13:84x 3 112.0 38.57 33.08 0:4670x 2 + 11:69x 4 100.1 38.86 31.66 0:5122x 2 + 8:35x 5 206.4 74.71 65.39 0:2480x 2 + 11:03x 6 138.4 50.96 43.53 0:3616x 2 + 14:09x 7 99.8 36.15 30.65 0:5169x 2 + 11:43x 1 2 3 4 5 6 7 2.5 3 3.5 4 4.5 x 10 4 Number of Wind Farms Expected Social Cost ($/hr) bidding cost functions bidding probability distributions Figure 4.2: The expected social cost versus the number of wind farms under two market architec- tures. with variedx i , and fit a quadratic function to the data. The characteristics of the wind farms are listed in Table 4.2. First, we compare the expected social costs under two market architectures with increasing numbers of wind farms, as shown in Fig. 4.2. When the number of wind farms isi, it means that wind farms 1 throughi are present. In either architecture, the expected social cost decreases as the number of wind farms increases. For any fixed number of wind farms, the proposed architecture always achieves a lower expected social cost than the conventional architecture. Next, we compare the share of wind power generation under two market architectures, as shown in Fig. 4.3. In either architecture, the share of wind power generation increases as the number of 79 1 2 3 4 5 6 7 0 50 100 150 200 250 300 Number of Wind Farms Share of Wind Power Generation (MW) bidding cost functions bidding probability distributions Figure 4.3: The share of wind power generation versus the number of wind farms under two market architectures. wind farms increases. For any fixed number of wind farms, the share in the proposed architecture is higher than that in the conventional architecture, due to the value of aggregation. We then compare the expected payoff of wind farm 1 under two market architectures, as shown in Fig. 4.4. The payoff of wind farmi in the conventional architecture is defined the same as that of a thermal unit: i = i x i C i (x i ), where i is the LMP. As the number of wind farms increases, the expected payoff of wind farm 1 in the conventional architecture decreases slightly, since there is increasing competition, yet the wind power penetration is not that high. The payoff of wind farmi in the proposed architecture is given by (4.12)–(4.13). When there are other wind farms, the expected payoff of wind farm 1 is evidently higher than when it is the only wind farm, again due to the value of aggregation. For any fixed number of wind farms, the expected payoff in the proposed architecture is always higher than that in the conventional architecture. This provides incentives for wind farms to participate in the proposed architecture. Lastly, we demonstrate that the proposed mechanism specified by (4.12)–(4.13) is incentive compatible. Consider the case in which all the 7 wind farms are present, and focus on wind farm 1. Assume that the capacity and the SD of generation are truthfully revealed. The only parameter wind farm 1 can misreport is the mean of generation. Fig. 4.5 depicts the expected payoff of wind farm 1 as the reported mean varies. The maximum value 649:0 $/hr is attained at 37:02 MW, which 80 1 2 3 4 5 6 7 0 100 200 300 400 500 600 700 800 900 1000 Number of Wind Farms Expected Payoff of Wind Farm 1 ($/hr) bidding cost functions bidding probability distributions Figure 4.4: The expected payoff of wind farm 1 versus the number of wind farms under two market architectures. 0 10 20 30 40 50 60 70 80 90 100 520 540 560 580 600 620 640 660 (37.02, 649.0) Reported Mean of Wind Farm 1 (MW) Expected Payoff of Wind Farm 1 ($/hr) Figure 4.5: The expected payoff of wind farm 1 versus the reported mean in the proposed archi- tecture. is exactly the actual mean ofW 1 . Therefore, truth-telling is the best strategy of wind farm 1. On the contrary, the LMP mechanism is not incentive compatible, in which agents may be better off by strategically misreporting its private information. 81 4.2 Revenue-Maximizing Objective We formulate a unified optimization problem in the mechanism design framework. For the revenue-maximizing objective, the solution is commonly called the optimal auction. In the seminal work on optimal auction design [78], a primal assumption is that the agents’ types are indepen- dent, which, however, may not be the case in our context. Since the wind farms are geographically close, the distributions are possibly correlated: conditioned on the accurate forecast of the local weather, each wind farm also has some coarse estimation of the others’, whereas the aggregator only has a prior joint distribution of the types. We adapt the work on correlated mechanism design [79, 80, 81, 82, 83] for the stochastic resource auction. We show that the aggregator may extract the full surplus by exploiting the correlation among the distributions. For theoretical contribution, we propose corollaries for special cases, and establish impossibility results when the constraints are strengthened. Then we provide a numerical example to illustrate the case where full surplus extraction is not achievable. 4.2.1 Problem Statement Consider an aggregator who wants to buy wind power (to be delivered at a given future time) from I wind farms (or simply agents), indexed byi = 1;:::;I. Agenti’s wind power generation is a random variableX i , the distribution of which can be parameterized by i . While the realization ofX i cannot be known a priori, agenti learns i (and hence the distribution ofX i ) by forecasting the local weather. The parameter i , referred to as agenti’s type, is his private information. We assume that i can take finitely many values such that i 2 i =f 1 i ;:::; m i i g for somem i , where i is referred to as agenti’s type space. Let i be j6=i j . The number of elements in i is denoted byn i = Q j6=i m j . Let = ( 1 ;:::; I ), i = ( 1 ;:::; i1 ; i+1 ;:::; I ), and similarly for the other vectors. While i is agenti’s private information, is drawn from a commonly known prior distribution (), which is treated as a probability mass function in this model. Moreover, 1 ;:::; I are not 82 necessarily independent. In fact, the main part of this work is to exploit the correlation among the agents’ types so as to extract the full surplus for the aggregator’s sake. The aggregator, either welfare-maximizing or revenue-maximizing, procures wind power from the agents through an auction. The agents have no marginal cost, and a realized generation X i has a value ofv(X i ) to the aggregator, wherev : [0;1)! R + is called the valuation function. Since the wind power generation is random, we propose a new auction paradigm, referred to as the stochastic resource auction, as shown in Fig. 4.6. In the DA market, nature draws according to the joint distribution(). Each agenti learns his own type i and updates his belief in the distribution of the others’ types, to the conditional distribution i ( i j i ). Then each agenti submits a bid ^ i as his reported type, which could be different from the true type i , to the aggregator. Based on the bid profile ^ , each agenti is selected as the wind power provider with probabilityp i ( ^ ) and makes a paymentt i ( ^ ) to the aggregator. The unselected agents leave the auction (and possibly resort to outside opportunities for profit, since they are still in the ex ante stage). In the RT market, upon the realization ofX i , the selected provideri gets paid an amount ofs i ( ^ ;X i ) from the aggregator. It is natural to prescribes i ( ^ ;X i ) =v(X i ). Thus, the payoff of agenti is u i ( ^ ;X i ) =p i ( ^ )v(X i )t i ( ^ ): In the stochastic resource auction, the selection and payment schemes define a direct revela- tion mechanism = fp;tg. We can focus on direct and truthful mechanisms without loss of generality by the revelation principle. While the mechanism is theoretically static, the proposed paradigm is aligned with the existing two-settlement market system. Moreover, when the ex post payment s i ( ^ ;X i ) is relaxed, we may derive a richer class of mechanisms with proper practical interpretations. In the context of auction design, there are mainly two classes of mechanisms: welfare- maximizing mechanisms and revenue-maximizing mechanisms. We first formulate the generic auction design problem, which is then specified for both scenarios. 83 Agents Aggregator () i ( i j i );8i ^ = ( ^ 1 ;:::; ^ I ) p = (p 1 ( ^ );:::;p I ( ^ )) t = (t 1 ( ^ );:::;t I ( ^ )) DA RT X = (X 1 ;:::;X I ) s = (s 1 ( ^ ;X 1 );:::;s I ( ^ ;X I )) Figure 4.6: The timeline of the stochastic resource auction with correlated types. The generic auction design problem is the following optimization problem: maximize p;t E[g(;p;t)] (4.14a) subject to (IC); (4.14b) (IR); (4.14c) X i p i () 1;8; (4.14d) p i () 0;8i;8; (4.14e) where the functiong represents the aggregator’s objective, which we call the objective function. The goal is to choosefp;tg to maximize the expectation ofg over the joint distribution of, subject to the incentive compatibility (IC) constraint (4.14b), the individual rationality (IR) constraint (4.14c), and the feasibility constraints (4.14d) and (4.14e). 84 We consider two notions of incentive compatibility: dominant strategy incentive compatibility (DSIC) and Bayesian incentive compatibility (BIC). The DSIC constraint states that truth telling is each agenti’s best strategy for all i : E[u i (;X i )j]E[u i ( ^ i ; i ;X i )j];8i;8;8 ^ i : (4.15) The BIC constraint is weaker, which states that truth telling is each agenti’s best strategy averaging over all i : E[u i (;X i )j i ]E[u i ( ^ i ; i ;X i )j i ];8i;8 i ;8 ^ i : (4.16) There are three notions of individual rationality, corresponding to the three stages at which voluntary participation may be relevant. The ex post IR constraint states that each agent has no incentive to withdraw from the auction after the others’ types are revealed: E[u i (;X i )j] 0;8i;8: (4.17) The interim IR constraint is weaker, which states that each agent has no incentive to withdraw from the auction after he learns his own type but before the others’ types are revealed: E[u i (;X i )j i ] 0;8i;8 i : (4.18) The ex ante IR constraint is the weakest, which states that each agent has no incentive to withdraw from the auction before he learns his own type: E[u i (;X i )] 0;8i: (4.19) 85 We define the social welfare as the expected value derived from the generation of the selected provider. Thus, we have the following objective function: g(;p;t) =E " X i p i ()v(X i ) # = X i p i () v( i ); where we define v( i ) =E[v(X i )], sinceE[v(X i )] is a function of i . Moreover, we adopt the DSIC constraint (4.15) and the ex post IR constraint (4.17). Then the welfare-maximizing problem is a linear program given by maximize p;t E " X i p i () v( i ) # (4.20a) subject to p i () v( i )t i ()p i ( ^ i ; i ) v( i )t i ( ^ i ; i );8i;8;8 ^ i ; (4.20b) p i () v( i )t i () 0;8i;8; (4.20c) X i p i () 1;8; (4.20d) p i () 0;8i;8: (4.20e) It can be shown that a variant of the Vickrey-Clarke-Groves (VCG) mechanism [42], which we call the stochastic VCG mechanism, is an optimal solution to the welfare-maximizing problem (4.20). Theorem 4.7. For each, let ~ p() be a solution to the following problem: maximize p() X i p i () v( i ) (4.21a) subject to X i p i () 1; (4.21b) p i () 0;8i: (4.21c) 86 For eachi and each, let ~ p i () be a solution to the following problem: maximize p() X j p j () v( i ) subject to X j p j () 1; p i () = 0; p j () 0;8j6=i: For eachi and each, define the payment scheme as ~ t i () = X j6=i ~ p i j () v( j ) X j6=i ~ p j () v( j ): (4.22) Then the stochastic VCG mechanism ~ =f~ p; ~ tg is an optimal solution to the welfare-maximizing problem (4.20). Proof. Since ~ p() maximizes (4.21a) subject to (4.21b) and (4.21c) for each , it follows that ~ p maximizes (4.20a) subject to (4.20d) and (4.20e). It remains to show thatf~ p; ~ tg satisfies (4.20b) and (4.20c). Suppose (4.20b) is not satisfied for somei, and ^ i . Then ~ p i () v( i ) ~ t i ()< ~ p i ( ^ i ; i ) v( i ) ~ t i ( ^ i ; i ): Substituting from (4.22) for ~ t i () and ~ t i ( ^ i ; i ), and noting that ~ p i () = ~ p i ( ^ i ; i ), we have X j ~ p j () v( j )< X j ~ p j ( ^ i ; i ) v( j ); which contradicts ~ p() solving (4.21). To verify (4.20c), we have ~ p i () v( i ) ~ t i () = X j ~ p j () v( j ) X j6=i ~ p i j () v( j ) = X j ~ p j () v( j ) X j ~ p i j () v( j ) 87 0; where the last inequality again follows from ~ p() solving (4.21). Therefore, the stochastic VCG mechanism ~ =f~ p; ~ tg solves the welfare-maximizing problem (4.20). Note that the above implementation is very robust, in the following two aspects. First, each agent’s best response depends on his own type only, whatever the others’ types or bids; in particu- lar, the best response does not depend on the prior distribution(). Second, the selection scheme ~ p does not depend on () either, which means that this mechanism always works, whether the types are independent or not. We define the aggregator’s revenue as the value derived from the generation of the selected provider minus the net payment made to the agents. Thus, we have the following the objective function: g(;p;t) =E " X i (p i ()v(X i ) +t i ()p i ()s i (;X i )) # = X i t i (): Moreover, we adopt the DSIC constraint (4.15) and the interim IR constraint (4.18). Then the revenue-maximizing problem is a linear program given by: maximize p;t E " X i t i () # (4.23a) subject to p i () v( i )t i ()p i ( ^ i ; i ) v( i )t i ( ^ i ; i );8i;8;8 ^ i ; (4.23b) E[p i () v( i )t i ()j i ] 0;8i;8 i ; (4.23c) X i p i () 1;8; (4.23d) p i () 0;8i;8: (4.23e) 88 Since (4.23c) is less constrained than (4.20c), it follows that the stochastic VCG mechanism is a feasible (though suboptimal in general) solution to the revenue-maximizing problem (4.23). To obtain the optimal solution, one can solve the linear program directly. It turns out that when there is “enough” correlation among the agents’ types, the optimal mechanism has an appealing property, which we will discuss next. 4.2.2 Main Results We first consider the optimal mechanism under full information, based on which we define full surplus extraction. We then define full correlation of the joint distribution of the types, under which the optimal mechanism for (4.23) extracts the full surplus. As theoretical contribution, we derive impossibility results to show that there do not exist “better” mechanisms. When the aggregator has full information about, then the revenue-maximizing problem is a family of problems indexed by, where the IC constraints are discarded: maximize p();t() X i t i () (4.24a) subject to p i () v( i )t i () 0;8i; (4.24b) X i p i () 1; (4.24c) p i () 0;8i; (4.24d) which is equivalent to the following problem: maximize p() X i p i () v( i ) (4.25a) subject to X i p i () 1; (4.25b) p i () 0;8i: (4.25c) 89 On the other hand, this family of problems is also equivalent to a single optimization problem: maximize p;t E " X i t i () # (4.26a) subject to p i () v( i )t i () 0;8i;8; (4.26b) X i p i () 1;8; (4.26c) p i () 0;8i;8: (4.26d) In the remaining of the work, we denote by = fp ;t g an optimal solution to problem (4.26), which also solves problem (4.24) and problem (4.25). Compared with problem (4.23), problem (4.26) omits the IC constraints, but strengthens the IR constraints. It is still true, though not obvious, that the optimal value of problem (4.23) is upper bounded by that of problem (4.26). Proposition 4.2. Let =f p; tg be an optimal solution to problem (4.23). Then E " X i t i () # E " X i t i () # : Proof. We have E " X i t i () # = X i E[E[ t i ()j i ]] X i E[E[ p i () v( i )j i ]] = X i E[ p i () v( i )] =E " X i p i () v( i ) # E " X i p i () v( i ) # =E " X i t i () # ; 90 where the first inequality follows from (4.23c); the second inequality follows from the fact that p () is an optimal solution to problem (4.25) for each; and the last equality follows from the equivalence between problem (4.24) and problem (4.25). Proposition 4.2 motivates the following definition. Definition 4.1 (Full Surplus Extraction). A mechanism =fp;tg is said to extract the full surplus, if E " X i t i () # =E " X i t i () # : Recall that we treat the joint distribution () and the conditional distribution i ( i j i ) as probability mass functions, and thatj i j = m i ;j i j = n i . Define the conditional distribution matrixA i for eachi as A i = 0 B B B @ i ( 1 i j 1 i ) i ( n i i j 1 i ) . . . . . . . . . i ( 1 i j m i i ) i ( n i i j m i i ) 1 C C C A ; where the rows are indexed by the elements in i , and the columns are indexed by the elements in i . Definition 4.2 (Full Correlation). The joint distribution() is said to have full correlation, ifA i has rankm i for alli. One special case of full correlation is perfect correlation, i.e., for anyi, knowing the value of i gives the exact value of i . In that case, eachA i has exactly one entry 1 in each row and 0s elsewhere. Under the full correlation condition, full surplus extraction is achievable. Interestingly, this can be proved by construction based on the stochastic VCG mechanism. Theorem 4.8. If() has full correlation, the optimal mechanism for (4.23) extracts the full sur- plus. 91 Proof. Let ~ =f~ p; ~ tg be a stochastic VCG mechanism specified in Theorem 4.7. Define h i ( i ) = X i i ( i j i )(~ p i () v( i ) ~ t i ()): Leth i = (h i ( 1 i );:::;h i ( m i i )) | . SinceA i has rankm i , there exists at 0 i = (t 0 i ( 1 i );:::;t 0 i ( n i i )) | such thatA i t 0 i = h i . Now letp = ~ p andt i () = ~ t i () +t 0 i ( i ). We show that =fp;tg is an optimal solution to problem (4.23) that extracts the full surplus. First,fp;tg satisfies (4.23b), sincef~ p; ~ tg satisfies (4.20b) andt 0 i ( i ) does not depend on i . To verify (4.23c), we have E[p i () v( i )t i ()j i ] = X i i ( i j i )(~ p i () v( i ) ~ t i ()t 0 i ( i )) = X i i ( i j i )(~ p i () v( i ) ~ t i ()) X i i ( i j i )t 0 i ( i ) =h i ( i )h i ( i ) = 0: Finally, the above equation and the equivalence between problem (4.21) and problem (4.25), shows that equalities hold everywhere in the proof of Proposition 4.2. Thus,fp;tg extracts the full sur- plus. Corollary 4.1. LetI = 2. Define the joint distribution matrixA as A = 0 B B B @ ( 1 1 ; 1 2 ) ( 1 1 ; m 2 2 ) . . . . . . . . . ( m 1 1 ; 1 2 ) ( m 1 1 ; m 2 2 ) 1 C C C A : IfA is square and invertible, the optimal mechanism for (4.23) extracts the full surplus. 92 Proof. SinceA is invertible, detA6= 0. Note thatA 1 is obtained by dividing each rowm ofA by the marginal distribution( m 1 ;). Thus, we have detA 1 = detA Q m 1 m=1 ( m 1 ;) 6= 0: SoA 1 is invertible. Similarly,A 2 is also invertible, and the full correlation condition is satisfied. The claim follows. Theorem 4.8 states that under certain conditions, there exists a DSIC and interim IR mechanism that extracts the full surplus. It is natural to ask whether there are mechanisms with more desirable properties. One direction is to strengthen the IR constraint, by replacing the interim IR constraint (4.23c) by the ex post IR constraint (4.17). It turns out that in general, this leads to an impossibility result, even if the IC constraint is relaxed. Lemma 4.1. If a mechanism =fp;tg is ex post individual rational and extracts the full surplus, thenfp;tg must be a solution to problem (4.26). Proof. Since =fp;tg is ex post IR, we have p i () v( i )t i () 0;8i;8; or E " X i t i () # E " X i p i () v( i ) # : Since =fp;tg extracts the full surplus, we have E " X i t i () # =E " X i t i () # =E " X i p i () v( i ) # : 93 Therefore, E " X i p i () v( i ) # E " X i p i () v( i ) # ; which means thatp() is a solution to problem (4.25) for each. It follows thatfp;tg is a solution to problem (4.26). Assumption 4.1. There exists somei who has at least two types i 6= ^ i such that v( i ) > v( ^ i ) andp i ( ^ i ; i )> 0 for some i . Theorem 4.9. Under Assumption 4.1, there does not exist a mechanism that is Bayesian incentive compatible, ex post individual rational, and extracts the full surplus. Proof. Suppose such a mechanism exists. By Lemma 4.1, it must be a solution to problem (4.26), hence denoted by =fp ;t g. By Assumption 4.1, there exists somei with i 6= ^ i such that v( i )> v( ^ i ) andp i ( ^ i ; i )> 0 for some i . Then E[p i ( ^ i ; i ) v( i )t i ( ^ i ; i )j i ] >E[p i ( ^ i ; i ) v( ^ i )t i ( ^ i ; i )j i ] = 0 =E[p i ( i ; i ) v( i )t i ( i ; i )j i ]; which violates the BIC constraint (4.16). Thus, there is no BIC and ex post IR mechanism that extracts the full surplus. Alternatively, we can strengthen the definition of full surplus extraction. A mechanism = fp;tg is said to ex post extract the full surplus, if X i t i () = X i t i ();8: Lemma 4.2. If a mechanism =fp;tg ex post extracts the full surplus, then it is ex post individual rational. 94 Proof. Since =fp;tg ex post extracts the full surplus, it solves (4.24) for each. Therefore, p i () v( i )t i () = 0;8i;8; which implies the ex post IR. Corollary 4.2. Under Assumption 4.1, there does not exist a mechanism that is Bayesian incentive compatible, and ex post extracts the full surplus. Proof. It follows by Lemma 4.2 and Theorem 4.9. Note that the impossibility results obtained so far do not depend on the structure of(), which assure us that Theorem 4.8 characterizes the best possible solution one could expect. 4.2.3 Example In this section, we validate Theorem 4.8 by providing a numerical example, which also illustrates the case where full surplus extraction is not achievable. Among the various models of wind power generation [84, 85, 86, 87], we adopt a simple yet realistic model. Assume that agenti’s generation isX i = i W 3 i , where i captures his technology (such as the turbine size and the energy conversion efficiency), and W i denotes the local wind speed. W i is a Rayleigh distributed random variable with parameter i > 0, whose probability density function is given by f(x; i ) = (x= 2 i )e x 2 =(2 2 i ) ; x 0: We assume that i ’s are fully correlated, whereas each i is independent of i and. Thus, when is common knowledge, we have one-dimensional types i = i that are fully correlated; when each i is private information, we have two-dimensional types i = ( i ; i ) that are not fully correlated. In the latter case, one cannot expect achieving the full surplus extraction, as verified by the following results. 95 LetI = 2 andv(X i ) =X i , i.e., each unit of wind power is worth one unit of money. It can be shown that v( i ) =E X i [v(X i )] = (3 p 2=2) i 3 i ; or simply v( i ) = i 3 i by normalization. Assume that 1 2f L 1 = 1; M 1 = 3; H 1 = 5g; 2 2 f L 2 = 2; M 2 = 4; H 2 = 6g, and the joint distribution matrix for is 0 B B B B B @ L 2 M 2 H 2 L 1 :25 :05 0 M 1 :05 :30 :05 H 1 0 :05 :25 1 C C C C C A : That is, the levels of wind speed at those two locations are highly positively correlated. It is easy to check that this matrix is invertible, so that 1 and 2 are fully correlated by Corollary 4.1. In the k-th scenario, i is independently distributed as i = 8 > > < > > : i +k with probability 1=2 i k with probability 1=2 ; where 1 = 190; 2 = 200. Herek characterizes the deviation of i . Whenk = 0, the types i = i are fully correlated; whenk> 0, the types i = ( i ; i ) are not fully correlated. We solve the linear programs (4.23) and (4.26) and compare the expected revenue, as shown in Fig. 4.7. Since the expected value of i does not change, the expected revenue under full information remains the same ask varies. Whenk = 0, the optimal mechanism for problem (4.23) extracts the full surplus; ask increases, its expected revenue decreases, since the uncertainty of i to the aggregator increases. 96 0 5 10 15 20 25 30 35 40 45 50 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 x 10 4 k Expected Revenue Full Optimal Figure 4.7: The expected revenue of the full information mechanism for problem (4.26) and the optimal mechanism for problem (4.23) versus the uncertainty in i ’s, whereas i ’s are fully corre- lated. 4.3 Discussion Motivated by the observation that it is more efficient for wind power producers to bid probability distributions of generation than bid cost functions, we propose the stochastic resource auction paradigm, which is aligned with the existing two-settlement system. In the aggregator’s problem, we propose two incentive compatible mechanisms. We then make several generalizations: general objective function, selecting multiple providers, and inclusion of fixed cost. In the ISO’s problem, we present a counterexample to show that the LMP-based mechanisms are subject to manipulation. We propose an alternative mechanism which is incentive compatible and thus induces the stochastic economic dispatch. The presented case study illustrates the advantages of the proposed architecture. For the revenue-maximizing objective, under the full correlation condition, there exists a dom- inant strategy incentive compatible and interim individual rational mechanism that extracts the full surplus. The impossibility results show that mechanisms with stronger properties do not exist. As the numerical example suggests, it is an interesting topic to quantify the revenue loss against the full information case, when the types are not fully correlated. One can also consider multiple winners and continuous types for extensions. In future work, we will investigate a dynamic version of this problem, by employing the dynamic mechanism design theory. 97 The presented work, along with the proposed future work, will potentially provide economic solutions for integrating renewables into the smart grid network. 98 Chapter 5 Stochastic Dynamic Pricing for Demand Response In most retail electricity markets, pricing to residential customers has followed the standard prac- tice of a flat rate structure. Utility companies (or distributors) charge household customers (or users) a fixed price per unit for electricity use regardless of the cost of supply at the time of con- sumption. As a variant, a tiered rate structure is often adopted that depends on the monthly energy usage. Such inflexible schemes are of no avail in incentivizing users to change their consumption patterns so as to reduce peak load. The development of smart grid technologies opens up the possibility of improving the effi- ciency of energy consumption. Dynamic pricing has been proposed recently. The idea is to coor- dinate demand response to the benefit of the overall system through financial incentives. The common approach of dynamic pricing is based on network utility maximization (NUM), a well- known framework to study network resource allocation problems [88] since the seminal work by Kelly [89]. A typical NUM problem is to maximize the sum of the users’ utilities subject to cer- tain linear constraints (e.g., flow and capacity constraints). The convex structure of the problem leads to dual decomposition methods which correspond to resource allocation via pricing. A tuto- rial on decomposition methods for NUM can be found in [90]. In the context of demand side management, a discrete-time model with a finite horizon is commonly adopted. A class of recent work studied designing time-varying prices that align individual optimality with social optimality [6, 91, 92, 54]. Such multistage models, regarded as dynamic NUM problems, are natural exten- sions of static NUM problems. Some more specific models include deadline differentiated pricing 99 [93], electric vehicle charging [94], etc. In practice, the so-called time-of-use plan is a basic imple- mentation of this idea, in which rates for periods with different demand levels are pre-established, typically highest during the peak load hours. However, all those models are deterministic, and thus the optimal prices, though time- dependent, are also deterministic. In reality, there is uncertainty in the system. The information sets are augmented as time advances. In such a stochastic setting, a deterministic price vector would not induce the socially optimal decisions. Rather, the optimal price process should evolve according to the up-to-date information; so do the optimal decisions. This is the motivation of our proposed model, called stochastic dynamic pricing that further generalizes dynamic NUM prob- lems. Demand response could also be done via interruptible power service contracts of varying reli- ability [95]. [96] studied dynamic pricing for a revenue-maximizing retailer. A novel real-time pricing scheme called monotonic marginal pricing is proposed in [97], while price discrimination is allowed. [98] is close to our work in spirit, though in a different context. Beyond proving the existence of an optimal price process, we develop a distributed algorithm and investigate the information structure. An important consideration in future grid systems is the possibility for customers to utilize storage technologies to hedge against significant price changes. Moreover, a variety of strategies can be utilized to mimic the behavior of storage, such as opportunistic utilization of electric vehicle availability and deferring appliance load [99] or air conditioning consumption [100, 101]. We consider a simple model for storage, in which each user has its own storage with a sufficiently large capacity. Inclusion of capacity limit, charging rate limit and round-trip efficiency is left for future work. The focus of this work is to develop an adaptive pricing mechanism in response to the presence of uncertainty. 100 5.1 User’s Problem: Stochastic Optimal Control Our first task is to understand how an individual user makes the optimal decisions to maximize his own payoff, given an exogenous price process. We formulate a stochastic optimal control problem, in which the user needs to decide how much power to buy and how much to consume (and therefore how much to store) at each time. We consider both the elastic and the inelastic demand cases. 5.1.1 Problem Statement The horizon of interest is typically one day that is discretized into T slots, indexed by t = 0; 1;:::;T 1. At each time t, the user buys z t 0 and consumes x t 0 amount of power. We callz t the demand andx t the consumption at timet. The user has a storage with a sufficiently large capacity. Lety t 0 be the storage level at the beginning of timet, withy 0 given. Denote by y T the storage level at the end of the horizon. Then the storage dynamics is given by y t+1 =y t +z t x t ;8t: Let t be the price of power at time t. We consider (y t ; t ) as the state variables at time t, and (x t ;z t ) as the decision variables. Definex =fx t ;8tg,y =fy t ;8tg,z =fz t ;8tg and =f t ;8tg. Assume that the price process is an exogenous Markov process which is independent of the decision variables (x;z). Conditioned on t , the probability density function of t+1 is denoted by f t (j t ). 5.1.2 Inelastic Demand Case In this case, the user’s demand at each timet is a fixed amountD t 0, so thatx t = D t for allt. Sincey t ;z t 0 for allt, the control constraint at timet given the state (y t ; t ) isz t (D t y t ) + , where we definex + = maxfx; 0g. The user’s problem is to minimize the expected cost: minimizeE " X t t z t # : (5.1) 101 We apply the dynamic programming algorithm to obtain the optimal solution to problem (5.1). Proposition 5.1. The optimal policy of problem (5.1) has a threshold form as shown in (5.4). Proof. Define the cost-to-go functions as J T (y T ; T ) = 0; (5.2a) J t (y t ; t ) = min zt(Dtyt) + Q t (y t ; t ;z t ); (5.2b) where Q t (y t ; t ;z t ) = t z t +E[J t+1 (y t +z t D t ; t+1 )]; and T is defined for notational convenience. By introducing the variabler t = y t +z t D t , we can write (5.2b) as J t (y t ; t ) = t (D t y t ) + min rt(ytDt) + G t (r t ; t ); (5.3) where G t (r t ; t ) = t r t +E[J t+1 (r t ; t+1 )]: It is clear that lim rt!1 G t (r t ; t ) =1, sinceJ t+1 (r t ; t+1 ) = 0 for all t+1 whenr t is suffi- ciently large. We will prove shortly thatG t (r t ; t ) is convex inr t . Thus, for a fixed t ,G t (r t ; t ) has a minimumS t ( t ) with respect tor t 0: S t ( t ) = arg min rt0 G t (r t ; t ): Then, in view of the constraint r t y t D t , a minimizing r t in (5.3) equals S t ( t ) if y t < S t ( t ) +D t , and equalsy t D t otherwise. Using the reverse transformationz t = r t +D t y t , 102 we see that the minimum in (5.2b) is attained atz t =S t ( t ) +D t y t ify t <S t ( t ) +D t , and at z t = 0 otherwise. We obtain an optimal policy in the following threshold form: z t (y t ; t ) = 8 > > < > > : S t ( t ) +D t y t ; y t <S t ( t ) +D t ; 0; y t S t ( t ) +D t : (5.4) To show that G t (r t ; t ) is convex in r t , we proceed to show the convexity of the cost-to- go functions J t inductively. Since J T (y T ; T ) is the zero function, it is convex in y T . Then G T1 (r T1 ; T1 ) is convex inr T1 , and an optimal policy at timeT 1 is given by z T1 (y T1 ; T1 ) = 8 > > < > > : S T1 ( T1 ) +D T1 y T1 ; y T1 <S T1 ( T1 ) +D T1 ; 0; y T1 S T1 ( T1 ) +D T1 : Furthermore, from (5.2b) we have J T1 (y T1 ; T1 ) = 8 > > < > > : T1 (S T1 ( T1 ) +D T1 y T1 ); y T1 <S T1 ( T1 ) +D T1 ; 0; y T1 S T1 ( T1 ) +D T1 ; which is convex iny T1 . Thus, given the convexity ofJ T , we can establish the convexity ofJ T1 . This argument can be repeated to show that for allt =T 2;:::; 0, ifJ t+1 is convex iny t+1 , then G t (r t ; t ) is convex inr t , and we have J t (y t ; t ) = 8 > > < > > : t (S t ( t ) +D t y t ) +E[J t+1 (S t ( t ); t+1 )]; y t <S t ( t ) +D t ; E[J t+1 (y t D t ; t+1 )]; y t S t ( t ) +D t ; which can be shown to be convex in y t . Thus, the optimality of the threshold policy (5.4) is proved. We now derive some important properties of the cost-to-go functions. Proposition 5.2. J t (y t ; t ) is decreasing iny t for allt. 103 Proof. Since J T (y T ; T ) = 0, J T (y T ; T ) is decreasing in y T . Assume that J t+1 (y t+1 ; t+1 ) is decreasing iny t+1 . Then for anyy t and ^ y t such that ^ y t > y t , we haveJ t+1 (^ y t +z t D t ; t+1 ) J t+1 (y t +z t D t ; t+1 ), so thatQ t (^ y t ; t ;z t )Q t (y t ; t ;z t ). Therefore, J t (^ y t ; t ) = min zt(Dt^ yt) + Q t (^ y t ; t ;z t ) min zt(Dtyt) + Q t (^ y t ; t ;z t ) min zt(Dtyt) + Q t (y t ; t ;z t ) =J t (y t ; t ): The claim follows by induction. Proposition 5.3. Iff t (j^ t ) has first-order stochastic dominance overf t (j t ) for allt and for all ^ t and t such that ^ t > t , thenJ t (y t ; t ) is increasing in t for allt. Proof. Since J T (y T ; T ) = 0, J T (y T ; T ) is increasing in T . Assume that J t+1 (y t+1 ; t+1 ) is increasing in t+1 . Then for any t and ^ t such that ^ t > t , the first-order stochastic dominance gives E[J t+1 (y t +z t D t ; t+1 )jy t ; ^ t ;z t ]E[J t+1 (y t +z t D t ; t+1 )jy t ; t ;z t ]; which impliesQ t (y t ; ^ t ;z t ) Q t (y t ; t ;z t ) and thereforeJ t (y t ; ^ t ) J t (y t ; t ). The claim fol- lows by induction. 5.1.3 Elastic Demand Case In this case, the user’s demand is represented by a utility functionu t (x t ) for each timet, which is concave, increasing and differentiable with lim xt!1 u 0 t (x t ) = 0. Sincex t ;y t ;z t 0 for allt, the 104 control constraint at timet given the state (y t ; t ) isx t 0;z t (x t y t ) + . The user’s problem is to maximize the expected net utility: maximizeE " X t (u t (x t ) t z t ) # : (5.5) Proposition 5.4. The optimal policy of problem (5.5) has a threshold form as shown in (5.8). Proof. Define the (negative) cost-to-go functions as J T (y T ; T ) = 0; (5.6a) J t (y t ; t ) = max xt0;zt(xtyt) + Q t (y t ; t ;x t ;z t ); (5.6b) where Q t (y t ; t ;x t ;z t ) =u t (x t ) t z t +E[J t+1 (y t +z t x t ; t+1 )]: By introducing the variabler t =y t +z t x t , we can write (5.6b) as J t (y t ; t ) = t y t + max xt0;rt(ytxt) + F t (x t ;r t ; t ) (5.7) = t y t + max xt0;rt(ytxt) + (G t (x t ; t ) +H t (r t ; t )); where F t (x t ;r t ; t ) =G t (x t ; t ) +H t (r t ; t ); G t (x t ; t ) =u t (x t ) t x t ; H t (r t ; t ) =E[J t+1 (r t ; t+1 )] t r t : First, we show inductively the concavity ofJ t (y t ; t ) iny t . SinceJ T (y T ; T ) = 0,J T (y T ; T ) is concave iny T . Assume thatJ t+1 (y t+1 ; t+1 ) is concave iny t+1 . ThenH t (r t ; t ) is concave inr t . 105 Also,G t (x t ; t ) is concave inx t . SoF t (x t ;r t ; t ) is concave in (x t ;r t ). Since partial maximization over a convex set preserves concavity, the second term on the right-hand side of (5.7) is concave in y t . Therefore,J t (y t ; t ) is concave iny t . Now we show that the optimal policy has a threshold form. Since lim xt!1 u 0 t (x t ) = 0, we have lim xt!1 G t (x t ; t ) =1 and lim rt!1 H t (r t ; t ) =1. Thus, for a fixed t ,G t (x t ; t ) has a maximum x t ( t ) with respect tox t 0: x t ( t ) = arg min xt0 G t (x t ; t ); andH t (r t ; t ) has a maximum r t ( t ) with respect tor t 0: r t ( t ) = arg min rt0 H t (r t ; t ): Let (^ x t (y t ; t ); ^ r t (y t ; t )) be a solution to the following optimization problem: maximize xt;rt u t (x t ) +E[J t+1 (r t ; t+1 )] subject to x t +r t =y t ; x t ;r t 0: Then, in view of the constraintr t y t x t , a maximizing (x t ;r t ) in (5.7) equals ( x t ( t ); r t ( t )) if y t x t ( t ) + r t ( t ); if y t > x t ( t ) + r t ( t ), there must be y t = x t +r t , so that (x t ;r t ) = (^ x t (y t ; t ); ^ r t (y t ; t )). Using the reverse transformationz t = r t +x t y t , we obtain an optimal policy in the following threshold form: x t (y t ; t ) = 8 > > < > > : x t ( t ); y t x t ( t ) + r t ( t ); ^ x t (y t ; t ); y t > x t ( t ) + r t ( t ); (5.8a) z t (y t ; t ) = 8 > > < > > : x t ( t ) + r t ( t )y t ; y t x t ( t ) + r t ( t ); 0; y t > x t ( t ) + r t ( t ); (5.8b) 106 O y t x t (y t ; t ) z t (y t ; t ) x t ( t ) x t ( t ) + r t ( t ) x t ( t ) + r t ( t ) Figure 5.1: The optimal policy (x t (y t ; t );z t (y t ; t )) of problem (5.5) as a function of y t for a fixed t . as shown in Fig. 5.1. Note that the curve of ^ x t (y t ; t ) ony t > x t ( t ) + r t ( t ) is plotted as linear for illustration purposes. In general, we can only claim that it is increasing iny t . 5.2 System Problem: Stochastic Dynamic Pricing Now we formulate the system problem, which involves multiple users served by a single distributor. The system objective is to maximize the expected social welfare. We show that there exists a price process such that when both the distributor and the users maximize their own payoffs, the resulting outcome is also socially optimal. 5.2.1 Problem Statement Consider a finite horizon, where the time is indexed byt = 0; 1;:::;T 1. There areN users, indexed byn = 1;:::;N. The user’s model is the same as that of the elastic demand case in the last section, except for the additional subscriptn in the corresponding variables. Specifically,x n;t is the consumption of usern at timet,y n;t is the storage level of usern at the beginning of timet, andz n;t is the demand of usern at timet. The storage dynamics of usern is given by y n;t+1 =y n;t +z n;t x n;t ;8t: 107 There is a single distributor who sellsz t 0 amount of power at each timet. We callz t the supply at timet, which must match the demand: X n z n;t =z t ;8t: The utility function of usern at timet is denoted byu n;t (x n;t ), which is concave, increasing and differentiable with lim xn;t!1 u 0 n;t (x n;t ) = 0. Note that the utility functions are deterministic. To model the uncertainty, we assume that the distributor has a stochastic cost functionc t (z t ;W t ) for each time t, which is driven by an exogenous stochastic processfW t ;8tg whose natural fil- tration is denoted byfF t ;8tg. Moreover, the function c t (z t ;W t ) with respect to z t is convex, increasing and differentiable with lim zt!1 c 0 t (z t ;W t ) =1. For example, the cost functionc t (z t ;W t ) = ((z t W t ) + ) 2 can be interpreted in the following way. The distributor owns a wind farm, which generatesW t amount of power for free at timet. On top of that, the distributor can procure deterministic power at a quadratic cost. Let d = fz t ;8tg be the collection of the decision variables of the distributor, and d n = fx n;t ;y n;t ;z n;t ;8tg be that of each usern. The system problem is to maximize the expected social welfare: maximize d;dn;8n E " X t X n u n;t (x n;t )c t (z t ;W t ) !# (5.9a) subject to X n z n;t =z t ;8t; (5.9b) y n;t+1 =y n;t +z n;t x n;t ;8n;8t; (5.9c) x n;t ;y n;t ;z n;t ;z t 0;8n;8t; (5.9d) x n;t ;y n;t ;z n;t ;z t 2F t ;8n;8t: (5.9e) 5.2.2 Dual Decomposition We call the solution to the system problem (5.9) an efficient allocation. The goal is to show that there exists a price process according to which the efficient allocation can be achieved when each 108 agent maximizes his own payoff. Such a price process is referred to as an optimal price process. To this end, we apply dual decomposition methods [102]. Define D =fdjz t 0;z t 2F t ;8tg and D n = 8 > > > < > > > : d n y n;t+1 =y n;t +z n;t x n;t ;8t; x n;t ;y n;t ;z n;t 0;8t; x n;t ;y n;t ;z n;t 2F t ;8t 9 > > > = > > > ; for all n. Note thatD[ ([ n D n ) is not the constraint set of the system problem (5.9). There is another coupling constraint (5.9b), which complicates the problem. Associate the Lagrange multipliers t with (5.9b). Note that = f t ;8tg is adapted to fF t ;8tg. The (partial) Lagrangian is defined as L(d;d 1 ;:::;d N ;) =E " X t X n u n;t (x n;t )c t (z t ;W t ) + t z t X n z n;t !!# : The dual function is defined as g() = max d2D;dn2Dn;8n L(d;d 1 ;:::;d N ;) =h() + X n h n (); where h() = max d2D E " X t ( t z t c t (z t ;W t )) # ; (5.10) h n () = max dn2Dn E " X t (u n;t (x n;t ) t z n;t ) # : (5.11) 109 That is, the dual functiong() decouples into the distributor’s problem (5.10) and user’s problem (5.11) for all n. Let d () be a solution to (5.10), and d n () be a solution to (5.11). The dual problem is then given by minimize h() + X n h n () (5.12a) subject to t 2F t ;8t: (5.12b) Since the system problem (5.9) is convex and satisfies the (refined) Slater’s condition, strong duality holds. That is, when is a dual optimal solution that solves (5.12),fd ( );d n ( );8ng is a primal optimal solution that solves (5.9). We state the result in the following. Proposition 5.5. If solves the dual problem (5.12), d ( ) solves the distributor’s problem (5.10), andd n ( ) solves the user’s problem (5.11) for alln, thenfd ( );d n ( );8ng solves the system problem (5.9). The dual decomposition has the following economic interpretation. When the dual optimal solution is chosen as the price process, according to which each agent maximizes his own payoff, the resulting outcome is also socially optimal. That is, the dual optimal solution is an optimal price process. We now give a more explicit form of . Proposition 5.6. Letfx n;t ;y n;t ;z n;t ;z t ;8n;8tg be a primal optimal solution that solves (5.9). Thenf t ;8tg with t =c 0 t (z t ;W t ) for allt is a dual optimal solution that solves (5.12). Proof. Associate the Lagrange multipliers n;t with (5.9c). Since strong duality holds, the KKT conditions provide necessary and sufficient conditions for optimality [34]: x n;t (u 0 n;t (x n;t ) n;t ) = 0;8n;8t; (5.13a) u 0 n;t (x n;t ) n;t 0;8n;8t; (5.13b) y n;t ( n;t n;t1 ) = 0;8n;8t; (5.13c) n;t n;t1 0;8n;8t; (5.13d) 110 z n;t ( n;t t ) = 0;8n;8t; (5.13e) n;t t 0;8n;8t; (5.13f) z t ( t c 0 t (z t ;W t )) = 0;8t; (5.13g) t c 0 t (z t ;W t ) 0;8t; (5.13h) X n z n;t z t = 0;8t; (5.13i) y n;t+1 y n;t z n;t +x n;t = 0;8n;8t; (5.13j) x n;t ;y n;t ;z n;t ;z t 0;8n;8t; (5.13k) x n;t ;y n;t ;z n;t ;z t ; t ; n;t ;2F t ;8n;8t: (5.13l) By (5.13h), t c 0 t (z t ;W t ) for allt. Suppose that t <c 0 t (z t ;W t ) for somet. It follows from (5.13g) thatz t = 0 and thereforez n;t = 0 for alln by (5.13i) and (5.13k). It is easy to check that if we replace t byc 0 t (z t ;W t ) and keep the other variables unchanged, the KKT conditions (5.13) are still satisfied. Thus,f t ;8tg with t =c 0 t (z t ;W t ) for allt is a dual optimal solution. 5.2.3 Offline Distributed Algorithm The dual decomposition (5.12) prompts an offline distributed algorithm to compute the optimal price process =f t ;8tg that is adapted tofF t ;8tg, the natural filtration forfW t ;8tg. In general, W t has an infinite support for all t; so does t . For computational tractability, assume thatF t is finite for allt, which can be done by quantization. Moreover, assume thath() andh n ()’s are differentiable with bounded gradients. Then the following gradient method can be used: t (k + 1) = " t (k) k z t ((k)) X n z n;t ((k)) !# + ;8t; (5.14) 111 wherek is the iteration index, and k > 0 is a diminishing step size satisfying lim k!1 k = 0; 1 X k=0 k =1: For example, we can choose k = 1=k. We emphasize that t (k) is a function ofF t . Under the above assumptions, the algorithm is guaranteed to converge [40]. That is,(k) will converge to the dual optimal solution ask!1. Note that x n;t ((k)) does not appear in (5.14) since it has been internalized by the user. In other words, each user only needs to report his optimal demandz n;t ((k)) in each iteration. To summarize, we present the following offline distributed algorithm, which computes the optimal price process at the beginning of the horizon: Algorithm 2 Offline Distributed Algorithm for Stochastic Dynamic Pricing Initialization: the system setsk = 0 and(0) a positive process that is adapted tofF t ;t2Tg, and then broadcasts(0) to the distributor and the users. repeat The distributor solves the distributor’s problem (5.10) and reports the solutionfz t ((k));8tg to the system; Each usern solves the user’s problem (5.11) and reports the solutionfz n;t ((k));8tg to the system; The system updates the price process according to (5.14) with k = 1=k, broadcasts the new price process(k + 1) to the agents, and setsk k + 1. until termination criterion is satisfied 5.2.4 Investigating the Information Structure We have conceptually provided a distributed algorithm to compute the optimal price process. For the algorithm to work, the embedded problems (5.10) and (5.11) have to be solved efficiently. Upon observing the constraint setsD andD n , we note that the distributer’s problem (5.10) decouples in t, so that it can be solved efficiently. The user’s problem (5.11), however, is a nontrivial stochas- tic program that is difficult to solve. The main issue is that the optimal price processf t ;8tg in 112 general has a sophisticated structure in terms of its joint distribution, even if the underlying pro- cessfW t ;8tg is simple enough. This suggests that it is important to understand the information structure of the stochastic processes involved in this model. Recall thatfW t ;8tg is an exogenous stochastic process, with the natural filtrationfF t ;8tg. The efficient allocationfx n;t ;y n;t ;z n;t ;z t ;8n;8tg, as a primal optimal solution to the system problem (2.7), is adapted tofF t ;8tg. The optimal price processf t ;8tg, as a dual optimal solution, is also adapted tofF t ;8tg. Denote the natural filtration forf t ;8tg byfG t ;8tg. One immediate observation is that fG t ;8tg is no more refined thanfF t ;8tg. That is,G t F t for all t, sinceG t is the smallest -algebra such that s 2 G t for all s t. Also,G t is not necessarily equal toF t . This is becausefF t ;8tg is the natural filtration for a general stochastic processfW t ;8tg. It is possible thatc t (z t ;W t ) = c t (z t ; ^ W t ) for someW t and ^ W t such thatW t 6= ^ W t , in which caseW t and ^ W t induce the same optimal price t and thereforeG t (F t . In the remainder of the subsection, we present a numerical example to demonstrate the fact that even ifW t ’s are independent,f t ;8tg is not necessarily a Markov process (with respect to its natural filtrationfG t ;8tg, and hencefF t ;8tg as well). This example also illustrates the advantage of stochastic dynamic pricing over deterministic dynamic pricing. Example 5.1. Consider n = 1 and T = 3. That is, there is only one user (served by a single distributor), so that we will omit the subscriptn in the corresponding variables. The user has a stationary utility functionu t (x t ) = log(x t + 1) fort = 0; 1; 2. The distributor has a cost function c t (z t ;W t ) = z 2 t + W t z t , where W 0 ;W 1 ;W 2 are independent Bernoulli random variables with success probabilityp = 1=2. While this problem can be solved by dynamic programming, it is too involved to give an analyt- ical representation of the optimal policy even for such a simple setting. Rather, since the number of sample paths is small (i.e., 2 3 = 8), we treat the stochastic convex program (2.7) as a large-scale static convex program, the solution to which gives the optimal values of the decision variables for each sample path. 113 Table 5.1: The Optimal Price Process in Example 5.1 W 0 W 1 W 2 0 1 2 000 0.8058 0.7474 0.6553 001 0.8058 0.7474 1 010 0.8058 1 0.7308 011 0.8058 1 1 100 1 0.7873 0.6824 101 1 0.7873 1 110 1 1 0.7311 111 1 1 1 We list the optimal price process in Table 5.1. In this example, the natural filtration generated byf 0 ; 1 ; 2 g is the same as that forfW 0 ;W 1 ;W 2 g. The result is intuitive. For instance, when- everW t = 1, we have t = 1 so thatz t = 0. This is true becauseu 0 t (0) = c 0 t (0; 1) = 1, which means that it cannot be optimal to buy any power (to be consumed either at the current time or in the future) when the cost is high, whatever the current storage level is. Clearly, we haveP ( 2 j 1 = 1; 0 = 0:8058)6= P ( 2 j 1 = 1; 0 = 1). This shows that the optimal price process is not a Markov process, which is a major challenge from the computational point of view. 5.3 Discussion We formulate a stochastic dynamic pricing framework in the context of retail electricity markets, as an extension of the deterministic dynamic NUM problems. We first consider the user’s problem and show that the optimal policy has a threshold form. We then proceed to the system problem, which involves multiple users served by a single distributor. We apply dual decomposition methods to show the existence of an optimal price process that incentives the agents to choose the socially optimal decisions. We develop a distributed algorithm and investigate the information structure of the involved stochastic processes via a numerical example. To illustrate the key idea, we have not taken into account additional constraints such as storage capacity limits and consumption quota. Those constraints can be easily incorporated into this model, and a similar argument follows. 114 The major challenge to implement stochastic dynamic pricing is the computational burden. In future work, we will develop solution methods for the embedded stochastic programs in the proposed algorithm. We will also address practical issues such as price volatility. 115 Chapter 6 Toward an Integrated Smart Grid After we obtain satisfactory results for each market, it remains a big problem how to integrate the two-level markets, which has not been addressed in the current literature. We study a network resource allocation problem in a hierarchical setting. Motivated by the results on hierarchical mechanism design, we discuss the challenges of the integration of the two-level markets. This is an open problem and the direction of our future work. 6.1 Hierarchical Mechanism Design As networks have become increasingly complex, so has the ownership structure. This means that traditional models and allocation mechanisms used for resource exchange between primary owners and end-users are no longer always relevant. Increasingly, there are middlemen, operators who buy network resources from primary owners and then sell them to end-users. Although middlemen play an important role in the distribution channel by matching supply and demand, they also potentially cause inefficiencies in network resource allocation. Consider the scenario of bandwidth allocation. Network bandwidth is primarily owned by Tier 1 ISPs (Internet Service Providers), who then sell it to various Tier 2 ISPs. Tier 2 ISPs then sell it further to Tier 3 ISPs, and so on. The presence of ISPs in the middle stages can potentially skew network resource allocation, and cause inefficiencies from a social welfare point of view. Similarly, in the case of wireless spectrum, primary users that acquire spectrum from the FCC and lease some of it to secondary users also play the role of middlemen in secondary spectrum markets. As another example, consider cloud computing services by providers such as IBM, Google, Amazon and others for enterprise end-users (e.g., enterprises having small computational or data center needs). This raises the key question regarding what incentive compatible or efficient hierarchical 116 mechanisms can be used in the presence of middlemen, and whether these two can be achieved together at all. Auctions as mechanisms for network resource allocation have received considerable attention recently. Following up on the network utility model proposed by Kelly [89], Johari and Tsitsiklis showed that the Kelly mechanism (with per-link bids) can exhibit up to 25% efficiency loss [103]. This led to a flurry of activity in designing efficient network resource allocation mechanisms, including the work of Maheshwaran and Basar [104], Johari and Tsitsiklis [105], Yang and Hajek [106], Jain and Walrand [107], Jia and Caines [108] among others [109, 110]. Most of the work focused on single-sided auctions for divisible resources, and is related to the approach of Lazar and Semret [111]. Double-sided network auctions for divisible resources were developed in [107]. One of the very few to focus on indivisible network resources is Jain and Varaiya [112] which proposed a Nash implementation combinatorial double auction. This is also the only work known so far that presents an incomplete information analysis of combinatorial market mechanisms [113]. All these mechanisms either involve network resource allocation by an auctioneer among mul- tiple buyers, or resource exchange among multiple buyers and sellers. Most of the proposed mech- anisms are Nash implementations, i.e., in which truth-telling is a Nash equilibrium but not neces- sarily a “dominant strategy” equilibrium. In reality, however, markets for network resources often involve middlemen. Often, they enable markets that do not exist due to information asymmetries, but that can also potentially cause inefficiencies. However, models with middlemen have not been studied much, primarily due to the difficulty of designing appropriate mechanisms. Even in eco- nomic and game theory literature, the closest related model is one that involves a resale among the same set of players after an auction, in which the winners can resell the acquired resources to the losers [114]. There is indeed some game-theoretic work on network pricing in a general topology. [115] studied a network formation game where the nodes wish to form a graph to route traffic among themselves. [116] examined how transit and customer prices and quality of service are set in a 3-tier network. However, such work focused on the pricing equilibrium, and problems like mechanism design were not studied. 117 In this work, we consider a multi-tier setting. A Tier 1 provider owns a homogeneous network resource and holds an auction to allocate this resource among Tier 2 operators, who in turn allocate the acquired resource among Tier 3 entities, and so on. Each end-user has a valuation for the resource as a function of acquired capacity, while the middlemen do not have any intrinsic valuation of the resource but a quasi-valuation which depends on the revenue gained from resale. Our goal is to design hierarchical auction mechanisms with desirable properties. We first consider hierarchical mechanism design for indivisible goods. We study a class of mechanisms wherein each sub-mechanism is either a first-price or VCG auction, and show that incentive compatibility and efficiency cannot be achieved simultaneously by such hierarchi- cal mechanisms. This seems to foretell a more general impossibility of achieving both incen- tive compatibility and efficiency in a hierarchical setting. We then study some representative sequential hierarchical mechanisms with both complete and incomplete information settings, and again observe the difficulty of achieving incentive compatibility and efficiency simultaneously. When the network resource is divisible, we propose two VCG-type mechanisms that employ two- dimensional bids, one with single-sided sub-mechanisms at all tiers, and one with double-sided sub-mechanisms at all tiers except Tier 1. We show that both mechanisms induce an efficient Nash equilibrium. 6.1.1 Problem Statement Consider a Tier 1 provider (e.g., the FCC or Google) who ownsC units of a homogeneous network resource. Such a good can be divisible (sold in arbitrary portions of the total amount) or indivisible (sold in integral units). Assume that there are K tiers in the hierarchical network. The Tier 1 provider auctions off the resource among the Tier 2 entities, referred to as the Tier 1 auction. Each Tier 2 entity then auctions off the good acquired in the Tier 1 auction to the Tier 3 entities, referred to as the Tier 2 auction, and in general at Tierk as the Tierk auction (for 1k<K). An example of a 3-tier network is shown in Fig. 6.1. We note that players other than the Tier 1 provider may own some of the resource. However, this does not affect their strategic considerations, and hence is ignored. 118 3 x 3 4 x 4 5 x 5 6 x 6 1 x 1 2 x 2 0 C Tier 1: Social Planner Tier 2: Middlemen Tier 3: End-Users 1 2 2 Figure 6.1: An example of a 3-tier network withN = 6. We consider the Tier 1 provider as the social planner (indexed by 0), who attempts to maximize the “social welfare”, which we will shortly define. This assumption is valid when the auctioneer is a governmental agency such as the FCC, and might still be reasonable even when the provider is a profit maximizer (since the two goals are not necessarily incompatible). The entities at other tiers are strategic players (indexed byi = 1;:::;N), among which the Tier k (for 1<k <K) entities are regarded as the middlemen, and the TierK entities as the end-users. The hierarchical model we consider is highly stylized, and each player can acquire the resource only from its parent in the upper tier; issues like routing and peering are not taken into account. The stylized model yields concrete results that help us in gaining an insight into the problem. In fact, even in this rather simplified model, we obtain some negative results, which suggest the difficulty of hierarchical mechanism design in more general settings. LetT (i) be the tier to which playeri belongs, andch(i) be the set of playeri’s children. Denote the capacity acquired by playeri byx i . Assume that each playeri has a quasilinear utility function u i = v i (x i )w i , wherev i () is his valuation function andw i is his payment. When playeri is an end-user,v i () is intrinsic; when playeri is a middleman,v i (x i ) = i c i (x i ), where i is his revenue gained from resale andc i () is his cost function, since middlemen do not derive utilities from the resource but may incur transaction costs. Denotex = (x 1 ;:::;x N ). We define the social welfareS() as S(x) = X i:T (i)=K v i (x i ) X i:1<T (i)<K c i (x i ); 119 the difference between the aggregate valuation derived by the end-users and the aggregate cost incurred by the middlemen. This is the aggregate social surplus generated by an allocationx. The social planner’s objective is to achieve an efficient allocation x = (x 1 ;:::;x N ) that solves the social welfare optimization (SWO) problem: maximize S(x) subject to X j2ch(0) x j C; X j2ch(i) x j x i ;8i : 1<T (i)<K; x i 0;8i: (6.1) The first two constraints state that the total allocation among the buyers in each auction cannot exceed the allocation acquired by their parent. The third constraint is to ensure non-negative allo- cations. In addition, if the resource is indivisible,x i ’s should be integers. Since the players are strategic and may misreport their private information, our goal is to design a mechanism that induces an efficient allocation maximizing the social welfare. Note that in a hierarchical setting, the social planner specifies the mechanisms to be used at all tiers which must then be used. This is quite reasonable when the social planner is, say the government, and has the power to regulate the market. Denote the mechanism by = ( 1 ;:::; K1 ), in which a common sub-mechanism k (for 1 k < K) is employed in the Tierk auctions. An auction (sub-mechanism) is a single-sided auction if buyers place bids and sellers do nothing; it is a double-sided auction if buyers place buy-bids and sellers place sell-bids. In our model, there is always only one seller in each auction. Before going further, we provide a brief discussion of the nature of the model and the difficul- ties of the problem. 1. The players at different tiers may bid simultaneously or sequentially, while the resources are always allocated from Tier 1 to TierK. This suggests that one might expect similar results between the two cases under certain conditions. 120 2. One key difficulty is that middlemen have no intrinsic valuations of the resource. We will introduce the notion of quasi-valuation functions as their “types”, which are related to the revenues gained from resale. As a result, middlemen cannot have dominant strategies (see [42] for the definition), and we will introduce a weaker notion of a dominant strategy. 3. The hierarchical mechanism is decentralized, with multiple auctions at each tier, and the social planner holds only one of theNM + 1 auctions. This makes the achievement of efficiency even more difficult. 4. For divisible resources, it is impossible for a player to report an arbitrary real-valued valu- ation function completely. Thus, we have to restrict the bid spaces to be finite-dimensional and focus on Nash implementation. 6.1.2 Hierarchical Auctions for Indivisible Resources When the resources are indivisible, we study a class of mechanisms wherein the common sub- mechanism at each tier is either a first-price auction or a VCG auction. In a first-price auction (denoted byF), the buyer with the highest bid wins the single unit good, and pays the amount of his bid to the seller. In a second-price auction, the highest bidder wins but pays only the second-highest bid. A second-price auction gives buyers an incentive to bid their true value while a first-price auction does not. A generalization of the second-price auction to multiple goods that maintains the incentive to bid truthfully is known as the Vickrey-Clarke-Groves (VCG) auction (denoted byV). The idea is that items are assigned to maximize the social welfare; then each player pays the “externality” imposed on the other players by his participation (see [42] for more details). Without loss of generality, we assume that middlemen have no transaction cost, since it can be directly incorporated into the valuation. 1 Before proceeding, we need to redefine some notions for the hierarchical setting. 1 We will consider cost functions for divisible resources though this elaboration is still not crucial. 121 Definition 6.1. A middlemani’s quasi-valuation function v i :Z + !R + specifies his revenue from resale for each possible allocation he may acquire, when all his children report their valuation functions (for end-users) or quasi-valuation functions (for middlemen) truthfully. The quasi-valuation v i (x i ) specifies the maximum that a middleman is willing to pay for an allocationx i . Note that this would depend on how much the middleman’s children are willing to pay for such an allocation. Given an allocation and a payment rule, the quasi-valuation function is well defined. Also, note the backward-recursiveness in the definition. We now define a dominant strategy as well as incentive compatibility in this new environment, both of which are weaker than the definitions in the standard setting. Definition 6.2. Given that all the playersch(i) report their valuation or quasi-valuation functions truthfully, a strategy is a hierarchical dominant strategy for player i if it maximizes his payoff regardless of what the others play. A dominant strategy yields the best payoff for a player regardless of what the others play. A player will always play such a strategy if it exists. Note that for an end-user, a strategy is a hierarchical dominant strategy if and only if it is a dominant strategy. Definition 6.3. A hierarchical mechanism is incentive compatible if it induces a hierarchical dom- inant strategy equilibrium wherein all the players report their valuation or quasi-valuation func- tions truthfully. Such equilibrium strategies will be referred to as truth-telling as a counterpart of the usual notion of truth-telling in the standard setting [42]. We remind the reader that a dominant strategy equilibrium is regarded as a strong solution concept of a game because it is independent of the information (or lack thereof) that a player may have about others. Thus, incentive compatible mechanisms are regarded as very desirable. We mainly focus on static auctions in a complete information environment. Later, we will also consider sequential auctions and the incomplete information setting, and through a case study show how equilibria for these settings can be derived. 122 Static Auctions with Complete Information We first study the hierarchical extension of the first-price auction, which we call the hierarchical first-price mechanism, in which 1 2 fF;Vg, 2 = = K1 = F, i.e., the Tier 1 sub- mechanism is a first-price or VCG auction, while all the others are first-price auctions. We show the existence of an efficient-Nash equilibrium in this mechanism. Proposition 6.1. Assume the social planner and each middleman have at least two children, i.e., the outdegree of each non-terminal node is at least two. Suppose a single indivisible good is to be allocated. 2 In the hierarchical first-price mechanism, there exists an efficient-Nash equilibrium. Proof. We construct such a strategy profile as follows. Suppose in the efficient allocation, the single good is transferred in this way: i 1 = 0! i 2 ! i 3 !! i K , whereT (i k ) = k for all 1<kK. Consider the strategy profile: b i = 8 < : v i K ; i =i 2 ;:::;i K ; v i K ; otherwise: Note thatv i K is the valuation (a scalar in this case) of the winning end-user. Clearly, this profile induces the efficient allocation that solves the SWO problem (6.1). It is also easy to check that it is an-Nash equilibrium: no one can gain more than by unilaterally deviating from his strategy. This proves the claim. Since the standard first-price auction is not incentive compatible, the hierarchical extension cannot be either. Moreover, as long as a first-price auction exists as a sub-mechanism, the entire hierarchical mechanism cannot be incentive compatible, since truth-telling is a weakly dominated strategy (always achieving a zero utility). We state this as a proposition. Proposition 6.2. The hierarchical first-price mechanism is not incentive compatible. More gener- ally, for any hierarchical mechanism , if there exists somek such that k =F, then cannot be incentive compatible. 2 For simplicity and to understand the essence of problems that arise in design, we focus on allocating a single unit. When multiple units present no additional complications, we consider them directly. 123 We now study the hierarchical extension of the VCG auction, which we call the hierarchical VCG mechanism, in which 1 = = K1 =V, i.e., each sub-mechanism is a VCG auction. We show the incentive compatibility of this mechanism. Proposition 6.3. Suppose multiple units of an indivisible good are to be allocated. The hierarchi- cal VCG mechanism is incentive compatible. Proof. According to Definition 6.3, we need to show that truth-telling is a hierarchical dominant strategy equilibrium. Consider the TierK 1 auction. Since this is a VCG auction, truth-telling is a dominant (and hence hierarchical dominant) strategy for each playeri withT (i) = K. For the purpose of backward induction, 3 assume each playeri withT (i) =k reports truthfully. Then the quasi-valuation function of each playeri withT (i) =k 1 can be equivalently viewed as an “intrinsic” valuation function by Definition 6.1. It follows that truth-telling is again a hierarchical dominant strategy for each playeri withT (i) =k 1 by Definition 6.2. Thus, all the players will report truthfully. The fact that the hierarchical first-price mechanism is not incentive compatible, but the hierar- chical VCG mechanism is, is not surprising since the non-hierarchical first-price and VCG mech- anisms respectively have these properties. However, unlike the non-hierarchical VCG mechanism, efficiency may not be achieved at the hierarchical dominant strategy equilibrium. We prove this surprising observation in the following proposition by providing a counterexample. Proposition 6.4. The hierarchical dominant strategy equilibrium in the hierarchical VCG mecha- nism may not be efficient. Proof. We provide a non-trivial counterexample. Consider the 3-tier network in Fig. 6.1 with C = 5. With the notation v = (v(1);v(2);v(3);v(4);v(5)), let the valuation functions of the end-users be v 3 = (10; 18; 24; 28; 30), v 4 = (20; 25; 29; 32; 34), v 5 = (15; 24; 32; 39; 45), v 6 = (16; 20; 24; 27; 29). Given 2 =V, the quasi-valuation functions of the middlemen are computed as v 1 = (10; 13; 15; 16; 15), v 2 = (15; 13; 16; 18; 19). Truth-telling is a hierarchical dominant 3 Here “backward” refers to the network topology, whereas the game itself is still static. 124 strategy equilibrium with the allocation (x 1 ;x 2 ;x 3 ;x 4 ;x 5 ;x 6 ) = (4; 1; 3; 1; 0; 1). However, the efficient allocation derived by (6.1) is (x 1 ;x 2 ;x 3 ;x 4 ;x 5 ;x 6 ) = (2; 3; 1; 1; 2; 1). Thus, the hierarchical dominant strategy equilibrium is not efficient. As shown above, quasi-valuation functions are not monotone in general, which suggests that it is very “unlikely” to be efficient for a hierarchical dominant strategy equilibrium in the hierarchical VCG mechanism. Moreover, though one may derive conditions on the valuation functions of the end-users under which efficiency can be achieved for simple cases (e.g., 3-tier network with a single unit), it is hard to obtain such conditions for generalK-tier networks with multiple units. A “limited” impossibility result follows immediately when we restrict our attention to first- price and VCG auctions as sub-mechanisms. Theorem 6.1 (Hierarchical Impossibility). Suppose we allocate a single indivisible good in aK- tier network (K 3). There does not exist an incentive compatible hierarchical mechanism with k 2fF;Vg (for 1 k < K) which induces an efficient hierarchical dominant strategy equilibrium. Proof. By Proposition 6.2, incentive compatibility cannot be achieved if there exists somek such that k =F. On the other hand, if k =V for allk, efficiency is not guaranteed in the hierarchical dominant strategy equilibrium by Proposition 6.4. This proves the claim. Our conjecture is that this “limited” impossibility theorem foretells a more general impossibil- ity result in hierarchical mechanism design with arbitrary sub-mechanisms at each tier. Sequential Auctions with Complete Information In this section, we consider a setting where the resource is allocated hierarchically via a sequential auction, i.e., auctions at various tiers do not take place simultaneously but sequentially. Design of such sequential auctions requires the theory of dynamic mechanism design, which is not well developed. Thus, we study some specific dynamic mechanisms to understand sequential hierar- chical mechanism design and for simplicity, focus on a 3-tier network. We define two types of sequential auctions. 125 Top-down auction (TD): In the first stage, Tier 2 players bid simultaneously. In the second stage, after observing all the previous bids, Tier 3 players bid simultaneously. Then the allocation is realized. Bottom-up auction (BU): In the first stage, Tier 3 players bid simultaneously which are observed by all players. In the second stage, Tier 2 players bid simultaneously, and then the allocation is realized. Recall that a player’s strategy in a game is a complete contingent plan that specifies how the player will act in every contingency in which he might be called upon to move. In TD, each middle- man’s strategy space is the same as that in the static game (which is identical to the action space), while each end-user’s strategy must specify one action for each possible set of the middlemen’s bids (which are observed when the end-users bid). In BU, however, each end-user’s strategy space is the same as that in the static game, while each middleman’s strategy must specify one action for each possible set of the end-users’ bids. We can deduce the equilibria in the sequential auctions by a proper “modification” of the equi- libria in the static auctions. Definition 6.4. A strategy profiles = (s 1 ;:::;s N ) in the sequential auction is an adaptation of the strategy (action) profilea = (a 1 ;:::;a N ) in the static auction ifs i ()a i for alli. That is, in an adaptation, the strategy of each player is independent of the contingency. We have the following proposition, the proof of which is trivial and thus omitted. Proposition 6.5. Assume the static hierarchical auction and the (TD or BU) sequential auction employ the same sub-mechanisms. For any Nash equilibrium in the static auction, its adaptation is also a Nash equilibrium in the sequential auction and induces the same allocation. Example 6.1. Consider VCG mechanisms as sub-mechanisms in the 3-tier network withv i ( v i ) as the valuations (quasi-valuations) of the Tier 3 (Tier 2) players. A Nash equilibrium in the static hierarchical auction is b i = 8 < : v i ; T (i) = 2; v i ; T (i) = 3: 126 The adaptations in TD and BU are b TD i = 8 < : v i ; T (i) = 2; v i ;8fb j g T (j)=2 ; T (i) = 3; b BU i = 8 < : v i ;8fb j g T (j)=3 ; T (i) = 2; v i ; T (i) = 3; which are respectively the Nash equilibria in the two sequential auctions. For dynamic games, however, subgame perfect Nash equilibria are more relevant. Recall that a strategy profile is a subgame perfect Nash equilibrium if it induces a Nash equilibrium in every subgame of the original game. We show that an adaptation (which is always a Nash equilibrium) may not be a subgame perfect Nash equilibrium. Example 6.2. Consider the same setting as in Example 6.1. It is easy to check that the adaptation in TD is also a subgame perfect equilibrium. However, the adaptation in BU is not. By a slight abuse of notation, for each playeri who is a middleman, let v i (fb j g T (j)=3 ) be his revenue function (that depends on the end-users’ bids). Then, a subgame perfect equilibrium in BU is b i = 8 < : v i (fb j g T (j)=3 ); T (i) = 2; v i ; T (i) = 3: On the equilibrium path, we have v i (fb j g T (j)=3 ) = v i (fv j g T (j)=3 ) = v i , the quasi-valuation func- tion. Sequential First-Price Auction with Incomplete Information We now consider hierarchical auctions with incomplete information. Specifically, we investigate a natural extension of first-price auctions, which we call the sequential first-price auction with incomplete information. Note that variants of VCG auctions for the incomplete information setting would be trivial due to the incentive compatibility. 127 Consider a 3-tier network with a single indivisible good to be allocated. There are two mid- dlemen, player 1 and player 2, withN 1 andN 2 children respectively. The end-users’ valuations are drawn from a commonly known prior distribution, and in particular, are i.i.d. random variables uniformly distributed on the interval [0; 1]. The stages of the game are as follows: 1. Nature draws a valuationv i U[0; 1] for each end-useri independently and reveals it to that player. 2. The end-users bid simultaneously, withb i = i (v i ) for playeri. 3. Each middleman learns the bids of his own children, but not those of the others. 4. The middlemen bid simultaneously, withb 1 = 1 (v 1 ) andb 2 = 2 (v 2 ) respectively. 5. The allocation and the payments are determined according to the mechanism 1 = 2 =F. We look for a perfect Bayesian equilibrium of this game. As we will see, the problem can be converted into an asymmetric first-price auction in the standard setting. Leti2f1; 2g. It can be shown that the equilibrium strategy of playeri’s children is(v) = N i 1 N i v. The prior distribution of playeri’s revenue isF i (v) = N i v N i 1 (N i 1)v N i , with the associated probability density functionf i (v) =N i (N i 1)(1v)v N i 2 . WheneverN 1 6=N 2 , 1 is a first-price auction with asymmetric bidders, i.e.,F 1 ()6=F 2 (). In equilibrium, we have 1 (0) = 2 (0) = 0; 1 (1) = 2 (1): (6.2) Let i := 1 i . We obtain the first-order condition for playeri: 0 i (b) = F i ( i (b)) f i ( i (b)) 1 j (b)b : (6.3) A solution to the system of differential equations (6.2)-(6.3) constitutes equilibrium strategies among the middlemen. While a closed-form expression is not available, we derive the properties of the equilibrium strategies indirectly. Assume thatN 1 >N 2 . It is easy to check that the distribution F 1 dominates F 2 in terms of the reverse hazard rate, i.e., f 1 (v) F 1 (v) > f 2 (v) F 2 (v) for all v 2 (0; 1). In 128 [117], it is proved that the “weak” player 2 bids more aggressively than the “strong” player 1, i.e., 1 (v)< 2 (v) for allv2 (0; 1). Clearly, this result leads to the inefficiency in the Tier 1 auction, and therefore to the inefficiency in the entire hierarchical mechanism. Again, this is a negative result for hierarchical mechanism design: efficiency is not guaranteed even when the end-users are symmetric and first-price auctions are held everywhere. 6.1.3 Hierarchical Auctions for Divisible Resources Often, network resources such as bandwidth and spectrum are available as (infinitely) divisible resources. We thus, now consider hierarchical mechanisms for a divisible resource. For simplicity of exposition, we focus on the 3-tier network as in Fig. 6.1, with some change of notation: let playeri be theith middleman and player (i;j) be the end-user who is thejth child of playeri. 4 The valuation functionv ij () of player (i;j) is assumed to be strictly increasing, strictly concave and continuously differentiable on [0;1), withv ij (0) = 0. The cost functionc i () of playeri is assumed to be strictly increasing, strictly convex and continuously differentiable on [0;1), with c i (0) = 0. The payoff of player (i;j) isu ij = v ij (x ij )w ij , wherew ij is the payment made by player (i;j). The payoff of playeri isu i = i w i c i (x i ), where i is playeri’s revenue andw i is the payment made by playeri. We define the endogenous budget balance condition: i = X j w ij ;8i: (6.4) 4 The results extend to the generalK-tier network albeit the notation is more complicated. 129 The social welfare optimization problem for the case of divisible resources (DIV-OPT) is maximize X (i;j) v ij (x ij ) X i c i (x i ) subject to X i x i C; [ 0 ] X j x ij x i ;8i; [ i ] x i ;x ij 0;8i;8(i;j); (6.5) where 0 and i ’s are the corresponding Lagrange multipliers (likewise for the following). The solution of the convex optimization problem is characterized by the Karush-Kuhn-Tucker (KKT) conditions: (c 0 i (x i ) + 0 i )x i = 0;8i; c 0 i (x i ) + 0 i 0;8i; (v 0 ij (x ij ) i )x ij = 0;8(i;j); v 0 ij (x ij ) i 0;8(i;j); 0 X i x i C ! = 0; X i x i C 0; i X j x ij x i ! = 0;8i; X j x ij x i 0;8i; 0 ; i ;x i ;x ij 0;8i;8(i;j): Our objective is to design a hierarchical mechanism that induces an efficient allocation as a solution of the DIV-OPT problem, despite the strategic behavior of the players. With divisible resources, however, it is impossible for a player to report an arbitrary real-valued valuation (or cost) 130 function completely. Thus, the mechanism must ask each player to communicate an approximation to the function from a finite-dimensional bid space, and dominant strategy implementation cannot be achieved here. Instead, we seek a static Nash implementation in a complete information setting. We propose two VCG-type mechanisms, one single-sided and one double-sided, both of which have two-dimensional bids that specify the unit price and the quantity. Such bid spaces are natural and used in many practical scenarios. Hierarchical Single-Sided VCG Mechanism We first propose the hierarchical single-sided VCG (HSVCG) mechanism. In the Tier 1 auction, player i reports a bid b i = ( i ;d i ), where i is the bid price and d i is the maximum quantity desired; in theith Tier 2 auction, player (i;j) reports a bidb ij = ( ij ;d ij ), where ij is the bid price andd ij is the maximum quantity desired. The allocation is then determined as follows. In the Tier 1 auction, the allocation ~ x = (~ x i ;8i) is a solution of the following optimization problem (HSVCG-1): maximize X i i x i subject to X i x i C; [ 0 ] x i d i ;8i; [ i ] x i 0;8i: (6.6) Let ~ x i = (~ x i l ;8l) denote the allocation as a solution of the above withd i = 0, i.e., when player i is not present. Then, the payment made by playeri is w i = X l6=i l (~ x i l ~ x l ): 131 In the ith Tier 2 auction (in which ~ x i has been determined), the allocation x i = ( x ij ;8j) is a solution of the following optimization problem (HSVCG-2): maximize X j ( ij i )x ij subject to X j x ij ~ x i ; [ i ] x ij d ij ;8j; [ ij ] x ij 0;8j: (6.7) Let x j i = ( x j ik ;8k) denote the allocation as a solution of the above withd ij = 0, i.e., when player (i;j) is not present. Then, the payment made by player (i;j) is w ij = i X k ( x ik x j ik ) + X k6=j ik ( x j ik x ik ); and playeri’s revenue equals exactly the total payment of his children, as in (6.4). The solution of (6.6) is characterized by the KKT conditions: ( i 0 i )x i = 0;8i; i 0 i 0;8i; 0 X i x i C ! = 0; X i x i C 0; i (x i d i ) = 0;8i; x i d i 0;8i; 0 ; i ;x i 0;8i; 132 and the solution of (6.7) is characterized by the KKT conditions (given a fixedi): ( ij i i ij )x ij = 0;8j; ij i i ij 0;8j; i X j x ij x i ! = 0; X j x ij x i 0; ij (x ij d ij ) = 0;8j; x ij d ij 0;8j; i ; ij ;x ij 0;8j: Theorem 6.2. The HSVCG mechanism induces an efficient Nash equilibrium. Proof. Letx = ((x i ;8i); (x ij ;8(i;j))) be an efficient allocation that solves (6.5). Then given a fixedi,v 0 ij (x ij )’s are equal andv 0 ij (x ij )c 0 i (x i ), for allj withx ij > 0. Consider the strategy profile: i = max j v 0 ij (x ij ),d i = x i , ij = v 0 ij (x ij ),d ij = x ij . It is easy to check thatx is also a solution of (6.6) and (6.7). It remains to show that the constructed strategy profile is a Nash equilibrium. Consider playeri, a middleman whose payoff isu i = i x i c i (x i ). Suppose he deviates by changing his bid to ( i ;d i ) with the resulting allocationx i . If it is possible for him to be better off, there must bex i x i (given his children’s demand fixed) and i i (otherwise he will get a zero revenue). Then, we haveu i i x i c i (x i ), and u i u i i (x i x i )c i (x i ) +c i (x i ) c 0 i (x i )(x i x i )c i (x i ) +c i (x i ) 0; 133 where the last inequality follows from convexity and monotonicity. Thus, he has no incentive to deviate. Consider player (i;j), an end-user whose payoff is u ij = v ij (x ij )v 0 ij (x ij )x ij . Suppose he deviates by changing his bid to ( ij ;d ij ) with the resulting allocationx ij . If it is possible for him to be better off, there must be ij ij (otherwise he will get a zero allocation). Then, u ij v ij (x ij )v 0 ij (x ij )x ij , and whetherx ij x ij orx ij <x ij , we always have u ij u ij =v ij (x ij )v ij (x ij ) +v 0 ij (x ij )(x ij x ij ) 0; where the inequality follows from concavity and monotonicity. Thus, he has no incentive to devi- ate. Hierarchical Double-Sided VCG Mechanism In the HSVCG mechanism, all the sub-mechanisms were single-sided auctions. We now pro- pose the hierarchical double-sided VCG (HDVCG) mechanism in which double-sided auctions are employed at all tiers except Tier 1. It seems that this mechanism provides more freedom for a middleman: besides a buy-bid, he can place an additional sell-bid. Nevertheless, it turns out that the two mechanisms are outcome-equivalent, in the sense that both mechanisms induce an efficient Nash equilibrium. We specify the HDVCG mechanism for the 3-tier network. The Tier 1 auction is single-sided: player i reports a bid b i = ( i ;d i ), where i is the bid price and d i is the maximum quantity desired. Each Tier 2 auction is double-sided: in the ith Tier 2 auction, player i reports a sell- bid a i = ( i ;q i ), where i is the sell-bid price and q i is the maximum quantity offered; player (i;j) reports a buy-bid b ij = ( ij ;d ij ), where ij is the buy-bid price and d ij is the maximum 134 quantity desired. The allocation is then determined as follows. In the Tier 1 auction, the allocation ~ x = (~ x i ;8i) is a solution of the following optimization problem (HDVCG-1): 5 maximize X i i x i subject to X i x i C; [ 0 ] x i d i ;8i; [ i ] x i 0;8i: (6.8) Let ~ x i = (~ x i l ;8l) denote the allocation as a solution of the above withd i = 0, i.e., when player i is not present. Then, the payment made by playeri is w i = X l6=i l (~ x i l ~ x l ): In the ith Tier 2 auction (in which ~ x i has been determined), the allocation x i = ( x ij ;8j) is a solution of the following optimization problem (HDVCG-2): maximize X j ( ij i )x ij subject to X j x ij minf~ x i ;q i g; [ i ] x ij d ij ;8j; [ ij ] x ij 0;8j: (6.9) Let x j i = ( x j ik ;8k) denote the allocation as a solution of the above withd ij = 0, i.e., when player (i;j) is not present. Then, the payment made by player (i;j) is w ij = i X k ( x ik x j ik ) + X k6=j ik ( x j ik x ik ); 5 Note that HDVCG-1 is identical to HSVCG-1. 135 and playeri’s revenue is i = X j ij x ij : Note that unlike HSVCG, the endogenous budget balance is not necessarily achieved in HDVCG; however, as we will see, it is ensured at the efficient Nash equilibrium we construct. The solution of (6.8) is characterized by the KKT conditions: ( i 0 i )x i = 0;8i; i 0 i 0;8i; 0 X i x i C ! = 0; X i x i C 0; i (x i d i ) = 0;8i; x i d i 0;8i; 0 ; i ;x i 0;8i; and the solution of (6.9) is characterized by the KKT conditions (given a fixedi): ( ij i i ij )x ij = 0;8j; ij i i ij 0;8j; i X j x ij minfx i ;q i g ! = 0; X j x ij minfx i ;q i g 0; ij (x ij d ij ) = 0;8j; x ij d ij 0;8j; i ; ij ;x ij 0;8j: 136 Theorem 6.3. The HDVCG mechanism induces an efficient Nash equilibrium, at which the endoge- nous budget balance is achieved. Proof. Letx = ((x i ;8i); (x ij ;8(i;j))) be an efficient allocation that solves (6.5). Consider the strategy profile: i = i = max j v 0 ij (x ij ), d i = q i = x i , ij = v 0 ij (x ij ), d ij = x ij . It is easy to check that x is also a solution of (6.8) and (6.9). Moreover, i = P j v 0 ij (x ij )x ij = P j w ij for all i. It remains to show that the constructed strategy profile is a Nash equilibrium. Consider playeri, a middleman whose payoff isu i = i x i c i (x i ). Suppose he deviates by changing his bid to (( i ;d i ); ( i ;q i )) with the resulting allocationx i . If it is possible for him to be better off, there must bex i x i (given his children’s demand fixed) and i i (otherwise he will get a zero revenue). Then, we haveu i i x i c i (x i ), and u i u i i (x i x i )c i (x i ) +c i (x i ) c 0 i (x i )(x i x i )c i (x i ) +c i (x i ) 0: Thus, he has no incentive to deviate. Consider player (i;j), an end-user whose payoff is u ij = v ij (x ij )v 0 ij (x ij )x ij . Suppose he deviates by changing his bid to ( ij ;d ij ) with the resulting allocationx ij . If it is possible for him to be better off, there must be ij ij (otherwise he will get a zero allocation). Then, u ij =v ij (x ij )v 0 ij (x ij )x ij , and whetherx ij x ij orx ij <x ij , we always have u ij u ij =v ij (x ij )v ij (x ij ) +v 0 ij (x ij )(x ij x ij ) 0: Thus, he has no incentive to deviate. 137 6.1.4 Discussions In this work, we introduced a hierarchical network resource allocation model. We developed a general hierarchical mechanism design framework for such models. Such a model and framework is novel and this is the first work on multi-tier auctions to our best knowledge. When the resource is indivisible, we investigated a class of mechanisms wherein each sub- mechanism is either a first-price or VCG auction. We showed that the hierarchical mechanism with a first-price or VCG auction at Tier 1, and first-price auctions at all other tiers is efficient but not incentive compatible and surprisingly, the hierarchical VCG auction mechanism is incentive compatible but not necessarily efficient. This seems to foretell a more general impossibility of achieving both incentive compatibility and efficiency in a hierarchical setting. We also studied some representative mechanisms for sequential auctions as well as the incomplete information setting, in which similar results can be obtained. When the resource is divisible, we propose two VCG-type mechanisms. The HSVCG mech- anism is composed of single-sided auctions at each tier, while the HDVCG mechanism employs double-sided auctions at all tiers except Tier 1. Both mechanisms induce an efficient Nash equilib- rium. Moreover, the HSVCG mechanism always achieves endogenous budget balance, while that is ensured only at an efficient equilibrium in the HDVCG mechanism. We note that the mechanisms we have designed can easily be extended to the setting where there are end-users even at intermediate tiers. The key results will remain unchanged for such a setting. Another natural question is whether more general classes of incentive compatible or efficient mechanisms can be designed than those wherein the sub-mechanisms are either first-price or VCG auctions. Indeed, this is an important question. But as we show for indivisible resources, consid- ering just these two, leads to hierarchical mechanisms that are either efficient or incentive compat- ible, but not both. We expect an impossibility result which claims the nonexistence of hierarchical mechanisms that are both incentive compatible and efficient. Proving such a conjecture requires new developments, which we shall consider in future work. 138 In future work, we will also consider more general network topologies wherein there may be more than one resource (e.g., bandwidth on multiple links, or bandwidth, storage and computation), and allow for sub-mechanism auctions with multiple sellers. Moreover, we may allow each Tierk player to participate in any of the Tierk 1 auctions as well. 6.2 Challenges of the Integration of the Two-Level Markets First, the hierarchical market structure requires the incentive consistency of the middlemen, i.e., the aggregators and the distributors who participate in both the primary market and the secondary mar- kets. For example, if the aggregator is assumed to be revenue-maximizing in the primary market, it would not make sense to assume that its objective in the secondary market I is to maximize the efficiency of renewable energy aggregation. The results on hierarchical mechanism design convey that the presence of middlemen can potentially skew the optimal allocation, and cause inefficiency from a social welfare point of view. Moreover, since aggregators are mostly financially but not physically linked to the generators, it is possible that there may be competition among the aggre- gators. Even for the demand side, nowadays consumers may have the option to choose between two or more distributors. Future work will include such competition in the modeling. Second, the primary and the secondary markets are evolving simultaneously and dynamically. The three markets are operated on different timescales. For example, the operating interval of the primary market is on the order of 5 minutes, while that of the secondary market II may be on the order of 1 hour. More importantly, the three markets interact with each other. When making decisions in the primary market, an aggregator or a distributor should also take into account the information of the secondary market, and vice versa. In the literature, when people study problems in the secondary market, it is often assumed that the spot market price in the primary market is exogenous. We note that this is just an approximation; in fact, aggregators and distributors may have enough market power to influence the prices. This complicates the incentives of the aggregators and the distributors, and hence the analysis of that. 139 Third, the timescale of the proposed two-level model is typically one day, which means the unit commitment problem has not been addressed at the moment. The unit commitment problem schedules the on-and-off times of the generating units day or week ahead, which involves binary decisions. In the future, we will formulate a mega-model which integrates inter-day and intra- day dispatch. We will investigate whether it can be decomposed into a day-ahead model and an intra-day model as the current one. Our ultimate goal is to develop a theory of smart grid market design. The ideal framework will not only facilitate the analysis and design of each submarket, but also promote the integration of the entire market. We believe that such work will also shed light on the market design of general networks. 140 References [1] D. S. Kirschen and G. Strbac, Fundamentals of power system economics. John Wiley & Sons, 2004. [2] M. Shahidehpour, H. Yamin, and Z. Li, Market operations in electric power systems. Insti- tute of Electrical and Electronics Engineers, Wiley-Interscience, 2002. [3] E. Y . Bitar, R. Rajagopal, P. P. Khargonekar, K. Poolla, and P. Varaiya, “Bringing wind energy to market,” Power Systems, IEEE Transactions on, vol. 27, no. 3, pp. 1225–1235, 2012. [4] S. Stoft, Power system economics: designing markets for electricity. Wiley-IEEE Press, 2002. [5] P. P. Varaiya, F. F. Wu, and J. W. Bialek, “Smart operation of smart grid: risk-limiting dispatch,” Proceedings of the IEEE, vol. 99, no. 1, pp. 40–57, 2011. [6] N. Li, L. Chen, and S. Low, “Optimal demand response based on utility maximization in power networks,” in Power and Energy Society General Meeting, 2011 IEEE, 2011, pp. 1–8. [7] B. Roberts, “Capturing grid power,” Power and Energy Magazine, IEEE, vol. 7, no. 4, pp. 32–41, 2009. [8] C. W. Gellings, The smart grid: enabling energy efficiency and demand response. The Fairmont Press, Inc., 2009. [9] W. Tang and R. Jain, “Game-theoretic analysis of the nodal pricing mechanism for elec- tricity markets,” in Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, 2013, pp. 562–567. [10] ——, “Pricing mechanisms for economic dispatch: a game-theoretic perspective,” under review, International Journal of Electrical Power & Energy Systems, 2015. [11] ——, “Dynamic economic dispatch among strategic generators with storage systems,” in Communication, Control, and Computing (Allerton), 2014 52nd Annual Allerton Confer- ence on, 2014, pp. 943–950. [12] ——, “Dynamic economic dispatch game: the value of energy storage,” under 2nd round revision, IEEE Transactions on Smart Grid, 2014. 141 [13] W. Tang, J. Qin, R. Jain, and R. Rajagopal, “Pricing mechanisms for multistage energy procurement,” submitted, Proceedings of the 6th IEEE International Conference on Smart Grid Communications, 2015. [14] W. Tang and R. Jain, “Stochastic resource auctions for renewable energy integration,” in Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on, 2011, pp. 345–352. [15] ——, “Market mechanisms for buying random wind,” to appear, IEEE Transactions on Sustainable Energy, 2015. [16] ——, “Buying random yet correlated wind power,” in Smart Grid Communications (Smart- GridComm), 2014 IEEE International Conference on, 2014, pp. 145–150. [17] W. Tang, R. Jain, and R. Rajagopal, “Stochastic dynamic pricing: utilizing demand response in an adaptive manner,” in Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on, 2014, pp. 6446–6451. [18] W. Tang and R. Jain, “Hierarchical auctions for network resource allocation,” in Game The- ory for Networks, 2nd International ICST Conference on, 2011. [19] ——, “Hierarchical auction mechanisms for network resource allocation,” Selected Areas in Communications, IEEE Journal on, vol. 30, no. 11, pp. 2117–2125, 2012. [20] McCullough Research. (2002) Congestion manipulation in ISO California. [Online]. Available: http://www.mresearch.com/pdfs/19.pdf [21] S. P. Karthikeyan, I. J. Raglend, and D. Kothari, “A review on market power in deregulated electricity market,” International Journal of Electrical Power & Energy Systems, vol. 48, pp. 139–147, 2013. [22] R. D. Green, “Increasing competition in the British electricity spot market,” The Journal of Industrial Economics, vol. 44, no. 2, pp. 205–216, 1996. [23] R. Baldick, R. Grant, and E. Kahn, “Theory and application of linear supply function equi- librium in electricity markets,” Journal of Regulatory Economics, vol. 25, no. 2, p. 143, 2004. [24] J. Weber and T. Overbye, “A two-level optimization problem for analysis of market bidding strategies,” in Power Engineering Society Summer Meeting, 1999. IEEE, vol. 2. IEEE, 1999, pp. 682–687. [25] B. Hobbs, C. Metzler, and J.-S. Pang, “Strategic gaming analysis for electric power systems: an MPEC approach,” IEEE Transactions on Power Systems, vol. 15, no. 2, pp. 638–645, 2000. [26] X. Hu and D. Ralph, “Using EPECs to model bilevel games in restructured electricity mar- kets with locational prices,” Operations research, vol. 55, no. 5, pp. 809–827, 2007. 142 [27] E. Hasan, F. D. Galiana, and A. J. Conejo, “Electricity markets cleared by merit order-part II: finding the market outcomes supported by pure strategy Nash equilibria,” IEEE Transactions on Power Systems, vol. 23, no. 2, pp. 361–371, 2008. [28] W. W. Hogan, “A market power model with strategic interaction in electricity networks,” The Energy Journal, vol. 18, no. 4, p. 107, 1997. [29] S. Borenstein, J. Bushnell, and S. Stoft, “The competitive effects of transmission capacity in a deregulated electricity industry,” The RAND journal of economics, vol. 31, no. 2, pp. 294–325, 2000. [30] A. Ehrenmann and K. Neuhoff, “A comparison of electricity market designs in networks,” Operations Research, vol. 57, no. 2, pp. 274–286, 2009. [31] H. Haghighat, H. Seifi, and A. R. Kian, “Pay-as-bid versus marginal pricing: the role of suppliers strategic behavior,” International Journal of Electrical Power & Energy Systems, vol. 42, no. 1, pp. 350–358, 2012. [32] B. Stott, J. Jardim, and O. Alsac, “DC power flow revisited,” IEEE Transactions on Power Systems, vol. 24, no. 3, p. 1290, 2009. [33] M. Hsu, “An introduction to the pricing of electric power transmission,” Utilities Policy, vol. 6, no. 3, pp. 257–270, 1997. [34] S. P. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, 2004. [35] California ISO. Data definitions for ISO public bid data. [Online]. Available: http://content.caiso.com/marketops/OASIS/pubbid/data/pbddocumentation.pdf [36] M. B. Cain and F. L. Alvarado, “Implications of cost and bid format on electricity market studies: linear versus quadratic costs,” in 2004 Large Engineering Systems Conference on Power Engineering. IEEE, 2004, pp. 2–6. [37] P. Wang, Y . Xiao, and Y . Ding, “Nodal market power assessment in electricity markets,” Power Systems, IEEE Transactions on, vol. 19, no. 3, pp. 1373–1379, 2004. [38] R. Baldick, “Electricity market equilibrium models: the effect of parametrization,” IEEE Transactions on Power Systems, vol. 17, no. 4, pp. 1170–1176, 2002. [39] S. Bose, D. W. Cai, S. Low, and A. Wierman, “The role of a market maker in networked Cournot competition,” in Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on, 2014, pp. 4479–4484. [40] D. P. Bertsekas, A. Nedi´ c, and A. E. Ozdaglar, Convex analysis and optimization. Athena Scientific, 2003. [41] D. Fudenberg and J. Tirole, Game theory. MIT Press, 1991. [42] A. Mas-Colell, M. Whinston, and J. Green, Microeconomic theory. Oxford, 1995. 143 [43] X. Han, H. Gooi, and D. S. Kirschen, “Dynamic economic dispatch: feasible and optimal solutions,” Power Systems, IEEE Transactions on, vol. 16, no. 1, pp. 22–28, 2001. [44] P. Attaviriyanupap, H. Kita, E. Tanaka, and J. Hasegawa, “A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function,” Power Systems, IEEE Transactions on, vol. 17, no. 2, pp. 411–416, 2002. [45] X. Xia and A. Elaiw, “Optimal dynamic economic dispatch of generation: A review,” Elec- tric Power Systems Research, vol. 80, no. 8, pp. 975 – 986, 2010. [46] Z. Li, W. Wu, B. Zhang, B. Wang, and H. Sun, “Dynamic economic dispatch with spin- ning reserve constraints considering wind power integration,” in Power and Energy Society General Meeting (PES), 2013 IEEE. IEEE, 2013, pp. 1–5. [47] D. C. Walters and G. B. Sheble, “Genetic algorithm solution of economic dispatch with valve point loading,” Power Systems, IEEE Transactions on, vol. 8, no. 3, pp. 1325–1332, 1993. [48] S. Hemamalini and S. P. Simon, “Dynamic economic dispatch using artificial immune sys- tem for units with valve-point effect,” International Journal of Electrical Power & Energy Systems, vol. 33, no. 4, pp. 868–874, 2011. [49] K. Vaisakh, P. Praveena, S. R. M. Rao, and K. Meah, “Solving dynamic economic dispatch problem with security constraints using bacterial foraging PSO-DE algorithm,” Interna- tional Journal of Electrical Power & Energy Systems, vol. 39, no. 1, pp. 56–67, 2012. [50] A. Rabiee, B. Mohammadi-Ivatloo, and M. Moradi-Dalvand, “Fast dynamic economic power dispatch problems solution via optimality condition decompositiona optimality con- dition decomposition,” Power Systems, IEEE Transactions on, vol. 29, no. 2, pp. 982–983, 2014. [51] Z. Li, W. Wu, B. Zhang, and B. Wang, “Decentralized multi-area dynamic economic dis- patch using modified generalized benders decomposition,” Power Systems, IEEE Transac- tions on, vol. PP, no. 99, pp. 1–13, 2015. [52] D. Liu, J. Guo, W. Wang, C. Liu, Y . Huang, and M. O’Malley, “A dynamic economic dis- patch method considering with the uncertainty and correlation of wind power,” in Power and Energy Society General Meeting (PES), 2014 IEEE. IEEE, 2014, pp. 1–5. [53] A. Soroudi, A. Rabiee, and A. Keane, “Stochastic real-time scheduling of wind-thermal generation units in an electric utility,” Systems Journal, IEEE, vol. PP, no. 99, pp. 1–10, 2015. [54] Q. Wang, M. Liu, and R. Jain, “Dynamic pricing of power in smart-grid networks,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on, 2012, pp. 1099–1104. [55] J. Qin and R. Rajagopal, “Dynamic programming solution to distributed storage operation and design,” in Power and Energy Society General Meeting (PES), 2013 IEEE. IEEE, 2013, pp. 1–5. 144 [56] F. C. Schweppe, R. D. Tabors, M. Caraminis, and R. E. Bohn, Spot pricing of electricity. Springer, 1988. [57] University of Washington. Power systems test case archive. [Online]. Available: http://www.ee.washington.edu/research/pstca/ [58] J. L. Sweeney, The California electricity crisis. Hoover Institution Press, 2002. [59] R. Rajagopal, E. Bitar, P. Varaiya, and F. Wu, “Risk-limiting dispatch for integrating renew- able power,” International Journal of Electrical Power & Energy Systems, vol. 44, no. 1, pp. 615–628, 2013. [60] A. J. Conejo, M. Carri´ on, and J. M. Morales, Decision making under uncertainty in elec- tricity markets. Springer, 2010. [61] J. M. Morales, A. J. Conejo, H. Madsen, P. Pinson, and M. Zugno, Integrating renewables in electricity markets: Operational problems. Springer Science+Business Media, 2014, vol. 205. [62] R. Jiang, J. Wang, and Y . Guan, “Robust unit commitment with wind power and pumped storage hydro,” Power Systems, IEEE Transactions on, vol. 27, no. 2, pp. 800–810, 2012. [63] D. Bertsimas, E. Litvinov, X. A. Sun, J. Zhao, and T. Zheng, “Adaptive robust optimization for the security constrained unit commitment problem,” Power Systems, IEEE Transactions on, vol. 28, no. 1, pp. 52–63, 2013. [64] P. Cramton and S. Stoft, “Why we need to stick with uniform-price auctions in electricity markets,” The Electricity Journal, vol. 20, no. 1, pp. 26–37, 2007. [65] G. Pritchard, G. Zakeri, and A. Philpott, “A single-settlement, energy-only electric power market for unpredictable and intermittent participants,” Operations research, vol. 58, no. 4-part-2, pp. 1210–1219, 2010. [66] J. M. Morales, A. J. Conejo, K. Liu, and J. Zhong, “Pricing electricity in pools with wind producers,” Power Systems, IEEE Transactions on, vol. 27, no. 3, pp. 1366–1376, 2012. [67] D. Bergemann and J. V¨ alim¨ aki, “The dynamic pivot mechanism,” Econometrica, vol. 78, no. 2, pp. 771–789, 2010. [68] Bonneville Power Administration. Wind generation & total load in the BPA balancing authority. [Online]. Available: http://transmission.bpa.gov/business/operations/wind/ [69] J. Qin and R. Rajagopal, “Price of uncertainty in multistage stochastic power dispatch,” in Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on, 2014, pp. 4065–4070. [70] Y . Zhao, J. Qin, R. Rajagopal, A. Goldsmith, and H. V . Poor, “Risky power forward contracts for wind aggregation,” in Communication, Control, and Computing (Allerton), 2013 51th Annual Allerton Conference on, 2013. 145 [71] I.-K. Cho and S. P. Meyn, “Efficiency and marginal cost pricing in dynamic competitive markets with friction,” Theoretical Economics, vol. 5, no. 2, pp. 215–239, 2010. [72] J.-J. Laffont and D. Martimort, The theory of incentives: the principal-agent model. Prince- ton University Press, 2009. [73] P. Bolton and M. Dewatripont, Contract theory. The MIT Press, 2005. [74] H. Tavafoghi and D. Teneketzis, “Optimal contract design for energy procurement,” in Com- munication, Control, and Computing (Allerton), 2014 52nd Annual Allerton Conference on, 2014, pp. 62–69. [75] I. Yang, D. S. Callaway, and C. J. Tomlin, “Dynamic contracts with partial observations: application to indirect load control,” in American Control Conference (ACC), 2014, 2014, pp. 1224–1230. [76] R. D. Zimmerman, C. E. Murillo-S´ anchez, and R. J. Thomas, “MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education,” Power Systems, IEEE Transactions on, vol. 26, no. 1, pp. 12–19, 2011. [77] National Renewable Energy Laboratory. The Eastern wind dataset. [Online]. Available: http://www.nrel.gov/electricity/transmission/eastern wind dataset.html [78] R. B. Myerson, “Optimal auction design,” Mathematics of operations research, vol. 6, no. 1, pp. 58–73, 1981. [79] J. Cr´ emer and R. P. McLean, “Optimal selling strategies under uncertainty for a discrim- inating monopolist when demands are interdependent,” Econometrica, vol. 53, no. 2, pp. 345–361, 1985. [80] ——, “Full extraction of the surplus in bayesian and dominant strategy auctions,” Econo- metrica, vol. 56, no. 6, pp. 1247–1257, 1988. [81] R. P. McAfee and P. J. Reny, “Correlated information and mecanism design,” Econometrica, vol. 60, no. 2, pp. 395–421, 1992. [82] M. H. Riordan and D. E. Sappington, “Optimal contracts with public ex post information,” Journal of Economic Theory, vol. 45, no. 1, pp. 189–199, 1988. [83] S. Bose and J. Zhao, “Optimal use of correlated information in mechanism design when full surplus extraction may be impossible,” Journal of Economic Theory, vol. 135, no. 1, pp. 357–381, 2007. [84] P. Gipe, Wind power: renewable energy for home, farm, and business, 2nd ed. Chelsea Green Publishing, 2004. [85] S. Mathew, G. S. Philip, and C. M. Lim, “Analysis of wind regimes and performance of wind turbines,” in Advances in Wind Energy Conversion Technology. Springer, 2011, pp. 71–83. 146 [86] E. C. Morgan, M. Lackner, R. M. V ogel, and L. G. Baise, “Probability distributions for offshore wind speeds,” Energy Conversion and Management, vol. 52, no. 1, pp. 15–26, 2011. [87] A. N. Celik, “Energy output estimation for small-scale wind power generators using weibull- representative wind data,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 91, no. 5, pp. 693–707, 2003. [88] S. Shakkottai and R. Srikant, Network optimization and control. Now Publishers Inc, 2008. [89] F. Kelly, “Charging and rate control for elastic traffic,” European transactions on Telecom- munications, vol. 8, no. 1, pp. 33–37, 1997. [90] D. Palomar and M. C. M. Chiang, “A tutorial on decomposition methods for network utility maximization,” IEEE Journal on Selected Areas in Communications, vol. 24, no. 8, p. 1439, 2006. [91] L. Chen, N. Li, L. Jiang, and S. H. Low, “Optimal demand response: Problem formulation and deterministic case,” Control and Optimization Methods for Electric Smart Grids, vol. 3, pp. 63–85, 2012. [92] P. Samadi, A.-H. Mohsenian-Rad, R. Schober, V . Wong, and J. Jatskevich, “Optimal real- time pricing algorithm based on utility maximization for smart grid,” in Smart Grid Com- munications (SmartGridComm), 2010 First IEEE International Conference on, 2010, pp. 415–420. [93] E. Bitar and S. Low, “Deadline differentiated pricing of deferrable electric power service,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on. IEEE, 2012, pp. 4991–4997. [94] L. Gan, U. Topcu, and S. Low, “Optimal decentralized protocol for electric vehicle charg- ing,” Power Systems, IEEE Transactions on, vol. 28, no. 2, pp. 940–951, 2013. [95] C.-W. Tan and P. Varaiya, “Interruptible electric power service contracts,” Journal of Eco- nomic Dynamics and Control, vol. 17, no. 3, pp. 495–517, 1993. [96] L. Jia and L. Tong, “Optimal pricing for residential demand response: a stochastic optimiza- tion approach,” in Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on, 2012, pp. 1879–1884. [97] W. G. Temple and R. T. B. Ma, “Monotonic marginal pricing: Demand response with price certainty,” in ACM GreenMetrics, in conjunction with ACM Sigmetrics 2013, 2013. [98] G. Wang, M. Negrete-Pincetic, A. Kowli, E. Shafieepoorfard, S. Meyn, and U. V . Shanbhag, “Dynamic competitive equilibria in electricity markets,” Control and Optimization Methods for Electric Smart Grids, vol. 3, pp. 35–62, 2012. [99] G. O’Brien and R. Rajagopal, “A method for automatically scheduling notified deferrable loads,” in American Control Conference (ACC), 2013. IEEE, 2013, pp. 5080–5085. 147 [100] S. Koch, J. L. Mathieu, and D. S. Callaway, “Modeling and control of aggregated hetero- geneous thermostatically controlled loads for ancillary services,” in Proceedings of the 17th Power Systems Computation Conference (PSCC), 2011. [101] H. Hao, B. M. Sanandaji, K. Poolla, and T. L. Vincent, “A generalized battery model of a collection of thermostatically controlled loads for providing ancillary service,” in Proceed- ings of the 51th Annual Allerton Conference on Communication, Control and Computing, 2013. [102] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computation. Old Tappan, NJ (USA); Prentice Hall Inc., 1989. [103] R. Johari and J. N. Tsitsiklis, “Efficiency loss in a network resource allocation game,” Math- ematics of Operations Research, vol. 29, no. 3, pp. 407–435, 2004. [104] R. T. Maheswaran and T. Basar, “Social welfare of selfish agents: motivating efficiency for divisible resources,” in Decision and Control, 2004. CDC. 43rd IEEE Conference on, vol. 2. IEEE, 2004, pp. 1550–1555. [105] R. Johari and J. N. Tsitsiklis, “Efficiency of scalar-parameterized mechanisms,” Operations Research, vol. 57, no. 4, pp. 823–839, 2009. [106] S. Yang and B. Hajek, “VCG-Kelly mechanisms for allocation of divisible goods: adapting VCG mechanisms to one-dimensional signals,” Selected Areas in Communications, IEEE Journal on, vol. 25, no. 6, pp. 1237–1243, 2007. [107] R. Jain and J. Walrand, “An efficient Nash-implementation mechanism for network resource allocation,” Automatica, vol. 46, no. 8, p. 1276, 2010. [108] P. Jia and P. E. Caines, “Auctions on networks: efficiency, consensus, passivity, rates of con- vergence,” in Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on. IEEE, 2009, pp. 8606–8611. [109] M. Bitsaki, G. D. Stamoulis, and C. Courcoubetis, “A new strategy for bidding in the network-wide progressive second price auction for bandwidth,” in Proceedings of the 2005 ACM conference on Emerging network experiment and technology. ACM, 2005, pp. 146– 155. [110] P. Maill´ e and B. Tuffin, “Multibid auctions for bandwidth allocation in communication net- works,” in INFOCOM 2004. Twenty-third Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 1. IEEE, 2004. [111] A. A. Lazar and N. Semret, “Design and analysis of the progressive second price auction for network bandwidth sharing,” Telecommunication Systems-Special issue on Network Eco- nomics, vol. 20, pp. 255–263, 1999. 148 [112] R. Jain and P. Varaiya, “A design for an asymptotically efficient combinatorial Bayesian mar- ket: generalizing the Satterthwaite-Williams mechanism,” in International Conf. on Game Theory, Stony Brook (July 2007), 2007. [113] ——, “An asymptotically efficient mechanism for combinatorial network markets,” submit- ted to Operations Research, 2011. [114] H. H. Cheng and G. Tan, “Asymmetric common-value auctions with applications to private- value auctions with resale,” Economic Theory, vol. 45, no. 1-2, pp. 253–290, 2010. [115] R. Johari, S. Mannor, and J. N. Tsitsiklis, “A contract-based model for directed network formation,” Games and Economic Behavior, vol. 56, no. 2, pp. 201–224, 2006. [116] S. Shakkottai and R. Srikant, “Economics of network pricing with multiple ISPs,” Network- ing, IEEE/ACM Transactions on, vol. 14, no. 6, pp. 1233–1245, 2006. [117] V . Krishna, Auction theory, 2nd ed. Academic Press, 2009. 149

Abstract (if available)

Abstract The electric power system is regarded as the largest and most complex machine ever built, in which supply and demand must be balanced on a second‐by‐second basis. Consequently, the electricity market has to be carefully designed to prevent a recurrence of the California electricity crisis. This task is even more demanding with the popularization of the smart grid. Our work attempts to address the following question: what smart mechanisms should a smart grid be equipped with, for a smart market that can be utilized but not manipulated by smart people? ❧ Based on a two‐level market model, we focus on the control, optimization and market design of the smart grid, where we incorporate three key elements: the increasing penetration of renewable generation, the increasing participation of demand response, and the fast development and deployment of energy storage systems. ❧ On the upper level is the primary market coordinated by the independent system operator. We consider the economic dispatch problem, and develop a game‐theoretic framework to investigate the market outcomes with strategic generators under locational marginal pricing. We then consider a dynamic extension, and investigate how the use of storage may affect the market structure and market outcomes. We also study a multistage energy procurement problem, and design incentivizing pricing mechanisms that facilitate efficient participation of the generators. ❧ On the lower level, there are two secondary markets. The secondary market I, when it exists, involves an aggregator who buys power from a group of generators and sells it to the primary market. In particular, we study how to design market mechanisms for buying wind power. We first consider the welfare‐maximizing objective, and propose the stochastic resource auction paradigm that elicits probability distributions of wind power generation. We then consider the revenue‐maximizing objective, and study how to extract the surplus given the correlation among wind power generation. ❧ The secondary market II mimics the retail electricity market. We study the distributor's problem how to utilize demand response in an adaptive manner. In a stochastic setting, the optimal pricing scheme should evolve according to the up‐to‐date information. We develop a distributed algorithm to compute the optimal price process that incentives the agents to choose the socially optimal decisions. ❧ Finally, we study a network resource allocation problem in a hierarchical setting. Motivated by the results on hierarchical mechanism design, we discuss the challenges of the integration of the two‐level markets. This is an open problem and the direction of our future work.

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Asset Metadata

Creator Tang, Wenyuan (author)

Core Title The smart grid network: pricing, markets and incentives

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/22/2015

Defense Date 05/12/2015

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

Tag demand response,economic dispatch,electricity market,OAI-PMH Harvest,renewable energy,smart grid

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Jain, Rahul (committee chair), Nayyar, Ashutosh (committee member), Savla, Ketan (committee member)

Creator Email tangwenyuan@gmail.com,wenyuan@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-604649

Unique identifier UC11301114

Identifier etd-TangWenyua-3676.pdf (filename),usctheses-c3-604649 (legacy record id)

Legacy Identifier etd-TangWenyua-3676.pdf

Dmrecord 604649

Document Type Dissertation

Format application/pdf (imt)

Rights Tang, Wenyuan

Type texts

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

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

Repository Name University of Southern California Digital Library

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