Efficient Auction Games: Theories, Algorithms and Applications in Smart Grids & Electric Vehicle Charging 9811526389, 9789811526381

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Zhongjing Ma Suli Zou

Efficient Auction Games Theories, Algorithms and Applications in Smart Grids & Electric Vehicle Charging

Efficient Auction Games

Zhongjing Ma Suli Zou •

Efficient Auction Games Theories, Algorithms and Applications in Smart Grids & Electric Vehicle Charging

123

Zhongjing Ma School of Automation Beijing Institute of Technology Beijing, China

Suli Zou School of Automation Beijing Institute of Technology Beijing, China

ISBN 978-981-15-2638-1 ISBN 978-981-15-2639-8 https://doi.org/10.1007/978-981-15-2639-8

(eBook)

© Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The issue of resource allocation problems among some entities has been a hot research topic in many fields, such as network optimization, computer science, and power systems. With the increasing number of entities, including resource consumers and suppliers, decentralized schemes become one of the key streams. However, decentralization may result in efficiency loss and disorder. This book focuses on the designs of auction games for decentralized and efficient resource allocation and developing the algorithms for achieving efficient equilibrium. This book studies several classes of resource allocation problems, say single-sided, double-sided, and hierarchical, which could cover most of the common resource allocation problems. Moreover, this book studies some special problems that occur in practice and formulates these practical problems as efficient resource allocation problems. The proposed auction mechanisms have some good properties like incentive compatibility and the existence of the efficient Nash equilibrium. The performances of proposed algorithms, such as convergence, computational complexity and efficiency, etc., are verified, respectively, in rigorous ways. Numerical simulation cases are also studied and analyzed in detail to verify the developed results. This book provides some approaches to formulating practical problems as appropriate resource allocation problems and designing auction games to achieve optimal solutions. It may benefit the researchers and engineers in universities and electrical power industry who are interested in mechanism design, system and control, game theory, and optimization and control in efficient resource allocation and power electricity systems. For example, it may benefit those interested in designing and developing optimal charging strategies of electric vehicles, utilization

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of energy resources, and economic operation and control of smart grids. This book may be useful to senior undergraduate and postgraduate students who are in a range of electrical engineering, computer sciences, and operation research as well. Beijing, China October 2019

Zhongjing Ma [email protected] Suli Zou [email protected]

Acknowledgements

This monograph would not be possible without those people who have encouraged and helped us during our researches. First, we would like to express our sincere gratitude to Prof. Peter Caines and Prof. Roland Malhame, who are the Ph.D. mentors of Zhongjing Ma. They led Zhongjing Ma to the field of optimization, system control, and game theory, from September 2002 to January 2009 at McGill University, Montreal, Canada. From January 2009, Zhongjing Ma went to the University of Michigan, Ann Arbor, working as a postdoctoral research fellow advised by Prof. Duncan Callaway and Prof. Ian Hiskens. The research during that time was in the direction of decentralized charging coordination of large-population PEVs in smart grids. From September 2015, Suli Zou visited the University of Michigan, Ann Arbor, dedicating to the study of decentralized coordination and the design of auction games for optimal charging of PEVs with Prof. Ian Hiskens. All of them provided us many insightful comments and contributed significantly to our research work. We would like to thank our collaborators who studied at Beijing Institute of Technology, Beijing, China, including Xiaokun Yin, Long Ran, Xingyu Shi, Yongjian Cai, Shan Liu, Nan Yang, Peng Wang, Xu Zhou, Fei Yang, etc. Moreover, we want to thank our team members, Prof. Zhigang Gao, Dr. Hongwei Ma, and Dr. Liang Wang, and other colleagues at the research institute of Electrical Engineering who have not only provided us with their valuable suggestions for our research but also enriched our research lives at Beijing Institute Technology for our warm team environment throughout our research activities. In addition, we want to express our appreciation to the editors, Jasmine Dou and Chandra Sekaran, and reviewers and other staff for their assistances. The financial supports are quite important for our researches, from the National Natural and Science Foundation of China under grant 61873303, State Grid of China and Beijing Institute of Technology Research Fund Program for Young Scholars.

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Last but not least, we would like to express our deepest appreciations to our friends and families. They have always provided us with their endless encouragement and supports.

Contents

1 Introduction . . . . . . . . . . . . . . . . 1.1 Backgrounds and Motivations 1.2 Contributions . . . . . . . . . . . . 1.3 Organization . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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of the Book . . . . . . . . . . . . . . . . . . ........................... ........................... ...........................

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Resource Allocation Under Auction Games . . . . . . . . . . . . . . . 2.3.1 Formulation of Divisible Resource Allocation Problems . 2.3.2 Auction-Based Mechanism Design . . . . . . . . . . . . . . . . 2.4 Decentralized Dynamic Implementation of NE of Auction Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A Brief Discussion on the NE Implementation Method . 2.4.2 NE Implementation of the PSP Auction Game . . . . . . . 2.5 Performance of Auction Mechanism: Efficiency and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Efficiency of the NE . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Convergence of the Proposed Algorithm . . . . . . . . . . . . 2.5.3 Algorithm with a Player Update Rule . . . . . . . . . . . . . . 2.5.4 A Discussion on the Bidder Drop Cases . . . . . . . . . . . . 2.6 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Case Study I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Case Study II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Case Study III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusions and Ongoing Researches . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Hierarchical Auction Games for Efficient Resource Allocation . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Resource Allocation Problems Under a Hierarchical Structure . 4.3 Implementation of Efficient NE Under Dynamic Process . . . . . 4.3.1 Auction-Based Mechanism for Local System Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Dynamic Resource Allocation of Hierarchical System . 4.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions and Ongoing Research Works . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Double-Sided Auction Games for Efficient Resource Allocation 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Resource Allocation Under Auction Mechanism . . . . . . . . . . 3.2.1 Resource Allocation Problems . . . . . . . . . . . . . . . . . 3.2.2 Double-Sided Auction Mechanism Design . . . . . . . . 3.3 Decentralized Efficient NE Implementation . . . . . . . . . . . . . 3.3.1 Study of Single-Sided Auctions with a Given Potential Quantity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Updates of the Potential Quantity . . . . . . . . . . . . . . . 3.3.3 Implementation of the Efficient NE . . . . . . . . . . . . . . 3.3.4 Main Results of the Proposed Algorithm . . . . . . . . . . 3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Case Study I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Case Study II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Large-Scale Elastic Load Management Under Auction Games . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Formulation of Coordination Problems of Large-Scale Elastic Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Auction-Based Mechanism for Load Coordination Problems . . 5.3.1 Auction Games of Individual Loads . . . . . . . . . . . . . . 5.3.2 Payment of Individual Loads Under PSP Auctions . . . 5.3.3 Payment of Individual Loads Under MCP Auctions . . . 5.4 Performance Analysis for Load Coordination Auction Games . 5.4.1 Payment Comparison Under PSP & MCP Auction Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.4.2 Best Bid Strategy of Individual Loads . . . . . . . . . . . 5.4.3 Nash Equilibrium Properties of Efficient Bid Profiles . 5.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Non-NE Property of Efficient Bids Under MCP Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 eN -NE Property of the Efficient Bids Under MCP Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusions and Ongoing Researches . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Economic Operations of Microgrid Systems Under Auction Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Formulation of Microgrid Economic Operation Problems . . . 6.2.1 Economic Operations of Microgrid Systems . . . . . . . 6.2.2 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . 6.3 Economic Operations in Connected Mode Under Auction Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Bid Profiles of Individual Units in Microgrid Systems 6.3.2 Resource Allocation Rule Subject to Bid Profiles of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Transfer Money of Individual Units Subject to Bid Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Payoff Functions of Individual Units . . . . . . . . . . . . . 6.3.5 Existence of Efficient NE . . . . . . . . . . . . . . . . . . . . . 6.3.6 Analysis on Price of Anarchy Under PSP Auction Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Implementation of NE Under Dynamic Process . . . . . . . . . . 6.4.1 Implementation Algorithm for NE . . . . . . . . . . . . . . 6.4.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . 6.5 Economic Operations of Microgrid in the Isolated Mode . . . 6.6 Conclusions and Ongoing Researches . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart Grid Under Auction Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Formulation of Vehicle-to-Grid Coordination Problems . . . . . . . 7.2.1 Coordination Capacity of Individual PEVs . . . . . . . . . . 7.2.2 Efficient V2G Coordinations for Frequency and Voltage Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 A Simulation for Frequency and Voltage Regulations . .

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7.3 Auction-Based Distributed Vehicle-to-Grid Coordination Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Bid Profiles for Regulation Auction Problems . . . . . . 7.3.2 Service Allocation Rule Subject to Bid Profiles of Individual Units . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Transfer Money of Agents Subject to Bid Profiles . . . 7.3.4 Payoff Functions of Individual Units . . . . . . . . . . . . . 7.3.5 NE Property of Efficient Bid Profiles . . . . . . . . . . . . 7.4 V2G Coordination Auction Games with Aggregated Players . 7.5 Implementation of Nash Equilibrium . . . . . . . . . . . . . . . . . . 7.6 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Efficient Charging Coordination for Electric Vehicles Under Auction Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Electric Vehicle Charging Coordination Formulation . . . . . . 8.2.1 Charging and Cost Models . . . . . . . . . . . . . . . . . . . . 8.2.2 Efficient Charging . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Distributed EV Charging Coordination Under a PSP Auction Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Bid Profiles of Individual Players . . . . . . . . . . . . . . . 8.3.2 Calculation of EV Payment and Payoff . . . . . . . . . . . 8.3.3 Related Work on EV Charging Games . . . . . . . . . . . 8.4 Efficiency of the Charging CoordinationP PSP Auction Game . 8.4.1 Verification of (8.17) when A
t2T dnt . . . . . . . . . P 8.4.2 Verification of (8.17) when 0  A\ t2T dnt . . . . . . 8.4.3 Existence of Efficient Nash Equilibrium . . . . . . . . . . 8.4.4 Analysis of Budget Balance at the Efficient NE . . . . . 8.4.5 Efficiency Loss of Single-Interval Auction Games . . . 8.5 PSP Auction Process for EV Charging . . . . . . . . . . . . . . . . . 8.5.1 An EV’s Best Bid with Respect to Other EVs . . . . . . 8.5.2 Update Mechanism for EVs . . . . . . . . . . . . . . . . . . . 8.5.3 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions and Ongoing Research . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

List of Figures

Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11

Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5

Updates of players’ bid profiles under Algorithm 2.1 . . . . . . . Updates of players’ bid profiles under Semret’s process . . . . . Updates of players’ bid profiles under Algorithm 2.1 with the players’ update rule stated in Theorem 2.3 . . . . . . . . Convergence steps w.r.t. termination parameter e under different methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updates of players’ bid profile by applying Algorithm 2.1 . . . Updates of players’ bid profile by applying Algorithm 2.1 with the players’ update rule stated in Theorem 2.3 . . . . . . . . Updates of players’ bid profile by applying Semret’s process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolutions of social welfare under different bid profile update methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence iteration steps under different methods for several repeated simulations . . . . . . . . . . . . . . . . . . . . . . . Evolution of the iteration steps with respect to population size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence iteration steps under different methods with respect to population size with termination criterion e ¼ 0:001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An illustration of NEs for the underlying double-sided auction game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matched prices and the updated potential quantity with respect to bid profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updates of the potential quantity and players’ allocations under Algorithm 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updates of players’ bid profile under Algorithm 3.1 . . . . . . . . Updates of the social welfare and the matched prices under Algorithm 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 3.6 Fig. 4.1 Fig. 4.2 Fig. 4.3

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5.3 5.4 5.5 5.6

Fig. 5.7 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 8.1

Updates of individual strategies of generators and load units by applying Algorithm 3.1 . . . . . . . . . . . . . . . . . . . . . . . An illustration of the topology of the hierarchical system . . . . Hierarchical mechanism framework . . . . . . . . . . . . . . . . . . . . Updates of suppliers’ allocation assigned by the provider and buyers’ bidding demands in a local system over the iteration steps under Algorithm 4.2 . . . . . . . . . . . . . . . . . . Updates of uniform bidding prices of local systems over the iteration steps under Algorithm 4.2 and evolution of players’ bidding prices of a specific local system at a specific iteration step . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updates of hierarchical system social welfare under and players’s bidding prices in a local system Algorithm 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specification of pðbÞ w.r.t. bid profile b . . . . . . . . . . . . . . . . . Evolutions of rN and eN with respect to the population size N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of total demand and generation . . . . . . . . . . . . . . A partition of the set of bid profiles Bn . . . . . . . . . . . . . . . . . Illustration of linear functions with inf and sup slopes . . . . . . An illustration of the clearing price b p and allocation bx n w.r.t.   b b ð b n ; bn Þ in case d n
dn . . . . . . . . . . . . . . . . . . . . . . . . . . . . An illustration of clearing price w.r.t. ðb b n ; bn Þ in case b d n \dn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generations of wind turbine and demands of inelastic loads. . Efficient allocation with respect to a retailed electricity price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PSP auction diagram for a microgrid system in connected mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An illustration of a bid profile specified in Definition 6.3 . . . . Illustration of a pair of distinct Nash equilibria. . . . . . . . . . . . Updates of individual bidding prices and the associated resource allocations under Algorithm 6.1 . . . . . . . . . . . . . . . . Apparent power constraint for individual PEV n . . . . . . . . . . An approximate Gaussian distribution of initial SOC values over the PEV populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . The efficient allocations to individual PEVs with respect to initial SOC values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inefficient NE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updates of players’ bid profiles and potential quantity under Q-PSP auction mechanism . . . . . . . . . . . . . . . . . . . . . . Illustration of bnt ðdnt ; AÞ with respect to dnt and A . . . . . . . . .

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. . 135 . . 136 . . 146 . . 147 . . 148 . . 155 . . 157 . . 162 . . 176 . . 180 . . 181 . . 187 . . 191 . . 215

List of Figures

Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6

P Illustration of b b nt and bnt when A
t2T dnt . . . . . . . . . . . . An illustration of partitioned subsets Ri ; i ¼ 0; . . .; 3 . . . . . . . Background demand and the aggregate optimal charging strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergent updates of Algorithm 8.1 . . . . . . . . . . . . . . . . . . . P Relationship between b b nt and bnt when 0  A\ t2T dnt . . .

xv

. . 216 . . 219 . . 227 . . 227 . . 230

List of Tables

Table 5.1 Table 6.1 Table 6.2 Table 7.1 Table 7.2

The values of rN and eN w.r.t. N . . . . . . . . . . . . . . . . . . . . . . List of key symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updates of the increment of agent payoff and the increment of system utility under Algorithm 6.1 . . . . . . . . . . . . . . . . . . Cost function parameters for aggregated players . . . . . . . . . . Evolution of constraint status of aggregated players . . . . . . .

. . 127 . . 142 . . 163 . . 190 . . 192

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Chapter 1

Introduction

1.1 Backgrounds and Motivations of the Book Over the last decade, smart grid systems, which integrate communication, control, and sensing technologies, have been deployed as hot research topics due to the stress operations of the power grid, increasing power demand, high penetration of renewable energy and environment requirements [1, 2]. The researches that aim at cost saving, environmental friendly, and intelligent operation of future grids cover economic dispatch and energy management in power grids, power markets, coordination of electric vehicles (EVs), demand response, secure problems, etc. [2–27]. Plenty of the above research problems could be regarded as electricity or service resource allocation problems, e.g., energy management and coordination of EVs. Beside the mentioned research issues in power electricity systems, resource allocation problems are also widely studied in many other fields, such as telecommunication networks, cloud computing, and traffic allocation, see [28–43] and references therein. A fundamental goal of resource allocation problems is to allocate the resources in an efficient way, say achieve a global objective such as maximizing the system’s social welfare. Possible allocation strategies could be implemented in centralized ways through solving a class of centralized (constrained) optimization problems. E.g. [6, 44–53] and the references therein. These centralized strategies require a central operator who is authorized to determine the allocation and the cost of the resource consumers. However, decentralization has an upward trend in smart systems in which autonomy, intelligence and distributed computing are the key characteristics. In addition, centralization is criticized for the heavy communication and computation and full information requirement, and the entities in the system may be autonomous and selfish, not be willing to share its private information with others and purse its own benefit instead of the overall welfare. Quite a few works have been dedicated to designing and developing resource allocation mechanisms with a variety of architectures and services in order to achieve efficient results in a decentralized way [29, 30, 33, 37, 41–43, 54–61]. Nevertheless, the challenge of coordinating many autonomous agents to achieve an optimal or near-optimal outcome is nontrivial. This © Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_1

1

2

1 Introduction

is the so-called “price of anarchy” [62, 63], which captures the lack of coordination in systems where agents are selfish and may have conflicted interests. Game theory (mostly noncooperative game), which is a branch of applied mathematics, has emerged to be a popular tool for decentralized control of multi-agent systems. It concerns how selfish and rational (strategic) agents make decisions in a conflict environment and provides a rich set of mathematical tools to model and analyze interactions among the agents [64–67]. A game is defined by a set of players (agents), a set of strategies for each player, and the payoffs for the players, and admits a concept of Nash equilibrium (NE) where no player has an incentive to deviate from its strategy with respect to its opponents’ strategies [68–71]. Overall, an individual player can receive no incremental benefit from changing strategies, assuming other players remain constant in their strategies. Roughly, the price of anarchy in a game is the system cost of the worst-case NE over the optimal system cost that would be achieved if the players were forced to coordinate. That is, the main issue is how to design a game or a mechanism that, despite strategic behaviors on the part of players and without a priori knowledge of the full information by the system, still achieves an allocation that optimizes the global objective. From economics, game theory has a wide application discipline such as engineering, computer science, and biology. Auction game [72, 73] is a game theoretic approach in which an object or service is exchanged on the basis of bids submitted by the bidders to an auctioneer. Auction is effective to allocate resource among rational players [30, 74] across many research fields, such as network resource allocation [36, 37, 65, 75, 76], cloud computing resource allocation [41–43, 77–82], and power resource allocation [54, 55, 83], wherein [65, 76] gave comprehensive surveys of auction approaches applied in wireless networks, [78, 79] described challenges and methodical analysis of auctionbased resource scheduling and algorithms in cloud computing, and [83] summarized auction bidding behaviors of players in electricity markets. In an auction, the players submit their individual bids, respectively, to the system (auctioneer) where an allocation, as well as a payment to each agent, is determined with respect to the collected bid profile. Thus, we seek to design an allocation and a payment rule to achieve efficiency. There are two main auction mechanisms, namely, the first- and second-price auctions. In first-price auction, an object or service is given to a bidder who submitted the highest bid and pays a price equal to the amount of bid. In second-price auction, an object or service is given to a bidder who submitted the highest bid and pays a price equal to the second highest amount of bid. Resource Allocation Mechanisms (Non-game-Based) and Their Efficiency Mechanism design for resource allocation problems is not a new problem, and one popular kind of the methods is to solve the resource allocation problems as optimization problems. The optimization methods, such as primal–dual method [84, 85], Lagrangian [86], and greedy algorithms [84, 87], are applied to allocate the resources efficiently. In the work of [84], a generalized greedy primal–dual algorithm was introduced for the problem of maximizing queueing network, which was formulated as convex optimization subject to linear and convex constraints. In this work, the solution was proved to be asymptotically optimal. Usually, as shown in

1.1 Backgrounds and Motivations of the Book

3

[87], greedy algorithms would obtain local optimum, not the global optimal solution when solving the combinatorial problems. At each iteration of the algorithm, some heuristic method is applied to implement local optimum, but it may yield an approximately global optimum within a reasonable time. In [85], resource allocation in an OFDM system was formulated as a convex optimization problem which might have multiple corresponding optimal solutions. To achieve the global optimum, a primal–dual-based algorithm was designed and the convergence and optimality of the algorithm was studied through a realistic OFDM simulator. In [86], the method of proximal augmented Lagrangian was applied to solve a class of convex resource allocation problems over a connected undirected network of n agents coupled by a linear resource equality constraint. An asymptotically optimal online algorithm, which is called NOVA in [88], was presented for the joint optimization of network resource allocation and video quality adaptation in a network supporting video clients streaming stored video. Reference [89] investigated device-to-device (D2D) overlaying communications with spectrum-power trading where D2D users consume transmit power to relay cell-edge cellular users for uplink transmission. The problem was modeled as a joint energy-efficient optimization problem can be solved with optimality by transforming the fractional objective function into a subtractive form. In addition, in classical assignment problems, Hungarian algorithm is a well-known combinatorial optimization method [90]. However, it is difficult to apply this algorithm to decentralized schemes since its common computation complexity is O(n 3 ). In [91], the author extended the Hungarian method to reduce the average computational complexity of an efficient implementation of the algorithm. In [92–94], Kelly et al. showed that when all the players are not strategic, the resource allocation problem can be solved efficiently according to the notion of proportional fairness per unit charge. Reference [95] suggests that, for strategic players with a single divisible good, the Kelly mechanism can result in an efficiency loss of up to 25%. Alternatively, the so-called celebrated Vickrey–Clarke–Groves (VCG) mechanisms have been widely applied to decentralized systems [75, 96, 97], due to the fact that (i) incentive compatibility (i.e., a player’s bids correspond to true valuations) is a dominant strategy and (ii) knowledge of others’ valuations cannot improve a player’s expected utility. In the allocation of infinitesimally divisible resources, the information required by traditional VCG mechanisms expands to the whole valuation functions of players. While it may be impractical in divisible resource allocation problems, the VCG mechanism reduces the complexity of auction mechanism design and of the decision-making itself. Game Theoretic Approaches for Efficient Resource Allocation The behaviors of individual resource entities could be modeled and analyzed via game theory, which forms a mechanism for solving resource allocation problems. Furthermore, game models, which rely on the local and low-dimensional information and individual preference of players to arrive efficient solutions, could provide solutions in a decentralized way based on solid theoretical foundations. • Noncooperative game methods for efficient resource allocation

4

1 Introduction

Resource allocation has very wide application scenarios, especially in communication networks and cloud computing. There are quite a couple of works in the literature that apply game theoretic approaches for resource allocation in communication systems, e.g. channel access [98–101], cognitive radio network [102, 103], D2D communications [104] and 5G [105, 106]. In noncooperative games, players are usually with conflict interests and the goal of a noncooperative game is to find the (Nash) equilibrium among players. In a specific resource allocation problem, the game formulation needs to model the set of players, the set of strategies and the individual payoff functions that are associated with the strategies of the whole population. Most of the game designed were showed to have the existence of the NE. Among these works, [98, 99] analyzed that game could be used as a unifying framework to study radio resource management in a variety of wireless networks, and presented a class of power control games aiming at minimizing the energy of the network terminals. A noncooperative power control game for multi-rate CDMA networks was studied in [100] to support heterogeneous services with different transmission rates, while in [101] control of both data transmission rate and power for mobile terminals were considered as the joint objective of the game. In a cognitive radio network wherein primary and secondary users coexist, [102] modeled the resource allocation problem between secondary users and primary users as S-modular games, which provided useful tools for multi-objective distributed algorithms in the context of radio communications. Reference [103] focused on the update of the transmission power and frequencies of the primary and secondary users and addressed a Stackelberg game model in which individual users attempt to hierarchically access to the wireless spectrum in individual optimization. A thorough analysis of the existence, uniqueness, and characterization of the Stackelberg equilibrium is conducted. The resource allocation in D2D communications was solved in [104] via a non-cooperative game, maximizing the spectral efficiency and minimizing the energy consumption simultaneously, and presented a distributed interference-aware energy-efficient resource allocation algorithm to achieve NE. In [105, 106], a communication framework based on 5G was studied and an energy-efficient power allocation was reached under a gamebased method. In [105], the resource allocation problem was divided into power allocation and channel allocation where the power allocation is addressed based on game theory and Dinkelbach’s algorithm. To reduce the computational complexity, the channel allocation is modeled as a three-dimensional matching problem, and solved by iterative Hungarian method with virtual devices. In [106], a two-layer game theoretic framework was formulated to maximize the energy efficiency. The outer layer allows each femtocell access point to maximize the data rate of its users by selecting the frequency band and the solution to this non-cooperative game can be obtained by using pure strategy NE. The inner layer ensures the energy-efficient user association method subject to the minimum rate and maximum transmission power constraints by using dual decomposition approach. Commonly, cloud computing enables users to share computing resources supplied by cloud providers. The resource may be regarded as virtual machines (VMs). Noncooperative game method has been applied extensively to study various issues in cloud computing. In [107], an extensive form game was formulated to consider the fairness

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5

among users and the resources utilization to address the on-demand resource management of cloud computing. Reference [108] formulated the multi-resource allocation problem as a game, where the players are different servers. Reference [109] studied efficient and fair allocation of multiple types of resources in an environment of heterogeneous servers in the presence of placement constraints. • Auction game approaches and mechanism design Auction is one of the most effective decentralized solutions to resource allocation problems. The flexibility of the auction model allows agents to necessarily cooperate and compete, to accomplish resource allocation efficiently. In the auction, the main challenges are how to allocate the resource and the pricing. What’s more, the message space for the bids and the payoff of players are also key issues to be designed. There are numerous works to study auction mechanisms for resource allocation in computing and communication systems. To name a few, [30, 36–38, 40–43, 77, 81, 110–122]. In these works, the auctions could be categorized as single-sided auctions [30, 110–113], double-sided auctions [38, 117–120], and other specific type auctions. The work of [36, 114, 115] studied a class of auction mechanisms for resource allocation in cognitive radio networks, while [77, 116] focused on the mobile cloud computing problems. In the work of [40, 43], a combinatorial double auction mechanism was offered for cloud computing as it was stated that combinatorial auction was needed for cloud computing since in practice it often demanded a bundle of heterogeneous VM instances for its successful execution. The proposed allocation problem in [43] was formulated as an integer linear programming model aiming at maximizing the total profit of users and providers, and the proposed model satisfied the desirable properties: truthfulness, fairness, economic efficiency, and allocation efficiency. Reference [37] studied a reverse iterative combinatorial auction for D2D communication resource allocation. In [41], a hierarchical auction structure was presented to formulate a framework with resource constraints such as limited bandwidth and dynamic structure through introducing a link quality indicator. The problem is the cloud computing resource allocation in multi-robot systems with ad hoc networks. The players bid the link quality and the bandwidth is allocated proportionally according to their bids and the payment is the product of the price and the allocation. The work in [121, 122] also adopted hierarchical schemes for network resource allocation. In [81], it represented an online combinatorial auction for cloud computing resource allocation to address the efficiency across the temporal domain and model dynamic provisioning of heterogeneous virtual machine types. The final result is an online auction framework that is truthful, computationally efficient, and guarantees a competitive ratio. Online auction mechanism was also designed in [42], where the bidders are allocated based on the highest bids of the quality of services. The main aim is to increase the profits of the provider and users from different criteria. In view of the advantages of VCG and auction mechanisms, some works focus on how to design VCG auctions by allowing players to submit low-dimensional bid strategies which are used to represent the partial valuation of individuals. As stated in the literature, VCG auction might be the only type of auction that simultaneously

6

1 Introduction

guarantees both truthfulness and absolute economic efficiency (social welfare maximization), through calculating the optimal allocation and a carefully designed pricing rule [81]. Nevertheless, it may suffer from the computationally infeasibility when the underlying allocation problem is NP-hard. In [34, 75, 96], one-dimensional message space of bid was applied such that the system could construct a surrogate valuation function for each player. In [34], a unique and efficient NE was achieved while the mechanisms traded off dominant-strategy implementation for ease in implementation. Both the mechanisms in [75, 96] also achieved uniqueness and efficiency, but they required the surrogate valuation function to be twice continuously differentiable. In [123, 124], the PSP mechanism was firstly designed with non-differentiable surrogate valuation functions and two-dimensional bids consisting of a unit price and a maximum demand were applied. Reference [35] generalized PSP mechanism to multiple route links and double auctions. Lots of researchers have studied the efficiency of the NE for these auction games in different application backgrounds, e.g., [125–130]. It is also worth to mention that the auction systems stated above are not Bayesian auction games as studied in [131–133], since the individual payoff of each player is unrelated to the information on the valuation functions of players, instead it is completely determined by the values of players’ bid strategies. Recently, some other (non-VCG) mechanisms have also been proposed to achieve efficiency. Some designed work could obtain efficiency by compromising on incentive compatibility. For example, an efficient game was designed in [33] for unicast service provisioning; a combinatorial and double auction was designed in [134, 135] for Gale–Shapley matching; an all-pay auction was proposed in [112] for wireless fading channel allocation; and a double-side auction was proposed in [119] for vertical handover decision. • Auction games for resource allocation in smart grids and electricity markets The scope of this book lies in efficient auction games and their applications in smart grids and electricity markets. Hence, in this part, we survey the literature in this field particularly. Auction is a widely used mechanism in electricity markets in the real world, [136], for various resource allocations, e.g., electricity, capacity, reserve, and ancillary services. In day-ahead deregulated electricity markets, the market clearing price (MCP) mechanism, which has been widely deployed in many regions, is able to dispatch electricity resources among generators economically, and set the wholesale price in distributed ways, see [137, 138]. Pay-as-bid auction, which is also referred to as discriminatory auction, is another well-known mechanism used in deregulated electricity markets [139, 140]. Besides, as stated in [141], Vickrey auction is also a promising way to dispatch electricity resources in an economic way, even though it has not been widely applied in practice. In the electricity markets, auctions including uniform price auction, discriminatory auction, and Vickrey auction [141–144], are a designed mechanism which allows the system to allocate resources in a distributed way [72, 145]. Some works dedicated to compare different auction types in electricity markets. For instance, [142–144, 146, 147] analyzed the performance of the uniform price and pay-as-bid pricing in an electricity market, such as the expected revenue

1.1 Backgrounds and Motivations of the Book

7

or payment and trade demand or supply, and the results vary depending on different scenarios. In this book, we compare the market clearing price and second-price auction for elastic load coordination in electricity markets and develop the results in the payment, incentive compatibility, and efficiency in the case of both finite population and large-scale population. Other auction mechanisms have been designed for resource allocation in electricity markets. Reference [148] presented a multi-period auction to deal with the transmission congestion, losses as well as inter-temporal operating constraints such as start-up costs, ramp rates, and minimum up and down times, included in the composite bid of generators. Within the proposed auction scheme, the information required is the bid strategy of bidders and some public network data. In [149], the authors formulated an auction-based optimization problem with a non-separable objective function and noncontinuous bid curves. It applied a combination theory based model, which is compared for minimum payment cost, to consider all possible combinations of suppliers who can participate in auction for hourly system demand. In [150], it adopted a different auction and settlement mechanism for energy and ancillary services from the currently used MCP to make the minimized cost identical with the payment cost. Reference [151] also studied payment cost minimization auction for electricity market using mixed-integer linear programming. In the work of [152], the problem of developing bidding strategies in an oligopolistic dynamic electricity market was studied. A double-sided auction was applied for the two-side participants and used Nash–Cournot strategies for the market participants (generating firms and load serving entities). The efficiency and competitiveness of the underlying double-sided auction were compared through simulation to those of supplier-only auction. Another efficient electricity double-sided auction mechanism was proposed in [153], to control market power and enhance the social welfare of the electricity market. The key issues of auctions, including the market clearing, payment rules, and transaction matching rules, accorded to a so-called social welfare contribution of each player, which gave the mechanism the ability to control the market power of some participants and guaranteed the budget balance. The work in [154] firstly adopted agent-based simulation approach to investigate the bidding optimization of a wind generation company in the wholesale day-ahead electricity markets, by considering the effect of short-term forecasting accuracy of wind power generation. The results in the paper demonstrated that improving wind forecasting accuracy helped to increase the net earnings of the wind generation company and increasing wind penetration within the investigation range could help reduce the market clearing price. Uncertainties were also considered in [155], wherein the advent of markets confronted each producer with the problem of efficient energy/capacity resource allocation. By assuming that the seller is a price taker, the auction problem could be formulated as a multistage mixed-integer stochastic optimization problem. Quite a few problems in smart grid systems could be solved through auctionbased mechanisms, e.g., energy management problems, grid service providing, and electric vehicle coordination, to name a few, [156–160]. The implementation of distributed energy resource management in smart grid operation was analyzed in [156], wherein the trading strategy adopted for the continuous double auction was a

8

1 Introduction

profit-maximizing adaptive bidding strategy based on risk and competitive equilibrium price prediction. The auctioneer managed the usage of distributed energy resources by receiving bids from buyers and asked from sellers. In [157, 158], auction-based approaches have been developed for EV charging coordination problems which could be formulated as an optimization problem. Since the problem could be regarded as a scalable resource allocation problem, the charging behaviors basically are the allocation of the electricity resource. Different allocation and payment rules are designed for EV charging under various settings. In [159], this paper presented a new approach to design reactive power capacity markets. Reactive power capacity was allocated using an optimization algorithm that matched capacity bids and system reactive power needs, for peak and low demand hours. Also, the distribution of costs of the reactive power capacity market accorded to the use of the service by each market participant. The work of [160] introduced McAfee’s secondprice sealed auction mechanism into multi-unit electric power transaction between competitive generators and large consumers, making the decision of electric power price between competitive generators and large consumers based on the inclusion of the transaction cost and power transmission cost. Nash Equilibrium Seeking Algorithms for Resource Allocation Auction Games The above survey mainly focused on the analysis of bidding processes as well as the existence and stability of NEs. It is in need of the algorithm or procedure to seek the NE of the proposed auction games, and the analyses of the convergence and convergence rate are important. Previously, heuristic and evolutionary approaches have been applied to solve the formulated auction games. These methods may have the difficulties to obtain solid theoretic analyses. In the work of [82], users calculated suitable price bid with their objective function during several rounds and repetitions and sent it to the auctioneer, and the auctioneer chose the winning user based on the suggested utility function. The endpoint of the procedure was the NE point and it was also demonstrated that forecasting Bayesian learning could help the auctioneer to reach resource equilibrium price. In [161], the authors used both optimization and justice method by considering a resource allocation problem for cloud computing. The problem was solved by applying the method of problem relaxation under binary integer programming and a game-based evolutionary mechanism was applied to reduce the amount of lost performance. In [43], a heuristic resource allocation algorithm with a quasi-linear time complexity was presented to reduce the high complexity of the proposed model. The results of evaluations confirmed the good agreement of the heuristic algorithm with the optimization model in terms of allocation performance. There are also many research works dedicated to the implementation of the efficient NE of resource allocation auction games, e.g., [115, 122, 162]. The algorithms have been presented in [115, 122, 162] for single-sided auctions to implement the NE. Moreover, in some other works, e.g., [124, 163–168], the convergence rate under certain proposed dynamic processes is analyzed. In [163], Ausubel et al. studied a combinatorial clock-proxy auction mechanism, which is an efficient protocol to conduct combinatorial auctions. As stated in [164, 165], the main idea to achieve the

1.1 Backgrounds and Motivations of the Book

9

efficiency is that the auctioneer computes an optimal allocation at each stage, and then the losers at this stage are allowed to increase their bids, respectively. In case no losers are eager to increase their bids, an efficient allocation is reached. In [164], the convergence rate is shown to be the order of O(B/ε), where B and ε represent the number of bundles and the termination criterion, respectively. In [124], Semret designed a dynamic process for divisible network resource, such that each player is allowed to asynchronously update its own bid with respect to others’ bid profile to maximize its payoff. At each iteration step, the resource may be reallocated from lower price players to the player who updates its bid at this step. As analyzed in their work, the truthful bid strategy of player is an ε-best response, and the value of ε determines the tradeoff between the convergence rate of the process and the performance of the implemented NE. More specifically, as shown in [168] and numerical simulations, the iteration steps to the NE is the order of O(N /ε) under Semret’s process, where N is the population size. In order to introduce extra information related to players’ marginal valuations, Tuffin et al. designed a multi-bid auction mechanism in [166, 167], such that each player submits a collection of bids simultaneously. The efficiency of the NE under this mechanism depends on the dimension of players’ bid profiles, that is, the higher the dimension of players’ bid strategy is, the more efficient the implemented NE is. That is to say, one step convergence of multi-bid mechanism engenders high computational, communication, and transmission costs, and heavy message space. In [168], Jia et al. analyzed a quantized PSP mechanism such that all the players update their bids simultaneously, and under certain conditions, the system converges to a quantized NE. The convergence speed is no longer related to the population size, and then it is feasible to apply this method to systems with large populations. This work is extended to double-side auction games in [169] and multilevel networks in [170]. However, in order to ensure the convergence to a nearly efficient quantized NE, the inefficient NEs have to be eliminated in advance.

1.2 Contributions In this book, we study different classes of resource allocation problems and explore the efficient auction games in solving them. The resource allocation problems are mainly categorized as single-type and multi-type resource allocation which may be further divided into single-sided, double-sided, etc. The first part of this book is about theories and algorithms, which assume general settings for the problems, and the second part is about the applications in smart grids and EV charging. Note that the second part is not just the corresponding application of the first part in practical scenarios. All of the application scenarios we choose have distinct characteristics, from both the theory part of this book and those works in the literature. • Decentralized mechanism for infinitesimal divisible resource allocation Previous work on how to efficiently allocate resources mainly focus on indivisible goods or centralized schemes. In this book, we start from single-type divisible resource allocation in a way of obtaining a global objective without centralization.

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1 Introduction

The feature of the divisible resource makes the set of the action or state continuous and infinite and the decentralization is motivated by its benefits, e.g., lower communication and calculation burden, privacy, and scalability. Since each of the resource buyers is selfish, strategic, and only cares about individual objectives instead of the overall welfare, efficiency loss will occur and this is referred to as “price of anarchy”. Being aware of the effectiveness of auction mechanisms in resource allocation, the main issue is the design of an auction game or mechanism such that, despite strategic behaviors of players and without a priori knowledge of the valuation function by the system, the system could still reach a status whose allocation optimizes the global objective. Our formulation of the single-type divisible resource allocation is motivated by the resource allocation in communication networks addressed by the so-called progressive second price (PSP) mechanism specified in [123, 124]. Tn auction mechanism we proposed employs a two-dimension bid space revealing, for each player, a unit price and the maximum quantity of resources it wants. The payment of each player follows the VCG rules with respect to its allocation broadcasted by the auctioneer. As mentioned, each player is selfish, acts strategically, and may have an incentive to misrepresent its bids. While under the proposed PSP auction game, (i) the incentive compatibility holds, i.e. a truthful bid is a (weakly) dominate strategy of the individual player; (ii) the NE for the underlying auction games is efficient and unique. It is still challenging to effectively implement the NE in a decentralized way, even though we know that a unique and efficient NE exists. In our book, we suppose that the system is aware of some rough information related to the valuations of players, say an upper bound and a lower bound of the gradients of players’ marginal valuations. Assisted with this pair of rough parameters, a decentralized algorithm is presented to implement the efficient NE. The algorithm proceeds as: at each iteration, a single player is randomly chosen to update its best bid under a constrained set of demands and submit the updated bid to the auctioneer. We verify the convergence and optimality of the proposed algorithm. Moreover, to enhance the convergence performance, we specify an update order for these players; then the system converges logarithmically to its equilibrium. From this point of view, this method surpasses those under the methods given in the literature, e.g. [124, 168]. Also, the underlying system is shown to converge to the efficient NE under the proposed process with the specific update sequence. • Double-sided auction games for efficient resource allocation Wide resource allocation scenarios characterized by both resource consumers and suppliers, which generates double-sided resource allocation problems. In this book, a double-sided auction model is designed to decide strategies for individual buyers (consumers) and sellers (suppliers). Each player (a buyer or a seller) submits a bid, including a unit price that it wants to pay/charge and a maximum amount of resources that it demands/supplies, to the auctioneer who then determines an allocation and a payment on the basis of players’ submitted bid profile. The same as the single-sided auction mechanism, the incentive compatibility holds under the double-sided auction mechanism with the VCG type payment, and the efficient

1.2 Contributions

11

bid profile is a Nash equilibrium (NE). The difference lies in the existence of infinite number of NEs in the underlying double-sided auction game, which brings difficulties for players to implement the efficient solution, especially in a decentralized way. This is because each player is greedy to buy/sell as many resources as possible to win the auction, with respect to a bid profile of its opponents, by maximizing its payoff, and then it may take too many steps for the system to reach a NE solution, which may not be efficient, or even the system may oscillate and hence can’t converge to any NE. To overcome this challenge, we divide the double-sided auction game into a pair of single-sided auction games which are coupled via a joint potential quantity of the resource, called potential quantity in this book. Thus, each player competes with its opponents in the corresponding single-sided auction under a given potential quantity which represents a common quantity of the resource traded in the system, i.e., a quantity of the resource allocated to all the buyers or generated by all the sellers. Based on this formulation, we propose an algorithm to achieve the efficient NE, where a buyer or a seller implements its best strategy with respect to a constraint on its bid as well as a given potential quantity. The potential quantity is then updated following a specified rule in the auctioneer accordingly. Following this procedure, the system converges to the efficient NE. Furthermore, it is verified that the system converges to the efficient NE within finite iteration steps in the order of O(ln(1/ε)) with ε representing the termination criterion of the algorithm. • Hierarchical auction games for efficient resource allocation In the above discussions, the resource is directly traded between the resource consumers (buyers) and the supplier(s). However under such trading structure, the large-scale resource system may become increasingly complex. Also, some resource allocation scenarios consist of multilayered components. As a result, hierarchical structures have been extensively considered in many research fields. The hierarchical formulation in this book is that a single system provider owns all the resources and assigns them to a set of suppliers, which auction off their allocated resources to their buyers, respectively. The higher level is the suppliers get resources from the provider and the lower level consists of several local system in each of which one supplier allocates its obtained resources to its buyers. To implement the efficient allocations for the underlying hierarchical system, we first design a PSP based auction mechanism for each local system and show that it possesses the advantages of the PSP mechanism as well. At the lower level, buyers act also strategically to maximize their own payoff, while suppliers are supposed to be price takers who get resources from the provider in a centralized way at the higher level. Indeed, suppliers could also act as strategic players which is similar to the work specified in Chap. 2. For purpose of simplicity, we adopt a centralized scheme for the higher level which does not affect the whole results. At the lower level, each local system converges to its own efficient Nash equilibrium (NE) by applying a dynamic algorithm, and the efficient resource allocation is achieved and the bidding prices of all the buyers in this local system are identical with each other at the equilibrium. For the resource allocation in the whole hierarchical system, we design an algorithm embedded with the algorithm implemented in each

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1 Introduction

local system, such that the resources assigned to the suppliers are readjusted via a dynamic hierarchical algorithm with respect to the bidding prices associated with the implemented equilibria of local systems and a few rough information related to individual buyers’ valuations. More specifically, at each iteration at the higher level, the provider appoints a specific supplier to increase its valuation by taking a certain quantity of resource from another supplier. It is shown that the social welfare increases with respect to the iteration and the system can converge to the efficient allocation solution under certain mild conditions. The convergence and optimality are achieved associated with the introduction of a pair of parameters related to buyers’ marginal valuation shapes, while note that the dynamic process for the whole hierarchical resource allocation system does not require complete information of individual buyers’ valuations. • Decentralized optimization method for large-scale elastic load management in electricity markets Auctions, e.g., uniform price auction and discriminatory auction, have been widely adopted in electricity markets and market clearing price (MCP) auctions are popular in many regions. In day-ahead deregulated electricity markets, MCP is able to dispatch electricity resources among generators economically, and set the whole sale price in distributed ways. In this book, we focus on the demand side resource allocation, that is, the elastic loads buy and pay for the electricity from the deregulated electricity markets concerning minimal and individual objective cost. As analyzed in the previous parts and in other works, PSP auctions possess promising properties of incentive compatibility and efficiency. We study the coordination of large-scale elastic loads in electricity markets by introducing the MCP and PSP auction mechanisms and compare the performance of them. The key issues of the performance comparison lie in the payment, incentive compatibility, and the efficiency of NE, especially the associated asymptotic phenomena as the scale size of the power systems increases. To our knowledge, we are the first to conduct analyses on the large-scale load coordination auction games. The following results are developed in the book: – The payment of an individual load under MCP auctions is always higher than that under PSP ones, and their difference vanishes asymptotically as the system scale increases. – The incentive compatibility holds under PSP auctions, but not under MCP ones. While it holds under MCP with respect to the efficient bid profile of other loads. – The efficient bid profile under PSP auctions is a NE, while that under MCP is an ε-NE which degenerates to a NE asymptotically as the system scale increases. • Economic operations of microgrid systems in a decentralized way This book studies the economic operations of the microgrid in a distributed way such that the operational schedule of each of units, like generators, load units, storage units, etc., in a microgrid system is implemented by autonomous agents. Different from the existed economic operation problems, the storage units are first considered which could supply and consume energy, and the microgrid system is considered as a whole to trade with the main grid. We don’t assume the status of the

1.2 Contributions

13

behaviors of the storage or the microgrid at some time, e.g., assume that the storage will discharge when its state of charge is more than 80%. The charging/discharging behaviors of the storage and the buying/selling decisions of the microgrid are determined by solving optimization problems. The underlying problem setting brings more challenges in solving the economic operation in a distributed way. In this problem, we apply the PSP auction mechanism to efficiently allocate the resource. Considering the economic operation for the microgrid systems, the generators play as sellers to supply energy and the load units play as the buyers to consume energy, while a storage unit, like battery, super capacitor, etc., may transit between buyer and seller, such that it is a buyer when it charges and becomes a seller when it discharges. This problem is different from the double-sided auction game specified in Chap. 3 due to that – In a connected mode, each individual unit competes against not only the other individual units in the microgrid but also the exogenous main grid possessing fixed electricity price and infinite trade capacity, that is to say, the auctioneer assigns the electricity among all individual units and the main grid with respect to the submitted bid strategies of all individual units in the microgrid in an economic way; – The bid strategy for a storage unit may charge or discharge with respect to the submitted bid strategies of other units in the microgrid and the electricity trade price of the main grid. That is to say, the auctioneer assigns the electricity among all individual units and the main grid with respect to the submitted bid strategies of all individual units in the microgrid in an economic way. We show that under mild conditions, the efficient economic operation strategy is a Nash equilibrium (NE) for the PSP auction games, and propose a distributed algorithm under which the system can converge to an NE. In this book, we also introduce the so-called notion of “price of anarchy”, for the game system under PSP mechanism, to measure the performance gap between the worst NE and the efficient one; We also show that the performance of worst NE can be bounded with respect to the system parameters, say the energy trading price with the main grid, and based upon that, the implemented NE is unique and efficient under some conditions. This work degenerates into a double-side PSP auction game analyzed by [35] in case that the microgrid is operated in an isolated mode, that is to say there is no energy exchange between the microgrid and the main grid, and no storage units are involved in the microgrid system. • Game-based decentralized methods for V2G coordination problems Plug-in electric vehicles (PEVs) could provide grid services such as frequency and voltage regulation, by coordinating their active and reactive power rates. However due to the autonomy of PEVs, it is challenging how to efficiently schedule the coordination behaviors among these units in a distributed way. In this book, we formulate the underlying coordination problems as a novel class of VCG-style auction games where players, power grid, and PEVs, do not report a full cost or valuation function but only a multidimensional bid signal: the maximum active

14

1 Introduction

and reactive power quantities that power grid wants and the maximum per-unit prices it is willing to pay, the maximum active and reactive power quantities that a PEV can provide and the minimum per-unit prices it asks. It proves the existence of the efficient NE for the V2G auction game, even though there may exist other inefficient NEs. In order to deal with the V2G coordination problems with largescale PEVs, a class of V2G coordination auction games is presented such that the behaviors of a collection of individual PEVs are represented by an aggregate player. Moreover, to compute the efficient NE, the algorithm proposed in [171] is adapted and extended to solve the underlying V2G auction games with constrained multidimensional bid strategies. As illustrated with the numerical examples, the game system converges to some Nash equilibria which is near to the efficient NE. • Decentralized method for EV charging coordination problems with coupling constraints Except the V2G services provided for the grid, EVs’ charging behaviors would bring unexpected effects to the power grid, e.g., new peak load, voltage drop, and cost loss. A novel class of auction games is formulated to study the coordination problems arising from charging a population of EVs over a finite horizon. Different from those analyzed above, the charging power of EVs at different time slots could be regarded as multi-type resources, and there exist coupling constraints among these resources. In the auction-based pricing mechanism, each EV submits a multidimensional bid related to the inter-temporal factor to the system. The system implements an optimal allocation with respect to the submitted bid profile, and broadcasts an associated system price to all the users. Due to the cross-elastic correlation among the intervals, the best updated response of users, at an instant, is determined by both the demand at this instant and the total demand over the whole horizon. This difficulty is addressed by partitioning the allowable set of bid profiles based on the total desired energy over the entire horizon. The use of the PSP auction mechanism ensures that incentive compatibility holds for the auction games. It is shown that the efficient bid profile over the charging horizon is a Nash equilibrium of the underlying auction game. In addition, it develops a dynamic dispatch algorithm for distributing the energy where all the individual EVs update their best bidding responses simultaneously with respect to others’ bid strategies. The auction-based charging coordination scheme is adapted to a receding horizon formulation to account for disturbances and forecast uncertainty.

1.3 Organization By exploiting the auction games for different types of resource allocation problems, it has super wide application scenarios. • In this chapter, we give the overall background and the motivation of this monograph, as well as the brief introduction of the main works in this monograph.

1.3 Organization

15

• In Chap. 2, the auction mechanisms for efficient single-type divisible resource allocation are formulated. A corresponding decentralized algorithm is proposed to guarantee the convergence to the equilibrium. • Chapter 3 considers a collection of double-sided resource allocation problems and formulates a double-sided auction game, which has an infinite number of Nash equilibriums. It proposes a decentralized algorithm which could converge to the efficient equilibrium. • In the case that suppliers get resources from a provider and allocate them to its buyers, a hierarchical auction game is built in Chap. 4 such that the proposed hierarchical system can converge to the efficient allocation. • Since auctions have been widely adopted in electricity markets, in Chap. 5, we study the coordination of large-scale elastic loads in deregulated electricity markets under PSP auctions and compare it to the widely used MCP auctions. The performance of the two auctions in load coordination is analyzed in three parts and it is pretty promising to apply both MCP and PSP auctions to the large-scale load coordination problems. • In Chap. 6, the economic operations of the microgrid in a distributed way are studied such that the operational schedule of each of units, like generators, load units, storage units, etc., in a microgrid system, is implemented by autonomous agents. This problem is single-type electricity resource allocation while it is not only double-sided since the storage unit which could be either buyer or seller is a special player. Under mild conditions, the efficient economic operation strategy is a Nash equilibrium (NE) for the PSP auction games, and it can be achieved by the proposed decentralized algorithm. • It studies in Chap. 7 the frequency and voltage regulations problems of V2G service. The problem is generalized to a two-type resource allocation problem and formulated as a class of VCG-style auction games, which are shown to have the efficient NE. In addition, an auction game with aggregate players is proposed for large-scale PEVs, and it extends the quantized-PSP mechanism to implement the efficient NE. • In Chap. 8, a class of auction games is formulated to study coordination problems arising from charging a population of electric vehicles (EVs) over a finite horizon. This is indeed a multi-type resource allocation with coupling constraints. It is shown that the efficient bid profile over the charging horizon is a Nash equilibrium of the underlying auction game and an update algorithm is designed for the proposed auction game.

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Chapter 2

Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

Abstract We study how to efficiently allocate the infinitesimal divisible resource under auction mechanism. In this chapter, we mainly focus on allocating a fixed amount of a single-type resource. A VCG-type auction mechanism is proposed with a two-dimensional bid, which specifies a per unit price and a maximum of the demand. Due to the absence of enough information related to the infinite-dimensional valuations of individual players in a single-bid strategy, it is challenging to implement the efficient Nash equilibrium (NE) in a dynamic way. In this chapter, we introduce a pair of parameters related to players’ valuations, and design a decentralized dynamic process assisted with this pair of values, such that at each iteration, a single player updates its best bid under a constrained set of demand. Under the proposed auction mechanism, we show the incentive compatibility, efficiency, and uniqueness of the NE. Furthermore, our method is guaranteed to converge to the efficient NE, and it presents the enhanced convergence performance compared with those methods proposed in the literature. Case studies are given to demonstrate the results developed in this work.

2.1 Introduction The issue of resource allocation among some entities has been a hot research topic in many fields, such as communication networks, traffic networks, and power electricity systems, see [1–12] and references therein. The resource(s) shall be allocated in a way to achieve a global objective (such as maximization of the system social welfare). However, in case each of the entities is autonomous and selfish, such entity is not willing to share its private information with others and attempts to pursue its own benefit instead of the overall welfare. Since the system operator is only given limited information and centralized allocation methods are criticized for the heavy communication and complex computation, alternatively it is desirable to design decentralized methods for resource allocation problems. As game theory has emerged to be a popular tool for decentralized control of multi-agent systems, a fundamental issue is how to design a game or a mechanism that, despite strategic behaviors on the part of players and without a priori knowledge of the valuation function by the system, © Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_2

25

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2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

still achieves an allocation that optimizes the global objective. Auction mechanism is effective to allocate resource among rational players in a decentralized way [3–5]. In this chapter, we intend to solve the resource allocation in the framework of auction games. Our formulation is motivated by the problem of resource allocation in communication networks addressed by the so-called progressive second price (PSP) mechanism specified in [13, 14]. We present an auction mechanism wherein each player is asked to reveal a bid signal consisting of a unit price and the maximum quantity of resources it wants. The auction employs a two-dimensional message space to coordinate the allocation among players and determines payments with respect to the bid profile collected and broadcasted by the auctioneer. However, each player is selfish, acts strategically, and may have an incentive to misrepresent its bids. As we show, under the presented auction games, (i) the incentive compatibility holds, i.e., a truthful bid is a (weakly) dominant strategy of the individual player; (ii) the NE for the underlying auction games is efficient and unique. While there exists a unique and efficient NE, it is also challenging to effectively implement the NE in a decentralized and dynamic way. In our work, we suppose that the system is aware of some rough information related to the valuations of players, say an upper bound and a lower bound of the gradients of players’ marginal valuations. Assisted with this pair of rough parameters, we design a decentralized dynamic process, such that, at each iteration, a single player is randomly chosen to update its best bid under a constrained set of demands. By adopting the presented process, the system converges to an equilibrium, and then converges to the efficient solution due to the uniqueness and efficiency of the NE. Moreover, to enhance the convergence performance, we specify an update order for these players; then the system converges logarithmically to its equilibrium. From this point of view, this method surpasses those under the methods given in the literature, e.g., [14, 15]. Still, the underlying system is shown to converge to the efficient NE under the proposed process with the specific update sequence. The rest of the chapter is organized as follows. Section 2.2 presents a brief overview of related studies, especially for the game-based mechanisms that lead to the efficient solution. In Sect. 2.3, we formulate a class of resource allocation problems as auction games, and in Sect. 2.4, we design a decentralized dynamic process for the underlying resource allocation problems. In Sect. 2.5, the efficiency and convergence of the designed auction system are analyzed. Numerical simulations are presented in Sect. 2.6 to demonstrate the results developed in this chapter. Finally, we summarize our results and give some extensions in Sect. 2.7.

2.2 Related Works For the purpose of comparison, here we will review some research works, related to our proposed auction mechanism, in the literature. In [16–18], Kelly et al. showed that when all the players are not strategic, the resource allocation problem can be solved efficiently according to the notion of proportional fairness per unit charge.

2.2 Related Works

27

Reference [19] suggests that, for strategic players with a single divisible good, the Kelly mechanism can result in an efficiency loss of up to 25%. Alternatively, Vickrey– Clarke–Groves (VCG) mechanisms have been widely applied to decentralized systems [20–22], due to the fact that (i) incentive compatibility (i.e., a player’s bids correspond to true valuations) is a dominant strategy and (ii) knowledge of others’ valuations cannot improve a player’s expected utility. In the allocation of infinitesimally divisible resources, the information required by traditional VCG mechanisms expands to the whole valuation functions of players. While it may be impractical in divisible resource allocation problems, the VCG mechanism reduces the complexity of auction mechanism design and of the decision-making itself. Auction games have been widely applied in resource allocation problems across many research fields, such as network resource allocation [11, 12, 20, 23] and power resource allocation [4, 5, 24], wherein [23] gave a comprehensive survey of auction approaches applied in wireless and mobile systems and [24] summarized auctionbidding behaviors of players in electricity markets. In an auction, players submit their individual bids, respectively, to the system (auctioneer) where an allocation, as well as a payment to each player, is determined with respect to the collected bid profile. Thus, we seek to design an allocation and a payment rule to achieve efficiency. In view of the advantages of VCG and auction mechanisms, many researchers have worked on how to design VCG auctions by allowing players to submit low dimensional bid strategies which are used to represent the partial valuation of individuals, e.g., [9, 10, 13, 14, 20, 21]. Specifically, in [9, 20, 21], one-dimensional bid was applied such that the system could construct a surrogate valuation function for each player. In [9], a unique and efficient NE was achieved while the mechanisms traded off dominant-strategy implementation for ease in implementation. Both the mechanisms in [20, 21] also achieved uniqueness and efficiency, but they required the surrogate valuation function to be twice continuously differentiable. In [13, 14], the PSP mechanism was firstly designed with non-differentiable surrogate valuation functions and two-dimensional bids consisting of a unit price and a maximum demand were applied. Reference [10] generalized PSP mechanism to multiple route links and double auctions. Lots of researchers have studied the efficiency of the NE for these auction games in different application backgrounds, e.g., [25–30]. It is also worth to mention that the auction systems stated above are not Bayesian auction games as studied in [31–33], since the individual payoff of each player is unrelated to the information on the valuation functions of players, instead it is completely determined by the values of players’ bid strategies. Recently, some other (non-VCG) mechanisms have also been proposed to achieve efficiency. Some designed work could obtain efficiency by compromising on incentive compatibility. For example, an efficient game was designed in [8] for unicast service provisioning; a combinatorial and double auction was designed in [34, 35] for Gale–Shapley matching; an all-pay auction was proposed in [36] for wireless fading channel allocation; and a double-side auction was proposed in [37] for a vertical handover decision. The analysis of bidding processes usually focuses upon the existence and stability of NEs. There are also many research works dedicated to design dynamic algorithms to implement the NE for auction games, e.g., [38–40]. Moreover, in some other

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works, e.g., [14, 15, 41–45], the convergence rate under certain proposed dynamic processes is analyzed. In [41], Ausubel et al. studied a combinatorial clock-proxy auction mechanism, which is an efficient protocol to conduct combinatorial auctions. As stated in [42, 43], the main idea to achieve the efficiency is that the auctioneer computes an optimal allocation at each stage, and then the losers at this stage are allowed to increase their bids, respectively. In case no losers are eager to increase their bids, an efficient allocation is reached. In [42], the convergence rate is shown to be the order of O(B/ε), where B and ε represent the number of bundles and the termination criterion, respectively. In [14], Semret designed a dynamic process for divisible network resource, such that each player is allowed to asynchronously update its own bid with respect to others’ bid profile to maximize its payoff. At each iteration step, the resource may be reallocated from lower price players to the player who updates its bid at this step. As analyzed in their work, the truthful bid strategy of player is an ε-best response, and the value of ε determines the tradeoff between the convergence rate of the process and the performance of the implemented NE. More specifically, as shown in [15] and numerical simulations, the iteration steps to the NE is the order of O(N /ε) under Semret’s process, where N is the population size. In order to introduce extra information related to players’ marginal valuations, Tuffin et al. designed a multibid auction mechanism in [44, 45], such that each player submits a collection of bids simultaneously. The efficiency of the NE under this mechanism depends on the dimension of players’ bid profiles, that is, the higher the dimension of players’ bid strategy is, the more efficient the implemented NE is. That is to say, one step convergence of multi-bid mechanism engenders high computational, communication, and transmission costs, and heavy message space. In [15] Jia et al. analyzed a quantized PSP mechanism such that all the players update their bids simultaneously, and under certain conditions, the system converges to a quantized NE. The convergence speed is no longer related to the population size, and then it is feasible to apply this method to systems with large populations. This work is extended to double-side auction games in [46] and multilevel networks in [47]. However, in order to ensure the convergence to a nearly efficient quantized NE, the inefficient NEs have to be eliminated in advance.

2.3 Resource Allocation Under Auction Games 2.3.1 Formulation of Divisible Resource Allocation Problems We consider a class of resource allocation problems such that a collection of players N = {1, . . . , N } shares Γ units of a divisible resource. Denote by xn the units of resource allocated by the system to player n who possesses a valuation vn (xn ) representing its benefits on xn . A collection of allocations among all the players, x ≡ (xn , n ∈ N ), is called admissible, if it satisfies the following:

2.3 Resource Allocation Under Auction Games

xn ≥ 0, ∀n ∈ N and

29



xn ≤ Γ,

n∈N

and denote by U the set of admissible allocations. Same as [10, 47], we consider the following assumption: Assumption 2.1 The valuation function vn , for all n ∈ N , is twice differentiable, increasing and strictly concave.  We define S(x) as the system social welfare such that S(x)  n∈N vn (x n ), and the objective of the system is to implement an efficient allocation, denoted by x ∗∗ ≡ (xn∗∗ , n ∈ N ), such that x ∗∗ = arg max S(x).

(2.1)

x∈U

Under Assumption 2.1, there exists a unique efficient solution x ∗∗ for the optimization problem which satisfies the following Karush–Kuhn–Tucker (KKT) necessary condition, [48]:   ∂ = λ, in case xn∗∗ > 0 ∗∗ vn (xn ) xn∗∗ = Γ, and ∗∗ ∂ xn ≤ λ, in case xn = 0 n∈N

(2.2)

where λ is a nonnegative constant. As specified in (2.2), the efficient solution can be implemented in case the complete valuations of all the players are revealed. However this may generate a high burden of communication and computation, and violate the privacy and autonomy of individual players. Alternatively, the auction-based schemes can be implemented without other players’ complete private information, and ensure a certain level of autonomy to individual players. In this chapter, we propose an auction-based mechanism to efficiently allocate the resource in a decentralized and dynamic method.

2.3.2 Auction-Based Mechanism Design In the auction mechanism for resource allocation, each player n, with n ∈ N , forms a two-dimensional bid below bn = (βn , dn ) ∈ Bn = [0, ∞) × [0, Γ ], which specifies the per unit price βn that player n is willing to pay and demand up to dn units of the resource. b ≡ (bn , n ∈ N ) denotes a bid profile of players, and b−n ≡ (b1 , . . . , bn−1 , bn+1 , . . . , b N ) represents the bid profile of player n’s opponents. An auction mechanism shall include (i) player n, n ∈ N submits a bid bn to the system subject to its individual preference, (ii) the system determines the allocation

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2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

to all the players with respect to the collected bid profile under certain allocation rules, and (iii) each player pays for the assigned resource under certain payment rules. Subject to a bid profile b, the system determines an allocation, denoted by x ∗ (b) = ∗ (xn (b), n ∈ N ), such that x ∗ (b) = arg max x∈U x≤d



βn xn .

(2.3)

n∈N

Motivated by the PSP mechanism, as introduced in [13, 25], we define the allocation and payment rules for the auction game, respectively, as below: ⎧ ⎡ ⎤+ ⎫ ⎨ ⎬  xn∗ (b) = min dn , ⎣Γ − dm ⎦ , ⎩ ⎭ m∈M n (b)    βm xm∗ (b(n) ) − xm∗ (b) , τn (b) =

(2.4) (2.5)

m =n

where in (2.4), Mn (b)  {m ∈ N ; s.t. βm > βn } ∪ {m ∈ N ; s.t. βm = βn and m < n}, and [x]+ = max{0, x}. In (2.5), τn (b) denotes the payment of player n and b(n) = ((βn , 0); b−n ) represents the bid profile when player n is absent from the auction. Remark on (2.4): The players are assigned with some units of resource in descending order of their bid prices, and they can get at most what they ask if there are enough units of resource left in the system. It is straightforward to verify that the allocation specified in (2.4) is a solution to the optimization allocation problem (2.3). Remark on (2.5): τn (b) can be expressed as the summation of all players’ valuations in case this player does not join the auction, minus the summation of all the other players’ valuations in case it joins the auction. Essentially, the payment of each related player is defined as the externality it imposes on the others through its participation. Straightforward from (2.4), the admissible allocation x ∗ (b) subject to a bid profile b, satisfies the following:  • In case n∈N dn ≤ Γ : xn∗ = dn for all n ∈ N , i.e., all players are fully allocated. • In case n∈N dn > Γ : there only exists at most one player m ∈ N such that xm∗ ∈ (0, dm ). Note that the valuation and the payment of individual players are coupled with each other and the conflict might be seen as a game where each player endeavors to maximize its own payoff independently. The payoff of an individual player involves the gains and reimbursement for participating in the auction games and indicates the satisfaction of an individual player in a specific environment [49]. Hence under the allocation and payment rules, the payoff function of player n with respect to a bid profile b, denoted by f n (b), is given as below:

2.3 Resource Allocation Under Auction Games

f n (bn , b−n )  vn (xn∗ (b)) − τn (b).

31

(2.6)

The proposed auction system is a strategic game N , {Bn }n∈N , { f n }n∈N , with the Nash equilibrium (NE) defined in the following. Definition 2.1 A collection of bid profiles b0 is a Nash Equilibrium (NE) of the auction game if f n (bn0 , b0−n ) ≥ f n (bn , b0−n ) for all bn ∈ Bn and n ∈ N . By (2.6), f n is associated with the outcome of the auction game under a certain bid profile b, which is crucial in the tradeoff between the benefit and reimbursement. Hence, we would like to design a dynamic process to set b and investigate the corresponding tradeoff. In the following, we present a dynamic method for the underlying auction-based mechanism, and analyze the efficiency and convergence of the auction system.

2.4 Decentralized Dynamic Implementation of NE of Auction Games In this section, we propose a novel method to implement the efficient NE for the underlying auction systems in Algorithm 2.1 with its performance to be analyzed in Sect. 2.5 later. Before that, for the purpose of clarity, we give a brief discussion on the PSP auction game and the proposed method below.

2.4.1 A Brief Discussion on the NE Implementation Method Under the PSP auction mechanism, each player implements its best response, with respect to bid profile of its opponents, by maximizing its payoff. Consequently, each player is greedy and asks for as many resource units as possible on condition that it wins the auction. Due to this, it may take too many steps for the system to implement the efficient NE solution. As stated in [15] and demonstrated with a simulation in Sect. 2.6.1, the iteration steps to implement NE under Semret’s method are inversely proportional to ε. Alternatively, in order to improve the convergence performance, we propose a novel method, such that the demand of each player is constrained with an upper bound which is determined by players’ bid profile and certain very rough information, a pair of scale-valued parameters, related to players’ (infinite-dimension) valuations. We analyze the properties of the implemented NE as well as the convergence performance under the proposed method. Note that, though we show the convergence of the proposed method by verifying the monotonic increasing property of the social welfare with respect to iteration steps, generally speaking, this property itself can’t reflect how well the convergence performance is. In fact, concerning the simulation studied in Sect. 2.6.1, it is straightforward to verify that this monotonic property also holds under Semret’s process. However, compared with our proposed method, it takes much more steps to implement the NE.

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2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

2.4.2 NE Implementation of the PSP Auction Game By adopting the method proposed in Algorithm 2.1, at each iteration step, a player asks to update its bid by maximizing its individual payoff under a constrained set of bids which is defined by the auctioneer with respect to the bid profile at last iteration. We suppose an upper bound and a lower bound of gradients of players’ marginal valuations are given as a pair of parameters for the auction systems; then assisted with this pair of parameters, our method presents the enhanced convergence performance compared with those methods proposed in the literature. We introduce a pair of scalar-valued parameters, denoted by (ρ, ¯ ρ), such that ρ¯ ≥ ρmax  max sup



 v (x) ,

(2.7a)

ρ ≤ ρmin  min inf



 v (x) ,

(2.7b)

l∈N x∈[0,Γ ]

l∈N x∈[0,Γ ]

l

l

that is to say, ρ¯ and ρ are, respectively, an upper bound and a lower bound of the gradients of players’ marginal valuations and could be far away from ρmax and ρmin . Remark on (2.7): ρ¯ and ρ can be specified by very rough evaluations on the players’ valuation shapes. Actually, as shown in Theorems 2.2 and 2.3 later, the values of ρ¯ and ρ don’t influence the convergence of the algorithm, but may effect the convergence time. Also as illustrated in numerical examples, the introduction of this pair of rough parameters significantly enhance the convergence performance. More particularly, the closer to (ρmax , ρmin ) the values of ρ¯ and ρ are, the faster the convergence speed is. In the following, we define a set of bids in (2.10) for each individual player with respect to an upper demand constraint as specified in (2.8) below. For player n, we define notion Dn (b) with respect to a bid profile b, such that   2 Dn (b)  xn + min dm + Γc (b), αΦn (b), βn , ρ¯

(2.8)

   with x = (xn , n ∈ N ) ≡ x ∗ (b), α ∈ (0, 1), Γc (b) ≡ max 0, Γ − l∈N dl representing the remaining resource that is not asked by the players yet, m ≡ mn (b) and Φn (b) specified in (2.9a) and (2.9b) below:   mn (b)  max m ∈ N /{n}; s.t. βm = min {βl } l∈N /{n} xl >0

Φn (b)  with [x]+ ≡ max{0, x}.

+ 1 1 βn − βm + ρ(dn − xn ) + ρΓ ¯ c (b) ρ¯ 2

(2.9a) (2.9b)

2.4 Decentralized Dynamic Implementation of NE of Auction Games

33

Now we define a set of bids for player n with respect to bid profile b, such that   n , dn ); s.t. dn ≤ Dn (b) . bn ≡ (β Bn (b)  

(2.10)

Note: Essentially, as mentioned before, in order to accelerate the convergence rate of the iteration process, Eq. (2.8) specifies an upper bound unit of resource that player n is allowed to ask. In the following, we will give a brief discussion below on those terms defined in (2.8): • xn is the allocation of player n under b following (2.4). • In (2.8), the upper bound of dn is set to be less than or equal to xn + dm + Γc (b). This makes sure that at most one player’s allocation can be effected by player n, since the selection of m is the lowest price player with nonzero allocation and Γc (b) is such units which are not allocated among players yet. • In order to avoid the oscillation of the system subject to the update of the bid strategy of player n, αΦn (b) is involved to specify a largest amount of resource which can be transferred from player m to player n, and 2βn /ρ¯ is used to make sure that player n cannot grasp too many units of resource from the system or another player. Based upon the notion specified above, we propose an asynchronous dynamic process for the auction games in Algorithm 2.1 below. At each iteration step, a single player implements its best response under a constrained set of bid demands as specified in (2.10). Algorithm 2.1 Asynchronous bid update algorithm. Require: A set of bidders N ; An initial bid profile b(0) ; A termination parameter ε; k ← 0; 1: while true do 2: Calculate allocation x (k) (b(k) ) following (2.4); (k+1) 3: A player n (k+1) = m, with m ∈ N , is randomly chosen to update its best response bm (k) w.r.t b , such that   (k) (k+1) f m (bm , b−m ) , := arg max bm bm ∈B m (b(k) )

4: 5: 6: 7: 8: 9: 10: 11: 12:

where n (k) represents the player who updates its bid at step k; (k+1) (k) b(k+1) ← (bm , b−m ) ε(k+1) ← b(k+1) − b(k) 1 ; if ∃ i < k, such that {n (i) , n (i+1) , · · · , n (k) } ⊇ N and ε(i+1) , · · · , ε(k+1) < ε then Break; else k ← k + 1; end if end while return {b(k) n }n∈N

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2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

Note: By the specification of NE given in Definition 2.1, the auction system reaches an equilibrium solution in case Algorithm 2.1 converges. In Sect. 2.5, we will analyze the efficiency of the implemented NE under the proposed auction game, as well as the convergence of Algorithm 2.1. We also give several simulations to verify the proposed method in Sect. 2.6.

2.5 Performance of Auction Mechanism: Efficiency and Convergence 2.5.1 Efficiency of the NE In [13], the authors show that a truthful bid of an individual player is an ε-best response, which implies the incentive compatibility. Under certain mild conditions specified in Algorithm 2.1, we have a similar property specified below. Lemma 2.1 (Incentive compatibility) Under Assumption 2.1 and by applying Algorithm 2.1, a truthful bid is a weakly dominant strategy for each individual player. Proof It is equivalent to show Lemma 2.1 by verifying that, for any bid strategy  bn ∈ Bn (b) of player n, we can find a truth bid strategy bnt = (βnt , dnt ) ∈ Bn (b), bn , b−n ). with βnt = vn (dnt ), such that f n (bnt , b−n ) ≥ f n ( The proof of Lemma 2.1 is given in Appendix. For player n and a bid profile of other players b−n , we define a set of bid strategies n (b−n ), such that of player n, denoted by B   n (b−n )  bn ≡ (βn , dn ) ∈ Bn ; s.t. x ∗ (bn , b−n ) = dn , B n where xn∗ (bn , b−n ) is given in (2.4). Lemma 2.2 Consider a player n and a bid profile of other players b−n ; then under Assumption 2.1 and by applying Algorithm 2.1, there exists a bid of player n in n (b−n ) which is a weakly dominant strategy with respect to b−n . B Proof The proof of Lemma 2.2 is given in Appendix. Thus, by applying Algorithm 2.1, we suppose that at iteration step k + 1, player (k+1) is the solution to m updates its best response bm(k+1) with respect to b(k) −m ; then bm the following optimization problem: bm(k+1) = arg max

bm ∈B m (b(k) )

where bm(k+1) = (vm (dm(k+1) ), dm(k+1) ).



 f m (bm , b(k) ) , −m

2.5 Performance of Auction Mechanism: Efficiency and Convergence

35

Define a bid profile, b∗ ≡ (bn∗ ; n ∈ N ), such that bn∗ ≡ (βn∗ , dn∗ ) = (vn (xn∗∗ ), xn∗∗ ), ∀n ∈ N ,

(2.11)

where x ∗∗ represents the efficient allocation specified in (2.2). By (2.2) and (2.4), we can have x ∗ (b∗ ) = x ∗∗ . Thus  βn∗

= λ, ≤ λ,

in case dn∗ > 0 , in case dn∗ = 0



dn∗ = Γ.

(2.12)

n∈N

We call Theorem 2.1 (Existence, efficiency, and uniqueness of NE) Under Assumption 2.1, the NE for the underlying auction games is unique and efficient, i.e., b∗ defined in (2.11) is the unique NE. Proof First, we could conclude that b∗ is an equilibrium solution with the proof details provided in the first part of Appendix; then, in the second part of Appendix, we will show the uniqueness of NE of the underlying auction game. In the rest of this section, we will study the performance of Algorithm 2.1.

2.5.2 Convergence of the Proposed Algorithm Before analyzing the convergence property of Algorithm 2.1, we introduce a lemma about the allocation evolution below. Lemma 2.3 Under Algorithm 2.1, xn(k+1) ≥ xn(k) holds at any iteration k ≥ 1, where player n is the player who updates its best response at step k. Proof The proof of Lemma 2.3 is given in Appendix. Theorem 2.2 (Convergence of Algorithm 2.1) Under Assumption 2.1 and by adopting Algorithm 2.1, the auction system converges. Proof Suppose that player n is the player who implements its best response at step k; then an individual player m ∈ N shall be appointed following (2.9a) with respect to b(k) . By xn(k+1) ≤ dn(k+1) ≤ Dn (b(k) ) with Dn (b(k) ) specified in (2.8), we have xn(k+1) ≤ (k) (k) xn + dm + Γc (b(k) ); then by (2.9a) and the allocation rule (2.4), we can obtain that (k+1) (k) xl = xl for all l ∈ N /{n, m}. Define δl(k)  xl(k+1) − xl(k) for all l ∈ N ; then we can verify that   (k) + (k) = Γc (b(k) ) ≡ Γ − dl ≥ 0. δn(k) + δm l∈N

(2.13)

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2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

We define ΔS  S(x (k+1) ) − S(x (k) ) as the change of the social welfare from step k to step k + 1. By Lemma 2.3, xn(k+1) ≥ xn(k) ; then we have the following discussions. (i). In case xn(k+1) = xn(k) . All the players’ allocation remains unchanged. It makes S(x (k+1) ) = S(x (k) ); then ΔS = 0. (ii). In case xn(k+1) > xn(k) . There are two cases as discussed in (ii.a) and (ii.b) below: (ii.a).

(k) = 0. We have In case δm

ΔS = vn (xn(k+1) ) − vn (xn(k) ); (ii.b).

then under Assumption 2.1, we have ΔS > 0. (k) < 0. We have the following: In case δm   (k) (k+1) ) − vm (xm ) ; ΔS = vn (xn(k+1) ) − vn (xn(k) ) − vm (xm then as verified in Appendix, we have ΔS > 0.

In conclusion, by (i) and (ii), we have that  ΔS

= 0, > 0,

in case xn(k+1) = xn(k) ; in case xn(k+1) > xn(k)

then by which together with that S(x) is bounded above by can converge under Algorithm 2.1.

 n∈N

vn (Γ ), the system

In summary, by Theorems 2.1 and 2.2, we can conclude that the auction game converges to the efficient NE b∗ by applying Algorithm 2.1. Nevertheless, in Sect. 2.5.3 below, we will specify an update sequence among all the players for Algorithm 2.1, and study the convergence performance under this procedure in Theorem 2.3. As demonstrated with simulations in Sect. 2.6 the system can converge to the equilibrium in fewer iteration steps under this specific procedure.

2.5.3 Algorithm with a Player Update Rule In the previous section, we analyzed the convergence property of the auction system under Algorithm 2.1 such that, at each iteration step, an individual player is randomly chosen to update its bid strategy. While the convergence rate of Algorithm 2.1 depends on the update sequence of players, we would like to improve the performance of Algorithm 2.1 by setting a specific update sequence, in Theorem 2.3, among these individual players. Theorem 2.3 (Convergence rate of Algorithm 2.1 with an update rule among players) We suppose, in Step 3.b in Algorithm 2.1, that the system can set a specific player n (k) to update its best bid as follows:

2.5 Performance of Auction Mechanism: Efficiency and Convergence

37

• If there exists m ∈ N such that xm(k) ∈ (0, dm(k) ); then n (k) := m; • Otherwise, if there exists m ∈ N such that xm(k) = 0 < dm(k) ; then n (k) := m; • Otherwise, n (k) := m ∈ arg max {βl(k) }, l∈N

and the algorithm terminates in case ε(k) < ε. Then, under the above update rule, the system converges to a bid profile, denoted by b0 , in at most K (N , ε) iteration steps, such that K (N , ε) is the order of O (N ln(1/ε)). Proof As verified in Appendix, we show that the auction system converges in K (N , ε) iteration steps such that  + K (N , ε) ≤ N

 N ln(Γ (ρ¯ + 1)/ε) , ln(ρ/( ¯ ρ¯ − ρ))

(2.14)

 is the order of O(N ), and x denotes the smallest integer larger than or where N equal to x. Hence we have the conclusion that K (N , ε) is the order of O (N ln(1/ε)). By (2.27) in the proof of Theorem 2.3 given in Appendix, we have b(k) − b0 1 ≤ ,

(2.15)

(k− N)/N  . Equation (2.15) implies that, for for any k, with ≡ Γ (ρ¯ + 1) (ρ¯ − ρ)/ρ¯ a given iteration steps k, it ensures that the game system can reach a bid profile which is in the neighbor of the equilibrium b0 with a distance less than or equal to at iteration step k. Theorem 2.4 By applying Algorithm 2.1 with the player update rule stated in Theorem 2.3, the implemented bid profile is the efficient NE in case the termination parameter ε vanishes. Proof The proof of Theorem 2.4 is given in Appendix.

2.5.4 A Discussion on the Bidder Drop Cases Some literature, e.g., [50] listed below, stated that a frequent starvation for the traded resources decreases the bidder’s interest and even may result in the drop of this bidder in the future auction rounds. In our framework, as shown in Lemma 2.1, the incentive compatibility property holds under the proposed mechanism; then under the concavity of valuation function of player n, the higher the submitted bid demand dn is, the lower the bid price βn tends to be. So if player n is too greedy to ask a high demand with a price lower than the others, this player may not be allocated with any resource. Also under Algorithm 2.1 the auctioneer randomly chooses a

38

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

player to update its bid; then player n may drop from the auction process if it isn’t selected for many rounds. As a consequence, though the proposed algorithm still converges, it reaches at a NE among the remaining players, and hence results in a certain efficiency loss since the dropped player has no allocations. To lower the occurrence of the unexpected bidder drops during the iteration rounds, instead of a random selection of a player to update its bid at an iteration step, the auctioneer may set a higher priority for such players, which has zero or low allocation, to update their bid. The player update rule stated in Theorem 2.3 is such a specific priority. Secondly, in the framework of our work, we suppose that all the players are rational. However, there might exist malicious or irrational bidders, who may construct a bid by shifting and randomizing the components of another bidder’s bid strategy, see [51], or just submit a bid under which its payoff function is not optimized. As a consequence, such bad-behavior bidders will cause a certain efficiency loss and unpleasant convergence property under the dynamic process proposed in our work. As ongoing research, we are willing to update our proposed auction mechanism to enhance its robustness.

2.6 Simulation Studies In this section, we illustrate the performances of the proposed method with several simulation examples.

2.6.1 Case Study I We consider an auction system with two players who share an identical marginal valuation such that vn (xn ) = [1 − xn ]+ , n = 1, 2 and Γ = 1, and set b1 = b2 = (0.2, 0.8) as an initial bid profile for players. Define ρ¯ = 2 and ρ = 0.5; set α = 0.99 and ε = 10−4 for the termination criterion of Algorithm 2.1. Note that the values of ρ¯ and ρ can be chosen in a pretty broad region as specified in (2.7), and α shall be set in the region of (0, 1). We first illustrate the simulation results of the auction system by adopting Algorithm 2.1 under the parameters specified above. The following is an associated update procedure. (S1) Initialize a bid profile b(0) s.t. b1(0) = b2(0) = (0.2, 0.8); (S2) Calculate the allocation by (4) and get x1(0) = 0.8, x2(0) = 0.2; (S3) Suppose, at iteration step 1, player 2 asks to implement its best response b2(1) = (β2(1) , d2(1) ), under the constraint d2(1) ≤ D2 (b(0) ) with D2 (b(0) ) = 0.3485 by (11); Note: The calculation and the role of D2 (b(0) ) are briefly stated below. (S4) Set b(1) = (b1(0) , b2(1) ) and ε(1) = b(1) − b(0) ;

2.6 Simulation Studies

39

(S5) (S2)–(S4) continues until the termination criterion is reached. Note: Discussions on the role of the upper bound constraint of Dm (b(k) ) in (S3) above are briefly stated below. At iteration step 0, the calculation of D2 (b(0) ) by (2.8) is given in the following: D2 (b(0) )

  2 = x2(0) + min d1(0) + Γc (b(0) ), αΦ2 (b(0) ), β2(0) ρ¯   0.3 2 , × 0.2 = 0.2 + min 0.8, 0.99 × 2 2 = 0.3485 with b(0) = ((0.2, 0.8), (0.2, 0.8)), Γc = 0, α = 0.99, Φ2 = 0.3/2, ρ¯ = 2; then the best response of player 2 under Algorithm 2.1 is b2(1) = (0.6515, 0.3485). The evolution of the updates of players’ bid profile under Algorithm 2.1 is displayed in Fig. 2.1, and as observed, the system converges to the efficient solution in 30 iteration steps. By applying Semret’s process, the best response of player 2 is (0.2 + ε, 0.8 − ε). Note that the efficient allocation solution for all players is x ∗ = (0.5, 0.5). That is to say, compared with 0.3485 given above under our method, player 2 asked too many units, say 0.8 − ε, under Semret’s process. Actually, under Semret’s process, each player only decreases its demand by ε at each iteration step until the system reaches the equilibrium. Consequently, under Semret’s process, the system terminates at the equilibrium after N (d (0) − x ∗ )/ε iteration steps which is equal to 2(0.8 − 0.5)/0.0001 = 6000 for this case. See Fig. 2.2 for the updates of players’ bid profile under Semret’s process. Figure 2.3 displays the updates of players’ bid profile under Algorithm 2.1 with the update rule stated in Theorem 2.3. As illustrated, the system terminates in 7 iteration steps which is less than 20, the upper bound steps specified in Theorem 2.3 and is also less than 30 iteration steps displayed in Fig. 2.1 for the evolution under Algorithm 2.1. Moreover, the convergence rates under Algorithm 2.1 and Semret’s process with respect to different values of ε are listed in Fig. 2.4, respectively. We can observe that: (i). The performance under Algorithm 2.1 is much better than that under Semret’s process; (ii). By applying Algorithm 2.1 with players’ update sequence defined in Sect. 2.5.3, the auction system can converge to the efficient equilibrium in certain iteration steps which is approximately in the order of the logarithm of the inverse value of termination parameter ε. This result is consistent with Theorem 2.3.

2.6.2 Case Study II In this case, consider a resource given a quantity Γ = 20. We specify a resource allocation problem such that a 20-unit divisible resource is allocated among N = 5

40

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

Bid demand of player 1 & 2

0.8

bid demand of player 1 bid demand of player 2

0.7

0.6

0.5

0.4

0.3

0

5

10

15

20

25

30

Iteration 0.7

bid price of player 1 bid price of player 2

Bid price of player 1 & 2

0.6 0.5 0.4 0.3 0.2 0.1

0

5

10

15

20

25

30

Iteration Fig. 2.1 Updates of players’ bid profiles under Algorithm 2.1

individual players. We suppose that the valuation functions of all the players share a common form of vn (xn ) = 2an (xn + 1)0.5 , for all n, with a1 = 1.9, a2 = 2.0, a3 = 2.1, a4 = 2.2, a5 = 2.4. The efficient allocation for this optimization problem is x ∗∗ = [2.991 3.416 3.873 4.355 5.365] such that the players share an identical marginal valuation 0.951 for all n and the corresponding social welfare is 47.56. We firstly initialize a bid profile (βn(0) , dn(0) ) such that βn(0) = vn (dn(0) ) and dn(0) = 1 for all n ∈ N , and set the termination parameter ε as 10−4 . Figures 2.5, 2.6, and 2.7 display the evolutions of players’ bid profiles under different methods, respectively. As we can observe, the convergence performances under these methods for this simulation case are essentially the same as that studied in

2.6 Simulation Studies

41

0.6

Bid price of player 1 & 2

0.55 0.5 0.276

0.45

0.2755 0.4

0.275

0.35

1500

1510

1520

0.3 0.25 bid price of player 1 bid price of player 2

0.2 0.15

0

1000

2000

3000

4000

5000

6000

Iteration Steps 0.85 bid demand of player 1 bid demand of player 2

Bid demand of player 1 & 2

0.8 0.75 0.7 0.65

0.725

0.6

0.7245 0.724

0.55

1500

1510

1520

0.5 0.45

0

1000

2000

3000

4000

5000

6000

Iteration

Fig. 2.2 Updates of players’ bid profiles under Semret’s process

Sect. 2.6.1, and consistent with the results developed in this work. It is interesting to mention that the social welfare under Algorithm 2.1 is monotonically increasing; while this monotonic property does not hold under Semret’s process. This is illustrated in Fig. 2.8. Under Algorithm 2.1 and Semret’s process, at each iteration step, a player is randomly chosen to update its own bid, respectively. So to further demonstrate the comparison between convergence iteration steps under Algorithm 2.1 and Semret’s process, we repeat the simulation for several times under these two dynamic processes, respectively. The resulting convergence steps are displayed in Fig. 2.9.

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2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

Bid demand of player 1 & 2

0.9 bid demand of player 1 bid demand of player 2

0.8 0.7 0.6 0.5 0.4 0.3

0

1

2

3

4

5

6

7

Iteration 0.7 bid price of player 1 bid price of player 2

Bid price of player 1 & 2

0.6 0.5 0.4 0.3 0.2 0.1

0

1

2

3

4

5

6

7

Iteration Fig. 2.3 Updates of players’ bid profiles under Algorithm 2.1 with the players’ update rule stated in Theorem 2.3

Note: We didn’t repeat the simulation under Algorithm 2.1 with the players’ update rule stated in Theorem 2.3, since the convergence step is identical for the repeated simulations due to the fixed update order of players stated in Theorem 2.3. In the simulation specified above, we adopted a common form of valuation function v(x), such that v(x) = 2a(x + 1)0.5 , with the valuation parameter a = 1.9, 2.0, 2.1, 2.2, 2.4 for n = 1, . . . , 5, respectively. We denote by an the valuation parameter for individual player n; then an can be used to represent the valuation type of player n.

2.6 Simulation Studies

43

600

Convergence Steps

500

Algorithm 1 Algorithm 1 with an update rule stated in Theorem 3 Semret process

400 300 200 100 0

0.1

0.05

0.02

0.01

0.005

0.002

0.001

Value of termination parameter (\varepsilon) Fig. 2.4 Convergence steps w.r.t. termination parameter ε under different methods

In order to demonstrate the results developed in our work in more complicated situations, we suppose that an , the valuation type of player n, is no longer a deterministic value as considered in previous cases. Instead we assume that {an ; n ∈ N } is a set of mutually independent random variables and, for demonstration simplicity, we consider that an , for all n ∈ N , is uniformly distributed over a region denoted by A . Moreover for the purpose of comparison we consider A = {1.9, 2.0, 2.1, 2.2, 2.4}. By applying Algorithm 2.1, we can obtain that the average iteration steps to converge to the equilibrium, denoted by E {K (N , ε)}, is approximately equal to 68 which isn’t much different with 60, the termination iteration steps for the auction system studied in Sect. 2.6.2.

2.6.3 Case Study III In previous case studies, we run several numerical simulations with a fixed population size, respectively. Here in this section, we would like to demonstrate the scalability of dynamic processes on the population size. For the purpose of demonstration, we simply consider the same form of the valuation functions adopted in Case Study II for all players, and suppose that the total units of resource Γ (N ) is proportional to the population size N , such that Γ (N ) = 5N , and use the other parameters applied in Case Study II.

44

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation 5.5

Bid Demands of Players

5 4.5 4 3.5 3

bid demand of player 1 bid demand of player 2 bid demand of player 3 bid demand of player 4 bid demand of player 5

2.5 2 1.5 1

0

20

40

60

80

100

120

Iteration Steps 1.8 bid price of player 1 bid price of player 2 bid price of player 3 bid price of player 4 bid price of player 5

Bid Prices of Players

1.6 1.4 1.2 1 0.8 0

20

40

60

80

100

120

Iteration Steps Fig. 2.5 Updates of players’ bid profile by applying Algorithm 2.1

Figure 2.10 displays the evolution of the converged iteration steps under Algorithm 2.1 with the players’ update rule given in Theorem 2.3. As illustrated, except of the population size of 2, the converged iteration step is approximately linear with respect to the population size N . This is essentially consistent with the statement given in Theorem 2.3. Moreover, the convergence iteration steps under Algorithm 2.1 and Semret’s process with respect to different population sizes are shown in Fig. 2.11. As illustrated, the iteration steps usually increase with respect to the population size N . However due to the random selection of players under Algorithm 2.1 and Semret’s process, the linear relation between the convergence steps and the population size doesn’t hold

2.6 Simulation Studies

45

5.5 5

Bid demand of players

4.5 4 3.5 3 2.5 bid demand of player 1 bid demand of player 2 bid demand of player 3 bid demand of player 4 bid demand of player 5

2 1.5 1

0

10

20

30

40

50

60

Iteration 1.8 bid price of player 1 bid price of player 2 bid price of player 3 bid price of player 4 bid price of player 5

1.7

Bid price of players

1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8

0

10

20

30

40

50

60

Iteration

Fig. 2.6 Updates of players’ bid profile by applying Algorithm 2.1 with the players’ update rule stated in Theorem 2.3

any longer, and as observed the convergence step of Algorithm 2.1 is even decreased from 190 to 176 as the population size goes from 25 to 30. Nevertheless, the convergence performance under Algorithm 2.1 greatly surpasses that under Semret’s process.

46

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

Fig. 2.7 Updates of players’ bid profile by applying Semret’s process

2.7 Conclusions and Ongoing Researches In this chapter, we formulated the resource allocation problem as a class of auctionbased games under which a decentralized dynamic algorithm is designed to implement its efficient NE. With the introduction of a pair of scalar-valued parameters which is related to players’ marginal valuation, a constrained set of demand is defined to achieve the monotonic increasing of social welfare and then the convergence of the algorithm. Moreover in order to improve the performance of the proposed method, we consider that the system can appoint a specific single player under the given rule

2.7 Conclusions and Ongoing Researches

47

50

Social Welfare

45 social welfare under Algorithm 1 social welfare under the improved algorithm 40

35

30

10

0

60

50

40

30

20

Iteration Steps

(a) Algorithm 1 and the improved algorithm 50 48 46

Social welfare

44 45

42 40

44.5

38 44 2500

36

2550

2600

34 32 30

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iteration

(b) Semret’s process Fig. 2.8 Evolutions of social welfare under different bid profile update methods

to update its own best bid response at each iteration step. We show the analysis on the NE properties of the auction mechanism, and evaluate the property of convergence and convergence rate of the proposed algorithm. It implies that the convergence behavior has largely relied on the number of players N , as well as the value of termination parameter ε of the proposed algorithm. Specifically, the upper bound of the convergent iteration is the order of O (N ln(1/ε)).

48

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation 6000

Algorithm 1 Semret's process

Convergence Steps

5000 4000 3000 2000 1000 0

1

2

3

4

5

6

7

8

9

10

Simulation Rounds Fig. 2.9 Convergence iteration steps under different methods for several repeated simulations

Iteration required for convergence

55 50 45 40 35 30 25 20 15

2

5

10

15

20

25

30

Population size N Fig. 2.10 Evolution of the iteration steps with respect to population size

We would like to extend our work in the following research directions: (i). Propose of a novel method such that all the players update their best bid responses simultaneously, and hence the system may converge to the efficient NE in iteration steps which are unrelated to the population size; (ii). Interesting to extend the proposed method to the double-side auction systems such that, besides the buyers who ask for the resource, the owners of the resource attend the auction as well; (iii). Design of dynamic algorithms for correlated multiple resource allocation auction games, and

2.7 Conclusions and Ongoing Researches

49

3000

Convergence Steps

2500

Algorithm 1 Algorithm 1 with an update rule stated in Theorem 3 Semret's process

2000 1500 1000 500 0

2

5

10

15

20

25

30

Population Size N Fig. 2.11 Convergence iteration steps under different methods with respect to population size with termination criterion ε = 0.001

rigorous analysis of the incentive compatibility and efficiency of the NE, and the convergence property of the proposed algorithms; (iv). Design of a novel mechanism, for the auction games with unexpected bidder drops, malicious players, etc., such that a certain nice performance on the efficiency loss and/or convergence rate can be guaranteed.

Appendices Proof of Lemma 2.1 n , dn ) ∈ Bn (b), we Given a bid profile b, we would like to show that for any  bn = (β can always find a bid bnt ∈ Bn (b), with βnt = vn (dnt ), such that bn , b−n ). f n (bnt , b−n ) ≥ f n ( xn the Denote by xnt the allocation of player n with respect to (bnt , b−n ), and  n = vn (dn ) allocation of player n with respect to ( bn , b−n ), respectively. Since β implies that  bn is incentive compatible, we only consider the non-truthful cases in (I) and (II) below: n < vn (dn ). (I). In case β xn ≤ dn . By Assumption 2.1, we have βnt ≥ vn (dn ) > Consider a bid bnt with dnt =  n by which together with (2.4), we have xnt ≥  β xn ; then we have xnt =  xn by xnt ≤

50

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

dnt =  xn . Since the allocation of player n under ( bn , b−n ) is same as that under bn , b−n ) and (bnt , b−n ) (bnt , b−n ), we have the individual payoffs of player n under ( are equal with each other. n > vn (dn ). (II). In case β n > vn (dn ) = In this case, let dnt = dn ≤ Dn (b). By Assumption 2.1, we have β

t t vn (dn ) = βn . For the given bid profile b, we first give the following analysis below in case xn < dn and Φn (b) > 0. By (2.4), it gives βn ≤ βm and Γc (b) = 0 due to xn < dn , with m specified in (2.9a). By (2.8), we have Dn (b) ≤ xn + αρ¯ (βn + ρ(dn − xn ) − βm ); then we have vn (xn ) − βnt ≤ ρ(d ¯ nt − xn ) ≤ βn + ρ(dn − xn ) − βm . It implies that βnt − βm ≥ vn (xn ) − βn − ρ(dn − xn ) ≥ 0. Also by (2.8), we can show that Dn (b) ≤ xn + dm , by which together with βnt ≥ βm and (2.4), we have that the allocation xnt = dnt . By n > βnt , the allocation  β xn = dn . Further the payoffs under ( bn , b−n ) and (bnt , b−n ) are identical with each other. By applying the similar technique applied above for the case of xn < dn and Φn (b) > 0, we can obtain the same result for all the other cases of xn and Φn (b). By (I) and (II), we can obtain the conclusion.

Proof of Lemma 2.2 n∗ , dn∗ ) the best bid response of player n with respect to b−n , Denote by  bn∗ (b−n ) ≡ (β ∗ bn∗ , b−n ). Consider another and  x as the allocation with respect to the bid profile (

∗ ∗ n , dn ) with β n = vn ( xn ) and dn =  xn ; and denote by  x the allocation with bid  bn = (β respect to ( bn , b−n ). xn∗ ≤ dn∗ , we have vn ( xn∗ ) ≥ vn (dn∗ ), which implies By the concavity of vn (·) and  ∗ n ; then by the allocation rule (2.4) and dn =  n ≥ β xn∗ , we have  xn ≥  xn∗ . It that β ∗ ∗ xn , since  xn ≤ dn =  xn . That is to say,  bn is a bid with full alloimplies that  xn =  bn∗ , we can obtain that  bn has the same cation. Since  bn has the same allocation as  bn is also a best response. payoff as  bn∗ ; then 

Proof of Theorem 2.1 • Proof of the existence and efficiency of the NE: Here we show the existence of the NE by verifying that the efficient bid profile b∗ satisfies Definition 2.1, that is to say bn∗ is a best response with respect to b∗−n for any n ∈ N in (i) and (ii) below, respectively: Recall that, by (2.11), x ∗ (b∗ ) = x ∗∗ where x ∗∗ represent the efficient allocation. bn and results a decrease of its allo(i). Suppose that player n deviates from bn∗ to  cation by δ > 0. Note: Here we have xn∗∗ > 0, since a player without any allocation

Appendices

51

cannot decrease its allocation. Denote by  x the allocation with respect to ( bn , b∗−n ); ∗∗ ∗ ∗∗ ∗∗ xm = xm since xi = di for all i ∈ N . then we can obtain that  xn = xn − δ and  Hence the difference of the payoffs of player n with respect to ( bn , b∗−n ) and b∗ , ∗ ∗ denoted by Δf n  f n ( bn , b−n ) − f n (b ), is given as below. xn ) − vn (xn∗∗ ) < 0, Δf n = vn ( where the inequality holds by Assumption 2.1 and  xn = xn∗∗ − δ. bn and results an increase of its (ii). Suppose that player n deviates from bn∗ to  xm ≤ xm∗∗ for those allocation by δ > 0; then we can obtain that  xn = xn∗∗ + δ and  xm = xm∗∗ for those m whose xm∗∗ = 0, since a player with m satisfying xm∗∗ > 0 and  xm∗∗ = 0 cannot decrease its allocation any more. Hence the difference of the payoffs of player n with respect to ( bn , b∗−n ) and b∗ is given as below. Δf n = vn ( xn ) − vn (xn∗∗ ) +



∗ ( ∗∗ ) < v (x ∗∗ )( βm xm − xm xn − xn∗∗ ) + λ n n

m =n



∗∗ ) ( xm − xm

m =n

= 0,

where the inequality holds by Assumption 2.1 and (2.12), and the second equality  xm − xm∗∗ ). holds by (2.11) and  xn − xn∗∗ = m =n ( In summary, by (i) and (ii), we have the conclusion. • Proof of the uniqueness of the NE: Suppose that there exists another NE, denoted by b0 , which is different from b∗ ; then b0 is inefficient and does not satisfy (2.12). Also by the definition of NE, bn0 is the best response of player n with respect to b0−n ; then, by Lemma 2.1, we have xn0 = dn0 for all n ∈ N . Since the allocation rule (2.4) is a solution to optimization problem (2.3), the bid profile b0 satisfies the KKT necessary condition, such that  = λ + σn , in case dn0 > 0 βn0 , ≤ λ + σn , in case dn0 = 0



dn0 ≤ Γ,

(2.16)

n∈N

where σn ≥ 0 is a parameter. Hence, with the comparison of (2.12), b0 satisfies the following cases:  dn0 < Γ . (I) In case n∈N  l , dl ), for bl = (β Define = Γ − n∈N dn0 and consider another bid strategy  u the allocation with respect to some l ∈ N , such that dl = dl0 + . Denote by  xl = dl and  xm = xm0 for all ( bl , b0−l ). By (2.4) and xn0 = dn0 for all n ∈ N , we have  m = l.

52

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

We define Δfl  fl (b0 ) − fl ( bl , b0−l ); then Δfl satisfies the following: xl ) < 0, Δfl = vl (xl0 ) − vl ( because Assumption 2.1 and  xl = dl = dl0 + > xl0 .  0 (II) In case n∈N dn = Γ . Since b0 does not satisfy (2.12), and by (2.16), there exists at least a player l, with l ∈ N , such that σl > 0. l , dl ), such that λ < β l < βl0 . Consider another bid for player l, denoted by  bl = (β 0  Under Assumption 2.1 and by Lemma 2.1, we have dl < dl by which, together with l , we can have xm0 = dm0 for all m = l and λ < β 

xl , xl0 <   0 xn =  xn ,

n∈N

xm0

= xm , ≥ xm ,

(2.17a) (2.17b)

n∈N

l in case βm0 > β otherwise

(2.17c)

We have the following analysis for Δfl : Δfl = vl (xl0 ) − vl ( xl ) +



l βm0 (xm0 −  xm ) < vl ( xl )(xl0 −  xl ) + β

m =l



  l xl0 −  ≤β xl + (xm0 −  xm ) = 0,



(xm0 −  xm )

m =l

m =l

where the first inequality holds by Assumption 2.1, (2.17a) and (2.17c); the second l = vl (dl ); and the last inequality holds by Assumption 2.1,  xl ≤ dl , (2.17a) and β equation holds by (2.17b). bl , b0−l ) which is contradicted (I) and (II) given above imply that fl (b0 ) < fl ( with that bl0 is the best response of player l. Hence, there does not exist another NE different from b∗ . Thus the NE for the underlying auction games is unique and efficient.

Proof of Lemma 2.3 by Contradiction Denote by bn the updated best response of player n at step k; then we have (k) (k) f n (bn , b(k) −n ) ≥ f n (bn , b−n ), ∀ bn ∈ Bn (b ).

(2.18)

Appendices

53

Suppose the updated allocation xn(k+1) satisfies that xn(k+1) < xn(k) ; then by the allocation rule (2.4), we can obtain that xm(k+1)

 = xm(k) , ≥ xm(k) ,

in case βm(k) > βn(k) , in case βm(k) ≤ βn(k)

(2.19)

for all m ∈ N /{n}; then the following holds: (k) (k) f n (bn , b(k) −n ) − f n (bn , b−n )  =vn (xn(k+1) ) − vn (xn(k) ) +

βm(k) (xm(k+1) − xm(k) )

m∈N /{n}

0 and (2.9b), we have Φn (b(k) ) = ρ1¯ (η + 21 ρΓ δn(k) < Φn (b(k) ), we have η + 21 ρΓ ¯ c (b(k) ) − ρδ ¯ n(k) > 0, by which together with (2.23) (k) and δm < 0 considered in case (ii.b), we can obtain that ΔS > 0.

Proof of Theorem 2.3 By Lemma 2.1, βl = vl (dl ) for all l ∈ N ; then by (2.7), we have b(k) − b0 1 = β (k) − β 0 1 + d (k) − d 0 1 ≤ (ρ¯ + 1)d (k) − d 0 1 .

(2.24)

where b0 ≡ (β 0 , d 0 ) represents the implemented Nash equilibrium. Firstly we can verify that, by the update rule stated in Theorem 2.3 and Lemma 2.2, after certain iteration steps, denoted by k˜ which is the order of O(N ), the whole resource is completely distributed and each player is fully allocated. That is to say, the following holds:

Appendices

55



˜

˜

˜

xl(k) = Γ, and xl(k) = dl(k) , ∀l ∈ N .

(2.25)

l∈N

˜ the player with the Following the update rule stated in Theorem 2.3, at step k, highest bidding price, say player m, is assigned by the system to do its best response. ˜ player m will take some Since the whole resource is completely distributed at step k, units of resource from another player, say player m , who possesses the lowest bid price. Player m  will update its bid strategy at step k˜ + 1. This single player turns to be fully allocated by Lemma 2.2 after the update at step k˜ + 1. By Lemma 2.3, at step k˜ + 1, there are two possible cases for player m : (i) it occupies certain units of resource from another player; (ii) its original allocation remains unchanged subject to the bid with demand equal to its allocation at last step. It can be verified that the updated bid profile is closer to the equilibrium in case (i); then in order to analyze the upper bound of the convergence rate to the equilibrium, we consider case (ii) in the following analysis. Note that the analysis, given in the above paragraph for the pair of iteration ˜ k˜ + 1), also holds for each pair of adjacent steps (k˜ + 2i, k˜ + 2i + 1), with steps (k, i = 0, 1, . . ., as well. For any pair of adjacent steps (k, k + 1) with k = k˜ + 2i, suppose that player n and player m are the specific players who update their best responses at step k and step k + 1, respectively. Equation (2.10) implies that |dn(k+1) − dn0 | + |dm(k+2) − dm0 | = |Dn (b(k) ) − dn0 | + |Dm (b(k+1) ) − dm0 | = |Dn (b(k) ) − dn0 | + |xm(k+1) − dm0 |; then by (2.8), xl(k) = dl(k) for all l ∈ N , yields |dn(k+1) − dn0 | + |dm(k+2) − dm0 |     1 1     ≤ dn(k) − dn0 + (βn(k) − βm(k) ) + dm(k) − dn0 − (βn(k) − βm(k) ) ρ¯ ρ¯ 2 = dn0 − dn(k) + dm(k) − dm0 − (βn(k) − βm(k) ) ρ¯  ρ 0 (k) (k) (dn − dn + dm − dm0 ), by Assumption 1, (2.2) and (2.7). ≤ 1− ρ¯ Thus the following holds:   |dn(k+1) − dn0 | + |dm(k+2) − dm0 | ≤ 1 − ρ/ρ¯ (|dn(k) − dn0 | + |dm(k) − dm0 |)

(2.26)

The above analysis holds for each pair of adjacent iteration steps (k, k + 1) with k = k˜ + 2i with i = 0, 1, . . .. Hence by (2.26) together with (2.24), we can derive that for any step k >  k,

56

2 Auction Mechanisms for Efficient Single-Type Divisible Resource Allocation

 k ˜ b(k) − b0 1 ≤ (ρ¯ + 1)d (k) − d 0 1 ≤ (ρ¯ + 1) 1 − ρ/ρ¯ d (k) − d 0 1  k ≤ Γ (ρ¯ + 1) 1 − ρ/ρ¯ , (2.27) where  k ≤ (k −  k)/N . Then by Theorem 2.3, the algorithm terminates when ε(k) < ε, which implies that the auction system converges in K iteration steps such that ln(Γ (ρ¯ + 1)/ε) . K ≤ k˜ + N ln(ρ/( ¯ ρ¯ − ρ))

Proof of Theorem 2.4 Suppose that the system terminates at iteration step k in case ε vanishes; then we have b(k+1) = b(k) . Hence β (k+1) = β (k) and d (k+1) = d (k) , and x (k+1) = x (k) by (2.4). As shown in Lemma 2.2, xn(k+1) = dn(k+1) holds, which gives xn(k) = dn(k) . Since at iteration k, player n updates its best bid response, by applying Algorithm 2.1 with the update rule stated in Theorem 2.3, we have xl(k) = dl(k) for all l ∈ N /{n}, and n =   i ∈ arg maxl∈N βl(k) . Suppose βn(k) = λ. These analyses are concluded as below: xl(k) = dl(k) = xl(k+1) = dl(k+1) , ∀l ∈ N ;

(2.28)

βl(k) = βl(k+1) ≤ λ, ∀l ∈ N .

(2.29)

By dn(k+1) ≤ Dn (b(k) ), (2.8), and (2.28), we have   2 (k) min dm + Γc (b(k) ), αΦn (b(k) ), βn(k) ≥ 0, ρ¯ +   with Γc (b(k) ) ≡ Γ − l∈N dl(k) , m ≡ mn (b(k) ) and Φn (b(k) ) specified in (2.9). (k) Hence by dm + Γc (b(k) ) > 0, ρ2¯ βn(k) > 0 and α ∈ (0, 1), we have (k)

Φn (b )



> 0, in case dn(k+1) < Dn (b(k) ) . = 0, in case dn(k+1) = Dn (b(k) )

For notational simplicity, we consider 1 (k) ϕn ≡βn(k) − βm ¯ c (b(k) ). + ρ(dn(k) − xn(k) ) + ρΓ 2

(2.30)

Appendices

57

By (2.28), (2.29) and ρΓ ¯ c (b(k) ) ≥ 0, we have ϕn ≥ 0; then by (2.9b) and (2.30), we have  > 0, in case dn(k+1) < Dn (b(k) ) ϕn . (2.31) = 0, in case dn(k+1) = Dn (b(k) ) In the following, we will show that dn(k+1) < Dn (b(k) ) does not hold by proof by contradiction. Suppose that dn(k+1) < Dn (b(k) ). By (2.31), we have ϕn > 0. Then by the definition of ϕn and dn(k) = xn(k) in (2.28), we have βn(k)



(k) , in case Γc (b(k) ) > 0 ≥ βm . (k) , in case Γc (b(k) ) = 0 > βm

n , dn ) of player n such that dn = dn(k+1) + ε ≤ Consider another bid  bn = (β (k) x the allocation with respect to the bid profile Dn (b ) with ε > 0. Denote by  ). ( bn , b(k) −n • In case Γc (b(k) ) > 0. We can find an ε such that ε < Γc (b(k) ); then we can obtain (k) that  xn = dn and  xm = xm . By the same technique applied in Theorem 2.2, each player l ∈ N /{n, m} will not change its allocation. Thus, we have the analysis below: (k+1) (k) (k+1)  (k) ) − vn ( xn ) + βm (xm −  xm ) Δf n  f n (bn(k+1) , b(k) −n ) − f n (bn , b−n ) = vn (x n (k+1) (k+1) (k+1) (k+1)   ) − vn (dn ), by xn = dn , xn = dn ,  xm = xm =vn (dn

dn(k+1)

(k) • In case Γc (b(k) ) = 0, βn(k) > βm . By Assumption 2.1, dn > dn(k+1) and βn(k) = (k) n ; then we can find an ε such that β n > βm βn(k+1) , we have βn(k) > β ; then we can (k+1) obtain that  xn = dn and  xm = xm − ε. Thus, we have the analysis below: (k) (k+1) xn ) + βm (xm −  xm ) Δf n =vn (xn(k+1) ) − vn ( (k+1) (k) (k+1) (k+1) ) − vn (dn ) + βm ε, by xn = dn(k+1) ,  xn = dn ,  xm = xm −ε =vn (dn

 (k+1) (k) − dn ) + βm ε, by Assumption 2.1 βm 0 . in case dl(k) = 0

(2.32)

 By Γc (b(k) ) = 0, we have Γ − l∈N dl(k) ≤ 0; then by which together with (2.28)  and l∈N xl(k) ≤ Γ , we can obtain that 

dl(k) = Γ.

(2.33)

l∈N

Since we suppose that the system converges at iteration step k, we have b(k+1) = b ; then we have (2.32) and (2.33) also hold for b(k+1) . Thus, by Lemma 2.1, it is equivalent to the sufficient conditions (2.2) of the efficient NE. (k)

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32. R. Gomes, K. Sweeney, Bayesnash equilibria of the generalized second-price auction. Games Econ. Behav. 86, 421–437 (2014) 33. M. Feldman nad, H. Fu, N. Gravin, B. Lucier, Simultaneous auctions are (almost) efficient, in Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, (New York, 2013), pp. 201–210 34. R. Jain, Designing a strategic bipartite matching market, in IEEE 46th Annual Conference on Decision and Control, (New Orleans, LA, 2007), pp. 139–144 35. R. Jain, P. Varaiya, The combinatorial seller’s bid double auction: an asymptotically efficient market mechanism (Research report, Department of Electrical Engineering and Computer Science, UC Berkeley, 2006) 36. J. Sun, E. Modiano, Fair allocation of a wireless fading channel: an auction approach. Wirel. Commun. 143, 297–330 (2007) 37. Olivier Marce, Hoang-Hai Tran, Bruno Tuffin, Double-sided auctions applied to vertical handover for mobility management in wireless networks. J. Netw. Syst. Manag. 22(4), 658–681 (2014). October 38. Junni Zou, Xu Hongwan, Auction-based power allocation for multiuser two-way relaying networks. IEEE Trans. Wirel. Commun. 12(1), 31–39 (2013) 39. L. Cao, W. Xu, J. Lin, K. Niu, Z. He, An auction approach to resource allocation in OFDMbased cognitive radio networks, in 75th IEEE Vehicular Technology Conference (VTC Spring) (Yokohama), pp. 1–5, 6–9 May 2012 40. D. Wu, Y. Cai, M. Guizani, Auction-based relay power allocation: pareto optimality, fairness, and convergence. IEEE Trans. Commun. 62(7), 2249–2259 (2014) 41. L. Ausubel, P. Cramton, P. Milgrom, The clock-proxy auction: A practical combinatorial auction design, in Combinatorial Auctions, eds. by P. Cramton, Y. Shoham, R. Steinberg (MIT Press, 2006) 42. D.C. Parkes, L.H. Ungar, Iterative combinatorial auctions: Theory and practice, in 17th National Conference on Artificial Intelligence (AAAI-00) (2000), pp. 74–81 43. L.M. Ausubel, P. Milgrom, Ascending auctions with package bidding. Front. Theor. Econ. 1, 1–42 (2002) 44. P. Maillé, B. Tuffin, Multibid auctions for bandwidth allocation in communication networks, in 23rd AnnualJoint Conference of the IEEE Computer and Communications Societies, vol. 1, pp. 54–65, 7–11 March 2004 45. Patrick Maillé, Bruno Tuffin, Pricing the internet with multibid auctions. IEEE/ACM Trans. Netw. 14(5), 992–1004 (2006) 46. P. Jia, P. Caines, Analysis of quantized double auctions with application to competitive electricity markets. INFOR: Inf. Syst. Oper. Res. 48(4), 239–250 (2010) 47. P. Jia, P.E. Caines, Analysis of decentralized quantized auctions on cooperative networks. IEEE Trans. Autom. Control 58(2), 529–534 (2013) 48. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004) 49. G. Owen, Game Theory, 3rd edn. (Academic, New York, 2001) 50. J.S. Lee, B.K. Szymanski, Auctions as a dynamic pricing mechanism for e-services, Service Enterprise Integration (Springer, US, 2007) 51. M. Abe, K. Suzuki, M+ 1-st price auction using homomorphic encryption, Public Key Cryptography (Springer, Berlin, 2002)

Chapter 3

Double-Sided Auction Games for Efficient Resource Allocation

Abstract With an effort to allocate divisible resources among suppliers and consumers, a double-sided auction model is designed to decide strategies for individual players in this chapter. Under the auction mechanism with the VCG-type payment, the incentive compatibility holds, and the efficient bid profile is a Nash equilibrium (NE). Different from the single-sided auction in the previous chapter, there exists an infinite number of NEs in the underlying double-sided auction game, which brings difficulties for players to implement the efficient solution. To overcome this challenge, we formulate the double-sided auction game as a pair of single-sided auction games which are coupled via a joint potential quantity of the resource. A decentralized iteration procedure is then designed to achieve efficient solution, where a single player, a buyer or a seller, implements his best strategy with respect to a given potential quantity and a constraint on his bid strategy. Accordingly, the potential quantity is updated with respect to iteration steps as well. It is verified that the system converges to the efficient NE within finite iteration steps in the order of O(ln(1/ε)) with ε representing the termination criterion of the algorithm.

3.1 Introduction The allocation of resources among a group of entities, including suppliers and consumers, has emerged to be a hot issue in many fields, such as telecommunication networks, power electricity systems, and cloud computing, see [1–5] and references therein. Due to heavy burdens on the communication and computation via centralized methods, decentralized schemes have gradually become the mainstream for the evolution of resource allocation problems. In case the entities are autonomous and selfish, they don’t desire to share their private information and would like to pursue their own benefits, respectively, instead of the whole system objective. Hence, it is challenging to design decentralized methods to motivate the entities to achieve a global objective. To consider the system efficiency, Vickrey–Clarke–Groves (VCG) mechanisms have been widely applied to decentralized systems [6–9], due to the fact that (i) the incentive compatibility holds, i.e., a truth-telling bid is a dominant strategy, and (ii) there exists an efficient equilibrium. However, it may be impractical © Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_3

61

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to apply the VCG mechanism in the allocation of infinitesimally divisible resources since the exchange of the whole infinite-dimensional preference functions of players is costly and may violate the privacy of players. Particularly, auction is effective for decentralized resource allocations among rational players, e.g., [2, 4, 10–12], such that players submit their individual bids, respectively, to the system which determines the resource allocation with respect to the collected bid profile. How to design the auction mechanism that can achieve the efficient resource allocation, despite the strategic behaviors and unawareness of revealing the private information of players, has become a fundamental issue. In view of the advantages of the VCG and auction mechanisms, many researchers have worked on how to design VCG auctions by allowing players to submit lowdimensional bid strategies which are used to represent the partial valuation of individuals, e.g., [2, 7, 8, 13–15]. In [2, 7, 8, 13], the authors proposed an auction game with one-dimensional bids to construct a surrogate valuation function for each player; while in [14, 15], the authors proposed a progressive second price (PSP) auction mechanism with a two-dimensional bid composed of a per unit price and a maximum quantity of the demand. Specifically, in [13], it achieved a unique and efficient NE while the mechanisms traded off the dominant-strategy implementation for ease in implementation. The mechanisms proposed in [7, 8] also achieved the uniqueness and efficiency, but they required the surrogate valuation function to be twice continuously differentiable. Iosifidis and Koutsopoulos [2] formulated a double-sided auction game for resource allocations in autonomous networks. As an extension of PSP mechanism, [1] generalized the PSP mechanism to double-sided auctions in telecommunication networks with multi-link routes. There are other works to study the market efficiency and properties of the NE for auction games based on the PSP mechanism in different application fields, e.g., [1, 16–20] and references therein. In this chapter, we study the resource allocation among suppliers and consumers in a double-sided auction framework, wherein suppliers and consumers are regarded as sellers and buyers, respectively. In the auction mechanism, each player (a buyer or a seller) submits a bid to the auctioneer who then determines a resource allocation and a payment on the basis of players’ submitted bid profile. In our formulation, a two-dimensional message is employed as each buyer (seller) is asked to reveal a bid signal including a unit price that he wants to pay (charge) and a maximum amount of resources that he demands (supplies). With respect to these bid information, players’ resource allocations are determined by maximizing a revealed system profit and their payments following the VCG mechanisms. The incentive compatibility holds, and the efficient strategy that maximizes the social welfare is a Nash equilibrium (NE). As shown in the chapter, besides the efficient NE, there exists an infinite number of other NEs for the underlying double-sided auction game. This brings difficulties to implement the efficient NE in a decentralized way. In our work, we divide the double-sided auction into a pair of single-sided auction games, say buyer-sided and seller-sided auction games, which are coupled via a joint potential quantity. Thus, each player competes with his opponents in the corresponding single-sided auction under a given potential quantity which represents a common quantity of the resource traded in the system, i.e., a quantity of the resource allocated to all the buyers or

3.1 Introduction

63

generated by all the sellers. Based on this formulation, we propose a novel dynamic process to implement the efficient NE, under which players update their strategies in the corresponding single-sided auction game with respect to a given potential quantity, and then the potential quantity is updated with respect to the bid profile of players as well. In each of the single-sided auction games, each player implements his best response, with respect to a bid profile of his opponents, by maximizing his payoff. In this case, each player is greedy and asks for as many resource units as possible on condition that he wins the auction. Due to this, it may take too many steps for the system to reach a NE solution, which may not be efficient, or even the system may oscillate and hence can’t converge to any NE. Alternatively, in order to improve the convergence performance, we firstly introduce a pair of parameters, say upper bounds of the gradients of buyers’ marginal valuation and sellers’ marginal cost, which reveal some rough information related to the (infinite-dimensional) valuation or cost functions of players; then, assisted with the given parameters, at each iteration step, a buyer and a seller update their best responses under a constraint on the bid demand and bid supply, respectively. The underlying auction system is guaranteed to converge to the efficient NE by applying the proposed method. Furthermore, it is verified that the system can converge to the efficient NE within finite iteration steps in the order of O(ln(1/ε)), where ε is the termination parameter of the algorithm. In the literature, some research works have been dedicated to the implementation of the efficient NE of resource allocation auction games [15, 21–28]. The algorithms have been presented in [21–23] for single-sided auctions to implement the NE. In [24, 25], an efficient protocol was designed to conduct combinatorial auctions, where the auctioneer computes an optimal resource allocation at each stage, and then the losers at this stage are allowed to increase their bids, respectively. As stated in [24], the convergence rate is shown to be the order of O(B/ε), where B and ε represent the number of bundles and the termination criterion, respectively. In [15], the players are allowed to sequentially update their own bid strategies with respect to the submitted bid profile to maximize their own individual payoff. At each iteration step, the resource may be reallocated from lower price players to the player who updates his bid at this step. Due to the deficiency of the enough information related to players’ marginal valuations in a single bid, the iteration steps to the NE is the order of O(1/ε). Alternatively, an auction mechanism with multi-bid profiles was adopted by Tuffin in [16, 26, 27] such that the efficient solution can be implemented in a single step asymptotically as the dimension of players’ submitted bid strategy goes to infinity. Compared with the multistep implementation before the system can converge to the equilibrium by applying our method, the one-step implementation of the equilibrium proposed by Tuffin, is based upon the infinite-dimensional bid submitted by each player. However, in practice, the individuals may not be willing to share their full private information with others, and the transmission of the complete information may create heavy communication burdens. In [28], the authors presented a quantized auction algorithm under which the system converges to a quantized NE by applying two-dimensional bids. It was shown under certain conditions that the system may converge very fast or oscillates indefinitely. This work was extended to

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double-sided auction games in [29] and to multilevel cooperative network systems in [30], such that the auction system can converge to a quantized NE near to the efficient one. To ensure the convergence to the efficient NE with the method proposed in [29], the collection of inefficient NEs is eliminated in advance. This kind of elimination isn’t required for our proposed method. In summary, compared with the main research works discussed above, our work proposes a dynamic algorithm for the double-sided auction games such that the system converges to the efficient NE, based upon two-dimensional bids submitted by individual players, within an amount of iteration steps which is bounded from above with a certain value. The rest of the chapter is organized as follows. In Sect. 3.2, we design a doublesided auction to formulate the resource allocation problems among a group of suppliers and customers. In Sect. 3.3, we present a novel update algorithm by adopting which the underlying auction system converges to the efficient NE. Numerical simulations are studied in Sect. 3.4 to demonstrate the main results developed in the chapter. Moreover, in Sect. 3.4.2, the developed work in this chapter is applied in the field of the optimal scheduling in power electricity systems. Finally, we summarize our work in Sect. 3.5.

3.2 Resource Allocation Under Auction Mechanism 3.2.1 Resource Allocation Problems We consider a class of resource allocation problems where a collection of consumers N requires an amount of a divisible resource supplied by a collection of suppliers M . We denote by xn and ym the demand of consumer n and the supply of supplier m, respectively, such that xn ≥ 0,

ym ≥ 0.

(3.1)

 Denote by z ≡ x, y), with x ≡ (xn , n ∈ N ) and y = (ym , m ∈ M ), a resource allocation of all the individuals. z is called admissible if it satisfies (3.1) and the  equality constraint of n∈N xn = m∈M ym . The set of admissible resource allocations is denoted by Z . Each consumer n ∈ N has a valuation on his allocation xn , denoted by vn (xn ), and each supplier m ∈ M has a cost on his allocation ym , denoted by cm (ym ). Assumption 3.2 We consider the following assumptions: • The valuation function vn , for all n ∈ N , is differentiable, increasing, and strictly concave. • The cost function cm , for all m ∈ M , is differentiable, increasing, and strictly convex.

3.2 Resource Allocation Under Auction Mechanism

65

We define W (z) as the system social welfare such that W (z) 



vn (xn ) −

n∈N



cm (ym ).

(3.2)

m∈M

The objective of the system is to implement an efficient allocation to maximize W (z), i.e., to determine an allocation z ∗∗ , such that z ∗∗ = arg max W (z).

(3.3)

z∈Z

Under Assumption 3.2, by the Karush–Kuhn–Tucker (KKT) optimality conditions and [31], there exists a unique efficient solution z ∗∗ for the optimization problem, such that  = λ, if xn∗∗ > 0  ∗∗ vn (xn ) (3.4a) ≤ λ, otherwise  = λ, if ym∗∗ > 0  ∗∗ cm (ym ) (3.4b) ≥ λ, otherwise   xn∗∗ = ym∗∗ (3.4c) n∈N

m∈M

with a constant value λ > 0.

3.2.2 Double-Sided Auction Mechanism Design We study the resource allocation among resource suppliers and consumers in a double-sided auction framework wherein resource suppliers and consumers are regarded as sellers and buyers, respectively. Buyer n submits a two-dimensional bid, denoted by bn , to the system: bn ≡ (βn , dn ) ∈ Bn = [0, ∞) × [0, ∞), ∀n ∈ N , where βn and dn represent a buying price per unit that buyer n would like to pay, and the maximum units of resources he demands, respectively; while seller m submits a bid, denoted by sm , to the system: sm ≡ (αm , h m ) ∈ Sm = [0, ∞) × [0, ∞), ∀m ∈ M , where αm and h m represent a selling price per unit that seller m would like to sell and the maximum units of resources he would like to supply, respectively.

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The bid profile of buyers is b ≡ (bn , n ∈ N ), and the bid profile of sellers is s ≡ (sm , m ∈ M ). Denote by r ≡ (b, s) a bid profile of all the players in the auction game. Let r −n ≡ r/{bn } and r −m ≡ r/{sm } represent the bid profile of all the players except buyer n and seller m, respectively. Thus, considering a bid of buyer n, say bn ∈ Bn , a bid profile r can be represented as r = (bn , r −n ), and similarly considering a bid of seller m, sm ∈ Sm , r can be represented as r = (sm , r −m ) as well. We further define a function U (·) on an admissible allocation z with respect to a bid profile r as the following: U (z, r) 



βn xn −

n∈N



αm ym ,

m∈M

which can be interpreted as the total income of the system with respect to z = (x, y). The auctioneer assigns an optimal allocation z ∗ (r) ≡ (x ∗ (r), y∗ (r)), such that z ∗ (r) = arg max U (z, r).

(3.5)

z∈Z x≤d, y≤h

We will specify the transfer money for each player, denoted by τl , l ∈ N ∪ M , following the so-called VCG mechanism introduced in [14, 16], such that   τn (r)  U (z ∗ (r (n) ), r (n) ) − U (z ∗ (r), r) − βn xn∗ (r) ,   τm (r)  U (z ∗ (r (m) ), r (m) ) − U (z ∗ (r), r) + αm ym∗ (r) ,

(3.6) (3.7)

where r (n) ≡ ((βn , 0); r −n ) is the bid profile when buyer n is absent from the auction, and r (m) ≡ ((αm , 0); r −m ) is the bid profile when seller m is absent from the auction. That is, under the VCG mechanism, the money transfer made by an individual player is the externality he imposes on others through his participation, and is defined as the difference of the summation of other players’ revenue in the situation that he is absent from the auction and that where he joins in the auction, Note: The transfer money made by buyer n and seller m can be interpreted as the payment that buyer n should pay and the opportunity cost that caused to seller m, respectively, [1]. Also, the opposite of the opportunity cost of a seller can be considered as the income of this seller. The payoff functions of the players are then specified in the following: f n (r)  vn (xn∗ (r)) − τn (r), f m (r) 

−cm (ym∗ (r))

∀n ∈N ,

− τm (r),

∀ m ∈ M.

Definition 3.1 A bid profile r 0 is a Nash equilibrium (NE) of the auction game if the following holds:

3.2 Resource Allocation Under Auction Mechanism

f n (bn0 , r 0−n ) ≥ f n (bn , r 0−n ), f m (sm0 , r 0−m )

≥

67

∀ bn ∈ Bn ,

f m (sm , r 0−m ),

∀ sm ∈ Sm ,

for each n ∈ N and m ∈ M . In Lemma 3.2, we will analyze some properties of the NE. Before that, we firstly specify a set of the truth-telling bids of buyers and sellers, respectively, and show the so-called incentive compatibility of the underlying auction game in Lemma 3.1. We define a specific set of (truth-telling) bids of buyer n, denoted by Bnt , such that   Bnt  bn ≡ (βn , dn ) ∈ Bn , s.t. βn = vn (dn ) ,

(3.8)

i.e., Bnt is composed of all those bids, of buyer n, the bid price of each of which is exact the marginal valuation of his bid demand. Similarly, we define a specific set of (truth-telling) bids of seller m, denoted by Smt , such that   Smt  sm ≡ (αm , h m ) ∈ Sm , s.t. αm = cm (h m ) ,

(3.9)

i.e., Smt is composed of all those bids, of seller m, the bid price of each of which is exact the marginal cost of his bid supply. Lemma 3.1 (Incentive Compatibility) Under Assumption 3.2, there always exists at least one truth-telling bid that is weakly dominant for each buyer or seller. Proof Please see Appendix for the proof. Consider a specific truth-telling bid profile, denoted by r ∗ ≡ (b∗ , s∗ ), such that bn∗ ≡ (βn∗ , dn∗ ) = (vn (xn∗∗ ), xn∗∗ ), sm∗

≡

(αm∗ , h ∗m )

=

(cm (ym∗∗ ), ym∗∗ ),

∀n ∈N , ∀ m ∈ M,

(3.10a) (3.10b)

where z ∗∗ = (x ∗∗ , y∗∗ ) represents the efficient resource allocation as specified in (3.3). By applying the KKT conditions, we can verify that z ∗ (r ∗ ) = z ∗∗ , i.e., the resource allocation with respect to r ∗ is efficient. We call r ∗ the efficient bid profile, and it satisfies 

= λ, ≤ λ,  = λ, ym∗ = h ∗m , αm∗ ≥ λ,   ∗ dn = h ∗m

xn∗

=

n∈N

dn∗ ,

βn∗

m∈M

if dn∗ > 0 , ∀n ∈ N otherwise if h ∗m > 0 , ∀m ∈ M otherwise

(3.11a) (3.11b) (3.11c)

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3 Double-Sided Auction Games for Efficient Resource Allocation

Fig. 3.1 An illustration of NEs for the underlying double-sided auction game

Lemma 3.2 Consider a bid profile r 0 , such that βn0 = β, αm0 = α, ∀n ∈ N , m ∈ M , with β ≥ α;   dn0 = h 0m = q 0 ; n∈N

(3.12a) (3.12b)

m∈M

then r 0 is a NE for the underlying double-sided auction game under Assumption 3.2. Moreover we have r 0 = r ∗ , in case β = α, i.e., r ∗ is the efficient NE. Also for any NE r 0 specified in (3.12), q ∗ ≥ q 0 . Proof Lemma 3.2 can be verified by applying the similar technique adopted in [19, 20]. We give an example of a double-sided auction with multiple NEs. For the purpose of demonstration, we consider a simple case involving N = 2 buyers and M = 2 sellers. Suppose that the valuations of buyers share a common form of vn (xn ) = an (xn + 1)0.5 with a1 = 2, a2 = 1, and the costs of sellers share a common form of cm (ym ) = a N +m ym2 with a3 = 0.05, a4 = 0.03. Figure 3.1 displays the efficient NE and an inefficient one. By solving the optimization problem (3.3), we can obtain that the efficient solution z ∗∗ that maximizes the social welfare given in (3.2) is (7.8, 1.2, 3.4, 5.6). The corre

sponding efficient bid profile r ∗ is (0.34, 7.8), (0.34, 1.2), (0.34, 3.4), (0.34, 5.6) which satisfies (3.12) with β ∗ = α ∗ = 0.34. By Definition 3.1, we have that r ∗ is a NE. Hence r ∗ is the efficient NE.

Consider another bid profile r † = (0.35, 7), (0.35, 1), (0.3, 3), (0.3, 5) which also satisfies (3.12) with β † = 0.35 > α † = 0.3. By Definition 3.1, it can be verified that r † is a NE, and q ∗ = 9 > q † = 8 which is consistent with Lemma 3.2. In Sect. 3.3, we propose a novel method to implement the efficient NE in a decentralized way.

3.3 Decentralized Efficient NE Implementation

69

3.3 Decentralized Efficient NE Implementation In order to achieve the efficient NE in a decentralized way, we design an update process to implement the efficient NE in Algorithm 3.1. Basically, the double-sided auction game is formulated as a pair of single-sided auction games coupled via a joint potential quantity such that the best responses of the players in each single-sided auction and the potential quantity are updated under certain regulations, respectively. Here we outline the organization of this section here. In Sect. 3.3.1, we design the resource allocation and payment rules for the players of each single-sided auction game with respect to a given potential quantity; and in Sect. 3.3.2, we present a method to update the potential quantity with respect to a bid profile of all the players. In Sect. 3.3.3, we formalize the algorithm, show that the underlying double-sided auction game converges to the efficient NE, and further quantify the iteration steps to converge to the efficient NE.

3.3.1 Study of Single-Sided Auctions with a Given Potential Quantity We denote by Γ the joint potential quantity of the resource for the pair of coupled single-sided auction games, such that all buyers share Γ units of the resource in the buyer-sided auction, and all sellers supply Γ units of the resource in the seller-sided auction. We design the resource allocation and payment rules for the players of the singlesided auction games, respectively, with respect to their bid profile and a given potential quantity. The resource allocation and the payment of buyer n, denoted by xn∗ and τn , respectively, are specified with respect to a bid profile of all the buyers b and a potential quantity Γ as below: xn∗ (b, Γ )





= min dn , Γ −

τn (b, Γ ) =





+  , di

(3.13a)

i∈N n (b)

  βi xi∗ (b(n) , Γ ) − xi∗ (b, Γ ) ,

(3.13b)

i=n

where Nn (b)  {i ∈ N ; s.t. βi > βn } ∪ {i ∈ N ; s.t. βi = βn and i < n}, and [u]+ represents max{0, u}; b(n) ≡ ((βn , 0); b−n ) is the bid profile when buyer n is absent from the auction. The resource allocation and the opportunity cost of seller m denoted by ym∗ and τm , respectively, are specified with respect to a bid profile of all sellers s and a potential quantity Γ as below:

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3 Double-Sided Auction Games for Efficient Resource Allocation

ym∗ (s, Γ )







= min h m , Γ −

τm (s, Γ ) =



+  , hj

(3.14a)

j∈M m (s)

  (−α j ) y ∗j (s(m) ) − y ∗j (s) ,

(3.14b)

j=m

where Mm (s)  { j ∈ M ; s.t. α j < αm } ∪ { j ∈ M ; s.t. α j = αm and j < m}, and s(m) ≡ ((αm , 0); s−m ) represents the bid profile when seller n is absent. Note: The payment of a player, defined in (3.13b) and (3.14b) for a buyer and a seller, respectively, is the externality he imposes on his opponents in the corresponding single-sided auction through his participation, hence is the VCG-style money transfer made in the buyer-sided and seller-sided auction systems with a common potential quantity. Lemma 3.3 Given a bid profile r ≡ (b, s) and a potential quantity Γ , the admissible allocation of buyers x ∗ (b, Γ ) obtained by (3.13a) satisfies the following:  • xn∗ = dn for all n ∈ N , if i∈N di ≤ Γ ; otherwise, there only exists at most one buyer n ∈ N such that xn∗ ∈ (0, dn ); and that of sellers y∗ (s, Γ ) obtained by (3.14a) satisfies the following:  • ym∗ = h m for all m ∈ M , if m∈M h m ≤ Γ ; otherwise, there only exists at most one seller m ∈ M such that ym∗ ∈ (0, h m ). Proof It is straightforward to verify Lemma 3.3 by the allocation rules given in (3.13a) and (3.14a).

3.3.2 Updates of the Potential Quantity We present a method to update the potential quantity with respect to a given bid profile of buyers, a bid profile of sellers and a potential quantity Γ . First we define a pair of scalar-valued parameters, ρ¯ and σ¯ , related to the valuation functions of buyers and cost functions of sellers, such that 1   v (x + δ) − v (x) , i i i∈N x,δ δ 1   c (y + δ) − c (y) , σ¯ ≥ max sup j j∈M y,δ δ j

ρ¯ ≥ max sup

(3.15a) (3.15b)

i.e., ρ¯ denotes an upper bound of the Lipschitz constants of buyers’ marginal valuations, and σ¯ denotes an upper bound of the Lipschitz constants of sellers’ marginal costs.

3.3 Decentralized Efficient NE Implementation

71

Note: A pair of scale-valued parameters, ρ¯ and σ¯ specified in (3.15a) and (3.15b), reveals pretty rough information of the valuation and cost functions of individual players. We then define the matched prices pb (r, Γ ) and ps (r, Γ ) with respect to r, with r ≡ (b, s), and Γ , such that pb (r, Γ )  min {βi , s.t. xi > 0},

(3.16a)

ps (r, Γ )  max{α j , s.t. y j > 0},

(3.16b)

i∈N

j∈M

where xi ≡ xi∗ (b, Γ ) and y j ≡ y ∗j (s, Γ ) are the allocations to buyer i and seller j, with respect to (r, Γ ), implemented in (3.13a) and (3.14a), respectively. The updated potential quantity, denoted by Γ(r, Γ ), with respect to r and Γ , is specified as the following: Γ(r, Γ ) = Q +

pb − ps , ρ¯ + σ¯

(3.17)

  where Q ≡ Q(r, Γ )  min{ i∈N xi , j∈M y j }, with xi ≡ xi∗ (b, Γ ) and y j ≡ y ∗j (s, Γ ), and pb ≡ pb (r, Γ ) and ps ≡ ps (r, Γ ) are given in (3.16), respectively. Note: The potential quantity represents a common quantity of the resource, for each of the single-sided auction games, which is updated in (3.17) with respect to the submitted bid profile of players and the given pair of parameters (ρ, ¯ σ¯ ). The matched prices and the update of the potential quantity are illustrated in Fig. 3.2, respectively. As shown in Fig. 3.2, pb represents the lowest bid price of buyers with positive allocations, while ps represents the highest bid price of sellers with positive allocations. The potential quantity Γ is updated with respect to ρ¯ and σ¯ , an upper bound of the Lipschitz constants of the marginal valuation function and marginal cost function, respectively.

Fig. 3.2 Matched prices and the updated potential quantity with respect to bid profile

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3 Double-Sided Auction Games for Efficient Resource Allocation

3.3.3 Implementation of the Efficient NE Given a bid profile r ≡ (b, s), and a pair of the potential quantities (Γ, Γ), an upper demand constraint for buyer n and an upper supply constraint for seller m, denoted by Dn and Hm , respectively, are defined as follows:  + xi , Dn (b, Γ, Γ)  xn∗ (b, Γ ) + Γ −

(3.18a)

i∈N

 + Hm (s, Γ, Γ)  ym∗ (s, Γ ) + Γ − yj ,

(3.18b)

j∈M

with xi ≡ xi∗ (b, Γ ) and y j ≡ y ∗j (s, Γ ) specified in (3.13a) and (3.14a), respectively. The constrained sets of bid profiles of buyer n and seller m, denoted by Tnb (b, Γ, Γ) and Tns (s, Γ, Γ), respectively, are defined as below: bn ≡ (vn (dn ), dn ) ∈ Bn ; dn ≤ Dn }, Tnb (b, Γ, Γ)  { sm ≡ (cm ( h m ),  h m ) ∈ Sm ;  h m ≤ Hm }. Tms (s, Γ, Γ)  {

(3.19a) (3.19b)

Thus the best responses of buyer n and seller m, denoted by bn∗ (r, Γ, Γ) and sets of bid profiles Tnb (b, Γ, Γ)

sn∗ (r, Γ, Γ), respectively, subject to the constrained and Tns (s, Γ, Γ), are specified in the following: bn∗ (r, Γ, Γ) 

 arg max

bn ∈T nb (b,Γ,Γ)

sm∗ (r, Γ, Γ)  arg max

sm ∈T ms (s,Γ,Γ)



 f n (bn , b−n ) ,

(3.20a)

 f m (sm , s−m ) .

(3.20b)

We develop an update procedure in Algorithm 3.1 to implement the efficient NE. Before that, we firstly specify an assignment of a buyer and a seller, denoted by  n (b, x) and m (s, y), with respect to (b, x) and (s, y), respectively. By Lemma 3.3,  n (b, x) and m (s, y) can be defined in the following: ⎧ ⎪ ⎨ν, if ∃ν ∈ N , s.t. xν ∈ (0, dν )  n (b, x) = ν, else if ∃ν ∈ N , s.t. xν = 0 < dν ⎪ ⎩ ν, else if ν ∈ arg maxl∈N {βl } ⎧ ⎪ ⎨μ, if ∃μ ∈ M , s.t. yμ ∈ (0, h μ ) m (s, y) = μ, else if ∃μ ∈ M , s.t. yμ = 0 < h μ ⎪ ⎩ μ, else if μ ∈ arg minl∈M {αl }

(3.21a)

(3.21b)

Note: By Lemma 3.3, we obtain that there only exists at most one buyer ν such that n (b, x) is assigned to be ν; else, find a buyer ν xν ∈ (0, dν ). Hence, if there is such a ν,

3.3 Decentralized Efficient NE Implementation

73

satisfying xν = 0 < dν , i.e., with zero allocation and positive bid demand, and assign  n (b, x) = ν. However the buyer satisfying xν = 0 < dν may not be unique; then in this case, a buyer will be arbitrarily chosen from the buyers satisfying xν = 0 < dν . If there is no buyer satisfying the above two cases,  n (b, x) is assigned to be ν such that ν ∈ arg maxl∈N . Also, in this case, there may also exist multiple buyers, and then  n (b, x) will be arbitrarily assigned among those buyers with the highest bid price. Similarly, the assignment of m (s, y) follows that of  n (b, x). In Algorithm 3.1, denote by r k ≡ (bk , sk ) the bid profile at iteration step k. Also we consider z k ≡ z k (r k , Γ k ), i.e., z k represents the allocation of the players at step k, and Γ k denotes the potential quantity at k. We denote by buyer n and seller m the pair of players who are assigned to implement their best responses at iteration step k, and bnk+1 , smk+1 represent the best responses of buyer n and seller m, respectively. Algorithm 3.1 Implementation of the efficient NE Require: Given an initial bid profile r 0 s.t. maxi∈N {βi0 } > min j∈M {α 0j };   Set an initial potential quantity s.t. Γ 0 < i∈N di0 and Γ 0 < j∈M h 0j ; Set k = 0 and ε0 > ε; Ensure: A bid profile r ≡ (b, s); 1: while εk > ε do 2: Update z k ≡ (x k , yk ) w.r.t. r k and Γ k by (3.13a) and (3.14a); 3: Assign a single buyer n ≡  n (bk , x k ) and a single seller m ≡ m (sk , yk ) by (3.21), respectively; 4: if xnk = 0 < dnk or ymk = 0 < h km then 5: Update Γ k+1 = Γ k ; 6: else   7: Update Γ k+1 = Γ r k , Γ k by (3.17); 8: end if   9: Update bnk+1 and smk+1 w.r.t r k , Γ k , Γ k+1 by (3.20); 10: Update bk+1 = (bnk+1 , bk−n ) and sk+1 = (smk+1 , sk−m ); 11: Set εk+1 = |Γ k+1 − Γ k | + r k+1 − r k 1 ; 12: Set k = k + 1; 13: end while

Note on Algorithm 3.1: Following the allocation rule of (3.13a) and (3.14a) adopted in Algorithm 3.1, for those buyers with equal bid price, the one with lower index will be allocated firstly. Nevertheless, in order to improve the fairness of the proposed method, this allocation rule could be properly revised as follows. At iteration step k, before the allocation among players with respect to the submitted bid profile, the system randomly sets a unique number for each of the players; then, for those buyers with equal bid price, the one, with the (randomly set) lower number, will be allocated firstly. Thus due to the randomness of the allocation sequence of players set at each iteration step, the proposed allocation method ensures fairness to some extent.

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3 Double-Sided Auction Games for Efficient Resource Allocation

Furthermore, it is worth to note that the main results developed in our work, like the convergence and efficiency of the implemented Nash equilibrium, aren’t effected by the above new allocation rule.

3.3.4 Main Results of the Proposed Algorithm In this section, we study the convergence, computational complexity, and global optimality of the proposed algorithm in Theorems 3.1, 3.2 and 3.3, respectively. Before these, we first specify a property of the best response of players by applying Algorithm 3.1 in Lemma 3.4 below. Lemma 3.4 Under Assumption 3.2 and by applying Algorithm 3.1, suppose that buyer n and seller m are assigned to implement their best responses at iteration n , dn ) ∈ Bnt and sm ≡ step k, respectively; then there exist truth-telling bids,  bn ≡ (β t h m ) ∈ Sm , such that ( αm ,  bn , bk−n ) = min f n (bn , bk−n ), and xn = dn , f n (

(3.22a)

f m ( sm , sk−m ) = min f m (sm , sk−m ), and ym =  hm ,

(3.22b)

bn ∈B n sn ∈S m

with xn ≡ xn∗ ( bn , bk−n , Γ k+1 ) and ym ≡ ym∗ ( sn , sk−m , Γ k+1 ). Proof We can verify the conclusion of (3.22a) by showing that, given bk−n and Γ k+1 , bn under which the allocation equals for any bid bn , we can always find another bid  the bid demand, such that the payoff under  bn is larger than or equal to that under bn . By Algorithm 3.1, bnk+1 is the best response of buyer n; then suppose that  bn , bk−n , Γ k+1 ). By the concavity of vn (·) under dn = xnk+1 , and consider xn ≡ xn∗ ( k+1 k+1 n = vn (xnk+1 ) ≥ vn (dnk+1 ) = βnk+1 ; then Assumption 3.2 and xn ≤ dn , we have β k+1 k+1  xn = xn = dn , i.e.,  bn is a bid with a full alloby dn = xn and (3.13a), we have  k bn is the same as that of cation. Moreover, under b−n and Γ k+1 , the allocation of  bn , bk−n ) and bk+1 are the same. It implies that  bn bnk+1 ; then the payoffs subject to ( is also the best response. Similarly, we can verify that (3.22b) holds for seller m as well. Note: At each iteration step k, a pair of truth-telling bid strategies  bn and sn specified in (3.22) represents a best response of buyer n and that of seller m, respectively. As shown in Lemma 3.4, buyer n and seller m are fully allocated under the bid sn , sk−m ), respectively. profiles ( bn , bk−n ) and ( Theorem 3.1 Under Assumption 3.2, the auction system converges to an equilibrium by adopting Algorithm 3.1.

3.3 Decentralized Efficient NE Implementation

75

Proof By Algorithm 3.1, the termination criterion εk+1 is set to be εk+1 = |Γ k+1 − Γ k | + r k+1 − r k 1 . In order to verify Theorem 3.1, it needs to verify that both Γ k and r k converge. By Lemma 3.4, a best response of each player is truthful and with a full allocation. That is to say, at each iteration step k, r k ≡ (bnk , smk ; n ∈ N , m ∈ M ), s.t. bnk = (vn (dnk ), dnk ), smk = (cm (h km ), h km ), and the associated collection of allocations, z k ≡ (x k , yk ), is given as x k = d k , x k = hk . Hence, to show this theorem is equivalent to show that both Γ k and z k converge to an equilibrium, respectively. The details are specified in (I) and (II) below, respectively. (I) To show the convergence of Γ k . By adopting Algorithm 3.1 and (3.17), for all k ≥ 0, the potential quantity is updated as  Γ

k+1

=

Γ k , if xnk = 0 < dnk or ymk = 0 < h km Γk +

pbk − psk , ρ+ ¯ σ¯

otherwise

(3.23)

since Q k = Γ k for all k ≥ 0 by Appendix (Proof of Q k = Γ k in Theorem 3.1), and where pbk ≡ pb (r k , Γ k ) and psk ≡ ps (r k , Γ k ). As verified in Appendix (Proof of pbk ≥ psk for all k ≥ 0 in Theorem 3.1), pbk ≥ psk holds, by which, together with (3.23), we have Γ k+1 ≥ Γ k ,

(3.24)

which implies that Γ k increases under Algorithm 3.1 with respect to the iteration steps. Next, we would like to show that Γ k is bounded from above. Define a notion Γ ∗ such that   dn∗ = h ∗m , Γ∗ = n∈N

m∈M

i.e., Γ ∗ represents the total amount of the resource under the efficient bid profile r ∗ . Then we will show, by proof of contradiction, that for any iteration step k, Γ k ≤ Γ ∗ . k k Suppose that Γ k > Γ ∗ at some step k. By Appendix (Proof kof Q = Γ in Theok k k k rem 3.1), we have Q = Γ , with Q = n∈N xn = m∈M ym ; then by Lemma 3.1 and (3.11), there must exist a buyer i with xik > 0 such that βik < λ and a seller j with y kj > 0 such that α kj > λ, where λ is a positive constant specified in (3.11). This, together with (3.16), implies that pbk < psk , which is contradicted with pbk ≥ psk shown in Appendix (Proof of pbk ≥ psk for all k ≥ 0 in Theorem 3.1). Hence, Γ k ≤ Γ ∗ always holds for all k, by which together with (3.24), we obtain that Γ k converges to a certain value Γ † . (II) To show the convergence of z k .

76

3 Double-Sided Auction Games for Efficient Resource Allocation

As shown in Appendix (Proof of (3.25) in Theorem 3.1), under Assumption 3.2, the following holds by adopting Algorithm 3.1: ≥ y kj , xik+1 ≥ xik and y k+1 j

(3.25)

for all i ∈ N and all j ∈ M . By the allocation rule given in (3.13a) and (3.14a), we have xnk ≤ Γ k for all n ∈ N and ymk ≤ Γ k for all m ∈ M . It implies that like Γ k , z k ≡ (x k , yk ) is bounded from above as well. Thus, z k converges to a certain value z † . In conclusion, the auction system converges to an equilibrium by adopting Algorithm 3.1. Furthermore, in Theorem 3.2, we give an upper bound of the convergence iteration steps. We firstly define a pair of parameters in the following:   1     v (x + δ) − vi (x) , ρ ≤ min inf i∈N x,δ δ i    1     σ ≤ min inf c (y + δ) − c j (y) . j∈M y,δ δ j 

(3.26a) (3.26b)

Theorem 3.2 Under Algorithm 3.1, the system converges to a bid profile, denoted by r ε , within K (ε) iteration steps which is the order of O(ln(1/ε)). Proof As verified in Appendix (Proof of Theorem 3.2), we show that the auction system converges in K (ε) iterations steps, such that K (ε) ≤  k   1  max ln(N θ (ρ¯ + 1)/ε), ln(M (σ¯ + 1)/ε) , + ln(θ )

(3.27)

ρ+ ¯ σ¯ where  k ≤ max{N , M}, θ ≡ ρ+ , represents a maximum value of the bid ¯ σ¯ −ρ−σ demand or bid supply of all the players, and a is the smallest integer larger than or equal to a.

Theorem 3.3 Under Algorithm 3.1, the double-sided auction system converges to the efficient NE in case the termination parameter ε vanishes. Proof We suppose that, the system converges to a bid profile, denoted by r k+1 , at iteration step k. In Appendix (Verification of (3.28) and (3.29) in Theorem 3.3), we can verify that the bid profile at the converged step k satisfies the following properties:  βik+1  α k+1 j

= λ, if dik+1 > 0 , ≤ λ, otherwise

∀i ∈ N ,

(3.28)

= λ, if h k+1 >0 j , ≥ λ, otherwise

∀j ∈ M,

(3.29)

3.3 Decentralized Efficient NE Implementation

77

  and i∈N dik+1 = j∈M h k+1 = Γ k+1 ; then by (3.4), (3.10), (3.28), and (3.29), j the implemented bid profile under Algorithm 3.1 is exactly the efficient NE for the underlying auction system.

3.4 Numerical Simulations 3.4.1 Case Study I As a numerical example, we consider a resource allocation problem among 6 consumers and 4 suppliers. Suppose that all the consumers share a common form of the valuation function vn (xn ) = 2an (xn + 1)0.8 , such that an , with n = 1, . . . , 6, equals 2, 2.1, 2.2, 2.3, 2.4, and 1.9, respectively, and all the suppliers possess a common form of the cost function cm (ym ) = bm (ym + 1)1.2 , such that am , with m = 1, . . . , 4, is equal to 1.1, 1.5, 1.4, and 1.6, respectively. The objective is to implement an allocation among all the consumers and suppliers to maximize the social welfare as defined in (3.2). By solving the optimization problem (3.3), we can obtain that the efficient consumption is x ∗∗ = [3.012 4.122 5.463 7.068 8.986 2.104] and the efficient supply is y∗∗ = [19.873 3.427 5.250 2.205], with identical marginal valuation for consumers and marginal cost for suppliers, i.e., vn (xn∗∗ ) = cm (ym∗∗ ) = 2.424, for all n ∈ N and m ∈ M . The corresponding maximum social welfare Wmax equals 41.15. By adopting Algorithm 3.1, we firstly consider an initial bid profile of players b0 and s0 such that βi0 = vi (di0 ) and α 0j = cj (h 0j ), with di0 = h 0j = 1 for all i ∈ N and j ∈ M. Figure 3.3 displays the evolution of the potential quantity and the players’ allocation with respect to iteration steps under Algorithm 3.1, respectively. As illustrated, the potential quantity Γ and the allocation of players z increase with respect to the iteration steps, respectively. This is consistent with the results developed in Theorem 3.1. As observed, the system converges in about 300 iteration steps less than 500 which is an upper bound convergence steps specified in (3.27). z k converges to the efficient allocation, and Γ k converges to the total quantity associated with the efficient allocation. These simulation results are consistence with the analysis developed in Theorems 3.2 and 3.3. In Fig. 3.4, it displays the updates of individual players’ bid profile under Algorithm 3.1, while Fig. 3.5 demonstrates the corresponding evolutions of the system social welfare and the matched prices pb and ps subject to the updated bid profile of players. As observed, like the evolution of the potential quantity, the system social welfare increases with respect to the iteration steps as well. The matched price pb of the buyers become identical with the matched prices ps of the sellers when the dynamic process terminates. The convergence behavior of the pair of matched prices ( pbk , psk ) is consistent with the convergence of the potential quantity illustrated in Fig. 3.3.

78

3 Double-Sided Auction Games for Efficient Resource Allocation 20

35 30

buyer 1 buyer 2 buyer 3 buyer 4 buyer 5

Players' allocation

Potential quantity

15 25 20 15 10 5

10

buyer 6 seller 1 seller 2 seller 3 seller 4

5

efficient quantity potential quantity 0

100

200

0

300

0

100

200

300

Iteration

Iteration

Fig. 3.3 Updates of the potential quantity and players’ allocations under Algorithm 3.1 3.5 buyer 1 buyer 2 buyer 3 buyer 4 buyer 5

15

10

buyer 6 seller 1 seller 2 seller 3 seller 4

5

0

0

100

200

300

Bid price of players

Bid demand/supply of players

20

buyer 1 buyer 2 buyer 3 buyer 4 buyer 5

3

2.5

buyer 6 seller 1 seller 2 seller 3 seller 4

2

1.5

0

Iteration

100

200

300

Iteration

Fig. 3.4 Updates of players’ bid profile under Algorithm 3.1

In summary, as demonstrated in the simulation results studied above, we can state that the bid profile of players, including buyers and sellers, converges to the efficient bid profile of the underlying double-sided auction games, and the system reaches the maximum social welfare.

3.4.2 Case Study II Auction mechanisms have been widely adopted as effective ways to implement the divisible resource allocation problems in different fields, like power electricity markets [32–37], cloud markets [38], financial markets [39], wireless networks [40–42], etc.

3.4 Numerical Simulations

79

45

2.8

2.6

Matched prices

Social welfare

40

35

30

25

maximum social welfare social welfare

0

100

200

Iteration

2.4

2.2 efficient matched price buyer matched price seller matched price

2

300

1.8

0

100

200

300

Iteration

Fig. 3.5 Updates of the social welfare and the matched prices under Algorithm 3.1

More specifically, we will briefly introduce the application of our proposed double-sided auction mechanism in power electricity markets below. The auction has been adopted to schedule the electricity generation in conventional electricity wholesale markets in many regions, e.g., [32, 33]. More recently, with the increasing penetration of distributed generators, like wind turbines and PVs, the demand response is involved to improve the utilizations of these renewable energy resources. As a consequence, certain double-sided auction mechanisms have been designed to effectively allocate electricity among generators and load units, [34–37]. Note that in these works, the specifications, such as the ramping rates of conventional generators, the topology of power systems and the capacities of transmission/feeder lines, etc., aren’t considered. In order to apply our proposed method to the underlying electricity allocation problems, we will give the necessary notions and specifications in the following. Denote by N and M the sets of load units and generators, respectively. Consider the valuation of load n on his demand xn , denoted by vn (xn ), is in the form of vn (xn ) = ξn log(xn + 1), [36], with ξn denoting a positive-valued parameter. Suppose that the generation cost of generator m, denoted by cm (ym ), is in a quadratic form on its supply ym , say cm (ym ) = am ym2 + bm ym which has been widely adopted in the literature, e.g., [43, 44] and references therein. Denote by bn ≡ (βn , dn ) the bid strategy of load n, with βn ($/MWh) and dn (MWh) representing, respectively, the per unit bid price load n is willing to pay and the maximum bid demand of the electricity. Denote by sm ≡ (αm , h m ) the bid strategy of generator m, with αm ($/MWh) and h m (MWh) representing the per unit price that generator m is willing to sell and the maximum bid supply it can generate, respectively. Following (3.13a) and (3.13b), we can specify the allocations and the payments of each load n with respect to a bid profile b ≡ (bn ; n ∈ N ) and a potential

3 Double-Sided Auction Games for Efficient Resource Allocation 2.5

0.8

2

0.6 0.4

Load 1 Load 2 Load 3 Gen 1 Gen 2 Gen 3

0.2 0

0

100

200

300

400

500

Bid price ($/MWh)

0

efficient quantity potential quantity

0

100

200

300

400

500

Iteration

(a) Allocation

(b) Potential quantity

200 150 100 50

100

1 0.5

600

Load 1 Load 2 Load 3 Gen 1 Gen 2 Gen 3

0

1.5

Iteration

250

0

Potential quantity

1

200

300

400

Iteration

(c) Bid price

500

600

600

1

Bid demand/supply (MWh)

Allocation (MWh)

80

0.8 0.6 0.4

Load 1 Load 2 Load 3 Gen 1 Gen 2 Gen 3

0.2 0

0

100

200

300

400

500

600

Iteration

(d) Bid demand/supply

Fig. 3.6 Updates of individual strategies of generators and load units by applying Algorithm 3.1

quantity Γ respectively. Similarly, the allocation and the payment of each generator m are obtained following (3.14a) and (3.14b) respectively. Following the above specifications of vn (·) and cm (·), we have that Assumption 3.2 is satisfied; then by Theorems 3.1 and 3.3, we can conclude that the formulated auction system converges to the efficient solution by applying Algorithm 3.1. For the purpose of demonstration, we give a numerical simulation here. Consider there are three generators and three load units, and the parameters are given as ξ ≡ (ξ1 , ξ2 , ξ3 ) = [50, 55, 56], a = [30, 33, 35], and b = [0, 0, 0]. Figure 3.6 displays the updates of individual strategies and the converged solution by applying Algorithm 3.1. This is consistent with the developed results.

3.5 Conclusions In this chapter, we studied the efficient resource allocation problem among suppliers and consumers. We formulated this problem as a VCG-type double-sided auction game, which possesses the incentive compatibility and the existence of the efficient NE. To implement the efficient NE, we proposed a novel dynamic process for the

3.5 Conclusions

81

underlying auction game. More specifically, the double-sided auction game is decomposed into two single-sided ones coupled via a so-called potential quantity, which represents the total trade quantity of the resource in the system. An iterative update scheme is then designed to implement the efficient NE based on the mechanisms designed in the single-sided auctions. At each iteration step, the auctioneer assigns a specific buyer and a specific seller to allow them update their strategies, respectively, and updates the potential quantity. Assisted with the given extra system information, a certain constraint is set on the bid demand of the assigned buyer and bid supply of the assigned seller, respectively. Under the proposed method, the potential quantity and the social welfare increase with respect to iteration steps, respectively. And we show that the underlying auction system converges to the efficient NE at which the system reaches the maximal social welfare. Furthermore, we verify that the convergence iteration steps are within a certain value which is the order of O(ln(1/ε)).

Appendices Proof of Lemma 3.1 It is equivalent to show the incentive compatibility for any buyer n ∈ N by verifying that, for any bid bn ≡ (βn , dn ) ∈ Bn , there exists a truth-telling bid bnt ≡ (βnt , dnt ) ∈ Bnt , say βnt = vn (dnt ), such that f n (bnt , r −n ) ≥ f n (bn , r −n ),

(3.30)

for any given bid profile r −n of other players. Denote by xn and xnt the allocations of buyer n with respect to (bn , r −n ) and t (bn , r −n ), respectively. We will show (3.30) in the following: (i) In case βn < vn (dn ). Consider a bid bnt such that dnt = xn ≤ dn . By Assumption 3.2, we have βnt ≥ vn (dn ) > βn . Then by (3.5), we have xnt ≥ xn . Also by xnt ≤ dnt = xn , we have xnt = xn . Hence, by the specification of the payoff functions of players, we have f n (bnt , r −n ) = f n (bn , r −n ). (ii) In case βn > vn (dn ). Consider a bid bnt such that dnt = dn ; then βn > vn (dn ) = vn (dnt ) = βnt . By (3.5), we have xnt ≤ xn . When xnt = xn , the payoffs f n (bnt , r −n ) = f n (bn , r −n ). Next we consider the case xnt < xn . The following holds: f n (bn , r −n ) − f n (bnt , r −n ) = vn (xn ) − vn (xnt ) + τn (bnt , r −n ) − τn (bn , r −n ) ≤ βnt (xn − xnt ) + U (z) − βn xn − U (z t ) + βnt xnt ≤ βnt (xn − xnt ) − βnt (xn − xnt ) = 0.

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3 Double-Sided Auction Games for Efficient Resource Allocation

In conclusion, by (i, ii) given above, we obtain that (3.30) holds. Following the same technique applied for buyers, we can also verify the incentive compatibility for any seller m ∈ M .

Proof of Q k = Γ k in Theorem 3.1   By adopting Algorithm 3.1, we have Γ 0 < i∈N di0 and Γ 0 < j∈M h 0j ; by which   together with (3.13a) and (3.14a), we have Γ 0 = i∈N xi0 = j∈M y 0j ; then Q 0 ≡    0 0 0 min i∈N x i , j∈M y j = Γ . At iteration step k ≥ 1, a buyer n asks to implement his best response in the buyersided auction. Denote by bk = (bnk , bk−n ) the updated bid profile at step k − 1 where k k k bk−n = bk−1 −n , and bn = (βn , dn ) represents the best response of buyer n. Denote by k k k k x = x(b , Γ ), i.e., x represents the allocation  of buyers.  We will show Q k = Γ k by verifying that i∈N xik = Γ k and j∈M y kj = Γ k in (I) and (II), respectively.  (I). To show i∈N xik = Γ k by proof of contradiction below.  k allocated quantity, we have i∈N xik ≤ Γ k ; then Firstly since  Γ isk the total k k i ∈ N , i.e., all suppose that i∈N xi < Γ k , by (3.13a),  we khavexi = dik for all the buyers are fully allocated. Hence i∈N xi = i∈N di < Γ k . By (3.18a), we have

+  xik−1 . Dnk−1 ≡ Dn (bk−1 , Γ k−1 , Γ k ) = xnk−1 + Γ k −

(3.31)

i∈N

We will show that dnk < Dnk−1 in (i)–(ii) below.  (i) In case Γ k − i∈N xik−1 < 0. The following inequalities hold: Γk
k

xnk−1  i∈N



xik−1 ≤ xnk−1 + +



i∈N /{n}

dik ,

i∈N /{n}

dik

dik−1

= dnk +



dik ;

i∈N /{n}

then it implies that dnk < xnk−1 . Also by (3.31), we have Dnk−1 = xnk−1 in case k−1 k < 0; then we have dnk < Dnk−1 . Γ − i∈N x i k (ii) In case Γ − i∈N xik−1 ≥ 0.

Appendices

83

We have the following: 

Γk >



dik = dnk +

dik−1 ≥ dnk +

i∈N /{n}

i∈N

which implies that dnk < Γ k −

i∈N /{n}



xik−1 . Also by (3.31), we have

xik−1 = Γ k −



xik−1 ,

(3.32)

i∈N /{n}

i∈N

in case Γ k −

xik−1 ,

i∈N /{n}



Dnk−1 = xnk−1 + Γ k −





xik−1 ≥ 0; then we have dnk < Dnk−1 .  k k By (i)–(ii) above, we have dnk < Dnk−1 . Also by i∈N  di < Γ , we can define     another bid profile for player n, denoted by bn ≡ βn , dn , such that i∈N

   dik . dn = dnk + ε, with 0 < ε < min Dnk−1 − dnk , Γ k − i∈N

   By ε < Γ k − i∈N dik , we have i∈N dik + ε = i∈N /{n} dik + dn < Γ k ; then by     (3.13a), we have  xn ≡ xn ( bn , bk−n ), Γ k = dn , and xi ( bn , bk−n ), Γ k = dik = xik for all i ∈ N /{n}. k k Hencek by thek payment of buyer n specified in (3.13b), we have τn (b , Γ ) =  τn (bn , b−n ), Γ . Thus we have f n (bk ) − f n ( bn , bk−n ) = vn (xnk ) − vn ( xn ) k = vn (dn ) − vn (dn ) < 0, where the last inequality holds by Assumption 3.2, which is contradicted with the  the best response of buyer n. It implies that i∈N xik < Γ k consideration that bnk is cannot be held. Hence i∈N xik = Γ k .  (II). By the same  technique applied in the proof of i∈N xik = Γ k given in (I) above, we can show j∈M y kj = Γ k as well. In summary, by the specification of Q k given in (3.17), we have Q k  min

 i∈N

xik ,

 j∈M

 y kj = Γ k .

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3 Double-Sided Auction Games for Efficient Resource Allocation

Proof of pbk ≥ psk for all k ≥ 0 in Theorem 3.1 Under Algorithm 3.1, the matched prices pb0 and ps0 subject to the initial bid profile are supposed to satisfy the inequality of pb0 ≥ ps0 ; then to show pbk ≥ psk holds for all k ≥ 0, it is equivalent to verify that pbk+1 ≥ psk+1 holds in case pbk ≥ psk with k ≥ 0. By (3.23) and the assumed pbk ≥ psk for some k ≥ 0, it can be shown that Γ k+1 ≥ Γ k .

(3.33)

Also suppose that at iteration step k, in the buyer-sided auction, buyer n updates his best response, denoted by bnk+1 .  By Appendix 3.5, we can have Γ k = i∈N xik , by which together with (3.18a) and (3.33), the following holds:  +   xik Dnk ≡ Dn bk , Γ k , Γ k+1 = xnk + Γ k+1 − i∈N

= xnk + Γ k+1 − Γ k .

(3.34)

By the concavity of vn under Assumption 3.2 and (3.15a), we have   ¯ k+1 − Γ k ). vn (Dnk ) = vn xnk + Γ k+1 − Γ k ≥ vn (xnk ) − ρ(Γ

(3.35)

Moreover, by (3.20a), the best response of buyer n satisfies dnk+1 ≤ Dnk ; then by which together with Assumption 3.2, we have βnk+1 ≥ vn (Dnk ). In the following, we will verify that ¯ k+1 − Γ k ) pbk+1 ≥ pbk − ρ(Γ

(3.36)

in (I)–(III) below. (I) In case xnk ∈ (0, dnk ). By (3.13a) and (3.16a), we have pbk = βnk . Also by (3.13a) we have  d k , in case {βik > βnk } or {βik = βnk , i < n} xik = i , 0, otherwise for all i ∈ N /{n}, and by which, together with βnk+1 ≥ vn (Dnk ), we can obtain that those buyers i ∈ N /{n}, such that xik = 0 and βik ≥ vn (Dnk ), can increase their own allocations, respectively. Thus by (3.16a), we have pbk+1 ≥ vn (Dnk ). Hence by (3.35), we get that vn (Dnk ) > vn (dnk ) − ρ(Γ ¯ k+1 − Γ k ) = βnk − ρ(Γ ¯ k+1 − Γ k ) = pbk − ρ(Γ ¯ k+1 − Γ k ),

Appendices

85

which implies that the inequality (3.36) holds by pbk+1 ≥ vn (Dnk ), and where the inequality holds by xnk < dnk , and the 2nd equality holds by pbk = βnk . (II) In case xnk = 0 < dnk . By (3.23), we have Γ k+1 = Γ k . By (3.34) and Γ k+1 = Γ k , we have Dnk = 0 by which together with dnk+1 ≤ Dnk , we have xnk+1 = dnk+1 = 0 = xnk . Hence we have pbk+1 = pbk which implies (3.36). (III) Other cases besides those considered in (I) and (II). By adopting Algorithm 3.1, we have xik = dik for all i ∈ N , i.e., all buyers are fully allocated, and βnk = maxi∈N {βik }; then pbk = mini∈N {βik } ≤ βnk . By βnk+1 ≥ vn (Dnk ), if vn (Dnk ) > pbk , we have pbk+1 = pbk ; then pbk+1 ≥ pbk − ρ(Γ ¯ k+1 − Γ k ) by Γ k+1 ≥ Γ k ; else, we have pbk+1 ≥ vn (Dnk ). Hence by (3.35) and pbk ≤ βnk , the inequality (3.36) holds. By adopting the same technique to verify (3.36) for the matched price on the buyer-sided auction game in (I)–(III) above, we can show the following inequality property for the matched price on the seller-sided auction game as well: psk+1 ≤ psk + σ¯ (Γ k+1 − Γ k ). By (3.23), Γ k+1 = Γ k or Γ k+1 = Γ k + ysis:

pbk − psk ; ρ+ ¯ σ¯

(3.37)

then we have the following anal-

• In case Γ k+1 = Γ k . By (3.36), (3.37) and pbk ≥ psk , we have pbk+1 ≥ psk+1 . pk − pk

s b • In case Γ k+1 = Γ k + ρ+ . We have pbk − ρ(Γ ¯ k+1 − Γ k ) = psk + σ¯ (Γ k+1 − ¯ σ¯ Γ k ); then by (3.36) and (3.37), we have pbk+1 ≥ psk+1 .

In summary, we have pbk+1 ≥ psk+1 on the condition that pbk ≥ psk , i.e., the conclusion holds.

Proof of (3.25) in Theorem 3.1 We will verify (3.25) in (I)–(II) below. (I). To show xik+1 ≥ xik for all i ∈ N . n , dn ) with β n = vn (dn ) and bn = (β Suppose that xnk+1 < xnk . Consider another bid  x the allocation with respect to the bid profile  b ≡ ( bn , bk+1 dn = xnk , and denote by  −n ). k+1 k ∗ k k By (3.24), (3.13a) and xn < xn = xn (b , Γ ), buyer n does not grab other buyers’ allocation under both bk+1 and  b. Since buyer n is assigned to implement his best response, he must satisfy one of the three cases specified in (3.21a) in sequence. Then by (3.13a), we obtain that those buyers whose bid prices satisfy βik > βnk , for some i ∈ N /{n}, must be fully allocated. In other words, their allocation cannot be increased. Since for all i ∈ N /{n}, βik = βik+1 holds, we obtain that only the buyers whose bid prices satisfy βik+1 ≤ βnk can increase the allocation. That is,

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3 Double-Sided Auction Games for Efficient Resource Allocation



xik+1 = xik and  xi = xik , xik+1 ≥ xik and  xi ≥ xik ,

in case βik+1 > βnk ; otherwise

(3.38)

then by the definition of the payoff function f n , the following holds: f n (bk+1 ) − f n ( b) = vn (xnk+1 ) − vn (xnk ) −



βi (xik+1 −  xi )

i∈N /{n}


 k; then d k+1 − d k 1 = k+1 k k+1 k k+1 k − h 1 = |h m − h m |. |dn − dn |, and h First in the buyer-sided auction, by (3.19a) and (i&ii), it gives 

d k+1 − d k 1 = xnk + Γ k+1 −

xik − dnk

i∈N pbk

= Γ k+1 − Γ k =

− psk . ρ¯ + σ¯

(3.39)

By the definition of pb and ps specified in (3.16), and (ii), we have  pbk

= 

psk

=

pbk−1 , βik ,

if βik ≥ pbk−1 , otherwise

psk−1 , α kj ,

if α kj ≤ psk−1 , otherwise

where i, j represent the buyer and the seller who update strategies at iteration step k − 1, respectively. By (3.39), we have ⎧ ⎨=

d k+1 − d k 1 ⎩≥

βik −α kj , ρ+ ¯ σ¯ βik −α kj ρ+ ¯ σ¯

in case βik < pbk−1 , α kj > psk−1

(3.40)

otherwise β k −α k

i j which implies that d k+1 − d k 1 is bounded by ρ+ from below. ¯ σ¯ By (3.25) and (ii) in earlier part of this section, we have d k+1 ≥ d k for any k with k ≥ k; then by which together with (3.40), at step k, we adopt

d k+1 − d k 1 =

βik − α kj ρ¯ + σ¯

(3.41)

to specify an upper bound for the convergence steps of the algorithm. By the specifications of ρ and σ given in (3.26), and (3.39), we can obtain that pbk−1 − psk−1 , ρ¯ + σ¯ p k−1 − psk−1 , >σ b ρ¯ + σ¯

βik−1 − βik > ρ α kj − α k−1 j

by which, together with (3.39), we have

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3 Double-Sided Auction Games for Efficient Resource Allocation

p k−1 − p k−1 s b βik − α kj < βik−1 − α k−1 − ρ+σ j ρ¯ + σ¯

− ρ + σ d k − d k−1 1 . = βik−1 − α k−1 j

(3.42)

Also by (3.16), we have βik−1 ≥ pbk−1 and α k−1 ≤ psk−1 which and (3.39) imply j that ≥ pbk−1 − psk−1 = (ρ¯ + σ¯ ) d k − d k−1 1 . βik−1 − α k−1 j

(3.43)

Then by (3.41) and (3.42), we have

<

d k+1 − d k 1  βik − α kj 1 ρ¯ + σ¯

d k − d k−1 1

 − (ρ + σ ) d k − d k−1 1 ,

(3.44)

by which together with (3.43), we have

d k+1 − d k 1
0 for all i ∈N. Insummary, we obtain that the updated bid profile bk+1 at step k satisfies (3.28) and i∈N dik+1 = Γ k+1 , and the matched price of buyer is specified as pbk+1 = λ. = psk ; then by the same technique of the analysis Since Γ k+1 = Γ k , we have pbk  k+1 = Γ k+1 as well. of b , we can verify (3.29) and j∈M s k+1 j

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Chapter 4

Hierarchical Auction Games for Efficient Resource Allocation

Abstract This chapter studies a class of auction-based resource allocation games under a hierarchical structure, such that each supplier is assigned a certain amount of resource from a single provider and allocates it to its buyers with auction mechanisms. To implement the efficient allocations for the underlying hierarchical system, we first design an auction mechanism, for each local system composed of a supplier and its buyers, which inherits the advantages of the progressive second price (PSP) mechanism. By employing a dynamic algorithm, each local system converges to its own efficient Nash equilibrium (NE), at which the efficient resource allocation is achieved and the bidding prices of all the buyers in this local system are identical to each other. After the local systems reach their own equilibria, respectively, the resources assigned to suppliers are readjusted via a dynamic hierarchical algorithm with respect to the bidding prices associated with the implemented equilibria of local systems. By applying the proposed hierarchical process, the formulated hierarchical system can converge to the efficient allocation under certain mild conditions. The developed results in this work are demonstrated with simulations.

4.1 Introduction As one of the most effective distributed schemes for resource allocation and assignment problems [1], auction mechanisms have been widely applied in many research fields, e.g., communication networks, power electricity, and transportation systems, [2–4]. Auctions only rely on local information and self-interests of individual agents to obtain efficient solutions, and show superiorities on low signal interaction and complexity. In [5], the authors propose a progressive second price (PSP) mechanism on the infinitesimally divisible resource allocation problems. The PSP mechanism is shown to be incentively compatible and can gain the efficient Nash equilibrium (NE), and has been applied in the fields of power grid and communication networks, see [6–8] and the references therein. To implement the efficient NE, in [5], a dynamic process is designed such that players are allowed to sequentially update their own bidding strategies with respect to the submitted bid profile to maximize their individual payoffs, respectively. However © Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_4

93

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4 Hierarchical Auction Games for Efficient Resource Allocation

due to the deficiency of enough information related to players’ (infinite-dimensional) valuation in a single bid, the system social welfare may not monotonically increase and the system may take many iteration steps to reach the efficient NE. Hence as discussed in [9], it may take many iteration steps for the system to converge to the efficient solution. In [10, 11], the authors propose a multi-bid mechanism to deal with the slow convergence of the single-bid mechanisms and showed that the efficient solution can be reached in a single step asymptotically as the dimension of players’ submitted bid profile goes to infinity. Different versions of a so-called quantized auction algorithm are presented in [12–14], and it is shown that under certain conditions, the system converges to a quantized NE close to the efficient one in a convergence speed with respect to the quantized level, in case the collection of inefficient NEs has been eliminated from the quantized set in advance. In the above discussions, the resource is directly traded between resource owners and buyers. However under such a trading structure, the large-scale resource system may become increasingly complex. In many different fields, their resource allocation problems consist of multilayered components, e.g., [15, 16]. As a result, hierarchical structures have been extensively applied in many research fields, such as communication networks and electricity systems [15, 17–19]. Under this multilayered structured system, the models and allocation mechanisms for resource allocation between resource owners and buyers have to be redesigned. Concerning the PSP style mechanisms discussed above, many research works have been dedicated to solving resource allocation problems in hierarchical structures. Reference [20] structures a hierarchical model such that a resale of resource can be implemented among the same set of buyers after an auction; that is, the winners can resell the acquired resources to the losers. In [21], the authors design a hierarchical PSP auction mechanism for network resource allocation and verify that the efficient solution is still a NE under the proposed hierarchical structure. A two-level cooperative network system is studied in [17] and the authors show that the system can converge to a quantized NE close to the efficient one. In this hierarchical structure, vertices in higher level system are regarded as cooperative suppliers which are adjusted through a so-called consensus algorithm, and vertices in lower level systems are considered as noncooperative buyers, which are aligned to a supplier in the higher level system. In this chapter, we formulate a class of divisible resource allocation problems as auction games under a hierarchical structure, and design a novel dynamic process to implement the efficient NE. In the underlying hierarchical auction system, a single system provider owns resources and assigns them to a set of suppliers, which auction off their allocated resources to their buyers, respectively. That is to say, suppliers act as a medium between the resource provider and buyers. Each buyer is allowed to obtain resources from one supplier; then each supplier together with its buyers constitutes a local system. Moreover, buyers are assumed to be strategic players, who incline to pursue their own benefits, respectively; while suppliers are price takers who get resources from the provider in a centralized way. We extend an auctionbased updating algorithm proposed in [9] to implement the optimal solution in local systems. At each iteration step, the superior supplier (auctioneer of a local system) assigns a specific single buyer to update its best response by maximizing its individual

4.1 Introduction

95

payoff. Thus the allocation problem over the whole hierarchical system is regarded as a distributed set of auction games. In the above paragraph, we stated a dynamic algorithm to implement the efficient solution for each local system. Concerning the resource allocation for the whole hierarchical system we design a novel algorithm embedded with the dynamic algorithm in each local system, such that the resources assigned to suppliers are readjusted via a dynamic hierarchical algorithm with respect to the bidding prices associated with the implemented equilibria of local systems and a few of rough information related to individual buyers’ valuations. More specifically, at each iteration step in the hierarchical algorithm, the system appoints a specific supplier to increase its valuation by taking a certain quantity of resource from another supplier. Note that the dynamic process for the whole hierarchical resource allocation system does not require complete information of individual buyers’ valuations. The convergence and efficiency properties are achieved associated with the introduction of a pair of parameters related to buyers’ marginal valuation shapes. In this work, we show that by applying the proposed process for the formulated hierarchical system, the social welfare increases over the iteration steps and the system can converge to the efficient allocation solution under certain mild conditions. The developed results in this work are demonstrated with simulations. The chapter is organized as follows. In Sect. 4.2, we formulate a class of divisible resource allocation problems under a hierarchical structure. In Sect. 4.3, we design the auction mechanism for the resource allocation problems of local systems. The auction mechanism and the Local-PSP algorithm of local systems is described in Sect. 4.3.1. In Sect. 4.3.2, we present a novel algorithm over the whole hierarchical system embedded with the Local-PSP algorithm and show that by applying the proposed algorithm, the hierarchical system can converge to the efficient solution. Numerical simulation is given in Sect. 4.4 to demonstrate the results developed in Sects. 4.3.1 and 4.3.2. Finally, we summarize our results and extensions in Sect. 4.5.

4.2 Resource Allocation Problems Under a Hierarchical Structure Consider a certain amount of divisible resource to be allocated to a set of buyers within a hierarchical system structure. The proposed model for the hierarchical resource allocation problems is illustrated in Fig. 4.1. In Fig. 4.1, the system provider, denoted by A0 , owns a type of resource with amounts of Γ and distributes it to a set of suppliers, denoted by {Ai , i ∈ N }. Suppose that each buyer can only obtain resources from one supplier; then this system can be divided into separate local systems, which is denoted by Li , with i ∈ N . In each Li , suppose that the set of buyers under supplier i is Mi , and denote the buyers of supplier i by Ai j , with j ∈ Mi . In this regard, suppliers participate in the resource allocation problems as a medium between the system provider and retail buyers.

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4 Hierarchical Auction Games for Efficient Resource Allocation

Fig. 4.1 An illustration of the topology of the hierarchical system

In this chapter, we focus on the optimal operation of the hierarchical system and present a self-scheduling model for the optimal participation of suppliers and buyers. In the following, we first formulate the optimal model for the whole system. Denote by yi and xi j the allocation acquired by supplier Ai and the allocation acquired by buyer Ai j , respectively. We define a valuation function of buyer Ai j , subject to the allocation xi j , denoted by vi j (xi j ). More specifically, in the rest of the chapter, we consider the following assumption. Assumption 4.1 The valuation function vi j (·) is twice continuously differentiable, increasing, and strictly concave. Note that the total allocation among all the buyers in the whole hierarchical system cannot exceed the amounts of resources that the provider owns and the allocation of each buyer is nonnegative; then we consider the following constraints:  

xi j ≤ Γ, and xi j ≥ 0.

(4.1)

i∈N j∈M i

We call x ≡ (xi j ; i ∈ N , j ∈ Mi ) admissible, if x satisfies the constraint (4.1). The set of admissible allocations is denoted by X . The social welfare, denoted by S(x), with respect to an admissible allocation x, is defined as below.   vi j (xi j ). (4.2) S(x)  i∈N j∈M i

Remark As considered in [21], in this chapter, we suppose that suppliers do not have any intrinsic valuations for the resource. The objective of the allocation problem of the whole system is to assign the efficient (or socially optimal) allocation to all the buyers, by maximizing the social welfare defined in (4.2). Denote by x ∗∗ ≡ (xi∗∗j ; i ∈ N , j ∈ Mi ) the efficient allocation; then x ∗∗  argmax S(x). x∈X

(4.3)

4.2 Resource Allocation Problems Under a Hierarchical Structure

97

By applying the Lagrange multiplier method and under Assumption 4.1, see [22], the efficient solution for the optimization problem is unique, and is specified as below:    = λ, in case xi∗∗j > 0  ∗∗ xi∗∗j = Γ. (4.4) vi j (xi j ) , and ≤ λ, in case xi∗∗j = 0 i∈N j∈M i

In each local system Li , the resources of supplier Ai are allocated to its local buyers based on a certain allocation rule specified in Sect. 4.3.1 later. Thus the hierarchical system is regarded as a distributed collection of auction games, such that buyers in the same local system play with each other to compete for a certain amount of resources obtained from the provider. The advantage of this hierarchical auction mechanism is that, in each local auction game, buyers’ bids and allocation are determined only by available local information, including the amount of resources assigned to the supplier and the collected bids in this local system. This hierarchical model may be motivated by the fact that an individual buyer in a large system cannot obtain complete information concerning all the buyers and auctioneers, but can observe those behaviors in the same local system.

4.3 Implementation of Efficient NE Under Dynamic Process We propose a dynamic process in Algorithm 4.2, for the underlying resource allocation problems under the hierarchical structure. Before that, we give a brief description of this process in (i)–(iv) below, and illustrate the information flow related to the process (i)–(iv) in Fig. 4.2. (i) The system initializes y ≡ (yi , i ∈ N ); (ii) In each local system Li , the resources of supplier Ai are allocated among its buyers {Ai j ; j ∈ Mi } under the following procedure: (a) Initialize a bid profile of buyers; (b) Ai assigns a specific buyer Ai j that is allowed to update its bid by maximizing its payoff; (c) Step (b) continues until no buyers update its bid any longer; (iii) The system adjusts the allocation among suppliers w.r.t. the bidding price updated in (ii); (iv) Steps (ii)–(iii) continues until the allocation among suppliers does not update any longer. Here we briefly outline the organization of this section. In Sect. 4.3.1, we formulate the auction mechanism for the resource allocation in local systems with respect to given amounts of resources, and the efficient NE of a local system is implemented

98

4 Hierarchical Auction Games for Efficient Resource Allocation

Fig. 4.2 Hierarchical mechanism framework

by applying a dynamic process specified in Algorithm 4.1. In Sect. 4.3.2, we present a dynamic process in Algorithm 4.2 for the hierarchical resource allocation system to adjust the allocation assigned to suppliers with respect to the updated solution implemented in local systems. The convergence property and convergence performance for the proposed algorithm are discussed in Corollaries 4.1 and 4.2, respectively.

4.3.1 Auction-Based Mechanism for Local System Resource Allocation 4.3.1.1

Resource Allocation of Local Systems with a Given Amount of Resources

In local system Li , consider that the allocation of supplier Ai is yi ; then we have the quantity constraints as below: 

xi j ≤ yi ≤ Γ, and xi j ≥ 0.

(4.5)

j∈M i

The allocations of buyers in Li with respect to the given quantity yi is denoted by  x i (yi ) ≡ (xi j (yi ), j ∈ Mi and yi ≤ Γ is satisfied. x i (yi ) is admissible if it satisfies the constraint (4.5), and the set of admissible allocations is denoted by Xi (yi ), that is to say, Xi (yi ) 

⎧ ⎨ ⎩

x ≡ (xi j ; j ∈ Mi ); s.t.

 j∈M i

xi j ≤ yi , and xi j ≥ 0 for all j ∈ Mi

⎫ ⎬ ⎭

.

(4.6) In this section, for notational simplicity, we consider x i (yi ) ≡ x i and xi j (yi ) ≡ xi j .

4.3 Implementation of Efficient NE Under Dynamic Process

99

Define the local social welfare, denote by Si (x i ), subject to an admissible allocation x i , such that Si (x i ) 



vi j (xi j ).

j∈M i

The supplier Ai , which behaves as an auctioneer in local system Li , aims to assign an optimal allocation to its buyers with respect to the given allocation yi , denoted by x i (yi ) ≡ xij (yi ), j ∈ Mi , such that x i (yi ) = argmax Si (x i ),

(4.7)

x i ∈X i (yi )

with Xi (yi ) specified in (4.6). Under Assumption 4.1 and by applying the Lagrange multiplier method, the optimization solution for the local problem defined in (4.7) is unique such that ⎧ −1 ⎨= v  (λi ), ij  −1 xi j (yi ) ⎩≤ v  (λi ), ij

in case xij (yi ) > 0 in case xij (yi ) = 0

,

and



xij (yi ) = yi ,

(4.8)

j∈M i

where [vi j ]−1 represents an inverse operator of the function vi j (·). We define the optimal local social welfare of Li , denoted by vi (yi ), as the (virtual) valuation function of Ai , that is, vi (yi ) 

max

x i ∈X i (yi )



vi j (xi j ).

(4.9)

j∈M i

Lemma 4.1 Under Assumption 4.1, vi (·) is continuously differentiable, increasing and strictly concave. Proof The proof of Lemma 4.1 is given in appendix. By the analysis given above, the efficient allocation of local systems can be implemented in case the system has complete information and can directly allocate the resources among buyers. However, in practice, each individual buyer may not want to share its private information with suppliers and other buyers, and does not permit the direct control from suppliers. Alternatively, in the rest of this section, we formulate the underlying resource allocation problems as PSP auction-based games; then apply the dynamic algorithm proposed in [9] to implement the efficient NE for each local system with respect to a certain amount of resource assigned by the provider.

100

4.3.1.2

4 Hierarchical Auction Games for Efficient Resource Allocation

PSP Auction Mechanism for Local Systems with a Given Amount of Resources

Suppose that supplier Ai is assigned with yi units of resources by the provider. Buyer Ai j , with j ∈ Mi , submits a two-dimensional bid below to Ai , bi j = (βi j , di j ) ∈ Bi j = [0, ∞) × [0, yi ],

(4.10)

which specifies the per unit price βi j that buyer Ai j is willing to pay, and demand up to di j units of the resources. The bid profile is defined as bi ≡ (bi j , j ∈ Mi ), and bi,− j ≡ (bi1 , . . . , bi, j−1 , bi, j+1 , . . . , bi,Mi ) is the profile of buyer Ai j ’s opponents in local system Li . Note shared by all the buyers in Li . that bi is the public information  We call x i (bi , yi ) = xi j (bi , yi ), j ∈ Mi an admissible allocation with respect to bi and yi , if the following constraints hold: 

xi j = yi , and 0 ≤ xi j ≤ di j .

(4.11)

j∈M i

The set of admissible allocations with respect to bi and yi is denoted by Ai (bi , yi ). Subject to a bid profile bi and a given allocation yi , supplier Ai determines an allocation profile x i∗ (bi , yi ), such that x i∗ (bi , yi ) = argmax



x i ∈A i (bi ,yi ) j∈M i

βi j xi j .

(4.12)

Under the PSP mechanism introduced in [5, 8], the allocation and payment rules for the auction game can be defined as below: ⎧ ⎨

⎡



⎤+ ⎫ ⎬ dik ⎦ , ⎭

xi∗j (bi , yi ) = min di j , ⎣ yi − ⎩ k∈K i j (bi )    ∗ ∗ βik xik (bi,(− j) , yi ) − xik (bi , yi ) , τi j (bi , yi ) =

(4.13) (4.14)

k = j

where in (4.13), Ki j (bi )  {k ∈ Mi ; s.t. βik > βi j } ∪ {k ∈ Mi ; s.t. βik = βi j and k < j} and [x]+ ≡ max{0, x}; while in (4.14), τi j (bi , yi ) denotes the payment (or transfer money) of buyer Ai j and bi,(− j) = ((βi j , 0), bi,− j ) represents the bid profile when buyer Ai j is absent from the local auction. The payoff of buyer Ai j , denoted by f i j (bi ), subject to the bid profile bi , is specified as   f i j (bi , yi )  vi j xi∗j (bi , yi ) − τi j (bi , yi ).

(4.15)

4.3 Implementation of Efficient NE Under Dynamic Process

101

Definition 4.1 A collection of bid profiles bi0 is a Nash equilibrium (NE) for the auction game of local system Li , if the following holds: 0 0 f i j (bi0j , bi,− j , yi ) ≥ f i j (bi j , bi,− j , yi ),

for all bi j ∈ Bi j and j ∈ Mi , i.e., bi0j represents a best response of buyer Ai j with 0 respect to bi,− j and a given resource quantity yi . 4.3.1.3

Implementation of Efficient NE for Local Systems

At each iteration step of this dynamic process, an appointed buyer updates its best response with respect to the bid profile of buyers in this local system given in last iteration step. To supply extra information related to the buyers’ marginal valuation shapes, a pair of scalar valued parameters (ρmax,i , ρmin,i ) with i ∈ N , say an upper bound and a lower bound of the gradients of buyers’ marginal valuation, are given as below:    ρmax,i ≥ max sup vil (xil ) , l∈M i    0 < ρmin,i ≤ min inf vil (xil ) .

(4.16a) (4.16b)

l∈M i

We will specify a constrained set in (4.19) for buyer’s bid demands; before that, we first define a few of notions, as below. For buyer Ai j , we define notion Di j (bi ) with respect to bid profile bi , such that  Di j (bi ; yi )  xi j + min dn + yic (bi ; yi ), αΦi j (bi ),

2 ρmax,i

 βi j ,

(4.17)

 +  with x i = (xi j , j ∈ Mi ) ≡ x i∗ (bi ), α ∈ (0, 1), yic (bi ; yi ) ≡ yi − l∈M i dil , and where n ≡ ni j (bi ) and Φi j (bi ) are specified in (4.18a) and (4.18b), respectively. ⎧ ⎨

⎫ ⎬ ni j (bi )  max n ∈ Mi /{ j}; s.t. βin = min {βil } ; ⎩ ⎭ l∈M i /{ j}

(4.18a)

xil >0

Φi j (bi ; yi ) 

1 ρmax,i

 + 1 ∗ βi j − βin + ρmin,i (di j − xi j ) + ρmax,i · xic (bi ) . 2 (4.18b)

We can define a constrained set of bid profiles Bi j (bi ; yi ) for buyer Ai j with respect to bid profile bi , in the following:   bi j ≡ (vi j (di j ), di j ); s.t. 0 ≤ di j ≤ Di j (bi ) . Bi j (bi ; yi )  

(4.19)

102

4 Hierarchical Auction Games for Efficient Resource Allocation

We introduce the following dynamic algorithm: Algorithm 4.1 Dynamic algorithm for local auction system Li . Require: A quantity of resource yi ; An initial bid profile bi(0) for buyers in local system Li ; k ← 0, j ← 0; 1: while true do   (k) (k) 2: Implement x i bi ; yi following (4.13); 3:

Set a buyer j ∈ Mi w.r.t. bi(k) and yi as follows:   (k) (k) • If there exists m ∈ Mi such that xim ∈ 0, dim ; then j ← m; (k) (k) Else, if there exists m ∈ Mi such that xim = 0 < dim ; then j ← m;  (k) Else, j ← m ∈ arg maxl∈M i βil ;

• • 4:

  Implement best response bi(k+1) of buyer Ai j w.r.t bi(k) ; yi , such that j bi(k+1) := j

 argmax  (k)

bi j ∈B i j bi ;yi

5: 6: 7: 8: 9: 10: 11: 12:

(k+1)

bi



    (k) f i j bi j , bi,− , with Bi j bi(k) ; yi specified in (4.19); j

  (k+1) (k) ← bi j , bi,− j ;

if bi(k+1) = bi(k) then Break; else k ← k + 1; end if end while (k) return {bi }n∈N

Theorem 4.1 Under Assumption 4.1 and by applying Algorithm 4.1 with a given allocation yi , the local system Li converges to the efficient NE bi with an identical price pi (yi ), such that  βij (yi )

= pi (yi ), in case di j > 0 . ≤ pi (yi ), in case di j = 0

(4.20)

We call pi (yi ) the uniform bid price of local system Li with respect to yi and it reflects the marginal valuation of supplier Ai , such that pi (yi ) = vi (yi ). Proof The proof of Theorem 4.1 is given in appendix.

4.3 Implementation of Efficient NE Under Dynamic Process

103

4.3.2 Dynamic Resource Allocation of Hierarchical System We design a dynamic process in Algorithm 4.2 to implement the efficient allocation for the hierarchical resource system. Before that we give some discussions on the motivation of the proposed algorithm in the following. At the efficient allocation of the hierarchical system, all the suppliers share an identical marginal valuation obtained by (4.4). Hence, due to the concavity of suppliers’ valuations shown in Lemma 4.1, the system may tend to the efficient solution if the resources are reallocated from the suppliers with lower marginal valuations to others with higher marginal valuations. Following the above discussions, in Algorithm 4.2, we choose a pair of suppliers (Ai , Am ), such that Ai is a supplier with the maximum marginal valuation and Am is a supplier with the minimum marginal valuation, and a certain amount of resources is reassigned from Am to Ai . However, if too many resources are transferred from Am to Ai , it may result in a consequence that the marginal valuation of Ai becomes larger than that of Am . To avoid this oscillation behavior, we set a specific amount of resources in (4.22) and (4.23) which would be transferred to Ai from Am . However, the valuations of suppliers are determined by the private valuations of their connected buyers, and the system only has very limited information related to buyers’ valuations, say ρmax and ρmin . Then based on ρmax and ρmin , we design this amount to guarantee that the reallocation behavior is feasible and the difference between the marginal valuations of the assigned pair of suppliers Am and Ai decreases. We first obtain a pair of scalar valued parameters over the whole hierarchical system below. ρmax ≡ max {ρmax, l },

(4.21a)

ρmin ≡ min {ρmin, l }.

(4.21b)

l∈N

l∈N

Suppose suppliers Ai is assigned to increase its allocation, then we define notion E i ( y) with respect to a given collection allocations among all suppliers y ≡ (yi , i ∈ N ), such that   2 E i ( y)  min ym , αΦi ( y), pi , ρmax

(4.22)

with y, α ∈ (0, 1), pi = pi (yi ), and where m ≡ mi ( y) and Φi ( y) are specified in (4.23a) and (4.23b) below, respectively: ⎧ ⎨

⎫ ⎬

mi ( y)  max m ∈ N /{i}; s.t. pm = min { pl } ; ⎩ ⎭ l∈N /{i} yl >0

(4.23a)

104

4 Hierarchical Auction Games for Efficient Resource Allocation

Φi ( y) 

1 ρmax

[ pi (yi ) − pm (ym )]+ .

(4.23b)

In Algorithm 4.2 below, we are ready to formalize the dynamic process for the whole hierarchical system which is briefly described in (i)–(iv) earlier. Algorithm 4.2 Dynamic resource allocation process of hierarchical system. Require: A quantity Γ ; An initial allocation y(0) ; An εstop to terminate iterations; Implement an uniform bid price p(0) w.r.t. y(0) under dynamic Algorithm 4.1 for all local systems; k ← 0, i ← 0, m ← 0, ε > εstop ; 1: while ε > εstop do  !  (k) (k) (k) pm (ym ) ; 2: Set a supplier i ∈ N w.r.t. y , such that i := min m ∈ argmax m∈N , ym >0

3: Set a supplier m ∈ N w.r.t. y(k) specified in (4.23a); (k+1) (k) (k+1) (k) := yi + E i ( yik ) and ym := ym − E i ( yik ); 4: Determine y(k+1) by setting yi (k+1) (k+1) 5: Determine p w.r.t. y under Algorithm 4.1 for all local systems; 6: Update ε := y(k+1) − y(k) 1 ; 7: k ← k + 1; 8: end while

In Corollary 4.1, we show that the hierarchical system can converge to the efficient allocation by applying the proposed dynamic method proposed in Algorithm 4.2. In Corollary 4.2, the convergence rate is discussed. We call y ≡ (yi , i ∈ N ) an admissible resource allocation among suppliers {Ai ; i ∈ N }, if 

yi = Γ, and yi ≥ 0, for all i ∈ N .

(4.24)

i∈N

The set of admissible allocations is denoted by Y . Corollary 4.1 (Convergence and efficiency under proposed method) Under Assumption 4.1 and by applying Algorithm 4.2, the hierarchical system can converge to the unique efficient NE. Proof The proof is shown Proof of Corollary 4.1 in appendix. Corollary 4.2 Under Assumption 4.1, the system by applying Algorithm 4.2 conlg(εstop /(N Γ )) with Q ∈ [1 − verges within K iteration steps, such that K ≤ lg(Q) αρmin , 1) and εstop representing a given termination parameter. ρmax Proof The proof is shown in Appendix.

4.4 Numerical Example

105

4.4 Numerical Example We study a numerical example considering a competitive market with a quantity Γ = 23 of a resource, composed of three suppliers, and each of the suppliers has a collection of buyers with size of the local systems equal to 3, 2, and 2, respectively. We further consider buyers’ valuations specified as vi j (xi j ) = 2ai j (xi j + 1)0.5 with parameters of a11 = 2, a12 = 2.15, a13 = 2.24, a21 = 2, a22 = 2.2, a31 = 1.9, a32 = 2.12. It implies that vi j (·) satisfies Assumption 4.1. The evolution of the hierarchical system under Algorithm 4.2 is illustrated in Figs. 4.3 and 4.4. More specifically, Fig. 4.3 displays the updates of suppliers’ allocation assigned by the provider and buyers’ bidding demands in a local system over the iteration steps under Algorithm 4.2, while Fig. 4.4 displays the updates of uniform bidding prices of local systems over the iteration steps under Algorithm 4.2 and the evolution of players’ bidding prices of a specific local system at a specific iteration step.

Fig. 4.3 Updates of suppliers’ allocation assigned by the provider and buyers’ bidding demands in a local system over the iteration steps under Algorithm 4.2

106

4 Hierarchical Auction Games for Efficient Resource Allocation

Fig. 4.4 Updates of uniform bidding prices of local systems over the iteration steps under Algorithm 4.2 and evolution of players’ bidding prices of a specific local system at a specific iteration step

Fig. 4.5 Updates of hierarchical system social welfare under and players’s bidding prices in a local system Algorithm 4.2

4.4 Numerical Example

107

In summary, as demonstrated with the numerical simulation, the hierarchical system converges to the efficient allocations among all the suppliers and all the buyers under Algorithm 4.2. After the dynamic process terminates, the social welfare reaches the maximal social welfare Smax of 57.473, and all the suppliers’ allocations converge to the efficient allocation and their uniform bid prices become identical. These results are consistent with Corollary 4.1. Figure 4.5 illustrates the updates of the system social welfare under Algorithm 4.2. The social welfare increases over the iteration steps and the system converges at about 45 iteration steps. It is consistent with Corollary 4.1.

4.5 Conclusions and Ongoing Research Works In this chapter, we introduce a hierarchical divisible resource allocation model in large-scale and complex systems, such that suppliers play as a medium between the provider and individual buyers. In a local system, which is composed of a supplier and its buyers, we design a PSP style auction mechanism where buyers behave as strategic players to pursue their individual best benefits. A dynamic process is also proposed to implement the efficient NE, of the local system, at which the efficient allocation is achieved and the bidding prices of buyers are identical with each other. In the high level, the provider assigns a certain amount of resource to suppliers, and this allocation is adjusted via a dynamic hierarchical algorithm with respect to some rough information of individual valuations and the implemented uniform bidding price from the local systems. The proposed update process guarantees the monotonic increasing of the system social welfare, and further the convergence to the efficient NE. The suppliers studied in this chapter are considered to be cooperative with each other. As ongoing researches, we would like to study the hierarchical allocation mechanism in more complex situation such that suppliers may have their own valuation function and work as strategic players when they grasp resources from the provider.

Appendices Proof of Lemma 4.1 Firstly the efficient solution for the optimization problem appeared in (4.7) can be specified as below:  vi j (xij )

= λi , in case xij > 0 ; ≤ λi , in case xij = 0

(4.25a)

108

4 Hierarchical Auction Games for Efficient Resource Allocation



xij = yi ;

(4.25b)

j∈M i

where λi is a nonnegative constant. Under Assumption 4.1, for the concerning the differentiability and increasing property of the valuation function v(·), the function specified in (4.9) can be constrained as a convex optimization problem; then there exists a unique solution for  (4.9) subject to an allocation yi of supplier A i , denoted by {x i j (yi ); j ∈ Mi }, that is  to say, {xi j (yi ); j ∈ Mi } = argmax j∈M i vi (yi ). (xi j ; j∈M i )∈X i (yi )

We will show the continuously differentiable, strictly increasing and concave properties of vi (yi ), respectively, as below. (I) To show the strictly increasing property of vi (yi ). Firstly we will show in the below, for each local system Li , that xij (yi† ) ≤ xij (yi‡ ) for any pair of distinct values (yi† , yi‡ ), such that yi† < yi‡ . Following the same discussion on (4.4), we have, by applying the Lagrange multiplier method and under Assumption 4.1, the efficient solution for Li with any provided amount of resource yi , denoted by xij (yi ), is unique, such that  vi j (xij (yi ))

= λi , in case xij (yi ) > 0 , ≤ λi , in case xij (yi ) = 0

with



xi∗∗j = yi .

(4.26)

j∈M i

Also under Assumption 4.1, vi j (xi j ) is positive and strictly decreases w.r.t. xi j ; then by (4.26), we can obtain that xij (yi† ) ≤ xij (yi‡ ), in case yi† < yi‡ , for all j ∈ Mi . Moreover there exists at lease a buyer Ais such that xis (yi† ) < xis (yi‡ ), since   ‡   † xi j (yi ) = yi† < yi‡ = xi j (yi ); then by these together with Assumpj∈M i

j∈M i

tion 4.1, we have vi (yi† ) =



     vi j xij (yi† ) < vi j xij (yi‡ ) = vi (yi‡ ), in case yi† < yi‡ .

j∈M i

j∈M i

(II) To show the strictly concave property of vi (yi ). By calculating the second derivative of vi (yi ), we have vi (yi ) =

⎧  ⎨ j∈Mi

"

v (x  ) ⎩ ij ij

# d xij 2 dyi

by (4.25), we can obtain that d 2 xij dyi2

= λi

 j∈M i

d 2 xij dyi2

d xij dyi

+ vi j (xij )

⎫ d 2 xij ⎬ dyi2 ⎭

, with vij ≡ vij (yi ), (4.27)

= 0 in case vi j (xij ) = λi ; then

, by which together with (4.27), we have

 j∈M i

vi j (xij )

Appendices

109

vi (yi ) =

⎧  ⎨ j∈Mi

v (x  ) ⎩ ij ij

"

# d xij 2 dyi

⎫  d 2 xij ⎬

+ λi

j∈Mi

dyi2 ⎭

=

⎧  ⎨ j∈Mi

v (x  ) ⎩ ij ij

"

# ⎫ d xij 2 ⎬ dyi

⎭

(4.28) where the last equality holds because

 j∈M i

d xij dyi

= 1.

In summary under Assumption 4.1 and by the fact there exists a buyer Ai j with d xij dyi

> 0, we have vi (xi ) < 0, i.e., vi (yi ) is strictly concave.

Proof of Theorem 4.1 Following Algorithm 4.1, suppose buyer Ai j in local system Li is assigned to implement its best response at step k + 1; then by (4.13), we can have xil(k) = dil(k) with l ∈ Mi /{ j}, i.e., all the other buyers except buyer Ai j , are fully allocated. We specify an individual buyer n ∈ Mi /{ j} as specified in (4.18a) with respect to bi(k) . (k) (k) By xi(k+1) ≤ di(k+1) ≤ Di(k) j j j (bi ; yi ) with Di j (bi ; yi ) specified in (4.17), we have (k+1) (k) (k) (k) (k) (k) xi j ≤ xi j + dn + yic (bi ; yi ); then by (4.18a), xin = din and the allocation rule (4.13), we can obtain that xil(k+1) = xil(k) for all l ∈ Mi /{ j, n}. Define δil(k+1)  xil(k+1) − xil(k) for all l ∈ Mi ; then we can verify that ⎡ (k+1) δi(k+1) + δin = yic (bi(k) ; yi ) ≡ ⎣ yi − j



⎤+ dil(k) ⎦ ≥ 0.

(4.29)

l∈M i

Define ΔSi  Si (x i(k+1) ) − Si (x i(k) ), as the change of the social welfare of Li from step k to step k + 1. As supposed, it is buyer Ai j to implement its best response ≥ xi(k) at step k + 1; then we can verify that xi(k+1) j j and consider the following cases below: = xi(k) • In case xi(k+1) j j . All the buyers’ allocation remains unchanged. It makes (k) (k+1) ); then ΔS = 0. Si (x i ) = Si (x i (k) (k+1) • In case xi(k+1) > x ≤ 0. There j i j . As supposed in Algorithm 4.1, we have δin are two cases discussed below: (k+1) * In case δin = 0. We have ΔSi = vi j (xi(k+1) ) − vi j (xi(k) j j ); then by the strictly increasing property of vi j under Assumption 4.1, we have ΔSi > 0. (k+1) < 0. We have * In case δin

  (k) (k+1) ) − vi j (xi(k) ) ΔSi = vi j (xi(k+1) j j ) − vin (x in ) − vin (x in

(4.30)

,

110

4 Hierarchical Auction Games for Efficient Resource Allocation

$ =

xi(k+1) j xi(k) j

vi j (xi j )d xi j −

$

(k) xin (k+1) xin

 vin (xin )d xin ,

(4.31)

(k+1) (k) ≤ xin . Moreover, under Assumption 4.1, the following inequalities with xin hold:

% & $ x+Δx % & 1 1 Δx vi j (x) − ρmax,i Δx ≤ vi (x)d x ≤ Δx vi j (x + Δx) + ρmax,i Δx , 2 2 x

(4.32) where ρmax,i and ρmin,i are defined in (4.16), respectively; then we can obtain that ΔSi

% & % & 1 1 (k+1) (k+1) (k) (k+1)  vi j (xi(k) + δin vin , ≥ δi(k+1) (xin ) − ρmax,i δin j j ) − ρmax,i δi j 2 2 by (4.30) and (4.32) % & 1 1 (k+1) (k) (k+1)  vi j (xi(k) , − vin (xin ) + ρmax,i δin j ) − ρmax,i δi j 2 2

(k+1) ≥ − δin

by (4.16), (4.17) and (4.29) % & 1 1 (k+1) (k) (k) (k+1) (k) (k+1) ≥ − δin + ρ (d − x ) − δ − β + δ βi(k) , ρ ρ min,i max,i max,i j ij ij ij in in 2 2 (k)

(k)

by (4.17) and xi j ≤ di j % & 1 (k+1) (k) (k) (k) (k+1) βi(k) , = − δin − β + ρ (d − x ) + y − ρ δ ρ min,i i j max,i ic max,i i j j in ij 2 by (4.29) (k+1) > 0, by (4.17), δin < 0 and α ∈ (0, 1).

In conclusion, we have that  ΔSi

= xi(k) = 0, in case xi(k+1) j j > 0, in case xi(k+1) > xi(k) j j

by which together with Si (x i ) is bounded above by



;

(4.33)

vi j (yi ), the system converges

j∈M i

to an equilibrium under Algorithm 4.1. Based upon the convergence analysis given above, by applying Algorithm 4.1, we can further show that the local system Li converges to the efficient NE of bi (yi ) with identical bidding prices by applying Algorithm 4.1, such that βij (yi )



= pi (yi ), in case di j > 0 . ≤ pi (yi ), in case di j = 0

Appendices

111

Hence by applying Algorithm 4.1, the local system Li converges to efficient its efficient resource allocation, such that NE and buyer Ai j , j ∈ Mi implements    argmax {xij ; j ∈ Mi } = j∈M i vi (yi ), with x i j ≡ x i j (yi ). (xi j ; j∈M i )∈X i (yi )

Suppose that supplier Ai increases its allocation from yi by a δi > 0. By applying Algorithm 4.1 local system Li converges to a new efficient NE and buyer Ai j , j ∈ Mi implements its efficient resource allocation by increasing its  allocation from xi j by a argmax δi j > 0, such that {xij + δi j ; j ∈ Mi } = j∈M i vi (yi + δi ). By (xi j ; j∈M i )∈X i (yi +δi )      xi j = yi and {xi j + δi j } = yi + δi , we have δi j = δi . Then we have j∈M i

j∈M i

j∈M i

⎛ 1 ⎝ δi

vi (yi ) = lim

δi →0

=

 j∈M i

=



δi →0

j∈M i



vi j (xij + ti j δi ) −

⎞ vi j (xij )⎠

j∈M i



  δi j 1    lim · · vi j (xi j + ti j δi ) − vi j (xi j ) δi →0 δi δi j δi j  · vi j (xi j ) δi →0 δi lim

j∈M i

= lim



 δi j · pi (yi ), since vi j (xi j ) = pi (yi ), δi

j∈M i

= pi (yi ), since

 δi j δi

= 1, for all δi > 0.

where vi j (xi j ) = pi (yi ) is obtained by comparing (4.8) and the uniform bid price defined in Theorem 4.1.

Proof of Corollary 4.1 Corollary 4.1 is shown in (I) and (II) below: (I). Proof of convergence of the proposed dynamic algorithm. Following Algorithm 4.2, suppose it is supplier Ai to increase allocation at step k + 1 and set supplier Am with respect to the allocation y(k) specified in (4.23a). Define δl(k+1)  yl(k+1) − yl(k) for all l ∈ N ; then by step (4) in Algorithm 4.2, we can verify that (k+1) = 0. δi(k+1) + δm

(4.34)

Define ΔS  S( y(k+1) ) − S( y(k) ) as the change of the social welfare from step k to step k + 1; we will show that ΔS ≥ 0 in the following cases.

112

4 Hierarchical Auction Games for Efficient Resource Allocation

– In case yi(k+1) = yi(k) . Each of the suppliers’ allocations remains unchanged, such that S( y(k+1) ) = S( y(k) ), i.e. ΔS = 0. – In case yi(k+1) > yi(k) , we have the following:   (k) (k+1) ) − vm (ym ) ΔS = vi (yi(k+1) ) − vi (yi(k) ) − vm (ym (k) $ yi(k+1) $ ym   = vi (yi )dyi − vm (yi )dyi . yi(k)

(k+1) ym

For supplier Al , l ∈ N , it is straightforward to verify that  1     min vlj (xl j ) ≤ vl (yl + δ) − vl (yl ) ≤ max vlj (xl j ) , j∈M i j∈M i δ with x l ≡ (xl j , j ∈ N ) and yi satisfied (4.9). Considering the definition of ρmin and ρmax in (4.21), we have ρmin ≤

 1   vl (yl + δ) − vl (yl ) ≤ ρmax , δ

(4.35)

by which together with Lemma 4.1: % & $ y+Δy % & 1 1 Δy vi (y) − ρmax Δy ≤ vi (y)dy ≤ Δy vi (y + Δy) + ρmax Δy , 2 2 y

where ρmax and ρmin are defined in (4.21), respectively; then we get % (k) vi (yi ) − % (k+1) (k) = δi vi (yi ) −  (k+1) (k) pi (yi ) − = δi (k+1)

ΔS ≥ δi

& % & 1 1 (k+1) (k+1)  (k) (k+1) + δm vm (ym ) − ρmax δm ρmax δi 2 2 & 1 1 (k+1) (k+1)  (k) − vm (ym ) − ρmax δi , by (4.34), ρmax δi 2 2  (k+1) (k) , by Theorem 4.1, pm (ym ) − ρmax δi

> 0,

where the last inequality is obtained by the following three conditions: – In case yi(k+1) > yi(k) , we have δi(k+1) > 0; – α ∈ (0, 1) – By yi(k+1) = yi(k) + E i ( y(k) ), we have δi(k+1) ≤ αΦi ( y(k) ), with E i ( y) defined in (4.22), by which together with Φi ( y) defined in (4.23b) and α > 0, ρmax > 0, we have (k) )− pi (yi(k) ) − pm (ym

1 ρmax δi(k+1) ≥ 0. α

Appendices

113



= 0, in case yi(k+1) = yi(k) ; ; then since S( y) is In conclusion, we have ΔS > 0, in case yi(k+1) > yi(k) .  bounded above by i∈N vi (Γ ), the system converges under Algorithm 4.2. (II). Proof of the efficiency of the implemented NE under the proposed dynamic algorithm. Suppose that, by applying Algorithm 4.2, the system converges to a collection of allocations denoted by y(k+1) , after step k; then y(k+1) = y(k) and consider i := n ∈ argmax { pn(k) }. Consider μ ≡ pi (yi(k) ), we have for all l ∈ N /{i}; then n∈N yn >0

the following holds: pl (yl(k) ) ≤ μ = pi (yi(k) ).

(4.36)

By Algorithm 4.2, we have yi(k+1) = yi(k) + E i ( y(k) ); then by the definition of E i ( y) in (4.22),   2 (k) min ym , αΦi ( y(k) ), pi (yi(k) ) = 0. ρmax where m ≡ mi ( y(k) ) and Φi ( y(k) ) are defined in (4.23a) and (4.23b), respec(k) , α, ρmax , pi (yi(k) ) > 0, we have Φi ( y(k) ) = 0. tively, by which together with ym (k) By the definition of Φi ( y ) in (4.23b), we have (k) ) = pi (yi(k) ). pm (ym

(4.37)

(k) By the definition of m in (4.23a), we have pl (yl(k) ) ≥ pm (ym ) for all l ∈ N /{i, m} with yl(k) > 0, by which together with (4.36) and (4.37), we have

pn (yn(k) ) and





= μ, in case yn > 0 , for all n ∈ N , ≤ μ, in case yn = 0

yn(k) = Γ . By Theorem 4.1, we have

n∈N

βn j (yn(k) ) and

  n∈N j∈M i



= μ, in case yn > 0 , for alln ∈ N and j ∈ Mn , ≤ μ, in case yn = 0

xn(k) j = Γ , which is equivalent to (4.4) of the efficient NE. Thus

the NE obtained by Algorithm 4.2 is efficient.

114

4 Hierarchical Auction Games for Efficient Resource Allocation

Proof of Corollary 4.2 Suppose that supplier Ai is set to increase its allocation while supplier Am is set to decrease its allocation; then by applying Algorithm 4.2, we have that (k) (k) > 0, and pi(k) ≥ pm . yi(k) , ym

(4.38)

Denote by y∗ the efficient allocation of the system; then by (4.38) and (4.4), we have  ∗ (ym ). vi (yi∗ ) = vm

(4.39)

To depict the difference between the updated allocation at an iteration step and y∗ , we explore the value of y(k+1) − y∗ 1 , and have that (k) ∗ − ym − E i ( y∗ )| +

y(k+1) − y∗ 1 = |yi(k) − yi∗ + E i ( y∗ )| + |ym

=

|yi(k)



(k) ∗ − yi∗ | + |ym − ym | − 2E i ( y∗ ) +



|yl(k) − yl∗ |

l =i,m

|yl(k)

− yl∗ |.

l =i,m

Hence, we have the following analysis:

y(k+1) − y∗ 1 (k) ∗ ≤|yi(k) − yi∗ | + |ym − ym |−

=|yi(k)

ρmax

(k) ( pi (yi(k) ) − pm (ym )+



|yl(k) − yl∗ |,

l =i,m

by (4.22), (4.23b), (4.38) α (k) ∗  ∗  (k) − + |ym − ym |− (v (y (k) ) − vi (yi∗ ) + vm (ym ) − vm (ym )) ρmax i i  (k) |yl − yl∗ |, by Theorem 4.1, (4.39) + yi∗ |

l =i,m

≤|yi(k)

α

(k) ∗ − yi∗ | + |ym − ym |−

 (k) αρmin (k) (k) ∗ (|yi − yi∗ | + |ym − ym |) + |yl − yl∗ |, ρmax l =i,m

by Assumption 3, (4.21)  (k) αρmin (k) ∗ =(1 − )(|yi(k) − yi∗ | + |ym − ym |) + |yl − yl∗ | ρmax l =i,m ≤Q y(k) − y∗ 1 , where Q is a certain value in the range of [1 −

αρmin , 1). ρmax

Appendices

115

This implies that y(k) − y∗ 1 is a contraction map; then, after an iteration step k, with k ≥ 1, we have

y(k) − y∗ 1 ≤ Q k y(0) − y∗ 1 ≤ Q k N Γ,

(4.40)

where the last inequality holds since yi(0) , yi∗ ∈ [0, Γ ], for all i ∈ N . We consider that the hierarchical system under Algorithm 4.2 terminates at a certain iteration step K in case y(K ) − y∗ 1 ≤ ε, with ε > 0; then by (4.40), it is straightforward to verify that the conclusion of the corollary.

References 1. V. Krishna, Auction Theory (Academic, Cambridge, 2009) 2. Y. Zhang, C. Lee, D. Niyato, P. Wang, Auction approaches for resource allocation in wireless systems: a survey. IEEE Commun. Surv. Tutor. 15(3), 1020–1041 (2013) 3. L. Wang, M. Liu, M.Q.H. Meng, Hierarchical auction-based mechanism for real-time resource retrieval in cloud mobile robotic system. In 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 2164–2169, Hong Kong, May-June 2014 4. N. Alguacil, J.M. Arroyo, R. García-Bertrand, Optimization-based approach for price multiplicity in network-constrained electricity markets. IEEE Trans. Power Syst. 28(4), 4264–4273 (2013) 5. A. Lazar, N. Semret, Design and analysis of the progressive second price auction for network bandwidth sharing. Telecommun. Syst. 13 (2001) 6. Xingyu Shi, Zhongjing Ma, An efficient game for vehicle-to-grid coordination problems in smart grid. Int. J. Syst. Sci. 46(15), 2686–2701 (2015) 7. Rahul Jain, Jean Walrand, An efficient Nash-implementation mechanism for network resource allocation. Automatica 46, 1276–1283 (2010) 8. Bruno Tuffin, Revisited progressive second price auction for charging telecommunication networks. Telecommun. Syst. 20(3–4), 255–263 (2002) 9. X. Shi, S. Zou, Z. Ma, A novel algorithm for sivisible resource allocations under PSP auction mechanism. In 26th Chinese Control and Decision Conference, pp. 1723–1728, Changsha, June 2014 10. Patrick Maillé, Bruno Tuffin, Pricing the internet with multibid auctions. IEEE/ACM Trans. Netw. 14(5), 992–1004 (2006) 11. P. Maillé, B. Tuffin, Multibid auctions for bandwidth allocation in communication networks. In 23rd AnnualJoint Conference of the IEEE Computer and Communications Societies, vol. 1, pp. 54–65, 7–11 Mar 2004 12. P. Jia, P.E. Caines. Probabilistic analysis of the rapid convergence of a class of progressive second price auctions. In IEEE 47th Annual Conference on Decision and Control, pp. 5476– 5481, 9–11 Dec 2008 13. P. Jia, P.E. Caines. Analysis of decentralized decision processes in competitive markets: quantized single and double-sided auctions. In IEEE 49th Annual Conference on Decision and Control, pp. 237–243, 15–17 Dec 2010 14. P. Jia, C.W. Qu, P.E. Caines, On the rapid convergence of a class of decentralized decision processes: quantized progressive second-price auctions. IMA J. Math. Control Inf. 26(3), 325– 355 (2009) 15. Y. Hua, Q. Zhang, Z. Niu, Resource allocation in multi-cell OFDMA-based relay networks. In 2010 Proceedings IEEE INFOCOM, pp. 1–9, San Diego (2010)

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16. A. Bidram, A. Davoudi, Hierarchical structure of microgrids control system. IEEE Trans. Smart Grid 3(4), 1963–1976 (2012) 17. P. Jia, P.E. Caines, Analysis of decentralized quantized auctions on cooperative networks. IEEE Trans. Autom. Control 58(2), 529–534 (2013) 18. M. Parvania, M. Fotuhi-Firuzabad, M. Shahidehpour, Optimal demand response aggregation in wholesale electricity markets. IEEE Trans. Smart Grid 4(4), 1957–1965 (2013) 19. W. Qi, Z. Xu, Z.J. Shen, Z. Hu, Y. Song, Hierarchical coordinated control of plug-in electric vehicles charging in multifamily dwellings. IEEE Trans. Smart Grid 5(3), 1465–1474 (2014) 20. Harrison H. Cheng, Guofu Tan, Asymmetric common-value auctions with applications to private-value auctions with resale. Econ. Theory 45(1–2), 253–290 (2010). October 21. W. Tang, R. Jain, Hierarchical auction mechanisms for network resource allocation. IEEE J. Sel. Areas Commun. 30(11), 2117–2125 (2012) 22. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)

Chapter 5

Large-Scale Elastic Load Management Under Auction Games

Abstract Auctions, e.g., market clearing price (MCP) auctions, have been widely adopted in electricity markets, and progressive second price (PSP) auctions are stated possessing promising properties of incentive compatibility and efficiency. In this work, we study the coordination of large-scale elastic loads in deregulated electricity markets under MCP and PSP auctions. To explore the performances of these auctions in the underlying problems, we focus on key issues of the payment comparison, incentive compatibility, and efficiency of Nash equilibrium (NE), and develop the following results: (i) The individual payment under MCP is always higher than that under PSP, and their difference vanishes asymptotically as the system scale increases; (ii) The incentive compatibility holds under PSP, and holds under MCP only with respect to others’ efficient bid profile; (iii) The efficient bid profile under PSP auctions is a NE, while that under MCP is an ε-NE which degenerates to a NE asymptotically as the system scale increases. With these analyses, we claim that it is pretty promising to apply both MCP and PSP auctions to the large-scale load coordination problems in deregulated electricity markets.

5.1 Introduction Since the 1980s, the need for enhanced efficiency in the generation, transmission, and consumption of electricity has led to a restructuring of the power industry, which had been under the centralized control of governments, see [1–3] and the references therein. China has also initiated a series of reforms to weaken the monopoly in electricity industries by introducing competition on both the generation and utility sides, and these reforms originally conceive of two successive stages: the separation of generation from the grid followed by the unbundling of transmission and distribution [4, 5]. The power industry has greatly benefited from restructuring and deregulation, such as lower generation cost, better services to customers and more effective capacity expansion planning [2]. As a consequence, the design of market rules and electricity price forecasting schemes has emerged as a hot research topic, see [6–8]. More recently, with the development of information technology and popularly installed smart meters, it is feasible to enhance the efficiency and the social © Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_5

117

118

5 Large-Scale Elastic Load Management Under Auction Games

welfare of electricity utilization by effectively coordinating large-scale elastic loads, e.g., air conditioners, heaters, plug-in electric vehicles [9–12]. The efficient solution can be effectively implemented in case that the system has complete information and can directly schedule the behaviors of all the loads. However, due to the autonomy property of individual loads and communication and computation overhead, it is impractical to coordinate large-scale loads directly. Alternatively, many researches have explored the coordination problem of elastic loads following certain pricing mechanisms, such as predetermined fixed retail price, e.g., [13], and dynamic price dependent upon system conditions, e.g., [14–16]. Auction, including uniform price auction, discriminatory auction, and Vickrey auction [17–20], is a designed mechanism which allows the system to allocate resources in a distributed way [21, 22]. In day-ahead deregulated electricity markets, the market clearing price (MCP) mechanism, which has been widely deployed in many regions, is able to dispatch electricity resources among generators economically, and set the wholesale price in distributed ways, see [23, 24]. The progressive second price (PSP) mechanism, which is a VCG-type auction, has been originally designed by Lazar and Semret in [25] to efficiently allocate divisible resources [26– 29]. As a VCG-type auction, the PSP mechanism is incentively compatible, i.e., the truthful bid is the best response, and under it, the efficient bid profile is a NE [27–29]. Therefore, the PSP mechanism shares its applications with MCP, in the spirit that they are able to dispatch electricity resources in an economic way, as stated in [17]. Following the above discussions, in this chapter, we implement an efficient solution to the underlying load coordination problems in distributed ways by introducing the MCP and PSP mechanisms, along with an analysis of the NE performance of the formulated auction games under each of the two mechanisms. To our knowledge, we are the first to conduct analyses on the large-scale load coordination auction games. Several critical issues under the PSP and MCP mechanisms are studied, including the payment comparison, incentive compatibility, and the NE property of the efficient bid profiles, especially the associated asymptotic phenomena as the scale size of the power systems increases. We show that the payment of an individual load under MCP auctions is always higher than that under PSP auctions, and the difference between them vanishes asymptotically as the scale size of power systems increases. Different from the PSP mechanism, the incentive compatibility does not hold for the auction games under the MCP mechanism, while a truthful bid is verified to be the best strategy with respect to the efficient bid profile of other loads. In [27, 29, 30], the efficient bid profile is shown to be a NE under PSP auction games; and it is shown in this chapter that the efficient bid profile under MCP auctions is an ε-NE of the formulated load coordination games, and becomes a NE asymptotically as the scale of the power systems increases. With the above analyses, we claim that it is promising to apply the MCP and PSP auctions to the coordination of elastic loads in deregulated electricity markets. The rest of the chapter is organized as follows: In Sect. 5.2, we formulate a class of coordination problems for elastic loads in the scenario of scalable power systems. Section 5.3 introduces the PSP and MCP auction mechanisms for the underlying load coordination problems, followed by the analysis of the NE performance of the

5.1 Introduction

119

auction games in Sect. 5.4. Some numerical simulations are given in Sect. 5.5 to demonstrate the results developed in the chapter. Section 5.6 concludes this chapter and sheds some light on the future research directions.

5.2 Formulation of Coordination Problems of Large-Scale Elastic Loads We study the economic coordination of a population of elastic loads N ={1, 2, . . . , N } in electricity markets. Denote by xn , with 0 ≤ xn ≤ Γn , an admissible demand of load n, where Γn , with Γn < Γ , represents the maximum demand of load n, and Γ is an upper limit of the maximum demands of all the loads. Define X as the set of admissible demands of load populations, such that X  {x ≡ (xn ; n ∈ N ); s.t. xn ∈ [0, Γn ]}. Besides the elastic loads discussed above, we define Q as a fixed normalized inelastic load demand with respect to the population size of elastic loads; then the total inelastic load of the system, denoted by D N , is given as D N = N Q. We further denote by vn (xn ) and c N (y) the valuation function of load n with respect to xn and the generation cost function of the power system with N elastic loads with respect to the total load consumption y, respectively;  then the total consumption can be expressed as y = D N + Σ(x) where Σ(x) ≡ n∈N xn . Consider the following assumptions which have been widely adopted in the literature, e.g., [31–33] and the references therein: • (A1). c N (y), is increasing, strictly convex and differentiable on y; • (A2). vn (x), n ∈ N , is increasing, strictly concave and differentiable on x. The system cost function, denoted by JN , with respect to demand x, is specified as below:  vn (xn ). (5.1) JN (x) = c N (D N + Σ(x)) − n∈N

The objective of the system is to assign the optimal allocation among all the loads, denoted by x ∗∗ , to minimize the system cost (5.1). That is, x ∗∗ = argmin JN (x), x∈X

and by applying the Lagrange multiplier method, x ∗∗ which is called efficient (socially optimal) can be specified by the following KKT conditions: 

∂ L N (x,λ) ∂ xn

≥ 0, xn ≥ 0, ∂ L N∂ x(x,λ) xn = 0 n ∀n ∈ N , xn − Γn ≤ 0, λn ≥ 0, (xn − Γn )λn = 0

(5.2)

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5 Large-Scale Elastic Load Management Under Auction Games

 where L N (x, λ)  JN (x) + n∈N λn (xn − Γn ) with λn representing the Lagrange multiplier associated with the constraint xn ≤ Γn , and hence ∂∂LxnN = cN (D N + Σ(x)) − vn (xn ) + λn . By the properties under Assumptions (A1, A2), the efficient allocation x ∗∗ is uniquely specified by (5.2). The efficient solution can be effectively implemented in case that the system has complete information and can directly schedule the behaviors of all the loads. However, in practice, individuals are unwilling to share their private information with others, and complete information may cause communication and computation overhead as well. To mitigate such risks, in the literature, auction mechanisms such as MCP auctions, have been widely adopted to dispatch electricity resources among generators and set the wholesale price in distributed ways without exposing individual private information. Moreover, due to its incentive compatibility, the PSP auction mechanism has been designed to efficiently allocate divisible resources in many fields, see [26–29]. Meanwhile, as pointed out by [17], the PSP mechanism can well achieve economic dispatch of electricity resources as the MCP mechanism.

5.3 Auction-Based Mechanism for Load Coordination Problems We firstly formulate the underlying load coordination problems as a class of auctionbased games in Sect. 5.3.1; then specify the payment rules under the PSP and MCP auction mechanisms in Sects. 5.3.2 and 5.3.3, respectively.

5.3.1 Auction Games of Individual Loads In the auction mechanism for the load coordination, each load n submits a twodimension bid bn , such that bn = (βn , dn ), with 0 ≤ dn ≤ Γn , which specifies the per unit price βn that load n is willing to pay and demands up to dn units of the electricity resources. We denote by Bn the set of bids of load n. Denote by xn (bn ), with 0 ≤ xn ≤ dn , an admissible allocation of load n with (b) as the set of the admissible allocation of respect to its bid bn , and define X (b)  {x ≡ (xn ; n ∈ load populations with respect to a bid profile b, such that X N ); s.t. xn ∈ [0, dn ]}. We define a function of load n, denoted by vn (xn ; bn ), with respect to a bid bn and vn (xn ; bn ) = βn min(xn , dn ); then by xn ≤ dn , an admissible allocation xn , such that  we have  vn = βn xn . It is regarded as a revealed valuation function in [27].

5.3 Auction-Based Mechanism for Load Coordination Problems

121

(b), denoted by JN (x; b), The cost with respect to an admissible allocation x ∈ X is defined in the following JN (x; b)  c N (D N + Σ(x)) −



 vn (xn ; bn ).

(5.3)

n∈N

The auctioneer assigns a collection of optimal demands with respect to b, denoted by x ∗ (b), to minimize the cost JN (·; b); and x ∗ (b) can be characterized by the following KKT conditions: 

∂ LN ∂ xn



∂LN ≥ 0, xn ≥ 0, x =0 ∂ xn n ∀n ∈ N , xn − dn ≤ 0, σn ≥ 0, (xn − dn )σn = 0

(5.4)

where σn is the Lagrange multiplier associated with the constraint xn ≤ dn , and ∂ LN = cN (D N + Σ(x)) − βn + σn . ∂ xn In Lemma 5.1 below, we specify a bid profile, denoted by b∗ , under which the efficient allocation can be obtained to minimize the system cost specified in (5.1). We call b∗ the efficient bid profile. At b∗ , all the loads with nonzero allocation share an identical bid price, while others with lower prices lose the auction. In the rest of the chapter, we analyze the efficiency of the two auctions through analyzing the NE property of b∗ . Lemma 5.1 Consider a bid profile b∗ ≡ (bn∗ = (βn∗ , dn∗ ); n ∈ N ) such that βn∗ = vn (xn∗∗ ) and dn∗ = xn∗∗ , with x ∗∗ specified in (5.2). Then the following conclusions hold: x ∗ (b∗ ) = x ∗∗ , and  βn∗

= cN (D N + Σ(d ∗ )), in case xn∗ > 0 . ≤ cN (D N + Σ(d ∗ )), in case xn∗ = 0

(5.5)

Proof Proof of Lemma 5.1 is given in Appendix. Considering a bid profile b, we define the payoff function of load n, denoted by f n (b), as below: f n (b)  vn (xn∗ (b)) − τn (b),

(5.6)

where τn (b) is the payment of load n with respect to b designed by the system under certain auction mechanisms.

5.3.2 Payment of Individual Loads Under PSP Auctions Under the PSP auction mechanism, see [25, 27], the payment (or transfer money) of an agent is that, the utility of the whole system, when this agent does not join the

122

5 Large-Scale Elastic Load Management Under Auction Games

auction game, minus the aggregated utility of the remaining system except for this agent in case he joins the auction game. That is to say, the payment of each agent is exactly the externality he imposes on the system through his participation. Hence psp the payment of load n under PSP auctions, denoted by τ N ,n (b), with respect to a bid profile b is given as follows: τ N ,n (b)  − JN∗ (b−n ) − [− JN∗ (b) − βn xn∗ (b)], psp

where for any bid profile b, b−n denotes the bid profile without load n’s participation, that is dn−n = 0. psp By (5.3), τ N ,n (b) is given as below:    psp τ N ,n (b) =c N (D N + Σ(x ∗ )) − c N D N + xm−n + βm (xm−n − xm∗ ), (5.7) m =n

m =n

with x ∗ ≡ x ∗ (b) and x −n ≡ x ∗ (b−n ). psp Denote by f N ,n (b) the payoff function under PSP mechanism; then by (5.6) and psp (5.7), f N ,n (b) is specified as below: f N ,n (b) = vn (xn∗ (b)) − τ N ,n (b). psp

psp

(5.8)

5.3.3 Payment of Individual Loads Under MCP Auctions In this section, before arriving at the definition of the payment of loads under MCP auctions, some notations are introduced first. We first give ρ(b) ≡ (ρn (b), n ∈ N ) with respect to a bid profile b, such that  ρn (b) =

βn , in case xn∗ > 0 , p, in case xn∗ = 0

(5.9)

where xn∗ ≡ xn∗ (b), and p represents a regulated price of electricity markets. The market clearing price with respect to b, denoted by p(b), is a uniform price such that the loads who submit a bidding price higher than p(b) are fully supplied and those who submit a price equal to p(b) is partially or fully allocated under the balance of demand and supply, see [23]. Hence p(b) can be specified as the following:

p(b)

⎧ ⎨= min ρn (b), ⎩∈

n∈N [cN (x ∗ ),

in case min ρn (b) = cN (x ∗ ) n∈N

min ρn (b)], in case min ρn (b) > cN (x ∗ )

n∈N

n∈N

with cN (x ∗ ) ≡ cN (D N + Σ(x ∗ )), see an illustration in Fig. 5.1.

,

(5.10)

5.3 Auction-Based Mechanism for Load Coordination Problems

(a)

123

(b)

Fig. 5.1 Specification of p(b) w.r.t. bid profile b

Thus, we can set a parameter η with 0 ≤ η ≤ 1 and further uniquely specify the market clearing price given in (5.10) with η as follows: p(b; η)  cN (D N + Σ(x ∗ )) + η( min {ρn (xn∗ )} − cN (D N + Σ(x ∗ ))). n∈N

(5.11)

Under the MCP mechanism, we specify the payment of load n under MCP mechmcp anism, denoted by τ N ,n (b; η), as below: τ N ,n (b; η)  p(b; η)xn∗ (b). mcp

(5.12)

mcp

Denote by f N ,n (b; η) the payoff function with respect to a bid profile b under MCP mechanism; then by (5.6) and (5.12), f N ,n (b; η) = vn (xn∗ (b)) − τ N ,n (b; η) = vn (xn∗ (b)) − p(b; η)xn∗ (b). mcp

mcp

(5.13)

5.4 Performance Analysis for Load Coordination Auction Games In this section, we first analyze the difference of payments and the best strategies of individual loads under MCP and PSP mechanisms in Sects. 5.4.1 and 5.4.2, respectively; then based upon these analyses, we show in Sect. 5.4.3, that the efficient bid profile b∗ given in Lemma 5.1 is an ε-NE of load coordination auction games under MCP mechanism.

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5 Large-Scale Elastic Load Management Under Auction Games

5.4.1 Payment Comparison Under PSP & MCP Auction Mechanisms For the analysis of the difference between the payments under MCP and PSP mechanisms, some notations are needed, as defined below: sup{cN } ≡ inf{cN } ≡

sup {cN (D N + y)},

(5.14a)

{cN (D N + y)}.

(5.14b)

y∈[0,N Γ ]

inf

y∈[0,N Γ ]

Then, the relationship between the two types of payments can be summarized in Theorem 5.1 below. Since the payment of individual agents is a key issue in auction games, it is interesting to discuss the difference between the payments under the PSP and MCP auctions. As shown in Theorem 5.1, loads are charged more under MCP auctions than under PSP auctions in the load coordination game, while this distinction is bounded above by a parameter associated with the system characteristics. Theorem 5.1 Under any bid profile b, the following inequalities hold: mcp

psp

0 ≤ τ N ,n (b; 0) − τ N ,n (b) ≤ σ N ,

(5.15)

where σ N = O(sup{cN }), i.e., σ N has the order of sup{cN }. mcp

psp

Proof For notational simplicity, we consider Δτ N ,n ≡ τ N ,n (b; 0) − τ N ,n (b). Thus, as shown in Appendix, we have 0 ≤ Δτ N ,n ≤ sup{cN }Γ 2 . Denote by σ N = sup{cN } · Γ 2 , and it is the order of sup{cN }; then the result is shown. Remark on Theorem 5.1: Consider that the marginal generation cost cN (·) is finite for any power systems; then under Assumption (A1), we can obtain that sup{cN } converges to zero asymptotically as N goes to infinity. Hence by Theorem 5.1, the difference between the payments under MCP and PSP mechanisms vanishes asymptotically as the system population approaches infinity.

5.4.2 Best Bid Strategy of Individual Loads Denote by Bn∗ (b−n ) the set of best responses of load n subject to b−n , where b−n is the bid profile of load n’s opponents, such that

  bn , b−n ) . Bn∗ (b−n )  bn ∈ Bn ; s.t. f n (bn , b−n ) = max f n (  bn ∈B n

(5.16)

The set of the best responses of load n with respect to the efficient bid profile of its opponents is written as Bn∗ (b∗−n ).

5.4 Performance Analysis for Load Coordination Auction Games

125

We also denote by Bnt the set of truthful bids of load n as below:   Bnt  bn ≡ (βn , dn ) ∈ Bn ; s.t. βn = vn (dn ) .

(5.17)

As stated in [25], the incentive compatibility property holds under the PSP auction mechanism, which implies Bn∗ (b−n )



Bnt = ∅,

(5.18)

i.e., there exists a truthful bid bnt that is a best response of load n with respect to b−n as well. As discussed above, under the PSP mechanism, the truthful bid is the best response of a load with respect to any collection of bid profiles of other loads. Different from the PSP mechanism, it is well known that incentive compatibility does not hold for the auction game under MCP mechanisms. Nevertheless, we show in Theorem 5.2 that a truthful bid is the best strategy in certain situations; that is, a truthful bid is the best response of load n only with respect to b∗−n , which is the efficient bid profile of other loads. Theorem 5.2 Under the MCP mechanism, the following holds: Bn∗ (b∗−n )



Bnt = ∅, for any η ∈ [0, 1],

(5.19)

i.e., there exists a truthful bid bnt ∈ Bnt , such that bnt is a best response of load n with respect to b∗−n . Proof It is sufficient to show (5.19) if we can show that, for any bid bn ∈ Bn , there always exists a truth-telling bid bnt ∈ Bnt , such that f N ,n (bnt , b∗−n ; η) ≥ f N ,n (bn , b∗−n ; η), mcp

mcp

(5.20)

which is verified in Appendix in details.

5.4.3 Nash Equilibrium Properties of Efficient Bid Profiles Recall that b∗ represents the efficient bid profile specified in Lemma 5.1. As analyzed in [27, 29], b∗ is a NE of the PSP auction games; then based upon the NE property of b∗ under the PSP mechanism and the results developed in the earlier parts in this chapter, we will show, in Theorem 5.3, that b∗ is an ε-NE of the formulated load coordination games under the MCP mechanism, and becomes a NE asymptotically as the load population goes to infinity. Before that we first define the ε-NE as below: Definition 5.1 (ε–Nash equilibrium) A bid profile b0 is an ε–Nash equilibrium (ε–NE for short), with ε ≥ 0, for the auction games if the following holds:

126

5 Large-Scale Elastic Load Management Under Auction Games

f n (bn0 , b0−n ) ≥ f n (bn , b0−n ) − ε,

(5.21)

for all bn ∈ Bn , that is to say, the auction game system is at an ε–NE with b0 , if any individual load n can benefit himself with a gain of at most ε by unilaterally deviating from his bid profile bn0 . An ε–NE becomes a NE in case ε = 0. By Theorem 5.1, together with (5.8) and (5.13), we have mcp

psp

mcp

psp

| f N ,n (b; η) − f N ,n (b)| = |τ N ,n (b; η) − τ N ,n (b)| ≤ σ N ;

(5.22)

with σ N = O(sup{cN }) in case η = 0; then by (5.22) and because b∗ is a NE under PSP auctions as verified in [27, 29], we can state that b∗ is an ε N –NE, with ε N = O(sup{cN }), under the MCP mechanism with η = 0. In Theorem 5.3, we study the NE property of the efficient bid profile b∗ in MCP auction-based games with any η ∈ [0, 1]. Theorem 5.3 The efficient bid profile b∗ is not a NE but an ε N –NE of MCP auction games with population size of N , for all η ∈ [0, 1], such that ε N = O([sup{cN }]2 ). Proof The proof is given in Appendix. Remark of Theorem 5.3. As stated in the analysis related to Theorem 5.1, sup{cN } converges to zero asymptotically as N goes to infinity. Hence the efficient bid profile b∗ of MCP auction games becomes a NE asymptotically as the system population goes to infinity. It implies that MCP auctions can achieve efficiency in large-scale load coordination problems.

5.5 Numerical Simulations Under Assumption (A1), we suppose c N = a N (D N + Σ(x))b , where a N is a coefficient depending on the population size N , and b is a constant larger than one denoting the convexity of generation cost. For the purpose of demonstration, we suppose that the inelastic load demand D N = N kW, and all the elastic loads are in the same type with a capacity of Γn = 10 kW for all n ∈ N ; then we can calculate a feasible pair of a N , b such that a N = 0.0937N −0.3 and b = 1.3. We consider the valuation function of load n as vn = 0.3xn0.9 which satisfies Assumption (A2).

5.5.1 Non-NE Property of Efficient Bids Under MCP Auctions We obtain by Theorem 5.3 that for any auction game with a finite population of load units, the efficient bid profile b∗ cannot be a NE, i.e., there must exist a bid  bn ∈ Bn mcp mcp such that f N ,n (b∗ ; η) < f N ,n ( bn , b∗−n ; η).

5.5 Numerical Simulations

127

Set N = 10 as an example and then a N = 0.0469. The efficient bid of load n is bn∗ = (βn∗ , dn∗ ) = (0.2236, 6.6020) for all n ∈ N . Consider an individual load n, n , dn ) = (0.2223, 7), while the others keep fixed as the whose bid changes to  bn = (β efficient bid profile, denoted by b∗−n . We can calculate that the allocation of load n xn = 5.0118 kW. Thus, the difference of payoffs subject with respect to ( bn , b∗−n ) is  mcp ∗ bn , b∗−n ), denoted by Δf n , is specified as: Δf N ,n = vn (dn∗ ) − p ∗ dn∗ − to b and ( xn ) +  p (η) xn = −0.0018 < 0, which implies that there exists a bid  bn specified vn ( mcp mcp bn , b∗−n ; η). above such that f N ,n (b∗ ; η) < f N ,n (

5.5.2 ε N -NE Property of the Efficient Bids Under MCP Auctions By Theorems 5.1 and 5.3, following the parameters above, we can calculate the value of σ N and ε N with respect to N , as summarized in Table 5.1. As we can observe from Table 5.1 and Fig. 5.2, ε N decreases much faster than σ N as the population size N increases, and they both tend to zero with the increasing of N . Table 5.1 The values of σ N and ε N w.r.t. N

N

σN

εN

2 5 10 100 1000

1.1247 0.2369 0.0729 0.0015 2.90 × 10−5

0.2632 0.0408 0.0074 4.87 × 10−6 1.96 × 10−9

(a) 4

(b)

0.8

The value of

The value of

3 2 1 0

1

0

1

2

3

0.6 0.4 0.2 0

0

1

lg(N)

Fig. 5.2 Evolutions of σ N and ε N with respect to the population size N

2

lg(N)

3

128

5 Large-Scale Elastic Load Management Under Auction Games

5.6 Conclusions and Ongoing Researches With the deregulation and reconstruction of electricity markets, consumers are allowed to participate in electricity markets directly, which, in turn, introduces challenges to the economic coordination and management of power systems. In this chapter, we introduce the MCP and PSP mechanisms to coordinate large-scale loads and implement the optimal solutions in a distributed way. To investigate the performances of MCP and PSP auctions in the large-scale load coordination problems, we study the issues including the payment comparison, incentive compatibility, and the NE property of the efficient bid profiles, especially the associated asymptotic phenomena as the scale size of the power system increases. It is shown that the difference between the payments of an individual load under MCP and PSP auctions vanishes asymptotically as the scale size of power systems increases, and the incentive compatibility holds under the MCP mechanism only with respect to the efficient bid profile of other loads. Furthermore, the efficient bid profile is an ε-NE of the formulated load coordination games under the MCP mechanism, and it becomes a NE asymptotically as the scale of the power systems increases. Supported with these analytical results, MCP and PSP mechanisms are both promising for the economic load coordination problems in the deregulated electricity market. As ongoing researches, we would like to extend the analysis developed in this chapter to the coordination of large-scale inter-temporal loads over multi-time intervals, such as plug-in electric vehicles, under PSP and MCP auction mechanisms. Also, the design of distributed dynamic algorithms is essential to allow the system to converge to the efficient coordination solution.

Appendices Proof of Lemma 5.1 By (5.2), λ∗n = 0 and βn∗ = vn (xn∗∗ ), we can obtain that βn∗



= cN (D N + Σ(x ∗∗ )), in case xn∗∗ > 0 , for all n ∈ N . ≤ cN (D N + Σ(x ∗∗ )), in case xn∗∗ = 0

(5.23)

Substitute dn∗ = xn∗∗ into (5.4), we have xn∗ − xn∗∗ ≤ 0, σn∗ ≥ 0, (xn∗ − xn∗∗ )σn∗ = 0.

(5.24)

xn∗ = 0 = xn∗∗ in case xn∗∗ = 0; then we will show that 0 ≤ xn∗ < xn∗∗ does not hold. By xn∗∗ > 0 and (5.23), we have

Appendices

129

βn∗ = cN (D N + Σ(x ∗∗ )). By xn∗ < xn∗∗ and (xn∗ − xn∗∗ )σn∗ = ∂  L (x, σ ; b) ≥ 0 in (5.4), we have ∂ xn N

(5.25)

0, we have σn∗ = 0, by which together with

cN (D N + Σ(x ∗ )) − βn∗ ≥ 0;

(5.26)

then by (5.25), we have cN (D N + Σ(x ∗ )) − cN (D N + Σ(x ∗∗ )) ≥ 0. However by the convexity of c N (y) and xn∗ < xn∗∗ , we have cN (D N + Σ(x ∗ )) − cN (D N + Σ(x ∗∗ )) < 0. Hence we get a contradiction. Thus, we can conclude that under the bid profile b∗ with bn∗ = (vn (xn∗∗ ), xn∗∗ ) for all n, the allocated demand x ∗ (b∗ ) as specified in (5.4) is efficient, say x ∗ (b∗ ) = x ∗∗ . Equation (5.5) holds by (5.23) and x ∗ (b∗ ) = x ∗∗ .

Proof of 0 ≤ Δτ N,n ≤ sup{cN }Γ 2 in Theorem 5.1 By (5.11), the clearing price with respect to b with η = 0, denoted by p 0 , is specified as below: p 0 ≡ p(b; 0) = cN (D N + Σ(x ∗ )), with x ∗ ≡ x ∗ (b).

(5.27)

Here, for notational simplicity, we consider g N (x) ≡ c N (D N + x); mcp

then g N (x) = cN (D N + x). psp

We consider Δτ N ,n ≡ τ N ,n (b; 0) − τ N ,n (b); then by (5.7) and (5.12), the following holds: Δτ N ,n = p 0 xn∗ − g N (Σ(x ∗ )) + g N



 xm−n − βm (xm−n − xm∗ ),

m =n

(5.28)

m =n

where x −n ≡ x ∗ (b−n ). Hence we have Δτ N ,n

   ≥ g N (Σ(x ∗ ))xn∗ − g N (Σ(x ∗ )) Σ(x ∗ ) − xm−n − βm (xm−n − xm∗ ), m =n

= g N (Σ(x ∗ ))

 m =n

≥ 0,

m =n

by (5.27), (5.28), and under Assumption (A1)  (xm−n − xm∗ ) − βm (xm−n − xm∗ ) m =n

130

5 Large-Scale Elastic Load Management Under Auction Games

where the last inequality holds since, for any load m, xm−n = xm∗ in case βm > g N (Σ(x ∗ )), and xm−n ≥ xm∗ otherwise. Moreover Δτ N ,n satisfies the following: Δτ N ,n ≤ g N (Σ(x ∗ ))xn∗ − g N



xm−n

   Σ(x ∗ ) − xm−n − βm (xm−n − xm∗ ),

m =n

≤

g N (Σ(x ∗ ))xn∗

m =n

by (5.27), (5.28), and under Assumption (A1)   − xm−n xn∗ − (βm − g N (Σ(x ∗ )))xm−n g N

m =n

≤

g N (Σ(x ∗ ))xn∗

m =n

−

g N



m =n

xm−n



xn∗ ,

since

 xm−n

m =n

  xm−n xn∗ , ≤ sup{cN } Σ(x ∗ ) −

≥ 0, in case βm ≥ g N (Σ(x ∗ )) = 0, otherwise (5.29)

m =n

with sup{cN } specified in (5.14a). We further analyze the value of Δτ N ,n in the following cases below:

 • In case min ql (xl∗ ) = cN (D N + Σ(x ∗ )), as illustrated in Fig. 5.3a: m =n xm∗ − l∈N   −n ∗ 2 m =n x m ≤ 0; then, by (5.29), we have Δτ N ,n ≤ sup{c N }[x n ] . ∗  ∗ • In case min ql (xl ) > c N (D N + Σ(x )), as illustrated in Fig. 5.3b: xm−n = xm∗ for l∈N

all m = n, since the demand required by load n is supplied by the generator; then, by (5.29), we have Δτ N ,n ≤ sup{cN }[xn∗ ]2 . Hence by the analysis given above, we have Δτ N ,n ≤ sup{cN }Γ 2 , for all b ∈ B and all n ∈ N , which implies the conclusion.

(a)

(b)

Fig. 5.3 Comparison of total demand and generation

Appendices

131

Proof of (5.20) in Theorem 5.2 Here, for notational simplicity, we consider w N (d) ≡ cN (D N +



dm∗ + d)

(5.30)

m =n p

p

p

We specify a truth-telling bid of load n, denoted by bn ≡ (βn , dn ) ∈ Bnt , such p p p that βn = vn (dn ) = w N (dn ). p By Lemma 5.1, we have p ∗ = βn = βm∗ for all m = n, where p ∗ represents the market clearing price in the efficient case, see an illustration in Fig. 5.4. The set of bids Bn can be partitioned into four disjoint regions, such that Bn =  ˙ Bni with Bni specified below: i=1,...,4

Bn1  {bn ∈ Bn ; s.t. dn ≤ dnp , βn ≥ p ∗ }, Bn2  {bn ∈ Bn ; s.t. dn > dnp , βn ≥ p ∗ }, Bn3  {bn ∈ Bn ; s.t. βn < p ∗ , βn ≤ w N (dn )}, Bn4  {bn ∈ Bn ; s.t. w N (dn ) < βn < p ∗ }, see Fig. 5.4 for an illustration; then we will analyze (5.20) in case bn ≡ (βn , dn ) ∈ Bni , with i = 1, . . . , 4 in (i)–(iv), respectively. (i) In case bn ∈ Bn1 . It is straightforward to verify that the clearing price  p w.r.t. (bn , b−n ) is in [w N (dn ), p ∗ ]; and the load n with bn is fully allocated. Hence for any fixed η ∈ [0, 1], the payoff of load n with (bn , b∗−n ) is the same as that with (bnt , b∗−n ) such that bnt = (vn (dn ), dn ) ∈ Bnt . (ii) In case bn ∈ Bn2 . p is specified as follows: For any bid bn ∈ Bn2 , the clearing price  

p∗ ,  p= βn ,

 in case m =n xm∗ (bn , b∗−n ) > 0 ,  in case m =n xm∗ (bn , b∗−n ) = 0

then the payoff of load n with respect to (bn , b∗−n ) is as follows: p xn∗ , f N ,n (bn , b∗−n ; η) = vn (xn∗ ) −  mcp

(5.31)

where x ∗ ≡ x ∗ (bn , b∗−n ) such that dn ≥ xn∗ > dn . p It is straight to have the payoff of load n with (bn , b∗−n ) as follows: p

f N ,n (bnp , b∗−n ; η) = vn (dnp ) − p ∗ dnp ; mcp

(5.32)

132

5 Large-Scale Elastic Load Management Under Auction Games

Fig. 5.4 A partition of the set of bid profiles Bn

then by (5.31) and (5.32), the following holds: Δf N ,n  f N ,n (bn , b∗−n ; η) − f N ,n (bnp , b∗−n ; η) mcp

mcp

mcp

≤ vn (xn∗ ) − vn (dnp ) − p ∗ (xn∗ − dnp ) ≤ vn (dnp )(xn∗ − dnp ) − p ∗ (xn∗ − dnp ) = 0, where the last inequality holds by Assumption (A2) and xn∗ > dn , and the last p equality holds since vn (dn ) = p ∗ . p

In (i) and (ii) above, we consider the bid bn such that βn ≥ p ∗ ; while in (iii) and (iv) below, we will consider another case for bn such that βn < p ∗ . Firstly in case of βn < p ∗ , if the allocated demand of load n satisfies that xn∗ (bn , b∗−n ) = 0, the associated payoff of load n equals the payoff subject to the truth-telling bid (vn (0), 0) ∈ Bnt . Hence, in the following we only consider those bids bn such that xn∗ (bn , b∗−n ) > 0. (iii) In case bn ∈ Bn3 . For any bn ≡ (βn , dn ) ∈ Bn3 and a truth-telling bid bnt ≡ (βn , [vn ]−1 (βn )) ∈ Bnt , p((bn , b∗−n ), η) = p((bnt , b∗−n ), η) = βn and the allocated demand to load n with (bn , b∗−n ) and (bnt , b∗−n ) are the same. It implies that the payoffs of load n with (bn , b∗−n ) and (bnt , b∗−n ) are the same as each other. (iv) In case bn ∈ Bn4 . For any bn ∈ Bn4 , the load n will be fully allocated with respect to the bid profile (bn , b∗−n ), and the clearing price  p≡ p ((bn , b∗−n ); η) = w N (dn ) + η(βn − w N (dn )); then for any fixed η ∈ [0, 1], we can find a truth-telling bid bnt ∈ Bnt , such that p . Moreover, we have the clearing price with bnt = (βnt , [vn ]−1 (βnt )) with βnt =  p as well, and thus we can further verify respect to the bid profile (bnt , b∗−n ) is  that dn ≤ xnt ≤ [vn ]−1 (βnt ), where xnt ≡ xn∗ (bnt , b∗−n ) represents the allocation to load n with (bnt , b∗−n ). Thus

Appendices

133

Δf N ,n  f N ,n (bn , b∗−n ; η) − f N ,n (bnt , b∗−n ; η) mcp

mcp

mcp

p (dn − xnt ) = vn (dn ) − vn (xnt ) −  p (dn − xnt ) ≤ vn (xnt )(dn − xnt ) −  p )(dn − xnt ) ≤ 0, = (vn (xnt ) −  p which holds by where the last inequality holds since dn ≤ xnt , and vn (xnt ) ≥  xnt ≤ [vn ]−1 (βnt , b∗−n ) and the concavity of vn under Assumption (A2).

Proof of Theorem 5.3 By Theorem 5.2, we consider the best response of load n with respect to b∗−n under the bn , b∗−n ) the optimal allocation MCP mechanism is a truthful bid. Denote by  x ≡ x ∗ ( ∗  p (η) the clearing price with respect to ( bn , b∗−n ). By with respect to (bn , b−n ) and  ∗  (5.6) and (5.12), the payoff of load n with respect to (bn , b−n ) under the MCP mcp bn , b∗−n ; η), is given below: mechanism, f N ,n ( mcp f N ,n ( bn , b∗−n ; η) = vn ( xn ) −  p (η) xn =



 xn

0

vn (x)d x −  p (η) xn ,

(5.33)

and the payoff of load n with respect to b∗ and f N ,n (b∗ ; η), is specified below: mcp

mcp f N ,n (b∗ ; η)

=

vn (dn∗ )

−

p ∗ dn∗



dn∗

= 0

vn (x)d x − p ∗ dn∗ ,

(5.34)

where by (5.10), we have p ∗ = w N (dn∗ ) with w N (d) defined in (5.30). mcp mcp mcp mcp bn , b∗−n ; η); then Δf N ,n satisfies the We consider Δf N ,n ≡ f N ,n (b∗ ; η) − f N ,n ( following analysis: mcp Δf N ,n

 =

dn∗

 xn

vn (x)d x − p ∗ (dn∗ −  xn ) − ( p∗ −  p (η)) xn .

(5.35)

We define a pair of linear functions, ν(x) and ν(x) below: ν(xn )  κ(xn − dn∗ ) + p ∗ ,

with κ  inf

inf {vn (xn )};

(5.36a)

ν(xn )  κ(xn − dn∗ ) + p ∗ ,

with κ  sup

sup {vn (xn )};

(5.36b)

n∈N xn ∈[0,Γn ] n∈N xn ∈[0,Γn ]

see Fig. 5.5 for an illustration. By (5.36) and under Assumption (A2), we have κ ≤ vn (xn ) ≤ κ < 0; then for any xn ∈ [0, dn∗ ], we have κ(xn − dn∗ ) ≤ vn (xn ) − vn (dn∗ ) ≤ κ(xn − dn∗ ), by which together with vn (dn∗ ) = p ∗ , we can obtain that

134

5 Large-Scale Elastic Load Management Under Auction Games

Fig. 5.5 Illustration of linear functions with inf and sup slopes

ν(xn ) ≤ vn (xn ) ≤ ν(xn ), for any xn ∈ [0, dn∗ ].

(5.37)

In the following, we will show in (I) that b∗ cannot be a NE, and in (II) that b∗ is an ε-NE respectively. bn such that (I) It is sufficient to show that b∗ is not a NE if there exists a bid  bn , b∗−n ). Δf n < 0 with respect to b∗ and ( Here we consider a truthful bid  bn , such that dn ≥ dn∗ .  By (5.17), we have βn (dn ) = vn (dn ) for all (βn , dn ) ∈ Bnt ; then under Assumption (A2), the following property holds: βn (x1 ) > βn (x2 ) > 0, with x1 < x2 ,

(5.38)

i.e., βn decreases with respect to dn for all (βn , dn ) ∈ Bnt . n ≡ βn (dn ) ≤ βn (dn∗ ) = βn∗ , since dn ≥ dn∗ , by which together Then we have β ∗ ∗ n ≤ βm∗ . It implies that the with βn = βm , m ∈ N /{n} in Lemma 5.1, we have β other loads will be allocated first before load n is allocated. Hence we have  xm = dm∗ .  ∗  xn ≤ dn∗ . By c N strictly increases on the total demand and βn ≤ βn , we have  ∗  n = By (5.10) and βn ≤ βm , we have, in this case, the clearing price  p (η) = β xn ), see an illustration in Fig. 5.6. w N ( mcp

Δf N ,n ≤

 d∗ n  xn

ν(x)d x − p ∗ (dn∗ −  xn ) − ( p∗ −  p (η)) xn ,

by (5.35) and (5.37), with ν(x)defined in (5.36a) 1 = − (w N (dn∗ ) − w N ( x n )2 , xn )) xn − κ(dn∗ −  2 since p ∗ = w N (dn∗ ) and  p (η) = w N ( xn ) 1 ≤ − inf{cN }(dn∗ −  xn ) xn − κ(dn∗ −  xn )2 , with inf {cN } given in (5.14b) 2  1 xn + κ(dn∗ −  xn ) (dn∗ −  xn ); = − 2 inf{cN } 2

Appendices

135

Fig. 5.6 An illustration of the clearing price  p and allocation  xn w.r.t. ( bn , b∗−n ) ∗  in case dn ≥ dn

mcp

then, we obtain that Δf N ,n < 0, in case  xn satisfies the follows:   xn ∈

 |κ| ∗ ∗ d ,d , 2 inf{cN } + |κ| n n

(5.39)

which implies that b∗ cannot be a NE for finite population MCP auction games. (II) we will show that b∗ is an ε-NE in (II.a) and (II.b). (II.a) In case dn ≥ dn∗ . Parallel with the analysis given in (I), we have the following: mcp

Δf N ,n ≥

=

≥ = =

 d∗ n  xn

ν(x)d x − p∗ (dn∗ −  xn ) − ( p∗ −  p(η)) xn ,

by (5.35) and (5.37), with ν(x) defined in (5.36b) 1 − (w N (dn∗ ) − w N ( xn )) xn − κ(dn∗ −  x n )2 , 2 since p ∗ = w N (dn∗ ) and  p(η) = w N ( xn ) 1 − sup{cN }(dn∗ −  xn ) xn − κ(dn∗ −  xn )2 , with sup {cN } given in (5.14a) 2  1 2 sup{cN } − xn + κ(dn∗ −  xn ) (dn∗ −  xn ) 2  2 sup{cN } + |κ| ∗ [sup{cN }]2 1  (2 sup{c N } + |κ|)  xn − − d [d ∗ ]2 2 2 sup{cN } + |κ| n 4 sup{cN } + 2|κ| n

≥ − ≥ −

[sup{cN }]2

[d ∗ ]2 4 sup{cN } + 2|κ| n [sup{cN }]2

4 sup{cN } + 2|κ|

Γ 2,

since dn∗ ≤ Γn < Γ, for alln ∈ N .

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5 Large-Scale Elastic Load Management Under Auction Games

Fig. 5.7 An illustration of clearing price w.r.t. ( bn , b∗−n ) in case dn < dn∗

d

(II.b) In case dn < dn∗ . n = βn (dn ) > βn (dn∗ ) = βn∗ , by which together By (5.38) and dn < dn∗ , we have β ∗ ∗ n > βm∗ . It implies that load n with βn = βm , m ∈ N /{n} in Lemma 5.1, we have β ∗  will be allocated first. We considered dn < dn , i.e., load n changes his bid to dn to xm = dm∗ , that is, all of the require less electricity resources; then we have  xn = dn ,  loads are fully allocated. n > βm∗ , we have  p (η), the clearing price with respect to ( bn , b∗−n ), By (5.10) and β is given as  p (η) = w N (dn ) + η( p ∗ − w N (dn )), see Fig. 5.7 for an illustration. mcp mcp bn , b∗−n ; η) ≤ f N ,n ( bn , b∗−n ; 0) holds for all η ∈ [0, 1], Hence we have f N ,n ( since  p (η) ≥  p (0) for all η ∈ [0, 1]. n , dn ) such that β n =  p (η); then, with η = 0, Consider another truthful bid  bn ≡ (β mcp  mcp  ∗ ∗ we have f N ,n (bn , b−n ; 0) = f N ,n (bn , b−n ; 0). p ≤ p∗ . Also we have dn ≥ dn∗ since  Thus by the above analysis, for any  bn such that dn < dn∗ , we have mcp mcp mcp mcp mcp bn , b∗−n ; 0) = f N ,n (b∗ ; η) − f N ,n ( bn , b∗−n ; 0) Δf N ,n ≥ f N ,n (b∗ ; η) − f N ,n (

n , dn ) such that β n =  with  bn ≡ (β p (η) and hence dn ≥ dn∗ . mcp  mcp  ∗ n =  Also f N ,n (bn , b−n ; η) = f N ,n (bn , b∗−n ; 0) for all η ∈ [0, 1], with β p (η) ≤ mcp mcp ∗ mcp  ∗ ∗ p ; then Δf N ,n = f N ,n (b ; η) − f N ,n (bn , b−n ; η). Hence by the analysis given in mcp

[sup{c }]2

N (II.a), we have Δf N ,n ≥ − 4 sup{c }+2|κ| Γ 2. N In summary, by (II.a) and (II.b), the following holds

mcp

Δf N ,n ≥ −

[sup{cN }]2 Γ 2, 4 sup{cN } + 2|κ|

for all η ∈ [0, 1], all  bn ∈ Btt and all n ∈ N , which implies the conclusion.

References

137

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26. P. Maillé, B. Tuffin, The progressive second price mechanism in a stochastic environment. Netnomics 5(2), 119–147 (2003) 27. R. Jain, J. Walrand, An efficient Nash-implementation mechanism for network resource allocation. Automatica 46, 1276–1283 (2010) 28. O. Marce, H.-H. Tran, B. Tuffin, Double-sided auctions applied to vertical handover for mobility management in wireless networks. J. Netw. Syst. Manag. 22(4), 658–681 (2014) 29. S. Zou, Z. Ma, X. Liu, Auction-based distributed efficient economic operations of microgrid systems. Int. J. Control 87(12), 2446–2462 (2014) 30. P. Jia, P. Caines, Analysis of quantized double auctions with application to competitive electricity markets. INFOR: Inf. Syst. Oper. Res. 48(4), 239–250 (2010) 31. E. Bompard, Y. Ma, R. Napoli, G. Abrate, The demand elasticity impacts on the strategic bidding behavior of the electricity producers. IEEE Trans. Power Syst. 22(1), 188–197 (2007) 32. V.P. Gountis, A.G. Bakirtzis, Bidding strategies for electricity producers in a competitive electricity marketplace. IEEE Trans. Power Syst. 19(1), 356–365 (2004) 33. D.S. Kirschen, Demand-side view of electricity markets. IEEE Trans. Power Syst. 18(2), 520– 527 (2003)

Chapter 6

Economic Operations of Microgrid Systems Under Auction Games

Abstract This chapter studies the economic operations of the microgrid in a distributed way such that the operational schedule of each of units, like generators, load units, storage units, etc., in a microgrid system is implemented by autonomous agents. In this problem, the divisible resource is electricity resource in the system and we apply the progressive second price (PSP) auction mechanism to efficiently allocate the resource. Considering the economic operation for the microgrid systems, the generators play as sellers to supply energy and the load units play as the buyers to consume energy, while a storage unit, like battery, supercapacitor, etc., may transit between buyer and seller, such that it is a buyer when it charges and becomes a seller when it discharges. This problem is different from the double-sided auction game specified in Chap. 3 due to the existence of the storage units. Furthermore, when the microgrid is in a connected mode, each individual unit competes against not only the other individual units in the microgrid but also the exogenous main grid possessing fixed electricity price and infinite trade capacity; that is to say, the auctioneer assigns the electricity among all individual units and the main grid with respect to the submitted bid strategies of all individual units in the microgrid in an economic way. Due to these distinct characteristics, the underlying auction games are distinct from those studied in the literature. We show that under mild conditions, the efficient economic operation strategy is a Nash equilibrium (NE) for the PSP auction games, and propose a distributed algorithm under which the system can converge to a NE. We also show that the performance of worst NE can be bounded with respect to the system parameters, say the energy trading price with the main grid, and based upon that, the implemented NE is unique and efficient under some conditions.

6.1 Introduction A microgrid system, see [1, 2] is a cluster of generators, storages, and loads which operates as a single controllable system that provides power and heat to its local area, and presents itself to the main grid as a single controllable unit. Recently more and more works have been dedicated to study the operation and management of the microgrid which ranges from centralized control, e.g., [3–5], to partially decentral© Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_6

139

140

6 Economic Operations of Microgrid Systems Under Auction Games

ized control method, e.g., hierarchical control in [6, 7], to fully decentralized control dependent upon local information to improve the steady-state and transient response of the microgrid, e.g., droop control in [8, 9], and PQ control discussed in [10, 11] and references therein. We study economic operations of the microgrid with dispatchable conventional (distributed) generators, controllable loads, and storages, like battery, supercapacitor, etc., with coordinated charging and discharging behavior, see [11, 12]. In a microgrid system, traditional micro source units can be renewable ones, e.g., photovoltaics and wind turbines, etc., and these kinds of energy can be predicted in advance ([13] presents a power forecasting module) and shall be fully consumed. There are critical loads which are inelastic and dispatched loads which are elastic and can be shed when necessary. Storage units can act as either elastic loads or dispatchable generators since they can charge or discharge. The efficient (or socially optimal) microgrid operation considered in this chapter is to allocate resources, with available information, among the units in the microgrid and power traded with main grids when the microgrid is connected with the main grid. Units in the microgrid in a low-voltage distribution network may belong to distinct autonomous individual owners. It makes agent-based microgrid operations feasible, e.g., [14, 15] where the authors presented an auction-based distributed algorithm with a fixed amount of energy, based on symmetric assignment problem for the optimal energy exchange. Actually, the agent-based methods had been widely applied in the economic dispatch problems in deregulated electricity markets, e.g., [16] and network bandwidth resource sharing problems, e.g., [17, 18] and references therein. In this chapter, we will adapt the so-called progressive second price (PSP) auction mechanism, which was proposed in [17, 19] to the microgrid economic operation problems. Under the PSP auction mechanism, the incentive compatibility holds, i.e., the truth-telling bid profile is the best response of agents. Hence the PSP auction can be considered as the extension of the second price auction proposed in [20–22] to allocate indivisible items with truth-telling bid profiles in an efficient way. Note that the incentive compatibility does not hold in general for other sealed auction mechanisms, see [23], like uniform market clearing price (MCP), e.g., [24], and pay as bid (PAD), e.g., [25], auction mechanisms, etc., which have been widely adapted to economically dispatch the generations in day-ahead deregulated electricity markets, [26]. In the allocation problems of divisible resources under the PSP auction mechanism, each buyer reports a two-dimension bid profile which is composed of a maximum amount of demand and an associated buying price, and is used to replace his complete (private) utility function, while each seller reports a two-dimension bid profile which is composed of a maximum amount of supply and an associated selling price, and is used to replace his complete (private) cost function. The transfer money (or payment) of an agent is that, the utility of the whole system, when this agent does not join the auction process, minus the aggregated utility of the remaining system except for this agent when he joins the auction process. That is to say, the payment of each agent is exactly the externality he imposes on the system through his participation. In [17], the authors verified that the efficient (truth-telling) bid profile

6.1 Introduction

141

is a Nash equilibrium (NE) of the PSP auction games for network resources. More recently much progress has been developed in PSP auction games. For example, Jain and Walrand extended the results developed in [17] from single side auctions to double-side auction games in the network resource allocations, [27]; A distributed algorithm, under quantized-PSP auction mechanism was proposed for power electricity sharing games in a short period of time, see [28, 29], following which the auction game system converges to the efficient power electricity allocations. This chapter is directly related to the work of Jain and Walrand [27] where the authors extended the work by Lazar and Semret [19] and Semret [32] to double-side PSP auction games and showed the existence of efficient NE by verifying that a player, either a seller or a buyer, can’t increase his individual payoff whenever the allocation to this player fluctuates from the efficient allocation due to the unilateral deviation of his bid profile from the efficient bid profile. In the microgrid economic operation problems formulated in our work, besides each elastic load as a buyer and each generator as a seller usually considered in the literature, each storage unit can play as either a buyer when it charges or a seller when it discharges; moreover, in a connected mode, the microgrid can trade with the main grid which is usually assumed to possess infinite (demand/supply) capacity of electricity resources with certain fixed (buying/selling) trading price. Consequently different from the double-side auction games formulated in Jain and Walrand [27], the microgrid economic operation auction games have the following specific characteristics: • In a connected mode, each individual unit competes against not only the other individual units in the microgrid but also the exogenous main grid possessing fixed electricity price and infinite trade capacity, that is to say, the auctioneer assigns the electricity among all individual units and the main grid with respect to the submitted bid strategies of all individual units in the microgrid in an economic way; • The bid strategy for a storage unit may charge or discharge with respect to the submitted bid strategies of other units in the microgrid and the electricity trade price of the main grid. Hence due to the above significant distinct characteristics of the underlying auction games, in the proof of the existence of efficient NE, except for the discussions on the fluctuations of the allocations to other buyers and sellers for a buyer and a seller, the remaining rigorous analysis on the variation of the individual payment, which are influenced by the storage units and energy trade between the microgrid and the main grid, are beyond the scope covered in Jain and Walrand [27] and other works in the literature. In this chapter, we also introduce the so-called notion of “price of anarchy”, for the game system under PSP mechanism, to measure the performance gap between the worst NE and the efficient one; then we specify the value of this notion with respect to the system parameters, say the energy trading price with the main grid. Based on the analysis, we can claim that the NE is unique and efficient under some conditions.

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6 Economic Operations of Microgrid Systems Under Auction Games

The work studied in the chapter degenerates into a double-side PSP auction game analyzed by Jain and Walrand [27] in case that the microgrid is operated in an isolated mode, that is to say, there is no energy exchange between the microgrid and the main grid, and no storage units are involved in the microgrid system. We propose the best response update algorithm such that individual players sequentially update their own best responses with respect to the bidding profiles of other players. We also show that the system converges to a NE following the proposed algorithm by verifying that the underlying auction games are ordinal potential games. However, there may exist other NEs different from the efficient NE. Actually, the system may not converge to the efficient NE following the algorithm proposed in the chapter. The organization of the chapter is as follows: In Sect. 6.2, we formulate a class of economic operations of the microgrid systems. In Sect. 6.3, the distributed economic operations for the microgrid under the PSP auction mechanism in the connected mode is presented. In Sect. 6.4, we propose a distributed best bid strategy update mechanism under which the system can converge to a NE. Economic operations in the isolated mode are briefly discussed in Sect. 6.5. Section 6.6 lists some future research works. For the sake of clarity, we list part of the key symbols in Table 6.1.

Table 6.1 List of key symbols (βi , di )

(α j , s j )

(bk , ak )

( ps , pb )

(x, y, z)

(es , eb )

Bid profile of load i βi : maximum (per unit) price load i to pay di : maximum power load i demanding Ask-bid profile of generator j α j : minimum price generator j to accept s j : maximum power generator j to supply Bid profile of storage k bk ≡ (βk , dk ): charging bid profile of storage k ak ≡ (αk , sk ): discharging bid profile of storage k Electricity price of the main grid ($/kWh) ps : selling electricity price of the main grid pb : buying electricity price of the main grid Coordination of individual units xi : allocated power of load i y j : supplied power of generator j z k ≡ (xk , yk ): charging & discharging of storage k Power flow between the microgrid and the main grid es : flow from the microgrid toward the main grid eb : flow toward the microgrid from the main grid

6.2 Formulation of Microgrid Economic Operation Problems

143

6.2 Formulation of Microgrid Economic Operation Problems Denote I , J and K a set of elastic loads, dispatchable generators, and storage units, respectively in a microgrid. We denote y j,t the power supplied by generator j at interval t ∈ T with T representing the whole operation interval of the microgrid, and xi,t the demand of elastic load i at t, such that 0 ≤ y j,t ≤ e j,max ,

and

0 ≤ xi,t ≤ ei,max

(6.1)

where e j,max and ei,max represent the maximal generation capacity of generator j and the maximal required capacity of load i, respectively. For each of storages k, with k ∈ K , we denote xk,t and yk,t the charging and discharging rate at interval t, respectively, and denote sock,t , such that socmin ≤ sock,t ≤ socmax ,

for all t,

(6.2)

the value of state of charge at the end of subinterval t, i.e., the ratio of the stored energy of a storage to its maximum attainable energy capacity; then subject to a coordination behavior (xk,t , yk,t ) during interval t, we have sock,t+1 = sock,t +

ΔT + (γ xk,t − yk,t ), Γk k

(6.3)

where Γk and γk+ , with 0 < γk+ < 1, represent the maximum attainable energy capacity and the charging efficiency of storage k, respectively, and ΔT denotes the uniform length of any subinterval t. We say a pair of (xk,t , yk,t ) admissible if xk,t yk,t = 0, Γk (socmax − sock,t ), ΔT γk+ Γk (sock,t − socmin ), ≤ A− k ≡ ΔT

(6.4a)

0 ≤ xk,t ≤ A+ k ≡

(6.4b)

0 ≤ yk,t

(6.4c)

Besides the dispatchable units specified above, there are intermittent renewable energy units and loads with inelastic demand in the microgrid, such that prior to a subinterval t, (i) the aggregated power generated by renewable resource units during the subinterval t, denoted by er,t , is predicted, and (ii) the total inelastic demand of those loads, denoted by ed,t , is given. For analytical simplicity er,t and ed,t are assumed to be constant during interval t. Moreover in order to maximize the social welfare and become as environmentally friendly as possible, we assume that the power generated by the renewable generators shall be fully utilized.

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6 Economic Operations of Microgrid Systems Under Auction Games

In the isolated mode, the units in the microgrid system interact with each other to minimize operational cost and no power flows between the main grid and the microgrid; while in the connection mode we suppose that the microgrid is permitted to exchange the electricity with the main gird bidirectionally, i.e., the power flow can be from the main grid toward the microgrid or vice versa. Denote ps,t and es,t the selling price and power flow toward the main grid from the microgrid, and pb,t and eb,t the buying price and power flow toward the microgrid from the main grid, respectively. To avoid the microgrid systems to make money by selling the bought electricity from the main grid, we suppose that ps,t < pb,t for all t. A collection of strategies of the microgrid system at ≡ (xt , yt , z t , eb,t , es,t ), during the interval t, is called an admissible strategy with respect to er,t , ed,t and sock,t , for all k, if at satisfies the constraints (6.1) and (6.4), and the power conservation law below:     y j,t + γk− yk,t + eb,t = ed,t + xi,t + xk,t + es,t (6.5) er,t + j

k

i

k

where γk− , with 0 < γk− ≤ 1, is the discharging efficiency of storage k. The set of admissible strategies is denoted S .

6.2.1 Economic Operations of Microgrid Systems We specify an operation cost of the microgrid, denoted by J , subject to a collection of admissible strategies at , as below: J (at ) =

 j∈J

c j (y j,t ) −

 i∈I

vi (xi,t ) +



wk (z k,t ) + pb,t eb,t − ps,t es,t

(6.6)

k∈K

where vi , c j and wk represent a utility function of load i, a cost function of generator j, and a cost function of storage k, respectively. The objective of the economic operation of the microgrid is to assign a socially optimal allocation to minimize the operation cost (6.6) over the set of admissible strategies S . We call the socially optimal allocation profile efficient. The underlying economic operation problems are optimization problems with inequality and equality constraints, and can be solved by the Lagrange multiplier methods. Let λ ≡ (λx , μ y , (λ, μ)z ) be the Lagrange multiplier corresponding to the inequality constraints (6.1), (6.2) and (6.4), with λx ≡ (λi ; i ∈ I ), μ y ≡ (μ j ; j ∈ J ), (λ, μ)z ≡ ((λk , μk ); k ∈ K ), and ν is the Lagrange multiplier corresponding to the equality constraint of (6.5); then it can be verified that the associated KKT conditions are (6.5) together with (6.7) specified below:

6.2 Formulation of Microgrid Economic Operation Problems

⎧ vi (xi,t ) − ν − λi ≤ 0, xi,t ≥ 0, (vi (xi,t ) − ν − λi )xi,t ⎪ ⎪ ⎪  ⎪ −c (y j,t ) + ν − μ j ≤ 0, y j,t ≥ 0, (−cj (y j,t ) + ν − μ j )y j,t ⎪ ⎪ ⎨ j ∂w   ∂wk k − − ν − λk ≤ 0, xk,t ≥ 0, − − ν − λk xk,t ⎪ ∂ xk,t ∂ xk,t ⎪ ⎪ ⎪   ∂wk ∂w k ⎪ ⎪ + ν − μk ≤ 0, yk,t ≥ 0, − + ν − μk yk,t ⎩ − ∂ yk,t ∂ yk,t ⎧ ⎪ ⎪ ei,max − xi,t ≥ 0, λi ≥ 0, (ei,max − xi,t )λi = 0 ⎨ e j,max − y j,t ≥ 0, μ j ≥ 0, (e j,max − y j,t )μ j = 0 , (A+ A+ ⎪ k − x k,t ≥ 0, λk ≥ 0, k − x k,t )λk = 0 ⎪ ⎩ (A− A− k − yk,t ≥ 0, μk ≥ 0, k − yk,t )μk = 0 − pb,t + ν ≤ 0, eb,t ≥ 0, (− pb,t + ν)eb,t = 0 . ps,t − ν ≤ 0, es,t ≥ 0, ( ps,t − ν)es,t = 0

145

=0 =0 = 0 , (6.7a) =0

(6.7b)

(6.7c)

In this chapter we consider the following properties: • (A1) vi (xi ), with i ∈ I , is increasing and strictly concave on xi ; • (A2) c j (y j ), with j ∈ J , is increasing and strictly convex on y j . • (A3) wk (xk,t , yk,t ) = δk (sock,t+1 − socr e f )2 , with socr e f ∈ [0, 1] and δk > 0. Under Assumptions (A1)–(A3), the underlying economic operation problems are convex optimization problems, hence the necessary KKT conditions (6.7) are also sufficient conditions for the optimality of the optimization problems, see [30]. Thus under Assumptions (A1)–(A3), there exists a unique solution and it is characterized by the KKT conditions specified in (6.7). Lemma 6.1 Suppose (eb∗ , es∗ ) is the energy exchange between the microgrid and the main grid subject to the efficient allocation (6.6) under Assumptions (A1)–(A3); then es∗ eb∗ = 0 in case ps < pb . Proof Let (eb∗ , es∗ ) be an efficient allocation and eb∗ es∗ = 0 and suppose that there exists another efficient allocation (eb† , es† ) and eb† es† = 0; then by the power conservation law, we can get that eb∗ − es∗ = eb† − es† . First we consider that es∗ ≥ 0, eb∗ = 0; then es∗ = es† − eb† and the following holds: ΔJ = J (eb† , es† ) − J (eb∗ , es∗ ) = ( pb eb† − ps es† ) − ( pb eb∗ − ps es∗ ) = ( pb − ps )eb† , >0

since es∗ = es† − eb†

Following the same technique, the above holds in case es∗ = 0, eb∗ ≥ 0.

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6 Economic Operations of Microgrid Systems Under Auction Games

6.2.2 A Simulation Example In this section, we will illustrate the performance of the auction-based algorithm with a few of simulation examples. For the purpose of demonstration we consider a microgrid composed of a wind turbine, a collection of inelastic loads, a diesel generator with generation capacity of 30kW with cost function c(yt ) = 0.07yt1.17 , see [31], an elastic load with utility function specified as v(xt ) = 0.115xt0.9 , a storage possessing a maximum capacity of 30kWh, and its cost function w(xt , yt ) = δ(socr e f − soct+1 )2 , with an initial SOC value as soc0 = 0.45, δ = 5, socr e f = 0.5. We also consider socmin = 0.1, socmax = 0.9, and charging and discharging efficiencies γ + = γ − = 98%. Ahead of any interval t the generation of wind turbine er,t and the aggregated demand of the collection of inelastic loads is illustrated in Fig. 6.1. We consider ps,t < pb,t for all t, such that ps,t = 0.09$/kWh and pb,t = 0.095$/kWh for all t. Figure 6.2 illustrates the efficient allocation profile for economic operation problems with respect to the specification given above. The efficient allocation is determined by the interaction of the electricity trade price and the marginal valuation (or cost). In case the marginal valuation of the load is relatively high, the system tends to satisfy the load’s requirements; while in case the marginal cost of the generator is relatively low, the generator tends to supply resources. The storage unit tends to charge when the buying price is relatively low and the resources supplied by generators are more than those demanded by loads, and discharges vise versa. If the buying electricity price is low, the system tends to buy resources from the main grid and if the selling electricity price is high, the system tends to sell resources to the main grid.

Fig. 6.1 Generations of wind turbine and demands of inelastic loads

6.2 Formulation of Microgrid Economic Operation Problems

147

Fig. 6.2 Efficient allocation with respect to a retailed electricity price

For example, as illustrated in Fig. 6.2, during time intervals 12:00–15:00, the inelastic demand is more than the renewable generation, so the allocation to the elastic load is little, while the generation supply, storage discharge, and the resources bought from the main grid are large. During time intervals 16:00–17:00, the renewable generation increases and then the load demand increases and the storage charges while the resources bought from the main grid decreases. At 17:00, there exist some resources to sell to the main grid. The efficient operations of the microgrid systems can be effectively implemented in case that the system controller has complete information and can directly schedule the behaviors of all individual units. However, in practice, the individual units may not want to share their private information with others or do not permit the system to directly control their units. In this situation, distributed coordination methods with limited information can be adapted.

6.3 Economic Operations in Connected Mode Under Auction Mechanism A typical microgrid is composed of generators, loads, and storage units which can be divided into three categories. The first category is buyers, i.e., (non-critical) elastic loads. The second category is sellers, i.e., (non-renewable) generators. The last category is storages, which can act as either buyers or sellers, depending on the operation condition of the microgrid. The main grid creates extra resources to balance the flow in the microgrid. An auction is a mechanism consisting of the unit (agents)

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6 Economic Operations of Microgrid Systems Under Auction Games

Fig. 6.3 PSP auction diagram for a microgrid system in connected mode

submitting bids and the auctioneer allocating resources to units based on their bids. This type of PSP mechanism is an extension of that studied in [27] where the units report a two-dimension bid parameters instead of reporting their types or complete utility functions. The auction mechanism is illustrated in Fig. 6.3.

6.3.1 Bid Profiles of Individual Units in Microgrid Systems As a buyer, a load i submits a (two-dimension) bid profile bi,t , such that bi,t = (βi,t , di,t ), with 0 ≤ di,t ≤ ei,max ,

(6.8)

of the revealed utility function

vi (xi,t ) = βi,t min(xi,t , di,t ), where specifies the maximum per unit price βi,t that the load i is willing to pay and demands up to di,t units of the electricity; then the corresponding admissible strategy xi,t of load i, with respect to bi,t , must satisfy: 0 ≤ xi,t ≤ di,t As a seller, a generator j specifies a (two-dimension) ask-bid a j,t , such that a j,t = (α j,t , s j,t ), with 0 ≤ s j,t ≤ e j,max ,

(6.9)

of the revealed cost function

c j (y j,t ) = α j,t min(y j,t , s j,t ), where α j,t is the minimum per unit price that generator j is willing to accept and can supply up to s j,t units of the electricity; then the corresponding admissible strategy y j,t of generator j, with respect to a j,t , must satisfies:

6.3 Economic Operations in Connected Mode Under Auction Mechanism

149

0 ≤ y j,t ≤ s j,t A storage can be either a buyer when it charges or a seller when it discharges. As a result, a storage k submits a (four-dimension) bid profile rk,t at interval t, with rk,t ≡ (bk,t , ak,t ), such that bk,t and ak,t are specified as follows: bk,t = (βk,t , dk,t ), with 0 ≤ dk,t ≤ A+ k ,

(6.10a)

ak,t = (αk,t , sk,t ), with 0 ≤ sk,t ≤

(6.10b)

A− k ,

with dk,t sk,t = 0, i.e., either dk,t or sk,t has to be zero valued, where (i) βk,t is the maximum per unit price that the storage k is willing to pay and can charge up to dk,t units of the electricity, and (ii) αk,t is the minimum per unit price that the storage k is willing to accept and can discharge up to sk,t units of the electricity. Note: The constraint of dk,t sk,t = 0 avoids the case that the storage unit k plays as a seller and a buyer simultaneously, for any admissible bid profile of storage unit k. Hence, the revealed utility function of the storage k with respect to the bid profile specified in (6.10) is given as follows:

wk (xk,t , yk,t ) = βk,t min(xk,t , dk,t ) − αk,t min(yk,t , sk,t ).

(6.11)

The corresponding admissible strategy xi,t of load i, with respect to bi,t , must satisfy: 0 ≤ xk,t ≤ dk,t , and 0 ≤ yk,t ≤ sk,t Remark For notational simplicity, in the rest of the chapter, we skip the time index t in case no confusions are involved.

6.3.2 Resource Allocation Rule Subject to Bid Profiles of Units Definition 6.1 Considering a collection of bid profiles (b, a, r ), we call a ≡ (x, y, z, eb , es ) an admissible allocation with respect to (b, a, r ), if the power conservation law (6.5) and the following constraints hold: 0 ≤ xi ≤ di 0 ≤ yj ≤ sj (0, 0) ≤ (xk , yk ) ≤ (dk , sk ) 0 ≤ es , eb

(6.12a) (6.12b) (6.12c) (6.12d)

The set of admissible allocations with respect to (b, a, r ) is denoted by A (b, a, r ).

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6 Economic Operations of Microgrid Systems Under Auction Games

We further define a function U on an admissible allocation a ≡ (x, y, z, eb , es ) with respect to a bid profile (b, a, r ) as the following: U (a) =

 i∈I

βi xi −

 j∈J

αj yj +



(βk γk+ xk − αk yk ) + ps es − pb eb .

(6.13)

k∈K

Remark U (a) can be interpreted as the total monetary income of the microgrid with allocation a = (x, y, z, eb , es ); then in a connected mode, each individual unit competes against not only the other individual units in the microgrid but also the exogenous main grid possessing fixed electricity price and infinite trade capacity, that is to say, the auctioneer assigns the electricity among all individual units and the main grid with respect to the submitted bid strategies of all individual units in the microgrid in an economic way studied below. The auctioneer will assign an optimal admissible allocation a∗ ≡(x ∗ , y ∗ , z ∗ , eb∗ , es∗ ) with respect to a collection of a bid profile (b, a, r ), such that a∗ = argmax {U (a)}.

(6.14)

a∈A (b,a,r )

Let ν be the Lagrange multiplier corresponding to the equality constraint, and λ = (λx , μ y , (λ, μ)z ) be the Lagrange multipliers corresponding to the inequality constraints, with λx ≡ (λi , i ∈ I ), μ y ≡ (μ j , j ∈ J ), (λ, μ)z ≡ ((λk , μk ), k ∈ K ) in the auction optimization (6.12) and (6.13); then a∗ can be characterized by the KKT conditions in (6.5), (6.7c) together with (6.15) given below:

βl − ν − λl ≤ 0, xl ≥ 0, (βl − ν − λl )xl = 0, with l ∈ I ∪ K , −αl + ν − μl ≤ 0, yl ≥ 0, (−αl + ν − μl )yl = 0, with l ∈ J ∪ K ,

(6.15a)

dl − xl ≥ 0, λl ≥ 0, (dl − xl )λl = 0, with l ∈ I ∪ K , sl − yl ≥ 0, μl ≥ 0, (sl − yl )μl = 0, with l ∈ J ∪ K .

(6.15b)

6.3.3 Transfer Money of Individual Units Subject to Bid Profiles Considering a collection of bid profiles, we will specify the so-called transfer money, denoted τ , for each of the loads, generators, and storages following the allocation way by the system auctioneer given in (6.14). Essentially, the transfer money of an individual unit can be interpreted as: the summation of all users’ utility functions when this unit doesn’t join the auction, minus the summation of all of the other units’ utility functions when this unit joins the auction. That is to say, the money transfer τ made by each of the related units, is exactly the externality he imposes on others through his participation.

6.3 Economic Operations in Connected Mode Under Auction Mechanism

6.3.3.1

151

Money Transfer of Load Units

Let

a∗(i) denote the solution to the allocation rule, defined in Sect. 6.3.2, but with, 0 ≤ xi ≤ di , the first constraint in the set of constraints (6.12) substituted with di = 0 for load i, i.e.,

a∗(i) =

argmax

a∈A ((

bi ,b−i ),a,r )

{U (a)}

(6.16)

i , d i ) with d i = 0. where

bi = (β The money transfer to be made by load i (the payment) with bid profiles (b, a, r ), denoted by τi (b, a, r ), is given by (6.17) below, a∗(i) ) − (U (a∗ ) − βi xi∗ ). τi (b, a, r ) = U (

(6.17)

with a∗ ≡ a∗ (b, a, r ) as specified in (6.14). 6.3.3.2

Money Transfer of Generator Units

Let

a∗( j) denote the solution to the allocation rule, defined in Sect. 6.3.2, but with the 2nd constraint in the set of constraints (6.12) substituted with s j = 0 for generator j, i.e.,

a∗( j) =

argmax a∈A (b,(

a j ,a− j ),r )

{U (a)}

(6.18)

where

a j = (

α j ,

s j ) with

s j = 0. The money transfer to be made by generator j (the payment) with bid profiles (b, a, r ), denoted by τ j (b, a, r ), is given (6.19) below, a∗( j) ) − (U (a∗ ) + α j y ∗j ). τ j (b, a, r ) = U (

(6.19)

Remark The negative value of τ j means money transfer to the generator j.

6.3.3.3

Money Transfer of Storage Units

Let

a∗(k) denote the solution to the allocation rule, defined in Sect. 6.3.2, but with the 3rd and 4th constraints in the set of constraints (6.12) substituted with dk = sk = 0 for storage k, i.e.,

a∗(k) =

argmax

{U (a)}

a∈A (b,a,(

rk ,r−k ))

k , d k ), (

αk ,

sk )) with d k =

sk = 0. where

rk = ((β

(6.20)

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6 Economic Operations of Microgrid Systems Under Auction Games

The money transfer to be made by storage k with bid profiles (b, a, r ), denoted by τk (b, a, r ), is given by (6.21) below, a∗(k) ) − (U (a∗ ) − (βk γk+ xk∗ − αk yk∗ )). τk (b, a, r ) = U (

(6.21)

Remark Since a storage can either charge or discharge, the value of τk , the money transfer to the storage k can be negative or positive.

6.3.4 Payoff Functions of Individual Units Subject to a collection of bid profiles, the payoff functions of individual units are specified by adapting the auctioneer’s optimal resource allocation and the money transfer mechanism: Load i has a payoff function u i (b, a, r ), such that u i (b, a, r ) = vi (xi∗ ) − τi (b, a, r ),

(6.22)

where xi∗ ≡ xi∗ (b, a, r ) represents the allocated demand for load i assigned by auctioneer with the bid profiles (b, a, r ), and τi (b, a, r ) is the money payed by the load i defined in (6.17). Generator j has a payoff function u j (b, a, r ), such that u j (b, a, r ) = −τ j (b, a, r ) − c j (y ∗j ),

(6.23)

where y ∗j ≡ y ∗j (b, a, r ) represents the allocated generation power for generator j assigned by auctioneer with (b, a, r ), and τ j (b, a, r ) is the money transfer of the generator j defined in (6.19). Storage k has a payoff function u k (b, a, r ), such that u k (b, a, r ) = −wk (xk∗ , yk∗ ) − τk (b, a, r ),

(6.24)

where xk∗ ≡ xk∗ (b, a, r ) and yk∗ ≡ yk∗ (b, a, r ) represent, respectively, charging and discharging rates for storage k assigned by auctioneer with the bid profiles (b, a, r ), and τk (b, a, r ) is the money transfer of the storage k defined in Sect. 6.3.3.3. The incentive compatibility holds under the PSP auction mechanism, see [17, 19], that is, the truth-telling bid profile, such that bi = (vi (di ), di ), with i ∈ I , a j = (cj (s j ), s j ), with j ∈ J ,  ∂wk ∂wk rk = − , dk , , sk , with k ∈ K , ∂dk ∂sk

(6.25a) (6.25b) (6.25c)

is a best bid strategy among all of the bid profiles for each of individual agents.

6.3 Economic Operations in Connected Mode Under Auction Mechanism

153

6.3.5 Existence of Efficient NE Definition 6.2 A collection of bid profiles b0 ≡ (b0 , a 0 , r 0 ) is a Nash equilibrium for the auction problems if the following holds: bi0 = argmax{u i (bi , b0−i )}, for all i ∈ I ,

(6.26a)

a 0j = argmax{u j (a j , b0− j )}, for all j ∈ J ,

(6.26b)

rk0 = argmax{u k (rk , b0−k )}. for all k ∈ K ,

(6.26c)

bi ∈B i

a j ∈B j

rk ∈B k

In Theorem 6.1, we will show that the bid profile with the efficient allocation is a NE; before that we first give Lemmas 6.2 and 6.3 below. Lemma 6.2 Suppose a∗∗ ≡ (x ∗∗ , y ∗∗ , z ∗∗ , eb∗∗ , es∗∗ ) is the efficient allocation, and consider a collection of bid profiles b∗ such that bi∗ = (vi (di ), di ), with di = xi∗∗ , a ∗j = (cj (s j ), s j ), with s j = y ∗∗ j , 

∂w ∂wk k , dk , , sk , with (dk , sk ) = (xk∗∗ , yk∗∗ ), rk∗ = − ∂dk ∂sk

(6.27a) (6.27b) (6.27c)

for all i ∈ I , j ∈ J and k ∈ K ; then a∗ (b∗ ) = a∗∗ , i.e., the allocation a∗ (b∗ ) is efficient. Proof Following the same technique applied in [27], Lemma 6.2 can be verified by substituting (6.27) into the KKT conditions of (6.7) and (6.15) for the socially optimal and auction allocation problems, respectively, and comparing these two KKT conditions. Lemma 6.3 Suppose a∗ is the allocation subject to (b∗ , a ∗ , r ∗ ) specified in (6.27); then  = ν ∗ + λl∗ , in case xl∗ > 0 βl∗ , for all l ∈ I ∪ K , (6.28a) ≤ ν ∗ + λl∗ , otherwise  ∗ ∗ ∗ ∗ = ν − μl , in case yl > 0 αl , for all l ∈ J ∪ K , (6.28b) ≥ ν ∗ − μl∗ , otherwise ⎧ ⎪ in case eb∗ > 0 ⎨= p b , ∗ where the system price ν = ps , in case es∗ > 0 ; and λl∗ , μl∗ are two param⎪ ⎩ ∈ ( ps , pb ), otherwise eters larger than or equal to zero.

154

6 Economic Operations of Microgrid Systems Under Auction Games

Proof Equation (6.28) can be verified by analyzing the KKT conditions given in (6.7) and (6.15). Theorem 6.1 Under Assumptions (A1)–(A3), the efficient bid profiles b∗ specified in (6.27) is a NE for the underlying auction game. Proof See Appendix for the proof of Theorem 6.1 in details. Due to the above significant distinct characteristics of the underlying auction games, in the proof of the existence of efficient NE analyzed in Theorem 6.1 above, except for the discussions on the fluctuations of the allocations to other buyers and sellers for a buyer and a seller, the remaining rigorous analysis on the variation of the individual payment, which are influenced by the storage units and energy trade between the microgrid and the main grid, are beyond the scope covered in [27] and other works in the literature. Note: The work studied in the chapter degenerates into a double-side PSP auction game analyzed by Jain and Walrand [27] in case that the microgrid is operated in an isolated mode, that is to say, there is no energy exchange between the microgrid and the main grid, and the storage unit is only allowed to discharge, i.e., the storage unit plays as a seller of the electricity resources.

6.3.6 Analysis on Price of Anarchy Under PSP Auction Mechanism In the last section, we show that there exists the efficient NE under PSP auction mechanism, however due to the non-uniqueness of NE, we would evaluate the performance under PSP auction mechanism. We define a notion of η to measure the performance gap between the worst NE and the efficient NE as below: η

J (a∗∗ ) max {J (a∗ (b))}

(6.29)

b∈B 0

where a∗∗ is the efficient allocation as specified in Lemma 6.2, and B 0 represents the set of NE bid profiles. In Corollary 6.1, we will specify the value of η with respect to the pair of parameters ( ps , pb ). Before that, we firstly define a set of bid profiles in Definition 6.3, and show in Lemma 6.4 that any NE has a form as specified in Definition 6.3, and then in Theorem 6.2, we give some properties for NEs. Definition 6.3 We define a bid profile, denoted by b0 (β), with respect to β, with β ∈ [ ps , pb ], such that the following properties hold:

6.3 Economic Operations in Connected Mode Under Auction Mechanism

155

Fig. 6.4 An illustration of a bid profile specified in Definition 6.3

⎧ ⎪ ⎨= β, βl0 ≥ β, ⎪ ⎩ < β, ⎧ ⎪ ⎨= α, αl0 ≤ α, ⎪ ⎩ > α,  er +

in case dl0 ∈ (0, el,max ) , in case dl0 = el,max otherwise

for all l ∈ I ∪ K ,

in case sl0 ∈ (0, el,max ) , for all l ∈ J ∪ K , in case sl0 = el,max otherwise  sl0 + eb0 = ed + dl0 + es0 ,

l∈J ∪K

(6.30a)

(6.30b) (6.30c)

l∈I ∪K

for some α with α ∈ [ ps , β]. Remark We will illustrate a bid profile satisfying Definition 6.3 in Fig. 6.4 below. For the purpose of demonstration, we consider a system composed of an elastic load i, a dispatchable generator j, and a storage unit k and suppose that er = ed . As displayed in Fig. 6.4, we construct a bid profile b such that pb > βi > α j = αk > ps and di = s j + sk with di < ei,max and sl < el,max for l = j or k. Then by the allocation rule, we can verify that b satisfies (6.30), and β = βi and α = α j < β. Lemma 6.4 Suppose that a collection of bid profiles b is a NE; then we have b = b0 (β) ∈ B 0 , for some β ∈ [ ps , pb ],  efficient in case β = α b is , inefficient, in case β > α eb0 = es0 = 0, in case β > α. Proof Proof of Lemma 6.4 is given in Appendix.

(6.31a) (6.31b) (6.31c)

156

6 Economic Operations of Microgrid Systems Under Auction Games

Theorem 6.2 Denote b0 (β ∗ ) as the efficient NE; then for any bid profile b0 (β) ∈ B 0 ,

), b0 (β) ∈ we have β ≥ β ∗ and α ≤ β ∗ . Moreover, for any pair of bid profiles b0 (β 0 B , we can show that

))) > J (a∗ (b0 (β))), J (a∗ (b0 (β

> β, if β

(6.32)

where a∗ (b0 (β)) denotes the allocation subject to bid profile b0 (β) ∈ B 0 ; Proof Denote b0 (β ∗ ) as the efficient NE; then by Lemma 6.4, α ∗ = β ∗ . We would like to show that if a bid profile b0 (β) ∈ B 0 , b0 (β) satisfies β ≥ β ∗ and α ≤ α ∗ . Under these conditions, if β ∗ = pb or β ∗ = ps , we can get that there exists one single NE which is efficient. We assume that β < β ∗ in case b0 (β) ∈ B 0 . Since there is only one efficient bid profile, b0 (β) is an inefficient NE; then by Lemma 6.4, b0 (β) satisfies (6.30) and β > α, eb0 = es0 = 0. Thus, β ≥ β ∗ holds in the following three cases and also α ≤ α ∗ by the same technique: • In case β ∗ = pb . By Lemma 6.3, eb∗ > 0. Since β < β ∗ and vn (·) is concave for all ∗ β ∗ = α ∗ , we have sm ≤ s n ∈ I ∪ K , we have dn ≥ dn∗ . Since α < β <  m for any m ∈ J ∪ K . By the conservation law er + sm∗ + eb∗ = ed + dn∗ , we have er +



sm < ed +

m∈J ∪K



m∈J ∪K

n∈I ∪K

dn , i.e., the conservation law is not sat-

n∈I ∪K

isfied. Then β ≥ β ∗ in this case. • In case β ∗ = ps . By β ∈ [ ps , pb ] in Lemma 6.4, we have β ≥ β ∗ . • In case β ∗ ∈ ( ps , pb ). By Lemma 6.3, eb∗ = es∗ = 0. Similar to the case β ∗ = pb , the conservation law is not satisfied. Then we have β ≥ β ∗ .

), b0 (β) ∈ B 0 with In case β ∗ ∈ ( ps , pb ), consider any pair of bid profiles b0 (β

> β. By Lemma 6.3, eb∗ = es∗ = 0. By Lemma 6.4, eb = es = 0 with respect to β inefficient NE. Then in this case, eb = es = 0 with respect to any NE. Denote b ≡

). Suppose that β

= β + Δβ with 0 < Δβ ≤ pb − β. Then by b ≡ b0 (β b0 (β) and

Lemma 6.5, all the players are fully allocated with respect to b and b. ym = ed + By eb = es = 0 and the conservation law, we have er +  n∈I ∪K

xn , i.e., er +

 m∈J ∪K

sm = ed +

vn (·) is concave, we can obtain that



m∈J ∪K

dn . By (6.30a), βn = vn (dn ) and

n∈I ∪K

d n ≤ dn , for all n ∈ I ∪ K , with dn ∈ (0, en,max ].

(6.33)

6.3 Economic Operations in Connected Mode Under Auction Mechanism

157

Together with (6.30b) and (6.30c), we have

sm ≤ sm , for all m ∈ J ∪ K , with sm ∈ (0, em,max ]; 

then



(

sm − sm ) =

m∈J ∪K

(6.34)

(d n − dn ) < 0, and the following holds:

n∈I ∪K

J (a∗ (

b)) − J (a∗ (b))    (c j (

s j ) − c j (s j )) − (vi (d i ) − vi (di )) + (wk (d k ,

sk ) − wk (dk , sk )) = j∈J

≥

i∈I



cj (

s j )(

sj − sj) −

j∈J

 i∈I

k∈K

 ∂wk vi (d i )(d i − di ) + (

sk )(

sk − sk ) ∂ yk k∈K

 ∂wk − (d k )(d k − dk ), under Assumptions (A1)–(A3) ∂ xk k∈K  

n (d n − dn ) β

αm (

sm − sm ) −

=

m∈J ∪K

n∈I ∪K



≥

α

(

sm − sm ) − β

m∈J ∪K

) = (

α−β



m∈J ∪K



n∈I ∪K

(

sm − sm ), by

(d n − dn ), by (6.30), (6.33), (6.34) 

(

sm − sm ) =

m∈J ∪K



(d n − dn )

n∈I ∪K

> 0,

and where the last inequality holds since

α β, which is consistent with b)) > J (a∗ (b)) in case β that U > U Theorem 6.2. Corollary 6.1 Denote b0 (β ∗ ) as the efficient NE; then ⎧ ⎨ 1,

in case β ∗ = pb or β ∗ = ps ; ∗∗ J (a ) η= , in case β ∗ ∈ ( ps , pb ); ⎩ J (a∗ (βsup ))

(6.35)

where a∗∗ denotes the efficient allocation, and βsup ≡ sup{β ∈ [ ps , pb ]; s.t. b0 (β) ∈ B 0 }. Proof It is straightforward to verify Corollary 6.1 by Theorem 6.2. Remark By Corollary 6.1, if the bid price under efficient bid profile is equal to pb or ps , the system only exists a unique NE which is efficient.

6.4 Implementation of NE Under Dynamic Process Up to now, we studied the existence of efficient NE. In this section, we will design an algorithm by adapting which the system can reach a NE which may not be efficient.

6.4.1 Implementation Algorithm for NE Firstly for notational simplicity, in Algorithm 6.1 below, we consider that ⎧ ⎪ ⎨bl , bl ≡ al−I , ⎪ ⎩ rl−I −J ,

with l = 1, . . . , I . with l = I + 1, . . . , I + J with l = I + J + 1, . . . , I + J + K

(6.36)

6.4 Implementation of NE Under Dynamic Process

159

Algorithm 6.1 NE implementation algorithm. Require: Initialize a collection of bid profiles b(0) ; n ← 0, Ψ ← false; 1: while Ψ = false do 2: for l = 1 : I + J + K do (n+1) 3: Implement a best response for player l, bl w.r.t. (n+1) (n+1) (n) (n) (b1 , . . . , bl−1 , bl+1 , . . . , b I +J +K ), by maximizing the payoff function of the player l, such that   (n+1) (n+1) (n+1) (n) (n) bl = argmax u l (bl ; b1 , . . . , bl−1 , bl+1 , . . . , b I +J +K ) ; bl ∈B l

4: 5: 6: 7: 8: 9: 10: 11: 12:

l ← l + 1; end for if b(n+1) = b(n) then Ψ = false; else Ψ = true; end if n ← n + 1; end while

Essentially by applying Algorithm 6.1, each of the players successively updates his own best response with respect to others’ bid profiles. This update procedure continues until no players updates his behavior any longer; then by the definition of NE specified in Definition 6.2, in case the procedure converges, the bid profile implemented following Algorithm 6.1 is a NE. Lemma 6.5 For any collection of bid profiles b, a truth-telling bid profile with full allocation of player l is a best response with respect to b−l . Proof We will show Lemma 6.5 with l ∈ I below, and following the same technique we can show it with l ∈ J and l ∈ K as well. Suppose that bl∗ ≡ (βl∗ , dl∗ ) is a best response for player l with respect to b−l , We denote a∗ and xl∗ as the system allocation and the allocation to player l, respectively a the system allocation with respect to bid with respect to (bl∗ , b−l ). We also denote

profiles ((vl (xl∗ ), xl∗ ), b−l ). Then, by the concavity property of vl (·) under Assumption (A1), we have vl (xl∗ ) ≥ vl (dl∗ ) = β,∗

(6.37)

a. and hence we have a∗ =

As a result we have the conclusion that (vl (xl∗ ), xl∗ ), the truth-telling bid profile with full allocation for buyer l, is a best response with respect to b−l .

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6 Economic Operations of Microgrid Systems Under Auction Games

Theorem 6.3 Under Assumptions (A1)–(A3), the auction game will converge to a NE by adopting Algorithm 6.1. Proof Firstly we define a function V (a) on resource allocation profile a, such that V (a) = −J (a), i.e., V (a) represents the overall valuation function of the whole microgrid with respect to a. Suppose it is the turn for agent l to update his best response with respect to a collection of bid profiles b−l ; denote bl∗ the best response of agent l; then we will show that V (a∗ (bl∗ , b−l )) > V (a∗ (b)),

in case u l (bl∗ , b−l ) > u l (b),

(6.38)

for all l ∈ I ∪ J ∪ K , where a∗ (b) denotes the allocation on bid profiles b. Before the proof of (6.38) given below, for notational simplicity, we consider that x,

y,

z,

eb ,

es ), a∗ (bl∗ , b−l ) ≡ (

a∗ (b) ≡ (x, y, z, eb , es ).

(6.39)

Proof of Inequality (6.38) in case l ∈ I , i.e., player l is a buyer. V (a∗ (bl∗ , b−l )) − V (a∗ (b))   xl ) − vl (xl ) + [vm (

xm ) − vm (xm )] − [c j (

y j ) − c j (y j )] = vl (

−



m=l

j∈J

[wk (

z k ) − wk (z k )] − pb (

eb − eb ) + ps (

es − es )

(6.40)

k∈K

which holds by the specification of system cost function given in (6.6). u l (bl∗ , b−i ) − u l (b) = vl (

xl ) − vl (xl ) + τl −

τl , by the payoff of buyer given in (6.22),   

m (

k γk+ (

β β xl ) − vl (xl ) + xm − xm ) −

α j (

yj − yj) + xk − xk ) = vl (

−



m=l

j∈J

k∈K

αk (

yk − yk ) − pb (

eb − eb ) + ps (

es − es ),

k∈K

by the payment of buyer given in (6.17). Thus in case u l (bl∗ , b−i ) > u l (b), we have that vl (

xl ) − vl (xl )    

m (

k γ + (

β β xm − xm ) +

α j (

yj − yj) −

αk (

yk − yk ) >− k xk − xk ) + m=l

j∈J

+ pb (

eb − eb ) − ps (

es − es ).

k∈K

k∈K

(6.41)

6.4 Implementation of NE Under Dynamic Process

161

Also we have (i) vm (

xm ) − vm (xm ) ≥ vm (

xm )(

xm − xm ), due to the concavity y j ) − c j (y j ) ≤ cj (

y j )(

yj − yj) property of vm under Assumption (A1); (ii) c j (

due to the convexity property of c j under Assumption (A2); and similarly (iii) ∂wk + ∂wk wk (

z k ) − wk (z k ) ≥ γ (

xk − xk ) − (

yk − yk ) due to the convexity (concav∂

xk k ∂

yk ity respectively) property of wk on xk (on yk respectively) under Assumption (A3) and xk yk = 0. By (i)–(iii) analyzed above and (6.40) and (6.41), we have V (a∗ (bl∗ , b−l )) − V (a∗ (b))   xl ) − vl (xl ) + vm (

xm )(

xm − xm ) − cj (

y j )(

yj − yj) ≥ vl (

m=l

j∈J

 ∂wk  ∂wk + γk+ (

xk − xk ) − (

yk − yk ) − pb (

eb − eb ) + ps (

es − es ) ∂

xk ∂

yk k∈K k∈K  

m )(

> (vm (

xm ) − β xm − xm ) − (cj (

yj) −

α j )(

yj − yj) m=l

j∈J

 ∂wk  ∂wk

k )γk+ (

+ ( −β xk − xk ) − ( −

αk )(

yk − yk ) ∂

xk ∂

yk k∈K

k∈K

≥0

m under the concavity xm ) ≥ vm (dm ) = β where the last inequality holds since (i) vm (

xm ≥ xm ; and similarly (ii) cj (

yj) ≤

α j under the convexity property of vm and

k

k under the concavity property of wk on

property of c j and

y j ≥ y j ; (iii) ∂w ≥β xk ∂

xk ∂wk and

xk ≥ xk ; (iv) ∂ yk ≤

αk under the convexity property of wk on

yk and

yk ≥ yk . Following the same technique in case l ∈ I , we can show (6.38) in case l ∈ J and l ∈ K respectively. End of proof of Inequality (6.38). Thus, by (6.38), together with that V (a) is bounded from above, the auction game can converge to a NE following the best response procedure proposed in Algorithm 6.1. Note: The convergence of Algorithm to NE was verified for one-side PSP auction games in [32]. Actually, the system may not converge to the efficient NE following our proposed algorithm. Jia and Caines [29] proposed a so-called quantized PSP (Q-PSP) auction mechanism, for a class of double-side auction games, by applying which the system can converge to some bid profile which is close to the efficient NE. As ongoing research, we may extend the work by Jia and Caines to the underlying microgrid economic operation auction games.

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6 Economic Operations of Microgrid Systems Under Auction Games

Fig. 6.6 Updates of individual bidding prices and the associated resource allocations under Algorithm 6.1

6.4 Implementation of NE Under Dynamic Process Table 6.2 Updates of the increment of agent payoff and the increment of system utility under Algorithm 6.1

Iteration step 1 2 3 4 5 6 7 8 9

163 Increment of agent payoff 0.0023 2.9226 7.1915 0.0145 0.0001 0.000003 0 0 0

Increment of system utility 0.0832 3.0023 7.6498 0.0249 0.0004 0.000006 0 0 0

6.4.2 Numerical Simulations In this section, we verify the implementation of NE developed in this chapter with numerical examples. We consider the microgrid economic operation problems specified in Sect. 6.2.2 during the time period of 16:00–17:00. The updates of best bid profiles following Algorithm 6.1 and the associated resources allocations are displayed in Fig. 6.6 where the negative and positive valued allocations to the storage unit represent discharging and charging, respectively. We can observe that the storage unit initially plays as a provider, and from the 3rd iteration step, it plays as a load. Moreover as illustrated in Fig. 6.6 and by Definition 6.3, the implemented NE is b0 (β) with β = ps ; then by Corollary 6.1, the implemented NE is efficient. Table 6.2 displays the updates of the system utility and individual payoff with respect to the update of the best bid profiles of individual players. We can observe that the system utility increases as each of individual units updates his individual best bid profile and the system converges to the NE in 9 iteration steps. This is consistent with Theorem 6.3.

6.5 Economic Operations of Microgrid in the Isolated Mode In this chapter, we studied the economic operations of the microgrid systems in the connected mode. Considering ps = 0, pb = +∞ and under Assumptions (A1)–(A3), we can obtain that es∗ = eb∗ = 0,

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6 Economic Operations of Microgrid Systems Under Auction Games

because: (i) no demand units will buy energy from the main grid with price of pb = +∞ since the marginal valuation of each demand is finite; and (ii) no supplier units will sell energy to the main grid with a price of ps = 0 since the marginal cost of each supplier is larger than zero. Consequently, in case ps = 0, pb = +∞ and under Assumptions (A1)–(A3), the economic operation problems for the microgrid system in the connected mode studied in the chapter degenerates to the economic operation problems for the microgrid system in the isolated mode.

6.6 Conclusions and Ongoing Researches In this chapter, we formulated a class of economic operation problems of the microgrid as a class of specific PSP auction games, and showed that the efficient coordination solution is a NE of the underlying auction games. We proposed a distributed method under which the system can converge to a NE which may not be efficient. It is also shown that the performance of the worst NE can be bounded with respect to the system parameters, say the energy trading price with the main grid, and based upon that, the implemented NE is unique and efficient under some conditions. As ongoing research, we may extend, to the underlying microgrid economic operation auction games, the work developed in [33] where the authors proposed a socalled quantized PSP (Q-PSP) auction mechanism, for a class of double-side auction games, by applying which the system can converge to a collection of bid profiles which is close to the efficient NE.

Appendices Proof of Lemma 6.4 We will show the statements of (6.31a)–(6.31c) in (i)–(iii) below, respectively. (i) Proof of (6.31a). We firstly show that βl , l ∈ I ∪ K , satisfies (6.30a) by proof of contradiction in the below. Following the same technique, we have αl , l ∈ J ∪ K , satisfies (6.30b). In case dl ∈ (0, el,max ), suppose that there are two players l1 , l2 ∈ I ∪ K such that dl1 ∈ (0, el1 ,max ), dl2 ∈ (0, el2 ,max ) and βl1 > βl2 . b is assumed to be a NE; then by Lemma 6.5, we have that all the players are fully allocated subject to b.

l1 , d l1 ) such that d l1 = dl1 + δ bl1 ≡ (β For player l1 , consider another bid profile

l1 > βl2 ; then, under (

bl1 , b−l1 ) and by the allocation rule, the with δ < dl2 and β xl2 , decreases by δ, and the allocation of l1 is d l1 , the allocation of l2 , denoted by

allocations of other players remain unchanged. Hence we have

Appendices

165

u l1 (bl1 , b−l1 ) − u l1 (

bl1 , b−l1 ) xl2 ) = vl1 (dl1 ) − vl1 (d l1 ) + βl2 (dl2 −

< vl1 (d l1 )(dl1 − d l1 ) + βl2 δ,

l1 δ + βl2 δ = −β < 0,

by convavity of vl1 (·) under Assumptions (A1, A3)

l1 > βl2 . It implies that bl1 is not the best where the last inequality holds since β response of player l1 with respect to b−l1 . This is contradicted with b is a NE. In case dl = el,max > 0, if βl < β, by the allocation rule, player l cannot be fully allocated, i.e., bl is not the best response of player l. In case dl = 0, by the allocation rule, we can get that βl < β. Secondly, we will show that β ∈ [ ps , pb ] and α ∈ [ ps , β]. Suppose β > pb . For player l, l ∈ I ∪ K with dl ∈ (0, el,max ), consider a bid

bl , b−l ) the allobl = ( pb , (vl )−1 ( pb )); then by the allocation rule, with respect to (

cation of l equals to (vl )−1 ( pb ). Since the resources that the main grid can supply or accept are infinite, the allocations of others remain unchanged. The change in the payoff of l satisfies the following: bl , b−l ) u l (bl , b−l ) − u l (

  = vl (dl ) − vl ((vl )−1 ( pb )) + pb (vl )−1 ( pb ) − dl      < vl (vl )−1 ( pb ) dl − (vl )−1 ( pb ) + pb (vl )−1 ( pb ) − dl , by convavity of vl (·) = 0, which is contradicted with the NE property of b; then β ≤ pb . Similarly, we can verify that α ≥ ps . If α > β, by the allocation rule, the allocations for all players are zero under bid profile b. It implies that bl cannot be the best response for any player l. Finally, (6.30c) holds by the conservation law and all of the players are fully allocated under b. (ii) Proof of (6.31b). In case β = α, we observe that (6.30) is consistent with Lemma 6.3 obtained by the KKT conditions of optimality to specify the features of efficient bid profiles, i.e., b is efficient. Since the efficient NE is unique, in case β > α, b is inefficient. (iii) Proof of (6.31c). While β > α, suppose es > 0. By the allocation rule, we bl = have α = ps . For player l, l ∈ I ∪ K with dl ∈ (0, el,max ), consider a bid

l > ps ; then under (

l , d l ) such that d l = dl + θ with θ < es and β bl , b−l ) and by (β the allocation rule, the allocation to player l is d l , the resources sold to the main grid decreases by θ , and the allocations to other players remain unchanged. Hence

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6 Economic Operations of Microgrid Systems Under Auction Games

u l (bl , b−l ) − u l (

bl , b−l ) = vl (dl ) − vl (d l ) + ps θ < vl (d l )(dl − d l ) + ps θ, by the concavity of vl (·), under Assumptions (A1, A3)

l θ + ps θ = −β < 0,

l > ps , which implies that bl is not the best where the last inequality holds since β response of l with respect to b−l . Hence we have es = 0. By the same technique adopted the above, we have eb = 0.

Proof of Theorem 6.1 It is sufficient to show that the efficient bid profile b∗ ≡ (b∗ , a ∗ , r ∗ ) is a NE if (1)–(3) below hold. (1) To show that bi∗ is a best response of buyer i with respect to b∗−i ≡ ∗ (b−i , a ∗ , r ∗ ). (1.I) Consider a buyer i with xi∗∗ > 0 (a buyer i with xi∗∗ = 0 cannot decrease his allocation), and, given the bids (b∗−i ) of the others as fixed, suppose buyer i changes his bid bi∗ to bi† to decrease his allocation xi∗∗ by a δ > 0. We denote a† = (x † , y † , z † , eb† , es† ) the resulting allocation, then a† satisfies the following analysis: • (R1) xi† = xi∗∗ − δ, and xl† = xl∗∗ , for all l ∈ K−i , i.e., the allocations of all the other buyers do not change, since all of them have already received the maximum quantity they ask for. • (R2) y †j ≤ y ∗∗ j , with j ∈ J , i.e., some of the sellers may sell less. • (R3) A storage unit k may be either a buyer or a seller; then 

xk† = xk∗∗ , yk† ≤ yk∗∗ , xk† = 0,

in case xk∗∗ > 0 , in case yk∗∗ > 0

i.e., when he is a buyer, his allocation doesn’t change; when he is a seller, he may sell less. • (R4) By Lemma 6.1, the microgrid can either buy electricity from the main grid or sell electricity to the main grid; then we have the following: 

es† ≥ es∗∗ , eb† ≤ eb∗∗ , es† ≥ 0,

in case es∗∗ > 0 , in case eb∗∗ > 0

i.e., in case the microgrid sells electricity to the main grid, he will sell more; while in case the microgrid buys energy from the main grid, the microgrid may buy less from the main grid and even sell electricity to the main grid.

Appendices

167

In (i) and (ii) below we will study Δτi , the payment variation of player i, with es∗∗ > 0 and eb∗∗ > 0, respectively. (i) In case es∗∗ > 0, i.e., the microgrid sells electricity to the main grid at efficient solution. By Lemma 6.3, we have ν ∗ = ps . The prices of the sellers and the storages discharging α ∗j , αk∗ ≤ ν ∗ , while the prices of the buyers and the storages charging βi∗ , βk∗ ≥ ν ∗ ; then • (R5) y †j = y ∗∗ j , for all j ∈ J , i.e., all the sellers don’t decrease their supply, since α ∗j ≤ ν ∗ . • (R6) (xk† , yk† ) = (xk∗∗ , yk∗∗ ), for all k ∈ K , i.e., all the storage units don’t change their allocation, since dk∗ = xk∗∗ and αk∗ ≤ ν ∗ . • (R7) es† = es∗∗ + δ, i.e., the electricity released from buyer i is sold to the main grid. By (6.17), (R1) and (R5)–(R7), we can obtain that the change in the payment of buyer i, denoted Δτi , satisfies the following: Δτi  τi (bi† , b∗−i ) − τi (b∗ ) = − ps es† + ps es∗∗ = −ν ∗ δ. (ii) In case eb∗∗ > 0, i.e., the microgrid buys electricity from the main grid. By Lemma 6.3, we have ν ∗ = pb and a† satisfies the following: • (R8) xl† = xl∗∗ , for all l ∈ I ∪ K /{i}, i.e., the allocations of all the buyers (except buyer i) and the storages charging remain unchanged. • (R9) eb† ≤ eb∗∗ , and yl† ≤ yl∗∗ , for all l ∈ J ∪ K such that αl∗ = pb , i.e., the main grid and the sellers (or the storages discharging) whose prices are the same as pb will sell less.   – In case eb∗∗ + yl∗∗ < δ and eb∗∗ + yl∗∗ ≥ δ: All of the units posl∈J ∪K αl∗ = pb

l∈J ∪K αl∗ ∈[ ps , pb ]

sessing selling price pb all decrease their supplies to 0, but the power in the system hasn’t balanced yet. Other sellers (or other storages discharging) whose selling prices are between ps and pb will sell less, and eb† ≥ 0 and es† = 0, i.e., the microgrid does not buy electricity from the main grid. The change in the payment of buyer i satisfies the following: Δτi  τi (bi† , b∗−i ) − τi (b∗ )  = pb (eb† − eb∗∗ ) +

≥ pb eb† − eb∗∗ +

(6.42) αl∗ (yl†

l∈J ∪K αl∗ ∈[ ps , pb ]



(yl† − yl∗∗ )

l∈J ∪K αl∗ ∈[ ps , pb ]

= − ν∗δ

−

yl∗∗ ) 

168

6 Economic Operations of Microgrid Systems Under Auction Games

where the last inequality holds because αl∗ ≤ pb and yl† − yl∗∗ ≤ 0 for all l ∈ J ∪ K ; while the last equality holds because ν ∗ = pb and the power conservation law.  yl∗∗ < δ: All the units above decrease their supplies to 0, – In case eb∗∗ + l∈J ∪K αl∗ ∈[ ps , pb ]

and the power conservation law is not satisfied, and the main grid will change his role to accept some energy. Now ν ∗ changes to ps . If the main grid changes his role, suppose he buys energy by a ε > 0, and 0 < ε < δ, the change in the payment of buyer i satisfies the following: Δτi  τi (bi† , b∗−i ) − τi (b∗ )  = pb (eb† − eb∗∗ ) +

(6.43) αl∗ (yl†

−

yl∗∗ )

−

ps es†

l∈J ∪K αl∗ ∈[ ps , pb ]

≥ pb (ε − δ) − ps ε ≥ − ps δ = − ν∗δ So by (6.42) and (6.43), we obtain that the change in the payment of buyer i always satisfies Δτi ≥ −ν ∗ δ. Moreover since vi is strictly increasing and concave, we get that Δvi = vi (xi∗∗ − δ) − vi (xi∗∗ ) < −δβi∗ ≤ −δν ∗ ≤ Δτi . Finally, by (6.22) and the analysis above, we have the change in the payoff of buyer i satisfies the following:     Δu i = vi (xi∗∗ − δ) − τi (bi† , b∗−i ) − vi (xi∗∗ ) − τi (b∗ ) = Δvi − Δτi < 0, i.e., the buyer i’ s payoff decreases if he unilaterally changes his bid profile such that his allocation decreases. (1.II) Suppose the buyer i, with xi∗∗ ≥ 0, changes his bid to bi† to increase his allocation xi∗∗ by a δ > 0, then a† satisfies the following discussions: • (R1)’ xi† = xi∗∗ + δ, and xl† ≤ xl∗∗ , with l ∈ K−i , i.e., the allocations of some of the other buyers may decrease. • (R2)’ y †j = y ∗∗ j , with j ∈ J , i.e., the allocations of all the sellers remain unchanged since s j = y ∗∗ j . • (R3)’ A storage unit k may be either a buyer or a seller; then 

xk† ≤ xk∗∗ , yk† = 0, yk† = yk∗∗ ,

in case xk∗∗ > 0 , in case yk∗∗ > 0

Appendices

169

i.e., when he is a buyer, his allocation may decrease; while when he is a seller, his allocation remains unchanged. • (R4)’ By Lemma 6.1, the microgrid can either buy electricity from the main grid or sell electricity to the main grid; then we have the following: 

es† ≤ es∗∗ , eb† ≥ 0, eb† ≥ eb∗∗ ,

in case es∗∗ > 0 , in case eb∗∗ > 0

i.e., in case the microgrid sells electricity to the main grid, he may sell less and even switches to buy electricity from the main grid; while in case the microgrid buys electricity from the main grid, the microgrid may buy more. We observe that the analysis in (R1)’–(R4)’ above is similar to the case of (R1)– (R4) that the buyer i decreases his energy, and we get the same conclusion. Thus, following the analysis in (1.I) and (1.II) above, we can get that, given the bids, b∗−i of all the other units as fixed, the best response of buyer i is the bid bi∗ . is a best response of seller j with respect to b∗− j ≡ (2) To show that a ∗∗ j ∗ ∗ ∗ (b , a− j , r ). Consider any seller j, j ∈ J . Following the similar techniques applied in (1) for buyers, we get that a j is the best response of seller j to bid of other units (b, a− j , r ) fixed and he gets y ∗∗ j . (3) To show that rk∗ is a best response of storage k with respect to b∗−k ≡ ∗ ). (b∗ , a ∗ , r−k Consider a storage k with xk∗∗ > 0 and any given fixed bid profile of the others b∗−k . If the storage k charges and changes his bid rk∗ to rk† to change his allocation, this is similar to the analysis of the buyers. If the storage k discharges and changes his bid to change his allocation, this is similar to the analysis of sellers. If the storage k charges and changes his bid to decrease his allocation by a δ > 0 to discharge, i.e., xk∗∗ + γk− yk† = δ, the change in the money transfer of the storage k is the same as that of buyer i. The change in the utility function of storage k is Δwk = wk (xk† , yk† ) − (xk∗∗ , yk∗∗ ) > αk∗ yk† + βk∗ xk∗∗ ≥ ν ∗ (xk∗∗ + yk† ) > ν ∗ δ ≥ −Δτk ,

with xk† = 0 and yk∗∗ = 0. Hence by (6.24), we obtain that the change in the payoff of storage k satisfies the following: Δu k = −Δwk − Δτk < 0, i.e., the payoff of storage unit k decreases. When the storage k discharges and changes his bid to increase his allocation to charge, we can also get the same conclusion in similar ways. Thus, following the analysis in (1)–(3), the efficient bid profile b∗ is a NE.

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Chapter 7

Efficient Vehicle-to-Grid (V2G) Coordination in Smart Grid Under Auction Games

Abstract Emerging plug-in electric vehicles (PEVs), as distributed energy sources, are promising to provide vehicle-to-grid (V2G) services for the power grid, like frequency and voltage regulations, by coordinating their active and reactive power rates. However, due to the autonomy of PEVs, it is challenging how to efficiently schedule the coordination behaviors among these units in a distributed way. In this chapter we formulate the underlying coordination problems as a novel class of VCGstyle auction games where players, power grid, and PEVs, do not report a full cost or valuation function but only a multi-dimensional bid signal: the maximum active and reactive power quantities that power grid wants and the maximum per unit prices it is willing to pay, the maximum active and reactive power quantities that a PEV can provide and the minimum per unit prices it asks. From this formulation, the underlying V2G problem is actually a two-type resource allocation problem featuring the active and reactive power as resources. We show the existence of an efficient Nash equilibrium (NE) for the underlying auction games, though there may exist other inefficient NEs. In order to deal with large-scale PEVs, we design games with aggregator players each of which submits bid profiles representing the overall utility for a collection of PEVs, and extend the so-called quantized-PSP mechanism to the underlying auction games to implement an efficient NE.

7.1 Introduction Nowadays plug-in electric vehicles (PEVs) have been achieving attractive developments since these vehicles may promisingly reduce the reliance on the exhaustible non-renewable energy sources; however improper or disordered charging coordination for large populations of PEVs will have significant impacts on the power grid, see [1–4]. Recently many research works has been dedicated to studying how to properly coordinate the PEV charging behaviors to mitigate the involved negative impacts, see [5–9] and references therein. To operate safely, the power grid has to be ensured with the amplitude of frequency and voltage close to their nominal values by balancing the production and consump-

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

tion of both active and reactive power,1 see [10–12]. PEVs can play as distributed electricity storages for the power grid, see [13], since they can deliver electricity to the power grid and consume electricity from the power grid. Nowadays quite a few researches have dedicated to the study how to apply vehicle-to-grid (V2G) structure for frequency regulations in smart grid, e.g., [14–18]. In practice, most of the reactive power loads are inductive, so power compensation for voltage regulation mainly involves adjusting capacitors, e.g., [19–22]. More recently, [23] proposed a V2G regulation model such that PEVs are applied to provide ancillary services of frequency and voltage regulations for power grid by coordinating their active and reactive power rates, and further formulated the underlying regulation problems as two joint optimization problems under different pricing and contract scenarios. In this chapter, we propose an auction-based distributed method for V2G coordination problems initialized by [23] under the progressive second price (PSP) auction mechanism which was firstly proposed by [24, 25] to efficiently allocate a single network divisible resource. Under the PSP auction mechanism, the incentive compatibility holds, i.e., the truth-telling bid profile is the best response of agent. Hence, the PSP auction can be considered as the extension of the second price auction proposed in [26–28] to allocate indivisible items with truth-telling bid profiles in an efficient way. It is worth to note that the incentive compatibility does not hold in general for other sealed auction mechanisms, like uniform market clearing price and pay as bid auction mechanisms etc., see [29, 30], which have been widely adapted to economically dispatch generations in day-ahead deregulated electricity markets, see [31]. In the allocation problems of a single divisible resource under the PSP auction mechanism, each agent only reports a two-dimension bid profile which is composed of a maximum amount of demand and an associated buying price, and is used to replace its complete (private) utility function. The transfer money (or payment) of an agent is that, the utility of the whole system, when this agent does not join the auction process, minus the overall utility of the remaining system except this agent when he joins the auction process, that is to say, the payment of each agent is exactly the externality he imposes on the system through his participation. In [25], the authors verified that the efficient incentive compatible bid profile is a Nash equilibrium (NE) for PSP network auction games. More recently many progresses have been developed in PSP auction games. For example, [32] extended the results from single-side auction games to double-side ones in the network resource allocations; [33–35] proposed a distributed algorithm, under quantized-PSP auction mechanism for power electricity sharing games, following which the auction game system converges to an efficient solution. In the V2G regulation problems, suppose that each of autonomous PEVs deals with the tradeoff between costs and benefits with respect to supplied active and reactive power service rates; while the power grid deals with the tradeoff between the benefits with the aggregated service rates provided by the PEV populations and the costs he needs to pay for the services. The players participating in the underlying 1 Balance

supply and demand of active power for frequency regulation and balance supply and demand of reactive power for voltage regulation.

7.1 Introduction

175

auction games, including power grid as a single buyer and autonomous PEVs as sellers, do not report a full cost or valuation function but only a multi-dimensional bid signal: the maximum active and reactive power quantities that power grid wants and the maximum per unit prices he is willing to pay, the maximum active and reactive power quantities that an autonomous PEV can provide and the minimum per unit prices he asks. In this chapter, we show the existence of an efficient NE for the V2G auction game, though there may exist other inefficient NEs. Moreover in order to deal with the V2G coordination problems with large-scale PEVs, we design a class of V2G coordination auction games with aggregator players each of which represents a collection of individual PEVs. To implement an efficient NE, in this chapter, we adapt and extend the PSP auction mechanism, which was proposed in [33] for double-side auction games, to the underlying V2G auction games with constrained multi-dimension bid strategies. As illustrated with the numerical examples, the game system converges to some NE which is near to an efficient NE. The organization of the chapter is as follows: In Sect. 7.2, we formulate a class of vehicle-to-grid coordination problems for frequency and voltage regulations. In Sect. 7.3, we propose a distributed V2G coordination method under the PSP auction mechanism and show the efficiency property of the underlying auction game. In Sect. 7.4, we designed V2G coordination auction games with aggregators each of which represents a collection of PEVs. We extend the so-called Q-PSP auction mechanism, to the underlying V2G auction games, in Sect. 7.5 which is illustrated with numerical examples. We list some ongoing research works in Sect. 7.6.

7.2 Formulation of Vehicle-to-Grid Coordination Problems In this section, we formulate a class of frequency and voltage regulations for power grid via charging coordination of PEVs, which was initialized in [23], such that an individual PEV can provide frequency regulation service by coordinating its charging/discharging active power and voltage regulation service by properly setting the phase angle difference between its charging current and power grid voltage. We denote by f the deviation of frequency from the nominal value in power grid and by P(f) the resulting imbalance between supply and demand with respect to f respectively; and denote by v the deviation of voltage from the nominal value in power grid and by Q(v) the resulting imbalance between supply and demand with respect to v, respectively. More specifically, in the chapter we consider that • P(f) > 0 (P(f) < 0 respectively) indicates that the supply is higher (lower respectively) than the demand with P(f) in power grid; • Q(v) > 0 indicates that power grid requires capacitive reactive power compensation; • Q(v) < 0 indicates that power grid requires inductive reactive power compensation.

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In the work, we suppose that (P(f), Q(v)) remains fixed over the next shorttime interval T . We denote by N the collection of electric vehicles which plug in the power grid and are willing to supply the V2G service. To simplify the analysis, we further assume that, as long as the vehicles agree to supply the service for the interval T , they will keep plugged in the power grid over that interval and provide the service as they agreed.

7.2.1 Coordination Capacity of Individual PEVs 7.2.1.1

Apparent Power Constraint

Consider a PEV charger with a power electronics AC/DC inverter which can adjust active and reactive power by coordinating In the magnitude of current of PEV n and θn the phase difference between the voltage of power grid and the current of PEV n. Let xn P and xn Q denote the active power and reactive power of PEV n; then we have xn P = Vs × In cosθn , xn Q = Vs × In sinθn ,

(7.1a) (7.1b)

where Vs represents the magnitude of voltage in power grid. We consider that • xn P > 0 and xn P < 0 represent that PEV n charges and discharges, respectively; • xn Q > 0 and xn Q < 0 represent that PEV n operates like a capacitive load and an inductive load, respectively. We consider the following nominal apparent power constraint for PEV n as below: xn2 P + xn2Q ≤ Sn2 ,

Fig. 7.1 Apparent power constraint for individual PEV n

(7.2)

7.2 Formulation of Vehicle-to-Grid Coordination Problems

177

where Sn denotes the nominal apparent power of the charger where PEV n plugs, as illustrated in Fig. 7.1, see [23]. For analytical simplicity, in this chapter, we only study the V2G coordination in case P(f), Q(v) > 0, i.e., the supply is higher than the demand and power grid needs capacitive reactive power compensation; then we consider xn P , xn Q ≥ 0,

(7.3)

for all n ∈ N , i.e., each PEV charges and operates like a capacitive load.

7.2.1.2

Active Power Constraint

We consider a maximum charging rate constraint such that   Φn max xn P ≤ xnmax  min ξ , − soc ) , (soc n n0 P n T

(7.4)

where ξn denotes a safe maximum charging rate, socnmin and socnmax represent the minimum and maximum state of charge (SOC) of PEV n, respectively, socn0 represents the initial SOC value of PEV n, and Φn denotes the battery energy capacity of PEV n, and T denotes the length of service interval T . Note: The specification in (7.4) can guarantee that the SOC of PEV n is in the range of [socnmin , socnmax ]. In summary, we call x n ≡ (xn P , xn Q ) an admissible active and reactive power coordination of PEV n over the interval T , if x n satisfies the constraints (7.2)–(7.4). We denote a local cost function of PEV n by gn (x n ; socn0 ), subject to an admissible coordination strategy x n ≡ (xn P , xn Q ) as follows: gn (x n ; socn0 ) = gn P (xn P ; socn0 ) + gn Q (xn Q ; socn0 ),

(7.5)

where gn P (xn P ; socn0 ) and gn Q (xn Q ; socn0 ) representing the battery degradation cost subject to charging rate xn P , xn Q , measures the cost related to the decrease of battery energy capacity, due to the battery resistance growth subject to the charging rate, which is analyzed in [36]. For notational simplicity, we may consider gn (x n ; socn0 ), gn P (xn P ; socn0 ) and gn Q (xn Q ; socn0 ) as gn (x n ), gn P (xn P ) and gn Q (xn Q ), respectively.

7.2.2 Efficient V2G Coordinations for Frequency and Voltage Regulations We define a valuation function of power grid, subject to the aggregated V2G coordination strategies x s with x s ≡ (xs P , xs Q ), denoted by vs (x s ; P, Q), with P ≡ P(f), Q ≡ Q(v) for notational simplicity, such that

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

vs (x s ; P, Q)  vs P (xs P ; P) + vs Q (xs Q ; Q).

(7.6)

As an example, we may consider a quadratic regulation form for valuation function of vs (x s ; P, Q), such that vs (x s ; P, Q) = −σs P (xs P − P)2 − σs Q (xs Q − Q)2 , with σs P , σs Q > 0. For notational simplicity, we may consider vs (x s ) ≡ vs (x s ; P, Q), vs P (xs P ) ≡ vs P (xs P ; P) and vs Q (xs Q ) ≡ vs Q (xs Q ; Q). Let X denote a collection of power grid strategies and admissible PEV coordination strategies, denoted by x ≡ (x s , x n ; n ∈ N ), if x n satisfies (7.2)–(7.4). By (7.5) and (7.6), we define the system valuation function, denoted by J (x), subject to an admissible coordination strategy x ∈ X , such that J (x)  vs (x s ) −

N 

gn (x n ),

(7.7)

n=1

such that x s ≡ (xs P , xs Q ) =

 N 

xn P ,

n=1

N 

 xn Q .

(7.8)

n=1

The objective of the optimal operation of V2G coordination problems is to assign efficient (or socially optimal) allocation, to all PEVs, to maximize the system valuation function (7.7) over the set of admissible strategies X . Let x ∗∗ denote the efficient allocation, i.e., x ∗∗ = argmax J (x), subject to (7.8).

(7.9)

x∈X

The Lagrange function for the optimization problem (7.9) subjected to (7.2)–(7.4) and the equality constraint (7.8) is given below: L(x s , x n , λ, μ) = vs (x s ) −

N  

gn P (xn P ) + gn Q (xn Q )



n=1



− xn P ) + λnS Sn2 − xn2 P − xn2Q  N   N    + μP xn P − xs P + μ Q xn Q − xs Q +

λn P (xnmax P

n=1

n=1

with x s ≡ (xs P , xs Q ), x n ≡ (xn P , xn Q ; n ∈ N ), λ ≡ (λn P , λnS ; n ∈ N ) and μ ≡ (μ P , μ Q ), where λn P is the Lagrange multiplier corresponding to the active power constraint of PEV n, λnS is the Lagrange multiplier corresponding to the apparent power constraint, μ P and μ Q are the Lagrange multipliers corresponding to the equality constraint related to active and reactive power defined in (7.8), respectively.

7.2 Formulation of Vehicle-to-Grid Coordination Problems

179

The following, together with the equality constraint (7.8), are the KKT conditions for the constrained optimization problems given in (7.9). vs P (xs∗∗P ) − μ∗P ≤ 0,

xs∗∗P ≥ 0,

xs∗∗P (vs P (xs∗∗P ) − μ∗P ) = 0

∗ vs Q (xs∗∗ Q ) − μ Q ≤ 0,

xs∗∗ Q ≥ 0,

 ∗∗ ∗ xs∗∗ Q (vs Q (x s Q ) − μ Q ) = 0

(7.10a) (7.10b) ∂ L(x s , x n , λ, μ) ≤ 0, ∂ xn P

xn∗∗P ≥ 0,

− gn Q (xn∗∗Q ) − 2λ∗nS xn∗∗Q + μ∗Q ≤ 0,

xn∗∗Q ≥ 0,

xn∗∗Q (−gn Q (xn∗∗Q ) − 2λ∗nS xn∗∗Q + μ∗Q ) = 0

∗∗2 2 xn∗∗2 P + x n Q ≤ Sn ,

λ∗nS ≥ 0,

∗∗2 λ∗nS (Sn2 − xn∗∗2 P − xn Q ) = 0

xn∗∗P ≤ xnmax P ,

λ∗n P ≥ 0,

∗∗ λ∗n P (xnmax P − xn P ) = 0

xn∗∗P

∂ L(x s , x n , λ, μ) = 0 ∂ xn P

(7.10c)

(7.10d) (7.10e) (7.10f)

∂ L(x s , x n , λ, μ) = −gn P (xn∗∗P ) − 2λ∗nS xn∗∗P − λ∗n P + μ∗P . ∂ xn P We consider the following assumptions in the chapter:

with

(A1) vs P (xs P ) and vs Q (xs Q ) are strictly increasing and concave on xs P , xs Q respectively; (A2) gn P (xn P ) and gn Q (xn Q ), with n ∈ N , are strictly increasing and convex on xn P , xn Q , respectively. Under Assumptions (A1) and (A2), the underlying regulation problems are convex optimization problems, hence the necessary KKT conditions are also sufficient conditions for the optimality of the optimization problems, see [37]. Thus under Assumptions (A1) and (A2), there exists a unique solution and it is characterized by the KKT conditions specified in (7.10) together with the equality constraint (7.8).

7.2.3 A Simulation for Frequency and Voltage Regulations We study the V2G coordination problems with N = 1000, and each of which possesses a battery with a capacity of Φn = 10 kWh. More specifically it is considered that the referred fixed active power P = 5 MW and reactive power Q = 2 MVar required for the power grid. As studied in the literature, e.g., [38], the distribution of typical travel miles of vehicles approximately satisfies a Gaussian distribution; then the distribution of initial SOC values of PEV batteries, denoted by {socn0 ; n ∈ N }, may approximately satisfy a Gaussian distribution, denoted by N (μ, γ ), as well, see [39, 40]. A Gaussian distribution with μ = 0.5 and γ = 0.15, illustrated in Fig. 7.2, is considered in the following simulations. We also consider that socnmin = 10% and socnmax = 90%.

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

Fig. 7.2 An approximate Gaussian distribution of initial SOC values over the PEV populations

We consider ξn = 10 kW, Sn = 10.5 kVA and ΔT = 15 min; then based upon the given specifications, we can specify the constraints of (7.2)–(7.4) for each individual √ PEV. We consider the cost functions of PEV n as gn P (xn P ; socn0 ) = 0.1 socn0 xn2 P √ and gn Q (xn Q ; socn0 ) = 0.05 socn0 xn2Q , which are strictly convex, and the valuation function of power grid as vs P (xs P ; P) = −0.12(xs P − P)2 and vs Q (xs Q ; Q) = −0.12(xs Q − Q)2 which are strictly concave. The efficient allocations to all PEVs, denoted by x ∗∗ , are displayed in Fig. 7.3, such that ⎧ max 2 ∗∗ in case socn0 ∈ [socnmin , 12%] x ∗∗2 + xn∗∗2 ⎪ ⎪ Q = Sn , x n P < x n P , ⎪ nP ⎨ max ∗∗2 2 ∗∗ xn∗∗2 in case socn0 ∈ (12%, 81%) P + x n Q < Sn , x n P < x n P , ⎪ ⎪ ⎪ Φ ⎩ x ∗∗2 + x ∗∗2 < S 2 , x ∗∗ = x max = n (socmax − soc ), in case soc ∈ [81%, socmax ] n0 n0 n nP n n nP nQ nP T

(7.11)

by which we obtain that at the efficient allocations, a PEV n, with n ∈ N , reaches its apparent power constraint Sn2 in case its SOC value lower than 12%, and reaches its maximum active power constraint xnmax P in case its SOC value higher than 81%. The corresponding aggregated active and reactive regulation service rates x ∗∗ s are specified as follows: x ∗∗ s

≡

(xs∗∗P , xs∗∗Q )

=

 N  n=1

xn∗∗P ,

N 

 xn∗∗Q

= (4.997 MW, 1.999 MVar) .

n=1

7.3 Auction-Based Distributed Vehicle-to-Grid Coordination Method The socially optimal V2G coordination problems can be implemented in case that the system controller has complete information and can directly schedule the behaviors of all individual PEVs and power grid. However, in practice, individual PEVs and power

7.3 Auction-Based Distributed Vehicle-to-Grid Coordination Method

181

Fig. 7.3 The efficient allocations to individual PEVs with respect to initial SOC values

grid may not want to share their private information with others and individual PEVs do not permit the system directly control their batteries. In this situation, decentralized coordination methods can be adapted. We propose an auction-based decentralized method for the underlying regulation problem, such that each of individual agents, either power grid or an individual PEV, responds with the best bid profile to the bid profiles of the others. The system is at an equilibrium in Nash sense if no single agent can benefit himself by unilaterally deviating from its individual bid profile.

7.3.1 Bid Profiles for Regulation Auction Problems 7.3.1.1

Bid Profile of Power Grid

As a buyer, power grid submits a (four-dimension) bid profile bs ≡ (bs P , bs Q ), such that

(bs P , bs Q ) ≡ (βs P , ds P ), (βs Q , ds Q ) ,

(7.12)

with ds P , ds Q ≥ 0, which specifies (i) the maximum per unit price βs P of active power that power grid is willing to pay and the demand up to ds P units of the active power; and (ii) the maximum per unit price βs Q of reactive power that power grid is willing to pay and the demand up to ds Q units of the capacitive reactive power.

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

As a result, the revealed valuation function of power grid with respect to the bid profile given in (7.12) is specified as follows:  vs (x s ) = βs P min(xs P , ds P ) + βs Q min(xs Q , ds Q ).

(7.13)



We set b−s ≡ (βs P , 0), (βs Q , 0) and Bs as the set of bid profiles of power grid.

7.3.1.2

Bid Profiles of Individual PEVs

As a seller, PEV n specifies a (four-dimension) bid profile an ≡ (an P , an Q ), such that

(an P , an Q ) ≡ (αn P , sn P ), (αn Q , sn Q ) ,

(7.14)

2 2 2 with 0 ≤ sn P ≤ xnmax P and sn P + sn Q ≤ Sn , which specifies (i) the maximum per unit price αn P of active power that PEV n asks and the supply up to sn P units of the active power; and (ii) the maximum per unit price αn Q of reactive power that PEV n asks and the supply up to sn Q units of the capacitive reactive power. As a result, the revealed cost function of PEV n, denoted by  gn , with respect to the bid profile specified in (7.14) is specified as follows:

 gn (x n ) = αn P min(xn P , sn P ) + αn Q min(xn Q , sn Q ).

(7.15)

7.3.2 Service Allocation Rule Subject to Bid Profiles of Individual Units Definition 7.1 Considering a collection of bid profiles c ≡ (bs , a), we call x ≡ (x s , x 1 , . . . , x N ) is an admissible allocation with respect to c, if the following constraints hold: (0, 0) ≤ (xs P , xs Q ) ≤ (ds P , ds Q ), (0, 0) ≤ (xn P , xn Q ) ≤ (sn P , sn Q ), for all n ∈ N .

(7.16a) (7.16b)

The set of admissible allocations with respect to c is denoted by A (c). We further define a function U on an admissible allocation x with respect to a bid profile c as the following: U (x) = (βs P xs P + βs Q xs Q ) −

N  (αn P xn P + αn Q xn Q ),

subject to (7.8).

n=1

(7.17)

7.3 Auction-Based Distributed Vehicle-to-Grid Coordination Method

183

The auctioneer assigns an optimal admissible allocation x ∗ with respect to a collection of bid profiles c, such that x ∗ (c) = argmax{U (x)}.

(7.18)

x∈A (c)

Let  λs P ,  λs Q denote the Lagrange multipliers corresponding to the quantity λn Q denote the Lagrange multipliers corconstraint of power grid (7.16a), and  λn P ,  μ Q denote the responding to the quantity constraint of PEV n (7.16b), and  μP ,  Lagrange multipliers corresponding to the equality constraint (7.8). The associated Lagrange function is given below:  ) =  λ, μ λs P (ds P − xs P ) +  λs Q (ds Q − xs Q ) L(x s , x n ,  + λn P (sn P − xn P ) +  λn Q (sn Q − xn Q ) + (βs P xs P + βs Q xs Q ) − + μP

 N 

 xn P − xs P

n=1

N   n=1

+ μQ

(αn P xn P + αn Q xn Q )  N 



 xn Q − xs Q

n=1



≡ with x s ≡ (xs P , xs Q ), x n ≡ (xn P , xn Q ; n ∈ N ),  λ≡  λs P ,  λs Q ,  λn P ,  λn Q and μ μ Q ; then the KKT conditions for the auction optimization problems are listed  μP ,  below: μ∗P −  λ∗s P ≤ 0, βs P −  βs Q −  μ∗Q −  λ∗s Q ≤ 0, λ∗n P +  μ∗P ≤ 0, − αn P −  − αn Q −  λ∗n Q +  μ∗Q ≤ 0, xs∗P xs∗Q xn∗P xn∗Q

≤ ds P , ≤ ds Q , ≤ sn P , ≤ sn Q ,

with u ∗s P =

N  n=1

xs∗P ≥ 0, xs∗Q xn∗P xn∗Q  λ∗s P  λ∗s Q  λ∗n P  λ∗n Q

u ∗n P and u ∗s Q =

N 

≥ 0, ≥ 0, ≥ 0, ≥ 0, ≥ 0, ≥ 0, ≥ 0,

xs∗P (βs P −  μ∗P −  λ∗s P ) = 0; xs∗Q (βs Q −  μ∗Q −  λ∗s Q ) = 0; xn∗P (−αn P xn∗Q (−αn Q

− λ∗n P +  μ∗P ) = 0; − λ∗n Q +  μ∗Q ) = 0;

(7.19a) (7.19b) (7.19c) (7.19d)

 λ∗s P (ds P − xs∗P ) = 0;  λ∗s Q (ds Q − xs∗Q ) = 0;

(7.19e)

 λ∗n P (sn P − xn∗P ) = 0;  λ∗n Q (sn Q − xn∗Q ) = 0.

(7.19g)

(7.19f) (7.19h)

u ∗n Q .

n=1

7.3.3 Transfer Money of Agents Subject to Bid Profiles For each bid profile c, we will specify the so-called transfer moneys for power grid and PEV n, denoted by τs (c) and τn (c), respectively, following the allocation way imple-

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

mented by the system auctioneer given in (7.18). Essentially, the payment/income of an individual unit can be expressed as the summation of all users’ utility functions when this unit didn’t join the auction, minus the summation of the all of other units’ utility functions when this unit joined the auction. That is to say, the money transfer τ made by each of related units is exactly the externality it imposes on others through its participation, just as in the VCG-style mechanism adapted in [32, 41], etc.

7.3.3.1

Money Transfer of Power Grid

Let x ∗ (c(s) ) denote the solution to the allocation rule, defined in Sect. 7.3.2, with c(s) = (b−s , a) for power grid, i.e., x ∗ (c(s) ) = argmax {U (x)}. x∈A (c(s) )

For notational simplicity, we rewrite x ∗ (c(s) ) as x ∗(s) . The money transfer to be made by power grid with bid profile c, denoted by τs (c), is given by (7.20) below,

τs (c) = U (x ∗(s) ) − U (x ∗ ) − (βs P u ∗s P + βs Q u ∗s Q ) ,

(7.20)

with x ∗ ≡ x ∗ (c) specified in (7.18).

7.3.3.2

Money Transfer of an Individual PEV

Let x ∗ (c(n) ) denote the solution to the allocation rule, defined in Sect. 7.3.2, with c(n) = (bs , a) substituted with sn P = sn Q = 0 for PEV n, i.e. x ∗ (c(n) ) = argmax {U (x)}. x∈A (c(n) )

For notational simplicity, we rewrite x ∗ (c(n) ) as x ∗(n) . The money transfer to be made by PEV n with bid profiles c, denoted by τn (c), is given by (7.21) below, τn (c) = U (x ∗(n) ) − (U (x ∗ ) + (αn P u ∗n P + αn Q u ∗n Q )).

(7.21)

7.3.4 Payoff Functions of Individual Units Subject to a collection of bid profiles, the payoff functions of individual units are specified by adapting the auctioneer’s optimal resource allocation and the money transfer mechanism.

7.3 Auction-Based Distributed Vehicle-to-Grid Coordination Method

185

Power grid has a payoff function, denoted by f s (c), such that f s (c)  vs (u ∗s ) − τs (c),

(7.22)

where u ∗s ≡ u ∗s (c) represents the allocation to power grid assigned by auctioneer with the bid profiles c, and τs (c) is the money payed by power grid defined in (7.20). Individual PEV n has a payoff function, denoted by f n (c), such that f n (c)  −τn (c) − gn (u ∗n ),

(7.23)

where u ∗n ≡ u ∗n (c) represents the allocated generation power for PEV n assigned by auctioneer with c, and τn (c) is the money transfer of PEV n defined in (7.21). Definition 7.2 A collection of bid profiles (bs0 , a10 , . . . , a 0N ) is a Nash equilibrium (NE) for the auction game if the following holds: f s (bs0 , a0 ) ≥ f s (bs , a0 ), for all bs ∈ Bs , f n (bs0 , an0 , a0−n ) ≥ f n (bs0 , an , a0−n ), for all an ∈ Bn , i.e., bs0 is a best response of power grid with respect to bid profiles a0 of PEV populations, and an0 represents a best response of PEV n with respect to (a0−n , bs0 ) 0 0 with a0−n ≡ (a10 , . . . , an−1 , an+1 , . . . , a 0N ).

7.3.5 NE Property of Efficient Bid Profiles We specify a collection of incentive compatible bid profiles, denoted by c∗ , such that

bs∗ = (vs P (xs∗∗P ), xs∗∗P ), (vs Q (xs∗∗Q ), xs∗∗Q ) ,

an∗ = (gn P (xn∗∗P ), xn∗∗P ), (gn Q (xn∗∗Q ), xn∗∗Q ) ,

(7.24a) (7.24b)

∗∗ for all n ∈ N , where x ∗∗ ≡ (xs∗∗P , xs∗∗Q , u ∗∗ n P , u n Q ; n ∈ N ) represents the efficient allocation specified in (7.9).

7.3.5.1

Existence of an Efficient NE

We will first show that the allocations subject to c∗ are efficient in Lemmas 7.1 and 7.2, and show in Theorem 7.1 that c∗ is a NE for the underlying auction games.

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

Lemma 7.1 Considering bid profile c∗ specified in (7.24), we have ⎧ ≥ μ∗P , in case xn∗∗P = 0, en∗∗ ≤ Sn2 ⎪ ⎪ ⎪ ∗ ∗∗ 2 ⎪ in case xn∗∗P ∈ (0, xnmax ⎨ = μP , P ), en < Sn ∗ ∗ ∗ ∗∗ max ∗∗ 2 in case xn P = xn P , en < Sn αn P = μ P − λn P , ⎪ ∗ ∗ ∗∗ ∗∗ 2 ⎪ = μ − 2λ x , in case xn∗∗P ∈ (0, xnmax ⎪ P nS n P P ), en = Sn ⎪ ⎩ ∗ ∗ ∗∗ ∗ ∗∗ max ∗∗ 2 = μ P − 2λnS xn P − λn P , in case xn P = xn P , en = Sn ⎧ in case xn∗∗Q = 0, en∗∗ ≤ Sn2 ⎨ ≥ μ∗Q , ∗ ∗ in case xn∗∗Q > 0, en∗∗ < Sn2 αn Q = μ Q , ⎩ ∗ ∗ ∗∗ = μ Q − 2λnS xn Q , in case xn∗Q > 0, en∗∗ = Sn2  in case u ∗∗ = μ∗P , sP > 0 βs∗P ∗ in case u ∗∗ ≤ μP , sP = 0  ∗ in case u ∗∗ = μQ , ∗ sQ > 0 βsQ ∗ in case u ∗∗ ≤ μQ , sQ = 0

(7.25a)

(7.25b)

(7.25c) (7.25d)

∗ ∗∗2 ∗ ∗ ∗ ∗ ∗ with en∗∗ ≡ xn∗∗2 P + x n Q , λ ≡ (λn P , λnS ; n ∈ N ) and μ ≡ (μ P , μ Q ) specified in (7.10).

Proof Lemma 7.1 can be verified by analyzing the KKT conditions (7.10) and c∗ defined in (7.24). Lemma 7.2 Suppose c∗ as the collection of bid profiles specified in (7.24); then x ∗ (c∗ ) = x ∗∗ , i.e., the allocation x ∗ (c∗ ) is efficient. Proof Lemma 7.2 can be verified by substituting (7.24) and (7.25) into the allocation rule (7.18), and then comparing the KKT conditions specified in (7.10) and (7.19) respectively. Theorem 7.1 Under Assumptions (A1) and (A2), the efficient bid profile c∗ specified in (7.24) is a NE for the underlying auction games. Proof The proof is given in the Appendix. In Theorem 7.1, we verified that there exists an efficient NE for the underlying V2G auction games; however as discussed below, there may exist other inefficient NEs beside the efficient one. In Fig. 7.4 we illustrate incentive compatible bid profiles for V2G auction games, such that s P ,  s Q ,   α1P =  α2P < β s1P + s2P = ds P , and  α1Q =  α2Q < β s1Q + s2Q = ds Q . (7.26) Remark For purpose of demonstration, we only consider V2G coordination problems with two PEVs.

7.3 Auction-Based Distributed Vehicle-to-Grid Coordination Method

187

Fig. 7.4 Inefficient NE

According to the KKT conditions, it is easy to check that the allocations to all players subject to bid profiles specified in (7.26) are inefficient, and by applying the technology used in the proof of Theorem 7.1, we can verify that the collection of bid profiles specified in (7.26), is a NE as well.

7.4 V2G Coordination Auction Games with Aggregated Players Until now, we studied the auction games for V2G coordination problems. However, it is infeasible to implement for the underlying auction games with high-penetration PEV populations in the large-scale power grid, since it may require enormous communication resources between individual PEVs and the power grid. To overcome this challenge, in this section, we define the notion of aggregated players (or aggregators), such that each aggregator can directly control the active and reactive power for the

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

collection of PEVs managed by this aggregator in an economic way, and plays the auction game with other aggregators and the power grid. By abuse of notation, we denote by xi P and xi Q the active power and reactive power of aggregator i, for any i ∈ I with I representing the set of aggregators. We also denote by Mi the collection of PEVs managed by aggregator i. x i ≡ (xi P , xi Q ) is an admissible power coordination of aggregator i, if the following constraints are satisfied: xi P ≥ 0, xi Q ≥ 0,  xnmax xi P ≤ ximax P  P , n∈M i

xi2P + xi2Q ≤ Si2 



(7.27a) (7.27b)

Sn2 ,

(7.27c)

n∈M i 2 where ximax P and Si are the max power charging rate of aggregator i and the nominal apparent power of the power electronic device for aggregator i, respectively. The cost function of aggregator i, denoted by G i (x i ), subject to an admissible coordination strategy x i is specified below as:

G i (x i ) = G i P (xi P ) + G i Q (xi Q )

(7.28)

where G i P (xi P ) and G i Q (xi Q ) denote the cost functions of aggregator i with respect to active power and reactive power, respectively. We suppose that each aggregator can directly control the active and reactive power for the collection of PEVs managed by this aggregator in an economic way; then G i P (xi P ) shall be defined as the minimal overall cost for all PEVs in Mi with the total supplied active power of xi P , i.e., G i P (xi P ) =



min

(xn P ;n∈M i )∈X i (xi P )

gn P (xn P )

(7.29)

n∈M i

with Xi (xi P ) specified in the following: Xi (xi P ) 

⎧ ⎨ ⎩



(xn P ; n ∈ Mi ) : s.t. xi P =

xn P , and xn P ≤ xnmax P

n∈M i

⎫ ⎬ ⎭

.

(7.30)

Similarly, G i Q (xi Q ) shall be defined as the minimal overall cost for all PEVs in Mi with the total supplied reactive power of xi P , i.e., G i Q (xi Q ) =

min

xi Q =Σn∈M i xn Q

 n∈M i

gn Q (xn Q ).

(7.31)

7.4 V2G Coordination Auction Games with Aggregated Players

189

Lemma 7.3 Under Assumption (A2), G i P (xi P ) and G i Q (xi Q ) are strictly increasing and convex. Proof Proof of Lemma 7.3 is given in the Appendix. By the specifications of V2G coordination problems with the collection of aggregators given in this section and Lemma 7.3, we can obtain that the analysis studied in Sect. 7.3 for auction games for individual PEVs still hold for the auction games for aggregators. As a consequence, we have the existence of an efficient NE for the aggregated auction games as stated in Corollary 7.1. Corollary 7.1 Under Assumptions (A1) and (A2), the efficient bid profile for the aggregated V2G coordination problems is a NE for the aggregated auction games.

7.5 Implementation of Nash Equilibrium In this section, we will discuss the implementation of an efficient Nash equilibrium for the underlying V2G coordination auction games by extending a so-called dynamical quantized-PSP (Q-PSP) mechanism, proposed in [33] for double-side auction games, under which the game system can converge to a NE close to an efficient NE and show that under the Q-PSP mechanism the system converges to a NE, in finite steps, which approximate the efficient NE up to a quantized level. Essentially the double-side auctions are formulated as two single-side auctions, say buyer-side and seller-sided auctions, coupled by a so-called “joint potential quantity”. We extend the Q-PSP mechanism to the V2G auction games as specified in Steps (1)–(4) below: (1) Initialize a collection of bid profiles for PEVs and power grid; (2) Set accumulated demand as the bid quantity of power grid, and the accumulated supply as the sum of the bid quantities of those PEVs whose bid price is less than the price of the power grid; (3) Set potential active (reactive respectively) power quantity as – Larger value of the accumulated active (reactive respectively) power demand and supply, encourages to supply or to demand more electricity in the market, when the active (reactive respectively) power selling price of any players whose constraint is not reached, is not equal to the buying price; – Smaller value of accumulated demand and supply, otherwise. (4) PEVs and power grid update their bid profiles w.r.t. potential quantity simultaneously as follows: – Each PEV updates its bid to maximize its own utility; then it bids a smaller value of the price of the power grid in the last iteration and

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

higher quantized price than its bid price in case the accumulated supply is less than potential quantity, and bids a smaller value of the price of the power grid in the last iteration and lower quantized price than its bid price, otherwise. – Power grid updates its bid to maximize its own utility; then it bids a larger value of the matched selling price and lowers quantized price than its bid price in case accumulated demand is less than potential quantity; and bids a larger value of the matched selling price and higher quantized price than its bid price, otherwise. (4) Go to Step (1) until joint potential quantity does not update anymore. Remark The matched selling price, appeared in Step (3) above, represents the highest price of PEV whose price is not higher than the price of the power grid. In the following, we study the numerical simulations for the V2G coordination games with four aggregators under the Q-PSP mechanism. For the purpose of demonstration we suppose that each aggregator possesses a common active power constraint of 9 MW and an apparent power constraint of 10 MVA, and consider the referred active and reactive powers as P = 30 MW and Q = 15 Mvar, respectively. By Lemma 7.3, the cost function of each aggregator is strictly convex under convexity assumption of cost functions of individual players. Here we suppose a simplified quadratic form for cost functions of aggregators, such that G i P (xi P ) = κi P xi2P and G i Q (xi Q ) = κi Q xi2Q with parameters given in Table 7.1. We consider that the (concave) valuation function of power grid has a common negative quadratic form as vs P = −1200(xs P − P)2 and vs Q = −1200(xs Q − Q)2 . The updates of players’ bid profiles and potential quantity under Q-PSP auction mechanism are displayed in Fig. 7.5, and the evolution of constraints of players is illustrated in Table 7.2 where 1 and 0 are used to represent the case that players reach and don’t reach at constraint, respectively. From Fig. 7.5 and Table 7.2, we can observe that the system converges to a NE close to an efficient NE.

7.6 Conclusions and Future Works In this chapter, we formulated a class of coordination problems of vehicle-to-grid to regulate the frequency and voltage of power grid, and further developed a distributed coordination method under the PSP double-sided auction mechanism such that the

Table 7.1 Cost function parameters for aggregated players Parameter of Player 1 Player 2 Player 3 aggregators κi P κi Q

98 50

135 83

156 93

Player 4 172 105

7.6 Conclusions and Future Works

191

Fig. 7.5 Updates of players’ bid profiles and potential quantity under Q-PSP auction mechanism

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

Table 7.2 Evolution of constraint status of aggregated players Iteration 1 2 3 4 5 6 7 8 step Player 1 Player 2 Player 3 Player 4

0 0 0 0

1 1 1 1

1 1 0 0

1 1 0 0

1 1 0 0

1 1 0 0

1 0 0 0

1 0 0 0

9

10

11

12

1 0 0 0

1 0 0 0

1 0 0 0

1 0 0 0

efficient coordination solution is a NE for the underlying auction games and its corresponding allocation is efficient. In order to deal with the V2G coordination problems with large-scale PEVs, we formulated the auction games with aggregators each of which can centrally coordinate the power rates of collection of individual PEVs. An efficient NE for the underlying auction games is implemented with numerical simulations by generalizing the so-called Q-PSP mechanism. As ongoing research, we would like to rigorously verify the implementation of an efficient NE for the V2G auction games with constrained multi-dimension bid strategy under the Q-PSP auction mechanism.

Appendices Proof of Theorem 7.1 It is sufficient to show that the efficient bid profile c∗ ≡ (bs∗ , a∗ ) is a NE if (1) and (2) below hold. (1) To show that bs∗ is a best response of power grid with respect to a∗ in (1.1)–(1.2) below. (1.1) Suppose that power grid decreases its allocation from xs∗∗P by a δs P ≥ 0 and decreases its allocation from xs∗∗Q by a δs Q ≥ 0, via deviating its bid profile bs∗ to another one bs† . We consider xs∗∗P > 0 in case δs P > 0 and xs∗∗Q > 0 in case δs Q > 0, since power grid with xs∗∗P = 0 can’t decrease its active power allocation and power grid with xs∗∗Q = 0 can’t decrease its reactive power allocation. Let x † = (xs†P , xs†Q , xn† P , xn†Q ; n ∈ N ) denote the allocation with respect to the bid profile c† = (bs† , a∗ ). By (7.17) and (7.18), we can obtain that the allocations to some PEVs decrease while the allocations to other PEVs are unchanged, in case the allocation to the power grid decreases. We denote by Δτs the change of transfer money of power grid by deviating from bs∗ to bs† ; then we have the following analysis:

Appendices

193

Δτs = τs (bs† , a∗ ) − τs (bs∗ , a∗ ) N    = αn∗ P (xn† P − xn∗∗P ) + αn∗Q (xn†Q − xn∗∗Q ) n=1 N  

≥

βs∗P

=

n=1 ∗ −βs P δs P

xn† P

−

xn∗∗P



+

βs∗Q

N  

xn†Q − xn∗∗Q



n=1

−

βs∗Q δs Q ,

where the 2nd equality is obtained just by taking differences of the payments of power grid, the inequality is obtained by comparing αn∗ P , βs∗P , αn∗Q and βs∗Q in (7.25), and the last equality follows by the equality constraint (7.8). The change in the valuation of power grid, denoted by Δvs , satisfies the following Δvs = vs P (xs∗∗P − δs P ) − vs P (xs∗∗P ) + vs Q (xs∗∗Q − δs Q ) − vs Q (xs∗∗Q ) < −vs P (xs∗∗P )δs P − vs Q (xs∗∗Q )δs Q = −βs∗P δs P − βs∗Q δs Q , where the 1st equality is the definition of the Δvs , the inequality is obtained by Assumption (A1), and the last equality is obtained by the definition of c∗ in (7.24). By the above analysis and (7.22), we have f (bs† , a∗ ) − f (bs∗ , a∗ ) = Δvs − Δτs < 0, i.e., the payoff of power grid decreases. (1.2) Suppose that the active or reactive power allocation to power grid increases via deviating from bs∗ to bs† . The allocation to power grid can’t increase anymore, since the allocation to each PEV reaches the maximum quantity it can provide under c∗ . (2) To show that an∗ is a best response of PEV n with respect to c∗−n ≡ (bs∗ , a∗−n ) in ∗∗2 ∗∗2 ∗∗2 2 2 (2.1) and (2.2) below in case xn∗∗2 P + x n Q < Sn and x n P + x n Q = Sn respectively. ∗∗2 ∗∗2 2 (2.1) In case xn P + xn Q < Sn . (2.1.i) Suppose that PEV n decreases its allocation from xn∗∗P by a δn P ≥ 0 and decreases its allocation from xn∗∗Q by a δn Q ≥ 0, via deviating its bid profile bn∗ to another one bn† . Then we consider xn∗∗P > 0 when δn P > 0 and xn∗∗Q > 0 when δn Q > 0, since PEV n with xn∗∗P = 0 can’t decrease its active power allocation and PEV n with xn∗∗Q = 0 can’t decrease its reactive power allocation. We define x † = (xs†P , xs†Q , xn† P , xn†Q ; n ∈ N ) as the allocation with respect to the bid profile c† = (bs∗ , an† , a∗−n ), then we note that x † satisfies the following analysis by (7.17) and (7.18): † † ∗∗ ∗∗ (1a) xn† P = xn∗∗P − δn P , u †m P = u ∗∗ m P ; x n Q = x n Q − δn Q , u m Q = u m Q , for all m = n, i.e., the active and reactive power allocations to other PEVs are unchanged.

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7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

(1b) xs†P ≤ xs∗∗P , the equality holds with δn P = 0; xs†Q ≤ xs∗∗Q , the equality holds with δn Q = 0, i.e. power grid buys less, when PEV n sells less. By the equality constraint (7.8), we have xs†P = xs∗∗P − δn P , xs†Q = xs∗∗Q − δn Q . By (7.21), we get the change in transfer money of PEV n is Δτn = τn (bs∗ , an† , a∗−n ) − τn (bs∗ , an∗ , a∗−n ) = βs∗P δn P + βs∗Q δn Q ≥ αn∗ P δn P + αn∗ Q δn Q ,

where the 2nd equality holds by (1a) and (1b), and the inequality holds by comparing αn∗ P , βs∗P , αn∗Q and βs∗Q in (7.25). Under Assumption (A2), gn P (xn P ) and gn Q (xn Q ) are increasing and strictly convex, then we get the change in the individual cost of PEV n is Δgn = gn P (xn∗∗P − δn P ) − gn P (xn∗∗P ) + gn Q (xn∗∗Q − δn Q ) − gn Q (xn∗∗Q ) > −gn P (xn∗∗P )δn P − gn Q (xn∗∗Q )δn Q = −αn∗ P δn P − αn∗Q δn Q , where the 1st equality is the definition of the Δgn , the inequality is obtained by Assumption (A2) and the 2nd equality is obtained by the definition of c∗ in (7.24). By the above analysis and (7.23), we have f (bs∗ , an† , a∗−n ) − f (bs∗ , an∗ , a∗−n ) = −Δgn − Δτn < 0, i.e., the payoff of PEV n decreases. (2.1.ii) Suppose that PEV n decreases its allocation from xn∗∗P by a δn P ≥ 0 and increases from xn∗∗Q by a δn Q > 0, via deviating its bid profile bn∗ to another one bn† . We consider xn∗∗P > 0 when δn P > 0, since PEV n with xn∗∗P = 0 can’t decrease its active power allocation. Define x † = (xs†P , xs†Q , xn† P , xn†Q ; n ∈ N ) as the allocation with respect to the bid profile c† = (bs∗ , an† , a∗−n ); then we note that x † satisfies the following analysis by (7.17) and (7.18): (2a) xn† P = xn∗∗P − δn P , u †m P = u ∗∗ m P , for all m = n, i.e., the active power allocations to other PEVs are unchanged; xn†Q = xn∗∗Q + δn Q , u †m Q ≤ u ∗∗ m Q , for all m = n,  † ∗∗ such that m =n (u m Q − u m Q ) = δn Q , i.e., the reactive power allocations to other PEVs decrease or are unchanged. (2b) xs†P ≤ xs∗∗P , the equality holds with δn P = 0; xs†Q = xs∗∗Q , i.e., power grid buys less active power when PEV n sells less and the reactive power allocation to the power grid is unchanged. By the equality constraint (7.8), we have xs†P = xs∗∗P − δn P . By (7.21), we get the change in the transfer money of PEV n is

Appendices

195

Δτn = τn (bs∗ , an† , a∗−n ) − τn (bs∗ , an∗ , a∗−n )  † αm∗ Q (u ∗∗ = βs∗P δn P − m Q − um Q ) m =n

≥ αn∗ P δn P − αn∗Q δn Q , where the 2nd equality is obtained just by taking differences of the transfer moneys of PEV n and considering (2b), the inequality is obtained by comparing the αn∗ P , ∗∗ βs∗P , αn∗Q and αm∗ Q with u ∗∗ m Q > 0 (since the allocation to PEV m with u m Q = 0 can’t decrease) in (7.25) and considering (2a). The change of the individual cost of PEV n, denoted by Δgn , satisfies the following analysis Δgn = gn P (xn∗∗P − δn P ) − gn P (xn∗∗P ) + gn Q (xn∗∗Q + δn Q ) − gn Q (xn∗∗Q ) > −gn P (xn∗∗P )δn P + gn Q (xn∗∗Q )δn Q = −αn∗ P δn P + αn∗Q δn Q , where the inequality is obtained by Assumption (A2) and the last equality is obtained by (7.24). By the analysis given above and (7.23), we have f (bs∗ , an† , a∗−n ) − f (bs∗ , an∗ , a∗−n ) = −Δgn − Δτn < 0, i.e., the payoff of PEV n decreases. (2.1.iii) Suppose that PEV n increases its allocation from xn∗∗P by a δn P ≥ 0 and decreases from xn∗∗Q by a δn Q > 0, via deviating its bid profile bn∗ to another one bn† . Then we consider xn∗∗Q > 0 with δn Q > 0, since PEV n with xn∗∗Q = 0 can’t decrease max ∗∗ its reactive power allocation, and xn P < xnmax P , since PEV n with x n P = x n P can’t increase its active power allocation. Define x † = (xs†P , xs†Q , xn† P , xn†Q ; n ∈ N ) as the allocation with respect to the bid profile c† = (bs∗ , an† , a∗−n ), then we note that x † satisfies the following analysis by (7.17) and (7.18):  † ∗∗ (3a) xn† P = xn∗∗P + δn P , u †m P ≤ u ∗∗ m =n (u m P − u m P ) = m P , for all m = n, such that δn P , i.e., the active power allocations to other PEVs decrease or are unchanged; xn†Q = xn∗∗Q − δn Q , u †m Q = u ∗∗ m Q , for all m = n, i.e., the reactive power allocations to other PEVs are unchanged. (3b) xs†P = xs∗∗P ; xs†Q ≤ xs∗∗Q , the equality holds with δn Q = 0, i.e., the active power allocation to power grid is unchanged and the power grid buys less reactive power when PEV n sells less. By the equality constraint (7.8), we have xs†Q = xs∗∗Q − δn Q . By (7.21), we get the change in transfer money of PEV n is

196

7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

Δτn = τn (bs∗ , an† , a∗−n ) − τn (bs∗ , an∗ , a∗−n )  † αm∗ P (u ∗∗ = βs∗Q δn Q − m P − um P ) m =n

≥ αn∗Q δn Q − αn∗ P δn P where the 2nd equality is obtained just by taking differences of the transfer moneys of PEV n and considering (3b), the inequality is obtained by comparing the αn∗ P , ∗∗ αm∗ P with u ∗∗ m P > 0 (since the allocation to PEV m with u m P = 0 can’t decrease), ∗ ∗ αn Q and βs Q in (7.25) and considering (3a). Under Assumption (A2), gn P (xn P ) and gn Q (xn Q ) are increasing and strictly convex, then we get the change in the individual cost of PEV n is Δgn = gn P (xn∗∗P + δn P ) − gn P (xn∗∗P ) + gn Q (xn∗∗Q − δn Q ) − gn Q (xn∗∗Q ) > gn P (xn∗∗P )δn P − gn Q (xn∗∗Q )δn Q = αn∗ P δn P − αn∗Q δn Q , where the inequality is obtained by Assumption (A2) and the 2nd equality is obtained by the definition of c∗ in (7.24). By the above analysis and (7.23), we have f (bs∗ , an† , a∗−n ) − f (bs∗ , an∗ , a∗−n ) = −Δgn − Δτn < 0, i.e. the payoff of PEV n decreases. (2.1.iv) Suppose that PEV n increases its allocation from xn∗∗P by a δn P ≥ 0 and increases its allocation from xn∗∗Q by a δn Q ≥ 0, via deviating its bid profile bn∗ to another one bn† . max max ∗∗ Then we consider u ∗∗ n P < x n P , since PEV n with x n P = x n P can’t increase its active power allocation. Define x † = (xs†P , xs†Q , xn† P , xn†Q ; n ∈ N ) as the allocation with respect to the bid profile c† = (bs∗ , an† , a∗−n ), then we note that x † satisfies the following analysis by (7.17) and (7.18):  † ∗∗ (4a) xn† P = xn∗∗P + δn P , u †m P ≤ u ∗∗ m =n (u m P − u m P ) = m P , for all m = n, such that δn P , i.e. the active power allocations to other PEVs decrease or are unchanged  with u m P = 0; xn†Q = xn∗∗Q + δn Q , u †m Q ≤ u ∗∗ m =n m Q , for all m = n, such that † (u ∗∗ m Q − u m Q ) = δn Q , i.e., the reactive power allocations to other PEVs decrease or remain unchanged. (4b) xs†P = xs∗∗P ; xs†Q = xs∗∗Q , i.e., the active and reactive power allocations to power grid are unchanged. By (7.21), we get the change in transfer money of PEV n is

Appendices

197

Δτn = τn (bs∗ , an† , a∗−n ) − τn (bs∗ , an∗ , a∗−n )   † † αm∗ P (u ∗∗ αm∗ Q (u ∗∗ =− m P − um P ) − m Q − um Q ) m =n

m =n

≥ −αn∗ P δn P − αn∗Q δn Q where the 2nd equality is obtained just by taking differences of the transfer moneys of PEV n, the inequality is obtained by comparing the αn∗ P , αm∗ P with u ∗∗ mP > 0 ∗ ∗ = 0 can’t decrease), α and α (since the allocation to PEV m with u ∗∗ mP nQ m Q with ∗∗ > 0 (since the allocation to PEV m with u = 0 can’t decrease) in (7.25) and u ∗∗ mQ mQ considering (4a) and (4b). The change of the individual cost of PEV n satisfies the following: Δgn = gn P (xn∗∗P + δn P ) − gn P (xn∗∗P ) + gn Q (xn∗∗Q + δn Q ) − gn Q (xn∗∗Q ) > gn P (xn∗∗P )δn P + gn Q (xn∗∗Q )δn Q = αn∗ P δn P + αn∗Q δn Q , where the inequality is obtained by Assumption (A2) and the 2nd equality is obtained by the definition of c∗ in (7.24). By the above analysis and (7.23), we have f (bs∗ , an† , a∗−n ) − f (bs∗ , an∗ , a∗−n ) = −Δgn − Δτn < 0, i.e., the payoff of PEV n decreases. ∗∗2 2 (2.2) In case xn∗∗2 P + x n Q = Sn . (2.2.i) Suppose that PEV n decreases its allocation from xn∗∗P by a δn P ≥ 0 and decreases its allocation from xn∗∗Q by a δn Q ≥ 0, via deviating its bid profile bn∗ to another one bn† . Following the similar analysis in (2.1.i), we can make a conclusion that the payoff of PEV n decreases in this case. (2.2.ii) Suppose that PEV n increases its allocation from xn∗∗P by a δn P ≥ 0, via deviating its bid profile bn∗ to another one bn† . Due to the constraint of apparent power in (7.2), the reactive power allocation to PEV n has to be decreased by some positive valued δn Q > 0, such that (xn∗∗P + δn P )2 + (xn∗∗Q − δn Q )2 ≤ Sn2 . ∗∗ max Then we consider xn∗∗P < xnmax P , since PEV n with x n P = x n P cannot increase its active power allocation, and xn∗∗Q > 0, since PEV n with xn∗∗Q = 0 cannot decrease its reactive power allocation. We define x † = (xs†P , xs†Q , xn† P , xn†Q ; n ∈ N ) as the allocation with respect to the bid profile c† = (bs∗ , an† , a∗−n ), then we note that x † satisfies the following analysis by (7.17) and (7.18):

198

7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

 † ∗∗ (5a) xn† P = xn∗∗P + δn P , u †m P ≤ u ∗∗ m =n (u m P − u m P ) = m P , for all m = n, such that δn P , i.e., the active power allocations to other PEVs decrease or are unchanged; xn†Q = xn∗∗Q − δn Q , u †m Q = u ∗∗ m Q , for all m = n, i.e., the reactive power allocations to other PEVs are unchanged. (5b) xs†P = xs∗∗P , i.e., the active power allocation to power grid is unchanged; xs†Q < xs∗∗Q i.e., the reactive power allocation to power grid decreases. By the equality constraint (7.8), we have xs†Q = xs∗∗Q − δn Q . By (7.21), we get the change in transfer money of PEV n is Δτn = τn (bs∗ , an† , a∗−n ) − τn (bs∗ , an∗ , a∗−n )  † ∗ αm∗ P (u ∗∗ =− m P − u m P ) + βs Q δn Q

(7.32)

m =n

≥ −μ∗P δn P + μ∗Q δn Q ,

(7.33)

where the 2nd equality is obtained just by taking differences of the transfer moneys of PEV n and considering (5b), the inequality is obtained by (5a) and (7.25) where αm∗ P , μ∗P , βs∗Q and μ∗Q are specified. Under Assumption (A2), gn P (xn P ) and gn Q (xn Q ) are strictly increasing and convex, we have Δgn = gn P (xn∗∗P + δn P ) − gn P (xn∗∗P ) + gn Q (xn∗∗Q − δn Q ) − gn Q (xn∗∗Q ) > αn∗ P δn P − αn∗ Q δn Q ,

by which together with (7.25), we have Δgn > (μ∗P − 2λ∗nS xn∗∗P )δn P − (μ∗Q − 2λ∗nS xn∗∗Q )δn Q .

(7.34)

By (7.32) and (7.34), we get f (bs∗ , an† , a∗−n ) − f (bs∗ , an∗ , a∗−n ) = −Δgn − Δτn < 2λ∗ns xn∗∗P δn P − 2λ∗ns xn∗∗Q δn Q . (7.35) By the apparent power constraint of (7.16), we have ∗∗2 2 xn∗∗2 P + x n Q = Sn ,

(xn∗∗P

+ δn P ) + 2

(xn∗∗Q

(7.36) − δn Q ) ≤ 2

Sn2 .

(7.37)

By (7.36) and (7.37), we have 2xn∗∗P δn P − 2xn∗∗Q δn Q + δn2 P + δn2 Q ≤ 0; then 2xn∗∗P δn P − 2xn∗∗Q δn Q < 0, by which together with (7.35), we obtain that f (bs∗ , an† , a∗−n ) − f (bs∗ , an∗ , a∗−n ) < 0, i.e., the payoff of PEV n decreases by deviating from an∗ to any other bid profile an† .

Appendices

199

(2.2.iii) Suppose that PEV n decreases its allocation from xn∗∗P by a δn P ≥ 0 and increases from xn∗∗Q by a δn Q > 0, via deviating its bid profile bn∗ to another one bn† . Following a similar analysis in (2.2.ii), we conclude that the payoff of PEV n decreases. Thus, following the analysis in (1) and (2), the efficient bid profile c∗ is a NE.

Proof of Lemma 7.3 Firstly we have the Lagrange function for the constrained optimization problem appeared in (7.29) is specified below: L(xn P , λ p , μ) =



gn P (xn P ) +

n∈M i



p

max

λn x n P − x n P

⎛ + μ ⎝xi P −

n∈M i



⎞ xn P ⎠ ,

n∈M i

(7.38) p

p

with λ p ≡ (λn , n ∈ Mi ), where μ and λn are the Lagrange multipliers corresponding to the equality and inequality constraints in (7.30), respectively. Equation (7.39) listed below, together with the equality constraint in (7.30), are the KKT conditions for the constrained optimization problems appeared in (7.29). gn P (xn♦P ) + λn P − μ∗ ≥ 0, p∗

xn♦P − xnmax P ≤ 0,

xn♦P ≥ 0,

  u ♦P gn P (xn♦P ) + λnp∗ − μ∗ = 0;

λnp∗ ≥ 0,





(7.39a)

= 0. λnp∗ xn♦P − xnmax P (7.39b)

Under Assumption (A2), the optimizations specified in (7.29) are constrained convex optimization problems; then thereexist a unique solution  for (7.29) with an ♦ aggregated active power xi P , denoted by xn P (xi P ); n ∈ Mi , that is to say, 

 xn♦P (xi P ); n ∈ Mi =

argmin (xn P ;n∈M i )∈X i (xi P )



gn P (xn P ).

n∈M i

In (1) and (2) below, we will show strictly increasing and convexity properties of G i P (xi P ) respectively. (1) To show the strictly increasing property of G i P (xi P ). Consider a pair of distinct valued (xi†P , xi‡P ), such that xi†P < xi‡P ; then it is straightforward to check that xn♦P (xi†P ) ≤ xn♦P (xi‡P ) for all n ∈ Mi , and more‡ † ♦ over there at least exists a PEV m such that u ♦ m P (x i P ) < u m P (x i P ), since   ‡ ‡ † † ♦ ♦ n∈M i x n P (x i P ) = x i P < x i P = n∈M i x n P (x i P ); then by these together with

200

7 Efficient Vehicle-to-Grid (V2G) Coordination in Smart …

Assumption (A2), we have 

G i P (xi†P ) =

     gn P xn♦P (xi†P ) < gn P xn♦P (xi‡P ) = G i P (xi‡P ),

n∈M i

n∈M i

in case xi†P < xi‡P . (2) To show the strictly convex property of G i P (xi P ). By calculate the second derivative of G i P (u i P), we have G iP (xi P ) =

⎧  ⎨ n∈M i



g  (x ♦ ) ⎩ nP nP

d xn♦P d xi P

2 + gn P



xn♦P

⎫  d2x♦ ⎬ nP , dx2 ⎭

with xn♦P ≡ xn♦P (xi P ).

iP

(7.40) d xn♦P = 0 in case By the KKT conditions of (7.39), we can obtain that d xi P     d2x♦   d2x♦ nP nP ∗ gn P xn♦P = μ∗ ; then gn P xn♦P = μ , by which 2 2 d x d x iP iP n∈M i n∈M i together with (7.40), we have G iP (xi P ) =

⎧  ⎨ ⎩ ⎧  ⎨

 gnP (xn♦P )

n∈M i

=

n∈M i

⎩

d xi P 

gnP (xn♦P )

d xn♦P d xn♦P d xi P

2 ⎫ ⎬ ⎭ 2 ⎫ ⎬ ⎭

+ μ∗

 d2x♦ nP 2 d x iP n∈M i

,

(7.41)

 dx♦ nP = 1. d xi P n∈M i In summary under Assumption (A2) and the fact that there exists a PEV n with d xn♦P > 0, we have G iP (xi P ) > 0, i.e., G i P (xi P ) is strict convex. d xi P Following the same technology above, we can show that G i Q (xi Q ) is strictly increasing and convex as well. where the last equality holds because

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Chapter 8

Efficient Charging Coordination for Electric Vehicles Under Auction Games

Abstract A novel class of auction games is formulated to study coordination problems arising from charging a population of electric vehicles (EVs) over a finite horizon. Different from those analyzed in the above chapters, the charging power of EVs at different time slots could be regarded as multi-type resources, and there exist coupling constraints among these resources, say the total charging power over the whole horizon should not exceed the battery size of EVs in this scenario. To compete for energy allocation over the horizon, each individual EV submits a multidimensional bid, with the dimension equal to two times the number of time-steps in the horizon. The use of the progressive second price (PSP) auction mechanism ensures that incentive compatibility holds for the auction games. Due to the cross elasticity of EVs over the charging horizon, the marginal valuation of an individual EV at a particular time is determined by both the demand at that time and the total demand over the entire horizon. This difficulty is addressed by partitioning the allowable set of bid profiles based on the total desired energy over the entire horizon. It is shown that the efficient bid profile over the charging horizon is a Nash equilibrium of the underlying auction game. An update mechanism for the auction game is designed. A numerical example demonstrates that the auction process converges to an efficient Nash equilibrium. The auction-based charging coordination scheme is adapted to a receding horizon formulation to account for disturbances and forecast uncertainty.

8.1 Introduction Vehicles that connect to the electricity grid to recharge, referred to generically as electric vehicles (EVs), offer a range of potential benefits, including reductions in reliance on liquid fuels and in pollutant emissions, and increased energy efficiency [1–3]. It is therefore, anticipated that EV sales will substantially increase over the next few years [4]. If such growth does eventuate, it will become necessary to account for EV charging patterns in grid operation, as argued in [4–7] and references therein. Accommodating large numbers of vehicles on the grid will require coordination of EV charging so that their power and energy requirements can be optimally and robustly satisfied. This is a challenging control problem. Work on analyzing EV © Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_8

203

204

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

charging schedules, and their effect on utilities, began in the 1980s [8]. Recent work is extensive and includes [9–11] which formulate EV charging control as constrained optimization problems, and [12] which develops an EV dispatch algorithm in the context of a day-ahead electricity market. Centralized coordination faces numerous difficulties, from computational complexity to the loss of EV decision-making autonomy. Many distributed coordination methods have been proposed to address those challenges, including [11, 13–22] and references therein. In a general sense, that work is structured around individual players determining their optimal charging strategy over the charging horizon with respect to either the total demand of the other players or the system (clearing) price, which is based on the total system demand. More specifically, a hierarchical structure is considered in [14, 15, 20, 21] for scheduling EV charging. Each EV determines its preliminary charging (load) profile by solving an individual optimization problem with respect to the latest forecast of the system clearing price. The clearing price is then updated to take into account the latest charging profiles of the individual EVs. For the update process designed in [14], the resulting strategies asymptotically approach a Nash equilibrium (NE) as the EV population increases to infinity. The resulting NE is nearly socially optimal. The market design in [21] considers both energy and reserve capacity,1 and also incorporates distribution network losses. It is shown that in such a market setting, the NE may not coincide with the socially optimal solution. To establish a tractable formulation, only energy markets will be considered in the remainder of the chapter. Also, network losses are not modeled. Most of the distributed methods cited above are quite distinct from the economic generation dispatch that underpins deregulated day-ahead electricity markets [23]. To economically dispatch generation, auction mechanisms, such as uniform marketclearing-price [24] and pay-as-bid [25], have been widely adopted in electricity markets around the world [26]. Each generating unit submits to the ISO their bids over the forward market period (typically 24 h), with bids consisting of pairings of minimal selling price and maximum supplied electricity for each market subinterval. The ISO dispatches the generation requirements among units based on their submitted bid profiles. However, these auction mechanisms do not achieve incentive compatibility and usually cannot attain an efficient solution [27]. In contrast, this chapter studies EV charging coordination over multiple time intervals under an incentive compatibility mechanism [28, 29]. The chapter utilizes a progressive second price (PSP) auction mechanism, designed by Lazar and Semret [30, 31] and initially applied in the allocation of network resources. In a single divisible resource allocation problem under the PSP auction mechanism, each player only reports a two-dimensional bid. This bid is composed of a maximum amount of demand and an associated buying price, and is used to replace the player’s complete (private) utility function. Under the PSP mechanism, the money transfer (or payment) of a player measures the externality that they impose on the system through their participation. This concept will be formalized in 1 Reserves refer to generator and/or load capacity that is available to compensate for sudden changes

in energy supply or demand.

8.1 Introduction

205

Sect. 8.3.2. As analyzed in [30, 31], the PSP auction mechanism is a VCG-style auction [32–34]. Therefore, incentive compatibility holds, ensuring that all players submit truth-telling bids, and resources are allocated efficiently. Under this mechanism, as verified in [31, 35] in the context of single-unit network resource allocations, the efficient bid profile is a NE. In formulating their bids, players must consider tradeoffs between energy costs that vary over the charging horizon, the benefit derived from the total acquired energy, and battery degradation. Individual EVs are, therefore, inter-temporal cross-elastic2 loads, as defined in [36]. This results in an auction-based allocation of a collection of divisible resources, where electric energy at each time-step of the horizon is a separate divisible resource. Consequently, each EV must submit a bid that has dimension double the number of divisible resources to be shared (equivalently double the number of time-steps in the charging horizon). Such auctions have received limited attention in the literature. It will be shown that a player’s marginal valuation for electric energy at a particular time is dependent upon both the amount of energy requested at that time and the total energy request over the entire charging horizon. A key contribution of the chapter is to show that the efficient set of EV bids over the charging horizon is a NE of the underlying auction game. However, due to the cross elasticity of a bid over the multi-step time horizon, it is infeasible to directly verify this NE property using analysis that is applicable for a single-resource auction game [31, 35]. An alternative approach is proposed in this chapter. In order to address cross elasticity, the set of bids of a player is partitioned into a collection of subsets, each of which is composed of bids that possess the same total desired electric energy over the horizon. Consequently, cross elasticity is eliminated for bids within each subset. For such bids, the marginal valuation at each time-step includes a variable part determined by the amount of energy requested at that time and a fixed part that is identical for all bids in that subset. With this construction, the NE property of the efficient solution can be established by verifying for each subset whether any player can benefit by unilaterally deviating from their efficient bid profile. The chapter is organized as follows: Sect. 8.2 establishes the problem structure by formulating a class of EV charging coordination problems over a multiple timestep horizon. Distributed charging under the PSP auction mechanism is introduced in Sect. 8.3. Section 8.4 shows that the efficient (truth-telling) bid profile is a NE of the underlying PSP auction game. In Sect. 8.5, an update process is designed to implement the PSP auction, and an example illustrates that this process converges to the efficient NE. Section 8.6 concludes the chapter and provides a discussion of the future work.

2 Cross

elasticity refers to the ability of EVs to move charging demand from onetime interval to another.

206

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

8.2 Electric Vehicle Charging Coordination Formulation 8.2.1 Charging and Cost Models This study focuses on coordinating the charging of a population of EVs, denoted by N , finite charging horizon T  {t0 , . . . , t0 + T − 1}, with t0 representing the initial time. For each EV, n ∈ N , the energy delivered 3 over the tth time interval is denoted by xnt , and the battery state of charge (SoC) evolves according to: sn,t+1 = snt +

1 xnt , Θn

(8.1)

where Θn is the battery capacity and snt is the normalized SoC for the nth EV at time t. An admissible charging strategy, x n ≡ (xnt , t ∈ T ), satisfies the constraints:  xnt

≥ 0, when t ∈ Tn , = 0, otherwise

with



xnt ≤ Γn ,

(8.2)

t∈T

where Tn ⊂ T denotes the charging horizon of the nth EV, Γn = Θn (s n − sn0 ) gives the maximum energy that it can receive, and 0 ≤ sn0 ≤ s n ≤ 1 give the normalized minimum (initial) and maximum SoC, respectively. The values for Tn , Γn and sn0 follow from the driving style and vehicle battery capacity, see for example [37]. The set of all possible admissible charging strategies is denoted by Xn . Also, define the collection of admissible charging strategies for all EVs by x ≡ (x n ; n ∈ N ), with its corresponding set being X . As specified in (8.2), the charging demand of each EV, n ∈ N , at each time interval is elastic, but charging over the time horizon is coupled through the maximum energy Γn that can be delivered during that time. As a consequence, EV coordination is inherently a problem of scheduling demand that exhibits inter-temporal cross elasticity [36]. The utility function of the nth EV, for a charging strategy x n , is given by: wn (x n ) = −

 t∈T

f n (xnt ) − δn



xnt − Γn

2

,

(8.3)

t∈T

where δn > 0 is a fixed parameter, and f n (·) denotes the battery degradation cost of the nth EV. Remarks • This work considers LiFePO4 lithium-ion batteries which have been widely used in a variety of EVs. Key characteristics of this type of battery, including state of health, growth of resistance, and cycle life, are affected by charging 3 It

is assumed that the energy is delivered at a constant rate (power) over each time interval, and that the time intervals are of unit length. Therefore, the charging rate is also given by xnt .

8.2 Electric Vehicle Charging Coordination Formulation

207

behavior over many cycles. A degradation cost model for this type of battery cell is formulated in [38], based on the evolution of battery cell characteristics developed in [39, 40]. This degradation model expresses the energy capacity loss per second of a cell with respect to the charging current and voltage. Using this model, the relationship between degradation cost and charging rate xnt can be developed. • The second term in (8.3) captures the penalty cost due to not fully charging the EV over the time horizon, with δn weighting the relative importance of delivering the maximum energy during charging [41]. • The utility function wn (x n ), therefore, establishes the tradeoff between the battery degradation cost and the benefit derived from delivering the full charge. For a collection of admissible charging strategies x, the system cost is given by: Js (x) =



 ct

Dt +

t∈T



 xnt

−

n∈N



wn (x n ),

(8.4)

n∈N

 where ct (·), Dt and Dt + n∈N xnt denote the generation cost, the aggregate inelastic background demand and the total demand at time t, respectively. It is assumed that a forecast for Dt is available over the charging horizon. For the example presented in Sect. 8.5.3, the demand profile D ≡ (Dt ; t ∈ T ) is given by the aggregate demand for a typical summer day in the Midwest ISO region of North America. The system cost function (8.4) considers tradeoffs between total generation cost, aggregate battery costs, and the penalty for deviating from full charging. This contrasts with the general literature, for example [14, 15, 18], where the objective is to achieve valley-filling. While valley-filling minimizes the total generation cost, it may result in high battery degradation costs across the EV population.

8.2.2 Efficient Charging It is desirable to determine the collection of efficient (socially optimal) charging strategies x ∗∗ that minimizes the system cost (8.4). This centralized EV charging coordination problem can be formulated as the following optimization problem: Problem 8.1 min Js (x),

x∈X

such that x satisfies constraints (8.2) for all n ∈ N . The efficient charging strategy x ∗∗ of Problem 8.1 can be characterized by its associated KKT conditions. Firstly, the Lagrangian can be written: L(x, λ) = Js (x) +

 n∈N

λn

 t∈T

 xnt − Γn ,

208

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

 where λn is the Lagrangian multiplier associated with the constraint t∈T xnt ≤ Γn from (8.2). The KKT conditions for Problem 8.1 are, therefore, given by: ∂ L(x, λ) ≥ 0, ∂ xnt  xnt − Γn ≤ 0,

xnt ≥ 0, λn ≥ 0,

t∈T

∂ L(x, λ)xnt = 0, ∂ xnt    λn xnt − Γn = 0,

(8.5a) (8.5b)

t∈T

for all t ∈ T and n ∈ N , where:    ∂ ∂  L(x, λ) = ct Dt + xnt − wn (x n ) + λn . ∂ xnt ∂ xnt

(8.5c)

n∈N

Assumptions: The following conditions apply throughout the remainder of the chapter: (A1) ct (y) is monotonically increasing, strictly convex, and differentiable on y; (A2) f n (x), for all n ∈ N , is monotonically increasing, strictly convex, and differentiable on x. Remarks • The generation cost ct (·) is widely assumed to be a convex function of total generation, see for example [42–44]. • The battery degradation cost f n (·) is governed by the chemical processes inherent in charging. It is shown in Fig. 7 of [39] that growth of battery resistance, hence the fade of battery energy capacity, is generally increasing and convex with respect to the charging rate. This provides some justification for (A2) since f n measures the cost related to the fade of battery capacity with respect to the charging rate. Lemma 8.1 The collection of efficient charging strategies x ∗∗ for Problem 8.1 is unique. Proof Under Assumptions (A1, A2), the cost function Js (x) is strictly convex and differentiable. Also, the constraints (8.2) determine a convex domain. Therefore, Problem 8.1 is a strictly convex optimization problem. Thus, there exists a unique solution. This centralized charging coordination strategy can only be effectively implemented when the system has complete information and can directly schedule the behavior of all EVs. In practice, however, individuals are often unwilling to share their private information with others. Furthermore, transmission of complete information may incur excessive communications, and centralized control might be computationally infeasible. Thus, this chapter focuses on the development of a distributed control method that is based on the progressive second price (PSP) auction mechanism, which has been applied in [30, 31, 35] for efficient allocation of single-unit network resources. However, because EVs are scheduled over a multiple time-step horizon, charging coordination is a multi-unit resource allocation problem.

8.3 Distributed EV Charging Coordination Under a PSP Auction Mechanism

209

8.3 Distributed EV Charging Coordination Under a PSP Auction Mechanism 8.3.1 Bid Profiles of Individual Players Each EV, n ∈ N , submits a 2T -dimensional bid, bn ≡ (bnt , t ∈ T ), where: bnt =(βnt , dnt ), with   ≥ 0, t ∈ Tn dnt ≤ Γn , dnt , and = 0, otherwise t∈T specifies the price βnt that player n is willing to pay for energy at time t and the maximum electrical energy dnt that is desired at that time. The corresponding feasible allocation x n ≡ (xnt , t ∈ T ) with respect to bn must satisfy: 0 ≤ xnt ≤ dnt , ∀t ∈ T .

(8.6)

Let Bn denote the allowable set of bids for player n, so that bn ∈ Bn . Each player’s revealed utility function, denoted by wn (x n (bn ); bn ), is defined as: wn (x n (bn ); bn ) 



βnt xnt .

t∈T

The revealed system cost with respect to a bid profile b ≡ (bn , n ∈ N ) is given by: J (x(b); b) =



 ct

Dt +

t∈T



 xnt

n∈N

−



wn (x n (bn ); bn ).

(8.7)

n∈N

Auction-based EV charging allocation can be written as the following optimization problem: Problem 8.2

J ∗ (b) =

min

Constraint (8.6)

J (x(b); b).

The objective of the auctioneer is to assign an optimal allocation x ∗ (b) with respect to a bid profile b to minimize the revealed system cost given by J . Unlike the single-sided auctions considered in [30, 31], the cross elasticity inherent in Problem 8.2 suggests that the optimal allocation x ∗ (b) over the charging horizon T depends upon both the bid profile b over the entire horizon and the generation cost ct (·). However, the following lemma shows that the optimal allocation x ∗t at time t is completely determined by the bid profile bt and ct (·) at only that time.

210

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

 Lemma 8.2 Suppose x ∗ (b) ≡ x ∗t , t ∈ T is the optimal allocation subject to bid profile b. Then: x ∗t (b) ≡ x ∗t (bt ),

for all t ∈ T .

Proof For notational simplicity, consider: S (bt ) ≡ {x t ; s.t. 0 ≤ xnt ≤ dnt , for all n ∈ N },     xnt − βnt xnt . h t (x t ; bt ) ≡ ct Dt + n∈N

n∈N

It follows from (8.7) that: min

Constraint (8.6)

J (x(b); b) = =



min

x t ∈S (bt ), t∈T



h t (x t ; bt )

t∈T

min h t (x t ; bt ),

x t ∈S (bt )

t∈T

where the last equality holds because the summation is separable in terms of x t . The desired conclusion follows. The optimal charging allocation x ∗ (b) of Problem 8.2 can be characterized by the associated KKT conditions. Firstly, the Lagrangian can be written: L a (x, σ ; b) = J (x(b); b) +



σnt (xnt − dnt ),

n∈N t∈T

where σnt is the Lagrangian multiplier associated with each constraint in (8.6). The KKT conditions for Problem 8.2 are given by: ∂ a L (x, σ ; b) ≥ 0, ∂ xnt xnt − dnt ≤ 0,

xnt ≥ 0, σnt ≥ 0,

∂ a L (x, σ ; b)xnt = 0, ∂ xnt (xnt − dnt )σnt = 0,

(8.8a) (8.8b)

for all t ∈ T and n ∈ N , where: ∂ a L (x, σ ; b) = ct ∂ xnt

 Dt +



 xkt

− βnt + σnt .

(8.8c)

k∈N

It is now possible to establish a connection between the optimal charging strategies given by Problems 8.1 and 8.2. Lemma 8.3 Consider a collection of bids,

8.3 Distributed EV Charging Coordination Under a PSP Auction Mechanism ∗ ∗ ∗ = (βnt , dnt )= bnt



∂ ∗∗ , wn (x ∗∗ ), x n nt ∂ xnt

211

(8.9)

for all n ∈ N and t ∈ T . Then, under Assumptions (A1, A2), x ∗ (b∗ ) = x ∗∗ . Also,  ∗ βnt

 ∗ = ct (Dt + k∈N dkt∗ ), if xnt >0  ∗  ∗ ≤ ct (Dt + k∈N dkt ), if xnt = 0,

(8.10)

for all n ∈ N and t ∈ T . Proof This lemma is essentially the so-called fundamental theorem of welfare economics [45]. Verification is provided in Appendix. Remarks Lemma 8.3 establishes two important properties: (i) It specifies a bid profile b∗ , given by (8.9), under which the optimal charging allocation x ∗ of Problem 8.2 is efficient. (ii) At bid profile b∗ , EVs with an allocation larger than zero share the same marginal price as generation, which is larger than or equal to the price of EVs with zero allocation. Incentive compatibility holds under the PSP auction mechanism [30, 31]. Therefore, a bid with price satisfying βnt = ∂d∂nt wn (d n ), for all t ∈ T , as is the case in (8.9), is the best choice among all possible bids. It follows from (8.3) that the truth-telling bid of the nth EV is given by:  βnt dnt ;



 dns

   dns . = − f n (dnt ) + 2δn Γn −

s∈T

(8.11)

s∈T

This implies that an EV’s marginal valuation at each time-step is determined  by both its electrical energy request dnt at that time and its total energy request s∈T dns over the entire multi-period charging horizon.

8.3.2 Calculation of EV Payment and Payoff The payment incurred by each EV will be specified with respect to the allocation law defined by Problem 8.2. Each EV’s payment is exactly the externality imposed on the system through its participation in the auction. For the nth EV, this is given by the system-wide utility when the nth EV does not join the auction process, minus the system-wide utility (but excluding the contribution of the nth EV itself) when the nth EV joins the auction. To express this payment, it is convenient to introduce a slight abuse of notation by writing the collection of bids as b ≡ (bn , b−n ), where b−n ≡ (bk , k ∈ N \ {n}). The payment of the nth EV, for a bid profile b, is then given by:

212

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

 ∗



∗

τn (b) = −J (0n , b−n ) − −J (b) −

 ∗ βnt xnt (b)

,

(8.12)

t∈T

where (0n , b−n ) denotes the bid profile without the nth EV’s participation, i.e., with the bid dnt replaced by dnt = 0 for all t ∈ T , and x ∗ (b) is the optimal charging allocation given by Problem 8.2, with respect to b. Thus by (8.7), (8.12) and Lemma 8.2, the payment of player n at time t can be expressed as: ⎛ τnt (bt ) = − ct ⎝ Dt + +







⎞ ∗,−n ⎠ xmt

m =n

 + ct

Dt +





 xkt∗ (bt )

k∈N

∗,−n ∗ βmt xmt − xmt (bt ) ,

m =n ∗,−n where xmt denotes the optimal charging allocation of EVs m = n given by Problem 8.2 with respect to (0n , b−n )t . (Recall from Lemma 8.2 that allocations at time t are unrelated to other times.) The total payment of player n is given by:

τn (b) =



τnt (bt ).

(8.13)

t∈T

The payoff function of the nth EV is given by the difference between the EV’s utility and its payment: u n (b) = wn (x ∗n (b)) − τn (b). (8.14) This payoff function provides the basis for defining a Nash equilibrium for the PSP auction game. Definition 8.1 A collection of bid profiles b0 is a Nash equilibrium for Problem 8.2 if: u n (b0n , b0−n ) ≥ u n (bn , b0−n ), for all bn ∈ Bn and for all n ∈ N . That is, no EV can benefit by unilaterally deviating from its bid profile b0n .

8.3.3 Related Work on EV Charging Games 8.3.3.1

Hierarchical EV Charging Games

Many of the distributed schemes that have been proposed for scheduling EV charging have adopted a hierarchical structure, see for example [14, 15, 20, 21]. Each EV first determines its optimal charging schedule with respect to the expected energy

8.3 Distributed EV Charging Coordination Under a PSP Auction Mechanism

213

price that is broadcasted by the ISO. The price is then updated based on the latest charging schedules attained by the population of EVs. This process repeats until price updates become negligible. Typically, the outcome of such a scheme approaches a NE asymptotically as the size of the EV population approaches infinity [14]. In contrast, the auction game formulated in this chapter achieves a NE for small collections of EVs.

8.3.3.2

Efficiency of NE for Auction Games

The tradeoff between efficiency and risk is considered in [22] in the context of scheduling load over a finite time horizon in oligopoly electricity markets. In noncooperative schemes, the system may suffer a loss, known as the price of anarchy, which relates to the difference between the efficient solution and the NE. The PSP auction proposed in this chapter circumvents that loss by ensuring the NE is efficient. Efficient games have been studied in a variety of related resource allocation problems. The energy consumption scheduling game formulated in [18] does not employ user utility and considers a linear payment rule that differs from VCG. It achieves a globally optimal NE that minimizes the total generation cost. The efficiency of single-unit network resource allocation is studied in [30, 31, 35], though EV charging coordination is essentially a multi-unit resource allocation problem.

8.4 Efficiency of the Charging Coordination PSP Auction Game This section establishes that the efficient solution given by Problem 8.1 is a NE of the PSP auction game. Budget balance under the PSP auction is considered in Sect. 8.4.4. Finally, it is shown in Sect. 8.4.5 that for a single-interval auction game, where T = 1, the efficient NE is unique and so there is no efficiency loss. Suppose b∗ is the bid profile specified in Lemma 8.3, such that the corresponding optimal charging allocation is efficient. It will be shown in this section that b∗ is a NE for the underlying auction game. By Definition 8.1, this implies: u n (b∗n , b∗−n ) ≥ u n (bn , b∗−n ),

(8.15)

for all bn ∈ Bn and for all n ∈ N . Due to the cross elasticity arising from the second term in (8.11), directly verifying that the efficient bid profile is the best for every player is infeasible. Accordingly, the approach developed in [35] for auction games of a single divisible resource is not applicable. In order to overcome this difficulty, an alternative approach will be developed. This involves partitioning the set of bid profiles Bn into a collection of subsets:

214

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

   Bn (A)  bn ∈ Bn ; s.t. dnt = A ,

(8.16)

t∈T

each of which is composed of admissible bid profiles that possess a common total A for the desired energy over the charging horizon T . The set of bids is then given by: Bn =



Bn (A),

A∈[0,Γn ]

  = ∅ whenever A = A.  For all bids in a particular subset noting that Bn ( A) Bn ( A) Bn (A), it follows from (8.11) that the marginal valuation price βnt at any time t includes a variable part − f n (dnt ) that is dependent upon the request dnt at that time, and a fixed part 2δn (Γn − A) that is identical for all bid profiles in Bn (A). Therefore, the cross elasticity over the time horizon is avoided. By Definition 8.1 and the specification of Bn (A), it is sufficient to show that b∗ is a NE, if for every fixed A ∈ [0, Γn ]: bn , b∗−n ), for all bn ∈ Bn (A), u n (b∗n , b∗−n ) ≥ u n (

(8.17)

and for all n ∈ N . This is easier to verify than (8.15), since the difficulty associated with cross elasticity is avoided when considering each specific subset Bn (A) with fixed A ∈ [0, Γn ].  ∗ , and It will be shown in Sect. 8.4.1 that  (8.17) holds when A ≥ t∈T dnt ∗ Sect. 8.4.2 considers the case where A < t∈T dnt . These results build on the following lemma. Lemma 8.4 Suppose βnt (dnt ; A) isthe bidding price given by (8.11) for player n at time t, but with the summation t dnt in the second term replaced by A. Then βnt (dnt ; A) satisfies the properties: 1 2 ; A) > βnt (dnt ; A) > 0, βnt (dnt

βnt (dnt ; A1 ) > βnt (dnt ; A2 ),

1 2 with dnt < dnt ,

for all A,

(8.18a)

with A1 < A2 ,

for all dnt .

(8.18b)

Proof By (8.11), βnt (dnt ; A) = 2δn (Γn − A) − f n (dnt ). It is straightforward to verify (8.18) under Assumption (A2). Lemma 8.4 implies that βnt (dnt , A) decreases with increasing dnt and A, as illustrated in Fig. 8.1.

8.4.1 Verification of (8.17) when A ≥

 t∈T

∗ dnt

The first step in verifying (8.17) is to show that all EVs m ∈ N \ {n} are fully  ∗ . This is established by Lemma 8.5. The main result allocated when A ≥ t∈T dnt then follows as Theorem 8.1.

8.4 Efficiency of the Charging Coordination PSP Auction Game

215

Fig. 8.1 Illustration of βnt (dnt ; A) with respect to dnt and A

For notational simplicity, let x ∗ and x denote the optimal allocations with respect bn , b∗−n ), respectively. to b∗ and (  ∗ Lemma 8.5 If A ≥ t∈T dnt , then ∗ , xmt = dmt

for all m ∈ N \ {n}.

∗ ∗ = 0 for any m ∈ N \ {n}, then xmt = xmt = 0 and the desired result Proof If dmt ∗ > 0 in the is obtained trivially for that m at time t. It will be assumed that dmt subsequent analysis. ∗ ∗ . From Lemma 8.4, since d nt ≥ dnt , (8.18a) gives: Case I, when d nt ≥ dnt

 βnt d nt ;



 ∗ dnt

 ∗ ; ≤ βnt dnt

t∈T

and because A ≥

 t∈T



 ∗ dnt

∗ , = βnt

t∈T

∗ dnt , (8.18b) gives:

 nt ≡ βnt (d nt ; A) ≤ βnt d nt ; β



 ∗ dnt

.

t∈T ∗ nt ≤ βnt when This implies β  , as ∗illustrated in Fig. 8.2, with equality∗ holding only ∗ bn , b−n )t and b∗t coincide, d nt = dnt and A = t∈T dnt . In that special case, the bids ( ∗ ∗ ∗ nt < βnt = dmt as desired. The following analysis considers β . Given so xmt = xmt ∗ ∗ ∗ the earlier assumption that dmt > 0, Lemma 8.3 indicates that βnt ≤ βmt . Hence, ∗ ∗ nt < βnt β ≤ βmt . x, σ ; ( bn , b∗−n )) = 0. Because σnt ≥ 0, (8.8c) If xnt > 0 then by (8.8a), ∂ x∂nt L a (   implies ct (Dt + k∈N xkt ) ≤ βnt . It follows that,

216

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

nt Fig. 8.2 Illustration of β ∗ and β nt when ∗ A ≥ t∈T dnt

 ct

Dt +



 xkt

 nt < ≤β

∗ βmt

=

ct

Dt +

k∈N



 xkt∗

,

(8.19)

k∈N

  xkt < k∈N xkt∗ . Therefore the total energy allocated to all and hence that k∈N the EVs at under the bid profile ( bn , b∗−n ). Furthermore, (8.19) gives time t decreases  ∗ xkt ) − βmt < 0, so (8.8c) implies that σmt > 0. The complementarity ct (Dt + k∈N ∗ for all m ∈ N \ {n}, which is the desired condition (8.8b) then ensures xmt = dmt result. ∗ ∗ xkt ≤ dkt∗ = xmt < dmt for some m ∈ If xnt = 0, then xkt for ∗all k ∈ N . Assume  xkt < k∈N xkt , and so N \ {n}. Then k∈N  ct

Dt +

 k∈N

 xkt


0, which is inconsistent with the assumption ∗ because of complementarity (8.8b). Therefore, xmt = dmt for all m ∈ N \ {n}, as desired.  ∗ ∗ ∗ xnt ≤ d nt < dnt = xnt , then k∈N xkt <  Case ∗(II), when dnt < dnt . Because x , and so (8.20) holds. It again follows from (8.8c) that σ > 0, and hence mt k∈N kt ∗ , as desired. the complementarity condition (8.8b) ensures that xmt = dmt

The main result for this section can now be established. Theorem 8.1 Under Assumptions (A1, A2), (8.17) holds when A ≥

 t∈T

∗ dnt .

Proof Considering the nth EV, the first step is to compute the difference between the payoff given by the optimal strategy u n (b∗ ) and that obtained from an alternative bn , b∗−n ). Using (8.14), these payoffs are given by, strategy u n (

8.4 Efficiency of the Charging Coordination PSP Auction Game

217

u n (b∗ ) = wn (d ∗n ) − τn (b∗ ), u n ( bn , b∗−n ) = wn ( x n ) − τn ( bn , b∗−n ).

(8.21a) (8.21b)

Hence, the difference is: bn , b∗−n ) Δu n  u n (b∗ ) − u n (   = wn (d ∗n ) − τn (b∗ ) − wn ( x n ) − τn ( bn , b∗−n )   nt β = wn (d ∗n ) − wn ( x n ) + J ∗ ( bn , b∗−n ) + xnt ∗

∗

− J (b ) −



∗ ∗ βnt dnt

 ,

t∈T

t∈T

where the final equality follows from (8.12). Straightforward analysis, using the ∗ for all m ∈ N \ {n}, gives: result from Lemma 8.5 that xmt = dmt ⎧ ⎛ ⎞ ⎨  ∗ xn ) + dmt + xnt ⎠ Δu n = wn (d ∗n ) − wn ( ct ⎝ Dt + ⎩ m =n t∈T    − ct Dt + dkt∗ . k∈N

Also, using Lemma 8.5, x) = Js (d ∗ ) − Js ( x n , d ∗−n ) Js (x ∗ ) − Js ( ⎧  ⎛ ⎞⎫  ⎬ ⎨   ∗ = dkt∗ − ct ⎝ Dt + dmt + xnt ⎠ ct Dt + ⎭ ⎩ m =n t∈T k∈N   − wk (d ∗k ) + wm (d ∗m ) + wn ( xn ) m =n

k∈N

⎧  ⎛ ⎞⎫  ⎬   ⎨ ∗ dkt∗ − ct ⎝ Dt + dmt + xnt ⎠ = ct Dt + ⎭ ⎩ t∈T

− = −Δu n .

k∈N

wn (d ∗n )

+ wn ( xn )

x), so Since x ∗ is the efficient allocation, Js (x ∗ ) ≤ Js ( x) − Js (x ∗ ) ≥ 0. Δu n = Js (

m =n

218

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

This implies that player n cannotbenefit by unilaterally changing its bid b∗n to any ∗ . other bid bn ∈ Bn (A) with A ≥ t∈T dnt

8.4.2 Verification of (8.17) when 0 ≤ A
0, = 0, otherwise,

for all t ∈ T .

(8.22)

Theorem 8.2 then shows that the bid of Lemma 8.6 remains optimal when the constraint (8.22) is relaxed. Using∗this result, Theorem 8.3 finally establishes that (8.17) . holds when 0 ≤ A < t∈T dnt

∗ ∗  ∗ ∗ Lemma  8.6∗ Consider a bid bn ≡ bn (A) ≡ (βnt , dnt ), t ∈ T , with A ∈ [0, t∈T dnt ), such that ∗ bn =

argmax

bn ∈B n (A) Constraint (8.22)

u n ( bn , b∗−n ).

(8.23)

xkt∗ , k ∈ N , t ∈ T ) denote the optimal allocations with respect to Let x ∗ ≡ ( ∗ ∗ ( bn , b−n ). Then, ∗

x ∗n = dn,

x ∗m = d ∗m for all m ∈ N \ {n}.

(8.24)

Furthermore, define the function, ⎛ gnt (d)  ct ⎝ Dt +



⎞ ∗ dmt + d ⎠ + f n (d).

(8.25)

m =n ∗ Then, under Assumptions (A1, A2), bn satisfies the property:

  ∗ gnt (dnt )

∗ = μ, when d nt > 0, ∗ = 0, ≥ μ, when d nt

for all t ∈ T ,

(8.26)

where μ is a constant. Proof The proof of Lemma 8.6 is given in Appendix. ∗ It will now be shown that the bid bn established in this lemma remains optimal when the constraint (8.22) is relaxed.

8.4 Efficiency of the Charging Coordination PSP Auction Game

219

Fig. 8.3 An illustration of partitioned subsets Ri , i = 0, . . . , 3

∗ Theorem 8.2 Suppose that bn is the optimal bid from Lemma 8.6. Then, under  ∗ ∗ ), bn satisfies: Assumptions (A1, A2) and with A ∈ [0, t∈T dnt ∗ bn , b∗−n ). bn = argmax u n ( bn ∈B n (A)

  ∗ nt , d nt ), t ∈ T ∈ Proof Assume that A ∈ [0, t∈T dnt ). Consider a bid bn ≡ (β ∗ Bn (A) such that d nt ≥ dnt for some t ∈ T . Then the desired result follows if it can be proven that bn cannot be the optimal bid profile in the subset Bn (A). Firstly, the following points hold:   ∗ ∗ > βnt (d nt ; t∈T dnt ) because A < t∈T dnt , by (8.18b). (i) βnt (d nt ; A)  ∗ + dnt ) increases with dnt under Assumption (A1). (ii) ct (Dt + m =n dmt  ∗ ∗ ∗ ∗ , dnt ) is the point at which βnt (dnt ; t∈T dnt ) and ct (Dt + m =n dmt + (iii) (βnt dnt ) coincide, by Lemma 8.3. Using (i)–(iii) together with (8.18a) gives, ∗ 1 < d nt , 0 < dnt

∗ 1 nt βnt 0.

Proof The proof is given in Appendix. Lemma 8.7 establishes that the budget balance for b∗ will always be in surplus (positive). This surplus ψ(b∗ ) can be used to reimburse the services provided by the system operator.

8.4.5 Efficiency Loss of Single-Interval Auction Games Given that the efficient bid profile b∗ is a NE, efficiency loss can only occur if there exists multiple NE. The following theorem shows that for single-interval auction games, i.e., an auction game with T = 1, the efficient NE is unique. Hence, for such games, there is no possibility of efficiency loss. In this case, the efficient bid profile degenerates to b∗ = ((βn∗ , dn∗ ), n ∈ N ). Theorem 8.4 b∗ = ((βn∗ , dn∗ ), n ∈ N ) is the unique NE for PSP auction games where T = 1. Proof Corollary 8.1 establishes that b∗ is a NE. Verifying Theorem 8.4 is equivalent to showing that every inefficient bid profile b cannot be a NE. This is undertaken in Appendix. Ongoing research is considering efficiency loss for EV charging auction games over a multiple time-step horizon.

8.5 PSP Auction Process for EV Charging Section 8.4 established the existence of efficient Nash equilibria, under appropriate conditions, for the underlying EV charging coordination game. This section develops a bid profile update process which motivates an algorithm for determining efficient NE. A process for determining an EV’s best bid, given the collection of bids for the other EVs, is presented in Sect. 8.5.1. It will be shown that this process can be formulated as a dynamic programming problem. Section 8.5.2 then provides an algorithmic description of the update mechanism that governs the underlying auction game. A numerical example is provided in Sect. 8.5.3 to demonstrate this update mechanism and show that it achieves an efficient NE.

8.5 PSP Auction Process for EV Charging

223

8.5.1 An EV’s Best Bid with Respect to Other EVs Recall that b∗n (b−n ) denotes the best bid of the nth EV with respect to the bid profiles b−n of all the other EVs, b∗n (b−n ) = argmax u n (bn , b−n ), bn ∈B n

where u n (bn , b−n ) is the individual payoff of the nth EV, as established in (8.14). However, due to the cross-temporal coupling arising from the summation term  d in truth-telling bids, as identified in (8.11), it is impractical to directly nt t∈T determine the best response b∗n (b−n ) that is incentive compatible. This can be addressed by finding the best response when bids are constrained to possess a common  total desired demand A = t∈T dnt , and then optimizing over A ∈ [0, Γn ]. The resulting optimization is given by, u n (b∗n , b−n ) = max

max u n (bn , b−n ).

A∈[0,Γn ] bn ∈B n (A)

The remainder of this section describes a dynamic programming approach to solve the inner optimal bidding problem that arises for each fixed total demand request A ∈ [0, Γn ]. The dynamics associated with the physical charging process (8.1) can be rewritten, sn,t+1 (bn , b−n ) = snt (bn , b−n ) +

1 xnt (bn , b−n ), Θn

(8.32)

with t ∈ T , and where xnt (bn , b−n ) denotes the allocated charging rate of the nth EV at time t, with respect to the bid profile (bn , b−n ). Lemma 8.8 Consider a bid bn ∈ Bn (A), for any fixed A ∈ [0, Γn ]. Then the payoff function of the nth EV has the summation form: u n (bn , b−n ) =



2 an (bt ) − δn Θn2 s n − snT ,

t∈T

where an (bt ) ≡ − f n (xnt (bt )) − τnt (bt ) and τnt (bt ) is defined in (8.13). Proof From (8.14), u n (b) = wn (x ∗n (bn )) − τn (b)   = − f n (xnt (bt )) − τnt (bt ) t∈T

− δn

 t∈T

xnt (bt ) − Γn

2

,

(8.33)

224

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

where equality holds by (8.3), Lemma 8.2, and (8.13). Then (8.33) follows directly from (8.32). Let Tt ≡ {t, ..., T − 1}, and define the value function vn (t, snt )≡vn (t, snt ; A, b−n ), for all t ∈ T as, vn (t, snt )  max

bn (T t )∈ B n (T t ,snt ;A)

T −1 

an (bk ) −

δn Θn2



sn −

2 snT

 ,

(8.34)

k=t

where an (bs ) is given in Lemma 8.8, and Bn (Tt , snt ; A) 

⎧

 (βnk , dnk ), k ∈ Tt s.t. βnk = − f n (dnk ) + 2δn (Γn − A) ⎪ ⎪ ⎪ ⎪ ! T −1 ⎨ and k=t dnk ≤ min A, Θn (s n − snt ) , (8.35) ⎪ ⎪ when t > 0 ⎪ ⎪ ⎩ Bn (A), when t = 0,

with Bn (A) defined in (8.16). The terminal value function is defined as,

2 vn (T, snT )  −δn Θn2 s n − snT .

(8.36)

Note that the set of bids of the nth EV over the interval Tt specified in (8.35) is defined in such a way that the total bidding demand over the whole interval T is guaranteed to equal A. The value function definition implies: vn (0, sn0 ; A, b−n ) = and, therefore,

max u n (bn , b−n ),

(8.37)

u n (b∗n , b−n ) = max vn (0, sn0 ; A, b−n ).

(8.38)

bn ∈B n (A)

A∈[0,Γn ]



∗ ∗ Let b∗n (Tt ) ≡ (βnk , dnk ), k ∈ Tt ≡ b∗n (Tt , snt ; A, b−n ) denote the best bid of the nth EV solving the optimization problem (8.34) over the interval Tt with respect to A and b−n , and let Πn∗ (t, snt ; A, b−n ) denote the total bidding demand over the interval Tt of the nth EV subject to the best bid b∗n (Tt , snt ; A, b−n ), Πn∗ (t, snt ; A, b−n ) 

T −1  k=t

∗ dnk (Tt , snt ; A, b−n ).

8.5 PSP Auction Process for EV Charging

225

Define Bn (t, snt ) ≡ Bn (t, snt ; A, b−n ), for any t ∈ T , as the set of bids at time t, such that Bn (t, snt ; A, b−n )  ⎧  ⎪ ⎪ (βnt , dnt ) s.t. βnt = − f n (dnt ) + 2δn (Γn − A) , ⎪ ⎪ ⎪ ⎪ and dnt + Πn∗ (t + 1, snt+1 ; A, b−n ) ⎪ ⎪ ⎪ ! ⎪ ⎨ ≤ min A, Θn (s n − snt ) ,  ⎪ (βnt , dnt ) s.t. βnt = − f n (dnt ) + 2δn (Γn − A) , ⎪ ⎪ ⎪  ⎪ ⎪ ∗ t+1 ⎪ and d + Π (t + 1, s ; A, b ) = A , ⎪ nt −n n n ⎪ ⎪ ⎩

when t > 0

when t = 0, (8.39)

where snt+1 is given by (8.32). As with the set of bids of the nth EV over the interval Tt specified in (8.35), the set of bids of at each time t ∈ T specified in (8.39) is defined such that the total bidding demand over the whole interval T is guaranteed to equal A. Theorem 8.5 The value function vn (t, snt ; A, b−n ) of the nth EV, with respect to a fixed A ∈ [0, Γn ] and a collection of bids b−n of the other EVs, can be implemented by solving the Bellman equation, vn (t, snt ; A, b−n ) =

maxt

bnt ∈B n (t,sn ;A,b−n )

  an (bt ) + vn (t + 1, snt+1 ; A, b−n ) ,

for t ∈ T , where Bn (t, snt ; A, b−n ) is defined in (8.39), an (bt ) is specified in Lemma 8.8, and vn (T, snT ) is given by (8.36). Proof It is straightforward to verify Theorem 8.5 by applying the optimality principle for the underlying optimization problem defined in (8.37) for the nth EV with respect to A and b−n . Finally, the nth EV’s overall optimal bid, with respect to the bid profiles b−n of the other EVs, is given by (8.38).

8.5.2 Update Mechanism for EVs The bid profile update process of Sect. 8.5.1 motivates a heuristic algorithm whereby each EV successively updates its optimal bidding strategy with respect to the latest available bidding strategies of all the other EVs. If this iterative process converges, the resulting bid profile will be a NE, as established by Definition 8.1. While the convergence of Algorithm 8.1 cannot currently be guaranteed, such guarantees have been established for related auction games. Various static iterative

226

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

Algorithm 8.1 NE implementation algorithm of EV charging Require: An initial bid profile b(0) ; An initial termination criterion ε > ε0 for some ε0 > 0; k ← 0; 1: while ε > ε0 do 2: for n = 1 : N do (k+1) (k+1) (k) 3: Determine the best response for the nth EV, b(k+1) , with respect to b1 , · · · , bn−1 , bn+1 , n (k)

· · · , b N , by maximizing the payoff function, (k+1)

b(k+1) = argmax u n (bn ; b1 n bn ∈B n

(k+1)

(k)

(k)

, · · · , bn−1 , bn+1 , · · · , b N ),

which can be achieved by applying the method developed in Section 8.5.1. 4: end for 5: ε ← ||b(k+1) − b(k) ||1 ; 6: k := k + 1; 7: end while

processes have been designed to determine the efficient NE for PSP auction games with a single type of resource, see for example [30, 31, 48]. Also, the so-called quantized-PSP auction mechanism was developed in [49–51] for electricity sharing games in a single time interval, with that process converging to a nearly efficient solution. Accordingly, ongoing research is studying the convergence properties of Algorithm 8.1 and the performance of the resulting NE for auction games where multiple resources are coupled.

8.5.3 Numerical Illustration To illustrate the auction-based coordination process, a numerical example will consider EV charging over a common time horizon T = 24 h, from 12:00 on one day to 12:00 the next day, with a time-step of ΔT = 1 h. The background demand for the example is shown in Fig. 8.4. For the purpose of demonstration, a small population of 5 vehicles will be considered. Each EV has a common battery capacity of 30 kWh and a common maximum SoC value s n = 0.9. Heterogeneity is introduced by letting the initial SoC values sn0 for the five EVs, prior to the charging interval, take the values s0 = [0.1 0.15 0.23 0.14 0.08] . 1.7

 The generation cost is given by ct (x t , Dt ) = 0.005 and the n∈N x nt + Dt 2 . Both of these functions are strictly battery degradation cost by f n (xnt ) = 0.002xnt convex. The weighting factor for the quadratic charging deviation cost of each EV is set to δn = 10 for all n ∈ N . The efficient∗∗EV charging trajectory, given by Problem 8.1, is shown in aggrega, t ∈ T ) in Fig. 8.4. In contrast, the distributed approach to chargtion ( n∈N xnt ing coordination, described by Algorithm 8.1, gives the update evolution shown in

8.5 PSP Auction Process for EV Charging

227

Fig. 8.4 Background demand and the aggregate optimal charging strategies

Fig. 8.5 Convergent updates of Algorithm 8.1

Fig. 8.5. At each iteration, all EVs determine their optimal bid profile by solving the dynamic programming formalized by Theorem 8.5. Figure 8.5 shows problem ∗ (b(k) ), t ∈ T ) obtained by the auctioneer solvthe aggregate allocation ( n∈N xnt ing Problem 8.2 with respect to the bid profile b(k) at the kth iteration. Comparing Figs. 8.4 and 8.5, it is clear that the auction game converges to the efficient charging solution.

228

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

At the  efficient strategy b∗ , the aggregated payments of the EV

 population ∗ τ (b ) = 23.44, and the increased generation cost is amount to n n∈N t∈T ct Dt +  !  ∗ ∗ k∈N x kt (b ) − ct (Dt ) = 22.48. The resulting surplus of 0.96, which is 4.1% of the total payments, can be used to reimburse the services provided by the system operator.

8.6 Conclusions and Ongoing Research As sales of electric vehicles (EVs) increase, the energy demanded by EV charging will begin to impact the power system operation. Accommodating large numbers of EVs on the grid will require coordination of their charging requirements. A distributed approach to coordinating EV charging, based on the progressive second price (PSP) auction mechanism, has been proposed. This auction mechanism ensures incentive compatibility. It is shown that the efficient (socially optimal) charging schedule is a Nash equilibrium (NE) of this auction. Individual EVs are capable of shifting their charging requirements in time across the charging horizon. Such cross elasticity complicates verification that the efficient solution is a NE. This has been addressed by partitioning players’ bids into subsets, each of which is composed of bids that possess the same total desired energy over the horizon. Consequently, cross elasticity is eliminated for bids within each subset. With this construction, the NE property of the efficient solution has been established by verifying for each subset that no player can benefit by unilaterally deviating from their efficient bid profile. An iterative update mechanism, based on dynamic programming, has been developed for the underlying auction game. This update mechanism was illustrated using a numerical example which showed convergence of the auction to an efficient NE. The developments in this chapter motivate various ongoing research directions. This current work shows that for a single-step time horizon, the efficient solution is the unique NE for the underlying game. It remains to verify whether efficiency loss may occur for multi-step time horizons due to the non-uniqueness of the NE. Work is also required to establish convergence properties of the auction update algorithm. The model underpinning the charging coordination scheme assumes that the EV population and the background demand are known with certainty prior to the charging period. The resulting auction game is static and the auction process can be undertaken off-line ahead of actual charging. In reality, the required information may be difficult to predict accurately, for example, EVs may come and go without any advanced warning. Consequently, the efficient NE determined ahead of the charging interval may be suboptimal relative to a solution that considers disturbances. A practical approach to addressing this deficiency is to adopt a receding horizon strategy [52]. This is the focus of ongoing research.

Appendices

229

Appendices Proof of Lemma 8.3 ∗ The bid price βnt =

∂ ∂ xnt

wn (x ∗∗ n ) can be written,

   ∗ ∗∗ ∗∗ − f n (xnt βnt = 2δn Γn − xnt ). t∈T

 ∗ ∗∗ ∗∗ Because βnt ≥  0 and f n (xnt )>  0 by Assumption A2, then ∗∗ t∈T xnt − Γn < 0. Together with t∈T x nt − Γn λn = 0 in (8.5b), this gives λn = 0. ∂ ∗ With βnt = ∂ xnt wn (x ∗∗ n ) in (8.5c), the complementarity condition (8.5a) gives, ∗ βnt

  ∗∗ = ct (Dt + n∈N xnt ),   ∗∗ ≤ ct (Dt + n∈N xnt ),

∗∗ when xnt >0 ∗∗ when xnt = 0,

(8.40)

∗ ∗∗ for all n ∈ N , t ∈ T . Substituting dnt = xnt into (8.8b) gives, ∗ ∗∗ ∗ ∗∗ − xnt ≤ 0, σnt∗ ≥ 0, (xnt − xnt )σnt∗ = 0. xnt

(8.41)

∗∗ ∗ ∗∗ ∗∗ ∗ = 0 then xnt = 0 = xnt . If xnt > 0 then (8.40) indicates that βnt = ct (Dt + If xnt  ∗∗ ∗ ∗∗ x ). The following argument shows that x < x cannot occur. nt nt n∈N nt ∗ ∗∗ < xnt , then, By the convexity property of ct (·) and if xnt

 ct

Dt +

 n∈N

 ∗ xnt

 − ct

Dt +



 ∗∗ xnt

< 0.

∗ ∗∗ < xnt then (8.41) indicates that σnt∗ = 0. Together with Also, if xnt this gives,       ∗ ∗∗ ct Dt + xnt xnt − ct Dt + ≥ 0, n∈N

(8.42)

n∈N ∂ La ∂ xnt

≥ 0 in (8.8),

n∈N

∗ ∗∗ ∗ ∗∗ < xnt cannot occur, implying xnt = xnt . In which contradicts (8.42). Hence, xnt ∗ ∗ ∗ summary, under the bid profile b , the associated optimal allocation x (b ) for Problem 8.2 is identical to the socially optimal solution to Problem 8.1. Also, (8.10) follows from (8.40) with x ∗ = x ∗∗ .

230

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

Fig. 8.6 Relationship ∗ when nt and βnt between β  ∗ 0 ≤ A < t∈T dnt

Proof of Lemma 8.6 ∗ ∗ If dnt = 0 then d nt = 0, according to (8.22). In that case, xnt = xnt = 0 trivially. The ∗ remainder of the proof assumes dnt > 0. bn , b∗−n ). From (8.18), The first step is to establish u n (

 nt  βnt (d nt ; A) > βnt d nt ; β



 ∗ dnt

,

t∈T

because A
βnt

∗ ; dnt

t∈T



 ∗ dnt

∗ . = βnt

t∈T

∗ ∗ ∗ nt > βnt , as illustrated in Fig. 8.6. Also, from Lemma 8.3, βnt ≥ βmt for Therefore, β ∗ nt > βmt . all m ∈ N \ {n}, so β ∗ ∗ ∗ ∗ Because xnt ≤ d nt < dnt = xnt and xmt ≤ dmt = xmt for all m ∈ N \ {n}, then,

 ct

Dt +

 k∈N

 xkt


0. From (8.8a), (8.8c), ct (Dt + k∈N ∗ σmt > 0, and hence from (8.8b) that xmt = dmt . σmt ≥ 0. Therefore, (8.43) implies ∗ nt + nt > βnt , ct (Dt + k∈N xkt ) − β σnt ≥ 0 implies σnt > 0, Likewise, because β and therefore, that xnt = d nt . Consequently, all players are fully allocated, and (8.24) is established. The payment of the nth player is given by (8.13), with the above argument indi∗,−n ∗ = xmt for all m ∈ N \ {n}. Therefore, cating that xmt

Appendices

231

τn ( bn , b∗−n ) =

⎧ ⎛ ⎞ ⎛ ⎞⎫ ⎬   ⎨ ∗ ∗ ⎠ . dmt + d nt ⎠ − ct ⎝ Dt + dmt ct ⎝ Dt + ⎭ ⎩ m =n

t∈T

m =n

The payoff of player n then follows from (8.14) as, u n ( bn , b∗−n ) = wn ( d n ) − τn ( bn , b∗−n ). ∗ The auctioneer needs to find an optimal bid bn that maximizes the payoff u n ( bn , b∗−n ). This is achieved by the following restatement of the optimization problem (8.23):

Problem 8.3

max u n ( bn , b∗−n ), bn

such that (8.22) is satisfied together with, 

d nt = A.

(8.44)

t∈T

The Lagrangian for Problem 8.3 is given by, d n , λ, μ) = u n ( bn , b∗−n ) + L s (



 ∗ λt d nt − dnt

t∈T

  +μ d nt − A , t∈T

where λt and μ are the Lagrangian multipliers associated with the constraints (8.22) and (8.44), respectively. The KKT conditions for Problem 8.3 are given by, ∂ s L ( d n , λ, μ) ≤ 0, ∂ dnt ∗ ≤ 0, d nt − dnt  d nt − A = 0,

d nt ≥ 0, λt ≥ 0,

∂ s L ( d n , λ, μ)d nt = 0, ∂ dnt ∗ (d nt − dnt )λt = 0,

t∈T

for all t ∈ T , and, ⎛ ⎞  ∂ s ∗ L ( d n , λ, μ) = − f n (d nt ) − ct ⎝ Dt + dmt + d nt ⎠ ∂ d nt m =n + λt + μ.

(8.45)

232

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

∗ Since d nt − dnt < 0, then λ∗t = 0, and so,

⎛  ∗ ∗ (dnt ) = f n (d nt ) + ct ⎝ Dt + gnt



⎞ ∗ ∗⎠ dmt + d nt ≥ μ,

m =n ∗ where gnt (dnt ) is defined in (8.25). If d nt > 0 then (8.45) equals zero, and so  ∗ (dnt ) = μ. Therefore, (8.26) holds. gnt Under Assumptions (A1, A2), the necessary KKT conditions are also sufficient ∗ for optimality. Consequently, (8.26) is sufficient for the optimal bid bn of player n.

Verification of Inequality (8.28) in Theorem 8.2 For notational simplicity,  x and x denote the optimal allocations with respect to bn , b∗−n ), respectively. ( bn , b∗−n ) and ( Consider the allocations of all EVs at time t1 with respect to the bid profile ∗ ∗ 2 < dnt < d nt , (8.18a) gives, ( bn , b∗−n )t1 . Because d nt1 < d nt 1

1

1

2 ∗ nt1  βnt1 (d nt1 ; A) > βnt1 (d nt β ; A) = βnt . 1 1 ∗ ∗ ∗ nt1 > ≥ βmt for all m ∈ N \ {n} when dnt > 0. Therefore, β Also, by Lemma 8.3, βnt ∗ βmt1 . Using an argument similar to that following (8.43), it is straightforward to show,

xnt1 = d nt1 ,

∗ xmt1 = dmt for all m ∈ N \ {n}. 1

(8.46)

Hence, at time t1 , all EVs are fully allocated with respect to ( bn , b∗−n )t1 . Similarly, ∗ with respect to ( bn , b−n )t1 ,  xnt1 = dnt1 ,

∗  xmt1 = dmt for all m ∈ N \ {n}. 1

(8.47)

By (8.46) and (8.47), the difference in the payments of the nth EV at time t1 with bn , b∗−n )t1 is given by, respect to ( bn , b∗−n )t1 and (     bn , b∗−n )t1 − τnt1 ( bn , b∗−n )t1 Δτnt1  τnt1 ( ⎛ ⎞  ∗ = ct ⎝ Dt1 + dmt + dnt1 ⎠ 1 ⎛

m =n

− ct ⎝ Dt1 +

 m =n

⎞ ∗ dmt + d nt1 ⎠ . 1

(8.48)

Appendices

233

For the nth EV at time t2 , the difference in payments with respect to ( bn , b∗−n )t2 ∗ and ( bn , b−n )t2 is given by,     bn , b∗−n )t2 − τnt2 ( bn , b∗−n )t2 Δτnt2  τnt2 ( ⎛ ⎞  = ct ⎝ Dt2 +  xmt2 +  xnt2 ⎠ ⎛

m =n

− ct ⎝ Dt2 + +





⎞ xmt2 + xnt2 ⎠

m =n ∗ βmt ( xmt2 −  xmt2 ). 2

(8.49)

m =n

The last term of (8.49) can be simplified by recalling from Lemma 8.3 that all EVs, ∗ k ∈ N , with dkt∗ 2 > 0 share the same value for β kt2 . Denoting that common value by ∗ ∗ xmt2 −  xmt2 ). βt2 allows the last term to be expressed as βt2 m =n ( It follows from (8.27d) that for the nth EV, the difference in payments at times bn , b∗−n )t and ( bn , b∗−n )t is, t = t1 , t2 , with respect to (



 Δτnt  τnt ( bn , b∗−n )t − τnt ( bn , b∗−n )t = 0, ∀t = t1 , t2 .

(8.50)

Thus, by (8.48)–(8.50), the difference in the payments of the nth EV with respect to bn , b∗−n ) satisfies, ( bn , b∗−n ) and ( Δτn  τn ( bn , b∗−n ) − τn ( bn , b∗−n ) = Δτnt1 + Δτnt2 .

(8.51)

The difference in utility of the nth EV, with respect to ( bn , b∗−n ) and ( bn , b∗−n ), is given by, x n ) − wn ( xn ) Δwn  wn (  2  2  xnt − Γn + δn xnt − Γn = −δn t∈T

t∈T

+ f n (d nt1 ) − f n (dnt1 ) + f n ( xnt2 ) − f n ( xnt2 ).

(8.52)

By (8.51) and (8.52), the difference in the payoff of the nth EV, subject to ( bn , b∗−n ) ∗ and ( bn , b−n ), becomes, Δu n  u n ( bn , b∗−n ) − u n ( bn , b∗−n ) = Δwn − Δτn .

(8.53)

∗ will be addressed, then the To establish (8.28), firstly the case with d nt2 = dnt 2  three cases dnt2 , dnt2 ∈ Ri , i = 1, 2, 3 will be considered separately.

234

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

∗ ∗ Case I,  dnt < dnt2 <  dnt2 = dnt 2 2 ∗ Because d nt2 = dnt , 2

 nt2 = β

∗ βnt2 (dnt , 2

A) > βnt2

∗ , dnt 2



 ∗ dnt

∗ . = βnt 2

t∈T ∗ ∗ Likewise, with d nt < dnt2 < d nt2 = dnt , 2 2 2 ∗ nt2 = βnt2 (dnt2 , A) > βnt2 (d nt β , A) = βnt . 2 2

A similar argument to that used to establish (8.46), (8.47) for t1 shows that all EVs bn , b∗−n )t2 and ( bn , b∗−n )t2 : are fully allocated at t2 with respect to both ( ∗ , xnt2 = d nt2 = dnt 2  xnt2 = dnt2 ,

∗ xmt2 = dmt for all m ∈ N \ {n}, 2 ∗  xmt2 = dmt for all m ∈ N \ {n}. 2

Substituting these allocations into (8.53) gives, Δu n = f n (d nt1 ) − f n (dnt1 ) + f n (d nt2 ) − f n (dnt2 ) ⎛ ⎛ ⎞  ∗ dmt + dnt1 ⎠ − ⎝ct ⎝ Dt1 + 1 ⎛

m =n

− ct ⎝ Dt1 + ⎛ + ct ⎝ Dt2 + ⎛ −ct ⎝ Dt2 +



⎞ ∗ dmt + d nt1 ⎠ 1

m =n



⎞ ∗ dmt + dnt2 ⎠ 2

m =n



⎞⎞ ∗ dmt + d nt2 ⎠⎠ 2

m =n

= gnt1 (d nt1 ) − gnt1 (dnt1 ) + gnt2 (d nt2 ) − gnt2 (dnt2 )  



 ∗  ∗ (d nt ) d nt1 − dnt1 + gnt (d nt ) d nt2 − dnt2 > gnt 1

1

2

2

= μ(d nt1 − dnt1 + d nt2 − dnt2 ) = 0, where the inequality holds due to the convexity of gnt (·) and the subsequent equality follows from (8.26). Therefore, (8.28) is satisfied in this case. Case II, d nt2 , dnt2 ∈ R1

Appendices

235

The initial step in showing (8.28) is to determine the allocations of all EVs at time bn , b∗−n )t2 . Firstly, consider d nt2 ∈ Int(R1 ). Then, t2 with respect to the bid profile ( ⎛ 1 nt2 > β nt β = ct ⎝ Dt2 + 2



⎞ ∗ 1 ⎠ dmt + d nt 2 2

m =n

⎛ > ct ⎝ Dt2 +



⎞ ∗ dmt + d nt2 ⎠ , 2

m =n

so it follows from the KKT conditions (8.8) that xnt2 = d nt2 . 1 Now consider the case with dnt2 = dnt2 , the upper boundary of R1 . In this case,  ∗ nt2 = ct (Dt2 + m =n dmt β + d nt2 ). Assume xnt2 < d nt2 . Then due to the convexity 2 of ct (·), ⎛ ct ⎝ Dt2 +



⎞

⎛

∗ dmt + xnt2 ⎠ < ct ⎝ Dt2 + 2

m =n



⎞ ∗ nt2 . dmt + d nt2 ⎠ = β 2

m =n

But (8.8) then implies σnt2 > 0 and therefore that xnt2 = d nt2 . Hence a contradiction, . so xnt2 = dnt 2  xmt2 = 0. If d nt2 ≥ k∈N dkt∗ 2 then it can be shown by contradiction that m =n xmt2 > 0 gives, Assuming m =n ⎛ ct

⎝ Dt2 +



⎞ xkt2 + xnt2 ⎠ >

 ct

k =n

Dt2 +



 dkt∗ 2

∗ , ≥ βmt 2

k∈N

for all m ∈ N \ {n}. But (8.8) then implies xmt2 = 0 for all m ∈ N \ {n}, hence a  nt2 < k∈N dkt∗ then it can be shown, once again contradiction. Alternatively, if d 2     by contradiction, that k∈N xkt2 = k∈N dkt∗ 2 . Consider k∈N xkt2 > k∈N dkt∗ 2 .  ∗ xkt2 ) > βmt for m  ∈ N \ {n}, with (8.8) implying xmt2 = 0, Then ct (Dt2 + k∈N 2  xkt2 < k∈N dkt∗ 2 , then ct (Dt2 + k∈N xkt2 ) < hence a contradiction. If k∈N ∗ ∗ , with (8.8) implying x = d . This leads to another contradiction, βmt mt 2 mt2 2    ∗ as k∈N xkt2 = m =n dmt + d nt2 > k∈N dkt∗ 2 . Summarizing, 2 xnt2 = d nt2 ,

 m =n

if d nt2
0, 



m =n

xmt2 + d nt2 =

m =n



dkt∗ 2 ,

k∈N

dkt∗ 2 ,

k∈N

xnt2 = d nt2 ,



xmt2 = 0,

(8.54a)  m =n

xmt2 + d nt2 ≥

 k∈N

dkt∗ 2 ,

236

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games



if d nt2 ≥

dkt∗ 2 .

(8.54b)

k∈N

Similarly, the above analysis also holds for the bid profile ( bn , b∗−n )t2 so, 

 xnt2 = dnt2 ,

m =n

if dnt2
0, 



m =n

m =n

if dnt2 ≥

 xmt2 = 0, 

 xmt2 + dnt2 =



dkt∗ 2 ,

k∈N

dkt∗ 2 ,

k∈N

 xnt2 = dnt2 ,



(8.55a) 

 xmt2 + dnt2 ≥

m =n



dkt∗ 2 ,

k∈N

dkt∗ 2 .

(8.55b)

k∈N

Substituting into (8.51) gives, ⎛ Δτn = ct ⎝ Dt1 + ⎛



⎞

∗ dmt + dnt1 ⎠ − ct ⎝ Dt1 + 1

m =n

+ ct ⎝ Dt2 +



⎞ ∗  xmt2 + dnt2 ⎠ + βt 2

m =n

⎛ − ct ⎝ Dt2 +

⎛



⎞



⎞ ∗ dmt + d nt1 ⎠ 1

m =n



( xmt2 −  xmt2 )

m =n

xmt2 + d nt2 ⎠ .

m =n

Because  xnt1 + xnt2 =  xnt1 +  xnt2 and xnt =  xnt for all t = t1 , t2 , it follows that  x =  x , and so (8.52) becomes, nt nt t t Δwn = f n (d nt1 ) + f n (d nt2 ) − f n (dnt1 ) − f n (dnt2 ). Three must be considered, depending on the relative values of dnt2 , d nt2  subcases ∗ and k∈N dkt2 . Case II.1, dnt2 <  dnt2
gnt (d nt )(d nt1 − dnt1 )

1  1∗  ∗ + f n (dnt2 ) + βt (d nt2 − dnt2 ) 2  ∗  ∗ = gnt1 (d nt1 )(d nt1 − dnt1 ) + gnt (dnt )(d nt2 2 2

(8.56a) (8.56b)

− dnt2 ),

(8.56c)

where (8.56a) holds by the specification of gnt (·) given in (8.25) and substitution from (8.54a) and (8.55a); (8.56b) holds by the convexity of gnt (·) together with (8.27a), and the convexity of f n (·) together with (8.29); and (8.56c) holds by (8.10) in Lemma 8.3 and (8.25). ∗ ∗  ∗  ∗ < dnt , so gnt (dnt ) > gnt (d nt ) due to the convexity of gnt2 (·). From (8.22), d nt 2 2 2 2 2 2 ∗  ∗  ∗ By construction, dnt1 > 0, so (8.26) gives gnt2 (dnt2 ) ≥ gnt (d nt ) = μ. Therefore, be1 1 cause (8.29) ensures d nt2 > dnt2 , (8.56c) gives, Δu n > μ(d nt1 − dnt1 + d nt2 − dnt2 ) = 0,

(8.57)

where the final equality holds by (8.27c). Case II.2, dnt2
gnt (d nt )(d nt1 − dnt1 ) + f n (dnt )(d nt2 − dnt2 ) 1 1 2 ⎞  ⎛   + ct Dt2 + dkt∗ 2 ⎝  xmt2 − dnt2 ⎠ dnt2 −

+

∗ βt 2



k∈N

m =n

 xmt2

(8.58b)

m =n  ∗ ∗ = gnt (d nt )(d nt1 − dnt1 ) + f n (dnt )(d nt2 − dnt2 ) 1 1 2 ∗ + βt (dnt2 − dnt2 ), 2

(8.58c)

where (8.58a) holds by (8.53) and (8.25); (8.58b) holds by the convexity of gnt (·) together with (8.27a), the convexity of f n (·) together with (8.29), and the convexity

238

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

of ct (·) using (8.55a); and (8.58c) holds by (8.10). Proceeding as in (8.56c), (8.57) yields Δu n > 0.  Case II.3, k∈N dkt∗ 2 ≤ dnt2 < d nt2 In this case, Δu n uses (8.54b) and (8.55b) to give, Δu n = gnt1 (d nt1 ) − gnt1 (dnt1 ) + f n (d nt2 ) − f n (dnt2 ) − ct (Dt2 + dnt2 ) + ct (Dt2 + d nt2 )  ∗ ∗ (d nt )(d nt1 − dnt1 ) + f n (dnt )(d nt2 > gnt 1 1 2 + ct (Dt2 + dnt2 )(d nt2 − dnt2 )

≥

 ∗ (d nt )(d nt1 gnt 1 1

⎛

− dnt1 ) ⎛

∗ ) + ct ⎝ Dt2 + + ⎝ f n (dnt 2



(8.59a)

− dnt2 ) (8.59b) ⎞⎞

∗ ∗ ⎠⎠ dmt + dnt 2 2

m =n

× (d nt2 − dnt2 )  ∗  ∗ (d nt )(d nt1 − dnt1 ) + gnt (dnt )(d nt2 − dnt2 ), = gnt 1 1 2 2

(8.59c) (8.59d)

where (8.59a) holds by (8.53) and (8.25); (8.59b) holds by the convexity of gnt (·) together with (8.27a), and the convexity of f n (·)and ct (·) together with (8.29); (8.59c) holds by the convexity of ct (·) with dnt2 ≥ k∈N dkt∗ 2 ; and (8.59d) holds by (8.25). Proceeding as in (8.57) yields Δu n > 0. Hence, Δu n > 0 whenever d nt2 , dnt2 ∈ R1 . Case III, d nt2 , dnt2 ∈ Int(R2 ) The situation where d nt2 ∈ R2 will be considered as two separate cases. Case III, 2 , the presented here, discusses d nt2 ∈ Int(R2 ), while Case IV addresses d nt2 = d nt 2 upper boundary of R2 . Consider the allocations of all EVs at time t2 with respect to the bid profile  ( bn , b∗−n )t2 . If d nt2 < k∈N dkt∗ 2 , then the argument presented in Case II can a  ∗ nt2 > βnt gain be used to show that k∈N xkt2 = k∈N dkt∗ 2 . Also, because β = 2  ∗  xnt2 = d nt2 . Similar outcomes hold for the bid ct (Dt2 + k∈N dkt2 ), (8.8) implies profile ( bn , b∗−n )t2 as d nt2 < dnt2 . Therefore, (8.54a) and (8.55a) are again applicable. 1 ∗ nt2 < β nt = ct (Dt + m =n dmt + However, if dnt2 > k∈N dkt∗ 2 , then because β 2 2 1 dnt2 ), there is no guarantee that xnt2 = dnt2 . Whether or not (8.54b) holds depends on  nt2 and ct (Dt2 + m =n the comparison between β xmt2 + d nt2 ). Similarly, for the bid ∗  profile (bnt2 , b−n,t2 ), there is no guarantee that (8.55b) holds. x t2 , depending on the relative Three subcases must x t2 and  be considered for values of dnt2 , d nt2 and k∈N dkt∗ 2 .  Case III.1, dnt2 < d nt2 < k∈N dkt∗ 2 Analysis of Δu n , in this case, is identical to that of Case II.1, so Δu n > 0.  Case III.2, dnt2 < k∈N dkt∗ ≤ d nt2 2

Appendices

239

  ∗ nt2 > βnt Because β , satisfying (8.8) for the nth EV results in k∈N xkt2 ≥ k∈N 2   dkt∗ 2 , with equality holding only if xnt = d nt2 = k∈N dkt∗ 2 and m =n xmt2 = 0. If 2 ∗ xkt2 ) > βmt for all m ∈ N \ {n}, with the inequality is strict, then ct (Dt2 + k∈N 2 (8.8) implying xmt2 = 0. Hence, 

dkt∗ 2 ≤ xnt2 ≤ d nt2 ,



xmt2 = 0.

(8.60)

m =n

k∈N

nt2 and ct (Dt2 + d nt2 ): The applicability of (8.54b) reverts to a comparison between β nt2 ≥ ct (Dt2 + d nt2 ) then it can be verified that (8.54b) holds. Thus, Δu n > 0, • If β since the analysis in this case is identical to that developed in Case II.2. nt2 < ct (Dt2 + d nt2 ), (8.54b) does not hold. Rather, Δu n can be established • If β using (8.53), (8.25), (8.55a) and (8.60), Δu n = gnt1 (d nt1 ) − gnt1 (dnt1 ) − δn + δn



xnt − Γn

t∈T

⎛

− ct ⎝ Dt2 + +

∗ βt 2





2



 xnt − Γn

2

t∈T

+ f n ( xnt2 ) − f n (dnt2 ) ⎞

 xmt2 + dnt2 ⎠ + ct (Dt2 + xnt2 )

m =n

 xmt2

m =n  ∗ ∗ > gnt (d nt )(d nt1 − dnt1 ) + f n (dnt )( xnt2 − dnt2 ) 1 1 2  ∗ + gnt (dnt )(dnt1 − d nt1 + dnt2 − xnt2 ) 2 2   ∗ ∗ ∗ + βt  xmt2 + ct (Dt2 + dmt + dnt ) 2 2 2

⎛

m =n

× ⎝ xnt2 −

m =n



⎞

 xmt2 − dnt2 ⎠

(8.61a)

m =n

 ∗  ∗ (d nt )(d nt1 − dnt1 ) + gnt (dnt )(dnt1 − d nt1 ) = gnt 1 1 2 2

(8.61b)

> μ(d nt1 − dnt1 ) + μ(dnt1 − d nt1 ) = 0,

(8.61c)

where (8.61a) holds by the convexity of gnt (·) together with (8.27a), the convexity of f n (·) together with (8.29) and (8.60), the convexity of ct (·) together with (8.55a)

240

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

 and (8.60), and the concavity of −δn ( t∈T xnt − Γn )2 together with Lemma 8.9  ∗  , b ∈ B (A) with d = A < d specified  below, recalling that b n n n t∈T nt t∈T nt ,  xnt − t∈T xnt = dnt1 + dnt2 − (d nt1 + xnt2 ) ≥ 0; (8.61b) holds and that t∈T  by (8.10) in Lemma 8.3 together with (8.25); and (8.61c) follows the same justification as (8.57) though using (8.27a). Lemma 8.9 Consider an allocation x n (b) ≡ (xnt , t ∈ T ) with respect to a bid pro  ∗ . Then: file b, such that t∈T dnt < t∈T dnt ∂ ∂ xnt

− δn



xnt − Γn

2

 ∗ > gnt (dnt ) > μ,

(8.62)

t∈T

for all t ∈ T , where gnt is defined in Lemma 8.6. Proof ∂ ∂ xnt

− δn



xnt − Γn

2

   = 2δn Γn − xnt

t∈T

t∈T

   ∗ > 2δn Γn − dnt ∗ + = βnt ⎛

t∈T ∗ f n (dnt )

= ct ⎝ Dt +



(8.63a) ⎞

∗ ∗⎠ ∗ dmt + dnt ) + f n (dnt

(8.63b) (8.63c)

m =n  ∗ = gnt (dnt ) > μ,

(8.63d) (8.63e)

   ∗ ; (8.63b) follows where (8.63a) holds because t∈T xnt ≤ t∈Tdnt < t∈T dnt ∗ ∂ ∗  ∗ ∗ from βnt = ∂dnt wn (d n ) = − f n (dnt ) + 2δn (Γn − t∈T dnt ); (8.63c) holds by (8.10) in Lemma 8.3; (8.63d) holds by the specification of gnt in (8.25); and (8.63e) holds by Lemma 8.6 and the convexity of gnt .  Case III.3, k∈N dkt∗ 2 ≤ dnt2 < d nt2   Using the same argument as in Case III.2 gives m =n xmt2 = m =n  xmt2 = 0. Analysis of Δu n depends on the relative values of dnt2 , xnt2 and d nt2 , keeping in mind from Lemma 8.4 that βnt2 (dnt2 , A) > βnt2 (d nt2 , A). • If xnt2 = d nt2 then  xnt2 = dnt2 must also hold. Analysis of Δu n in this case is identical to that developed in Case II.3. xnt2 < d nt2 then  xnt2 = dnt2 ≤ xnt2 . Analysis of Δu n follows that of Case I• If dnt2 ≤ II.2. xnt2 <  xnt2 , and Δu n satisfies, • If xnt2 < dnt2 < d nt2 then  ∗ (d nt )(d nt1 − dnt1 ) + f n ( xnt2 )( xnt2 −  xnt2 ) Δu n > gnt 1 1

Appendices

241

+ ct (Dt2 +  xnt2 )( xnt2 −  xnt2 )  ∗ + gnt2 (dnt2 )(dnt1 − d nt1 )    xnt2 −  xnt ( xnt2 ) + 2δn Γn −

(8.64)

t∈T

xnt2 )( xnt2 −  xnt2 ) > f n (  + ct (Dt2 +  xnt2 )( xnt2 −  xnt2 )    + 2δn Γn − xnt2 −  xnt ( xnt2 ),

(8.65)

t∈T

where (8.64) holds by the convexity of gnt1 (·) together with (8.27a), the convexity  of f n (·) and ct (·), the concavity of −δn (Γn − t∈T xnt )2 and Lemma 8.9; and (8.65) makes use of (8.61b). Further analysis uses  xnt ≤ dnt for all t ∈ T to give,        xnt ≥ 2δn Γn − 2δn Γn − dnt t∈T

t∈T

nt2 , = f n (dnt2 ) + β where the equality follows from (8.11). Because  xnt2 > 0 and   xnt2 ). Therefore, (8.8) gives βnt2 ≥ ct (Dt2 + 

(8.66) 

xmt2 m =n 

= 0,

   2δn Γn − xnt2 ) dnt ≥ f n (dnt2 ) + ct (Dt2 +  t∈T

xnt2 ) + ct (Dt2 +  xnt2 ). ≥ f n (

(8.67)

xnt2 , (8.65) and (8.67) ensure Δu n > 0. Because xnt2 <  Hence, Δu n > 0 whenever d nt2 , dnt2 ∈ Int(R2 ). 2  , dnt2 ∈ R2 Case IV, d nt2 = d nt 2   ∗ ∗ In this case, βnt2 = βnt , so (8.8) ensures that k∈N xkt2 = k∈N dkt∗ 2 and dnt ≤ 2 2 2 xnt2 ≤ d nt2 .  Case IV.1, dnt2 < k∈N dkt∗ 2 Using the same argument as in Case III, (8.55a) is again applicable. xnt2 = dnt2 , Δu n can be established by, If xnt2 >   ∗ ∗ (d nt )(d nt1 − dnt1 ) + f n (dnt )( xnt2 − dnt2 ) Δu n > gnt 1 1 2  ∗ + gnt (dnt )(dnt1 − d nt1 + dnt2 − xnt2 ) 2

2

∗ + βt ( xnt2 − dnt2 ) 2  ∗  ∗ = gnt (d nt )(d nt1 − dnt1 ) + gnt (dnt )(dnt1 − d nt1 ) 1 1 2 2

(8.68b)

> 0,

(8.68c)

(8.68a)

242

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

where (8.68a) holds by the convexity of gnt (·) together with (8.27a), the  convexity xnt2 >  xnt2 and (8.55a), and the concavity of  −δn ( t∈T xnt − of f n (·) together with bn ,  bn ∈ Bn (A) with t∈T dnt = A < Γn )2 together with Lemma 8.9, recalling that  ∗ d ; (8.68b) holds by (8.10) in Lemma 8.3 together with (8.25); and (8.68c) t∈T nt follows from (8.61b). xnt2 = dnt2 , Δu n is given by, If xnt2 <   ∗ (d nt )(d nt1 − dnt1 ) + f n (dnt2 )( xnt2 − dnt2 ) Δu n > gnt 1 1  ∗ ∗ + gnt2 (dnt2 )(dnt1 − d nt1 ) + βt2 ( xnt2 − dnt2 )  + 2δn (Γn −  xnt )(dnt2 − xnt2 )

(8.69a)

t∈T ∗ xnt2 − dnt2 ) + βt ( xnt2 − dnt2 ) > f n (dnt2 )( 2    + 2δn Γn −  xnt (dnt2 − xnt2 )

(8.69b)

t∈T

> 0,

(8.69c)

where (8.69a) holds by the convexity of gnt (·) together with  (8.27a), the 2convexity of f n (·) together with (8.55a), and the concavity of −δ n ( t∈T x nt − Γn )together ∗ bn ∈ Bn (A) with t∈T dnt = A < t∈T dnt ; with Lemma 8.9, recalling that bn ,  ∗  (8.69b) uses (8.68b); and (8.69c) uses (8.66) together with βnt2 > βt2 .  Case IV.2, dnt2 ≥ k∈N dkt∗ 2 Using the same argument as in Case III, (8.55b) is applicable. Then similar to the analysis of Case IV.1, Δu n > 0. Case V, d nt2 , dnt2 ∈ R3 ∗ ∗ nt2 < β nt2 < βnt In this case, β , so (8.8) ensures that xmt2 =  xmt2 = dmt for all m ∈ 2 2 ∗ 4 xnt2 < dnt . Hence, (8.51) becomes, N \ {n}, and xnt2 ≤  2 Δτn = ⎛ ct ⎝ Dt1 + ⎛ + ct ⎝ Dt2 +

 m =n



⎞

⎛

∗ dmt + dnt1 ⎠ − ct ⎝ Dt1 + 1

⎞

 m =n

⎛

∗ dmt + xnt2 ⎠ − ct ⎝ Dt2 + 2

m =n



⎞ ∗ dmt + d nt1 ⎠ 1

⎞ ∗ dmt + xnt2 ⎠ . 2

m =n

Using (8.52) and (8.70) in (8.53) gives,

4 The

equality xnt2 =  xnt2 = 0 can occur if ct (Dt2 +



m =n

∗ )≥β nt2 > β nt2 . dmt 2

(8.70)

Appendices

243

Δu n = gnt1 (d nt1 ) − gnt1 (dnt1 ) + gnt2 ( xnt2 ) − gnt2 ( xnt2 )  2  2  xnt − Γn + δn xnt − Γn − δn t∈T

t∈T

t∈T

t∈T

 

 ∗ > −μ dnt1 − d nt1 − gnt (dnt )  xnt2 − xnt2 2 2  2  2 − δn  xnt − Γn + δn xnt − Γn ,

(8.71)

where the inequality holds by the convexity of gnt (·) together with (8.27a) for the ∗ xnt2 < dnt for the second term. first term, and with xnt2 ≤  2 Using (8.62) from Lemma 8.9 together with (8.27a), the concavity of  xnt2 > xnt2 gives, −δn ( t∈T xnt − Γn )2 , and  −δn



 xnt − Γn

2

+ δn

t∈T



xnt − Γn

2

t∈T



 ∗ > gnt (dnt ) dnt1 − d nt1 +  xnt2 − xnt2 2 2

 

 ∗ > μ dnt1 − d nt1 + gnt (dnt )  xnt2 − xnt2 . 2

(8.72)

2

Thus, it follows from (8.71) and (8.72) that Δu n > 0 whenever d nt2 , dnt2 ∈ R3 . In summary, the analysis presented in Cases I-V shows that inequality (8.28) holds ∗ . for all d nt2 ≥ dnt 2

Proof of Lemma 8.7 The total aggregate payment made by all EVs is given by (8.13), 

τn (b∗ ) =

n∈N



τnt (b∗t ),

n∈N t∈T

with ⎛ τnt (b∗t ) = − ct ⎝ Dt + +





⎞ ∗,−n ⎠ xmt + ct

m =n ∗ βmt

∗,−n xmt

−

∗ xmt )

! ,

 Dt +



 xkt∗

k∈N

(8.73)

m =n

where xkt∗ denotes the optimal charging allocation of the kth EV given by Problem 8.2 ∗,−n with respect to b∗t , and xmt denotes the optimal charging allocation of EV m = n ∗,−n ∗ = xmt with respect to (0n , b∗−n )t . It follows from the KKT conditions (8.8) that xmt for all m = n. Therefore, (8.73) simplifies to,

244

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games

 τnt (b∗t ) = ct

Dt +



⎛

 xkt∗

− ct ⎝ Dt +

 m =n

∗ xmt
× xnt xkt∗ k∈N n∈N t∈T



   ct Dt + k∈N xkt∗ − ct Dt  ∗  = × xnt ∗ x k∈N kt t∈T n∈N 

 

 ∗ ct Dt + = xkt − ct Dt . t∈T

k∈N

Proof of Theorem 8.4 It will be shown initially that for each EV, n ∈ N , there always exists a best rebn∗ (b−n ) ≡ sponse bn ≡ (βn , dn ) under which it is fully allocated, xn = dn . Denote by ∗ ∗ (βn , dn ) the best response of the nth EV with respect to b−n , bn , b−n ), bn∗ (b−n ) = argmax u n ( bn ∈B n

bn∗ , b−n ). As stated in Sect. 8.3.1, incentive and let  x ∗ be the allocation with respect to ( compatibility holds, so a bid with price satisfying βn = wn (dn ) is the best choice among all possible bids. Thus, it will be assumed that all bids are truthful.

Appendices

245

n , d n ) with β n = wn ( Consider another bid bn = (β xn∗ ) and d n = xn∗ , and denote xn∗ ≤ d n∗ by x the allocation with respect to ( bn , b−n ). The concavity of wn (·) and  ∗  ∗ ∗ n ≥ β n . ensure that wn ( xn ) ≥ wn (dn ), which implies that β Assume xn < d n = xn∗ . Then, ⎛ c ⎝ D +



⎞

⎛

xm∗ + x n ⎠ < c ⎝ D +

m =n



⎞ n . n∗ ≤ β xm∗ + xn∗ ⎠ ≤ β

m =n

But (8.8c) then implies σn > 0 and so xn = d n . Hence a contradiction so xn ≥ d n . xn = d n = xn∗ . That is to say, the bid bn results in full allocation. But xn ≤ d n , so bn∗ , it follows that bn has the same payoff as bn∗ . Since bn has the same allocation as Therefore, bn is also the best response. It may be concluded that for each EV, there exists the best response with a full allocation. Hence, the remainder of the proof assumes that EVs consider their best response with full allocation. Suppose there exists another NE, denoted by b0 , which differs from b∗ . By Lemma 8.3, b0 is inefficient and does not satisfy (8.10). Also, according to Definition 8.1, bn0 is the best response of the nth EV with respect to b0−n . From the above argument regarding full allocation, it may be assumed that xn0 = dn0 for all n ∈ N . Moreover, since b0 satisfies (8.8):  βn0

 = c (D + k∈N xk0 ) + σn , if xn0 > 0  ≤ c (D + k∈N xk0 ) + σn , if xn0 = 0,

(8.74)

where σn ≥ 0. Since xn0 = dn0 and b0 is inefficient, at least one EV, l ∈ N , must satisfiy σl > 0. l , d l ), such that c (D + another bid for the lth EV, denoted by bl = (β  Consider 0 0 x be the allocation with respect to ( bl , b0−l ). From (8.11), k∈N x k ) < βl < βl . Let  0 0 the concavity of wn (·) gives dl < dl . Then, because xm = dm0 for all m = l, and   l , it follows that: c (D + k∈N xk0 ) < β xl . 0 ≤ xl0 <

(8.75a)

Because xl > 0, the KKT conditions (8.8) require:  c



D+



 xk

l . ≤β

(8.75b)

for all m = l,

(8.75c)

k∈N

Furthermore, (8.8) implies:  xm

= xm0 , ≤ xm0 ,

l , if βm0 > β otherwise,

246

8 Efficient Charging Coordination for Electric Vehicles Under Auction Games



and



xk ≥

k∈N

xk0 .

(8.75d)

k∈N

Hence, the difference of payoffs between b0 and ( bl , b0−l ), given by Δu l = 0 0 u l (b ) − u l ( bl , b−l ), satisfies the following analysis:

 xl ) − fl (xl0 ) − δl (xl0 − Γl )2 − ( xl − Γl )2 Δu l = fl (       0 xk − c D + xk +c D+ +



k∈N

βm0 (xm0

k∈N

− xm )

m =l

xl )( xl − xl0 ) + 2δl ( xl − Γl )( xl − xl0 ) < fl (        0 xk xk − xk +c D+ k∈N

l +β

 (xm0 − xm )

k∈N

k∈N

(8.76a)

m =l



xl − xl0 ) ≤ fl (d l ) + 2δl (d l − Γl ) (        0 xk xk − xk +c D+ k∈N

l +β

 (xm0 − xm ) m =l

 

= c D +

 k∈N

≤ 0,

k∈N

(8.76b)

 xk

k∈N

 l −β



k∈N

xk −



 xk0

(8.76c)

k∈N

(8.76d)

where (8.76a) holds because of Assumptions (A1, A2) and (8.75c); (8.76b) follows from Assumption (A2), xl ≤ d l and (8.75a); (8.76c) uses (8.11); and finally (8.76d) uses (8.75b) and (8.75d). bl , b0−l ) which contradicts the statement This analysis implies that u l (b0 ) < u l ( 0 that bl is the best response of the lth EV with respect to b0−l . Hence, no NE exist apart from b∗ . The NE for single-interval auction games is unique and efficient.

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48. Bruno Tuffin, Revisited progressive second price auction for charging telecommunication networks. Telecommun. Syst. 20(3–4), 255–263 (2002) 49. P. Jia, P. Caines, Analysis of a class of decentralized dynamical systems: rapid convergence and efficiency of dynamical quantized auctions. IMA J. Math. Control Inf. 27(3), 329–372 (2010) 50. P. Jia, P. Caines, Analysis of quantized double auctions with application to competitive electricity markets. INFOR: Inf. Syst. Oper. Res. 48(4), 239–250 (2010) 51. P. Jia, P.E. Caines, Analysis of decentralized quantized auctions on cooperative networks. IEEE Trans. Autom. Control 58(2), 529–534 (2013) 52. E.F. Camacho, C. Bordons, Model Predictive Control, 2nd edn. (Springer, New York, 2004)

Chapter 9

Conclusions and Future Work

9.1 Conclusions This book focuses on auction games to efficiently allocate resources. It is divided into two parts: theory part and applications in smart grids and electric vehicle charging. In the theory part, the book studies three classes of resource allocation problems, say single-sided, double-sided, and hierarchical, designing corresponding auction games and algorithms to achieve an efficient equilibrium in a decentralized way. The followed part is the application of auction games in the problems of elastic loads management in electricity markets, economic operation of the microgrid, V2G service regulation, and EV charging coordination. Different from the general singlesided, double-sided, and hierarchical resource allocation problems considered in the theory part, all of those application problems have special characteristics, e.g., in the problem of economic operation of the microgrid, there exist three-side participants. In the following, we conclude this book in seven parts. • Infinitesimal divisible resource allocation has challenges in developing decentralized mechanisms. We studied the efficient and decentralized method for singletype single-sided resource allocation under the auction mechanism. A so-called PSP auction mechanism was proposed following the VCG type payment, say the payment of each related player was defined as the externality it imposed on the others through its participation. The message space applied is two-dimensional with a bid that specifies per unit price and a maximum of the demand. By introducing a pair of scaler-valued parameters, i.e., the upper and lower bounds of players’ marginal valuation, we designed an algorithm at each iteration of which, one player updated individual best bid under a constrained set of demand. The monotonic increase of social welfare and then the convergence of the algorithm could be verified. Moreover, in order to improve the performance of the proposed method, the update sequence of players could be determined related to the bid profile of the players. We show the analysis of the NE properties of the auction mechanism, and evaluate the property of convergence and convergence rate of the proposed algorithm. The result is that the convergence behavior largely relied on © Springer Nature Singapore Pte Ltd. 2020 Z. Ma and S. Zou, Efficient Auction Games, https://doi.org/10.1007/978-981-15-2639-8_9

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the number of players N , as well as the value of termination parameter ε of the proposed algorithm. Specifically, the upper bound of the convergent iteration is the order of O (N ln(1/ε)). • In order to address the trade flow among suppliers and consumers, we modeled it as a double-sided auction game under the VCG payment. This mechanism possessed the properties of the incentive compatibility and the existence of an efficient NE. There exist infinite NEs for this auction game even though the efficient NE is unique. To achieve an efficient NE and avoid the inefficient ones, we proposed a novel dynamic process for the underlying auction game. More specifically, the double-sided auction game was decomposed into two single-sided ones coupled via a so-called potential quantity, which represented the total trade quantity of the resource in the system. A pair of sequential algorithms were proposed for the buyerside and seller-side auction game. At each iteration of the proposed algorithm, the auctioneer assigns a specific buyer and a specific seller to allow them to update their strategies, respectively, and updates the potential quantity. Assisted with the given extra system information, a certain constraint is set on the bid demand of the assigned buyer and bid supply of the assigned seller, respectively. Under the proposed method, the potential quantity and the social welfare increase with respect to iteration steps, respectively. And we show that the underlying auction system converges to an efficient NE at which the system reaches the maximal social welfare. Furthermore, we verify that the convergence iteration steps are within a certain value which is the order of O(ln(1/ε)). • A hierarchical resource allocation model in large-scale and complex systems was formulated in this book, wherein suppliers play as a medium between the provider and individual buyers. In a local system formed by a supplier and its buyers, we designed a PSP-style auction mechanism to regard each local system as a singlesided auction. We presented a dynamic process to calculate an efficient NE of each local system. At the local equilibrium, the efficient allocation is achieved and the bidding prices of buyers are identical with each other. In the high level, the provider assigned a certain amount of resource to suppliers, and this allocation was adjusted via a dynamic hierarchical algorithm with respect to some rough information of individual valuations and the implemented uniform bidding price from the local systems. The proposed update process guarantees the monotonic increase of the system social welfare, and further the convergence to an efficient NE. • In the electricity market, auction mechanism such as uniform bid pricing has been widely applied for supply side management. We studied the auction mechanisms for large-scale elastic loads which are allowed to participate in electricity markets directly with the deregulation and reconstruction. Demand side participation introduces challenges to the economic coordination and management of power systems. We studied two auction mechanisms, the MCP and PSP mechanisms, to coordinate large-scale loads and achieve the optimal solution in a distributed way. The issues of the payment comparison, incentive compatibility, and the NE property of the efficient bid profiles were investigated, as well as the associated asymptotic phenomena as the scale size of the power system increases. It is shown that the difference between the payments of an individual load under MCP and PSP auctions vanishes asymptotically as the scale size of power systems increases,

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and the incentive compatibility holds under the MCP mechanism only with respect to the efficient bid profile of other loads. Furthermore, the efficient bid profile is an ε-NE of the formulated load coordination games under the MCP mechanism, and it becomes a NE asymptotically as the scale of the power systems increases. • The economic operation of the microgrid was formulated as a class of resource allocation problems wherein three-side participants traded with each other. We modeled the problem as a class of specific PSP auction games, following the VCG type payments, and showed that the efficient coordination solution is a NE of the underlying auction games. We proposed a distributed method under which the system can converge to a NE which may not be efficient. It is also shown that the performance of the worst NE can be bounded with respect to the system parameters, say the energy trading price with the main grid, and based upon that, the implemented NE is unique and efficient under some conditions. • We studied the coordination of vehicle-to-grid services to regulate the frequency and voltage of power grid, and further developed a distributed mechanism under the PSP double-sided auction mechanism. The underlying problem is actually a twotype resource allocation and the message space of each player is four-dimensional. It is shown that the efficient coordination solution is a NE for the underlying auction games and its corresponding allocation is efficient. Considering largescale PEVs in the V2G coordination problem, an aggregator was introduced to centrally coordinate the power rates of collection of individual PEVs. The efficient NE for the underlying auction games is implemented with numerical simulations by generalizing the so-called Q-PSP mechanism. • Accommodating large-scale of EVs on the grid required coordination of their charging requirements. We proposed the PSP based auction mechanism for the allocation of multi-type resources, which represent the charging power of EVs over the multi-time horizon. The challenge is that individual EVs are capable of shifting their charging requirements in time across the charging horizon, which brings complication in verifying that the efficient solution is a NE. This auction mechanism ensures incentive compatibility and the existence of an efficient NE. In order to implement an efficient NE, an extended dynamic programming algorithm was proposed for the underlying auction game. This update mechanism was illustrated using a numerical example which showed convergence of the auction to an efficient NE. Besides, a practical approach to addressing the inaccurate forecast of the EV population and the background demand is to adopt a receding horizon strategy. Consequently, an efficient NE determined ahead of the charging interval may be suboptimal relative to a solution that considers disturbances.

9.2 Future Work It lists a few possible future research topics in the following: • In many parts of this book except the EV charging coordination problem, we study static and deterministic models that don’t consider the forecast and disturbances in the system. Also, in the practical scenarios of smart grids and EV coordina-

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tion problems, renewable generation receives more and more attention, which introduces random effects to the system. Therefore, a natural extension to the associated optimization methods would incorporate uncertain supply and demand forecast errors into the optimization process. Moreover, it is necessary to implement feasible and optimal solutions in case of the hard constraints on the systems, e.g., the capacity constraints on the transformers in the transmission systems and the feeder lines of the distribution networks, and the feasible range of the nodes of the distribution network, etc. In this book, we mainly focus on the VCG type auction games to solve the resource allocation problems. We have put an eye on the combinatorial auction mechanism for network optimization which could be modeled as multi-type resources allocation. The challenge is that the coupling relationship exists among the resources and the decentralized algorithm may result in a local optimum. The problem studied in Chap. 8 belongs to this kind of problem. As ongoing researches, it is worth to design decentralized algorithms and verify the convergence to an efficient NE. In this book, the optimization generalized are all convex optimization, i.e., the payoffs of players and constraints are all convex. However, non-convex payoffs and non-convex constraint sets are also popular in practical scenarios. As ongoing researches, we are interested to develop auction mechanisms for non-convex optimization and guarantee the convergence to at least a local optimum. The efficiency loss, in this case, is also studied. In the auction games in this book, we all suppose that all the players are rational, that is, they aim at maximizing individual benefits. However, as briefly mentioned in Chap. 2, frequent starvation for the traded resources decreases the bidder’s interest and even may result in the drop of this bidder in the future auction rounds. Moreover, there might exist malicious or irrational bidders, who may construct a bid by shifting and randomizing the components of another bidder’s bid strategy, or just submit a bid under which its payoff function is not optimized. As a consequence, such bad-behavior bidders will cause a certain efficiency loss and unpleasant convergence property under the dynamic process proposed in our work. As ongoing research, we are willing to update our proposed auction mechanism to enhance its robustness. As an interesting direction, it may be also worth to explore the effect of these unexpected behaviors to enhance its robustness. The implementation of the NE of the auction games rely on the form of the payoffs of the players. Sometimes the payoffs of players or the gradient information could not be identified, but the value of the payoff could be observed. For example, the payoff is influenced by uncertainties whose distribution or expectation and moment information is unknown. In such cases, it is difficult to analyze incentive compatibility and determine the bid strategy. In the future, it is interesting to explore the auction mechanism for the unknown payoff functions.