Estimation of the Shapley Value of a Peer-to-peer Energy Sharing Game Using Multi-Step Coalitional Stratified Sampling


One of the main objectives of a peer-to-peer energy market is to efficiently manage distributed energy resources while creating additional financial benefits for the participants. Cooperative game theory offers such a framework, and the Shapley value, a cooperative game payoff allocation based on the participants’ marginal contributions made to the local energy coalition, is shown to be fair and efficient. However, its high computational complexity limits the size of the game. In order to improve this peer-to-peer cooperative scheme’s scalability, this paper investigates and adapts a stratified sampling method for the Shapley value estimation. It then proposes a multi-step sampling strategy to further reduce the computation time by dividing the samples into incremental parts and terminating the sampling process once a certain level of estimation performance is achieved. Finally, selected case studies demonstrate the effectiveness of the proposed method, which is able to scale up the game from 20 players to 100 players.

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Correspondence to Liyang Han.

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recommended by Associate Editor Soohee Han under the direction of Editor-in-Chief Keum-Shik Hong. This work was supported in part by the Engineering and Physical Sciences Research Council under Grants EP/N03466X/1, EP/S000887/1, and EP/S031901/1, and in part by the Oxford Martin Programme on Integrating Renewable Energy.

Liyang Han received his B.S degree in energy engineering from Zhejiang University, China, in 2012, and his M.S. degree in civil engineering from Stanford University, CA, USA, in 2014. He recently finished his Ph.D. defense at the University of Oxford and is expecting to confirm his Ph.D. degree by October 2020. He joined the Energy Analytics and Markets (ELMA) group at the Technical University of Denmark (DTU) in July 2020 as a postdoctoral researcher. His research interests include prosumer-centric energy markets and data markets.

Thomas Morstyn received his B.Eng. (Hon.) degree from the University of Melbourne, Australia, in 2011, and his Ph.D. degree from the University of New South Wales, Australia, in 2016, both in electrical engineering. He is a Lecturer in Power Electronics and Smart Grids with the School of Engineering at the University of Edinburgh, and he is also a visiting fellow with the Oxford Martin School at the University of Oxford. His research interests include multi-agent control and market design for integrating distributed energy resources into power system operations.

Malcolm McCulloch received his B.Sc. (Eng.) and Ph.D. degrees in electrical engineering from the University of the Witwatersrand, Johannesburg, South Africa, in 1986 and 1990, respectively. In 1993, he joined the University of Oxford, Oxford, U.K., to head up the Energy and Power Group, where he is currently an Associate Professor with the Department of Engineering Science. He is active in the areas of electrical machines, transport, and smart grids. His work addresses transforming existing power networks, designing new power networks for the developing world, developing new technology for electric vehicles, and developing approaches to integrated mobility.

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Han, L., Morstyn, T. & McCulloch, M. Estimation of the Shapley Value of a Peer-to-peer Energy Sharing Game Using Multi-Step Coalitional Stratified Sampling. Int. J. Control Autom. Syst. (2021).

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  • Cooperative game theory
  • energy management
  • energy storage
  • P2P energy sharing
  • Shapley value