Charging Coordination via Non-cooperative Games

  • Zhongjing MaEmail author


This chapter develops a strategy to coordinate the charging of autonomous plug-in electric vehicles (PEVs) using concepts from non-cooperative games. The foundation of the chapter is a model that assumes PEVs are cost-minimizing and weakly coupled via a common electricity price. At a Nash equilibrium, each PEV reacts optimally with respect to a commonly observed charging trajectory that is the average of all PEV strategies. This average is given by the solution of a fixed point problem in the limit of infinite population size. The ideal solution minimizes electricity generation costs by scheduling PEV demand to fill the overnight non-PEV demand “valley”. The chapter’s central theoretical result is a proof of the existence of a unique Nash equilibrium that almost satisfies that ideal. This result is accompanied by a decentralized computational algorithm and a proof that the algorithm converges to the Nash equilibrium in the infinite system limit. Several numerical examples are used to illustrate the performance of the solution strategy for finite populations. The examples demonstrate that convergence to the Nash equilibrium occurs very quickly over a broad range of parameters, and suggest this method could be useful in situations where frequent communication with PEVs is not possible. The method is useful in applications where fully centralized coordination is not possible, but where optimal or near-optimal charging patterns are essential to system operation.


  1. 1.
    P. Denholm, W. Short, An evaluation of utility system impacts and benefits of optimally dispatched plug-in hybrid electric vehicles. Technical Report NREL/TP-620-40293 (National Renewable Energy Laboratory, 2006)Google Scholar
  2. 2.
    S. Rahman, G.B. Shrestha, An investigation into the impact of electric vehicle load on the electric utility distribution system. IEEE Trans. Power Deliv. 8(2), 591–597 (1993)CrossRefGoogle Scholar
  3. 3.
    F. Koyanagi, Y. Uriu, Modeling power consumption by electric vehicles and its impact on power demand. Electr. Eng. Jpn. 120(4), 40–47 (1997)CrossRefGoogle Scholar
  4. 4.
    F. Koyanagi, T. Inuzuka, Y. Uriu, R. Yokoyama, Monte Carlo simulation on the demand impact by quick chargers for electric vehicles, in Proceedings of the IEEE Power Engineering Society Summer Meeting, vol. 2 (1999), pp. 1031–1036Google Scholar
  5. 5.
    D.S. Callaway, I.A. Hiskens, Achieving controllability of electric loads. Proc. IEEE 99(1), 184–199 (2011)Google Scholar
  6. 6.
    T. Lee, Z. Bareket, T. Gordon, Stochastic modeling for studies of real-world PHEV usage: driving schedule and daily temporal distributions. IEEE Trans. Veh. Technol. 61(4), 1493–1502 (2012)Google Scholar
  7. 7.
    N.P. Padhy, Unit commitment-a bibliographical survey. IEEE Trans. Power Syst. 19(2), 1196–1205 (2004)CrossRefGoogle Scholar
  8. 8.
    S. Borenstein, J.B. Bushnell, F.A. Wolak, Measuring market inefficiencies in California’s restructured wholesale electricity market. Am. Econ. Rev. 92(5), 1376–1405 (2002)Google Scholar
  9. 9.
    J.B. Bushnell, E.T. Mansur, C. Saravia, Vertical arrangements, market structure, and competition: an analysis of restructured US electricity markets. Am. Econ. Rev. 98(1), 237–266 (2008)Google Scholar
  10. 10.
    B.E. Hobbs, Linear complementarity models of Nash-Cournot competition in bilateral and POOLCO power markets. IEEE Trans. Power Syst. 16(2), 194–202 (2002)Google Scholar
  11. 11.
    L.B. Cunningham, R. Baldick, M.L. Baughman, An empirical study of applied game theory: transmission constrained Cournot behavior. IEEE Trans. Power Syst. 17(1), 166–172 (2002)Google Scholar
  12. 12.
    E. Pettersen, Managing end-user flexibility in electricity markets (Fakultet for samfunnsvitenskap og teknologiledelse, Institutt for industriell økonomi og teknologiledelse, NTNU, 2004)Google Scholar
  13. 13.
    A.B. Philpott, E. Pettersen, Optimizing demand-side bids in day-ahead electricity markets. IEEE Trans. Power Syst. 21(2), 488–498 (2006)Google Scholar
  14. 14.
    K.C. Border, Fixed Point Theorems with Applications to Economics and Game Theory (Cambridge University Press, Cambridge, UK, 1985)Google Scholar
  15. 15.
    Ettore Bompard, Yuchao Ma, Roberto Napoli, Graziano Abrate, The demand elasticity impacts on the strategic bidding behavior of the electricity producers. IEEE Trans. Power Syst. 22(1), 188–197 (2007)Google Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    F.S. Wen, A.K. David, Strategic bidding for electricity supply in a day-ahead energy market. Electr. Power Syst. Res. 59, 197–206 (2001)Google Scholar
  18. 18.
    J.G. Wardrop, Some theoretical aspects of road traffic research, in Proceedings of the Institution of Civil Engineers, Part 2, vol. 1 (1952), pp. 325–378Google Scholar
  19. 19.
    M. Huang, P.E. Caines, R.P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions, in Proceedings of the 42th IEEE International Conference on Decision and Control (Maui, Hawaii, 2003), pp. 98–103Google Scholar
  20. 20.
    M. Huang, P.E. Caines, R.P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: individual-mass behaviour and decentralized epsilon-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)Google Scholar
  21. 21.
    D.R. Smart, Fixed Point Theorems (Cambridge University Press, London, UK, 1974)Google Scholar
  22. 22.
    D.P. Bertsekas, Dynamic Programming and Optimal Control, vol. I (Athena Scientific, Nashua, 1995)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.School of AutomationBeijing Institute of TechnologyBeijingChina

Personalised recommendations