Journal of Global Optimization

, Volume 70, Issue 2, pp 329–346 | Cite as

On an equilibrium problem with complementarity constraints formulation of pay-as-clear electricity market with demand elasticity

  • Elisabetta Allevi
  • Didier Aussel
  • Rossana Riccardi


We consider a model of pay-as-clear electricity market based on a Equilibrium Problem with Complementarity Constraints approach where the producers are playing a noncooperative game parameterized by the decisions of regulator of the market (ISO). In the proposed approach the bids are assumed to be convex quadratic functions of the production quantity. The demand is endogenously determined. The ISO problem aims to maximize the total welfare of the market. The demand being elastic, this total welfare take into account at the same time the willingness to pay of the aggregated consumer, as well as the cost of transactions. The market clearing will determine the market price in a pay-as-clear way. An explicit formula for the optimal solution of the ISO problem is obtained and the optimal price is proved to be unique. We also state some conditions for the existence of equilibria for this electricity market with elastic demand. Some numerical experiments on a simplified market model are also provided.


Electricity market Generalized Nash equilibrium problem Elastic demand 

Mathematics Subject Classification

49J40 49J53 91B26 49J52 


  1. 1.
    Aussel, D., Cervinka, M., Marechal, M.: Deregulated electricity markets with thermal losses and production bounds: models and optimality conditions. RAIRO Oper. Res. 50, 19–38 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aussel, D., Bendotti, P., Pištěk, M.: Nash equilibrium in a pay-as-bid electricity market: part 1—existence and characterization. Optimization 66(6), 1013–1025 (2017)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aussel, D., Bendotti, P., Pištěk, M.: Nash equilibrium in a pay-as-bid electricity market part 2—best response of a producer. Optimization 66(6), 1027–1053 (2017)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Aussel, D., Correa, R., Marechal, M.: Spot electricity market with transmission losses. J. Ind. Manag. Optim. 9, 275–290 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Escobar, J.F., Jofré, A.: Monopolistic competition in electricity networks with resistance losses. Econ. Theory 44(1), 101–121 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gabriel, S.A., Conejo, A.J., Fuller, J.D., Hobbs, B.F., Ruiz, C.: Complementarity Modeling in Energy Markets. International Series in Operations Research & Management Science, vol. 180. Springer, New York (2013)MATHGoogle Scholar
  9. 9.
    Henrion, R., Outrata, J.V., Surowiec, T.: Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market. ESAIM Control Optim. Calc. Var. 18, 295–317 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hu, M., Fukushima, M.: Variational inequality formulation of a class of multi-leader–follower games. J. Optim. Theory Appl. 151, 455–473 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hu, X., Ralph, D.: Using EPECs to model bilevel games in restructured electricity markets with locational prices. Oper. Res. 55(5), 809–827 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Klemperer, P.D., Meyer, M.A.: Supply function equilibria in oligopoly under uncertainty. Econometrica 57, 1243–1277 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Leyffer, S., Munson, T.S.: Solving multi-leader–common-follower games. Optim. Methods Softw. 25(4), 601–623 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  15. 15.
    Outrata, J.V.: A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38(5), 1623–1638 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Outrata, J.V.: A note on a class of equilibrium problems with equilibrium constraints. Kybernetika 40, 585–594 (2003)MathSciNetMATHGoogle Scholar
  17. 17.
    Pang, J.-S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader–follower games. Comput. Manag. Sci. 1, 21–56 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Reny, P.: Non-cooperative Games: Equilibrium Existence. The New Palgrave Dictionary of Economics, vol. 8 set, 2nd edn. Palgrave Macmillan, Basingstoke, Hampshire New York (2008)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Economics and ManagementUniversità degli Studi di BresciaBresciaItaly
  2. 2.PROMES UPR CNRS 8521University of Perpignan Via DomitiaPerpignanFrance

Personalised recommendations