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electricity markets; bidding; noncooperative games; equilibrium constraint; EPEC; optimality condition; co-derivative; random demand
Modeling several competitive leaders and followers acting in an electricity market leads to coupled systems of mathematical programs with equilibrium constraints, called equilibrium problems with equilibrium constraints (EPECs). We consider a simplified model for competition in electricity markets under uncertainty of demand in an electricity network as a (stochastic) multi-leader-follower game. First order necessary conditions are developed for the corresponding stochastic EPEC based on a result of Outrata. For applying the general result an explicit representation of the co-derivative of the normal cone mapping to a polyhedron is derived. Then the co-derivative formula is used for verifying constraint qualifications and for identifying $M$-stationary solutions of the stochastic EPEC if the demand is represented by a finite number of scenarios.
[1] B.  Blaesig: Risikomanagement in der Stromerzeugungs- und Handelsplanung. Aachener Beiträge zur Energieversorgung, Band  113. Klinkenberg, Aachen, 2007.
[2] A. L.  Dontchev, R. T.  Rockafellar: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J.  Optim. 7 (1996), 1087–1105. DOI 10.1137/S1052623495284029 | MR 1416530
[3] A.  Eichhorn, W.  Römisch: Mean-risk optimization models for electricity portfolio management. Proceedings of the 9th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS 2006), Stockholm, , , 2006.
[4] J. F.  Escobar, A.  Jofre: Oligopolistic competition in electricity spot markets, Dec. 2005. Available at
[5] F.  Facchinei, J.-S.  Pang: Finite-dimensional Variational Inequalities and Complementarity Problems, Vol.  I and II. Springer-Verlag, New York, 2003. MR 1955648
[6] R.  Fletcher, S.  Leyffer, D.  Ralph, and S.  Scholtes: Local convergence of SQP  methods for mathematical programs with equilibrium constraints. SIAM J.  Optim. 17 (2006), 259–286. DOI 10.1137/S1052623402407382 | MR 2219153
[7] R.  Garcia-Bertrand, A. J.  Conejo, and S.  Gabriel: Electricity market near-equilibrium under locational marginal pricing and minimum profit conditions. Eur. J. Oper. Res. 174 (2006), 457–479. DOI 10.1016/j.ejor.2005.03.037
[8] B. F.  Hobbs, J.-S.  Pang: Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints. Oper. Res. 55 (2007), 113–127. DOI 10.1287/opre.1060.0342 | MR 2290891
[9] B.  F.  Hobbs, C.  Metzler, and J.-S. Pang: Strategic gaming analysis for electric power networks: An MPEC  approach. IEEE Power Engineering Transactions 15 (2000), 638–645. DOI 10.1109/59.867153
[10] X.  Hu, D.  Ralph: Using EPECs to model bilevel games in restructured electricity markets with locational prices. Optimization Online, 2006 ( MR 2360950
[11] X.  Hu, D.  Ralph, E. K.  Ralph, P.  Bardsley, and M. C. Ferris: Electricity generation with looped transmission networks: Bidding to an ISO. Research Paper No.  2004/16, Judge Institute of Management, Cambridge University, 2004.
[12] M.  Kočvara, J. V.  Outrata: Optimization problems with equilibrium constraints and their numerical solution. Math. Program.  B 101 (2004), 119–149. DOI 10.1007/s10107-004-0539-2 | MR 2085261
[13] S.  Leyffer, T. S.  Munson: Solving multi-leader-follower games. Preprint ANL/MCS-P1243-0405, Argonne National Laboratory, Mathematics and Computer Science Division, April 2005.
[14] Z.-Q.  Luo, J.-S.  Pang, and D.  Ralph: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, 1997. MR 1419501
[15] B. S.  Mordukhovich: Variational Analysis and Generalized Differentiation. Vol.  1: Basic Theory, Vol.  2: Applications. Springer-Verlag, Berlin, 2006. MR 2191745
[16] B. S.  Mordukhovich, J. V.  Outrata: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J.  Optim. 18 (2007), 389–412. DOI 10.1137/060665609 | MR 2338444
[17] J. V.  Outrata: A note on a class of equilibrium problems with equilibrium constraints. Cybernetica 40 (2004), 585–594. MR 2120998 | Zbl 1249.49017
[18] J. V.  Outrata: On constrained qualifications for mathematical programs with mixed complementarity constraints. Complementarity: Applications, Algorithms and Extensions, M. C.  Ferris et al. (eds.), Kluwer, Dordrecht, 2001, pp. 253–272. MR 1818625
[19] J. V.  Outrata, M.  Kočvara, and J.  Zowe: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht, 1998.
[20] J.-S.  Pang, M.  Fukushima: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 1 (2005), 21–56. DOI 10.1007/s10287-004-0010-0 | MR 2164953
[21] M. V.  Pereira, S.  Granville, M. H. C.  Fampa, R.  Dix, and L. A.  Barroso: Strategic bidding under uncertainty: A binary expansion approach. IEEE Transactions on Power Systems 20 (2005), 180–188. DOI 10.1109/TPWRS.2004.840397
[22] D.  Ralph, Y.  Smeers: EPECs as models for electricity markets. Power Systems Conference and Exposition (PSCE), Atlanta, 2006, , , .
[23] R. T.  Rockafellar, R. J.-B.  Wets: Variational Analysis. Springer-Verlag, Berlin, 1998. MR 1491362
[24] Stochastic Programming. Handbooks in Operations Research and Management Science, Vol. 10. A.  Ruszczyński, A.  Shapiro (eds.), Elsevier, Amsterdam, 2003. MR 2051791
[25] H.  Scheel, S.  Scholtes: Mathematical programs with equilibrium constraints: Stationarity, optimality and sensitivity. Math. Oper. Res. 25 (2000), 1–22. DOI 10.1287/moor. | MR 1854317
[26] A.  Shapiro: Stochastic programming with equilibrium constraints. J. Optim. Theory Appl. 128 (2006), 221–243. DOI 10.1007/s10957-005-7566-x | MR 2201897 | Zbl 1130.90032
[27] A.  Shapiro, H.  Xu: Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation. Optimization Online 2005 ( MR 2412074
[28] Y.  Smeers: How well can one measure market power in restructured electricity systems?. CORE Discussion Paper 2005/50 (2005), Center for Operations Research and Econometrics (CORE), Louvain.
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