Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market

Henrion, René; Outrata, Jiří; Surowiec, Thomas

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 295-317.

We consider an equilibrium problem with equilibrium constraints (EPEC) arising from the modeling of competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called M-stationarity conditions are derived. This first requires a structural analysis of the problem, e.g., verifying constraint qualifications. Second, the calmness property of a certain multifunction has to be verified in order to justify using M-stationarity conditions. Third, for stating the stationarity conditions, the coderivative of a normal cone mapping has to be calculated. Finally, the obtained necessary conditions are made fully explicit in terms of the problem data for one typical constellation. A simple two-settlement example serves as an illustration.

Classification : 90C30, 49J53
Mots clés : equilibrium problems with equilibrium constraints, epec, M-stationary solutions, electricity spot market, calmness

[1] N. Biggs, Algebraic Graph Theory. Cambridge University Press, Cambrige, 2nd edition (1994). | MR | Zbl

[2] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000). | MR | Zbl

[3] J.B. Cardell, C.C. Hitt and W.W. Hogan, Market power and strategic interaction in electricity networks. Resour. Energy Econ. 19 (1997) 109-137.

[4] S. Dempe, J. Dutta and S. Lohse, Optimality conditions for bilevel programming problems. Optimization 55 (2006) 505-524. | MR | Zbl

[5] A.L. Dontchev and R.T. Rockafellar, Characterization of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 7 (1996) 1087-1105. | MR | Zbl

[6] J.F. Escobar and A. Jofre, Monopolistic competition in electricity networks with resistance losses. Econ. Theor. 44 (2010) 101-121. | MR | Zbl

[7] R. Henrion and W. Römisch, On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling. Appl. Math. 52 (2007) 473-494. | MR | Zbl

[8] R. Henrion, J. Outrata and T. Surowiec, On the coderivative of normal cone mappings to inequality systems. Nonlinear Anal. 71 (2009) 1213-1226. | MR | Zbl

[9] R. Henrion, B.S. Mordukhovich and N.M. Nam, Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20 (2010) 2199-2227. | MR | Zbl

[10] B.F. Hobbs, Strategic gaming analysis for electric power systems : An MPEC approach. IEEE Trans. Power Syst. 15 (2000) 638-645.

[11] X. Hu and D. Ralph, Using EPECs to model bilevel games in restructured electricity markets with locational prices. Oper. Res. 55 (2007) 809-827. | MR | Zbl

[12] 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).

[13] D. Klatte and B. Kummer, Nonsmooth Equations in Optimization. Kluwer, Academic Publishers, Dordrecht (2002). | MR | Zbl

[14] D. Klatte and B. Kummer, Constrained minima and Lipschitzian penalties in metric spaces. SIAM J. Optim. 13 (2002) 619-633. | MR | Zbl

[15] Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge (1996). | MR | Zbl

[16] B.S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of extremal problems. Soviet Mathematics Doklady 22 (1980) 526-530. | Zbl

[17] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, Basic Theory 1, Applications 2. Springer, Berlin (2006). | MR | Zbl

[18] B.S. Mordukhovich and J. Outrata, On second-order subdifferentials and their applications. SIAM J. Optim. 12 (2001) 139-169. | MR | Zbl

[19] B.S. Mordukhovich and J. Outrata, Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18 (2007) 389-412. | MR | Zbl

[20] J.V. Outrata, A generalized mathematical program with equilibrium constraints. SIAM J. Control Opt. 38 (2000) 1623-1638. | MR | Zbl

[21] J.V. Outrata, A note on a class of equilibrium problems with equilibrium constraints. Kybernetika 40 (2004) 585-594. | EuDML | MR | Zbl

[22] J.V. Outrata, M. Kocvara and J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints. Kluwer Academic Publishers, Dordrecht (1998). | MR | Zbl

[23] S.M. Robinson, Some continuity properties of polyhedral multifunctions. Math. Program. Stud. 14 (1976) 206-214. | MR | Zbl

[24] S.M. Robinson, Strongly regular generalized equations. Math. Oper. Res. 5 (1980) 43-62. | MR | Zbl

[25] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer, Berlin (1998). | MR | Zbl

[26] V.V. Shanbhag, Decomposition and Sampling Methods for Stochastic Equilibrium Problems. Ph.D. thesis, Stanford University (2005).

[27] C.-L. Su, Equilibrium Problems with Equilibrium Constraints : Stationarities, Algorithms and Applications. Ph.D. thesis, Stanford University (2005).

[28] J.J. Ye and X.Y. Ye, Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22 (1997) 977-997. | MR | Zbl