Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T03:53:00.115Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  19 January 2011

René Henrion ,
Jiří Outrata  and
Thomas Surowiec
René Henrion
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany. henrion@wias-berlin.de
Jiří Outrata
Affiliation:
Institute of Information Theory and Automation, 18208 Praha 8, Czech Republic; outrata@utia.cas.cz
Thomas Surowiec
Affiliation:
Humboldt University Berlin, Unter den Linden 6, 10099 Berlin, Germany; surowiec@math.hu-berlin.de
Get access

Abstract

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.

Keywords

Equilibrium problems with equilibrium constraintsEPECM-stationary solutionselectricity spot marketcalmness
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012 , pp. 295 - 317
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

N. Biggs, Algebraic Graph Theory. Cambridge University Press, Cambrige, 2nd edition (1994).
J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000).
Cardell, J.B., Hitt, C.C. and Hogan, W.W., Market power and strategic interaction in electricity networks. Resour. Energy Econ. 19 (1997) 109–137. Google Scholar
Dempe, S., Dutta, J. and Lohse, S., Optimality conditions for bilevel programming problems. Optimization 55 (2006) 505–524. Google Scholar
Dontchev, A.L. and Rockafellar, R.T., Characterization of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 7 (1996) 1087–1105. Google Scholar
Escobar, J.F. and Jofre, A., Monopolistic competition in electricity networks with resistance losses. Econ. Theor. 44 (2010) 101–121. Google Scholar
Henrion, R. and Römisch, W., On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling. Appl. Math. 52 (2007) 473–494. Google Scholar
Henrion, R., Outrata, J. and Surowiec, T., On the coderivative of normal cone mappings to inequality systems. Nonlinear Anal. 71 (2009) 1213–1226. Google Scholar
Henrion, R., Mordukhovich, B.S. and Nam, N.M., 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. Google Scholar
Hobbs, B.F., Strategic gaming analysis for electric power systems : An MPEC approach. IEEE Trans. Power Syst. 15 (2000) 638–645. Google Scholar
Hu, X. and Ralph, D., Using EPECs to model bilevel games in restructured electricity markets with locational prices. Oper. Res. 55 (2007) 809–827. Google Scholar
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).
D. Klatte and B. Kummer, Nonsmooth Equations in Optimization. Kluwer, Academic Publishers, Dordrecht (2002).
Klatte, D. and Kummer, B., Constrained minima and Lipschitzian penalties in metric spaces. SIAM J. Optim. 13 (2002) 619–633. Google Scholar
Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge (1996).
Mordukhovich, B.S., Metric approximations and necessary optimality conditions for general classes of extremal problems. Soviet Mathematics Doklady 22 (1980) 526–530. Google Scholar
B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, Basic Theory 1, Applications 2. Springer, Berlin (2006).
Mordukhovich, B.S. and Outrata, J., On second-order subdifferentials and their applications. SIAM J. Optim. 12 (2001) 139–169. Google Scholar
Mordukhovich, B.S. and Outrata, J., Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18 (2007) 389–412. Google Scholar
Outrata, J.V., A generalized mathematical program with equilibrium constraints. SIAM J. Control Opt. 38 (2000) 1623–1638. Google Scholar
Outrata, J.V., A note on a class of equilibrium problems with equilibrium constraints. Kybernetika 40 (2004) 585–594. Google Scholar
J.V. Outrata, M. Kocvara and J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints. Kluwer Academic Publishers, Dordrecht (1998).
Robinson, S.M., Some continuity properties of polyhedral multifunctions. Math. Program. Stud. 14 (1976) 206–214. Google Scholar
Robinson, S.M., Strongly regular generalized equations. Math. Oper. Res. 5 (1980) 43–62. Google Scholar
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer, Berlin (1998).
V.V. Shanbhag, Decomposition and Sampling Methods for Stochastic Equilibrium Problems. Ph.D. thesis, Stanford University (2005).
C.-L. Su, Equilibrium Problems with Equilibrium Constraints : Stationarities, Algorithms and Applications. Ph.D. thesis, Stanford University (2005).
Ye, J.J. and Ye, X.Y., Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22 (1997) 977–997. Google Scholar