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

René Henrion; Jiří Outrata; Thomas Surowiec

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 2, page 295-317
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Henrion, René, Outrata, Jiří, and Surowiec, Thomas. "Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 295-317. <http://eudml.org/doc/272823>.

@article{Henrion2012,
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.},
author = {Henrion, René, Outrata, Jiří, Surowiec, Thomas},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {equilibrium problems with equilibrium constraints; epec; M-stationary solutions; electricity spot market; calmness; EPEC},
language = {eng},
number = {2},
pages = {295-317},
publisher = {EDP-Sciences},
title = {Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market},
url = {http://eudml.org/doc/272823},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Henrion, René
AU - Outrata, Jiří
AU - Surowiec, Thomas
TI - Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 2
SP - 295
EP - 317
AB - 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.
LA - eng
KW - equilibrium problems with equilibrium constraints; epec; M-stationary solutions; electricity spot market; calmness; EPEC
UR - http://eudml.org/doc/272823
ER -

References

top
  1. [1] N. Biggs, Algebraic Graph Theory. Cambridge University Press, Cambrige, 2nd edition (1994). Zbl0797.05032MR347649
  2. [2] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000). Zbl0966.49001MR1756264
  3. [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. [4] S. Dempe, J. Dutta and S. Lohse, Optimality conditions for bilevel programming problems. Optimization55 (2006) 505–524. Zbl1156.90457MR2274922
  5. [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. Zbl0899.49004MR1416530
  6. [6] J.F. Escobar and A. Jofre, Monopolistic competition in electricity networks with resistance losses. Econ. Theor.44 (2010) 101–121. Zbl1231.91114MR2646897
  7. [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. Zbl1164.90379MR2357576
  8. [8] R. Henrion, J. Outrata and T. Surowiec, On the coderivative of normal cone mappings to inequality systems. Nonlinear Anal.71 (2009) 1213–1226. Zbl1176.90568MR2527541
  9. [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. Zbl1208.49010MR2650845
  10. [10] B.F. Hobbs, Strategic gaming analysis for electric power systems : An MPEC approach. IEEE Trans. Power Syst.15 (2000) 638–645. 
  11. [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. Zbl1167.91357MR2360950
  12. [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. [13] D. Klatte and B. Kummer, Nonsmooth Equations in Optimization. Kluwer, Academic Publishers, Dordrecht (2002). Zbl1173.49300MR1909427
  14. [14] D. Klatte and B. Kummer, Constrained minima and Lipschitzian penalties in metric spaces. SIAM J. Optim.13 (2002) 619–633. Zbl1042.49020MR1951038
  15. [15] Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge (1996). Zbl1139.90003MR1419501
  16. [16] B.S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of extremal problems. Soviet Mathematics Doklady22 (1980) 526–530. Zbl0491.49011
  17. [17] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, Basic Theory 1, Applications 2. Springer, Berlin (2006). Zbl1100.49002MR2191745
  18. [18] B.S. Mordukhovich and J. Outrata, On second-order subdifferentials and their applications. SIAM J. Optim.12 (2001) 139–169. Zbl1011.49016MR1870589
  19. [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. Zbl1145.49012MR2338444
  20. [20] J.V. Outrata, A generalized mathematical program with equilibrium constraints. SIAM J. Control Opt.38 (2000) 1623–1638. Zbl0968.49012MR1766433
  21. [21] J.V. Outrata, A note on a class of equilibrium problems with equilibrium constraints. Kybernetika40 (2004) 585–594. Zbl1249.49017MR2120998
  22. [22] J.V. Outrata, M. Kocvara and J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints. Kluwer Academic Publishers, Dordrecht (1998). Zbl0947.90093MR1641213
  23. [23] S.M. Robinson, Some continuity properties of polyhedral multifunctions. Math. Program. Stud.14 (1976) 206–214. Zbl0449.90090MR600130
  24. [24] S.M. Robinson, Strongly regular generalized equations. Math. Oper. Res.5 (1980) 43–62. Zbl0437.90094MR561153
  25. [25] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer, Berlin (1998). Zbl0888.49001MR1491362
  26. [26] V.V. Shanbhag, Decomposition and Sampling Methods for Stochastic Equilibrium Problems. Ph.D. thesis, Stanford University (2005). 
  27. [27] C.-L. Su, Equilibrium Problems with Equilibrium Constraints : Stationarities, Algorithms and Applications. Ph.D. thesis, Stanford University (2005). 
  28. [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. Zbl1088.90042MR1484692

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.