| Title:
|
On $M$-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling (English) |
| Author:
|
Henrion, René |
| Author:
|
Römisch, Werner |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
52 |
| Issue:
|
6 |
| Year:
|
2007 |
| Pages:
|
473-494 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
electricity markets |
| Keyword:
|
bidding |
| Keyword:
|
noncooperative games |
| Keyword:
|
equilibrium constraint |
| Keyword:
|
EPEC |
| Keyword:
|
optimality condition |
| Keyword:
|
co-derivative |
| Keyword:
|
random demand |
| MSC:
|
90C15 |
| MSC:
|
91B26 |
| MSC:
|
91B52 |
| idZBL:
|
Zbl 1164.90379 |
| idMR:
|
MR2357576 |
| DOI:
|
10.1007/s10492-007-0028-z |
| . |
| Date available:
|
2009-09-22T18:31:28Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134691 |
| . |
| Reference:
|
[1] B. Blaesig: Risikomanagement in der Stromerzeugungs- und Handelsplanung. Aachener Beiträge zur Energieversorgung, Band 113.Klinkenberg, Aachen, 2007. |
| Reference:
|
[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. MR 1416530, 10.1137/S1052623495284029 |
| Reference:
|
[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. |
| Reference:
|
[4] J. F. Escobar, A. Jofre: Oligopolistic competition in electricity spot markets, Dec. 2005.Available at http://ssrn.com/abstract=878762. |
| Reference:
|
[5] F. Facchinei, J.-S. Pang: Finite-dimensional Variational Inequalities and Complementarity Problems, Vol. I and II.Springer-Verlag, New York, 2003. MR 1955648 |
| Reference:
|
[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. MR 2219153, 10.1137/S1052623402407382 |
| Reference:
|
[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. 10.1016/j.ejor.2005.03.037 |
| Reference:
|
[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. MR 2290891, 10.1287/opre.1060.0342 |
| Reference:
|
[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. 10.1109/59.867153 |
| Reference:
|
[10] X. Hu, D. Ralph: Using EPECs to model bilevel games in restructured electricity markets with locational prices.Optimization Online, 2006 (www.optimization-online.org). MR 2360950 |
| Reference:
|
[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. |
| Reference:
|
[12] M. Kočvara, J. V. Outrata: Optimization problems with equilibrium constraints and their numerical solution.Math. Program. B 101 (2004), 119–149. MR 2085261, 10.1007/s10107-004-0539-2 |
| Reference:
|
[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.. |
| Reference:
|
[14] Z.-Q. Luo, J.-S. Pang, and D. Ralph: Mathematical Programs with Equilibrium Constraints.Cambridge University Press, Cambridge, 1997. MR 1419501 |
| Reference:
|
[15] B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. Vol. 1: Basic Theory, Vol. 2: Applications.Springer-Verlag, Berlin, 2006. MR 2191745 |
| Reference:
|
[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. MR 2338444, 10.1137/060665609 |
| Reference:
|
[17] J. V. Outrata: A note on a class of equilibrium problems with equilibrium constraints.Cybernetica 40 (2004), 585–594. Zbl 1249.49017, MR 2120998 |
| Reference:
|
[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 |
| Reference:
|
[19] J. V. Outrata, M. Kočvara, and J. Zowe: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints.Kluwer, Dordrecht, 1998. |
| Reference:
|
[20] J.-S. Pang, M. Fukushima: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games.Comput. Manag. Sci. 1 (2005), 21–56. MR 2164953, 10.1007/s10287-004-0010-0 |
| Reference:
|
[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. 10.1109/TPWRS.2004.840397 |
| Reference:
|
[22] D. Ralph, Y. Smeers: EPECs as models for electricity markets.Power Systems Conference and Exposition (PSCE), Atlanta, 2006, , , . |
| Reference:
|
[23] R. T. Rockafellar, R. J.-B. Wets: Variational Analysis.Springer-Verlag, Berlin, 1998. MR 1491362 |
| Reference:
|
[24] : Stochastic Programming. Handbooks in Operations Research and Management Science, Vol. 10.A. Ruszczyński, A. Shapiro (eds.), Elsevier, Amsterdam, 2003. MR 2051791 |
| Reference:
|
[25] H. Scheel, S. Scholtes: Mathematical programs with equilibrium constraints: Stationarity, optimality and sensitivity.Math. Oper. Res. 25 (2000), 1–22. MR 1854317, 10.1287/moor.25.1.1.15213 |
| Reference:
|
[26] A. Shapiro: Stochastic programming with equilibrium constraints.J. Optim. Theory Appl. 128 (2006), 221–243. Zbl 1130.90032, MR 2201897, 10.1007/s10957-005-7566-x |
| Reference:
|
[27] A. Shapiro, H. Xu: Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation.Optimization Online 2005 (www.optimization-online.org). MR 2412074 |
| Reference:
|
[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. |
| . |
