A MPCC-NLP Approach for an Electric Power Market Problem


The electric power market is changing-it has passed from a regulated market, where the government of each country had the control of prices, to a deregulated market economy. Each company competes in order to get more cli.e.nts and maximize its profits. This market is represented by a Stackelberg game with two firms, leader and follower, and the leader anticipates the reaction of the follower. The problem is formulated as a Mathematical Program with Complementarity Constraints (MPCC). It is shown that the constraint qualifications usually assumed to prove convergence of standard algorithms fail to hold for MPCC. To circumvent this, a reformulation for a nonlinear problem (NLP) is proposed. Numerical tests using the NEOS server platform are presented.

Share and Cite:

H. Rodrigues, M. Monteiro and A. Vaz, "A MPCC-NLP Approach for an Electric Power Market Problem," Smart Grid and Renewable Energy, Vol. 1 No. 1, 2010, pp. 54-61. doi: 10.4236/sgre.2010.11009.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] M. Willrich, “Electricity Transmission Policy for America: Enabling a Smart Grid, End to End,” The Electricity Journal, Vol. 22, No. 10, 2009, pp. 77-82.
[2] “Third Benchmarking Report on the Implementation of the Internal Electricity and Gas Markets,” Tech Republic 1, Commission Draft Staff Paper, March 2004.
[3] B. F. Hobbs and F. A. M. Rijkers, “Strategic Generation with Conjectured Transmission Price Responses in a Mixed Transmission Pricing System,” IEEE Transmissions on Power Systems, Vol. 19, No. 2, 2004, pp. 707-879.
[4] Y. Smeers, “How Well Can Measure Market Power in Restructured Electricity Systems,” Tech Republic, University Catholique De Louvain, November 2004.
[5] S. A. Gabriel and F. U. Leuthold, “Solving Discretely- Constrained MPEC Problems with Applications in Electric Power Markets,” Energy Economics, Vol. 32, 2010, pp. 3-14.
[6] M. Crappe, “Electric Power Systems,” Wiley-ISTE, 2009.
[7] C. B. Metzler, “Complementarity Models of Competitive Oligopolistic Electric Power Generation Markets,” Master’s Thesis, The Johns Hopkins University, 2000.
[8] Z.-Q. Luo, J.-S. Pang and D. Ralph, “Mathematical Programs with Equilibrium Constraints,” Cambridge University Press, Cambridge, 1996.
[9] L. N. Vicente and P. H. Calamai, “Bilevel and Multilevel Programming: a Bibliography Review,” Journal of Global Optimization, Vol. 5, 1994, pp. 291-306.
[10] S. Dempe, “Annotated Bibliography on Bilevel Program- ming and Mathematical Programs with Equilibrium Con- straints,” Optimization, Vol. 52, No. 3, 2003, pp. 333-359.
[11] M. C. Ferris and J. S. Pang, “Engineering and Economic applications of Complementarity Problems,” SIAM Review, Vol. 39, No. 4, 1997, pp. 669-713.
[12] M. F. Ferris and F. Tin-Loi, “Limit Analysis of Frictional Block Assemblies as a Mathematical Program with Com- plementarity Constraints,” International Journal of Mech- anical Sciences, Vol. 43, 2001, pp. 209-224.
[13] J. S. Pang, “Complementarity Problems,” In: Publishers, K.A., Ed., Handbook in Global Optimization, R. Horst and P. Pardalos, Boston, 1994.
[14] J. F. Rodrigues, “Obstacle Problems in Mathematics Phy- Sics,” Elsevier Publishing Company, Amsterdam, 1987.
[15] M. C. Ferris and F. Tin-Loi, “On the Solution of a Minimum Weight Elastoplastic Problem Involving Dis- placement and Complementarity Constraints,” Computer Methods in Applied Mechanics and Engineering, Vol. 174, 1999, pp. 107-120.
[16] K. P. Oh and P. K. Goenka, “The Elastohydrodynamic Solution of Journal Bearings under Dynamic Loading,” Transactions of ASME, Vol. 107 ,1985, pp. 389-395.
[17] A. U. Raghunathan and L. T. Biegler, “Mathematical Pro- grams with Equilibrium Constraints (Mpecs) in Process Engineering,” Tech Republic, CMU Chemical Engine- ering November 2002.
[18] O. Drissi-Kaitouni and J. Lundgren, “Bilivel Origin-Dest- ination Matrix Estimation Using a Descent Approach,” Tech Republic LITH-MAT-R-1992-49, Linköping Institute of Technology, Department of Mathematics, Sweden, 1992.
[19] M. Stohr, “Nonsmooth Trust Region Methods and their Applications to Mathematical Programs with Equilibrium Constraints,” Shaker Verlag, Aachen, 2000.
[20] C. Ballard, D. Fullerton, J. B. Shoven and J. Whalley, “A General Equilibrium Modelfor Tax Policy Evaluation,” National Bureau of Economic Research Monograph, Chicago, 1985.
[21] H. Ehtamo and T. Raivio, “On Aapplied Nonlinear and Bilevel Programming for Pursuit-Evasion Games,” Journal of Optimization Theory and Applications, Vol. 108, 2001, pp. 65-96.
[22] J. V. Outrata, “On Necessary Optimal Conditions for Stackelberg Problems,” Journal of Optimization Theory and Applications, Vol. 76, 1993, pp. 305-320.
[23] J. C. Carbone, C. Helm and T. F. Rutherford, “Coalition formation and International Trade in Greenhouse Gas Emission Rights,” October 2003.
[24] G. Hibino, M. Kainuma and Y. Matsuoka, “Two-Level Mathematical Programming for Analysing Subsidy Options to Reduce Greenhouse-Gas Emissions,” Tech Republic, Laxenburg ,1996.
[25] M. Kocvara and J. V. Outrata, “Optimization Problems with Equilibrium Constraints and Their Numerical Solution,” Mathematical Programs, Series B, Vol. 101, 2004, pp. 119-149.
[26] H. S. Rodrigues and M. T. Monteiro, “Solving Mathe- matical Program with Complementarity Constraints (MPCC) with Nonlinear Solvers,” Recent Advances in Optimization, Lectures Notes in Economics and Mathe- matical Systems.
[27] J. F. Bard, “Convex Two-Level Optimization,” Mathema- tical Programming, Vol. 40, No. 1, 1988, pp. 15-27.
[28] J. Outrata, M. Kocvara and J. Zowe, “Nonsmooth App- roach to Optimization Problems with Equilibrium Cons- traints,” Kluwer Academic Publishers, Dordrecht, 1998.
[29] M. C. Ferris, S. P. Dirkse and A. Meeraus, “Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimi- zation,” Tech Republic, Oxford University Computing Laboratory, July 2002.
[30] H. Scheel and S. Scholtes, “Mathematical Programs with Complementarity Constraints: Stationarity, Optimality and Sensitivity,” Mathematics of Operations Research, Vol. 25, 2000, pp. 1-22.
[31] S. Scholtes, “Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints,” SIAM Journal Optmization, Vol. 11, No. 4, 2001, pp. 918-936.
[32] B. F. Hobbs, C. B. Metzler and J.-S. Pang, “Strategic Gaming Analysis for Electric Power Systems: An MPEC Approach,” IEEE Transactions on Power Systems, Vol. 15,No. 2, 2000, pp. 638-645.
[33] V. Krishna and V. C. Ramesh, “Intelligent agents in Neg- otIations in Market Games, Part 2: Application,” IEEE Transactions on Power Systems Vol. 13, No. 3, 1998, pp. 1109-1114.
[34] O. Alsa and B. Scott, “Optimal Load Flow with Steady- State Security,” IEEE Transactions on Power Systems, Vol. 93, No. 3, 1973, pp. 745-751.
[35] S. Leyffer, Macmpec, 2010. www.mcs.nl.gov/leyffer/ macmpec
[36] Lancelot, 2010. www.numerical.rl.ac.uk/lancelot/blurb.html
[37] H. Benson, A. Sen, D. Shano and R. Vanderbei, “Interior- Point Algorithms Penalty Methods and Equilibrium Prob- lems,” Tech Republic, Operations Research and Financial Engineering Princeton University, October 2003.
[38] NEOS, 2010. www-neos.mcs.ang.gov/neos

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.