American Institute of Mathematical Sciences

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July  2015, 11(3): 985-998. doi: 10.3934/jimo.2015.11.985

Electricity day-ahead markets: Computation of Nash equilibria

Margarida Carvalho 1, , João Pedro Pedroso 1, and  João Saraiva 2,

1. 

Faculdade de Ciências, Universidade do Porto and INESC TEC, Rua do Campo Alegre, 4169-007 Porto, Portugal, Portugal
2. 

Faculdade de Engenharia, Universidade do Porto and INESC TEC, Rua Dr. Roberto Frias, 4200 - 465 Porto, Portugal

Received  May 2012 Revised  June 2014 Published  October 2014

In a restructured electricity sector, day-ahead markets can be modeled as a game where some players - the producers - submit their proposals. To analyze the companies' behavior we have used the concept of Nash equilibrium as a solution in these multi-agent interaction problems. In this paper, we present new and crucial adaptations of two well-known mechanisms, the adjustment process and the relaxation algorithm, in order to achieve the goal of computing Nash equilibria. The advantages of these approaches are highlighted and compared with those available in the literature.
Keywords: Electricity market, Nash equilibria, adjustment process, combinatorial optimization, auctions/bidding., relaxation algorithm.
Mathematics Subject Classification: Primary: 91A80; Secondary: 90C27, 65K05, 90C5.
Citation: Margarida Carvalho, João Pedro Pedroso, João Saraiva. Electricity day-ahead markets: Computation of Nash equilibria. Journal of Industrial and Management Optimization, 2015, 11 (3) : 985-998. doi: 10.3934/jimo.2015.11.985
References:
[1]

P. Bajpai and S. N. Singh, Fuzzy adaptive particle swarm optimization for bidding strategy in uniform price spot market, Power Systems, IEEE Transactions on, 22 (2007), 2152-2160. doi: 10.1109/TPWRS.2007.907445.

[2]

A. G. Bakirtzis, N. P. Ziogos, A. C. Tellidou and G. A. Bakirtzis, Electricity producer offering strategies in day-ahead energy market with step-wise offers, Power Systems, IEEE Transactions on, 22 (2007), 1804-1818. doi: 10.1109/TPWRS.2007.907536.

[3]

T. Barforoushi, M. P. Moghaddam, M. H. Javidi and M. K. Sheikh-El-Eslami, Evaluation of regulatory impacts on dynamic behavior of investments in electricity markets: A new hybrid dp/game framework, Power Systems, IEEE Transactions on, 25 (2010), 1978-1986. doi: 10.1109/TPWRS.2010.2049034.

[4]

L. A. Barroso, R. D. Carneiro, S. Granville, M. V. Pereira and M. H. C. Fampa, Nash equilibrium in strategic bidding: A binary expansion approach, Power Systems, IEEE Transactions on, 21 (2006), 629-638. doi: 10.1109/TPWRS.2006.873127.

[5]

M. Carvalho, J. P. Pedroso and J. Saraiva, Nash equilibria in electricity markets, In The Proceedings of the VII ALIO/EURO Workshop on Applied Combinatorial Optimization,, pages 153-156, Porto, Portugal, May 2011.

[6]

A. Conejo and F. Prieto, Mathematical programming and electricity markets, TOP, 9 (2001), 1-54. doi: 10.1007/BF02579062.

[7]

J. Contreras, M. Klusch and J. B. Krawczyk, Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets, Power Systems, IEEE Transactions on, 19 (2004), 195-206. doi: 10.1109/TPWRS.2003.820692.

[8]

F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR: A Quarterly Journal of Operations Research, 5 (2007), 173-210. doi: 10.1007/s10288-007-0054-4.

[9]

D. Fudenberg and J. Tirole, Game Theory, MIT Press, Cambridge, MA, 5th edition, 1996.

[10]

B. A. Gomes, Simulador dos operadores de mercado e sistema num mercado de energia eléctrica considerando restriçøes intertemporais, Master's thesis, Faculdade de Engenharia da Universidade do Porto, Portugal, 2005.

[11]

E. Hasan and F. D. Galiana, Fast computation of pure strategy Nash equilibria in electricity markets cleared by merit order, Power Systems, IEEE Transactions on, 25 (2010), 722-728. doi: 10.1109/TPWRS.2009.2037153.

[12]

B. F. Hobbs, C. B. Metzler and J. -S. Pang, Strategic gaming analysis for electric power systems: An MPEC approach, Power Systems, IEEE Transactions on, 15 (2000), 638-645. doi: 10.1109/59.867153.

[13]

J. Krawczyk and J. Zuccollo, Nira-3: An improved Matlab package for finding Nash equilibria in infinite games, Computational Economics, 5 (2006), 24644. URL http://mpra.ub.uni-muenchen.de/1119/.

[14]

K. Lee and R. Baldick, Tuning of discretization in bimatrix game approach to power system market analysis, Power Engineering Review, IEEE, 22 (2002), p55. doi: 10.1109/MPER.2002.4311811.

[15]

M. V. Pereira, S. Granville, M. H. C. Fampa, R. Dix and L. A. Barroso, Strategic bidding under uncertainty: A binary expansion approach, Power Systems, IEEE Transactions on, 20 (2005), 180-188. doi: 10.1109/TPWRS.2004.840397.

[16]

D. Pozo and J. Contreras, Finding multiple Nash equilibria in pool-based markets: A stochastic EPEC approach, Power Systems, IEEE Transactions on, 26 (2011), 1744-1752. doi: 10.1109/TPWRS.2010.2098425.

[17]

D. Pozo, J. Contreras, Á. Caballero and A. de Andrés, Long-term Nash equilibria in electricity markets, Electric Power Systems Research, 81 (2011), 329-339. doi: 10.1016/j.epsr.2010.09.008.

[18]

J. Saraiva, J. P. da Silva and M. P. de Leão, Mercados de Electricidade - Regulaçcão de Tarifação de Uso das Redes, FEUPedições, 2002.

[19]

Y. S. Son and R. Baldick, Hybrid coevolutionary programming for Nash equilibrium search in games with local optima, Evolutionary Computation, IEEE Transactions on, 8 (2004), 305-315. doi: 10.1109/TEVC.2004.832862.

[20]

A. Vaz and L. Vicente, A particle swarm pattern search method for bound constrained global optimization, Journal of Global Optimization, 39 (2007), 197-219. doi: 10.1007/s10898-007-9133-5.

show all references

References:
[1]

P. Bajpai and S. N. Singh, Fuzzy adaptive particle swarm optimization for bidding strategy in uniform price spot market, Power Systems, IEEE Transactions on, 22 (2007), 2152-2160. doi: 10.1109/TPWRS.2007.907445.

[2]

A. G. Bakirtzis, N. P. Ziogos, A. C. Tellidou and G. A. Bakirtzis, Electricity producer offering strategies in day-ahead energy market with step-wise offers, Power Systems, IEEE Transactions on, 22 (2007), 1804-1818. doi: 10.1109/TPWRS.2007.907536.

[3]

T. Barforoushi, M. P. Moghaddam, M. H. Javidi and M. K. Sheikh-El-Eslami, Evaluation of regulatory impacts on dynamic behavior of investments in electricity markets: A new hybrid dp/game framework, Power Systems, IEEE Transactions on, 25 (2010), 1978-1986. doi: 10.1109/TPWRS.2010.2049034.

[4]

L. A. Barroso, R. D. Carneiro, S. Granville, M. V. Pereira and M. H. C. Fampa, Nash equilibrium in strategic bidding: A binary expansion approach, Power Systems, IEEE Transactions on, 21 (2006), 629-638. doi: 10.1109/TPWRS.2006.873127.

[5]

M. Carvalho, J. P. Pedroso and J. Saraiva, Nash equilibria in electricity markets, In The Proceedings of the VII ALIO/EURO Workshop on Applied Combinatorial Optimization,, pages 153-156, Porto, Portugal, May 2011.

[6]

A. Conejo and F. Prieto, Mathematical programming and electricity markets, TOP, 9 (2001), 1-54. doi: 10.1007/BF02579062.

[7]

J. Contreras, M. Klusch and J. B. Krawczyk, Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets, Power Systems, IEEE Transactions on, 19 (2004), 195-206. doi: 10.1109/TPWRS.2003.820692.

[8]

F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR: A Quarterly Journal of Operations Research, 5 (2007), 173-210. doi: 10.1007/s10288-007-0054-4.

[9]

D. Fudenberg and J. Tirole, Game Theory, MIT Press, Cambridge, MA, 5th edition, 1996.

[10]

B. A. Gomes, Simulador dos operadores de mercado e sistema num mercado de energia eléctrica considerando restriçøes intertemporais, Master's thesis, Faculdade de Engenharia da Universidade do Porto, Portugal, 2005.

[11]

E. Hasan and F. D. Galiana, Fast computation of pure strategy Nash equilibria in electricity markets cleared by merit order, Power Systems, IEEE Transactions on, 25 (2010), 722-728. doi: 10.1109/TPWRS.2009.2037153.

[12]

B. F. Hobbs, C. B. Metzler and J. -S. Pang, Strategic gaming analysis for electric power systems: An MPEC approach, Power Systems, IEEE Transactions on, 15 (2000), 638-645. doi: 10.1109/59.867153.

[13]

J. Krawczyk and J. Zuccollo, Nira-3: An improved Matlab package for finding Nash equilibria in infinite games, Computational Economics, 5 (2006), 24644. URL http://mpra.ub.uni-muenchen.de/1119/.

[14]

K. Lee and R. Baldick, Tuning of discretization in bimatrix game approach to power system market analysis, Power Engineering Review, IEEE, 22 (2002), p55. doi: 10.1109/MPER.2002.4311811.

[15]

M. V. Pereira, S. Granville, M. H. C. Fampa, R. Dix and L. A. Barroso, Strategic bidding under uncertainty: A binary expansion approach, Power Systems, IEEE Transactions on, 20 (2005), 180-188. doi: 10.1109/TPWRS.2004.840397.

[16]

D. Pozo and J. Contreras, Finding multiple Nash equilibria in pool-based markets: A stochastic EPEC approach, Power Systems, IEEE Transactions on, 26 (2011), 1744-1752. doi: 10.1109/TPWRS.2010.2098425.

[17]

D. Pozo, J. Contreras, Á. Caballero and A. de Andrés, Long-term Nash equilibria in electricity markets, Electric Power Systems Research, 81 (2011), 329-339. doi: 10.1016/j.epsr.2010.09.008.

[18]

J. Saraiva, J. P. da Silva and M. P. de Leão, Mercados de Electricidade - Regulaçcão de Tarifação de Uso das Redes, FEUPedições, 2002.

[19]

Y. S. Son and R. Baldick, Hybrid coevolutionary programming for Nash equilibrium search in games with local optima, Evolutionary Computation, IEEE Transactions on, 8 (2004), 305-315. doi: 10.1109/TEVC.2004.832862.

[20]

A. Vaz and L. Vicente, A particle swarm pattern search method for bound constrained global optimization, Journal of Global Optimization, 39 (2007), 197-219. doi: 10.1007/s10898-007-9133-5.

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