Electricity spot market with transmission losses

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April 2013, 9(2): 275-290. doi: 10.3934/jimo.2013.9.275

Electricity spot market with transmission losses

Didier Aussel ^1, , Rafael Correa ^2, and Matthieu Marechal ^1,

1.	Lab. PROMES, UPR 8521, Université de Perpignan, Perpignan, France, France
2.	Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile

Received May 2011 Revised April 2012 Published February 2013

Abstract
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In order to study deregulated electricity spot markets, various models have been proposed. Most of them correspond to a, so-called, multi-leader-follower game in which an Independent System Operator (ISO) plays a central role. Our aim in this paper is to consider quadratic bid functions together with the transmission losses in the multi-leader-follower game. Under some reasonable assumptions we deduce qualitative properties for the ISO's problem. In the two islands type market, the explicit formulae for the optimal solutions of the ISO's problem are obtained and we show the existence of an equilibrium.

Keywords: generalized Nash equilibrium., transmission loss, Spot market.

Mathematics Subject Classification: Primary: 91A10, 90C26; Secondary: 91B2.

Citation: Didier Aussel, Rafael Correa, Matthieu Marechal. Electricity spot market with transmission losses. Journal of Industrial and Management Optimization, 2013, 9 (2) : 275-290. doi: 10.3934/jimo.2013.9.275

References:

[1]	E. J. Anderson and H. Xu, Supply function equilibrium in electricity spot markets with contracts and price caps, J. Opt. Theory Appl., 124 (2005), 257-283. doi: 10.1007/s10957-004-0924-2.
[2]	R. Baldick, Electricity market equilibrium models: The effect of paramatrization, IEEE Transactions on Power Systems, 17 (2002), 1170-1176.
[3]	R. Baldick and W. Hogan, Capacity constrained supply function equilibriam models of electricity markets: Stability, nondecreasing constraints, and function space iterations, Paper PWP-089, Univ. California Energy Institute, 2001.
[4]	M. Bjørndal and K. Jørnsten, The deregulated electricity market viewed as a bilevel programming problem, J. Global Optim., 33 (2005), 465-475. doi: 10.1007/s10898-004-1939-9.
[5]	F. Bolle, Supply function equilibria and the danger of tacit collution: The case of spot markets for electricity, Energy Economics, 14 (1992), 94-102.
[6]	J. F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour, SIAM Rev., 40 (1998), 228-264. doi: 10.1137/S0036144596302644.
[7]	C. J. Day, B. F. Hobbs and J.-S. Pang, Oligopolistic competition in power networks: A conjectured supply function approach, IEEE Transactions on Power Systems, 17 (2002), 597-607.
[8]	A. Downward, G. Zakeri and A. B. Philpott, On Cournot equilibria in electricity transmission networks, Oper. Res., 58 (2010), 1194-1209. doi: 10.1287/opre.1100.0830.
[9]	J. Eichberger, "Game Theory for Economists," Academic Press, Inc., San Diego, CA, 1993.
[10]	J. F. Escobar and A. Jofré, Equilibrium analysis for a network model, in "Robust Optimization-Directed Design," Nonconvex Optim. Appl., 81, Springer, New York, (2006), 63-72. doi: 10.1007/0-387-28654-3_3.
[11]	J. F. Escobar and A. Jofré, Monopolistic competition in electricity networks with resistance losses, Econom. Theory, 44 (2010), 101-121. doi: 10.1007/s00199-009-0460-2.
[12]	M. Fukushima and J.-S. Pang, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Manag. Sci., 2 (2005), 21-56; [Erratum: Comput. Manag. Sci., 6 (2009), 373-375]. doi: 10.1007/s10287-004-0010-0.
[13]	R. J. Green and D. M. Newbery, Competition in the British electricity spot market, J. of Political Economy, 100 (1992), 929-953.
[14]	R. Henrion, J. Outrata and T. Surowiec, Strong stationary solutions to equilibrium problems with equilibrium constraints with applications to an electricity spot market model, preprint, 2010.
[15]	B. F. Hobbs and J.-S. Pang, Strategic gaming analysis for electric power systems: An MPEC approach, IEEE Trans. Power Systems, 15 (2000), 638-645.
[16]	B. F. Hobbs and J.-S. Pang, Spatial oligopolistic equilibria with arbitrage, shared resources, and price function conjectures, Math. Program. Ser. B, 101 (2004), 57-94. doi: 10.1007/s10107-004-0537-4.
[17]	B. F. Hobbs and J.-S. Pang, Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints, Oper. Research, 55 (2007), 113-127. doi: 10.1287/opre.1060.0342.
[18]	X. Hu and D. Ralph, Using EPECs to model bilevel games in restructured electricity markets with locational prices, Oper. Res., 55 (2007), 809-827. doi: 10.1287/opre.1070.0431.
[19]	P. D. Klemperer and M. A. Meyer, Supply function equilibria in oligopoly under uncertainty, Econometrica, 57 (1989), 1243-1277. doi: 10.2307/1913707.
[20]	R. B. Myerson, "Game Theory. Analysis of Conflict," Harvard University Press, Cambridge, MA, 1991.
[21]	N. K. Nair and L. X. Zhang, SmartGrid: Future networks for New Zealand power systems incorporating distributed generation, Energy Policy, 37 (2009), 3418-3427.
[22]	V. Nanduri and D. K. Das, A survey of critical research areas in the energy segment of restructured electric power markets, Electrical Power and Energy Systems, 31 (2009), 181-191.
[23]	A. Rudkevich, Supply Function Equilibrium in Power Markets: Learning All the Way, TCA Technical Paper, (1999), 1299-1702.
[24]	B. Willems, I. Rumiantseva and H. Weigt, Cournot versus Supply Functions: What does the data tell us?, Energy Economics, 31 (2009), 38-47.

show all references

References:

[1]	E. J. Anderson and H. Xu, Supply function equilibrium in electricity spot markets with contracts and price caps, J. Opt. Theory Appl., 124 (2005), 257-283. doi: 10.1007/s10957-004-0924-2.
[2]	R. Baldick, Electricity market equilibrium models: The effect of paramatrization, IEEE Transactions on Power Systems, 17 (2002), 1170-1176.
[3]	R. Baldick and W. Hogan, Capacity constrained supply function equilibriam models of electricity markets: Stability, nondecreasing constraints, and function space iterations, Paper PWP-089, Univ. California Energy Institute, 2001.
[4]	M. Bjørndal and K. Jørnsten, The deregulated electricity market viewed as a bilevel programming problem, J. Global Optim., 33 (2005), 465-475. doi: 10.1007/s10898-004-1939-9.
[5]	F. Bolle, Supply function equilibria and the danger of tacit collution: The case of spot markets for electricity, Energy Economics, 14 (1992), 94-102.
[6]	J. F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour, SIAM Rev., 40 (1998), 228-264. doi: 10.1137/S0036144596302644.
[7]	C. J. Day, B. F. Hobbs and J.-S. Pang, Oligopolistic competition in power networks: A conjectured supply function approach, IEEE Transactions on Power Systems, 17 (2002), 597-607.
[8]	A. Downward, G. Zakeri and A. B. Philpott, On Cournot equilibria in electricity transmission networks, Oper. Res., 58 (2010), 1194-1209. doi: 10.1287/opre.1100.0830.
[9]	J. Eichberger, "Game Theory for Economists," Academic Press, Inc., San Diego, CA, 1993.
[10]	J. F. Escobar and A. Jofré, Equilibrium analysis for a network model, in "Robust Optimization-Directed Design," Nonconvex Optim. Appl., 81, Springer, New York, (2006), 63-72. doi: 10.1007/0-387-28654-3_3.
[11]	J. F. Escobar and A. Jofré, Monopolistic competition in electricity networks with resistance losses, Econom. Theory, 44 (2010), 101-121. doi: 10.1007/s00199-009-0460-2.
[12]	M. Fukushima and J.-S. Pang, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Manag. Sci., 2 (2005), 21-56; [Erratum: Comput. Manag. Sci., 6 (2009), 373-375]. doi: 10.1007/s10287-004-0010-0.
[13]	R. J. Green and D. M. Newbery, Competition in the British electricity spot market, J. of Political Economy, 100 (1992), 929-953.
[14]	R. Henrion, J. Outrata and T. Surowiec, Strong stationary solutions to equilibrium problems with equilibrium constraints with applications to an electricity spot market model, preprint, 2010.
[15]	B. F. Hobbs and J.-S. Pang, Strategic gaming analysis for electric power systems: An MPEC approach, IEEE Trans. Power Systems, 15 (2000), 638-645.
[16]	B. F. Hobbs and J.-S. Pang, Spatial oligopolistic equilibria with arbitrage, shared resources, and price function conjectures, Math. Program. Ser. B, 101 (2004), 57-94. doi: 10.1007/s10107-004-0537-4.
[17]	B. F. Hobbs and J.-S. Pang, Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints, Oper. Research, 55 (2007), 113-127. doi: 10.1287/opre.1060.0342.
[18]	X. Hu and D. Ralph, Using EPECs to model bilevel games in restructured electricity markets with locational prices, Oper. Res., 55 (2007), 809-827. doi: 10.1287/opre.1070.0431.
[19]	P. D. Klemperer and M. A. Meyer, Supply function equilibria in oligopoly under uncertainty, Econometrica, 57 (1989), 1243-1277. doi: 10.2307/1913707.
[20]	R. B. Myerson, "Game Theory. Analysis of Conflict," Harvard University Press, Cambridge, MA, 1991.
[21]	N. K. Nair and L. X. Zhang, SmartGrid: Future networks for New Zealand power systems incorporating distributed generation, Energy Policy, 37 (2009), 3418-3427.
[22]	V. Nanduri and D. K. Das, A survey of critical research areas in the energy segment of restructured electric power markets, Electrical Power and Energy Systems, 31 (2009), 181-191.
[23]	A. Rudkevich, Supply Function Equilibrium in Power Markets: Learning All the Way, TCA Technical Paper, (1999), 1299-1702.
[24]	B. Willems, I. Rumiantseva and H. Weigt, Cournot versus Supply Functions: What does the data tell us?, Energy Economics, 31 (2009), 38-47.

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