American Institute of Mathematical Sciences

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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

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 & 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.  doi: 10.1007/s10957-004-0924-2.  Google Scholar

[2]

R. Baldick, Electricity market equilibrium models: The effect of paramatrization,, IEEE Transactions on Power Systems, 17 (2002), 1170.   Google Scholar

[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, (2001).   Google Scholar

[4]

M. Bjørndal and K. Jørnsten, The deregulated electricity market viewed as a bilevel programming problem,, J. Global Optim., 33 (2005), 465.  doi: 10.1007/s10898-004-1939-9.  Google Scholar

[5]

F. Bolle, Supply function equilibria and the danger of tacit collution: The case of spot markets for electricity,, Energy Economics, 14 (1992), 94.   Google Scholar

[6]

J. F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour,, SIAM Rev., 40 (1998), 228.  doi: 10.1137/S0036144596302644.  Google Scholar

[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.   Google Scholar

[8]

A. Downward, G. Zakeri and A. B. Philpott, On Cournot equilibria in electricity transmission networks,, Oper. Res., 58 (2010), 1194.  doi: 10.1287/opre.1100.0830.  Google Scholar

[9]

J. Eichberger, "Game Theory for Economists,", Academic Press, (1993).   Google Scholar

[10]

J. F. Escobar and A. Jofré, Equilibrium analysis for a network model,, in, 81 (2006), 63.  doi: 10.1007/0-387-28654-3_3.  Google Scholar

[11]

J. F. Escobar and A. Jofré, Monopolistic competition in electricity networks with resistance losses,, Econom. Theory, 44 (2010), 101.  doi: 10.1007/s00199-009-0460-2.  Google Scholar

[12]

M. Fukushima and J.-S. Pang, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,, Comput. Manag. Sci., 2 (2005), 21.  doi: 10.1007/s10287-004-0010-0.  Google Scholar

[13]

R. J. Green and D. M. Newbery, Competition in the British electricity spot market,, J. of Political Economy, 100 (1992), 929.   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10107-004-0537-4.  Google Scholar

[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.  doi: 10.1287/opre.1060.0342.  Google Scholar

[18]

X. Hu and D. Ralph, Using EPECs to model bilevel games in restructured electricity markets with locational prices,, Oper. Res., 55 (2007), 809.  doi: 10.1287/opre.1070.0431.  Google Scholar

[19]

P. D. Klemperer and M. A. Meyer, Supply function equilibria in oligopoly under uncertainty,, Econometrica, 57 (1989), 1243.  doi: 10.2307/1913707.  Google Scholar

[20]

R. B. Myerson, "Game Theory. Analysis of Conflict,", Harvard University Press, (1991).   Google Scholar

[21]

N. K. Nair and L. X. Zhang, SmartGrid: Future networks for New Zealand power systems incorporating distributed generation,, Energy Policy, 37 (2009), 3418.   Google Scholar

[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.   Google Scholar

[23]

A. Rudkevich, Supply Function Equilibrium in Power Markets: Learning All the Way,, TCA Technical Paper, (1999), 1299.   Google Scholar

[24]

B. Willems, I. Rumiantseva and H. Weigt, Cournot versus Supply Functions: What does the data tell us?,, Energy Economics, 31 (2009), 38.   Google Scholar

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.  doi: 10.1007/s10957-004-0924-2.  Google Scholar

[2]

R. Baldick, Electricity market equilibrium models: The effect of paramatrization,, IEEE Transactions on Power Systems, 17 (2002), 1170.   Google Scholar

[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, (2001).   Google Scholar

[4]

M. Bjørndal and K. Jørnsten, The deregulated electricity market viewed as a bilevel programming problem,, J. Global Optim., 33 (2005), 465.  doi: 10.1007/s10898-004-1939-9.  Google Scholar

[5]

F. Bolle, Supply function equilibria and the danger of tacit collution: The case of spot markets for electricity,, Energy Economics, 14 (1992), 94.   Google Scholar

[6]

J. F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour,, SIAM Rev., 40 (1998), 228.  doi: 10.1137/S0036144596302644.  Google Scholar

[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.   Google Scholar

[8]

A. Downward, G. Zakeri and A. B. Philpott, On Cournot equilibria in electricity transmission networks,, Oper. Res., 58 (2010), 1194.  doi: 10.1287/opre.1100.0830.  Google Scholar

[9]

J. Eichberger, "Game Theory for Economists,", Academic Press, (1993).   Google Scholar

[10]

J. F. Escobar and A. Jofré, Equilibrium analysis for a network model,, in, 81 (2006), 63.  doi: 10.1007/0-387-28654-3_3.  Google Scholar

[11]

J. F. Escobar and A. Jofré, Monopolistic competition in electricity networks with resistance losses,, Econom. Theory, 44 (2010), 101.  doi: 10.1007/s00199-009-0460-2.  Google Scholar

[12]

M. Fukushima and J.-S. Pang, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,, Comput. Manag. Sci., 2 (2005), 21.  doi: 10.1007/s10287-004-0010-0.  Google Scholar

[13]

R. J. Green and D. M. Newbery, Competition in the British electricity spot market,, J. of Political Economy, 100 (1992), 929.   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10107-004-0537-4.  Google Scholar

[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.  doi: 10.1287/opre.1060.0342.  Google Scholar

[18]

X. Hu and D. Ralph, Using EPECs to model bilevel games in restructured electricity markets with locational prices,, Oper. Res., 55 (2007), 809.  doi: 10.1287/opre.1070.0431.  Google Scholar

[19]

P. D. Klemperer and M. A. Meyer, Supply function equilibria in oligopoly under uncertainty,, Econometrica, 57 (1989), 1243.  doi: 10.2307/1913707.  Google Scholar

[20]

R. B. Myerson, "Game Theory. Analysis of Conflict,", Harvard University Press, (1991).   Google Scholar

[21]

N. K. Nair and L. X. Zhang, SmartGrid: Future networks for New Zealand power systems incorporating distributed generation,, Energy Policy, 37 (2009), 3418.   Google Scholar

[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.   Google Scholar

[23]

A. Rudkevich, Supply Function Equilibrium in Power Markets: Learning All the Way,, TCA Technical Paper, (1999), 1299.   Google Scholar

[24]

B. Willems, I. Rumiantseva and H. Weigt, Cournot versus Supply Functions: What does the data tell us?,, Energy Economics, 31 (2009), 38.   Google Scholar

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