Nash-Cournot Equilibria in Power Markets on a Linearized DC Network with Arbitrage: Formulations and Properties

Abstract

Extending a prior arbitrage-free model of Hobbs (2001), this article presents two models of an electric power market with arbitrage on a linearized DC network with a affine price functions. The two models represent a decentralized system involving bilateral contracts between producers and consumers in which the system operator's role is limited to providing transmission services. The two models differ in how arbitrage is handled. In the first model, the producers anticipate the effect of arbitrage upon prices at different locations (Stackelberg assumption), and therefore treat the arbitrage amounts as decision variables in their profit maximization problems. In the second model, the firms take the arbitrage quantities as inputs in their problems (Cournot assumption), and the arbitrager solves a separate profit maximization problem that takes the electricity prices and the transmission costs as inputs. In each model, we adopt a Nash-Cournot equilibrium as the solution concept for the game among producers. We show that the resulting equilibrium problems can be formulated as monotone mixed linear complementarity problems. Based on such a formulation, we obtain existence,uniqueness,and various quantitative properties of the equilibrium solutions to the models. It is also demonstrated that these two models of a bilateral market yield the same prices, producer outputs, and profits as a model of Cournot competition in a “Poolco” system,in which a system operator runs a centralized auction and buys all production, and then resells it to consumers. This result implies that Cournot competition among producers yields the same outcomes for two distinct market designs. Finally, we present a numerical example to illustrate the theoretical results.