Numerical Solutions to Nash–Cournot Equilibria in Coupled Constraint Electricity Markets

Abstract

Abstract-A numerical method based on a relaxation algorithm and the Nikaido-Isoda function is presented for the calculation of Nash-Cournot equilibria in electricity markets. Nash equilibrium is attained through a relaxation procedure applied to an objective function, the Nikaido-Isoda function, which is derived from the existing profit maximization functions calculated by the generating companies. We also show how to use the relaxation algorithm to compute, and enforce, a coupled constraint equilibrium, which occurs if regulatory, generation, and distribution (and more) restrictions are placed on the companies and entire markets. Moreover, we use the relaxation algorithm to compute players' payoffs under several player configurations. This is needed for the solution of our game under cooperative game theory concepts, such as the bilateral Shapley value and the kernel. We show that the existence of both depends critically on demand price elasticity. The numerical method converges to a unique solution under rather specific but plausible concavity conditions. A case study from the IEEE 30-bus system, and a three-bus bilateral market example with a dc model of the transmission line constraints are presented and discussed.

An evolutionary algorithm based on Nash Dominance for Equilibrium Problems with Equilibrium Constraints

“…This function as given in Eqn. 3 is a mathematical tool that plays a key role in the study of NE problems [5], [12], [20], [24]. Consider again two strategy profiles {x, y} ∈ S, then the interpretation of Ψ(x, y) is as follows: each summand shows the increase in payoff a player will receive by unilaterally deviating and playing a strategy y while other players play according to x.…”

Towards an understanding of tradeoffs between regional wealth, tightness of a common environmental constraint and the sharing rules

“…We see that Wallonia's usage of energy per worker is the highest and the revenue generated by a unit of labour is the lowest. 4 In the rest of this paper we will propose and analyse a few models to help politicians decide about the energy sharing rules that could lead to a socially preferred equilibrium. 5 …”

“…If, in addition, the amount of energy to be used by the entire country is restricted 7 , a constraint needs to be added to the agents' joint strategy space. In result, the input and output will be 4 The last column of Table II provides a rather complex and non standard measure of "efficiency", which is the revenue (in Euros) obtained out of one boe by one unit of labour. Here, Wallonia may appear more "economic" than the rest of the country in such per unit of labour terms.…”

“…Many electricity market and environmental problems belong to this class see e.g., [9], [10], [4] where this concept has been applied in microeconomic contexts; see [7] for an international economics application. In general, CCE allows modelling of a situation in which the actions of one player condition how 'big' the actions of other players can be.…”

“…The problem's setup in this paper is conceptually similar to that of [9], [10], [15], [4], [12], [18], [7] and also [5]. The common feature is that all those papers deal with coupled constraints games, in which competitive agents maximise their utility functions subject to constraints upon their joint strategy space.…”