Existence of Energy Market Equilibria with Convex and Nonconvex Players

Motivated by examples from the energy sector, we consider a special type of market equilibrium problems (MEPs) involving players both with convex and nonconvex strategy spaces or objective functions. Such problems naturally occur in settings in which energy trading is combined with aspects of the actual energy transport through networks. We propose an algorithm that determines if an equilibrium of a given MEP exists and that computes a market equilibrium in case of existence. Moreover, we provide both a uniqueness and a non-existence result for MEPs that include players with a unique best response. Finally, we test the proposed algorithm on well-known energy market instances from the power and gas literature. There, nonconvexities - either due to integrality conditions or nonlinearities - mainly arise from considering the transmission system operator as an additional player who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that an equilibrium often exists, especially for the case of continuous nonconvexities in the context of gas market problems.