Complementarity and Game-Theoretical Models for Equilibria in Energy Markets: Deterministic and Risk-Averse Formulations

Electricity and natural gas transmission and distribution networks are subject to regulation in price, service quality, and emission limits. The interaction of competing agents in an energy market subject to various regulatory interventions is usually modeled through equilibrium problems that ensure profit maximization for all the agents. These types of models can be written in different manners, for example, by means of mixed complementarity problems, variational inequalities, and game-theoretical formulations. More generally, we consider energy markets both in deterministic and stochastic settings and explore theoretical relations between the various formulations found in the literature and in practice. Our analysis shows that the profit-maximization complementarity model is equivalent to a game with agents minimizing costs if the setting is deterministic or risk neutral. On the other hand, when the agents exhibit risk aversion which is natural in this type of markets, the equivalence no longer holds. This gives rise to an interesting economical interpretation. As a complement to our theoretical study, and for the European natural gas market with deterministic data, we present some numerical results showing the impact of market power on equilibrium prices.