On Nash equilibria in duopolistic power markets subject to make-whole uplift

We consider a duopolistic power market in which firms bid their offer functions while abiding by capacity constraints. Minimal cost allocation decisions and marginal cost prices are determined through the solution of two optimization problems, the first a mixed-integer linear program (the unit commitment problem) and the second a related linear program (the economic dispatch problem) that has constraints derived from the solution of the commitmemt problem. Traditional marginal-cost pricing mechanisms have relied on using the dual variables associated with the supply-demand constraints from the dispatch problem. Unfortunately, such mechanisms fail to cover the cost of start-up and energy costs faced by dispatched generators. In existing markets, one attempt at a remedy is by providing every dispatched generator with an expost out of market settlement called a make-whole uplift. Given the presence of such an uplift mechanism, this paper seeks to characterize the nature of equilibria in duopolistic settings. The two main conclusions of our analysis are: 1) Pure Nash equilibria are identified when one firm can supply the entire demand; 2) Otherwise, it is shown that mixed-strategy Nash equilibria exist, and these equilibria can be characterized by two coupled differential equations. These results provide a framework for further analysis of electricity markets, and in particular the true value of make-whole uplift payments in a strategic setting.