A Stochastic Control Model for Electricity Producers

Abstract

Modern electricity pricing models include a strong reversion to a long run mean and a number of non-local operators to encapsulate the discontinuous price behavior observed in such markets. However, incorporating non-local processes into a stochastic control problem presents significant analytical challenges. The motivation for this work is to solve the problem of optimal control of the burn rate for a coal-powered electricity plant. We first construct a pricing model that is a good general representative of the class of models currently used for electricity pricing as well as a model for the supply of fuel to the plant. Under this model, we state the control problem of maximizing the expected discounted revenue until the first time at which the plant runs out of fuel. Deriving the HJB equation for this control problem results in a partial integro-differential equation, which does not t the classical theory of viscosity solutions. Building o of work by Barles and Imbert on viscosity solutions for non-local processes, we extend their theory to apply to non-local processes which also include a mean-reversion component. We first show that the value function for the control problem is a solution to this HJB equation. In our main result, we prove a comparison principle for viscosity solutions which uses a slightly more regular structure of the non-local operators to relax some of the assumptions of Barles and Imbert. Using this comparison principle, we are able to show that the value function is in fact the unique solution to the HJB equation. Thus, we have the desired result that solving the HJB equation is equivalent to solving the control problem, giving us a direct method for finding the optimal control policy for the electricity producer.