Optimal Installation of Solar Panels with Price Impact: a Solvable Singular Stochastic Control Problem

Abstract: We consider a price-maker company which generates electricity and sells it in the spot market. The company can increase its level of installed power by irreversible installations of solar panels. In absence of the company's economic activities, the spot electricity price evolves as an Ornstein-Uhlenbeck process, and therefore it has a mean-reverting behavior. The current level of the company's installed power has a permanent impact on the electricity price and affects its mean-reversion level. The company aims at maximizing the total expected profits from selling electricity in the market, net of the total expected proportional costs of installation. This problem is modeled as a two-dimensional degenerate singular stochastic control problem in which the installation strategy is identified as the company's control variable. We follow a guess-and-verify approach to solve the problem. We find that the optimal installation strategy is triggered by a curve which separates the waiting region, where it is not optimal to install additional panels, and the installation region, where it is. Such a curve depends on the current level of the company's installed power, and is the unique strictly increasing function which solves a first-order ordinary differential equation (ODE). Finally, our study is complemented by a numerical analysis of the dependency of the optimal installation strategy on the model's parameters.