When are Lossy Energy Storage Optimization Models Convex?

We examine a class of optimization problems involving the optimal operation of a single lossy energy storage system, where energy losses occur during charging and discharging. These inefficiencies typically lead to a nonconvex set of feasible charging and discharging power profiles. In this paper, we derive an equivalent reformulation of this class of optimization problems by eliminating the charging and discharging power variables and recasting the problem entirely in terms of the storage state-of-charge variables. We show that the feasible set of the proposed reformulation is always convex. We also provide sufficient conditions under which the objective function of the proposed reformulation is guaranteed to be convex. The conditions provided both unify and generalize many existing conditions for convexity in the literature.