Tight MIP Formulations for Optimal Operation and Investment of Storage Including Reserves

Fast and accurate large-scale energy system models are needed to investigate the potential of storage to complement the fluctuating energy production of renewable energy systems. However, standard Mixed-Integer Programming (MIP) models that describe optimal investment and operation of these storage units, including the optional capacity to provide up/down reserves, do not scale well. To improve scalability, the integrality constraints are often relaxed, resulting in Linear Programming (LP) relaxations that allow simultaneous charging and discharging, while this is not feasible in practice. To address this, we derive the convex hull of the solutions for the optimal operation of storage for one time period, as well as for problems including investments and reserves, guaranteeing that no tighter MIP formulation or better LP approximation exists for one time period. When incorporating this convex hull into a multi-period formulation and including it in large-scale energy system models, the improved LP relaxations can better prevent simultaneous charging and discharging, and the tighter MIP could positively affect the solving time. We demonstrate this with illustrative case studies of a unit commitment problem and a transmission expansion planning problem.