Optimization Formulations for Storage Devices with Disjoint Operating Modes

We consider a storage device, such as a pumped storage hydroelectric generator, that has a state of charge together with mutually exclusive and disjoint charging and generating modes. We develop valid inequalities for a storage model that uses binary variables to represent the charging and generating modes. To investigate the model, we consider two contexts, stand-alone and large-scale. The stand-alone context involves the hydroelectric generator purchasing or selling electricity based on known or forecast prices. We consider properties of an optimization formulation with an objective that evaluates the profit from the sale of net generation and value of stored energy, present conditions for the optimum of the continuous relaxation of this optimization formulation to have binary values for the charging and generation commitment variables, and demonstrate the result numerically with a small example system. Analysis of the stand-alone context helps to explain why the combination of features in the storage model results in a difficult problem. The large-scale context embeds the model into a unit commitment and dispatch formulation for multiple generators. For several large-scale test cases, we numerically verify that the valid inequalities can improve the computation compared with the standard model in the literature.