Inner Approximation of Minkowski Sums: A Union-Based Approach and Applications to Aggregated Energy Resources: Preprint

Abstract

This paper develops and compares algorithms to compute the inner approximation of Minkowski sum of convex polytopes. We consider flexibility from distributed re- sources, such as solar photovoltaic (PV) inverters, demand side resources, etc. A polytopic representation of a single inverter's feasible operating region is proposed first. The aggregate flexibility can be computed using the Minkowski sum. Homothet and Zonotope-based approaches have been explored in literature. We show that as heterogeneity increases, such approaches lead to conservative estimates. Hence, we show how to exploit union-based Minkowski sum computation through successive homothetic decomposition of polytopes. While the union-based approach can lead to dimensionality issues, we show how to limit the complexity by predefining a candidate set. Efficiency, accuracy and trade-offs have been analyzed. Numerical examples have been presented to illustrate the effectiveness of the proposed algorithm.