The energy distance for ensemble and scenario reduction

Abstract

Scenario reduction techniques are widely applied for solving sophisticated dynamic and stochastic programs, especially in energy and power systems. We propose a new method for ensemble and discrete scenario reduction based on the energy distance which is a special case of the maximum mean discrepancy (MMD). We discuss the choice of energy distance in detail, especially in comparison to the popular Wasserstein distance which is dominating the scenario reduction literature. The energy distance is a metric between probability measures which allows for powerful tests for equality of arbitrary multivariate distributions or independence. Thanks to the latter, it to a suitable candidate for ensemble and scenario reduction problems. The theoretical properties and considered examples indicate clearly that the reduced scenario set tend exhibit better statistical properties for energy distance than a corresponding reduction with respect to the Wasserstein distance. We show applications to binary trees and a real data based examples for electricity demand profiles.