The energy distance for ensemble and scenario reduction

TL;DR

This paper introduces a new scenario reduction method based on the energy distance, a metric for probability measures, demonstrating its advantages over Wasserstein distance in energy system applications.

Contribution

The paper proposes using the energy distance for ensemble and scenario reduction, offering better statistical properties than Wasserstein distance in energy-related problems.

Findings

Energy distance-based reduction yields more statistically representative scenarios.

The method outperforms Wasserstein distance in energy system applications.

Applications include electricity demand and price data analysis.

Abstract

Scenario reduction techniques are widely applied for solving sophisticated dynamic and stochastic programs, especially in energy and power systems, but also used in probabilistic forecasting, clustering and estimating generative adversarial networks (GANs). We propose a new method for ensemble and 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 that allows for powerful tests for equality of arbitrary multivariate distributions or independence. Thanks to the latter, it is a suitable candidate for ensemble and scenario reduction problems. The theoretical properties and considered examples indicate clearly that the reduced scenario sets tend to exhibit better statistical properties for the energy distance than a corresponding reduction with respect to the Wasserstein distance. We show applications to a Bernoulli random walk and two real data based examples for electricity demand profiles and day-ahead electricity prices.