Scenario reduction techniques are widely applied for solving sophisticated dynamic and stochastic programs, especially in energy and power systems, but are also used in probabilistic forecasting, clustering and estimating generative adversarial networks. 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. 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.

This article is part of the theme issue ‘The mathematics of energy systems’.

场景缩减技术广泛应用于解决复杂的动态和随机程序，特别是在能源和电力系统中，但也用于概率预测、聚类和估计生成对抗网络。我们提出了一种基于能量距离的集成和场景缩减的新方法，能量距离是最大平均差异的特例。我们详细讨论了能量距离的选择，特别是与流行的 Wasserstein 距离相比，后者在场景减少文献中占主导地位。能量距离是概率度量之间的度量，它允许对任意多元分布的相等性或独立性进行强大的测试。由于后者，它是集成和场景减少问题的合适候选者。理论特性和考虑的例子清楚地表明，与相对于 Wasserstein 距离的相应减少相比，减少的场景集往往表现出更好的能量距离统计特性。我们展示了伯努利随机游走的应用以及电力需求概况和日前电价的两个基于数据的真实示例。

这篇文章是主题问题“能量系统的数学”的一部分。