Distributionally Robust Optimization

Distributionally Robust Optimization and its Applications in Power System Energy Storage Sizing

This book introduces the mathematical foundations of distributionally robust optimization (DRO) for decision-making problems with ambiguous uncertainties and applies them to tackle the critical challenge of energy storage sizing in renewable-integrated power systems, providing readers with an efficient and reliable approach to analyze and design real-world energy systems with uncertainties. Covering a diverse range of topics, this book starts by exploring the necessity for energy storage in evolving power systems and examining the benefits of employing distributionally robust optimization. Subsequently, the cutting-edge mathematical theory of distributionally robust optimization is presented, including both the general theory and moment-based, KL-divergence, and Wasserstein-metric distributionally robust optimization theories. The techniques are then applied to various practical energy storage sizing scenarios, such as stand-alone microgrids, large-scale renewable power plants, bulk power grids, and multi-carrier energy networks. This book offers clear explanations and accessible guidance to bridge the gap between advanced optimization methods and industrial applications. Its interdisciplinary scope makes the book appealing to researchers, graduate students, and industry professionals working in electrical engineering and operations research, catering to both beginners and experts.

Distributionally Robust Optimization

This chapter presents a class of distributionally robust optimization problems in which a decision-maker has to choose an action in an uncertain environment. The decision-maker has a continuous action space and aims to learn her optimal strategy. The true distribution of the uncertainty is unknown to the decision-maker. This chapter provides alternative ways to select a distribution based on empirical observations of the decision-maker. This leads to a distributionally robust optimization problem. Simple algorithms, whose dynamics are inspired from the gradient flows, are proposed to find local optima. The method is extended to a class of optimization problems with orthogonal constraints and coupled constraints over the simplex set and polytopes. The designed dynamics do not use the projection operator and are able to satisfy both upper- and lower-bound constraints. The convergence rate of the algorithm to generalized evolutionarily stable strategy is derived using a mean regret estimate. Illustrative examples are provided.

Distributionally Robust Learning