Sub-differentiability in Convex and Stochastic Optimization Applied to Renewable Power Systems

Inserting renewable power systems in the electric grid is a key challenge of the energy transition. However, such systems introduce new engineering problems, due to the erratic behavior of renewable energy sources. In this thesis, we study how techniques from convex and stochastic optimization can be applied, and extended, to address some of these problems. The manuscript is divided in two parts. In the first part, Convex and stochastic optimization for renewable power systems, we focus on techniques for designing and assessing energy management systems. We start with a benchmark of optimal control methods for managing a prosumer microgrid, and we highlight, on a large testbed, the potential gains of methods based on stochastic dynamic programming. Then, in a more theoretical chapter, we investigate the differentiability properties of parametric value functions, introduced for solving a class of multistage stochastic optimization problems parametrized by an upstream decision. Lastly, we apply our previous results to the management of a photovoltaic power plant constrained by day-ahead commitment rules. We showcase significant gains compared to state-of-the-art techniques. In the second part, Numerical methods in generalized convexity, we study potential applications of the so-called one-sided linear couplings - a class that encompasses the Fenchel coupling of (standard) convex analysis. We start by extending the mirror descent algorithm. Then, turning to the Capra (constant along primal rays) coupling as a particular case, we provide explicit formulations for the Capra sub-differential of the l₀ pseudonorm. Lastly, we discuss the difficulties that arise when trying to use Capra convexity to solve sparse optimization problems. Although we do not directly address energy problems, we contribute to an original viewpoint on sparse optimization, whose applications in statistics and signal processing have a huge impact on all engineering fields. (author)