Robust linear programming; optimal sizing of an hybrid energy stand-alone system

Abstract

This is a summary of the author PhD thesis supervised by Marie-Christine Costa and Alain Billionnet, and defended the 17th of December 2013 at the Conservatoire National des Arts et Metiers (CEDRIC laboratory) and Ensta-Paristech in Paris. The thesis is written in French and is available from the author upon request at plpoirion@gmail.com. This work deals with Two-stages linear robust programs and the optimal sizing of an hybrid energy stand-alone system under demand and production uncertainties. The thesis is divided into two parts: first we study the problem from a theoretical perspective, then we apply our results to a “real-world” problem and run some numerical experiments with realistic data. Robust optimization is an approach to study problems with uncertain data that does not rely on a prerequisite precise probability model but on mild assumptions on the uncertainties involved in the problem. In two-stage robust optimization, the variables of the problem are split in two parts: the first-stage variables concern the decisions that must be taken before the realization of the uncertain data; the secondstage variables, or recourse variables can be computed only once the uncertainty has been “revealed”. In this setting, we studied, in this thesis, the case where the recourse variables are continuous and the first stage variables are integer or mixed-integer. To solve the problem we first studied the so-called Recourse problem obtained when the values of the first-stage variables are fixed. We proved that when a specific hypothesis holds, we can reformulate the recourse problem as a MILP whose constraints does not depend of the first stage variables. In order to solve the robust problem, we use a Bender’s decomposition on a linearization of the problem having exponentially many constraints. In the literature, the previously so-called full recourse hypothesis, which assumes that the recourse problem has always a solution for any values of the uncertain parameters and the first stage variable, is generally assumed to hold. When it is not the case,weproved that there exists a polynomial transformation of the robust problem into one that satisfies the hypothesis. We performed some tests on the transformed problem to prove that the new problem is also tractable. Finally, we studied the complexity of the robust problem.