Optimal power flow of a distribution system based on increasingly tight cutting planes added to a second order cone relaxation

Abstract

Convex relaxations of the optimal power flow (OPF) problem have received a lot of attention in the recent past. In this work, we focus on a second-order cone (SOC) relaxation applied to an OPF based on a branch flow model of a radial and balanced distribution system. We start by examining various sets of conditions ensuring the exactitude of such a relaxation, which is the main focus of the existing literature. In particular, we observe that these sets always include a requirement on the objective to be a minimization of a function increasing with the branch flow apparent powers. We consider this hypothesis to be at odds with what is to be expected of an active distribution system and demonstrate in specific case studies its counterproductive impact. We continue by introducing an objective function allowing distributed generations and storages (DGS) to take advantage of the benefits they bring to the power system as a whole. As this entails the possibility for the relaxation not to be exact, we describe and prove the theoretical convergence to optimality of an algorithm consisting in adding an increasingly tight linear cut to the SOC relaxation. In order to allow the attainment of a solution satisfying the network constraints in a finite number of steps, we continue by introducing a tailored termination criterion. Afterwards, we investigate the ability of our algorithm to obtain a satisfactory solution on several case studies, spanning various network sizes, number of nodes equipped with DGs and their level of penetration. We then conclude on the benefits brought about by this approach and reflect on its limits and the opportunities for further improvements.