How to Learn the Optimal Clique Decompositions in Solving Semidefinite Relaxations for OPF

Abstract

The Optimal Power Flow (OPF) problem is a central optimization problem in power systems. Its global resolution is a challenge since it is highly nonconvex and NP-hard. Semidefinite Programming (SDP) is a powerful tool to progress towards global optimality as semidefinite relaxations provide tight lower bounds for the OPF problem. However, solving semidefinite relaxations for large power networks is very costly, because it is required to exploit its sparsity for achieving this aim. One efficient way to exploit sparsity for the OPF problem is to use clique decomposition techniques along with state-of-the-art interior point algorithms. Yet many clique decompositions can be computed for the same sparse SDP problem, their performance can significantly varies in practice. In this context, it is crucial to identify a good decomposition, where by good we mean a decomposition that allows to solve the SDP relaxation of the OPF problem in a small amount of time. At the moment, it is not possible in the literature to find a systematic analysis that allows to characterize in detail the properties of a good decomposition, the works proposed so fare relies on the basic assumption that there is a trade-off between the size and the number of the cliques: a decomposition with only one large clique is problematic because of memory issues but a decomposition with many tiny cliques is not advisable either as it implies lots of linking constraints, which slows down the resolution. In this work, we propose to use machine learning techniques to understand what are the characteristics of a good clique decomposition. More precisely, we propose to identify the relevant features to describe a good clique decomposition, using both classification and regression approaches. The results show that the decomposition identified with the proposed techniques are comparable with the state of the art.