Convex relaxation for optimal power flow problem: Mesh networks

This paper is concerned with a fundamental resource allocation problem for electrical power networks. This problem, named optimal power flow (OPF), is nonconvex due to the nonlinearities imposed by the laws of physics, and has been studied since 1962. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over cyclic (transmission) networks. Given an arbitrary weakly-cyclic network with cycles of size 3, it is shown that the injection region is convex in the lossless case and that the Pareto front of the injection region is convex in the lossy case. It is also proved that the SDP relaxation of OPF is exact for this type of network. Moreover, it is shown that if the SDP relaxation is not exact for a general mesh network, it would still have a low-rank solution whose rank depends on the structure of the network. Finally, a heuristic method is proposed to recover a rank-1 solution for the SDP relaxation whenever the relaxation is not exact.