Convexification of optimal power flow problem by means of phase shifters

This paper is concerned with the convexification of the optimal power flow (OPF) problem. We have previously shown that this highly nonconvex problem can be solved efficiently via a convex relaxation after two approximations: (i) adding a sufficient number of virtual phase shifters to the network topology, and (ii) relaxing the power balance equations to inequality constraints. The objective of the present paper is to first provide a better understanding of the implications of Approximation (i) and then remove Approximation (ii). To this end, we investigate the effect of virtual phase shifters on the feasible set of OPF by thoroughly examining a cyclic system. We then show that OPF can be convexified under only Approximation (i), provided some mild assumptions are satisfied. Although this paper mainly focuses on OPF, the results developed here can be applied to several OPF-based emerging optimization problems for future electrical grids.