Nondegeneracy and Inexactness of Semidefinite Relaxations of Optimal Power Flow

Abstract: The Optimal Power Flow (OPF) problem can be reformulated as a nonconvex Quadratically Constrained Quadratic Program (QCQP). There is a growing body of work on the use of semidefinite programming relaxations to solve OPF. The relaxation is exact if and only if the corresponding optimal solution set contains a rank-one matrix. In this paper, we establish sufficient conditions guaranteeing the nonexistence of a rank-one matrix in said optimal solution set. In particular, we show that under mild assumptions on problem nondegeneracy, any optimal solution to the semidefinite relaxation will have rank greater than one, if the number of equality and active inequality constraints is at least twice the number of buses in the network. The sufficient condition holds for arbitrary network topologies (including tree networks). We empirically evaluate the practical implications of these results on several test cases from the literature.