Investigation of Non-zero Duality Gap Solutions to a Semidefinite Relaxation of the Optimal Power Flow Problem

Recently, a semi definite programming relaxation of the power flow equations has been applied to the optimal power flow problem. When this relaxation is "tight" (i.e., the solution has zero duality gap), a globally optimal solution is obtained. Existing literature investigates sufficient conditions whose satisfaction guarantees zero duality gap solutions. However, there is limited study of non-zero duality gap solutions. By illustrating the feasible spaces for optimal power flow problems and their semi definite relaxations, this paper investigates examples of non-zero duality gap solutions. Results for large system models suggest that non-convexities associated with small subsections of the network are responsible for non-zero duality gap solutions.