Examining the limits of the application of semidefinite programming to power flow problems

The application of semidefinite programming (SDP) to power system problems has recently attracted substantial research interest. Specifically, a recent SDP formulation offers a convex relaxation to the well-known, typically nonconvex “optimal power flow” (OPF) problem. This new formulation was demonstrated to yield zero duality gap for several standard power systems test cases, thereby ensuring a globally optimal OPF solution in each. The first goal of the work here is to investigate this SDP algorithm for the OPF, and show by example that it can fail to give a physically meaningful solution (i.e., it has a nonzero duality gap) in some scenarios of practical interest. The remainder of this paper investigates an SDP approach utilizing modified objective and constraints to compute all solutions of the nonlinear power flow equations. Several variants are described. Results suggest SDP's promise as an efficient algorithm for identifying large numbers of solutions to the power flow equations.