Inexact Convex Relaxations for AC Optimal Power Flow: Towards AC Feasibility

Andreas Venzke, Spyros Chatzivasileiadis, Daniel K. Molzahn

Published 2019-02-13Version 1

Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted significant interest as in several instances they provably yield the global optima to the original non-convex problems. If the relaxation fails to be exact, the optimality gap is often small, i.e., the relaxation's objective function value is close to the objective value of the best known AC-feasible point. This work studies inexact solutions to the quadratic convex (QC) and semidefinite (SDP) relaxations, providing a comprehensive analysis on the resulting distances to both AC-feasibility and local optimality. To this end, we propose two empirical distance metrics which complement the optimality gap. In addition, we investigate two methods to recover AC-feasible solutions: penalization methods and warm-starting of non-convex solvers. For the PGLib OPF benchmarks, we show that i) despite an optimality gap of less than 1%, several test cases still exhibit substantial distances to AC-feasibility and local optimality; ii) the optimality gaps do not strongly correlate with either distance. We also show that several existing penalization methods may fail to recover AC-feasible or near-globally optimal solutions and that benefits of warm-starting of local solvers strongly depend on test case and solver and are not correlated with the optimality gaps.