Minimal Sparsity for Second-Order Moment-SOS Relaxations of the AC-OPF Problem

AC-OPF (Alternative Current Optimal Power Flow) aims at minimizing the operating costs of a power grid under physical constraints on voltages and power injections. Its mathematical formulation results in a nonconvex polynomial optimization problem which is hard to solve in general, but that can be tackled by a sequence of SDP (Semidefinite Programming) relaxations corresponding to the steps of the moment-SOS (Sum-Of-Squares) hierarchy. Unfortunately, the size of these SDPs grows drastically in the hierarchy, so that even second-order relaxations exploiting the correlative sparsity pattern of AC-OPF are hardly numerically tractable for large instances — with thousands of power buses. Our contribution lies in a new sparsity framework, termed minimal sparsity, inspired from the specific structure of power flow equations. Despite its heuristic nature, numerical examples show that minimal sparsity allows the computation of highly accurate second-order moment-SOS relaxations of AC-OPF, while requiring far less computing time and memory resources than the standard correlative sparsity pattern. Thus, we manage to compute second-order relaxations on test cases with thousands of power buses, which we believe to be unprecedented.