Ex post conditions for the exactness of optimal power flow conic relaxations

Convex relaxations of the optimal power flow (OPF) problem provide an efficient alternative to solving the intractable alternating current (AC) optimal power flow. The conic subset of OPF convex relaxations, in particular, greatly accelerate resolution while leading to high-quality approximations that are exact in several scenarios. However, the sufficient conditions guaranteeing exactness are stringent, e.g., requiring radial topologies. In this short communication, we present two equivalent ex post conditions for the exactness of any conic relaxation of the OPF. These rely on obtaining either a rank-1 voltage matrix or self-coherent cycles. Instead of relying on sufficient conditions a priori, satisfying one of the presented ex post conditions acts as an exactness certificate for the computed solution. The operator can therefore obtain an optimality guarantee when solving a conic relaxation even when a priori exactness requirements are not met. Finally, we present numerical examples from the MATPOWER library where the ex post conditions hold even though the exactness sufficient conditions do not, thereby illustrating the use of the conditions.