Existence of strong Lagrange duals to certain optimal power flows

In this paper, we consider the non-convex optimal power flow (OPF) problem. We apply the recently proposed continuous-time gradient dynamics approach to solve OPFs and study their convergence properties. This approach is appealing because it has a naturally distributed structure. We numerically show, for a three-bus OPF example, that the gradient dynamics locally converges to a saddle point (the primal dual optimum by definition) for the associated Lagrangian, whereas the semi-definite programming (SDP) dual approach yields a non-zero duality gap. This suggests that there are certain OPFs for which strong Lagrange duality holds, although their SDP duals fail to maintain a zero duality gap.