Network Topologies Guaranteeing Zero Duality Gap for Optimal Power Flow Problem

We have recently shown that the optimal power flow (OPF) problem with a quadratic cost function can be solved in polynomial time for a large class of power networks, including IEEE benchmark systems, due to their physical properties. In this work, our previous zero-duality-gap result is extended to OPF with arbitrary convex cost functions, and then it is proved that adding phase shifters to the network makes it possible to solve OPF efficiently. More precisely, it is first shown that the Lagrangian dual of OPF can be used to find a globally optimal solution of OPF (i.e., the duality gap is zero) if and only if a linear matrix inequality (LMI) optimization has a specific solution. Since both the dual of OPF and this LMI problem might be expensive to solve for a large-scale network, the sparsity structure of the power network is exploited to significantly reduce the computational complexity of solving these problems. Furthermore, it is proved that adding multiple controllable phase shifters (with variable phases) to certain lines of the power network simplifies the verification of the duality gap. Interestingly, if a sufficient number of phase shifters are added to the network, the duality gap becomes zero for OPF, provided load over-satisfaction (over delivery) is allowed. This result implies that every network topology can be modified by the integration of phase shifters to guarantee the solvability of OPF in polynomial time for all possible values of loads, physical limits and convex cost functions.