Nonsmooth optimization for optimal power flow over transmission networks

The optimal power flow (OPF) of a power transmission network is a NP-hard optimization problem with nonlinear equality and inequality constraints on the bus voltages. The existing nonlinear solvers often fail in yielding a feasible solution. Semi-definite relaxation (SDR) could provide an optimal solution only when the optimal solution of the relaxed semi-definite program (SDP) is of rank-one, which does not hold in general. Otherwise, the solution found by SDR is infeasible. Very recently, high-order semi-definite relaxation has been used to find the global optimal solution but such approach leads to an explosive growth of the variable dimension and so could be applied to test OPF with very small networks with 2, 3 and 5 buses, where there are only 2, 3 and 5 voltages variables. In this paper, we adapt our previously developed nonsmooth optimization algorithm to address this difficult OPF problem, which is an iterative process to generate a sequence of improved solution that converges to an optimal solution. Each its iteration calls a SDP of a moderate dimension. Preliminary simulations for difficult OPF problems of networks with a large number of buses are provided to show the efficiency of our approach.