On exact and near optimal power flow solutions for microgrid applications

Control of inverter-based microgrids is typically accomplished through a hierarchical control architecture, where the lower level controllers are designed to ensure convergence to a desired equilibrium determined by the upper level controller. Selecting the equilibrium can be cast as an optimal power flow (OPF) problem. The OPF problems associated with microgrid applications often involve non-monotone cost functions and nontrivial reactive power cost, posting new challenges for OPF problems that have not been completely addressed in the literature. In this paper, we formulate a distributed OPF problem motivated by microgrid control applications. We first derive new conditions on the exact SDP convex relaxation on the OPF problems with quadratic cost functions on the active and reactive power. For microgrids with tree topology, we show that under mild conditions on the reactive power bounds, a near global optimal solution can always be found for the OPF problem with those nontrivial cost functions. Such results are not only important for microgrid applications, but also provide new perspectives and insights for general OPF problems. The effectiveness of the proposed algorithm is demonstrated through numerical examples.