Recent advances in optimization techniques provide opportunities to better analyze and operate electric power systems. Due to the potential for finding globally optimal solutions, significant research interest has focused on the application of convex relaxation techniques to many problems in the field of electric power systems. This presentation discusses a semidefinite relaxation of the non-convex AC optimal power flow (OPF) problem, which seeks to minimize the operating cost of an electric power system subject to both engineering inequality and network equality constraints. The convex semidefinite relaxation is capable of finding globally optimal solutions to many OPF problems. By exploiting power system sparsity, semidefinite relaxations of practically sized OPF problems are computationally tractable.
The semidefinite relaxation is exact (i.e., yields the globally optimal solution to the original non-convex problem) for many but not all OPF problems. For practical problems where the semidefinite relaxation is not exact, results show small active and reactive power mismatches at the majority of load buses while only small subsets of the network exhibit significant mismatch. This suggests that the relevant non-convexities in these problems are isolated in small subsets of the network. Examination of the feasible spaces for small test cases illustrates such non-convexities and explains the semidefinite relaxation’s failure to be exact. Finally, results from the application of higher-order “moment” semidefinite relaxations show promise in obtaining globally optimal solutions to these small test cases.
Daniel Molzahn is a Dow Sustainability Fellow at the University of Michigan. He received the B.S., M.S. and Ph.D. degrees in Electrical Engineering and the Masters of Public Affairs degree from the University of Wisconsin–Madison, where he was a National Science Foundation Graduate Research Fellow. His research interests are in the application of optimization techniques and policy analysis to electric power systems.