Introduction to the power flow equations and moment/sum-of-squares relaxations of optimal power flow problems

摘要

Recent advances in computational methods related to the power flow equations provide opportunities to better analyze and operate electric power systems. This presentation begins by overviewing the power flow equations and discussing the state-of-the-art in solution techniques. After this overview, this presentation will discuss recent work on convex relaxations of the power flow equations. With the ability to globally solve many power system optimization problems (most notably the optimal power flow (OPF) problem used to determine an optimal operating point), significant research interest has focused on convex relaxations of the power flow equations. Existing relaxations globally solve many OPF problems. However, there are practical problems for which existing relaxations fail to yield physically meaningful solutions. A hierarchy of moment/sum-of-squares relaxations globally solve many problems for which existing relaxations fail. The moment relaxations, which take the form of semidefinite programs, are developed from the Lasserre hierarchy for generalized moment problems. Increasing the order in this hierarchy results in tighter relaxations at the computational cost of larger semidefinite programs. Recent computational improvements enable faster solution of larger OPF problems. These advances include exploiting network sparsity, selectively targeting the application of the computationally intensive higher-order moment constraints, relaxing some of the semidefinite programming constraints to second-order cone programming constraints, and using a complex analogue of the Lasserre hierarchy.