Recent Advances in Computational Methods for the Power Flow Equations

Abstract

The power flow equations are at the core of most of the computations for designing and operating electric power systems. The power flow equations are a system of multivariate nonlinear equations which relate the power injections and voltages in a power system. A plethora of methods have been devised to solve these equations, starting from Newton-based methods to homotopy continuation and other optimization-based methods. While many of these methods often efficiently find a high-voltage, stable solution due to its large basin of attraction, most of the methods struggle to find low-voltage solutions which play significant role in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in methods for solving power flow equations to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to the power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Groebner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction.

13 pages, 2 figures. Submitted to the Tutorial Session at IEEE 2016 American Control Conference

References in corpus (5)

• Strong SOCP Relaxations for the Optimal Power Flow Problem
• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Convex Relaxations of Optimal Power Flow Problems: An Illustrative Example
• Solution of Optimal Power Flow Problems using Moment Relaxations Augmented with Objective Function Penalization
• Toward Topologically Based Upper Bounds on the Number of Power Flow Solutions

Cited by in corpus (5)

• Convex Restriction of Power Flow Feasibility Sets
• Lossy DC Power Flow
• Optimal Power Flow: An Introduction to Predictive, Distributed and Stochastic Control Challenges
• Multistability and anomalies in oscillator models of lossy power grids
• Hypergraph-Based Fast Distributed AC Power Flow Optimization