The Geometry of Real Solutions to the Power Flow Equations

Abstract

Analyses of power engineering networks rely on an understanding of the power flow equations, a system of quadratic equations relating power injections and voltages at each node. Finding the number of nontrivial, real solutions to these equations is an active area of study. In this paper we propose that in general for a fixed topology, a single, univariate polynomial whose coefficients are polynomials in the network susceptances suffices for determining the number of nontrivial, real solutions to the power flow equations. We then exploit this polynomial to gain geometric insight to the solution regions for a fixed network where a solution region is the region in the space of susceptances wherein the number of real solutions to the system is some fixed constant. This type of analysis allows easier insight to robustness analysis for varying susceptances. We observe that some solution regions resemble thickened hyperplanes and therefore appear as stripes in our figures.