The Distribution of the Number of Real Solutions to the Power Flow Equations

TL;DR

This paper introduces a new homotopy continuation method to analyze the distribution of real solutions to power flow equations, revealing how these solutions vary with electrical parameters and graph structures.

Contribution

It presents a novel monodromy and parameter homotopy approach for efficiently finding all solutions and analyzing their distribution in power flow equations.

Findings

Power flow equations often have fewer real solutions than complex bounds.

Low load levels can lead to more real solutions than random polynomials.

Certain graph structures can achieve maximum real solutions or infinitely many.

Abstract

In this paper we study the distributions of the number of real solutions to the power flow equations over varying electrical parameters. We introduce a new monodromy and parameter homotopy continuation method for quickly finding all solutions to the power flow equations. We apply this method to find distributions of the number of real solutions to the power flow equations and compare these distributions to those of random polynomials. It is observed that while the power flow equations tend to admit many fewer real-valued solutions than a bound on the total number of complex solutions, for low levels of load they tend to admit many more than a corresponding random polynomial. We show that for cycle graphs the number of real solutions can achieve the maximum bound for specific parameter values and for complete graphs with four or more vertices there are susceptance values that give infinitely many real solutions.