Investigating the Maximum Number of Real Solutions to the Power Flow Equations: Analysis of Lossless Four-Bus Systems

TL;DR

This paper investigates the maximum number of real solutions to power flow equations in lossless four-bus systems, using algebraic geometry techniques to conjecture an upper bound of 16 real solutions, which is less than the total complex solutions.

Contribution

It introduces a novel approach to counting real solutions in four-bus power systems and conjectures a strict upper bound, advancing understanding of solution multiplicity in power flow models.

Findings

Conjecture that four-bus lossless systems have at most 16 real solutions.

Explicit parameter examples achieve the 16 real solutions bound.

The maximum number of complex solutions is 20, but real solutions are fewer.

Abstract

The power flow equations model the steady-state relationship between the power injections and voltage phasors in an electric power system. By separating the real and imaginary components of the voltage phasors, the power flow equations can be formulated as a system of quadratic polynomials. Only the real solutions to these polynomial equations are physically meaningful. This paper focuses on the maximum number of real solutions to the power flow equations. An upper bound on the number of real power flow solutions commonly used in the literature is the maximum number of complex solutions. There exist two- and three-bus systems for which all complex solutions are real. It is an open question whether this is also the case for larger systems. This paper investigates four-bus systems using techniques from numerical algebraic geometry and conjectures a negative answer to this question. In particular, this paper studies lossless, four-bus systems composed of PV buses connected by lines with arbitrary susceptances. Computing the Galois group, which is degenerate, enables conversion of the problem of counting the number of real solutions to the power flow equations into counting the number of positive roots of a univariate sextic polynomial. From this analysis, it is conjectured that the system has at most 16 real solutions, which is strictly less than the maximum number of complex solutions, namely 20. We also provide explicit parameter values where this system has 16 real solutions so that the conjectured upper bound is achievable.