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The power flow equations model the steady-state relationship between thepower injections and voltage phasors in an electric power system. By separatingthe real and imaginary components of the voltage phasors, the power flowequations can be formulated as a system of quadratic polynomials. Only the realsolutions to these polynomial equations are physically meaningful. This paperfocuses on the maximum number of real solutions to the power flow equations. Anupper bound on the number of real power flow solutions commonly used in theliterature is the maximum number of complex solutions. There exist two- andthree-bus systems for which all complex solutions are real. It is an openquestion whether this is also the case for larger systems. This paperinvestigates four-bus systems using techniques from numerical algebraicgeometry and conjectures a negative answer to this question. In particular,this paper studies lossless, four-bus systems composed of PV buses connected bylines with arbitrary susceptances. Computing the Galois group, which isdegenerate, enables conversion of the problem of counting the number of realsolutions to the power flow equations into counting the number of positiveroots of a univariate sextic polynomial. From this analysis, it is conjecturedthat the system has at most 16 real solutions, which is strictly less than themaximum number of complex solutions, namely 20. We also provide explicitparameter values where this system has 16 real solutions so that theconjectured upper bound is achievable.

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