  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.
