  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.
