On the Distribution of Real-Valued Solutions to the Power Flow Equations

The number and nature of real-valued solutions to the nonlinear power flow equations are not completely understood and there are no proven simple methods for computing all solutions. It has been noted that in practice the equations tend to admit many fewer real-valued solutions than what might be estimated using traditional bounds. In this paper we examine a means to calculate the distribution of number of solutions as a function of multiple parameters. This method relies on a reduction technique that resolves to an equation in one variable for which the calculation of real-valued solutions is straightforward. Importantly we examine the fundamental question of whether a real-valued solution in one variable necessarily implies all variables will be simultaneously realvalued. We present topologies for which this favorable property is retained and topologies for which it fails.