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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., Comment: 13 pages

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.

Analyses of power engineering networks rely on an understanding of the power flow equations, a system of quadratic equations relating power injections and voltages at each node. Finding the number of nontrivial, real solutions to these equations is an active area of study. In this paper we propose that in general for a fixed topology, a single, univariate polynomial whose coefficients are polynomials in the network susceptances suffices for determining the number of nontrivial, real solutions to the power flow equations. We then exploit this polynomial to gain geometric insight to the solution regions for a fixed network where a solution region is the region in the space of susceptances wherein the number of real solutions to the system is some fixed constant. This type of analysis allows easier insight to robustness analysis for varying susceptances. We observe that some solution regions resemble thickened hyperplanes and therefore appear as stripes in our figures.

Operation and planning of electric grids involve the analysis of certain power flow equations, which exhibit multiple solutions. One solution generally corresponds to a preferred operating condition, while other less desirable solutions are often useful for assessing dynamic angular stability and static voltage stability margins. The number and nature of solutions to these network equations has been studied for decades, yet without revealing a complete understanding. In light of observations that the number of solutions encountered in practice is smaller than the number predicted by the usual mathematical bounds, the most recent theoretical papers on this topic have sought to produce more accurate bounds on the number of solutions, accounting for network structure. In this paper we further refine the analysis to describe distributions of solutions. We present algebraic results for a certain small system and empirical results for others.

In this paper we present an algorithm which provably finds all voltage solutions to the power flow equations on tree networks. Our algorithm uses elimination theory to reduce the equations to univariate polynomials whose roots govern the voltages of the entire tree. The algorithm utilizes the sparse structure of the tree to decrease its computational cost. We show that the runtime of the algorithm depends on the degrees of the vertices in the tree. This implies that in physically motivated distribution networks, our algorithm runs much faster than in the worst-case scenario, scaling more like the number of real solutions as opposed to the number of complex solutions. Finally, we compare our algorithm to common approaches to finding all voltage solutions, including homotopy continuation methods and Grobner basis methods. We show that our algorithm outperforms these other methods in the low-, medium-, and high-degree settings and scales better with the number of buses.