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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.