Efficiently finding all power flow solutions to tree networks

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.