The holomorphic embedding method applied to the power-flow problem

The Holomorphic Embedding Load-Flow Method (HELM) solves the power-flow problem to obtain the bus voltages as rational approximants, that is, a ratio of complex-valued polynomials of the embedding parameter. The proof of its claims (namely that: 1) it is guaranteed to find a solution if it exists; 2) it is guaranteed to find only a high-voltage (operable) solution; and 3) that it unequivocally signals if no solution exists) are rooted in complex analysis and the theory developed by Antonio Trias and Herbert Stahl. HELM is one variant of the holomorphic embedding method (HEM) for solving nonlinear equations, the details of which may differ from those available in its published patents. In this paper we show that the HEM represents a distinct class of nonlinear equation solvers that are recursive, rather than iterative. As such, for any given problem, there are an infinite number of HEM formulations, each with different numerical properties and precision demands. The objective of this paper is to provide an intuitive understanding of HEM and apply one variant to the power-flow problem. We introduce one possible PV bus model compatible with the HEM and examine some features of different holomorphic embeddings, giving step-by-step details of model building, germ calculation, and the recursive algorithm.