2015/09/08 by Antonio Trias, Trias, Antonio · 1 citation
Engineering · Mathematics · #14H50 #14H81 #30B10 #30B40 #30B70 #30E10 #94C99 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Electrical engineering #FOS: Mathematics #Numerical methods for differential equations #Optimal Power Flow Distribution #Power System Optimization and Stability #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1509.02421
openalex publication_date 2015/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Holomorphic Embedding Load-Flow Method (HELM) was recently introduced as a novel technique to constructively solve the power-flow equations in power grids, based on advanced complex analysis. In this paper, the theoretical foundations of the method are established in detail. Starting from a fundamental projective invariance of the power-flow equations, it is shown how to devise holomorphicity-preserving embeddings that ultimately allow regarding the power-flow problem as essentially a study in algebraic curves. Complementing this algebraic-geometric viewpoint, which lays the foundation of the method, it is shown how to apply standard analytic techniques (power series) for practical computation. Stahl's theorem on the maximality of the analytic continuation provided by Padé approximants then ensures the completeness of the method. On the other hand, it is shown how to extend the method to accommodate smooth controls, such as the ubiquitous generator-controlled PV bus.