Fundamentals of the Holomorphic Embedding Load-Flow Method

Abstract

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.

17 pages, 1 figure

Cited by in corpus (4)

• Approximate Analytical Solutions of Power Flow Equations Based on Multi-Dimensional Holomorphic Embedding Method
• Learning to Solve AC Optimal Power Flow by Differentiating through Holomorphic Embeddings
• Root Distribution in Padé Approximants and its Effect on Holomorphic Embedding Method Convergence
• Implications of Stahl's Theorems to Holomorphic Embedding Pt. 2: Numerical Convergence