Fundamentals of the Holomorphic Embedding Load-Flow Method

TL;DR

The paper provides a detailed theoretical foundation for the Holomorphic Embedding Load-Flow Method (HELM), connecting complex analysis and algebraic geometry to solve power-flow equations in power grids effectively.

Contribution

It establishes the mathematical underpinnings of HELM, including projective invariance, algebraic curves, and analytic continuation, enhancing the method's theoretical rigor and practical applicability.

Findings

Theoretical foundations of HELM are rigorously developed.

Application of algebraic geometry and complex analysis to power-flow problems.

Extension of HELM to include generator-controlled PV buses.

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 Pad\'e 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.