Root Distribution in Pad\'e Approximants and its Effect on Holomorphic Embedding Method Convergence

TL;DR

This paper investigates how the distribution of roots in Padé approximants influences the convergence of the Holomorphic Embedding Method in solving nonlinear equations in power systems, revealing a link to electrostatic equilibrium distributions.

Contribution

It proves that Padé roots converge to electrostatic equilibrium distributions, explaining the convergence behavior of HEM through potential theory and root distribution analysis.

Findings

Root distribution converges to electrostatic equilibrium on branch cuts.

The convergence factor relates to the logarithmic capacity of branch cuts.

Numerical experiments confirm theoretical convergence predictions.

Abstract

The requirement for solving nonlinear algebraic equations is ubiquitous in the field of electric power system simulations. While Newton-based methods have been used to advantage, they sometimes do not converge, leaving the user wondering whether a solution exists. In addition to improved robustness, one advantage of holomorphic embedding methods (HEM) is that, even when they do not converge, roots plots of the Pad\'e approximants (PAs) to the functions in the inverse-$\alpha$ plane can be used to determine whether a solution exist. The convergence factor (CF) of the near-diagonal PAs applied to functions expanded about the origin is determined by the logarithmic capacity of the associated branch cut (BC) and the distance of the evaluation point from the origin. However the underlying mechanism governing this rate has been obscure. We prove that the ultimate distribution of the PA roots on the BC in the complex plane converges weakly to the equilibrium distribution of electrostatic charges on a 2-D conductor system with the same topology in a physical setting. This, along with properties of the Maclaurin series can be used to explain the structure of the CF equation We demonstrate the theoretical convergence behavior, with numerical experiments.