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The requirement for solving nonlinear algebraic equations is ubiquitous inthe field of electric power system simulations. While Newton-based methods havebeen used to advantage, they sometimes do not converge, leaving the userwondering whether a solution exists. In addition to improved robustness, oneadvantage of holomorphic embedding methods (HEM) is that, even when they do notconverge, roots plots of the Pad\'e approximants (PAs) to the functions in theinverse-$\alpha$ plane can be used to determine whether a solution exist. Theconvergence factor (CF) of the near-diagonal PAs applied to functions expandedabout the origin is determined by the logarithmic capacity of the associatedbranch cut (BC) and the distance of the evaluation point from the origin.However the underlying mechanism governing this rate has been obscure. We provethat the ultimate distribution of the PA roots on the BC in the complex planeconverges weakly to the equilibrium distribution of electrostatic charges on a2-D conductor system with the same topology in a physical setting. This, alongwith properties of the Maclaurin series can be used to explain the structure ofthe CF equation We demonstrate the theoretical convergence behavior, withnumerical experiments.

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