Embedding AC Power Flow in the Complex Plane Part I: Modelling and Mathematical Foundation

TL;DR

This paper introduces a novel complex plane modeling framework for AC power flow with voltage control and exponential load models, utilizing analytic continuation and Pade approximants to improve stability analysis.

Contribution

It develops two approaches for voltage control modeling in the complex plane and applies analytic continuation theory to enhance power flow analysis accuracy.

Findings

Pade approximants effectively predict voltage collapse proximity.

The framework preserves holomorphicity in complex power flow modeling.

Mathematical foundations include Stahl's theory and convergence rates of Pade approximants.

Abstract

Part I of this paper embeds the AC power flow problem with voltage control and exponential load model in the complex plane. Modeling the action of network controllers that regulate the magnitude of voltage phasors is a challenging task in the complex plane as it has to preserve the framework of holomorphicity for obtention of these complex variables with fixed magnitude. The paper presents two distinct approaches to modelling the voltage control of generator nodes. Exponential (or voltage-dependent) load models are crucial for accurate power flow studies under stressed conditions. This new framework for power flow studies exploits the theory of analytic continuation, especially the monodromy theorem for resolving issues that have plagued conventional numerical methods for decades. Here the focus is on the indispensable role of Pade approximants for analytic continuation of complex functions, expressed as power series, beyond the boundary of convergence of the series. The zero-pole distribution of these rational approximants serves as a proximity index to voltage collapse. Finally the mathematical underpinnings of this framework, namely the Stahl's theory and the rate of convergence of Pade approximants are explained.