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

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 Padé 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 Padé approximants are explained.