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Part I of this paper embeds the AC power flow problem with voltage controland exponential load model in the complex plane. Modeling the action of networkcontrollers that regulate the magnitude of voltage phasors is a challengingtask in the complex plane as it has to preserve the framework of holomorphicityfor obtention of these complex variables with fixed magnitude. The paperpresents two distinct approaches to modelling the voltage control of generatornodes. Exponential (or voltage-dependent) load models are crucial for accuratepower flow studies under stressed conditions. This new framework for power flowstudies exploits the theory of analytic continuation, especially the monodromytheorem for resolving issues that have plagued conventional numerical methodsfor decades. Here the focus is on the indispensable role of Pade approximantsfor analytic continuation of complex functions, expressed as power series,beyond the boundary of convergence of the series. The zero-pole distribution ofthese rational approximants serves as a proximity index to voltage collapse.Finally the mathematical underpinnings of this framework, namely the Stahl'stheory and the rate of convergence of Pade approximants are explained.

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