Theoretical Study of Non-Iterative Holomorphic Embedding Methods for Solving Nonlinear Power Flow Equations: Algebraic Property

Solving nonlinear algebraic equations is a core task in many fields of engineering and the sciences. Iterative methods such as Newton's method have been popular in solving power flow equations. Recently, the non-iterative holomorphic embedding (HE) methods that employ polynomial embedded systems have been proposed to solve power flow equations. It is advantageous for the solution functions obtained by the HE method to possess an algebraic property. The present paper points out that algebraic solution functions can have several desired properties for the HE method to work. This paper also exemplifies that the algebraic property cannot be ensured by the elimination process, which corrects an assertion in the literature. It is hence essential to study required conditions for ensuring the existence of algebraic solution functions. The present paper is devoted to the theoretical foundation of HE methods using a polynomial embedded system and develops a novel general theorem and sufficient conditions to ensure the required algebraic property. Examples and numerical results are presented to better explain the derived theoretical results.