Approximate Analytical Solutions of Power Flow Equations Based on Multi-Dimensional Holomorphic Embedding Method

Index Terms—Holomorphic embedding method, multi- dimensional, power flow calculation, analytical solution. I. INTRODUCTION AST growing electricity markets and relatively slow upgrades on transmission infrastructure have made many power systems occasionally operated closer to their voltage stability limits. However, many blackouts, such as the western North America blackout in July, 1996 [1], and Indian power system blackout in July, 2012 [2] resulted from voltage collapses have led to enormous societal and financial losses. To prevent voltage collapses, many utilities have deployed different assessment and early awareness systems on the transmission grids. However, the accurate situational awareness and the corresponding preventive control will only be achieved when the control center have the fast, reliable and accurate tools to directly process these measured data preferably in real-time. Power flow analysis on a power system is necessary for utilities to provide the voltage stability assessment, situational

This work was supported in part by NSF CURENT Engineering Research Center and NSF grant ECCS-1610025. C. Liu, B. Wang, X. Xu and K. Sun are with Department of EECS, University of Tennessee, Knoxville, TN, USA (email: cliu48@utk.edu, bwang@utk.edu, xxu30@vols.utk.edu, kaisun@utk.edu,) C. L. Bak is with Department of Energy, Aalborg University, Aalborg, Denmark (email: clb@et.aau.dk) awareness and preventive control. Theoretically, power-flow equations (PFEs) of power systems are a set of non-linear algebraic equations reduced from an enormous number of detailed differential equations by neglecting the fast electromagnetic dynamics. Traditionally, to solve AC PFEs, many iterative numerical methods have been adopted by commercialized power system software, including the Gauss- Seidel method, Newton-Raphson method and fast decoupled method. A major concern on these methods is that the numerical divergence of their iterations is often interpreted as the happening of voltage collapse but, theoretically speaking, does not necessarily indicate the non-existence of a power flow solution. Also, there is a probability for these numerical methods to converge to the non-physically existing ghost solutions [3]. These concerns influence the performances of these iterative, numerical methods in real-time applications. The holomorphic embedding power flow method (HELM) was firstly proposed by A. Trias in [3]-[5], which is a non- iterative method to analyze the power flows. The basic idea of HELM is to design a holomorphic function and adopt its analytical continuation in the complex plane to find the solution of the power flow equations as a power series form about an embedded complex variable. Recently, many derivative algorithms and applications based on HELM have developed [6]-[9], such as the HELM with non-linear static load models [10], the HELM used in AC/DC power systems [11], using HELM to find the unstable equilibrium points [12], [13], network reduction [14], the analysis of saddle-node bifurcation [15], [16] and the applications of real-time voltage stability assessment [17]. In this paper, a novel multi-dimensional holomorphic embedding method (MDHEM) is proposed to obtain the approximate analytical solution to PFEs, by finding a germ solution in the space of physical solutions and arbitrarily embedding each load or groups of loads of respective scales. Therefore, the approximate analytical solutions to the PFEs in the high-dimensional solution space become achievable. Based on the proposed MDHEM, the algebraic variables of a power network in all operating conditions can be prepared offline and evaluated online

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