A Multi-Dimensional Holomorphic Embedding Method to Solve AC Power Flows




Liu, Chengxi
Wnag, Bin
Xu, Xin
Sun, Kai
Shi, Di
Bak, Claus Leth

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Institute of Electrical and Electronics Engineers


It is well-known that ac power flows of a power system do not have a closed-form analytical solution in general. This paper proposes a multi-dimensional holomorphic embedding method that derives analytical multivariate power series to approach true power flow solutions. This method embeds multiple independent variables into power flow equations and hence, can, respectively, scale power injections or consumptions of selected buses or groups of buses. Then, via a physical germ solution, the method can represent each bus voltage as a multivariate power series about symbolic variables on the system condition so as to derive approximate analytical power flow solutions. This method has a non-iterative mechanism unlike the traditional numerical methods for power flow calculation. Its solution can be derived offline and then evaluated in real time by plugging values into symbolic variables according to the actual condition, so the method fits better into online applications, such as voltage stability assessment. The method is first illustrated in detail on a 4-bus power system and then demonstrated on the IEEE 14-bus power system considering independent load variations in four regions.


This version is the accepted manuscript. See final published version at https://doi.org/10.1109/ACCESS.2017.2768958. © 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.


holomorphic embedding method, multi-dimensional holomorphic embedding method, power flow calculation, voltage stability


Liu, C., Wang, B., Xu, X., Sun, K., Shi, D., & Bak, C. L. (2017). A Multi-Dimensional Holomorphic Embedding Method to Solve AC Power Flows. IEEE Access, 5, 25270-25285. doi:10.1109/ACCESS.2017.2768958


Electrical and Computer Engineering