### Abstract

The Holomorphic Embedding Load-Flow Method (HELM) solves the power-flow problem to obtain the bus voltages as rational approximants, that is, a ratio of complex-valued polynomials of the embedding parameter. The proof of its claims (namely that: 1) it is guaranteed to find a solution if it exists; 2) it is guaranteed to find only a high-voltage (operable) solution; and 3) that it unequivocally signals if no solution exists) are rooted in complex analysis and the theory developed by Antonio Trias and Herbert Stahl. HELM is one variant of the holomorphic embedding method (HEM) for solving nonlinear equations, the details of which may differ from those available in its published patents. In this paper we show that the HEM represents a distinct class of nonlinear equation solvers that are recursive, rather than iterative. As such, for any given problem, there are an infinite number of HEM formulations, each with different numerical properties and precision demands. The objective of this paper is to provide an intuitive understanding of HEM and apply one variant to the power-flow problem. We introduce one possible PV bus model compatible with the HEM and examine some features of different holomorphic embeddings, giving step-by-step details of model building, germ calculation, and the recursive algorithm.

Original language | English (US) |
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Journal | IEEE Transactions on Power Systems |

DOIs | |

State | Accepted/In press - Dec 10 2015 |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Energy Engineering and Power Technology

### Cite this

*IEEE Transactions on Power Systems*. https://doi.org/10.1109/TPWRS.2015.2503423

**The Holomorphic Embedding Method Applied to the Power-Flow Problem.** / Rao, Shruti; Feng, Yang; Tylavsky, Daniel; Subramanian, Muthu Kumar.

Research output: Contribution to journal › Article

*IEEE Transactions on Power Systems*. https://doi.org/10.1109/TPWRS.2015.2503423

}

TY - JOUR

T1 - The Holomorphic Embedding Method Applied to the Power-Flow Problem

AU - Rao, Shruti

AU - Feng, Yang

AU - Tylavsky, Daniel

AU - Subramanian, Muthu Kumar

PY - 2015/12/10

Y1 - 2015/12/10

N2 - The Holomorphic Embedding Load-Flow Method (HELM) solves the power-flow problem to obtain the bus voltages as rational approximants, that is, a ratio of complex-valued polynomials of the embedding parameter. The proof of its claims (namely that: 1) it is guaranteed to find a solution if it exists; 2) it is guaranteed to find only a high-voltage (operable) solution; and 3) that it unequivocally signals if no solution exists) are rooted in complex analysis and the theory developed by Antonio Trias and Herbert Stahl. HELM is one variant of the holomorphic embedding method (HEM) for solving nonlinear equations, the details of which may differ from those available in its published patents. In this paper we show that the HEM represents a distinct class of nonlinear equation solvers that are recursive, rather than iterative. As such, for any given problem, there are an infinite number of HEM formulations, each with different numerical properties and precision demands. The objective of this paper is to provide an intuitive understanding of HEM and apply one variant to the power-flow problem. We introduce one possible PV bus model compatible with the HEM and examine some features of different holomorphic embeddings, giving step-by-step details of model building, germ calculation, and the recursive algorithm.

AB - The Holomorphic Embedding Load-Flow Method (HELM) solves the power-flow problem to obtain the bus voltages as rational approximants, that is, a ratio of complex-valued polynomials of the embedding parameter. The proof of its claims (namely that: 1) it is guaranteed to find a solution if it exists; 2) it is guaranteed to find only a high-voltage (operable) solution; and 3) that it unequivocally signals if no solution exists) are rooted in complex analysis and the theory developed by Antonio Trias and Herbert Stahl. HELM is one variant of the holomorphic embedding method (HEM) for solving nonlinear equations, the details of which may differ from those available in its published patents. In this paper we show that the HEM represents a distinct class of nonlinear equation solvers that are recursive, rather than iterative. As such, for any given problem, there are an infinite number of HEM formulations, each with different numerical properties and precision demands. The objective of this paper is to provide an intuitive understanding of HEM and apply one variant to the power-flow problem. We introduce one possible PV bus model compatible with the HEM and examine some features of different holomorphic embeddings, giving step-by-step details of model building, germ calculation, and the recursive algorithm.

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U2 - 10.1109/TPWRS.2015.2503423

DO - 10.1109/TPWRS.2015.2503423

M3 - Article

JO - IEEE Transactions on Power Systems

JF - IEEE Transactions on Power Systems

SN - 0885-8950

ER -