# Newton Power Flow Methods for Unbalanced Three-Phase Distribution Networks

^{*}

## Abstract

**:**

## 1. Introduction

- Radial or weakly meshed (radial network with a few simple loops) structure:In general, a transmission network is operated in a meshed structure, whereas a distribution network is operated in a radial structure where there are no loops in the network and each bus is connected to the source via exactly one path.
- High $R/X$ ratio:Transmission lines of the distribution network have a wide range of resistance R and reactance X values. Therefore, $R/X$ ratios in the distribution network are relatively high compared to the transmission network.
- Multi-phase power flow and unbalanced loads:A single-phase representation is used for power flow analysis on transmission network which is assumed to be a balanced network. Unlike the transmission network, a distribution network must use a three-phase power flow analysis due to the unbalanced loads.
- Distributed generations:Unlike conventional power plants connected to the transmission network, DGs have fluctuating power output that is difficult to predict and control since it is strongly dependent on weather conditions.

- Modification of conventional power flow solution methods [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]:Methods in this category are generally a proper modification of existing methods such as GS, NR and FDLF.
- Backward–forward sweep (BFS)-based algorithms [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61]:BFS-based algorithms generally take an advantage of the radial network topology. The method is an iterative process in which at each iteration two computational steps are performed, a forward and a backward sweep. The forward sweep is mainly the node voltage calculation and the backward sweep is the branch current or power, or the admittance summation.

## 2. Power System Model

- $|{V}_{i}|$ : the voltage magnitude
- ${\delta}_{i}$ : the voltage phase angle
- ${P}_{i}$ : the active power
- ${Q}_{i}$ : the reactive power

#### 2.1. Load Model

- Constant power (PQ):The powers (P and Q) are independent of variations in the voltage magnitude $\left|V\right|$:$$\frac{P}{{P}_{0}}=1,\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\frac{Q}{{Q}_{0}}=1$$
- Constant current (I):The powers (P and Q) vary directly with the voltage magnitude $\left|V\right|$:$$\frac{P}{{P}_{0}}=\frac{\left|V\right|}{|{V}_{0}|},\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\frac{Q}{{Q}_{0}}=\frac{\left|V\right|}{|{V}_{0}|}$$
- Constant impedance (Z):The powers (P and Q) vary with the square of the voltage magnitude $\left|V\right|$:$$\frac{P}{{P}_{0}}={\left(\frac{\left|V\right|}{|{V}_{0}|}\right)}^{2},\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\frac{Q}{{Q}_{0}}={\left(\frac{\left|V\right|}{|{V}_{0}|}\right)}^{2}$$
- Polynomial (Po):The relation between powers (P and Q) and voltage magnitudes $\left|V\right|$ is described by a polynomial equation:$$\frac{P}{{P}_{0}}={a}_{0}+{a}_{1}\frac{\left|V\right|}{|{V}_{0}|}+{a}_{2}{\left(\frac{\left|V\right|}{|{V}_{0}|}\right)}^{2},\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\frac{Q}{{Q}_{0}}={b}_{0}+{b}_{1}\frac{\left|V\right|}{|{V}_{0}|}+{b}_{2}{\left(\frac{\left|V\right|}{|{V}_{0}|}\right)}^{2}$$$${a}_{0}+{a}_{1}+{a}_{2}=1,\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}{b}_{0}+{b}_{1}+{b}_{2}=1$$
- Exponential:The relation between powers (P and Q) and voltage magnitudes $\left|V\right|$ is described by an exponential equation:$$\frac{P}{{P}_{0}}={\left(\frac{\left|V\right|}{|{V}_{0}|}\right)}^{n},\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\frac{Q}{{Q}_{0}}={\left(\frac{\left|V\right|}{|{V}_{0}|}\right)}^{n}$$

#### 2.2. Load Connection

#### 2.3. Generator Model

- The constant power factor model (PQ bus):The active power P output and power factor $pf$ are specified and the reactive power Q is determined by these two variables.
- The variable reactive power model (PQ bus):The active power P output is specified and the reactive power Q is determined by applying a predetermined polynomial function.
- The constant voltage model (PV bus):The active power P output and voltage magnitude $|V|$ are specified.

#### 2.4. Transformer Model

- Divide the self admittance matrix of the primary by ${\alpha}^{2}$: $\frac{{Y}_{pp}^{abc}}{{\alpha}^{2}}$
- Divide the self admittance matrix of the secondary by ${\beta}^{2}$: $\frac{{Y}_{ss}^{abc}}{{\beta}^{2}}$
- Divide the mutual admittance matrices by $\alpha \beta $: $\frac{{Y}_{ps}^{abc}}{\alpha \beta}$, $\frac{{Y}_{sp}^{abc}}{\alpha \beta}$

## 3. Power Flow Problem

## 4. Newton Power Flow Solution Methods

**power**or

**current**-mismatch functions and designate the unknown bus voltages as the problem variables $\overrightarrow{x}$.

#### 4.1. The Power Mismatch Function

#### 4.2. The Current Mismatch Function

#### 4.2.1. Polar Current Mismatch Version (NR-c-pol)

#### 4.2.2. Representation of PV Buses for NR-c-pol

## 5. Numerical Experiment

#### 5.1. Single-Phase Problems

#### 5.2. Three-Phase Problems

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BFS | backward-forward sweep |

DG | distributed generation |

DSO | distribution system operators |

FDLF | fast-decoupled load flow |

GS | Gauss–Seidel |

NR | Newton power flow method |

NR-p-pol | polar power mismatch version of NR |

NR-p-car | Cartesian power mismatch version of NR |

NR-p-com | complex power mismatch version of NR |

NR-c-pol | polar current mismatch version of NR |

NR-c-car | Cartesian current mismatch version of NR |

NR-c-com | complex current mismatch version of NR |

TSO | transmission system operator |

SG | smart grid |

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**Figure 1.**Wye and delta connections for three-phase loads [68].

**Figure 2.**Combination of power converters and energy sources [69].

**Figure 4.**Computed voltage magnitude of DCase69. (

**a**) Computed voltage magnitude $|V|$; (

**b**) Difference between proposed methods and existing method [48] for the computed voltage magnitude.

**Figure 7.**Convergence result for different load models (constant power (PQ) and polynomial (Po)) in DCase69.

**Figure 10.**Convergence result for different load models (constant power (PQ) and polynomial (Po)) in DCase13.

**Table 1.**Network bus type. i: index of the bus; ${N}_{g}$: number of generator buses; N: total number of buses in the network.

Bus Type | Number of Buses | Known | Unknown |
---|---|---|---|

slack node or swing bus | 1 | $|{V}_{i}|,\phantom{\rule{4pt}{0ex}}{\delta}_{i}$ | ${P}_{i},\phantom{\rule{4pt}{0ex}}{Q}_{i}$ |

generator node or PV bus | ${N}_{g}$ | ${P}_{i},\phantom{\rule{4pt}{0ex}}|{V}_{i}|$ | ${Q}_{i},\phantom{\rule{4pt}{0ex}}{\delta}_{i}$ |

load node or PQ bus | $N-{N}_{g}-1$ | ${P}_{i},\phantom{\rule{4pt}{0ex}}{Q}_{i}$ | $|{V}_{i}|,\phantom{\rule{4pt}{0ex}}{\delta}_{i}$ |

Transformer Connection | Self Admittance | Mutual Admittance | |||
---|---|---|---|---|---|

Bus $\mathit{P}$ | Bus $\mathit{S}$ | ${\mathit{Y}}_{\mathit{pp}}^{\mathit{abc}}$ | ${\mathit{Y}}_{\mathit{ss}}^{\mathit{abc}}$ | ${\mathit{Y}}_{\mathit{ps}}^{\mathit{abc}}$ | ${\mathit{Y}}_{\mathit{sp}}^{\mathit{abc}}$ |

Wye-G | Wye-G | ${Y}_{I}$ | ${Y}_{I}$ | $-{Y}_{I}$ | $-{Y}_{I}$ |

Wye-G | Wye | ${Y}_{II}$ | ${Y}_{II}$ | $-{Y}_{II}$ | $-{Y}_{II}$ |

Wye-G | Delta | ${Y}_{I}$ | ${Y}_{II}$ | ${Y}_{III}$ | ${Y}_{III}^{T}$ |

Wye | Wye-G | ${Y}_{II}$ | ${Y}_{II}$ | $-{Y}_{II}$ | $-{Y}_{II}$ |

Wye | Wye | ${Y}_{II}$ | ${Y}_{II}$ | $-{Y}_{II}$ | $-{Y}_{II}$ |

Wye | Delta | ${Y}_{II}$ | ${Y}_{II}$ | ${Y}_{III}$ | ${Y}_{III}^{T}$ |

Delta | Wye-G | ${Y}_{II}$ | ${Y}_{I}$ | ${Y}_{III}^{T}$ | ${Y}_{III}$ |

Delta | Wye | ${Y}_{II}$ | ${Y}_{II}$ | ${Y}_{III}^{T}$ | ${Y}_{III}$ |

Delta | Delta | ${Y}_{II}$ | ${Y}_{II}$ | $-{Y}_{II}$ | $-{Y}_{II}$ |

Methods | Test Cases | |||||
---|---|---|---|---|---|---|

DCase33 | DCase69 | |||||

Iter | Time | $||\mathit{F}(\overrightarrow{\mathit{x}}){||}_{\infty}$ | Iter | Time | $||\mathit{F}(\overrightarrow{\mathit{x}}){||}_{\infty}$ | |

NR-p-pol [8] | 3 | 0.0123 | $7.4675\times {10}^{-6}$ | 4 | 0.0131 | $5.5875\times {10}^{-9}$ |

NR-p-car | 3 | 0.0067 | $1.0433\times {10}^{-6}$ | 3 | 0.0069 | $8.1777\times {10}^{-6}$ |

NR-p-com | 6 | 0.0058 | $6.4610\times {10}^{-6}$ | 7 | 0.0060 | $4.0138\times {10}^{-6}$ |

NR-c-pol | 3 | 0.0087 | $1.4291\times {10}^{-9}$ | 3 | 0.0090 | $8.5226\times {10}^{-9}$ |

NR-c-car | 3 | 0.0073 | $1.3954\times {10}^{-9}$ | 3 | 0.0077 | $1.9503\times {10}^{-8}$ |

NR-c-com | 7 | 0.0068 | $5.3792\times {10}^{-6}$ | 10 | 0.0084 | $2.7697\times {10}^{-6}$ |

BFS [43] | 7 | 0.0102 | $1.0454\times {10}^{-6}$ | 7 | 0.0104 | $7.7770\times {10}^{-6}$ |

Methods | Test Cases | |||||
---|---|---|---|---|---|---|

DCase13 | DCase37 | |||||

Iter | Time | $||\mathit{F}(\overrightarrow{\mathit{x}}){||}_{\infty}$ | Iter | Time | $||\mathit{F}(\overrightarrow{\mathit{x}}){||}_{\infty}$ | |

NR-p-pol [8] | 3 | 0.0116 | $1.5571\times {10}^{-9}$ | 2 | 0.0134 | $3.4150\times {10}^{-7}$ |

NR-p-car | 3 | 0.0067 | $6.7018\times {10}^{-9}$ | 2 | 0.0069 | $1.1627\times {10}^{-7}$ |

NR-p-com | 5 | 0.0055 | $5.0957\times {10}^{-7}$ | 3 | 0.0055 | $5.3394\times {10}^{-7}$ |

NR-c-pol | 3 | 0.0087 | $6.9974\times {10}^{-11}$ | 2 | 0.0094 | $3.9750\times {10}^{-8}$ |

NR-c-car | 3 | 0.0073 | $8.1499\times {10}^{-11}$ | 2 | 0.0079 | $4.0339\times {10}^{-8}$ |

NR-c-com | 5 | 0.0067 | $3.5585\times {10}^{-7}$ | 3 | 0.0065 | $7.4985\times {10}^{-7}$ |

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**MDPI and ACS Style**

Sereeter, B.; Vuik, K.; Witteveen, C.
Newton Power Flow Methods for Unbalanced Three-Phase Distribution Networks. *Energies* **2017**, *10*, 1658.
https://doi.org/10.3390/en10101658

**AMA Style**

Sereeter B, Vuik K, Witteveen C.
Newton Power Flow Methods for Unbalanced Three-Phase Distribution Networks. *Energies*. 2017; 10(10):1658.
https://doi.org/10.3390/en10101658

**Chicago/Turabian Style**

Sereeter, Baljinnyam, Kees Vuik, and Cees Witteveen.
2017. "Newton Power Flow Methods for Unbalanced Three-Phase Distribution Networks" *Energies* 10, no. 10: 1658.
https://doi.org/10.3390/en10101658