# On the Matricial Formulation of Iterative Sweep Power Flow for Radial and Meshed Distribution Networks with Guarantee of Convergence

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## Abstract

**:**

## 1. Introduction

#### 1.1. General Context

#### 1.2. Motivation

#### 1.3. Literature Review

#### 1.4. Contributions and Scope

- The reformulation of the classical backward–forward power-flow method for application to distribution networks with the ability to handle radial and meshed configurations by rewriting the branch variables into nodal variables using the branch-to-node incidence matrix.
- The parametric independence of the power-flow formulation, as no assumptions about relations reactance/resistance, are required in the proposed matricial formulation.
- The possibility of guaranteeing convergence under well-defined voltage conditions by applying the Banach fixed-point theorem to the recursive solution, which only requires that the short-circuit current be more significant to the load current in all the nodes to ensure the convergence of the algorithm.
- The presentation of the proposed matricial formulation in an intuitive manner to introduce students of electrical engineering to power-flow analysis by providing the MATLAB/OCTAVE algorithm for solving a small test feeder as a numerical example comprising radial and meshed structures.

#### 1.5. Organization of the Document

## 2. Matricial Power-Flow Formulation

- Calculates the total current demands at all the loads with assumed known voltages.
- Determines all the currents that flow in all the branches of the network by applying the first Kirchhoff’s law at each node.
- Calculates the voltage drops by starting from the source and using an ordering stage that defines the layers at which nodes are located.
- Updates the voltage profile in all the demand nodes and repeats all the stages until the convergence tolerance is reached.

**Definition**

**1**

- ${\mathcal{A}}_{i,j}=1$ if the line i is connected to the node j and its current leaves from this node.
- ${\mathcal{A}}_{i,j}=-1$ if the line i is connected to the node j and its current arrives at this node.
- ${\mathcal{A}}_{i,j}=0$ if the line i is not connected to the node j.

**I**) and all the branch currents, let us apply Kirchhoff’s second law at each node while considering the current directions defined in Table 1. We thus obtain

**Remark**

**1.**

## 3. Convergence Analysis

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Theorem**

**1**

**Proof.**

## 4. Numerical Example

- √
- From lines 1 to 36, all the numerical information in the test system is presented and transformed into a per-unit representation.
- √
- Lines 37 to 48 comprise the branch-to-node incidence matrix, impedance matrix, and all the constant components for evaluating the recursive power-flow formula.
- √
- From lines 49 to 59, the iterative procedure for solving the power-flow problem while considering a matricial formulation is implemented while considering a convergence tolerance of approximately $\u03f5=1\times {10}^{-10}$. It should be noted that lines 53 and 54 present the required calculations for determining the total grid power losses using branch variables, i.e., $\mathbb{J}$ and $\mathbb{E}$, and the line impedance matrix $\mathbb{Z}$.
- √
- Lines 60 to 65 comprise the $\theta $ coefficients for all the loads for proving that each of them fulfills the Banach fixed-point theorem condition.

## 5. Test Systems

#### 5.1. 33-Three-Node Test Feeder

#### 5.2. 69-Node Test Feeder

## 6. Computational Validation

#### 6.1. Results for the 33- and 69-Node Test Feeders

- √
- The proposed approach is the most efficient in terms of processing times as compared with the classical approaches as it only takes 1.32 ms for completion in the case of the 33-node test feeder and 5.37 ms for completion in the case of the 69-node test feeder, which implies that it is at least seven times faster than the NR method, which is the most used approach in research and the industry.
- √
- The classical GS method presents the worse performance in terms of processing times and number of iterations. However, its accelerated version with the $\alpha $-coefficient presents a significant improvement in its performance. In the case of the 33-node test feeder, this method is approximately ten times faster, and in the case of the 69-node test feeder, it is at least 18 times faster, in terms of the total processing time required to solve the power-flow problem.
- √
- The NR and LM methods belong to the same class as both comprise the use of Jacobian matrices. Therefore, their performance with respect to the number of iterations and the processing times is very similar. Even if the proposed approach requires double the number of iterations, in terms of processing times, we can confirm that the MBF is the best numerical method for distribution power-flow analysis as compared with the NR and LM approaches.
- √
- In terms of the power losses calculation, it is important to mention that all the numerical methods listed in Table 1 arrive at the same solution with negligible errors of estimation (lower than $1\times {10}^{-10}$). This implies that each of them is suitable for implementation; nevertheless, our proposed method, i.e., the MBF reformulation, is the most desirable approach due to its speediest performance.

#### 6.2. Inclusion of Voltage-Controlled Nodes

**Remark**

**2.**

**Remark**

**3.**

## 7. Conclusions and Future Works

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. MATLAB/OCTAVE Code for VC Nodes

**Figure A1.**MATLAB implementation of the matricial iterative sweep power flow while considering VC nodes.

## References

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**Figure 2.**Electrical configuration of a radial/mesh distribution network (adapted from [31]).

**Figure 4.**Behavior of the $\theta $-coefficient in the 13-node test feeder in the case of radial and mesh configurations.

**Figure 8.**Behavior of the $\theta $-coefficient: (

**a**) 33-node test system, and (

**b**) 69-node test feeder.

Branch Number i | Send Node j | Receiving Node k |
---|---|---|

1 | 1 | 2 |

2 | 1 | 3 |

3 | 2 | 4 |

4 | 3 | 4 |

5 | 3 | 5 |

6 | 4 | 5 |

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kvar] |
---|---|---|---|---|---|

1 | 2 | 0.3968 | 0.5290 | 2000 | 1600 |

2 | 3 | 0.4232 | 0.5819 | 3000 | 400 |

2 | 4 | 0.4761 | 0.9522 | 2000 | −400 |

4 | 5 | 0.2116 | 0.2116 | 1500 | 1200 |

1 | 6 | 0.5819 | 0.5819 | 4000 | 2700 |

6 | 7 | 0.4232 | 0.5819 | 5000 | 1800 |

6 | 8 | 0.5819 | 0.5819 | 1000 | 900 |

7 | 9 | 0.5819 | 0.5819 | 600 | −400 |

1 | 10 | 0.5819 | 0.5819 | 1000 | 900 |

10 | 11 | 0.4761 | 0.6348 | 1000 | −1100 |

10 | 12 | 0.4232 | 0.5819 | 1000 | 900 |

12 | 13 | 0.2116 | 0.2116 | 2100 | −800 |

Tie-lines | |||||

3 | 9 | 0.2116 | 0.2116 | — | — |

8 | 11 | 0.2116 | 0.2116 | — | — |

5 | 13 | 0.4761 | 0.6348 | — | — |

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 0.0922 | 0.0477 | 100 | 60 | 17 | 18 | 0.7320 | 0.5740 | 90 | 40 |

2 | 3 | 0.4930 | 0.2511 | 90 | 40 | 2 | 19 | 0.1640 | 0.1565 | 90 | 40 |

3 | 4 | 0.3660 | 0.1864 | 120 | 80 | 19 | 20 | 1.5042 | 1.3554 | 90 | 40 |

4 | 5 | 0.3811 | 0.1941 | 60 | 30 | 20 | 21 | 0.4095 | 0.4784 | 90 | 40 |

5 | 6 | 0.8190 | 0.7070 | 60 | 20 | 21 | 22 | 0.7089 | 0.9373 | 90 | 40 |

6 | 7 | 0.1872 | 0.6188 | 200 | 100 | 3 | 23 | 0.4512 | 0.3083 | 90 | 50 |

7 | 8 | 1.7114 | 1.2351 | 200 | 100 | 23 | 24 | 0.8980 | 0.7091 | 420 | 200 |

8 | 9 | 1.0300 | 0.7400 | 60 | 20 | 24 | 25 | 0.8960 | 0.7011 | 420 | 200 |

9 | 10 | 1.0400 | 0.7400 | 60 | 20 | 6 | 26 | 0.2030 | 0.1034 | 60 | 25 |

10 | 11 | 0.1966 | 0.0650 | 45 | 30 | 26 | 27 | 0.2842 | 0.1447 | 60 | 25 |

11 | 12 | 0.3744 | 0.1238 | 60 | 35 | 27 | 28 | 1.0590 | 0.9337 | 60 | 20 |

12 | 13 | 1.4680 | 1.1550 | 60 | 35 | 28 | 29 | 0.8042 | 0.7006 | 120 | 70 |

13 | 14 | 0.5416 | 0.7129 | 120 | 80 | 29 | 30 | 0.5075 | 0.2585 | 200 | 600 |

14 | 15 | 0.5910 | 0.5260 | 60 | 10 | 30 | 31 | 0.9744 | 0.9630 | 150 | 70 |

15 | 16 | 0.7463 | 0.5450 | 60 | 20 | 31 | 32 | 0.3105 | 0.3619 | 210 | 100 |

16 | 17 | 1.2890 | 1.7210 | 60 | 20 | 32 | 33 | 0.3410 | 0.5302 | 60 | 40 |

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 0.0005 | 0.0012 | 0 | 0 | 3 | 36 | 0.0044 | 0.0108 | 26 | 18.55 |

2 | 3 | 0.0005 | 0.0012 | 0 | 0 | 36 | 37 | 0.0640 | 0.1565 | 26 | 18.55 |

3 | 4 | 0.0015 | 0.0036 | 0 | 0 | 37 | 38 | 0.1053 | 0.1230 | 0 | 0 |

4 | 5 | 0.0251 | 0.0294 | 0 | 0 | 38 | 39 | 0.0304 | 0.0355 | 24 | 17 |

5 | 6 | 0.3660 | 0.1864 | 2.6 | 2.2 | 39 | 40 | 0.0018 | 0.0021 | 24 | 17 |

6 | 7 | 0.3810 | 0.1941 | 40.4 | 30 | 40 | 41 | 0.7283 | 0.8509 | 1.2 | 1 |

7 | 8 | 0.0922 | 0.0470 | 75 | 54 | 41 | 42 | 0.3100 | 0.3623 | 0 | 0 |

8 | 9 | 0.0493 | 0.0251 | 30 | 22 | 42 | 43 | 0.0410 | 0.0475 | 6 | 4.3 |

9 | 10 | 0.8190 | 0.2707 | 28 | 19 | 43 | 44 | 0.0092 | 0.0116 | 0 | 0 |

10 | 11 | 0.1872 | 0.0619 | 145 | 104 | 44 | 45 | 0.1089 | 0.1373 | 39.22 | 26.3 |

11 | 12 | 0.7114 | 0.2351 | 145 | 104 | 45 | 46 | 0.0009 | 0.0012 | 39.22 | 26.3 |

12 | 13 | 1.0300 | 0.3400 | 8 | 5 | 4 | 47 | 0.0034 | 0.0084 | 0 | 0 |

13 | 14 | 1.0440 | 0.3450 | 8 | 5.5 | 47 | 48 | 0.0851 | 0.2083 | 79 | 56.4 |

14 | 15 | 1.0580 | 0.3496 | 0 | 0 | 48 | 49 | 0.2898 | 0.7091 | 384.7 | 274.5 |

15 | 16 | 0.1966 | 0.0650 | 45.5 | 30 | 49 | 50 | 0.0822 | 0.2011 | 384.7 | 274.5 |

16 | 17 | 0.3744 | 0.1238 | 60 | 35 | 8 | 51 | 0.0928 | 0.0473 | 40.5 | 28.3 |

17 | 18 | 0.0047 | 0.0016 | 60 | 35 | 51 | 52 | 0.3319 | 0.1114 | 3.6 | 2.7 |

18 | 19 | 0.3276 | 0.1083 | 0 | 0 | 9 | 53 | 0.1740 | 0.0886 | 4.35 | 3.5 |

19 | 20 | 0.2106 | 0.0690 | 1 | 0.6 | 53 | 54 | 0.2030 | 0.1034 | 26.4 | 19 |

20 | 21 | 0.3416 | 0.1129 | 114 | 81 | 54 | 55 | 0.2842 | 0.1447 | 24 | 17.2 |

21 | 22 | 0.0140 | 0.0046 | 5 | 3.5 | 55 | 56 | 0.2813 | 0.1433 | 0 | 0 |

22 | 23 | 0.1591 | 0.0526 | 0 | 0 | 56 | 57 | 1.5900 | 0.5337 | 0 | 0 |

23 | 24 | 0.3460 | 0.1145 | 28 | 20 | 57 | 58 | 0.7837 | 0.2630 | 0 | 0 |

24 | 25 | 0.7488 | 0.2475 | 0 | 0 | 58 | 59 | 0.3042 | 0.1006 | 100 | 72 |

25 | 26 | 0.3089 | 0.1021 | 14 | 10 | 59 | 60 | 0.3861 | 0.1172 | 0 | 0 |

26 | 27 | 0.1732 | 0.0572 | 14 | 10 | 60 | 61 | 0.5075 | 0.2585 | 1244 | 888 |

23 | 28 | 0.0044 | 0.0108 | 26 | 18.6 | 61 | 62 | 0.0974 | 0.0496 | 32 | 23 |

28 | 29 | 0.0640 | 0.1565 | 26 | 18.6 | 62 | 63 | 0.1450 | 0.0738 | 0 | 0 |

29 | 30 | 0.3978 | 0.1315 | 0 | 0 | 63 | 64 | 0.7105 | 0.3619 | 227 | 162 |

30 | 31 | 0.0702 | 0.0232 | 0 | 0 | 64 | 65 | 1.0410 | 0.5302 | 59 | 42 |

31 | 32 | 0.3510 | 0.1160 | 0 | 0 | 11 | 66 | 0.2012 | 0.0611 | 18 | 13 |

32 | 33 | 0.8390 | 0.2816 | 14 | 10 | 66 | 67 | 0.0047 | 0.0014 | 18 | 13 |

33 | 34 | 1.7080 | 0.5646 | 19.5 | 14 | 12 | 68 | 0.7394 | 0.2444 | 28 | 20 |

34 | 35 | 1.4740 | 0.4873 | 6 | 4 | 68 | 69 | 0.0047 | 0.0016 | 28 | 20 |

33-node Test Feeder | |||
---|---|---|---|

Method | Proc. time [ms] | Iterations | Losses [p.u] |

GS | 441.974 | 2313 | 2.110 |

AG$(\alpha =1.82)$ | 38.555 | 227 | 2.110 |

NR | 10.751 | 5 | 2.110 |

LM | 10.882 | 5 | 2.110 |

MBF | 1.323 | 10 | 2.110 |

69-node Test Feeder | |||

GS | 31107.756 | 49031 | 2.422 |

AG$(\alpha =1.92)$ | 1662.691 | 2455 | 2.422 |

NR | 38.303 | 5 | 2.422 |

LM | 42.719 | 5 | 2.422 |

MBF | 5.369 | 10 | 2.422 |

Line | R [$\mathbf{\Omega}$] | X [$\mathbf{\Omega}$] | ${\mathit{Y}}_{\mathbf{shunt}}$ [$\mathsf{\mu}$S] | Current [A] |
---|---|---|---|---|

1–2 | 5.3323 | 26.6616 | 193.7618 | 200 |

1–3 | 3.9358 | 19.6788 | 146.5028 | 400 |

2–4 | 3.9358 | 19.6788 | 146.5028 | 500 |

3–4 | 6.7289 | 33.6444 | 241.0208 | 400 |

Node | ${\mathit{P}}_{\mathit{g}}$ [MW] | ${\mathit{Q}}_{\mathit{g}}$ [MVAr] | ${\mathit{P}}_{\mathit{d}}$ [MW] | ${\mathit{Q}}_{\mathit{d}}$ [MVAr] | V [pu] |
---|---|---|---|---|---|

1 (slack) | — | — | 50 | 30.99 | 1.00$\angle 0$ |

2 (load) | 0 | 0 | 170 | 105.35 | 1.00$\angle 0$ |

3 (load) | 0 | 0 | 200 | 123.94 | 1.00$\angle 0$ |

4 (VC) | 318 | — | 80 | 49.58 | 1.02$\angle 0$ |

Node | DigSILENT | MBF | Node | DigSILENT | MBF |
---|---|---|---|---|---|

1 | $1.000\angle {0.00}^{\circ}$ | $1.000\angle {0.00}^{\circ}$ | 3 | $0.969\angle {-1.87}^{\circ}$ | $0.969\angle {-1.87}^{\circ}$ |

2 | $0.982\angle {-0.98}^{\circ}$ | $0.982\angle {-0.98}^{\circ}$ | 4 | $1.020\angle {-1.52}^{\circ}$ | $1.020\angle {-1.52}^{\circ}$ |

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## Share and Cite

**MDPI and ACS Style**

Montoya, O.D.; Gil-González, W.; Giral, D.A. On the Matricial Formulation of Iterative Sweep Power Flow for Radial and Meshed Distribution Networks with Guarantee of Convergence. *Appl. Sci.* **2020**, *10*, 5802.
https://doi.org/10.3390/app10175802

**AMA Style**

Montoya OD, Gil-González W, Giral DA. On the Matricial Formulation of Iterative Sweep Power Flow for Radial and Meshed Distribution Networks with Guarantee of Convergence. *Applied Sciences*. 2020; 10(17):5802.
https://doi.org/10.3390/app10175802

**Chicago/Turabian Style**

Montoya, Oscar Danilo, Walter Gil-González, and Diego Armando Giral. 2020. "On the Matricial Formulation of Iterative Sweep Power Flow for Radial and Meshed Distribution Networks with Guarantee of Convergence" *Applied Sciences* 10, no. 17: 5802.
https://doi.org/10.3390/app10175802