# The Equivalence between Successive Approximations and Matricial Load Flow Formulations

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## Abstract

**:**

## 1. Introduction

## 2. SA Load Flow

**Remark**

**1.**

**Remark**

**2.**

## 3. MBF Load Flow

**Definition**

**1**

- ${\mathcal{A}}_{i,j}=1$ when branch i connects bus j and its current is leaving bus i;
- ${\mathcal{A}}_{i,j}=-1$ when branch i connects bus j and its current is arriving bus j;
- ${\mathcal{A}}_{i,j}=0$ when branch i has no connection with bus j.

**Remark**

**4.**

## 4. Demonstration of the Equivalence

**Remark**

**5.**

## 5. Test Systems and Comparative Methods

## 6. Numerical Validation

- √
- The conventional GS approach exhibits the worst performance in terms of processing times and the number of iterations required to solve the load flow problem in both test feeders. In addition, the accelerated version of the GS approach allows for improvement in its performance (i.e., the GS behavior) by reducing processing times by about $91.28$ % and $94.66\%$ for the 33- and 69-bus distribution grids, while the number of iterations is reduced by about $90.19\%$ and $94.99\%$ respectively;
- √
- The classical NR and LM load flow approaches have the same numerical performance regarding the total iterations required to solve the load flow problem in both test feeders with 5 iterations in both cases. The low number of iterations in these methods is attributed to the fact that both work with the Jacobian matrix, which contains information about the direction of the maximum change of load flow equations, a situation that does not occur for the remainder of the load flow methods, as these do not use information of the derivatives of the load flow equations. With regard to processing times, the NR and LM methods are very similar, as in the case of the system with 33 buses, the difference is lower than $0.15$ ms, and in the case of the system composed of 69 buses, this difference is about $4\phantom{\rule{3.33333pt}{0ex}}41$ ms;
- √
- The numerical results for the 33- and 69-node test feeders obtained for the SA and the MBF load flow methods, i.e., the equivalent load flow approaches, is the same regarding the number of iterations, as these take 10 iterations to solve the load flow problem in both test feeders however, regarding the processing times, we can observe that the SA is the faster approach as demonstrated in [12]. Even if both methods are equivalent, the SA approach calculates the admittance matrix, i.e., $\mathbb{Y}$, using the direct method by adding the inverse of the conductance at this line, while the MBF method uses matricial calculations for the products between the incidence matrix and the primitive matrices, which consumes additional processing times.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

^{®}” presented by the student María Camila Herrera-Briñez to the Electrical Engineering Program of the Engineering Faculty at Universidad Distrital Francisco José de Caldas as a partial requirement for the Bachelor in Electrical Engineering.

## Conflicts of Interest

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**Figure 1.**Electrical configuration of the test feeders: (

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

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

**Figure 2.**Voltage profiles in the distribution networks under analysis: (

**a**) 33-bus test system and (

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

**Table 1.**Load flow results for both test feeders. Successive approximations (SA); Newton–Raphson (NR); Gauss–Seidel (GS); matricial backward/forward (MBF); and Levenberg–Marquardt (LM).

Method | Proc. Time (ms) | Iterations | Losses (p.u) |
---|---|---|---|

Test system composed of 33 buses | |||

GS | 441.973960 | 2313 | 2.109785 |

AG $(\alpha =1.82)$ | 38.555403 | 227 | 2.109785 |

NR | 10.751203 | 5 | 2.109785 |

LM | 10.881656 | 5 | 2.109785 |

MBF | 1.322962 | 10 | 2.109785 |

SA | 0.518957 | 10 | 2.109785 |

Test system composed of 69 buses | |||

GS | 31107.756292 | 49031 | 2.421523 |

AG $(\alpha =1.92)$ | 1662.690792 | 2455 | 2.421523 |

NR | 38.303088 | 5 | 2.421523 |

LM | 42.719055 | 5 | 2.421523 |

MBF | 5.369374 | 10 | 2.421523 |

SA | 2.488095 | 10 | 2.421523 |

Iteration | SA | MBF |
---|---|---|

1 | 8.8223$\times {10}^{-2}$ | 8.8223 $\times {10}^{-2}$ |

2 | 8.0437 $\times {10}^{-3}$ | 8.0437 $\times {10}^{-3}$ |

3 | 7.5688 $\times {10}^{-4}$ | 7.5688 $\times {10}^{-4}$ |

4 | 7.5134 $\times {10}^{-5}$ | 7.5134 $\times {10}^{-5}$ |

5 | 7.1145 $\times {10}^{-6}$ | 7.1145 $\times {10}^{-6}$ |

6 | 7.0636 $\times {10}^{-7}$ | 7.0636 $\times {10}^{-7}$ |

7 | 6.6881 $\times {10}^{-8}$ | 6.6881 $\times {10}^{-8}$ |

8 | 6.6400 $\times {10}^{-9}$ | 6.6400 $\times {10}^{-9}$ |

9 | 6.2870 $\times {10}^{-10}$ | 6.2870 $\times {10}^{-10}$ |

10 | 6.2418 $\times {10}^{-11}$ | 6.2418 $\times {10}^{-11}$ |

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**MDPI and ACS Style**

Herrera-Briñez, M.C.; Montoya, O.D.; Alvarado-Barrios, L.; Chamorro, H.R.
The Equivalence between Successive Approximations and Matricial Load Flow Formulations. *Appl. Sci.* **2021**, *11*, 2905.
https://doi.org/10.3390/app11072905

**AMA Style**

Herrera-Briñez MC, Montoya OD, Alvarado-Barrios L, Chamorro HR.
The Equivalence between Successive Approximations and Matricial Load Flow Formulations. *Applied Sciences*. 2021; 11(7):2905.
https://doi.org/10.3390/app11072905

**Chicago/Turabian Style**

Herrera-Briñez, María Camila, Oscar Danilo Montoya, Lazaro Alvarado-Barrios, and Harold R. Chamorro.
2021. "The Equivalence between Successive Approximations and Matricial Load Flow Formulations" *Applied Sciences* 11, no. 7: 2905.
https://doi.org/10.3390/app11072905