# A Sequential Hybridization of Genetic Algorithm and Particle Swarm Optimization for the Optimal Reactive Power Flow

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## Abstract

**:**

## 1. Introduction

## 2. Metaheuristic Methods and Their Hybridization

#### 2.1. The Genetic Algorithms

#### 2.2. Particle Swarm Optimization

#### 2.3. Sequential Hybridization of Metaheuristics

## 3. Mathematics Setting

_{ij}is the conductance between the i-th and j-th nodes, ${\mathrm{V}}_{\mathrm{i}}$ the voltage at the i-th node, θ

_{i}is the angle at the i-th node, and θ

_{ij}= θ

_{i}− θ

_{j}. The sum in relation (1) runs over the total number of nodes n.

_{ij}is the susceptance of the pair of nodes i and j, and, ${\mathrm{P}}_{\mathrm{i}}^{\mathrm{g}}$ ${\mathrm{P}}_{\mathrm{i}}^{\mathrm{l}}$ and the active powers generated and consumed at i-th node, respectively. ${\mathrm{Q}}_{\mathrm{i}}^{\mathrm{g}}$ and ${\mathrm{Q}}_{\mathrm{i}}^{\mathrm{l}}$ are, respectively, the generated and consumed reactive powers at the i-th node. Here, ${\mathrm{Q}}_{\mathrm{i}}^{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ is the reactive power of the compensator at node i.

_{g}, n

_{T}, and n

_{comp}are the number of generators, transformers, and compensators, respectively. The transformation ratio of the i-th transformer is denoted by a

_{i}. Subscripts ‘min’ and ‘max’ are relevant to the minimum and maximum of the considered variables.

## 4. Illustration

- Step 1: Carry out a load flow study to determine an operating point (see node voltage profile and active losses by Fast Decoupled Load Flow Method (FDLF) [1])
- Step 2: Initialization of the genetic algorithm and production of an initial population with the following parameters:
- maximum number of iterations;
- population size;
- probability of crossover;
- probability of mutation.

- Step 3: Application of the three operators of the genetic algorithm (selection, crossover, and mutation) to have the new individuals until the stopping criterion is reached.
- Step 4: Injection of the best solution obtained by GA as an initial population of the PSO method.
- Step 5: Initialization of the PSO program with the following parameters:
- initial weight Wmax;
- final weight Wmin;
- maximum number of iterations itmax;
- weighting factor C1 = C2;
- population size nind;
- number of variables nvar.

- Step 6: Applying the operators of the PSO method (speed and position of updates).

#### 4.1. The Electrical Network IEEE14 Nodes

#### 4.2. The Algerian Western Wetwork with 102 Nodes

#### 4.3. Results Analysis

## 5. Conclusions

- development of the sequential hybridization of GA and PSO to improve the execution time, to have minimum losses in minimum time;
- development of other types of hybridization such as integrative hybridization.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The one-line diagram of the Algerian Western Network with the three voltage levels 400 kV (blue lines)/220 kV (red lines)/60 kV (green lines).

**Figure 4.**The variation of the voltage on the nodes in different cases before and after the optimization for GA, PSO, and GA–PSO hybridization in the 400 kV electrical network.

**Figure 5.**The voltage of nodes in different cases before and after the optimization for GA, PSO, and GA–PSO hybridization in the 220 kV electrical network.

**Figure 6.**The voltage of nodes from the FDLF and the different optimization methods (GA, PSO, and GA–PSO hybridization) in the case of the 60 kV electrical network: (

**a**) The voltage of the nodes numbers 30 to 65; (

**b**) The voltage of the nodes numbers 66 to 102.

Valuemin | Value Max | FDLF | GA | PSO | GA–PSO Sequential Hybridization | |
---|---|---|---|---|---|---|

${Q}_{1}^{g}(MVAR)$ | −16.78 | −16.78 | −0.7 | −2.41 | ||

${Q}_{2}^{g}(MVAR)$ | −40 | 50 | 42.737 | 42.73 | 39.94 | 38.5 |

${Q}_{3}^{g}(MVAR)$ | 0 | 40 | 23.603 | 23.60 | 32.78 | 21.95 |

${Q}_{6}^{g}(MVAR)$ | −6 | 24 | 13.141 | 13.14 | −1.86 | 23.72 |

${Q}_{8}^{g}(MVAR)$ | −6 | 24 | 18.062 | 18.06 | 23.91 | 21.10 |

a_{1} (p.u) | 0.9 | 1.1 | 0.978 | 1.077 | 0.978 | 1.026 |

a_{2} (p.u) | 0.9 | 1.1 | 0.969 | 1.007 | 0.969 | 1.00 |

a_{3} (p.u) | 0.9 | 1.1 | 0.932 | 1.043 | 0.932 | 1.012 |

FDLF | AG | PSO | GA–PSO Sequential Hybridization | |
---|---|---|---|---|

Active losses (MW) | 13.39 | 12.54 | 12.57 | 12.52 |

Reduction in MW | 0.85 | 0.82 | 0.87 | |

Reduction (%) | 6.34 | 6.12 | 6.49 | |

Execution time (s) | 3.21 | 23.54 | 4.33 |

Number of charges nodes | 92 |

Number of nodes of generations | 10 |

Number of lines | 119 |

Number of transformers | 14 |

Voltage | Minimal Value (p.u) | Voltage (p.u) |
---|---|---|

400 kV | 0.9 | 1.1 |

220 kV | 0.9 | 1.1 |

60 kV | 0.9 | 1.1 |

Variables | Minimal Value | Maximal Value |
---|---|---|

${\mathrm{a}}_{\mathrm{i}}$ | 0.9 | 1.1 |

${\mathrm{Q}}_{1}^{\mathrm{g}}$ | −170 | 350 |

${\mathrm{Q}}_{6}^{\mathrm{g}}$ | −240 | 270 |

${\mathrm{Q}}_{12}^{\mathrm{g}}$ | −60 | 100 |

${\mathrm{Q}}_{13}^{\mathrm{g}}$ | −90 | 180 |

${\mathrm{Q}}_{20}^{\mathrm{g}}$ | −80 | 400 |

${\mathrm{Q}}_{22}^{\mathrm{g}}$ | −35 | 60 |

${\mathrm{Q}}_{24}^{\mathrm{g}}$ | −80 | 400 |

${\mathrm{Q}}_{39}^{\mathrm{g}}$ | −15 | 48 |

${\mathrm{Q}}_{51}^{\mathrm{g}}$ | −8 | 38 |

${\mathrm{Q}}_{55}^{\mathrm{g}}$ | −20 | 30 |

Number of Nodes | FDLF | GA | PSO | GA–PSO Sequential Hybridization |
---|---|---|---|---|

1 | −236.56 | −118.01 | −118.78 | −170.53 |

6 | −76.28 | −203.16 | 590.04 | −200 |

12 | 78.47 | 72.16 | −44.91 | 75.61 |

13 | −46.1 | −84.42 | −58.40 | −32.93 |

20 | 7.72 | 109.00 | −29.71 | −25.8 |

22 | −24.07 | 11.18 | 66.56 | −3.29 |

24 | −45.18 | −5.58 | −77.74 | 37.91 |

39 | 116.85 | −8.31 | 139.29 | 38.86 |

51 | −20.44 | 4.41 | 87.2 | 20.43 |

55 | 6.97 | 25.75 | −2.09 | −0.48 |

Number of Nodes | FDLF | GA | PSO | GA–PSO Sequential Hybridization |
---|---|---|---|---|

02→06 | 0.96 | 1.00 | 1.1 | 1.06 |

03→23 | 0.96 | 0.99 | 1.08 | 0.98 |

06→30 | 0.98 | 1.00 | 0.99 | 1.00 |

07→31 | 0.99 | 0.99 | 0.96 | 0.96 |

08 →32 | 0.98 | 1.00 | 0.99 | 0.98 |

08→33 | 0.95 | 0.99 | 0.99 | 0.94 |

09→34 | 0.98 | 0.99 | 0.99 | 1.01 |

10→35 | 0.98 | 0.99 | 1.09 | 1.05 |

11→36 | 0.99 | 0.99 | 1.01 | 1.01 |

12→37 | 0.96 | 1.00 | 0.95 | 1.02 |

12→38 | 0.99 | 0.99 | 0.94 | 0.94 |

13→39 | 1.07 | 0.99 | 1.01 | 0.97 |

14→40 | 0.95 | 1.00 | 1.02 | 1.05 |

15→41 | 0.98 | 0.99 | 1.05 | 1.03 |

18→52 | 0.98 | 1.00 | 1.01 | 0.99 |

19→77 | 1.00 | 0.99 | 1.01 | 1.01 |

21→61 | 0.97 | 1.00 | 0.98 | 1.06 |

24→86 | 0.97 | 0.99 | 1.06 | 0.91 |

26→87 | 0.99 | 1.00 | 0.98 | 1.02 |

FDLF | GA | PSO | GA–PSO Sequential Hybridization | |
---|---|---|---|---|

Active loses (MW) | 51.06 | 36.60 | 45.07 | 29.19 |

Reduction (MW) | 14.46 | 5.98 | 21.87 | |

Reduction (%) | 28.3% | 11.7% | 42.8% | |

Execution time(s) | 26.70 | 35.29 | 36.42 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cherki, I.; Chaker, A.; Djidar, Z.; Khalfallah, N.; Benzergua, F.
A Sequential Hybridization of Genetic Algorithm and Particle Swarm Optimization for the Optimal Reactive Power Flow. *Sustainability* **2019**, *11*, 3862.
https://doi.org/10.3390/su11143862

**AMA Style**

Cherki I, Chaker A, Djidar Z, Khalfallah N, Benzergua F.
A Sequential Hybridization of Genetic Algorithm and Particle Swarm Optimization for the Optimal Reactive Power Flow. *Sustainability*. 2019; 11(14):3862.
https://doi.org/10.3390/su11143862

**Chicago/Turabian Style**

Cherki, Imene, Abdelkader Chaker, Zohra Djidar, Naima Khalfallah, and Fadela Benzergua.
2019. "A Sequential Hybridization of Genetic Algorithm and Particle Swarm Optimization for the Optimal Reactive Power Flow" *Sustainability* 11, no. 14: 3862.
https://doi.org/10.3390/su11143862