# Optimization of the Voltage Total Harmonic Distortion in Multilevel Inverters by Using the Taguchi Method

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multilevel Inverter Output Voltage Signal

#### 2.2. Genetic Algorithm

**Population initialization:**This is randomly generated and consists of possible solutions to the problem, bounded by the constraints previously defined.**Fitness Function:**The initial population is evaluated over fitness function to compare one with the other.**Selection:**This process selects the parents (individuals) from the initial population that makes crossover and permutation to find the next generation.**Crossover:**Consists of selecting two parents from a population based on fitness function and then changing and improving parent parts to create a new individual.**Mutation:**This process applies random changes to introduce diversity in the population genes.**Reinsertion:**The children from the new generation replace some members of the current generation. The new one entirely replaces the current generation.

#### 2.3. Taguchi Method

#### 2.4. Taguchi Results Analysis

#### 2.4.1. Five-Level Results

#### 2.4.2. Seven-Level Results

## 3. Results

#### 3.1. Five-Level Implementation Results

#### 3.2. Seven-Level Implementation Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**A quarter-cycle of a seven-level signal voltage, each line aligns the relative magnitude with its respective $\theta $.

No. | Parameter | Code | Level 1 | Level 2 | Level 3 | Level 4 |
---|---|---|---|---|---|---|

1 | Migration | A | Forward | Both | - | - |

2 | Population size | B | 50 | 100 | 150 | 200 |

3 | Fitness scaling function | C | Proportional | Rank | Top | Shift linear |

4 | Selection function | D | Uniform | Tournament | Roulette | Stochastic uniform |

5 | Elite count | E | 1 | 5 | 10 | 15 |

6 | Crossover fraction | F | 0.3 | 0.5 | 0.7 | 0.9 |

7 | Mutation function | G | Uniform | Constraint dependent | Adaptive feasible | Gaussian |

8 | Crossover function | H | Single point | Two-point | Arithmetic | Scattered |

9 | Hybrid function | I | None | Fmin-search | Pattern-search | Fminunc |

GA Solver Parameters | |||||||||
---|---|---|---|---|---|---|---|---|---|

Experiment | A | B | C | D | E | F | G | H | I |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

4 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

5 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 3 | 4 |

6 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 3 |

7 | 1 | 2 | 3 | 3 | 4 | 4 | 1 | 1 | 2 |

8 | 1 | 2 | 4 | 4 | 3 | 3 | 2 | 2 | 1 |

9 | 1 | 3 | 1 | 2 | 3 | 4 | 1 | 2 | 3 |

10 | 1 | 3 | 2 | 1 | 4 | 3 | 2 | 1 | 4 |

11 | 1 | 3 | 3 | 4 | 1 | 2 | 3 | 4 | 1 |

12 | 1 | 3 | 4 | 3 | 2 | 1 | 4 | 3 | 2 |

13 | 1 | 4 | 1 | 2 | 4 | 3 | 3 | 4 | 2 |

14 | 1 | 4 | 2 | 1 | 3 | 4 | 4 | 3 | 1 |

15 | 1 | 4 | 3 | 4 | 2 | 1 | 1 | 2 | 4 |

16 | 1 | 4 | 4 | 3 | 1 | 2 | 2 | 1 | 3 |

17 | 2 | 1 | 1 | 4 | 1 | 4 | 2 | 3 | 2 |

18 | 2 | 1 | 2 | 3 | 2 | 3 | 1 | 4 | 1 |

19 | 2 | 1 | 3 | 2 | 3 | 2 | 4 | 1 | 4 |

20 | 2 | 1 | 4 | 1 | 4 | 1 | 3 | 2 | 3 |

21 | 2 | 2 | 1 | 4 | 2 | 3 | 4 | 1 | 3 |

22 | 2 | 2 | 2 | 3 | 1 | 4 | 3 | 2 | 4 |

23 | 2 | 2 | 3 | 2 | 4 | 1 | 2 | 3 | 1 |

24 | 2 | 2 | 4 | 1 | 3 | 2 | 1 | 4 | 2 |

25 | 2 | 3 | 1 | 3 | 3 | 1 | 2 | 4 | 4 |

26 | 2 | 3 | 2 | 4 | 4 | 2 | 1 | 3 | 3 |

27 | 2 | 3 | 3 | 1 | 1 | 3 | 4 | 2 | 2 |

28 | 2 | 3 | 4 | 2 | 2 | 4 | 3 | 1 | 1 |

29 | 2 | 4 | 1 | 3 | 4 | 2 | 4 | 2 | 1 |

30 | 2 | 4 | 2 | 4 | 3 | 1 | 3 | 1 | 2 |

31 | 2 | 4 | 3 | 1 | 2 | 4 | 2 | 4 | 3 |

32 | 2 | 4 | 4 | 2 | 1 | 3 | 1 | 3 | 4 |

Code | Parameter | Five-Level Inverter Parameters |
---|---|---|

A | Migration direction | Both |

B | Population size | 100 |

C | Fitness scaling function | Rank |

D | Selection function | Tournament |

E | Elite count | 15 |

F | Crossover fraction | $0.5$ |

G | Mutation function | Constraint dependent |

H | Crossover function | Two-point |

I | Hybrid function | Fminunc |

Code | Parameter | Five-Level Inverter |
---|---|---|

A | Migration direction | Forward |

B | Population size | 200 |

C | Fitness scaling function | Rank |

D | Selection function | Uniform |

E | Elite count | 5 |

F | Crossover fraction | $0.5$ |

G | Mutation function | Uniform |

H | Crossover function | Scattered |

I | Hybrid function | None |

5-Level MLI | 7-Level MLI | |
---|---|---|

${\theta}_{1}$ | ${13.406}^{\circ}$ | ${8.692}^{\circ}$ |

${\theta}_{2}$ | ${41.915}^{\circ}$ | ${27.896}^{\circ}$ |

${\theta}_{3}$ | - | ${49.817}^{\circ}$ |

Theoretical THD | $15.299\%$ | $10.432\%$ |

Instrument | Manufacturer | Reference |
---|---|---|

Oscilloscope | Tektronix | MDO3024 |

Power sources | TOELLNER and Tektronix | 8951 and PWS4602 |

Power quality analyzer | Fluke | 43B |

Characteristic | Value |
---|---|

Voltage | 127 Vrms |

Current | $2.0$ A |

Frequency | 60 Hz |

Power | $1/20$ HP |

Speed | 1550 RPM |

Levels in the CHBMLI | Theoretical THD Voltage Value | Experimental THD Voltage Value |
---|---|---|

Five | $15.299\%$ | $15.3\%$ |

Seven | $10.432\%$ | $10.3\%$ |

Paper | Angle Values | THD |
---|---|---|

This paper | ${\theta}_{1}={13.406}^{\circ}$${\theta}_{2}={41.915}^{\circ}$ | $15.3\%$ (Experimental) |

[14] | ${\theta}_{1}={17.02}^{\circ}$${\theta}_{2}={43.01}^{\circ}$ | $16.14\%$ (Simulation) |

[35] | ${\theta}_{1}={14.33}^{\circ}$${\theta}_{2}={42.10}^{\circ}$ | $15.35\%$ (Simulation) |

Paper | Angle Values | THD |
---|---|---|

This paper | ${\theta}_{1}={8.692}^{\circ}$${\theta}_{2}={27.896}^{\circ}$${\theta}_{3}={49.817}^{\circ}$ | $10.3\%$ (Experimental) |

[14] | ${\theta}_{1}={12.00}^{\circ}$${\theta}_{2}={26.93}^{\circ}$${\theta}_{3}={55.49}^{\circ}$ | $11.57\%$ (Simulation) |

[35] | ${\theta}_{1}={8.08}^{\circ}$${\theta}_{2}={28.35}^{\circ}$${\theta}_{3}={50.18}^{\circ}$ | $10.47\%$ (Simulation) |

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## Share and Cite

**MDPI and ACS Style**

Lopez, A.R.; Sosa, J.M.; Sámano, C.; De León-Aldaco, S.E.; Aguayo-Alquicira, J.; López-Santos, O.
Optimization of the Voltage Total Harmonic Distortion in Multilevel Inverters by Using the Taguchi Method. *Machines* **2024**, *12*, 7.
https://doi.org/10.3390/machines12010007

**AMA Style**

Lopez AR, Sosa JM, Sámano C, De León-Aldaco SE, Aguayo-Alquicira J, López-Santos O.
Optimization of the Voltage Total Harmonic Distortion in Multilevel Inverters by Using the Taguchi Method. *Machines*. 2024; 12(1):7.
https://doi.org/10.3390/machines12010007

**Chicago/Turabian Style**

Lopez, Adolfo R., José M. Sosa, Cristian Sámano, Susana Estefany De León-Aldaco, Jesus Aguayo-Alquicira, and Oswaldo López-Santos.
2024. "Optimization of the Voltage Total Harmonic Distortion in Multilevel Inverters by Using the Taguchi Method" *Machines* 12, no. 1: 7.
https://doi.org/10.3390/machines12010007