# Estimation of the Equivalent Circuit Parameters in Transformers Using Evolutionary Algorithms

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## Abstract

**:**

## 1. Introduction

## 2. The Single-Phase Transformer Equivalent Circuit

${R}_{1}$: primary winding resistance; |

$j{X}_{1}$: primary winding reactance; |

${R}_{2}^{\prime}$: secondary winding resistance with respect to the primary side; |

$j{X}_{2}^{\prime}$: secondary leakage reactance with respect to the primary side; |

${R}_{c}$: core resistance; |

$j{X}_{m}$: magnetizing reactance. |

## 3. Optimization Algorithms

#### 3.1. The Genetic Algorithm

Algorithm 1: GA |

Require:${I}_{1},$${I}_{2}^{\prime}$ and ${V}_{2}^{\prime}$; 1: for i = 1 to NP do2: Create ${X}_{i}=[{R}_{1},{X}_{1},{R}_{2}^{\prime},{X}_{2}^{\prime},{R}_{c},{X}_{m}]$; 3: end for 4: for i = 1 to NP do 5: Evaluate (12); 6: end for 7: for gen = 1 to NG do 8: Select NP individuals based on fitness from X; 9: Apply crossover operator to individuals selected to generate NP children; 10: Apply mutation operator to the NP children; 11: Keep the NP children and discard the NP individuals in X, just keeping the best solution to replace the worst child; 12: end forEnsure: ${X}_{sol}=[{R}_{1},{X}_{1},{R}_{2}^{\prime},{X}_{2}^{\prime},{R}_{c},{X}_{m}]$. |

#### 3.2. Particle Swarm Optimization

Algorithm 2: PSO |

Require:${I}_{1},$${I}_{2}^{\prime}$ and ${V}_{2}^{\prime}$; 1: for i = 1 to NP do2: Create $Particl{e}_{i}=[{R}_{1},{X}_{1},{R}_{2}^{\prime},{X}_{2}^{\prime},{R}_{c},{X}_{m}]$; 3: end for4: for i = 1 to NP do5: Evaluate (12); 6: end for7: while g < NGs do8: for i = 1 to NP do9: for j = 1 to D = (6) do10: ${r}_{1}$, ${r}_{2}$ = $rand[0,1]$; 11: Update velocity (13); 12: Update position (14); 13: end for14: if $f\left(Particl{e}_{i}^{g}\right)\le f\left(pbes{t}_{i}^{g-1}\right)$ then15: $pbes{t}_{i}^{g}=Particl{e}_{i}^{g}$; 16: end if17: end for18: if $f\left(Particl{e}_{i}^{g}\right)\le f\left(gbes{t}_{i}^{g-1}\right)$ then19: $gbes{t}_{i}^{g}=Particl{e}_{i}^{g}$; 20: end if21: end whileEnsure: $Particl{e}_{sol}=[{R}_{1},{X}_{1},{R}_{2}^{\prime},{X}_{2}^{\prime},{R}_{c},{X}_{m}]$. |

#### 3.3. The Gravitational Search Algorithm

Algorithm 3: GSA |

Require:${I}_{1},$${I}_{2}^{\prime}$ and ${V}_{2}^{\prime}$; 1: for i = 1 to NP do2: Create $mas{s}_{i}=[{R}_{1},{X}_{1},{R}_{2}^{\prime},{X}_{2}^{\prime},{R}_{c},{X}_{m}]$; 3: end for4: for i = 1 to NP do5: Evaluate (12); 6: end for7: for t = 1 to NGs do8: $G\left(t\right)=Goexp\left(-\alpha \frac{t}{NGs}\right)$; 9: $best\left(t\right)=mi{n}_{j\in [1,\dots ,NP]}fi{t}_{j}\left(t\right)$; 10: $worst\left(t\right)=ma{x}_{j\in [1,\dots ,NP]}fi{t}_{j}\left(t\right)$; 11: ${M}_{i}={M}_{ii}={M}_{ai}={M}_{pi}$; 12: ${m}_{i}\left(t\right)=\frac{fi{t}_{i}\left(t\right)-worst\left(t\right)}{best\left(t\right)-worst\left(t\right)}$; 13: ${M}_{i}=\frac{{m}_{i}\left(t\right)}{{\sum}_{j=1}^{NP}{m}_{j}\left(t\right)}$; 14: for i = 1 to NP do15: for j = 1 to Kbest do16: if $i\ne j$ then17: ${F}_{ij}\left(t\right)=G\left(t\right)\frac{{M}_{i}\left(t\right)}{Rij{\left(t\right)}^{R}+\u03f5}(mas{s}_{j}\left(t\right)-mas{s}_{i}\left(t\right))$; 18: end if19: end for20: end for21: for i = 1 to NP do22: for j = 1 to d = 6 do23: ${a}_{i}^{d}=\frac{{F}_{i}^{d}}{Mii\left(t\right)}$; 24: ${v}_{i}^{d}(t+1)=rand[0,1]{v}_{i}^{d}\left(t\right)+{a}_{i}^{d}$; 25: $mas{s}_{i}^{d}(t+1)=mas{s}_{i}^{d}\left(t\right)+{v}_{i}^{d}(t+1)$; 26: end for27: end for28: end forEnsure: $mas{s}_{sol}=[{R}_{1},{X}_{1},{R}_{2}^{\prime},{X}_{2}^{\prime},{R}_{c},{X}_{m}]$. |

## 4. Simulation and Results

#### 4.1. The Single-Phase Transformer: 4 kVA, 50 Hz, 250–125 V

#### 4.2. The Single-Phase Transformer: 15 kVA, 50 Hz, 2400–240 V

#### 4.3. The Single-Phase Transformer: 33 MVA, 60 Hz, 230,000/**$\sqrt{3}$**–34,500 V

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GA | genetic algorithm; |

PSO | particle swarm optimization; |

GSA | gravitational search algorithm; |

STPECP | single-phase transformer equivalent circuit parameters; |

RMS | root mean square. |

## References

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**Figure 3.**Convergence curves of the 4 kVA single-phase transformer using the GA, PSO and the GSA methods.

**Figure 5.**Convergence curves of the 15 kVA single-phase transformer using the GA, PSO and the GSA methods.

**Figure 7.**Convergence curves of the 33 MVA single-phase transformer using the GA, PSO and the GSA methods.

**Table 1.**Statistical values of the SPTECPs using the optimization methods for the single-phase transformer, 4 kVA, 250–125 V at 50 Hz.

Methods | PSO | GA | GSA | ||||
---|---|---|---|---|---|---|---|

Stats | Fitness | AE (%) | Fitness | AE (%) | Fitness | AE (%) | |

Best | 6.4729 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 18.16 | 7.4670 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 10.93 | 7.9168 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 12.26 | |

Mean | 7.1537 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 4.73 | 7.6007 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 10.00 | 8.8683 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 5.72 | |

Medium | 7.1537 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 9.41 | 7.5987 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 10.91 | 8.8683 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.78 | |

Worst | 7.9736 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 24.28 | 7.7175 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 7.43 | 9.6510 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 8.45 | |

St. dev. | 3.2688 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | - | 6.7636 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | - | 4.7937 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | - | |

Wilcoxon rank-sum test with 95% confidence | + | + |

**Table 2.**Parameters obtained with the optimization methods for the single-phase transformer 4 kVA, 250–125 V at 50 Hz.

Parameters | ${\mathit{R}}_{1}(\Omega )$ | ${\mathit{X}}_{1}(\Omega )$ | ${\mathit{R}}_{2}^{\prime}(\Omega )$ | ${\mathit{X}}_{2}^{\prime}(\Omega )$ | ${\mathit{R}}_{\mathit{c}}(\Omega )$ | ${\mathit{X}}_{\mathit{m}}(\Omega )$ | AE (%) | |
---|---|---|---|---|---|---|---|---|

Methods | ||||||||

Ref. [4] | 0.4 | 0.2 | 0.4 | 2 | 1500 | 750 | - | |

GA | 0.3413 | 0.1879 | 0.4183 | 2.4543 | 1405 | 707.6 | - | |

GA error (%) | 14.6783 | 6.0567 | 4.5633 | 22.7133 | 6.3222 | 5.6458 | 10.00 | |

PSO | 0.3763 | 0.2150 | 0.4023 | 2.0256 | 1327 | 738.26 | - | |

PSO error (%) | 5.9125 | 7.5183 | 0.5742 | 1.2815 | 11.5444 | 1.5653 | 4.7327 | |

GSA | 0.3751 | 0.1904 | 0.3904 | 2.3839 | 1482.4333 | 745.9333 | - | |

GSA error (%) | 6.225 | 4.785 | 2.3883 | 19.1983 | 1.1711 | 0.5422 | 5.7183 |

Variables | ${\mathit{I}}_{1}$ (A) | ${\mathit{I}}_{2}^{\prime}$ (A) | ${\mathit{V}}_{2}^{\prime}$ (V) | Efficiency (%) | AE (%) | |
---|---|---|---|---|---|---|

Methods | ||||||

Ref. [4] | 14.0813 | 13.6893 | 235.8759 | 83.9990 | - | |

GA | 13.9666 | 13.7435 | 236.7835 | 93.1954 | - | |

GA error (%) | 0.8145 | 0.3958 | 0.3848 | 10.9483 | 3.1358 | |

PSO | 13.9698 | 13.7410 | 236.7577 | 93.1524 | - | |

PSO error (%) | 0.7921 | 0.3778 | 0.3738 | 10.8971 | 3.1102 | |

GSA | 13.9558 | 13.7451 | 236.8283 | 93.3009 | - | |

GSA error (%) | 0.8909 | 0.4077 | 0.4038 | 11.0738 | 3.1941 |

**Table 4.**Statistical values of the SPTECPs using the optimization methods for the single-phase transformer 15 kVA, 2400–240 V at 50 Hz.

Methods | GA | PSO | GSA | ||||
---|---|---|---|---|---|---|---|

Stats | Fitness | AE (%) | Fitness | AE (%) | Fitness | AE (%) | |

Best | 1.8178 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 12.3039 | 1.92 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 11.8855 | 1.8542 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 11.7523 | |

Mean | 1.8299 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 11.8772 | 2.0875 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 7.8132 | 1.8936 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 9.8612 | |

Medium | 1.8305 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 12.0800 | 2.0849 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 8.0294 | 1.8911 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 10.9429 | |

Worst | 1.8500 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 11.7215 | 2.2897 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 8.8045 | 1.9326 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 9.6136 | |

St. dev. | 7.4936 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | - | 8.7936 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | - | 1.8433 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | - | |

Wilcoxon rank-sum test with 95% confidence | + | + |

**Table 5.**Parameters obtained with the optimization methods for the single-phase transformer 15 kVA, 2400–240 V at 50 Hz.

Parameters | ${\mathit{R}}_{1}(\Omega )$ | ${\mathit{X}}_{1}(\Omega )$ | ${\mathit{R}}_{2}^{\prime}(\Omega )$ | ${\mathit{X}}_{2}^{\prime}(\Omega )$ | ${\mathit{R}}_{\mathit{c}}(\Omega )$ | ${\mathit{X}}_{\mathit{m}}(\Omega )$ | AE (%) | |
---|---|---|---|---|---|---|---|---|

Methods | ||||||||

Ref. [4] | 2.45 | 3.14 | 2 | 2.2294 | 105,000 | 9106 | - | |

GA | 2.0038 | 2.5440 | 1.5060 | 2.0531 | 105,730 | 9176 | - | |

GA error (%) | 18.2121 | 18.9820 | 24.6983 | 7.9095 | 0.6952 | 0.7662 | 11.8772 | |

PSO | 2.0568 | 2.9236 | 1.5504 | 2.2139 | 104,473 | 9130 | - | |

PSO error (%) | 16.0503 | 6.8928 | 22.48 | 0.6953 | 0.5016 | 0.2595 | 7.8132 | |

GSA | 2.0075 | 2.7198 | 1.5103 | 2.1694 | 104,453 | 9103 | - | |

GSA error (%) | 18.0612 | 13.3811 | 24.4833 | 2.6913 | 0.5206 | 0.0297 | 9.8612 |

**Table 6.**Electrical variables at full load of a single-phase transformer 15 kVA, 2400–240 V, at 50 Hz.

Variables | ${\mathit{I}}_{1}$ (A) | ${\mathit{I}}_{2}^{\prime}$ (A) | ${\mathit{V}}_{2}^{\prime}$ (V) | Efficiency (%) | AE (%) | |
---|---|---|---|---|---|---|

Methods | ||||||

Ref. [4] | 6.2 | 6.2 | 2383.8 | 98.5 | - | |

GA | 6.2128 | 6.1834 | 2377.4751 | 98.5928 | - | |

GA error (%) | 0.2072 | 0.2673 | 0.2674 | 0.0920 | 0.2085 | |

PSO | 6.2113 | 6.1815 | 2376.6855 | 98.5528 | - | |

PSO error (%) | 0.1829 | 0.2983 | 0.2985 | 0.0536 | 0.2083 | |

GSA | 6.2130 | 6.1864 | 2377.3001 | 98.5783 | - | |

GSA error (%) | 0.2091 | 0.2187 | 0.2727 | 0.0795 | 0.1950 |

**Table 7.**Statistical values of the SPTECPs using the optimization methods for the single-phase transforme 33 MVA, 230,000/$\sqrt{3}$–34,500 V at 60 Hz.

Methods | GA | PSO | GSA | ||||
---|---|---|---|---|---|---|---|

Stats | Fitness | AE (%) | Fitness | AE (%) | Fitness | AE (%) | |

Best | 4.1138 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 8.3152 | 4.0597 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 4.5892 | 4.1520 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 4.5154 | |

Mean | 4.2171 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 2.6105 | 4.3105 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 1.3703 | 4.3586 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 1.3415 | |

Medium | 4.2202 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 2.5620 | 4.3196 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 3.3515 | 4.3646 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 3.0648 | |

Worst | 4.3153 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 3.6331 | 4.5475 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 4.6009 | 4.5626 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 5.4431 | |

St. dev. | 5.0994 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | - | 1.0156 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | - | 1.1556 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | - | |

Wilcoxon rank-sum test with 95% confidence | + | + |

**Table 8.**Parameters obtained with the optimization methods for the single-phase transformer 33 MVA, 230,000/$\sqrt{3}$–34,500 V at 60 Hz.

Parameters | ${\mathit{R}}_{1}(\Omega )$ | ${\mathit{X}}_{1}(\Omega )$ | ${\mathit{R}}_{2}^{\prime}(\Omega )$ | ${\mathit{X}}_{2}^{\prime}(\Omega )$ | ${\mathit{R}}_{\mathit{c}}(\Omega )$ | ${\mathit{X}}_{\mathit{m}}(\Omega )$ | AE (%) | |
---|---|---|---|---|---|---|---|---|

Methods | ||||||||

IEM-Condumex | 0.835 | 37.5 | 0.835 | 37.5 | 728,504 | 513,610 | - | |

GA | 0.8724 | 38.99 | 0.8540 | 35.81 | 729,277 | 515,267 | - | |

GA error (%) | 4.4790 | 3.9645 | 2.2754 | 4.5155 | 0.1061 | 0.3226 | 2.6105 | |

PSO | 0.8105 | 37.15 | 0.8074 | 37.16 | 727,823 | 514,097 | - | |

PSO error (%) | 2.9341 | 0.9184 | 3.2814 | 0.8995 | 0.0934 | 0.0948 | 1.3703 | |

GSA | 0.8408 | 38.25 | 0.8002 | 37.13 | 727,743 | 514,170 | - | |

GSA error (%) | 0.6946 | 1.992 | 4.1677 | 0.9741 | 0.1044 | 0.1090 | 1.3415 |

**Table 9.**Electrical variables at full load of a single-phase transformer 33 MVA, 230,000/$\sqrt{3}$–34,500 V at 60 Hz.

Variables | ${\mathit{I}}_{1}$ (A) | ${\mathit{I}}_{2}^{\prime}$ (A) | ${\mathit{V}}_{2}^{\prime}$ (V) | Efficiency (%) | AE (%) | |
---|---|---|---|---|---|---|

Methods | ||||||

IEM-Condumex | 247.926 | 247.728 | 131,060 | 99.46 | - | |

GA | 247.9332 | 247.7348 | 131,053.0316 | 99.4567 | - | |

GA error (%) | 0.0025 | 0.0027 | 0.0053 | 0.0033 | 0.0035 | |

PSO | 247.9329 | 247.7345 | 131,052.887 | 99.4765 | - | |

PSO error (%) | 0.0028 | 0.0026 | 0.0054 | 0.0166 | 0.0069 | |

GSA | 247.9347 | 247.7364 | 131,053.8826 | 99.4722 | - | |

GSA error (%) | 0.0035 | 0.0034 | 0.0047 | 0.0123 | 0.0060 |

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## Share and Cite

**MDPI and ACS Style**

Ascencion-Mestiza, H.; Maximov, S.; Mezura-Montes, E.; Olivares-Galvan, J.C.; Ocon-Valdez, R.; Escarela-Perez, R.
Estimation of the Equivalent Circuit Parameters in Transformers Using Evolutionary Algorithms. *Math. Comput. Appl.* **2023**, *28*, 36.
https://doi.org/10.3390/mca28020036

**AMA Style**

Ascencion-Mestiza H, Maximov S, Mezura-Montes E, Olivares-Galvan JC, Ocon-Valdez R, Escarela-Perez R.
Estimation of the Equivalent Circuit Parameters in Transformers Using Evolutionary Algorithms. *Mathematical and Computational Applications*. 2023; 28(2):36.
https://doi.org/10.3390/mca28020036

**Chicago/Turabian Style**

Ascencion-Mestiza, Hector, Serguei Maximov, Efrén Mezura-Montes, Juan Carlos Olivares-Galvan, Rodrigo Ocon-Valdez, and Rafael Escarela-Perez.
2023. "Estimation of the Equivalent Circuit Parameters in Transformers Using Evolutionary Algorithms" *Mathematical and Computational Applications* 28, no. 2: 36.
https://doi.org/10.3390/mca28020036