# Application of Statistical Distribution Models to Predict Health Index for Condition-Based Management of Transformers

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## Abstract

**:**

## 1. Introduction

## 2. Transformer Health Index Estimation Model

_{2}), methane (CH

_{4}), acetylene (C

_{2}H

_{2}), ethylene (C

_{2}H

_{4}), ethane (C

_{2}H

_{6}), carbon monoxide (CO), and carbon dioxide (CO

_{2}). The oil quality analysis (OQA) included dielectric breakdown voltage, interfacial tension, color, acidity, and water content. The furanic compound analysis (FCA) that consisted of 2-furfuraldehyde (2-FAL) was also used in the study to consider the in-service ageing of the solid insulation. It is important to note that this study does not consider any abnormal data due to the unusual operating environment, internal electrical faults, and electromagnetic interference during in-service.

#### 2.1. Estimations Distribution Parameters Estimation

#### 2.2. Condition Data Estimation

#### 2.3. Health Index Model Based on Scoring Algorithm

## 3. Case Study

#### 3.1. Implementation of Statistical Distribution Models to Transformer CBM Data

#### 3.2. Distribution Parameters and Condition Data Estimations

_{2}deviates from the computed H

_{2}during the first two years, between year 4–7 and 17–21, as shown in Figure 7a. Both of the predicted and computed H

_{2}maintain in “very good” condition for 25 years. The predicted CH

_{4}still follows the decrement trend of the computed CH

_{4}, regardless of the deviation, as seen in Figure 7b. The predicted and computed CH

_{4}remains in “very good” condition for 25 years. A few of the predicted CO show reasonable agreement with the computed CO between year 4 and 23, as shown in Figure 7c. The deviation between the predicted and computed CO occurs between year 1–3 and year 24–25. The predicted CO maintains in “very good” condition during the first seven years and later transits to the “good” condition. The computed CO maintains in “very good” during the first six years. Between year 7 and 23, it is in “good” condition. The computed CO reinstates to the “very good” condition after 23 years. The majority of the predicted CO

_{2}deviates from the computed CO

_{2}, as shown in Figure 7d. The predicted CO

_{2}is in “very good” condition during the first two years. It is in a “good” condition between year 3 and 7. After seven years, the predicted CO

_{2}remains in a “fair” condition. The computed CO

_{2}is in “very good” condition during the first three years. From year 4 to 6, the computed CO

_{2}is in “good” condition. It enters a “fair” condition after year 6. It reinstates to “good” condition between year 21 and 23, and later transits to “very good” condition. The predicted C

_{2}H

_{4}is close to computed C

_{2}H

_{4}during the first 24 years, as shown in Figure 7e. It deviates from computed C

_{2}H

_{4}at year 25. Predicted and computed C

_{2}H

_{4}both maintain in “very good” condition for 25 years. Apparent deviation between predicted and computed C

_{2}H

_{6}, as shown in Figure 7f. The predicted and computed C

_{2}H

_{6}are in “very good” condition for 25 years. Similarly, the predicted C

_{2}H

_{2}shows a clear deviation from the computed C

_{2}H

_{2}, as shown in Figure 7g. The predicted C

_{2}H

_{2}is in “good” condition during the first 10 years. After year 8, the predicted C

_{2}H

_{2}remains in “fair” conditions until 25 years. On the other hand, the computed C

_{2}H

_{2}is in “good” condition during the first eight years. From year 10 to 19, the computed C

_{2}H

_{2}is a “fair” condition. After year 19, it remains in “good” condition and later transits to “fair” condition after year 23.

_{2}H

_{2}, CO, and CO

_{2}fitting normal distribution. C

_{2}H

_{6}has the highest ${R}^{2}$ with 0.7155 and CO

_{2}has the lowest ${R}^{2}$ with 0.2375. The exponential-based model was chosen for the curve fitting process for all dissolved gas parameters data, except for C

_{2}H

_{4}, which was curve fitted by the power-based model. The justification of the chosen distributions for dissolved gas parameters data is the same as the oil quality and furanic compound parameters data.

## 4. Conclusions

_{2}, and C

_{2}H

_{2}under study could be represented by the normal distribution. The Weibull distribution is suitable for representing the IFT, acidity, water content, H

_{2}, CH

_{4}, C

_{2}H

_{6}, and C

_{2}H

_{4}. It is found that SDM can be utilized to estimate the HI of transformers while using individual condition parameters data. The predicted accuracy is subject to the obtainability of the data at various ages. Predominantly, the trends of the predicted HI are close to the computed HI. The hypothesis testing from the results using Chi-square shows that the ${X}^{2}$ value of HI data is 12.94, where it falls outside the rejection area at 0.05 significance level. The overall average percentages of absolute errors in training and validation regions are 0.65% and 2.17%, respectively. The predicted HI of transformers based on SDM yields accuracy of about 97.83%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

AI | Artificial intelligence |

CBM | Condition-based management |

CDF | Cumulative distribution function |

CH_{4} | Methane |

C_{2}H_{2} | Acetylene |

C_{2}H_{4} | Ethylene |

C_{2}H_{6} | Ethane |

CO | Carbon monoxide |

CO_{2} | Carbon dioxide |

CDF | Cumulative distribution function |

2-FAL | 2-Furfuraldehyde |

g/cm^{3} | gram per cubic centimeter |

H_{2} | Hydrogen |

HI | Health index |

ICDF | Inverse cumulative distribution function |

KOH/g | mass of potassium hydroxide per grams |

kV | kilo-volt |

OLS | Ordinary least square |

MLE | Maximum likelihood estimate |

mg | milligrams |

mN/m | millinewton per metre |

MOM | Method of moments |

Probability distribution function | |

ppm | parts-per-million |

ppb | parts-per-billion |

WLS | Weighted least square method |

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**Figure 2.**Normal distribution probability plot for dielectric breakdown voltage from year (

**a**) 1 to 3; (

**b**) 4 to 6; (

**c**) 7 to 9; (

**d**) 10 to 12; and, (

**e**) 13 to 15.

**Figure 3.**Weibull distribution probability plot for acidity from year (

**a**) 1 to 3; (

**b**) 4 to 6; (

**c**) 7 to 9; (

**d**) 10 to 12; and, (

**e**) 13 to 15.

**Figure 4.**Parameters estimation for dielectric breakdown voltage based on the normal distribution model.

**Figure 6.**Comparison between computed and predicted (

**a**) dielectric breakdown voltage; (

**b**) water content; (

**c**) interfacial tension; (

**d**) color; (

**e**) acidity; and, (

**f**) 2-furfuraldehyde.

**Figure 7.**Comparison between computed and predicted (

**a**) hydrogen; (

**b**) methane; (

**c**) carbon monoxide; (

**d**) carbon dioxide; (

**e**) ethylene; (

**f**) ethane; and, (

**g**) acetylene.

Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

Dataset | 9 | 22 | 26 | 30 | 32 | 37 | 33 | 49 | 52 | 52 | 56 | 77 | 76 | 61 |

Year | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | Total | ||

Dataset | 113 | 116 | 86 | 103 | 87 | 65 | 55 | 37 | 31 | 10 | 7 | 1322 |

Transformer Age | Dielectric Breakdown Voltage | Acidity | ||
---|---|---|---|---|

$\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | |

1 | 61.444 | 24.587 | 0.0323 | 0.9318 |

2 | 64.000 | 20.459 | 0.0220 | 1.8741 |

3 | 59.808 | 24.832 | 0.0227 | 1.6902 |

4 | 63.862 | 22.696 | 0.0298 | 1.6803 |

5 | 59.250 | 22.489 | 0.0457 | 1.8153 |

6 | 57.460 | 21.898 | 0.0384 | 1.7849 |

7 | 65.727 | 19.391 | 0.0617 | 1.9102 |

8 | 52.265 | 27.099 | 0.0810 | 1.6304 |

9 | 49.308 | 24.218 | 0.0588 | 1.2918 |

10 | 50.654 | 19.531 | 0.0712 | 1.2977 |

11 | 56.000 | 21.916 | 0.0880 | 1.4233 |

12 | 52.605 | 18.849 | 0.0716 | 1.1990 |

13 | 46.324 | 19.959 | 0.0727 | 1.3391 |

14 | 50.656 | 21.168 | 0.0714 | 0.9885 |

15 | 48.566 | 21.355 | 0.0850 | 1.0715 |

Transformer Age | Dielectric Breakdown Voltage | Acidity | ||
---|---|---|---|---|

$\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | |

16 | 43.877 | 20.170 | 0.0719 | 1.2458 |

17 | 42.790 | 19.970 | 0.0779 | 1.2175 |

18 | 41.729 | 19.771 | 0.0844 | 1.1898 |

19 | 40.695 | 19.574 | 0.0914 | 1.1627 |

20 | 39.686 | 19.379 | 0.0991 | 1.1363 |

21 | 38.702 | 19.187 | 0.1073 | 1.1105 |

22 | 37.743 | 18.996 | 0.1163 | 1.0852 |

23 | 36.807 | 18.807 | 0.1259 | 1.0605 |

24 | 35.895 | 18.619 | 0.1364 | 1.0364 |

25 | 35.005 | 18.434 | 0.1478 | 1.0129 |

Parameter | Fitted Distribution | Master Curve Equation | ${\mathit{R}}^{2}$ |
---|---|---|---|

Dielectric breakdown voltage | Normal | y = exp(4.204 − 0.026x + 0.0001x^{2}) | 0.8379 |

Water content | Weibull | y = exp(2.181 + 0.423x − 0.0004x^{2}) | 0.6167 |

Interfacial tension | Weibull | y = exp(3.468 − 0.056x + 0.002x^{2}) | 0.3602 |

Color | Normal | y = 0.321(1 + x)^{0.727} | 0.9044 |

Acidity | Weibull | y = 0.015(1 + x)^{0.531} | 0.6224 |

2-FAL | Normal | y = 9.088x^{1.459} | 0.4674 |

Parameter | Fitted Distribution | Master Curve Equation | ${\mathit{R}}^{2}$ |
---|---|---|---|

H_{2} | Weibull | y = exp(4.491 + 0.021x − 0.005x^{2}) | 0.5986 |

CH_{4} | Weibull | y = exp(3.121 − 0.056 + 0.001x^{2}) | 0.4785 |

CO | Normal | y = −305.6exp^{0.017x} | 0.2375 |

CO_{2} | Normal | y = 2000exp^{0.035x} | 0.5168 |

C_{2}H_{4} | Weibull | y = 2.183x^{0.732} | 0.6714 |

C_{2}H_{6} | Weibull | y = exp(4.090 − 0.135 + 0.002x^{2}) | 0.7155 |

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## Share and Cite

**MDPI and ACS Style**

Mohd Selva, A.; Azis, N.; Shariffudin, N.S.; Ab Kadir, M.Z.A.; Jasni, J.; Yahaya, M.S.; Talib, M.A.
Application of Statistical Distribution Models to Predict Health Index for Condition-Based Management of Transformers. *Appl. Sci.* **2021**, *11*, 2728.
https://doi.org/10.3390/app11062728

**AMA Style**

Mohd Selva A, Azis N, Shariffudin NS, Ab Kadir MZA, Jasni J, Yahaya MS, Talib MA.
Application of Statistical Distribution Models to Predict Health Index for Condition-Based Management of Transformers. *Applied Sciences*. 2021; 11(6):2728.
https://doi.org/10.3390/app11062728

**Chicago/Turabian Style**

Mohd Selva, Amran, Norhafiz Azis, Nor Shafiqin Shariffudin, Mohd Zainal Abidin Ab Kadir, Jasronita Jasni, Muhammad Sharil Yahaya, and Mohd Aizam Talib.
2021. "Application of Statistical Distribution Models to Predict Health Index for Condition-Based Management of Transformers" *Applied Sciences* 11, no. 6: 2728.
https://doi.org/10.3390/app11062728