# Power Transformer Diagnosis Based on Dissolved Gases Analysis and Copula Function

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Copula Function and Common Factor Analysis

#### 2.1. Copula Function

_{1}, x

_{2},…, x

_{n}of marginal distributions F

_{1}, F

_{2},…, F

_{n}, then there is a copula function C, for any X ϵ R:

_{1}, F

_{2},…, F

_{n}are continuous, then C is unique, and the copula function C can be derived from Equation (1):

#### 2.2. Common Factor Analysis

^{−1}sets other off-diagonal elements of the inverse matrix of the correlation matrix R to 0, and the KMO statistic is:

_{ij}and q

_{ij}are the correlation matrix R and the ith row and jth column elements of the image correlation matrix, and the KMO statistic is between 0 and 1. When the sum of squares of correlation coefficients among all variables is much larger than the sum of squares of partial correlation coefficients, the closer the KMO value is to 1, the stronger the correlation between variables, the more suitable the original variables are for factor analysis. This is vice versa when the sum of squares of the simple correlation coefficients is close to 0.

_{0}and the alternative hypothesis Ha of the Bartlett sphericity test are that the variances of the k samples are equal and that there are at least one group of samples with unequal variances, respectively. Its statistic can be calculated by the following formula:

_{i}is the data size of the i-th sample, ${s}_{i}^{2}$ the variance of the i-th sample, ${s}_{p}^{2}$ is the combined variance of the samples, which is defined as:

## 3. Copula Function Selection and Parameter Fitting

#### 3.1. Bayesian Estimation of θ

#### 3.2. Goodness of Fitting

## 4. Study Cases

#### 4.1. Factor Analysis of Dissolved Gas in Oil

#### 4.2. Marginal Distribution Fitting

#### 4.3. Copula Function Selection and Fitting

#### 4.4. Transformer Status Diagnosis of Power Transformer

## 5. Conclusions

- (1)
- Compared with other combinations of fault characteristic gases, the copula function CDF boundary whose marginal variable is hydrocarbon gas can separate the transformer healthy and faulty status, and the joint probability of the dissolved gas in oil is correlated with the transformer healthy or defective status.
- (2)
- When the CDF result of the dissolved gas in the oil in the copula function is close to 0.8, the fluctuation of its gas concentration leads to a sharp change in the probability that the data diagnosis results are defective or healthy.
- (3)
- Based on the correlation between the joint probability of dissolved gas in the oil and the transformer state (healthy or defective), the nearest neighbor algorithm can be used to evaluate the correlation between the data of the unknown state and the known state data in the copula function, and thus the unknown data state can be evaluated.
- (4)
- The binary copula function selected in this paper has a low degree of freedom and avoids deep network training. The copula function is used to separate the correlation structure between the marginal distribution and random variables, which simplifies the multivariate probability modeling process and is beneficial to field applications and implementation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Symbols and Acronyms

AIC | Akaike information criterion |

BIC | Bayesian information criterion |

CDF | cumulative distribution probability |

DGA | dissolved gas analysis |

KMO | Kaiser–Meyer–Olkin |

NSE | Nash coefficient |

RMSE | root mean square error |

THC | total hydrocarbons |

k | the sample size |

R | correlation matrix |

Q | anti-image correlation matrix |

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**Figure 2.**Copula joint distribution of different variable combinations (Methane vs. Hydrogen, Methane vs. Ethane).

**Figure 4.**Copula function of the transformer oil color spectrum data fitting with methane and ethane. (

**a**) Frank copula, (

**b**) Gaussian copula, (

**c**) Gumbel copula, (

**d**) T copula, and (

**e**) Clayton copula.

Dissolved Gases | Common Factor Loading #1 | Common Factor Loading #2 |
---|---|---|

H_{2} | 0.271 | 0.460 |

CH_{4} | 0.916 | 0.301 |

C_{2}H_{6} | 0.993 | 0.085 |

C_{2}H_{4} | 0.774 | 0.602 |

C_{2}H_{2} | 0.088 | 0.746 |

THC (total hydrocarbons) | 0.886 | 0.457 |

Copula Function Type | RMSE | NSE | Parameters of Fitting |
---|---|---|---|

Gaussian | 1.1549 | 0.9767 | 0.8301 |

T | 0.9455 | 0.9844 | 0.9001, 7.8203 |

Frank | 0.7690 | 0.9897 | 15.9219 |

Clayton | 1.8818 | 0.9382 | 1.3189 |

Gumbel | 0.9886 | 0.9826 | 3.6969 |

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**MDPI and ACS Style**

Zhang, X.; Zhu, H.; Li, B.; Wu, R.; Jiang, J. Power Transformer Diagnosis Based on Dissolved Gases Analysis and Copula Function. *Energies* **2022**, *15*, 4192.
https://doi.org/10.3390/en15124192

**AMA Style**

Zhang X, Zhu H, Li B, Wu R, Jiang J. Power Transformer Diagnosis Based on Dissolved Gases Analysis and Copula Function. *Energies*. 2022; 15(12):4192.
https://doi.org/10.3390/en15124192

**Chicago/Turabian Style**

Zhang, Xiaoqin, Hongbin Zhu, Bo Li, Ruihan Wu, and Jun Jiang. 2022. "Power Transformer Diagnosis Based on Dissolved Gases Analysis and Copula Function" *Energies* 15, no. 12: 4192.
https://doi.org/10.3390/en15124192