# Power Transformer Fault Diagnosis Based on Dissolved Gas Analysis by Correlation Coefficient-DBSCAN
## 3 Improved CCDBSCAN Algorithm  
### Introduction of Correlation Coefficient  
The correlation coefficient matrix \(R\) is defined as the matrix formed by the correlation coefficients of each parameter vector and all other vectors in the dataset, i.e.,  
\[R=\left[ \begin{array}{cccc}r_{11}&r_{12}&\cdots&r_{1n}\\ r_{21}&r_{22}&\cdots&r_{2n}\\ \vdots&\vdots&&\vdots\\ r_{n1}&r_{n2}&\cdots&r_{nn}\end{array} \right] \tag{3}\]  
\[r_{ij}=\frac{\text{cov}(t_{i},t_{j})}{\sqrt{D(t_{i})}\sqrt{D(t_{j})}} \tag{4}\]  
where, in Equation (4), \(R\) is the correlation coefficient of transformer oil characteristic gas vector \(t_{i}\) and \(t_{j}\); \(\text{cov}(t_{i},t_{j})\) is the covariance of \(t_{i}\) and \(t_{j}\); \(D(t_{i})\) and \(D(t_{j})\) are the covariance of \(t_{i}\) and \(t_{j}\), respectively. The classification of faults in Publication 60,599 is according to the main types of faults that can be reliably identified by the equipment after the fault has occurred in service [27]:  
Partial discharge (PD): under the electric field, partial discharge will be triggered in the area with weak insulation performance in the transformer insulation system [28]. Partial discharge is electric discharge that only partially bridges the insulation between conductors. Corona partial discharge is evidenced by the formation of x-wax [29]. Discharges of low energy (D1): due to sparking, treeing and tracking evidenced by significant paper punctures, carbonization of paper surface plus carbon particles in oil. Discharges of high energy (D2): with power follow through, evidenced by extensive carbonization, metal fusion, and possible tripping of the equipment. Thermal faults below 300 \({}^{\circ}\)C (T1): evidenced by brownish paper. If paper has carbonized, thermal faults above 300 \({}^{\circ}\)C but below 700 \({}^{\circ}\)C (T2). Thermal faults above 700 \({}^{\circ}\)C (T3): evidenced by oil carbonization, metal coloration, or fusion. For the power transformers, the reasons for its failure include the rationality of its structure design, the quality of its insulation performance, and more importantly, the various stresses it needs to bear in the process of work. These stresses include all kinds of over-voltage and -heating, that is, thermal stress and electrical stress. DGA data of transformers provides information of electric and thermal stress of oil immersed power transformers [30]. Oil and paper are decomposed due to electrical and thermal stresses. Both are insulation materials of transformer. These two stresses can lead to the breakdown of insulating materials and release of gas decomposition products [31]. [PERSON] [32] described the types of faults associated with these gases. The decomposition of transformer oil produces hydrogen (H\({}_{2}\)), methane (CH\({}_{4}\)), acetylene (C\({}_{2}\)H\({}_{2}\)), ethylene (C\({}_{2}\)H\({}_{4}\)) and ethane (C\({}_{2}\)H\({}_{6}\)). CH\({}_{4}\) and C\({}_{2}\)H\({}_{6}\) are related to low-temperature oil breakdown, while C\({}_{2}\)H\({}_{4}\) is related to high-temperature oil breakdown, C\({}_{2}\)H\({}_{2}\) is related to discharge, and H\({}_{2}\) is related to partial discharge. In order to illustrate the existence of DGA gas content correlation of the same fault transformer and explain the existing problems when clustering directly using correlation, 12 pieces of DGA data were selected in the typical application case of power grid equipment state detection technology prepared by the operation and maintenance department of State Grid Corporation of China [33]. Fault types include T1, T2, T3, PD, D1, D2, and each fault includes two. First, calculate the percentage content of hydrogen and various hydrocarbon gases in the total gas of 12 data, and convert the absolute content of gas into the relative content, and then the correlation analysis is carried out according to Equation (4); finally, the correlation coefficient matrix \(R\) is obtained. The line chart of DGA gas content of 12 fault transformers is shown in Figure 1.  
Calculate the correlation according to Equation (4).