## 3 Case Study
### Distribution Parameters and Condition Data Estimations
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. Table 5 summarizes the representative distributions for each of the dissolved gas parameters data. Based on the results, the majority of the dissolved gas parameters data fit Weibull distribution, except for C\({}_{2}\)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. Figure 8 shows the predicted HI obtained by statistical model in Figures 6 and 7 for a period of 25 years. It is observed that most of the predicted HI values are close to the computed HI. Based on Figure 9, there are considerably small deviations for the predicted HI at year 10, 19, and 24. The HI at year 17 recorded the highest deviation. Further \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Fitted Distribution** & **Master Curve Equation** & \(R^{2}\) \\ \hline 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 = -305.6 exp\({}^{0.017x}\) & 0.2375 \\ CO\({}_{2}\) & Normal & y = 2000 exp\({}^{0.035x}\) & 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 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The representative distribution for dissolved gas parameters data. hypothesis testing to measure the best-of-fit between the computed and predicted HI was performed while using the Chi-square statistic, as seen in Equation (24),  
\[X^{2}=\sum_{n=1}^{25}\frac{(P_{n}-C_{n})^{2}}{C_{n}} \tag{24}\]  
where \(n\) is the total year of the transformer in term of age, \(C_{n}\) is the computed HI at \(n\) year, \(P_{n}\) is the predicted HI at \(n\) year, and \(X^{2}\) is a Chi-square statistic coefficient with degree of freedom, \(n-1\). The significance level \(\alpha\) was set to 0.05, thus the rejection area fell after the critical value, which is 13.85. The \(X^{2}\) of HI is 12.94, where, at \(\alpha=0.05\), it falls outside the area of rejection. Figure 7: _Cont._  
Figure 8: Comparison between computed and predicted health index. Figure 7: Comparison between computed and predicted (**a**) hydrogen; (**b**) methane; (**c**) carbon monoxide; (**d**) carbon dioxide; (**e**) ethylene; (**f**) ethane; and, (**g**) acetylene. The average percentage error between the predicted and computed HI was performed based on Equation (25). \[Average\ percentage\ error\ (\%)=\ \frac{\sum_{n=1}^{25}\left(\frac{|C_{n}-P_{n}|}{|C_{n}|} \times 100\%\right)}{n} \tag{25}\]  
where \(C_{n}\) is the computed HI, \(P_{n}\) is the predicted HI, and \(n\) is the age of the transformer.