## 3 Case Study
### Distribution Parameters and Condition Data Estimations  
Next, the distribution parameters for two-parametric Weibull and normal were computed based on (5)-(18). Table 2 tabulates an example of Weibull and normal distribution parameters fittings for dielectric breakdown voltage and acidity. For dielectric breakdown voltage, the distribution parameters for two-parametric Weibull and normal distribution parameters are shown in Fig. 3.  
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.  
voltage data, the mean \(\mu\) shows an apparent linear decrement trend as the transformer age increases. The standard deviation \(\sigma\) only shows a slight decrement trend with the increment of transformer age. For the acidity data, the \(\alpha\) initially fluctuates between 0.022 and 0.0323 for year 1-4. It starts to increase significantly as the transformer age increases from year 4 to year 10. After year 10, it stabilizes between 0.0712 and 0.085 after year 10. The \(\beta\) for year 1 is relatively low as compared to other data and it is due to the poor fittings of Weibull distribution as shown in Figure 2a. Nonetheless, a decrement pattern is observed for the \(\beta\) as the transformer age decreases. Next, the distribution parameters from year 16 to 25 were fitted and extrapolated while using the curve fitting process based on the WLS method, as shown in Figures 4 and 5. It is quite difficult to obtain high \(R^{2}\) for all of the fittings due to the large variation of the distribution parameters. However, this limitation needs to be considered in this study in order to obtain the representative model for the transformer population. It is important to be noted, due to the nature of scoring and weighting HI technique used in this study, the variations of the individual condition parameters data will be less sensitive, since the calculation itself is based on aggregation method, whereby some of the values have a small effect on the overall model itself. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Transformer Age**} & \multicolumn{2}{c}{**Dielectric Breakdown Voltage**} & \multicolumn{2}{c}{**Acidity**} \\ \cline{2-5}  & \(\mu\) & \(\sigma\) & \(\alpha\) & \(\beta\) \\ \hline
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 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computed distribution parameters for dielectric breakdown voltage and acidity. Figure 4: Parameters estimation for dielectric breakdown voltage based on the normal distribution model. The exponential-based model was chosen for the curve fitting process, since it can provide the highest \(R^{2}\) when compared to other models. For dielectric breakdown voltage data, the fittings of \(\mu\) and \(\sigma\) exponentially decrease as the transformer age is increased. Based on the extrapolation, the \(\mu\) and \(\sigma\) at year 25 are 35.005 and 18.434, respectively, as shown in Table 3. For the acidity data, the fitting of \(\alpha\) increases exponentially as the transformer age increases. On the other hand, the fitting of the \(\beta\) shows a slight decrement trend. Table 3 presents the estimated distribution parameters for dielectric breakdown voltage and acidity from year 16 to 25.