## 4 Results and Discussion  
This section presents the results obtained with the Weibull plots implemented in this study. Comparisons of the resulting maximum likelihood estimations (MLE) presented with a truncated Newton method (TNC) optimizer for each category of distribution-transformer failures with a 96% confidence interval (CI) are shown in Table 3.  
By identifying beta and alpha parameters and checking the goodness of fit, decisions can be made about the nature of the failure and its prevention. Table 4 provides information about the interpretation of the Weibull-analysis data consi \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Transformer Category** & **No. of Failures** & **Parameter** & **Point Estimate** & **Standard Error** & **Lower CI** & **Upper CI** \\ \hline
11 kV to 400 V & 4 & Alpha Beta & 11.4954 & 4.41194 & 5.41794 & 24.3902 \\
100 kVA & & Beta & 1.37384 & 0.555899 & 0.621607 & 3.0364 \\ \hline
11 kV to 400 V & 3 & Alpha Beta & 12.7236 & 3.20768 & 7.7628 & 20.8546 \\
160 kVA & & Beta & 1.48876 & 0.410304 & 0.86743 & 2.55515 \\ \hline
11 kV to 400 V & 18 & Alpha Beta & 20.573 & 2.40769 & 16.3562 & 25.8771 \\
250 kVA & & Beta & 2.10485 & 0.426645 & 1.41477 & 3.13153 \\ \hline
11 kV to 400 V & 7 & Alpha Beta & 22.1934 & 2.19919 & 18.2758 & 26.9508 \\
400 kVA & & Beta & 4.01613 & 1.28084 & 2.14951 & 7.50373 \\ \hline
11 kV to 400 V & 3 & Alpha Beta & 28.1318 & 3.12802 & 22.623 & 34.9819 \\
630 kVA & & Beta & 5.52089 & 2.44696 & 2.31601 & 13.1607 \\ \hline
33 kV to 400 V & 37 & Alpha Beta & 15.2267 & 1.41542 & 12.6905 & 18.2696 \\
100 kVA & & Beta & 1.8654 & 0.229805 & 1.46525 & 2.37485 \\ \hline
33 kV to 400 V & 44 & Alpha Beta & 16.3943 & 1.33623 & 13.9738 & 19.234 \\
160 kVA & & Beta & 1.9358 & 0.240369 & 1.51763 & 2.46919 \\ \hline
33 kV to 400 V & 42 & Alpha Beta & 17.9555 & 1.20517 & 15.7422 & 20.48 \\
250 kVA & & Beta & 2.41537 & 0.303852 & 1.88757 & 3.09075 \\ \hline
33 kV to 400 V & 19 & Alpha Beta & 15.2044 & 2.49815 & 11.0183 & 20.981 \\
400 kVA & & Beta & 1.46352 & 0.28023 & 1.00557 & 2.13002 \\ \hline
33 kV to 400 V & 8 & Alpha Beta & 16.9269 & 3082971 & 10.8641 & 26.3731 \\
630 kVA & & Beta & 1.62459 & 0.49219 & 0.897143 & 2.94189 \\ \hline
33 kV to 400 V & 4 & Alpha Beta & 10.4237 & 3.3206 & 5.5829 & 19.4618 \\
800 kVA & & Beta & 1.66328 & 0.639442 & 0.782936 & 3.53352 \\ \hline
33 kV to 400 V & 6 & Alpha Beta & 16.7628 & 2.74205 & 12.1649 & 23.0986 \\
1000 kVA & & Beta & 2.62885 & 0.880714 & 1.36332 & 5.06914 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the Weibull analysis of DT-failure data. parameters [27]. The shape parameter (beta) of the above Weibull distributions of the DT failures provides insight into the nature of the failure. In the above plots, a beta value less than 1 indicates that the failure rate was high at the beginning and decreased over time, which is known as infant mortality or early-life failure in the bathtub curve. In addition, a beta value equal to 1 indicates that the failure rate was constant over time, which is known as random failure. A beta value greater than 1 indicates that the failure rate was low at the beginning and increased over time, which is known as wear-out failure or late-life failure. The scale parameter (alpha) of the above Weibull distributions of DT failures provides insight into the overall reliability of the system. A larger alpha value indicates a larger time to failure and higher reliability, whereas a smaller alpha value indicates a shorter time to failure and lower reliability. Table 4 has been developed to include the above information on actual failure data of the distribution transformers of the case-study region of the case-study organization considering related alpha and beta values in the literature. Table 5 provides the log likelihood, AIC, BIC, and AD for the goodness of fit of the Weibull model to the actual failure data set of the distribution transformer.