## 4 Case Study and Results Discussion
### Analysis 1: Covariate Weight Sensitivity Analysis
#### 4.2.1 Reliability Metrics  
The results of the following figures are delineated following this format: _Case MX: Solver Approach-Scaler Method-Bounds Determination._ Figure 3 and Table 8 display the cluster boundaries and centroids produced for the five cases. It is evident that cases M5 resembled the reference case, case M1. Conversely, case M2 displayed a noticeably different data segmentation due to its optimization parameters. The remaining cases exhibited comparable results to case M1. Figure 4 shows considerable dispersion among the conditional reliability states in case M2 and case M3, with state 3 in case M2 incurring a more significant penalty. Two factors could contribute to this outcome. Firstly, it was noted that a lower weight was assigned to \(\gamma_{\text{R.Die}}\) in both cases, as seen in Table 5, while the other cases shared a similar magnitude. Secondly, the exclusive application of the MinMax scaler to these cases and the following non-scaling of the data points before entering them into the PHM may also have affected these results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **S1** & **S2** & **S3** & **Limit 1** & **Limit 2** & **Limit 3** \\ \hline M1 & 2.80 & 2.99 & 2.92 & 0–2.90 & 2.90–2.93 & 2.93–\(\infty\) \\ M2 & 4.51 & 5.14 & 7.015 & 0–4.77 & 4.77–5.66 & 5.66–\(\infty\) \\ M3 & 12.66 & 14.36 & - & 0–13.08 & 13.08–\(\infty\) & - \\ M4 & 1.81 & 1.92 & - & 0–1.87 & 1.87–\(\infty\) & - \\ M5 & 1.09 & 1.14 & 1.18 & 0–1.13 & 1.13–1.15 & 1.15–\(\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Centroids and limits for each case. Figure 3: \(f(\gamma,z)\) for all cases with their respective centroids and cluster limits. Comparing the results from the GA and IPOPT solvers revealed significant differences in conditional reliability results. The GA tended to produce more optimistic reliability curves, leading to less similarity with the reference case. This discrepancy suggested a potential non-optimal fit in this analysis. Further investigation underscored the crucial role of the scaler in determining the state dispersion and curve shapes, as evidenced in cases M1, M4, and M5. Overall, the dispersion of the reliability curves appeared to be influenced by the choice of the scaler, while the length of the conditional reliability seemed to depend on the optimization method used. Similar trends were observed for the RUL in Figure 5.  
Table 9 presents the optimal time values derived from the cost function, confirming these trends. The case most closely aligned with the reference case was M2, where states 2 and 3 displayed values that were very similar. This suggested that combining IPOPT with the NS transformation could result in policies similar to those of case M1. Additionally, all the generated cases tended to show times greater than those of case M1. Figure 4: Results of the conditional reliability for each case. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cases** & **State 1** & **State 2** & **State 3** \\ \hline M1 & 9000 & 8500 & 8000 \\ M2 & 13,000 & 10,500 & 5500 \\ M3 & 18,000 & 15,500 & - \\ M4 & 16,500 & 15,000 & - \\ M5 & 24,500 & 23,500 & 23,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Optimal time (in hours) taking into account the results of the cost function for each case. Finally, Figure 6 shows the decision policies for all cases. In the reference case M1, power transformer 13 required immediate intervention at the 20,000 h mark, as it surpassed the worst conditional warning-limit graph. Similarly, transformers 14 and 15 were in the caution zone approximately at the 18,000 h and 20,000 h marks, indicating that preparations for their replacement should be made soon. The remaining transformers did not require intervention, as they were operating correctly. These insights allowed for predictive planning of interventions. In case M2, despite being the best solution in terms of log-likelihood, did not generate a reasonable policy compared to case M1.