## 5 Case Study
### Case Calculation and Analysis
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Defective** & \
\begin{tabular}{c} **Defective** \\ **Components** \\ \end{tabular} & **Severity** & **Fit Value \(m\)** & **Fit Value \(\mathbf{t_{0}}\)** & **Fit Value \(\mathbf{p}\)** \\ \hline \multirow{6}{*}{
\begin{tabular}{c} Cooling \\ system \\ (strong oil \\ circulation) \\ \end{tabular} } & Fan & serious & 2.5015 & 3574.2238 & 0.0902 \\ \cline{2-6}  & & commonly & 2.6389 & 4967.3247 & 0.0613 \\ \cline{2-6}  & \multirow{2}{*}{Control box} & serious & 2.4655 & 5485.2396 & 0.5002 \\ \cline{2-6}  & & commonly & 2.5912 & 5712.4332 & 0.1327 \\ \cline{2-6}  & & critical & 3.8266 & 7961.4850 & 0.7721 \\ \cline{2-6}  & \multirow{2}{*}{cooler} & serious & 2.2119 & 5221.8332 & 0.1170 \\ \cline{2-6}  & & commonly & 2.2826 & 5227.6844 & 0.0539 \\ \cline{2-6}  & & critical & 4.6117 & 6287.6012 & 0.3421 \\ \cline{2-6}  & \multirow{2}{*}{oil pump} & serious & 2.3529 & 4906.7916 & 0.5605 \\ \cline{2-6}  & & commonly & 3.8059 & 4717.6453 & 0.0801 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter estimation table for the defect mode distribution of each component. Figure 6: The probability density function of the failure of various components in a strong oil circulation cooling system: (**a**) PDF of the fan; (**b**) PDF of the control box; (**c**) PDF of the cooler; (**d**) PDF of the submersible oil pump. Based on the parameters derived from the fitted Weibull distribution and taking into account the dynamic imperfect maintenance optimization factors, the aforementioned maintenance interval optimization model was formulated. Employing the Sequential Least Squares Programming (SLSQP) algorithm [49; 50], and comparing it with the actual maintenance interval plan of 720 h, a maintenance schedule spanning 9 years after operation (within a single major overhaul cycle) was devised, as depicted in Figure 7. This schedule comprises a total of 5 preventive maintenance plans, which are categorized as follows: (a) actual scenario, (b) fan, (c) control box, (d) cooler, and (e) submersible pump. It is observed that, by incorporating the aspects of imperfect maintenance enhancement, the overall number of maintenance plans within a single overhaul cycle is reduced, and the frequency progressively increases over time, thereby aligning more effectively with the aging characteristics of long-life products such as power transformers. Furthermore, due to the shape parameter m and scale parameter t0 of the control box being virtually identical to those of the cooler under both normal and severe conditions, after applying the rounding-up procedure as per Formula 6, we found that the maintenance interval plans for (c) and (d) in Figure 7 are found to be the same. Figure 7: Comparison of maintenance interval planning for cooling system components under a single overhaul cycle: (**a**) actual cooling system maintenance interval under a single overhaul cycle; (**b**) maintenance interval of fan components under a single overhaul cycle; (**c**) maintenance interval of control box components under a single overhaul cycle; (**d**) maintenance interval of cooler components under a single overhaul cycle; (**e**) maintenance interval of submersible oil pump components under a single overhaul cycle. The cost comparison before and after the optimization of the maintenance cycle is presented in Table 5. Compared to the actual maintenance plan cost under the 720-h cycle, it is evident that the maintenance cost for the strong oil circulation cooling system of a single power transformer has been reduced from the original CNY 1,980,689 to CNY 1,159,816 over one major overhaul cycle. This represents a significant reduction of 41.443%. Importantly, this cost reduction has been achieved while ensuring that the failure rates for defect modes across all severity levels remain within acceptable ranges.