## 6 Conclusions  
We propose an optimization method for maintenance intervals constrained by failure rate, taking into account imperfect maintenance factors. This approach addresses the lack of maintenance historical data and non-dynamic maintenance costs in current power industry maintenance planning. The method begins by modeling the fault distribution of power transformers using a two-parameter Weibull distribution. Bayesian methods based on Gibbs and M-H sampling are employed to fit the distribution parameters of power transformers, largely solving the problem of Weibull distribution parameter estimation under small sample fault data of transformers. The estimated parameters are evaluated using Q-Q diagrams and other methods, with results indicating a good fitting effect suitable for calculating the number of failures in subsequent maintenance interval optimization. In the maintenance interval optimization model, an imperfect maintenance improvement factor that dynamically changes as the maintenance cycle progresses is introduced. This allows for dynamic consideration of maintenance costs as the number of repairs increases, more accurately reflecting the maintenance interval of long-life products such as power transformers under imperfect maintenance conditions. The objective function aims to minimize the maintenance cost per unit time during the transformer's life cycle. When constructing the objective function, the change in the severity of defect modes on the cost of corrective maintenance is considered, and a maintenance interval optimization model is established with fault rate as a constraint. The method was validated on the strong oil circulation cooling system of a 220 KV oil-immersed main transformer in the northern power grid of China. The maintenance cost of a single power transformer's strong oil circulation cooling system was reduced from CNY 1,980,689 to CNY 1,159,816. As shown in Table 6, we compared our approach with selected publications from the past three years. This comparison demonstrates that our method not only integrates the relationship between dynamic maintenance costs and imperfect maintenance factors but also accounts for parameter identification in small sample sizes. This allows engineers to develop economically viable plans even with limited sample data. Our method establishes a more scientific maintenance cycle for transformers at the subsystem level, significantly reducing maintenance labor and costs throughout the lifespan of transformers. The data required for this method is fundamental information that should be routinely recorded in the power industry's maintenance practices, including defect time, defect type, repair cost, and unreliability. Consequently, this approach applies to optimizing the maintenance cycle of power transformers in various countries and regions. The cases presented in studies [51; 52] serve as examples of this applicability. The method's effectiveness in reflecting the dynamic nature of maintenance costs is primarily based on the number of defects occurring during the maintenance cycle. However, it is important to note that additional factors, such as inflation of maintenance and replacement costs over time, should also be taken into consideration for a more comprehensive analysis. Figure 9: Failure rate curve with maintenance times. This approach can be particularly valuable for power utilities and maintenance planners worldwide, as it offers a robust framework for optimizing maintenance schedules while accounting for the specific operational conditions and economic factors of different regions. By implementing this method, organizations can potentially achieve substantial cost savings and improved reliability in their transformer maintenance programs. Future research could focus on incorporating additional economic factors and refining the model to account for regional variations in maintenance practices and cost structures. This would further enhance the method's applicability and accuracy across diverse global contexts. Not applicable.