# Life Cycle Cost Estimation and Analysis of Transformers Based on Failure Rate
## 3 Results
### Failure Rate Analysis
These three periods can be summarized as the failure period, accidental failure period, and failure loss period, and meet the classic bathtub curve. In this study, the Weber distribution is considered as the theoretical basis of the reliability analysis and life test, the distribution is used as a descriptive function of the transformer failure rate distribution, and the parameters to be estimated are estimated in the Weber distribution. Based on the actual operational and maintenance data of the transformer and the data on defects and failures, determine the parameters and distribution functions of the assumed distribution. The failure rate of transformers based on Weibull distribution can be written as [32]:  
\[f(x;\lambda,k)=\frac{k}{\lambda}(\frac{x}{\lambda})^{k-1}(x\geq 0) \tag{10}\]  
where \(k\) is the shape parameter and \(\lambda\) is the scale parameter. When \(k<1\), the failure shows a downward trend; when \(k=1\), the failure rate is constant; and, when \(k>1\), the failure rate shows an upward trend. The Levenberg Marquardt algorithm [33] can provide numerical solutions for minimizing numerical nonlinearity (local minimization). This algorithm can combine the advantages of the Gaussian Newton algorithm and gradient descent method by modifying parameters during execution, and improve the shortcomings of both (such as the absence of the inverse matrix of the Gaussian Newton algorithm or the initial value being too far from the local minimum). Assume the form of an unknown nonlinear relationship is expressed as follows:  
\[y=f(x_{1},x_{2}\ldots x_{n};a_{1},a_{2}\ldots a_{m}) \tag{11}\]  
where \(x\) represents the independent variable and \(a\) represents the parameter to be estimated. Firstly, we need to expand the nonlinear relationship based on [PERSON]'s formula. Substitute in the \(i\)-th observation value: \(f(x_{i},a)=y_{i}\). And assign an initial value to all parameters a: \(a^{(0)}=(a_{1}^{(0)},a_{2}^{(0)}\ldots,a_{\rm m}^{(0)})\). This results in the following expression:  
Figure 3: Structural diagram of transformer fault tree. \[f(x_{i},a)=f(x_{i},a^{(0)})+\frac{\partial f(x_{i},a)}{\partial a_{1}}(a_{1}-a_{1 }^{(0)})+\frac{\partial f(x_{i},a)}{\partial a_{2}}(a_{2}-a_{2}^{(0)})+\ldots \frac{\partial f(x_{i},a)}{\partial a_{\rm m}}(a_{\rm m}-a_{\rm m}^{(0)}) \tag{12}\]  
According to the principle of least mean squares, let the \(Q\) function be the following:  
\[Q=\!\!\sum_{i=1}^{n}\left\{y_{i}-\left[f(x_{i},a^{(0)})+\sum_{j=1}^{m}\frac{ \partial f(x_{i},a)}{\partial a_{j}}(a_{j}-a_{j}^{(0)})\right]\right\}^{2}+d \sum_{j=1}^{m}\left(a_{j}-a_{j}^{(0)}\right)^{2} \tag{13}\]  
Let the \(Q\) function take the partial derivative of all unknown parameters and set the partial derivative to 0. We obtain an iterative form of the equation system, substitute a set of initial parameter values \(a\) and corresponding resistance factor \(d\), set the corresponding pre- and post-iteration stop thresholds, and, when the accuracy requirements are met, stop the iteration. We can consider the last iteration value as the estimated value of the parameter. The operation and maintenance methods of transformers and the occurrence of failures will directly affect their operational reliability, maintenance costs, and maintenance costs, thereby affecting the entire life cycle cost. The growth rate of equipment operation and maintenance costs may change over time. In some years, cost growth may be relatively slow, while in other years it may accelerate. These changes may be influenced by equipment usage, maintenance quality, or other external factors. Drawing a trend chart of equipment operation and maintenance costs over time is helpful in analyzing the overall trend and possible patterns of data. By analyzing the trend of data and using the method of data fitting, the operation and maintenance costs of equipment under different operating years are predicted.