# Reliability Analysis of Small-Sample Failure Data for Random Truncation High-Voltage Relay
## 3 Application Cases  
Moreover, 56 electric vehicles with Type I high-voltage relays were subjected to random truncation reliability tests simultaneously. If the high-voltage relay failed, a new high-voltage relay was used as a replacement. A total of 22 electric vehicle samples generated eight relays fault data as shown in Table 1, with \(T\) being the test time for each relay. The testing time for another 34 samples is shown in Table 2.  
distribution, using the total least squares method, and according to Formulas (7) and (8), Weibull distribution parameters can be estimated. Then, according to Formulas (9) and (10), the linear correlation test and d-test are performed. The parameter estimation values and test values are shown in Table 6.  
When the significance level \(\alpha=0.1\), the critical values of the linear correlation coefficient \(\hat{\rho}_{0.1}\) and the critical values of the d-test method \(D_{p,\alpha}\) can be obtained through a table search, which are \(\hat{\rho}_{0.1}=0.9000\), \(D_{p,\alpha}=0.51\), respectively. As shown in Table 6, both Weibull distribution functions obtained from the two empirical distribution functions pass the linear correlation coefficient test and the d-test. Therefore, the root mean square error and relative root mean square error methods are introduced to optimize the reliability model, the results of which are also shown in Table 6.  
Table 6 shows that based on the root mean square error value and relative root mean square error, the priority order of the two empirical distribution functions is 3, 5. The conclusion drawn from the values of the root mean square error and the relative root mean square error is consistent; that is, the Weibull distribution model estimated based on \(F_{3}(t_{i})=\frac{i}{n+1}\) is the best. The optimal Weibull distribution parameters are as follows:  
\[\left\{ \begin{array}{l}\hat{\eta}=70,623\\ \hat{\beta}=0.5347\end{array} \right.\]  
The optimal Weibull distribution is  
\[F(t)=1-\exp\left[-(t/70,623)^{0.5347}\right]\]  
According to Equation (13), the MTBF of the high-voltage relay is  
\[ \begin{array}{l}MTBF=\int_{0}^{\infty}tf(t)dt=\int_{0}^{\infty}tdF(t)=tF(t) |_{0}^{\infty}-\int_{0}^{\infty}F(t)dt\\ =\eta\Gamma(1+1/\beta)=125,741.04\ \text{km}\end{array} \]  
Comparing the maximum likelihood estimation method with the method proposed in this paper, the maximum likelihood function of the Weibull distribution is:  
\[ \begin{array}{l}L(t_{i};\eta,\beta)=\prod\limits_{i=1}^{r}f(t_{i};\eta,\beta )\prod\limits_{i=1}^{N-r}R(t_{i};\eta,\beta)\\ =\frac{\beta^{r}}{\eta^{r\beta}}\left(\prod\limits_{i=1}^{r}t^{\beta-1}\right) ^{\beta-1}\exp\left[-\sum\limits_{i=1}^{n}\left(t/\eta\right)^{\beta}\right] \end{array}  \tag{14}\]  
where \(r\) is the number of failures and \(N\) is the sum of the number of failures and truncations. After logarithmic differentiation calculations, the estimated values of Weibull distribution parameters can be obtained as follows:  
\[\left\{ \begin{array}{l}\hat{\eta}^{\prime}=91,085\\ \hat{\beta}^{\prime}=0.576\end{array} \right.\]  
The \(MTBF^{\prime}=\eta^{\prime}\Gamma(1+1/\beta^{\prime})\) obtained by the maximum likelihood method is 144112.08 km. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Empirical** & **\(\beta\)/\(\eta\) of Weibull** & **Linear Correlation** & **D-Test Value** & **Root Mean Square** \\
**Distribution** & **Distribution** & **Coefficient** & **D-Test Value** & **Error/Relative Root** \\
**Function** & **D-Test Value** & **D-Test Value** & **Mean Square Error** \\ \hline \(F_{3}(t_{i})=\frac{\mu}{n+1}\) & 0.5347/70,623 & 0.952 & 0.0166 & 0.0116/0.1501 \\ \(F_{3}(t_{i})=\frac{\mu}{n+2}\) & 0.6288/35,586 & 0.922 & 0.0293 & 0.0175/0.2361 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Weibull distribution parameter estimation of relay test fault data.