# Statistical Power Analysis in Reliability Demonstration Testing: The Probability of Test Success
## 5 Case Study  
The Probability of Test Success, \(P_{\text{Is}}\), enables objective assessment of different test types and scenarios. In addition, it allows the reliability engineer to identify feasible regions of test designs and enables an optimal test for reliability demonstration. In order to demonstrate these benefits, a reliability demonstration test for a high-voltage battery of a battery-electric vehicle is planned and explained here. The lifetime of the battery is the accumulated energy throughput (ETP) for which a reliability requirement of \(R_{\text{r}}=95\%\), \(C_{\text{r}}=90\%\) and \(t_{\text{r}}=400\,\text{MWh}\) is to be demonstrated. Because of the battery technology and use case being a very recent development, no usable tests of prior product generations can be used as prior knowledge. However, during research and development a simulation model was established for the prediction of the ETP using the driving profiles of several measurements in the field. Therefore, the simulation was able to yield an estimation of the lifetime of the battery. The exemplary simulation results are taken from [62]. Using the simulation model and a failure criterion of \(80\%\) state of health, the Weibull distributed failure behavior of the battery can be found. The parameters are \(T=596.37\,\text{MWh}\) and \(b=9.598\) (see Table 3). This failure distribution can be then used as prior knowledge for the reliability demonstration test planning of the battery. As a first step, the feasible region of test design is analyzed using the exact method of the SR test as well as the approximate method for the EoL test. Because an MLE has to be \begin{table}
\begin{tabular}{l l} \hline \hline
**Calculation Method** & **Key Findings** \\ \hline \multirow{7}{*}{General method} & \(\bullet\) Applicable for all tests \\  & \(\bullet\) Most precise \\  & \(\bullet\) Most flexible \\  & \(\bullet\) Test costs and test time can be calculated \\  & \(\bullet\) High calculation effort due to bootstrap \\  & \(\bullet\) Though precise, calculation effort is unnecessary for SR tests \\  & \(\bullet\) Only an approximation \\ \hline \multirow{7}{*}{Approximate method (only for EoL tests)} & \(\bullet\) Fastest calculation \\  & \(\bullet\) Simple to implement \\  & \(\bullet\) Easy to calculate \\ \cline{1-1}  & \(\bullet\) Very good approximation for large sample sizes \\ \cline{1-1}  & (can replace general method) \\ \cline{1-1}  & \(\bullet\) Good approximation for small sample sizes \\ \cline{1-1}  & \(\bullet\) Good approximation for both censored and uncensored tests \\ \hline \multirow{7}{*}{Exact method (only for SR tests)} & \(\bullet\) Exact, thus no approximation \\ \cline{1-1}  & \(\bullet\) Very fast calculation \\ \cline{1-1}  & \(\bullet\) Eassiest to implement \\ \cline{1-1}  & \(\bullet\) Should always be used for SR tests \\ \hline \multirow{7}{*}{Test simulation method} & \(\bullet\) Good for very large sample sizes \\ \cline{1-1}  & \(\bullet\) Suffers from bias amplification \\ \cline{1-1}  & \(\bullet\) Very good for SR tests (coincides with general method) \\ \cline{1-1}  & \(\bullet\) Should only be used in special cases \\ \cline{1-1}  & \(\bullet\) Not usable for strongly censored EoL tests \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the Comparison of Methods of Calculating of the Probability of Test Success, \(P_{\text{Is}}\). \begin{table}
\begin{tabular}{l l} \hline \hline
**Requirement** & **Prior Knowledge** \\ \hline \(R_{\text{r}}=95\%\) & \(T=596.37\,\text{MWh}\) \\ \(C_{\text{r}}=90\%\) & \(b=9.598\) \\ \(t_{\text{r}}=400\,\text{MWh}\) & \(\to s=0.086\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reliability Requirement and Prior Knowledge Stemming From the Simulation. used for test evaluation, the MLE is used for both the approximate method and the general method.