## 4 Case Analysis
### Case of Multiple Faults  
Due to the high integration of the device of the protection system, the fault of a component in the merging unit, protection device, and intelligent terminal is likely to cause other component faults in the current device. To verify the accuracy of the GBDT diagnostic model under multiple faults, the multiple fault types shown in Table 15 are considered.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\) & \(f_{11}\) & \(f_{12}\) \\ \hline Number of samples & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\ \hline Number of correct diagnoses for GBDT & 9 & 8 & 10 & 9 & 9 & 9 & 10 & 10 & 9 & 9 & 10 & 10 \\ \hline \end{tabular}
\end{table}
Table 13: Fault sample distribution and diagnosis results when two false alarms occur.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline  & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\) & \(f_{11}\) & \(f_{12}\) \\ \hline Number of samples & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\ \hline Number of correct diagnoses for GBDT & 8 & 8 & 9 & 8 & 9 & 7 & 10 & 9 & 9 & 7 & 10 & 10 \\ \hline \end{tabular}
\end{table}
Table 14: Fault sample distribution and diagnosis results when three false alarms occur.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline  & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\) & \(f_{11}\) & \(f_{12}\) \\ \hline Number of samples & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\ \hline Number of correct diagnoses for RF & 9 & 10 & 10 & 9 & 10 & 10 & 10 & 10 & 10 & 9 & 10 & 10 \\ \hline Number of correct diagnoses for RF & 7 & 8 & 9 & 9 & 9 & 10 & 9 & 9 & 8 & 7 & 9 & 10 \\ \hline Number of correct diagnoses for RF & 5 & 6 & 9 & 9 & 10 & 10 & 9 & 7 & 9 & 6 & 9 & 9 \\ \hline \end{tabular}
\end{table}
Table 12: Fault sample distribution and diagnosis results when one false alarm occurs.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline  & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\) & \(f_{11}\) & \(f_{12}\) \\ \hline Number of samples & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 \\ \hline Number of correct diagnoses for GBDT & 8 & 8 & 9 & 8 & 9 & 7 & 10 & 9 & 9 & 7 & 10 & 10 \\ \hline \end{tabular}
\end{table}
Table 15: Type of multiple fault.  
Four types of fault samples in Table 15 are selected from the sample set for training based on the GBDT model formed in Section 3.2. The sample distribution and diagnosis results are shown in Table 16.  
It can be seen from Table 16 that the diagnostic accuracy of this method for multiple faults is as high as 91.8367%. Considering the low probability of multiple faults in the actual operation environment, this method has a reliable fault diagnosis ability.  
Taking \(f_{13}\)--\"Main DSP module failure of merging unit, Sampling DSP module failure of merging unit\" as an example, when the fault occurs, the fault feature information is shown in Table 17.  
The diagnosis steps are the same as those in Section 4.1, and the output result set \(\text{R}=\{f_{1},f_{2},f_{3},f_{4},\ldots,f_{13},f_{14},f_{15},f_{16}\}=\{0,0, 0,0,\ldots,1,0,0,0\}\). The fault is diagnosed as the fault of \"Main DSP module failure of merging unit, Sampling DSP module failure of merging unit\", and the diagnosis is correct.