## 6 Conclusions  
This paper illustrates the importance of the mother wavelet for fault classification in electrical systems. When faults occur in transmission line systems and transformers, the coefficients in three phases and a positive sequence are detected. The behavior of the coefficients is dependent on the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Mother** \\ **Wavelet** \\ \end{tabular} } & \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Coefficient from the DWT of the Current Signal**} \\ \cline{3-4}  & & \(P_{\text{max}}\) & **Result** \\ \hline db2 & \begin{tabular}{c} Fault \\ Normal \\ \end{tabular} & \(2.80\times 10^{-2}\) & Fault \\ \hline sym2 & \begin{tabular}{c} Fault \\ Normal \\ \end{tabular} & \(2.80\times 10^{-2}\) & Fault \\ \hline bior31 & \
\begin{tabular}{c} Fault \\ Normal \\ \end{tabular} & \(8.04\times 10^{-2}\) & Fault \\ \hline coif1 &
\begin{tabular}{c} Fault \\ Normal \\ \end{tabular} & \(6.78\times 10^{-2}\) & Fault \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients from the DWT of the current signals for fault classification (relay 2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Mother** \\ **Wavelet** \\ \end{tabular} } & \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Coefficient from the DWT of the Differential Current Signal**} \\ \cline{3-5}  & & \(P_{\text{max}}\) & **Result** \\ \hline db2 & \begin{tabular}{c} Fault \\ Normal \\ \end{tabular} & \(3.92\times 10^{-2}\) & \(1.25\times 10^{-2}\) & \(3.92\times 10^{-1}\) & Internal \\ \hline sym2 & \begin{tabular}{c} Internal \\ External \\ \end{tabular} & \(3.92\times 10^{-2}\) & \(1.25\times 10^{-2}\) & \(3.92\times 10^{-1}\) & Internal \\ \hline bior31 & \
\begin{tabular}{c} Internal \\ External \\ \end{tabular} & \(2.70\times 10^{-2}\) & \(9.46\times 10^{-3}\) & \(2.70\times 10^{-2}\) & Internal \\ \hline coif1 &
\begin{tabular}{c} Internal \\ External \\ \end{tabular} & \(1.66\times 10^{-3}\) & \(2.10\times 10^{-3}\) & \(2.10\times 10^{-3}\) & External \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coefficients from the DWT of the current signals for fault classification (relay 1). fault type, fault angle, and fault position. Phase faults have higher coefficients than phase unfaults. In addition, fault conditions can be detected by a positive sequence. The mother wavelets, i.e., Daubechies (db), symlets (sym), biorthogonal (bior), and Coiflets (coif), are used to compare the coefficient values and behaviors. Figures 5-8 show the coefficient values and behaviors. Each mother wavelet has a similar behavior, but its value is not the same. The same behavior can be obtained by different fault classification algorithms in data simulation, as shown in Figure 9. The data are divided into three parts: 1. algorithm design--50% (888 data), 2. data testing--25% (444 data points), and 3. case study--25% (444 data points). For the case study of the different mother wavelets, the accuracy of the results is summarized in Table 3. The faults in the case study are discriminated by using the proposed algorithm. It is found that there is an average accuracy of internal fault detection (relay 1) of 96.12% and of external fault detection (relay 2) of 99.07%. The mother wavelet types of Daubechies (db2) and symlets (sym2) provide the highest accuracy under condition 1, while the mother wavelet type of biorthogonal (bior3.1) provides the highest accuracy under condition 2, as shown in Figure 10.  
The results from the study illustrated the performance of different mother wavelets on the fault classification algorithm. The different mother wavelets provide a different level of accuracy on the fault classification on a transformer, while they do not show a significant impact on fault classification in the transmission line. Thus, this mother wavelet is also one of the factors that must be considered in order to select the suitable mother wavelet for an application.