## 4 FRA, Statistical Analysis
### Statistical Indicators and Results
\[CC_{(X,Y)}=\frac{\sum_{i=1}^{N}X(i)\times Y(i)}{\sqrt{\sum_{i=1}^{N}[X(i)]^{2} \times\sum_{i=1}^{N}[Y(i)]^{2}}} \tag{1}\]  
\[ASLE_{(X,Y)}=\frac{\sum_{i=1}^{N}\lvert 20 log_{10}Y_{i}-20 log_{10}X_{i}\rvert}{N} \tag{2}\]  
\[SD_{(X,Y)}=\sqrt{\frac{\sum_{i=1}^{N}[Y(i)-X(i)]^{2}}{N-1}} \tag{3}\]  
\[MSE=\frac{\sum_{i=1}^{N}(Y(i)-X(i))^{2}}{N} \tag{4}\]  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{N}\left(X(i)-Y(i)\right)^{2}}{N}} \tag{5}\]  
\[DABS=\frac{\sum_{i=1}^{N}\lvert y(i)-x(i)\rvert}{N} \tag{6}\]  
\[COVAR(x,y)=\frac{1}{N}\sum_{i=1}^{N}(x(i)-\overline{x}(i)(y(i)-\overline{y}(i)) \tag{7}\]  
\[MM=\frac{\sum_{i=1}^{N}min(Xi_{i},Y\bar{i})}{\sum_{i=1}^{N}max(Xi_{i},Y\bar{ i})} \tag{8}\]  
where:  
\(x_{i}\): is the data set from the first FRA measurement (signature of normal condition)  
\(y_{i}\): is the data set from the second FRA measurement (signature of fault condition)  
\(N\): the number of data set  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fault Type**} & \multicolumn{8}{c}{**Statistical Indicators**} \\ \cline{2-13}  & \multicolumn{3}{c}{**CC**} & \multicolumn{3}{c}{**ASLE**} & \multicolumn{3}{c}{**SD**} & \multicolumn{3}{c}{**MSE**} \\ \cline{2-13}  & LF & MF & HF & LF & MF & HF & LF & MF & HF & LF & MF & HF \\ \hline SC at Tap 2 & 1.00 & 0.94 & 1.00 & 1.48 & 2.54 & 0.45 & 1.65 & 3.06 & 0.73 & 2 \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**Statistical Indicators**} \\ \cline{2-13}
**Fault Type** & \multicolumn{3}{c}{**CC**} & \multicolumn{3}{c}{**ASLE**} & \multicolumn{3}{c}{**SD**} & \multicolumn{3}{c}{**MSE**} \\ \cline{2-13}  & **LF** & **MF** & **HF** & **LF** & **MF** & **HF** & **LF** & **MF** & **HF** & **LF** & **MF** & **HF** \\ \hline Turn to turn SC & 1.00 & 1.00 & 0.99 & 0.79 & 1.59 & 0.94 & 0.91 & 1.60 & 1.59 & 0.84 & 2.55 & 2.54 \\ \hline Coil to coil SC & 1.00 & 1.00 & 0.97 & 1.30 & 2.61 & 1.57 & 1.50 & 2.61 & 2.46 & 2.25 & 6.82 & 6.06 \\ \hline Turn to ground SC & 1.00 & 1.00 & 0.96 & 0.00 & 0.13 & 1.61 & 0.01 & 0.22 & 2.88 & 0.00 & 0.05 & 8.28 \\ \hline Coil to ground SC & 1.00 & 1.00 & 0.98 & 0.83 & 1.55 & 1.26 & 0.94 & 1.56 & 1.91 & 0.88 & 2.42 & 3.67 \\ \hline Neutral to ground SC & 1.00 & 1.00 & 1.00 & 0.01 & 0.01 & 0.24 & 0.01 & 0.01 & 0.67 & 0.00 & 0.00 & 0.45 \\ \hline Phase to phase SC & 0.99 & 1.00 & 0.83 & 3.41 & 7.58 & 3.58 & 4.01 & 7.59 & 5.85 & 7.07 & 7.61 & 7.19 \\ \hline Open circuit & \(-0.93\) & \(-1.00\) & 0.91 & 71.59 & 26.59 & 2.23 & 28.97 & 28.97 & 4.62 & 5313.25 & 839.05 & 21.37 \\ \hline \hline \multicolumn{13}{c}{**Statistical Indicators**} \\ \cline{2-13}
**Fault Type** & \multicolumn{3}{c}{**DABS**} & \multicolumn{3}{c}{**RMSE**} & \multicolumn{3}{c}{**COVAR**} & \multicolumn{3}{c}{**MM**} \\ \cline{2-13}
**LF** & **MF** & **HF** & **LF** & **MF** & **HF** & **LF** & **MF** & **HF** & **LF** & **MF** & **HF** \\ \hline Turn to turn SC & 0.79 & 1.59 & 0.94 & 0.91 & 1.60 & 1.59 & 0.07 & 0.41 & 0.05 & 1.07 & 1.05 & 1.03 \\ \hline Coil to coil SC & 1.30 & 2.61 & 1.57 & 1.50 & 2.61 & 2.46 & 0.06 & 0.40 & 0.03 & 1.13 & 1.09 & 1.04 \\ \hline Turn to ground SC & 0.00 & 0.13 & 1.61 & 0.01 & 0.22 & 2.88 & 0.08 & 0.42 & 0.08 & 1.00 & 1.00 & 1.05 \\ \hline Coil to ground SC & 0.83 & 1.55 & 1.26 & 0.94 & 1.56 & 1.91 & 0.07 & 0.41 & 0.05 & 1.08 & 1.05 & 1.04 \\ \hline Neutral to ground SC & 0.01 & 0.01 & 0.24 & 0.01 & 0.01 & 0.67 & 0.08 & 0.41 & 0.07 & 1.00 & 1.00 & 1.01 \\ \hline Phase to phase SC & 3.41 & 7.58 & 3.58 & 4.01 & 7.59 & 5.85 & 0.03 & 0.38 & 0.02 & 1.41 & 1.32 & 1.11 \\ \hline Open circuit & 71.59 & 26.59 & 2.23 & 72.89 & 28.97 & 4.62 & \(-0.34\) & \(-0.44\) & 0.15 & 7.14 & 1.85 & 1.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistical indicators for normal and faulty FRA signatures of the tested IM with star stator winding.