## 4 Fault Coefficient Feature Extraction in Complex Electrical Engineering
### Fault Coefficient Feature Extraction in Symmetrical Short Circuit Fault  
Now, let us further analyze a more complex symmetrical short circuit fault in IEEE 39-BUS New England power system. This time BUS-18 is subjected to a three-phase short circuit fault. By BPA simulations, the node positive sequence voltages have been calculated. In this simulation, let's introduce a stronger white Gaussian stochastic noise \(N(0,0.06^{2})\).  
Based on the fundamental of feature extraction about fault coefficient, the initial characteristic values, initial characteristic vectors, squared loadings and rotation squared loadings can also be calculated. Among them, the initial characteristic value of the first fault factor is 1.83004634, the variance percentage is 0.8828, and the cumulative variance percentage is also 0.8828. For the second fault factor, the initial characteristic value of the first fault factor is 0.24305527, the variance percentage is 0.1172, and the cumulative variance percentage is 1.0000. Obviously, the first fault factor is just what we are seeking.  
The fault coefficient of all fault factors can be obtained through the same approach (see Table 3). Therefore, the general form of the first fault factor can be written as:  
\[\begin{split}\text{Fault1}=0.074346\text{BUS1}+0.193963\text{BUS2}+0.195029\text{BUS3}+0.169430\text{BUS4}\\ +0.088094\text{BUS5}+0.106626\text{BUS6}+0.090224\text{BUS7}+0.139135 \text{BUS8}\\ +0.042615\text{BUS9}+0.130098\text{BUS10}+0.117286\text{BUS11}+0.130002 \text{BUS12}\\ +0.123483\text{BUS13}+0.115265\text{BUS14}+0.160039\text{BUS15}+0.156472 \text{BUS16}\\ +0.349494\text{BUS17}+0.380180\text{BUS18}+0.080870\text{BUS19}+0.09479 \text{BUS20}\\ +0.173913\text{BUS21}+0.162841\text{BUS22}+0.104185\text{BUS23}+0.239691 \text{BUS24}\\ +0.122016\text{BUS25}+0.234122\text{BUS26}+0.245774\text{BUS27}+0.109912 \text{BUS28}\\ +0.109377\text{BUS29}+0.137135\text{BUS30}+0.156919\text{BUS31}+0.064740 \text{BUS32}\\ +0.041073\text{BUS33}+0.061483\text{BUS34}+0.039357\text{BUS35}+0.08774 \text{BUS36}\\ +0.133753\text{BUS37}+0.083942\text{BUS38}+0.258362\text{BUS39} \end{split} \tag{17}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**BUS** & **1** & **2** & **3** & **4** & **5** \\ \hline BUS-1 & 0.074346 & 0.052964 & \(-0.001593\) & \(-0.000682\) & \(-0.005143\) \\ BUS-2 & 0.193963 & 0.012428 & \(-0.007633\) & \(-0.007139\) & \(-0.012111\) \\ BUS-3 & 0.195029 & 0.039759 & \(-0.006921\) & \(-0.006016\) & \(-0.012461\) \\ BUS-4 & 0.169430 & \(-0.011105\) & \(-0.007275\) & \(-0.007172\) & \(-0.010351\) \\ BUS-5 & 0.088094 & 0.160465 & 0.000814 & 0.003357 & \(-0.007109\) \\ BUS-6 & 0.106626 & 0.068563 & \(-0.002489\) & \(-0.001293\) & \(-0.007299\) \\ BUS-7 & 0.090224 & 0.073833 & \(-0.001669\) & \(-0.000420\) & \(-0.006341\) \\ BUS-8 & 0.139135 & \(-0.062473\) & \(-0.007449\) & \(-0.008164\) & \(-0.007946\) \\ BUS-9 & 0.042615 & 0.066201 & 0.000078 & 0.001137 & \(-0.003320\) \\ BUS-10 & 0.130098 & 0.068635 & \(-0.003453\) & \(-0.002218\) & \(-0.008750\) \\ BUS-11 & 0.117286 & 0.075571 & \(-0.002734\) & \(-0.001416\) & \(-0.008030\) \\ BUS-12 & 0.130002 & 0.026162 & \(-0.004623\) & \(-0.004025\) & \(-0.008303\) \\ BUS-13 & 0.123483 & 0.080095 & \(-0.002864\) & \(-0.001468\) & \(-0.008460\) \\ \hline \end{tabular}
\end{table}
Table 3: The fault coefficient in symmetrical short circuit fault.  
In the former expression, the real fault location BUS-18 certainly has also been identified. In addition, the noise influence for fault coefficients is presented in Figure 3. In this simulation, the total average deviation level is 0.0016. Therefore, the fault coefficient feature is completely reliable.