## 4 Fault Coefficient Feature Extraction in Complex Electrical Engineering
### Fault Coefficient Feature Extraction in Asymmetrical Short Circuit Fault  
In the IEEE 39-BUS New England power system, an asymmetrical short circuit fault breaks out suddenly, and BUS-18 is the actual fault location. By BPA simulations, one can get the node negative sequence voltages. Meanwhile, one has also introduced a white Gaussian stochastic noise \(N(0,0.003^{2})\), and carries out detailed analysis about these original electrical information vectors.  
According to the fundamental of feature extraction about the fault coefficient, one can calculate the initial characteristic values, initial characteristic vectors, squared loadings and rotation squared loadings. Among them, the initial characteristic value of the first fault factor is 0.01563622, the variance percentage is 0.9852, and the cumulative variance percentage is also 0.9852. For the second fault factor, the initial characteristic value of the first fault factor is 0.00008786, the variance percentage is 0.0056, and the cumulative variance percentage is 0.9908. For the third fault factor, the initial characteristic value of the first fault factor is 0.00007585, the variance percentage is 0.0048, and the cumulative variance percentage is 0.9956. For the fourth fault factor, the initial characteristic value of the first fault factor is 0.00007056, the variance percentage is 0.0044, and the cumulative variance percentage is 1.0000. Actually, only the first fault factor needs to be extracted in this simulation.  
Figure 1: IEEE 39-BUS New England power system.  
Furthermore, we calculate the fault coefficient of all fault factors, as shown in Table 2. Because we have focused on the first fault factor, and the general form of the first fault factor is:  
\[\begin{split}\text{Fault1}=0.080184\text{BUS1}+0.162261\text{BUS2}+0.266219 \text{BUS3}+0.127872\text{BUS4}\\ +0.101886\text{BUS5}+0.094243\text{BUS6}+0.060710\text{BUS7}+0.092 570\text{BUS8}\\ +0.056463\text{BUS9}+0.105765\text{BUS10}+0.096482\text{BUS11}+0.106 159\text{BUS12}\\ +0.109040\text{BUS13}+0.128760\text{BUS14}+0.178810\text{BUS15}+0.204 963\text{BUS16}\\ +0.337853\text{BUS17}+0.520984\text{BUS18}+0.092290\text{BUS19}+0.057 301\text{BUS20}\\ +0.144469\text{BUS21}+0.120420\text{BUS22}+0.106459\text{BUS23}+0.224 937\text{BUS24}\\ +0.146205\text{BUS25}+0.228639\text{BUS26}+0.269287\text{BUS27}+0.13 0908\text{BUS28}\\ +0.110627\text{BUS29}+0.069968\text{BUS30}+0.058711\text{BUS31}+0.06 5727\text{BUS32}\\ +0.063605\text{BUS33}+0.044253\text{BUS34}+0.091074\text{BUS35}+0.04 7710\text{BUS36}\\ +0.082034\text{BUS37}+0.069996\text{BUS38}+0.003890\text{BUS39}\end{split} \tag{15}\]  
In the first fault factor, we will concentrate on the coefficient feature. For all of these coefficients in Fault1, the coefficient of BUS-18 is 0.520984, which is the biggest one. Consequently, we come to the conclusion that BUS-18 is just the fault BUS. In the meantime, the expression of the first fault factor without the interference of white Gaussian stochastic noise has also been obtained through the same approach, namely  
\[\begin{split}\text{Fault1}^{\prime}=0.072173\text{BUS1}+0.168284\text{BUS2}+0.282617\text{BUS3}+0.138272\text{BUS4}\\ +0.092896\text{BUS5}+0.088251\text{BUS6}+0.076103\text{BUS7}+0.073602\text{BUS8}\\ +0.037873\text{BUS9}+0.095754\text{BUS10}+0.093253\text{BUS11}+0.091824 \text{BUS12}\\ +0.105044\text{BUS13}+0.130054\text{BUS14}+0.168642\text{BUS15}+0.20 2942\text{BUS16}\\ +0.347645\text{BUS17}+0.521645\text{BUS18}+0.107902\text{BUS19}+0.063419 \text{BUS20}\\ +0.150062\text{BUS21}+0.116477\text{BUS22}+0.115762\text{BUS23}+0.193652 \text{BUS24}\\ +0.153635\text{BUS25}+0.204371\text{BUS26}+0.263681\text{BUS27}+0.14 4346\text{BUS28}\\ +0.126838\text{BUS29}+0.086465\text{BUS30}+0.052879\text{BUS31}+0.056809 \text{BUS32}\\ +0.067171\text{BUS33}+0.051807\text{BUS34}+0.078961\text{BUS35}+0.06 2526\text{BUS36}\\ +0.092181\text{BUS37}+0.087179\text{BUS38}+0.012862\text{BUS39}\end{split} \tag{16}\]  
Figure 2: The noise influence in asymmetrical short circuit fault.