# Research on Misalignment Fault Isolation of Wind Turbines Based on the Mixed-Domain Features
## 3 Fault Isolation of Transmission System Based on PSO-SVM
### The Principles of SVM  
SVM was proposed by [PERSON] in 1995. The principle of SVM can be briefly described as follows [7]. The giving training samples \((x_{1},y_{1})\), \((x_{2},y_{2})\),..., \((x_{l},y_{l})\) (where \(x_{i}\in R^{n}\), \(y_{i}\in\{1,-1\}\), and \(l\) is the number of training samples) can be separated by a hyper plane, and the hyper plane can be expressed as:  
\[\omega x+b=0, \tag{18}\]  
where \(\omega\) and \(b\), respectively, represent the normal vector and the constant term. When \(\omega\) and \(b\) are the best, it means the optimal hyper plane has been found, which makes the distance between the positive and negative samples the largest. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Type** & \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) & \(\mathbf{P_{4}}\) & \(\mathbf{P_{5}}\) & \(\mathbf{P_{7}}\) & \(\mathbf{P_{7}}\) & \(\mathbf{P_{8}}\) \\ \hline \multirow{6}{*}{0} & 0.3109 & 0.3429 & 0.1709 & 0.0747 & 0.0476 & 0.072 & 0.097 & 0.0521 \\  & 0.3166 & 0.352 & 0.1553 & 0.0784 & 0.0477 & 0.0688 & 0.0575 & 0.0176 \\  & 0.3186 & 0.3498 & 0.1657 & 0.0814 & 0.0478 & 0.0452 & 0.069 & 0.0172 \\  & 0.3256 & 0.341 & 0.2031 & 0.0895 & 0.0564 & 0.037 & 0.0933 & 0.0148 \\  & 0.3302 & 0.3386 & 0.1669 & 0.0748 & 0.0574 & 0.0522 & 0.0788 & 0.0129 \\ \hline \multirow{6}{*}{1} & 0.1484 & 0.3676 & 0.3623 & 0.193 & 0.1201 & 0.0863 & 0.0477 & 0.1012 \\  & 0.1489 & 0.3669 & 0.3678 & 0.2267 & 0.0938 & 0.0481 & 0.028 & 0.1249 \\  & 0.1513 & 0.3678 & 0.3662 & 0.1839 & 0.1077 & 0.0665 & 0.0517 & 0.2216 \\  & 0.1558 & 0.3533 & 0.3675 & 0.2133 & 0.1479 & 0.0683 & 0.0637 & 0.1977 \\  & 0.1574 & 0.3601 & 0.3639 & 0.157 & 0.0807 & 0.062 & 0.2031 & 0.1429 \\ \hline \multirow{6}{*}{2} & 0.2335 & 0.3067 & 0.2325 & 0.1467 & 0.0843 & 0.0763 & 0.1386 & 0.3648 \\  & 0.2361 & 0.3024 & 0.3679 & 0.289 & 0.1934 & 0.1157 & 0.1594 & 0.0478 \\ \cline{1-1}  & 0.2423 & 0.3665 & 0.3417 & 0.2574 & 0.1862 & 0.0897 & 0.0395 & 0.0279 \\ \cline{1-1}  & 0.2455 & 0.3476 & 0.3679 & 0.3096 & 0.1083 & 0.0873 & 0.0337 & 0.0536 \\ \cline{1-1}  & 0.2487 & 0.3298 & 0.3134 & 0.2565 & 0.1254 & 0.0804 & 0.1309 & 0.3447 \\ \hline \multirow{6}{*}{3} & 0.1456 & 0.2285 & 0.2912 & 0.3322 & 0.2521 & 0.1672 & 0.2001 & 0.1162 \\  & 0.1061 & 0.2784 & 0.3393 & 0.2876 & 0.2762 & 0.0813 & 0.2059 & 0.1677 \\ \cline{1-1}  & 0.1074 & 0.2581 & 0.3357 & 0.3582 & 0.1606 & 0.1101 & 0.1396 & 0.1396 \\ \cline{1-1}  & 0.1082 & 0.2853 & 0.3271 & 0.3022 & 0.1685 & 0.1765 & 0.0751 & 0.1556 \\ \cline{1-1}  & 0.111 & 0.3268 & 0.3654 & 0.2737 & 0.2681 & 0.0886 & 0.0773 & 0.1014 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The partial data of the energy entropy of the first eight IMF components of IEMD. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **FC** & **MSF** & **VF** \\ \hline \multirow{6}{*}{2} & \(-\)440 & 8,587,500 & 8,393,500 \\  & \(-\)443 & 8,652,700 & 8,456,200 \\  & \(-\)475 & 9,280,300 & 9,053,800 \\  & \(-\)447 & 8,707,000 & 8,507,200 \\  & \(-\)451 & 8,785,300 & 8,581,700 \\ \hline \multirow{6}{*}{3} & \(-\)376 & 7,338,600 & 7,196,600 \\  & \(-\)355 & 7,033,700 & 6,907,400 \\ \cline{1-1}  & \(-\)388 & 8,190,100 & 8,039,200 \\ \cline{1-1}  & \(-\)369 & 7,400,400 & 7,264,100 \\ \cline{1-1}  & \(-\)346 & 6,958,600 & 6,838,700 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._For linearly inseparable problems, the samples can be mapped to the hyper plane by the kernel function \(K(x_{i},y_{j})\) to realize linear separability. Its objective function is as follows:  
\[ \begin{split} W(a)=&\sum_{i=1}^{l}a_{i}-\frac{1}{2} \sum_{i=1}^{l}\sum_{j=1}^{l}a_{i}a_{j}y_{i}y_{j}K(x_{i},x_{j}),\\ & s.\text{t.}\sum_{i=1}^{l}a_{i}y_{i}=0,\ 0\leq a_{i}\leq C,\end{split}  \tag{19}\]  
where \(a_{i}\) is the Lagrange operator; \(C\) is the punishment factor, and its basic function is to control the penalty of the wrong samples; \(K(x_{i},x_{j})\) is the kernel function, and its basic function is to transform the vectors of low-dimensional to the inner product in the high-dimensional.