## 3 Fault Diagnosis Method
### Trend Feature Analysis of Decomposed Data  
A novel method of trend feature analysis is proposed for extracting trend feature vectors in this part. The three-phase current \(I_{A/B/C}\) varying with wind speed \(v\) of GSC under 22 states are decomposed by variational mode in \(K\) levels, and each phase current gets \(K\) levels of modes:  
\[A_{xu}=\Big{\{}\ A_{xu1},\ \ \cdots,\ \ A_{xuK}\ \Big{\}}, \tag{12}\]  
\[B_{xu}=\Big{\{}\ B_{xu1},\ \ \cdots,\ \ B_{xuK}\ \Big{\}}, \tag{13}\]  
\[C_{xu}=\Big{\{}\ C_{xu1},\ \ \cdots,\ \ C_{xuK}\ \Big{\}}, \tag{14}\]  
where \(A\), \(B\) and \(C\) denote the each phase of GSC in Figure 1; \(x=1,2,\cdots,22\) denotes fault code; \(A_{xu}\), \(B_{xu}\) and \(C_{xu}\) are the modes sets of three-phase current after decomposing; \(A_{xuk}\), \(B_{xuk}\) and \(C_{xuk}\) (\(k=1,2,\ldots,K\)) are the \(k\)-th mode coefficient series of three-phase current modes sets, respectively. Extracting \(E_{Axk}\), \(E_{Bxk}\) and \(E_{Cck}\) the feature energy of each mode of three-phase current can be expressed as:  
\[E_{Axkj}=\sum\limits_{j=1}^{n}\big{|}A_{xuk}(j)\big{|}^{2},x=1,2,\cdots,22,k=1,2,\ldots,K, \tag{15}\]  
\[E_{Bxkj}=\sum\limits_{j=1}^{n}\big{|}B_{xuk}(j)\big{|}^{2},x=1,2,\cdots,22,k=1,2,\ldots,K, \tag{16}\]  
\[E_{Cxkj}=\sum\limits_{j=1}^{n}\big{|}C_{xuk}(j)\big{|}^{2},x=1,2,\cdots,22,k=1,2,\ldots,K, \tag{17}\]  
where \(E_{Axk}\), \(E_{Bxk}\) and \(E_{Cck}\) are the feature energy of each mode of three-phase current, respectively. \(n\) is the total number of coefficients at each mode coefficient series. The each open-circuit fault of IGBT modules would have a great impact on the feature energy in each mode of three-phase current. Therefore, the feature energy vectors \(E_{A}\), \(E_{B}\) and \(E_{C}\) could be constructed by the feature energy of each mode. \[E_{Ax}=\Big{[}\ E_{Ax1}\ \ E_{Ax2}\ \ \ \ldots\ \ E_{AxK}\ \Big{]},x=1,2,\cdots,22, \tag{18}\]  
\[E_{Bx}=\Big{[}\ E_{Bx1}\ \ E_{Bx2}\ \ \ \ldots\ \ E_{BxK}\ \Big{]},x=1,2,\cdots,22, \tag{19}\]  
\[E_{Cx}=\Big{[}\ E_{Cx1}\ \ E_{Cx2}\ \ \ \ldots\ \ E_{CxK}\ \Big{]},x=1,2,\cdots,22, \tag{20}\]  
The three-phase current vary according to the wind speed. The same varieties occur to the feature energy of each mode. Thus, when the open-circuit happens in the GSC, the varied feature energy vectors would bring difficulties to the following data analysis. It is necessary to normalize the feature energy vectors. Let:  
\[F_{Axk}=\frac{E_{Axk}}{\sum\limits_{l=1}^{K}E_{Axl}+\sigma},x=1,2,\cdots,22,k= 1,2,\ldots,K, \tag{21}\]  
\[B_{xu}=\Big{\{}\ B_{xu1},\ \ \cdots,\ \ B_{xuK}\ \Big{\}}, \tag{13}\]0\[B_{xu}=\Big{\{}\ B_{xu1},\ \ \cdots,\ \ B_{xuK}\ \Big{\}}, \tag{13}\]1  
where \(\sigma\) is a tiny real number to avoid the erroneous judgment when the phase current is zero, and minimize impact on final classification results. The normalized feature energy vectors can be expressed as:  
\[F_{Ax}=\left[ \begin{array}{cccc}F_{Ax1}&F_{Ax2}&\ldots&F_{AxK}\end{array}  \right],x=1,2,\cdots,22, \tag{24}\]  
\[F_{Bx}=\left[ \begin{array}{cccc}F_{Bx1}&F_{Bx2}&\ldots&F_{BxK}\end{array}  \right],x=1,2,\cdots,22, \tag{25}\]  
\[F_{Cx}=\left[ \begin{array}{cccc}F_{Cx1}&F_{Cx2}&\ldots&F_{CxK}\end{array}  \right],x=1,2,\cdots,22, \tag{26}\]  
Then, the factors of normalized feature energy vectors can be the function about the part factors of trend feature vectors. Let:  
\[I_{x,k=\left(F_{A\text{\tiny{C}}k}\right)}^{\frac{1}{p}},x=1,2,\cdots,22,k=1, 2,\ldots,K, \tag{27}\]  
\[I_{x,k+K=\left(F_{B\text{\tiny{C}}k}\right)}^{\frac{1}{p}},x=1,2,\cdots,22,k=1,2,\ldots,K, \tag{28}\]  
\[I_{x,k+2K=\left(F_{C\text{\tiny{C}}k}\right)}^{\frac{1}{p}},x=1,2,\cdots,22,k=1,2,\ldots,K, \tag{29}\]  
where \(p\) is a positive real number. The value of \(p\) could be confirm in an optimal range through several experiments. \(\left\{ \begin{array}{cccc}I_{x,1}&I_{x,2}&\ldots&I_{x,3K}\end{array} \right\}\) is the part factors set of trend feature vectors, which could be used to judge which the phases are open-circuit.