# Research on Key Technology of Wind Turbine Drive Train Fault Diagnosis System Based on Digital Twin
## 4 Fault Diagnosis Method Based on Digital Twin
### Feature Extraction Based on Improved Variational Modal Decomposition (IVMD)
After the decomposition, the kurtosis of each Instantaneous Mode Function (IMF) is calculated using Formula (3), and the highest value is deemed the kurtosis value for the current combination of parameters \((k,\alpha_{s})\). Then, using the parameters \((k,\alpha_{s+\Delta\alpha})\), the signal is decomposed again by VMD, and the kurtosis values of each IMF are recalculated. The maximum kurtosis value is then taken as the kurtosis value for \((k,\alpha_{s+\Delta\alpha})\). This cycle continues until \(\alpha_{s}=\alpha_{e}\). The value of \(\alpha\) corresponding to the maximum kurtosis value in the obtained sequence is taken as the optimal \(\alpha\) for that mode number \(k\). Subsequently, the mode number is set to \(k+1\), and the cycle continues until \(k=K_{e}\). Then, a series of locally optimal parameters can be obtained. The mathematical expression is represented as follows:  
\[ \begin{array}{c}\overset{\rightarrow}{e}_{s}^{k}=\max(e_{s}^{1},e_{s}^{2}, \ldots,e_{s}^{k})\\ \overset{k}{global}=(\overset{k}{e}_{b^{\prime}}\overset{k}{e}_{b+\Delta\alpha ^{\prime}}^{k}\cdots\overset{\rightarrow}{e}_{s}^{k},\ldots,\overset{k}{e}_{ e}^{k})\\ \overset{k}{global}=\operatorname*{argmax}(\overset{k}{global})\end{array}  \tag{23}\]  
After obtaining a series of locally optimal parameter pairs, decompose the signal using the locally optimal parameter combinations. Then, the energy loss coefficient (ELC) of the modal components is calculated, and the parameters with the least energy loss are selected as the globally optimal VMD decomposition parameters. The formula for calculating ELC is as follows:  
\[ELC=\frac{\left\|f(t)-\underset{k=1}{\overset{K}{\sum}}u_{k}\right\|_{2}^{2}} {\left\|f(t)\right\|_{2}^{2}} \tag{24}\]  
where \(f(t)\) represents the signal, and \(u_{k}\) is the modal component  
The modal component signals obtained after preprocessing the vibration signals above retain the frequency characteristics of the original signals to the maximum extent and eliminate the noise and other irrelevant signals accordingly. In order to better realize fault state recognition, it is also necessary to carry out effective feature extraction on the preprocessed modal component signals. The purpose of feature extraction is to extract representative features from the original vibration signals and to use fewer low-dimensional feature vectors to represent a large amount of high-dimensional raw data. Due to the complexity of vibration signals in the drive train, the original Normalized Dispersion Entropy (NDE) [21] cannot effectively describe the signal characteristics; hence, the Normalized Composite Multi-scale Dispersion Entropy (NCMDE) algorithm is introduced to describe signal features. For a discrete sequence \(x[n]\) of length \(N\), it is first coarse-grained, as shown in Equation (25), to obtain its coarse-grained sequence under scale \(\tau\), denoted as \(\left\{x_{1}^{\tau},x_{2}^{\tau},\ldots,x_{\tau}^{\tau}\right\}\). \[x_{k}^{\tau}[m]=\left\{\underset{i=\tau(m-1)+k}{\overset{\tau m+k-1}{\sum}} x[i]\right\}/\tau,k\in[1,\tau],m\in[1,N/\tau] \tag{25}\]The distribution probability of each scale in the coarse-grained sequence is calculated. The calculation steps are consistent with the original entropy calculation method. The distribution probability of the coarse-grained sequence under scale \(\tau\) is obtained, denoted as \(\left\{p_{1}^{\tau},p_{2}^{\tau},\ldots,p_{\tau}^{\tau}\right\}\). The mean distribution probability is calculated, and the NCMDE value is obtained, as shown in Equation (26). \[\left\{ \begin{array}{l}NCMDE(xt,c,m,d,\tau)=\frac{1}{\ln(C^{m})}\sum\limits_ {i=1}^{m}\overline{p}_{\tau}[\pi_{i}]\ln(\overline{p}_{r}[\pi_{i}])\\ \overline{p}_{r}=mean(p_{k}^{\tau})\end{array} \right. \tag{26}\]