# Misalignment Fault Prediction of Wind Turbines Based on Combined Forecasting Model
## 2 Forecasting Method and Application  
### Multivariate Grey Model  
In 1982, Professor [PERSON] of Huazhong University of Science and Technology proposed the grey system theory, which has good adaptability to small samples and uncertain systems [26]. Because the GM (1,1) model only uses single time series data and it cannot reflect the interaction between multiple variables, some scholars have proposed a MGM (1, \(n\)), where \(n\) is the number of variables. The MGM (1, \(n\)) is not a simple combination of GM (1,1), but a generalization from the univariate GM (1,1) to the multivariate case by solving \(n\) differential equations simultaneously [27].  
It is assumed that \(x_{i}^{(0)}(k)\), \((i=1,2,\ldots,n;k=1,2,\ldots,m)\) is \(n\) sets of time series data, and \(k\) is the number of each data set.  
The process of establishing a MGM (1, \(n\)) prediction model is as follows:  
(1) Accumulate the original data to generate a new sequence data \(x_{i}^{(1)}(k)\):  
\[x_{i}^{(1)}(k)=\sum_{j=1}^{k}x_{i}^{(0)}(j) \tag{1}\]  
(2) A system of \(n\)-ary first-order ordinary differential equations can be used to express MGM (1, \(n\)):  
\[\left\{\begin{array}{l}\frac{\mathrm{d}x_{1}^{(1)}}{\mathrm{d}t}=a_{11}x_{1 }^{(1)}+a_{12}x_{2}^{(1)}+\cdots+a_{1n}x_{n}^{(1)}+b_{1}\\ \frac{\mathrm{d}x_{2}^{(1)}}{\mathrm{d}t}=a_{21}x_{1}^{(1)}+a_{22}x_{2}^{(1)} +\cdots+a_{2n}x_{n}^{(1)}+b_{2}\\ \vdots\\ \frac{\mathrm{d}x_{1}^{(1)}}{\mathrm{d}t}=a_{n1}x_{1}^{(1)}+a_{n2}x_{2}^{(1)} +\cdots+a_{m}x_{n}^{(1)}+b_{n}\end{array}\right. \tag{2}\]  
Equation (2) can be written in matrix form as:  
\[\frac{\mathrm{d}\mathbf{X}^{(1)}(t)}{\mathrm{d}t}=\mathbf{A}\mathbf{X}^{(1)}(t)+\mathbf{B} \tag{3}\]\[\text{where, }\mathbf{A}=\left[\begin{array}{cccc}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ a_{n1}&a_{n2}&\ldots&a_{nm}\end{array}\right]\mathbf{X}^{(1)}(t)=\left[\begin{array}[ ]{c}x_{1}^{(1)}(k)\\ x_{2}^{(1)}(k)\\ \vdots\\ x_{n}^{(1)}(k)\end{array}\right]\mathbf{B}=\left[\begin{array}{c}b_{1}\\ b_{2}\\ \vdots\\ b_{n}\end{array}\right]\text{.}\]
3. By the method of least squares, the parameter matrices \(\mathbf{A}\) and \(\mathbf{B}\) can be estimated. Assume that \(\mathbf{a}_{i}=\left[a_{11},a_{22},\ldots,a_{in},b_{i}\right]^{\mathsf{T}},(i=1,2,\ldots,n)\). The corresponding matrix can be expressed as: \[\hat{\mathbf{a}}_{i}=\left(\mathbf{L}^{\mathsf{T}}\mathbf{L}\right)^{-1}\mathbf{L}^{\mathsf{T }}\mathbf{Y}_{i}\] (4) where, \[\mathbf{Y}_{i}=\left[\begin{array}{c}x_{i}^{(0)}(2),x_{i}^{(0)}(3),\ldots,x_{i }^{(0)}(m)\end{array}\right]^{\mathsf{T}}\text{, }\qquad\mathbf{L}=\left(\mathbf{L}_{1},\mathbf{L}_{2},\ldots,\mathbf{L}_{j},\ldots,\mathbf{L}_{n}, \mathbf{I}\right)\text{,}\] \[\mathbf{L}_{j}=\left[\begin{array}{c}(x_{j}^{(1)}(2)+x_{j}^{(1)}(1))/2\\ (x_{j}^{(1)}(3)+x_{j}^{(1)}(2))/2\\ \vdots\\ (x_{j}^{(1)}(m)+x_{j}^{(1)}(m-1))/2\end{array}\right]\text{and }\mathbf{I}\text{ is the unit matrix.}\]
4. The predicted values of MGM (1, \(n\)) can be obtained. \[\hat{\mathbf{X}}^{(1)}(k)=e^{\mathbf{A}(k-1)}\mathbf{X}^{(1)}(1)+\hat{\mathbf{A}}^{-1}(e^{\hat {\mathbf{A}}(k-1)}-\mathbf{I})\cdot\hat{\mathbf{B}}\text{, }(k=1,2,\ldots)\] (5) \[\hat{\mathbf{X}}^{(0)}(1)=\mathbf{X}^{(0)}(1)\] (6) \[\hat{\mathbf{X}}^{(0)}(k)=\hat{\mathbf{X}}^{(1)}(k)-\hat{\mathbf{X}}^{(1)}(k-1)\text{, }(k=2,3,\ldots)\] (7) When \(n=1\), MGM (1, \(n\)) model is transformed into GM (1,1) model.