# Misalignment Fault Prediction of Wind Turbines Based on Combined Forecasting Model
## 4 Fault Prediction Results and Discussion  
### Fault Prediction Results and Discussion of Vibration Signals  
#### 4.1.1 The Results of IMGM (1, \(n\))  
The MGM (1, \(n\)) is suitable for the prediction of few samples and the required number of modelling is usually between 5 and 60 [52, 53, 54]. Therefore, the collected 60 vibration signal samples are divided into the first 45 samples and the last 15 samples according to the ratio of 3:1. The first 45 samples are used to construct an IMGM (1, \(n\)) prediction model. The indexes of the posteriori error test calculated from the fitted values are listed in Table 5.  
It can be shown from Table 5 that the models of MGM (1,9) and IMGM (1,9) belong to the qualification grade, so they can be used for prediction. Because the \(C\) value of IMGM (1,9) is smaller and the \(P\) value of IMGM (1,9) is larger, IMGM (1,9) is more suitable for predicting the kurtosis of vibration signals than MGM (1,9). Figure 4 shows the prediction results of MGM (1,9) and IMGM (1,9) of vibration signals.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature Vector** & **Feature** & **Index** \\ \hline Current features library (11 dimensions) & Time-domain & root mean square, kurtosis, kurtosis index \\ \cline{2-3}  & Frequency-domain & mean square frequency, center of gravity \\  & & frequency, frequency variance \\  & Time-frequency domain & 5 sample entropy obtained by 4-layer \\  & & decomposition of DTCWT \\ \hline \hline \end{tabular}
\end{table}
Table 4: The indexes of feature vector for current signal.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(C\) & \(P\) & **Precision Grade** \\ \hline MGM (1,9) & 0.4952 & 0.8667 & Qualified \\ IMGM (1,9) & 0.4880 & 0.9048 & Qualified \\ \hline \hline \end{tabular}
\end{table}
Table 5: The precision grade of the Multivariate Grey Model (MGM (1,9)) and IMGM (1,9) for vibration signal.  
In Figure 4, the predicted and fitted values of MGM (1,9) and IMGM (1,9) both fluctuated up and down around their actual values. The forecasting evaluation indexes of IMGM (1,9) and MGM (1,9) are listed in Table 6. For the fitted values, the RMSE and \(R^{2}\) of IMGM (1,9) is nearly the same to that of MGM (1,9), while the MAE of IMGM (1,9) is better than that of MGM (1,9). For the predicted values, the RMSE, MAE, and running time of the IMGM (1,9) are smaller than those of MGM (1,9). The \(R^{2}\) of IMGM (1,9) is larger than that of MGM (1,9). Therefore, IMGM (1,9) improves the prediction accuracy of MGM (1,9) for misalignment fault vibration signal of wind turbine. This is because when predicting the last 15 points, the data used to establish the MGM (1,9) model has always been the first 45 points, and the data has not been updated with the increase prediction steps. However, the modelling data of IMGM (1,9) is updated by adding the actual value corresponding to the previous step of the current prediction point. The modelling data is updated every time the model output one predicted value. Hence, the IMGM (1,9) can be one of the single prediction models input to the combined forecasting model for vibration signal.