# Wind Turbine Power Curve Modeling with a Hybrid Machine Learning Technique
## 3 Proposed Wind Power Curve Model
### Results of Wind Turbine Power Curve Modeling  
According to the steps described in Section 3.3, the original wind data are clustered by FCM at first. Then, in each cluster, if the Mahalanobis distance (Equation (24)) is larger than 10, the sample will be considered as outliers. The refined data and outliers are presented in Figure 3. From Figure 3, most outliers can be detected from the original wind data by FCM-based outlier detection model. Figure 3: Different kinds of data in four datasets. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Indicator** & **Calculation** & **Description** \\ \hline MAE & \(MAE=\sum\limits_{l=1}^{L}\frac{\left|y_{l}-\hat{y}_{l}\right|}{L}\) & The smaller the better \\ RMSE & \(RMSE=\sum\limits_{l=1}^{L}\frac{\left(y_{l}-\hat{y}_{l}\right)^{2}}{L}\) & The smaller the better \\ NMAPE & \(NMAPE=\frac{1}{L}\sum\limits_{l=1}^{L}\frac{\left|y_{l}-\hat{y}_{l}\right|}{ \max\left\{y_{l},-\hat{y}_{l}\right\}}\times 100\%\) & The smaller the better \\ \(R^{2}\) & \(R^{2}=1-\frac{\sum_{l=1}^{L}\left(y_{l}-\hat{y}_{l}\right)^{2}}{\sum_{l=1}^{L} \left(y_{l}-\hat{y}\right)}^{2}\) & The bigger the better \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance evaluation metrics and the corresponding definitions. From Table 4, in Dataset A, the optimal numbers of hidden nodes for ELM, WELM, and ORELM are 120, 120 and 80, respectively. According to the results in Table 5, 40, 40, and 40 were selected as the optimal numbers of hidden nodes in ELM, WELM, and ORELM, respectively, for Dataset B. From Table 6, the optimal ones are 120, 120, and 200 for Dataset C, while from Table 7, they are 20, 60, and 80 for Dataset D. After the optimal numbers of hidden nodes in ELM-based models were decided, the trained models were used to forecast the power output of a wind turbine at an unknown wind speed. The results of ELM, WELM, and ORELM in different datasets are presented in Tables 8 and 9. According to the Step 4 of the proposed hybrid model described in Section 3.3, the wind power forecasts of the optimal ELM, WELM, and ORELM on the validation set are used as inputs, together with the corresponding wind speed, to train SVR. Then, the forecasted power output can be obtained by the trained SVR. The results are presented in Tables 8 and 9. Moreover, the results of the other ten compared models (five parametric models and five nonparametric models) are also shown in Tables 8 and 9. According to Table 8, the proposed hybrid model provides better power forecasts than the other compared models in Dataset A. Moreover, the majority of nonparametric models outperform parametric models. However, 5-PLF performs better than ANFIS. In five parametric models, 5-PLF is the best one. With the exception of the hybrid model, among the five nonparametric models, no model outperforms the others in terms of four error indicators. Specifically, WELM performs the best in terms of MAE and NMAPE, while the best one is BPNN in terms of RMSE and \(R^{2}\). From the results on Dataset B and Dataset D presented in Tables 8 and 9, respectively, the best model is also the proposed hybrid model in terms of all error indicators. Similar to the results in Dataset A, most nonparametric models perform better than parametric models. Among all parametric models, 5-PLF usually generates better power forecasts. According to the results in Dataset C presented in Table 9, the proposed hybrid model is superior to the other models in terms of MAE, RMSE, and NMAPE, while BPNN is the best model in terms of \(R^{2}\). Moreover, in terms of MAE and RMSE, ELM, WELM, and ORELM perform better than the other compared models. However, in terms of NMAPE and \(R^{2}\), ELM, WELM, and ORELM perform worse than the other compared models. According to the results in Tables 8 and 9, the average ranks of all wind turbine power output estimation models were computed, and these are presented in Figure 4. From Figure 4, according to different error indicators, the average ranks of wind turbine power output models are also different.