# Wind Turbine Power Curve Modelling with Logistic Functions Based on Quantile Regression
## 4 Case Study
### Experimental Results
#### 4.3.3 Results for WTPC Modelling  
This part compares the proposed QRLF method (5P-QRLF optimized by PSO) with 5 PL, RVM and QRNN to comprehensively evaluate the model performance. In order to avoid the impact of individual cases, we randomly select five wind turbines from each wind farm for testing. On the one hand, MAPE and NRMSE are utilized to evaluate the deterministic fitting accuracy of the aforementioned fitting methods; on the other hand, we use PICP, PINAW and NC to test the performance of the predicted CI. Table 5 lists the average values of the fitting results of each wind farm, and Figure 9 shows the detailed information of one of the test wind turbines. Figure 8: Results of outlier filtering by GP-Filter, DBSCAN-Filter and QRLF-Filter, where the black points are the raw data of power outputs, and the orange points are the power outputs after outlier filtering. In Table 5, 5 PL is selected as the benchmark for deterministic WTPC modelling because it has been proved to have good fitting accuracy in previous studies [9]. As mentioned in Section 2.1, 5 PL can only be utilized for deterministic curve fitting, therefore, we cannot obtain the \(PICP_{90\%}\), \(PINAW_{90\%}\) and \(NC_{90\%}\) of 5 PL. KVM is selected because it can significantly increase the calculation speed while maintaining the fitting accuracy of GP [27]. We can make the following conclusions from the test results listed in Table 5. (1) both KVM and QRLF have good nonlinear mapping ability in deterministic power curve fitting, and their average MAPE and NRMSE are lower than the benchmark (5 PL). (2) For interval predictions, QRLF can significantly reduce the width of predicted CI, while \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Wind** \\ **Farm** \\ \end{tabular} & **Methods** & _MAPE_ & **NRMSE** & \(PICP_{90\%}\) & \(PINAW_{90\%}\) & \(NC_{90\%}\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} WF1 \\ (2 MW) \\ \end{tabular} } & 5 PL & 6.26\% & 1.68\% & N/A & N/A & N/A \\ \cline{2-6}  & RVM & 5.73\% & 1.62\% & 0.91 & 0.46 & 0.49 \\ \cline{2-6}  & QRNN & 7.93\% & 1.83\% & 0.83 & 0.31 & 0.38 \\ \cline{2-6}  & QRLF & 5.95\% & 1.65\% & 0.82 & 0.27 & 0.29 \\ \hline \multirow{3}{*}{\
\begin{tabular}{c} WF2 \\ (2 MW) \\ \end{tabular} } & 5 PL & 11.44\% & 2.21\% & N/A & N/A & N/A \\ \cline{2-6}  & RVM & 9.12\% & 2.19\% & 0.90 & 0.81 & 0.89 \\ \cline{2-6}  & QRNN & 15.88\% & 2.34\% & 0.79 & 0.43 & 0.54 \\ \cline{2-6}  & QRLF & 9.28\% & 2.19\% & 0.91 & 0.36 & 0.39 \\ \hline \multirow{3}{*}{
\begin{tabular}{c} WF3 \\ (5 MW) \\ \end{tabular} } & 5 PL & 10.28\% & 1.72\% & N/A & N/A & N/A \\ \cline{2-6}  & RVM & 8.05\% & 1.65\% & 0.88 & 0.61 & 0.69 \\ \cline{2-6}  & QRNN & 13.99\% & 2.49\% & 0.61 & 0.39 & 0.67 \\ \cline{2-6}  & QRLF & 8.84\% & 1.72\% & 0.93 & 0.39 & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fitting results of different wind turbine power curve (WTPC) modelling methods. Figure 9: Fitting results of WT03 in WF1 where the black points are the measured power outputs, the green curves are the deterministic fitting curves and the orange curves are the boundaries of the predicted CI. maintaining high coverage probabilities. As listed in Table 5, the proposed method has almost the highest \(\mathit{PICP}_{90\%}\) and the lowest \(\mathit{PINAW}_{90\%}\) compared with RVM and QRNN. As a result, the proposed QRLF provides the most suitable predicted CI and has the lowest \(\mathit{NC}_{90\%}\). (3) From the fitting results in Table 5, there is no obvious correlation between the fitting accuracy and the installed capacity of wind turbine. Moreover, the performance rankings of the aforementioned fitting methods have not changed with the wind turbine installed capacity. More details can be obtained from Figure 9.  
In Figure 9, during the training process, RVM assumes that the wind power follows the same Gaussian prior distribution, and thus the predicted CI in different wind speed ranges have similar width.