# Wind Turbine Power Curve Modelling with Logistic Functions Based on Quantile Regression
## 4 Case Study
### Experimental Results
#### 4.3.1 Results for Parameter Selection and Optimization  
Before power curve fitting, we eliminate the outliers by using the method introduced in Section 3.1 to reduce the interference of power outliers on model structure determination. Then, the fitting results of three selected wind turbines in WF3 are listed in Table 3.  
In Table 3, \(NC_{90\%}\) is \(NC_{\alpha}\) at the confidence level of 0.9; 4P-QRLF and 5P-QRLF are QRLF with four (five) model parameters; we can get the deterministic fitting curves when \(\alpha\) is set to 0.5. For each type of QRLF based method, PSO, WOA and Adam optimization algorithm are used to estimate the model parameters, respectively. As mentioned in Section 2.4.3, we repeat each optimization algorithm five times to reduce the fitting error caused by local minimum. Table 4 lists the detailed information of the control parameter for optimization algorithms, and Figure 6 shows the values of QR cost function (expressed in Equation (8)) of WT02 during the training process. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Control Parameters** \\ \hline PSO & particle number = 20; inertia weight (\(\omega\)) = 0.8; \\  & acceleration constants (\(c_{1}\), \(c_{2}\)) = 2 \\ \hline WOA & search agent number (whales population) = 40 \\ \hline Adam & exponential decay rates (\(\gamma_{1}\), \(\gamma_{2}\)) = 0.9; learning rate = 0.0002 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Control parameters of the optimization algorithms. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Fitting** & **Optimization** & \multicolumn{3}{c}{**WT02**} & \multicolumn{3}{c}{**WT08**} & \multicolumn{3}{c}{**WT10**} \\ \cline{3-11}
**Method** & **Algorithm** & **MAPE** & **NRMSE** & \(NC_{90\%}\) & **MAPE** & **NRMSE** & \(NC_{\alpha}\) & **MAPE** & **NRMSE** & \(NC_{90\%}\) \\ \hline \multirow{3}{*}{4P-QRLF} & PSO & 18.89\% & 2.04\% & 0.53 & 9.12\% & 1.77\% & 0.51 & 10.30\% & 1.94\% & 0.44 \\ \cline{2-11}  & WOA & 19.10\% & 1.98\% & 0.62 & 9.31\% & 1.81\% & 0.47 & 9.90\% & 1.49\% & 0.46 \\ \cline{2-11}  & [PERSON] & 38.50\% & 2.40\% & 0.72 & 22.25\% & 2.37\% & 0.49 & 20.99\% & 2.33\% & 0.57 \\ \hline \multirow{3}{*}{5P-QRLF} & PSO & 12.57\% & **1.52\%** & **0.44** & 7.42\% & **1.49\%** & **0.39** & **7.00\%** & **1.37\%** & **0.38** \\ \cline{2-11}  & WOA & **9.92\%** & 1.92\% & 0.57 & **7.31\%** & 1.61\% & 0.43 & 8.23\% & 1.56\% & 0.41 \\ \cline{2-11}  & Adam & 27.85\% & 1.96\% & 0.77 & 9.09\% & 1.81\% & 0.45 & 12.44\% & 1.86\% & 0.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitting results of different model parameters and optimization algorithms. The model with the lowest fitting error is indicated by bold numbers. As shown in Figure 6, WOA has the fastest convergence speed in both 4P-QRLF and 5P-QRLF, followed by PSO. However, Adam algorithm is difficult to converge, especially in the optimization process of 4P-QRLF. After 1000 iterations, the values of QR cost function of 4P-QRLF optimized by PSO, WOA and Adam are 72.6, 72.7 and 98.3, and the values of 5P-QRLF are 60.6, 62.2 and 83. Although the test results might be inconsistent in the repeated experiments, they generally have the same trend. We can draw the following conclusions from the results in Table 3 and Figure 6. (1) Similar to the conclusions of previous studies on 4 PL and 5 PL [11], 5P-QRLF can reduce the lack-of-fit error of 4P-QRLF in asymmetry curve fitting. The results show that 5P-QRLF has better performance in both deterministic and probabilistic WTPC modelling. (2) WTPC optimized by PSO and WOA has higher fitting accuracy than Adam algorithm. The main reason is that Adam is difficult to converge during the optimization process. (3) Although the convergence speed of PSO is lower than that of WOA, it has the lowest fitting error among the above three optimization algorithms, especially in probabilistic WTPC modelling (listed in Table 3). In addition, similar conclusions can be obtained when the confidence level \(\alpha\) is set to other values, such as 0.95 or 0.85.