## 3 Results and Discussion
### Extreme Loads at the Blade Root  
Finally, the maximum flapwise bending moment of blade root in chord coordinate (CMF_ROOT) and the maximum edgewise bending moment of blade root in chord coordinate (CME_ROOT) was calculated respectively by the PSO-ELPM model. Each of them was computed ten times independently. Figures 8 and 9 give the iterative processes, from which the extreme CMF_ROOT and the extreme CME_ROOT was obtained. The extreme values of CMF_ROOT and CME_ROOT are shown in Table 1. The status parameters of the wind turbine, while the extreme loads were reached, are given to highlight in Table 2.  
From Table 1, it can be seen that the results using PSO-ELPM are well in agreement with those using FOCUS, with a difference of less than 10%. Especially for the extreme CMF_ROOT, the error was lower than 5%. It shows that PSO-ELPM can be used for predicting the extreme load. Moreover, PSO-ELPM needs much less computational cost than FOCUS. That is, PSO-ELPM only needs about 50 min to get the extreme CMF_ROOT (ten times calculation) while FOCUS needs about 12 h (one-time calculation). Hence, PSO-ELPM is very suitable for preliminary blade design, and even optimization design. Furthermore, from Figures 8 and 9, it is clear that the status parameters of the wind turbine changed with iteration. When the maximum load was obtained, the corresponding values of these \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{3}{c}{**1.5 MW**} & \multicolumn{3}{c}{**2.0 MW**} \\ \cline{2-7}  & **FOCUS** & **PSO-ELPM** & **Error** & **FOCUS** & **PSO-ELPM** & **Error** \\ \hline CMF\_ROOT (kNm) & 4176.45 & 4197.57 & 0.51\% & 8084.68 & 7928.81 & \(-\)1.93\% \\ CME\_ROOT (kNm) & 2571.43 & 2741.36 & 6.61\% & 4542.79 & 4352.48 & \(-\)4.19\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The extreme values of CMF_ROOT and CME_ROOT. \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline  & \multicolumn{3}{c}{**1.5 MW**} & \multicolumn{3}{c}{**2.0 MW**} \\ \cline{2-7}  & **FOCUS** & **PSO-ELPM** & **Error** & **FOCUS** & **PSO-ELPM** & **Error** \\ \hline CMF\_ROOT (kNm) & 4176.45 & 4197.57 & 0.51\% & 8084.68 & 7928.81 & \(-\)1.93\% \\ CME\_ROOT (kNm) & 2571.43 & 2741.36 & 6.61\% & 4542.79 & 4352.48 & \(-\)4.19\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Extreme bending moments and their corresponding status parameters with considering the constraint conditions about the \(\beta_{m}\). parameters were immediately determined in PSO-ELPM. This is an advantage of PSO-ELPM while FOCUS and other statistical models cannot do it. This will be very useful to analyze the conditions while the extreme loads occur. If we can avoid this condition by control or other methods, the extreme loads will largely reduce. From Table 2, we can see that the values of \(\gamma\) were all negative. This reveals that the negative yaw angle made it easy to generate the extreme bending moments. For the pitch angle, it is always close to the \(\beta_{r}\), which indicates that smaller pitch angle will obtain larger values for the extreme bending moments. This can also be used to explain why the control rules always increase the pitch angle when the wind speed exceeds the rated value. For the rotor speed, the extreme CMF_ROOT does not get its upper bound while the extreme CME_ROOT does. The reason may be the function of the constraints, which limits the rotor speed corresponded to the pitch angle and the wind speed. To further certify the above speculation, the constraint conditions about the \(\beta_{m}\) are not considered. In this case, \(\beta_{2}\) will be decided by the mechanical property of the wind turbine and meet Equation (44). \[\beta_{u}\geq\beta_{2}\geq\beta_{l} \tag{44}\]  
Table 3 gives the extreme bending moments at the root without considering the constraint conditions about the \(\beta_{m}\) and their corresponding status parameters of the wind turbine calculated by PSO-ELPM. We can see that \(\Omega\) gets the upper bound at the extreme loads. These results indicate that the above speculation is correct.