# Effects of Blade Extension on Power Production and Ultimate Loads of Wind Turbines
## 3 Steady Power Curve and Operational Loads
### Steady Operational Loads  
The wind rotor trust \(T\) and the blade-bending moment \(M_{y}\) of turbines with blade extension and the original design are shown in Figures 7 and 8, respectively. Both the loads show a significant increase in the low wind speed region due to the blade extension. The maximum loads appear near the rated wind speed. For more details, the ultimate loads from the steady calculation for the turbines with blade extension and the original design are shown in Table 5. As can be observed, the ultimate loads are significantly increased due to the blade extension. To reveal the increase mechanism, the loads are analyzed under the ultimate conditions. Figure 8: Bending moment of the blade root for turbines with different blade lengths. Figure 7: Thrust of the wind rotor for turbines with different blade lengths. The trust of the wind rotor \(T\) can be calculated by:  
\[T=\frac{1}{2}\rho V_{m}^{2}C_{T}A \tag{5}\]  
where \(V_{m}\) is the wind speed corresponding to the maximum thrust force, which is generally around the rated wind speed, \(C_{T}\) is trust coefficient and \(A\) is swept area. For clarification, the variation of the quantities due to the blade extension are shown in Table 6. As can be observed, the increase percentage of the thrust force is significantly lower than that of the swept area due to the decrease of \(V_{m}\). In our case, the former percentage is 50\(\sim\)65% of the latter one. Table 5 shows that after blade extension, the bending moment \(M_{y}\) of blade root increase significantly due to the increase in the swept area. Theoretically, for a wind turbine with blade extension,  
\[M_{y1}=\int_{0}^{R_{0}}F_{x}(r)\cdot r\mathrm{d}r+\int_{R_{0}}^{R_{1}}F_{x}(r )\cdot r\mathrm{d}r=M_{y0}+\Delta M_{y} \tag{6}\]  
\[T_{1}=\int_{0}^{R_{0}}F_{x}(r)\mathrm{d}r+\int_{R_{0}}^{R_{1}}F_{x}(r)\mathrm{ d}r=T_{0}+\Delta T \tag{7}\]  
where \(F_{x}(r)\) is the distribution of the trust force along the blade, \(R_{0}\) is the radius of the original wind rotor and \(R_{1}\) is the radius of wind rotor with blade extension. After the blade extension, the bending moment and thrust are increased by \(\Delta M_{y}\) and \(\Delta T\), respectively. As shown in Figure 9, the arm of \(\Delta T\) for the blade root is much larger than the mean arm of \(T_{0}\). As a result, the increase percentage of \(M_{y}\) on blade root is slightly greater than that of thrust \(T\), and specifically, the former one is about 110\(\sim\)130% of the latter one. It is worth noting that no linear relationship can be observed between AEP and the rotor area. Some other steady loads of the wind turbine are shown in Table 5. Quantitatively, the variation of wind rotor trust is close to that of the thrust of hub and tower bottom and overturning moment of tower bottom. Recall the coordinate system shown in Figures 1 and 9; the thrust of the wind rotor, hub and tower are almost equivalent and hence they have almost the same increase ratio under the same wind conditions. Since the tower height is constant, the variation of overturning moment of tower bottom is similar to that of the thrust of the wind rotor. We can see that the thrust and bending moment at yaw bearing all increase due to the blade extension. In addition, the variation of these loads with blade extension is close to that of the trust of wind rotor. It is also worth noting that the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Blade Length** & **37.5 m** & **38.5 m** & **39 m** & **39.5 m** \\ \hline \(V_{m}\) (m/s) & 11.2 & 10.9 (97.32\%) & 10.8 (96.43\%) & 10.6 (94.64\%) \\ \(C_{T}\) & 0.551 & 0.572 (103.81\%) & 0.574 (104.17\%) & 0.587 (106.53\%) \\ \(A\) (m\({}^{2}\)) & 4668.7 & 4914.1 (105.26\%) & 5039.1 (107.93\%) & 5165.7 (110.65\%) \\ \(T\) (kN) & 197.6 & 204.5 (103.36\%) & 206.6 (104.56\%) & 208.7 (105.60\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The wind speed and thrust coefficient under the maximum thrust force for different blade lengths.