# Effects of Blade Extension on Power Production and Ultimate Loads of Wind Turbines
## 4 Dynamic Load Evaluations
### Ultimate Loads in Gust Wind Condition  
Gust wind condition includes extreme coherent gust with direction change (ECD), extreme wind shear (EWS) and extreme operating gust (EOG). For the extreme coherent gust with direction change (ECD) [23], the magnitude \(V_{cg}\) is 15 m/s. The wind speed should be defined by:  
\[V(z,\ t)=\left\{ \begin{array}{ll}V(z)&for\ \ \ t\leq 0\\ V(z)+0.5V_{cg}(1-\cos(\pi t/T))for\ \ \ 0\leq t\leq T\\ V(z)+V_{cg}&for\ \ t\geq T\end{array} \right. \tag{14}\]  
where \(T=10\) s is the rise time and the wind speed \(V(z)\) is given by the normal wind profile model in Equation (11). The rise in wind speed should be assumed to occur simultaneously with the wind direction change \(\theta\) from \(0^{\circ}\) up to and including \(\theta_{cg}\), where the magnitude of \(\theta_{cg}\) is defined by:  
\[\theta_{cg}(V_{hub})=\left\{ \begin{array}{ll}180^{\circ}&for\ \ \ V_{hub}<4\ m/s\\ \frac{720^{\circ}(\text{m/s})}{V_{\text{lnb}}}&for\ \ \ 4\ m/s<V_{hub}<V_{ref} \end{array} \right. \tag{15}\]  
Then, the simultaneous direction change is given by:  
\[\theta(t)=\left\{ \begin{array}{ll}0^{\circ}&for\ \ \ t<0\\ \pm 0.5\theta_{cg}(1-\cos(\pi t/T))&for\ \ 0\leq t\leq T\\ \pm\theta_{cg}&for\ \ t>T\end{array} \right. \tag{16}\]  
The extreme wind shear (EWS) [23] includes the vertical shear and the horizontal shear. Transient (positive and negative) vertical shear is defined by:  
\[V(z,\ t)=\left\{ \begin{array}{ll}V_{hub}\Big{(}\frac{z}{z_{hub}}\Big{)}^{ \alpha}\pm\Big{(}\frac{z-z_{hub}}{D}\Big{)}\Big{(}2.5+0.2\beta\sigma_{1}\Big{(} \frac{D}{\Lambda_{1}}\Big{)}^{1/4}\Big{)}(1-\cos(2\pi t/T))&for\ \ \ 0\leq t\leq T\\ V_{hub}\Big{(}\frac{z}{z_{hub}}\Big{)}^{\alpha}&\text{otherwise}\end{array} \right. \tag{17}\]  
Transient horizontal shear is defined by:  
\[V(z,\ t)=\left\{ \begin{array}{ll}V_{hub}\Big{(}\frac{z}{z_{hub}}\Big{)}^{ \alpha}\pm\Big{(}\frac{y}{D}\Big{)}\Big{(}2.5+0.2\beta\sigma_{1}\Big{(}\frac{D }{\Lambda_{1}}\Big{)}^{1/4}\Big{)}(1-\cos(2\pi t/T))&for\ \ \ 0\leq t\leq T\\ V_{hub}\Big{(}\frac{z}{z_{hub}}\Big{)}^{\alpha}&\text{otherwise}\end{array} \right. \tag{18}\]  
where \(\alpha\) and \(\beta\) are constant values, equal to 0.2 and 6.4 respectively. For the extreme operating gust (EOG) [23], the hub height gust magnitude \(V_{gust}\) should be given by:  
\[V_{gust}=Min\left\{1.35(V_{e1}-V_{hub}),3.3\Bigg{(}\frac{\sigma_{1}}{1+0.1 \frac{D}{\Lambda_{1}}}\Bigg{)}\right\} \tag{19}\]  
where \(D\) is the rotor diameter, \(V_{e1}\) is the hub-height 1-year extreme mean wind speed, and \(\Lambda_{1}\) is the turbulence scale parameter, which should be given by:  
\[\Lambda_{1}=\left\{ \begin{array}{ll}0.7z&for\ \ z\leq 60\ m\\ 42\ m&for\ \ z\geq 60\ m\end{array} \right. \tag{20}\]  
where \(z\) is the height above ground. The simulation duration is 80 s for ECD wind condition and 60 s for EWS and EOG wind condition. Gust starts at 10 s. The turbine operating conditions include power production (DLC 1.4 and DLC 1.5 in IEC 61400-1 standard), power production plus loss ofelectrical grid connection (DLC 2.3 in IEC 61400-1 standard) and normal shutdown (DLC 4.2 in IEC 61400-1 standard). The tower bottom ultimate loads of turbines with blade extension and the original design in gust wind condition are shown in Table 11. The table shows that loads at tower bottom all increase after the blade extension. Among them, the ultimate load of \(M_{y}\) shows an obvious increasing trend with the blade extension, and the percentage increase of \(M_{y}\) is 15.43%. Five load cases which have the highest \(M_{y}\) at tower bottom of turbines with blade extension are compared with that of the original design, and the histogram is shown in Figure 14. The figure shows that all \(M_{y}\) in each group have an increasing trend with the blade extension. The ultimate load of \(M_{y}\) at tower bottom of original blade turbine occurs at the wind speed of 11.5 m/s, while that of blade extension turbines occurs at the wind speed of 13.5 m/s.