# Effects of Blade Extension on Power Production and Ultimate Loads of Wind Turbines
## 2 Evaluation Method and Transformation Scheme
### Turbine Model and Transformation Scheme  
Here, we use the model of a IIA 1.5 MW wind turbine to investigate the influence of blade extension on the turbine performance and ultimate loads. The model is obtained from the turbine manufacturing enterprise and is sophisticatedly established in the design \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Wind Conditions** & **Operating Conditions** & \(\begin{array}{c}V_{sub}\\ \text{(m/s)}\end{array}\) & \(\begin{array}{c}\text{\bf Yaw Error}\\ \text{(deg)}\end{array}\) & \(\begin{array}{c}\text{\bf Starting Azimuth Angle}\\ \text{(deg)}\end{array}\) & \(\begin{array}{c}\text{\bf Design Load Case}\end{array}\) \\ \hline NTM & \begin{tabular}[]{c}Power production plus \\ occurrence of fault, \\ emergency shutdown \\ \end{tabular} & 3–25 & — & — & 2.1, 2.2, 5.1 \\ \hline ETM & Power production & 3–25 & \(-8,0,8\) & — & 1.3 \\ \hline EOG & \begin{tabular}[]{c}Power production plus loss of \\ electrical grid connection, \\ normal shutdown \\ \end{tabular} & 3–25 & \(-8,0,8\) & 0-90 & 2.3, 4.2 \\ \hline ECD & Power production & 3–25 & \(-8,0,8\) & 0-90 & 1.4 \\ \hline EWS & Power production & 3–25 & \(-8,0,8\) & 0-90 & 1.5 \\ \hline EWM & Idling with loss of electrical & 42.5 & \(-8,0,8\) & — & 6.1 \\ \cline{2-6}  & \
\begin{tabular}[]{c}Idling with excess of electrical \\ power consumption \\ \end{tabular} & 42.5 & 0–330 & — & 6.2 \\ \cline{2-6}  &
\begin{tabular}[]{c}Idling with extreme \\ yaw misalignment \\ \end{tabular} & 34 & \(-20\)–20 & — & 6.3 \\ \cline{2-6}  & Idling with pitch failure & 34 & \(-8,0,8\) & — & 7.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design load cases. Figure 1: Coordinate systems for different loads: (**a**) Coordinate system for blade loads; (**b**) Coordinate system for hub loads; (**c**) Coordinate system for tower loads. procedure. The blade length and hub height of the turbine model are 37.5 m and 61.5 m, respectively. For clarification, the parameters of the wind turbine are shown in Table 2. As can be observed, the electrical losses follow a liner model, where the electrical power output \(P_{e}\) can be calculated by [24]:  
\[P_{e}=\varepsilon(P_{\mathrm{s}}-L_{0}) \tag{1}\]  
where \(\varepsilon\) is efficiency and \(L_{0}\) is no-load power loss. Considering the technical difficulty and the cost, extension is performed at the tip of the blade. The 0-36 m part of the blade remains unchanged and the left 1.5 m part is extended by 1 m, 1.5 m and 2 m, respectively. The extension part is carefully designed to ensure that the blade shape is smooth. Both the original and modified blades are shown in Figure 2.  
Due to the extension of the blades, the generator torque-speed curve of the turbine should be adjusted to track the optimal tip-speed ratio. In the original model of the wind turbine (without blade extension), the torques are determined by a look-up table way in GH-Bladed. As is shown in Figure 3, a linear transition is adopted in the phases with minimum or maximum rotational speed. Specifically, in the cut-in phase, generator torque-speed curve is set to linearly increase to the 1200 rpm, and when the generator speed reaches 1600 rpm, the generator torque-speed curve is set to linearly increase to the maximum generator torque. Between the two linear parts of the curve, the wind turbine will track its optimal tip-speed ratio and the generator torques are given by:  
\[Q_{d}=K_{opt}\Omega^{2} \tag{2}\]  
where \(Q_{d}\) is generator torque, \(\Omega\) is measured generator speed, and \(K_{opt}\) can be calculated as:  
\[K_{opt}=\pi\rho R^{5}C_{Pmax}/2\lambda_{opt}^{3}G^{3} \tag{3}\]  
where \(\rho\) is air density, \(R\) is rotor radius, \(\lambda_{opt}\) is the optimal tip-speed ratio, \(C_{Pmax}\) is power coefficient at the optimal tip-speed ratio, and \(G\) is the gearbox ratio. The \(C_{P}\)-\(\lambda\) curve can be calculated in the \"Performance coefficients\" module in GH-Bladed. \(\lambda_{opt}\) and \(C_{Pmax}\) are the values corresponding to the highest point of \(C_{P}\)-\(\lambda\) curve.