# Collaborative Optimization of Aerodynamics and Wind Turbine Blades
## 4 MDO of the Wind Turbine Blade
### Optimization Objective  
In the framework of the collaborative optimization of the aerodynamics and structure of the blades of a wind turbine, the design variables circulate between the system level and the discipline level. As discussed in the discipline analysis section above, the chord length and twist angle are the main factors governing aerodynamic performance, while the chord length, twist angle, and ply thickness are the main factors governing structural performance. As a result, chord length, twist angle, and ply thickness are selected as the design variables. The scatter relating to chord length and twist angle in Table 2 can be regressed by a third-order polynomial. Therefore, regarding the aerodynamic discipline, four design variables can be selected for the chord length, denoted as \(a_{i}\) (\(i=1,\,2,\,3,\,4\)). In the same way, the number of design variables relating to twist angle is selected as four, denoted as \(b_{i}\) (\(i=1,\,2,\,3,\,4\)). For the ply thickness, since the area around the blade root contributes a lot in withstanding external loads on the blade, limited sections around the blade root are optimized. Considering the dominant role of the spar in resisting external loads, in this study, four spar ply thicknesses are selected as the design variable relating to the structure discipline, denoted as \(d_{i}\) (\(i=1,\,2,\,3,\,4\)). In a CO framework, the optimization objective at the system level is a combination of performance indices relating to both disciplines. In the study, the performance index from the aerodynamic discipline is selected as torque since it is closely positively related to the power coefficient of the wind turbine. The performance index from the structural discipline is selected as the mass of the blade, while the stress and deflection of the blade are taken as  
Figure 19: Stress distribution on the blade. (**a**) Stress on windward surface of blade; (**b**) stress on leeward surface of blade. Figure 20: Deflection of the blade. the constraints. It is understandable that in the case of guaranteed strength and stiffness, a blade with a lower mass can lower the manufacturing cost. Thus, the optimization objective at the system level is designed as  
\[J=\max\{F=k_{1}D-k_{2}M\} \tag{15}\]  
where \(D\) denotes the torque of the blade; \(M\) is the mass of the blade; and \(k_{i}\) is the weight of discipline \(i\) in the objective. Unlike multi-objective optimization, CO performs a hierarchical optimization. At the discipline level, the optimization objective is not related to the discipline performance but discrepancies associated with design variables. Since different design variables exert different influences on the discipline performance, the effects of design variables are analyzed. To this end, parameter sensitivity analysis is performed to determine the effects of the design variables in the aerodynamic and structural discipline analyses. The principles adopted in the sensitivity analysis are described in [50]. Four chord length \(a_{i}\), four twist angles \(b_{i}\), and four spar ply thicknesses \(d_{i}\) (\(i=1,2,3,4\)) are evaluated to determine their importance. For the aerodynamic discipline, design variable effects on the torque of the blade determined by sensitivity analysis are presented in Figure 21a. For structural discipline, the effects of design variables on the mass of the blade determined by sensitivity analysis are presented in Figure 21b. Based on the results of the sensitivity analysis, one can select important variables as the disciplinary design variables. The two following optimization objectives at the discipline level can be obtained as  
\[f_{1}=(b_{4}^{*}-b_{4})^{2}+(b_{3}^{*}-b_{3})^{2}+(b_{2}^{*}-b_{2})^{2}+(a_{4 }^{*}-a_{4})^{2} \tag{16}\]  
for the aerodynamics discipline, and  
\[f_{2}=(d_{1}^{*}-d_{1})^{2}+(d_{2}^{*}-d_{2})^{2}+(a_{4}^{*}-a_{4})^{2}+(d_{3 }^{*}-d_{3})^{2}+(d_{4}^{*}-d_{4})^{2} \tag{17}\]  
for the structure discipline.