# Development of a CFD-Based Wind Turbine Rotor Optimization Tool in Considering Wake Effects
## 3 Results and Discussion
### Rotor Optimizations  
#### 3.4.1 Optimization Setup  
This section describes the optimization problem used in present rotor aerodynamic design. With the optimization objectives of AEP and ATP, and the design variables of chord length, twist angle, relative thickness and rotational speed, the NSGA-II optimization method [55] is used to optimize the wind turbine rotor. In the NSGA-II optimization algorithm, the binary coding, crossover mutation and gene mutation are built to avoid premature phenomena during the optimization process. To better understand the optimization process, a sketch of the optimization diagram is given in Figure 12. The design variables are the blade geometrical data as well as the rotational speed. The \"Bspline\" curve is applied to reduce the number of control points. The CFD solver is embedded in the optimization loop which is a time-consuming part of the whole design process. The number of necessary iterations is determined by the NSGA-II optimization routine. Each evolution contains 30-50 populations and each population takes about 60 min. Figure 11: Comparisons between the observed data and RANS numerical for Cp, Ct, Power, and normalized power for 2 MW V80 wind turbines. (**a**) Power; (**b**) Ct.  
Figure 10: Contours of the stream-wise velocity for eight wind turbines. The design objective is to maximize the annual energy production (AEP) and minimize the rotor thrust force (ATP, annual thrust production), thus the design objectives are to minimize both 1/AEP and ATP  
subject to \[X\in R^{n},\] (20)  
\[x_{k}^{L}\leq x_{k}\leq x_{k}^{UI},\;k=1,\ldots,n \tag{21}\]  
The vector \(X\) contains a total of \(n\) variables that are real numbers, \(R\). The design variables, \(X=[x_{1},x_{2},\ldots,x_{n}]\), are the control points that define the chord, twist angle and relative thickness as a function of blade span. The upper U and lower L notations, \(x_{k}^{L}\leq x_{k}\leq x_{k}^{UI}\), are the upper and lower limits of the control points. The 3 rd-order Bsplines [56] are used to parameterize the chord, twist angle and relative thickness distributions, see Figure 13. The design is performed from a position at a radius of 0.337R to the tip of the blade. Thus, three control points for the chord, and two control points each for the twist angle and relative thickness distributions at the outboard of the blade are optimized. The 3 rd-order Bspline curve is defined by  
\[q_{i}(u)=\sum\limits_{i=0}^{n}N_{i,p}(u)P_{i}\;\;\;\;\;\;\;\;\; \;u_{a}<u<u_{b},\;i=1,2,\cdots,n \tag{22}\] \[N_{i,0}(u)=\left\{ \begin{array}{ll}1&\mbox{if }u_{i}\leq u\leq u_{i +1}\\ 0&\mbox{otherwise}\end{array} \right.\] \[N_{i,p}(u)=\frac{u-u_{i}}{u_{i+p}-u_{i}}N_{i,p-1}(u)+\frac{u_{i+ p+1}-u}{u_{i+p+1}-u_{i+1}}N_{i+1,p-1}(u)\] \[u_{0}=u_{1}=\cdots=u_{p}=0\] \[u_{i+p}=u_{i+p-1}+|p_{i}-p_{i-1}|,\;i=1,2,\cdots,n-1\] \[u_{n+p}=u_{n+p+1}=\cdots=u_{n+p+3}=1\]  
where \(q_{i}(u)\) is a vector-valued function of the independent variable, \(u\). \(\{P_{i}\}\) are the control points, and the \(\{N_{i,p}(u)\}\) are the 3 rd degree Bspline basis functions. The \(i\)th Bspline basis function of \(p\)-degree (order \(p\)+1). \(u_{i}\) is the knot vector. Figure 12: Wind turbine rotor aerodynamic optimization diagram. To calculate the AEP and ATP, it is necessary to combine the power curve with the probability density of wind. The Weibull distribution function defining the probability density can be written in the following form  
\[f(V_{i}<V<V_{i+1})=\exp\left(-\left(\frac{V_{i}}{A}\right)^{k}\right)-\exp\left( -\left(\frac{V_{i+1}}{A}\right)^{k}\right) \tag{23}\]  
where \(A\) is the scaling parameter, \(k\) is the shape factor and \(V_{i}\) is the wind speed distribution.