## 2 Mathematical Model for Aerodynamic Shape Optimization
### PARSEC Parametrization  
In the previous literature for the representation of different types of airfoils, the parameterization method was used to transform the shape of the airfoil with a minimum number of control points for aerodynamic optimization purposes. [PERSON] [21] developed a parameterization scheme called parametric section (PARSEC), which represents the shape of an airfoil using an unknown linear combination of the base functions. In the present study, the NREL S-809 was selected as the reference airfoil for parameterization and optimization purposes due to its high lift-to-drag ratio and better aerodynamic performance in comparison with low-chamber and high-thickness airfoils, as established in the literature [28]. The NREL S-809 airfoil is achieved by solving a set of linear equations using twelve different airfoil geometry parameters. The PARSEC method uses these twelve geometric parameters because of the design variables that control the configuration of the airfoil. All twelve airfoil design parameters in geometrical and tabular form are demonstrated in Figure 2 and Table 1, respectively. Figure 3 compares the baseline airfoil with the airfoil generated by the PARSEC method. The NREL S-809 airfoil is divided into two parts,--the upper and lower sections--and both the upper and lower sections are represented by the sixth-order polynomials listed in Equations (12) and (13), respectively:  
\[y_{u}=\sum_{i=1}^{6}a_{up}^{i}x^{i-(1/2)} \tag{12}\]  
\[y_{l}=\sum_{i=1}^{6}a_{lo}^{i}x^{i-(1/2)} \tag{13}\]  
where \(y_{u}\) and \(y_{l}\) signify the y-coordinates of the upper and lower sections, respectively, \(x\) is the normalized chord-wise coordinate, and \(a_{up}^{i}\) and \(a_{lo}^{i}\) are the unknown coefficients for the upper and lower sections, to be determined by using the airfoil design parameters as follows:  
\[C_{up}\times a_{up}=b_{up} \tag{14}\]  
\[C_{lo}\times a_{lo}=b_{lo} \tag{15}\]  
where  
\[C_{up}=\left[\begin{array}{ccccc}1&1&1&1&1\\ p_{3}^{1/2}&p_{3}^{3/2}&p_{3}^{5/2}&p_{3}^{7/2}&p_{3}^{9/2}&p_{3}^{11/2}\\ \frac{1}{2}&\frac{3}{2}&\frac{5}{2}&\frac{7}{2}&\frac{1}{2}\\ \frac{1}{2}p_{3}^{-1/2}&\frac{2}{2}p_{3}^{1/2}&\frac{5}{2}p_{3}^{3/2}&\frac{7 }{2}p_{3}^{5/2}&\frac{9}{2}p_{3}^{7/2}&\frac{11}{2}p_{3}^{9/2}\\ -\frac{1}{4}p_{3}^{-3/2}&\frac{3}{4}p_{3}^{-1/2}&\frac{15}{4}p_{3}^{1/2}&\frac {35}{4}p_{3}^{3/2}&\frac{63}{4}p_{3}^{5/2}&\frac{9}{4}p_{3}^{9/2}\\ 1&0&0&0&0&0\end{array}\right] \tag{16}\]  
Figure 1: National Renewable Energy Laboratory (NREL) S809 airfoil generated by class shape transformation (CST) parameterization.  
\[C_{lo}=\left[\begin{array}{cccccc}1&1&1&1&1&1\\ p_{6}^{1/2}&p_{6}^{3/2}&p_{6}^{5/2}&p_{6}^{7/2}&p_{6}^{9/2}&p_{6}^{11/2}\\ \frac{1}{2}&\frac{3}{2}&\frac{5}{2}&\frac{7}{2}&\frac{7}{2}&\frac{9}{2}&\frac{ 1}{2}\\ \frac{1}{2}p_{6}^{-1/2}&\frac{3}{2}p_{6}^{5/2}&p_{6}^{7/2}&\frac{7}{2}&p_{6}^{ 7/2}&\frac{1}{2}p_{6}^{9/2}\\ -\frac{1}{4}p_{6}^{-3/2}&\frac{3}{4}p_{6}^{-1/2}&\frac{1}{4}p_{6}^{1/2}&\frac{ 3}{4}p_{6}^{9/2}&\frac{63}{4}p_{6}^{5/2}&\frac{9}{4}p_{6}^{6/2}\\ 1&0&0&0&0&0\end{array}\right] \tag{17}\]  
\[b_{up}=\left[\begin{array}{c}(p_{9}+p_{10})/2\\ p_{4}\\ \tan((p_{11}-p_{12})/2)\\ 0\\ p_{5}\\ \sqrt{2p_{1}}\end{array}\right] \tag{18}\]  
\[b_{lo}=\left[\begin{array}{c}(p_{9}-p_{10})/2\\ p_{7}\\ \tan((p_{11}+p_{12})/2)\\ 0\\ \sqrt{2p_{2}}\end{array}\right] \tag{19}\]