# Design and Testing of a LUT Airfoil for Straight-Baded Vertical Axis Wind Turbines
## 2 Basis and Method of Design for the LUT Airfoil
### LUT Airfoil Optimization and Design Method  
Optimization algorithm and numerical simulation are widely used in design of the airfoil for VAWTs. Therefore, in this study, the sequential quadratic programming (SQP) method [25; 26] was adopted to design the LUT airfoil, because it can properly solve the problem of constrained optimization, maintain the approximate second-order drop speed, and effectively handle constraint conditions. The optimization process is detailed in the following. The optimization problem can essentially be boiled down to a nonlinear programming problem, as follows:  
\[\left\{ \begin{array}{ll}Min&f(\mathrm{X})\\ \begin{array}{ll}s.t.&c_{i}(\mathrm{X})=0&i=1,2,\cdots,m_{e}\\ &c_{i}(\mathrm{X})\geq 0&i=m_{e}+1,\cdots,m\end{array}\end{array} \right. \tag{1}\]  
where \(f(\mathrm{X})\) is the objective function, \(c_{i}(\mathrm{X})\) is the constraint condition, \(m_{e}\) is the number of constraint conditions of the equation, and \(m\) is the number of total constraint conditions. Figure 1: Schematic of an H-type Vertical Axis Wind Turbines (H-VAWT): (**a**) 3D schematic; and (**b**) 2D cross-section. The rotor of the H-VAWT turns counterclockwise; the azimuth angle \(\theta\) is defined to help describe the active conditions of an H-VAWT. R is the radius of the rotor in units of mm. \(\omega\) is the rotation angular velocity of the rotor in units of rad/s. For the convenience of derivation, the Lagrange function of Equation (1) is:  
\[L(\mathrm{X},\lambda)=f(\mathrm{X})-\sum_{i=1}^{m_{e}}\lambda_{i}\cdot c_{i}( \mathrm{X}) \tag{2}\]  
The gradient vector of this function of \(\mathrm{x}\) and the Hesse matrix are denoted, respectively, by:  
\[\
abla_{x}L(\mathrm{X},\lambda)=\
abla f(\mathrm{X})-\sum_{i=1}^{m_{e}}\lambda _{i}\
abla c_{i}(\mathrm{X}) \tag{3}\]  
\[\
abla_{x}^{2}L(\mathrm{X},\lambda)=\
abla^{2}f(\mathrm{X})-\sum_{i=1}^{m_{e} }\lambda_{i}\
abla^{2}c_{i}(\mathrm{X}) \tag{4}\]  
For \(\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{m})\), the gradient vector and Hesse matrix of the Lagrange function of \(\mathrm{X}\) and \(\lambda\), corresponding to \(\
abla L(\mathrm{X},\lambda)\) and \(\
abla^{2}L(\mathrm{X},\lambda)\), respectively, are defined as follows:  
\[\
abla L(\mathrm{X},\lambda)=\left[ \begin{array}{cc}\
abla f(\mathrm{X})- \
abla c(\mathrm{X})\lambda\\ -c(\mathrm{X})\end{array} \right] \tag{5}\]  
\[\
abla^{2}L(\mathrm{X},\lambda)=\left[ \begin{array}{cc}\
abla_{x}^{2}L( \mathrm{X},\lambda)&-\
abla c(\mathrm{X})\\ -\
abla c(\mathrm{X})^{T}&0\end{array} \right] \tag{6}\]  
where \(\
abla c(\mathrm{X})\) is the Jacobian matrix for the vector function at \(\mathrm{X}\), that is, the matrix of \(\mathrm{n}\times\mathrm{m}\) with the partial derivative \(\frac{\partial c_{i}(\mathrm{X})}{\partial x_{i}}\) of the \((i,j)\) element. The first-order Taylor of the gradient vector is expanded to:  
\[\
abla L(\mathrm{X},\lambda)=\
abla L\Big{(}\mathrm{X}^{k},\lambda^{k}\Big{)} +\
abla^{2}L\Big{(}\mathrm{X}^{k},\lambda^{k}\Big{)}\left[ \begin{array}{c} \mathrm{X}-\mathrm{X}^{k}\\ \lambda-\lambda^{k}\end{array} \right] \tag{7}\]  
where \(k\) represents the current iteration. It is a necessary condition that the gradient is equal to zero for the optimal solution of the Lagrange function, that is, \(\
abla L(\mathrm{X},\lambda)=0\). Substituting Equations (5) and (6) into the first-order Taylor expansion of the gradient vector produces:  
\[\left[ \begin{array}{cc}\
abla_{x}^{2}L\Big{(}\mathrm{X}^{k},\lambda^{k} \Big{)}&-\
abla c\Big{(}\mathrm{X}^{k}\Big{)}\\ -\
abla c\Big{(}\mathrm{X}^{k}\Big{)}^{T}&0\end{array} \right]\left[ \begin{array} []{c}\mathrm{X}-\mathrm{X}^{k}\\ \lambda-\lambda^{k}\end{array} \right]=\left[ \begin{array}{c}-\
abla f\Big{(} \mathrm{X}^{k}\Big{)}+\
abla c\Big{(}\mathrm{X}^{k}\Big{)}\lambda^{k}\\ c\Big{(}\mathrm{X}^{k}\Big{)}\end{array} \right] \tag{8}\]  
By defining \(\mathrm{d}=\mathrm{X}-\mathrm{X}^{k}\) and replacing \(\
abla_{x}^{2}L\Big{(}\mathrm{X}^{k},\lambda^{k}\Big{)}\) with the Hessen matrix \(B^{k}\), we can obtain:  
\[\left[