## 3 Methodology
### Boundary Handling Methods  
The boundary of studied wind farm is irregular, which is shown in Figure 2, as indicated by the black curved lines. Two different grid densities, i.e., coarse 20 \(\times\) 30 and refined 40 \(\times\) 60 grid, are employed, as shown in Figure 2a,b, respectively. The grid cell lengths for the two different divisions are 200 and 100 m, respectively. It is defined that the feasible grid cells are those that are inside the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Direction** & **k** & **c** & **\(\omega\)** & **Direction** & **k** & **c** & **\(\omega\)** \\ \hline
0\({}^{\circ}\)\(\sim\)15\({}^{\circ}\) & 2 & 9 & 0 & 180\({}^{\circ}\)\(\sim\)195\({}^{\circ}\) & 2 & 9 & 0.01 \\
15\({}^{\circ}\)\(\sim\)30\({}^{\circ}\) & 2 & 9  
wind farm boundary that can be the potential wind turbine locations, while the infeasible grid cells are those outside the wind farm boundary that cannot be the positions for placing wind turbines. In order to guarantee that the wind turbines are placed within the wind farm boundary for final solutions, the infeasible grid cells are combined into a number of larger rectangles with different sizes. These artificially selected rectangles are made to facilitate the optimization coding, and they are determined by the boundary defining point \(P_{1}\) (red color asterisk points in Figure 2) and one of the four vertices of the whole wind farm boundary as the rectangle diagonal. In this manner, the infeasible grid regions can be determined by the range of row and column indices of these rectangles. Reflected by the number of boundary defining points for the two different grid densities, there are 25 rectangles in total for the 20 \(\times\) 30 grid, while there are 43 rectangles for the 40 \(\times\) 60 grid. As can be anticipated, the number of infeasible rectangles increases with more a densified grid. More infeasible rectangles that are inappropriate for locating wind turbines will enable the optimization to be subject to a more complex boundary constraint, which leads to an increasingly high computational burden. Given the significant increase in computational cost, grid cells that are denser than 40 \(\times\) 60 are not included in the comparative study of this paper. Unlike the boundary representation using the grid method, Figure 3 shows an alternative way to represent the irregular boundary while using the coordinate method. To facilitate the mathematical coding of the wind farm boundary representation, the left portion of the boundary curve is approximated by interpolated polylines. In order to increase the accuracy of the interpolation polylines, the left portion boundary is divided into two interpolated curves by polynomial approximation: \(y_{1}(x)\) (blue color in Figure 3) and \(y_{2}(x)\) (green color in Figure 3). These two interpolation polylines are connected with the interpolation points (shown as the asterisk points in the figure). As can be seen, the difference between the original boundary and the interpolated polylines is small, which justifies the accuracy of this approximation process. In comparison, the right portion of the boundary is directly the polyline \(p\) ((X1, Y1), (X2, Y2),..., (X7, Y7)), which can be determined by connecting these vertex  
Figure 2: Schematic of the boundary handling technique for the optimization study using the grid method with: (**a**) coarse 20 \(\times\) 30 grid and (**b**) refined 40 \(\times\) 60 grid. points (shown as the dot points in Figure 3). For \(i\)-th wind turbine with the \(x\) coordinate \(x_{i}\), define a feasible interval of y values that is located inside the wind farm boundary as:  
\[\mathcal{C}(x_{i})=\{z\mid y_{\ell}(x_{i})\leq z\leq y_{u}(x_{i})\} \tag{11}\]  
where \(y_{\ell}(x_{i})\) is the lower bound and \(y_{u}(x_{i})\) is the upper bound at the coordinate \(x_{i}\), and \(X=x_{1}y_{1}x_{2}y_{2}\ldots x_{N}y_{N}\) is comprised of the coordinates of wind turbines (with 2N variables in total).