## 2 Numerical Method
### Optimization of Terrain Model BTS
#### 2.3.2 Weight Allocation of Evaluation Indexes  
Because the evaluation of BTS at the entrance of the terrain model mainly focuses on speed-up effect, the equivalent wind attack angle and the increase ratio of turbulence intensity, the specific evaluation indexes of effects of the terrain model BTS are defined as follows:  
\[V_{x}=\sum\limits_{k=1}^{N}\frac{v_{k}-v_{0k}}{v_{0k}}\cdot\frac{\Delta l_{k}}{ H_{w}} \tag{7}\]  
\[A_{x}=\sum\limits_{k=1}^{N}\frac{a_{k}\cdot\Delta l_{k}}{H_{w}} \tag{8}\]  
\[I_{x}=\sum\limits_{k=1}^{N}\frac{I_{k}-I_{0k}}{I_{0k}}\cdot\frac{\Delta l_{k}} {H_{w}} \tag{9}\]  
where \(N\) is the total amount of monitoring points. \(v_{k}\) is the downward wind speed of the \(k^{th}\) monitoring point from the ground and \(v_{0k}\) is the wind speed at the \(k^{th}\) monitoring point of the wind speed entrance. \(\Delta l_{k}\) is the vertical distance between the \(k^{th}\) monitoring point and the \(\left(k-1\right)^{th}\) monitoring point. \(H_{w}\) is the total height of the monitoring points. \(a_{k}\) is the wind attack angle of the \(k^{th}\) monitoring point from the ground. \(I_{k}\) is the turbulence intensity at the \(k^{th}\) monitoring point and \(I_{0k}\) is the turbulent intensity at the \(k^{th}\) monitoring point from the ground at the wind speed entrance. The two-dimensional CFD simulation was used to optimize the parameters of the Witzzinsky curve and horizontal line. For the indexes such as the speed-up ratio of wind speed, equivalent wind attack angle, and turbulence intensity increase ratio at the boundary of the terrain model, the weight allocation based on the correlation degree proposed by [PERSON] ([PERSON] et al. [45]) was adopted to determine the BTS curve parameters for a given total length of the transition section ([PERSON], et al. [32]). There are \(m=13\) schemes and \(n=4\) indexes that have an influence on the comprehensive evaluation of each scheme. They can compose the matrix \(X=\left(x_{ij}\right)_{m\times n}\). In this paper, the evaluation indexes are all cost indexes, which means that the smaller the index is, the better the result is. Standardize values in each column of \(X=\left(x_{ij}\right)_{m\times n}\) using min-max standardization (shown in formula (10) to map each index of the matrix into [0,1]. After conversion, the dimensionless matrix is \(X^{\prime}=\left(x_{ij}\right)_{m\times n}\). Then, select the maximum in each column of \(X^{\prime}=\left(x_{ij}\right)_{8\times 6}\) to obtain the optimal objects \(S_{0}^{+}\) and select the minimum in each column to get the worst object \(S_{0}^{-}\). The correlation coefficient of each scheme with the optimal object and the worst object in each index can be calculated by Equations (11) and (12),  
\[x^{\prime}=\frac{x-\min}{\max-\min} \tag{10}\]  
\[\zeta_{0^{+}i}(j)=\frac{1}{2\Delta_{0^{+}ij}+1} \tag{11}\]  
\[\zeta_{0^{-}i}(j)=\frac{1}{2\Delta_{0^{-}ij}+1} \tag{12}\]  
where \(\Delta_{0^{+}ij}=\left|x_{0^{+}j}-x_{ij}\right|\) and \(\Delta_{0^{-}ij}=\left|x_{0^{-}j}-x_{ij}\right|\). The larger \(\zeta_{0^{+}i}(j)\) is, the larger the correlation coefficient between scheme to be evaluated and the optimal scheme is in terms of index \(j\). On the other hand, the performance of \(\zeta_{0^{-}i}(j)\) is absolutely opposite. Assume the weight vectors of \(n\) indexes are \(\omega=\left(\omega_{1},\omega_{2},\ldotsldots,\omega_{n-1},\omega_{n}\right)\) respectively and \(f(\omega)=\sum\limits_{j=1}^{n}\omega_{j}^{2}[1-\zeta_{0^{+}i}(j)]^{2}+\sum \limits_{j=1}^{n}\omega_{j}^{2}\zeta_{0^{-}i}(j)^{2}\) represents the square sum of the weighted distance betweenthe optimal solution and the worst one. Therefore, the smaller \(f(\omega)\) is, the better scheme \(i\) is. Establish the multi-objective programming model as follows:  
\[