# An Optimization Framework for Wind Farm Design in Complex Terrain
## 2 Wind Farm Modelling
### Site Condition  
Due to terrain effects, i.e., the effect of topographic changes on the flow filed over a complex terrain [9], different locations at a complex terrain site will experience different wind conditions. Using onsite wind measurements and standard wind resource assessment tools, such as WAsP [5] and WindPRO [20], one could get the essential information on location-specific site conditions, i.e., wind resource and terrain effects, for any given inflow wind direction sector.  
For flat or moderately complex terrain, linear models, such as the conventional WAsP [6], provide reasonable results. For more complex terrain, nonlinear models, such as WAsP CFD [21], are needed. In the current framework, we assume that the location-specific values of wind resource and terrain effects have been obtained by using a wind resource assessment tool.  
Considering a rectangle area covering the interested area for a wind farm design, we first discretize the area into a number of grids, defined by a range of \(x\) coordinates \([x_{1},\ x_{2},\ \dots,\ x_{N_{x}}]\) and a range of \(y\) coordinates \(\left[y_{1},\ y_{2},\ \dots,\ y_{N_{y}}\right]\). For a certain number of wind direction sectors defined by a range of far field inflow wind directions \(\left\lvert\theta_{1}^{\infty},\ \theta_{2}^{\infty},\ \dots,\ \theta_{N_{y}}^{\infty}\right\rvert\), we can then obtain the relevant sector wise values of wind resource and terrain effects variables, such as Weibull-A, Weibull-k, frequency, speed up factor, terrain induced turning angle, and mean wind speed, at a given height above the ground. Note that Weibull-A is the scale parameter and Weibull-k is the shape parameter of Weibull distribution, which is the most commonly used distribution for wind speed [4]. Values of variables that do not depend on inflow wind direction, such as elevation and overall mean wind speed, are also obtained.  
Using linear interpolation, we can then easily estimate the values of these variables for any arbitrary position \((x,\ y)\) in this area, under any far field inflow wind direction, \(\theta^{\infty}\), if wind direction dependent variables are concerned. The variables important for estimating AEP are listed in Table 1.  
In Table 1, speed up is defined as the ratio between the local wind speed, \(v\), at location \((x,\ y)\) and the far field wind speed, \(v^{\infty}\), at the same height above the ground for a given far field inflow  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Unit** & **Notation** \\ \hline Elevation & m & \(z(x,\ y)\) \\ Weibull-A & m/s & \(A(x,\ y,\theta^{\infty})\) \\ Weibull-k & - & \(k(x,\ y,\theta^{\infty})\) \\ Frequency & -/\({}^{\circ}\) & \(freq(x,\ y,\theta^{\infty})\) \\ Speed up & - & \(sp(x,\ y,\theta^{\infty})\) \\ Turning angle & \({}^{\circ}\) & \(tu(x,\ y,\theta^{\infty})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Site condition variables important for AEP (annual energy production) estimation.  
wind direction, \(\theta^{\infty}\), while turning angle, \(tu\), denotes the difference between the local wind direction, \(\theta\), and the far field inflow wind direction, \(\theta^{\infty}\). Thus, they can be computed as:  
\[sp(x,\,y,\theta^{\infty})=v(x,\,y,\theta^{\infty})/v^{\infty}(\theta^{\infty}), \tag{1}\]  
\[tu(x,\,y,\theta^{\infty})=\theta(x,\,y,\theta^{\infty})-\theta^{\infty}. \tag{2}\]  
As a demonstration, four interpolated site condition variables are shown in Figure 1 for a real wind farm site in complex terrain. Note that the results on wind resource and terrain effects are imported from WAsP CFD simulations and the wind farm on this site will be used in Sections 4 and 5 as a test case.