# An Optimization Framework for Wind Farm Design in Complex Terrain
## 2 Wind Farm Modelling
### Wake Model  
Modelling the wake effects of wind turbines is essential for AEP estimation, but quite challenging for wind farms in complex terrain due to the complex interplay between terrain and wake flow [22]. While the CFD can model the wake flow field of a wind farm in complex terrain with a reasonable accuracy [23], its high computational cost makes it unsuitable for wind farm design optimization, since typical optimization algorithms require a large number of design evaluations. Several studies have proposed a few fast wake models for wind farms in complex terrain: [PERSON] et al. proposed a virtual particle wake model [24] and applied it in wind farm layout optimizations [12; 13]; [PERSON] and [PERSON] developed an adapted Jensen wake model in [14]; [PERSON] et al. [25] proposed a wake model by solving a simplified variation of the Navies-Stokes equations, which yields results with a reasonable accuracy and has a much less computational cost when compared with full CFD simulations. In the current framework, the wake flow field of a wind farm in complex terrain is modelled by coupling the adapted [PERSON] wake model with the terrain flows obtained by CFD simulations,  
Figure 1: Interpolated site condition variables for a wind farm site in complex terrain: (**a**) elevation; (**b**) mean wind speed at 67 m height above the ground; (**c**) speed up at 67 m height under the far field inflow wind direction of 180\({}^{\circ}\); (**d**) Weibull-k at 67 m height under the far field inflow wind direction of 164\({}^{\circ}\). i.e., flow fields of the wind farm site without wind turbines, such as those imported from WAsP CFD simulations. The adapted [PERSON] wake model assumes the wake centerline of a wind turbine's wake follows the terrain at the same height above the ground along the local inflow wind direction, while its wake zone expands linearly and its wake deficit develops following the same rule as the original [PERSON] wake model [14]. The schematic of this wake model is shown in Figure 2.  
Considering the wake effect between an upwind turbine, \(i\), located at \((x_{i},\ y_{i})\) and a downwind turbine, \(j\), located at \((x_{j},\ y_{j})\) for a given far field inflow condition: \(v^{\infty}\) and \(\theta^{\infty}\), we can calculate the downwind distance, \(d^{down}_{ij}(\theta^{\infty})\), and the crosswind distance, \(d^{cross}_{ij}(\theta^{\infty})\), from the upwind turbine to the location of the downwind turbine. The downwind distance can be computed by integration following the wake centerline (the red line in Figure 2) based on the elevation, \(z(x,\ y)\), and the local inflow wind direction, \(\theta_{i}(\theta^{\infty})=\theta^{\infty}+tu(x_{i},\ y_{i},\theta^{\infty})\), while the crosswind distance can be easily obtained by considering the coordinates of these two turbines and the local inflow wind direction of the upwind turbine. Then, the local inflow wind speed at turbine \(i\) and the effective wake deficit turbine \(i\) caused on turbine \(j\) can be calculated by:  
\[v_{i}(v^{\infty},\theta^{\infty})=sp(x_{i},\ y_{i},\theta^{\infty})\cdot v^{ \infty}, \tag{3}\]  
\[\Delta v_{ij}(v^{\infty},\theta^{\infty})=\frac{A^{ol}_{ij}(\theta^{\infty})} {A^{rotor}_{j}}\cdot v_{i}(v^{\infty},\theta^{\infty})\cdot\Bigg{(}\frac{1- \sqrt{1-C_{T}(v_{i}(v^{\infty},\theta^{\infty}))}}{1+\alpha\cdot d^{down}_{ij }(\theta^{\infty})/R_{i}}\Bigg{)}, \tag{4}\]  
in which \(A^{ol}_{ij}(\theta^{\infty})\) is the overlapping area between the wake zone of turbine \(i\) and the rotor area of turbine \(j\); \(A^{rotor}_{j}=\pi R_{j}^{2}\) is the rotor area of turbine \(j\); \(C_{T}(\cdot)\) represents the thrust coefficient function of the wind turbine; \(\alpha\) denotes the wake decay coefficient, and \(R_{i}\) represents the rotor radius of turbine \(i\).