## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling
The analytical solution to the first order model, together with the semi-analytical calibration of the involved constants, is shown in Equations (26)-(28):  
\[u_{x}(x,r)=-\frac{U_{\infty}}{9}\left(c_{T}Ax^{-2}\right)^{\frac{1}{3}}\left[ r^{\frac{3}{2}}(3c_{1}^{2}c_{T}Ax)^{-\frac{1}{2}}-\left(\frac{35}{2\pi}\right)^{ \frac{3}{10}}\left(3c_{1}^{2}\right)^{-\frac{1}{5}}\right]^{2} \tag{26}\]  
\[u_{r}(x,r)=\frac{U_{\infty}}{3}(c_{T}A)^{\frac{1}{3}}x^{-\frac{5}{3}}r\left[ r^{\frac{3}{2}}(3c_{1}^{2}c_{T}Ax)^{-\frac{1}{2}}-\left(\frac{35}{2\pi}\right)^{ \frac{3}{10}}\left(3c_{1}^{2}\right)^{-\frac{1}{5}}\right]^{2} \tag{27}\]  
\[R_{W}(x)=\left(\frac{35}{2\pi}\right)^{\frac{1}{5}}(3c_{1}^{2})^{\frac{1}{5}} (c_{T}Ax)^{\frac{1}{3}} \tag{28}\]  
where \(u_{x}\) [m/s] is the axial wind speed magnitude at some axial (\(x\) [m]) and radial (\(r\) [m]) position within the wake, \(u_{r}\) [m/s] is the radial wind speed magnitude at some axial and radial position within the wake, \(c_{T}\) is the thrust coefficient of the WT, \(A\) [m\({}^{2}\)] is the rotor swept area, \(c_{1}=(D/2\alpha_{1})^{5/2}x_{0}^{-5/6}\), \(D\) [m] is the rotor diameter, \(\alpha_{1}=(105/2\pi)^{1/5}(c_{T}A)^{1/3}\), \(x_{0}=\left((D/2\alpha_{1})^{-3}(\beta_{1}/(U_{\infty}-U_{m}))^{3/4}-1\right)^ {-1}\Delta x\), \(U_{m}=U_{\infty}(1-\beta_{1}c_{1}^{-4/5}(x_{0}+\Delta x)^{-2/3})\) is a measured axial velocity that depends on \(\Delta x\) [m], which represents a distance measured from the WT rotor, and \(\beta_{1}=(1/9)(3)^{-2/5}(35/2\pi)^{3/5}(c_{T}A)^{1/3}\). The second order model of [PERSON] included the previously neglected distance terms of the form \(x^{-7/3}\) and its analytical solution, together with additional assumptions, resulted in a very large model that is not described here for the sake of brevity. The updated version of the [PERSON] model [287] included two major improvements: an approach to merge wakes, which is necessary to quantify the performance of wind farms, and a proper calibration of the model with full-scale experiments. The updated first order model is shown in Equations (29) and (30):\[u_{x}(x,r)=-\frac{U_{\infty}}{9}(c_{T}A(x+x_{0})^{-2})^{\frac{1}{3}}\left[r^{ \frac{3}{2}}(3c_{1}^{2}c_{T}A(x+x_{0}))^{-\frac{1}{2}}-\left(\frac{35}{2\pi} \right)^{\frac{3}{10}}(3c_{1}^{2})^{-\frac{1}{5}}\right]^{2} \tag{29}\]  
\[R_{\mathrm{W}}(x)=\left(\frac{35}{2\pi}\right)^{\frac{1}{5}}(3c_{1}^{2})^{ \frac{1}{5}}(c_{T}A(x+x_{0}))^{\frac{1}{3}} \tag{30}\]  
where \(c_{1}=(105/2\pi)^{-1/2}(d_{1}D/2)^{5/2}(c_{T}Ax_{0})^{-5/6}\), \(x_{0}=9.6D/((2R_{9.6}/d_{1}D)^{3}-1)\), \(d_{1}=\left((1+1/\sqrt{1-c_{T}})/2\right)^{1/2}\), \(R_{9.6}\) is the wake radius at 9.6 rotor diameters downstream from the WT, which was calculated empirically form the Vindeby offshore wind farm data and expressed by \(R_{9.6}=a_{1}\exp(a_{2}c_{T}^{2}+a_{3}c_{T}+a_{4})(b_{1}I_{a}+1)D\), where the \(a_{i}\) coefficients are resumed in [287]. In order to calculate the effective wind speed of merged wakes in wind farms, [PERSON] considered the calculation of the mean effective wind speed reaching the rotor of a WT, given by either \(U_{\mathrm{eff}}=1/A\int U\mathrm{d}A\) or \(U_{\mathrm{eff}}^{2}=1/A\int U^{2}\mathrm{d}A\), where the wind shear was assumed to be described by the logarithmic laws. The effective wind speed within the wind farm was calculated using a linear superposition of wakes, as shown in Equation (31):  
\[U_{\mathrm{W}}^{\mathrm{eff}}=U_{\mathrm{eff}}-\sum\
olimits_{i=1}^{N_{u}}u_{ x_{i}} \tag{31}\]  
where \(N_{u}\) is the number of upstream WT. A second approach was suggested by using the effective kinetic energy reaching the WTs, as shown in Equation (32):  
\[U_{\mathrm{W}}^{\mathrm{eff}}=\sqrt{\frac{1}{A}\int\left(U_{\mathrm{eff}}- \sum\
olimits_{i=1}^{N_{u}}u_{x_{i}}\right)^{2}\mathrm{d}A} \tag{32}\]  
Finally, like the Ainslie wake model, the Larsen models do not account for the ground effects or the tower effects in its basic formulations. Another wake modeling approach was suggested by [PERSON] _et al_. in 2006 [283].