## 2 Methodological Framework
### Analytical Wake Models  
Two analytical wake models are used in this study, the Jensen [4] and the Larsen [5]. The Jensen model is based on the description of a single wake behind a rotor. Here the normalized wind velocity deficit as presented in [6] is used \(\delta U=\frac{U_{\infty}-V}{U_{\infty}}\), at a distance \(x\) behind a single wind turbine with a thrust coefficient \(C_{T}\) is found by  
\[\delta U=\frac{1-\sqrt{1-C_{T}}}{\left(1+\frac{2 kx}{D}\right)^{2}}, \tag{6}\]  
where \(k\) is the wake expansion coefficient, \(D\) the rotor diameter of the wind turbine and \(V\) the wind velocity in the wake at position \(x\) (Figure 2). The wake expansion coefficient \(k\) is found by \(k=A_{j}\ln(z_{h}/z_{0})\). Where \(A_{j}\) is a constant equal to \(A_{j}=0.5\), \(z_{h}\) the wind turbine hub height and \(z_{0}\) is the effective roughness height. When multiple wakes influence the velocity at a position, the total normalized velocity deficit \(\delta U_{tot.}\), is found by  
\[\delta U_{tot.}=\sqrt{\sum\delta U_{j}^{2}}, \tag{7}\]  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Description** & \begin{tabular}{c} **Power Production** \\ **Estimation** \\ \end{tabular} & **Thrust Coefficient** & **Thrust Distribution** \\ \hline ACD (2008) & \begin{tabular}{c} Velocity at \\ hub height \\ \end{tabular} & \(C_{T}\) static & \begin{tabular}{c} Uniform, Parabolic \\ and Polynomial \\ \end{tabular} \\ \hline ACD (2016) & \
\begin{tabular}{c} Averaged value \\ over the rotor \\ \end{tabular} &
\begin{tabular}{c} \(C_{T}(U_{d,i})\) update \\ at each iteration \\ \end{tabular} & Undistributed \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main differences between the ACD (2008) and the ACD (2016) methods.  
where \(\delta U_{j}\) is the normalized wind velocity deficit from the \(j\)-th wind turbine.  
The Larsen model is based on turbulent boundary layer equations and a similarity assumption. By neglecting different terms in the governing equations, two different versions of the model are presented in [5]. Here we will be using the first order wake model in which the normalized velocity deficit is described by  
\[\delta U=\frac{\left(C_{T}Ax^{-2}\right)^{1/3}}{9}\left[r_{x}^{3/2}\left(3c_{1 }^{2}C_{T}Ax\right)^{-1/2}-\left(\frac{35}{2\pi}\right)^{3/10}\left(3c_{1}^{2} \right)^{-1/5}\right]^{2}, \tag{8}\]  
where \(r_{x}\) is the radial distance at a position \(x\) downstream of the rotor and \(c_{1}\) is a computed parameter. The parameter \(c_{1}\) is found by  
\[c_{1}=\left(\frac{D}{2}\right)^{5/2}\left(C_{T}Ax_{0}\right)^{5/6}, \tag{9}\]  
where  
\[x_{0}=\frac{9.5D}{(2R_{95}/D)^{3}}-1, \tag{10}\] \[R_{95}=0.5\left(Rnb-\min(z_{h},Rnb)\right),\] (11) \[Rnb=\max\left(1.08D,1.08D+21.7D(Tl_{h}-0.05)\right). \tag{12}\]  
\(Tl_{h}\) is the ambient turbulence intensity at hub height (\(z_{h}\)), \(Rnb\) is a constant dependent on the rotor diameter and the ambient turbulence intensity at hub height. A notable difference between the [PERSON] and [PERSON] models is that the wake deficit for the [PERSON] model is not uniform, as is the case for the [PERSON] model, but differs radially in the cross-section of the wake, due to \(r_{x}\). The effect of multiple wakes on the velocity field at a position is also taken into account by Equation (7).  
The power production at each wind turbine is found by the velocity at hub height minus the velocity deficit calculated by either the [PERSON] or [PERSON] wake model.