## 2 Methodological Framework
### ACD Method  
The ACD method is a way to represent the wind turbine's effect on the wind flow in a simulation. To do so, a thrust force calculated from the 1D momentum theory is applied to a porous disc, which in turn acts as a momentum sink.  
For the ACD (2016) method presented in [30] the thrust force \(F_{i}\) at each cell of the disc is calculated from  
\[F_{i}=C_{T}^{\prime}\frac{1}{2}\rho\left(\frac{U_{d,i}}{1-a_{i}}\right)^{2}A_{ i}, \tag{1}\]  
where \(U_{d,i}\) is the velocity of the flow at \(i\)-th cell of the disc, \(a_{i}\) is the axial induction factor calculated for each individual cell of the disc, \(A_{i}\) is the surface area of the cell facing the undisturbed wind flow direction and \(C_{T}^{\prime}\) is a modified thrust coefficient dependent on the velocity at the disc \(U_{d,i}\) and \(\rho\) is the air density set to 1.225 kg m\({}^{-3}\). In most cases wind turbine manufacturers offer \(C_{T}\) as a function of \(U_{\infty}\), the undistributed wind velocity. This \(C_{T}\) is reasonable for the first wind turbine of the row but not for the downstream wind turbines were the flow has been disturbed. Hence, in the present case a \(C_{T}\) which is a function of \(U_{d}\), the velocity at the disc, is needed [31]. This function can be established from the 1D momentum theory by combining the definition of the trust coefficient \(C_{T}\) and the axial induction factor \(a\)  
\[C_{T} =4a(1-a), \tag{2}\] \[U_{d} =(1-a)U_{\infty}. \tag{3}\]  
Figure 1: Methodological framework.  
Hence from Equations (2) and (3) the following is obtained  
\[U_{d}=U_{\infty}\left(1-\frac{1}{2}\left(1-\sqrt{1-C_{T}\left(U_{\infty}\right)} \right)\right). \tag{4}\]  
The power production is estimated by finding the average induction factor over the disc as \(\overline{u}=\frac{1}{N}\sum_{i=1}^{N}a_{i}\), and then for each velocity \(U_{d,i}\) over the disc an undisturbed wind velocity \(U_{\infty,i}\) is found using the following equation  
\[U_{\infty,i}=\frac{U_{d,i}}{1-\overline{u}}. \tag{5}\]  
For each undisturbed wind velocity, a power is found using the power curve. This power is then averaged over the disc. The data for the thrust coefficient curve \(C_{T}\left(U_{\infty}\right)\) is supplied by [32] and the data for the power curve is provided by [33].  
The main differences between the ACD (2016) method and the ACD (2008) method presented in [28] are shown in Table 1. Even though the two ACD methods seem quite similar, there are however some crucial differences. The equation used to find the thrust at each cell is slightly different between the ACD (2016) and ACD (2008) method. In the ACD (2016) method the thrust coefficient updates at each iteration depending on the wind velocity at the rotor position (\(U_{d,i}\)), whereas in the ACD (2008) method the thrust coefficient is kept constant during the simulation and is found by the undisturbed wind velocity. This static behavior of the ACD (2008) method and its limitations has been already pointed in [28]. It should be noted that the ACD (2008) method in WindSim has not only an ACD with a uniform distribution but also has ACDs with a polynomial and a parabolic thrust distribution. These were not considered in this study as the power production of the wind turbine for the ACD (2008) is assessed solely from the local velocity at the disc (\(U_{d}\)) at hub height. Hence for distributions other than the uniform the power production of the first wind turbine varied significantly from the measured data.