## 2 Materials and Methods  
In this study, the [PERSON]'s method is used to model the wake effect. As shown in Figure 1, when wind passes through a rotor blade, its speed falls and the waking zone spreads from the wind flow, similar to a cone [63; 64; 65]. The radius of this cone can be calculated. As shown in Figure 1, if the wind turbine zone is located downstream of the wake cone, the wake and the downstream turbine will overlap. In case the region is generated by an upstream wind turbine, the wind speed will experience reduction [66; 6]. Additionally, Figure 2 presents the wake shadow area. \[R_{w}=R_{r}+kx \tag{1}\]  
In the above equation, \(R\) is rotor's radius, \(x\). is the distance of turbines, and \(k\) is the expansion rate that takes a value between 0.04 to 0.08 which is calculated by [67; 68]:  
\[k=\frac{0.5}{\log\left(\frac{z}{z_{0}}\right)} \tag{2}\]  
where \(z\) is the hub height and \(z_{0}\) is the roughness length. Wind flow velocity of the wake is computed as [69; 70]:  
\[V_{single}=V_{0}\left(1-\left(1-\sqrt{1-C_{T}}\right)\left(\frac{R_{r}}{R_{w} }\right)^{2}\left(\frac{A_{shad}(ud)}{A_{0}}\right)\left(\frac{V_{z}}{V_{ref}} \right)\right),\ k=\frac{0.5}{\log\left(\frac{z}{z_{0}}\right)} \tag{3}\]  
where \(A_{0}\) is the area created by the rotor blade system and \(A_{shad}(ud)\) refers to an area covered by the downstream turbine and the shaded area of the upstream turbine, which is calculated by Equation (4) [71]:  
\[A_{shad}(ud)=\left\{ \begin{array}{cc}\pi R^{2}&d<R_{w}-R\\ \frac{1}{2}\ R^{2}(\theta_{r}-\sin\theta_{r})+\frac{1}{2}\ R_{w}^{2}(\theta_{ w}-\sin\theta_{w})&R_{w}-R<d<R_{w}+R\\ 0&d>R_{w}+R\end{array} \right.\]  
Figure 1: The wake effect produced by a downstream unit of shadow caused by the existence of an upstream wind turbine unit. Figure 2: Wake shadow area [71]. \[\theta_{r}=2\cos^{-1}\frac{R^{2}+d^{2}-{R_{w}}^{2}}{2 Rd_{w}}\theta_{w}=2\cos^{-1} \frac{{R_{w}}^{2}+d^{2}-R^{2}}{2R_{w}d} \tag{4}\]  
For calculating wind speed originated from multiple upstream turbines reaching a downstream wind turbine, the following equation is used (Equation (5)):  
\[V_{d}=V_{in}\left(1-\sum\left(1-\sqrt{1-C_{T}}\right)\left(\frac{R}{R_{w}} \right)^{2}\left(\frac{A_{shad}(ud)}{A_{0}}\right)\left(\frac{V_{z}}{V_{ref}} \right)\right) \tag{5}\]  
Shaded wind turbines and the number of shaded units are different and they both depend on the wind direction and the geometrical layout of wind turbines. It is illustrated in the following figure for two different directions of wind. As shown in Figure 3, the area covered by the vortex cone changes with wind direction [51]. The wake factor can be defined as [37; 72]:  
\[C_{w}=\frac{Total\ power\ of\ wind\ farm\ considering\ wake\ effect}{Total\ power\ of\ wind\ farm\ without\ wake\ effect} \tag{6}\]  
which can be formulated as:  
\[C_{w}=\frac{\sum_{j=1}^{n}\sum_{i=1}^{m}P\left(V_{ij}\right)}{mn\times P_{n}} \tag{7}\]  
where \(m\) is the number of turbines in a row, \(n\) is the number of rows, and \(V_{ij}\) is the wind speed fraction perpendicular to the \(i\)-th turbine in the \(j\)-th row. In addition, \(P_{n}\) stands for the power generated by a turbine with wind speed of \(V_{ij}\). When the wake factor is equal to one, it means there is no wake effect. The following relation is used to obtain wind speed at any hub height [66; 73]:  
\[V_{z}=V_{ref}\frac{\log\frac{z}{z_{0}}}{\log\frac{z_{ref}}{z_{0}}} \tag{8}\]  
where \(V_{ref}\) represents the reference wind speed and \(z_{0}\) is the roughness length. The cost function considered in this study is the same as [PERSON] et al. [40]. Through this function, the annual plant's total cost by using the number of turbines (\(N_{t}\)) can be calculated with decent accuracy. \[Cost_{tot}=N_{t}\left(\frac{2}{3}+\frac{1}{3}e^{-0.00174N_{t}^{2}}\right) \tag{9}\]  
Figure 3: The effect of wind direction on the wake effect by upstream wind turbines for two different wind directions [51].