## 2 Modelling and Methodology
### Two-Dimensional Analytical Wake Model  
The popular PARK model was initially proposed by [PERSON] [22] and was further tuned by [PERSON] et al. [23] for the clustered wind turbine research. The improved PARK model has been mostly  
Figure 1: Diagram of the irregular wind farm boundary representation. applied in literature for the wind farm layout optimization study [24]. Figure 2 shows the general properties of the PARK model, which assumes a linear wake propagation behind the upstream wind turbine rotor and notice that the radial wind speed distribution inside the wake is set to be constant. The wake-affected wind speed at the downstream \(x\) distance (horizontally in Figure 2) towards the dominant wind direction is given by:  
\[v_{x}=v_{0}\left[1-2a\left(\frac{r_{0}}{r_{0}+\alpha\cdot x}\right)^{2}\right] \tag{7}\]  
where \(v_{0}\) is the undisturbed wind speed, \(v_{x}\) is the wake wind speed, and \(r_{0}\) is the downstream rotor radius. According to the actuator disk theory, \(r_{0}\) is calculated based on the wind turbine rotor radius \(R\) as:  
\[r_{0}=R\sqrt{\frac{1-a}{1-2a}} \tag{8}\]  
in which, \(a\) is axial induction factor calculated as the percentage of reduction from free-stream wind speed to the rotor-plane wind speed. In Equation (8), \(\alpha\) is an entrainment constant, which is determined by the relative surface roughness length (\(z_{0}\)) and wind turbine hub height (\(l\)) in the empirical mathematical form as [25]:  
\[\alpha=\frac{0.5}{\ln\left(\frac{h}{z_{0}}\right)} \tag{9}\]  
In Figure 2, \(r_{x}\) is the wake radius which is linearly expressed with the form of:  
\[r_{x}=r_{0}+\alpha\cdot x \tag{10}\]  
As mentioned above, [PERSON]'s wake model is one-dimensional assuming the wake velocity only related to the axial distance towards the dominant wind direction. Further research showed that the wind speed variation in the wake occurs not only in the axial direction (horizontal wind direction in Figure 2), but in the radial wind direction (vertical wind direction in Figure 2) as well. It has been reported that the wind speed profile exhibits an approximately Gaussian axisymmetric shape [26; 27]. Hence, a two-dimensional Gaussian wake model was applied for the wind farm layout optimization in this paper and since the model was based on the [PERSON]'s model, it was named Gaussian-Jensen wake model. The new wake model equation is given by:  
\[u_{x}=A\left(\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{r^{2}}{2\sigma^{2}}}\ \right)+B \tag{11}\]  
Figure 2: Diagram of one-dimensional [PERSON]’s wake model. where \(u_{x}\) is the wake wind speed in the Gaussian-Jensen wake model and \(\sigma\) is the standard deviation. According to reference [28], the following equations of equality are formulated:  
\[\left\{ \begin{array}{c}\sigma=\frac{r_{x}}{2.58}\\ A\left(\frac{2.58}{r\sqrt{2\pi}}e^{-2.58^{2}}\right)+B=v_{0}\\ A+2B\cdot r=2r\cdot v_{x}\end{array} \right. \tag{12}\]  
by solving the equation with extra simplification, the arguments of \(A\), \(B\), and \(\sigma\) are obtained as:  
\[\left\{ \begin{array}{c}\sigma=\frac{r_{x}}{2.58}\\ B=v_{0}\\ A=2r_{x}\cdot(v_{x}-v_{0})\end{array} \right. \tag{13}\]  
The final wake velocity expression for the Jensen-Gaussian wake model is represented by below formula, and the comparison of wind speed contours between the [PERSON]'s wake model and the Jensen-Gaussian wake model is shown in Figure 3. As can be seen, Figure 3a shows the [PERSON]'s model with the constant wind speed in the radial direction while the speed in the radial direction is variable for the Jensen-Gaussian wake model in Figure 3b.