## 3 Methodology
### Objective Function Evaluation Method  
According to the wind turbine positions (as represented in the form of Cartesian coordinates), the velocity deficits of all wind turbines for different incoming wind directions are calculated based on the introduced analytical wake model. For the Weibull distribution wind condition, it has been confirmed that only the scaling parameter \(c\) is modified by the wake effect, which is given by [32]:  
\[c_{i}(\theta)=c(\theta)\times(1-V\textit{el\_def}_{i}),\ i=1,2\ldots,N \tag{14}\]  
where \(c(\theta)\) is the scale parameter on the wind direction \(\theta\) and \(\textit{Vel\_def}_{i}\) is the velocity deficit on \(i\)-th wind direction. The expected individual wind turbine power output \(P_{1}\) under the Weibull distribution wind condition can be represented by:  
\[P_{i}=\int\limits_{0}^{360}p_{\theta}(\theta)\int\limits_{0}^{\infty}P(v) \frac{k(\theta)}{c_{i}(\theta)}\bigg{(}\frac{v}{c_{i}(\theta)}\bigg{)}^{k( \theta)-1}\exp\Bigg{(}-\bigg{(}\frac{v}{c_{i}(\theta)}\bigg{)}^{k(\theta)} \Bigg{)}\mathrm{d}\theta\mathrm{d}v \tag{15}\]  
where \(p_{\theta}(\theta)\) is the probability of wind blowing on the wind direction \(\theta\), \(k(\theta)\) is the shape parameter on the wind direction \(\theta\), and \(P(v)\) is the wind turbine power output on wind speed \(v\). By the discretization of both wind speeds and wind directions, the individual wind turbine power can be mathematically formulated, which consists of two parts. When the wind speed is beyond the rated speed, the expected power \(P_{1}\) is given by:  
\[P_{1}=P_{rated}\sum\limits_{l=1}^{N_{\theta}}\Bigg{\{}\omega_{l-1}\Bigg{(}\exp \Bigg{(}-\bigg{(}\frac{2v_{rated}}{c_{i}(l-1)+c_{i}(l)}\bigg{)}^{k}\Bigg{)}- \exp\Bigg{(}-\bigg{(}\frac{2v_{cut-out}}{c_{i}(l-1)+c_{i}(l)}\bigg{)}^{k} \Bigg{)}\Bigg{)}\Bigg{\}} \tag{16}\]  
where \(N_{\theta}\) is the total number of divided wind direction sectors and \(\omega_{l-1}\) is the probability of a specific wind direction interval of (\(\theta_{l-1}\), \(\theta_{l}\)). \(c_{i}(l-1)\) and \(c_{i}(l)\) are the wake-affected scale parameter of the corresponding wind directions, respectively. Between the cut-in and the rated wind speeds, the expected power output \(P_{2}\) is given by:  
\[\begin{split} P_{2}=\sum\limits_{j=1}^{N_{\psi}+1}0.2963\Big{(} \frac{v_{j-1}+v_{j}}{2}\Big{)}^{3}\sum\limits_{l=1}^{N_{\theta}}\bigg{\{} \omega_{l-1}\bigg{(}\exp\bigg{(}-\Big{(}\frac{2v_{j-1}}{c_{i}(l-1)+c_{i}(l)} \bigg{)}^{k}\bigg{)}\bigg{)} \tag{17}\]  
where \(N_{\
u}\) is the total number of wind speed discretization bins and \(\omega_{l-1}\) is the probability of the wind direction amongst the interval of (\(\theta_{l-1}\), \(\theta_{l}\)). The individual wind turbine power is calculated by the aggregation of the two separate power parts by [35]:  
\[P_{i}=P_{1}+P_{2} \tag{18}\]By substituting the wind farm cost (_cost_) and the total expected power output (\(P_{\text{tot}}\)) into the CoE equation, the substituted CoE function is given by:  
\[\text{CoE}=\left(-0.1539\times P_{\text{r}}-0.001\times N+2\times P_{\text{r}} \times N+0.2504\right)/\sum_{i=1}^{N}P_{i} \tag{19}\]  
where \(P_{\text{r}}\) is the wind turbine nominal power and \(N\) is wind turbines number. The optimization function can be mathematically represented as follows and the minimum distance between wind turbines, \(d_{min}\), is set to be 7D (D is wind turbine diameter) in this paper. \[ \begin{array}{l}\text{Minimize}:\\ \text{CoE}=cost/P_{\text{tot}}\\ \\ \text{Subject to}:\\ \sqrt{\left(x_{j}-x_{i}\right)^{2}+\left(y_{j}-y_{i}\right)^{2}}\geq d_{ \text{min}}\left(\text{for }\forall\;i\text{ and }j,\;i\
eq j\right)\end{array}  \tag{20}\]  
Besides the objective function (CoE) that is optimized in the paper, it should also be mentioned that the wind farm design methods tested could be incorporated into any objectives of the wind farm layout optimization problem so long as the objective function is mathematically formulated. For instance, the wind farm fatigue load is estimated with a stochastic model in reference [36].