## 2 Modelling and Methodology
### Objective Function Fitness Evaluation  
In this paper, the optimization objective was to maximize the total wind farm energy production (\(P_{total}\)) by taking into account the wake interactions between wind turbine using the wake effect model. Under the simplified wind condition with discrete wind speed and wind direction, the individual wind turbine power output can be directly calculated based on the wake-effected wind speed \(v_{i}\) and the wind turbine power curve. For the realistic wind condition of Weibull distribution with continuous wind speed variation, the means of discretization evaluation method was applied for calculating the wake-affected wind turbine power output as follows. The discretization method tries to discretize the wind speed into a number of small bins in the same manner as the division of wind direction into several sectors. When the wind turbine produces constant rated power, the produced power \(P_{1}\) under Weibull distribution equals to \(P_{rated}\int_{0}^{2\pi}p(\theta)d\theta\int_{v_{rated}}^{v_{cut-out}}\frac{k( \theta)}{c(\theta)}\Big{(}\frac{v}{c(\theta)}\Big{)}^{k(\theta)-1}e^{-(\frac{v }{c(\theta)})^{k(\theta)}}dv\). By further operations of discretizing the wind speed region and wind direction region, the discrete mathematical expression is given by:  
\[P_{1}=P_{rated}\sum_{l=1}^{N_{\theta}}\omega_{l-1}\left\{e^{-(\frac{2\ \sigma_{rated}}{c(l-1)+c(l)})}\frac{k(l-1)+k(l)}{2}-e^{-(\frac{2\ \sigma_{total}}{c(l-1)+c(l)})}\frac{k(l-1)+k(l)}{2}\right\} \tag{19}\]  
in which, \(P_{rated}\) is the wind turbine rated power, \(v_{rated}\) is the rated wind speed, \(v_{cut-out}\) is the cut out wind speed. The wind direction region [0, 2\(\pi\)] is uniformly divided into \(N_{\theta}\) bins with a constant interval of \(2\pi/N_{\theta}\). Each divided wind direction bin has been assigned with a value (\(\omega\)) to indicate the probability of occurrence that wind comes amongst that particular wind direction sector, and the dividing points are marked as \(\theta_{0},\ \theta_{1},\ \ldots,\ \theta_{N_{\theta}-1}\) with corresponding probability values as \(\omega_{0},\ \omega_{1},\ \ldots,\ \theta_{N_{\theta}-1}\). The indices of shape parameter and the scale parameter in each wind direction sector are represented by \(k\) and \(c\), respectively. When the produced power increases with the wind speed, the power \(P_{2}\) under Weibull distrubution equals to \(\int_{0}^{2\pi}p(\theta)d\theta\int_{v_{cat-in}}^{v_{rated}}0.2963\cdot v^{3 }\frac{k(\theta)}{c(\theta)}\Big{(}\frac{v}{c(\theta)}\Big{)}^{k(\theta)-1}e^ {-(\frac{v}{c(\theta)})^{k(\theta)}}dv\). By further discretization of the wind speed region and wind direction region, the discrete mathematical expression is represented by:  
\[P_{2}=\sum_{j=1}^{N_{v}}0.2963\cdot\left(\frac{v_{j-1}+v_{j}}{2}\right)^{3}\sum _{l=1}^{N_{\theta}}\omega_{l-1}\left\{e^{-(\frac{2\,v_{j-1}}{c(l-1)+c(l)})} \frac{k(l-1)+k(l)}{2}-e^{-(\frac{2\,v_{j}}{c(l-1)+c(l)})}\frac{k(l-1)+k(l)}{2} \right\} \tag{20}\]  
apart from the discretization of wind direction, the region between cut in wind speed and rated wind speed is divided into \(N_{v}\) bins with the constant interval of \((v_{rated}-v_{cut-in})/N_{v}\). The dividing points of the wind speed region were assumed to be \(v_{0},\ v_{1},\ldots,\ v_{N_{v}-1},v_{N_{v}}\) and the forward difference method was applied to calculate the integral of the power output function. Evidently, the calculation accuracy is dependent on the interval density of the discretization. Just like the means of calculating the integral with calculus theory, a smaller discretization interval will increase the accuracy of the wind farm power output calculation by approximation. The final individual wind turbine power was calculated by summing up the separate power parts as: \(P_{i}=P_{1}+P_{2}\).