## 3 Results and Discussion
### Weibull Distribution Wind Scenario  
Finally, a realistic wind condition of Weibull distribution was studied for the wind farm layout optimization. The properties of the Weibull distribution are shown in Table 2. As can be seen, 360\({}^{\circ}\) wind direction was equally divided into 24 sections with a 15\({}^{\circ}\) interval. In each of the 24 sections, the wind speed variation was represented by the Weibull distribution, where the probability of occurrence of wind blowing amongst that section is indicated by \(\omega\). From the table, it is obvious that the predominated wind direction is from 75\({}^{\circ}\) to 105\({}^{\circ}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Direction** & \(\mathbf{k}\) & \(\mathbf{c}\) & \(\mathbf{\omega}\) & **Direction** & \(\mathbf{k}\) & \(\mathbf{c}\) & \(\mathbf{\omega}\) \\ \hline
0\({}^{\circ}\)\(\sim\)15\({}^{\circ}\) & 2 & 9 & 0 & 180\({}^{\circ}\)\(\sim\)195\({}^{\circ}\)  
Figure 12 shows the optimization results of total wind farm power output and total power losses under Weibull distribution wind condition. Though the total power output still showed a linear increasing trend with the increasing number of wind turbines, the change of total power losses showed a very different pattern from above cases. In general, the total power losses increased but at some particular numbers of wind turbines (e.g., 41 and 42 wind turbines and 45 and 46 wind turbines), the change of power losses was trivial. The optimal wind farm layout of 40 optimized wind turbines is shown in Figure 13 and the predominant wind direction is from bottom side to the top side. As can be seen, the wind turbines were staggered towards the dominant wind direction to avoid the wake effect as much as possible. Most of the wind turbines were distributed along or near to the wind farm boundary, and part of the inner wind farm area was free of wind turbines in order to enlarge the distance of wind turbines between each other along the dominant wind direction. The minimum distance apart for any two wind turbines was 7 turbine diameters and the average distance of all wind turbines was 35.2 turbine diameters. In order to demonstrate the effectiveness of the wind farm layout optimization, the results of optimized wind farm layout and rule of thumb layout (fixed spacing on the cross-wind direction) were compared for the Weibull distribution. Figure 14 shows the schematic of wind farm rule of thumb layout, which had fixed spacing of adjacent wind turbines on the cross-wind direction. The potential wind turbine placements are shown by the dashed lines in Figure 14. Table 3 shows the comparison of total wind farm power output between the rule of thumb layout and optimized layout of wind farm with different number of wind turbines. Evidently, the optimized wind farm layout obtained a larger power production than the rule of thumb layout under this wind characteristic. By comparing to the rule of thumb layout, the power output of the optimized layout improved approximately 17.3%. The properties of optimized wind farm layouts for all tested different wind scenarios are tabulated and compared in Table 4. As can be seen, the minimum distance between wind turbines were the same as  
Figure 12: Results of wind farm layout optimization with different number of wind turbines under Weibull distribution wind condition: (**a**) wind farm power production and (**b**) wind farm power losses. Figure 13: Optimal wind farm layout with 40 wind turbines under Weibull distribution wind condition. the predefined proximity constraint distance (7D) for all scenarios except for the scenario of variable wind speeds and directions, which was slightly larger (7.28D). As the complexity of wind scenario increased, the average distance of wind turbines decreased, indicating a more compact distribution of wind turbines locations.