## 2 Case Studies  
The optimization of a power system with an irregular wind farm of 25 wind turbines was used to describe the impact of power loss on the economics of the power system. The AeroDyn 2.5 MW wind turbines were selected as energy inputs to the power system, having a rotor with a diameter of 103.2 m, and the curve of the power with the wind speed is shown in Figure 4. The wind turbine group was randomly distributed within a range of 10 km \(\times\) 10 km, and the distance between any two wind turbines was greater than 10 times the diameter of the rotor. The power loss caused by the wake effect between the wind turbines was neglected due to the extremely dispersed distribution of the wind turbines. The offshore substation (OS) was fixed in the middle of the wind farm with coordinates (0.5 km, 0.5 km), and the coordinates of the onshore connection point (OCP) were (0.5 km, \(-0.5\) km). The medium-voltage and high-voltage power networks were constructed with 30 kV and 220 kV, respectively. The available cable contained eight cross-sectional areas (mm\({}^{2}\)) of 50, 70, 120, 150, 185, 240, 400, and 500, respectively. The cost and basic technical parameters of the cable were obtained from the reference [7]. The resistances and reactances of the cables as a function of the cross-sections are shown in Figure 5. In the literature, the price of the cable was denominated in Polish Zloty (PLN), which was converted to USD denominated (USD) at a rate of 1 PLN = 0.2625 USD. The power of the offshore substation was 75 MVA. The wind speed distribution of the wind farm was set according to the Class II wind farm of the GL2010 design standard, wherein Weibull shape parameters and annual mean wind speed distributions were 2 and 8.5 m/s. In the illustration adopted, the population size was taken as 100, the largest evolutionary generation was 5000, and the crossover and mutation rates were 0.9 and 0.01, respectively. A total of three scenarios were set up for the optimization of the power system. Script 1 employs a traditional step-by-step design approach in which the connection topology was generated by the MST and then the cable cross-section was designed according to the principle of maximum current carrying capacity. Script 2 also used MST to generate a topology, but it optimized the cross-section of the cable. Script 3 performed a coupling optimization of topology and cable cross sections. All scripts are run for a maximum of 5000 function evaluations and converged to the final solutions after about 3500 generations. The optimal topology and cable cross section schemes obtained from the above three scripts are given in Figures 6-8. The length comparison of the different cross-section cables used in the three script optimal solutions is shown in Figure 9. A comparison of the key economic parameters of the power systems is given in Table 1.  
Figure 4: Power curve of AeroDyn 2.5 MW and Weibull wind speed distribution. Scenario 1 used MST to generate a connection topology, thus obtaining the topology with the shortest total cable length. The optimal topology formed four branches, and the number of wind turbines in the branch was 4, 6, 9, and 6, respectively. From the perspective of cost composition, the energy loss cost in the 20-year operation period was the main expenditure part of the power system, accounting for 49.3% of the total cost, while the medium-voltage cable cost and substation cost only accounted for 15.9% and 14.7%, respectively, indicating that the power system obtained by the traditional step-by-step design method can cause a serious energy and economic losses. The main reasons are as follows: a large number of cables with small cross-sections were used in the topology according to the maximum current-carrying criterion, so that the overall resistance level of the topology remained large, resulting in a large amount of energy loss.