## 4 Numerical Results
### Regular Wave  
In C1-3, FAST was utilized to calculate floating wind turbine dynamic responses with T0-4. Total calculation time was 900 s with former 600 s transient results being removed. As a result, 600-900 s floating wind turbine dynamic responses were analyzed. Since both wind and wave are unidirectional, wind turbine sway, heave, and roll movements are smaller than those in surge, pitch, and yaw directions. Thus, wind turbine surge, pitch, and yaw movements are presented and discussed. Figure 7 presents wind turbine surge movements from 600-900 s in C1-3. It can be obviously seen that wind turbine surge movement with the original mooring system T0 \begin{table}
\begin{tabular}{c c c c c} \hline
**Case Name** & **Wave Height (m)** & **Wave Period (s)** & **Wind Speed (m s\({}^{-1}\))** & **Wave Type** \\ \hline C1 & 6.00 & 10.00 & 8 & Regular \\ C2 & 2.56 & 7.00 & 8 & Regular \\ C3 & 4.00 & 8.00 & 8 & Regular \\ C4 & 6.70 & 8.6 & 8 & Irregular \\ C5 & 3.66 & 9.7 & 8 & Irregular \\ C6 & 9.14 & 13.6 & 8 & Irregular \\ \hline \end{tabular}
\end{table} 
Table 2: Environmental condition settings. Figure 6: Developed mooring systems with (**a**) three fairleads and (**b**) six fairleads. was larger than the developed mooring systems T1-4. Wind turbine surge movement with the original mooring system oscillated near 12 m, while those using the developed mooring systems oscillated near 8 m. For the developed mooring systems T1-3 in C1-3, it can be observed that wind turbine surge movements of T2 and T3 almost coincided with each other. However, the wind turbine surge movement of T1 was larger than those of T2 and T3; quantitatively, surge movement mean value of T1 was approximately 0.5 m larger than those in T2 and T3. This implies that the developed mooring systems can effectively control floating wind turbine surge movement when under regular waves. Mean surge displacement was reduced by approximately 33.33%. Meanwhile, with the increase of the included angle \(\beta\), the developed mooring system with three fairleads performed better and better. Mean surge displacement reduced 0.4 m every time when \(\beta\) increased 10\({}^{\circ}\). From the numerical results in Figure 7, it can be said that the included angle of 40\({}^{\circ}\) (T3) for the developed mooring system with three fairleads performed better than 20\({}^{\circ}\) and 30\({}^{\circ}\), which should be recommended. For T3 and T4 in C1-3, it can be seen that floating wind turbine surge movement with T4 was larger than that with T3, and it was even larger than that with T1. By contrast with the surge movement with T0, the surge movement with T4 was smaller (see Figure 7). This tells us that the developed mooring system with six fairleads could also reduce floating wind turbine dynamic responses, while it performed worse than that with three fairleads. Mean surge movement with T4 was 0.7 m larger than that with T3. Thus, after comparing with the developed mooring system with six fairleads, it can be safely said that T3 was the final optimal configuration for the developed mooring system in this study. Figure 8 shows wind turbine pitch movements with T0-4 under C1-3. It can be seen that pitch movement of T0 was obviously larger than those of T1-4, especially in Figure 8b, implying that the developed mooring systems could effectively reduce wind turbine pitch movements by approximately 7% when under regular waves. In C1-3, the wind turbine pitch movement times series with T1-3 almost coincided with each other, indicating that  
Figure 7: Wind turbine surge movement time series in (**a**) C1, (**b**) C2, and (**c**) C3. the included angle between two mooring lines in the same group had a negligible effect on pitch movement control. The pitch movement of T4 was slightly larger than that of T3 with a difference of 0.03\({}^{\circ}\).