## 3 Numerical Simulations
### Simulation Results and Analysis of the Proposed System

Table 2: Structural properties of the platform and float. Figure 12: Description of the bottom-fixed platform with 2 WECs: (**a**) 3D view; (**b**) Top view; (**c**) Side view. The rotational damping proportional to the rotational velocity of the float was applied to the center of rotation as the PTO system. The optimal extraction coefficient was obtained as 100 kN-m/(rad/s) through a parametric study for the regular wave [19]. The wave power generated by the PTO system was proportional to the square of the rotational velocity of the float, as shown in the following equation. \[P_{PTO}(t)=B_{PTO}\cdot\left\{\dot{\theta}(t)\right\}^{2} \tag{24}\]  
The distance between hinge points was set to 3 m. This distance does not affect the pitch RAO of the float significantly when the platform is fixed [19]. The frequency-dependent hydrodynamic coefficients considering the interaction between the two floats were obtained using multi-body WAMIT. The interactive hydrodynamic coefficients are the same because the two floats have the same shape and size. The natural frequency of the platform was obtained by modal analysis, and the total number of vibration modes was 708. Table 3 lists the natural frequencies up to the fifth mode. In this study, all vibration modes were considered, and the equation of motion of the platform was expressed using 708 SDOFs. The results without WECs were first compared to identify the dynamic response characteristics of the platform. Figure 13 shows the time histories of \(x\)-axis displacement at the top of Leg 1 and stress at the bottom of Leg 1. The stress shown here represents the sum of axial stress and bending stress acting on each element of the structure. According to the figure, the displacement and stress increased significantly due to the influence of the floats. The response of the platform was affected by the horizontal and vertical constraint forces from the floats. The negative (compressive) stress was caused by external forces from the floats and the weight of the platform. The former particularly plays a significant role in the increase in dynamic stress. As the displacement increases, the bending stress of each element also increases, resulting in a similar trend to the time series of displacement and stress. The results underscore that this kind of fully coupled dynamic analysis is essential for a reliable estimation of the fatigue life of the system. Figure 14 shows the distribution of the maximum absolute value of the displacement and stress of each element from the seabed to the deck. The displacement at the top had  
Figure 13: Time histories of dynamic responses of Leg 1: (**a**) \(x\)-axis displacement at the top; (**b**) Stress at the bottom. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **1 st Mode** & **2 nd Mode** & **3 rd Mode** & **4 th Mode** & **5 th Mode** \\ \hline Natural frequency [rad/s] & 0.4026 & 0.4059 & 3.1132 & 6.3066 & 6.3739 \\ Natural period [s] & 15.60 & 15.48 & 2.02 & 1.00 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Natural frequency of the platform. the same value in all legs because the stiffness of the deck was considerable. In addition, the stresses on legs 1 and 2 (front legs) are the same because the structure/loading is \(x\)-axis-symmetric. Similarly, the stresses on legs 3 and 4 (rear legs) were the same. The displacements were zero at the sea bottom because the structure was fixed there and increased monotonically toward the top, as shown in Figure 14a. The top displacement was increased significantly (\(\sim\)10 times) with the two floats compared to the no-float case. The maximum stresses occurred at the top and the bottom of the foundation structure. The significantly increased stress at the top was attributed to the dynamic coupling with the WECs, as shown in Figure 14b. The magnitude of the stress was the smallest in the middle of the leg because of the smallest degree of bending.