## 2 Model Statement
### Equations of Motions in the FOWT  
In this study, the input is unidirectional regular waves. Besides, Airy wave theory is considered to explain its surface dynamics [25]:  
\[z(t)=A\sin(\omega t)=A\sin(2\pi ft)=A\sin(\frac{2\pi}{\lambda}ct) \tag{1}\]  
where the propagation velocity and the wave amplitude from _SWL_ to the wave crest may be defined as \(c=\lambda f\) and \(A\), respectively. Also, the wavelength (\(\lambda\)) is calculated by measuring the distance between two crests. Equation (1) expresses the temporal variation of a wave as a macroscopic representation of the oscillating motion of water particles at a specific point [26]. A variable for the representation of the spatial dimension in the wave's front direction may be represented for transferring the oscillating behavior to any point on the wave's surface. Hence, Equation (1) may be rewritten as follows [27]:  
\[z(x,t)=A\sin(\frac{2\pi}{\lambda}(ct-x)) \tag{2}\]  
having the wave number as \(k=2\pi/\lambda\), Equation (2) can be written as:  
\[z(x,t)=A\sin(\omega t-kx)=\frac{H}{2}\sin(\omega t-kx) \tag{3}\]  
where \(H\) is wave height between the wave trough and wave crest. After describing the unidirectional regular waves, the equations of the FOWT's motions are described. The complete nonlinear equations of motion in time-domain for the floating wind turbine with a platform equipped by OWCs may be given by:  
\[M_{ij}(q,u,t)\ddot{q}_{j}=f_{i}(q,\dot{q},u,t) \tag{4}\]  
where \(M_{ij}\) and \(u\) are the inertia mass components and the control inputs respectively. \(q\) is the system's states and \(t\) is time. \(\dot{q}_{j}\) is the second time derivative of the \(j\)th DOFs and \(f_{i}\) is the component of the force function linked to \(i\)th DOF. Also, the parameter of the equation's right-hand side is the generalized forces imposing to the system consisting of the aerodynamic load on the blades and the nacelle, hydrodynamic force on the platform, elastic and servo forces. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Platforms’ size (W \(\times\) L \(\times\) H) & 40 m \(\times\) 40 m \(\times\) 10 m \\ Each OWC’s size (W \(\times\) L \(\times\) H) & 5 m \(\times\) 5 m \(\times\) 10 m \\ Draft, Free board for both platforms & 4 m, 6 m \\ Water displacement for the simple barge & 6400 m\({}^{3}\) \\ Water displacement for the barge with OWCs & 6000 m\({}^{3}\) \\ Mass, Including Ballast & 5,452,000 kg \\ CM Location below SWL & 0.281768 m \\ Roll Inertia about CM & 726,900,000 kg\(\cdot\)m\({}^{2}\) \\ Pitch Inertia about CM & 726,900,000 kg\(\cdot\)m\({}^{2}\) \\ Yaw Inertia about CM & 1,453,900,000 kg\(\cdot\)m\({}^{2}\) \\ Anchor (Water) Depth & 150 m \\ Separation between Opposing Anchors & 773.8 m \\ Unstretched Line Length & 473.3 m \\ Neutral Line Length Resting on Seabed & 250 m \\ Line Diameter & 0.0809 m \\ Line Mass Density & 130.4 kg/m \\ Line Extensional Stiffness & 589,000,000 N \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Standard barge and the 40 WC-based barge platforms’ features. The system's frequency-dominant equations of motion may be described by:  
\[I_{FOWT}(\omega)\ddot{q}+B_{FOWT}(\omega)\dot{q}+C_{FOWT}q=\vec{f}_{FOWT}(\omega )+\vec{f}_{PTO}(\omega) \tag{5}\]  
here \(I_{FOWT}\) is the inertia matrix, \(B_{FOWT}\) is the damping matrix and \(C_{FOWT}\) includes stiffness coefficients. \(\vec{f}_{FOWT}(\omega)\) stands for the hydrodynamic force and viscous drag of the waves on the platform. \(\vec{f}_{PTO}(\omega)\) symbolizes as the load induced by the Power-Take-Off (PTO) equipment.