## 2 Mathematical Equations of the Coupled Dynamics Theory of a Moored FWT  
Equations of motion of a moored FWT can be derived based on the [PERSON]'s equation [20; 21; 22], taking into consideration of all nonlinearities and system's DOFs (Degrees of Freedom). [PERSON]'s equation is generally based on the principle that all generalized active forces and inertial forces acting on the complex system of the coupled rotor-nacelle assembly, the tower and the support platform are balanced. The kinetics equations for the support platform include contributions from the platform mass and inertia, gravity, hydro-restoring, hydrodynamics and the reaction forces of the mooring system, which can be written as  
\[F_{i}^{Platform}(q)=-\big{(}M_{ij}+A_{ij}\big{)}\hat{q}_{j}-\int\limits_{0}^{t }K_{ij}(t-\tau)\hat{q}_{j}(\tau)d\tau-C_{ij}^{Hydrostatic}q_{j}+F_{i}^{Waves}( q)+F_{i}^{Lines}(q)+\rho gV\delta_{ij}, \tag{1}\]where \(M_{ij}\), \(A_{ij}\) and \(C_{ij}^{Hydrostatic}\) are the (\(i\), \(j\)) component of the platform mass matrix, impulsive hydrodynamic added-mass matrix and hydrostatic-restoring matrix, respectively; \(q_{j}\) is the \(j\)th DOF of the platform motions; \(F_{i}^{Waves}\) represents the total excitation load on the support platform from incident waves; \(K_{ij}\) is the (\(i\), \(j\)) component of the matrix known as the wave-radiation-retardation kernel or impulse-response functions; the last term on the right-hand side of Equation (1) represents the buoyancy force of the platform derived from Archimedes' principle, in which \(\rho\) is the water density, \(g\) is the gravity acceleration, \(V\) is the displaced volume of water by the immersed part of the platform, \(\delta_{i3}\) is the (\(i\), \(3\)) component of the Kronecker-Delta function; \(F_{i}^{Lines}\) is the \(i\)th component of the load on platform from all mooring lines. It can be noted that the first three terms on the right-hand side of Equation (1) are all linear to the platform acceleration, velocity and displacement, respectively. \(F_{i}^{Waves}\) and \(F_{i}^{Lines}\) can be linear or nonlinear, depending on the requisite modelling complexity. For a generic problem, usually up to second-order wave forces are considered for the potential flow theory. It is sometimes supplemented by the Morison-strip theory in the case there exist slender braced members in the floating platform structure (which is also nonlinear). \(F_{i}^{Lines}\) is usually modelled by a quasi-static method, e.g., the improved catenary theory [22], to calculate the tension force of every mooring line and them make a summation of them. When no portion of the line rests on the seabed, the analytical formulation of each line is as follows:  
\[x_{F}(F_{H},F_{V})=\frac{F_{H}}{w}\left\{ln\left[\frac{F_{V}}{F_{H}}+\sqrt{1+ \left(\frac{F_{V}}{F_{H}}\right)^{2}}\right]-ln\left[\frac{F_{V}-wL}{F_{H}}+ \sqrt{1+\left(\frac{F_{V}-wL}{F_{H}}\right)^{2}}\right]\right\}+\frac{F_{H}L}{ EA}, \tag{2}\]  
\[z_{F}(F_{H},F_{V})=\frac{F_{H}}{w}\left[\sqrt{1+\left(\frac{F_{V}}{F_{H}} \right)^{2}}-\sqrt{1+\left(\frac{F_{V}-wL}{F_{H}}\right)^{2}}\right]+\frac{1} {EA}\left(F_{V}L-\frac{wL^{2}}{2}\right), \tag{3}\]  
where \(x_{F}\) and \(z_{F}\) are respectively the horizontal and vertical coordinates of the fairlead position relative to the anchor, \(F_{H}\) and \(F_{V}\) are the horizontal and vertical components of the effective tension in the mooring line at the fairlead, \(w\) is the mass of the mooring line per unit length, \(L\) is the total unstretched mooring line length, and \(EA\) is the sectional stiffness of the mooring line. It is worth noting that Equations (2) and (3) should be solved iteratively in the local coordinate system via the Newton-Raphson method, etc. When the environment conditions are severe or if the floating platform has large responses, high-fidelity mooring line models are recommended for the mooring analysis, especially for prediction of the mooring line loads. The existing high-fidelity models include FEA-based models and lumped-mass models.