# Passive Vibration Control of a Semi-Submersible Floating Offshore Wind Turbine
## 2 Modeling the Dynamics of Semi-Submersible FOWT
### Model Parameters Identification  
Before the proposed dynamic model is applied in the optimization of TMD parameters, its unknown parameters should be identified first. In the proposed dynamic model, there are totally 11 unknown parameters. However, \(K_{\text{p}}^{\text{sg}}\) and \(K_{\text{sg}}^{\text{p}}\) therein are simplified to one parameter due to the symmetry of the mooring stiffness. Finally, the ten parameters, all of which need to be identified and evaluated properly, are given by  
\[\mathbf{Q}=\left[I_{\text{t}},K_{\text{p}}^{\text{p}},K_{\text{sg}}^{\text{sg }},K_{\text{sg}}^{\text{p}},K_{\text{t}},D_{\text{t}},D_{\text{p}}^{\text{ pg}},D_{\text{sg}}^{\text{sg}},D_{\text{sg}}^{\text{pg}}\right] \tag{17}\]  
Because the tower and RNA are treated as a single rigid body and hinged to the platform, their inertia \(I_{\text{t}}\) should be re-identified in the simplified model. The parameter values that can be calculated exactly from the practical model are listed in Table 2.  
In this study, the vertical pretension of each mooring line is equivalent to a mass fixed at the fairlead point. Consequently, the mass, the inertia and the height of the center of gravity of the platform in Table 2 are modified accordingly. In order to achieve parameter identification efficiently, the proposed model is established in Matlab Simulink. The Levenberg-Marquardt (LM) algorithm, which combines the advantages of the gradient descent algorithm and the Gauss-Newton approach [22], is employed in the identification process. The LM approach is capable of searching the minimum value of the sum of squared errors (SSE) effectively. The definition of SSE between outputs from the proposed model and FAST is given by  
\[SSE(\mathbf{Q})=\frac{1}{2}\underset{j=1}{\overset{m}{\sum}}\underset{i=1}{ \overset{n}{\sum}}w_{j}\big{[}y_{j}(t_{i})-f_{j}(t_{i},\mathbf{Q})\big{]}^{2} \tag{18}\]  
where \(w_{i}\) denotes weight for each variables. Since the magnitude of TTD is smaller than the pitch and surge motions, the weights of TTD, pitch and surge are set as 10, 1 and 1, respectively. \(m\) denotes the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(m_{\text{p}}\) & \(1.3661\times 10^{7}\) kg & \(I_{\text{p}}\) & \(9.4614\times 10^{9}\) kg\(\cdot\)m\({}^{2}\) \\ \(m_{\text{t}}\) & \(5.9911\times 10^{5}\) kg & g & \(9.806\) N/m\({}^{2}\) \\ \(R_{\text{p}}\) & \(13.46\) m & \(A_{\text{sg}}^{\text{sg}}\) & \(6.49\times 10^{6}\) kg \\ \(R_{\text{t}}\) & \(60.29\) m & \(A_{\text{sg}}^{\text{sg}}\) & \(-8.51\times 10^{7}\) kg\(\cdot\)m\({}^{2}\) \\ \(H_{\text{c}}\) & \(10\) m & \(A_{\text{sg}}^{\text{p}}\) & \(-8.51\times 10^{7}\) kg\(\cdot\)m\({}^{2}\) \\ \(H_{\text{Rwr}}\) & \(77.6\) m & \(A_{\text{p}}^{\text{p}}\) & \(7.21\times 10^{9}\) kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Known parameters of the dynamic model. number of selected output responses, and \(n\) is the number of data samplings. \(y_{j}(t_{i})\) is the \(j\)th response obtained from FAST at time \(t_{i}\), and \(f_{j}(t_{i},\textbf{Q})\) is the results of the proposed model adopting parameter vector **Q**. Similar to [12], since the wind and wave loads are not considered in the proposed model, the free-decay motion with a \(5^{\circ}\) initial pitch and inactive TMD is selected as the test case for identifying parameters. The surge and pitch of platform, and TTD are employed as the targets of comparison to fit in the FAST results. Specifically, in the FAST simulation, deactivate all the DOFs except the surge and pitch of platform and the first fore-aft bending of tower, run FAST and record the response time-histories. In the simplified model, switch off the DOF of TMD and execute in the same condition as FAST. During parameter identification, the initial values are supposed to influence the efficiency and tendency of the optimization. To achieve a quick convergence, it is important to assign appropriate initial values.