# Passive Vibration Control of a Semi-Submersible Floating Offshore Wind Turbine
## 2 Modeling the Dynamics of Semi-Submersible FOWT  
### Simplified Dynamic Model for Semi-Submersible FOWT  
The semi-submersible foundation is gradually becoming the mainstream of FOMTs. In consideration of availability, the offshore code comparison collaboration continuation (OC4) DeepCwind semi-submersible foundation, one of the 5 MW FOWT benchmark model in the IEA-OC4 project, is selected as the research object in this study. The IEA-OC4 project is a program jointly conducted by the International Energy Agency (IEA) and various international research institutes to validate the quality of various FOWT analysis tools. In the OC4-DeepCwind system, the open accessed 5 MW offshore wind turbine designed by National Renewable Energy Laboratory (NREL) [16] is mounted on the foundation. Properties and model diagram of the OC4-DeepCwind model [17] are shown in Table 1 and Figure 1, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Rating & 5 MW \\ Rotor configuration & Upwind, 3 blades \\ Cut-in, rated, cut-out wind speed & 3 m/s, 11.4 m/s, 25 m/s \\ Hub height above sea level (SL) & 90 m \\ Total draft below SL & 20 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the OC4-DeepCwind model. In order to achieve effective loads reduction, a TMD device was planned to install in the OC4-DeepCwind system. For a floating wind turbine system, the TMD device can be installed in the nacelle/tower or platform, and the direction of TMD motion can be fore-aft or side-side. Since the fore-aft fatigue loads of horizontal axis wind turbines are much larger than the side-side loads, we focus on the fore-aft TMD installed in the nacelle/tower in this work (see Figure 2). To simplify the problem in this preliminary study, an ideal TMD device with constant stiffness and damping was adopted. In addition, the TMD track was assumed linear and frictionless. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Platform mass (including ballast) & \(1.3473\times 10^{7}\) kg \\ Elevation of tower base above SL & 10 m \\ Tower and tower-top mass & 599,718 kg \\ Height of tower & 77.6 m \\ Number of mooring lines & 3 \\ Depth to fairleads below SL & 14 m \\ Nacelle dimensions & 18 m \(\times\) 6 m \(\times\) 6 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 1: Diagram of DeepCwind semi-submersible. Figure 2: Schematic of the OC4-DeepCwind simplified dynamic model. To find out the optimal parameters of the TMD device, combining an optimization algorithm within the frame of FAST was the first choice. However, due to the limitation of the existing computational resources, the time cost of FAST was unacceptable to finish an onerous optimization procedure. Nevertheless, with respect to the floating systems, previous studies have demonstrated that only a few degrees of freedom are responsible for the most of system responses. Thus, to acquire the optimal TMD configurations for the OC4-DeepCwind model efficiently, it is helpful to establish a simplified mathematical model with limited degree of freedoms to replace the complex fully coupled model in FAST. For instance, [PERSON] et al. [7] established a 3-DOF (i.e., platform pitch, tower tilt and TMD motion) dynamic model for a large-type floating wind turbine, in which both the damping coefficient and the recovery stiffness coefficient are considered linear. For the spar-type floating wind turbine in the OC3 project, [PERSON] et al. [11; 12] obtained a higher accuracy by establishing the dynamic model accounting for 5 DOFs, i.e., the platform surge, heave, pitch, tower tilt and TMD translation. A schematic of the OC4-DeepCwind simplified dynamic model is illustrated in Figure 2. Following aforementioned studies, the preconditions and assumptions required to establish an efficient dynamic model are listed below:  
1. The semi-submersible FOWT was modeled as a multi-body dynamic system, in which the platform, tower (include rotor and nacelle), and the TMD were considered as rigid bodies.