## 8 Real-Life Application (Cost for Floating Offshore Wind Turbine Support Structures (FOWTs))  
Wind energy technology is growing rapidly at commercial scales and is one the mature means of renewable energy generations [99]. At the coastlines, higher wind speed resources are available and they provide consistent wind power. Therefore, offshore wind turbine technologies require floating platforms to harvest the wind energy in deep waters instead of using fixed structures which are suitable for shallow waters [100]. This section provides a real life application for a group of optimization algorithms to evaluate FOWTs support structure cost based on the method reported in [100]. A spar buoy platform is one of the structures that is currently used in offshore wind projects and has been selected in this study for cost analysis as shown in Figure 14.  
The approach used in this paper is to define a design parameterization for the support structure as well as a linearized frequency domain dynamic model which evaluates the stability of the floating system using the NREL 5 MW offshore wind turbine [100; 101]. The support structure parametrization pattern used in this work provided the spar buoy platforms with a limited number of design variables. Hence, four design parameters are defined to shape the platform and mooring system: a central cylinder with a draft, HI, cylinder radius, RI, top tapper ratio, TI, and a catenary mooring system, XM for spar buoy platforms. Note that a free board (FB) of 5 m is used for all the spar buoy platform designs as a constant parameter. The geometric design variables are summarized in Table 6 and Figure 15. In this work, the mooring system design variable, XM defines the angled taut line and slack line configurations for the spar buoy platform. Three lines are used for this platform and they are located at the middle the cylinder height as presented in Figure 15. The anchor position is defined by the mooring variation of XM from positioning under the platform to the horizontal spread of the lines. The mass properties for each spar buoy platform are determined using the platform and mooring design parameters. The mass of support structure is calculated using a wall thickness of 50 mm using a steel density of 8050 kg/m\({}^{3}\). Figure 14: Floating Offshore Wind Turbine (FOWT) with a spar buoy support structure. In this study, the cost of the FOWT design is defined based on three main components which include the mooring line, the 5 MW wind turbine, and the floating platform. To determine the stability of the floating system in a given environmental condition, the linearized characteristics of the hydrodynamics and aerodynamics of the FOWT must be collected in a frequency domain dynamic model. For floating platform hydrodynamics, the added mass, damping, and hydrostatic loads are calculated using WAMIT. The wind turbine dynamic characteristics are kept constant in this study. However, FAST is used to calculate the linearized dynamic properties for the NREL 5 MW offshore wind turbine. These properties include linearized stiffness, mass, and damping matrices of the offshore wind turbine in a given wind speed. Like the floating platform hydrodynamics, linearized mooring line properties can be added to the frequency domain dynamic model using a quasi-static mooring subroutine of FAST. All the calculated coefficients and loads are collected into a 6 \(\times\) 6 system stiffness, mass, damping to generate the linearized frequency domain equation of motion for a FOWT Equation (1). The stability of the system is evaluated by calculating the complex response amplitude operator (RAO) for all modes of motion. More details about the dynamic model and platform parameterization can be found in [101]. \[-\omega^{2}M_{total}(\omega)\hat{Z}(\omega)+i\omega B_{total}(\omega,\zeta)\hat {Z}(\omega)+C_{total}\hat{Z}(\omega)=\hat{X}\ (\omega) \tag{1}\]  
\[\left[RAO_{1}(\omega)::\ RAO_{6}(\omega)\right]=\left[-\omega^{2}M_{total}( \omega)+i\omega B_{total}(\omega,\zeta)+C_{total}\right]^{-1}\hat{X}(\omega) \tag{2}\]