## 8 Real-Life Application (Cost for Floating Offshore Wind Turbine Support Structures (FOWTs))
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Description** & **Min.** & **Max.** \\ \hline HI & Central cylinder draft & 2 m & 150 m \\ RI & Central cylinder radius & 3 m & 25 m \\ TI & Top tapper ratio & 0.2 & 2 \\ XM & Mooring System & 0.2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Geometric design variables of spar buoy platform. Figure 15: Design characteristics of a spar buoy platform including height, radius, and taper ratio of the cylinder as well as platform’s free board. The performance of the FOWT in this study is defined as the standard deviation of nacelle acceleration and is shown below:  
\[\sigma_{anac}(\omega)=\sqrt{\int\limits_{0}^{\infty}\left|RAO_{anac}(\omega) \right|^{2}S(\omega)\;d\omega} \tag{3}\]  
where \(S(\omega)\) is the wave spectral density, and \(RAO_{anac}(\omega)\) is the fore-aft nacelle acceleration response amplitude operator. The complex forms of the nacelle displacement \(RAO\) and nacelle acceleration \(RAO\) are given by Equations (4) and (5):  
\[RAO_{g_{nac}}\left(\omega\right)=\left(RAO_{1}(\omega)+RAO_{5}(\omega)\;z_{ nac}\right) \tag{4}\]  
\[RAO_{a_{nac}}(\omega)=-\omega^{2}(RAO_{1}(\omega)+RAO_{5}(\omega)z_{nac}) \tag{5}\]  
where \(z_{nac}\) is the wind turbine hub height. In Equations (4) and (5) the \(RAO_{anac}(\omega)\) includes the aero dynamic effects of the wind turbine implicitly from the linearized dynamic properties of the wind turbine. \(Z_{nac}RAO_{anac}\) The objective function to optimize in this study is the support structure cost including mooring system, anchor and floating platform costs:  
\[min\;f(x):\;C(x)=C_{Platform}\;(x)+C_{Mooring}\;(x)+C_{Anchor}\;(x) \tag{6}\]  
The floating platform cost is defined as a function of the design parameters as shown in Table 7. In this study, the platform cost contains the materials, manufacturing, and installation costs using a constant cost factor of $2.5/kg [100; 101]. The mooring system cost is calculated using the length of the mooring lines and the maximum steady state mooring line tension with a factor of $0.42/m kN. The anchor cost is the third component of the cost model that includes the installation and technology costs. The drag embedment anchor technology is selected in this study for catenary mooring system. Therefore, the installation cost for each anchor is defined as 100 $/anchor/kN and the technology cost is based on 5000 $/anchor. In this study, a number of constraints are also applied to the objective function. To avoid expensive design configurations, a constraint limits the cost of the platform to less than $9 M. The performance metric \(RAO_{anac}(\omega)\) is also limited to less than 1. In addition, to avoid the floating platform turning over, the steady state pitch angle of the platform should be less than 10\({}^{\circ}\). The expression of this constraint is given in the following equation:  
\[\zeta_{5}=\frac{F_{thrust}\cdot z_{nac}+M_{moorings}-M_{ballast}}{\rho g\%z_{ CB}-M_{l}gz_{CG}+\rho g\;I_{xx}-C_{moorings5}+C_{5,1}z_{fair}} \tag{7}\]  
where \(F_{thrust}\) is the steady thrust load, \(M_{moorings}\) is the mooring line pitching moment, \(M_{ballast}\) is the pitching moment from the ballast mass stabilizers, \(\forall\) is the platform displacement, \(z_{CB}\) is the center of buoyancy of the platform, \(M_{t}\) is the total mass of the FOWT, \(z_{CG}\) is the center of gravity location, \(I_{xx}\) is the water plane moment of inertia in platform pitch motion, \(C_{moorings5}\) is the mooring lines stiffness in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & \(f^{\star}\)**Min** & **NFE** & **CPU Time** \\ \hline BA & 5,533,111.9378 & 1000 & 121.1719 \\ CSA & 4,426,006.2921 & 1000 & 123.5938 \\ FFA & 4,811,655.8121 & 1000 & 131.8125 \\ FFA & 4,727,085.732 & 1000 & 132.875 \\ ABC & 3,817,499.5279 & 951 & 119.2969 \\ GWO & 4,395,036.2793 & 1000 & 123.5313 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Summary of comparison result for support structure cost for FOWTs application.