## 2 Aero-Hydro-Servo-Elastic Model  
### Aerodynamic Model  
To model the transient rotor loading for a FOWT the unsteady Blade Element Method (UBEM) is used. The unsteady flow around a FOWT is caused by different factors. The factors which are taken into account in this paper are the unsteady and uneven wind field with turbulence, the translational velocity of the TLP and tower. The structural dynamics of the blade are not taken into account, as this is assumed to be small compared to the surge motion. Furthermore, the yaw motion of the wind turbine is not taken into account in the aerodynamic calculations. The theory presented in this paper regarding the modeling of unsteady aerodynamics of wind turbines is mainly by [PERSON] from [21]. The BEM arises from a combination of two theories; the 2D blade element theory and the 1D momentum theory. In BEM the blade is discretized spanwise, and the forces are calculated for each section, which is summarized at the end to calculate the forces on the whole rotor. To model the unsteady effects in BEM three unsteady dynamic models are applied, these are a dynamic stall model, a dynamic inflow model, and a tower-interference model. The main objective of the BEM method is to calculate the relative velocity the blade sections experience. The relative velocity at a blade section can be found by  
\[\mathbf{V}_{rel}=\mathbf{V}_{0}+\mathbf{V}_{Tbw}+\mathbf{V}_{TLP}+\mathbf{V}_{ Rot}+\mathbf{W} \tag{1}\]where \(\mathbf{V}_{rel}\in\mathbb{R}^{2}\) is the relative velocity at a blade section, \(\mathbf{V}_{0}\in\mathbb{R}^{2}\) is the free stream velocity of the wind, \(\mathbf{V}_{TLP}\in\mathbb{R}^{2}\) and \(\mathbf{V}_{Tow}\in\mathbb{R}^{2}\) is the translational velocity of the TLP and tower respectively, \(\mathbf{V}_{Rot}\in\mathbb{R}^{2}\) is the rotational velocity at the blade section and \(\mathbf{W}\) is the induction velocity. The velocity vectors can be visualized in Figure 1.  
The flow angle can be found from the two components of the relative wind speed. By a geometrical inspection of the velocity triangle, the flow angle can be calculated as  
\[\tan(\phi)=\frac{\mathbf{n}_{y}\mathbf{V}_{rel}}{-\mathbf{n}_{x}\mathbf{V}_{ rel}} \tag{2}\]  
where \(\phi\) is the flow angle, \(\mathbf{n}_{y}\) and \(\mathbf{n}_{x}\) are the direction vectors vectors \(\mathbf{n}_{y}\) = [0 1] and \(\mathbf{n}_{x}\) = [1 0]. From this the blades angle of attack can be found as  
\[\alpha=\phi-(\beta+\theta) \tag{3}\]  
where \(\beta\) is the blade sections pitch angle and \(\theta\) is the blade sections geometrical twist. As the yaw movement of the wind turbine is neglected, the induced velocity can be described by two induction factors  
\[a=\left(\frac{4F\sin^{2}(\phi)}{\
u C_{n}}+1\right)^{-1} \tag{4}\]  
\[a^{\prime}=\left(\frac{4F\sin(\phi)\cos(\phi)}{\
u C_{t}}-1\right)^{-1} \tag{5}\]  
where \(a\) and \(a^{\prime}\) are the axial and tangential induction factors respectively. \(\
u\) is the solidity factor, which describes the fraction of the blade section covers in the rotor annulus. \(F\) is Prandtl's tip loss factor, which describes the loss from the tip wake at the blade section. \(C_{n}\) and \(C_{t}\) are the normal and tangential coefficients. These are found by the projection of the lift and drag coefficients, which can be found by using airfoil data for a given angle of attack. The expression for the axial coefficient in Equation (4) hold for values lower than approximately 0.2, as beyond this the momentum theory does no longer hold, therefore the [PERSON]'s correction is used in this case. The tangential and normal force on the blade section can be calculated as  
\[FT_{j,k}=\frac{1}{2}\,\rho_{a}\,\mathbf{n}_{1}\,\mathbf{V}_{rel}\,C_{t}\,c\,dr \tag{6}\]  
\[FN_{j,k}=\frac{1}{2}\,\rho_{a}\,\mathbf{n}_{2}\mathbf{V}_{rel}\,C_{n}\,c\,dr \tag{7}\]  
where \(FT\) and \(FN\) are the tangential and normal force on blade \(j\), at blade section, \(k\). \(\rho_{a}\) is the density of the air. \(c\) and \(dr\) is the chord length and spanwise length of the blade section respectively.