**2. Theoretical Background**

Different from offshore oil and gas platform, a floating wind turbine experiences the aerodynamic load and hydrodynamic load simultaneously. These loads are both important in the system design at the same order. The basic theories used in the combined system analysis are described in this section.

### *2.1. Aerodynamic Loads and Aerodynamic Damping*

Blade element momentum (BEM) theory is one of the efficient and most commonly used methods for calculating induced loads on wind turbine blades [44]. In this theory, the wind turbine blade is divided into many sections along the span direction, and these sections are called blade elements. The wind turbine blade is simplified as a finite element that is superimposed in the radial direction, so the three-dimensional aerodynamic characteristics of the wind turbine blade can be obtained by integrating the aerodynamic characteristics of the element in the radial direction [42].

The axial velocity *V*<sup>2</sup> and the circumferential velocity *V*<sup>3</sup> at the rotor plane are calculated as follows:

$$V\_2 = (1 - a)V\_1 \tag{1}$$

$$V\_3 = \Omega r \left(1 + a'\right) \tag{2}$$

where *V*<sup>1</sup> is the incoming wind velocity, *a* is the axial induction factor, and *a* is the angular induction factor. The axial velocity and the circumferential velocity vector are combined to obtain the resultant velocity *V*0. The lift force (*L*) is formed perpendicular to the resultant velocity, and the drag force (*D*) is formed in the same direction as the incoming velocity. They can be calculated by the following equations:

$$L = \frac{1}{2}\rho V\_0^2 A \mathbb{C}\_L \tag{3}$$

$$D = \frac{1}{2}\rho V\_0^2 A \mathbb{C}\_D \tag{4}$$

where *A* is the rotor sweeping area, *ρ* is the air density, *CL* is the lift coefficient, and *CD* is the drag coefficient.

The lift coefficient *CL* and drag coefficient *CD* of the aerofoil are projected in the normal and tangential directions, respectively, to obtain the normal force coefficient *CN* and the tangential force coefficient *CT*, as shown in the following equations:

$$\mathbf{C}\_{N} = \mathbf{C}\_{L}\cos\varphi + \mathbf{C}\_{D}\sin\varphi\tag{5}$$

$$\mathbf{C}\_{T} = \mathbf{C}\_{L}\sin\varphi - \mathbf{C}\_{D}\cos\varphi\tag{6}$$

Finally, the thrust force and torque on the blade section are calculated by the following equations:

$$dT = \frac{1}{2}\rho V\_0^2 c \mathbf{C}\_N dr\tag{7}$$

$$dM = \frac{1}{2}\rho V\_0^2 c \mathcal{C}\_T r dr\tag{8}$$

where *c* is the aerofoil chord length.

When studying aerodynamic loads, the influence of aerodynamic damping cannot be ignored. Analyzed at the micro level, the aerodynamic damping of the wind turbine essentially comes from the relationship between the aerodynamic load of the wind turbine blades and the inflow wind velocity. Without considering the platform motion and elastic structure deformation [45], the following equation can be used:

$$V\_0 = V\_1 \sqrt{(1-a)^2 + \left[\frac{\Omega r}{V\_0}(1+a')\right]^2} = V\_1 f\_1 \tag{9}$$

The factor *f*<sup>1</sup> indicates that the relative inflow wind velocity at the blade is simultaneously affected by the axial induction factor, angular induction factor, structural deformation of the wind turbine, and rotational velocity [46].

Considering the relative movement of the platform, the relative velocity is written as follows:

$$V\_0 = V\_1 f\_1 - L \dot{\mathbf{x}}\_5 \cos(\mathbf{x}\_5) - \dot{\mathbf{x}}\_1 \tag{10}$$

where *<sup>x</sup>*<sup>5</sup> is the pitch motion of the platform, . *<sup>x</sup>*<sup>5</sup> is the pitch velocity of the platform, and . *x*1 is the surge velocity of the platform.

From Equation (7), the total thrust *T* on the blade can be obtained as follows:

$$T\approx V\_0^2 \cos\left(V\_1 f\_1 - L\dot{x}\_5 \cos\left(\dot{x}\_5\right) - \dot{x}\_1\right)^2\tag{11}$$

Ignoring the velocity components above the second order, the following equation is obtained:

$$\begin{array}{l} T \approx (V\_1 f\_1)^2 - 2V\_1 f\_1 L \dot{\mathbf{x}}\_5 \cos \left( \dot{\mathbf{x}}\_5 \right) - 2V\_1 f\_1 \dot{\mathbf{x}}\_1\\ \approx (V\_1 f\_1)^2 - 2V\_1 f\_1 L \dot{\mathbf{x}}\_5 - 2V\_1 f\_1 \dot{\mathbf{x}}\_1 \end{array} \tag{12}$$

The last two terms in the above equation represent aerodynamic damping and can be added to the left side of the equation of motion. The aerodynamic thrust is proportional to the square of the wind velocity, and the aerodynamic damping is proportional to the first power of the wind velocity. This shows that when the wind velocity is not particularly high, the effect of aerodynamic damping on the offshore floating wind turbine may be more obvious, and the relationship between them is positive. However, as the wind velocity increases, the influence of aerodynamic thrust may be much greater than that of aerodynamic damping. Of course, the aerodynamic damping force of offshore floating wind turbines is affected not only by wind velocity but also by other complex factors, such as aerodynamic induction factor. Thrust coefficient may even be affected by pitch angle and stall effect. Since F2A and AQWA were used for simulation in this study, the largest difference between them is whether considering aerodynamic damping.
