## 3 Effect of Wave on Load and Aerodynamic Torque of Wind Turbines
### Effect of Regular Waves on Blade Load and Aerodynamic Torque  
The relative wind speed on the wind turbine blade is the superposition of the wind speed and the blade's own motion speed. That is, when the tower top moves forward and backward, the relative wind speed of the blade will increase or decrease accordingly. The wind speed under the influence of tower motion can be expressed as  
\[V= V_{0}-\dot{u}_{top} \tag{9}\]  
where \(V_{0}\) is the original wind speed; \(\dot{u}_{top}\) is the tower top motion speed in the direction of wind speed. In the area where the wind blows, it is generally considered that the direction of wave propagation is consistent with the wind direction. Therefore, in this paper, we assume that the wind load and wave load acting on the wind turbine and the tower are in a consistent direction. The aerodynamic load of the wind turbine is directly related to the relative wind speed of the blade, and the former will change with the latter. When the load of the wind wheel acting on the tower changes, the tower deforms to varying degrees, which affects the relative wind speed. Therefore, there exists a dynamic coupling effect between the wind wheel and the tower during the whole vibration process of the wind turbine. Without considering the blade-tower coupling effect, the root shear force of the rotating blade is the sum of the total inertial force of the blade, which can be expressed as [33]  
\[\{Q_{B}(t)\}=\sum_{i=1}^{n}m_{Bi}\ddot{w}_{Bi}(t), \tag{10}\]  
where \(\{Q_{B}(t)\}\) is the blade root shear force; \(m_{Bi}\) is the mass of the blade element, and \(\ddot{w}_{Bi}(t)\) is the acceleration of the blade element. When considering the blade-tower coupling effect, the effective shear force of a blade can be written as [34]  
\[\left\{\overline{Q}_{B}(t)\right\}=\sum_{i=1}^{n}m_{Bi}(\ddot{w}_{Bi}(t)+\ddot {u}_{top}(t)), \tag{11}\]  
where \(\ddot{u}_{top}(t)\) is the acceleration of the tower top. Applying Equation (10) to Equation (11), we have  
\[\left\{\overline{Q}_{B}(t)\right\}=\sum_{i=1}^{n}m_{Bi}\ddot{w}_{Bi}(t)+\ddot {u}_{top}\sum_{i=1}^{n}m_{Bi}=\{Q_{B}(t)\}+\ddot{u}_{top}M_{B}, \tag{12}\]  
where \(M_{B}\) is blade mass. In Equation (12), the inertia effect of the tower top on the blade is considered, and the coupled equation can be decoupled by the method of additional mass in the process of solving. So, the shear force of a single blade is affected by its own mass distribution and acceleration, as well as the acceleration of the tower top. In other words, when considering the blade-tower coupling effect, the vibration of the tower top caused by the wave will add a shear force of the same frequency to the blade root in the same direction. The relationship between the shear force of the blade root flap-wise direction \(\overline{Q}_{Bx}\) and the blade root flap-wise load \(M_{Bx}\) can be written as [35]  
\[M_{Bx}=\int_{r_{0}}^{R}\overline{Q}_{Bx}rdr=\int_{r_{0}}^{R}(Q_{Bx}+\ddot{u}_{ top}M_{B})rdr, \tag{13}\]  
where \(r_{0}\) is the hub radius; \(R\) is the wind wheel radius; \(r\) is the distance between the blade element and blade root. The change of the relative wind speed of the blade will lead to the obvious change of the aerodynamic torque produced by the wind wheel. \[ \begin{array}{ll}T_{aero}&=\frac{1}{2}\rho_{air}\pi R^{3}C_{T}(V,\Omega, \beta)V^{2}\\ &=\frac{1}{2}\rho_{air}\pi R^{3}C_{T}(V,\Omega,\beta)(V_{0}-\dot{u}_{top})^{2} \end{array} , \tag{14}\]  
where \(\rho_{air}\) is air density; \(C_{T}\) is the torque coefficient; \(\Omega\) is wind wheel speed, and \(\beta\) is pitch angle.