# Control of a Variable Blade Pitch Wind Turbine Subject to Gust Wind and Actuators Saturation
## 2 Materials and Methods  
### Wind Generator Dynamic Model  
A simplified dynamic model of the wind turbine and its drive train based on the two-mass system is illustrated in Figure 1. The drive train consists of a rotor, a generator, and a gearbox. The mathematical model based on this configuration has been used to design linear and nonlinear controller [33; 34].  
Figure 1: Wind turbine model.  
With the assumptions that the moment of inertia of the gearbox and the damping effect of the blades can be ignored, the equations of motion are derived using Lagrange dynamics and written in the following form:  
\[\begin{cases}J_{r}\dot{\omega}_{r}=T_{a}-T_{ls}-B_{r}\omega_{r}\\ J_{g}\dot{\omega}_{g}=T_{hs}-B_{g}\omega_{r}-T_{em}\\ \dot{\beta}=-\frac{1}{\tau}\beta+\frac{1}{\tau}\beta_{d}\end{cases} \tag{1}\]  
where \(J_{r}\), \(\omega_{r}\) represents the moment of inertia and the angular velocity of the rotor, \(T_{ls}\) is the torque on the low-speed shaft, \(B_{r}\) is rotor damping coefficient, and \(T_{a}\) is the aerodynamic torque, it and can be modeled as  
\[T_{a}=\frac{1}{2}\rho\pi\;R^{3}C_{q}(\lambda,\beta)V^{2} \tag{2}\]  
where \(\rho\) is the air density, \(R\) is the rotor radius, \(V\) is the wind speed, \(C_{q}(\lambda,\beta)\) is the torque coefficient in which \(\beta\) is the blade pitch angle, and \(\lambda\) is tip speed ratio, and it can be written as  
\[\lambda=R\frac{\omega_{r}}{V} \tag{3}\]  
\(J_{g}\), \(\omega_{g}\) represents the moment of inertia and the angular velocity of the generator, \(B_{g}\) is generator damping coefficient, \(T_{em}\) is the electromagnetic torque, and \(T_{hs}\) is the high-speed shaft torque. It is worthwhile to mention that the third equation in Equation (1) represents the electromechanical pitch systems model where \(\tau\) is a time delay constant and \(\beta_{d}\) is the desired pitch angle. It is worth mentioning without going into many details that the flux-weakening control technique, which is considered one of the most practical solutions [35; 36; 37; 38; 39; 40], is used to limit the generator speed, to achieve an extended constant power range, to eliminate the use of multiple gear ratios, and to reduce the power inverter volt-ampere rating.  
The low-speed shaft torque can be written as  
\[T_{ls}=K_{ls}(\theta_{r}-\theta_{T_{ls}})+B_{ls}(\omega_{r}-\omega_{ls}) \tag{4}\]  
and for an ideal gearbox with a ratio \(n_{g}\), the following relationship holds [41]:  
\[n_{g}=\frac{T_{ls}}{T_{hs}}=\frac{\omega_{g}}{\omega_{ls}}=\frac{\theta_{g}}{ \theta_{ls}} \tag{5}\]  
Using Equations (1)-(5), the nonlinear model of the wind generator can be written in the following form.  
\[\begin{cases}\dot{\omega}_{r}=\frac{1}{\tau}\Big{[}-K_{ls}\theta_{r}+\frac{ K_{ls}}{n_{g}}\theta_{g}-(B_{ls}+B_{r})\omega_{r}+\frac{B_{ls}}{n_{g}}\omega_{g}+T_{a} \Big{]}\\ \dot{\omega}_{g}=\frac{1}{\tau_{g}}\Big{[}\frac{K_{ls}}{n_{g}}\theta_{r}-\frac {K_{ls}}{n_{g}^{2}}\theta_{g}+\frac{B_{ls}}{n_{g}}\omega_{r}-(\frac{B_{ls}}{n _{g}^{2}}+B_{g})\omega_{g}-T_{em}\Big{]}\\ \dot{\beta}=-\frac{1}{\tau}\beta+\frac{1}{\tau}\beta_{d}\end{cases} \tag{6}\]  
where, \(x=[\theta_{r}\;\theta_{g}\;\omega_{r}\;\omega_{g}\;\beta]^{T}\) represents the state vector, and \(u=[T_{em}\;\beta_{d}]^{T}\) is the the control input vector. Note that the torque coefficient \(C_{q}(\lambda,\beta)\) can be approximated as follows [41]:  
\[C_{q}(\lambda,\beta)=\frac{1}{\lambda}\Bigg{[}c_{1}(\frac{c_{2}}{\lambda_{1}} -c_{3}\beta-c_{4})e^{\frac{-c_{5}}{\lambda_{1}}}+c_{6}\lambda\Bigg{]} \tag{7}\]where  
\[\frac{1}{\lambda_{1}}=\frac{1}{\lambda+0.08\beta}-\frac{0.035}{1+\beta^{3}} \tag{8}\]