# A Maximum Power Point Tracking Control Method Based on Rotor Speed PDF Shape for Wind Turbines
## 2 Principle of MPPT Control  
Figure 1 illustrates the schematic diagram of the direct-drive wind energy conversion system. In this system, the prime mover is connected to the shaft of the wind generator's rotor, whereas the stator is linked in either case to the utility grid or standalone loads by a suitable power electronic interface. The MPPT signal is controlled by the wind power generation control system. The wind turbine converts wind energy into mechanical energy and then generates electric energy through the operation of generators. The energy conversion of the former can be realized by the dynamic model of the wind turbine drivetrain, which can be modeled as the two-mass model, as shown in Figure 2. In this simplified mechanical dynamic model, the individual lumped inertia of the wind wheel and the generator is considered, but the damping of the drive shaft is ignored. The shafting dynamic model of wind turbines is simplified as follows:  
\[T_{t}-T_{g}=J_{t}\frac{d\omega_{t}}{dt}+J_{g}\frac{d\omega_{g}}{dt} \tag{1}\]  
This is actually a linear combination of the acceleration at the wind wheel end and generator end. In large-scale wind turbines, the rotational inertia of the generator is much smaller than that of the wind wheel, and the stiffness of the transmission shaft is extremely large. Thus, the shafting dynamic model can be further written as follows:  
\[T_{t}-T_{g}=J\frac{d\omega_{t}}{dt} \tag{2}\]  
where \(J\) is the equivalent moment of inertia of the wind turbine and \(J=J_{t}+J_{g}\). Figure 1: The schematic diagram of direct-drive wind energy conversion system. Figure 2: Two-mass drive-train model. In this model, \(J_{t}\) is the moment inertia of rotor; \(J_{g}\) is the moment inertia of generator; \(\omega_{t}\) is the rotor speed of wind turbines; \(\omega_{g}\) is the mechanical speed of the generator rotor; \(T_{t}\) is the mechanical torque of wind turbines; \(T_{g}\) is the electromagnetic torque of the generator system; \(K_{s}\) represents the equivalent stiffness of the two-mass model. According to aerodynamic analysis, the mechanical power (\(P_{t}\)) captured by the wind turbine from the wind energy is:  
\[P_{t}=0.5\rho\pi R^{2}v^{3}C_{p}(\lambda,\beta) \tag{3}\]  
where \(\rho\) is the air density, \(R\) is the radius of the wind wheel, \(v\) is the incoming wind speed, and \(C_{p}\) is the power coefficient of the wind turbine, which is determined jointly by the blade tip speed ratio (\(\lambda\)) and pitch angle (\(\beta\)). The tip speed ratio is an important parameter affecting blade performance, which is defined as the ratio of the linear velocity of the blade tip to the incoming wind speed, namely:  
\[\lambda=\frac{\omega_{t}R}{v}. \tag{4}\]  
So, the mechanical torque of wind turbines can be expressed:  
\[T_{t}=\frac{P_{t}}{\omega_{t}}=0.5\rho C_{p}(\lambda,\beta)\pi R^{5}\frac{1}{ \lambda^{3}}{\omega_{t}}^{2}. \tag{5}\]  
Generally, \(C_{p}\) is considered a nonlinear function of \(\lambda\) and \(\beta\), which is calculated by empirical formula [17,18]:  
\[\left\{ \begin{array}{l}C_{p}(\lambda,\beta)=c_{1}\Big{(}\frac{c_{2}}{ \lambda_{i}}-c_{3}\beta-c_{4}\beta^{c_{5}}-c_{6}\Big{)}e^{\frac{C_{p}}{\lambda _{i}}}\\ \frac{1}{\lambda_{i}}=\frac{1}{\lambda+c_{8}\beta}-\frac{c_{9}}{\beta^{3}+1} \end{array} \right. \tag{6}\]  
where the coefficients (\(c_{1},c_{2},\cdots,c_{9}\)) are related to the rotor type of wind turbines. When the wind speed changes, MPPT control method is used to change the rotor speed of the generator by variable speed constant frequency devices, so that the wind turbine keeps running at the optimal tip speed ratio (\(\lambda_{opt}\)). That is, \(C_{p}\) is always maintained at or close to its maximum value (\(C_{p\ max}\)) to capture more energy. The OT control method [19] maintains a specific relationship between electromagnetic torque and rotational speed according to the curve of \"\(T_{t}-\omega_{t}\)\" so that the rotor speed tends to its optimal value. When the wind turbine runs at \(\lambda_{opt}\), the power coefficient is \(C_{p\ max}\).