# A Maximum Power Point Tracking Control Method Based on Rotor Speed PDF Shape for Wind Turbines
## 3 MPPT Control Based on PDF Shape
### OT Control Based on PDF Shape  
According to the principle of maximum wind energy capture, the capture of more wind energy is achieved by changing the rotor speed of the generator to keep the wind turbine working at \(\lambda_{opt}\). Therefore, the optimal rotor speed of the wind turbine (\(\omega_{t\ opt}\)) is selected as the tracking goal, the actual \(\omega_{t}\) is selected as the system output, and the electromagnetic torque (\(T_{g}\)) of the generator is used as the controller output. Ignoring the tracking error between \(T_{g}\) and \(T_{g}\)*, let \(x(t)=\omega_{t}(t)\), \(y(t)=\omega_{t}(t)\), \(u(x)=T_{g}\), and the state equation of wind power generation system is established as follows:  
\[\left\{\begin{array}{ll}\frac{dx(t)}{dt}=\frac{\rho\pi R^{5}\ C_{T}(\lambda, \beta)}{\lambda^{2}}x^{2}(t)-\frac{1}{2}u(x)+\varepsilon(t)\\ y(t)=x(t)\end{array}\right. \tag{25}\]  
where \(\varepsilon(t)\) is the interference caused by wind speed uncertainty and other random factors during operation.  
If Equation (25) is written in the form of Equation (12), the partial coefficients of the controller:  
\[a_{0}=a_{1}=0,\ a_{2}=\frac{\rho\pi R^{5}\ C_{T}(\lambda,\beta)}{2J},\ a_{3}=a_{4}= \cdots=a_{r}=0. \tag{26}\]  
Due to the large stiffness of the transmission shaft, \(\omega_{t}=\omega_{g}\) is the default. The controller gain of the system is produced by Equations (24) and (25), which is:  
\[u(x)=I\left\{-c_{1}\omega_{g}+\left(\frac{\rho\pi R^{5}C_{T}\left(\overset{ \wedge}{\lambda},\beta\right)}{2J\overset{\wedge}{\lambda}}-2c_{2}\right) \omega_{g}{}^{2}-3c_{3}\omega_{g}{}^{3}-\cdots-(l+1)c_{l+1}\omega_{g}{}^{l}\right\} \tag{27}\]  
where \(C_{T}(\overset{\wedge}{\lambda},\beta)\). is the estimated value of aerodynamic torque coefficient, and \(C_{T}(\overset{\wedge}{\lambda},\beta)=C_{p}(\overset{\wedge}{\lambda},\beta) /\overset{\wedge}{\lambda}\). In addition, \(dx(t)/dt<0\) can be obtained by substituting Equation (27) into the first-order inertial system of (25), so the stability of the system can be guaranteed, and the convergence of OT method is also verified.  
In the actual control operation, the calculation of some variables is usually replaced by estimated values. Thus, Equation (27) is converted to the following:  
\[\left\{\begin{array}{ll}u(x)=\overset{\wedge}{T}_{t}-J\overset{\wedge}{ \underset{i=0}{\sum}}(i+1)c_{i+1}\omega_{g}{}^{i}\\ \overset{\wedge}{T}_{t}=\frac{\rho\pi R^{5}C_{T}\left(\overset{\wedge}{\lambda },\beta\right)}{2\overset{\wedge}{\lambda}}\omega_{g}{}^{2}\.\end{array}\right. \tag{28}\]  
When the wind turbine system runs in a steady state, it works under the optimal blade tip ratio, that is, \(\overset{\wedge}{T}_{t}=T_{g}{}^{*}=K_{opt}\omega_{g}{}^{2}\). The \(T_{g}{}^{*}\) of the wind turbine system can be determined as follows:  
\[T_{g}{}^{*}=K_{opt}\omega_{g}{}^{2}-J\overset{\wedge}{\underset{i=0}{\sum}}( i+1)c_{i+1}\omega_{g}{}^{i}. \tag{29}\]  
This design still focuses on tracking the maximum power point (MPP), but also takes into account the stochastic distribution of wind speed. This novel control design can track the goal in a small range instead of tracking a certain point, avoiding the damage to the system caused by frequent fluctuation of rotor speed.