# Wind Turbines Around Cut-In Speed: Startup Optimization and Behavior Analysis Reported to MPP
## 5 Mathematical Model of Dual Powered Asynchronous/Synchronous Motor
### Stationary Mode  
Upon reaching the optimal mechanical angular speed \(\omega_{OPTIM}=72,138\) rad/s, at the time moment \(t=60\) s, the system operates in the \(MPP\) of the turbine at the wind speed of 3 m/s. The frequency and voltage being constant, they stabilize at the values:  
\[f=\frac{72.138}{2\cdot\pi}=11.481[\mathrm{Hz}];\ \omega=2\cdot\pi\cdot f=2\cdot\pi \cdot 11.481=72.137\ [\mathrm{Hz}] \tag{110}\]  
\[U_{S}=13.8\cdot f=13.8\cdot 11.481=158.44\ [\mathrm{V}] \tag{111}\]  
the regime becomes stationary. In this case, at constant tension, the equation of motion becomes:  
\[136\frac{d\omega}{dt}\cdot\omega=5.692\cdot 10^{5}[(3/\omega)-1.7488 \cdot 10^{-2}]\cdot e^{-41.495(3/\omega)}\cdot 3^{3}-158.44\cdot 8695.6\cdot \sin\theta\] \[\frac{d\theta}{dt}+\omega+72.137=0\] \[\omega(0)=72.1371\] \[\theta(0)=0.01\]  
obtaining the variation in the power at the generator, Figure 11b:  
\[P_{EG-b}=158.44\cdot 8695.6\cdot\sin\theta \tag{113}\]  
The fluctuations in value are significantly more evident during the process of ramping up the turbine to achieve maximum power, in contrast to the variations observed in a steady state (see Figure 11a,b).  
The oscillations are, in this case, smaller compared to the previous case and the power to the generator is:  
\[\frac{P_{EG-2}}{P_{WT-2}}=\frac{79,135}{65,946}=1.2 \tag{114}\]  
higher than that of the turbine.  
The oscillations are pronounced in both regimes due to the fact that the electrical losses in the winding resistances and in the iron have been neglected. From the power characteristic \(WT\), \(P_{WT}(V,\omega)\), at \(\omega_{OPTIM}\):  
\[P=5.692\cdot 10^{5}[(3/\omega)-1.7488\cdot 10^{-2}]\cdot e^{-41.495(3/ \omega)}\cdot 3^{3} \tag{115}\]  
the power developed by the \(WT\) results:  
\[P_{WT}(3,72.138)=65,946 \tag{116}\]  
The power developed by the \(WT\) is during the process of bringing the \(WT\) to the optimal constant speed, because the wind speed is constant, at 3 m/s. The power output of the generator oscillates around the power developed by \(WT\). However, the power developed by the synchronous motor is variable during the process.  
The variations over time of the mechanical angular speed and load angle, as illustrated in Figure 9, reveal several key aspects:At the beginning of the transient process, the mechanical angular speed, \(\omega\), and the charge angle, \(\vartheta\), vary linearly over time;
* At the end of the transient process, the mechanical angular speed, \(\omega\), and the charge angle, \(\vartheta\), vary oscillating over time;
* The oscillations of the load angle, \(\vartheta\), are more pronounced in cases where the motor is powered at variable voltage and frequency;
* The turbine enhances the fluctuations of the load angle, which are reflected in the variations in power output and current.  
The fluctuations in the load angle, and consequently in the power output, are further intensified by the temporal variations in voltage and frequency, which are influenced by the regulated stator flow. In conclusion, by changing the voltage and frequency of the generator, in synchronous mode, there is a significant disturbance in the Electrical Power System. Consequently, the asynchronous generator regime is used to mitigate these disturbances.  
The modification of the voltage and frequency of the generator is imposed by the wind speed variable in time and the capture of a maximum wind energy at these wind speeds can only be achieved when the _WT_ operates at the point of maximum power, at the optimal mechanical angular speed.  
The wind turbine consistently functions in dynamic mode at variable speeds, adapting to the fluctuations in wind speeds that occur over time.