# Wind Turbines Around Cut-In Speed: Startup Optimization and Behavior Analysis Reported to MPP
## 4 Operating the Turbine by Switching the Generator to Engine Mode
### Operation at Frequency Equality \(f_{1}\) = \(f_{2}\) = \(f\)  
The stator and rotor are connected to the same frequency network \(f_{1}\). From the slippage definition relationship:  
\[s=\frac{n_{1}-n}{n_{1}} \tag{45}\]  
or:  
\[s=\pm\frac{f_{2}}{f_{1}} \tag{46}\]  
results:  
\[n=n_{1}\bigg{(}1\mp\frac{f_{2}}{f_{1}}\bigg{)} \tag{47}\]  
Take the sign--at the same sequence of the phases in the stator as the phases in the rotor, and at an inverse sequence of phases, we take the + sign. The result is a synchronous motor regime with \(n=0\) at the same phase sequence and \(n=2n_{1}\) at an inverse phase sequence. At the reverse sequence of phases, the rotor speed is \(n=2n_{1}=2f/p_{1}=2f/2=f\); therefore, the \(MAS\)\(\omega\), at the shaft of the electric motor results in the form:  
\[\omega=2\cdot\pi\cdot n=2\cdot\pi\cdot f \tag{48}\]  
The relationship between the voltage time variation \(U(t)\) = \(k\cdot t\) and the supply frequency of the double-powered asynchronous motor must be aligned with the temporal changes in rotational speed to prevent the occurrence of saturation phenomena. Figure 4: Connecting the stator and rotor to the same, time-varying frequency network. Considering that the process takes place at a constant stator flux of value \(\Psi_{S}=2.1963\) Wb and at a load angle, \(\theta\), also constant, \(\theta=30^{\circ}\), the power to be generated by the asynchronous motor, \(P_{ENGINE}\), is given by the equation of motion:  
\[136\frac{d\omega}{dt}\cdot\omega=5.692\cdot 10^{5}\left[\frac{3}{\omega}-1.7488 \cdot 10^{-2}\right]\cdot e^{-41.495\frac{3}{\omega}}\cdot 3^{3}+P_{ENGINE} \tag{49}\]  
It is required that at the wind speed of 3 m/s, the optimal _MAS_\(\omega_{OPTIM}=72,138\) rad/s has to be obtained within a time interval of one minute, resulting in:  
\[\frac{d\omega}{dt}=\frac{72.138}{60}=1.2023\ [\mathrm{rad./s}^{2}] \tag{50}\]  
and the time variation in the MAS, \(\omega(\mathrm{t})=1.2023\) t.  
In this case, the equation of motion becomes:  
\[136\cdot 1.2023\cdot 1.2023\cdot t=5.692\cdot 10^{5}\left[\frac{3}{1.2023 \cdot t}-1.7488\cdot 10^{-2}\right]\cdot e^{-41.495\left(\frac{3}{1.2023 \cdot t}\right)}\cdot 3^{3}+P_{ENGINE} \tag{51}\]  
resulting in the necessary power that the double-powered asynchronous motor must develop:  
\[P_{ENGINE}(t)=136\cdot 1.2023\cdot 1.2023\cdot t-5.692\cdot 10^{5}\left[ \frac{3}{1.2023\cdot t}-1.7488\cdot 10^{-2}\right]\cdot e^{-41.495\left(\frac{3}{1.2023 \cdot t}\right)}\cdot 3^{3} \tag{52}\]  
At the increasing slope of the mechanical angular speed, \(\omega\), of 1.2023 rad/s\({}^{2}\) the motor is energized up to \(t_{M}=15.83\) s, at which point the power developed by the asynchronous motor becomes zero, as seen from Figure 5. This result is also confirmed by solving the equation:  
\[s=\pm\frac{f_{2}}{f_{1}} \tag{46}\]0  
The variations over time of the engine and turbine power outputs are illustrated in \[s=\pm\frac{f_{2}}{f_{1}} \tag{46}\]1 At the time moment \(t_{M}\), the _MAS_\(\omega\) has the value \(\omega(\mathrm{t})=1.2023\cdot 15.83=19.032\) rad/s and, further, the turbine rotor is accelerated only by the action of the wind. The representation of the process is derived from equations characterized by powers of the following form:  
\[s=\pm\frac{f_{2}}{f_{1}} \tag{46}\]2  
or:  
\[ \begin{split} 136\frac{d\omega}{dt}\cdot\omega=5.692\cdot 10^{5} \cdot\left(\frac{3}{\omega}-1.7488\cdot 10^{-2}\right)\cdot e^{-41.495\frac{3}{ \omega}}\cdot 3^{3}\\ \omega(0)=19.032\end{split}  \tag{55}\]  
Given that \(\omega(10.12)=72.148\) rad/s, it can be concluded that the optimal mechanical angular speed, denoted as \(\omega_{OPTIM}\), is attained at 10.12 s, as illustrated in Figure 6.  
Figure 5: Evolution in engine and turbine power over time. At a wind speed of 3 m/s, the optimal mechanical angular speed, \(\omega_{OPTIM}\), is obtained in 25.95 s, lower than initially prescribed.