# Wind Turbines Around Cut-In Speed: Startup Optimization and Behavior Analysis Reported to MPP
## 2 The Mathematical Model of the Double-Powered Induction Generator  
The wind turbine operates at the point of maximum power by changing the load on the double-powered induction generator.  
The wind turbine on which the analysis is based is Fuhrhander FL MD 70, equipped with double-powered induction generators, with the following nominal data [31]:  
* Nominal power \(P_{N}=P_{EG}=1.5\) MW;
* Nominal voltage, \(U_{N}\)--690 V;
* Nominal current, \(I_{N}=P_{EG}/3U_{N}=1\),500,000/3 \(\cdot\) 690 = 724.64 A;
* Nominal rotation, \(n_{N}\)--1500 rpm;
* Maximum rotation, \(n_{max}\)--1800 rpm;
* Equivalent moment of inertia, \(J\)--136 [kgm\({}^{2}\)].  
The generator's parameters are established based on the nominal data \(P_{N}\), \(U_{N}\), \(I_{N}\). The short-circuit impedance, \(Z_{sc}\), is calculated from the ratio between the nominal voltage, \(U_{N}\) and the short-circuit current \(I_{sc}\), considering \(I_{sc}\cong 9\cdot I_{N}\).  
\[Z_{sc}=\frac{U_{N}}{9\cdot I_{N}}=\frac{690}{9\cdot 724.64}=0.1058\ [\Omega] \tag{2}\]  
\[Z_{sc}=\sqrt{R_{sc}^{2}+X_{sc}^{2}}=0.1058\ [\Omega] \tag{3}\]  
Short-circuit resistance, \(R_{sc}\), is calculated from short-circuit power [33]:  
\[P_{sc}\cong 0.05\cdot P_{N} \tag{4}\]  
\[3\cdot R_{sc}\cdot I_{N}^{2}=0.05\cdot P_{N}=0.05\cdot 1,500,000=75,000\ [W] \tag{5}\]  
Obtaining \(R_{sc}=4.7610\cdot 10^{-2}\).  
The short-circuit reactance, \(X_{sc}\), is calculated from the short-circuit impedance:  
\[X_{sc}^{2}=Z_{sc}^{2}-R_{sc}^{2}=0.1058^{2}-\left(2.3805\cdot 10^{-2}\right)^{ 2}=1.0627\cdot 10^{-2}\ [\Omega] \tag{6}\]  
\[X_{sc}=X_{1}+X_{2}=\sqrt{1.0627\cdot 10^{-2}}=0.10309\ [\Omega] \tag{7}\]  
\[X_{1}=\omega\cdot L_{1}=X_{2}=\omega\cdot L_{2}=\frac{0.10309}{2}=5.1545\cdot 1 0^{-2}=314\ \mathrm{L} \tag{8}\]  
Short-circuit inductances: \(L_{1}=L_{2}=L=1.6416\cdot 10^{-6}\ [\mathrm{H}]\).  
The magnetization reactance, \(X_{M}\), is calculated using the current from idle operation: \(I_{no\text{-}load}=0.05\cdot I_{N}\).  
\[X_{M}\cong\frac{U_{N}}{I_{no\text{-}load}}\cong\cdot\frac{690}{0.05\cdot 724.64 }=19.044\ [\Omega] \tag{9}\]  
\[X_{M}=\omega\cdot L_{M}=314\cdot L_{M}=19.044\ [\Omega] \tag{10}\]  
Magnetization inductance: \(L_{M}=6.0650\cdot 10^{-2}\ \mathrm{H}\).  
With these parameters of the generator, the stator and rotor equations are:  
\[U_{S}=\left(R_{1}+j\cdot X_{1}\right)\cdot I_{S}+j\cdot X_{M}\cdot\left(I_{R }+I_{S}\right) \tag{11}\]  
\[U_{R}=\left(\frac{R_{2}}{s}+j\cdot X_{2}\right)\cdot I_{S}+j\cdot X_{M}\cdot I _{R} \tag{12}\]where s is the slip, \(s=(n_{1}-n)/n_{1}\), \(n_{1}\) is the speed of the rotating field, and \(n\)--the rotational speed of the rotor.  
The rotor parameters \(R_{2}\) and \(X_{2}\) are reduced to the stator in the form of  
\[R_{2}=k^{2}\cdot R_{2-REAL}\quad X_{2}=k^{2}\cdot X_{2-REAL} \tag{13}\]  
where \(k=N_{1}/N_{2}\), \(N_{1}\) is the number of turns per phase in the stator, and \(N_{2}\) the number of turns per phase in the rotor.  
Rotor tension \(U_{R}\) is obtained from the real rotor tension \(U_{R}\)--\(REAL\) by multiplying by \(k\) and dividing by sliding \(s\):  
\[U_{R}=\frac{k\cdot U_{R-REAL}}{s} \tag{14}\]  
Since the rotor is wound, a rotor tension can be imposed \(U_{R}\) to obtain the desired rotational speed.