## 3 Mathematical Models of the Induction Generator and the Three-Blade Horizontal Axis Turbine
### The Mathematical Model of the Induction Generator  
An analysis was conducted on wind turbines equipped with induction generators with the following nominal data [1; 31; 41]:  
* The moment of inertia, \(J\) = 0.34 kgm\({}^{2}\);
* The nominal power \(P_{N}\) = \(P_{EG}\) = 55 kW;
* The rated voltage \(U_{N}\) = 400 V;
* The nominal current \(I_{N}\) = 98 A;
* The number of poles \(2p_{1}\) = 2;
* Nominal speed \(n_{N}\) = 2940 rpm;
* The nominal sliding \(s_{N}\) = 0.02. The parameters of the generator are established based on the nominal data, at the nominal frequency \(f_{N}\) = 50 Hz. The nominal operating slip is  
\[s_{N}=\frac{n_{1}-n}{n_{1}}=\frac{3000-2940}{3000}=0.02 \tag{6}\]  
\(n_{1}\) represents the rotational speed of the spinning field, while \(n\) is the rotor's rotational speed. The rotor resistance, denoted as \(R_{R}\), is determined from the power balance equation  
\[s_{N}\cdot P_{N}=3\cdot R_{R}\cdot I_{N}^{2} \tag{7}\]  
or  
\[0.02\cdot 55000=3\cdot R_{R}\cdot 98^{2} \tag{8}\]  
with the solution \(R_{R}=3.8179\times 10^{-2}\)\(\Omega\). The stator resistance is equal to the rotor resistance: \(R_{S}=R_{R}=3.8179\times 10^{-2}\)\(\Omega\). The short-circuit reactance, \(X_{sc}\), is derived from the nominal current, \(I_{N}\), as follows:  
\[I_{N}=\frac{230}{\sqrt{\left(3.8179\cdot 10^{-2}+\frac{3.8179\cdot 10^{-2}}{0.0 2}\right)^{2}+{X_{sc}}^{2}}} \tag{9}\]  
or  
\[98=\frac{230}{\sqrt{\left(3.8179\cdot 10^{-2}+\frac{3.8179\cdot 10^{-2}}{0.0 2}\right)^{2}+{X_{sc}}^{2}}} \tag{10}\]  
with the solution \(X_{sc}=1.31\)\(\Omega\). The rotor parameters \(R_{R}\) and \(X_{R}\) are minimized at the stator in the following manner:  
\[R_{R}=k^{2}\cdot R_{R-REAL};R_{R}=k^{2}\cdot X_{R-REAL} \tag{11}\]  
where \(k=N_{1}/N_{2}\), \(N_{1}\) is the number of turns per phase in the stator, and \(N_{2}\) is the number of turns per phase in the rotor. The magnetizing reactance, \(X_{M}\), is regarded as significantly larger in comparison to the short-circuit reactance. With these parameters of the generator, the stator current \(I_{S}\) and rotor current \(I_{R}\) at the stator voltage \(U_{S}\) are  
\[\underline{I_{S}}=\frac{U_{S}}{R_{S}+\frac{R_{R}}{s}+j\cdot X_{sc}}=\frac{U_{ S}}{3.8179\cdot 10^{-2}+\frac{3.8179\cdot 10^{-2}}{s}+j\cdot 1.31} \tag{12}\]  
\[\underline{I_{R}}=-\underline{I_{S}} \tag{13}\]  
The electromagnetic power of the induction generator at the shaft is  
\[P_{EG}=3\cdot\frac{U_{S}^{2}\left(\frac{R_{R}}{s}\right)}{\left(R_{S}+\frac{R _{R}}{s}\right)^{2}+1.31^{2}}=-\frac{3\cdot 230^{2}\left(\frac{3.8179\cdot 10^{-2}}{ s}\right)}{\left(3.8179\cdot 10^{-2}+\frac{3.8179\cdot 10^{-2}}{s}\right)^{2}+1.31^{2}} \tag{14}\]  
By substituting the slip \(s\) with the mechanical angular velocity \(\omega\), the following results are obtained:  
\[s=\frac{\omega_{1}-\omega}{\omega_{1}}=1-\frac{\omega}{\omega_{1}}=1-\frac{ \omega}{314}=\left(1-3.1847\cdot 10^{-3}\cdot\omega\right) \tag{15}\]  
The power characteristic of the induction generator, \(P_{EG}\), is obtained:  
\[s_{N}\cdot P_{N}=3\cdot R_{R}\cdot I_{N}^{2} \tag{7}\]0  
at \(U_{N}=230\)\(V_{f}=50\) Hz. Figure 4 illustrates the generator's power characteristics. The coordinates of the generator's nominal power point are nominal power \(P_{N}=P_{EG}=55\) kW. The nominal slip is  
\[s_{N}\cdot P_{N}=3\cdot R_{R}\cdot I_{N}^{2} \tag{7}\]1  
the nominal mechanical angular velocity is \(\omega_{N}=320.28\) rad./s. The converter positioned between the generator and the grid facilitates the operation of the wind system across a broad range of speeds, frequencies, and voltages by maintaining a constant steady-state flow:  
\[s_{N}\cdot P_{N}=3\cdot R_{R}\cdot I_{N}^{2} \tag{7}\]2  
The ratio of voltage to frequency serves as an indicator of the saturation level of the magnetic core.