**3. One-Mass Shaft Wind Station Model**

Induction machine equation is

1 <sup>3</sup>

 u

*p w* (1)

<sup>9</sup> <sup>7</sup> <sup>3</sup> <sup>8</sup> 1

æ ö - - ç ÷ <sup>+</sup> <sup>+</sup> è ø

*pitch pitch <sup>c</sup> <sup>c</sup> <sup>c</sup>*

q

(2)

1 5

1

æ ö - + ç ÷

è ø (3)

= (4)

l

l q

2 *P AC* = r

The *C <sup>p</sup>* curve and equation are shown in Fig. 1 and given by equation (2) and (3)

1 2 34 6 9 3

ç ÷ - + + è ø è ø

0.44 125( 0.002) 6.94 *C e <sup>p</sup>*

l

æ ö ç ÷

*C cc c c ce c*

<sup>=</sup> -- - æ ö

8

*c*

lq

rameters are given in Table 1.

32 Advances in Wind Power

where *R* is blade radius.

**Figure 1.** Curve of *Cp* for different tip speed ratios λ .

1

1

1

*pitch pitch*

 q

*p pitch pitch*

where*ρ* is air density, *A*is area of turbine, *Cp*is power coefficient and *υw*is wind speed.

q

è ø æ ö = + -× ç ÷

where*θpitch* is blade pitch angle, *λ*is the tip speed ratio described by equation (4). The pa‐

*M w R v* w l

 q

1 16.5 0.002

$$T\_e - T\_m = J \frac{d\,\phi\_m}{dt} + C\phi\_m \tag{5}$$

Where, *Tm*is the mechanical torque, *Te*is the generator torque, *C*is the system drag coeffi‐ cient and *J* is the total inertia.

Table 1 shows the parameters of the one-mass shaft turbine model and induction generator.


**Table 1.** Parameters of one- mass shaft turbine model and generator.

#### **4. Two-Mass Shaft Induction Machine Model**

This model is used to investigate the effect of the drive train or two-mass shaft, i.e., the masses of the machine and the shaft, according to the equation (8) [3], [4]. In this equation,*Jt* is wind wheel inertia, *JG*is gear box inertia and generator's rotor inertia connected through the elastic turbine shaft with a *κ* as an angular stiffness coefficient and *C* as an angular damping coefficient.

The angular shaft speed *ωt* can be obtained from equations (6) and (7) [1], [3], [4].

*TG*is the torque of the machine, *Tt*is the turbine torque, *δt*is the angular turbine shaft angle, *δG*is the angular generator shaft angle, *ν*is the inverse of the gear box ratio and *JG*and*Jt* are the inertia of the machine shaft and turbine shaft, respectively.

The Parameters, defined above, are given in Table 2.

This model is described as equation (8).

$$T\_G = J\_G \frac{d\phi\_G}{dt} - \frac{\kappa}{\nu} (\delta\_\iota - \delta\_{G\mathbb{B}}) - \frac{C}{\nu} (\alpha\_\iota - \alpha\_{G\mathbb{B}}) \tag{6}$$

$$T\_t = J\_t \frac{d\phi\_t}{dt} + \kappa(\delta\_t - \delta\_{GB}) + C(\phi\_t - \phi\_{GB}) \tag{7}$$

$$
\begin{pmatrix} \phi\_{G} \\ \dot{\phi}\_{\iota} \\ \phi\_{\iota} \\ \phi\_{\iota} \\ \phi\_{\iota} \end{pmatrix} = \begin{pmatrix} -\nu^{2}.C & -\nu.C & -\nu^{2}.\kappa & \nu\kappa \\ \hline J\_{G} & -J\_{G} & -J\_{G} & J\_{G} \\ \hline J\_{\iota} & -C & -\nu\kappa & -\kappa \\ J\_{\iota} & J\_{\iota} & -J\_{\iota} & J\_{\iota} \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} \phi\_{G} \\ \phi\_{\iota} \\ \delta\_{G} \\ \delta\_{\iota} \\ \delta\_{\iota} \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ \overline{J\_{G}} & 1 \\ 0 & \overline{J\_{\iota}} \\ 0 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} T\_{G} \\ T\_{\iota} \end{pmatrix} \tag{8}
$$


**Table 2.** Parameters of two-mass shaft model.

#### **5. Induction Machine and Kloss Theory**

In a single-fed induction machine, the torque angular speed curve of equation (12) [1] is nonlinear, but by using the Kloss equation (13), equations (9), (10), and (11), this curve is lin‐ early modified [1], [2] as shown in Fig. 2. Therefore, the effect of frequency changes in wind power stations can be derived precisely by equation (12) and approximately using equation (13), as shown in Figs. 2–6.

$$G = \pm \sqrt{\frac{\left(\frac{f\_s}{f\_b}\right)^2 \left(R\_s^{\prime 2} + X\_{ss}^{\prime 2}\right)}{\left(X\_m^{\prime 2} - X\_{ss}^{\prime} X\_{rr}^{\prime}\right)^2 \left(\frac{f\_s}{f\_b}\right)^2 + R\_s^{\prime 2} X\_{rr}^{\prime 2}}}\tag{9}$$

$$\mathbf{s}\_k = \mathbf{R}\_r' \mathbf{G} \tag{10}$$

$$T\_k = \frac{\frac{\int\_s}{f\_b} X^2 \,\_mGV\_s^{\; 2}}{\left(R\_s + G\left(\frac{f\_s}{f\_b}\right)^2 \left(X\_m \,^2 - X\_{ss}X\_{rr}'\right)\right)^2 + \left(\frac{f\_s}{f\_b}\right)^2 \left(X\_{ss} + GR\_sX\_{rr}'\right)^2} \tag{11}$$

$$T\_s = \frac{\frac{f\_s}{f\_b} X\_m^{\prime 2} R\_s^{\prime} s V\_s^{\prime 2}}{\left(R\_s + G\left(\frac{f\_s}{f\_b}\right)^2 \left(X\_m^{\prime 2} - X\_{ss} X\_m^{\prime}\right)\right)^2 + \left(\frac{f\_s}{f\_b}\right)^2 \left(X\_m + GR\_s X\_m^{\prime}\right)^2} \tag{12}$$

$$T\_e = 2T\_k \frac{s}{s\_k}; \text{ s$$

**Figure 2.** Electrical torque (nonlinear and linear) versus speed (slip).

the elastic turbine shaft with a *κ* as an angular stiffness coefficient and *C* as an angular

*TG*is the torque of the machine, *Tt*is the turbine torque, *δt*is the angular turbine shaft angle, *δG*is the angular generator shaft angle, *ν*is the inverse of the gear box ratio and *JG*and*Jt* are

( )( ) *<sup>G</sup>*

( )( ) *<sup>t</sup>*

.. . 1 0

<sup>1</sup> . <sup>0</sup>

*C C T*

 k

*J J JJ T*

υ 1/80 *JG* **[kg.m2]** .5 *Jt* **[kg.m2]** 1 *C* **[Nm/rad2]** 1e6 κ **[Nm/rad]** 6e7

In a single-fed induction machine, the torque angular speed curve of equation (12) [1] is nonlinear, but by using the Kloss equation (13), equations (9), (10), and (11), this curve is lin‐ early modified [1], [2] as shown in Fig. 2. Therefore, the effect of frequency changes in wind power stations can be derived precisely by equation (12) and approximately using equation

*t t G*

ç ÷ æ ö æ ö æ ö ç ÷ ç ÷ ç ÷ ç ÷ --- æ ö ç ÷ ç ÷ = + ç ÷ ç ÷ç ÷ ç ÷ ç ÷ ç ÷ ç ÷è ø ç ÷ ç ÷ ç ÷ ç ÷ç ÷ ç ÷ è ø è ø ç ÷

*J J JJ <sup>J</sup>*

nk

 n k nk

*G t t tt G t*

è ø è ø

 n

> w w

 w

 w

& (8)

 d

 d

 ww

= - -- - (6)

0 0

*t*

*J*

*G*

0 0

(7)

*G G t GB t GB*

*t t t GB t GB*

= +-+ kd d

dd

*<sup>d</sup> <sup>C</sup> T J*

k

n

*<sup>d</sup> T J <sup>C</sup>*

1 0 00

*t t*

0 1 00

*dt* w

*dt* w

2 2

*C C*

 n

n

n

w

&

w

w

w

**Table 2.** Parameters of two-mass shaft model.

(13), as shown in Figs. 2–6.

**5. Induction Machine and Kloss Theory**

*G G G G GG*

æ ö - --

The angular shaft speed *ωt* can be obtained from equations (6) and (7) [1], [3], [4].

the inertia of the machine shaft and turbine shaft, respectively.

The Parameters, defined above, are given in Table 2.

This model is described as equation (8).

damping coefficient.

34 Advances in Wind Power

Equations (11) and (12) are given in per unit, but the associated resistances are in ohms.

**Figure 3.** Mechanical and linear electrical torque versus slip.

**Figure 4.** Mechanical and electrical torque versus frequency curves per unit with *V*sag = 10% .

**Figure 5.** Mechanical and electrical torque versus frequency per unit with *V*sag = 20%.

Equations (11) and (12) are given in per unit, but the associated resistances are in ohms.

**Figure 3.** Mechanical and linear electrical torque versus slip.

36 Advances in Wind Power

**Figure 4.** Mechanical and electrical torque versus frequency curves per unit with *V*sag = 10% .

**Figure 6.** Mechanical and electrical torque versus frequency per unit with *V*sag = 50%.

Figs. 3, 4, 5, and 6 illustrate that for lower wind speeds of 6 and 10 m/s, as the synchronous frequency *f <sup>s</sup>* and *V* sag change, the *T <sup>e</sup>* and *Tm* values of the rotor change in the same direction as the frequency of the network, as shown in Tables III, IV, V, and VI. These figures and ta‐ bles give the results for *V* sag = 0% (i.e., only the frequency changes), 10%, 20%, and 50%. However, for a higher wind speed of 13 m/s, as *f <sup>s</sup>* and *V* sag change, the *T <sup>e</sup>* and *Tm* values of the rotor change in the opposite direction to the changes in the frequency of the network.

For small changes in the slip according to the Kloss approach in equation (13), the torque changes as follows [2]:

$$T\_{m1} = T\_{m0} + K\_\alpha \Delta \phi \tag{14}$$

Then:

$$T\_{m1} = 2\frac{T\_k}{s\_k} \left( 1 - \frac{\alpha \rho\_{m0} + \Delta \alpha \rho}{\alpha \rho\_{\prec}} \right) \tag{15}$$

and

$$K\_a = \frac{\partial T}{\partial \alpha} = \frac{\partial T}{\partial \lambda} \cdot \frac{\partial \lambda}{\partial \alpha} \tag{16}$$

or

$$K\_a = \frac{1}{\alpha\_{M0}} \left( \frac{1}{2} \rho R^4 \upsilon\_{a0} \frac{\partial C\_p}{\partial \mathcal{L}} \Big|\_{\lambda\_0, \nu\_0} - T\_{M0} \right) \tag{17}$$

Thus, the new angular operation speed[2] is

$$
\Delta \phi = \frac{-T\_{m0} + 2\frac{T\_k}{s\_k} - 2\frac{T\_k}{s\_k}\frac{\phi\_{m0}}{\phi\_e}}{k\_a + 2\frac{T\_k}{s\_k\phi\_e}}\tag{18}
$$


**Table 3.** Analytical MATLAB results for different frequencies.


**Table 4.** Analytical MATLAB results for *Vsag*= 10%.

For small changes in the slip according to the Kloss approach in equation (13), the torque

a w

0

è ø

w

æ ö + D = - ç ÷

*k m*

*e*

w

 w

l

0 0

*M*

0

w

w

0 ,0

 ln

*k km*

*k ke k k e*

w

ω*<sup>m</sup> pu Te pu* ω*<sup>m</sup> pu Te pu* ω*<sup>m</sup> pu Te pu* .96050 -.1157 1.0005 -.1064 1.0405 -.0974 .9621 -.5337 1.0021 -.491 1.0421 -.4493 .9631 -.7863 1.0035 -.8122 1.0439 -.8331

*p*

*Rv T*

*C*

 l

¶ ¶¶ = = × ¶ ¶¶ (16)

è ø ¶ (17)

(14)

(15)

(18)

*TTK m m* 1 0 = +D

<sup>1</sup> 2 1

*k*

*T T <sup>K</sup>*

w

4

w

<sup>0</sup> 2 2

+

υ*<sup>w</sup> f <sup>s</sup>*= 48 *f <sup>s</sup>* = 50 *f <sup>s</sup>*= 52

*T T <sup>T</sup> s s <sup>T</sup> <sup>k</sup> s*

a


2

æ ö ¶ <sup>=</sup> ç ÷ -

r

*m*

 lw

a

*M*

w

w

D =

*K*a

Thus, the new angular operation speed[2] is

**Table 3.** Analytical MATLAB results for different frequencies.

*m*

*<sup>T</sup> <sup>T</sup> s*

changes as follows [2]:

38 Advances in Wind Power

Then:

and

or


**Table 5.** Analytical MATLAB results for *Vsag*= 20%


**Table 6.** Analytical MATLAB results for *Vsag*= 50%

#### **6. Simulation of wind generator with frequency change**

During turbulence and changes in the grid frequency, the torque speed (slip) curves change in such a way that as the frequency increases, the torque is increased at low wind speeds; 6 and 10 m/s, in contrast to Fig. 6 and decreases at a high speed of 13 m/s [2], as shown in Table 7 and Figs. 7–15.


**Table 7.** Simulink simulation results for one- and two-mass shaft models

Figs. 7–15 show the electrical torque and mechanical speed of the induction machine for the one- and two-mass shaft turbine models at wind speeds of 6, 10, and 13 m/s to vali‐ date Table 7.

**Figure 7.** Electrical torque when = 48 and = 6m/s.

**Figure 8.** Electrical torque when *f <sup>s</sup>*= 50 and υ*w*= 6m/s.

**Figure 9.** Electrical torque when *f <sup>s</sup>*= 52 and υ*w*= 6m/s.

Figs. 7–15 show the electrical torque and mechanical speed of the induction machine for the one- and two-mass shaft turbine models at wind speeds of 6, 10, and 13 m/s to vali‐

date Table 7.

40 Advances in Wind Power

**Figure 7.** Electrical torque when = 48 and = 6m/s.

**Figure 8.** Electrical torque when *f <sup>s</sup>*= 50 and υ*w*= 6m/s.

**Figure 10.** Electrical torque when *f <sup>s</sup>*= 48 and υ*w*= 10m/s.

**Figure 11.** Electrical torque when *f <sup>s</sup>*= 50 and υ*w*= 10m/s.

**Figure 12.** Electrical torque when *f <sup>s</sup>*= 52 and υ*w*= 10m/s.

**Figure 13.** Electrical torque when *f <sup>s</sup>*= 48 and υ*w*= 13m/s.

**Figure 11.** Electrical torque when *f <sup>s</sup>*= 50 and υ*w*= 10m/s.

42 Advances in Wind Power

**Figure 12.** Electrical torque when *f <sup>s</sup>*= 52 and υ*w*= 10m/s.

**Figure 14.** Electrical torque when *f <sup>s</sup>*= 50 and υ*w*= 13m/s.

**Figure 15.** Electrical torque when *f <sup>s</sup>*= 52 and υ*w*= 13m/s.
