**8. New Simulation of Induction Machine**

Figs. 33 and 42 show the results of new simulation of the induction machine model illustrat‐ ed in Fig. 43 [1]. The new simulation, which has no limiters and switches, is used because at high grid voltage drop-down or sag, the Simulink induction model does not yield realis‐ tic results.

**Figure 43.** Induction machine Model in *dqo* system

**Figure 41.** Torque-time in per unit while *Vsag*= 50% and υ*w* = 10m/s, *f <sup>s</sup>*= 52

58 Advances in Wind Power

**Figure 42.** Torque-time in per unit while *Vsag*= 50% and υ*w* = 13m/s, *f <sup>s</sup>*= 52 in new simulation of wind generator

The new simulation of induction machine is in *dqo* system and synchronous reference frame simulation on the stator side; *n* (Transfer coefficient) is assumed to be 1. Circuit theory is used in this simulation, and it does not have saturation and switch blocks, unlike the MAT‐ LAB–SIMULINK Induction block. In Fig. 43, *L <sup>M</sup>* is the magnetic mutual inductance, and *r* and*L* are the ohm resistance and self-inductance of the *dqo* circuits, respectively. The ma‐ chine torque is given by equation (19). In this equation, *i <sup>d</sup>* ,*qs*and*λ<sup>d</sup>* ,*qs*, the current and flux pa‐ rameters, respectively, are derived from linear equations (20)–(23); they are sinusoidal because the voltage sources are sinusoidal.

$$T\_a = \left(\frac{3}{2}\right) \left(\frac{P}{2}\right) \left(\mathcal{A}\_{ds}i\_{qs} - \mathcal{A}\_{qs}i\_{ds}\right) \tag{19}$$

Where *P* is poles number, *λds*and *λqs*are flux linkages and leakages, respectively, and *i qs*and *i ds* are stator currents in *q* and *d* circuits of *dqo* system, respectively.

Then *i* matrix produced by the *λ* matrix is given by equation (20).

$$
\begin{bmatrix}
\boldsymbol{\lambda}\_{q\text{obs}} \\
\boldsymbol{\lambda}\_{q\text{obs}}'
\end{bmatrix} = \begin{bmatrix}
\boldsymbol{K}\_{s}\boldsymbol{L}\_{s}\left(\boldsymbol{K}\_{s}\right)^{-1} & \boldsymbol{K}\_{s}\boldsymbol{L}\_{sr}'\left(\boldsymbol{K}\_{r}\right)^{-1} \\
\boldsymbol{K}\_{r}\left(\boldsymbol{L}\_{sr}'\right)^{T}\left(\boldsymbol{K}\_{s}\right)^{-1} & \boldsymbol{K}\_{r}\boldsymbol{L}\_{r}'\left(\boldsymbol{K}\_{r}\right)^{-1}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{i}\_{q\text{obs}} \\
\boldsymbol{i}\_{q\text{obs}}'
\end{bmatrix} \tag{20}
$$

where the inductance matrix parameters are given by (21), (22), (23).

$$\left(K\_s L\_s \left(K\_s\right)^{-1}\right) = \begin{bmatrix} L\_{ls} + L\_M & 0 & 0\\ 0 & L\_{ls} + L\_M & 0\\ 0 & 0 & L\_{ls} \end{bmatrix} \tag{21}$$

$$\left(K\_r L'\_r \left(K\_r\right)^{-1}\right) = \begin{bmatrix} L'\_{lr} + L\_M & 0 & 0\\ 0 & L'\_{lr} + L\_M & 0\\ 0 & 0 & L'\_{lr} \end{bmatrix} \tag{22}$$

$$\left(K\_s L\_w' \left(K\_r\right)^{-1} = K\_r \left(L\_w'\right)^T \left(K\_s\right)^{-1} = \begin{bmatrix} L\_M & 0 & 0\\ 0 & L\_M & 0\\ 0 & 0 & 0 \end{bmatrix} \tag{23}$$

The linkage and leakage fluxes are given by (24) to (29).

$$\mathcal{A}\_{qs} = L\_{ls}\dot{\imath}\_{qs} + L\_{\mathcal{M}} \left(\dot{\imath}\_{qs} + \dot{\imath}'\_{qr}\right) \tag{24}$$

$$\mathcal{A}\_{\rm ds} = L\_{\rm ls} i\_{\rm ds} + L\_{\rm M} \left( i\_{\rm ds} + i\_{\rm dr}' \right) \tag{25}$$

$$\mathcal{A}\_{\alpha\alpha} = L\_{\text{la}} \mathbf{i}\_{\alpha} \tag{26}$$

$$
\lambda\_{qr}' = L\_{lr}'i\_{qr}' + L\_M \left( i\_{qs} + i\_{qr}' \right) \tag{27}
$$

$$\mathcal{X}'\_{dr} = L'\_{lr}\mathbf{i}'\_{dr} + L\_M \left(\mathbf{i}\_{ds} + \mathbf{i}'\_{dr}\right) \tag{28}$$

$$
\mathcal{X}'\_{\alpha r} = L'\_{\text{lr}} \mathfrak{i}'\_{\alpha r} \tag{29}
$$

To create the torque in equation (19), it is necessary to determine the currents in equations (30)–(33) from the stator and rotor currents by using current meters.

$$\mathbf{v}\_{qs} = r\_s \mathbf{i}\_{qs} + \alpha \mathbf{\hat{\lambda}}\_{ds} + \frac{d\mathbf{\hat{\lambda}}\_{qs}}{dt} \tag{30}$$

$$\mathbf{v}\_{\rm ds} = r\_s \mathbf{i}\_{\rm ds} - \alpha \mathbf{\mathcal{A}}\_{qs} + \frac{d \, \mathbf{\mathcal{A}}\_{\rm ds}}{dt} \tag{31}$$

$$\mathbf{v}'\_{qr} = r'\_r \mathbf{i}'\_{qr} + \left(\alpha - \alpha\_r\right) \mathcal{X}'\_{dr} + \frac{d\mathcal{X}'\_{qr}}{dt} \tag{32}$$

$$\mathbf{v}'\_{dr} = r'\_r \mathbf{i}'\_{dr} - \left(\alpha - \alpha\_r\right) \boldsymbol{\lambda}'\_{qr} + \frac{d\boldsymbol{\lambda}\_{dr}}{dt} \tag{33}$$

## **9. Conclusion**

( ) ( )

*KL K KL K i K L K KL K i*

1 1

0 0

0 0

*M*

*L*

*ls*

*lr*

0 0 0 0 0 00

*qs ls qs M qs qr* =+ + *Li L i i* ( ¢ ) (24)

*ds ls ds M ds dr* =+ + *Li L i i* ( ¢ ) (25)

¢ ¢¢ ¢ =+ + *Li L i i* ( ) (27)

¢ ¢¢ ¢ =+ + *Li L i i* ( ) (28)

¢ ¢¢ = *L i* (29)

*os ls os* = *L i* (26)

*L*

é ù

ê ú ë û

*L*

0 0

0 0

é ù + ê ú = +

ë û

é ù ¢ +

¢ ë û

0 0

0 0

(20)

(21)

(22)

(23)


( ) ( ) ( )

1 1 . *qdos ss s s sr r qdos <sup>T</sup> qdor qdor r sr s rr r*

é ù é ù ¢ é ù <sup>=</sup> ê ú ê ú ê ú ¢ ¢ ë û ê ú ¢ ¢ ë û ë û

*ls M ss s ls M*

*lr M rr r lr M*

*T s sr r r sr s M*

ê ú ¢ ¢ = = ê ú

ê ú ¢ ¢ = +

*L L KL K L L*

*L L KL K L L*

where the inductance matrix parameters are given by (21), (22), (23).

( ) <sup>1</sup>

( ) <sup>1</sup>

The linkage and leakage fluxes are given by (24) to (29).

l

l

l

l

l

l*or lr or*

*qr lr qr M qs qr*

*dr lr dr M ds dr*


( ) ( ) ( ) 1 1

*KL K K L K L* - -


l

60 Advances in Wind Power

l

As frequency changes and voltage sag occurs because of turbulence in wind stations in ridethrough faults, the system's set point changes. The theoretical and simulation results results are similar for one mass shaft and two mass shaft turbine models. At lower wind speeds; 6 and 10 m/s, the directions of the changes in the new working point are the same as those of the frequency changes. At a higher wind speed; 13 m/s, the directions of these changes are opposite to the direction of the frequency changes. Simulation results of high grid voltage sag with SIMULINK induction block has error and new simulation of wind induction gener‐ ator in synchronous reference frame is presented without error and in 50% voltage sag, new simulation of wind generator model has higher precision than that in 10% and 20% voltage sags; however, this model can simulate wind generator turbulence with voltage sags higher than 50%. Although results of new simulation of induction machine with wind turbine for 50% voltage sag and frequencies 50 and 52 have been presented in this chapter.

#### **10. Nomenclature**

*P* =Generator power

*ρ* =Air density

*A*=Turbine rotor area

*Cp* =Power Coefficient *υ<sup>w</sup>* =Wind speed *θpitch* =Pitch angle *Te* =Electrical torque *Tm* =Mechanical torque J = Inertia *ω<sup>m</sup>* =Mechanical speed *C* =Drag coefficient *ν* =Gear box ration *R* =Blade radius *Rs* = Stator resistance *L <sup>s</sup>* = Stator inductance *L <sup>m</sup>* = Mutual inductance *L* ′ *<sup>r</sup>* = Rotor inductance *R* ′ *<sup>r</sup>* = Rotor resistance *p* =Pole pairs *κ* =Stiffness *λr*,*<sup>s</sup>* =Rotor and stator flux *Kr*,*<sup>s</sup>* =Rotor and stator park transformation in synchronous reference frame *i <sup>r</sup>*,*<sup>s</sup>* =Rotor and stator current

*vr*,*<sup>s</sup>* =Rotor and stator voltage
