11. Mechanical subsystem modeling

The wind turbine mechanical subsystem is known as the drive train. It comprises of a blade pitching component, a hub with blades, a rotor shaft and a gearbox (Figure 7).

The mechanical model of the wind turbine will be modeled based on two lumped masses assumptions: the gear box mass and the wind wheel mass.

The induction generator equation of motion as given [35] is defined as:

$$H\_{\rm g} \times \frac{dW\_{\rm g}}{dt} = T\_{\rm e} + \frac{T\_m}{n} \tag{13}$$

where Te is Electromagnetic torque,Tm is Mechanical torque,Tw is wind torque.

Since the wind turbine shaft and generator are linked utilizing a gearbox, the shaft of the turbine is not viewed as stiff. Hence there will be movement in the shaft. The equation of motion of the drive train shaft is computed as

$$H\_m \times \frac{d\mathcal{W}\_m}{dt} = T\_w - T\_m \tag{14}$$

Figure 7. Mechanical subsystem of a wind turbine.

ρ = the density of air.

level (H) as shown in Equation (5).

Wind Solar Hybrid Renewable Energy System

Density of air can be conveyed as a component of the turbine's rise above sea

where ρ<sup>0</sup> = 1.225 kg/m3 which is the air density at sea level at temperature

Cp = 0.593 (Betz law). The rotor power coefficient is a function of both the tip speed ratio "λ" and the blade pitch angle "β" (in degrees). The blade pitch angle is characterized as the angle between the blade cross-area and the plane of rotation. It alludes to changing the attack angle to best suited angles to adjust the rotation speed

> <sup>λ</sup> <sup>¼</sup> wm � <sup>R</sup> v

<sup>¼</sup> Cpð Þ� <sup>λ</sup>; <sup>β</sup> <sup>1</sup>

<sup>¼</sup> Cpð Þ� <sup>λ</sup>; <sup>β</sup> <sup>1</sup>

The power coefficient Cp can be expressed as shown in Eq. (11).

1 y

wm is angular velocity of the rotor, R blade radius, and "wm\*R" is the blade tip speed.

T = 298 K. The power extracted from wind is defined as [32],

of the blades hence adjusting generated power. Tip speed ratio is defined in Eq. (7),

The rotor torque is therefore defined as,

And, A, the area covered by the blade

Tw <sup>¼</sup> PBLADE wm

Substitute Eq. (9) into Eq. (8), giving Eq. (10).

Tw <sup>¼</sup> PBLADE wm

Cpð Þ¼ λ; β c<sup>1</sup> � c<sup>2</sup> �

1 <sup>y</sup> <sup>¼</sup> <sup>1</sup>

where gamma "y" is given as [32],

0.4; c4 is 5; c5 is 21; and c6 is 0.0068.

(Eq. (10)).

54

PBLADE ¼ Cpð Þ� λ; β Pw ¼ Cpð Þ� λ; β

<sup>ρ</sup> <sup>¼</sup> <sup>ρ</sup><sup>0</sup> � <sup>1</sup>:<sup>194</sup> � <sup>10</sup>�<sup>4</sup> � <sup>H</sup> (5)

1 2

<sup>2</sup> � <sup>v</sup><sup>3</sup> � <sup>A</sup> � <sup>ρ</sup>

<sup>2</sup> � <sup>v</sup><sup>3</sup> � <sup>π</sup> � <sup>R</sup><sup>2</sup> � <sup>ρ</sup>

<sup>A</sup> <sup>¼</sup> <sup>π</sup> � <sup>R</sup><sup>2</sup> (9)

� e �c6

<sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>3</sup> (12)

wm

wm

� <sup>c</sup><sup>3</sup> � <sup>β</sup> � <sup>c</sup><sup>4</sup> � <sup>β</sup><sup>x</sup> � <sup>c</sup><sup>5</sup> 

<sup>λ</sup> <sup>þ</sup> <sup>0</sup>:08<sup>β</sup> � <sup>0</sup>:<sup>035</sup>

where c1–c6 are the aerodynamic coefficients given as c1 is 0.5176; c2 is 116; c3 is

The simulation schematic of the aerodynamic modeling is shown in Figure 6. The equations used are (Eq. (7)) for the Lambda, the beta is the pitch angle for the wind turbine blades. The gamma function block uses (Eq. (11)), the power coefficient is modeled using (Eq. (11)) and the wind torque is modeled using

� <sup>v</sup><sup>3</sup> � <sup>A</sup> � <sup>ρ</sup> (6)

(7)

(8)

(10)

<sup>y</sup> (11)

Figure 8. Simulation schematic diagram of the mechanical model.

Tm is given by

$$T\_m = K \times \frac{\theta}{n} + D \times \frac{W\_\text{g} - W\_m}{n} \tag{15}$$

$$\frac{d\theta}{dt} = W\_{\text{g}} - W\_{m} \tag{16}$$

• When streaming toward the network, the stator currents are positive.

Modeling and Simulation of a 10 kW Wind Energy in the Coastal Area of Southern Nigeria…

• The stator and rotor windings are as far as the mutual effect with the rotor is

• The stator openings show no considerable differences of the rotor inductances

• The rotor openings show no considerable variations of the stator inductances

In order to simulate the induction generator in SIMULINK; the three-phase supply is converted into a two-phase supply using the help of "Parks Transformation Matrix" where the flux linkage is taking as a staple variable. After conversion, both phases are

performing power system dynamic studies of induction generator, two models exist;

• A complete model which comprises of electromagnetic transients both in the rotor and the stator circuits, it contains four electromagnetic variables. This is

The modeling of the asynchronous generator can be done by finding an equation that relates Vds, Vqs, the stator direct and quadrature axis voltages, to Ids, Iqs, the stator direct and quadrature axis currents. The 0dq reference outline model takes positive currents while rotating at synchronous speed and can be represented using

Vds ¼ �Rs � Ids <sup>þ</sup> ws � <sup>φ</sup>qs � <sup>d</sup>φds

Vqs ¼ �Rs � Iqs � ws � <sup>φ</sup>ds � <sup>d</sup>φds

φds ¼ Xs � Ids þ Xm � Iqr (17) φqs ¼ Xs � Iqs þ Xm � Iqr (18) φdr ¼ Xr � Idr þ Xm � Ids (19) φqr ¼ Xr � Iqr þ Xm � Iqrs (20)

dt (21)

dt (22)

• The simplified model neglects the stator transients which contains two electromagnetic state variables. This is also known as the third order model.

• Magnetic hysteresis can be neglected as well as saturation effects.

called d-axis and q-axis. The conversion process is not covered here. When

• The real and reactive powers are positive.

DOI: http://dx.doi.org/10.5772/intechopen.85064

with rotor position.

with rotor position.

concerned, are set sinusoidally along the air-gap.

• The stator and rotor windings are symmetrical.

known as the fifth order model.

12.1 Model including stator transients

the below equations [34].

• Magnetic fluxes

• Voltages

57

• Also neglected is the capacitance of all the windings.

where, n is the gear ratio; θ is the angle between the turbine rotor and the generator rotor; Wm is the speed of the turbine; Wg is the speed of the generator; Hm is the turbine inertia constant; Hg is the generator inertia constant; K is drive train stiffness; and D is damping constant [35] (Figure 8).

## 12. Generator model

An induction generator (asynchronous generator) is used in this model. This is because of its high reliability and low maintenance compared to synchronous generators. The power captured by the drive train of the turbine is converted to electrical power which takes the form of an alternating current. The induction generator has three-phase stator armature windings (AS, BS, CS) and three-phase rotor windings (AR, BR, CR). The external stationary part is known as the stator and the rotor the internal rotating part of the generator. The rotor is placed on bearings fixed to the stator. At the point when the wind torque exerted on the rotor is enough to drive it beyond synchronous speed, there is electrical energy generated.

When modeling the induction machine the following assumptions are made as described by the mathematical modeling of induction generator for power systems principles.

Modeling and Simulation of a 10 kW Wind Energy in the Coastal Area of Southern Nigeria… DOI: http://dx.doi.org/10.5772/intechopen.85064


In order to simulate the induction generator in SIMULINK; the three-phase supply is converted into a two-phase supply using the help of "Parks Transformation Matrix" where the flux linkage is taking as a staple variable. After conversion, both phases are called d-axis and q-axis. The conversion process is not covered here. When performing power system dynamic studies of induction generator, two models exist;


#### 12.1 Model including stator transients

The modeling of the asynchronous generator can be done by finding an equation that relates Vds, Vqs, the stator direct and quadrature axis voltages, to Ids, Iqs, the stator direct and quadrature axis currents. The 0dq reference outline model takes positive currents while rotating at synchronous speed and can be represented using the below equations [34].

#### • Magnetic fluxes

$$
\rho\_{ds} = X\_s \times I\_{ds} + X\_m \times I\_{qr} \tag{17}
$$

$$
\rho\_{q\text{s}} = X\_{\text{s}} \times I\_{q\text{s}} + X\_m \times I\_{qr} \tag{18}
$$

$$
\rho\_{dr} = X\_r \times I\_{dr} + X\_m \times I\_{ds} \tag{19}
$$

$$
\rho\_{qr} = X\_r \times I\_{qr} + X\_m \times I\_{qrs} \tag{20}
$$

#### • Voltages

$$V\_{ds} = -R\_s \times I\_{ds} + \omega\_s \times \rho\_{qs} - \frac{d\rho\_{ds}}{dt} \tag{21}$$

$$V\_{q^s} = -R\_s \times I\_{q^s} - \omega\_s \times \varphi\_{ds} - \frac{d\varphi\_{ds}}{dt} \tag{22}$$

Tm is given by

Simulation schematic diagram of the mechanical model.

Wind Solar Hybrid Renewable Energy System

Figure 8.

12. Generator model

generated.

principles.

56

Tm ¼ K �

stiffness; and D is damping constant [35] (Figure 8).

dθ

θ n þ D �

where, n is the gear ratio; θ is the angle between the turbine rotor and the generator rotor; Wm is the speed of the turbine; Wg is the speed of the generator; Hm is the turbine inertia constant; Hg is the generator inertia constant; K is drive train

An induction generator (asynchronous generator) is used in this model. This is

When modeling the induction machine the following assumptions are made as described by the mathematical modeling of induction generator for power systems

because of its high reliability and low maintenance compared to synchronous generators. The power captured by the drive train of the turbine is converted to electrical power which takes the form of an alternating current. The induction generator has three-phase stator armature windings (AS, BS, CS) and three-phase rotor windings (AR, BR, CR). The external stationary part is known as the stator and the rotor the internal rotating part of the generator. The rotor is placed on bearings fixed to the stator. At the point when the wind torque exerted on the rotor is enough to drive it beyond synchronous speed, there is electrical energy

Wg � Wm n

dt <sup>¼</sup> Wg � Wm (16)

(15)

$$V\_{dr} = \mathbf{0} = -R\_r \times I\_{dr} + \boldsymbol{w}\_s \times \boldsymbol{s} \times \boldsymbol{q}\_{qr} - \frac{d\boldsymbol{q}\_{dr}}{dt} \tag{23}$$

$$V\_{qr} = 0 = -R\_r \times I\_{qr} + w\_s \times \mathbf{s} \times \boldsymbol{\varrho}\_{dr} - \frac{d\boldsymbol{\varrho}\_{qr}}{dt} \tag{24}$$

Here, the sub-indexes (s, r) represent the rotor and stator quantities and the subindexes (d, q) represent the d- and q-axis in the synchronous rotating reference. The rotor voltages Vdr and Vqr are equated to zero because current is fed into the stator. The variable ϕ represents the magnetic linkage flux, ws represent the synchronous rotor speed and wg represent the generator rotor speed. The slip of the rotor "s" is given as

$$s = \frac{w\_s - w\_\text{g}}{w\_s} \tag{25}$$

The electrical parameters Rs, Xs, Xm, Rr and Xr represent the stator resistance and reactance, mutual reactance and rotor resistance and reactance, respectively.

The electrical torque is given as

$$T\_e = \wp\_{qr} \times I\_{dr} - \wp\_{dr} \times I\_{qr} \tag{26}$$

River State (N6.67, E8.48). Monthly data averages on wind speed, temperature and humidity for the last 9 years (January 2009–December 2018) measured at a height of 10 m was gotten from the World Weather Center and presented in Figure 10 and

Modeling and Simulation of a 10 kW Wind Energy in the Coastal Area of Southern Nigeria…

At rated wind speed of 3.2 m/s (Table 1) the relationship between power and the speed of turbine is shown in Figure 13. The results of the simulation are as

Tables 1 and 2.

Figure 10.

Figure 9.

59

Simulation schematic diagram of the induction generator subsystem.

14. Simulation result

Average wind speed in Ogoja.

DOI: http://dx.doi.org/10.5772/intechopen.85064

The power generated by the wind turbine is expressed as

$$P = P\_{\text{active}} + Q\_{\text{reactive}} \tag{27}$$

$$P\_{\text{active}} = V\_{ds} \times I\_{ds} + V\_{q^s} \times I\_{q^s} \tag{28}$$

$$Q\_{reactive} = V\_{qs} \times I\_{ds} - V\_{ds} \times I\_{qs} \tag{29}$$

#### 12.2 Model neglecting stator transients

Neglecting the stator transients reduces the overall order of the model and increases the size of the system that can be simulated. In this model, the rate of change of stator flux linkage is dismissed. The terms dϕds/dt and dϕqs/dt in Eqs. (21) and (22) will be neglected. Eqs. (17)–(29) are used in the modeling of the induction generator subsystem. The simulation schematic is shown in Figure 9.

The model is designed using attributes of steady-state power of a turbine. There is infinite drive train stiffness and friction factor and turbine inertia are joined to the turbine. Eq. (30) gives the output power of the turbine,

$$P\_m = C\_p \times (\lambda, \beta) \times \frac{\rho A}{2} \times v\_{wind}^3 \tag{30}$$

where, Pm is mechanical output power (W).

The mechanical power in per unit is expressed in Equation (31).

$$P\_{m\_{-}pu} = k\_p \times C\_{p\_{-}pu} \times v\_{wind\_{-}pu'}^3 \tag{31}$$

where pu is per unit.

## 13. Case study

Wind speed attributes are considered stronger in coastal areas and offshore as stated in Section 2 above. This study utilized data from Ogoja community in Cross Modeling and Simulation of a 10 kW Wind Energy in the Coastal Area of Southern Nigeria… DOI: http://dx.doi.org/10.5772/intechopen.85064

#### Figure 10. Average wind speed in Ogoja.

Vdr <sup>¼</sup> <sup>0</sup> ¼ �Rr � Idr <sup>þ</sup> ws � <sup>s</sup> � <sup>φ</sup>qr � <sup>d</sup>φdr

Vqr <sup>¼</sup> <sup>0</sup> ¼ �Rr � Iqr <sup>þ</sup> ws � <sup>s</sup> � <sup>φ</sup>dr � <sup>d</sup>φqr

Here, the sub-indexes (s, r) represent the rotor and stator quantities and the subindexes (d, q) represent the d- and q-axis in the synchronous rotating reference. The rotor voltages Vdr and Vqr are equated to zero because current is fed into the stator. The variable ϕ represents the magnetic linkage flux, ws represent the synchronous rotor speed and wg represent the generator rotor speed. The slip of the rotor "s" is given as

> <sup>s</sup> <sup>¼</sup> ws � wg ws

reactance, mutual reactance and rotor resistance and reactance, respectively.

Neglecting the stator transients reduces the overall order of the model and increases the size of the system that can be simulated. In this model, the rate of change of stator flux linkage is dismissed. The terms dϕds/dt and dϕqs/dt in Eqs. (21) and (22) will be neglected. Eqs. (17)–(29) are used in the modeling of the induction

The model is designed using attributes of steady-state power of a turbine. There is infinite drive train stiffness and friction factor and turbine inertia are joined to

> 2 � <sup>v</sup><sup>3</sup>

wind (30)

wind\_pu<sup>0</sup> (31)

generator subsystem. The simulation schematic is shown in Figure 9.

Pm <sup>¼</sup> Cp � ð Þ� <sup>λ</sup>; <sup>β</sup> <sup>ρ</sup><sup>A</sup>

Pm\_pu <sup>¼</sup> kp � Cp\_pu � <sup>v</sup><sup>3</sup>

Wind speed attributes are considered stronger in coastal areas and offshore as stated in Section 2 above. This study utilized data from Ogoja community in Cross

The mechanical power in per unit is expressed in Equation (31).

the turbine. Eq. (30) gives the output power of the turbine,

where, Pm is mechanical output power (W).

where pu is per unit.

13. Case study

58

The power generated by the wind turbine is expressed as

The electrical torque is given as

Wind Solar Hybrid Renewable Energy System

12.2 Model neglecting stator transients

The electrical parameters Rs, Xs, Xm, Rr and Xr represent the stator resistance and

Te ¼ φqr � Idr � φdr � Iqr (26)

P ¼ Pactive þ Qreactive (27)

Pactive ¼ Vds � Ids þ Vqs � Iqs (28) Qreactive ¼ Vqs � Ids � Vds � Iqs (29)

dt (23)

dt (24)

(25)

River State (N6.67, E8.48). Monthly data averages on wind speed, temperature and humidity for the last 9 years (January 2009–December 2018) measured at a height of 10 m was gotten from the World Weather Center and presented in Figure 10 and Tables 1 and 2.
