# Wind Turbine

Turbine that converts wind motion into electrical energy

• Library:
• Simscape / Driveline / Engines & Motors

## Description

The Wind Turbine block represents a wind turbine that converts wind motion into mechanical rotational energy. You specify the incident wind velocity and collective blade pitch as inputs, and you can optionally output the thrust acting on the turbine. The block calculates the coefficients of power and torque using a table lookup such that

where:

• βRef is the reference pitch angle.

• λRef is the reference tip speed ratio.

• CP,Ref and CT,Ref are the Power coefficient table and Thrust coefficient table parameters, respectively.

• λSmooth is the smoothed tip speed ratio.

The block uses this equation as the basis for the instantaneous tip speed ratio

`$\lambda =\frac{R\omega }{V},$`

where:

• R is the Turbine radius parameter.

• ɷ is the differential angular velocity between the shaft and the case.

• V is the incident air velocity on the rotor. This value is the physical signal input port V.

The block uses this equation to describe the smoothed version of the instantaneous tip speed ratio equation

`${\lambda }_{Smooth}=\frac{R\omega V}{\left({V}^{2}+{V}_{Thr}^{2}\right)},$`

where VThr is the Velocity threshold parameter. The block uses these equations as a basis for the power and thrust

`$\begin{array}{l}Power=1}{2}{C}_{P}\rho A{V}^{3}=Torque\cdot \omega \\ Thrust=1}{2}{C}_{T}\rho A{V}^{2}\end{array}$`

where:

• ρ is the Air density parameter.

• A is the area of the circle swept by the turbine blades, and A = πr2

To relate the block parameters to the wind turbine mechanical power rating, determine the wind turbine power at the peak power coefficient and the rated wind speed. The rated power corresponds to the block parameters using this equation

`$Powe{r}_{rated}=0.5{C}_{P,max}\rho A{V}_{rated}^{3},$`

where:

• CP,max is the peak power coefficient. This is the maximum value in the Power coefficient table, Cp(β,TSR) parameter.

• Vrated is the rated wind speed. Rated wind speeds are typically 10 to 15 m/s. Wind turbine controller designs may alter strategy at this wind speed to maintain the rated power.

• A is the rotor swept area, where A = πr2.

The block uses numerically smoothed equations for the thrust, power, and torque, such that

`$\begin{array}{l}Powe{r}_{Smooth}=\frac{{C}_{P}\rho A{\left({V}^{2}+{V}_{thr}^{2}\right)}^{3/2}}{2}\\ Torqu{e}_{Smooth}=\frac{Powe{r}_{Smooth}}{\sqrt{{\omega }^{2}+{\left({V}_{Thr}/R\right)}^{2}}}\\ Thrus{t}_{Smooth}=1}{2}{C}_{T}\rho AV\sqrt{{V}^{2}+{V}_{Thr}^{2}}\end{array}$`

When < VThr, the block smoothly saturates the power to a near-zero threshold proportional to VThr3

### Assumptions and Limitations

The block generates torque and power only for positive angular velocities.

## Ports

### Inputs

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Physical signal input associated with the incident wind velocity, in m/s.

Physical signal input associated with the pitch angle of the turbine blades, in deg.

### Outputs

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Physical signal output port associated with the axial force that the wind applies to the turbine blades, in N.

#### Dependencies

To enable this port, select Model thrust.

### Conserving

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Mechanical rotational conserving port associated with the wind turbine shaft.

Mechanical rotational conserving port associated with the wind turbine casing.

## Parameters

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Whether to model the thrust forces that the turbine encounters. Selecting this parameter enables port T.

Distance from the turbine hub center to the blade tips.

Reference collective blade pitch angle. The length of this vector defines the number of rows is the Power coefficient table, Cp(β,TSR) and Thrust coefficient table, Ct(β,TSR) parameters.

Reference tip speed ratio (TSR). TSR is the ratio of the blade tip speed to wind speed. The length of this vector defines the number of columns in the Power coefficient table, Cp(β,TSR) and Thrust coefficient table, Ct(β,TSR) parameters. The block supports negative TSR values. The values must be strictly monotonically increasing.

Power coefficients for a given pitch angle and tip speed ratio. Each row corresponds to an element in the Pitch angle vector, β parameter, and each column corresponds to an element in the Tip speed ratio vector, TSR parameter. The default parameter value is ```[0.0000, 0.0161, 0.1446, 0.3865, 0.5009, 0.4404, 0.2545, 0.0002, -0.1384; 0.0016, 0.0173, 0.1079, 0.2676, 0.3779, 0.4111, 0.3838, 0.3176, 0.2753; 0.0027, 0.0197, 0.1151, 0.2606, 0.3469, 0.3558, 0.3069, 0.2222, 0.1718; 0.0054, 0.0283, 0.1315, 0.2364, 0.2589, 0.2022, 0.0929, -0.0455, -0.1204; 0.0109, 0.0536, 0.1320, 0.1256, 0.0120, -0.1752, -0.4017, -0.6435, -0.7655; -0.01 * ones(1, 9)]```.

Thrust coefficients for a given pitch angle and tip speed ratio. Each row corresponds to an element in the Pitch angle vector, β parameter, and each column corresponds to an element in the Tip speed ratio vector, TSR parameter. The default parameter value is ```[0.0000, 0.2451, 0.6740, 0.9616, 1.0000, 0.9882, 0.8430, 0.0002, -0.1341; 0.0016, 0.2541, 0.5942, 0.8582, 0.9561, 0.9754, 0.9599, 0.9092, 0.8669; 0.0027, 0.2705, 0.6113, 0.8503, 0.9339, 0.9407, 0.8993, 0.8016, 0.7239; 0.0054, 0.3215, 0.6475, 0.8205, 0.8482, 0.7727, 0.5565, -0.0450, -0.1171; 0.2035, 0.4335, 0.6485, 0.6348, 0.2129, -0.1684, -0.3701, -0.5712, -0.6681; -0.05 * ones(1, 9)] ```.

#### Dependencies

To enable this parameter, select Model thrust.

Constant density of air.

Threshold velocity smoothing location. This parameter smooths the torque, thrust, and power when the rotational speed approaches or crosses 0. The block applies more smoothing over greater velocity ranges as you increase the value of this parameter.

## References

[1] Manwell, J. F., J. G. McGowan, and A. L. Rogers. Wind Energy Explained: Theory, Design and Application. 1st ed. Wiley, 2009. https://doi.org/10.1002/9781119994367.

## Version History

Introduced in R2022b