# Wind Turbine

Turbine that converts wind kinetic energy into rotational motion

Since R2022b

Libraries:
Simscape / Driveline / Engines & Motors

## Description

The Wind Turbine block represents a wind turbine that converts wind motion into mechanical rotational energy. Wind turbines harness wind energy for electricity generation. Wind turbine development focuses on enhancing the efficiency, reliability, and cost-effectiveness of individual turbines, while wind turbine farm development involves the strategic placement of multiple turbines to optimize energy capture. You can use the block to simulate individual wind turbines and entire wind farms. You can analyze the turbine performance, power generation, and interactions in a wind farm, or the effect of different turbine geometries, configurations, control algorithms, and layout designs on wind farm performance and energy output.

You specify the incident wind velocity and collective blade pitch as inputs, and you can optionally output the thrust acting on the turbine. You can include the effects of thrust and inertia. Parameterize the block using tabulated power and thrust coefficients or airfoil lift and drag coefficients.

### Parameterize by Power and Thrust Coefficients

When you set Parameterization to ```Tabulated data for power and thrust coefficients```, the block calculates the coefficients of power and torque using table lookups, 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

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

where VThr is the Wind 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,λ) 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

where ωThr is the Rotational velocity threshold parameter. When ɷ < Thr, the block smoothly saturates the power to zero.

The block asserts Cp(λ=0)≅ 0. Generated power equals zero when the rotor rotational velocity is zero, and a non-zero value of Cp(λ=0) affects the start-up torque. The start-up torque relates to Cp(λ=0) such that

`$Torqu{e}_{Startup}=\frac{\rho A{C}_{P}\left(\lambda =0\right){|V|}^{3}}{2{\omega }_{Thr}}.$`

Your model may be sensitive to this start-up torque behavior if you simulate braking the rotor in strong winds.

### Parameterize by Airfoil Lift and Drag Coefficients

When you set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```, you can parameterize the lift and drag coefficients and the airfoil geometry for a given blade element. The default values represent an NREL 5 MW reference wind turbine. The block treats the propeller as a continuous disc. Conservation of momentum applies to the air that crosses the disc when the block calculates the induced velocity, vi. The block uses the induced velocity to find the magnitude and direction of the total flow velocity at a vector of radial locations along the blade, which it then uses to find lift and drag based on the lift and drag coefficient lookup tables. These quantities are specific to this parameterization:

• TMT — Thrust calculated by momentum theory

• vi — Axial flow velocity induced by the motion of the wind turbine blades

• vax — Axial velocity at the blade location

• TBET — Thrust calculated by blade element theory

• QBET — Torque calculated by blade element theory

• e — Nondimensional location of the root cutout as given by the first element of the Nondimensional radial location vector, r parameter

• Ω — Wind turbine rotational velocity

• Cl,Cd — Element-wise coefficients of the lift and drag, respectively

• ϕ(y) — Flow angle at a given point along the blade

• a = -vi/v — Axial induction factor

• a'=ω/2R — Angular induction factor

• ${\lambda }_{r}=\frac{Ry\Omega }{\sqrt{{v}^{2}+{v}_{Thr}^{2}}}$ — Smoothed local tip speed ratio at each blade element

The block uses momentum theory to define a smoothed thrust equation such that

`${T}_{MT}=\frac{{C}_{T}\rho \pi {R}^{2}{\left({v}^{2}+{v}_{Thr}^{2}\right)}^{2}}{2},$`

where the block uses the Glauert correction in the turbulent wake state when a > 0.4, such that

`${C}_{T}=\left\{\begin{array}{cc}4a\left(1-a\right)\cdot \mathrm{sign}\left(v\right)& a\le 0.4\\ \left(\frac{8}{9}±\frac{4}{9}a+\frac{14}{9}{a}^{2}\right)\cdot \mathrm{sign}\left(v\right)& a>0.4\end{array}$`

The smoothed axial induction factor is

`$a=\frac{-{v}_{i}\left(v+\text{sign}\left(v\right)\cdot \left(1e-6\right)/{v}_{Thr}\right)}{{v}^{2}+{v}_{Thr}^{2}}.$`

The block interpolates the values from the Nondimensional radial location vector, r parameter to find y. Then the block interpolates the lift and drag coefficients to find Cl(y) and Cd(y) based upon the tabulated angle of attack and lift and drag coefficients. The block uses blade element theory to calculate the thrust and torque such that

`$\begin{array}{l}{T}_{BET}=\frac{{N}_{blades}\rho {D}^{2}}{4}\underset{e}{\overset{1}{\int }}\left({v}_{r}^{2}+{v}_{ax}^{2}\right)\cdot \left({C}_{l}\mathrm{cos}\varphi \left(y\right)+{C}_{d}\mathrm{sin}\varphi \left(y\right)\right)\cdot c\mathrm{dy}\\ {Q}_{BET}=\frac{{N}_{blades}\rho {D}^{3}}{8}\underset{e}{\overset{1}{\int }}\left({v}_{r}^{2}+{v}_{ax}^{2}\right)\cdot \left({C}_{l}\mathrm{sin}\varphi \left(y\right)+{C}_{d}\mathrm{cos}\varphi \left(y\right)\right)\cdot cy\mathrm{dy}\end{array}$`

where:

`$\begin{array}{c}{v}_{r}=Ry\Omega \left(1+a\text{'}\right)\\ {v}_{ax}=v\left(1-a\right)\end{array}$`

The block performs this integration across the each discrete blade element. The block discretizes y according to the specification in the Number of blade elements parameter and calculates the angular induction factor at each blade element as

`$a\text{'}=-\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{4}{{\lambda }_{r}^{2}}a\left(1-a\right)}.$`

### 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 Output thrust.

### Conserving

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Mechanical rotational conserving port associated with the wind turbine shaft.

## Parameters

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### Wind Turbine

Whether to parameterize the wind turbine by thrust and power coefficients or airfoil lift and drag coefficients.

#### Dependencies

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 in the Power coefficient table, Cp(β, λ) and Thrust coefficient table, Ct(β, λ) parameters.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for power and thrust coefficients```.

Reference tip speed ratio (λ). λ 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, λ) and Thrust coefficient table, Ct(β, λ) parameters. The block supports negative λ values. The values must be strictly monotonically increasing.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for power and thrust coefficients```.

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, λ parameter. The default parameter value is ```[0.0010, 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)]```. The block asserts Cp(λ=0)≅ 0.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for power and thrust coefficients```.

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, λ 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, set Parameterization to ```Tabulated data for power and thrust coefficients``` and select Output thrust.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

Radial location for a given set of blade dimensions. `1` is equivalent to the radius of the blade. The first element of this vector defines e, the root cutout.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

Blade twist angles, θ, for a given radial location. The elements of this vector correspond one-to-one with the Nondimensional radial location vector, r parameter.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

Chord length normalized by radius for a given radial location along the blade. The elements of this vector correspond one-to-one with the Nondimensional radial location vector, r parameter and the columns of the Airfoil lift coefficient table, Cl(α,r) and Airfoil drag coefficient table, Cd(α,r) parameters.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

Angle of attack range. The elements of this vector correspond one-to-one with the rows of the Airfoil lift coefficient table, Cl(α,r) and Airfoil drag coefficient table, Cd(α,r) parameters.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

Airfoil lift coefficients for a given angle of attack and radial location along the blade. The rows in this matrix correspond one-to-one with the Airfoil angle of attack vector, α parameter. The columns in this matrix correspond one-to-one with the Nondimensional radial location vector, r parameter.

The default value is ```[0, 0, 0, 0, 0, 0, 0; 0, .397, .547, .547, .735, .788, .749; 0, .642, .685, .685, .695, .67, .659; 0, .757, .816, .816, .828, .797, .783; 0, .762, .832, .832, .846, .813, .798; 0, .68, .756, .756, .771, .739, .724; 0, .532, .609, .609, .624, .596, .581; 0, .337, .411, .411, .426, .403, .39; 0, .114, .182, .182, .195, .179, .169; 0, -.12, -.061, -.061, -.05, -.06, -.067; 0, -.349, -.302, -.302, -.294, -.295, -.299; 0, -.557, -.523, -.523, -.518, -.512, -.512; 0, -.727, -.708, -.708, -.706, -.693, -.689; 0, -.842, -.838, -.838, -.839, -.82, -.814; 0, -.886, -.895, -.895, -.898, -.875, -.866; 0, -.839, -.858, -.858, -.862, -.838, -.829; 0, -.685, -1.013, -1.013, -.815, -.869, -.958; 0, -.311, -.8496, -.8496, -.8284, -.8284, -.711; 0, .137, .288, .288, .444, .521, .442; 0, 1.368, 1.458, 1.458, 1.442, 1.358, 1.382; 0, 1.7825, 1.398, 1.398, 1.354, 1.311, 1.428; 0, 1.904, 1.265, 1.265, 1.076, .962, .926; 0, 1.903, 1.258, 1.258, 1.064, .95, .804; 0, 1.69, 1.146, 1.146, .98, .884, .763; 0, 1.323, .932, .932, .81, .74, .656; 0, .88, .657, .657, .582, .54, .495; 0, .449, .362, .362, .326, .304, .291; 0, .124, .092, .092, .072, .053, .053; 0, -.118, -.15, -.15, -.17, -.198, -.199; 0, -.348, -.379, -.379, -.399, -.434, -.436; 0, -.549, -.578, -.578, -.596, -.637, -.64; 0, -.702, -.727, -.727, -.743, -.787, -.79; 0, -.787, -.807, -.807, -.821, -.864, -.868; 0, -.782, -.797, -.797, -.806, -.847, -.85; 0, -.664, -.673, -.673, -.679, -.711, -.714; 0, -.41, -.547, -.547, -.735, -.788, -.749; 0, 0, 0, 0, 0, 0, 0]```.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

Airfoil drag coefficients for a given angle of attack and radial location along the blade. The rows in this matrix correspond one-to-one with the Airfoil angle of attack vector, α parameter. The columns in this matrix correspond one-to-one with the Nondimensional radial location vector, r parameter.

The default value is ```[.35, .0602, .0267, .0267, .0202, .0185, .0198; .35, .1107, .0968, .0968, .0943, .0945, .0955; .35, .3045, .2876, .2876, .2848, .2809, .2807; .35, .5355, .5232, .5232, .5215, .5112, .5086; .35, .7685, .7656, .7656, .766, .7485, .7427; .35, .9788, .9882, .9882, .9911, .9665, .9574; .35, 1.1499, 1.173, 1.173, 1.1787, 1.1476, 1.1355; .35, 1.2716, 1.3084, 1.3084, 1.3168, 1.2805, 1.2656; .35, 1.3378, 1.3875, 1.3875, 1.3984, 1.3582, 1.341; .35, 1.346, 1.407, 1.407, 1.4201, 1.3774, 1.3587; .35, 1.2964, 1.3664, 1.3664, 1.3811, 1.3376, 1.3181; .35, 1.1918, 1.2676, 1.2676, 1.2833, 1.2409, 1.2212; .35, 1.0376, 1.1156, 1.1156, 1.1315, 1.0919, 1.0731; .35, .8429, .9187, .9187, .9341, .899, .882; .35, .6215, .6904, .6904, .7042, .6754, .661; .35, .3932, .4503, .4503, .4616, .4405, .4295; .35, .1861, .2388, .2388, .2237, .1983, .1785; .35, .0931, .0718, .0718, .0287, .0287, .0111; .35, .0113, .0087, .0087, .0065, .0057, .0052; .35, .0393, .0192, .0192, .0262, .0255, .015; .35, .3998, .2689, .2689, .228, .1987, .2379; .35, .8441, .5843, .5843, .5149, .4813, .4294; .35, 1.2873, .897, .897, .7901, .7396, .6452; .35, 1.6401, 1.1686, 1.1686, 1.0378, .9781, .8664; .35, 1.836, 1.3647, 1.3647, 1.2333, 1.1796, 1.0693; .35, 1.8347, 1.4621, 1.4621, 1.3587, 1.3297, 1.2438; .35, 1.6334, 1.4544, 1.4544, 1.4063, 1.4202, 1.3809; .35, 1.3879, 1.3938, 1.3938, 1.3985, 1.4512, 1.4565; .35, 1.3795, 1.3798, 1.3798, 1.381, 1.4294, 1.4345; .35, 1.3114, 1.3063, 1.3063, 1.3041, 1.3464, 1.3512; .35, 1.1864, 1.1763, 1.1763, 1.1709, 1.2057, 1.2099; .35, 1.0102, .9962, .9962, .9883, 1.0144, 1.0179; .35, .7935, .7771, .7771, .7676, .7845, .7871; .35, .5532, .5364, .5364, .5264, .5346, .5363; .35, .3147, .3, .3, .2912, .2922, .2931; .35, .1144, .1051, .1051, .0995, .0969, .0971; .35, .0602, .0267, .0267, .0202, .0185, .0198]```.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

#### Dependencies

To enable this parameter, set Parameterization to ```Tabulated data for airfoil lift and drag coefficients```.

### Environment and Dynamics

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

Whether to simulate inertia due to the motion of the rotor. The block applies inertia at port R.

Inertia of the windmill rotor.

#### Dependencies

To enable this parameter, select Model inertia.

Initial rotational velocity at port R.

#### Dependencies

To enable this parameter, select Model inertia.

Constant density of air.

Wind velocity at which the block applies smoothing.

Rotational velocity at which the block applies smoothing. This parameter smooths the torque 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] Buhl Jr., Marshall L. “New Empirical Relationship between Thrust Coefficient and Induction Factor for the Turbulent Windmill State.” National Renewable Energy Lab (NREL), Golden, CO (United States), No. NREL/TP-500-36834 (2005).

[2] Jain, Palash, Jayant Sirohi, and Christopher Cameron. “Design, Analysis, and Testing of a Passively Deployable Autorotative Decelerator.” Journal of Aircraft 59, no. 1 (January 2022): 272–77. https://doi.org/10.2514/1.C036509.

[3] Jonkman, Jason. “Definition of a 5-MW Reference Wind Turbine for Offshore System Development.” National Renewable Energy Lab (NREL), Golden, CO (United States), no. No. NREL/TP-500-38060 (2009).

[4] 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

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