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

$$\begin{array}{l}{C}_P=tablelookup\left({\beta }_{Ref},{\lambda }_{Ref},C_{P,Ref},\beta ,{\lambda }_{Smooth},\text{interpolation = linear,extrapolation = nearest}\right)\\ C_T=tablelookup\left({\beta }_{Ref},{\lambda }_{Ref},C_{T,Ref},\beta ,{\lambda }_{Smooth},\text{interpolation = linear,extrapolation = nearest}\right)\end{array}$$

where:

• β_Ref is the reference pitch angle.

• λ_Ref is the reference tip speed ratio.

• C_P,Ref and C_T,Ref are the Power coefficient table and Thrust coefficient table parameters, respectively.

• λ_Smooth 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 V_Thr is the Wind velocity threshold parameter. The block uses these equations as a basis for the power and thrust

$$\begin{array}{l}Power=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.C_P\rho A{V}^3=Torque\cdot \omega \\ Thrust=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.C_T\rho A{V}^2\end{array}$$

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

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

where:

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

• V_rated 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

$$\begin{array}{l}Power=\frac{\rho A{C}_P{\left|V\right|}^3}{2}\\ Torqu{e}_{Smooth}=\{\begin{array}{cc}0& \text{if }\omega <-{\omega }_{Thr}\\ \frac{Power}{\sqrt{{\omega }^2+{\omega }_{Thr}^2}}& \text{if }\omega >0\end{array}\\ Powe{r}_{Smooth}=Torqu{e}_{Smooth}\omega \\ Thrus{t}_{Smooth}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.C_T\rho A\left(V^2+V_{Thr}^2\right)\end{array}$$

where ω_Thr is the Rotational velocity threshold parameter. When ɷ < -ɷ_Thr, the block smoothly saturates the power to zero.

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

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

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, v_i. 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:

• T_MT — Thrust calculated by momentum theory

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

• v_r —Radial velocity at the blade location

• v_ax — Axial velocity at the blade location

• T_BET — Thrust calculated by blade element theory

• Q_BET — Torque calculated by blade element theory

• N_blades — Number of propeller blades

• 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

• R — Wind turbine blade radius

• C_l,C_d — Element-wise coefficients of the lift and drag, respectively

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

• a = -v_i/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

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

The smoothed axial induction factor is

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

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 C_l(y) and C_d(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

$$\begin{array}{l}{T}_{BET}=\frac{N_{blades}\rho D^2}{4}{\displaystyle \underset{e}{\overset{1}{\int }}\left(v_r^2+v_{ax}^2\right)\cdot \left(C_l\cos \varphi (y)+C_d\sin \varphi (y)\right)}\cdot c\mathrm{dy}\\ Q_{BET}=\frac{N_{blades}\rho D^3}{8}{\displaystyle \underset{e}{\overset{1}{\int }}\left(v_r^2+v_{ax}^2\right)\cdot \left(C_l\sin \varphi (y)+C_d\cos \varphi (y)\right)}\cdot cy\mathrm{dy}\end{array}$$

where:

$$\begin{array}{c}{v}_r=Ry\Omega (1+a\text{'})\\ v_{ax}=v(1-a)\end{array}$$

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

$$a\text{'}=-\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{4}{{\lambda }_r^2}a(1-a)}.$$

Assumptions and Limitations

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

Examples

Wind Turbine

Model, parameterize, and test a wind turbine with a supervisory, pitch angle, MPPT (maximum power point tracking), and derating control. When you run the plot function, it generates a plot of the state transitions, normalized physical quantities such as the wind speed, wind turbine rotation speed, generator power, and pitch angle.

• Open Model

Wind Turbine Blade Optimization

Adjust the twist and chord length along a wind turbine blade to optimize power output. It combines a Simscape™ Driveline™ test harness model with the fminsearch optimization function. The wind turbine model uses blade element momentum (BEM) theory, including wake rotation. By putting the test harness into Fast Restart, you can enable rapid optimization iterations. In this case, the optimization increases the power output from 4.3 MW to 5.9 MW. Other products available for performing this type of optimization with Simscape Driveline models include the Optimization Toolbox™ and Simulink® Design Optimization™. These products provide predefined functions to manipulate and analyze blocks using either GUIs or a command line approach.

• Open Live Script

Wind Turbine Driveline with Vibrations

A wind turbine rotor, controller, and flexible drive shaft with transverse vibrations. Click the button below to open the model.

• Open Live Script

Ports

Inputs

V — Incident wind velocity, m/s

physical signal

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

β — Pitch angle, deg

physical signal

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

Outputs

T — Thrust, N

physical signal

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

R — Turbine shaft

mechanical rotational

Mechanical rotational conserving port associated with the wind turbine shaft.

Parameters

Wind Turbine

Parameterization — Wind turbine parameterization option

Tabulated data for power and thrust coefficients (default) | Tabulated data for airfoil lift and drag coefficients

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

Dependencies

Turbine radius — Blade tip radius

80 m (default) | positive scalar

Distance from the turbine hub center to the blade tips.

Pitch angle vector, β — Reference pitch angle

[0, 3, 5, 10, 20, 90] deg (default) | vector of length M

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.

Tip speed ratio vector, λ — Reference tip speed ratio

[0, 2, 4, 6, 8, 10, 12, 14, 15] (default) | vector of length N

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 coefficient table, Cp(β,λ) — Power coefficient table

M-by-N matrix

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 C_p(λ=0)≅ 0.

Dependencies

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

Thrust coefficient table, Ct(β,λ) — Thrust coefficient table

M-by-N matrix

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.

Number of blades — Number of blades

3 (default) | positive integer

Number of wind turbine blades.

Dependencies

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

Nondimensional radial location vector, r — Radial location

[.1, .17, .23, .36, .5, .63, 1] (default) | vector

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 vector, θ_tw(r) — Blade element twist

[13.31, 13.31, 11.48, 9.01, 6.54, 4.19, .11] deg (default) | vector

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.

Normalized chord length vector, c(r)/R — Blade element chord length

[.068, .074, .076, .069, .061, .053, .023] (default) | vector

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.

Airfoil angle of attack vector, α — Angle of attack

[-180 : 10 : 180] deg (default) | vector of length M

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 coefficient table, Cl(α,r) — Airfoil lift coefficients

M x N matrix

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 coefficient table, Cd(α,r) — Airfoil drag coefficients

M x N matrix

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.

Number of blade elements — Number of blade elements

20 (default) | positive integer

Number of blade elements for a single blade.

Dependencies

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

Environment and Dynamics

Output thrust — Option to model turbine thrust output

off (default) | on

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

Model inertia — Inertia option

off (default) | on

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

Rotor inertia — Windmill rotor inertia

3.5e7 kg*m^2 (default)

Inertia of the windmill rotor.

Dependencies

To enable this parameter, select Model inertia.

Initial rotor rotational velocity — Rotor initial angular velocity

0 rad/s (default)

Initial rotational velocity at port R.

Dependencies

To enable this parameter, select Model inertia.

Air density — Air density

1.225 kg/m^3 (default) | positive scalar

Constant density of air.

Advanced

Wind velocity threshold — Wind velocity smoothing threshold

0.01 m/s (default) | positive scalar

Wind velocity at which the block applies smoothing.

Rotational velocity threshold — Angular velocity smoothing threshold

0.01 rad/s (default) | positive scalar

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. 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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2022b

R2024a: Parameterize block using airfoil lift and drag coefficients

To parameterize the block using airfoil lift and drag coefficients, set Parameterization to Tabulated data for airfoil lift and drag coefficients.

R2023a: Performance Updates

Port C is no longer required and has been removed.

The default value for the Power coefficient table, Cp(β, λ) has been updated. The block asserts C_p(λ=0)≅ 0.

R2023a: New Inertia Option

Include inertia within the Wind Turbine block by selecting Model inertia.