# Wind Turbine

**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*is the reference pitch angle._{Ref}*λ*is the reference tip speed ratio._{Ref}*C*and_{P,Ref}*C*are the_{T,Ref}**Power coefficient table**and**Thrust coefficient table**parameters, respectively.*λ*is the smoothed tip speed ratio._{Smooth}

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

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

where:

*ρ*is the**Air density**parameter.*A*is the area of the circle swept by the turbine blades, and*A*=*πr*^{2}

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:

*C*is the peak power coefficient. This is the maximum value in the_{P,max}**Power coefficient table, Cp(β,λ)**parameter.*V*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._{rated}*A*is the rotor swept area, where*A = πr*.^{2}

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

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

where *ω _{Thr}* is the

**Rotational velocity threshold**parameter. When

*ɷ*<

*-ɷ*, the block smoothly saturates the power to zero.

_{Thr}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*(λ=0) affects the start-up torque. The start-up torque relates to

_{p}*C*(λ=0) such that

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

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*— Thrust calculated by momentum theory_{MT}*v*— Axial flow velocity induced by the motion of the wind turbine blades_{i}*v*—Radial velocity at the blade location_{r}*v*— Axial velocity at the blade location_{ax}*T*— Thrust calculated by blade element theory_{BET}*Q*— Torque calculated by blade element theory_{BET}*N*— Number of propeller blades_{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*— Element-wise coefficients of the lift and drag, respectively_{l},C_{d}*ϕ(y)*— Flow angle at a given point along the blade*a*= -*v*— Axial induction factor_{i}/v*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*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

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

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.

## Ports

### Inputs

### Outputs

### Conserving

## Parameters

## 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.

## Extended Capabilities

## Version History

**Introduced in R2022b**