# Wind Turbine

Implement model of variable pitch wind turbine

• Library:
• Simscape / Electrical / Specialized Power Systems / Renewables / Wind Generation

## Description

The Wind Turbine block models the steady-state power characteristics of a wind turbine. The stiffness of the drive train is infinite and the friction factor and the inertia of the turbine must be combined with those of the generator coupled to the turbine. The output power of the turbine is given by the following equation.

 ${P}_{m}={c}_{p}\left(\lambda ,\beta \right)\frac{\rho A}{2}{v}_{\text{wind}}^{3},$ (1)

where:

 Pm Mechanical output power of the turbine (W) cp Performance coefficient of the turbine ρ Air density (kg/m3) A Turbine swept area (m2) vwind Wind speed (m/s) λ Tip speed ratio of the rotor blade tip speed to wind speed β Blade pitch angle (deg)

Equation 1 can be normalized. In the per unit (pu) system we have:

`${P}_{m\text{_pu}}={k}_{p}{c}_{p\text{_pu}}{v}_{\text{wind_pu}}^{3},$`

where:

 Pm_pu Power in pu of nominal power for particular values of ρ and A cp_pu Performance coefficient in pu of the maximum value of cp vwind_pu Wind speed in pu of the base wind speed. The base wind speed is the mean value of the expected wind speed in m/s. kp Power gain for cp_pu=1 pu and vwind_pu=1 pu, kp is less than or equal to 1

A generic equation is used to model cp(λ,β). This equation, based on the modeling turbine characteristics of [1], is:

`${c}_{p}\left(\lambda ,\beta \right)={c}_{1}\left({c}_{2}/{\lambda }_{i}-{c}_{3}\beta -{c}_{4}\right){e}^{-{c}_{5}/{\lambda }_{i}}+{c}_{6}\lambda ,$`

with:

`$\frac{1}{{\lambda }_{i}}=\frac{1}{\lambda +0.08\beta }-\frac{0.035}{{\beta }^{3}+1}.$`

The coefficients c1 to c6 are: c1 = 0.5176, c2 = 116, c3 = 0.4, c4 = 5, c5 = 21 and c6 = 0.0068. The cp-λ characteristics, for different values of the pitch angle β, are illustrated below. The maximum value of cp(cpmax= 0.48) is achieved for β = 0 degrees and for λ = 8.1. This particular value of λ is defined as the nominal value (λ_nom).

The Simulink® model of the turbine is illustrated in the following figure. The three inputs are the generator speed (ωr_pu) in pu of the nominal speed of the generator, the pitch angle in degrees, and the wind speed in m/s. The tip speed ratio λ in pu of λ_nom is obtained by the division of the rational speed in pu of the base rotational speed (defined below) and the wind speed in pu of the base wind speed. The output is the torque applied to the generator shaft.

## Ports

### Input

expand all

Generator speed based on the nominal speed of the generator, specified as a scalar, in pu.

Pitch angle, specified as a scalar

Wind speed, specified as a nonnegative scalar, in m/s.

### Output

expand all

Mechanical torque of the wind turbine, returned as a scalar, in pu of the nominal generator torque. The nominal torque of the generator is based on the nominal generator power and speed.

## Parameters

expand all

The nominal output power in watts (W).

The nominal power of the electrical generator coupled to the wind turbine, in VA. This parameter is used to compute the output torque in pu of the nominal torque of the generator.

The base value of the wind speed, in m/s, used in the per-unit system. The base wind speed is the mean value of the expected wind speed. This base wind speed produces a mechanical power that is usually lower than the turbine nominal power.

Power gain kp at base wind speed in pu of the nominal mechanical power. kp is less than or equal to `1`.

The rotational speed at maximum power for the base wind speed. The base rotational speed is in pu of the base generator speed. For a synchronous or asynchronous generator, the base speed is the synchronous speed. For a permanent-magnet generator, the base speed is defined as the speed producing nominal voltage at no load.

The pitch angle beta, in degrees, used to display the power characteristics. Beta must be greater than or equal to zero.

Click to plot the turbine power characteristics for different wind speeds and for the specified pitch angle beta.

## Examples

The illustration below shows the mechanical power Pm as a function of generator speed, for different wind speeds and for blade pitch angle β = 0 degrees. This figure is obtained with the default parameters (base wind speed = 12 m/s, maximum power at base wind speed = 0.73 pu (kp = 0.73), and base rotational speed = 1.2 pu).

## References

[1] Siegfried Heier, “Grid Integration of Wind Energy Conversion Systems,” John Wiley & Sons Ltd, 1998, ISBN 0-471-97143-X