Documentation

# Wind Turbine

Implement model of variable pitch wind turbine

## Library

Simscape / Electrical / Specialized Power Systems / Renewables / Wind Generation

## Description

The model is based on the steady-state power characteristics of the 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:

 ${P}_{m\text{_pu}}={k}_{p}{c}_{p\text{_pu}}{v}_{\text{wind_pu}}^{3},$ (2)

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:

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

with

 $\frac{1}{{\lambda }_{i}}=\frac{1}{\lambda +0.08\beta }-\frac{0.035}{{\beta }^{3}+1}.$ (4)

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.

## Parameters

Nominal mechanical output power

The nominal output power in watts (W). Default is `1.5e6`.

Base power of the electrical generator

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. Default is `1.5e6/0.9`.

Base wind speed

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 which is usually lower than the turbine nominal power. Default is `12`.

Maximum power at base wind speed

The maximum power at base wind speed in pu of the nominal mechanical power. This parameter is the power gain kp already defined. Default is `0.73`.

Base rotational speed

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. Default is `1.2`.

Pitch angle beta to display wind turbine power characteristics

The pitch angle beta, in degrees, used to display the power characteristics. Beta must be greater than or equal to zero. Default is `0`.

Display wind turbine power characteristics

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

## Inputs and Outputs

`Generator speed (pu)`

Simulink input of the generator speed in pu based on the nominal speed of the generator.

`Pitch angle (deg)`

Simulink input of the pitch angle.

`Wind speed (m/s)`

Simulink input of the wind speed in m/s.

`Tm (pu)`

Simulink output of the mechanical torque of the wind turbine, in pu of the nominal generator torque. The nominal torque of the generator is based on the nominal generator power and speed.

## Examples

The mechanical power Pm as a function of generator speed, for different wind speeds and for blade pitch angle β = 0 degrees, is illustrated below. 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