Implement model of variable pitch wind turbine
Simscape / Electrical / Specialized Power Systems / Electrical Machines
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.
$${P}_{m}={c}_{p}(\lambda ,\beta )\frac{\rho A}{2}{v}_{\text{wind}}^{3},$$ | (1) |
where:
P_{m} | Mechanical output power of the turbine (W) |
c_{p} | Performance coefficient of the turbine |
ρ | Air density (kg/m^{3}) |
A | Turbine swept area (m^{2}) |
v_{wind} | 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:
P_{m}_{_pu} | Power in pu of nominal power for particular values of ρ and A |
c_{p}_{_pu} | Performance coefficient in pu of the maximum value of c_{p} |
v_{wind_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. |
k_{p} | Power gain for c_{p}_{_pu}=1 pu and v_{wind_pu}=1 pu, k_{p} is less than or equal to 1 |
A generic equation is used to model c_{p}(λ,β). This equation, based on the modeling turbine characteristics of [1], is:
$${c}_{p}(\lambda ,\beta )={c}_{1}\left({c}_{2}/{\lambda}_{i}-{c}_{3}\beta -{c}_{4}\right){e}^{-{c}_{5}/{\lambda}_{i}}+{c}_{6}\lambda ,$$
with:
$$\frac{1}{{\lambda}_{i}}=\frac{1}{\lambda +0.08\beta}-\frac{0.035}{{\beta}^{3}+1}.$$
The coefficients c_{1} to c_{6} are: c_{1 }= 0.5176, c_{2 }= 116, c_{3 }= 0.4, c_{4 }= 5, c_{5 }= 21 and c_{6 }= 0.0068. The c_{p}-λ characteristics, for different values of the pitch angle β, are illustrated below. The maximum value of c_{p}(c_{pmax}= 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.
The illustration below shows the mechanical power P_{m} 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 (k_{p} = 0.73), and base rotational speed = 1.2 pu).
[1] Siegfried Heier, “Grid Integration of Wind Energy Conversion Systems,” John Wiley & Sons Ltd, 1998, ISBN 0-471-97143-X
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