Documentation |
Implement model of variable pitch wind turbine
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.
$${P}_{m}={c}_{p}(\lambda ,\beta )\frac{\rho A}{2}{v}_{\text{wind}}^{3},$$ | (1-1) |
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 1-1 can be normalized. In the per unit (pu) system we have:
$${P}_{m\text{\_pu}}={k}_{p}{c}_{p\text{\_pu}}{v}_{\text{wind\_pu}}^{3},$$ | (1-2) |
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:
$${c}_{p}(\lambda ,\beta )={c}_{1}\left({c}_{2}/{\lambda}_{i}-{c}_{3}\beta -{c}_{4}\right){e}^{-{c}_{5}/{\lambda}_{i}}+{c}_{6}\lambda ,$$ | (1-3) |
with
$$\frac{1}{{\lambda}_{i}}=\frac{1}{\lambda +0.08\beta}-\frac{0.035}{{\beta}^{3}+1}.$$ | (1-4) |
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 degree 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 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 which is usually lower than the turbine nominal power.
The maximum power at base wind speed in pu of the nominal mechanical power. This parameter is the power gain k_{p }already defined.
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.
If this parameter is checked, the turbine power characteristics are displayed for different wind speeds and for the specified pitch angle beta.
Simulink input of the generator speed in pu based on the nominal speed of the generator.
Simulink input of the pitch angle.
Simulink input of the wind speed in m/s.
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.
The mechanical power P_{m} as a function of generator speed, for different wind speeds and for blade pitch angle β = 0 degree, 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 (k_{p} = 0.73) and base rotational speed = 1.2 pu).