Wind_Energy_Essentials_Lecture 8

# 593 27 derived for axial incompressible flow ref

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Unformatted text preview: K. Johnson, L. Pao, M. Balas, and L. Fingersh, Control of Variable Speed Wind Turbines, IEEE Control Systems Magazine, June 2006 Betz’ Law • Betz’ Limit (1919): C p , Betz = 16 ≈ 0.593 27 • Derived for axial, incompressible flow • Ref: Burton, et. al, Wind Energy Handbook, 2001 A Figure from Wikipedia Entry on Betz’ Law 15 11/9/2010 Betz’ Law: Proof Sketch • Ref: Burton, et. al, Wind Energy Handbook, 2001 • Show by conservation of mass and energy balances that wind speed at rotor = (v1 + v2)/2 2 • Extracted Power = Av ⋅ 1 ρ v12 − v2 = 1 ρAv13 1 + v2 − 2 4 v1 ( ) ( ( ) −( ) ) v2 2 v1 v2 3 v1 • Power coefficient is maximized when v2/v1 = 1/3 and maximal value is Cp = 16/27 Figure from Wikipedia Entry on Betz’ Law Rigid Body Model • Simple model for a variable speed turbine &amp; Jω = τ a − τ g ω, τa τg = • where – J:= rotor + drivetrain inertia – τg:= generator torque P – τa= aerodynamic torque = captured ω ρAv 3C p ( β , λ ) 2ω • Model derived from Newton’s second law • Actuators: β and τg • Sensor: ω 16 11/9/2010 Rigid Body Model • The rigid body model is simple but it neglects or simplifies important details – Flexible modes: drivetrain, blade and towerbending – Turbine aerodynamics • Drivetrain flexibility and tower fore/aft modes are important in the control law design – These modes can be excited by control actions – Excitation of these modes can cause drivertrain and/or tower failure – Higher fidelity, but more complex models can be constructed that include these two modes. Fatigue, Aerodynamics, Structures and Turbulence (FAST) • “Medium” complexity nonlinear simulation code – AeroDyn: Aerodynamics modeled using bladeelement/momentum theory rather than CFD – Main degrees of freedom: Rotor position, nacelle yaw, drivetrain flexibility, tower fore/aft and side/side bending, blade flapwise and edgewise bending • Software is freely available: – http://wind.nrel.gov/designcodes/simulators/fast/ – Jonkman and Buhl, FAST User’s Guide, 2005 17 11/9/2010 FAST: Additional Feat...
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