Post on 06-Feb-2018
Turbine aero-servo-elasticityMorten Hartvig Hansenmhha@risoe.dtu.dkRis DTU
2011Wind Turbine Control Symposium2 Ris DTU, Technical University of Denmark
Outline Modal dynamics of three-bladed wind turbines
Modes at standstill What happens when the turbine is rotating?
Methods for analysis Concept of periodic mode shapes Backward and forward whirling Splitting of frequencies Multiple frequencies for single mode
Closed-loop aero-servo-elastic eigenvalue and frequency-domain analysis HAWCStab2: A linear aero-servo-elastic time-invariant model Example: Analysis and tuning of collective and cyclic pitch controller Outlook: Reduced order models for controllers
2011Wind Turbine Control Symposium3 Ris DTU, Technical University of Denmark
Why consider modal dynamics of wind turbines?
Accumulated load spectrum for a 1.4 MW turbine
2011Wind Turbine Control Symposium4 Ris DTU, Technical University of Denmark
Talk to your neighbor: Name and order these modes after frequency
A B
E
D
H
C
GF
2011Wind Turbine Control Symposium5 Ris DTU, Technical University of Denmark
First 8 standstill mode shapes of 600 kW turbine
1 2
5
4
8
3
76
2011Wind Turbine Control Symposium6 Ris DTU, Technical University of Denmark
WHAT HAPPENS WHEN THE ROTOR IS ROTATING?
2011Wind Turbine Control Symposium7 Ris DTU, Technical University of Denmark
Robert P. Colemans original system
NACA Wartime Report L-308 Simplification
2011Wind Turbine Control Symposium8 Ris DTU, Technical University of Denmark
Linear equations of motion
2011Wind Turbine Control Symposium9 Ris DTU, Technical University of Denmark
FIRST SOME THEORYExample to be continued ...
2011Wind Turbine Control Symposium10 Ris DTU, Technical University of Denmark
Eigenvalue analysis of rotary systems
Governing equation of free vibrations with periodic system matrices
On first order form
where
Insertion of solution used in traditional eigenvalue analysis yields
which can never be fulfilled unless .
Warning: The so-called snapshot eigenvalue analysis computing periodicazimuth dependent eigenvalues is nonsense.
2011Wind Turbine Control Symposium11 Ris DTU, Technical University of Denmark
Seek a transformation to obtain time-invariance
Seek a periodic matrix for transformation to new coordinates
Insertion into the first order equation yields
Time invariant matrix
Modal expansion of response using eigensolutions of
Periodic mode shapes
2011Wind Turbine Control Symposium12 Ris DTU, Technical University of Denmark
Floquet analysis of rotary systems (1)First order equations
where the N x N system matrix is T-periodic
Definition of fundamental matrix of N linearly independent solutions
obtained by numerical integration with N linearly independent initial conditions
Shift of the time t by one period to t + T yields
which shows that also is a fundamental solution matrix.
2011Wind Turbine Control Symposium13 Ris DTU, Technical University of Denmark
Floquet analysis of rotary systems (2)Only N linearly independent solutions can exist in a N-state linear system, hence the latter can be constructed as a linear combination of the former:
whereby the so-called monodromy matrix is derived as
A constant non-singular matrix is then defined as
whereby the fundamental solution matrix can be reconstructed as
where is a periodic matrix
Stability analysis:Hence, the exponential decay or growth of vibrations are given by the real part of the eigenvalues of .
2011Wind Turbine Control Symposium14 Ris DTU, Technical University of Denmark
Lyapunov-Floquet (L-F) transformationFor any constant non-singular matrix , a transformation is defined as
The time derivative of the transformation matrix is
From previous slide, the transformed system equation is derived as
Insertion of and into the transformed system equation yields
2011Wind Turbine Control Symposium15 Ris DTU, Technical University of Denmark
L-F transformation Choice of RChoice of the constant non-singular matrix to make T-periodic
Choosing such that yields
Matrix is computed from the monodromy matrix as
If all eigenvectors of are linearly independent then
where are the corresponding eigenvalues of .
2011Wind Turbine Control Symposium16 Ris DTU, Technical University of Denmark
L-F transformation Frequency indeterminacy L-F transformed time-invariant system equation
If has a diagonal Jordan form then and therefore do too:
whereand
and
The eigenvalues of are the complex logarithm of the eigenvalues of
where is an arbitrary integer leading to the frequency indeterminacy
2011Wind Turbine Control Symposium17 Ris DTU, Technical University of Denmark
Coleman transformation for B-bladed rotors with B > 2
where is an analytical function of time:
where
As shown later, the new multi-blade coordinates of the Coleman transformation will describe the rotor motion in the ground fixed frame of reference.
only
for
Bev
en
and
2011Wind Turbine Control Symposium18 Ris DTU, Technical University of Denmark
Coleman transformed equation
Skjoldan, P. F. and Hansen, M. H., On the similarity of the Coleman
and Lyapunov-Floquettransformations for modal analysis of bladed rotor structures, J. of
Sound and Vibration (2009), Vol. 327, pp. 424439.
The Coleman transformation is a special case of the L-F transformation.
Time-invariant matrix if the rotor is isotropic, i.e., the blades are identical and purely azimuth dependent
coupling to surroundings.
Advantage: Frequencies are determined and defined in the ground fixed frame:
2011Wind Turbine Control Symposium19 Ris DTU, Technical University of Denmark
BACK TO THE EXAMPLE
2011Wind Turbine Control Symposium20 Ris DTU, Technical University of Denmark
Rotor motion described in ground fixed frameColeman transformation into multi-blade coordinates
Center of gravity for all three rotor blades of equal mass
Assume
2011Wind Turbine Control Symposium21 Ris DTU, Technical University of Denmark
Original linear equations of motion
2011Wind Turbine Control Symposium22 Ris DTU, Technical University of Denmark
Coleman transformed equations of motion
2011Wind Turbine Control Symposium23 Ris DTU, Technical University of Denmark
Campbell diagram (frequency - rotor speed)
Rotor speed [rad/s]
Nat
ura
l fr
equen
cies
[H
z]
2011Wind Turbine Control Symposium24 Ris DTU, Technical University of Denmark
Campbell diagram for asymmetric support
Rotor speed [rad/s]
Nat
ura
l fr
equen
cies
[H
z]
2011Wind Turbine Control Symposium25 Ris DTU, Technical University of Denmark
Modal dynamics of 3-bladed turbines during operation
2011Wind Turbine Control Symposium26 Ris DTU, Technical University of Denmark
Coleman transformation into multi-blade coordinates
Physical blade coordinates described by multi-blade coordinates
All blade coordinates transformed to the ground fixed frame
2011Wind Turbine Control Symposium27 Ris DTU, Technical University of Denmark
Modal response measured on blade
The eigenvalue solution for the multi-blade coordinates
Insertion into Coleman transformation yields response in the rotor frame:
where
2011Wind Turbine Control Symposium28 Ris DTU, Technical University of Denmark
Frequency response measured on blade
Insertion into yields response for the DOFs of blade k:
where
Harmonic excitation in a point on the ground fixed structure
2011Wind Turbine Control Symposium29 Ris DTU, Technical University of Denmark
BACK TO THE EXAMPLE
Asymmetric supportSymmetric support
2011Wind Turbine Control Symposium30 Ris DTU, Technical University of Denmark
Modal amplitudes of rotor modes
Mode 3: Symmetric mode
Mode 4:Pure backward whirling mode
Mode 5:Pure forward whirling mode
Mode 4:Backward whirling modewith forward whirling component
Mode 5:Forward whirling modewith backward whirling component
Mode 3: Symmetric mode
2011Wind Turbine Control Symposium31 Ris DTU, Technical University of Denmark
Frequency response due to harmonic excitation of horizontal rotor support DOF
2011Wind Turbine Control Symposium32 Ris DTU, Technical University of Denmark
Frequency response due to harmonic excitation of horizontal rotor support DOF
2011Wind Turbine Control Symposium33 Ris DTU, Technical University of Denmark
Runanimations
2011Wind Turbine Control Symposium34 Ris DTU, Technical University of Denmark
Anisotropic rotors
Skjoldan, P. F. and Hansen, M. H., On the similarity of the Coleman and Lyapunov-Floquettransformations for modal analysis of bladed rotor structures, J. of Sound and Vibration(2009), Vol. 327, pp. 424439.
Example: G1 = 1.1Gb and G2 = G3 = 0.95Gb
Simple 5 DOF system considered:
2011Wind Turbine Control Symposium35 Ris DTU, Technical University of Denmark
Anisotropic rotor:Tilt response to excitation of tilt DOF
2011Wind Turbine Control Symposium36 Ris DTU, Technical University of Denmark
OPEN- AND CLOSED-LOOP AERO-SERVO-ELASTICITY
2011Wind Turbine Control Symposium37 Ris DTU, Technical University of Denmark
Aeroelastic model Nonlinear kinematics based on co-
rotational Timoshenko elements.
Blade Element Momentum coupled with unsteady aerodynamics based on Leishman-Beddoes.
Uniform inflow to give a stationary steady state that approximates the mean of the periodic steady state.
Analytical linearization about the stationary steady state that include the linearized coupling terms from the geometrical nonlinearities.
2011Wind Turbine Control Symposium38 Ris DTU, Technical University of Denmark
Linear open-loop aeroelastic equations
= elastic and bearing degrees of freedom= aerodynamic state variables= forces due to actuators and wind disturbance
Coupling to
structural states
Open-loop first order equations
2011Wind Turbine Control Symposium39 Ris DTU, Technical University of Denmark
Closed-loop aero-servo-elastic equations
Present closed-loop equations
Additional output matrices in HAWCStab2
Present PI controller states
2011Wind Turbine Control Symposium40 Ris DTU, Technical University of Denmark
Example: Collective and cyclic pitch controllers
Lead angle,
2011Wind Turbine Control Symposium41 Ris DTU, Technical University of Denmark
Closed-loop aero-servo-elastic equations
Tuning parameters
Filters andintegrators
2011Wind Turbine Control Symposium42 Ris DTU, Technical University of Denmark
Lead angle from open-loop analysis
NREL 5 MW turbine28 deg
BW
edge
FW e
dge
Tow
er m
odes
2011Wind Turbine Control Symposium43 Ris DTU, Technical University of Denmark
Open and closed-loop wind shear response
2011Wind Turbine Control Symposium44 Ris DTU, Technical University of Denmark
Aero-servo-elastic modes and damping
2011Wind Turbine Control Symposium45 Ris DTU, Technical University of Denmark
HAWC2 simulations at 17 m/s with NTM
2011Wind Turbine Control Symposium46 Ris DTU, Technical University of Denmark
Validation of transfer functions with HAWC2Generator torque to generator speed (LSS)
2011Wind Turbine Control Symposium47 Ris DTU, Technical University of Denmark
Validation of transfer functions with HAWC2Collective pitch to generator speed (LSS)
2011Wind Turbine Control Symposium48 Ris DTU, Technical University of Denmark
Summary HAWCStab2 can be used for performing open-loop and closed-loop
eigenvalue and frequency-domain analysis of three-bladed turbines:
Controller equations are still hardcoded. A suitable interface is under consideration, for example based on DLLs as in HAWC2.
Full order analyses can be performed both inside or outside HAWCStab2 by writing out system matrices for each operation point.
Reduced order modelling capabilities are currently performed outside HAWCStab2. Automated procedures for obtaining models with desired details will be implemented in HAWCStab2, or in Matlab scripts.
HAWCStab2 is a common tool for both control engineers and mechanical engineers:
It can provide first-principle models for model-based controllers.
It can explain phenomena observed in load simulations.
Turbine aero-servo-elasticityOutlineWhy consider modal dynamics of wind turbines?Talk to your neighbor: Name and order these modes after frequencyFirst 8 standstill mode shapes of 600 kW turbineWhat happens when the rotor is rotating?Robert P. Colemans original systemLinear equations of motionFirst some theoryEigenvalue analysis of rotary systemsSeek a transformation to obtain time-invarianceFloquet analysis of rotary systems (1)Floquet analysis of rotary systems (2)Lyapunov-Floquet (L-F) transformationL-F transformation Choice of RL-F transformation Frequency indeterminacy Coleman transformation for B-bladed rotors with B > 2 Coleman transformed equationBack to the exampleRotor motion described in ground fixed frameOriginal linear equations of motionColeman transformed equations of motionCampbell diagram (frequency - rotor speed)Campbell diagram for asymmetric supportModal dynamics of 3-bladed turbines during operationColeman transformation into multi-blade coordinatesModal response measured on bladeFrequency response measured on bladeBack to the exampleModal amplitudes of rotor modesFrequency response due to harmonic excitation of horizontal rotor support DOFFrequency response due to harmonic excitation of horizontal rotor support DOFSlide Number 33Anisotropic rotorsAnisotropic rotor:Tilt response to excitation of tilt DOF Open- and closed-loop aero-servo-elasticITYAeroelastic modelLinear open-loop aeroelastic equations Closed-loop aero-servo-elastic equations Example: Collective and cyclic pitch controllersClosed-loop aero-servo-elastic equations Lead angle from open-loop analysisOpen and closed-loop wind shear responseAero-servo-elastic modes and dampingHAWC2 simulations at 17 m/s with NTMValidation of transfer functions with HAWC2Generator torque to generator speed (LSS)Validation of transfer functions with HAWC2Collective pitch to generator speed (LSS)Summary