Morten Hartvig Hansen [email protected] Risø –...

48
Turbine aero-servo-elasticity Morten Hartvig Hansen [email protected] Risø – DTU

Transcript of Morten Hartvig Hansen [email protected] Risø –...

  • Turbine aero-servo-elasticityMorten Hartvig [email protected] 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

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    Talk to your neighbor: Name and order these modes after frequency

    A B

    E

    D

    H

    C

    GF

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    First 8 standstill mode shapes of 600 kW turbine

    1 2

    5

    4

    8

    3

    76

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    WHAT HAPPENS WHEN THE ROTOR IS ROTATING?

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    Robert P. Colemans original system

    NACA Wartime Report L-308 Simplification

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    Linear equations of motion

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    FIRST SOME THEORYExample to be continued ...

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    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.

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    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

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    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.

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    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 .

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    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:

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    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

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    Original linear equations of motion

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    Coleman transformed equations of motion

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    Campbell diagram (frequency - rotor speed)

    Rotor speed [rad/s]

    Nat

    ura

    l fr

    equen

    cies

    [H

    z]

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    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

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    Coleman transformation into multi-blade coordinates

    Physical blade coordinates described by multi-blade coordinates

    All blade coordinates transformed to the ground fixed frame

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    Modal response measured on blade

    The eigenvalue solution for the multi-blade coordinates

    Insertion into Coleman transformation yields response in the rotor frame:

    where

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    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

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    Runanimations

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    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

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    OPEN- AND CLOSED-LOOP AERO-SERVO-ELASTICITY

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    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

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    Example: Collective and cyclic pitch controllers

    Lead angle,

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    Closed-loop aero-servo-elastic equations

    Tuning parameters

    Filters andintegrators

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    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

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    Aero-servo-elastic modes and damping

  • 2011Wind Turbine Control Symposium45 Ris DTU, Technical University of Denmark

    HAWC2 simulations at 17 m/s with NTM

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    Validation of transfer functions with HAWC2Generator torque to generator speed (LSS)

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    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