POLI di MI tecnicolanotecnicolanotecnicolano ESTIMATION OF DAMPING FOR WIND TURBINES OPERATING IN CLOSED LOOP C.L. Bottasso, S. Cacciola, A. Croce Politecnico

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ESTIMATION OF DAMPING FOR WIND TURBINES

OPERATING IN CLOSED LOOP

C.L. Bottasso, S. Cacciola, A. CrocePolitecnico di Milano, Italy

S. GuptaClipper Windpower Inc., USA

EWEC 2010 Warsaw, Poland, April 20-23, 2010

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POLITECNICO di MILANO Poli-Wind Research Lab

OutlineOutline

• Introduction and motivation

• Approach: modified Prony’s method for linear time periodic systems

• Applications and results:

- Simulation models

- Library of procedures for modes of interest

- Examples: tower, rotor and blade modes

• Conclusions and outlook

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Introduction and MotivationIntroduction and Motivation

Focus of present work: estimation of damping in a wind turbine

Applications in wind turbine design and verification:• Explaining the causes of observed vibration phenomena • Assessing the proximity of the flutter boundaries• Evaluating the efficacy of control laws for low-damped modes• …

Highlights of proposed approach:• Closed loop: damping of coupled wind turbine/controller

system• Applicable to arbitrary mathematical models (e.g., finite

element multibody models, modal-based models, etc.)• In principle applicable to a real wind turbine in the field

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Introduction and MotivationIntroduction and MotivationPrevious work:

• Linear Time Invariant (LTI) systems: Hauer et al., IEEE TPS, 1990; Trudnowski et al., IEEE TPS 1999However: wind turbines are characterized by periodic

coefficients (vertical/horizontal shear layer, up-tilt, yawed flow, blade-tower interaction, etc.)

• Linear Time Periodic (LTP) systems: Bittanti & Colaneri, Automatica 2000; Allen IDETC/CIE

2007However: methods well suited only whencharacteristic time τ (time to half/double) much larger than period T (1rev): τ ≫T

Typically not the case for WT problemsE.g.: damping of tower fore-aft modes ▶

Proposed approach: transform LTP in equivalent/approximate LTI, then use Prony’s method (standard for LTI analysis)

T 5.5 sec

τ1 3.45 sec, 1st fore-aft tower

mode

τ2 0,96 sec, 2nd fore-aft tower

mode

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OutlineOutline




- Simulation models




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ApproachApproachLTP system:

x. = A(ψ)x + B(ψ)u

A(ψ) = closed-loop matrix (accounts for pitch-torque controller)

u = exogenous input (wind), constant in steady conditions

Fourier reformulation (Bittanti & Colaneri 2000):

A(ψ) = A0+Σi(Aissin(i ψ)+Aiccos(i ψ))

B(ψ) = B0+Σi(Bissin(i ψ)+Biccos(i ψ))

1. Approximate state matrix: A(ψ) ≈ A0

2. Transfer periodicity to input term (remark: arbitrary amplitude)

Obtain linear time invariant (LTI) system:

x. = A0x + Ub(ψ)

where b(ψ) = exogenous periodic input

Remark: no need for model generality, just good fit with measures

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ApproachApproach

Given reformulated LTI system

x. = A0x + Ub(ψ)

use standard Prony’s method (Hauer 1990; Trudnowski 1999):

1. Trim and perturb with doublet (or similar, e.g. 3-2-1-1) input

2. Identify discrete time ARX model (using Least Squares or Output Error method) with harmonic input

3. Compute discrete poles, and transform to continuous time (Tustin transformation)

4. Obtain frequencies and damping factors

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OutlineOutline




- Simulation models




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Cp-Lambda highlights:

• Geometrically exact composite-ready beam models

• Generic topology (Cartesian coordinates+Lagrange multipliers)

• Dynamic wake model (Peters-He, yawed flow conditions)

• Efficient large-scale DAE solver

• Non-linearly stable time integrator

• Fully IEC 61400 compliant (DLCs, wind models)

Cp-Lambda (Code for Performance, Loads, Aero-elasticity by Multi-Body Dynamic Analysis):Global aero-servo-elastic FEM model

• Rigid body

• Geometrically exact beam

• Revolute joint

• Flexible joint

• Actuator

ANBA (Anisotropic Beam Analysis) cross sectional model

Compute sectional stiffness

Recover cross sectional

stresses/strains

Simulation ModelsSimulation Models

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Controller

Sensor models Virtual plant Cp-Lambda model

WindMeasureme

nt noise

SupervisorStart-up, power production,

normal shut-down, emergency shut-down, …

Pitch-torque controller

Simulation EnvironmentSimulation Environment

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Excitations (inputs)

Applications and ResultsApplications and Results

Response (outputs)

Definition of best practices for the identification of modes of interest:

For each mode:

• Consider possible excitations (applied loads, pitch and/or torque inputs) and outputs (blade, shaft, tower internal reactions)

• Verify presence of modes in response (FFT)

• Verify linearity of response

• Perform model identification

• Verify quality of identification (compare measured response with predicted one)

Compiled library of mode id procedures:

In this presentation:

• Tower fore-aft mode

• Rotor in-plane, blade first edge modes

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Excitation: doublet of hub force in fore-aft direction

Example: Damping Estimation of Fore-Aft Tower Modes


Output: tower root fore-aft

bending moment

Verification of linearity of response

Doublets of varying intensity to verify linearity

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First tower mode

Second tower mode1P

Verification of linearity of response and presence of modes

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◀ Time domain▼ Frequency domain

• Excellent quality of identified models (supports hypothesis A(ψ) ≈ A0)

• Necessary for reliable estimation

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Estimated damping ratios for varying wind speed



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Excitation: doublet of • In-plane blade tip

force• Generator torque

Example: Damping Estimation of Blade Edge and Rotor In-Plane

Modes


ModesFirst blade

edgewise mode

Quality of identified model, using blade root bending

Rotor in-plane mode

Rotor in-plane mode

Quality of identified model, using shaft torque

Outputs: • Blade root bending

moment• Shaft torque

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Modes


Modes◀ Little sensitivity to used output (blade bending or shaft torque)

Rotor in-plane mode

Blade edge mode

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OutlineOutline




- Simulation models




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ConclusionsConclusionsProposed a method for the estimation of damping in wind

turbines: • Modified Prony’s method (accounts for periodic nature of

wind turbine models)• Good quality model identification is key for reliable damping

estimation• Compiled library of mode id procedures (need specific

inputs/outputs for each mode)• Fast and robust

Outlook:• Riformulation leading to Periodic ARX, and comparison• Effect of turbulence (simulation study):

- Turbulence as an excitation- Turbulence as process noise (filter error method)

• Verify applicability in the field (theoretically possible)

Documents

POLI di MI tecnicolanotecnicolanotecnicolano ESTIMATION OF DAMPING FOR WIND TURBINES OPERATING IN CLOSED LOOP C.L. Bottasso, S. Cacciola, A. Croce Politecnico