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PO
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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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POLITECNICO di MILANO Poli-Wind Research Lab
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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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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POLITECNICO di MILANO Poli-Wind Research Lab
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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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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POLITECNICO di MILANO Poli-Wind Research Lab
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
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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Example: Damping Estimation of Fore-Aft Tower Modes
Example: Damping Estimation of Fore-Aft Tower Modes
First tower mode
Second tower mode1P
Verification of linearity of response and presence of modes
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Example: Damping Estimation of Fore-Aft Tower Modes
Example: Damping Estimation of Fore-Aft Tower Modes
◀ 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
Example: Damping Estimation of Fore-Aft Tower Modes
Example: Damping Estimation of Fore-Aft Tower Modes
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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
Example: Damping Estimation of Blade Edge and Rotor In-Plane
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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Example: Damping Estimation of Blade Edge and Rotor In-Plane
Modes
Example: Damping Estimation of Blade Edge and Rotor In-Plane
Modes◀ Little sensitivity to used output (blade bending or shaft torque)
Rotor in-plane mode
Blade edge mode
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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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POLITECNICO di MILANO Poli-Wind Research Lab
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)