POLItecnico
di MIlano
ESTIMATION OF DAMPING FOR
WIND TURBINES OPERATING IN
CLOSED LOOP
C.L. Bottasso, S. Cacciola, A. Croce
Politecnico di Milano, Italy
S. Gupta
Clipper Windpower Inc., USA
EWEC 2010
Warsaw, Poland, April 20-23, 2010
Damping Estimation of Wind Turbines
Outline
• 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
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Introduction 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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Poli-Wind Research Lab
Introduction and Motivation
Damping Estimation of Wind Turbines
Previous work:
•
Linear Time Invariant (LTI) systems:
Hauer et al., IEEE TPS, 1990; Trudnowski et al., IEEE TPS 1999
•
However: 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 2007
However: methods well suited only when
characteristic time τ (time to half/double)
much larger than period T (1rev): τ ≫T
τ2 0,96 sec, 2nd fore-aft tower mode
τ1 3.45 sec, 1st fore-aft tower mode
T
5.5 sec
Typically not the case for WT problems
E.g.: damping of tower fore-aft modes ▶
Proposed approach: transform LTP in equivalent/approximate LTI, then use
Prony’s method (standard for LTI analysis)
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Outline
• 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
POLITECNICO di MILANO
Poli-Wind Research Lab
Approach
LTP system:
Damping Estimation of Wind Turbines
.
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
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Approach
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
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Outline
• 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
POLITECNICO di MILANO
Poli-Wind Research Lab
Simulation Models
Cp-Lambda highlights:
Damping Estimation of Wind Turbines
• 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)
Compute
sectional
stiffness
• Rigid body
ANBA (Anisotropic Beam Analysis)
cross sectional model
• Geometrically exact beam
Recover cross
sectional
stresses/strains
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• Revolute joint
• Flexible joint
• ActuatorPoli-Wind Research Lab
Damping Estimation of Wind Turbines
Simulation Environment
Measurement
noise
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Wind
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Applications and Results
Definition of best practices for the identification
of modes of interest:
For each mode:
Excitations (inputs)
• 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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Response (outputs)
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Example: Damping Estimation of
Fore-Aft Tower Modes
Doublets of varying intensity to verify linearity
Excitation: doublet of hub
force in fore-aft direction
Verification of linearity of response
Output: tower root
fore-aft bending
moment
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Example: Damping Estimation of
Fore-Aft Tower Modes
Verification of linearity of response and presence of modes
First tower mode
1P
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Second tower mode
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
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
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Example: Damping Estimation of
Fore-Aft Tower Modes
Estimated damping ratios for varying wind speed
POLITECNICO di MILANO
Poli-Wind Research Lab
Example: Damping Estimation of
Blade Edge and Rotor In-Plane Modes
Damping Estimation of Wind Turbines
Quality of identified model, using blade root bending
First blade
edgewise mode
Excitation: doublet of
• In-plane blade tip force
• Generator torque
Rotor in-plane
mode
Quality of identified model, using shaft torque
Outputs:
• Blade root bending moment
• Shaft torque
Rotor in-plane
mode
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Example: Damping Estimation of
Blade Edge and Rotor In-Plane Modes
Rotor in-plane mode
◀ Little sensitivity to used output
(blade bending or shaft torque)
Blade edge mode
POLITECNICO di MILANO
Poli-Wind Research Lab
Damping Estimation of Wind Turbines
Outline
• 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
POLITECNICO di MILANO
Poli-Wind Research Lab
Conclusions
Damping Estimation of Wind Turbines
Proposed 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)
POLITECNICO di MILANO
Poli-Wind Research Lab