1. ESTIMATION OF DAMPING FOR WIND TURBINES OPERATING IN CLOSED LOOP C. L
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 Warsaw, Poland, April 20-23, 2010

2. 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

3. 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

4. Introduction and Motivation
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
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)
τ2 0,96 sec, 2nd fore-aft tower mode
τ sec, 1st fore-aft tower mode
T 5.5 sec

5. 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

6. Approach LTP 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))
Approximate state matrix: A(ψ) ≈ A0
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

7. Approach Given reformulated LTI system x. = A0x + Ub(ψ)
use standard Prony’s method (Hauer 1990; Trudnowski 1999):
Trim and perturb with doublet (or similar, e.g ) input
Identify discrete time ARX model (using Least Squares or Output Error method) with harmonic input
Compute discrete poles, and transform to continuous time (Tustin transformation)
Obtain frequencies and damping factors

8. 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

9. Simulation Models 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 compliant (DLCs, wind models)
Cp-Lambda (Code for Performance, Loads, Aero-elasticity by Multi-Body Dynamic Analysis):
Global aero-servo-elastic FEM model
Compute sectional stiffness
ANBA (Anisotropic Beam Analysis) cross sectional model
Rigid body
Geometrically exact beam
Revolute joint
Flexible joint
Actuator
Recover cross sectional stresses/strains

10. Simulation Environment
Sensor models
Virtual plant Cp-Lambda model
Measurement noise
Wind
Supervisor
Start-up, power production, normal shut-down, emergency shut-down, …
Pitch-torque controller
Controller

11. Applications and Results
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
Excitations (inputs)
Response (outputs)

12. 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

13. Example: Damping Estimation of Fore-Aft Tower Modes
Verification of linearity of response and presence of modes
First tower mode
Second tower mode 1P

14. 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

15. Example: Damping Estimation of Fore-Aft Tower Modes
Estimated damping ratios for varying wind speed

16. Example: Damping Estimation of Blade Edge and Rotor In-Plane Modes
Quality of identified model, using blade root bending
Excitation: doublet of
In-plane blade tip force
Generator torque
First blade edgewise mode
Rotor in-plane mode
Outputs:
Blade root bending moment
Shaft torque
Quality of identified model, using shaft torque
Rotor in-plane mode

17. Example: Damping Estimation of Blade Edge and Rotor In-Plane Modes
◀ Little sensitivity to used output (blade bending or shaft torque)
Blade edge mode

18. 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

19. Conclusions
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)