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 2010 Warsaw, Poland, April 20-23, 2010

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

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

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 τ1 3.45 sec, 1st fore-aft tower mode T 5.5 sec

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

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

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. 3-2-1-1) 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

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

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 61400 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

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

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)

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

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

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

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

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

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

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

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)