Report

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 POLITECNICO di MILANO 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 POLITECNICO di MILANO • Revolute joint • Flexible joint • ActuatorPoli-Wind Research Lab Damping Estimation of Wind Turbines Simulation Environment Measurement noise POLITECNICO di MILANO 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 POLITECNICO di MILANO 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 POLITECNICO di MILANO 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