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Karl Stol

A Comparison of Multi-Blade Coordinate Transformation and Direct Periodic Techniques for Wind Turbine Control Design. Karl Stol. Hans-Georg Moll HTWG Konstanz, Germany Gunjit Bir National Renewable Energy Laboratory Hazim Namik The University of Auckland. Wind Energy Symposium

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Karl Stol

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  1. A Comparison of Multi-Blade Coordinate Transformation and Direct Periodic Techniques for Wind Turbine Control Design Karl Stol Hans-Georg Moll HTWG Konstanz, Germany Gunjit Bir National Renewable Energy Laboratory Hazim Namik The University of Auckland Wind Energy Symposium AIAA Aerospace Sciences Meeting Orlando, Florida 5-8 January 2009

  2. Outline • Motivation • Multi-Blade Coordinate Transformation • Modal Analysis Comparison • Control Design Comparison • Conclusions and Recommendations 2

  3. Motivation • Wind turbine equations of motion contain periodic coefficients • Multi-blade coordination transformation (MBC) is a common technique for 3-bladed rotors • Modal Analyses & Control Designs • No recorded evaluation of performance loss due to the time-invariant assumption • No comparisons made to direct periodic techniques (almost) MBC Time Invariant Model Periodic Model 3

  4. Deterministicwind field • Non-uniform wind field • Wind shear • Nacelle tilt and yaw error • Tower shadow/reflection Sources of Periodic Coefficients • Structural • 2-bladed rotors • Flexible 3-bladed rotors • Gravity • Aerodynamic Dealing with Periodicity • Ignoring or averaging coefficients directly • Multi-blade coordinate transformation • Direct periodic methods 4

  5. Outline • Motivation • Multi-Blade Coordinate Transformation • Modal Analysis Comparison • Control Design Comparison • Conclusions and Recommendations 5

  6. Multi-Blade Coordinate Transformation • Linear state-space WT model in mixed frame: periodic over azimuth angle where states, individual blade pitch, outputs • Transformation of coordinates in rotating frame: collective cosine-cyclic (tilt) sine-cyclic (yaw) 6

  7. Transformation of state-space model: MBC 7

  8. After MBC AmplitudeSpectra Harmonic • Observations of MBC filtering: • Periodic entries of may contain all harmonics of rotor speed (1p, 2p, 3p, etc.) • Only multiples of 3p remain after MBC Before MBC AmplitudeSpectra Harmonic 8

  9. Outline • Motivation • Multi-Blade Coordinate Transformation • Modal Analysis Comparison • Control Design Comparison • Conclusions and Recommendations 9

  10. Modal Analysis Comparison Floquet Analysis • Approaches: Averaging MBC Eigenanalysis Averaging Eigenanalysis 10

  11. Turbine properties: 5-MW NREL baseline turbine • Two operating conditions: 11

  12. Results: Normal Operating Case • First 5 modes: • Largest difference = 0.5% in damping ratio of higher mode 12

  13. Results: Extreme Operating Case • First 5 modes: • Largest difference = 21% in damping ratio • Not significant enough to affect stability conclusions 13

  14. Outline • Motivation • Multi-Blade Coordinate Transformation • Modal Analysis Comparison • Control Design Comparison • Conclusions and Recommendations 14

  15. Control Design Comparison • Normal operating conditions, above rated wind speed • is weakly periodic, but not necessarily or • Individual blade pitch (IBP) for fatigue load mitigation • Common use of MBC 15

  16. Control design approaches: DirectPeriodic Design Control Gain MBC Periodic Design Averaging LTI Design 16

  17. Periodic linear quadratic regulation (PLQR) Minimises a quadratic cost function: Direct Periodic Design: pitch usage output regulation • Resultant full-state feedback control law: 17

  18. Linear quadratic regulation (LQR) Minimises a quadratic cost function: Linear time-invariant (LTI) design using MBC: • Full-state feedback control law: • Transformation back to mixed frame for implementation: A periodic feedback gain! 18

  19. Control Design Test Cases • Objective 1: reduce shaft bending fatigue • Common IBP objective • Regulate 2 measured outputs: • Objective 2: reduce tower fore-aft bending fatigue • Chosen to provide larger periodic variations after MBC • Allow cyclic pitch only (no collective pitch) • Regulate 1 measured output: 19

  20. open-loop closed-loop Results: Objective 1 (shaft bending) Closed-loop pole locations: Feedback gains for blade 1: Flap asym. (prog.) Imag(s) [rad/s] Flap cosine-cyclic Flap coll. Flap sine-cyclic Tower f-a kij Flap asym. (reg.) Rotor Flap collective Azimuth angle  [deg] Real(s) [rad/s] 20

  21. Simulation results with 5-MW FAST model: • Steady wind speed input • 0.0 to 0.2 step in vertical shear exponent at 30 sec. LSS yaw moment [kNm] Baseline Blade pitch [deg] IBP Time [s] Time [s] • No improvement in performance for direct periodic design (PLQR) 21

  22. Results: Objective 2 (tower fore-aft) • Objective chosen from observing significant periodic variations in for the tower-fore aft EOM cosine-cyclic pitch bij collective pitch sine-cyclic pitch Azimuth angle  [deg] • Potential for performance loss due to averaging 22

  23. Simulation results: • Direct PLQR not desirable due to unavoidable coupling with speed regulation • No way of penalizing collective pitch usage • PLQR after MBC is the ideal theoretical approach • Still no significant improvement in performance using PLQR • Averaging after MBC does not degrade performance 23

  24. Outline • Motivation • Multi-Blade Coordinate Transformation • Modal Analysis Comparison • Control Design Comparison • Conclusions and Recommendations 24

  25. Conclusions and Recommendations • MBC redistributes harmonics in periodic equations of motion • 3p, 6p, etc. harmonics exist after MBC but with very small amplitudes • For modal analyses, averaging after MBC is acceptable • Normal operating case showed no difference to Floquet results • Extreme operating case (high yaw, idling) showed small differences: maximum of 21% difference in damping ratio • For control designs, averaging after MBC is also acceptable • Two different control objectives were investigated • To reduce shaft bending fatigue alone, a time-invariant IBP controller is adequate • MBC or periodic gains are not necessary 25

  26. Worst case scenarios were sought to test robustness of MBC • Other operating conditions or control objectives may exist that show averaging after MBC is not desirable • Recommendation: • Perform MBC regardless of the situation for 3-bladed rotors • Check harmonic spectra of transformed equations of motion • If necessary, use Floquet or PLQR after MBC (not directly) Questions? 26

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