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

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

3. 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 Periodic Model Time Invariant Model MBC (almost)

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

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

6. 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)

7. 7 Transformation of state-space model: MBC

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

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

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

11. 11 Turbine properties: 5-MW NREL baseline turbine Two operating conditions: Normal OperationExtreme Operation BasisTypical control design pointIEC load case 6.2a, idling Hub-height wind speed18 m/s50 m/s Vertical shear exponent0.2 Yaw error 00 60  Rotor speed12.1 rpm (rated)0.1 rpm Collective blade pitch 15  90  Generator torque/powerrated0

12. 12 Results: Normal Operating Case Largest difference = 0.5% in damping ratio of higher mode Mode Floquet Modal Analysis Eigenanalysis after MBC Eigenanalysis before MBC Freq. [Hz] Damp. Ratio [%] Freq. [Hz] Damp. Ratio [%] Freq. [Hz] Damp. Ratio [%] Blade 1st flap regressive0.2987.20.2987.30.7461.5 Tower 1st side-side0.320.60.320.60.330.4 Tower 1st fore-aft0.338.10.338.10.336.8 Blade 1st flap collective0.5270.20.5270.20.5270.1 Blade 1st flap progressive0.6960.90.7060.70.7461.9 First 5 modes:

13. 13 Results: Extreme Operating Case Largest difference = 21% in damping ratio  Not significant enough to affect stability conclusions Mode Floquet Modal Analysis Eigenanalysis after MBC Eigenanalysis before MBC Freq. [Hz] Damp. Ratio [%] Freq. [Hz] Damp. Ratio [%] Freq. [Hz] Damp. Ratio [%] Tower 1st fore-aft0.313.10.313.10.322.3 Tower 1st side-side0.330.320.33 Blade 1st flap mode 10.6515.10.6512.90.6714.2 Blade 1st flap mode 20.6713.80.6716.70.6714.2 Blade 1st flap mode 30.7012.70.7012.00.6813.5 First 5 modes:

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

15. 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

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

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

18. 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!

19. 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:

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

21. 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. Time [s] LSS yaw moment [kNm] Blade pitch [deg] Baseline IBP  No improvement in performance for direct periodic design (PLQR) Performance MeasureBaseline IBP Load Controller LQR after MBCPLQR RMS speed error1.001.121.18 LSS tilt moment FDEL1.000.55 LSS yaw moment FDEL1.000.390.38 RMS pitch rate1.0049.549.3

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

23. 23 Simulation results:  PLQR after MBC is the ideal theoretical approach Performance MeasureBaseline IBP Load Controller LQR after MBCPLQRPLQR after MBC RMS speed error1.001.061.131.06 Tower fore-aft FDEL1.000.570.630.57 RMS pitch rate1.0029.230.529.1  Direct PLQR not desirable due to unavoidable coupling with speed regulation  No way of penalizing collective pitch usage  Still no significant improvement in performance using PLQR  Averaging after MBC does not degrade performance

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

25. 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

26. 26 Questions? 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)