Use of Kane’s Method for Multi-Body Dynamic Modelling and Control of Spar-Type Floating Offshore Wind Turbines
Abstract
:1. Introduction
- The paper demonstrates in detail the use of the recently emerged Kane’s method in deriving the equations of motion of flexible multi-body systems like FOWTs. The method presented here is general and applies to any wind turbine or mechanical system. Such detailed discussion on Kane’s method for flexible multi-body modeling is unavailable in the literature.
- The paper details the powerful vector approach brought about by Kane’s method, which allows the formulation of the dynamics of the complex wind turbine system relatively easy.
- The paper further demonstrates the installation method of an external damper in the FOWT using Kane’s method. The purpose of this exercise is to emphasize the ease of using Kane’s method in augmenting/coupling the system equations with an auxiliary device. Again, the steps presented here are general and apply to any auxiliary device.
2. Methodology
2.1. Assumptions
2.2. Coordinate Systems
2.3. Kinematics
2.4. Kinetics and Kane’s Equation of Motion
2.5. Wave–Current Interaction Model
2.5.1. Regular Wave on Current
2.5.2. Irregular Waves on Current
2.6. Aerodynamic Loads
2.7. Hydrodynamic Loads
2.8. Mooring Dynamics Model
2.9. Structural Control—Passive TMDI Installed on Tower-Top
2.9.1. TMDI Parameter Optimization Using a Simplified Model
- A 2-DOF system with the TMDI represents the wind turbine placed on top of the tower as shown in Figure 7.
- The wind turbine is subjected to white noise.
- The mass of the blades, the hub, and the nacelle are lumped on top of the tower and its base is fixed.
- The inerter is hooked between the mass of the damper and the tower at an arbitrary height from the base of the tower.
- The primary structure, in this case, the wind turbine tower, does not offer any damping.
3. Results and Discussion
3.1. Benchmarking against FAST v8
3.2. Performance of the Passive TMDI
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Low speed shaft-skew angle (rad) | |
Low speed shaft-tilt angle (rad) | |
Height of the wind turbine tower (m) | |
Blade cone angles (rad) | |
Blade pitch angles (rad) | |
Vertical distance from the MSL to the platform CM (m) | |
Vertical distance from the MSL to the platform reference point (m) | |
Downwind distance from the tower-top to the nacelle CM (m) | |
Vertical distance from the tower-top to the nacelle CM (m) | |
Lateral distance from the tower-top to the nacelle CM (m) | |
Distance from yaw axis to rotor apex [3 blades] (m) | |
Vertical distance from the tower-top to the rotor shaft (m) | |
Lateral distance from the tower center line to the rotor shaft (m) | |
Distance from rotor apex to hub mass [positive downwind] (m) | |
Hub Radius | |
Generator direction | |
Gear Box ratio | |
Generator inertia about HSS (kg m) | |
First fore-aft tower bending-mode | |
Second fore-aft tower bending-mode | |
First side-to-side tower bending-mode | |
Second side-to-side tower bending-mode | |
Generator azimuth angle | |
Drive-train torsion | |
First flapwise bending-mode of blade | |
Second flapwise bending-mode of blade | |
First edgewise bending-mode of blade | |
Platform surge | |
Platform sway | |
Platform heave | |
R | Platform roll |
P | Platform pitch |
Y | Platform yaw |
Fore-aft | |
Side-to-side | |
Out-of-plane | |
In-plane | |
Platform inertia for roll tilt rotation about the platform CM (kg m) | |
Platform inertia for yaw rotation about the platform CM (kg m) | |
Platform inertia for pitch tilt rotation about the platform CM (kg m) | |
Yaw bearing mass (kg) | |
Nacelle-yaw spring constant (N-m/rad) | |
Nacelle-yaw damping constant (N-m/(rad/s)) | |
Nacelle inertia about yaw axis (kg m) | |
Hub inertia about rotor axis (kg m) | |
Flexural length of the blade | |
High speed shaft | |
Low speed shaft | |
Drive-train torsional spring (N-m/rad) | |
Drive-train torsional damper (N-m/(rad/s)) | |
Tuned mass damper | |
Tuned mass damper inerter |
Appendix A. Symbols
Platform surge DOF | |
Platform sway DOF | |
Platform heave DOF | |
Platform roll DOF | |
Platform pitch DOF | |
Platform yaw DOF | |
First tower fore-aft bending mode DOF | |
Second tower fore-aft bending mode DOF | |
First tower side-to-side bending mode DOF | |
Second tower side-to-side bending mode DOF | |
Nacelle yaw DOF | |
Generator azimuth angle DOF | |
Drive-train torsional flexibility DOF | |
First flapwise bending mode for blade DOF | |
Second flapwise bending mode for blade DOF | |
First edgewise bending mode for blade DOF | |
Tuned mass damper DOF | |
Tower side-to-side rotation (rad) | |
Tower fore-aft rotation (rad) | |
Structural pre-twist of blades (rad) | |
Blade out-of-plane rotation (rad) | |
Blade in-plane rotation (rad) | |
Tower mass per unit length (kg/m) | |
Nacelle mass (kg) | |
Hub mass (kg) | |
Blade per unit length of blade (kg/m) | |
First fore-aft tower mode shape | |
Second fore-aft tower mode shape | |
First side-to-side tower mode shape | |
Second side-to-side tower mode shape | |
First flapwise blade i mode shape | |
Second flapwise blade i mode shape | |
First edgewise blade i mode shape | |
Normalised hydrodynamic-added-mass coefficient in Morison’s equation | |
Normalised mass (inertia) coefficient in Morison’s equation | |
Normalised viscous-drag coefficient in Morison’s equation | |
Vertical distance of TMDI centre of mass from tower top | |
b | Inertance constant of the TMDI |
Mass of the TMDI | |
Linear stiffness of the TMDI | |
Linear damping ratio of the TMDI | |
Natural frequency of the TMDI |
Appendix B. Kinematics
Appendix B.1. Coordinate Systems
Appendix B.2. Position Vectors
Appendix B.3. Angular Velocities
Appendix B.4. Linear Velocities
Appendix B.5. Partial Angular Velocities
Appendix B.6. Partial Linear Velocities
Appendix B.7. Angular Accelerations
Appendix B.8. Linear Accelerations
Appendix C. Kinetics
Appendix C.1. Platform
Appendix C.2. Tower
Appendix C.3. Yaw Bearing
Appendix C.4. Nacelle
Appendix C.5. Hub
Appendix C.6. Blades
Appendix C.7. Drivetrain
Appendix D. Coupling TMDI to FOWT
Appendix D.1. Position Vector
Appendix D.2. Linear Velocity Vector
Appendix D.3. Partial Linear Velocities
Appendix D.4. Linear Accelerations
Appendix D.5. Kinetics and Kane’s Equations of Motion
Appendix E. Blade and Tower Deflection Shapes
Appendix F. Model Verification Results
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Baseline turbine properties | |
Rating | 5 MW |
Rotor orientation, no of blades | Upwind, 3 blades |
Rotor diameter, hub diameter | 126 m, 3 m |
Hub height | 90 m |
Cut-in, rated, cut-out wind speeds | 3 m/s, 11.4 m/s, 25 m/s |
Cut-in, rated rotor speeds | 6.9 RPM, 12.1 RPM |
Rated tip speed | 80 m/s |
Control | Variable speed, collective pitch |
Drivetrain | 3-stage gearbox |
Overhang, shaft tilt, precone | 5 m, 5, 2.5 |
Tower properties | |
Elevation of tower base from SWL | 10 m |
Elevation of tower top from SWL | 87.6 m |
Floating platform properties | |
Depth of platform base below SWL | 120 m |
Elevation of platform top above SWL | 10 m |
Depth to top of taper below SWL | 4 m |
Depth to bottom of taper below SWL | 12 m |
Platform diameter above taper | 6.5 m |
Platform diameter below taper | 9.4 m |
Response | Mean | Min | Max | Std. | ||||
---|---|---|---|---|---|---|---|---|
FAST | DM | FAST | DM | FAST | DM | FAST | DM | |
Blade OOP displacement (m) | 5.283 | 5.296 | 4.396 | 4.366 | 5.971 | 6.037 | 0.448 | 0.477 |
Blade IP displacement (m) | −0.559 | −0.565 | −1.085 | −1.090 | −0.038 | −0.046 | 0.354 | 0.353 |
Tower-top FA displacement (m) | 0.460 | 0.458 | 0.436 | 0.431 | 0.481 | 0.478 | 0.011 | 0.011 |
Tower-top SS displacement (m) | −0.051 | −0.051 | −0.058 | −0.062 | −0.040 | −0.040 | 0.003 | 0.003 |
Nacelle yaw angle () | 2.2E-3 | 2.2E-3 | 7.1E-4 | 6.1E-4 | 3.8E-3 | 3.8E-3 | 1.0E-3 | 1.0E-3 |
Rotor speed (RPM) | 11.910 | 11.899 | 11.840 | 11.834 | 11.970 | 11.959 | 0.035 | 0.035 |
Platform surge (m) | 25.075 | 24.872 | 20.380 | 20.009 | 30.830 | 30.837 | 3.430 | 3.564 |
Platform sway (m) | −0.319 | −0.309 | −0.359 | −0.372 | −0.261 | −0.253 | 0.026 | 0.027 |
Platform heave (m) | −0.589 | −0.611 | −0.961 | −0.968 | −0.377 | −0.417 | 0.133 | 0.131 |
Platform roll (rad) | 4.4E-3 | 4.4E-3 | 3.7E-3 | 3.4E-3 | 5.1E-3 | 5.3E-3 | 2.95E-4 | 4.45E-4 |
Platform pitch (rad) | 0.085 | 0.086 | 0.079 | 0.079 | 0.090 | 0.090 | 3.1E-3 | 2.9E-3 |
Platform yaw (rad) | 3.9E-3 | 3.9E-3 | 3.5E-3 | 3.6E-3 | 4.4E-3 | 4.5E-3 | 2.1E-4 | 2.1E-4 |
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Sarkar, S.; Fitzgerald, B. Use of Kane’s Method for Multi-Body Dynamic Modelling and Control of Spar-Type Floating Offshore Wind Turbines. Energies 2021, 14, 6635. https://doi.org/10.3390/en14206635
Sarkar S, Fitzgerald B. Use of Kane’s Method for Multi-Body Dynamic Modelling and Control of Spar-Type Floating Offshore Wind Turbines. Energies. 2021; 14(20):6635. https://doi.org/10.3390/en14206635
Chicago/Turabian StyleSarkar, Saptarshi, and Breiffni Fitzgerald. 2021. "Use of Kane’s Method for Multi-Body Dynamic Modelling and Control of Spar-Type Floating Offshore Wind Turbines" Energies 14, no. 20: 6635. https://doi.org/10.3390/en14206635