Specialization in Ocean Energy

1. Specialization in Ocean Energy MODELLING OF WAVE ENERGY CONVERSION António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2014

2. PART 3 MODELLING OF OSCILLATING BODY WAVE ENERGY CONVERTERS

3. Isolated: Pico, LIMPET, Oceanlinx Fixed structure Oscillating Water Column (with air turbine) In breakwater: Sakata, Mutriku Floating: Mighty Whale, BBDB Heaving: Aquabuoy, IPS Buoy, Wavebob, PowerBuoy, FO3 Floating Oscillating body (hydraulic motor, hy-draulic turbine, linear electricgenerator) Pitching: Pelamis, PS Frog, Searev Heaving: AWS Submerged Bottom-hinged: Oyster, Waveroller Shoreline (with concentration): TAPCHAN Fixed structure Overtopping (low head water turbine) In breakwater (without concentration): SSG Floating structure (with concentration): Wave Dragon Wave Energy Converter Types RESONANCE

4. Thesixmodesofoscillationof a rigid body

5. Characteristicscales Mostships “Large” WECs Inviscidlinearizeddiffractiontheoryapplicable

6. Wavefieldof a single heaving body • m = body mass • mg = body weight • In theabsenceofwavesmg = buoyancy force and • We ignore mooring forces (maybeconsidered later) • In thedynamicequations, weconsideronlydisturbances to equilibriumconditions; body weight does notappear

7. Wavefieldof a single heaving body • Wavefield I: Incidentwavefield • satisfiesbottomconditionand free-surfacecondition • Wavefield II: Diffractedwavefielddue to thepresenceofthefixed body • satisfiesbottomconditionand free-surfacecondition • Wavefields I + II: • satisfiesalsoconditiononfixed body wettedsurface due to wavefields I and II due to wavefields I and II

8. Wavefieldof a single heaving body • Wavefield III: Radiatedwavefieldofmoving body • satisfiesbottomcondition, free-surfaceconditionandconditiononwettedsurfaceofheaving body due to wavefield III due to wavefield III

9. Hydrostaticrestoring force If, in theabsenceofincidentwaves, the body isfixedat , thebuoyancy force does not balance the body weight. Thedifferenceis a hydrostaticrestoring force . For smalldisplacement , itis area

10. Dynamicequation for heaving body motion excitation mass PTO radiation acceleration hydrostatic/restoring

11. Frequency-domain analysis of wave energy absorption by a single heaving body output input LINEAR SYSTEM Our WEC is a linear systemifthe PTO is linear Linear PTO: linear springand/or linear damper springstiffness dampingcoef.

12. Frequency-domain analysis of heaving body Incidentwave Thesystemis linear: Complex amplitudes

13. Frequency-domain analysis of heaving body Decomposeradiation force coefficient: radiationdampingcoef. addedmass Exercise Show thattheradiationdampingcoefficientBcannotbe negative.

14. Frequency-domain analysis of heaving body Thehydrodynamiccoefficients are related to eachother: Haskindrelation: body Kramers-Kronigrelations:

15. Frequency-domain analysis of heaving body Calculationofhydrodynamiccoefficients: • They are functionsoffrequency • Analyticalmethods for simplegeometries: sphere, horizontal cylinder, plane vertical and horizontal walls, etc. • CommercialcodesbasedonBoundary-Element-Method BEM for arbitrarygeometries, severaldegreesoffreedomandseveral bodies: WAMIT, ANSYS/Aqua, Aquaplus, …

16. Absorbedpowerandpower output Instantaneouspowerabsorbedfromthewaves = vertical force componentonwettedsurface times vertical velocityof body Instantaneouspoweravailable to PTO = force of body on PTO times vertical velocityof body

17. Conditions for maximumabsorbedpower Given body, fixedwavefrequencyand amplitude velocity in phasewithexcitation force

18. Conditions for maximumabsorbedpower Separateinto real andimaginaryparts: radiationdamping = PTO damping resonancecondition Analogy

19. Capture orabsorptionwidth Avoidusingefficiencyofthewaveenergyabsorptionprocess, especially in the case of “small” devices. Capture orabsorptionwidth Incident waves L capture width Maybelargerthanthephysicaldimensionofthe body

20. Axisymmetricheaving body Haskindrelation: (deepwater)

21. Axisymmetricheaving body Maximum capture width for anaxisymmetricheavingbuoy Maximum capture width for anaxisymmetricsurgingbuoy

22. Incident waves Axisymmetricheaving body Max. capture width Incident waves Axisymmetric surging body Axisymmetric body with linear PTO

23. Exercise 3.1 Hemisphericalbuoy in deepwater Dimensionlessquantities

24. No springK = 0

25. Reproduce the curves plotted in the figures by doing your own programming. • Compute the buoy radius a and the PTO damping coefficient C that yield maximum power from regular waves of period T = 9 s. Compute the time-averaged power for wave amplitude . • Assume now that the PTO also has a spring of stiffness K that may be positive or negative. Compute the optimal values for the damping coefficient C and the spring stiffness K for a buoy of radius a = 5 m in regular waves of period T = 9 s. Explain the physical meaning of a negative stiffness spring (in conjunction with reactive control).

27. Exercise 3.2. Heaving floater rigidly attached to a deeply submerged body WaveBob, Ireland

28. Time-domain analysis of a single heaving body • If the power take-off system is not linear • then the frequency-domain analysis cannot be employed. • This is the real situation in most cases. • In particular, even in sinusoidal incident waves, the body velocity is not a sinusoidal function of time. • In such cases, we have to use thetime-domain analysisto model the radiation force.

29. Time-domain analysis of a single heaving body • When a body is forced to move in otherwise calm water, its motion produces a wave system (radiated waves) that propagate far away. • Even if the body ceases to move after some time, the wave motion persists for a long time and produces an oscillating force on the body which depends on the history of the body motion. • This is a memory effect.

30. Time-domain analysis of a single heaving body This dependence can be expressed in the following form: see later why How to obtainthememoryfunction ? Take Weobtain Changing the integration variable from to , we have

31. Time-domain analysis of a single heaving body Since the functions A, B and are real, we may write Note that, sinceiffinite,theintegralsvanish as , whichagreeswith Assume to beanevenfunction Invert Fourier transform

32. Time-domain analysis of a single heaving body Thishas to beintegrated in the time domainfrominitialconditions

33. Time-domain analysis of a single heaving body Note: sincethe “memory” decaysrapidly, theinfinite integral can bereplacedby a finite integral. In most cases, threewaveperiods (about 30 s) isenough. • Integrationprocedure: • Set initialvalues (usually zero) • Compute therhsat time • Compute fromtheequation • Set • Compute etc. Adopted time steps are typicallbetween 0.01 s and 0.1 s Theconvolution integral must becomputedatevery time step

34. Wave energy conversion in irregular waves • Real ocean waves are not purely sinusoidal: they are irregular and largely random. • In linear wave theory, they can be modelled as the the superposition of an infinite number of sinusoidal waveletswith different frequencies and directions. • The distribution of the energy of these wavelets when plotted against the frequency and direction is the wave spectrum. • Here, we consider only frequency spectra.

35. Wave energy conversion in irregular waves A frequencyspectrumis a function isis the energy content within a frequency band of width equal to df

36. Wave energy conversion in irregular waves A frequencyspectrumis a function isis the energy content within a frequency band of width equal to df

37. Wave energy conversion in irregular waves The characteristics of the frequency spectra of sea waves have been fairly well established through analyses of a large number of wave records taken in various seas and oceans of the world. Godaproposed the following formula for fully developed wind waves, based on an earlier formula proposed by Pierson and Moskowitz

38. Wave energy conversion in irregular waves

39. Wave energy absorption from irregular waves In computations, it is convenient to replace the continuum spectrum by a superposition of a finite number of sinusoidal waves whose total energy matches the spectral distribution. Divide the frequency range of interest into N small intervals of width and set Simulation of excitation force in irregular waves or

40. Wave energy absorption from irregular waves Oscillating body withlinear PTO and linear dampingcoefficientC . Averagedpowerover a long time: Simulation of excitation force in irregular waves Note that:

41. Wave energy absorption by 2-body oscillating systems In singe-body WECs, the body reacts against the bottom. In deep water (say 40 m or more), this may raise difficulties due to the distance between the floating body and the sea bottom, and also possibly to tidal oscillations. • Two-body systems may then be used instead. • The energy is converted from the relative motion between two bodies oscillating differently. • Two-body heaving WECs: Wavebob, PowerBuoy,AquaBuoy

42. Wave energy absorption by 2-body oscillating systems • The coupling between bodies 1 and 2 is due firstly to the PTO forces and secondly to the forces associated to the diffracted and radiated wave fields. • The excitation force on one of the bodies is affected by the presence of the other body. • In the absence of incident waves, the radiated wave field induced by the motion of one of the bodies produces a radiation force on the moving body and also a force on the other body.

43. Wave energy absorption by 2-body oscillating systems Linear system. Frequencydomainanalysis Decomposeradiation force:

44. Wave energy absorption by 2-body oscillating systems. Linear system. Frequency domain analysis Relationshipsbetweencoefficients: radiationdamping force andexcitation force Axisymmetricsystems:

45. Wave energy absorption by 2-body oscillating systems. Non-linear system. Time domain analysis Excitation forces:

46. Exercise 3.3. Heaving two-body axisymmetric wave energy converter Bodies 1 and 2 are axisymmetricand coaxial. Thedraughtdof body 2 islarge: The PTO consistsof a linear damper, and no spring.

47. Exercise 3.3. Heaving two-body axisymmetric wave energy converter

48. Exercise 3.3. Heaving two-body axisymmetric wave energy converter Discusstheadvantagesandlimitationsof a waveenergy converter basedonthisconcept

49. Oscillating systems with several degrees of freedom Thetheory can begeneralized to single bodies withseveraldegreesoffreedomorgroupsof bodies. For the general theory, seethebookby J. Falnes

50. Time-domain analysis of a heaving buoy with hydraulic PTO • Hydraulic circuit: • Conventional equipment • Accommodates large forces • Allows energy storage in gas accumulators (power smoothing effect) • Relatively good efficiency of hydraulic motor • Easy to control (reactive and latching) • Adopted in several oscillating-body WECS • PTO is in general highly non-linear (time-domain analysis)