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

2. MODELLING OF OSCILLATING BODY WAVE ENERGY CONVERTERS
PART 3
MODELLING OF OSCILLATING BODY WAVE ENERGY CONVERTERS

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

4. The six modes of oscillation of a rigid body

5. Characteristic scales
Most ships
“Large” WECs
Inviscid linearized diffraction theory applicable

6. Wave field of a single heaving body
m = body mass
mg = body weight
In the absence of waves mg = buoyancy force and
We ignore mooring forces (may be considered later)
In the dynamic equations, we consider only disturbances to equilibrium conditions; body weight does not appear

7. Wave field of a single heaving body
Wave field I: Incident wave field
satisfies bottom condition and free-surface condition
Wave field II: Diffracted wave field due to the presence of the fixed body
satisfies bottom condition and free-surface condition
Wave fields I + II:
satisfies also condition on fixed body wetted surface
due to wave fields I and II
due to wave fields I and II

8. Wave field of a single heaving body
Wave field III: Radiated wave field of moving body
satisfies bottom condition, free-surface condition and condition on wetted surface of heaving body
due to wave field III

9. Hydrostatic restoring force
If, in the absence of incident waves, the body is fixed at , the buoyancy force does not balance the body weight.
The difference is a hydrostatic restoring force
For small displacement , it is
Hydrostatic restoring force
area

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

11. Our WEC is a linear system if the PTO is linear
Frequency-domain analysis of wave energy absorption by a single heaving body
LINEAR SYSTEM
input
output
Our WEC is a linear system if the PTO is linear
Linear PTO: linear spring and/or linear damper
damping coef.
spring stiffness

12. Frequency-domain analysis of heaving body
Incident wave
The system is linear:
Complex amplitudes

13. Frequency-domain analysis of heaving body
Decompose radiation force coefficient:
added mass
radiation damping coef.
Exercise
Show that the radiation damping coefficient B cannot be negative.

14. Frequency-domain analysis of heaving body
The hydrodynamic coefficients are related to each other:
body
Haskind relation:
Kramers-Kronig relations:

15. Frequency-domain analysis of heaving body
Calculation of hydrodynamic coefficients:
They are functions of frequency
Analytical methods for simple geometries: sphere, horizontal cylinder, plane vertical and horizontal walls, etc.
Commercial codes based on Boundary-Element-Method BEM for arbitrary geometries, several degrees of freedom and several bodies: WAMIT, ANSYS/Aqua, Aquaplus, …

16. Absorbed power and power output
Instantaneous power absorbed from the waves = vertical force component on wetted surface times vertical velocity of body
Instantaneous power available to PTO = force of body on PTO times vertical velocity of body

17. Conditions for maximum absorbed power
Given body, fixed wave frequency and amplitude
velocity in phase with excitation force

18. Conditions for maximum absorbed power
Separate into real and imaginary parts:
radiation damping = PTO damping
resonance condition
Analogy

19. May be larger than the physical dimension of the body
Capture or absorption width
Avoid using efficiency of the wave energy absorption process, especially in the case of “small” devices.
Incident waves
capture width L
May be larger than the physical dimension of the body

20. Axisymmetric heaving body
Haskind relation:
(deep water)

21. Axisymmetric heaving body
Maximum capture width for
an axisymmetric heaving buoy
Maximum capture width for
an axisymmetric surging buoy

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

23. Hemispherical buoy in deep water
Exercise 3.1
Hemispherical buoy in deep water
Dimensionless quantities

24. No spring K = 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 the time-domain analysis to 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:
How to obtain the memory function ?
see later why
Take
We obtain
Changing the integration variable from to , we have
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, since if finite, the integrals vanish as , which agrees with
Invert Fourier transform
Assume to be an even function

32. Time-domain analysis of a single heaving body
This has to be integrated in the time domain from initial conditions

33. Time-domain analysis of a single heaving body
Note: since the “memory” decays rapidly, the infinite integral can be replaced by a finite integral. In most cases, three wave periods (about 30 s) is enough.
Integration procedure:
Set initial values (usually zero)
Compute the rhs at time
Compute from the equation
Set
Compute etc.
Adopted time steps are typicall between 0.01 s and 0.1 s
The convolution integral must be computed at every 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 wavelets with 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 frequency spectrum is a function
is is the energy content within a frequency band of width equal to df

36. Wave energy conversion in irregular waves
A frequency spectrum is a function
is is 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.
Goda proposed 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.
Simulation of excitation force in irregular waves
Divide the frequency range of interest into N small intervals
of width and set or

40. Wave energy absorption from irregular waves
Simulation of excitation force in irregular waves
Oscillating body with linear PTO and linear damping coefficient C . Averaged power over a long time:
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. Frequency domain analysis
Decompose radiation force:

44. Wave energy absorption by 2-body oscillating systems.
Linear system. Frequency domain analysis
Relationships between coefficients: radiation damping force and excitation force
Axisymmetric systems:

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 axisymmetric and coaxial.
The draught d of body 2 is large:
The PTO consists of 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
Discuss the advantages and limitations of a wave energy converter based on this concept

49. Oscillating systems with several degrees of freedom
The theory can be generalized to single bodies with several degrees of freedom or groups of bodies.
For the general theory, see the book by 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)

51. Time-domain analysis of a heaving buoy with hydraulic PTO

52. Time-domain analysis of a heaving buoy with hydraulic PTO

53. Time-domain analysis of a heaving buoy with hydraulic PTO

54. Underdamping and overdamping
Time-domain analysis of a heaving buoy with hydraulic PTO
Underdamping and overdamping

55. Time-domain analysis of a heaving buoy with hydraulic PTO
PTO power

56. The smoothing effect decreases for more energetic sea states
Time-domain analysis of a heaving buoy with hydraulic PTO
The smoothing effect decreases for more energetic sea states

57. Phase control by latching
Time-domain analysis of a heaving buoy with hydraulic PTO
Phase control by latching
Kjell Budall ( )
Johannes Falnes
Pioneers in control theory of wave energy converters.
They introduced the concept of phase- control by latching:
J. Falnes, K. Budal, Wave-power conversion by power absorbers. Norwegian Maritime Research, 6, 2-11, 1978.

58. Phase control by latching
Time-domain analysis of a heaving buoy with hydraulic PTO
Phase control by latching
How to achieve phase-control by latching in a floating body with a hydraulic power-take-off mechanism?
Introduce a delay in the release of the latched body.
How?
Increase the resisting force the hydrodynamic forces have to overcome to restart the body motion.
Phase-control by latching: release the body when

59. G control of flow rate of oil through hydraulic motor
Time-domain analysis of a heaving buoy with hydraulic PTO
Phase control by latching
Two control variables
G control of flow rate of oil through hydraulic motor
R release of latched body

60. Regular waves
No latching
R = 1

61. Regular waves
Optimal latching
R > 1

62. NO LATCHING
OPTIMAL LATCHING

63. Irregular waves, Te = 9 s

64. Irregular waves
Te = 9 s
No latching
R = 1

65. Irregular waves
Te = 9 s
Optimal latching
R > 1

66. NO LATCHING
OPTIMAL LATCHING

67. Latching control May involve very large forces
May be less effective in two-body WECs

68. MODELLING OF OSCILLATING BODY WAVE ENERGY CONVERTERS
END OF
PART 3
MODELLING OF OSCILLATING BODY WAVE ENERGY CONVERTERS