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

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

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

The six modes of oscillation of a rigid body

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

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

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

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

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

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

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

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

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.

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

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, …

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

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

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

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

Axisymmetric heaving body Haskind relation: (deep water)

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

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

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

No spring K = 0

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).

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

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.

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.

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

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

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

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

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.

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

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

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

Wave energy conversion in irregular waves

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

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:

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

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.

Wave energy absorption by 2-body oscillating systems Linear system. Frequency domain analysis Decompose radiation force:

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

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

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.

Exercise 3.3. Heaving two-body axisymmetric wave energy converter

Exercise 3.3. Heaving two-body axisymmetric wave energy converter Discuss the advantages and limitations of a wave energy converter based on this concept

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

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)

Time-domain analysis of a heaving buoy with hydraulic PTO

Time-domain analysis of a heaving buoy with hydraulic PTO

Time-domain analysis of a heaving buoy with hydraulic PTO

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

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

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

Phase control by latching Time-domain analysis of a heaving buoy with hydraulic PTO Phase control by latching Kjell Budall (1933-89) 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.

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

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

Regular waves No latching R = 1

Regular waves Optimal latching R > 1

NO LATCHING OPTIMAL LATCHING

Irregular waves, Te = 9 s

Irregular waves Te = 9 s No latching R = 1

Irregular waves Te = 9 s Optimal latching R > 1

NO LATCHING OPTIMAL LATCHING

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

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