hydrodynamics ofarrays of heaving … ofarrays of heaving point absorbers ashank sinha dissertação...

HYDRODYNAMICS OFARRAYS OF HEAVING POINT

ABSORBERS

Ashank Sinha

Dissertação para obtenção do Grau de Mestre em

Engenharia e Arquitectura Naval

Orientadores: Prof. Carlos Guedes Soares, Dr. Debabrata Karmakar

Júri

Presidente: Prof. Yordan Garbatov

Orientador: Prof. Carlos Guedes Soares

Co-orientador: Dr. Debabrata Karmakar

Vogal: Prof. Serge Sutulo

Decembro 2015

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DEDICATION

Dedicated to Dadaji and Dadi Ma

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ACKNOWLEDGMENTS

I express my sincere thanks and appreciation to my supervisor Professor Carlos Guedes Soares for his

guidance, support and help during the tenure of the present research work.

I also sincerely appreciate and thank Dr. Debabrata Karmakar for his co-operation and help right from the

inception of the problem to the final preparation of the manuscript. I am highly indebted to him as he has

taken keen interest and offered valuable suggestions whenever needed.

I also thank all the faculty members of the CENTEC for their technical suggestions at various times.

Lastly, I extend my gratitude to my parents and brothers for always standing by me and supporting me for

this endeavor and thank my friends for all the great moments we spent.

Ashank Sinha

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ABSTRACT

This work presents a detailed study about hydrodynamics of point absorber wave energy converters. It

includes the development of a linear frequency domain model to evaluate the performance of arrays of

heaving point absorbers. The hydrodynamic parameters are obtained with WAMIT. The wave climate

chosen covers the most probable sea states. Three floater shapes with varying waterline drafts are examined

to determine the most suitable shape and dimension. Further, the performance of several array

configurations (linear, grid, circular, concentric) of point absorbers is evaluated. The effect of wave heading

is also taken into account. A linear external damping coefficient is applied to enable power extraction and a

supplementary mass is added to the system to tune the point absorber to the incoming wave conditions.

These coefficients are optimized according to a few constraints to maximize the power absorption by the

array.

Moreover, the performance of point absorbers attached to a floating platform is analyzed and the equation

of motion is developed for the same. Lastly, the performance of a point absorber in shallow water is

evaluated at a location on the Portuguese coast. This is done by transforming the wave spectra for shallow

depths.

Keywords: wave energy; arrays of point absorbers; floating platform; shallow water.

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RESUMO

Este trabalho presenta uma detalhada análise hidrodinâmica dos conversores de energia das ondas do tipo

“point absorber”. O estudo inclui o desenvolvimento de um modelo no domínio da frequência linear de

forma a avaliar a performance de sistemas de tipo “point absorber” no modo de arfagem. Os parâmetros

hidrodinâmicos são obtidos com WAMIT. O clima das ondas escolhido compreende os estados do mar

mais prováveis. Três formas de flutuadores com calados variáveis são examinadss para determinar a forma

e as dimensões mais adequadas para um “point absorber”. Além disso, a performance de várias

configurações (linear, grelha, circular, concêntrica) de sistemas de “point absorbers” é avaliada. O efeito da

direcção das ondas é também considerado. Um coeficiente de amortecimento linear externo é aplicado para

permitir a extracção de energia e uma massa suplementar é adicionada ao sistema para ajustar o point

absorber às condições das ondas. Os coeficientes vão ser optimizados de acordo com algumas restrições

para obter a máxima absorção de energia no sistema.

Além disso, a performance dos “point absorbers” ligados a uma plataforma flutuante é analisada e a relativa

equação de movimento é desenvolvida. Finalmente, a performance de um “point absorber” em águas pouco

profundas é avaliada num local na costa de Portugal, através da transformação dos espectros das ondas para

baixas profundidades.

Palavras-chave: energia das ondas, sistemas de tipo point absorber; plataforma flutuante; águas pouco

profundas.

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TABLE OF CONTENTS

DEDICATION II

ACKNOWLEDGMENTS III

ABSTRACT IV

RESUMO V

TABLE OF CONTENTS VI

LIST OF FIGURES IX

LIST OF TABLES X

CHAPTER 1 1

OVERVIEW OF OCEAN WAVE ENERGY DEVICES 1

1.1 OCEAN WAVE ENERGY 1

1.2 CLASSIFICATION OF WECs 3 1.2.1 Classification According to Distance from Shoreline 3 1.2.2 Classification According to Type of Technology 4

1.2.2.1 Attenuator 4 1.2.2.2 Overtopping 5 1.2.2.3 Oscillatory Water Column (OWC) 6 1.2.2.4 Oscillatory Wave Surge Converter (OWSC) 8 1.2.2.5 Point Absorbers 8 1.2.2.6 Pressure Differential 11

1.2.3 Others 11

CHAPTER 2 12

PERFORMANCE OF A SINGLE POINT ABSORBER 12

2.1 INTRODUCTION 12

2.2 FREQUENCY DOMAIN MODELLING 12 2.2.1 Equation of motion of a heaving point absorber 12 2.2.2 Power absorption 14 2.2.3 Phase control 15 2.2.4 WAMIT 16

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2.2.4.1 Geometry Description 16

2.3 WAVE CLIMTE 17

2.4 CASE STUDY 18

2.5 RESULTS AND DISCUSSIONS 18 2.5.1 Hydrodynamic coefficients for different floater shapes 18 2.5.2 Hydrodynamic coefficients for different floater drafts 19

2.6 CONCLUSIONS 20

CHAPTER 3 21

PERFORMANCE OF ARRAYS OF POINT ABSORBERS 21

3.1 INTRODUCTION 21

3.2 CASE STUDY 22 3.2.1 Arrangements of array of point absorbers 22 3.2.2 Numerical Modelling 23 3.2.3 Power absorption 24 3.2.4 Floater Motion Restrictions 25

3.2.4.1 Slamming restriction 25 3.2.4.2 Stroke restriction 26 3.2.4.3 Force restriction 27

3.3 OPTIMIZATION 28 3.3.1 Sequential Quadratic Programming 28

3.4 RESULTS AND DISCUSSIONS 29 3.4.1 Floaters in different arrangements 29 3.4.2 Matrix 31 3.4.3 Total power absorption 35 3.4.4 Effect of wave headings 35

3.5 CONCLUSIONS 37

CHAPTER 4 39

PERFORMANCE OF POINT ABSORBERS ATTACHED TO A FLOATING PLATFORM 39

4.1 INTRODUCTION 39

4.2 CASE STUDY 39 4.2.1 Platform Geometry 39 4.2.2 Numerical Modelling 40 4.2.3 Constrained Parameters 41

4.2.3.1 Slamming Restriction 41 4.2.3.2 Stroke Restriction 41 4.2.3.3 Force Restriction 42

4.3 RESULTS AND DISCUSSIONS 42

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4.3.1 Single Floater 42 4.3.2 Multiple Floaters 44 4.3.3 Total power absorption and efficiencies 46

4.4 CONCLUSIONS 47

CHAPTER 5 49

PERFORMANCE IN SHALLOW WATER 49

5.1 INTRODUCTION 49

5.2 STANDARD SPECTRAL MODELS 50

5.3 NUMERICAL MODELLING 52 5.3.1 Equations of motion 52 5.3.2 Hydraulic power take-off system 53

5.4 CASE STUDY: AGUÇADOURA 53

5.5 RESULTS AND DISCUSSIONS 55 5.5.1 Effect of water depth on the spectrum 55 5.5.2 Effect of water depth on the significant wave height 55 5.5.3 Effect of water depth on the scatter diagram 55 5.5.4 Effect of water depth on the wave power resource 56 5.5.5 Effect of water depth on the power absorption and reactive control parameters 57

5.6 CONCLUSIONS 61

CONCLUSIONS 62

REFERENCES 64

APPENDIX A 68

APPENDIX B 71

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LIST OF FIGURES

Figure 1.1: Wind Blowing Over Fetch of Water Produces Waves (EPRI (2005)). .........................................2 Figure 1.2: Wave Energy Levels in [kW/m] (FEM (2009)). ...........................................................................2 Figure 1.3: Attenuator-the Pelamis. .................................................................................................................4 Figure 1.4: Overtopping: the Wave Dragon (Kofoed et al. (2006)).................................................................5 Figure 1.5: Overtopping: the wave plane (WavePlane (2010)). ......................................................................6 Figure 1.6: OWC: (a) the Limpet and (b) the Pico Power Plant. .....................................................................7 Figure 1.7: Water particle trajectories on a slopping beach. ............................................................................8 Figure 1.8: Point absorbers: (a) the Powerbuoy, (b) the Aquabuoy and (c) the AWS. ....................................9 Figure 1.9: Point Absorber: (a) The PS FrogMk5, (b) the Searev and (c) the nodding Duck. ...................... 10 Figure 2.1: Schematic representation of phase control based on Falnes (2002). ........................................... 15 Figure 2.2: JONSWAP wave amplitude spectrum for 8 sea states. ............................................................... 18 Figure 2.3: Three floaters shape (a) hemisphere-cylinder floater (b) cone-cylinder floater (c) hemisphere floater............................................................................................................................................................. 18 Figure 2.4: Comparison of different floater shapes. ...................................................................................... 19 Figure 2.5: Comparison of different floater drafts. ........................................................................................ 20 Figure 3.1: cone-cylinder floater. .................................................................................................................. 22 Figure 3.2: Arrangements of arrays of point absorbers. ................................................................................ 23 Figure 3.3: Response of floaters in different arrangements. .......................................................................... 31 Figure 3.4: Array power absorption for different wave headings. ................................................................. 36 Figure 3.5: Power absorption for linear and grid array. ................................................................................. 37 Figure 4.1: Point absorbers combined with a floating platform. ................................................................... 39 Figure 4.2: Power absorbed with changing draft of the floater. .................................................................... 43 Figure 4.3: Configurations of a single floater attached to a floating platform. .............................................. 43 Figure 4.4: Power absorbed for different floater positions. ........................................................................... 44 Figure 4.5 Different configurations of floaters attached to a floating platform. ............................................ 44 Figure 4.6: Response of each floater attached to a floating platform. ........................................................... 45 Figure 4.7: Power absorbed by floaters in different configurations. ............................................................. 47 Figure 4.8: Efficiency of each floater for different sea states. ....................................................................... 47 Figure 5.1: The limited depth function in the TMA spectrum. ...................................................................... 51 Figure 5.2: Hydraulic PTO model. ................................................................................................................ 53 Figure 5.3: Scatter diagram for Agaçudoura. ................................................................................................ 54 Figure 5.4: TMA spectral shapes for different water depths. ........................................................................ 55 Figure 5.5: Effect of TMA transformations on the reduction of significant wave heights. ........................... 56 Figure 5.6: Probability distribution versus sH for different water depths. ..................................................... 56 Figure 5.7: Power matrices for different water depths. ................................................................................. 58 Figure 5.8: PTOb matrices for different water depths. ..................................................................................... 59 Figure 5.9: PTOk matrices for different water depths. ..................................................................................... 60 Figure A.1: Concentric array of floaters – configuration c2. ......................................................................... 68 Figure A.2: Bottom- mounted cylindrical platform connected with point absorbers. ................................... 69 Figure A.3: Response of each floater. ........................................................................................................... 69

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LIST OF TABLES

Table 2.1: Reference Sea States..................................................................................................................... 17 Table 3.1: Average power matrix (kW) for different array arrangements. .................................................... 32 Table 3.2: External damping coefficient matrix (ton/sec) for different array arrangements.......................... 33 Table 3.3: Supplementary mass coefficient matrix (tons) for different array arrangements.......................... 34 Table 3.4: Total average power absorption for different array arrangements ................................................ 35 Table 3.5: Ranking of arrays for different wave headings. ............................................................................ 37 Table 4.1: Radiation damping versus external damping coefficients (tons/sec) ............................................ 46 Table 5.1: Incident average annual wave power reduction due to water depth effects .................................. 57 Table 5.2: Reduction in wave power absorption due to water depth effects ................................................. 61 Table A.1: Total power absorption for SS-5 for different concentric arrangements. .................................... 69 Table A.2: Difference in total power absorption in linear and grid arrangements as compared to concentric array c2. .......................................................................................................................................................... 70

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CHAPTER 1

OVERVIEW OF OCEAN WAVE ENERGY DEVICES

1.1 OCEAN WAVE ENERGY

Renewable energy such as: solar and wind energy have been extensively studied during the last few

years. However, one energy source which has remained relatively untapped to date is ocean wave energy.

Ocean wave energy has several advantages over other forms of renewable energy since waves are more

constant, more predictable and have higher energy densities enabling devices to extract more power from

smaller volumes at reduced costs and lower visual impact (Brekken (2009)).

Two primary types of ocean energy have been identified as: the mechanical and thermal energy. The

mechanical contribution is produced by the rotation of the earth and the moon’s gravitational influence. The

rotation of the earth creates wind on the ocean surface that forms waves, while the gravitational pull of the

moon creates coastal tides and currents. On the other hand, thermal energy is derived from the sun, which

heats the surface of the ocean while the depths remain colder. This temperature difference allows energy to

be captured and converted to electric power.

Ocean waves are a form of concentrated solar energy which is transferred through complex wind-wave

interactions. The ocean waves are a result of the effects of earth’s temperature variation due to solar

heating, combined with a multitude of atmospheric phenomena - the earth’s rotation and the gravitational

interaction with the moon. This consequently generates wind currents in global scale. Wave generation,

propagation and direction are directly related to wind currents, wind speed, time duration of the wind

blowing and the respective area over the ocean where it is blowing. This area in which a particular set of

waves is developed depends on the size of the pressure fronts involved (fetch) and water depth (see Fig.

1.1).

Figure 1.2 maps the kilowatt per meter (kW/m) crest length of wave power flux around global

coastlines. The crest length is measured from one crest, or peak, to the next. Note that wave energy

increases as the latitudes increase north or south.

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Figure 1.1: Wind Blowing Over Fetch of Water Produces Waves (EPRI (2005)).

In general, wave energy is harnessed by the movements of the device, either floating on the surface of

the ocean or operating below the sea level. The energy is captured by the oscillating motions at the wave

frequency. Many different techniques to convert wave energy into electric power have been studied in the

recent years and consequently various types of technologies aiming to harvest ocean waves have been

proposed. Therefore, a spread of possibilities is available to date, but none seems to be predominant. In the

literature, different classifications can be found; some of them relative to the generation type or to the

power conversion available today. However, in the present thesis classifications relative to the distance

from shoreline: shoreline, near shore and offshore wave energy converters (WEC) (CRES (2006)); and by

the technology type: point absorber, oscillating water column, oscillating waves surge converter,

overtopping, attenuator, submerged pressure differential and others (AEA (2006) and FEMP (2009)) are

discussed.

Figure 1.2: Wave Energy Levels in [kW/m] (FEM (2009)).

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Since WECs are devices which have to operate continuously harvesting ocean waves, they have to fulfil

general requirements such as: survivability, serviceability and practical installation. Hence, WECs can be

designed to survive 1 to 50 year storms or to move to the ‘fail safe zone’ thus avoiding extreme loading

during storms. Moreover, WECs also need to meet requirements such as: (a) good power capture over the

range of most common incident wave frequencies, considering the ‘phase or complex conjugate control’

(Budal and Falnes (1980); Ringwood (2008); Whittaker and Folley (2005), and (b) ensure harmony

between device and power take off (PTO) to achieve maximum efficiency (Ringwood (2008)).

Most of the WECs are oscillating systems, such as Pelamis (Pelamis Wavepower (2010)), and the

PowerBuoy (Ocean Power Technologies (2010)). These wave energy converters absorb ocean energy while

simultaneously generating waves (Falnes and Budal (1978)). Hence, their hydrodynamic problem is usually

studied as combination of the diffraction and radiation problems (Budal and Falnes (1980)). Consequently,

approaches such as the dry and wet oscillator models and WAMIT can be used to study these devices.

WAMIT was employed by (Buchner (2010); De Backer et al. (2009); McCabe et al. (2006); Ricci et al.

(2007)) to study WECs operating in isolation and in array and has proved to be an efficient and accurate

tool for this type of analysis.

Moreover, non-oscillating devices such as the Wave dragon (Kofoed et al. (2006)) and the WavePlane

(WavePlane, 2010) absorb ocean energy by capturing the water volume of overtopped waves in a basin and

creating a hydraulic head (Beels et al. (2010a)). They are studied by a time-dependent mild-slope equation

and the Boussinesq model. A mild-slope equation was used by (Beels et al. (2010b)) to analyze the Wave

Dragon.

1.2 CLASSIFICATION OF WECs

1.2.1 Classification According to Distance from Shoreline

A classification according to distance from shoreline can be established as: shoreline, near shore and

offshore WECs according to CRES (2006). Shoreline WECs are fixed or embedded to the shoreline,

having the advantage of easier installation and maintenance. In addition, shoreline devices do not require

deep-water moorings or long lengths of underwater electrical cable. However, they would experience a

much less powerful wave regime. Furthermore, the deployment of such schemes could be limited by the

requirements for shoreline geology, tidal range, preservation of coastal scenery etc. The most advanced

class of shoreline devices is the oscillating water column. Versions of these devices are: the Limpet, the

European Pilot Plant on the island of Pico in the Azores and the Wavegen.

Near shore WECs are deployed at moderate water depths around 20- 30 m, at distances up to around

500 m from the shore. They have nearly the same advantages as shoreline devices and at the same time are

exposed to higher wave power levels (CRES (2006)). The example of near shore wave energy device is

WaveRoller (WaveRoller (2010)).

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Another type is offshore WECs, which exploits more powerful wave regimes available in deep water

with water depths deeper than 40 m before energy dissipation mechanisms have had a significant effect

(CRES (2006)). In order to extract the maximum amount of energy from the waves, the devices need to be

at or near the surface (i.e. floating), so they usually require flexible moorings and electrical transmission

cables.

Early overseas designs to harness ocean wave energy concentrated on small, modular devices, yielding

high power output when deployed in arrays. In comparison to the previous multi-megawatt designs, these

small size devices were rated at a few tens of kilowatts each. More recent designs for offshore devices have

also concentrated on small, modular devices. Some of the most promising ones are: the McCabe Wave

Pump (CRES 2006) and the Pelamis.

1.2.2 Classification According to Type of Technology

Wave energy technology is rapidly growing and varies widely in application of conversion devices.

Energy conversion devices can be divided in six principal groups: attenuators, overtopping, oscillating

water column (OWC), oscillating wave surge converters (OWSC), point absorbers and pressure differential

devices.

1.2.2.1 Attenuator

An Ocean wave energy converter which is a line absorber is called an attenuator if it is aligned parallel

to the prevailing direction of wave propagation, which effectively rides the waves (Falnes (2007) and

FEMP (2009)). When the attenuator rides the waves, differing heights of waves along the length of the

device causes flexing where the segments connect. This flexing in the segments of the device produces

forces and moments which are captured in the form of hydraulic pressure, which in turn is converted into

energy. An advantage of the attenuator is that it has less area perpendicular to the waves, and hence it

experiences lower forces (AEA (2006)).

Figure 1.3: Attenuator-the Pelamis.

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One of the most known attenuator is the Pelamis. The Pelamis (Fig. 1.3) is a long and narrow (snake-

like) semi-submerged structure which points into the waves. The Pelamis is composed of long cylindrical

pontoons connected by three hinged joints. This device can exploit relative yaw and pitch motions between

sections to capture ocean energy (Ringwood (2008)). Those motions are used to pressurize a hydraulic

piston arrangement and then turn into a hydraulic turbine/generator to produce electricity (EPRI (2005); Al-

Habaibeh et al. (2009)). Other examples of WECs which are classified as attenuators can also be found in

the literature, like the McCabe wave pump, the Ocean Wave Treader and the Wave Treader

(GreenOceanEnergy (2010)) and the Waveberg (Waveberg (2010)).

1.2.2.2 Overtopping

Another option to convert wave energy is by capturing the water that is close to the wave crest. To

achieve this principle, the overtopping devices are designed with long arms and a wall placed between

them. Large arms channel the waves focusing them at the centre wall over which waves topple into a

storage reservoir, which is at a level higher than the average free-surface level of the surrounding sea. The

reservoir creates a head of water and this potential energy is converted into electric energy by a

conventional low-head hydraulic water turbine. Amongst many different types of water turbines, the low-

head “Kaplan” turbine is the most common choice.

The advantage of the overtopping principle is that turbine technology has already been in use in the

hydropower industry for long time and is well understood (Powertech (2009)). However, the disadvantage

is the strong non-linear hydrodynamics of overtopping devices and therefore the hydrodynamic problem of

the overtopping principle cannot be addressed by linear water wave theory (Falcão (2010)).

One of the novel technologies using this principle is the Wave Dragon (Fig. 1.4). The Wave Dragon is

an offshore converter developed in Denmark, whose slack moored floating structure consists of two wave

reflectors focusing the incoming waves towards a doubly curved ramp, a set of low-head hydraulic turbines

(multiple modified Kaplan-Turbines) and a raised reservoir (Kofoed et al. (2006)). The low head hydraulic

turbines convert this low-pressure head into electricity using direct-drive low-speed permanent magnet

generators (EPRI (2005)).

Figure 1.4: Overtopping: the Wave Dragon (Kofoed et al. (2006)).

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Another device which harnesses ocean waves by overtopping waves is the WavePlane (Fig. 1.5). This is

a V-shaped construction anchored with the tip facing the incoming waves. In front, below the surface line,

the wave plane is equipped with an artificial beach that makes the capture of the wave energy more

efficient. The wave plane is symmetrical in its construction. The water from the waves is caught in different

heights through an inlet divided into separate levels. Further the water is let into this inlet tangentially

through the turbine pipe. Through this process it is sought to maintain as much of the water’s kinetic energy

as possible in consideration of the manageable volume. The kinetic energy is used to generate a rotating

stream of water in the turbine pipe. At the end of the turbine pipe which is at the end of the two “legs in the

V-shape” the turbine pipe bends back and downwards. It is able to bring the end of the drive train outside

the turbine pipe. From the turbine the water is led back into the sea. By means of a gear box the turbine

runs a generator, which is connected to the electrical grid (Wave energy centre (2010)). Other devices

developed by this principle are: the Floating Wave Power Vessel (FWPV) (Powertech (2009)), the Tapered

Channel Wave Power Device (TAPCHAN) (Powertech (2009)), the Seawave Slot-Cone Generator (SSG),

the SSG offshore installation (Waveeenergy (2010)), and the WaveBlanket (Gatti (2010)).

Figure 1.5: Overtopping: the wave plane (WavePlane (2010)).

1.2.2.3 Oscillatory Water Column (OWC)

An Oscillating Water Column is a steel or concrete structure with a chamber presenting at least two

openings, one in communication with the sea and the other with the atmosphere. Under the action of waves

the free surface inside the chamber oscillates and displaces the air above the free surface. The air is thus

forced to flow through a turbine. The turbine is usually a bi-directional ‘wells’ turbine, which makes use of

airflow in both directions, on the compression and decompression of the air to extract the ocean power

(ISSC (2006); EPRI (2005); AEA (2006); Al-Habaibeh (2009)). For an OWC an optimum power take-off

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can be achieved if the wave periods are close to the natural period of the water column. Therefore the

design must be tuned to the site-specific wave climate.

These devices too as in the case of overtopping devices can be floating or fixed to the shore. Fixed

structures are located on the shoreline or near shore, standing on the sea bottom (with foundations like

gravity based structures or fixed to the rocky cliff) while floating structure are placed offshore with a

partially submerged chamber with air trapped above a column of water (EPRI (2005)).

Shore line devices have the advantage of an easier installation and maintenance, and do not require

deep-water moorings and long underwater electrical cables. However, less energetic wave climate at the

shoreline is seen to be the main problem. Falcão (2010) suggested that this disadvantage can be partly

compensated by natural wave energy concentration due to refraction and/or diffraction.

Figure 1.6: OWC: (a) the Limpet and (b) the Pico Power Plant.

OWC is one of the wave energy devices which have been studied extensively. The Limpet OWC (Fig.

1.6 (a)) is one that is working on this technology. This is a grid-connected, shoreline-based OWC, with a

rated power of 500 kW. The Limpet used a unique construction method, where construction of the concrete

column structure occurred behind a rock wall, which was then removed using explosives. Unfortunately,

several complications arose due to the presence of debris near and underneath the structure, and the overall

performance of the device was found to be highly dependent on the shape and depth of the seafloor around

the device. The OWC drives a pair of Wells turbines, and provides around 22 kW of power (annual

average), peaking near 150 kW.

Another WEC which is also classified as OWC is the Pico Power Plant. This is an OWC power plant

rated at 400 kW, was installed on the shoreline of the island of Pico, in the Azores, Portugal. The plant

(Fig. 1.6 (b)) uses a concrete structure, mounted on the seabed/shoreline, counting with Wells turbines used

as power take-off. Built in 1995-1999, various problems caused to stop the operation of the device. Testing

resumed again in 2005 and the plant was connected to the local power grid. Unfortunately, the presence of

mechanical resonance in the structure prevented the plant from operating at optimum power levels, limiting

8

it to power production to 20-70 kW range. Other oscillating water columns are: The Wave Energy

Conversion Actuator (WECA) the Osprey OWC, the Port Kembla, the Sakata OWC, the Mighty Whale, the

Orecon MRC1000, the SPERBOY and the OE Buoy.

1.2.2.4 Oscillatory Wave Surge Converter (OWSC)

According to AEA (2006) an OWSC is also a wave surge/focusing device, but extracts the energy that

exists in waves caused by the movement of water particles within them. This principle is then applied for

water depths less than the half of the wave length, where the particle trajectory has large displacement on

the longitudinal direction (see Fig. 1.7).

Figure 1.7: Water particle trajectories on a slopping beach.

One example of this type of technology is the OYSTER. The Oyster is a wave energy converter that

interacts efficiently with the dominant surge forces encountered in the near shore wave climate at depths of

10 to 15 m. The Oyster concept consists of a large buoyant bottom-hinged oscillator that completely

penetrates the water column from the ocean surface to the seabed. The surging action of waves on the

oscillator drives hydraulic pistons which pressurize fresh water causing it to be pumped to shore through

high pressure pipelines. The onshore hydroelectric plant converts the hydraulic pressure into electrical

power via a Pelton wheel, which runs an electrical generator. The low pressure return-water passes back to

the device in a closed loop via a second pipeline. Another device which operates as an OWSC is the

WaveRoller.

1.2.2.5 Point Absorbers

Point absorber is a WEC oscillating with either one or more degrees of freedom. It can either move with

respect to a fixed reference, or respect to a floating reference (Backer et al. (2007)). A point absorber which

is a floating structure moves relative to its own components due to the wave action (e.g., a floating buoy

inside a fixed cylinder) therefore, this relative motion is used to drive an electromechanical or hydraulic

energy converter.

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For a wave energy converter (WEC) device, to be a point absorber, the linear dimension has to be much

smaller than the prevailing wave length. As general rule, to consider a WEC as a point absorber, its

respective diameter should be preferably in the range of five to ten percent of prevailing wavelengths

(Falnes and Lillebekken (2003)). Point absorbers devices, can be classified according to the degree of

freedom from which they capture the ocean energy. A brief explanation of the respectively mode of capture

are described below.

a) Heaving systems

Figure 1.8: Point absorbers: (a) the Powerbuoy, (b) the Aquabuoy and (c) the AWS.

Heaving point absorber are the ones that harness ocean wave energy based on heave motion while the

remaining movements are restricted by the mooring system which is fixed to the ocean bottom. During the

past years, heaving point absorbers have been studied and developed to capture ocean wave energy because

of the simplicity in their hydrodynamic problem and similarity with the well-known buoys. Moreover, most

of these devices are similar in nature with a difference in the PTO system.

An example of a point absorber device is the Powerbuoy, developed by Ocean Power Technologies

(Fig. 1.8 (a)). The construction involves a floating structure with one component relatively immobile, and a

second component with movement driven by wave motion (a floating buoy inside a fixed cylinder). The

relative motion is used to drive electromechanical or hydraulic energy converters. A PowerBuoy

demonstration unit rated at 40 kW was installed in 2005 for testing offshore from Atlantic City, New

Jersey, U.S. testing in the Pacific Ocean is also being conducted, with a unit installed in 2004 and 2005 off

the coast of the Marine Corps Base in Oahu, Hawaii.

Another example is The AquaBuoy (Fig. 1.8 (b)) (Weinstein et al. (2004)) being developed by the

AquaEnergy Group, Ltd. is a point absorber that is the third generation of two Swedish designs that utilize

the wave energy to pressurize a fluid that is then used to drive a turbine generator. The vertical movement

of the buoy drives a broad, neutrally buoyant disk acting as a water piston contained in a long tube beneath

the buoy. The water piston motion in turn elongates and relaxes a hose containing seawater, and the change

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in hose volume acts as a pump to pressurize the seawater. The AquaBuoy design has been tested using a

full-scale prototype, and a 1 MW pilot offshore demonstration power plant is being developed offshore at

Makah Bay, Washington.

Other point absorbers that have been tested at prototype scale include the Archimedes Wave Swing

(AWS) (Fig. 1.8 (c)) (Prado (2008)). This device is quite different relative to the ones described above. The

AWS is a submerged structure unlike the Aquabuoy and the Powerbuoy which are floating on the ocean

surface. The technology of the AWS consists of an air-filled cylinder that moves up and down as waves

pass over. This motion relative to a second cylinder fixed to the ocean floor is used to drive a linear

electrical generator. A 2 MW capacity device has been tested offshore of Portugal (Gardner (2005)). Other

point absorber devices are the Swedish heaving buoy, with linear electrical generator, the Norwegian

heaving buoy and the IPS buoy (Edinburgh University).

b) Pitching systems

Pitching systems are other oscillating-body systems in which the energy conversion is based on relative

rotation (mostly pitch) rather than translation. Examples of this point absorber type are: The PS FrogMk5

(McCabe (2006)) which consists of a large buoyant paddle with an integral ballasted handle hanging below

it (Fig. 1.9 (a)). Waves act on the blade of the paddle and the ballast beneath provides the necessary

reaction. When the WEC is pitching, power is extracted by partially resisting the sliding of a power-take-

off mass which moves in guides above sea level.

Figure 1.9: Point Absorber: (a) The PS FrogMk5, (b) the Searev and (c) the nodding Duck.

Another example is the nodding Duck created by Stephen Salter, from the University of Edinburgh and

the Searev (Babarit et al. (2005)) wave energy converter, developed at Ecole Centrale de Nantes, France, is

a floating device enclosing a heavy horizontal-axis wheel serving as an internal gravity reference (Fig. 1.9

(b)). The centre of gravity of the wheel being off-centred, this component behaves mechanically like a

pendulum. The rotational motion of this pendular wheel relative to the hull activates a hydraulic PTO

which, in turn, sets an electrical generator into motion. Major advantages of this arrangement are that: (a)

11

all the moving parts (mechanical, hydraulic, electrical components) are sheltered from the action of the sea

inside a closed hull, and (b) the choice of a wheel working as a pendulum involves neither end-stops nor

any security system limiting the stroke.

1.2.2.6 Pressure Differential

This is a submerged device typically located nearshore and attached to the seabed. The motion of the

waves causes the sea level to rise and fall above the device therefore, inducing a pressure differential in the

device. The alternating pressure can then pump fluid through a system to generate useful energy (AEA

(2006)).

One example of this technology is the CETO. The CETO units are fully submerged and permanently

anchored to the sea floor meaning that there is no visual impact as the units are out of sight. This also

assists in making them safe from the extreme forces that can be present during storms. They are self-tuning

to tide, sea state and wave pattern, making them able to perform in a wide variety of wave heights and in

any direction. CETO units are manufactured from steel, rubber and hypalon materials, all proven for over

20 years in a marine-environment (Carnegie (2010)). Another device operating by the same principle is the

WaveMaster (Powertech Labs Inc. (2009)).

1.2.3 Others

There are different kinds of WECs which cannot fit in the previous classifications of WECs. The

anaconda is one of these devices. This is all-rubber WEC which operates in a completely new way. When

the anaconda is operating, the wave squeezes the tube (at the bow) and starts a bulge running. But as it runs

the wave runs after it, squeezing more and more, so the bulge gets bigger and bigger. The bulge runs in

front of the wave where the slope of the water (pressure gradient) is highest. In effect the bulge is surfing

on the front of the wave concentrating the wave power over a wide frontage, at the end of the tube, which

can be used to drive a turbine to generate electricity (Anaconda (2010)). Other WECs are: the BioWave

(BioWave (2010)), The Netherlands, WaveRotor (EcoFyes (2010)) and the TETRON (Carbon trust

(2010)).

12

CHAPTER 2

PERFORMANCE OF A SINGLE POINT ABSORBER

2.1 INTRODUCTION

A linear frequency domain model has been employed to compute the behaviour of a heaving point

absorber system. The hydrodynamic parameters are obtained with WAMIT, a software package based on

boundary element method. A linear external damping coefficient is applied to enable power absorption and

a supplementary mass is introduced to tune the point absorber to the incoming wave conditions. Three

floater shapes are examined with different waterline drafts to obtain the most suitable shape and dimension

of the point absorber to be used for designing the array of point absorbers.

2.2 FREQUENCY DOMAIN MODELLING

2.2.1 Equation of motion of a heaving point absorber

In this section, the response of a point absorber, oscillating in a harmonic wave with respect to a fixed

reference is discussed. The motion of the point absorber is restricted to the heave mode only. In equilibrium

position the floater has a draft d. Due to the vertical wave action, the floater has a position z from its

equilibrium position. The equation of motion of this point absorber can be described by Newton’s second

law:

2

2 ,ex rad res damp tund zm F F F F Fdt

= + + + + (2.1)

where m is the mass of the floater, 2

2

d zdt

the floater acceleration, exF the exciting wave force and radF the

radiation force. The radiation force can be decomposed with linear theory into a linear added mass term and

a linear hydrodynamic damping term given by

( ) ( )2

2 .rad a hydd z dzF m b

dtdtω ω= − − (2.2)

The hydrostatic restoring force resF is the buoyancy force bF minus the gravity force gF which corresponds

to the spring force. With a linear spring constant k , the hydrostatic restoring force can be expressed as

13

( ) ,res b gF F F V t mg kzr= − = − = − (2.3)

where ( )V t is the instantaneous, submerged floater volume. The spring constant or hydrostatic restoring

coefficient is given by Eq. (2.4) with wA as the waterline area.

wk gAr= (2.4)

The term dampF defined in Eq. (2.1) is the external damping force exerted by the power take-off (PTO)

system and tunF is the tuning force to control the phase of the floater. The damping and tuning forces are

determined by the power take-off and control mechanism respectively, and are in practical applications

typically non-linear. However, for simplicity they have been assumed to be linear. Hence, the damping

force is given by

,damp extdzF bdt

= (2.5)

where extb is the linear damping coefficient originating from the PTO used to enable power extraction.

A linear tuning force can be realised for instance by means of a supplementary mass term (see Vantorre

et al. (2004)) or an additional spring term (see Ricci et al. (2007)). In the present study, a supplementary

mass term has been used. The supplementary inertia is realized by adding two equal masses at both sides of

a rotating belt. In that way, the inertia of the system can be increased without changing the draft of the

floater. For practical applications, the tuning force, associated with the supplementary mass, is more likely

to be realized by a control mechanism, such as by the PTO, rather than by adding real masses to the device.

The tuning force associated with the equation of motion is expressed by

2

sup 2 .tund zF mdt

= (2.6)

Taking into account all the considerations, the equation of motion of a heaving point absorber can be re-

written as

( ){ } ( ) ( ){ } ( ) ( ) ( )2

sup 2 , .a hyd ext ex

d z t dz tm m m b b kz t F t

dtdtω ω ω+ + + + + = (2.7)

The steady solution of Eq. (2.7) is given by

( )sin ,A motz z tω β= + (2.8)

14

where Az and motβ are of the form

( ) ( )

( )( ){ } ( ){ },

2 22sup

,( )

ex AA

a hyd ext

Fz

k m m m b b

ωω

ω ω ω ω=

− + + ⋅ + + ⋅ (2.9)

( )( )

( )( ) 2sup

tan hyd ext

a

b b

k m m m

ω ωβ

ω ω

+ ⋅= −

− + + ⋅ (2.10)

( )( )( )( ) ( )( )

( )( )( )( ) ( )( )

1 2sup2

sup

1 2sup2

sup

tan for , if 02 2

3tan for , if 02 2

hyd exta

a

hyd exta

a

b bk m m m

k m m m

b bk m m m

k m m m

ω ω p pβ ω ωω ω

βω ω p pβ ω ω

ω ω

−

−

+ ⋅− − < < − + + ⋅ >

− + + ⋅= + ⋅

− < < − + + ⋅ <− + + ⋅

(2.11)

exmot Fβ β β= − (2.12)

where Az is the amplitude of heave motion, β is the phase lag and corresponds to the counter clockwise

angle from the positive real axis, motβ is the phase angle of the floater motion and exFβ is the phase angle of

the heave exciting force.

2.2.2 Power absorption

A harmonically oscillating body is assumed, with velocity v, and subjected to a force ( )F t :

( ) cos( t )A FF t F ω β= + (2.13)

( ) cos( t )A vv t v ω β= + (2.14)

The power averaged over a period T can be expressed as:

1 cos( )2av A A F vP F v β β= − (2.15)

The average absorbed power of a point absorber is equal to the average excited power minus the average radiated power: , ex, rad,abs av av avP P P= − (2.16)

According to Eq. (2.15), the average exciting power can be expressed as:

15

, ,1 cos( )2ex av ex A A F vP F v β β= − (2.17)

with F vγ β β= − the phase shift between F̂ and v̂ . Combining Eq. (2.2) and Eq. (2.14) gives the average

radiated power:

(2.18)

Hence, the average power absorption is given by:

(2.19)

or, alternatively, ,abs avP can be expressed as the power absorbed by the power take-off system: (2.20)

The power absorbed in irregular is obtained by the linear superposition of floater responses.

2.2.3 Phase control

Generally, the natural frequency of a point absorber system is higher than the wave frequency. The

natural frequency can be decreased by adding supplementary mass, as explained by a flywheel

mechanically coupled with the vertical motion of the floater, or by an additional spring term with a negative

spring coefficient. The effect of this tuning is shown in Figure 2.1. The solid line shows the water

elevation. This line would correspond to the floater position if its mass were negligible. The dashed line

illustrates the position of the floater in case the inertia of the point absorber is increased so that the natural

frequency of the device corresponds to the wave frequency. This is called ‘optimal’ control (tuning).

Water elevation

Position of resonating buoy (with large inertia)

Position of latched buoy (with small inertia)

2 2 2,

1 12 2abs av ext A ext AP b v b zω= =

Figure 2.1: Schematic representation of phase control based on Falnes (2002).

2,

1P2rad av hyd Ab v=

2, ,

1 1cos( )2 2abs av ex A A F v hyd AP F v b vβ β= − −

16

In practical applications it might be difficult to realize the tuning by changing the supplementary inertia

dependent on the incoming waves. The tuning force, as described in Eq. (2.6) can also be delivered by the

power take-off system. In that case, it would be required to return some energy back to the sea during some

small fractions of each oscillation cycle and benefit from this during the remaining time. For this reason

‘optimum control’ is also denoted by ‘reactive control’. It is clear that in order to obtain this optimum

control in practice, a reversible energy- converting mechanism with very low conversion losses is required.

It will be shown in this thesis that the required tuning forces and associated instantaneous power levels

might be much larger than the damping force and the corresponding power absorption values, respectively.

Hence, these tuning forces will influence the design of the power take-off system and might possibly result

in an uneconomic solution. The tuning forces can be limited. However, this can be associated with large

power losses, depending on the restrictions and the sea states, which will be illustrated in the next chapters.

Another phase control technique is ‘latching’. A mechanism holds the floater in a fixed position when it

has reached an extreme excursion, i.e. when the velocity equals zero. The floater is released again at a

certain time (approximately one quarter of the natural period nT before the next extremum in the exciting

force occurs). The motion of a point absorber subjected to latching control is illustrated with the dash-

dotted line in Figure 2.1. Latching induces a non-linear response of the point absorber.

2.2.4 WAMIT

WAMIT is a program developed for the computation of wave loads and motions of floating or

submerged offshore structures. It is based on linear (and second-order) potential theory. WAMIT solves the

diffraction and radiation problem for a given geometry and for given frequencies and returns the first order

hydrostatic and hydrodynamic parameters. The version employed in this work (v6) is restricted to first

order potential theory only.

2.2.4.1 Geometry Description

Two different approaches are possible to discretize the body surface. The first method is the low-order

method (or panel method). The second method is the higher-order method. In the low-order method, the

body surface is approximated with small quadrilateral panels. The velocity potential is assumed constant in

each panel. Hence, the integral equations with the velocity potential as unknown, consist of a set of

piecewise constant integrals that must be satisfied at the centroid of each panel. The panels are described

by the coordinates of each vertex and the coordinate list is generated using Rhinoceros, a NURBS-based 3-

D modelling software.

A more efficient method is the higher order method, where the velocity potential is represented by

continuous B-splines and the body surface by smooth continuous surfaces, called ‘patches’. The patches

can be described analytically or by means of B-spline functions. This higher-order option generally leads

to more accurate results than the low-order method for the same CPU time. However, CPU time is

17

generally not an issue in the present work, since the considered bodies are small and axisymmetric. Due to

the symmetry, only a quarter of the body needs to be modelled, requiring very few computation time.

Hence, both methods have been applied, but no significant differences were found.

2.3 WAVE CLIMTE

The calculations are performed for eight different sea states of the floater measurements as described in

Table 2.1. The first sea state covers sH from 0.00 to 0.50 m, the second sea state covers the range between

0.50 m and 1.00 m, and so on. The combinations of significant wave height sH and peak period pT are

representative of the North Sea. The wave amplitude spectra, ( )A

S fζ is based on the parameterized

JONSWAP spectrum (see Liu et al. (1997)) and is of the form

( )4

2 4 5 5exp ,4

s

A

ps s p

fS f H f f

fβ

ζ α γ− − =

(2.20)

with γ as the peak enhancement factor ( )3.3γ = , the peak frequency pf , sα and sβ are given by

0.0624 ,0.1850.230 0.0336

1.9

sαγ

γ

=

+ − +

(2.21)

2

2 2

( )exp .

2p

sp

f ff

βs

−= −

(2.22)

Table 2.1: Reference Sea States

Sea state sH [m] pT [s]

1 0.25 5.24

2 0.75 5.45

3 1.25 5.98

4 1.75 6.59

5 2.25 7.22

6 2.75 7.78

7 3.25 8.29

8 3.75 8.85

The value of the spectral width parameters depends on the frequency

for 0.07

. for 0.09

p

p

f ff f

s<

= ≥ (2.23)

18

The calculated wave amplitude spectra corresponding to eight sea states are shown in Fig. 2.2.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.000

5

10

15

20

25

S ζ,Α(ω

) [m

2 /rad]

ω (rad)

Hs = 0.25 m Hs = 0.75 m Hs = 1.25 m Hs = 1.75 m Hs = 2.25 m Hs = 2.75 m Hs = 3.25 m Hs = 3.75 m

Figure 2.2: JONSWAP wave amplitude spectrum for 8 sea states.

2.4 CASE STUDY

The performance of a single point absorber will be evaluated for three floater shapes namely - a

hemisphere-cylinder floater, a cone-cylinder floater with cone having an apex angle of 90° and a

hemisphere floater, each with waterline radius of 2.5 m. The cone-cylinder and hemisphere-cylinder floater

have an extended cylindrical part of 0.5 m as shown in Fig. 2.3.

(a)

(b)

(c)

Figure 2.3: Three floaters shape (a) hemisphere-cylinder floater (b) cone-cylinder floater (c) hemisphere floater.

Moreover, the calculations are performed for three cone-cylinder floater drafts d = 3m, 3.5m, and 4.5m.

2.5 RESULTS AND DISCUSSIONS

2.5.1 Hydrodynamic coefficients for different floater shapes

In Figs. 2.4(a)-(c), the hydrodynamic added mass, damping coefficients and heave excitation forces for

the three floater shapes namely hemisphere-cylinder floater, cone-cylinder floater and hemisphere floater

are presented, respectively. The hydrodynamic added mass, damping coefficient and excitation force for the

cone-cylinder floater is observed to be higher as compared to the other floater shapes, which indicates that

the cone-cylinder floater is a better wave absorber.

19

In Fig. 2.4(d), the wave power extracted by the different floaters is plotted versus significant wave

height. As suggested from Figs. 2.4(a-c), the cone-cylinder floater is a better wave absorber as compared to

hemisphere-cylinder floater and hemisphere floater as it has better hydrodynamic properties and

coefficients. The average power absorbed by the cone-cylinder floater is observed to be higher as compared

to other two floaters. However, the difference is not huge. The hemisphere floater performs slightly better

than the hemisphere-cylinder floater.

0,05 0,10 0,15 0,20 0,25 0,300

10

20

30

mas

s (to

n)

Frequency (Hz)

cone-cylinder hemisphere-cylinder hemisphere

0,05 0,10 0,15 0,20 0,25 0,300

10

20

b hyd (

tons

/sec

)

Frequency (Hz)


(a): Added mass coefficients. (b): Hydrodynamic damping coefficients.

0,05 0,10 0,15 0,20 0,25 0,300

40

80

120

160

200

F ext,A

(kN/

m)

Frequency (Hz)


0 1 2 3 40

40

80

120

P abs (

kW)

Hs (m)


(c): Heave exciting force per wave amplitude. (d): Power absorbed by different floaters.

2.5.2 Hydrodynamic coefficients for different floater drafts

In Figs. 2.5(a)-(d), the hydrodynamic added mass, damping coefficients, heave excitation forces and

power absorption for the cone-cylinder floater with drafts d = 3 m, 3.5 m, and 4.5 m are presented,

Figure 2.4: Comparison of different floater shapes.

20

respectively. The conical part remains the same for all the cases and only the height of the top cylindrical

part increases.

0,05 0,10 0,15 0,20 0,25 0,300

10

20

30

mas

s [to

n]

Frequency [Hz]

d-3m d-3.5m d-4.5m

0,05 0,10 0,15 0,20 0,25 0,300

5

10

15

20

25

b hyd [

tons

/sec

]

Frequency [Hz]

d-3m d-3.5m d-4.5m

(a): Added mass coefficients. (b): Hydrodynamic damping coefficients

0,05 0,10 0,15 0,20 0,25 0,300

40

80

120

160

200

F ext,A

[kN/

m]

Frequency [Hz]

d-3m d-3.5m d-4.5m

0 1 2 3 40

40

80

120

P abs [

kW]

Hs [m]

d-3m d-3.5m d-4.5m

(c): Heave exciting force per wave amplitude. (d): Power absorbed by different floaters.

Figure 2.5: Comparison of different floater drafts.

As seen from the above graphs, a smaller draft is associated with a larger added mass, hydrodynamic

damping coefficient and the amplitude of the heave exciting force which ultimately leads to larger power

absorption and is confirmed by Fig 2.5(d).

2.6 CONCLUSIONS

By means of a linear frequency domain model, the behavior of a heaving point absorber is simulated

and the absorbed power is calculated. The hydrodynamics parameters of the oscillating floaters are derived

with BEM code WAMIT. Three floater shapes namely – cone-cylinder, hemisphere-cylinder and

hemisphere are compared in terms of their hydrodynamic properties and the cone-cylinder floater performs

slightly better than the others followed by hemisphere in terms of wave power extraction. Also, the draft of

the floater was varied and the power absorption rises for smaller drafts of the floater. Therefore, the cone-

cylinder floater with draft of 3 m is used to study the performance of arrays of point absorbers in the next

chapters.

21

CHAPTER 3

PERFORMANCE OF ARRAYS OF POINT ABSORBERS

3.1 INTRODUCTION

The point absorbers can also operate in arrays to produce considerable amounts of power from ocean

waves similar to wind energy farms. The array of point absorber devices which have been developed

consist of a large structure containing multiple closely spaced oscillating bodies such as Wave Star,

Manchester Bobber and FO3. Several theoretical models were developed by researchers in order to deal

with the waves and interacting bodies. Budal (1977), Evans (1980) and Falnes (1980) adopted the point

absorber approximation to derive the expressions for the maximum power that can absorbed by an array of

point absorbers. This approximation relies on the assumption that the bodies are small compared to the

incident wave lengths so that the wave scattering within the array can be neglected while calculating the

interactions. A theory accounting more accurately for the wave body interactions is the plane wave

approximation which is based on the assumption that the bodies are widely spaced relative to the incident

wavelengths, so that the radiated and circular scattered waves can be locally approximated by plane waves

(see Simon (1982), McIver (1984) and McIver and Evans (1984)). This theory is not suitable for the closely

spaced bodies as the wide spacing requirement is not fulfilled except for very short wave lengths. Contrary

to the point absorber approximation, the plane wave approximation is also suitable to study the power

absorption by an array in sub-optimal conditions as scattering might be relatively important in that case.

In the present study, the performance of the multiple point absorbers in the absorption of wave energy

is analysed for several arrangements of floater arrays. The results obtained for circular and concentric

arrangement will be compared with linear and grid arrangement (Backer et al. (2009)). Apart from the total

amount of power absorbed by multiple bodies, attention is paid to the individual power absorption and the

difference in performance of floaters within a certain configuration. The efficiency of each floater in

different arrays is also calculated to compare the performance of each floater. For multiple bodies optimal

control parameters are not only dependant on the incoming waves, but also on the position and the

behaviour of other floaters. The control parameters are optimized for individual floaters. The variation of

the individual floater control parameter is analysed to see the uniformity of the control forces. Finally, the

wave heading angle is also varied to see which configuration performs better for different angles of

incidence.

22

3.2 CASE STUDY

The shape of the floater considered for the array of point absorbers is cone-cylinder with a cone of apex

angle of 90° and a cylindrical upper part as shown in Fig. 3.1. The diameter of the floater is considered to

be 5 m and an equilibrium draft of 3 m.

3.2.1 Arrangements of array of point absorbers

In order to analyse the arrangement of multiple point absorbers, the study is carried out for four

different arrays of floater arrangements. An array of 12 cone-cylinder floaters is considered forming a part

of the WEC. The four array arrangements considered are linear, grid circular and concentric for the analysis

on the performance of the array of heaving point absorbers as shown below in Fig. 3.2.

In Fig. 3.2(a), a linear arrangement for the array of 12 floaters with 6 floaters at the top and 6 floaters at

the bottom is presented. The floaters are placed in a rectangular fixed structure with four supporting

columns at the edges with each floater at a distance of 6 m in vertical direction and 3 m in horizontal

direction from each other. The inter-spacing between the two successive rows and successive columns is 11

m and 8 m respectively. The influence of the columns is neglected and the incoming waves propagate in the

direction of the x-axis. The floaters are assumed to oscillate in the heave mode. These assumptions will be

valid for all the arrangements of floaters.

In Fig. 3.2(b), an arrangement for the array of 12 floaters in a staggered grid is considered. The floaters

are placed in a square fixed structure with four supporting columns at the edges. The inter-spacing between

the two successive rows is 6.5 m.

0.5 m

5 m

2.5 m

1

2

3

4

5

6

7

8

9

10

11

12

6m

3m

Figure 3.1: cone-cylinder floater.

y

x

23

(a): Linear array of floaters.

(b): Grid array of floaters.

(c): Circular array of floaters (d): Concentric array of floaters.

Figure 3.2: Arrangements of arrays of point absorbers.

In Fig. 3.2(c), a circular arrangement for the array of 12 floaters having a radius of 24 m is presented.

The floaters are placed in a square fixed structure with four supporting columns at the edges and each

floater is kept at a distance of 21.5 m from the center. In Fig. 3.2(d), a concentric arrangement for the array

of 12 floaters is presented. The outer floaters have a radius of 24 m, while the inner ones have a radius of

12 m.

3.2.2 Numerical Modelling

The equation of motion for N multiple bodies, oscillating in heave in a regular wave with angular

frequencyω can be expressed based on the linear theory in frequency domain, and is given by

1

2

3

4

5

6

7

8

9

10

11

12

21.5m

1

2

3

4

5

6

7

8

9

10

11

12 1.5m

4m

8m

24 m

21.5 m

9.5 m

1

3

7

10

9

5

12

11

4 2

6

8

24

( ) ( )2

sup ,a ext hyd exM M M Z j B B Z KZ Fω ω− + + + + + =d d d d

(3.1)

where Zd

is the complex amplitude of the floater positions, M is the mass matrix of the floaters and K is

the matrix with hydrostatic restoring coefficients or stiffness matrix. The added mass matrix and

hydrodynamic damping matrix are denoted by aM and hydB , respectively. They are both symmetric N N×

matrices with the hydrodynamic interaction coefficients on the non-diagonal positions. The vector exFd

containing the complex amplitudes of the heave exciting forces and the hydrodynamic parameters aM ,

hydB and exFd

are calculated based on the BEM approach using WAMIT. Since the natural frequency of the

floaters is generally smaller than the incident wave frequencies, the floaters are often tuned towards the

characteristics of the incident wave to augment power extraction. In the present study, a tuning force

proportional to the acceleration has been implemented by means of a supplementary mass matrix,

sup sup ,jNM m I= with sup

jm as the supplementary mass for thj floater and NI as the N N× identity matrix.

The linear external damping matrix extB enabling power extraction is given by

,jext ext NB b I= (3.2)

where jextb is the external damping for thj floater.

3.2.3 Power absorption

The average power absorbed in a regular wave is given by

2 20.5 .abs ext AP b zω= (3.3)

The response in irregular long-crested waves is obtained by superimposing the response in regular waves.

The wave amplitude of those regular wave components is derived from the JONSWAP spectrum obtained

by

2 ( ) .AA Sζζ ω ω= ∆ (3.4)

The spectrum of the amplitude of the floater position is defined as

( ) ( ),

2 2, ,

, 2,

,2A i

j jA i A ij

z iA i

z zS Sζω ω

ωζ= =

∆ (3.5)

where i corresponds to thi frequency, from a range of 40 frequencies, ranging between 0.22 rad/sec to

1.885 rad/sec with 0.0427ω∆ = rad/sec. Assuming Rayleigh distribution of the floater motion amplitudes,

some characteristic values such as the significant amplitude of the floater motion can be obtained as

, ,

0

2 .A sign

j jz iz S dω

∞

= ∫ (3.6)

25

In irregular waves, the available power over the diameter D of the point absorber is expressed as

( ) ( ),0

.Aavail D gP D gC S dζr ω ω ω

∞

= ∫ (3.7)

The absorbed power in a regular wave is given in Eq. (3.3). By applying linear superposition of the floater

responses, the total power absorption in irregular waves for each floater j can be obtained by

( )2

2

0

,A

jj j A

abs extA

zP b S dζω ω ωζ

∞ =

∫ (3.8)

All the floaters are equipped with their own power take-off system. The absorption efficiency of each

floater, denoted by jη is defined as the ratio of the absorbed power to the incident wave power within the

device width given by

,

.j

j abs

avail D

PP

η = (3.9)

It may be noted that, in order to maximize the power absorbed, the two external parameters extb and supm

have to be optimized. The optimization carried out in this thesis is described later in this chapter. Further,

several restrictions are also introduced, in order to avoid unrealistic solutions, such as extremely large

floater motions.

3.2.4 Floater Motion Restrictions

3.2.4.1 Slamming restriction

Slamming is a phenomenon that occurs when the buoy re-enters the water, after having lost contact with

the water surface. The floater experiences a slam, which may result in very high hydrodynamic pressure

and loads. These impacts have a very short duration, with a typical order of magnitude of milliseconds.

However, they may cause local plastic deformation of the material. Fatigue may also be observed by

repetitive slamming pressures and can be responsible for structural damage of the material. For this reason,

a restriction has been implemented, requiring that the significant amplitude of each floater position relative

to the free water surface elevation should be smaller than a fraction α of the draft d of the floater:

( ),

,j j

A signz dζ α− ≤ ⋅ (3.10)

where jz .is the position of the floater j , jζ is the water elevation at the centre of the floater j , and α is a

parameter which is arbitrarily chosen. In the present study, α is chosen as 1. The slamming constraint is a

weak restriction, however, for bodies with a very small draft, this may transform into a quite stringent

restriction. In order to implement this restriction in the numerical model, the motion of the point absorber

relative to the wave needs to be known. Considering a harmonic wave component cosA tζ ω , this relative

floater motion can be written as

26

( ), cos cos ,j j jrel wave A mot Az z t tω β ζ ω= + − (3.11)

where jmotβ is the phase angle of the thj floater position and j

Az is the heave amplitude of thj floater.

Using the trigonometric transformations, the relative floater motion is obtained as

( ), cos cos sin sin .j j j j jrel wave A mot A A motz z t z tβ ζ ω β ω= − − (3.12)

Hence, the relative motion amplitude is of the form

( ) ( )2 2

, , cos sin .j j j j jA rel wave A mot A A motz z zβ ζ β= − + (3.13)

The significant value of this relative motion amplitude can be determined as given in Eqns. (3.5) and (3.6)

is obtained as

2, , , , , , / 2 ,j j

z rel wave i A rel wave iS z ω= ∆ (3.14)

, , , , , ,0

2 .j jA rel wave sign z rel wave iz S dω

∞

= ∫ (3.15)

The velocity of the floater relative to the vertical velocity of the water surface ,rel wavev can be expressed as

( ), sin sin .j j jrel wave A mot Av z t tω ω β ωζ ω= − + + (3.16)

The amplitude of this relative velocity is given by

( ) ( )2 2

, , cos sin .j j j j jA rel wave A mot A A motv z zω β ζ β= − + (3.17)

This slamming restriction might require a decrease in the tuning parameter supm and/or increase of the

external damping coefficient extb . Not only the occurrence probability of slamming will be reduced by this

measure, but also the magnitude of the associated impact pressures and loads will drop, since they are

dependent on the impact velocity of the body relative to the water particle velocity and this impact velocity

will decrease when the control parameters of the floaters are adapted according to the constraints imposed.

3.2.4.2 Stroke restriction

In practice, many point absorbers are very likely to have restrictions on the floater motion, such as

restriction imposed by the limited height of the rams in case of a hydraulic conversion system or by the

limited height of a platform structure enclosing the oscillating point absorbers. Therefore, a stroke

constraint is implemented, imposing a maximum value on the significant amplitude of the body motion of

the form

, , ,max .jA sign A signz z≤ (3.18)

27

In the present study, a maximum value of the significant amplitude is chosen to be 2.50 m. Further, the

position spectrum is defined as:

2, , / 2 .j j

z i A iS z ω= ∆ (3.19)

The significant amplitude of the floater motion can be obtained as

, ,0

2 .j jA sign z iz S dω

∞

= ∫ (3.20)

Hence, the constraint on the significant amplitude on the floater motion is not an absolute constraint, but a

restriction associated with a statistical exceedance probability. The implementation of constraints on the

body motion leads to smaller body motions (assumption for the linear model). In practice, when the

maximum stroke is nearly reached, an additional mechanism will have to brake the floater motion.

3.2.4.3 Force restriction

In some cases, the optimal control parameters for maximum power absorption result in very large

control forces. The tuning force, in particular, might become very high and can even be a multiple of the

damping force. If this tuning force is to be delivered by the PTO, it might result in a very uneconomic

design of the PTO system. For this reason, it is interesting to study the response of floaters in case the total

force is restricted. The force spectrum is given by

2, , / 2 ,j j

F i A iS F ω= ∆ (3.21)

and the significant amplitude of the force is defined as

, F,0

2 .j jA sign iF S dω

∞

= ∫ (3.22)

Then the significant amplitude of the damping and the tuning force, respectively, for floater j are given by

( ),, , F

0

2 ,bext A

j jbext A signF S dω ω

∞

= ∫ (3.23)

( )sup,sup, ,

0

2 .m A

jm A sign FF S dω ω

∞

= ∫ (3.24)

A restriction imposed on the significant amplitude of the total force is expressed as

sup

2 2tot, , , , , , .

ext

jA sign b A sign m A signF F F= + (3.25)

In the present study, the significant value of the total control force on each of the floater is limited to

200kN.

28

3.3 OPTIMIZATION

The optimization of power absorption by an array of point absorbers is necessary in order to achieve the

maximum efficiency and also to make the device economically viable. The control parameters supjm and j

extb ,

are modified to obtain the maximum power for a corresponding sea state. The optimization of average

power is done keeping in mind the constraints, which are non-linear. For multiple bodies optimal control

parameters are not only dependant on the incoming waves, but also on the position and behaviour of the

other floaters. The optimization can be either carried out for a specific array, for a particular sea state, or for

each of the floaters in an array (individual optimization) with separate control values for each of them. The

latter gives better results as in Backer et al. (2009).

In the present study, all the floaters are individually optimized for all the sea states corresponding to the

wave climate, i.e. for every floater separate values of supjm and j

extb are determined. In the case of 12 floaters

and 8 sea states, 12*8*2 = 144 control parameters are determined. While carrying out the optimization, all

the constraints are also taken into account as discussed earlier. For the constrained optimization, there are

two techniques - first one is indirect approach in which the problem is solved by transforming it into an

unconstrained problem. The second approach is to handle the constraints without transformation which is a

direct approach. In the present study, we use the direct approach method of Sequential Quadratic

Programming (SQP) as the quadratic programming sub-problem implemented in MATLAB.

3.3.1 Sequential Quadratic Programming

Sequential Quadratic Programming (SQP) is one of the most successful methods for the numerical

solution of constrained nonlinear optimization problems. It relies on a profound theoretical foundation and

provides powerful algorithmic tools for the solution of large-scale technologically relevant problems. The

application of the SQP methodology to nonlinear optimization problems (NLP) considered is of the form

minimize (x) over x ,subject to ( ) 0, ( ) 0.

nfh x g x

∈= ≤

(3.26)

where : nf → is the objective functional, the functions : n mh → and : n pg → describe the

equality and the inequality constraints. The basic idea of SQP methodology is to model non-linear problem

(NLP) at a given approximate solution, say, by a quadratic programming sub-problem, and then to use the

solution to this sub-problem to construct a better approximation. This process is iterated to create a

sequence of approximations that, it is hoped, will converge to a solution. The key to understanding the

performance and theory of SQP is the fact that, with an appropriate choice of quadratic sub-problem, the

method can be viewed as the natural extension of Newton and quasi-Newton methods to the constrained

optimization setting. Thus one would expect SQP methods to share the characteristics of Newton-like

methods, namely, rapid convergence when the iterates are close to the solution but possible erratic

29

behaviour that needs to be carefully controlled when the iterates are far from a solution. While this

correspondence is valid in general, the presence of constraints makes both analysis and implementation of

SQP methods significantly more complex. A drawback of SQP algorithm for individual tuning is that it

might converge to a local maximum, depending on the initial conditions. Hence, the choice of initial

conditions is very important. The control parameters from previously done work have been used as initial

values from which individual control parameters are obtained.


In this section, the numerical results are presented for power absorption, control parameters, significant

amplitude of motion for each of the 12 floaters and also their efficiencies in different array arrangements.

The results have been calculated for all the sea states in the wave climate and all the control

parameters jextb and sup

jm have been optimized to get the maximum absorbed power for each case.

3.4.1 Floaters in different arrangements

The results presented in this section are for each of the 12 floaters in different arrangements for the sea

state 5 ( 2.25sH m= and 7.22spT = ) and wave heading of 0° . These results give us an idea of the

behaviour of each floater in terms of power absorption, control parameters, amplitude of motion,

efficiencies and control forces on each of them.

Fig. 3.3(a) shows the average power absorption by each floater for different array configurations. The

average power absorbed is most uniformly distributed in the circular arrangement and also for each radii of

the concentric arrangement. In linear and grid arrangements, the absorbed power is found to be unevenly

distributed, with the front floaters performing better and extracting higher power as compared to the rear

ones. Figs. 3.3(b-d) present the significant amplitude of motion and the control parameters for each floater.

In the case of circular arrangement it is observed that the control parameters are uniform for all the floaters.

Also for concentric arrangement, the control parameters are stable for floaters arranged in different radii.

However, for linear and grid arrays the control parameters vary from one floater to the other. The control

parameters being uniform for the 12 floaters will lead to efficient power take-off system. Hence, the control

mechanism to maximize power extraction can be achieved easily and efficiently.

Figs. 3.3(e-g) shows the significant amplitude of the control forces and the total control force exerted on

each floater in different array arrangements. For the given sea state and wave heading angle in

consideration, each floater in all the arrangements reach the maximum control force limit of 200kN.

Further, for circular arrangement and for floaters placed in each radii of the concentric arrangement, the

control forces on each of the floater are uniformly distributed in contrast to the linear and grid

arrangements. Hence, the floater WEC system is more uniform and efficient in circular and concentric

arrangements as compared to linear and grid arrangements. The absorption efficiency for each floater in

different array configuration is presented in Fig. 3.3(h). As stated above, for circular and concentric arrays,

30

the floaters are equally efficient. Whereas for linear and grid arrangements, the front floaters perform better

than the ones at the rear.

1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

40

50

P abs (

kW)

Floater Number

circular linear grid concentric

1 2 3 4 5 6 7 8 9 10 11 120.0

0.4

0.8

1.2

1.6

2.0

z sign (

m)

Floater Number


(a): Average power absorption. (b): Significant amplitude of motion.

1 2 3 4 5 6 7 8 9 10 11 120

20

40

60

80

100

120

b ext (

ton/

s)

Floater Number


1 2 3 4 5 6 7 8 9 10 11 120

20

40

60

80

100

120

140

msu

p (to

n)

Floater Number


(c): External damping coefficients. (d): Supplementary mass coefficients.

1 2 3 4 5 6 7 8 9 10 11 120

20

40

60

80

100

120

140

F bext

,A,s

ign (k

N)

Floater Number


1 2 3 4 5 6 7 8 9 10 11 120

20

40

60

80

100

120

140

F msu

p,A,

sign (

kN)

Floater Number


(e): Significant amplitude of the damping force. (f): Significant amplitude of the tuning force.

31

1 2 3 4 5 6 7 8 9 10 11 120

100

200F to

t,A,s

ign (k

N)

Floater Number


1 2 3 4 5 6 7 8 9 10 11 120.0

0.2

0.4

0.6

0.8

1.0

η

Floater Number


(g): Significant amplitude of total control force. (h): Absorption efficiency of each floater.

Figure 3.3: Response of floaters in different arrangements.

3.4.2 Matrix

In this section, the matrix for average power absorption and control parameters j

extb and supjm are shown in

Tables 3.1-3.3 respectively. The values are calculated for wave heading 0θ = ° for each floater in different

arrangements and for all the sea states in the wave climate as shown in Sec. 2.4. All the control values in

Tables 3.2 and 3.3 have been optimized to give the corresponding maximum power absorption values

shown in Table 3.1.

Table 3.1 shows the average power matrix for all the arrangements considered. The power absorbed by

each floater is evaluated for every sea state for concentric, circular, grid and linear arrangements of floaters.

It is observed that the power values increases with higher wave height, as expected.. It is further observed

that in linear and grid arrangement, the front floaters perform much better than the rear floaters and there is

a considerable difference in the power values, whereas in the circular and concentric arrangements the

power is more uniformly distributed among the floaters, for all the sea states considered. The floaters have

been represented by F(j), with j being the floater number.

In Table 3.2, the external damping coefficient matrixes jextb are obtained for concentric, circular, grid

and linear array arrangements. It is observed that, the damping coefficient increases with higher waves in

linear and grid arrangements whereas in circular and concentric arrangements the highest values are in the

moderate sea state. In Table 3.3, the supplementary mass coefficient matrix supjm is obtained for concentric,

circular, grid and linear array of floater arrangements. It is observed that the control values increase with

higher waves for circular and concentric arrangements, whereas for linear and grid arrangements, the

highest values are in the moderate sea state.

32

Table 3.1: Average power matrix (kW) for different array arrangements.

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,63 0,58 0,58 0,55 0,55 0,53 0,53 0,39 0,39 0,55 0,55 0,655,72 5,5 5,5 5,62 5,62 4,99 4,99 4,01 4,01 5,44 5,44 5,93

17,26 17,25 17,25 16,81 16,81 14,2 14,2 14,34 14,34 16,02 15,97 15,8426,49 25,97 25,97 24,44 24,44 22,68 22,68 23,47 23,47 24,08 24,02 23,6333,39 32,63 32,63 31,27 31,27 30,64 30,64 31,16 31,16 30,96 30,92 30,4940,51 39,81 39,81 38,92 38,92 38,62 38,62 38,52 38,52 37,73 37,7 37,3248,36 47,73 47,73 47,06 47,06 46,6 46,6 45,9 45,9 44,61 44,59 44,2356,32 55,46 55,46 55,08 55,08 53,59 53,59 53 53 51,28 51,26 50,46

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,67 0,67 0,54 0,54 0,43 0,43 0,38 0,38 0,36 0,36 0,23 0,236,49 6,49 5,17 5,17 4,1 4,1 3,8 3,8 3,52 3,52 2,32 2,32

18,21 18,21 15,45 15,45 13,6 13,6 13,21 13,21 11,74 11,74 8,46 8,4626,83 26,83 24,04 24,04 22,57 22,57 21,69 21,69 19,65 19,65 16,64 16,6434,52 34,52 31,64 31,64 29,91 29,91 28,33 28,33 25,75 25,75 22,76 22,7642,43 42,43 39,2 39,2 36,95 36,95 34,72 34,72 31,75 31,75 28,65 28,6550,56 50,56 46,82 46,82 44,01 44,01 41,25 41,25 37,95 37,95 34,79 34,7958,37 58,37 54,1 54,1 50,8 50,8 47,65 47,65 44,17 44,17 41,12 41,12

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,72 0,62 0,57 0,45 0,48 0,53 0,47 0,53 0,57 0,68 0,32 0,376,53 5,58 5,72 4,72 5,08 5,37 4,9 5,21 5,22 6,43 3,07 3,47

18,13 16,64 17,74 16,33 16,85 16,58 15,67 15,5 14,36 17,26 9,78 10,4927,92 26,49 27,06 25,55 26,15 24,72 24,19 23,35 21,97 25,14 17,56 18,1336,5 35,02 34,95 32,89 33,95 31,2 32,24 29,83 28,12 31,85 23,6 24,21

44,63 43,07 42,53 39,92 41,42 37,59 40,67 36,3 34,31 38,55 29,56 30,2752,61 50,98 50,09 47,02 48,93 44,16 49,35 42,94 40,65 45,37 35,75 36,5659,98 58,35 57,18 53,76 56,07 50,48 57,82 49,39 46,8 51,89 42,05 42,92

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,45 0,61 0,61 0,52 0,52 0,41 0,6 0,6 0,52 0,52 0,55 0,554,95 5,66 5,66 5,18 5,18 3,89 5,61 5,61 5,24 5,24 5,4 5,4

16,55 16,21 16,21 16,08 16,08 12,04 16,19 16,19 16,5 16,5 16,04 16,0425,18 25,13 25,13 24,68 24,68 19,69 25,33 25,33 25,5 25,5 24,16 24,1632,83 33,5 33,5 31,96 31,96 25,94 33,68 33,68 33,13 33,13 30,87 30,8740,92 41,89 41,89 39,06 39,06 32,2 41,78 41,78 40,45 40,45 37,47 37,4749,42 50,31 50,31 46,25 46,25 38,71 49,79 49,79 47,8 47,8 44,22 44,2257,8 58,33 58,33 53,19 53,19 45,25 57,33 57,33 54,82 54,82 50,75 50,75

Circular arrangement

Linear arrangement

Grid arrangement

Concentric arrangement

33

Table 3.2: External damping coefficient matrix (ton/sec) for different array arrangements.

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 79,5 81,1 81,1 88,2 88,2 83,3 83,3 87 87 86,2 86,2 79,70,75 90 94,1 94,1 97,4 97,4 91,3 91,3 99 99 95 95 89,41,25 103,8 104,1 104,1 103,7 103,7 110,4 110,4 114,8 114,8 105,9 106,1 103,21,75 85,6 87,7 87,7 92,1 92,1 99,2 99,2 98,9 98,9 93,9 94,2 91,22,25 72,3 74,8 74,8 78,9 78,9 82,1 82,1 82 82 80,3 80,4 77,42,75 59,2 61,2 61,2 63,6 63,6 65,2 65,2 66 66 66,6 66,6 64,13,25 45,9 47,2 47,2 48,5 48,5 49,6 49,6 51,2 51,2 53,5 53,5 51,93,75 32,7 27,1 27,1 34,1 34,1 24,2 24,2 37,6 37,6 41,1 41,1 34,4

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 26,3 26,3 27,5 27,5 28,9 28,9 28 28 27,2 27,2 27,3 27,30,75 25,5 25,5 26,7 26,7 28,2 28,2 26,8 26,8 26,2 26,2 26,3 26,31,25 41,4 41,4 36 36 33,1 33,1 31,8 31,8 27,3 27,3 23,6 23,61,75 63,2 63,2 61,7 61,7 61,5 61,5 59,7 59,7 55,9 55,9 52,2 52,22,25 78,1 78,1 78,7 78,7 79,4 79,4 77,7 77,7 75,2 75,2 75,5 75,52,75 89,4 89,4 90,9 90,9 91,8 91,8 90,6 90,6 89 89 91,8 91,83,25 99,1 99,1 100,8 100,8 101,9 101,9 101 101 100,3 100,3 104,7 104,73,75 109,2 109,2 110,9 110,9 112 112 111,5 111,5 111,7 111,7 117,1 117,1

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 26,7 27,6 27,5 30,2 25,4 26,7 24,6 25,8 28 24,5 26,3 25,60,75 27,2 28,2 26 28 24,5 25,9 23,4 25,1 28,1 24,5 25,8 25,41,25 42,2 40,4 42,1 39,9 41,4 37,9 37,8 36 32,8 39,3 24,6 24,91,75 64,9 65,3 65,6 64,6 66 60,8 64 60 56,7 60,7 53,2 542,25 79,1 80,5 80,4 80,3 81,9 76 82,4 75,9 72,4 74,8 74,9 74,92,75 89,4 91,2 91,1 91,4 93,4 87,2 95,7 87,6 83,9 85,5 90,2 89,63,25 98,2 100,2 99,9 100,4 102,7 96,7 106,3 97,4 93,9 94,9 102,4 101,53,75 107,5 109,6 109 109,5 112,1 106,6 116,8 107,8 104,7 104,9 114,4 113,3

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 93,1 81,1 81,1 86,3 86,3 82,8 82,3 82,3 88,4 88,4 85,4 85,40,75 101,8 91,5 91,5 96,3 96,3 92,8 91,9 91,9 97,5 97,5 94,8 94,81,25 107,4 105,4 105,4 107,4 107,4 116,5 105,9 105,9 107,2 107,2 106,2 106,21,75 92,3 90,2 90,2 92,9 92,9 109,6 90 90 91,5 91,5 93,1 93,12,25 76,5 73,3 73,3 78 78 95,9 73,1 73,1 76,1 76,1 79,6 79,62,75 59,9 57,3 57,3 63,7 63,7 80,5 57,7 57,7 61,5 61,5 66,3 66,33,25 44,1 42,7 42,7 50,2 50,2 65,5 43,8 43,8 47,8 47,8 53,5 53,53,75 29,2 29,2 29,2 37,6 37,6 51,3 31 31 35 35 41,6 41,6


Linear arrangement

Grid arrangement


34

Table 3.3: Supplementary mass coefficient matrix (tons) for different array arrangements.

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 26 26,5 26,5 25 25 23,8 23,8 27 27 24,3 24,3 260,75 27,1 26,6 26,6 23,5 23,5 24,1 24,1 25,5 25,5 23,7 23,7 25,51,25 41,8 41,5 41,5 39,1 39,1 33,4 33,4 35,8 35,8 37,4 37,3 36,41,75 64,2 63,5 63,5 61,3 61,3 61,1 61,1 63,8 63,8 61,3 61,5 59,52,25 77,9 77,8 77,8 77,7 77,7 79,9 79,9 81,7 81,7 77,3 77,4 74,72,75 88,8 89,4 89,4 90,8 90,8 93,3 93,3 94,2 94,2 88,9 89,1 85,93,25 98,7 99,8 99,8 101,8 101,8 104 104 104,2 104,2 98,8 98,9 95,43,75 109,4 127,4 127,4 112,9 112,9 142,2 142,2 114,1 114,1 109,1 109,2 126,3

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 82,9 82,9 81,4 81,4 81,2 81,2 85,8 85,8 83,8 83,8 85,2 85,20,75 92,6 92,6 90,8 90,8 92,2 92,2 95,8 95,8 93,3 93,3 96 961,25 98,9 98,9 106,7 106,7 114,5 114,5 115,5 115,5 117,3 117,3 124,6 124,61,75 84,3 84,3 93,4 93,4 99,9 99,9 103,4 103,4 110,2 110,2 125,4 125,42,25 70 70 77,7 77,7 83,7 83,7 89,1 89,1 97,3 97,3 112 1122,75 56 56 62,6 62,6 68,3 68,3 74,3 74,3 82,6 82,6 95,1 95,13,25 42,8 42,8 48,6 48,6 54 54 60,2 60,2 67,9 67,9 77,8 77,83,75 30,2 30,2 35,8 35,8 41 41 46,9 46,9 54 54 61,1 61,1

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 78,1 77,3 84,4 88,1 90,7 86,7 90,5 84,4 79,7 83,4 82,8 81,40,75 88,3 88,6 95,5 98,7 100,4 94,8 98,9 93,3 87,9 92,1 92,4 90,91,25 99,4 105,2 102,5 107,7 107,3 104,3 108,7 106,6 107 100,7 120,1 118,81,75 81,3 86,1 85,2 90,2 89,9 91,8 95,2 95,2 96,8 88,2 118,9 115,42,25 65,7 69,3 69,9 75,2 74 79,4 78,1 82,1 84,4 75,5 105,8 1022,75 52,2 54,9 56,1 61,2 59,3 66,6 60,6 68,7 71,5 63 89,6 863,25 39,9 42 43,4 48,3 45,6 54,2 44,3 55,8 59,3 51,2 73,4 70,33,75 28,6 30,2 31,9 36,5 33 42,7 29,1 43,8 47,9 40,1 58,1 55,6

Hs F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F120,25 24,9 25 25 24,7 24,7 24,6 25,7 25,7 24,9 24,9 25,3 25,30,75 23,2 25 25 24,1 24,1 24,4 25,9 25,9 24,7 24,7 25 251,25 40,3 38,5 38,5 38,7 38,7 28,5 38,8 38,8 39,9 39,9 38 381,75 65,1 63,9 63,9 63,5 63,5 56,8 64,2 64,2 64,6 64,6 61,6 61,62,25 82,4 80,7 80,7 79,6 79,6 76 80,6 80,6 80,7 80,7 77 772,75 95,3 92,9 92,9 91,1 91,1 89,7 92,3 92,3 92,4 92,4 88,3 88,33,25 105,8 103 103 100,8 100,8 101 102 102 102,1 102,1 97,9 97,93,75 116 113,2 113,2 110,7 110,7 112,6 111,8 111,8 111,9 111,9 108,1 108,1


Linear arrangement

Grid arrangement


35

3.4.3 Total power absorption

The total average power absorbed, which is the sum of the power absorbed by each floater of the array,

is shown for different configurations in Table 3.4. These values are the maximum power that can be

absorbed by different arrays based on the constraints specified. It is observed that the concentric

arrangement performs better for all the sea states, except SS-1, closely followed by the circular

arrangement. The values marked in red are the maximum value for an array, for each sea state. The rank of

the arrangements in terms of total average power absorption (for wave heading 0θ = ° ) is - concentric >

circular > grid > linear.

Table 3.4: Total average power absorption for different array arrangements

Linear Grid Circular Concentric

0.25 5.21 6.31 6.48 6.47

0.75 50.79 61.30 62.77 63.02

1.25 161.32 185.32 190.29 190.64

1.75 262.83 288.22 291.34 294.48

2.25 345.80 374.36 377.16 385.04

2.75 427.42 458.82 465.01 474.41

3.25 510.78 544.41 556.36 564.86

3.75 592.41 626.68 643.58 651.92

3.4.4 Effect of wave headings

In Figs. 3.4(a-e) the variation of total average power absorption for different array arrangements are

plotted for various wave heading angles. It is observed that the concentric and circular arrangements

perform better than linear and grid arrangements, for all the wave headings except 90° For wave heading

angle of 90° , the linear arrangement performs better than other arrays. For circular and concentric

arrangements the total power absorption remains the same for different wave headings angles because of

symmetry. Whereas for linear arrangement the total power absorption increases with change in wave

heading θ from 0° to 90° . The grid array absorbs the maximum power for wave headings of 60° and 90°

Figs. 3.5(a-b) show the effect of change in wave heading angle on total power absorption for linear and grid

arrays, respectively. In table 3.5, various array arrangements have been ranked according to the total power

absorption for different wave headings, with concentric array performing the best closely followed by

circular array.

36

0 1 2 3 40

200

400

600

P abs (

kW)

Hs (m)


0 1 2 3 40

200

400

600

P abs (

kW)

Hs (m)


(a): For 0° wave heading. (b): For 30°wave heading.

0 1 2 3 40

200

400

600

P abs (

kW)

Hs (m)


0 1 2 3 40

200

400

600P ab

s (kW

)

Hs (m)


(c): For 45° wave heading. (d): For 60° wave heading.

0 1 2 3 40

200

400

600

P abs (

kW)

Hs (m)


(e): For 90° wave heading.

Figure 3.4: Array power absorption for different wave headings.

37

0 1 2 3 40

200

400

600

P abs (

kW)

Hs (m)

θ = 0ο

θ = 30ο

θ = 45ο

θ = 60ο

θ = 90ο

0 1 2 3 40

200

400

600

P abs (

kW)

Hs (m)

θ = 0ο

θ = 30ο

θ = 45ο

θ = 60ο

θ = 90ο

(a): linear array. (b): grid array.

Figure 3.5: Power absorption for linear and grid array.

Table 3.5: Ranking of arrays for different wave headings.

Wave

heading

Linear Grid Circular Concentric

0° 4 3 2 1

30° 4 3 2 1

45° 4 3 2 1

60° 3 4 2 1

90° 3 4 3 2

3.5 CONCLUSIONS

The arrays of heaving point absorbers in circular and concentric arrangement are analysed to understand

the performance of arrays of point absorbers in conversion of wave energy. The linear and grid

arrangement of the floaters is also analysed and compared with the circular and concentric arrangement of

the floaters. The numerical simulations for the determination of hydrodynamic forces and coefficients are

obtained using WAMIT. The floaters are restricted in their motion on the basis of slamming, stroke and

force constraints restrictions. The optimal control parameters, external damping coefficient ( jextb ) and

supplementary mass ( supjm ), are determined to maximize the power absorption by each floater. Sequential

Quadratic Programming (SQP) method is used to optimize the performance of the floaters. The following

conclusions are drawn from the results obtained for different arrangements of the floaters:

38

• The concentric and circular arrangements perform better in terms of power absorption as compared to

the linear and grid arrangements.

• In the linear and grid arrangements, the absorbed power is unevenly distributed among the floaters,

with front floater being more efficient, whereas in the circular and concentric arrangements, the power

distribution is uniform.

• The control parameters jextb , sup

jm and the corresponding control forces are uniform in circular and

concentric arrangements which make the WEC system easier to control and more efficient for

maximum wave power extraction.

• The circular and concentric arrangements absorb more power than grid and linear arrangements for

wave heading angles of 0° , 30° , 45° and 60° for all the sea states in the wave climate

39

CHAPTER 4

PERFORMANCE OF POINT ABSORBERS ATTACHED TO A FLOATING PLATFORM

4.1 INTRODUCTION

In the present study, the performance of multiple point absorbers attached to a floating platform is

analysed for different arrangements of floaters. The floater draft and the distance between the floater and

the platform are varied to see the effect on the power absorption. The control parameters are optimized for

individual floaters for each configuration. For multiple bodies optimal control parameters are not only

dependant on the incoming waves, but also on the position and the behaviour of the other floaters in the

array and in this case, as well as on the position and the behaviour of the floating platform. Apart from the

total amount of power absorbed by multiple bodies, attention is paid to the individual floater power

absorption and the difference in performance between the floaters by calculating the efficiency of each

floater. Further, the variation of wave heading angles is taken into account as well.

4.2 CASE STUDY

4.2.1 Platform Geometry

The calculations have been performed for heaving point absorbers moving with respect to a floating

platform with overall dimensions 10m 10m 1m.× × The hydrodynamic characteristics of the platform are

calculated with boundary element method (BEM) software. The floating platform geometry considered in

the present study is shown in Fig. 4.1 connected to cone-cylinder floaters on each side.

Figure 4.1: Point absorbers combined with a floating platform.

40

4.2.2 Numerical Modelling

Time domain models are necessary for the analysis of wave energy conversion systems due to the

presence of nonlinearities which are contributed in varying degrees by fluid compressibility, power take

off system and control mechanisms. However, for the purpose of simplicity, all the nonlinearities have

been neglected in the present analysis and the study is carried out to access the effects of a floating

platform for wave energy extraction by the array of point absorbers. Thus, a frequency domain model for

the point absorber attached to a fixed and a floating platform is presented and described in detail in this

section.

The frequency domain modelling for a point absorber attached to a fixed and a floating platform is

presented and described in detail. The floaters are assumed to be oscillating in heave mode only and also

that there is absence of back influence. The modelling for a fixed platform case has already been

performed before and now it l be translated for a floating platform. In the case of a heaving point absorber

oscillating with respect to a floating platform, Eq. (2.7) describing the motion of a point absorber has to be

modified accordingly. Since the generator and the supplementary mass move together with the platform,

the forces associated with the control parameters, extb and supm , are dependent on the floater velocity

relative to the platform and the floater acceleration relative to the platform, respectively. Hence, the

equation of motion of the floater relative to the floating platform is given by

{ }22 2

sup2 2 2( ) ( ) ( ),plat plata hyd ext ex

d z dzd z d z dz dzm m m b b kz Fdt dt dtdt dt dt

ω ω ω

+ + − + + − + = (4.1)

where platz is the displacement of the floating platform which is obtained in a similar way by solving the

equation of motion of the heaving platform, as done before for a floater. The hydrodynamic coefficients of

the platform are obtained using WAMIT. Rearranging Eq. (4.1) gives

{ } { }22

sup sup2 2( ) ( ) ( ) .plat plata hyd ext ex ext

d z dzd z dzm m m b b kz F m bdt dtdt dt

ω ω ω+ + + + + = + + (4.2)

Considering ,ex rF to be the modified exciting force as a result of the presence of a floating platform, which

is given by

2

, sup 2( ) .plat platex r ex ext

d z dzF F m b

dtdtω= + + (4.3)

In order to obtain the steady state solution of the floater motion, the amplitude, , ,ex r AF and phase shift, ,ex rFβ

of ,ex rF needs to be determined. Considering ( ),

platj tplat plat Az z e ω β+= and ,( t )

, , ,Fex rj

ex r ex r AF F e ω β+= the

complex amplitude of ,ex rF can be expressed as

41

,, , ,

Fex rjex r ex r AF F e β

=

2, , sup ,

F plat platexj j jex A ext plat A plat AF e j b z e m z eβ β βω ω= + −

( )( )

2, , sup ,

2, , sup ,

cos sin cos

sin cos sinex

ex

ex A F ext plat A plat plat A plat


F b z m z

j F b z m z

β ω β ω β

β ω β ω β

= − −

+ + − (4.4)

The phase angle ,ex rFβ can be computed as

,

2, , sup ,

2, , sup ,

sin cos sintan .

cos sin cosex

ex r

ex

ex A F ext plat A plat plat A platF


F b z m zF b z m z

β ω β ω ββ

β ω β ω β

+ −= − −

(4.5)

So, the amplitude of heave and the phase angle of the floater relative to the platform can be calculated the

same way as in Eqns. (2.9) – (2.12).

4.2.3 Constrained Parameters

4.2.3.1 Slamming Restriction

In order to implement the restriction in the numerical model, the motion of the point absorber relative

to the wave needs to be known. So the presence of a floating platform doesn’t change the slamming

restriction and is given by Eq. (3.10).

4.2.3.2 Stroke Restriction

In case of a floating platform, the stroke restriction reduces the probability that the oscillating point

absorber hits the platform. Therefore, a stroke constraint is implemented imposing a maximum on the

significant value of the amplitude of the floater motion relative to the platform motion, which is given by

, , ,max( ) ( ) .plat A sign plat A signz z z z− ≤ − (4.6)

The motion of the floater relative to the platform is given by

,cos( ) cos( ).plat A mot plat A platz z z t z tω β ω β− = + − + (4.7)

2 2, ,( ) ( cos cos ) ( sin sin )plat A A mot plat A plat A mot plat A platz z z z z zβ β β β− = − + −

2 2, ,2 cos( ).A plat A A plat A mot platz z z z β β= + − − (4.8)

In the present study, a maximum value of the significant relative amplitude is chosen to be 2 m. The

position spectrum is defined as

2, ,( ) / 2 .z i plat A iS z z ω= − ∆ (4.9)

The significant amplitude of the floater motion is obtained as

42

, ,0

( ) 2 .plat A sign z iz z S dω∞

− = ∫ (4.10)

4.2.3.3 Force Restriction

Both the control force and the force due to power absorption are dependent on the platform motions.

The significant amplitude of the damping and the tuning force are given by

2

2, ,

0

( )12 ( ) ,2ext

plat Ab A sign ext

A

z zF b S dζω ω ω

ζ

∞ − =

∫ (4.11)

\ sup

22

, , sup0

( )12 ( ) .2

plat Am A sign

A

z zF m S dζω ω ω

ζ

∞ − =

∫ (4.12)

A restriction will be imposed on the significant amplitude of the total control force which is expressed as

sup

2 2tot, , , , , , .

extA sign b A sign m A signF F F= + (4.13)


In this section, the results for power absorption, control parameters, significant amplitude of motion of

floaters heaving with respect to a floating platform are presented and discussed in detail. The study is

carried out for different sea states in the wave climate, and the control parameters extb and supm is optimized

to get the maximum absorbed power for each configuration of the floater arrangement.

4.3.1 Single Floater

The analysis of a single floater heaving with respect to a floating platform is performed. Firstly, the

draft of the cone- cylinder floater is varied to see the effect on the average power absorption. Three

different drafts of d = 3 m, 3.5 m and 4 m of the cone cylinder floater is considered, each with the same

conical lower part but with increasing height of the top cylinder. In Fig. 4.2, the average power absorbed

by a cone-cylinder floater with different drafts is presented.

It is observed that as the draft of the floater increases, the power absorption is reduced. This is perhaps

due to the difference in hydrodynamic properties with change in drafts. However, the difference is not so

great. For SS-5, the floater with d = 3 m absorbs around 6% and 16% more power than the ones with

higher drafts of d = 3.5 m and d = 4 m, respectively. The slamming constraint becomes less stringent with

increasing draft, but its effect is not prominent because of the already imposed stronger stroke and force

restrictions on the floater motions.

43

1 2 30

20

40

60

P abs (

kW)

Hs (m)

d = 3 d = 3.5 d = 4

Figure 4.2: Power absorbed with changing draft of the floater.

(a) (b) (c)

Figure 4.3: Configurations of a single floater attached to a floating platform.

Next, the distance between the platform and the floater is varied to analyse the change in power

absorption with change in the spacing. The distance from the edge of the platform to the centre of the

floater is assumed to be y´. The three configurations considered with y´ = 5 m, 7.5 m and 10 m are shown

in Figs. 4.3 (a-c).

In Fig. 4.4, the average power absorbed by the floaters in different configurations is shown. The floater

placed at y´ = 10 m performs better than the other configurations for all the sea states. The configurations

with y´= 7.5 m and y´ = 5 m are closer in terms of their performance for lower wave heights. However, the

former outperforms the latter for higher sea states. For SS-5, the floater placed at y´ = 10 m absorbs about

12% and 23% more power as compared to the floaters placed at y´ = 7.5 m and y´ = 5 m, respectively.

1 1

1

44

1 2 30

20

40

60

y´ = 10m y´ = 7.5m y´ = 5m

P abs (

kW)

Hs (m)

Figure 4.4: Power absorbed for different floater positions.

4.3.2 Multiple Floaters

(a)

(b)

(c)

Figure 4.5 Different configurations of floaters attached to a floating platform.

The performance of multiple floaters attached to a floating platform is evaluated for different

configurations of multiple floaters j as shown in Figs. 4.5(a-c) with j = 4, 6, and 8 respectively. The

control parameters extb and supm for each of the floaters, is optimized to get the maximum absorbed power

for sH = 2.25m and pT = 7.22s. The calculations are performed for 0° wave heading.

The wave is considered to be propagating along the positive x-axis, according to the co-ordinate system

shown next to the configuration as in Fig. 4.5(c). The floaters are arranged alternatively at distances y´=

7.5 m and y´ = 10 m on the sides of the platform, for different configurations. The study is performed for

1 2

3

4

5 6

1 2

3

4

5 6

7

8

1 2

3

4

y

x

45

four, six and eight floaters covering all the sides of the rectangular platform to analyse the performance of

each floater, placed at different points relative to the floating platform.

1 2 3 4 5 6 7 80

10

20

30

40

50

60

P abs (

kW)

Floater Number

j = 8 j = 6 j = 4

1 2 3 4 5 6 7 80,0

0,5

1,0

1,5

2,0

2,5

j = 8 j = 6 j = 4

(z-z

plat) A,

sign (

m)

Floater Number

(a): Power absorbed by each floater. (b): significant amplitude of relative motion.

1 2 3 4 5 6 7 80

40

80

120

160

j = 8 j = 6 j = 4

b ext (

ton/

s)

Floater Number

1 2 3 4 5 6 7 8

0

20

40

60

j = 8 j = 6 j = 4

msu

p (to

n)

Floater Number

(c): External damping coefficients. (d): Supplementary mass coefficients.

Figure 4.6: Response of each floater attached to a floating platform.

The time-averaged power absorption, the significant amplitude of floater motion relative to the floating

platform, the external damping coefficients and the supplementary mass coefficients for each of the

floaters in all the three configurations as shown in Figs. 4.5 (a-c) are presented in Figs. 4.6(a-d),

respectively. For the sea state SS-5, the floaters in configuration 4.5(c) absorb more power than the other

two configurations. It is observed that the floaters at position 2 and 5 absorb the most power, followed by

floaters number 3 and 4. The floaters 1, 6, 7 and 8 do not perform well. This study helps us to determine

the efficient positions to attach the floaters to a floating platform. It can be seen from Figs. 4.6(b-d) that the

values for the significant amplitude of relative motion, the external damping coefficients and the

46

supplementary mass coefficients follow a similar pattern for different configurations and every floater

responds differently to absorb the maximum power.

The values for external damping coefficients are higher for floaters 1, 2, 5 and 6, that is, for the floaters

on the top and the bottom side of the platform. The corresponding supplementary mass coefficients are

higher for the floaters 3, 4, 7 and 8, that is for the floaters attached on the left and right side of the

platform.

Table 4.1: Radiation damping versus external damping coefficients (tons/sec)

Floater

number hydb extb

1 20.99 131.60

2 19.78 109.91

3 20.99 56.09

4 19.78 47.26

5 20.99 105.14

6 19.78 127.76

7 20.99 38.11

8 19.78 38.62

The comparison of the radiation damping and external damping coefficients for different floaters are

presented in Table 4.1. It is observed that there is a substantial difference in the values of the radiation

damping and external damping coefficients, the latter being much higher. Moreover, the values for

radiation damping coefficients are equal for alternate floaters and do not differ much as compared to the

nearby floaters, in contrast to the external damping coefficients. The external damping coefficients are the

optimized value for each floater for a particular sea state. The values shown in table 2 are for SS-5.

4.3.3 Total power absorption and efficiencies

In Fig. 4.7, the total average power absorbed for 0° wave heading angle is shown for the different

configurations. This is the maximum value of power absorption by each configuration obtained by

optimizing the control parameters for each sea state, based on the constraints. The power absorbed by the

different configurations follow a similar pattern, with configuration as in Fig. 4.5(c) extracting the most

power as there is more number of floaters.

In order to ascertain the performance of each floater position, the absorption efficiency of each of the

floater is calculated in the case of the configuration as in Fig. 4.5(c) for different sea states (SS3-SS6). In

Fig. 4.8, it can be seen that the floaters numbered 2 and 5 perform the best, followed by floaters 3 and 4.

The floaters 1, 6, 7 and 8 are the least efficient, for all the sea states.

47

1 2 30

100

200

300

j = 8 j = 6 j = 4

P abs,

tota

l (kW

)

Hs (m)

Figure 4.7: Power absorbed by floaters in different configurations.

1 2 3 4 5 6 7 80,0

0,2

0,4

0,6

0,8

1,0

η

Floater Number

SS 3 SS 4 SS 5 SS 6

Figure 4.8: Efficiency of each floater for different sea states.

Finally it should be noted that while this work is concerned with the dynamics of the floaters, for the

system to operate, it is also necessary to deal in detail with the geometry of the oscillating arm (Calvario et

al. (2015)) and with the way in which the motion is transmitted to a hydraulic system as is being studied in

(Gaspar et al. (2015)).

4.4 CONCLUSIONS

The motion of heaving floaters with respect to a floating platform for different configurations are

analysed to understand the performance of the point absorbers in the extraction of wave energy. The cone-

cylinder floater shape is chosen with an apex angle of 90° with a top cylindrical part. The numerical

simulations for the determination of hydrodynamic forces and coefficients are obtained using WAMIT.

The floaters are assumed to be equipped with a linear power-take off mechanism consisting of a linear

48

damping and a linear tuning force. The performance of different configurations of floater arrangements is

assessed numerically in a frequency domain model. The floaters are restricted in their motion on the basis

of slamming, stroke and force constraints. The slamming restriction reduces the occurrence probability of

emergence events. The stroke constraint limits the point absorber stroke length and the force constraint

limits the total force that is required to realise the control and the damping of the system and the maximum

total control force that can be exerted on each of the floater.

The optimal control parameters extb and supm are determined to maximize the power absorption by each

floater for each corresponding sea state. The time-averaged power absorption values, the significant

amplitude of the relative motion between the floater and the platform, the external damping coefficient and

the supplementary mass coefficients were evaluated for each floater in different configurations to

understand the performance of each floater position with respect to the floating platform. The efficiency of

each floater position is also evaluated to determine the best floater positions. In addition, the effect of wave

heading angle is also taken into account and it is observed that the total power absorption for eight-floater

configuration does not vary much for majority of the sea states. The present study can be used to

efficiently design and install array of multiple point absorbers with floating structures for wave power

extraction.

49

CHAPTER 5

PERFORMANCE IN SHALLOW WATER

5.1 INTRODUCTION

The offshore floating devices have attracted interest due to the higher levels of annual average incident

wave power that occur in deep water and the minimal topographical constraints for their deployment. An

area that has received less focus is seabed mounted active near shore devices, where near shore is defined

as having a water depth of between 10 and 20 m and is typically located at a distance between 0.5 and 2.0

km from the coastline. It would appear that this may be due to fact that, the near shore has a lower annual

average incident wave power than in deep water, even though the exploitable resource is often only 5-10%

less. However, a near shore site will have a lower average incident wave power but it does possess a

number of attractive characteristics. A near shore location will reduce both the cost and power losses in the

cable bringing power back to shore. A near shore location may also reduce the costs of installation and

maintenance and also increase the plant availability by utilisation of smaller weather windows for repair

and maintenance. The operations and maintenance costs, accounting for perhaps 40% of the net present

cost of a wave energy converter, may be a significant issue (Henry et al. (2010); Folley et al. (2005)).

Shallow water also filters out the largest waves, potentially reducing the maximum loads required for

survival, although placement within the breaking wave zone may negate this advantage and may even

cause the maximum loads to increase. Balancing the costs and the benefits, it is possible to determine the

ideal water depth for the installation of the device for a specific location.

In order to assess the finite water depth effects, one needs to consider two aspects. Firstly, we need to

represent the limited water depth wave spectrum correctly. In fact, the existing parametric sea spectra have

been derived for deep water; however, the wave characteristics are modified as they approach shallow

water. Secondly, wave climatology databases exist for many coastal areas, but typically obtained

from wave measurements with floaters located at water depths larger than 70 m. It is therefore

necessary to transform the wave climatology scatter diagram from deep water to shallow water depths. The

most accurate numerical method to assess the near shore sea state characteristics consists of applying wave

models which propagate in time and space accounting for the wave generation by wind action, refraction

and frequency shifting due to depth effects and due to currents, nonlinear wave-wave interactions,

dissipation effects due to white capping and wave breaking and diffraction effects. The free software

50

SWAN is probably the most well-known wave model computer code (Bouws et al. (1985)). When reliable

wave measurements are not available, wave propagation models are in fact recommended to assess site

specific characteristics prior to the design and installation of a wave energy converter.

The preliminary objective of the present work is to obtain a general overview of the wave power

resource (wave energy flux) reduction as the water depth reduces and then to determine the reduction of

average power extraction by a WEC device. The latter aspect is interesting, since, on one hand, it is

expected that the largest reduction of the resource occurs for the higher sea states, but on the other

hand the power production of wave energy converters is usually limited to some level, which means

that above a specific value of the wave heights the converted power does not increase. For this

reason, it might happen that the reduction of wave energy flux is not so important for the annual energy

production. This procedure, being practical, gives the possibility to understand which might be the best

operational location for a device at an early design stage.

5.2 STANDARD SPECTRAL MODELS

There are many wave spectra used for waves offshore in deep water, i.e. when the wavelength is larger

than twice the water depth. A fundamental spectrum is the Pierson- Moskowitz spectrum, which describes

the wave spectra for fully developed sea, or fully arisen sea (FAS), when a constant wind blowing

infinitely long cannot increase the energy in the waves, but the energy transfer is balanced by dissipation.

This spectrum is a one parameter spectrum completely described by the wind speed given by

4

00.74( / )2 5( ) ,PMS g e ω ωω α ω −−= (5.1)

where 0.0081α = is Phillip’s constant, g the acceleration due to gravity, 0 19.5/g Uω = and 19.5U the wind

speed at the height 19.5 m above still water level. Mostly the sea state is not fully developed as the wind

speed and direction change, the fetch is too short, or the wind duration is not long enough, especially for

strong winds and high waves. The two parameter spectra for developing seas can be used where the wave

height and frequency are the parameters. This was originally proposed by Bredtschneider and offers more

flexibility, because the energy of the spectra can be placed at arbitrary locations on the frequency axis with

the corresponding significant wave height. Such spectra belonging to the PM-family are also, somewhat

incorrectly, referred to as PM-spectra. The widely used spectra are the ISSC spectra (ISSC = International

Ship Structures Congress 1964).

The two-parameter spectrum still gives too little freedom to reproduce realistic spectra of developing

sea. Hasselmann et al. (1973) published the five-parameter JONSWAP spectrum, which was one of the

results from the Joint North Sea Wave Project.

212 ( )

( ) ( ) ,p

peJONSWAP PMS S

ω ω

s ω ω

ω ω γ − −

= (5.2)

51

where γ is the peak enhancement factor, ω the angular frequency, pω the modal angular frequency

(peak of spectrum), ( ) as ω s= if pω ω< , the standard deviation of the peak enhancement factor to the

left and ( ) bs ω s= if pω ω> , the standard deviation of the peak enhancement factor to the right. The

parametersα and 0ω of ( )PMS ω also needs to be chosen. The JONSWAP spectrum is in common use for

design of drilling platforms in the offshore industry because it offers more flexibility with its five

parameters, and can produce realistic spectra. The parameters are then chosen from wave statistics

combined with systematic parameter fitting.

While the JONSWAP spectrum originally was developed for developing sea in deep water, the waves

in shallow areas are often waves coming in from deeper areas into an area where the waves are much

affected by the limited water depth. For such cases, the modified JONSWAP spectrum in shallow water

called the TMA spectrum is used as in Hughes et al. (1977). It is based on the fact that low-frequency or

equivalently, long-period waves must have a limited height in shallow water. Therefore the spectrum is

multiplied by a function of limited depth ( , )hφ ω given by

2

2

0.5( / ) , if / 1,

( , ) 1 0.5(2 / ) , if 1 / 2.

1, if / 2.

h g h g

h h g h g

h g

ω ω

φ ω ω ω

ω

<= − − ≤ <

≥

(5.3)

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

φ(ω,

h)

ω

Figure 5.1: The limited depth function in the TMA spectrum.

In Fig. 5.1, the limited depth function is varied with respect to frequency and a similar observation

follows as described above in Eq. (5.3). The expression for the TMA spectrum (Guedes Soares et al.

(1995)) is of the form

52

24

2 5 5 1( ) ( , ) exp ln( )exp .4 2 ( )

p pTMA

p

S g hω ω ω

ω α ω φ ω γω s ω ω

− − = − + −

(5.4)

5.3 NUMERICAL MODELLING

5.3.1 Equations of motion

The equation of motion of an oscillating point absorber, connected to a PTO performing reactive

control, can be described by Newton’s second law as

2

,2 .ex rad res PTO reactived zm F F F Fdt

= + + + (5.5)

The above equation is the same as the one described in the earlier chapters with an additional term

,PTO reactiveF , which is the control force exerted by the PTO on the floater and is composed of a position term

and a velocity term, given by -

, .PTO reactive PTO PTOdzF k z bdt

= + (5.6)

The coefficients PTOk and PTOb are not real physical quantities but control parameters. Hence ,PTO reactiveF is

generated as a linear feedback from measurements of the float movement. This is a causal implementation

of the reactive control, as the parameters are not changed wave-wave, but tuned according to the current

sea state. The maximal absorption of ocean energy is achieved when the velocity of the point absorber is in

phase with the force of the incident waves. However, the frequency of the ocean waves varies, the

condition of maximal absorption is not automatically fulfilled and a control strategy is therefore necessary.

If the energy only flows from the ocean to the electrical grid, the control strategy is typically referred to as

resistive or passive. On the other hand if the energy flow is bi-directional, the control strategy is referred to

as reactive control (Sinha et al. (2015)).

Taking into account all the considerations, the equation of motion of a heaving point absorber, coupled

with a hydraulic PTO system able to perform a reactive control feedback, can be re-written as –

{ } { } { }2

2( ) ( ) ( ) ( ).a hyd PTO PTO exd z dzm m b b k k z t F

dtdtω ω ω+ + + + + = (5.7)

The coefficients PTOk and PTOb are the optimal coefficients maximizing the average absorbed power by the

floater, where PTOk is a negative spring term and PTOb the positive external damping coefficient. This

optimization will include the PTO bandwidth and force limit in order to avoid unrealistic solutions such as

extremely large floater motions.

53

5.3.2 Hydraulic power take-off system

The hydraulic PTO system used in the present study is based on a hydrostatic transmission principle,

which is quite analogous to the system suggested for Salters Duck (Salter et al. (1974)). An illustration of

the PTO is presented in Fig. 5.2. This system consists of a symmetric cylinder operated in closed-circuit

with a swash plate variable displacement axial-piston pump/motor powering a generator.

Figure 5.2: Hydraulic PTO model.

The bi-directional flow is converted to a uni-directional rotation by the closed-circuit pump/motor

capable of both positive and negative swash-plate angles. The study does not cover the dynamic modelling

and design of the internal control loops of the PTO. Instead a second order approximation of the closed

loop behaviour of the cylinder-force control is used as in Hansen et al. (2011) with a PTO bandwidth of

6p rad/s and damping factor of 0.6. The cylinder is limited to give a maximum significant force of 200kN.

5.4 CASE STUDY: AGUÇADOURA

Aguçadoura wave climate scatter diagram is chosen in the present study as a case study to analyse the

finite depth effects on the energy resource of a location. Aguçadoura located near Póvoa de Varzim, north

of Porto in Portugal, is an important testing and demonstrating area for wave energy converters.

The Aguçadoura wave farm, designed to use the three Pelamis wave energy converters to convert the

motion of the ocean surface waves into electricity, was the world's first wave farm. It is located 5 km

offshore. The plotted scatter diagram as shown in Fig. 5.3, is a result of statistical analysis and were

generated using the three-hour consecutive significant wave height and wave period time sequences

resulting from simulations with the SWAN model for the entire time interval (2009–2011), as in Silva et

al. (2013). The scatter diagrams shown in Fig. 5.3 have been obtained for three different water depths of 79

m, 58 m and 20 m respectively. In the scatter diagram, for sH = 0.5 m, the values less than 0.5 m and for

sH = 1 m values > 0.5 m and ≤ 1m were taken into account and so on, also the same for pT .The values

are the probabilities of occurrences of different sea states expressed in percentages of the total number of

occurrences. The sea states were structured into bins of 0.5 s × 0.5 m ( )p sT H∆ ×∆ .

54

Depth:79m

Hs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 180,5 0,01 0,02 0,03 0,02 0,04 0,03 0,11 0,12 0,13 0,14 0,22 0,041 0,01 0,09 0,03 0,21 0,74 1,41 1,65 1,14 2,47 2,64 1,58 0,89 0,72 0,19 0,06 0,01 0,01 0,01

1,5 0,07 0,49 1,09 2,93 5,89 6,83 3,38 1,91 1,57 0,76 0,21 0,08 0,062 0,01 0,32 2,28 6,29 5,12 3,88 3,23 0,79 0,22 0,16 0,09

2,5 0,26 1,29 3,21 3,93 3,09 1,29 0,16 0,10 0,063 0,32 1,18 2,71 3,48 1,17 0,19 0,03 0,03

3,5 0,03 0,34 0,84 2,70 1,49 0,30 0,11 0,024 0,02 0,03 0,30 1,48 1,15 0,22 0,06 0,01

4,5 0,01 0,03 0,59 1,04 0,28 0,13 0,035 0,09 0,87 0,50 0,30 0,03

5,5 0,12 0,52 0,196 0,06 0,19 0,28 0,03

6,5 0,09 0,18 0,027 0,23 0,02

7,5 0,09 0,018 0,03

Depth:58mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 0,03 0,01 0,02 0,03 0,10 0,03 0,10 0,02 0,17 0,20 0,19 0,19 0,26 0,081 0,02 0,08 0,11 0,64 1,48 1,69 1,52 2,58 2,90 1,98 0,88 0,88 0,18 0,07 0,01 0,06

1,5 0,04 0,40 1,20 2,65 6,00 6,87 3,39 1,60 1,73 0,78 0,30 0,09 0,06 0,012 0,02 0,03 0,23 2,39 6,30 5,10 3,68 3,20 0,74 0,28 0,17 0,07

2,5 0,09 1,15 3,28 3,98 3,45 1,24 0,20 0,08 0,063 0,01 0,32 1,05 2,44 3,28 1,20 0,17 0,04 0,01

3,5 0,04 0,19 0,69 2,75 1,48 0,37 0,09 0,034 0,03 0,18 1,18 1,31 0,24 0,03 0,01

4,5 0,44 1,01 0,37 0,20 0,025 0,09 0,73 0,62 0,26 0,02

5,5 0,12 0,34 0,236 0,03 0,12 0,41

6,5 0,07 0,227 0,11 0,04

7,5 0,038

Depth:20mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 0,06 0,02 0,01 0,03 0,11 0,16 0,17 0,07 0,22 0,33 0,18 0,26 0,31 0,09 0,021 0,04 0,08 0,20 0,79 1,72 2,07 1,68 3,20 3,96 2,35 1,39 1,01 0,38 0,08 0,07 0,03

1,5 0,02 0,29 1,03 2,24 5,84 7,21 4,31 2,69 2,73 0,91 0,39 0,14 0,08 0,012 0,10 1,18 4,51 4,68 4,78 3,48 1,09 0,22 0,13 0,09

2,5 0,02 0,44 1,87 3,89 4,15 1,38 0,20 0,08 0,043 0,06 0,30 1,39 3,65 1,68 0,26 0,04 0,04

3,5 0,07 0,24 2,29 1,43 0,37 0,09 0,024 0,01 0,60 1,38 0,43 0,04 0,02

4,5 0,18 0,90 0,53 0,26 0,025 0,33 0,46 0,37

5,5 0,07 0,20 0,316 0,13 0,30

6,5 0,17 0,047 0,03

7,58

Tp(s)

Tp(s)

Tp(s)

Figure 5.3: Scatter diagram for Agaçudoura.

55


5.5.1 Effect of water depth on the spectrum

The transformation effect on the spectral shape is plotted in Fig. 5.4 for peak period of 12 s. It can be

seen that the peak of the spectrum reduces with decrease in water depth.

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,40,0

0,1

0,2

0,3

0,4S/

H s2 (m2 s/

m2 )

ω (rad)

depth = 79m depth = 58m depth = 20m

Figure 5.4: TMA spectral shapes for different water depths.

5.5.2 Effect of water depth on the significant wave height

Fig. 5.5 illustrates the effect of water depth on the significant wave height for different peak periods of

the scatter diagram. The study shows that the significant wave height reduces as the water depth decreases

and the result is normalized by the deep water value ,s deepH . The significant wave heights are calculated

using zero order spectral moments. It can be seen that there is a considerable reduction of the significant

wave height on moving to shallower water depths.

5.5.3 Effect of water depth on the scatter diagram

Due to water depth decrement, the sea states having the highest significant wave height are less

probable, and on the other hand the probability of occurrence of the sea states having lower heights is

increased with limited water depth. This is the main characteristic that stands out from the three scatter

diagrams; the classes of higher significant wave height sH decrease their probability, and those of lower

sH increase their probability, as water depth decreases. This aspect may be interesting for wave energy

converters, since usually the power take off system is limited in terms of nominal power and therefore the

system will not use the full wave energy power resource of the largest sea states. The water depth effect on

56

the probability distribution of the significant wave heights is plotted in Fig. 5.6. The probability is more

concentrated around the lower heights between 1 m and 2 m.

150 125 100 75 50 25 00,0

0,2

0,4

0,6

0,8

1,0

H s/Hs,

deep

Depth (m)

Tp = 5s Tp = 7s Tp = 10s Tp = 12s Tp = 15s

Figure 5.5: Effect of TMA transformations on the reduction of significant wave heights.

0 2 4 6 80,0

0,1

0,2

0,3

prob

abilit

y

Hs (m)

depth = 79m depth = 58m depth = 20m

Figure 5.6: Probability distribution versus sH for different water depths.

5.5.4 Effect of water depth on the wave power resource

Once the new spectral shapes are calculated, it is possible to compute how the power resource of each

sea state is transformed by passing from deep to shallower waters. The values of wave power available are

calculated using the equation of deep water energy flux which is given by

2

2 ,64 e s

gP T Hrp

= (5.8)

where r is the water density, g the acceleration due to gravity, eT the average energy period and sH the

significant wave height. The total annual incident power for each depth is calculated by summing the mean

incident wave power of all the sea states weighted by their probability of occurrences in one year. It can

57

be seen from Table 5.1 that more than 50% of the available wave power is lost as the water depth changes

from 79 m to 20 m, and around 10% at 59 m depth. Hence, there is a significant reduction of the wave

power resource with decrease in water depth. The result is quite similar to the one obtained as in Folley et

al. (2006).

Table 5.1: Incident average annual wave power reduction due to water depth effects

Water depth Power available (kW/m) % incident power lost

79 m 31.20 0.0

58 m 27.87 10.67

20 m 14.14 54.70

5.5.5 Effect of water depth on the power absorption and reactive control parameters

The final step is to study the water depth effects on the wave power extracted by the single floater

WEC. The power matrices for each water depth have been developed as shown in Fig. 5.7. The WEC

power matrices depend on the water depth, not so much because of the device’s hydrodynamics, but

because the wave spectra are modified as the wave propagates from deep to shallow water. The reactive

control parameters PTOb and PTOk have been optimized for each sea state in the scatter diagram to give the

maximum power absorption values for all the water depths, keeping in mind the PTO bandwidth and

maximum force constraints. The PTOb and PTOk matrices for different water depths are shown in Fig. 5.8

and Fig. 5.9 respectively.

From the power matrix as shown in Fig. 5.7, it can be seen that there is a considerable reduction of

absorbed power as we move from deeper to shallow water, especially for the sea states with longer periods.

Also, it can be observed that the maximum absorbed power is distributed around different peak period

values. This affirms the importance of considering the scatter diagram for the correct water depth when

designing and fine tuning the WEC. The average power absorbed by the floater is calculated by summing

the power absorbed for all the sea states weighted by their probability of occurrences. It can be seen from

Table 5.2 that around 50% less power is extracted at water depth of 20 m than at 79 m. This is a significant

reduction of extracted wave power. The power absorption reduces by around 7% on moving to a depth of

58 m. From the matrix, it is observed that there is a considerable difference in values as we move to

shallow waters, for sea states with longer periods. For water depth of 20 m, the optimal PTOb values are less

as compared to the other water depths and are the highest for water depth of 79 m.

58

Depth:79mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 0,8 2,1 2,9 4,8 5,9 6,7 7,3 7,2 6,9 6,71 6,25 5,61 3,3 5,8 12 15 17 17 18 17 17 16 15 14,5 13,3 12 11 9,8 8,59 7,6

1,5 29 29 29 28 27 25 23 22,4 20,5 18 16 15 13,22 41 39 38 35 32 30,5 27,9 24 22 20 17,9

2,5 48 44 41 38,9 35,5 31 28 26 22,73 54 50 47,3 43,2 38 34 31 27,6

3,5 64 59 56 51,1 45 41 37 32,74 74 68 64,7 59 51 47 43 37,8

4,5 77 73,5 67 58 53 49 42,95 75,1 65 60 55

5,5 73 66 61 53,46 80 73 67 58,6

6,5 79 73 63,97 79 69,2

7,5 85 74,58 79,8

Depth:58mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 1,4 2,1 2,9 3,8 4,8 5,9 6,7 7 7,1 6,9 6,4 6,13 5,56 4,81 8,4 12 15 17 17 18 17 17 16 14 13,3 11,9 10 9,1 8,1 6,97

1,5 29 29 29 28 27 24 22 20,7 18,5 16 14 13 10,7 9,22 41 40 39 37 34 30 28,2 25,2 21 19 17 14,5

2,5 48 43 38 36 32 27 24 22 18,43 58 52 46 43,8 39 33 29 26 22,5

3,5 62 55 51,8 46,1 39 35 31 26,64 64 59,9 53,3 45 40 36 30,7

4,5 60,6 51 46 41 34,95 67,9 57 51 46 39,2

5,5 63 57 516 70 62 56

6,5 68 617 66 56,7

7,5 718

Depth:20mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 1,4 2,1 2,7 3,2 3,6 4 4,2 4,2 4 3,5 2,9 2,68 2,27 1,8 1,141 8,2 11 13 14 13 13 12 11 8,8 7,2 6,53 5,43 4,2 3,6 3,2 2,63

1,5 25 23 22 20 18 14 12 10,5 8,7 6,8 5,8 5 4,15 3,52 28 25 20 16 14,6 12 9,3 8 6,9 5,69

2,5 32 26 21 18,7 15,4 12 10 8,8 7,243 32 25 22,9 18,9 15 12 11 8,81

3,5 30 27,2 22,4 17 15 13 10,44 31,6 25,9 20 17 15 12

4,5 29,6 23 19 17 13,75 25 22 19

5,5 28 24 216 27 23

6,5 25 20,57 27

7,58

Tp(s)

Tp(s)

Tp(s)

Figure 5.7: Power matrices for different water depths

59

Depth:79m

Hs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 180,5 22 20 18 15 15 20 30 39 47,2 50,67 56,92 651 22 22 18 22 30 38 46 53 60 73 85,4 90,78 100,6 113 120 126 134,1 141

1,5 45 54 62 71 78 93 107 113 124,6 140 149 157 168,22 63 79 87 102 116 122 134,2 151 162 171 184,3

2,5 90 104 117 122,9 135,1 153 164 174 188,93 101 114 119,5 131,3 149 160 171 186,6

3,5 97 108 114 125,2 142 153 164 180,44 92 102 107,6 118,2 134 145 156 172,4

4,5 96,2 101,1 111,1 126 137 148 163,65 104,2 119 129 139 154,7

5,5 111 121 1316 105 114 123 137,9

6,5 107 116 130,27 110 123,2

7,5 104 116,68 110,6

Depth:58mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 22 20 18 17 15 15 20 25 30 37 43,3 45,85 50,16 551 20 18 22 30 38 45 53 59 71 80 84,15 91,33 100 104 107 110,9

1,5 45 54 62 70 77 91 102 107,1 116,4 128 134 139 145,6 1512 63 71 79 87 100 112 118 128,6 143 150 157 166,5

2,5 89 103 115 120,9 132,3 148 157 166 177,33 88 101 113 119,3 131 148 158 168 181,1

3,5 97 109 115,1 126,9 144 155 166 180,44 104 109,7 121,3 139 150 161 176,8

4,5 115,2 132 144 155 171,55 108,9 126 137 148

5,5 119 130 1426 113 124 135

6,5 117 1287 122 138,3

7,5 1168

Depth:20mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 22 20 18 17 15 13 13 15 17 19 20 20,44 21 21 21,981 20 18 19 24 29 33 37 39 42 43,5 43,95 44,55 45 45 45 45,44

1,5 39 45 50 54 57 61 62,9 63,66 64,72 66 66 66 66,87 682 65 69 74 77,9 79,11 81,02 83 84 85 85,81

2,5 77 84 88,6 90,48 93,54 97 99 100 102,13 90 95,8 98,38 102,7 108 110 113 115,7

3,5 100 103,5 109,1 116 119 123 126,94 106,4 113,3 122 126 130 135,9

4,5 115,7 126 131 1365 128 135 141

5,5 130 137 1446 138 146

6,5 147 157,77 147

7,58

Tp(s)

Tp(s)

Tp(s)

Figure 5.8: PTOb matrices for different water depths.

60

Depth:79m

Hs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 180,5 77 113 126 144 151 154 160 164 167 168 171 1741 77 96 126 133 136 138 140 142 143 145 147 148 150 153 155 157 160 163

1,5 122 122 123 123 123 124 124 125 127 130 132 134 1382 105 104 103 103 103 103 104 107 110 113 117

2,5 86 85 84 84,7 85,7 89 91 94 98,13 70 69 69,6 70,5 73 75 78 82,3

3,5 58 57 57,6 58,4 61 63 65 69,34 49 48 48 48,7 51 53 55 58,6

4,5 40 40,4 41 43 44 46 49,95 34,8 36 38 40 42,7

5,5 31 33 346 27 28 30 32,1

6,5 25 26 28,17 23 24,7

7,5 20 21,98 19,5

Depth:58mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 96 113 126 136 144 151 154 157 160 164 168 169 172 1761 113 126 133 136 138 140 142 144 147 150 152 155 159 162 164 168

1,5 122 122 123 123 124 126 129 130 134 139 142 146 151 1552 105 105 104 104 106 108 110 113 119 123 127 133

2,5 87 88 90 91,5 94,9 101 105 110 1163 73 73 75 76,4 79,6 85 90 94 101

3,5 61 63 64 67 72 77 81 87,64 53 53,9 56,7 62 66 70 76,3

4,5 48,2 53 57 60 66,55 41,4 46 49 53 58,3

5,5 40 43 466 34 37 40

6,5 33 367 32 35,7

7,5 288

Depth:20mHs(m) 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 9 10 10,5 11,5 13 14 15 16,5 18

0,5 96 112 125 135 143 150 155 159 162 168 173 174 178 181 1871 112 125 134 138 142 146 150 153 160 165 168 171 176 178 180 183

1,5 127 130 134 138 142 149 156 158 163 169 172 175 178 1802 124 128 137 145 148 154 161 165 168 172

2,5 114 124 133 137 144 153 157 161 1663 112 121 126 134 143 149 153 159

3,5 110 115 124 134 140 146 1524 105 114 126 132 138 145

4,5 105 117 124 130 1385 109 116 123

5,5 102 109 1166 102 109

6,5 102 1117 96

7,58

Tp(s)

Tp(s)

Tp(s)

Figure 5.9: PTOk matrices for different water depths.

61

Table 5.2: Reduction in wave power absorption due to water depth effects

Water depth Power absorbed (kW) % power lost

79 m 33.57 0.0

58 m 31.11 7.33

20 m 16.83 49.86

The values in the PTOk matrix are the absolute values of PTOk , actual values are just negative of the one

shown in the matrix. From the matrix, it can be seen that there is a considerable difference in values as we

move to shallow waters, especially for sea states with longer periods. For water depth of 20 m, the optimal

PTOk values are higher as compared to the other water depths.

5.6 CONCLUSIONS

The present study uses a simplified method to represent the wave spectrum modification as the water

depth reduces, namely the TMA method and presents a wave power resource reduction analysis, as well as

the impact on the average power absorbed by a floating point absorber. Most of the recently proposed

wave energy converters are planned to be installed in relatively shallow water depths where the wave

energy resource is reduced compared to deep water. The case study uses the annual wave statistics from

Aguçadoura, located at north-west coast of Portugal. The scatter diagram representing the annual wave

statistics modifies significantly as the water depth reduces from 79 m to 58 m and to 20 m. The classes of

highest significant wave height reduce their probability of occurrence, while those of lower sH increase

their probability. The consequence is that the annual wave power resource reduces by 11% at 58 m and

55% at 20 m as compared to water depth of 79 m. The reactive control strategy is implemented for a

single-floater point absorber WEC, with optimal control parameters PTOb and PTOk maximizing the

extracted wave power. The power matrices are determined for each water depth and it is found that the

absorbed power reduces by 7% for 58 m and 50% for 20 m water depth as compared to 79 m. There is

considerable reduction in absorbed power for sea states with longer periods and maximum power is

distribution changes with water depth. The optimal control parameters also vary greatly for different water

depths for sea states with longer periods. The present study can be useful for a preliminary cost-benefit

analysis of the most convenient depth to install a floating point absorber wave energy converter.

62

CONCLUSIONS

In the present thesis, a detailed hydrodynamic study on the array of heaving point absorbers is

presented. Initially, three floater shapes namely – cone-cylinder, hemisphere-cylinder and hemisphere are

compared in terms of their hydrodynamic properties and the cone-cylinder floater performs slightly better

than the others followed by hemisphere floater in terms of wave power extraction. Also, the draft of the

floater was varied and the power absorption rises for smaller drafts of the floater. Therefore, the cone-

cylinder floater is used to study the performance of arrays of point absorbers.

Subsequently, several arrays of heaving point absorbers (12 floaters) in different arrangements namely

- linear, grid, circular and concentric arrangements are analysed to understand the performance of the point

absorbers in absorption of wave energy in different configuration. The floaters are restricted in their

motion on the basis of three restrictions – slamming, stroke and force constraints. The slamming restriction

reduces the occurrence probability of emergence events. The stroke constraint limits the point absorber

stroke length and the force constraints limits the total force that is required to realise the control and the

damping of the system and the maximum total control force that can be exerted on each of the floater. The

optimal power take-off parameters jextb and sup

jm are determined to maximize the power absorption by each

floater for every corresponding sea state. Sequential Quadratic Programming (SQP) method is used to

carry out the optimization problem. Following were the results obtained:

• The concentric and circular arrangements perform better in terms of power absorption values as

compared to the linear and grid arrangements. Moreover, the absorbed power is found to

unequally distributed among the floaters in linear and grid arrangements in contrast to the circular

and concentric arrangements. The rear floaters in linear and grid arrangements are less efficient

than in circular arrangements.

• The control parameters jextb and sup

jm are more uniform for circular arrangements which make the

control easier and efficient for the different power take-off system.

• The circular and concentric arrangements perform better than grid and linear arrangements for all

the sea states and for all the wave headings except 90 , for which linear arrangement out performs

the others.

Moreover, the motion of heaving floaters with respect to a floating platform for different

configurations is analysed. The efficiency of each floater position is also evaluated to determine the best

floater positions. In addition, the effect of wave heading angle is also taken into account and it is observed

63

that the total power absorption for eight-floater configuration does not vary much for majority of the sea

states. This study can be used to efficiently design and install array of multiple point absorbers with

floating structures for wave power extraction.

Finally, the performance of a point absorber WEC in shallow water is evaluated. The case study uses

the annual wave statistics from Aguçadoura, located at north-west coast of Portugal. The scatter diagram

representing the annual wave statistics modifies significantly as the water depth reduces from 79 m to 58 m

and to 20 m. The classes of highest significant wave height reduce their probability of occurrence, while

those of lower sH increase their probability. The consequence is that the annual wave power resource

reduces by 11% at 58 m and 55% at 20 m as compared to water depth of 79 m. The reactive control

strategy is implemented for a single-floater point absorber WEC, with optimal control parameters PTOb and

PTOk maximizing the extracted wave power. The power matrices are determined for each water depth and it

is found that the absorbed power reduces by 7% for 58 m and 50% for 20 m water depth as compared to 79

m. There is considerable reduction in absorbed power for sea states with longer periods and maximum

power distribution changes with water depth. The optimal control parameters also vary greatly for different

water depths, especially for sea states with longer periods. This study can be useful for a preliminary cost-

benefit analysis of the most convenient depth to install a floating point absorber wave energy converter.

64

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APPENDIX A

PERFORMANCE OF CONCENTRIC ARRAYS OF POINT ABSORBERS ATTACHED TO A BOTTOM MOUNTED PLATFORM

In order to analyze the concentric array of point absorbers, the study is carried out for three different

concentric arrangements c1, c2 and c3, with floaters placed at different radii. The configuration c1 has an

outer radius of 18 m and an inner radius of 9 m, the configuration c2 has an outer radius of 24 m and an

inner radius of 12 m and the configuration c3 has an outer radius of 30 m and inner radius of 15 m. The

configuration c2 is shown in Figure A.1.

Figure A.1: Concentric array of floaters – configuration c2.

The floaters are attached, by means of an arm, to a central cylindrical bottom mounted platform as

shown in Figure A.2. This platform has a radius of 5 m and height of 50 m, mounted on the ocean bed

floor.

1

2

4

5 6

7

8

9

10

11

12

3

24 m

12 m

69

Some results obtained are shown below in Figs. A.3 (a)-(b) and Tables A.1 and A.2.

1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

40

50

60

70

P ext (

kW)

Floater Number

c1 c2 c3

1 2 3 4 5 6 7 8 9 10 11 120,0

0,2

0,4

0,6

0,8

1,0

η

Floater Number

c1 c2 c3

(a): Power extracted by each floater. (b): Efficiency of each floater.

Figure A.3: Response of each floater.

Table A.1: Total power absorption for sea state SS-5 for different concentric arrangements.

Power c1 c2 c3 linear grid

extP (kW)

501.21

504.63 491.75 429.3 464.22

Figure A.2: Bottom- mounted cylindrical platform connected with point absorbers.

y

x

Water line Floater

Platform

Sea bed

70

Table A.2: Difference in total power absorption in linear and grid arrangements as compared to concentric array c2.

Sea State Linear (in %) Grid (in %) SS-1 20.80 3.39 SS-2 20.94 4.58 SS-3 20.72 6.84 SS-4 16.73 7.69 SS-5 14.91 8.01 SS-6 13.59 7.83 SS-7 12.49 7.44 SS-8 11.43 7.03

71

APPENDIX B

LIST OF PUBLICATIONS

Sinha, A., Karmakar, D. & Guedes Soares, C. , Effect of floater shapes on the power take-off of wave energy converters, Renewable Energies Offshore, Guedes Soares, C.(Ed.), Taylor & Francis Group, London, UK, 2015, pp. 375-382. Sinha, A., Karmakar, D. & Guedes Soares, C., Numerical modelling of an array of heaving point absorbers, Renewable Energies Offshore, Guedes Soares, C.(Ed.), Taylor & Francis Group, London, UK, 2015, pp. 383-391. Sinha, A., Karmakar, D. & Guedes Soares, C. Hydrodynamic analysis of array of point absorbers combined with a large floating platform. In Proceedings of European Wave and Tidal Energy Conference (EWTEC), Nantes, France, 6 – 11 September 2015. Sinha, A., Karmakar, D. & Guedes Soares, C. Shallow water effects on a hydraulic power take-off WEC with reactive control. In Proceedings of 8th OWEMES Conference, Rome, 8-9 October 2015. Sinha, A., Karmakar, D. & Guedes Soares, C. Hydrodynamic behaviour of concentric array of point absorbers attached to a bottom-mounted platform. In Proceedings of 8th OWEMES Conference, Rome, 8-9 October 2015. Sinha, A., Karmakar, D. & Guedes Soares, C. (Submitted). Performance of circular and concentric arrays of heaving point absorbers. Journal of Renewable Energy. Gaspar, J.F., Kamarlouei, M., Sinha, A., Xu, H., Calvário, M., Faÿ, F.-X., Robles, E., & Guedes Soares, C. (Submitted). Speed control of power take-off electric drive for wave energy converters. Journal of Renewable Energy.

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