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AN ABSTRACT OF THE DISSERTATION OF

Emmanuel B. Agamloh for the degree of Doctor of Philosophy in

Electrical and Computer Engineering presented on October 19, 2005.

Title: A Direct-Drive Wave Energy Converter with Contactless Force TransmissionSystem

Abstractapproved:______________________________________________________

Alan K. Wallace Annette R. von Jouanne

Commonly proposed ocean wave energy converters (OWEC) use inefficient

and maintenance demanding intermediate hydraulic and pneumatic systems. We

propose a novel rotary direct-drive OWEC that eliminates these intermediate

stages. The new device employs a contactless force transmission system (CFTS)

comprising a “piston” and a “cylinder” to spin a conventional rotary generator.

We present an analytical model of the OWEC and also propose a numerical

technique that performs a coupled fluid-structure interaction simulation of the wave

energy device in a 3-D numerical wave flume using a computational fluids

dynamics (CFD) code. In previous investigations, the motion of floating bodies

was prescribed rather than determined. The current method represents a significant

advancement in that it determines the motion of the buoy from the dynamic

solution of the fluid flow problem and the dynamic buoy motion problem. The

technique was extended to assess the performance of two neighboring buoys and

their interference effects.



A Direct-Drive Wave Energy Converter with Contactless Force TransmissionSystem

by

Emmanuel B. Agamloh

A DISSERTATIONsubmitted to

Oregon State University

in partial fulfillment of

the requirements for thedegree of

Doctor of Philosophy

Presented October 19, 2005Commencement June 2006



Doctor of Philosophy dissertation of Emmanuel B. Agamloh presented onOctober 19, 2005.

APPROVED:

Co-Major Professor, representing Electrical and Computer Engineering

Co-Major Professor, representing Electrical and Computer Engineering

Director of the School of Electrical Engineering and Computer Science

Dean of the Graduate School

I understand that my dissertation will become part of the permanent collection of

Oregon State University libraries. My signature below authorizes release of my

dissertation to any reader upon request.

Emmanuel B. Agamloh, Author



ACKNOWLEDGEMENTS

Now, let us give all power and glory and honor and praise unto our Lord

and Savior Jesus Christ, who is able to do exceedingly abundantly above all that we

ask or think, according to the power that works in us. It is by the grace of God that

this work has been done and I thank God for His blessings over all these years.

I also thank my advisors, Dr. Alan K. Wallace and Dr. Annette R. von

Jouanne for their guidance and support throughout my program; this work would

not have been possible without their vision, synergy and shared passion for wave

energy. I have been truly blessed with their knowledge, research skills, counsel,

accessibility, friendship and the congenial research environment they have created

for me and the rest of the group in the Motor Systems Resource Facility (MSRF). I

sincerely appreciate their kindness towards me and my family and for welcoming

us into their homes for well-deserved breaks from work. In this regard sincere

thanks are also due to Pat Wallace and Dr. Alex Yokochi.

I also thank Dr. Andreas Weisshaar, Dr. Kartikeya Mayaram and Dr. Joseph

Nibler for their work on my program committee. I thank them for their valuablesuggestions and comments during my program meetings and for their patience in

reading through my dissertation and providing useful comments.

I would like to thank Manfred Dittrich for his expertise and for sacrificing

some of his lunch breaks to discuss design issues with me and also for

implementing all the hardware in this work.

I have learned from fruitful technical discussions with past and present

graduate and undergraduate students of the MSRF. In particular, I thank Dr. Andre

Ramme, Dr. Aleksandr Nagornny, Fuminao Kinjo, Dr. Jifeng Han, Xioalin Zhou,

Ken Rhinefrank, and Dave Eveland.



I appreciate Ferne and staff members of the EECS office, as well as Todd

and his computer support team. I thank them for all their help. I also thank all

EECS professors for their inspirational teaching.

Dr. Deborah Pence and Dr. S.C.S. Yim were helpful at certain stages of the

project. I thank them for eagerly answering questions that I encountered in my

work.

My sincere thanks also go to the Johnson pastoral families and members of

the United Pentecost Church in Albany, Oregon for being our church family and

for praying for me.

Finally, I thank my wife Jane and our children David and Audrey for being

there, waiting for me to come back from the laboratory. Their support is

immeasurable! And so is the support and encouragement I received from my

mother and brothers and sisters. I also acknowledge the encouragement I received

from friends such as Moses, Justice, Enoch and Gboloo.

The financial support provided by the National Science Foundation and

Oregon Sea Grant for this work is gratefully acknowledged.



DEDICATION

To grandma Lydia K. Ocansey and dad Stephen N. Agamloh.



TABLE OF CONTENTS

Page

1. INTRODUCTION ………………………………………… 1

1.1 Background ………………………………………….. 1

1.2 Motivation for this work……………………………… 1

1.3 Literature Review…………………………………….. 3

1.3.1 The Oscillating Water Column ………………. 4

1.3.2 Buoy Point-absorbers ………………………... 61.3.3 Overview of Some Existing Prototype Devices 8

1.4 Direct-Drive Wave Energy Converters …………….... 10

1.5 Contribution of This Dissertation ……………………. 11

1.6 Organization of This Dissertation ……………………. 12

2. BASIC ANALYTICAL CONCEPTS AND DEFINITIONS .. 14

2.1 Introduction ………………………………………….. 14

2.1 Basic parameters of ocean waves ……………………. 14

2.3 Linear wave theory …………………………………... 16

2.3.1 Governing Equation …………………………. 16

2.3.2 Boundary Conditions ………………………… 172.3.3 Solution of Boundary Value Problem ……….. 19

2.4 Wave Energy Resource ……………………………… 20

2.4.1 Wave Energy and Power …………………….. 202.4.2 Wave Energy Resource off Oregon Coast …… 22

2.5 Wave forces on structures …………………………… 24

3. ANALYTICAL MODEL OF THE OWEC ..………………... 25

3.1 Introduction .…………………………………………. 25

3.2 Mathematical description of system ………………… 263.2.1 Single float system ………………………….. 263.2.2 Double float system …………………………. 30



TABLE OF CONTENTS (Continued)

Page

3.3 Simulation of the Buoy using Matlab/Simulink …….. 32

3.3.1 Introduction ………………………………….. 32

3.3.2 Simulation Results ………………………….... 33

3.4 Control of OWECs …………………………………… 38

3.5 Summary ……………………………………………... 42

4. COUPLED FLUID-STRUCTURE-INTERACTIONMODELING …………………………………………………. 43

4.1 Introduction …………………………………………... 434.2 Numerical Method …………………………………… 44

4.2.1 Governing Equations ………………………… 444.2.2 The Numerical Grid ………………………...... 46

4.2.3 Boundary Conditions ………………………… 474.2.4 Generation of Waves ………………………... 48

4.3 Coupling Procedure ………………………………….. 49

4.4 Results ……………………………………………….. 53

4.4.1 Free Surface ………………………………….. 53

4.4.2 Response characteristics and Power Capture … 53

4.5 Array of OWECs ……………………………………... 60

4.5.1 Introduction …………………………………... 60

4.5.2 Double buoy array of OWECs ……………….. 61

4.5 Summary ……………………………………………... 66

5. DESIGN AND TESTING OF NOVEL DIRECT-DRIVE

OWEC WITH CFTS …………………………………………. 68

5.1 Introduction ………………………………………….. 68

5.2 The CFTS …………………………………………….. 70

5.2.1 Description and Proof-of-concept of CFTS …. 70

5.2.2 Design of the CFTS …………………………. 715.2.3 Finite Element Analysis of the CFTS ……….. 72



TABLE OF CONTENTS (Continued)

Page

5.2.4 Laboratory Testing of the CFTS ……………... 76

5.3 Buoy System Design …………………………………. 81

5.3.1 Types of Buoys ………………………………. 815.3.2 Inner Module ………………………………… 82

5.3.3 Outer Float …………………………………… 845.3.4 Complete Assembly …………………………. 85

5.4 Wave Flume Testing ………………………………… 86

5.5 Summary …………………………………………….. 90

6. SUMMARY OF RESULTS …………………………………. 92

6.1 Analytical results …………………………………….. 92

6.1.1 Roller Screw and CFTS Design and Analysis .. 926.1.2 Analytical Simulation of OWEC …………….. 94

6.2 Numerical Results ……………………………………. 94

6.2.1 Introduction …………………………………... 946.2.2 Power Capture Width ………………………... 95

6.2.3 Full Scale Prototype …………………………. 976.2.4 Double Buoy Array ………………………….. 98

6.3 Experimental Results ………………………………… 99

6.3 Summary …………………………………………….. 100

7. CONCLUSION ……………………………………………… 101

7.1 Conclusions ………………………………………….. 101

7.2 Recommendations for future work …………………... 103

BIBLIOGRAPHY ……………………………………………………. 104

APPENDICES ……………………………………………………….. 110



LIST OF FIGURES

Figure Page

1.1 The Oscillating Water Column ………………………………. 4

1.2 Working principle of Wells turbine ………………………….. 5

1.3 Wave pattern of two interfering waves as seen from above …. 6

1.4 Capture width of a heaving symmetrical cylinder

of diameter d ………………………………………………… 7

2.1 The basic parameters of a sinusoidal wave ………………….. 15

2.2. The water wave boundary value problem ……………………. 172.3. Oregon coast wave energy resource …………………………. 23

3.1 SDOF heave motion of cylindrical buoy …………………….. 26

3.2 Equivalent circuit for the PMSG …………………………….. 30

3.3 Schematic for the double buoy system ………………………. 31

3.4 Comparison of response amplitude operators ………………... 33

3.5 Float component of Simulink model …………………………. 34

3.6 Oscilloscope Capture of CFTS under constant thrust ………… 35

3.7 Simulation of CFTS under constant thrust …………………… 36

3.8 Typical simulation results of CFTS buoy underreciprocating wave action ……………………………………. 37

3.9 Typical no-load voltage of CFTS buoy with clutch …………. 38

3.10 Buoy response and power captured(a) without latching control (b) with latching control ………. 40

3.11 OWEC amplitude and phase control scheme ………………… 41

4.1 Control Volume ……………………………………………… 45

4.2 Mesh around a 3D cylindrical buoy …………………………. 47

4.3 2D representation of Boundaries of Solution domain ……….. 47

4.4 Block diagram of Coupling Algorithm ………………………. 50



LIST OF FIGURES (Continued)

Page

4.5 Heave displacement with different numerical

time-integration schemes …………………………………….. 53

4.6 Free Surface Capturing (single buoy) ………………………... 54

4.7 Typical instantaneous power of OWEC ……………………... 55

4.8 Instantaneous power of OWEC in different waves ………….. 56

4.9 In-line force (x-direction) ……………………………………. 57

4.10 Vertical force (z-direction) …………………………………... 57

4.11 Total pressure and viscous forces in vertical (z) direction …... 584.12 Buoy heave displacement with different damping coefficient . 58

4.13 Velocity vectors in heave oscillation ………………………… 59

4.14 Mesh around a double buoy system …………………………. 61

4.15 Buoy Spacing Evaluation Parameter ………………………... 62

4.16 In-line wave force on buoy array ……………………………. 63

4.17 Instantaneous power of buoy array …………………………... 64

4.18 Heave displacement of double buoy array …………………… 65

4.19 Free Surface Capturing of double buoy array ………………... 66

5.1 Solid model of Buoy with CFTS …………………………….. 69

5.2 CFTS proof of concept development stages …………………. 71

5.3 Design configurations of CFTS ……………………………… 72

5.4 Flux 2D Finite Element Modeling …………………………… 73

5.5 FEMM 2D Finite Element Modeling ………………………... 74

5.6 FEMM 2D axis symmetric computation of Thrust ………….. 74

5.7 Permanent magnet generators ……………………………….. 76

5.8 CFTS, Measurement of conversion efficiency ………………. 78

5.9 Generator #1: Speed (RPM) …………………………………. 79



LIST OF FIGURES (Continued)

Page

5.10 Generator #1: Current ………………………………………... 79

5.11 Generator #1: Voltage versus power output …………………. 80

5.12 Generator #1: System Efficiency versus power output ……… 80

5.13 Basic types of buoys …………………………………………. 81

5.14 Inner Module with piston and aluminum support ………….... 83

5.15 Buoy Outer Float …………………………………………….. 84

5.16 Back-iron PVC Housing ……………………………………... 85

5.17 OWEC with CFTS on display in the MSRF …………………. 86

5.18 OWEC with CFTS during wave tank testing ………………… 87

5.19 Generator no-load voltage during wave tank testing ………… 88

5.20 Generator voltage, current and power duringwave tank testing …………………………………………….. 89

5.21 Generator voltage, current and power showing irregular motionof the shaft system caused by irregular wave excitation …….. 90

6.1 Comparison of simulation and experimental results under

constant thrust (a) Generator current (b) Generator voltage .. 936.2 Operation with clutch T=2.5s, Hs=0.145, Rload =75ohm

(a) generator load current (b) instantaneous output power …. 94

6.3 Power absorption width from numerical model ……………... 96

6.4 Full scale power output of OWEC …………………………... 98



LIST OF TABLES

Table Page

3.1 Comparison of Simulated and Testing for Thrust = 627N …... 36

5.1 Comparison of FEMM data for Piston/CylinderConfigurations for ¾ shaft CFTS ……………………………. 75

5.2 Validation of peak axial thrust from Finite

Element Modeling and experimental test ……………………. 76

5.3 Wave Flume Test Results of OWEC ………………………… 89

6.1 Typical Conversion Efficiency ………………………………. 96

6.2 Froude Scaling Law ………………………………………….. 97

6.3 Wave flume test results ………………………………………. 100



LIST OF APPENDICES

Page

A. PERMANENT MAGNET GENERATOR

PARAMETERS ……………………………………………… 111

B. MODEL SCALING ………………………………………….. 112

C. ON SURFACE WAVES AND MASS

CONSERVATION …………………………………………... 114

D. EQUATION OF BUOY MOTION ………………………….. 116



A DIRECT-DRIVE WAVE ENERGY CONVERTER WITH

CONTACTLESS FORCE TRANSMISSION SYSTEM

1. INTRODUCTION

1.1 Background

Ocean energy can occur in five main forms, namely, ocean thermal energy

(temperature gradient), marine current, salinity gradients, waves and tidal energy.This work relates primarily to ocean wave energy, which is the conversion of the

up-and-down motion of ocean waves into electrical energy. The world’s first wave

energy device patent was registered in 1799 by Girard, in Paris. Since then, wave

energy research has experienced periods of booms and periods of relative quiet.

During the oil crisis in the 1970’s, there was a strong motivation to develop

alternative sources of energy. Consequently, wave energy research activity

increased and some significant wave energy device components that we see today,

including the Well’s turbine, were invented during that period. However, after the

crisis abated, the interest in wave energy declined and only resurfaced whenever

energy prices soared. Today, the driving factors for wave energy research include

environmental concerns, the need to diversify energy sources and generally, the

good wave energy resource potential.

1.2 Motivation for this work

A significant portion of the world’s energy requirements can be met in a

sustainable and environmentally benign manner, through renewable energy sources



2

including wave energy. While it has been demonstrated for many years that it is

possible to harness the energy from waves efficiently, ocean wave energy converter(OWEC) technology is currently not used for any significant commercial

production of electricity. A review of existing technologies shows that more

research and development is needed in the area of direct drive conversion

techniques, such as proposed in this work, to ensure that devices are reliable,

survivable, maintainable and efficient. This research into direct drive approaches

can be enhanced by ongoing efforts in materials science and engineering, aimed at

developing advanced materials that can withstand the harsh ocean environment.

In the quest for efficient prototypes there is a need for increased

understanding of the dynamics of ocean energy device operation and the

simplification of the energy extraction process. The simulation of these devices

enhances the research and development efforts by enabling the analysis of devices

in various stages of the design process. Simulation also enables the analysis of

large devices that cannot be otherwise analyzed without actually building those

devices.

The analysis of OWECs has been largely based on linear wave theory in

which time domain or frequency domain solutions are obtained usually with

assumptions of small amplitude oscillations (linearity). However, practical

OWECs require large amplitude oscillations and non-linear effects become an

issue. Also, these devices are usually “tuned” to take advantage of resonance. The

time domain solutions required have been enhanced with the availability of

commercial computational fluid dynamics (CFD) software, to iteratively solve the

Navier-Stokes equations. The iterative nature of these fluids software packages also

makes it convenient to include the effects of the generators or power take-off

mechanisms (PTOs) of the wave energy device, which is generally a non-linear

phenomenon.



3

1.3 Literature Review

The ocean wave energy research and development process has involved

both theoretical and experimental research. There have also been prototype

developments and these prototypes are at various stages of development, ranging

from small scale research devices developed in laboratories to full scale models

deployed for sea testing. A detailed account on the types of prototypes and their

various classifications can be found in [1]-[3]. One of several ways the devices are

classified is based on their location with respect to the shoreline.

Depending on the distance between the conversion devices and the

shoreline, wave energy systems can be classified as shoreline, nearshore and

offshore extraction systems. Shoreline devices are devices fixed to or embedded in

the shoreline. These devices have the advantages of easier access for installation

and maintenance and they do not need deep-water moorings or underwater

electrical cables. However, at the shoreline, the power of the waves can be

significantly reduced by bottom friction with the rough seabed.

The offshore devices are exposed to the more powerful waves available in

deep water. These devices can be located at or near the surface and so they usually

would require moorings and submarine electrical cables for power transmission to

the shore. The nearshore devices are variants of either shoreline or offshore

devices that are located close to the shore. For instance, if floating devices have to

be bottom mounted, then they are sited close enough to the shore, where the depth

of water is reasonable for such mounting requirements.

Offshore devices are typically freely floating buoy point-absorbers while

the shoreline devices are typically oscillating water column devices (OWC). The

OWC and the point absorber theory have been the subject of significant theoretical

and experimental research since the early 1970s. Following the invention of the



4

Well’s turbine, many researchers have done theoretical and experimental

investigations on the turbine and findings are available in literature. Almost aroundthe same period the pioneering work on the theory of point-absorbers began. The

following subsections review the literature on these two concepts which form the

basis of most of the ocean wave energy converters (OWECs).

1.3.1 The Oscillating Water Column

An OWC system has a partially submerged hollow air chamber, which

opens to the sea under the still water line as shown in Fig 1.1. As waves enter the

chamber the air in the column is forced through a turbine; when the waves retreat,

the air is sucked back into the chamber, passing through the turbine again [4]. The

most common turbine for this kind of application is the “Wells turbine”, a self-

rectifying axial flow type turbine made of symmetrical airfoils sections.

Fig. 1.1 The Oscillating Water Column (OWC)



5

According to airfoil theory, an isolated airfoil at an angle of incidence

to afree stream will generate a lift force normal to the direction of flow. In a viscous

flow, the airfoil will also experience a drag force in the direction of the free stream.

These forces can be resolved into tangential and axial components. As shown in

Fig 1.2, even though the axial force produced is oscillating with respect to the

direction of air flow, the tangential force on the rotor is always in the same

direction. Consequently, the turbine will rotate in the same direction irrespective of

the direction of air flow through it.

Fig. 1.2 Working principle of Wells turbine

The basic principles of operation and an interactive approach to the design

of the Wells turbine has been described in [4]. In several theoretical investigations

[5]-[7], the parameters that affect the aerodynamic performance of the turbine are

discussed in detail. These parameters include the angle of airflow incidence,

solidity of rotor, blade chord Reynolds number and hub-to-tip ratio. An

experimental investigation of the turbine in [6], also confirmed these parameters as



6

important for the performance of the Wells turbine. In addition to the aerodynamic

parameters relating to the Wells turbine, the geometry of the OWC chamber andpower take off mechanism also affect the overall performance of the turbine.

Evans [8] demonstrated an approach for optimizing a two dimensional device with

respect to the OWC geometry and turbine parameters, using numerical modeling,

while Tindall [9] performed an optimization study of the Wells turbine using

turbine parameters and the induction generator parameters. Thus, the overall

performance of the OWC depends on a complex combination of hydraulic,

pneumatic, aerodynamic and electrical power-take-off issues.

1.3.2 Buoy Point-Absorbers

A point-absorber is a device whose dimension is very small compared to the

wavelength. A significant contribution to the theory of point absorbers was made

by Budal and Falnes [10]-[13], and Evans [14]. Falnes [13] considered the

absorption of waves as a destructive interference between an incident wave and a

radiated wave as shown in Fig. 1.3.

Fig. 1.3 Wave pattern of two interfering waves as seen from above [13]



7

Falnes [13] and Evans [14] showed that a submerged circular cylinder

making small harmonic oscillations in small amplitude waves can be efficient inabsorbing the wave energy in an incident regular wave. They showed that the

maximum power that can be absorbed by a heaving axis-symmetric body is equal

to the incident wave power contained in a wave front of width /2 of regular

waves of wavelength equal to . This width has been termed the “capture” width or

“absorption” width of the device. Falnes [11], [13] and Evans [8] further showed

that theoretically, only 50% of the incident energy can be captured by an axis-

symmetric body in heave oscillation mode. Evans [8] extended these results to

include non-axisymmetric bodies under some assumptions made on the body

geometry. In the case of non symmetric bodies such as Salter Duck or symmetric

bodies in two modes of oscillation a 100% power absorption is possible. This is

confirmed by the high efficiency reported for the Duck, which is a typical non

symmetric device developed in the 1970s.

Fig. 1.4 Capture width of a heaving symmetrical cylinder of diameter d



8

As shown in Fig 1.4, it is theoretically possible for a point absorber to

extract energy from a capture width much larger than its own physical dimensions.However, the smaller the device, the larger the amplitude it has to oscillate in order

to capture maximum energy. Absorbers that are oscillating at amplitudes higher

than that of the incident wave are said to be in resonance with the incident waves.

For an incident wave of a given frequency, there is an optimum phase and

amplitude of oscillation at which maximum energy capture occurs. But, the real

sea is a spectrum of many frequencies and therefore devices have to be

continuously tuned at least to the most dominant frequency.

The phase control of point absorbers to maximize the power capture was

first proposed by Budal and Falnes [17] in a technique called latching. An

electronic circuit was used to provide pulses to a locking magnet and a spring in

order to lock the buoy when its speed is zero at its highest or lowest positions. The

buoy is released by another pulse after a specified time delay. Further work in this

area was carried out by Korde [18] and others, but the problem of control in real

seas remains a fairly complex one that requires further work as discussed in

Chapter 3.

1.3.3 Overview of Some Existing Prototype Devices

The world’s first commercial wave power plant is the LIMPET 500, an

OWC system mounted on the cliffs of the Islay island in Scotland by the company

Wavegen Ltd. It has been connected to the UK’s national grid and it generates a

peak power of about 500 kW. Another type of the OWC technology has been

reported by Energetech Ltd. in Australia [19]. Unlike the LIMPET, it uses a

variable pitch turbine instead of the Wells turbine and it has a parabolic wall behind

the OWC to focus the wave energy on to a collector and associated plant. Though

the OWC type device is primarily designed for shoreline application, there are



9

floating variants like the Mighty Whale. This device was developed by the Japan

Marine Science and Technology Center (JAMSTEC) [ 20].The Pelamis is a nearshore device developed by Ocean Power Delivery in

Edinburgh, Scotland [21]. This is a semi-submerged, articulated structure

composed of cylindrical sections linked by hinged joints. As the waves peak and

trough, the sections of the Pelamis act as a pump and move hydraulic fluid through

hydro-turbine generators. The power generated from each segment runs to an

underwater substation and then to land via a submersible electric cable. A 750kW

device of the Pelamis type that has been deployed at the European Marine Energy

Center (EMEC) in Orkney is the only other known commercial plant connected to

the UK grid.

Several other prototypes that have been developed are either going through

laboratory testing, sea trials or at permitting stages for commercial deployment.

The AquaBuoy is an offshore buoy system which uses the up and down motions of

ocean waves to cause pressure changes which draw seawater into a hose pump.

This pressurized water is expelled into a collecting line leading to a turbine, which

generates electricity [22]. Other prototypes such as the PowerBuoy, developed by

Ocean Power Technologies Ltd. also use hydraulic systems that are coupled to

submerged floats to produce power from heaving motion.

The Archimedes Wave Swing (AWS) consists of a cylindrical, air filled

chamber (the “Floater”), which can move vertically with respect to the cylindrical

“Basement” which is fixed to the sea bed [23]. The air within the “Floater” ensures

buoyancy. When a wave passes over the top of the device, it alternatively

pressurizes and depressurizes the Floater, and it moves up and down with respect to

the Basement. This relative motion is then used to produce electricity in a linear

generator.



10

1.4 Direct-Drive Wave Energy Converters

The typical period of waves in the ocean is about 10s or a frequency of

about 0.1Hz. However, conventional electric generators operate at frequencies of

about 60Hz. In order to connect the slow moving waves to high speed generators,

most of the devices that have recently been proposed have used hydraulic or

pneumatic intermediate power conversion systems. Under this arrangement, the

slow motion action of the waves is used to pump a high pressure working fluid

through a hydraulic motor. The motor then spins a generator at the required speed.

A direct drive device couples the slow motion of the waves to the electric

generator, which is usually a specially designed linear generator or a rotary

generator with some kind of mechanical or magnetic form of thrust transmission

and amplification of speed [24].

Linear generators for direct-drive OWECs have been proposed in [2], [25],

[26]. These devices have simple mechanical construction and few moving parts.

However, their dimensions could be relatively large and this can be explained by

Faraday’s Law. The induced voltage in a generator is E = N d φ /dt , where E is the

induced voltage in the generator windings, N is the number of turns in the winding,

and d φ /dt is the time rate of change of magnetic flux in the generator. Thus, to

increase the induced voltage in a generator, it is necessary to either increase the

relative speed of the magnetic field or increase the amount of magnetic flux while

avoiding saturation of the magnetic circuits. In the linear generator, the amount of

flux is increased by increasing the dimension of the magnetic system to make up

for the slow motion. In addition, the large forces that are generated in these devices

must be supported by auxiliary structures. These devices could therefore have very

large dimensions.



11

Traditionally, electric power generators have been efficient rotary systems

(sometimes with efficiency as high as 98%), although rotary OWECs may notnecessarily be more efficient than those with linear generators. However, high

speed rotary generators can be made more compact for a given power rating due to

the high speed of rotation and this can result in smaller overall dimensions for the

floating device. The rotary generators can also be readily available off-the-shelf

than their linear counterparts. Device developers, apparently desiring to use these

advantages, have resorted to hydraulic or pneumatic intermediate power conversion

systems, which are often inefficient and costly and could create maintenance

concerns in the ocean environment. These hydraulic systems also require not only

seals to prevent the ingress of sea water, but also such “continuously working”

seals are subjected to large stresses, that can create maintenance problems for these

systems. Thus, although rotary generators have desirable advantages their

application in ocean wave energy extraction systems with hydraulic intermediate

systems have limited these advantages.

1.5 Contribution of This Dissertation

The contribution of this work is the development of a novel rotary direct

drive device that will help eliminate the intermediate hydraulic and pneumatic

systems. This was done by employing a contactless force transmission system

(CFTS) comprising a “piston” and a “cylinder” and a ball screw to spin a

conventional rotary generator. The CFTS is a magnetic reluctance force system

made up of axially magnetized neodymium iron boron (NdFeB) permanent

magnets and is designed to enable direct coupling of a floating buoy to the power-

take-off (PTO) components. It has been suggested that the CFTS may have several

other applications, for instance in the replacement of hydraulic actuators.



12

Also, an analytical model of the OWEC based on small amplitude wave

theory was developed to assess the energy extraction of the device withconsideration of the power take-off mechanism (PTO). The conditions for

maximum power extraction were presented. Furthermore, a numerical technique

that employs a computational fluids dynamics (CFD) code is proposed to extend

the inherent limitations of the analytical method in assessing the power output of an

OWEC. The numerical model is a coupled fluid-structure interaction simulation of

the wave energy device using a Reynolds-Averaged Navier Stokes (RANS) solver.

A 3D numerical wave flume was created and the OWEC was excited with waves

created in the tank. In previous investigations, the motion of floating bodies was

prescribed rather than determined. The current method determines the motion of

the buoy from the dynamic solution of the fluid flow problem and the dynamic

buoy motion problem. Finally, the interference effects of an array of buoys have

been evaluated for the case of two buoys with varying separation distance to

wavelength ratio. This effort is aimed at determining the optimum spacing of

buoys for a wave park comprising several modules.

1.6 Organization of This Dissertation

This dissertation is organized as follows; Chapter 1 provides the motivation

and contributions of this work. The chapter gives a brief literature review of

pertinent work as well as an overview of existing technology of wave energy

extraction devices, including a comparison of relative advantages of using either

rotary or linear generators for wave energy devices. Chapter 2 discusses basic

analytical concepts and definitions in wave energy extraction research including a

brief review of linear wave theory. Chapter 3 presents an analytical model of the

buoy system and its simulation, using Matlab/Simulink. The devices simulated



13

include a single float and double float system with a CFTS system and permanent

magnet synchronous generator. Finally in this chapter, the complex problem of control of wave energy devices is introduced. The coupled fluid-structure

interaction problem is formulated and solved in Chapter 4. The chapter begins with

an introduction of the problem and goes on to discuss the solution algorithm and

procedures used to simulate the OWEC. Chapter 5 is devoted to the design of the

novel direct drive OWEC with CFTS. The chapter describes the CFTS and the

overall buoy system. The design procedure and finite element analysis and

optimizations are presented. Also laboratory testing of the CFTS as well as wave

flume testing of the OWEC are presented. The summary of the analytical,

numerical and experimental results are presented in Chapter 6 and conclusions and

recommendations for further work are given in Chapter 7. The dissertation

includes appendices of additional derivations and explanations.



14

2. BASIC ANALYTICAL CONCEPTS AND DEFINITIONS

2.1 Introduction

In this chapter, some basic concepts and definitions are presented. The

chapter begins with the fundamental parameters of ocean waves and goes on to

discuss linear wave theory. The theory assumes small amplitude displacements of

the ocean surface elevation and is no longer valid when the amplitudes are large.

However, it remains a very useful first approximation to ocean surface wave

behavior. In addition, based on linear theory, an expression is derived in this

chapter for the incident power in ocean surface waves in terms of the basic wave

parameters. Finally, the forces on a structure in waves are discussed. The

Morrison equation as it applies to a structure that is free to oscillate in waves is

given.

2.2 Basic Parameters of Ocean Waves

Ocean surface waves are non-linear and random in time and space.

However for basic understanding and for most applications they can be considered

to propagate with an approximately sinusoidal profile with three fundamental

parameters. These are the wave period, T, defined as the time it takes two

successive wave crests or troughs to pass a fixed point, wave height, H, defined as

the vertical distance between the crest and trough and mean water depth, h, defined

as the distance between the still water line and the sea bed.



15

Fig 2.1 The basic parameters of a sinusoidal wave

The wave length, , is the horizontal distance between two successive wave crests

or two successive wave troughs. The wavelength is related to the wave period and

water depth by the dispersion relationship ( )khgk tanh2=ω as shown in section

2.3.3, where = 2 /T is the angular frequency, g is acceleration due to gravity and

k = 2 / is the wave number. In one wave period the crest travels a wavelength,hence the speed of the wave C = /T = /k . The group velocity C g or the velocity

at which a packet of waves travels across the ocean is / k .

Real sea conditions are stochastic and are often described with statistical

parameters such as the significant wave height H s and the energy period, T e. The

significant wave height is defined as the average crest to trough height of one third

of the highest waves and this was chosen to be closest to wave heights reported by

visual observers. The energy period (or peak wave period) is defined as the

reciprocal of the frequency at which the peak of the wave spectrum occurs [27].



16

2.3 Linear Wave Theory

2.3.1 Governing Equation

The linear wave theory is reviewed here in general terms. A more detailed

discussion of this subject can be found in [28]-[30]. The linear water wave

problem is usually formulated with simplifying assumptions of an incompressible

(constant density ρ ), irrotational (curl of velocity vector = 0), inviscid fluid

(viscosity negligible). From the irrotationality condition and mass conservation

(see Appendix C) the following are true for the velocity vector u,

0=×∇ u (2.1a)

0=⋅∇ u (2.1b)

Then a velocity potential function φ can be defined, from which the velocity field

u(u,v,w) can be derived as φ −∇=u or

xu

∂

∂−=

φ ,

yv

∂

∂−=

φ ,

zw

∂

∂−=

φ (2.2)

With equations 2.2 the continuity equation in (2.1b) can also be expressed in terms

of the velocity potential by the Laplace equation in 2.3, that must be satisfied in the

fluid domain, subject to approprite boundary conditions. This makes the water

wave problem a boundary value (BV) problem as shown in Fig 2.2.

02

2

2

2

2

22

=∂

∂+

∂

∂+

∂

∂=∇=∇⋅∇

z y x

φ φ φ φ φ (2.3)

The main difficulty of solving this BV problem lies in the specification of the

conditions at the free surface, which is constantly changing and is in fact the

solution to be determined for the problem. If the amplitude of oscillation is small,

however, the free surface can be assumed to be the still water line, which is

constant.



17

η

02=∇ φ

Fig 2.2. The water wave boundary value problem

The momentum equation for the flow can be expressed by the Bernoulli equation

)()(2

1 222t C gz

p zvu

t =+++++

∂

∂−

ρ

φ (2.4)

where C(t) may depend on time but not on space variables.

2.3.2 Boundary Conditions

At the fixed bottom boundary, since the sea-bed is impermeable, the normal

component of the velocity is zero or,

0=∂

∂−=

−= h z z

w

φ (2.5)

For the free surface there are two boundary conditions; the kinematic free

surface boundary condition (KFSBC) and the dynamic free surface boundary



18

condition (DFSBC). The KFSBC can be derived from the time variation of z =

(x,y,t)as

( )t y x z y

v x

ut

w

,,η

η η η

=∂

∂+

∂

∂+

∂

∂=

(2.6)

Physically, the KFSBC means that particles on the free surface must always remain

there and not cross the water surface. The DFSBC maintains a constant pressure on

the free surface and is derived from equation 2.4 by taking the pressure at the free

surface as zero (the gauge pressure).

)(21

222

t C g z y xt

=+

∂

∂+

∂

∂+

∂

∂+∂

∂− η φ φ φ φ (2.7)

Equations (2.6) and (2.7) are non-linear, and they make the solution of the BV

problem a non-trivial one. For small amplitude waves some linearizing

assumptions [29], [30] enable these conditions to be written respectively as

follows:

0≅∂

∂=

zt

w

η (2.8a)

0

1

≅∂

∂−=

zt g

φ η (2.8b)

or combined to obtain:

01

0

2

2

=∂

∂+

∂

∂

≅=η

φ φ

z zt g

(2.9)

The wave propagation is periodic in time and space and therefore the periodic

lateral boundary conditions are given by

( ) ( )t xt x ,, λ φ φ += , ( ) ( )T xt x += λ φ φ ,, (2.10)



19

2.3.3 Solution of Boundary Value Problem

By using the method of separation of variables and representing the velocitypotential as a product of three variables φ =X(x)Z(z)T(t), we have, from equation

(2.3),

2

2

2

2

211

k dz

Z d

Z dx

X d

X −=−= (2.11)

where k is a constant called the wave number. For small amplitude waves, the task

now is to find a general solution of equation (2.11) that satisfies the boundary

conditions given in equations (2.5), (2.9) and (2.10). The surface displacement,

velocity potential and dispersion relationship from the solution are given by [29],

[30]:

)cos(2

t kx H

ω η −= (2.12a)

)sin(cosh

)(cosh

2t kx

kh

zhk g H ω

σ φ −

+−= (2.12b)

( )khgk tanh2=ω (2.12c)

These equations are important results of the solution of the water wave

problem. Thus, the wave frequency and wave number (or wavelength) cannot be

chosen arbitrarily but rather they must be related through the third expression in

equation 2.12c called the dispersion relationship. The particle velocities and local

acceleration can now be obtained from the above expressions. For instance, the

horizontal velocity and the corresponding local acceleration are given by;

)cos(cosh

)(cosh

2t kx

kh

zhk H

xu ω ω

φ −

+=

∂

∂−= (2.13)

)sin(cosh

)(cosh

2

2 t kxkh

zhk H

t

uu ω ω −

+−=

∂

∂−= (2.14)



20

The pressure at any depth z below the free surface is made up of hydrostatic and

dynamic components and is given by

)( zK ggz p pη ρ ρ +−= ;kh

zhk zK p

cosh

)(cosh)(

+= (2.15)

2.4 Wave Energy Resource

2.4.1 Wave Energy and Power

The total energy per unit width of a wave can be computed as the sum of

the potential and kinetic energy. Their average values over one wavelength are

given by

=

λ η

ρ λ

0

2

2

1dxgPE =

16

2 H g ρ (2.16a)

−

+=

λ η ρ

λ 0

22)(

2

1

h

dxdzvuKE =16

2 H g ρ (2.16b)

The total energy is KE PE E += =8

2 H g ρ . The rate at which work is done on a

vertical section through the water, perpendicular to the direction of the waves is the

energy flux F l or the incident power

−

⋅=

η

h

Dl udz pF = −

⋅

η

η ρ h

p udz zK g )( (2.17)

where pD is the dynamic pressure at depth z and u is the velocity. Substituting

expressions in equations (2.12) and (2.13) into (2.17) and integrating over a wave

period we obtain the incident wave power crossing the section as

dt dzt kxkh

zhk H zK g

T dt F

T P

T

h

p

T

lw ⋅⋅−+

== −

)cos(cosh

)(cosh

2)(

11

00

ω ω η ρ η



21

+=

kh

kh

k gH

2sinh

21

2

1

8

1 2 ω ρ (2.18)

The speed with which energy is being transmitted by the wave packet or the group

speed can be written as

+=

kh

kh

k C g

2sinh

21

2

1 ω and the phase speed is

k C = . For

deep waterk

C g ⋅=2

1and for shallow water C C g = . Generally, deep sea is

defined as one with a depth to wavelength ratio 5.0≥λ

h. Shallow water is also

defined as that with ratio 05.0≤λ h . Water depths that are between these two limits

are known as intermediate depths. The wavelengths can be expressed in terms of

basic parameters by approximations from the dispersion relationship asπ

λ 2

2gT =

for deep sea and ghT =λ for shallow water. From the above expression for deep

sea wavelength, the incident wave power in deep sea expressed in [W/m] of crest

length is given by

π

ρ

32

22TH gPw = (2.19)

The expression for incident wave power is often given in terms of the

statistical parameters for sea spectra and used in the form of P = 0.49H s2T e

(kW/m), where H s is the significant wave height and T e is the peak or energy

period.



22

2.4.2 Wave Energy Resource off Oregon CoastIn order to ascertain how much energy a wave energy device will be able to

convert to useful energy, the potential power available to the device at the site it

would be located must first be determined. Also, the estimation of the incident

wave power is an important step in the design of wave energy devices because if

the equipment is rated too high it will be under utilized most of the time; on the

other hand if it is rated too low it will be unable to capture much of the available

energy or may be damaged. As part of this project, wave energy resource

assessment was carried out using 10 years information from data buoy along the

Oregon coast. Fig 2.3a shows a typical diagram for joint distribution of wave

heights and periods for a typical year. This is typical for the Oregon coast, with the

most dominant parameters being H s =1.5m and T z = 8s. Fig 2.3b shows the

seasonal average incident wave power along the Oregon coast. This profile

matches closely the load profile of the Pacific Northwest which has a

predominantly heating driven demand in the winter months and relatively low air-

conditioning requirements in the summer months.



23

(a)

0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10 11 12

Months

W a v e P o w e r , k W / m

(b)

Fig 2.3. Oregon coast wave energy resource

(a) typical joint distribution of wave height and periods(b) Typical seasonal average wave power over 10 years



24

2.5 Wave Forces on Structures

The wave forces on structures in the ocean can be calculated using

Morrison’s Equation. The equation, originally proposed for estimating the

horizontal force on structures, can be applied to a buoy that is free to oscillate in

waves is a linear superposition of two independent flow fields due to the wave

motion (with the effects of structure motion neglected) and due to structure motion

in otherwise still water [33]

z z AC uu AC z AC u AC F D D D D I A I M M '−+−= (2.20)

where u and u are the vertical water particle velocity and acceleration respectively,

z and z are the velocity and acceleration of the buoy, A I and A D are projected

areas normal to the direction of propagation, D M

C C , are inertia and drag

coefficients, ', D A

C C are added mass and drag coefficients for the oscillating

cylinder in still water. These coefficients can be determined from system

identification experiment. For a vertically oriented cylindrical buoy, the

coefficients2

4 d A I

ρπ = and d A D

2

ρ = are proportional to the projected areas

normal to the direction of the force component, d being the diameter of the cylinder

and ρ is density of water . (Note that the force is expressed as force per unit

length).



25

3. ANALYTICAL MODEL OF OWEC

3.1 Introduction

This chapter discusses an analytical model of an OWEC concept, the design

of which will be described in Chapter 5. The mathematical model of the OWEC is

based on small amplitude linear wave theory explained in Chapter 2. The model

consists of a floating buoy and a power take off mechanism comprising a linear

damper as provided by the braking effects of a permanent magnet generator. The

power output of the generator is modeled by means of its electrical equivalent

circuit. The OWEC has two main components; the inner module and an outer float

and two configurations are modeled.

The first configuration comprises a single float system with the inner

module rigidly moored and the outer float free to respond to wave excitation. For

the sake of physical construction, this arrangement is only suitable for mounting in

shallow waters and requires the use of rigid shafts and swivel joints to allow for

motion in all degrees of freedom. The second configuration is a two float system

with the inner module floating as well as the outer module. The inner module is

equipped with a damper plate and a slack mooring by means of tethers to the sea

bed. As the inner module also floats, it is thus self adjusting and therefore both the

inner module and outer float can maintain the required reference level with respect

to the ball screw mechanism at all times. This is especially useful during high tides

in the real ocean environment.

As the waves hit the floats, both of them respond with different levels of

heave amplitude. The relative velocity of these two floats activates the PTO that is

connected between them. The floats are designed in such a way, that the resonance

frequency of the inner module is outside the normal wave excitation frequency



26

ranges while that of the outer float lies within the most dominant wave frequencies.

The drag plate ensures that the vertical drag on the inner float restricts its motion inheave, so as to increase the relative motion between the two floats.

3.2 Mathematical Description of the System

3.2.1 Single Float System

The schematic diagram of a single float buoy system with rigidly mounted

inner module is shown in Fig 3.1. Considering the inertia and drag forces and

pressure forces mentioned in sections 2.3 and 2.5 of Chapter 2, the equation of

motion of the single float OWEC buoy in Fig 3.1, in a single degree of freedom

(SDOF) heave mode is given by (see Appendix D for derivation)

)cos(0 σ ω +=++ t F cz zb zmv

(3.1)

Fig 3.1 SDOF heave motion of cylindrical buoy

where mv = (m+a) is the virtual mass of the body including the added mass a, b is

the damping of the buoy comprising the hydrodynamic damping of the waves, b1,



27

and that provided by the PTO generator bG, c is the spring constant, F = F 0

cos(ω t+σ ) is the exciting force from the waves, z = z0 cos(ω t) is the heavedisplacement. It can be shown [35]-[37] that the exciting force amplitude and

heave when oscillations are small and linear conditions are assumed are given by

( )( ) kDs ebac H

F −+−=

2

1

22

1

22

02

ω ω (3.2a)

( )( ) ( )

kD

G

s ebbamc

bac H z

−

+++−

+−=

2

1

22

1

22

22

1

22

0

)(2 ω ω

ω ω (3.2b)

The added mass a, hydrodynamic damping b1 , and the spring constant c are

given by McCormick [31] for a cylindrical buoy. From the above expressions, it is

clear that the buoy will be heaving with maximum amplitude if the following

conditions are true;

Gbb =1 and 2)( ω amc += (3.3)

The parameters a, b1 , c, m relate to the physical geometry of the buoy while bG

relates to the PTO damping mechanism. Considering the power across the damper

as a product of the damping force and velocity z zbP G ⋅= , the maximum power

corresponding to the conditions in equation (3.3) is given by [34]

1

2

0

max8b

F P = (3.4)

The damping constant of the generator is determined from the following

considerations. The relationship between the torque on the shaftscrewT and the

axial forcescrewF for the roller/ball screw is given by,

f

scew

screw

lF T

πη 2= (forward driving) (3.5a)



28

b

scew

screw

lF T η

π 2= (back driving) (3.5b)

where l = screw lead [m/rev], η f , η b are the forward and back drive efficiencies of

the ball screw respectively. The generator basically acts like a brake, opposing the

rotation with a torque on the shaft that can be expressed as [40]

0T K T T

+Ω= (3.6)

where T 0 = the loss torque [Nm], K T = the braking torque coefficient of the

generator [measured in Nm/rad/s], Ω = angular velocity of the shaft. For instance if

a permanent magnet synchronous generator (PMSG) is used, as in this project, the

introduction of the constant K T effectively assumes a linear magnetic circuit with

no saturation of the rotor and stator iron. With the relatively large effective air gaps

(of the magnets themselves) that are common in PMSGs, this assumption would

not lead to significant errors. However for control purposes the approach should be

different.

The PTO force during the upstroke is then given by

)(2

0α

π GmT screw I T K

l

F ++Ω= (3.7)

where for the roller screwl

zπ 2

=Ω , l being the screw lead and z is the velocity

of the float in the case of a single float system or is the relative velocity of the two

floats in the double float system,dt

d Ω=α is the angular acceleration of shaft, and

I mG is the moment of inertia of generator. The generator damping coefficient and

inertia are given by

22

=

lK b T G

π and

22

=

l I I

GmG

π (3.8)



29

Because the generator is decoupled, during the down stroke, there is no axial force

from the PTO on the buoy. The generator “free wheels” i.e it is decelerated by theelectrical load connected to it, its own inertia and that of the shaft through the

unidirectional clutch. We then have F screw = 0, or,

0=+ T I Gα (3.9)

The output mechanical power or input power to the permanent magnet generator is

calculated as a product of the torque and mechanical speed is given by

)()()( t t T t Pout Ω= (3.10)

The average shaft power is given by

Ω+Ω=Ω=

Tz

T

Tz

z z

dt t T t K T

dt t t T T

Pav0

0

0

)())((1

)()(1

Ω=

Tz

T

z

dt t K T

0

2)((

1(3.11)

If linear response is assumed, )cos(0 t z z ω = and

2

0

2

22

2

1 z

lK Pav T ω

π

= .

The equivalent circuit of the PMSG is shown in Fig 3.2. The voltage acrossa phase of the generator windings can be expressed as

dt

d

dt

di Lir v

jf j

j j j j

λ ++−= , (3.12)

where r j = phase resistance, i j = current of j-th phase, jf λ =flux linkage in phase j

due to the permanent magnet, j L = phase inductance.



30

Fig 3.2 Equivalent circuit for the PMSG

The peak value of the induced emf of the PMSG is dependent on speed and can be

expressed as Ω⋅== f

jf

j K dt

d E

λ . The currents can be obtained by rearrangement

and integration of equation 3.12, noting that load Riv 11 = . For our example, the

generator parameters were determined from test measurements with Rs = 0.43 ohm,

X s = 0.19 ohm for generator #1. Other specifications of the generators employed are

given in the Appendix.

3.2.2 Double Float System

The double buoy system comprises two floats made up of an outer float #1

and a central spar float #2 (with a drag plate) as shown in Fig 5.17 in Chapter 5. A

schematic representation for analytical modeling is shown in Fig 3.3, where c3

represents an optional mechanical return spring if required. The relative motions

between the two floats activate the PTO damper, bG, that is located between the two



31

floats and which represents the operation of the generator. The force is transmitted

from the outer float to the PTO encased in the inner float by the CFTS which,having been measured in test to have a very high efficiency is assumed “ideal” or

lossless in this analysis.

Fig 3.3 Schematic for the double buoy system

The equations of motion of the two floats are respectively (these are derived

by considering all forces on buoy, in the same ways as for single system),

232213111111)()()( zc zb z I zcc zbb z I am GGGG −−−++++++

)cos(10 σ ω += t F (3.13a)

131123222222)()()( zc zb z I zcc zbb z I am GGGG −−−++++++

)cos(20 σ ω += t F (3.13b)



32

where I G is the effective linear inertia produced by the rotation of the generator and

ball screw shaft system. As before, during the down stroke, there is no axial forcefrom the PTO on the buoy. The generator is decelerated by its load, own inertia and

that of the shaft through the unidirectional clutch. We then have,

1111111 )( zc zb zam +++ )cos(10 σ ω += t F (3.14)

2222222 )( zc zb zam +++ )cos(20 σ ω += t F (3.15)

where F 01 and F 02 are the excitation forces of the outer and inner floats

respectively. The excitation forces are identical to the expression in equation 3.2.

The inner module has a pontoon and a drag plate attached to its base in order todamp vertical motions. The added mass of this float is therefore essentially that of

the rectangular pontoon and drag plate. Note here again that in the above

expressions as well as in Fig 3.3, the parameters (ai , bi , ci ; i =1,2) with the

appropriate suffixes, are the added mass, hydrodynamic damping and spring

constant of the respective floats.

3.3 Simulation of the CFTS Buoy using Matlab/Simulink

3.3.1 Introduction

The expressions from the analytical model of the buoy in section 2.6 were

programmed in Matlab/Simulink for computation of the response and power

output. The computed heave response (with zero PTO damping) was compared to

the response amplitude operator output from a frequency domain analysis code,

SML, developed by Boston Marine Consulting for a spar buoy of radius 20m and

draft of 100m in water depth of 830m and there was agreement as shown in Fig

3.4. This also validated our approximations used for the added mass and



33

hydrodynamic damping of the buoy in our Matlab code and in the Simulink model

for the computation of the power output.

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

Wave Period (T), secs

R e s p o n s e M a g

n i t u d e

matlab code

SML

Fig 3.4 Comparison of response amplitude operators

3.3.2 Simulation Results

A component of the simulink model describing the float and clutch blocks

of the single float systems is shown in Fig. 3.5. The entire model is made up of

generator, buoy/PTO, unidirectional clutch and rectifier sub-blocks.



34

Fig 3.5 Float component of Simulink model

In order to ensure the model accuracy to reciprocating inputs, the model was first

checked with a constant speed input in order to compare with experimental results

performed in the MSRF laboratory.



35

Fig 3.6 Oscilloscope Capture of CFTS under constant thrust

Fig. 3.6 shows a scope capture of experimental tests for an applied thrust of 627N

at a linear speed of 0.26m/s, with the generator connected to a resistive load of 15

ohms. The line voltage waveform is in green with a value of 109V peak-to-peak at

frequency of 205Hz and the other waveforms are line currents with value of

approximately 4A peak-to-peak. Fig 3.7 shows a typical waveform of the simulink

model predictions for the same conditions as in the test experiment mentioned

above. The output of the generator is V=128V, I = 4.28A, f=208Hz. The

difference can be explained by error in capturing the oscilloscope measurements at

the peak values and second order effects neglected in the simulations.



36

0.18 0.19 0.2 0.21 0.22

-50

0

50

25

-25

Time, s

V o l t a g e a n d C u r r e n t , V p p ,

I p

p

currentvoltage

V=128VI=4.28Af=208Hz

Fig 3.7 Simulation of CFTS under constant thrust

A comparison of measured and simulated results for the constant thrust of

627N are shown in Table 3.1. There is generally good agreement between the

experimental and simulation results except for the values of voltage which are

lower from experimental testing as explained above.

Table 3.1: Comparison of Simulated and Testing for Thrust = 627N

LoadLinear

SpeedLine Voltage, Vpp Current, App Frequency, Hz

ohms m/s Simulation Test Simulation Test Simulation Test

5 0.12 57.8 50 6.00 5.7 93.4 98

10 0.20 98.4 80 4.92 4.8 158.7 158

15 0.26 128.0 109 4.28 4.00 208.0 205

20 0.30 149.2 130 3.72 3.6 238.1 240.4



37

A typical voltage output from reciprocating action (without a unidirectional clutch)from a wave with T = 2.5, Hs = 0.15m and 20-ohm generator load, from simulation

is shown in Fig 3.8.

0 1 2 3-50

0

50

-25

25

V o l t a

g e ,

V p p

time, s

0 1 2 3-1.5

-1

-0.5

0

0.5

1

1.5

time, s

C u r r e n t , A

(a) voltage (b) current

Fig 3.8 Typical simulation results of CFTS buoy

under reciprocating wave action

The no-load voltage of the generator during operation in waves with a

unidirectional clutch action on the shaft under the same wave conditions as the

above case is given in Fig 3.9. During free-wheeling, the voltage produced is zero

as the clutch disengages generator from the rotation and the generator is

decelerated. Also, unlike operation under the reciprocating action, with a clutch the

voltage time area is less symmetrical. This is similar to results obtained during

experimental testing in waves (see Fig 5.19).



38

11 12 13 14 15 16-300

-200

-100

0

100

200

300

Time, s

V o l t a g e ,

V

Fig 3.9 Typical no-load voltage of CFTS buoy with clutch

3.4 Control of OWEC

The purpose of control is to ensure that the OWEC oscillates at an optimum

phase and amplitude in order to maximize the power captured. From equation

(3.3), the optimum condition under which the buoy heaves with the maximum

amplitude is attained whenGbb =1

and the frequency of the excitation of the

incident waves, , equals the natural frequency of oscillation of the buoy, given by

the expression

amcn+

=ω (3.16)

The parameterwgAc ρ = is the buoyancy stiffness of the body and is

determined by shape of the body or its water-plane area Aw. The added mass, a



40

10 15 20 25 30 35 40-2

0

2

1

-1 z ,

f v ,

d i s p

time

10 15 20 25 30 35 400

0.2

0.4

P o w e r

time

velocityforcedisp

(a)

10 15 20 25 30 35 400

0.5

1

1.5

P o w e r

time

10 15 20 25 30 35 40-2

0

2

1

-1 z ,

f v ,

d i s p

time

velocityforcedisp

(b)

Fig 3.10 Buoy response and power captured

(b) without latching control (b) with latching control



41

For optimal control, it has been shown [12] that there is need for energy to

be returned from the PTO to the waves during part of the oscillation cycle. If themethod of maximizing the absorbed power is based on phase control, then the

implementation of such a control strategy requires that the velocity of the body be

continually measured and compared to the optimal and then the difference used as

an error signal for the control system. But as given in equation 3.17, the optimal

velocity of the buoy is a function of the excitation force and this requires that the

excitation force must be known in advance. Several schemes have been discussed

in [12], [18].

r R2

1

Fig 3.11 OWEC amplitude and phase control schemeA- incident wave amplitude, f e - excitation force

coefficient, F e - excitation force, Rr – radiationresistance, u – velocity, uopt – optimal velocity,

F u – load force [12].

In the above scheme in Fig. 3.11, the excitation force is determined fromincidence wave measurements at some distance away from the device. The

excitation force is used to calculate the optimal velocity, while the actual velocity is



42

measured for comparison. The signal difference between the optimal velocity and

the actual velocity is input to the controller to generate the correct load force.

3.5 Summary

An analytical model of a ball screw based OWEC has been presented. Two

configurations of the device have been considered. The first configuration is the

single float type device with a rigid mooring, most suitable for installation in

shallow water. The other configuration is made up of two modules that are capable

of floating by themselves; the inner module and an inner float. The inner module is

equipped with a drag plate and flexible mooring by means of tethers. The device is

thus able to adjust itself during periods of high tides. For modeling purposes

different sets of equations are presented for these configurations, since the forces

acting on them are different. However, for practical purposes, one hardware design

of the OWEC as shown in Fig.5.17 can be used with minor modifications that

address mooring requirements to represent the two configurations. There was

generally good agreement between analytical simulations and experimental results.

Also, the control of OWECs to maximize power captured is introduced in

this chapter. A Matlab/Simulink model has been presented to demonstrate the

principle of latching control. The average power captured by the device increases

significantly when it is controlled than when it is not controlled. However,

latching control is not an optimal type of control. For a more accurate phase

control, the excitation force of the waves must be known. A number of proposed

schemes are currently available [12], but the subject needs further research.



43

4. COUPLED FLUID-STRUCTURE INTERACTION MODELING

4.1 Introduction

Until recently, analysis of ocean wave energy extraction devices has been

largely based on linear potential flow theory where the equation of motion is solved

either in the time domain or frequency domain usually with assumptions of small

amplitude oscillations (linearity). However, in situations where large amplitude

oscillations occur, non-linear effects become an issue (note that wave energy

devices should be tuned to resonance for effective operation). With the availability

of commercial computational fluid dynamics (CFD) software, the Reynolds

Averaged Navier-Stokes Equations (RANSE) based techniques which incorporate

fluid viscosity can be used to simulate fluid-structure-interaction (FSI) and include

the viscous effects (these effects are sometimes ignored [15],[34],[39]). The ability

of these codes to support complex grid generation either from their own command

language or to interface with grid generation software can be very useful in the

future as the shape of ocean buoys/energy extraction devices become more

complex. Also, the iterative nature of these solvers can enhance the inclusion of the

effects of the PTO of the wave energy device which is essentially a non-linear

phenomenon. Furthermore, these CFD codes can also be useful in evaluating

multiple buoy systems enabling the ‘shadowing’ effects of buoys to be fully

investigated.

In this chapter, a commercial CFD code has been employed to simulate a

heaving buoy ocean wave energy extraction device. Unlike in other FSI

simulations where the motion of floating bodies are prescribed, in this work, the

buoy is excited by waves that are generated in a 3D numerical wave tank (NWT)

and is free to move in response to the waves. The present work is limited only to



46

fluid and solid mechanics problems. One of its unique features that was employed

in this work is the ability of the user to generate the grid in blocks with any givenblocks not necessarily matching at their interface. Also, there are features that

enable two blocks of grid to slide against each other, thus a part of the grid that

contains the buoy can be moved or regenerated while the far field grids may remain

stationary. The moving of the grid in the fixed global coordinate systems where the

RANSE equations are expressed requires the space conservation law, which

ensures that the volume of the cells is conserved, even though their shapes change

as a result of moving the grid. This law is expressed as

0=− S

s

V

dsvdV dt

d (4.3)

4.2.2 The Numerical Grid

A part of the numerical 3D solution grid is shown in Fig 4.2. The overall

size of the domain is 12m x 3m x 3m and the cylindrical buoy has a diameter of

0.6m and an equilibrium draft of 0.5m. The mesh is made up of three main blocks;

the middle block is a finer mesh that moves with the buoy. This block is

regenerated at every time step to maintain its structural integrity as the buoy moves

up and down. This buoy-fitted block is flanked on each side by relatively coarse

“stationary” meshes that form part of the solution domain but do not form part of

the mesh regeneration procedure at each time step. Furthermore, the "stationary”

mesh has variable size with larger sizes towards the wall boundaries to provide

damping effect of the waves. The numerical grid is generated using the command

language of Comet.



47

Z Y

X X

Fig 4.2 Mesh around a 3D cylindrical buoy

4.2.3 Boundary Conditions

The solution domain is bounded by a wave maker on the left wall boundary.

At the wave maker boundary, the horizontal velocity of motion of the boundary is

imposed on the water particle velocities at the boundary.

Fig 4.3 2D representation of Boundaries of Solution domain



48

The other boundaries of the solution domain are solid walls where no-slip boundary

conditions are applied. The no-slip condition ensures that the fluid moving over asolid surface does not have velocity relative to the surface at the point of contact.

For the floating buoy, the no slip condition also means that the vertical heave

velocity is imposed on the water particles at the boundary of the buoy.

4.2.4 Generation of Waves

The waves were generated by a piston type wave maker located at the left

boundary of the solution domain. The plunger moves sinusoidally with the

function t as sin= . For most wave tanks, the opposite end of the wavemaker is a

typical beach which absorbs the waves that are generated, in order to prevent their

reflection back into the solution domain. This means either non-reflecting

boundaries have to be used or a damping/dissipation zone is added to the solution

domain for damping the waves. The non-reflecting boundary option was not

feasible in the current work and it was thought that adding a damping zone would

increase computational demands. Therefore, in this simulation, the far field

boundary is located far enough and simulation time is chosen in such a way to

avoid such reflections. Also, the cell volumes located towards the boundaries are

made larger and this provided some damping of the waves at the boundaries,

adequate enough to avoid wave reflections.

A linear propagating wave generated by a piston wave maker has velocity

potential and surface elevation given by [44], [29]

)cos()(cosh)2sin2(

sinhtanh4t kxh zk khkh

khkhs

ω ω φ −++= (4.4)

)sin(2sinh2

sinh4),(

2

0

t kxkhkh

khs

t t x

z

ω φ

η −+

=∂

∂−=

=

(4.5)



49

where s is stroke of the wave maker and k = wave number of the generated wave.

The expression in equation 4.5 was used as analytical check on the surfaceelevation of the waves generated by the wave maker without any structure in the

tank.

4.3 Coupling Procedure

The flow around a floating cylinder has been investigated in previous

works, where the motion of the floating body has been prescribed [43], [46]. In the

current work, the motion of the wave energy buoy is not prescribed but rather

obtained from the dynamic system solution of the fluid flow interacting with the

buoy structure with due consideration of the damping of the power take off

mechanism.

The coupling procedure is shown in the block diagram in Fig 4.4. The

problem is defined with a finite volume solution domain grid using the CFD code,

as well as in the set of files that constitute the body motion dynamics block. In this

problem, the mesh moves as the buoy heaves within the fluid. The instantaneous

position of the buoy is determined from the set of add-on codes, implemented in

FORTRAN. These additional codes interact with the solver to determine the

displacements in order to move the buoy to its new position and also to update the

boundary conditions. In order to maintain the structural integrity of the fluid mesh,

it is regenerated at every time step. For this reason, the solution domain mesh has

been generated in a way to enhance mesh regeneration. The grid consists of a fine

mesh fitted to the body (regenerated at every time step) and flanked on each side by

a relatively coarse mesh that is not regenerated. These blocks are connected

together using the explicit connectivity function of the Comet code.



50

Fig 4.4: Block diagram of Coupling Algorithm

The motion of the buoy is governed by Newton’s second law of motion:

ext gwave F F F U m ++= (4.6)

where waveF = wave forces on the buoy , gF = weight of the buoy, ext F = external

force on the buoy, m = mass of the buoy, U

= acceleration of buoy. Following[48], the wave force on the buoy can be calculated as the integration of the pressure

field and viscous stresses on the instantaneous wetted surface of the buoy by the

following expression:

=

+−=

n

j

j j j jwave S n pF 1

)( τ (4.7)

The force component gF is the weight of the buoy which acts in the negative z-

direction and the force ext F is any external force that comes into play for the

particular degree of freedom. In this problem, examples of this force are PTO

force, or restraint forces as provided by tethers and moorings. For this ocean wave



51

energy buoy, the PTO device is represented as a damper. Therefore, the PTO force

on the buoy and the pneumatic power across the damper are respectively given byequation (4.8). Note here that the pneumatic power is calculated as the product of

the vertical heave velocity and the damping force.

U bF d pto= , U U bP d pto

⋅⋅= (4.8)

where bd = power absorbing damping coefficient and U = velocity, in this

particular case vertical heave velocity.

The equation 4.6 is essentially, a set of three translational equations in each

of the three coordinate directions x, y, z. For non-spherical buoys such as the buoyunder consideration in this study, any possible rotation is important because it

could affect the translational motions. Therefore, in addition to the translational

equations, the angular momentum has to be solved to yield additional equations.

Since the buoy was constrained to heave only, all rotational motions were curtailed

and this step was unnecessary. For the equation 4.6, expressed in the form,

F U m = the left side of the equation can be integrated from t n-1 to t n exactly. The

right hand side requires approximation of some mean value of F (which

incorporates the convective and diffusive fluxes and source terms) over the interval

of integration. Consequently, the velocity and displacements in equation 4.6 can be

written as follows,

F m

t U U nn

ˆ1

∆+=

+ (4.9)

U t U U nn

ˆ1

⋅∆+=+

(4.10)

where F ˆ = average value of the resultant force and U = average value of the

velocity,1+nU and

1+nU are displacement and velocity of the current time step,

nU andnU are are displacement and velocity of the previous time step and t ∆ is a



52

carefully chosen time step (chosen in accordance with the CFL criteria). There are

numerous methods by which F ˆ and U can be approximated. Azcuetta [48]

approximated the average values of velocity and force with the following

expressions:

( )1

2

1ˆ−

+= nn F F F (4.11)

( )nn U U U +=

+12

1 (4.12)

This integration scheme employs the trapezoidal rule for determining the velocity

and the Crank-Nicholson integration method for determining the displacement.

The widely used Newmark time integration scheme approximates the velocity and

displacement as follows

+

−∆+=

++ 112

1nnnnn U U t U U γ γ (4.13)

+

−∆+∆+=

++ 1

2

12

1n N n N nnnnn U U t U t U U β β (4.14)

The hybrid Crank-Nicholson method proposed in [48] was compared to the

unconditionally stable Newmark method, with parameters N = 0.5, = 0.25 and

both time integration methods were found to be stable and gave similar results as

shown in Fig 4.5.



53

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 5 10 15 20 25 30

Time, s

H e a v e , m

Newmark hybrid Crank-N

Fig 4.5: Heave displacement with different numerical time integration schemes

4.4 Results

4.4.1 Free Surface

The air and water in the solution domain are replaced by an “effective fluid”

whose properties depend on the physical properties of the constituent fluids and a

scalar indicator function, known as the volume fraction, c. The volume fraction is

assigned a value of one for cells filled with water and a value of zero for cells filled

with air. The transport equation 4.15 is solved for the volume fraction using

computed current values of the velocity field:

0)( =⋅−⋅+⋅ dS vvcdV cdt

d

S

s

V

(4.15)



54

where v is the velocity of the particle, vs is the characteristic velocity of the fluid, V

is the cell volume andS

is the area of the cell face. The high resolution interfacecapturing technique is used for capturing the deformation of the free surface [50].

The HRIC method is based on a convective transport of the scalar quantity c which

indicates the presence of either air or water. The scheme is a non-linear blend of

upwind and downwind differencing schemes designed to provide a sharp interface

between the two fluids. In Fig 4.6, the cylindrical buoy is shown in the NWT with

a capture of the free water surface at some four time instants during the simulation.

Fig 4.6: Free Surface Capturing (single buoy)



56

0

10

20

30

40

50

60

0 10 20 30

Time, s

P o w e r , W

Hs=0.6m

Hs=0.15m

Fig 4.8: Instantaneous power of OWEC in different waves

The computed forces for a wave condition of Hs = 0.15, T = 1.5s. are shown

in Figs. 4.9 and 4.10. Though there are some fluctuations in both the in-line and

the vertical force, the horizontal force which is induced by the oscillatory ambient

flow has the same frequency as the exciting waves and seems to have less

fluctuation than the vertical force which is more strongly non-linear. A magnified

view of a typical vertical force for T = 2.5s, Hs= 0.6m is shown in Fig 4.11 for two

different damping conditions. The graph shows the strong non linearity of the

vertical force. Also, the heave responses corresponding to these two damping

situations as shown in Fig 4.12 show differences in heave displacement amplitudes.



58

0 2 4 6 8 101000

1100

1200

1300

1400

1500

1600

1700

time,s

V e r t i c a l f o r c e s ,

N

bd=600

bd=1000

Fig 4.11: Total pressure and viscous forces in vertical (z) direction.

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 5 10 15 20 25 30 35

time, s

h e a v e , m

bd=1000 bd=600

Fig 4.12: Heave displacement with two damping coefficient (Hs = 0.6m, T = 2.5s)



59

As expected, as the buoy becomes heavily damped by the power extractionmechanism, its motion is reduced in amplitude. The response characteristic of the

buoy is non-linear and shows some interaction of the incident wave frequency and

low frequency oscillations of the WEC as shown by the uneven peaks of the heave

response.

The velocity vectors shown in Fig 4.13 indicate the flow around the buoy.

As the buoy moves up, naturally the water particles will tend to move into the void

created by the buoy, hence we see the velocity vectors pointing up as shown in the

figure. The opposite effect takes place when water particles are moved ahead of

the buoy as it moves down. Thus the velocity vectors point downwards and are

displaced from the buoy. This velocity vector distribution is similar to results

obtained by Yueng and Ananthakrishna [46] under prescribed oscillation.

(a) buoy moving upwards (a) buoy moving downwards

Fig 4.13: Velocity vectors in heave oscillation



60

4.5 Array of OWECs

4.5.1 Introduction

The primary purpose for which OWECs will be utilized in the near future is

for electricity production to feed into a grid or to feed isolated island communities.

In order to obtain the required amount of power to feed into the grid, several

modules of OWECs will have to be interconnected in a wave park. This is

particularly true for locations with low to moderate wave climates, where only

small devices of the order of 1-10kW can be designed. For these areas, several of

smaller devices will have to be interconnected to obtain commercially significant

power to feed into the grid. Even for areas where there are high waves, there are

(structural) limitations on the power that can be extracted per unit volume of a

given device [12], and modules of devices still have to be connected together to

achieve higher output for grid connection.

The electrical interconnection of these plants will invariably follow the

experiences of offshore wind energy. Thus, the module devices located in a given

wave park will be interconnected to a central collection point and the output will be

rectified into direct current, transmitted to the shore by submarine cables and at the

shoreline, the dc will be inverted to alternating current for grid connection. The

main challenge is to determine how neighboring buoys/devices will interact hydro

dynamically and how their power output will increase or attenuate as a result of the

interference effects from other devices. Budal [10] and Evans [15] showed through

theoretical calculations, based on linear wave theory, that there is indeed

interference between two or more point absorbers heaving in close proximity.

More recently, McIver [51], using point-absorber theory and interaction theory,

presented results that include surging devices. In general, the hydrodynamic



61

interference between neighboring bodies can be constructive under certain wave

conditions and destructive under others.This section investigates the performance of an array of buoys, heaving

under the excitation of waves, with the investigation being limited to only two

buoys. It is expected that this will be extended in the future to multiple buoy

systems.

4.5.2 Double buoy array of OWECs

The system comprises two identical buoy OWECs heaving in response to

incident waves in a NWT. The objective here is to investigate the effects of

interactions between the devices and to determine how their performance compares

to their operation as isolated OWECs. The dimension of the NWT is 15m x 3m x

3m and the buoys are each of diameter 0.6m and length 1m, neutrally buoyant. The

buoys are initially spaced at a physical distance of 3m and as the wavelength is

changed this relative spacing to wavelength also changes. A typical grid

surrounding the two buoys is shown in Fig 4.14.

Fig 4.14: Mesh around a double buoy system

The effect of the variation of the spacing between the two buoys is evaluated with a

parameter expressed as a ratio of the spacing between buoy divided by the



62

wavelength. With the range of periods considered, this ratio is in the range of 0.2

to 1 as shown in Fig 4.15.

1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Period, s

s p a c i n g / l a n d a

Fig 4.15: Buoy Spacing Evaluation Parameter

The effects of the separation between the two buoys can be observed in the in-line

wave forces on the two buoys. Fig 4.16 shows these forces on the two cylindrical

buoys as a result of wave excitation. These figures correspond to H s = 0.15m and

T=1.5s and T=2.5s respectively for (a) and (b) or a ratio of spacing to wavelength

of about 0.85 and 0.4 respectively. As the wavelength is increased the forces on

the second buoy lag by a wider margin. Also, the amplitude changes in Fig. 4.15

could be due to interference of the two buoys.



63

-1000

-500

0

500

1000

0 5 10 15 20 25 30

time, s

F x ,

N

buoy#1 buoy#2

(a)

-800

-600

-400

-200

0

200

400

600

800

0 5 10 15 20 25 30 35 40 45

time,s

F o r c e ( F x ) , N

buoy#1 buoy#2

(b)

Fig 4.16: In-line wave force on buoy array

The location of two buoys apart also helps smooth out the instantaneous power

output of the plant. As shown in Fig 4.17 the shift in phase of the two systems

enables the troughs in instantaneous power of one device to be filled by peaks of

the other device. As the number of devices connected increases this smooth power

output improves the ultimate quality of the power that is injected into the grid.



64

0.0

0.2

0.4

0.6

5 15 25time, s

P o w e r , W

buoy#1 buoy#2

Fig 4.17: Instantaneous power of buoy array

The heave displacements in Fig 4.18 show some interaction between the two

buoys. In the cases presented the amplitude of motion of the second buoy is higherthan the first buoy. The first buoy has experienced a reduction in amplitude

compared to the isolated case, while the second has experienced increase in

amplitude over the isolated case. The case in 4.7 (c) shows severe interactions of

the displacements compared to (a) and (b). These cases correspond to wave

conditions of H s = 0.15m and wave periods of 1.5s, 2s and 2.5s respectively.



65

-0.06

-0.03

0.00

0.03

0.06

2 7 12 17 22 27

Time, s

D i s p l a c e m e n t ,

m

buoy#1

buoy#2

(a)

-0.20-0.15

-0.10-0.050.000.050.100.150.20

2 4 6 8 10 12 14 16 18 20 22

Time, s

D i s p l a c e m e n t , m

buoy #1

buoy #2

(b)

-0.60

-0.40

-0.20

0.00

0.20

0.40

0 5 10 15 20 25 30

Time,s

D i s p l a c e m e n t , m

buoy#1

buoy#2

(c)

Fig 4.18: Heave displacement of double buoy array



66

The free surface capture of the array is shown in Fig 4.19, with the two

buoys initially neutrally buoyant.

Fig 4.19: Free Surface Capturing of double buoy array

4.6 Summary

A numerical method to demonstrate the use of FSI to simulate an OWEC

device. The CFD code employed, COMET is a finite volume code that has been

widely used and validated in previous published works and provides efficient

algorithms and mathematical models for the simulation of continuum mechanics

problems.



67

The buoy investigated in this work has a simple cylindrical shape, but the

structure of existing device prototypes is more complex than smooth cylinders.However, the procedures developed in this work can be used in applying

commercially available fluids solvers to simulate FSI and determine the power

output of an OWEC device. For more complex buoy shapes, further work would

be needed to use external mesh generation algorithms and programs to create more

complex grids that replicate the shape of real buoys and then import these grids into

commercial codes to solve.

The work was expanded to include multiple buoy systems to determine

spacing requirements, similar to what pertains with wind energy. However, it

turned out that this required more computational resources.



68

5. DESIGN AND TESTING OF DIRECT-DRIVE OWEC WITH

CONTACTLESS FORCE TRANSMISSION SYSTEM

5.1 Introduction

One of the major thrust areas of this work has been the investigation and

design of a novel wave energy buoy concept with a focus on the simplification of

the ocean wave energy extraction processes. This means avoiding systems that use

intermediate hydraulics or pneumatics stages and promoting the concept of a direct-

drive approach that allows generators to respond to the slow movement of the

ocean waves either directly (linear generator systems) or indirectly through

efficient thrust transmission system (rotary generator systems). The direct drive

approach requires innovative systems not only for efficient conversion of the slow

motion of waves into high rotary speeds for power take off (PTO) systems, but also

an effective means of transmitting the force from the waves onto PTO mechanisms.

In this chapter, a system is proposed, which employs magnetic fields for contact-

less mechanical thrust transmission. This system has enhanced the design of a new

direct drive ocean wave energy extraction device using ball screw to act as a

mechanical gear system for fast speed and torque transmission. Although the

contactless force transmission system (CFTS) was developed for the above

mentioned buoy, it has been suggested that it could have several potential

applications in industry and therefore an extensive investigation and optimization

of this system has been carried out. The following sections describe the system and

the optimization and testing of the CFTS as well as the design and testing of the

OWEC with CFTS.



69

Fig. 5.1 Solid model of Buoy with CFTS

Fig. 5.1 shows a solid model of the buoy with CFTS. The complete device

consists of an outer buoy “ float ” inside which is a ferromagnetic cylinder which

slides with respect to an inner module which contains the PTO components,

comprising the piston, ball screw and permanent magnet synchronous generator.

The inner module is completely sealed. The buoyancy force on the outer cylinder

is transmitted through the wall of the inner module to the ball nut by the magnetic

fields of the CFTS. The generator is coupled to the screw shaft by means of a

unidirectional clutch and it can be located either at the top of the inner module (see

Fig 5.17) or at the base of the module (see Fig 5.1) without change in performance.

The main consideration in the location of the generator is the requirements for

accessibility.



70

5.2 The Contactless Force Transmission System

5.2.1 Description and Proof-of-concept of CFTS

The contactless force transmission system is a tubular reluctance system

made up of alternating pole permanent magnets and soft iron pole pieces. The

neodymium-iron-boron (NdFeB) magnets are axially magnetized and configured in

a "piston" with two opposing poles squeezing magnetic flux through a central pole

piece through the back-iron "cylinder" which is mounted on the buoy. The

magnetic field distribution of the system is dependent on the position of both the

piston and cylinder and is shown by the finite element plots in section 5.2.3.

Generally, a reluctance force is generated when there is axial displacement of the

back iron cylinder with respect to the central poles of the piston. The force tends to

restore the cylinder to a position of minimum magnetic energy. This means that, as

the cylinder is moved up by the waves, the piston tends to follow it thereby moving

the nut. This results in rotary motion of the shaft. It is important to note that the

reluctance force that is developed is totally transmitted to the piston through the

magnetic field of the permanent magnets on the shaft. As there is no contact

between the shaft system and the buoy system, the former can be completely

enclosed and sealed to prevent the ingress of sea-water and other unwanted

material. Also the tubular construction of the contact-less system, with ring type

permanent magnets provides a centering effect on the ball nut thus reducing side

loading on the shaft. This is especially helpful for ball screw systems that perform

less satisfactorily under side loads.

The proof of concept of the CFTS and ball screw system was developed

through three stages, with 5/8, 3/8 and 3/4-inch diameter ball screws with screw

lead of 5mm as shown in Fig 5.2. Each design had different implementations of

either the piston or cylinder or both. Initially, the piston and cylinder were



71

implemented with small pieces of cylindrical magnets embedded in PVC (Fig 5.2a)

but these were found to have inadequate holding force. A more compact designwith ring magnets was proposed and another proof-of-concept design was

implemented with a non-salient cylindrical back iron (Fig 5.2b). In the third stage

of development, several configurations of implementation of the cylindrical piston

and back iron were investigated and a suitable design based on the criteria of

maximizing the peak axial thrust was selected. In the final prototype, friction was

also reduced by using UHMW, a teflon-like material, on both the piston and

cylinder components.

(a) (b) (c)

Fig 5.2: CFTS proof of concept development stages(a) prototype #1 – cylindrical magnets with 5/8’ ball screw

(b) prototype #2 – ring type magnets with 3/8’ ball screw(c) prototype #3 – ring type magnets with 3/4’ ball screw

5.2.2 Design of the CFTS

A number of configurations of permanent magnet arrangement and design

of the back iron were investigated for optimum transmission of thrust as shown in

Fig 5.3. Out of these, two configurations stand out as the best possible

arrangement. Due to the absence of salient poles to concentrate the flux through the



72

back-iron, the non-salient back-iron construction had the lowest value of axial

thrust. For the salient constructions, three variants were possible but out of theseconfigurations, Design #2 and #4 provided superior characteristics. Also, the thrust

characteristics of Design #2 and Design #3 were not significantly different.

Fig 5.3: Design configurations of CFTS

5.2.3 Finite Element Analysis of the CFTS

Finite Element Analysis (FEA) was done to optimize the magnetic circuit

design. For the optimization process, the finite element package Flux-2D from

Magsoft Corporation, with its parameterization and translating air gap features was

used (see Fig. 5.4). Also a finite element magnetics freeware package (FEMM)

developed by David Meeker, was used for comparison. For a given optimum pole

thickness and magnet dimensions, the best two configurations of permanent magnet

and pole piece arrangements (Designs #2 and 4 in Fig 5.3) were investigated for the

“piston” as shown in Fig 5.5. The main difference between these two

configurations is the size of the central pole piece. In Design #2, the central pole

piece is twice the size of the outer pole pieces. In a conventional tubular machine,



73

this would have been the arrangement of pole pieces to create a symmetrical system

of equal flux linkage to all phases in order to produce balanced two or three phasevoltages. As there are no such requirements in this application, the Design #4,

which has all pole pieces of the same size is also possible and has been

investigated.

Fig 5.4: Flux 2D Finite Element Modeling

As shown in the FEA results on Fig 5.6, the peak thrust of the Design #4 is

higher than that of Design #2. The peak thrust is obtained at a displacement

approximately equal to one pole dimension. However, the thrust characteristics of

Design #2 are wider than that of Design #4, with high thrusts distributed over a

wider range of axial displacement.



74

(a) Design #2 (b) Design #4

Fig 5.5: FEMM 2D Finite Element Modeling

CFTS Reluctance Force

0

100

200

300

400

500

600

700

800

900

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Axial displacement, inches

A x i a l F o

r c e , N

Design #4 Design#2 Design #1

Fig 5.6: FEMM 2D axis symmetric computation of Thrust



75

The difference in the two characteristics is interesting and can be attributed

to saturation of the central pole in Design #4 compared to Design #2 and the effectsof flux leakage. In Design #4, the effects of saturation of the central pole make the

thrust lower compared to Design #2 at higher displacements. On the other hand,

the large central pole piece and consequently larger dimensions in Design #2 allow

for increased leakage which generally reduces the flux density and thrust.

Depending on the required application, either curve can be chosen either to

maximize the peak thrust (Design #4) or to allow adequate vertical travel (Design

#2).

For the implementation of the back iron cylinder, the salient back iron

designs developed larger reluctance force compared to the non-salient cylindrical

back iron, as shown in Fig 5.6. The thrust of the non-salient cylindrical back iron

was about 2-3 times lower than that of the salient back-iron due to the influence of

saliency on the magnetic reluctance. The peak thrust comparison is in Table 5.1.

Table 5.1: Comparison of FEMM data for Piston/CylinderConfigurations for ¾ shaft CFTS

Design

Configuration

Peak

Thrust, N

Remarks

Design #1 343 Non-salient back iron

Design #2 763 Salient back iron type1



The FEA results were validated with experimental testing to determine thepeak output thrust. As shown in Table 5.2, there is strong agreement between the

FEA results and the experimental results.



76

Table 5.2: Validation of peak axial thrust fromFinite Element Modeling and experimental test

5.2.4 Laboratory Testing of the CFTS

The laboratory testing of the CFTS was carried out in the MSRF by

applying known thrust to the cylinder and measuring the electrical output of the

permanent magnet generator. Two permanent magnet generators, shown in Fig.

5.7, were investigated for this project.

(a) Generator #1 (b) Generator #2

Fig 5.7: Permanent magnet generators

Peak Axial Force (N)

Prototype

FEA ModelPrediction

Test

3/8 Shaft 122 117.6

3/4 Shaft 900 894.3



77

Generator #1 was the generator intended for use in the buoy. However, in

order to increase the travel of the outer float, it was thought that the generator thatwould be used should fit into the inner module so that the outer float would be able

to slide past it. Generator #2 fits well into the inner module and was used for the

wave tank testing, though its impedance was very high and there was significant

voltage drop across its windings when it is loaded.

The laboratory setup for the testing is shown in the Fig 5.8. A known thrust

is obtained by attaching weights to the outer cylinder and releasing it to accelerate

under gravity. The speed measurement was obtained from the oscilloscope capture

of the output waveform by measuring its frequency and using equation for the

speed of a synchronous generator

p

f ns

120= (5.1)

where p is the number of poles and f is the frequency and ns is the synchronous

speed of the generator. From the calculated speed, the axial velocity is obtained

from the formula

ldt

dz π 2⋅=Ω [rad/s], using the screw lead, where Ω is the

mechanical speed of rotation of the shaft and z is the vertical heave displacement.

The input power to the system is the product of the applied thrust and linear

velocity and the output power is measured directly as the electrical power

dissipated in the resistances that were connected across the generator.



78

Fig 5.8: CFTS, Measurement of conversion efficiency

Figs 5.9-5.12 show test result of generator #1 during the laboratory testing to

determine the system efficiency. The Fig 5.9 shows the shaft speed of the generator

under load and during no load operation. Under no-load, the higher speeds result in

higher losses and consequently a non-linear speed-thrust characteristic. Under

load, the generator speed much lower and is more linear with thrust. The current,

as expected increases fairly linearly with the applied thrust as shown in Fig 5.10.

The overall system efficiency is greater than 50% for the 10-ohm load but falls as

the electrical load is reduced. The screw system has some inherent value of system

static friction and this becomes less significant as the shaft is loaded with high

thrust and it spins more freely. As shown in Fig 5.12, the generator is able to

develop an output RMS power of about 150W. Similar curves were obtained forgenerator #2, except that its high impedance, resulted in significant voltage drops

and lower power output.



79

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800

Thrust, N

R P M

20-ohm

no-load

15-ohm

10-ohm

5-ohm

Fig 5.9: Generator #1: Speed (RPM)

0.00

0.50

1.00

1.50

2.00

2.50

0 200 400 600 800

Thrust, N

C u r r e n t , A r m s

5-ohm

10-ohm

15-ohm

20-ohm

Fig 5.10: Generator #1: Current



80

0

10

20

30

40

50

60

0 50 100 150

Power Output, W

V o l t a g e ,

V r m s 5ohm

10ohm

15ohm

20ohm

Fig 5.11: Generator #1: Voltage versus power output

0

10

20

30

40

50

60

70

0 50 100 150

Power Output, W

E f f i c i e n c y , p u

5ohm

10ohm

15ohm

20ohm

Fig 5.12: Generator #1: System Efficiency versus power output



81

5.3 Buoy System Design

5.3.1 Types of Buoys

The buoys that are deployed in ocean environments can be either of the

subsurface types or surface types. The subsurface buoys are submerged while

surface buoys float on the surface. Most of the ocean wave energy extracting buoy

type devices are surface type devices, because at the surface, they are exposed to

the repetitive excitation of waves (as we go deeper, the effect of the waves

diminish). The basic types of surface buoys are shown in Fig 5.13. Based on their

shape, these buoys can be classified as discuss buoys, characterized by large water

plane area and small draft or spar buoys that have a small water plane area and

large draft. The discuss buoys are wave surface followers with relatively small

displacements. The spar buoys however can have large displacements but are often

surface decoupled due to their small water plane area for wave excitation [35].

Fig 5.13: Basic types of buoys



82

The intermediate buoys combine the features of these two basic buoy types

to achieve large heave displacements with little roll motion. These buoys haverelatively large water plane areas that provide adequate buoyancy for energy

extraction. The buoyancy force is essentially a spring force with stiffness given

by wgAc ρ = , where Aw is the water plane area.

For OWECs devices that are extremely large and heavy, the PTOs are often

mounted under water or directly on the sea bed. This is particularly true for large

linear generators. When the devices are smaller, they can be mounted in the floats

on the sea surface or under water on the sea bed. The devices that are mounted on

the sea bed are potentially safe from “hazards” that are usually present on the

surface; such as damage from vessels and strong waves, PTO cable damage and

entanglement, etc. However, these devices are not readily accessible for

maintenance. The opposite is true for devices that are mounted on the sea surface;

they have better accessibility but are exposed to the hazards mentioned above.

5.3.2 Inner Module

The OWEC with CFTS has been designed with a spar type buoy as the

inner module and an intermediate buoy as the float. The inner module is

completely sealed and contains the ball screw, generator and the piston. For design

purposes, the outer diameter of the inner module is determined by the magnet

dimensions. The effective airgap between the piston and the outer cylinder is

determined by the wall thickness of the inner module and associated clearances. A

large gap allows the use of a thicker PVC tube that will be rigid enough to prevent

bending. However, a larger gap significantly reduces the thrust that can betransmitted by the magnetic fields. Therefore a trade-off between a rigid tube and a

high thrust is required. In the current design a PVC tube of a ¼ inch thickness was

found to satisfy the design requirements mentioned above (see Fig 5.14). Other



83

materials, such as aluminium, were investigated but were found to have some

disadvantages. Though an aluminium tube of half the thickness of the PVC tubecould have provided the required stiffness, the eddy currents that are induced in this

material tend to oppose the motion of the piston even at low speeds of operation.

Also it was observed that even a small dent on the tube due to impact (this may

occur during installation or transportation) could also restrict motion of the piston

inside the tube. The PVC material, being plastic in nature, was thus found to be

more suitable for this project. However, for larger devices the flexing or buckling

of PVC tube may have to be overcome with the use of advanced composite

materials such as carbon fiber or kevlar.

Fig 5.14: Inner Module with piston and aluminum support

The total length of the inner module is 5.5 feet. Fig 5.14 shows components

of the inner module including the piston on an aluminum base and the PVC

cylinder.



84

5.3.3 Outer Float

The Fig 5.15 shows the outer buoy, made from PVC barrel and coveredwith ½ inch thick PVC top and bottom plates. The top and bottom plates are

sealed with silicon gaskets placed in grooves in the plates and are held down by 12

equally spaced aluminum bolts and nuts in order to distribute the load. The inner

part of the buoy has been machined from a PVC block to provide housing for the

back-iron cylinder. The Fig 5.16 shows the PVC housing completely covering the

back-iron cylinder. The PVC material thickness has been increased in order to

provide the required support for the top and bottom plates as well as being able to

hold an o-ring for sealing the buoy.

Fig 5.15: Buoy Outer Float

The diameter and length of the buoy are both approximately 2 feet. The size of the

outer float is determined by limiting its weight to the maximum holding force of



85

the CFTS so that the buoy/cylinder would remain coupled to the inner module/

piston under normal conditions of operation. Therefore the weight of the buoy wasfixed at 165 pounds, well below the 200 pounds required to decouple the cylinder

from the piston.

Fig 5.16: Back-iron PVC Housing

5.3.4 Complete Assembly

Fig 5.17 shows the complete assembly of the OWEC with CFTS assembled

in the main test laboratory of the MSRF. The system is equipped with a base plate

that serves as a mooring system for the OWEC. The plate has three studs on which

weights are placed to keep the assembly in place during testing in the wave flume.

The plate is made of aluminum and enables the entire length of the system to be

adjustable from 6-7 feet. A swivel joint with a variable angle of swing is also

mounted on the shaft to allow the buoy to move in all six degrees of freedom.



86

The PTO generator is located at the top of the inner module in a transparent

PVC tube with end caps holding the power cable. However the current design issuch that the generator can be located at the base of the inner module without any

change in performance.

Fig 5.17: OWEC with CFTS on display in the MSRF

5.4 Wave Flume Testing of OWEC with CFTS

The wave flume that was employed for tank testing is located in

Springfield, OR and is about 7 feet deep, 30 feet wide and 110 feet long and tapers

Inner

module

Generator

Float

Mooringsystem



87

to a typical beach. The are two sets of hydraulically driven wave makers that are

activated in sequence to create irregular waves of about 4 feet height and about4seconds dominant period. The OWEC was tested in irregular waves in order to

prove the design concept. Further testing is planned in the O.H. Hinsdale Wave

Research Laboratory, where various incident wave frequencies can be applied to

determine the optimum operation point of the device. Fig 5.18 shows the OWEC

under testing in the Springfield wave flume.

Fig 5.18: OWEC with CFTS during wave tank testing



88

The oscilloscope capture in Fig 5.19 shows the no-load voltage of the electric

generator during the up-stroke and down stroke portions of the wave cycle.Because of the uni-directional clutch the generator free-wheels on the down stroke

and no voltage is generated. The peak to peak voltage of the generator is about

400V. Fig 5.20 shows a typical oscilloscope capture of the OWEC operating into

a 75-ohm load, showing waveforms for the voltage (green), current (blue) and

power (red). The peak output power under load is about 69W. Although it fits into

the inner module nicely as required, the generator’s synchronous reactance is very

high and the voltage drop across it was also high. It is anticipated that with a

generator of relatively lower impedance, the output power will greatly improve.

The possibility of modifying this generator by rewind, to reduce the synchronous

reactance is currently being pursued.

Fig 5.19: Generator no-load voltage during wave tank testing



89

Fig 5.20: Generator voltage, current and power during wave tank testing

Table 5.3 shows other test results under different electrical loading conditions in

the wave flume testing.

Table 5.3: Wave Flume Test Results of OWEC

Load

Resistance

ohm

Voltage

(Vpp)

V

Current

(Ipp)

A

Power

(Wp)

W

20 16 0.5 6

30 35 0.7 18.4

50 52 0.6 23.4

75 65 0.6 29.3



90

The Fig. 5.21 shows a typical waveform showing the irregular motion of the

shaft system due to the irregular wave excitation. The irregular strokes thatresulted from these wave conditions sometimes tend to stall the shaft rotation on

the upstroke cycles where it should be spinning. It is proposed that this can be

overcome with a good dynamic control system.

Fig 5.21: Generator voltage, current and power showing irregular motion of the

shaft system caused by irregular wave excitation

5.5 Summary

A CFTS has been designed to transfer thrust from an outer cylindrical back

iron to an inner piston. The device enabled the design of a new OWEC in which an

outer float was coupled to a PTO mechanism, comprising the piston and a ball



91

screw shaft with a mounted rotary generator. Because the coupling is through

magnetic means, the inner module containing the piston as well as the generatorwas completely sealed. The design of the CFTS progressed through three

prototype development stages. Several design options of the final prototype of the

CFTS have been compared through finite element simulations and experimental

testing, with good agreement. The optimal design of the CFTS chosen was based

on the criterion of maximum peak axial thrust.

Based on the chosen CFTS configuration, a new OWEC was designed and

successfully tested in irregular waves in a flume. The test results show a peak

power output of about 50W. Further testing is planned in the O.H. Hinsdale Wave

Research Laboratory under different excitation frequencies and wave heights.



92

6. SUMMARY OF RESULTS

6.1 Analytical Results

6.1.1 Roller Screw and CFTS Design and Analysis

The performance of the screw and CFTS is dependent on a number of

parameters that include the screw properties (particularly the pitch), the piston and

cylinder holding force capability and the generator damping coefficient. With the

high overall system efficiency achieved (50-60%), the OWEC with a ball screw has

been demonstrated as an efficient device for ocean wave energy conversion. The

central component in the buoy system, the CFTS is made up of a piston and

cylinder, and has been designed to optimize the axial thrust. The piston was

designed with NdFeB ring type magnets of dimensions 100mm outer diameter,

50mm internal diameter and 25mm thickness. A peak axial thrust of 900 N was

achieved from testing compared to finite element analysis predictions of 894 N.

The buoy system with the CFTS was modeled together with a coupled

permanent magnet synchronous generator for reciprocal testing in the MSRF

Laboratory. The generator was modeled with its simple equivalent circuit, ignoring

mutual inductances. The simulations results of the system characterization of the

CFTS are generally comparable to the experimental results and as shown in Fig 6.1,

with the simulation results slightly higher than experimental results due to second

order effects ignored in the simulation.



93

0

1

2

3

4

5

6

7

0 5 10 15 20 25

Load, ohms

C u r r e n t , A

testing simulation

(a)

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

frequency, Hz

V o l t a g e

, V

t esting sim ulat ion

(b)

Fig 6.1: Comparison of simulation and experimental results under constant thrust

(a) Generator current (b) Generator voltage



94

6.1.2 Analytical Simulation of OWEC

The analytical modeling of the OWEC in the presence of waves was basedon linear wave theory by considering the inertia and drag forces on the buoy. The

added mass and hydrodynamic damping were taken from closed form expressions

derived by [31]. The floating component of the analytical model was first validated

with a frequency domain code, SML [52] for a floating spar buoy prior to

application to the OWEC. The results obtained from analytical simulations are also

comparable to experimental tank test results as shown in Fig 6.2.

0 0.5 1 1.5-1

-0.5

0

0.5

1

Time, s

C u r r e n t , A

0 0.5 10

5

10

15

20

25

30

Time, s

P o w e r , W

(a) (b)

Fig 6.2:Operation with clutch T=2.5s, Hs=0.145, Rload =75ohm(a) generator load current (b) instantaneous output power

6.2 Numerical Results

6.2.1 Introduction

For the numerical model, the emphasis was on the fluid-structure interaction

analysis and therefore the equivalent circuit of the generator was not included in the

analysis. The power extraction mechanism was represented as a linear damper



95

system and the energy capture across the damper was therefore essentially the

pneumatic energy. For such a system, the accuracy of the model is dependent onthe losses in the electrical generator. If these losses are significant, then the results

would differ from the real situation by a wide margin. Due to the fact that losses in

generator are generally small, and the desire of this work to establish a method to

simulate the OWEC using computational fluids dynamics software, this approach

was found adequate. In the future the equivalent electrical circuit can be included

in the simulations in order to obtain the actual electrical power output.

6.2.2 Power Capture Width

The power capture or power absorption width of an OWEC is defined as

w

cap

capP

P=λ (6.1)

where Pcap is the power captured by the heaving buoy and Pw is the maximum

power in the incident waves. The power absorption width of the single buoy

system of the numerical model is given in Fig 6.3 as a function of the kd , where k is

the wave number and d is buoy diameter. As shown in the figure, the capture

width increases gradually as the resonance frequency for the device is approached

and then drops with frequency. However, the capture width that was achieved is

well below the maximum capture width of a point absorber in heave defined as

/2 , where is the wavelength. With optimized buoy geometry, for example, a

cylindrical buoy with a semi-sphere or conical base it would be possible to increase

the power absorption width as reported in [36].



96

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1 1.2

kd

C a p t u r e W i d t h , m

cap width max cap width

Fig 6.3: Power absorption width from numerical model

The numerical model produced conversion efficiency that is shown in the

Table 6.1. The maximum efficiency achieved was 37% under resonance conditions.

This is consistent with the theory of point- absorbers [12].

Table 6.1: Typical Conversion Efficiency

T Hs P PinConversion

Efficiency

s m W W %

1.5 0.15 5 20 24.7

2.0 0.15 10 27 37.0

2.5 0.15 4.5 34 13.3

2.5 0.60 50 540 9.3

3.0 0.15 1.4 41 3.5

4.0 0.15 0.6 54 1.1



97

6.2.3 Full Scale Prototype

In order to determine an appropriate rating and dimensions of full scale orprototype devices that would be constructed for ocean testing, scaling laws are used

to establish similarity or equivalence between the model and the prototype. The

Froude’s law is one of the most widely used scaling laws. The Froude number is

defined as [53]

d

r gD

uF

2

= (6.2)

where u is the fluid velocity, g is acceleration due to gravity and Dd is a

characteristic dimension, usually the diameter of the buoy. Assuming a model

scale factor of p, the Froude scaling law can be determined by equality of F r

between model and prototype parameters as shown in Table 6.2. For example, to

get the speed of the prototype we multiply the model speed by the square root of

the scale factor ( p>1). An additional explanation of the Froude model scaling

procedure is given in the Appendix B.

Table 6.2: Froude Scaling Law

Parameter Model Prototype

Length 1 p

Time 1 p0.5

Speed 1 p0.5

Force 1 p3

Power 1 p3.5

Based on computations using Table 6.1, a full scale model of buoy diameter

d = 6m and length l = 10m , under a typical wave height of Hs = 1.5m will develop



98

a power output as shown in Fig 6.4, with a nominal of about 30kW . The output of

the OWEC will increase with the increase in wave height as shown in Fig. 4.8 andthe actual sizing of the OWEC for sea operation would be based on this output and

the probability of exceedance curves with various wave heights of the particular

location where the OWEC will be installed. The actual device could be uprated by

a factor of 2-3 to capture the energy in higher waves than 1.5m, depending on the

anticipated sea states in the test location. The actual device could therefore be rated

at about 100kW.

Power Output

0

5

10

15

20

25

30

35

3 5 7 9 11 13 15

Period, s

P o w e r , k W

Fig 6.4: Full scale power output of OWEC

6.2.4 Double Buoy Array

FSI simulations for the interaction of two buoys was presented to

investigate the performance of one buoy in proximity to the other, with varying

wave conditions. Since the separation distance between the buoys was fixed,

changing the wavelength changes the separation distance to wavelength ratio which

is similar to changing the physical distance between the buoys. There appears to be



99

interaction between the two buoys as the wavelength is changed. However, the

investigation is excessivelly computationally intensive and more simulation runsare required to draw conclusions.

6.3 Experimental Results

The experimental testing in a wave flume was done to assess the

hydrodynamic performance of the OWEC under wave excitation. With a variable

wave input parameters, the performance of the device can be ‘tuned’ so as to match

the frequency of the incident waves to the resonance frequency of the oscillating

device, in order to achieve resonance. However, in the real seas the waves are

irregular and the performance in irregular waves is necessary in order to predict the

device’s open sea performance. The wave flume testing was done in waves of fixed

amplitude and there was no means of varying either the wave height or the wave

period. The device was thus operating sub-optimally and the power captured, 69W,

was not the maximum power possible for the given design. It is expected that with

the testing in the O.H. Hinsdale Wave Research Laboratory, it will be possible to

vary the incident wave heights and frequency to achieve resonance tuning.

Additional results from the tank testing for other generator loading

conditions are given in Table 6.3. These results are sub-optimal because the input

frequency and wave height of the flume could not be adjusted to match the

resonance frequency of the device. However, for a device of this scale, the

preliminary results reported here are remarkeable.



100

Table 6.3: Wave flume test results

Load Voltage CurrentOhms Vp Ip

PowerWp

20 16 0.5 6

30 35 0.7 18.4

50 52 0.6 23.4

75 65 0.6 29.3

6.4 Summary

In this chapter, the analytical, numerical and experimental results have been

discussed. Several stages of the design and analysis stages employed in this work

have been supported either by finite element or analytical simulations. These

stages have been validated with experimental testing where applicable. Within the

assumptions made, analytical results and experimental results have been

comparable. The numerical results have demonstrated an effective method for

assessing the power output of OWECs using CFD. The numerical results have

been scaled to obtain the full scale prototype device for ocean testing off the

Oregon coast. The method for investigating the interaction between buoys in close

proximity has also been proposed. The investigation requires additional

computational resources.



101

7. CONCLUSIONS AND FURTHER WORK

7.1 Conclusions

A novel OWEC device with a CFTS has been proposed as a direct drive

approach to ocean wave energy extraction. The device utilizes an efficient linear-

to-rotary conversion system of the traditional ball screw. The novelty of the device

lies partly in the design of a CFTS that couples an external float (buoy) to an inner

piston through a magnetic field. With this arrangement, the inner piston that

contains the critical power-takeoff components can be fully sealed for their

operational life. The OWEC device and its main components have been

investigated through finite element and analytical modeling, with a generally good

agreement between the model results and actual testing results. The device has been

successfully tested in the Motor Systems Resource Facility (MSRF) and in irregular

waves in a wave flume in Springfield, OR. A peak power output of about 69W was

achieved. Further testing is planned for the O.H. Hinsdale Wave Research

Laboratory.

A numerical method has been proposed for a coupled fluid-structure

interaction modeling, using computational fluid dynamic software to simulate an

OWEC device in the form of a heaving cylindrical buoy in order to assess its power

output. The structure of existing buoy prototypes are more complex than simple

smooth cylinders. However, the procedures developed in this work can be used in

applying commercially available fluids solvers to simulate FSI and determine the

power output of an OWEC device. With the numerical model, it has been

determined from scaling laws that an average power of about 30kW can be

extracted from a buoy of diameter 6m located off the Oregon coast in the most

prevalent wave conditions of the coast. The actual device could however be rated at



102

about 100kW to capture waves bigger than the nominal 1.5m waves. The numerical

modeling work was expanded to include multiple buoy systems to determinespacing requirements. However, this required a larger computational effort and

more simulation runs are required.

The dissertation started with an overview of the state-of-the-art in wave

energy extraction and the issues currently of interest to researchers. The

introductory chapter stated that most of the existing ocean wave energy extracting

devices are hydraulic based systems that would require significant maintenance,

which is inconvenient in the ocean environment. The need for direct-drive systems

was thus articulated in this chapter. One of the objectives of this work was to

propose a direct-drive alternative that eliminates the hydraulic intermediate

systems. This objective was achieved with the design and testing of the novel

OWEC with CFTS. Chapter 2 discussed the basic definitions and concepts of

linear wave theory, that form the basis for an analytical model that was discussed in

Chapter 3. The coupled FSI model was discussed in Chapter 4. This type of

modeling is important for ocean wave energy research, not only because ocean

wave phenomenon is non-linear but also because it can directly enhance the

inclusion of non-linear effects of PTO systems into the iterations of the fluid flow

solvers. The design of the novel OWEC was discussed in Chapter 5 and a

summary of the results given in Chapter 6. Chapter 6 also includes power output

estimation for a full scale prototype, using Froude’s scaling laws. The conclusions

and recommendations for further work are given in Chapter 7.



103

7.2 Recommendations for Further Work

The control of OWECs to maximize their power output remains an area that

requires significant attention. The use of vector control techniques in the control of

devices is requires further investigation. This technique is already available,

matured and is employed in industrial drives including those for wind turbines.

The numerical and analytical models also require further work in several

areas. First, there is need for further work to develop a coupled numerical model of

the OWEC that includes the electrical modeling of the generator with energy

extraction in both regular and irregular waves. Also, the current work could be

expanded to include all six degrees of freedom for both analytical and numerical

models and finally, the multiple buoy systems need further investigation for arrays

that are uniform or staggered.

These additional investigations can lead to development of wave energy

extraction buoys of significant size to be deployed in commercially significant

ocean energy wave parks.



104

BIBLIOGRAPHY

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110

APPENDICES



111

A. PERMANENT MAGNET GENERATOR PARAMETERS

Generator #1

Manufacturer AMETEK

Type Brushless DC

Rated Voltage 270V

Phase 3

RPM 12000

Rs, Xs 0.43, 0.19

Generator #2

Manufacturer MARVILOR MOTORS

Type BS073A00010T.00

Phase 3

BEMF 241V

Peak Stall Torque 13.6Nm

Continuous Stall Torque 2 Nm

KT 0.71 Nm/A

Max RPM 5600

Insulation Class F

Resolver 2T8



112

B. MODEL SCALING

In this Appendix, we expand the concept of model scaling for clarity and

better understanding. If we build a 10th

scale model of a device that has a model

dimension of 0.2m and obtained a power output of say 3W. The full scale prototype

will have a dimension of 0.2m x 10 = 2m. What will its power output be?

In order for the performance of the full scale device to be determined (or

estimated) from the model scale device, we need to ensure that they are “similar”.

One of the tools used to compare the model and the prototype is the Froude

Number (Fr). Thus the F r must be the same for model and prototype. The Fr is a

dimensionless number defined as

d

r gD

uF

2

= (B.1)

where u is the fluid velocity, g is acceleration due to gravity and Dd is a

characteristic dimension, for example the diameter of the device. Equating the F r

for model and prototype and using the subscripts m for model and p for prototype

we have

md

m

gD

u2

= pd

p

gD

u2

(B.2)

Note gravity is same for both. From B.2,

md

pd

m

p

D

D

u

u= (B.3)

Since the scale factor is 10 thenmd

pd

D

D=

110 = 10. Therefore p

m

p

u

uλ = , where λ p

is the length scale factor. This means that the ratio of the prototype velocity to the

model velocity equals the square root of the scale factor. Other derived quantities



113

can be obtained using the length and velocity; for instance if we use in general L

for length scale, since we can compute time as a ratio of length scale to velocityscale

pm p uu λ = and pm p L L λ = (B.4)

pm

pm

pm

p

p

p t u

L

u

Lt λ

λ

λ === (B.5)

Other quantities can be computed from B.4 and B.5 as derived quantities.

For a more practical example, let us take the most dominant period of the Oregon

Coast to be 10s. Then for testing a 10th

scale model that we plan for the dominant

period of the coast, we are looking at exciting the model with waves of period

st

t p

m 16.310

10

10=== .

Through dimensional analysis we can get expression for the power

as7

pm p PP λ = . For the 3W scenario mentioned above, our full scale prototype

will produce about7

103= pP = 9.5kW and not 30W. A more detailed discussion

of this subject can be found in [3] and [53].



114

C. ON SURFACE WAVES AND MASS CONSERVATION

In this appendix we review the source of waves and derive the continuity

equation. Surface waves are generated mainly by the wind blowing over the

surface of water. Other wave generation mechanisms include under-water

landslides, tides caused by gravitational forces, ship motion and moving pressure

disturbances. The main restoring force is gravity. Once generated, these waves can

propagate several miles in deep water. In deep water, bottom friction is small (but

not necessarily negligible) and as there are no forces to dissipate them these waves

keep propagating until they reach the shoreline where they break and lose energy.

Simply stated, linear wave theory is concerned about solving the surface

elevation of the waves that were generated somewhere else, enter a domain of

interest and come out of it. In this domain, there are no sources and Laplace’s

equation holds in this domain. The Laplace equation is a direct formulation of the

continuity equation with the additional assumption of irrotational flow, because

then the velocity vector can be represented as a gradient of a single-valued scalar

velocity potential function.

The continuity equation is derived by considering that mass must be

conserved in the volume of Fig C.1. The accumulated mass in the volume and the

net mass inflow through the cube of sides dx, dy, dz must be the same.

dxt

uu

∂

∂+

ρ ρ u ρ

Fig. C.1. Small cube in a fluid



115

Considering change in density after time dt the mass accumulated is

dtdxdydzt

dxdydzdt t

dxdydz∂

∂−=

∂

∂+−⋅ ρ ρ ρ ρ (C.1)

Considering the change in mass flow in x-direction

( ) ( )dtdxdydz

t

udxdydzudxdydzdt

t

uu

∂

∂=⋅−

∂

∂+

ρ ρ

ρ ρ (C.2)

There are similar expressions to equation C.2 for directions y and z, simply by

replacing u with the component velocities of y and z direction (v and w ). The total

change in mass flow during time dt is then summing (C.2) for all directions, or

( ) ( ) ( )dxdydzdt

z

u

y

u

x

u

∂

∂+

∂

∂+

∂

∂ ρ ρ ρ (C.3)

Equating C.1 and C.3 we get

( ) ( ) ( )0=

∂

∂+

∂

∂+

∂

∂+

∂

∂

z

u

y

u

x

u

t

ρ ρ ρ ρ (C.4)

Equation C.4 is the exact conservation of mass equation. Expanding it yields

01

=

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂

z

w

y

v

x

u

z

w

y

v

x

u

t

ρ ρ ρ ρ

ρ

(C.5)

Using the total derivative notation C.5 can be re-written as

01

=∂

∂+

∂

∂+

∂

∂+

z

w

y

v

x

u

Dt

D ρ

ρ

For incompressible fluids where there are no variations in density, 0= Dt

D ρ and

therefore 0=∂

∂+

∂

∂+

∂

∂

z

w

y

v

x

uor 0=⋅∇ u . We then say there is non-divergent flow.



116

D. EQUATION OF BUOY MOTION

In this Appendix we expand some important derivations which have been

summarized in the main text. All symbols have been defined previously in the main

text. Considering only a SDOF oscillation in the heave mode, the forces on the

buoy are as follows inertia and viscous forces denoted as f 1, the pressure forces

denoted as f 2, the weight denoted as W . Other forces such as the power take of

force are ignored for now. All the parameters and symbols in this section were

defined in Chapters 2 and 3.

z z AC z AC ww AC w AC f w Dw Aw Dw M

2'

2221

ρ ρ ρ ρ −−+=

4321 F F F F +++= (D.1)

Now, F3 and F4 are the reaction forces on the vertical cylinder as it moves in still

water; F1 and F2 are the inertial and drag forces due to the wave motion; w and w

are the vertical water particle velocity and acceleration respectively; z and z are

the velocity and acceleration of the buoy respectively. The pressure force at the

bottom of the buoy at the time when the buoy is accelerating at a small

displacement z upwards (coordinate z D z +−=1) is given by

)()(2 zK gA A z Dg f pww η ρ ρ ++−−= (D.2)

or using the large draft approximation (see further explanation below),

kD

ww egA A z Dg f −+−= η ρ ρ )(2

(D.3)

The weight of the buoy is

DgAmgW w ρ == (D.4)

where m is the mass of buoy. Ignoring the force of the PTO on the buoy the sum of

forces is given by adding D.1, D.2, D.3, D.4 or



117

ww Dw Aw Dw M A z Dg z z AC z AC ww AC w AC )(2

'222

−+−−+ ρ ρ ρ ρ ρ

==−+− zmF W egA kD

wη ρ (D.5)

Following [33], we linearize the drag forces by assuming their dependence on the

velocity, thus

wwww )(3

80

π = (D.6)

z z z z )(

3

8⋅= ω

π

(D.7)

where kDe

H w

−⋅= ω

20

is the amplitude of the velocity. Then the components of

equation D.1 ( f 1) are simplified as follows:

==− we

H AC ww AC kD

w Dw D23

8

22ω

π

ρ ρ

kDkD

W D ed we H

AC −−

== η ω ρ π

)2

(3

4(D.8)

kDkDw M w M eme AC w AC −− == η η ρ ρ ''

22(D.9)

za z AC w A =

2

ρ (D.10)

zb z z AC z z AC w Dw D =⋅= )(

3

8

2'

2' ω

π

ρ ρ (D.11)

where z AC bW D ⋅= ω ρ

π '

3

4is a function of the heave and frequency.

Considering that DgAW w ρ = , we have

zmegAgzA zb zaed emkD

ww

kDkD =+−−−+

−−− η ρ ρ η η '' (D.12)

or



118

zmgzA zb zaegAed em w

kD

w

kDkD =−−−++

−−− ρ η ρ η η '' (D.13)

or denoting wgAc ρ =

kDkDkDeced emcz zb zam −−−

++=+++ η η η '')( (D.14)

Computing the right hand side terms at x = 0, we note that

)cos()cos(2

t t H

⋅=⋅= ω ξ ω η , )sin( t ⋅−= ω η and )cos(2 t ⋅−= ω ξ ω η . Then

putting these into the RHS of the equation which holds the excitation force terms

we have kDkDkDkD et d t mceced em −−−−⋅−⋅−=++ ξ ω ω ω ω η η η )]sin()cos()''[('' 2

.

It can be shown that )cos()]sin()cos()''[( 02 ϕ ω ξ ω ω ω ω +⋅=⋅−⋅− − t F et d t mc kD ,

where 2222

0 )''( ω ω ξ d mceF kD

+−=− and

−=

−

2

1

''tan

ω

ω ϕ

mc

d . (Note that in

the model we used am ≈'' and b d ), therefore

)cos()(0 ϕ ω +⋅=+++ t F cz zb zam (D.15)

The PTO and other forces can be considered in the discussions above. In each case

the coefficients a, b, c will then change but the procedure is similar.

Also, note McCormick [31] determined from strip theory that

d Ag

b

n

2

3

22

8ω

ρ π = (D.16)

where A is the ratio of the wave amplitude and amplitude of motion andn

ω is

natural frequency of the buoy.

Further Explanation : Drag Linearization [33]

If we represent the heave velocity as Θ⋅= cos0 z z ω then z z

is equivalent to

writing



119

Θ=ΘΘ ∞

=

na

n

n coscoscos

0

. The coefficients ΘΘΘΘ=

π

π

2

0

coscoscos1

d nan , and

0=na for n = 0,2,4 . Alsoπ 3

81 =a ,

π 15

83 =a

π 105

85 =a , etc

This is the source of coefficient for equations D.6 and D.7.

Further Explanation: Large Draft Approximation

We know that2

)cosh( x x ee

x−

+= and

2)sinh(

x x ee x

−−

= therefore

khkh

kzkhkzkh

khkh

kzkhkzkh

pee

eeee

ee

ee

kh

zhk zK

−

−−

−

−−+

+

+=

+

+=

+=

cosh

)(cosh)( . For deep water, (h/L

0.5) 0→−khe , kze zKp →)( . Also, kze

kh

zhk →

+

sinh

)(sinh

Equivalently for large draft, D, for the purposes of evaluating pressure at the

bottom of the cylinder, ( D z D z −≈+−=1 ), kDe zKp −→)(

This is the source of expression for D.3.

For more on hydrodynamics and ocean engineering wave mechanics, thereader is referred to [28]-[35].



120

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