Application of wave generator theory to the

development of a Wave Energy Converter

by

Maila Sepri

Bachelor of Science in Engineering

Swarthmore College

2005

A thesis submitted to Florida Institute of Technology

in partial fulfillment of the requirements for the degree of

Master of Science

In Ocean Engineering

Melbourne, Florida September 2008

© Copyright 2008 Maila Sepri All Rights Reserved

The author grants permission to make single copies ________________________

We the undersigned committee hereby approve the attached thesis

Application of wave generator theory to the

development of a Wave Energy Converter

by Maila Sepri

as fulfilling in part the requirements for the degree of

Master of Science in Ocean Engineering.

__________________________________

E.D. Thosteson, Ph.D., P.E. Assistant Professor, Ocean Engineering

__________________________________ S.L. Wood, Ph.D., P.E.

Assistant Professor, Ocean Engineering

__________________________________ H.M. Gutiérrez, Ph.D., P.E. Associate Professor, Mechanical and Aerospace Engineering

__________________________________

G.A. Maul, Ph.D. Professor and Head of Department, Marine and Environmental Systems

iii

Abstract

Application of wave generator theory to the development of a

Wave Energy Converter

by

Maila Sepri

Principal Advisor: E.D. Thosteson, Ph.D., P.E.

This project was intended as a the initial step towards developing a wave

energy converter (WEC) capable of providing at least a quarter-Watt of power to a

small aquatic environmental monitoring sensor package being developed by Ocean

Research and Conservation Association (ORCA) in Ft. Pierce, Florida. The focus of

this project was to apply theoretical wave mechanics equations and basic wave

generation technology to improving the power capture design of a basic direct drive

WEC. A single WEC with a linear induction generator as its power take-off

mechanism was designed and constructed, along with two interchangeable wave

absorbers. The base absorber design, a cylindrical buoy comparable to the

Archimedes Wave Swing, was intended to harness primarily potential energy. The

second absorber design employs a unique buoy with a hyperbolic curve on its

windward face, specially shaped to take advantage of both the kinematic energy in

water wave particles and the wave’s potential energy.

In order to estimate the available wave power and wave parameters at

deployment sites around the Indian River Lagoon on the east coast of Florida, a

computer model based on meteorological hindcasting and wave generation equations

was created. The model predicted that sufficient energy is available year-round for

iv

supporting the low-power sensor, given at least a 5% efficiency of input wave power to

output power across the load. Comparison to wave measurements made in the lagoon

show that, on average, the model overestimates the power by 38%. Nevertheless,

dependence of the hindcasting data on daily averages recorded some distance from the

coast suggests that more thorough calibration from in situ wind data should be

conducted.

A design wave of 10cm height and 1.2-second period traveling in 1m of water

depth was selected to determine the dimensions of the second buoy’s curved face.

Upon deployment, the WEC successfully logged the power output of the system in

multiple wave conditions. A set of data in which the wave conditions experienced by

both buoys were consistent was used to form a direct comparison of their

performances. The deployment conditions consisted of 8.48cm wave heights at

approximately 1.5-second periods in 1.7m water depth. Contrary to predictions, the

specially-designed buoy in fact showed wave absorption performance inferior to that

of the cylinder. Hydrodynamic effects, including smaller buoyant forces for a given

wave height, were observed to interfere with the curved buoy’s intended mode of

operation. Detailed modeling of the constructed system’s mechanical frequency

response is necessary for tuning the electrical system and developing an unbiased

analysis of the advantage of incorporating wave mechanics into WEC designs. In

addition, adaptive load and generator circuitry must be incorporated before the WEC

will be capable of achieving the efficiency needed to support a sensor system load.

v

Table of Contents List of Keywords.................................................................................................... vi List of Figures....................................................................................................... vii List of Tables ....................................................................................................... viii List of Abbreviations.............................................................................................. ix List of Symbols ....................................................................................................... x Preface ................................................................................................................. xii Acknowledgements.............................................................................................. xiv Dedication............................................................................................................ xv

1. Introduction.................................................................................................... 1 1.1 Existing WEC Designs ............................................................................ 1 1.2 Design Constraints .................................................................................. 6 1.2.1. Sensor Suite Requirements ............................................................... 6 1.2.2. Wave Energy Available in the IRL ................................................... 7 1.2.3. Challenges in Wave Energy Converter Design ................................ 19

1.3 Objective and Scope .............................................................................. 22 2. Design Concepts ........................................................................................... 23 2.1 Wave Generators and Absorbers ............................................................ 23 2.2 Application of Wave Theory to Buoy Design.......................................... 26 2.3 Mechanical Components ....................................................................... 32 2.3.1. Hydrodynamics ............................................................................. 34 2.3.2. Materials Selection......................................................................... 41 2.3.3. Dimensions ................................................................................... 42

2.4 Electronics Design................................................................................. 44 2.4.1. Power Generation Components...................................................... 50 2.4.2. Data Collection Circuitry ............................................................... 56

2.5 Software................................................................................................ 63 2.5.1 PIC Microcontroller Code.................................................................. 63 2.5.2 Post-processing Matlab® Code........................................................... 66

3. Construction ................................................................................................. 67 4. Testing Procedures ........................................................................................ 74 4.1. Calibration ............................................................................................ 74 4.1.1. ADC analog input block calibration................................................ 74 4.1.2. Frequency testing........................................................................... 76

4.2. Deployment .......................................................................................... 80 5. Results and Conclusions ................................................................................ 82 5.1. Practical Observations ........................................................................... 82 5.2. Testing Conditions ................................................................................ 84 5.3. WEC Performance Results..................................................................... 87

6. Recommendations for Future Work............................................................... 93

References ............................................................................................................ 98 Appendices......................................................................................................... 103

vi

List of Keywords

Alternative energy sources

Direct drive

Evanescent waves

Hindcasting

Hydrodynamics

Impedance matching

Indian River Lagoon

Linear generator

Sustainability

Wave absorber

Wave Energy Converter (WEC)

Wave mechanics

vii

List of Figures

Figure 1: Commercial WEC design drawings. .......................................................... 2

Figure 2: AWS scale prototype and OPT PowerBuoy ............................................... 3

Figure 3: ORCA sensor suite ................................................................................... 6

Figure 4: Example map from IRL wave power distribution model............................. 9

Figure 5: Pixel-scale image of Melbourne Harbor. .................................................. 13

Figure 6: Example of daily wind variation.............................................................. 14

Figure 7: PAFB 2006 wind speed histogram and density distribution....................... 15

Figure 8: Cumulative distribution of annual wind speeds ........................................ 16

Figure 9: Seasonal wave power maps ..................................................................... 17

Figure 10: Piston paddle, flap-type paddle, and plunger wavemakers ....................... 23

Figure 11: Difference between proposed buoy shapes.............................................. 24

Figure 12: Water particle trajectories under progressive waves at different depths..... 26

Figure 13: Block diagram of the WEC system......................................................... 33

Figure 14: Populated PCB ..................................................................................... 45

Figure 15: WEC printed circuit board schematic and layout.................................... 46

Figure 16: Block diagram of inputs and outputs to major circuitry components ........ 49

Figure 17: Magnetic gearing in a transverse flux machine........................................ 51

Figure 18: Winding the generator coil .................................................................... 68

Figure 19: Linear Inductance Generator box .......................................................... 70

Figure 20: Mechanical and structural components .................................................. 71

Figure 21: Buoy manufacturing processes............................................................... 73

Figure 22: Frequency calibration test configuration................................................. 77

Figure 23: Capacitor charging profiles and DC/DC converter power output plots. ... 79

Figure 24: Deployment site.................................................................................... 80

Figure 25: Frame assembly and deployment........................................................... 81

Figure 26: Floating levels of the buoys during deployment ...................................... 82

Figure 27: Incident wave analysis from 13 August 2008.. ........................................ 85

Figure 28: Capacitor voltage and DC/DC converter power output .......................... 88

viii

List of Tables

Table 1: PAFB 2006 wind speed and frequency of occurrence ................................. 16

Table 2: Seasonal wave height and power characteristics......................................... 17

Table 3: Comparison of modeled and measured wave conditions ............................ 19

Table 4: Comparison between standing wave coefficients of piston and exponential

curve wavemakers..................................................................................... 31

Table 5: Oscillation table calibration trial results..................................................... 79

Table 6: Deployment log ....................................................................................... 84

Table 7: Comparative performance characteristics for the cylindrical and hyperbolic

curve buoys .............................................................................................. 87

ix

List of Abbreviations

ADC Analog-to-Digital Converter

CEM Coastal Engineering Manual

CTD Conductivity (Salinity), Temperature, and Depth sensor

FDS Fully-Developed Seas

IRL Indian River lagoon

NWS National Weather Service

ORCA Ocean Research and Conservation Association

PAFB Patrick Air Force Base

PIC Programmable Interrupt Controller

PTO Power Take-Off

RMS Root-Mean-Square

RTC Real-Time clock

WEC Wave Energy Converter

x

List of Symbols

Ap Progressive wave coefficient

C Capacitance

CD Wind drag coefficient

Cm Standing wave coefficient

co Deep-water wave celerity

cg Group celerity

∀ Volumetric displacement

ε Induced voltage (electromotive force)

F Force (subscripted for identification)

g Acceleration due to gravity

h water depth

Hs Significant wave height, esp. determined by time-series analysis

Hmo Significant wave height, esp. determined by spectral frequency analysis

k Wave number

kp Progressive wave number

ks Standing wave component

λ Wavelength (EM wave)

L Inductance, Wavelength (Water wave)

m Mass (subscripted for identification)

mA-h Milliamp-hours

φ Velocity potential of a water wave

Φ Magnetic flux

ρ Density, specifically of lagoon water

R Real resistance (electrical or mechanical analog)

σ Wave frequency

S Stroke length of wave generator/absorber, Buoyant spring constant

T Wave period

Tp Peak wave period

xi

u Water particle velocity in the x-direction

U10 Wind speed at 10m altitude

w Water particle velocity in the z-direction

x Displacement along the X-axis (horizontal) direction

X Reactance, Fetch

ξ Water particle trajectory in z-direction

ζ Water particle trajectory in x-direction

y Displacement along the Y-axis (horizontal) direction

z Displacement along the Z-axis (vertical) direction, esp. height of a water

particle within the water column or the buoy with respect to the still water

line.

Z Impedance

xii

Preface

Concerns over climate change and fossil fuel availability have channeled

attention toward revising systems of energy usage around the world. Numerous

proposals for alternative energy solutions have been presented, ranging from wind

turbines and solar panels to biofuels and tidal and wave energy converters. Naturally,

these solutions are all application-dependent. While biofuels offer a renewable fuel

alternative for mobile vehicle engines, they are not viable for applications far from

filling stations, such as remote sensor packages. Similarly, solar and wind power

technologies have advanced to commercial availability, but some characteristics make

them non-ideal for the application targeted in this project. Firstly, both solar and

wind power devices must have the majority of their components above the surface of

the water, and the initial deployment locations in the Indian River Lagoon see a good

deal of recreational traffic. Any visible devices are thus susceptible to vandalism,

whether malicious or simply curiosity-driven, and weather. Specifically pertaining to

solar panels, bird-induced fouling and salt accumulation require constant cleaning to

allow sunlight to penetrate the housing. Another drawback to solar panels is that they

collect energy only for small angles of incidence; this limits the collection time to

approximately 5.5 hours per day for non-tracking solar panels. If the panel cover is

clouded by bird excretions or salt deposits, the energy collection during the key hours

will be further inhibited. Developing alternatives to these energy capture methods

may make the difference between creating an economical sensor and one that is

prohibitively expensive for widespread use.

While marine sources of stored energy also include tides and currents, this

project concerns only wave energy. This decision was made partly to focus the

xiii

project, but also because waves have a more pervasive effect throughout the entirety of

the lagoon and its tributaries than do currents or tides.

Taking advantage of localized renewable energy sources averts the energy loss

inherent in power transmission over long distances. Moreover, such a configuration

makes use of previously untapped mesoscale renewable energy supplies, which may be

on the scale of just tens of Watts or less, but which still provide sufficient power to

satisfy individual applications. For example, running a remote sensor suite on

traditional battery power would require frequent travel to the field site or a permanent

power line connecting the site to shore. This configuration would demand a larger

sum of energy than providing the low-power instrument with its own robust recharger,

based on renewable technology. Optimizing a wave energy converter design to take

advantage of the small ripples and chop in the Indian River Lagoon would make use

of readily available local reserves of energy.

Although coastal Florida’s wave and wind levels do not make it a candidate

for megawatt-scale generator research, small-scale development environments provide

a platform for economical testing and polishing of technology in natural conditions.

Improvements in wave energy absorption demonstrated as a result of testing this

project’s apparatus would allow it to serve as a scale model for enhancing a larger

system’s performance.

xiv

Acknowledgements

Dr. Edith Widder, Ocean Research and Conservation Association

Dr. Eric Thosteson

Dr. Stephen Wood

Dr. Héctor Gutiérrez

Edward W. and Lee Hill Snowdon Administrative Fund

Gertrude E. Skelly Foundation

Mr. Tony Cimaglia

Mr. Benjamin Burns

Mr. Doug Bowlus

Mr. Ed Martin

Mr. Bill Bailey

Mr. Bill Battin

Mr. Charley Phelps

Mr. Kevin Donnelly

Dr. Geoffrey Swain

Dr. Lee Harris

Dr. David Fleming

Dr. Lee Caraway

Dr. Kenneth Fulton and Family

Mr. Michael Vergalla

Mr. Anthony Tedeschi

Mr. Anass Jerrari

Ms. Kelli Zargiel

Dr. Paavo Sepri

Mrs. Ann Sepri

Ms. Ashley Naimaster

Mr. Abraham Stephens

xv

Dedication

To my parents, for their devotion to my multiple activities, encouragement of any

interest I decide to adopt, and for their patience…

and

to Tiia, my sister and best friend, for all our laughter.

1

1. Introduction

1.1 Existing WEC Designs

Although U.S. patents making use of wave energy date back to the 1880’s,

interest in wave energy converters (WECs) gained speed after the oil crisis of the

1970s. Over the past 3 decades a number of schemes have been proposed for capturing

wave energy and converting it into a useable form (see Appendix A). Developments

have increased pace in the past few years, buoyed by recent technological

improvements. A few WEC designs have progressed past the prototyping stage to be

integrated into a power grid [17]. A majority of the designs employ hydraulics or

pneumatics to spin turbines or pumps and power conventional generators. Design

drawings for four of these devices are shown in Figure 1. WaveGen’s Limpet 500 uses

the oscillating level of water in an inverted chamber to force air through a Wells

turbine at the roof [21]. Similarly, the OWEL (Offshore Wave Energy, Ltd) WEC

utilizes wave front forces to compress air in the preceding trough through a chamber,

and the rushing air in turn rotates a turbine [30]. Another, the Pelamis by Ocean

Power Delivery, Ltd., depends directly on wave steepness to pump pressurized oil

through hydraulic motors [31]. Aalborg University’s wave overtopping system Wave

Dragon channels waves into an artificial shoal and traps water from the breaking wave

crests at a higher elevation, then releasing it down to sea level through turbines [42].

The explanation behind implementing pneumatics or hydraulics in rotary

wave generators lies in the slow reciprocation speed of waves; gearing is needed to

increase the direct drive linear speed from only 0.5 to 2m/s to a typical generator

2

speed near 1500rpm [24]. Drawbacks to the pneumatics and hydraulics designs

include complexity and efficiency. The number of moving parts required for gearing

increases maintenance costs, and unavoidable part-loading (operation out of

resonance) drastically reduces turbine efficiency. In addition, hydraulic seal durability

has been drawn into question for variable velocity applications [24].

(a)

(b)

(c)

(d)

Figure 1: Commercial WEC design drawings, from their website descriptions. (a) Limpet

500 [21]. (b) OWEL [30]. (c) Pelamis [31]. (d) Wave Dragon [42].

The most commonly proposed alternative to conventional generators is the

linear inductance generator. This device employs permanent magnets to induce

electromagnetic fields on coils, exciting electrical current. Since any change in the

magnetic field creates a current in the wire, mechanical coupling to wave motion

ensures that some amount of power is transferred to the circuitry. This design offers

the advantage of fewer moving parts to be threatened by biofouling and corrosion, and

also less complexity in the generator stage. Unlike a rotating motor, the linear motion

of a magnet through a coil is one of the most basic concepts in electromagnetic theory.

Other benefits include high force density in the generator and reasonable efficiency at

3

low speeds [32]. Nevertheless, higher velocities are preferable because they induce

higher voltages on the coils, and many aspects of linear generator design determine

their performance efficiencies.

Despite a plethora of studies on linear generators for WECs [19, 20, 23, 24, 32,

39, 43, 41, 45], only two systems are notably beyond the design stage: the Archimedes

Wave Swing (AWS) and Ocean Power

Technologies’ PowerBuoy. These devices,

made by AWS Ocean Energy, Ltd., and

Ocean Power Technologies (OPT),

respectively, share generator geometries, but

their absorber components operate on

different principles. The moving component

of the AWS is a cylindrical, air-filled,

pressurized chamber that rises and plunges

according to the pressure head created by

waves propagating above it. The Power Take-

Off (PTO) system is a linear synchronous

generator using multiple permanent magnets

on the translator [32]. The relative motion of

the translator (coupled to the “floater”

cylinder) to the base section (secured to the

sea-floor) induces current in the inductor

located in the base. Creative positioning of

the magnets, coils, and core material on this

translator are being explored for increased

efficiency of the generator. Scale prototypes

of the AWS have verified model predictions,

and a pilot plant is currently being studied off the coast of Portugal (Figure 2). OPT’s

PowerBuoy, which has undergone field testing in both the Atlantic and Pacific

Oceans, pierces the water surface and depends on wave heights’ lifting of the buoy to

drive its linear generator [28]. While little has been published to reveal the

(a)

(b)

Figure 2: (a) AWS scale prototype

before deploy-ment [32], and (b) OPT

PowerBuoy [28]

4

composition and operation of the PowerBuoy’s generator, OPT has acted as the

assignee in multiple patents on piezoelectric power conversion systems, such as that in

[11].

Literature has not yet concluded on the comparative efficiency of linear

generators versus that of rotary generators in WECs. An industry standard procedure

for rating efficiencies must be established for the blossoming WEC designs to resolve

the question of optimal topology. Meanwhile, it is instructive to weigh other factors.

Only two other challenges have been presented in literature regarding linear

generators. These are weight requirements and the increased difficulty in transferring

power to a grid due to amplitude and frequency fluctuations in the voltage [10].

Firstly, large quantities of permanent magnets needed for large-scale generation add

substantial weight to the translator. The WEC must be designed to accommodate or

make use of this weight. Secondly, rotary generators are intended to operate at a rated

voltage, with power generation levels thus determined by current output, but linear

generators will vary in both voltage and current. This complicates the powering of a

battery charger circuit, which requires a clean DC signal, so a rectifier and DC-DC

converter must be used.

One way to interpret the wide variety of WEC designs discussed above is that

each wave energy capture mechanism is designed around the PTO technology

available. Logically, if rotary generators are to be used, the motion of the wave must

be converted into rotary motion. For linear generators, linear motion is needed. The

challenge to WECs is that wave energy consists of both types of motion: vertical

(linear) displacement of the surface as a wave propagates through the water mass, and

elliptical motion of the water particles themselves. An energy analysis of a wave shows

that wave energy consists of equal parts hydrostatic (potential) energy and kinetic

energy. Simply harvesting the potential energy stored by the water surface, as does the

AWS, neglects to use the dynamic portion of the energy. Using only the wave’s

velocity loses the hydrostatic portion of the energy.

This concept indicates that, while efficiency of the generator is crucial to the

system’s performance, the other requirement for efficiency is effective transfer of

kinetic and potential energy from the wave to the absorption device. Fortunately,

5

wave absorption is not a newborn field. Numerous attempts have been made to

dissipate wave energy in laboratory wave tanks by using the reverse technology of

wave generators. The original motivation behind these efforts was to minimize noise

from reflections and reduce settling time between experiments by removing wave

energy at the end of the wave channel. The development also shows that computations

nearly identical to wave generation equations may be applied to wave absorption [98].

Conveniently, wave generators have become quite efficient. A logical next step, then,

is to design a WEC that operates in the reverse of a wave generator. In this case, a

variety of wave generator geometries are available, including both piston- and flapper-

paddle generators, as well as plunger-type generators. In the wave absorber scheme,

the motor that conventionally drives the wave generation paddle or plunger becomes

the electrical generator driven by the paddle or plunger. Integrating the PTO with the

absorber then becomes a study on inverse-engineering efficient wave generators. This

thesis hypothesizes that focusing the design process on wave generation technology

should improve wave energy capture efficiency.

6

1.2 Design Constraints

This section discusses the conditions that determined basic design decisions,

motivated by the intended application.

1.2.1. Sensor Suite Requirements

The marine monitoring sensor suite for which this WEC system is intended is

being developed from the component level upwards by Ocean Research and

Conservation Association (ORCA) in order to create an affordable platform for

collecting oceanographic data. Its first generation version includes the sensors and

supporting circuitry to record water conductivity, temperature, depth, flow magnitude

and direction, optical backscatter (turbidity), wave height, wave period, package GPS

location, orientation, and speed of sound. Additional capabilities are planned for

future versions. Currently, resolution and regularity of data collection in the field is

severely limited by the power available to remote sensor systems. The purpose of this

system is to make spatially and temporally dense sampling economically and

logistically feasible for the scientific and educa-

tional communities by providing ample data

storage, regular data transmission and low

power consumption.

In the current design, a continuous

supply of 0.25W enables a pair of Li-Polymer

batteries to support the system for 2.5 days,

sampling at a rate of once every five minutes

and transmitting data via cellular

communications every hour. Since the sensor

suite can support six of these batteries and

deployment durations of at least a month are

desired, the package will require a lengthy

power supply tether if a local (renewable energy) battery recharger is not implemented.

Figure 3: ORCA sensor suite

7

While initial tests are being conducted on boat dock pilings in the Indian River

Lagoon (IRL) near Fort Pierce, Florida, future deployments in remote locations will

require reliable alternatives to a land-based power tether. The device in this project

was designed as a stand-alone mechanism that would eventually support a short tether

for feeding power to a battery charger within the sensor suite.

As per the power requirements mentioned above, the equivalent continuous

supply required to maintain operation is 0.25W. This is a time-averaged value; up to

6W is required during sampling and data transmission, and only a few milliwatts

during the intervals between. Similarly, wave power is neither continuous nor

smooth, so a WEC serving the sensor package would be expected to smooth the

supply voltage and store excess power as it becomes available so that reserves are

available when waves dwindle.

1.2.2. Wave Energy Available in the IRL

In order to gage the feasibility of harnessing the equivalent of 0.25W

continuously in the intended deployment region, it was first necessary to quantify the

wave power available in the region. For instance, useful estimates include the energy

content of the predominant IRL wave climate, the mean energy value, the energy

content of peak and lull wave events, the number of hours per day during which these

conditions are likely to occur, and seasonal variation statistics. The following sections

detail the computer model written to arrive at these estimates. First, a few definitions

will be helpful.

1.2.2.1 Wave Parameters and Theory

As detailed in [7], wave energy is dependent upon the height of the wave. The

potential energy is the work involved in lifting a volume of water of a given density

from the trough’s center of gravity to the height of the crest’s center of gravity; kinetic

energy is the energy of water particles moving at a given velocity. Both the potential

and kinetic components of the energy are proportional to the square of the wave

height, so the average energy per unit surface area is given by

8

1 2

8

1gHE ρ=

where ρ is the water density, g is gravitational acceleration, and H is the wave height

from trough to crest. Note that wave energy is described in terms of energy density:

the amount of energy per unit area of water surface, integrated over the depth of the

water. While wave energy is dependent upon only wave height (not depth or period),

most of the energy is concentrated in the region between the surface and a depth equal

to half the wavelength. For this reason, and for power calculation, the height, period,

and water depth must be known in WEC design. Energy flux, or power per unit

width of wave crest, is the amount of energy per unit time that a wave carries over a

line of unit width parallel to its crest.

2 gEcP =

where Cg is the speed of energy transmission, called the group velocity. This value

depends on the ratio of depth to wavelength. In shallow water, the group velocity is

the same as the phase celerity of individual waves, but as the depth increases to greater

than 5% of the wavelength, the group celerity begins to approach half the phase speed.

To put a physical scale on the power predicted by these equations into familiar terms,

a 0.75m, 6-second wave breaking in 1m of sea water, not uncommon on Florida’s

Atlantic coast during the winter [27], carries over 2.04kW of power per meter of wave

front. During the summer, a typical 0.25m, 8-second wave carries just over 0.23kW.

In conclusion, in order to calculate the wave power in a certain area, the region’s

typical wave heights, periods, and water depth must be known. Design depth for the

wave absorber should be based upon half the wavelength of the predominant wave

climate.

1.2.2.2 Modeling the IRL Wave Energy

Installing sensors in multiple locations and collecting data over a period of

time offers a straightforward and reliable method for determining typical local wave

climates, but at the current time an extensive set of wave data is not available for the

IRL. Meteorological data, on the other hand, is accessible, so an alternative method is

9

to use historical meteorological data to hindcast, or mathematically model, the wave

conditions corresponding to the wind record. Sample data must be collected to verify

the model’s predictions, yet the model is useful for both creating initial estimates of

energy magnitudes and discerning trends and relationships between locations in close

proximity. A convenient way to present results from these models is to generate color-

coded maps of the IRL showing available wave power, in W/m of wave front, at each

location, such as in Figure 4. The full MathWorks Matlab® code for completing the

maps can be found in Appendix C, but the general equations and concepts will be

discussed here so that the procedures followed are clear.

Figure 4: Example map of IRL wave power distribution, in W/m,

generated using wind data hindcasting models

The equations used to calculate wave height based on wind data are provided

in the Coastal Engineering Manual (CEM), compiled by the United States Army

Corps of Engineers [32]. The equations are both theoretically- and empirically-based,

and they state that the main input arguments to the hindcasting model are the water

depth, d, 10-m elevation wind speed, U10, and straight-line wind fetch, X. Due to the

empirical coefficients in these formulas, SI units must be used. Equations for wave

growth that take fetch into account were needed because of the limitations imposed by

10

the barrier island on one side of the lagoon and the mainland on the other. The

significant wave height, Hs, is defined as the highest 1/3 of the waves, and is given by

3 2

1

2

2

210134

×= −

*

*s

u

gX

g

u. H

where g is gravitational acceleration and *u is the friction velocity related to the

standard 10-m elevation wind speed by

4 2

10* *UCu D=

with wind stress drag coefficient CD calculated as below.

5 ( ) *U..*. CD 100350110010 +=

If the wind speed is measured at an altitude other than the standard reference height of

10m, or if it is measured inland or for only a short time interval, equations and plots

are given to adjust the value, but the author refers interested readers to Part II, Chapter

2 of the CEM for further information. The peak period is given by

6 3

1

27510

=

*

*p

u

gX

g

u.T

The CEM equations account for depth, steepness, and fetch limitations on the wave

height and period. With the exception of a deep shipping channel that runs along the

axis of the lagoon, the water depth of the IRL averages to slightly over 1.5m. For

simplicity and also conservatism in energy estimates, 1m was chosen as the water

depth for all predictions. This imparts a factor of safety into the calculations, since

larger waves generated in 1.5m water depth would break upon encountering a shoal of

1m water depth. Since studies have refuted the effect of bottom friction on wave

generation, shallow-water wave generation is calculated in the same way as in deep

water, with the exception that the peak period’s upper bound is given by

7 2

1

, 78.9

=

g

hT SWp .

11

In the case that the period in 6 is greater than that of 7, the Tp,SW value should

replace Tp, and the measured fetch X should be replaced by a dimensionless fetch

X derived from the new peak period and Equation 6.

8

3

*751.0ˆ

=

u

gTX

p

Since winds often change frequently, a criterion for assessing the applicability

of these equations is the duration for which wind traveling at U10 traversed the fetch X.

Intervals shorter than t(x,u) given below yield duration-limited waves.

9 33.034.0

67.0

*23.77 ),(

gU

Xuxt

a

=

This criterion is not automatically tested in the Matlab® code, but the result is printed

so that the user may check the wind record to verify that the wind blew at the input

speed and direction for a sufficient amount of time. The code does ensure that wave

height predictions do not exceed depth and steepness breaking limitations by applying

the relations in 10 and 11, respectively.

10 hH d 6.0lim_ = 1

11 7

lim_

LH s =

One important point to note is that this depth limitation is not the same as the

dispersion equation depth limitation based on the value of kh, explained further in

Section 2.2 below. This means that waves predicted by the model will not exceed the

breaking criterion, but they are not necessarily deep water waves. This condition must

be tested independently when the shape of the waves predicted by this model is of

concern.

Since the above formulas all assume that the waves are fetch-limited, the fully-

developed sea (FDS) wave height is calculated very approximately for comparison.

1 This depth-limitation coefficient of 0.6 is used for random waves. The less conservative coefficient accepted for monochromatic waves is 0.78, but the use of significant wave height inherently implies random wave conditions.

12

12 g

U. H FDS

2

10270=

After significant wave heights and peak periods have been assigned to the locations of

interest, the wave energy and power can be computed using 1 and 2. In the

process, the group velocity of the waves must be computed. The dispersion equation

relates wavelength and period at a given depth and requires iteration to solve for the

wavelength.

13 )tanh(2 khgk=σ

In 13, σ is the wave frequency equal to 2/T, and k is the wave number, given by

2/L. The wave celerity is given simply by wavelength L divided by wave period T,

14 T

Lc =

and the group velocity by

15 nccg =

where

16

+=

)2sinh(

21

2

1

kh

khn .

The factor n accounts for the phenomenon that energy propagates at the speed of the

group of waves, which may be different than the phase speed of individual waves. The

energy travels only as fast as the foremost boundary of the group, so the power is

limited by the group velocity. In deep water (h/L > ½), Equation 16 simplifies to

0.5, while in shallow water (h/L < 1/20), n approaches unity.

Equation 16 concludes the inventory of formulas included in the modeling

code, so before the results of the simulations are presented, an account of the fetch-

and wind velocity- determination is needed. The maps generated by the simulation

are intended to show the power distribution over the entire lagoon, so fetch is very

direction- and location- dependent. In order to assign each point in the lagoon an X-

value for a given wind velocity, a basic image-processing program was written to find

the straight-line distance from each pixel on a map to the shoreline in the direction of

the wind. First a screen capture is taken of a Google Earth satellite image of the

13

region of interest in the lagoon. By enabling the “Postal Code Boundaries” layer

(within the US Government menu), yellow lines highlighting the coastlines are

conveniently added to the image, so that the program will be able to pick out the land

from the water. One caveat is that the image must be aligned such that the north-

south compass direction is exactly vertical while the screen shot is taken. The image is

then saved as a bitmap, and the actual east-west and north-south distances (in meters)

spanned by the image must be noted using the ruler tool in Google Earth. These

dimensions provide the program with a scale upon which the fetches are calculated.

For some maps, a good deal of pixel-level editing must precede application of the

fetch-calculating code. For instance, the barrier island’s eastern coast must be

highlighted red rather than yellow in order to decrease computation time and increase

color resolution within the lagoon on the final power maps. In addition, intricate

coastlines with capes and north/south-oriented corners confuse the program unless a

distinct coastline-land-coastline pattern is drawn (see Figure 5).

Figure 5: Pixel-scale image of Melbourne Harbor showing (a) the original intricate coastline

highlighted with yellow corners and (b) edited so that all corners and jetties have gaps

between yellow pixels to create a distinct coastline-land-coastline pixel pattern.

When these steps are completed, the program uses the yellow borders to create

a black and white image (black land and white water) and then assigns a fetch value to

each pixel in the water by following a search in the direction of the approaching wind

until it finds the first coastline pixel.

The last parameter required to run the program is wind velocity. As stated

earlier, useful wave energy estimates would include predominant, peak, and lull

14

conditions, as well as the frequency with which those conditions should be expected.

A study conducted over 30 years by the National Weather Service (NWS) at the

Melbourne International Airport [100] and another conducted over 66 years by the

National Climatic Data Center (NCDC) [98] provided monthly summaries of speed

and direction in wind rose format, and a year’s worth of digital wind data from Patrick

Air Force Base (PAFB) was used to create histograms of the wind speeds and their

yearly duration. The PAFB data was recorded every five minutes throughout 2006. A

qualitative comparison of the PAFB data with the NWS wind roses showed (a) that

the wind distribution on the barrier island (PAFB) does not differ greatly from that of

the mainland (NWS) and (b) that 2006 did not deviate greatly from the 30-year

average [24]. Another important conclusion derived from the digital data is that the

minimum duration to achieve fetch-limited conditions should be checked when

conditions are predicted, because both wind speed and direction can vary greatly over

the span of a day. As seen in Figure 6, variations of 7 m/s and 150˚ may occur in just

a few hours.

0

1

2

3

4

5

6

7

8

9

Time of Day

Wind Speed (m/s)

-40

10

60

110

160

210

260

310

360

Wind direction

(degrees from North)

Wind Speed

Wind Direction

Figure 6: Example of daily wind variation, plotted from

PAFB digital data from 7 June 2006.

This minimum duration varies primarily with the direction of the wind, as approaches

parallel to the length of the lagoon (maximum possible fetch) will need time to

15

generate their maximum power potential. If the minimum duration is not met,

changes in wind will simply be assumed to reduce the predicted energy, as the CEM

monograms do not account for such small-scale fetch and time parameters. A

normalized histogram compiling data from the entire year is shown in Figure 7.

0 2 4 6 8 10 12 14 16 180

0.02

0.04

0.06

0.08

0.1

0.12

Windspeed (m/s)

Percentage of Hours Annually

Figure 7: PAFB 2006 wind speed histogram and density distribution

Aptly similar in shape to the JONSWAP spectrum for fetch-limited, deep-water

waves, the histogram shows the number of hours over the course of 2006 that PAFB

experienced winds of the given speed. The mode is easily seen as the peak in Figure 7;

it represents the wind speed that occurs most often annually. This value corresponds

to the predominant wave climate mentioned above. The power mean is given as the

speed that generates the mean value of wave energy flux density, calculated as the first

moment of the wave power histogram.2 Peak and lull periods of energy content are

integral in determining the survivability and dependability, respectively, of the WEC,

and should not be omitted from any long-term study of WEC feasibility. However,

2 The power histogram was determined by plotting on a log-log scale the wave energy versus wind speed for a range of wind speeds and a given average fetch, and then measuring the slope in order to assign an exponent to the wind speed/wave energy relationship. This exponent was determined to be 1.3 for a fetch of 1.5km and 2.38 for 4.5km.

16

these events were not simulated extensively with the modeling in this project, as the

requisite statistical analysis of daily patterns of wind durations is beyond the scope of

the project. In a related strain, it is worth noting that no tropical events occurred in

the region in 2006, so the extreme weather conditions are not factored into this data.

The number of bins into which the histogram is divided greatly affects the number of

hours corresponding to each speed, so a cumulative probability density function

method is useful in quantifying the number of hours per year that winds of at least a

given speed are likely to occur. The results are given in Figure 8 and Table 1.

0 2 4 6 8 10 12 14 16 180

1000

2000

3000

4000

5000

6000

7000

8000

9000

Wind speed (m/s)

Hours per year

Figure 8: Cumulative distribution of annual wind speeds

Table 1: PAFB 2006 wind speed and frequency of occurrence

Wind speed (m/s)

Annual hours during which speed is greater

Average daily hours during which speed

is greater Mode 3.15 4665 12.7 Power mean 4.11 3050 8.35

Using the wind directions from the 30-year study’s monthly wind roses to associate

months with a season, the year’s worth of digital data was divided into winter

(November – February), spring (March – April), and summer/fall (May – October). A

17

fourth season is not distinguished here due to the similarity of the fall months’ wind

directions to those of the summer months’. The wind speed corresponding to the first

moment of each season’s wave power distribution was used with the MATLAB®

program to generate the wave power maps in Figure 9. This wind speed represents the

wind that creates the average power seen in the location annually.

Figure 9: Seasonal wave power maps. (a) Winter prevalent conditions of 4.76 m/s winds

from the north yield power up to 65 W/m2. (b) Spring prevalent conditions of 4.34 m/s

winds from the southeast yield power up to 27 W/m2. (c) Summer prevalent conditions of

4.11 m/s winds from the east yield power up to 13 W/m2.

Table 2: Seasonal wave height and power characteristics

Annual Spring Summer/Fall Winter Wind Direction E SE E N Mean Power Wind Speed (m/s) 4.11 4.34 4.11 4.75 Hs (m) 0.10 0.13 0.10 0.18 Tp (s) 1.2 1.3 1.2 1.6 Mean Condition Max Power (W/m) 13 27 13 65

It is important to note that the equations used to generate the maps do not

account for diffraction of waves around barriers or bathymetric effects like refraction

over sloped terrain, so the boundaries between regions sheltered by outcroppings of

18

land and those less sheltered are defined with artificial sharpness. Other factors that

affect wave parameters include temperature gradients, reflection, and tides.

1.2.2.3. Comparison to Field Measurements

In order to verify that the wave properties predicted by the hindcasting model

do indeed correspond to actual conditions in the IRL, a series of water level

measurements was conducted. A 0.5m-long Ocean Sensor Systems Wave Logger was

deployed on a piling at various locations around the lagoon in the vicinity of

Melbourne, FL. This instrument is a user-calibrated capacitive-type wave gage with a

programmable data collection frequency. It stores the stream of water level data to a

Compact Flash Card in text form in a CSV file. The wind and wave directions, GPS

coordinates, and sample time were documented during each deployment, and the

average wind speed for that time was recorded from the NWS webpage. In addition,

the wind speeds from the hours preceding the data collection time were noted in order

to determine if the minimum wave-generating wind duration had been achieved.

The documented wave-generating wind speeds and directions were then used

as input conditions in the models, and the output peak periods, significant wave

heights, and minimum durations were compared to the measured data. In order to

extract these characterizing properties from the wave gage water level data, the CSV

file had to be re-saved as an Excel (.xls) file and reformatted using the

convertOSS_WEC.m Matlab® program given in Appendix C. The time series was

then run through a zero up-crossing routine to identify individual waves, and a

cumulative distribution was created to determine the height of the largest third of the

waves. The frequency of peak energy content, and thus the peak period, was

determined by generating a wave energy spectrum. The period associated with the

peak in the energy spectrum was selected as the peak period. The results of the

comparison are presented in Table 3. The comparison reveals that the model is useful

for targeting a very approximate range of wave heights and periods for given weather

conditions. Wave periods generally corresponded within 15%, but percent differences

in the 40% range were common with the wave heights. Notably, the modeling

conducted here overestimated wave heights, causing the average overestimation of

19

wave power to be 38%. Nevertheless, trends were preserved, so WEC designs based

on modeling wave parameters may be improved after a sufficient amount of wave data

is collected for the specific site of deployment.

Table 3: Comparison of modeled and measured wave conditions

Location/ Date

Generating Wind direction

Avg. Wind Speed (m/s)

Hs (m) measured

Hs (m) model

T (s) measured

T (s) model

Anchorage/ Aug 13

130 3.5 0.108 0.141 1.52 1.59

Front Street/ Aug 17

115 2.6 0.065 0.076 0.93 1.16

Front Street/ Aug 21 108 5.3 0.055 0.094 1.05 1.2

An important consideration is the source of the wind data used to hindcast the

waves; the daily average speed and direction from the NWS record at the Melbourne

Airport may not represent actual on-site meteorological conditions. For true

calibration of the model, multiple data sets must be collected that include in situ wind

records for the period of time leading up to wave data collection.

1.2.3. Challenges in Wave Energy Converter Design

WECs face a unique set of challenges to implementation and efficiency. Like

marine structures of all kinds, they are subject to the extreme biofouling and corrosion

elements in the aquatic environment. Not all marine structures, however, rely on

constant unimpeded movement for efficiency, as WECs do. Lancaster University’s PS

Frog addresses this by minimizing external moving components, by encasing all power

take-off (PTO) mechanisms within a closed hull [13]. For designs in which this is not

possible, bio-resistant materials and coatings should be used for long-term

deployments. As this project is a proof-of-concept, a brief materials study was

conducted before purchasing construction materials, but long-term performance under

loading design will be left for future generations of the WEC. Exposure to continual

wave forces and episodic storm forces make robustness and simplicity of repair a

major design consideration.

20

A second hurdle, and one that has been studied extensively, is tuning the WEC

system to efficiently use a broad spectrum of wave frequencies, heights, and shapes.

As was mentioned briefly above with conventional rotational generators, any dynamic

system has a characteristic resonance frequency at which it operates most efficiently.

Resonance means that, as the exciting force acting on the system approaches a

particular frequency, the system reacts with the greatest possible amplitude of

oscillation. When no damping is present, an excitation force matching the system’s

natural frequency will produce undamped motion at the same frequency, such that the

resonant and natural frequencies of the system coincide. However, damping from the

water’s viscosity, system friction and loading, etc., will always be present, so the

damped natural frequency is slightly lower than the resonant frequency. The motion

is still maximized when the system’s actual natural frequency and the excitation

frequency are equal. This sensitivity to excitation frequency can both help and hinder

wave energy capture. If the WEC has a natural frequency in water that is close to that

of the approaching waves, maximum oscillation speeds and therefore maximum

power transfer will occur [11]. However, for wave periods dissimilar to the resonance

period, efficiency will drop. Ocean waves are rarely monochromatic, and the wave

climate of any location is not constant over time. For the WEC designer this creates

an unavoidable trade-off between optimizing the resonance frequency for maximum

performance efficiency at peak frequency and optimizing for fairly high efficiency over

a broad range of frequencies [3, 8, 9]. The optimal design would balance high

efficiency at the resonant frequency with fairly efficient operation over a wide

bandwidth. An ideal solution, although one that has yet to be successfully

implemented, is to enable the WEC to adjust its natural frequency in real time to

match current wave conditions. The models and measurements detailed above were

used to select an appropriate characteristic frequency, but attempts to optimize for or

adapt to multiple frequencies is beyond the scope of this project.

Another challenge, and the one that has been specifically addressed with this

project, is capturing the most energy possible from an approaching wave. As

discussed in Section 1.1, many current designs make use of either the hydrostatic or

the dynamic portions of the wave’s energy. A wave generator, however, supplies both

21

components of energy to the generated wave. This project proposes the use of inverse

wave generator theory so that the maximum possible energy may be absorbed for

conversion to electricity.

22

1.3 Objective and Scope

In an effort to improve wave energy capture efficiency, the wave absorber

shape of a basic direct drive WEC was modified using knowledge of wave mechanics

and wave generators. A single PTO section consisting of a linear induction generator

and its supporting circuitry was designed and built to accommodate interchangeable

buoys. Both a cylindrical buoy shaped like the AWS’ floater and a plunger buoy with

a specially curved face were constructed. The curve was contoured to match the

hyperbolic profile of water particle displacements in an approaching wave of height

0.1m and period 1.2 seconds in a water depth of 1m. The two buoys were then

deployed in same location during similar conditions in order to ascertain whether the

customized shape enhances wave energy absorption. In this way, the system focuses

upon testing the wave absorber efficiency’s effect on WEC power generation. The

device was designed as a stand-alone, watertight mechanism such that future versions

may be connected to the sensor suite with a short tether, over which power can be fed

to a battery charger.

23

2. Design Concepts

2.1 Wave Generators and Absorbers

In deciding to use wave generation technology to design wave absorbers, a few

design options were available. The two main classes of wave generators are paddles

(including both piston and flapper types) and plungers [41]. As seen in Figure 10,

paddle-type generators are simple boards that translate while positioned vertically (in

the case of the piston) or that pivot about the end attached to the ground (in the case of

the flap). The plunger type generally consists of a wedge-shaped block that is thrust

downwards into the water’s surface and pulled up again at a desired frequency. The

angle the wedge face makes with the vertical wall is denoted β.

Figure 10: Piston paddle, flap-type paddle, and plunger wavemakers

As the motion of the plunger is similar to that of a buoy bobbing at the surface, this

wave generator design is similar to that of the AWS’ and PowerBuoy’s absorbers, and

their cylindrical body shapes provide the control model for this project’s wave absorber

comparison. The WEC’s linear inductor PTO is coupled to the vertical motion of the

buoy, as shown in Figure 11. As the wave moves the buoy up and down, the magnet

24

is driven through a coil in the box above the buoy. This WEC is anchored to a piling

during deployment, with the buoy submerged and its top resting on the surface. An

advantage discussed earlier is shielding the WEC from of both storm forces and

curious vandals by positioning the entire PTO under the surface, like the AWS. For

simplicity of construction and deployment, that geometry was not used in this

prototype, but the same technology may be inverted and submerged in a subsequent

version, after the concepts have been successfully demonstrated. The floater of the

AWS was designed with survivability in mind; it is actually submerged at least 6m

below the surface so that a large passing wave pushes it downwards under the

hydrostatic pressure, rather than lifts it to the point of structural damage. Also shown

in Figure 11 is the proposed change to the buoy shape, which centers upon buoy

movement being directly coupled to wave motion. Any improvement in energy

capture displayed by the modified floater shape would suggest that increased emphasis

on wave mechanics would be a beneficial consideration for direct-drive WECs like the

AWS.

Figure 11: Difference between proposed buoy shapes

In general, plunger wavemakers are designed for single-direction wave

generation. In the new buoy design, all sides except the wave-absorbing face are flat in

order to minimize transmission of wave energy back to the water in other directions.

The rod restrains the buoy to vertical movement, and thus the wave energy

retransmitted to the water through radiation from the reverse side of the buoy is kept

to a minimum. Eventually, giving the buoy freedom to rotate on the rod that connects

25

it to the translator would allow the buoy to rotate as wave direction shifts without

requiring the entire apparatus to be moved. However, due to the potential

complication of the buoy’s yaw dynamics, the direction was fixed in the prototype to

avoid admitting too many variables into the performance analysis. The primary

reasoning behind the hyperbolic curve-shaped profile of the new wedge buoy is to

better match the wavemaker surface with the natural particle trajectories in waves.

Hence the hyperbolic profile is derived from theoretical equations of water particle

trajectories in wave mechanics.

In any wave generator geometry, the forced displacement of water transfers

both potential and kinetic energy to the water particles. However, the boundary

conditions at the paddles’, cylinder’s, and wedge-shaped plunger’s faces do not match

the elliptical trajectories of water particles under the progressive wave at a figurative

“infinite” distance from the wave generator. For this reason, some of the power put

into a wave generation system is lost in creating vertical standing, or evanescent, waves

that compensate for the mismatch at the boundary. In essence, some of the wave

energy approaching the paddle will be diffused into creating vertically propagating

standing waves that oscillate in front of the paddle. These evanescent waves decay

with distance from the paddle in the same way that particle movement decays with

depth in progressive waves. A quantitative explanation of this is given in the next

section.

26

2.2 Application of Wave Theory to Buoy Design

Many of the most energetic waves generated by IRL wave simulations based

on meteorological hindcasting have depth-to-wavelength ratios that qualify them as

transitional depth waves, rather than as shallow- or deep-water waves. The deep

depth regime criterion is important during the WEC design process because it

determines the shape of water particle movements. Particles in a wave do not translate

significantly as the wave propagates; they travel in elliptical trajectories given by

17

∫

∫

−+

==

−+

−==

)cos(sinh

)(sinh

2

)sin(sinh

)(cosh

2

tkxkh

zhkHwdt

tkxkh

zhkHudt

σξ

σς

where ζ and ξ are the time-variant positions in the x- and y-directions, respectively.

The depth of the water particle, z, becomes increasingly negative as depth from the

surface at z=0 increases. The time-invariant terms in these equations are the

horizontal and vertical axes lengths. As kh→0 in shallow water, ξ shrinks relative to

ζ , so the ellipses become increasingly horizontal. In deep water, however, the

trajectories are circular, and axis magnitude decreases exponentially with depth.

Figure 12: Water particle trajectories under progressive waves at different depths [8]

These particle trajectories come into consideration when deciding on the type of wave

generator to use. Piston paddles force equal horizontal displacement of water particles

27

over the entire depth, thus creating waves more akin to shallow water-type waves than

do flapper paddles. In the absorber regime, it is therefore logical that piston paddles

should better absorb energy from shallow water waves than from deep water waves.

The following equation can be used to predict the stroke of a piston paddle generating

a wave of height H:

18 ( )

hkhk

hk

S

H

pp

p

22sinh

12cosh2

+

−=

Dean and Dalrymple state that linear theory for plunger wavemakers making small

vertical motions and with small angle β is the same as that for piston paddle

wavemakers. This is logical because as a triangular wedge oscillates vertically, the

proximal water particles all experience forced horizontal displacement of equal

magnitude, regardless of depth, as with a piston paddle. The cylindrical buoy has β=0,

and the curve-faced buoy has a small β at the deepest section. Using 18 for the latter

buoy, however, would result in an underestimated height-to-stroke prediction because

it does not accurately represent the effect of the upper portion of the hyperbolic curve-

faced buoy, where β is large. Due to this section the buoy’s profile has some similarity

to a flapper paddle, which displaces surface particles more than particles near the sea

floor. The flapper paddle has a height-to-stroke ratio given by

19 hkhk

hkhkhk

hk

hk

S

H

pp

ppp

p

p

22sinh

1coshsinhsinh4

+

+−

=

These equations are derived from the Laplace equation with the dynamic and

kinematic free surface boundary conditions, the bottom boundary condition, and the

lateral boundary condition imposed by the wavemaker. The following equation,

derived in Appendix E, expresses the height-to-stroke ratio of an exponential decay-

faced plunger optimized for strictly deep water waves:

20 ( )

( ) ( )1222sinh

sinh

2 −+

+= − hk

p

hk

pp

p pp ehkehkhk

hk

S

H

If, however, the plunger’s curved face is shaped to match exactly the water particle

trajectories’ magnitudes in shallow-water or transitional regime waves, a hyperbolic

curve results of the form

28

21 ( )( )

( )hkzhkS

zSp

p

sinh

cosh)(

+=

Evidence that this stroke profile should perfectly match the water particle trajectory in

a linear wave is shown when the S(z) in 21 is substituted into the equation relating

generated waveforms and the progressive wave heights far from the paddle (see

Appendix E); the height-to-stroke ratio is exactly equal to one. Since basic

wavemaker theory is based upon volumetric displacement of water by the wavemaking

body, a plunger whose stroke displaces the same volume of water from a certain

control volume as a progressive wave would move naturally from that control volume

needs to travel only the same distance as the water particles it is exciting.

For a given stroke S, a flapper paddle creates smaller amplitude wave than a

piston paddle, since it displaces half the water volume. This means that in the inverse,

the same wave will create a greater stroke on a flapper paddle absorber than on a

piston paddle absorber. By extension, a concave-faced paddle (or plunger) displaces

even less water per stroke than a flapper paddle and therefore would undergo a greater

stroke when absorbing a given wave. For a 13cm wave in a meter of water with a 1.3-

second period (generated by a northerly 5m/s wind in the IRL), the flapper paddle

stroke would be between 10 and 11cm while the piston paddle stroke would be slightly

over 7cm. An exponential decay plunger traverses a stroke of 13.7cm, and the

hyperbolic curve plunger has a stroke equal to the wave height, 13cm. The average

power needed to generate such waves with a comparable wavemaker is given by

Equation 2. For this example, the result is approximately 20 W/m.

Since kh=2.419 (between /10 and ) in example above, this is a transitional

water depth wave, and the particle motions will be more like those created by the

generator of Equation 21 than any of the other three. The reason behind the

hyperbolic curve buoy traversing a greater stroke than the cylinder is that, in the wave

generator scheme, its smaller internal volume requires a larger stroke to displace the

amount of water contained in the wave peak. The exponential decay plunger

displaces even less volume and therefore needs a greater stroke length, regardless of

the wave type. Nevertheless, the theory in this project recommends the hyperbolic

decay buoy over the exponential decay buoy for intermediate wave types. While a

29

linear generator does depend on translator speed to spike its voltage, matching the

wave mechanics should take maximum advantage of both the kinetic and potential

energy in the waves. In the absorber scheme, if the wave transmits energy to the buoy

as it would another water particle, the buoy will gain both the kinetic and potential

components of the wave energy. If, however, the wave encounters the buoy as a

disruptive solid body, wave forces vary over the extent of the buoy’s surface.

Momentum is instead transferred back to the water particles as the wave energy is

partially dissipated in diffraction and radiation of waves off the buoy. These concepts

are discussed further in the Hydrodynamics section below, and they explain why the

cylindrical buoy should dissipate more energy per unit absorption width than the

hyperbolically curved block. For the similar shapes of the exponential and hyperbolic

curves, we look to evanescent wave generation to determine the better choice. The

hyperbolic curve-shaped absorber face is an attempt to minimize evanescent wave

generation by matching the face movement with water particle movement. The

exponential decay buoy is specifically targeted for deep-water waves. Ideally the front

face of a wave generator, be it a paddle or a plunger, would move identically to the

motion of water particles to create deep water wave particle motions directly in front

of the paddle. This condition would not produce evanescent waves. While elliptical

motion is impossible for a flat piston or flapper face, a specially curved face should

better match the trajectories, since the face displaces neighboring water particles by the

same amount as particles move far from the paddle. By making the remaining faces of

the plunger vertical, little energy will be transferred back into wave energy on the other

sides of the buoy.

In order to verify that the hyperbolic curve profile associated with water

particles in a progressive wave will minimize evanescent wave generation, the

wavemaker theory velocity potential, Φ, was calculated for a paddle with a stroke

s(z)=S*cosh(kp(h+z))/sinh(kph) for z=0 at the surface and increasingly negative z with

depth. For comparison, calculations for the exponential decay profile s(z) = S*exp(kpz)

were also performed.

Before the equations for velocity potential are presented, a side note

concerning these stroke equations should be discussed. In each of the three cases, s(z)

30

represents the maximum stroke width at a given depth, and not the equation defining

the curve of the plunger’s face. If f(z,t) represents the curve of the buoy’s geometry,

s(z)=f(z, T/2 ) - f(z,0). In other words, s(z) represents the difference between the buoy

face’s positions at the crest and at the trough. For a wedge plunger, f(z,t)=S1/h[z-

η(t)+h], but the stroke width resulting from a sinusoidal wave of amplitude H/2 is a

constant: s(z) = S1H/h. Thus the wedge-shaped piston emulates a piston paddle with

stroke s(z) =S2=S1H/h. As it turns out, for both curved buoy geometries, the difference

between the face positions over time results in a stroke width defined by the same form

of equation, simply with a different magnitude value. This means that the buoy must

be dimensioned such that a stroke width of magnitude S, matching the particles’

horizontal displacements, is a result of the difference in face positions at the crest and

trough of the design waves’ height.

The velocity potential is defined by

22 ∑

∞

=

+−

+−+=

1

)cos()])((cos[))(exp(

))sin())(cosh(

n

ssn

ppp

tzhnkxnkC

txkzhkA

σ

σφ

where subscripts p are associated with progressive waves and subscripts s with standing

waves, and

23

∫

∫

−

−

+

+−

=0

2

0

))((cosh

))(cosh(2

)(

h

pp

h

p

p

dzzhkk

dzzhkzS

A

σ

and

24

∫

∫

−

−

+

+

=0

2

0

)))(((cos)(

)))((cos(2

)(

h

ss

h

s

m

dzzhmkmk

dzzhmkzS

C

σ

Applying the paddle wavemaker velocity potential equations to the curved plunger is

valid because of the condition that linear theory closely matches the theory of piston

wavemakers for plungers with a small β and small vertical motion [7]. Since all terms

31

within the summation in Equation 22 correspond to standing (evanescent) waves,

any stroke S(z) that results in all Cm terms approaching zero will minimize the standing

wave magnitude. The Matlab® code that performed this test is given in Appendix C.

Under a given set of wave conditions, inserting s(z) = S for the profile of a wedge-

shaped plunger, s(z) =S*exp(kpz) for an exponential decay plunger, and

s(z)=S*cosh(kp(h+z))/sinh(kph) for a hyperbolically curved plunger shows that the Cm

values are smaller for the exponential decay paddle than for a straight piston paddle.

The magnitude of this reduction depends upon how closely the wave conforms to the

deep water regime. Under the mean power conditions predicted with the IRL

simulations, the difference was on the order of 2.5 to 12.5 times smaller, as shown in

Table 4. Because these incident waves were of the transitional regime, the

hyperbolically curved plunger conformed to water particle motion even more aptly

than the exponential decay plunger. The Cm values resulting from the hyperbolically

curved plunger were 6 orders of magnitude smaller than the piston paddle. A

numerical integration routine was used to perform these calculations, but the results

were made independent of the trapezoidal integration resolution by running the code

with finer and finer resolution until the output did not change with a finer grid. This

also suggests that the Cm values of the hyperbolically curved plunger might have been

zero if not for rounding in the numerical computation process.

Table 4: Comparison between standing wave coefficients of piston and exponential curve

wavemakers for various wave conditions and integration resolutions in 1m water

depth with a resolution of 0.0001m.

Piston

Exponential Curve

Hyperbolic curve

Cm(1) Cm(2) Cm(3) Cm(4) Cm(5) Cm(6) Cm(7) Cm(8)

0.0563 -0.0039 0.0011 -0.0004 0.0002 -0.0001 0.0001 -0.0001 -0.0076 -0.0005 -0.0002 -0.0001 0.0 0.0 0.0 0.0

T1 = 1.3s

H1 = 0.13m 0.64e-5 -0.09e-5 0.02e-5 -0.04e-5 0.02e-5 -0.01e-5 0.01e-5 0.0

0.0432 -0.0034 0.0009 -0.0004 0.0002 -0.0001 0.0001 0.0 -0.0165 -0.0008 -0.0004 -0.0001 -0.0001 0.0 0.0 0.0

T2 = 1.6s H2 = 0.18m

0.31e-5 -0.18e-5 0.05e-5 -0.04e-5 0.03e-5 -0.02e-5 0.01e-5 0.0 0.0540 -0.0037 0.0010 -0.0004 0.0002 -0.0001 0.0001 0.0 -0.0043 -0.0004 -0.0001 0.0 0.0 0.0 0.0 0.0

T3 = 1.2s

H3 = 0.1m 0.44e-5 -0.04e-5 0.04e-5 -0.01e-5 0.0 0.0 0.0 0.0

32

2.3 Mechanical Components

Mechanically, the WEC operates by coupling the movement of a magnet with

the vertical movement of the plunger suspended at the surface of the waves. As the

buoy plunger moves up and down with passing waves, it lifts the long vertical rod

above it, as shown in Figure 11. This rod slides through the bearings on the piling-

secured frame, which in turn supports the electronics box containing the linear

generator’s coil. Thus as the buoy moves up and down, the magnet moves vertically

through the coil. The primary source of resistance to this movement are friction from

the polyethylene bearings acting on the aluminum rod, which is caused by the bending

moment on the buoy’s rod due to the wave forces on the buoy. In the simplest

formulation, if the vertical wave forces acting on the buoy are able to overcome the

friction forces derived from the horizontal wave forces, the plunger oscillates in heave

and thus drives current through the generator. However, in any case where the body is

sufficiently large in comparison to the wavelength, the wave kinematics will not be

constant over the entire body, and the absorber’s motion will not automatically match

the magnitude and phase of the wave; many other forces and physical factors work in

conjunction to determine the system’s characteristic equation of motion. In order to

predict the performance of the plunger under a given excitation force, a complete

formulation of these physical properties must be completed. Since the modified

plunger shape was initially designed with the presumption that the face would move in

phase with the water particles, these details cannot be ignored. A block diagram of the

system’s individual components is shown in Figure 13. The following section presents

a general acknowledgement of the forces acting on the WEC system and estimates the

values of many physical coefficients based on calculations or literature values. The

general conclusion is that the parameters are impossible to accurately enumerate

without some preliminary physical testing, especially for the specially curved buoy, so

the system was designed as if the body diameter is small enough in relation to the

wavelength for it to be interpreted as a water particle. Afterwards, a discussion of the

materials selection and dimensions of the system are given to explain some of the

values in the section preceding it.

Figure 13: Block diagram of the W

EC system

showing input and output states of each component. The wind speed and direction

determine the wave height, length, and period in a given water depth. Those wave parameters determine the position and velocity

of the absorber buoy over time, which in turn govern the current induced in the coil by the magnet. The magnetic field induced by

the coil’s back EMF interacts with the permanent magnet to also influence the position and velocity of the buoy, since it is directly

coupled to the magnet. This total current in the coil is limited by the load current and by the voltage of the capacitor it is

charging. The circuitry performs according to the amount of power supplied to it.

Absorber

Radiation forces

Load

v(t), i(t)

Capacitor v(t), Load i(t)

Coil v(t), i(t)

Wind U,

θ

Water d

WEC

Wave

H, L, T

)(

),(

tz

tz

&

Fu (reaction due to coil B-field)

Wind/water

interaction

Power take-off

(generator)

Circuitry

34

2.3.1. Hydrodynamics

The wave absorber is constrained to move vertically (in the z-direction) only.

The sum of the forces acting in heave has been broken down in [11] as

25 ufvbreo FFFFFFzm +++++=&&

where mo is the buoy’s mass, z&& is the vertical acceleration, Fe, Fr, and Fb are the wave’s

excitation, radiation, and buoyant forces acting on the cylinder, Fv is the viscous force

imparted by skin drag of the fluid moving past the body, Ff is the sum of the frictional

forces opposing the excitation motion, and Fu is the load force imparted by the

generator as it delivers energy to the circuitry. A frequency-dependent form of the

above equation isolates the absorbed components from the excitation and radiated

components:

26 rem

mm FFui

SuRumi ˆˆˆˆˆ +=++

ωω

Although the subscripts m generally signify mechanical coefficients, load impedances

are also included in these terms. The carats in 26 recognize that the variables have

complex values. As will be expanded upon below, viscous and frictional power

dissipation and the real portion of the load determine Rm, and the buoyant force, along

with capacitive loads and any restoring springs built into the hardware, contribute to

Sm. While masses control the value of mm, it will be seen later that inductive loads are

also included in this term.

The splitting of the linear wave’s forces into three components takes advantage

of superposition to simplify calculations. Fe symbolizes the excitation force, which

accounts for the incident wave force and the diffraction forces resulting when the

incident wave strikes a stationary (non-heaving) body with force Fi, and diffracts

around it, causing the diffraction force Fd to act on the body. The incident component

is usually enumerated according to the Froude-Krylov approximation [46], and in

cases where the body’s diameter is much smaller than the wavelength, the diffraction

component is negligible. Otherwise, the diffraction component Fd must be calculated

as the surface integral of the derivative of the diffracted wave’s velocity potential with

respect to time:

35

27 dSnt

F Dd

ˆ∫∫

∂

∂−=

φρ

where dS indicates the integral is carried out over the body’s surface. The WEC’s

design conditions of 0.1m wave heights and 1.2-second periods in 1m water depths

dictate that its absorber width-to-wavelength ratio will be 0.1936. Fortunately, this

value is just slightly under the threshold of 0.2 that allows diffraction forces to be

ignored [46]. Numerical computations have been published for the Froude-Krylov

excitation forces on simple body geometries, such as a cylinder [10]. The results are

presented as plots of the transfer function relating incident wave amplitude to the

excitation force, with respect to wave frequency. Although the wave frequencies and

water depth in the computations were clearly outside the range of those used in this

project, a rough value for Fe acting on the WEC, given the design wave parameters,

was estimated roughly by locating the point on the plot’s x-axis with a kh value

identical to the WEC design wave’s. For the WEC design conditions, kh = 2.817.

Through back-calculation it was determined that the plot in [10] should be read at

ω=1.09 rad/s. This value was 400kN per meter of wave amplitude, given the WEC’s

capture width-to-water depth ratio and the wave frequency. Hence 40kN acts on the

bottom surface of the cylindrical absorber buoy when it encounters a 0.1m wave of a

1.2 second period. Because no literature is available for heave dynamics of irregular

shapes like the hyperbolic curve, a separate set of excitation calculations was not

performed, and the excitation force on both buoys is assumed to be equal.

The radiation force, Fr, and the buoyant spring force, Fb, account for the forces

reacting on a floating body as it oscillates in flat water. (Note that the linear

combination of diffraction and radiation forces is used to describe the forces acting on

a body in heave in linear wave conditions.) Fr includes the phenomena of “added

mass” and radiation damping. Added mass is a convenient way to interpret the

additional force needed to accelerate the fluid surrounding a body when it is, in turn,

accelerating. It is a function of geometry only and has the form

28 ( )zmiRF arr&ω+−=

36

where ma is the added mass and R is the radiation resistance (damping term).

Radiation resistance accounts for the power carried away from the vicinity of the buoy

by radiated waves. Added mass, on the other hand, accounts for the potential and

kinetic energy stored in the water particles adjacent to the buoy as radiation waves are

created. These two coefficients have been estimated in literature for vertical cylinders

in various water depths [10, 35, 37, 47], but the widely varying methods of applying

and reporting the input parameters make comparison between the sources difficult.

To achieve consistency in the face of the multiple formulations, the method and source

[10] used to estimate the excitation force above were again used for determining Fr.

These values are Rr = 105 N·s·m-1 and ma = 2.5×105kg. For the wave parameters given,

the maximum vertical velocity is 0.2618ms-1, and thus the maximum magnitude of the

sinusoidal radiation force is |Fr|=343.7kN, acting against vertical motion.

Again, similar numerical models for the hyperbolic curve’s radiation forces do

not exist, but it is worth setting forth the desired conditions for Rr and ma to yield

optimum power absorption. It has been shown that the maximum power absorption is

only 50% of the incident power if a symmetrical buoy moves with only one mode of

motion, such as heave, but that this limit does not necessarily apply to an

asymmetrical body in heave [12]. Maximum power is absorbed (although not

necessarily used by the load) when the buoy’s velocity and the excitation force act in

phase with each other. This is the optimum phase condition, and in general, it means

that the body’s position lags the water surface height, and therefore heave force of the

wave. A prerequisite for the optimum phase condition is that the system’s natural

frequency and the wave frequency must be equal. Using 26 and 28, the system

impedance can be expressed as

29 ( ) ( )

−+++=

ωω m

amrme S

mmiRRz

F

ˆ

ˆ

&

The minimum impedance to velocity occurs when the imaginary components in 25

cancel. This means that, at the excitation frequency of the wave, ω0, the relation

between the body mass, the added mass, and the coefficient accounting for the

buoyant spring and reactive load forces should be

37

30 ( )am

m

mm

S

+=0ω

For a symmetrical body able to absorb only half the wave power, the mechanical and

radiated impedances must be complex conjugates, implying that Rm and Rr must be

equal. For an asymmetrical body, it is simply desired that Rr be minimized to allow

for minimum radiated power. The optimum added mass value depends on the body

mass and total spring force for any given frequency. Minimizing the reactive

component also avoids phase shifting of the force with respect to the velocity. Any

phase shift present not only brings the system out of resonance but also paves the way

for the evanescent modes to dissipate energy as they try to match the boundary

conditions of the incident and radiated waves.

For the WEC’s cylindrical buoy to float in equilibrium with the top resting at the

surface of the water, mm = 73.95kg, while mm = 23.4kg for the hyperbolic curve buoy.

These masses are determined by the mass of the water displaced by the buoys’

volumes, and they are in part balanced by the weight of the rod and hardware above it.

Additional mass may be added via fishing weights in chambers inside the buoys to

ensure that they rest at the water level. If the buoys are deployed with lower mass, the

buoys will not float completely submerged, but the reduced mass would allow for

greater acceleration. For this device, the buoyant spring force Fb is fundamentally tied

to the body mass, as it, too, depends on the floating level of the body. Fb is the

hydrostatic force acting to restore a heave-displaced body to its equilibrium level in the

water. It is therefore equal to the weight of the displaced water and takes the form

31 dbdb zSgF =∀= ρ

where d∀ is the additional (positive) or reduced (negative) volume submerged when

the body is displaced by a distance zd from its equilibrium floating position. This force

is zero if the buoy moves in phase with the incident wave, but it changes in proportion

to the submergence depth of a buoy moving out-of-phase with the wave. The buoyant

force has therefore been expressed in three dimensions as a spring force dependent on

body geometry, where the spring constant for each of the directions is given by the

coefficients (or equations) in the 6x6 matrix bS . For a body moving strictly in heave,

38

all but the S3,3 coefficients equal zero, and S3,3 = Sb in 31 and forms a portion of Sm in

29. Other contributions to Sm may include a mechanical spring if present in the

design, and capacitance in the load. S3,3 is a function of the horizontal planar area, so

for the cylindrical buoy in heave,

32 db zrgF 2πρ−=

where the negative sign indicates that the force acts in the positive (upwards) direction

when negative values of zd (signifying submersion below equilibrium at z=0) occur.

For the hyperbolic curve, the equation is more complicated due to the increase in

planar area as distance from the bottom face increases:

33 ( )∫−

+⋅−=dzz

z

b dzzSbwgF0

0

)(ρ

where w is the capture width equal to the cylinder’s 2r, the stroke width S(z) is the

curve defined by 21, and b is the constant depth of the additional rectangular area

that contains the buoy rod at the back of the buoy. For consistency with 21, z must

be defined as zero at the top of the buoy, which was designed to coincide with the

water’s surface. Again, z becomes increasingly negative as depth from the surface

increases. For the case in which the buoy floats above the water’s surface, 33

defines z0 as the negative-quantity z-value on the buoy that coincides with the surface

at equilibrium, and zd is measured from this reference point with the same sign

convention as in 32. The effect of these equations is that a 1cm submersion imparts

a 14.51N force on the cylinder, while the same submersion on the curved buoy

floating 0.34m below the top imparts only 4.66N upwards. For the case that the buoys

float with their top faces level to the water surface and also move out of phase with the

wave, they will be overtopped. No buoyant force acts when they are overtopped, since

the submerged volume is unchanged. Instead, a hydrostatic force from the added

water pressure further forces the buoy downwards, and an increased viscous force

occurs as the buoy rises to the surface again.

The viscous force, Fv, is synonymous with the total drag on the object and

consists of skin friction and pressure distribution components. Both components are

linked to the speed of the fluid passing over the surface, and therefore Fv may be

39

modeled as a damping force as in 26. The skin friction is the integral of the shear

force over the surface area of the body. Since shear force is proportional to the

velocity gradient in Newtonian fluids like water, the velocity field near the surface of

the buoy determines the skin friction. Skin friction overwhelms pressure forces at low

speeds (Reynolds numbers less than 1). The pressure distribution portion of the drag is

caused by the pressure difference between the bottom face of the buoy and the top as

the body moves downward through the water column; if it moves at a higher speed

than the water particles, stagnation will occur at the bottom face, and the pressure

difference will impede motion. Skin friction plays a negligible role for Reynolds

numbers higher than 105, so at higher velocities, pressure differences dominate. The

Reynolds number for the cylindrical buoy is 1.06×105 at the maximum vertical

velocity, and it is 3.17×104 for the hyperbolic curve. These values are near the

pressure-dominated end of the transition region. Fortunately, total drag has been

determined to be of the form

34 2

2vACF pDD ρ=

where CD is a coefficient based on body geometry and Ap is the planar area of the body

facing the fluid traveling at velocity v with respect to the body. For a cylinder with a

length-to-diameter ratio near 1.0, CD = 0.93 for Re>105 [26]. With Ap=r2=0.1464m2,

ρ =1010kg·m-3, and vmax=0.2618m·s-1, the maximum drag force should be 4.94N. This

is fairly negligible in comparison to the other forces acting on the buoy. A drag

coefficient for the bottom of the hyperbolic curve buoy is not readily available, but it

was estimated as that for a rectangular sold (CD = 1.0), yielding a maximum drag force

of 1.16N.

The friction force Ff is due to contact of the aluminum rod with the

polyethylene bearings in the WEC frame. Friction force is quantified by

35 NFf µ=

where µ is the coefficient of static friction and N is the normal force, caused by the

bending moment on the buoy’s rod due to horizontal wave forces on the buoy. Based

on empirical equations for wave loading on a vertical wave barrier, a 0.1m wave

40

height with a 1.2-second period in 1.0m of depth will impart 218.6N of force per unit

width of barrier. Wave barriers have a similar geometry to the WEC in that they

extend partway down the water column. The equation for wave force per unit width

of barrier, Fmo, is given in [5] by

36 7.0

386.0

)cosh(

)sinh(

−

=

=

pL

w

omo

pp

p

oo

h

wFF

hkk

hkgHF ρ

where w is the depth of submersion of the barrier’s bottom face from the still water

line. Moment calculations about the support points on the frame were then used to

approximate the horizontal support forces acting on the aluminum rod to maintain a

total moment of zero. The two moment arms were dictated by the height of the frame

and the distance of the top of the buoy from the bottom of the frame. The total

friction, using µ=0.16 for UHMW polyethylene against steel, was estimated around

56N.

The final force for which we must account is the electrical reaction force of the

load, denoted Fu in 25. As the energy from the wave is converted into electricity in

the generator, an opposing voltage, called the back electromagnetic force (EMF),

develops to resist changes in current in the inductor. The force actually exerted upon

the buoy-coupled magnet depends on the counter-magnetic field generated by the

current running through the inductor. Since the self-capacitance of the coil is

negligible at the design operation frequencies (see Section 2.4.1), the generator circuit

can be modeled as series RL circuit also in series with the back EMF, signified by a

variable voltage source. The output voltage available to the load, vout, is thus given by

37 outemf vdt

diLiR =−− intε

where Rint is the internal resistance of the inductor. The physical background behind

this force and estimated values are discussed more thoroughly in the Electronics

41

Design section below, but the most important feature is its proportionality to the

magnet’s velocity:

38 zK zemf&=ε

Kz is determined by the dimensions of the coil and properties of the magnet. As

indicated by 26 above, the load is expected to contribute both a real and an

imaginary component to the system’s response, thus determining the value of the

current traveling through and from the generator. The generator and its circuitry

should be designed to hold real and reactive power proportional to the buoy’s heave

velocity. In an ideal case, the back EMF would induce the proper amount of reaction

force to keep the magnet moving in resonance with the mechanical portion of the

system.

The conclusion derived from these very rough estimates prior to testing was

that if the hyperbolically curved buoy were to be able to absorb the incident wave

energy without losing half of the power in radiation waves, as was expected from the

symmetrical cylinder, the excitation force should be sufficient to drive the buoy. The

cylinder is much less likely to be immune to the radiation forces, and therefore the

excitation forces were expected to be insufficient to overcome the forces opposing

oscillation. In addition, the radiation impedance term proposes a physical explanation

for why the standing wave coefficients calculated above indicate that the greater stroke

of an exponential decay buoy would not in fact yield more energy than a hyperbolic

curve buoy matched to the water particle trajectories.

2.3.2. Materials Selection

Topics of concern when selecting materials included biofouling, corrosion, and

robustness during extreme weather events. For this prototype version, cost,

machinability, ease of deployment, and ease of procurement played large roles as well.

A complete parts list of raw materials can be found in Appendix B, but a brief

discussion of each component is given here to explain design decisions. The majority

of the components shown in Figure 11 are composed of Ultra Corrosion-Resistant

Architectural Aluminum (Alloy 6063). Aluminum was selected for its light weight

42

and machinability, and the 6063 alloy is the most suitable for outdoor and humid

environments. Steel is much more corrosion-resistant than aluminum and should be

used for long-term deployments, but for this prototype testing, the light-weight

material’s maneuverability was preferable.

The low-friction bearings through which the generator rod and buoy rod move

are Ultra-High Molecular Weight Polyethylene tubes. This material is a thermoplastic

whose molecules are composed of extremely long polymer chains, giving it high

impact strength and abrasion resistance. It is self-lubricating and has a coefficient of

friction similar to Teflon, allowing for low-resistance movement. Another important

trait is its low moisture absorption.

A Pelican Case (size specification 1300) was purchased to that house the PTO

and circuitry. Made from Polypropylene copolymer, it has an o-ring seal, making it

watertight. A hole will be drilled into the top and bottom to accommodate the rod

that drives the generator magnet, and an aluminum fitting with an o-ring seals the

electronics from the elements. The magnet itself is a Neodymium iron boron (NdFeB,

or rare earth) magnet encased in epoxy to protect against corrosion. The connection

between the buoy rod and the generator shaft is made with polyurethane fasteners to

damp vibrations.

The purpose of the plunger is to absorb energy as if it were a water particle, so

the buoy is designed to be nearly neutrally buoyant. While the polystyrene foam

comprising the buoy itself is positively buoyant, fishing weights were added as needed

to the weight of the aluminum rod, magnet, and fittings in order to offset the

buoyancy. The buoy is coated with a layer of 6-oz. weight E-glass fiberglass with

epoxy resin fiberglass to protect it from water absorption and deformation.

2.3.3. Dimensions

The dimensions of the WEC are determined by the dimensions of the waves in

which it will be operating. The WEC was designed to operate normally for waves up

to 0.25m in height. This constraint was selected based on measurements of from the

lagoon under average conditions; while average seasonal wave heights were as given

43

in Table 3, the measured spectrum had a sizeable component of up to 0.25m. This

means that the prototype is not designed to be tested for efficiency in extreme weather

conditions, but it should be able to survive them.

Since wave energy propagates to a depth equal to only half the wavelength,

creating the buoy reaching deeper than this depth would cause extra fluid friction to

retard its motion. Not extending deep enough would waste a portion of the energy

available. In water of the 1m design depth, setting the buoy depth to 0.5m below the

surface covers the full energy region for waves with periods up to 0.8s. Since periods

shorter than this are either unlikely or not the energetic waves targeted for absorption,

we are not concerned with the additional drag on the buoy for waves under 0.8s. The

0.4318m buoy width was selected as the greatest diameter cylindrical buoy that could

be manufactured on the Florida Tech CNC machine, and the capture widths of the

two buoys needed to be equal to perform a power absorption comparison free of any

nonlinearities in the WEC’s ability to report wave energy excitation.

Axial stress in the translator rod supporting the buoy was calculated to ensure

they would not fail under the static weight of the WEC and the dynamic loading of the

waves. Given the horizontal wave loading calculated above for average conditions

and the dimensions of the frame, the round translator rod is expected to experience

surface flexural stresses due to bending moments of up to 2.758×108N·m-2, below the

yield stress of the aluminum by a safety factor of 5.2. Further calculations for the

extreme wave height loadings were not pursued since the testing of the hypothesis

requires waves similar to the design wave.

44

2.4 Electronics Design

In accordance with the scope of the project, the circuitry in this prototype

version of the WEC is intended to measure the comparative power output between the

two buoy shapes. As such, the generator must serve as a transducer between

mechanical energy in the buoy and electrical energy measured in circuitry. Given this

objective and the need to keep the project’s scope at a reasonable level, the circuit was

not optimized to collect sufficient power to support the ORCA sensor suite. In future

versions, more emphasis on the generator and circuitry design phase can greatly

improve the power generation levels. Another feature to note is that the power

measurement circuitry is supported by two batteries: one to power the circuit during

data collection and another to maintain the real-time clock’s user-set initial time and

date even while power to the circuit is turned off. The circuitry power is supplied via a

9V battery connected to a LM1117 low-dropout linear voltage regulator, which then

distributes 5V power and logic levels to the rest of the circuit. The clock reserve

battery is a 3.6V C-cell Lithium/thionyl chloride battery rated at 8.5Ah and a 20-year

shelf-time. An obvious goal for future versions is to achieve self-sufficiency using

power collected from the waves, but in the current version it is important to ensure

that data collection can continue in the event of minimal or zero power generation.

The circuit draws 96mA at startup, 78mA during capacitor charging, and 114mA

while the DC/DC converter is operating. Averaging this to 100mA continual

operation, the 9V battery should be able to support the circuit through four hours of

continuous data acquisition and storage before its voltage drops to 6.2V, the minimum

voltage needed for the LM117 to function.

Aside from these limitations, the circuitry was designed keeping in

consideration the eventual goal of powering the sensor suite’s battery charger. The

Texas Instruments bq2057 Li-Ion/Li-Polymer Charge Management ICs require a DC

voltage between 4.5V and 15V, so the convenient value of 5.0V was selected as the

output across the load resistor that substitutes for the charger in this prototype. The

measurement circuitry serves to record the generator voltage, load voltage, and the

45

voltage across two current sense resistors, along with the time of each measurement.

While the current sense resistor on the output of the DC/DC converter simply

reiterates the current passing through the user-selectable load, another resistor on the

input of the DC/DC converter is needed to determine the coil’s raw power generation.

The real power output of the DC/DC converter is the most applicable result in the

long-term, but in the case that no appreciable difference in real power is observed

between the wave absorbers, the input-side current sense resistor offers a second

measurement opportunity, independent of the remaining power management circuitry

and programming. In this way, the real-time power output can be compared with the

wave conditions recorded on a real-time wave gage, allowing for calculation of the

WEC system’s efficiency. The following sections describe in detail the design decisions

involved in building the generator and selecting the individual components integrated

into the printed circuit board (PCB). For a full inventory of parts, please see

Appendix B. A photo of the manufactured and populated PCB is shown below in

Figure 14, and the PCB schematic, with electrically connected traces labeled for

neatness purposes, is shown in Figure 15. A block diagram of the inputs and outputs

of each major circuitry component is given in Figure 16.

Figure 14: Populated PCB

Figure 15: (a) Schem

atic of WEC power generation and measurement printed circuit board

Figure 15: (b) Schematic of WEC power generation and measurement printed circuit board (contd.)

Figure 15: (c) Board layout of WEC power generation and measurement printed circuit board

Figure 16: Block diagram of inputs and outputs to major circuitry components. The position and velocity of the magnet within

the coil determine the induced current and voltage across the coil. These values may also be limited if the voltage on the capacitor

is higher than the coil’s voltage spike, and by the current being allowed to flow through the DC/DC converter. The capacitor’s

voltage governs when the load will be switched into the circuit, and it ultimately governs the amount of input power available to

the DC/DC converter. This power determines the operating efficiency of the DC/DC converter and thus the output power.

Generator

assembly

Rectifier

Capacitor

Load i(t)

)(

),(

tz

tz

&

Coil v(t), i(t)

Coil |v(t)|-1.2V,

|i(t)|

Capacitor v(t)

vin(t), iin(t)

Load

v(t), i(t)

Capacitor v(t)

Microcontroller/

Relay

DC/DC

Converter

50

2.4.1. Power Generation Components

Magnet and Coil

The power take-off element of this WEC is, as described previously, a linear

generator consisting of a simple coil of insulated copper magnet wire and a disc-

shaped magnet that translates through the center of the coil. The precedent for this

generator topography has been set theoretically in literature [20, 23, 24, 32, 43] and

practically by the Archimedes Wave Swing and the PowerBuoy. The linear topology is

attractive for its minimum of maintenance-intensive moving and interfacing

components, as compared to rotational motor, hydraulic, and pneumatic systems.

The gears and sealing in those conventional systems serve to interface the low

reciprocating speeds of the waves to the high, ~1500rpm speeds required in rotary

electric generators. The high speed of the field-providing components with respect to

the armature, or power-producing components, is the basic tenet behind generating

enough voltage to drive loads. Faraday’s Law of Induction states that the

electromotive force, or induced voltage, ε, generated across a closed circuit is

proportional to the time rate of change of the magnetic flux, ΦB, through the circuit.

39 t

B

∂

Φ∂=ε

While generators containing electromagnets use alternating current to alter the field

quickly over time, permanent-magnet generators rely on the movement of the magnet

with respect to the circuit to achieve the change in flux. Thus voltage generated is

proportional to magnet velocity in these systems. As such, the useful force (denoted as

Fu above) acting on the plunger buoy due to the generator has been modeled as a

damper in literature [10, 24]. The direction of the force always acts to retard the

motion, since Lenz’s law holds that the induced electromotive force will act to oppose

the change in flux that caused it. A closed circuit can be approximated by a single

loop in a tightly-wound coil, and thus a coil with N loops subjected to the same

magnetic flux will generate N times the voltage level.

Various schemes for creating higher-frequency electrical signals from the low

frequencies of wave-excited linear motion have been evaluated for their practicality

51

and airgap shear stress production ability. These innovative topologies include

longitudinal- and transverse-flux permanent magnet linear machines and the similar

Vernier hybrid machine [24]. These generators rely on alignment of iron teeth on the

translator with magnets of alternating polarities on the stator to electrically gear the

magnetic field frequencies to higher frequencies, as shown in Figure 17.

These studies generally hold that the

innovative geometries lend themselves

to both higher shear stresses within the

gap between the magnets and the coil

and higher normal (attraction) forces

between the magnets and the steel

armature. Hence the power-generating

potential is increased, but the increased

danger of contact between translator

and stator requires a heavier, sturdier

support structure. In comparison, a

simple air-cored tubular machine with

no iron on the stator avoids the risk of

magnetic forcing hindering the translator’s motion, but at the expense of shear force

density and thus power generation. The lack of iron to concentrate the flux causes the

magnetic field to decay quickly with distance from the surface of the magnet; increased

magnetic material is needed to increase flux in the coils. Recognizing this drawback,

the air-cored tubular machine, or simple disc magnet translating through a coil, was

chosen to simplify construction and physical analysis. Commercial linear generators

with the combination of low speed and low torque are not readily available, so the

physically basic magnet-in-a-coil geometry was implemented.

The initial design for the linear generator was based upon a U.S. patent [19]

and the circuit observed in a shake-powered flashlight. Calculations for the inductor’s

dimensions were then made to target the stroke of the magnet and the frequency at

which the oscillation would occur. In order to extract the most power from the coil,

the power take-off component’s impedance should be complex conjugate of the

Figure 17: Magnetic gearing in a transverse

flux machine [24]

52

mechanical impedance of the system [12]. The real part of that mechanical impedance

includes the viscous and frictional resistance, along with any radiated resistance. The

mechanical reactance includes the buoyant spring force, radiation reactance (added

mass) and body mass. From 29 it can be seen that, for a given excitation frequency,

any load capacitance must add to the mass terms, while load inductance add to the

buoyant spring force to complete Sm. Controlling these load values can allow for the

condition in 30 to be met.

It should be made clear that the circuit in the current WEC was designed

without knowledge of the added mass or even the final buoyant force and body mass

values. Instead it was assumed that the capacitive and inductive components of the

load must resonate at the excitation frequency to maintain system resonance. The

ramifications of this design on total system efficiency depend on each buoy’s

mechanical characteristics and will be discussed further in the Results section. For

any RLC circuit, the resonance frequency is given by

40 LC

f rπ2

1=

Given the predominate wave climate in the IRL, the target resonance frequency range

was between 0.67 and 2 Hz. The appropriate combination of inductance and

capacitance was thus calculated for that frequency range using

41 22

2

1

2

2

4 cLC

πλλ −

=

where λi are the wavelengths of the electricity at the limits of the frequency range.

Hence the LC value targeted was 0.05s2. This is a very high product of inductance and

capacitance, as typical inductor values are on the order of microhenries, while

capacitors’ maximum rated voltages drop quickly for values above a few millifarads.

Capacitors present a trade-off between high capacitance and the voltage level at which

they suffer physical damage. Supercapacitors small enough to be inserted on a PCB

can be purchased with values on the order of 5 Farads, but their limiting voltages are

less than the 5V that is desirable to keep the charging circuit operating (please see

DC/DC converter section below). A variety of supercapacitors rated for 6.3V (25%

factor of safety) were purchased with values between 0.47F and 1F. A capacitance on

53

the order of 0.5F then requires an inductor of approximately 0.1H. A guide manual

for winding coils and transformers [4] was used to determine the correct dimensions

for achieving such a high inductance value. The following criteria were used to

determine the dimensions:

• Inductor length is inversely proportional to inductance.

• Inductor cross-sectional area is proportional to inductance.

• Inductance is proportional to the square of the number of turns in the coil.

• Capacitance of the coil increases with the voltage difference between adjacent

turns.

• Resistance (and therefore real power loss) increases with the gage of the wire.

• The number of turns per inch is limited by the thickness of the wire, but adding

layers to the coil decreases the magnetic flux density acting on the more distant

coils.

Certain parameters were dictated by the geometry of the problem. The length

of the coil should not exceed the expected wave height, as this would waste decreased

inductance on turns that are not exposed to appreciable magnetic field changes. In the

end, the length was determined by the safe range of motion for the oscillation testing

machine, 0.635m (2.5 in). This was also appropriate for the size of the design wave

height, as a wave 0.1m (4 in) in height would allow the magnet to travel past the ends

of the coil, thus producing the maximum exposure to changing fields. The cross-

sectional area was maximized by using the magnet with the largest diameter available

for purchase. Rare earth, or Neodymium-Iron-Boron (NdFeB), magnets were used for

their high flux density, economy, and better mechanical resistance to brittle fracture as

compared to Samarium-Cobalt magnets. Two 1.5-inch outer diameter, 0.375-inch

thickness, N40-strength ring magnet (with a 0.5-inch inner diameter to accommodate

the supporting axle) were purchased, and the coil was wound around a 1.625-inch

outer diameter PVC former to minimize the airgap thickness. A bank-winding

geometry, in which successive layers are wrapped atop the layers closest to the former

in a wine bottle stack format, before the first layer is completed, minimizes the voltage

54

difference and therefore capacitance between adjacent turns. The number of turns

needed to achieve the desired inductance was determined by the following equation

for bank-winding inductance from [4]

42 cba

NaL

1093

2.0 22

++=

where a= 1.625” (inner diameter of coil), b=2.5” (coil length), c=0.1” (estimated radial

depth of winding for 10 layers of 30AWG wire), and L= 5×104µF (factor of safety built

in for possible losses). This yielded N=1640 windings, and thus 2000windings were

completed to compensate for inexact winding. The bank-winding geometry was

roughly attempted, but the hand-winding procedure was not precise, so some added

degree of capacitance was expected. In the end, tests and calculations placed the

inductor as having somewhere between 0.10 and 0.25 H of inductance, with a

resistance of 175Ω and a capacitance of 0.35µF. The impedance of a non-ideal

inductor is modeled as a series combination of the inductor and a resistor, all in

parallel with the capacitance. Hence the total source impedance in Ohms is

43 ( )[ ] ( )582 101.61075.81

25.0175−− ×+×−

+=

ωωω

j

jZs

As will be discussed in the Results section, this impedance cannot include the

supercapacitor capacitance, although that was the intention during the design phase.

The supercapacitor was located downstream of the rectifier (discussed next), and

therefore could not discharge across the inductor to complete the oscillation cycle.

The implication of this is that excitation forces on the order of kilohertz, rather than

the design operating frequency of 0.83Hz, would be needed to raise the source

impedance significantly above 175Ω, so the source impedance is defined as 175Ω with

no phase shift.

Rectifier

The full-wave rectifier serves to invert the negative voltages generated during

the downward stroke of the magnet in the coil. The ST Microelectronics L6210 was

selected for its low forward voltage drop Schottky diodes (only 0.65 to 0.8V loss per

pair at 100mA). In addition, the 50V reverse voltage per diode and the maximum

55

current rating of 4A are more than enough to ensure the rectifier can withstand the

conditions supplied to it by the generator. The rectified current is connected in parallel

with a 22000µF capacitor such that any charge generated in the coil is stored until an

appropriate voltage is achieved to power the DC/DC converter. The supercapacitor’s

value was calculated in conjunction with the inductor’s value with the intention of

matching the target wave excitation frequency, as described above.

DC/DC Converter

The Texas Instruments TPS61202 DC/DC voltage converter was chosen for

its potential to operate at high efficiencies, its fixed 5V output, its wide range of input

voltages (especially its low 0.5V startup voltage), its low quiescent current of 55µA or

less, its ability to automatically switch between a boost converter and a voltage

regulator, and its advertised applicability in battery-power circuitry.

The microcontroller operates a relay switch that latches the converter into and

out of the circuit based on the voltage present on the inductor/capacitor component.

Since the DC/DC converter’s efficiency falls quickly from its rated 90% when the

input voltage drops under 3V, the relay detaches the converter (and thus the load as

well) from the generator circuitry when the magnet’s movement fails to maintain at

least 3V on the capacitor. This allows the capacitor to charge unloaded until it reaches

a voltage level at which it can again support the load. Although the data sheet lists the

minimum startup voltage as 0.5V, preliminary tests showed that the converter draws a

surge of current during startup, which the capacitor could not sustain while carrying a

low potential. For this reason, the microcontroller waits until 5V has been stored on

the capacitor before connecting the converter and load back into the circuit. This

scheme also ensures that the converter will spend virtually zero time operating as a

Down Conversion Mode regulator, which dissipates more power than a boost

converter.

The constant 5V output drops to 0V when the switch releases the converter for

the capacitor to charge, and it latches right back to 5V when the switch is restored.

The tradeoff between the latching voltage input value and constant operation should

be further investigated to achieve a constant DC voltage supply from the output. One

56

caveat is that the efficiency of the converter drops greatly for output currents 10mA

and under.

The primary drawback of the TPS61202 is that it is available only in the

miniscule 3mm×3mm QFN-10 package. While this is intended as a space-saving

attribute, it makes soldering to a footprint onto a PCB very difficult. For prototyping a

SchmartboardTM surfboard was purchased, which is specially designed such that extra

reserves of solder on the leads facilitate soldering to small-pitch chips.

The use of a 2N3904 NPN transistor on the relay controlling the DC/DC

converter input pin allows the device to be switched on and off in the PIC®

microcontroller’s software. This configuration produces a ~0.6V drop across the

transistor, causing the relay to receive 4.4V rather than 5V on the pin that controls its

internal magnetic switch. While this type of voltage drop on a low-side load NPN

configuration is generally undesirable, the RSB-5-S relay functions normally down to

3.5V, so the configuration is acceptable. In this way, a high signal from the PIC’s

control line turns the DC/DC converter on while a low signal turns it off.

2.4.2. Data Collection Circuitry

Microcontroller

Due to familiarity with the family of microcontrollers, the Microchip®

PIC18F-series was selected as the central controller for data collection and component

latching on the PCB. The 18F4523 was selected for its 12-bit analog-to-digital

converter (ADC). Other chips in the 18F series provide only 10 bits of resolution on

their internal ADCs. Microchip’s MPLAB® Integrated Development Environment

(IDE) was used in to write the microcontroller programs in the C language, using

MPLAB’s C18 compiler libraries. The microcontroller runs on a 20MHz crystal

oscillator, allowing for 200ns instruction cycles.

Programming and Communication

Interfaces for both the MPLAB ICD 2 and PICkit 2 are designed into the

circuit board to provide programming interface versatility for future users. These

57

interfaces allow the PIC microcontroller to be programmed without the need to

physically remove the chip from and replace it into its 40-pin socket on the board.

These programmers are also useful when operated in debug mode, in which they can

step through a PIC program to display the function of each line of code.

The serial data line on the PIC is accessible for initializing the RTC and

monitoring the data storage messaging during debugging. A Maxim MAX233 logic

level converter switches the 0-5V logic of the PIC to the ±15V logic of a PC, and the

serial data is fed out through a DB9 connector interface. This may be accessed using

the HyperTerminal serial port communications software available with any Windows

operating systems prior to Vista.

Load and Current Sense Resistors

A pair of resistors forming a voltage divider and two separate 10Ω current

sense resistors serve as the prototype’s load and current sensing elements, respectively.

The resistors are connected to the board via Molex connector housings so that the load

values can be changed as wave conditions vary during testing. The series load

resistors used during deployments were 4.7kΩ each This value was settled upon taking

the ADC’s saturation level and the DC/DC converter’s output level into account.

Controlled sinusoidal excitation of the magnet with oscillations near 1Hz showed that

both saturated and low duty cycle power outputs were possible with a ~10kΩ load,

depending on the magnitude. With 5V across a 10kΩ load, 0.5mA of current can be

expected to flow. This translates to a 2.5mW saturation level for power dissipation.

The bq2057 consumes 300mW of power itself, so eventually a larger current is desired

to increase the power output capacity. This can be accomplished by reducing the load

resistance. A 500Ω load allows for 50mW of power, while a 100Ω saturates at

250mW. The caveat with increased output capacity is that it causes the supercapacitor

to drain more quickly, thus switching the DC/DC converter out of the circuit more

frequently. Under manual excitation, a 500Ω load provided a subjectively selected

good balance between increased output power and appropriate operation durations.

However, with steady sinusoidal velocity excitation, the magnet’s speed was not high

enough to maintain capacitor voltage for such a low load, and a 10kΩ load was

58

needed before the versatility of both saturated and unsaturated test conditions was

possible. The middle ground achieved by these operating conditions was desired to

allow for fast initial analysis of which buoy shape yields more constant power capture;

with the current hardware, 10kΩ was necessary to make apparent which of the buoys

can support converter operations for a longer duration in a given set of wave

conditions.

The PIC’s ADC saturates at 5V, so no amplification is needed for measuring

the load resistor, which experiences essentially the entire 5V drop across it. With 5V

running across 500Ω, a 10mA current can be expected. This creates a 10mV drop

across the 1.0Ω current sense resistor. A specialized current-sense operational

amplifier with a fixed gain of 100V/V is used as a high-side amplifier to increase the

current sensing signal to 1V. This only uses a fifth of the ADC’s resolution capability,

but increasing the amplification further risks increasing noise levels. The sense resistor

is a 5% tolerance carbon film resistor, meaning that exactly 10mA traveling across it

could produce between 9.5 and 10.5mV. With the gain of 100 and the 12-bit, 0-5V

ADC, this translates to readings anywhere between 778 and 860 counts out of 4095,

without even taking into account the uncertainty in the load resistor, current level or

the DC/DC converter output voltage. Filtering and averaging the data removed time-

varying noise from the data, but in future versions, more expense can be applied

toward a higher-accuracy sense resistor.

The MAX4372 operational amplifier was chosen for its intended application

as a precision current sensing amplifier (to ensure minimal uncertainty is contributed

in the amplification stage), its low 30µA current consumption, and its wide signal

input range that includes voltages near zero. Its high-side configuration prevents

ground path interference on loads that require grounded signals, such as the battery

charger. To ensure the output signal could actually sink to the low voltage rail during

times of no current flow, a negative voltage supply was connected through a pull-down

resistor to the output pin of the op amp. A MAX889 was selected for the task of

voltage inversion due to its availability, and its (absolute value) minimum output

voltage of -2.5V was configured to minimize ambient current flow. Power dissipation

added by the MAX889 is approximately 36mW.

59

While the load resistor with the known DC/DC converter output voltage

applied across it is, in itself, sufficient for measuring the power dissipated by the load,

keeping the current sense resistor in place allows for the option of connecting the

rectifier output directly to the resistor series. This unregulated measurement mode

would allow for analysis of the generator’s passive ability to support the battery

charger, without being subject to the switch control regime or the power requirements

involved in operating the DC/DC converter. While this configuration was not

explored in this set of tests, the feature is explained here for future investigation.

Filtering, Buffering, A/D Conversion

As mentioned above, 12-bit ADC was specially selected to gain higher

resolution on the power measurement data. Higher-resolution ADCs, however,

cannot automatically guarantee higher accuracy data if the noise level on the input

signal is too high or if the sampling frequency it too high to support the signal’s

source’s impedance.

Filtering aids in noise reduction. Since the LC circuit was designed for

resonance with waves between 0.67 and 2Hz, the data recording frequency fr must be

no lower than 4Hz to avoid aliasing. A low-pass RC filter must be added before A/D

conversion to minimize the effect of high-frequency noise on the data. A general rule

for RC filtering is to select a filtering cutoff frequency fc that is 10 times greater than the

highest data frequency, thus suppressing any higher-frequency signals by more than

3dB. A 50Hz fc was selected to reliably support fr up to 5Hz. To achieve an RC time

constant of 0.02s, a 200kΩ resistor and a 1µF capacitor was used for each filter. The

sampling frequency fs was then designed to be approximately 500Hz, or 10 times larger

than fc, and thus suppressed by 20dB (to a tenth of its amplitude). After the circuit has

been completed, calculating the standard deviation of data recorded during DC input

conditions indicates whether or not noise contributes a low enough uncertainty to the

recorded value to merit using a higher-resolution ADC. The completed PCB’s

standard deviation was measured to be zero given a DC signal, since no variation in

ADC output was recorded at a given input voltage.

60

While the filtering minimizes the effects of high-frequency noise, it presents

another problem for the ADC. ADCs contain internal capacitors that must fully

charge in order to maintain availability of the correct input voltage to the internal

comparators as they loop through calculations and identify the voltage value. The PIC

data sheet warns that source impedance on the analog line should not exceed 2.5kΩ in

order to avoid voltage divider effects with the internal impedance of these ADC

capacitors. With the RC filter on the signal line, the impedance would rise above

230kΩ for even the lowest design frequency waves. For this reason a unity-gain op

amp was added to the back of each filter as a buffer to reduce the signal’s source

impedance into the ADC, and thus supply more current to the internal ADC

capacitors. The MCP604 op amp is capable of sourcing 22mA continuously, meaning

that the capacitors with maximum 30pF of capacitance in the PIC’s ADCs could

charge from zero to 5V in less than a nanosecond. This is orders of magnitude shorter

than the maximum signal acquisition time selectable for the PIC’s ADC, which is

16µs. The A-to-D comparator calculations take a minimum of 10.4µs. Hence the

entire process uses 26.4µs per conversion. Using a sampling frequency of 518Hz

means that four A-to-D conversions (coil voltage, load voltage, DC/DC converter

input current, and DC/DC converter output current) are requested only every 1.9ms,

or one every 0.483ms. Hence the ADC has 0.483ms to carry out an operation that

requires only 26.4µs. The buffer thus makes it possible for the ADC to accurately use

its 12 bits to measure dynamic signals at the requested sampling frequency.

A 12-bit ADC would ideally separate the full-scale input voltage neatly into 212

= 4096 bins, each signifying 1/4096th of the full scale. For the circuit’s 0 to 4.096V full

scale input, each ADC count should signify 1mV. This 1mV resolution was verified

on the small scale by matching an increase in the recorded ADC value with the

number of millivolts by which a controlled DC input signal was raised. Nevertheless,

a broader-range calibration is needed to ensure proper interpretation of the data, so a

calibration of the entire analog input block to each ADC was carried out. Due to the

dependence of the ADCs’ operating characteristics on actual input impedance values,

this had to be conducted after the printed circuit board’s manufacture and population,

not on the prototype breadboard, since any resistance in the PCB traces may alter

61

intermediate voltages slightly. The calibration of each of the four analog in put blocks

to the PIC was carried out by generating a controlled voltage (for the coil- and load-

monitoring ADCs) or controlled current (for the current sense resistor-monitoring

ADCs) and fitting a linear equation to the corresponding ADC values as monitored

over HyperTerminal. The complete step-by-step calibration procedures are detailed

for repeatability in the Testing Procedures section below.

Real-Time Clock (RTC)

In order to correlate the measured power generated with the wave power

acting to excite the buoy, the ST Microelectronics M41T80 was added to the board.

This RTC was selected for compatibility with the ORCA sensor suite, so that future

integration into the control scheme will be less complicated. It communicates with the

PIC over I2C, and it has a pin dedicated to outputting a user-programmable frequency

square wave. A PIC program was written which allows the user to initialize the

RTC’s time setting via HyperTerminal®. The Lithium C-cell battery then allows it to

retain this setting after the circuit has been powered down. The M41T80 supplies the

PIC with the year, month, day, hour, minute, second, and hundredth of a second. In

binary coded decimal (BCD) format. This data is read every time the PIC records an

averaged data point.

DataStorage

A Logomatic V1.0 by Sparkfun Electronics was employed to capture data

written directly to the serial bus to an SD card. A configuration file stored on the SD

card instructs the Logomatic to operate at the desired UART mode and baud rate.

The Logomatic offers ADC capabilities, as well, but these channels were unused in

this application. As such, the only signals required for the power measurement circuit

are the data transmit, file stop, and file reset lines. The stop and reset lines remain

high until the end of a sampling period, at which point they are dipped low for a short

time before returning to high to begin another sample set. These lines are controlled

by PIC output pins, which are then run through 2N3904 NPN transistors. The stop

and reset lines use pull-up resistors on board the Logomatic, so connecting the NPNs’

62

collectors to the stop/reset pins and their emitters to ground allows the voltage at the

pin to dip to zero when the transistor is turned on by the PIC. This dip in voltage

closes the text file being written to the SD card, ensuring validity of the data recorded.

63

2.5 Software

Brief mention has been made of individual software routines when the

hardware associated with them was presented. In this section a full description and

user’s guide to the PIC microcontroller programs and the MATLAB® post-processing

routines is given. The code for these programs can be found in Appendices C and D.

2.5.1 PIC Microcontroller Code

The PIC code was written, compiled, and debugged in Microchip’s

programming interface, MPLAB IDE, version 8.00. Given that the PIC18 family of

microcontrollers was used, the C18 Toolsuite (which includes the MPLAB C18

Compiler, MPLIB Librarian, MPLINK Object Linker, MPASM Assembler) was

selected during project file configuration. The only other project configuration setting

required was declaring 18f4523.lkr as the linker file, since the hardware

configuration bits were programmed into the code. The code, then, is written in a C-

based language defined by the C18 Libraries. The documentation for these libraries

and development tools is available on Microchip’s website [22].

The project consists of seven source files: WECfinal.c, SetClock.c,

ConfigRTCSq.c, ReadClock.c, I2CRandomRead.c, I2CByteWrite.c, and

ConvertTime.c. The main() function is contained in WECfinal.c, and the

functions defined in the other files are called from within the main function. The main

function also calls an interrupt service routine (ISR) called RTC_sample_ISR(),

which is defined within the same WECfianl.c file. Global variables that define the

sampling and record frequencies, record length, and resistor values installed in the

PCB are set at the beginning of this file as well. This means that when a change in

sampling frequency, record length, or current sense resistor value is desired, the

program must be recompiled, and the PIC reprogrammed.

The general structure of the code sends the PIC into a continuous loop of ADC-

sampling and data-writing directly after initial setup procedures. Each iteration of the

loop performs time-averaging on samples from the four ADC ports, reads the clock’s

64

time, and writes the time and data to the serial port. This loop runs for the duration

requested by the record length global variable before sending a signal to the Logomatic

to reset and create a new file on the SD card. Within the loop, each iteration’s timing

is controlled by a flag set within the ISR. The ISR initiates on every falling edge of a

controlled-frequency square wave signal sent from the RTC to the PIC; a counter in

the ISR keeps track of the number of samples collected before setting the flag to

indicate the data should be averaged and written to the serial port.

Upon initial power-up of the entire circuit, the PIC sends power to various

peripheral devices on the board, and the PIC routine prompts the user to initialize its

real-time clock. At this juncture in the code, the SetClock() function is called. The

initialization is necessary if the battery that preserves the RTC’s time setting has been

removed from the circuit at any time since its previous use.3 To complete this

operation, the PCB must be connected to a PC’s serial or USB port via the DB9

connector on the PCB. Running the Hilgraeve. Inc., Windows-compatible software

HyperTerminal, connected with 115200 baud, 8 data bits, 1 stop bit, and no parity or

flow control, allows the prompt “Set Clock YYMMDDHHMMSS>” to be displayed on

the PC screen upon circuitry power-up. The user must then respond with twelve digits

corresponding to the date and time before a carriage return. Since these values are

relayed directly to the RTC via I2C communications, they must fall within the

boundaries of the RTC’s registers’ 00-99 year (YY) range, 01-12 month (MM) range,

01-31 day (DD) range, 00-23 hour (HH) range, or 00-59 minute (MM) and second (SS)

range. If incompatible values are entered, the routine will freeze, and power must be

cycled to the circuitry. The SetClock() function is nearly identical to that written

by Tony Cimaglia for implementation of the RTC in the ORCA sensor suite.

3 Note that this function call should be commented out during the final compilation and programming before WEC deployment in order to avoid the need of user-input whenever power is cycled to the circuit. The RTC’s battery connection must be maintained during and subsequent to programming, but it is important to make this coding change before deployment to ensure that sampling is not completely halted during an unintended power cycle.

65

Next the main() function calls the ConfigRTCSq() function in order to set

the frequency of the square wave generated by the RTC, which in turn drives the

interrupt prompting the PIC to sample its ADCs. As mentioned above, the frequency

is hard-coded into the WECfinal.c file. The value used for testing during this project

was 512 Hz. All communications with the RTC are conducted via the PIC’s I2C bus.

Thus both the SetClock()and ConfigRTCSq() functions call the

I2CByteWrite() function to relay initializations and settings. In addition, the

main loop gathers current time data from the RTC over the I2C bus using the

I2CRandomRead() function. ConvertTime() then reformats the binary-coded

decimal (BCD) data into decimal format for recording. These three functions are also

heavily based on Tony Cimaglia’s sensor suite code.

As a final note on the PIC code, it should be emphasized that seamless

sampling at the prescribed data rate relies on calculated pairing of appropriate

sampling and recording frequencies. The 512Hz sampling rate and 8Hz record

frequency were chosen to accommodate the RC filter cutoff frequency of 50Hz.

However, the amount of time required for the PIC to carry out all the operations

within one iteration determines the upper bound of the sampling rate that the RTC

may request. Using the formulae on pages 232 and 233 of the PIC18f4523’s data

sheet, it was determined that four ADC acquisitions would require a minimum time of

105.6µs to complete. This is well within the 1.9ms window allowed by the 512Hz

interrupt operation. Nevertheless, every eighth of a second, all the averaging, clock

reading, and serial port writing operations in the main loop must be completed within

this 1.9ms window, or the data will be sampled but not recorded. That these

operations can be completed at this frequency is proven by the complete data record

supplied during testing. The RTC has the capability of outputting square waves of

certain frequencies, ranging as high as 32.768kHz. It should be noted that both the

ADC’s settling and acquisition times and the PIC’s ability to complete commands in

the allotted time must be taken into account when raising the sampling frequency in

pursuit of higher accuracy data.

66

2.5.2 Post-processing Matlab® Code

The main goal of the post-processing code is to determine the efficiency of the

WEC given a text file containing the generator data time series and the comma-

separated value (.csv) file from the OSSI Wavelogger containing the corresponding

wave time series. The names of these files are input as strings, along with the water

depth in the test location, as the arguments of the primary post-processing function,

defined in efficiency_spectrum.m. The files are imported into the Matlab®

environment as matrices and separated into individual array variables. The data from

the Wavelogger must be specially reformatted; this operation is completed by calling

the convertOSS_WEC() function. The function also plots the water level time series

in centimeters and outputs a seven-column matrix containing the date, time, and water

level in centimeters. The compiled time arrays generated by the WEC and

Wavelogger data files are compared to determine if they started recording data

simultaneously. This implies that the RTCs on each instrument must be perfectly

synced prior to deployment so that the time series data may be compared. Then, if

one instrument was powered on before the other, the disparity between the records’

start times is displayed to indicate that the user should edit the input files.

The routine then converts the ADC values into voltage and current values using

the calibration coefficients hard-coded into the script. From these values the

instantaneous power into and out of the DC/DC converter, the DC/DC converter’s

efficiency, the percentage of the time that the converter is being used, and the total

accumulated charge are calculated. The wave height is analyzed in the frequency

domain to identify peak frequencies and total energy into the system (via the area

under the wave power spectrum). The power out of the system is computed as the

average power emitted from the DC/DC converter over the course of the sampling

period. The efficiency of the system is therefore the average power generated divided

by the average incident wave power. The frequency domain analysis enables

determination of whether the WEC’s target frequency range dominates the excitation

forces. If needed, a subset of the total data record may be selected for a comparison to

be made between the efficiencies of the separate buoys under the same conditions.

67

3. Construction

All components of the WEC were machined at the Florida Tech Machine

shop or assembled in the Underwater Technologies Laboratory on the Florida Tech

campus. For fine-pitched soldering, the help of Tony Cimaglia and the ORCA

microscope were requested. This section details manufacturing procedures so that

future work on the project may be consistent or include improvements.

Coil

The coil dimensions and bank-winding geometry were designed according to

the equations for inductance given in section 2.4.1. Since the magnet would

eventually need to move freely through the inside axis of the coil in as close proximity

to the wire as possible, the wound wire needed to be epoxied and the former removed.

Initially a lathe was proposed to feed the wire at a controlled speed onto a former, but

students attempting this method discovered it was very apt to break the wire or cause

the shielding to split and short the coil. The coil-winding machine located at the

Applied Magnetics Laboratory in Melbourne may eventually be a good solution, but

because it does not have an operator or a manual, the hand-winding method was used

to quickly create a preliminary test coil. This test coil yielded sufficient energy so that

the method was used to quickly, albeit rudimentarily, wind the final coil used in

testing.

The former was constructed from a 1.625” outer-diameter PVC pipe covered

with Lithium grease and wrapped in wax paper to create lubrication for eventual

removal from within the coil. The wrapped pipe was inserted into two wide cardboard

68

rings, positioned apart from each other by the desired length of the coil. These served

as endstops to facilitate layering of the bank-wound wire, as shown in Figure 18. A

sturdier shielded 12-inch segment of

wire was soldered and shrink-

wrapped to the end of the magnet

wire before winding commenced to

give the coil’s electrical connections

reinforcement against wear and tear

(visible as the shorter, yellow wire in

the prototype shown in Figure 18).

The first 20 windings were wrapped

on the first layer to establish a base

that wouldn’t slide as additional

layers were wound into the

“crevices” between the wires on the layer below. After 2 layers had been wound

around a short segment of the pipe, a layer of West System epoxy (Fast Hardener) was

brushed on to begin giving the wires a medium in which to set. Although true bank-

winding, with clean layering of wires in the crevices of the supporting layer’s wires,

was the ideal goal, the hand-winding method degenerated quickly to simply striving to

keep tight windings as close to the first end of the pipe segment as possible and

working slowly toward the other end, applying epoxy along the way. The number of

windings was counted as winding progressed in order to match the design variable N.

One important feature was maintaining flat, 90° ends of the coil in order to keep

mating with the other generator pieces as smooth as possible.

After the epoxy had cured for 24 hours, the PVC pipe was slipped out of the

center of the coil, leaving the freely standing coil to mate with its supporting pieces. A

connector housing was crimped onto the ends of the free leads to provide for clean and

reliable attachment to the PCB.

Figure 18: Winding the generator coil

69

PCB

The schematic in Figure 15 was drawn in CadSoft Computer, Inc.’s EAGLE

Layout Editor. The traces were routed using the software’s Autorouter capabilities,

using a vertical default orientation for component-side traces and horizontal default

orientation for the bottom side. A ground plane was also added to the bottom side to

reduce noise. The excellon- and gerb274x- CAM files were generated to the

specifications of Advanced Circuits’ printed circuit board manufacturing, and the files

were checked for design rules compatibility using the FreeDFM online service. After

manufacturing, the components were soldered to the board in-house. The RS232 port

was then connected to a USB port on a computer via a serial/USB converter cable,

and the microcontroller was programmed using the PICkit2. The RTC time was

initialized, and then its separate battery attached to preserve the time setting. Plastic

adhesive-based stand-offs were screwed to holes in the PCB to allow for mounting

inside the Pelican case that houses the electronics.

LIG box

The electronics and coil are housed within a Pelican Case 1300 watertight and

corrosion-proof box, which was modified with a hole in the top and bottom to allow

the magnet to travel vertically through the coil. The challenge was designing a

completely sealed path for the magnet to travel while maintaining close proximity to

the magnet wire of the coil. A pair of o-ring-sealed bulkheads was fashioned from a

Type II PVC cylinder using a lathe in order to both support the coil and pressure-seal

the electronics from the weather elements to which the magnet is exposed (Figure 19).

The threading on the bulkhead endcap and its corresponding nut was accomplished

using a CNC mill. Closed-cell foam gasket material seals the junctures between the

bulkheads and the tubes supporting the coil and between these tubes and the coil. This

creates a smooth, 1.625-inch diameter tunnel through which the 1.5-inch diameter

magnet can freely travel. The magnet has ten inches of travel before the frame

carrying the Pelican case limits its motion.

The assembly design makes it possible for the coil to be replaced with a

maximum of 2 other fittings needing to be re-manufactured: the two disks interfacing

70

the coil with the white polyethylene tubes. As the tubes are 2.365 inches in outer

diameter, a new interfacing disk would require a 1.625-inch hole through the middle, a

2.365-inch indentation on one face, and another indentation of the new coil’s outer

diameter on the other face. When these are all assembled, the pressure from the

latched Pelican Case and the aluminum frame holding it ensure sealing of the

bulkhead junctures.

(a)

(b)

(c)

(d)

Figure 19: Linear Inductance Generator box with (a) bulkhead seal design drawings, (b)

bulkhead inside box, (c) circuit board and coil inside box, and (d) closed box displaying

through-hole magnet path.

71

Frame

The square-channel aluminum for the main frame was cut to dimension using

a mitre saw and then welded together with the solid rectangular bars of aluminum that

comprise the crossbars. The 1.5-inch holes in the crossbars were milled out using a

CNC machine. Inch-long segments of polyethylene with a 1.1-inch inner diameter

were press-fitted into the holes to serve as low-friction bearings, keeping the 1-inch

diameter, buoy-driven aluminum rod oscillating vertically. The cage stabilizing the

LIG box allows the Pelican Case to be slipped into place from above and a crossbar to

be screwed across the top to apply enough pressure for the coil to stay in alignment.

Neoprene rubber pads on the inner faces of the cage also help in maintaining

alignment. Slits for the stainless steel banding were milled into the 6-inch-wide C-

channel for attaching to the piling, and the 22-inch section was bolted to the backplate

of the frame.

(a)

(b)

(c)

(d)

Figure 20: Aluminum frame, C-channel strapped to piling, and magnet on its support rod

72

The translator rod holding the magnet consists of two segments of 1-inch diameter

solid aluminum rod, each threaded at one end axially with a ¼”-20 tap. A threaded

rod passes through the hole in the ring magnet and into the tapped holes. At the top

and bottom of this rod, through the cross-sections, are tapped a ¼”-20 hole so that

polyurethane rubber rectangles can be screwed in tangentially to serve as vibration

relief while coupling the translator to the buoy rod via the horizontal braces. The top

horizontal brace was welded to the rod driven by the buoy, but for convenience in

assembly and dismantling, the bottom support was screwed into the rod.

Buoy

The buoys were carved from polystyrene foam blocks using a CNC mill in the

Florida Tech Machine Shop. The G-code files were generated via MasterCam

software from the .igs files saved from Autodesk Inventor drawings. Each buoy was

manufactured in two halves to allow for an ABS plastic tube to be screwed into place

in the center. This 1.5-inch OD tube, which stretches the entire length of the buoy,

serves to both guard the foam from damage as the aluminum rod is inserted and to

secure the buoy at the intended angle of incidence. A pin inserted through a hole in

the rod and a hole in the aluminum dowel aligns the buoy. In future versions, the

buoy may be left free to rotate so that the entire apparatus would not need to be shifted

as wave direction shifts, but for this prototype, the angle variable needed to be

eliminated.

The foam was then covered with 6-oz. E-glass fabric and DOW Chemical

Company’s 331 Epoxy Resin. Epoxy needed to be used for the first coat, a least,

because polyester resin melts polystyrene foam. Due to the simplicity of the

geometries, a careful handworking layup process was sufficient for obtaining a suitably

smooth surface, without the need for vacuum bagging or creating a female mold.

Sanding of the final forms was necessary before applying the yellow polyester gel coat,

and a clear coat was also applied on the outer surface.

In order to add receptacles to hold weights, four holes were drilled into the top

of each buoy using a Dremel Tool to cleanly pierce the fiberglass and an extended-

73

length drill bit to reach 14 inches into the foam. PVC pipes (2x 1.5” diameter and 2x

2” diameter), capped on one end and threaded on the other were inserted into the

holes, and expanding watertight glue was used to secure them in place. A threaded

endcap is screwed into place after the weights are inserted to seal the receptacles.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 21: Buoy CNCing (a-b), fiberglassing (c-e), gelcoating (f), weight receptacle

insertion (g), and finished product (h-i).

74

4. Testing Procedures

4.1. Calibration

4.1.1. ADC analog input block calibration

This section lists the steps by which the post-processing coefficients for each of

the four ADCs were determined. As mentioned in the electronics segment above, a 0-

4.096V, 12-bit ADC ideally assigns an additional count for every 4.096V/4096 ≈

1.0mV jump in voltage, but true operation can only be ascertained in the final circuit.

This is especially important for the current-sense measurements because the ADC

voltage readings are indirect measurements of the current signals and depend on the

operation characteristics of the intermediate amplifiers. The ADC values were viewed

via HyperTerminal, and the corresponding voltages and currents driving the ADC

values were recorded to yield the coefficients displayed in Appendix F.

First, to calibrate the coil/supercapactor voltage ADC, the voltage divider was

kept intact, with 1MΩ as the upper load and 2.2MΩ as the lower load, but the

DC/DC converter surfboard was removed from the circuit. This isolated the testing

from the effects of the relay switching on when the coil achieved the threshold voltage.

A perfect follower transistor circuit (eg. containing a LF411 op amp, a TIP102

Darlington transistor, and a potentiometer) was used to apply a controlled voltage to

the supercapacitor terminals. Intervals of 0.5V between 0 and 5.6V were found to be

appropriate. The voltage was allowed to settle before each ADC digital value and the

corresponding voltage on the coil were recorded.

75

The load voltage ADC was calibrated in the same method as the coil, except

that different components were removed from the board during the process. The

controlled voltage (0 to 5.0V in steps of 0.5V), was applied directly across the voltage

divider, so that the DC/DC converter remained removed, thus avoiding forcing

voltage backwards across it. For safety, the supercapacitor was also removed. Rather

than the typical 2kΩ-magnitude loads used during operation, higher 1MΩ or 680Ω

resistors were inserted into the load voltage divider to so that the perfect follower

circuit was able to source sufficient current at the high end of its voltage range.

The current-sense ADCs’ calibrations were slightly more involved. The load

current calibration requires that the DC/DC converter remain uninstalled and a 10Ω

resistor be inserted as the sense resistance. Depending on the range and accuracy

being tested, a pair of identical resistors between 560Ω and 1kΩ was loaded into the

load slots. An ammeter was installed in series between the power supply and the high

side of the current sense amplifier. The power supply unit was set to 5.0V with the

output current-limited, starting at 0mV. The current over the 560Ω resistor pair was

varied between 0 and 4.1mA, resulting in 41mV across the 10Ω sense resistor. The

100x gain op amp thus supplied up to 4.1V to the ADC, which saturates at 4.096V.

The calibration coefficient was determined by recording the current measured along

with the corresponding ADC value.

Using higher load resistance values limits the current to below this ADC-

determined saturation current of 4.096mV, since the 5.0V setting actively determines

the power output from the supply at times when the impedance is so high that it draws

less than the set current limit. It was determined that testing the current-sense

components with a few varying load resistance values was necessary to ensure that the

high-side resistor setup behaves consistently, especially at low current values. This

precaution was taken because current-limiting the power supply to such low

amperages was observed to prevent the preset 5.0V from being supplied, as well. This

means that, while the ammeter-measured current may be known, the “high side” in

the load resistor series could be below its intended operational voltage. When it is

very low, perhaps within 1.3mV of ground, the op amp is unable to amplify the signal

dependably. The MAX4372 op amp has a 0.3mV typical input offset voltage, and

76

although its full-scale common-mode input range is listed at 0-28V, this means that the

difference between the input pins can be as low as zero, but the voltage value at either

input pin may be read as 0.3mV when the actual voltage level lies in the range of 0-

0.3mV.

In order to calibrate the ADC measuring the current entering the DC/DC

converter, the DC/DC converter was removed, and a 1.2k to 1.8k resistance value was

inserted between pins 4 and 5 of the DC/DC converter header. This allowed the input

voltage to drain to ground without the uncertainty of the DC/DC converter’s

adaptable input impedance. An ammeter was set in series between the current-limited

power supply and the 10Ω sense resistor’s high side. Again, the current was varied

between 0 and 4.1mA (or, again, the maximum current dictated by the resistance)

using the power supply’s control knobs. After the ADC values and current levels were

recorded, the circuit was tested with the DC/DC converter in place to verify that its

input impedance never dropped below the lowest resistance tested. If so, the ADC

values would have climbed out of the bounds of the calibration data, and testing of a

lower resistance value would have been required to accommodate the higher current

draw.

4.1.2. Frequency testing

Due to the emphasis on frequency targeting in the design of the WEC, it was

very important to test the functioning of the PCB module under a variety of controlled

frequencies to understand its performance before deployment. The LC circuit was

designed for resonance over a bandwidth of 0.67 to 2 Hz, so operation over this

frequency band needed to be verified. Of equal importance, the maximum current

load sustainable at each frequency needed to be determined in order to prevent

saturation of the DC/DC converter output during deployment. With the existing

rectification circuit, saturation during eventual battery-charging applications would

simply mean that available energy would be drained or not collected, but the battery

would constantly receive energy. However, during the absorber performance

comparison, it would mean the inability to distinguish which absorber actually

77

captured more energy. To avoid saturation, the load resistance must simply be

reduced. This issue was discovered when initial circuit testing with simple hand-

controlled excitation saturated the capacitor with various high-resistance loads.

During frequency testing, the original objective was to determine the load resistance

value that would give a 50% ratio of on-to-off time during the record. This would

allow the appropriate resistance value to be installed during deployment, based on

measurement of the current wave frequency and magnitude. In the end, the trials

allowed for targeting of the load resistance’s magnitude, but due to the conclusions

derived in the process, fine-tuning of the resistance was not pursued.

In order to characterize the load capabilities of the LC circuit under controlled

frequency conditions, the PCB was tested with the oscillation table in the Florida Tech

MAE department. The oscillation velocity is controlled using the frequency and

voltage amplitude settings on a signal generator. The coil was secured to the table,

and the magnet, attached to its aluminum rod, was affixed to the moving plate on the

oscillating platform, as shown in

Figure 22.

Three main observations

resulted from the trials.

• The ability of the PCB’s

capacitor to charge is heavily

affected by the specific

combination of frequency and

stroke length, rather than

either alone. However,

because rather extreme

conditions (i.e. high energy,

long period waves) in the lagoon would be needed to reach the low frequency

threshold of this coil, wave heights will be the more limiting factor. Higher

frequencies do create higher magnetic flux derivatives, allowing the capacitor to

charge faster, but high frequency oscillations of low magnitude cannot provide a

sufficient voltage spike to charge the capacitor to 5V. Thus the DC/DC converter

Figure 22: Frequency calibration test setup

78

never turns on for low-magnitude oscillations. The detriment of a lower frequency

is pronounced for a given wave height, but the WEC is much more likely to

experience short-period waves with too little height than tall waves with too low a

frequency.

• A mild drift in the table’s position over time affected the capacitor’s charging

profile enough to yield appreciable differences between two trials of the same input

settings. As the oscillation’s zero-position approached the end of the coil, the

charging time for the capacitor decreased. Quantifiable patterns were not derived

from these observations, but variation by up to 65% between trials did demonstrate

that the circuitry component values, excitation frequency, and excitation

magnitude alone cannot predict the circuit’s performance. Median position also

plays a governing role in the capacitor’s ability to charge. For this reason it is

important to ensure that both buoys begin their trials at the same equilibrium

position with respect to the coil.

• Due to the high variability of the circuit’s performance under similar sets of inputs,

the goal of a 50% on-to-off ratio for determining load resistance was traded for

determining a load that is capable of causing saturation under some conditions and

of discharging under other, only slightly modified, conditions. This selection of

time in operation should contribute to more reliable detection of differences in

power take-off abilities than using current measurements alone. A load value of

10kΩ was selected.

A table of results of the oscillation table testing is given below. The input test

conditions, the average power, and the percentage of time that the DC/DC converter

was outputting power are included. An example comparing two capacitor charging

profiles is plotted with the output power in Figure 23. The test conditions that

generated the plots are a 1Hz oscillation with a 7.70cm stroke length and a 1.5Hz,

5.13cm stroke, both draining into a 10kΩ load. For the combinations shown, the

higher frequency induced a higher percentage of power generation time, but as can be

seen in Table 5, some pairs of trials exhibited greater dependence on a longer stroke

length than on frequency. An overall efficiency based on the force needed to

79

accelerate the magnet, the magnet’s velocity through the coil, and the output power

indicated that a very low efficiency near 0.26% was exhibited by the generator system.

Improvements in generator efficiency are discussed in Chapter 6 below.

Table 5: Oscillation table calibration trial results

Frequency (Hz) 0.75 0.75 0.85 0.85 1 1 1.5

Amplitude (cm) 8.98 10.26 7.92 9.06 6.74 7.70 5.13

Load resistance = 10kΩ

Record time (s) 124.89 162.05 103.36 121.45 61.65 234.65 138.43

Avg. power (mW) 0.059 1.509 0.087 1.518 0.080 0.095 0.237

“On” time ratio 5.31% 99.92% 7.48% 99.90% 6.85% 8.57% 19.88%

Load resistance = 1kΩ

Record time (s) 110.62 102.99 72.72 --- 57.22 61.77 93.52

Avg. power (mW) 0.546 1.52 0.877 --- 0.77 01.194 1.623

“On” time ratio 3.77% 11.20% 6.07% --- 5.35% 8.73% 11.94%

0 20 40 60 80 100 120-1

0

1

2

3

4

5

time (s)

DC/D

C converter output

Power out (mW)

Coil Voltage (V)

0 20 40 60 80 100 120

-1

0

1

2

3

4

5

time (s)

DC/D

C converter output

Power out (mW)

Coil Voltage (V)

Figure 23: Capacitor charging profiles and DC/DC converter power output plots showing

the difference in performance between 1Hz, 7.7cm oscillations (left) and 1.5Hz, 5.1cm

oscillations (right).

80

4.2. Deployment

The primary field testing of the wave energy converter was conducted on a

piling near the eastern coast of the Indian River Lagoon in Brevard County,

approximately four miles north of Sebastian Inlet. This location was chosen for the

convenience of an accessible dock nearby. Because of the eastern shoreline location,

the low wind climate of the summer, and the need to obtain wave heights as large as

the design wave of the hyperbolic curve, it was necessary to select days for testing

during which the wind originated from an at least partially western direction. Data

was collected on 3, 12, 13, and 26 August 2008.

During setup, the C-channel was strapped to

the piling using two 152-sized hose clamps threaded

through slits on the legs of the C-channel. Originally

¾” stainless steel Pelco banding and buckles were

used, but the need to shift the height of the apparatus

on the piling with tides and with the differing floating

levels of the two buoys proved the hose clamps to be

more convenient with the current hardware. The frame

was bolted using four ½”-13 bolts through the

backplate, which contains threaded holes. The frame

was oriented toward the direction of the approaching

waves. By standing on a ladder, it was possible for a

single person assemble the remainder of the apparatus.

The magnet’s translator rod and the rod onto which

the buoy is inserted were coated with lithium grease,

threaded through the polyethylene bearings and in turn

coupled to each other with the bottom horizontal

brace, as shown in Figure 25. The magnet was also

screwed into place on the rod, but the top rubber

stoppers and screws affixing the buoy direction could not be tightened into place until

the electronics box was inserted. The long buoy rod was then inserted into the tube at

Figure 24: Deployment site

Deployment site

Sebastian Inlet

81

the buoy’s center and secured with a plate threaded into the bottom of the rod. At this

point it was possible to verify that the buoy was floating at the correct height, such that

its movement was not limited more often by its top extent than its bottom extent, or

vice versa.

Figure 25: Frame assembly and deployment

The WEC circuitry was preprogrammed with the RTC time synced to the

wave gage’s RTC and preserved with the 3.6V battery attached. The PCB was

screwed into its place in the box on land, and immediately preceding deployment, the

main power battery was attached. After the box was set in its place in the frame and

the final rubber stoppers and screws installed, the official start time was recorded such

that premature data points could be ignored in the data record.

An Ocean Sensor Systems Wave Logger II with a 0.5m staff was deployed on

the dock adjacent to the WEC’s piling. Although measuring in the immediate

proximity is preferable to the dock’s approximately five-meter distance, measuring

from the same piling would have made obtaining data unaffected by the WEC’s

movements impossible. Data was gathered in 20-minute-long files, and the wave

direction noted. After 2-3 files had been collected for one buoy, the WEC was

disassembled only to the point where exchanging the buoys was possible, and a new

set of WEC and wave gage files was collected for the second buoy.

82

5. Results and Conclusions

5.1. Practical Observations

A few design modifications became necessary during final planning and the

early stages of deployment. Initially, the buoys were intended to float completely

submerged, with the top faces level with the water surface. After the final weight of

the buoys and hardware was recorded, it became evident that their own weights would

not nearly offset the buoyancy provided by displacing the full volumes, and added

weight would be needed to submerge them to the desired depth. The hyperbolic curve

buoy was indeed deployed with enough weight added (36lbs) via fishing sinkers to

nearly submerge it properly, but the added inertia prevented almost all movement of

the buoy. For this reason the final deployment conditions were modified such that the

buoys both weighed the same amount (and therefore displaced the same volume of

water) rather than floated at the same level. For this to occur, 4.25lbs of weight were

added to the hyperbolic curve, bringing the finished weight of each buoy to 13lbs.

(a)

(b)

(c)

Figure 26: Floating levels of the buoys. (a) Curved buoy with 21 lbs

added. (b) Curved buoy with 4.25 lbs added to bring it to the cylindrical

buoy's weight. (c) Cylindrical buoy without weight added

83

Not sinking the cylindrical buoy does not affect its expected performance, as it was

designed to simply harvest the potential component of the wave energy. By riding

atop the wave surface, the cylinder experiences strictly potential energy excitation.

This modified geometry, however, does compromise the theory behind the hyperbolic

curve’s performance because it cannot float at the level for which its curve is intended

to match the water particle movements. Nevertheless, it is unquestionable that the

hydrodynamic effect of the added inertia impedes the motion of the curved buoy, so a

modified experiment was necessary to create a closer contest. Clearly, adding the even

higher weight needed to sink the cylinder would impede its motion to a greater degree,

rendering this WEC ineffective for such a comparison.

The conclusion gained from this segment is that other hydrodynamic

properties (including body mass, added mass, and the change in buoyant force with

respect to time) overwhelm kinematic effects for bodies floating in heave. Some

suggestions for changing the buoy shapes or adding a mechanical spring force in order

to revisit the issue of inertia versus proper submersion are given in Chapter 6 below.

Nevertheless, other valuable and more quantitative conclusions were possible from

testing the adapted setup, and these are discussed next.

84

5.2. Testing Conditions

Enumerating the testing conditions must include both circuitry component

values and the wave climate under which the WEC was deployed. As shown in Table

6, the initial capacitance values were changed for subsequent deployments. This is

because the capacitor charge and discharge rates under hand-driven excitation were so

steep during the first deployment that a more gradual charging profile was desired for

distinguishing between the two buoys’ performances during more favorable wave

climates (see Figure 28). Because the 0.47F supercapacitor was unable to charge to

5.0V during oscillation table trials, a median value of 22000µF was used for the

remainder of the deployments. The values that remained constant for all trials were

the inductance (measured as being between 0.1H and 0.25H), and the load resistance

of 10kΩ. The approximate water depth was 1.2m in the immediate vicinity of the

piling, except for during the last deployment on 26 August 2008, when storm water

runoff increased the depth to 1.7m.

Table 6: Deployment log

Date Wind Direction

Avg. Wind speed4 (m/s)

C (µF)

Tp (s)

Hs (cm)

Hmo (cm)

Avg. Wave power (W/m)

3.08.08 190˚ 2.41 4700 1.15 0.66 0.57 0.037 12.08.08 215˚ 2.91 22000 1.34 9.44 7.10 6.794 13.08.08 240˚ 4.96 22000 1.66 15.24 11.71 24.668 26.08.08 220˚ 4.25 22000 1.55 8.73 8.48 11.625

From the wave records gathered by the Ocean Sensor Systems Wave Logger, a

time series analysis and a spectral analysis were conducted in Matlab® to characterize

the input wave conditions. First, the time series of water level data was de-trended to

remove dependence on tides from the spectral analysis. Then a zero-up-crossing

routine was conducted on the data to determine the most numerous periods and wave

4 Recorded from the National Weather Service log of daily wind from the Melbourne International Airport [28]. Given the variability of wind speed over the course of the day and across the sixteen miles that separate the deployment site from the airport, these wind values can only characterize the general wind conditions from the day

85

heights during the course of the deployment. The histograms for data collected on 13

August 2008 are displayed in Figure 27.

13:55:12 14:09:36 14:24:00 14:38:24 14:52:480

10

20

30

40

time (hh:mm:ss)

Eta (cm)

0 500 1000 1500 2000 2500 3000-20

-10

0

10

20

30

time (s)

Eta (cm)

0 0.5 1 1.5 2 2.5 30

100

200

300

400

500

T (s) 0 0.1 0.2 0.3 0.4

0

50

100

150

200

250

300

350

H (m) 0 2 4 60

0.5

1

1.5

2x 10

-5

frequencies (Hz)

Sxx

Figure 27: Incident wave analysis from 13 August 2008. Raw and de-trended time series

(top) wave period histogram (left), wave height histogram (center), and wave power

spectrum (right).

Encouragingly, the histogram of wave heights shows a similar shape to the

narrow-banded Rayleigh distribution, indicating that in the IRL, targeting the WEC’s

absorber and circuitry resonance frequencies to the concurrent wave climate should

improve energy conversion. The narrow-banded spectrum observation is supported by

the steep peak in the power spectrum. A bimodal spectrum, for instance, would

require targeting both “predominate” wave properties, and a broad-banded spectrum

would similarly indicate that a large range of LC values would need to be active

simultaneously to take advantage of most of the available energy. This is possible with

switched capacitance control, but the task should be simpler to tune when vastly

divergent values are not required. The significant wave height, Hs, or the average

86

height of the tallest third of the waves, was determined using a cumulative count from

this histogram. The variance of the wave displacement was used to calculate the

significant wave height, Hmo, spectrally. Slight differences yielded by the two methods

for determining significant wave height were logged. The peak wave period was

determined from the frequency at which the power spectrum peaked.

The measurements allowed for the incident wave power to be calculated as

being between 37mW and over 24W per meter of wave front during various days of

testing. Approximately 6W was measured repeatedly. While the low end of this

range is much too weak to support the ORCA sensor suite, all but one day of data

collection registered energy levels that should be able to support the sensor if the WEC

were eventually optimized to achieve 5% efficiency. Without disregarding the

admittedly selective choice of deployment dates and times according to wave energy

potential, this does verify the meteorological hindcasting model’s assertion that

reasonable efficiency can be expected to support the sensor even during the annual

summer lull in wave climate.

Another important observation from the extended wave records during each

deployment is that the significant wave height in the IRL is very sensitive to shifting

wind speeds and tides. This is evident in the plot of the raw water level data in Figure

27. For this reason it is important to take the short-term incident wave climate into

careful consideration when determining the comparative performance between the

cylindrical and hyperbolic curve buoys.

87

5.3. WEC Performance Results

In order to arrive at an unbiased comparison between the two buoys’

performances, the 20-minute wave record associated with each set of WEC data was

divided into subsets, and a data windowing routine was conducted on each to

determine the shorter-term wave properties through spectral analysis. This method

contributes to both a more accurate overall wave climate calculation and the ability to

select segments of data during which both buoys were truly subjected to the same

significant wave heights. Table 7 and Figure 28 show the results of this comparison.

Table 7: Comparative performance characteristics for the cylindrical and hyperbolic curve

buoys

Cylinder Buoy Hyperbolic Curve Buoy

Hmo = 8.49cm

Tp = 1.50s

Hmo = 8.47cm

Tp = 1.66s

Average capture width incident power (mW)

4707 5333

Average power output (mW) 0.1251 0.0446

WEC efficiency (%) 0.00266% 0.00084%

Percent of time operating 8.30% 3.11%

mA-h generated per hour of deployment 0.053 0.029

Hmo = 16.44cm

Tp = 1.495s

Hmo = 13.62cm

Tp = 1.268s

Average capture width incident power (mW)

18543 10234

Average power output (mW) 0.140 0.0636

WEC efficiency (%) 0.00076% 0.00062%

Percent of time operating 9.87% 4.50%

mA-h generated per hour of deployment

0.412 0.189

88

0 200 400 600 800 1000 1200-15

-10

-5

0

5

10

15

20

25

time (s)

Eta (cm)

Power out (mW)

Coil Voltage (V)

0 200 400 600 800 1000 1200-5

0

5

10

15

20

time (s)

Eta (cm)Power out (mW)Coil Voltage (V)

Figure 28: Capacitor voltage and DC/DC converter power output backdropped by the wave

displacement time series acting on it during the deployment. Results from the clinder buoy

are shown in the top, while the comparatively slower charging profile of the curved buoy is

shown on the bottom.

89

The first comparison relates nearly identical excitation conditions, although

the curve buoy experienced slightly more incident power. The second comparison

shows to less similar excitation conditions, this time with the cylindrical buoy

experiencing more incident energy. Clearly the higher efficiency, longer active

operation time, and denser charging profile over the same timescale indicate that the

cylindrical buoy was able to absorb energy from the waves more effectively than the

hyperbolic curve in both the trials. Although these results were not anticipated by

theory, they were not surprising given the observed motions of the respective buoys.

Direct measurements of oscillation magnitude were not collected, but the cylindrical

buoy visibly oscillated with greater frequency, if not amplitude, under the same wave

excitation conditions. For objective comparison, a buoy position sensor should be

implemented.

These results beg for an explanation for the specially-designed buoy’s inferior

performance. As mentioned above, the inertial term governs the buoy’s acceleration.

A high body mass will prevent substantial acceleration given a certain combination of

external forces. This issue was skirted by setting the buoys’ weights equal, leaving

added mass and buoyant force considerations to govern the effect of the excitation

force from Equation 25 on the buoy’s acceleration. Nevertheless, the inertia of

individual water particles is low, and a large solid mass of any shape cannot move

with the dynamics of a water particle. Modifying the composite density of the heaving

body affected the buoyant force behavior; the m in the zm && term could not be modified

without affecting the forces on the right-hand-side of the equation. Without a constant

buoyant force, the natural frequency becomes very dependent on the changing depth

of submersion of the buoy. For the cylindrical buoy, its large horizontal cross-

sectional area at every z-coordinate means that any small change in water surface level

it encounters before beginning its heave motion immediately submerges a large

volume of the buoy. The buoyant spring force acting on the cylinder is therefore

proportionally greater than that acting on the curve by the difference between

submerged volume, given a certain rise in water level. A comparison of kinematic

contribution to wave absorption would therefore need both buoys to have equal

horizontal cross-sectional areas at every z-coordinate.

90

Additional suggestions for resolving the hydrodynamic issues affecting the

heaving buoy are presented in the next section, but the overall performance of the

WEC should also be mentioned here. The overall efficiency did not nearly approach

the 5% target set by the wave climate and the sensor suite’s power requirements, but

the ability to obtain a measurable power output from the simple, rugged apparatus

gives practical evidence that the waves in the IRL offer a promising source of power

for small-scale electronics. A nearly 7000% increase in efficiency is needed to support

the ORCA sensor suite during the summer months. The first component upon which

improvements should focus is the power take-off section. While further developments

to advance the performance can also be carried out, some revisions to the existing

setup should be completed first.

It was recognized during the data analysis phase that the intended generator

resonance and maximum power transfer impedance matching did not in fact occur in

the current hardware configuration. Because the rectifier prevented discharge of the

capacitor back to the inductor, current flows toward the capacitor only, and only when

the voltage induced by the magnet in the coil is greater than the capacitor’s voltage by

at least two diode drops. At other times, when the wave excitation forces are too low

or the capacitor voltage is too high, the generator terminals form an open circuit, and

any energy generated by the coil is dissipated by its internal resistance. Due to the

small self-capacitance of the coil, the voltage is likely to spike quickly, so voltage

spikes sufficient to overcome the diode drop may be quite common even for small

waves. However, their durations are short-lived as the current flows into a large

capacitance, and without a significant imaginary component to the generator’s

impedance, no power can be preserved as reactive power. Adding the supercapacitor

in parallel with the coil upstream of the rectifier would accomplish this purpose.

Tuning the LC circuit to be allow for total system resonance with the wave excitation

force, as in 30, would allow for optimum operation of the WEC. In this case, a

second, much smaller capacitor would need to replace the downstream capacitor to

power the load circuitry. This smaller capacitor would only need to be large enough

to sustain the DC/DC converter and its load over the course of half a wave period,

until the next spike in voltage occurs.

91

The open-circuit impedance of the coil, or source impedance, is, as stated

above, 175Ω. When the coil voltage spikes sufficiently, the capacitor and remaining

downstream circuitry become the load. Due to the downstream capacitor, the value of

this load depends on the frequency of the coil signal, which in the current

configuration is solely dependent on the magnet’s traveling frequency. Consider the

22000µF capacitor used for testing and the design wave frequency of 5.236rad/s.

Assuming that the buoy were to travel at the wave frequency, the value of the load

impedance would be 8.66Ω whenever the DC/DC converter is switched into the

circuit as coil voltage spikes. (This calculation assumes the measured 40% DC/DC

converter average efficiency, an average capacitor voltage of 4.25V, and a load resistor

of 10kΩ.) If the DC/DC converter is switched out of the circuit during the spike, the

capacitor-dominated load impedance is negligibly higher at 8.68Ω. In both cases the

load is phase-shifted by -90˚. The implications of these impedance levels are that most

of the energy is dissipated within the coil because the load impedance consisting of the

supercapacitor and DC/DC converter circuitry is so much lower than the source

impedance, consisting of the coil. Maximum power transfer for any electrical system

occurs when the source and load have impedances that are complex conjugates of

each other. Hence the need to return a portion of the reactive energy to the

mechanical system in order to compensate for reactive impedance from added mass

and buoyancy creates a trade-off between total system resonance and circuitry power

transfer. In combination with the very important issue of creating a truly resonant LC

circuit, this optimization should be considered in future versions. If the mechanical

components can be designed such that the system resonance coincides with the

excitation force, designing the LC circuit to have no resultant reactive component at

the excitation frequency optimizes both components. Mechanical tuning, however, is

likely to be harder than electrical tuning, and therefore the optimum power transfer is

liable to be sacrificed for a control method that allows for better absorption and thus

higher total power generation. This is accomplished by adding [C(ω)]-1 to the

numerator and L(ω) to the denominator in 30 to bring the system oscillation back to

ω0. In addition, the load must contain a method of compensating for the DC/DC

92

converter being removed during periods of low capacitor voltage, such that load

impedance levels are maintained.

An improvement in efficiency of the PCB can be achieved by replacing the

relay circuitry with a MOSFET to control the DC/DC converter’s activation. The

highest DC/DC converter efficiencies are achieved when the power supplied to it is

sufficient to maintain an output current above 100mA. The efficiency of the

TPS61202 drops extremely quickly for low output currents and low input voltages.

The latter issue was solved by using the relay and microcontroller code, but in a self-

sufficient version of the WEC, low power consumption by the control circuitry will

determine whether the generated power will be sufficient to maintain the higher output

currents needed for 90% efficiency and higher. The DC/DC converter efficiencies

ranged in general between 30% and 60% during data collection.

93

6. Recommendations for Future Work

In order to facilitate future improvements, this section enumerates areas of

development that this current research indicates will be most crucial to increasing the

WEC’s efficiency enough to support the ORCA sensor suite.

Real-time tuning for wave conditions, directions, and tides.

Currently the load resistance consists of a pair of resistors inserted into the

circuit with pin headers that can only be interchanged manually. As indicated by the

frequency calibrations, the value of the load resistance determines how quickly the

capacitor discharges, and therefore, the load value would ideally shift for different

wave conditions and generator frequencies. A maximum power point tracking

algorithm, which adjusts the source and load impedances in real time by switching the

signal frequencies on resistors, capacitors, and inductors as needed according to the

power output of the DC/DC converter, is a promising avenue for tuning the circuit’s

performance.

In order to adapt to changing directions without manual rotation of the

apparatus on the piling, any directional buoys should be made self-orienting, such as

with vanes attached in the downstream direction.

Another process that must be made automatic is adjustment to tidal levels.

Because of the sensitivity of the coil to the magnet’s median position in its oscillation,

the small effect o tidal rise affects the efficiency of the system appreciably. Since

decreasing the length of a coil also increases its inductance, a favorable option seems

to be attaching multiple shorter coils axially and switching each coil into the circuit

94

when its level is appropriate for the floating buoy’s position. Another solution to the

tidal issue is to affix the entire generator and frame to a floating ring that encircles the

piling such that they all move with the tides, and thus the absorber could always float

at the same position in relation to the coil. This adaption would also prepare the

system’s design for operation as a free-floating buoy in the open ocean, where no

pilings are available for anchoring.

Deployment considerations

Changes that would vastly improve the convenience of deployment include

easy shifting of the device height on the piling and access to data and the circuitry

during WEC operation.

Currently the hose clamps must be loosened and re-tightened in order to slide

the device as a unit vertically on the piling with tidal changes. Although multiple

holes were drilled into the backboard to give the frame mobility with respect to the C-

channel, unscrewing and re-screwing the bolts is not in fact more convenient, and it

allows only for integral, rather than continuous, selection of height. Milling slits into

the backboard to serve as tracks for bolts secured in the backboard would allow the

frame to be moved easily while a longer-term solution to tidal changes is developed.

The addition of a segment of 0.95” diameter aluminum rod to the bottom of

the buoy-support rod will allow for smooth, continuous movement in larger wave

climates than is now possible. This can be done by milling a short rod to the correct

diameter and threading a ¼”-20 hole into each end axially, so that a threaded rod can

secure the extension at its top end, and the buoy’s bottom plate can still be screwed in

at its bottom to hold the buoy in place.

Modifying the electronics box to allow for accessibility to the circuit board

without any disassembly of the WEC frame during operation would prevent data loss

due to circuitry issues during deployments. Rotating the box to allow its lid to open

from the side, rather than the top, is a good first step. Orienting the circuit board such

that both batteries, as well as the SD card and programming headers, are accessible

without removing the PCB from the box would increase convenience, as well.

95

Ultimately, streaming data to an LCD display visible from the outside of the box or

even transmitting data wirelessly would greatly reduce the difficulty of deployment.

Buoy shape modifications

Since the primary focus of this research was to determine the performance

differences between a buoy that takes advantage of solely potential wave energy and

one that also takes advantage of kinetic wave energy, suggestions for modifications to

create a comparison unbiased by other hydrodynamic effects are made here. Although

thorough modeling of all hydrodynamic parameters is recommended before any

significant expense is committed to manufacturing, the primary issue is hypothesized

to be the horizontal cross-sectional area differences between the buoys. While the

total volumetric displacement for two buoys of equal weight is the same, the cylinder

displaces all this volume over only 6.6cm of submersion, while the hyperbolic curve

does so with its bottom floating 16cm under the surface. In order to make the buoyant

force acting on each buoy the same for any wave height, the cross-sectional area at

every level of the hyperbolic curve should be made equal to those on the cylinder,

meanwhile maintaining the special curve at the forward face. This requires an

interesting geometry of the back face of the curved buoy, or a cavity in the middle of

the cylinder to provide reduced inertia and buoyant force at that level. Furthermore,

the back sides of the buoy should be identical in order for diffraction effects not to

interfere with the comparisons.

The primary issue to be dealt with concerning the hydrodynamics is the

inertial force involved in sinking the buoys. Two possible solutions are adding a

mechanical spring to impart a continual submersion force on the buoy and creating

holes in the centers of the buoys to reduce the body mass. The first solution provides a

method for substituting the spring’s restoring force for the restoring force currently

served by the buoy’s weight, thereby maintaining submersion while reducing inertia.

The second solution proposes a scheme in which the buoy contains vertical holes

penetrating its entire depth. Such holes would allow water to fill the tubes at the

trough of the wave but stay at the same inertial level as the buoy rises at the crest, thus

emptying out of the tube partially. While a solid body cannot move with the dynamics

96

as the same volume of individual water particles, this solution would reduce the

movement-minimizing inertial force on the buoy, with minor effects of increased

viscous drag and an altered added mass.

Another naval architecture-related improvement is the distribution of added

weights in the buoys, particularly in the hyperbolic curve. Space considerations

dictated that the weight receptacles be installed at the top of the buoys and behind the

zy plane separating the curve from the volume that encloses the rod. The end result,

however, was a natural moment about the y-axis of the hyperbolic curve buoy, since

the center of gravity and center of buoyancy are not vertically aligned. This adds to

the frictional force acting on that supportive rod’s bearings, thus further impeding

motion.

Again, detailed hydrodynamic modeling should be carried out prior to

modified field experimentation. In addition, a useful test for determining whether

wave mechanics-based design is worth further investigation is wave tank testing with

the design wave. While the IRL served as a platform for testing in true deployment

conditions, the wave mechanics of waves other than the design wave are not expected

to contribute to improved operation. Isolating the test to the design wave would

provide an important evaluation of the kinetic energy-absorbing ability of the

hyperbolic curve.

For improved absorption analysis, a buoy position sensor should also be

incorporated such that the movement of the buoy can be isolated from the operation of

the electronics, allowing each component’s effect on the other to be determined and

optimized for cooperative performance.

Power take-off issues

In addition to the points made in the previous section’s circuitry analysis, a

few improvements could increase the actual conversion from mechanical to electrical

energy. Two identical ring magnets oriented in the same direction were used in this

version of the WEC. By orienting these magnets in opposite directions, the change in

magnetic flux is twice as great over the course of the same stroke length. Incidentally,

97

new magnets must be purchased if the two rings are allowed to align and attach,

because N40 NdFeB magnets of this size are too strong to wrest apart afterwards

without damage. By extension, increasing the number of magnets and alternating

poles further increases the change in magnetic flux over a single stroke length.

During a redesign of the generator stage, other geometries may be considered.

Although the expense and bulk of high-flux linear generators such as Vernier hybrid

machines or transverse flux machines is not suggested until other aspects of the WEC

design are addressed, gearing the translator’s motion to a commercial rotational motor

may be a useful intermediate step for focusing analysis on the wave absorber

component. The concept of a linear generator is ultimately attractive for reducing

complexity of the mechanical coupling, but as in any engineering design, multiple

iterations of the design spiral are needed before all the specially-designed components

operate together as intended.

98

References

1. Aggidis, G.A., A. Bradshaw, M.J. French, A.P. McCabe, J.A.C. Meadowcroft,

and M.B. Widden. “PS Frog MK 5 WEC Developments & Design

Progress.” World Renewable Energy Congress (2005): 1199-1204.

2. Archimedes Wave Swing. Retrieved on 23 May 2007.

<http://www.awsocean.com/ >

3. Beyene, A., and J.H. Wilson. “Parameter Variation and Part-Load Efficiencies of

Wave Energy Conversion.” Journal of Energy Engineering, v.132, n.1

(2006): 15-26.

4. Build Your Own Coils and Transformers. New Delhi: BPB Publications, 1992.

5. Burcharth, H.F. and Hughes, S.A. “Fundamentals of Design.” Coastal Engineering

Manual, Part VI-5.1. U.S. Army Corps of Engineers: Washington, DC,

(2002)..

6. Climatic Wind Data for the United States. Asheville: NOAA National Climatic Data

Center (1998).

7. Dean, R.G., and R.A. Dalrymple. Water Wave Mechanics for Engineers and Scientists.

V.2, Hackensack: World Scientific, 1991.

8. Duckers, L.J., F.P. Lockett, B.W. Loughridge, A.M. Peatfield, M.J. West, and

P.R.S. White. “Optimisation of the Clam Wave Energy Converter.”

Renewable Energy, v.5, n.5-8 (1994): 1464-1466.

9. Duclos, G., A. Babarit, and A.H. Clément. “Optimizing the Power Take Off of a

Wave Energy Converter With Regard to the Wave Climate.”

Transactions of the ASME, v.128 (2006): 56-64.

10. Eriksson, M., J. Isberg, and M. Leijon. “Hydrodynamic modeling of a direct drive

wave energy converter.” International Journal of Engineering Science, v.43

(2005): 1377-1387.

99

11. Epstein, M.Y. “Piezoelectric Generation of Electrical Power from Surface Waves

on Bodies of Water Using Suspended Weighted Members.” United States

Patent 5578889 (1996).

12. Falnes, J. Ocean Waves and Oscillating Systems. Linear Interactions Including Wave-

Energy Extraction. UK: Cambridge University Press, 2002. ISBN

0521782112.

13. French, M.J., and R. Bracewell. “PS Frog: A Point-Absorber Wave Energy

Converter Working in a Pitch/Surge Mode.” Fifth International

Conference on Energy Options; the role of alternatives in the world energy scene

(1987): 198-200.

14. Gerber, J.S., and G.W. Taylor. “Installation of a Scaleable wave energy

Conversion System in Oahu, Hawaii.” Proceedings of the Thirteenth

International Offshore and Polar Engineering Conference (2003): 361-367.

15. Henderson, R. “Design, simulation, and testing of a novel hydraulic power take-

off system for the Pelamis wave energy converter.” Renewable Energy,

v.31 (2006): 271-283.

16. Hudson, J.A., D.C. Phillips, and N.J.M. Wilkins. “Review: Materials aspects of

wave energy converters.” Jounal of Materials Science, v.15 (1980): 1337-

1363.

17. Jones, A.T., and A. Westwood. “Recent Progress in Offshore Renewable Energy

Technology Development.” IEEE Power Engineering society General

Meeting, v.2 (2005): 2017-2022.

18. Kofoed, J.P., P. Frigaard, E. Friis-Madsen, and H.C. Sørensen. “Prototype testing

of the wave energy converter wave dragon.” Renewable Energy, v.31

(2006): 181-189.

19. Konotchick, J.A. “Linear Motion Electric Power Generator.” United States Patent

5347186 (1996).

20. Leijon, M., O. Danielsson, M. Eriksson, K. Thorburn, H. Bernhoff, J. Isberg, J.

Sundberg, I. Ivanova, E, Sjöstedt, O. Ågren, K.E. Karlsson, and A.

Wolfbrandt. “An electrical approach to wave energy conversion.”

Renewable Energy, v.31 (2006): 1309-1319.

100

21. Limpet. Retrieved on 23 May 2007.

<http://www.wavegen.co.uk/what_we_offer_limpet.htm>

22. Microchip. Retrieved 29 May 2008. <www.microchip.com>

23. Mueller, M.A., and N.J. Baker. “Direct drive electrical power take-off for offshore

marine energy converters.” Proceedings of the Institution of Mechanical

Engineers v.219, Part A3 (2005): 223-234.

24. Mueller, M.A., and N.J. Baker. “A low-speed reciprocating permanent magnet

generator for direct drive wave energy converters.” Power Electronics,

Machines, and Drives, n.487 (2002): 468-473.

25. Mulligan, C. Is there a Relationship between Larval Supply, Wind, and Barnacle

Recruitment in a Subtropical Estuary?. Master’s Thesis, Florida Institute

of Technology, 2007.

26. Munson, B.R., Young, D.F, and Okiishi, T.H. Fundamentals of Fluid Mechanics, 4th

Ed. USA: Johen Wiley & Sons, Inc, 2002.

27. Naimaster, A. Sebastian Inlet Wave Gage Historical Data. Retrieved 28 July 2007.

<http://research.fit.edu/wavedata/historical_data.html>

28. National Weather Service Observed Weather Report Archives. Retrieved 1-18 August

2008. <http://www.weather.gov/climate/index.php?wfo=mlb>

29. Ocean Power Technologies. Retrieved on 20 June 2008.

<http://www.oceanpowertechnologies.com/products.htm>

30. OWEL. Retrieved on 23 May 2007. <http://owel.co.uk/owel.htm>

31. Pelamis. Retrieved on 23 May 2007.

<http://www.oceanpd.com/Pelamis/default.html>

32. Polinder, H., M.E.C. Damen, and F. Gardner. “Linear PM Generator System for

Wave Energy Conversion in the AWS.” IEEE Transactions on Energy

Conversion, v.19, n.3 (2004): 583-589.

33. Resio, D., Bratos, S., and Thompson, E. “Meteorology and Wave Climate.” In

Coastal Engineering Manual, Part II, Chapter II-2, Engineer Manual 1110-2-

1100. Washington, DC: U.S. Army Corps of Engineers, 2002.

34. Retzler, C. “Measurements of the slow drift dynamics of a model Pelamis wave

energy converter.” Renewable Energy, v.31 (2006): 257-269.

101

35. Sabuncu, T., and Calisal, S. “Hydrodynamic Coefficients for Vertical Circular

Cylinders at Finite Depth.” Ocean Engineering, v.8 (1981): 25-63.

36. Simmons, W.J. “Use of Scrap Tires to Harness Wave Energy.” Proceedings of the

First International Conference on Ocean Energy Recovery (1989): 129-139.

37. Smith, G.H., and L. Johanning. “Mooring Research to Support Marine Energy

Development.” Sea Technology, v.47, n.4 (2006): 23-26.

38. Techet, A.H. “Derivation of Added Mass around a Sphere.”13.012 Hydrodynamics,

Fall 2005. (Massachusetts Institute of Technology: MIT

OpenCourseWare), http://ocw.mit.edu (Accessed 15 June 2008).

License: Creative Commons BY-NC-SA

39. Thorburn, K., H. Bernhoff, and M. Leijon. “Wave energy transmission system

concepts for linear generator arrays.” Ocean Engineering, v.31 (2004):

1339-1349.

40. Vining, J. “Ocean Wave Energy Conversion.” Report, University of Wisconsin –

Madison, 2005.

41. Wang, S. “Plunger-type Wavemakers: Theory and Experiment.” Journal of

Hydraulic Research, v.12, n.3 (1974): 357-388.

42. Wave Dragon. (2005) Retrieved on 23 May 2007.

<http://www.wavedragon.net/index.php?option=com_content&task=

view&id=6&Itemid=5>

43. Wolfbrandt, A. “Automated Design of a inear Generator for Wave Energy

Converters—A Simplified Model.” IEEE Transactions on Magnetics v.42,

n.7 (2006): 1812-1818.

44. Wolfe, D.E. “Capturing the Energy of Ocean Waves.” 37th Intersociety Energy

Conversion Engineering Conference (2002): 460-463.

45. Xiang, J., P.R.M. Brooking, and M.A. Mueller. “Control Requirements of Direct

Drive Wave Energy Converters.” Proceedings of IEEE TENCON ’02

(2002): 2053-2056.

102

46. Yue, D.K.P

“Lecture 13: Unsteady Motion - Added Mass.” 13.021 Marine

Hydrodynamics, Spring 2005. (Massachusetts Institute of

Technology: MIT OpenCourseWare), http://ocw.mit.edu

(Accessed 15 June 2008). License: Creative Commons BY-

NC-SA

“Lecture 22: Wave Forces on a Body.” 13.021 Marine Hydrodynamics,

Spring 2005. (Massachusetts Institute of Technology: MIT

OpenCourseWare), http://ocw.mit.edu (Accessed 15 June

2008) License: Creative Commons BY-NC-SA

47. Yueng, R.W. “Added mass and damping of a vertical cylinder in finite-depth

waters.” Applied Ocean Research, v.3, n.3 (1981)

Appendices

Appendix A: Table of WEC Companies [from 40] ...............................................105

Appendix B: Parts List ...........................................................................................114

Appendix C: Matlab® Code ..................................................................................118

C.1: IRL Wave Power Modeling .......................................................................118

C.1.1: detect_land_red.m ....................................................................118

C.1.2: fetch_calc_red.m.......................................................................123

C.1.3: wave_calc.m...................................................................................126

C.1.4: energy_calc.m..............................................................................128

C.1.5: L_calc.m..........................................................................................130

C.1.6: do_it_all.m...................................................................................130

C.1.7: energy_slope.m ...........................................................................131

C.1.8: calc_hist.m...................................................................................133

C.1.9: convertOSS.m ................................................................................134

C.1.10: process_wave_data.m .............................................................136

C.2: Wave Properties .........................................................................................138

C.2.1: newton_raphson.m.......................................................................138

C.2.2: sine_particles.m.......................................................................139

C.2.3: sine_particles_with_time.m...............................................141

C.3: Wavemaker Optimization...........................................................................143

C.3.1: stroke.m..........................................................................................143

C.3.2: find_standing.m .........................................................................144

C.3.3: paddleswath.m..............................................................................146

C.3.4: Smatrix.m .......................................................................................147

C.4: WEC Design Calculations..........................................................................149

C.4.1: buoy_def.m .....................................................................................149

C.4.2: find_best_magnet.m..................................................................150

C.4.3: gen_dims.m .....................................................................................152

C.4.4: force_moment_calc.m ...............................................................152

C.5: Postprocessing Routines.............................................................................153

C.5.1: freq_calibration.m..................................................................153

C.5.2: efficiency_spectrum.m...........................................................155

C.5.3: convertOSS_WEC.m .....................................................................161

C.5.4: spectral_anal.m .........................................................................163

C.5.5: heights.m .......................................................................................166

Appendix D: PIC Microcontroller Code................................................................168

D.1: WECfinal.c...................................................................................168

D.2: ConfigRTCsq.c.............................................................................174

D.3: ConvertTime.c............................................................................176

D.4: I2CByteWrite.c ..........................................................................177

D.5 I2CRandomRead.c........................................................................178

D.6: ReadClock.c .................................................................................180

D.7: SetClock.c....................................................................................181

Appendix E: Derivation of H/S..............................................................................183

E.1 Exponential Decay Paddle...........................................................................183

E.2 Hyperbolic Curve Paddle.............................................................................186

Appendix F: Calibration Results............................................................................187

Glossary .................................................................................................................190

Appendix A: Table of WEC Companies [from 40]

Company

Name

Product

Special Notes

Base

Location

Projects

Website

374's Electric

Power

Corporation

Ocean Surf

Energy System

s

Stoughton,

MA

www.374electric.com/welcome.

htm

ABS Alaskan

Small water

generators

(micro-hydro

turbines)

Alternative energy and rem

ote

power products

Fairbanks,

AK

Residential consumers

www.absak.com

Aqua Energy

Group, Ltd.

"AquaBuOY"

Wave Energy

Converter

Point-absorber incorporating a

hose-pump which uses water as

the hydraulic fluid

Mercer

Island, WA

1MW Power Plant - Clallum

County Public Utility

(Makah Bay, WA)

www.aquaenergygroup.com

Float

Incorporated

"Pneumatically

Stabilized

Platform

" (PSP)

Floating Ocean Real Estate -

uses an OWC to extract energy

from waves

San Diego,

CA

www.floatinc.com

Florida Hydro

Power and

Light

Company

Offshore Gulf

Stream Current

Energy

Palatka, FL

www.floridahydro.com

GCK

Technology,

Inc.

"Gorlov Helical

Turbine"

Power take-off for ocean and

tidal currents: rotates in same

direction regardless of water

flow direction, received Edison

Patent Award

San Antonio,

TX

www.gcktechnology.com/GCK

Hawaii

Energy Dept.

Ocean Thermal

Energy

Conversion

(OTEC Energy)

Honolulu, HI

5MW Pre-C

ommercial Plant

www.hawaii.gov/dbedt/ert/b

ib/bib_otec.htm

l

HydroVenturi

"Rochester

Venturi"

Tidal current electrical

generation: no moving

mechanical or electrical parts

underwater, water flow

pressure reduction brings in air

which is used to generate

electricity

San

Francisco, CA

60kW Dem

o Unit North of

England

www.hydroventuri.com

Independent

Natural

Resources,

Inc. (IN

RI)

"SEADOG

Pump"

Point-absorber incorporating a

piston for pumping water or air

into a turbine

Eden Prairie,

MN

1/32-scale prototype was

tested in the wave tank of

Texas A&M University

www.inri.us

Kinetic

Energy

System

s

"Hydrokinetic

Generator",

"KESC

Bowsprit",

"KESC Tidal

Generator"

Tidal current energy: meant for

flood and ebb tides, 600kW

with products ranging 35%-65%

efficiency

Ocala, FL

www.kineticenergysystem

s.com

Marine

Development

Associates,

Inc.

OTEC

Saratoga, CA

Project RATAK: 5-10 M

W

OTEC for the Gov't of the

Marshall Islands; OTEC

Development Plan Review

for Gov't of Taiw

an;

Assessm

ent for Philippine

Gov't; MDA's Island Nation

OTEC Program

www.m

arinedevelopmentinc.c

om/ocean_e nergy.htm

Marine

Innovation &

Technology

Ocean Current

Farm

Berkeley, CA

www.m

inifloat.com/ocean.htm

Mo-T.O

.P.S

Oceanic

Power

System

s

"Wind Goose"

OTEC variant

Del Rio, TX

www.isfind.com

OCEES,

International-

Ocean

Engineering

and Energy

System

s OTEC

Honolulu, HI

www.ocees.com/mainpages/

otec.htm

l

Ocean M

otion

International

LLC (OMI)

"WavePump"

Point absorber array whose

buoys pump seawater through a

hydro-turbine generator as it

descends in a wave trough

Colorado and

Oregon

Functioning 1/20th scale

model of new

OMI

WavePump design is

unveiled in Dana Point, CA

(June 2002); OMI submits

application to Hawaii

Renew

able RFP (April 2004),

submits application to

present investm

ent

opportunity at the Novem

ber

2003 NREL Industry Growth

Forum

www.oceanmotion.w

s

Ocean Power

Technologies,

Inc.

"PowerBuoy"

Near Shore W

ave Energy Point

Absorber: passed the rigorous

Environmental Assessm

ent

process to install units in

Hawaii, Initial Public Offering

(IPO) AIM

Market of the

London Stock Exchange

(“AIM

-OPT”) on October 21,

2003

West

Trenton, NJ;

VIC

,

Australia

Partnering with Iberdrola

S.A

. in Spain and Total in

France; Partnering with US

Navy in Hawaii; Contracts

with Lockheed M

artin Corp.

and New

Jersey Board of

Public Utilities

www.oceanpowertechnologies

.com

Ocean W

ave

Energy

Company

"Ocean W

ave

Energy

Converter"

Completed bench top trials with

full size components under a

Small Business Innovation

Research contract from the US

Coast Guard

Bristol, RI

www.owec.com

Ocenergy

Near Shore,

Offshore W

ave

Energy for

Hydrogen

Production

Norw

alk, CT

www.ocenergy.com

ReE

nergy

Group PLC

(“ReE

nergy”)

"Oases"

Uses wave power generation

technology as a cost effective

method for desalination

UK (US: San

Diego, CA)

Contracts in M

exico,

California, Peru and

Morocco

www.renergypacific.com

Scientific

Applications

& Research

Associates

(SARA) Inc.

Point Absorber

with

Magnetohydrody

namic G

enerator

Magnetohydrodynamic (MHD)

Generator; Alm

ost no moving

parts. No gears, no levers, no

turbines, no drive belts, no

bearings, etc.

Huntington

Beach, CA

DOE contract Phase II SBIR

Program: design, construct,

and dem

onstrate a 50-100

kW M

HD unit

www.sara.com/energy/WEC.

htm

l

Sea Solar

Power

International

OTEC

University of Maryland tests

confirm

heat transfer with 3.4%

total cycle efficiency etc.

(perfect Carnot Cycle efficiency

is 7.4%)

Baltim

ore,

MD

www.seasolarpower.com

SeaVolt

Technologies

"Wave Rider"

Near Shore Point Absorber:

uses hydraulics for power take-

off

Berkeley, CA

www.seavolt.com

Tidal Electric,

Inc.

Tidal Lagoons

Uses a rubble mound

impoundment structure and

low-head hydroelectric

generating equipment situated a

mile or more offshore in a high

tidal range area. Shallow tidal

flats provide the most

economical sites

West

Sim

sbury,

CT;

Anchorage,

AK; London,

UK

60 M

W Swansea Bay project,

UK: measures 5 km2 in area

and about a mile offshore;

project agreem

ent with

Liaoning Province, China

www.tidalelectric.com

UEK

Corporation

Underwater

Electric Kite

Hydro kinetic turbines for

Current, Tidal, OWC Energy;

no dams or im

poundments

required since no foundation is

necessary

Annapolis,

MD

DOE contract SBIR

/DOE

DE-FG02-00ER82930;

Contract with Ontario Power

Generation to test the hydro

kinetic 'Twin Turbines'

uekus.com

Verdant

Power

Instream Energy

Generation

Technology

(IEGT)

Tidal Current Energy: free-flow

hydropower technology or

kinetic hydro energy system

s Arlington, VA

In conjunction with G

CK

Technology, 1MW Tidal Site

at Merrimack River, MA.

Uses GCK's Gorlov Helical

Turbine; Testing in the

Potomac River, Carderock,

MD; Prototype in New

York

City's East River

www.verdantpower.com

Wader, LLC

Hydrocratic

Generator

Salinity Gradient Energy

Laguna

Beach, CA

www.w

aderllc.com

Seapower

Pacific Pty.

Ltd.

(Shareholders:

Renew

able

Energy

Holdings Plc

(REH) - UK;

Pacific Hydro

Ltd. -

Australia;

Carnegie)

"CETO" W

ave

power converter

Near Shore Point Absorber:

wave crests depress a disk

which delivers pressurized

water to shore where energy

conversion takes place

West Perth,

Western

Australia

100kW in-sea prototype,

2005

www.carnegiecorp.com.au/O

perations/Renew

able_Wave_E

nergy_Project_2004.htm

l

Blue Energy

Canada

The Davis

Turbine

(Vertical Axis

Turbine for

Tidal Currents)

Proof of Concept Review: US

Arm

y Corps of Engineers, the

National Research Council of

Canada, et al - RW Beck

(Engineering) Inc., Sept. 2005

Alberta,

Canada

Proposed tidal energy project

for Scotland's Pentland Firth

www.bluenergy.com

Wavem

ill

Energy Corp.

"Wavem

ill"

shoreline and nearshore

applications with patented

suction chamber and enclosed

surge wall

Dartmouth

(Halifax),

Nova Scotia,

Canada

August 2001: scaled model of

the ESW W

avem

ill®

www.w

avem

ill.com

China New

Energy (CNE)

Tidal Energy,

Ocean Current,

Wave Energy,

Thermal Energy,

Salinity

Gradients

CNE is a non-profit research

network founded by

Guangzhou Institute of Energy

Conversion, Chinese Academ

y

of Sciences, while jointly under

the direction of State M

inistry

of Science & Technology, State

Economic & Trade Commission

and Chinese Academ

y of

Sciences

Guangzhou,

China

www.new

energy.org.cn/engli

sh/ocean/index.htm

Wave Dragon

"Wave Dragon"

Overtopping wave energy

converter uses large collector

arm

s to funnel water to a

Kaplan turbine which turns a

PM generator

København,

Denmark

January 2005: 20 kW W

ave

Dragon prototype taken

offline after one year and

nine months of continuous

real sea testing

www.w

avedragon.net

WavePlane

Production

A/S

"WavePlane"

Overtopping wave energy

converter uses a fly-w

heel-tube

for power take-off

Gentofte,

Denmark

September 2002 to April

2003: WavePlane underwent

3:10 sea tests in Japan by

NKK

www.w

aveplane.com

DAEDALUS

Inform

atics

"WECA" (Wave

Energy

Conversion

Activator)

20kW OWC device

Athens,

Greece

www.daedalus.gr

Hydam

Technologies

Ltd.

"McC

abe Wave

Pump" (MWP)

Point absorber with two

rectangular steel pontoons

which move in relation to a

stationary central raft. The

hinges of the pontoons drive a

hydraulic power take-off

system

.

Kerry, Ireland

Has received funding from

the Irish M

arine Institute

n/a

Clearpower

Technology

Ltd. (also

listed as

Wavebob Ltd.

and duQuesne

Environmenta

l Ltd.)

"Wavebob"

Self-reacting point absorber that

extracts power from the relative

movem

ent of tw

o floating

bodies that have different heave

frequency responses. This

property enables the Wavebob

to utilize energy from more

wave frequencies than

conventional single buoy point

absorbers.

Dublin,

Ireland

1/50th and 1/20th scale tests

conducted at the Hydraulics

and M

aritime Research

Centre (U

CC, Cork) and the

large wave channel of the

German Coastal Defence

Centre (H

anover University

and the Technical University

of Braunschweig)

www.clearpower.ie

National

Institute of

Ocean

Technology

(NIO

T)

"Backward Bent

Ducted Buoy"

(BBDB)

OWC device with variable

resistance induction generator

Vizhinjam,

Kerala (India)

1/13 prototype gives air

power / wave power

conversion above 60%

(started Dec. 1990 with

improvem

ents added over

time)

www.niot.res.in/m1/Wave.htm

S.D

.E. Energy

Ltd.

Offshore wave

energy

OWC / Overtopping device

works by forcing waves into

cavity separated from hydraulic

oil by mem

brane. Incoming

waves pressurize the oil which

drives a hydraulic generator.

Tel Aviv,

Israel

Israeli government has

authorized S.D

.E. to produce

and sell 4MW of electricity

for 20 years, at 5.25 cents per

kWh. Project is approved

with partial financing by the

Chief Scientist of Israel.

www.sde.co.il

JAMSTEC

"Mighty W

hale"

OWC device with three

induction generators

Yokosuka,

Japan

28 M

arch 1998: deployment

of 110 kW prototype at

Gokasho Bay, Japan funded

by Japan's Science and

Technology Agency

www.jamstec.go.jp/jamstec/

MTD/Whale

Ecofys

"Wave Rotor"

Orbital currents in waves and

tidal currents induce

hydrodynamic lift which turns a

set of blades around a vertical

axis

Utrecht, The

Netherlands

Oct. and Nov. 2004: 1/10

scale model tested at NaREC

in Blyth (0.5 M

W model to

be installed in UK)

www.ecofys.com

Neptune

System

s

Magnetohydrody

namic (MHD)

generator in a

buoy wave

energy converter

and tidal current

energy converter

Uses a 20 m diameter

superconducting magnet

solenoid (SMES) which

produces a magnetic field

strength of 5 Tesla with energy

content ~100 G

J. The tidal

generator can deliver 4 M

W

power installed at a 3 m/s tidal

current velocity.

Breda, The

Netherlands

www.neptunesystem

s.net

Hammerfest

Stroem

AS

"Tidekraft"

Underwater turbine props

similar to wind turbine props

collect tidal current energy

Hammerfest,

Norw

ay

25 Sept. 2005, 'The Blue

Concept' project: prototype

installed at Kvalsundet

producing 21 GWh per year

www.e-tidevannsenergi.com

Ing Arvid

Nesheim

Point absorber

Absorbs energy from vertical,

pivotal, horizontal backwards

and forw

ards (to-and-fro)

motion via a hydraulics system

Vollen,

Norw

ay

www.anwsite.com

WAVEenergy

AS

"Seawave Slot-

Cone Generator

(SSG)"

Overtopping wave energy

converter with multi-stage

turbine (M

ST)

Norw

ay

Has received funding from

the European Commission

FP-6-2004-Energy-3 (7 Apr.

2005) and the Norw

egian

Research Council to develop

the MST turbine (25 Jan

2005)

www.w

aveenergy.no/index.htm

Sea Electrical

Generators,

Ltd.

Point absorber

Made from cheap recycled

plastic (polyethylene,

polypropylene), meant to be

replaced every five years

Russia

easy-energy.iatp.org.ua

Vortex

Oscillation

Technology,

Ltd. in

partnership

with The

Engineering

Business Ltd.

(Northumberl

and, England)

"Stingray"

Oscillating wings extract energy

from tidal currents

Moscow,

Russia

2004: 5 M

W installation off

the coast of Scotland

www.vortexosc.com/index.ph

p, www.engb.com

Yakov Kolp

"Sea W

ave

Energy

Converter

(SWEC)"

Wave oscillation generator

with capacity from kilo to

megawatt range

Russia

www.rvf.ru/engl/expo-

yakor.php

Seabased

Energy AB

Point Absorber

with Linear

Generator

Linear permanent magnet

generator with large number of

poles and NdFeB

magnets that

allow for high magnetic

excitation with smaller magnets

Uppsala,

Sweden

Working with the Division

for Electricity and Lightning

Research at Uppsala

University, Sweden

www.seabased.com

Sea Power

International

AB

Overtopping

wave energy

converter

Near shore W

EC

Stockholm

,

Sweden

www.seapower.se

Naturalist

Wave Power

Plant AB

Hydraulic-based

offshore W

EC

Waves rotate joints of chassis

which pressurizes mineral oil.

This drives a hydraulic

generator.

Ankara,

Turkey

www.dalgaenerjisi.com/ana-

english.asp,

www.w

ipo.int/pctdb/en/fetc

h.jsp?LANG=E

NG&DBSELECT=PCT&SER

VER_TYPE=19&SORT=11492

88-

KEY&TYPE_FIELD=256&ID

B=0&ID

OC=630415&ELEME

NT_SET=IA

,WO,TTL-

EN&RESULT=1&TOTAL=1

&START=1&DISP=25&FOR

M=SEP-0/HITNUM,B-

ENG,D

P,M

C,PA,A

BSUM-

ENG&QUERY=wo%2f020751

51

AWS Ocean

Energy Ltd.

"Archim

edes

Wave Swing"

Air-filled, submerged point

absorber uses a linear

generator. W

ave crests depress

the device, and troughs force it

upwards.

Ross-shire,

England

24 M

ay 2004: 2MW

installation of prototype off

of Portugal

www.awsocean.com,

www.w

aveswing.com

Embley

Energy Ltd.

"Sperboy"

Floating buoy with multiple

oscillating water columns of

different lengths to utilize a

larger range of wavelengths

England

The Carbon Trust is

evaluating the Sperboy

through their Marine Energy

Challenge programme

info@sperboy.com,

www.thecarbontrust.co.uk/ct

marine2/Page1.htm

Marine

Current

Turbines, Ltd.

(MCT) and IT

Power

"SEAFLOW"

marine current

turbine

Tidal currents drive tw

o blades

around a horizontal axis like an

underwater wind turbine

Hampshire,

UK

31 M

ay 2003: The Carbon

Trust sponsors construction

of a 300kW experim

ental

turbine 3km offshore from

Lynmouth, Devon, 2003-3-

122-1-2

www.itpower.co.uk,

www.m

arineturbines.com

Ocean Power

Delivery, Ltd.

"Pelamis"

Near shore cylindrical structure

whose hinges drive hydraulic

motors as the power-take off

Edinburgh,

Scotland, UK

2.25MW Prototype: three

Pelamis P-750 machines

located 5km off the

Portuguese coast

www.oceanpd.com

ORECon Ltd.

"MRC" System

Multiple oscillating water

column

Plymouth,

UK

June 2005: FEED for a 2MW

pre-production prototype

www.orecon.com

Offshore

Wave Energy

Ltd. (O

WEL)

"Grampus"

Long inlets trap and compress

the air in wave troughs.

Reservoirs accumulate the

compressed air which drives a

turbine.

Portsm

outh,

UK

Future project: tank-testing

18 metre long physical model

www.owel.co.uk

The Wave

Power Group

at the

University of

Edinburgh

"Salter's Duck"

and "Sloped IPS

Buoy"

Both are point absorbers using

hydraulics power take-off

Edinburgh,

Scotland, UK

Salter's Duck project no

longer funded, Sloped IPS

Buoy applying for EPSRC

programme funding

www.m

ech.ed.ac.uk/research

/wavepower/index.htm

Wavegen

"LIM

PET"

Shoreline OWC

Northumber-

land, England

June 2004: first generation of

Breakwater Turbine installed

on the Lim

pet plant

www.w

avegen.co.uk

Appendix B: Parts List

Function

Distributor

Part Number

Total Price

Hardware

6063 Al round rod, 1" dia., 8' length

Buoy rod, generator translator

McM

aster-Carr

1640T37

$42.53

6063 Al square tube, 1" x 1" x 6' Q

ty:

4

Main frame

McM

aster-Carr

88875K33

82.88

6063 Al rectangular bar, 1/2" x 2" x

8'

Main frame crossmembers

McM

aster-Carr

89755K62

54.05

6063 Al U-channel, 2" base x 1/2"

leg x 8' length

LIG Box cage, bottom support for

buoy

McM

aster-Carr

9001K39

26.75

Al 6061 round tube, 1" ID x 1.5"

OD, 12" length

Segment fits around buoy rod to

strengthen connection to translator

McM

aster-Carr

9056K271

15.04

UHMW polyethylene tube,

0.97"IDx1.5"ODx1' length

Low-friction bearings for buoy and

translator rods, pressed into holes in

frame

McM

aster-Carr

8705K141

6.11

UHMW Polyethylene tube, 1.625"

ID 2.365" OD, 1' length

Risers for coil inside Pelican Case

McM

aster-Carr

8705K74

14.02

ABS plastic tube, 1"ID x 1.5"OD x 5'

length

Tube inside buoy

McM

aster-Carr

1839T22

40.10

Type I PVC rod, 4" OD, 12" length

Generator bulkheads

McM

aster-Carr

8745K28

26.52

Neoprene/EPDM/SBR Closed-cell

Foam Rubber, Sheet 3/8" thick, 1'

length

Coil gaskets, box frame padding

McM

aster-Carr

8647K45

13.40

Polyurethane sheet, 6" x 6" x 1/2",

60A hardness

Vibration relief coupler between

translator and buoy rod

McM

aster-Carr

8716K242

23.23

Al 6061-T6 RH Thread 1/4"-20

fully threaded rod, 4" long, pack of

10

Compress bulkhead fittings to coil,

connect halves of magnet rod through

magnet

McM

aster-Carr

93225A874

8.62

Al Hex-head 1/4"-20 cap screw, 3"

length, pack of 10

Various connections

McM

aster-Carr

93306A558

7.15

Size -339 o-rings, Qty: 4

Seal bulkhead fittings to Pelican Case

ORCA

Donated

Pelican case 1300, silver, with foam Linear Inductance Generator housing

Techni-Tool, Inc.

6031300SILVER

38.34

NdFeB ring magnets, 1.5” OD, Qty: 2

Magnet in generator

Super Magnet Man

R1111

16.00

1-lb. Spool of #30 AWG Enameled

Magnet Wire

Coil

Force Field

0930

17.00

West System epoxy

To set coil

UTL

Available

Polystyrene foam

Buoy

FIT Machine shop

Donated

6-oz E-glass fabric

Buoy cover

Fiber Glass Florida

Donated by

MFP

DER 331 epoxy resin and hardener

Buoy cover

Fiber Glass Florida

19248 Donated by

MFP

3M K-1 Scotchlite Bubbles

Buoy cover

Fiber Glass Florida

15.00

Various screws, 1” length

Secure ABS tube inside foam

UTL

Available

Polyester Gelcoat Series 944

Buoy coating

Progress Plastics

Donated by

MFP

U-channel (6” base, 2” legs))

Piling mount

FIT machine shop

Donated

316 SS ¾” band

Piling mounting band

Pelco Utility Products

US-8124-06 105.35/100’

316 SS Buckle Qty: 4

Piling mounting band

Pelco Utility Products

US-8124-56

79.25/100

Compression Spring

Provide return force for

Reid Supply

CX-39, CX-47

CX-49, CX-50

15.95

Electronics

PIC18f4523

Microprocessor

Microchip

PIC18F4523-E/P

Sampled

Logomatic Serial SD Datalogger

Data recording

SparkFun Electronics

WIG-00752

59.95

256MB SD card

Storage space

Available

DC/DC converter

Creates 5V output

Texas Instruments

TPS61202DRC

Sampled

Chip scale (QFN) 10,32 surfboard,

.5mm pitch

DC/DC converter surfboard

SchmartBoard

202-0017-01

9.99

Serial access Real Time Clock with

alarm

Global timekeeper for data collection

consistency

ST Microelectronics

M41T80M6E

Sampled

9V battery clip

Main power connection

IC 3.3V 150MA LDO REG SOT-

23-5

Main power voltage regulator

Digikey Electronics

296-11021-1-ND

0.84

CONN DB9 FEMALE .318" R/A

NICKEL

Serial connection for HyperTerminal

Digikey Electronics

182-809FE-ND

4.46

CONN HEADER 2POS .100

VERT GOLD Qty: 7

Load and current-sense resistor

connectors

Digikey Electronics

WM2700-ND

4.20

CONN TERM FEMALE 22-

30AWG GOLD Qty: 30

Load and current-sense resistor

connectors

Digikey Electronics

WM2624-ND

8.58

CONN HOUS 2POS .100

W/RAMP/RIB Qty:10

Load and current-sense resistor

connectors

Digikey Electronics

WM2000

1.31

CONN HEADER .100 SNGL STR

36POS

Header to be soldered into Logomatic

Digikey Electronics

929647-02-36-

ND

3.31

CONN HEADER .100 SNGL

R/A 36POS

RTC and PICkit2 headers

Digikey Electronics

929550-01-36-

ND

3.96

0.5W, 1ohm, 5% tolerance carbon

film resistor Qty: 2

Digikey Electronics

1.0H-ND

0.26

2.2uH inductor, axial, 10%

tolerance

DC/DC converter inductor

Digikey Electronics

M9977-ND

1.13

1F, 6.3V supercapacitor

Digikey Electronics

604-1018-ND

6.15

0.68F, 6.3V supercapacitor

Digikey Electronics

604-1017-ND

6.15

0.47F, 6.3V supercapacitor

Digikey Electronics

604-1016-ND

6.15

L6210 IC diode bridge Schottky

dual 16DIP

Rectifier

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497-3646-ND

2.97

0.1uF ceramic capacitor, 100V Qty: 4

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P4910-ND

1.92

22pF capacitor Qty: 2

Microcontroller oscillator capacitors

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490-3709-ND

0.56

1uF electrolytic capacitor, Qty: 9

Digikey Electronics

P824-ND

1.26

10uF electrolytic capacitor, Qty: 5

Digikey Electronics

P5309-ND

0.85

20.0MHz crystal oscillator, 18pF,

HC49/US

PIC oscillator

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300-6042-ND

0.70

32.786kHz crystal oscillator

RTC oscillator

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X801-ND

2.70

2-pin screw terminal block, 0.2"

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Power and Supercapacitor connectors

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ED1609-ND

0.92

Green LED, 3mm T-1

DC/DC converter ON indicator

Digikey Electronics

LG 3330-LP-1-0-

10-BULK-ND

0.09

Blue LED, 3mm T-1

Power indicator

Digikey Electronics

160-1600-ND

0.60

RJ11 jack, PCB right angle, no

panel stops

Microchip universal programmer

connector

Digikey Electronics

A24907-ND

2.02

IC REF PREC VOLT

MICROPWR 8-SOIC

4.096 voltage reference for ADCs

Digikey Electronics

LM4140CCM-

4.1-ND

3.58

DIODE ZENER 5W 5.6V DO-15

Supercapacitor protection diode

Digikey Electronics

1N5339BTPMSC

T-ND

0.39

MOSFET P-CH 20V 2.4A SOT-23

Digikey Electronics

NTR4101PT1G

OSCT-ND

0.36

IC OPAMP QUAD SNGL

SUPPLY 14DIP Qty: 2

ADC buffers

Digikey Electronics

MCP604-I/P-ND

3.16

CONN SOCKET STRIP 50PIN

.100 STR

Digikey Electronics

ED6350-ND

7.26

MAX233

TTL/RS232 logic level converter

Maxim Integrated

Products

MAX233CPP

Sampled

Single Pole, Double Throw Relay

Switches DC/DC converter into/out

of circuit

All Electronics

RSB-5

2.25

2N3904 NPN Transistor, Qty: 3

Digikey Electronics

2N390FS-ND

0.33

MAX4372, Qty: 2

Current sense op amp

Maxim Integrated

Products

MAX4372HESA

Sampled

1N4148 Switching diode, Qty: 2

Reverse polarity protection diode

Digikey Electronics

1N4148FS-ND

0.08

1N4001 Rectifier

Power selector diode or RTC

Digikey Electronics

1N4001FSCT-

ND

0.11

1N4007 Rectifier

Power selector diode or RTC

Digikey Electronics

1N4007FSCT-

ND

0.10

Resistors:

7 x 1k

4 x 20k

1 x 100k

1 x 1M

1 x 2.2M

Available

PCB adhesive standoffs

Available

Appendix C: Matlab® Code

C.1: IR

L W

ave Pow

er M

odeling

C.1.1: detect_land_red.m

Detecting land from a bitmap image screen-captured from Google Earth (modified to highlight coastlines) and saving it as a black-and-white

figure file

function[b_w_map, IRLfeatures, atlantic_flag] = detect_land_red(image_file)

%For example, image_file could be 'big_map/fort_pierce_all_red.bmp'

load_str = ['imread(', 'image_file',')']

map = eval(load_str);

figure;

image(map);

jpixels = size(map,1); %number of rows

ipixels = size(map,2); %number of columns

%*********** Initializations *****************

%Define IRLfeatures contains the IRLwidth IRLstart_pixel and IRLstop_pixel]of each row;

% initialize all to zero

%Also, initialize the indexed output fetch map to 0 and the RGB output fetch map

% to white for debugging; default = not over the atlantic ocean

IRLfeatures = zeros(jpixels, 3);

fetch = zeros(jpixels, ipixels);

atlantic_flag = zeros(jpixels, ipixels);

b_w_map = ones(jpixels, ipixels, 3);

%Define yellow border as anything within the RGB thresholds set here

yellow_threshold = [140 140 79];

red_threshold = [255 0 0 ];

purple_threshold = [128 0 255];

%******************* Algorithm ********************

%Run through each pixel and find the IRL coastlines

for (j=1:jpixels)

for (i=1:ipixels)

if(i == 1) %assume pixels (:,1) are over land

IRLflag = 0;

atlantic_flag(j,i) = 0;

last_yellow = -10;

yellow_counter = 0;

end

%Purple pixels prevent fetch from being calculated near a jetty

if(map(j,i,1)==purple_threshold(1) && map(j,i,2)==purple_threshold(2) && map(j,i,3)==purple_threshold(3))

b_w_map(j,i,:)=0; %black

while(~(map(j,i,1)~=red_threshold(1) && map(j,i,2)~=red_threshold(2) && map(j,i,3)~=red_threshold(3)))

atlantic_flag(j,i) = 1;

IRLflag = 0;

b_w_map(j,m,:)=0; %black

% if((map(j,i,1)>yellow_threshold(1)) && (map(j,i,2)>yellow_threshold(2)) &&

%

(map(j,i,3)<yellow_threshold(3)))

% b_w_map(j,m,:)=0; %black

% else

% b_w_map(j,m,:)=1; %white

% end

i=i+1;

end

end

%first red pixel marks atlantic coastline

if(map(j,i,1)==red_threshold(1) && map(j,i,2)==red_threshold(2) && map(j,i,3)==red_threshold(3))

for m=i:ipixels

b_w_map(j,m,:)=1; %white

atlantic_flag(j,m) = 1;

IRLflag = 0;

end

break;

end

if (~IRLflag) %if current pixel is not located over the IRL, map it as black and check if it's a coastline

b_w_map(j,i,:) = 0; %black

fetch(j,i) = -20; %lowest value in fetch calculation

% if current pixel is yellow, it's either the 1st yellow pixel

% on the west coast, a random pixel, or a consecutive yellow pixel on the east coast

if ((map(j,i,1)> yellow_threshold(1)) && (map(j,i,2)> yellow_threshold(2)) && (map(j,i,3)<

yellow_threshold(3)))

if(last_yellow == i-1) %if 2 yellow pixels occur consecutively on the east side of the lagoon

last_yellow = i;

b_w_map(j,i,:) = 0; %black

fetch(j,i) = -20;

continue; %don't reflip the flag to 'over the IRL

elseif(i>1 && i<ipixels-1 && j>1 && j<jpixels-1) %for non-border pixels

if(proximo(map,i,j,yellow_threshold(1),yellow_threshold(2),yellow_threshold(2))) %random

%yellow pixel

%disp('proximo yellow, not in IRL')

b_w_map(j,i,:) = 0; %black

fetch(j,i) = -20;

continue; %go on to the next pixel

else %we are on the west side of the lagoon, so:

if(IRLfeatures(j,2)==0)

IRLfeatures(j,2) = i; %set the start pixel for row j as the first yellow pixel

end

IRLflag = 1; %flip the flag to over the IRL

last_yellow = i;

end

end

end

else %current pixel is located over the IRL

%if the current pixel is yellow, it's either a pixel on the

%eastern IRL coast, a corner, or a random pixel in the IRL

if ((map(j,i,1)> yellow_threshold(1)) && (map(j,i,2)> yellow_threshold(2)) && (map(j,i,3)<

yellow_threshold(3)))

if(last_yellow == i-1) %keep track of consecutive yellow pixels

last_yellow = i;

b_w_map(j,i,:) = 0; %black

fetch(j,i) = -20;

continue; %skip the rest b/c the flag shouldn't be reversed again

else %if the current pixel is the first yellow (coastline) pixel, flip the flag

b_w_map(j,i,:) = 0; %black

fetch(j,i) = -20;

last_yellow = i;

IRLflag = 0; %now we are not over the lagoon anymore

IRLfeatures(j,3) = i; %set the end pixel for row j as the last yellow pixel

end

end

end

end

end

figure;

image(b_w_map);

title('Land and water');

% load_str2 = ['save ',BWmap_name];

% eval(loadstr2);

%***************** Subfunctions ********************

function [is_alone] = proximo(map, i, j, yellow_R_threshold, yellow_G_threshold, yellow_B_threshold)

if (...

((map(j-1,i-1,1)>yellow_R_threshold) && (map(j-1,i-1,2)> yellow_G_threshold) && (map(j-1,i-1,3)<

yellow_B_threshold)) ||...

((map(j-1,i,1)> yellow_R_threshold) && (map(j-1,i,2)>yellow_G_threshold) && (map(j-1,i,3)<

yellow_B_threshold)) ||...

((map(j-1,i+1,1)> yellow_R_threshold) && (map(j-1,i+1,2)> yellow_G_threshold) && (map(j-1,i+1,3)<

yellow_B_threshold)) ||...

((map(j,i-1,1)>yellow_R_threshold) && (map(j,i-1,2)> yellow_G_threshold)&& map(j,i-1,3)<

yellow_B_threshold)) ||...

((map(j,i+1,1)>yellow_R_threshold) && (map(j,i+1,2)>yellow_G_threshold) && (map(j,i+1,3)<

yellow_B_threshold)) ||...

((map(j+1,i-1,1)>yellow_R_threshold) && (map(j+1,i-1,2)> yellow_G_threshold) && (map(j+1,i-1,3)<

yellow_B_threshold)) ||...

((map(j+1,i,1)>yellow_R_threshold)&&(map(j+1,i,2)>yellow_G_threshold) && (map(j+1,i,3)<

yellow_B_threshold)) ||...

((map(j+1,i+1,1)> yellow_R_threshold)&&(map(j+1,i+1,2)> yellow_G_threshold) && (map(j+1,i+1,3)<

yellow_B_threshold))...

)

is_alone = 0;

else

is_alone = 1;

end

function [is_corner] = corner_check(map, i, j, yellow_R_threshold, yellow_G_threshold, yellow_B_threshold)

%if all the pixels above or all the pixels below (but not both) are NOT

%yellow, return 1 (IS a corner); else return 0 (ISN'T a corner)

if (xor( ...

( ...

((map(j-1,i-1,1)> yellow_R_threshold) && (map(j-1,i-1,2)> yellow_G_threshold) && (map(j-1,i-1,3)<

yellow_B_threshold)) ||...

((map(j-1,i,1)>yellow_R_threshold) &&(map(j-1,i,2)>yellow_G_threshold) &&(map(j-1,i,3)<

yellow_B_threshold)) ||...

((map(j-1,i+1,1)> yellow_R_threshold) && (map(j-1,i+1,2)> yellow_G_threshold) && (map(j-1,i+1,3)<

yellow_B_threshold)) ...

),...

(...

((map(j+1,i-1,1)> yellow_R_threshold) && (map(j+1,i-1,2)> yellow_G_threshold) && (map(j+1,i-1,3)<

yellow_B_threshold)) ||...

((map(j+1,i,1)>yellow_R_threshold) &&(map(j+1,i,2)>yellow_G_threshold) &&(map(j+1,i,3)<

yellow_B_threshold)) ||...

((map(j+1,i+1,1)> yellow_R_threshold) && (map(j+1,i+1,2)> yellow_G_threshold) && (map(j+1,i+1,3)<

yellow_B_threshold))...

)...

) ...

)

disp('corner1')

is_corner = 1;

else

disp('corner0')

is_corner = 0;

end

C.1.2: fetch_calc_red.m

Calculates fetch from each pixel (on the B&W map created above) to the nearest land in the direction from which the wind is blowing.

Requires the three matrices returned above, plus a image scale (in m) in both the N-S and E-W directions, and the direction from which the

wind blows

function[fetch] = fetch_calc_red(b_w_map, IRLfeatures, atlantic_flag, jlength, ilength, theta)

jpixels = size(b_w_map,1); %number of rows

ipixels = size(b_w_map,2); %number of columns

iscale = ilength/ipixels; %in meters/pixel

jscale = jlength/jpixels;

%Now calculate and plot fetch

for (j=1:jpixels)

for (i=1:ipixels)

moves = 0; %reset row deviation counter each time we move to a new pixel

b=j; %(b,a) is the target shorline pixel; start the search at the current pixel

a=i;

if(b_w_map(j,i,1)==0 && b_w_map(j,i,2)==0 && b_w_map(j,i,3)==0)

fetch(j,i)=-1;

continue;

elseif(atlantic_flag(j,i)==1)

for(c=i:ipixels)

if((theta>45 && theta<135) || (theta>225 && theta<315))

fetch(j,c) = abs(0.075*ilength*sin(theta));

else

fetch(j,c) = abs(0.075*jlength*cos(theta));

end

end

i=ipixels; %skip the rest of the row for now, just for speed's sake

continue;

else %pixel is over the IRL; determine the fetch for this pixel

IRLfeatures(j,1) = IRLfeatures(j,1)+1; %increment the width of the IRL for row j

%while the we don't land on a black (coast) pixel or go over

%the ocean

while(b_w_map(b,a,:)==1)

if(atlantic_flag(b,a)==0)

moves = moves+1;

%keep track of the # of rows above/below j we move

% in finding the coast

% move right/left by the corresponding num of pixels

if(theta == 90)

b=j;

a = i+moves;

elseif(theta==270)

b=j;

a = i-moves;

elseif((theta>45 && theta<135) || (theta>225 && theta<315))

b = j-moves*sign(cos(theta*pi/180)); %move up/down by a row

a = i+moves*(round(sign(sin(theta*pi/180))*abs(tan(theta*pi/180))));

else

a = i+moves*sign(sin(theta*pi/180)); %move right/left by a row

b = j-moves*(round(sign(cos(theta*pi/180))*abs(cot(theta*pi/180))));

end

% deal with pixels off the map, leaving RGB white to show borders are not

% accounted for

if(a<1 || a>ipixels || b<1 || b>jpixels)

fetch(j,i) = -20;

break;

end

else

break;

end

end

end

% when we've reached a coastline pixel, record the fetch

fetch(j,i) = sqrt(((b-j)*jscale)^2 + (((a-i)*iscale)^2));

end

end

figure;

fetch_colors = colormap;

fetch_colors(1,3)=0;

colormap(fetch_colors);

imagesc(fetch);

colorbar('eastoutside');

title('Brevard IRL Fetch, wind from %d degrees');

C.1.3: wave_calc.m

Uses fetch, windspeed, and an average depth to calculate the significant wave height, peak period, fully-developed sea height, and FDS

required duration on the maps created in AB.1.1 and AB.1.2

function[wave_heights, Tp, H_FDS, t_req_max] = wave_calc(fetch_matrix, b_w_map, windspeed, avg_depth)

% arguments: Wind Speed, Fetch, and Depth

% returns: a matrix of wave heights and a column of wave periods, printed

% to an image

%*******Initializations*********

jpixels = size(fetch_matrix,1); %number of rows

ipixels = size(fetch_matrix,2); %number of columns

Tp = zeros(jpixels, ipixels);

wave_heights = ones(jpixels, ipixels);

H_FDS = ones(jpixels, ipixels);

t_req_max = 0;

g = 9.8;

Ua = windspeed; %10m elevation windspeed

F = fetch_matrix;

d = avg_depth;

Cd = 0.001*(1.1+0.035*Ua);

Ustar = sqrt(Cd*Ua^2);

Tp_SW = 9.78*(d/g)^(1/2);

for(j=1:jpixels)

for(i=1:ipixels)

if(b_w_map(j,i,:)==0)

wave_heights(j,i) = 0;

H_FDS(j,i) = 0;

else

t_req = 77.23*F(j,i)^0.67/(Ua^0.34*g^0.33);

if(t_req > t_req_max)

t_req_max = t_req;

end

H_FDS(j,i) = 0.27*Ua^2/g;

wave_heights(j,i) = Ustar^2/g * 4.13*10^(-2) * (g*F(j,i)./Ustar^2).^(1/2);

Tp(j,i) = Ustar/g * 0.751 * (g*F(j,i)./Ustar^2).^(1/3);

if(Tp_SW < Tp(j,i))

Tp(j,i) = Tp_SW;

%disp('depth-limited at %d,%d', j,i)

non_dim_F = (g*Tp(j,i)/(Ustar*0.751))^3;

wave_heights(j,i) = Ustar.^2./g .* 4.13*10^(-2) * (non_dim_F).^(1/2);

end

if(wave_heights(j,i)>0.6*d)

%disp('steepness-limited at %d,%d', j,i)

wave_heights(j,i) = 0.6*d;

end

end

end

end

%t_req_max

figure;

fetch_colors = colormap;

fetch_colors(1,3)=0;

colormap(fetch_colors);

imagesc(wave_heights);

colorbar('eastoutside');

title('Brevard IRL Wave Heights, %d degrees, %d m/s');

figure;

colormap(fetch_colors);

imagesc(Tp);

colorbar('eastoutside');

title('Brevard IRL Peak Period, %d degrees, %d m/s');

C.1.4: energy_calc.m

Uses wave conditions calculated above to determine energy content and power of waves for each pixel on the map.

function[E, P] = energy_calc(wave_heights, Tp, atlantic_flag)

g=9.81; %m/s^2

rho = 1000; %kg/m^3

d = 1; %m

[wavelengths,k] = L_calc(Tp,d);

Co = wavelengths./Tp;

n = 1/2*(1 + 2.*k.*d./sinh(2.*k.*d));

Cg = n.*Co;

for j=1:size(wave_heights,1)

for i=1:size(wave_heights,2)

if(wave_heights(j,i) > wavelengths(j,i)*1/7 || wave_heights(j,i) > 0.78*d)

H_limited(j,i) = (wavelengths(j,i)*1/7 < 0.78*d)*wavelengths(j,i)*1/7 ...

+ (wavelengths(j,i)*1/7 >= 0.78*d)*0.78*d;

else

H_limited(j,i) = wave_heights(j,i);

end

end

end

E = 1/8*rho*g*H_limited.^2;

P = E.* Cg;

maxP=max(max(P));

for j=1:size(P,1)

for i=1:size(P,2)

if atlantic_flag(j,i)==1

P(j,i)=.15*maxP;

end

end

end

figure;

fig_colors = colormap;

fig_colors(1,3)=0;

colormap(fig_colors);

imagesc(E);

colorbar('eastoutside');

title('Fort Pierce IRL Wave Energy, wind from %d degrees at %d m/s');

figure;

colormap(fig_colors);

imagesc(P);

colorbar('eastoutside');

title('Fort Pierce IRL Wave Power, wind from %d degrees at %d m/s');

C.1.5: L_calc.m

Matrix version of dispersion equation solver for L and k, given h and an [m x n] matrix T. Called in energy_calc.m above.

function[L,k] = L_calc(T,h)

CLOSE_ENOUGH = 1e-10; % threshold for ending iterations

g=9.81; % defines acceleration due to gravity in m/s^2

Lo=g*(T.^2)./(2*pi); % defines Lo from the dispersion eq'n, same dimensions as T

L=Lo; % initializes L, the wavelength matrix, which has dim of T

diff=ones(size(L)); % initializes the difference variable to a (relatively) large value

n=0;

N=0;

for i=1:size(diff,1),

for j=1:size(diff,2),

while diff(i,j) > CLOSE_ENOUGH, % this is the loop for iterative application of dispersion eq'n

L2 = Lo.*tanh((2*pi*h)./L);

diff=abs(L2-L); % diff tells how close the previous loop's and the current

% loop's L vlaues are

L=L2; % updates L

n=n+1;

end % breaks out of loop when diff<0.001

N=N+n;

end

end

k=(2*pi)./(L);

C.1.6: do_it_all.m

Calls all the above IRL wave energy functions in the correct order to avoid confusion with arguments needed. The initial mapreading function

must be called before running this m-file, since that operation is only needed once.

fetch = fetch_calc_red(bw_map, IRL_features, atl_flag,54000,28250,42.4);

sprintf('finished fetch. entering wave height calc.')

[wave_heights, Tp, H_FDS, t_req_max] = wave_calc(fetch, bw_map,8.4,1);

[E, P] = energy_calc(wave_heights, Tp, atl_flag);

C.1.7: energy_slope.m

Allows for an estimation of wave power in a depth of 1m given either 4.5km or 1.5km fetchand an array of wind speeds; plots

wave power versus wind speed on a log-log plot. Intended as a generalized estimate of wave power for various locations in the

lagoon.

function[wave_H, P, Tp]=energy_slope(windspeed, fetch)

%fetch = 4500 for optimistic, 1500 for conservative

g = 9.8;

rho=1000;

Ua = windspeed; %10m elevation windspeed,array

F = fetch;

d = 1;

Cd = 0.001.*(1.1.*ones(size(Ua))+0.03.*Ua);

Ustar = sqrt(Cd.*Ua.^2);

Tp_SW = 9.78*(d/g)^(1/2);

t_req = 77.23*F^0.67./(Ua.^0.34.*g^0.33);

H_FDS = 0.27.*Ua.^2./g;

wave_H = Ustar.^2/g * 4.13*10^(-2) .* (g*F./Ustar.^2).^(1/2);

Tp = Ustar./g * 0.751 .* (g*F./Ustar.^2).^(1/3);

for i=1:length(Tp)

if(Tp(i)>Tp_SW)

sprintf('period-limited at %d', i)

Tp(i) = Tp_SW;

non_dim_F = (g*Tp(i)./(Ustar.*0.751)).^3;

wave_H = Ustar.^2./g .* 4.13*10^(-2) .* (non_dim_F).^(1/2);

end

if(wave_H(i)>0.6*d)

sprintf('depth-limited at %d', i)

wave_H(i) = 0.6*d;

end

end

[wavelengths,k] = L_calc(Tp,d);

Co = wavelengths./Tp;

n = 1/2*(1 + 2.*k.*d./sinh(2.*k.*d));

Cg = n.*Co;

for i=1:length(wave_H)

if(wave_H(i) > wavelengths(i)*1/7 || wave_H(i) > 0.78*d)

H_lim(i) = (wavelengths(i)*1/7 < 0.78*d)*wavelengths(i)*1/7 ...

+ (wavelengths(i)*1/7 >= 0.78*d)*0.78*d;

else

H_lim(i) = wave_H(i);

end

end

E = 1/8*rho*g*H_lim.^2;

P = E.* Cg;

loglog(Ua,P)

title('Wave Power vs. Wind Speed');

xlabel('Wind Speed (m/s)')

ylabel('Wave Power (W/m)')

C.1.8: calc_hist.m

Prints histograms of the wind speed in a data set of wind speeds recorded every 5 minutes and loaded as a data file into Matlab®;

intended for use with the data extracted from text files of PAFB wind records for 2006

function[month_mins, daily_mins, total_month_hrs,xmonth, monthPcentrd] =

calc_hist(Month_m,total_days_in_month,days_of_data)

[month_mins,xmonth]=hist(Month_m,30);

month_mins=month_mins*5;

daily_mins=month_mins/days_of_data; %size(month,2); %240 for year_m, 278 for yrdata

total_month_hrs=total_days_in_month.*daily_mins/60;

stem(xmonth, total_month_hrs)

title('___ Wind Distribution')

xlabel('Windspeed (m/s)')

ylabel('Hours per month')

figure;

total_month_hrs_norm = total_month_hrs/(24*total_days_in_month);

stem(xmonth, total_month_hrs_norm)

title('___ Normalized Wind Distribution')

xlabel('Windspeed (m/s)')

ylabel('Relative Hours')

figure;

for i = 1:length(month_mins)

cumulative(i) = sum(total_month_hrs(1:i));

end

plot(xmonth,cumulative(length(cumulative))-cumulative)

title('Cumulative Distribution of Annual Wind')

xlabel('Windspeed (m/s)')

ylabel('Hours')

xmonthpwr=xmonth.^(1.3); %exp=1.3 for 1.5km fetch, 2.3797 for 4.5km fetch

monthPcentrd=(sum(xmonthpwr.*total_month_hrs)./sum(total_month_hrs))^(1/1.3);

C.1.9: convertOSS.m

Reformats a .xls file containing raw OSSI Wave Logger data into a new .xls file containing columns with the date, collection time,

time during sampling, and water level in cm.

function convertOSS(Hz)

%This function inputs a text file of water level "counts" recorded by the

%capacitance wave gage and converts them to cm with a formula given by

%Ocean Sensor Systems. A time stamp for only the first count was originally

%taken from the text file; it is used to generate time stamps for all data

%such that there is one column of time stamps and one column of water level

%in cm.

%Hz is the fequency at which the wave gage was configured to sample

%Input file

matrix = xlsread('WLOG_000.xls');

a = size(matrix);

%Omit column 13, which has all zeroes (due to formatting of inputted

%matrix)

for i = 1:12

new(:,i) = matrix(:,i);

end

%Concatenate all the rows of water level data into one large row

new2 = new(1,:);

for i = 2:a(1)

new2 = [new2,new(i,:)];

end

%Transpose water level data to a column of length b

new2 = new2';

%Convert from counts to cm

new2 = new2.*50./4096;

b = length(new2);

%Add timestamps for all water level data points from timestamp given at

%beginning of data series

date(1) = datenum([2007 8 11 15 57 0]);

for i = 2:b

%1/864000 is the fraction of one day that 1/10 of a second constitutes

%1/20 of a second is how often data are sampled

date(i) = date(i-1) + (1./(86400*Hz));

end

%Transpose the timestamps into a column

date = date';

%Change from julian to vector date

date2 = datevec([date]);

%Create matrix with vector date and column of water levels

final = [date2,new2];

size(final)

%Write new csv file with data in proper format

dlmwrite('final.csv',final,',');

%Plot data for visual inspection

plot(date,new2)

set(gca,'Box','Off');

set(gca,'FontSize',10);

datetick('x',13,'keepticks','keeplimits');

C.1.10: process_wave_data.m

Performs zero-up-crossing analysis on a time series of water levels and returns the significant wave height and peak period and

prints the amplitude spectrum

function [N, wave_heights, zero_ups, periods, freqs, amp_spec] = process_wave_data(eta, t, Hz)

% Zero up-crossing wave height analysis

n=1;

m=2;

for i = 1:length(eta)

if(eta(1)<0)

if(eta(i)>0)

z1=i;

break

end

end

if(eta(1)>0)

if(eta(i)<0)

z1=i;

break;

end

end

end

zero_ups(1) = (t(z1)+t(z1+1))/2;

last_upcrossing = z1;

for i = z1:(length(eta)-2)

if( (eta(i+1) > 0) & (eta(i) < 0) )

zero_ups(m) = (t(i+1) + t(i))/2;

periods(m-1) = zero_ups(m)-zero_ups(m-1);

m = m+1;

segment = eta(last_upcrossing:i);

%look for crests

crests(n) = max(segment);

%look for troughs

troughs(n) = min(segment);

last_upcrossing = i+1;

n = n+1;

end

end

N = n-1

wave_heights = crests-troughs;

wave_heights = wave_heights';

zero_ups = zero_ups';

crests = crests';

troughs = troughs';

periods = periods';

figure;

subplot(2,1,1), plot(wave_heights);

subplot(2,1,2), plot(t, eta);

xlabel('t (s)');

figure;

[freq_heights, x_heights] = hist(wave_heights,40)

subplot(2,1,1), bar(x_heights, freq_heights);

title('Hs');

for i = 1:length(freq_heights)

cumulative(i) = sum(freq_heights(1:i));

end

subplot(2,1,2), plot(x_heights, cumulative);

Hs_threshhold = 2/3*cumulative(length(cumulative))

figure;

hist(periods,30)

title('Tp');

% Energy spectrum generation

Ts = 1/Hz;

ne = length(eta);

Tr = t(ne-1)-t(1);

fr = 1/Tr;

fN = 1/(2*Ts);

freqs = [0, fr:fr:fN, -fN+fr:fr:-fr];

Eta = eta(1:length(freqs));

sig_spec = fft(Eta)/ne;

amp_spec = abs(sig_spec);

figure;

stem(freqs, amp_spec);

C.2: W

ave Properties

C.2.1: newton_raphson.m

Newton-Raphson method for solving dispersion equation to calculate wavelength of linear water waves; performs same function as

L_calc.m above.

function[kp] = newton_raphson(T,h)

CLOSE_ENOUGH = 1e-10;

g = 9.81; %gravitational accel

k = 0.5; %initialize k

k_next = 0.8; %intialize k_next to greater than k+CLOSE_ENOUGH

sigma = 2*pi/T; %wave frequency

n=0;

while abs(k-k_next)>CLOSE_ENOUGH,

k = k_next;

f = sigma^2-g*k*tanh(k*h); %want this to equal 0

df_dk = -g*(k*(h*(1/(cosh(k*h))^2))+tanh(k*h));

k_next = k - f/df_dk;

n = n+1;

end

%display results, inputs, and number of cycles

kp=k;

C.2.2: sine_particles.m

Calculates and illustrates particle vertical and horizontal displacement at a given time in a water column over 4 wavelengths for a

linear wave

function[x_disp,y_disp,z,x] = sine_particles(T,h,H)

%water particle positions over 4 wavelengths at t=T/2

% in y-direction, for z=0;

% in x-direction, for x=0:L/2

kp = newton_raphson(T,h)

L=2*pi/kp;

x=[0:.01:4*L];

z=[H/2:-.01:-h];

t=0.5*T %Note that since we are modeling over more than a wavelength, the choice of t is not

important

for i = 1:length(x)

for j=1:length(z)

% deep water (linear) ellipses of wave particle movement

x_disp(j,i) = -H/2*cosh(kp*(h+z(j)))/sinh(kp*h).*sin(kp.*x(i)-2*pi/T*t);

y_disp(j,i) = H/2*sinh(kp*(h+z(j)))/sinh(kp*h).*cos(kp.*x(i)-2*pi/T*t);

end

end

for i=1:length(z)

if z(i)>0

continue;

else

p=i;

break;

end

end

p

for j=p:length(z)

plot(x,h+z(j)+y_disp(j,:))

hold on;

end

ylabel('z (m)')

xlabel('x (m)')

title('water particle vertical displacements at t=T/2 for x=0 to 4L')

figure;

for i = 1:length(x)/4+2 %only one wavelength, plus 2 columns to get the full effect

plot(x(i)+x_disp(p:length(z),i),z(p:length(z)),'b', x(i), H/2*cos(kp*x(i)-2*pi/T*t), 'r')

hold on;

end

ylabel('z (m)')

xlabel('x (m)')

title('water particle x-displacement at t=T/2 for x=0 to L')

figure;

for j = p:length(z)

for i = 1:5:1.5*(length(z)-p)

plot(x(i)+x_disp(j,i),h+z(j)+y_disp(j,i),'b', x(i), h+ H/2*cos(kp*x(i)-2*pi/T*t),'r')

hold on;

end

end

ylabel('z (m)')

xlabel('x (m)')

title('water particle positions at t=T/2 for x=0 to L')

C.2.3: sine_particles_with_time.m

Same as sine_particles.m except additionally illustrates the movement of particles over 0.1 seconds.

function[x_disp,y_disp,z,x] = sine_particles(T,h,H)

%water particle positions over 4 wavelengths at t=T/2

% in y-direction, for z=0;

% in x-direction, for x=0:L/2

kp = newton_raphson(T,h)

L=2*pi/kp;

x=[0:.01:4*L];

z=[H/2:-.01:-h];

t=0.5*T %Note that since we are modeling over more than a wavelength, the choice of t is not

important

for i = 1:length(x)

for j=1:length(z)

% deep water (linear) ellipses of wave particle movement

x_disp(j,i) = -H/2*cosh(kp*(h+z(j)))/sinh(kp*h).*sin(kp.*x(i)-2*pi/T*t);

y_disp(j,i) = H/2*sinh(kp*(h+z(j)))/sinh(kp*h).*cos(kp.*x(i)-2*pi/T*t);

end

end

for i=1:length(z)

if z(i)>0

continue;

else

p=i;

break;

end

end

p

for j=p:length(z)

plot(x,h+z(j)+y_disp(j,:))

hold on;

end

ylabel('z (m)')

xlabel('x (m)')

title('water particle vertical displacements at t=T/2 for x=0 to 4L')

figure;

for i = 1:length(x)/4+2 %only one wavelength, plus 2 columns to get the full effect

plot(x(i)+x_disp(p:length(z),i),z(p:length(z)),'b', x(i), H/2*cos(kp*x(i)-2*pi/T*t), 'r')

hold on;

end

ylabel('z (m)')

xlabel('x (m)')

title('water particle x-displacement at t=T/2 for x=0 to L')

figure;

for j = p:length(z)

for i = 1:5:1.5*(length(z)-p)

plot(x(i)+x_disp(j,i),h+z(j)+y_disp(j,i),'b', x(i), h+ H/2*cos(kp*x(i)-2*pi/T*t),'r')

hold on;

end

end

ylabel('z (m)')

xlabel('x (m)')

title('water particle positions at t=T/2 for x=0 to L')

C.3: W

avemak

er O

ptim

ization

C.3.1: stroke.m

Calculates the stroke needed to generate waves with parameters h, T, H using a flapper paddle

function[Sflap, Spiston, Sexp, Sexp1, Shyp] = stroke(h,T,H)

%Calculates stroke needed to generate waves with parameters h,T,H with a

%flapper paddle, piston paddle, exponential decay, and hyperbolic curve

%plunger

g=9.8;

kp = newton_raphson(T,h)

Sflap=H/(4*(sinh(kp*h)/(kp*h))*(kp*h*sinh(kp*h)-cosh(kp*h)+1)/(sinh(2*kp*h)+2*kp*h));

Spiston = H/(2*(cosh(2*kp*h)-1)/(sinh(2*kp*h)+2*kp*h));

z = [-h:.001:0];

arrrrgh = 4*kp.*sinh(kp*h)./(sinh(2*kp*h)+2*kp*h).*(trapz(z,exp(kp*z).*cosh(kp*(h+z))));

Sexp = H/arrrrgh;

arrrgh1 = 4*kp.*sinh(kp*h)./(sinh(2*kp*h)+2*kp*h).*(exp(-kp*h)*(2*kp*h+exp(2*kp*h)-1))/(4*kp);

Sexp1=H/arrrgh1;

Shyp=H;

kh = [0:.01:6];

H_S_flap = 4.*(sinh(kh)./kh).*(kh.*sinh(kh)-cosh(kh)+1)./(sinh(2*kh)+2*kh);

H_S_piston = 2*(cosh(2*kh)-1)./(sinh(2*kh)+2*kh);

H_S_exp = 4*kp.*sinh(kh)./(sinh(2*kh)+2*kh).*(exp(-kh).*(2*kh+exp(2*kh)-1))/(4*kp);

H_S_hyp = 1;

figure;

plot(kh, H_S_flap, 'b', kh, H_S_piston, 'r', kh, H_S_exp, 'g', kh, H_S_hyp, 'c')

ylabel('H/S')

xlabel('kh')

legend('flap', 'piston', 'exponential decay', 'hyperbolic')

figure;

plot(kh, 1./H_S_flap, 'b', kh, 1./H_S_piston, 'r', kh, 1./H_S_exp, 'g', kh, 1/H_S_hyp, 'c' )

ylabel('S/H')

xlabel('kh')

legend('flap', 'piston', 'exponential decay', 'hyperbolic')

C.3.2: find_standing.m

Demonstration that Standing (Evanescent) Wave Components are Negligible for Exponential Decay-Shaped Paddles, and that

results improve for higher-resolution integration

function[H,ks,kp,Sz]=find_standing(T,h,n,S)

%Calculates evanescent wave coefficients for exponential decay paddle with

%stroke Sz; Based on Ch.6 Dean & Dalrymple

%Also calculates significant wave height generated and standing and

%progeressive

%T = incoming wave period

%h = water depth

%n = number of terms of the standing wave desired

%S = max stroke length

kp=newton_raphson(T,h); %progressive wave component

g=9.81;

kh=0.0001:0.0001:2*n*pi; %all frequencies for at least as many cycles as desired terms

y1=-tan(kh); % = sigma^2*h/(g*k*h), for k=ks

plot(kh,y1,'b') %the places where y1 and y2 cross are standing wave k's

hold on;

s=2*pi/T; %sigma

y2=s^2*h./(g*kh);

plot(kh,y2,'r');

hold off;

m=1;

for j=1:length(kh) %find crossing frequencies

if(y1(j)<y2(j));

continue;

else

break;

end

end

for i=j:length(kh)

if(y1(i)>=y2(i))

ks(m)=kh(i)/h;

continue;

else

if(y1(i-1)>=y2(i-1) && y1(i)<y2(i))

m=m+1;

if(m>n)

break;

end

end

continue;

end

end

ks

z=[0:-.0001:-h];

Sz=S*exp(kp*z); %exponential decay paddle

figure;

plot(Sz,z);

for m=1:n

C(m)=trapz(z,1/2*Sz.*(s*cos(ks(m)*(h+z))))/(ks(m)*trapz(z,(cos(ks(m)*(h+z))).^2));

end

C

Ap = -trapz(z,Sz./2*s.*cosh(kp*(h+z)))/(kp*trapz(z,(cosh(kp*(h+z))).^2))

H = -2*Ap./g*s*cosh(kp*h)

C.3.3: paddleswath.m

Illustration of Lateral Boundary Condition at Face of Exponential-Decay-Shaped Plunger Buoy, given deep-water sinusoidal

waves

function[dS]=paddleswath(T,h,S,H)

% Calculates the position of the exponential decay buoy face for all depths

% at t=0, t=T/4, and t=T/2

% Plots paddle face movement (and therefore particle displacement) over

% time at the surface and at half-depth.

% Plots maximum differnce in face position for all depths

kp = newton_raphson(T,h);

z = [H/2:-.01:-h]; %Plots show +eta to sea floor at -h

t = [0:.01:2*T]; %Plots over 2 periods

eta = H/2*cos(2*pi/T*t); %Sine wave at surface

%in general, s(z,t) = S*exp(kp*(z-eta));

S0t=S*exp(kp*(0-eta)); %Buoy face at top of buoy, wrt t

S0_5t=S*exp(kp*(-0.5*h-eta)); %Buoy face at buoy half-way down, wrt t

Sz0=S*exp(kp*(z-H/2)); %Buoy face when buoy top is at +H/2, wrt z

SzT_2=S*exp(kp*(z+H/2)); %Buoy face when buoy top is at -H/2, wrt z

SzT_4=S*exp(kp*(z+0)); %Buoy face when buoy top is at 0, wrt z

dS = Sz0-SzT_2; %Difference in face displacement betw. crest and trough

plot(Sz0,z, SzT_4,z,SzT_2,z); %Plot face position at various times

axis([0,S,-h,H/2]);

title('Sinusoidal Buoy Motion in Z-axis');

ylabel('z (m)');

xlabel('S(z) (m)');

legend('t=0', 't=0.25T', 't=0.5T');

figure;

plot(t,S0t,'b',t,S0_5t,'r'); %Plot face position at given z over 2 periods

title('Strokes at z=0 and z=-0.5h wrt time');

ylabel('S (m)');

xlabel('t (s)');

legend('Top of buoy', 'Half-way down buoy');

figure;

title('Extent of the paddle stroke wrt z');

plot(dS,z);

C.3.4: Smatrix.m

Same as AB.6, except calculated with more resolution

function[s,z,t]=Smatrix(T,h,S,H)

%Plots the surface of an exponential decay paddle over time, and

%shows particle movement diameters with depth given sinusoidal

%movement of the buoy

kp = newton_raphson(T,h);

z = [H/2:-.01:-h];

t = [0:.01:4*T];

eta = H/2*(cos(2*pi/T*t));

w = H/2*2*pi/T*cos(2*pi/T*t); % vertical velocity = derivative of eta wrt t

s = zeros((H/2+h)/.01,4*T/.01); % initialize stroke matrix

for j=1:length(z) % rows are increasing depth

for i=1:length(t) % columns are increasing time

s(j,i) = S*exp(kp*(z(j)-eta(i)));

end

end

plot(s,z); %plot the face of the buoy at each time

axis([0,0.5,-h,H/2+0.1]);

title('Buoy/Paddle face position over 4 periods');

ylabel('z (m)');

xlabel('x (m)');

figure;

plot(t,s(51,:),t,s(76,:),t,s(length(z),:));

%Note: all s-values over S are invalid, and therefore the highest row in s

%to hold totally valid values is -eta

hold on;

% for n=1:8

% plot(t,(n+.5*((-1)^n-1))*pi*.215-(-1)^n*.215*acos(1-t/.215)-sqrt(2*.215*t-t.^2),'m');

% end

title('Water particle movement at various depths given sinsoidal buoy movement');

xlabel('t (s)');

ylabel('stroke (m)');

legend('z=-0.25m', 'z=-0.5m', 'z=-1m');

hold off;

S_extent = [max(s,[],2)-min(s,[],2)];

figure;

plot(S_extent,z);

title('Depth vs. Magnitude of horizontal particle movement at paddle face');

C.4: W

EC D

esign Calculation

s

C.4.1: buoy_def.m

Creates an excel file defining the profile of the buoy; to be used with Autodesk Inventor as the source of the widths defining the buoy

profile

function[curve] = buoy_def(h,T,H)

z=[-h:.01:0];

kp = newton_raphson(T,h);

L = 2*pi/kp;

d_min = L/2;

[Sflap, Spiston, Sexp, Sexp1, Shyp] = stroke(h,T,H);

curve = Shyp*cosh(kp*(h+z))/sinh(kp*h);

curve = curve';

% xlswrite('buoy_curve.xls', 'param name', 'A1');

% xlswrite('buoy_curve.xls', 'value', 'B2');

% xlswrite('buoy_curve.xls', 'unit', 'C1');

xlswrite('buoy_curve1.xls', curve, 'B1:B100');

xlswrite('buoy_curve1.xls', 'm', 'C1:C103');

xlswrite('buoy_curve1.xls', kp, 'B101:B101');

xlswrite('buoy_curve1.xls', L, 'B102:B102');

xlswrite('buoy_curve1.xls', d_min, 'B103:B103');

xlswrite('buoy_curve1.xls', 'ul', 'C101:C101');

C.4.2: find_best_magnet.m

Used for calculating BHmax strength and mass of a ring magnet given its dimensions and N-rating

function[V,BHmax1_V, BHmax2_V, mass] = find_best_magnet(OD, ID, L, N, mm_or_in)

% V returns in cubic meters

% BHmaxes return in KJ

% masses return in grams

if mm_or_in == 'mm'

elseif mm_or_in == 'in'

ID = ID*25.4;

OD = OD*25.4;

L = L*25.4;

else

return;

end

A = ((OD/2)^2-(ID/2)^2)*pi;

V = A*L*10^-9;

switch N

case 35,

BHmax1_V = 263*V;

BHmax2_V = 279*V;

case 38

BHmax1_V = 287*V;

BHmax2_V = 303*V;

case 40

BHmax1_V = 303*V;

BHmax2_V = 318*V;

case 42

BHmax1_V = 318*V;

BHmax2_V = 334*V;

case 45

BHmax1_V = 342*V;

BHmax2_V = 358*V;

case 50

BHmax1_V = 374*V;

BHmax2_V = 406*V;

otherwise

BHmax1_V = 0;

BHmax2_V = 0;

end

mass = V*7500*10^3;

C.4.3: gen_dims.m

Calculates the number of turns and turns per inch per layer for the linear generator, given the ideal inductance, former diameter,

coil length and depth, and number of layers permissible

function[] = gen_dims(L,a,b,c,layers)

%b=[1:.1:5];

Nsqd = L*(3*a+9.*b+10*c)/(0.2*a^2);

N = sqrt(Nsqd)

TPI = N./b

TPIpLayer = TPI./layers

%plot(b,TPIpLayer);

R = 306/(1000*36)*b

C.4.4: force_moment_calc.m

Calculates wave forces on a partial-depth wave barrier, based on equations from the CEM, section VI-5-52

function[Fmo] = force_moment_calc(w, h, rho, Hs, Tp, r_wave_rod)

%w = barrier penetration depth,

%h = water depth

%r_wave_rod = distance from SWL to first support

g=9.8;

[Lp,kp] = L_calc(Tp,h);

Fo = rho*g*Hs*sinh(kp*h)/(kp*cosh(kp*h));

Fmo = Fo*(w/h)^(0.386*(h/Lp)^-0.7);

w_h = w/h

h_Lp = h/Lp

M = Fmo*r_wave_rod;

C.5: Postprocessing Rou

tines

C.5.1: freq_calibration.m

Displays the power output and proportion of the record time that the DC/DC converter was powered during a trial of the

oscillation table, given the oscillation frequency and distance and the record of the circuit’s data.

function[speed, mA_h_WEC, WEC_avg_pwr, on_ratio] = freq_calibration(WEC_file, xls_sheet, excitation_freq, travel)

%*****************Load Files*****************

%xls files entered as strings, for example,

%'test_data/front_street_180408_WEC.xls'.

%import the data/text file from the WEC into excel and read

%here. The user interface import is used for retrieving non-strictly-numerical data

speed = 2*travel*excitation_freq;

load_str = ['xlsread(', 'WEC_file',',','xls_sheet',')'];

WEC_data = eval(load_str);

pause;

Rsense = 10;

Rsense1 = 10;

WEC_freq = 8;

load_slope = 488.9; %ADC counts per V

sense_slope = 938.24; %ADC counts per mA

sense_offset = 266; %ADC counts at 0 current

sense1_slope = 1006; %ADC counts per mA

coil_slope = 674.52; %ADC counts per V

count_Vload = 0;

%************Retrieve date and initialize time ******

hour_W = WEC_data(3:length(WEC_data)-1,1);

minutes_W = WEC_data(3:length(WEC_data)-1,2);

seconds_W = WEC_data(3:length(WEC_data)-1,3);

file_string = [xls_sheet];

WEC_time = ((hour_W-hour_W(1))*3600)+((minutes_W-minutes_W(1))*60)+(seconds_W-seconds_W(1)); %time elapsed since

data collection start, in seconds

ADCsense = WEC_data(3:length(WEC_data)-1,6);

ADCload = WEC_data(3:length(WEC_data)-1,7);

ADCcoil = WEC_data(3:length(WEC_data)-1,4);

ADCsense1 = WEC_data(3:length(WEC_data)-1,5);

Isense = (ADCsense-sense_offset)./sense_slope; %in mA (from calibration and 0V ICs)

Vload = ADCload./load_slope; %slope in ADC counts per volt, 0 offset

Vcoil = ADCcoil./coil_slope; %slope in ADC counts per volt, 0 offset

Isense1 = (ADCsense1)./sense1_slope; %in mA (from calibration and 0V ICs)

P1 = Isense1.*Vcoil; %mW into DC/DC converter

P2 = Isense.*Vload; %mW out of DC/DC converter

DCDCefficiency = P2./P1;

comparison_data = [Vcoil'; Isense1'; Isense'; Vload'; P1'; P2'; DCDCefficiency']'

load_str = ['save ', file_string,'_data comparison_data'];

eval(load_str);

mA_h_WEC = sum(Isense1)/(WEC_freq*(WEC_time(length(WEC_time)))); %hours*milliamps, per hour of sampling

for i = 1:length(comparison_data)

if comparison_data(i,4)>4.90

count_Vload = count_Vload + 1;

end

end

on_ratio = count_Vload/length(comparison_data)*100;

WEC_charge = sum(P2/1000)/(WEC_freq); %total energy collected, in N*m=V*A*s

WEC_n = length(WEC_time);

Tr = WEC_time(WEC_n-1)-WEC_time(1);

WEC_avg_pwr = WEC_charge/Tr*1000; %mW

str =['avg speed = ', num2str(speed),' (in/s)'];

disp(str)

str =['Tr = ', num2str(Tr),' (s)'];

disp(str)

str =['total energy (charge)= ', num2str(mA_h_WEC),' (mA-s per second of sampling)'];

disp(str)

str =['avg_pwr = ', num2str(WEC_avg_pwr),' (mW)'];

disp(str)

str =['ON ratio = ', num2str(on_ratio),'%'];

disp(str)

%Plot shows how coil output corresponds to input wave time series

figure;

plot(WEC_time, P2, 'm', WEC_time, Vcoil, 'r');

title('DC/DC Power Out and Coil Voltage vs. time');

legend('Power out (mW)', 'Coil Voltage (V)');

C.5.2: efficiency_spectrum.m

Conducts the WEC performance and spectral analysis calculations given .xls files containing the OSS wave staff record and the

WEC data record

function[Hs, Hmo, Tp, wg_avg_power, WEC_avg_power, efficiency, on_ratio] = efficiency_spectrum(WEC_file,

WEC_worksheet, wave_file, water_depth)

%*****************Load Files*****************

%Files entered as strings, for example,

%'test_data/front_street_180408_WEC.xls'.

%Both files should be xls files; import the csv file from wg into Excel and

%process here; import the data/text file from the WEC into excel and read

%here. The user interface import is used for retrieving non-strictly-numerical data

load_str = ['xlsread(', 'WEC_file', ', WEC_worksheet' ,')'];

WEC_data = eval(load_str);

load_str = ['uiimport(', 'wave_file',')'];

eval(load_str);

pause;

Rsense = 10;

Rsense1 = 10;

WEC_freq = 8;

load_slope = 488.9; %ADC counts per V

sense_slope = 938.24; %ADC counts per mA

sense_offset = 266; %ADC counts at 0 current

sense1_slope = 1006; %ADC counts per mA

coil_slope = 674.52; %ADC counts per V

density = 1015; %lagoon water, in kg/m^3

capture_width = 0.4318; %meters

n_seg = 1024; %number of data points per spectrum in wave analysis

%************Retrieve date and initialize time ******

day_W = WEC_data(1,1);

month_W = WEC_data(1,2);

year_W = WEC_data(1,3);

hour_W = WEC_data(3:length(WEC_data)-1,1); %remove final row in case of partial data

minutes_W = WEC_data(3:length(WEC_data)-1,2);

seconds_W = WEC_data(3:length(WEC_data)-1,3);

year_wg = char(textdata(1,1))-0; %turns 'Y07' into an array of 3 ascii values

year_wg = 2000+[(year_wg(2)-48)*10+(year_wg(3)-48)]; %converts to a decimal year

month_wg = char(textdata(1,2))-0;

month_wg = [(month_wg(2)-48)*10+(month_wg(3)-48)];

day_wg = char(textdata(1,3))-0;

day_wg = [(day_wg(2)-48)*10+(day_wg(3)-48)];

hour_wg = char(textdata(1,4))-0;

hour_wg = [(hour_wg(2)-48)*10+(hour_wg(3)-48)];

mins_wg = char(textdata(1,5))-0;

mins_wg = [(mins_wg(2)-48)*10+(mins_wg(3)-48)];

secs_wg = char(textdata(1,6))-0;

secs_wg = [(secs_wg(2)-48)*10+(secs_wg(3)-48)];

freq_wg = char(textdata(1,7))-0;

freq_wg = [(freq_wg(2)-48)*10+(freq_wg(3)-48)];

clear data;

if (day_W ~= day_wg) | (month_W ~= month_wg) | (year_W ~= year_wg)

display('Dates do not match');

disp('WEC: ')

WEC_date_str = ['day: ',num2str(day_W),' month: ',num2str(month_W),'year: ',num2str(year_W)];

%hour_W(1),minutes_W(1),seconds_W(1)

disp(WEC_date_str)

disp('wg:')

wg_date_str = ['day: ',num2str(day_wg),' month: ',num2str(month_wg),'year: ',num2str(year_wg)];

%hour_wg(1),mins_wg(1),secs_wg(1)

disp(wg_date_str)

end

date_string = [num2str(day_W),'_',num2str(month_W),'_',num2str(year_W)];

wg_final = [num2str(day_W),'_',num2str(month_W),'_',num2str(year_W),'_',WEC_worksheet,'_','wg.csv'];

wg_data = convertOSS_WEC(wave_file,freq_wg,wg_final,year_wg,month_wg,day_wg,hour_wg,mins_wg,secs_wg);

WEC_time = ((hour_W-hour_W(1))*3600)+((minutes_W-minutes_W(1))*60)+(seconds_W-seconds_W(1)); %time elapsed since

data collection start, in seconds

wg_time = ((wg_data(:,4)-wg_data(1,4))*3600)+((wg_data(:,5)-wg_data(1,5))*60)+(wg_data(:,6)-wg_data(1,6)); %time

elapsed since data collection start, in seconds

delta_t = ((hour_wg(1)*3600+mins_wg(1)*60+secs_wg(1))-(hour_W(1)*3600+minutes_W(1)*60+seconds_W(1)));

%************ Separate data into variables ***********

wave_data = wg_data(:,7); %wave height in cm

wg_time = wg_time(~isnan(wave_data)); %remove data indeces with NANs in the wave gage data

wave_data = wave_data(~isnan(wave_data)); %remove data indeces with NANs in the wave gage data

ADCsense = WEC_data(3:length(WEC_data)-1,6);

ADCload = WEC_data(3:length(WEC_data)-1,7);

ADCcoil = WEC_data(3:length(WEC_data)-1,4);

ADCsense1 = WEC_data(3:length(WEC_data)-1,5);

%**************Calculate WEC performance results*************

Isense = (ADCsense-sense_offset)./sense_slope; %in mA (from calibration and 0V ICs)

Vload = ADCload./load_slope; %slope in ADC counts per volt, 0 offset

Vcoil = ADCcoil./coil_slope; %slope in ADC counts per volt, 0 offset

Isense1 = (ADCsense1)./sense1_slope; %in mA (from calibration and 0V ICs)

P1 = Isense1.*Vcoil; %mW into DC/DC converter

P2 = Isense.*Vload; %mW out of DC/DC converter

DCDCefficiency = P2./P1;

comparison_data = [Vcoil'; Isense1'; Isense'; Vload'; P1'; P2'; DCDCefficiency']';

% limit the data to full cap charge cycles

i=1;

while(Vcoil(i)>=3)

i=i+1;

end

j=length(Vcoil);

while(Vcoil(j)>=3)

j=j-1;

end

comparison_data = comparison_data(i:j,:);

Isense = Isense(i:j);

Vload = Vload(i:j);

Vcoil = Vcoil(i:j);

Isense1 = Isense1(i:j);

P1 = P1(i:j);

P2 = P2(i:j);

DCDCefficiency = DCDCefficiency(i:j);

WEC_time = WEC_time(1:j-i+1);

WEC_outfile = [date_string,'_',WEC_worksheet,'_WEC.csv'];

load_str = ['dlmwrite(','WEC_outfile',',','comparison_data',')'];

eval(load_str);

count_Vload = 0;

for i = 1:length(comparison_data)

if comparison_data(i,4)>4.90

count_Vload = count_Vload + 1;

end

end

on_ratio = count_Vload/length(comparison_data)*100; % Percentage of time DC/DC converter was outputting power

mA_h_WEC = sum(Isense1)/(WEC_freq*(WEC_time(length(WEC_time)))); %hours*milliamps, per hour of sampling

WEC_charge = sum(P2/1000)/(WEC_freq); %total energy collected, in N*m=V*A*s

WEC_n = length(WEC_time);

Tr = WEC_time(WEC_n-1)-WEC_time(1);

WEC_avg_power = WEC_charge/Tr*1000; %mW

%***************Incoming wave analysis*****************

eta = (detrend(wave_data))/100; %in m

figure(1);

subplot(2,1,2),plot(wg_time, eta*100);

xlabel('time (s)')

figure; %Plot how coil output corresponds to input wave time series

%plot(wg_time, eta*100, 'b', WEC_time, P2, 'm', WEC_time, Vcoil, 'r'); %use when wave record is shorter than WEC

record

plot(wg_time(1:round(length(P2)*10/8)), eta(1:round(length(P2)*10/8))*100, 'b', WEC_time, P2, 'm', WEC_time, Vcoil,

'r');

title('Eta, DC/DC Power Out, and Coil Voltage vs time');

legend('Eta (cm)', 'Power out (mW)', 'Coil Voltage (V)');

xlabel('time (s)');

%*** Time series analysis ***

[wave_heights, T, Hs] = heights(eta, wg_time);

figure;

plot(wave_heights);

title('Wave Heights (m)');

figure;

hist(wave_heights, 20);

title('Wave Heights Histogram (Hmode)');

xlabel('H (m)')

figure;

hist(T, 20)

title('Wave Period Histogram (Tmode)')

xlabel('T (s)')

%*** Frequency analysis ***

[Tp, Lp, kp, Hmo, variance] = spectral_anal(eta, wg_time, water_depth, P2, Vcoil, n_seg);

Hmo_avg = mean(Hmo);

Tp_avg = mean(Tp);

Lp_avg = mean(Lp);

kp_avg = mean(kp);

var_avg = mean(variance);

wg_energy = var_avg * 1010 * 9.81; %J/m^2

Cg = 1/2*(1+(2*kp_avg*water_depth)./sinh(2*kp_avg*water_depth))* Lp_avg/Tp_avg;

wg_avg_power = wg_energy*Cg*capture_width*1000; %(in mW)

efficiency = WEC_avg_power/wg_avg_power;

str = ['WEC record started ',num2str(delta_t),'s before wavegage record'];

disp(str)

str = [' WG Record time = ',num2str( wg_time(1024)-wg_time(1) ),' (s)'];

disp(str)

str = ['Hs = ', num2str(Hs),' (m)'];

disp(str)

str = ['Hmo = ', num2str(Hmo_avg),' (m)'];

disp(str)

str = ['Tp = ', num2str(Tp_avg),' (s)'];

disp(str)

str = [' Variance = ', num2str(var_avg),' (m^2)'];

disp(str)

str = [' Incident wave energy = 1/8*rho*g*Hmo^2 = ',num2str(wg_energy),' (J/m^2)'];

disp(str)

str = [' Wave avg power per m wavefront = ', num2str(wg_avg_power/capture_width),' (mW)'];

disp(str)

str = [' WEC avg excitation power = ', num2str(wg_avg_power),' (mW)'];

disp(str)

str =['Total energy = sum(Power*dt) = ', num2str(WEC_charge),' (J)'];

disp(str)

str =[' WEC avg power output = ', num2str(WEC_avg_power),' (mW)'];

disp(str)

str =['WEC efficiency = ', num2str(efficiency*100),'%'];

disp(str)

str =[' ON ratio = ', num2str(on_ratio),' (%)'];

disp(str)

figure;

subplot(2,1,1), stem(wg_freqs, wg_sig_spect, 'b');

subplot(2,1,2), stem(wg_freqs, wg_amp_spect, 'b');

title('Signal and Amplitude Spectra for Incident Wave Power')

xlabel('Hz')

C.5.3: convertOSS_WEC.m

Reformats the wavegage record into a time and a water surface data vector

function[final] = convertOSS_WEC(file,Hz,outfile,yr,month,day,hr,min,sec)

%This function inputs a text file of water level "counts" recorded by the

%capacitance wave gage and converts them to cm with a formula given by

%Ocean Sensor Systems. A time stamp for only the first count was originally

%taken from the text file; it is used to generate time stamps for all data

%such that there is one column of time stamps and one column of water level

%in cm.

%Hz is the fequency at which the wave gage was configured to sample

%Input file

load_str = ['xlsread(', 'file',')'];

matrix = eval(load_str);

a = size(matrix);

%Omit column 13, which has all zeroes (due to formatting of inputted

%matrix)

for i = 1:12

new(:,i) = matrix(:,i);

end

%Concatenate all the rows of water level data into one large row

new2 = new(1,:);

for i = 2:a(1)

new2 = [new2,new(i,:)];

end

%Transpose water level data to a column of length b

new2 = new2';

%Convert from counts to cm

new2 = new2.*50./4096;

b = length(new2);

%Add timestamps for all water level data points from timestamp given at

%beginning of data series

date(1) = datenum([yr month day hr min sec]);

for i = 2:b

%1/864000 is the fraction of one day that 1/10 of a second constitutes

%1/20 of a second is how often data are sampled

date(i) = date(i-1) + (1./(86400*Hz));

end

%Transpose the timestamps into a column

date = date';

%Change from julian to vector date

date2 = datevec([date]);

%Create matrix with vector date and column of water levels

final = [date2,new2];

%size(final);

%Write new csv file with data in proper format

load_str = ['dlmwrite(','outfile',',','final',')'];

eval(load_str);

%Plot data for visual inspection

figure;

subplot(2,1,1),plot(date,new2)

title('Eta (cm) vs time, first raw from OSS converter, then from upcrossing code');

xlabel('time (hh:mm:ss)')

set(gca,'Box','Off');

set(gca,'FontSize',10);

datetick('x',13,'keepticks','keeplimits');

C.5.4: spectral_anal.m

Divides the wavegage record into segments, windows it, and determines the wave properties based on FFT analysis.

function [Tp, Lp, kp, Hmo, variance, first_data_point, last_data_point] =

spectral_anal(eta, t, water_depth, P2, Vcoil, n_seg,a, b)

n = length(eta);

ni = (floor(n/n_seg))*4-3 %number of "subspectra" with 75% overlap

Tr = t(n_seg)-t(1);

Ts = Tr/n_seg;

fr = 1/Tr;

fN = 1/(2*Ts);

wg_freqs = [0, fr:fr:fN, -fN+fr:fr:-fr];

pos_freqs = [0, fr:fr:fN];

my_window = blackmanharris(n_seg);

window_area = mean(my_window);

my_window = my_window/window_area;

var_threshhold = 0.0005;

for i = 0:1:ni-1

%*** Window the data subset and create the subspectrum ***

current_eta = [eta(1+i*n_seg/4:(i+4)*n_seg/4)].*my_window;

if( i == a-1 )

str = ['first at ',num2str(1+i*n_seg/4) ];

first_data_point = (1+i*n_seg/4);

end

if ( i == b)

str = ['last at ',num2str((i+4)*n_seg/4 )];

last_data_point = (i+4)*n_seg/4;

end

%current_P2 = [P2(1+i*1024*10/8:(i+1)*1024*10/8)];

sig_spect =fft(current_eta)/n_seg;

amp_spect = abs(sig_spect);

pwr_spec = sig_spect.*conj(sig_spect);

pwr_spec_cont = pwr_spec/fr; %divide by fr for continuous spectrum

%*** Various methods for calculating variance; should agree ***

wg_pwr_spec_var = sum(pwr_spec_cont)*fr; %variance of eta(t) = time-avg wave energy/rho*g =

% auto-correlation of t=0 (in m^2)

wg_var_trap = trapz(wg_freqs,pwr_spec_cont);

eta_mean = sum(current_eta)/n_seg;

eta_var = sum((current_eta-eta_mean).^2)/n_seg;

eta_var_function = var(current_eta);

if(abs(wg_pwr_spec_var-wg_var_trap)> var_threshhold |...

abs(wg_pwr_spec_var-eta_var)> var_threshhold | ...

abs(wg_pwr_spec_var-eta_var_function)> var_threshhold)

disp('********** Check variance **********');

pause;

end

current_Hmo = sqrt(8*wg_pwr_spec_var);

[Tp_value, Tp_index] = max(pwr_spec_cont);

Tp_current = abs(1/wg_freqs(Tp_index));

while(isinf(Tp_current) | Tp_current > 5)

pwr_spec_cont(Tp_index) = 0;

[Tp_value, Tp_index] = max(pwr_spec_cont);

Tp_current = abs(1/wg_freqs(Tp_index));

end

[current_Lp,current_kp] = L_calc(Tp_current,water_depth);

%[wave_heights,periods, current_Hs ] = heights(current_eta, t);

if( mod(i,30)==0)

figure;

% subplot(3,2,3), hist(wave_heights, 15);

% title('Wave Heights Histogram (Hmode)');

% xlabel('H (m)')

% subplot(3,2,4), hist(periods, 15);

% title('Wave Period Histogram (Tmode)')

% xlabel('T (s)')

%Plot shows how coil output corresponds to input wave time series

%plot(wg_time, eta*100, 'b', WEC_time, P2, 'm', WEC_time, Vcoil, 'r');

subplot(3,1,1), plot(t(1:length(current_eta)), (current_eta)*100, 'b');

title('Eta, DC/DC Power Out, and Coil Voltage vs time');

legend('Eta (cm)')%, 'Power out (mW)', 'Coil Voltage (V)');

xlabel('time (s)');

subplot(3,1,3),stem(pos_freqs, 2*pwr_spec(1:n_seg/2+1), 'b')

title('Incident Wave Power Spectrum')

xlabel('frequencies (Hz)');

ylabel('Sxx');

end

variance(i+1) = wg_pwr_spec_var;

Hmo(i+1) = current_Hmo;

Tp(i+1) = Tp_current;

Lp(i+1) = current_Lp;

kp(i+1) = current_kp;

end

C.5.5: heights.m

Zero up-crossing analysis of wave time series

function [wave_heights, periods, Hs ] = heights(eta, t)

n=1;

m=2;

for i = 1:length(eta)

if(eta(1)<0)

if(eta(i)>0)

z1=i;

break

end

end

if(eta(1)>0)

if(eta(i)<0)

z1=i;

break;

end

end

end

zero_ups(1) = (t(z1)+t(z1+1))/2;

last_upcrossing = z1;

for i = z1:(length(eta)-2)

if( (eta(i+1) > 0) & (eta(i) < 0) )

zero_ups(m) = (t(i+1) + t(i))/2;

periods(m-1) = zero_ups(m)-zero_ups(m-1);

m = m+1;

segment = eta(last_upcrossing:i);

%look for crests

crests(n) = max(segment);

%look for troughs

troughs(n) = min(segment);

last_upcrossing = i+1;

n = n+1;

end

end

N = n-1;

wave_heights = crests-troughs;

wave_heights = wave_heights';

zero_ups = zero_ups';

crests = crests';

troughs = troughs';

periods = periods';

sorted_heights = sort(wave_heights, 'ascend');

Hs_threshhold = round(2/3*length(sorted_heights));

Hs = mean(sorted_heights(Hs_threshhold:end));

Appendix D: PIC

Microcontroller Code

D.1:

WECfinal.c

Contains main loop

#include <p18f4523.h>

#include <delays.h>

#include <usart.h>

#include <stdio.h>

#include <adc.h>

#include <math.h>

#include <stdlib.h>

#include <timers.h>

#include <portb.h>

#include "wec.h"

/****

ADDS INTERRUPTS to SAMPLING FREQ CONTROL, DATA LOGGER (115200 baud), RTC and ADC

****/

// ************* CONFIGURATION BITS **************** //

#pragma config OSC = HS

// oscillator selection

#pragma config FCMEN = OFF

// fail safe clock monitor

#pragma config IESO = OFF

// internal external oscillator switch over

#pragma config PWRT = ON

// power up timer

#pragma config BOREN = OFF

// brown out reset

#pragma config WDT = OFF

// watchdog timer

#pragma config MCLRE = ON

// MCLR enable

#pragma config LPT1OSC = OFF

// T1 oscillator enable

#pragma config PBADEN = OFF

// port B A/D enable

#pragma config CCP2MX = PORTC

// CCP2 mux

#pragma config STVREN = OFF

// stack overflow reset

#pragma config LVP = OFF

// low voltage ICSP

// *********** VARIABLE DECLARATIONS *********** //

// **** User-defined constant variables **** //

int sample_freq = 512;

//sampling frequency in Hz; use 512 for a wave up to 5Hz in freq., with fc = 50 on

filter

int T_record = 1800;

// record length in seconds

int record_freq = 8;

int bits_to_shift = 6;

//bits_to_shift = log base 2 of (sample_freq/record_freq)

float Rsense = 10;

// in Ohms

float Rsense1 = 10;

// **** Data variables **** //

int volatile ADCsense, ADCload, ADCcoil, ADCsense1;

unsigned long volatile ADCsense_sum, ADCload_sum, ADCcoil_sum , ADCsense1_sum;

int ADCsense_write = 0, ADCload_write = 0, ADCcoil_write = 0, ADCsense1_write = 0;

// **** Clock variables **** //

unsigned char volatile hundredths;

unsigned char volatile hours;

unsigned char volatile minutes;

unsigned char volatile seconds;

unsigned char volatile day;

unsigned char volatile month;

unsigned char volatile year;

// **** Sample rate Variables **** //

unsigned far ram char DateHeader[14];

unsigned far ram char WriteBuffer[60];

volatile int sample_counter = 0;

volatile char WriteFlag = 0;

long j = 0;

int samples_per_avg;

int n;

int k;

// *************************************************** //

// **** ISR declaration **** //

void RTC_sample_ISR(void);

#pragma code high_vector = 0x08 //high-priority interrupt

void interrupt_at_high_vector(void)

_asm GOTO RTC_sample_ISR _endasm

#pragma code

#pragma interrupt RTC_sample_ISR

void RTC_sample_ISR(void)

while (BusyADC());

SetChanADC( ADC_CH4 );

//Note that Vref is on CH3

while (BusyADC());

ConvertADC();

while (BusyADC());

ADCsense1 = ReadADC();

while (BusyADC());

SetChanADC( ADC_CH0 );

while (BusyADC());

ConvertADC();

while (BusyADC());

ADCsense = ReadADC();

while (BusyADC());

SetChanADC( ADC_CH1 );

while (BusyADC());

ConvertADC();

while (BusyADC());

ADCload = ReadADC();

while (BusyADC());

SetChanADC( ADC_CH2 );

while (BusyADC());

ConvertADC();

while (BusyADC());

ADCcoil = ReadADC();

ADCsense_sum += ADCsense;

ADCload_sum += ADCload;

ADCcoil_sum += ADCcoil;

ADCsense1_sum += ADCsense1;

sample_counter++;

if( ADCcoil < 1981) // 3V on coil. 660.23 ADC counts/volt from calibration

PORTBbits.RB5 = 0;

//De-activate switch, removing DC/DC converter from circuit

else if ( ADCcoil > 3301 ) // 5V on coil. 660.23 ADC counts/volt from calibration

PORTBbits.RB5 = 1;

//else maintain current operation

if(sample_counter == samples_per_avg-1)

sample_counter = 0;

WriteFlag = 1;

INTCONbits.INT0IF = 0;

// clear INT0 interrupt flag

// ******************* //

// ************* MAIN PROGRAM ************* //

void main (void)

OpenPORTB( PORTB_PULLUPS_ON );

TRISCbits.TRISC3 = 0; // configure SCL as an output for I2C master mode

TRISBbits.TRISB5 = 0; // DC/DC converter switch controller (NPN low-side load: 1=on, 0=off)

TRISBbits.TRISB4 = 0; // Logomatic reset pin 37 (NPN high-side load: 1=5V on load, 0=0V on load )

TRISBbits.TRISB3 = 0; // Logomatic stop pin 36 (NPN high-side load)

TRISBbits.TRISB2 = 0; // Logomatic power pin 35 (PNP low-side load: 1=off, 0=on)

TRISBbits.TRISB1 = 0; // Enable for 4.096V reference; 0=off, 5V=on

TRISBbits.TRISB0 = 1; // input square wave from RTC - INT0

PORTBbits.RB1 = 1;

// enable voltage reference

PORTBbits.RB3 = 0;

// start with reset and stop pins hi on Logomatic (off)

PORTBbits.RB4 = 0;

PORTBbits.RB5 = 0;

// start with DC/DC converter removed from circuit (switch off)

n = record_freq * T_record; //number of samples written to memory

samples_per_avg = sample_freq/record_freq;

//

ADCON1 = 00011010;

//Set first 5 ADC ports to analog (automatic)

PORTBbits.RB2 = 0;

// turn Logomatic on

SetClock();

//set time on RTC if it has lost power (comment out otherwise)

ConfigRTCSq(sample_freq);

OpenUSART(USART_TX_INT_OFF & USART_RX_INT_OFF & USART_ASYNCH_MODE &

USART_EIGHT_BIT & USART_CONT_RX & USART_BRGH_HIGH, 10);

OpenADC(ADC_FOSC_32 & ADC_RIGHT_JUST & ADC_0_TAD,

ADC_CH0 & ADC_INT_OFF & ADC_REF_VREFPLUS_VSS, 10);

ADCsense_sum = 0;

ADCload_sum = 0;

ADCcoil_sum = 0;

ADCsense1_sum = 0;

ReadClock();

ConvertTime();

k=1;

while(k<20)

Delay10KTCYx(800);

// delay at power up to allow Logomatic to prepare for receiving metadata

k=k+1;

//end of while

sprintf(DateHeader,"%02d %02d 20%02d\r",day,month,year);

//not enough time to write entire data string during loop at fs=512, frecord=4, so write base info at top of file

while( BusyUSART() );

TXSTAbits.SENDB = 0;

putsUSART(DateHeader);

while( BusyUSART() );

putrsUSART("hours mins seconds Vcoil Vsense1 Vsense Vload\r");

// Initialize external INT0 interrupt, which is always high priority

INTCONbits.INT0IF = 0;

// Start with INT0 interrupt flag clear

INTCON2bits.INTEDG0 = 0;

// Interrupt on falling edge

INTCONbits.INT0IE = 1;

// Enable INT0 interrupt

INTCONbits.GIE = 1;

// Global interrupt enable

INTCONbits.RBIE = 0;

// disable RB interrupt

INTCONbits.TMR0IE = 0;

// disable TMR0 interrupt

while(1)

//

PORTBbits.RB2 = 0;

// turn Logomatic on

if(WriteFlag)

WriteFlag = 0;

j++;

ADCsense_write = ADCsense_sum >> bits_to_shift;

ADCload_write = ADCload_sum >> bits_to_shift;

ADCcoil_write = ADCcoil_sum >> bits_to_shift;

ADCsense1_write = ADCsense1_sum >> bits_to_shift;

ADCsense_sum = 0;

ADCload_sum = 0;

ADCcoil_sum = 0;

ADCsense1_sum = 0;

ReadClock();

ConvertTime();

sprintf(WriteBuffer,"%02d %02d %02d.%02d %d %d %d %d\r", hours, minutes, seconds, hundredths,

ADCcoil_write, ADCsense1_write, ADCsense_write, ADCload_write);

while( BusyUSART() );

TXSTAbits.SENDB = 0;

putsUSART(WriteBuffer);

while( BusyUSART() );

if (j == n)

INTCONbits.INT0IE = 0;

// Disable INT0 interrupt

PORTBbits.RB3 = 1;

//stop file

Delay10KTCYx(5000);

// 1s delay

PORTBbits.RB4 = 1;

//reset

Delay10KTCYx(5000);

// 1s delay

PORTBbits.RB3 = 0;

//stop file

PORTBbits.RB4 = 0;

//reset

j = 0;

sample_counter = 0;

INTCONbits.INT0IE = 1;

// Enable INT0 interrupt

//end of if

//end of while

//end of main

D.2:

ConfigRTCsq.c

Programs square wave output pin’s frequency

#include <p18f4523.h>

#include <i2c.h>

#include "wec.h"

void ConfigRTCSq( int Hz )

char RS3_0;

int i; //for debugging

//Configure frequency of square wave output on RTC pin 7

switch(Hz)

case 1:

RS3_0 = 0b11110000;

break;

case 2:

RS3_0 = 0b11100000;

break;

case 4:

RS3_0 = 0b11010000;

break;

case 8:

RS3_0 = 0b11000000;

break;

case 16:

RS3_0 = 0b10110000;

break;

case 32:

RS3_0 = 0b10100000;

break;

case 64:

RS3_0 = 0b10010000;

break;

case 128:

RS3_0 = 0b10000000;

break;

case 256:

RS3_0 = 0b01110000;

break;

case 512:

RS3_0 = 0b01100000;

break;

case 1024:

RS3_0 = 0b01010000;

break;

default:

RS3_0 = 0b00000000;

//no square wave

break;

// Configure and open the I2C port

OpenI2C( MASTER, SLEW_ON);

//RTC communicates at 400kHz, so slew rate control must be enabled

SSPADD = 0x0B;

//RTC address = 0xD0 = 0b11010000

//SQWE (Square wave enable) address is 0x0A bit 6

I2CByteWrite( 0xD0, 0x0A , 0b01000000 );

//SQW address = 0x13 = 19;

I2CByteWrite( 0xD0, 0x13 , RS3_0 );

i = 3;

CloseI2C();

D.3:

ConvertTime.c

Reformats time data from BCD to decimal format

/****

Adapted for PIC18F from code written by Tony Cimaglia of Ocean Recon for a dsPIC30

****/

#include "wec.h"

void ConvertTime(void)

extern unsigned volatile char hundredths;

extern unsigned volatile char hours;

extern unsigned volatile char minutes;

extern unsigned volatile char seconds;

extern unsigned volatile char day;

extern unsigned volatile char month;

extern unsigned volatile char year;

hundredths = ((hundredths & 0b11110000)>>4)*10+(hundredths & 0b00001111);

// convert hundredths from BCD to decmial

seconds = ((seconds & 0b01110000)>>4)*10+(seconds & 0b00001111); // convert seconds from BCD to decmial

minutes = ((minutes & 0b01110000)>>4)*10+(minutes & 0b00001111); // convert minutes from BCD to decmial

hours = ((hours & 0b00110000)>>4)*10+(hours & 0b00001111); // convert hours from BCD to decmia

day = ((day & 0b00110000)>>4)*10+(day & 0b00001111); // convert day from BCD to decmial

month = ((month & 0b00010000)>>4)*10+(month & 0b00001111); // convert month from BCD to decmial

year = ((year & 0b11110000)>>4)*10+(year & 0b00001111); // convert year from BCD to decmial

D.4:

I2CByteWrite.c

Writes a byte of data from the RTC to the PIC via the I2C bus

/* ****

Based on EEByteWrite() written by Tony Cimaglia of Ocean Recon

Adapted for PIC18F from dsPIC. Intended for use in I2C Master mode

Writes one byte of data to a one-byte adrressable I2C device

**** */

#include <p18f4523.h>

#include <i2c.h>

#include "wec.h"

void I2CByteWrite(

unsigned char control1, //address byte of device

unsigned char address, //address location on device

unsigned char data ) //data to be written to the location

IdleI2C();

// waits for I2C port to become idle

StartI2C();

// I2C start condition

while(SSPCON2bits.SEN);

// wait until end of start condition

IdleI2C();

WriteI2C(control1);

// send the slave address, control code

while(SSPSTATbits.BF);

//wait until address is transmitted

while(SSPCON2bits.ACKSTAT);

//test for ACK condition received

IdleI2C();

WriteI2C(address);

// send the address where the data should be written

while(SSPSTATbits.BF);

while(SSPCON2bits.ACKSTAT);

IdleI2C();

WriteI2C(data);

//send data to be written

while(SSPSTATbits.BF);

while(SSPCON2bits.ACKSTAT);

IdleI2C();

StopI2C();

// start the stop condition

while(SSPCON2bits.PEN);

// wait until stop condition is over

IdleI2C();

D.5

I2CRandomRead.c

/* Adapted for use with a PIC18F from code written by Tony Cimaglia for a dsPIC30. Reads one byte of data from a

one-byte address device */

#include <p18f4523.h>

#include <i2c.h>

#include "wec.h"

unsigned char I2CRandomRead( unsigned char control,

// address byte of decive with 0 in last bit for "write"

//unsigned char address ) address of loaction in device

unsigned char data = '0';

// variable to hold read data

IdleI2C();

// waits for I2C port to become idle

StartI2C();

// I2C start condition

while(SSPCON2bits.SEN);

// wait until end of start condition

IdleI2C();

// waits for I2C port to become idle

WriteI2C(control);

// send the slave address (plus write condition LSB)

while(SSPSTATbits.BF);

//wait until address is transmitted

while(SSPCON2bits.ACKSTAT);

//test for ACK condition received

IdleI2C();

// waits for I2C port to become idle

WriteI2C(address);

// send the slave address

while(SSPSTATbits.BF);

while(SSPCON2bits.ACKSTAT);

IdleI2C();

StopI2C();

// start the stop condition

while(SSPCON2bits.PEN);

// wait till stop condition is over

control +=1;

// set last bit of control to 1 to indicate read action

IdleI2C();

// waits for I2C port to become idle

StartI2C();

// I2C start condition

while(SSPCON2bits.SEN);

// wait until end of start condition

IdleI2C();

// waits for I2C port to become idle

WriteI2C(control);

// send the slave address (plus read condition)

while(SSPSTATbits.BF);

while(SSPCON2bits.ACKSTAT);

IdleI2C();

// waits for I2C port to become idle

data = ReadI2C();

// read a data byte from the bus

NotAckI2C();

// send not acknowlage bit to end reading

while(SSPCON2bits.ACKSTAT);

IdleI2C();

StopI2C();

// start the stop condition

while(SSPCON2bits.PEN);

// wait until stop condition is over

IdleI2C();

// wait until the bus is idle

return data;

// return the data

D.6:

ReadClock.c

//

Adapted for PIC18F from code written by Tony Cimaglia for

//

//

a dsPIC30 for Ocean Research and Conservation Association

//

//

This program reads the RTC.

//

#include <p18f4523.h>

#include <i2c.h>

#include "wec.h"

void ReadClock(void)

extern unsigned volatile char hundredths;

extern unsigned volatile char seconds;

extern unsigned volatile char minutes;

extern unsigned volatile char hours;

extern unsigned volatile char day;

extern unsigned volatile char month;

extern unsigned volatile char year;

// Configure and open the I2C port

OpenI2C( MASTER, SLEW_ON);

//RTC communicates at 400kHz, so slew rate control must be enabled

SSPADD = 0x0B;

/*

(I2C_ON & I2C_IDLE_STOP & I2C_CLK_HLD & I2C_IPMI_DIS & I2C_7BIT_ADD

& I2C_SLW_DIS & I2C_SM_DIS & I2C_GCALL_DIS & I2C_STR_DIS &

I2C_NACK & I2C_ACK_DIS & I2C_RCV_DIS & I2C_STOP_DIS & I2C_RESTART_DIS

& I2C_START_DIS, 0x07);

*/

hundredths

= I2CRandomRead(0b11010000, 0);

// read hundredths from clock

seconds

= I2CRandomRead(0b11010000, 1);

// read seconds from clock

minutes

= I2CRandomRead(0b11010000, 2);

// read minutes from clock

hours

= I2CRandomRead(0b11010000, 3);

// read hours from clock

day

= I2CRandomRead(0b11010000, 5);

// read day from clock

month

= I2CRandomRead(0b11010000, 6);

// read month from clock

year

= I2CRandomRead(0b11010000, 7);

// read year from clock

CloseI2C();

D.7:

SetClock.c

// **** Adapted for the PIC18f from Tony Cimaglia's SetClock() function

//

written for the dsPIC30 for Ocean Recon **** //

#include <p18f4523.h>

#include <stdio.h>

#include <usart.h>

#include <i2c.h>

#include "wec.h"

void SetClock(void)

char ClockBuffer[13] = 0,0,0,0,0,0,0,0,0,0,0,0,0;

unsigned int DataIn = 0;

char j = 0;

char TxData[28];

unsigned char hours;

unsigned char minutes;

unsigned char seconds;

unsigned char day;

unsigned char month;

char year;

/******************* Set up UART *******************/

OpenUSART(USART_TX_INT_OFF & USART_RX_INT_OFF & USART_ASYNCH_MODE &

USART_EIGHT_BIT & USART_CONT_RX & USART_BRGH_HIGH, 10);

/***************************************************/

TXSTAbits.SENDB = 0;

sprintf(TxData, "\r\nSet Clock\r\nYYMMDDHHMMSS>\0");

while(BusyUSART());

putsUSART(TxData);

while(BusyUSART());

j = 0;

// Get Incomming Time Data

while(DataIn != 0x0D)

while(!PIR1bits.RCIF);

DataIn = RCREG;

PIR1bits.RCIF = 0;

putcUSART( DataIn ); //check the time being entered

while(BusyUSART());

ClockBuffer[j] = DataIn - 48;

//Turn ascii into decimal

j = j + 1;

CloseUSART();

//Convert decimal to BDC format

year

= ClockBuffer[0] << 4 | ClockBuffer[1];

month = ClockBuffer[2] << 4 | ClockBuffer[3];

day

= ClockBuffer[4] << 4 | ClockBuffer[5];

hours = ClockBuffer[6] << 4 | ClockBuffer[7];

minutes = ClockBuffer[8] << 4 | ClockBuffer[9];

seconds = ClockBuffer[10] << 4 | ClockBuffer[11];

// configure and open the I2C port

OpenI2C( MASTER, SLEW_ON);

//RTC communicates at 400kHz, so slew rate control must be enabled

SSPADD = 0x0B;

I2CByteWrite(0xD0,0x07,year);

I2CByteWrite(0xD0,0x06,month);

I2CByteWrite(0xD0,0x05,day);

I2CByteWrite(0xD0,0x03,hours);

I2CByteWrite(0xD0,0x02,minutes);

I2CByteWrite(0xD0,0x01,seconds);

CloseI2C();

183

Appendix E: Derivation of H/S

E.1 Exponential Decay Paddle

The water surface displacement for large x (far from the paddle) is given by

( ) ( ) ( ))cos2

)coscosh 1

0

txkH

txkhkg

A

tgppp

p

z

σσσφ

η −=−−

=∂∂

==

where

∫

∫

−

−

+

+−

=0

2

0

))((cosh

))(cosh(2

)(

h

pp

h

p

p

dzzhkk

dzzhkzS

A

σ

and zk pSezS =)(

Substituting and canceling the cosine term,

( )2

cosh

))((cosh

))(cosh(2

)(

- 0

2

0

Hhk

dzzhkgk

dzzhkzS

p

h

pp

h

p

=

+

+−

∫

∫

−

− σσ

From the hyperbolic cosine properties,

( )

( )[ ]( )

2 cosh

1)(2cosh2

1

)(cosh

2

0

0

2 Hhk

dzzhk

dzzhke

gk

Sp

h

p

h

p

zk

p

p

=

++

+

∫

∫

−

−σ

( )

( )( )

2 cosh

2

)(2sinh

2

1

)(cosh

2

0

0

2 Hhk

zk

zhk

dzzhke

gk

Sp

hp

p

h

p

zk

p

p

=

+

+

+

−

−∫σ

184

( )

( ) ( )2

cosh

2

2sinh

)(cosh

0

2 Hhk

k

hkhk

dzzhke

gk

Sp

p

pp

h

p

zk

p

p

=

+

+

∫−σ

Substituting the dispersion equation into the first coefficient and solving for H/S,

( )hkgk pp tanh2 =σ

( )[ ]( )

( ) ( )S

Hhk

k

hkhk

dzzhke

hk p

p

pp

h

p

zk

p

p

cosh

2

2sinh

)(cosh

tanh2

0

=

+

+∫−

( )( ) ( )

S

Hdzzhke

hkhk

hkk

h

p

zk

pp

pp p )(cosh2sinh

sinh4

0

=+

+ ∫−

In order to solve the final integral, convert the hyperbolic cosine into its

exponential form:

( )∫−

+0

)(cosh h

p

zkdzzhke p

∫−

+−+

+=

0 )()(

2

h

zhkzhkzk

dzee

epp

p

∫−

−−

+=

0

2

h

zkhkzkhkzk

dzeeee

epppp

p

∫−

−

+=

0 2

2

h

hkzkhk

dzeee ppp

02

22

1

h

hk

p

zkhk

zek

eep

pp

−

−

+=

−−+= −

−hk

p

hkhk

p

hk

p

ppp

hek

ee

k

e

20

22

1

2

185

( )

+−= −− hkhk

p

hk

pp

p

heek

e 21

22

1

( )

+−=

+−=

−−−−

p

hk

p

hkhk

p

hk

p

hkhk

k

hekee

k

hekee pppppp

4

2

4

21

2

( )

−+=

−−

p

hk

p

hk

k

ehke pp

4

12

2

186

E.2 Hyperbolic Curve Paddle

Starting once again from

( )2

cosh

))((cosh

))(cosh(2

)(

- 0

2

0

Hhk

dzzhkgk

dzzhkzS

p

h

pp

h

p

=

+

+−

∫

∫

−

− σσ

and substituting

( )( )( )hk

zhkSzS

p

p

sinh

cosh)(

+=

yields

( )( )( ) ( )( )

( )2

cosh

))((cosh

coshsinh

cosh

2

0

2

0

Hhk

dzzhkgk

dzzhkhk

zhkS

p

h

pp

h

p

p

p

=

+

++

∫

∫

−

− σ

σ

Since this leaves cosh2 terms in both the numerator’s and denominator’s integrals,

we cancel them to yield

( ) ( )2

coshsinh

1

2

2 Hhk

hkgk

Sp

pp

=σ

( ) S

H

hkgk pp

tanh

2

=σ

According to the dispersion relation,

( )hkgk pp tanh2 =σ

and therefore

1=S

H

187

Appendix F: Calibration Results

Current into DC/DC converter (Rsense = 10Ω, Vsense = 100mV max)

Rload = 1130Ω Applied Current (mA)

0 0.52 0.75 1.08 1.51 2.05 2.5 3.05 3.56 4.05 4.14

Vout op amp 0.26 0.685 0.869 1.12 1.515 2.05 2.49 3.03 3.53 4.02 4.09

ADC value 252 673 855 1105 1486 2019 2455 2986 3487 3961 4031

Rload = 1338Ω Applied Current (mA)

0 0.25 0.49 0.75 1.04 1.52 2.02 2.51 3.01 3.51 4.1

Vout op amp 0.258 0.465 0.65 0.842 1.064 1.534 2.02 2.51 2.99 3.49 4.09

ADC value 250 452 634 824 1041 1505 1991 2473 2954 3444 4030

Rload = 1973Ω Applied Current (mA)

0 0.25 0.5 0.75 1.01 1.25 1.5 2.01 2.25 2.53

Vout op amp 0.258 0.433 0.592 0.774 1.026 1.271 1.516 2.08 2.25 2.53

ADC value 263 423 577 757 1005 1245 1487 1988 2221 2494

ADC Value vs. Applied Current

y = 939.36x + 131.23

R2 = 0.9985

y = 942.55x + 118.14

R2 = 0.9992y = 932.81x + 106.3

R2 = 0.9970

1000

2000

3000

4000

5000

0 1 2 3 4 5Applied current (mA)

ADC value

Rload = 1338

Rload =1120

Rload = 1973

Linear (Rload = 1338)

Linear (Rload =1120)

Linear (Rload = 1973)

Average slope: 938.24 counts/mA All four 12-bit ADCs were programmed to use a voltage reference of 4.096V as the ceiling value, 4096 counts. In addition to the 10Ω sense resistor tests shown above and on the following page, calibration for a 1Ω sense resistor was also conducted. However, the low currents present during field testing create too low a voltage across this smaller resistance for the operational amplifier to boost dependably. Due to the 0.3V input offset voltage of the op amp, nonlinear operation was observed for currents less than 2mA on both current sense resistors, regardless of the attached load resistance value. Performance data for the 1Ω tests is available in electronic format in the file “\data processing\ADC calibrations.xls.”

188

Current out of DC/DC converter (Rsense = 10Ω, 100mV max)

Rload = 1178Ω Applied Current (mA) 0 0.5 0.75 1 1.5 2 2.5 3 3.5 4 4.05

Vout op amp 0 0.459 0.738 1.019 1.535 2.03 2.52 3.03 3.53 4.03 4.08

ADC value 0 448 723 1000 1508 2008 2492 2995 3492 3983 4031

Rload = 1493Ω Applied Current (mA) 0 0.25 0.5 0.75 1 1.25 1.5 2 2.5 3 4.04

Vout op amp 0.002 0.18 0.473 0.762 1.033 1.282 1.534 2.02 2.53 3.03 4.08

ADC value 0 174 462 747 1013 1258 1507 1999 2501 2997 4028

Rload = 1780Ω Applied Current (mA) 0 0.25 0.5 0.75 1 1.25 1.5 2 2.5 3.01

Vout op amp 0 0.19 0.489 0.781 1.033 1.281 1.533 2.02 2.52 3.04

ADC value 0 187 478 766 1013 1258 1506 2000 2495 3003

Isense1 ADC Value vs. Applied Current

y = 1006x - 13.098

R2 = 0.9995

y = 1006.1x - 18.732

R2 = 0.9996

y = 1001.1x - 13.164

R2 = 0.9999

0

1000

2000

3000

4000

5000

0 1 2 3 4 5

Applied current (mA)

ADC value

Rload = 1178

Rload = 1493

Rload = 1780

Linear (Rload = 1780)

Linear (Rload = 1493)

Linear (Rload = 1178)

Average Slope = 1006 counts/mA

189

Voltage on Capacitor (5.4V max)

Rload = 1130Ω Vcap total (V) 0 0.5 1.01 1.5 2.00 2.51 3.02 3.50 4.00 4.50 5.03 5.60

ADC value 0 332 672 1005 1351 1690 2038 2359 2701 3034 3394 3780

V at divider (1M+2.2M)

0 0.319 0.644 0.96 1.28 1.6 1.942 2.24 2.56 2.88 3.22 3.58

Vlow/Vtotal - 0.639 0.640 0.64 0.64 0.637 0.643 0.64 0.64 0.64 0.640 0.64

Voltage across Load Resistors (Output of DC/DC Converter) (5.0V max)

Voltage divider 666Ω+662Ω Vload total (V) 0 0.5 1 1.5 2.00 2.50 3.04 3.50 4.00 4.55 5.04

ADC value 0 242 485 729 974 1216 1464 1707 1949 2218 2460

V at divider 0 0.25 0.49 0.74 0.99 1.23 1.48 1.73 1.97 2.25 2.49

Vlow/Vtotal - 0.5 0.49 0.493 0.495 0.492 0.487 0.494 0.493 0.495 0.494

Voltage divider 1kΩ+1kΩ Vload total (V) 0 0.5 1.003 1.5 2.03 2.50 3.01 3.50 4.00 4.50 5.04

ADC value 0 241 488 732 996 1228 1479 1720 1967 2212 2460

V at divider 0 0.249 0.5 0.748 1.01 1.245 1.5 1.74 1.99 2.24 2.49

Vlow/Vtotal - 0.498 0.499 0.499 0.498 0.498 0.498 0.497 0.498 0.498 0.494

ADC values vs Voltage at component

y = 674.52x

R2 = 1

y = 487.03x

R2 = 0.9999

y = 490.82x - 1.3938

R2 = 1

0

500

1000

1500

2000

2500

3000

3500

4000

0 1 2 3 4 5 6Applied voltage (V)

ADC value

Capacitor VoltageVload @ Rload = 1.2kVload @ Rload = 2kLinear (Capacitor Voltage)Linear (Vload @ Rload = 1.2k)Linear (Vload @ Rload = 2k)

Average slopes:

Capacitor voltage: 674.52 counts/V Load voltage: 488.9 counts/V

190

Glossary

Active (real) power: Useful component of power that is dissipated by the purely

real component of impedance (resistance), distinguished from reactive

power by its net transfer of energy in one direction over time.

Added mass: The inertia added to a system when a body surrounded by a fluid

accelerates and must therefore accelerate the fluid occupying the space

and nearby spaces.

Airgap Shear Stress: The force per unit surface area occurring in the small space

between the rotor and stator in a generator, resulting from the interaction

between the magnetic field-producing component and the inductive

armature. This quantity determines the power factor of the

motor/generator, and it is limited by the material properties of the flux-

carrying component and the heating, reactance, etc. of the current-

carrying component.

Apparent power: Imaginary component of complex powr, defined as the product of

voltage and current in an electrical system.

Bandwidth: Segment in the frequency spectrum across which the frequencies of

interest span, especially the range of frequencies passed by a signal

processing filter or the range of frequencies over which significant power

is available in a wave spectrum.

Capture (width) ratio: Quotient of extracted power divided by the power incident

on the cross-section of the buoy.

Complex power: Total power in a system, incluing that component not dissipated

over time, but stored in the imaginary impedance, unusable until

converted to real power.

Damping coefficient: Constant of proportionality describing the energy dissipated

proportional to velocity in a mechanical system and proportional to

current in an electrical system, such as by viscous forces and resistances.

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Direct drive linear generator: A pwer-generating mechanism that transfers power

without any gear or belt intermediary reductions, allowing for reduced

frictional losses and reduced noise, but requiring a special motor and

control system due to the relatively high torque at lower speeds than

conventional motors.

Excitation (vs Radiation): Component of wave forces acting on an initially

stationary floating body due to both the incident wave pressure

distribution and the pressure distribution of diffracted waves bending

around the body.

Fourier transform: Mathematical analysis technique of characterizing a signal’s

components involving conversion from the time domain to the frequency

domain.

Harmonic: Integer multiple of the frequency of interest.

Heave: Linear vertical motion of a floating body.

Inductive coupling: The transfer of energy from one circuit component to another

through a shared magnetic field, allowing a change in current in one

device to induce a change in the current flowing through the other device.

Impedance matching: Designing a circuit’s load such that its impedance is equal to

the complex conjugate of its source’s impedance in order to achieve

maximum power transfer by minimizing the total impedance.

Natural frequency: The oscillation frequency to which a system’s motion defaults

when the system has been displaced and released, dependent on the total

mass and spring force acting in the direction of oscillation.

Pitch (of a magnet): Width of an individual magnet in an assembly of multiple

magnets aligned axially.

Pitch (ship motion): Rotation of a floating body about its transverse axis.

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Power factor: A coefficient between 0 and 1 that describes the phase difference

between voltage and current in a complex (AC) system; it is the ratio of

the average power to the

Radiation (vs Excitation): Forces acting on a floating body as a result of its

oscillation in the water column; including added mass and radiation

resisteance effects.

Reactance: The opposition presented by a pure coil or capacitor to the flow of

alternating current through it; diminishes current from an alternating EMF

by setting up an opposing EMF, without wasting energy.

Reactive power: Component of power that is alternately stored by complex

impedance elements and then released back into the AC circuit.

Reluctance: Resistance to magnetization

Resonance frequency: The excitation frequency that produces the largest output

signal for a given input signal magnitude. Closely tied to the natural

frequency of the system, but slightly lower depending on the damping

levels.

Shunt: A device that allows electrical current to pass around another point in the

circuit (opposite of a fuse). In the case of a shunt resistor, it allows for

indirect measurement of currents too high for an ammeter or in places

where an ammeter is inconvenient by inserting a minimal resistance

across which the voltage can be measured instead.

Stator: The stationary component of a motor or generator, which may contain

either the field windings or the armature, depending on the geometry of

the machine.

Sway: Linear lateral movement of a floating body.

Translator: The moving component of a linear generator, analogous to the rotor on

a conventional generator.

Wave spectrum: Description of the wave energy distribution across a range of

wave frequencies, used to describe the wave climate in a given location.