Digital Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology 287
Modelling and Experimental
Verification of Direct Drive Wave
Energy Conversion
Buoy-Generator Dynamics
MIKAEL ERIKSSON
ACTA
UNIVERSITATIS
UPSALIENSIS
UPPSALA
2007
ISSN 1651-6214
ISBN 978-91-554-6850-7
urn:nbn:se:uu:diva-7785
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To my son Philip
List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
II
III
IV
V
VI
VII
VIII
IX
Danielsson, O., Eriksson, M., Leijon, M., "Study of a Longitudinal Flux Permanent Magnet Linear Generator for Wave Energy
Converters" International Journal of Energy, 2006, 30:1130-1145
Leijon, M., Danielsson, O., Eriksson, M., Thorburn, K., Bernhoff, H., Isberg J., Sundberg, J., Ivanova, I., Sjöstedt, E., Ågren,
O., Karlsson, K.E., Wolfbrandt, A., "An electrical approach to
wave energy conversion" Renewable energy 31, 2006, pp. 13091319
Thorburn, K., Karlsson, K-E.,Wolfbrandt, A., Eriksson, M., Leijon, M., "Time stepping finite element analysis of variable speed
synchronous generator with rectifier" Applied Energy 83, 2006,
pp. 371-386
Eriksson, M., Thorburn, K., Bernhoff, H., Leijon, M. "Dynamics of a Linear Generator for Wave Energy Conversion" OMAE
2004, 23:th International Conferance on Mechanics and Arctic
Engineering, 2004, Vancouver, British Columbia, Canada, June
20-25
Eriksson, M., Isberg, J., Leijon, M., "Hydrodynamic Modelling
of a Direct Drive Wave Energy Converter" International Journal
of Engineering Science 43, 2005, pp. 1377-1387
Eriksson, M., Isberg, J., Leijon, M., "Theory and Experiment
of an Elastically Moored Cylindical Buoy", IEEE Journal of
Oceanic Engineering, vol 31, NO. 4, October 2006, pp. 959-963
Isberg, J., Eriksson, M., Leijon, M., "Instantaneous Energy Flux
in Fluid Gravity Waves" Submitted to Phys. Rev. E, Dec 2006
Waters, R., Stålberg, M., Danielsson, O., Svensson, O., Gustafsson, S., Strömstedt, E., Eriksson, M., Sundberg, J., Leijon, M.,
"Experimental results from sea trials of an offshore wave energy
system" Applied Physics letter 90, 034105 2007
Eriksson, M., Waters, R., Svensson, O., Isberg, R., Leijon, M.,
"Wave power absorption: Experiment and Simulation" Submitted
to Journal of Applied Physics, March 2007
5
X
XI
Wolfbrandt, A., Eriksson, M., Isberg, J., Leijon, M., "Simulation of a Linear Generator for Wave Power Absorption -PartI:
Modeling" Submitted to IEEE Transaction on Energy Conversion,
March 2007
Eriksson, M., Wolfbrandt, Waters, R., Svensson, O., Isberg, J.,
Leijon, M., "Simulation of a Linear Generator for Wave Power
Absorption -Part II: Verification", Submitted to IEEE Transaction
on Energy Conversion, March 2007
The author has also contributed with inputs to the following paper which not
is relevant for this thesis or the results are present in some of the included
papers.
A
B
C
D
E
Stålberg, M., Waters, R., Eriksson, M., Danielsson, O., Thorburn, K., Bernhoff, H., Leijon, M. "Full-Scale Testing of PM
Linear Generator for Point Absorber WEC" Sixth European Wave
Energy Conferance, August 29th-September 2nd, 2006, Glascow,
Scotland
Danielsson, O., Leijon, M., Thorburn, K., Eriksson, M., Bernhoff, H., "A Direct Drive Wave Energy Converter - Simulations
and Experiment" OMAE 2005, "24:th International Conference
on Mechanics and Arctic Engineering, Halkidiki, Greece, 12-17
June 2005
Danielsson, O., Thorburn, K., Eriksson, M., Leijon, M., "Permanent Magnet Fixation Concepts for Linear Generator" Fifth European Wave Energy Conferance, 17-20 September, Cork, Ireland
Bolund, B., Segergren, E., Solum, A., Lundström, L.,
Lindblom, A., Thorburn, K., Eriksson,M., Nilsson, K.,
Ivanova, I.,Danielsson, O., Eriksson, S., Bengtsson, H., Sjöstedt,
E., Isberg, J., Sundberg, J., Bernhoff, H., Karlsson, K-E.,
Wolfbrandt, A., Ågren, O., Leijon, M. "‘Rotating and Linear
Synchronous Generators for Renewable Electric Conversion - an
Update of the Ongoing Research Projects at Uppsala University"
Procedings from NORPIE 2004 conference 14-16 June 2004,
Trondheim, Norway
Bolund, B., Thorburn, K., Sjöstedt, E., Eriksson, M., Segergren,
E., Leijon, M., "Upgrading Generators with new Tools and High
Voltage Technology" International Journal on Hydropower and
Dams Vol. 11, issue 3, May 2004, pp. 104-108
Reprints were made with permission from the publishers.
6
Glossary of symbols
Symbol
SI unit
Quantity
σ
γ
φ
φ0
φd
λ
μ0
μr
η
ρ
ν
ω
A
B
D
E
H
I
Jf
Jm
M
v
a
A f ac
d
E
f
Fr
Fe
Fem
H
A/V m
Conductivity
Nm/s
Generator damping factor
m2 /s
Velocity potential
m2 /s
Velocity potential for incident wave
m2 /s
Velocity potential for diffracted wave
m
Wave length
V s/Am
Permeability of vacuum
Relative permeability
m
Wave elevation
Kg/m3
Density
m2 /s
Cinematic viscosity
rad/s
Angular frequency
Tm
Magnetic vector potential
T
Magnetic flux density
C/m2
Displacement field
V/m
Electric field
A/m
Magnetic field
A
Current
A/m2
Free current density
A/m2
Magnetization current density
A/m
Magnetization
m/s
Velocity
m
Buoy radius
Active stator length
m
Buoy draft
J
Energy
Hz
Frequency
N
Radiation force
N
Excitation force
N
Electromagnetic force
m
Wave height
7
Hs
m
Significant wave height
h
m
Water depth
J
W/m
Energy flux
Jpm
A/m2
PM-current density
Js
A/m2
Source current density
k
1/m
Wave number
ku
N/m
Spring constant upper endstop
kl
N/m
Spring constant lower endstop
kw
N/m
Spring constant wire
ks
N/m
Spring constant
L
H
Inductance
ls
m
Stator length
lp
m
Piston length
ma
kg
Added mass
m∞
kg
Added in infinity frequency limit
m
kg
Mass
P
W
Power
patm
Pa
Atmospheric pressure
p
Pa
Pressure
R
Ω
Resistance
Rr
kg/s
Radiation resistance
S
m2 /Hz
Wave power spectrum
Tp
s
Energy period
Wc
J
Co-energy
Z
kg/s
Radiation impedance
Bold symbols denotes a vector quantity and hat denotes a Fourier transformed quantity.
8
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Wave energy conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Energy absorption with point absorbers . . . . . . . . . . . . . . . . . . . . .
2.1 Overview of the proposed concept . . . . . . . . . . . . . . . . . . . . . .
2.2 Linear generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Energy absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Potential wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Buoy water interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Time series and power spectrum . . . . . . . . . . . . . . . . . . . . . . .
3.4 Wave energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Validity of potential wave theory . . . . . . . . . . . . . . . . . . . . . . .
4 Generator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Modelling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Field based modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Circuit theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Calculation of Impedances . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Mechanical damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Buoy geometry and damping factor . . . . . . . . . . . . . . . . . . . . .
5.2 Performance of simulation approaches . . . . . . . . . . . . . . . . . . .
5.3 Energy absorption with rectified armature voltage . . . . . . . . . .
6 Experiment and verification of models . . . . . . . . . . . . . . . . . . . . . .
6.1 Experimental generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Offshore experiment I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Offshore experiment II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Energy absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Experiment and modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Svensk sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
13
14
17
17
19
21
25
25
27
29
30
33
35
35
36
38
40
40
43
43
45
47
49
49
50
52
57
57
58
59
61
63
67
69
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
71
1. Introduction
Today’s huge energy consumption is largely based on fossil fuels, like coal
and oil. These sources have a limited lifetime and are not environmentally
friendly. Every year, carbon dioxide is released into the air, which contributes
to the greenhouse effect. It is necessary to find alternative energy sources if
we want to have the same welfare in the future. One of the major problems so
far in extracting energy from renewable sources is to reach energy prices that
can compete with the low cost of fossil fuels.
The unfriendly environment that a wave power device is exposed to makes it
harder to find a device with good economical potential compared for example
to wind power, even if the energy density is higher. On the other hand wave
power is an unused renewable energy source with huge potential. For Sweden
it has the benefit to follow the season variation of energy demands. In wave
power there are a great variety of different concepts, some better than others.
With research and experience these many concepts will converge to a few with
better potential than the others. This is a process that takes time but as the time
proceeds we get closer to a concept which is economically competitive.
1.1 Wave energy conversion
What makes wave energy interesting is that it has the highest energy density
among all renewable energy sources [1]. Further, it is available up to 90%
of the time at a given site, this can be compared with solar and wind power
which tend to be available just 20-30% of the time [2]. The higher degree of
utilization for wave power can be explained by considering waves as energy
a storage created by winds. Even when it has stopped blowing waves remain
for a long time, and waves can travel very long distances with very small energy losses. Another advantage for countries like Sweden, where autumns and
winters are very energy demanding periods, the natural seasonal variability of
wave energy follows the electrical demands. For solar energy is it the opposite.
If wave energy is so interesting and oceans contain so much environmental
friendly energy why are there no large commercial plants? From a technical
point there are many challenges to overcome in the attempt to extract energy
from waves and to reach economic feasibility. Some of the concepts have to
struggle with low velocities, typically 0.1Hz. A generator requires 500 times
higher frequency, typically [1]. This puts high demands on the generators and
gearboxes.
11
Figure 1.1: Illustration of different principles for wave energy absorption [3]. 1. Vertical oscillating buoy moving relative sea bottom, 2. Vertical oscillating buoy moving
relative a damping plate, 3. Heaving and pitching buoy moving relative a damping
plate, 4. Two buoys moving relative each other, 5. Floating device with a oscillatory
rotation motion around a fixed axle, 6. A flap following the water motion back and
forth, 7. Vertical oscillating buoy moving relative a fixed chamber, 8. The wave elevation causes pressure difference which drives a turbine, 9. The same principle as 8,
for offshore use. 10. The wave causes pressure difference which drives a turbine 11.
The waves are led into a narrowing channel; it makes an increase in wave amplitude.
Some of the water is spilling in to a reservoir where a turbine is place at the outlet.
Another factor, the irregular behaviour of ocean waves makes it difficult
to obtain maximum efficiency of a device over the entire range of excitation
frequency. Furthermore, the irregularities cause a power fluctuation that puts
12
higher demands on generators; they have to handle electrical overloads. The
peak power can be ten times larger than the average power. Power fluctuations
also cause problems for grid companies. But the most challenging is the survivability all concepts have to face in the extreme conditions that the oceans
offer. The wave power devices are exposed to salt water, biological growth
and under some periods very rough wave conditions. The structural loading in
the event of extreme weather conditions, such as hurricanes, may be as high
as 100 times the average loading [1]. In the design of a power device there are
two extremes; an underestimation or an over estimation of the design loads
for a device. In the first case total or partial destruction of the device can be
expected. In the latter case, very high construction costs thus making the technology non competitive.
Extracting energy from the ocean waves are not something new. The first
technique was patented more than 200 years ago. The first patent is from
France in 1799 by Girard and son. From 1855 to 1973 there are 340 patents
documented in Great Britain [1], which have carefully been documented by
Leishman and Scobie [4]. In figure 1.1 different types of conversion principles are illustrated schematically, number one illustrates the principle that is
studied in this thesis.
It is during the last 30 years that major’s steps have been taken, it begun
in the mid 1970: s during the oil crisis. Norway was one on the first countries and one of the leading in wave power research. Research has also been
conducted in Sweden. The two best known concepts are the IPS buoy [5] and
the hose pump [6]. So far most researchers have concentrated on the hydrodynamic aspects of different converters and not so much on how to transform
the motion created by a wave to electric energy. Independently on the chosen
technology it is crucial to have an overall solution from wave absorber to grid
connection. As this thesis will show the absorption is very much dependent
on the extraction of electric power.
1.2 Potential
The available energy in the world’s oceans is estimated to be between 200 and
5000GW, mostly found in offshore locations [7]. This can be compared to the
world’s technical potential of hydro power which is estimated to 2200GW of
which only 25% is exploited [7]. Most energy is found at the coastlines of
countries facing the major oceans, see figure 1.2. The wave climate around
the Swedish coast is different with smaller waves and less energy compared
to the major oceans, which put higher demand on the extraction technology.
In an article from 1979 [8] the total annual energy flux on the Swedish coast
was estimated to 40-60TWh, of that 3-15TWh was estimated to be utilizable. In a article from 2002 the technically available resource is approximately
5-10TWh [1]. The amount utilizable energy is very much dependent on the
13
60
50
50
40
30
20
30
100
50
15
20
30
40
50
70
40
30
40
40
100 50
10
15
70
50
15
20
15
20
15
15
10
20
40
30
100
50
20
40
40
40
60
70
100
20
20
50
100
Figure 1.2: Approximate yearly average global distribution of wave power levels in
kW per meter of wave front [9].
technology. Some devices delivers power only if the waves are large enough,
i.e. smaller waves will not contribute. Studies of the technically potential for
Swedish climates has been made [10, 11]. The total electricity consumption
is approximately 150ThW/year, where about half of the energy comes from
hydropower and the other half from nuclear power, and approximately 1%
comes from wind power.
1.3 Overview of the thesis
This work is a part of a larger project carried out at Uppsala University, which
aims to investigate and develop the wave energy concept described in chapter
2. Within the project two theses have been written, one focused on the linear
generator written by Oskar Danielsson [12]. He has designed and verified calculations of our first prototype of the linear generator [13, 14, 15]. Further, he
has studied longitudinal end effects [16]. The second thesis is about system
aspects and is written by Karin Thorburn [17]. She has studied connections of
many generators in a farm [18, 19, 20, 21, 22]. The focus on this thesis lies
in modelling the converter, from incident waves to output power. Different
modelling approaches have been tested due to accuracy and simulation time.
The ability to absorb energy, the dynamic behaviour and damping has been
studied to understand the limitations and possibility with this type of concept.
The hydrodynamic models used have the limitation to work for a converter in
operation condition, not for extreme waves and survivability.
The second chapter gives the idea behind and an explanation of the concept
which this thesis is based on. Further, an introduction to linear generators
and different known methods for increasing the power absorption for a point
absorber are explained and discussed.
14
The third chapter introduces the reader to potential wave theory, which has
been used trough out the thesis. Potential wave theory is a well known theory
used to simulate wave-body interaction and to calculate the energy content in
waves. Moreover, equations for calculating the instantaneous power flux in a
wave are presented. These equations are derived from potential wave theory
for infinite depth; the derivation is presented in paper [23].
The fourth chapter presents the different model approaches that are used
for modelling the generator. Further, an explanation of the approaches and the
kind of simulation they are applicable to are discussed.
The following chapters summarize and discuss the results presented in the
papers. The different modelling approaches are presented and some of the
more important results are highlighted. They are followed by a discussion. Finally, a summary of the most important conclusions of the thesis is given, followed by a summary of the papers and the contribution of the author are given.
The goal with the thesis is to clarify the dynamic behaviour and absorption of
the proposed concept for wave energy conversion and to investigate important
parameters for an effective system.
15
2. Energy absorption with point
absorbers
The concept studied in the thesis is a point absorber [24] connected to a direct driven linear generator located at the sea floor. This section presents an
overview of the proposed concept describing the different parts from buoy to
grid connection. Moreover, a short explanation is presented of how energy is
absorbed from the ocean waves as different techniques used to increase the
energy absorption.
2.1 Overview of the proposed concept
The proposed concept studied in this thesis is of type 1 in figure 1.1, a direct
driven linear generator connected to a heaving buoy (point absorber) [25, 26].
Figure 2.1 illustrates a farm of converters where one converter unit is opened
and the different parts are visible.
Figure 2.1: Illustration of a farm of converters where one converter unit is opened and
the different parts are visible.
17
Each converter consists of a floating buoy, from the buoy a rope is connected to the piston of a linear direct driven generator located at the seabed.
A spring is attached from the lower piston end to a concrete foundation on
the sea bottom. In wave crests the buoy and piston moves upwards and energy
is stored as potential and mechanical energy in the piston and spring respectively. In a wave trough the stored energy accelerates the piston downwards.
The piston is covered with rows of permanent magnets of alternating polarity.
When it moves relative to the stator an alternating magnetic field in the stator
coils induces a voltage giving rise to currents in the conductors. The current
in turn affects the piston with a force opposite to the direction of motion. Controlling the power output from the generator makes it possible to affect the
dynamics of the whole system. A concept using direct drive generators results
in a armature voltage that varies in both amplitude and phase, due to the irregular ocean waves and the oscillating motion. Figure 2.2 shows experimental
data from the test site in Lysekil, which clearly illustrates the irregular armature voltage and the output power. Variations are on both the short time scale,
a few seconds, and the long timescale, hours and days.
Phase voltage (V)
75
50
25
0
−25
−50
−75
110
115
120
Time (s)
125
130
(a)
4
Power (kW)
3
2
1
0
110
115
120
Time (s)
125
130
(b)
Figure 2.2: Example of experimental data from Lysekil, measured by Olle Svensson.
a) Phase voltage from a generator in real waves. b) Generated power for voltage in a).
The figure shows short time scale variations in power as variations in both amplitude
and frequency for the voltage.
18
Short time scale variations correspond to irregularities from one wave to the
next wave, whereas a long time scale variation depends on changes in weather.
The voltage has to be refined before connection to grid. The refinement consists of converting the irregular voltages to voltages with constant frequency
and amplitude corresponding to the grid. At the first stage we intend to rectify
the voltage from the unit to a constant DC voltage with passive diode rectifiers. The constant DC voltage is then inverted and transformed to the main
grid frequency and amplitude [25].
The proposed concept consists of many small wave power converters connected into a cluster of many hundred or even thousand units [20]. Each unit is
connected at the same DC level and the current from each unit is added before
inverting to grid frequency. Connecting a large number of units will smooth
out the power fluctuations on the short time scale. Further, a failure to one or
a few units will not affect the power production considerably, as would be the
case for a few larger units.
The wave energy converters are designed to have their eigenfrequencies
different from the frequency spectrum of incident waves; i.e. buoy is intended
to follow the wave elevation. In future designs it is possible to include different
control system such as latching or some kind of power control system with
wave prediction algorithms to increase the energy absorption, see section 2.3.
2.2 Linear generator
In many applications that involve linear/straight motions it can be advantageous to use linear generators or actuators to reduce the complexity of the
mechanical interface. Today there exist a number of different linear machines
[27, 28]. Wave power devices are traditionally using conventional high speed
rotating generators for energy conversion. This requires a system which converts the slow linear/rotation motion of the wave energy absorber to a high
speed rotating motion. Hydraulic systems and gearboxes are used for that purpose [29]. An alternative would be direct driven energy converters, such approach requires low speed generators and in some cases they have to be linear
[30, 31, 32, 33, 34, 35].
The first electric linear motor was developed more than 100 years ago in
the USA. However, linear generators are fairly new for wave energy applications. The idea with direct drive generators is to reduce the complexity of the
mechanical interface and there by reduce the number of movable parts and
to minimize the mechanical losses. In this way the need for maintenance will
hopefully be kept to a minimum. The mechanical interface is in this way replaced with an electrical interface. The electrical interface can be expected to
have a longer life time and require less maintenance.
Within this project a special type of generator has been developed to fit
the demands and required properties for an energy converter that is properly
19
(a)
(b)
Figure 2.3: Illustration of the main parts in the linear generator, a. The four sided
piston ant the stators, b. Cross section of piston and stator.
working and adapted to the wave motion. The generator has to handle electric
overloads and have a high damping factor for low speeds. A high damping factors means that the generator develops a high counteracting electromagnetic
force that will damp the piston motion.
20
The generator developed, is a three phase linear synchronous generator with
surface mounted permanent magnets [13, 14, 36, 37, 38]. The piston (or rotor
for a traditional rotating generator) is covered with rows of permanent magnets
of alternating polarity. Moreover, the magnets used are Nd-Fe-B magnets that
have very high remanence, about four times higher than ferrite magnets. The
magnet rows are separated with aluminum spacers, see figure 2.3. The stator
is made of laminated electrical non-oriented steel sheets and isolated copper
conductors. The conductors are wound in slots (holes) in the stator steel and
forms closed loops or coils.
A low speed generator is larger in size than a high speed generator for the
same power and voltage rating. An increase in size implies more magnets and
stronger attractive forces between piston and stator. As the attractive forces
increase rapidly with the decreased air-gap this puts very high demands on
precision and a robust supporting structure.
2.3 Energy absorption
Absorbing wave energy for conversion means that energy has to be removed
from the waves. This is the same as producing a wave that interferes destructively with the incoming waves; it can be argued that a good wave absorber is
closely related to its ability to generate waves [39]. Figure 2.4 illustrates the
principle how energy is absorbed from an incident wave in this 2D example.
The oscillating absorber is of infinite length and the incident waves are plane
parallel, see figure 2.4a. If the buoy oscillates in heave a symmetric radiated
wave is created, figure 2.4b, if it is in resonance with the incident wave and the
amplitude is optimal it has been shown that it is possible to absorb 50% of the
energy in the incident wave [40, 41, 42]. For a buoy oscillating in pitch motion, an unsymmetrical radiated wave is created, see figure 2.4c. In this case
it is possible to absorb more than 50% of the incident wave energy [41, 42].
With a combination of heave and pitch it is possible to absorb all incident energy, se figure 2.4d. Experiment with Salter’s duck has shown that absorption
of more than 80% of the energy in a monochromatic incident wave is possible
[43].
The parameters that are connected to the ability to absorb energy are excitation force, radiation impedance and damping. The first two parameters are
dependent on the buoy geometry. The last parameter, the damping, has to do
with the generator characteristics and how energy is extracted from the generator.
By using linear theory for buoy-water interaction and assuming that the
rope never slacks, 100% active piston/stator area, and that the electromagnetic
damping force is proportional to the piston velocity [41], gives that absorbed
energy can be written as,
21
(a)
(b)
(c)
(d)
Figure 2.4: a. Incident wave b. Symmetric wave radiation by heaving ocsillation c.
Antisymmetric wave generation by pitching oscillation d. Superposition of the generated waves on the incident wave results in complete absorption of the incident wave.
[39, 42]
2
γ F̂e 1
Pa =
2 (γ + Rr )2 + (ω m + ω ma ˘ks /ω − ρ gπ a2 )2
(2.1)
Where γ is the damping factor, R̂r the radiation resistance, m piston and
buoy mass, ma the added mass, a the buoy radius, ks the spring constant, and
F̂e the excitation force (see chapter 3 for further explanation the hydrodynamic
parameters). By letting,
ω m + ω ma ˘ks /ω − ρ gπ a2 = 0,
(2.2)
maximizes the absorbed power Pa . From the relation 2.2 the natural frequency of the converter is expressed as,
ω=
22
k + ρ gπ a2
m + ma
(2.3)
Figure 2.5: Wave elevation and vertical displacement of buoy position as function
of time [39]. a. Wave elevation, b.Vertical displacement of latched buoy (artificial
resonance), c. Vertical displacement of oscillating buoy in resonance.
The natural frequency of the converter has only to do with the buoy/piston
mass and the added mass of the buoy (the added mass depends on the buoy
shape). When the wave frequency coincides with the natural frequency of the
converter we have resonance in the system.
A vertical axis-symmetric point absorber (a heaving body which has a much
smaller diameter than the wave length) absorbs at most the energy transported
by the incident wave front of width equal to the wave length divided by 2π
[41]. This width is named the absorption width. By tuning the natural frequency of the system to coincide with the wave frequency, the system will be
in resonance. This is called phase control. In resonance there is a phase difference of 90 degrees between wave and buoy motion and the excitation force is
in phase with the buoy velocity. The natural frequency can be tuned by varying mass and spring constant, as discussed above. For a real wave with a broad
wave spectrum it will be difficult to use the resonance optimally. Instead an
artificial resonance can be obtained by something called latching. In latching
a lock and release mechanism forces the motion to have a phase difference of
90 degree to incident wave. A comparison of buoy motion for a resonant and
a latched system is given in figure 2.5.
The second parameter to optimize is the oscillator’s amplitude; this is adjusted by the damping factor. A larger damping will decrease the oscillator
amplitude and the velocity. In the proposed concept the damping is dependent
on the electric load. By changing load and in turn the power outtake it will be
possible to control the absorption.
Latching could be implemented by a heavy non linear damping, i.e. to have
a very large damping that almost makes the piston to stand still for low velocities. When the accelerating forces becomes too high and the buoy motion
is out of phase the damping decreases to a normal operation condition. Such
non-linear behaviour could be achieved using power electronics. There are a
number of control strategies how to control the phase and amplitude to optimize the power extraction from the waves [44]. The most advanced strategies
23
predict the incident wave for some seconds in advance to calculate optimal
motion for maximal energy absorption.
Simulations of wave energy converters similar to the proposed concept
show an increase in power absorption for a system with motion control. A
simulation of a heaving buoy moving relative a fixed reference made by H. Eidsmoen [45], showed that the year average power production was three times
higher for a system with a phase controlled motion than without. Furthermore, a controlled system gave a more steady power output. In an article by
K. Nielsen [46], simulation and experiment of a point absorber with hydraulic
power take off, showed that latching increased the absorbed power 50% in
regular waves but with only 4-8% for irregular waves.
24
3. Wave Theory
Modelling water waves is a very complex task. Simple problems quickly become very computational demanding. Depending on required accuracy and
purpose of the simulations different approximations can be made to simplify
the theories. Trough out this thesis small amplitude linear potential wave theory has been used. For a wave energy converter in normal operation condition
this theory valid, see section 3.5. To model survivability of a converter in
rough conditions other theories have to be adopted.
3.1 Potential wave theory
Potential wave theory is frequently used to describe water waves with small
amplitude compared with its wave length [47, 48, 41, 49]. The theory can be
derived from the conservation of mass and momentum equations;
∂ρ
+ ∇ · (ρ v) = 0
∂t
(3.1)
∂v
1
1
+ (v · ∇)v = − ∇Ptot + ν ∇2 v + f,
∂t
ρ
ρ
(3.2)
Where v is the velocity vector of the flowing fluid element, ρ the density,
Ptot the pressure of the fluid, ν the cinematic viscosity constant, and f external forces per unit volume. In this case, with a gravity wave, f = (0, 0, ρ g).
Moreover, the following assumptions are made,
• Ideal fluid
An ideal fluid has no viscosity, i.e. ν = 0. This will result in no friction
losses in the fluid.
• Incompressibility
For an incompressible fluid the continuity equation 3.1 gives ∇ · v = 0
• Irrotational
It is also assumed that the fluid is rotational free, i.e. ∇ × v = 0.
For an irrotational motion the velocity can be written as the gradient of a
velocity potential, v = ∇φ , inserting this into the incompressible continuity
equation results in the Laplace equation for the potential,
∇2 φ = 0.
(3.3)
25
The boundary conditions are the Cinematic Free Surface Boundary Condition (CFSBC), the Dynamic Free Surface Boundary Condition (DFSBC),
Bottom Boundary Condition (BBC) and Periodic Lateral Boundary Condition (PLBC). An illustration of the geometry and definition of parameters of
the boundary value problem is found in figure 3.1. Moreover, surface tension is neglected. The DFSBC is derived from Bernoulli’s equation. Letting
the pressure at the surface, p = 0, which follows from the transformation,
p → p − patm , gives the DFSBC,
∂φ 1
(3.4)
∇φ · ∇φ + gη = 0 at y = 0
∂t 2
The CFSBC is derived by noting that a fluid particle at the surface remains
there. Mathematically this is written,
D
∂
{y − η (x,t)} = {y − η (x,t)} + v · {y − η (x,t)} = 0
Dt
∂t
Which gives,
(3.5)
∂φ
∂η ∂φ ∂η
=
+
(3.6)
∂y
∂t
∂x ∂x
At the seabed, which is a rigid surface, there can be no normal velocity,
hence,
∂φ
=0
(3.7)
∂z
By assuming small amplitude compared to wavelength and water depth,
boundary conditions 3.4 and 3.6 can be linearized,
∂φ
+ gη = 0 at y = 0
∂t
(3.8)
∂φ
∂η
=
at y = 0
(3.9)
∂y
∂t
An analytical solution to the linearized boundary value problem is derived
by using separation of variables and applying PLBC, φ (x,t) = φ (x + λ ,t) =
φ (x,t + T ). The solution for a particular frequency ω is given by,
φω =
Hg cosh(k(h + z))
sin(kx − ω t)
2ω cosh(kh)
(3.10)
Where ω is the wave angular frequency, k is the wave number, z is the depth,
H is twice the amplitude, and h is the distance from surface to bottom. The
surface wave elevation is,
ηω (x,t) =
26
H
sin(kx − ω t)
2
(3.11)
Figure 3.1: Definition of parameters for the boundary value problem.
3.2 Buoy water interaction
The buoy-wave interaction can be divided into two separate problems, where
each problem gives rise to a force. These two forces are excitation force and
radiation damping force. The excitation force is calculated with the buoy kept
fixed with a constant draft. An incident harmonic plane parallel wave impinge
on the buoy, see figure 3.2a. The velocity potentials is solved for the boundary
value problem, where the total potential is the sum of the potential corresponding from the incident wave and the diffracted potential created by the
fixed buoy. The excitation forces for a harmonic wave with frequency ω is
calculated by integrating the pressure over the wet surface S of the buoy. The
expression for excitation force in the frequency domain for oscillation mode
j is given by,
F̂e, j (ω ) = iωρ
S
(φ̂0 + φ̂d )n j dS = fˆe, j (ω )η̂ (ω )
(3.12)
Where ω is the angular velocity of the incident wave, n j is the normal for
oscillating mode j, φ̂0 is the incident velocity potential, φ̂d is the diffracted
wave potential, fˆe, j (ω ) is the excitation force factor, and η̂ (ω ) is the wave
elevation.
To determine the radiation force it is assumed that the buoy oscillates with
the angular velocity ω in the absence of incident waves, see figure 3.2b. The
solution to the boundary value problem gives the radiated velocity potential
φr . As in the excitation problem the radiation force is calculated by integrating
the pressure over the wet surface S of the buoy. The alternative representation
is by multiply the radiation impedance, Z j j , with the buoy velocity, v̂ j . The
expression for radiation force in the frequency domain for oscillation mode j
is given by,
F̂r, j = iωρ
S
(φ̂r n j )dS = −Z j j v̂ j
(3.13)
27
The buoy water interaction is determined by the excitation force and radiation force, the excitation force will drive the buoy while the radiation force
will damp the buoy motion. To give an intuitive understanding of the radiation
impedance we divide it into a real and imaginary part,
Z j j = R j j + iX j j = R j j + iω m j j
(3.14)
Where the real part is radiation resistance and the imaginary part is the
added mass multiplied with the angular frequency. The radiation resistance
gives rise to a counteracting force on the buoy which is in phase with the buoy
velocity and is associated with active power losses. This force will damp the
buoy motion; the energy loss in kinetic energy is transformed into a radiated
outgoing wave. As an example, consider a free oscillating buoy. This buoy
will create an outgoing wave; the energy content in this wave comes from loss
of the buoys kinetic energy. After some time all buoy energy has been transformed into waves and the buoy stands still. The so called added mass ma has
the dimension of mass and corresponds to a volume of water that is moved
when the buoy is moving. The added mass is a function of angular velocity
and it is also dependent of the buoy shape. A large added mass give the buoy
more inertia, the eigenfrequency of the buoy is close related to added mass.
For simpler geometries such as cylinders and spheres it is possible to give
closed expression for the excitation force factor and radiation impedance [50],
the hydrodynamical parameters can also be calculated using software such as
WAMIT [51]. The buoy-water interaction presented is a linear system, where
the excitation force factor and radiation impedance is transfer functions relating the acting forces to incident wave elevation and buoy velocity. Until now
the theories is for the frequency domain. In the time domain multiplications
and transfer functions becomes convolutions, denoted ∗, and impulse response
functions respectively. The excitation force in the time domain is written as,
Fe,t (t) = ft (t) ∗ vt (t)
(3.15)
And the radiation force is given as,
Fr,t (t) = −Zt (t) ∗ vt (t)
(3.16)
In general the added mass does not vanish in the infinite frequency limit.
This causes problems when one wishes to apply Fourier transforms. To remedy this problem, the added mass in the infinite limit frequency, m∞ , is subtracted from the added mass and is treated separately [41]. The added mass
and radiation resistance are closely related to each other, from the radiation
resistance it is possible to calculate the added mass and vice versa. They are
related by Kramers-Kronig relations [41]. This gives the possibility to express
the radiation force as a function of buoy velocity or buoy acceleration,
Fr,t (t) = −k(t) ∗ vt (t) − m∞ v̇t (t) = −h(t) ∗ v˙t (t) − m∞ v̇t (t)
28
(3.17)
(a)
(b)
Figure 3.2: (a) Excitation problem, (b) Radiation problem.
Where
k(t) =
2
π
h(t) =
2
π
∞
0
∞
0
R̂r (ω ) cos(ω t)d ω = −
2
π
∞
0
ω [ma (ω )˘m∞ ] sin(ω t)d ω (3.18)
(ma (ω ) − m∞ ) cos(ω t)d ω = −
2
π
∞
R̂r (ω )
0
ω
sin(ω t)d ω (3.19)
For a more thorough reading about body wave interaction see for example
[52] or [53].
3.3 Time series and power spectrum
An ocean wave can be represented in different ways depending on the field of
application. In simulations it is in most cases preferable to use raw wave data,
i.e. time series of wave elevation. The other way of representing a wave is by
its wave power spectrum. However this neglects any phase information. The
power spectrum can be calculated from the wave elevation data. The wave
power spectrum S( f ) is defined as,
29
S( f ) =
2
|
T0
T0
η (t)e2π f t dt|2 ,
0
(3.20)
where T0 is the total measured time. In this form the power spectrum gives
no information of the direction the waves is propagating. This information can
also be included, then this is called the directional power spectrum [49]. From
the power spectrum different moments can be calculated, the n:th moment of
a spectrum mn is defined as,
mn =
∞
0
f n S( f )d f
(3.21)
Where f is the wave frequency. The moments give statistical information
of the wave. Important quantities are the significant wave height, Hs , and the
energy period, Tp , defined as,
√
Hs = Hm0 = 4 m0
Tp = Tm0−1 = (
m−1
)
m0
(3.22)
(3.23)
An older definition of significant wave height is the average height of the
highest third of the waves, H1/3 .
3.4 Wave energies
The energy in an ocean wave can be divided into two parts, kinetic and potential. The kinetic energy is due to the water fluid motion, and potential energy is
due to the wave elevation. The total wave energy content in a plane harmonic
wave on deep water is half kinetic and half potential energy. The total energy
can be calculated from the wave power spectrum, as
E = ρg
∞
0
S( f )d f
(3.24)
The wave energy transport denoted J , is the energy passing a fictious vertical wall of unit width is defined as,
J = ρg
0
−∞
pvx dz
(3.25)
Where vx is the horizontal particle velocity and p is the pressure. The average energy transport is calculated from significant wave height and the energy
period, by,
J = cTp Hs2
30
(3.26)
Where the constant c = ρ g/32π . The energy can be considered to be transported with the group velocity. The relation between energy transport and
wave energy is given by,
J = vg E
(3.27)
In paper [23] we have derived an expression for the instantaneous energy
flux, J(t), from time series of the wave elevation for waves of arbitrary shape.
Furthermore have we derived an expression for the energy flux vector j =
( jx , jy , jz ). The energy flux vector corresponds to the Poynting vector in electrodynamics. The expression is derived using linear wave theory for infinite
depth. The expression for the energy flux is given by. The energy transport
through a vertical plane perpendicular to the wave is given as a double convolution,
ρ g2 ∞ ∞ |t | + |t | η (t − t)η (t − t)dt dt (3.28)
2π −∞ −∞ t 2 + t 2
In figure 3.4 is a plot of the energy flux for a harmonic wave. As can be
seen the energy transport seems always to be positive, which as well is proven
in paper [23].
The expression for the energy flux vector for a wave travelling in the positive x-direction is given by,
J(t) =
jx = ρ g
∞
−∞
K(−z/g,t − t)η (t )dt ·
∞
−∞
M(−z/g,t − t)η (t )dt jy = 0
jz = ρ g
∞
−∞
K(−z/g,t − t)η (t )dt ·
(3.29)
(3.30)
∞
−∞
L(−z/g,t − t)η (t )dt (3.31)
Where the integration kernels K, L, and M are given by,
2
1
−τ
K(ζ , τ ) = e 4ζ
2 πζ
(3.32)
2
τ
−τ
L(ζ , τ ) = √ 3/2 e 4ζ
4 πζ
(3.33)
|τ |
|τ | − 4τζ2 2√ζ x2
1
−
e
e dx
(3.34)
2πζ 2πζ 3/2
0
The last two factors of the M kernel is called Dawson’s integral, for which it
exist efficient numerical computation algorithms [54]. The modulus of power
flux vector for a harmonic wave is shown in figure 3.3. Moreover the flux
vector for an enlargement of figure 3.3 is shown in figure 3.4.
M(ζ , τ ) =
31
Figure 3.3: (a) Wave elevation, (b) Integrated energy flux, (c) Horizontal enery flux.
Figure 3.4: Enlargement of the energy flux in figure 3.3
32
3.5 Validity of potential wave theory
Linear potential wave theory assumes that the wave height is much smaller
than the wave length. The fluid motion generated by any moving object must
also be correspondingly small. Figure 3.5 roughly indicates in what region the
theory is valid. In the figure H is the wave height, a is the typical diameter of
the object, and λ is the wave length. When H/a is large, so that the particle
paths are long compared to the object diameter, flow separation occurs and
this changes substantially the flow predicted by the inviscid theory. This area
is marked ’viscous’ in the figure. When λ /a is large, so that the wavelength
is much larger than the typical diameter of the object, the wave field is little
modified by the object and wave diffraction is relatively unimportant. This
are is marked ’diffraction’ in the figure. The square in figure 3.5 illustrates
the domain of interests for simulation this thesis is based on. Typical wave
periods are 4-8 seconds, wave heights is 0-4m, and buoy diameters 2-5m. As
can be seen the potential wave theory is well suited for modeling the buoy
wave interaction.
Figure 3.5: The figure illustrates domains where viscous and diffraction forces are of
importance. The square indicates the domain of intrests for simulation in this thesis
[52].
33
4. Generator Theory
In the thesis three approaches of generator modelling have been used. The
different models are of varying complexity and are suitable for different kinds
of simulations. The simple model is very fast and rough while the most complex model is computationally demanding and very accurate. The models are:
Field based model, Equivalent circuit model, and Mechanical model.
4.1 Modelling approaches
• Field based modelling: In the Field based model the time dependent electromagnetic field solved by a FE method. External electric circuit for load
and motion equation is coupled to the field calculations and these are solved
simultaneously in time. From the fields are losses, electromagnetic force
etc calculated. This modelling gives a near complete picture of the generator and its properties. Moreover, from the fields and current distributions
a temperature analysis can be made. This approach of modelling the generator is very computationally heavy and is useful mainly to design new
generators.
• Circuit theory: In the equivalent circuit model the generator is represented
by a system of four coupled windings. Three of them represent the armature
phase winding and the fourth is fictious representing the field winding or in
this case the permanent magnets. Inductances in the coils are pre-calculated
by stationary FE solutions of the field based models. Moreover, the inductances are piston position and electric load dependent. This approach of
modelling the generator gives the same accuracy as the field based models
but is much faster. This approach opens up for system analysis of complex
system, such as studies of different load conditions and farms etc.
• Mechanical Damping: In this model the electric system, generator and load,
is translated into a mechanical damping function. With this approach only
the mechanical system can be studied, such as energy absorption, prediction of energy production and design of the mechanical structure as springs
and end-stops. This way of modeling is very fast and large amount of data
can be produced that statistical investigations of energy absorption etc can
be based on.
35
4.2 Field based modeling
The fundamental and governing equations in electromagnetics is Maxwell’s
equations,
∇×E = −
∂B
∂t
∇×H = Jf +
∂D
∂t
(4.1)
(4.2)
∇·B = 0
(4.3)
∇·D = ρf
(4.4)
Where E is the electric field, D is the electric flux density, H is the magnetic
field, and B is the magnetic flux density. J and ρ are current and charge density
respectively. In addition to Maxwell’s equation material properties is needed,
these can be formulated by the following constitutive equations,
B = μ H,
(4.5)
J = σE
(4.6)
Where μ is the non-linear permeability, represented by a B-H curve that
can be found in data sheets from manufactures of electrical steel. The conductivity is represented by σ , which is a scalar constant here. For numerical
computational reasons the magnetic flux density is written as a magnetic vector potential, B = ∇ × A. Restricting the field problem to two dimensions an
expression for the field can be derived [55, 56],
1
∂ Az
∇·
∇Az + σ
(4.7)
= Js + Jpm
μr μ0
∂t
Where Az is the z-component of the magnetic vector potential. The current
densities in stator windings are denoted Js , whereas Jpm is a fictous current
density that models the permanent magnets. The field calculations are coupled to external algebraic equations describing the electric load connected to
the generator, the external circuit includes factors which compensates for end
effects which are not included in the two dimensional field formulations. The
external equations are,
Ia + Ib + Ic = 0
Ua + Rs Ia + Lsend
36
∂ La
∂ Ib
−Ub − Rs Ib − Lsend
= Vab
∂t
∂t
(4.8)
(4.9)
∂ Lc
∂ Ib
−Ub − Rs Ib − Lsend
= Vcb ,
(4.10)
∂t
∂t
where Ua , Ub , and Uc are the phase voltages, Ia , Ib , and Ic are the phase
currents, Vab and Icb are the line voltages, Rs is the phase resistance, and is the
coil end inductance.
The field problem is solved for a rotating generator. To speed up the simulation time a moving boundary in the air gap is adopted, this condition is used
to simulate that the rotor rotates inside the stator. By using this method we do
not have to re-mesh for every time step. It has the limitation that the air gap
is constant resulting in that no excentrities can be studied. Moreover, cyclic
boundary conditions are used that makes it possible to only simulate one pole
of the generator. Figure 4.1 shows the mesh and the resulting field for one
calculation pole/cell/domain.
The induced counteracting electromagnetic force which is present when
power is delivered by the generator is calculated from the change in magnetic
field energy in the air gap, the force is expressed as,
Uc + Rs Ic + Lsend
dWc
(4.11)
dx
Where Wc is the co-energy in the airgap and x denotes the piston position.
Fem =
(a)
(b)
Figure 4.1: Illustration of one calculation domain, (a) The mesh, (b) The resulting
field
37
4.3 Circuit theory
In the equivalent circuit theory the generator is represented by inductances,
resistances and voltages sources. The circuit representation a generator can be
seen as a system of four coupled windings. The first three windings represent
the stator coils, the fourth winding which is a fictitious winding represent the
permanent magnets. Figure 4.2 shows the equivalent circuit for a Y-connected
three phase generator; in this particular case Y-connected impedance, consisting of inductance in series with a resistance, acts as a load. The terminal
voltage, U, for the circuit can be written under any load condition as,
dL
dI
+ω
I+E
(4.12)
dt
dθ
Where R is the resistance matrix, I is the current vector, L is the inductance
matrix, and E is the induced emf. These parameters are defined as,
U = RI + L
⎡
Laa Lab Lac
⎤
⎢
⎥
L = ⎣ Lba Lbb Lbc ⎦
Lca Lcb Lcc
⎡
Ea
⎤
⎡
La f
(4.13)
⎤
d ⎢
⎥
⎥
⎢
E = ⎣ Eb ⎦ = τω
⎣ Lb f ⎦ I f
dθ
Ec
Lc f
(4.14)
I = [Ia , Ib , Ic ]T
(4.15)
U = [Ua ,Ub ,Uc ]T
(4.16)
I = diag {Ra , Rb , Rc }
(4.17)
The machine inductances are represented by a matrix L with nine elements,
where the three diagonal element are the three phases self inductance and the
off diagonal element representing different combinations of the mutual inductances. The inductance matrix is pre-calculated by stationary FEM solutions
the field problem for a particular generator design. The inductance matrix is
electric load and piston position dependent and is stored as a database. In
paper [57] shows how the inductances can look like for a generator with a certain stator winding configuration. In section 4.3.1 follow a brief explanation
of how the impedances are calculated.
The permanent magnets are modelled by fictitious windings supplied from
a current source with a constant current, I f . The current is linearly proportional to coercivity of the magnet times the magnet height in the direction of
magnetization.
38
This representation is for a conventional rotating generator where rotor angular velocity ω gives the rotor angle as, θ = ω t + θ0 . The transformation
and approximation of the rotating to a linear generator is here made with a
scale factor τ (θ ), it is defined as active stator length divided with total stator
length. The scale factor reduces the induced emf when the piston leaves the
stator. Additional equations are needed to give a closed electric circuit, these
equations represents the load and approximates the end-effects, which is not
incorporated in the inductance model. For example for a Y-connected load the
additional equations are given by,
Ia + Ib + Ic = 0
(4.18)
ΔUab + (Rs + Rload )ΔIab + (Lsend + Lload )
∂
ΔIab = 0
∂t
(4.19)
ΔUbc + (Rs + Rload )ΔIbc + (Lsend + Lload )
∂
ΔIbc = 0
∂t
(4.20)
Where ΔUab ≡ Ua −Ub , ΔUbc ≡ Ub −Uc , ΔIab ≡ Ia − Ib , and ΔIbc ≡ Ib − Ic .
The end effects are included as inductances as described in the previous section. The electromagnetic force developed to counteract the motion is calculated as,
Fem =
Tdev
,
r
(4.21)
where r is the inner radius of the stator and Tdev is the developed torque.
The electromagnetic torque is calculated by a method based on energy conservation principle. Tdev is expressed by,
1
dW
Tdev =
(U j I j − Rs I 2j ) −
(4.22)
∑
ωr a,b,c
dt
Where ωr is mechanical angular speed of the rotor, ω = ω /p, p is the pole
number, and W is stored magnetic energy.
Figure 4.2: Definition of impedances in circuit representation of a three phase generator connected to a load.
39
4.3.1 Calculation of Impedances
The inductance coefficients describe the relation between flux linkage and
currents. The method used for calculating the inductance matrix in the simulation used in papers is the Enhanced Incremental Energy Method, EIEM
[58, 59, 60]. The method requires stationary finite element solution under current perturbation for each machine winding and provides detailed inductance
variations as function of rotor position. After the operating solution has been
computed the differential reluctivity is evaluated. This computation is carried
out in each step of the non-linear FE-analysis as a part of a Newton iteration
process.
The linear generator is seen as a rotating machine with a very large radius.
The inductance is considered as a function of rotor position angle, θ , which
is expressed as a fourier serie. Hence, the derivative of the inductances with
respect to θ can easily be obtained. For a more detailed explanation see paper
[57].
4.4 Mechanical damping
In contradistinction to the very accurate generator models described above a
simple generator model has been developed, this model is very rough and acts
as a non-linear damper in the mechanical system. The damping is a function
of velocity, load and piston position. The damping force is derived from the
relation between force, power and piston velocity,
P = F ẋ
(4.23)
F = γ ẋ
(4.24)
Where g is the damping constant or function for the linear or the non-linear
case respectively. For a constant velocity the damping is proportional to active
power in the circuit and it is expressed as,
γ = 3∑
1 ui 2
Ri ẋ
(4.25)
where γ is the damping, Ri is the resistance, ui the voltage, and ẋ is the
piston velocity. The effects on the force when the piston leaves the stator in
taken into account by a factor which is the quote between active piston length
and stator length, this expressed as,
⎧
⎪
⎨ 0
A f ac =
1
⎪
⎩ 1 1
ls
40
if
|x| ≥ 12 (l p + ls )
|x| ≤ 12 (l p − ls )
if
else
2 (l p + ls ) − |x|
(4.26)
Where ls and l p is the stator and piston length respectively. The damping
force is then expressed as,
Fem = A f ac γ ẋ
(4.27)
The total absorbed power is given by,
P = γ A f ac ẋ2
(4.28)
41
5. Modelling
Development and improvement of the next generation wave energy converters
is very much dependent on computer simulations. This chapter presents in
brief the developed models of the converter used in the thesis. Furthermore,
some results based on simulations are presented.
5.1 Buoy geometry and damping factor
To achieve high energy absorption it is necessary to adopt the buoy motion in
an optimal way to fit present wave climate. The dynamics is totally determined
by the buoy shape, mass of moving parts and the damping. A simplified linear
model of the system is presented in paper [61]. The model includes heave
motion only and the buoy and piston is connected by a stiff rod. Further, the
generator represents by a constant damping factor. The buoy amplitude in the
time domain, z, is given as,
z = H ∗ η,
(5.1)
where H is a transfer function describing the wave energy converter dynamics,
η is wave elevation, the ∗ denotes convolution. The transfer function in the
frequency domain is given by,
Ĥ =
fˆe
,
−ω (ma + m) + iω (γ + R̂r )ρ gπ a2 ks
(5.2)
where ma is added mass, m is the total mass of piston and buoy, γ is damping
factor, a is buoy radius, ks is spring constant, R̂r is the radiation resistance,
and fˆe is the excitation force. The hydrodynamic parameters ma and Fe are
pre-calculated using commercial software WAMIT [51].
For a converter of the size studied in this thesis it is very difficult to reach
resonance by an increase in mass. Another possibility is to use a buoy with a
smaller diameter, see figure 5.1a. Moreover, the buoy shape has very little impact on the power absorption as long as the system is off resonance, at least for
the studied shapes that were a cylinder with flat-, conical- and spherical bottom. This has to be studied more carefully before conclusions can be drawn.
The different studied buoy geometries have only significant impact for frequencies near the resonance frequency. The highest resonance and narrowest
peak was obtained for a
43
25
Flat−shaped bottom
Cone−shaped bottom
Spherical−shaped bottom
22.5
Power Capture Ratio (%)
20
17.5
15
12.5
10
7.5
5
2.5
0
0
10
20
30
(a)
50
60
γ (kNs/m)
70
80
90
100
70
80
90
100
(b)
2.5
30
R=0.2m, d=0.7m
R=0.5m, d=0.7m
R=1.0m, d=0.7m
R=1.5m, d=0.7m
R=1.5m, d=0.4m
R=2.0m, d=0.4m
R=2.0m, d=0.3m
2
1.75
27.5
25
22.5
Power Capture Ratio (%)
2.25
abs(H) (m/Hz)
40
1.5
1.25
1
0.75
20
R=0.2m, d=0.7m
R=0.5m, d=0.7m
R=1.0m, d=0.7m
R=1.5m, d=0.7m
R=1.5m, d=0.4m
R=2.0m, d=0.4m
R=2.0m, d=0.3m
17.5
15
12.5
10
7.5
0.5
5
0.25
0
0
2.5
1
2
3
ω (rad/s)
(c)
4
5
6
0
0
10
20
30
40
50
60
γ (kNs/m)
(d)
Figure 5.1: (a) Transfer function H for cylindrical buoy with flat-, conic- and
spherical- bottom., (b) to (a) corresponding power capture ratio as function of damping
factor for ω = 1. , (c) Transfer function H for cylindrical buoy with different radius.,
(d) to (c) corresponding power capture ratio as function of damping factor for ω = 1.
Unpublished work.
cylinder with spherical bottom, lowest and widest peak for a cylinder with flat
bottom, see figure 5.1a. To reach a high absorption it is necessary to damp
the buoy motion as effectively as possible, the damping is proportional to the
generated power. The damping factor has to be increases with buoy radius
for obtain optimal energy absorption as long as the converter is off resonance
5.1c,d. Simulations show that the resonance frequency of the energy converter
and the frequency of characteristic wave spectrum coincide when the buoy diameter is small, for a flat bottom buoy these coincide for a radius of approximately 0.2m, this for an incident wave with Te ≈ 6s. For larger buoy radius
the resonance frequency increases.
44
5.2 Performance of simulation approaches
Within this thesis a number of different simulation approaches has been used
to simulate the wave energy converter in operation. Common to all approaches
is that potential wave theory is used to model the buoy wave interaction. Hydrodynamic parameters such as radiation resistance and added mass for the
studied buoy geometry are pre-calculated using WAMIT [51] or from analytical expressions derived by Bhatta [50]. It should be mentioned again that all
models used for a converter are in normal operation, for survivability more
advanced models for the buoy water interaction have to be used. The difference in the models lies in the generator/load and the coupling between these
different parts. The generator models are presented in detail chapter 4.1. Different approaches are suited for different type of simulation and in different
stage of the research. The most computational heavy model is also the one
giving the most information. The different models are illustrated in figure 5.2
and explained below.
• Approach I (Field based model)
This model is based on two equation of motion, one for the buoy and for
the piston. The buoy is restricted to move in heave only. The buoy-wave
interaction is given by the excitation force coefficient and the radiation
impedance for the given buoy geometry, these parameters are given in the
frequency domain. When the distance between buoy and piston is shorter
than the rope, the motion equation decouples. The generator is modelled
by the field based model, see chapter 4. To the generator an electric circuit
is connected, which is a π -link and a resistive load. This model is illustrated in figure 5.2. This result in an expanded stiffness matrix, where the
external equations are added rows. External equations include the electric
load, the coupled motion equation for buoy and piston. The hydrodynamic
forces acting on the buoy are included, the excitation force is calculated in
advance since it depends only on the incident wave elevation, but the damping force is a convolution of the impulse response function and the buoy
velocity history. For the implicit formulated system semi-implicit RungeKutta time integration methods of different orders are used. It can handle
events, i.e. the simulations can be interrupted at arbitrary times; equations
can be replaced by other equations and parameters can be set to new values.
This gives a possibility to use different equations in different phases of the
simulation. For example the motion equation can be decoupled when there
is a slack on the wire or it can simulate a diode when it turns on and off.
• Approach II (Equivalent Circuit model)
This model is basically the same as the previous. The difference lies in the
generator model. In this case the generator is represented by an inductance
matrix, which is pre-calculated from stationary FE solutions to the field
problem, see chapter 4. The inductances are load and velocity dependent.
• Approach III (Non-linear damping)
45
Figure 5.2: Illustration of the different simulation approaches.
This model implemented in the commercial software SIMULINK. The
buoy is restricted to move in heave only; its equation of motion is coupled to the equation of motion for the piston, when the distance is smaller
than the rope length they decouple. The generator is modelled as a nonlinear damper, see chapter 4. This time dependent problem solves by using
SIMULINK’s numerical solvers. In this case the hydrodynamic parameters
have to be given as an impulse response function, see paper [62] for details.
• Approach IV (Linear damping)
46
This mathematical model is linear; the buoy position in frequency domain
calculates as a product of a transfer function and incident wave elevation.
In time domain becomes the transfer function an impulse response function
and the product a convolution. The transfer function describes the converter
and buoy water interaction. That converter is modelled as a damper in parallel with a spring. The connection between buoy and piston is a stiff rod
in this case.
5.3 Energy absorption with rectified armature voltage
As was discussed in chapter 2, the voltage has to be refined and tuned into
a constant frequency and amplitude before connection to the grid. When the
generator is connected to the grid via a rectifier the damping factor will behave
differently compared to a resistive load. This affects also the dynamics and energy absorption. The field based simulation tool was equipped with a rectifier,
this to study the energy absorption and buoy dynamics. In paper [63] the rectifier model is presented. It is integrated in a time-stepping finite element simulation environment where the generator and circuit equations are solved simultaneously. Moreover, the model handles bidirectional piston speeds. In the
following study of energy absorption and buoy dynamics Archimedes principle is used to simulate the buoy-wave interaction.
A generator connected to a rectifier delivers power only when the armature
voltage is higher than the DC voltage. The no-load voltage is proportional to
piston velocity, i.e. the piston has to move faster than the velocity corresponding to the velocity for the DC voltage to deliver power. Even if the accelerating piston force tries to increases the piston velocity it is kept almost constant.
Such a force increases the currents in the machine and generates a higher electromagnetic damping force, the behaviour is visible in figure 5.3 and 5.4, for
a sinusoidal incident wave the piston moves with almost constant velocity as
long as the generator delivers power.
The DC level can be used to affect the dynamics and thus the power absorption, by increasing the DC voltage a higher piston speed is needed for the
generator to deliver power and vice versa. The effect the DC voltage has on
the power absorption was studied in paper [64]. It was found that there are different optimal DC levels for waves of different frequencies and that this level
is very important for an effective absorption, see figure 7.1. We have to keep in
mind that the buoy water interaction is modelled with Archimedes principle,
which is a rough approximation, but it indicates anyhow that the DC level is
important and it is something to study in more detailed.
47
1.5
1
Speed (m/s) Position (m)
c
0.5
b
0
-0.5
-1
a
-1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Wave time (s)
Vertical wave speed
Buoy speed
Wave position
Piston speed
Active part
Piston position
Figure 5.3: Illustration of WEC dynamics for the case of rectification, incident wave
is a sinusodial wave with T=4.5s.
Figure 5.4: Instantaneous generator power.
48
6. Experiment and verification of
models
Altogether three experiments have been conducted, one in the laboratory and
two offshore. The first experimental setup was to verify the generator design
tool and to get experience how to build and to see difficulties in building a linear generator. The first of the two offshore experiments was to get experience
with the forces acting on the experimental setup and to verify hydrodynamic
simulations. This experiment was followed up with an installation of a full
scale wave energy converter, where it is possible to verify all models, from
buoy wave interaction to consumed power by load resistance. The preparation
and a description of the two off-shore experiments and the wave measuring
buoy are presented in papers [65, 66, 67].
6.1 Experimental generator
During the year 2002-2003 the first prototype of the linear generator designed
and built [13, 36, 37]. The experiment generator was intended to be used for
verifying the generator design and simulation tool. Another important issue
was to learn how to build such new machine, the best way to identify real
problems as we saw with this construction was to build a generator. During
the construction work a few of what we saw as difficulties turned out to be
simple and nothing to worry about.
In the generator the piston moves vertically, it is fixed with eight bearings
which run along rails mounted on the central pillar. The pillar is fastened at the
top and bottom with a supporting structure, which holds the four stator packages. The structure has to be very strong to keep the airgap constant because
of the strong attractive forces between piston and stators. The experimental
setup is illustrated in figure 6.1.
The generator was designed to have a nominal output power of 10kW for
the constant piston speed of 0.7m/s. Under that criterion was the generator
optimized to have a high efficiency and a low load angle and to have small
cogging. The electromagnetic efficiency at nominal output power is 86%, the
load angle is 6.6 degree, and the power fluctuation is 1.3%. At an overload of
300% the load angle barely exceeds 12 degree [37].
The experiment performed on the generator showed a good agreement with
simulations. Static measurements of the magnetic field in the airgap have been
49
Figure 6.1: Illustration of the experimental generator.
measured and compared with simulations. Moreover is has also been made
dynamic measurements, where the piston has been pulled trough the stators
with a constant velocity [36, 68].
6.2 Offshore experiment I
In the first offshore experiment a buoy was moored to a concrete foundation
via rope and springs, as illustrated in figure 6.2. Data of the setup is in table
6.1. In the experiment the rope force and the buoy acceleration were measured
by a force sensor and an accelerometer respectively. Close to the experiment
setup the wave elevation was measured simultaneous with the force and acceleration measurements. All data was transmitted to a base station onshore.
This offshore experiment was performed mainly to get experience with the
unfriendly conditions at sea and to validate the hydrodynamic modelling for
further design of a full scale wave energy converter. Moreover, the experiment
was also intended to get knowledge of the magnitude of the forces acting on
the buoy. This experiment is described in paper [62].
A mathematical model of the experiment setup was developed and implemented in the commercial program SIMULINK. In his model the buoy was
allowed to move in two dimensions, heave and surge. The rope was modelled
as a very stiff spring which also could slack if the distance between the buoy
50
Table 6.1: Main features of experimental setup for the first offshore experiment.
Parameter
Value
Buoy type
Flat cylinder
Diameter
3.0m
Height
0.8m
Draft
0.25m
Mass
1000kg
Water depth
25m
Spring constant
7064N/m
Mass of spring unit
400kg
Figure 6.2: Illustration of experimental setup for the first offshore experiment.
and concrete foundation was shorter than the length of the rope. Moreover,
the excitation force was calculated both with and without contribution from
the diffracted velocity potential. Neglecting of the diffracted potential is valid
only for small buoy diameters compared with the wave length; it is called
the Froude-Krylov approximation [41]. The overall agreement between simu51
lation and experiment is very good. The Froude-Krylov approximation gives
reasonable good agreement to the experimentally rope force, but for the acceleration it is very bad. Including the second degree of freedom, surge motion,
gives almost identical results as for heave motion only. Figure 6.3 shows the
different approximations compared with experimental data of the wire force.
Figure 6.3: Comparison between simulated and measured wire force spectra, for different model approximations.
6.3 Offshore experiment II
In the second experiment a full scale wave energy power plant was installed
at sea. See figure 6.4 for an illustration of the experimental setup. Its main
features are described in table 6.2. The wave energy converter is installed at a
dept of 25m, 2.0km off the Swedish west coast outside Lysekil. A 2.9km sea
cable connects the power plant to an onshore base station where the converted
electric energy is consumed by a Δ-connected resistive load. The load resistance is adjustable, which allows us to study the absorption of wave energy
for different loads.
At the base station the three phase voltages over the resistive load are measured in time with a sampling frequency of 50Hz. The voltages measurements
are recorded simultaneously with the wave elevation.The wave elevation is
measured with a an omni directional wave rider [69, 66]. The waves are measured approximately
52
Figure 6.4: Illusteration of experimental setup for the second offshore experiment.
Table 6.2: Main features of WEC
Parameter
Value
Parameter
Value
Buoy type
Flat cylinder
ls
1264mm
Diameter
3.0m
Pnom
10kW
Height
0.8m
vnom
0.67m/s
Weight
1000kg
ws
400mm
Draft
0.4m
Slot/(pole and phase)
6/5
Vnom
200V
ku
243kN/m
Rg
0.45Ω
kl
215kN/m
Lg
5.5mH
kw
450kN/m
lp
1867mm
ks
6.2kN/m
80m from the wave energy converter. All measurements take place during 20
minutes every hour. The converter was in operation during three month in
spring 2006.
During the time the converter was in operation it was exposed to a varying wave climate, from very calm conditions to very energetic waves. To see
what impact the electric load condition has on the ability to absorb energy the
resistive load has been varied in three steps from 2.2Ω to 10Ω. In the first pa53
per of R. Waters et. al. [70] the very first results was presented on the energy
absorption for different wave climates and electric loads.
All together more than 2000 sets of data were analyzed. This gives a good
statistical sample of the converter performance and can be used for verification
of the mathematical models.
Two papers have been written on validation of models using this setup
[71, 72]. The first paper concerns the non-linear damping model presented
in section 4.4. This model is very fast, which allow us to simulate and compare with all data presented in paper [70]. This gives good statistical information about the accuracy to experiment, this for a varying wave climate and for
different loads. It has to be mentioned that it is only possible to study time
integrated parameters, due to the distance between the experimental setup and
the wave elevation measuring buoy. Inputs to the simulations are the measured
wave elevation and the still water level. Output is time series of absorbed energy.
The overall agreement between simulation and experiment are very good.
Best agreement is for Rload = 2.2Ω and Rload = 4.9Ω , there is some discrepancy for Rload = 10Ω. Table 6.3 shows the correlation, m, and scattering, r,
between simulated and experimental power, this with and without compensation for water level changes. In figure 7.2 the absorbed and simulated power
is plotted against incident wave power, each point corresponds to the average
absorbed power during 20 minutes. The inset shows correlation between sim-
Figure 6.5: Absorbed and simulated power as function of time with load resistence
Rload = 2.2Ω.
54
Table 6.3: Correlation between simulated and experimental energy absorption.
Rload
r
r waterlevel comp.
m
m waterlevel comp.
2.2
0.9935
0.9935
0.9352
0.9499
4.9
0.9926
0.9926
0.9999
1.0000
10.0
0.9906
0.9905
1.3562
1.2664
ulated and experiment measured average power. Further, in figure 6.5 Shows
the captured power as function of time. The results presented in these plots
are for Rload = 2.2Ω , see paper [71] for the whole study.
The second paper concerns validation of the equivalent circuit representation of the generator coupled to the potential wave theory used to describe
buoy-wave interaction; the model is presented in paper [73] and was described
briefly in section 4. This validation is based on data from the data set used in
previous validation. From that data set eight time series of wave elevation and
corresponding measured load voltages have been chosen. The data sets are
chosen so that the power content in the waves close to 5, 10 and 19kW/m. To
each power level waves with similar energy period (TE ) and significant wave
height (Hs ) for respective load resistance have been chosen.
The agreement is very good, largest discrepancies is for Rload = 10Ω. Figure 6.6 gives statistical information in form of a box plot, showing the median,
1:st and 3:rd quintile for Rload = 2.2Ω. The whole study is found in paper [72].
In that paper a simulation was made to see what impact a buoy of larger diameter would have on the energy absorption, figure 6.7. An increase in buoy
diameter results in an increase of absorbed power independent of load resistance. For the particular case of Rload = 2.2Ω the an absorption can be doubled
by increase of the diameter by a factor of two, from 3m to 6m.
55
Figure 6.6: Experimental and simulated data presentad as a boxplot, for Rload = 2.2Ω.
Figure 6.7: Simulated power absorption as function of buoy radius.
56
7. Discussion
In this section a follows discussion based on results presented in the papers.
The first section concerns energy absorption for a point absorber off resonance. In the next section the different models and the verification of them are
discussed. The last section presents ideas of interesting topics for future work.
7.1 Energy absorption
This particular concept is based on a point absorber operating off resonance.
There are different reasons to why such approach has been chosen for study.
First of all, it is very difficult to move the eigenfrequency of the studied absorber so it coincides with the frequency spectrum of the incident wave. The
only possibility is by using absorbers with small diameters. A point absorber
in resonance oscillate with higher amplitude than the incident waves, this can
cause problem because of the limited length of the piston stroke for a linear
generator. Resonance can also be obtained by latching, but this increases the
complexity of the converter. For a point absorber operating off resonance it
seems that the buoy shape is not of significant importance, at least for the geometries studied. It is at least important for point absorbers operating in resonance; in that case the resonance peak is affected by the buoy geometry. More
important for good energy absorption for point absorbers operating off resonance is the bottom surface of the buoy, in relation to the damping. In general
a larger bottom area requires larger damping for optimal energy absorption.
To get a high damping requires that the electric power extraction from the generator is high. In the case studied here, when the generator is stand alone and
connected to a resistive load, a high damping is achieved when the load resistance is small. This implies large currents in the machine and the generator
has to handle overloads.
So far, we have studied the energy absorption for a resistive load, with exception for paper [64]. This will affect the buoy dynamics and thereby the energy absorption, the rectification makes the damping non-linear. Preliminary
results indicate that the DC-level is of importance for an effective absorption.
Figure 7.1 shows the energy absorption as function of DC-voltage for three
harmonic waves with different frequencies.
57
Figure 7.1: Power absorption as function of DC-level for three different sinusodial
incident waves.
7.2 Experiment and modelling
The full scale experiment with the installed wave energy converter has been
very important for this work and the whole project. This experiment has given
us knowledge how to build such a generator to withstand the unfriendly environment and the demands on precision.
In the thesis, linear potential wave theory has been used to describe the
buoy-wave interaction. A very good overall agreement between experiments
and simulations is achieved. Best agreement is achieved for high damping.
Figure 7.2 shows the simulated and experimental absorption for Rload = 2.2Ω.
Largest discrepancies is for the Rload = 10Ω, when the generator exerts the
lowest damping. Possible explanations are that the mechanical losses are
larger relative the extracted power and that the buoy velocity relative the
water is higher resulting in non linear hydrodynamic coupling. Further, the
pitch motion seems not to be of importance for the simulations and can be
neglected, for the studied depth of 25m.
The two most interesting generator models are the non-linear damping and
the equivalent circuit. The first model is very simple and can be implemented
for example in SIMULINK. It is easy to do modifications and get to get good
agreement with the real world. The second model is more complicated to implement and it needs a pre-calculated data base for the generator properties.
When one has the data base this model has a huge potential, it is fast and gives
the same accuracy as the field based model. With this approach it will be possible to do system analysis of a farm of converters. Some improvement of the
model has to be made, as explained earlier the linear generator is simulated
as a rotating generator with large diameter. This approximation neglects the
edge effects which is present when the piston leaves the stator.
58
Figure 7.2: Comparison between experimentally mesured and simulated power absorption for different wave climates, with a load resistence Rload = 2.2Ω.
A model for the rectification of armature voltage has been developed for
the FE-based model, some simulations on what effect it has on the energy
absorption was presented in paper [64]. This study is very rough and has to
be made more carefully. To do this, the rectifier model has to be implemented
in the equivalent circuit model, this to reduce CPU time. The developed field
based model with rectifier [63] is only equipped with a very simple model
for buoy-wave interaction. The simulations will be too computationally heavy
for making such study if it was equipped with the potential wave theory for
buoy-wave interaction. Simulations indicate the DC level is of importance for
effective energy absorption.
7.3 Suggestions for future work
So far has models been developed and verified against experimental data. The
absorption has almost been simulated for resistive loads. A more careful investigation of the DC level dependence on energy absorption has to be made.
As indicates in paper [64], the DC level is of importance for optimal energy
absorption. Another important issue is the connection of two or more converter units. Here there are interesting questions to answer, e.g. how many
converters have to be connected to reduce power fluctuations on the delivered
electric power to a certain level etc.
59
8. Conclusion
This thesis concerns the development of simulation tools, which can be used
for design of new and more efficient wave energy converters. Linear potential wave theory has been used for modeling buoy-wave interaction. Comparisons with extensive experimental results show high accuracy, and indicate
that the model is well suited for the design of wave energy converters of the
studied type. Furthermore, the fast calculation times of the model open up
for extensive studies in optimization of the converters to specific wave climates, and for the studies of farms of interconnected wave energy converters.
Best agreement is obtained for nominal load conditions. For the studied case
where the converter is installed at a water depth of 25m, only heave mode has
to be considered as the contribution from surge is negligible. Moreover, the
diffracted potential has to be included for calculation the excitation force, i.e.
the Froude-Krylov approximation gives large discrepancies between simulations and experiments. The radiation force, which is included in the models,
is of less importance if the generator operates and delivers power at nominal
load.
For this concept with relatively low mass of moving parts and the studied
buoy geometries, the resonance is obtained only for small buoy diameters,
approximately 0.4m.
The full scale offshore wave energy converter installed at the Swedish west
coast proves that the concept works and delivers power. On average the nominal power is six times lower than peak power.
61
9. Summary of papers
• Paper I: Study of a Longitudinal Flux Permanent Magnet Linear Generator for Wave Energy Converters [37]
In this paper a linear generator for wave energy absorption is presented together with an analysis of different design parameters. Moreover the first
experimental results are presented, mainly to verify the numerical simulations. The author was involved in the design and construction of the experimental setup and in some of the measurements.
International Journal of Energy Research, 30, pp. 1130-1145, 2006
• Paper II: An Electrical Approach to Wave Energy Conversion [38]
This paper is a more general paper presenting and describe the wave energy
concept studied at Uppsala University. The research on the linear generator
is summarized, from simulation and design to experiments. The author has
contribution with parts of experimental setup and calculations.
Renewable Energy 31, pp. 1309-1319, 2006
.
• Paper III: Time Stepping Finite Element Analysis of a Variable Speed
Synchronous Generator with Rectifier [63]
In this paper a rectifying model is presented. The model is integrated in a
time stepping finite element environment where electromagnetic fields and
circuit equation describing the rectifier and load are solved simultaneously.
The author contributed with simulations.
Applied energy 83, pp. 371-386, 2006
• Paper IV: Dynamics of a Linear Generator for Wave Energy Conversion [64]
In this paper the energy absorption and the dynamics of a wave energy
converter were studied for a case of a passive diode bridge rectifier as the
load. Moreover, the impact of the DC level for optimum absorption was
studied for incident harmonic waves. Most of the work in this paper was
carried out by the author.
Reviewed conference paper. Presented 2004 at the 23:rd OMAEconference by the author.
• Paper V: Hydrodynamic Modelling of a Direct Drive Wave Energy
Converter[61]
63
In this paper the wave energy absorption by a point absorber is studied.
The absorption has been studied for different buoy radii and damping coefficients both for as system in resonance and off resonance. The study is
made for wave energy converters in a typical Swedish climate. Moreover,
the absorption in real wave climates has been studied as well. Most of the
work in this paper was carried out by the author.
International Journal of Engineering Science 43, pp. 1377-1387, 2005
• Paper VI: Theory and Experiment on an Elastically Moored Cylindrical Buoy [62]
In this paper simulations based on of linear wave theory for the wave/buoy
interaction were compared with full scale ocean experiment. A cylindrically shaped buoy moored in a concrete foundation via rope and springs,
similar to a wave energy converter were tested. Simulated buoy acceleration and rope tension were compared to experiments. Most of the work in
this paper was carried out by the author.
IEEE Journal of Oceanic Engineering vol. 31 No. 4, pp.959—963, 2006
• Paper VII: Instantaneous Energy Flux in Gravity Waves [23]
In this paper the instantaneous power flux and power densities has been
derived for an arbitrary ocean wave on infinite depth. The derivation is
based on potential linear wave theory. Numerical solutions, figures, and
smaller text contributions were made by the author.
Submitted to Physics Review Letter E, Dec 2006
• Paper VIII: Experimental Results from Sea Trials of an Offshore Wave
Energy System [70]
This paper presents results from the first sea trials of a full scale wave
energy converter. The data are collected for 3 month, during that time has
the converter experienced a varying wave climate. Furthermore the load
resistance has been varied to study its effect on the energy absorption. The
author gave ideas and help with the data treatment.
Applied Physics Letter 90, 034105, 2007
• Paper IX: Wave Power Absorption: Experiment and Simulation [71]
In this paper the validity of the mathematical model of the wave energy
converter was compared to results from sea trials of the converter. Lot of
data was simulated and compared for different electrical loads and wave
climates, which gave good statistical ground for validating the model. Most
of the work in this paper was carried out by the author except for experimental work.
Submitted to Journal of Applied Physics, March 2007
64
• Paper X: Simulation of a Linear Generator for Wave Power Absorption
- Part I: Simulation [73]
In this paper a model is presented of a wave energy converter, where the
hydrodynamic modelling of buoy/water interaction is coupled to an electromagnetic model of the generator. The buoy/water interaction is modelled
with linear potential wave theory. Three different approaches of modelling
the generator have been compared re simulation time and accuracy. Implementation of hydrodynamic part was performed by the author, the coupling
between hydrodynamic- and electromagnetic problem was a collaboration.
Submitted to IEEE Transaction on Energy Conversion, March 2007
• Paper XI: Simulation of a Linear Generator for Wave Power Absorption - Part II: Verification [72]
In this paper the model presented in the first paper of the two companion
papers have been validated to a full scale trials on a wave energy converter
installed at sea. Most text and preparation of data for simulation was made
by the author.
Submitted to IEEE Transaction on Energy Conversion, March 2007
65
10. Svensk sammanfattning
Energibehovet ökar stadigt i världen allt medan tillgången på fossila bränslen
minskar. Lägg där till de skadliga effekter som naturen utsätts för då fossila
bränslen eldas. I dag blir det allt populärare att satsa på miljöriktiga alternativ.
Vågkraft är hittills en outnyttjad förnyelsebar energikälla. Det som gör vågenergi extra intressant är dess höga energidensitet, högst av alla förnyelsebara
energikällor. För Sverige har våg energin även fördelen att tillgången följer
säsongens behov, på hösten och vintern när behovet av energi är som störst
finns det mest vågor. Detta kan t.ex. jämföras med solenergi, där behovet av
energi är som minst på sommaren då solen lyser som mest.
Vid Uppsala universitet startade 2002 ett projekt där ett koncept för energiutvinning ur vågorna ska undersökas. Konceptet bygger på en linjär direktdriven generator som är kopplad till en boj på vattenytan, som följer vågrörelsen. Varje aggregat är har förhållandevis liten effekt, genom att koppla
ihop fler aggregat till stora kluster av hundratals och kanske tusentals erhålles
en betydande energiproduktion.
Denna avhandling handlar till stor del av utveckling av modeller samt modellering och verifiering av ett vågkraftverk i drift. Modellerna ska sedan kunna
användas som hjälpmedel till design av framtida generationer av vågkraftverk.
Gemensamt för alla modeller är att potentialteori har använts för att simulera
våg/boj interaktionen. Det som skiljer de olika modellerna åt är kopplingen
till generatorn och hur själva generatorn modelleras. Den enklaste modellen,
som är helt linjär, beskrivs generator som en mekanisk dämpare i parallell
med en fjäder. Dämpningen karakteriseras av en dämpfaktor. Denna modell
har använts för att undersöka olika boj dimensioner och några få geometriers
inverkan på energiabsorptionen. Vidare så har den potentiella våg teorin verifierats mot experimentella försök, för den här typen av våg aggregat. I det
första experimentet jämfördes uppmätta wire krafter mot simulerade krafter.
Det visade på en mycket god överensstämmelse. Denna studie visade även på
att bojens sidorörelse hade en mycket liten inverkan och att det räcker att endast simulera bojens rörelse i en dimension. Däremot räcker det inte att bortse
från diffraktionen vid beräkning av excitationskraften (Frode-Krylov approximationen). I den andra jämförelsen användes experimentell data från ett fullskarligt vågkraftverk. Vågkraft aggregatet installerades våren 2006, utanför
Sveriges västkust vid Lysekil. Aggregatet är anslutet via en 2.9km lång sjökabel till en varierbar resistiv last placerad på land. I nära anslutning till vågkraft
aggregatet mäts vågorna samtidigt som inducerad spänning över lasten. Under
67
de tre månader som aggregatet var i drift samlades data in, där lasten varierades. Dessa data har sedan använts för att verifiera modellerna. På det hela
taget är överensstämmelsen mellan simulering och experiment mycket god.
Bäst är fallet med en hårt dämpad generator, vilket är mest likt driftfallet.
68
11. Acknowledgment
Det finns många personer att tacka. Först skulle jag vilja tacka min huvudhandledare Professor Mats Lejon för allt stöd och intressanta idéer, samt att
jag fått chansen att jobba i vågkraftprojektet. Sen vill jag även tacka mina två
assisterande handledare Dr. Jan Isberg och Dr. Arne Wolfbrandt för ett mycket bra och lärorikt samarbete. Dr. Karl Erik Karlsson tackas också för hans
insats vid utvecklingen av vårat generator simulerings program, och Dr. Anna
Wolfbrandt för ett bra och givande samarbete.
Utan Gunnel Ivarsson, Tomas Götschl och Ulf Ring så hade allt på avdelningen varit mycket svårare. T.ex. skulle jag aldrig fått några pengar på grund av
felaktigt ifyllda reseräkningar, ingen som fixar mina kraschade hårddiskar och
ingen att konsultera när det gäller byggnationer... Tack så mycket:)
Jens Engström och Dr. Björn Bolund ska tackas för genomläsning och kommentarer vid avhandlingsskrivandet.
Dr. Oskar Danielsson måste bland annat tackas för hjälp med å, ä och ö. Rafael
Waters för intressanta diskussioner och bra kommentarer vid genomläsning av
artiklar.
Jag vill även tacka alla på avdelningen och övrig personal på Ångström.
Några som det inte går att glömma att tacka är lunch kompisarna, det roligaste
på hela dagen är luncherna..
Sist men inte minst vill jag tacka min familj för allt. Mia för allt stöd, och
Philip för all lek och fotbollsspelande.
Om jag har glömt någon så tackar jag även dom...
69
Bibliography
[1] A. Clement, P. McCullen, A. Falcao, A. Fiorentino, F. Gardner, K. Hammarlund,
G. Lemonis, T. Lewis, K. Nielsen, S. Petroncini, P. Schild M.-T. Pontes, B.-O.
Sjostrom, H. C. Sorensen, and T. Thorpe. Wave energy in europe: current status
and perspectives. Renewable and Sustainable Energy Reviews, 6:405–431,
2002.
[2] Robin Pelc and Rod M. Fujita. Renewable energy from the ocean. Marine
policy, 26:471–479, 2002.
[3] G.M. Hagerman and T. Heller. Wave energy: a survey of twelve near-term technologies. Proceedings of the international Renewable Energy Conference,
pages 98–110. Honolulu, Hawaii, 18-24 September 1988.
[4] J.M. Leishman and G. Scobie. The development of wave power – a techno
economical study. Dept. of Industry, NEL Report, EAU M25, 1976.
[5] G. Fredriksson. Ips wave power buoy. Wave Energy R&D, Cork, Ireland,
1993 EUR 15079, 1992.
[6] B-O Sjöström. The past, the present, and the future of the hose-pump wave
energy converter. First European Wave Energy Symposium. Edinburgh,
CEC 1994 EUR 15571 EN, 1993.
[7] L.J. Duckers. Wave energy; crests and troughs. Renewable energy, 5, part
II:1444–1452, 1994.
[8] L. Bergdahl, L. Claesson, C. Falkemo, J. Forsberg, and A. Rylander. The
swedish wave energy research programme. Symposium on wave energy utilization, pages 1–31. Gothenburg 30 October - 1 November 1979.
[9] T.W. Thorpe. An overview of wave energy technologies: status, performance,
and costs. article. ETSU report, 1999.
[10] Hans Berhoff, Elisabeth Sjöstedt, and Mats Leijon. Wave energy resources
in sheltered sea areas: A case study of the baltic sea. Renewable Energy,
31:2164–2170, 2006.
[11] Urban Henfridsson, Viktoria Neimane, Kerstin Strand, Robert Kappar, Hans
Berhoff, Oskar Danielsson, Mats Leijon, Jan Sundberg, Karin Thorburn, Ellerth
71
Ericsson, and Karl Bergman. Wave energy potential in the baltic sea and the danish part of the north sea, with reflections in the skagerrak. Renewable Energy,
32:2069–2084, 2007.
[12] Oskar Danielsson. Wave Energy Conversion -Linear Synchronous Permanent Magnet Generator. Phd thesis, Uppsala University, ISBN 91-554-66834, 2006.
[13] O. Danielsson, M. Leijon, and E. Sjöstedt. Detailed study of the magnetic circuit
in a longitudinal flux permanent-magnet synchronous generator. IEEE Transaction on Magnetics, 9:2490–2495, 2006.
[14] O. Danielsson, M. Stålberg, and M. Leijon. Verification of cyclic boundary
condition and 2d field model for synchronous pm linear generator. Submitted
to IEEE Transaction on Magnetics, 2006.
[15] Oskar Danielsson, Mikael Eriksson, and Mats Leijon. Study of a longitudinal
flux permanent magnet linear generator for wave energy converters. International Journal of Energy, 30, 2006.
[16] Oskar Danielsson and Mats Leijon. Analytical model of flux distribution in linear pm synchronous machines including longitudinal end effects. IEEE Transaction on Magnetics. Accepted July 2006.
[17] Karin Thorburn. Electric Energy Conversion System: Wave Energy and
Hydropower. Phd thesis, Uppsala University, ISBN 91-554-6617-6, 2006.
[18] Karin Thorburn and Mats Leijon. Analytical and circuit simulations of linear
generators in farm. IEEE PES Transmission and Distribution. Dallas, USA,
21-24 May, 2006.
[19] Karin Thorburn and Mats Leijon. Farm size comparison with analytical model
of linear generator wave energy converters. Ocean Engineering, Volume 34,
Issues 5-6:908–916, 2007.
[20] K. Thorburn, H. Berhoff, and M. Leijon. Wave energy transmission systems
concepts for linear generator arrays. Ocean engineering 31, pages 1339–1349,
2004.
[21] Karin Thorburn and Mats Leijon. Ideal analytical expression for linear generator
flux at no load voltage. Conditionally accepted for publication in Journal of
Applied Physics, August, 2006.
[22] Karin Thorburn, Karin Nilsson, Oskar Danielsson, and Mats Leijon. MAREC
06, London, UK, 6-10 March, 2006.
[23] Jan Isberg, Mikael Eriksson, and Mats Leijon. Instantaneous energy flux in fluid
gravity waves. Submitted to Phys. Rev. E, Dec 2006.
72
[24] K. Budal and J. Falnes. A resonant point absorber of ocean-wave power. Nature,
256:478–479, 1975.
[25] Mats Leijon, Hans Bernhoff, Olov Agren, Jan Isberg, Jan Sundberg, Marcus
Berg, Karl-Erik Karlsson, and Arne Wolfbrandt. Multiphysics simulation of
wave energy to electric energy conversion by permanent magnet linear generator. IEEE Transaction on Energy Conversion, 20(1):219–224, 2005.
[26] Oskar Danielsson, Mats Leijon, Karin Thorburn, Mikael Eriksson, and Hans
Bernhoff. A direct drive wave energy converter -simulations and experiment.
Proceedings of, OMAE. Halkidiki, Greece, 2005.
[27] I. Boldea and S.A. Nasar. Linear electric actuators and generators. IEEE Transaction on Energy Conversion, 14 no. 3:712–717, 1999.
[28] G.W. McLean. Review of recent progress in linear motors. IEE Proceedings
B, 135:380–416, 1988.
[29] E. Spooner and M. A. Mueller. Comperative study of linear generators and
hydraulic systems for wave energy conversion. article. ETSU report, 2001.
[30] H. Polinder, B.C. Mecrow, A.G. Jack, P.G. Dickinson, and M.A. Mueller. Conventional and tfpm linear generators for direct-drive wave energy conversion.
IEEE Transaction on Energy Conversion, 20(2):260–267, 2005.
[31] H. Polinder, F. Gardner, and B. Vriesma. Linear pm generator for wave energy conversion in the aws. ICEM 28-30 August, Espoo, Finland, 309–313,
2000, pages 309–313, 2000.
[32] M. A. Mueller. Electrical generators for direct drive energy converters. IEE
Proceedings on Generation, Transmission and Distribution, 149 issue
4:446–456, February 2002.
[33] N.J. Baker and M. A. Mueller. Direct drive wave energy converters. Rev. Energ.
Ren. :Power engineering, pages 1–7, 2001.
[34] M. A. Mueller and N.J. Baker. A low speed reciprocating permanent magnet
generator for direct drive wave energy converters. Conferance publication,
Power electronics, machines and drives, pages 468–473, April 2002.
[35] M. A. Mueller, N.J. Baker, and E. Spooner. Electrical aspects of direct drive
wave energy converters. pages 235–242, 2000. Proceedings of Fourth European
Wave Energy Conference, Aalborg, Denmark.
[36] Oskar Danielsson, Karin Thorburn, Mikael Eriksson, and Mats Leijon. Permanent magnet fixation concepts for linear generator. Fifth European Wave
Energy Conferance. Cork, Ireland, 2003.
73
[37] Oskar Danielsson, Mikael Eriksson, and Mats Leijon. Study of a longitudinal
flux permanent magnet linear generator for wave energy converters. International Journal of Energy Research, 30:1130–1145, 2006.
[38] Mats Leijon, Oskar Danielsson, Mikael Eriksson, Karin Thorburn, Hans Bernhoff, Jan Isberg, Jan Sundberg, Irina Ivanova, Elisabeth Sjöstedt, Olov Ågren,
Karl-Erik Karlsson, and Arne Wolfbrandt. An electrical approach to wave energy conversion. Renewable energy, 31:1309–1319, 2006.
[39] J .Falnes and K. Budal. Wave - power conversion by point absorber. Norwegian
Maritime Research, 6 No. 4:2–11, 1978.
[40] D.V. Evans. A theory for wave-power absorption by oscillating bodies. Journal
of fluid mechanics, 77 part 1:1–25, 1976.
[41] Johannes Falnes. Ocean waves and oscillating systems. Cambridge. ISBN
0 521 78211 2, 2002.
[42] J.
Falnes,
Principles
for
Capuring
of
Energy
from
Ocean
Waves,
Phase
Control
and
Optimum
Oscillations,
http://www.phys.ntnu.no/instdef/prosjekter/bolgeenergi/index-e.html.
[43] S.H. Salter. Wave power. Nature, 249:720–724, 1974.
[44] Umesh A. Korde. Active control in wave energy conversion. Sea Technology,
pages 47–52, July 2002.
[45] H Eidsmoen. Tight-moorde amplitude-limited heaving-buoy wave-energy converter with phase control. Applied Ocean Research, 20:157–161, 1998.
[46] K. Nielsen and C. Plum. Point absorber - numerical and experimental results.
pages 219–226.
[47] J.J. Stoker. Water Waves: The Mathematical Theory with Applications.
Wiley Classics Library. ISBN 0-471-57034-6, 1992.
[48] J. Billingham and A.C. King. Wave motion. Cambridge University Press. ISBN
0-521-63-450-4, 2006.
[49] R.G. Dean and R.A. Dalrymple. Water Wave Mechanics for Engineers and
Scientists, Advanced series on ocean engineering 2:ed. Prentice Hall, Inc.
ISBN 981-02-0420-5, 1991.
[50] D.D. Bhatta and M. Rahman. On scattering and radiation problem for a cylinder in water of finite depth. International Journal of Engineering Science,
41:931–967, May 2003. Vol. 41, Issue 9.
[51] WWW.WAMIT.com.
74
[52] O.M. Faltinsen. Sea loads on ships and offshore strructures. Cambridge
university press. ISBN 0-521-45870-6, 1998.
[53] C.M. Linton and P. McIver. Mathematical techniques for wave/structure
interaction. Chapman and Hall. ISBN 1-58488-132-1, 2001.
[54] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical
Recipes in C. Cambridge University Press. ISBN 0-521-75033-4, 1992.
[55] K. Hameyer and R. Belmans. Electrical machines and devices. Advances in
electrical and electronic engineering. WIT Press, 1999.
[56] O. Craiu, N. Dan, and E.A. Badea. Numerical analysis of permanent magnet
dc motor performance. IEEE Transaction on Magnetics, 31(6):3500–3502,
1995.
[57] Anna Wolfbrandt. An efficient method for long time simulation of linear synchronous generators. Submitted to IEEE Transaction on Energy Conversion.
[58] N.A. Demerdash and T.W. Nehl. Determination of iductances. IEEE Trans.
Magn., 18:1052–1054, 1982.
[59] T.W. Nehl, F.A. Faud, and N.A. Demerdash. Inductances by energy pertubation.
IEEE Trans. Power App. Syst., 101:4441–4451, 1982.
[60] M. Gyimesi and D. Ostergaard. Inductance computation by incremantal finite
element analysis. IEEE Transaction on Magnetics, 35:1119–1122, 1999.
[61] Mikael Eriksson, Jan Isberg, and Mats Leijon. Hydrodynamic modelling of a
direct drive wave energy converter. International Journal of Engineering
Science, 43:1377–1387, 2005.
[62] Mikael Eriksson, Jan Isberg, and Mats Leijon. Theory and experiment of an
elastically moored cylindrical buoy. IEEE Journal of Oceanic Engineering,
31(4):959–963, 2006.
[63] Karin Thorburn, Karl-Erik Karlsson, Arne Wolfbrandt, Mikael Eriksson, and
Mats Leijon. Time stepping finite element analysis of variable speed synchronous generator with rectifier. Applied Energy, 83:371–386, 2006.
[64] Mikael Eriksson, Karin Thorburn, Hans Bernhoff, and Mats Leijon. Dynamics
of a linear generator for wave energy conversion. OMAE 2004, 23:th International Conference on Mechanics and Arctic Engineering, 2004. Vancouver,
British Columbia, Canada, June 20-25.
[65] S. Gustavsson, O. Svensson, J. Sundberg, H. Bernhoff, M. Leijon, O. Danielsson, M. Eriksson, K .Thorburn, K. Strand, U. Henfridsson, E. Ericsson, and
75
K. Bergman. Experiment at islandsberg on the west coast of sweden in preparation of the construction of a pilot wave power plant. 6th EWTEC conference
in Glasgow. August 29 - September 2, 2005.
[66] S. Gustavsson. Measuring of waves at islandsberg and litterature survey of wave
measurment technology. Master’s degree project, UPTE F04048. Division
for electricity and lightning research, Uppsala University, Sweden, June, 2004.
[67] Magnus Stålberg, Rafael Waters, Oskar Danielsson, Olle Svensson, Karin Thorburn, and Mats Leijon. On-site experiments of wave energy to electric energy
conversion by permanent magnet linear generator. Submitted to IEEE Trans-
actions on Energy Conversion.
[68] Magnus Stalberg, Rafael Waters, Mikael Eriksson, Oskar Danielsson, Karin
Thorburn, Hans Berhoff, and Mats Leijon. Full-scale testing of pm linear generator for point absorber wec. 6th EWTEC conference in Glasgow. August
29 - September 2, 2005.
[69] WWW.datawell.nl.
[70] Rafael Waters, Magnus Stålberg, Oskar Danielsson, Olle Svensson, Stefan
Gustafsson, Erland Strömstedt, Mikael Eriksson, Jan Sundberg, and Mats Leijon. First experimental results from sea trials of a novel wave energy system.
Applied Physics Letter, 90(034105), 2007.
[71] Mikael Eriksson, Rafael Waters, Olle Svensson, Jan Isberg, and Mats Leijon.
Wave power absorption: Experiment and simulation. Submitted to Journal of
Applied Physics, march 2007.
[72] Mikael Eriksson, Anna Wolfbrandt, Rafael Waters, Olle Svensson, Jan Isberg,
and Mats Leijon. Simulation of a linear generator for wave power absorption part ii: Verification. Submitted to IEEE Transaction on Energy Conversion,
March 2007.
[73] Anna Wolfbrandt, Mikael Eriksson, Jan Isberg, and Mats Leijon. Simulation of
a linear generator for wave power absorption -parti: Modelling. Submitted to
IEEE Transaction on Energy Conversion, March 2007.
76
Acta Universitatis Upsaliensis
Digital Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology 287
Editor: The Dean of the Faculty of Science and Technology
A doctoral dissertation from the Faculty of Science and
Technology, Uppsala University, is usually a summary of a
number of papers. A few copies of the complete dissertation
are kept at major Swedish research libraries, while the
summary alone is distributed internationally through the
series Digital Comprehensive Summaries of Uppsala
Dissertations from the Faculty of Science and Technology.
(Prior to January, 2005, the series was published under the
title “Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology”.)
Distribution: publications.uu.se
urn:nbn:se:uu:diva-7785
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2007