electric energy conversion systems: wave energy and hydropower
TRANSCRIPT
ACTAUNIVERSITATISUPSALIENSISUPPSALA2006
Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 202
Electric Energy ConversionSystems: Wave Energy andHydropower
KARIN THORBURN
ISSN 1651-6214ISBN 91-554-6617-6urn:nbn:se:uu:diva-7081
To Stefan, the best supporter
List of Papers
This thesis is based on the following papers, which are referred to in the textby their Roman numerals.
I Leijon M., Danielsson O., Eriksson M., Thorburn K., BernhoffH., Isberg J., Sundberg J., Ivanova I., Sjöstedt E., Ågren O., Karls-son K.-E., and Wolfbrandt A. An electrical approach to wave en-ergy conversion. Renewable Energy, 31(9), pp. 1309–1319, July2006.
II Danielsson O., Thorburn K., Sjöstedt E., and Leijon M.Simulated response of a linear generator wave energy converter.ISOPE’04, Toulon, France, 23–28 May, 2004.
III Eriksson M., Thorburn K., Bernhoff H., and Leijon M. Dynam-ics of a linear generator for wave energy conversion. OMAE’04,Vancouver, Canada, 20–25 June, 2004.
IV Thorburn K., Bernhoff H., and Leijon M. Wave energy transmis-sion system concepts for linear generator arrays. Ocean Engineer-ing, 31(11-12), pp 1339–1349, August 2004.
V Thorburn K., Eriksson M., Karlsson K.-E., Wolfbrandt A., andLeijon M. Time stepping finite element analysis of variable speedsynchronous generator with rectifier. Applied Energy, 83(4), pp.371–386, April 2006.
VI Thorburn K. and Leijon M. Analytical and circuit simulations oflinear generators in farm. 2005/2006 IEEE PES Transmission &Distribution, Dallas, USA, 21–24 May, 2006.
VII Thorburn K. and Leijon M. Farm size comparison with analyt-ical model of linear generator wave energy converters. Acceptedfor publication in Ocean Engineering, May, 2006.
VIII Thorburn K. and Leijon M. Ideal analytical expression for lin-ear generator flux at no load voltage. Conditionally accepted forpublication in Journal of Applied Physics, August, 2006.
IX Henfridsson U., Neimane V., Strand K., Kapper R., Bernhoff H.,Danielsson O., Leijon M., Sundberg J., Thorburn K., EricssonE., and Bergman K. Wave energy potential in the Baltic Sea andthe Danish part of the North Sea, with reflections on the Skager-rak. Submitted to Renewable Energy, March, 2006.
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X Thorburn K., Nilsson K., Danielsson O., and Leijon M. Gen-erators and electrical systems for direct drive energy conversion.MAREC’06, London, UK, 6–10 March, 2006.
XI Thorburn K. and Leijon M. Case study of upgrading potentialfor a small hydro power station. Renewable Energy, 30(7), pp.1091–1099, June 2005.
XII Bolund B., Thorburn K., Sjöstedt E., Eriksson M., Segergren E.,and Leijon M. Generator upgrade potential using new tools andhigh voltage technology. Journal on Hydropower and Dams, IssueThree, pp. 104–108, 2004.
The author has also contributed with minor inputs to the following conferencepapers (not in appendix).
A Danielsson O., Thorburn K., Sjöstedt E., Eriksson M., and Lei-jon M. Permanent magnet fixation concepts for linear genera-tor. 5th EWTEC, Cork, Ireland, 17–20 September, 2003. Non-reviewed.
B Bolund B., Segergren E., Solum A., Perers R., 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., ÅgrenO., and Leijon M. Rotating and linear synchronous generators forrenewable electric energy conversion – an update of the ongoingresearch projects at Uppsala University. NORPIE’04, Trondheim,Norway, 14–16 June, 2004. Non-reviewed.
C Danielsson O., Leijon M., Thorburn K., Eriksson M., and Bern-hoff H. Dynamics of a linear generator for wave energy conver-sion. OMAE’05, Halkidiki, Greece, 12–17 June, 2005. Reviewed.
D Gustafsson S., Svensson O., Sundberg J., Bernhoff H., Leijon M.,Danielsson O., Eriksson M., Thorburn K., Strand K., Henfrids-son U., Ericsson E., and Bergman K. Experiments at Islandsbergon the west coast of Sweden in preparation of the construction ofa pilot wave power plant. 6th EWTEC, Glasgow, UK, 28 August– 3 September, 2005. Non-reviewed.
E Stålberg M., Waters R., Eriksson M., Danielsson O., ThorburnK., Bernhoff H., and Leijon M. Full-Scale Testing of PM LinearGenerator for Point Absorber WEC. 6th EWTEC, Glasgow, UK,28 August – 3 September, 2005. Non-reviewed.
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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1 Wave energy concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 Hydropower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Generator modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Field based models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.2 Circuit equivalent models . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Rectifier modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Analytical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Fourier series expansion . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2 Simulated examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Wave energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Concept description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Force models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Cogging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Rectification of generator output . . . . . . . . . . . . . . . . . . . . . . . 373.5 Farm connections and transmission . . . . . . . . . . . . . . . . . . . . . 373.6 Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Hydropower upgrading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 High voltage generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Small-scale hydropower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Economical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Upgrading of Swedish hydropower . . . . . . . . . . . . . . . . . . . . . 40
5 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.1 Linear generator wave energy converter . . . . . . . . . . . . . . . . . . 497.2 Hydropower upgrading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8 Svensk sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Nomenclature and abbreviations
A Tm Magn. vector pot. pkW e Price per kW
B T Magnetic field q Slots per pole
D C/m2 Displacement field qc C Total charge
E V/m Electric field r % Interest rate
H A/m Magnetizing field R W Resistance
j A/m2 Free current dens. V , v(t) V Voltage
Bt T Field in tooth Vd m3 Airgap volume
Bd T Airgap field Wd J Airgap energy
c Cables per slot wp m Pole width
C F Capacitance wt m Tooth width
e(t) V EMF x m Transl. position
F0 N Spring pre-tension x m/s Transl. speed
Fb N Buoy force a % Utilisation factor
Fem N EM force am rad Harm. phase shift
Fes N End stop force q rad Electric angle
Fs N Spring force l Wb-turns Flux linkage
g m/s2 Gravity L e Value of inst. kW
hm m Harm. amplitude m Vs/Am Permeability
I, i(t) A Current r kg/m3 Density
k N/m Spring constant rc C/m3 Charge density
L H Inductance rr Wm Resistivity
m kg Mass s A/Vm Conductivity
n No of years F Wb Magnetic flux
p No of poles W rad/s Transl. ang. freq.
AC Alternating Current EMF Electromotive force
DC Direct Current OWC Oscillating Water Column
FEM Finite Element Method PM Permanent Magnet
HV High Voltage LV Low Voltage
The term rotor is used as a reference to a rotating machine, and translatorrefers to a linear generator.
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Preface
The division for Electricity and Lightning Research at the Ångstrom Labora-tory in Uppsala has a good record within the lightning field. Since 2000, whenMats Leijon was appointed as professor in Electricity, has the division alsodeveloped within renewable energy research. Several fields are investigated;wave energy, hydropower, wind power and energy conversion from water cur-rents, as well as the related energy storage issue.
Professor Mats Leijon worked as a research leader within ABB, until 2001,where an experimental approach was used. One way to succeed with researchis to build a prototype, based on thorough research and calculations, and test itto verify the research. This approach has been adopted and a wave energy con-version prototype was successfully installed in March 2006. Similar projectsare pending for most of the other research areas.
The author was registered as a PhD student within wave energy in October2002. Three PhD areas were formed then, with three students. Mikael Eriks-son has been working mainly with the hydrodynamics and force interactionsof the buoy related to the wave energy converter and Oskar Danielsson hasfocused on the generator; the stator, the translator and the electromagneticinduction. The author’s field was defined as farm interconnection and powertransmission. A project regarding hydropower was run in parallel initially,which resulted in two publications (paper XI and XII).
It has been increasingly clear over the years that all three wave energy areasare vast, and could each be the scope of research for another dozen of PhD stu-dents (which will hopefully be the case). Several questions – perhaps obviousdepending on the reader’s background – rest therefore unanswered today.
A lot of theoretical and experimental work lies ahead, which is both annoy-ing and exciting. Annoying, as each PhD student aims for the stars and setsout to get it all done, and exciting as the project will go on and incorporatemore people and it will therefore develop in other ways than today’s plan.
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"The sun’s rays shower as much energy on the earth’s surface in oneminute as the entire human race utilizes in one year. Despite the presenceof this bountiful and unusual flow of energy, a large part of the strugglesof the human race are concerned with acquiring and controlling sourcesof power. Evidently our state of development in the utilization of poweris still rather crude."
(...)
"Some day the photochemical approach to energy utilization will eitherbe solved or definitely proved impracticable. In view of our own energyresources it may seem foolish to start working on it now. But it maynot be too early to start. If we wait too long we may be caught shortas energy supplies dwindle. Moreover, many parts of the world alreadysuffer from insufficient energy. Many international problems might dis-appear if every group of people could fully utilize the energy falling onits roof-tops."
Furnas, 1941 [1]
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Acknowledgments
There are several people that I would like to thank for various reasons. First,my supervisor Prof. Mats Leijon, who encouraged me to go back to the Uni-versity after a couple of years in the industry, and with whom I have had sev-eral interesting discussions. Further, my assistant supervisor Dr. Arne Wolf-brandt has done a great job with the simulation environment with Dr. Karl-Erik Karlsson, and they are dedicated to explaining how it works.
All financiers, enabling our research, are given a warm thank you. Amongthem, over the years, are: the Swedish Energy Agency (STEM), the Ångpan-neföreningen Research Foundation, the J Gust Richert Research Foundation,Draka Kabel AB, the Vargön Smältverk AB Foundation, the Gothenburg En-ergy AB Research Foundation, the Helge Ax-son Johnson Foundation, the CFEnvironmental Research Foundation, Vattenfall AB and Uppsala UniversityFaculty of Natural Sciences and Technology. Since 2004 I was financed bythe Swedish Centre for Renewable Electric Energy Conversion (Centrum förFörnybar Elenergiomvandling, CFE), which is funded by the Swedish EnergyAgency (Energimyndigheten) and the Swedish Governmental Agency for In-novation Systems (VINNOVA). I have been the secretary of CFE and wouldlike to thank the members of the steering committee for their interest and en-couragement to our work.
Mrs. Gunnel Ivarsson, Mr. Ulf Ring and Mr. Thomas Götschl are grate-fully acknowledged for their assistance with various problems; administrative,practical and computer issues respectively. Further, I would like to express mygratitude to all colleagues at Electricity and Lightning Research for interestingdiscussions, and for helping me with proof reading.
Finally I would like to thank my family and my fiancé for their support andinspiration.
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1. Introduction
There are different types of energy in nature, and one type is the kinetic energythat is related to natural movements. Water in motion can be converted intoanother type of energy, electricity, which we use daily. The energy in movingwater in rivers can be converted with hydropower, and the water can be storedas potential energy in dams. Ocean waves are induced by wind, and the energyin waves can also be converted into electricity.
This thesis deals mainly with energy conversion from ocean waves, withfocus on the electric output and issues related to how this output can be trans-mitted to the electric grid. Some work relates to upgrading of hydropower bysubstituting conventional generators for cable wound high voltage generators.The focus of this thesis lies on the most recent work, as the earlier work waspresented also in a Licentiate Thesis in December 2004, [2].
1.1 Wave energy conceptsThere has been several wave energy conversion projects in the world overthe years. One development period was in the late 1970’s, after the oil crisis.Reviews of several of these concepts are found in [3] – [7]. The conceptsmay be categorized in three groups – OWC, wave activated bodies (includingpoint absorbers) and overtopping devices – and those who cannot be placed inany of these are grouped as not classified. These classifications were used inthe WaveNet report [8] and a schematic illustration of the three categories isshown in Fig. 1.1, where also a point absorber is shown.
An OWC, or oscillating water column, extracts the energy via an air turbine.A hollow pipe, or other structure, is placed in the waves and the wave willpress or suction the air through the turbine.
Wave activated bodies use the relative motion of the waves to drive a gener-ator. Two bodies on the ocean surface can be connected with a joint, in whichhydraulic oil is pumped in response to the motion. Alternatively, a flap thatmoves with the waves can be used. Point absorbers are buoys that are placedon the ocean surface.
An overtopping device is a container, into which the waves spill water withhigh potential energy. The water is led back into the ocean via a turbine similarto a hydropower turbine.
The concept behind the work in this thesis is based on a point absorber and alinear generator, [9] – [21]. A generator is placed on the seabed and its moving
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(a) OWC (b) Wave activated
(c) Overtopping (d) Point absorber
Figure 1.1: Schematics of wave energy conversion types.
generator part, the translator, is connected to the buoy with a rope. Springsbetween the translator and the generator foundation improve the generatorperformance and end stops prevent the translator from separating from thegenerator. Different aspects of this ocean wave energy conversion system arepresented in paper I-X, [19] – [28].
1.2 HydropowerTwo of the papers in this thesis, XI [29] and XII [30], deal with electricalupgrading of old hydropower plants. This is a relatively small field withinrenewable energy research, but can nevertheless contribute with a substantialincrease in power production, [31].
Several hydropower plants need revision in the near future as most of thehydropower plants were built 30 – 50 years ago. If the overall efficiency ofa large plant (500 MVA) is increased by 1% we add 5 MVA to the powerproduction. The increase in energy production per annum is approximately 20GWh. This figure can be compared with wind power where a new wind energyconverter can be rated at 2 MVA. If this converter has a utilization factor of30 % (i.e. the plant runs at nominal speed 30 % of the hours per year), theenergy production amounts to 5 GWh. Such an electrical upgrading of a largehydropower plant would hence correspond to a new installation of four largewind energy converters.
A hydropower plant consists of several parts, of which the generator andthe electrical system constitutes a small part, see also Fig. 1.2.
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Reservoir
Dam
Outflow
Powerhouse
Transformer
Generator
TurbineIntake Control gates Penstock
Figure 1.2: Overview of hydroelectric power plant
The most visible part of the hydropower station is the dam. Dams are im-portant structures and the planning work is crucial. All extremes must be ac-counted for as a failure would have devastating consequences. The dam isone of the reasons why new installations of hydropower in Sweden (and mostother countries) is unlikely. The ecological impact is large. Both the flora andfauna are affected as large areas are submerged into the water. It is also pos-sible that animal habitats are disturbed during the construction work. Socialfactors must also be taken into account. More than one million people mustleave their homes in China, as a result of the Three Gorges hydropower instal-lation. The Three Gorges station is the largest in the world and it will deliverapproximately 18.2 GW, [32], when completed.
Watercourses guide the water through the hydropower plant. Above the tur-bine the water is led through a vertical pipe, and below the turbine the wateris directed to the downstream outlet. The downstream pipes can sometimesbe long, and even interconnect two upstream dams to one common outletriver. The shape of the watercourses affects the hydrodynamic properties ofthe system. An overall upgrading of a hydropower plant includes revision ofthe watercourses to optimize the water flow through the turbine.
There are three main types of turbines, [33]; Kaplan, Francis and Peltonturbines. The Pelton turbines are used for the highest heads (100 – 1800 m)and Kaplan for the lowest (5 – 50 m). The Francis turbines are used in a widespectrum of head heights; from 20 m to 800 m.
The electrical generator is a fairly small part of the investment of a hy-dropower plant. And there have been few breakthroughs in generator designsince the first constructions a century ago. The rotor of the generator is placedon the turbine shaft. Today the turbine and generator are placed vertically, butin the early days the shaft was horizontal. The rotational speed is usually 100– 300 rpm, and this speed determines the number of electrical poles.
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2. Theory
2.1 Generator modellingTwo main tracks are detected within generator modelling today; the first is afield approach and the second is based on the analytical Park transform, [34]– [39]. These approaches are often used for different purposes.
Firstly, a model can be used to design a new device with correct physi-cal properties. A method where field equations are solved in accordance withphysical correlations is probably the more appropriate here. It is possible touse a circuit approach for new design also, but such models are normally sim-plified and non-linearities (such as magnetic saturation) will then not be takeninto account.
The second purpose of a model is to represent a generator, which has al-ready been built, and which makes use of parameters obtained from short-circuit and other tests. For this purpose a circuit equivalent model is probablyadequate.
The detailed finite element (FE) modelling of both hydropower generatorsand linear wave energy generators, in the work behind this thesis, is based onthe same physical principles: the field equations.
2.1.1 Field based modelsOne way to describe a generator is to use the Maxwell’s equations for electro-magnetic coupling and induction, Eqs. 2.1 – 2.4.
Ð ·D = rc (2.1)
Ð ·B = 0 (2.2)
Ð×E = −
µBµ t
(2.3)
Ð×H = j+µDµ t
(2.4)
B is the magnetic flux density, D refers to the displacement field, E is theelectric field and H is the magnetizing field. On the right hand sides, rc refersto charge density and j is the free current density.
To obtain a field model of a generator, a magnetic circuit on a rotor is re-quired. This magnetic circuit can originate from electromagnets (i.e. a rotorwith field windings) or permanent magnets mounted on the rotor. A stator
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with stator windings is modelled outside the rotor. An alternating voltage isinduced in the stator, when the rotor moves in relation to the stator windings,in accordance with Faraday’s law.
One approach to solve the field model numerically is to use a finite elementmethod, FEM. This method has been developed and has been presented inliterature, see e.g. [40] and [41], and refer to [42] for an overview.
A finite element calculation is initiated with a definition of a geometry.For generators a two-dimensional geometry is most frequently chosen, witha cross-section normal to the axis of rotation. An error is then introduced,in which the losses in the end windings constitutes the main part. A circuitcan be added to the FEM model to compensate for the end windings, and themodelling error by doing so is small compared with the difficulties associatedwith a full three-dimensional simulation. Other external circuits can also becoupled to the FE model, which are handled outside the FE environment, [43]– [46].
The simulations of the generator are based on mathematical models of mag-netic fields and there are three main model simplifications in the generatormodel. Firstly the displacement field µD/µ t has been neglected due to the low(<100 Hz) frequencies. Secondly the magnetic field is solved for a two dimen-sional section of the translator and the stator. The last simplification concernsthe end effects of the stator windings, which are modelled as impedances in anelectric circuit. With these simplifications Maxwell’s equations can be com-bined into Eq. (2.5).
sµAz
µ t−Ð ·
(1mÐAz
)= −sÐV (2.5)
This formulation is also used in, for example, [43]. s is the conductivity, Az
is the vertical component of the magnetic potential and m is the permeability.The V on the right hand side is an applied voltage, which couples the externalcircuits to the generator model.
FE simulations were used in simulations behind the results in paper I-III, V,VIII, X–XII
2.1.2 Circuit equivalent modelsA circuit equivalent of an electromagnetic object contains resistances, R, ca-pacitances, C, inductances, L and sources for voltage, V , and current, I. Theconfiguration of these parameters is regulated in differential equation rela-tions.
When a generator is modelled as circuit equivalents one generally assumesthat the generator rotor moves at a constant speed. Material parameters areassumed to be linear (e.g. constant magnetic permeability, m). Several circuitelements can be identified and the degree of complexity of the model is deter-mined by the need for detail in the model.
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An example of a method for determination of parameters in lumped circuitmodels, and of how electric circuits are coupled to mechanical circuits, isfound in [47]. An example of the process is found in [48].
The resistance, R, can be modelled simply as rrl/Al , where rr is resistivity,l is length of the conductor and Al is the cross-sectional conductor area. Idealgeneralized equations for inductance and capacitance are described further asfollows, [47].
InductanceThe flux linkage, l , is defined as
l =∫
SB ·nda, (2.6)
where the magnetic field B is integrated over the surface S. If a perfect con-ductor is assumed the terminal voltage, v, is obtained as
v =dldt
. (2.7)
l is a function of the induced current, i, and the position, x, of the movingpart (rotor or translator) of the system, i.e. l = l (i,x). Eq. 2.7 can thereforebe rewritten as
v =dldt
=µlµ i
didt
+µlµx
dxdt
. (2.8)
If it is assumed that the system is electrically linear, i.e. the flux linkage islinear with the current, l becomes
l = L(x)i. (2.9)
Eqs. 2.8 and 2.9 combined gives the terminal voltage as
v = L(x)didt
+ idLdx
dxdt
. (2.10)
The inductance L is a function of rotor position, which is intuitively reason-able. Parameters such as saturation, stator slots, small variations in air gapdistance and other imperfections will affect the inductance of the machine.
CapacitanceA capacitance may be derived in analogy with the previous section. Startingwith the total charge qc on one of the equipotential bodies in the circuit, thecurrent can be written as
i =dqdt
. (2.11)
The charge, qc, depend on material properties, geometry and applied volt-age, hence q = q(v,x) (where v and x may be functions of time). Therefore can2.11 be expanded to
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i =dqdt
=µqµv
dvdt
+µqµx
dxdt
. (2.12)
If it is assumed again that the system is electrically linear the charge can berewritten as
q = C(x)v. (2.13)
Eqs. 2.12 and 2.13 are combined into
i = C(x)dvdt
+ vdCdx
dxdt
. (2.14)
The capacitance, C, depend on geometry.
2.2 Rectifier modellingRectifiers are used for electricity conversion, from AC to DC. Power semicon-ductor devices such as diodes and thyristors can be used for this purpose; themost simple component is the diode.
A diode conducts when it is subjected to a voltage drop in the forward di-rection, and ideally the diode blocks the current when the voltage is reversed.Two diodes are needed per phase to rectify, one diode is used for the posi-tive and the other for the negative half period. For a full three-phase rectifierbridge six diodes are mounted and the positive and negative direct current,DC, leads are connected, respectively. See Fig. 2.1. A bipolar DC output isthereby obtained, where the net voltage is Vdc+ +Vdc−.
ABC
+Vdc+
-Vdc-
a1 b1 c1
a2 b2 c2
Figure 2.1: Six pulse diode rectifier connected to phase A, B and C
In the wave energy conversion concept used in the present work a rectifieris needed to enable interconnection of several generating units in a farm. Adetailed description of the rectifier model used with the FEM simulations ispresented in paper V. The results obtained were compared with results fromrotating machine simulations, [49].
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2.3 Analytical theoryThe ocean wave energy source is irregular, and it is therefore not straightfor-ward to predict the generator output without simulations. An ideal analyticaltheory has therefore been developed, which may lead to deeper understanding,or which can be developed further to serve as a tool for system design. Res-onances and unwanted harmonics can be avoided if transmission equipment(e.g. cables and filters) and power converters are designed properly, e.g. [50].This theory evolves in paper VI–VIII.
Assume a linear translator where permanent magnets (PM’s) are mountedwith alternating polarity, [24, 25]. This translator is then moved vertically, seeFig. 2.2 where a sinusoidal motion is illustrated. However, as the ocean wavesare irregular it is not realistic to expect that the translator moves sinusoidally.Any continuous function can be described as an infinite Fourier series, whichmeans that several superimposed waves form an arbitrary motion pattern. Anexample of this is found in Fig. 2.3. The terms "wave motion" and "waveposition" hereafter refer to the translator motion and position.
Translator position
Time
Translator with permanent magnets
Stator with cable windings
Ω
h
Translator speed
Time
Ωhwp
0
h
-h
Figure 2.2: Cross section of linear generator. The translator with permanent magnetsmoves sinusoidally in the most simple case.
The amplitudes associated with the harmonics are denoted hm and the fun-damental angular frequency of the translator motion is W. Then the positionand speed of the translator, x and x, can be written as:
x(t) =¤
äm=1
hm sin(mWt +am) (2.15)
x(t) = W¤
äm=1
mhm cos(mWt +am). (2.16)
W is the fundamental angular frequency of the translator motion based on thewave period T , W = 2p/T and hm is the harmonic amplitude. The harmonicindex is m and the phase shift is am for each harmonic.
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-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6Time [s]
Wave
he
igh
t[m
]
x(t)
h1
h2
h3 h4h5
α =α =α1 2 3=0α4
α5
Figure 2.3: Example of series expanded wave. The harmonic waves have frequenciesthat are multiples of the fundamental frequency, and phase shifts are included.
If the pole pair width, wp, is defined as the distance from one north pole tothe next (see Fig. 2.2), then the electric angle q is
q(t) =2pwp
¤
äm=1
hm sin(mWt +am) =¤
äm=1
bm sin(mWt +am). (2.17)
A mathematical amplitude bm = 2phm/wp is introduced here.Assuming that the flux, F, lags the motion by angle d , the flux (a function
of q ) may be written as,
F(t) = Ft cos(q +d ) =Ft cos
[¤
äm=1
bm sin(mWt +am)+d
](2.18)
where Ft is approximated here by a constant describing the amplitude of thetotal flux. The angle d is related to the load angle, i.e. how much the voltageshifts during full load with respect to the no load voltage. From this pointonwards d is set to zero, which means that the no load case is considered, i.e.where the voltage equals the electromotive force (EMF). The magnitude ofthe flux, Ft , is identified as
Ft = Bt ·wt ·d (2.19)
i.e. the product of Bt , the magnetic field in a tooth (obtained from numericalfinite element, FE, simulations)[16, 12, 40, 51], and the cross sectional areathrough which the flux lines are guided (defined as the width of a stator tooth,wt , times the width of the stator stack, d). Further, p is the total number ofpoles, q is the number of slots per pole and phase and c is the number ofcables in a slot, which gives the total number of turns, N as
N = p ·q · c. (2.20)
24
The last step is to derive Eq. (2.18) to obtain the EMF according to Faraday’slaw:
e(t) = −NdFdt
= NFt
[¤
äm=1
mWbm cos(mWt +am)
]×
sin
[¤
äm=1
bm sin(mWt +am)
]. (2.21)
The EMF expression in Eq. (2.21) may be used for numerical Fourier trans-forms to find the frequency contents. An alternative is to calculate the Fourierseries based on the wave fundamental W, as shown in the following section.
2.3.1 Fourier series expansionThe target for the following derivation is to write the flux in Eq. (2.18) as aseries expansion,
F(t) =¤
än=−¤
AneinWt . (2.22)
A complete solution is presented in paper VIII, and is not duplicated here. Theresults are highlighted, nevertheless. The solution to An is
An = Ft
¤
Ôm=1
[¤
äpm=−¤
eipmamJpm(bm)
], (2.23)
with contributions that fulfils¤
äm=1
mpm = n and¤
äm=1
pm = 2q. (2.24)
Jpm(bm) refers to Bessel functions, see e.g. [52].This series expansion may be turned into a positive sum of cosines, where
the terms are calculated as
F(t) =¤
än=0
Bn cos(nWt +bn), (2.25)
where
Bn =
[ä∗
¤
Ôm=1
Jpm(bm)cos
(¤
äm=1
pmam
)]2
+
+
[ä∗
¤
Ôm=1
Jpm(bm)sin
(¤
äm=1
pmam
)]2
1/2
(2.26)
bn = (p+)arctan
[äÔJpm(bm)sin(ä pmam)
äÔJpm(bm)cos(ä pmam)
]. (2.27)
p is added if the arctan denominator is negative.
25
The asterisk (*) sum is defined in Eq. (2.24) and an offset is obtained forn = 0, where b0 = 0 and
B0 = ä∗
[¤
Ôm=1
Jpm(bm)
]cos
(¤
äm=1
pmam
). (2.28)
Accordingly the EMF is derived as
e(t) = −NdF(t)
dt= −iWNFt
¤
än=−¤
nAneinWt (2.29)
= −W¤
än=1
nBn sin(nWt +bn) (2.30)
where An, Bn and bn are given by Eqs. (2.23), (2.26) and (2.27).The harmonic content in the EMF for each multiple of the fundamental
wave frequency is thereby obtained directly from Eq. (2.30).
2.3.2 Simulated examplesTwo simulated examples are used to illustrate the theory presented here. Inthe first example an analytical simulation with a circuit is compared with adynamic finite element (FE) simulation. The second example shows how thetheoretical model can be used, and results from a range of simulations arediscussed.
FE simulation comparisonA linear test generator has been defined and simulation parameters were cho-sen to somewhat correspond to wave energy converters for Swedish waters. Aselection of generator parameters are presented in Table 2.1.
Parameter Value in simulations
wt 8 mm
wp 100 mm
d 400 mm
p 100
q 6/5
c 6
T 5 s
h 1 m
Bt 1.53 T
Table 2.1: Parameters for example study.
26
Generator
Resistance Inductance
Load
R
Rload
L
e(t)
+
-
v(t)
+
-
i(t)
Figure 2.4: Circuit
Dynamic FE simulation
The generator was designed with a finite element calculation tool where thegenerator voltage is used as a design target, which affects the machine length.This design is done in a stationary mode, i.e. the translator is locked in spacerelative the stator. The design process is usually iterated several times until themost desired performance is obtained. Other geometric parameters are fixed,as well as material properties. A range of generator parameters are found inTable 2.1. The magnetic flux in the stator tooth, Bt , was obtained from a no-load simulation, and is used for analytical calculations.
It is possible to simulate the generator in a dynamic mode, which has beendone for the designed generator. The generator windings has a resistance of0.37 ohms per phase and an inductance of 11.5 mH at nominal speed (0.8 m/s)and a resistive load of 3.1 ohms per phase was connected to the generator.
Analytical and circuit simulation
The data from Table 2.1 was inserted into Eq. 2.29 with m = 1 in An, Eq.(2.23),(i.e. for a perfectly sinusoidal motion), which gives the ideal EMF forthe machine. That is, for a no-load case. To compare the results with the FEcalculations a load situation is formed by adding a resistive circuit to the EMFmodel, see Fig. 2.4. There will be a voltage drop over the cable windings aswell as over the load, and a phase shift occurs due to the winding inductance.
The differential equation for the circuit with respect to the load voltage iscalculated as:
v(t) = e(t)−R · i(t)−L ·
didt
(2.31)
v(t) = Rload · i(t) (2.32)dvdt
=Rload
L
[e(t)− v(t)
(R
Rload+1
)](2.33)
The reactance is described with a differential equation here, rather than X =wL, as the frequency varies.
27
-300
-200
-100
0
100
200
300
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Vo
ltag
e[V
]
Time [s]
Figure 2.5: Voltages from comparison simulation. The EMF from the analytical simu-lation has the highest amplitude (green), then the corresponding terminal voltage v(t)(red), and the FE simulated load voltage (blue).
-80
-60
-40
-20
0
20
40
60
80
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Cu
rre
nt
[A]
Time [s]
Figure 2.6: Currents, analytical (red) and FE simulated (blue).
EMF and load voltage are shown with the FE simulated voltage in Fig. 2.5.The corresponding currents, powers and analytical flux are are found in Figs.2.6, 2.7 and 2.8 respectively.
Interpretation of the simulation
The load voltages from the analytical and from the FE simulations are quitesimilar, as are the corresponding currents. An offset in the power levels revealsa discrepancy of some 4% for the time averages.
Three simulated wavesHeight comparison
Another simulated example shows two waves, Fig. 2.9, the first with a heightof 25% of the second wave. The aim is to illustrate the harmonic content of the
28
0 0.2 0.4 0.6 0.8 1 1.2 1.40
5
10
15
20
25
30
Po
wer
[kW
]
Time [s]
(Time averages,~4% difference)
Figure 2.7: Powers, momentary and time averages, analytical (dashed) and FE simu-lated (solid).
-4
-3
-2
-1
0
1
2
3
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Time [s]
Flu
x[m
Wb],
spe
ed
[ m/ s
]
Figure 2.8: An illustration of the flux F(t) (solid) in the analytical and circuit calcu-lation, plotted with the translator speed (dashed).
magnetic flux and EMF as derived in the previous sections. The fundamentalfrequency is W0 = 2p/5 rad/s, i.e. the period is T = 5s.
The waves illustrate the motion of the translator, and the correspondingfluxes and voltages are found in Fig. 2.10.
One can identify that the highest EMF is obtained for the highest velocity,i.e. for the larger wave when the slope is steepest (right before t = 5s). Thisspeed is never obtained for the smaller wave and the voltage is therefore moremodest. The flux patterns differ between the two waves for that same reason.
29
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6Time [s]
Wave
he
igh
t[m
]
(a) Small wave
-0.5
0
0.5
-1.5
-1
1
0 1 2 3 4 5 6Time [s]
Wave
he
igh
t[m
]
(b) Large wave
Figure 2.9: Two waves, dashed are fundamental and dotted graphs are first harmon-ics (double fundamental frequency). Solid graphs show the sum, which is the wave.hsmall = [0.225 0.05]m, hlarge = [0.9 0.2]m, a = [0 0.8]rad.
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 1 2 3 4 5 6Time [s]
Flu
x[m
Wb]
(a) Small wave flux
Fl u
x[m
Wb]
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 1 2 3 4 5 6Time [s]
(b) Large wave flux
-80
-60
-40
-20
0
20
40
60
80
Volta
ge
[V]
0 1 2 3 4 5 6Time [s]
(c) Small wave EMF
-400
-300
-200
-100
0
100
200
300
400
Volta
ge
[V]
0 1 2 3 4 5 6Time [s]
(d) Large wave EMF
Figure 2.10: Flux and EMF for the waves in Fig. 2.9. There are two graphs in eachdiagram with good overlap: dashed represents the ideal graph as per Eqs. (2.18) and(2.21), and the solid graph shows the series expanded solutions, Eqs. (2.25) and (2.30).
30
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Multiples of Omega (n)
Flu
xh
arm
on
i ccon
ten
t[m
Wb]
(a) Flux harmonics, small wave
-2
0
2
4
6
8
10
12
14
Flu
xha
rmon
i cco
nte
nt
[10
Wb
]- 4
0 20 40 60 80 100 120Multiples of Omega (n)
(b) Flux harmonics, large wave
0
2
4
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40Multiples of Omega (n)
Em
fh
arm
on
i cco
nte
nt
[V]
(c) EMF harmonics, small wave
0
5
10
15
20
25
30
35
40
45
Em
fh
arm
on
i cco
nte
nt
[V]
0 20 40 60 80 100 120Multiples of Omega (n)
(d) EMF harmonics, large wave
-5
-4
-3
-2
-1
0
1
2
0 5 10 15 20 25 30 35 40Multiples of Omega (n)
Arg
um
ent
vari
at ion
[ rad
]
(e) Argument, small wave
-5
-4
-3
-2
-1
0
1
2
0 20 40 60 80 100 120Multiples of Omega (n)
Arg
um
ent
vari
at ion
[ rad
]
(f) Argument, large wave
Figure 2.11: Harmonics, or rather, values of Bn, WnBn and bn in Eqs. (2.25) – (2.30).
31
The peaks in the flux plots represent passages of north and south magneticpoles in relation to one phase. Further, as the translator velocity is higher forthe larger wave, we will see higher harmonics present for flux and EMF forthe larger wave. See Fig. 2.11, where also the phase is plotted against themultiples of the fundamental frequency.
Frequency comparison
In the final simulation two waves with identical harmonic amplitudes, but withdifferent fundamental frequencies (W= 2p/4 andW= 2p/5), have been com-pared. Illustrations are found in Fig. 2.12.
-1.5
-1
-0.5
0
0.5
1
0 1 2 3 4 5 6
Time (s)
(a) Waves xi(t) and harmonics
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 1 2 3 4 5 6
Time (s)
(b) Fluxes, Fi(t)
-400
-300
-200
-100
0
100
200
300
400
0 1 2 3 4 5 6
Time (s)
(c) EMFs, ei(t)
-5
-4
-3
-2
-1
0
1
2
0 20 40 60 80 100 120
Multiples of Omega (n)
(d) Arguments bi,n vs n
-2
0
2
4
6
8
10
12
14
0 20 40 60 80 100 120
Multiples of Omega (n)
(e) Fi,n vs n
0
10
20
30
40
50
60
0 20 40 60 80 100 120
Multiples of Omega (n)
(f) ei,n vs n
0 5 10 15 20 25 30
Frequency, Hz
-5
-4
-3
-2
-1
0
1
2
(g) bi,n vs frequency f (Hz)
-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30
Frequency, Hz
(h) Fi,n vs f (Hz)
0 5 10 15 20 25 30
Frequency, Hz
0
10
20
30
40
50
60
(i) ei,n vs f (Hz)
Figure 2.12: Argument and harmonics for flux and EMF for waves with T=4 s (i =1: red, black) and T=5 s (i = 2: blue, green). a) First (dashed) and second multiple(dotted). d)-f) n refers to multiple.
32
Interpretation of the simulations
General peaks in flux and EMF appear for the same multiples of the funda-mental frequency in the height comparison. The maximum peaks do not nec-essarily coincide with respect to harmonic number n, however, as the EMF isscaled with n, see Eq. (2.30). A variation in wave height will affect the numberof harmonics so that higher amplitudes return higher numbers of harmonics.
The EMF is also scaled with the fundamental angular frequency, W, whichmeans that the exact graphs for the EMFs in Fig. 2.11 are valid for W =2p/5 rad/s. Another fundamental frequency will scale the EMF amplitude,and a scaling also occurs in the frequency domain (but the number of harmon-ics is the same), which is seen in the frequency variation example.
33
3. Wave energy
One reason why wave energy is of interest as a renewable energy source isthe potential, which has been estimated in paper IX to 56 TWh in the BalticSea. A sample of real data from Ölands södra grund is found in paper II. Theenergy content of waves in those waters is quite modest compared to e.g. theNorwegian and Portuguese coasts. An often-used measure is the energy flux[kW/m], which refers to the energy transported in each metre of wave front,per second. The annual average in the Baltic is about 5 kW/m whereas thefigure can be as high as 100 kW/m in some areas, [53], [54].
The idea behind this wave energy research is to convert ocean wave energyto electricity with a simple system of direct drive generating units in farms.Today the solution comprises a buoy on the ocean surface connected by a ropedirectly to the generator on the seabed. The generator is linear, which meansthat the motion of the translator is up and down (in contrast to a conventionalgenerator with a rotating rotor). Linear generators are used in a few other waveenergy concepts, see e.g. [55] and [56].
A fundamental description of the research is found in paper I, where thework is motivated with theory, simulations and experimental results.
3.1 Concept descriptionThe system for energy conversion from ocean waves used in this thesis isbased on a point absorber and a linear generator, [57].
To broaden the understanding of the model, the linear generator systemused is briefly presented. A buoy on the ocean surface pulls a translator up bya rope. Springs connecting the foundation (on the seabed) and the lower endof the translator are also energized from the buoy. Surface mounted permanentNeodymium Iron Boron (Nd2Fe14B) magnets, [10], on the translator induce avoltage in the stator windings, which are positioned outside the translator. Seealso [9], in which the system is described, and [21], where the power outputis investigated at different DC voltage levels.
The concept also includes transmission and grid integration and severalunits can be interconnected to increase the total power output, [22]. In Fig.3.1 (a) a schematic generator is shown, and an example of a farm connectionand cable transmission is illustrated in (b).
35
Foundationon seabed
Generatorenclosure
Translatorwith PMs
Stator withcable windings
Spring
Rope
Buoy
Oceanwave
20
-100
m
2-
6m
(a) Power take-off
Ocean Land
Converter
Transformer
Grid
Generator
Transmissioncable
Rectifier
(b) Transmission schematics, [22]
Figure 3.1: Permanent magnet linear generator for wave energy conversion, (a) powertake-off and (b) electric connections and transmission, [22].
3.2 Force modelsAll parts in the power take-off system contribute with forces that either en-ergize or damp the motion of the translator. The forces are presented in Eqs.(3.1-3.6), [9].
mx = Fb +Fem +Fes +Fs +mg (3.1)
Fb = rgVb (3.2)
Fs = −F0 − kDx (3.3)
Fem =dWddx
(3.4)
P =dWddt
(3.5)
where
Wd =∫ B2
d2m0
dVd . (3.6)
m is the mass of the translator and the buoy, in the equations above, if the buoyforce is greater than zero, or the mass of the piston only. Fb is the buoy force,Fem denotes the electromagnetic forces, Fes is an end stop force, modelled asa stiff spring to prevent the piston from leaving the generator, and Fs is thespring force. r denotes the water density, g represents gravity and Vb is thevolume of the water displaced by the buoy. z is the position of the wave andx is the piston position, along the same vertical axis. F0 is the spring charge,k denotes the spring constant, and Fem is calculated from the electromagneticfield in the airgap. The energy in the airgap, Wd , is integrated from functions
36
of the magnetic flux in the airgap, Bd , and the permeability m0, over the airgapvolume Vd .
Such a configuration generates voltages and currents that not only varyin amplitude and frequency, but also in phase orientation as the translatorreverses direction, as indicated in Fig. 3.1. A corresponding rotating phe-nomenon would be a rotor that spins in the clockwise direction first, thenstops, and when it starts again it rotates in the anti-clockwise direction.
3.3 CoggingA phenomenon, called cogging here, appears if the distance between two mag-nets of the same polarity is a multiple of the distance between six stator slots.If a magnet opposes a stator tooth the field is locked in the steel material.When the translator moves the field stays in the tooth until the next tooth iscloser. Then the field lets go and the magnet is jerked towards the next tooth.See also [15].
When all magnets follow the same pattern the motion is affected and thetranslator no longer moves smoothly. Cogging causes excessive fatigue to ma-terials and introduce spikes in the power output.
3.4 Rectification of generator outputThe generator terminal voltage is far from sinusoidal, both due to the stochas-tic nature of the waves and due to the linear generator setup itself (the transla-tor stops and reverses). See e.g. Fig. 6 in paper I or Fig. 2.5. Power electroniccomponents, [58], [59], are required to shape the electricity into somethinguseful, and a diode rectifier can be connected as a first step.
There are different levels for semiconductor modelling, as well as for gen-erators, and the model types are suitable for different purposes. FE modelsof e.g. diodes are aimed at diode development, [60], [61], and lumped circuitmodels are used in circuit simulation software, [62]. The most simple modelof a diode, a switched resistance, [49] – [64], is often used in external circuitscoupled to FE simulations.
The introduction of a diode in a circuit affects the voltages and currents onboth sides of the diode, see paper V where a rectifier model was included asan external circuit in detailed FE calculations.
3.5 Farm connections and transmissionIt is possible to interconnect several linear generators in farms provided thateach unit has a rectifier and the interconnection is done in parallel on the DC
37
side. The DC voltage will then serve as a regulation parameter, as seen inpaper III, and the power variations from each generator appears as variationsin current injections. The power output from one unit varies widely over awave period, whereas the power variations from several units in a farm areless severe, paper VII.
The units can be interconnected in farms in several ways using transformersand / or power electronic components, illustrated in paper IV and IX. A cableis used for the transmission of power to the electrical grid onshore, [65] – [67].
Paper IV shows some transmission options, and in paper V a more detailedsimulation procedure for connecting a rectifier to the FE simulations is pre-sented.
3.6 Experimental workTwo experimental generators were built. The first was aimed for laboratoryuse, and the experimental data in the verification in paper I was obtained fromthis setup, see Fig. 3.2.
Figure 3.2: Prototype in laboratory. The blue structure supports the stator windingsand the translator is pulled by a motor, seen to the right.
The second generator is a part of a full scale test unit, which was installednear Islandsberg on the Swedish west coast, in March 2006.
38
4. Hydropower upgrading
4.1 High voltage generatorsThe generators found in most hydro power plants today are conventional syn-chronous machines in which the voltage level cannot exceed 25 kV due toinsulation limits. This means that a high current, I, is needed in the statorwindings to obtain a high output power. Since the resistive losses and mechan-ical forces are proportional to I2, a lower current is desirable. A transformer isalso a necessity, when a lower generator voltage is used, as transmission volt-ages can be of magnitude of several hundred kV. The transformer step alsocontributes to the losses.
It is possible to increase the voltage substantially by using a PowerformerTM
generator, described in [68]–[71], due to the use of insulated cables in the sta-tor windings, [72]. A number of transformers in a power system can therebybe removed completely, or replaced by cheaper and more efficient autotrans-formers, whose function is described in [73]. Generator and transformer lossescan be cut by several per cent. In Fig. 4.1 a traditional single line diagram anda Powerformer diagram are presented. A cable wound high voltage generatorhas been built for 155 kV and can theoretically be designed for voltages ashigh as 400 kV.
Generator Transformer
Surge arrester
HV Breaker
Powerformer
Grid
Generatorbreaker
HV Breaker
Surge arresters
Grid
Figure 4.1: Single line diagams, conventional generator (top) and Powerformer
39
4.2 Small-scale hydropowerIt is of interest to discuss the small hydropower plants even if the energy pro-duction from each station is small. Historically, the plants have grown in sizeover the years, and several small plants were installed 50 - 100 years ago. Seealso [74]. This implies, that if they have not been restored already, the needfor refurbishment is high. It might even be time for a second restoration of theoldest plants. [75]
A recent study, [29], shows that an upgrade to new technology results intwo advantages. Firstly, the number of components is reduced, resulting in areduction of cost compared with replacement of these components during aconventional restoration. Secondly, the power production is increased as theavailable power is converted more efficiently. The annual energy productionfrom plant the in the study, where the total power is 18 MW, is estimated toincrease more than 5%, from 75 to 79.2 GWh.
4.3 Economical modellingFor hydropower plants that are due for an upgrading one can easily motivatea change to a more efficient system. It might not be that simple to justify anupgrading for a fairly new installation, even if it is possible.
Parameters that are taken into account in the economical models are reduc-tion in losses, reduction of number of components, reduction of maintenance,and increased system efficiency. These parameters are combined, and the for-mula for capitalized cost may be used
L= a ·H · pkW ·
1− (1+ r)−n
r. (4.1)
Here L is the value of one installed kW, a is the utilization factor, H=8760 isthe number of hours in one year, pkW is the price per kWh, r is interest rate,and n the number of years for the investment. This equation can be used asa first estimate to determine how much an investment is allowed to cost withrespect to the increase in energy production.
4.4 Upgrading of Swedish hydropowerA study was conducted in 2003 regarding the upgrading potential for hy-dropower in Sweden. This study resulted in two reviewed papers, XI and XII,and a report to the Swedish Energy Agency (STEM), [76].
The work was done as a cooperation between Electricity and LightningResearch at Uppsala University and Luleå Technical University.
40
5. Summary of papers
I: An electrical approach to wave energy conversion, [19]This paper describes the work with renewable kinetic electric energy conver-sion at Uppsala University, applied to wave energy. Direct drive is promoted toreduce maintenance cost and the theoretical foundation is presented. Resultsare provided from simulations and experiments.
Renewable Energy, vol. 31, no. 9, July 2006. The author’s research is usedin the paper, but there was no direct participation in the writing. The paperwas number 11 on the TOP25 list for Renewable Energy in January-March2006. Refereed journal publication, published.
II: Simulated response of a linear generator wave energy converter, [20]The behaviour of a linear generator wave energy converter with a point ab-sorber is investigated in this paper for an authentic wave climate, i.e. the gen-erator is simulated for waves other than the design wave. It is found that theoutput power characteristics for waves smaller than the design wave is similarto the output for the nominal wave, only lower in amplitude. A limiting func-tion is found for waves higher than the design wave, as a built-in feature of thesystem. The system can therefore, hopefully, limit the destructive overloads,e.g. during a storm.
Oral presentation at the ISOPE-2004 conference in Toulon, by O. Dani-elsson. The author participated in the evaluation work. Refereed conferencepublication.
III: Dynamics of a linear generator for wave energy conversion, [21]A rectifier is included in the FEM simulations as it is a part of the system andhas a great impact on the electric parameters. The induced voltage determineswhether the diode conducts or blocks; if the induced voltage is higher than theDC voltage the diode conducts. The DC voltage level is therefore essential forthe current (i.e. power) output from the generator, and is varied in simulationsin this paper, as well as simulated ocean wave heights. Optimum DC voltagesare identified for three wave heights.
Oral presentation at OMAE-2004 in Vancouver by M. Eriksson. The authorcontributed with text, layout and evaluation. Refereed conference publication.
41
IV: Wave energy transmission system concepts for linear generatorarrays, [22]Transmission and farm interconnection aspects are discussed in this paper.Four transmission options and four connection schemes are presented andevaluated with examples. The conclusion is that small farms are most econom-ical with small transmissions. This means that as little equipment as possibleis placed at sea. Large installations, on the contrary, might gain from powercomponents offshore.
Ocean Engineering Volume 31, number 11–12, August 2004. The authorinitiated and completed the paper. The paper was third on the TOP25 listfor Ocean Engineering in July-September 2004. Refereed journal publication,published.
V: Time stepping finite element analysis of variable speed synchronousgenerator with rectifier, [23]Direct driven variable speed synchronous generators will have a variable out-put. It is therefore necessary to convert the voltages and currents to make themgrid compatible. A rectifier model that handles variable voltages and variabledirections (i.e. for linear generators) is presented in this paper. Simulations areused to illustrate the rectifier behaviour.
Applied Energy volume 83, number 4, April 2006. The author is main au-thor. Refereed journal publication, published.
VI: Analytical and circuit simulations of linear generators in farm, [24]A mathematical model is used to describe the ideal generator EMF in a PSpicesimulation environment. An inductance and a resistance were placed in seriesto model the generator impedance. Five such generators were modelled andconnected to rectifiers, and the output was analyzed for five cases where theamplitudes and frequencies were varied between the generators.
Presented by the author with a poster at the 2005/06 IEEE PES Transmis-sion and Distribution Conference in Dallas, USA, 23 May 2006. The authoris main author. Refereed conference publication.
VII: Farm size comparison with analytical model of linear generator waveenergy converters, [25]The material in this paper builds on paper VI, and has been extended to includea comparison with ten generators. A smoother power output can be detectedwith more generators if the generators’ translator motions are well spaced.
This paper was accepted for publication in May 2006 in Ocean Engineering.The author is the main contributor. Refereed journal publication.
42
VIII: Ideal analytical expression for linear generator flux at no loadvoltage, [26]The harmonic content in the voltage generated from a linear generator varieswidely over a wave period. An ideal expression for the voltage harmonics hasbeen derived with Fourier series expansions in this paper, so that the harmon-ics are expressed as multiples of the fundamental translator frequency. Theknowledge of the harmonic content is useful for the design of the transmis-sion system and electrical filters, primarily to avoid resonance. The mathemat-ical model from papers VI and VII is used, and is also compared with finiteelement simulations.
Conditionally accepted for publication in Journal of Applied Physics in Au-gust, 2006. The author is main author. Refereed journal publication.
IX: Wave energy potential in the Baltic Sea and the Danish part of theNorth Sea, with reflections on the Skagerrak, [27]A cooperation project with the Swedish utility Vattenfall resulted in a surveyof the potential for wave energy in the Baltic and the Danish part of the NorthSea. Several aspects of wave energy are addressed, such as ecological impact,energy and efficiency calculations, geological impact (i.e. seabed conditions)and some case studies are used as illustrations.
Submitted to Renewable Energy in March 2006. The author contributed tothe transmission aspects of the case studies. Refereed journal publication.
X: Generators and electrical systems for direct drive energy conversion,[28]Continuous speed variations will be a challenge for direct drive generatorsaimed for renewable energy conversion, as an as high efficiency as possible isdesired for all load conditions. The efficiencies for two generators, one linearand one rotating, have been mapped in simulations with respect to translatorvelocity or rotor speed in this paper. It is concluded that both generators, withdata from experimental set-ups, exhibit good efficiencies for part load (halfspeed) up to severe overload (2.5 times nominal speed); >70% for the rotatingand >80% for the linear generator.
Oral presentation (by the author) at MAREC’06 in London, UK, 9 March2006. The author is corresponding and contributed mainly to work on lineargenerators. Refereed conference publication.
XI: Case study of upgrading potential for a small hydro power station,[29]It is possible to increase power output from small hydropower plants by opti-mizing the use of the available power. A new generator and electrical systemcan further reduce losses. In this paper a Swedish small hydropower plant isused as a reference, a new generator is simulated and a new system is pro-
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posed. The annual increase in energy production is estimated to 4.2 GWh forthe total generator power of 18 MW.
Renewable Energy volume 30, number 7, June 2005. The author is mainauthor. Refereed journal publication, published.
XII: Upgrading generators with new tools and high voltage technology,[30]Several hydropower plants are investigated and new generators are simulated.Losses are estimated and the conclusion is that losses may be cut by up to 66% compared with losses today.
Journal on Hydropower and Dams, Volume Eleven, Issue Three, 2004. Aspaper (XI), this paper resulted from a study in 2003. The author contributedwith simulations and evaluation work. Refereed journal publication,published.
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6. Discussion
One of the first issues that one encounters when working with direct drivesystems is that the frequency is far from constant. This means that most con-ventional models are out of scope, and it is also essential to investigate theoperating conditions for experimental material.
It is important to know the purpose of the simulation work to choose theright model. Detailed models require longer computing times than simplermodels, and the results correspond to the level of detail. That is why it is use-ful to use detailed FE simulations to gain knowledge of the device properties.FE simulated results can act as a "target" for simpler models if real exper-imental results are lacking (which was the case during the first half of thework). It is also possible to continue to use the FE simulations as targets evenif experimental data are present, as long as the FE model has been properlyverified.
Paper I includes simulated and measured data, which serves as a verificationof the FE software used. This verification is reproduced in Fig. 6.1.
0 2 4 6 8 10 12 14 16 18 20
-8
-6
-4
-2
0
2
4
6
8
Time (s)
Volta
ge
(V)
Va
VbVc
(a) Experiment, three-phase voltage
-2
-4
-6
-8
0
4
2
6
8
0 2 4 6 8 12 14 16 18 2210 20
Voltage
(V)
Time (s)
Va
VbVc
(b) Simulation, three-phase voltage
Figure 6.1: Verification of simulation with experimental results. From [19].
The energy output is probably more interesting than the technology from acommercial perspective. An attempt to estimate the annual energy productionis presented in paper II, although the models used will need a revision to be ofgreat significance for the future. The research has moved on in several fields,e.g. [77], [78], and an updated hydrodynamic model is essential.
One significant result for the control of a linear generator with a rectifier isfound in paper III, see Fig. 6.2. The power output is highly dependent on thelevel of the DC voltage on the DC bus. This means that there is an optimal DC
45
voltage for all wave conditions, which will serve as a target in a future controlalgorithm.
T=3.0s T=4.5s T=6.5s
Po
we
r(k
W)
20
25
15
10
5
DC-voltage (kV)
00.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Figure 6.2: Output power for three wave periods: 3.0, 4.5 and 6.5s as functions of DClevel. Each x represents a full physics simulation, [21]. The shorter wave period (3.0s)returns the highest power as the translator moves fast.
An overview of system transmission options was presented in paper IV. It isimplied that the farm size and the distance to the electrical grid are importantparameters in choosing transmission system. What has yet to be addressed arethe power quality and reliability issues with respect to transmission scheme.Offshore wind power also deals with these questions, e.g. [79] – [81].
Rectification was studied in paper V and an illustration of how a diode in acircuit affects the voltage and currents on both of its sides is found in Fig. 6.3.This means, for example, that the behaviour of fields in the generator changeswhen a diode is connected to the generator terminals.
Rectifier models are commonly used in simulation software (e.g. PSpice),and such diode models were used in applied farm simulations in paper VI andVII to see what happens when several units are interconnected to the sameDC bus. The studied examples constitute a basic overview of the generic farmbehaviour, and it is indicated that the power fluctuations are reduced with anincreased number of units in the farm.
One step further was taken in paper VIII to calculate the harmonics in thegenerated voltage. This model can be developed further as only the no-loadvoltage is calculated. Another need for improvement is to include the statorwinding ratio in the model. Today is the ratio only used as a scaling, but in-cluding the ratio would improve the effects of cogging and the flux harmonics.To include the ratio the idealized assumption that F = cosq must be revised,and also include harmonics, [82]. The analytical model, and its series expan-sion, can also be expanded with the force equations, Eqs. (3.1) – (3.4), to usethe wave motion, rather than the translator motion.
An estimate of the wave energy potential in the Baltic Sea is presented inpaper IX, which was a co-operation between the Swedish utility Vattenfall andthe wave energy group in Uppsala. One insight from the project was that the
46
(a) Currents (b) Voltages
Figure 6.3: AC (solid) and DC currents and voltages, at constant piston speed, [23].The corresponding AC graphs without the rectifier are sinusoidal.
ocean waves in Swedish waters have not been measured carefully comparedwith, e.g., the waves in the USA. It is likely that new hind casting models,verified with long-term measurements, would provide more reliable results ofthe overall Swedish potential.
The direct drive concept, where the source motion drives the generator rotor(or translator) directly, results in the varying induction shown in several illus-trations through the thesis. It is almost without meaning to talk about nominalspeed for these generators as they always run at part load (=slow) or overload(=fast). It is, however, important that the generators have a wide frequencyoptimum, and this aspect has been simulated in paper X for a linear, see Fig.6.4 and a rotating machine.
81
82
83
84
85
86
87
88
89
LG
Effi c
i en
cy
[%]
0.5 1 1.5 2 2.5Speed [p.u.]
Figure 6.4: Efficiency with respect to translator speed for 13 kW linear generator. [28]Nominal speed at 1 p.u. is 0.8 m/s
47
Hydropower is a large contributor to the Swedish energy mix. A study donein 2003 showed that the electrical efficiency of a hydropower plant could beincreased significantly. For plants with generators rated at a power less than15 MVA the increase can be as high as 5%. The efficiency for larger plantscan be increased by 1% or more. This is described in paper XI and XII. Arevision of the waterways and turbines may lead to even higher increases inenergy production.
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7. Conclusion
7.1 Linear generator wave energy converterA wave energy system with a linear generator, a point absorber and springs canbe used for renewable electric energy conversion. Such a device will producecurrents and voltages with varying frequencies and amplitudes in accordancewith Faraday’s law. See Fig. 7.1. Therefore must the output be rectified withpower electronic components. The DC output then allows for several units tobe interconnected in a farm, to increase the total output power and reduce thenumber of transmission cables (if found economically favourable).
-10 -8 -6 -4 -2 0 2 4 6 8 10
-150
-100
-50
0
50
100
150
Vo
ltag
e(V
)
Time (s)
23 May 2006, 07:50:50 PM
Figure 7.1: Example of raw experimental data from the Lysekil test plant, three phaseload voltage. Unpublished work by courtesy of Olle Svensson.
An analytical model for identifying the harmonics in the induced no-loadvoltage has been proposed and verified with FE simulations and indirectlywith experimental results. This model can be used in system design for farm-to-grid connection, to avoid resonance frequencies in filters, cables and powerelectronic equipment.
7.2 Hydropower upgradingAn upgrading of the hydropower generator to a new high voltage generator,with a cable wound stator, can increase the efficiency substantially and elimi-
49
nate the need for an intermediate transformer for connection to the power grid.The exclusion of the transformer reduces space consumption in the plant, re-duces environmental impact and reduces losses. One circuit breaker, the gen-erator breaker, which must handle high currents, is also excluded.
Resistive losses also decrease as the system with a high voltage generatorimplies that the current is low, and the ohmic losses are proportional to I2. Thecooling system can thereby be reduced in size, and cooling costs decrease.
An upgrading to a high voltage generator renders a substantial gain, in all.For a small plant the electric efficiency alone may increase with 5%.
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8. Svensk sammanfattning
Tillgång på el är en viktig fråga för dagens - och morgondagens - samhälle.Kraven är höga på energiomvandling från nya och nygamla energikällor:miljöpåverkan och kostnad ska minimeras. Det är i sammanhanget viktigt attskilja på energikällan (=fysiken) och omvandlingen (=tekniken) eftersomkällan kan vara energirik och en teknik med låg effektivitet eller högmiljöpåverkan används. Denna påverkan är ofrånkomlig, men den kanminimeras, och en annan teknik kan medföra en högre effektivitet. Fysikenbör man inte döma ut, bara tekniken.
Två energikällor utgör basen i arbetet bakom den här avhandlingen: energini havsvågor samt den potentiella energin i älvar. Vågkraften kan anses vara nyeftersom den inte finns i kommersiell drift ännu, medan vattenkraften använtsi över ett sekel.
Vågkraften står idag inför en spännande framtid. Flera grupper har inlettsamarbeten med industriella partners för att kommersialisera sina produkter,och koncepten bygger på fundamentalt olika tekniska principer. Den tekniskalösningen som används i det här arbetet baseras på en punktabsorbator (boj)på vattenytan som kopplas direkt med en lina till translatorn ("rotorn") i enlinjärgenerator på havsbotten. Mellan generatorfundamentet och translatornsitter en fjäder som förbättrar rörelsen. Då en vågtopp lyfter bojen dras trans-latorn genom statorn (som är fast i fundamentet) och fjädern lagrar energi.I vågdalen bidrar fjädern till en ökad hastighet hos translatorn. Permanent-magneter är monterade på translatorn, och på så sätt induceras en spänning itrefaslindningarna i statorn.
Eftersom vågorna är oregelbundna, och på grund av aggregatets konstruk-tion, så varierar frekvensen och amplituden kontinuerligt hos den alstradeelen. Effekten varierar från noll i translatorns vändlägen till ett maxvärde närhastigheten är som högst. För att jämna ut effektvariationerna vill man kopplaflera aggregat i en park, och för detta kopplas en likriktare till varje generator.Ett transmissionssystem kan sedan designas för att överföra elen till elnätet påland.
Flera aspekter av vågkraftkonceptet beskrivs i artiklarna i avhandlingen,med fokus på likriktning, sammankoppling och parksimuleringar. En ana-lytisk härledning som kan utgöra grunden till ett verktyg för systemdesignpresenteras också.
Vattenkraften har byggts ut i Sverige i över hundra år och de största anläg-gningarna togs i drift på 1950-80-talen. Idag behöver flera av dessa rustas upp
51
och en av delarna som kan uppgraderas är generatorn. En studie har visat attden elektriska verkningsgraden hos vattenkraftanläggningar kan höjas med enkabellindad högspänningsgenerator. För stora kraftverk kan det röra sig om1% medan mindre anläggningar kan förbättra verkningsgraden med upp till5%.
52
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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 202
Editor: The Dean of the Faculty of Science and Technology
A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally through theseries Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.(Prior to January, 2005, the series was published under thetitle “Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology”.)
Distribution: publications.uu.seurn:nbn:se:uu:diva-7081
ACTAUNIVERSITATISUPSALIENSISUPPSALA2006