UPTEC ES05 005

Examensarbete 20 p September 2005

Experimental Test Set-up for Wave Energy Converter Linear Generator Magnus Stålberg

Sammanfattning Ett flertal av världens länder har enats om att minska sina koldioxidutsläpp. Samtidigt ökar världens energibehov med ökande levnadsstandard hos befolkningen och på grund av växande folkmängd. Det finns också indikationer på att världens oljeproduktion börjar bli otillräcklig i förhållande till den alltjämt ökande efterfrågan. Alternativ energi har blivit slagord i kampen mot koldiodidutsläppen, men vad är egentligen alternativ energi? Med alternativ energi menar man oftast förnybar energi och vanliga exempel på el- och värmeproduktion från förnybar energi är vattenkraft, biobränslen, vindkraft och solenergi. Ungefär hälften av den elenergi som vi använder i Sverige varje år (2005) kommer från vattenkraft, biobränslen och vindkraft tillsammans, resten kommer från kärnkraft. För att underlätta för förnybara energikällor att bli ekonomiskt goda alternativ för elenergiproducenterna finns idag ”gröna elcertifikat”. Systemet med elcertifikat är ett subventionssystem som bygger på att vi alla betalar lite extra på elräkningen i form av något öre per kWh köpt el. Pengarna går till elproducenter som producerar el från förnybara energikällor så att de kan räkna med en lägre investeringskostnad när de investerar i till exempel ett vindkraftverk eller en panna för fliseldning. På detta sätt blir det lite billigare för elproducenterna att välja en förnybar energikälla för sin nya elproduktion. I ett projekt vid Centrum för Förnybar Elenergiomvandling vid Uppsala Universitet bedrivs forskning på vågkraft, eller vågenergi. Projektet går ut på att utveckla ett koncept för att producera el från havets vågor, se fig.(1), sid 4. Världshavens vågor innehåller mycket stora mängder energi som skulle kunna utnyttjas för att producera el. En anläggning som producerar el från vågenergi kallas för ett vågkraftverk. Ett sådant kan utformas på många sätt. Sedan 1970-talets perioder med mycket höga oljepriser, de så kallade oljekriserna, har flera koncept för att producera el från vågenergi tagits fram. Inget av dessa projekt har hittills lyckats bli kommersiellt lönsamma eller lösa problemen med hållbarhet; när vågorna går höga slås en anläggning lätt sönder av de starka krafterna från vattnet. Därför är vågkraftverkets överlevnadsförmåga, översatt från engelskans survivability, en viktig fråga när man konstruerar ett vågkraftverk. Projektet om vågenergi som bedrivs i Uppsala har som mål att ta fram en teknik som kan producera billig elenergi från havsvågor. I konceptet tänker man sig att många hundra små vågkraftverk kopplas samman i vågkraftparker ute till havs, se fig.(2), sid 4. På det sättet kan enheterna massproduceras vilket sänker kostnaderna. Om parkerna med vågkraftverk ligger långt från kusten syns de inte från land. Vågorna är dessutom större ute på djupa vatten på stora hav, så det är där den mesta energin finns att hämta och där blir lönsamheten förmodligen också bäst. Den uppfinning som är hjärtat i vågkraftverket, kallas för en linjärgenerator. Vågornas rörelse omvandlas här till elektricitet på enklast möjliga sätt och med endast en rörlig del. Eftersom generatorn, som till skillnad från generatorer som används i de flesta andra sammanhang, rör sig upp och ned istället för att snurra runt, kan man koppla den direkt med en lina till en stor boj på havsytan. Eftersom generatorn är tänkt att stå i en vattentät inneslutning på ett betongfundament på havsbotten, så är den skyddad från stora kraftiga vågor som uppstår vid storm.

Sammanfattningsvis kan alltså konceptet för vågkraft beskrivas med att en stor boj på havsytan kopplas samman med en linjär generator på havsbotten med hjälp av en lina. Linjärgeneratorn står i en vattentät inneslutning och hundratals, eller fler, vågkraftverk kopplas samman i parker. Från vågkraftparken dras en överföringskabel in till land och med hjälp av elektronik kan man anpassa strömmen så att den kan kopplas till elnätet. Den här rapporten behandlar ett steg i utvecklingen av vågkrafttekniken. För att designa en effektiv och billig generator på ett snabbt och noggrant sätt utnyttjas datorberäkningar. Med hjälp av dessa beräkningar, eller simuleringar som det också kallas, kan en enda person få mycket detaljerad och precis förhandsinformation om egenskaperna hos en generator man eventuellt ska bygga. Detta är en stor fördel jämfört med att göra beräkningar för hand, något som skulle ta många veckor och dessutom ge ett mer osäkert resultat. För att veta om datorberäkningarnas resultat är riktiga så behöver det bevisas på något sätt, simuleringen behöver verifieras. Detta görs genom att jämföra resultaten från datorsimuleringen med resultaten från mätningar på den existerande generatorn. Om resultaten stämmer bra överens så vet man att datormodellen ger en god beskrivning av verkligheten. Då kan det noggranna och tidsbesparande beräkningsprogrammet sedan användas när man designar generatorer till nya vågkraftverk. För verifikation av datorprogrammet har ett första exemplar av den nya linjärgeneratorn byggts. Uppgiften i detta examensarbete har varit att färdigställa en experimentuppställning för prototypen så att verifikation av datorprogrammets resultat kan göras. En motor med tillhörande drivsystem har installerats och ett program har skrivits för att styra motorn från en vanlig dator. Motorn är kopplad till den rörliga delen av linjärgeneratorn, rotorn, som därmed kan dras upp och ned genom att köra motorn fram och tillbaka. Under en sådan körning, när rotorn dragits upp genom den fasta delen, statorn, av linjärgeneratorn en gång har producerad ström och spänning mätts. Det exakta utseendet, geometrin, hos linjärgeneratorn har matats in i beräkningsprogrammet och en simulering har gjorts. Resultat från simuleringen har jämförts med mätningar från experimentet. Överensstämmelsen är mycket god vilket visar att datormodellen är bra. Resultat av detta examensarbete är också att den motor med tillhörande drivsystem som installerats fungerar bra. Styrningen till motorn har uppfyllt ställda krav på prestanda och de delar som designats och byggs för att möjliggöra experimentet har fungerat enligt förväntan.

Nomenclature and abbreviations r Br Bs r BR r D r Er Facc r FC r Felec.magn r Fgravity r Fmotor (t ) r Frope r Fsprings r H r J r l r v r τ r τ start r

τ pitch r τ nom

Droll GR H Ic IL IL,M Im I0 I1 I2 I2’ OVmax P Peddy Pfriction,windage Physteresis Pin Pout Pres,loss AC BJT DAQ DC EM emf FEM GTO IGBT LG

T

Magnetic field density

T

Stator magn. field density

T

Rotor magn. field density

C/m

Electric displacement field

V/m

Electric field strength

N

Acceleration force

N

Rotor conductor force

N

Electromagnetic force

N

Force of gravity

N

Rope force on motor side

N

Total rope force

N

Total spring force

A/m 2

Magnetizing field

A/m

Current density

m m/s Nm Nm

Effective length of conductor Periphery speed Rotor torque Starting torque

Nm

Max. torque, (pitching moment)

Nm

Nominal torque

m

Diameter of pulley Gear ratio Wave height Core loss component of I0 LG line current Motor line current Magnetizing component of I0 Exciting component of I1 Stator current Rotor current Load component of I1 Max. output control voltage Number of poles Eddy current loss Friction- and windage loss Hysteresis loss Input power Mechanical output power Ohmic loss

m A A A A A A A A V W W W W W W

Alternating Current Bipolar Junction Transistor Data Acquisition Direct Current Electromagnetic Electromotive Force Finite Element Method Gate Turn-Off Insul. Gate Bipolar Transistor Linear Generator

Pres,rot Pres,stat Protational P3Φ,inst. Q Rrotor RS,M R1 R2 S T ULN Vcore VF V1 Xl1 Xl2 a bm fcore fmech fnom gc k ksp m msp nsp s spitch tq

x(t ) x& (t ) &x&(t ) x& rope (t )

Ω δ η ϕ ρ µr µ0 ωelec ωmech ωnom ωslip ωsync LL LN mmf MOSFET ND WEC PA PM PWM

W W W W VAr Ω Ω Ω Ω VA s V m3 V V Ω Ω m ohm-1 Hz Hz Hz ohm-1 N/m kg N % % s m m/s

Rotor resistive loss Stator resistive loss Total rotational loss 3-phase instantaneous power Reactive power Rotor resistance of motor Stat. winding res. of motor Rotor resistance per phase Stator resistance per phase Apparent power Period time of wave LG phase voltage Volume of iron core Forward voltage drop Stator phase voltage Stator leakage reactance Standstill rotor leakage react. Core lamination thickness Magnetizing susceptance Freq. of iron core flux change Motor rotational speed Motor nominal speed Core loss conductance Machine constant Spring constant Mass of LG rotor Pretension of spring Number of springs Slip Slip at pitching (max.) mom. Turn-off time interval LG rotor position LG rotor speed

2

m/s

LG rotor acceleration

m/s

Speed of rope on motor side

rad/s degr % degr C/m3 H/m H/m rad/s rad/s rad/s rad/s rad/s

Angular frequency of wave Load angle Efficiency Power factor angle Charge density Relative permeability Permeability of free space Ang. freq. of power supply Mechanical speed Nominal speed Slip speed Synchronous speed Line-to-Line Line-to-Neutral Magnetomotive Force Metal Oxide Semiconductor Field-Effect Transistor Non-Drive Wave Energy Converter Point Absorber Permanent Magnet Pulse Width Modulation

Contents 1. INTRODUCTION……………………………………………………………………………………….. 3 1.1 Background…………………………………………………………………………………………. 3 1.2 The concept…………………………………………………………………………………………. 3 2. THEORY ………………………………………………………………………………………………… 5 2.1 Induction motors…………………………………………………………………………………….. 5 2.1.1 Principle of operation………………………………………………………………………….. 5 2.1.2 Stator…………………………………………………………………………………………… 7 2.1.3 Rotor…………………………………………………………………………………………… 8 2.1.4 Power factor…………………………………………………………………………………… 8 2.1.5 Losses…………………………………………………………………………………………. 8 Ohmic losses………………………………………………………………………………………………… 8 Hysteresis losses……………………………………………………………………………………………. 9 Eddy-current losses………………………………………………………………………………………… 10 Friction and windage losses………………………………………………………………………………. 11 Total rotational losses……………………………………………………………………………………... 11 2.1.6 Efficiency……………………………………………………………………………………….11 2.1.7 Equivalent circuit………………………………………………………………………………. 11 2.1.8 Variation of speed of induction motors………………………………………………………... 12 Change in number of poles………………………………………………………………………………...12 Variable line voltage…………………………………………………………………………….………… 12 Variable rotor resistance………………………………………………………………………………….. 13 Variable frequency…………………………………………………………………………………………. 13 2.2 Frequency conversion……………………………………………………………………………….. 13 2.2.1 Non-controllable switches……………………………………………………………………... 13 Diodes………………………………………………………………………………………………………...13 Thyristors……………………………………………………………………………………………………. 14 2.2.2 Controllable switches………………………………………………………………………….. 16 GTO:s………………………………………………………………………………………………………... 16 IGBT:s……………………………………………………………………………………………………….. 16 2.2.3 PWM, Pulse width Modulation………………………………………………………………... 17 2.2.4 Protective circuits……………………………………………………...………...……….……. 18 2.3 Permanent magnet linear synchronous generators……………………………………………….….. 19 2.3.1 The simulation tool, Magic……………………………………………………...……….…….. 20 2.3.2 Maxwell’s equations……………………………………………………...………...……….…. 20 2.3.3 Direct drive benefits……………………………………………………...………...……….…. 21 2.3.4 Reference note……………………………………………………...………...………...……… 21 2.4 Dynamic forces in experimental test set-up…………………………………………………….…… 21 2.4.1 Forces……………………………………………………...………...………...……….……… 21 2.4.2 Gearing……………………………………………………...………...………...………...…… 23 2.4.3 Mechanical frequency……………………………………………………...………...………....23 2.4.4 LabVIEW control voltage……………………………………………………...……….……... 23 3. EXPERIMENTAL TEST SET-UP……………………………………………………...………...……… 24 3.1 Variable speed drive……………………………………………………...………...……….………. 25 3.2 Induction motor……………………………………………………...………...………...……….….. 25 3.3 Conversion of rotating movement to linear movement……………...………...………...……….….. 25 3.4 LabVIEW motor control……………………………………………………...………...……….…... 25 3.5 Hardware……………………………………………………...………...………...………...……….. 26 3.6 Measurements……………………………………………………...………...………...……….……. 26 3.6.1 Three-phase portable DAQ system……………………………………………………...……...26 3.6.2 Load……………………………………………………...………...………...……….………... 28

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4. EXPERIMENT……………………………………………………...………...………...……….……….. 29 5. RESULTS……………………………………………………...………...………...………...……….…... 30 5.1 Phase voltage……………………………………………………...………...………...……….…….. 30 5.2 Current……………………………………………………...………...………...………...……….…. 31 5.3 Phase voltage and current……………………………………………………...………...……….….. 31 5.4 Three-phase instantaneous power……………………………………………………...……….…… 32 5.5 Qualitative results……………………………………………………...………...………...……….... 33 6. DISCUSSION……………………………………………………...………...………...………...……….. 34 7. CONCLUSION……………………………………………………...………...………...………...……… 35 8. ACKNOWLEDGEMENTS……………………………………………………...………...……….…….. 35 9. REFERENCES……………………………………………………...………...………...………...……… 36 Data sheet ACS800 frequency converter………………………………………………Appendix 1 Data sheet induction motor ………….….……………………………………………. Appendix 2 Data sheet NI-DAQmx 6229 PCI…………………………………………………….. Appendix 3 Data sheet NI USB 6009……………………………………………………………… Appendix 4 Circuit diagram of measurement circuits………………………………………………Appendix 5

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1. Introduction 1.1 Background Very simple schemes were used more than 100 years ago to convert the motion of water waves [Boyle 1996] into useful power. Not until the oil crises of the 1970:s, the development of schemes to convert ocean wave energy into electricity began. Since then, several wave power concepts have been developed but no commercial plant has yet been installed. Renewable energy sources, such as hydroelectric power and biomass make considerable contribution to the world commercial energy production of today. In Sweden (2005), about half of the electricity we use each year is primarily produced from these sources, the rest is produced from nuclear energy. The need for energy supplies increases each year in most developed and developing countries. This is primarily due to increasing standards of living. A peak in the world oil production1 is expected in a relatively close future, which will lead to even higher oil prices than today. Oil supplies, primarily the oil used for producing transportation fuels, will have to be replaced with other energy sources. New schemes for economically competitive conversion of renewable energy into electricity, for which the worldwide need is substantial, therefore need to be developed. Wave power and current power (e.g. tidal energy, ocean currents and rivers) represent, so far almost unexploited, sources of renewable energy. These large-potential sources have high power density, which make them interesting from an economic point of view. Parts to be used in the first2 commercial wave power project have been ordered for delivery. The project is a 2.25 MW-farm, which will consist of three wave energy converters of 750 kW each. The farm will be placed five kilometres off the northern coast of Portugal. The cost at which electricity will be produced from the farm is not specified and if the project will be successful is yet to be determined.

1.2 The concept At the Swedish Centre for Renewable Electric Energy Conversion, Division for Electricity and Lightning Research at Uppsala University, a new concept for conversion of ocean wave power into electricity is being developed. The concept [Leijon 2005] is based on a threephase, synchronous, permanent magnet linear generator placed in a watertight steel enclosure standing on the sea floor, see fig.(1).

1 2

http://www.peakoil.net/ http://www.oceanpd.com

3

Figure (1). The concept for a linear generator wave energy converter being developed at the Swedish Centre for Renewable Electric Energy Conversion, Uppsala. ( Illustration by Oskar Danielsson ).

The only moving part of the generator is called translator and is pulled upwards by means of a buoy, a point absorber, floating on sea level. When the translator moves upwards, springs, which are connected between the concrete foundation and the translator, are stretched and thus energy is stored. In wave troughs, the potential energy stored in the springs is used to enhance the electricity production by pulling the translator down. The wave energy converters [Eriksson 2004] may be connected in farms to increase the annual energy output, see fig.(2). The alternating current induced in each wave energy converter, WEC, is rectified to DC and transmitted via DC cable to land where it is inverted to AC, transformed and finally connected to the grid.

Figure (2). Illustration of a farm of linear generator wave energy converters.

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At the Division for Electricity and Lightning Research, a first prototype of a synchronous PM linear generator has been built. A steel structure for a lab test set-up has previously also been built, but so far experimental testing of the linear generator has not been possible to do. The aim of this degree project is to modify the experimental test set-up, add the necessary components and test the system so as to make it possible to test the LG prototype at moderate to high translator speeds. It would then be possible to make measurements of the performance of the LG and compare with simulated results from the generator simulation tool that is used at the department. Comparing simulated results of the LG performance with experimental results is also an aim of this degree project. Several components have to be added to the experimental test set-up, electric equipment has to be installed and tested and new computer equipment and measuring electronics have to be ordered, arranged and/or installed. Two of the necessary components are an induction motor and a frequency converter motor drive. The frequency converter is used to operate the motor at variable rotational speeds and different rotational directions. The motor is used to pull the LG translator up and down in the test setup, simulating the force of a point absorber. In this thesis, some theory of induction motors, variable-speed operation of induction motors and of a few power electronic components is reviewed in sections 2.1 and 2.2. This is to give a better understanding and a deeper insight into the technology of variable speed three-phase induction motor operation. To give an introduction to linear generator PM technology, a short theoretical review is given in section 2.3.

2. Theory 2.1 Induction motors There are two main types of induction machines [Sarma 1996], namely squirrel cage motors and slip ring motors. The squirrel-cage induction motor has been the standard choice for industrial applications for decades. Its rugged design, little need for maintenance and the possibility to use easy direct start-up have made it dominate the market for small as well as large rotating machines. In this degree project a squirrel-cage induction motor is used to pull the translator of a linear, synchronous, permanently magnetized generator to be used in a wave energy converter, in order to simulate energy conversion from ocean waves. The speed of the motor is controlled using variable frequency, a speed control technology explained in section 2.1.8. 2.1.1 Principle of operation The word induction in induction motor [Chapman 1999] refers to the way the magnetisation of the rotor is achieved. The rotor winding may consist of either a copper winding or solid bars of aluminium, copper or brass. No matter the choice of design of the rotor, the magnetisation of the rotor is achieved through induction. The current in the stator winding creates a rotating magnetic field with a frequency ωelec. The mechanical synchronous speed, ωsync, is related to the frequency of the power supply and the number of poles, P, in the machine according to Eq.(1).

ω sync =

ω elec ⎛ P⎞ ⎜ ⎟ ⎝2⎠

( 1)

An electromotive force, emf, is then created in each of the rotor conductors. This emf is given by Eq.(2).

5

(

)

r r r emf = v × BS • l

( 2)

r r where v is the periphery speed vector of the rotor conductor, BS the stator magnetic field r density vector and, l , the effective length of the conductor. The emf drives a current, i, r through the conducting bars, that according to Lenz’s law creates a magnetic field, BR , that opposes the direction of the magnetic field from the stator. Due to the difference between the frequency of the magnetic field in the stator, ω sync , and the physical rotational frequency of

the rotor, ωmech, the current-carrying conductors of the rotor moves in a magnetic field making an angle of 90 degrees to the field lines. This results in a force [Halliday 2001] on the r conductors of the rotor due to the time-varying magnetic field crossing them. This force, Fc , is equal to

(

r r r Fc = i ⋅ l × BS

)

( 3)

where i is the magnitude of the current in the conductor. i is taken to be positive in the r r direction of l . The force Fc is thus directed in the tangential direction of the rotor. The resulting rotor electromechanical torque [Chapman 1999] is the cross product between the r r rotor magnetic field density vector, BR , and the stator magnetic field density vector, BS , according to r

(

r

r

τ = k ⋅ B R × BS

)

( 4)

where k is a machine-specific constant. When the mechanical speed of the rotor, ω mech , increases due to the torque, the relative difference between the stator magnetic field frequency, ω sync , and the rotational frequency of the rotor, ω mech , decreases and thus the amplitude of the induced emf in the rotor is reduced. An equilibrium is reached where the rotational speed of the rotor differs from the frequency of the rotating magnetic field in the armature only by the slip, s. The slip may be expressed in terms of the synchronous speed and the mechanical speed. The slip, s, is defined as

s=

ω sync. − ω mech. ω slip = ω sync. ω sync.

( 5)

where the rotor’s slip speed, ω slip , is defined as

ω slip = ω sync. − ω mech

( 6)

The slip is positive for motors and is usually a few percent in standard operating conditions.

6

The theoretically derived torque-speed curve is usually given to facilitate the understanding of an induction-machine’s characteristics as a motor or generator. The torque-speed curve will not be derived in this thesis, see e.g. [Alfredsson 1996] for further reading. Important r characteristics [Alfredsson 1996] are e.g. starting torque, τ start , maximum torque, which is r r also called pitching moment, τ pitch , nominal torque, τ nom , nominal speed or full-load speed,

ω nom , and slip at the pitching moment (maximum torque), s pitch . In fig.(3) a typical torquespeed curve of a squirrel-cage induction motor is shown. Notice the maximum torque, here denoted by Mmax, the starting torque, Mstart, and the torque at zero slip, s = 0.

Figure (3). Typical torque-speed curve of a squirrel-cage induction motor. Here, the max. torque is denoted by Mmax, the starting torque by Mstart and the nominal torque by Mn. ( Figure is taken from [Alfredsson 1996] )

2.1.2 Stator The stator of an induction motor is laminated using sheet metal. This reduces eddy current losses. The three-phase winding is made from copper wire with a rectangular or circular cross section. Depending on the choice of cross-section, different materials may be used for the insulation. Laquer is used for round cross-sections and laquer with additional impregnated glassfiber can be used with rectangular cross-sectional wire. The conductors are placed in slots [Sarma 1996] in the stator steel and the conductors are divided into sections called phase belts. For a three-phase machine the number of phase belts per pole pair (one pole pair is two poles) is usually six. A phase belt contains all the conductors of one phase in each pole. Each phase belt thus contains a different number of conductors depending on if the winding is a one- or two-layer (double-layer) winding. A more common terminology is the number of slots per pole per phase, q, being the same as the number of slots in each phase belt. A single-layer winding has only one conductor in each slot of the stator whereas double-layer windings have two conductors in each slot, the latter being the most common in large three-phase machines. Motors of moderate size are mass-produced and even the conductors may be wound using winding machines. Motors are usually designed for 50 or 60 Hz standard usage, but special features [Elfving 1993] can be added depending on the intended use of the motor, e.g. protection to chemicals, or external fans for improved cooling at low-speed and high-torque mechanical load. Large machines have ventilation channels [Alfredsson 1996] for air-cooling between separate sections of the stator whereas smaller machines in the range up to about 90 kW are mounted in an aluminium frame for sufficient cooling via heat conduction. A fan is often placed directly on the ND-side (Non-Drive) of the shaft of the motor and provides the necessary cooling airflow passing the interior and exterior of the motor. The main purpose of the aluminium or steel frame is to provide mechanical support for the bearings carrying the rotor shaft.

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2.1.3 Rotor The rotor [Sarma 1996] is laminated from sheet metal to minimize eddy-current losses. Since the rotor rotates with a frequency only separated from the frequency of the rotating magnetic field in the stator by the slip, usually in the range of 2-6 %, thicker sheets can be used to laminate the rotor than the stator. The rotor may carry different types of windings. The most common type of rotor consists of laminated sheet metal carrying short-circuited, solid aluminium, brass or copper bars. These motors are called squirrel-cage induction motors due to the rotor’s similarity with a squirrel cage. The squirrel-cage induction motor require very little maintenance due to the absence of slip rings, present in other types of motors e.g. DCmotors (commutator), non-PM synchronous machines or slip-ring (wound-rotor) induction motors. The wound-rotor induction motor represents another approach. This concept features a laminated core but carries a three-phase winding instead of the solid bars of the rotor of the squirrel-cage motor. Slip rings may be used in external speed control and will be further explained in section 2.1.8. In order for the machine to produce torque, the number of poles in the rotor has to equal that of the stator. 2.1.4 Power factor Induction motors consume reactive power [Alfredsson 1996]. The power factor [Sarma 1996] may be derived by making simplifications of the induction motor equivalent circuit to r determine the total impedance, z , per phase of the motor. The impedance’s real part, i.e. the r resistive part, makes an angle with z and this angle is the power factor angle, ϕ . The power factor is the cosine of this angle, namely cos ϕ, and this usually being low (often close to 0.80.85) in induction machines is the reason to the need for power factor correction. The power factor gives information about how much of the supplied power that has to be reactive power, Q. 2.1.5 Losses Electromechanical energy conversion is always associated with certain losses [Alfredsson 1996] yielding a limited efficiency. The types of losses that are associated with induction machines are • Ohmic (copper) losses • hysteresis losses • eddy-current losses • friction and windage losses. Hysteresis losses and eddy-current losses are usually called iron losses. Depending on the design of an arbitrary motor/generator, the different losses may be of different importance. Ohmic losses The main losses of an induction motor in standard operation are ohmic losses, Pres,loss, in the stator and rotor windings. These depend on the level of mechanical load on the motor and thus on the square of the current according to

Pres ,loss = RS , M ⋅ I L

2

( 7)

RS,M, is the resistance in the stator winding. IL, is the line current. At no-load operation, friction losses and ohmic losses in the stator winding dominate.

8

Hysteresis losses The size of the hysteresis losses [Chapman 1999] depends on the shape of the hysteresis curve or BH-curve, which in turn is material-specific. If an external magnetic field with a magnetic r field intensity vector H , unit A/m, is applied to a piece of ferromagnetic material, a magnetic r r flux density, B (unit T, tesla), arise within the material. The field strength H may be produced e.g. by a winding carrying a current, I, as in a stator tooth of the armature in an electric motor. r The size of the magnetic field density, B , depend on the relative permeability, µ r (H/m), of the material and the permeability of free space, µ 0 , according to Eq.(8). r r B = µr ⋅ µ0 ⋅ H

( 8)

r When the H -field is applied, the randomly organised magnetic domains of the material is r aligned with the direction of the applied field and thus creating a B -field in the material. The magnetic domains require energy to change their alignment direction and therefore they are r not completely randomised when the H -field is removed, e.g. when the current in the r armature winding ceases. The resulting B -field within the material is called the residual flux r density or the remanence of the material and can be seen for H = 0 in fig.(4). A material is suitable for use in permanent magnets if it has high remanence. In order for the magnetic field r density B to reach zero again a coercive magnetomotive force, mmf, is needed and this has to r be applied in the opposite direction of the B -vector. Another way of “destroying” the residual flux is to feed energy in some form into the magnetic domain system. In this way a permanent magnet may be destroyed (demagnetised) if exposed to electric chock, mechanical impact or high temperature. r If an H -field is applied in the opposite direction compared to the above, the same thing will r happen apart from that the direction of the produced B -field will be the opposite. Due to the phenomenon of hysteresis, energy will always be dissipated as heat when reversing the magnetisation direction of an iron core. r The size of the B -field produced increases relatively rapidly to a certain level, the saturation r level. Above the saturation level the B -field increases very moderately. The saturation level is reached when all of the magnetic domains are aligned. The saturation level is materialdependant, but for common sheet metal for use in electric machine core-laminations the saturation level is around 1.8 T.

9

B

H

Figure (4). Typical hysteresis loop with direction and remanence indicated. (Figure is taken from http://sound.westhost.com/xfmr11-1.gif)

The result of the above is that hysteresis losses are unavoidable in electric machines that use (laminated) steel in the stator and/or rotor. The hysteresis loss [Sarma 1996] is dependent upon the area enclosed by the hysteresis curve (BH-curve), the frequency, fcore, at which the iron core flux direction changes, the total iron core volume, Vcore, and may be calculated using Eq. (9)

r r Physteresis = f core ⋅ Vcore ⋅ ∫ H dB

( 9)

r r where the curve integral of HdB is the area enclosed by the hysteresis loop in fig.(4).

Eddy-current losses Eddy currents [Halliday 2001] appear as a result of time-varying magnetic fields. An emf is then induced in the conductor, or electrically conducting material, and this emf drives a current which creates a magnetic field that opposes the direction of the change in the primary magnetic field. This phenomenon is stated in Lenz’s Law. The currents being induced are called eddy currents and appear in any conducting material that is subjected to a time-varying magnetic field. The eddy-current losses, Peddy, appearing are dissipated as heat and may be calculated [Sarma 1996] using the expression

Peddy = k e ⋅ Vcore ⋅ f core ⋅ a 2 ⋅ Bmax 2

2

( 10)

where ke is a material-constant (iron), a the core lamination thickness and Bmax the maximum magnetic field density. Vcore and f is the same as in Eq.(9). It should be noted that using Eq.(10) calculating the eddy-current loss, saturation is neglected and the eddy-current paths are assumed to be purely resistive. The eddy-current losses may be reduced to very low levels using sheet metal for core laminations. Usually 0.35 mm, 0.5 mm or 0.65 mm sheets are used for electric machine stator laminations. The sheets are insulated electrically from each other using e.g. laquer or very thin plastic sheets. The thinner the sheets the more the number of sheets to be used for a certain volume of core, hence the economic value of reducing eddy-current losses has to be carefully considered. 10

Friction and windage losses Friction and windage losses [Chapman 1999] are proportional to the cube of the rotational speed, or mechanical speed ω mech , of the rotor i.e.

Pfriction , windage ∝ ω mech

3

( 11)

The windage losses appear as a result of the viscosity of air creating drag when the rotor rotates in the machine housing. Friction losses result from friction in bearings supporting the ends of the rotor shaft. If a slip-ring induction motor or e.g. a wound-rotor synchronous machine with slip-ring excited rotor winding is used, excessive friction losses appear due to the coal brushes’ physical contact with the slip-rings. Total rotational losses The hysteresis losses and eddy-current losses depend on the frequency of the time-varying magnetic field according to Eq.(9) and Eq.(10) respectively. Since the frequency of the magnetic field in the rotor is low at low slip, the rotor losses (hysteresis and eddy-current losses) decrease with decreasing slip. The friction and windage losses increase cubically with the mechanical speed according to Eq.(11), yielding a nearly constant value of the total rotational losses of an induction machine.

Protational = Physteresis ,rotor + Peddy ,rotor + Pfriction , windage ≈ constant

( 12)

2.1.6 Efficiency The efficiency of a motor is the ratio of output mechanical power to input electrical power. The input power may be expressed per phase using the line-to-line voltage VLL, the line current IL and the power factor cos ϕ. Then the efficiency may be written as

η=

Pout Pout = Pin 3 ⋅ VLL ⋅ I L ⋅ cos ϕ

( 13)

Losses are subtracted from the input power to yield the mechanical output power. With the losses appearing approximately in the following order, the output power is Pout = Pin − (Pres , stat + Physteresis + Peddy + Pres ,rot + Protational )

(14)

2.1.7 Equivalent circuit An equivalent circuit [Sarma 1996] of the induction motor is shown in fig.(5). [Chapman 1999] describes the induction machine as a rotating transformer. Comparing the equivalent circuit of a transformer, see e.g. [Chapman 1999] or [Sarma 1996], to the equivalent circuit of the induction motor in fig.(5), similarities are obvious.

11

Figure (5). Per phase equivalent coupled circuit of an induction motor. ( Picture is taken from [Sarma 1996] ).

Here V1 is the phase voltage, I1 the stator current, R1 and Xl1 the stator resistance and stator leakage reactance respectively, I’2 the load component of the stator current, I0 the exciting component of the stator current, Ic the core loss component of the exciting current and Im, the magnetizing component of the exciting current. gc is the core loss conductance and bm is the magnetizing susceptance. Xl2 is the standstill rotor leakage reactance, R2 the rotor resistance, s the slip and I2 is the rotor current. E1 is the ideal stator winding voltage and E2 the ideal induced rotor voltage. 2.1.8 Variation of speed of induction motors Induction motors are most often used in constant-speed applications but can be run at different speeds [Sarma 1996]. Variation of speed can be achieved in a number of ways but some methods result in extensive losses and poor operational performance. In this section, a brief overview of some ways of achieving variable speed operation will be shortly presented. In the experimental set-up of the degree project, an induction motor is operated at variable speed using variable frequency. Below, this method will be explained in more detail. Change in number of poles This can be achieved by using two different windings in the stator. This makes two different synchronous speeds possible and thus the motor can be run at two different speeds. To avoid having to change the connections in the rotor winding, preferably a squirrel-cage induction motor is used for this type of speed regulation. Variable line voltage Another way of varying the motor speed is to use variable terminal voltage [Chapman 1999]. r As shown in Eq.(15), the resulting torque, τ , of an induction motor depends on the phase voltage according to

3 ⋅ Vth ⋅ 2

r

τ =

Rrotor s

2 ⎤ ⎡⎛ R2 ⎞ ω sync ⋅ ⎢⎜ Rth + ⎟ + ( X th + X rotor )2 ⎥ s ⎠ ⎦⎥ ⎣⎢⎝

(15)

where Vth, Rth and Xth is the thevenin equivalent voltage, resistance and reactance of the simplified equivalent circuit in fig.(6) respectively. This type of speed control can only be used within a limited range of rotational speeds and is only used on small induction machines.

12

Figure (6). Simplified equivalent circuit of an induction machine. ( Figure is from taken Chapman 1995 ).

Variable rotor resistance This method [Sarma 1996] can only be used in wound-rotor induction motors with slip rings. An external resistance can be connected in series or in parallel with the rotor circuit and a variable resistance in series with the rotor circuit provides the possibility of varying the rotational speed. The speed at which the maximum torque appears is changed when changing the rotor resistance. This enables high torque characteristics at different rotational speeds. The drawback [Elfving 1993] is low efficiency and a high sensitivity to variations in mechanical load. Variable frequency Variable speed drives change the mechanical frequency of the motor by changing the voltage and angular frequency of the power supply. This can be achieved by converting the supply voltage to a DC-voltage (e.g. by using diodes or thyristors) and then converting the DCvoltage to AC with a different angular frequency by using power electronic components such as GTO:s or IGBT:s. A variable speed drive provides the motor with an arbitrary synchronous frequency, which can be changed instantaneously to a different value. This way, the slip may remain the same whereas the rotational speed of the rotor is changed.

2.2 Frequency conversion Motor drives use frequency converter technology to provide the user with an energy-efficient and accurate way of controlling the speed of one or several motors. The basic operation is rectification of the AC supply to DC. This is followed by the conversion of DC to an AC having an arbitrary frequency and amplitude set by a control programme. The mechanical load may be e.g. pumps or fans in the pulping industry or in a chemical plant. Motor drives are available in a wide range of sizes and are limited in size only by the rating of the power semiconductor devices used in the frequency converter. Different requirements exist for the rectifier part and the frequency inverter part of the frequency converter. Simple diodes can be used for rectification, but more advanced components are required to convert DC to AC. In this section a few of the most common power semiconductor devices are shortly presented. 2.2.1 Non-controllable switches A diode conducts in one direction and cannot be controlled whereas a thyristor can be switched on, but not off. The classification of power semiconductor devices separate between components with full switching possibilities and those with limited or no switching possibilities. In this section the so-called non-controllable diode and thyristor will be shortly presented. Diodes Diodes [Alfredsson 1996] are the simplest of the various kinds of power electronic components on the market. Diodes are used in different applications e.g. AC-to-DC

13

converters. They are also used in different kinds of protective electronic circuits. Important characteristics of the diode are shown in fig.(7).

Figure (7). Diode: a).Symbol. b).Characteristics. c). Idealized characteristics. (Picture is taken from [Mohan 1995] ).

Diodes consist of a simple pn-junction made from p-doped and n-doped silicon. Ideally it conducts in one direction and blocks in the other direction. The diode is an entirely passive component. Instead the efficiency is high, typically about 99,5%. The diode, and other semiconductor components, are in reality e.g. characterized by a forward voltage drop, see VF in fig.(7).b). This is to be kept as low as possible since it creates losses during the conducting phase. The forward voltage drop of a diode is usually about 0,7-2,5 V. There is a limit in the amplitude of an inverse voltage [Thorborg 1993] applied to the diode before it conducts in the reverse direction. This is called the reverse breakdown voltage or the brake-over voltage. In normal operation it is essential to ensure that the reverse breakdown voltage is never reached since it may severely harm the device. Part of the total losses of a diode is the leakage current, which runs through the diode when it operates in the reverse-blocking region, see fig.(7).b), but the magnitude of the voltage still is below the reverse breakdown voltage. The leakage current is in the order of magnitude of 100mA depending on diode temperature and material. When the diode begins to conduct it can be considered to be ideal, but the same is not valid for the cease of current in the reverse-blocking state. The diode is assigned a reverse recovery time, which is the time from when the current reaches zero, reversing for an instant and then becoming zero again (compare with the thyristor turn-off time interval shown in fig.(9)). A few different types of diodes [Mohan 1995] are Schottky diodes, Fast-recovery diodes and Line-frequency diodes. These feature low forward voltage drop, short reverse-recovery time and large reverse-recovery time respectively. These features are useful in specific applications but further explanation lies beyond the scope of this thesis. Further information is found in e.g. [Mohan 1995] or [Thorborg 1993]. Thyristors A thyristor consists of a pnpn-junction [Alfredsson 1996]. The anode is the positive terminal and is the p-doped end of the junction. The mid-terminal, the gate, is connected to the second p-doped area of the thyristor and the cathode is the negative terminal, the n-doped end of the junction.

The operation of a thyristor is similar to the operation of a diode apart from the built-in possibility of switching [Mohan 1995] to the on-state in the forward blocking direction. 14

Diodes have no forward blocking direction since it always conducts when a positive voltage, larger than the voltage drop, is applied between the diode’s anode and cathode, see fig.(7).a). The switching to the on-state is achieved using the mid terminal G, the gate, shown in the thyristor circuit symbol in fig.(8).a). A gate current pulse igniting the thyristor changes the thyristor from its off-state to its on-state in the forward blocking direction. This makes it conduct as a diode though the conduction losses being slightly higher. Since thyristors can block the current in both directions, there is also a forward breakdown voltage similar to the reverse breakdown voltage. These breakdown voltages [Alfredsson 1996] can be very high (up to 10kV), which together with the ability of handling very large currents (up to 4kA) make the thyristor the component with the highest power rating among the power semiconductor devices available on the market.

Figure (8). Thyristor: a).Symbol. b).Characteristics. c). Idealized characteristics. ( Figure is taken from [Mohan 1995] ).

An important characteristic of the thyristor is the reverse recovery time, trr. This is the time the current reverses at turn-off before becoming zero again. The reverse recovery time is indicated in fig.(9). The turn-off time interval, tq, is perhaps an even more important characteristic. The turn-off time interval is the time from zero crossing of current to zero crossing of voltage over the thyristor terminals. Before the turn-off time interval has passed, the thyristor cannot block a forward voltage applied to its terminals. The turn-off time interval is indicated in fig.(9). For use in high-frequency switching applications, a very limited turnoff time interval is required. A typical value [IVA 2002] of thyristor efficiency is 98.5%.

15

Figure (9). Illustration of the reverse recovery time (top) and the turn-off time interval (bottom) ( Picture is taken from [Mohan 1995] ).

2.2.2 Controllable switches There are several types of controllable switches. Two of the most commonly used types will be presented in this section. The switching of controllable semiconductor devices require drive circuits to perform pulse width modulation, PWM. To limit the impact of harmful voltage and current transients, different types of protective circuits may also have to be used in combination with controllable or non-controllable switches. The snubber circuit is one such circuit and a brief overview of the main fields of application of such circuits is given in section 2.2.4. GTO:s A GTO (Gate Turn-Off) [Alfredsson 1996] is a controllable thyristor, i.e. a thyristor capable of blocking not only the first part of the positive half-period of the current in the forward blocking direction, but any part of the positive half-period. This is possible since the GTO can be forced out of the conducting phase by feeding it with a reverse current pulse via the mid terminal, the gate.

The utilisation of GTO:s in switching applications may require the use of so-called snubber circuits. Further explanation of such protective circuits will be given in section 2.2.4. GTO:s can be used in high frequency switching up to about 10kHz and at high voltages and high currents up to about 4,5kV and a few thousand amperes respectively. IGBT:s An IGBT (Insulated Gate Bipolar Transistor) provides full switching possibilities in both the forward and the reverse blocking state. IGBT:s is the device of choice for frequency converter applications, at least in the 400-690V voltage range. The most common application is in DC/AC-inverters. In fig.(10), the IGBT circuit symbol, characteristics and idealized characteristics are shown. Note that the terminals of the IGBT are named in accordance with transistor terminology (collector, C, and emitter, E) rather than that for diodes and thyristors.

16

Figure (10). IGBT: a).Symbol. b).Characteristics. c).Idealized characteristics. ( Figure is taken from [Mohan 1995] ).

IGBT:s can be used in high frequency (up to 10kHz) switching applications and combines the low power drive-circuit requirements of the voltage-controlled MOSFET (Metal Oxide Semiconductor Field Effect Transistor) with the low voltage drop of the current-controlled BJT (Bipolar Junction Transistor). A small voltage-drop yields a lower conduction powerloss. Similar to the MOSFET, the IGBT is a voltage-controlled device. Also regarding the current, the IGBT operates similar to both the MOSFET and the BJT, namely that the size of the current can be limited by controlling the size of the gate voltage and current respectively. This characteristic of an IGBT is shown in fig.(10).b). As for the GTO, snubber circuits and inductances limiting the current derivative may have to be used to ensure sufficient protection of the device. See further section 2.2.4. 2.2.3 PWM, Pulse width Modulation Pulse width modulation [Elfving 1993] is often used in frequency inverters. In motor drive applications up to about 300kW, IGBT:s are very often used for the commutation of voltage. In the DC-to-AC part of a frequency converter the semiconductor components are controlled from a drive circuit. This is a vital part of the frequency inverter technology since it, in the end, will decide the performance of the motor drive.

The basic operation of an inverter using PWM is to chop the positive and negative DCvoltage into square waveforms with varying pulse widths. The resulting average voltage is the output voltage to the driven object, i.e. an almost perfect sine waveform. The higher the switching frequency in the inverter the more accurate the waveform will be. The development in inverter technology has aimed for an increasing switching frequency in order to reach more

17

accurate waveforms. Fig.(11) shows a simple example of the schematic build-up of a sine wave using PWM.

Figure (11). Build-up of a sine wave using PWM. ( Picture is taken from http://www.andrew.cmu.edu/user/zke/pwm_module/pwm_example.png )

Switching can be made in many different ways and new modulation schemes are constantly developing. A few examples of modulation schemes are mid-point modulation, sine modulation and star modulation. Mid-point modulation was one of the first modulation schemes for PWM. The content of the fifth and seventh harmonic becomes large which yield large motor torque pulsations, especially at low rpm.

In sine modulation the pulse width is varied sinusoidally every half period. The fifth and seventh harmonic is suppressed which yield a much smoother motor torque than in mid-point modulation. Sine modulation was introduced in the beginning of the eighties and still in the beginning of the nineties, it was a very common modulation scheme. In star modulation the optimum pulse width pattern is calculated and relatively fast power semiconductor components are controlled by microprocessors in drive circuits. Many different types of star modulation exist. Using star modulation, the ratio of frequency to voltage can be changed depending on the momentary changes in mechanical load. High dynamic controlling properties are achieved using star modulation. 2.2.4 Protective circuits When an IGBT [Mohan 1995] is turned on or off, transients appear resulting from the high voltage or current derivatives. These high derivatives may be harmful to circuit components

18

which stresses the need for protective circuits. Protective circuits can be so-called snubbers or snubber circuits. Snubber circuits can be used to: • limit high voltages during turn-off transients • limit high currents during turn-on transients • limit the current derivative, di/dt, through the device at turn-on • limit the voltage derivative, dU/dt, across the device at turn-off • shape the voltage or current during turn-on and turn-off. Increasing costs and complexity must always be taken into consideration when adding snubber circuits of any kind to a power electronic converter circuit. The benefit is less stress on the components, which increases the lifetime. When high voltages are to be switched, many components may have to be connected in series to reduce the voltage drop across each semiconductor component. This may also require additional circuit components to guarantee that the voltage drop over each power semiconductor component is the same. This is important to ensure synchronous switching of the controllable power semiconductor components.

2.3. Permanent magnet linear synchronous generators The linear generator tested in this degree project is a three-phase, permanent magnet synchronous linear generator, see fig.(12). It is four-sided and the stator sides are wound using 16 mm2 PVC-insulated cable. To reduce eddy-current losses, the stator sides are made from laminated steel. The stator steel sheets are laser cut for small-tolerance manufacturing. The maximum frequency of induced current is lower in the linear generator proposed than in conventional, rotating, counterparts. This motivates the rather large sheet thickness of 1mm (compare with section 2.1.5). To reduce cogging effects, a fractional winding of 6/5 slots per pole and phase has been used. The magnet material used for the translator is sintered NdFeB and the magnets are surface-mounted. The magnets are mounted with alternating polarity, resulting in a change of magnetic flux direction when the translator moves through the stator. Different permanent magnet fixation concepts were evaluated by [Danielsson 2003].

Figure (12).Schematic illustration of a three-phase permanent magnet synchronous linear generator. (Illustration by Oskar Danielsson).

19

It has been shown [Leijon 2005] that three-phase permanent magnet linear generators can be designed using the FEM. The simulation tool, Magic, used in the design process of the prototype tested in this project, has previously been used when designing high voltage hydroand turbo generators. It has also been used in designing high voltage PM machines. In these applications the programme has previously been verified by experiments on actual machines. Simulated results on a three-phase linear generator show that even a machine of 10 kW size may be of economic interest for large-scale ocean wave energy conversion. 2.3.1 The simulation tool, Magic The basis for the simulation tool is a FEM-based module for solving the system of coupled field equations. The model is two-dimensional but takes e.g. three-dimensional coil-end effects into account. The equations are based on Maxwell’s fundamental electromagnetic field equations (see section 2.3.2). The time-stepping module [Ivanova 2004] for transient analysis is made using a semi-implicit Runge-Kutta method. The computation time [Bolund 2004] is reduced using cyclic (periodic) boundary conditions instead of changing the airgap triangular mesh used for transient analysis applications. This requires separate coordinate systems for the translator and the stator. The mesh is coarse in simple geometric domains of the problem and detailed in the airgap and in the stator teeth. The simulation tool can e.g. give preliminary steady-state results of fields and currents. For more accurate results, a transient analysis has to be performed. These calculations are more time-consuming and employ the periodic boundary conditions mentioned above. A typical transient, constant translator speed simulation on a linear generator of the same size as the experimental set-up LG prototype tested in this thesis, takes less than one hour to perform with the computers currently available for simulation on the Division for Electricity and Lightning Research (year 2005). 2.3.2 Maxwell’s Equations In the simulation tool the generator is modelled using the four fundamental equations [Wangsness 1986] of electricity, namely Maxwell’s equations, Eq.(16)-Eq.(19).

r r ∂B ∇× E = − ∂t

(16)

r r ∇× H = J

(17)

r ∇•B = 0

(18)

r ∇•D = ρ

(19)

r r r r E is the electric field, B is the magnetic flux density, H is the magnetizing field, J is the r current density, D is the displacement field and ρ is the charge density. Eq.(16) is Faraday’s Law, which describes that any change in a magnetic field results in an electric field. Eq.(17) is known as Ampere’s Law and states that the curl of the magnetizing field is equal to the current density. Ampere’s Law connects the current flowing through a surface, to the line integral over a continous, closed curve surrounding the surface. Eq.(18) is called Gauss’ Law and states that any magnetic field line must complete a continous curve. It also states that there are no magnetic monopoles. Eq.(19) States that the divergence of the displacement field equals the charge density.

20

The numerical implementation of Maxwell’s equations in the simulation tool is complemented by calibrated material properties. These are e.g. sheet metal density, hysteresis properties, permeability and magnet material properties such as remanence. 2.3.3 Direct drive benefits [Danielsson 2005] has shown that PM linear generators used in direct driven point absorber WEC:s have the potential of reducing the need for complex power take off systems. This is due to the simplicity of the design using few moving parts. Mechanical overloads are handled electrically rather than mechanically, which further has the potential of reducing the number of parts used in the design and thus reducing the maintenance costs.

Simulations by [Danielsson 2004] show that a direct driven PA WEC with a PM linear generator placed in a sheltered site at deep water in the Baltic Sea can produce about 45 000 kWh of electric energy annually. In the simulations, an estimate of losses in the transmission system of about 5 % is included. Hydrodynamic effects are excluded in the study. 2.3.4 Reference note According to [Boldea 1997], the FEM is not used for making preliminary design of linear PM generators or actuators, but only for design refinements. Instead, analytic methods are recommended and are said to “…yield quick insight into the various phenomena while requiring much less computation time.”3. This is counter to the method used in this thesis, where the use of the FEM for not only preliminary, but final, high-accuracy design of a threephase linear PM generator has had a central part.

2.4 Dynamic forces in test set-up operation To estimate the operational performance of the experimental test set-up drive train, a theoretical analysis was made of the system using a sine wave as motion for the translator. Note the in the actual experiment, as well as in the computer simulation, the translator motion was constant. 2.4.1 Forces A total of five forces are acting on the linear generator translator during operation. These are shown in fig.(13) below. Frope

m

Fgravity Fsprings Felec.magn. Facc Figure (13). Forces acting on linear generator translator during test set-up operation. 3

Boldea.I, Linear Electric Actuators and Generators, p.92

21

The resultant force, which has to be applied to the rope connecting the motor to the translator, is given by the time-dependent function r r r r r Frope (t ) = Facc (t ) + Felec.magn (t ) + Fgravity + Fsprings (t )

(20)

The positive direction is taken to be upwards. The acceleration may be derived from the translator position as a function of time. If the buoy, and thus the translator, is assumed to follow an incident sinusoidal wave, the position, x(t), of the translator is x(t ) =

H ⋅ sin (Ωt ) 2

(21)

2π . T is the period time of the sine wave in seconds and, H, the significant wave T height in meters from trough to crest. The translator speed then is where, Ω =

x& (t ) =

H ⋅ Ω ⋅ cos(Ωt ) 2

(22)

The acceleration of the translator is the second time-derivative of the position x, namely &x&(t ) = −

H 2 Ω ⋅ sin (Ωt ) 2

(23)

The counteracting electromagnetic force, Felec.magn, is influenced by the size of the electrical load, which is connected to the generator terminals. A small resistance will allow a large current to flow and so the extracted power to increase. The force will thus vary with time as the translator moves through the stator and may be approximated using Eq. (24). r Pmomentary U LN ,1 (t i ) ⋅ I L ,1 (t i ) U LN , 2 (t i ) ⋅ I L , 2 (t i ) U LN ,3 (t i ) ⋅ I L ,3 (t i ) = + + Felec.magn (t i ) = x& (t ) x& (t ) x& (t ) x& (t )

(24)

The values of phase voltage in each phase, ULN,1,2,3(t), and of line current, IL,1,2,3(t), in Eq.(24) are momentary values at time, ti. The acceleration force, Facc, may be written using the expression for acceleration in Eq. (25): r H Facc (t ) = m&x&(t ) = −m Ω 2 ⋅ sin (Ωt ) 2

(25)

The force of gravity on the translator is constant and is r Fgravity = mg

(26)

The total spring force can be varied using different number of springs. The pretension, msp, can also be changed. The total spring force can be written as

22

r Fsprings (t ) = n sp ⋅ (k sp ⋅ x(t ) + m sp )

(27)

where nsp, ksp and msp are the number of springs used, the spring constant and the pretension respectively. The pretension is defined as the spring force at x(t) = 0. 2.4.2. Gearing The gear ratio, GR, between the motor and the generator translator is constant and is GR =

x& rope (t ) x& (t )

=4

(28)

This means that the rope speed at the motor side, x& rope (t ) , of the gear is four times greater than the LG translator speed, x& (t ) , see fig.(14). It also results in that the rope force on the motor side of the gear is one fourth of the translator pulling force, namely r r Frope (t ) Fmotor (t ) = GR

(29)

2.4.3 Mechanical frequency The relation between maximum motor rotational speed, fmech, wave height, H, and period time, T, for the sine wave is

f mech

=

⎛H⎞ GR ⋅ ⎜ ⎟ ⋅ Ω ω 1 1 GR ⋅ x& max (t ) ⎝2⎠ = ⋅ = mech = ⋅ = 2π 2π 2π ⎛ Droll ⎞ ⎛ Droll ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠

1 GR ⋅ H ⋅ Ω 1 GR ⋅ H GR ⋅ H ⋅ = ⋅ ⋅ 2π = 2π 2π Droll Droll T ⋅ Droll

(30)

where Droll is the diameter of the grooved aluminium roll used to wind and rewind the rope which connects the motor to the generator translator. 2.4.4 LabVIEW control voltage Eq. (31) is used for calculating the correct control voltage to be used in the LabVIEW control programme. The period time, T, can be set directly in the LabVIEW control programme whereas the amplitude of the control DC voltage is related to the parameters H, T , D roll and GR of Eq.(30) by the expression V DC ,control = OVmax ⋅

f mech OVmax GR ⋅ H OVmax ⋅ GR ⋅ H = ⋅ = [V] f nom f nom T ⋅ Droll T ⋅ Droll ⋅ f nom

(31)

OVmax is the maximum output voltage on the PCI plug-in card. The OVmax is 10 volts and corresponds to 100 % of the nominal motor speed set on the motor drive. fnom is the nominal mechanical speed of the motor in hertz. The parameter, fnom, can be set on the motor drive

23

local control unit and is usually the motor nominal speed specified on the nameplate. Deriving the expression for VDC,control, the gear ratio, GR, between the motor and the generator translator has been taken into account.

3. Experimental test set-up In this degree project, an experimental lab test set-up for a linear generator prototype was made ready for testing. The aim being full-speed testing, several components had to be added, modified or designed from scratch. The most important components of the experimental test set-up are shown in fig.(14).

Motor and grooved aluminium roll

Figure (14). Main components of the experimental test set-up.

A frequency converter was installed and a control programme was implemented in LabVIEW. From the control signals sent from LabVIEW, the frequency converter runs the motor on which an aluminium grooved roll is mounted. The pulling-rope is wound and rewound onto the aluminium roll as the motor shaft rotates clockwise and counter clockwise respectively. The rope is guided via pulleys through the gearing, see fig.(14), so as to pull the linear generator translator upwards when the rope is wound onto the roll. See also fig.(15).

Load Measuring equipment.

AC output

Figure (15). Experimental test set-up.

The grooved aluminium roll was modified to fit a steel structure with ball bearings and also to be able to fit the motor shaft. A steel structure holding the roll bearings was designed, ordered and assembled during the work of this project. The motor and steel structure was finally fitted

24

onto a concrete foundation. Springs and additional pulleys also had to be added to be able to conduct the first tests. As precautionary measures an electrical breaker was connected, ventilation pipes mounted surrounding the springs and boards of perspex® were put in to prevent injuries from squeezing.

3.1 Variable speed drive An ACS800 frequency converter motor drive, delivered by ABB, was used to control the motor in the experimental set-up. The ACS800 employs IGBT frequency inverter technology. A data sheet of the ACS800 can be found in appendix 1.

3.2 Induction motor The induction motor used to pull the translator in the experimental test set-up, is a 75 kW, 2130 rpm, delta-connected motor operating at 400 V nominally. The weight of the motor is 335 kg and it is mounted onto a concrete foundation. A data sheet of the induction motor can be found in appendix 2.

3.3 Conversion of rotating movement to linear movement The grooved aluminium roll is shown in fig.(16). The roll is mounted directly onto the shaft of the induction motor. It is used to wind and rewind the rope connecting the motor to the translator of the linear generator. The rope runs through several pulleys mounted on the translator as well as in a steel structure surrounding the linear generator. The rope is tied to a horizontal steel beam at the stationary end and is clamped by the aluminium roll at the moving end.

Figure (16). Grooved aluminium roll for winding and rewinding the rope connecting the motor, via the gearing, to the translator of the linear generator.

3.4 LabVIEW motor control The programme controlling the frequency converter motor drive was written in LabVIEW version 7.1. LabVIEW is a graphical programming language suitable for simple as well as advanced data acquisition and control applications. A graphical user interface is created automatically as the application programme is written. The number of channels that can be used for control and data acquisition is limited by the hardware used with the computer. A LabVIEW programming tool called DAQ Assistant (Data Acquisition Assistant) was used to communicate with the hardware to send digital and analogue signals to the frequency

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converter motor drive. The DAQ Assistant automatically configures the analogue or digital input- or output channels from given parameter choices. In this project an existing LabVIEW programme was modified to control the induction motor. The first version of the control programme was written in 2005 by the author and another student during a course in Computer Based Measurement Techniques at Uppsala University. The programme handles three output signals that are required to successfully control the frequency converter. Two digital signals control turning the motor on and off and setting the forward/reverse rotational direction respectively and an analogue output signal specifies the rotational speed of the motor. The analogue output signal is limited, by the hardware, to ten volts, corresponding to 100% of the motor nominal speed specified in the frequency converter operational parameters.

3.5 Hardware To send and receive digital and analogue input and output signals a NI-DAQmx 6229 PCI plug-in card was used. NI means National Instruments, the company selling the specific product. DAQ means Data Acquisition and refers to the field of application. mx is a type of product code and PCI refers to what technical standard that is used in the connection system and in computer language. The term plug-in, in plug-in card, means that it can easily be connected to the back of a PC. To learn more about data acquisition systems and products related to data acquisition, see e.g. the website of National Instruments4. The NI-DAQmx 6229 PCI plug-in card features a total of four analogue output channels, 48 digital I/O channels, including two counter I/O channels, and 24 analogue input channels. Analogue signals may range from zero to ten volts. For more information about this product, see the National Instruments’ website. The relatively advanced nature of the PCI plug-in card compared to the specific application in this project is motivated since considerably more advanced measurements will be made in the near future. The data sheet of the NI -DAQmx 6229 PCI plug-in card is found in appendix 3.

3.6 Measurements 3.6.1 Three-phase portable DAQ system Measurements of voltage and current were made using a portable data acquisition system. A laptop is connected to a measuring-hub, which can acquire three-phase voltages and currents. Data is automatically stored on the hard drive of the laptop. The length of the data sequence stored as well as the sampling rate can be changed in the LabVIEW data acquisition programme running on the laptop. In fig.(17) a schematic connection diagram of the threephase DAQ system is shown.

4

www.ni.se

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Figure (17). Scheme of the portable three-phase DAQ system.

The wire connection box seen in fig.(17) carries one measuring resistor per phase. This is where the voltage- and current sensors are connected. The instantaneous values of current and voltage are sampled by the USB 6009 logger (see data sheet in appendix 4) via measurement circuits a, b and c, see appendix 5. The values are then transferred to the LabVIEW DAQ programme. The internal connection diagram of the wire connection box can be seen in fig.(18). Note the measurement resistors of 0.25 Ω each. The current sensors are connected to the 470 kΩ resistors on each side of the measurement resistors and the voltage sensors are connected between the right side of the measurement resistors and the neutral point of the load Y-connection.

Figure (18). Internal circuit of the three-phase, DAQ system wire connection box.

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3.6.2 Load The electric load was built in a student project work at the Division for Electricity and Lightning Research during the summer of 2005, see fig.(19). The load consists of eight 20 ohm clusters per phase mounted on an aluminium heatsink. Each cluster consists of two 10 ohm resistors connected in series. Thus it is possible to achieve a large number of different ohmic values of the resistance per phase by connecting the clusters in series, parallel or combinations of both. The schematic circuit diagram of the load can be seen in fig.(18) (bottom left).

b).

a). Figure (19) Experimental set-up three-phase load a). Front view. b). Tilted top view.

In section 5, results are presented from measurements made at 3.84 ohms per phase. This value represents the measured value of the total load resistance per phase, Rload, and includes both the load resistors, a power cable, connection wires and the measurement resistors, Rmeas, of the three-phase DAQ system, see table (1). Load resistance in ohms

Rload 3.591 +/- 7 %

Rmeas 0.25 +/- 1%

Total 3.841 +/- 7%

Table (1). Values of the load resistance per phase.

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Connection of the resistors is made using ordinary banana plugs enabling fast changing of load resistance. The load is connected to the portable three-phase DAQ system by an ordinary three-phase, 16 A, power cable, see fig.(19) b) top left.

4. Experiment The translator of the linear generator was pulled upwards one half cycle, simulating the first half of a wave period of an incoming ocean wave. In real ocean conditions, when the buoy is lifted by the wave, the translator of the linear generator is pulled upwards by the rope connecting them. The springs, which connect the translator to the concrete foundation, are thus stretched. In the experiment, two 97 mm diameter springs, each of length 650 mm at rest, were connected to the linear generator translator via pulleys mounted on the experimental setup concrete foundation. At the lower translator position (wave trough) the springs remain slack and in the translator top position, a total spring force of approximately 2000 Newton is reached. The translator speed was controlled in real time from the LabVIEW programme, using a graphical slide controller. In the experiment the generator was connected to a three-phase electrical load of 3.84 ohms per phase, see fig.(19). Three-phase voltage and current were measured simultaneously using the portable three-phase DAQ system described in section 3.6.1. The generator prototype design parameters, and a few values of parameters used in the experiment, are shown in table 2. These values are the parameter values used in the simulations. Parameter Value No. of sides of the generator 4 Stator width 400 Stator length 750 Stator height 91 Translator length 650 Airgap width 7 Pole width 50 Cable diameter 8 Cable cross sectional area 16 Translator speed 0.86 Significant height of artificial wave 1.38 Wave period time of artificial wave 3.3 Value of load resistance in simulation 3.66 Table 2. Linear generator prototype design parameters

Unit mm mm mm mm mm mm mm mm mm2 m/s m s Ω

Due to the programmers’ choice of parameter inputs in the simulation tool interface, the demands on output voltage and apparent power had to be changed successively until the correct stator- and translator length was achieved. Therefore, in the simulation, the period time, T, of the wave acting on the LG translator was changed until the correct constant speed was achieved. From this, the apparent power and voltage was changed successively until the correct load resistance was achieved.

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5. Results The expected performance of the linear generator could be determined by running transient analysis simulations in the linear generator application module of the simulation tool. Approximately 25-30 simulations of about 50 minutes each were necessary to reach the target value of load resistance and of stator/translator length corresponding to the LG prototype geometry. In the simulations, the wave acting on the LG was an artificial wave having constant vertical speed. The generator performance characteristics, calculated in the simulation, are compared with measurements in the following sections, namely sections 5.1 to 5.5. Comparisons between simulations and measurements are shown in fig.(20) to fig.(23).

5.1 Phase voltage A comparison between simulated and measured phase voltage is shown in fig.(20).

Figure (20). Comparison between simulated and measured phase voltage. Simulation: Black line. Experiment: Red line

Simulations agree well with experimental results both in the phase voltage waveform and in the value of maximum voltage amplitude. The measured phase voltage curve (red) has been shifted in time so that the voltage peaks of the two curves occur at the same instant. The envelope of the voltage and current in fig.(20) and fig.(21) respectively, is primarily due to that the translator is shorter than the stator. Full induction only occurs for 100 mm of the

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translator motion since the translator is 100 mm shorter than the stator. This also results in that only part of the stator windings is active during the motion of the translator.

5.2 Current A comparison between simulated and measured current is shown below in fig.(21).

Figure (21). Comparison between simulated and measured current. Simulation: Black line. Experiment: Red line

Simulations agree well with experimental results both in the current waveform and in the value of maximum amplitude, which is 16.5 A. The measured current curve (red) has been shifted in time so that the highest peaks of the two curves occur at the same instant.

5.3 Phase voltage and current In fig.(22), the results of fig.(20) and fig.(21) are shown compiled in the same diagram. As before, simulated results are shown in black and measurements are presented in red. The voltage curves are the curves with the higher amplitude and the current curves are the curves with the lower amplitude.

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Figure (22). Comparison between simulated and measured line-to-neutral voltage and current. Simulation: Black line. Experiment: Red line

Fig.(22) is shown to give an impression of how power is generated in the machine. The instantaneous values of voltage and current can be multiplied to give the output power. Since the load is purely resistive, the voltage and current will be exactly in phase, which is confirmed by the curves in fig.(22).

5.4 Three-phase instantaneous power The instantaneous three-phase power was simulated. From the measurements of three-phase line-to-neutral (phase) voltage and current, the instantaneous three-phase power was calculated using Eq.(32). P3Φ ,inst . = abs(Va (ti )) ⋅ abs(ia (ti )) + abs (Vb (ti )) ⋅ abs (ib (ti )) + abs (Vc (ti )) ⋅ abs (ic (ti ))

(32)

A comparison between simulated and measured three-phase instantaneous power is shown below in fig.(22).

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Figure (23). Comparison between simulated and measured three-phase instantaneous power. Simulation: Black line. Experiment: Red line

Simulations agree well with experimental results. The value of simulated maximum threephase power is in close proximity to measured maximum instantaneous power. The time shifting of the measured voltage and current curves shown in fig.(20) and fig.(21) result in the power curve shown in fig.(23). No additional time shifting has been made plotting fig.(23).

5.5 Qualitative results The accuracy of the values of measured parameters has been determined, see table 3. The resistance of the three-phase load in the experimental set-up was determined by measuring current and voltage when feeding the load from a lab DC power supply. Two Fluke 83 III digital multimeters were used to measure current and voltage respectively. 22 combinations of current and voltage were fed through the load to increase the accuracy in the determination of the load resistance. Parameter Accuracy Unit Load resistance +/- 7 % Ohm Translator speed +/- 1.7 % m/s Phase voltage and current +/- 2 % V 3-phase DAQ system +/- 2 % V and A Table 3. The determined accuracy of parameters in the experiment.

The maximum speed of the translator was determined from the electric output frequency. The shortest quarter of a wave period was found to be the rising of voltage before the highest

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voltage peak in fig.(20). Since, in this LG, the translator speed is 0.1 m/s per Hz of electric frequency in the output and the sample time in the measurements were 1 ms, the speed could be determined with an accuracy of 1.7 % as stated in table 3. The 3-phase DAQ system is calibrated using Fluke multimeters and has an accuracy of +/- 2 %. This explains why the accuracy of measured voltage and current in table 3 is +/- 2 %. The operational performance of the motor control system was satisfactory. The experimental test set-up as a whole worked according to expectations. The springs, which were attached between the translator and the experimental test set-up concrete foundation, provided the necessary spring force to reduce the effects of cogging forces when translator moving downwards at low speed. The LabVIEW control programme worked according to expectations and fulfilled the necessary requirements for conducting the experiment.

6. Discussion The value of load resistance used in the simulation is 3.66 ohms per phase, which was intended to correspond precisely to the actual resistance of the load in the experimental set-up. Later, when the load resistance was determined with higher accuracy, it was discovered that the actual resistance was slightly lower, namely 3.84 +/-7 % ohms per phase, than the value used in the simulation. The relatively low accuracy in the value of load resistance could be increased if other, better, equipment would be used. E.g. the three-phase DAQ system could have given an accuracy of the load resistors of something in the range of +/- 2 % instead of the +/- 7 % achieved when using two Fluke multimeters. In fig.(20) and fig.(21), differences between simulated results and experimental results appear. The differences appear at time t ≈ 0.5s and t ≈ 1.5s and are probably due to the controlling of the motor speed. The speed of the LG translator has probably deviated from the constant speed used in the computer simulation at several instants, both before and after the voltage and current peaks seen in fig.(20) and fig.(21). If this is correct, it would explain, in accordance with Faraday’s Law, why the measured phase voltage drops below the simulated phase voltage in the beginning and in the end of the cycle since the speed was lower than in the simulation. The same explanation would in that case apply also to the fact that the number of voltage peaks in the cycle differs between simulated and experimental results. The speed in the beginning and in the end of the cycle was simply too low in the experiment to induce voltages, and currents successively, in the stator windings. The asymmetri near the peak of the measured three-phase power in fig.(23) is probably due to that one of the LG phases was connected the wrong way during the experiment. This could be confirmed plotting all the phase voltages in the same diagram. In future experimental work at the Division for Electricity and Lightning Research, the experiment will be remade using the correct connection of the phases. One of the most important results is the comparison between the numerical and experimental results of the amplitude of the phase voltage. This shows the validity of the simulation tool numerical results, i.e. that the induction is as large as the simulation predicts. Hence the possible difference in translator speed on either side of the voltage peak ( see fig.(20) ) is expected not to influence the quality of the comparison between the amplitude of simulated and measured phase voltage and current.

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Calibration of the control system, which is necessary to be able to test the generator in constant cycling, lies beyond the scope of this thesis. The LabVIEW control programme has the possibility of generating sine waves instead of a constant control voltage. To make constant cycling possible, a position transducer has to be connected to the LG translator and some further programming has to be made in LabVIEW. This lies within the future work with the experimental test set-up.

7. Conclusion Experimental results compare well with numerical calculations. This was an expected result since the simulation tool has been tested against experiments in other applications. The amplitude and shape of e.g. the phase voltage show good correspondence with numerical results, see fig.(20). The frequency of the phase voltage is varying in accordance with translator speed and the amplitude is varying in accordance with the translator speed and the fraction of total translator length that is inside of the stator. The overall operational performance of the experimental test set-up was satisfactory. Using the ACS800 frequency converter motor drive, the rotational speed of the motor could be changed arbitrarily from the LabVIEW control programme. The speed of the linear generator translator could thus be changed and set to different constant values. The precautionary measures taken before experiments were conducted have been shown to work satisfactorily. The electrical breaker installed between the control system and the frequency converter motor drive was employed at one instant during an experiment and it showed satisfactory performance. The springs installed to store potential energy when the translator was pulled upwards, was sufficiently large to prevent cogging effects during operation. The steel structure mounted on the same concrete foundation as the electric motor provided sufficiently rigid support for the roll mounted on the motor shaft. Pulleys mounted on the translator of the generator, on the concrete foundation of the experimental set-up and those mounted in the steel structure supporting the generator, worked according to expectations.

8. Acknowledgements Olle Svensson is greatfully acknowledged for patiently helping out with electronics issues and for designing the portable three-phase measurement system. Great support and considerations on the written material were given by the supervisor, Karin Thorburn, and the opponent, Rafael Waters. A big thanks to Ulf Ring, Stefan En and Stefan Gustafsson for helping out with some parts manufacturing and helping to solve practical problems arising while getting the experimental test set-up ready for operation. Anders Hildingson participated in the programming of the LabVIEW control programme and Simon Tyrberg built the electrical load for the test set-up. Dr. Arne Wolfbrandt and Dr. Karl-Erik Karlsson are acknowledged for developing the simulation tool and Thomas Götschl for always giving a helping hand when it comes to computer problems. The author would also like to greatfully acknowledge CF Environmental Fund for supporting this degree project by giving personal financial support. The following are acknowledged for financially supporting the wave energy project: The Vargön Smältverk AB Research Foundation, Draka Kabel AB, the Engkvist Research Foundation, CF Environmental Fund, The Gothenburg Energy Research Foundation and the Ångpanneföreningen Research Foundation.

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9. References Alfredsson Alf, Jacobsson Karl-Axel, Rejminger Anders, Sinner Bengt, Elkrafthandboken-Elmaskiner, första upplagan, Liber AB 1996, (Swedish) Boldea I, Nasar Syed A, Linear Electric Actuators and Generators, Cambridge University Press 1997 Bolund, Björn, High Voltage Generators for Generation and Storage of Electric Energy, Licentiate Thesis, UURIE 293-04L, Uppsala University 2004 Boyle Godfrey, Renewable Energy – Power for a Sustainable Future, The Open University, Oxford University Press 1996 Chapman, Stephen J, Electric Machinery Fundamentals, 3rd edition, McGraw-Hill, cop. Boston 1999 Danielsson Oskar, Leijon Mats, Thorburn Karin, Eriksson Mikael, Bernhoff Hans, A Direct Drive Wave Energy Converter – Simulations and Experiments, Proceedings of OMAE 2005: 24th International Conference on Mechanics and Arctic Engineering, Halkidiki, Greece, 12-17 June 2005 Danielsson Oskar, Sjöstedt Elisabeth, Thorburn Karin and Leijon Mats, Simulated Response of a Linear Generator Wave Energy Converter, ISOPE-2004, Toulon, France, 23–28 May 2004 Danielsson Oskar, Thorburn Karin, Eriksson Mikael, Leijon Mats, Permanent Magnet Fixation Concepts for Linear Generator, 5th European Wave Energy Conference, Cork, Ireland, Sep 17-19, 2003 Elfving Gunnar, ABB Handbok Industri, Wallin och Dalholm Tryckeri AB, Lund 1993, (Swedish) Eriksson Mikael, Thorburn Karin, Bernhoff Hans, Leijon Mats, Dynamics of a linear generator for wave energy conversion, Proceedings of OMAE04, 23rd International Conference on Offshore Mechanics and Arctic Engineering, OMAE2004-51205, Vancouver 2004 Halliday David, Resnick Robert, Walker Jearl, Fundamentals of Physics, 6th Edition, John Wiley and Sons, Inc, 2001 IVA-Projektet Energiframsyn, Vindkraft till lands och till sjöss, Kungliga Ingenjörsvetenskapsakademien, IVA, Multitryck, Eskilstuna 2002, (Swedish) Ivanova, Irina Alexeevna, Simulation of Linear Permanent Magnet Octagonal Generator for Sea Wave Energy Conversion, Licentiate Thesis, UURIE 295-04L, Uppsala University 2004 Leijon Mats, Bernhoff Hans, Ågren Olov, Isberg Jan, Sundberg Jan, Berg Marcus, Karlsson Karl Erik and Wolfbrandt Arne, Multiphysics Simulation of Wave Energy to Electric Energy Conversion by Permanent Magnet Linear Generator, IEEE Transactions on Energy Conversion, Vol. 20, NO. 1, March 2005 Mohan Ned, Undeland Tore M, Robbins William P, Power Electronics – Converters, Applications, and design, Second edition, John Wiley and Sons, Inc, 1995 Sarma, Mulukutla S, Electric Machines – Steady State Theory and Dynamic Performance, second edition, Northeastern University, PWS Publishing Company 1996 Thorborg, Kjeld, Power Electronics – in Theory and Practice, Studentlitteratur, Lund 1993 Wangsness, Roald K, Electromagnetic Fields, 2nd Edition, john Wiley & Sons 1986 http://www.oceanpd.com/ http://www.peakoil.net/

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Appendix 1

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Appendix 2

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Appendix 3

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Appendix 4

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Appendix 5 Circuit diagram of each of the measurement circuits a,b and c in fig.(16).

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