Analysis of Loss Components in aSynchronous Generator underNon-ideal Operating Conditions

Joona Rajamäki

School of Electrical Engineering

Thesis submitted for examination for the degree of Master ofScience in Technology.Espoo 30.3.2019

Supervisor

Prof. Anouar Belahcen

Advisor

Dsc. (Tech.) Sahas BikramShah

Copyright c⃝ 2019 Joona Rajamäki

Aalto University, P.O. BOX 11000, 00076 AALTOwww.aalto.fi

Abstract of the master’s thesis

Author Joona RajamäkiTitle Analysis of Loss Components in a Synchronous Generator under Non-ideal

Operating ConditionsDegree programme Automation and Electrical EngineeringMajor Electrical Power and Energy Engineering Code of major ELEC3024Supervisor Prof. Anouar BelahcenAdvisor Dsc. (Tech.) Sahas Bikram ShahDate 30.3.2019 Number of pages 56 Language EnglishAbstractThe wide use of high-power salient pole synchronous generators (SPSG) makes theunderstanding of losses and efficiency important. Factors such on-site conditionsand applications affect the power losses of an SPSG. Recently, the effect of torqueoscillation on the power losses of an SPSG have been studied and it was shown thattorque oscillations may generate additional losses. Furthermore, it is well known, thatcurrent unbalance and temperature generate additional losses. All the mentionedeffects may be present in on-site use of an SPSG.

This thesis determines the effects of torque pulsations, unbalance phase currentsand temperature rise on power losses of an SPSG. The different effects were simulatedby in-house finite element software FCSMEK. In order to determine the effects ofnon-ideal conditions on the losses, the limitations for the simulation cases were defined.IEC standards effectively restrict the maximum temperatures and current unbalancesof an SPSG. However, the torsional vibrations are not directly limited. Therefore,the measured torque profile of an internal combustion engine (ICE) was used in orderto determine the torque oscillation frequencies and amplitudes. The results indicatethat current unbalance and temperature increase power losses significantly. The lowfrequency torque pulsation of an ICE had less effect on the power losses probablydue to the low amplitude of the torsional vibration. The temperature rise increasesthe losses linearly. However, constant temperature was assumed, thus meaning thatthe effect of the losses on the final temperature was not accounted for. The effect oftemperature rise on losses is significant, but more predictable and the effect can belimited by cooling. Furthermore, results show that the negative sequence current maycause significant additional losses. Further investigation could involve introducingthe losses into a thermal network to determine the total temperature rise.Keywords electrical machine, FEM, power losses, synchronous generator, torsional

oscillation

Aalto-yliopisto, PL 11000, 00076 AALTOwww.aalto.fi

Diplomityön tiivistelmä

Tekijä Joona RajamäkiTyön nimi Analyysi epäideaalisen toimintaympäristön vaikutuksesta avonapaisen

sähkögeneraattorin häviökomponentteihinKoulutusohjelma Automation and Electrical EngineeringPääaine Electrical Power and Energy Engineering Pääaineen koodi ELEC3024Työn valvoja ja ohjaaja Prof. Anouar BelahcenPäivämäärä 30.3.2019 Sivumäärä 56 Kieli EnglantiTiivistelmäAvonapaisten sähkögeneraattorien (SPSG) laaja käyttö tekee niiden häviökomponent-tien ja hyötysuhteen tutkimisesta tärkeää. Käyttösovellukset ja on-site -olosuhteetvaikuttavat SPSG:n häviöihin. Tutkimuksissa on huomattu, että erityisesti matala-taajuuksinen momentin oskillointi saattaa lisätä SPSG:n häviöitä. Lisäksi on-sitekäytössä esiintyvä sähkökuorman epätasapaino ja korkeat lämpötilat saattavat lisätähäviöitä

Tässä diplomityössä määritetään vääntövärähtelyn, sähkökuorman epätasapainonja lämpötilan vaikutusta SPSG:n häviöihin. Eri tapaukset laskettiin numeerisestielementtimenetelmällä FCSMEK -ohjelmistoa käyttäen. Edellämainittujen epäide-aalisten olosuhteiden raja-arvot määritettiin IEC standardien avulla. Standarditrajoittavat SPSG:n käyttölämpötioja ja virran epätasapainoa. Momentin oskillaatioil-le ei ole sitä suoraan rajoittavaa standardia, joten FEM malleissa käytettiin mitattuadieselkoneen (ICE) momenttikäyrää, josta erotettiin eri taajuuskomponentteja. Tulok-set osoittivat, että lämpötilan nousu ja sähkökuorman epätasapaino lisäävät häviöitämerkittävästi. Simuloinnissa käytetty matalataajuuksinen sinimuotoisesti oskilloivamomentti lisäsi häviöitä vain vähän. Tämä saattoi johtua momentin matalasta ampli-tudista. Lämpötilan nousu kasvatti häviöitä lineaarisesti. Simulaatio sisälsi oletuksen,että kasvavat häviöt eivät lisää lämpötilaa, ja että lämpötila on tasainen läpi SPSG:ngeometrian. Lisäys häviöissä oli merkittävä, mutta lineaarisuudesta johtuen helpostiennustettava ja rajoitettavissa esimerkiksi lisäämällä jäähdytystä. Lisätutkimuksiatarvitaan häviöiden vaikutuksesta koneen lopulliseen lämpötilaan.Avainsanat sähkökone, FEM, tehohäviö, Avonapainen sähkögeneraattori,

vääntövärähtely

5

PrefaceThis thesis was written at ABB in Helsinki during the spring and winter 2018-2019

I want to thank my instructor Sahas Bikram Shah for his advice, comments andguidance throughout the whole process of writing a thesis. A sincere thanks for mysupervisor Prof. Anouar Belahcen for all the discussions, advice and motivation.Thanks to all my colleagues at ABB for providing me an interesting topic, tools anda wide range of invaluable discussion about electromechanics and life.

I would also like to thank all my friends at Aalto University for making my timehere extremely fun.

Otaniemi, 30.3.2019

Joona Rajamäki

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ContentsAbstract 3

Abstract (in Finnish) 4

Preface 5

Contents 6

Symbols and abbreviations 8

1 Introduction 101.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Literature study 122.1 Salient pole synchronous generator . . . . . . . . . . . . . . . . . . . 122.2 Electromagnetic equations . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Losses in synchronous generators . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Resistive losses . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Iron losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Torsional vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Voltage unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Temperature rise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Methods 293.1 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Magnetic vector potential . . . . . . . . . . . . . . . . . . . . 293.1.2 Elements and boundary conditions . . . . . . . . . . . . . . . 303.1.3 Time-stepping analysis . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Simulation in FCSMEK . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Results and discussion 374.1 Machine parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Nominal point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Non-ideal operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.1 Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.2 Torque oscillations . . . . . . . . . . . . . . . . . . . . . . . . 464.3.3 Temperature rise . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Summary and future work 52

7

References 54

8

Symbols and abbreviations

Symbolsµ Permeability [V s/Am]σ Conductivity [S/m]A Cross-sectional area [m2]B Magnetic flux density [T ]c Loss coefficientD Electric flux density [C/m2]E Electric field strength [V/m]f Frequency [Hz]fs Supply frequency [Hz]fv1 Oscillation frequency [Hz]ft Slot harmonic frequencyH Magnetic field strength [A/m]In Nominal current [A]J Current density [A/m2]kr Eddy current coefficientl Length [m]n Number of turns in series with coilns Synchronous speed [1/min]p Number of pole pairsPcu Resistive loss in copper [W ]Pfe Iron loss [W ]Phy Hysteresis loss [W ]Ped Eddy current loss [W ]Pex Excess loss [W ]Q Number of slotsRp Phase resistance [Ω]Sn Cross sectional area of a coilVi Volume of iron [m3]

Operators∇× curl operator∇· divergence operatorddt

derivative with respect to variable t

∂

∂tpartial derivative with respect to variable t∑

i sum over index iA · B dot product of vectors A and B

9

AbbreviationsAC Alternating currentAVR Automatic voltage regulatorDC Direct currentFEA Finite element analysisFEM Finite element methodICE Internal combustion engineSPSG Salient pole synchronous generator

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1 Introduction

1.1 BackgroundPower loss in electrical machines is an important field of study for the machinemanufacturer and operator. One reason for this is that losses can raise the temperatureof machines. Since this temperature rise affects the maximum power output, itrepresents an important design parameter. In extreme cases of temperature rise, theinsulation of the machine will deteriorate, thus decreasing generator life expectancyand increasing maintenance costs (Pyrhonen et al. 2013). Power loss and efficiency isimportant, especially for large megawatt-size machines, since even a slight increasein efficiency will lead to large savings in energy and costs over time.

In islanded industrial networks, power is typically generated by synchronousgenerators coupled to an internal combustion engine (ICE). Since the torque in ICEsis produced by combustion in the cylinders, the total torque can vary considerably,leading to torque peaks during combustion in each cylinder. Therefore, a torqueoscillation is present during normal operation of ICE, which also causes fluctuationin the active and reactive power (Falcone et al. 2003, Casado et al. 2017). Recently,low frequency torque oscillation have been found to cause significant additional lossesin electrical machines (Arkkio et al. 2018a). Moreover, the torque oscillations cancause unwanted torsional stress on the coupling and the shaft of the machine (Joyceet al. 1978).

When connected to the grid, synchronous generators are subject to unbalancesin the load or other unwanted phenomena, especially if the generator is operated inan islanded system. According to IEC standard 60034-1, a synchronous generatormust be capable of operating continuously under an unbalance system where theratio of negative sequence current component (I2) to the rated current (In) does notexceed 8%. The negative sequence component induces a counter-rotating field inthe air gap, thus decreasing the performance of the machine, generating additionalresistive losses, and increasing unwanted thermal stress in the synchronous generator(Debruyne et al. 2017). Moreover, unbalanced phase current can cause mechanicalissues, such as vibration and torsional pulsation (Lee 1999).

1.2 Aim of the thesisThe aim of this work is to determine the effects of torque oscillations and grid-inducedunbalance on the losses of a salient pole synchronous generator. To accomplish thisgoal, the power losses in synchronous generators will be simulated with an actualmeasured torque profile of an ICE using 2-D finite-element software. In addition totorque oscillations, the effects of negative sequence voltage or current are simulatedto determine the losses. The simulation uses a 12-MVA 8-pole synchronous generator.The goal is to quantify the effects of torque oscillation and unbalanced voltage on thepower losses of the machine. These results can be used to determine whether torqueoscillation and unbalance load are a significant loss sources for this specific machine.

1.3 Outline of the thesisChapter 1 introduces the objectives of the thesis. Chapter 2 highlights the basicinformation of electrical machines and losses. Additionally, Chapter 2 discusses theeffects of non-ideal operation on electrical machines. Chapter 3 presents the methodsand software used to obtain the losses in different cases of non-ideal operation.Chapter 4 presents and discusses the simulation results. Chapter 5 summarizes thebackground, methods and results and give some ideas for future work.

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2 Literature studyIn order to simulate the effects of non-ideal operation conditions on the lossesof a synchronous generator, it is important to understand the source of losses insynchronous machines, as well as the structure and applications for synchronousmachines. Therefore, this chapter presents an overview of electrical machines. InSection 2.1, the construction of a salient pole synchronous generator is presentedin order to identify the reasons and consequences of non-ideal operation. Section2.2 presents the electromagnetic relations needed for estimating the losses. Section2.3. introduces different loss components in order to understand the mechanisms andthe origin of these loss components, as well as methods for loss computation. Therest of the sections discuss the non-ideal operation conditions which are simulated inChapter 4.

2.1 Salient pole synchronous generatorThis section presents the operation principles and construction of a salient polesynchronous generator (SPSG). Electrical machines work on the energy conversionprinciple using the electromagnetic field. Electrical machines are used both in awide range of industrial applications, including fans and pumps, as well as forgenerating electrical energy in power plants. Generally, electrical machines arecategorized based on their operation principle into DC machines and AC machines.Furthermore, the AC machines include synchronous machines and induction machines.Synchronous machines can be used either as motors or as generators. A commontype of synchronous machine is the SPSG. A salient pole rotor is a specific type ofrotor which contains poles composed of electrical steel sheets. When the rotor ofa SPSG is magnetized and rotated using an external engine, it creates a rotatingmagnetic field, which induces voltage in the stator winding. A magnetic field is alsocreated in the stator winding by the magnetomotive force. SPSGs coupled with ICEsas prime movers are used for energy generation in islanded systems and as emergencyunits for small networks. SPSGs rotate in synchronism with the network frequency,with the rotation speed depending on the number of poles in the rotor, as shown inthe following equation:

Ns = 60f

p(1)

Where f is the grid frequency and p is the number of pole pairs.The structure of the SPSG is illustrated in Figure 1. The stationary part of

the SPSG is the stator, which is bolted to a rigid welded steel frame. The statorcore is made of 0.5-mm or 0.35mm thick laminated steel sheets which are stackedto form an iron core. The steel sheets are insulated to decrease circulating currents.Silicon iron is predominately used in SPSGs because of its high induction at low fieldstrength, not only making it a good conductor for magnetic flux with low coercivefield strength, but also resulting in a narrow hysteresis loop and lower losses. The

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electrical properties and power losses in the iron are discussed more thoroughly inSection 2.1.2. The armature winding of high power machines are frequently made ofwound copper bars. The copper in the stator winding can be divided into severalsmaller conductors to decrease the skin effect, thus decreasing the losses in the copper(Khang et al. 2014). The whole stator is impregnated with epoxy in order to increasemechanical strength and thermal conductivity.

The rotor consists of a shaft, poles fixed on the shaft, exciter and fan. Thepoles connected on the rotor are made of 2-mm laminated steel sheets, which arestacked to form an iron core similarly as in the stator. In order to run the machineat synchronous speed, the rotor poles are excited by conducting DC current to thefield winding. When the current flows in the winding, a magnetic field is formedwhich then couples with the stator magnetic field through the air gap. When therotor is rotated by an ICE, the stator field lags behind, generating torque. The anglebetween the stator and rotor magnetic fields is called the power angle. Since salientpole rotor machines have a non-uniform air gap, the air gap length, and hence thereluctance, changes according to the rotor position. Additionally, the reluctance alsovaries because of the slots and teeth in the stator core. Reluctance variations affectthe flux density distribution of the air-gap. The rotor poles rotating at synchronousspeed induce a sinusoidal flux, where high frequency ripple is present due to theslot-tooth effect (Sen 2007).

Figure 1: Construction of a salient pole synchronous generator

To improve the machine performance during transients, damper windings areplaced on the surface of the rotor poles. Damper windings are a set of short-circuitedcopper bars near the air gap, which are used to damp the unwanted counter-rotatingfields and the fluctuation of rotation speed caused by pulsating torque loads. Inaddition, damper winding can be also used to start the generator as a asynchronousmachine (Pyrhonen et al. 2013).

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For the machine to operate at a synchronous speed, the rotor poles need tobe excited. Excitation of the rotor poles is essential for operation of synchronousgenerators and motors, which distinguishes them from other types of electricalmachines (Pyrhonen et al. 2013). High-voltage synchronous generators frequentlyuse a brushless excitation system. A separate excitation machine is placed on thesame shaft as the main machine for supplying power to the main machine. When therotor of the excitation machine rotates, a three-phase alternating current is induced.The current is rectified to a direct current by a diode bridge, and then supplied tothe rotor winding.

An automatic voltage regulator (AVR) is used for adjusting the excitation currentin changing load conditions and for controlling the reactive power of the mainmachine. The excitation power is taken from the line, through a voltage transformer.In addition, a pair of permanent magnet poles can be installed on the exciter toensure self-excitation when no other power sources are available.

2.2 Electromagnetic equationsThe previous section described the structure and operation of synchronous generator.This section presents the fundamental electromagnetic equations needed for analysisand loss computation in electrical machines. The mechanical energy is converted toelectrical energy by electromagnetic coupling through the air-gap of the machine.Maxwell’s equations describe the relationships between electricity and magnetism,which makes them essential for understanding the electromagnetic relations andFEM theory presented later in the thesis.

In electromagnetic problems, time-varying magnetic fields generate an electricfield and vice versa. The set of equations that describe the time relationship betweenmagnetism and electricity are known as Maxwell’s equations (Luomi 1993). Thedifferential form is expressed follows:

∇ × E = −∂B

∂t(2)

∇ × H = J + ∂D

∂t(3)

∇ · B = 0 (4)∇ · D = ρ (5)

(6)

where,

E is the electric field strengthH is the magnetic field strengthD is the electric flux densityB is the magnetic flux densityJ is the electric current densityρ is the electric charge density

15

The variables are related by the material properties:

D = ϵE (7)B = µH (8)J = σE (9)

where ϵ is the permittivity, µ is the permeability and σ is the conductivity ofthe medium. The integral forms of Equations 2-5 can be derived by using Gaussand Stokes theorems. For Equations 2 and 3 Stokes theorem is used to achieve thecirculation of E and H on path s. Often, in electromechanical problems, we canassume the quasi-static state, meaning that the conductivities in the materials arelarge, thus making the displacement currents negligible

˛S

E · ds = −ˆ

S

∂B

∂t· dS (10)

˛S

H · ds =ˆ

S

(J + ∂D

∂t) · dS (11)

And using Gauss theorem on a region V and its surface S for Equation 4 and 5we achieve

˛S

D · dS =ˆ

V

ρdV (12)

˛S

B · dS = 0 (13)

Equations 2 and 10 illustrates how time varying magnetic field gives rise to anelectric field. Equations 3 and 11 represent the Ampere’s circuit law with addeddisplacement current.

2.3 Losses in synchronous generatorsThe previous section presented the Maxwell’s equation most often used for analyzingelectrical machines. This section presents the losses in electrical machines, with themain focus being on resistive losses and iron losses. A common way of defining theloss components in synchronous generators is to categorize them by their origin andmechanisms. Most of the losses generated in the machine are resistive losses, ironlosses on the pole surface and iron losses in the stator core. Resistive losses in thestator and rotor represent approximately 50-60% of the total losses in a generator.Losses in the stator iron core and the rotor pole surface make up approximately 20%of the total losses (J. Tervaskanto 2018). A relatively large portion of the lossesconsist of mechanical losses: friction in the bearing and windage of the rotating rotor.However, mechanical losses are often considered as constant throughout the differentoperation points of the machine. Therefore, mechanical losses are not examined inthis thesis.

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2.3.1 Resistive losses

Resistive losses can be divided into losses in the stator winding, rotor bars, rotorwinding and additional resistive losses. In the rotor winding, resistive losses aregenerated by the DC current used to magnetize the poles. An alternating currentgenerates additional losses which consist of two main components. Firstly, skin effectand eddy currents, which cause uneven current distribution in the conductor, thusleading to additional losses. Secondly, additional losses are induced in the statorcoils due to the stray fields in the machine. Resistive losses in the rotor bars aregenerated because of the air-gap flux pulsation.

DC lossesCurrent flowing in a conductor results in losses due to the conductivity of the material:

Pcu = RI2 = l

σAI2 (14)

where σ is the conductivity, A is the cross-sectional area, and l is the length. Theconductivity, and therefore the losses, of the material depends on the temperaturethat it operates in. Winding temperatures in electrical machines can rise up to140C. For more accurate loss estimation the conductivity can be scaled to any giventemperature θ by a linear approximation:

σ = σ20[1/(1 + α(θ − 20C))] (15)

where α is the temperature coefficient of the conductor. For three-phase electricalmachines with perfectly balanced phases the losses can be estimated as:

Pcu = 3RpI2n (16)

where In is the nominal current, and Rp is the phase resistance. Often, the lossestimation in electrical machines is estimated according to IEC standard 60034-2-1,where the reference operating temperatures are defined to be 95C for temperatureclass B machines, or 115C for temperature class F machines. However, the maximumwinding temperatures can be 135C or 155C leading to increased losses accordingto Equations 15 and 14.

AC lossesIn addition to the DC losses, the time variation of the currents and magnetic field inthe electrical machine induces eddy currents in the windings and other conductiveparts of the machine. Essentially two phenomena are related to eddy currents inthe conductors of electrical machines. Firstly, the skin- and proximity effect whichare caused by the alternating stator current inside the conductors, and neighboringconductors (Islam 2010). Secondly, the varying stray fields that cross the conductor

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in the stator slots and winding ends (Sadarangani 2006). The distribution of eddycurrents in these two cases are illustrated in Figure 2. The origin of the currentsis governed by Equations 2 and 9. In other words, when a conducting materialexperiences a time-varying magnetic field, a current will be induced in it. Moreover,according to Lenz’s law, eq. 3 the current induces an opposing magnetic field. Fromthese equations can say that an alternating current in the conductor produces analternating magnetic field, which in turn produces an electric field that opposes thechange in current density. The opposing electric field is strongest in the middle ofthe conductor forcing the electrons on the edge of the conductor.

Figure 2: (a) Skin effect by the conductors own alternating current. (b) Proximityeffect caused by an external field (Sadarangani 2006).

The uneven current distribution in the conductor leads to changes of resistancein the conductor in radial direction (Sadarangani 2006). Therefore, the losses in theconductors are higher than just the DC resistance, and a separate AC resistancecoefficient should be estimated for higher accuracy. The AC resistance can becalculated as similar to Equation 15, but adding a specific eddy current coefficientto it kr.

Rac = krl

Aσ(17)

The eddy current factor kr is the ratio of the AC and DC resistances in the slotembedded part of the winding (Pyrhonen et al. 2013).

kr = RAC

RDC

(18)

The above equations are often used to estimate the resistive losses in the coreregion of the stator, and the magnetization losses of the rotor. The end windingeffects and the losses on the rotor pole surface, and damper bars are discussed in thenext section

Additional lossesThe additional losses include loss components not included in the previously presented

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losses. Karmaker (1992) presents that these losses are a combination of the losses inthe following areas of the machine:

• Loss in stator teeth, yoke iron due to stray flux

• Loss in the end plates and winding ends

• Loss in the rotor pole surface

• Loss from constructional sources i.e rotor eccentricity

The losses in the winding ends, and end plates are generated by the stray fluxespenetrating the end area in axial direction, generating eddy current losses. Theend winding losses are affected especially by the geometry of the winding end. Thelosses in the winding ends can be estimated by calculating the end winding leakageinductance (Pyrhonen et al. 2013).

The rotor pole surface and rotor bar losses are generated by the harmonics in theair-gap flux. The harmonics can be divided into time harmonics which vary by time,and space harmonics. Sources for time harmonics are, for example, the supply of themachine. The space harmonics are generated by non-sinusoidal distribution of thecoils and the machine slotting. The permeance variations of the stator slot-toothgenerate ripple in the air-gap flux density according to the size of the slot opening.Furthermore, the coils are spatially displaced resulting in permeance variations whichis a function of the space angle. The time and space harmonics cause pulsation ofthe air-gap flux which induces eddy current losses on the rotor poles, and especiallyon the damper bars (Ladjavardi et al. 2006). Additionally, the harmonics can beinduced by the unbalanced phase currents or constructional eccentricity of the rotor(Bruzzese & Joksimovic 2011, Ranlöf & Lundin 2010).

2.3.2 Iron losses

Previous section presented the resistive losses in the machine which occur whencurrent flows in the stator copper, and the additional losses due to time-varyingcurrents and magnetic fields. This section presents the magnetic properties of iron,and the origin of losses in stator core, and rotor poles. Losses in the electrical steelare called iron losses or core losses.

Most of the iron losses are generated in stator core and on the rotor pole surfacewhere the ferromagnetic iron core experiences time-varying magnetic field. The totaliron losses can be divided into hysteresis loss, eddy current loss and excess loss

Pfe = Phy + Ped + Pex. (19)

Hysteresis loss is the power consumed by the magnetic domains during thechange of the magnetic orientation, whereas the classical eddy current losses arerisen because of time-varying magnetic fields induce currents in the iron. Excesslosses are considered to be caused by local, microscopic eddy currents around the

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moving magnetic domain walls (Boon & Robey 1968). The iron losses are related tothe magnetic domains, and domain walls separating the domains. The ferromagneticsteel sheets consist of several magnetic domains, where the magnetic moment pointsin different directions as seen in Figure 3. Between the domains, inside the wall area,the direction of the magnetization changes gradually to match the direction next toit. This means that the domain walls are not actual walls, but a small area wherethe direction of the magnetization changes (Pyrhonen et al. 2013).

Figure 3: The movement domain walls under external magnetic field. In (a) noexternal field is present. In (b) (c) and (d ) H increases and the Weiss domains alignwith the magnetic field. (Pyrhonen et al. 2013).

Hysteresis lossesHysteresis losses are generated when a time-varying magnetic field is applied to aferromagnetic material, and the direction of magnetization of the magnetic domainchanges. The ferromagnetic material consists of magnetic domains which increaseor decrease in size and direction depending on the direction of the external fieldH. The movement of magnetic domains is well illustrated in Figure 3 However, themovement of the domain walls is not continuous, but it depends on the impurities ordefects in the material around the walls. The magnetization direction jumps betweenlocal energy minimums leading to nonlinear relation of B and H and causing lossesin the material. The energy consumed in the domain during the change of themagnetization direction is the hysteresis loss. Often the B − H relation is describedby a hysteresis loop, which shows the relation of magnetic flux density and magneticfield strength. The hysteresis loop also shows the saturation of the magnetic fluxdensity, and the amount of remanence magnetism in the material when the magneticfield strength is reduced to zero. An example of a hysteresis loop can be seen inFigure 4.

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Figure 4: An example hysteresis loop. When increasing H the flux density B followsa non-linear curve. When H is decreased back to zero, a remanence flux is presentin the material (Sen 2007).

Eddy current lossesThe time-varying magnetic field of the machine induces eddy currents in the steelsheets of the stator. Generally, these so called classical eddy currents are generatedsimilarly as the resistive losses: The changing magnetic field induces a current inthe material as seen from the Equations 2 and 3. When the flux density rapidlychanges, a voltage is induced around it’s path. The material resistivity causes powerloss according to I2R.

The classical eddy current model assumes uniform magnetic flux density in thematerial as shown in Figure 5 (a). However, the microscopic structure of the materialcause additional eddy current losses, also known as the excess loss.

Excess lossesExcess losses can be considered to be any losses which are not explained by theeddy current, or hysteresis losses. (Roshen 2007) presents that the excess losses areactually microscopic eddy current losses induced around the moving domain wallsduring the magnetization of the material. The domain walls make up only a smallfraction of the total area of the material, which means that the magnetization insidethe wall area has to be much larger than the average magnetization of the material.Therefore, the eddy currents around the walls have to be also larger than the classicaleddy current model with uniform magnetization suggests. Figure 5 (b) shows theinduced microscopic eddy currents.

The iron losses in this section were presented as separate components to showthe main mechanisms of the loss components. However, in reality the iron lossesare interconnected by the electromagnetic equations so that the amount of losses

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Figure 5: Figure (a) shows the classical eddy current distribution in steel sheetthickness d. Figure (b) shows the effect of the domain walls to eddy currents (Boon& Robey 1968).

also affect the field solution. Additionally, the eddy currents and hysteresis lossesare coupled through the material properties non-linear material properties. Theeffects of iron losses on the field solution were studied by Dlala et al. (2010) and itwas found that including the iron losses in the field solution affect the total losses.Several methods for solving the field equations are presented extensively by Pippuri(2010) and Rasilo (2012). Often, the three dimensional equations can be simplifiedinto a 1-D problem by assuming that the magnetic field strength and flux densitypenetrate the steel sheets in parallel direction, leading to solution which is dependenton thickness of the sheet. Additionally, the magnetic flux density can be assumed tobe uniform around the thickness of the lamination.

Experimental iron loss modelsA method for iron loss estimation, as shown by (Steinmetz 1892), is to use experi-mental data and the information of frequency and flux density of the material:

pfe = CfαBβmax (20)

Where the coefficients C,α and β are determined by fitting the model intomeasurement data and Bmax is the maximum flux density. The losses are calculatedby frequency components which are then added together in order to obtain the totallosses. Additionally, the Steinmetz equation can be extended to include the hysteresislosses and eddy current losses. The Jordan extension assumes that hysteresis lossesare proportional to hysteresis loop at low frequencies (Krings & Soulard 2010). TheSteinmetz-Jordan equation can be written as:

22

Pfe = ChyfB2max + Cclf

2B2max (21)

However, this approach still neglects the excess loss component. Therefore,method including excess losses is presented by (Bertotti et al. 1988), where thetotal losses are statistically segregated. The experimental coefficients require asingle measurement of hysteresis loss and excess loss which can be determined frommeasuring total losses. With the measurement data the losses can be scaled to anyflux density and any frequency. In this model, the classical eddy current loss is theloss of an ideal material, with no domain structure and uniform magnetization. Theclassical eddy current loss can be estimated when the lamination thickness d andconductivity σ are known. For all of the losses the maximum flux density value Bmaxis needed, which can be computed or estimated. In all of the following three casesthe c refers to a empirical coefficient which is determined by measurements data.

Ped = π2σ(d)2

6 (Bmaxf)2 (22)

The measured hysteresis loss is used to determine coercivity of the material. Thisway the hysteresis loss per cycle chy can be determined

Phy = chyfB2max (23)

The excess loss coefficient cex is determined by parameters such as the mobilitycoefficient, and global microstructural properties of the materials which can bemeasured.

Pex = cex(fBmax)1,5 (24)

This empirical approach does not require heavy computing processes. With a setof measurement data, and the estimation of the magnetic flux densities, the lossescan be determined with relatively good accuracy.

Using the above approach, it’s assumed the magnetic field is unidirectional andpurely sinusoidal, and that the flux density is uniform along the thickness of thelamination. However, in reality, the flux density inside the material is nonuniformcausing the eddy current term in Equation 22 to be dependent on the B-H relation.Likewise, the hysteresis and excess losses are affected by the skin effect, which is notincluded in Equations 23 and 24. Additionally, the influence of losses on the fieldsolution are left out, causing some error in the loss estimation (Dlala et al. 2010).Moreover, the statistical iron loss equations presented here do not take into accountthe losses due to rotational flux component, or the losses in the end shields and theframe caused by flux leakages in the machine which affect especially the stator teethand yoke (Pippuri 2010).

23

A comparison of several iron loss models has been made by Krings & Soulard(2010), where it was shown that the Steinmetz model and the loss separation modelare suitable for rough loss estimation and comparison between materials. Themethods are simple to implement in FEM, where the maximum flux densities aresolved in each element, and the result can be used to determine the losses in post-processing. Additionally, little information is needed of the iron material prior to theloss estimation.

2.4 Torsional vibrationsAny shaft or drive train is subject to torsional vibrations which cause mechanicalstress and electrical losses in generators. Torsional vibrations involve rotating masses,which are connected to each other by a shaft or coupling with a finite stiffness. Whenthe torque applied to the drive train is not perfectly steady, it will cause torsionaltwisting along the shaft. The torsional vibration is especially harmful when thevibration frequency coincides with the natural frequency of the system leading tohigh vibration amplitudes and mechanical stress. The natural frequencies of thedrive train of an SPSG is determined by the coupling between engine and generator,the electromagnetic effects and the inertias of the whole system. The drive trainwith one coupling can be described as a two mass system as illustrated in Figure6. The system includes two springs and two dampers, leading to a system with twonatural frequencies.

Figure 6: A two-mass model for engine and a generator including the electromagneticeffects. dm and km are the spring and damping constants of the rotor. Where dc andkc are for coupling. Jc and Jm are the rotor and engine inertias. Picture modifiedfrom (E. G. Hauptmann 2013).

The first, purely mechanical, sprig-damper system is the coupling, and the secondis generated by the electromagnetic effect in the air-gap. The electromagnetic springeffect is generated by torsional vibration which generates additional currents in therotor bars, which consequently produce torque towards the rotor (E. G. Hauptmann2013). The torque acts as a mechanical springs and damper, which define a naturalfrequency for the system together with the rest of the drive train. The magneticdamping and spring constants can be determined numerically with an impulse

24

simulation (Repo et al. 2008). The effects of low frequency torque oscillations andthe electromagnetic effect on the losses of several electrical machines were studied byArkkio et al. (2018a). The authors show that at low torque oscillation frequenciesthe mechanical damping is negligible, which may lead to large torsional vibrationamplitudes. Therefore, a low frequency torque pulsation may cause significantadditional losses, especially if the oscillation frequency is close to the first naturalfrequency of the system. For large machines the first natural frequency is usuallyvery low, some Hertz, and for smaller machines some tens of Hertz.

The torsional excitaion can be generated by an ICE, the generator, or fromthe electrical network. In an ICE the firing process in a single cylinder does notonly produce a torque peak, but also requires torque to return back to the firingposition. Therefore, torque oscillations are present in normal operation of an ICE.The fundamental frequency of the torque oscillation depends on the type and numberof the cylinders, and the speed of the engine. For example, in a four-stroke, 18cylinder and 750 rpm engine each cylinder fired once per two revolutions, leading toa fundamental oscillation component of 6.25 Hz. The low oscillation frequency maycoincide with the first natural frequency of a large generator (Arkkio et al. 2018a).The torsional vibrations induce current harmonics at two frequencies:

f = fs ± fv, (25)

where fs is the supply frequency and fv is the frequency of the oscillating torque.Higher oscillation amplitude leads to higher harmonics, thereby increasing losses. Dueto their higher frequency, the harmonic currents are influenced more by the skin effect,decreasing the effective area of the conductor and increasing losses. Additionally, theeddy current losses of the conductor are increased at higher frequencies.

Several methods for controlling torsional vibrations have been studied. In Arkkioet al. (2018b) the authors managed to reduce the losses of induction motors andsynchronous motors under torsional vibrations by injecting harmonic currents to thestator winding using a power converter. The method decreased the additional lossesgenerated by the torsional vibration.

2.5 Voltage unbalancePower systems may experience voltage unbalances due to unbalanced loads, largesingle phase loads, or blown fuses. Electrical machines may also generate unbalanceif the number of winding turns is not equal, or if the rotor is misaligned (von Jouanne& Banerjee 2001). The unbalanced voltages give rise to several problems in electricalmotors and generators. Even a minor unbalance in voltages may cause severaltimes higher unbalance in the currents leading to additional losses and heating.Furthermore, the unbalanced conditions induce counter rotating field in the air-gap,creating pulsations in the air-gap flux, generating additional losses on the rotor polesurfaces and damper bars (Hsu 1997, Sahu et al. 2017). There are several possibleunbalance conditions such as: Under, or overvoltage in one or two phases, phase-angledisplacement, or any combination of the mentioned.

25

A three phase system can be presented by symmetrical components, wherethree phasors represent the phases of the system. In a perfectly balanced system,as illustrated in Figure 7, the magnitudes of the voltages are sinusoidal, equal inmagnitude and the phases are 120 apart. An unbalanced system can include unequalvoltage magnitudes, shifted phase angles or distortion between phases. A method foranalyzing unbalance in system is to divide the components into zero sequence U0,positive sequence U1 and negative sequence U2 voltages:

Figure 7: Symmetrical components of an balanced three phase system. The magni-tudes are equal and the displacement between the phasors is 120

⎡⎢⎣U0U1U2

⎤⎥⎦ = 13

⎡⎢⎣1 1 11 a a2

1 a2 a

⎤⎥⎦⎡⎢⎣Ua

Ub

Uc

⎤⎥⎦ , (26)

(27)

Ua,Ub and Uc are the stator phase voltages. And,

a = ej 2π3 (28)

Using the above equations, any three phase system can be divided into positive,negative and zero sequence components. In electrical machines the zero sequencecomponent can be neglected, as normally there is no neutral path for zero sequencecurrent. Therefore, an unbalanced system contains positive and negative sequencecomponents (von Jouanne & Banerjee 2001). The positive sequence componentrepresents the three equal phasors which are displaced by 120 and with the samephase sequence as the original phasors. A perfectly balanced system only has apositive sequence component which generates positive torque.

26

A negative sequence current will flow in a unbalanced system. The phase order ofthe negative sequence component being opposite of the positive sequence component,creating counter-rotating flux in the air-gap. The generated flux rotates againstthe opposite direction as the positive flux, reducing the speed and torque output.Additionally, the negative sequence voltage can generate a large negative sequencecurrent because the negative sequence impedance is small (von Jouanne & Banerjee2001).

The positive and negative fluxes rotate at the synchronous frequency in oppositedirections. The positive and negative sequence fluxes add constructively and destruc-tively twice per revolution, leading to twice-line-frequency air-gap flux pulsationsand currents on the rotor pole surfaces. These currents induce additional losses, andthe effect is seen especially on the rotor bars, which have good conductivity whencompared to the surrounding rotor core material. Additionally, the negative sequencecomponent generates additional losses on the stator winding. As seen from theEquation 16, the total stator winding losses are increased because of the unbalancebetween phases.In extreme case, when one phase carries the nominal current, thelosses are three times higher compared to perfectly balanced case (Sahu et al. 2017).

The effects of negative sequence voltage on the core losses of an induction machinewas studied by Daniel Donolo et al. (2016) and it was found that the losses caused bythe negative sequence component are negligible when compared to the losses causedby the positive sequence component. Additionally, similar studies for synchronousgenerators under current unbalance show marginal increase in excess losses. The ironlosses are not affected since the induction of the stator lamination and the frequencyare not affected by the unbalance (Debruyne et al. 2017). Therefore, the iron lossescan be initially assumed to be constant.

A maximum negative sequence current has been defined by IEC in order to limitthe unwanted effects in rotating electrical machines. According to IEC 60034-1 themaximum continuous negative sequence current for an indirect cooled SPSG mustbe under 8% of the nominal, and 5% for direct cooled SPSGs. The same standarddefines a maximum negative sequence current in fault conditions as:

( I2

In

)2t (29)

Where t is the fault duration. For Directly cooled SPSGs the maximum value is15 seconds, and for indirectly cooled the value is 20 seconds.

A more general definition for voltage unbalance levels can be found in IEC Elec-tromagnetic compatibility standard 61000-2-2, which defines a maximum unbalanceas the ratio between the positive sequence voltage and negative sequence voltageU2U1

to be 2%. However, generally the terminal voltage of synchronous generators isadjusted by the AVR, meaning that for SPSGs the applicability of the of voltageunbalance may not be as reasonable as the current unbalance.

In Brekken & Mohan (2007) the authors studied methods for reducing the air-gap flux pulsation of a doubly-fed wind generator by injecting a negative sequencecomponent to the rotor current. It was found that the compensating negative

27

sequence current can, not only reduce the unwanted torque and power pulsationsbut also compensate the current unbalances and reduce mechanical wear.

2.6 Temperature riseThe winding temperature of an SPSG is an important factor when considering thelosses and the thermal stress of the insulation. The resistive losses in the stator andthe rotor make up a significant part of the losses of an SPSG, and the losses increaseapproximately linearly with temperature. Additionally, high winding temperaturescause unwanted ageing of the insulation (Pyrhonen et al. 2013). Thermal classesfor electric insulators have been defined in IEC 60085. The standard defines themaximum continuous temperature of the conductors, including electrical machines.The total temperature of the machine is the combination of ambient temperatureand the temperature rise, which is defined by the power losses and cooling propertiesof the machine.

The thermal classes and their letter designations are illustrated in Table 1.

Table 1: Thermal classes according to IEC

Thermal class C Letter designation Reference temperature C90 Y -105 A -120 E -130 B 95155 F 115180 H 135200 N -220 R -

In SPSGs and other electrical machines the most common thermal classes areclass B and class F. Since the temperature may vary in different parts of the windingdepending on the cooling and structure of the machine, the thermal class temperatureis considered to be the highest temperature, or so called hot-spot, of the winding.Often, the hot-spot temperature is measured between the stator winding and statorcore, at the end winding area. The temperature of the hot-spot can be 10-20 Khigher than the average temperature Pyrhonen et al. (2013).

Additionally, IEC 60034-2 defines reference winding temperatures used for losscalculation for temperature classes B,F,H seen in Table 1. However, the real on-sitewinding temperatures may be significantly higher, thus generating resistive losses.The operation temperature of the winding is affected by the total losses and ambienttemperature. For example, the stator slots, the air-gap geometry and variablefrequency drives induce harmonic currents which increase the resistive losses and ironlosses (Fouladgar & Chauveau 2005). Additionally, the torsional vibrations discussedin Section 2.3.1 induce harmonic currents according to Equation 25. Furthermore,

28

the unbalanced line voltage can cause additional losses, leading to temperature riseand additional losses.

2.7 Chapter summarySalient pole synchronous generators are often used for energy production. Whengenerators are combined with ICE as a prime mover, they can be used in islandednetworks or as an emergency backup power source in remote locations. Section 2.1introduced the construction and basic operation principles of synchronous generators.Section 2.2 described the electromagnetic equations essential for numerical analysisused later in this thesis. Section 2.3 presented the electromagnetic losses of themachine. Sections 2.4, Section 2.5 and Section 2.6 discussed three cases of non-idealoperation and their origin. This information will be used late in Chapter 4 to calculatethe effects of these non-idealities on power losses.

29

3 MethodsThis chapter discusses the Finite element method which is widely used for solvingboundary value problems. In this thesis we are using a two-dimensional model inorder to solve the field equations and estimate the losses. Section 3.1. discusses thebasics of Finite element method used to estimate the solutions of Maxwells equations.Section 3.2. describes the operation of a 2-D FEM software FCSMEK used fornumerical computation.

3.1 Finite element methodElectrical machines work on the energy conversion through the electromagneticfield in the air gap. The energy conversion process is based on the solution orapproximation of the field equations. Traditionally, the field equations are solvedby using experimental data gathered over a long period of time, therefore allowingthe problem to be reduced into a simple form (Luomi 1993). However, nowadayswe are able to directly solve the field equations numerically by using computationalmethods. This section covers the basics of Finite Element Method (FEM) which isused to solve the field equations for 2-D radial flux machines.

In finite element method a large problem area is divided into several smallerareas, or elements, and a function is used to approximate the solution within eachelement. Section 3.1.1. presents the mesh and the concept of elements in whichthe field equations are estimated by shape functions. Section 3.1.2 discusses themagnetic vector potential which is used for simplifying the field equations. Section3.1.3. presents the discretization of the partial differential equations which is done inorder to solve the equations numerically.

3.1.1 Magnetic vector potential

The magnetic vector potential is used for reducing the vector problem into a scalar,and the method widely used for solving two-dimensional magnetic fields (Luomi1993). The definition of magnetic vector potential A

B = ∇ × A (30)

Assuming that the conductors in the problem area have a good conductivity, andthe displacement current is small leads to ∂D

∂t= 0. Combining the Equations 8 and 3

with the magnetic vector potential, and placing reluctivity µ = 1/ν leads to:

∇ × (ν∇ × A) = J (31)

Which can be formulated in 2-D into:

∇ · (ν∇Az) = −J (32)

30

The magnetic vector potential is solved by using a polynomial as shown in thenext section.

3.1.2 Elements and boundary conditions

Previous section presented the concept of magnetic vector potential which allow us toestimate the current densities and flux densities in electrical machines. This sectionpresents methods to estimate the magnetic vector potential by using shape functionsin small elements inside the problem domain.

In FEM process the problem area is divided into small elements, in which themagnetic vector potential is estimated. The process of dividing the area into elementsis called meshing, and the area with several elements is called a mesh. In 2-D, themesh is a two-dimensional presentation of the electrical machine in radial directionwhere the third axis, the length of the machine, is assumed to be infinite. However, asdiscussed in section 2.2, the end winding effects should be include in loss estimation.Often in 2-D FEM, the end winding inductance is used to obtain the end windinglosses. A single triangular finite element in a global plane is illustrated in Figure 8.The corners of the elements are called nodes, and a single element can share a nodewith other elements.

A shape function is a continuous function which is defined over a single element.Shape functions allow us to express the estimate of the unknown function as a set ofunknown scalar coefficients. Ultimately the shape functions are used to find the bestpossible solution for the unknown vector potential Equation 33. The shape functionsof the whole problem area form a global matrix to which the boundary conditionsand current sources are applied for solving the equations.

Figure 8: A triangular element. xi and yi are the coordinates of the nodes in themesh

After the mesh of the problem area is created magnetic vector potential is estimatedin each element by polynomial leading to

A(x) ≈n∑

j=1Nj(x, y)aj, (33)

31

where Nj is the global shape function which has a non-zero value in the elementsin which node i belongs. aj is the nodal value of nodal point j and n is the totalnumber of nodes. The necessary boundary conditions are applied, and the equationsare solved to obtain the result.

3.1.3 Time-stepping analysis

Within the created mesh we are able to solve the magnetic vector potential in eachelement with the aid of the shape functions. However, additional methods are neededfor taking the rotation of the rotor and induced currents into account during thesimulation. The rotor motion induces harmonics in the air-gap, which consequentlyaffect the air-gap torque and losses of the machine. Therefore, to take the effectsinto account in FEM simulations, time-stepping analysis (TSA) is used. In TSAthe rotor motion is simulated by modifying the mesh in the air-gap in small stepscorresponding the angular speed of the machine as presented by Arkkio (1987). Thetime-dependence of the field equations is solved by discretization of the equationsusing the Crank-Nicholson method:

Ak+1 = 12

[∂A

∂t|k+1 + ∂A

∂t|k

]∆t + Ak (34)

In order to evaluate the time dependence of the field, the vector potentials areestimated at each time step. Each period of line frequency must be divided intoseveral hundreds of time steps in order to take the slot-effect into account (Arkkio1987).

A potential difference is induced in the winding of a electrical machine by the linevoltage, which also generates the magnetic field. By coupling the circuit equationsand field equations, the magnetic vector potential, voltages and currents in the statorand rotor are coupled. The line voltage or current is used as the source term forsolving the magnetic vector potential. Voltage equations for rotor and stator areused in order to include the source terms in FEM (Arkkio 1987). The current densityin a coil can be obtained from the equation

J = nl

S(35)

Where n is the number of turns and S is the cross-sectional area of the coil.Equation 35 combined with the Equation 32 gives the magnetic vector potential inthe coil side.

3.2 Simulation in FCSMEKPrevious section described the FEM used to numerically estimate the solution ofelectromagnetic equations in the generator. This section presents FCSMEK software,which is used for implementing the FEM for analysis of radial flux electrical machines.

32

FCSMEK is a collection of programs designed for 2-D finite element analysis ofelectrical machines. The simulation process is illustrated in Figure 9.

The machine parameters and material constants are given as an input for thesoftware, which then creates a geometry of the machine based on these parameters.After the geometry is defined, it is divided into finite elements with a routine called"Mesh". All the materials in the geometry are described in terms of permeability andconductivity. The basic electromagnetic equations shown in Section 3.1 are solvedtogether with the circuit equations of the stator and rotor. The non linearity of therotor and stator materials are defined by their B-H curves. The B-H curve of thestator material is illustrated in Figure 10 and the rotor material in Figure 11 (Arkkio1987).

0 5 10 15

H [A/m] ×104

0

0.5

1

1.5

2

2.5

B [

T]

Figure 10: B-H curve of stator iron

33

0 5 10 15

H [A/m] ×104

0

0.5

1

1.5

2

2.5

B [

T]

Figure 11: B-H curve of rotor pole material

The equations are discretized as shown in Equation 34 in order to perform theTSA. The SYDC routine is used for computing the initial values of synchronousmachines for TSA. The non-linear equations obtained from FEM are solved usingNewton-Raphson method. With the initial values TSA starts from steady stateinstead of zero field, thus saving computation time. The motion of the rotor ismodeled by modifying the mesh in the air-gap at each time-step (Arkkio 1987).

The steel sheets are assumed as non-conducting material, so that the solutionof magnetic flux density is not affected by the conductivity of iron. Finite elementmethod is used to solve the magnetic flux density in the nodes at each time step,and the magnetic flux density is then used to solve the losses in post processing. Theiron losses are calculated in post-processing by using the Steinmetz-Jordan modelpresented in Chapter 2.

The resistive losses include the stator DC loss and AC loss of the winding. In therotor, the resistive losses include the damper bar losses and the field winding losses. NoAC losses are generated in the field winding since the magnetizing current is DC. Thetemperature dependence of the conductivity of the materials is included as discussedin Section 2.2. The absolute rotor and stator temperatures can be defined separatelyin FCSMEK initialization file. The stator temperature influences the conductivity ofthe stator winding, while the rotor temperature influences the conductivity of thefield winding and the damper bars. Because of the 2-D representation used for FEanalysis, the end winding effects are not automatically included. Therefore, The endwinding effects are taken into account by estimating the end winding inductance

34

and using it as a parameter of the stator circuit equation as illustrated in Figure 12.Additionally, the rotor ends are not included in the 2-D FEA. Therefore, the rotorcircuit equations are used in order to include the impedance of the rotor bars, andthe connection rings used to short-circuit the rotor bars in the rotor end region.

3.3 Chapter summaryThe 2-D finite element method presented in this chapter is a powerful tool foranalyzing 2-D radial flux machines. The problem area is be divided into smallelements, where the magnetic vector potential is solved. Section 2.1 presented thebasics of FEM. Section 2.2 introduced the FE process of the in-house softwareFCSMEK, used for simulations in Chapter 4.

35

Figure 9: Flowchart of FCSMEK calculation process

36

Figure 12: The coupling of FEM to the machine circuit equations

37

4 Results and discussionThis chapter presents the machine parameters, simulation parameters and resultsobtained from FEM. Section 4.1 introduces the basic machine parameters used forthe simulations. Section 4.2 observes the losses in the nominal operation point. Theresults obtained in the subsequent sections are compared to the results obtainedfrom nominal point simulation. Section 4.3 presents the non-ideal operation cases,and the losses obtained.

4.1 Machine parametersThis thesis studies the effects of non-ideal operation conditions on the loss componentsof a salient-pole synchronous generator. A 2-D model of the machine is used inFEM. The studied machine is a three phase, 50-Hz, 11-kV and 12-MVA SPSG witha star connection. The cross-section of the machine is illustrated in Figure 13. Theparameters of the machine are listed in Table 2.

Figure 13: Cross section of the machine

The stator and rotor core are assumed to have zero electrical conductivity, andthe core losses are solved during post-processing. The stator winding and the damperbars are modeled as material with a conductivity of 57 × 106 S

mat 20C. The machine

has a two-layer winding in the stator.Initially, the MESH program is used to generate a FE mesh of the 2-D geometry of

the generator. The periodic boundary condition is used to find the smallest possiblegeometry in order to decrease the total computation time. The mesh consisted of4578 elements and 2804 nodes. The mesh is illustrated in Figure 4.1. After generatingthe mesh, the SYDC program was used to obtain the initial values as shown in Table3.

38

Table 2: Machine parametersParameter ValueRated voltage [kV] 11Rated Current [A] 638.4Rated power [MVA] 12Frequency [Hz] 50Number of poles 8Speed [RPM] 750Number of stator slots 108Nominal power factor 0.8Rotor bars per pole 7IEC Thermal class BStator winding phase DC resistance at 20 C [Ω] 31.3 × 10−3

Stator end winding phase reactance [Ω] 463.1 × 10−3

Figure 14: Finite element mesh of the smallest symmetry section of the machine atno load conditions

The generated mesh and the results from SYDC were used for TSA in order toobtain the characteristics such as losses, current and voltage waveforms and air-gaptorque. In order to reach steady state operation during the simulation, a sufficientnumber of supply periods is chosen. Number of time steps per period is chosen to be300 in order to include the slot-effect. Because of the high number of simulations run,first order finite elements are used to decrease the total computation time. By default,the stator and rotor temperatures during the simulations are set to 20C. Becausethe mechanical losses are assumed constant in all simulations, only electromagneticlosses are discussed.

39

Table 3: Results obtained from SYDC

Parameter ValueTerminal voltage [kV] 11Terminal Current [A] 643Power factor 0.8Air-gap torque [kN] 125.4Apparent power [kVa] 12251Shaft power [kW] -9853Rotor voltage [V] 198.6

4.2 Nominal pointThe nominal point simulation assumes balanced sinusoidal phase voltages and constantspeed. TSA was run for 35 supply periods. The resulting current and torquewaveforms are illustrated in Figure 16. As shown in the figure, the induced phasecurrents in the stator winding are not completely sinusoidal because of the shapeof the rotor poles and slot-tooth effect. Additionally, the sixth harmonic in theair-gap torque can be seen in the torque waveform and FFT. Figure 15 illustratesthe magnetic flux density distribution throughout the smallest symmetry sectionof the machine. It can be seen that the flux density is highest near the side of therotor pole. This occurs because of rotor magnetic field drags the stator field. Thedifference is the electrical angle. The losses obtained from the TSA in nominal pointoperation are illustrated in Figure 17. The figure shows that the iron losses generatedin the stator core account for 30% of the total losses, whereas the resistive losses instator and rotor winding make up 49% of the total losses.

40

Figure 15: Flux contours and flux density across the symmetry section in nominalpoint

41

0.65 0.66 0.67 0.68 0.69 0.7

time [s]

-20

-10

0

10

20

Am

plit

ude [kV

]

Phase voltages

Ua

Ub

Uc

0.65 0.66 0.67 0.68 0.69 0.7

time [s]

-1000

-500

0

500

1000

Am

plit

ude [A

]

Phase currents

Ia

Ib

Ic

200 400 600 800

Frequency [Hz]

0

1

2

3

4

Am

plit

ude [kN

m]

Dynamic torque FFT

0.65 0.66 0.67 0.68 0.69 0.7

Time [s]

-135

-130

-125

-120

Am

plit

ude [kN

m]

Torque

Figure 16: Balanced operation with constant speed

St. resistive Rotor damper Rotor core Stator core Field winding res.0

5

10

15

20

25

30

35

% o

f to

tal lo

ss

Losses in nominal point

Figure 17: Loss components in nominal point operation

4.3 Non-ideal operationIn this section, the non-ideal operation conditions discussed in Chapter 2 are imple-mented into FEM. The simulated cases include grid unbalance, torque oscillationsand temperature rise. Unbalance and temperature rise parameters are chosen asdescribed by IEC standards in order to obtain realistic results. However, for tor-sional vibration, no limiting standard is used. Instead, a torque profile of an ICE isexamined, and used for TSA.

42

4.3.1 Unbalance

This section discusses the effects of voltage and current unbalances on the studiedSPSG. The simulations were conducted by using voltage or current waveforms asFCSMEK input and running the TSA. The rotor speed is assumed to be constant.Two separate cases of unbalance were studied. First, the voltage unbalance varyingbetween -4%-4% in one phase. Second case was the current unbalance with -8% to8% unbalance in one phase. In both cases, the unbalance was incremented by 1%between the simulations. The current unbalance is chosen to be U2 < 8% accordingto IEC 60034-1 for SPSGs. In both cases, no zero sequence component is presentand the percentage of unbalance is the ratio of the negative sequence component (I2)divided by the nominal current, or nominal voltage.

Voltage unbalanceVoltage unbalance simulations were run for 35 supply periods to limit the effects oftransients on the results. Each period was simulated for 300 time-steps. Figure 18demonstrates the effects of negative sequence voltage on the air-gap torque and thecurrent waveform. In the example case, a 2% voltage unbalance induces approximatelya 30% unbalance in the current. As a result the current density in the rotor baris increased by 14%. Additionally, the twice-the-line frequency torque pulsationgenerated by the negative sequence flux interfering with the positive sequence fluxcan be seen in the air-gap torque FFT. The slot-effect is seen in the torque FFT at300Hz. For clarity, the 0Hz torque component is not shown in the FFT.

The losses obtained from the TSA are presented in Figure 20. The rotor speed isassumed constant, and the losses are compared to the simulation without unbalance.As shown, stator resistive losses increased by 5%. Additional stator resistive lossesare generated by the higher current flow in one of the phases. The figure shows thatthe negative sequence component significantly increases rotor damper losses. At 4%unbalance, the rotor damper losses and rotor core losses increase by 122% and 3.4%,respectively. The increase can be attributed to the counter rotating flux induced bythe negative sequence voltage. The pulsation in the air-gap flux generates losses,most of which occur in the rotor bars because of their good conductivity compared tothe rotor poles. Furthermore, the rotor damper winding is short circuited, whereasthe rotor core is made of laminated steel sheets, thus decreasing the eddy currentlosses. The rotor core losses are mostly generated on the pole surface, as shown inFigure 19.

The increase in stator core losses and field winding losses are small

43

0.65 0.66 0.67 0.68 0.69

time [s]

-20

-10

0

10

20

Am

plit

ude [kV

]

Phase voltages

Ua

Ub

Uc

0.65 0.66 0.67 0.68 0.69

time [s]

-1000

-500

0

500

1000

Am

plit

ude [A

]

Phase currents

Ia

Ib

Ic

200 400 600 800

Frequency [Hz]

0

2

4

6

8

Am

plit

ude [kN

m]

Dynamic torque FFT

0.65 0.66 0.67 0.68 0.69

Time [s]

-160

-140

-120

-100

Am

plit

ude [kN

m]

Torque

Figure 18: Results with 2% negative sequence voltage. The current waveform issignificantly affected by the voltage unbalance. The air-gap torque FFT clearly showstwice-the-line frequency torque pulsation caused by the negative sequence voltage.

-4 -2 0 2 4

I2/In [%]

-1

0

1

2

3

4

5

6

7

8

Incre

ase

in

lo

sse

s %

0

20

40

60

80

100

120

140

Incre

ase

in

lo

sse

s %

Losses: Unbalance

st. resistive

rotor core

field winding

st. core

total

rt. damper

Figure 20: The effect of negative sequence component on losses. The right axis showsthe rotor damper winding losses

44

Figure 19: The pole surface losses generated by a negative sequence voltage

The losses affected by the voltage unbalance are presented in Table 4. The effectof unbalance on field winding loss is small. Therefore, the field winding loss is notpresented in the table.

Table 4: Increase in losses compared to nominal point [%]

U2 St. resistive Rt. bar Rt. core Total-4 4.9 122.8 3.4 7.2-3 2.7 69.0 2.0 4.1-2 1.2 30.1 1.0 1.8-1 0.3 7.7 0.3 0.40 0 0 0 01 0.3 7.7 0.01 0.42 1.2 30.1 0.4 1.73 2.7 69.0 1.0 3.94 4.9 122.8 2.0 7.0

The results show that even a small negative sequence voltage can cause significantadditional losses in the machine, thus decreasing the efficiency. The negative sequence

45

component increase losses in all parts of the machine. The most significant increaseis seen in the damper winding.

Current unbalanceIn this section a negative sequence current was introduced into TSA in order tosimulate the effects of current unbalance on the machine. Because the source is setto a current source, the voltage is calculated by FCSMEK. The TSA was run forfour supply periods with each period consisting of 300 steps. The current unbalancesimulations had less transients, thus reducing the total periods needed for simulations.The voltage and torque waveforms obtained from the TSA are illustrated in Figure21.

The amplitude of the twice-the-line frequency component in the air-gap torquecan be seen in the results. Unlike the voltage unbalance case, the torque waveformis distorted because the current waveform is not perfectly aligned with the initialvalues used for the TSA. The losses obtained using TSA in Figure 22. The losses arecompared to the case without any unbalance. Overall, the effect of current unbalanceon losses is less than that in voltage unbalance case. The losses in the rotor bars aregenerated by air-gap flux pulsations, which induce currents in the rotor bars.

0.1 0.11 0.12

time [s]

-20

0

20

Am

plit

ud

e [

kV

]

Phase voltages

Ua

Ub

Uc

0.1 0.11 0.12

time [s]

-1000

0

1000

Am

plit

ud

e [

A]

Phase currents

Ia

Ib

Ic

500 1000 1500 2000

Frequency

0

2

4

Am

plit

ud

e [

kN

m]

Dynamic torque FFT

0.1 0.11 0.12

Time [s]

-140

-130

-120

-110

Am

plit

ud

e [

kN

m]

Torque

Figure 21: The effect of negative sequence current on voltage and torque waveforms

46

-8 -6 -4 -2 0 2 4 6 8

I2/In [%]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0

1

2

3

4

5

6

Incre

ase

in

lo

sse

s %

Losses: Unbalance

st. resistive

rt. bar

rotor core

field winding

st. core

total

Figure 22: The effect of negative sequence current on losses. The right axis showsrotor damper winding losses

On the simulation region, the rotor bar losses increase up to 5.5%. The increasein rotor core losses, stator resistive losses and stator core losses is small. The totallosses increase 0.4% at 8% unbalance.

4.3.2 Torque oscillations

This section presents the results obtained from FE simulations with oscillating loadtorque. First, different frequencies of torque oscillation are used for TSA to determinethe first natural frequency of the SPSG. When the natural frequency is known, atorque profile of an ICE is studied in order to find a harmonic torque componentnear of the natural frequency. The extracted component is then used as input torquein TSA to determine the effects of low frequency torque oscillation occurring nearthe natural frequency, on the losses of the SPSG. This method allows us to simulatethe effects of a real ICE on the power losses of the machine. The simulations assumethat most of the losses occur at low frequency torque oscillations, and that the effectof high frequency oscillations on the losses are minor.

The simulations for determining the natural frequency are done by introducing asinusoidal, high amplitude torque for the TSA. In order to see the effects of torqueoscillations on the SPSG, the speed of the rotor is solved using the torque as an inputvalue. As discussed in Section 2.3, the torsional vibration amplitude may grow largeif the torque oscillation frequency coincides with the natural frequency of the SPSG,thus increasing the losses. The frequency of torque oscillation is varied between 1-15Hz, as the natural frequency of large machines is relatively low. The torque oscillationfrequency was increased by 1 Hz per simulation. Ultimately, when searching for the

47

natural frequency, the amplitude of the torque oscillation is arbitrary since the lossesare only compared to find the natural frequency of the SPSG, and no conclusions ofthe actual losses are made. The results are illustrated in Figure 23.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

Lo

sse

s [

kW

]

Total loss

X: 4

Y: 370.1

Figure 23: Simulation results with various sinusoidal torque profiles

The peak of the losses suggest that the natural frequency of the machine is closeto 4 Hz. The results include a 0.5 Hz margin of error because of the 1 Hz resolution.The result is reasonable, as the natural frequency is generally excepted to be onlysome Hertz for large machines.

Next step was to extract the frequency components from a measured ICE torqueprofile. The ICE is a four-stroke engine operating at 750 rpm. At this rotationspeed the cylinders fire once in 0.16 seconds. This leads to fundamental frequencyof 6.25 Hz. The dynamic part of the torque for one rotation, and the harmoniccomponents of 6.25 Hz are illustrated in Figure 24. The fundamental componenthas the amplitude of 1.927 kN. The highest peak is at the 14th harmonic at 87,5 Hzhaving an amplitude of 22.9 kN. The fundamental component oscillating componentis 1.5% of the nominal torque

The fundamental 6.25 Hz component is added as sinusoidal oscillation to thestatic 125.4 kN torque. Additionally, to see the effects of small torsional oscillationat the natural frequency, the fundamental amplitude is set to oscillate at 4 Hz. Theprofiles were used as input torque for FCSMEK simulation and TSA was run for 50periods and 300 time-steps per period. The average losses are calculated over onecycle of torque oscillation, i.e. 0.16 seconds and the losses are compared to the lossesin nominal point. The results are illustrated in Figure 25. The results show that the

48

effect of the torque oscillation is small. The stator resistive losses increase by 0.57%and the rotor damper winding losses show an increase of 0.62%. The results show aincrease of 0.07% in the total losses.

0 10 20 30

Harmonic order

0

5

10

15

20

25

Am

plit

ude [kN

m]

Frequency

0 0.05 0.1 0.15

Time [s]

-80

-60

-40

-20

0

20

40

60

80

Torq

ue [kN

m]

Dynamic torque

Figure 24: The torque profile of a 20 cylinder ICE without the static component

49

St. resistive Rotor damper Rotor core Stator core Field winding Total-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Loss incre

ase [%

]

Losses under torque oscillations

Figure 25: Comparison of losses

4.3.3 Temperature rise

This section presents the effects of temperature rise on the losses of the studiedSPSG. This thesis only focuses on the effect of ambient temperature on the losses.Therefore, the effects are studied separately from the other non-ideal conditions. TheTSA was run for four periods of line frequency and 300 steps per period. The aimis to study the losses around the maximum operating temperature defined by IEC.Therefore, the simulation temperatures and resulting losses were compared to thethermal class reference temperature. The temperature of the stator and rotor areseparately defined for TSA. The temperature of the stator was assumed to be 16%higher than the temperature of the rotor at all temperatures. This estimation isbased on the tests made on similar machines. The simulation results are illustratedin Figure 26.

The resistive losses increase linearly as expected, and effect is highest on the statorwinding. The higher stator winding losses are explained by the higher temperature,and the higher current when compared to the rotor. The stator core materialis modeled as non-conductive, meaning that the core losses are not affected bythe temperature rise. The results are somewhat linear as excepted as a linearapproximation is used for the resistivity of copper. However, some non-linearity canbe seen especially in the field winding losses at 44%-46% and 48%-50%. The variationcan be explained by the rounding of the temperature values between simulations.

As shown in Figure 26, The total losses increase 5.8% when compared to the

50

38 40 42 44 46 48 50 52 54 56 58

Increase in temperature of reference [%]

0

2

4

6

8

10

12

14

Incre

ase in losses [%

]

St. resistive loss

Rotor bar loss

Field winding loss

St. core loss

Total loss

Figure 26: Losses compared to the losses in reference temperature

losses in reference temperature. The increase is approximately 0.11% per degree oftemperature rise. The simulations assumed that the stator and rotor temperaturesare constant throughout their geometry of the machine. As discussed in Section2.5, this would mean that the measured hot-spot temperature can be 10-20 degreeshigher than the average temperature used in the simulations.

The results show that the temperature rise increase losses of the machine. Whenthe machine is operated close to the thermal class maximum, the losses increase sig-nificantly when compared to the thermal class reference temperature, thus generatingthermal stress.

4.4 DiscussionUnbalance increase losses in stator windings and rotor bars. The 4% negative sequencecomponent increased power losses in stator winding by 4.9% and in the rotor barsby 122%. Total power loss increase was 7.2% The iron losses and stator core lossesare unaffected. The results are as expected. In Sahu et al. (2017) and von Jouanne& Banerjee (2001) the authors showed similar results in the stator losses. Accordingto Daniel Donolo et al. (2016) the iron losses were not expected to change undernegative sequence voltage. The results show that the negative sequence componentsaffect especially the rotor bars.

The effects of fundamental oscillation component of the ICE on the losses werefound to be insignificant. The amplitude of the oscillating torque component wasonly 1.6% of the nominal torque which is very small when compared to the studyconducted by Arkkio et al. (2018a) where the oscillating component was up to 50%

51

of the nominal torque. Unlike the unbalance, the torsional vibrations are not directlyrestricted by any standard. Traditionally, the electromagnetic spring and damperconstants are determined by an impulse test E. G. Hauptmann (2013). In thisthesis we determined the natural frequency by sweeping different torsional oscillationfrequencies using TSA, and finding a peak in electromagnetic losses.

The temperature rise increases losses linearly. The total losses increase approxi-mately 0.11% per degree. When approaching the thermal class temperature limit,the increase in total losses is 5.8%. The temperature rise is strictly limited by IEC60085. To achieve higher power output and less losses, the average temperature ofthe machine should be as low as possible.

52

5 Summary and future workThis thesis has determined the effects of torque oscillations and grid-induced unbal-ance on the losses of a salient pole synchronous generator. The effects of non-idealoperation conditions on the power losses of a SPSG were simulated using the finiteelement method. For the simulations, three non-ideal operation conditions were inves-tigated: current and voltage unbalance, torque oscillations and increased operatingtemperature.

Unbalance was simulated by adding a negative sequence component to thecurrent or voltage, and running TSA for obtaining the losses. The voltage unbalancesimulations included various negative series components to test the effects of single-phase over and undervoltage. The results obtained from these simulations were asexpected. A notable increase in losses was observed in rotor bars due to interferencecaused by the counter-rotating air-gap flux. Additionally, resistive losses in the statorwinding increased due to uneven current distribution. It was found that for thestudied SPSG, a 2% voltage unbalance induces approximately 30% of the currentunbalance. At 2% voltage unbalance, the increase in total losses was 1.8%. Generallythe voltage of an SPSG is adjusted by an AVR to reach balanced phase voltages.Therefore, the effects of current unbalance were also studied. The effect of currentunbalance on rotor bar and stator winding losses were similar as in the voltageunbalance case. The current unbalance simulation was run with forced sinusoidalcurrent waveform, thus meaning that the voltage waveform depends on the currentwaveform. This posed a problem in FCSMEK simulation as the simulations had aphase shift between the initial values and the input current, causing distortion inthe voltage waveform. A 8% current unbalance increased total losses by 0.5%.

The effects of torque oscillation on the losses were studied by introducing anoscillating torque to the TSA. First, the first natural frequency of the SPSGwas determined by running TSA with varying torque oscillation frequencies witharbitrary magnitude. The natural frequency of the machine was found to be closeto 4 Hz. Secondly, the torque profile of an ICE was examined in order to separatethe frequency components and their magnitudes. It was assumed that low frequencytorque oscillation induce power losses. Therefore, the fundamental torque componentof the studied ICE was used as input for the TSA in order to determine the powerlosses. The results showed that the effect of torque oscillation on losses was negligible.These results were not as expected, as it has been shown that, especially low-frequencytorque oscillations can induce additional power losses. Nevertheless, the results canbe explained by the low amplitude of the oscillating component, or by the oscillatingtorque frequency not being at exactly the natural frequency of the SPSG.

Temperature rise simulations were carried by increasing the rotor and statortemperatures. The losses in the conductors increased linearly as expected. When thetemperature was increased by 58%, the total losses rose by 5.8% compared to the IECreference temperature. The simulations included two major assumptions. Firstly,FCSMEK allows only the rotor and stator temperatures which are not constant, butactually vary across the geometry of the machine. Often, the hot-spot temperatureof the machine can be 10-20 degrees higher than the average value, thus indicating

53

that the losses may vary. Secondly, a constant temperature was assumed, thoughthe losses most likely would raise the machine temperature. Power losses increasethe temperature of machines, thereby increasing resistive losses.

Unbalance and torsional vibration produce air-gap flux pulsations which generateadditional losses and mechanical effects. To reduce these effects in a SPSG, controlmethods can be used for injecting negative sequence component current to the rotoror stator current.

Future work could investigate the effect of losses on machine temperature byinputting the losses into a thermal network to determine the actual temperaturerise. The work would provide valuable information of the relation between lossesand the temperature of the machine. Additionally, different harmonic components inthe torque profile of the ICE could be used in TSA for identifying further sources ofpower loss.

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