1466508248transformers_2

K14690 Cover 6/8/12 9:59 AM Page 1 C M Y CM MY CY CMY K

Recent catastrophic blackouts have exposed major vulnerabilities in the existing generation,transmission, and distribution systems of transformers widely used for energy transfer,measurement, protection, and signal coupling. As a result, the reliability of the entire powersystem is now uncertain, and many blame severe underinvestment, aging technology, and aconservative approach to innovation.

Composed of contributions from noted industry experts around the world, Transformers:Analysis, Design, and Measurement offers invaluable information to help designers andusers overcome these and other challenges associated with the design, construction,application, and analysis of transformers. This book is divided into three sections to addresscontemporary economic, design, diagnostic, and maintenance aspects associated with power,instrument, and high-frequency transformers.

Power Engineering

TOPICS COVERED INCLUDE

Design considerations

Capability to withstand short circuits

Insulation problems

Stray losses, screening, and local excessiveheating hazard

Shell type and superconductingtransformers

Links between design and maintenance

Component-related diagnostics andreliability

Economics of life-cycle cost, design review,and risk-management methods

Parameter measurement and prediction

This book is an essential tool for understanding and implementing solutions that will ensureimprovements in the development, maintenance, and life-cycle management of optimizedtransformers. This will lead to enhanced safety and reliability and lower costs for the electricalsupply. Illustrating the need for close cooperation between users and manufacturers oftransformers, this book outlines ways to achieve many crucial power objectives. Among these,the authors focus on the growing demand for transformer miniaturization, increased transmittedpower density, and use of advanced materials to meet the requirements of power materialsrunning under higher operational frequencies.

Suggesting ways to redirect resources and exploit new technologiessuch as computationalmodeling softwarethis book presents relatively inexpensive, simple, ready-to-implementstrategies to advance transformers, improve power system integrity, reduce environmentalimpact, and much more.

ANALYSIS, DESIGN, AND MEASUREMENT

EDITED BY

Xose M. Lpez-FernndezH. Blent Ertan Janusz Turowski

TRANSFORMERSLpez-Fernndez

ErtanTurowski

TRANSFORMERS


TRANSFORMERS

ANALYSIS, DESIGN, AND MEASUREMENTK14690

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an informa business


EDITED BY

Xose M. Lpez-FernndezH. Blent Ertan Janusz Turowski

TRANSFORMERS

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MATLAB is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This books use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB software.

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iii

Contents

Preface............................................................................................................................................. viiAcknowledgments..........................................................................................................................ixEditors...............................................................................................................................................xiContributors.................................................................................................................................. xiii

Part I Power TransformersMarceli Kazmierski and Janusz Turowski

1 Selected Problems of Transformers Capability to Withstand Short Circuits............3Wadysaw Pewca

2 Insulation Problems of HV Power Transformers........................................................... 31Andrey K. Lokhanin

3 Stray Losses, Screening, and Local Excessive Heating Hazard in Large Power Transformers..........................................................................................................................45Janusz Turowski

4 Superconducting Transformers.......................................................................................... 95Jan K. Sykulski

5 Challenges and Strategies in Transformer Design...................................................... 109S.V. Kulkarni

6 Large-Shell-Type Transformers: Aspects of Design.................................................... 139Xose M. Lpez-Fernndez

7 Transformer Design Review: A Link between Design and Maintenance Stages.........................................................................................................175Ryszard Sobocki

8 Life Management of Transformers.................................................................................. 191Marceli Kazmierski

9 Power Transformer Acceptance Tests.............................................................................. 215Ryszard Malewski

10 Functional and Component-Related Diagnostics for Power Transformers, a Basis for Successful Transformer Life Management............................................ 239Adolf J. Kachler and Ivanka Hhlein

iv Contents

11 Sources, Measurement, and Mitigation of Sound Levels in Transformers............. 265Jeewan Puri

12 Power Transformers Fault Diagnostics by Parks Vector Approach........................ 287A.J. Marques Cardoso and Lus M.R. Oliveira

13 Transformer Reliability: A Key Issue for Both Manufacturers and Users.............. 311Adolf J. Kachler

14 Economics in Transformer Management: Focus on Life Cycle Cost, Design Review, and the Use of Simple Bayesian Decision Methods to Manage Risk....... 325Kjetil Ryen

Part II Instrument TransformersElzbieta Lesniewska

15 Applications of Field Analysis during Design Process of Instrument Transformers..............................................................................................349Elzbieta Lesniewska

16 CAD System-Boundary Integral Equation Method for 3D Electric Field Analysis of Voltage Transformers.................................................................................... 381Ivan Yatchev and Radoslav Miltchev

17 Instrument Transformers Insulation Behavior to High-Voltage and High-Frequency Transients....................................................................................... 399A. Ibero

18 Transfer of HF Disturbances through Instrument Transformers.............................409Wieslaw Jalmuzny and Andrzej Koszmider

Part III High-Frequency TransformersH. Blent Ertan

19 High-Frequency Transformer Materials.........................................................................429Ralph Lucke

20 Powder Core Materials.......................................................................................................447Arcan F. Dericioglu

21 Measurement and Prediction of Parameters of High-Frequency Transformers.....................................................................................461H. Blent Ertan, Erdal Bizkevelci, and Levent B. Yalner

22 High-Voltage, High-Frequency Transformer Design................................................... 487Paul Lefley and Philip Devine

vContents

23 Coreless PCB Transformers...............................................................................................505Jess Doval-Gandoy and Moiss Pereira Martnez

24 Planar Transformers........................................................................................................... 557Frederick E. Bott

vii

Preface

Historically,.transformers.have.been.designed.and.widely.used.to.transfer.energy,.and.for.measurement,.protection,.electrical.isolation,.and.signal.coupling..Most.transformers.were.designed.for.low-frequency.operations,.and.despite.a.long.history.of.low-frequency.trans-former.theory.and.design,.many.stray.field.problems.have.yet.to.be.resolved..On.the.other.hand,.there.has.been.a.significant.increase.in.the.operating.frequency.of.many.switched.mode.power.supplies.to.several.hundreds.of.kilohertz.and,.in.some.cases,.even.up.to.a.few.megahertz..However,.in.the.megahertz.region,.several.problems.arise.related.to.both.the.windings.and.the.standard.core.materials..Therefore,.the.recent.use.for.some.applications.of.coreless.transformers.is.an.alternative.to.transformers.with.a.core.

In.the.case.of.power.transformers,.which.are.key.and.costly.components.in.networks,.the.failure.of.a.single.transformer.can.have.a.major.impact.on.the.safety,.reliability,.and.cost.of.electrical.supply..Their.maintenance.or,.more.generally,.managing.their.life.cycle,.along.with. improved.reliability.and.reduction.of.scheduled.outages,.assumes.more.and.more.importance.due.to.economic.compulsions..Both.users.and.manufacturers.play.an.impor-tant.role.in.the.life.cycle.management.of.power.transformers,.and.hence.close.cooperation.between.them.is.essential..Recent.catastrophic.blackouts.have.exposed.major.vulnerabili-ties. in. existing. generation,. transmission,. and. distribution. systems.. Severe. underinvest-ment,.aging.technology,.and.a.conservative.approach.to.innovation.are.being.blamed.for.the.current.sorry.state.of.affairs..Resources.need.to.be.directed.into.technologies.that.have.the.potential.to.improve.the.integrity.of.the.system..High-temperature.superconductivity.has.such.potential;.it.also.has.a.positive.environmental.impact.as.it.reduces.the.losses.as.well.as.the.size.and.weight.of.the.power.devices.significantly.

Interest.in.increasing.operational.frequencies.remains.alive.since.there.are.two.impor-tant.trends.influencing.the.development.of.transformers:.miniaturization.and.an.increase.in.transmitted.power.density..Therefore,.power.applications.of.high-frequency.products.comprise. transformers. in. switch-mode. power. supplies,. power. chokes,. and. electronic.lamp.ballast.devices..The.latter.includes.transformers.for.wind.power.plants,.contact-less.chargers.for.electrical.cars.(safety!),.and.contact-less.energy.transmission.to.robotic.arms.(unlimited.twisting.in.all.directions).

The.demand.for.power.materials.running.under.higher.operating.frequencies.can.now.be.met.by.highly.advanced.materials.such.as.ferrites..Metallic.powders.and.amorphous.metals.are.also.becoming.increasingly.important.for.use.in.this.area.

Solutions.for.measuring.currents.and.voltages.needed.for.protection.and.monitoring.in.power.systems.have.been.achieved.using.instrument.transformers..New.solutions.such.as. electronic. instrument. transformers. (sensors). are. also. being. successfully. employed..Because.of.their.superior.technical.performance,.current.and.voltage.electronic.transform-ers.(sensors).are.expected.to.play.an.important.role.in.MV.systems.in.the.future..In.this.area.of. instrument.transformers,.problems.of.propagation.of.disturbances.to.secondary.circuits.occur.depending.on.the.design.features..While.the.principle.of.operation.of.instru-ment.transformers.is.the.same.as.that.of.power.transformers,.the.problems.concerned.with.the.design.are.quite.different.

The.advances. in.computational. tools.are.promoting.the.development.of.sophisticated.modeling.software.for.the.solution.of.3D.fields.which.is.supporting.the.design.of.transform-ers.in.the.last.few.decades..However,.these.advances.have.not.always.been.accompanied.

viii Preface

by. relatively. inexpensive,. simple,. and. rapid. engineering. methods,. such. as. those. estab-lished.for.regular.industrial.design.use.

This.book.offers.invaluable.information.for.designers.and.users.of.transformers.to.over-come. some. of. the. difficulties. highlighted. in. the. preceding. text.. The. material. has. been.compiled. by. an. international. group. of. experts. who. met. at. the. International. Advanced.Research.Workshop.on.Transformers.(ARWtr)..ARWtr.meetings.are.organized.every.third.year. to. provide. an. opportunity. for. specialists. from. industrial,. academic,. and. research.backgrounds.to.engage.in.an.intense.exchange.of.practical.knowledge.and.the.establish-ment.of.collaboration.links.on.new.trends.and.issues.of.high-frequency.instrument.and.power.transformers..The.workshop.was.initiated.in.2004.in.Vigo,.Galicia,.and.was.held.in.Baiona,.Galicia,.in.2007.and.in.Santiago.de.Compostela,.Galicia,.in.2010..The.next.work-shop.will.be.held.in.2013.

This.book.(consisting.of.24.chapters).covers.a.wide.range.of.subjects,.including.contem-porary.economic,.design,.diagnostics,.and.maintenance.aspects.of.power,.instrument,.and.high-frequency.transformers.

It.is.hoped.that.this.book.will.serve.as.an.invaluable.resource.material.to.those.working.with.transformers.

Xose M. Lpez-FernndezARWtr General Chairman

For.MATLAB.and.Simulink.product.information,.please.contact:

The.MathWorks,.Inc.3.Apple.Hill.DriveNatick,.MA,.01760-2098.USATel:.508-647-7000Fax:.508-647-7001E-mail:[email protected]:.www.mathworks.com

ix

Acknowledgments

We.would.like.to.thank.the.CIGRE.Spanish.and.Portuguese.National.Committees,.CIGRE.Study. Committee. A2. Transformers,. and. IEEE. PES. Spanish. for. supplying. the. mate-rial.presented.in.this.monograph,.which.was.stimulated.by.the.International.Advanced.Research. Workshop. on. Transformers. (ARWtr).. We. would. also. like. to. acknowledge. the.support. provided. by. the. following. institutions. and. companies:. Universidade. de. Vigo,.Technical. University. of. Lodz,. Spanish. Ministry. of. Science. and. Technology,. Xunta. de.Galicia. Autonomous. Government,. EFACEC. Energa,. ARTECHE,. WEIDMANN,. Red.Elctrica.de.Espaa.(REE),.Rede.Nacional.de.Portugal.(REN),.Machinen.Fabrik.(MR),.Asea.Brown. Boveri. (ABB),. HSP. Hochspannungsgerte. GmbH,. ENPAY,. Gas. Natural. Fenosa,.IBERDROLA,. Doble. Engineering,. NYNAS,. OMICRON,. DuPont,. Unitronics. Electric,.Megger,.General.electric.(GE).Energy,.KELMAN,.ENERGY.Support,.Computer.Simulation.Technology.(CST),.Ingeniera.de.Diseo.Electrotcnico.(INDIELEC),.and.CEDRAT..Finally,.we.express.our.gratitude.to.Patricia.Penabad-Durn.and.Casimiro.Alvarez-Mario.from.Universidade.de.Vigo,.Spain,.for.their.invaluable.help.during.the.preparation.of.this.book.

xi

Editors

Xose M. Lpez-Fernndez.received.his.MSc.in.electrical.engineering.and.his.European.PhD.(with.first.class.honors).in.electrical.engineering.from.Vigo.University,.Vigo,.Spain,.in.1992.and.1997,.respectively..He.is.currently.a.professor.in.the.Department.of.Electrical.Engineering,.University.of.Vigo,.and.was.a.visiting.professor.at.the.University.of.Artois,.Bthune,. France.. Professor. Lpez-Fernndez. is. the. founder. and. general. chairman. of.the. International. Advanced.Research. Workshop.on.Transformers. (ARWtr)..He.has. cur-rently.been.entrusted.with.the.responsibility.of.leading.research.projects.for.Spanish.and.Portuguese.utilities.and.power.transformer.manufacturers..His.research.interests.include.the. design. aspects. of. electrical. machines.. Dr.. Lpez-Fernndez. received. the. Alfons.Hoffmanns.Medal.from.the.Polish.Power.Engineering.Society.in.2004..He.is.a.member.of.the.IEEE-PES.Transformers.Committee.as.well.as.CIGRE.

H. Blent Ertan.received.his.BS.and.MS.in.electrical.and.electronics.engineering.(EEE).in.1971.and.1973,.respectively,.from.the.Middle.East.Technical.University.(METU).in.Ankara,.Turkey,.and.his.PhD.from.the.University.of.Leeds,.United.Kingdom,.in.1977..He.then.joined.METU.as.a.staff.member.in.1977.

Professor. Ertan. is. currently. a. staff. member. in. the. Department. of. Electrical. and.Electronics.Engineering.at.METU..His.research.interest.is.in.the.area.of.electrical.machine.design.and.electrical.drives.

Dr..Ertan.has.served.as.consultant.to.a.number.of.companies.manufacturing.electrical.motors.and.drives.in.Turkey..He.has.directed.more.than.20.industry-financed.research.proj-ects.and.many.other.projects.financed.by.the.Turkish.Scientific.and.Technological.Research.Institute. (TUBITAK).. He. has. served. as. the. assistant. director. of. TUBITAK. Information.Technologies.and.Electronics.Research.Institute.since.1999.and.led.the.Intelligent.Energy.Conversion.Systems.Research.Group.at.TUBITAK-SPACE.Institute.for.8.years.

Dr..Ertan.is.one.of.the.editors.of.the.book.entitled.Modern Electrical Drives,.published.in.2000.by.Kluwer.Academic.Publishers..He.has.published.more.than.100.articles.in.technical.journals.and.conferences..Currently,.he.holds.four.international.patents.

Dr..Ertan.is.a.member.of.IEEE.and.IET,.and.is.the.founder.of.the.International.Conference.on.Electrical.Machines.and.Power.Electronics..He.is.also.a.member.of.the.international.steering.committee.of.the.International.Conference.on.Electrical.Machines..He.is.an.edito-rial.board.member.of.Electromotion Journal.and.Advanced Technology of Electrical Engineering and Energy. (China)..He.also.served.as.the.chairman.of.the.board.of.trustees.of.Mustafa.Parlar.Education.and.Research.Foundation.and.is.currently.a.board.member.

Janusz Turowski,.PhD,.DSc.electrical.engineering,.has.served.as. full.professor. (retired.in.2003).in.electrical.machines.and.applied.electromagnetics.at.the.Institute.of.Electrical.Machines. and. Transformers. (now. Institute. of. Mechatronics. and. Information. Systems),.Technical.University.of.Lodz. (TUL),.Poland..From.1999. to.2010,.he.was.a. full.professor.and.dean.at.the.Academy.of.Humanities.and.Economics.at.TUL..He.received.an.honor-ary.degree. from. the.University.of.Pavia,. Italy,. in.1998..He. is.also.a. full.member.of. the.International. Academy. of. Electrotechnical. Sciences,. a. member. of. CIGRE,. and. a. senior.member.of.IEEE..He.served.as.the.director.of.IEMT.from.1973.to.1992..He.has.authored.

xii Editors

and.coauthored.over.300.scientific.publications,.including.20.books,.cited.over.980.times..He.has.also.supervised.18.PhD.theses.

Dr..Turowski.served.as.a.consultant.of.Polish.ministers.and.transformer.works.in.Poland,.India,.China,.Australia,.and.Canada..He.is.the.past.president.and.an.honorary.member.of.the.Polish.Association.of.Theoretical.and.Applied.Electrotechnical.Sciences.(PTETiS)..He.is.also.a.member.(1978).and.vice-chairman.(19992003).of.the.Electrical.Committee.of.the.Polish.Academy.of.Sciences.and.the.chairman.(19792001).and.honorary.chairman.(2001).of.the.International.Symposium.on.Electromagnetic.Fields.in.Electrical.Engineering.(ISEF)..He.was.awarded.the.2004.Silver.Medal.by.the.International.Academy.of.Electrotechnical.Sciences.For.a.prominent.contribution.to.the.development.of.the.world.science,.education.technology.and.international.co-operation.

xiii

Contributors

Erdal BizkevelciInformation.Technologies.and.Electronics.

Research.InstituteAnkara,.Turkey

Frederick E. BottNoctiluca.Ltd.Bournemouth,.United.Kingdom

A.J. Marques CardosoDepartment.of.Electromechanical.

EngineeringUniversity.of.Beira.InteriorCovilh,.Portugal

and

Departamento.de.Engenharia.Electrotcnica.e.de.Computadores

Instituto.de.TelecomunicaesCoimbra,.Portugal

Arcan F. DericiogluDepartment.of.Metallurgical.and.Materials.

EngineeringMiddle.East.Technical.UniversityAnkara,.Turkey

Philip DevineDepartment.of.EngineeringUniversity.of.LeicesterLeicester,.United.Kingdom

Jess Doval-GandoyDepartment.of.Electronics.TechnologyUniversity.of.VigoVigo,.Spain

H. Blent ErtanDepartment.of.Electrical.and.Electronics.

EngineeringMiddle.East.Technical.UniversityAnkara,.Turkey

John FothergillDepartment.of.EngineeringUniversity.of.LeicesterLeicester,.United.Kingdom

Ivanka HhleinSiemens.Power.TransformerandPower.Transformer.Factory.NurembergNuremberg,.Germany

A. IberoElectrotcnica.Arteche.HermanosMungia,.Spain

Wieslaw JalmuznyDepartment.of.Applied.Electrical.

Engineering.and.Instrument.Transformers

Technical.University.of.LodzLodz,.Poland

Adolf J. KachlerTransformer.ConsultantQuality.MM,.Testing.and.DiagnosticsNuremberg,.Germany

Marceli KazmierskiTransformer.DivisionInstitute.of.Power.EngineeringLodz,.Poland

Andrzej KoszmiderDepartment.of.Applied.Electrical.

Engineering.and.Instrument.Transformers

Technical.University.of.LodzLodz,.Poland

S.V. KulkarniDepartment.of.Electrical.EngineeringIndian.Institute.of.Technology.BombayMumbai,.India

xiv Contributors

Paul LefleyDepartment.of.EngineeringUniversity.of.LeicesterLeicester,.United.Kingdom

Elzbieta LesniewskaDepartment.of.Instrument.Transformers.

and.Electromagnetic.CompatibilityInstitute.of.Electrical.Power.EngineeringTechnical.University.of.LodzLodz,.Poland

Andrey K. LokhaninRussian.Electrotechnical.InstituteMoscow,.Russia

Xose M. Lpez-FernndezDepartment.of.Electrical.EngineeringUniversidade.de.VigoVigo,.Spain

Ralph LuckeFit-CeramicsMiesbach,.Germany

Ryszard MalewskiMalewski.Electric,.Inc.Montreal,.Quebec,.Canada

Radoslav MiltchevDepartment.of.Computer.Systems.and.

InformaticsUniversity.of.ForestrySofia,.Bulgaria

Lus M.R. OliveiraDepartamento.de.Engenharia.

Electrotcnica.e.de.ComputadoresInstituto.de.TelecomunicaesCoimbra,.Portugal

and

Instituto.Superior.de.EngenhariaUniversidade.do.AlgarveFaro,.Portugal

Moiss Pereira MartnezDepartment.of.Electronics.TechnologyUniversity.of.VigoVigo,.Spain

Wadysaw PewcaTransformer.DivisionInstitute.of.Power.EngineeringLodz,.Poland

Jeewan PuriTransformer.Solution.Inc.Matthews,.North.Carolina

Kjetil RyenMulticonsult.ASSki,.Norway

Ryszard SobockiPolish.Consultants.SocietyZabrze,.Poland

Jan K. SykulskiSchool.of.Electronics.and.Computer.

ScienceUniversity.of.SouthamptonSouthampton,.United.Kingdom

Janusz TurowskiInstitute.of.Mechatronics.and.Information.

SystemsTechnical.University.of.LodzLodz,.Poland

Levent B. YalinerInformation.Technologies.and.Electronics.

Research.InstituteAnkara,.Turkey

Ivan YatchevDepartment.of.Electrical.ApparatusTechnical.University.of.SofiaSofia,.Bulgaria

Part I

Power Transformers

Marceli Kazmierski and Janusz Turowski

Introduction

The.history.of.power. transformers. is.part.of. the.history.of.all.phenomena.and.devices.based.on.electromagnetic.induction..An.important.step.forward.was.the.moment,.when.the.transformer.production.has.entered.on.the.energy.industry.market.

It.all.started. in.1883.with.one.of. the.first. implementations. to. the. transmission. line.a.single-phase.15.kVA,.1500/300.V.transformer,.by.K..Zipernowski,.M..Deri,.O..Blathy.(Ganz,.Budapest).and.others.

The.first.real.industrial.three-phase.15.kVA,.1500/300.V.power.transformer.was.built.and.patented.in.1891.by.Michal.Doliwo-Dobrowolski,.was.Polish.descent,.born.in.Petersburg.(Russia).and.worked.at.in.AEG.Berlin..The.history.of.contemporary.transformer.industry.and.science.originated.then.with.Doliwo-Dobrowolski.121.years.ago.

The.material.presented.in.Part.I.is.probably.the.most.competent.report.on.the.state.of.the.art.in.this.field.and.will.hopefully.become.an.indispensable.manual.for.practicing.engi-neers.in.industry.and.in.the.field.of.power.generation.and.distribution.of.electric.energy..The.Part.I.is.divided.into.24.chapters.and.deals.with.design,.diagnostics.and.maintenance,.economy.and.on-site.transformer.management.

Most.of.the.problems.discussed.are.closely.linked.and.overlapping;.hence.this.subdivi-sion.is.rather.formal,.flexible,.and.not.very.precise.

This.book,.prepared.as.post-conference.work,.provides.one.of.the.most.contemporary.treatments.of.modern.transformers..The.scientists,.who.work.at.the.same.time.as.manufac-turers.of.large.transformers.and.consultants.of.most.outstanding.transformer.works.and.

2 Power Transformers

power.utilities,.have.also.contributed.to.this.monograph..The.contribution.are.provided.mainly.from.researchers.and.practicing,.experienced.engineers.and.should.be.useful.to.users.of.both.very.large.power.transformers.and.very.small.HF.units..It.can.also.be.con-sidered.as.a.convenient.practical.handbook,.which.deals.with.many.difficult.problems.that.have.been.resolved.successfully.

Fifteen.outstanding.international.experts.have.contributed.to.this.part.of.the.book..They.are.in.alphabetic.order.as.follows:

A..J..Cardoso.(with.Luis.M..R..Oliveira).on.fault.diagnotics.by.Parks.Vector.Approach..Adolf.J..Kachler.on.Transformer.reliability:.a.key.issue.for.both.manufacturers.and.users.and. (jointly. with. Ivanka. Hhlein). on. Functional. and. component-related. diagnostics.for. power. transformers,. a. basis. for. successful. transformer. life. management;. Marceli.Kazmierski. on. Life. management. of. transformers;. Andrey. K.. Lokhanin. on. Insulation.problems.of.HV.power.transformers;.Xose.M..Lopez-Fernandez.on.large-shell-type.trans-formers,.Malewski;.on.Power.transformer.acceptance.tests,.Wladyslaw.Pewca.on.Selected.problems. of. transformers. short. circuit. withstand. capability;. Jeewan. Puri. on. sources,.measurement.and.mitigation.of.sound.levels.in.transformers,.Kjetil.Ryen.on.Economics.in.transformer.management:.focus.on.life.cycle.cost,.design.review,.and.the.use.of.sim-ple.Bayesian.decision.methods.to.manage.risk;.Ryszard.Sobocki.on.Transformer.design.review:.link.between.design.and.maintenance.stages;.S..V..Kulkarni.on.Challenges.and.strategies. in. transformer. design;. Jan. Sykulski. on. Superconducting. transformers;. and.Janusz.Turowski.on.Stray.losses,.screening,.and.local.excessive.heating.hazard.in.large.power.transformers.

The.editors.of.Part. I.would. like. to.express. their.gratitude.to. the.Authors,.Workshops.Organizers.and.to.the.Publisher.for.their.excellent.work.

31Selected Problems of Transformers Capability to Withstand Short Circuits

Wadysaw Pewca

1.1 Introduction

Standard.CEI.IEC.60076-5.Power Transformers Part 5: Ability to Withstand Short-Circuit.pro-vides.a.basis.for.assessing.short.circuit.withstand.capability.by.calculation.methods.that.are.considered.to.be.as.reliable.as.the.dynamic.tests..The.currently.available.Annex.A.to.the.standard.provides.information.on.the.theoretical.evaluation.of.a.transformers.ability.to.withstand.the.dynamic.effects.of.short.circuit.and.serves.as.a.guidebook.that.provides.methodologies. for. evaluating. short. circuit. withstand. ability.. It. can. be. concluded. from.Annex.A.that.the.analysis.can.be.carried.out.using.simple.methods.such.as.calculating.stresses.and.confronting.them.with.their.permissible.values,.regardless.of.deformations.and.changes.that.take.place.during.short.circuit.forces.[1].

The.criteria. for. the.assessment.of.short.circuit.strength,. in. this.case,.are.of.a.contrac-tual. nature. and,. as. a. general. rule,. apply. only. to. the. technological. and. constructional.solutions.used.by.a.manufacturer,.taking.into.account.a.specific.power.range..Advanced..computer-assisted. methods,. which. are. available. now,. permit. more. precise. calculations.and.assessments,.taking.into.account.the.deformities.of.the.windings,.their.essential.phys-ical.properties.and.constructional.solutions,.as.well.as.some.aspects.of.production.process..It.should.be.expected.that. the.development.of. these.methods.will.subsequently. lead.to.the.creation.of.a.more.generalized.calculation.method.of.verifying.short.circuit.withstand.capability.than.the.one.in.Annex.A.

This. chapter. presents. selected. computer-assisted. methods. of. analysis. of. short. circuit.withstand.capability.that.take.into.account.deformations,.the.structure.of.the.windings,.

CONTENTS

1.1. Introduction.............................................................................................................................31.2. Radial.Forces,.Stresses,.and.Deformations.........................................................................4

1.2.1. Analytical.Method.of.Calculation............................................................................41.2.2. Numerical.Method.of.Calculation......................................................................... 111.2.3. Buckling.Stresses...................................................................................................... 16

1.3. Axial.Forces,.Stresses,.and.Deformation........................................................................... 201.3.1. Axial.Winding.Vibration......................................................................................... 201.3.2. Critical.Axial.Forces................................................................................................. 26

1.4. Conclusions............................................................................................................................ 27References........................................................................................................................................28

4 Transformers: Analysis, Design, and Measurement

and. the.physical.properties.of. the.winding.materials..The. results.obtained.using. these.methods.were.compared.with.the.available.test.results.and,.in.some.cases,.also.with.the.results.that.could.be.obtained.by.using.simplified.methods.

1.2 Radial Forces, Stresses, and Deformations

1.2.1 AnalyticalMethodofCalculation

The.starting.point.in.the.analysis.of.withstand.capability.of.windings.against.radial.short.circuit.forces.is.the.coil.of.outer.winding.subjected.to.tensile.forces,.the.calculation.model.of.which.is.shown.in.Figure.1.1..Generally,.in.addition.to.the.discussed.winding,.there.can.also.be.other.parts.such.as.regulating.winding.or.bandage.that.increase.the.transformers.strength.to.withstand.radial.forces..Provided.the.coil.was.correctly.produced.and.there.are.no.gaps.between.its.conductors.and.the.outer.bandage,.stress.k.and.deformations.k.for.its.particular.conductors.(coils).can.be.calculated.with.(2n..2).set.of.equations.connected.with.the.conditions.of.static.equilibrium.[2]:

.k

k k k k

c

q p p RS

=

+

( )1 . (1.1a)

Rk

Rk

qk

qn

q1 qk

qk

qk

Pk

PkPk1

Pk1

k n2 n1 n32

2

1

1

FIGURE1.1Electrodynamic.conductor.load.qk.and.interaction.on.its.adjacent.terns.and.their.insulation.for.stretched.coils:.1,.axial.spacers;.2,.bandage.


. k k k k k kR R = + +1 1 . (1.1b)

whereqk.represents.the.unitary.radial.electrodynamics.force.acting.on.k-th.conductor,.pkpk..1.represents.the.unitary.reaction.forces.coming.from.the.insulation.k.and.(k..1)Rk.represents.the.mean.radius.of.k-th.conductor,.kk.+.1.represents.the.relative.deformation.of.conductors.k.and.(k.+.1)k,.k.represents.the.relative.deformation.and.the.thickness.of.k-th.insulation

In.addition,.Equation.1.1a.is.supplemented.by.the.boundary.conditions.of.the.outer.conduc-tors.applying.to.stresses.on.the.insulation.of.the.outer.conductor.p0.and.outer.bandage.pn..This.is.usually.based.on.the.assumption.that.the.conditions.are.as.follows:

. p pn0 0= = . (1.2)

Provided.the.mechanical.characteristics.(stresses..in.the.function.of.deformity.).are.lin-ear,.the.distribution.of.stresses.on.conductors.and.isolations.can.be.calculated.analytically;.however,.this.solution.has.no.practical.value.because.constructional.materials.of.windings.demonstrate. substantial. physical. nonlinearity. (Figure. 1.2).. Furthermore,. economic. con-siderations.as.well.as.dimensional.limitations.of.transformers.make.it.necessary.to.apply.stresses.that.are.exerted.also.in.the.nonlinear.parts.of.such.materials.

For.practical.reasons,.the.analysis.has.to.take.into.account.stresses.and.deformities.of.the.nonlinear.character,.as.well.as.the.phenomenon.of.accumulation.of.deformities,.which.can.occur.in.exploitation.or.during.dynamic.tests.after.consecutive.short.circuits.that.can.lead.to.the.transformers.failure.or.its.negative.test.result..For.plastic.materials.such.as.Cu,.Al.the.phenomenon.of.accumulation.is.widely.known.and.can.be.described.using.the.fol-lowing.relationship,.which.is.also.illustrated.in.Figure.1.3.[2]:

.i

i

IBA I

=

= 2

1( ) . (1.3)

wherei.represents.permanent.conductor.set.as.a.result.of.succeeding.stress.maxI.represents.multiple.of.the.conductor.loadingA,.B.represents.empirical.coefficients

Comprehensive.investigations.show.that.the.coefficients.depend.largely.on.the.relationship.max/0,2.and.also.on.the.initial.hardness.of.the.conductors,.loading.(dynamic.or.static),.and,.to.a.lesser.extent,.on.the.temperature..Coefficient.A,.monotonically,.but.sharply,.increases.with.max.while.B..1.and.has.for.max..0.2.a.relatively.flat.maximum..A.few.approximate.values.of.A.and.B.(listed.in.Table.1.1.[2]).obtained.in.the.dynamic.load.tests,.all.of.which.had.20.impulses,.with.max.=.const..and.1020.ms.duration,.illustrate.quantitative.relation-ships.that.we.have.to.deal.with..The.specified.values.indicate.that.a.short.circuit.resulting.in.the.average.radial.stress.close.to.0.2.creates,.when.repeated,.a.risk.of.unacceptable.level.of.permanent.set.of.the.winding.

Figure.1.4.explains.the.method.of.using.formula.1.1a.for.computations..The.coil.conduc-tors.interact.during.a.short.circuit,.as.described.earlier..As.a.result,.the.postshort.circuit.


stresses.lead.to.(see.Figure.1.6).a.permanent.set.in.a.coil,.which.needs.to.be.included.when.calculating. the. cumulating. effect. in. a. winding.. The. points. 1. and. 2. correspond. to. the.mechanical.state.of.a.selected.coil.conductor.after.the.first.(1).or.second.(2).short.circuit,.whereas.12.and.22.represent.the.related.stresses.in.the.coil.conductors.

The.chosen.results.of.calculations.obtained.by.using.the.program,. the.essential.algo-rithm.elements.of.which.have.been.presented.earlier,.are.shown.in.Figures.1.5.through.1.7..The.calculations.were.made.for.the.stretched.coil.of.winding.HV.of.a.GSU.transformer.of.great.power,.consisting.of.24.paper-insulated.Cu.conductors,.and.the.coefficient.of.filling.the.coil.with.copper.was.40%,.when.there.were.axial.cooling.ducts,.or.50%,.when.there.

0.00(a)

40.00

80.00

120.00

(M

pa)

(0.1%)

0.00(b) 5.00 10.00 15.00 20.00 25.000.00

10.00

20.00

30.00

40.00

1

2

(%)

(M

Pa)

FIGURE1.2Strain/stress.curves.for.(a).copper.conductor,.proof.stress.0.2.=.100.MPa,.and.(b).transformer1,.board;.2,.paper.


(i = 3)

(i = 10)

i = 1, 2, ..., 10

n

FIGURE1.3Variation.of.the.conductor.permanent.set.at.repeated.tensile.stresses.of.same.maximum.value..Also.effect.of.increased.stresses.

TABLE1.1

Coefficients.Related.to.Formula.(1.3).for.a.Selected.Case

max/0.2 A B

0.7 0.025 0.51.0 0.41 1.01.1 1.0 0.9

Source:. Kozowski,.M..and.Pewca,.W.,.J. Eur. Trans. Electr. Power (ETAP),.6(4),.259,.1996.

22

3

12

11

E2 E E3E

21

E

1 2 332

1222

k= 11

FIGURE1.4Approximation.of.cumulating.effect.of.permanent.deformations.used.in.calculations.for.k-conductor.


were.no.such.ducts..In.more.general.terms,.the.results.of.calculation.were.shown.in.rela-tive.units,.which.in.relation.to.stresses.means.referring.them.to.the.value.of.proof.stresses.0,2.of.conductor.material.or.average.stresses.m.of.the.coil,.calculated.in.a.classic.way,.that.is,.from.the.relation.given.as.follows:

.m

m

c

q Rs

=

0

2. (1.4)

0.0(a)

0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

4

3

2

1XA

0.2

0.0(b)

60

40

20

0

20

40

60

0.2 0.4

12

34

4

m

0.6 0.8

x

1.0A

100(%)

FIGURE1.5Distribution,.along.the.width.of.the.coil,.of.stresses.in.the.coil.conductors.for.different.ratios.m/0.2:.1,.0.353;.2,.0.9;.3,.1.0;.4,.1.1.(a).at.short.circuit.state.and.(b).after.short.circuit.state:.solid line,.coil.with.inner.axial.ducts;.dashed line,.without.axial.ducts.

0.00.0

0.4

0.8

1.2

1.6

1

m0.2

2

0.5 1.0 1.5 2.0

m

FIGURE1.6Effect.of.the.m/0.2.ratio.on.the.coil.conductor.stress.difference./m.at.surge.current.(1).and.after.the.short.circuit.ceases.(2)..(Solid line,.coil.with.inner.axial.ducts;.dashed line,.without.axial.ducts).


whereq0.is.the.maximum,.unitary.radial.load.for.the.inner.conductor.of.the.coilRm.is.the.average.radius.of.the.coilsc.is.the.cross.section.of.the.conductor

The.increase.of.the.ratio.of.mean.tensile.stress.(m).in.the.coil.conductor.to.the.apparent.proof.stress.of. the.Cu.conductor. (0.2). results. in.a.more.even.distribution.of.stresses.().along.the.width.of.the.coil..In.consequence,.the.maximum.stress.rises.more.slowly.than.the.mean.one..It.is.accompanied.by.decreasing.increments.of.pressure.on.the.wire.insulation.(p).of.maximum.value.(about.2%.of.m).appearing.in.the.middle.of.the.width.of.the.coil..Local.pressure.increases.appear.at.axial.spacers,.which.determine.the.width.of.the.cooling.duct.and.are.caused.by.limited.filling.of.the.coil.circumference..The.calculations.show.the.insignificant.effect.of.inner.cooling.ducts.on.the.stress.distribution.

The.mechanical.condition.of.the.coil,.following.the.first.short.circuit.fault,.is.illustrated.in.Figures.1.5.and.1.6..After.a.short.circuit.of.low.m/0.2.ratiohere.0.353,.curve.1the.stress.in.the.coil.disappears..Despite.uneven.distribution.of.the.stress..under.the.short.circuit.(Figure.1.5a,.curve.1),.the.maximum.stress,.appearing.in.an.inner.conductor,.results.in.no.plastic.strain.of.the.copper.conductor.and.leaves.no.permanent.deformations.of.the.coil.conductors.

As.can.be.seen.in.Figure.1.5,.with.higher.values.of.the.m/0.2.ratio,.the.plastic.strains.appear.in.a.inner.conductors.of.the.coil.and.straighten.it,.leading.to.a.more.even.profile.of.the.stress.along.the.width.of.the.coil..However,.this.is.accompanied.by.permanent.sets.in.plastically.strained.conductors..In.consequence,.the.short.circuit.leaves.new,.differently.distributed.stresses:. Inner.conductors,.plastically.strained.during.the.short.circuit. fault,.are. compressed. by. outer. conductors,. elastically. strained. and. also. stretched,. tending. to.restore.their.original.dimensions..Stress.difference..of.coil.conductors,.already.left.after.the.first.fault,.is.significant.(Figure.1.5).

0.00

1

1

2

2

10th

1st

m

(%)

3

4

0.5 1.0 1.5 2.0

0.2

FIGURE1.7Strains.at. short.circuit. (solid lines).and.permanent.sets. (dotted lines).of. inner. (1).and.outer. (2).coil. conductors.shown.for.the.1st.(reference.designation.*).and.10th.short.circuits.(reference.designation.),.as.a.function.of.the.m/0.2.ratio.


Figure.1.6.is.a.somewhat.different.presentation.of.the.phenomena.and.allows.their.quan-titative.evaluation..For.m/0.2..0.35.the.short.circuit.leaves.no.permanent.sets.in.coil.con-ductors..For.0.35..1,.both.the.dynamic.strain.and. the.permanent.set. still. increase.and.become.higher,.when.compared.with. the.first.short.circuit..The.difference.between.the.dynamic.strain.and.the.permanent.set.decreases.in.the.course.of.consecutive.short.circuits:.The.permanent.set.increases.faster.and.differs.slightly.only.after.the.10th.short.circuit.dynamic.strain..Due.to.the.cumulating.effect,.the.increase.of.permanent.set.is.bigger.after.the.10th.short.circuit.than.the.increase.of.dynamic.strain.after.the.first.short.circuit.

The.coil.deformation.(illustrated.in.Figure.1.7). is.a.decisive.factor.for.the.transformer.short. circuit. withstand. capability.. The. appearance. of. the. deformation. leads. to. the. fol-lowing.two.kinds.of.danger:.disruption.of.wire.weld.connection.(due.to.dynamic.strain),.and. an. inadmissible. decrease. of. the. major. insulation. dielectric. strength.due. to. signifi-cant.widening.of.the.inner.oil.duct.at.the.outer.winding.(permanent.set)..Weld.failures.can.be.expected.at.1%.strain,.so.at.m.only.slightly.exceeding.0.2.(Figure.1.7)..Permanent.sets.of.this.kind.are.not.permitted.because.of.the.hazard.to.the.dielectric.strength.of.the.insulation.system..Such.a.permanent.set. level. in.large.power.transformers.(240420.kV).means.widening.of.the.first.cooling.duct,.the.one.that.is.close.to.the.winding,.by.as.much.as.68.mm,.up.to.1416.mm..This.would.result.in.a.decrease.of.the.dielectric.strength.to.85%.of.the.original.value..Such.a.transformer.should.be.also.rejected.after.a.short.circuit.test:.Its.posttest. impedance.voltage.increase.would.be.between.4%.and.5%.[2,3]..This.means.that.the.permanent.set.of.its.outer.winding.is.limited.by.the.required.dielectric.strength.of.major. insulation..Acceptable. lowering.by.5%.of.dielectric.strength.corre-sponds. to. a. permanent. set. of. about. 0.2%,. which,. considering. the. cumulating. effect,.requires.the.m/0.2.ratio.to.be.below.0.7.

Summing. up. the. results. of. the. analysis. presented,. one. can. come. to. the. following.conclusions:

. The.existence.of.plastic.deformations.of.inner.coil.conductors,.under.tensile.short.circuit.forces,.should.be.considered.as.a.rule.for.large.power.transformers.

. Dynamic. plastic. strains. result. in. permanent. sets. of. conductors. and. leave. post-fault.stresses.in.the.coils.of.the.winding.


. The.limit.of.deformations.of.a.winding.stretched.under.a.short.circuit. is.deter-mined.by.an.acceptable.decrease.in.the.dielectric.strength.of.major.insulation..The.decrease.is.caused.by.the.oil.axial.duct.at.the.outer.winding.becoming.danger-ously.wide.due.to.the.wire.permanent.set..The.permissible.value.can.be.estimated.to.be.0.2%..With.deformations.of.this.kind,.the.ability.of.wire.welds.to.withstand.a.short.circuit.is.still.sufficient,.provided.they.were.properly.made.

. If.the.permanent.set.of.a.wire.is.to.stay.within.permissible.limits,.the.m/0.2.has.to.be.below.0.7.(provided.the.winding.was.properly.made)..Insufficient.compact-ness.and.rigidity.of.conductor.paper.insulation.and.insufficient.winding.process.means.uncontrolled.decrease.of.this.limit.

. The.apparent.elastic.limit.0.2.of.a.wire.depends.on.the.conditions.in.which.the.winding.was.made.. It. is.not.possible. to.estimate. the.winding.strength.without.knowing.the.ultimate.value.of.0.2.

The.analytical.calculation.method.presented.here.is.used.in.the.assessment.of.withstand.capability.of.outer.windings.subjected.to.radial.tensile.forces.and,.after.appropriate.modi-fications,.can.be.extended.to.inner.winding.compressed.by.short.circuit.forces..The.expe-rience.of.IEnOT.shows.that.because.of.the.complexity.of.phenomena.taking.place.in.the.compressed. windings,. the. numerical. method. proves. to. be. a. more. effective. method. of.analysis.in.this.case..It.is.also.a.more.general.method,.applicable.not.only.to.compressed.windings.but.also.to.stretched.ones,.regardless.of.the.shape.of.the.winding.

1.2.2 NumericalMethodofCalculation

In. the. numerical. method. applied. by. IEnOT,. the. coil. of. the. winding. in. the. core. trans-former.is.discretized.in.a.flat.configuration.with.finished.elements.in.a.manner.as.shown.in.Figure.1.8.

P v

W1

W8

W9

W10W11 W12

W13

W14

W2

W3W4 W5

W6

A

A

4

1

6

x

z

h

h

d

d

W7

a*z.d

PvPv

Pv Pv

Pv

Pv

5y

u

v

W

2

5

FIGURE1.8Discrete.model.of.coil.sector:.1,.coil.conductors;.2,.support.strips;.3,.curvilinear.elements.representing.coil.con-ductor;.4,.springy.elements.modeling.insulation;.5,.dimensions.of.coil.conductors;.6.


It. is. possible. to. analyze. only. a. chosen. segment. of. the. coil. limited. by. the. axis. of..symmetryas.demonstrated.earlieror.its.whole.circumference,.regardless.of.the.shape.of. the.coil. itself..The.conductors.are.represented.by.curvilinear.plane.elements.and.the.insulation.of.the.coil.(distancing.axial.spacers,.bundle.insulation.on.the.conductors).with.springy.elements..These.elements.are.described.with.parameters.representing.their.real.dimensions.and.nonlinear.mechanical.characteristics.(compare.Figure.1.2).of.the.materi-als.from.which.they.were.made..The.radial.load.of.the.coil.can.be.continuous.or.concen-trated,.and.its.distribution.along.the.circumference.and.on.particular.conductors.in.the.radial.direction.can.be.of.any.value.or.direction.(stretching,.compressing,.circumferential.tension).. It. is.also.possible.to.load.the.coil.with.displacements.(the.so-called.kinematics.loading),.which.is.sometimes.useful.in.defining.mechanical.characteristics.of.the.coils.or.glued.conductors.CTC..The.algorithm.of.the.program.takes.into.account.for.such.gener-ally.defined.loading.conditions.the.possibility.of.bending.of.the.coil.conductors.occurring.between.support.strips,.which.makes.it.necessary.to.split.the.conductors.into.layers.(usu-ally.about.10,.with.the.radial.dimension.of.the.conductor.amounting.to.a.few.millimeter),.for.which.the.distribution.of.stresses.during.bending.in.each.layer.can.already.be.consid-ered.to.be.invariable.

One. of. the. most. essential. characteristics. of. the. discussed. software,. which. bring. the.algorithm.of.calculation.close.to.real.conditions,.is.the.introduction.of.springy.elements.of.unilateral.constraints.(realized.by.the.appropriate.adjustment.of. their.characteristics,.for.example,.by.compressing.and.stretching).and.the.possibility.of.calculating.permanent.sets.after.the.first.short.circuit.(Figure.1.4the.loading.cycle.0-1-1)..Modeling.the.springy.elements.according.to.the.unilateral.constraints.also.makes.it.possible.to.include.the.fol-lowing.fact.in.the.analysis.of.withstand.capability:.Insulationin.case.of.typical.produc-tion.technology,.without.using.gluing.substancescan.transfer.only.radial.compressive.forces,.and.in.case.of.stretching.forces,.it.is.subject.to.layering..Of.course,.in.the.discussed.algorithm,.computer.calculations.are.carried.out.by.applying.the.iteration.method.with.an.appropriately.adjusted.step.of.load.increment.q..For.each.calculation.step,.the.lineariza-tion.of.incremental.equilibrium.equation.is.performed,.which.is.presented.in.the.follow-ing.matrix.form.[4]:

. [ ( )]( ) ( ) ( ) ( )K K f qepi i i i+ =+ + 1 1 . (1.5a)

whereKep(i) .is.the.elastic-plastic.stiffness.matrixK(i) ( ).is.the.stress.stiffness.matrixf(i.+.1).is.an.increment.of.generalized.displacements.vectorq(i.+.1).is.an.increment.of.loads.vector

Matrix.Kep(i) .and.K(i) .are.calculated.on.the.basis.of.displacements.and.stresses,.which.are.calculated.at.the.beginning.of.the.(i.+.1)-th.step.

The.incremental.equation.system.is.solved.according.to.the.variable.stiffness.method..The.total.displacement.vector.f(i).and.loads.vector.q(i).are.determined.by.the.accumula-tion.of.respective.vectors.increment.according.to.the.relationship:

.f fi j

j

i( ) ( )

=

=

1

. (1.5b)


.

q qi j

j

i( ) ( )

=

=

1

. (1.5c)

The.algorithm.of.the.computer.calculation.method.helps.to.establish,.in.a.known.way,.the.critical.load.qcr.occurring.during.a.loss.of.stability..If.the.so-called.proportional.load.q.is.used.for.the.i-th.step.of.calculation.and.critical.load.can.be.defined.with.the.relationship,

. q qcri i( ) ( )

= . (1.6a)

where..represents.a.multiplier.of.the.comparative.load,.the.problem.of.initial.stability.on.i-th.step.of.calculation.can.be.expressed.as.follows:

. [ ( )]( )( ) ( )K Kepi i

= 0 . (1.6b)

where..is.the.pattern.of.stability.loss..The.Equation.1.6.can.be.solved.by.Stodels.or.Lanczos.methods,.and.the.limiting.load.qcr.is.obtained.from.formula:

.

q qcr crj

j

n

=

=

( )1

. (1.6c)

where.n.is.the.step.of.calculation,.at.which.the.singularity.of.matrix.is.not.investigated.and.no.solution.for.Equation.1.5.is.obtained..There.are.also.other.known.methods.of.calculat-ing.the.critical.load.occurring.during.the.loss.of.stability;.however,.the.authors.experience.indicates.that.the.best.results.can.be.achieved.with.the.so-called.deflection.method,.the.practical.use.of.which.will.be.presented.in.Section.1.2.3.

The.examples.of.uses.of. the.described.numerical.method.for.calculating.stresses.and.deformations.of.the.windings.subjected.to.radial.short.circuit.forces.are.shown.in.Figures.1.9.through.1.11.

The. results. of. calculations. presented. in. Figures. 1.10. and. 1.11. also. seem. interesting..These. calculations.were. carried.out. for. the.Working.Group.WG19.12. [5],. in. connection.with. their. debate. concerning. calculating. stresses. in. the. conductors. of. layer. windings,.which.would.be.a.reliable.basis.for.the.assessment.of.strength.of.radial.short.circuit.forces.in.this.construction..There.was.still.concern.that.in.this.type.of.winding,.especially.for.units.of.the.largest.power,.the.existing.cooling.ducts.between.the.layers.may.significantly.impede.the.transfer.of.radial.forces.between.them.and.it.may.not.be.appropriate.to.rely.on.mean.stresses.in.the.assessment.of.withstand.capability.(calculated.with.the.formula.1.4.for.such.a.construction).

Calculations.were.carried.out.for.a.certain.step-up.unit,.the.dimension.of.which.are.com-patible.with.the.power.of.approx..600.MVA,.provided.that.representatives.of.the.producer.spe-cializing.in.constructions.of.this.kind.accept.the.following.parameters.for.the.winding.HV:

. The.number.of.layers.in.the.winding4,.each.wound.with.CTC.conductor.with.dimensions.21..(8.4..1.5).mm,.with.bundle.insulation.(paper).0.56.mm/side..Proof.stress.for.the.conductor.equals.0.2.=.210.MPa

. The.inner.diameter.of.the.winding.is.1696.mm,.and.the.outer.1847.mm..The.thick-ness. of. the. layer. (the. same. as. radial. dimension. of. insulated. conductor. CTC).amounts.to.19.7.mm.and.the.axial.cooling.duct.up.to.6.mm


2 mmW 1312

W 342

W245

W 98

W 50 W 197 W 246 W 1216

FIGURE1.9Winding.deformation.of.the.distribution.transformer.with.rectangular.legs.at.three-phase.fault.

N22

(a)

N32

16mm69mm

N232 N253 N925Y

XZ

N935

0.4087E+02 (mm)

N22

(b)

N32

16.2 mm 51mmN925 Y

XZ

N935

0.4123E+02 (mm)

FIGURE1.10HV.layer.winding.deformation.of.large.power.GSU.Transformer,.caused.by.short.circuit.tensile.radial.forces.(black line,.the.winding.without.deformation;.gray line,.deformed.winding),.(a).at.the.amplitude.of.short.circuit.current.and.(b).after.short.circuit.state.


. Mean.tensile.stresses.for.the.whole.winding.equal.m.=.282.MPa..These.data.are.characterized.by.a.high.value.of.mean.tensile.stresses.m.in.relation.to.proof.stress.0.2. of. the. coil. conductor. material. (relation. m/0.2.. 1.34),. which. basically. con-tradicts. the. recommendations. presented. in. the. summary. of. Section. 1.2.1.. Such.relations.of.stresses.were.intended.to.exaggerate.the.phenomenon.of.transferring.loads.by.the.neighboring.layers.of.the.winding,.in.order.to.obtain.clear.conclu-sions.of.calculations.

Figure.1.10.demonstrates.that.considerable.radial.deformations.of.layers.take.place.in.the.winding,.and.in.particular.inner.layer.(approx..16.8.mm.in.short.circuit.condition.and.approx..16.2.mm.after.the.first.short.circuit),.which.refers.both.to.dynamic.deformation.taking.place.when.impact.short.circuit.current.flows.through.the.winding.and.permanent.(plastic).sets.remaining.in.the.winding.when.the.short.circuit.current.disappears..Moreoveras.Figure.1.10b.showswhen.the.short.circuit.current.disappears,.two.inner.layers.of.the.winding.concerned.are.buckled..This.is.caused.by.strongly.deformed.outer.layers.pressing.on.these.two.layers..Such.big.deformations.are.unacceptable.because.of.the.stress.exerted.on.the.main.insulation.system.[2,3].and.because.they.can.lead.to.a.mechanical.damage.of.the.insulation.arrangement.itself..Due.to.considerable.plastic.sets.occurring.after.the.first.circuit,.one.can.expecttaking.into.account.the.relation.(Equation.1.3)that.equally.large,.quickly.increas-ing.permanent.sets.will.occur.after.consecutive.short.circuits..In.the.light.of.these.results,.based.on.design.data.and.physical.properties.of.the.windings,.one.can.unmistakably.ascer-tain.that.the.winding.concerned.does.not.have.sufficient.withstand.capability.against.radial.short. circuit. forces.. The. obtained. result. confirms. the. conclusions. and. recommendations.

600.00

(M

Pa)

400.00

200.00

200.000 10 20 30 40

Conductor no

50

0.00

FIGURE1.11Stresses. in. HV. winding. conductors. of. large. power. GSU. Transformer,. caused. by. short. circuit. tensile. radial.forces;.,.the.amplitude.of.short.circuit.current;.,.after.short.circuit.state.result.of.simplified.calculation,.whose.deformation.is.ignored.


presented.in.the.summary.of.Section.1.2.1..It.would.not.be.possible.to.draw.such.explicit.conclusions,. if.the.assessment.of.withstand.capability.of.the.winding.was.based.only.on.the. comparison. between. stresses. m. and. 0.2. and. do. not. take. in. account. deplacements..Although.stresses.in.the.discussed.case.significantly.exceed.stresses.0.2,.there.are.still.no.grounds.to.draw.conclusions.on.the.scale.of.risks,.and.it.is.not.possible.to.establish.the.maxi-mum.stresses.in.the.inner.layer.of.the.winding,.at.the.dissipating.gap.

The.results.shown.in.Figure.1.11.give.an.answer.to.the.problem.presented.by.the.Working.Group..One.can.conclude.on.the.basis.of.their.findings.that.in.short.circuit.condition,.a.considerable.equalization.of.stress.distribution.on.the.layers.occurs.in.the.analyzed.wind-ing,.which.in.light.of.the.observations.presented.in.Section.1.2.1.confirms.that.plastic.sets.take.place,.as.illustrated.in.Figure.1.10;.for.such.a.large.assumed.relation.m/0.2.(approx..1.34).the.maximum.stresses.in.the.conductor.of.the.inner.layer.are.only.about.17%.higher.than.the.average.stresses.m.approved.by.the.designer.and.much. lower. than.the.maxi-mum.stresses.(approx..550.MPa).that.occur.in.a.nondeformed.coil..Relatively.low.values.of.stresses,.which.occur.after.the.short.circuit.current.disappears.in.relation.to.the.results.presented.in.Figure.1.5b.for.a.disc.coil.winding,.can.be.explained.by.a.buckling.of.two.inner.layers.as.well.as.probably.big.plastic.sets.of.the.layers.

In.connection.with.the.aforementioned.numbers.concerning.the.distribution.of.stresses,.one. can. also. jump. into. a. misleading. conclusion. that. since. such. considerable. equaliza-tion. of. stresses. took. place. in. the. coil. and. the. theoretical. maximum. stresses. are. nearly.twice.as.great.as.the.calculated.ones,.withstand.capability.against.radial.forces.of.the.ana-lyzed.winding.seems.not.to.be.at.risk..The.danger.of.such.erroneous.interpretation.can.only.occur.when.the.assessment.of.withstand.capability.is.based.merely.on.the.values.of.stresses.in.a.simplified.way,.not.taking.into.account.the.deformation.of.windings.in.the.short.circuit.condition.and.when.that.short.circuit.disappears.

1.2.3 BucklingStresses

According.to.the.authors.experience,. the.best.results. in.calculating.buckling.stresseswhich.were.mentioned.earlierwere.achieved.with.the.deflection.method..Generally,.this.method.is.based.on.the.analyses.of.coil.deformation.(solving.only.statics,.Equation.1.5).in.the.function.of.increasing.load..The.purpose.of.the.method.is.to.find.the.calculation.step.from.which.the.deformations.of.the.winding.become.asymmetrical.and.quickly.increase,.and. in. case. of. further,. even. insignificant. load. increment,. the. shape. of. the. deformation.changes..As.numerical.tests.demonstrated,.this.process.has.its.periodicity,.and.after.the.first.critical.load.and.when.the.shape.of.the.coil.changes,.the.coil.becomes.more.fixed,.and.a.further.load.increment.makes.it.possible.to.calculate.the.second.buckling.pattern.and.the.corresponding.second.critical. load..Theoretically,. the.process.can.be.continued,.but.because.of.the.finite.exactness.of.calculations.and.the.numerical.dimensions.of.the.calcu-lation.model.of.the.coil,.sensible.results.are.usually.obtained.only.in.the.phase.between.the.second.and.the.third.critical.load..For.practical.reasons,.of.course,.only.the.first.critical.load.and.its.corresponding.buckling.pattern.are.essential.

The. described. algorithm. of. calculating. critical. loads. is,. therefore,. entirely. consistent.with.the.general.theory.describing.the.phenomenon.of.the.loss.of.stability,.as.published.in.many.textbooks.on.the.stability.of.constructions..However,.the.deflection.method.can.be.successfully.applied.only.when,.just.as.it.was.shown.in.the.nonlinear.calculation.model.of.the.coil.presented.earlier,.it.uses,.among.other.things,.curvilinear.plane.elements,.taking.into.account.nonlinear.physical.properties.of.materials..As.it.is.known,.in.a.typical.linear.calculation.model.based.on.rectilinear.rod.and.springy.elements,.in.which.the.materials.


have.linear.characteristics,.the.load.increment.does.not.permit.achieving.the.aforemen-tioned. condition. of. the. loss. of. stability.. In. such. a. model,. deformations. always. remain.linearly.dependent.on.the.load.

The.results.obtained.with.use.of.the.deflection.method,.which.are.presented.in.the.fol-lowing,. for.coils.whose.models. in.scale.1:1.were.subjected. to.dynamic. tests. in.order. to.experimentally.establish.radial.critical.loads.at.a.loss.of.stability..However,.the.results.for.the.unsupported.circular.ring.of.rectangular.cross.section.will.be.shown.first..The.shape.of.the.ring.resembles.a.single.turn.of.a.coil.and.its.stability.is.theoretically.worked.out,.whereas.the.critical.load.cr.is.defined.with.the.relation:

.cr

E hR

=

4

2

. (1.7)

whereE.is.the.modulus.of.elasticity.of.copperh.is.the.radial.dimension.of.the.conductorR.is.the.mean.ring.radius

Confronting. the. results. of. computer. calculations. with. the. results. obtained. using. the.Equation. 1.7,. taking. into. account. the. buckling. patterns,. which. for. rings. are. commonly.known,.will.permit.assessing.the.possibilities.and.accuracy.of.the.method.applied.to.cal-culate. the. critical. load.. It. will. also. be. possible. to. assess. the. effect. of. simplifications. in.the. representation. of. the. coil. (1/4. of. the. circumference.or. the.whole. circumference).on.the.accuracy.of.calculations..For.analytic.and.computer.calculations,.the.following.equa-tion.was.used:.E.=.1.103..105.MPa,.h.=.3.68.mm,.R.=.171.5.mm;.the.results.of.calculations.are.shown.in.Figures.1.12.and.1.13.and.Table.1.2.

The. obtained. results. of. the. numerical. test. for. unsupported. ring. confirm. the. correct-ness. of. the. presented. method. of. calculating. buckling. stresses,. both. in. relation. to. the.

N98

N50

(a)

Y

XZ

Step 110

Step 105

Step 111

Step 80Step 105 Step 80

Step 110

Step 111N366

Y

Z X

N310

(b)

R

FIGURE1.12Deformations. of. the. ring. for. different. steps. of. radial. load. (gray line:. step. 80. . m.=.8.8.MPa,. step105..m.=.11.55.MPa,. step. 110. . m.=.12.1.MPa,. step. 111. . m.=.12.21.MPa;. black line:. unsupported. and. unloaded. ring)..(a). model. representing. a. quarter. of. the. circumference. of. the. ring. and. (b). model. representing. the. whole.circumference.of.the.ring.


value. of. critical. stresses. themselves. and. to. the. buckling. pattern. (usually. presented. for.the.first.two.patterns)..For.understandable.reasons,.the.greatest.exactness.was.achieved.for.the.first.buckling.pattern.and.when.the.coil.was.represented.with.its.full.circumfer-ence..Representing.the.coil.with..of.the.circumference,.which.was.sometimes.necessary,.diminishes. the. size. of. the. calculation. model. and. the. size. of. collections. with. recorded.results,.and.also.reduces.the.time.of.calculations,.which.makes.the.calculations.less.reli-able.and.somewhat.overstates.the.calculated.critical.loads.

The.aforementioned.numerical.test.is.supplemented.by.the.results.shown.in.Table.1.3.and. in. Figures. 1.14. and. 1.15,. referring. to. the. real. coils,. for. which. dynamic. tests. [3,6,7].were. carried. out.. Because. different. publications. often. refer. to. the. Fischers. formula. [8].when.discussing.the.phenomenon.of.the.loss.of.stability,.one.form.of.which.is.defined.by.Equation.1.8,.it.seems.appropriate.to.include.this.simplified.method.in.the.comparisons:

.cr

Ez

hR

=

482

2

. (1.8)

where.z.is.the.number.of.supports.on.winding.circumference.

0.10

0.05

0.00N50

N980.05

0.10

0.150.00(a) 5.00 10.00 15.00 20.00 25.00

sr(MPa)

R (m

m)

0.16

0.12

0.08

N310

N3660.04

0.000.00(b) 5.00 10.00 15.00 20.00 25.00

sr(MPa)

R (m

m)

FIGURE1.13Radial.displacement.of.chosen.knots.of.the.calculation.ring.model.in.function.of.loading,.defined.with.average.compressive.stresses.m.(position.of.knots.on.the.ring.is.shown.in.Figure.1.12)..(a).model.representing.a.quarter.of.the.circumference.of.the.ring.and.(b).model.representing.the.whole.circumference.of.the.ring.

TABLE1.2

Critical.Load.kr.for.Unsupported.Ring

Pattern of Buckling

Value of Critical Load crr (MPa) Calculated According to

Deflection Method Representing ( of the Circumference)

The Whole Circumference

Formula (1.7)

1 14.96 11.55 11.02 15.18 13.31 11.03 No.data 15.07 No.data


TABLE1.3

Value.of.Buckling.Stresses.for.Selected.Models

Symbol of Models

Test Results in (MPa)

Calculated Results in (MPa), According to

Deflection Method Fischers Rule [2]MR32 74.4 70.4 31.7M16Ba 50.5 56.6 40.4Del.Vecc.1b 88.4 109.0 123.3Del.Vecc.2b 90.4 99.0 69.4

Source:. Pewca,.W..(in.Polish),.Energetyka,.1,.25,.2002.a. Model.was.made.of.ctc.conductor.[6,7].b. Del.Vecchios.model.tests.[12].

(a)

Y

XZ(b)

FIGURE1.14Test.and.calculation.results.of.model.MR32..(From.Pewca,.W..[in.Polish],.Energetyka,.1,.25,.2002.).(a).Model.after.destructive.dynamic.test.and.(b).calculated.instability.form.(dotted line).for.inner.turn.of.coil.

16 B

(a)

u=0.185 mm

(b)

u

FIGURE1.15Test.and.calculation.results.of.model.M16B..(From.Pewca,.W..[in.Polish],.Energetyka,.1,.25,.2002;.Kozowski,.M..et.al.,.Selected.short-circuit.strength.problems.in.power.transformer,.CIGRE.Report.12-05,.18,.1980.).(a).model.after.destructive.dynamic.test.and.(b).calculated.instability.form.for.turn.of.coil.


Summing. up. the. results. presented,. one. can. say. that. the. numerical. method. of. ana-lyzing. the. strength. of. windings. to. radial. short. circuit. forces. allows. bringing. the. cal-culation.model.of. the.coil. (winding).close.to.real.solutions,.by.taking.into.account.the.nonlinear. characteristics. of. construction. materials. and. specific. character. of. insulation.elements.transferring.not.only.compressive.forces.but.also.curvatures.of.conductors..The.algorithm.of.method.presented.is.universal.and.can.be.used.both.for.the.windings.com-pressed. by. radial. short. circuit. forces. and. for. the. stretched. ones.. It. allows. calculating.deformations.of.the.winding.subjected.to.short.circuit.condition.and.after.the.condition.disappears,.which.is.especially.important.for.the.proper.assessment.of.withstand.capa-bility.of.windings.

It.should.be.emphasized.that.the.exactness.of.calculating.buckling.stress.is.considerable,.especially.when.the.analyzed.coil.of.the.winding.is.represented.with.the.full.circumfer-ence..When.simplified.models.are.applied.(i.e.,.representing..of.the.circumference).the.exactness.of.calculation.is.theoretically.worse,.yet.it.is.satisfactory.for.practical.solutions,.as.is.confirmed.with.short.circuit.tests.

1.3 Axial Forces, Stresses, and Deformation

1.3.1 AxialWindingVibration

Reduced.stiffness.of.disk.windings.of.core-type.transformers.for.axial.direction.of.short.circuit. forces,. combined.with. the. risk.of. the.occurrence.of. resonance.phenomena,.war-rants. accurate. methods. to. be. applied. for. analyzing. short. circuit. withstand. capability..Additional.reasons.that.support.this.view.include.the.nonlinearity.of.the.characteristics.of.materials.and.time.variability.of.electrodynamic.forces.acting.on.the.coils.of.windings..A.typical.time.variability.of.electrodynamic.axial.component.force.acting.on.the.end.coil.of.the.winding.LV.of.a.unit.of.large.power.is.shown.in.Figure.1.16a.[9]..This.time.variability.is.characterized.by.the.existence.of. two.harmonics:.50.Hzexisting.in.a.period.of. time.when.a.unidirectional.component.of.short.circuit.current.flows.through.the.winding.and.harmonic.of.100.Hz.occurring.at.steady.state.of.short.circuit.current..Figure.1.16b,.on.the.other.hand,.shows.the.distribution.of.amplitudes.of.these.axial.forces.acting.on.particular.coils.of.this.winding.

The.simplified.methods.of.analyzing.withstand.capability.of.the.winding.against.axial.forces.take.into.account.only.the.first.maximum.amplitude.of.electrodynamic.forces.for.each.coil,.approaching.the.problem.in.accordance.with.the.rules.of.static.method,.and.con-sidering.the.winding.as.not.deformed..Stresses.in.insulation.established.in.this.way.are.not.sufficiently.objective.data.for.a.reliable.assessment.of.real.short.circuit.risks.

Time.variability.of.axial.short.circuit.forces.and.reduced.stiffness.of.the.windings.(cable.paper.and.pressboard.take.a.significant.part.in.the.height.of.these.windings).show.that.it.is.not.possible.to.ignore.the.vibrations.of.the.coils,.which.are.not.synchronous.in.every.point.of.the.winding..An.accurate.calculation.method.must.necessarily.include.in.the.process.of.vibration.also.the.forces.that.are.different.from.electrodynamic.forces.acting.on.the.coil.(see.Figure.1.17b).and.comprising.static.compressive.forces.

The. computer. program. developed. in. IEnOT. for. analyzing. the. dynamic. strength. of.winding.to.axial.short.circuit.forces.takes.into.account.all.components.of.the.forces.as.well.as. the. earlier-discussed. quality. of. insulation. (see. Section. 1.2.2). permitting. transferring.


only.compressive.forces.[9]..The.algorithm.of.this.program.is.based.on.the.winding.model.presented. in. Figure. 1.17. and. on. the. assumption. that. the. electrodynamic. forces. will. be.analyzed. in.a.flat.configuration,.which.allows.obtaining. the.exactness.of.calculation.of.axial. forces. that. is. satisfactory. for.practical. reasons. [9,10].. In. the.assumed.model. radial.spacers.are.represented.with.nonlinear.springy.elements.and.the.coils. themselves.with.point-concentrated.masses.

Calculations. realized. with. the. program,. the. algorithm. of. which. is. presented. in. this.chapter,. begin. with. estimating. the. static. compressive. forces. and. their. distributions. on.every.winding..To.achieve.this.aim,.the.equation.of.the.static.balance.[10].is.used,.linear-ized.for.particular.calculation.steps:

. K Z F S Ti i

pr(i i i( ) ( )( ) ) = ++ 1 1+ . (1.9a)

whereKi().is.the.matrix.of.stiffness.dependent.on.coil.displacements.Z(i.+.1).is.the.increment.of.displacement.vectorFpri( )+1 .is.the.vector.of.pressing.forcesSi.is.the.vector.of.flexibility.forcesTi.is.the.vector.of.friction.forces

Time (s)0.00 0.04

(a)0.08 0.12 0.1

200.00

160.00

120.00

80.00

40.00

0.00

F e(kN)

Coil no(b)100 120806040200

200.00

150.00

100.00

50.00

0.00

50.00

100.00

F e (k

N)

150.00

FIGURE1.16Axial.electrodynamic.forces.acting.on.the.coils.of.winding.LV.of.large.power.GSU.transformer..(From.Pewca,.W..[in.Polish],.Energetyka,.11,.492,.1998.).(a).time.characteristic.of.a.force.acting.on.the.end.coil.and.(b).distribution.of.amplitudes.of.axial.force.per.the.winding.coils.


The.input.data.establishing.the.distribution.of.a.force.pressing.on.particular.coils.com-prise:.a).the.assumed.characteristics.of.compressing.pressboard,.from.which.radial.spac-ers.were.made.and.characteristics.of.bundle.insulation.on.the.conductors.of.the.coil.(as.shown.in.Figure.1.2b),.and.b).the.percentage.of.compression.of.the.winding.insulation.on.the.height,.as.assumed.by.the.designer..Example.stress.distributions.in.insulation.com-ing.from.these.forces.for.a.transformer,.which.was.taken.into.consideration.earlier,.were.shown.in.Figure.1.18..These.calculations.were.realized.for.two.significantly.different.tech-nological.alternative.version,.namely.[9]:

. 1..High.static.pressing.degree.of.the.insulation9%.of.its.height,.by.using.rigid.sup-ports.at. the.ends.of.winding.and.wound.coils.with. low.tension. (force.pressing.conductor.to.the.axial.distance.strips.about.2.6.kN)

. 2..Limited.static.pressing.degree.of.the.insulationabout.7%.of.its.height,.less.rigid.ends.supports,.and.force.pressing.the.conductor.to.the.axial.strips.at.about.25.5.kN

The.fact.that.compressive.static.forces.are.not.directly.assumed.in.the.analysis.presented,.but. are. calculated. from. constructional. data. of. the. winding. and. data. of. materials. and.

Fn CnTn

Tn1

Ti

T3

T2

T1

Cn1

Ci

C3

C2

C1

Fn1

Fi

F3

F2

F1 1

2

3

i

n1

n

k1

k2

k3

k4

ki

ki+1

kn+1

kn1

kn

(a)

Fi

Si

Si +1

mi TiRiBi

(b)

FIGURE1.17Winding.model.for.dynamic.analysis..(From.Pewca,.W.[in.Polish],.Energetyka,.11,.492,.1998;.Borkowski,.W..et.al.,.Non-linear.analyse.of.dynamic.withstandability.of.transformer.winding.under.axial.short-circuit.forces,.Report of Eighth International, Symposium on Short-Circuit Currents in Power Systems,.pp..149154,.1998.).(a).base.diagram.of.coil.arrangement.and.(b).calculated.forces.acting.on.i-coil.end.insulation..mi,.mass.of.i-coil;.i,.coefficient.of.dry.friction;.ci,.viscous.friction.coefficient;.ki,.stiffness.of.insulation.element;.Fi,.axial.electrodynamics.force;.Ti,.dry.friction.force;.Ri,.viscous.friction.force;.Bi,.force.of.inertia;.Si,.Si.+.1;.reaction.force.of.i-.and.(i.+.1)-insulation.elements.


production.process,.is.an.important.element.of.the.calculation.methods.discussed,.bring-ing. the. algorithm. close. to. real. solutions.. Another. essential. element. of. the. algorithm. is.the. inclusion. of. the. distribution. of. compressive. forces. on. the. coils. in. the. calculations..Experience.indicates.that.too.small.static.compressive.force.(or.its.diminishing.during.ser-vice).in.relation.to.axial.electrodynamic.forces.is.one.of.the.most.frequent.causes.of.trans-former.short.circuit.damages..Physically,.they.are.caused.in.such.cases.by.breaking.contact.between.the.coil.and.the.adjacent.radial.spacers. in. the.process.of. their.vibrations..This.initiates.the.so-called.Hammer.Effect,.deepening.the.axial.gaps.in.the.winding.and.initiat-ing.the.process.of.dynamic.destruction.of.the.winding.structures.(among.other.things,.the.radial.displacement.of.radial.spacers.can.happen)..Identifying.this.calculation.process.is.an.objective.element.of.the.assessment.of.short.circuit.withstand.capability.of.the.winding.against.axial.short.circuit.forces..To.confront.this,.the.assessment.of.short.circuit.strength.of.the.winding.only.on.the.basis.of.values.of.stresses.in.insulation.(mechanical.withstand.of.the.pressboard.and.cable.paper.to.compressive.stresses.is.considerable),.which.is.used.in.simplified.models,.does.not.seem.to.be.convincing.

In.order. to. identify. the.moment.of.detachment.of.coils. from.radial.spacers,. for.every.calculation.step.and.for.every.springy.element.the.program.analyses.the.direction.of.force.Si.(see.the.markings.from.Figure.1.17b),.and.even.if.one.springy.element.a.stretching.force.occurs,.that.is.[10],

. Si < 0 . (1.9b)

The.calculations.are. interrupted,. for.example,.when. there. is.a. signal. that.a.destruction.process.is.beginning.in.the.winding.and.the.compressive.force.resulting.from.design.data.and.assumed.production.process. is. too.small.. If. this. force.can.be. further. increased. for.the.winding.(i.e.,.by.improving.with.technological.ways.the.stiffness.of.the.insulation.or.increasing.the.percentage.of.compression.of.the.winding).and.the.process.of.calculations.can.be.brought.to.an.end.in.the.assumed.time.period.of.short.circuit,.withstand.capability.of.the.winding,.from.the.point.of.view.of.the.static.compressive.force,.is.appropriate.

Figure.1.19.illustrates.time.characteristic.of.the.forces.in.the.end-yoke.insulation.for.the.considered.unit.and.the.assumed.earlier.two.versions.of.its.production.process..For.ver-sion.1,.in.which.the.winding.is.strongly.compressed.(see.Figure.1.18),.there.is.no.risk.of.

Coil no

120.00

100.00

80.00

60.00

40.00

20.00

0.00

2 1

14.00 16.00 18.00 20.00 22.00 24.00 (MPa)

26.00

FIGURE1.18Stress.distribution.in.the.insulation.of.the.winding.at.static.pressing,.the.technology.covering.version.(1).and.(2)..(From.Pewca,.W..[in.Polish],.Energetyka,.11,.492,.1998.)


detachment.of.the.end.insulation.from.the.winding.in.the.whole.analyzed.period.of.time..Such.a.risk.could.occur.in.case.of.variant.2,.in.the.initial.phase.of.a.short.circuit..However,.the. assumed. for. the. calculations. increased. friction. forces. on. axial. strips. stopped. the.movement.of.the.end.coil.downward.and.only.because.of.this.it.was.possible.to.avoid.its.detachment.from.the.yoke.insulation..The.presented.example.illustrates.how.significant.is.the.influence.of.technologies.on.short.circuit.withstand.capability.of.transformers,.which.in.quality.relations.is.well.known.

Other.essential.parameters.that,.apart.from.stresses.in.insulation,.are.taken.into.account.in.the.assessment.of.short.circuit.strength.include.displacements.of.the.coils,.in.particu-lar.the.end.ones,.cooperating.with.the.end.compressive.insulation..The.algorithm.of.the.discussed. program. establishes. these. parameters. on. the. basis. of. known. calculations. in.mechanics,.linearized.for.particular.calculation.steps,.matrix.equations.[10]:

. M Z C Z K Z F S Ted + + = +( ) ( ) . (1.10)

whereM.is.the.matrix.of.inertiaC.is.the.matrix.of.dampingFed.is.the.vector.of.electrodynamic.forcesZ,.Z,.Z.is.the.vector.of.displacements.and.its.first.and.second.derivatives

Example.results.of.calculating.displacements.of.end.coils.and.stresses.in.insulation,.which.were.obtained.with.the.discussed.method.for.the.assumed.earlier.versions.of.technologi-cal.solutions,.are.shown.in.Figures.1.20.and.1.21.

1800.00

1600.00

1400.00

1200.00

1000.00

2000.00

2200.00

2400.00

S (kN)

0.00(a)

0.04 0.08 0.12 0.16Time (s)

0.00

500.00

1000.00

1500.00

2000.00

S (kN)

0.00 0.05(b)

0.10 0.15 0.20 0.25Time (s)

FIGURE1.19Forces.in.end.insulation.of.the.winding.as.functions.of.time.for.versions.1.(a).and.2.(b).


Calculation.results.presented.in.the.previous.figure.for.end.coils.also.confirm.the.essen-tial.influence.of.production.process.on.values.of.coil.displacements.in.short.circuit.con-dition.. During. a. strong. static. compression. of. the. winding. (version. 1). the. displacement.of.these.coils.are.slight,.and.it.should.be.predicted.that.they.will.not.have.a.destructive.effect.on.the.end.insulation.of.the.analyzed.winding..For.technological.version.2,.in.which.

1.00

0.00

d (m

m)

1.00

2.00

4.00

3.00

5.000.00

(a)0.04 0.08

Time (s)0.12 0.16

10.00

5.00

d (m

m)

0.00

5.00

10.00

15.00

20.000.00 0.05 0.10

Time (s)(b)0.15 0.20 0.25

FIGURE1.20Axial.displacement.of.end.coils.for.version.1.(a).and.2.(b)..(From.Pewca,.W..[in.Polish],.Energetyka,.11,.492,.1998.)

50.00

40.00

30.00

20.00

10.00

0.000.00 0.05 0.10 0.15 0.20 0.25

Time (s)

(M

Pa)

FIGURE1.21Pressure.in.radial.spacers.at.half.of.the.winding.height,.version.2..(From.Pewca,.W..[in.Polish],.Energetyka,.11,.492,.1998.)


nearly.one-third.of.the.compressive.forces.were.diminished.(comp..Figure.1.18),.over.three-fold.increase.of.the.amplitude.of.vibrations.in.the.initial.phase.of.the.short.circuit.occurs..Although.the.absolute.value.of.displacement.of. the.end.coils.does.not.seem.to.be.criti-cal,.the.ultimate.assessment.of.short.circuit.strength.should.take.into.account.the.conse-quences.of.such.displacements.in.the.design.review.of.end.insulation..In.the.same.variant,.increased.stress.on.radial.spacers.should.also.be.expected.in.the.middle.of.the.height.of.the.winding,.which.is.documented.in.Figure.1.21.

Summing. up. the. presented. basic. elements. of. the. algorithm. of. the. program. for. the.assessment.of.withstand.capability.against. axial. short. circuit. forces.and.obtained.with.this.method.results,.one.can.draw.the.following.conclusions:

. Static.forces.compressing.the.winding.and.stiffness.of.radial.insulation.areapart.from.the.values.of.electrodynamics.forcesbasic.parameters,.which.have.an.effect.on.the.short.circuit.withstand.strength.of.the.winding.

. Maintaining.constant.mechanical.contact.between.the.coils.of. the.winding.and.the.adjacent.radial.spacers.and.the.value.of.the.amplitudes.of.coil.vibrations,.in.particular.of.the.end.coils,.can.be.accepted.as.appropriate.criteria.for.the.assess-ment.of.short.circuit.withstand.capability.of.coil.windings.

For.stresses.in.radial.spacers,.which.are.affected.by.the.aforementioned.factors,.it.is.dif-ficult.to.establish.precise.critical.values.based.on.the.physical.properties.of.materials..In.fact,.practice.shows.that.even.in.dynamically.damaged.windings,.radial.spacers.do.not.demonstrate.significant.signs.of.mechanical.spoil.(including.cases.when.they.were.dam-aged. by. arcing).. However,. the. values. of. these. stresses. are. useful. in. checking. whether.there.is.no.risk.of.the.coil.loss.stability.for.axial.direction,.which.is.connected.with.the.tilting.effect.

1.3.2 CriticalAxialForces

Axial.loading.of.a.winding,.due.to.short.circuit.forces,.can.be.destructive.for.the.coil,.espe-cially.because.of.their.unacceptable.tilting.effect..Critical.axial.load.at.which.the.effect.for.the.coil.occurs.is.difficult.to.determine.analytically.and.as.a.rule.requires.experimental.investigation.on.models..Figure.1.22.illustrates.a.special.arrangement.for.determining.this.critical.axial.load,.which.was.used.in.Institute.of.Power.Engineering.[3].

Tests. were. performed. on. winding. models. with. different. coil. parameters,. and. their.results.permit.determining.empirical.coefficients.in.a.formula.for.the.tilting.critical.force.Fcr.referred.to.in.literature.[11].and.also.to.propose.a.new.formula.[3]:

.F

zD

m b Acr 2 7 0 3. . . (1.11)

wherez.is.the.number.of.terns.(conductors).in.a.coilD.is.the.coil.mean.diameterb.is.the.conductor.radial.dimensionA.is.the.radial.spacer.widthm.is.the.number.of.spacers.on.the.coil.girt


The.static. tests.were.followed.by.dynamic.tests.on.coils.of. identical.design.and.dimen-sions..Specially.designed.arrangements.presented.on.Figure.1.23.were.used.in.these.tests..It.was.demonstrated.that.the.critical.axial.forces,.determined.statically.and.dynamically,.are.similar..The.difference.was.below.6%.and.only.in.one.case.reached.15%.

1.4 Conclusions

. 1..The. results.of. the.analysis.presented.proved. that,. regardless.of. the. type.of. the.winding.(coil,.layer).of.the.core-type.transformer,.the.average.stresses.in.the.coil.conductors.m.and.proof.stress.0.2.of. the.material. from.which.they.were.made.

FIGURE1.22Arrangement.with.hydraulic.press.for.static.evaluation.of.critical.axial.forces.

FIGURE1.23Arrangement.for.dynamic.evaluation.of.critical.axial.forces..Supply.coils.and.model.after.testing.


are.the.most.essential.parameters.determining.the.withstand.capability.of.these.windings.against.radial.short.circuit.forces.

. 2.. In.case.m..0.2,.a.phenomenon.of.cumulating.of.deformations.after.consecutive.stresses.can. take.place,. indicating. insufficient. strength.of. the.winding. to.with-stand.radial.tensile.short.circuit.forces..The.relation.of.stresses.which.should.be.considered.as.completely.safe.is.m/0.2..0.7.

. 3..Simplified. methods. of. calculating. buckling. stresses. for. windings. subjected. to.radial. compressive. forces. do. not. permit. achieving. the. accuracy. of. calculations.that. is. satisfactory. for. design. purposes.. Much. better. results. are. obtained. with.numerical.methods,.which.take.into.account.deformations.and.nonlinear.physical.and.geometrical.properties.of.the.windings.

. 4..The.criterion.of.permissible.stresses.in.insulation,.especially.when.they.are.calcu-lated.with.simplified.methods,.is.not.an.objective.and.sufficiently.precise.parameter.for.assessing.withstand.capability.of.the.windings.against.axial.short.circuit.forces.

. 5..The.analysis.of.winding.strength.to.withstand.the.axial.component.of.short.circuit.forces.should.be.carried.out.taking.into.account.the.dynamics.of.the.phenomena.connected.with.time.variability.of.electrodynamic.force,.nonlinearity.of.physical.materials,. and. the. occurring. deformations.. The. basic. criteria. of. the. assessment.of.withstand.capability.include.the.following:.(1).the.correctness.of.adjusting.the.static.compressive.force.eliminating.the.hammer.effect;.(2).value.of.maximum.amplitude.of.axial.deformation.of.the.winding,.which.can.be.accepted.for.a.given.system.of.compressing.the.winding;.and.(3).stresses.in.insulation.radial.spacers.are.smaller.than.the.critical.ones.causing.tilting.effect.

. 6..Applying. appropriate. numerical. calculation. methods. in. the. analysis. permits.making.a.correct.assessment.of.short.circuit.withstand.capability.of.the.windings.based.on.the.aforementioned.criteria.and.taking.into.account.essential.design.and.technological.solutions.

References

. 1.. CIGRE.WG,.12.06.(1982).Final.Report.of.WG.06.of.Study.Committee.12.(Transformers),.Electra,.No.82.

. 2.. Kozowski.M,.Pewca.W.(1996).Short-circuit.performance.of.a.stretched.transformer.winding.with.regard.to.its.actual.mechanical.characteristics..Journal of European Transactions on Electrical Power (ETAP),.6(4),.259265.

. 3.. Kozowski.M,.Pewca.W.(1994).Improvement.in.estimation.criteria.of.transformer.short-circuit.withstand.capability..Archiwum Elektrotechniki,.XII(3),.577590.

. 4.. Borkowski.W,.Pewca.W,.Szymczyk.E,.Nieczorek.M.(1996).Numerical.analysis.of.short-circuit.withstandability.of.transformer.windings.under.radial.forces..Seventh International Symposium on Short-Circuit Currents in Power Systems,.Warsaw,.Poland,.Report.2.12,.pp..16.

. 5.. Fyvie.J.(2002).The.short.circuit.performance.of.power.transformer..The.Final.Report.of.WG12.19,.Electra.25,.205.

. 6.. Pewca.W.(in.Polish,.2002).Loss.of.stability.and.buckling.stresses.for.transformer.inner.wind-ings..Energetyka,.1,.2531.

. 7.. Kozowski. M,. Pewca. W,. Marciniak. M,. Weretynski. W. (1980). Selected. short-circuit. strength.problems.in.power.transformer..CIGRE.Report.12-05,.pp..18.


. 8.. Fischer. E. (1952). Die. Festichkeit. der. inneren. Roehre. von. Transformatorenwicklungen..Elektrotechnische Zeitschrift,.73.

. 9.. Pewca. W. (in. Polish,. 1998). Influence. the. clamping. forces. and. winding. support. stiffness. on.winding.short-circuit.axial.force.results..Energetyka,.11,.492496.

. 10.. Borkowski.W,.Pewca.W.(1998).Non-linear.analyse.of.dynamic.withstandability.of.transformer.winding. under. axial. short-circuit. forces.. Report of Eighth International Symposium on Short-Circuit Currents in Power Systems,.Brussels,.Belgium,.pp..149154.

. 11.. Hiraiishi.K.(1971).Mechanical.strength.of.transformer.windings.under.short-circuit.conditions..IEEE Transactions,.Pas-90,.23902391.

. 12.. Del.Vecchio.RM,.Poulin.B,.Anuja.R.(1999).Radial.buckling.strength.calculation.and.test.com-parison.for.core.form.transformers..CIGRE Tran

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