chapter 4 br

8/20/2019 Chapter 4 Br

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IEEE PES General Meeting, Tampa FL

June 24-28, 2007Coner!n"ia #ra$ileira %e &uali%a%e %e Energia

Santo$, S'o Paulo, (go$to )-8, 20071

C*apter 4 Mo%eling o +onlinear Loa%

Contriutor$ S. T$ai, /. Liu, an% G. . C*ang

Tutorial on 1armoni"$ Mo%eling an% Simulation


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Chapter outline

• Introduction• Nonlinear magnetic core sources• Arc furnace

• 3-phase line commuted converters• Static var compensator • Cycloconverter


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Introduction

• The purpose of harmonic studies is to quantifythe distortion in voltage and/or current aveformsat various locations in a poer system!

• "ne important step in harmonic studies is to

characteri#e and to model harmonic-generatingsources!

• Causes of poer system harmonics $ Nonlinear voltage-current characteristics

$ Non-sinusoidal inding distri%ution

$ &eriodic or aperiodic sitching devices

$ Com%inations of a%ove


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Introduction 'cont!(

• In the folloing) e ill present the harmonicsfor each devices in the folloing sequence*

+! ,armonic characteristics

! ,armonic models and assumptions

3! .iscussion of each model


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Chapter outline

• Introduction• +onlinear magneti" "ore $our"e$• Arc furnace



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Nonlinear agnetic Core Sources

• ,armonics characteristics

• ,armonics model for transformers

• ,armonics model for rotating machines


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,armonics characteristics of iron-corereactors and transformers

• Causes of harmonics generation $ Saturation effects

$ "ver-e0citation temporary over-voltage caused %y reactive poer un%alance un%alanced transformer load asymmetric saturation caused %y lo frequency magneti#ing current transformer energi#ation

• Symmetric core saturation generates odd harmonics• Asymmetric core saturation generates %oth odd and even

harmonics

• The overall amount of harmonics generated depends on $ the saturation level of the magnetic core $ the structure and configuration of the transformer


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,armonic models for transformers

• ,armonic models for a transformer* $ equivalent circuit model

$ differential equation model

$ duality-%ased model

$ 1IC 'geomagnetically induced currents( saturationmodel


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2quivalent circuit model 'transformer(

• In time domain) a singlephase transformer can %erepresented %y anequivalent circuit referring allimpedances to one side of

the transformer • The core saturation is

modeled using a pieceiselinear appro0imation ofsaturation

• This model is increasinglyavaila%le in time domaincircuit simulation pacages!


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.ifferential equation model 'transformer(

• The differential equations descri%e the relationships %eteen $ inding voltages

$ inding currents

$ inding resistance

$ inding turns

$ magneto-motive forces

$ mutual flu0es

$ leaage flu0es

$ reluctances

•Saturation) hysteresis) and eddy current effects can %e ellmodeled!

• The models are suita%le for transient studies! They may also%e used to simulate the harmonic generation %ehavior ofpoer transformers!

+

=

N NN N N

N

N

N NN N N

N

N

N

i

i

i

dt

d

L L L

L L L

L L L

i

i

i

R R R

R R R

R R R

v

v

v

2

1

21

22221

11211

2

1

21

22221

11211

2

1


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.uality-%ased model 'transformer(

• .uality-%ased models arenecessary to represent multi-legged transformers

• Its parameters may %ederived from e0periment data

and a nonlinear inductancemay %e used to model thecore saturation

• .uality-%ased models aresuita%le for simulation of

poer system lo-frequencytransients! They can also %eused to study the harmonicgeneration %ehaviors

Magneti" "ir"uit Ele"tri" "ir"uit

agnetomotive4orce '4( Ni

2lectromotive 4orce'42( E

4lu0 Current I

5eluctance 5esistance R

&ermeance Conductance

4lu0 density Current density

agneti#ing force

H

&otential difference

V

φ

ℜ

ℜ/1 R/1

A B /φ = A I J /=


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1IC saturation model 'transformer(

• 1eomagnetically induced currents1IC %ias can cause heavy half cyclesaturation

$ the flu0 paths in and %eteen core)tan and air gaps should %eaccounted

• A detailed model %ased on 3. finiteelement calculation may %enecessary!

• Simplified equivalent magnetic circuitmodel of a single-phase shell-type

transformer is shon!• An iterative program can %e used to

solve the circuitry so that nonlinearityof the circuitry components isconsidered!

F

~AC

DC

Rc1 Ra1

Ra4

Ra4’

Rt4

Rc3

Rc2

Rc2

Ra3

Rt3


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5otating machines

• ,armonic models for synchronous machine

• ,armonic models for Induction machine


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Synchronous machines

• ,armonics origins* $ Non-sinusoidal flu0 distri%ution The resulting voltage harmonics are odd and usually

minimi#ed in the machine6s design stage and can %enegligi%le!

$ 4requency conversion process Caused under un%alanced conditions

$ Saturation Saturation occurs in the stator and rotor core) and in the

stator and rotor teeth! In large generator) this can %eneglected!

• ,armonic models $ under %alanced condition) a single-phase inductance issufficient

$ under un%alanced conditions) a impedance matri0 isnecessary


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7alanced harmonic analysis

• 4or %alanced 'single phase( harmonic analysis) asynchronous machine as often represented %y asingle appro0imation of inductance

$ h* harmonic order

$ * direct su%-transient inductance

$ * quadrature su%-transient inductance

• A more comple0 model

– a: 8!9-+!9 'accounting for sin effect and eddy currentlosses(

– Rneg and X neg are the negative sequence resistance and

reactance at fundamental frequency

( )[ ]2/"" qd h L Lh L +=

"d L

"q L

neg neg a

h jhX Rh Z +=


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:n%alanced harmonic analysis

• The %alanced three-phase coupled matri0 model can %eused for un%alanced netor analysis

– Z s=( Z o+2 Z neg )/3

– Z m=( Z o− Z neg )/3

– Z o and Z neg are #ero and negative sequence impedance at h th

harmonic order

• If the synchronous machine stator is not precisely %alanced)the self and/or mutual impedance ill %e unequal!

=

smm

m sm

mm s

h

Z Z Z

Z Z Z

Z Z Z

Z


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Induction motors

• ,armonics can %e generated from $ Non-sinusoidal stator inding distri%ution

Can %e minimi#ed during the design stage

$ Transients

,armonics are induced during cold-start or load changing $ The a%ove-mentioned phenomenon can generally %e

neglected

• The primary contri%ution of induction motors is toact as impedances to harmonic e0citation

• The motor can %e modeled as $ impedance for %alanced systems) or

$ a three-phase coupled matri0 for un%alanced systems


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,armonic models for induction motor

• 7alanced Condition $ 1enerali#ed .ou%le Cage odel

$ 2quivalent T odel

• :n%alanced Condition


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1enerali#ed .ou%le Cage odel forInduction otor

5s

;<s

5c

;


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2quivalent T model for Induction otor

• s is the full load slip at fundamental frequency) and h is theharmonic order

• =-6 is taen for positive sequence models• =>6 is taen for negative sequence models!

5s ;h


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:n%alanced model for Inductionotor

•The %alanced three-phase coupled matri0 model can %e used for un%alancednetor analysis

– Z s=( Z o+2 Z pos)/3

– Z m=( Z o− Z pos)/3

– Z o and Z pos are #ero and positive sequence impedance at hth harmonic order

• ?8 can %e determined from 5s8

;<s8

5m8

8!95r8

'->3s(

;<m8

;<r8

5m8

8!95r8

'@-3s(

;<m8

;<r8

=

smm

m sm

mm s

h

Z Z Z

Z Z Z

Z Z Z

Z


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June 24-28, 2007

Coner!n"ia #ra$ileira %e &uali%a%e %e Energia


Chapter outline

• Introduction• Nonlinear magnetic core sources• (r" urna"e



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June 24-28, 2007



Arc furnace harmonic sources

• Types* $ AC furnace

$ .C furnace

• .C arc furnace are mostly determined %y its AC/.C converter and the characteristic is morepredicta%le) here e only focus on AC arc

furnaces


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June 24-28, 2007



Characteristics of ,armonics 1enerated %y Arc 4urnaces

• The nature of the steel melting process isuncontrolla%le) current harmonics generated %yarc furnaces are unpredicta%le and random!

• Current chopping and igniting in each half cycleof the supply voltage) arc furnaces generate a

ide range of harmonic frequencies

a


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June 24-28, 2007



,armonics odels for Arc 4urnace

• Nonlinear resistance model• Current source model• oltage source model

• Nonlinear time varying voltage source model• Nonlinear time varying resistance models• 4requency domain models• &oer %alance model


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June 24-28, 2007



Nonlinear resistance model

a

simplified to

• 5+ is a positive resistor • 5 is a negative resistor

• AC clamper is a current-controlled sitch• It is a primitive model and does not consider the time-varyingcharacteristic of arc furnaces!

modeled as


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June 24-28, 2007



Current source model

• Typically) an 2A4 is modeled as a current source forharmonic studies! The source current can %e represented%y its 4ourier series

' an and bn can %e selected as a function of $ measurement

$ pro%a%ility distri%utions

$ proportion of the reactive poer fluctuations to the active

poer fluctuations!• This model can %e used to si#e filter components and

evaluate the voltage distortions resulting from the harmoniccurrent in;ected into the system!

( ) ∑ ∑+= ∞

=

∞

=1 co&&in

n nnn L t nbt nat i ω ω


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June 24-28, 2007



oltage source model

• The voltage source model for arc furnaces is aThevenin equivalent circuit!

$ The equivalent impedance is the furnace loadimpedance 'including the electrodes(

$ The voltage source is modeled in different ays* form it %y ma;or harmonic components that are non

empirically

account for stochastic characteristics of the arc furnaceand model the voltage source as square aves ithmodulated amplitude! A ne value for the voltage

amplitude is generated after every #ero-crossings of thearc current hen the arc reignites


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June 24-28, 2007



Nonlinear time varying voltage sourcemodel

• This model is actually a voltage source model• The arc voltage is defined as a function of the arc

length

– V ao *arc voltage corresponding to the reference arc lengthl o)

– (t )* arc length time variations

• The time variation of the arc length is modeled ithdeterministic or stochastic las! $ .eterministic*

$ Stochastic*

( ) ( ) ( ) l aoV t l aV =

( ) ( ) ( )t !l l t l o ω &in12 +−=

( ) ( )t Rl t l o −=


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June 24-28, 2007



Nonlinear time varying resistance models

• .uring normal operation) the arc resistance can %emodeled to follo an appro0imate 1aussiandistri%ution

σ is the variance hich is determined %y short-termpercepti%ility flicer inde0 &st

• Another time varying resistance model*

– R 1* arc furnace positive resistance and R 2 negative resistance

– * short-term poer consumed %y the arc furnace

– *i% and *#x are arc ignition and e0tinction voltages

( ) ( )RA+D22co&RA+D1n2 π σ −+= R Ra"#

2

2

2

2

2

1

R

V

R

V $

V R

e%ig

ig

−+

=


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June 24-28, 2007



&oer %alance model

' " is the arc radius

• e0ponent n is selected according to the arccooling environment) nB8) +) or • recommended values for e0ponent m are 8) +

and

•K1) K2 and K3 are constants

2

2

321 i

"

&

dt

d" " & " &

m

n

+=+


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June 24-28, 2007



Chapter outline


• -p*a$e line "ommute% "on3erter$• Static var compensator • Cycloconverter


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June 24-28, 2007



Three-phase line commuted converters

• ine-commutated converter is mostly usualoperated as a si0-pulse converter or configuredin parallel arrangements for high-pulseoperations

• Typical applications of converters can %e found in AC motor drive) .C motor drive and ,.C lin


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June 24-28, 2007



,armonics Characteristics

• :nder %alanced condition ith constant output current andassuming #ero firing angle and no commutation overlap) phase acurrent is

h B +) 9) D) ++) +3) !!!

$ Characteristic harmonics generated %y converters of any pulsenum%er are in the order of n = 1, 2, ··· and p is the pulse num%er of the converter

• 4or non-#ero firing angle and non-#ero commutation overlap) rmsvalue of each characteristic harmonic current can %e determined%y

– ' 'µ)α( is an overlap function

),-co&(.co&/)0( µ α α π α µ +−= h ' I I d h

∑ +=h

ha t hh I t i )&in()/2()( 11 δ ω

1±= pnh


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June 24-28, 2007



,armonic odels for the Three-&haseine-Commutated Converter

• ,armonic models can %e categori#ed as $ frequency-domain %ased models

current source model

transfer function model

Norton-equivalent circuit model harmonic-domain model

three-pulse model

$ time-domain %ased models models %y differential equations

state-space model


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June 24-28, 2007



Current source model

• The most commonly used model for converter is to treat itas non sources of harmonic currents ith or ithoutphase angle information

• agnitudes of current harmonics in;ected into a %us aredetermined from

$ the typical measured spectrum and

$ rated load current for the harmonic source ' I "ated (

• ,armonic phase angles need to %e included hen multiplesources are considered simultaneously for taing theharmonic cancellation effect into account!

θ h) and a conventional load flo solution is needed forproviding the fundamental frequency phase angle) θ 1

sp sph"ated h I I I I −−⋅= 1/

)( 11 sp sphh h −− −+= θ θ θ θ


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June 24-28, 2007



Transfer 4unction odel

• The simplified schematic circuit can %e used todescri%e the transfer function model of a converter ' * the ideal transfer function ithout considering

firing angle variation and commutation overlap

' ϕ0c and ϕ0ac) relate the dc and ac sides of theconverter

• Transfer functions can include the deviation termsof the firing angle and commutation overlap

• The effects of converter input voltage distortion orun%alance and harmonic contents in the output dccurrent can %e modeled as ell

#baV (V d#d# 000 0 =∑= ϕ ϕ

ϕ ϕ a)b)#i(i d#a# == ϕ ϕ ϕ 0 0


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June 24-28, 2007



Norton-2quivalent Circuit odel

• The nonlinear relationship %eteenconverter input currents and its terminalvoltages is

$ I E are harmonic vectors

• If the harmonic contents are small) one

may lineari#e the dynamic relationsa%out the %ase operating point ando%tain* 4 = 56* + 4

– 56 is the Norton admittance matri0representing the lineari#ation! It alsorepresents an appro0imation of the

converter response to variations inits terminal voltage harmonics orun%alance

– 4 = 4 b 7 56* b 'Norton equivalent(

)(VI * =


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June 24-28, 2007



,armonic-.omain odel

• :nder normal operation) the overall state of the converter isspecified %y the angles of the state transition

$ These angles are the sitching instants corresponding to the Ffiring angles and the F ends of commutation angles

• The converter response to an applied terminal voltage ischaracteri#ed via convolutions in the harmonic domain

• The overall dc voltage

– V )p: + voltage samples

Φ p* square pulse sampling functions – H : the highest harmonic order under consideration

• The converter input currents are o%tained in the samemanner using the same sampling functions!

∑ ∑ Φ=∑ Φ⊗== ==

H

h

H

n

n p

h p

p p p d V V V

1

2

10

12

10


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June 24-28, 2007



Chapter outline


• 3-phase line commuted converters• Stati" 3ar "ompen$ator • Cycloconverter


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June 24-28, 2007



,armonics characteristics of TC5

• ,armonic currents are generated for any conduction intervalsithin the to firing angles

• Gith the ideal supply voltage) the generated rms harmoniccurrents

– h = 3, 5, 7, ·HH) σ is the conduction angle) and L R is theinductance of the reactor

−

−=

)1(

&in)co&()&in(co&4)( 21 hh

hhh

L

V I

Rh

α α α α

πω α


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June 24-28, 2007



,armonics characteristics of TC5 'cont!(

• Three single-phase TC5s are usually in deltaconnection) the triplen currents circulate ithinthe delta circuit and do not enter the poersystem that supplies the TC5s!

• Ghen the single-phase TC5 is supplied %y anon-sinusoidal input voltage

$ the current through the compensator is proved to %ethe discontinuous current

+


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June 24-28, 2007

Coner!n"ia #ra$ileira %e &uali%a%e %e EnergiaSanto$, S'o Paulo, (go$to )-8, 2007

43

,armonic models for TC5

• ,armonic models for TC5 can %e categori#ed as $ frequency-domain %ased models

current source model

transfer function model

Norton-equivalent circuit model

$ time-domain %ased models models %y differential equations

state-space model


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44

Current Source odel

∑ +=

h

hhh t h I t i )&in()( θ ω

+


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June 24-28, 2007


45

Norton equivalence for the

harmonic poer floanalysis of the TC5 for the

h-th harmonic

Norton-2quivalent odel

• The input voltage is un%alanced and no coupling%eteen different harmonics are assumed

1)( −− = eqeqh L jhω Y heqheqh L jh IVΙ −=− )/( ω

hhh V φ ∠=V hhh I θ ∠=I )&in/( σ σ π −= Req L L


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46

Transfer 4unction odel

• Assume the poer system is %alanced and isrepresented %y a harmonic Thvenin equivalent

• The voltage across the reactor and the TC5 current can%e e0pressed as

' 58CR B5R S can %e thought of TC5 harmonic admittancematri0 or transfer function

R S* & *=

S8CR R R R *5*5 ==


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June 24-28, 2007


47

Time-.omain odel

s

+ #

#

R+ #

#

V

Li

v

L

s

Ldt

didt

dv

+

+−=

1

)1

(

1odel +

odel

i L

R R

L

v

dt

di ,

+−=


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June 24-28, 2007


48

Chapter outline


• 3-phase line commuted converters• Static var compensator • C"lo"on3erter

, i Ch t i ti f


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June 24-28, 2007


49

,armonics Characteristics ofCycloconverter

• A cycloconverter generates very comple0 frequencyspectrum that includes side%ands of thecharacteristic harmonics

• 7alanced three-phase outputs) the dominantharmonic frequencies in input current for

$ F-pulse

$ +-pulse

$ p = 6 or p= 12 ) and m = 1, 2, ….

• In general) the currents associated ith theside%and frequencies are relatively small andharmless to the poer system unless a sharplytuned resonance occurs at that frequency!

oih * * pm * 2)1( ±±=

oih * * pm * )1( ±±=


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June 24-28, 2007

,armonic odels for the Cycloconverter

• The harmonic frequencies generated %y acycloconverter depend on its changed outputfrequency) it is very difficult to eliminate themcompletely

• To date) the time-domain and current sourcemodels are commonly used for modelingharmonics

• The harmonic currents in;ected into a poer

system %y cycloconverters still present achallenge to %oth researchers and industrialengineers!

chapter 4 br

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