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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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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, 20072
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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IEEE PES General Meeting, Tampa FL
June 24-28, 2007Coner!n"ia #ra$ileira %e &uali%a%e %e Energia
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Chapter outline
• Introduction• +onlinear magneti" "ore $our"e$• Arc furnace
• 3-phase line commuted converters• Static var compensator • Cycloconverter
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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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June 24-28, 2007Coner!n"ia #ra$ileira %e &uali%a%e %e Energia
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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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Chapter outline
• Introduction• Nonlinear magnetic core sources• (r" urna"e
• 3-phase line commuted converters• Static var compensator • Cycloconverter
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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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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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,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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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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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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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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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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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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&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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Chapter outline
• Introduction• Nonlinear magnetic core sources• Arc furnace
• -p*a$e line "ommute% "on3erter$• Static var compensator • Cycloconverter
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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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,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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,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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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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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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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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,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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Chapter outline
• Introduction• Nonlinear magnetic core sources• Arc furnace
• 3-phase line commuted converters• Stati" 3ar "ompen$ator • Cycloconverter
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,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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,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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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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Current Source odel
∑ +=
h
hhh t h I t i )&in()( θ ω
+
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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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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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48
Chapter outline
• Introduction• Nonlinear magnetic core sources• Arc furnace
• 3-phase line commuted converters• Static var compensator • C"lo"on3erter
, i Ch t i ti f
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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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,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!
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