1 chapter 4 modeling of nonlinear load organized by task force on harmonics modeling &...
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1
Chapter 4 Modeling of Nonlinear Load
Chapter 4 Modeling of Nonlinear Load
Organized by
Task Force on Harmonics Modeling & Simulation
Adapted and Presented by Paulo F Ribeiro
AMSC
May 28-29, 2008
Contributors: S. Tsai, Y. Liu, and G. W. Chang
2
Chapter outlineChapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter
3
IntroductionIntroduction
• The purpose of harmonic studies is to quantify the distortion in voltage and/or current waveforms at various locations in a power system.
• One important step in harmonic studies is to characterize and to model harmonic-generating sources.
• Causes of power system harmonics – Nonlinear voltage-current characteristics
– Non-sinusoidal winding distribution
– Periodic or aperiodic switching devices
– Combinations of above
4
Introduction (cont.)Introduction (cont.)
• In the following, we will present the harmonics for each devices in the following sequence:
1. Harmonic characteristics
2. Harmonic models and assumptions
3. Discussion of each model
5
Chapter outlineChapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter
6
Nonlinear Magnetic Core Sources Nonlinear Magnetic Core Sources
• Harmonics characteristics
• Harmonics model for transformers
• Harmonics model for rotating machines
7
Harmonics characteristics of iron-core reactors and transformers
Harmonics characteristics of iron-core reactors and transformers
• Causes of harmonics generation– Saturation effects– Over-excitation
temporary over-voltage caused by reactive power unbalance unbalanced transformer load asymmetric saturation caused by low frequency magnetizing current transformer energization
• Symmetric core saturation generates odd harmonics • Asymmetric core saturation generates both 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
8
Harmonic models for transformersHarmonic models for transformers
• Harmonic models for a transformer:– equivalent circuit model
– differential equation model
– duality-based model
– GIC (geomagnetically induced currents) saturation model
9
Equivalent circuit model (transformer)Equivalent circuit model (transformer)
• In time domain, a single phase transformer can be represented by an equivalent circuit referring all impedances to one side of the transformer
• The core saturation is modeled using a piecewise linear approximation of saturation
• This model is increasingly available in time domain circuit simulation packages.
10
Differential equation model (transformer)Differential equation model (transformer)
• The differential equations describe the relationships between – winding voltages– winding currents– winding resistance– winding turns– magneto-motive forces– mutual fluxes– leakage fluxes– reluctances
• Saturation, hysteresis, and eddy current effects can be well modeled.
• The models are suitable for transient studies. They may also be used to simulate the harmonic generation behavior of power transformers.
NNNNN
N
N
NNNNN
N
N
N
i
i
i
dt
d
LLL
LLL
LLL
i
i
i
RRR
RRR
RRR
v
v
v
2
1
21
22221
11211
2
1
21
22221
11211
2
1
11
Duality-based model (transformer)Duality-based model (transformer)
• Duality-based models are necessary to represent multi-legged transformers
• Its parameters may be derived from experiment data and a nonlinear inductance may be used to model the core saturation
• Duality-based models are suitable for simulation of power system low-frequency transients. They can also be used to study the harmonic generation behaviors
Magnetic circuit Electric circuit
Magnetomotive Force (FMM) Ni
Electromotive Force (FEM) E
Flux Current I
Reluctance Resistance R
Permeance Conductance
Flux density Current density
Magnetizing force
H
Potential difference
V
/1 R/1
AB / AIJ /
12
GIC saturation model (transformer)GIC saturation model (transformer)
• Geomagnetically induced currents GIC bias can cause heavy half cycle saturation– the flux paths in and between core,
tank and air gaps should be accounted
• A detailed model based on 3D finite element calculation may be necessary.
• Simplified equivalent magnetic circuit model of a single-phase shell-type transformer is shown.
• An iterative program can be used to solve the circuitry so that nonlinearity of the circuitry components is considered.
F
~AC
DC
Rc1 Ra1
Ra4
Ra4’
Rt4
Rc3
Rc2
Rc2
Ra3
Rt3
13
Rotating machinesRotating machines
• Harmonic models for synchronous machine
• Harmonic models for Induction machine
14
Synchronous machinesSynchronous machines
• Harmonics origins:– Non-sinusoidal flux distribution
The resulting voltage harmonics are odd and usually minimized in the machine’s design stage and can be negligible.
– Frequency conversion process Caused under unbalanced conditions
– Saturation Saturation occurs in the stator and rotor core, and in the
stator and rotor teeth. In large generator, this can be neglected.
• Harmonic models– under balanced condition, a single-phase inductance is
sufficient– under unbalanced conditions, a impedance matrix is
necessary
15
Balanced harmonic analysisBalanced harmonic analysis
• For balanced (single phase) harmonic analysis, a synchronous machine was often represented by a single approximation of inductance
– h: harmonic order– : direct sub-transient inductance– : quadrature sub-transient inductance
• A more complex model
– a: 0.5-1.5 (accounting for skin effect and eddy current losses)
– Rneg and Xneg are the negative sequence resistance and reactance at fundamental frequency
2/""qdh LLhL
"dL
"qL
negnega
h jhXRhZ
16
Unbalanced harmonic analysisUnbalanced harmonic analysis
• The balanced three-phase coupled matrix model can be used for unbalanced network analysis
– Zs=(Zo+2Zneg)/3– Zm=(ZoZneg)/3
– Zo and Zneg are zero and negative sequence impedance at hth harmonic order
• If the synchronous machine stator is not precisely balanced, the self and/or mutual impedance will be unequal.
smm
msm
mms
h
ZZZ
ZZZ
ZZZ
Z
17
Induction motorsInduction motors
• Harmonics can be generated from– Non-sinusoidal stator winding distribution
Can be minimized during the design stage
– Transients Harmonics are induced during cold-start or load changing
– The above-mentioned phenomenon can generally be neglected
• The primary contribution of induction motors is to act as impedances to harmonic excitation
• The motor can be modeled as– impedance for balanced systems, or– a three-phase coupled matrix for unbalanced systems
18
Harmonic models for induction motorHarmonic models for induction motor
• Balanced Condition– Generalized Double Cage Model
– Equivalent T Model
• Unbalanced Condition
19
Generalized Double Cage Model for Induction MotorGeneralized Double Cage Model for Induction Motor
Rs jXs
RcjXm
jXr
R1(s)
jX1
R2(s)
jX2
Stator
Excitation branch
At the h-th harmonic order, the equivalent circuit can be obtained by multiplying h with each of the reactance.
mutual reactance of the 2 rotor cages
2 rotor cages
20
Equivalent T model for Induction MotorEquivalent T model for Induction Motor
• s is the full load slip at fundamental frequency, and h is the harmonic order
• ‘-’ is taken for positive sequence models
• ‘+’ is taken for negative sequence models.
Rs jhXs
Rc jhXm
jhXr
Rrsh
h
shsh
1
21
Unbalanced model for Induction MotorUnbalanced model for Induction Motor
• The balanced three-phase coupled matrix model can be used for unbalanced network analysis
– Zs=(Zo+2Zpos)/3 – Zm=(ZoZpos)/3
– Zo and Zpos are zero and positive sequence impedance at hth harmonic order• Z0 can be determined from Rs0 jXs0
Rm0
20.5Rr0
(-2+3s)
jXm0
2jXr0
2
Rm0
20.5Rr0
(4-3s)
jXm0
2jXr0
2
smm
msm
mms
h
ZZZ
ZZZ
ZZZ
Z
22
Chapter outlineChapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter
23
Arc furnace harmonic sourcesArc furnace harmonic sources
• Types:– AC furnace
– DC furnace
• DC arc furnace are mostly determined by its AC/DC converter and the characteristic is more predictable, here we only focus on AC arc furnaces
24
Characteristics of Harmonics Generated by Arc FurnacesCharacteristics of Harmonics Generated by Arc Furnaces
• The nature of the steel melting process is uncontrollable, current harmonics generated by arc furnaces are unpredictable and random.
• Current chopping and igniting in each half cycle of the supply voltage, arc furnaces generate a wide range of harmonic frequencies
(a)
25
Harmonics Models for Arc Furnace Harmonics Models for Arc Furnace
• Nonlinear resistance model
• Current source model
• Voltage source model
• Nonlinear time varying voltage source model
• Nonlinear time varying resistance models
• Frequency domain models
• Power balance model
26
Nonlinear resistance modelNonlinear resistance model
(a)
simplified to
• R1 is a positive resistor
• R2 is a negative resistor
• AC clamper is a current-controlled switch
• It is a primitive model and does not consider the time-varying characteristic of arc furnaces.
modeled as
27
Current source modelCurrent source model
• Typically, an EAF is modeled as a current source for harmonic studies. The source current can be represented by its Fourier series
• an and bn can be selected as a function of– measurement
– probability distributions
– proportion of the reactive power fluctuations to the active power fluctuations.
• This model can be used to size filter components and evaluate the voltage distortions resulting from the harmonic current injected into the system.
1 0cossin
n nnnL tnbtnati
28
Voltage source modelVoltage source model
• The voltage source model for arc furnaces is a Thevenin equivalent circuit.– The equivalent impedance is the furnace load
impedance (including the electrodes)
– The voltage source is modeled in different ways: form it by major harmonic components that are known
empirically account for stochastic characteristics of the arc furnace
and model the voltage source as square waves with modulated amplitude. A new value for the voltage amplitude is generated after every zero-crossings of the arc current when the arc reignites
29
Nonlinear time varying voltage source modelNonlinear time varying voltage source model
• This model is actually a voltage source model
• The arc voltage is defined as a function of the arc length
– Vao :arc voltage corresponding to the reference arc length lo,
– k(t): arc length time variations
• The time variation of the arc length is modeled with deterministic or stochastic laws.– Deterministic:
– Stochastic:
00 laoVtklaV
tDlltl o sin12
tRltl o
30
Nonlinear time varying resistance modelsNonlinear time varying resistance models
• During normal operation, the arc resistance can be modeled to follow an approximate Gaussian distribution
is the variance which is determined by short-term perceptibility flicker index Pst
• Another time varying resistance model:
– R1: arc furnace positive resistance and R2 negative resistance
– P: short-term power consumed by the arc furnace– Vig and Vex are arc ignition and extinction voltages
RAND22cosRAND1ln2 RRarc
2
2
2
2
2
1
R
V
R
VP
VR
exig
ig
31
Power balance modelPower balance model
• r is the arc radius
• exponent n is selected according to the arc cooling environment, n=0, 1, or 2
• recommended values for exponent m are 0, 1 and 2
• K1, K2 and K3 are constants
22
321 i
r
K
dt
drrKrK
mn
32
Chapter outlineChapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter
33
Three-phase line commuted convertersThree-phase line commuted converters
• Line-commutated converter is mostly usual operated as a six-pulse converter or configured in parallel arrangements for high-pulse operations
• Typical applications of converters can be found in AC motor drive, DC motor drive and HVDC link
34
Harmonics CharacteristicsHarmonics Characteristics
• Under balanced condition with constant output current and assuming zero firing angle and no commutation overlap, phase a current is
h = 1, 5, 7, 11, 13, ...
– Characteristic harmonics generated by converters of any pulse number are in the order of n = 1, 2, ··· and p is the pulse number of the converter
• For non-zero firing angle and non-zero commutation overlap, rms value of each characteristic harmonic current can be determined by
– F(,) is an overlap function
)]}cos([cos/{),(6 hFII dh
h
ha thhIti )sin()/2()( 11
1pnh
35
Harmonic Models for the Three-Phase Line-Commutated ConverterHarmonic Models for the Three-Phase Line-Commutated Converter
• Harmonic models can be categorized as– frequency-domain based models
current source model transfer function model Norton-equivalent circuit model harmonic-domain model three-pulse model
– time-domain based models models by differential equations state-space model
36
Current source modelCurrent source model
• The most commonly used model for converter is to treat it as known sources of harmonic currents with or without phase angle information
• Magnitudes of current harmonics injected into a bus are determined from – the typical measured spectrum and
– rated load current for the harmonic source (Irated)
• Harmonic phase angles need to be included when multiple sources are considered simultaneously for taking the harmonic cancellation effect into account. h, and a conventional load flow solution is needed for
providing the fundamental frequency phase angle, 1
spsphratedh IIII 1/
)( 11 spsphh h
37
Transfer Function Model Transfer Function Model
• The simplified schematic circuit can be used to describe the transfer function model of a converter
• G: the ideal transfer function without considering firing angle variation and commutation overlap
• G,dc and G,ac, relate the dc and ac sides of the converter
• Transfer functions can include the deviation terms of the firing angle and commutation overlap
• The effects of converter input voltage distortion or unbalance and harmonic contents in the output dc current can be modeled as well
cbaVGV dcdc ,,, ,
a,b,ciGi dcac , ,
38
Norton-Equivalent Circuit Model Norton-Equivalent Circuit Model
• The nonlinear relationship between converter input currents and its terminal voltages is
– I & V are harmonic vectors
• If the harmonic contents are small, one may linearize the dynamic relations about the base operating point and obtain: I = YJV + IN
– YJ is the Norton admittance matrix representing the linearization. It also represents an approximation of the converter response to variations in its terminal voltage harmonics or unbalance
– IN = Ib - YJVb (Norton equivalent)
)(VI f
39
Harmonic-Domain ModelHarmonic-Domain Model
• Under normal operation, the overall state of the converter is specified by the angles of the state transition– These angles are the switching instants corresponding to the
6 firing angles and the 6 ends of commutation angles
• The converter response to an applied terminal voltage is characterized via convolutions in the harmonic domain
• The overall dc voltage
– Vk,p: 12 voltage samples p: square pulse sampling functions
– H: the highest harmonic order under consideration
• The converter input currents are obtained in the same manner using the same sampling functions.
H
h
H
n
np
hpk
pppkd VVV
1
2
1,
12
1,
40
Chapter outlineChapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter
41
Harmonics characteristics of TCRHarmonics characteristics of TCR
• Harmonic currents are generated for any conduction intervals within the two firing angles
• With the ideal supply voltage, the generated rms harmonic currents
– h = 3, 5, 7, ···, is the conduction angle, and LR is the inductance of the reactor
)1(
sin)cos()sin(cos4)(
21
hh
hhh
L
VI
Rh
42
Harmonics characteristics of TCR (cont.)Harmonics characteristics of TCR (cont.)
• Three single-phase TCRs are usually in delta connection, the triplen currents circulate within the delta circuit and do not enter the power system that supplies the TCRs.
• When the single-phase TCR is supplied by a non-sinusoidal input voltage
– the current through the compensator is proved to be the discontinuous current
tt
tthhLh
V
ti hhh
and 0 ,0
)],cos()[cos()(
1
h
hhs thVtv )sin()(
43
Harmonic models for TCRHarmonic models for TCR
• Harmonic models for TCR can be categorized as– frequency-domain based models
current source model transfer function model Norton-equivalent circuit model
– time-domain based models models by differential equations state-space model
44
Current Source ModelCurrent Source Model
h
hhh thIti )sin()(
tt
tthhLh
V
ti hhh
and 0 ,0
)],cos()[cos()(
1
by discrete Fourier analysis
45
Norton equivalence for the harmonic power flow analysis of the TCR for the h-th harmonic
Norton-Equivalent ModelNorton-Equivalent Model
• The input voltage is unbalanced and no coupling between different harmonics are assumed
1)( eqeqh LjhY heqheqh Ljh IVΙ )/(
hhh V V hhh I I )sin/( Req LL
46
Transfer Function ModelTransfer Function Model
• Assume the power system is balanced and is represented by a harmonic Thévenin equivalent
• The voltage across the reactor and the TCR current can be expressed as
• YTCR=YRS can be thought of TCR harmonic admittance matrix or transfer function
R SV s V
STCRRRR V Y V Y I
47
Time-Domain ModelTime-Domain Model
sSc
c
RSc
c
VLi
v
L
s
Ldt
didt
dv
10
0)1
(
10Model 1
Model 2i
L
RR
L
v
dt
di T
48
Chapter outlineChapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter
49
Harmonics Characteristics of CycloconverterHarmonics Characteristics of Cycloconverter
• A cycloconverter generates very complex frequency spectrum that includes sidebands of the characteristic harmonics
• Balanced three-phase outputs, the dominant harmonic frequencies in input current for – 6-pulse
– 12-pulse
– p = 6 or p= 12, and m = 1, 2, ….
• In general, the currents associated with the sideband frequencies are relatively small and harmless to the power system unless a sharply tuned resonance occurs at that frequency.
oih kffpmf 2)1(
oih kffpmf 6)1(
50
Harmonic Models for the Cycloconverter Harmonic Models for the Cycloconverter
• The harmonic frequencies generated by a cycloconverter depend on its changed output frequency, it is very difficult to eliminate them completely
• To date, the time-domain and current source models are commonly used for modeling harmonics
• The harmonic currents injected into a power system by cycloconverters still present a challenge to both researchers and industrial engineers.
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