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## Chapter 4 Modeling of Nonlinear Load

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### ... also be used to simulate the harmonic generation behavior of power transformers. ... Duality-based models are necessary to represent multi-legged transformers ... – PowerPoint PPT presentation

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Title: Chapter 4 Modeling of Nonlinear Load

1
Chapter 4 Modeling of Nonlinear Load
Tutorial on Harmonics Modeling and Simulation
• Contributors S. Tsai, Y. Liu, and G. W. Chang

2
Chapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter

3
Introduction
• 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.)
• In the following, we will present the harmonics
for each devices in the following sequence
• Harmonic characteristics
• Harmonic models and assumptions
• Discussion of each model

5
Chapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter

6
Nonlinear Magnetic Core Sources
• Harmonics characteristics
• Harmonics model for transformers
• Harmonics model for rotating machines

7
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 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)
• 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)
• 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.

11
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

12
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.

13
Rotating machines
• Harmonic models for synchronous machine
• Harmonic models for Induction machine

14
Synchronous machines
• Harmonics origins
• Non-sinusoidal flux distribution
• The resulting voltage harmonics are odd and
usually minimized in the machines 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 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

16
Unbalanced harmonic analysis
• The balanced three-phase coupled matrix model can
be used for unbalanced network analysis
• Zs(Zo2Zneg)/3
• Zm(Zo?Zneg)/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.

17
Induction 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 motor
• Balanced Condition
• Generalized Double Cage Model
• Equivalent T Model
• Unbalanced Condition

19
Generalized Double Cage Model for Induction Motor
Stator
mutual reactance of the 2 rotor cages
Excitation branch
2 rotor cages
At the h-th harmonic order, the equivalent
circuit can be obtained by multiplying h with
each of the reactance.
20
Equivalent 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.

21
Unbalanced model for Induction Motor
• The balanced three-phase coupled matrix model can
be used for unbalanced network analysis
• Zs(Zo2Zpos)/3
• Zm(Zo?Zpos)/3
• Zo and Zpos are zero and positive sequence
impedance at hth harmonic order
• Z0 can be determined from

22
Chapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter

23
Arc 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
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

25
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 model
simplified to
modeled as
• 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.

27
Current 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.

28
Voltage 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 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

30
Nonlinear 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

31
Power balance model
• r is the arc radius
• exponent n is selected according to the arc
cooling environment, n0, 1, or 2
• recommended values for exponent m are 0, 1 and 2
• K1, K2 and K3 are constants

32
Chapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter

33
Three-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 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

35
Harmonic 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 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

37
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

38
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)

39
Harmonic-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.

40
Chapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter

41
Harmonics 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

42
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

43
Harmonic 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 Model
by discrete Fourier analysis
45
Norton-Equivalent Model
• The input voltage is unbalanced and no coupling
between different harmonics are assumed

Norton equivalence for the harmonic power flow
analysis of the TCR for the h-th harmonic
46
Transfer 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
• YTCRYRS can be thought of TCR harmonic
admittance matrix or transfer function

47
Time-Domain Model
Model 1
Model 2
48
Chapter outline
• Introduction
• Nonlinear magnetic core sources
• Arc furnace
• 3-phase line commuted converters
• Static var compensator
• Cycloconverter

49
Harmonics 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.

50
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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