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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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