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Tutorial on Harmonics Modeling and Simulation. Chapter 4 Modeling of Nonlinear Load. Contributors: S. Tsai, Y. Liu, and G. W. Chang. Chapter outline. Introduction Nonlinear magnetic core sources Arc furnace 3-phase line commuted converters Static var compensator Cycloconverter.

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chapter outline
Chapter outline
  • Introduction
  • Nonlinear magnetic core sources
  • Arc furnace
  • 3-phase line commuted converters
  • Static var compensator
  • Cycloconverter
  • 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
introduction cont
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
chapter outline5
Chapter outline
  • Introduction
  • Nonlinear magnetic core sources
  • Arc furnace
  • 3-phase line commuted converters
  • Static var compensator
  • Cycloconverter
nonlinear magnetic core sources
Nonlinear Magnetic Core Sources
  • Harmonics characteristics
  • Harmonics model for transformers
  • Harmonics model for rotating machines
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
harmonic models for transformers
Harmonic models for transformers
  • Harmonic models for a transformer:
    • equivalent circuit model
    • differential equation model
    • duality-based model
    • GIC (geomagnetically induced currents) saturation model
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.
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.
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
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.
rotating machines
Rotating machines
  • Harmonic models for synchronous machine
  • Harmonic models for Induction machine
synchronous machines
Synchronous 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
balanced harmonic analysis
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
unbalanced harmonic analysis
Unbalanced harmonic analysis
  • The balanced three-phase coupled matrix model can be used for unbalanced network analysis
    • Zs=(Zo+2Zneg)/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.
induction motors
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
harmonic models for induction motor
Harmonic models for induction motor
  • Balanced Condition
    • Generalized Double Cage Model
    • Equivalent T Model
  • Unbalanced Condition
generalized double cage model for induction motor
Generalized Double Cage Model for Induction Motor


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.

equivalent t model for induction motor
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.
unbalanced model for induction motor
Unbalanced model for Induction Motor
  • The balanced three-phase coupled matrix model can be used for unbalanced network analysis
    • Zs=(Zo+2Zpos)/3
    • Zm=(ZoZpos)/3
    • Zo and Zpos are zero and positive sequence impedance at hth harmonic order
  • Z0 can be determined from
chapter outline22
Chapter outline
  • Introduction
  • Nonlinear magnetic core sources
  • Arc furnace
  • 3-phase line commuted converters
  • Static var compensator
  • Cycloconverter
arc furnace harmonic sources
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
characteristics of harmonics generated by arc furnaces
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
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
nonlinear resistance model
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.
current source model
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.
voltage source model
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
nonlinear time varying voltage source model
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:
nonlinear time varying resistance models
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
power balance model
Power balance model
  • ris 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
chapter outline32
Chapter outline
  • Introduction
  • Nonlinear magnetic core sources
  • Arc furnace
  • 3-phase line commuted converters
  • Static var compensator
  • Cycloconverter
three phase line commuted converters
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
harmonics characteristics
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
harmonic models for the three phase line commutated converter
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
current source model36
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
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
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)
harmonic domain model
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.
chapter outline40
Chapter outline
  • Introduction
  • Nonlinear magnetic core sources
  • Arc furnace
  • 3-phase line commuted converters
  • Static var compensator
  • Cycloconverter
harmonics characteristics of tcr
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
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
harmonic models for tcr
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
current source model44
Current Source Model

by discrete Fourier analysis

norton equivalent model
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

transfer function model46
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
  • YTCR=YRS can be thought of TCR harmonic admittance matrix or transfer function
time domain model
Time-Domain Model

Model 1

Model 2

chapter outline48
Chapter outline
  • Introduction
  • Nonlinear magnetic core sources
  • Arc furnace
  • 3-phase line commuted converters
  • Static var compensator
  • Cycloconverter
harmonics characteristics of cycloconverter
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.
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.