1. Chapter 4 Modeling of Nonlinear Load
Contributors: S. Tsai, Y. Liu, and G. W. Chang
Organized by
Task Force on Harmonics Modeling & Simulation
Adapted and Presented by Paulo F Ribeiro
AMSC
May 28-29, 2008
Certain devices exhibit nonlinear input and output characteristics which make the prediction of its behavior difficult. We are familiar with linear system, for nonlinear devices, we can still understand its characteristics through linearized modeling. In this chapter, we will look at different nonlinear devices and see how to model them under harmonic studies.
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2. Chapter outline Introduction Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
In today’s talk, we will briefly introduce some common models of different devices for harmonic studies. Among the harmonics generating devices, we select the most significant ones listed on this slide. There are other devices such as PCs, TVs, energy efficient lights, and battery chargers also generate currents rich in odd harmonics, but the harmonic magnitudes are usually small and the phase angles are diversified. Harmonic modeling for their group effects generally requires statistical or probabilistic approaches. These topics will not be discussed here.
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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
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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
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5. Chapter outline Introduction Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
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6. Nonlinear Magnetic Core Sources
Harmonics characteristics
Harmonics model for transformers
Harmonics model for rotating machines
First, we will introduce the harmonic models for non-linear magnetic core sources. We will briefly introduce its general characteristics, and then we will look into different ways to model transformers and rotating machines
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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
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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
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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.
The equivalent model for transformer in time domain, a single phase transformer can be represented on the right.
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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.
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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
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
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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.
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13. Rotating machines Harmonic models for synchronous machine
Harmonic models for Induction machine
Since synchronous machines and induction motors have different harmonic characteristics, we will illustrate the harmonic models for synchronous and induction machines.
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14. Synchronous machines Harmonics origins: Harmonic models
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
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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: (accounting for skin effect and eddy current losses)
Rneg and Xneg are the negative sequence resistance and reactance at fundamental frequency
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16. 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.
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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
The model is similar to that of the synchronous machines as we previously talked about.
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18. Harmonic models for induction motor
Balanced Condition
Generalized Double Cage Model
Equivalent T Model
Unbalanced Condition
For induction motor, we can generally model it from 2 aspects: under balanced condition and unbalanced condition
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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.
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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.
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21. 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
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22. Chapter outline Introduction Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
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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
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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
Figure (a) shows the typical voltage-current characteristics of the furnace arc
Figure (b) shows the resulting arc current and voltage when a sinusoidal source supplies the furnace
The last figure shows the typical spectrum of arc furnace current. It can be seen that besides integer harmonics, the arc furnace currents are also rich in inter-harmonics.
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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
Although an arc furnace can be modeled simply as an inductor in series with a resistor, many complex models were proposed to more precisely represent its characteristics and to study its impacts on power systems. Here is the list of 7 models to be covered here. Next, we will briefly introduce each one of them.
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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.
The simplified V-I characteristics can be modeled into the circuit as shown. Several characteristics are listed.
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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.
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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
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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:
Deterministic: Dl is the maximum variation of arc length, and  is an angular frequency in the range of 0.5 to 25 Hz, which is the popular flicker frequency range of an arc furnace.
Stochastic: R(t) is a band-limited white noise. Its frequency band is in the range of the flicker frequency of the arc furnace.
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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
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31. Power 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
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32. Chapter outline Introduction Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
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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
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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
the ac harmonic currents generated by a six-pulse converter include all odd harmonics except triplens
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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
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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
For harmonic current magnitude, the subscript 'sp' indicates the typical harmonic current spectrum of the source
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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
the dc voltage is directly computed by summing each input phase voltage, V, multiplied by its corresponding transfer function.
I is the input current of the converter of each phase
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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)
In the iterative harmonic analysis, the converter is usually represented by a fixed harmonic current source at each iteration. For better convergence, the Norton equivalent can be used for the converter.
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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.
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40. Chapter outline Introduction Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
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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
(b) shows a typical current waveform generated by a TCR branch, where the dash line represents the supply voltage.
All odd harmonics are present in the input current for the TCR
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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
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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
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44. Current Source Model
by discrete Fourier analysis
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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
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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
YTCR=YRS can be thought of TCR harmonic admittance matrix or transfer function
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47. Time-Domain Model Model 1 Model 2
Model 1: a single-phase SVC that includes a fixed capacitor in parallel with the TCR. The power system is represented by its Thévenin equivalent. s is the switching function
Model 2: two anti-parallel thyristors are replaced by two time-varying resistors, Rt1 and Rt2. An equivalent resistance, RT, is then used to represent these resistors. RT becomes zero when any of the thyristors is conducted
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48. Chapter outline Introduction Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
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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.
the first term specifies the characteristic harmonics from a p-pulse converter and the second term represents sidebands of each of the dominant characteristic harmonics, which may not be integers
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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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