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
1
Chapter outline
•
•
•
•
•
•
Introduction
Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
2
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
3
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
4
Chapter outline
•
•
•
•
•
•
Introduction
Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
5
Nonlinear Magnetic Core Sources
•
Harmonics characteristics
•
Harmonics model for transformers
•
Harmonics model for rotating machines
6
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
7
Harmonic models for transformers
•
Harmonic models for a transformer:
–
–
–
–
equivalent circuit model
differential equation model
duality-based model
GIC (geomagnetically induced currents) saturation
model
8
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.
9
Ip
Rp
Lp
Ls
+
Rs
Is
+
Iex
+
Vin
-
Vm
Lm
Rm
-
Im
Vout
-
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
 v1   R11 R12
v  R
 2    21 R22
    

  
v N   R N 1 R N 2
 L11 L12
L
L22
  21
 


 L N1 L N 2








R1N   i1 
R2 N   i 2 
   
 
R NN  i N 
L1N   i1 
L2 N  d  i 2 
  dt   
  
L NN  i N 
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.
10
Duality-based model (transformer)
•
•
•
Duality-based models are
necessary to represent multilegged 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
11
Magnetic circuit
Electric circuit
Magnetomotive
Force (FMM) Ni
Electromotive Force
(FEM) E
Flux 
Current I
Reluctance 
Resistance R
Permeance
1/ 
Conductance 1 / R
Flux density
Current density
Magnetizing force
H
Potential difference
V
B  / A
J I/A
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.
12
Rc2
Rc1
Ra1
Ra4’
AC
~
DC
F
Ra4
Ra3
Rc3
Rt4
Rc2
Rt3
Rotating machines
•
Harmonic models for synchronous machine
•
Harmonic models for Induction machine
13
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
14
Balanced harmonic analysis
•
For balanced (single phase) harmonic analysis, a
synchronous machine was often represented by
a single approximation of inductance


Lh  h L"d  L"q / 2
•
– h: harmonic order
– L"d : direct sub-transient inductance
– L"q : quadrature sub-transient inductance
A more complex model
Z h  h a Rneg  jhX neg
– 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
15
Unbalanced harmonic analysis
•
The balanced three-phase coupled matrix model
can be used for unbalanced network analysis
 Zs
Z h  Z m
Z m
Zm
Zs
Zm
Zm 
Z m 
Z s 
– 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.
16
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
17
Harmonic models for induction motor
•
Balanced Condition
– Generalized Double Cage Model
– Equivalent T Model
•
Unbalanced Condition
18
Generalized Double Cage Model for
Induction Motor
Stator
Rs
mutual reactance of the 2 rotor cages
jXs
jXr
R1(s)
R2(s)
jX1
jX2
jXm
Rc
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.
19
Equivalent T model for Induction Motor
Rs
jhXs
jhXr
Rc
•
•
•
jhXm
Rr
sh
sh 
h  1  s 
h
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.
20
Unbalanced model for Induction Motor
•
The balanced three-phase coupled matrix model can be used for
unbalanced network analysis
 Zs
Z h  Z m
Z m
•
Zm
Zs
Zm
Zm 
Z m 
Z s 
– Zs=(Zo+2Zpos)/3
– Zm=(ZoZpos)/3
– Zo and Zpos are zero and positive sequence impedance at hth
R
jX
harmonic order
Z0 can be determined from
s0
s0
Rm0
21
0.5Rr0
(-2+3s)
Rm0
2
2
0.5Rr0
(4-3s)
jXm0
2
jXr0
2
jXm0
2
jXr0
2
Chapter outline
•
•
•
•
•
•
Introduction
Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
22
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
23
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
(a)
24
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
25
Nonlinear resistance model
(a)
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.
26
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


n 1
n0
i L t    a n sin nt   bn cos nt
•
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.
27
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
28
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
Va l0   k t Vao l0 
– 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:
l t   lo  Dl 21  sin t 
l t   lo  Rt 
29
Nonlinear time varying resistance models
•
During normal operation, the arc resistance can
be modeled to follow an approximate Gaussian
distribution
Rarc  R    2 ln RAND1  cos2RAND2 
•
–  is the variance which is determined by short-term
perceptibility flicker index Pst
Another time varying resistance model:
R1 
Vig2
Vig2
2
Vex
P

R2
R2
– 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
30
Power balance model
K3 2
dr
K1r  K 2 r

i
m

2
dt r
n
•
•
•
•
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
31
Chapter outline
•
•
•
•
•
•
Introduction
Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
32
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
33
Harmonics Characteristics
•
Under balanced condition with constant output current and
assuming zero firing angle and no commutation overlap, phase a
current is
ia (t )   ( 2 I1 / h) sin(h1t   h )
h
h = 1, 5, 7, 11, 13, ...
– Characteristic harmonics generated by converters of any pulse
number are in the order of h  pn  1

•
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
I h  6 I d F ( ,  ) /{h[cos   cos(   )]}
– F(,) is an overlap function
34
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
35
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
I h  I rated  I hsp / I1sp
 h   hsp  h(1  1sp )
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
36
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
Vdc   G , dcV ,   a, b, c

•
•
i  G , ac idc ,   a,b,c
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
37
Norton-Equivalent Circuit Model
•
•
The nonlinear relationship between
converter input currents and its terminal
voltages is I  f (V)
– 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)
38
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
H 2H h
n
The overall dc voltage Vd  12
 Vk , p   p    V  p
p 1
h 1 n 1
k, p
– 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.
39
Chapter outline
•
•
•
•
•
•
Introduction
Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
40
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
4V1
I h ( ) 
 L R
 cos  sin(h )  h cos(h ) sin  


2


h(h  1)
– h = 3, 5, 7, ···,  is the conduction angle, and LR is the
inductance of the reactor
41
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 vs (t )  Vh sin(ht  h )
h
– the current through the compensator is proved to be
the discontinuous current

 
 V1
[cos(
h



)

cos(
h

t


)],

t


h
h

 h h L


i (t )  

 
 
 0,
0t 
and
t





42
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
43
Current Source Model

 
 V1
[cos(
h



)

cos(
h

t


)],

t

h
h
 hL


i (t )   h

 
 
 0,
0t 
and
t




by discrete Fourier analysis
i h (t )   I h sin(ht   h )
h
44
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
Yh eq  ( jh Leq ) 1
Vh  Vhh
45
Ι heq  Vh /( jhLeq )  I h
I h  I hh
Leq  LR /(  sin  )
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
VR  s VS
I R  YR VR  YTCR VS
•
YTCR=YRS can be thought of TCR harmonic
admittance matrix or transfer function
46
Time-Domain Model
Model 1
 dvc 
0
1 v   0
 dt  
c
 di    ( 1  s ) 0     1
  ic   L
 c   LS L R

 S
 dt 
Model 2

Vs


di v R  RT
 
i
dt L
L
47
Chapter outline
•
•
•
•
•
•
Introduction
Nonlinear magnetic core sources
Arc furnace
3-phase line commuted converters
Static var compensator
Cycloconverter
48
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
f h  ( pm  1) f i  2kfo
– 12-pulse f h  ( pm  1) f i  6kfo
– 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.
49
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.
50