Download Chapter 2-AMSC

Chapter 2
Harmonics and Interharmonics Theory
Contributors: G. W. Chang and A. Testa
Organized by
Task Force on Harmonics Modeling & Simulation
Adapted and Presented by Paulo F Ribeiro
AMSC
May 28-29, 2008
1
Outline
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•
•
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•
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Introduction
Fourier Series and Analysis
Basic Definition of Harmonic and Interharmonic
Quantities
Harmonic and Interharmonic Indices
Power Factor under Distorted Situation
Power System Response to Harmonics and
Interharmonics
Solutions to Harmonics and Interharmonics
Summary
2
Introduction
•
With the widespread proliferation of nonlinear
devices such as power electronics and arc
furnace loads, significant amounts of harmonic
and interharmonic currents are being injected
into the power system.
•
Harmonic and interharmonic currents not only
disturb loads that are sensitive to waveform
distortion, but also cause many undesirable
effects on power system components.
3
Introduction
•
Harmonic Sources（nonlinear loads）
– Single-phase loads: fluorescent lights, personal computers
– Three-phase loads: arc furnaces, ac/dc converters
4
Introduction
5
Introduction
•
Harmonics and interharmonics are usually
defined as periodic steady-state distortions of
voltage and/or current waveforms in the power
system.
•
In the harmonics and interharmonics polluted
environment, the theory regarding harmonic and
interharmonic quantities needs to be defined to
distinguish from those quantities defined for the
system fundamental frequency.
6
Fourier Series and Analysis
•
Periodic Function
f (t )  f (t  hT ), h  0,  1,  2, ... Period: T
•
Orthogonal Function { h (t )}
  (t ) (t ) dt   0, i  j
 i

j
 , i = j
e.g.
{1, cos0t , ..., cosh0t , ..., sin0t , ..., sinh0t , ...} T / 2  t  T / 2
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Fourier Series and Analysis

1
f (t )  a0  {ah cos(h0t )  bh sin(h0t )} 0  2 / T
2
h 1

f (t )  c0   ch sin(h0t  h )
h 1
By the use of orthogonal relations, we have
2 T /2
2 T /2
a

T / 2 f (t ) cos(h0t )dt
a0  T / 2 f (t )dt
h
T
T
2 T /2
bh  T / 2 f (t ) sin(h0t )dt
T
c0  a0 / 2
ch 
ah2  bh2
8
h  tan 1 (ah / bh )
Fourier Series and Analysis
•
Fourier analysis is a process of the de-composition
for any distorted period wave shape into a
fundamental and a series of harmonics.
•
Advantages of Fourier series and analysis:
-Useful for studying electrical networks which contain
non- sinusoidal voltages and currents
- The frequency components are harmonics of the
fundamental frequency
- For linear networks, treat each harmonic separately by
using phasor analysis (frequency domain), then combine
the results and convert back to the time domain
waveform
9
Fourier Series and Analysis
•
Complex Form
f (t ) 

jh  0 t
c
e
 h
h  0,  1,  2, ...
h  
1 T /2
ch  T / 2 f (t )e  jh0t dt
T
•
Waveform Symmetry
- Even function: f (t )  f (t ) (no sine terms)
- Odd function: f (t )   f (t ) (no cosine terms)
- Half-wave symmetry: f (t )   f (t  T / 2) (no even harmonics)
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Fourier Series and Analysis
•
•
Fourier Transform
F ( ) = - f (t )e - jt dt
f (t ) 
1 
jt
d
 F ( )e
2
Discrete Fourier Transform: T frequency-domain spectrum
and the time-domain function are both periodic sampled
functions with N samples per period, Fourier transform
pair becomes
N 1
F (k)  
n 0
N 1
f (nT )e  j 2kn / N
f ( nT )   F ( k)e j 2kn / N
k, n = 0, 1, ..., N-1,
  2 / T T  T / N
k 0
F (k) is the so-called spectrum of f (nT ) which is
assumed to be one cycle of a periodic signal.
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Fourier Series and Analysis
•
The angular frequency resolution of the spectrum is
determined by the length of the signal as   2 / T . Thus,
if T is selected as one period of f (nT ) , the outcome
spectrum will only show components that are integer
multiples of the fundamental frequency, which are defined
as harmonics.
•
If the data length is selected as p cycles (p>1 and is an
integer) of the fundamental, the frequency resolution will
change to   2 / pT . This implies that once we use more
than one fundamental cycle to perform the DFT, it also
becomes possible to obtain components at frequencies
that are not integer multiples of the fundamental.
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Fourier Series and Analysis
•
The non-integer order components, according to the
IEC-61000-2-1 definition, are called interharmonics.
•
The DFT is often used in harmonics and interharmonics
measurement. Fast Fourier transform (FFT) algorithms
are very fast methods for performing the DFT
calculations.
•
There are pitfalls of the aliasing, the spectral leakage,
and the picket-fence effect, when applying FFT for
harmonics and interharmonics computations.
13
Basic Definition of Harmonic and
Interharmonic Quantities
•
The definition of a harmonic can be stated as: A
sinusoidal component of a periodic wave having a
frequency that is an integer multiple of the fundamental
frequency.
•
The interharmonics are defined as those components
with frequencies between two consecutive harmonics
or those components whose frequencies are not
integer multiples of the fundamental power frequency.
•
One special subset of interharmonics that have
frequency values that are less than that of the
fundamental frequency is called sub-harmonics.
14
Basic Definition of Harmonic and
Interharmonic Quantities
•
One major source of harmonics in the power system is the
static power converter. Under ideal operating conditions,
the current harmonics generated by a p-pulse linecommutated converter can be characterized by I h  I1 / h
and
h  pn  1, n  1,2 , ...
e.g. p  6 , h  5,7 ,11,13,17 ,19,...
Such harmonics are usually termed as characteristic
harmonics.
•
Non-characteristic harmonics are typically categorized as
those integer frequency components other than
characteristic ones.
15
Basic Definition of Harmonic and
Interharmonic Quantities
•
The power electronic equipment with double conversion
systems that connects two AC systems with different
frequencies through a DC link can be an interharmonic
source. Variable speed drives, HVDC, and other static
frequency converters are typical examples of this class of
sources. Other sources of interharmonics include timevarying loads such as welder machines and arc furnaces.
•
There are various causes that could lead to the
interharmonic components. One example is a signal that
actually contains in the frequency domain with a
component whose frequency is non-integer multiples of the
fundamental frequency.
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Basic Definition of Harmonic and
Interharmonic Quantities
•
There are cases where the interharmonic components are
produced by the picket-fence effect when applying the FFT,
due to sampling the signal with a spectral leakage. The picketfence effect occurs when the analyzed waveform includes
spectral components which are not an integer multiple of the
FFT fundamental frequency (i.e. the reciprocal of the window
length in time). Such effect may lead to a situation where the
frequency resolution (i.e. the sampled frequency interval) of
the spectrum is not observable for certain frequencies.
•
A frequency component lying between two FFT consecutive
harmonics will affect these two harmonic magnitudes and also
may cause the spectral leakage.
17
Basic Definition of Harmonic and
Interharmonic Quantities
v(t )  2 sin(60  2t )  sin(180  2t )
 sin(280  2t )  sin(550  2t )
Frequency resolution = 30 Hz
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Basic Definition of Harmonic and
Interharmonic Quantities
Interharmonics: fi  ( p1m  1) f s  p2nf0
+
3 Phase
AC Supply
vd
Cd
Induction
Motor
Diode Bridge DC Link
Rectifier
Filter
PWM
Inverter
p1: the pulse number of the rectifier section
p2: the pulse number of the output section
m and n: integers
fs: the power frequency
f0: the inverter output frequency
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Basic Definition of Harmonic and
Interharmonic Quantities
•
Sub-harmonics have frequency
values that are less than that of
the fundamental frequency.
Lighting flicker is one indication of
the presence of interharmonics
around the fundamental power
frequency (including subharmonics), which is due to the
voltage fluctuations with
frequencies being much less than
the system fundamental
frequency. A well-known source of
the voltage fluctuations that cause
light flicker is the arc furnace.
20
v(t )  (V  V sin  m t ) sin  s t
Electric Quantities Under Nonsinusoidal Situation
•
•
•
•
•
•
•
•
Instantaneous voltage and current
Instantaneous power
Average power
RMS voltage and current
Apparent power
Reactive power
Distortion power
Total power factor
21
Electric Quantities Under Nonsinusoidal Situation
•
Instantaneous Voltage and Current


h 1
h 1


h 1
h 1
v(t )   vh (t )   2Vh sin(h0t   h )
i (t )   ih (t )   2 I h sin(h0t   h )
•
Instantaneous and Average Power
p(t )  v(t )i(t )
1 T
0 p(t )dt
T


P   V h I h cos( h   h )   Ph  P1  PH
P
h 1
h 1
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Electric Quantities Under Nonsinusoidal Situation
•
RMS Voltage and Current
Vrms 
•
 2
1 T 2
2
2
0 v (t )dt   Vh  V1  VH
T
h 1
 2
1 T 2
2
2
I rms 
0 i (t )dt   I h  I1  I H
T
h 1
Apparent, Reactive, and Distortion Power
S  Vrms I rms
S 2  V1 I1 2  V1 I H 2  V H I1 2  V H I H 2
2
S H  VH I H  PH2  DH
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Electric Quantities Under Nonsinusoidal Situation
•
Power at sinusoidal situation
(V1I1) 2  P12  Q12
P1  V1I1 cos(1  1)
•
Q1  V1I1 sin(1  1)
Total power factor
P
PF 
S
No consensus in the definition and physical
meaning on reactive and distortion power.
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Phase Sequences of Harmonics and
Interharmonics
•
Phase sequences of harmonics
vah (t ) 
2Vh sin(h0t   h )
vbh (t )  2Vh sin(h0t  2h / 3   h )
vch (t )  2Vh sin(h0t  2h / 3   h )
•
For variable frequency drives and motors
with fluctuating loads, interharmonics can
have either positive or negative sequence
and are rarely zero sequence. The
general rule is that the sequence of the
interharmonc component is the same as
that of the supply system harmonic
components being modulated.
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Harmonic
Order
Phase
Sequence
1
+
2
-
3
0
4
+
5
-
6
0
.
.
3h-1
-
3h
0
3h+1
+
Phase Sequences of Harmonics and
Interharmonics
3h-1: negative sequence
3h: zero sequence
3h+1:positive sequence
h = 1, 2,…
Negative
sequence
26
Zero
sequence
Positive
sequence
Harmonic Indices
•
•
•
•
•
•
•
Total Harmonic Distortion (THD)
Total Demand Distortion (TDD)
Telephone Influence Factor (TIF)
VT and IT Products
C-Message Weighted Index
Transformer K-Factor and Harmonic Loss Factor
Distortion Power Factor
27
Harmonic Indices
•
Total Harmonic Distortion (THD)

THDV 
•

2
 Vh
h2
THD I 
V1
2
 Ih
h2
I1
Total Interharmonic Distortion (TIHD)
2
 Ii
2
 Vi
TIHD V 
i
TIHD I 
V1
i
I1
Λ is the set of all interharmonics components under
considerations.
28
Harmonic Indices
•
Total Demand Distortion (TDD)

TDD 
2
 Ih
h 2
IL
I L is the maximum demand load current (15 or 30
minute demand) at fundamental frequency at the
point of common coupling (PCC), calculated as the
average current of the maximum demands for the
previous twelve months.
29
Harmonic Indices
•
Telephone Influence Factor (TIF)

TIFV 

2
 ( whVh )
h 1
 ( wh I h )
TIF I 
Vrms
2
h 1
I rms
wh is a weighting accounting for audio and inductive
coupling effects at the h-th harmonic frequency.
•
VT and IT Products
V T 


I T 
( whVh ) 2


( wh Ih )2
h 1
h 1
TIFV  Vrms  V  T
TIFI  I rms  I  T
30
Harmonic Indices
•
C-Message Weighted Index


2
 (chVh )
CV 
 (c h I h )
h 1
CI 
Vrms
2
h 1
I rms
wh  5ch f h
•
K-Factor and Harmonic Loss Factor
hmax

K  factor   I h ( pu) 2 h 2
h 1
2
2
 h ( I h / I1)
FHL  h 1
hmax
2
 ( I h / I1)
h 1
31
Harmonic Indices
•
Distortion Power Factor (PFD)
P1  PH
P
PF  
S Vrms I rms
Vrms  V1 1  (THDV ) 2
PF 
I rms  I1 1  (THD I ) 2
[1  ( PH / P1 )] P1
V1I1 1  (THDV ) 2 1  (THD I ) 2
P1
1
 PF1  PFD
PF 

V1I1
1  (THD I ) 2
PF  PFD
32
Harmonic Indices
THDI (%)
PFD
10
30
0.995
0.958
50
0.894
70
100
125
0.819
0.707
0.625
150
0.555
33
Power System Response to
Harmonics and Interharmonics
•
Power System Response to Harmonics and Interharmonics
- Series resonance
- Parallel resonance
- Distributed resonance
bus m
bus 1
bus 2
Vh  Zh  I h
Original
Network
Z
bus 3
h
bus
Reference
34

bus p
bus M
k
Harmonic
Source
XC
Zf
Power System Response to
Harmonics and Interharmonics
XL
Series
resonance
XC
Ih
Power h 
System r
XC / X L
Power
System hr 
XC

XL
XL
Ih
Parallel
resonance
XC
Substation
Distributed
resonance
XL
Ih
XC
35
MVASC
MVARCAP
Power System Response to
Harmonics and Interharmonics
Parallel Resonace
11KV
150MVA 50Hz
Harmonics
source
400V
Network
inductance
Capacitance
of the
Capacitor
bank
Harmonics
Variable
Speed
Drive
36
Power System Response to
Harmonics and Interharmonics
Series Resonance
11KV
11KV
150MVA 50Hz
Transformer
,
Inductance s
400V
400V
Harmonics
Variable
Speed
Drive
Harmonics
source
Capacitor
bank
37
Capacitance
of the
Capacitor
bank
Solutions to Harmonics and Interharmonics
•
Remedial methods
- Passive Filters
- Phase Multiplication
- Special Designed Transformer (e.g. zig-zag)
- Active Filters
•
Preventive method
- Harmonic Standards
* IEEE 519-1992
* IEC 61000-3-6
38
Remedial Methods
•
•
Series filter – characterized as a parallel
resonant and blocking type with a high
impedance at its tuned frequency
Parallel filter – characterized as a series
resonant and trap type with a low impedance at
its tuned frequency
L
C
L
+
+
ih
+
ih
C
Series filter
Parallel filter
39
+
Remedial Methods
Passive filter
PCC
Utility
Common Bus
Non-linear
Load
Consumer
AC
Load
Power Supply Distribution
System
Transformer
Lnear Load
Capacitors /
Harmonic Filter
40
Remedial Methods
Phase Multiplication
M
h=5, 7, 11, 13, 17,
19, 23, 25
h=11, 13,
23, 25
M
41
Remedial Methods
Phase Multiplication
ia1 
2 3
1
1
I d (cos   cos 5  cos 7
2 N
5
7
1
1
 cos11  cos13 . . .)
11
13
ia 2 
2 3
1
1
I d (cos   cos 5  cos 7
2 N
5
7
1
1
 cos11  cos13 . . .)
11
13
ia  ia1  ia 2

42
2 3
1
1
I d (cos   cos11  cos13 . . .)
N
11
13
Remedial Methods
Special
Designed
Transformer
43
Remedial Methods
44
Remedial Methods
Active filter
va isa
vb isb
ila
ilan
ilb
ilbn
Nonlinear
vc isc
ilc
ilcn
Load
ifc ifb ifa
Active Power Filter
ilcl ilbl ilal
Linear Load
45
Remedial Methods
Active filter
va i sa
vb i sb
vc i sc
ila
id +
ilb
vd
ilc
_
ifc ifb ifa
Active Power
Filter
46
Preventive Methods (Harmonic Standards)
IEEE 519-1992 Current Distortion Limits for General Distribution Systems
47
Preventive Method (Harmonic Standards)
IEEE 519-1992 Recommended Voltage Distortion Limits
48
Preventive Method (Harmonic Standards)
IEC 61000-3-6: Compatibility levels for harmonic voltages (in
percent of the nominal voltage) in LV and MV power systems
Odd harmonics non
multiple of 3
Order h
Harmonic
voltage %
5
7
11
13
17
19
23
25
>25
Odd harmonics
multiple of 3
Order h
Harmonic
voltage %
Even harmonics
Order
h
6
3
5
2
5
9
1,5
4
3,5
15
0,3
6
3
21
0,2
8
2
>21
0,2
10
1,5
12
1,5
>12
0,2 +
1,3‧(25/h)
NOTE – Total harmonic distortion (THD): 8%.
49
Harmonic
voltage %
2
1
0,5
0,5
0,5
0,2
0,2
Preventive Method (Harmonic Standards)
IEC 61000-3-6:Indicative values of planning levels for harmonic voltage
(in percent of the nominal voltage) in MV, HV and EHV power systems
Odd harmonics non
multiple of 3
Order
h
5
7
11
13
17
19
23
25
>25
Harmonic voltage
%
MV
HV-EHV
5
4
3
2,5
1,6
1,2
1,2
1,2
0,2 +
2
2
1,5
1,5
1
1
0,7
0,7
0,2 +
0,5(h/25)
0,5(h/25)
Odd harmonics multiple
of 3
Order
h
3
9
15
21
>21
Harmonic
voltage %
MV
HV-EHV
4
1,2
0,3
0,2
0,2
2
1
0,3
0,2
0,2
Even harmonics
Order
h
2
4
6
8
10
12
>12
Harmonic
voltage %
MV
HV-EHV
1,6
1
0,5
0,4
0,4
0,2
0,2
1,5
1
0,5
0,4
0,4
0,2
0,2
NOTE – Total harmonic distortion (THD): 6,5% in MV networks 3% in HV networks.
50
Preventive Method (Harmonic Standards)
IEC 61000-3-6
Illustration of basic voltage quality
concepts with time/location statistics
covering whole system
51
Illustration of basic voltage quality
concepts with time statistics relevant
to one site within the whole system
Summary
•
•
Fourier Series and Analysis
•
•
•
Harmonic and Interharmonic Indices
•
Solutions to Harmonics and Interharmonics
Basic Definition of Harmonic and Interharmonic
Quantities
Power Factor under Distorted Situation
Power System Response to Harmonics and
Interharmonics
52