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Conferência Brasileira de Qualidade de EnergiaSantos, São Paulo, Agosto 5-8, 2007
1
Chapter 2Harmonics and Interharmonics Theory
Chapter 2Harmonics and Interharmonics Theory
Contributors: G. W. Chang and A. Testa
Tutorial on Harmonics Modeling and Simulation
IEEE PES General Meeting, Tampa FLJune 24-28, 2007
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OutlineOutline
• 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
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IntroductionIntroduction
• 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.
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IntroductionIntroduction
• Harmonic Sources ( nonlinear loads)– Single-phase loads: fluorescent lights, personal computers
– Three-phase loads: arc furnaces, ac/dc converters
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IntroductionIntroduction
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IntroductionIntroduction
• 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.
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• Periodic Function
• Orthogonal Function
e.g.
Fourier Series and AnalysisFourier Series and Analysis
... ,2 ,1 ,0 ),()( hhTtftf
ji
jidttt ji = ,
,0)()(
...} ,sin ..., ,sin..., ,cos ..., ,cos ,1{ 0000 thttht
)}({ th
2/2/ TtT
Period: T
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Fourier Series and AnalysisFourier Series and Analysis
1000 )}sin()cos({
2
1)(
hhh thbthaatf T/20
100 )sin()(
hhh thcctf
2/
2/0 )(2 T
T dttfT
a 2/
2/ 0 )cos()(2 T
Th dtthtfT
a
2/
2/ 0 )sin()(2 T
Th dtthtfT
b
2/00 ac 22hhh bac )/(tan 1
hhh ba
By the use of orthogonal relations, we have
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Fourier Series and AnalysisFourier 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
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Fourier Series and AnalysisFourier Series and Analysis
• Complex Form
• Waveform Symmetry
- Even function: (no sine terms)
- Odd function: (no cosine terms)
- Half-wave symmetry: (no even harmonics)
h
tjhhectf 0)(
... ,2 ,1 ,0 h
2/
2/0)(
1 TT
tjhh dtetf
Tc
)()( tftf )()( tftf
)2/()( Ttftf
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Fourier Series and AnalysisFourier Series and Analysis
• Fourier Transform
• 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
is the so-called spectrum of which is assumed to be one cycle of a periodic signal.
--= dtetfF tj )()(
deFtf tj)(2
1)(
1
0
/2)()(N
n
NknjeTnfkF
1
0
/2)()(N
k
NknjekFTnf NTT /
k, n = 0, 1, ..., N-1,
T /2
)( kF )( Tnf
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• The angular frequency resolution of the spectrum is determined by the length of the signal as . Thus, if T is selected as one period of , 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 . 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.
T/2 )( Tnf
pT/2
Fourier Series and AnalysisFourier Series and Analysis
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• 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.
Fourier Series and AnalysisFourier Series and Analysis
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• 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.
Basic Definition of Harmonic and Interharmonic Quantities
Basic Definition of Harmonic and Interharmonic Quantities
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Basic Definition of Harmonic and Interharmonic Quantities
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 line-commutated converter can be characterized by
and
e.g.
Such harmonics are usually termed as characteristic harmonics.
• Non-characteristic harmonics are typically categorized as those integer frequency components other than characteristic ones.
, ...2,1n1pnh ,,...,,,,, , 1917131175h6p
hIIh /1
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Basic Definition of Harmonic and Interharmonic Quantities
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 time-varying 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
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 picket-fence 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.
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Basic Definition of Harmonic and Interharmonic Quantities
Basic Definition of Harmonic and Interharmonic Quantities
)2550sin()2280sin(
)2180sin()260sin(2)(
tt
tttv
Frequency resolution = 30 Hz
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Basic Definition of Harmonic and Interharmonic Quantities
Basic Definition of Harmonic and Interharmonic Quantities
+
-
PWMInverter
InductionMotor
Diode BridgeRectifier
DC LinkFilter
3 Phase AC Supply
dv Cd
021 )1( nfpfmpf si p1: the pulse number of the rectifier section p2: the pulse number of the output section m and n: integers fs: the power frequencyf0: the inverter output frequency
Interharmonics:
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Basic Definition of Harmonic and Interharmonic Quantities
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 sub-harmonics), 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.
ttVVtv sm sin)sin()(
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Electric Quantities Under Non-sinusoidal Situation
Electric Quantities Under Non-sinusoidal Situation
• Instantaneous voltage and current
• Instantaneous power
• Average power
• RMS voltage and current
• Apparent power
• Reactive power
• Distortion power
• Total power factor
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Electric Quantities Under Non-sinusoidal Situation
Electric Quantities Under Non-sinusoidal Situation
• Instantaneous Voltage and Current
• Instantaneous and Average Power
1 10 )sin(2)()(
h hhhh thVtvtv
1 10 )sin(2)()(
h hhhh thItiti
)()()( titvtp
T dttpT
P 0 )(1
Hh
hh
hhhh PPPIVP
1
11)cos(
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Electric Quantities Under Non-sinusoidal Situation
Electric Quantities Under Non-sinusoidal Situation
• RMS Voltage and Current
• Apparent, Reactive, and Distortion Power
221
1
20
2 )(1
Hh
hT
rms VVVdttvT
V
221
1
20
2 )(1
Hh
hT
rms IIIdttiT
I
rmsrms IVS
221
21
211
2HHHH IVIVIVIVS
22HHHHH DPIVS
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Electric Quantities Under Non-sinusoidal Situation
Electric Quantities Under Non-sinusoidal Situation
• Power at sinusoidal situation
• Total power factor
No consensus in the definition and physical meaning on reactive and distortion power.
21
21
211 )( QPIV
)sin( 11111 IVQ)cos( 11111 IVP
S
PPF
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Phase Sequences of Harmonics and Interharmonics
Phase Sequences of Harmonics and Interharmonics
• Phase sequences of harmonics
• 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.
)sin(2)( 0 hhah thVtv
)3/2sin(2)( 0 hhbh hthVtv
)3/2sin(2)( 0 hhch hthVtv
Harmonic Order
Phase Sequence
1 +
2 -
3 0
4 +
5 -
6 0
. .
3h-1 -
3h 0
3h+1 +
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Phase Sequences of Harmonics and Interharmonics
Phase Sequences of Harmonics and Interharmonics
3h-1: negative sequence3h: zero sequence3h+1:positive sequence
h = 1, 2,…Positivesequence
Negativesequence
Zerosequence
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Harmonic IndicesHarmonic 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
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Harmonic IndicesHarmonic Indices
• Total Harmonic Distortion (THD)
• Total Interharmonic Distortion (TIHD)
Λ is the set of all interharmonics components under considerations.
1
2
2
V
V
THD hh
V
1
2
2
I
I
THD hh
I
1
2
V
V
TIHD ii
V
1
2
I
I
TIHD ii
I
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Harmonic IndicesHarmonic Indices
• Total Demand Distortion (TDD)
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.
L
hh
I
I
TDD
2
2
LI
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Harmonic IndicesHarmonic Indices
• Telephone Influence Factor (TIF)
is a weighting accounting for audio and inductive coupling effects at the h-th harmonic frequency.
• VT and IT Products
rms
hhh
V V
Vw
TIF
1
2)(
rms
hhh
I I
Iw
TIF
1
2)(
V T w Vh hh
( ) 2
1
I T w Ih hh
( )2
1
TVVTIF rmsV TIITIF rmsI
hw
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Harmonic IndicesHarmonic Indices
• C-Message Weighted Index
• K-Factor and Harmonic Loss Factor
rms
hhh
V V
Vc
C
1
2)(
rms
hhh
I I
Ic
C
1
2)(
hhh fcw 5
1
22)(h
h hpuIfactorK
max
max
1
21
1
21
2
)/(
)/(
h
hh
h
hh
HL
II
IIh
F
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Harmonic IndicesHarmonic Indices
• Distortion Power Factor (PFD)
rmsrms
HF IV
PP
S
PP
1
21 )(1 Vrms THDVV
21 )(1 Irms THDII
2211
11
)(1)(1
)]/(1[
IV
HF
THDTHDIV
PPPP
211
1
)(1
1
I
FTHDIV
PP
FDF PP 1
FDF PP
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Harmonic IndicesHarmonic Indices
THDI (%) PFD
10 0.995
30 0.958
50 0.894
70 0.819
100 0.707
125 0.625
150 0.555
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Power System Response to Harmonics and Interharmonics
Power System Response to Harmonics and Interharmonics
• Power System Response to Harmonics and Interharmonics - Series resonance
- Parallel resonance
- Distributed resonance
hhh IZV h
busZ
OriginalNetwork
HarmonicSource
Reference
k
CX fZ
pbus1bus
2bus
3bus
mbus
Mbus
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Power System Response to Harmonics and Interharmonics
Power System Response to Harmonics and Interharmonics
Ih
XCXL
PowerSystem LCr XXh /
Ih
XC
XL
SystemPower
CAP
SC
L
Cr MVAR
MVA
X
Xh
IhXC
XLSubstationDistributedresonance
Seriesresonance
Parallelresonance
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Power System Response to Harmonics and Interharmonics
Power System Response to Harmonics and Interharmonics
Parallel Resonace
11KV 150MVA 50Hz
400V
Harmonics
VariableSpeedDrive
Capacitanceof the
Capacitorbank
Harmonicssource
Networkinductance
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Power System Response to Harmonics and Interharmonics
Power System Response to Harmonics and Interharmonics
Series Resonance
Capacitorbank
11KV 150MVA 50Hz
400V
Harmonics
VariableSpeedDrive
Capacitanceof the
Capacitorbank
Harmonicssource
Transformer
Inductance,s
11KV
400V
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Solutions to Harmonics and InterharmonicsSolutions 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
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Remedial MethodsRemedial 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
Parallel filterSeries filter
ih+ +
L
C
+ +ih
L
C
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Remedial MethodsRemedial Methods
Non-linear LoadUtility
Lnear Load
Capacitors / Harmonic Filter
Consumer
PCC
Power SupplySystem
DistributionTransformer
Common Bus
ACLoad
Passive filter
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Remedial MethodsRemedial Methods
h=5, 7, 11, 13, 17, 19, 23, 25
h=11, 13,23, 25
M
M
Phase Multiplication
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Remedial MethodsRemedial Methods
). . . 13cos13
111cos
11
1
7cos7
15cos
5
1(cos
2
321
da IN
i
). . . 13cos13
111cos
11
1
7cos7
15cos
5
1(cos
2
322
da IN
i
). . . 13cos13
111cos
11
1(cos
32
21
d
aaa
IN
iii
Phase Multiplication
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Remedial MethodsRemedial Methods
SpecialDesignedTransformer
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Remedial MethodsRemedial Methods
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Remedial MethodsRemedial Methods
Active filter
Nonlinear
Load
Active Power Filter
sai
sbi
sci
lai
lbi
lci
fci fbi fai
av
bv
cv
Linear Load
lci lbi lail l l
lain
lbin
lcin
0 0.01 0.02 0.03 0.04 0.05-50
0
50 ILa
0 0.01 0.02 0.03 0.04 0.05-50
0
50 Isa
0 0.01 0.02 0.03 0.04 0.05-20
0
20 Ifa
0 0.01 0.02 0.03 0.04 0.05-20
0
20 In
0 0.01 0.02 0.03 0.04 0.05-2000
0
2000 P
(sec)
(a)
(b)
(c)
(d)
(e) p p f f
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Remedial MethodsRemedial Methods
id +
_
vd
Active PowerFilter
sai
sbi
sci
lai
lbi
lci
fci fbi fai
a
b
c
v
v
v
Active filter
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Preventive Methods (Harmonic Standards)
Preventive Methods (Harmonic Standards)
IEEE 519-1992 Current Distortion Limits for General Distribution Systems
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Preventive Method (Harmonic Standards)Preventive Method (Harmonic Standards)
IEEE 519-1992 Recommended Voltage Distortion Limits
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Preventive Method (Harmonic Standards)Preventive Method (Harmonic Standards)
Odd harmonics non multiple of 3
Odd harmonics multiple of 3
Even harmonics
Order h Harmonic voltage %
Order h Harmonic voltage %
Order h Harmonic voltage %
5
7
11
13
17
19
23
25
>25
6
5
3,5
3
2
1,5
1,5
0,2 +
1,3‧(25/h)
3
9
15
21
>21
5
1,5
0,3
0,2
0,2
2
4
6
8
10
12
>12
2
1
0,5
0,5
0,5
0,2
0,2
NOTE – Total harmonic distortion (THD): 8%.
IEC 61000-3-6: Compatibility levels for harmonic voltages (in percent of the nominal voltage) in LV and MV power systems
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Preventive Method (Harmonic Standards)Preventive Method (Harmonic Standards)
Odd harmonics non multiple of 3
Odd harmonics multiple of 3 Even harmonics
Order h
Harmonic voltage %
Order h
Harmonic voltage %
Order h
Harmonic voltage %
MV HV-EHV MV HV-EHV MV HV-EHV
5
7
11
13
17
19
23
25
>25
5
4
3
2,5
1,6
1,2
1,2
1,2
0,2 +
0,5(h/25)
2
2
1,5
1,5
1
1
0,7
0,7
0,2 +
0,5(h/25)
3
9
15
21
>21
4
1,2
0,3
0,2
0,2
2
1
0,3
0,2
0,2
2
4
6
8
10
12
>12
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.
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
IEEE PES General Meeting, Tampa FLJune 24-28, 2007
Conferência Brasileira de Qualidade de EnergiaSantos, São Paulo, Agosto 5-8, 2007
51
Preventive Method (Harmonic Standards)Preventive Method (Harmonic Standards)
Illustration of basic voltage qualityconcepts with time/location statisticscovering whole system
Illustration of basic voltage quality concepts with time statistics relevantto one site within the whole system
IEC 61000-3-6
IEEE PES General Meeting, Tampa FLJune 24-28, 2007
Conferência Brasileira de Qualidade de EnergiaSantos, São Paulo, Agosto 5-8, 2007
52
SummarySummary
• 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