ieee pes general meeting, tampa fl june 24-28, 2007 conferência brasileira de qualidade de energia...

IEEE PES General Meeting, Tampa FLJune 24-28, 2007

Conferência Brasileira de Qualidade de EnergiaSantos, São Paulo, Agosto 5-8, 2007

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Chapter 2Harmonics and Interharmonics Theory

Chapter 2Harmonics and Interharmonics Theory

Contributors: G. W. Chang and A. Testa

Tutorial on Harmonics Modeling and Simulation



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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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• Harmonic Sources （ nonlinear loads）– Single-phase loads: fluorescent lights, personal computers

– Three-phase loads: arc furnaces, ac/dc converters



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• 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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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 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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• 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 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




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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.




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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




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• 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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• 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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• 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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)2550sin()2280sin(

)2180sin()260sin(2)(

tt

tttv

Frequency resolution = 30 Hz



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+

-

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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• 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


• 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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• 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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• 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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• 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

• 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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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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• 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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• 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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• 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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• 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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• 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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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 - 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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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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Parallel Resonace

11KV 150MVA 50Hz

400V

Harmonics

VariableSpeedDrive

Capacitanceof the

Capacitorbank

Harmonicssource

Networkinductance



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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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Non-linear LoadUtility

Lnear Load

Capacitors / Harmonic Filter

Consumer

PCC

Power SupplySystem

DistributionTransformer

Common Bus

ACLoad

Passive filter



41


h=5, 7, 11, 13, 17, 19, 23, 25

h=11, 13,23, 25

M

M

Phase Multiplication



42


). . . 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



43


SpecialDesignedTransformer



44




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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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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



48

Preventive Method (Harmonic Standards)Preventive Method (Harmonic Standards)

IEEE 519-1992 Recommended Voltage Distortion Limits



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Odd harmonics non multiple of 3

Odd harmonics multiple of 3

Even harmonics

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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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



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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



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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

ieee pes general meeting, tampa fl june 24-28, 2007 conferência brasileira de qualidade de energia...

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