TP125_Full_Content.pdf

8/10/2019 TP125_Full_Content.pdf

1/85

IEEE

ow r

Engineering Society

Tutorial on Harmonics Modeling and

Simulation

D

TP 125 0


2/85

Tutorial on

Harmonics Modeling and Simulation

IEEEPowerEngineering Society TaskForceonHarmonics Modeling and Simulation

IEEEPowerEngineering SocietyHarmonics

Working Group

Sponsoredby theLife Long Learning Subcommittee of the

IEEE

Power EngineeringEducation Committee


3/85

IEEE Power

Engineer ing Society

Tutorial On

H RMONICS

MODELING AND SIMULATION

Abstractingis permittedwithcreditto the

source.

For other copying, reprint,or republicationpermission,

write to the IEEECopyright

Manager

IEEEService Center,445HoesLane, Piscataway,NJ 08855-1331.

All rightsreserved. Copyright

1998

byThe Institute of Electrical andElectronicsEngineers, Inc.

IEEE

CatalogNumber:

98TP125-0

Additionalcopiesof

this

publication

are

available from

IEEE

Operations Center

P. O.Box 1331

445

HoesLane

Piscataway, NJ 08855-1331 USA

1-8oo-678-IEEE IndividuallMember

Orders

1-800-701-

IEEE

Institutional

Orders

1-732-981-0060

1-732-981-9667 FAX)

email: [email protected]


4/85

FOREWORD

Theproblemof power system harmonics is not new.Utilities recognizedthe consequences ofhannonics in

the 1920s and early 1930swhen distorted voltage and currentwaveforms were observed on transmission lines. At

that time, the major concernswerethe effects of harmonics on electric machines, telephone interference and power

capacitorfailures. Althoughsuchconcerns stillexisttoday,harmonics are becoming a seriousproblem, potentially

damaging consumer loads as well as power delivery equipment because of the substantial increase

of

harmonic

producing loads in recentyears.

Significant efforts have been

made

in the past two decades to improve the management of harmonics in

power systems. Standards for

harmonic control

have been established. Sophisticated instruments for harmonic

measurements are readily available. Thearea of power system harmonic analysis has also experienced significant

developments and well-accepted

component models

simulation methods and analysis procedures for conducting

harmonic studies have been established. Harmonic studies arebecoming an importantcomponentof power system

analysis anddesign.

The progress in the area of power system harmonic modeling and simulation and the need of practicing

engineers to upgrade their harmonic analysis skills were recognized by the Power System Harmonics Working

Group of the IEEE Power Engineering Society and the Harmonics Working Group of the IEEE Industry

ApplicationsSociety.Under the

sponsorship

of the Transmission andDistribution Committeein the IEEE-PESand

the Power SystemsEngineeringCommittee in the IEEE-lAS, theHarmonics Modeling and Simulation Task Force

of

the PES Harmonics Working

Group

and the lAS Harmonics Working Group have developed this tutorial on

harmonicsmodelingand simulation. Thepurpose of the tutorial is to summarize the developments in the area

from

both theoretical as well as application perspectives. Latest and proven techniques for harmonic modeling and

simulation are discussed along with

case

studies.

By

focusing on the practical aspects of applying harmonic

modeling and simulationtheories,the tutorial is expectedto provide readerswith a sound theoreticalbackgroundas

well as practicalguidelinesfor harmonic analysis.

We begin the tutorial withan introduction to the objectives and key issues of harmonics modeling and

simulation.The theoryof Fourieranalysis is discussed for applications in powersystemharmonicanalysis.Detailed

discussion of modelingof electricnetworks andcomponents including harmonic-producing devices then follows

in

several papers. Various network

solution

techniques for harmonic power flow and frequencyscan calculationsare

summarizedand casestudies are

used

to demonstrate the practical aspects of harmonicanalysis.Threeharmonictest

systems arepresented.Finally, the areas thatstillneedfurtherresearch and developmentare discussed

in

the closing

commentsof this tutorial.

This tutorialmaterialhasdrawnon theconsiderable expertise of the HarmonicsWorkingGroups andtheir

task forces. The contributors have

generously

donatedtheir timeand effort to what we believe will be a valuable

reference work on the subject. In

addition guidance

and encouragement of Mr. Tom Gentile, Chair of the PES

HarmonicsWorking Group andDr.Mack

Grady

Chair of the General Systems Subcommittee of the PES T D

Committeemade our task much easier. Dr. M.E.El-Hawary, Chair of Life Long Learning Subcommittee of the

IEEE Power Engineering Education

Committee

provided generous support to many aspects of this activity. We

wish to takethis opportunityto thankallcontributors for theireffort

in

completingthis task.

MarkHalpin,TutorialEditor

WilsunXu,TutorialOrganizer andChair PESHarmonics Modelingand SimulationTaskForce

SatishRanade, Past

Chair PES Harmonics

Modeling andSimulation TaskForce

PauloF. Ribeiro,ViceChair PES

Harmonics

Modeling andSimulation TaskForce

iii


5/85

Tutorial Contents

I ~ H ~ i i ~ ~ t r j i l ~ ~ ~ i w i l ; 1 ' j ~ 1 @ ~ j ~ I I i . 4 J i ~ j ~ l m ~ ~ ~ ~ ~ ~ m ~ m l f : { ~ i m t , ~ l i l l i l ~ . 1 ~ I ~ a l , . i l i j i l t ~ ~ W 1 I M J ~ ~ ~ ~ ~ I f . . : ~ ~ ; ~ ~ ~ ~ ~ ~ j : ~ 1

Forward M. Halpin, W. Xu,

S. Ranade

G. Chang

8

P. Ribeiro

15

G. Chang,W.Xu

28

Y. Liu, Z. Wang

35

T. Ortmeyer, M.F.

43

Akram,

T

Hiyama

M. Halpin, P.

49

Ribeiro, J.J. Dai

C. Hatziadoniu

55

w Xu, S.Ranade 61

M. Halpin,

67

R. Burch

W Xu

71

M.Halpin

78

79

1.

An

Overview

of

Harmonics Modeling

and

Simulation

2. Harmonics

Theory

3. Distribution System andOther Elements

Modeling

4. Modeling of Harmonic Sources: Power

Electronic

Converters

5. Modeling ofHannonic Sources -

Magnetic

Core Saturation

6. Harmonic

Modeling

ofNetworks

7. Frequency-Domain Harmonic Analysis

Methods

8.

Time

DomainMethodsfor the Calculation of

Harmonic

Propagation and Distortion

9.

Analysis

ofUnbalanced Harmonic

Propagation

inMultiphase Power Systems

10.Harmonic Limit Compliance Evaluations Using

IEEE519-1992

11

Test Systems for

Harmonics

Modeling and

Simulation

Conclusions

AuthorBiographies

v

Ranade, W. Xu 1


6/85

Chapter 1

AN OVERVIEW OF HARl\10NICS

MODELING

AND SIMULATION

s

J. Ranade

NewMexico StateUniversity

LasCruces, NM,USA

W.Xu

University ofAlberta

Edmonton,

Alberta,

Canada

1 3 S 7 9 11 13 1S 17 19 21 23 2S 27 29 31

Harmonic Order

Figure1.1. A harmonic (amplitude) spectrum.

: :

-.-.

0.0164

.0123.0082

Harmonic spectrum

Time

Sees

0.0041

Fundamental

-1

-1.5

o

0.80

0.70

0.90

8 0.60

::I

i

0.50

0.40

0.30

0.20 .

0.10

-0.5

Figure 1.2. Synthesis ofawaveform from harmonics.

The waveform is aperiodic where theFourier series is an

approximation [4].

0.5

1.5

1.00

The waveform is aperiodic but can be expressed as a

trigonometric series [3]. In this case the components in

the Fourier series that are not

integral

multiples of the

power

frequency

are sometimes

called

non-integer

harmonics.

1.1 Introduction

f t =C

o+

:LCacos nmt+9a

(1.1)

11=1

where Co is the dc value of the function. ll is

the peak

value of the nTh harmonic component and ll is its phase

angIe.

A plot of

normalized

harmonic amplitudes eric} is

called the

harmonic magnitude spectrum

as illustrated in

Figure 1.1.

The

superposition of harmonic components to

createtheoriginal waveform is shown

in

Figure 1.2

Domain of Application: In general one can think

of

devices that

produce distortion as exhibiting a nonlinear

relationship between voltage and

current.

Such

relationships can lead to several

forms

of distortion

summarized

as:

Fourier

Series:Theprimary

scopeofharmonics modeling

and simulation is in the study of periodic, steady-state

distortion. The Fourier series for a regular, integrable,

periodic function f(t), ofperiodT seconds and fundamental

frequency f=lrr

Hz, or

eo=21tfradls,

can

bewritten

as

[3]:

Distortion of sinusoidal voltageandcurrent

waveforms

caused by harmonics is one of the major power quality

concerns in

electric power

industry. Considerable

efforts

have been

made

in recent years to improve the

management

of harmonic distortions in power

systems.

Standards for harmonic control have been established.

Instruments for harmonic measurements are widely

available. The areaof powersystemharmonic analysis has

also

experienced

significant advancement

[1,2].

Well

accepted component models, simulation methods and

analysis

procedures for conducting systematic harmonic

studies havebeen

developed.

In this chapter wepresentan

overview of the

harmonics

modeling andsimulation issues

andalsoprovide anoutline of this tutorial.

1.2 Fourier Series

and

Power System Harmonics

A periodic steady-state exists and the distorted

waveform has a Fourier series with fundamental

frequency equal

to

power

system frequency.

A periodic steady state exists and the distorted

waveform has a Fourier series with fundamental

frequency

that is a sub-multiple of power system

frequency.

The first case is

commonly

encountered and there ar

several advantages

to using the decomposition in tenns o

harmonics.

Harmonics

have a

physical

interpretation anda

intuitive appeal Since the

transmission

network is

usually

modeled

as a linear

system,

the

propagation

of eac

harmonic

can

be studied

independent of

the others. Th

number of

harmonics

to be

considered

is

usually

small

which

simplifies computation. Consequences such as

losse

1


7/85

can be related to harmonic components and

measures

of

waveform quality can he developed in terms of harmonic

amplitudes.

Certain types of

pulsed

or modulated

loads

create

waveforms

corresponding to

the

secondcategory.

The

third

category can occur in certain pulse-width modulated

systems. Some

practicalsituations such

as

arc furnaces and

transformer inrush

currents

correspond

to the

fourth

case.

DC arc furnaces utilize conventional multiphase rectifiers

but the underlying process of melting is

not

a stationary

process. When reference is made to harmonics in this

instance

it corresponds

to the periodic waveform that

would be obtained if furnace conditions were to be

maintained constant over a period of time. Harmonics

modeling can lend insight into some of the potential

problemsbut transient studiesbecomevery

important.

The Origin of Harmonics: Main sources of harmonics in

conventionalpowersystems are summarized

below.

1. Devices involving electronic switching:

Electronic

power processingequipment utilizes switching devices.

The switchingprocess is generally, but notnecessarily,

synchronized to the ac voltage.

2. Devices with nonlinear voltage-current relationships:

Iron-core

reactors

and arcing loads are typical examples

of

such

devices. When excited

with

a periodic input

voltage the nonlinear v-i curve leads to

the

generation

of harmoniccurrents:

Distortion Indices: The mostcommonlyused measure of

the quality of a periodic waveformis the total harmonic

distortion 1lID .

TIID

=

J

I

Cl

(1.2)

IEEE Std. 519 [5] recommends limits on voltage and

current THD

values. Other such as telephone

interference factor (TIF)

and

leT product are used to

measure telephone interference. The K-faetor indices are

usedto describe the

impact

of harmonics onlosses

and

are

useful in de-rating equipment suchas

transformers.

Harmonics in Balanced

and Unbalanced Three-Phase

Systems:

In

balanced three-phase

systems

and under

balanced operating

conditions, harmonics

In

each phase

have specificphaserelationships. For example, in the case

of the third

harmonic,

phase b

currents

would

lag those in

phasea by 3x120 or 360, andthose in phase c wouldlead

by the same amount. Thus, the third harmonics have no

phase shift and appear as

zero-sequence

components.

Similar

analysis

shows that fifth harmonics appear

to

be

of

negative sequence, seventh are of positive sequence, etc.

System impedances must be appropriately modeled based

on

the

sequences.

The magnitudes and phase angles (in particular) of

three-phaseharmonicvoltages andcurrentsare sensitiveto

network or load unbalance. Even for small deviations from

balanced conditions at the fundamental frequency it ha

been noted that harmonic unbalance can be significant. I

the

unbalanced

case

line currents

and

neutral

currents ca

contain all orders of harmonics and contain components o

all sequences. Three-phase power electronic converters can

generate non-characteristic under unbalancedoperation.

1.3 Harmonics Modeling and Simulation

The

goal of harmonic studies

is

to quantify

the

distortio

in

voltage

and current waveforms at

various

points in a

power system. The results are useful for evaluating

corrective

measures and

troubleshooting

harmonic cause

problems. Harmonic studies can

also determinetheexistenc

of dangerousresonant conditionsand verifycompliancewit

harmonic

limits.

The

need

for

a harmonic

study

may

b

indicated by excessive

measured distortion

in existin

systems or by installation of harmonic-producing

equipment. Similar

to

other

power systems studies th

harmonics study consistsof the following

steps:

Definition of harmonic-producing equipment an

detennination of models for

their-representanon,

Determination of the models to represent othe

components in

the

system

including

external

networks.

Simulation of the

system

for various scenarios.

Many

models

have been proposed for representing

harmonic

sources as well as linear

components. Variou

network harmonic solution algorithms have also bee

published. In the following sections, we briefly

summariz

the well-accepted methods for harmonic

modeling

an

simulations.

Other chapters in this tutorial will

expandupo

these

ideas and illustrate how to set

up

studies in typica

situations.

1.4 Nature

and

Modeling ofHarmonic Sources

The

mostcommon

model

for harmonic sources is

in

th

form

of

a harmonic

current

source, specified

by

it

magnitude and phase

spectrum The

phase is

usually

define

with

respect

to the

fundamental

component

of

the tennina

voltage.

The

data can be obtained form an idealjze

theoretical

model or from

actual

measurements. In man

cases,

the

measured

waveforms

provide a

more

realisti

representation of

the

harmonic

sources to be modeled. Thi

is

particularly

true i the system has significant unbalance

or

if

non-integer

harmonics are present When a system

contains

a single

dominant source

of

harmonics the phas

spectrum is

not

important However, phase angles must b

represented whenmultiple sources are present A commo

method is to

modify

the phase spectrum

according

to

th

phase angle of the fundamental frequency voltage seen b

the load.

Ignoring

phase angles does

not always resultin

th

worst case .

More detailed models become necessary if voltag

distortion is significant or if voltages are unbalanced. Ther

2


8/85

are three basic approaches that can

be taken

to develop

detailed

models:

Develop

analytical formulas for the Fourier series as a

function

of

terminal voltage and operating parameters

for the device.

Develop analytical models for device operation and

solve for device current waveform by a suitable

i

terati

vemethod.

Solve for device steady state current

waveform

using

time domainsimulation.

Advanced models require design data for the device.

For example, for a mediumpower ASD it is necessary to

specify

parameters

such as

transformer

data, de link data

and motor parameters. Apart

from

potentially higher

accuracy, an important advantage of such detailed models

is that the usercan specifyoperating

conditions,

e.g.,motor

speed

in a

drive, rather than

spectra.

In the

analysis

of distribution and

commercial

power

systems one may

deal

with a harmonic source that is an

aggregateof

many

sources.Such a sourcecanbe modeled

bymeasuring the aggregate spectrum. It is

very

difficult to

develop

a

current source type model analytically

based

on

the

load composition

data.

Reference

[7] has pointed out

that the aggregate waveforms can be much less distorted

thanindividual device waveforms.

Harmonie sources may also exhibit time-varying

characteristics.

Since

standards and practice

permit

harmonic guidelines to be violated for

short

periods of

time,

including

the time-varying characteristics of

hannonic sources can be useful and can present a more

realistic

picture

of actual distortions. More research is

needed

in

this

area

[8].

NonlinearVoltage-Current Sources:

The most

common

sources in

this

category

are

transformers

( due to their

nonlinear magnetization requirements), fluorescent

and

other gas discharge lighting, and devices such

as

arc

furnaces. In all

cases there

exists a

nonlinear

relationship

between

the

current

and

voltage. The

harmonic

currents

generated by these

devices

can be significantly affected by

the waveforms and peak values of supply voltages.

It

is

desirable to represent

the

devices with their actual

nonlinear v-i characteristicsin harmonic studies, instead of

as voltageindependent harmoniccurrent

sources.

Power Electronic Converters: Examples of power

electronic devices are adjustable speed

drives,

HVDC

links, and static var compensators. Compared to the

non

linear v-i

devices, harmonics from these converters are

less

sensitive to supply voltage variation and distortion.

Harmonic current source models are therefore commonly

used

to represent these devices. As discussed before, the

phase angles of the current sources are functions of the

supply

voltage

phase angle. They must

be

modeled

adequately

for

harmonic analysis involving more than one

source. The devices are

sensiuve

to

supply

voltage

unbalance. For large

power

electronic devices

such

as

HVDC terminals and transmission level

SVCs,

detailed

three-phase models

may

be needed.

Factors such as tiring

angle dependent

harmonic

generation

and

supply voltage

unbalance are taken into account in themodel. Thesestudies

normally scan through various possible device operating

conditions and filter performance,

Rotating Machines: Rotating machines can be

a harmonic

source

as

well.

The

mechanism

of

harmonic

generation

in

synchronous machines is

unique.

It cannot be described by

using either the nonlinear v-i device model or the power

electronic switching model. Only the

salient

pole

synchronous

machines

operated

under unbalanced

conditions can generate harmonics with

sufficient

magnitudes. In this case, a

unbalanced

current experienced

by the generator induces a second harmonic current in the

field winding, which in

tune

induces a third harmonic

current in the stator. In a similar

manner,

distorted system

voltage

can cause the

machines

to produce harmonics.

Models to represent

such mechanisms have

been

proposed

[1]. For the cases of saturation-caused harmonic generation

from

rotatingmachines, the

n o n i n e 3 ~ i

modelcan

be

used.

High frequency sources:

Advances

in

power electronic

devices

have created

the potential

for a wide range of new

power

conversion

techniques. The electronic ballast for

fluorescent lighting is oneexample. In general, these systems

employ high

frequency

switching to achieve greater

flexibility in power conversion. With proper design, these

techniques can

be

used to reduce the low frequency

harmonics.

Distortion is

created

at the switching

frequency,

which is

generally above20 kHz.

At

such

high frequency,

current distortion generally does not penetrate far into the

system

but

the

possibility

of

system

resonance

at

the

switching frequency can stillexist

Non-integer harmonic sources:

There exist

several

power

electronic systems

which produce distortion at

frequencies

that are harmonics of a base frequency other than 60 Hz.

There

are also devices that produce distortion at discrete

frequencies

that are not

integer

multiples of the base

frequency. Some devices have

waveforms

that donotsubmit

to a Fourier or trigonometric series

representation.

Lacking

standard terminology, we will call these non-harmonic

sources.

Modeling of this

type

of harmonic

sources

has

attracted

many

research interests

recently.

1.5Network and LoadModels

NetworkModel:

The main difficulty

in

setting

upa

network

model is to determine

howmuch

the network needs to

be

modeled.

1be extent of

network

representation is limited by

available data and computing

resources.

The following

observations

can bemade:

For industrial power systems connected to strong or

dedicated three-phase distribution feeders it is generally

3


9/85

The above observations are not guaranteed

rules,

but are

based on common practice. Perhaps

the

best way to

determine the extent

of

network

modeling

needed is to

perform

a

sensitivity

study;

i,e.,

one

can progressively

expandthe network

model

until the

results

do not

change

significantly. In many

harmonic studies involving

industrial plants,

the supply

system is represented as a

frequency-dependent driving-point impedance at the point

of common coupling.

OverheadLinesand UndergroundCables: Modeling

of

lines and cables over a wide range of frequencies

is

relatively

well

documented

in

literature [9]. Typical

lines

or cablescanbe

modeled by muItiphase

coupled

equivalent

circuits. For balancedharmonicanalysis themodels can be

fur ther s impl if ied into single-phase.

pi-circuits

using

positive

and

zero sequence data. The main issues

in

modeling these components are the

frequency dependence

of per-unit length series

impedance and

the long line

effects.

As a result ,

the

level

of

detail

of

their models

dependson the line lengthand harmonic order:

sufficient to

model

two

transtormations

from

the

load

point. Generally, transformer impedances dominate.

Branch circuits should be modeled if

they

connect to

power factorcorrection capacitors ormotors. Although

capacitance of overhead lines is

usually

neglected,

cable capacitanceshould

be modeled

for cables longer

than 500

feet.

Large industrial facilities are

served

at sub

transmission and even transmission voltage.

In

this

case

it

is important to

model

at

least

a portion of the

HVIEHV network

if

the facility

has

multiple

supply

substations. If it has only one supply substation,

utilities

mayprovide the driving-point impedance seen

by the

facility.

Distributionfeeders (at least in theUSandCanada) are

unbalanced and loads are often served from single

phase laterals. Shunt capacitors are extensively used.

Thus it becomes mandatory to model

the

entire

feeder,

and

sometimes

adjacent feeders aswell.

In

industrial

systems and utility distribution

systems

where

line lengths are short

it

is customary to use

sequence

impedances.

Capacitanceis

usually

neglected

except

in the

case of

long

cableruns,

An

estimate

of line-length beyond

whicb

long line

models should be used is 1501nmiles foroverhead line

and 90In

miles

for underground

cable,

where

n

is the

harmonicnumber.

Skin effect correction is important in EHV systems

because line resistance is the principal source of

damping.

windings are used to mitigate harmonics.

The

phase shift

associated

with transformer connections

must

be accounted

for inmultiplesource systems.

Other considerations include the nonlinear characteristic

of core loss resistance, the winding stray capacitance and

core saturation.

Harmonic

effects due to

nonlinear

resistanc

are small

compared

to the nonlinear inductance. Effects o

stray capacitanceare usually noticeable only for frequencies

higher than 4 kHz. The saturation characteristics can

b

represented

as

a harmonic

source using

the

nonlinear v-

model if saturation-caused harmonic generation is o

concern.

Passive Loads: Linear passive loads have a significan

effect on system frequency response primarily near resonan

frequencies. As in other power system studies it is only

practical to model an aggregate

load

for

which

reasonably

good

estimates

(MW and MYAR)

are USUally

readil

available. Such an aggregate model should include the

distributionor service transformer.

At

power frequencies

the

effect of distribution transformer impedance is not o

concern in the

analysis

of the high voltage network. A

harmonic frequencies the

impedance

gf..

the transfonnerca

be

comparable to

that

of motor loadS, because inductio

motors appear as locked-rotor impedances at thes

frequencies.

A general model thus appears as in Figure 1.3. To

characterize this model properly, it is necessary to know

the

typical composition of

the load. Such

data

are

usually

no

easily available.The following models havebeen suggeste

in literature(n represents the harmonic order):

ModelA : ParallelR,L with R

=

V

2

/

(P); L=

V

2

/(2

1Cf

Q)

This

model

assumes

that

the total reactive

load is assigned t

an inductor L. Because a

majority

of reactive powe

corresponds

to induction motors, this

model

is no

recommended.

Model B : Parallel

R,L

with

R =V

2

/ (k*P), L =

V

2

/ (21tfk*Q) ; k= .1h+.9

Model C : Parallel R,L in series with transfonnerinductanc

Ls

where

R=V

21P;

L=nR/(21tf 6.7*(QIP)-.74);Ls=

.073

b R

Model

C

is

derived

from

measurements

on medium

voltag

loads using audio frequency ripple generators. Th

coefficients cited above

correspond to one

set of studie

[10], and

may

not be appropriate for all loads. Loa

representation for harmonic analysis is an active researc

area.

Transformers: In most applications, transformers are

modeled as

a

series impedance with resistance

adjusted

for

skin effects. This is because adequate

data is usually

not

available. Three-phase transformer connections may

provide 30phase shift.

Other connections

such as zigzag

4


10/85

Figure 1.3:

Basic Load

Model.

Large

Rotating Loads:

In synchronous and induction

machines

the

rotating

magnetic field created

by

a stator

harmonic rotates

at a speed significantly different

from

that

of the

rotor,

Therefore at

harmonic frequencies the

impedance approaches

the negative sequence impedance.

In the case of

synchronous machines the inductance

is

usually taken to

be

either thenegative sequence impedance

or

the

average

of

direct

and quadrature sub-transient

impedances. For

induction

machines the inductance

is

taken

to be the locked rotor inductance. In each

case the

frequency-dependence

of resistances

can be significant.

The resistance

normally

increase

in

the

form

n

where n is

the harmonic order and

the parameter a

ranges

from 0.5

1.5.

Most

motors

are delta-connected

and therefore

do

not

provide a path forzero-sequence harmonics.

1.6HarmonicSimulation

It is appropriate to

note that

a large

number ofharmonic

related

problems encountered

in practice

involve systems

with

relativelylow distortion and often a single

dominant

harmonic source. In these cases simplified resonant

frequency

calculations, for

example, canbe

performed

by

hand [5] and distortion calculations can be made

with

a

simple spreadsheet For larger systems

and

complicated

harmonic producing

loads,

more

fonnal

harmonic power

flow analysis

methods

are needed. In

this

section,

techniques presently being used for harmonics studies are

reviewed. These

techniques

vary

in

terms

of

data

requirements,

modeling complexity, problem

formulation,

and solution algorithms.Newmethods arebeingdeveloped

and published. .

Mathematically,

the

harmonic study

involves

solving

the

network

equation

for each

harmonic written

in

matrix

form

as

[Iml

= [YnJ[VnJ m=I,2 ... n (1.3)

where [Yml represents the

nodal admittance

matrix, [lml is

thevector of

source

currents and [V

ml

is the

vector

of bus

voltages for

harmonic number m. In

more advanced

approaches thecurrent source vector becomes a

function

of

busvoltage.

Frequency

Scan:The frequency .scan

is

usually the first

step

in a hannonicstudy.A frequency or impedance scanis

a plot of the driving point (Thevenin) impedance at a

system

bus versus frequency. The bus of interest is one

where a harmonic source exists. For simple system this

impedance can be obtained

from

an

impedance diagram.

More

formally,

the Thevenin impedance can be calculated

by

injecting a

1

per

unit

source

at

appropriate

frequency

into

the bus of interest. The other currents are set to zero and

(1.3) is solved for bus voltages.

These

voltages equal

the

drtving-point and transfer impedances. The calculation is

repeated over the harmonic frequency range of interest.

Typically,

a

scan is developed for

both

positive and

zero

sequence networks.

If

a harmonic source

is

connected to

the

bus

of

interest,

the harmonic

voltage at

the bus is

given

by

the harmonic

current multiplied by the harmonic impedance. The

frequency

scan thus gives a visual

picture

of

impedance

levels and potential voltagedistortion. It

is

a very effective

tool to detect resonances which appear as peaks (parallel

resonance) and valleys

(series

resonance)

in

the plot of

impedance

magnitude vs.

frequency.

Simple Distortion Calculations: In the simplest harmonic

studies harmonic sources are represented as current source

specified

by their current spectra. Admittance

matrices are

then constructed

and

harmonic

voltage

components

are

calculated from

(1.3).

The

hannomc current

components

have a magnitude

determined

from the typical harmonic

spectrum and rated load current for theharmonic producing

device.

where n

is

the

harmonic order and the

subscript spectrum

indicates

the typical harmonic spectrumof the element To

compute indices such as

THD the

nominal

bus voltage

is

used.

For

the

multiple harmonic source cases

it is importantto

also

model

the phase angle of

harmonics.

A fundamenta

frequency power-flow

solution

is

needed, because

the

harmonic

phase angles are functions of the fundamenta

frequency phaseangleas

follows:

9

n

= D spedIUm

n(8

1

-9

1

spectzum}

where 9

1

is

the

phase

angleof the harmonic source currem. a

the

fundamental

frequency. 9

n

-specllUm is the

phase

angle of the

n-th

harmonic

current

spectrum. Depending

on

the phas

angles used, the effects of multiple

harmonic sources

ca

either add or cancel. Ignoring phase

relationships

may

therefore,

lead to

pessimistic

or optimistic

results.

Harmonic Power

Flow Methods:

The

simple

distortion

calculation discussed above is

the basis

for

most

harmoni

study software

and is

useful in many practical cases.

Th

main disadvantage of the method is the use of typical

spectra. This

prevents

an

assessment

of non-typica

operatingconditions. Such conditions include

partial

loadin

of

harmoruc-producing devices,

excessive

distortion an

unbalance. To explore suchconditions theuser

must

develo

5


11/85

typical spectra tor each condition when

using

the

simplified method. The disadvantages have

prompted

the

development

of

advanced

harmonic

analysis methods.

The

goal is to model the physical

aspects

of

harmonic

generation from the

device

as a function of actual system

conditions.

The

general

idea is to create a

model

for the

harmonic

producing device in the form

HereV

t,

V

2

, , V

n

are

harmonic

voltage

components,

It, 1

2

,

..., In, are corresponding harmonic currentcomponents and

C represents multiple operating and design parameters.

Equation (1.4)permits

the

calculation of

harmonic currents

from voltages and includes power flow constraints. The

total

procedure

is

to simultaneously solve

(1.3) and (1.4).

One

of

the well-known methods is the so called

harmonic iteration method [11,12]. Equation (1.4)is first

solved using an estimated supply voltage. The resulting

current

spectrum

is used in (1.3) to calculate the supply

voltage.

This iterative

process is repeated until

convergence is achieved. Reliable convergence is achieved

although

difficulties

may

occur when

sharp

resonances

exist Convergence can be improved by including a

linearized

model

of

(1.4) in

(1.3).

A

particular advantage of

this decoupled approach is that device models in the

form of (1.4) can be in a closed form, a time

domain

model, or in anyother suitable form.

Another method

is to solve (1.3) and (1.4)

simultaneously using Newton type

algorithms.

This

method requires that

device models

be

available in

closed

form

whereinderivatives can

be

efficiently computed

[13].

The

various methods above can

be

extended, with a

significant increase in computational burden, to the

unbalanced case. Both (1.3) and

(1.4) are

cast in

a multi

phase

framework [11,14].

Such an approach can have

several advantages. The first is the modeling

of zero

sequence current flow. Second is the capabiltty of

addressing

non-charaeteristic

harmonics.

Finally,

it

is appropriate

to

note

that

harmonic studies

can be performed in the time

domain The idea

is to run a

time-domain

simulation untila steady state is reached. The

challenge is first to identify that a

steady-state

has indeed

been

achieved.

Secondly, in lightly

damped

systems

techniques are needed

to obtain the

steady-state conditions

within a

reasonable amount of computation time.

References

[14,15] provide examples

of

such

methods.

1.6Summary

Harmonic

studies

are

becoming an

important

componentof power system planning and design. In using

software to analyze practical conditions it is important to

understand the assumptions made and the

modeling

capabilities. Models and methods used

depend

upon system

complexity anddata

availability.

The

purpose

of thistutorial

is to suggest what is

required

to set up

harmonics

studies

with emphasis on modelingand simulation.

This overview has attempted to summarize

key

ideas

from chapters that follow. The propagation of harmonic

current in a power

system,

and the

resulting voltage

distortion,

depends

on the characteristics of harmonic

sources as well as the

frequency

response of system

components. Characteristics of

various harmonic

sources

and consideration in their

modeling

have been summarized.

Component modeling has been described. Different

approaches to conductanalysis werediscussed in a common

framework.

Subsequent chapters of this tutorial will expand

uponeachof these topicsandprovided

illustrative

examples.

Acknowledgments

This chapter was adapted from a paper developed by the

Task

Forceon

Harmonics Modeling and Simulation [1].

References

1. Task force on Harmonics Modeling and Simulation,

The

modeling

and simulation of

the

propagation of

harmonics in electricpower networks PartI : Concepts,

models and simulation techniques,t

IEEE

Tranasactions

on Power Delivery, Vol.l l, No.1, January 1996, pp.

452-465.

2. Task

force on Harmonics Modeling and Simulation,

The modeling and simulation of the propagation of

harmonics in electric power networks Part II : Sample

systems and

Examples, IEEE Tranasactions

on

Power

Delivery, Vol.I 1,No.1

t

January

1996,

pp. 466-474.

3.

A.

Guillemin,

The

Mathematics

Circuir

Analysis

JohnWileyand Sons,INC., NewYork, 1958.

4. Corduneanu, Almost Periodic Functions,

John Wiley

(Interscience), New

York,

1968.

5. IEEE

Recommended

Practices and Requirements for

Harmonic Control

in

Electric Power Systems, IEEE

Standard519-1992,

IEEE,

NewYork, 1992.

6.

Emanuel,

A,E, Janczak,

J

Pillegi, D.O.,Gulachenski,E.

M.,

Breen,

M., Gentile, TJ. , Sorensen, D., Distribution

Feeders with Nonlinear Loads in the

Northeast

USA

Part l-Vojtage

Distortion

Forecast, IEEE

Transactions

on Power Delivery, Vol.

10,No.1,

January 1995, pp.340

347.

7. Mansoor, Grady, W.M,

Staats,

P. T.,Thallam, R. S.

Doyle, M. T., Samotyj, Predicting the net hannonic

currents from large numbers

of distributed

single-phase

computer

loads:

IEEE Trans.

on. Power Delivery, Vol.

10,

No.4,

Oct.. 1995,

pp.

2001-2006.

8. Capasso, Lamedica, R, Prudenzi, A, Ribeiro, P, F

Ranade, S. J., .. Probabilistic

Assessment

of Harmonic

Distortion Caused by Residential Loads,

Proc. ICHPS

IV,

Bologna, Italy.

6


12/85

9. Dommel, Electromagnetic Transients Program

Reference Manual EMTP Theory

Book ,

Prepared

tor Bonneville Power Administration, Dept. of

Electrical Engineering,Universityof British Columbia,

Aug. 1986.

10. ClORE Working Group 36-05, Harmonics,

Characteristics, Parameters, Methods of Study,

Estimates of Existing Values in the Network, Electra,

No. 77, July 1981,pp. 35-54.

II .

W. Xu, J.R. Jose and H.W. Dommel, A Multiphase

Harmonic Load Row Solution Technique ,

IEEE

Trans. on Power Systems, vol. PS-6, Feb. 1991, pp.

174-182.

12.

Sharma,

V, Fleming,

R.I.,

Niekamp,

L., An

iterative

Approach for Analysis of Harmonic Penetration in

Power Transmission Networks,

IEEE

Trans. on

Power

Delivery, Vol. 6,

No.4,

October 1991, pp.

1698-1706.

13. D. Xia and G.T. Heydt, Harmonic Power Row

Studies, Part I - Fonnulation and Solution, Part

IT

Implementation and Practical

Application , IEEE

Transactions on Power Apparatus and

Systems,

Vol.

PAS-lOl,

June

1982,pp.1257-1270.

14. Lombard, X., Mahseredjian, J., Lefebvre, S., Kieny,

C., Implementation of a new

Harmonic

Initialization

Methodin

EMlP,n

Paper94-

8M

438-2 PWRD, IEEE

Summer Power Meeting, San Francisco, Ca., July

1994.

IS.

Semlyen,

A., Medina, A., Computation of the

Periodic Steady State in

Systems with

Nonlinear

Components Using a Hybrid Tune and Frequency

Domain Methodology, Paper 95-WM

146-1

PWRS,

IEEEWinter Power Meeting, New York, NY, Jan.

1995.

7


13/85

Chapter

2

HARMONICS THEORY

Gary

W.

K.Chang

Siemens PowerTransmission Distribution

BrooklynPark, MN,USA

for

a l l ~ .

The

smallest

constant

T

that satisfies

(2.1)

is calle

thepenod

of

the function. By

iteration of

(2.1),we

have

f t = f t+hT ,

h=O, l,

2,

....

(2.2)

Let function

.f t be

periodic with period

T,

then thi

function canberepresented by the trigonometric series

1

f(1) = 2

lJo

+ L { a h c o s h ~ ) + ~ s i n h a J o l } ,

(2.3

h=l

w ~ e r e

(i)o

=

2nl T . A series

such

as (2.3) is calle

trigonometric Fourier series. It

can

be rewritten

as

(2.7

(2.6

(2.5

f t

= L c , . e j h ~ ,

h = - -

where for

h =0,

1, 2,

...,

1

I

T

'

2

.

ia = (t)e-jhmoldt.

T

-T/2

Orthogonal Functions

A

set

of

functions

{tph (t)}

is

called

orthogonal on

a

interval

a< t h t =

I,.J2V

hsin(hfl'ot+

~ , (2.18)

h=l h=l

i t) =Lih(t) =I ,-Iit, sin(hOJot + ~ ,

(2.19)

h=l h=l

where the

de terms

are

usually ignored

for

simplicity, V

h

and t, are rms

values

for h-th order of

harmonic

voltage

and

current, respectively.

The instantaneous power

isdefined as

p t)

=

v t)i t),

(2.20)

andthe

average

power over oneperiod T of p t) is defined

as

Most

elements and

loads

in a power

system

respond the

same in both positive and negative half-cycles. The

produced voltages and currents havehalf-wavesymmetry.

Therefore,

harmonics

of

even orders

are

not

characteristic.

Also, triplens

(multiples

of third harmonic)

always

can be

blocked by

using

three-phase

ungrounded-wye

or delta

transformer

connections

in a

balanced system,

because

triplens are

entirely

zero sequence. For these reasons,

even-ordered and triplens are often

ignored

in harmonic

analysis.

Generally, the frequencies of interests for

harmonic

analysis

are limited to the50th

multiple.

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

lh

=

I I I

h and h =

pn

1 (characteristic

harmonics)

where

n

= 1, 2, ... and

p

is an integral multiples of six.

I f

1)

the

converter input

voltages

are unbalanced or

2) unequal

commutating reactances exist between

phases

or 3)

unequally

spaced firing pulses

are present in the converter

bridge, then the converter will produce non-characteristic

harmonics in addition to thecharacteristic

harmonics.

Non

characteristic harmonics are those that are not integer

multiples of the

fundamental

power

frequency.

The harmonic frequencies that arenot integral multiples

of the fundamental power frequency are usually called

interharmonics. A major source

of

interharmonics is

the

cycloconverter [2].

One

special

subsetof inter

harmonicsis

called

sub-harmonics.

Sub-harmonics have

frequency

values that arelessthan

that

of the

fundamental

frequency.

lighting

flicker

is

one indication of

the

presence of

sub

harmonics. A

well-known source

of

flicker

is the arc

furnace [3].

Electric Quantities Under Nonsinusoidal

Situation

When steady-state harmonies are present, instantaneous

voltageand current canbe represented

by

Fourier series as

follows:

(2.21)

where P is the

average

power contributed

by

th

fundamental

frequency Component and other harmoni

components, as shownin (2.22). In the

next

section, we als

will show the relationship between

the

power

factor

an

some

harmonic

distortion

indices.

10

lIT

= p t)dt.

T

0

I f

we substitute (2.18) and (2.19) into

(2.20)

andmakeuse

of the orthogonal

relations

of

(2.7),it can

beshown

that

p = I,VIJhcos 8h-8i1

=

LJ;. (2.22)

h=l h=l

p

p = - ,

s

(2.29


16/85

Phase

Sequences

of Harmonics

For a

three-phase balanced system under nonsinusoidal

conditions, the Iz th order of harmonic voltage of each

phase can be expressed as

vaJ, t

=

.fiv,r sin(hCtJot + ()h)' 2.30

vbh (I )

=

.fiv

h

sin{hCtJot - 2hn

/3+

6

h

,

(2.31)

vch t)

=

.fiv

h

sin haJol +2hn /3

+

6

h

. (2.32)

Therefore, the

harmonic

phase

sequence

in a

balanced

three-phase system has thepatternshown inTable 1.1.

Table

1.1.

Harmonic Phase

Sequences in

aBalanced

Three-Phase

PowerSystem

Harmonic

Phase

Order

Sequence

1

+

2

-

3

0

4

+

5

-

6

0

Observing Table 1.1, we find that the negative and zero

sequences are also present in the system, and

all triplens

are entirely

zero

sequence. The

above

simple phase

sequence pattern

does

not hold for

the unbalanced

system,

because harmonicsof each order contain the threedifferent

sequences. It requires amore complicated analysis [9].

The definitions in (2.18) - (2.24) are also suitable for

three-phase balanced

system.

However,

for

theunbalanced

system,

the apparent power needs

to

be

redefined and

the

consensus has yetto

be

reached.

Reference

[10]

provides

some practical power definitions under unbalanced

conditions.

2.4 Harmonic Indices

In

harmonic analysis there are

several

important

indices

used to

describe the effects of harmonics on power system

components and communication systems. This section

describes the definitions of those harmonic indices in

common use

[11-13].

Total

Harmonic Distortion (Distortion Factor)

The mostcommonly used harmonic

index

is

f ~

Jill

THn- = h=2

or

THD h=2 (2.33)

1

which is defined as the ratio of the

rms

value of the

harmonic components to

the

rms value of the

fundamental

component and usually expressed in percent

This

index is

used to measure the deviation of a periodic

waveform

containing harmonics from a perfect sinewave. For a

perfect sinewave at fundamental frequency, the THD is

zero. Similarly, the measures

of

individual

harmonic

distortion for voltage and current at Iz th order aredefined as

vh IV

I

and

lhlll'

respectively.

Total

Demand

Distortion

The total demanddistortion (TOO) is the totalharmonic

currentdistortiondefinedas

TDD=V

6.

lh (2.34)

I

L '

where I L is the maximum

demand

load current (15-

or

30

minute demand) at fundamental

frequency

at the point of

common coupling

(Pee),

calculated

as

the

average

current

of the maximum demands for the previous twelve months.

The concept of TOO is particularly

relevant

in the

application of IEEEStandard519.

Telephone Influence Factor

Telephone influence factor (TIF) is a measure used to

describe

the

telephone noise originating

from

harmonic

currents and voltages in power systems. TIF is adjusted

based on the sensitivity of the telephone system and the

human

ear

to noises

at

various

frequencies.

Itis

defined

as

00 00

L(Wh

Vh)2

L(wh

l

h)2

Tl F

v

= h=l or TIF] = h=l , (2.35)

V

m u

Inns

where

wh is a weighting accounting for

audio

and

inductive coupling effects at the

h-th

harmonic

frequency.

Obviously,

TIF

is

a variation of the previously defined

THD wherethe root of the sumof the squares is weighted

using

factors thatreflectthe response

in

the

voice band.

VeT and IT Products

Another distortion index

that

gives

a

measure

o

harmonic

interference on audio circuits

similar

to

TIF

i

the V-Tor IT product,whereV is rms voltage in volts, I is

rIDS current in amperes, and T is the TIF. In practice

telephone interference is often expressed as VT or IT

which

is

defined as

_ 00

V

T=

L(W V,,)2 or I T =

L(whlh)2,

2.36

h=1 h= l

where

wh is the

same as previously

described.

I f

kVT

o

kl-T

is

used,

then

the

index

must

be multiplied by

a facto

of

1000. EQuation

(2.36)

refers

to the fact that the indexi

a product Of harmonic voltageor

harmonic

current andth

corresponding telephone influence factor. Observin

(2.35)and (2.36),we fmd

that

TIF

v

Vmrs=V T

and

TIFf I

rms=I T. (2.37)

C-Message Weighted Index

The C-message weighted tndexis similartoTIF, excep

that each weighting

Ch is

used in place of wh .

Th

weighting is derived from listening tests to indicate th

relative annoyance or speechimpairment by an interfering

11


17/85

herelation between TIF weight and C-message weight

IS

signal of

frequency

f

as heard through a -SOO-type

telephone

set.

This index is defined

as

by substituting (2.23) and (2.24) into (2.33). The total

powerfactor in (2.29)becomes

p

pltot

= 1

(2.43)

Yi/lv l + (THDy /1(0)2 + (THD[ /1(0)2

most ~ a s e s only very s m ~ l portion of average powerof P

IS

contributed by harmonics and total

harmonic voltage

distortion is lessthan 10%. Thus (2.43) can

be

expressed as

11 1

pltot

=-

;==================

VIII

~ 1

+

(THD[/lOO)2

=cos(6

1

- pI

dist (2.44)

In (2.44), the first term, cos(6

1

-

1

,

is

known

as the

displacement power factor, and the second term P dist' is

defined

as

the

distortion

power factor.

Because the

displacement powerfactor is

always not greater than one,

we

have

P/'ot s pIdist (2.45)

O b v i ~ s l y

.for single-phase nonlinear loads

with high

current distortton, the total power factor is poor. It also

should be noted that adding power factor correction

c p ~ ~ r s

to such load is

l i k e l y _ ~ 9

cause

resonance

conditions. An alternative to improve-the distortion power

factor

is

using passive or active filter to cancel

harmonics

producedby nonlinear loads.

2.5 PowerSystem Response to Harmonics

In comparison

with

the load, a power system is stiff

enough

to

withstand considerable amounts of

harmonic

currents without causing problems. This means

that

the

~ y s t e m impedance is smaIler compared to the load

A powersystem ~ t s e l f is not a significant source

of harmomcs. However, It becomes a contributor of

problems bywayof resonancewhensevere

distortion

exists.

Assuming

all

nonlinear loads can

be

represented as

r m ~ n i c current i n j e c t i o ~ the harmonic,voltage at each

bus

In

a power system can

be

obtained by

solving

the

following Impedance

matrix

or nodal admittance

equations

for all ordersof harmonics

under

consideration:

V

h

=

Z.

III

(2.46)

or

(2.38)

r C/

=--------

v

=- - - -

}\. h

=

5Chfh'

2.39

where

fh is the frequency of the h-th orderharmonic.

Transformer K-Factor

Transformer K-factor is an index used to calculate the

derating

of standard

transformers

when

harmonic

currents

are

present[14]. The K-factor is

defined

as

00

L h

2

(lh I /})2

K

=

2.40

L lh

ll

l )2

h=1

where

h

is the

harmonic

order and lhlll is the

corresponding individual harmonic current distortion.

(2.40) is calculated based on the assumption

that

the

transformer Winding eddy current loss produced by each

harmonic current component is proportional to the square

of the

harmonic

order and the square

of

magnitude

of

the

harmonic component

The K-rated transformer is constructed to withstand

more voltage distortion than standard transformers. The

K-faetor actually

relates to

the

excessive heat

that

must

be dissipated by the transfonner. It is considered in the

design andinstallation stagefor nonlinear loads,

and

it

is

used as a specification for new or replacement power

source equipment Table 2

shows

typical commercially

available K

-rated transformers,

where

all regular

transformers fall into K-l category.

Table 2:CommerciallyAvailable K-RatedTransfonners

Cate20ry

K-4

K-9

K-13

K-20

K-30

K-40

DistortionPowerFactor

When voltage and currentcontain harmonics, it can be

shown

[15] that

Vnn.r =V +

(THDv

/1(0)2

(2.41)

and

I

h

=

Y

b

V

h

t (2.47)

where

V

h

is the vector consisting of the

h-th harmonic

voltage

at

each bus

that

is to

be

determined.

Z. is

the system

harmonic impedance matrix, Y

b

is the system harmonic

admittance mattix, and I

is the vector

of measured

or

estimated harmonic currents representing the harmonic

generating loads

at

connected

busses.

In (2.46), Z. can

be

obtained by using a

Z.bus

building

algorithm for each harmonic of interestor from the inverse

of Y

h

in (247). but the harmonic effects ondifferent powe

system components and loads need to

be

properly modeled

[16].

Approaches for harmonic

analysis

based on (2.46) or

(2.47)

are commonly called curreet injection

methods.

1bese

approaches

are usually -used

in conjunwon

with

fundamental

frequency

load flow

computations. Through

providing

the network

harmonic impedance or

admittance

and harmonic currents injected by nonlinear loads for al

12


18/85

Power

System

Figure 2.2. Parallel Resonance

When parallel resonance occurs

in the

circuit

of

Figure

2.2,

the resonant

frequency

canbe determined by

hr

=

J

Xc

= MVA

sc

, (2.49)

XL MVAR

cAP

where MYA

sc

is the

short-circuit MVA at

the

harmonic

generating

load connection point to the system

and

MYAR

CAP is

MVAR rating of the

capacitor.

It

should

be

understood that this approximatiqI : s

only accurate for

systems

with high XIR ratios.

Another resonant

scheme

is

shown

in

the distribution

network of Figure 2.3. If some of the

feeder inductance

appears between

groups

of smaller capacitor

banks,

the

system may present

a

combination of

many series and

parallel resonant circuits, although the

resonant effects

are

somewhat less

than

that caused by one

large resonant

element. For this type of

resonance problem,

more

sophisticated

harmonic analysis programs must be

employed to predict the harmonic

characteristics

of the

system.

Substation

i

frequency. When parallel resonance exists on the power

system, significant voltage distortion and current

amplification

may

occur.

The

highly distorted bus voltage

may

cause distorted currents flowing

in

adjacent circuits.

Theamplitied current may result in equipment

failure.

XL

Figure 2.3. Distributed

Resonance

2.6Solutionsto Harmonics

Passive

harmonic

tilters are

an effective mitigation

method

for

harmonic problems.

The

passive

filter is

generally designed to provide a path to

divert

the

troublesome harmonic currents in the

power system.

Two

common

types

of filters are the series

and

the shunt

filters. The series fIlter is characterized as a parallel

resonant and blocking type whichhas a high impedance at

its tuned frequency.

The

smoothing reactor

used in

power

electronics

device

is an example.

The

shunt

filter is

characterized

asa series

resonant

and

trap

type

which

has a

low impedance at its tuned frequency. The single tuned

LC filter is the most

common design

in

power systems.

More

detailed information on

harmonic

tilter

design

and

applications

canbe

found

in

[12,17].

Harmonic

currents

in a

power system

can

also

be

reduced

by providing a phase

shift

between nonlinear loads on

13

Figure

2.1. Series

Resonance

ParallelResonance

Figure 2.2

shows the

circuit topology in

which

parallel

resonance

is

likely

to

occur. Parallel

resonance

occurs

when

the

parallel inductive

reactance

and

the parallel

capacitive reactance

of the system are

equal

at certain

frequency,

and

the parallel combination appears to be a

very

large impedance

to the

harmonic source.

The

frequency where

the

large impedance occurs

is theresonant

harmonics under consideration, the individual and total

harmonic voltage

distortions

at each bus can be

determined. Reterence [16]

also describes

some

other

harmonic analysis methods.

Observing (2.46),

we see that

system harmonic

impedance plays

an

important

role in the

system response

to harmonics, especially when resonance

occurs in

the

system.

Resonance

is defined

as an amplification

of power

system

response to

a

periodic

excitation when

the

excitation

frequency

is

equal

to a natural

frequency

of

the

system.

For a

simple

LC

circuit excited by a

harmonic

current, the

inductive and

capacitive

reactance seen from

the

harmonic

current source are equal

at the resonant

frequency Ir =

1/

(2rc.J LC .

In a

power system,

most significant resonance

problems are caused

by a large capacitor installed for

displacement power factor

correction or voltage

regulation

purposes.

The resonant

frequency

of the

system inductive reactance and

the capacitor reactance

often occurs near fifth

or seventh

harmonic. However,

resonant

problems

occurring at

eleventh or

thirteenth

harmonic are not

unusual.

There are two

types

of

resonances

likely

to

occur

inthe

system: series

andparallel

resonance.

Series resonance

is a low

impedance

to the

flow of

harmonic current,

and parallel

resonance

is a high

impedance tothe

flow

of

harmonic

current

Series Resonance

As

shown

in

Figure 2.1,

if the capacitor bank

is in

series

with the

system reactance

and creates a low impedance

path

to the

harmonic current,

a series

resonance

condition

may result. Series resonance

may

cause

high voltage

distortion levels between the

inductance and the capacitor

in

the

circuit

due to

the harmonic current c-oncentrated

in

the

low

impedance

path it sees.

Series

resonance

often

causes capacitor

or fuse failures because of

overload.

The

series resonant condition

isgiven

by

h;=J

Xc , (2.48)

XL

where h;

is

the

harmonic order

of

resonant frequency.

XL

x,

L Power

~ y s t ~


19/85

different branches. One popular method called phase

multiplication is to operate separate six-pulse static

converters (12-pulse and higher) in series on the de side

and in parallel on the ac side through thephase-shifting

and

L\-Y

transformers [18] so that there is self

cancellation of some harmonics. Sometimes, a specially

designed transformer (zigzag) is used to trap triplen

harmoniccurrents and to prevent

the

currents

flowing

back

to the source from

the nonlinear

load. This Zigzag

transformer is usually designed to provide a low harmonic

impedance between its windings compared to the source

harmonic impedance. Thus there are circulating harmonic

currents between the nonlinear load and the transformer.

Active filtering techniques [19] have drawn great

attention in recent years.

By

sensing the nonlinear load

harmonic voltages and/or currents, active filters use either

1) injected harmonics at 180degrees out of phasewith the

load harmonics

or

2) injected/absorbed current bursts to

hold the voltage waveform

within

an acceptable tolerance.

These approaches provide effective filtering of harmonics

and eliminate some adverse effects of passive filters

such

as component agingand resonanceproblems.

Harmonic standards provide useful preventive solutions

to harmonics. Recent standards such as IEEE 519-1992

[11] and

lEe

1000-3-2 [20] emphasizeplacement

of limits

on harmonic

currents

produced by nonlinear loads for

customersand network bus harmonicvoltage distortionfor

electricutilities.

2.7 Summary

For harmonic studies, Fourier series

and

Fourier

analysis are fundamental concepts. Many

FFr

algorithms

have been implemented for DFf computations on

measuring harmonics.

In nonsinusoidal situations, the conventional electric

quantities used in sinusoidal environment need to be

redefined. However,powerdefinitionsaswellas harmonic

phasesequences underunbalanced three-phase

systems

are

still

under

investigation. Several hannonic indices have

been defined for the evaluation

of

harmonic effects on

powersystem components andcommunication systems.

To predict precisely the power system response to

harmonics requires accurate models for power system

elements and harmonic-generating loads. A simple

technique for hannonic

aDalysis

is

the

current injection

method,

which is perfonned in the frequency domain.

Other analysis

nletbods

include time domain and

f r ~ e n y t i m e domain techniques. Solutions to harmonics

can be classified as remedial and preventive. Passive and

active filters are widely-used remedial solutions, and

harmonic standards provide the best solution before actual

harmonic

problems occur.

References

1. A V. Oppenheim and R. W. Schafer, Discrete-Time

Signal

Processing, Prentice-Hall, lnc., Englewood

Cliffs,

NJ, 1989.

2. R. F. Chu and

J. J. Bums, Impact

of Cycloconverter

Harmonics,

IEEE Trans. on Industry Applications,

Vol. 25, No.3,May/June 1989,pp.427435.

3. R. C. Dugan, Simulation of Arc Fmnace Power

Systems,

IEEE

Trans.

on Industry Applications,

IA

16(6),Nov/Dec1980, pp.813-818.

4. A. E. Emanuel, Powers inNonsinusoidal Situations - A

Review

of

Definitions and Physical Meaning, IEEE

Trans.

on Power Delivery,

Vol. 5,

No.3,

July 1990,

pp.1377-1389.

5. A. E. Emanuel, On the Definition of

Power

Factor and

Apparent Power in Unbalanced Polyphase Circuits,

IEEE Trans. on Power Delivery, Vol. 8, No.3, JUly

1993, pp.841-852.

6. L. S. Czarnecki, Misinterpretations of Some Power

Properties of Electric Circuits,

IEEE

Trans.

on Power

Delivery,

Vol.9, No.4, October 1994, pp.1760-1769.

7. P. S. Filipski, Y. Baghzouz, andM. D.Cox, Discussion

of Power Definitions Contained in the IEEE

Dictionary, IEEE Trans. on Power Delivery, Vol. 9,

No.3, July 1994, pp.1237-1244.

8. Nonsinusoidal

Situations:

Effects on the Performance

of Meters

and

Definitions of Power, IEEE

Tutorial

Course90 EH0327-7-PWR, IEEE, New

York,

1990.

9. K. Srinivasan, Harmonics and Symmetrical

Components, PowerQuality Assurance, Jan/Feb 1997.

10. IEEE Working Group on Nonsinusoidal Situations

Practical Definitions for Powers in Systems with

Nonsinusoidal Waveforms

and

Unbalanced

Loads:

A

Discussion,

IEEE

Trans.

on

P-ower Delivery, Vol. 11

No.1, January 1996,pp. 79-101.

11. Recommended

Practices and

Requirements

for

Harmonic Control

in Electric

Power Systems, IEEE

Standard519-1992, IEEE,NewYork,

1993.

12. J. Arrillaga, D. A Bradley, and P. S.

Bodger, Power

System Harmonics, John Wiley & Sons, New York

1985.

13. G. T.

Heydt

Electric Power Quality, Stars in a Circle

Publications,WestLafayette, IN, 1991.

14. IEEE Recommended Practice for Establishing

Transformer Capability When Supplying Nonsinusoidal

Load

Currents,

ANSIllEEE Standard C57.110-1986

IEEE,NewYork,1986.

15. W. M. Grady and R. J. Gilleskie, Harmonics and How

They Relate to Power Factor,

Proceedings

of PQA93,

San Diego,CA, 1993.

16. Task Force on Harmonics Modeling and Simulation

Modeling

and

Simulation of the Propagation

o

Harmonics

in

Electric Power Networks Part I

Concepts, Models and Simulation Techniques,

IEEE

Trans.

onPower

Delivery, Vol.l l, No.1,January 1996

pp.452-465.

17. E. W.

Kimbark.

Direct

Current

Transmission, Vol. 1

John Wiley

Sons,NewYork,1971.

18. N. Mohan,T. M. Undeland, and W. P.

Robbins, Power

Electronics - Converters, Applications, and Design

JohnWiley Sons.

New

York. 1995.

19. W.M.

Grady,

M.

J. Samotyj, and

A H.

Noyola,

Surve

of Active Power Line Conditioning Methodologies,

IEEE Trans. on Power Delivery, Vol. 5, No.3,

July

1990,pp.1536-1542.

limits for Harmonic Current Emmisions, Internationa

Electroteehnical CommissionStandard lEe 1000-3-2, March

1995.

14


20/85

ClL\.PTER 3

DISTRIBUTION SYSTEM AND

OTHER

ELEMENTS

MODELING

Paulo

F.

Ribeiro

BWXTechnologies, Inc.

Naval Nuclear Fuel Division

Lynchburg, VA 24505-0785

3.1

Introduction

One difficulty in calculating

harmonic voltages

and

currents throughout

a

transmission system

is the

need

for an

adequate

equivalent to represent the

distribution

system

and

consumers'

loads

fedradially fromeach

busbar.

It

has become

evident that the use of

equivalents

without a

comprehensive check on the

effect

of

all

impedances

actually present can lead to inaccurate

estimation

of

harmonic

voltages and currents in the

transmission

system.

Onthe

other hand,

it

is

not practicable to obtain

and

represent

all the

system

details.

A

detailed

analysis

of

distribution systems,

loads

and

other system

elements

is

carriedout,

models discussed

and a

simple

but more realistic approach

adopted.

It

consists

basically

of representing the

dominant

characteristics

of the

network

using altemative

configurations

and

models.

Simpler

equivalents for

extended networks

are

also

suggested.

the

transmission

and

distribution systems

should

be

used only for

remote

points.

(7)

For

distribution system

studies

all

the

elements

may

be

assumed

to be

uncoupled

three-phase branches

with

nomutuals, but

allowing unbalanced

parameters

per phase.

A

distribution system

comprises

a

number

of

loads

conveniently supplied

by

circuits from

thenearest

distribution

point.

The

distribution circuit configuration depends

on the

particular

load

requirements. IB general,

a

considerable

number

of

loads

are

located

so close

together

and

supplied

from the

main distribution point

that tRey-can be

considered as

a whole. For

the

majority

of

installations, whether supplying

a small factory, domestic/commercial consumers,

or a

large

plant, a

simple

radial system isused[I].

A

typical distribution

network

is

shown

in

Figure 3.1

- -..- .

Figure

3.1.

Typical distribution system configuration

- .......... . .....----38OkV

I3.StV

69kV

;-T:r

1

13.ski l

---.--.-

: , , 1

1

. .

. .

:

: :

:r

.

.

. .

- - ~ I - - - - - - - - - - - 230kV

69kV

............-. : : , 1

1

.

l l lloads p.f,c,

.

..

+

can.

~

identical circuits

A simplified dominant

configuration

can

be

derived

as

illustrated in

Figure

3.2,basedOD thebasic

assumptions.

This

arrangement would represent

the dominant

characteristics

(impedances)

of

the

supply

circuit

fed radially from each

transmission

busbar.

(1)

Distribution

lines and

cables

(say, 69-33kV,

for

example) should

be

represented

by

an

equivalent

pi.

Forshort lines, estimate thetotal

capacitance

at each

voltage

leveland

connect

it at the

termination

buses.

(2)

Transformers between

distribution voltage level

should

be represented

by

an

equivalent

element

(3) As

the

active

power

absorbedby

rotating

machines

does

not correspond to a

damping value,

the

active

and

reactive

power

demand

at the

fundamental

frequency

may not be used

straightforwardly.

Alternative

models for

load

representation should

be

used according

to their

cOmposition

and

characteristics.

(4)

Power

factor

correction (PFC)

capacitance

should be

estimatedas

accurately

as

possible

and

allocated

at

thecorresponding voltage

level.

(5) Other

elements, such as

transmission

line inductors,

tilters

and

generators

should

be represented

according to their actual

configuration and

composition.

(6)

The representation should

be

more

detailed nearer

thepointsof interest.

Simpler equivalents,

eitherfor

3.2 General

Considerations

Although

further

considerations

leading

to simpler

equivalents

are

given

later, the

basic assumptions used

in

this

chapter are

as

follows:

15


21/85

230kV

60kV 69kV 13.8k

addition

of

load

can result in

either

an

increase

or

decrease

in

harmonic flow.

Figure 3.2.

Dominant arrangement

Transmission

System3-phase

Representation

In order to simplify the manipulation of the distribution

system, load and other element

data,

the following procedure

based on the

configuration of

Figure 3.2

is suggested.

The

dotted lines in Figure 3.3 mean

different

possibilities of

connecting the load or

other elements such

as

compensators

tilters, generators, etc.. Thetotal equivalent impedance is then

calculated at

each

harmonic frequency in star-grounded and

connected to

the transmission

busbar

as

a shunt

element.

Consequently,

there

is

no

alteration

of

the

dimension

of

the

transmission

system matrix,

See

illustration

in

Figures

3.3

and

3.4. A

composition

of

different

arrangements can be

represented at the samebusbar.

..............................................

Load&t

Other

Elemems

Figure3.3. Distribution system, loadsandothel'

elements

3.3 The

Modeling

ofLoads

In

this

section, the

modeling

of individual

elements

is

discussed in detail. Considering

that

there

is some

disagreement regarding which harmonic

models

are best for

loads, transformers, generators, etc [2], various

proposed

models are

discussed. Also simpler equivalents

for

disUibution and

transmission systems at

relJlote pointsof the

area of

interest arediscussed.

Consumers'

loads

playa

very

important

part

in the

harmonic

network characteristic. They constitute not

only the main

element of the

damping component but may

affect the

resonance conditions,

particularly

at

higher

frequencies.

Indeed, measurements [3]

have

shown

that

maximum

plant

conditions

resulted

in a

lowering

of the

impedance

at the

lower frequencies, but

cause

an

increase

at

higber frequencies.

Mahmoud and Shultz [4] observed in simulations that the

Distribution system

and otherelements

Figure3.4. Overallsystemrepresentation

Consequently, an adequate representation of the system loads

is

needed.

However, it is

very hard

to

obtain detailed

information

about this. Moreover,

as

Ule

general loads consist

of an aggregate number of

components,

it is difficult to

establish a

modelbased

on theoretical analysis.

The necessity of practical measurements on distribution

points, at 13.8kV for example,

together with detailed

information

of the network: under study, is vital for the

understanding and

establishment of a

realistic

model.

Attempts to deduce

a

model from measurements

have

been

made.

See

Bergea1

et

al

[5] and Baker [6]. However, more

comprehensive measurements and system data

are

needed.

Although

practical experience

is

still insufficient

to

guarantee

thebestmodel,system studieshave toproceed

with

whatever

information is available. Thus,load characteristics are

looked

at

in detailand alternative

models

developed-in

.the

following

sections.

A typical composition of consumers'plantmay

be

as shown in

Table

3.1. From

Table

3.1, it seems evident

that

there

are

basically

two sorts of loads - resis

TP125_Full_Content.pdf

Documents