HARMONICS IN POWER SYSTEMS: THEIR CAUSES, EFFECTS AND MITIGATION
George J. Wakileh
Department of Electrical and Computer Engineering
New Mexico State University
Las Cruces, NM 88003
Abstract | Harmonics have existed in power systems for
many years. However, the issue has recently received added
signicance by the simultaneous setting of two trends: the
electric utility's increased use of capacitor banks attempting
an improved power factor, and the industry's widespread application of power-electronic converters seeking higher system reliability and eciency.
This paper provides a discussion of the issue of harmonics
in electric power systems. Their origins, eects and methods
of reduction are reviewed providing a quantitative analysis
when possible. Pertinent equations are developed.
form is composed of sinusoidal waves of dierent frequencies; a
basic waveform at the fundamental frequency and a number
of sinusoids or harmonic components with frequencies that are
integer multiples of the fundamental.
This section is aimed at developing and presenting some basics that will lead to a better understanding of the issue of harmonics.
A. Representation of Harmonics
The Fourier series represents an eective way to study and
analyze harmonic distortion. It allows inspecting the various
I. INTRODUCTION
constituents of a distorted waveform through decomposition.
Generally, any periodic waveform can be expanded in the
LECTRIC utilities are always concerned about having a
high power factor which has several advantages. This is, form of a Fourier series as
however, accompanied by the industry's increasing use of variable speed drives and electronic equipment which, as sources
1
X
of undesirable phenomena, can interact with power factor
f
(t) = A0 + (Ah cos h!0 t + Bh sin h!0 t)
correction capacitor banks to result in voltage and current amh=1
plication.
The steep development that the industry of solid-state elec1
X
tronics has undergone lately, resulted in the introduction of
Ch cos(h!0 t + h )
(1)
=
A
0+
\delicate" appliances which are more sensitive to the quality
h
=1
of power supplied by electric utilities, in comparison with the
\robust" devices of the older days. At the same time, these where
electric appliances result in the distortion of the steady-state f (t) is a periodic function of frequency f0 , angular frequency
ac current and voltage waveforms.
!0 = 2f0 and period T = 1=f0
This dilemma has set the basis for paying considerable atten- C1 cos(!0 t + 1 ) represents the fundamental component
tion to the quality of electric power and seriously addressing the C cos(h!0 t + h ) represents the hth harmonic of amplitude
issue of current and voltage distortion, a major form of which h
Ch , frequency hf0 and phase h
RT
is harmonic distortion.
1
Harmonics are components of a distorted periodic waveform A0 = T2 R0T f (t) dt
whose frequencies are integer multiples of the fundamental fre- Ah = T R0 f (t) cos(h!0 t) dt
2 T
quency.
Bh = p
T 0 f (t) sin(h!0 t) dt
This paper provides a quantitative discussion of harmonics.
2
2
h Section II provides some preliminaries. The Fourier series is Ch = Ah ?+1 B
presented. Characteristics of harmonics are illustrated. Equa- h = ? tan ABhh .
tions pertinent to distortion measures are developed. The use
of capacitor banks for reactive power supply and power factor B. Characteristics of Harmonics in Power Systems
correction is discussed. Resonance is explained.
Symmetry:
Origins of harmonic distortion are dealt with in Section III.
o Odd symmetry is characterized by f (?t) = ?f (t) and
Section IV is aimed at quantifying the eects of harmonic distorresults in no cosine terms in the waveform Fourier series
tion on power system equipment and loads. Impacts on capacexpansion [1].
itor banks, transformers and rotating machines are examined.
o Even symmetry is characterized by f (?t) = f (t) and reFinally, Section V is devoted to the reduction or suppressults in the waveform Fourier series expansion having
sion of power system harmonics. Particular attention on how
only cosine terms [1].
to attain this in converters, capacitor banks, transformers and
o Waveforms with half-wave symmetry, f (tT=2) = ?f (t),
rotating machines is given.
have zero dc components and result in the cancellation of
even-order (2, 4, 6, ...) harmonics. This feature leads us
II. FUNDAMENTALS OF HARMONICS
to ignore even harmonics in power systems [2] since they
consist of bilateral components that produce voltages
Sinusoidal waveforms are preferred since a sine wave contains
and currents which are half-wave symmetric.
no harmonics thus reducing iron losses and resulting in an in In a balanced three-phase system, the individual frequency
creased eciency. Moreover, machine, transformer and electric
harmonic components are of entirely positive, negative or
appliance designs assume a sinusoidal supply thus simplifying
zero sequence. The Fourier series representation of phase
design calculations. However, a sinusoidal waveform is somevoltages reveals that:
thing ideal and cannot be realized in practice. A distorted wave-
E
o fundamental as well as fourth, seventh, ... harmonics have
positive sequence
o second, fth, ... harmonics have negative sequence
o triplen (third, sixth, ...) harmonics have zero sequence.
The property that linear networks in balanced power systems have their responses to dierent harmonics independent of others, enables us [1] to treat each harmonic separately. That is, construct the equivalent circuit for each
harmonic (in the frequency domain) and solve for currents
and voltages. The total response is obtained by adding the
harmonic components in the time domain.
C. Measures of Harmonic Distortion
A distorted periodic current or voltage waveform expanded
into a Fourier series is expressed as follows
1
X
Ih cos(h!0 t + h )
i(t) =
V1 , I1 represent the fundamental peak voltage and current, respectively.
With
v(t) =
h=1
and
Vh cos(h!0 t + h )
the apparent power is
v
u
1
uX
S = Vrms Irms = t Vh 2rms Ih 2rms
h=1
Vh rms Ih rms cos(h ? h )
p
p
P = 2 Vh Ih cos(h ? h )
h=1
h=1
(7)
h=1
p
1
1X
=
v
u
1
p
uX
Irms = t Ih 2rms = I1 rms 1 + (CDF )2
= V1 rms I1 rms 1 + (V DF )2 1 + (CDF )2
where
th harmonic peak current
Ih is the hth
Vh is the h harmonic peak voltage
h is the hth harmonic current phase
h is the hth harmonic voltage phase
!0 is the fundamental angular frequency, !0 = 2f0
f0 is the fundamental frequency, f0 = 60 Hz.
Accordingly, the following relationships for active and reactive
power apply. Active power is
1
X
(6)
h=1
h=1
1
X
v
u
1
p
uX
Vrms = t Vh 2rms = V1 rms 1 + (V DF )2
p
= S1 1 + (V DF )2 1 + (CDF )2
(8)
where S1 is the apparent power at the fundamental frequency
and the power factor is
1p
pf = PS = SP p
1
1 + (V DF )2 1 + (CDF )2
= pfdisp pfdist
pfdisp = SP1
1p
1 + (V DF )2 1 + (CDF )2
pfdist = p
(2) where
(9)
pfdisp is the displacement power factor
pfdist is the distortion power factor.
Reactive power is dened as
Noting that when harmonics are present, apparent power S is
not only comprised of active power P and reactive power Q,
Q = 2 Vh Ih sin(h ? h )
distortion power (voltamperes) D is dened [1], [2] to account
for the dierence, where
h=1
D2 = S 2 ? (P 2 + Q2 )
(10)
1
X
=
(3)
Vh rms Ih rms sin(h ? h )
Finally, harmonics generate telephone interference through
h=1
inductive coupling. The I:T product, used to measure telephone
Voltage distortion factor V DF , also known as voltage total har- interference, is dened as [3]
monic distortion THDV , is dened as
v
u
1
1X
v
u
1
uX
1
V DF = V t Vh2
1
h=2
1
uX
(4)
I:T = t
h=1
(Ih Th )2
(11)
where
Analogously, current distortion factor CDF , further known as Th is the telephone interference weighting factor at the hth harmonic.
current total harmonic distortion THDI , is dened as
v
D. Calculation of Distortion
u
1
uX
Harmonic studies are conducted to inspect the impacts of
(5)
CDF = I1 t Ih2
1
non-linear devices and analyze certain harmonic situations.
h=2
They are aimed at detecting resonance and calculating distorwhere
tion factors. An impedance scan, also known as a frequency
scan, is a plot of the magnitude of driving point impedance at
the bus of interest versus harmonic order or harmonic frequency
and is useful in identifying resonance conditions. A dip occurring in the impedance value implies series resonance. Parallel
resonance, on the other hand, is identied by a sharp rise in
the impedance value. A harmonic power ow study further calculates fundamental and harmonic line currents and bus voltages. The outcome of dierent solutions could nally be examined through the simulation of various ltering alternatives.
In simple systems, harmonic problems can be analyzed using a
spreadsheet. Harmonic Power Flow Analysis Programs
are further commercially available to analyze large systems and
allow the detailed modeling of the dierent types of harmonicproducing devices.
E. Capacitor Banks
Switched at the bus of interest, capacitor banks provide a
means of increasing the voltage and supplying reactive power
thus correcting the system power factor [4]. Capacitor banks in
parallel with an inductive load supply this load with reactive
power thus reducing the system's reactive and apparent power,
hence increasing its power factor. Moreover, capacitor current
causes voltage rise which results in lower line losses and voltage
drops leading to an improved eciency and voltage regulation
[4]. Referring to the power triangle shown in Figure 1, the
reactive power delivered by the capacitor bank is
QC = Q1 ? Q2 = P (tan 1 ? tan 2 )
impedance is low and a small exciting voltage results in a huge
current. Analogously, parallel resonance occurs in a parallel
RLC circuit with equal inductive and capacitive reactances, so
that the circuit admittance is low and a small exciting current
develops a large voltage [1]. From the above, resonance occurs
when
X = !L = X = 1
L
C
!C
that is, at a resonant frequency of
! = p 1 red/sec or f =
p1
Hz
(13)
2 LC
Consider a power system represented by its Thevenin equivalent
as shown in Figure 2. With the resistance neglected and for a
base voltage equal to the rated voltage
1
1
(14)
Xspu = SCMV
A and XCpu = CAPMV A
LC
pu
and the system's resonant frequency is
f = 2p1L C = p !0
2 Xspu =XCpu
s
= f0
where
r
r
XCpu = f SCMV Apu
Xspu 0 CAPMV Apu
pu
(15)
f0
is the system's fundamental frequency, 60 Hz
SCMV Apu is the short circuit MVA at the bus dened by
Equation 14
(12) CAPMV Apu is the capacitor MVA rating as dened by Equa-
= P [tan(cos?1 pf1 ) ? tan(cos?1 pf2 )]
tion 14 or approximated by Equation 12.
where
Equation 15 gives the implication that a system having
P is the real power delivered by the system and absorbed by SCMV A = 25 pCAPMV A will undergo fth harmonic resthe load
onance (f=f0 = 25 = 5).
Q1 is the load reactive power
S1 is the load apparent power
Q2 is the system's reactive power with the capacitor bank conI
Ls
nected
S2 is the system's apparent power with the capacitor bank
IC
connected
j Xs I C
IC
pf1 is the original power factor, pf1 = cos 1 = P=S1
Vs
pf2 is the improved power factor following the connection of
j Xs I
Z
Vs
the capacitor bank, pf2 = cos 2 = P=S2
Vbus
V
C
C
QC is the reactive power delivered by the capacitor bank.
I
Fig. 2. A power system with a shunt switched capacitor bank.
QC
S1
S2
Writing an equation for the phase voltage rise at the bus due
to
the connection of the capacitor bank, the system's resonant
Q2
frequency can be re-expressed as
φ1
φ2
f0
= p fo
(16)
f = 2p1L C = p
P
Vbus pu
s
Vbus =jVbus rated j
Introducing the resonance order hr , dened as hr = f=f0 , Equation 16 becomes
Fig. 1. Power triangle for a power factor correction capacitor bank.
(17)
Vbus = 1
Q1
pu
h2r
which tells that a 0.04 pu bus voltage rise due to the addition of
F. Resonance
a capacitor
p results in resonance at the fth harmonic frequency
Series resonance occurs in a series RLC circuit having (hr = 1= 0:04 = 5) and a 0.02ppu voltage rise corresponds to
equal inductive and capacitive reactances, so that the circuit the seventh harmonic (hr = 1= 0:02 = 7:07).
Harmonics in power systems have been known since the adoption of alternating current as a means for electric energy transmission. They have, however, been magnied nowadays with
the increased use of non-linear devices. A non-linear device
produces non-sinusoidal current when supplied with a sinusoidal
voltage, and vice versa [1]. As sources of harmonics, non-linear
devices can be classied as:
Traditional (Classical) types:
{ Transformers
{ Rotating machines
{ Arc furnaces.
Modern (Power-Electronic) types:
{ Fluorescent lamps
{ Electronic controls and switched-mode power supplies
widely used these days in industry and modern oce
electronic equipment
{ Thyristor-controlled devices which include:
Rectiers
Inverters
Static VAR compensators
Cycloconverters
HVDC transmission.
A. Transformers
Power transformers are sources of harmonics since they use
magnetic materials that are operated very close to and often in
the non-linear region for economic purposes [1]. This results in
the transformer magnetizing current being non-sinusoidal and
containing harmonics (mainly third) even if the applied voltage
were sinusoidal [5]. The converse is also true: if the magnetizing current were sinusoidal, the voltage could not be so. In
three-phase transformers, a delta or ungrounded wye connection
blocks the ow of zero sequence triplen harmonic currents.
B. Rotating Machines
Rotating machines are considered sources of harmonics because [1], [2] the windings are embedded in slots which can
never be exactly sinusoidally distributed so that the mmf is distorted. However, and as will be seen later, coil spanning in
three-phase machines is used to reduce fth and seventh harmonics. Moreover, large generators are usually connected to
power grids through delta-connected transformers thus blocking the ow of third harmonic currents. Generally, harmonics
produced by rotating machines are considered negligible compared to those produced by other sources [1].
However, a negative value of E and a greater than 90 ring angle put the converter in the dc/ac inversion (ac power supply
or discharging) mode. Higher pulse number converters consist
usually of two or more six-pulse converters operated through
parallel transformers with some carefully selected phase shifts
[1].
1
3
5
Ideal Transformer
III. SOURCES OF HARMONICS
a
b
c
E
4
6
2
Fig. 3. A three-phase six-pulse line-commutated converter.
Consider two identical six-pulse converters operating in parallel through Y ? Y and Y ? transformers as depicted in
Figure 4. The turns ratios are such that the same voltage magnitude is applied to both converters [1]. With the Y ? Y sixpulse converter phase current having a rectangular waveform,
Iy would be expressed as
Iy = I1 sin !0 t ? I5 sin 5!0 t ? I7 sin 7!0 t + I11 sin 11!0 t
+ I13 sin 13!0 t + : : :
With the voltage applied to the -connected converter lagging
by 30 , and assuming equal current magnitudes
Id = I1 sin(!0 t ? 30 ) ? I5sin(5!0 t ? 150 ) ? I7sin(7!0 t ? 210 )
+ I11 sin(11!0 t ? 330 ) + I13 sin(13!0 t ? 390 ) + : : :
Y
Iy
Y
I
Y
HVS
LVS
∆
I
d
C. Fluorescent Lights
In a uorescent lamp, the voltage builds up in each half cycle
[1], [2] till ignition occurs. The lamp then appears as a negative
resistance, the current being limited by the non-linear reactive Fig. 4. A 12-pulse converter comprising two parallel 6-pulse ones.
ballast. The current is thus distorted. Moreover, electronic
ballasts used nowadays produce harmonics of patterns similar
Complying with the ANSI standard for Y ? transformers,
to and sometimes dierent from those produced by the older high side positive sequence voltage or current should lead the
core and coil ballasts.
respective low side by 30 . High side negative sequence voltage
or
current, on the other hand, should lag the low voltage side
D. Power Converters
one by 30 . So, on the high side, Iy is the same in pu. On
Three-phase converters produce harmonics because of the the other hand, Id positive and negative sequence harmonics
commutation of dc-side current between the three ac phases are shifted by + 30 and - 30 , respectively. Accordingly,
[1], [2]. Line-commutated converters are so called since they get IY = I1 sin !0 t ? I5 sin 5!0 t ? I7 sin 7!0 t + I11 sin 11!0 t
+ I13 sin 13!0 t + the voltage necessary for commutation from the network. A six+ I1 sin(!0 t ? 30 + 30 ) ? I5 sin(5!0 t ? 150 ? 30 )
pulse three-phase converter, shown in Figure 3, can be operated
? I7 sin(7!0 t ? 210 + 30 ) + I11 sin(11!0 t ? 330 ? 30 )
as an ac/dc rectier (dc power supply or charger) if E is positive.
+ I13 sin(13!0 t ? 390 + 30 ) + = 2 fI1 sin !0 t + I11 sin 11!0 t + I13 sin 13!0 t + g pu
which manifests the following well-dened \characteristic" harmonics 11; 13; 23; 25; 35; 37; or 12n 1 where n = 1; 2; 3; .
The sequence 12n 1 gave rise to the name 12-pulse converter.
In general, a p-pulse converter produces harmonics of the order
[1], [2]
h = pn 1 ; n = 1; 2; 3; (18)
Six-pulse converters, accordingly, tend to produce harmonics of
the order 5; 7; 11; 13; 17; 19; . Moreover, the resultant nearrectangular-shaped phase current pulses have a theoretical harmonic content of
Ih = 1
(19)
For resonance at the hth harmonic frequency
! = ! = h! = p 1
h
and
Ls C
0
r
Vs = ?| Vs Ls = ?| Zc V = ?|A V
VCh = |!CR
f s
Rs C
Rs s
s
(21)
where
VCh is the hth harmonic capacitor voltage, equivalently the capacitor voltage at resonance
p
ZC is the characteristic impedance dened as ZC = Ls =C
I1 h
Af is the amplication factor dened as Af = ZC =Rs .
Practical values are, however, less due to the smoothing eect Equation 21 points out that harmonics corresponding or close
of commutation [2].
to the resonant frequency are amplied.
IV. EFFECTS OF HARMONIC DISTORTION
As mentioned earlier, the increasing use of non-linear devices
and loads is causing increased harmonic distortion problems
in power systems. This section is dedicated to reviewing and
discussing harmonic impacts on power systems equipment and
loads, providing a quantitative analysis when possible.
A. Thermal Losses in a Harmonic Environment
Harmonics have the eect of increasing equipment copper,
iron and dielectric losses and thus the thermal stress. Equipment derating [1] is a preventive requirement in this case. However, it is perceived that this may not work [6] since it is likely
that \signs hung on the transformer warning not to load it over
a certain percentage will be gone after a period. People will
then load the unit up to its nameplate rating."
B. Harmonic Eects on Power Systems Equipment
Harmonics result in increased losses and equipment loss-oflife. Triplen harmonics result in the neutral carrying a current
which might equal or exceed the phase currents even if the loads
are balanced. This dictates the derating or oversizing of neutral
wires [1], [6], [7]. Moreover, harmonics-caused resonance might
damage equipment. Harmonics further interfere with protective
relays, metering devices, control and communication circuits,
and customer electronic equipment [1], [2]. Sensitive equipment
would experience maloperation or component failure [7].
B.1 Capacitor Banks
Harmonics aect capacitor banks in the following manner:
Capacitors are overloaded by harmonic currents, since the
fact that their reactance decreases with frequency makes
them act as sinks for harmonics. Also, harmonic voltages
produce large currents causing capacitor fuses to be blown
[1], [2].
Harmonics tend to increase the dielectric losses. Additional
heating and loss-of-life are direct consequences [1], [2].
Capacitors combine with source inductance to form a parallel resonant circuit, the resonant frequency of which is
given by Equation 13, 15 or 16. In the presence of resonance, harmonics are amplied. The resulting voltages
highly exceed the voltage rating and the consequence is
capacitor damage or blown fuses [1], [2].
Harmonic amplication can be best explained as follows. Considering the power system of Figure 5, one can express the bus
voltage when the capacitor is switched as
Vs
0 = ?|XC Vs =
VC = Vbus
Zs ? |XC
1 ? !2 LsC + |!0 CRs
0
Rs
Ls
IC
Vs
Vbus
IC
Vs =
Vbus
j Xs I C
R s IC
C
V
C
Fig. 5. Capacitor Switching.
ANSI/IEEE standard 18-1980 \Shunt Power Capacitors" [8]
states that capacitors can be continuously operated in a harmonic environment provided that:
reactive power does not exceed 135 % of rating
peak current does not exceed 180 % (superseded to 130 %)
of rated peak current
peak voltage does not exceed 120 % of rated
rms voltage does not exceed 110 % of rated.
B.2 Transformers
Transformers operating in a harmonic environment suer [1],
[2]:
Increased load losses which comprise copper losses and
stray (winding
losses given by [9], [10]
P eddy-current)
2 2
Pe = Pe1 1
h=1 h Ih pu , where Pe1 is the winding eddycurrent loss at the fundamental frequency. The increase
in the latter is the most signicant factor in determining
the additional transformer core heating losses due to nonlinear loads [9], [10].
Increased hysteresis and eddy-current losses. It is worth
noting here that these losses greatly increase with frequency and the increase in eddy-current loss due to harmonics exceeds that of hysteresis loss. However, according
to [9], temperature rise in the core due to the increased
iron losses is less critical than in the windings.
The possibility of resonance between the transformer inductance and power factor correction capacitors. Harmonic
amplication was covered in Subsection IV-B.1 and is dened by Equation 21.
Increased insulation stress due to the increased peak volt(20)
age.
These losses result in transformer heating [1], [2] and a corresponding loss-of-life. This brings us to the conclusion that,
in the presence of harmonics, transformers should be derated.
Once again, care should be taken here of the note recently made
on equipment derating.
ANSI/IEEE standard C57.12.00-1987 \General Requirements for Liquid-Immersed Distribution, Power and Regulating
Transformers" [11] poses the following limits on transformers
operating in a harmonic environment:
current distortion factor should not exceed 5 %
steady-state rms voltage should not exceed 110 % of rated
at no-load and 105 % of rated at rated load. This is equivalent to saying
p that the voltagepdistortion factor should not
exceed 0:21 at no-load and 0:1025 at rated load.
B.3 Rotating Machines
The eects of harmonics on rotating machines can be summarized as follows:
Copper and iron losses are increased resulting in heating [1],
[2]. Calculations of losses proceed in the same manner as
for transformers. IEEE 519 [3] says that generator ratings
should be determined in consultation with manufacturers.
Pulsating torques [2] are produced due to the interaction
of the harmonics-generated magnetic elds and the fundamental. These result in a higher audible noise.
B.4 Protection, Communication and Electronic Equipment
Harmonics aect protection and control equipment, metering devices, communication circuits and electronic loads in the
following manner:
Harmonics aect the interruption capability of circuit
breakers.
Relays whose operation is governed by the voltage/current
peak or zero voltage are aected by harmonics. Electromechanical relays time delay characteristics are altered in the
presence of harmonics. Ground relays cannot distinguish
between zero sequence and third harmonic currents resulting in erroneous tripping.
Metering and instrumentation devices exhibit a dierent
response to non-sinusoidal signals.
Harmonics result in interference with telephone circuits
through inductive coupling.
Through the shifting of zero crossing, harmonics impair the
operation of electronic equipment and control circuits.
Harmonics interfere with customer loads. This is of special
concern in computer systems [1].
Harmonics shorten the life of incandescent lamps [2].
References [1], [2], [12], [13], [14], [15] tackle this topic in more
detail.
V. SOLUTIONS TO HARMONIC PROBLEMS
Solutions to harmonic problems are categorized as preventive
and remedial [1].
Preventive solutions are those policies sought for at discretion
to avoid harmonics and their consequences. These include:
Phase cancellation or harmonic control in power converters.
Developing procedures and methods to control, reduce or
eliminate harmonics in power systems equipment; mainly
capacitors, transformers and generators.
Attempting at keeping harmonics at a low \damage-free" level,
standards are further developed setting limits on the level of
individual frequency harmonics and/or harmonic distortion factors.
Remedial solutions are those techniques recoursed to aiming at
overcoming existing harmonic problems. They include:
The use of lters.
Circuit detuning which involves the reconguration of feed-
ers or relocation of capacitor banks to overcome resonance.
Harmonics can be eciently reduced through the use of a passive lter [1] which consists, basically, of a series combination
of a capacitor and a reactor tuned to a specic harmonic frequency. Filters provide a low impedance \trap" to a harmonic
to which the lter is tuned. Theoretically, the lter has a zero
impedance at the tuning frequency thus absorbing the harmonic
of interest. Shown in Figure 6, typical harmonic lters are [1],
[2]:
Series-tuned lters: A series-tuned lter consists of a series combination of a capacitor and a reactor and is tuned
to low harmonic frequencies. At the tuned harmonic,
the capacitor and the reactor have equal reactances and
the lter has a purely resistive impedance. The lter's
impedance is capacitive for lower harmonics and inductive
for higher harmonics, a consequence of which is aggravating the impedance below the lowest tuned frequency [12].
Double band-pass lters: A double band-pass lter is a
series combination of a main capacitor, a main reactor and
a tuning device [2] which consists of a tuning capacitor and
a tuning reactor connected in parallel. The impedance of
such a lter is low at two tuned frequencies.
Damped (High-pass) lters: A damped lter consists of
a capacitor in series with a parallel combination [2], [12]
of a reactor and a resistor. It provides a low impedance
for a moderately wide range of frequencies. When used
to eliminate high order harmonics (17th and above), it is
referred to as a high-pass lter, providing a low impedance
for high frequencies but stopping low ones.
Detuned (Anti-resonant) lters: A detuned lter is tuned
below [1] a characteristic harmonic (usually tuned to the
fourth harmonic), thus absorbing some of the harmonic but
not as much as a higher tuned one.
Bus
Bus
Bus
(a)
(b)
(c)
Fig. 6. Typical harmonic lters: (a) Series-tuned (b) Double bandpass (c) Damped.
References [2], [12] report on the details of lters; their congurations and design tactics.
A. Power Converters
It was seen in Subsection III-D that harmonics of pulse converters constructed through the operation of lower pulse number converters can be eliminated through the proper selection of
phase shifts. This is called phase cancellation [1] or phase multiplication. Analysis revealed that fth, seventh, seventeenth,
nineteenth, harmonics are eliminated in two six-pulse converters operating in parallel or series with 0 and - 30 phase
shifts. This means that a twelve-pulse converter has a lower
harmonic impact than two six-pulse units of a comparable size.
Grady [1] further points out that two twelve-pulse converters
operating in parallel or series through + 7.5 and - 7.5 phase
shifts eliminate eleventh and thirteenth harmonics. A consequential conclusion is to use converters with higher number of
pulses.
B. Capacitor Banks
Examination of Equation 15 leads us to the conclusion that:
Relocating capacitors changes the source-to-capacitor inductive reactance thus avoiding parallel resonance with the
supply.
Varying the reactive power output of a capacitor bank will
alter the resonant frequency.
Capacitors can be designed to trap a certain harmonic by
employing a tuning reactor whose inductive reactance is equal
to [1], [2] the capacitive reactance of the capacitor at the tuned
frequency. Parallel resonance involving a capacitor and a source
inductance is achieved when
XCr = XhC 1 = Xsr = hr Xs1
r
XLn is the reactor's
inductive reactance at the tuned frequency,
p
XLn = L1 =C1
XL 1 is the reactor's inductive reactance at the fundamental
frequency
XCn is the p
capacitor's reactance at the tuned frequency,
XCn = L1 =C1
hn is the tuning order, alternatively the harmonic order to
which the capacitor is tuned or which is to be ltered
is the tuned frequency, fn = hn f0 .
Expressed dierently, Equation 24 becomes
fn
r
The voltage appearing across the capacitor would be
VC 1
?|XC 1
XC 1 =XL 1
h2n
Vbus 1 = |(XL1 ? XC 1 ) = XC 1 =XL 1 ? 1 = h2n ? 1
and
s1
or
1
red/sec
fr = hr f0 = 2pL1 C Hz
s1 1
(26)
p
L1 =C1
VCn
?|XCn
Vbus n = R + |(XLn ? XCn ) = ?| R
that is, at a resonant frequency of
!r = hr !0 = pL1 C
r
Xs 1
C1
hn = ffn = p1 = X
(25)
XL 1 = h r XL 1
! 0 L 1 C1
0
which claries that a reactorpwith XL1 = 0:04 XC 1 represents a
fth harmonic lter (hn = 1=0:04 = 5).
(22) where
= ?| XR0 = ?|hn Qf
(27)
VC 1 is the fundamental component of the voltage across the
capacitor
is the fundamental component of the voltage at the bus
r
r
is the capacitor voltage at the tuned frequency
SCMV Apu
is
the bus voltage at the tuned frequency p
hr = ffr = ! pL1 C = XXC 1 = CAPMV
Apu
0
s1
0
s1 1
is the lter's characteristic reactance, X0 = L1 =C1 =
XLn = XCn
where
Q
is the lter's quality factor dened as Qf = XL 1 =R.
f
XCr is the capacitor's reactance at resonance
XC 1 is the capacitor's reactance at the fundamental frequency Being sensitive to peak voltages, however, the capacitor needs
to be able to withstand the total peak voltage across it. That
hr is the harmonic order activating resonance
is,
it needs to have a voltage rating equal to the algebraic sum
Xsr is the source inductive reactance at resonance
Xs1 is the source inductive reactance at the fundamental fre- of the fundamental and tuned harmonic voltages [1], [2].
quency
VC = VC 1 + VCn = IC 1 XC 1 + ICn XCn
(28)
fr is the resonant frequency, fr = hr f0
f0 is the fundamental frequency, f0 = 60 Hz
However, since it is likely that a capacitor tuned to a certain
CAPMV Apu is the capacitor rating in pu MVA, harmonic
will absorb other harmonics, a safety measure would
CAPMV Apu = 1=XC 1 pu
be
to
let
the
capacitor have a voltage rating of
SCMV Apu is the bus short circuit capacity in pu MVA,
SCMV Apu = 1=Xs 1 pu .
hX
max
Tuning the capacitor to a certain harmonic, alternatively, deVC =
ICh XCh
(29)
signing the capacitor to trap (lter) a certain harmonic, requires
h=1
the addition of a series reactor. At the tuned harmonic
C. Transformers
XLn = hn XL1 = XCn = XhCn1
Harmonics are reduced through the ingenuity of transformer
connection. Delta-connected transformers prevent the ow of
so that the tuned frequency is
zero sequence triplen harmonics, thus acting as two-way lters
[2] protecting both the source and load sides of a power system.
!n = hn !0 = pL1 C red/sec
Harmonic currents can be damaging to transformers designed
1 1
to operate at 60 Hz. Oversizing the neutral conductor and derating the transformer [7] represents a short-term solution. Dini
or
[10] reexamines a formula for the derating factor based on the
(24) equations derived in [9].
fn = hn f0 = 2p1L C Hz
1 1
K-rated transformers [6], [10] are specically designed to tolwhere
erate harmonics and have the following features [6]:
Equation 22 can be rewritten as
Vbus 1
VCn
(23) Vbus n
X0
They have lower than normal ux densities and can, thus,
tolerate overvoltages coupled with circulating harmonic
currents.
They employ an electromagnetic shield between the primary and secondary windings of each coil, thus attenuating
higher frequency harmonics.
They provide a neutral with twice the size of a phase conductor, to account for increased neutral currents due to the
ow of triplen harmonics.
Windings are designed with several smaller sizes parallel
conductors, therefore reducing skin eect at higher frequency harmonics.
They use insulated and transposed conductors resulting in
reduced
P losses.
2 2
The sum 1
h=1 h Ih pu is designated as the K-factor [10] in the
Underwriters Laboratories (UL) Standards and is a useful term
for describing the additional heating that occurs in a transformer supplying non-linear loads. Transformers specically
designed for use with non-linear loads are marked [10] \Suitable for non-sinusoidal current load with K-factor not to exceed
: : : ", where standardized K-factor ratings are 4, 9, 13, 20, 30,
40, 50.
D. Rotating Machines
Although distribution and chording result in reducing the
fundamental component of the induced emf, this negative eect
is more than outweighed when considering the purpose they
were intended for, namely harmonics. Obviously, they have the
advantage of reducing harmonic voltages. Recalling that
Kp h = sin( h
2)
(30)
where
Kp h is the pitch factor at the hth harmonic
Kp 1 is the pitch factor at the fundamental frequency,
Kp1 = sin( 2 ) < 1 is the coil span in elect., = s
s is the coil span in slots
it is noticed that any one harmonic can be completely eliminated
through selecting a coil span (fractional pitch) that results in
the respective pitch factor being zero.
360 or 720
)
=
0
for
=
(31)
Kph = sin( h
2
h
h
That is to say, a coil span of 4=5 pole pitch (144 elect.) results
in eliminating the fth harmonic. The third harmonic is suppressed through using a coil of 2=3 pole pitch (120 elect.) span.
Furthermore, a coil span of 5=6 pole pitch (150 elect.) greatly
reduces fth and seventh harmonics (Kp 5 = Kp 7 = 0:2588).
VI. CONCLUSIONS
This paper reected an eort dedicated towards providing a
\quantitative-directed" analysis of harmonics in electric power
systems. Their causes, eects and methods of reduction were
discussed. Needed beyond this point is industry-academia collaborative research intended at:
\Hanging numbers" on the eects of harmonics on power
systems equipment and loads, with special consideration
being given to those harmonics produced by the newlyintroduced power semiconductor devices. Recommendations seeking the control of these harmonics are further
appreciated.
Developing equipment that enable the accurate and precise
monitoring and measurements of harmonics.
Designing and developing low cost active lters which are
devoid of the disadvantages of conventional passive lters,
namely:
{ The ltering characteristics being dependent on the
source impedance.
{ Aggravating the impedance below the lowest tuned harmonic.
{ Being inadequate for ltering non-characteristic harmonics (dierent from the lter's tuned frequency), such as
those produced by cycloconverters.
References
[1] W. M. Grady, W. H. Kersting, D. Osborn, N. R. Prasad, S. J. Ranade
and H. A. Smolleck, Power Factor Correction and Power System
Harmonics, Spring Short Course Series on Electric Power Systems
and Harmonics, a short course held at the Department of Electrical
and Computer Engineering, New Mexico State University, March 15{
18, 1993.
[2] IEEE Working Group on Power System Harmonics, Power System Harmonics, IEEE Power Engineering Society Tutorial Course,
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[3] IEEE Recommended Practices and Requirements for Harmonic
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[11] General Requirements for Liquid-Immersed Distribution, Power
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George J. Wakileh received the B.Sc. in Electrical Engineering
from the University of Jordan, Amman, Jordan in 1986. He
worked for the Jordanian Royal Scientic Society/Renewable
Energy Research Center from October 1988 to December 1992.
George received the M.S. in Electrical Engineering from New
Mexico State University in December 1993 and is currently a
Ph.D. student at the Department of Electrical and Computer
Engineering, Kansas State University.