8/10/2019 Newton Raphson Harmonic Pf
1/126
;
A DETAILED DERIVATION OF
-. .
NEWT ON-RAPHSON BASED HARM ONIC POWER FLOW/
A Thesis Presented to
The Faculty of the Russ C ollege of Engineering and T echnology
Ohio University
In Partial Fulfillment
of the Requirements for the Degree
M aster of Science
David Charles Heidt,
.--
i
June,
994
8/10/2019 Newton Raphson Harmonic Pf
2/126
ACKNOWLEDGMENTS
The author would like to thank all those who helped to provide the guidance
needed to complete this thesis including Dr. Herman Hill and Dr. Brian Manhire of Ohio
University and Dr. Jerry Heydt of Purdue University.
The author would also like to thank the Electrical and Computer Engineering
Department at Ohio University for providing the financial support necessary to pursue the
Master of Science degree.
Lastly the author would like to thank family and friends who were patient
supportive and understanding through the many hours of sacrifice.
8/10/2019 Newton Raphson Harmonic Pf
3/126
TABLE OF CONTENTS
Page
LIST OF TABLES i
LIST OF FIGURES. . . vii
LIST OF SYMBOLS
......................................................................................................
.
ABSTRACT..
mil
CHAPTER
1
CONTRIBUTIONS OF TH IS THESIS
.................................................
CHAPTER CAUSES
ND
DETRIMENTAL EFFECTS OF POW ER
SYSTEM HARMO NICS..
5
2 1
Definitions Related to Harmon ics..
. 5
2 2
The Cause of Harmonics in Power S ystems..
8
2 3
Detrimental Effects of Harmonics on Power System s..
18
CHAPTER
3
A NEWTON RAPHSON BASED HARMONIC POWER
FLOW..
. 22
3 1
Introduction..
22
3 2
Solving Nonlinear Algebraic Equations by New ton Raphson
Methods..
25
3 3 A
Brief Derivation of a Newton Raphson Based Conven tional
Power Flow..
33
3 4 A
Detailed Derivation of a Newton Raphson Based Harmonic
Power Flow..
.
3
9
3 5
Netw ork M odels for the Newton Raphson based Conventional
and Harmonic Pow er Flows .............................
....
.. ..
53
Introduction..
53
Transmission Lines..
. .
53
Transformers.. .
58
Generators..
60
Induction Motors.. 6
1
8/10/2019 Newton Raphson Harmonic Pf
4/126
ther Conventional Loads 66
as Discharge Lighting Loads 68
3 .6 Simplifications to the Newton Raphson Based Harmonic Power
low Equation Set 72
CHAPTER 4 DIRECT SOLUTION OF POWER SYSTEM HARMONIC
OLTAGES 76
HAPTER EXAMPLES 79
5.1 Introduction 9
.2 Example One 90
.3 Example Two 92
.4 Example Three 101
CHAPTER 6 CONCLUSIONS NDFUTURE WORK 103
.1 Conclusions 103
.2 Future Work 106
IST OF REFERENCES 108
APPENDIX 112
8/10/2019 Newton Raphson Harmonic Pf
5/126
LIST OF TABLES
Table Page
3 1 Sequence of a Fundamental Frequency Signal and its Harmonics in a
Completely Balanced Three Phase ac Pow er System 2 4
3 2 Skin Effect Table 7
5 1
Fundamental F requency
60Hz)
ystem G enerator Data
5 2
Data for the Cond uctors of the
230 Kv
Transmission Line o f Figures
5 2 8 2
5 3
Data for the Ground Wires of the
230Kv
Transmission Line of Figure
5 2 8 3
4
Impedance Data for the
230 Kv
Transmission Line of Figure 5 2
83
5
Per Unit Impedance Data for the
230 Kv
Transmission Line of Figure
5 2
83
6
Data for Polynomial Curve Fits
9 6
8/10/2019 Newton Raphson Harmonic Pf
6/126
LIST OF FIGUR ES
Figure Page
2.1.
Generalized Six-Pulse Converter..
9
2.2.
A Simple Circuit Diagram o f a Fluorescent Lamp Placed in Series with
an Inductive Ballast..
12
2.3.
Typical B-H Curve used to Describe the Characteristics of the M aterial
used for a Transformer Core I 5
2.4.
n Example of Actual B-H Curves.. I6
.1 . Geom etrical Interpretation of the Newton-Raphson Me thod..
.27
3.2.
n
Example of How the Next Approximation to
x ) =
can be Far
Away from the Desired Solution x if Iteration
xr
is Close to the
Minimum of a Function
x )
8
3.3 . Illustration of the Tw o Systems Given by Equation Sets 3.7) and 3.8 ). 30
3.4 . Single-Line Diagram at Bus
i
of a General n-Bus Pow er System 34
3.5.
The Equivalent Positive-Sequence-Impedance Diagram of Figure 3.4 .. 34
3.6.
General n-Bu s Power System Arranged to Form the Fundamental
requency Admittance M atrix.
.36
3.7.
Fundamental Frequency Admittance Diagram of a Two-Bu s Power
ystem.. .3
6
3.8.
Typical ac-Side Line Current Waveform of an ac Current to dc Current
onverter.. .40
3.9.
a) A Bus Containing a G enerator and a Harm onic Producing
HP)
Device.
b) n
Equivalent Representation of a), Separated into Tw o
Buses Connected by a Short Circuit SC) 4
8/10/2019 Newton Raphson Harmonic Pf
7/126
General n-Bus Power System Arranged to form the
kth
Harmonic
Frequency Admittance Matrix.. 47
General Representation of a Power System at the
kth
Harmonic
Frequency, with a Harmonic Producing
HP)
Load at Bus m.. 48
a). A General Representation of a Power System at the
kth
Harmonic
Frequency, with a Conventional C) Load at Bus i. b). Equivalent
Representation of a), with the Conventional Load Modeled as a Series
-L Combination.. .50
Pi-Equivalent Model of a Long Transmission Line. 4
i-Equivalent Model of a Short Transmission Line.
56
Power Flow Model of a Transformer 59
Harmonic Power Flow Model of a Synchronous Machine for the
kth
armonic Frequency.. 62
Per-Phase Fundamental Frequency Equivalent Model for an
nduction Machine Phase a Shown)
63
Harmonic Power Flow Model of an Induction Motor for the kth
armonic Frequency.. 65
Suggested Harmonic Power Flow Model of an Unknown Conventional
oad for the
kth
Harmonic Frequency..
67
General n-Bus Power System Arranged to form the
kth
Harmonic
Admittance Matrix. Shunt Impedances used to Model Conventional
Buses at the kth Harmonic Frequency, are Included when Forming
he
kth
Harmonic Frequency Admittance Matrix..
73
Representation of a Harmonic Producing Device at Bus
m,
as an
deal Current Source at the
kth
Harmonic Frequency.
77
Example Two-Bus Power System, with a Generator at Bus One and
Fluorescent Lighting FL) Load at Bus Two
..80
onfiguration of the 23 Kv Transmission Line in Figure 5 1
.8 1
Impedance Diagram of the Power System in Figure 5.1, for
undamental Frequency
60Hz )
Analysis 85
8/10/2019 Newton Raphson Harmonic Pf
8/126
8/10/2019 Newton Raphson Harmonic Pf
9/126
LIST OF SYMBOLS
In order to eliminate confbsion, frequencies of interest and ite ration numbers will
be represen ted by superscripts. In addition, bus numbers, phase and phase sequences will
be represen ted by subscripts. Fo r clarity, the most commonly used superscripts and
subscripts used in this thesis are explained below. O ther superscripts and subscripts are
used, but are explained as they are presented.
superscripts
1)
Fundamental frequency.
3),
5), 7),
Harmonic frequencies.
f)
The frequency of interest f
1,
odd).
k)
harmonic frequency k 3, ko dd )
h) The highest harmonic frequency of interest h
>
1,
h o d )
r general iteration number.
subscripts
i
conventional nonharmonic producing) bus.
m harmonic producing) bus.
j
conventional or harmonic bus j
=
i
or m) .
a
Phase
a
b
Phase b.
c
Phase
c
8/10/2019 Newton Raphson Harmonic Pf
10/126
Direct.
Quadrature.
n Neutral.
Fo r this thesis, all power system values shall be expressed in per-unit, instead o f
conventional SI units. In addition, all currents and voltages are rms values only. The most
commonly seen variables in this thesis are explained below. Other variables appear in this
thesis, but are explained as they are presented .
A voltage.
A current entering a conventional bus from ground .
A current entering a harmonic producing) bus to ground.
Time.
A hndam ental or harmonic component of a voltage Fourier Series.
A hndam ental or harmonic component of a current Fourier Series.
Fundamental angular frequency.
Angular frequency.
Active power.
Reactive power.
Apparent power.
Complex power.
Distortion power.
8/10/2019 Newton Raphson Harmonic Pf
11/126
A fundamental f = 1) or harmonic f > 1,
f
odd) frequency current entering bus j
j = i o r m ) from ground into the transmission netwo rk. p = I:)/ '.
?
A hndamental f
=
1) or harmonic f > 1 odd) frequency current entering bus j
j
=
i o r
m
rom the transmission network to ground .
=
gg'/g-
7'
A fundamental f = 1) or harmonic f > 1,
f
odd) frequency voltage at bus j
j = i o r m ) y n= yw /q ) .
Z Impedance. = R + X
R Resistance.
X Reactance.
L Inductance.
Capacitance.
Admittance. =
y g
=
g
+ b
IFfll
A fundamental f = 1) or harmonic f >
1,f
odd) frequency admittance matrix.
I z ~ I
A fundamental f = 1) or harmonic f > 1, odd ) frequency impedance matrix.
IlWI
A fundamental f = 1) or harmonic f
>
1,
f
od d) frequency current vector.
fl I
A fundamental f = 1) or harmonic f > 1,
f
odd) frequency voltage vector
9
The total number of harmonic frequencies of interest.
M
The total number of harmonic producing ) buses in the pow er system considered.
Unknown value.
L Permeability.
A
n
incremental change.
p.u. Per unit.
rad. Radians
8/10/2019 Newton Raphson Harmonic Pf
12/126
ABSTRACT
Heidt, David Charles. M . S., Ohio University, June
1994 A
Detailed Derivation of a
New ton-Raphson Based H armonic Pow er Flow. Major Professor: Dr. Herman Hill.
The ideal electric pow er system contains only elements that genera te, transmit, o r
receive undistorted hn dam ental frequency voltages and currents. Several devices are
responsible for introducing harmonics of pure hndam ental frequency waveforms into
pow er systems. If significant enough in amplitude, these harmonics can have a detrimental
effect on the performance and life of power system elements.
A
tool commonly used to analyze the power system under normal balanced three-
phase sinusoidal steady -state conditions, is called a pow er flow. How ever, to this date, the
only power flow s which are readily available in textbook form, assume pure hndam enta l
frequency voltages and currents throughout the power system.
It is the primary intent o f this thesis, to tak e an available Newton -Raphson based
Harmonic Pow er Flow algorithm, and provide a textbook type derivation. This
algorithm eliminates the assumption of purely sinusoidal voltages and curren ts in a three-
phase power system operating under normal balanced steady-state conditions.
8/10/2019 Newton Raphson Harmonic Pf
13/126
CHAPTER CONTRIBUTIONS OF THIS THESIS
Chapter two focuses on the cause and detrimental effects of power system
harmonics. The chapter begins by discussing pertinent definitions related to power system
harmonics. Several equations in this introductory section and throughout this thesis paper,
are written in Fourier Series form. The method of Fourier Series is a convenient and
appropriate way to express the presence of harmonics in a power system.
The next section of chapter two describes some of the most common sources of
harmonics in power systems today. Power electronic devices are by far the most
significant source of harmonics in power systems today, not only because of the nature of
these devices, but also because of their rapidly growing usage
[4]
Several other devices
such as gas discharge lighting, transformers, and arc furnaces are also significant harmonic
sources in power systems.
Chapter two is concluded by describing several of the detrimental effects that
harmonics can have on power systems. Here it is described how harmonics can cause
several devices such as relays and computers to misoperate. This section also discusses
why harmonics can lead to a reduction in the efficiency and life of several other devices
such as motors and transmission lines. It is also discussed how power system harmonics
can be amplified significantly depending upon the power system configuration and why
triplen harmonics i.e., the third harmonic and its integer multiples) represent a special and
rather complicated problem. Chapter two provides sufficient information as to what power
system harmonics are, how they originate, and why they cannot be ignored.
The primary focus of chapter three is to derive a Newton-Raphson based
Harmonic Power Flow. Chapter three begins by defining both the conventional and
8/10/2019 Newton Raphson Harmonic Pf
14/126
harmonic power flows. This introduction describes the differences between conventional
and harmonic power flows. In addition, this introduction states why a conventional power
flow cannot be used to obtain information about power system harmonics.
The next section of chapter three discusses the solution of nonlinear algebraic
equations by the Newton-Raphson method, along with some of the most common
problems encountered when using this method. One needs to be aware of how the
Newton-Raphson method can fail for the problem to be analyzed.
Chapter three continues by providing the derivation for the Newton-Raphson
based Conventional Power Flow. These equations are then used to help derive the
Newton-Raphson based Harmonic Power Flow. This is meant to show that the Newton-
Raphson based Harmonic Power Flow is a logical extension of the Newton-Raphson
based Conventional Power Flow. The derivation begins by describing the number of
equations required for the Newton-Raphson based Harmonic Power Flow and explaining
briefly how they are obtained. The derivation continues by listing all required equations
and providing a detailed explanation for the logic behind each equation as they are
provided. The equation set obtained is derived in a general format, and is to be treated the
same as any other nonlinear algebraic equation set which is to be solved by the Newton-
Raphson method. The detailed derivation of the Newton-Raphson based Harmonic Power
Flow, represents the author s main contribution to knowledge. This derivation is in a much
more general format than in reference [I] In addition, intricate details provide more
insight than is available in reference [ I ] To the best of the author s knowledge, reference
[ I ]
had provided the best derivation of the Newton-Raphson based Harmonic Power Flow
to date.
The following section of chapter three provides network models for the Newton-
Raphson based Conventional and Harmonic Power Flows. Network models for some of
the most common power system elements is provided. It is stressed that the hndamental
frequency power system representation will change for each harmonic frequency of
8/10/2019 Newton Raphson Harmonic Pf
15/126
interest. A different admittance matrix is required for each frequency to be examined. The
equations for modeling one harmonic source (gas discharge lighting) is derived. All
harmonic producing loads cannot be modeled in the same way however, since the current
entering each different type of harmonic load will possess a different Fourier Series
The last section of chapter three discusses possible simplifications to the Newton-
Raphson based Harmonic Power Flow equation set. Since this derivation is provided in a
general format, it is possible to model every bus in the power system the same way that
harmonic producing buses are modeled. In other words, for every bus in the power
system, the Fourier Series of the current entering each bus could be expressed as a
hnction of the Fourier Series of the voltage at the respective bus, and of any parameters
which describe this distorted waveform. The simplifications possible result when this
approach is used only for harmonic producing busses. The derivation of the Newton-
Raphson based Harmonic Power Flow provided in this thesis does not assume that a
simplified approach will always be used, whereas reference [ I ] does.
Chapter four briefly discusses the most commonly used method for determining
power system harmonics voltages called the Current Injection Technique. In order to find
the power system harmonic voltages, this method employs the direct (non-iterative)
solution of the harmonic frequency admittance matrices. As the voltage distortion levels
increase throughout the power system, the less effective the Current Injection Technique
becomes at determining the power system harmonic voltages [I] In addition, the Current
Injection Technique is generally effective only when the power system is a simple radial
network [ l 1
Chapter five provides three examples to help increase the reader s knowledge of
the Newton-Raphson based Harmonic Power Flow. The same two bus power system is
analyzed
in
all three examples. In addition, the goal is to obtain the same information in all
three examples. In the first example, the Newton-Raphson based Conventional Power
Flow is used to obtain the kndamental frequency voltages in the power system. The
8/10/2019 Newton Raphson Harmonic Pf
16/126
Current Injection Technique is then used t o determine the harmonic voltages in this power
system.
The second exam ple in chapter five uses the Newton-Raph son based Harmonic
Power Flow to determine the hndamental and harmonic frequency voltages in this same
power system. For each bus in this example, the Fourier Series of the cu rrent entering the
bus, is modeled as a hn ct io n o f the Fourier Series of the voltage at this bus, and of any
parame ters which describe this distorted cu rrent waveform. In this case, harmonic
frequency admittance matrices include only the admittance information for transmission
lines and transformers, as is true for the hn dam ent al frequency admittance matrix.
The last example in chapter five uses the simplified approach which always
assumed t o be tru e in reference
[ I ] .
Therefore, all buses which do not contain harmonic
sources are treated as an impedance load at harmonic frequencies. In this case, harmonic
frequency admittance matrices include admittance information for transmission lines,
transformers, and impedance loads (for nonharmonic producing loads). The hndamental
frequency admittance matrix, however, still only includes admittance information for
transmission lines and transformers. C hapter five also represents th e au thor s contribution
to knowledge. To the best of the author s knowledge, detailed examples on how t o
perform a Newton-R aphson based Harmonic Pow er Flow are not available.
Chapter six completes the thesis by listing appropriate conclusions and suggestions
for future work.
8/10/2019 Newton Raphson Harmonic Pf
17/126
CHAPTER
--
CAUSES AND DETRIMENTAL EFFECTS OF POWER SYSTEM
HARMONICS
2.1 Definitions Related to Harmonics
Harm onic producing loads have a non-sinusoidal load current that is periodic and
can be expressed as a Fourier Series. The distorted load current passing through the
pow er system results in distorted bus voltages that are periodic and can also be expressed
as a Fourier Series.
All
frequency terms higher than the hndamental frequency will be
referred to as the harmonics.
Fo r this thesis completely balanced pow er systems will be assum ed so that even
harmonics may be ignored. A property of a completely balanced power system is that even
harmonic currents and voltages will not exist [20] . For this thesis harmonic frequencies
shall be assum ed to be odd integer multiples of a
6 Hz
kndam ental frequency. By
definition of a Fourier Series any voltage v t) or current i t) n this balanced power
system must be an odd function as shown below f 1 odd).
i t)
=
cV in oot P
Also for this completely balanced power system the active power P and reactive
power will be defined as follows
f
1
f
odd).
8/10/2019 Newton Raphson Harmonic Pf
18/126
In addition to the above equations the apparent power S is a scalar quantity given
by equation 2 . 5 ) .Th e root mean square of the voltage and current is defined by equations
2 . 6 )and 2 . 7 ) espectively f
1
odd) .
Harmonic distortion of a voltage
v t )
and a current i t) is defined by equations
2 . 8 )and 2 . 9 ) espectively f
1 odd .
Th e definition of power fac tor is given below.
In th e special case when
both
v t ) and i t ) contain fbndamental frequency
components only equation 2 . 1 1 )will be valid.
S2
p
Q 2
2 . 1 1 )
However when either v t ) or i t ) contains harmonic frequency components
equation 2 . 1 1 ) s no longer valid. term that is used to account for the discrepancy is
called the distortion power D as given below.
8/10/2019 Newton Raphson Harmonic Pf
19/126
D = /m 2.12)
The physical origination of in power systems is an extremely controvers ial issue
[3] and will not be discussed in this thesis The concept of distortion power is mentioned
here only for the sake of completeness
8/10/2019 Newton Raphson Harmonic Pf
20/126
2 . 2 The Cause of Harmonics in Power Systems
The generation of harmonics in pow er systems today, results from a number of
major sou rces. However, power electronic devices are by far the most significant sources
of harmonics
[4]
[ 5 ]
For example, power electronic six-pulse converter devices used as rectifiers (to
convert ac current to dc current) and a s inverters (to convert dc current to ac current) are
growing in usage because they are more efficient and economical than conventional
converter systems such as motor-generator sets [ 6 ] A few of their applications include
he1 cells and batteries, s tatic var generators, and adjustable-speed drives for motors.
A generalized six-pulse converter configuration is shown in F igure 2 1 When a
squa re wave switching scheme is used, each switch changes state only twice for a
designated switching time period. Whether the converter is used as an inverter or as a
rectifier, significant ac side harmonics of the hndam ental current will result. Under
balanced conditions, the characteristic ac side harmonic currents produced by a six-pulse
converter are k
= 6b
b
=
1 2 3 . [ 6 ] The magnitude of these harmonics decrease
inversely proportional to their harmonic order, i.e., = I '/k, where
I )
is the
hndamental frequency current magnitude, and is a harmonic curren t magnitude.
Uncharacteristic harmonics may also be produced under unbalanced conditions, where the
magnitudes depend upon the amount of unbalance [ 7 ]
A switching scheme for the six-pulse converter which is recently becoming more
popular is called the pulse-width modulated (PWM) scheme. For the PWM scheme, the
switches change state for a designated switching time period at a higher frequency than the
squa re wave switching scheme. Basically, the higher the switching frequency, the greate r
the number of lower order harmonics that can be eliminated. How ever, the PWM scheme
results in the generation of higher frequency ac side harmonics. These harmonics can be
even greate r in amplitude than the lower frequency ac side harmonics generated by the
squa re wave switching scheme. These higher frequency harmonics and their amplitudes
8/10/2019 Newton Raphson Harmonic Pf
21/126
Figure 2. 1 . Generalized Six Pulse Converter.
Source: Boost [8]
p.273.
8/10/2019 Newton Raphson Harmonic Pf
22/126
depend upon the switching frequency of the PWM converter. In addition, a PWM
converter produces less hndamental frequency current than a converter with a square
wave switching scheme.
However, the PWM scheme does have advantages over the square wave switching
scheme. For example, the main advantage of the PWM scheme is that a large number of
lower order harmonics can be eliminated. This results in a more sinusoidal ac side
waveform than a square wave switching inverter can produce. In addition, a square wave
switching inverter is not capable of regulating the ac output voltage magnitude. Therefore,
the dc input voltage must be adjusted in order to control the magnitude of the
inverter-
output voltage [8]. Since the PWM inverter allows amplitude control of the ac output
voltage from within the device, this feature can result in a simpler and cheaper power
inverter [8].
It should be noted that 12-pulse converters are actually much more common,
particularly in new designs. One example of where 12-pulse converters are more
advantageous than 6-pulse converters is in High-Voltage
DC HVDC)
transmission
applications. In order to meet high voltage requirements, and to reduce the number of
harmonics produced, it is necessary to use 12-pulse converters, instead of 6-pulse
converters. It is noted that under balanced conditions, the characteristic ac side harmonic
currents produced by a 12-pulse converter will be = 12b 1 b = 1,2,3,. [6]. The
magnitudes of the remaining harmonics will be the same as for a six-pulse converter.
Fuel cells and batteries are highly likely to grow in usage in power systems as
energy sources during utility peak loading conditions [6]. It is desirable to operate the
most efficient power generating plants such as nuclear and the newer high-efficiency coal-
fired plants) at their rated capacity at all times. Due to the time of day and weather
conditions, the utility load demand will not remain constant. In order to meet peak loading
conditions, either oil- or gas-fired generators can be used, but are expensive to operate
because of the high cost of he l.
n
alternative is to store the energy generated by the
8/10/2019 Newton Raphson Harmonic Pf
23/126
more efficien t pow er generating plants in batteries o r fuel cells during low-load conditions,
and t o use this surplus during peak-loading conditions. However, since both batteries and
fuel cells produce a dc voltage, an inverter is required to connect them to the utility
system.
converte r system can be designed such that th e ac current can be quickly
controlled in magnitude and phase leading or lagging) with respect to the ac voltage of
the device. Such a converter is commonly termed a static var generator, and can be used
for power factor correction applications [8]. Industrial loads such as arc furnaces can
cause very rapid changes in power factor . In this case, a fast and efficient static var
generator would be more desirable to use than a slower conventional power-factor-
correction capacitor bank.
Pow er electronic converter devices are commonly used as adjustable-speed drives
for motors [9]. For instance, a common configuration for an ac m otor drive is made up of
a rectifier ac to dc) supplying an inverter dc to ac). By controlling the direction of the
power flow, this adjustable-speed drive configuration can allow precise con trol over the
motoring and braking of the ac motor [ lo ].
Gas discharge lighting such as fluorescent, mercury arc, mercury vapor, neon,
xenon, and high pressure sodium) represent a major source of harmonics, particularly in
metropolitan areas [ l l ] . In particular, fluorescent lighting may constitute up to twenty-five
percent o f the total customer load, and may exceed thirty percent [12]. The wide usage of
fluorescent lamps as compared to other lamp types is due to their high energy efficiency.
Greater stability in the operation o f conventional
OHz
gas discharge lamps is
provided by an inductive ballast placed in series with the lamp [ l 11. simple circuit
diagram of a fluorescent lamp placed in series with an inductive ballast is shown in Figure
2.2, where
v
s the ac source supply.
8/10/2019 Newton Raphson Harmonic Pf
24/126
8/10/2019 Newton Raphson Harmonic Pf
25/126
The efficiencyof the lamp can be h rt he r increased twenty to thirty percent by
using a high frequency electronics solid-state) ballast which converts an incoming 6 Hz
ac input to a higher frequency in the 25 to
4 kHz
ange [6].
Under balanced conditions, all odd harmonics of the hndam ental ac current are
produced by a series lamp and ballast combination. The actual m agnitudes of these
harmonics depends greatly upon the manufacturer models o f both the lamp and ballast
type. In particular, solid -state ballasts tend to result in higher amplitudes for each of the ac
current harmonics produced [12]. Depending upon the lamp and ballast combination used,
a third harmonic ranging from seven to eighty-seven percent of the hndamental ac current
has been measured [12]. Many other odd harmonics can be also qu ite significant in
amplitude. Typical third and fifth harmonic currents produced are 21 and percent of the
fbndamental ac cu rrent respectively [ l 11.
Even if the harmonics produced by individual lamps are small in amplitude, each of
the respective harmonics produced by the individual lamps tend t o be in phase and
additive. This is due to the fact that these harmonics arise from distortion of a 6OHz
fbndamental current [13].
Arc fbrnaces, which are most often used for the meltdow n of scrap metals, are a
source of a number of problems in the power system today. The use of arc fbrnaces is
expected to increase, since the technology is improving [14].
In an arc fbm ace, graphite electrodes are lowered into a basin containing scrap
metal, in order to strike electric arcs between the electrodes and the scrap. The heat
generated by the electric arcs results in meltdown o f the scrap metal [15].
Under balanced conditions, typical arc fbrnace third, fifth, seventh, and ninth
voltage harmonics produced are 20, 10, 6, and 3 percent of the hndamenta l respectively.
Unbalanced conditions can lead to significant even harmonics, and additional
magnification of the odd voltage harmonics [14].
8/10/2019 Newton Raphson Harmonic Pf
26/126
It is noted that as the pool of molten metal grows, the arc becomes more stable.
This in turn results in much steadier currents, with much less distort ion, and less harmonic
activity.
The harmonic problem in power systems today is also due to a change in design
philosophy. In the past, power system devices tended to be underrated or overdesigned.
Now, in order t o be more competitive, power system devices are more critically designed
[4] Because o f this design philosophy, iron core devices such as transformers are more
likely to becom e sa turated.
The most common curve used to describe the characteristics of the material used
for a transformer core is the B-H curve as shown in Figure 2 3 [16] Figure 2 4 provides
an example of some actual B-H curves.
The
B-H
curve represents the relationship between the magnetic field B), and the
magnetic field strength H).When there is little change in the value of B for large changes
in H, the transform er is said to be saturated.
Modern high-voltage transformers with grain oriented steel cores saturate typically
somewhere above 1 0 to 1 2 imes the rated magnetic flux in the core [17] Magnetic flux
represents the surface integral of the magnetic field. In general, manufacturer ratings are
given somewhere around the point of saturation. Operation of the transformer beyond
these ratings drives the core into satura tion, and resul ts in rapidly increasing levels of
harmonic currents [ I ]
It is impractical to design transformers without saturation, and as a consequence,
harmonic currents produced by a transformer are unavoidable.
The significant harmonic currents produced by a transformer under balanced
conditions, are all odd harmonics of the hndamental frequency exciting current. The
magnitudes of these harmonic currents depends greatly upon the magnitude of the voltage
at the transform er terminals. However, typical magnitudes for the third, fifth, seventh, and
8/10/2019 Newton Raphson Harmonic Pf
27/126
Fig 1-1 0. Hysteresis loop; hysteresis loss is proportional to the loop area shaded).
Figure
2.3 .
Typical
B H
Curve used to Describe the Characteristics of the Material used
for
a
Transformer Core.
Source: Fitzgerald 1161
p.2
1
8/10/2019 Newton Raphson Harmonic Pf
28/126
10 0 10 20 30 40
50
70 90 110130150170
H turndm
Fig.
1-6. 6 -H loops for
M-5
grain oriented electrical steel
0.012
in thick. Only the
top
halves
of
the loops are shown here.
Armco
I ~ c .
Figure 2 4 An Example of Actual
B H
Curves.
Source: Fitzgerald [16] p.
15.
8/10/2019 Newton Raphson Harmonic Pf
29/126
ninth harmonic currents produced by a saturated transformer are 50, 20, 5 and 2.6
percent of the fundamental exciting current respectively [18].
Even in the absence of other nonlinearities in the power system, transformer
harmonic currents can reach significant levels [19].
Transformer harmonics can be of great concern, since transformers are widely used
in power systems, and play an integral role in power transmission.
Considerable effort is taken to design rotating machines such that the harmonic
currents they produce are negligible [16]. Special care is taken to eliminate triplen i.e., the
third harmonic and its integer multiples) and fifth and seventh harmonics of the
fundamental frequency exciting current, where other harmonics are taken to be
insignificant [20].
Saturation of the stator and rotor teeth can lead to the generation of odd
harmonics, but these are also generally insignificant in amplitude [21]. For example, rotor
third harmonics produced under saturated conditions, typically have values lower than one
percent of the rated stator current 1211.
Unbalanced operating conditions may also result in odd harmonic voltages at the
machine terminals of a generator. For example, a third harmonic voltage of magnitude
greater than six percent of the fundamental can occur. However, these harmonics are
greatly dependent upon the length of the line that the generator is feeding [22].
8/10/2019 Newton Raphson Harmonic Pf
30/126
2 3
Detrimental Effects of Harmonics
Harm onics have been linked to a number o f problems in the power system,
including communication interference. Very often , comm unication lines used for the
transmission of signals share the same path as power lines used for transmitting and
distributing electrical energy
[20]
Current flowing in the power lines results in m agnetic
and electric fields which induce currents or voltages) in the nearby communication lines
[26]
The interference introduced depends greatly upon the separation distance between
the power and communication lines which are often closely in parallel with each other),
and the frequency particularly the harmonics after the fundamental) and magnitude of the
induced currents voltages)
[20]
The inductive reactance of the ac power system composed of generators,
transmission lines, transformers, etc.) and the capacitive reactance of widely used
capacitor banks for power factor correction), and insulated cables, produce resonance
conditions [25] The resonant frequency of an inductive-capacitive LC) circuit occurs
when the inductive reactance equals the capacitive reactance.
If the power system appears to be a very low impedance for instance, to a
harmonic source) at the resonant frequency, then this condition is termed series resonance .
Likewise, if the system appears to be a very high impedance at the resonant frequency,
then this condition is termed parallel resonance. In either case, if the resonant frequency of
the power system happens to be close to one of the frequencies generated by a harmonic
source in the system, the result may be the flow of high harmonic curren ts or the
appearance of high harmonic voltages)
[26]
Although a system resonance condition does not produce harmonic curren ts or
voltages, small currents voltages) generated by a harmonic source in the power system
can be amplified significantly by a resonance condition .
Since switching the configuration of the capacitor banks for a different power
factor) changes the capacitive reactance of the system, more than one resonant frequency
8/10/2019 Newton Raphson Harmonic Pf
31/126
will exist [27]. In addition, since motor loads appear to be primarily inductive at harmonic
frequencies, changes in the m otor loads on the system, can result in a shift of the resonant
frequency [25].
In a three-phase system, triplen harmonics i.e ., the third harmonic and its integer
multiples) represen ts a special and rather com plicated problem. The objective here will be
to keep the discussion as simple as possible, yet to introduce a couple of realistic examples
of how triplen harmonics cause problems in pow er systems. Therefore, consider the case
of a wye-connected load, that of a delta-connected load.
Three-phase power systems are operated under balanced conditions as much as
possible), where the hndam ental frequency line currents are 120 degrees ou t of phase
with respect t o each other. For example, in a balanced three-phase system, the
hndam ental frequency line currents can be:
I:) I( ) cos Wt)
I;) I( )
cos Wt 120)
I: I ( ) cos 0t 120)
The third harmonic of the line currents above assumed to be generated by a nonlinear
device in the power system) would be:
I:)
C O S ~ W ~ ) )
I ( ~ )
O S ~ W ~ ) )
I?)
I ( ~ ) O S ~ W ~1200)) I ( ~ ) O S ~ W ~ ) )
I;)
I ( ~ ) O S ~ W ~1200
1
I ( ~ ) O S ~ W C ) )
In a wye-connected load, these third harmonic currents would be in phase and additive in
the neutral wire. Ideally, it is desired that all currents should be ze ro in the neutral wire. If
significant enough in amplitude, these third harmonic currents would lead to increased
heating of the neutral wire, and therefo re shorten its lifespan. The absence of a neutral
wire will result in circulating triplen harmonic currents in a delta-connected load. Again, if
significant in amplitude, these third harmonic cu rrents will lead to increased heating of the
elements in the delta-connected load, and therefore shorten their lifespan.
8/10/2019 Newton Raphson Harmonic Pf
32/126
Motors and generators (both induction and synchronous) experience increased
heating due to iron and copper losses at harmonic frequencies. Harmonics can cause
rotating machinery to have a pulsating torque output. This results in mechanical
oscillations which can lead to rotor shaR fatigue and accelerated aging of the shaft and
connected mechanical parts. In particular, harmonics can cause or enhance phenomena
called cogging (a refbsal to start smoothly) or crawling (very high slip) in induction
machines [25]. The net effect of harmonics on rotating machinery is a reduction in both
efficiency and life.
The m ajor effects of harmonics on transformers is that current harmonics cause an
increase in copp er losses and stray flux losses, and voltage harmonics cause an increase in
iron losses [25 ]. As a result, there will be an increase in transform er heating, which in turn
can shorten the life of the unit.
Harmonics flowing through transmission lines (power cables) will lead to
additional power losses due to the skin effect . As a result, the effective alternating
current resistance (R,,), will be raised above the direct current resistance (R,,), especially
for large co nduc tors. Therefore, when the current flowing through the cable contains a
high harmonic conten t, the equivalent RaC or the cable is raised even higher, which will
amplifjr the 12Ra, oss as illustrated in equation (2.1 9) . No te that superposition of power is
valid as long as all terms to be added, are at different frequencies. On the right hand side
of equation (2.19 ), each successive term is a t a frequency which is an odd integer multiple
of the first.
Hot spo ts are created along the cable by maximums of the overall current and voltage
due to standing wave phenomena. It is at these points, that excessive damage to the cable
and its insulation can occur [42].
In order to measure the effective power drawn by industrial loads, watt-hour
meters a re typically used. Depending upon the type of meter used, and the harmonics
8/10/2019 Newton Raphson Harmonic Pf
33/126
present and their magnitudes) in the current and voltage waveforms, a significant
overestimation or possible underestimation) of the real power drawn by the industrial
load can take place [30] Therefore, large industrial loads that are significant harmonic
producers could appear to have a higher or lower) power factor than they actually do.
For example, induction watt-hour meters are designed to operate in the presence
of purely sinusoidal voltage and current waveforms, and are calibrated in this way [3
11
The watt-hour meter will indicate the real average) power drawn by sensing the voltage,
the current, and the phase relation between the voltage and current. However, in the
presence of harmonics, the true phase relationship between the distorted current and
voltage waveforms becomes unclear, thus leading to errors in the power readings [32]
Several other devices are sensitive to the presence of harmonics, such as relays and
computers. relay is a device which will perform one or more switching actions based on
the information received from the power system, and is often used in power system
protection schemes [2] Negative-sequence current relays are oRen used to detect small
fault currents [33] This negative-sequence overcurrent relay can be used to trip a breaker
coil to interrupt operation) when the negative-sequence component of the fault current
exceeds a certain value. The presence of high negative-sequence harmonic currents, may
cause this relay to operate for a fault current with a negative-sequence component that is
much lower than the specified value. In a balanced power system, the fiRh harmonic is
entirely negative-sequence.
In order to prevent malhnction or damage to a computer, manufacturers typically
speci@ that the voltage distortion due to harmonics) is to be less than
5
percent of the
hndamental voltage [34]
8/10/2019 Newton Raphson Harmonic Pf
34/126
8/10/2019 Newton Raphson Harmonic Pf
35/126
in this chapter, only power systems operating in a completely balanced three-phase ac
mode will be considered. If each phase of the three-phase power system contains identical
fundamental and harmonic voltages and currents, the power system is said to be
completely balanced [I] property of a completely balanced power system is that even
harmonic currents and voltages will not exist
[20]
With completely balanced conditions
assumed, a single line admittance impedance) diagram of the power system may be used
for each harmonic considered, corresponding to the correct sequence given by Table 3 1
In addition, the admittances impedances) of each single line diagram must be scaled
according to the harmonic frequency.
If unbalanced conditions exist, fundamental and harmonic currents can each
contain positive-, negative-, and zero-sequence components. The method of symmetrical
components must then be used to simplifL analysis, but will not be considered in this
thesis.
For this thesis, buses which do not contain harmonic producing devices will be
referred to as conventional buses. In addition, buses which do contain harmonic producing
devices will be referred to as harmonic buses.
8/10/2019 Newton Raphson Harmonic Pf
36/126
Table 3.1. Sequence o f a Fundamental Frequency
Signal
and its Harmonics in a
Completely Balanced Three-Phase ac Power System.
Source: Westinghouse Electric Corporation 1201
p.759.
Sequence
Poai
tive
ero
Negative
Poeitive
ero
Negative
Poeitive
Harmonic
9
2
23
25
27
29
3
eta
Harmonic
6
7
9
3
5
17
Sequence
Pwi tive
Zero
Negative
Poeitive
ero
Negative
Positive
Zero
Negative
8/10/2019 Newton Raphson Harmonic Pf
37/126
8/10/2019 Newton Raphson Harmonic Pf
38/126
8/10/2019 Newton Raphson Harmonic Pf
39/126
Figure
3 .1 .
Geometrical Interpretation of the Newton-Raphson Method.
Source: Choma [37]
p.3
14.
8/10/2019 Newton Raphson Harmonic Pf
40/126
Figure 3.2.
An
Example of How the Next Approximation to x ) an be Far Away
From the Desired Solution xs , if Iteration xr is Close to the Minimum o f a Function x ) .
Source: Choma [37] p.3
15
8/10/2019 Newton Raphson Harmonic Pf
41/126
For two systems, x = C and
By
= D , if the elements of A and differ by little, and
those of
C
and D differ by little, then the elements of the solution vectors x and y will
also differ by litt le if the system is stable. As an example of an unstable system, consider
the eq uation sets (3.7) and (3.8 ) below.
x + y = l (3.7a)
x 1.00001y
=
0
(3.7b)
x + y = l (3.8a)
x 0 . 9 99 99 ~ 0 (3 .8b)
Although these tw o systems above differ by little, the solution set fo r equations
(3.7) is (x
=
100,00 1 and y
=
-100,00 0), and the solution set for equations (3.8) is
( x = -99,999 and y = 100 ,000 ). Each equation set represents an effort to find a position
(x ,y ) at which the tw o lines will intersect, as shown in Figure 3.3. Therefore , a slight shift
of either line, can move the point of intersection quite significantly.
The above problems with using the Newton-Raphson method a re not necessarily
separable, and may occur simultaneously.
Therefore, the Newton-Raphson method for solving nonlinear algebraic equation
sets is not perfect, and one needs to be aw are of how this method can fail for the problem
to be analyzed.
For nonlinear algebraic equa tions with N unknowns as in equation (3.9) below,
the method of Newton-Raphson may be used to solve the equation set if
N
f r
=
f (Ar ,Br ...........N r ) = O
r
=
g(Ar B ,
............,
)
=
0
8/10/2019 Newton Raphson Harmonic Pf
42/126
0 99999~
Figure
3 .3 .
Illustration
of
the
Two
Systems Given by Equation Sets
3.7)
and 3.8).
8/10/2019 Newton Raphson Harmonic Pf
43/126
Using these equations, a Jacobian matrix
I J ) I
is formed by equation 3.10).
Numerical values for the partial derivatives are obtained by using the values of the
unknowns at the rt iteration.
The error in the unknowns,
AA
AB .......
N is
found by equation 3.1 1). The
column matrix on the right side of equation 3.1 I), contains numerical values for equation
set 3.9) at the rt iteration.
The next approximation to the roots of equation set 3.9) is obtained from
equation 3.12).
8/10/2019 Newton Raphson Harmonic Pf
44/126
The itera tion process on the unknowns continues until I l E I BI I
.... and IAN E ~ nless the solution set diverges or the number of iterations has reached
the maximum number allowed
8/10/2019 Newton Raphson Harmonic Pf
45/126
3.3 - A Brief Derivation of a Newton-Raphson Based Conventional Power Flow
The equations for a Newton-Raphson based Conventional Power Flow or
Conventional Power Flow) are derived by first observing the hndamental frequency
current flow at bus
i
of a general
n
-bus power system. The power system is shown in
Figure 3.4. According to Table 3.1, only the positive-sequence-impedance diagram is
required to properly represent this power system, as shown in Figure
3 5
Using Kirchhoff s Current Law KCL) at bus i
,
esults in equation 3.13).
Taking the conjugate of equation 3.13), and multiplying through by the
hndamental frequency voltage at bus
i , ~ l )
~ ) / 4 ,
esults in equation 3.14) or its
equivalent, equation 3.15).
~ ( 1 )f z ) ) * ~ ( 1 ) 7 ~ ) ) ~~ ( 1 )? I)*
3.14)
-
(1) - ( 1 )
S ,
- Li
i 3.15)
The terminology of equation 3.15) is explained below.
-
s, )= The hndamental frequency three-phase complex generated power flowing into the
ith
bus from the power source s).
s:) =
The hndamental frequency three-phase complex power flowing out of the
ith
bus
towards the load buses.
3;)
=
The hndamental frequency three-phase complex power flowing out of ith bus
towards the transmission network.
The individual hndamental frequency complex powers of equation 3.15) can be
resolved into their rectangular components as shown below.
-
( )
p 1) jQ(1)
s
Gi
Gi
-
s;) p 1) Q(
Li x
-
1) - p 1) Q(1)
Ti -
Ti
i
8/10/2019 Newton Raphson Harmonic Pf
46/126
8/10/2019 Newton Raphson Harmonic Pf
47/126
By inspection of equations (3.19, (3.16), (3.17), and (3.18 , equations (3.19) and (3.20)
are formed.
P = pi
')
(3.19)
(1)
-
(1)
Qa
-
Qfi
+
Q
(3.20)
A simple rearrangement of equations (3.19) and (3.20) results in the equations to
be written at each bus i in an n-bus power system for the Conventional Power Flow. el
and
a )
epresent the net fundamental frequency real and reactive power flows at each bus
i
= -
P ) e)
P
=
(3.2
1)
el)
- ) Q;)
Q; =
(3.22)
Depending upon the bus type, the terms e nd
Q /
will be known or unknown
values in the Conventional Power Flow. Their final values will fall between a prespecified
minimum and maximum value for each.
q
8/10/2019 Newton Raphson Harmonic Pf
48/126
Fundamental frequency
positive-sequence
transmission network
which only contains
transformers and
transmission lines. ll
loads and generation
are external.
Figure 3 .6 .
General -Bus Power System Arranged to Form the Fundamental Frequency
Admittance Matrix.
Source: Gross [2] p.258.
referen e
Figure 3.7. A Fundamental Frequency Admittance Diagram o f a Tw o-Bus Power System.
Source: Gross [2] p.259.
8/10/2019 Newton Raphson Harmonic Pf
49/126
Equations 3.23) and 3.24) can be written in matrix form, to obtain equations 3.25),
3.26), 3.27), and 3.28).
Every entry of the matrix
IF
is in the general form of 7;:
y F / a :
Equation 3.18) may now be written in terms of the transmission current j:); ,
which was formed when creating 1F' 1
Equation 3.18) above may be separated into its rectangular components, as shown
below.
8/10/2019 Newton Raphson Harmonic Pf
50/126
Q:)
= vl) y ~in(8')q at)
Therefo re, the nonlinear algebraic equations that are to be written at each bus i in
the C onventional Pow er Flow, and solved using the New ton-Raphson technique, are
formed by rewriting equations (3.21) and (3.2 3) as shown below.
The above two equa tions describe the real and reactive power cond itions at each
bus
i
in the power system. They will be referred to as the real and reactive pow er
mismatch equations.
The known and unknown variables in equations (3.3 1) and (3.32) depend upon the
bus type. As stated before,
e'
nd
Q;)
are always prespecified numerical values at every
bus in the pow er system.
t
the swing (or slack bus), the hndamental frequency voltage is
assumed to be ql)1/@ p.u . as a reference, while
e
and
Q
are the unknowns.
How ever, equations (3.31) and (3.32) do not need to be w ritten at the slack bus, since
there are no restrictions placed upon e and Q: and ql)nd 4 re known. At all load
buses, both E)nd
Q;)
are known and usually zero for each, while vl'nd 8'' re
unknown . Lastly, at gen erator (voltage-controlled) buses, P
8/10/2019 Newton Raphson Harmonic Pf
51/126
8/10/2019 Newton Raphson Harmonic Pf
52/126
Figure 3 8 Typical ac Side Line Current Waveform of an ac Current to dc C urrent
Converter.
Source: Mohan [ 6 ] p.57.
8/10/2019 Newton Raphson Harmonic Pf
53/126
However, the following derivation will show that conventional and harmonic buses
are basically treated the same way in the Harmonic Pow er Flow, even though the difficulty
of their bus models may not be the same. The goal here is to make this point as obvious as
possible for two reasons. First, once the simplified conventional bus model is understood ,
the more detailed harmonic bus models will be easier to understand . Second, it will be
easier to understand how improved conventional bus models may be incorporated into the
Harmonic Power Flow.
Following this derivation, simplifications to the Harmonic Pow er F low equation
set will be discussed. These simplifications are possible only when the simplified
conventional bus model as discussed above is used.
In order to solve for the hndam enta l frequency unknowns in an
n
-bus power
system, 2 n 1) equa tions are required. Like the Conventional Power Flow, there is no
need to solve for any hnd am enta l frequency unknowns at the slack bus, since ql and q
are know n, and because there a re no restrictions placed upon el nd
Q: .
Also,
2 n) q )
equa tions are required to solve for every harmonic unknown in an
n -bus pow er system, when
q
harmonic frequencies are of interest. For this thesis, all
frequencies of interest greater than the hn dam ental, are referred to as the harmonics.
Up to two equations may be required to solve for the p arameters which are used to
describe the distorted current waveform entering each harmonic producing bus m .
Therefore, for
M
harmonic buses, up to
2 M )
additional equations may be required fo r
the Harm onic Power Flow. Therefore, the total number of equ ations which may be
required to solve for all of the unknowns in the Harmonic Power Flow is
2 n
1
2 n ) q ) 2 M ) .
brief description of the Harmonic Power Flow equations is now given, before
explaining them in further detail.
First, 2 n 1) equations are provided by describing the real and reactive power
condition s at each bus in the power system, as a Fourier Series. Like the Conventional
8/10/2019 Newton Raphson Harmonic Pf
54/126
Pow er Flow, these equations will be referred to as the real and reactive power mismatch
equa tions in the Harmonic Pow er Flow. These equations will allow complex power
generated and load) to be specified at each bus in the power system, exactly as was
desired with the Conventional Pow er F low.
As will be shown, a convenient way to obtain the additional 2 n) q) 2 M )
equations required for the Harmonic Power Flow is to apply Kirchhoff s current law
KCL) at every bus in the power system.
Applying KCL at every bus in the power system for each harmonic frequency of
interes t, will result in n) q) equations. Separating each of these equations into its real and
imaginary components, results in 2 n) q) equations for the Harmonic Pow er Flow. These
equa tions will be referred to as the harmonic frequency real and imaginary current
mismatch equations.
Lastly, 2 M ) equations are obtained from the real and imaginary components of
the fundamental frequency KCL equation formed at each harmonic bus. These equations
will be referred to as the fundamental frequency real and imaginary current mismatch
equations.
Like the Conventional Power Flow, the real and reactive power mismatch
equations will be formulated such that all complex power will be treated as either
generation, transmission, or load complex power. A lthough these equations will contain
the hndamental and all harmonic frequency components, generated complex power will
be assumed to contain no harmonic components. Generators synchronous machines) are
generally not considered to be significant harmonic producing devices, and are therefore
not m odeled as harmonic sources in the Harmonic Power Flow [I] .
Genera tors and harmonic devices both fulfill important, yet different roles in the
Harmonic Power Flow. Therefore, unnecessary complications may be avoided , if
generators and harmonic devices are placed at different buses in the power system
impedance diagrams. In addition, it will generally be easier if different types of harmonic
8/10/2019 Newton Raphson Harmonic Pf
55/126
devices are placed at separa te buses. For example, a gas discharge lighting load is not
treated the same as an ac current to dc current converter, in the Harmonic Power Flow.
When necessary, a single bus may be separated into two buses connected by a short circuit
(or a very small impedance). Figure 3.9 provides a simple example of how a single bus
containing a genera tor and a harmonic producing device, can be separated into two buses
connected by a short circuit.
The refore , the general form of the real and reactive power mismatch equations at
every harmonic bus m are given by equations (3.33 ) and (3.34). The letter h refers to the
highest harmonic to be considered in the Harmonic Pow er Flow, and the letter
represents the frequency of interest (fbndamental or harmonic).
In add ition, the general form of the real and reactive power mismatch equations at
every conventional bus are given by equations (3.35 ) and (3.36).
Only odd terms appear in equations (3.33) through (3.36) above, since even
harmonics do not exist in a completely balanced pow er system. Therefore,
1f
odd
The slack bus in the Harmonic Pow er Flow is treated essentially the same as for
the Conventional Pow er Flow. The hndamental frequency voltage is assumed to be
V
=
l / g p u
as a reference, while
El
nd
Qz
are the unknowns. How ever,
equations(3.35) and (3.36 ) do not need to be w ritten at the slack bus, since there a re no
restrictions placed upon
and Q and v nd 4 )re known.
8/10/2019 Newton Raphson Harmonic Pf
56/126
Figure 3 9 a)
A
Bus Containing a Generator and a Harmonic Producing@Pevice. b)
An
Equivalent Representation of a), Separated into Two Buses Connected by a Short
Circuit SC).
8/10/2019 Newton Raphson Harmonic Pf
57/126
8/10/2019 Newton Raphson Harmonic Pf
58/126
8/10/2019 Newton Raphson Harmonic Pf
59/126
8/10/2019 Newton Raphson Harmonic Pf
60/126
Figure 3 . 1
1
A General Representation of a Power System at the kt Harmonic
Frequency, with a Harmonic Producing HP) Load at Bus
m .
8/10/2019 Newton Raphson Harmonic Pf
61/126
for this thesis.
As stated before, conventional buses can be modeled exactly the same way as
harmonic buses are modeled in the Harmonic Power Flow. This approach, however is
generally not used, in order to simplifjr the Harmonic Power Flow. Instead, each
conventional bus is modeled as a complex power demand at the hndamental frequency,
and is modeled as an impedance (that has a linear voltage-current characteristic) at the
harmonic frequencies. At each harmonic frequency, a different impedance is used to
model the device(s) at the conventional bus. These harmonic impedances are easily
estimated, and are discussed for several conventional devices in this thesis. Therefore, it is
easy to obtain the harmonic currents entering a conventional bus, and to obtain the
harmonic complex power drawn by a conventional bus.
For example, Figure 3.12(a), shows a general representation of a power system at
the kt harmonic frequency, with a conventional
(C)
load at bus i The current
ZEI? g E / p j * , represents the
kt
harmonic current entering the conventional load at bus
i The kt harmonic voltage at bus i is represented as y ' ~ ( ~ / 4 ~ ) .ssuming that the
kt harmonic impedance model at conventional bus i is a series
R-L
combination as in
Figure 3.12(b), the current g: is found by Ohm s law to give equation (3.43).
g
y ( k )
(3.43)
R + j 2 @
By substitution of equation (3.43) into
f
I , at the
kt
harmonic frequency, the
harmonic complex load power for the conventional load of Figure 3.12(b) is given by
equations (3.44) and (3.45).
8/10/2019 Newton Raphson Harmonic Pf
62/126
8/10/2019 Newton Raphson Harmonic Pf
63/126
The additional 2 n ) q ) 2 M ) equations required for the Harmonic Pow er Flow,
are conveniently obtained by applying KCL at every bus in the power system. As stated
before, applying KCL at every bus for each harmonic frequency of interest, will give
n) q) equations . Separa ting each of these equations into its real and imaginary
components, results in 2 n) q) equations for the Harmonic Pow er Flow. These equations
are referred t o as the harmonic frequency real and imaginary curren t mismatch equations.
Lastly, 2 M ) equations are obtained from the real and imaginary components of
the hndam ental frequency KCL equation formed at each harmonic bus. These equations
are referred to as the hndam ental frequency real and imaginary current mismatch
equations.
By observation of Figure 3.1 1, application of KCL at a harmonic bus m for any
frequency of interest hndam ental or harmonic), results in equation 3.46), where
1,
and is odd ordered.
Also, by observation of Figure 3.12 a), application of KCL at a conventional bus
i
or any harmonic frequency of interest, gives equation 3.47), where
k
3, and
k
is
odd ordered.
The current
j
i
or m) is simply a KCL equation that is written at every bus
in the power system, using nodal analysis. These KCL equations are then used to produce
the hndamental f I), or harmonic f
>
1, odd frequency admittance matrix
/PI
Therefore, equations 3.46) and 3.47) can be written as equations 3.48) and 3.49)
respectively. Again,
1
k 3 and and k are odd ordered.
8/10/2019 Newton Raphson Harmonic Pf
64/126
8/10/2019 Newton Raphson Harmonic Pf
65/126
3.5 Network Models for the Newton-Raphson Based Conventional and Harmonic Power
Flows
Introduction
This section describes how to model a power system elements for the New ton-
Raphson based Conventional and Harmonic Power Flow s. The power system transmission
network composed of transmission lines and transformers, must be modeled a t the
fundamental and harmonic frequencies. Harmonic frequency shunt impedance models, for
the most common conventional loads, such as synchronous machines and induction
motors is presented here.
The harmonic load m odel for a gas discharge lighting load is discussed. All
harmonic devices cannot be modeled in the same way, how ever, since the current entering
each different type of harmonic load will possess a different Fourier Series.
For the Conventional Pow er Flow, all impedances are positive-sequence.
However, for the H armonic Power Flow, harmonic impedances may be positive-,
negative-, o r zero-sequence, depending upon the harmonic fi-equency. The harmonic
frequencies presented in Table 3.1 , apply to all power system netw ork and device
impedances discussed here.
Transmission Lines
For the H armonic Power Flow, three-phase transmission lines are modeled by a
single-phase pi-equivalent with the correct phase sequence, for both the fundam ental and
harmonic frequencies. This model is also used for the Conventional Power Flow . The most
important factors to be considered for the pi-equivalent model in Figure 3.13, are line
length and skin effect. The long line pi-equivalent model in Figure 3.13 is recomm ended
for distances longer than five percent of the wavelength of the highest harmonic of
interest[2] Knowing that c/f where c is the speed of light c
3.00
x
lo
mls ) , and
8/10/2019 Newton Raphson Harmonic Pf
66/126
z,
sinh(@) ohms
RC Series resistance per unit length.
~ Series inductance per unit length.
d :Shunt capacitance per unit length
@:shunt conductance per unit length.
d:
Length of the transmission line.
Figure 3; 13. Pi-Equivalent Model o a Long Transmission Line.
Source: Grady El], p.24.
8/10/2019 Newton Raphson Harmonic Pf
67/126
8/10/2019 Newton Raphson Harmonic Pf
68/126
8/10/2019 Newton Raphson Harmonic Pf
69/126
8/10/2019 Newton Raphson Harmonic Pf
70/126
8/10/2019 Newton Raphson Harmonic Pf
71/126
Figure 3 1 5 Power Flow Model of a Transformer.
Source: Grady
[I]
p 30
8/10/2019 Newton Raphson Harmonic Pf
72/126
based Harmonic Pow er Flow.
s suggested by Figure 3.15, tap changing transform ers must be used when
necessary, such as for modeling phase shift. For either wye-delta or delta-wye
connections, the high voltage side leads the low voltage side by thirty degrees for the
positive-sequence networks. Likewise, for the negative-sequence network, high voltage
side lags the low voltage side by thirty degrees. The zero-sequence ne twork will introduce
no phase shift.
Generators
For each harmonic frequency, an acceptable model for a synchronous generator is
to directly scale the hndamental frequency negative-sequence inductance reactance of the
generator with frequency [I ]. The resistive component of the hndam ental frequency
negative-sequence impedance is generally much smaller in comparison to th e reactive
com ponent. In addition, in the absence of elaborate information, the resistive component is
assumed to remain constant for all harmonic frequencies, and negligible.
Negative-sequence hndam ental frequency stator currents rotate at twice
synchronous speed, as seen from the rotor. The resultant flux is forced into paths o f low
permeability, which do not link any rotor circuitry. These paths a re characterized by the
direct and quadrature subtransient inductances
L;
and L: respectively. By definition, the
hndam ental frequency negative-sequence reactance of a synchronous machine is given by
equation 3.55) [20]
x: [(xi) ' (x;) ']/2
3.55)
where:
(xi)' Direct axis subtransient reactance
(xi) '
Quadrature axis subtransient reactance
8/10/2019 Newton Raphson Harmonic Pf
73/126
Then, by observation of equation (3.59, the findamental frequency negative-
sequence inductance of a synchronous machine is given by equation (3.56). This is the
inductance met by findamental frequency negative-sequence currents flowing into the
stator winding of a synchronous machine.
iq [(L; ) ' (~:) ']/2
(3.56)
When harmonic currents flow from the network into the stator windings of a
synchronous generator, they also create a flux rotating at multiples of synchronous speed,
where the direction may be the same or in opposition to the rotor s direction of rotation
[38]. In addition, these fluxes are also forced into paths characterized by the subtransient
inductances. Therefore, the average inductance which appears to be met not only by
negative-sequence fbndamental frequency stator currents, but also by harmonic frequency
stator currents is given by equation (3.56) [I].
Then, the reactance used to model the synchronous machine as in Figure 3.16, for
the
kth
harmonic frequency is given by equation
3.57),
where
k
is the harmonic frequency
of interest, and
x )
is given by equation (3.55).
x
q
(3.57)
Induction Motors
The induction machine is one of the most commonly used motor loads in power
systems today. An adequate harmonic frequency model of the induction machine is the
same one proposed for the synchronous machine [ l]. However, if the direct axis and
quadrature axis subtransient reactances are unknown, then a harmonic impedance model
can be obtained, based on the per-phase fbndamental frequency model shown in Figure
3.17 [I]
The magnetizing inductance L, is generally ignored, since it is large in comparison
with the other terms.
8/10/2019 Newton Raphson Harmonic Pf
74/126
8/10/2019 Newton Raphson Harmonic Pf
75/126
ST
=The per
ph se
fundamental frequency stator voltage.
Tz =The per
ph se
fundamental frequency stator current.
r, =The resistance of the stator winding.
L =The leakage inductance of the stator winding.
L =The magnetizing inductance
L =Th e leakage inductance of the rotor winding.
r, =The resistance of the rotor winding.
s =The slip = a, o r / @ ,
w =The angular frequency in radians per second.
Figure
3 17
A
Per-Phase Fundamental Frequency Equivalent Model for an Induction
Machine Phase a Shown).
Source: Grady [ I ] p.33.
8/10/2019 Newton Raphson Harmonic Pf
76/126
8/10/2019 Newton Raphson Harmonic Pf
77/126
Figure
3 18
Harmonic Power Flow Model o f an Induction Motor for the
k h
Harmonic
Frequency
Source: Grady
[I] p 36
8/10/2019 Newton Raphson Harmonic Pf
78/126
Other Conventional Loads
It is difficult to represent many conventional load buses at harmonic frequencies,
since their exact composition is usually unknown. It is suggested, that in the absence of
specific information, a conventional load bus must be modeled as a shunt resistor in
parallel with an inductor or a capacitor [I] . The resistance and the reactance can be
determined by the hndamental frequency active and reactive power demand of the
conventional load. Therefore, the values of R ~ nd LS~, are obtained from equations
3.62), and 3.63) below, and are used in the Harmonic Power Flow model of Figure
3.19. Note that if
Qz
is negative, then a capacitance
c
should be used instead of an
inductance
LS~,;.
In the absence of elaborate information, the hndamental frequency resistance is
assumed to remain constant for all harmonic frequencies
[I] .
The hndamental frequency
reactance is scaled directly with frequency.
Therefore, the impedance used to model an unknown conventional load at all
harmonic frequencies is given by equations 3.64) and 3.65) below.
~ f R L 3.64)
lo d
L
kX l) 3.65)
8/10/2019 Newton Raphson Harmonic Pf
79/126
Figure 3.19 . Suggested Power Flow M odel of an Unknown Conventional Load for the
kt Harmon ic Frequency
Source: Grady
[I] p.37.
8/10/2019 Newton Raphson Harmonic Pf
80/126
8/10/2019 Newton Raphson Harmonic Pf
81/126
were taken. The term B in equation (3.6 8) is simply a scaling variable which is
multiplied across the entire equation.
The Fourier Series expansions for odd powers of voltage is completely given by
equation (3.69 ). No te that equation (3.69) is simply a shortcut method o f raising equation
(3.67) to an odd power, and expanding and rearranging this equation to a form suitable for
the Harmonic Power Flow.
(3.69)
The terms
L
and
A
are given by equations (3 .70 ) and (3.71) respectively.
AL
=
8i2
.
..............+did (3.71)
Since even harmonics are not considered here, the values of A A 2 ....... A d and h are
odd integers only. In addition, since -h
Ld
h , negative values of
Ld
are converted to
positive values via the trigonom etric identities below.
cos(x) = cos(-x), and in(x) = sin(-x)
(3.72)
After directly substituting equation (3.69) into equation (3.68) and expanding,
equation (3.68) will assume the form given by equation (3.73 ). No te that all harmonic
terms greater than
h
are simply ignored.
From equation (3.73), the terms gZVrnd
g2A
re directly substituted into
equations (3.50a), and (3.50b ) respectively.
Tw o suggested methods are given by equations (3.74) and (3 .7 9 , which can be
used to initialize the fbndamental and harmonic voltage magnitudes at each bus in the
power system, in the absence of a better method. For both initialization methods, the angle
8/10/2019 Newton Raphson Harmonic Pf
82/126
8/10/2019 Newton Raphson Harmonic Pf
83/126
8/10/2019 Newton Raphson Harmonic Pf
84/126
3 6
Simplifications to the Newton-Raphson Based Harmonic Power F low Equation Set
As stated in the previous section, since nonlinear components are the source of
harmonics in pow er systems, it is of great importance in the Harm onic Power Flow, to
accurately model the nonlinear relationship between the current and voltage waveforms at
each bus containing a harmonic device. For this purpose, it is convenient to express
distorted voltages and currents as Fourier Series. For the H armonic Power Flow, the
Fourier Series of the current entering a harmonic bus is expressed as a function of the
Fourier Series of the voltage at this bus, and of any parameters which describe this
distorted current waveform.
Conventional buses can be modeled exactly the same way as harmonic buses are
modeled in the Harmonic Power Flow. This approach, however is generally not used, in
order t o simplify the Harmonic Power Flow. Instead, each conventional bus is modeled as
a complex power demand at the hndamental frequency, and is modeled as an impedance
that has a linear voltage-current characteristic) at the harmonic frequencies. At each
harmonic frequency, a different impedance is used to model the dev ice s) at the
conventiona l bus. This is the approach which is assumed in reference [I].
In the previous section, the Harmonic Pow er Flow was derived in a general
format. Therefore, both harmonic and conventional buses were approached in the same
way . However, when the simplified modeling approach discussed above) is used for
conventiona l buses, two basic simplifications can be made when forming the Harmonic
Pow er Flow equation set. These simplifications result when the kt harmonic power
system impedance diagram is arranged according to Figure 3 .20 instead of Figure 3 o),
when forming the kt harmonic frequency admittance matrix i.e.,
lY(* I
Although
simplified, the resulting equation set will be equivalent to the set formed by following the
previous section. Note that no information is lost as a result of these simplifications.
Rather, the same information is simply restated in a different way.
8/10/2019 Newton Raphson Harmonic Pf
85/126
8/10/2019 Newton Raphson Harmonic Pf
86/126
First, at every conventional bus modeled as a shunt impedance for each harmonic
frequency, as in Figure 3.12), the real and reactive power mismatch equations only need to
be written with fbndamental frequency components. Therefore, equations 3.3 5), 3.36),
3.39), and 3.40) are combined to give equations 3.81) and 3.82) below.
In addition, by forming the
kt
harmonic frequency admittance matrix according to
Figure 3.20, the currents i ) and g: of Figure 3.12 and equation 3.47), are both
automatically included in the
K L
equations which are written. Therefore, the harmonic
frequency real and imaginary current mismatch equations to be written at each
conventional bus modeled as a shunt impedance for each harmonic frequency), are also
simplified. Hence, equations 3.5 1a) and 3.5 1b) are rewritten as equations 3.83) and
3.84) respectively, where k
3,
and k is odd ordered.
F)y k)in(a ) y
Note that all other equations discussed in the previous section, will remain
unchanged.
In the Examples chapter, the equations discussed in the previous section are used
to form the Harmonic Power Flow equation set, for a two bus power system. Then, this
equation set is again formed, using the two simplifications discussed in this section. It will
be shown that both equations sets have an identical solution set.
Although it may be then concluded that the two equation sets are equivalent, it is
very possible that they may converge at a different rate. In addition, depending upon the
8/10/2019 Newton Raphson Harmonic Pf
87/126
initial conditions the two equation sets may converge to a different solution set. However
no attemp t will be made by the au thor of this thesis to prove this hypothesis.
8/10/2019 Newton Raphson Harmonic Pf
88/126
8/10/2019 Newton Raphson Harmonic Pf
89/126
us - k)
1
Figure 4 1 Representation o f
a
Harmonic Producing Device
at
Bus m as an Ideal Current
Source at
the
kth Harmonic Frequency.
8/10/2019 Newton Raphson Harmonic Pf
90/126
In the examples chapter the Current Injection Technique is used to find the third harmonic
voltages in a two bus pow er system with a gas discharge lighting load.
Project IEEE-5 19 [25] states that the assumption which permits the Current
Injection Technique to be used is that the pow er system voltages are not distor ted.
However Project IEEE-5 19 also states that the Current Injection Technique is generally
accurate if the voltage distortion levels in the pow er system are less than ten percen t.
Basically as the voltage distortion levels in the power system increase the less effective
the Current Injection Technique becomes at modeling the relationship between harmonic
voltages and currents in the power system.
The purpose of this chapter was not meant to be a comparison between the
Harmonic Pow er Flow and the Current Injection Technique. Rather the idea was to
present a simple method which can be used in place of the Harmonic Power Flow in
certain instances. Very often a simplified solution is adequate in simple radial netw orks.
However this is usually not the case in nonradial netw orks or when the magnitudes of
power system harmonic voltages are significant [I].
8/10/2019 Newton Raphson Harmonic Pf
91/126
CHAPTER 5
EXAMPLES
5 1 Introduction
The example system under consideration is as shown in Figure 5.1 . The complex
pow er generated at bus one is carried over a one half mile,
230Kv
transmission line, to a
large fluorescent lighting load drawing
6MW
and
OMVARS
at bus tw o. Fo r simplicity, it
will be assum ed that the presence of the fluorescent lighting load will only introduce third
harmonic currents and voltages into the power system.
It is noted that this exam ple pow er system is not entirely realistic. The au thor of
this thesis constructed this example power system based on readily available information.
In addition, simplicity was desired in ord er to keep the examples manageable. However,
the solution methods presented in the examples are correct.
For all impedance data, the system base values are given below in equation 5.1).
The generato r data of Table 5.1 is supplied in per-unit form on the system bases
chosen.
The configuration of the
230Kv
transmission line is as shown in Figure 5.2. No te
that due to lack of information, some dimensions had t o be approximated. In Tables 5.2
and 5.3 , the data for the transmission line conductors and ground wires is given. Using
Tables 5 .2 and 5.3, and the equations in the Appendix, the necessary impedance data for
the
230Kv
transmission line was obtained, and is given in Table 5.4. Because the
8/10/2019 Newton Raphson Harmonic Pf
92/126
Figure
5.1.
Example Two-Bus Power System,
with
a Generator at Bus One and a
Fluorescent Lighting FL) Load at Bus Two.
ur 2 ur generaor3 8Kv
Rotor
s3+ ,4 VL , ~ f r a t s a f r a t 4 Q- Q* H
MVA) kV)
kg)
(Hz) Mvar) Mvar) Type Poles s)
Salient
0 5 mile - 230Kv transmission line
X d
XL
Xi
X
i X i X
X Xm
R,
R, R
impedance
data
in on generator
ratings)
j 0 0 2 ~
Table
5 .1 .
Fundamental Frequency
60Hz
System Generator Data.
Source: Gross
[2], p.251.
8/10/2019 Newton Raphson Harmonic Pf
93/126
Figure
5 2
Configuration of the 23 Kv Transmission Line in Figure
5 1
Source: Westinghouse Electric Corporation [20] p 590
8/10/2019 Newton Raphson Harmonic Pf
94/126
8/10/2019 Newton Raphson Harmonic Pf
95/126
Electrical characteristics of overhead ground wires
Part
A :
Alumoweld strand
Table 5 .3. Data for the Ground Wires of the
23 Kv
Transmission Line of Figure 5.2 .
Source: Gonen [41]
p.658.
I
freauencv positive-sequence impedance
I
zero-sequence impedance I
Strand
A W G )
NO. 10
Table
5.4.
Impedance Data for the
23 Kv
Transmission Line of Figure 5 2
60 Hz
reactance
for -lt rad iw
Induct~ve Capac~tive
n / m l
Mn.
mi
0.777 0.1
392
frequency I positive-sequence impedance I zero-sequence impedance
60
Hz
geometric
mean radius
it
0.001650
Res~stancc.Rim1
Small currents
25C 25C
oc 60Hz
8 870 8 870
Table 5 .5 . Per-Unit Impedance Data for the 230
Kv
Transmission Line o f Figure 5 .2 .
75 of cap.
75C 75C
oc 60
Z
10.440 10.670
Hz)
6 0
per unit
0.043415 + 0.258128
per
unit
0.082514 + 0.507372
8/10/2019 Newton Raphson Harmonic Pf
96/126
transmission line of Figure 5.1 is only one half mile in length, and the highest harmonic
i.e., current or voltage) of interest is the third, the short transmission line equations of
Chapter 3 can be used, with shunt capacitance ignored. The transmission line data of Table
5.4 is converted to per-unit, by using the chosen system base values, and is given in Table
5.5.
As an example, the 60Hz posi
Top Related