newton raphson harmonic pf

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


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.


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


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


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


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


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


8/126


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


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.


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'


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


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


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.


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


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


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


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.


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).


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.


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


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


21/126

Figure 2. 1 . Generalized Six Pulse Converter.

Source: Boost [8]

p.273.


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


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.


24/126


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].


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


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


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.


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].


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


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.


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


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]


34/126


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.


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


37/126


38/126


39/126

Figure

3 .1 .

Geometrical Interpretation of the Newton-Raphson Method.

Source: Choma [37]

p.3

14.


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


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


42/126

0 99999~

Figure

3 .3 .

Illustration

of

the

Two

Systems Given by Equation Sets

3.7)

and 3.8).


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).


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


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


46/126


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


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.


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.


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


51/126


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.


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


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


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.


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).


57/126


58/126


59/126


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 .


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).


62/126


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.


64/126


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


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.


67/126


68/126


69/126


70/126


71/126

Figure 3 1 5 Power Flow Model of a Transformer.

Source: Grady

[I]

p 30


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


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.


74/126


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.


76/126


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


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)


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.


80/126


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


82/126


83/126


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.


85/126


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


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.


88/126


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.


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].


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


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.


93/126

Figure

5 2

Configuration of the 23 Kv Transmission Line in Figure

5 1

Source: Westinghouse Electric Corporation [20] p 590


94/126


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


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

newton raphson harmonic pf

Documents