POWER QUALITY: HARMONICS IN POWER SYSTEMS

Chhaya B. Shukla B.S., California State University, Sacramento, 1998

PROJECT

Submitted in partial satisfaction of the requirements for the degree of

MASTER OF SCIENCE

in

ELECTRICAL AND ELECTRONIC ENGINEERING

at

CALIFORNIA STATE UNIVERSITY, SACRAMENTO

FALL 2009

ii

POWER QUALITY: HARMONICS IN POWER SYSTEMS

A Project

by

Chhaya B. Shukla

Approved by: __________________________________, Committee Chair Dr. Turan Gonen __________________________________, Second Reader Dr. Salah Yousif ____________________________ Date

iii

Student:

Chhaya B. Shukla

I certify that this student has met the requirements for format contained in the University format

manual, and that this project is suitable for shelving in the Library and credit is to be awarded for

the Project.

__________________________, Graduate Coordinator ________________ Dr. Preetham Kumar Date Department of Electrical and Electronic Engineering

iv

Abstract

of

POWER QUALITY: HARMONICS IN POWER SYSTEMS

by

Chhaya B. Shukla

The increased use of nonlinear electronic equipment has become a concern in most

utility power systems. Nonlinear loads draw current discontinuously during the cycle of

the input voltage waveform and produce low power factors when harmonics are taken

into account. This increases line current and can limit the available capacity of branch

circuits. In addition, harmonic currents can cause heating in utility and facility

transformers. Modern personal computers and other information technology equipments

utilize “switching regulators” or switch mode power supplies, to convert utility AC

power to regulated DC power. These switching regulators and switch mode power

supplies generate high third and fifth harmonic current. If the equipments are not

properly designed or rated, equipment will often malfunction when harmonics are present

in an electrical system and that equipment can be personal computer in business

environment or an ultrasonic imaging machine in a hospital. To eliminate this harmful

effect, in depth study of power system analysis is required.

v

In this project, study of power quality and detailed analysis of harmonics is performed.

This project will look at causes and effects of harmonics in power systems. In depth

analysis is performed and mathematical model and software simulation for passive

harmonic filter is developed to design inexpensive solution for the utilities and power

industries. And the results will be compared with the industry standards.

_______________________, Committee Chair Dr. Turan Gonen _______________________ Date

vi

ACKNOWLEDGMENTS

The author would like to acknowledge Dr. Turan Gonen, Professor of Electrical

Engineering at California State University, Sacramento, for his guidance, supervision and

patience in evaluating this project, as well as mention of his excellent instruction in the

area of Power Engineering at California State University, Sacramento.

The author appreciative of Dr. Salah Yousif, Professor of Electrical Engineering at

California State University, Sacramento, for his excellent instruction in the area of Power

Engineering at California State University, Sacramento as well as being a reader of this

project.

The author would also like to acknowledge Dr. Preetham Kumar, Graduate

Coordinator and Professor of Electrical Engineering at California State University,

Sacramento, for his guidance and in completion of this project.

vii

TABLE OF CONTENTS Page

Acknowledgments ……………………………………………………………………vi

List of Tables …………………………………………………………………………ix

List of Figures …………………………………………………………………………x

Chapter

1. INTRODUCTION …………………………………………………………………1

1.1 Introduction ………………….…………………………………………….…1

1.2 Statement of the Problem …………………………………………………….1

1.3 Project Structure ……………………………………………………………….2

2. LITERATURE REVIEW ………………………………………………………….3

2.1 Introduction ……………………………………………………….………….3

2.2 Harmonics Generation …………………………………………….……….…6

2.3 Effects of Harmonic ……………………………………………………………8

2.4 Harmonics Analysis …………………………………………….…………….9

2.5 Harmonics Mitigation and Filters ………………………………….…….….…9

3. MATHEMATICAL MODEL ………………………….……………….….………14

3.1 Measurements of Electrical Power Quality ……………………….…….….…14

3.2 Power in Passive Elements ……………………………………….…….….…17

3.3 Capacitor Banks and PF Correction ……………………………….…….….…18

3.4 Short circuit Capacity or MVA or KVA … ……………………….…….….…19

3.5 Passive Filter …………………………………………………….…………….19

viii

4. HARMONIC FILTER DESIGN ………………………………….…………………21

4.1 A Single Tuned Notch Filter .…………………………….…….……………21

4.2 MATLAB…………………………………………………………………….31

5. CONCLUSION…………………………………………………………………….32

Appendix A Basic Definitions ………………………………………………………33

Appendix B MATLAB Program ……………………………………………………34

Appendix C MATLAB Calculations ………………………………………………37

References ……………………………………………………………………………39

ix

LIST OF TABLES Page

1. Table 4.1 Comparison table for evaluating filter duty limit………………………30

x

LIST OF FIGURES Page

1. Figure 2.1 Current distortion caused by nonlinear resistance ………………………4

2. Figure 2.2 Fourier series representation of a distorted waveform…………………. 5

3. Figure 2.3 General flow of harmonic currents in a power system …………………7

4. Figure 2.4 Power factor capacitors can alter the direction of flow of the harmonic component of the current ………………………………………………………….7

5. Figure 2.5 Creating a fifth-harmonic notch filter and its affect on the system

Response …………………………………………………………………………11

6. Figure 2.6 Harmonic filter for high voltage ………………………………………12

7. Figure 2.7 Automatic detuned filter capacitor banks ……………………………13

8. Fig.4.1 Low-voltage filter configuration …………………………………………22

1

Chapter 1

INTRODUCTION

1.1 Introduction

Normally, power systems are designed to operate at frequencies of 50 or 60Hz.

Although certain types of loads produce current and voltage signal with frequencies that

are integer multiples of the 50 or 60 Hz fundamental frequency. These higher

frequencies are called electrical pollution that is known as power system harmonics.

Harmonics causes obstruction to the normal operation of the equipment or the system.

Studying their causes can help develop protective schemes for harmonic isolation and

also clearance of harmonics. Harmonics are caused by various reasons such as

saturation, switching and winding connections in transformers, shunt capacitors

resonance and nonlinear loads like switching mode power supply, wind and solar power

generation. Harmonics analysis involved the calculation of system parameters. In this

project how the capacitor bank parameters contribute to develop the harmonics and

recalculating the value of the capacitor bank can help resolve the system harmonics.

Hence, harmonics analysis of a power system forms an important aspect of a reliable

system design.

1.2 Statement of Problem

This project involves in designing the harmonic filter to eliminate the harmonic

from the system. The single tuned notch filter is designed. The theoretical analysis is

presented in chapter 2.

2

The harmonic analysis of power system involves the calculation of power factor,

frequency responses, capacitor bank size, filter reactor size, evaluating filter duty

requirements, fundamental duty requirements, harmonic duty requirements, harmonic

currents and voltage parameters. Calculation of the peak voltage, RMS voltage, RMS

currents and kvar values then compared with the standard limitations.

1.3 Project Structure

Project is consists of five parts. Chapter 1 has the introduction, statement of

problem and Project Structure. Chapter 2 covers Literature Review. In Chapter 3 The

Mathematical Model, where measurements of electrical power quality are described.

Chapter 4 being the Application of the theory presented in chapter 3 to the given

problem. Then the results are compared with the standards. In addition, the Matlab

Program used to analyze the problem is presented along with results in the Appendix.

Chapter 5 covers the conclusion of the project.

3

Chapter 2

LITERATURE REVIEW

2.1 Introduction

In linear circuits current is directly proportional to the voltage. However, in

nonlinear circuits current is not proportional to the applied voltage. Figure 2.1 shows this

concept by applying voltage to a nonlinear resistor where the voltage and current vary as

shown in the curve. As we can see the voltage is perfectly sinusoidal but the resultant

current waveform is distorted. Now as we increase the voltage by just few percentages

may cause the current to double the value and takes the different shapes. This is the

source of most harmonics in a power system.

The distorted waveform can be a sum of sinusoidal signals. When the waveform is

identical, it can be shown as a sum of pure sine waves where the frequency of each

sinusoid is an integer multiple of the fundamental frequency of the distorted wave. This

multiple is called a harmonic of fundamental. The sum of the sinusoidal is called the

Fourier series. Figure 2.2 shows Fourier series of a distorted waveform. Here the

fundamental frequency is the frequency of the power system. That is 60 Hz and the

multiples that are 120Hz, 180Hz, 240Hz, 300Hz called second, third, fourth and fifth

harmonics respectively. The combine waveform shows the result of adding the

harmonics on to the fundamental.

4

Figure 2.1 Current distortion caused by nonlinear resistance [1]

5

Figure 2.2 Fourier series representation of a distorted waveform [8]

6

2.2 Harmonics Generation

There are different types of loads that generate harmonics in power systems.

The linear time-invariant loads are designed such a way so that the sinusoidal

voltage results in a sinusoidal flow of current. These loads have constant steady-state

impedances during the applied sinusoidal voltage. When the voltage is increased, the

current increases in direct proportion. The transformers and rotation machines are the

examples of this kind of loads when operated in normal condition.

In nonlinear load, the applied sinusoidal voltage does not result in a sinusoidal flow

of current. These loads are not constant impedances during the entire cycle of the applied

sinusoidal voltage. For example, wind and solar power generation, switching mode

power supplies, computers, copy machines and television sets.

In utility distribution feeders and industrial plant power systems, the main tendency

is for the harmonic currents to flow from the harmonic producing load to the power

system source. This is shown in Figure 2.3. The impedance of the power system is

normally the lowest impedance seen by the harmonic currents. That means the bulk of

the current flows in to the source. The source of harmonics can be located by using this

general tendency of the harmonic current flow. The power quality meters can be used to

measure the harmonic currents in each branch starting at the beginning of the circuit and

trace the harmonics to the source.

7

Figure 2.3 General flow of harmonic currents in a power system

The power factor correction capacitors can alter this flow pattern. For example, adding a

capacitor to this circuit as shown in the following circuit may draw a large amount of harmonic

current into that portion of the circuit as shown in figure 2.4.

Figure 2.4 Power factor capacitors can alter the direction of flow of the harmonic component of the current.

8

2.3 Effect of Harmonics

Harmonics practically effect to every equipment in the power system. The effect of

voltage distortion is divided in three major categories, the thermal stress, the dielectric

stress and load disruptions.

Heating effects: Harmonic current flowing in the circuits cause heating effects in the

conductors. Especially eddy current losses are proportional to the square of the

frequency. Some harmonics, notably the 5th, are negative sequence or backward rotating

and tease can increase losses by inducing even higher frequency currents in machine

rotors.

Interference: Harmonics can cause interference to communications systems, protections

systems and signaling circuits due to electromagnetic induction or to the flow of the

ground currents.

Resonance: Harmonics generated in one part of circuit may increase the resonance

effects in another part of the circuit. Some resonance can be dangerous if the

magnification is large because of high circuit Q-factor or low damping.

Even harmonics: Even harmonics may cause asymmetrical magnification and can lead

to saturation.

Some more adverse effects of harmonics listed as follows:

- Malfunction in electronics devices and computer equipments

- Errors in measurements

- Overheating and over stressing of insulations

- Lamp flicker when harmonic pulses involved.

9

- Sometimes machine vibrates

- Blowing out of small auxiliary devices like fluorescent lamp capacitors.

2.4 Harmonics Analysis

The first step in solving harmonic related problem is to perform an analysis to

determine the specific needs of power system. The analysis then applied to study of

resonant conditions and harmonic filter design. The in-depth study is involved because

of the interaction between harmonics producing source and power system, the limitations

of modeling equipments in the power system and need to check for the accuracy.

2.5 Harmonics Mitigation and Filters

The harmonics is becoming a bigger concern now a day with the increase nonlinear

load in the power system. There are multiple ways to control the harmonics as follow:

- Find the nonlinear load and reduce the harmonic current

- Add filter to remove the harmonic current or block the harmonic current from

entering to the system

- Modify the system frequency response to avoid harmful interaction with

harmonic current.

10

Passive filters

Nonlinear load produces harmonic currents that can travel to other part of the power

systems and eventually goes back to the source. As we review that harmonics current can

damage power systems many ways. One of the ways to block this unwanted

characteristic of the system is to block it by using filters.

There are two types of filters, active and passive filter. The interest of this project

is to design a single tuned “notch filter” since it is sufficient for the application and

importantly it is inexpensive. Figure 2.5 shows configuration of the filter, equivalent

circuit of the filter and the frequency response of the filter. This filter has two

advantages, it suppresses the harmonics and increases power factor. This filter is tuned

slightly lower than the harmonic to be filtered to provide a safely margin in case there is

some change in the system parameters that may raise the notch frequency. Figure 2.6

shows the picture of harmonic filter for high voltage and figure 2.7 shows an automatic

detuned filter capacitor banks used in the industries.

11

Figure 2.5 Creating a fifth-harmonic notch filter and its affect on the system response [2]

12

Figure 2.6 Harmonic filter for high voltage [7]

13

Figure 2.7 Automatic detuned filter capacitor banks[7]

14

Chapter 3

MATHEMATICAL MODEL

3.1 Measurements of Electrical Power Quality [1]

It is important to perform analytical analysis of the system in order to understand

the status of the system and then the solution can be calculated to resolve the harmonics

conditions. In this section in depth study is performed to calculate the system parameters.

3.1.1 RMS Voltage and Current

The expressions for the RMS voltage and current are

(3.1)

and

(3.2)

Here it is assumed that and are also given in RMS.

3.1.2 Distribution Factor

The total harmonic distortion is defined as

(3.3)

or

(3.4)

15

Where is the harmonic voltage at harmonic frequency h in RMS, V1 is the rated

fundamental voltage in RMS, and h is the harmonic order. H=1 corresponds to the

fundamental frequency.

Similarly

(3.5)

or

(3.6)

Where is the harmonic current at harmonic frequency h in RMS and

is the rated fundamental current in RMS.

The RMS voltage and current can now be expressed in the terms of THD as

(3.7)

and

(3.8)

3.1.3 Active and Reactive Power

(3.9)

(3.10)

16

The real power is

(3.11)

The reactive power is

(3.12)

3.1.4 Apparent Power

Based on the aforementioned formulas for voltage and current, the apparent power is

(3.13)

or

(3.14)

3.1.5 Power Factor

For purely sinusoidal voltage and current, the average power is

(3.15)

or

(3.16)

Where

(3.17)

17

(3.18)

(3.19)

Simplifying

(3.20)

3.2 Power In Passive Elements

3.2.1 Power In A Pure Resistance:

Real (or active) power dissipated in a resistor is give by

(3.21)

Where is the resistance at the hth harmonic. 3.2.2 Power in pure Inductance

Power in pure Inductance can be calculated as

(3.22)

where

(3.23)

(3.24)

Thus

18

(3.25)

(3.26)

3.2.3 Power in Pure Capacitance

Power in pure capacitance is

(3.27)

Here negative sign indicates that the reactive power is delivered to the load

and

(3.28)

(3.29)

(3.30)

(3.31)

3.3 Capacitor Banks and PF Correction

Power delivered by the capacitor bank Qc is

(3.32)

(3.33)

19

Where P is the real power delivered by the system and absorbed by the load, Q1 is the

load’s reactive power, and Q2 is the system’s reactive power after the capacitor bank

connection.

Here an important situation to observe from the following equation that current is

inversely proportional to power factor.

(3.34)

3.4 Short circuit Capacity or MVA or KVA

Where a new circuit is to be added to an existing bus in a complex power system,

short circuit capacity or MVA (or kVA) a data provide the equivalent impedance of the

power system up to that bus. The three-phase short circuit MVA is determined from

(3.35)

3.5 Passive Filter

The Notch filter provides PF correction in addition to harmonics suppression.

(3.36)

The actual fundamental frequency compensation provided by a derated capacitor bank is

found from

(3.37)

20

The fundamental frwquency current of the capacitor back is

(3.38)

The equivalent single-phase reactance of the capacitor bank is

(3.39)

The reactance of the filter reactor is found from

(3.40)

where is the tuned harmonic. The fundamental frequency current of the filter becomes

(3.41)

Since the filter draws more fundamental current than the capacitor alone, the supplied var

compensation is larger than the capacitor rating and is found from

(3.42)

21

Chapter 4

HARMONIC FILTER DESIGN

4.1 A Single Tuned Notch Filter:

A single tuned notch filter is designed for a facility and applied at 480 V bus. The

load where the filter is installed is 1200kVA with power factor of 0.75 lagging. The total

harmonic current produced by the load is 30% of the fundamental current and has

maximum of 25% of 5th harmonic. This facility has 1500kVA transformer with 6% of

impedance. The 5th harmonic voltage distortion on the utility side of the transformer is

1.0% of the fundamental when there is no load.

The harmonic filter is designed step-by-step as shown below.

Step 1. Selection of a tuned frequency for the filter: The tuned frequency is selected

based on the harmonic characteristics of the loads where the power is applied. According

to the nature of the single tuned filter, the filtering needs to start at the lowest harmonic

frequency generated by the load. In this design that is the fifth harmonics.

The filter will be tuned little below the harmonic frequency to allow for the

tolerances in the filter components and the variations in the system impedance. This

protects the filter from acting as a direct short circuit for the offending harmonic current,

reducing duty on the filter components. It also reduces duty on the filter components and

minimizes the possibility of destructive harmonic resonance if the system parameters

change and cause the turning frequency to shift.

22

Figure 4.1 Low voltage filter configuration. [3]

23

In this design, the filter is tuned to the 4.7th harmonic. The notch filter is shown in

the figure.

Step 2. Calculation of capacitor bank size and the resonant frequency: In general, the

filter size is based on the load reactive power requirement for power factor correction. In

this case when an existing power factor correction capacitor is turned in to a harmonic

filter, the capacitor size is given. From there the reactor size is selected to tune the

capacitor to the required frequency. Although, depending to the tuned frequency, the

voltage rating of the capacitor bank need to be higher than the system voltage to allow for

the voltage rise across the reactor. That is why it is better to change out the capacitor

anyway. In this design, there is no capacitor installed and the desired power factor is

96%. That is the net reactive power from the filter required to correct from 75 to 96

percent power factor.

Reactive power demand for a 75 percent power factor would be

(4.1)

Reactive power demand for a 96 percent power factor would be

(4.2)

Required compensation from the filter:

(4.3)

For a normal 480 V system, the net wye-equivalent filter capacitive reactance is determined by

24

(4.4)

is the difference between the capacitive reactance and the inductive reactance at

the fundamental frequency:

(4.5)

For tuning at the 4.7th harmonic,

(4.6)

The desired capacitive reactance can be determined by

(4.7)

Here it is not known whether the filter capacitor can be rated at 480 V same as the

system or will have to be rated one step higher at 600 V. To calculate this reactance at a

480 V rating, the capacitor would have to be rated

(4.8)

Same way at 600 V, the capacitor would have to be rated 682 kvar. Now the filter will be

designed using a 480 V capacitor rated 450 kvar, which is a commonly available size

near the desired value. For this capacitor rating,

(4.9)

25

Step 3. Calculating filter reactor size: The filter reactor size can now be selected to tune

the capacitor to the desired frequency that is frequency at the 4.7th harmonic or 282 Hz.

The filter reactor size is computed from the wye-equivalent capacitive reactance,

calculated in step 2, as shown below:

(4.10)

or

(4.11)

The reactor size can also be computed by solving for L in the following equation:

(4.12)

Where

Step 4. Evaluate filter duty requirements. Now evaluating filter duty requirements

typically involves capacitor bank duties. These duties include peak voltage, current, kvar

produced and rms voltage. IEEE Standard 18-1992, IEEE standard for shunt power

capacitor is used as the limiting standard to evaluate these duties. Calculation of the

duties are kind of lengthy, therefore they are divided in to three steps as follow.

(1) Computation of fundamental duty requirements

(2) Harmonic duties

(3) rms current and peak voltage duties

26

step 5. Calculation of fundamental duty requirements: In this step, a fundamental

frequency operating voltage across the capacitor bank is determined as follow:

The apparent reactance of the combined capacitor and reactor at the fundamental

frequency is

(4.13)

The fundamental frequency filter current is

(4.14)

The fundamental frequency operating voltage across the capacitor bank is

(4.15)

This is fundamental voltage across the capacitor and it should be less than 110 percent of

the capacitor rated voltage.

Since the filter draws more fundamental current than the capacitor alone, the actual

reactive power produced is larger than the capacitor rating, that is

(4.15)

Step 6. Calculation of harmonic duty requirements: We need to calculate the maximum

harmonic current expected in the filter. This calculation is divided in two parts: the

harmonic current produced by the nonlinear load and the harmonic current from the

utility side.

27

Step 6a. The nonlinear load produces 25 percent fifth harmonic of the fundamental

current, the harmonic current produced by the load will be

(4.16)

Step 6b. The harmonic current contributed to the filter from the source side is calculated

as follow. Here we are going to assume that the one percent fifth harmonic voltage

distortion present on the utility system is limited to only by the impedances of the service

transformer and the filter. The utility impedance is being neglected.

Fundamental frequency impedance of the service transformer:

(4.17)

The fifth harmonic impedance of the service transformer:

(4.18)

The harmonic impedance of the capacitor back:

(4.19)

The harmonic impedance of the reactor:

(4.20)

The voltage distortion of the utility system is given as 0.01 pu, then the amount of fifth

harmonic current contributed to the filter from the source side is

(4.21)

28

= 46.5 A (4.22)

Step 6c. The maximum harmonic current is the sum of the harmonic current produced

by the load and that contributed from the utility side:

(4.22)

Step 6d. The harmonic voltage across the capacitor:

(4.23)

(4.24)

Step 7. Calculate total rms current and peak voltage requirements. These quantities are

calculated as follows:

Total rms current passing through the filter:

(4.25)

This is the total rms current rating required for the filter reactor.

The peak voltage across the capacitor is the summation of the harmonic and the

fundamental components.

(4.26)

29

(4.27)

The rms voltage across the capacitor

(4.28)

(4.29)

The total kvar seen by the capacitor

(4.30)

(4.31)

(4.32)

Step 8. Evaluate capacitor rating limits. The duties for the proposed filter capacitor are

compared to the IEEE standard limits shown in the table 4.1. This would be a very

marginal application because the capacitor duties are essentially at the maximum limits.

There is no tolerance for any deviation in assumptions or increases in service voltage. A

480V capacitor will likely have a short life in this application. When this happens, a

capacitor rated for higher voltage must be used. At 600V, the equivalent capacitor rating

would be

(4.33)

The nominal rating of 700 kvar with the reactor values computed in step 3 will give

the same filter within normal manufacturing tolerances. The 600V capacitor would be

well within its rating in this application.

30

Table 4.1. Comparison Table for Evaluating Filter Duty Limit ________________________________________________________________________ Duty Definition Limit, % Actual Values Actual Values, %

Peak Voltage 120 575V/480V 119

RMS Voltage 110 508V/480V 106

RMS current 180 698A/541A 129

Kvar 135 614kvar/450kvar 136 ________________________________________________________________________

Step 9. Evaluation of the filter frequency response. Now the filter frequency response

need to evaluate to make sure that the filter is not generating a new resonance at a

frequency that could cause additional issues. The harmonic at the parallel resonance

below the notch frequency is calculated as follows:

(4.34)

Step10. Evaluating the result with the specified tolerance:

Generally the tolerance for the capacitors are +15% and +/- 5% for the inductance.

These tolerances sometimes can make big difference and can create harmful resonance.

31

For this reason, the final step is to check the filter design for the multiple extreme

situations.

Step 1 to 10 shows the single tuned filter design. When single tune filter cannot

control harmonics to the desired level, then we may need to design multiple filters. For

example, for fifth, seventh and eleventh harmonic filter may be needed for some large

loads. The same procedure needs to follow with one additional step. That is, the

reactive power requirement needs to be divided between the filter stages.

4.2 MATLAB

The harmonic filter is also simulated using the following Matlab code and the code and

the calculated results are shown in Appendix B and Appendix C respectively.

32

Chapter 5

CONCLUSION

The wide spread utilization of power electronic devices has significantly increased

the number of harmonic generating apparatus in the power systems. This harmonics

cause distortions of the voltage and current waveforms that have adverse effects on

electrical equipment. This harmonics effect on power systems can be summarized as

increase losses of devices, equipment heating and loss-of-life, and interference with

protection, control and communication circuits as well as customer loads. Harmonics are

one of the major power quality concerns. The estimation of harmonic from nonlinear

loads is the first step in a harmonic analysis and this may not be straightforward task.

To eliminate this situation, the harmonic study analysis becomes an important and

necessary task for engineers in almost every industrial project.

In this project, the analytical analysis of the system parameters was performed to

understand the status of the system. After understanding the status of the system, as a

solution, the passive filter was designed to eliminate the unwanted and harmful

harmonics condition. The mathematical model was developed and the results ware

compared with the IEEE standard. The calculations were performed both by hand and

verified by MATLAB simulations.

33

APPENDIX A

Basic Definitions

Harmonics: Sinusoidal voltages or currents having frequencies that are an integer

multiples of the fundamental frequency at which the supply system is designed to

operate.

Total Harmonic Distortion (THD): The ratio of the root-mean-square (RMS) of the

harmonic content of the RMS value of the fundamental quantity, expressed as a percent

of the fundamental.

Nonlinear Load: An electrical load which draws current discontinuously or whose

impedances varies throughout the cycle of the input AC voltage waveform.

Harmonic Distortion: Periodic distortion of the sign wave.

Voltage fluctuation: A series of voltage changes or cyclical variation of the voltage

envelope.

Frequency Domain: An increase or decrease in the power frequency. Its duration varies

from few cycles to several hours.

Passive Filter: A combination of inductors, capacitors and resistors designed to eliminate

one or more harmonics. The most common filter is inductor in series with a shunt

capacitor, which short-circuits the major distorting harmonic component from the power

system.

Active Filter: Any of a number of sophisticated power electronic devices for eliminating

harmonic distortion.

34

APPENDIX B

MATLAB Program

%harmonic filter deasign % input data V = 480 %voltage kV = V/1000 VAl = 1200 %kVA load PFl = 0.75 %pf load lagging pfd = 0.96 % desired pf f = 60 % frequency Vhpu = 0.01 V1 = 450 V2 = 600 % facility trnaformer ratings VAt = 1500 %kVA tranformer MVAt = VAt/1000 % MVA h = 4.7 % harmonic tuning needed Xt = 0.06 Ihpu = 0.25 %reactive power demand % kvar load VARl = VAl*sin(acos(PFl)) % kavar desired VARd = VAl*sin(acos(pfd)) %actual kVAR kvar = VARl-VARd %wye- equivalent filter capacitive reactance Xf = kV^2*1000/kvar % for fundamentla frequency Xl = Xf Xcap = (h^2*Xl)/(h^2-1)

35

Xl1 = Xcap/h^2 %for fundamentle inductive reactance L = Xl1/(2*pi*f) % calculation of fundamental duty requirements Xfun = abs(Xl1-Xcap) %the fundamental frequency filter current Ifun = (V/sqrt(3))/(Xfun) %the fundamental frequency filter voltage Vcap = sqrt(3)* Ifun* Xcap %actual reactive power because of fundamental current kvarfun = sqrt(3)* Ifun* kV Ih = Ihpu* VAl/(sqrt(3)* kV) ffun = (Xt*kV^2)/MVAt %fifth harmonic impedance Xth = h*ffun %harmonic impendence of capacitor Xcaph = Xcap / h % harmonic impedance Xlh = h * Xl1 %5th harmonic current Ihu = Vhpu * V / (sqrt(3) * (Xth - Xcaph + Xlh)) %total current Iht = Ih + Ihu %harmonic voltage caross the capacitor Vcaph = sqrt(3)*Iht * Xcaph

36

% total rms current through the filter Irms = sqrt(Ifun^2 + Iht^2) Vcappeak = Vcap + Vcaph % rms voltage across the capacitor Vrms = sqrt( Vcap^2 + Vcaph^2) %kvar at capacitor kvarc = sqrt(3)* Irms * Vrms %Capacitor rating at 600 V kvar2 = V1 * V2^2 / V^2

37

APPENDIX C

MATLAB Calculations

V = 480 kV = 0.4800 VAl = 1200 PFl = 0.7500 pfd = 0.9600 f = 60 Vhpu = 0.0100 V1 = 450 V2 = 600 VAt = 1500 MVAt = 1.5000 h = 4.7000 Xt = 0.0600 Ihpu = 0.2500 VARl = 793.7254 VARd = 336.0000 kvar = 457.7254 Xf = 0.5034 Xl = 0.5034 Xcap = 0.5272 Xl1 = 0.0239 L = 6.3310e-005 Xfun = 0.5034 Ifun = 550.5581 Vcap = 502.7596 kvarfun = 457.7254 Ih = 360.8439 ffun = 0.0092 Xth = 0.0433 Xcaph = 0.1122 Xlh = 0.1122 Ihu = 63.9794

38

Iht = 424.8233 Vcaph = 82.5406 Irms = 695.4057 Vcappeak = 585.3002 Vrms = 509.4901 kvarc = 6.1367e+005 kvar2 = 703.1250

39

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