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Transcript
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
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
ii
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
Department of Electrical and Electronic Engineering
iii
________________
Date
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.
iv
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
v
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.
vi
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
vii
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
viii
LIST OF TABLES
Page
1.
Table 4.1 Comparison table for evaluating filter duty limit………………………30
ix
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
8.
Fig.4.1 Low-voltage filter configuration …………………………………………22
x
……………………………13
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.
presented in chapter 2.
The theoretical analysis is
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
becomes
is the tuned harmonic. The fundamental frequency current of the filter
(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
determined by
is
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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