Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
An Experimental Investigation Based On
Mathematical and Software Modeling Of Total
Harmonic Distortion in Personal Computer
M. Shahzad Bajwa, Aslam. Pervez Memon, Jamshed A. Ansari, M.Tarique Bhatti
Abstract — The penetration of sensitive and nonlinear loads
like Personal computers (PC’s) increases distortion levels and
can cause severe problems to power systems. Personal
computers are nonlinear devices that cause flow of nonsinusoidal current with harmonics at input power supply.
Production of harmonics in personal computers has special
interest to power quality (PQ) due to fact that personal
computers are usually concentrated in large groups. Therefore
it has become increasingly important to address their influence
with good level of accuracy. This paper presents the results of an
experimental investigation that was conducted to develop
mathematical model related to harmonics and concentration of
personal computers. Curve fitting toolbox of MATLAB software
has been used for developing mathematical model and then
comparison is done for measured results of experimental work
with developed mathematical models. The findings of this
experimental work are helpful to achieve the best curve fit model
for total harmonic distortion as well as individual current
harmonics, which helps to determine the magnitude of
harmonics for cluster of personal computers. Due to highly
nonlinear behavior of personal computer and big data tables for
harmonic measurement, it is very difficult to predict the effect of
harmonics for power quality of distribution system. Therefore it
is always required to choose the best curve fit for mathematical
models. The results obtained in this paper will help in accurate
modeling of harmonics due to cluster of personal computer.
Index Terms — Point of Common Coupling (PCC), Power
Quality (PQ), Total Harmonic Distortion (THD), Personal
Computer(PC’s), Curve Fitting.
multiples of the fundamental frequency (50 or 60Hz)
[5].Harmonics cause distortion and distorted wave can be
represented as shown in Fig 1.
Fig. 1 Distorted waveform breaks into sinusoids
Personal computers are electronic and nonlinear loads to
AC supply system because they have a power supply design
known as a switch mode power supply (SMPS), conducting
current for part of a cycle through multiple paths in order to
efficiently convert alternating current to direct current, which
get down the quality of the electricity supply system
[6].SMPS in computer is responsible for production of
harmonics. In SMPS based load, nonlinear current is
produced due to charging and discharging of dc-link capacitor
in power supply [7]. A general SMPS block diagram and
equivalent SMPS circuit model of personal computer is
shown in Fig.2 and Fig.3
I. INTRODUCTION
In AC power distribution systems, harmonics are
produced when the current waveform is distorted by non
linear loads. This cause power quality (PQ) problem [1]. The
problem resort to massive fault occurrences and economic
losses to power supply system. Therefore power system
harmonics have become one of the important investigations
[2-3]. Effects of harmonic in AC power supply system have
always been present, but they are not in the limelight until
recently due to the intensified usage of power electronic
gadgets such as personal computer. Good power quality
means less distortion and fewer harmonic in the voltage and
current sources [4]. As harmonic in power system can be
defined as the sinusoidal waveforms with frequencies
Muhammad Shahzad Bajwa, Aslam Pervez Memon, Department Of
Electrical Engineering, QUEST Nawabshah and Jamshed Ahmed Ansari,
Department of Electrical Engineering, Sukkur IBA, Muhammad Tarique
bhatti, Department of Electronic Engineering QUEST Nawabshah
Email:shahzadbajwa80@yahoo.com. Manuscript received February 03,
2016; revised on March 21 and May 16, 2016; accepted on June 03, 2016.
Page 62
Fig. 2 General block diagram of SMPS load [6]
Fig. 3 Equivalent SMPS circuit model of PC [7]
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
Information Technology facilities are having number of
computers as high as five hundred numbers or more than that.
In such facilities SMPS used in computers plays important
role for deciding harmonic contents. It is therefore necessary
to make an experimental investigation of measuring
harmonics indices and make mathematical equations for
individual and total harmonics distortion indices with respect
to increasing non-linear load (electronic load PC’s.)
To compute and quantify the harmonic pollution the
following harmonic indices are most commonly used [08-10].
A. Current and Voltage Harmonic
Mathematically
In
(1)
I1
Where Ih is value of harmonic current in percentage and In is
value of nth harmonic current and I1 is value of fundamental
or RMS (Root Mean Square).
For the harmonic voltages, I can be replaced with V.
B.
Measurement of Harmonic Distortion In Equipment
Degree of distortion: This will cause equipment damage
or malfunction i.e. THDv
Equipment Emission: How equipment will effects the
supply mains i.e. THDi .
THD: The distortion in voltage or current waveform is
quantified by total harmonic distortion (THD). THD is used
to define level of harmonic content in alternating signals.
THDi =
K1
K>1
(4)
Where AF,K is attenuation factor and IKN is k order total
N
harmonic current of N loads and IK1
is k order harmonic
current for single loads. The value of DF and AF in between
0 and 1, low value means high attenuation.
III. METHODOLOGY AND EXPERIMENTAL SETUP
II. CRITERIA TO ANALYZE HARMONIC
DISTORTION
Ih =
IN
AF,K = N∗IKN
In this experiment, harmonics measurements of non
linear load i.e. PC’s was conducted in computer laboratory by
power quality analyzer (PQA) instrument at QUEST
Nawabshah. Measuring harmonics in terms of individual and
total Harmonic distortion (THD) and power quantities such as
Power factor, active power, reactive power, apparent power,
input voltage and current waveforms are recorded for
analyzing purpose.
In experiment 24 numbers of PC’s are connected
gradually one by one by pressing switches/button of AC
supply mains and magnitudes of harmonics are observed and
recorded. So that harmonic distortion indices as mentioned in
section II can be computed and calculated.
PC’s data
P-4 cpu-3.00 GHz, Power supply of ATX type 220v, Monitor
(CRT)-17 Inch, Power range-140w-200w, 256 MB of RAM.
2
√∑∞
n=2 Irms,n
(2)
Irms
Where
∞
2
Irms = √∑ Irms,n
n=1
Fig. 4 View of computer laboratory in Electrical
department QUEST Nawabshah
Similarly for THDv , I can be replaced with V.
C.
Evaluation of Harmonic Distortion
To evaluate the harmonic quantification two terms are
generally used [11-13]. One is diversity means partial
cancellation of harmonic current among different loads due to
dispersion in harmonic current phase angles.
Second is attenuation describes as the reduction in
harmonic magnitude due to share system impedance.
These two terms can be mathematically written as
DF,K =
|∑N
n=1 IK,n |
N
∑N
n=1 IK
K>1
(3)
Where DF,K is diversity factor and IK,n is k order phasor
magnitude of harmonic current of nth load and IKN is simply
magnitude of harmonic current.
Page 63
Fig. 5 Connection of PQA and switching board
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
A. Functionality of PQA instrument:
The features of PQA are as shown in Fig.6, Fig.7 and Fig.8
Interpolation: It is used for connecting the data dots or
points. The function obtained by Interpolation cannot be use
as general function f(x) since it’s really a collection of small
f(x) s, in which one point is connected to the next .it does not
work very well for data that has built in random error or
scatter.
Curve Fitting: In curve fitting we find approximate function
that may not pass through all data points .Approximation is
usually preferable for smoothing noisy data.
An example of interpolation and curve fitting for straight
line as shown in Fig.9 and Fig.10
Fig. 6 Functions in PQA
Fig. 9 Graph of function with interpolation
Fig. 7 Recording screen of PQA
Fig. 10 Graph of function with curve fit
In actual practice there is not only straight line function
is used for curve fit, there are lots of functions such as higher
polynomials, exponential, trigonometric etc.
So how do we define good fit and how we choose
appropriate model?
Fig. 8 Measured Data of PQA
B. Curve Fitting Technique
For set of data points obtained from an experimental or
simulation base worked. it is assumed that there is some
functions f(x) that pass through data points and we have to
find that function.
There are two types of curve fitting methods [14]
(1) interpolation
(2) curve fitting
Page 64
What makes a particular model a ‘good’ fit?
Most popular criteria for choosing the ‘best’ fit
Residual: Difference between the measured value and fit
function f(x).
ei = (xi) − f(xi) is called the residual or error related with fitted
data for the data pair (xi, f(xi)). ei is the vertical distance
between the known data and the fitted function.
residual = data – fit
For straight line, residual or error may write in this way
error = ∑𝑛𝑖=1(𝑦𝑖 − (𝑎𝑥𝑖 + 𝑏))2
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
So we can say criteria for choosing the ‘best’ fit , in which
sum of squares of residual values (also called least squares
criterion)= ∑ni=0 e2i , known as the sum of squares due to error
(SSE) is minimum .The Value closer to 0 show a better fit and
then approximation is equivalent to interpolation.
A. Experimental Results of Voltage and Current
Waveforms
IV. RESULTS AND DISCUSSIONS
Section A present the discussion of the experimental
results obtained from measurement of harmonics. Section B
explains the graphical and mathematical representation of
individual harmonics and THD with the help of curve fitting
toolbox of MATLAB software. In section C the calculated
and develop mathematical expressions have been compared.
A. Results Of PQA Measurement
PC’s numbering from PC1 to PC24 are connected to the
AC supply mains gradually at a interval of two computers and
then wave-forms of supply voltage and input current, values
of power indices, harmonic spectrum of odd harmonics in
current significant up to 15th harmonics and THD in current
and voltage, have been captured and recorded in real time for
observations.
Fig. 11 One PC
Fig. 12 Twenty four PC
B. Experimental Results of Active Power, Reactive Power,
Apparent Power, Power Factor and Displacement Power
Factor Values
Fig. 13 One PC
Fig. 14 Twenty four PC
Table 1. PQA Measurement Results
Page 65
No. of PC’s
Input
Supply
Voltage
(volts)
Input
Current
(Ampere)
Power
Factor
Displacement
Power factor
Active
Power
(Watt)
Reactive
Power
(Var)
Apparent
power
(VA)
Crest factor
for voltage
Crest factor
for current
1
209.5
0.870
-0.77
-1.00
-140
118
183
1.4
2.2
2
205.0
1.715
-0.77
-1.00
-266
219
344
1.4
2.2
4
200.8
3.74
-0.79
-0.99
-0.58k
0.46k
0.74k
1.4
2.0
6
196.9
5.57
-0.78
-0.99
-0.90k
0.71k
1.15k
1.4
2.1
8
226.1
6.81
-0.78
-0.99
-1.18k
0.97k
1.53k
1.4
2.1
10
223.2
8.67
-0.78
-0.99
-1.49k
1.21k
1.92k
1.4
2.1
12
219.0
9.66
-0.77
-0.99
-1.64k
1.34k
2.11k
1.4
2.1
14
188.9
12.93
-0.80
-0.99
-1.93k
1.45k
2.41k
1.4
2.0
16
194.3
14.45
-0.80
-0.99
-2.25k
1.69k
2.82k
1.3
2.0
18
196.8
15.84
-0.79
-0.99
-2.48k
1.92k
3.13k
1.4
2.1
20
205.7
17.49
-0.79
-0.99
-2.82k
2.22k
3.59k
1.4
2.1
22
192.1
20.26
-0.80
-0.99
-3.09k
2.34k
3.88k
1.3
2.0
24
190.4
22.22
-0.80
-0.99
-3.41k
2.52k
4.24k
1.3
2.0
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
C. Analysis of Measured Results
The waveforms results of input voltage and current gives
information that supply voltage is nearly sinusoidal has no
significant impacts of PC’s or nonlinear load up to certain
level .However input current wave shape is characterized by
a pulsed current. The input current is non-sinusoidal, it
contains of 2 pulses over a cycle. The result is that such a
current waveform contains high level of harmonic distortion.
The crest factor of input supply voltage waveforms is 1.3
lower than 1.414 when number of PC’s increases. It means
concentration of PC’s increases distortions in input supply
voltage waveforms .it is observed that crest factor of voltage
can be much better predictor of THDi than THDv .
As crest factor for input supply current is higher that is
2.2 for single PC from 1.414 this mean there is harmonic
distortion in current .By increasing number of PC’s it
decrease up to 2.0 .
In table.4.1 the negative active power sign shows that
flow of power from loads to source and if power is negative
then it mean power factor is also negative sign because RMS
values of current and voltage is positive. It can be concluded
that “-” and “+” not show the leading and lagging Power
Factor. In measured results power factor is almost 0.77 and at
some point it differ very high. As per IEEE standards power
factor must be between 0.90 and 0.95. But in this case power
factor is quite less than 0.90 that means the reactive power
drawn from supply is very high and it needs to be reduced.
Because the current pulses are centered in the voltage period,
the displacement power factor is almost one or 0.9.
Fig. 19 9th harmonic
Fig. 20 11th harmonic
Fig. 21 13th harmonic
Fig. 22 15th harmonic
E. When Twenty Four(24) PCs Are Connected To AC
Supply Mains
D. Experimental Results of Odd Harmonics
When One (01) PC Is Connected To AC Supply Mains
Fig. 15 1st harmonic
Fig. 17 5th harmonic
Page 66
Fig. 23 1st harmonic
Fig. 24 3rd harmonic
Fig. 25 5th harmonic
Fig. 26 7th harmonic
Fig. 16 3rd harmonic
Fig. 18 7th harmonic
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
F. Experimental Results Of Maximum, Minimum And
Average Value Of 𝑇𝐻𝐷𝑖
Fig. 27 9th harmonic
Fig. 28 11th harmonic
Fig. 31 One PC
Fig. 32 Twenty four PC
G. Experimental results of THDV values
Fig. 29 13th harmoni
Fig. 30 15th harmonic
Fig. 33 One PC
Fig. 34 Twenty four PC
Table 2. PQA measured results of IHDi , THDi , THDv
No.
of
PC’s
%
Magnitude
Of 3rd
harmonic
%
Magnitude
Of 5th
harmonic
%
Magnitude
Of 7th
harmonic
%
Magnitude
Of 9th
harmonic
%
Magnitude
Of 11th
harmonic
%
Magnitude
Of 13th
harmonic
%
Magnitude
Of 15th
harmonic
%
Magnitude
Of 𝐓𝐇𝐃𝐢
%
Magnitude
Of 𝐓𝐇𝐃𝐯
1
57.2
28.3
8.6
11.3
8.6
1.5
5.4
66.3
1.6
2
55.9
22.3
6.1
11.6
5.5
1.6
3.9
62.5
2.0
4
55.2
20.2
5.6
11.1
8.5
1.9
3.4
60.1
2.2
6
56.1
23.1
2.7
10.0
5.4
2.1
4.9
62.2
2.1
8
56.1
23.3
2.0
9.3
4.8
2.3
4.2
62.1
2.0
10
56.1
22.9
1.9
8.6
4.0
2.8
4.3
61.6
2.2
12
56.2
23.3
1.1
7.9
1.2
4.3
2.5
61.8
2.2
14
53.8
16.6
7.5
7.9
1.4
3.9
2.5
57.4
2.4
16
54.1
16.6
6.3
8.4
1.1
3.9
2.2
57.9
3.7
18
54.5
18.2
5.3
8.6
1.8
3.9
2.9
58.7
3.3
20
54.4
18.1
5.0
8.0
1.2
4.0
1.7
58.6
3.6
22
53.1
14.5
6.6
6.8
1.4
4.0
1.3
53.2
4.0
24
51.3
10.6
9.0
5.8
2.5
3.7
0.4
55.9
4.3
Page 67
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
Table 3. PQA measured results of phase angle of IHDi
No. of
PC’s
Phase Angle
Of 3rd
harmonic
Phase Angle
Of 5th
harmonic
1
145
-48
2
158
-26
4
163
-22
6
162
8
161
10
Phase Angle
Of 7th
harmonic
Phase Angle
Of 9th
harmonic
Phase Angle
Of 11th
harmonic
165
23
-172
59
-50
-114
85
-97
-100
62
-78
109
-121
-74
87
-27
-85
113
-79
-85
82
-29
-88
111
-85
-76
88
162
-27
-89
112
-90
-60
90
12
161
-29
-96
112
-145
-68
104
14
165
-20
-51
116
-120
-65
106
16
165
-21
-57
117
-106
-69
103
18
164
-22
-61
114
-105
-66
100
20
164
-22
-59
116
-166
-61
108
22
165
-22
-41
123
171
-57
112
24
165
-28
-27
133
146
-55
-92
H. Analysis Of Measured Results Of 𝐼𝐻𝐷𝑖 ,𝑇𝐻𝐷𝑖 , 𝑇𝐻𝐷𝑣
Diversity Effect:
For example, from Table .2 and Table .3
For 2 PC , 10PC and combine 12 PC, the diversity factor for
5th harmonic will be
DF5 =
23.3 ∗ 9.66
22.9 ∗ 8.67 + 22.3 ∗ 1.715
DF5 =
225.08
198.54 + 38.24
DF5 =
225.08
236.78
DF5 = 0.95
For 2 PC and 22 PC and combined 24 PC, the diversity factor
for 5th harmonic will be
DF5 =
22.22 ∗ 10.6
20.26 ∗ 14.5 + 22.3 ∗ 1.715
DF5 =
235.5
293.7 + 38.24
DF5 =
235.5
355.7
DF5 = 0.66
The diversity factor value is between 0 and 1, small value
of DF show significant amount of cancellation due to the
circulation of harmonic currents among individual loads.
Similarly
Attenuation Effect:
Attenuation factor for one PC and 24 PC is connected to AC
supply mains, the attenuation factor for 5 th and 15th harmonic
can be calculated as
Page 68
Phase Angle
Of 13th
harmonic
2.48
2.48
0.08
0.08
AF5
= 0.220∗(24) = 5.28
AF15
= 0.042∗(24) = 1.008 = 0.08
=
Phase Angle
Of 15th
Harmonic
0.5
The attenuation factor value is between 0 and 1, small
value of AF shows more cancellation of current harmonics.
In similar manner, we can calculate value of DF and AF
of any number of PC’s.
With the operation of increasing nonlinear loads, the
injected harmonic current magnitudes and phase angles vary
in a random way.
Attenuation and diversity are two key factors that are
used to evaluate magnitudes of harmonics, so we can say that
total distortion in current is not the arithmetic sum of
harmonic current magnitudes as traditional methods describe,
it can significantly overestimate the cumulative harmonic
currents produced by PC’s loads.
Among odd harmonics, 3rd harmonic is not much effected
due to diversity effects, diversity in phase angle much more
appeared for higher harmonics. Mostly cancellation of
individual current harmonics occurs above from 7th harmonic.
If the concentration of nonlinear loads (PC’s) is not large in
cluster form, then they do not possess any severe problems for
distribution feeders, but if PC’s are increases then the
cumulative magnitudes of harmonic are dangerous which may
impacts on distribution transformer.
If the load power increases, the shape of current
waveforms is affected. As a result the pulse of current is taller
and high. A change is occurred in magnitudes of current
harmonics and also change in phase angle. The increase of
power level also effected for attenuation factor.
The non-sinusoidal current in PC’s generate high
magnitudes of harmonics, due to half wave symmetry the
current harmonics are in odd magnitudes.
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
I.
Graphical Representation and Mathematical Modeling
of Results
The experiment which carried out in this research work
shows the results of harmonics in magnitudes ,but for
engineering application of these results the convenient
method is to show the 𝐼𝐻𝐷𝑖 ,𝑇𝐻𝐷𝑖 , 𝑇𝐻𝐷𝑣 function with
respect to number of PC’s with mathematical expressions.
Therefore curve fitting toolbox of MATLAB is used. it is
possible to show the trend of odd IHDi ,THDi , THDv with
increasing PC’s loads.
From the graph shown in Fig.36 5th harmonic decrease as
PC’s connected to AC supply mains increase. So afterward it
reaches to zero.
Using curve fitting technique, the function between
magnitude of 5th harmonic and number of PC’s from Table.2
can be written as
I5 = 0.001239N6PC-0.001111N5PC+0.03816N4PC
-0.6282N3PC+5.0826N2PC -17.56NPC +41.56
(6)
>> goodnesspoly6
a.
Graphical representation of 3rd harmonic
sse:
3rd Harmonic
1.8365
60
c.
59
Graphical Representation Of 7th Harmonic
7th Harmonic
10
58
9
57
7
Magnitude %
Magnitude %
8
56
55
54
53
6
5
4
3
2
52
1
51
0
50
2
4
Fig. 35
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Graphical representation of 3rd harmonic
rd
From the graph shown in Fig.35 3 harmonic magnitude
is at high percentage and after decrease in constantly manner
by the application of increasing PC’s load.
Using curve fitting technique, the function between
magnitude of 3rd harmonic and number of PC’s from Table.2
can be written as
I3 = 0.002869N6PC -0.0002706N5PC+0.009613N4PC
-0.1617N3PC +1.304N2PC-4.525NPC+60.68
(5)
Where NPC shows number of PC’s.
>> goodnesspoly6
sse:
1.8365
b.
Graphical Representation Of 5
th
Harmonic
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Fig. 37 Graphical representation of 7th harmonic
From the graph shown in Fig.37 indicates that the
magnitude of 7th harmonic first decreases and at some points
again increase with increase of PC’s rather than for 3rd and 5th
harmonics.
Using curve fitting technique, the function between
magnitude of 7th harmonic and number of PC’s from Table.2
can be written as
I7 = 0.00655N6PC+0.003636N5PC-0.003653N4PC
+0.0831N3PC-0.6375N2PC+0.7058NPC+7.8444
(7)
>> goodnesspoly6
sse:
d.
5th Harmonic
2
16.1732
Graphical Representation Of 9th Harmonic
30
9th Harmonic
20
18
25
16
14
Magnitude %
Magnitude %
20
15
10
12
10
8
6
4
5
2
0
0
2
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Fig. 36 Graphical representation of 5th harmonic
Page 69
2
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Fig. 38 Graphical representation of 9th harmonic
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
From the graph shown in Fig.38 indicates that 9th
harmonic magnitude decrease in cubic functional way as PC’s
increase.
Using curve fitting technique, the function between
magnitude of 9th harmonic and number of PC’s from Table.2
can be written as
From the graph shown in Fig.40 13th harmonic increase
rapidly and then remains constant with further increase of
PC’s.
Using curve fitting technique, the function between
magnitude of 13th harmonic and number of PC’s from Table.2
can be written as
I9=-0.00122N3PC+0.04904N2PC-0.7495NPC+12.68
I13 = -0.007463N2PC+0.30344NPC+0.8987
(8)
>> goodnesscubic
>> goodnesspoly2
sse:
sse:
e.
2.7824
Graphical Representation Of 11th Harmonic
g.
1.4743
Graphical Representation Of 15th Harmonic
11th Harmonic
15th Harmonic
10
9
9
8
8
7
7
Meagnitud %
10
Manitude %
(10)
6
5
4
6
5
4
3
3
2
2
1
2
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
0
Fig. 39 Graphical representation of 11th harmonic
2
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Fig. 41 Graphical representation of 15th harmonic
th
From the graph shown in Fig. 39 11 harmonic
magnitude linearly decrease and reaches to zero and then
remains constant for further addition of electronic loads.
Using curve fitting technique, the function between
magnitude of 11th harmonic and number of PC’s from Table.2
can be written as
I 11 = -0.0001655N4PC +0.009842N3PC -0.17N2PC
+0.4942 NPC +7.199
(9)
>> goodnesspoly4
sse:
f.
The magnitude of 15th harmonic decreases in constant
manner with increase of PC’s.
Using curve fitting technique, the function between
magnitude of 13th harmonic and number of PC’s from Table.2
can be written as
I 15 = 0.007311N6PC-0.0005843N5PC+0.01793N4PC
-0.2632N3PC+1.867N2PC-5.735NPC+9.575
(11)
>> goodnesspoly6
sse:
1.2223
10.2610
h.
Graphical Representation Of 13th Harmonic
Graphical Representation Of 𝑇𝐻𝐷𝑖
13th Harmonic
5
THDi in current
70
4.5
68
4
66
3.5
Magnitude %
Magnitude %
64
3
2.5
2
62
60
58
1.5
56
1
54
0.5
52
0
2
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Fig. 40 Graphical representation of 13th harmonic
Page 70
50
2
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Fig. 42 Graphical representation of THDi
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
From the graph shown in Fig.42 indicate impact of
individual current harmonics on total harmonic distortion in
current (THDi ). THDi is decrease with the increase of
electronic or PC’s loads, though the curve is not smooth
,because the background distortion of power supply is not
fixed. From function THDi = f(NPC ) is not continuous, it is
decreasing up to some PC’s and almost constant for higher
PC’s . Is looks like there is kind of saturation in THDi values,
when number of PCs is increased. However, the distortion
current is constant (√∑ 𝐼𝑛2 ).The values of I3, I5,I7 ,I9 ,I11 ,I13
and I15 also decline, although their absolute values are almost
constant.
Now the THDi and number of PC’s of table.2 can be
written with help of curve fitting as
THDi THDi = 0.001798N6PC-0.001408N5PC+0.04274N4PC
-0.6299N3PC+4.591N2PC-15NPC+77.73
(12)
>> goodnesspoly6
sse:
12.8395
The evaluation of current harmonics (odd) are derived for
mathematical modeling proved that the decrease in THDi with
respect to increase the number of PC’s.
According power quality analyzer Fluke 43B manual
instructions, if THDi is below from 20%, then it is acceptable
limit. Now obtained measured results of total harmonic
current distortion is more than 20% which may makes affect
on the neutral line conductor. A harmonic filter necessary
required to install in order to eliminate the harmonics
injection into this system.
For illustration purpose, when one PC and 24 PC’s
connected at PCC. The harmonic current spectra in Table 3
shows cancellation among harmonics due to phase diversity,
the result is that the current which is flowing through the PCC
has lower value of THDi as compared to individual computer
taken separately.
i.
The relation b/w number of PC’s and THDv can be
written as from table.2
THDv = 0.00424N2PC+0.005218NPC+1.827
(13)
>> goodnesspoly2
sse:
0.9013
j. Verification Of Mathematical Model
Now the derived mathematical expressions by curve fitting
techniques are verified by comparing the measured results
with practical results.
THDi =
√I2rms,3 +I2rms,5 +I2rm,7 +I2rms,9 +I2rms,11 +I2rms,13 +I2rms,15
Irms
For example when 8 personal computers switched on
from AC supply mains
Irms = 6.81 A
The values of odd harmonics are as
3rd =56.1% * 6.81 = 3.8204
5th =23.3% * 6.81 = 1.5867
7th =2.0 % * 6.81 = 0.1362
9th =9.3 % * 6.81 = 0.6333
11th =4.8% * 6.81 = 0.3268
13th =2.3% * 6.81= 0.1566
15th =4.2% * 6.81= 0.2860
Now if we put these odd harmonics values in above Eq.
THDi =
√(3.8204)2 + (1.5867)2 + (0.1362)2 + (0.633)2 + (0.3268)2 + (0.1566)2 + (0.2860)2
6.81
THDi =17.74/6.81=0.618
%THDi=0.618*100=61.8 %
Above Eq. can be modified as
THD
Graphical Representation Of 𝑇𝐻𝐷𝑣
=√
THDv in voltage
10
2
2
2
2
2
2
2
Irms,5
Irms,15
Irms,3
Irms,7
Irms,9
Irms,11
Irms,13
+ 2 + 2 + 2 + 2
+ 2
+ 2
2
Irms
Irms
Irms
Irms
Irms
Irms
Irms
9
8
2
2
2
THD= √I32 + I52 + I72 + I92 + I11
+ I13
+ I15
Magnitude %
7
6
THD= √56.12 + 23.32 + 2.02 + 9.32 + 4.82 + 2.32 + 4.22
5
4
THDi= 61.8 %
3
2
1
0
2
4
6
8
10
12
14
16
18
20
Number of PCs connected to AC supply mains
22
24
Fig. 43 Graphical representation of THDv
From the graph shown in Fig.43 THDv is increased with
increase in electronic loads.
Page 71
Now I3,I5,I7,I9,I11,I13 and I15 are magnitudes of current
harmonic in percentage.
The measured value of THDi is 62.1 % and calculated
value is 61.8%,difference between measured and calculated
value is 0.25 which in terms of percentage is 0.40%.This
small difference is due to neglect the odd harmonics higher
from 15th harmonics. The closed results between developed
ISSN – 1999-4974
Bahria University Journal of Information & Communication Technologies Vol. 9, Issue 1, June 2016
mathematical model and measured results prove the validity
of measured harmonics by PQA. It plays vital role for analysis
of the odd current harmonics. Now the value of THDi from
derived expression of Eq.(4.8) of THDi with the help of curve
fitting toolbox is 62.67 % .The difference b/w derived value
and calculated value is 0.82 which in terms of percentage is
1.3%.
observed under standard limit. This project can provide an
extension for further investigation in the field of nonlinear
load by connecting some more new nonlinear load
collectively. The applications of this particular mathematical
model can be extended to the power systems with much larger
number of buses. Also filtration work can be included in
laboratory work as well as in software models to reduce
harmonics.
For odd current harmonic calculation
REFRENCES
Now the Eq.(5) to Eq.(11) can be used as for odd
harmonic calculation.
For example for 10 PC’s ,the values of odd harmonics as
given below
I3 =56.07 %
I5 = 23.16 %
I7 =2.02 %
I9 =8.86 %
I11 =3.33 %
I13 =3.18 %
I15 =3.83 %
As derived Eq.(5) to Eq.(11) obtained by the curve fitting
technique give very close results to the experimental
measured values with less than 2% error which is acceptable.
This confirm the authenticity of developed model.
In same way values of IHDi ,THDi , THDv can be
calculated for any number of PC’s.
V. CONCLUSION
This research work shows the analysis of nonlinear load
concentration effecting on total harmonic distortion. The
simple method to detect the harmonics and model
mathematically in nonlinear load at PCC is discussed with
experiment and computed by the software means. Measured
results shows that magnitude of THDi is decreased when
nonlinear loads (PC’s) connected to the laboratory is
increased in concentration. Although individual single PC
harmonic current distortions are approximately constant .The
extensive measurements during experimental work provide
insight in the behavior of harmonics into PC. Diversity factor
and attenuation factor phenomena in PC clusters can be used
for analysis of harmonic voltage and harmonic current
characteristics .The more attenuation of harmonic current as
harmonic increased from the 3rd harmonic. However, the 3rd
harmonic experiences only slight attenuation. The harmonic
spectrum for current are more dominated by high magnitudes
of low order odd harmonics such as 3rd, 5th and 7 th which
decay rapidly with frequency. THDv is found under standard
limit but it can be high if more nonlinear load will be
connected. Curve fitting technique helped us to quantify the
harmonic magnitudes and also show the values of harmonics
within very short time by developed mathematical expression.
Verification of results shows the accuracy of using the curve
fitting technique to mathematical model. Filtration
methodology can be easily designed to reduce harmonic
content in voltage and current if accurate harmonic generated
phenomena are known. In this work voltage harmonic is
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ISSN – 1999-4974