International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 9 - Jun 2014
Harmonics Reduction in Cascade H-Bridge
Multilevel Inverters Using GA and PSO
Aman Parkash#1, S.L. Shimi#2, S. Chatterji#3
#1
M.E. (student), #2Assistant Professor, #3Professor & Head
Department of Electrical Engineering, National Institute of Technical Teachers’ Training and Research,
Chandigarh - 160019, India
Abstract— This paper presents the mitigation of total harmonic
distortions (THD) in cascade H-bridge multilevel inverter.
Selective harmonic elimination pulse width modulation (SHEPWM) switching method is used to calculate the values of
switching angles from the solution of non-linear transcendental
equations. These non-linear complex equations is minimized by
the using of particle swarm optimization (PSO) and genetic
algorithm (GA) techniques, and also have been compared the
simulating and analyzing results related to harmonics.
carrier-based PWM [21], Periodic Carrier
frequency modulation (PCFM), random carrier
frequency modulation
(RCFM) [22],
and
switching through FPGA [23]. Disadvantages of
high frequency switching are produces the
electromagnetic interferences, switching power
losses, The second category includes low frequency
switching which utilizes the fundamental
Keywords— Multilevel inverter, particle swarm optimization
component frequency and produce a staircase
(PSO), genetic algorithm (GA), SHE-PWM.
voltage waveform besides reasonable sinusoidal.
I. INTRODUCTION
In this paper, the investigators have performed
The multilevel inverters have received attention simulation for seven levels and eleven levels
in the area of high power rating and medium cascade H-bridge multilevel inverter, and used the
voltage applications. The multilevel inverter has selective harmonic elimination PWM switching
overcome the limitations of conventional two method
PWM
[24],
[25].
voltage level converters. The advantages of Fundamental frequency is utilized in selective
multilevel inverter are low electromagnetic harmonic
interference, higher power quality, lower switching
losses and higher voltage capability [1]. Mainly the
three types of multilevel inverter topologies used
are diode clamp multilevel inverter, flying capacitor
multilevel inverter and cascade multilevel inverter
with separate dc sources [ 2], [3]. Among these
converter topologies, the cascade H-bridge
multilevel inverters has received special attention
due to its circuit layout modularity, packaging and
simplicity of switching control to avoids bulky and
lossy resistor-capacitor-diode snubbers. Switching
techniques in multilevel inverter can be used in
two categories: high frequency switching and low
frequencies switching. Low frequency switching
includes space vector control (SVC) [4], [5], and
selective harmonic elimination pulse width
modulation (SHE-PWM) [6], [7]. High frequencies
Fig. 1 Single phase cascade multilevel inverter
switching includes the space vector PWM [8]-[10],
[11]-[12], phase shifted PWM (PSPWM) [13], [14],
In this paper, the investigators have performed
[15], level shifted PWM (LSPWM) [13], [16]-[19], simulation for seven levels and eleven levels
third harmonic injection PWM (TIPWM) [20],
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 9 - Jun 2014
cascade H-bridge multilevel inverter, and used the
selective harmonic elimination PWM switching
method
PWM
[24],
[25].
Fundamental frequency is utilized in selective
harmonic elimination PWM switching method for
controlling the gate signals of switching device.
Switching angles are solved from the non-linear
transcendental equations. For generating the
optimized staircase voltage waveform, the
optimized switching angles are obtained using by
the particle swarm optimization (PSO) and genetic
algorithm (GA) techniques.
II. PROBLEM FORMULATION
By using Fourier series expansion and from Fig.2,
the cascade multilevel staircase voltage waveform
is expended as below in equation Eq.1.
(
)
∑
(
)
( )
(
(
))
( )
is given by:
∑
(
(
(
)
(
)
(
)
))
( )
Where,
=
…..=
L is the number of dc sources for each full H-bridge
inverter cell,
N is the number of conducting angles,
n =1, 3, 5, … odd harmonics (2N-1).
L numbers of variables (switching angles)
have constrains and bounds with
⁄ . A set of L+2 harmonics
equations and including one fundamental voltages
equation for equal and constant source is obtainable
from Eq.3. In three-phase power system, triplen
harmonics are automatically cancelled from line-to
line voltage.
In SHE-PWM, the desired value is assigned for
fundamental component and all other except triplen
harmonics components are equated to zero.
is given by:
(
(
(
(
Fig. 2 Generated multilevel inverter staircase voltage waveform
∑ (
)
(
ISSN: 2231-5381
)
(
)
)
(
)
)
)
(
(
)
(
(
)
)
:
:
)
)
(
(
)
)
(
)
(4)
Where,
⁄(
⁄ ),
Modulation index,
,
L is the number of dc sources.
The main challenge is that to solve non-linear
transcendental Eq. 4. Newton-Raphson method [26]
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 9 - Jun 2014
have been used but this method have need to
require the best guess for variables, and takes more
time than artificial intelligence techniques .The
objective function is important for optimizing the
conducting angles. Maintaining the fundamental
components at pre specified value and mitigating
the selected order harmonics except triplen
harmonics. Minimizing of objective function with
pre defined constraints to obtain the optimized
conducting angles as given below:
(
)
(|
| |
|
|
|
|)
|
(5)
.Each particle refine its
search through its present velocity, previous
experience and the neighboring particles. The best
particle i is found so far is called personal best, best
particle vector denoted by vector
, and the best position in the
entire swarm is called global best and is denoted by
. The updated
velocities and positions are updated by using Eq. 7
and Eq. 8 as given below:
(
)
( )
( ))
(
(
( ))
(7)
(
)
( )
(
)
(8)
Where,
is the inertia weight
and
are the
cognitive and social parameters, respectively.
Random values and
are uniformly distributed
III. PSO TECHNIQUE
within range [0, 1]. Moreover, if Eq 9. and Eq. 10
PSO was first introduced in 1995 [27], as are satisfied, then the system will be guaranteed to
described by James Kennedy and Russell Eberhart, converge to a stable equilibrium point [28], [29].
Particle swarm algorithm imitates humans (or
(
)
(9)
insects) behaviour.
(
)⁄ )
(10)
⁄
(6)
A. Solution Using PSO
Fig.3 Example of particle movements
. Individuals interact with another while learning
from their experience, and gradually the population
members moves into better regions of the problem
space. The individuals in the population space are
called particles. Each particle in the entire swarm
has velocity and acceleration. The sociological
behaviour, which modelled by PSO techniques is
used to guide the swarm, and finding the best
solution in the search space. Each particle is
determined by the two vectors in D-dimensional
search space: the position vector
and velocity vector
ISSN: 2231-5381
In proposed PSO technique, the main challenge is
that solving the non-linear transcendental equation
and finding the best -optimized solution by using
selective harmonic elimination pulse width
modulation (SHE-PWM) switching method. A set
of L+2 harmonics equation is reduced to zero and
including the one fundamental voltages equation at
pre-specified value. Where, L is number of dc
sources. Hence the objective function is minimized
by applying the proposed PSO technique. The
objectives function and constrains and bounds are
given in Eq. 5 and Eq.6. The procedure for solving
the problem using PSO is given in flow chart shown
in Fig. 4.
B.GA Technique
Genetic algorithm is a directed search algorithms
[30]. It was developed by John Holland; University
of Michigan in 1970.The basic philosophy of the
genetic algorithm theory was inspired by Darwin's
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 9 - Jun 2014
theory of evolution which states that the survival of
an organism is affected by rule "the strongest
species that survives". Darwin also stated that the
the best solution. Selected individuals are called the
parents that contribute to the population at the next
generation.
3) Crossover: combination of two parents to form children
for the next generation. Crossover
is
controlled by
crossover rate.
4) Mutation: Random changes to individual parents to form
children. Mutation is controlled by mutation rate.
D. Solution Using GA
The procedure for solving the problem using GA is
given in flow chart of Fig.5.
Fig.4 Flow chart proposed PSO technique
survival of an organism can be maintained through
the process of reproduction, crossover and mutation.
Darwin's concept of evolution is then adapted to
computational algorithm to find solution to a
problem called objective function in natural fashion.
A solution generated by genetic algorithm is called
a chromosome.
Population: collection of chromosome.
Chromosome: It composed from genes, and its
value can be either, binary, numerical symbols or
characters.
C. Genetic Algorithm process
1) Initial population: is generated by many individual
solutions randomly.
2) Selection: Individual solutions are selected through a
fitness based process. Certain selection methods rate the
fitness function of each solution and preferentially select
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Fig. 5 Flow chart proposed GA technique
IV. SIMULATION RESULTS
All simulating results and work is done on
MATLAB 2012b package. Selective harmonic
elimination pulse width modulation (SHE-PWM)
switching method for controlling the cascade
multilevel
inverter,
and
the
non-linear
transcendental trigonometric Eq. 4 and objective
fitness function are solved and minimized by
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applying both of proposed PSO and GA techniques
respectively. The simulating results are discussed
for 7 levels and 11 level three-phase inverter with
separate constant dc sources (
=
…=
) .Each separate source has 12 volts ,
and two case ( for 7 and 11 level inverter) are
studied for modulation index range from 0.4 to 1.
TABLE I
VALUE OF MODULATION INDEX FOR BOTH 7 AND 11-LEVEL
INVERTER WHERE THE THD IS LOWEST
No. of Level
Modulation Index at where the THD is
Lowest.
GA
PSO Technique
Technique
7
0.4750
0.8500
11
0.700
0.6250
A.7 Level Inverter Results
1) Convergence Characteristics: Using proposed PSO and
GA techniques, the convergence characteristics of the fitness
function value is shown in Fig. 6 and Fig. 7. The result is
shown that the solution is converged in about 60 and 40
iterations respectively. The best score is approximately equal
to mean score in Fig.6. The best value and mean value are
given in table 2.
Fig.7 Convergence characteristics using GA technique
TABLE II.
BEST AND MEAN VALUES USING BY BOTH PSO AND GA
TECHNIQUES FOR BOTH 7 AND 11-LEVEL INVERTER
RESPECTIVELY
No.
of
Lev
el
PSO Technique
GA Technique
Modula
ting
Index
Best
Fitness
Value
Mean
Fitness
Value
Modu
lating
Index
Best
Fitnes
s
Value
Mean
Fitnes
s
Value
7
0.475
57.68
57.68
0.85
61.23
90.22
11
0.70
24.03
24.036
0.62
36.53
36.53
2) Modulation index versus switching angles: The values of
switching angles (degree) are displayed in Fig. 8 and 9 using
both PSO and GA techniques.
Fig.6 Convergence characteristics using PSO technique
Fig. 8 Modulation angles and switching angles using PSO technique
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Fig. 11 7-level phase-to-phase voltage (in volts) using GA technique
Fig.9 Modulation angles and switching angles using GA technique
3) Phase-to-phase voltages: The phase-to-phase voltages
using both PSO and GA techniques are displayed in Fig. 10
and Fig.11. Peak voltage is calculated by Eq. 11.
(
)
4) Line-to-voltages: Then line to line voltages are displayed in
Fig.12 and Fig.13, and peak voltage is calculated by Eq. 11.
Inner axis of Fig. 10 shows the 1 cycle portion for further
analyzing the produced harmonic components and also
calculates the total harmonic distortion (THD).
(11)
Fig. 12 7-level line-to-voltages (in volts) using PSO technique
Fig.10 7-level phase-to-phase voltage (in volts) using PSO technique
Fig.13 7-level line-to-line voltage (in volts) using GA technique
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5) Modulating index versus objective value: The
characteristics of modulation index and objective values are
displayed in Fig. 14.
7) Modulating index versus THD: Fundamental components
and harmonics profile are displayed in Fig.16.
Fig. 16 Frequency versus Magnitude (in Percent)
Fig. 14 Modulating index versus objective value.
6) Modulating index versus THD: The characteristics of
modulation index and THD (%) are displayed in and Fig.15.
B.11-Level Inverter Results
1) Convergence characteristics: Using proposed PSO and
GA techniques, the convergence characteristics of the fitness
function value is shown in Fig. 17 and Fig. 18. This result is
shown that the solution is converged about 60 and 40
iterations respectively. The best score is approximately equal
to mean score in Fig. 17. The best value and mean value are
given in table 1.
Fig.15 Modulating index versus thd .
Fig. 17 Convergence characteristics using PSO technique
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3) Phase-to-phase voltages: The phase-to-phase voltages
using both PSO and GA techniques are displayed in Fig. 21
and Fig. 22. Peak voltage is calculated by Eq. 11.
Fig.18 Convergence characteristics using GA technique
2) Modulation index versus switching angles: The values of
switching angles (degree) are displayed in Fig. 19 and 20
using both PSO and GA techniques.
Fig.21 11-level phase-to-phase voltage (in volts) using PSO technique
Fig.19 Modulation angles and switching angles using PSO technique
Fig.22 11-level phase-to-phase voltage (in volts) using GA technique
Fig. 20 Modulation angles and switching angles using GA technique
ISSN: 2231-5381
4) Line-to-voltages: Then line to line voltages are displayed in
Fig.23 and Fig.24, and peak voltage is calculated by Eq. 11.
Inner axis of Fig. 23 shows that 1 cycle portion for further
analyzing the produced harmonic components and also for
calculating the total harmonic distortion (THD).
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Fig.23 11-level line-to-line voltages (in volts) using PSO technique
Fig. 25 Modulating index versus objective value
6) Modulating index versus THD: Fundamental components
and harmonics profile are displayed in Fig.26.
Fig.24 11-level line-to-line voltages (in volts) using GA technique
5) Modulating index versus objective value: The
characteristics of modulation index and objective values are
displayed in Fig. 25.
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Fig. 26 Modulating index versus THD
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TABLE III
HARMONICS PROFILE FOR 7 AND 11-LEVEL INVERTER USING BOTH PSO AND GA TECHNIQUES
AI
Techniqu
e
Magnitude of harmonic(%) up to 19 th order for 7Level Inverter (modulating Index, ma= 0.475)
Magnitude of harmonic (%)up to 19 th order for 11Level Inverter (modulating Index, ma= 0.7)
Harmonics Profile
Harmonics Profile
Harmoni
c orders
Even
Harmonic
Magnitude
Harmoni
c orders
Odd
Harmonic
Magnitude
Harmoni
c orders
Even
Harmonic
Magnitude
Harmoni
c orders
Odd
Harmonic
Magnitude
0th
dc
Component
(2.0924)
1st
Fundamental
component
(100)
0th
dc
Component
(2.0981)
1st
Fundamental
component
(100)
2nd
1.17
3rd
0.48
2nd
1.73
3rd
0.59
th
0.33
5
th
0.21
4
th
0.22
5
th
1.36
6th
0.20
7th
0.79
6th
0.34
7th
0.18
0.06
th
0.15
th
0.16
th
0.07
4
8
GA
th
10
th
12
th
14
th
16
th
18
th
9
11
th
13
th
15
th
17
th
0.37
19
th
0th
dc
Component
(2.0604)
2nd
4th
6
th
8
th
10
th
12
th
14
th
16
th
18
th
8
9
10
th
12
th
14
th
16
th
1.32
18
th
1st
Fundamental
component
(100)
0th
dc
Component
(2.1124)
1st
Fundamental
component
(100)
1.16
3rd
0.48
2nd
1.99
3rd
0.81
0.25
5th
0.12
4th
0.43
5th
0.10
0.31
7
th
0.96
6
th
0.70
7
th
2.83
9
th
8
th
9
th
0.04
0.26
0.49
0.15
0.18
0.03
0.25
0.52
0.19
0.17
0.27
11
th
13
th
15
th
17
th
19
th
1.43
0.23
0.17
0.23
0.14
1.35
0.23
0.15
0.26
1.25
7) Frequency versus Magnitude: Fundamental components
and harmonics profile are displayed in Fig.27.
10
th
12
th
14
th
16
th
18
th
11
th
0.94
13
th
0.24
15
th
0.20
17
th
0.37
0.62
19
th
2.20
0.08
0.43
0.22
0.28
0.36
0.12
0.33
0.14
0.09
0.30
11
th
1.04
13
th
0.24
15
th
0.10
17
th
0.03
19
th
0.87
V. COMPARISION BETWEEN 7 AND 11-LEVEL
INVERTER REGARDING HARMONIC PROFILE
A. Comparison of both 7 and 11-level cascade multilevel
inverters using proposed PSO technique
The Fig. 28, is displayed the 7-level and 11-level
inverters harmonic profile for line-to-line voltage
with fundamental magnitude (in percent) , and inner
axis bar figure displays clearly the harmonic profile
up 19th order harmonics and mitigated the total
harmonic distance (THD) down to 6.6273% and
4.9210 %
for 7-level and 11-level inverter
Fig.27 Frequency versus magnitude (in percent)
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respectively. The comparative results of all
harmonic contents have been tabulated in table III.
The conclusion from observing the results and
table III shows that, percentage of 3rd , 5th, 13th,
17th, and 19th order harmonics are well reduced in
7-level inverter in comparison to 11-level inverter,
while 7th , 9th, 11th, 15th order harmonics are well
reduced in 11-level converter in comparison to 7level inverter.
Fig.29 Comparison of harmonic contents using GA technique.
VI.COMPARISION BETWEEN PROPOSED PSO AND GA
TECHNIQUE REGARDING COMPUTATIONAL TIME
The computational time for each modulating
index for solving and optimizing the objective
function have been displayed in Fig. 30 and Fig. 31
using PSO and GA techniques for both 7-level and
11-level inverter respectively .The computational
Fig.28 Comparison harmonic contents using PSO technique
results shows that the speed performance of PSO is
3.207 times faster than GA techniques for 7-level
B.Comparison of both 7 and 11-level cascade multilevel Inverter and 3.024 times faster than GA techniques
inverter using proposed GA technique
for 11-level Inverter.
The Fig. 29, is displayed the 7-level and 11-level
inverters harmonic profile for line-to-line voltage
with fundamental magnitude (in percent), and inner
axis bar figure displays clearly the harmonic profile
up 19th order harmonics and mitigated the total
harmonic distance (THD) down to 6.6082% and
4.9081 % for 7-level and 11-level inverter
respectively. The comparative results of all
harmonic contents have been tabulated in table III.
The conclusion of observing the results and
table III is show that, percentage of 3rd , 7th, and 13th
order are well reduced in 7-level inverter in
contrast to 11-level inverter, while 5th , 9th, 11th, 15th
, 17th ,and 19th order harmonics are well reduced by
11-level converter in contrast to 7-level inverter.
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Fig.30 Computational time for 7 level inverter
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TABLE IV
COMPARISON OF CALCULATED THD BY REAL CODE, OBTAINED THD FROM MATLAB FFT ANALYSIS TOOL, AND
COMPUTATIONAL TIME
Obtained THD by MATLB FFT
Tool , Calculated THD by Real
Time Code, and Computational
Time
Switching Angles (degree)
Modulating
Index
By FFT
Tool
By Real
Code
Computati
onal Time
(secs)
0.475 (PSO 7-level )
4.17
17.40
34.13
-
-
6.62
6.52
0.2520
0.85 (GA 7 level)
4.45
17.64
34.08
-
-
6.60
6.55
0.8082
0.700 (PSO 11-level
5.42
10.48
21.32
28.88
45.04
4.92
4.82
0.2491
0.6250 (GA 11level)
2.79
11.07
18.04
26.02
36.17
4.90
4.83
0.7534
REFERENCES
[1]
Fig. 31 Computational time for 11-level inverter
VI. CONCLUSION
The PSO and GA techniques for harmonics
elimination have been compared for 7 and 11-level
constant dc source cascade H-bridge inverter
respectively. Optimized angles have been obtained
by solving the SHE problem .The total harmonic
distortion (THD) for line-to-line voltages have been
reduced more by using GA techniques rather than
PSO for both 7 and 11-level inverter respectively,
but PSO is converged much faster than GA. At
some modulation index, the value of harmonics
generated is more in GA rather than using PSO.
ISSN: 2231-5381
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