Electrical Power and Energy Systems 64 (2015) 699–707
Contents lists available at ScienceDirect
Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
Optimized switching scheme of cascaded H-bridge multilevel inverter
using PSO
Vivek Kumar Gupta, R. Mahanty ⇑
Department of Electrical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India
a r t i c l e
i n f o
Article history:
Received 17 October 2013
Received in revised form 14 July 2014
Accepted 23 July 2014
Available online 23 August 2014
Keywords:
Particle swarm optimization (PSO)
Cascaded H-bridge multilevel inverter
Harmonics
Sinusoidal pulse width modulation (SPWM)
a b s t r a c t
The paper uses particle swarm optimization (PSO) to determine the optimum switching angles of
cascaded H-bridge multilevel inverter (CH-MLI) so as to produce the required fundamental voltage and
reduce the harmonic content. This is done by solving the transcendental equations characterizing the
harmonic content. The validity of the proposed method is verified through simulation studies for
three-phase, five-level CH-MLI. To compare the results obtained using PSO, the simulation studies have
been extended for three-phase, five-level CH-MLI using sinusoidal pulse width modulation (SPWM). The
results obtained using PSO are found superior as compared to SPWM in terms of total harmonic distortion
at different modulation indices.
Ó 2014 Elsevier Ltd. All rights reserved.
Introduction
Multilevel inverters (MLIs) produce a desired output voltage
from several levels of DC voltages as inputs. By taking sufficient
number of DC sources, a nearly sinusoidal voltage waveform can
be obtained. MLIs have been receiving increasing attention in
recent years for high voltage and high power applications [1–6].
To control the output voltage and reduce undesired harmonics,
sinusoidal pulse width modulation (SPWM) and space vector modulation techniques have been conventionally used in MLIs [7,8].
Methods such as selective harmonic elimination (SHE) or programmed pulse width modulation (PWM) techniques have also
been used extensively, wherein specific higher order harmonics
such as 5th, 7th, 11th and 13th are suppressed in the output voltage of the inverter [9–13]. The major intricacy associated with such
methods is to solve the nonlinear transcendental equations characterizing the harmonics, which can be solved by iterative techniques such as Newton–Raphson method. However, this method
is not suitable in cases involving a large number of switching
angles if good initial guess is not available. Another approach based
on mathematical theory of resultant, wherein transcendental
equations that describe the SHE problem are converted into an
equivalent set of polynomial equations and then mathematical
theory of resultant is utilized to find all possible sets of solutions
for the equivalent problem has also been reported [14]. However,
as the number of harmonics to be eliminated increases (up to five
⇑ Corresponding author. Tel.: +91 542 2575388.
E-mail address:
[email protected] (R. Mahanty).
http://dx.doi.org/10.1016/j.ijepes.2014.07.072
0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
harmonics), the degrees of the polynomials in the equations
become so large that solving them becomes very difficult. Recently,
this problem has been solved by stochastic optimization methods
based on genetic algorithm (GA) approach in a simpler manner
[15]. The GA has been successfully applied to MLIs to find all possible set of solutions for switching angles. However, the quality of
solution deteriorates using GA as the level of inverter increases in
MLI. Moreover, the convergence speed of the GA algorithm is low
and each step to find the switching angles is a time consuming process. In this paper particle swarm optimization (PSO) [16–20] has
been applied to solve the SHE problem of cascaded H-bridge MLI
(CH-MLI).
Cascade H-bridge MLI
The basic circuit of a single-phase m-level CH-MLI is shown in
Fig. 1 [4,5]. It consists of (m 1)/2 cells connected in series in each
phase. Each cell consists of single-phase H-bridge inverter with
separate DC source. There are four active devices in each cell and
a cell can produce three voltage levels 0, Vdc/2 and Vdc/2. When
switches S1 and S2 of one H-bridge inverter are closed, the output
voltage is Vdc/2 and when switches S3 and S4 are closed, the output voltage is +Vdc/2. When either the switches S1 and S3 or the
switches S4 and S2 are closed, the output voltage is 0. Higher
output voltage levels can be obtained by connecting these cells
in cascade. The phase voltage Van in a CH-MLI is the sum of voltages
of individual cells.
V an ¼ V 1 þ V 2 þ V 3 þ þ V m
ð1Þ
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V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699–707
S1
Particle swarm optimization
S3
PSO is an intelligent algorithm which relies on exchanging
information through social interaction among particles [16–20].
The PSO conducts search using swarm of particles randomly generated initially. Each particle i (i = 1 to swarm size) possesses a current position xi = [xi1 xi2 . . . xid] and a velocity ui = [ui1 ui2 . . . uid],
where d is the dimension of search space. The position of the particle represents a possible solution of the problem. The velocity
indicates the change in the position from one step to the next. Each
particle memorizes its personal best position (pbest) which corresponds to the best fitness value in the searched space. Each particle
can also access the global best position (gbest) that is the overall
best place found by one member of the swarm. The particles profit
from their own experiences and previous experience of other particles during the exploration, to adjust their velocity, in direction
and amount. The concept of a moving particle is illustrated in
Fig. 3. The basic concept behind the PSO technique is to change
the velocity of each particle towards its pbest and gbest positions
at each time step. This means that each particle tries to modify
its current position and velocity according to the distance between
its current position and pbest and the distance between its current
position and gbest. In its canonical form, PSO is modeled as follows:
Let x and u denote the coordinates (position) and flight speed
(velocity) of a particle respectively in a search space. The position
of the ith particle is represented as
+
Vdc1
-
S4
S2
S1
S3
S4
S2
S1
S3
S4
S2
+
Vdc2
-
+
Vdcn
-
xi ¼ ½xi1 xi2 . . . xid in the d-dimensional space:
The best previous position of the ith particle is recorded and represented as
pbesti ¼ ½pbesti1 pbest i2 . . . pbestid
Fig. 1. Single-phase m-level CH-MLI.
The index of the best particle among all the particles in the group is
represented as
where V1, V2, V3, . . . , Vm are the output voltages of each cell.
The output voltage of a CH-MLI is shown in Fig. 2. This can be
represented as
VðtÞ ¼
1
X
ðan sin nan þ bn cos nan Þ
ð2Þ
n¼1
m
4V dc X
cos nak
np k¼1
and all switching angles must satisfy the condition
p
0ha1 ha2 ham h :
2
ð3Þ
The even harmonics are zero due to quarter wave symmetry of the
output voltage (bn = 0).
Fig. 2. Output voltage waveform of CH-MLI.
The velocity of the ith particle is represented as
ui ¼ ½ui1 ui2 . . . uid
The modified velocity and position of each particle can be calculated using the current velocity and the distance from pbestid and
gbestd as given in (4) and (5).
where
an ¼
Gbest ¼ ½gbest 1 gbest2 . . . gbest d
kþ1
u1d
¼ ukid þ c1 r 1 ðpbest id xkid Þ þ c2 r 2 ðgbestd xkid Þ
ð4Þ
k
kþ1
xkþ1
1d ¼ xid þ uid
ð5Þ
i ¼ 1; 2; 3; . . . m
where m is the number of particles in group, k is the number of iterations (generation), d is the number of dimensions corresponds to
number of members of each particle, ukþ1
is the velocity of member
id
d of particle i at iteration k + 1, ukid is the velocity of member d of par-
Fig. 3. Concept of modification of searching points.
V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699–707
701
Fig. 4. (a) Simulink model of three-phase five-level CH-MLI, (b) sub-circuit of three-phase five-level CH-MLI using PSO and (c) sub-circuit of R–L load.
ticle i at iteration k, umin
6 uid 6 umax
, xkþ1
is the position of member
d
d
id
d of particle i at iteration k + 1, xkid is the position of member d of
particle i at iteration k, c1 is the constant weighing factor corre-
sponding to pbest, c2 is the constant weighing factor corresponding
to gbest, r1 and r2 are the random numbers between 0 and 1, pbestid
is the local best position of member d of particle i.
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V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699–707
Fig. 5. Sub-circuit of pulse generation block using PSO (for phase A).
umax determines the fitness of the regions to be searched
between the present position and target position. If umax is too
high, particles may fly past good solutions. If umax is too small, particles may not explore sufficiently beyond local solutions. In many
experiences with PSO, it is often set at 10–20% of the dynamic
range of the variable in each dimension. The constants c1 and c2
represent the weighing of the stochastic acceleration terms that
pull each particle towards the pbest and gbest position. Low values
of pbest and gbest allow particles to roam far from the target
regions before being tugged back. On the other hand, high values
Fig. 6. Switching pulses of three-phase CH-MLI (for phase A).
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V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699–707
(V)
(V)
(V)
(s)
Fig. 7. Line voltages three-phase CH-MLI using PSO at modulation index 0.85: (a) Vab, (b) Vbc and (c) Vca.
Fig. 8. Harmonic spectrum of line voltage Vab using PSO at modulation index 0.85.
of pbest and gbest result in abrupt movement towards target
regions. The acceleration constants c1 and c2 are often set to 1.8
according to past experiences. The velocity and position update
are described by (4) and (5). A new velocity for each particle is
based on the particle’s previous velocity, the particle’s location at
which the best fitness has been achieved so far and the population
global location at which the best fitness has been achieved so far.
The constants c1 and c2 provide the correct balance between exploration and exploitation (individuality and sociality). Acceleration is
weighted by a random term, with separate random numbers being
generated for acceleration toward pbest and gbest locations. The
random numbers provide stochastic for the particles velocities in
order to simulate the real behavior of the birds in a flock. Fig. 3
shows the concept of modification of searching points described
by (4) and (5).
An inertia weight parameter w is introduced in order to
improve the performance of the original PSO model. This parameter plays the role of balancing the global search and local search
capability of PSO. A better chance of finding the global optimum
within reasonable number of iterations can be achieved by incorporating this parameter into the velocity update given in (4), as
follows:
ukþ1
¼ wukid þ c1 r1 ðpbestid xkid Þ þ c2 r 2 ðgbestd xkid Þ
id
ð6Þ
Suitable selection of inertia weight provides a balance between global and local exploration abilities and thus require less iterations to
find the optimal solution. The inertia weight often decreases linearly from about 0.9 to 0.4 during a run. In general, the inertia weight
is set according to the following equation:
w ¼ wmax
wmax wmin
iter
iter max
ð7Þ
where wmax and wmin are the maximum and minimum values of
inertia weight, and iter and itermax are current and maximum values
of iteration.
Objective function formulation
The objective function
has been chosen to get the optimized
so that the relative fundaswitching angles ai i ¼ 1; 2; . . . ; ðn1Þ
2
mental component V 01 is equal to the desired voltage and the lower
order harmonics are equal to zero, where n is the number of levels
of the inverter.
The harmonic elimination problem is converted into optimization problem and is rewritten as
ðn 1Þ
Fitness pbest ai ; i ¼ 1; 2; . . . ;
2
0 ðn 1Þ
¼ w1 V 1
M
2
þ
13
X
k¼2mþ1
w2mþ1 V 02mþ1
ð8Þ
704
V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699–707
Fig. 9. Sub-circuit of three-phase five-level CH-MLI using SPWM.
Fig. 10. Sub-circuit of pulse generation block using SPWM.
V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699–707
705
(V)
(V)
(V)
(s)
Fig. 11. Line voltages of three-phase CH-MLI using SPWM at modulation index 0.85: (a) Vab, (b) Vbc and (c) Vca.
where m = 1, 2, . . . , 6, M is the modulation index and
wi i ¼ 1; 2; . . . ; ðn1Þ
are positive weights which give more impor2
tance to fundamental component than harmonic elimination.
Using the objective function (8), the PSO is used to find the opti
.
mal ai i ¼ 1; 2; . . . ; ðn1Þ
2
PSO algorithm for cascaded multilevel inverter
The proposed PSO algorithm for CH-MLI is as follows:
Step 1: Initialization
For each particle:
h
i
– Initialize the position ai ð0Þ ¼ ai1 ð0Þai2 ð0Þ aiðn1Þ ð0Þ of each
2
particle with random angles that respect the constraints such
that 0ha1 ha2 h haðn1Þ hp2 .
2
h
i
ð0Þ
– Initialize the velocity v ai ð0Þ ¼ v ai 1ð0Þv ai 2ð0Þ v ai ðn1Þ
2
of each particle to random values;
– Initialize the best fitness local_best_fitness of particle i.
End for
– Initialization of the best fitness global_best_fitness of the
swarm.
Loop
{
For each particle
Step 2: Objective function evaluation
– Compute the Fitness_pbesti value of each particle i of the
swarm using (8)
Step 3: Personal best position updating
If
{
Fitness_pbesti < local_best_fitnessi
Then local_best_fitnessi = Fitness_pbesti and alocal_best = ai
End
Step 4: Global best position updating
If
{
Fitness_pbesti < global_best_fitness
Then global_best_fitnessi = Fitness_pbesti and aglobal_best = ai
}
End
Step 5: Position and velocity updating
vaii = wvai + c1r1(alocal_best ai) + c2r2(aglobal_best ai)
ai = ai + vai
}
End
Until a sufficiently good fitness value is reached.
Simulation studies
A three-phase, five-level CH-MLI with R–L load is shown in
Fig. 4(a). The CH-MLI consists of eight switching devices with
two separate DC sources per phase. The supply voltage is 11 kV,
50 Hz. The load inductance is 50 mH and the load resistance is
50 X. The model has two blocks; one block representing the subcircuit of three-phase, five-level CH-MLI using PSO is shown in
Fig. 4(b) and the other block representing the sub-circuit of R–L
load is shown in Fig. 4(c).
Three-phase CH-MLI using PSO
A sub-circuit block designed to generate switching pulses for
three-phase, five-level CH-MLI using PSO (for phase A) is shown
in Fig. 5. The switching pulses are generated for the switches of
three-phase, five-level CH-MLI on the basis of a program written
in MATLAB. Modulation index has been introduced in the program
based on PSO algorithm that generates the optimum switching
angles of CH-MLI. The intention of introducing modulation index
is to control the output voltage of CH-MLI. The switching pulses
generated using Fig. 5 (for phase A) is shown in Fig. 6. Switching
pulse with magnitude ‘1’ indicates that the switch is ‘on’ and the
magnitude ‘0’ indicates that the switch is ‘off’. Similarly switching
pulses for other phases are also generated. These switching pulses
are used for turning ‘on’ and ‘off’ the switches of CH-MLI. Fig. 7
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V.K. Gupta, R. Mahanty / Electrical Power and Energy Systems 64 (2015) 699–707
Fig. 12. Harmonic spectrum of line voltage Vab using SPWM at modulation index 0.85.
Table 1
THDs (%) for three-phase five-level CH-MLI using PSO and SPWM.
Modulation index
THD (%) using PSO
THD (%) using SPWM
0.85
0.875
0.9
0.925
0.95
0.975
1
1.025
1.05
1.075
1.1
1.125
1.15
1.175
1.2
7.21
6.34
6.17
5.91
5.68
5.13
4.84
4.69
4.18
4.34
4.04
3.76
3.48
3.16
2.81
13.21
13.04
12.76
12.48
12.16
11.84
11.41
10.94
10.69
10.38
9.61
9.15
8.64
8.27
7.56
sub-circuit of three-phase, five-level CH-MLI using SPWM. This
consists of six modules of H-bridges. The sub-circuit of pulse generation block using SPWM is shown in Fig. 10. The line voltages Vab,
Vbc and Vca of three-phase, five-level cascaded CH-MLI are shown in
Fig. 11 and the harmonic spectrum of one of the line voltages Vab
using SPWM at modulation index 0.85 is shown in Fig. 12. The
THD of Vab at this modulation index is 13.21%.
Comparison between three-phase CH-MLI using PSO and SPWM
Table 1 gives the comparison of THDs using PSO and SPWM at
different values of modulation indices. Fig. 13 shows the graphical
representation of THD versus modulation index for the threephase, five-level CH-MLI using PSO and SPWM. It is observed that
as the modulation index increases, THD of output voltage of threephase CH-MLI decreases for both PSO and SPWM, however, the
THD of output voltage using PSO is less as compared to SPWM at
all values of modulation indices. Hence the quality of the output
voltage is better in PSO as compared to SPWM.
Conclusion
Fig. 13. THD versus modulation index for three-phase five-level CH-MLI using PSO
and SPWM.
shows the line voltages Vab, Vbc and Vca of three-phase, five-level
cascaded CH-MLI and Fig. 8 shows the harmonic spectrum of one
of the line voltages Vab using PSO at modulation index 0.85. It is
evident from the harmonic spectrum that the lower order harmonics reduce significantly. The total harmonic distortion (THD) of Vab
at this modulation index is found to be 7.21%. The THD further
reduces at increased values of modulation indices.
In this paper, PSO has been used to eliminate some selected
lower order harmonics of CH-MLI. In PSO, there is only one fitness
value which move towards the global optimal point in each iteration. This makes the PSO method computationally faster. The convergence of the PSO is better than the other evolutionary methods.
The PSO converges to the global or near global point with respect
to the last function. The voltage harmonics minimization of fivelevel CH-MLI inverter using PSO includes velocity equations, equality and inequality constraints and creation of initial position. The
application of velocity calculation in PSO is a powerful strategy
to improve the global searching ability. Simulations have been carried out for three-phase five-level CH-MLI using PSO and SPWM
and their FFT analysis has been carried out for different modulation
indices. The simulation results show that the THD is less in the output voltage using PSO as compared to SPWM.
Hence PSO can be used for improving the harmonic elimination
problem of MLIs. The proposed method exhibits advantages in
terms of switching frequency and high output voltage quality.
The present study shows that PSO is suitable for MLIs optimal
design. However, practical implementation of the proposed
scheme requires further study specially under varying load
conditions.
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To compare the results of three-phase, five-level CH-MLI using
PSO, the three-phase, five-level CH-MLI has been simulated using
one of the widely used PWM techniques, SPWM. Fig. 9 shows the
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