International Journal on Electrical Engineering and Informatics - Volume 8, Number 2, June 2016
An Efficient Method for THD Minimization in Multilevel Inverter’s
Rahim Shamsi Varzeghan, Mohammad Reza Jannati Oskuee, Alireza Eyvazizadeh
Khosroshahi, and Sajad Najafi-Ravadanegh
Smart Distribution Grid Research Lab.
Electrical Engineering Department of Azarbaijan Shahid Madani University, Tabriz, Iran
s.najafi@azaruniv.edu
Abstract: In this paper, minimization of total harmonic distortion (THD) is discussed for the
output voltage of multilevel inverter. THD minimization is an efficient method in reduction of
the harmonic components of the inverter’s output voltage. In multilevel inverters, the switching
angles can be selected in a way that the output voltage THD will be minimized. Given that
THD minimization is an optimization problem, intelligent algorithm is found to be an
appropriate alternative in this regard. Shuffled Frog Leaping Algorithm and Harmony Search
Algorithm are employed to find optimum switching angles to generate desired voltage value in
the possible minimum THD. The obtained results of two algorithms are compared with each
other to determine that which algorithm is more efficient in this regard. Also, both simulation
and experimental results indicate superiority of this approach over the published work using
GA in this concept. The experiments are conducted on a seven-level inverter to validate the
feasibility of presented approach.
Keywords: Shuffled Frog Leaping Algorithm (SFLA), Harmony Search Algorithm (HS),
Multilevel Inverter, Minimization of Total Harmonic Distortion, Phase Voltage
1. Introduction
In the recent years, power electronics engineers have centralized their attention to
multilevel inverters. Compared to conventional two level inverters, stepwise output voltage is
the basic advantage of multilevel inverters. This advantage will have some results such as
better Electromagnetic Capability (EMC), lower switching losses, higher power quality,
reduction of dv/dt stresses, lower total harmonic distortion (THD), needlessness of a
transformer at distribution voltage and lower rating on power semiconductor switches [1-5].
Due to the advantages mentioned above for multilevel inverters, they are employed in many
applications such as: distributed generation [6], micro grids [7-8], FACTs devices [9], High
Voltage Direct Current (HVDC) [10], and electrical vehicles [11-12]. Multilevel inverters are
mainly classified into three configurations which are the flying capacitor [13], diode clamped
[14-15] and cascaded multilevel inverters [16]. Multilevel inverters are divided into two
categories from the aspect of DC source value, which are called symmetric and asymmetric
topologies. For DC voltage sources, capacitors, batteries and renewable energy sources can be
implemented. In symmetric topology all DC sources have the same value but in asymmetric
topology they have different values. Asymmetric multilevel inverters generate higher number
of voltage steps for a definite number of switches compared to symmetric ones. The stepwise
output voltage of multilevel inverters composed by a number of DC voltage sources [17]. An
increase in the number of steps, will lead to generation of a near sinusoidal output voltage of
multilevel inverter. This results in a considerable reduction in output voltage THD.
Nevertheless, problems such as voltage unbalance, circuit layout and voltage clamping, limit
the possible number of levels. Consequently, reducing the THD of output voltage waveform is
a vital issue in designing useful and efficient multilevel inverters. Hence, improving the output
waveform quality and minimizing THD has become the subject of many recent papers. The
performance of multilevel converters depends on its switching strategy. To qualify the output
voltage waveform of multilevel inverters, different modulation strategies have been proposed.
th
st
Received: July 10 , 2015. Accepted: June 11 , 2016
DOI: 10.15676/ijeei.2016.8.2.4
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Rahim Shamsi Varzeghan, et al.
Some of popular switching strategies are space vector modulation, minimization of the total
harmonic distortion, pulse width modulation (PWM), selective harmonic elimination (SHE),
and sinusoidal pulse width modulation (SPWM) [18–23]. Also in [24] a technique has been
proposed based on optimal minimization of THD (OMTHD). Selective harmonic elimination
approach minimizes or eliminates some low-order harmonics and achieves to the fundamental
component. But in THD minimization approach all of the harmonic components are considered
together to be minimized while having the fundamental component. Commonly an
optimization algorithm is employed to search for the best switching angles by which the
minimum possible THD will be achieved. In this paper THD minimization is applied to the
phase voltage. Shuffled Frog Leaping Algorithm (SFLA) and Harmony Search (HS) Algorithm
which are two new approaches of optimization algorithms are developed and employed to
solve the problem. Then the simulation and experimental results are compared together and
with other studies. Provided comparison validate that the obtained results are better in this
work. The rest of the paper is organized as follows. Section II gives a brief description to the
staircase output voltage of a seven-level inverter. Section III describes THD minimization in
more details. In section IV and section V, SFLA and HS Algorithms are described, respectively.
In section VI and VII simulation and experimental result are given and compared, respectively.
Finally section VIII is devoted to conclusion.
2. Output voltage of multilevel inverter
Figure 1 shows a typical stepped waveform of the phase voltage of a seven-level inverter
with equal DC sources.
Vref
3Vdc
2Vdc
Vdc
-Vdc
α1 α2 α3
π
π/2
2π
ωt
-2Vdc
-3Vdc
Figure 1. Output Phase voltage of 7-level inverter
Assuming the waveform, only three angles α1, α2, and α3 are required to determine the
mentioned waveform. Regarding to switching angles, such a waveform can be demonstrated in
terms of step function ( t). For positive half cycle of the waveform, we have:
i 3
i 3
i 1
i 1
Vap  Vdc  u  t   i    Vdc  u  t  π   4i  
0  t  
(1)
For negative half cycle in which
is lagging in
  t  2
radians we have:
(2)
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An Efficient Method for THD Minimization in Multilevel Inverter’s
Considering (1) and (2) the whole waveform can be obtained from following:
0  t  
  t  2
Vap

Va  

Van
(3)
Applying Fourier analysis to
, such a waveform yields the following expression for
harmonic components of the phase voltage and so the rms value of fundamental calculation
(for odd n):

Va  
n 1
2 2
n
Vdc sin(
)(cos  na1   cos(na2 )  cos(na3 ))Sin(n t )
n
2
(4)
This is an odd function and it only contains odd harmonics, i.e. even harmonics are
eliminated. From the following equation, it is concluded that the maximum possible value of
the fundamental component is obtained when α1, α2, and α3 are all equal to zero:
Va1 (max) 
pu
Va1
1
 (cos(a1 )  cos(a2 )  cos(a3 ))
Va1 (max) 3
(5)
The phase voltage rms value can be easily calculated by considering the waveform in figure 1:
2
1
2

Vrms 
Va2 dt  Vdc ((a2  a1 )  4(a3  a2 )  9(  a3 ))

2 0

2
(6)
The ratio of the sum of the RMS value of power of all harmonic components to the RMS
value of power of the fundamental frequency component is defined as THD. So the following
equation gives the THD value.

THD phase 
V
n2
Va1
2
an
2

 Van 

 1
 Va1 
(7)
Finally, for the phase-voltage THD the following analytical expression is obtained:
THDphase  (

 [a2  a1 ]  4[a3  a2 ]  9[ 2  a3 ]
4 (cos(a1 )  cos(a2 )  cos(a3 )) 2
) 1
(8)
3. THD Minimization
THD Minimization technique aims to determine the optimum switching angles while
generating the fundamental component with the possible minimum THD value. This is a
problem that an optimization algorithm is used to solve it. In this paper SFLA and HS
algorithms are used which are general purpose stochastic global search algorithms. These
algorithms do not need functional derivative information to search for the solutions to solve the
problem.
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Rahim Shamsi Varzeghan, et al.
The objective function is defined as follows to solve the problem:
Objective :
10 | V1*  V1 | THDPhase
(9)
V1* is the required fundamental component in per unit and substitute from equation (5). It
*
varies from zero to one. The first term of the objective function, | V1
 V1 | , is the absolute
value of error, which is required for adjusting the fundamental harmonic. A weighting factor of
ten has been applied to terms of error to increase the importance of fundamental component.
The value of errors factor can be eristic and is tentative. The fundamental component reaches a
nearly to the desired as well as the possible minimum THD in the output with the presented
weighting factor. The following basic constraint must be satisfied with switching angles which
are the solutions of aforementioned objective function:
0  a1  a2  a3 

(10)
2
The obtained results from both SFLA and HS are compared together to determine which
one is more efficient to solve THD minimization problem. And also, a comparison is provided
with the published paper which uses GA in this regard. The comparison will finally confirm
that the attained results in this work cause lower phase voltage THD value.
4. Shuffled Frog Leaping Algorithm (SFLA)
In the SFLA, there are several frogs with the same structure but different adaptabilities and
each of them represents a possible solution to an optimization problem [25]. The population of
frogs is shared into some frog memeplexes according to particular rules and each memeplex
represents a type of meme. Frogs in the memeplex conduct local survey of solution space
according to particular strategies which permit the meme exchange between local individuals.
The initial population is made of F frogs generated randomly and the fitness value f (i) for the
ith frog can be evaluated and sorted in descending order to form memeplexes. The entire
population of F frogs is divided into m memeplexes and each memeplex consists of n frogs. To
avoid local optima, a sub memeplex is carried out in each memeplex, which contains frogs
chosen on the basis of their respective fitness value. If the fitness is better, it is chosen easier.
The worst solutions are updated and replaced with a better one. The calculations carry on for a
definite number of iterations. Then, for global information exchange all frogs are shuffled. The
local survey and the shuffling procedure continue until a pre-defined convergence criterion is
met.
Figure 2 shows the flowchart of SFLA.
5. Harmony Search Algorithm
Harmony search is a recently developed music-based metaheuristic optimization algorithm
[26]. It was roused by the observation that the goal of music is to search for a impeccable state
of harmony. Specifically, the process by which the musicians (who have never played together
before) quickly refine their individual improvisation through adjusting the pitches of their
instruments resulting in a pleasing harmony. This harmony in music is similar to find the
optimality in an optimization process. Simple concept, few parameters to adjust, and easy
implementation make HS as the major competitor of other evolutionary algorithms. In music
improvisation, each player sounds any pitch within the possible range, together making one
harmony vector. If all the pitches make a good harmony, that experience is stored in each
player’s memory, and the possibility to make a good harmony is increased next time. Figure 3
represents the optimization procedure of the HS algorithm.
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An Efficient Method for THD Minimization in Multilevel Inverter’s
Figure 2. SFLA flowchart
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Rahim Shamsi Varzeghan, et al.
Figure 3. HS flowchart
6. Simulation Results
To confirm the attained results, a simulation is carried out for a 7-level inverter with equal
DC sources according to the waveform shown in figure 1. SFLA and HSA are applied to
minimize the Eq. (9), for finding the optimal switching angles. The optimum switching angles
(degree), simulated values for phase voltage THD and line voltage THD versus modulation
index for SFLA and HS algorithms are plotted in Figure 4 and Figure 5, respectively.
292
An Efficient Method for THD Minimization in Multilevel Inverter’s
(a)
(b)
(c)
Figure 4. (a) Optimal Switching Angles, (b) Phase voltage THD and (c) Line voltage THD;
Obtained by SFLA
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Rahim Shamsi Varzeghan, et al.
(a)
(b)
(c)
Figure 5. (a) Optimal Switching Angles, (b) Phase voltage THD and (c) Line voltage THD;
Obtained by HSA
As explained before, different optimum switching angles are found by different algorithms.
Figure 6(a) shows the obtained values from simulation for normalized amplitude of
fundamental component for both algorithms versus the modulation index. Phase Voltage THD
for both algorithms versus the modulation index is plotted in Figure 6(b).
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An Efficient Method for THD Minimization in Multilevel Inverter’s
(a)
(b)
Figure 6. (a) The normalized value of fundamental component, (b) Phase voltage THD; versus
Modulation index, for both algorithms
Figure 6(a) shows that SFLA is more powerful and mostly maintained fundamental
component value close to the desired, despite that HS seems that is not able to find the
switching angles generating the desired value. To corroborate the efficiency of obtained
switching angles the phase voltage THD from both algorithms are shown in Figure 6 (b). The
comparison shows that values of phase voltage THD for SFLA are always lower than value of
phase THD voltage using HSA. The switching angles, related to obtained minimum value of
phase voltage THD for SFLA and HSA are given in Table 1 and Table 2, respectively.
Table 1. (SFLA)
*
1
V
0.84
'
1
'
a
a2
8.366
27.00
a3
'
50.4
Phase THD%
Line-To-Line THD%
11.21
9.84
Table 2. (HSA)
*
1
V
0.85
'
1
'
a
a2
8.166
27.06
a3
'
48.67
Phase THD%
Line-To-Line THD%
11.57
9.74
From the above tables, it is inferred that the minimum phase voltage THD value attained
from using SFLA is lower than the THD value achieved by HSA. Also, the least phase voltage
THD value obtained from using SFLA is lower than the phase THD value given in [27] using
GA to solve THD minimization problem. So, for THD minimization of phase voltage, SFLA
seems to be more effective. As a case study the related phase voltage and Line voltage for the
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Rahim Shamsi Varzeghan, et al.
V1* values given in Table I and II, and also harmonic content of them are shown in Figure 7 and
Figure 8 for both SFLA and HS algorithms, respectively.
(a)
(b)
(c)
(d)
Figure 7. (a) phase voltage, (b) harmonic spectra of phase voltage, (c) Line voltage and (d)
harmonic spectra of Line voltage; Using SFLA
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An Efficient Method for THD Minimization in Multilevel Inverter’s
(a)
(b)
(c)
(d)
Figure 8. (a) phase voltage, (b) harmonic spectra of phase Voltage, (c) Line voltage and (d)
harmonic spectra of Line voltage; Using HSA
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Rahim Shamsi Varzeghan, et al.
Figure 7 and 8, validate the ability of obtained switching angles in generation of phase
voltage with minimum possible THD, for both algorithms. Compared to Figure 8 (b), at Figure
7 (b) the amplitude of harmonics, mostly are lower. So, SFLA is more efficient than HSA to
minimize the phase voltage THD.
7. Experimental Results
The output phase voltage and the output line-to-line voltage of the seven-level inverter
prototype are shown in Figure 9(a) and (b), respectively. This is related to the case that
switching angles are adjusted using SFLA algorithm. Figure 9 is related to table 1.
(a)
(b)
Figure 9 a) Output phase voltage, b) Output Line voltage; Related to table 1
In Figure 10(a) and (b) the output phase voltage and the output line voltage of the
seven-level inverter case study are presented. The switching angles are obtained from
employing HS algorithm. Figure 10 relates to table 2.
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An Efficient Method for THD Minimization in Multilevel Inverter’s
(a)
(b)
Figure 9 a) Output phase voltage, b) Output Line voltage; Related to table 2
The waveform exhibited in Figure 9 and 10 corroborate the simulation results, which
validates the feasibility of obtained results and accuracy of the study. Form this paper, it is
concluded that SFLA is more efficient to solve the phase THD minimization problem. Because,
the minimum phase voltage THD achieved by SFLA is the least in comparison with HS and
GA algorithms.
8. Conclusion
Minimization of Total Harmonic Distortion (THD) for phase voltage of multilevel inverter
was studied in this work. SFLA and HSA were the two optimization algorithms which were
applied to solve the problem. Simulation and experimental results for both of the algorithms
were presented for a seven-level inverter. The results of the two aforementioned algorithms
were compared together and also with a work published using GA. It is shown that both HSA
and SFLA algorithms provide better switching angles to minimize the phase voltage THD.
From the attained results it is inferred that SFLA has more efficiency to minimize phase
voltage THD while providing the fundamental component.
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Rahim Shamsi Varzeghan was born in Varzeghan, Iran in 1987 and
received his B.Sc. degree in Electrical Power Engineering from University of
Tabriz, Tabriz, Iran in 2011. He received the M.S. degree at the azarbaijan
Shahid Madani University, Tabriz, Iran in 2015. He researches on power
electronics and application of evolutionary algorithms in power systems. His
main research interests include frequency response analyses (FRA) of large
power transformers and fault detection on power transformer windings.
Mohammad Reza Jannati Oskuee was born in Tabriz, Iran, in 1988. He
received his B.Sc. degree in electrical power engineering from University of
Tabriz, Tabriz, Iran and M.Sc. degree from Azarbaijan Shahid Madani
University, Tabriz, Iran, graduating with first class honors, where he is
currently working towards the Ph.D. degree at Electrical Engineering
Department. In 2014, he was the recipient of the Best Student Researcher
Award of the Azarbaijan Shahid Madani University. His major research
interests include: smart grids, power system planning and operation, power
electronics, power system dynamics and FACTs devices.
Alireza Eyvazizadeh Khosroshahi received the B.S. degree in electrical
engineering from Islamic Azad University of Tabriz, Iran, in 2012. He
received the M.S. degree at the department of electrical and computer
engineering, University of Tabriz, Iran, in 2015. He is currently working as a
power electrical engineer in Behin Sazehaye Sanati Deniz Company (Deniz
Industrial Group). His research interests include design, control and
reliability analysis of power electronic converters and energy conversion
systems.
Sajad Najafi Ravadanegh was born in Iran, in 1976. He received the B.Sc.
degree from the University of Tabriz, Tabriz, Iran, and the M.Sc. and Ph.D.
degrees from the Amirkabir University of Technology, Tehran, Iran, in 2001,
2003, and 2009, respectively, all in electrical engineering. He is currently an
Associate Professor with the Electrical Engineering Department, Azarbaijan
Shahid Madani University, Tabriz, where he is responsible for the Smart
Distribution Grid Research Laboratory. His current research interests
include smart distribution networks and microgrids optimal operation and
planning, power system stability and control, power system controlled islanding, optimization
algorithms applications in power systems, nonlinear dynamic, and chaos. He has
authored/co-authored of over 40 technical papers.
302