Technique to Determine the Optimised Harmonic Switching Angles of a Cascaded Multilevel Inverter for Minimum Harmonic Distortion
Content
IETE Journal of Research
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Technique to Determine the Optimised Harmonic Switching
Angles of a Cascaded Multilevel Inverter for Minimum
Harmonic Distortion
Manuscript Type:
Date Submitted by the Author:
Original Article
24-Aug-2015
Sharma, Angshuman; Tezpur University, Electronics & Communication
Engineering Dept.
Bardalai, Aroop; Assam Engineering College, Electrical Engineering
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Keywords:
TIJR-2015-1247
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Complete List of Authors:
IETE Journal of Research
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Manuscript ID:
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Journal:
Harmonic distortion , Inverters, Multilevel systems, Switching frequency
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Multilevel inverters have received considerable attention from industries
and researchers for its high power and voltage applications. Various
switching techniques have been suggested for improving the quality and
performance of inverters. One of the conventional techniques for
implementing the switching algorithm in these inverters is Optimised
Harmonic Stepped Waveform (OHSW). However, this technique involves
the major problem of solving nonlinear and complex equations, which
indicates a possibility of multiple solutions. This paper describes a novel
technique that uses the simple arithmetic sequence of natural numbers to
determine the optimised switching angles of a single phase cascaded
multilevel inverter of any number of levels and fed by equal dc sources.
The basic objective was to avoid the laborious process of solving the nonlinear equations using complex switching algorithm for finding the optimal
solution of the switching angles. This technique is implemented to calculate
the optimised switching angles of a 9-level cascaded inverter that reduces
the total harmonic distortion to below 9%.
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Abstract:
URL: https://mc.manuscriptcentral.com/tijr E-mail: [email protected]
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Figure 1. Odd quarter wave symmetric 9-level cascaded inverter waveform.
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Figure 2. Optimised harmonic stepped voltage waveform of a nine-level inverter.
153x79mm (96 x 96 DPI)
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Figure 3. Column chart showing the amplitudes of the harmonic components of 9-level inverter.
174x120mm (96 x 96 DPI)
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Figure 4. Column chart showing the comparison of the THD between the three multilevel inverters.
80x60mm (96 x 96 DPI)
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Technique to Determine the Optimised Harmonic
Switching Angles of a Cascaded Multilevel Inverter
for Minimum Harmonic Distortion
Angshuman Sharma and AroopBardalai
Angshuman Sharma was with Electrical Engineering Department, Assam Engineering College, Guwahati – 781013, Assam, India. He is now with the
Department of Electronics & Communication Engineering, Tezpur University, Tezpur - 784028, Assam, India (corresponding author, phone: 03712273199; +91-9707475263, fax: 03712-267005; e-mail: [email protected]).
AroopBardalai is with the Electrical Engineering Department, Assam Engineering College, Guwahati – 781013, Assam, India. (e-mail:
[email protected]).
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ABSTRACT
Multilevel inverters have received considerable attention from industries and researchers for its high power and voltage
applications. Various switching techniques have been suggested for improving the quality and performance of inverters. One
of the conventional techniques for implementing the switching algorithm in these inverters is Optimised Harmonic Stepped
Waveform (OHSW). However, this technique involves the major problem of solving nonlinear and complex equations,
which indicates a possibility of multiple solutions. This paper describes a novel technique that uses the simple arithmetic
sequence of natural numbers to determine the optimised switching angles of a single phase cascaded multilevel inverter of
any number of levels and fed by equal dc sources. The basic objective was to avoid the laborious process of solving the nonlinear equations using complex switching algorithm for finding the optimal solution of the switching angles. This technique
is implemented to calculate the optimised switching angles of a 9-level cascaded inverter that reduces the total harmonic
distortion to below 9%.
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Keywords:
Arithmetic sequence; multilevel inverter; natural number, optimised harmonic stepped waveform; switching angles; total harmonic
distortion.
1. INTRODUCTION
levels, the number of switches also will increase in number.
Hence the voltage stresses and switching losses will also
increase [6]. So the number of achievable voltage levels is
limited by voltage unbalance problem, voltage clamping
requirement, circuit layout, and packaging constraints [7].
The multilevel inverter can yield operating characteristics
such as high voltage, high power level and high efficiency
without the use of transformers [2,8]. It is recently applied
in static synchronous compensators, active filters, reactive
power compensation applications [9], photovoltaic power
conversion, uninterruptible power supplies and magnetic
resonance imaging. Furthermore, one of the growing
applications for multilevel inverter is electric and hybrid
motor drives.
The multilevel inverters are mainly classified as Diode
Clamped [4,10], Flying Capacitor [11] and Cascaded Hbridge multilevel inverter with separate dc sources
(SDCSs)[2,12-14].The cascaded multilevel inverter was first
proposed in 1975 [1`3]. The cascaded multilevel inverter is
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The concept of multilevel inverters has revolutionised
inverter technology. A multilevel voltage source inverter
divides the main dc supply voltage into several smaller dc
sources which are used to synthesise an ac voltage into a
staircase or stepped approximation of the desired sinusoidal
waveform [1]. Among the significant advantages of
multilevel configuration is the harmonic reduction in the
output voltage waveform without increasing switching
frequency or decreasing the inverter power output [2-4]. The
so-called multilevel starts from three levels. The multilevel
inverter topology can overcome many limitations of the
standard bipolar inverter. Output voltage and power increase
with number of levels. Increasing the output voltage does
not require an increase in the voltage rating of individual
force commutated devices. If the multilevel inverter output
increases to infinite level, the harmonic content of the output
voltage is reduced to zero [5]. But for increasing voltage
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made of several H-bridge inverters, each powered by a
separate dc source which may be obtained from batteries,
fuel cells, or solar cells. It synthesises a desired voltage from
several independent DC voltage sources, such that the
synthesised voltage waveform corresponds to the sum of the
inverter outputs. Since this topology consist of series power
conversion cells, the voltage and power level may be easily
scaled. The cascaded inverter control method is easier to
implement when compared to other multilevel inverters due
to circuit layout flexibility, absence of transformer, extra
clamping diode or voltage balancing capacitor [15], and
easy adjustment of the number of output voltage levels by
adding or reducing H-bridge cells [16]. This configuration
has recently become very popular in ac power supplies and
adjustable speed drive applications [1,12,13].
An important key in designing an effective and efficient
multilevel inverter is to ensure that the total harmonic
distortion (THD) in the output voltage waveform is small
enough [17,18]. With more voltage levels, the multilevel
waveform becomes smoother with low harmonic content,
but with many levels, the design becomes more complicated
with more components and a more complicated controller
for the inverter is required [19]. Power electronics
researchers have suggested several switching strategies,
such as sinusoidal or “sub-harmonic” natural pulse width
modulation (SPWM) [20,21], selective harmonic-eliminated
pulse width modulation (SHE PWM) [20,22], space-vector
modulation (SVM) [23,24], optimised harmonic-stepped
waveform (OHSW) [3,10-12], and optimal minimisation of
THD (OMTHD) [25], to eliminate or minimise the harmonic
content in multilevel waveforms comprising a specific
number of levels.
The OHSW technique is very suitable for a multilevel
inverter circuit [26].In this method, the goal is to conduct
potential elimination of low order harmonics; when this goal
cannot be achieved, the highest possible harmonics
optimisation is desired [16]. The challenge associated with
such techniques is to obtain the optimised harmonic
switching angles through analytical solutions of non-linear
transcendental equations that contain trigonometric terms
which naturally exhibit multiple sets of solutions. Attention
has previously been focused on using the numerical iterative
methods and the evolutionary search algorithms for solution
of the non-linear complex equations. However, each of them
has their own advantages and disadvantages.
In this paper, the optimums witching angles for a
cascaded multilevel inverter are determined using a simple,
fast, efficient and reliable technique that does not require to
solve the complex non-linear equations at all, in order to
achieve minimum harmonic distortion of the output voltage
waveform. The method focuses on quarter wave symmetric
multilevel inverter waveform having equal step height, i.e.
fed by equal dc sources. It uses the arithmetic sequence of
natural numbers to determine the step spaces and henceforth
the switching angles of the cascaded multilevel inverter.
This technique involves simpler formulation and can be
used with multilevel inverters having any odd number of
levels. A 9-level cascaded inverter is considered in this
paper for analysis and the optimised harmonic switching
angles are calculated.
2.
OPTIMISED HARMONIC STEPPED
WAVEFORM TECHNIQUE (OHSW)
A general odd quarter wave symmetric 9-level cascaded
inverter waveform is represented in Figure 1. To achieve the
9-level waveform, four separate dc sources are required. V1
to V4 are dc voltage supplies from separated dc sources.
Figure 1. Odd quarter wave symmetric 9-level cascaded
inverter waveform.
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Considering equal amplitude of all dc sources, i.e.,
V1=V2=V3=V4=E, the expression of the amplitude of the
fundamental and harmonic components of the waveform are
given as:
∑
for odd n
Hn(α) =
(1)
0 for even n
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Hence the Fourier series of the output voltage waveform
is given as:
On
Vout() = ∑
∑
(2)
where
E
dc voltage supply;
n
odd harmonic order;
s
number of dc sources;
optimised harmonic switching angle, which must
satisfy the condition: , ! , " , #
!
For determining the Fourier series of the 9-level output
voltage waveform, four switching angles, , ! , " , , need
to be known. Mathematically, four equations are required to
be set up to solve these switching angles. Unfortunately,
these equations are nonlinear as well as transcendental in
nature, which indicates a possibility of multiple solutions.
Moreover, the estimated solutions must be less than π/2.
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Several methods have been suggested for solving these
equations which can be broadly categorised into two sets.
The first group attempts to solve these equations using the
Newton-Raphson (N-R) iterative method [27]. Iterative
methods mainly depend on an initial guess. Moreover,
divergence problems are likely to occur, especially when the
number of inverter levels is more [28]. Although the N-R
method is conveniently fast, it can only find one set of
solutions. Chiassonet. al. [29] derived analytical expressions
using the mathematical Resultant Theory to compute the
optimum switching angles with the exact range of
modulation index (M). These expressions were polynomials
of 22nd degree which were difficult and time consuming to
derive, and for any change of levels or voltage inputs, new
expressions were required [30]. Homotopy algorithm is
another approach applied to determine one set of solutions
[31]. Overall, all methods included in the first group do not
suggest any optimum solutions for a particular M.
The second group finds solutions that deal with
eliminating the lower order harmonics completely. In this
case, all evolutionary search algorithms can be regarded as
suitable choices. These approaches are applicable for
problems that deal with any number of levels, with simple
derivation of analytical expressions. But these approaches
involve extensive computing [10,16,28,32,33] and are not as
fast as the first group of methods.
One of the major problems in electric power quality is the
harmonic contents. There are several methods of indicating
the quantity of harmonic contents. The most widely used
measure is the total harmonic distortion (THD) [3]. The
THD evaluates the extent of harmonic contents in the output
waveform [7]. THD is mathematically given by,
'
$ ∑(
&)' %
&
(3)
THD =
* 1
$ ∑(
&)'+ ∑
'
,-.
/0 2
& 0)*
∑1
0)* ,-.
/0
(4)
3.
t=
PROPOSED TECHNIQUE
The technique that has been developed for the
determination of step spaces is based on the simple
Arithmetic Sequence of Natural Numbers, which is 1, 2, 3,
4,………, n. If time is considered as the reference, then for
the 1st positive quarter wave, this technique assigns 1 unit of
Then,
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Therefore, output voltage THD of the presented waveform
can be calculated. Theoretically, to get exact THD, infinite
harmonics need to be calculated. However, practically, it is
not possible. Therefore, certain number of harmonics is
calculated. It relies on how precise the THD is needed.
On
where H1is the amplitudes of the fundamental component,
whose frequency is ω0 and H(n) is the amplitude of the nth
harmonic at frequency nω0.
Substituting H1 and H(n) in the above equation, we have
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THD =
time for the 1st step space, 2 units of time for the 2nd step
space, 3 units of time for the 3rd step space and so on till the
last step space is completed at π/2.
Since the 9-levelwaveform of Figure 1 is considered to be
quarter-wave symmetric, so Fourier analysis of the first
quarter wave from 0 to π/2 is sufficient to compute the
amplitude of the fundamental and odd harmonic components
of the complete waveform. There are five steps in the 1st
quarter wave of the waveform, which indicates that there are
4 H-bridge cells in the 9-level cascaded inverter. Following
this technique, the step spaces can be assigned as follows: 1st
step space is assigned 1 unit of time, 2nd step space is
assigned 2 units of time, 3rd step space is assigned 3 units of
time, 4th step space is assigned 4 units of time, and the 5th
and final step space of the 1st quarter wave is assigned 5
units of time.
The procedure can be further described as follows. Let t1,
t2, t3, t4 are the switching instants of the four H-bridge cells
of the cascaded 9-level inverter. Initially, all the H-bridge
cells are in the OFF state and will continue to be in the OFF
state for 1 unit of time till t1 is reached. At the instant t1, the
1st H-bridge cell is switched ON. The 2nd H-bridge cell is
switched ON after 2 units of time at the instant t2. Now both
the 1st and the 2nd H-bridge cells are in the ON state for the
next 3 units of time till the instant t3 is reached. At the
instant t3, the 3rd H-bridge cell is switched ON and the 1st,
2nd and 3rd H-bridge cells operate simultaneously for the
next 4 units of time till the instant t4 is reached. At the
instant t4, the 4th and the last H-bridge cell is switched ON.
Now all the four H-bridge cells operate simultaneously
for the next 5 units of time when finally π/2 is reached and
the first quarter wave is accomplished.
∴ Total number of units assigned for all the step spaces of
the 1st quarter wave = 1 + 2 + 3 + 4 + 5 = 15
For the waveform of Figure 1, let the frequency be 50Hz,
so that the time period of the complete wave is,
T = 1/50 second = 0.02 second.
∴ Time for half wave = 0.01 second.
∴ Time for quarter wave = 0.005 second.
Let tbe the time for each unit. Since there are total 15
units assigned in the 1st quarter wave which has a time
period of 0.005 seconds, therefore, we can write,
15t = 0.005 second
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5.556
s
6
t = 0.000333 s
∴1st step space, ∆t1 = t = 0.000333 s
2nd step space, ∆t2 = 2t = 0.000667 s
3rd step space, ∆t3= 3t = 0.000999 s
th
4 step space, ∆t4 = 4t = 0.001333 s
5th step space, ∆t5 = 5t = 0.001667 s
t1 = 0 + ∆t1 = 0.000333 s
t2 = t1 + ∆t2 = 0.001 s
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t3 = t2 + ∆t3 = 0.002 s
t4 = t3 + ∆t4 = 0.003333 s
Thus the 4thH-bridge cell is switched on at instant t4 =
0.003333 s and is operated for ∆t5 = 0.001667 s when the 1st
quarter wave is completed at 0.005 s.
!
Since we know, angle = ωt = 9
=
!
!
5.5!
8
9 0.000333, ! =
!
!
5.5!
9 0.001,
" =
9 0.002, =
9 0.003333
5.5!
5.5!
Thus the optimised harmonic switching angles of the four
H-bridge cells in the first quarter wave are
= 0.0333π, ! = 0.1π, " = 0.2π, = 0.3333π in
radian
or, = 6°, ! = 18°, " = 36°, = 60° in degrees
These switching angles were used to generate the
optimised harmonic stepped voltage waveform of the ninelevel inverter operating at 50 Hz frequency and fed by four
equal dc sources of magnitude E each, so as to maintain
equal step height. The waveform is depicted in Figure 2,
where 1,2,3,4 and 5 indicate the 1st, 2nd, 3rd, 4th and 5th step
space respectively.
4.
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RESULTS AND DISCUSSION
Let the 9-level cascaded inverter, shown in Figure 2, be
fed by four identical dc sources of 100V each. The output
voltage waveform is controlled by four switching angles ,
! , " and . Performing Fourier analysis, the amplitude of
the fundamental and odd harmonic components and the
Total Harmonic Distortion (THD) can be easily calculated.
Using the nonlinear equation system (1) and the optimised
harmonic switching angles, , ! , " and , obtained in the
previous section, the amplitude of the fundamental
component, H1 is calculated and is found to be 414.40 V.
The same equation system also allows us to calculate the
amplitudes of the odd harmonic components. The Column
chart of Figure 3 shows the amplitudes of the fundamental
and odd harmonic components up to the 63rd harmonic.
Results show that the proposed technique does not eliminate
the harmonics, but minimises it satisfactorily. Further, the
amplitude of each odd harmonic component is reduced
below 4.1% of the amplitude of the fundamental component
for the 9-level inverter. By substituting the amplitudes of the
harmonic components in (4), the output voltage THD,
calculated up to63rd harmonic, is found to be 8.99%.
This technique has also been applied to determine the
optimised harmonic switching angles of 11-level and 13level cascaded inverters and study their respective harmonic
distribution. Assuming identical operating conditions, it is
found that the amplitude of the fundamental component and
the THD are 514.41V and 8.14% respectively in case of 11level cascaded inverter and 614.20V and 7.99% respectively
in case of 13-level cascaded inverter. Studying the results
obtained for the 9-level, 11-level and 13-level cascaded
inverters, it is observed that the output voltage of the
inverters increase as the number of levels increase while
their THD decrease with higher number of levels. This can
be seen from the column chart of Figure 4 which compares
the output voltage THD of the 9-level, 11-level and 13-level
cascaded inverters.
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Figure 2. Optimised harmonic stepped voltage waveform of a nine-level inverter.
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450
400
Voltage Amplitude
350
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250
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1
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9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63
Harmonic Order
Figure 3.Column chart showing the amplitudes of the harmonic components of9-level inverter.
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9.2
9
8.8
8.6
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THD
mathematicaltechnique,involving simpler calculations, was
presented to determine the switching angles of a cascaded
multilevel inverter without extensive derivation of analytical
expressions.The proposed technique includes the Arithmetic
Sequence of natural numbers to assignthe step spaces and
henceforth to determine the optimised harmonic switching
angles of the cascaded multilevel inverter having any odd
number of levels. The technique holds good for multilevel
inverters having quarter wave symmetric waveform and fed
by equal dc sources. As an example, it was used to solve the
switching angles of a nine-level cascaded inverter and the
output voltage THD was found to be 8.99%. Results show
that the proposed technique does not eliminate the
harmonics, but minimises it satisfactorily. Further, this
technique has also been applied to 11-level and 13-level
cascaded inverters and their output voltage THDs are found
to be 8.14% and 7.99% respectively, which indicates that
the THD decrease with higher number of levels.
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8.4
8.2
8
7.8
7.6
7.4
11-level
13-level
Figure 4.Column chart showing the comparison of the THD between
the three multilevel inverters
5. CONCLUSION
The goal of this technique was to calculate the optimised
harmonic switching angles for which the multilevel
waveform exhibits minimum harmonic distortion. The idea
was to avoid the tedious process of solving non-linear and
complex equations to find the optimised harmonic switching
angles.
A
fast,
efficient
and
reliable
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9-level
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REFERENCES
1.
Tim Cunnyngham, “Cascade Multilevel Inverters for
Large Hybrid-Electric Vehicle Applications with
Variant DC Sources,” M.S. Thesis, Dept. Elect. Eng.,
Univ. Tennessee, Knoxville, TN, 2001.
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2.
3.
4.
5.
6.
7.
15.
20.
21.
22.
23.
24.
25.
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19.
On
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May 2009.
AroopBardalai received his B.E.
degree in Electrical Engineering from
Assam Engineering College (AEC),
Guwahati, Assam, India in 1984. After
a brief period in BRPL as an executive
trainee, he joined AEC in 1985 as a
Lecturer in Electrical Engineering
Department. Subsequently he obtained Master’s degree from
Indian Institute of Science, Bangalore, India in 1988 and
was awarded PhD from Gauhati University, Guwahati,
Assam, India in 2008. He is presently working as an
Associate Professor in the Department of Electrical
Engineering, AEC. During his long academic career, apart
from offering various courses and laboratories, he has
guided numerous projects of practical importance for under
graduate and graduate students. He has also been
instrumental in developing laboratories in the Department,
steering the examination process for a long time, and
involved in hostel administration. He has been an active
member of the Institution of Engineers (India), Assam State
Centre.
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Angshuman Sharma was born in
Assam, India in 1989. He received his
B.E. degree in Electrical Engineering
from Jorhat Engineering College
(JEC), Jorhat, Assam, India in 2012,
and his M.E. degree in Power Systems
from Assam Engineering College
(AEC), Guwahati, Assam, India in 2014. Soon thereafter, he
joined Tezpur University, Tezpur, Assam, India and is
presently working as an Assistant Professor in Electrical
Engineering in the Dept. of ECE. His current research
interests include multilevel inverters, analysis and control of
power electronics devices, application of power electronics
in power system, solid state transformers and robotics.
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Technique to Determine the Optimised Harmonic Switching
Angles of a Cascaded Multilevel Inverter for Minimum
Harmonic Distortion
Manuscript Type:
Date Submitted by the Author:
Original Article
24-Aug-2015
Sharma, Angshuman; Tezpur University, Electronics & Communication
Engineering Dept.
Bardalai, Aroop; Assam Engineering College, Electrical Engineering
ie
Keywords:
TIJR-2015-1247
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Complete List of Authors:
IETE Journal of Research
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Manuscript ID:
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Journal:
Harmonic distortion , Inverters, Multilevel systems, Switching frequency
w
Multilevel inverters have received considerable attention from industries
and researchers for its high power and voltage applications. Various
switching techniques have been suggested for improving the quality and
performance of inverters. One of the conventional techniques for
implementing the switching algorithm in these inverters is Optimised
Harmonic Stepped Waveform (OHSW). However, this technique involves
the major problem of solving nonlinear and complex equations, which
indicates a possibility of multiple solutions. This paper describes a novel
technique that uses the simple arithmetic sequence of natural numbers to
determine the optimised switching angles of a single phase cascaded
multilevel inverter of any number of levels and fed by equal dc sources.
The basic objective was to avoid the laborious process of solving the nonlinear equations using complex switching algorithm for finding the optimal
solution of the switching angles. This technique is implemented to calculate
the optimised switching angles of a 9-level cascaded inverter that reduces
the total harmonic distortion to below 9%.
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Abstract:
URL: https://mc.manuscriptcentral.com/tijr E-mail: [email protected]
Page 1 of 11
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Figure 1. Odd quarter wave symmetric 9-level cascaded inverter waveform.
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Figure 2. Optimised harmonic stepped voltage waveform of a nine-level inverter.
153x79mm (96 x 96 DPI)
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Figure 3. Column chart showing the amplitudes of the harmonic components of 9-level inverter.
174x120mm (96 x 96 DPI)
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Figure 4. Column chart showing the comparison of the THD between the three multilevel inverters.
80x60mm (96 x 96 DPI)
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Technique to Determine the Optimised Harmonic
Switching Angles of a Cascaded Multilevel Inverter
for Minimum Harmonic Distortion
Angshuman Sharma and AroopBardalai
Angshuman Sharma was with Electrical Engineering Department, Assam Engineering College, Guwahati – 781013, Assam, India. He is now with the
Department of Electronics & Communication Engineering, Tezpur University, Tezpur - 784028, Assam, India (corresponding author, phone: 03712273199; +91-9707475263, fax: 03712-267005; e-mail: [email protected]).
AroopBardalai is with the Electrical Engineering Department, Assam Engineering College, Guwahati – 781013, Assam, India. (e-mail:
[email protected]).
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ABSTRACT
Multilevel inverters have received considerable attention from industries and researchers for its high power and voltage
applications. Various switching techniques have been suggested for improving the quality and performance of inverters. One
of the conventional techniques for implementing the switching algorithm in these inverters is Optimised Harmonic Stepped
Waveform (OHSW). However, this technique involves the major problem of solving nonlinear and complex equations,
which indicates a possibility of multiple solutions. This paper describes a novel technique that uses the simple arithmetic
sequence of natural numbers to determine the optimised switching angles of a single phase cascaded multilevel inverter of
any number of levels and fed by equal dc sources. The basic objective was to avoid the laborious process of solving the nonlinear equations using complex switching algorithm for finding the optimal solution of the switching angles. This technique
is implemented to calculate the optimised switching angles of a 9-level cascaded inverter that reduces the total harmonic
distortion to below 9%.
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Keywords:
Arithmetic sequence; multilevel inverter; natural number, optimised harmonic stepped waveform; switching angles; total harmonic
distortion.
1. INTRODUCTION
levels, the number of switches also will increase in number.
Hence the voltage stresses and switching losses will also
increase [6]. So the number of achievable voltage levels is
limited by voltage unbalance problem, voltage clamping
requirement, circuit layout, and packaging constraints [7].
The multilevel inverter can yield operating characteristics
such as high voltage, high power level and high efficiency
without the use of transformers [2,8]. It is recently applied
in static synchronous compensators, active filters, reactive
power compensation applications [9], photovoltaic power
conversion, uninterruptible power supplies and magnetic
resonance imaging. Furthermore, one of the growing
applications for multilevel inverter is electric and hybrid
motor drives.
The multilevel inverters are mainly classified as Diode
Clamped [4,10], Flying Capacitor [11] and Cascaded Hbridge multilevel inverter with separate dc sources
(SDCSs)[2,12-14].The cascaded multilevel inverter was first
proposed in 1975 [1`3]. The cascaded multilevel inverter is
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The concept of multilevel inverters has revolutionised
inverter technology. A multilevel voltage source inverter
divides the main dc supply voltage into several smaller dc
sources which are used to synthesise an ac voltage into a
staircase or stepped approximation of the desired sinusoidal
waveform [1]. Among the significant advantages of
multilevel configuration is the harmonic reduction in the
output voltage waveform without increasing switching
frequency or decreasing the inverter power output [2-4]. The
so-called multilevel starts from three levels. The multilevel
inverter topology can overcome many limitations of the
standard bipolar inverter. Output voltage and power increase
with number of levels. Increasing the output voltage does
not require an increase in the voltage rating of individual
force commutated devices. If the multilevel inverter output
increases to infinite level, the harmonic content of the output
voltage is reduced to zero [5]. But for increasing voltage
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made of several H-bridge inverters, each powered by a
separate dc source which may be obtained from batteries,
fuel cells, or solar cells. It synthesises a desired voltage from
several independent DC voltage sources, such that the
synthesised voltage waveform corresponds to the sum of the
inverter outputs. Since this topology consist of series power
conversion cells, the voltage and power level may be easily
scaled. The cascaded inverter control method is easier to
implement when compared to other multilevel inverters due
to circuit layout flexibility, absence of transformer, extra
clamping diode or voltage balancing capacitor [15], and
easy adjustment of the number of output voltage levels by
adding or reducing H-bridge cells [16]. This configuration
has recently become very popular in ac power supplies and
adjustable speed drive applications [1,12,13].
An important key in designing an effective and efficient
multilevel inverter is to ensure that the total harmonic
distortion (THD) in the output voltage waveform is small
enough [17,18]. With more voltage levels, the multilevel
waveform becomes smoother with low harmonic content,
but with many levels, the design becomes more complicated
with more components and a more complicated controller
for the inverter is required [19]. Power electronics
researchers have suggested several switching strategies,
such as sinusoidal or “sub-harmonic” natural pulse width
modulation (SPWM) [20,21], selective harmonic-eliminated
pulse width modulation (SHE PWM) [20,22], space-vector
modulation (SVM) [23,24], optimised harmonic-stepped
waveform (OHSW) [3,10-12], and optimal minimisation of
THD (OMTHD) [25], to eliminate or minimise the harmonic
content in multilevel waveforms comprising a specific
number of levels.
The OHSW technique is very suitable for a multilevel
inverter circuit [26].In this method, the goal is to conduct
potential elimination of low order harmonics; when this goal
cannot be achieved, the highest possible harmonics
optimisation is desired [16]. The challenge associated with
such techniques is to obtain the optimised harmonic
switching angles through analytical solutions of non-linear
transcendental equations that contain trigonometric terms
which naturally exhibit multiple sets of solutions. Attention
has previously been focused on using the numerical iterative
methods and the evolutionary search algorithms for solution
of the non-linear complex equations. However, each of them
has their own advantages and disadvantages.
In this paper, the optimums witching angles for a
cascaded multilevel inverter are determined using a simple,
fast, efficient and reliable technique that does not require to
solve the complex non-linear equations at all, in order to
achieve minimum harmonic distortion of the output voltage
waveform. The method focuses on quarter wave symmetric
multilevel inverter waveform having equal step height, i.e.
fed by equal dc sources. It uses the arithmetic sequence of
natural numbers to determine the step spaces and henceforth
the switching angles of the cascaded multilevel inverter.
This technique involves simpler formulation and can be
used with multilevel inverters having any odd number of
levels. A 9-level cascaded inverter is considered in this
paper for analysis and the optimised harmonic switching
angles are calculated.
2.
OPTIMISED HARMONIC STEPPED
WAVEFORM TECHNIQUE (OHSW)
A general odd quarter wave symmetric 9-level cascaded
inverter waveform is represented in Figure 1. To achieve the
9-level waveform, four separate dc sources are required. V1
to V4 are dc voltage supplies from separated dc sources.
Figure 1. Odd quarter wave symmetric 9-level cascaded
inverter waveform.
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Considering equal amplitude of all dc sources, i.e.,
V1=V2=V3=V4=E, the expression of the amplitude of the
fundamental and harmonic components of the waveform are
given as:
∑
for odd n
Hn(α) =
(1)
0 for even n
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Hence the Fourier series of the output voltage waveform
is given as:
On
Vout() = ∑
∑
(2)
where
E
dc voltage supply;
n
odd harmonic order;
s
number of dc sources;
optimised harmonic switching angle, which must
satisfy the condition: , ! , " , #
!
For determining the Fourier series of the 9-level output
voltage waveform, four switching angles, , ! , " , , need
to be known. Mathematically, four equations are required to
be set up to solve these switching angles. Unfortunately,
these equations are nonlinear as well as transcendental in
nature, which indicates a possibility of multiple solutions.
Moreover, the estimated solutions must be less than π/2.
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Several methods have been suggested for solving these
equations which can be broadly categorised into two sets.
The first group attempts to solve these equations using the
Newton-Raphson (N-R) iterative method [27]. Iterative
methods mainly depend on an initial guess. Moreover,
divergence problems are likely to occur, especially when the
number of inverter levels is more [28]. Although the N-R
method is conveniently fast, it can only find one set of
solutions. Chiassonet. al. [29] derived analytical expressions
using the mathematical Resultant Theory to compute the
optimum switching angles with the exact range of
modulation index (M). These expressions were polynomials
of 22nd degree which were difficult and time consuming to
derive, and for any change of levels or voltage inputs, new
expressions were required [30]. Homotopy algorithm is
another approach applied to determine one set of solutions
[31]. Overall, all methods included in the first group do not
suggest any optimum solutions for a particular M.
The second group finds solutions that deal with
eliminating the lower order harmonics completely. In this
case, all evolutionary search algorithms can be regarded as
suitable choices. These approaches are applicable for
problems that deal with any number of levels, with simple
derivation of analytical expressions. But these approaches
involve extensive computing [10,16,28,32,33] and are not as
fast as the first group of methods.
One of the major problems in electric power quality is the
harmonic contents. There are several methods of indicating
the quantity of harmonic contents. The most widely used
measure is the total harmonic distortion (THD) [3]. The
THD evaluates the extent of harmonic contents in the output
waveform [7]. THD is mathematically given by,
'
$ ∑(
&)' %
&
(3)
THD =
* 1
$ ∑(
&)'+ ∑
'
,-.
/0 2
& 0)*
∑1
0)* ,-.
/0
(4)
3.
t=
PROPOSED TECHNIQUE
The technique that has been developed for the
determination of step spaces is based on the simple
Arithmetic Sequence of Natural Numbers, which is 1, 2, 3,
4,………, n. If time is considered as the reference, then for
the 1st positive quarter wave, this technique assigns 1 unit of
Then,
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Therefore, output voltage THD of the presented waveform
can be calculated. Theoretically, to get exact THD, infinite
harmonics need to be calculated. However, practically, it is
not possible. Therefore, certain number of harmonics is
calculated. It relies on how precise the THD is needed.
On
where H1is the amplitudes of the fundamental component,
whose frequency is ω0 and H(n) is the amplitude of the nth
harmonic at frequency nω0.
Substituting H1 and H(n) in the above equation, we have
w
%*
ie
THD =
time for the 1st step space, 2 units of time for the 2nd step
space, 3 units of time for the 3rd step space and so on till the
last step space is completed at π/2.
Since the 9-levelwaveform of Figure 1 is considered to be
quarter-wave symmetric, so Fourier analysis of the first
quarter wave from 0 to π/2 is sufficient to compute the
amplitude of the fundamental and odd harmonic components
of the complete waveform. There are five steps in the 1st
quarter wave of the waveform, which indicates that there are
4 H-bridge cells in the 9-level cascaded inverter. Following
this technique, the step spaces can be assigned as follows: 1st
step space is assigned 1 unit of time, 2nd step space is
assigned 2 units of time, 3rd step space is assigned 3 units of
time, 4th step space is assigned 4 units of time, and the 5th
and final step space of the 1st quarter wave is assigned 5
units of time.
The procedure can be further described as follows. Let t1,
t2, t3, t4 are the switching instants of the four H-bridge cells
of the cascaded 9-level inverter. Initially, all the H-bridge
cells are in the OFF state and will continue to be in the OFF
state for 1 unit of time till t1 is reached. At the instant t1, the
1st H-bridge cell is switched ON. The 2nd H-bridge cell is
switched ON after 2 units of time at the instant t2. Now both
the 1st and the 2nd H-bridge cells are in the ON state for the
next 3 units of time till the instant t3 is reached. At the
instant t3, the 3rd H-bridge cell is switched ON and the 1st,
2nd and 3rd H-bridge cells operate simultaneously for the
next 4 units of time till the instant t4 is reached. At the
instant t4, the 4th and the last H-bridge cell is switched ON.
Now all the four H-bridge cells operate simultaneously
for the next 5 units of time when finally π/2 is reached and
the first quarter wave is accomplished.
∴ Total number of units assigned for all the step spaces of
the 1st quarter wave = 1 + 2 + 3 + 4 + 5 = 15
For the waveform of Figure 1, let the frequency be 50Hz,
so that the time period of the complete wave is,
T = 1/50 second = 0.02 second.
∴ Time for half wave = 0.01 second.
∴ Time for quarter wave = 0.005 second.
Let tbe the time for each unit. Since there are total 15
units assigned in the 1st quarter wave which has a time
period of 0.005 seconds, therefore, we can write,
15t = 0.005 second
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5.556
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t = 0.000333 s
∴1st step space, ∆t1 = t = 0.000333 s
2nd step space, ∆t2 = 2t = 0.000667 s
3rd step space, ∆t3= 3t = 0.000999 s
th
4 step space, ∆t4 = 4t = 0.001333 s
5th step space, ∆t5 = 5t = 0.001667 s
t1 = 0 + ∆t1 = 0.000333 s
t2 = t1 + ∆t2 = 0.001 s
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IETE Journal of Research
t3 = t2 + ∆t3 = 0.002 s
t4 = t3 + ∆t4 = 0.003333 s
Thus the 4thH-bridge cell is switched on at instant t4 =
0.003333 s and is operated for ∆t5 = 0.001667 s when the 1st
quarter wave is completed at 0.005 s.
!
Since we know, angle = ωt = 9
=
!
!
5.5!
8
9 0.000333, ! =
!
!
5.5!
9 0.001,
" =
9 0.002, =
9 0.003333
5.5!
5.5!
Thus the optimised harmonic switching angles of the four
H-bridge cells in the first quarter wave are
= 0.0333π, ! = 0.1π, " = 0.2π, = 0.3333π in
radian
or, = 6°, ! = 18°, " = 36°, = 60° in degrees
These switching angles were used to generate the
optimised harmonic stepped voltage waveform of the ninelevel inverter operating at 50 Hz frequency and fed by four
equal dc sources of magnitude E each, so as to maintain
equal step height. The waveform is depicted in Figure 2,
where 1,2,3,4 and 5 indicate the 1st, 2nd, 3rd, 4th and 5th step
space respectively.
4.
ee
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RESULTS AND DISCUSSION
Let the 9-level cascaded inverter, shown in Figure 2, be
fed by four identical dc sources of 100V each. The output
voltage waveform is controlled by four switching angles ,
! , " and . Performing Fourier analysis, the amplitude of
the fundamental and odd harmonic components and the
Total Harmonic Distortion (THD) can be easily calculated.
Using the nonlinear equation system (1) and the optimised
harmonic switching angles, , ! , " and , obtained in the
previous section, the amplitude of the fundamental
component, H1 is calculated and is found to be 414.40 V.
The same equation system also allows us to calculate the
amplitudes of the odd harmonic components. The Column
chart of Figure 3 shows the amplitudes of the fundamental
and odd harmonic components up to the 63rd harmonic.
Results show that the proposed technique does not eliminate
the harmonics, but minimises it satisfactorily. Further, the
amplitude of each odd harmonic component is reduced
below 4.1% of the amplitude of the fundamental component
for the 9-level inverter. By substituting the amplitudes of the
harmonic components in (4), the output voltage THD,
calculated up to63rd harmonic, is found to be 8.99%.
This technique has also been applied to determine the
optimised harmonic switching angles of 11-level and 13level cascaded inverters and study their respective harmonic
distribution. Assuming identical operating conditions, it is
found that the amplitude of the fundamental component and
the THD are 514.41V and 8.14% respectively in case of 11level cascaded inverter and 614.20V and 7.99% respectively
in case of 13-level cascaded inverter. Studying the results
obtained for the 9-level, 11-level and 13-level cascaded
inverters, it is observed that the output voltage of the
inverters increase as the number of levels increase while
their THD decrease with higher number of levels. This can
be seen from the column chart of Figure 4 which compares
the output voltage THD of the 9-level, 11-level and 13-level
cascaded inverters.
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Figure 2. Optimised harmonic stepped voltage waveform of a nine-level inverter.
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450
400
Voltage Amplitude
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9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63
Harmonic Order
Figure 3.Column chart showing the amplitudes of the harmonic components of9-level inverter.
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8.8
8.6
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THD
mathematicaltechnique,involving simpler calculations, was
presented to determine the switching angles of a cascaded
multilevel inverter without extensive derivation of analytical
expressions.The proposed technique includes the Arithmetic
Sequence of natural numbers to assignthe step spaces and
henceforth to determine the optimised harmonic switching
angles of the cascaded multilevel inverter having any odd
number of levels. The technique holds good for multilevel
inverters having quarter wave symmetric waveform and fed
by equal dc sources. As an example, it was used to solve the
switching angles of a nine-level cascaded inverter and the
output voltage THD was found to be 8.99%. Results show
that the proposed technique does not eliminate the
harmonics, but minimises it satisfactorily. Further, this
technique has also been applied to 11-level and 13-level
cascaded inverters and their output voltage THDs are found
to be 8.14% and 7.99% respectively, which indicates that
the THD decrease with higher number of levels.
ie
8.4
8.2
8
7.8
7.6
7.4
11-level
13-level
Figure 4.Column chart showing the comparison of the THD between
the three multilevel inverters
5. CONCLUSION
The goal of this technique was to calculate the optimised
harmonic switching angles for which the multilevel
waveform exhibits minimum harmonic distortion. The idea
was to avoid the tedious process of solving non-linear and
complex equations to find the optimised harmonic switching
angles.
A
fast,
efficient
and
reliable
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May 2009.
AroopBardalai received his B.E.
degree in Electrical Engineering from
Assam Engineering College (AEC),
Guwahati, Assam, India in 1984. After
a brief period in BRPL as an executive
trainee, he joined AEC in 1985 as a
Lecturer in Electrical Engineering
Department. Subsequently he obtained Master’s degree from
Indian Institute of Science, Bangalore, India in 1988 and
was awarded PhD from Gauhati University, Guwahati,
Assam, India in 2008. He is presently working as an
Associate Professor in the Department of Electrical
Engineering, AEC. During his long academic career, apart
from offering various courses and laboratories, he has
guided numerous projects of practical importance for under
graduate and graduate students. He has also been
instrumental in developing laboratories in the Department,
steering the examination process for a long time, and
involved in hostel administration. He has been an active
member of the Institution of Engineers (India), Assam State
Centre.
w
ly
On
Angshuman Sharma was born in
Assam, India in 1989. He received his
B.E. degree in Electrical Engineering
from Jorhat Engineering College
(JEC), Jorhat, Assam, India in 2012,
and his M.E. degree in Power Systems
from Assam Engineering College
(AEC), Guwahati, Assam, India in 2014. Soon thereafter, he
joined Tezpur University, Tezpur, Assam, India and is
presently working as an Assistant Professor in Electrical
Engineering in the Dept. of ECE. His current research
interests include multilevel inverters, analysis and control of
power electronics devices, application of power electronics
in power system, solid state transformers and robotics.
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