RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
Harmonic Reduction Technique for a Cascade
Multilevel Inverter
Jagdish Kumar1, Biswarup Das2, and Pramod Agarwal2
1
Indian Institute of Technology Roorkee, Roorkee, India
Email: jkb70dee@iitr.ernet.in
2
Indian Institute of Technology Roorkee, Roorkee, India
Email: {biswafee, pramgfee}@iitr.ernet.in
Abstract—In this paper, an optimization technique is
proposed to compute switching angles at fundamental
frequency switching scheme by solving non linear
transcendental equations (known as selective harmonic
elimination equations), thereby eliminating certain
predominating lower order harmonics, and simultaneously,
control over magnitude of output voltage of a multilevel
inverter is achieved. As these equations are nonlinear
transcendental in nature, there may exist simple, multiple or
even no solution for a particular value of a modulation
index. The proposed scheme is implemented in such a way
that all possible solutions are obtained without knowing
proper initial guess at the solutions. Moreover, this
technique is suitable for higher level of multilevel inverters
where other existing methods fail to compute the switching
angles due to more computational burden. For the values of
modulation indices where multiple solutions exist, the
solutions which produce least THD in the output voltage is
chosen. A significant decrease in THD is obtained by
considering multiple solution sets instead of taking a single
set of solution. The computational results are shown
graphically for better understanding and to prove the
effectiveness of the method. An experimental 11-level
cascade multilevel inverter is employed to validate the
computational results.
Index Terms—Cascade multilevel inverter, modulation
index, objective function, total harmonic distortion (THD),
sequential quadratic programming (SQP).
I. INTRODUCTION
Multilevel inverters are more advanced and latest type
of power electronic converters that synthesize 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 synthesized.
In comparison with the hard-switched two-level pulse
width modulation inverters, multilevel inverters offer
several advantages including their capabilities to operate
at high voltage with lower voltage stress per switching,
high efficiency and low electromagnetic interferences [1],
[3] etc.
To synthesize multilevel output ac voltage using
different levels of dc inputs, semiconductor devices must
be switched on and off in such a way that desired
fundamental is obtained with minimum harmonic
1 Corresponding author: Tel. No. +91-9410747840, Email Addresses:
jkb70dee@iitr.ernet.in, jk_bishnoi@yahoo.com
© 2009 ACADEMY PUBLISHER
distortion. There are different approaches for the
selection of switching techniques for the multilevel
inverters [4]-[9], one of the important techniques is
selective harmonic elimination (SHE) method. In SHE
technique, certain predominating lower order harmonics
are eliminated whereas higher order harmonics are
filtered using suitable filter. Switching angles are
computed by solving the SHE equations, but it is difficult
to solve SHE equations because of their nonlinear
characteristics. Due to nonlinear nature, solution of these
equations may be simple, multiple or even no solution for
a particular value of modulation index (m). A big task is
how to get all possible solutions when they exist using
simple and less computationally complex method. Once
these solutions are obtained, the solutions having least
THD are selected for switching purpose.
In [4], [5], iterative numerical techniques such as
Newton- Raphson method have been implemented to
solve the SHE equations producing only one solution set,
and even for this a proper initial guess and starting value
of m for which solutions exist are required. Some
techniques as discussed in [6], [7], here SHE equations
are first converted into polynomial equations, and then
the resulting polynomial equations are solved using
theory of resultants of polynomials and the theory of
symmetrical polynomials, producing all possible
solutions. A difficulty with these approaches is that for
higher levels of multilevel inverter the order of
polynomials becomes very high, thereby making the
computations of solutions of these polynomial equations
very complex. Optimization techniques based on Genetic
Algorithm (GA) and Particle Swarm Optimizations
(PSO) have been discussed in [8], [9] for computing
switching angles only for 7-level multilevel inverters.
The implementation of these approaches requires proper
selection of certain parameters such as population size,
mutation rate, initial weight etc. It becomes difficult to
select these parameters for higher level multilevel
inverters. To circumvent above mentioned problems, a
simple optimization technique based on sequential
quadratic programming (SQP) is proposed in this paper to
solve SHE equations which produces all possible
solutions. The proposed technique is implemented in such
a way that all possible solutions for any number of Hbridges connected in series are computed by using any
arbitrary initial guess with negligible computational
effort. A complete analysis for an 11-level inverter using
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International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
five H-bridges per phase in series is presented, and it is
shown that for a range of m, switching angles can be
computed to produce the desired fundamental voltage
along with elimination of 5th, 7th, 11th, and 13th order
harmonic components. The computational results are
validated through experiments.
II. CASCADE MULTILEVEL INVERTER
Cascade Multilevel Inverter (CMLI) is one of the most
important topology in the family of multilevel inverters.
It requires least number of components with compare to
diode-clamped and flying capacitors type multilevel
inverters. It has modular structure with simple switching
strategy and occupies less space [1] - [3].
The CMLI consists of a number of H-bridge inverter
units with separate dc source for each unit and is
connected in cascade or series as shown in Fig. 1. Each
H-bridge can produce three different voltage levels: +Vdc,
0, and –Vdc by connecting the dc source to ac output side
by different combinations of four switches S1, S2, S3, and
S4. The ac output of each H-bridge is connected in series
such that the synthesized output voltage waveform is the
sum of all of the individual H-bridge outputs.
S1
va
S2
+
Vdc _
v a1
S3
S4
S1
S2
v a2
+
Vdc _
S3
S4
S1
S2
+
Vdc _
v a3
S3
S4
cascade and using proper modulation scheme, a nearly
sinusoidal output voltage waveform can be synthesized.
The number of levels in the output phase voltage is 2s+1,
where s is the number of H-bridges used per phase. Fig. 2
shows an 11-level output phase voltage waveform using
five H-bridges. The magnitude of the ac output phase
voltage is given by van = va1+va2+va3+va4+va5 [2].
In general, when s number of H-bridges per phase is
connected in cascade, the Fourier series expansion of the
staircase output voltage waveform is given by (1).
v an ( wt ) =
∞
4Vdc
(cos (kα 1 ) + ...
k =1, 3, 5 kπ
∑
+ cos (kα s )) sin (kwt )
III. PROBLEM FORMULATION AND SELECTIVE HARMONIC
ELIMINATION EQUATIONS
For a given fundamental peak voltage V1, it is required
to determine the s switching angles such that the selection
of one angle is used to determine V1 and remaining (s-1)
angles are used to eliminate the same number of
harmonics (generally lower order harmonics), and also all
switching angles should be in the range 0 ≤ α1 < α2 < … <
α5 ≤ π/2. In three-phase power system, triplen harmonics
are canceled out automatically in line-to-line voltages as
a result only non-triplen odd harmonics are present in
line-to-line voltages [3], [4].
For an 11-level cascade inverter, there are five Hbridges per phase i.e. s = 5 or five degrees of freedom are
available; one degree of freedom is used to control the
magnitude of the fundamental output voltage and the
remaining four degrees of freedom are generally used to
eliminate 5th, 7th, 11th, and 13th order harmonic
components as they dominate the total harmonic
distortion [5].
The modulation index, m is defined as the ratio of the
fundamental output voltage to the maximum obtainable
voltage (maximum voltage is obtained when all switching
angles are zero). From (1), the relation between m and
switching angles is given as:
cos(α 1 ) + cos(α 2 ) + ... + cos(α 5 ) = 5m
S1
(1)
(2)
S2
v a4
+
Vdc _
S3
S4
S1
S2
v a5
+
Vdc _
n
S3
S4
Figure 2. Output voltage waveform of an 11-level CMLI.
Figure 1. Configuration of a single-phase 11-level CMLI.
By connecting the sufficient number of H-bridges in
© 2009 ACADEMY PUBLISHER
182
Similarly from (1), expressions for 5th, 7th, 11th, and
th
13 harmonic components (scaled values) are given as:
RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
optimization toolbox [12].
cos(5α1 ) + cos(5α 2 ) + ... + cos(5α 5 ) = h5
V. COMPUTATIONAL RESULTS
cos(7α 1 ) + cos(7α 2 ) + ... + cos(7α 5) = h7
cos(11α1 ) + cos(11α 2 ) + ... + cos(11α 5 ) = h11
By implementing the proposed method, all possible
solution sets for an 11-level CMLI were computed and a
complete analysis is also presented. Starting with any
random initial guess all solution sets were computed by
incrementing m in small steps from 0 to 1. The nature of
the results obtained is shown (only in the range of m
where solutions occur) in Fig. 3. By using preliminary
computed results from Fig. 3, complete solution sets were
computed as shown in Fig. 4. It can be seen from the Fig.
4 that the solutions do not exist at lower and upper ends
of the modulation indices and also for m = [0.3800
0.4400], [0.7300 0.7310], and [0.7330 0.7470]. Multiple
solution sets exist for m = [0.5050 0.5800], [0.6120
0.7000]. Even some solutions existing in very narrow
range of m = [0.3760 0.3790], [0.5470 0.5490], [0.7320]
were also obtained by implementing the proposed
method, hence this demonstrate the capability of
proposed method in computing all possible solution sets.
cos(13α1 ) + cos(13α 2 ) + ... + cos(13α 5 ) = h13 (3)
Equations (2) and (3) in combined form are known as
SHE equations in terms of five unknowns α1, α2, α3, α4,
and α5; provided that h5, h7, h11, h13 are identically zero.
For the given value of m (from 0 to1), it is required to get
complete and all possible solutions for the switching
angle (0 to π/2) when the solutions exist, with minimum
computational burden and complexity.
Following objective function is formulated to solve
SHE equations using optimization technique:
Φ (α1 ,α 2 , α 3 ,α 4 , α 5 ) = h5 + h7 + h11 + h13 (4)
2
2
2
2
Now, (4) is to be minimized subject to equality
constraints given by (2) and 0 ≤ α1 < α2 < … < α5 ≤ π/2.
Feasible solutions exist only where objective function (4)
is identically zero.
IV. PROPOSED OPTIMIZATION TECHNIQUE
To solve the SHE equations using optimization
technique, a proper initial guess at the solution is
required. In general it is difficult to make initial guess i.e.
what value one should start with. In the proposed
technique no such specific initial guess is required i.e.
one can start with any random initial guess and can obtain
required information about the nature of the solutions.
Once initial knowledge about the solutions is known,
different solution sets (in case of multiple solutions) can
be obtained by using switching angles thus obtained as
initial guess.
The algorithm for the optimization technique is as
follows:
1) Assume any random initial guess for switching
angles such that 0 ≤ α1 < α2 < … < α5 ≤ π/2.
2) Set m = 0;
3) Calculate the objective function and check
equality constraints.
4) Accept the solution if Φ is zero and equality
constraints are satisfied, else drop the solution.
5) Increment m by a very small value and repeat
step 4).
6) Plot the switching angles thus obtained as a
function of m and observe the nature of the
solutions. Different solution sets appear.
7) Take one solution set at a time and compute all
switching angles set for the whole range of m,
where solutions exist, by taking solutions
obtained in step 6 as initial guess.
8) Complete other solution sets in similar way.
To implement the above algorithm, the sequential
quadratic programming (SQP) method [10], [11] has been
used and been implemented by using MATLAB
Figure 3. Solution sets computed with an arbitrary initial guess.
Figure 4. Exact solution sets for 11-level CMLI.
For each of the multiple solution sets as computed
above, total harmonic distortion (THD) in percent is
computed according to (5), the set of switching angles
among multiple solutions sets which produce least THD
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RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
is selected and termed as combined solution. The THD
plots for different solution sets along with combined
solution are plotted as a function of m in Fig. 5. It can be
seen from the Fig. 5 that there is a significant decrease in
THD if one uses all possible solution sets for determining
the combined solution instead of using only one solution
set as reported in [3]-[5]. For example, if one computes
THD produced due to all possible solution sets at m =
0.5470 (at this value of modulation index, there are three
solution sets), the difference in THD for the solution sets
having highest and lowest THD is about 3%.
2
THD =
2
V17 + V19 + ... + V49
V1
2
× 100
Figure 5. THD of different solution sets.
(5)
Where V17, V19 …V49 are magnitudes of 17th, 19th, 49th
order harmonic components (non-triplen odd harmonics
only) respectively, and V1 is magnitude of fundamental
voltage.
One of the key features of the proposed method is that
the switching angles can be obtained for those values of
modulation indices for which solutions of sets of
equations given by (2) and (3) do not exist, thus
producing continuous solution for the complete range of
m from 0 to 1. In this case, the objective function given
by (4) is minimized (instead of making it zero) subjected
to equality constraint given by (2) with switching angles
range 0 ≤ α1 < α2 < … < α5 ≤ π/2. Variation of the
objective function with m is shown in Fig. 6. The
switching angles corresponding to least THD for the
values of m where solutions exist, and corresponding to
least objective function for the values of m where
solutions do not exist are plotted in Fig. 7. The plot for
switching angles in Fig. 7 is plotted only for the range of
m where THD is below certain limits.
Figure 6. Variation of objective function.
VI. EXPERIMENTAL RESULTS
A prototype single-phase 11-level CMLI has been built
using 400V, 10A MOSFET as the switching device. Five
separate dc supply of 13V each is obtained using step
down transformers with rectifier unit. Pentium 80486
processor based PC with clock frequency 2MHz with
timer I/O card is used for firing pulse generation. Firing
pulses to the switching devices are given through a delay
circuit which provides 5µsec delay to avoid any short
circuit due to simultaneous conduction of devices in the
same leg of H-bridges.
In order to validate the analytical and simulated results,
an 11-level single-phase output voltage at m = 0.5480 (for
this value of m three solution sets exist) was synthesized
at fundamental frequency (f = 50Hz). The fundamental
voltage V1 produced was 45.34V (peak) as calculated
using (1) and (2). For each of the multiple solution set,
THD in line to line voltage was computed as per (5).
Figure 7. Switching angles.
The experimentally produced phase voltage along with
its harmonic spectrum corresponding to lowest THD is
shown in Fig. 8, and in Fig. 9 phase voltage with
corresponding harmonic spectrum for the solution having
highest THD is depicted. The harmonic spectrums of
synthesized phase voltage show that 5th, 7th, 11th, and 13th
harmonics are almost absent as predicted analytically.
The THD in line to line voltage as computed analytically
and experimentally are: 5.70% and 5.25% respectively
for the solution set producing least THD in the output
voltage. Corresponding data for the solution set having
highest THD are 8.28% and 7.03%. The above data show
that experimental results are in close agreement with
theoretical values. The triplen harmonics are present in
Figs. 8 and 9 due to the fact that the synthesized wave
form is a single-phase one.
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© 2009 ACADEMY PUBLISHER
RESEARCH PAPER
International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009
[5]
[6]
[7]
Figure 8. Phase voltage and harmonic spectrum for the solution set
having smallest THD at m = 0.5480.
[8]
[9]
[10]
[11]
Figure 9. Phase voltage and harmonic spectrum for the solution set
having highest THD at m = 0.5480.
[12]
VII. CONCLUSION
The selective harmonic elimination method at
fundamental frequency switching scheme has been
implemented using the optimization technique that
produces all possible solution sets when they exist. In
comparison with other suggested methods, the proposed
technique has many advantages such as: it can produce all
possible solution sets for any numbers of multilevel
inverter without much computational burden, speed of
convergence is fast, it can produce continuous solutions
for the complete range of modulation index thereby
giving more flexibility in control action etc. The proposed
technique was successfully implemented for computing
the switching angles for 11-level CMLI. A complete
analysis for 11-level inverter has been presented and it is
shown that a significant amount of THD reduction can be
attained if all possible solution sets are computed.
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