Harmonic Optimization of Multilevel Converters Using Genetic Algorithms
Abstract—
In this paper, a genetic algorithm (GA) optimization
technique is applied to multilevel inverter to determine
optimum switching angles for cascaded multilevel inverters for
eliminating some higher order harmonics while maintaining
the required fundamental voltage. This technique can be
applied to multilevel inverters with any number of levels; as an
example in this paper, a 7-level inverter is considered, and the
optimum switching angles are calculated offline to eliminate
the 5th and the 7th harmonics. Then, these angles are used in
an experimental setup to validate the results.
I.
I
NTRODUCTION
Multilevel inverters have been drawing growing attention
in the recent years especially in the distributed energy
resources area due to the fact that several batteries, fuel
cells, solar cell, wind, and microturbines can be connected
through a multilevel inverter to feed a load or the ac grid
without voltage balancing problems. Another major
advantage of multilevel inverters is that their switching
frequency is lower than a traditional inverter, which means
they have reduced switching losses.
The output waveforms of multilevel inverters are in a
stepped form; therefore they have reduced harmonics
compared to a square wave inverter. To reduce the
harmonics further, different multilevel sinusoidal PWM and
space-vector PWM schemes are suggested in the literature
[1,2]; however, PWM techniques increase the control
complexity and the switching frequency. Another approach
to reduce the harmonics is to calculate the switching angles
in order to eliminate certain order harmonics. Chiasson et.
al. [3-5] derived analytical expressions using the
mathematical Resultant Theory to compute the optimum
switching angles. These expressions were polynomials of
22
nd
degree, are difficult and time consuming to derive, and
for any change of levels or voltage inputs, new expressions
are required.
In this paper, a general genetic algorithm (GA) approach
will be presented, which solves the same problem with a
simpler formulation and with any number of levels without
extensive derivation of analytical expressions.
GA is a search method to find the maximum of functions
by mimicking the biological evolutionary processes. There
are only a few examples of GA applications for power
electronics in the literature [6-8], but none on GA applied to
multilevel inverters.
II.
C
ASCADED MULTILEVEL INVERTERS
The cascaded multilevel inverter is one of several
multilevel configurations. It is formed by connecting several
single-phase H-bridge converters in series as shown in Fig.
1 for a 7-level inverter. Each converter generates a square
wave voltage waveform with different duty ratios, which
together form the output voltage waveform as in Fig. 2. A
three-phase configuration can be obtained by connecting
three of these converters in Y or ∆.
For harmonic optimization, the switching angles
θ
1
,
θ
2
,
and
θ
3
(for a 7-level inverter) shown in Fig. 2, have to be
selected so that certain order harmonics are eliminated.
III. G
ENETIC
A
LGORITHM
(GA)
Genetic algorithm is a computational model that solves
optimization problems by imitating genetic processes and
the theory of evolution. It imitates biological evolution by
using genetic operators like reproduction, crossover,
Burak Ozpineci
1
1
Oak Ridge National Laboratory
Knoxville, TN USA
Email: burak@ieee.org
Leon M. Tolbert
1,2
, John N. Chiasson
2
2
The University of Tennessee
Knoxville, TN USA
Email: tolbert@utk.edu, chiasson@utk.edu
V
1
V
2
V
3
V
ao
Fuel Cell
Module
a
o
Fuel Cell
Module
Fuel Cell
Module
Q
11
D
11
Q
12
D
12
Q
13
D
13
Q
14
D
14
Q
21
Q
22
D
22
Q
23
D
23
Q
24
Q
n1
Q
n2
D
n2
Q
n3
D
n3
Q
n4
D
21
D
24
D
n1
D
n4
C
dc1
D
FC1
C
dc2
D
FC2
C
dc3
D
FC3
Fig. 1. 7-level cascaded multilevel inverter
Prepared by the Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, managed by UT-Battelle for the U.S. Department of Energy unde
contract DE-AC05-00OR22725.
r
.
The submitted manuscript has been authored by a contractor of the U.S
Government under Contract No. DE-AC05-00OR22725. Accordingly, the
U.S. Government retains a non-exclusive, royalty-free license to publish
from the contribution, or allow others to do so, for U.S. Government
purposes.
2004 35th Annual IEEE Power Electronics Specialists Conference Aachen, Germany, 2004
0-7803-8399-0/04/$20.00 ©2004 IEEE. 3911
mutation, etc.
Optimization in GA means maximization. In cases where
minimization is required, the negative or the inverse of the
function to be optimized is used.
To minimize a function,
using GA, first,
each x
(
k21
x,,x,xf K
)
i
is coded as a binary or floating-point string of length
m. In this paper, a binary string is preferred, e.g.
[]
[]
[]
010111110x
1111000101x
0100110001x
k
2
1
K
KKKKK
K
K
=
=
=
(1)
The set of {x
1
, x
2
,…,x
k
} is called a chromosome and x
i
are
called genes. The algorithm works as follows:
B. Initialize population
Set a population size, N, i.e. the number of chromosomes
in a population. Then initialize the chromosome values
randomly. If known, the range of the genes should be
considered for initialization. The narrower the range, the
faster GA converges.
Population, P=
(2)
N,kN,2N,1
2,k2,22,1
1,k1,21,1
x,,x,x
x,,x,x
x,,x,x
K
KKKK
K
K
C. Evaluate each chromosome
Use a cost function specific to the problem at hand to
evaluate the fitness value (FV) of each chromosome,
()
()
k21
k21
x,,x,xfFV
x,,x,xf
1
FV
K
K
−=
=
or
(3)
Add all the
FV
s to get the total fitness. Divide each
FV
by
the total
FV
and find the weight/probability of selection,
p
i
,
for each chromosome. The integer part of the product,
p
i
N
gives the number of descendents (offspring) from each
chromosome. At the end, there should be
N
descendent
chromosomes. If the number of descendents calculated is
less then
N
, the rest of the descendents are found randomly
considering the reproduction probabilities,
p
i
of each
chromosome.
D. Crossover Operation
0 θ
1
θ
3
θ
2
V
1
V
1
+V
2
+V
3
V
1
+V
2
V
1
V
2
V
3
π
2π
-V
1
-(V
1
+V
2
+V
3
)
-(V
1
+V
2
)
π/2
Fig. 2. 7-level cascaded multilevel inverter waveform generation
A floating number (between 0 and 1) for each
chromosome is assigned randomly. If this number is smaller
than a pre-selected
crossover probability
, this chromosome
goes into
crossover
. The chromosomes undergoing
crossover are paired randomly. In this case assume
x
1
and
x
2
are paired. The crossing point is randomly selected, assume
3 in this case.
Then, before crossover,
[
]
[
1111000101x
0100110001x
2
1
K
K
=
=
]
(4)
and after crossover,
[
]
[
0100100101x
1111010001x
2
1
K
K
=
=
]
(5)
As seen above, the bits after the 3rd one are exchanged.
E. Mutation Operation:
A floating number (between 0 and 1) for each bit is
assigned randomly. If this number is smaller than a pre-
selected
mutation probability
, this bit mutates. Assume that
the 2nd and 4th bits of
x
1
and 2nd, 3rd and 5th bits of
x
2
need to be mutated.
Then, before mutation and after crossover,
[
]
[
0100100101x
1111010001x
2
1
K
K
=
=
]
(6)
and after mutation,
[
]
[
0100101000x
1111011011x
2
1
K
K
=
=
]
(7)
Finally, the new population is ready for another cycle of
genetic algorithm. The algorithm runs a certain number of
times as required by the user. At the end, the chromosome
with the maximum
FV
is the answer.
IV. F
ORMULATING THE PROBLEM
GA algorithm explained in the previous section is the
same for any application. There are only a few parameters to
be set for GA to work. The steps for formulating a problem
and applying GA are as follows:
1- Select binary or floating point strings.
2- Find the number of variables specific to the problem;
this number will be the number of genes in a chromosome.
In this application, the number of variables is the number of
controllable switching angles, which is the number of H-
bridges in a cascaded multilevel inverter.
A 7-level inverter requires three H-bridges; thus, each
chromosome for this application will have three switching
angles, i.e. {
θ
1
,
θ
2,
θ
3
}.
3- Set a population size and initialize the population. In
this application a population size of 20 is selected. Higher
population might increase the rate of convergence but also
increases the execution time. The selection of optimum
sized population requires some experience in GA.
The population in this paper has 20 chromosomes, each
containing three switching angles. The population is
initialized with random angles between 0 and 90 degrees
2004 35th Annual IEEE Power Electronics Specialists Conference Aachen, Germany, 2004
3912
taking into consideration the quarter-wave symmetry of the
output voltage waveform.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
0
20
40
60
80
Switching angles
θ
1
, θ
2
, a nd θ
3
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
0
0.1
0.2
0.3
0.4
Cost function
Modulation index, M
θ
2
θ
3
Fig. 3. Solutions for
θ
1
,
θ
2
, and
θ
3
and the cost function
4- The most important item for the GA to evaluate the
fitness of each chromosome is the cost function. The
objective of this study is to minimize some harmonics;
therefore the cost function has to be related to these
harmonics. As an example assume that the 5th and 7th
harmonics at the output of a 7-level inverter have to be
minimized. Then, the cost function,
f
can be selected as the
sum of these two harmonics normalized to the fundamental,
()
1
75
321
V
VV
100,,f
+
×=
θθθ
. (8)
where
θ
i
are the switching angles and
V
n
are the
n
th order
voltage harmonics.
For each chromosome, a multilevel output voltage
waveform (Fig. 2) is created using the switching angles in
the chromosome and the required harmonic magnitudes are
calculated using FFT techniques.
The fitness value,
FV
is calculated for each chromosome
inserting (8) in (3). In this case,
()
1
75
321
V
VV
100,,FV
+
×−=
θθθ
(9)
is used. The switching angle set producing the max
FV
is
the best solution of the first iteration.
5- GA is usually set to run for a certain number of
iterations (100 in this case) to find an answer. After the first
iteration,
FV
’s are used to determine new offspring as
explained in Section II. These go through crossover and
mutation operations and a new population is created which
goes through the same cycle starting from
FV
evaluation.
Sometimes, GA can converge to a solution much before
100 iterations are completed. To save time, in this paper, the
iterations have been stopped when the cost function goes
below 1 in which case the sum of the 5th and the 7th
harmonics is negligible compared to the fundamental. As
seen in Fig. 3, GA resulted in cost functions even smaller
than 0.4.
Note that after these iterations, GA finds one solution;
therefore, it has to be run as many times as the number of
solutions required to cover the whole modulation index
range.
The MATLAB source code required to solve this problem
for any number of levels and up to any number of
harmonics is given in the Appendix at the end of the paper.
The MATLAB GA Optimization Toolbox still needs to be
downloaded from [9].
V.
R
ESULTS
For the 7-level inverter, switching angles, which
minimize the 5th and 7th harmonics, are shown in Fig. 3.
Note that this plot is similar to the one in [3] but has more
solutions. The reason for this is that in [3], the solution only
includes angles that result in zero 5th and 7th order
harmonics. In this paper, however, as seen in the bottom
plot of Fig. 3, any solution that yields a cost function less
than 1 is accepted. This means that if low harmonics are
tolerable, then a wider solution space is available.
In this figure for certain modulation indices, several sets
of solutions are available. Either of these solutions can be
used to minimize the selected harmonics. Another
possibility [3] is to calculate THDs for each solution set and
use the set that gives the lowest THD.
As can be observed in Fig. 3, for some modulation
indices, no solution sets are available. This means that for
those modulation indices, either there is not a solution or
GA could not find one. The former reason is more of a
possibility than the latter.
Fig. 4 shows the experimental 7-level voltage waveform
for
M
=0.42. Fig. 5, on the other hand, shows the first 15
harmonics of the waveform in Fig. 4. As seen in this figure,
the 5th and the 7th harmonics of the voltage waveform are
negligible.
Fig. 6 shows the optimum switching angles when this
technique is applied to a 9-level inverter (4 H-bridges, 4
switching angles) to minimize the 5th, 7th, 11th, and 13th
harmonics. In this case, for the angles GA selected, the cost
function is bound at 0.2, which much lower than the
previous case.
θ
1
2004 35th Annual IEEE Power Electronics Specialists Conference Aachen, Germany, 2004
3913
Cost fu nction
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
0
20
40
60
80
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
0
0.05
0.1
0.15
0.2
θ
2
θ
3
θ
4
Modulation index, M
Sw itc h in g a n gl es
θ
1
, θ
2
, θ
3,
and θ
4
Fig. 6. Solutions for
θ
1
,
θ
2
,
θ
3
, and
θ
4
and the cost function.
Phase Voltage, V
Time, s
Fig. 4. Experimental output voltage waveform
VI.
C
ONCLUSIONS
The comparison of the results in this paper to similar
work in the literature shows that the GA approach for the
harmonic optimization of multilevel inverters works
properly. As in this approach, GA can be applied to any
problem where optimization is required; therefore, it can be
used in many applications in power electronics. A freely
available MATLAB GA optimization toolbox [9] can be
used for any optimization needs. When the toolbox is used,
the only programming required for the GA application is
given in the appendix below.
R
EFERENCES
[1] L.M Tolbert, F. Z. Peng, T.G. Habetler, “Multilevel PWM methods at
low modulation indices,” IEEE Transactions on Power Electronics,
15(4), July 2000, pp. 719 – 725.
[2] L.M. Tolbert, T.G. Habetler, “Novel multilevel inverter carrier-based
PWM method,” IEEE Transactions on Industry Applications, 35(5),
Sept.-Oct. 1999, pp. 1098 – 1107.
[3] J. Chiasson, L. M. Tolbert, K. McKenzie, Z. Du, “Eliminating
harmonics in a multilevel converter using resultant theory,”
Conference Proceedings of IEEE Power Electronics Specialists
Conference, 2002, vol. 2, 503– 508.
[4] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, Z. Du, “A Complete
Solution to the Harmonic Elimination Problem,” IEEE Transactions
on Power Electronics, 19(2), March 2004, pp. 491 – 499.
[5] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, Z. Du, “A Unified
Approach to Solving the Harmonic Elimination Equations in
Multilevel Converters,” IEEE Transactions on Power Electronics,
19(2), March 2004, pp. 478 – 490.
[6] B. Ozpineci, J. O. P. Pinto, L. M. Tolbert, “Pulse-width optimization
in a pulse density modulated high frequency AC-AC converter using
genetic algorithms,” Conference Proceedings of IEEE International
Conference on Systems, Man, and Cybernetics, 2001, pp. 1924 –
1929.
[7] A. I. Maswood, S. Wei, M. A. Rahman, “A Flexible Way to Generate
PWM-SHE Switching Patterns Using Genetic Algorithms,”
Conference Proceedings of IEEE Applied Power Electronics
Conference and Exposition (APEC), 2001, pp. 1130– 1134.
[8] M. J. Schutten, D. A. Torrey, “Genetic Algorithms for Control of
Power Converters,” Conference Proceedings of IEEE Power
Electronics Specialists Conference, 1995, pp. 1321– 1326.
[9] C. Houck, J. Joines, M. Kay, The Genetic Algorithm Optimization
Toolbox (GAOT) for Matlab 5,
http://www.ie.ncsu.edu/mirage/GAToolBox/gaot.
θ
1
5
th
7
th
a
k
/a
1
Frequency, Hz
Fig. 5. Normalized (with respect to the fundamental) FFT vs
frequency.
2004 35th Annual IEEE Power Electronics Specialists Conference Aachen, Germany, 2004
3914
A
PPENDIX
A. Matlab source code for the menu
% Main menu to select the number of levels of the inverter,
% the number of solution sets required, the maximum
% number of iterations, the minimum error tolerable, and
% up to what number of harmonics will be minimized.
function [p] = mainmenu(px)
clc
disp(sprintf('1-Change the number of levels : %g',px(1)));
disp(sprintf('2-Change number_of_points : %g',px(2)));
disp(sprintf('3-Change the maximum number of iterations: %g',px(3)));
disp(sprintf('4-Change the minimum error (percent) : %g',px(4)));
disp(sprintf('5-eliminate (6n+1) harmonics. n : %g',px(5)));
p=input('Make a selection (1-4) or Press enter to run : ');
if (isempty(p)==1),p=0;end
B. Matlab source code for the main program
% Finds the optimum angles to minimize up to the 6n+-1th harmonics using
GA.
% Uses functions
% "fitness.m" to evaluate the fitness of the population
% "mainmenu.m" to display an options menu
%
% If the percentage of the sum of the selected harmonics is less than
% "minh", the solution is accepted, if not it is not accepted
%
% If some solutions are not accepted, then "number of points" will be
% different than the "number of iterations", otherwise they will be the
% same.
%
% For n cascaded converters, the number of levels is 2n+1
clear
global harm_n
p=100;
k=7;nop=2000;mnoi=20000;minh=2;harm_n=8;
px=[7 2000 20000 2 8];
while p~=0
p=mainmenu(px);
switch p
case 0
case 1
k=(input('number of levels (default 7)= ')-1)/2;
if isempty(k),k=3;end
px(1)=2*k+1;
case 2
nop=input('number_of_points (default 2000): ');
if isempty(nop),nop=2000;end
px(2)=nop;
case 3
mnoi=input('maximum number of iterations (default 20000): ');
if isempty(mnoi),mnoi=20000;end
px(3)=mnoi;
case 4
minh=input('minimum error (default 2): ');
if isempty(minh),minh=2;end
px(4)=minh;
case 5
harm_n=input('eliminate (6n+1) harmonics. n (default 8): ');
if isempty(harm_n),harm_n=2;end
px(5)=harm_n;
otherwise
disp('Error 01:Enter a value between 0 and 4')
disp('Press enter to continue')
pause
end
end
k=(px(1)-1)/2;
nop=px(2);
mnoi=px(3);
minh=px(4);
for i=1:k
pool(i,:)=[.01 .99];
end
file='fitness';
n=100;
out=zeros(1,k+3);
it=1;
it_all=0;
flag1=0;
VV=125*ones(1,k);
while (it<nop)
it_all=it_all+1;
if it_all==mnoi,return,end
if flag1==1
it=size(out,1)+1;
flag1=0;
else
it;
end
% Create a random initial population of size 20.
initPop=initializega(20,pool,file);
for i=1:20
initPop(i,1:k)=sort(initPop(i,1:k));
end
[x endPop] = ga(pool,file,[],initPop,[1e-6 1 1],'maxGenTerm',n);
if (-x(k+1)<minh)
MM=x;
N=4096;
[fftvaoo vaoo]=modulation(N,MM,VV);
vvv=abs(fftvaoo(2));
sumhh=0;
for g=1:harm_n
h55=abs((fftvaoo(6*g+1+1))); %(6n+1)th harmonic
h77=abs((fftvaoo(6*g-1+1))); %(6n-1)th harmonic
sumhh=sumhh+h55+h77;
end
xx=sort(x(1:k))*90;
out(it,:)=[xx(1:k) vvv/sum(VV) -x(k+1) 100*(sumhh)/vvv]; %The best found
it=it+1;
flag1=0;
else
flag1=1;
end
if flag1==0
figure(1)
subplot(2,1,1),plot(0:N/2-1,abs(fftvaoo(1:N/2)));
axis([0,50,0,(abs(fftvaoo(2))*1.01)])
subplot(2,1,2),plot(vaoo)
hold, plot(abs(fftvaoo(2))*sin(2*pi*(1:N)/N),'r'),hold
figure(2)
subplot(3,1,1)
for i=1:k
plot(out(:,k+1),out(:,i),'.'),
if i==1,hold,end
if i==k,hold off,end
end
subplot(3,1,2),
plot(out(:,k+1),sum(out,2)/(k*90),'.'),hold,
plot(out(:,k+1),4/pi-out(:,k+1),'r'),
plot(out(:,k+1),1-out(:,k+1),'g'),hold,ylabel('overall M')
subplot(3,1,3),
plot(out(:,k+1),(out(:,k+2)),'.'),hold on
plot(out(:,k+1),(out(:,k+3)),'.r'),hold off, ylabel('cost function')
end
drawnow
end
C. Matlab source code for the fitness evaluation
% Evaluates the fitness of the chromosomes
function [sol, val] = fitness(sol,options)
global harm_n
M=sol;
N=4096; %N is a power of 2
kk=max(size(sol))-1;
V=125*ones(1,kk);
[fftvao v]=modulation(N,M,V);
2004 35th Annual IEEE Power Electronics Specialists Conference Aachen, Germany, 2004
3915
