A Hybrid Genetic Algorithm for Selective Harmonic

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A Hybrid Genetic Algorithm for Selective Harmonic Elimination PWM AC/AC
Converter Control
Article in Electrical Engineering · March 2007
DOI: 10.1007/s00202-006-0003-9
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Electr Eng (2007) 89: 285–291
DOI 10.1007/s00202-006-0003-9
O R I G I NA L PA P E R
Mohamed S. A. Dahidah · M. V. C. Rao
A hybrid genetic algorithm for selective harmonic elimination PWM
AC/AC converter control
Received: 10 May 2005 / Accepted: 19 November 2005 / Published online: 14 March 2006
© Springer-Verlag 2006
Abstract This paper presents an optimal solution for eliminating pre-specified orders of harmonics from the output
waveform of AC/AC converter. The main challenge of solving the associated nonlinear equations, which are transcendental in nature and therefore have multiple solutions, is the
convergence and therefore an initial point selected considerably close to the exact solution is required. The paper
discusses an efficient hybrid real coded genetic algorithm
(HRCGA) that reduces significantly the computational burden resulting in a fast convergence. An objective function
describing a measure of effectiveness of eliminating selected
orders of harmonics, while controlling the fundamental is
derived and a comparison of different operating points is reported. It is observed that the modulation index can reach
unity value. The theoretical findings are verified through simulation results using the PSIM software package.
Keywords Genetic algorithms · Selective harmonic elimination (SHE) · Pulse width modulation (PWM) · AC/AC
converter
1 Introduction
The developments recently achieved in the field of power
electronics made it possible to improve the performance of
electrical system utilities. Some of the advantages offered by
using such devices are fast response, compactness and low
power demands of control circuitry. On the other hand, control by switching is often accompanied by extra losses due to
time harmonics presented in output voltage waveform, which
leads to poor power factor. An AC voltage regulator is used as
one of the power electronics system to control an output AC
M. S. A. Dahidah (B) · M. V. C. Rao
Faculty of Engineering and Technology,
Multimedia University,
75450, Ayer Keroh Lama-Melaka, Malaysia
E-mails: [email protected]; [email protected]
Tel.: +606–2523176
Fax: +606–2316552
voltage due to its simplicity and the ability of controlling large
amount of power economically [1]. The pulse-width modulation (PWM) AC chopper has been suggested as an alternative
to conventional AC regulator due to improved power factor
and the possibility of eliminating low -order harmonics with
a wide voltage control range.
The harmonic elimination methods which introduced to
AC converters are similar to those employed in PWM inverters or in AC to DC converters. The common characteristic of
the selective harmonic elimination–pulse–width modulation
(SHE–PWM) method is that the waveform analysis is performed using Fourier theory. Sets of nonlinear transcendental
equations are then derived, and the solution is obtained using
an iterative procedure, mostly by Newton–Raphson method
[2]. This method is derivative-dependent and may end in
local optima; further, a judicious choice of the initial values
alone will guarantee convergence [3]. Another approach uses
Walsh functions [4] where solving linear equations, instead
of solving nonlinear transcendental equations, optimizes the
switching angles.
A number of cost functions have been studied in the
literature such as eliminating selected harmonics [2–6], minimizing THD [6] or minimizing motor losses [7]. All abovementioned methods are off-line techniques that suffer from
the requirement of the large memory to accompany the different operating points.
The on-line real-time elimination techniques, which are
outside the scope of this paper, are presented in several publications where nonlinear curves of the nonlinear transcendental equations are replaced with piecewise representations,
with each curve made up of one or more straight line segments
[7]. It also has been reported in [8] that by using accurate modeling techniques it is possible to define an analytic expression
describing the law governing the switching angles of PWM
that is implemented using microprocessors or digital signal
processing (DSP) [8].
In contrast, microprocessor-based techniques require a
large memory, whereas most of the computational effort of
the microprocessor is spent on tedious timing tasks associated
with the generation of the switching signals for individual
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M. S. A. Dahidah and M. V. C. Rao
Fig. 1 a Circuit of Pulse-width modulation (PWM) AC converter. b Output voltage
phases of the inverter. The application of neural-network
technology has been adapted to allow construction of a simple
microprocessorless PWM that realizes optimal switching
angles for the inverters, for any value of the modulation
index [9].
The objective of this paper is to introduce a minimization
technique assisted with a hybrid genetic algorithm in order to
reduce the computational burden associated with the nonlinear transcendental equations of the SHE–PWM method. An
accurate solution is guaranteed even for a number of switching angles that is higher than other techniques would be able
to calculate for a given computational effort. The proposed
algorithm could calculate angles even beyond the point where
other methods fail to converge and hence it seems to be a
promising method for applications when overmodulation is
required. It is observed that the modulation index can reach
unity without resulting in failure of convergence.
in the PWM form at the load terminals. By the proper choice
of PWM switching angles, the fundamental component can
be controlled and a selected low order of harmonics can be
eliminated.
Setting up N switching angles per-quarter cycle allow the
elimination of (N − 1) low-order harmonics and the remaining angle is used to control the fundamental component.
Consider for example, an output voltage of AC converter
with N = 5 pulses perquarter cycle as in the case of Fig. 1b.
Fourier series can easily express the general output voltage
of the inverter as follows:
2 Circuit analysis and problem formulation
vo (ωt) =
vo (ωt) =
∞
An cos nωt + Bn sin nωt
(1)
n=1
Owing to the PWM waveform characteristics of odd function
symmetry and halfwave symmetry, the output voltage can be
reduced to:
∞
Bn sin nωt
(2)
n=1,3,5,...
The basic power circuit of a PWM AC converter that consists
of two bi-directional switches (SW1 and SW2 ) is illustrated
in Fig. 1a. The series connected switch (SW1 ) regulates the
power delivered to the load, and the parallel switch (SW2 )
provides the freewheeling path to discharge the stored energy
when the series one is turned off [1]. When a switching function shown in Fig. 1b, is applied, the output voltage appears
where, Bn is the Fourier coefficient
Hence, one can write the magnitude of the harmonic components including the fundamental as:
5
2
sin 2αi
i
B1 = 1 −
(−1) αi −
π
2
i=1
(3)
A hybrid genetic algorithm
5 2 sin(n − 1)αi
sin(n + 1)αi
Bn =
−
π
n−1
n+1
287
(4)
i=1
where n = 3, 5, . . . , 2N − 1, N is the number of switching
angles per-quarter cycle (i.e. N = 5 in this case), and αi is
the ith switching angle.
An objective function describing a measure of effectiveness of eliminating selected order of harmonics by controlling
the fundamental, i.e. eliminating low order of harmonics is
specified as
F(α1 , α2 , . . . , α N )
(5)
= (B1 − m i )2 + B32 + B52 + · · · + Bn2
where m i is the modulation index which is defined as m i =
(V1 /Vm ), noting that V1 and Vm are the values of the fundamental component and the peak of the input voltage, respectively
Minimizing (5), that is subject to the constraint of (6),
the optimal switching angles are generated and consequently
selected harmonics are eliminated. These switching angles
are generated for different values of modulation index and
then stored in look-up tables to be used to control the converter for certain operating point.
π
0 ≤ α1 ≤ α2 ≤ · · · ≤ α N ≤
(6)
2
3 Genetic algorithm
The genetic algorithm (GA) is a search mechanism based
on the principle of natural selection and population genetics,
which is transformed by three genetic operators: selection,
crossover and mutation. Each string (chromosome) has a possible solution to the problem being optimized and each bit (or
group of bits) represent a value or some variable of the problem (gene). These solutions are classified by an evaluation
function, giving better values, or fitness, to better solutions.
Each solution must be evaluated by the fitness function to
produce a value. In selection, a number of selected exact
copies of chromosomes in the current population become a
part of the offspring. In crossover, randomly selected subsections of two individual chromosomes are swapped to produce
the offspring. In mutation, randomly selected genes in chromosomes are altered by a probability equal to the specified
mutation rate.
3.1 Simple crossover
This kind of crossover operation is analogous to that of the
binary implementation. The basic one is one-cut point crossover. Let two parents be x = [x1 , x2 , . . . , xn ] and y =
[y1 , y2 , . . . , yn ]. They are crossed after the kth position; the
resulting offsprings are given by;
(7)
x = [x1 , x2 , . . . , xk , yk+1 , yk+2 , . . . , yn ]
(8)
y = [y1 , y2 , . . . , yk , xk+1 , xk+2 , . . . , xn ]
The further extensions of one-point crossover are
multi-cut-point, and uniform crossover.
3.2 Arithmetic crossover
The basic concept of this kind of operator is borrowed from
the convex set theory [10]. Simple arithmetic operators are
defined as the combination of two vectors (chromosomes)
given as:
x = λx + (1 − λ)y
x = (1 − λ)x + λy
(9)
(10)
where λ is a uniformly distributed random variable between
zero and one.
3.3 Dynamic mutation
Michaelewicz [10] proposed the mutation operator also called
nonuniform mutation. It is designed for fine-tuning capabilities aimed at achieving high precision. For a given parent
x, if the element xk of it is selected for mutation, the resulting offspring is x = [x1 . . . xk . . . xn ], where xk is randomly
selected from the two possible choices;
xk = xk + (t, xkU − xk ) or xk = xk − (t, xk − xkL ) (11)
The function (t, d x)returns a value in the range [0, d x]
such that the value of (t, d x) approaches zero as t increases.
This property causes the operator to search the space uniform
initially (when t is small) and very locally at later stages. The
function (t, d x) is given by;
d
(t, d x) = d x · r · 1 − t T
(12)
where r is a random number from [0,1]. T is the maximal generation number and d is a parameter determining the degree
of nonuniformity (usually assumed as 2).
3.4 Solution acceleration technique
The convergence of the real coded GA can be accelerated
greatly by assuming that the best solution in the population
is closest to the global optimum. If it is true, then searching
the solution spacing this neighborhood will produce solutions to the global optimum. This can be accomplished by
remapping the population after each competition stage in the
GA algorithm so that all solutions are moved a random distance towards the best solution at that generation. This results
in more solutions in the neighborhood of the minimum and
hence allows a more thorough search of its surrounds [4].
Mathematically, the solution-acceleration technique can
be described as:
x = xb + r (xb − x)
(13)
where xb = best solution vector (best individual) in the population,
x = solution vector (individual) to be remapped,
x = remapped solution vector, and
r = uniformly distributed random number between zero and
one.
288
M. S. A. Dahidah and M. V. C. Rao
4 Hybrid genetic algorithm implementation
4.2 Phase-2 (direct search optimization method) algorithm
In general, local search techniques have the advantage of
solving the problem quickly, though their results are very
much dependent on the initial starting point; therefore they
can be easily trapped in a local optimum. On the other hand,
genetic algorithm samples a large search space, climbs many
peaks in parallel, and is likely to lead search towards the
most promising area. However, GA difficulties in a fining–
tuning of local search; it spends most of the time competing
between different hills, rather than improving the solution
along a single hill that the optimal point locates. Hence using
the advantage of both the local and GA techniques, the search
algorithm can be improved both effectively and efficiently
[4,10].
The proposed hybrid genetic algorithm combines a standard real coded GA and the second phase of conventional
search technique. Real coded GA takes the place of the first
phase of the search providing the potential near optimum
solution, and phase-2 of a search technique using optimization by direct search and systematic reduction of the size of
search region. Phase-2 algorithm is applied to rapidly generate a precise solution under the assumption that the evolutionary search has generated a solution near the global
optimum.
After the phase-1 is halted, satisfying the halting condition
described in the previous section, optimization by direct
search and systematic reduction of the size of search region
method is employed in the phase-2. In the light of the solution
accuracy, the success rate, and the computation time, the best
vector obtained from the phase-1 is used as an initial point
for the phase-2.
The optimization procedure based on direct search and
systematic reduction in search region is found effective in
solving various problems in the field of nonlinear programming [10,11]. This direct search optimization procedure is
implemented as follows:
Michaelewicz [10] indicates that for real valued numerical
optimization problems floating-point representation outperform binary representations because they are more consistent, more precise and lead to faster execution. For most
application of genetic algorithms to constrained optimization problems, the real coding technique is used to represent a solution to a given problem. Hence, real coded
algorithm is considered as phase-1 [4] and it is implemented
as follows:
1. A population of N P trail solution is initialized. Each solution is taken as a real valued vector with their dimensions
corresponding to the number of variables (α1 , α2 , . . . ,
α N ). The initial components of αi are selected in accordance with a uniform distribution ranging between 0
and 1.
2. The fitness score for each solution vector αi is evaluated,
after converting each solution into corresponding switching instants αa using upper and lower bounds.
3. Roulette wheel based selection method is used to produce
N P offspring from parents.
4. Arithmetic crossover and non-uniform mutation operators are applied to offspring to generate next generation
parents.
5. The algorithm proceeds to step 2, unless the best solution
does not change for a pre-specified interval of generations.
Table 1 Hybrid genetic algorithm parameters
Phase-1
Phase-2
Population size (N P ) = 80
Max. no. of generations = 300
Crossover probability = 0.2
Mutation probability = 0.06
No. of trails (N S ) = 80
Max. no. of generations = 300
Range reduction factor = 0.01
Range = 0.5
β = 0.3
100
Switching angles (degrees)
4.1 Phase-1 (real coded) algorithm
1. The best solution vector obtained from the first phase of
the hybrid algorithm is used as an initial point α(0) for
phase-2 and an initial range vector is defined as
R(0) = RMF × Range
(14)
where Range is defined as the difference between the
upper and lower bounds (the upper and lower bounds for
each switching angle here are π/2 and 0, respectively)
and RMF is a range multiplication factor. The value of
RMF varies between 0.0 and 1.0.
2. N S trail solution vectors around α(0) are generated using
following relationship,
αi = α(0) + α(0) ·∗ rand(1, n)
(15)
∗
where, αi is the ith trail solution vector, (· ) represents
element-by-element multiplication operation, and rand
80
α3
60
α2
40
α1
20
0
0.1
0.2
0.3
0.4 0.5 0.6 0.7
Modulation index ( mi )
0.8
Fig. 2 Switching angles versus modulation index (N = 3)
0.9
1
A hybrid genetic algorithm
289
Fig. 3 The spectrum of the output waveform (case I)
100
Switching angles ( degrees)
α5
80
α4
60
α3
α2
40
α1
20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Modulation index (m i )
Fig. 4 Switching angles versus modulation index (N = 5)
(1, n) is a random vector, whose element value varies
from – 0.5 to 0.5.
3. For each feasible trail solution vector find objective function value and find the trail solution set, which minimizes
F(α) and equate it to α(0) as follows:
α(0) = αbest
(16)
where αbest is the trail solution set with minimum F(α).
4. Reduce the range by an amount given by R(0) = R(0) ∗
(1 − β). Where β is the range reduction factor, whose
typical value is 0.05.
5. The algorithm proceeds to step 2, unless the best solution
does not change for a pre-specified interval of generations.
5 Results and discussion
To verify the validity of the proposed algorithm, a program
was developed using the software package MATLAB 6.0
[12]. The program is run for a number of independent trials
on Pentium-IV computer operating at 1.4 GHz clock speed.
The parameters of GA such as crossover and mutation probability, population size and number of generations are usually selected as common values given in the literature or by
means of a trial and error process to achieve the best solution
set [3]. The parameters thus selected for the implementation
of HRCGA are tabulated in Table 1.
290
M. S. A. Dahidah and M. V. C. Rao
Fig. 5 The spectrum of the output waveform (case II)
0.25
Case I (N = 3)
HLF (p.u.)
0.2
0.15
0.1
0.05
0
Case II (N = 5)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Modulation index (mi )
Fig. 6 Harmonic loss factor (HLF) variations versus modulation index
The program is run for different values of modulation
index as well as different number of pulses per quarter cycle
of the output voltage including odd and even numbers. The
calculated switching angles are then simulated using the software package PSIM [13]. The results are presented in the
following sections.
5.1 Case I: eliminating the 3rd and 5th harmonic
In this case, three switching angles per quarter cycle (N = 3)
are considered. The fundamental component was maintained
at 0.75 p.u. Figure 2 shows the calculated switching angles for
the entire range of modulation indices (0.02 ≤ m i ≤ 0.99).
Fig. 3 illustrates the spectrum of the output voltage waveform where the absence of 3rd and 5th harmonics is clearly
evident.
5.2 Case II: eliminating the 3rd, 5th, 7th, and 9th harmonics
Five switching angles per quarter cycle (N = 5) are chosen
in this case that is aimed to eliminate the 3rd, 5th, 7th, and
9th order of harmonics while maintaining the fundamental
component at 0.60 p.u. The solution sets for the switching
angles for the entire range of modulation indices are illustrated in Fig. 4 & 5. With reference to Fig. 5, it can be clearly
seen that the selected harmonics are totally eliminated. It is
worth noting that the first significant harmonic in this case
A hybrid genetic algorithm
(i.e. 11th harmonic) is relatively high, as the fundamental is
pulled down.
5.3 Performance index
In order to indicate the usefulness and effectiveness of the
proposed technique, a quality factor has to be chosen. Several
quality factors have been reported in [2] such as Harmonic
loss factor (HLF), second order distortion factor (DF2 ) and
total harmonic distortion (THD). With the fact that most of
the loads under consideration here are induction motors; The
harmonic equivalent circuit of an induction machine can be
assumed to be its total leakage reactance at the harmonic
frequency, hence HLF is considered in this paper. The HLF,
which is proportional to total rms harmonic current, can be
defined by (17). For practical reasons, the HLF is calculated
up to 31st order of harmonics.
31
1 Vn 2
HLF =
p.u.
(17)
V1
n
n=2
where V1 and Vn are the amplitudes of the fundamental and
the nth harmonics, respectively. Furthermore, this factor is
calculated for both case I and case II and for the whole range
of the modulation index as depicted in Fig. 6. As it can be
obviously seen from Fig. 6, a poor HLF is obtained when the
converter is operated at low modulation index. However, at
higher modulation index, the HLF improve tremendously. On
the other hand, the higher the number of pulses per-quarter
cycle the lower the HLF is. However, the switching losses
would be increased if higher number of pulses are considered.
6 Conclusion
A method to generate optimal switching angles in order to
eliminate a certain order of harmonics is introduced in this
paper. A two phase genetic algorithm, namely the real-coded
and direct search is proposed to overcome the computational
burden and to ensure the accuracy of the calculated angles.
The algorithm was developed using MATLAB software and
is run for a number of times independently to ensure the
feasibility and the quality of the solution. This method was
found to be superior to conventional techniques that may fail
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291
to converge if higher pulses per quarter-wave are sought. It
is also proved to be an appropriate technique when an overmodulation operation is demanded since it still finds the solution for modulation index as high as unity. In order to prove
the feasibility and effectiveness of the proposed method,
the algorithm is tested with different operating points/cases
including three- and fine-pulse per quarter waveforms. The
harmonic loss factor (HLF) was chosen as performance index for optimality and is calculated for different operating
points. Good agreement between the theoretical findings and
the simulation results is achieved.
Acknowledgements The authors gratefully acknowledge the support
of Multimedia University (MMU) for this work.
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