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PD6.5 - CMP-667
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Bat-Inspired Optimization Approach for the Brushless DC Wheel
Motor Problem
Teodoro C. Bora1, Leandro dos S. Coelho2,3, and Luiz Lebensztajn4
1Graduação em Engenharia de Controle e Automação, Pontifícia Universidade Católica do Paraná, PR 80215-901 Brasil
2Laboratório de Automação e Sistemas, PPGEPS-PUCPR, Pontifícia Universidade Católica do Paraná, PR 80215-901 Brasil
3Departamento de Engenharia Elétrica, Universidade Federal do Paraná, PR 80215-901 Brasil
4Laboratório de Eletromagnetismo Aplicado, Escola Politécnica da Universidade de São Paulo, SP 05508-900 Brasil
This paper presents a metaheuristic algorithm inspired in evolutionary computation and swarm intelligence concepts and
fundamentals of echolocation of micro bats. The aim is to optimize the mono and multiobjective optimization problems related to the
brushless DC wheel motor problems, which has 5 design parameters and 6 constraints for the mono-objective problem and 2
objectives, 5 design parameters and 5 constraints for multiobjective version. Furthermore, results are compared with other
optimization approaches proposed in the recent literature, showing the feasibility of this newly introduced technique to high nonlinear
problems in electromagnetics.
Index Terms— Evolutionary computation, Optimization, Brushless machines.
I. INTRODUCTION
HE BAT ALGORITHM (BA) is a very interesting
approach recently proposed by Yang [1]. It is based on the
nature behavior of micro bats when looking for food. These
bats use echolocation to guide their search. This feature is
idealized and implemented by the BA to optimization
problems. This idealized idea of bats is that they generate
sound waves with some given frequencies and pulse rates.
When they tend to get close to its prey, the pulse rate
increases, while the loudness decreases. BA algorithm mimics
this idea, i.e., the design parameters are the bat position and
the prey is the objective.
In this way, it is possible to say that the BA tries to merge
the main features of two different optimization algorithms: the
Particle Swarm Optimization (PSO) and the Simulated
Annealing (SA), as shown in [1]. This paper not only
investigates the BA ability for mono-objective optimization
problems, but also proposes a BA extension to multiobjective
tasks. A Brushless DC (BLDC) Motor Wheel Optimization
Problem, an already well known problem [2], is used as
benchmark for the mono-objective and multiobjective
optimization problem.
To extend the BA algorithm for multiobjective optimization
tasks, some algorithm of the classic Non-Dominated Sorting
Genetic Algorithm – Version II (NSGA-II) are merged in BA
context. NSGA-II is one of the most efficient multiobjective
algorithms, with several applications in a lot of application
done in electromagnetic problems, for instance an induction
motor, as shown in [4].
In general, the use of the Fast Non-Dominated and
crowding distance assignment procedures guarantees an elitist
and efficient way to make the selection procedure. In the
following sections a best overview of the mono BA and multi
BA are given, besides the BLDC Motor problem is used as
benchmark, whose optimization results are presented and
compared with other algorithms, as shown in [2].
II. BRUSHLESS DC MOTOR OPTIMIZATION BENCHMARK
BLDC Motors are a well known and widely studied
application, and in this context offers a lot of results to be
compared. Some of these deals with the shape optimization by
response surface method [7], or multiobjective as decreasing
of cogging torque and increasing of torque as objectives by 3D
Equivalent Magnetic Circuit Network Method [8], or by
genetic algorithm in [9].
The analyzed problem has a Matlab model, which can be
taken in [2] for research proposes, where 78 nonlinear
equations are implemented with 5 design variables and 6
constraints to optimize in monobjective case, or with 5
constraints in multiobjective case. This mathematical model
can be completely depicted in [5].
The problem consists to design a Brushless DC Wheel
Motor by maximizing the efficiency with five optimization
parameters: bore stator diameter (), magnetic induction in
the air gap (), current density in the conductors (),
magnetic induction both in the teeth () and back iron (),
while respect some constraints involving the: total mass
(), inner diameter (), magnetics maximum current
() and temperature () and, finally the determinant
( used in the slot height calculation and which depends
on 5 design parameters.
(1)
T
Manuscript received July 1, 2011. Corresponding author: Luiz
Lebensztajn (e-mail: leb@pea.usp.br)
Digital Object Identifier inserted by IEEE
PD6.5 - CMP-667
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For the BLDC multiobjective counterpart, the total mass
constraint is turned to a minimization objective.
III. THE MONO-OBJECTIVE BA
Fig. 1 presents the proposed BA implementation
pseudocode for the mono-objective approach. For a prescribed
iteration , a set of bats is generated. Each
bat is defined by your position (, velocity (, frequency
, loudness and the rate of pulse emission ). In
order to tune the frequency, and are defined as
lower and upper bound emission frequency. is a bat position,
which can be one taken randomly among the best solutions or
from the entire population. Finally, and are parameters to
be set before the run. In this work, their values are assigned to
0.9, for both cases. The variable
is the same of in (4),
but after the use of (5), it is stored in the same memory
position but denoted, mathematically as
only to avoid
misunderstand. The variable x* is the current global best
location, which is located after comparing all the solutions
among all the bats.
While (< maximum number of iterations)
For
Generate using (2), (3) and (4)
If
Select one among the best solutions and generate a local
solution around this one, using (5)
Else
Select randomly a solution and generate a local solution
around this one, using (5)
End if
Evaluate the bats
If
Increase and reduce , using (6)
End if
End for
Rank bats to find the best solutions in population.
Find the best bat
End while
Fig. 1. Proposed mono-objective BA pseudocode based on [1].
One can realize that the proposed algorithm uses the same
dynamics of a Particle Swarm Optimization (PSO) [10], but
the loudness and rate of pulse make the BA works like the
standard PSO combined with an intensive local search, as
given in [1], which is very similar to some SA [11] approaches
for local search.
(2)
(3)
(4)
(5)
where
(6)
IV. PROPOSED MULTIOBJECTIVE BA
The mono-objective BA seems to have together the good
dynamic of PSO with the intensive local search of SA. In the
multiobjective optimization framework the tradeoff between
exploitation and exploration is even harder to be handled.
NSGA-II is the most important genetic algorithm in this
context because is able to do this balance with a cheap, in
relation to other algorithms of the same kind, computation
time in an elitist structure. So, the idea is to merge a mono-
objective algorithm (BA) in the multiobjective environment
(NSGA-II).
So, the main procedures in NSGA-II to maintain diversity
and execute an elitist selection in acceptable processing time
are maintained, i.e., the fast non-dominating sort and crowding
distance assignment are adapted in BA algorithm.
The proposed multiobjective BA follows the pseudocode
presented in Fig. 2. It is possible to say that it is based on the
mono-objective BA with the selection structure of the NSGA-
II. In this way, it allows to provide the multiobjective
optimization problem the both good features of these
algorithms, i.e., the good and fast convergence of BA and the
parameter free NSGA-II operators to ensure elitism and
diversity.
While (< maximum number of iterations)
For
Generate using (2), (3) and (4)
If
Select a non-dominated (first front) solution and
generate a local solution around this one, using (5)
Else
Select randomly a solution and generate a local solution
around this one, using (5)
End if
Evaluate the bats
End for
Sort the solution in ranks (Elitism) with the Fast Non-
Dominated Sort Algorithm [3] generating
While
Crowding – Distance Assignment () [3]
End while
Sort in the descending order in relation to the
crowding distance values
Increase and reduce , using (6)
Find the best bat
End while
Fig. 2. The proposed multiobjective BA pseudocode.
It is also important to note that there are many ways in
which a solution should dominate or not any other one in the
same population. The multiobjective BA uses Fast non-
dominating sort algorithm, i.e., the same dominance NSGA-II
criterion. Thus a solution is said to dominate another
solution if, and only if both of the following conditions
PD6.5 - CMP-667
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are satisfied:
(7)
where and are some given solution vectors in the
search space , is the objective function of
the ones. Finally operator denotes that the solution
is better (minor or greater for minimization/maximization)
than for some objective.
The proposed selection for the BA multiobjective algorithm
is shown in Fig. 3 and implies in a two stage procedure: (i)
first ranks enough to complete the future generation are
selected, and (ii) the last rank selected is cropped such that
only the solution more distant of others are selected to next
generation. The first stage guarantees the exploitation, while
the second one tries to maintain diversity in the next
population.
Fig. 3. NSGA-II selection structure implemented in BA.
V. THE OPTIMIZATION PROBLEM AND RESULTS
Before deal with a real-world problem, we will analyze a
benchmark optimization problem, proposed by Deb [3]: the
function ZDT1, as given in equation 8.
The multiobjective problem was also run 30 times, with 250
generation and 100 bats and the results are shown in Table I.
(8)
Both algoritms were run with 100 individuals and 300
iterations. One can observe that both methods are able to attain
the theoretical Pareto-set of this problem. As expected the
multi-BA has a worst spread due to its harmonic solution
pursuit characteristic.
In this work the BLDC motor problem is also optimized.
The corresponding scripts of objective functions are provided
in [2] for research purposes. The analytical problem model has
78 non-linear equations. Electric, magnetic and thermal
phenomena are taken into account.
Table II shows a comparison between several algorithms for
the mono-objective problem, where the result presented for the
BA in the best found after 30 runs. The metrics for these BA’s
runs are presented in Table III. The mean of the best results
gives an efficiency of 95.2 % which is very close to the best
result. The ratio standard deviation-mean for the efficiency is
also very small, as can be observed in Table III. Even though
the best efficiency found is not the best one, the constraint
handle was well performed by penalty in the BLDC motor
design. This feature can be modified by the adjustment of run
parameters (maximum frequency, and constants) or
penalty coefficients.
TABLE I
MULTIOBJECTIVE BA VS NSGA-II - 30 RUNS - ZDT1 FUNCTION
Algorithm
Generational distance (ϒ)
Spread (
Bat
Mean
0.043134
1.527840
Std
0.025122
0.451859
Best
0.001536
0.673935
NSGA-II
Mean
0.023381
0.528440
Std
0.001031
0.039576
Best
0.020606
0.465538
Fig. 4. NSGA-II vs Multi-BA for ZDT1
TABLE II
THE OPTIMIZATION RESULTS FROM [2] AND THE BAT ALGORITHM (BA)
Method
SQP
[2]
GA
[2]
GA &
SQP [2]
ACO
[2]
PSO
[2]
BA
201.2
201.5
201.2
201.2
202.1
202.2
0.6481
0.6480
0.6481
0.6481
0.6476
0.6535
2.0437
2.0602
2.0615
2.0437
2.0417
2.0514
1.8
1.799
1.8
1.8
1.8
1.8
0.8959
0.8817
0.8700
0.8959
0.9298
0.9792
95.32
95.31
95.31
95.32
95.32
95.31
90
3380
1644
1200
1600
1590
15
15
15
15
15
14.95
125
125
125
125
125
130.5
76
76
76
76
76
81.5
238.9
239.2
238.9
238.9
239.8
240.3
95.35
95.21
95.31
95.35
94.98
94.95
0.0235
0.0251
0.0246
0.0235
0.0253
0.0254
TABLE III
THE MONO-OBJECTIVE CASE USING BA IN 30 RUNS (BEST RESULTS)
Mean
95.23
14.97
151.91
88.11
242.85
94.93
0.0249
Standard Deviation
0.0568
0.2905
43.8935
9.3930
3.2893
1.5040
0.0091
The multiobjective problem was also run 30 times, with 250
generation and 100 bats. The results are shown in Table IV,
where the best harmonic feasible solution of all run is
presented. Fig. 5 shows the Pareto's front after one of these
runs by using the Bat Algorithm (BA) in the proposed
multiobjective version. The same problem was solved by the
NSGA-II and Fig. 6 shows the Pareto set.
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TABLE IV
THE MULTIOBJECTIVE CASE USING BA IN 30 RUNS (BEST RESULTS)
Objectives
Constraints
95.083
13.887
147.853
80.375
232.852
100.224
0.017
Fig. 5. The multiobjective case: the Pareto front for BA+NSGAII
Fig. 6. The multiobjective case: the Pareto front for the NSGA-II.
It is interesting to compare BA with the original NSGA-II,
given their correlation discussed before. For this purpose, we
will analyze one of the solutions: the best harmonic feasible
solution, which is given by the one closest to the harmonic
mean of all solution in the Pareto frontier. The harmonic mean
could be calculated with respect to the mass or to the
efficiency.
Table V shows the best harmonic solution for both methods,
after 30 runs. In first case the minor mass solution is taken,
and it gets clear that NSGA-II found a harmonic solution with
best weight than BA. In the second case, the efficiency was
emphasized and in this context BA performed better, but with
a greater mass, as expected. Also the multiobjective solution,
particularly for BA, seems to have a good quality, i.e., Pareto
set also shows high diversity.
VI. CONCLUSION
The BA approach is a new approach for optimization
problems which seems to be a robust tool. In addition,
regarding to the analyzed problem, it is also possible to state
that BA is competitive when its results are compared against
several traditional multiobjective optimization methods. It
could be explained by the fact that it merges some features of
two different optimization algorithms: PSO and SA. It has
shown a good performance in both analyzed multiobjective
optimization problems, merging the features of PSO, SA and
NSGA-II has also provided a high quality Pareto Front.
TABLE V
THE MULTIOBJECTIVE CASE USING BA IN 30 RUNS
Method
BA
NSGA-II
Minor
Mass
Major
Efficiency
Minor
Mass
Major
Efficiency
187.0768
201.4504
186.0083
194.8235
0.6636
0.6716
0.6992
0.6862
2.5054
2.1735
3.2406
2.3231
1.7466
1.7682
1.7731
1.8000
1.6000
0.8350
1.5342
1.1555
94.7759
95.2780
94.2212
95.0944
12.6082
14.9680
11.6094
13.9127
132.6306
148.9984
175.5168
162.4990
80.0924
82.5324
102.7703
84.4048
223.6776
240.8918
225.2693
234.3519
107.6317
94.9854
112.2216
99.3174
0.0131
0.0262
0.0343
0.0226
The results show a good trend for the mono-objective BA:
the efficiency is always close to 95.23% when solving the
BLDC motor problem. This result is, of course, dependent of
parameter tuning. This fact is extended for all this kind of
stochastic algorithms like PSO and SA. At the same time, no
general rule exists, as given by the No-Free Lunch Theorem.
In this context, a lot of self-tuning techniques have been
proposed, as for instance, for Tabu Search in [6] or Simulated
Annealing [11].
REFERENCES
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