IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012 947

Bat-Inspired Optimization Approach for the Brushless DC Wheel MotorProblem

Teodoro C. Bora�, Leandro dos S. Coelho���, and Luiz Lebensztajn�

Graduação em Engenharia de Controle e Automação, Pontifícia Universidade Católica do Paraná, PR 80215-901, BrasilLaboratório de Automação e Sistemas, PPGEPS-PUCPR, Pontifícia Universidade Católica do Paraná, PR 80215-901, Brasil

Departamento de Engenharia Elétrica, Universidade Federal do Paraná, PR 80215-901, BrasilLaboratório 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 fundamen-tals of echolocation of micro bats. The aim is to optimize the mono and multiobjective optimization problems related to the brushlessDC wheel motor problems, which has 5 design parameters and 6 constraints for the mono-objective problem and 2 objectives, 5 designparameters, and 5 constraints for multiobjective version. Furthermore, results are compared with other optimization approaches pro-posed in the recent literature, showing the feasibility of this newly introduced technique to high nonlinear problems in electromagnetics.

Index Terms—Brushless machines, evolutionary computation, optimization.

I. INTRODUCTION

T HE BAT ALGORITHM (BA) is a very interesting ap-proach recently proposed by Yang [1]. It is based on the

nature behavior of micro bats when looking for food. These batsuse echolocation to guide their search. This feature is idealizedand implemented by the BA to optimization problems. This ide-alized idea of bats is that they generate sound waves with somegiven frequencies and pulse rates. When they tend to get close toits prey, the pulse rate increases, while the loudness decreases.BA algorithm mimics this idea, i.e., the design parameters arethe bat position and the prey is the objective.

In this way, it is possible to say that the BA tries to merge themain features of two different optimization algorithms: the Par-ticle Swarm Optimization (PSO) and the Simulated Annealing(SA), as shown in [1]. This paper not only investigates the BAability for mono-objective optimization problems, but also pro-poses a BA extension to multiobjective tasks. A Brushless DC(BLDC) Motor Wheel Optimization Problem, an already wellknown problem [2], is used as benchmark for the mono-objec-tive and multiobjective optimization problem.

To extend the BA algorithm for multiobjective optimizationtasks, some algorithm of the classic Non-Dominated SortingGenetic Algorithm—Version II (NSGA-II) are merged in BAcontext. NSGA-II is one of the most efficient multiobjective al-gorithms, with several applications in a lot of application donein electromagnetic problems, for instance an induction motor,as shown in [4].

In general, the use of the Fast Non-Dominated and crowdingdistance assignment procedures guarantees an elitist and effi-cient way to make the selection procedure. In the following sec-tions a best overview of the mono BA and multi BA are given,besides the BLDC Motor problem is used as benchmark, whoseoptimization results are presented and compared with other al-gorithms, as shown in [2].

Manuscript received July 06, 2011; revised October 25, 2011; acceptedNovember 04, 2011. Date of current version January 25, 2012. Correspondingauthor: L. Lebensztajn (e-mail: leb@pea.usp.br).

Digital Object Identifier 10.1109/TMAG.2011.2176108

II. BRUSHLESS DC MOTOR OPTIMIZATION BENCHMARK

BLDC Motors are a well known and widely studied applica-tion, and in this context offers a lot of results to be compared.Some of these deals with the shape optimization by responsesurface method [7], or multiobjective as decreasing of coggingtorque and increasing of torque as objectives by 3D EquivalentMagnetic Circuit Network Method [8], or by genetic algorithmin [9].

The analyzed problem has a Matlab model, which can betaken in [2] for research proposes, where 78 nonlinear equa-tions are implemented with 5 design variables and 6 constraintsto optimize in monobjective case, or with 5 constraints in mul-tiobjective case. This mathematical model can be completelydepicted in [5].

The problem consists to design a Brushless DC Wheel Motorby maximizing the efficiency with five optimization param-eters: bore stator diameter , magnetic induction in the airgap , current density in the conductors , magnetic induc-tion both in the teeth and back iron , while respectsome constraints involving the: total mass , inner diam-eter , magnetics maximum current and tempera-ture and, finally the determinant used in the slotheight calculation and which depends on 5 design parameters

(1)

For the BLDC multiobjective counterpart, the total mass con-straint is turned to a minimization objective.

III. THE MONO-OBJECTIVE BA

Fig. 1 presents the proposed BA implementation pseudocodefor the mono-objective approach. For a prescribed iteration (t), aset of bats is generated. Each bat is definedby your position , velocity , frequency , loudness

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948 IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012

Fig. 1. Proposed mono-objective BA pseudocode based on [1].

and the rate of pulse emission . In order to tune the fre-quency, and are defined as lower and upper boundemission frequency. is a bat position, which can be one takenrandomly among the best solutions or from the entire popula-tion. Finally, and are parameters to be set before the run. Inthis work, their values are assigned to 0.9, for both cases. Thevariable is the same of in (4), but after the use of (5), itis stored in the same memory position but denoted, mathemat-ically as only to avoid misunderstand. The variableis the current global best location, which is located after com-paring all the solutions among all the bats.

One can realize that the proposed algorithm uses the samedynamics of a Particle Swarm Optimization (PSO) [10], but theloudness and rate of pulse make the BA works like the stan-dard PSO combined with an intensive local search, as given in[1], which is very similar to some SA [11] approaches for localsearch

(2)

(3)

(4)

(5)

(6)

IV. PROPOSED MULTIOBJECTIVE BA

The mono-objective BA seems to have together the gooddynamic of PSO with the intensive local search of SA. In themultiobjective optimization framework the tradeoff betweenexploitation and exploration is even harder to be handled.NSGA-II is the most important genetic algorithm in this con-text because is able to do this balance with a cheap, in relationto other algorithms of the same kind, computation time inan elitist structure. So, the idea is to merge a mono-objectivealgorithm (BA) in the multiobjective environment (NSGA-II).

Fig. 2. The proposed multiobjective BA pseudocode.

So, the main procedures in NSGA-II to maintain diversityand execute an elitist selection in acceptable processing timeare maintained, i.e., the fast non-dominating sort and crowdingdistance assignment are adapted in BA algorithm.

The proposed multiobjective BA follows the pseudocode pre-sented in Fig. 2. It is possible to say that it is based on themono-objective BA with the selection structure of the NSGA-II.In this way, it allows to provide the multiobjective optimiza-tion problem the both good features of these algorithms, i.e.,the good and fast convergence of BA and the parameter freeNSGA-II operators to ensure elitism and diversity.

It is also important to note that there are many ways in which asolution should dominate or not any other one in the same pop-ulation. The multiobjective BA uses Fast non-dominating sortalgorithm, i.e., the same dominance NSGA-II criterion. Thus asolution is said to dominate another solution if, andonly if both of the following conditions are satisfied:

(7)where and are some given solution vectors in thesearch space, , is the objective functionof 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 algorithmis shown in Fig. 3 and implies in a two stage procedure: (i) firstranks enough to complete the future generation are selected, and(ii) the last rank selected is cropped such that only the solutionmore distant of others are selected to next generation. The firststage guarantees the exploitation, while the second one tries tomaintain diversity in the next population.

BORA et al.: BAT-INSPIRED OPTIMIZATION APPROACH FOR THE BRUSHLESS DC WHEEL MOTOR PROBLEM 949

Fig. 3. NSGA-II selection structure implemented in BA.

Fig. 4. NSGA-II vs Multi-BA for ZDT1.

TABLE IMULTIOBJECTIVE BA VS NSGA-II—30 RUNS—ZDT1 FUNCTION

V. THE OPTIMIZATION PROBLEM AND RESULTS

Before deal with a real-world problem, we will analyze abenchmark optimization problem, proposed by Deb [3]: thefunction ZDT1, as given in (8).

The multiobjective problem was also run 30 times, with 250generation and 100 bats and the results are shown in Table I.

(8)

Both algoritms were run with 100 individuals and 300 itera-tions. One can observe that both methods are able to attain the

TABLE IITHE OPTIMIZATION RESULTS FROM [2] AND THE BAT ALGORITHM (BA)

TABLE IIITHE MONO-OBJECTIVE CASE USING BA IN 30 RUNS (BEST RESULTS)

TABLE IVTHE MULTIOBJECTIVE CASE USING BA IN 30 RUNS (BEST RESULTS)

theoretical Pareto-set of this problem. As expected the multi-BAhas a worst spread due to its harmonic solution pursuit charac-teristic.

In this work the BLDC motor problem is also optimized.The corresponding scripts of objective functions are providedin [2] for research purposes. The analytical problem model has78 non-linear equations. Electric, magnetic and thermal phe-nomena are taken into account.

Table II shows a comparison between several algorithms forthe mono-objective problem, where the result presented for theBA in the best found after 30 runs. The metrics for these BA’sruns are presented in Table III. The mean of the best results givesan efficiency of 95.2% which is very close to the best result.The ratio standard deviation-mean for the efficiency is also verysmall, as can be observed in Table III. Even though the best effi-ciency found is not the best one, the constraint handle was wellperformed by penalty in the BLDC motor design. This featurecan be modified by the adjustment of run parameters (maximumfrequency, and constants) or penalty coefficients.

The multiobjective problem was also run 30 times, with 250generation 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 usingthe Bat Algorithm (BA) in the proposed multiobjective version.The same problem was solved by the NSGA-II and Fig. 6 showsthe Pareto set.

It is interesting to compare BA with the original NSGA-II,given their correlation discussed before. For this purpose, wewill analyze one of the solutions: the best harmonic feasible

950 IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012

Fig. 5. The multiobjective case: The Pareto front for BA+NSGAII.

Fig. 6. The multiobjective case: the Pareto front for the NSGA-II.

solution, which is given by the one closest to the harmonic meanof all solution in the Pareto frontier. The harmonic mean couldbe 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, andit gets clear that NSGA-II found a harmonic solution with bestweight than BA. In the second case, the efficiency was empha-sized and in this context BA performed better, but with a greatermass, as expected. Also the multiobjective solution, particularlyfor BA, seems to have a good quality, i.e., Pareto set also showshigh diversity.

VI. CONCLUSION

The BA approach is a new approach for optimization prob-lems which seems to be a robust tool. In addition, regarding tothe analyzed problem, it is also possible to state that BA is com-petitive when its results are compared against several traditionalmultiobjective optimization methods. It could be explained bythe fact that it merges some features of two different optimiza-tion algorithms: PSO and SA. It has shown a good performance

TABLE VTHE MULTIOBJECTIVE CASE USING BA IN 30 RUNS

in both analyzed multiobjective optimization problems, mergingthe features of PSO, SA and NSGA-II has also provided a highquality Pareto Front.

The results show a good trend for the mono-objective BA: theefficiency is always close to 95.23% when solving the BLDCmotor problem. This result is, of course, dependent of param-eter tuning. This fact is extended for all this kind of stochasticalgorithms like PSO and SA. At the same time, no general ruleexists, as given by the No-Free Lunch Theorem. In this context,a lot of self-tuning techniques have been proposed, as for in-stance, for Tabu Search in [6] or Simulated Annealing [11].

REFERENCES

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