Multiobjective Particle Swarm Approach for the Design of a Brushless ...

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010

Multiobjective Particle Swarm Approach for the Design of a Brushless DC Wheel Motor Leandro dos Santos Coelho1 , Leandro Zavarez Barbosa2 , and Luiz Lebensztajn2 Automation and Systems Laboratory, LAS/PPGEPS, Pontifical Catholic University of Paraná, 80215-901 Curitiba, Paraná, Brazil Laboratório de Eletromagnetismo Aplicado, LMAG-PEA, Escola Politécnica da Universidade de São Paulo, 0550-900 São Paulo, SP, Brazil The roots of swarm intelligence are deeply embedded in the biological study of self-organized behaviors in social insects. Particle swarm optimization (PSO) is one of the modern metaheuristics of swarm intelligence, which can be effectively used to solve nonlinear and non-continuous optimization problems. The basic principle of PSO algorithm is formed on the assumption that potential solutions (particles) will be flown through hyperspace with acceleration towards more optimum solutions. Each particle adjusts its flying according to the flying experiences of both itself and its companions using equations of position and velocity. During the process, the coordinates in hyperspace associated with its previous best fitness solution and the overall best value attained so far by other particles within the group are kept track and recorded in the memory. In recent years, PSO approaches have been successfully implemented to different problem domains with multiple objectives. In this paper, a multiobjective PSO approach, based on concepts of Pareto optimality, dominance, archiving external with elite particles and truncated Cauchy distribution, is proposed and applied in the design with the constraints presence of a brushless DC (Direct Current) wheel motor. Promising results in terms of convergence and spacing performance metrics indicate that the proposed multiobjective PSO scheme is capable of producing good solutions. Index Terms—Brushless machines, optimization methods.

I. INTRODUCTION

N RECENT decades, many bio-inspired algorithms have been developed for optimization of electromagnetic devices, such as evolutionary algorithms and swarm intelligence paradigms. These methods have attracted a great deal of attention, because of their high potential for optimization problems in environments, which have been resistant to solution by classical mathematical programming techniques. The design of electromagnetic devices provides many optimization problems involving multiple objectives (multiobjective optimization problems, MOPs), which should be optimized simultaneously. There is a set of compromise solutions in multiobjective optimization problem and none of the corresponding trade-offs can be said to be better than the others in the absence of preference information. A considerable number of multiobjective algorithms have been proposed for solving MOPs because they can deal simultaneously with a set of possible solutions in a single run instead of a series of separate runs as in the traditional optimization techniques. In recent years, there have been several attempts to apply PSO to MOPs [1]–[6], including applications in electromagnetics [7]–[9]. Furthermore, Sierra and Coello [10] presented a recent survey about PSO design to MOPs. PSO is a stochastic optimization technique of swarm intelligence developed by Eberhart and Kennedy [11] that is inspired by the sociological behavior of swarm of bees, flock of birds and/or school of fish during their food-searching activities. At

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Manuscript received December 21, 2009; accepted February 13, 2010. Current version published July 21, 2010. Corresponding author: L. Lebensztajn (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2044145

a fundamental level, the key difference between PSO and classical genetic algorithms is the way in which new samples are generated. In genetic algorithms, new samples are produced by some recombination of selected ‘parent’ solutions, which may then go on to replace members of the population. In PSO, new samples (particles) are generated by perturbation of existing solutions. Generally, PSO approaches present reduced memory requirement, computationally effective and are easier to implement when compared to typical evolutionary algorithms. PSO is based on the hypothesis that members of a swarm can profit from their past experiences and the experiences of other particles. During the exploration of the search space, each particle has access to two kinds of information: the best potential solution (gbest, global best) that it has encountered and the best potential solution encountered by its vicinity (pbest, personal best). The aim of this paper is to propose an enhanced multiobjective PSO (EMOPSO) approach based on Pareto dominance, archiving external and truncated Cauchy distribution to the design with the constraints presence of a brushless DC wheel motor. We compare our results with those generated by another multiobjective PSO (RNMOPSO) proposed by Raquel and Naval [12]. RNMOPSO incorporates the concept of nearest neighbor density estimator for selecting the global best particle, mutation operator and the constraint-handling technique from the NSGA-II [13]. The EMOPSO proposed in this paper is inspired in [12]. The remainder of the paper is organized as follows. In Section II, the fundamentals of EMOPSO are detailed. The analyzed problem is presented in Section III. Analysis of optimization results are given in Section IV. Finally, the conclusion is presented in Section V. II. FUNDAMENTALS OF EMOPSO The RNMOPSO algorithm could have difficulties in striking a balance between exploration and exploitation. Hence, the

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DOS SANTOS COELHO et al.: MULTIOBJECTIVE PARTICLE SWARM APPROACH FOR THE DESIGN OF A BRUSHLESS DC WHEEL MOTOR

global search ability of PSO algorithm becomes more restricted. To address this problem, some improvements proposed in EMOPSO are made on RNMOPSO as described. The EMOPSO uses a truncated Cauchy distribution to update the social and cognitive factors. EMOPSO does not use the mutation operator. The implementation of EMOPSO is based on following steps: , with positions and i) Initialize a swarm (population), velocities using a generator of random solutions based on uniform distribution. Set the initial value of counter of iterations (generations), ; ii) Evaluate the particles and store nondominated ones in an archive with size . Pareto-dominance concept is used to evaluate the fitness of each particle and thus determine which particles should be selected to store in the archive of non-dominated solutions. The archive absorbs superior current non-dominated solutions and eliminates inferior solutions in the archive through interacting with the generational population, for every iteration. iii) Compute the crowding distance of each member of ; iv) Sort in descending crowding distance order; v) Randomly select the gbest for the swarm form a specified top portion (e.g., top 20%) for the sorted , and store its position in gbest. vi) Update velocities and positions according to

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TABLE I OPTIMIZATION VARIABLES

(1) (2) is the inertia weight; stands for the velocity of the -th particle, stands for the position of the -th particle of population, and represents the best previous position of the -th particle. Positive constants and are the cognitive and social factors, respectively, which are the acceleration constants responsible for varying the particle velocity towards pbest and gbest, respectively. Index represents the index of the best particle among all the particles in the swarm. Variables and are two random numbers using truncated Cauchy distribution in range [0,1]. vii) Evaluate the particles in swarm; viii) Insert all new nondominated solution into if they are not dominated by any of the stored solutions; ix) Update the iteration counter, ; x) Return to Step (iii) until a criterion is met. In this work, a maximum number of iterations (generations), , is adopted. It is important to emphasize that the Cauchy distribution (see Step vi) is used in EMOPSO to introduce large variations in swarm of particles considering rate due to its ability to perform longer jumps with higher probability. The Cauchy density function is similar to that of Gaussian density function, but it approaches towards the axis so slowly that its variance tends to infinity. The one dimension Cauchy density function centered at origin is defined by where

,

(3)

Fig. 1. The geometry of the brushless DC wheel motor.

where is the scale parameter. In this paper, in EMOPSO design, the parameter is adopted equal to 1, and it is used a truncated (normalized) value of between [0, 1]. III. THE ANALYZED PROBLEM The optimization problem is the design of a brushless DC wheel motor for a race solar car [14], which has 10 optimization variables and 6 constraints. Fig. 1 shows the geometry of the motor. Table I shows the optimization variables. On [14], the authors propose three different benchmarks with only one objective: maximize the efficiency. The first one has 5 optimizaand the others are fixed. The tion variables authors solve this version of the optimization problem and propose two other optimization problems: the second one, which concerns all the variables except the number of poles (fixed to 6). On the third proposed benchmark, every optimization variables are used. The adopted model for the benchmarks is analytical. This is very useful in the first steps of the design process. So, saturation, eddy current in the magnets and other effects are neglected. A natural extension of the problem was proposed on [15]: it consists on a multiobjective problem, which consists on the maximization of the efficiency and the minimization of the mass. They proposed a multiobjective problem with five optimization variables, with some constraints. Here we have

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adopted some minor changes on the multiobjective problem: the constraints are maintained and we take into account every optimization variables described on Table I, except the number of poles (fixed to 6). The great majority of the optimization variables are associ, which is linked to the ated to the magnetic circuit unless converter. In this problem, the constraints play a very important role and they have a very different nature. There are sizing constraints, like the internal diameter and the total axial length, but there is also a thermal constraint and some magnetic constraints, like the average magnetic induction in the stator back yoke iron and electrical constraints, like the current rise time. This problem is typically a multidisciplinary optimization one. that will represent the Two objective functions are defined: mass and that will represent the efficiency. The multiobjective problem could be written as

TABLE II SPACING AND EUCLIDIAN DISTANCES INDEXES FOR RNMOPSO AND EMOPSO APPROACHES USING VALUES OF PARETO FRONT SOLUTIONS OBTAINED IN 30 RUNS

(4) Submitted to: mm mm

mm (5) Other details of this problem could be obtained on [14] and some results with the 5 variables problem with several multiple optimization methodologies could be seen on [15]. IV. RESULTS The optimization problem is the design of a brushless DC wheel motor for a race solar car [14], which has 10 optimization variables and 6 constraints (see Fig. 1). The objective functions are the minimization of the mass and the maximization of the efficiency . In this work, we include a penalty function and to punish unfeasible solutions given by (5). The on penalty function measures how much the solution has violated the constraints. Due to a weight, only feasible solutions could become a nondominated solution. We adopted the following control parameters for RNMOPSO [3] and EMOPSO for the motor design: population (swarm) size particles, factors , size of archive , and iterations. The experiments were conducted for 30 independent runs to evaluate the performance of RNMOPSO and EMOPSO in MATLAB (MathWorks) environment. All the programs were run under Windows XP on a 3.2 GHz Pentium IV processor with 2 GB of random access memory. In comparative studies, the choice of performance metrics is very important. The optimization goal of multiobjective algorithms is (i) to minimize distance between the generated and Pareto front, (ii) to obtain a good distribution and (iii) to obtain a good spread. Thirty independent simulation runs with different initial conditions were performed to evaluate the performance of RNMOPSO and EMOPSO on the brushless DC wheel motor design.

Fig. 2. Pareto set points using RNMOPSO and EMOPSO.

Table II shown the spacing and Euclidian distances indexes for RNMOPSO and EMOPSO approaches. RNMOPSO presented minor value of mean spacing between the objective functions and , but the EMOPSO obtained best value in terms of mean normalized Euclidian distance between and . On other hand, simulation results were presented in Fig. 2 showed and the non-dominated solutions (30 runs) obtained by EMOPSO dominated the solutions obtained by RNMOPSO, i.e., typically for a given mass value, the EMOPSO solution has always a better efficiency than the correspondent RNMOPSO solution. The distribution of the nondominated solutions along the variable space is another important point of analysis. Fig. 3 shows the histograms of the optimization variables for the nondominated solutions obtained by the EMOPSO, except of the magand the average magnetic innetic induction on the teeth duction in the rotor back iron . All the non-dominated solutions have the following characteristic: the value of is very close to 1.8 T and the value of is very close to 1.6 T, as expect from classical electrical machines design, so they are not shown in the histograms. From Fig. 3, we can observe that there is net trend towards 0.67 T on the non-dominated solutions when the magnetic induction in the airgap is analyzed. The same conclusion could be observed when the airgap value is analyzed: a typical value is 0.3 mm, i.e., good machines must have a very low airgap, as expected. The optimization variables that are showed in the histograms have a ratio standard deviation-mean in the range 1.9% (for ) ). The histogram of the current density shows to 25% (for

DOS SANTOS COELHO et al.: MULTIOBJECTIVE PARTICLE SWARM APPROACH FOR THE DESIGN OF A BRUSHLESS DC WHEEL MOTOR

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of EMOPSO algorithm was compared with RNMOPSO. The solution sets of Pareto of both methods are competitive on a problem with high dimension and several constraints. EMOPSO outperforms RNMOPSO when the mean normalized Euclidian is used, but RNMOPSO presents a more uniform Pareto set due to the better mean space. In the future we would like to investigate and compare the robustness and efficiency of EMOPSO for different MOPs in electromagnetics with higher number of objectives. REFERENCES

Fig. 3. Histogram of some optimization variables.

an interesting conclusion: for quite every value of , it is possible to build an optimal motor in the efficiency and mass sense. Once the Pareto set is obtained, a decision maker algorithm, see for example [18], could be adopted. V. CONCLUSION PSO is an optimization technique inspired by social behavior observable in nature. PSO has shown to be an efficient and robust optimization algorithm in MOPs [12]. In this context, archiving has been studied in design of evolutionary algorithms and swarm intelligence approaches to solve MOPs [16], [17]. In these methods, the non-dominated (best) solutions of each generation are kept in an external population, called archive. In this paper, the RNMOPSO and EMOPSO approaches based on archiving are compared. The EMOPSO is inspired in the RNMOPSO because the concept of nearest neighbor density estimator for selecting the global best particle is also used. The main difference between the methods is: the EMOSO has no mutation operator, but it has a normalized Cauchy distribution in range [0,1] used on the cognitive and social factors. To show the validity and efficiency of the proposed optimization algorithm for the analyzed case, the effectiveness

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