5, MAY 2013 2105Multiguiders and Nondominate Ranking Differential Evolution Algorithm for Multiobjective Global Optimization of Electromagnetic Problems Nyambayar Baatar , Minh-Trien Pha
Trang 1IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 2105
Multiguiders and Nondominate Ranking Differential Evolution Algorithm for Multiobjective Global Optimization of Electromagnetic Problems
Nyambayar Baatar , Minh-Trien Pham , and Chang-Seop Koh , Senior Member, IEEE
College of Electrical and Computer Engineering, Chungbuk National University, Cheongju, Chungbuk 361-763, Korea
University of Engineering and Technology, Vietnam National University, Hanoi 100000, Vietnam
The differential evolution (DE) algorithm was initially developed for single-objective problems and was shown to be a fast, simple algorithm In order to utilize these advantages in real-world problems it was adapted for multiobjective global optimization (MOGO) recently In general multiobjective differential evolutionary algorithm, only use conventional DE strategies, and, in order to optimize performance constrains problems, the feasibility of the solutions was considered only at selection step This paper presents a new multi-objective evolutionary algorithm based on differential evolution In the mutation step, the proposed method which applied multiguiders instead of conventional base vector selection method is used Therefore, feasibility of multiguiders, involving constraint optimization problems, is also considered Furthermore, the approach also incorporates nondominated sorting method and secondary population for the nondominated solutions The propose algorithm is compared with resent approaches of multiobjective optimizers in solving multi-objective version of Testing Electromagnetic Analysis Methods (TEAM) problem 22.
Index Terms—Differential evolution, multiguiders, multiobjective optimization, nondominated ranking, Testing Electromagnetic
Analysis Methods (TEAM) problem 22.
I INTRODUCTION
I N ENGINEERING application, optimization problems
in-volving multiple objectives together with constraints are
popular Therefore, many multiobjective global optimization
(MOGO) algorithms have been proposed In order to apply the
DE algorithm for solving MOGO problems, the original scheme
has to be modified since the multiobjective problems do not
consist of single solution Instead, in multiobjective
optimiza-tion, a set of different solutions should be founded and called
Pareto-optimal front There are two issues when designing a
multiobjective evolutionary algorithm: population diversity and
survivor selection The first issue is directly related to the
ques-tion of how to guide the search towards the Pareto-optimal front
[1] The second one addresses the question of which individual
will be kept during the evolution process
In the past, a wide variety of evolutionary algorithms (EAs)
have been used to solve multiobjective optimization problems
[2] However, from the several types of EAs available, few
re-searchers have attempted to extend DE [3] to solve
multiobjec-tive optimization problems DE has been very successful in the
solution of a variety of continuous (single-objective)
optimiza-tion problems in which it has shown a great robustness and a
very fast convergence These are precisely the characteristics
of DE that make it attractive to extend to solve multiobjective
optimization problems DE has been adapted to solve MOGO
in several ways In the early approaches (PDE [4] and GDE
[5]), only the concept of Pareto dominance was used to
com-pare the individuals The candidate replaced its com-parent only if
it (weakly) dominated it Otherwise, it was discarded This is
a rather strict demand, especially when the number of
objec-tives is high Many subsequent approaches (PDEA [6], MODE
Manuscript received November 21, 2012; revised December 26, 2012;
ac-cepted January 08, 2013 Date of current version May 07, 2013 Corresponding
author: C.-S Koh (e-mail: kohcs@cbnu.ac.kr).
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.2013.2240285
[7], and NSDE-DCS [8]) used nondominated sorting and/or the crowding distance metric to calculate the fitness of individuals Therefore we proposed multiguiders nondominated ranking differential evolution algorithm (MG-NRDE) for solving MOGO problems
In mutation step, the proposed method introduces new base vector selection method for constrained multiobjective opti-mization by adopting multiguiders Additionally, the approach also incorporates nondominated sorting method [9] and sec-ondary population for the nondominated solutions to archive Pareto front solutions The proposed algorithm is compared with recent approaches of multiobjective optimizers in solving multiobjective version of TEAM problem 22
The remainder of this paper is organized as follows Section II provides fundamentals of the MOGO problems and DE algo-rithm In Section III we described the proposed multiguiders nondominated ranking DE in detail TEAM problem 22 and comparison results are provided in Section IV Finally Section V contains our conclusions
II FUNDAMENTALS OFMOGO PROBLEMS ANDDE Some fundamentals and basic definitions related to this work are introduced in following subsections
A MOGO Problems
A general MOGO problem contains a number of conflicting objectives, for example, to be minimized and optional con-straints to be satisfied Mathematically, a MOGO problem is formulated as follows:
Minimize subject to
(1) where is the vector of design variables, and are the numbers of the objectives and constraints, respectively
In practical applications, there is no solution that can mini-mize all of the objectives simultaneously As a result, mul-0018-9464/$31.00 © 2013 IEEE
Trang 22106 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013
tiobjective optimization problems tend to be characterized by a
family of alternatives that must be considered equivalent in the
absence of information concerning the relevance of each
objec-tive relaobjec-tive to the others [1] These alternaobjec-tives are referred to
as Pareto optimal solutions
A multiobjective optimization problem, on the other hand,
has a set of optimal solutions which every member is not
domi-nated by others All the members are optimal from the viewpoint
of one (or more) objective(s), but none of them is optimal for all
the objectives The choice among the Pareto-optimal solutions
belongs to a designer’s decision [1], [9]
The target of a MOGO algorithm, therefore, is to converge to
the true Pareto-front and to provide a good distribution of the
solutions on the entire Pareto-front
B Deferential Evolution Algorithm
The differential evolution algorithm is a novel parallel direct
search method, which utilizes parameter vectors as a
The crucial idea behind DE is a scheme for generating trial
parameter vectors DE generates new parameter vectors by
adding a weighted difference vector between two
popula-tion members to a third member Currently, there are several
variants of DE The particular variant described throughout
this section is the “DE/rand/1/bin” scheme Each individual
) is a -dimensional vector with parameter values determined randomly and uniformly in
the search space
(2) For each target vector a mutant vector
) is generated by mutation
Note that indexes have to be different from each other, is
called base vector index, and , and are difference vector
indexes
After mutation, the binominal crossover is applied to generate
the trial vector The specified process is shown in (3)
otherwise
(3) where a trial vector is generated from the mutant vector
and its target vector based on probabilistic parameter
selection Cr is a user-specified crossover factor in the range
[0,1) and is a randomly chosen integer in the range [1,
] to ensure that the trial vector will differ from its
corre-sponding target vector by at least one parameter
if
The fitness value of each trial vector is compared
to that of its corresponding target vector in the current
population, and one with better fitness will be selected for next
iteration The selection operation is expressed in (4)
III PROPOSED MULTIGUIDERS NONDOMINATED
RANKINGDE ALGORITHM
In order to provide good Pareto front, the suggested algorithm incorporates nondominated ranking and multiguiders methods The proposed multiguiders nondominated ranking DE (MG-NRDE) algorithm keeps two populations: the main population which is the target population (used to search Pareto optimum solutions) and external population (to archive nondominated solutions and provide guiders)
Additionally, in mutation step, we considered feasibility of solution when we select the guiders; this action will be taking into account in case of performance constraint problems Furthermore, when number of external solutions exceeds its maximum value, an improved pruning method [10] is used to remove the solutions with small crowding distance, one by one until number of solutions equals to its maximum value
Step 1: Initialize the target population.
• Generate target population with randomly uniform, and set the iteration counter
• Evaluate all objective function and constraint values, and apply nondominated sorting to rank the all individ-uals in the current population, and calculate crowding distance in each rank
• Store nondominated solutions into the external archive , Calculate the crowding distance in objective space (objective crowding distance) for all members in
Step 2: Generate mutant populations
• Randomly select the first guider (in conventional DE would be called base vector) from the top 10% of solution with the big crowding distances
• Second guider required only when the fist guider is not extreme solution [1]
• Between the two nondominated solutions beside in Pareto-front, as shown in Fig 1, the one with bigger crowding distance is selected as the second guider And the feasibility of the guiders must be considered And then generated by using (5)
(5) where is -th mutant vector and are guiders
and are randomly selected difference
random numbers in [0, 1]; however, is the uniform distribution and is Gaussian distribution , when the is an extreme solution
Step 3: Crossover operation.
• Generate trial vector using binominal crossover ex-pressed in (3)
Step 4: Selection operator.
• Combine target population and trial population
into 2 population
• Update the number of feasible solutions
method for all feasible solutions to select number
of individuals for the next iteration
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Fig 1 Selection of two guiders and population ranking.
— If : the combination of feasible solutions
and solutions with lowest sum of constraint
violation solutions will be selected the next iteration
— In case of unconstrained problem: From the
com-bined 2 population, individuals with lower
rank will be will be survived for the next iteration
— Note that Rank1 solutions in the Fig 1 are the Pareto
front solutions in the current iteration
— If the number of solutions in the Rank1 is greater than
, we apply improved pruning method to remove
the solutions with small crowding distance [10]
Step 5: Update nondominated solutions.
• External archive absorbs superior current
nondomi-nated solutions and eliminates inferior ones using
con-strained domination
• If the number of solutions in A is bigger than
predeter-mined value, reduce it by repeating following process:
— calculate the crowding distance for all members in ;
— remove the nondominated solutions which have small
crowding distance by using pruning method [3]
Step 6: Termination check.
• If termination condition (maximum number of iteration)
is not satisfied, go to Step 2, otherwise terminate
IV OPTIMIZATIONRESULT
In order to validate the proposed algorithm, we adopted
three parameter version of multiobjective TEAM problem 22
reported in [11] The TEAM problem 22 is an optimal design
of a superconducting magnetic energy storage device (SMES)
(SMES) configuration shall be optimized with respect to the
following objectives
• The stored energy in the device should be 180 MJ
• The magnetic field must not violate a certain physical
con-dition which guarantees superconductivity
• The stray field (measured at a distance of 10 meters from
the device) should be as small as possible
Multiobjective TEAM problem 22 is expressed in (6)
Minimize
subject to
(6)
Fig 2 SMES Configuration and Design variables.
Fig 3 Critical curve of the superconductor The true critical curve is in contin-uous black The dotted line represents the linear approximation to the previous curve used as quench condition limit in this paper.
TABLE I
V ARIABLE R ANGES AND V ALUES U SED
where the reference stored energy is MJ and are current density and magnetic flux density in the -th coil TEAM problem 22 is composed of two coils with opposite cur-rent densities The first coil is charged to store the energy and the second should be designed to diminish the high magnetic stray caused by the first coil The classical configuration of the SMES device can be obtained from Fig 2
The superconducting material should not violate the quench condition that links together the value of the current density and the maximum value of magnetic flux density, as shown in Fig 3 The critical curve has been approximated by constraints in (6) The range of the three design variables which defines the size and position of the outer coil of the device and other fixed pa-rameters are shown in Table I
Optimal result of proposed algorithm is compared with those from MultiGuiders Cross-search MOPSO [1] and Gaussian MOPSO [12] The optimization parameters are set exactly same as in [1] for all algorithms, i.e., 30 individuals, maximum number of iteration 200, and external archive of 100
Trang 42108 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013
Fig 4 Pareto-front obtained by MGC-MOPSO, G-MOPSO, and MG-NRDE.
TABLE II
C OMPARISONS OF E XTREME S OLUTIONS
TABLE III
C OMPARISONS OF S PACING M ETRIC U SING C ROWDING D ISTANCE
Fig 4 shows the obtained Pareto front solutions of
G-MOPSO, MGC-MOPSO and MG-NRDE respectively
In order to show the difference more clearly, every four others
of Pareto front solution is shown in the figures The solutions
obtained by MG-NRDE are more uniform distribution than
those of MGC-MOPSO and G-MOPSO Furthermore, it is also
proven in comparison of spacing metric by using crowding
distances of Pareto front solutions
Graphically we can see the proposed MG-NRDE obtained
better uniformly distributed Pareto front solutions while
G-MOPSO and MGC-MOPSO obtained Pareto fronts with
some crowding and discontinuity But we can see the lower part
of Pareto front obtained by proposed method is not showing
the good results than other methods except the extreme
so-lution However, the proposed MG-NRDE algorithm shows
better result in the middle and upper part of the Pareto front
Numerically, we compared the extreme solutions in Table II
Comparison of space metric using crowding distance is shown
in the Table III It is revealed that Pareto front obtained by
MG-NRDE has better distribution than these who obtained
by G-MOPSO and MGC-MOPSO Additionally, the proposed
MG-NRDE has smaller standard deviation in the Pareto front It
can be said that the proposed algorithm can find better solutions
regarding the multiple objectives
V CONCLUSION
In this paper, the multiguiders nondominated ranking differ-ential evolution algorithm (MG-NRDE) is developed for mul-tiobjective optimization problems The proposed algorithm is compared with recent approaches of multiobjective optimizers
in solving multiobjective version of TEAM problem 22 Our proposed approach was able to produce results that competi-tive with respect to other approaches such us MGC-MOPSO and G-MOPSO The comparison results show the advantageous behavior of the proposed algorithm in locating Pareto optimal solutions regarding the multiple objectives without missing the extreme solutions Furthermore, a better uniformly distribution
of solutions was obtained by the proposed algorithm
ACKNOWLEDGMENT This work was supported by the Basic Science Research Pro-gram through the NRF of Korea funded by the Ministry of Ed-ucation, Science, and Technology (2011-0013845)
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