In this method, the objective function of electromagnetic problem is interpolated by using adaptive Taylor Kriging, in which the covariance parameter is obtained by Maximum Likelihood Es
Trang 1IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 2061
A Global Optimization Algorithm for Electromagnetic Devices by Combining Adaptive Taylor Kriging and Particle Swarm Optimization
Bin Xia , Minh-Trien Pham , Yanli Zhang , and Chang-Seop Koh
College of ECE, Chungbuk National University, Chungbuk 361-763, Korea University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam School of Electrical Engineering, Shenyang University of Technology, Liaoning 110870, China
This paper presents an efficient optimization strategy which employs adaptive Taylor Kriging and Particle Swarm Optimization (PSO) In this method, the objective function of electromagnetic problem is interpolated by using adaptive Taylor Kriging, in which the covariance parameter is obtained by Maximum Likelihood Estimation (MLE) And then, PSO is used to search for optimal solutions of electromagnetic problem The proposed algorithm is verified its validity by analytic functions and TEAM (Testing of Electromagnetic Analysis Method) problem 22.
Index Terms—Adaptive Taylor Kriging, maximum likelihood estimation, particle swarm optimization, TEAM problem 22.
I INTRODUCTION
P ERFORMANCE ANALYSIS of electromagnetic device
usually involves computationally expensive finite element
analysis through finding solutions of electromagnetic fields So
far, the optimization problems in electromagnetic devices are
typified by features that present difficulties to most
determin-istic search algorithm, such as the existence of multiple local
minima PSO with their ability to search more globally is better
suited for exploring complicated objective function landscapes
The high computational cost of evaluating the objective
func-tion in such problems, however, means that directly use of a PSO
is often not feasible or is impractical, owing to their general
re-quirement for a large number of objective function evaluations
[1], [2]
Thus, it is important to consider the possible means of
re-ducing the cost of the analysis One technique that has recently
attracted significant attention, called surrogate modeling [3], is
the focus of this paper In this technique, the objective
func-tion is evaluated indirectly by interpolated funcfunc-tions The
ap-proximate function must have low computational cost, high
ac-curacy, and good interpolation performance So far, Kriging, a
spatial statistical technique, is now popular for the
electromag-netic design optimization The Kriging surrogate model is
dif-ferent from random optimization method, in which the
numer-ical analysis of electromagnetic fields is carried out in each
itera-tion, and it is also different from response surface model (RSM)
with fixed parameterized polynomial It is used by the
semi-pa-rameterization to construct the response model [4], [5]
According to the different drift functions, Kriging models are
generally divided into Simple Kriging, Ordinary Kriging and
Universal Kriging Universal Kriging is a non-stationary
geosta-tistical method and its drift function is a general linear function
Due to complexity of equation calculation, it is seldom
inves-Manuscript received November 21, 2012; accepted December 29, 2012 Date
of current version May 07, 2013 Corresponding author: C.-S Koh (e-mail:
kohcs@chungbuk.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.2013.2238907
tigated Recently, Taylor expansion is used to approximate the drift function of Kriging, which is called Taylor Kriging, and it has the strongest approximation potentials and the best perfor-mance [6]
In this paper, adaptive Taylor Kriging is developed to sim-ulate the objective function, in which the Gaussian covariance parameters are estimated by Maximum Likelihood Estimation (MLE) [7] And then PSO is employed to get the optimal values
of objective function [8]
II MLE ASSISTEDADAPTIVETAYLORKRIGING
A Taylor Kriging Principles
The response function of a deterministic computer experi-ment is given by
(1)
It is a Kriging model, where is the position vector with
of regression coefficients, is called a drift function showing the average behavior of response , and is a random error term with Thus, a Kriging model is
a combination of a linear regression and a stochastic process with mean 0 and variance
derivatives up to the th order at point , which is the
expansion of the drift function at is given as follows:
(2) where is a vector between and , and is Lagrange
of Taylor expansion The expression is given as follow:
(3) Suppose sample points with corresponding observed
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Trang 22062 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013
at the unknown point can be estimated through a linear
combination of the observed values by Kriging as follows:
(4) where the coefficients are Kriging weights
In Kriging, the Best Linear Unbiased Predictor is used to
se-lect coefficients by satisfying conditions as follows:
1) The estimator (4) must be unbiased
2) The variance of estimator errors between the estimator and
the true response should be minimized
The above requirements can be satisfied by minimizing a
problem as follows:
(5) The variance between and as defined in (1) and
(4) can be calculated as:
(6)
de-rived:
(7) Finally, because the basic functions are obtained by the Taylor
expansion, this model is called Taylor Kriging, and the
cor-responding equations are formed by Lagrange multipliers
as follows:
(8) Therefore, the Kriging weights can be obtained from (7) and
(8) Taylor Kriging is one of Universal Kriging However, if
the order of Taylor expansion is given, the basis function in its
drift function can be obtained, so Taylor Kriging overcome the
difficult that the base function in Universal Kriging needed to
be specified
B Covariance Function
In the Taylor Kriging equations, the covariance function is
not defined yet In Kriging models, selecting a proper
covari-ance function is a crucial problem The research on covaricovari-ance
function mainly focuses on two approaches; one is the known
correlation function such as the spherical model, the thin elastic
plates model, the cubic model and so on It aims to find the
best correlation function which best approximate the
distribu-tion of sampled response values However, in real engineering
problem, with the given correlation parameter of the covariance
function, the existence covariance function with fixed
coeffi-cient maybe not the best approximation for the distribution of
sampled response value
In order to find the best covariance function, another ap-proach is called Gaussian covariance function, it is trying to find the optimal correlation parameter of the covariance func-tion And the optimal correlation parameter of the covariance function is estimated by using MLE To apply this method, at first, we assume generally the distribution of sampled response values is Gaussian distribution For two samples and , the Gaussian covariance function is defined as follows
(9)
param-eter vector, which influences the effect of covariance function along direction
If the variance of is , the covariance between two samples is:
(10)
In order to estimate the best correlation parameter , MLE is used, and the likelihood function for N sampled response values, which follows Gaussian distribution is defined as:
(11)
, and the correlation function matrix
So, for simplicity, maximizing the logarithm of likelihood functions is defined as follows
(12) Then the maximum likelihood estimators of and are
(13-a) (13-b)
By substituting the mean and variance into (12), and elimi-nating the constants, the maximize likelihood function becomes function only of correlation parameter as follows
(14)
In order to maximize this problem, because of difficulty of likelihood function, global version of PSO is used, which is known as the accuracy and fast convergence
After the optimal parameter is found, the covariance func-tion is defined Therefore, Taylor Kriging model is completely defined Any unknown response of point will be calculated
by (4)
Trang 3XIA et al.: A GLOBAL OPTIMIZATION ALGORITHM FOR ELECTROMAGNETIC DEVICES 2063
Fig 1 Flow chart of global optimization algorithm.
III GLOBALOPTIMIZATIONALGORITHMEMPLOYING
MULTIPLEITERATION ANDGRADUALREFINEMENT
When evolutionary algorithms are used to solve optimization
problems, explicit fitness functions may not exist With the help
of interpolation capability, the Taylor Kriging model is applied
to build surrogate fitness function to guide further searching of
optimal solution
In the optimization strategy, the Latin Hypercube sampling
(LHS) technique is used to obtain sample points in the design
space Based on the optimal result obtained from previous
it-eration, the design space is reduced and new sample points of
current iteration are gradually inserted to the approximate
ob-jective function, so that the efficiency and simulation accuracy
will be improved The proposed optimization algorithm is
sum-marized as follows:
Step 1: Define the initial design space and generate initial
sampling points by LHS in the whole design space
Step 2: Calculate corresponding objective values
Step 3: Construct the response surface by Taylor Kriging
model
Step 4: Find the current optimal point by PSO, and check the
convergence, stop and output the result when the
error of the current optimal point and the previous
one is very small (less than )
Step 5: Reduce design space by adaptive factor of 0.618
around the current optimal point [4]
Step 6: Generate new sampling points by the LHS in the
re-duced design space, and go to Step 2.
In the algorithm, the iteration repeats until the optimal point
converges The flow chart is shown in Fig 1
IV ANALYTICEXAMPLES
A Mathematical Examples
The true accuracy of a surrogate model can only be
deter-mined if the true function, which it is attempting to
approxi-mate, is also known and is available for comparison Almost all
electromagnetic optimization design problems are non-analytic,
meaning it is impossible to measure the accuracy of any
surro-gate model constructed Therefore, in this paper, one analytic
test problem is selected as:
Fig 2 Optimization process of analytic function (a) true response surface; (b) initial 25 sampling points; (c) 2nd iteration; (d) 3rd iteration; (e) Kriging response surface; (f) normalized root mean squared error for different sampling points.
(15) where the true global maximum exists at
with the function value of 8.1061
Firstly, Generating 25 initial sampling points in the whole design space by LHS, and the best parameter vector is found
Therefore, the Gaussian covariance function is confirmed Fig 2 shows the constructed response surface by Taylor Kriging model and the distribution of sample points at each iteration At the initial iteration, 25 sample points are obtained
by the LHS in the whole design space, and corresponding response values are calculated by the sample points And then the Kriging response surface is constructed to give an
in Fig 2(b) Then, the design space is adaptively reduced, and
7 and 13 additional sampling points are inserted the design space, respectively, as shown in Fig 2(c) and (d) After three
is obtained and corresponding Kriging response surface is shown in Fig 2(e)
In other to assess its accuracy, the normalized root mean squared error (NRMSE) will be used for comparing the surro-gate model with the true function, and it is defined as follows
(16)
Trang 42064 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013
Fig 3 Configuration of the SMES device.
where is the predicted value at by Taylor Kriging,
is the true value at , and the mesh of test points about
Fig 2(f) shows NRMSE for different number of sample
points From the results, the estimation error of the NRMSE
decreases as the number of sample points increases
Cross-vali-dation becomes more reliable, resulting in a higher accuracy of
Kriging interpolation This verifies Kriging claim to be a very
flexible for highly nonlinear functions
B TEAM Workshop Problem 22
TEAM workshop problem 22 is classified as three-parameter
and eight-parameter problem Fig 3 shows the design
parame-ters of TEAM workshop problem 22 In this paper, we consider
the three design parameters problem The objective function of
the problem takes into account both the energy requirement (E
should be as closed as possible to 180 MJ) and the minimum
stray field requirement ( evaluated along 22 equidistant
points along line a and line b in Fig 3 as small as possible)
Therefore, the single objective function is defined as follows
[9]
(17)
as:
(18)
Limits of three design parameters are specified in [4] This
op-timization problem has several feasible regions, and nonlinear
multi-objective function that is consist of the error of energy to
be stored and stray field It is not easy to find optimum solution
of this problem using typical optimization techniques
There-fore, proposed global optimization by Taylor Kriging and PSO
is employed to resolve the difficulties
Electromagnetic analysis of TEAM problem is performed
by MAXWELL 12.0 27 sampling points generating by LHS
is used as initial sample set We can find a global optimum
which is superior to other methods such as GA and simulated
annealing (SA) as given in Table I The global optimum is
and corresponding objective func-tion value is 0.0870 The objective funcfunc-tion value of optimum
design is decreased by 65.66% compared with initial design
TABLE I
C OMPARISON OF O PTIMIZATION R ESULTS FOR TEAM P ROBLEM 22
The results and comparison with other methods are given in Table I
V CONCLUSION
In this paper, a global optimization strategy employing mul-tiple iterations and gradual refinement is proposed The Taylor Kriging model with Gaussian covariance function is used as in-terpolation function approximate to the objective function Then the best parameter of Gaussian covariance function is success-fully found by MLE, and the optimal point of problem or ob-jective function is obtained Through the applications to numer-ical example and TEAM problem 22, the proposed optimization strategy is computationally efficient with less sampling points and higher computation efficiency
ACKNOWLEDGMENT
This work was supported by the Basic Science Research Pro-gram through NRF of Korea funded by the Ministry of Educa-tion, Science, and Technology (2011-0013845)
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