2901 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 5, SEPTEMBER 1998

An Universal Tabu Search Algorithm for Global Optimization of Multimodal Functions with Continuous Variables in Electromagnetics

Yang Shiyou, Ni Guangzheng, Li Yan Zhejiang University, Hangzhou, 3 10027, China Shenyang Polytechnic University ,Shenyang, 1 10023, China

Tian Baoxia Li Ronglin Zhejiang University, Hangzhou, 3 10027, China

Abstract-An universal tabu search algorithm for the optimization of functions with continuous variables is presented based on the tabu method so far used. Essentially, the improvements include the transition criterion for different states, the determination of the step vector, the cancellation of tabu list, the stop criteria, and the restart from the optimum etc.. The numerical performances of the present algorithm are investigated using a benchmark problem and the geometry optimization of the multisection arc pole shoe in large salient pole synchronous generators.

Index terms-Inverse problem, Global optimization, Stochastic algorithm, SA algorithm, Tabu search technique.

I . INTRODUCTION

Recently, the probabilistic heuristic algorithms have been used as efficient and general approaches for the global optimization of multimodal functions with continuous variables in electromagnetics. These algorithms include simulated annealing (SA), genetic algorithm (GA), and tabu search method, all of which are initially developed for combinatorial optimization problems. Only a few applications of the tabu algorithm have been found in electromagnetics compared to SA and GA, and the tabu method used in these applications is directly come from that proposed by N.HU in[ 11 without any adaptation. Investigations on benchmark problems and analysis have revealed that the tabu algorithm proposed by HU (hereafter called HUtabu) is often trapped in local optima, even if used to solve the test functions give in [ 11 only with a simple variation of the algorithm parameter. So HUtabu has some drawback in solving global optimization problems.

An Universal tabu method (Utabu) for the global optimization of multimodal functions with continuous variables is developed based on the thorough study of the available tabu search method and the improvements on HUtabu. Numerical results on a benchmark problem and practical applications show that Utabu is more general and powerful than HUtabu and has advantages both in efficiency and accuracy over the SA algorithm. And in the final, the

Manuscript received November 3, 1997 Yang Shivou, NI Guangzheng, e-marl ee-officemema Z I U edu cn fax

+86-571-7951625 This work was sponsored by the National Natural Science Foundation of

China under project No 59505008

detail of the comparison results about large salient synchronous generators for different pole arc geometry is investigated using the filed computation and Utabu.

11. UNIVERSAL TABU SEARCH METHOD

The details about HUtabu are referred to[l]. Hereafter is only about the improvements made in this paper. For the convenience of explaining, we restrict ourselves to the problem. min f (x) R N + R , R N = {Xja, 5 Y, I b , , ~ = 1,2, . , N . 1 A. Transition Criterion for D&%rent States

In HUtabu, only moves with decrement in objective functions are accepted, the algorithm has no ‘climb up’ property. It is no surprising that HUtabu is often trapped in local optima. Thus for Uiabu, the best solution of feasible moves generated in neighbors of the current state is selected as the new current one whether the value of its objective function is better than that of the current state or not. So the Utabu not only has the ‘climb up’ property but also possesses the advantage of using the information of the feasible moves in the neighbors of the current state to guide the search process.

B. Determination of Step Vector

For practical applications, the length of the interval for different variable directions is not always the same, so it is not reasonable to use the same length of intervals for different directions Thus in Utabu, 1)the intervals of different variable directions are firsi transformed into [0,1]. 2)then an universal step vector H is given using

Thirdly, a feasible random move from the current sate in the direction j for the neighbor h, is realized using

here p , = O5(b, - a , ) ( ~ = 1,2, . . . ,N) , a , ,b, are the inferior and superior bounds for the j-th variable, R is a random parameter out of the interval of [-1,1].

{ Hlh, =h,-l / e , ( l = 1,2,...,1. ;Ck=lOOOi’(’-~);hl = l )} . 3)

Y = X + R p , h, (1)

C. About Tabu List

To explain clearly, let X be the current state, Y be the best

0018-9464/98$10.00 0 1998 IEEE

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solution of feasible moves generated in neighbors of X. Now Y is accepted as new current one, the corresponding step h, is stored into the tabu list in HUtabu to prevent the method to move back to X, and the algorithm begins a new cycle iteration. For the following reasons it is not reasonable to use tabu list in the optimization of functions with continuous variables:

1) not only the transfer form Y to X (denoted using Xlk in what follows) is forbidden, but also to all states in the h, neighbor of Y ,

2) the moving back from Y to X,k can not be avoided if Xlk is located in some other neighbor hJ of Y;

3) with the increment of the elements in tabu list, the number of feasible moves generated in the current state reduces, thus the information used to guide searches also reduces, correspondingly the possibility for the algorithm to be trapped in local optima increases.

4) the cycle occurs in the combinatorial optimization for tabu method can be avoided effectively in the case of the optimization of multimodal functions with continuous variables by moves of different size neighbors

So no tabu list is introduced in Utabu

D. Stop Criteria

The Utabu has two termination criteria to decide when the global optima has been searched. The first one is: after every cycle of iteration, the search is stopped if

where f,' is the best solution of objective function searched in the k-th cycle, andf,,, is the best one of objective function searched so far

The other termination criterion stops the search when the number of consecutive moves with no improvements in the best objective function searched so far is lager than a threshold Khoid

E Restavtfrom the Optimum

In HUtabu, the state with the best objective hnction searched so far, denoted using X,,,,, is used as the new

current one for beginning the iteration of every new cycle. Due to the fact that .Yc1,,, may not change for some consecutive cycles, which means that the search too often restarts from the same point, not allowing a proper exploration of the total hnction domain. In this case the algorithm is likely to be trapped in local optima. On the other hand, if the search does not restart from the best point reached, the information on this best point may be far away from the current point, and again, the probability to end the search in a local minimum is very high. So Utabu restarts

The above leads to the following Utabu algorithm described in pseudo-Pascal.

Procedure Utabu method begin

k:=O; n:=O; initialize (X, Xopt,fopt,H); stopcriterion:=false; while stopcriterion:=false do begin

k.=k+l; n : =n+ 1 ; for j = l to N do

begin for i=l to r do

begin generate a feasible move, Y, , from X

(using (1) 1, end

let X* be the best move among the feasible moves; x . = x * . if f(X*)< fopt then

begin x .=x*. opt. >

fop,:=f(X*); end

if j=1 then begin

end fk* = f (x*) ;

else if f(X*)< f k * then begin

end f k * .= f(X*);

end if k>Kt(hold then stopcriterion.=true, if terminate test (2) is past then stopcriterion:=true; if n2N, then

begin

end x. =Xopt,

end end

111. NUMERICAL VALIDATION

To validate the effectiveness of Utabu method, the following test function is first selected from [2] as a benchmark problem.

,,-I

1 f ( x ) = k,{sin(?rk,x,) + ~ ( x , - /c,)'[1+ /c,sin2(nk,x,+,)l

(3 1 + (x, - k5)'[1 + k , sm2(nk,x , , )]} + 5~(x, ,5 ,100,4)

where k3=0 S,k4=3,kj=l,k6=S ,k7=2, X={XrRS 1 -5Ix,S5,1=1,2, ... ,5}, the penalty hnction u is

I

from the optimum reached after every consecutive cycles N,. defined by

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k(x, -U)" 0% >a> u(x, ,a, k , m) = 0 (V--a<x, < a ) (4) i k(-x, -a)"' (VX/ < a )

The function has roughly 1 55 local minima. The global one

The computational results using different optimization

From Table I it can be seen that

is X,,,=( 1,1,1,1 , l)T andf,,,=O.

algorithms are shown in Table I .

(1) the Utabu method converges to the global minimum; (2) Hutabu algorithm converges to a local minimum; (3) the introduction of the tabu list in Utabu algorithm

(Algorithm* in Table I ), other thing being equal, leads the algorithm to be trapped in some local minimum;

(4) the number of iterations used by executing Utabu is only about 14% of that by executing SA algorithm, whilst the optimal results obtained by means of this method is slightly better than those by means of SA algorithm.

To show the effectiveness of the Utabu for engineering problems, the present algorithm is used to solve the geometry optimization of the multisection arc pole shoe in large salient pole synchronous generators. The details about this problem are referred to[?]. To make this paper self-contained, here give the mathematical model of the optimization problem and the geometry parameters to be optimized, i.e.

max B, , (XI s.t e,. - e,,, 5 0

THF - THF, 50 SCR - SCR, 2 0 x; - x;, 5 0

( 5 )

where Bfl is the amplitude of the fundamental component of the flux density in the air gap, e, is the distortion factor of a sinusoidal voltage of the machine at no load, THF is the telephone harmonic factor, SCR is the short circuit ratio.

The corresponding geometry parameters to be optimized, denoting using X, are depicted in Fig.1. The optimization results using different algorithms are given in Table I1 From these results it IS significant that

TABLE 1 OPTIMIZATION RESULTS OF TEST FUNCTION USING DIFFERENT

A , L G E X t M S -** * - -_--**-- w-se-*I *

HUtabu Utabu Algor ick * SA ^ _ _ _

(-2008193, ( I 0000012 ( I 2q11315, ( I 000875, -2.335685, 0 9999807. 0 6201007, 1.001558,

XO*t -1.429575, 1.0000056, 1 3630451, 0.998166. -2.691851, 1.0000009. 0.9999316, 1.004652, -0.497615) I.0000028) 0.7076457) 1.001612)

foLd 0.94835 16 5 . 5 6 9 ~ 10.'' 7 . 0 2 3 ~ lo-* 1 . 2 ~ 390013

Algorithm* is formed by only introducing a tabu list in Utabu 30239 ,,--*-

No Iteration -4056ppsw 54270 Note :

TABLE 11 OPTIMIZATION RESULTS OF A PRACTICAL PROBLEM

ESA 140000 1197 SUMT 7 1 0 8 8

SA 10347 1197 HUtabu 1348 1102 Utabu 2168 1197

Note ESA is the abbreviation of Exhaustive Search Algorithm

--- de-*-%%->>m-* * wmH

" No Iteration - BfiQu) II_

w-zwm*u**n _V*--w_ -

Fig. 1 The Schematic diagram of the multisection arc pole shoes

(1) the present Utabu algorithm is more powerful and efficient for searching the global optima than HUtabu in the optimization of functions with continuous variables;

(2) HUtabu is trapped in a local optimum; (3) Utabu algorithm presented here is a strong contender

to SA algorithm in the sense that the number of iterations used by executing Utabu is only about 20% of that by executing SA algorithm, whilst the optimal results obtained by means of Utabu are slightly better than those by means of SA algorithm;

(4) other thing being equal, the possibility to be trapped in local optima for Utabu with an introduced tabu list for the optimization of functions with continuous variables increases.

IV. APPLICAT~ON

In this section, the details about the large salient pole synchronous generators with the multisection arc pole shoe geometry is investigated by means of the numerical optimization using Utabu ;and the parameter comparisons on different machines.

Table 111 is the geometry optimization results of a 300 MW hydrogenerator and Table IV is the performance comparison of the machine with different pole shoe geometry, i.e. the conventional one-piece arc pole shoe structure and the optimized multisection arc shoe geometry. From these results it is very clear that

(1) the amplitude of the fundamental component of the flux density in the air gap under the optimized pole shoes, other thing being equal, is higher about 6.4% than that under the traditional pole shoes;

optimized multisection arc pole shoe geometry is smaller than that of the machine with a traditional pole shoe;

(3) the differences of the parameters SCR and THF under two different pole shoe geometry are negligible;

(2) the transient parameter. 1: ~ of the machine with the

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(4) the ratio, noted using U, of the leakage flux to the total one of the machine with the optimized multisection arc pole shoe geometry is reduced compared to that of the machine with a traditional pole shoe;

Table V and Table VI is the typical results of a 125 MW hydrogenerator with different pole shoe structures Table VI1 and Table VIII is the optimized results of a 200 MW machine.

From these results it can be deduced that (1) the ratio of the leakage flux to the total one of the

machine with the optimized multisection arc pole shoe geometry is reduced to 97% of that of the machine with the traditional pole shoe;

(2) the fundamental component of the flux density in the air gap of the machine with optimized multisection arc pole shoes, other thing being equal, is raised 7% compared to that of the traditional pole shoe structure;

(3) the transient parameters, X ; , for the optimized multisection arc pole shoe machine is reduced;

(4) the differences of the parameters SCR and THF under two different pole shoe geometry are also negligible.

In order to determine the optimal values of the geometry parameter K, i.e. the ratio of the length of the pole arc with an equal air gap length to the total length of the pole arc, the optimization values of this parameter for five different machines are presented in Table IX From the results it can be known that the optimal values of this parameter are in the region of [0.5,0 61.

Additionally, the study on different pole shoe geometry machines has also revealed that the strength of the multisection arc pole shoe is stronger than that of the traditional pole shoe.

can be made (1) other thing being equal, the amplitude of the

fundamental component of the flux density in the air gap can be raised to 6%-10% by using the multisection arc pole shoe geometry;

(2) the machine with optimized multisection arc pole shoes has more excellent transient performances;

(3) the strength of the multisection arc pole shoe is stronger than that of a traditional pole shoe;

(4) the optimal value of the ratio of the length of the pole arc with an equal air gap length to the total length of the pole arc IS 0.5-0.6

i From what just analyzed above, the following conclusions

TABLE 111

TABLE V GEOMETRY OPTIMIZATION RESULTS OF A 125 MW HYDROGENERATOR

Parameter Rl(m) R2(m) Xl(in) Yl(ni) X2(m) Yz(m) Value O!X! 0-9104 0 1736b 5 2 7 9

~ _ ~ * x - ~ - ~ * ~ ~ ~ --* ~ SA v c ~~ l"-*---* *-,,s

P

TABLE Vi

-- Pole arc shape o-*-- Bfi(T) X'd(pU) CCRP~,(%) THF(O/L

~

Traditional type 1210 0 934 l d ^ 3 4 i 1 106 0 155 0 812

TABLE VI1 GEOMETRY OPTIMIZATION RESULTS OF A 200 MW HYDROGENERATOR

Parameter Rl(iii) Rz(in) Xl(m) Yl(m) Xz(m) Y2 (m) Value 0 0763 1 6978 , 0 3 9 8 0 1 9 9 8 5 -c04-04157

~~~ ~ - - - ~ ~ ~ ) I ) ~ ) x *-* w___-I.-x-aM--

m~ ~" ~ -*

V. CONCLUSION

The numerical results on the benchmark problem and practical engineering applications as well as the analysis demonstrate that

(1)the present algorithm is more powerful and efficient than the tabu search algorithm so far used in global optimizations, and the number of iterations of the present method is only about 20% of that of SA algorithm, whilst the optimal results are slightly better than those of SA algorithm;

(2)the details of comparison results for different pole shoe geometry of large salient pole synchronous generators given in this paper may be useful for the engineering applications.

REFERENCES

[ l ] Nantang HLI Tabu Search with Random Moves for Globally Optimal design, hit lournal Numer Method Engineering, Vol 35,

[2] A Dekkers and E Arts, Global Optimization and Simulated Annealing, Math Programming, 50( 1991),367-393

[3] Tang Renyuaii, Yaiig Shiyou, et al , Combined Strategy of Improved Simulated Annealing and Genetic Algorithm for Inverse Problem, IEEE

[4] R R Saldanha, J A Vasconcelos A N Moreira, G B Alvarenga, Optimization of the Cross-Section Shape of a Ridge Waveguide Using the Ellipsoid and the Tabu Search Algorithm, IEEE Trans Magn Vol32, No 3,pp1254-1257, 1996

1055--1070(1990)

Trsnq Magn Vol 22 No 3 ~ ~ 1 3 2 6 1 3 2 9 1996

PERFORMANCE COMPARISON OF THE 300 MW HYDROGENERATOR l____l- P

Pole arc shape CT BfI(T) x'd(pu) SCR- e"('??) THF(%) Traditioiialtype I 1 9 1022 0 3 3 11685 0 169 0 113 Optimizedshape 1 1 7 1086 0 3 1 11665 0 184 0 125

--vm --------*vW"-P-