Genetic Algorithm-Based Optimization Used in Rolling Schedule

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JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2008, 15(2) : 18-22

Genetic Algorithm-Based Optimization Used in Rolling Schedule YANG Jing-ming,

CHE Hai-jun,

DOU Fu-ping,

ZHOU Tao

(Institute of Electrical Engineering, Yanshan University, Qinhuangdao 006004. Hebei, China)

Abstract: A genetic algorithm-based optimization was used for 1 370 mm tandem cold rolling schedule, in which the press rates were coded and operated. The superiority individual is reserved in every generation. Analysis and comparison of optimized schedule with the existing schedule were offered. It is seen that the performance of the optimal rolling schedule is satisfactory and promising. Key words: tandem cold rolling; rolling schedule! energy consumption; genetic algorithm; optimization

To meet the challenges of international competition, a rolling mill must be able to produce high quality products in a cost effective manner. Therefore, the rolling schedule is an important but difficult task. The most important factor in tandem rolling mill is the reasonable distribution of the total thickness deformation. The optimized schedule should lead to improved thickness and product quality, and reduced energy ons sump ti on^'-^^. Genetic algorithm (GA) is an efficient and parallel method that can perform global searching. It can acquire and accumulate the knowledge of search space automatically in the search process, and control the search process self-adaptively to achieve optimizationc4’. In the present study, a method with the energy-saving objective and GA optimization to obtain the least energy consumption within the certain constraints was presented.

1 Genetic Algorithm G A is a group of random optimization methods, which explains the objective to the adaptability of biology to the environment, regarding the optimized variable to the individual of the biology species. After the operators of parent selection, crossover, and mutation, the performance of every individual is evaluated by the fitness function. Then, the new individuals are evaluated, and this evolutionary cycle is repeated from generation to generation until an adequate solution is found. In the present study, the procedure for a G A is Biography:YANG Jing-ming(l957-), Male, Doctor, Professor1

described as follows : (1) Define a fitness function for the optimization problem. ( 2 ) Encode the variables into binary codes, and then combine the individual binary code for each variable together into a binary chain, as a combined single variable. ( 3 ) Initialize a population as the population for the first generation. (4) Evaluate each individual in the population. ( 5 ) Implement the genetic operations: selecting, crossover, and mutation. ( 6 ) Rank the population. (71 Delete the lowest-ranked genomes in the population, and keep high ranked individuals. (8) Repeat Step ( 5 ) to Step ( 7 ) until the evolutionary termination criterion is satisfied. When compared with the traditional optimized algorithm, the advantages of G A are that the operated objective is not a single answer, but a group of feasible answers1 it only requires the objective function and the fitness function relatively without the use of other information and knowledge; it can reach the global optimization and impossible to get into the local minimum. Therefore, G A is applied for the optimization of large-scale and nonlinear function.

2 Models of Rolling Process 2.1

Objective function and constraints There are few objective functions for the rolling

E-mail: yangjmaysu. edu. cni

Revised Date: December 18, 2006

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Genetic Algorithm-Based Optimization Used in Rolling Schedule

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schedule, and here, the roll power function is selected to attain the minimum roll power schedule through the optimization,

(1) where, F is total power cost; N,is roll power of stand i ; h,-, , h, are entry and exit thicknesses a t stand i, respectively; m is total number of stands, and here m=5. The constraint condition of tandem cold rolling mills may be divided into the equipment factors and the technical factors. The technical factor constraints can be described as follows: ~ r n i n < ~ t
Gtrnex

(4)

where E , , n,, t , are reduction rate, roller speed, and tension at stand i, respectively; and E,,, , nmln,t,,, and E,,, , n m s x ,t,, are lower and upper limits of the reduction rate, the roller speed, and the tension, respectively. The equipment factor constraints include: Pr
where

1 ~ ~ / [ 2 ( l f u ) f 3 ~ ~ ] (13) ) p is friction coefficient; a is influence coefficient of the lubricant’s quality and category, and when the coal oil emulsion is selected as the lubricant, a is 1. 0 ; and u is exit speed of the strip. ,u=a{O. 07-0.

2.3

Forward slip The main factor affecting the forward slip is the reduction rate E , ; therefore, the forward slip ratio, f , is calculated from the Bland-Ford forward slip formula. f=O. 001 5 f 0 . 222oi+0. 2 2 2 ~ : (14) 2.4

Roll torque and power The total roll torque M can be described as fol-

lows : (15) M=M, f M, M, fMd where Mf , M , , and Md are addictive friction torque, no-load motion torque, and kinetic torque, respectively. Based on the results of the roll speed and the roll torque, the roll power at each stand is estimated as: N,=O. 103Mn, (16) where n, is rotational speed of the motor shaft.

+

3 Genetic Algorithm-Based Optimization 2. 2

Rolling force model The rolling force, p , , can be obtained based on the Bland-Ford-Hill force calculation formula (Bland and Ford, 1948, 1952). p , =B,L’,Q,K,K/I ooo (8) where I>‘,= dR’(h,-, - h , )

( 9) (10)

C = 16 ( 1- ‘u’ ) / x E (11) B, is average width of the strip; L‘, is distortion length of the deformed work roll; Q, is stress state coefficient; K is average yield stress; K T is tension coefficient; R’ is deformed work roll ratio and is described according t o Hitchcock’ s equation; R is

work roll radius; ’u is Poisson’s ratio of the work roll; and E is Young’s modulus of the work roll. According to Hill’s experimental results, Q, is defined as Eqn. (12) :

3. 1

Initialization of population When using GA to solve an optimization problem, the first step is to determine how to represent a solution of the problem as a chromosome, which can then be submitted meaningfully to crossover and mutation operators. T o ensure the perfect shapeC5’, a constant small thickness reduction is given at the last stand, such as 0. 05 mm. For a five-stand tandem cold rolling mill as shown in Fig. 1 , since the exit thickness h6 and the reduction thickness at the last

Fig. 1 Configuration of plate mill

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five stands is fixed, only the exit gauges at the first three stands h, , h 3 , h, need to be confirmed. In the optimization of the rolling schedule, the reduction rate distributed proportion A is represented as the chromosome:

A=Cai

(17)

,US]

where a l , u 2 , and a3 are proportion coefficient. For such a multiple variable encoding problem, each variable should be encoded first, and then linked together as a chain. According to the empirical method, the reduction rate of the single mill cannot be larger than 45% , and therefore, can be valued at random in the scale [ 10, 453. After iterative calculation, Q , is transformed as the reduction rate at stand i , which can meet the finished product requirement. T h e process is as follows: Firstly, suppose the average reduction of every stand is 5 calculated using Eqn. (18). Then, the approximate exit thickness of every stand can be defined. h 1/3 -€-l-l/l$] (18)

h,+l = h , ( l - E ) , i = 1 , 2 , 3 (19) Secondly, the distributed proportion a , is required to confirm the reduction rate E , approximately CEqn. (2011. 3

E,=u,

3

Z E /, =ZI a , , i=1,2,3 ,=I

(20)

Therefore, it can get the h,,,(n is the time of iteration, n = 1 firstly) from the initial gauge h , . If h the condition 11-4 1 < A is satisfied, the iteration h4.n will be completed. On the contrary, one adds n , and E , is corrected as follows: When the iteration h4,n-1 f h , , it is comprehended as follows: (21)

and (22)

Thus, the corrected E ~ , . is like Eqn. (23). h4

I ,

I

(23) h,,".-I After attaining E,,, it can get the first three exit thicknesses again. When the estimation is satisfied, the exit thicknesses are obtained at the same time. T h u s , a group of exit thickness is obtained as soon as the chromosome is defined. Then, the roll force, the roll gap, the forward slip, the roll torque, and the exit speed of every stand can be calEj,"=1-(l--Ei,~-1)

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culated using the mathematical model.

3.2

Evaluation

In the GA, a fitness function is used to measure the performance of the individual at the optimization. The higher the fitness value is, the more possibility is the individual selected for the next generation. On the contrary, the individual may be eliminated. In the present study, the total power cost function [Eqn. ( l ) ] is employed as the fitness function to evaluate the chromosome individual.

3.3

Operators

Starting from the initial population, through random selection , single-point crossover and mutation operators, a group of more adaptable individuals are generated, and furthermore, the search space is close to the optimization region increasingly. The rolling parameters are recalculated by the new generation; therefore, it is the objective value. After the evolution from one generation t o another, the final generation is the most adaptable one and it seeks the best solutions at last. T h e GA has a high capacity of searching in global because of the randomness in the process of the special continued evolution, so that it can avoid the local optimization. The individual which is the most advantageous chromosome in the past dynasties will be output as the result of the final optimization. T h e whole evolutionary process will cease once the termination criterion is met. In this research, the termination criterion is the total number of trials, 1 000.

4 Optimization and Results T h e basic idea of the optimization is that the reduction rate distribution coefficients of the first three stands are first defined on the basis of the initial data of the strip, such as the width, the initial thickness, and the exit gauge, and then optimized with the GACG1.When the power cost attains the minimum, the exit thicknesses of the five stands are fixed, and the roll force, the roll gap, the forward slip, the roll torque, and the exit speed of every stand can be calculated. T h e optimization method was applied to 1370 five-stand tandem cold mill in one steelworks of Tangshan. Table 1 shows the mill parameters. T h e strip condition was: the initial gauge of the strip is 2. 25 mm, the exit gauge is 0 . 5 mm, and the

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Genetic Algorithm-Based Optimization Used in Rolling Schedule

Table 1

Parameters of 1370 tandem cold mill Stand number

Mill parameters Rated power/kW Motor rotated speed/(r

min-')

1

2

3

4

5

2 572.5

2 572.5

2 572.5

3 675

3 675

135/305

175/375

225/445

250/500

250/500

Max roll force/ kN

20 000

20 000

20 000

20 000

20 000

Work roll diameter/mm Backup roll diameter/mm

520. 7

528. 0

526.8

529.7

531. 6

1380

1 380

1310

1310

1310

strip width is 900 mm. In GA optimization, the initial population size was assigned a s 20, in other words, t h e individual was 20 in every generation. T h e total number of trials was 1 000. T h e crossover and mutation rates were set a s 0. 8 and 0. 01, respectively. In the evolution process, the minimum objective value and the corresponding individual in every generation were recorded. The ultimate minimum objective value is 9 006.8 kW. From Fig. 2 , i t can be seen that the individuals in the population are convergent in the optimization. T h e power distributions at five stands for different schedules a r e shown in Fig. 3. T h e variety of the average and the optimized value of the objective in the trail process are shown in Fig. 4. Table 2 shows the comparison of the result between

4 000

Rated pow 3 000

B

2 2P

2000

1 000

2 3 The number of stands

4

5

Power distributions for different schedules

Fig. 3 9600 I

- Objective value Average value

B

9 400

eb

2

9200

k

9 000 4

0

Fig. 2

8 12 Individual

16

20

0

Table 2

Fig. 4

800

1000

Variety of average and optimized value

Comparison of optimized schedule with existing schedule ~

Rolling schedule

600

Times of iterative

Object value of hundredth

~~

400

200

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Exit gauge/mm

Reduction/%

Roll force/kN

Roll power/kW

1

Initial Optimized

1.710 1.904

24.00 15.37

6 334 5 431.4

1750 1 011. 6

2

Initial Optimized

1.190 1.348

30.41 29. 19

8 090 10 021

2 315 2 325.2

3

Initial Optimized

0. 850 0. 893

28.57 33.78

8 066 10 231

2 293 2 513.4

1

Initial Optimized

0. 580 0.550

31. 76 38.40

9 015 10 596

3 043. a 3 156.6

5

Initial Optimized

0. 500

13.79 9. 09

6 855 6 478.8

1929.8 1 322. 3

Total power/kW

Initial Optimized

0. 500

11 331.6 10 329.1

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the GA optimization and the empirical method. It can be seen from the table that the optimized schedule consumes less power than the experience as follows : Nold-Nnew xloo/,/-11 331.6-10 329. 1,8. 8s% A= 11 331. 6 NOld

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Conclusion

The genetic algorithm has been presented to optimize the rolling schedule. It has the superior ability of searching in global, and therefore, the optimized rolling schedule can make full use of the equipment and reduce the total rolling consumption. The optimized schedule improves the energy saving. At the same time, the optimal time is shorter than traditional time because of the parallel computation. Therefore, this method is adapted for online application.

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References:

c11 C2l

c31

c41 c51

C61

LIANG Guo-ping. The Optimum Distribution of Loads on Rolling Mills [J]. Iron and Steel, 1980, 15(1): 42 (in Chinese). Wang D D, Tieu A K , de Boer F G. Toward a Heuristic Optimum Design of Rolling Schedule for Tandem Cold Rolling Mills [J]. Engineering Appl of Artificial Intelligence, 2000, 13(4) : 397. LU Ji-min. WE1 Li-qun. Optimizing Process of Steel Strip Hot Continuous-Rolling Rules [J]. Journal of Shanghai Technical College of Metallurgy, 1989, lO(3): 1 (in Chinese). Sarker R, Liang K H , Newton C. A New Multiobjective Evolutionary Algorithm [J]. Eur J Oper Res, 2002, 140(1): 12. DI Hong-shuang, XU Jian-zhong, GONG Dian-yao. Effect of Load Distribution on Strip Crown in Hot String Rolling [J]. J Master Sci Technol, 2004, 20(3): 330. Hsiao-Lan Fang, Chung-Hsiu Tsai. A Genetic Algorithm Approach to Hot Strip Mill Rolling Scheduling Problems [A]. IEEE, eds. The 1998 IEEE 10th International Conference on Tools With Artificial Intelligence [C]. Piscataway: IEEE, 1998. 264.

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