Multi-Objective Load Distribution Optimization for Hot Strip Mills

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JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2013, 20(2) : 27-32, 61

Multi-Objective Load Distribution Optimization for Hot Strip Mills JIA Shu-jin 1 ' 2 ,

LI Wei-gang 2 ' 3 ,

LIU Xiang-hua 4 ,

DU Bin1·2

( 1 . Department of Automation and Key Laboratory of System Control and Information Processing, Shanghai Jiao Tong University, Shanghai 200240, China; 2. Automation Division of Research Institute, Baosteel Group Corporation, Shanghai 201900, China; 3. College of Information Science and Engineering, Northeastern University, Shenyang 110819, Liaoning, China; 4. Research Institute of Science and Technology, Northeastern University, Shenyang 110819, Liaoning, China) Abstract: Load distribution is a key technology in hot strip rolling process, which directly influences strip product quality. A multi-objective load distribution model, which takes into account the rolling force margin balance, roll wear ratio and strip shape control, is presented. To avoid the selection of weight coefficients encountered in single objective optimization, a multi-objective differential evolutionary algorithm, called MaximinDE, is proposed to solve this model. The experimental results based on practical production data indicate that MaximinDE can obtain a good pareto-optimal solution set, which consists of a series of alternative solutions to load distribution. Decision-makers can select a trade-off solution from the pareto-optimal solution set based on their experience or the importance of ob­ jectives. In comparison with the empirical load distribution solution, the trade-off solution can achieve a better per­ formance, which demonstrates the effectiveness of the multi-objective load distribution optimization. Moreover, the conflicting relationship among different objectives can be also found, which is another advantage of multi-objective load distribution optimization. Key words : hot strip mill; load distribution; multi-objective optimization

Load distribution is critical for the hot rolling process, which not only directly affects product quality such as strip thickness and shape precision, but also has a major impact on rolling energy, roll consump­ tion, production stability and efficiency, etc1-1-1. Since the 1960s, load distribution of hot strip mills have gone through several stages such as experience form method, power curve method, load distribution fac­ tor method, and artificial intelligence method1-2-1. Many scholars have worked in this field. Ref. [ 3 ] presented an immune genetic algorithm to optimize load distribution of hot strip mills, and built an opti­ mal load distribution model combining with strip shape. Ref. [4] proposed an improved particle swarm opti­ mization algorithm with the mutation in weighted gradient direction based on the evaluation of the fit­ ness variance to optimize load distribution of hot strip mills. A minimum variance objective function of shape and gauge control was proposed in the con­ dition of ensuring good shape. Ref. [ 5 ] proposed an

improved adaptive search area particle swarm opti­ mization and applied it to the optimum design of load distribution for hot strip mills. These papers have a common defect; load distribution of hot strip mills is a multi-objective optimization problem, but it is usually converted into a single objective one by weighted-sum method. However, the weight coefficients are not easy to determine in practice, especially when the objec­ tives have different order of magnitudes. To over­ come this defect, Ref. [ 6 ] proposed a multi-objective differential evolution, and then this algorithm was applied to the load distribution calculation of hot strip mills. This paper further explores the advanta­ ges of multi-objective load distribution optimization, and proposes a new multi-objective differential evo­ lution algorithm, called MaximinDE, to optimize the load distribution problem for hot strip mills.

1

Multi-Objective Load Distribution Model A reasonable load distribution can provides a good

Foundation Item:Item Sponsored by National Natural Science Foundation of China (50974039) BiographyJIA Shu-jin(1982—), Male, Doctor Candidate ; E-mail: [email protected]; Received Date: September 8, 2011

Journal of Iron and Steel Research, International

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starting point for the control of strip shape and strip thickness, hence, load distribution optimization tak­ ing into account strip shape has become a research hot focus. This paper builds a multi-objective load distribution model considering rolling force margin balance, roll wear ratio and strip shape control. 1.1

Decision variables of load distribution Objective functions and constraint conditions of load distribution optimization have to do with the exit thicknesses of finishing mill stands directly or indirectly, so the exit thickness of each stand can be taken as the decision variable of the load distribution optimization problem. To avoid aimless search, take the result of em­ pirical load distribution method[7] as the initial value of optimization calculation; h° = H 0 exp Ci~ VCi+4:Ci
2d

CCn—Ci

In

H0

'+C 2 ln

Ho]

(1)

where, h° is the initial exit thickness of stand i-, H0 is the slab thickness; h„ is the product target thick­ ness; C\ , C2 are the statistics coefficient of actual production; a„ is an initial parameter; and ψι is the cumulative energy distribution coefficient of stand i. 1. 2

Optimization objectives of load distribution Rolling force margin balance, roll wear ratio con­ trol and strip shape control are taken to be the optimiza­ tion objectives. The upstream three stands take rolling force margin balance as the optimization objective, and should provide rolling reduction as far as possi­ ble within the equipment capacity, while the down­ stream four stands focus on roll wear ratio control and strip shape control. In view of the special process of schedule free rolling (SFR) of hot rolling, and the factors influencing strip shape, a multi-objective load distribution model is built. This model mainly takes into account the following three aspects. 1) Rolling force margin balance In view of the factors such as the fluctuation of strip thickness and the difficulty to bit slab into the first stand F l , the rolling reduction of F l should be somewhat smaller, while the second and third stands should make full use of the equipment capacity to give bigger rolling reductions as far as possible. Hence, rolling force margin balance of the upstream three stands is taken to be the first objective ( / i ) : /1 = (P1-K1P2)2 + (P2-K2P3)2 (2) where, P ; is the rolling force of stand i, i— 1, 2, 3;

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Ki and K2 are the rolling force margin balance ratios. 2) Roll wear control Roll wear control is a key technology of SFR of hot rolling. It hopes that the roll wear of the down­ stream four stands should be distributed according to a certain ratios to prevent excessive roll wear of individual pass, which is beneficial to the roll change cycle of a rolling schedule and surface quality of hot strips :

f2=i(™i-Gi'-wi+1y+k-w* i=4

(3)

i=i

where, it»; is the midpoint wear of stand i, whose formula can be referred to Ref. [8 — 9] ; G, is the roll wear ratio and / 2 is the second objective. 3) Strip shape control For downstream stands, the difference of en­ trance relative crown and exit relative crown of each stand should meet the strip flatness dead zone condi­ tion and ensure a good strip shape between stands. /3=Σ[(α/Α;-α/Η±Λ)2]

(4)

;=4

where, H, and hi are the entry and exit thicknesses of stand i, respectively; C'H and Ch are the entry and exit crowns of stand i; A, is the optimal adjustment, whose value is relative to the allowable difference of entrance relative crown and exit relative crown of stand i; and / 3 is the third objective. Based on the above analysis, the multi-objective load distribution model is given by: min/=(/! ,fz,f3) r O^P^Pn, s. t. 0 < I , < i m (5) lHi+1
2 Multi-Objective Load Distribution Optimization 2.1

Main procedure of MaximinDE Fig. 1 shows the schematic diagram of MaximinDE. Step 1 : Initialize the population of JV individu­ als and store them in P 0 ; Generation counter i = 0 . Step 2 : Generate the offspring population Q, through differential evolution algorithm1-10-1. 1 ) Mutation. For each individual JC, in P , , a mutant vector v; is generated according to Vi=xrl+F(xr2—xr3) (6) where i, rx, r 2 , r3 E {1, ··· » N) are mutually differ­ ent integers; F > 0 is a real parameter. 2) Crossover. A trial vector uiz={ua , ···» uiD) is generated with

Issue 2

Multi-Objective Load Distribution Optimization for Hot Strip Mills

Step 5: Set ί = ί + 1 , and go back to Step 2, un­ til a termination criterion is met.

Non-dominated sorting based on maximin fitness Fi Fi

^ Q,

Fk

Others

I

• 29 ·

Modified maxi) fitness sorting

—=Φ/

-Delete

R,

Fig. 1 Schematic diagram of MaximinDE _ ί Vij , if [rawCR~\ or where, j = l» 2, · · · , D; randb(j) is the j-th evalua­ tion of a uniform random number generator within [ 0 , 1 ] ; CR is a user-defined crossover constant in the range of [ 0 , 1 ] ; rnbr(ï) Ç { 1 , · · · , D) is a ran­ domly chosen index which ensures that w,· gets at least one parameter from v;. 3) Selection. In single objective optimization, the better solution from JC, and «, is selected as the ith individual j , in the offspring population Q,. In multi-objective optimization, it is not known which solution is better until all candidates are assigned non-domination levels. Therefore, u{ is first added to the offspring population, i. e. , J;=M;. Step 3 : Store both the parent population P, and the offspring population Qt in an intermediate popu­ lation Rt. Note that R, should be twice the size of Pt. Since all parent and offspring population mem­ bers are included in Rt, elitism is ensured. Step 4 : Construct the next generation of popula­ tion P,+i according to the elitism strategy procedure. 1) Non-dominated sorting procedure. Calculate the modified maximin fitnesses for all individuals in Rt ; Get the non-dominated solutions ( F i ) based on the modified maximin fitness values ( less than or equal to 0) and assign them a rank 1; Recalculate the modified maximin fitnesses for the remaining in­ dividuals in Rt, get the non-dominated solutions ( F 2 ) and assign them a rank 2; This procedure is continued until the count of solutions in all sets from Fi to Fk is larger than the population size N. 2) Diversity-maintenance procedure. All solu­ tions in sets Fi to Fk-i are filled in the new popula­ tion P,+\ for the next generation; Calculate the mod­ ified maximin fitnesses for the solutions in set Fk, and sort them according to their maximin fitnesses in ascending order; Then choose a part of the solutions which rank ahead to fill P,+ i.

2. 2 Elitist strategy based on modified maximin fit­ ness function The MaximinDE is based on the maximin fit­ ness function1-11-1, which can be defined as follows : »(*,·) = max { min [/j(*,·)— / ; ( * , ) ] } j=l,-,N,

j^i

1=1,-,m

(8) where, N is the population size; m is the number of objectives; /<(*,·) is the l-t\\ objective value of the individual JC,. There are two interesting properties that make the maximin fitness function so appealing to multiobjective optimization. 1) The maximin fitness of a dominated solution is greater than zero, and the maximin fitness of a unique non-dominated solution is less than zero. Hence, the maximin fitness func­ tion can be used to determine the Pareto domination. 2) The maximin fitness function rewards diversity and penalizes clustering of non-dominated solutions, which means no additional diversity maintenance mechanism is needed. However, there are also two drawbacks when apply the maximin fitness function to diversity maintenance. 1) If the objectives have different order of magni­ tudes, the maximin fitness according to Eqn. (8) has bias. 2) The maximin fitnesses of non-dominated solutions are affected by the dominated solutions. Hence, the objectives should be normalized and the dominated solutions should be eliminated before cal­ culating their maximin fitnesses used for diversity maintenance. The formula for the modified maximin fitness function can be expressed as follows: fl

(·*,)

=

L/i (*i)

' /maximi„(*;)=

7/,mmJ/(/i,max

max

f Z.min)

{ min [ / / ( * , ■ ) — / / ( * ; ) ] }

(9) where, /i jma x, /;,min are the Z-th maximum and minimum objective values in the current population; / , (x ; ) is the l-th normalized objective value of solution x, ; and N S denotes a non-dominated solution set. In this study, a two-dimensional array, which is used to store the minimum fitnesses, is introduced to reduce the complexity for calculating the modified maximin fitnesses repeatedly. This method makes the elitist strategy with O (mNz ) complexity. 2. 3 Multi-objective load distribution optimization procedure The steps of multi-objective load distribution

Journal of Iron and Steel Research, International

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Vol. 20

optimization based on M a x i m i n D E are as follows: Step 1: Initialize input p a r a m e t e r s ( s l a b s P D I , technological p a r a m e t e r s , e t c ) . Step 2 : Determine the initial exit thickness of each stand t h r o u g h empirical formula. Step 3 : Call M a x i m i n D E procedure to search the Pareto-optimal solutions within decision space. In this s t e p , a constraint handling technique p r o ­ posed in Ref. [ 1 2 ] is used to g u a r a n t e e the feasibility of Pareto-optimal solutions. Step 4 : Repeat Step 3 until a termination crite­ rion is met. If t e r m i n a t e d , t h e n o u t p u t t h e P a r e t o optimal solution set.

3

Experimental Results and Discussion

Fig. 3

Pareto-optimal solution set of load distribution

10

Take a typical carbon strip from Baosteel 1880 m m H S M for example. The steel grade is Q235B, and strip width, t h i c k n e s s , target t h i c k n e s s , and target crown are 1 2 2 5 , 4 1 . 5 , 3. 2 5 , and 0. 04 m m , respectively. T h e rolling length in the last stand F7 is 769 m. Set MaximinDE algorithm control parameters : pop­ ulation size J V = 2 0 0 , C R = 0 . 9, F = 0 . 5 , the m a x i m u m evolution generation T max = 100. Set load balance ra­ tio Ki and roll wear ratio G; as follows: Ki = 0 . 9 , K 2 = l . 0 7 5 , G 4 = 0 . 8 , G 5 = l . 05 a n d G 6 = l . 1. After 100 g e n e r a t i o n s , the Pareto-optimal solu­ tion set of load distribution is obtained. Fig. 2 shows the evolution curves of objective function 1 , 2 , 3 w i t h generations. After 50 g e n e r a t i o n s , the t h r e e objective functions begin to reach steady s t a t e , which indicates the convergence of MaximinDE is fast. Fig. 3 s h o w s the Pareto-optimal solution set in objective space at t h e end of evolution. Fig. 4 and Fig. 5 show the 2D Pareto-optimal solution set of /Ί vs fz ' fï vs fi > respectively. N o t e t h a t the more right the points location, the smaller will be the third optimization objective (except the horizontal and 35

- 25 - 10<^ - 15

-5 20

40 60 Generalion

80

100

Fig. 2 Objective function 1, 2, 3 vs generation

Fig. 4 2D Pareto-optimal solution set of / i vs f3 10 8

i

6

1

4 2 V<"»*A«,„,„„

0.1

0.2

-

'

-

0.3

' ■ ■

0.4

-

■

0.5

Fig. 5 2D Pareto-optimal solution set of f2 vs fs vertical axis objective). A s s h o w n in Fig. 3 , when objective 1 increases, objective 2 t e n d s to decrease; w h e n objective 1 increases, objective 3 tends to in­ crease (see more clearly in Fig. 4) ; w h e n objective 2 increases, objective 3 t e n d s to decrease ( s e e more clearly in Fig. 5 ) . It indicates that t h e r e are conflict­ ing relationships between /Ί and / 2 , fz and f3, r e ­ spectively. In order to balance the three objectives, a compromise is needed. T h e r e f o r e , load distribu­ tion problem has no unique optimal solution, but a set of alternative solutions. Decision-makers can se­ lect a trade-off solution based on their experience or

Multi-Objective Load Distribution Optimization for Hot Strip Mills

Issue 2

Decision-makers can select an arbitrary group of {en , Ü2 t 03} according to their preference. In this p a p e r , set ai = 0 . 3 , 02 = 0. 1, a3 = 0 . 6. Calculate the weighted-sum objective function / for all the solu­ tions in the Pareto-optimal solution s e t , and t h e n sort these solutions according to / in ascending or­ der. T a b l e 1 lists 20 typical alternative solutions in t h e Pareto-optimal solution s e t , w h e r e /i; represents the exit thickness of stand i. Select the solution t h a t has t h e m i n i m u m value of / as the trade-off solution (indicated in bold faces). T a b l e 2 lists the load dis­ tribution results of t h e trade-off solution. F o r com­ p a r i s o n , T a b l e 3 lists the ones of the empirical solu­ tion.

the importance of objectives. In practice, some decision-makers may be used to optimize load distribution w i t h weighted-sum a p ­ proach. H e n c e , a decision-making m e t h o d based on weighted-sum approach is also proposed to select the trade-off solution. T h e weighted-sum objective function can be defined as follows: min/=ai · a3

Jlmax J3

J lmin

~a2

J2 / 2max

/2min J2n

J 3min

J 3max

(10)

./3min

• 31 ·

w h e r e , ai, 02> and a 3 are t h e weight coefficients of three objectives, respectively, and they should meet the relation; α ι + α 2 + ο 3 = 1; /.min and / i m , x ( i = l , 2 , 3) are the minimum and the m a x i m u m values corre­ sponding to the i-th objective, which can be obtained from the Pareto-optimal solution set.

A s s h o w n in Fig. 6 , rolling forces of the trade­ off solution in stands F2 and F3 are larger than those of the empirical solution to make full use of the equipment

Table 1 Some optimal solutions in Pareto-optimal solution set No.

Ai

ht

A3

hi

A5

A6

/l

h

h

/

1

24.50

13.69

8.17

5.50

4.36

3.63

32

0.151

0.227

0.035

2

22.81

13.18

8.07

5.49

4.36

3.62

113

0.263

0. 105

0.043

3

24.45

13.69

8.43

5.51

4. 35

3. 60

890

0.352

0.007

0.052

4

24.39

13.71

8.52

5.51

4.36

3.60

752

0.409

0.000

0.059

5

24.13

14.83

8.24

5.54

4.38

3.64

8 244

0.233

0.067

0.069

6

23.66

14.83

8. 17

5.54

4.35

3.64

3 793

0.046

0.987

0.084

7

23.01

14.81

8.16

5.53

4.35

3.64

17 446

0.162

0. 180

0.103

8

24.53

13.79

8.45

5.38

4.38

3.58

13 200

0.038

1.223

0. 135

9

24.45

13.97

8.46

5.61

4.31

3.63

4519

0.025

2.328

0. 169

10

21.91

14.83

8.14

5.50

4.35

3.63

6 902

0.024

2.519

0. 190

11

20.79

13.84

8.05

5.49

4.30

3.63

18 930

0.019

2.635

0.245

12

21.58

14.59

8. 10

5. 52

4.31

3.63

5 325

0.021

4.737

0.324

13

21.39

14.76

8.08

5.50

4.31

3. 63

24 835

0.010

4.061

0.358

14

20.20

14.09

8.02

5.49

4.30

3.63

21634

0.009

4.877

0.397

15

20.40

14.12

8.04

5.48

4.27

3.62

29502

0.008

4.796

0.423

16

18.87

13.42

7.95

5.47

4.28

3.62

35 928

0.006

5.520

0.494

17

19.87

14.35

8.07

5.50

4.30

3.62

26 399

0.007

6.818

0.539

18

18.71

13.23

7.89

5.45

4.26

3. 62

45 472

0.004

6.166

0.573

19

19.98

14.76

8.08

5.50

4.30

3.62

54415

0.002

6.976

0.660

20

19.53

14.83

8.09

5.51

4.30

3.63

60 405

0.002

7.912

0.743

Table 2 Exit thickness/mm Reduction percent/ % Rolling force/kN

Fl

F2

F3

F4

F5

F6

F7

24.50

13.69

8.17

5.50

4.36

3.63

3.25

41.0

44. 1

40.3

32.7

20.7

16.7

10.5

19441

21651

20167

16316

11383

8 942

7197

Table 3 Exit thickness/mm Reduction percent/ % Rolling force/kN

Load distribution results of trade-off solution

Load distribution results of empirical solution

Fl

F2

F3

F4

F5

F6

F7

22.18

13.08

8.53

6.00

4.48

3.65

3.25

46.5

41.0

34.8

2,9.7

25.4

18.5

11.0

21921

20 298

18505

16263

13751

10 503

7 876

Journal of Iron and Steel Research, International

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Vol. 20

F5 goes beyond the limit of wavy edges. T h e r e f o r e , it can be concluded t h a t the trade-off solution outperforms the empirical solution, and ef­ fectively improve the quality of the strip product.

4 Fig. 6 Rolling force in experience and MaximinDE load distribution mode capacity, while the contrary is the case in stands F5 to F7 so as to control roll wears and strip shape ef­ fectively. Fig. 7 shows a comparison of roll wears between the trade-off and empirical solutions after rolling this strip. Roll wears of t h e trade-off solution in stands F5 to F7 are more uniform t h a n those of empirical solution. F u r t h e r m o r e , it should be noted t h a t the rolling force in stand F4 of the trade-off so­ lution is very similar w i t h t h a t of the empirical solu­ t i o n , but the roll wear of the trade-off solution in stand F4 is larger t h a n t h a t of t h e empirical solution. T h i s is because t h e rolling length ( 4 5 4 m ) of the trade-off solution is longer t h a n t h a t (417 m ) of the empirical solution in stand F 4 . A s s h o w n in Fig. 8 , relative crown differences of the trade-off solution are basically in the limit of center buckle and wavy e d g e s , while t h e one of the empirical solution in stand 2.5 n Experience " ■ MaximinDE %,,

1 ) T h i s paper p r e s e n t s a multi-objective load distribution optimization m e t h o d for hot strip mills. M a x i m i n D E can not only avoid the selection of weight coefficients encountered in single objective load distribution optimization, but also can find the conflicting relationship among different objectives, which is another advantage of multi-objective load distribution optimization in comparison w i t h single objective load distribution optimization. 2) M a x i m i n D E s h o w s a flexible ability for load distribution optimization and a favorable perform­ ance in the evolution process. A stable Pareto-optimal solution set can be obtained quickly, which is beneficial to the practical application. Decision-mak­ ers can select a desired solution based on their expe­ rience or the importance of objectives. 3) E x p e r i m e n t a l results confirm t h e feasibility of t h e multi-objective load distribution optimization, t h e trade-off solution is effective in t h e control of rolling force margin balance, roll wear ratio and strip s h a p e , and t h e n achieve a good foundation, for t h e S F R of h o t rolling. References: [1]

1 0

I·

a 0.5 0

Conclusions

LI Hai-jun, XU Jian-zhong, W A N G Guo-dong, et al. Improve­ ment on Conventional Load Distribution Algorithm in Hot Tan­ dem Mills [ I ] , Journal of Iron and Steel Research» Internation­ a l , 2007, 1 4 ( 2 ) : 36.

1

2

3

4 S>tand

5

6

7

[2]

Z H A N G Jin-zhi. Analysis and Summarization of Load Distribu­ tion [ J ] . Wide and Heavy P l a t e , 2004, 1 0 ( 3 ) : 14 (in Chi­

Fig. 7 Roll wears in experience and MaximinDE load distribution mode

nese) . [3]

W A N G Yan, LIU Jing-lu, SUN Yi-kang. Immune Genetic Al­ gorithm ( I G A ) Based Scheduling Optimization for Finisher [ J ] , Journal of University of Science and Technology Beijing, 2002,

0.04

2 4 ( 3 ) : 339 (in Chinese). [4]

W A N G Jian-hui, X U Lin, YAN Yong-liang, et al. Improved PSO and Its Application to Load Distribution Optimization of

to =8

H o t Strip Mills [ J ] . Control and Decision, 2 0 0 5 , 2 0 ( 1 2 ) : 1379 (in Chinese).

e -0.02 Ί -0.04 OS

-0.06

[5]

♦ Center buckle x Wavy edges ■ Experience * MaximinDE

4 5 Stand Fig. 8 Relative crown difference in experience and MaximinDE load distribution mode

Y A O Feng, Y A N G Wei-dong, Z H A N G Ming. Improved PSO and Its Application to Load Distribution Optimization of Hot Strip Mills CJ]. Journal of University of Science and Technolo­ gy Beijing, 2009, 3 1 ( 8 ) : 1061 (in Chinese).

[6]

Y A O F e n g , Y A N G Wei-dong, Z H A N G Ming. Multi-Objective Differential Evolution Used for Load Distribution of Hot Strip Mills [ J ] . Control Theory and Application, 2010, 2 7 ( 7 ) : 897 (in Chinese).

(Continued on Page 61)

Issue 2

T h e r m o d y n a m i c s a n d M a g n e t o s t r i c t i o n of F e - C o - B - S i - N b - C B u l k M e t a l l i c G l a s s e s

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·

The authors thank to S Y Chen for his valuable contribution to magnetostriction measurement. References [1]

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[6]

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