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Intelligence based modeling and optimisation oftandem cold rolling processDadong WangUniversity of Wollongong
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Recommended CitationWang, Dadong, Intelligence based modeling and optimisation of tandem cold rolling process, Doctor of Philosophy thesis, Faculty ofEngineering, University of Wollongong, 2002. http://ro.uow.edu.au/theses/1830
Intelligence Based Modeling and Optimisation of
Tandem Cold Rolling Process
A thesis submitted in fulfillment of the requirements for the award of the degree
DOCTOR OF PHILOSOPHY
from
University of Wollongong
by
DADONG WANG
BE, ME (USTB)
Faculty of Engineering
2002
I
Declaration
I, Dadong Wang, declare that this thesis, submitted in fulfillment of the requirements for
the award of Doctor of Philosophy, in the Faculty of Engineering, University of
Wollongong, Australia, is wholly my own work unless otherwise referenced or
acknowledged. The document has not been submitted for qualifications at any other
academic institution.
Dadong W A N G
Nov. 24, 2002
n
Acknowledgements
The author wishes to take this opportunity to thank everyone who has helped make this
thesis a reality.
First, many thanks to my supervisor, Professor A. Kiet Tieu, the Discipline Leader of
Mechanical Engineering at the University of Wollongong for his invaluable guidance,
excellent supervision. It is not often that you find someone who combines excellence of
academy with continuous encouragement over your Ph.D. study.
The author would like to acknowledge my co-supervisor, Dr. F. G. de Boer, for his
supervision and reviewing my publications.
Thanks are also due to Mr. Giovanni D'Alessio at Packaging Products, BHP Steel, for
his assistance during my stay at BHP Steel five stand cold mill.
The author wishes to express his thanks to Dr. W. Y. D. Yuen at Steel Research
Laboratories, BHP Steel, for his advice, discussions and regular meetings.
The author would also like to extend my thanks and appreciation to The State Key
Laboratory of Rolling and Automation, Northeast University, China, during my visiting
study in 1998.
m
A special mention must go to the other members of the Rolling Technology research
group steered by Prof. A. Kiet Tieu for their informative discussions and exchanges that
benefit this thesis.
The author would like to acknowledge the financial support from the Australian
Research Council (ARC) (Collaborative) and BHP Steel Research Laboratories during
the course of this research.
The author is grateful to the University of Wollongong for providing me scholarships
through which this research is made possible.
My heartfelt and sincere thanks to my families for their sacrifices, endless support and
continuous encouragement.
IV
Publications
1. Publications Relevant to My Ph.D. Topic (1998 - now)
I. Contribution in Journals:
(1) D. D. Wang, A. Kiet Tieu, F. G. de Boer, B. Ma and W. Y. D. Yuen, "Toward A
Heuristic Optimum Design of Rolling Schedules for tandem Cold Rolling mills",
International Journal of IF AC, Engineering Applications of Artificial Intelligence,
Vol. 13, No. 4, July 2000, pages 397 - 406.
(2) Yingjian Liu, A. Kiet Tieu, D. D. Wang and W. Y. D. Yuen, "Friction
Measurement in Cold Rolling", Journal of Materials Processing Technology, Vol.
Ill, Issues 1-3, April 2001, pages 142-145.
EL Conference Papers
(1) D. D. Wang, A. K. Tieu, Giovanni D'Alessio, F. G. de Boer and W. Y. D. Yuen,
"Theoretical Analysis and Intelligent Optimisation for Unsteady Processes in
Tandem Cold Rolling", The Third Australasian Congress on Applied Mechanics,
Sydney, Australia, 20th - 22nd February 2002, pages 315-320.
(2) D. D. Wang and A. Kiet Tieu, "Numerical Simulation and Optimisation of
Threading Process in Tandem Cold Rolling", Proceedings of the Second
International Conference on Mechanics of Structures, Materials and Systems,
Australia, 14-16 February 2001, pages 363-368.
V
(3) D. D. Wang and A. Kiet Tieu, "Numerical Simulation and Analysis of Acceleration
Process in Cold Rolling", Proceedings of the Second International Conference on
Mechanics of Structures, Materials and Systems, Australia, 14-16 February 2001,
pages 339-344.
(4) D. D. Wang, A. Kiet Tieu, Giovanni D'Alessio, F. G. de Boer, and W. Y. D. Yuen,
"Development of A Computational Model for Shape Analysis and Intelligent Set
up of Jack Forces in Cold Rolling", International Symposium on Advanced
Forming and Die Manufacturing Technology (AFDM'99), Pusan, Korea, Sep. 7-9,
1999, pages 149-154.
(5) D. D. Wang, A. Kiet Tieu, F. G. de Boer, and W. Y. D. Yuen, "Evolutionary
Optimisation of Rolling Schedule for the Setup of a Tandem Cold Rolling Mill",
Proceedings of The 7th International Conference on Steel Rolling (Steel Rolling
'98), Makuhari, Japan, Nov. 9-11, 1998, pages 139-144.
2. Publications in Other Fields (1993 - 1997)
L Contribution in Journals:
(1) "Harmony Theory Yields Robust Machine Fault Diagnostic System Based on
Learning Vector Quantization Classifier", International Journal of IFAC,
Engineering Applications of Artificial Intelligence, Vol. 9, No. 5, 1996, pages 487-
498. (with Peter Tse)
VI
(2) "The Recognition of Wear Particles in Lubricating Oil Using Neural Networks",
Journal of University of Science and Technology Beijing, Vol. 3, No. 1, 1996, P.
R. China, pages 26-30. (with Debing Yang and Jinwu Xu)
(3) "Running Machine Condition Forecasting Based on Multiple Parameters Using
Adaptive Neural Networks", Special Issue of International Journal of Mathematical
Modeling and Scientific Computing, Vol. 6, 1996, (ISSN 1067-0688), USA. (with
Peter Tse and Jinwu Xu)
(4) "An Integrated Technique for Process Fault Diagnosis Based on Neural/Harmonic
Models", Special Issue of International Journal of Mathematical Modeling and
Scientific Computing, Vol. 6, 1996, (ISSN 1067-0688), USA. (with Peter Tse)
(5) "A Failure Diagnosis Expert System", Journal of University of Science and
Technology Beijing, Vol. 15, No. 4, 1993, P. R. China, pages 389-393. (with Jinwu
Xu)
(6) "A Developing Tool of Expert System", Computer Science, Technology and
Application, Vol. 2, 1993, P. R. China, (with Jinwu Xu)
IL Conference Papers:
(1) "Enhanced LVQ Fault Classifier", Proceedings of CMDTC-97, pages 220-227,
Beijing, 1997. (with Debin Yang and Jinwu Xu)
(2) "Investigation Into Genetic Algorithms Based Cluster and Applications in Machine
Fault Diagnosis", Proceedings of CMDTC-97, Beijing, 1997, pages 238-243. (with
Jinwu Xu)
vn
(3) "Computational Intelligence Based Machine Fault Diagnosis", Proceedings of
IEEE International Conference on Industrial Technology (IEEE ICIT96),
Shanghai, China, 2-6 December 1996, pages 465-469. (with Jinwu Xu and Debing
Yang)
(4) "Intelligent Monitoring of Sheet Forming Process", Proceedings of IEEE
International Conference on Industrial Technology (IEEE ICIT'96), Shanghai,
China, 2-6 December 1996, pages 455-459. (with Debing YANG and Jinwu XU)
(5) "Fault Detection Based on Evolving LVQ Neural Networks", Proceedings of 1996
IEEE International Conference on Systems, Man and Cybernetics, Volume 1 of 4,
Beijing, China, October 14-17, 1996, pages 255-260. (with Jinwu XU)
(6) "A Hybrid Neural Networks Based Machine Condition Forecaster and Classifier by
Using Multiple Vibration Parameters", 1996 IEEE International Conference on
Neural Networks (ICNN '96), Vol. 4, Washington, DC, USA, June 2-6, 1996,
pages 2096-2100. (with Peter Tse)
(7) "Improving Learning Vector Quantization Classifier in Machine Fault Diagnosis
by Adding Consistency", proceedings of 1995 IEEE International Conference on
Neural Networks (ICNN1 95), Vol. 2, Australia, Nov. 27 - Dec. 1, 1995, pages 927-
931. (with Peter Tse)
(8) "Classification of Image Texture Inherited with Overlapped Features Using
Learning Vector Quantization", proceedings of Second International Conference on
Mechatronics and Machine Vision in Practice, Hong Kong, Sept. 1995, pages 286-
290. (with Peter Tse and Jinwu Xu)
(9) "The Diagnosis Oriented Expert System Shell", Proceedings of CMDTC-93,
China, November 1993. (with Jinwu Xu)
vm
HI. Technical Reports:
Wang D. D. and Peter Tse, "Investigations into Intelligent Systems for the Diagnostics
and Control of Manufacturing Machinery", research report, Strategic Research Grant
No. 7000262, July 1996, City University of Hong Kong, Hong Kong.
IX
Abstract
The requirements for the manufacturers of steel have been increased in many respects in
recent years. The harsh competition between the manufacturers requires a continuous
reduction of the production costs and a successive improvement of product quality. The
work presented herein is a part of efforts towards maximizing rolling mill throughput
and minimizing processing costs and crop losses by computational intelligence based
process modeling and optimisation in tandem cold rolling.
It is the first time that the computational intelligence based optimisation has been
applied to the rolling schedules of tandem cold rolling mills as is shown in this thesis.
Rolling schedule is essential for a tandem cold strip mill operation. This involves the
determination of inter-stand gauges, tensions and rolling speeds at individual stands. As
the rolling process is examined and shown to be a nonlinear programming problem
incorporated the complex interactions of multiple rolling parameters such as reductions,
roll force and torque, rolling speeds and tensions. Each of the parameters can be set to a
value in its admissible data range of float type, which is determined by threading
conditions. The rolling schedule is a set of such rolling parameters. Therefore, different
combinations of these rolling parameters generate numerous rolling schedules.
Although some empirical rolling schedules may be reasonably satisfactory, however, it
is hard to declare which schedule is the best solution. In this study, an intelligent
searching mechanism is introduced to optimize the rolling schedule by assessing rolling
constraints and the combined cost function of tension, shape and power distribution.
The optimisation results have been compared with current rolling practices based on
X
empirical models. It is shown that the proposed model can significantly reduce the
power distribution cost, maximize the safe level of strip tension, and obtain good strip
shape. The proposed model is generic for complex engineering problem optimisation,
and is capable for Multiple-Objective Problem Solving (MOPS).
A hybrid model of Influence Coefficient Method and Spring-Beam Method is proposed
and verified for strip shape analysis and prediction. From the point of view of product
quality, shape is an important parameter in the rolling process. Bad shape is a frequent
cause for threading delay, rejection or rectification by costly means. Particularly for
cold strip, the thinner the strip, the greater is the difficulty in maintaining good shape,
which requires the optimum relationship of roll force, strip tension, work roll crowns
and/or bending force. A physically based shape predictive model has been established,
on the basis of which, a heuristic optimisation approach is employed to search for the
optimum setting of the bending forces for those mill stands with bending systems, and
to optimize reduction schedule for those without roll bending system. The proposed
model has also been extended to optimize shape problem during threading by using
optimal roll ground crown or optimum roll bending force. A comparison is made
between results predicted by the proposed model and test results obtained on a
production mill.
The control of head and tail off-gauge is becoming increasingly important for many
companies in the steel industry so as to maximize yield from their rolling operations. As
there are various kinds of fluctuations in unsteady states such as threading, acceleration,
deceleration, and tailing out processes, and these fluctuations always accompany with
XI
non-linearity and hysteresis, it is very difficult to accurately model, and then optimize
and/or control these processes. Therefore, dynamic simulation and analysis of mill
performance during these transient rolling phases, is vital in order to ensure that
thickness is controlled within the desired tolerance. The significant advantage to be
gained from the simulation is that the logic errors potentially existing in a rolling mill
operation could be corrected and the mill performance could be improved.
Innovative models for the unsteady rolling process simulation have been presented. The
threading and acceleration processes have been dynamically simulated. The results
generated from the simulation have been compared with the measurements of the strip
produced from a production mill. The interactive relationships between roll force, exit
thickness and inter-stand tensions are presented for both threading and acceleration
processes. The speed dependent rolling parameters for acceleration process are analyzed
and taken into account in the simulation. Deceleration and tailing-out processes are also
addressed, as they are similar to acceleration and threading processes, respectively. The
proposed model has also been examined with respect to changes in initial conditions
such as friction between work rolls and strip, strip properties and work roll profile.
The proposed models and simulations have been implemented and verified with plant
data. The numerical simulations together with their result analysis will be beneficial to
fine-tune the rolling process parameters and are effective in achieving significant cost
savings and product quality improvements.
XII
Table of Contents
Declaration I
Acknowledgement II
Publications IV
Abstract IX
Table of Contents XII
List of Figures XVIII
List of Tables XXIV
Nomenclature XXV
Chapter 1 Introduction 1
1.1 Main Research Obj ectives 2
1.2 Overview of This Research 3
1.3 Organisation of the Thesis 4
1.4 Summary of Contributions 7
Chapter 2 Fundamentals of Tandem Cold Rolling and Optimisation Theory 11
2.1 Process Description and Analysis in Tandem Cold Rolling 12
2.1.1 Tandem Cold Rolling Mills 13
2.1.2 Automatic Control Systems for Tandem Cold Rolling Mills 16
2.1.2.1 AGC Control System 17
2.1.2.2 Strip Profile, Shape, Flatness and Automatic
Flatness Control System 19
2.1.3 Description of Tandem Cold Rolling Processes 23
2.1.3.1 Rationale of Cold Rolling 23
2.1.3.2 Five Phases of Tandem Cold Rolling 27
2.2 Overview of Classical Optimisation 31
2.2.1 Concept of Optimisation 31
2.2.2 Three Components of Optimisation 32
2.2.2.1 Objective Function 32
2.2.2.2 Variables 33
2.2.2.3 Constraints 33
2.2.3 Classical Optimisation Hierarchy 34
2.3 Summary and Concluding Remarks 35
Chapter 3 Principle of Evolutionary Optimisation and Its Applications in
Engineering 37
3.1 Introduction 38
3.1.1 Composition of Evolutionary Optimisation 40
3.1.2 Genetic Algorithms 41
3.1.2.1 Concept and Origin of Genetic Algorithms 41
3.1.2.2 Generic Operators 42
3.1.2.3 Mathematical Foundation of Genetic Algorithms 45
3.2 Comparison of Evolutionary and Conventional Optimisation 58
3.3 GA's Applications in Steel Industry 61
3.3.1 Rolling Scheduling 63
3.3.2 Production Planning and Optimisation 64
3.3.3 Rolling Mill Optimisation 71
3.3.4 Steel Products Design 72
3.3.5 Other Fields in Steel Industry 72
3.4 Summary and Concluding Remarks 74
Chapter 4 Evolutionary Optimisation of Rolling Scheduling for
Tandem Cold Rolling Mills 76
4.1 Introduction 77
4.2 Basic Scheduling for Tandem Cold Rolling Mills 80
4.3 Optimised Scheduling 82
4.3.1 Cost Functions 83
4.3.2 Constraint Checks 97
4.3.3 Genetic Algorithm (GA) Based Optimisation 100
4.3.3.1 Genetic Encoding and Operation for
Rolling Scheduling Problem 100
4.3.3.2 Optimisation Parameters and Results 104
4.3.3.3 Discussion of the Results 115
4.4 Summary and Concluding Remarks 119
Chapter 5 Shape Analysis and Optimisation in Tandem Cold Rolling 121
5.1 Introduction 121
5.2 Roll Pressure and Profile Model 128
5.2.1 Roll Pressure Distribution 128
5.2.2 Roll Pressure Analysis 129
5.3. Roll Deformation Theory and Derivation of Equations 131
5.3.1 Initial Roll and Strip Profile 131
5.3.1.1 Incoming Strip Profile 131
5.3.1.2 Work/Backup Roll Crown 131
5.3.2 Static Equilibrium of the Work/Backup Rolls 132
5.3.2.1 Static Force Equilibrium 132
5.3.2.2 Static Bending Moment Equilibrium 133
5.3.3 Derivation of Roll Deformation Compatibility Equations 134
5.3.3.1 Calculation of Influence Coefficients 134
5.3.3.2 Deformation Compatibility Condition for Contact of
Backup and Work Rolls 136
5.3.3.3 Deformation Compatibility Equation for Contact of
Work Roll and Strip 138
5.3.4 Objective Function of Strip Shape Optimisation 142
5.4 Equation Set Solving 144
5.5 Flowchart of Shape Analysis 145
5.6 Model Validation and Numerical Experimental Results 146
5.7 Optimisation of Shape under Threading Condition 153
5.8 Summary and Concluding Remarks 157
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes
in Tandem Cold Rolling 158
6.1 Introduction 159
6.2 Threading and Acceleration Processes 163
XVI
6.2.1 Threading Process 163
6.2.2 Speed Dependent Variables Analysis during Acceleration 165
6.3 Modeling of Dynamic Threading and Acceleration Processes 168
6.3.1 Modeling of Threading Process 168
6.3.1.1 Motor Speed Droop Consideration 168
6.3.1.2 Threading Process Modeling 169
6.3.2 Modeling of Acceleration Process 173
6.3.2.1 AGC Control during Acceleration 173
6.3.2.2 Modeling of Acceleration Process 175
6.4 Numerical Simulation of Threading Process 176
6.4.1 Flow Chart of Threading Simulation 176
6.4.2 Simulation Condition and Results 179
6.4.2.1 Threading Simulation Parameters 179
6.4.2.2 Threading Simulation Results 179
6.5 Numerical Simulation of Acceleration Process 194
6.5.1 Flow Chart of Acceleration Simulation 194
6.5.2 Acceleration Simulation Parameters 195
6.5.3 Acceleration Simulation Results 195
6.6 Other Relevant Issues 202
6.6.1 Deceleration and Tailing out processes 202
6.6.1.1 Deceleration Process 202
6.6.1.2 Tailing Out Process 204
6.6.2 Effect of Friction on Unsteady Rolling Processes 206
6.6.3 Effect of Strip Hardness and Entry Thickness 211
6.6.4 Effect of Work Roll Profile 212
6.6.5 Strip Shape Consideration 212
6.7 Summary and Concluding Remarks 212
Chapter 7 Conclusions and Recommendations 215
7.1 Summary of Main Conclusions 215
7.2 Recommendation for Future Research 217
References 220
Appendix A Classical Optimisation Hierarchy 251
Appendix B Overview of GA's Applications in Other Industries 265
Appendix C Additional Results of Threading and Acceleration Simulations 277
List of Figures
Figure 2.1 A typical five-stand tandem cold rolling mill 14
Figure 2.2 A typical mill drive system 15
Figure 2.3 A typical mill stand arrangement 16
Figure 2.4 Central control pulpit of a tandem rolling mill 17
Figure 2.5 Main control loops of a typical AGC system of
a 5-stand tandem cold mill 18
Figure 2.6 A typical automatic flatness control system 22
Figure 2.7 Schematic of the strip and work roll interface 24
Figure 2.8 Typical speed diagram of a tandem cold rolling mill 28
Figure 2.9 Classical optimisation tree 35
Figure 3.1 The roulette-wheel selection 43
Figure 3.2 An illustration often encoded variables 46
Figure 3.3 Comparison of searching efficiency of three types of schemes 58
Figure 3.4 Papers collected in Inspec Database relevant to engineering applications
of Genetic Algorithms 62
Figure 4.1 A typical rolling scheduling procedure for the setup of
a tandem cold rolling mill 80
Figure 4.2 Transverse profile of rolled strip 89
Figure 4.3 Deflection of the neutral axis due to point loads 91
XIX
Figure 4.4 Deflection due to a bending moment 93
Figure 4.5 Interference between work rolls beyond the edges of the strip 95
Figure 4.6 Roll forces for different schedules 106
Figure 4.7 Power distributions for different schedules 107
Figure4.8 Work roll speeds for different schedules 107
Figure 4.9 Tension forces for different schedules 108
Figure 4.10 Torque at each stand for different schedules 108
Figure 4.11 Relationship between deflection of work roll and roll force 110
Figure 4.12 Relationship between deflection of backup roll and roll force 110
Figure 4.13 Deflection of backup roll due to bending Ill
Figure 4.14 Deflection of backup roll due to shear Ill
Figure 4.15 Work roll/backup roll flattening 112
Figure 4.16 Deflection of backup roll due to the effect of Poisson' s ratio 112
Figure 4.17 Deflection of work roll due to bending 113
Figure 4.18 Deflection of work roll due to shear 113
Figure 4.19 Flattening at work roll/strip 114
Figure 4.20 Deflection of work roll due to the effect of Poisson's ratio 114
Figure 5.1 Edge buckle (long-edge shape) 123
Figure 5.2 Center buckle (long-middle shape) 123
Figure 5.3 Roll bending systems 123
Figure 5.4 CVC rolls with the roll shifting (a) neutral crown; (b) positive crown;
and (c) negative crown 124
Figure 5.5 Force distribution model for rolling mills with bending systems 129
XX
Figure 5.6 Crowned roll 132
Figure 5.7 Influence coefficient calculation model 135
Figure 5.8 Deformation compatibility condition of the work roll and strip 139
Figure 5.9 Flow chart of strip shape analysis 146
Figure 5.10 Solution time for different element lengths 148
Figure 5.11 Convergence performance (convergence tolerance = 0.001) 148
Figure 5.12 Measured work roll surface temperature distribution (Degree C)
at Standi 150
Figure 5.13 Thermal crown of the work roll at Stand 1 150
Figure 5.14 Real profile of the work roll at Stand 1 151
Figure 5.15 Wear profile of the work roll at Stand 1 151
Figure 5.16 Comparison of calculated and actual strip thickness at Stand 1 exit 152
Figure 5.17 Comparison of actual and predicted strip flatness at Stand 1 exit 152
Figure 5.18 Comparison of calculated and actual strip thickness at Stand 2 exit 153
Figure 5.19 Setup values for exit thickness to maintain the same flatness level with
different work roll crowns at Stand 1 156
Figure 5.20 Flatness of exit strip under different roll bending forces (KN) 156
Figure 6.1 Roll gap model diagram 164
Figure 6.2 Motor speed droop and control 169
Figure 6.3 BISRA and Davy-United gauge-meter control model 175
Figure 6.4 A practical gauge-meter control model 175
Figure 6.5 Flow chart for the threading process simulation of
a tandem cold mill 178
Figure 6.6 Comparison of threading forces from the simulation with actual
measurements 181
Figure 6.7 Tension establishment process during threading 182
Figure 6.8 Dynamic process of tension establishment after the strip head enters the
second stand 182
Figure 6.9 Dynamic process of tension establishment after the strip head
enters the third stand 183
Figure 6.10 Dynamic process of tension establishment after the strip head
enters the fourth stand 183
Figure 6.11 Dynamic process of tension establishment after the strip head
enters the fifth stand 184
Figure 6.12 Stand 1 threading force variation after the strip head enters Stand 2 185
Figure 6.13 Stand 2 threading force variation after the strip head enters Stand 2 185
Figure 6.14 Stand 2 threading force variation after the strip head enters Stand 3 186
Figure 6.15 Stand 3 threading force variation after the strip head enters Stand 3 186
Figure 6.16 Stand 3 threading force variation after the strip head enters Stand 4 187
Figure 6.17 Stand 4 threading force variation after the strip head enters Stand 4 187
Figure 6.18 Stand 4 threading force variation after the strip head enters Stand 5 188
Figure 6.19 Stand 5 threading force variation after the strip head enters Stand 5 188
Figure 6.20 Overview of the strip thickness variation at each of all five stands 189
Figure 6.21 Exit thickness variation at Stand 1 after the strip head enters Stand 2 .. ..190
Figure 6.22 Exit thickness variation at Stand 2 after the strip head enters Stand 2 .. ..190
Figure 6.23 Exit thickness variation at Stand 2 after the strip head enters Stand 3 .. ..191
Figure 6.24 Exit thickness variation at Stand 3 after the strip head enters Stand 3 191
XXII
Figure 6.25 Exit thickness variation at Stand 3 after the strip head enters Stand 4 .. ..192
Figure 6.26 Exit thickness variation at Stand 4 after the strip head enters Stand 4 .. ..192
Figure 6.27 Exit thickness variation at Stand 4 after the strip head enters Stand 5 .. ..193
Figure 6.28 Exit thickness variation at Stand 5 after the strip head enters Stand 5 .. ..193
Figure 6.29 Flow chart for the acceleration process simulation of
a tandem cold mill 194
Figure 6.30 Rolling speed variation at Stand 5 during acceleration 198
Figure 6.31 Variation of mean friction coefficient during acceleration
at all 5 stands 198
Figure 6.32 Comparison of roll forces from the simulation
with actual measurements 199
Figure 6.33 Roll force variation during acceleration 199
Figure 6.34 Exit thickness variation at Stand 1 during acceleration
(time delay = 0.05 seconds) 200
Figure 6.35 Exit thickness variation at Stand 5 during acceleration
(time delay = 0.05 seconds) 200
Figure 6.36 Exit thickness variation at Stand 1 during acceleration
(time delay = 0.1 seconds) 201
Figure 6.37 Exit thickness variation at Stand 5 during acceleration
(time delay = 0.1 seconds) 201
Figure 6.38 Roll force variation during deceleration 203
Figure 6.39 Exit thickness variation during deceleration 204
Figure 6.40 Roll force variation during tailing out 205
Figure 6.41 Exit thickness variation during tailing out 206
Figure 6.42 Roll force variation at Stand 1 during threading with different friction
Conditions applied 209
Figure 6.43 Exit thickness variation at Stand 1 during threading
with different friction conditions applied 209
Figure 6.44 Roll force variation at Stand 5 during threading
with different friction conditions applied 210
Figure 6.45 Exit thickness variation at Stand 5 during threading
with different friction conditions applied 210
XXIV
List of Tables
Table 3.1 Terminology comparison between the natural evolution and
Evolution algorithms 39
Table 3.2 Four initial individuals in the first generation 43
Table 3.3 Individuals, schema and crossover 56
Table 4.1 Comparison of empirical rolling schedule and optimized schedules 106
Table 4.2 Optimized schedules based on perturbed empirical
schedule (4-40) 118
Table 4.3 Variations of the optimized schedules based on
the perturbed empirical schedule (4-40) 119
Table 5.1 Numerical experiment conditions 147
Table 6.1 Threading simulation parameters 179
Table 6.2 Acceleration simulation parameters 195
Nomenclature
Automatic Flatness Control
Automatic Gauge Control
Cross-sectional area (Equation (4-24))
Acceleration rates at the ith stand
Constants determined by experiments
Continuously Variable Crown
AGC control compensation coefficient
Constants (Equations (5-13) to (3-17))
Backup roll crown at its vertical centerline
Crown at the ith element of backup roll
Backup roll crown
Semi-height of roll gap at the /'* element of the strip
Input coefficients (Equations (3-2) and (3-3))
Work roll crown at its vertical centerline
Crown at the i'h element of work roll
Work roll crown
Work roll diameter
Backup roll diameter
Diameter of the backup roll bearing
Motor droop parameter at the i stand
Tension rate of change
Evolutionary Algorithms
Elastic Foundation Method
Young's modulus of strip
Young's modulus of work roll
Finite Element Method
Total load exerted on top backup roll bearing chocks
Summation of fitness values of all individuals (Equation (3-12))
Average fitness of all individuals
Fitness of the ith individual
Forward slip factor (Equation (4-3))
Forward slip ratio at the /'* stand of the mill
Forward slip ratio at the (i + l)th stand
Average fitness of all individuals in the population
Fitness function
Genetic Algorithms
Motor base torque at Stand i
Work roll motor torque at Stand i
Maximum work roll torque
Cost function
Average entry thickness of the strip
XXVII
Annealed strip thickness
Entry thickness of the strip at the strip centerline
Thickness at the strip edges
Strip entry thickness
Strip thickness at the entry of the (i + l)'h stand
Entry thickness of the i"' element of the strip
Entry thickness of the strip at the position of a distance x from
the strip centerline
Exit strip thickness
Average exit thickness of the strip
Strip thickness at the exit of Stand i
Strip thickness at the exit of the (i + \)th stand
Exit thickness of the ith element of the strip
Lower and upper limits of exit strip thickness
Exit thickness at the strip center
Exit thickness of the rolled strip at the position of a distance x
away from the strip centerline
Influence Coefficient Method
The second moment of area
Moment of inertia of the roll barrel about the horizontal axis
Moment of inertia of the roll neck about the horizontal axis
Shear modulus of the beam (Equation (4-24))
XXVIII
Bending force exerted between work rolls at the left-hand side
Bending force exerted between work rolls at the right-hand side
Rolling mill elastic modulus
Combined stand flexibility including mill modulus, bearing and
screw elastic coefficients
Elastic contact coefficient between the backup and work rolls
(Equations (5-21))
Combined elastic deformation coefficient of work roll surface
and strip (Equation (5-270)
Elastic deformation coefficient of the work roll surface
Yield stresses at entry and exit points of roll bite
Average yield stress
Work roll width
Distance between the centerlines of two backup roll bearings
Distance between two jacks of the work roll
Distance between the centerline of the backup roll bearing and
roll barrel
Distance of the work roll bearing centerline from roll barrel
Distance between two neighboring stands
Length of backup roll barrel
Length of work roll barrel
Length of the schema S
Multiple-Objective Problem Solving
XXIX
Bending moment
Plastic modulus of the strip
Normal Tension Control
Total number of mill stands
Total number of samples (Equation (6-15))
Total number of schemata
Number of individuals at time (t +1)
Backup bearing oil film thickness
Roll force (Equation (4-10))
Constants (Equations (4-18) and (4-19))
Probability of an individual being one of the individuals of the
schema S (Equation (3-19))
Crossover rate
Total roll force at Stand i
Probability of a schema losing its individuals (Equation (3-15))
Contact pressure between work rolls at the left-hand side
Mutation rate
Maximum roll force
Specific roll force at the center point of the strip
Probability for the schema S to keep its individuals
Contact pressure between work rolls at the right-hand side
Power at the ith mill stand (Equation (4-2))
P( /) Roll gap concentrated loads
p( /) Unit contact load between the work roll and strip
p(x) Specific roll force between work roll and strip at position x
p{xsl) Probability for an individual to be chosen (Equation (3-5))
p(z) Point load
Qx, Q2 Constants (Equations (4-18) and (4-19))
Q{ /') Inter-roll concentrated loads
q( i) Unit contact load between the work roll and backup roll
q(x) Inter-roll specific force at position x
q(z) Point load
R Work roll radius
R' Deformed work roll radius (Equation (4-15))
r Strip thickness reduction
f Mean reduction (Equation (4-12))
SBM Spring and Beam Method
S Schema
So No-load roll gap
TOS Totally Ordered Set
Ts Mean tension stress of the strip (Equation (5-42))
tb, t f Front and back tensions (Equation (4-17))
tfi Tension between two adjacent stands (the i'h and (7 + 1)
stands)
XXXI
t'rmm > '/max Lower and upper tension limits at Stand /
t" Exit tension at time at time («) at Stand i
t"+1 Exit tension at time (n + 1) at Stand i
tsh (i) Exit tension at the i'h portion of the strip
**(*), t^fi) Calculated tension stresses at the ith element of the strip in the
k'h and (k + \),h iterations (Equation (5-42))
u Friction coefficient
u{x) Profit function
V Rolling speed (m/s)
VH"+l Strip velocity at the entry of the (i + 1)* stand of the mill
Vh" Strip velocity at the exit of the ith stand of the mill.
Vj Input strip velocity
v Poisson's ratio of the strip
vB Peripheral speed of the bearing
v0l. Work roll speed at the ith stand during threading
v. Output strip velocity
v" Work roll speed at the /'* stand of the mill
v"+1 Strip velocity at the exit of the (/' + l)'h stand
vroll Poisson's ratio of the work roll
W Strip width
WX,W2, W3 Weighting constants (Equations (4-1), (4-17), (4-22) and (4-33))
XXXII
xs'' The /* individual in the g'h generation (Equation (3-4))
YBL Rigid body motion of the backup roll on the left-hand side
YB(i) Deflection of the backup roll at the Ith element due to bending
and shear plus the roll axis rigid body motion
YBL Rigid body motion of the backup roll at the left-hand side
YBR Rigid body motion of the backup roll on the right-hand side
YWL Rigid body motion of the work roll on the left-hand side
YWR Rigid body motion of the work roll on the right-hand side
Yw(i) Deflections of the work rolls at the ith element due to bending
and shear plus the roll axis rigid body motion
y(x) Roll deflection (Equations (4-23), (4-24), (4-25) and (4-26))
yb (x) Total backup roll-axis deflection
yw (x) Deformed roll profile
ywb (o) Total roll interference at the strip center
ywb (x) Interference between work and backup rolls
yws (x) Work roll flattening at the contact area of work roll and strip
y„ (x) Interference between work rolls
ZBfv(i) Local contact deformation at the center of the *'* element of the
backup roll
ZWB(i) Local contact deformation at the center of the ith element of the
work roll
Zwso Local contact deformation at the center of the work roll
XXXIII
Zws (/) Local contact deformation of the work roll at the location of the
ith element (Equation (5-23))
Zswo Local contact deformation at the center of the strip
Zsw{i) Local contact deformation of the strip at the location of the i'h
element (Equation (5-23))
a(i, /) Influence coefficient defined as the deflection of the ith element
due to the influence of the bending and shear of the unit load at
the /'* element
aB(i, j) Influence coefficient for the backup roll
aw (i, j ) Influence coefficient for the work roll
£ Convergence tolerance
t, Precision of the solution generated by GA (Equation (3-1))
e Mean value of the natural strain (Equation (5-37))
e(i) Natural strain distribution due to the stand reduction at the ith
portion of the strip
A Constant (Equation (4-14))
r\ Backup roll bearing oil viscosity
8(S) Defining length of the schema S
<Jh Standard thickness deviation
co' Work roll rotational speed at Stand / (Equation (4-2))
y Shape optimization parameter
*F Bearing radial clearance
ASL Relative change of screw down on the left-hand side
AS0 No-load gap adjustment
ASR Relative change of screw down on the right-hand side
tsZ Side-shifting value of work roll
Ah Exit thickness error
At Time increment
Ax Length of each of divided portions of work roll or backup roll
Chapter 1 Introduction 1
Chapter 1
Introduction
Steel is vital to our everyday life. We depend on steel for food and clothes
manufacturing, housing and traveling. Even industries producing other materials like
glass, aluminum and plastics, all need steel. In May of 1995, the steel industry
published its Vision "Steel: A National Resource for the Future". In this vision, steel is
proposed to be the material of choice for manufacturing in the 21st century. Actually, the
steel industry has enjoyed the position of "material of choice" in a number of markets
for decades [Attwood and Miner, 1998]. However, the only thing that is constant in our
industry today is "change". Externally, steel is facing the threats from competing
materials such as aluminum, plastics and glass etc. One of the prime examples is the
material competition in the automotive and packaging markets. Internally, competition
within the steel industry is becoming aggressive. The keys for steel industry to succeed
in staying competitive among all materials, and for individual manufacturers in the steel
industry to survive, are continuous reduction of the production costs and continuous
improvement of the product quality. The research work addressed in this thesis is one of
the efforts in reducing the cost and improving the quality of steel products.
This chapter introduces the main objectives of this study, presents an overview of the
research, outlines the organization of the thesis, and summarizes the contributions of
this research.
Chapter 1 Introduction 2
1.1 MAIN RESEARCH OBJECTIVES
The main objectives of the research in this thesis include the following aspects:
• Develop a heuristic searching based optimisation procedure for rolling schedule of a
tandem cold rolling mill so as to maximise the throughput of the mill, unify the
power distribution, maximise safe level of strip tension, and enable the strip to be
rolled with a perfect shape.
• Investigate and implement a theoretical predictive model for shape analysis of cold
strip rolling. The model shall be flexible to deal with the optimisation of pre-setting
of both rolling force and roll bending force, especially for threading process so as to
minimise the possibility of bad shape on the strip nose, thus, reduce threading delay.
• Dynamically simulate unsteady rolling processes including threading, acceleration,
deceleration and tailing-out of a tandem cold rolling mill, especially simulate the
process of tension establishment, the changes of exit strip thickness deviation, and
roll force at each stand. Analyse the interactions of rolling parameters and the effect
of strip properties, roll profile and frictions on strip off-gauge during unsteady
rolling.
Chapter 1 Introduction 3
1.2 OVERVIEW OF THIS RESEARCH
The research presented in this thesis focuses on the modeling, theoretical analysis,
optimisation and control of tandem cold rolling process under both steady and unsteady
rolling conditions. Computational intelligence based heuristic searching mechanism is
employed to optimize the rolling process parameters.
The research starts from the rolling schedule optimisation with rolling constraints
considered. Firstly, cost functions for the optimisation are defined taking into account
the tension, the optimal shape, and power distribution. Secondly, validity checks
including all rolling constraint satisfaction conditions are conducted taking into account
the roll force and torque limitations, work roll speed references, strip exit thickness,
threading conditions and tension force limitations. Genetic Algorithm is the core in the
searching of the optimized solution, which is the combination of a set of rolling
parameters other than one parameter or several individual parameters.
The investigation is then conducted towards producing quality strip with good shape
especially for thin strip such as tinplate. An approach for shape analysis and prediction
is proposed which combines the strengths of both Influence Coefficient Method (ICM)
and Spring and Beam Method (SBM). The approach consists of deriving the physical
equations in a form suitable for efficient computer solution. It is also extended to the
presetting of the bending force for those mills with bending systems, and to optimize
reduction schedule for those without roll bending systems.
Chapter 1 Introduction 4
To analyze the off-gauge situation of rolled strip, both threading and acceleration
processes are comprehensively modeled and dynamically simulated. The simulation
includes the gradual establishment process of inter-stand tension between two
neighboring stands, the roll force and the exit thickness fluctuations during threading
process. For the acceleration, the speed dependent parameters are simulated with
increasing rolling speed. The correlation between the exit strip thickness and other
rolling parameters is analyzed and graphically illustrated. As opposite processes of
threading and acceleration, tailing out and deceleration processes are also addressed.
The interactions of rolling parameters and the effect of strip properties, roll profiles and
frictions on unsteady rolling processes are also discussed.
To verify both the optimized and simulated results, they are compared with the real data
collected from steel plants. The comparison shows they correspond well with the actual
situation and the improvements of both the throughput of the mill and the strip quality
in terms of on-gauge and shape have been achieved.
1.3 ORGANIZATION OF THE THESIS
The dissertation is organized according to the research objectives described in Section
1.1 and the easy-to-follow principle. It contains seven chapters.
Chapter 1 Introduction 5
Chapter 1 gives a brief introduction of the significance and innovation of the research,
and the main objectives. This chapter also presents an overview of the research and a
summary of achievements.
Chapter 2 provides the background information of the research including rolling and
optimisation theories. The intention is to help understand what are the research
objectives. The geometry, driving systems, and control systems of a typical five-stand
tandem cold rolling mill are detailed. The models of strip deformation and shape are
reviewed and the functions of the control systems are also examined. The tandem cold
rolling process is dissected into five phases including threading, acceleration, full speed
operation, deceleration and tailing out. Each phase is briefly analyzed and the features
associated with individual phases are addressed. Then, a review is conducted on
conventional optimisation including its three basic components. An optimisation tree is
also outlined with each of its branches briefed.
In Chapter 3, the methodology of optimisation employed in this research is detailed.
Evolutionary optimisation is comprehensively introduced including its origin, concept
and composition. The genetic operators and mathematical foundation of Genetic
Algorithms (GA) are presented. The advantages and disadvantages of the evolutionary
optimisation and the conventional optimisation are compared. Finally, a literature is
surveyed on the engineering applications of Genetic Algorithms, especially in the steel
industry over the last two decades.
Chapter 1 Introduction 6
In Chapter 4, basic scheduling for tandem cold rolling mills is described including
individual steps in the scheduling procedure such as the determination of reduction
pattern, tension force, roll force, forward slip ratio, torque and roll speed, and validity
check of rolling constraints. Emphasis is placed on the demonstration of the proposed
optimisation scheduling. Cost functions are defined and rolling constraints are
considered. The principle of GA-based searching for the optimisation is introduced and
a step-by-step procedure for rolling schedule optimisation is presented. Numerical
experiments have been conducted and the results from a case study are compared with
those based on semi-empirical formula.
Chapter 5 describes a theoretical predictive model that has been implemented to analyze
thin strip shape during cold rolling. Four typical shape models are briefed including
Elastic Foundation Method (EFM), Influence Coefficient Method (ICM), Finite
Element Method (FEM), and Spring and Beam Method (SBM). Conventional shape
control means is also reviewed. The special rolling condition during threading is
considered. An optimisation procedure for the presetting of roll force and/or roll
bending force is developed. The models for roll deflection/deformation in the roll bite
and the roll profile are proposed. The effect of both work roll crown and roll bending
force on strip shape has been examined. Based on perfect shape condition during
threading, more accurate threading parameters are determined. The results generated
from numerical experiments are compared with those from a steel plant. Proposals to
improve shape during threading are also addressed.
Chapter 1 Introduction 7
Chapter 6 presents the models for unsteady process simulation. This chapter first
reviews previous work in the area, then theoretically dissects the threading and
acceleration processes. Attention has been paid on the disturbances associated with the
processes and analysis of the sources resulted in these disturbances. Numerical
simulation procedure including the detailed flow chart is highlighted and the simulation
results for both threading and acceleration are detailed and compared with actual plant
data. For the acceleration, speed dependent parameters are also analyzed. Deceleration
and tailing out processes are addressed. The interactions of rolling parameters and the
effect of strip properties, work roll profiles and frictions on unsteady rolling processes
are also discussed.
Chapter 7 concludes the thesis by summarizing the research results and achievements.
Important issues for future work are also raised.
1.4 SUMMARY OF CONTRIBUTIONS
A computational intelligence based approach for the optimisation of tandem cold rolling
processes has been proposed and implemented. Case studies have been conducted and
results are compared with real data from steel plants. The comparison of results shows
the proposed approach is working well and has potential to be applied in the steel
industry. The main contribution from this research is summarized as follows:
Chapter 1 Introduction 8
• It is the first time that the computational intelligence based optimisation has been
applied to the rolling scheduling of tandem cold rolling mills in this thesis. Cost
functions have been well defined with most of important factors directly related to
product quality, energy cost, production efficiency, and rolling constraints
considered. This work has been published on International Journal of IF AC,
Engineering Applications of Artificial Intelligence, Vol. 13, No. 4, July 2000.
• A generic model for complex engineering problem optimisation has been developed.
The model is capable for Multiple-Objective Problem Solving (MOPS) and non-
convex cost optimisation. The significant difference from other optimizations is that
the approach is employed to search for a combination solution of a set of
parameters, not individual parameters. The method named Totally Ordered Set
(TOS) is utilized to deal with multiple constraints. TOS means that there is a full list
of the preferences of all constraints, and the priority of each constraint is measured
by a weight.
• A hybrid theoretical model for the shape analysis and prediction of thin strip such as
tinplate in tandem cold rolling has been implemented. The model proposed
combines the strengths of conventional Influence Coefficient Method (ICM) and
Spring and Beam Method (SBM). Meanwhile, the model adopted in the shape
analysis consists of deriving the physical equations in a form suitable for efficient
computer solution. This makes the model developed more flexible, accurate and
convenient for practical use.
Chapter 1 Introduction 9
• A GA-based heuristic optimisation procedure for the pre-setting of reductions and/or
roll bending forces for threading process in tandem cold rolling has been proposed
to minimize threading time delay, and to improve strip shape quality and production
safety. The concept of shape optimisation parameter is introduced to guide the
searching for more accurate threading parameters in their intervals of individual
empirical parameters.
• Unsteady rolling processes are dynamically modeled and simulated with most of the
disturbances in the processes considered. The fluctuations of interactive rolling
parameters including tensions, roll force and exit thickness against time are
graphically presented to outline what is happening during the unsteady rolling.
Motor speed droop during threading and speed dependent rolling parameters in
acceleration are modeled and taken into account in the simulation. Some parameters,
which may have effect on the unsteady rolling processes, have also been addressed,
such as friction coefficient, steel grade, and entry thickness of strip. To the best of
the author's knowledge, a complete simulation for threading and acceleration
processes has not been published anywhere, and therefore this work is a new
contribution.
• The proposed models and simulations have been implemented in C/C++. The results
generated from the proposed models and simulations have been compared with plant
data from steelworks to verify their effect and efficiency. Based on the proposed
process models, the numerical simulations conducted in this research in combination
with their result analysis are beneficial to fine-tune the rolling process parameters
Chapter 1 Introduction 10
and are effective in achieving significant cost savings and product quality
improvements.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 11
Chapter 2
Fundamentals of Tandem Cold Rolling and
Optimisation Theory
Technological process model forms the basis of high performance process control and
ensures high mill availability and safe, automated operation. The wide range of validity
and the high accuracy of process models for the rolling process allow for a precise mill
set-up, control and optimisation under all operational conditions, resulting in tight
product tolerances in both gauge and shape of cold rolled strip.
As other industrial problems, the rolling schedule and process control can also be
reduced in the final analysis to the problem of determining the largest or smallest value
of one or more objective functions of several variables [Beveridge and Schechter,
1970]. In most aspects of industrial life, continual improvement is an important feature.
Thus we may want to get the high production from given raw materials, the greatest
profit from a fixed investment, and so on. Optimisation is a formal presentation of these
ideas. The optimisation, seemingly simple, is in fact a complex one. This can be
mastered by viewing the components of the optimisation, the hierarchy of the
optimisation before carrying out a practical task.
This chapter first gives an introduction of the fundamentals of cold rolling process, then
reviews the classical optimisation including its concept, components and hierarchy.
3 0009 03295332 0
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 12
2.1 PROCESS DESCRIPTION AND ANALYSIS IN TANDEM COLD ROLLING
During the last half century, our understanding of the cold rolling process has been
considerably enhanced through the development of mathematical models describing the
process. This section focuses on a brief introduction of tandem cold rolling mills and the
description of the tandem cold rolling process.
According to "Dictionary of Metallurgy", "tandem mill" is defined as "arrangement of
rolling mills, in direct line, allowing the metal to pass from one set of rolls into the
next". Cold rolling is defined as "rolling metal at a temperature below the softening
point of the metal to create strain hardening (work-hardening)". This definition is the
same as cold reduction, except that the working method is limited to rolling. Cold
rolling changes the mechanical properties of strip and produces certain useful
combinations of hardness, strength, stiffness, ductility and other characteristics known
as tempers. Basically, there are six stages by which a steel component is made (UK
Steel Association, http://www.uksteel.org.uk/). They are:
(1) Raw material processing,
(2) Steelmaking,
(3) Casting,
(4) Primary forming,
(5) Manufacturing, fabrication and finishing, and
(6) Product applications.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 13
Cold rolling is one of the important links in Stage (5). It is a process by which the sheet
metal or strip stock is introduced between rollers and then compressed and squeezed.
There are some advantages of the cold rolling. First of all, work hardening can increase
steel strength; meanwhile large deformation (50 to 90%), excellent surface finish,
tolerance on thickness and shape can be achieved.
2.1.1 Tandem Cold Rolling Mills
Tandem cold rolling mills are named by their stand arrangement and designated as
"sheet mills" if they reduce hot band to sheet gauge or "tin mills" if they roll the same
incoming material to tinplate stock. A typical five-stand tandem cold rolling mill is
illustrated in Figure 2.1.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 14
Figure 2.1 A typical five-stand tandem cold rolling mill
The five-stand tandem cold rolling mill consists of five 4-high conventional mill stands.
Each stand consists of two housings being joined by spacer bars to form a rigid closed
unit [Roberts, 1978; Ginzburg, 1989]. The work roll neck at drive side is shaped for
attachment to long spindle driving each roll individually. The pinion gearbox divides
the motor torque to provide equal power to the top and bottom work rolls as shown in
Figure 2.2. The gearbox consists of a double helical pinion and gear. The work roll
chucks are fitted into a recess between the top and bottom back-up chucks. The work
roll bearings are anti-friction spherical rollers and are contained in the work roll chucks
as demonstrated in Figure 2.3.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 15
Sactop M Balancing System
fl
/
=L
£i««rDmNhani(ii scramem
J t^fier SpiniS* Boijncing Syilem
Lower Spindle- latafiflg iptem
'Hassinijf'iwt
Figure 2.2 A typical mill drive system
The backup rolls provide rigid support to prevent the work roll bending or flexure. The
material of the back-up rolls is softer than the work rolls. The back-up rolls are
supported on oil film bearings and support the total rolling loads. The mill housing
windows have wear plates on the vertical planes and encase the roll stack, ensuring roll
alignment and vertical position. To prevent lateral movement of the work roll
assemblies, retaining latches are used to tie chucks to the mill housings. The mill screws
are threaded through the upper crossbeam of the mill housing and are electrically driven
by motors and worm gear drives. They are used essentially to adjust the roll gap and
rolling force acting between the mill housing and back-up chucks. Nowadays, hydraulic
capsules are being used in increasing number to provide better gauge control.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 16
Figure 2.3 A typical mill stand arrangement
When roll strip, the strip coil is fed into the mill via an uncoiler. As the strip exits the
final mill stand, it is wound tightly onto a coiler which is an expanding tension reel
mandrel that maintains constant tension during the rolling operation.
2.1.2 Automatic Control Systems for Tandem Cold Rolling Mills
Since the first computer controlled cold mill was commissioned at the Port Talbot
works of British Steel in the mid-1960's, a dramatic growth in automation system
installations has occurred on new and revamped mills [Duval et al., 1991; Carlton et al.,
1992]. The two main automatic control technologies of cold mills are: Automatic Gauge
Control (AGC) and Automatic Flatness Control (AFC).
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 17
In order to produce perfectly uniform thickness including the strip edge, it is important
to maintain the strip gage accuracy in both the longitudinal and the lateral directions.
The longitudinal thickness accuracy is achieved by Automatic Gauge Control (AGC)
system, while the lateral thickness is controlled by Automatic Flatness Control (AFC)
system. Both control systems are integrated and operating on process control computers
located at a so-called central control pulpit as shown in Figure 2.4.
Figure 2.4 Central control pulpit of a tandem rolling mill
2.1.2.1 A G C Control System
Figure 2.5 shows a typical A G C architecture of a tandem cold rolling mill [Duval et al.,
1991].
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 18
Feedforward Stand 1
Tension 3-4
Figure 2.5 Main control loops of a typical A G C system of a 5-stand tandem cold mill
The main functions of the above A G C system include:
(1
(2
(3
(4
(5
(6
(7
(8
(9
inter-stand tension control by screwdowns,
vernier feedback AGC,
monitor feedback AGC,
feed-forward AGC at Stand 1,
feedback AGC at Stand 1,
mass flow control into Stand 2,
inter-stand tension control by speed,
oil film compensation (all five stands),
gauge-meter constant gap control (Stand 1), and
(10) eccentricity compensation (Stand 1).
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 19
Stand 3 is a pivot stand on the tandem cold mill, this means that the roll speed of Stand
3 will not be modified by the AGC system during rolling process. Normal tension
control (NTC) is employed to maintain tension without disturbing mass flow. To ensure
a constant surface finish, the force on Stand 5 must remain constant. Therefore, a
tension regulator using the speed of Stand 5 is employed instead of the screw-down
tension control. Vernier feedback AGC uses the signal of delivery x-ray gauge
deviation to correct the gauge out of Stand 5. Stand 1 feed-forward AGC tracks the
entry thickness gauge signal so as to follow incoming gauge variations and correct these
defects using Stand 1 hydraulic screw-down. The function of Stand 1 feedback AGC is
to correct Stand 1 delivery thickness gauge variations. Mass flow into and out of Stand
2 is kept constant by adjusting the rolling speed of Stand 1.
2.1.2.2 Strip Profile, Shape, Flatness and Automatic Flatness Control System
From the point of product quality, shape is an important parameter in the rolling
process. It is determined from the profile of the strip at the entry to and exit from the
roll bite. Any mismatch between the strip entry and exit profiles will result in a variation
in elongation and hence a poor strip shape when the exit tension is applied. This means,
if the percentage reduction in the strip thickness varies across the strip width, a
transverse variation in elongation will result. Thus, the strip will exhibit buckles. If
however, the percentage reduction in the strip thickness is constant across the strip
width there will be no transverse variation in stress and the strip will have perfect shape
[Sabatini and Yeomans, 1968; O'connor and Weinstein, 1972; and Yuen and Wechner,
1990].
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 20
The strip profile is a measure of the variation in strip thickness across the strip width. A
typical strip profile is characterised by a gradual change in thickness across the central
section of the strip and a rapid change in thickness in a region within 50mm of each
edge.
Flatness is a measure of the ability of the strip to lay flat when placed on a level surface
with no externally applied loads. It is related to shape in that any transverse variation in
stress may result in a buckled strip when the tension applied during rolling is removed.
Over last decades, the physically based shape model has been evolved, which was
initiated by J. G. Wistreich at BISRA in London [Wistreich, 1968; Sabatini and
Yeomans, 1968]. The shape model was developed into a mature simulation tool for cold
rolling analysis at the Imperial College, London, in the early 1970's [Bryant, 1973]. It
was further developed by BHP Steel [Edwards and Johnston et al., 1976], Industralia
Automation Services (IAS) in Australia [Cresdee and Edwards et al., 1991] and The
Broner Group in the UK. Spooner [Spooner and Bryant, 1976], Hsu [Hsu, 1987] and
Chen and Zhou [Chen and Zou, 1987] also analyzed the deformations in rolling mill
stands and described their combination with some form of metal deformation models to
generate a shape analysis tool suitable for cold rolling analysis and controls. Tozawa
and Ishikawa et al. [Ishikawa, 1980; Tozawa and Ishikawa et al., 1982; Ishikawa,
Yukawa and Tozawa, 1987] extended the two dimentional, classical mechanics
approach of Von Karman and Orowan to encompass the transverse dimension, thereby
enabling a prediction of profile change in the roll gap. Cresdee and Edwards et al.
improved the computational efficiency of Tozawa's analysis and considered models for
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 21
deformation external to the roll gap and proposed a closed-form solution for the
longitudinal stress distribution at each side of the roll gap [Cresdee and Edwards et al.,
1991].
Strip flatness control system is designed to make improvement across the width of the
strip. The flatness control problem is usually divided into two areas: the control of strip
crown in the early part of the hot strip mill, and the control of strip shape or flatness at
the delivery end of the hot strip mill and in the cold mill. Both objectives, however, can
be achieved by essentially the same rolling technique - that of proper control of mill
crowns [Stone and Gray, 1965].
Automatic flatness control system is supplied for the flatness measurement and closed-
loop control at the delivery side of Stand 5 [Trautwig, 1999]. The system interfaces with
the analog mill electronics, the work roll bending system, spray cooling system, screw-
down drives and a setup computer. Basic components of the flatness system are shown
in Figure 2.6. The system includes flatness measurement roll, and flatness PC for
processing measurement, carrying out closed-loop control and generating coil reports.
The flatness measurement roll is usually designed as a solid-body. Mechanically
isolated transducer pockets allow sensitive measurement without bias from the roll body
flexing. The roll system employs an optical signal transmission between the rotor and
the stationary part, which significantly reduces maintenance requirements. Based on the
strip width, the flatness measurement roll is configured with a number of zones. For
example, if strip width is 1066.8mm (42 inches), the measuring roll can be configured
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 22
with 10 zones, 50.8mm (2-inch) wide, in the center and 11 zones, 25.4mm (1-inch)
wide, on both sides. The small zones provide for higher resolution at the edges.
Figure 2.6 A typical automatic flatness control system
Usually, the hardware platform for flatness measurement and control is an industrial P C
based, real-time UNIX system. The closed-loop control software runs on a slot
processor board with its own proprietary operating system. This provides excellent
communication capabilities to other automation systems. The control software converts
the measured radial force values from the sensors into their corresponding elongation or
flatness distribution. Edges are considered based on coverage of the measurement
pocket. The flatness distribution is compared to a set-point curve and the difference is
stored as the flatness deviation. The flatness deviation corresponds to the tension
distribution in the strip. Based on the knowledge of relations between the tension
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 23
distribution and roll gap, the flatness control ensures a correct roll gap profile at all
times. A mathematical algorithm then uses deviations from set-points to generate
commands for roll shifting, bending, tilting, cooling, and all the other actuators of the
rolling mill.
2.1.3 Description of Tandem Cold Rolling Processes
Cold rolling process is complex as it is nonlinear and time varying. The parameters with
the process model are uncertain and dependent on each other [Geddes et al., 1998].
Complete knowledge of the rolling process is the basis of any rolling problem solving.
This session addresses the rationale of cold rolling and theoretically dissects the tandem
cold rolling process.
2.1.3.1 Rationale of Cold Rolling
The function of a cold-rolling mill is to reduce the ingoing strip, normally supplied as a
coil at room temperature, by 50 to 90%. The reduction of strip thickness is caused by
high compressive stresses in the contact region between work roll and strip. In this
region, plastic deformation occurs and slip takes place between the work roll surface
and strip. Significant tension stress exists in the length of strip between any two stands
and it reduces the rolling force. Figure 2.7 shows the schematic of the strip and work
roll interface [Ginzburg, 1989].
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 24
a\
• 1
\ H
1
[ f': "1 5^A \/^ \ \ a^-/ii?wx 1
or v to /?*
t orTT 1 A
(p L'to[
Ito J to ' \
xf R 7
v /z
t
Figure 2.7 Schematic of the strip and work roll interface
At the heart of the rolling process is the deformation of the strip in the roll bite. One of
the most important components in the deformation is rolling force. It has a dominating
effect on the accuracy of thickness, on the quality of the profile and flatness of the
rolled strip. There are many formulae to calculate rolling force [Wang, 1999]. However,
they are based on different assumptions. In cold rolling, the principal parameters
affecting the resistance to deformation are work hardening and friction in the roll bite.
Roll flattening also plays an important role in cold rolling due to a high resistance to
deformation. Moreover, the effect of strip tension becomes substantial as cold rolling is
conducted with greater specific tensions in comparison with that in hot rolling. These
features of cold rolling process are usually taken into account to a different degree in the
methods for calculating rolling force.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 25
In 1948, Bland and Ford [Bland and Ford, 1948] developed a model of strip
deformation, which obtained a good agreement with experimental data and formed the
basis for the approximate solutions by Bryant et al. [Bryant, 1973]. The model takes the
following into account: friction, tension and material yield stress variation in the arc of
contact. Graphical methods for calculating rolling force based on the Bland and Ford's
solution was proposed by Ford et al. [Ford, 1951], using a constant mean value of yield
stress along the roll gap, and assuming that the mean yield stress in a plane strain is
constant along the roll gap. In 1953, Stone developed a method to calculate the rolling
force considering the effect of tension and flattening [Stone, 1953]. It is based on a
simplified analysis of deformation during rolling with dry slipping friction. Roberts
[Roberts, 1965] proposed a method based on the assumption that deformation occurs in
the roll bite, the rolling pressure is equal to the resultant resistance to deformation of the
material being rolled. Under these circumstances, the rolling force is merely the product
of the projected area of contact and rolling pressure. Wusatowski [Wusatowski, 1969]
proposed a method for cold rolling by determining the resistance to deformation from
the work-hardening curves. In 1978, Roberts derived empirical equations for rolling
force and torque calculation in temper rolling based on experimental data [Roberts,
1978]. In the same year, Alexander designed a Fortran program from Orowan's
formulation which solves Von Karman's equation for rolling force [Alexander et al.,
1987]. In the last decade, some researchers continue their effort to improve the method
of rolling force calculation for different circumstances. Fleck et al. [Fleck et al., 1987,
1992] derived an accurate model for rolling thin strip where both the plastic
deformation of the strip and the elastic deformation of the roll are realistically
accounted for. Pauskar et al. developed a microstructure dependent flow stress model
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 26
for the accurate prediction of rolling force using the finite element method [Pauskar et
al., 1997].
It should be noted that some of the above methods are based on several simplified
assumptions such as the existence of a constant effective coefficient of friction along
each arc of contact. Those methods are usually used in the design of new rolling mills,
and in the determination of rolling capability limits of existing mills, the coefficient of
friction in the roll bite and the compressive yield strength of the strip under actual
rolling conditions. The accuracy and calculation time of these equations may or may not
satisfy the requirements of real time control. Generally, the rolling force can be
represented as a function of work roll diameter, strip width, material chemical
composition, metallurgical characteristics, friction, work hardening, strain, strain rate,
reduction, entry and exit tensions [Bland and Ford, 1948 and 1951], etc. In practice,
rolling force is measured using load cells first, and then a rolling force equation is
derived and revised with empirical coefficients based on the measurement and with the
fluctuations of specific rolling conditions. A typical formula based on Bland and Ford's
solution is of a form
P= f(W,k(r),R,H,h,u,Ts,ts,K) (2-1)
where P is the rolling force; W is the mean width of strip; R is the radius of work rolls;
K is an empirical coefficient specific to a certain mill; k(f) is the average yield stress
evaluated at the mean reduction f \ Ts and ts are back and front tension stress,
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 27
respectively; H, h are the strip entry and exit thickness respectively; and u is friction
coefficient.
The above formula has been employed widely in cold rolling in some research papers
published in IEEE Transactions [Sekiguchi et al., 1996; Cho et al, 1997; Geddes et al.,
1998; Guo, 2000], and as well in steel industries such as the Fukuyama Works of NKK
in Japan [Jimba et al., 1990; Hikino et al., 1977], Dofasco's tandem cold mill in Canada
[Sekiguchi et al., 1996], Pohang Iron and Steel Company, POSCO, Korea [Cho et al,
1997; Hwang et al., 1996].
2.1.3.2 Five Phases of Tandem Cold Rolling
Tandem cold rolling process is composed of threading, acceleration, run speed
operation, deceleration and tailing out. Figure 2.8 illustrates a typical speed diagram for
tandem cold rolling [McDonald and Spooner et al., 1993]. It shows the threading is the
first phase of the rolling. Following the threading, dynamic control loops (AGC and
AFC) are closed and corresponding actuators are activated. Then the mill speed is
increased during acceleration phase. Once the mill speed is up to the preset maximum
mill speed, the speed control system will keep the rolling speed in the vicinity of the
maximum speed. This phase is defined as run speed operation. Towards the end of a
coil, the mill is decelerated to the preset de-threading speed preparing the strip for
tailing out. This session addresses individual phases and the disturbances accompanying
these phases and control measures being taken in cold rolling.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 28
V(m/s)
Full speed rolling
Deceleration
Figure 2.8 Typical speed diagram of a tandem cold rolling mill
2.1.3.2.1. Threading Phase
The threading takes place at a relatively low speed and involves alignment of the feed
coil at mill entry and sequencing of stand controls until tension is established between
the last stand and the coiler (reel) [Edwards, 1973]. The threading speed is usually as
low as about 40-130m/min. During threading, there is no front tension. Accompanied
with strip entry thickness and hardness deviations, the threading process is associated
with fluctuations and complex interactions occurring between stands of a tandem cold
rolling mill. The theoretical analysis of threading process is detailed in Chapter 6.
2.1.3.2.2 Acceleration Phase
Acceleration phase begins when the mill starts to accelerate and the dynamic control
systems function to maintain the rolled strip thickness, flatness and surface finish with
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 29
specification in exit of the last stand. Inter-stand tension and strip shape must also be
constrained to prevent strip from tearing or looping. With the latest technology in
mechanical, electrical, hydraulic and computer systems, currently acceleration can have
a value of more than 2m/s .
During the acceleration phase, the strip thickness decreases as the mill speed is
increasing [Sims and Arthur, 1952]. This is because of the change of the speed
dependent variables such as oil film thickness of the backup roll bearing, coefficient of
friction between work roll and strip, and inter-stand tension fluctuation. The theoretical
dissection of the acceleration phase is addressed in Chapter 6.
2.1.3.2.3 Full Run Phase (The Constant Speed Roiling Phase)
Full run operation phase starts when the constant rolling speed is achieved and ends
when the rolling speed is reduced for the strip tailing out. In this phase, the rolling mill
works in a relatively stable state comparing to the threading phase. The control
techniques for the rolling mill in the full run phase are also relatively well developed.
Currently the average body gage deviation sigma (cr) is less than 0.40% [Hennes,
1997]. With the advancements in mechanical, electrical, hydraulic and computer
systems, especially advanced control techniques, the strip delivery speed can achieve a
value of more than 30m/s for a 5-stand tandem cold rolling mill and more than 40m/s
for a 6-stand tandem cold mill. During the constant speed rolling phase, once a
disturbance such as the strip entry thickness occurs, corresponding control loops will be
in the position of alleviating or eliminating its effect on the strip delivery thickness.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 30
2.1.3.2.4 Deceleration Phase
Deceleration begins when the rolling speed decreases to prepare for the strip tailing out
and ends when the rolling speed achieves the tailing out speed. Opposite to the
acceleration process, in the deceleration phase, the strip thickness increases as the mill
is decelerated [Sims and Arthur, 1952]. The main causes of the thickness deviation may
be a decrease of the thickness of the oil film separating the roll neck and the chock as
the mill speed decreases, this leads directly to a change of the roll gap position.
Meanwhile, the coefficient of friction increases as the mill speed is decreased, this
results in an increase of the roll force, which, in turn, leads to an increase of the rolled
strip thickness.
2.1.3.2.5 Tailing Out Phase
Tailing out phase begins when the tail of the rolled strip leaves the uncoiler and ends
when the tail leaves the last stand. For a smooth tailing out process, the tailing out speed
is as low as that during threading. Because of the lack of back tension, the tailing out
process is usually accompanied with fluctuations of inter-stand tensions, rolling force
and strip thickness. When the strip tail leaves the uncoiler, the back tension of Stand 1 is
zero. The tension between Stands 1 and 2 is removed when the strip tail exits from
Stand 1. Such process continues until the strip tail leaves the last stand of the tandem
mill. The whole tailing out process ends when the strip is fully coiled to the coiler.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 31
Similar to the threading process, the deviation of strip thickness is affected by several
factors such as a lack of the back tensions and other disturbances from stand and strip
such as strip hardness and entry thickness variations.
2.2 OVERVIEW OF CLASSICAL OPTIMISATION
This section focuses on introduction of optimisation including its concept and
components, and overview of the classical optimisation. Next chapter will review one of
new branches in optimisation area, Evolutionary Optimization, and its engineering
applications. The comparison between the classical optimisation and Evolutionary
Optimisation will also be conducted in the next chapter.
2.2.1 Concept of Optimisation
An optimisation problem begins with a set of independent variables or parameters, and
includes conditions and restrictions that define acceptable values of the variables, and
finally ends with finding a set of values of variables or parameters that minimize or
maximize the objective function to achieve the best result from a given situation [Gill,
Murry, and Wright, 1981]. In particular, mathematical models are often developed in
order to analyse and understand complex phenomena. Optimisation is used in this
context to determine the form and characteristics of the model that corresponds most
closely to reality.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 32
2.2.2 Three Components of Optimisation
Irrespective of the application, any optimisation problems are made up of three basic
ingredients: objective functions, variables and constraints.
2.2.2.1 Objective Function
An objective function is what we want to maximize or minimize. Any problem
investigated in an optimisation analysis will have as its objective of the improvement of
the system or systems. For instance, in a manufacturing process, we may want to
maximize the profit or minimize the cost. In fitting experimental data to a user-defined
model, we may want to minimize the total deviation of observed data from predictions
based on the model. In designing an automobile panel, we may want to maximize the
strength. Almost all optimisation problems have a single objective function. The two
interesting exceptions are:
(1) No objective function. In some cases (for example, design of integrated circuit
layouts), the goal is to find a set of variables that satisfies the constraints of the
model. The user does not particularly want to optimize anything, so there is no
reason to define an objective function. This type of problems is usually called a
feasibility problem.
(2) Multiple objective functions. Often, the user would actually like to optimize a
number of different objectives at once. For instance, in the panel design problem, it
would be beneficial to minimize weight and maximize strength simultaneously.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 33
Usually, the different objectives are not compatible; the variables that optimize one
objective may be far from optimal for the others. In practice, problems with
multiple objectives are reformulated as single-objective problems by either forming
a weighted combination of different objectives or else replacing some of the
objectives by constraints. These approaches and others are described in the section
on multi-objective optimisation.
2.2.2.2 Variables
Variables are a set of unknowns that affect the value of the objective function. For the
manufacturing problem, the variables may include the amounts of different resources
used or the time spent on each activity. In fitting-the-data problem, the unknowns are
the parameters that define the model. In the panel design problem in Section 2.2.2.1, the
variables used define the shape and dimensions of the panel. The variables are essential.
If there are no variables, the objective function and the problem constraints can not be
defined.
2.2.2.3 Constraints
A set of constraints that allow the unknowns to take on certain values but exclude
others. Although optimisation studies may be required to achieve the best result from a
given situation, meaning the absolute best, this may not always be attainable due to the
imposition of restrictions. For example, for the manufacturing problem, it does not
make sense to spend a negative amount of time on any activity, so we constrain all the
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 34
"time" variables to be non-negative. In the panel design problem, w e would probably
want to limit the weight of the product and to constrain its shape.
2.2.3 Classical Optimisation Hierarchy
The hierarchy of classical optimisation [NEOS, 1996; Chong and Zak, 1996] can be
illustrated as an optimisation tree shown in Figure 2.9. It introduces the different sub-
fields of optimisation. The connections between the tree's pathways mirror the
relationships between these different areas.
Basically, the optimisation is classified into continuous optimisation, in which all the
variables are allowed to take values from sub-intervals of the real life, and discrete
optimisation, in which some or all of the variables are required to have integer values.
The discrete optimisation includes integer programming and stochastic programming.
The continuous optimisation is composed of Unconstrained and Constrained
Optimisations. The Unconstrained Optimisation comprises of Nonlinear Equations,
Nonlinear Least Squares, Global Optimisation, and Non-differentiable Optimisation.
The Constrained Optimisation consists of Linear Programming, Semi-definite
Programming, Nonlinearly Constrained, Bound Constrained, Quadratic Programming,
Network Programming, and Stochastic Programming. In Appendix A, these two
categories of optimisation, and the major algorithms of each branch are addressed
individually.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 35
Optimisation
3L Discrete Optimisation
3L Integer Programming
^L Continuous Optimisation
-$-
Constrained Optimisation
±. Stochastic Programming ^
Linear Programming ^.
Nonlinearly Constrained ^
Bound Constrained <-
Network Programming <r
M. Unconstrained Optimisation
Nonlinear Equations <r
Nonlinear Least Squares ^
Global Optimisation <r
Non-differentiable Optimisation <=-
^> Semi-definite Programming
V Quadratic Programming
Figure 2.9 Classical optimisation tree
2.3 SUMMARY AND CONCLUSION REMARKS
A typical tandem cold rolling mill is introduced including its structure and automatic
control systems. The geometry and driving system of the tandem cold rolling mill have
been covered. The main loops of AGC and AFC are presented to illustrate their
functions in achieving good quality of strip. This may assist in explaining the
background of the research in this thesis.
Chapter 2 Fundamentals of Rolling Process and Optimisation Theory 36
Tandem cold rolling process is complex and requires careful coordination and attention
to detail before any effort is made to improve the mill performance. The cold rolling
rationale is addressed, and the deformation model of strip in the roll bite is reviewed. As
one of the difficult technical aspects of cold rolling, shape problem is also presented.
Shape models are reviewed and traced back to its origins. Tandem cold rolling process
is also addressed from the point of view of speed. The five phases including threading,
acceleration, full speed operation, deceleration and tailing out are described. The
features and disturbances associated with individual phases of the above are also briefly
dissected.
This chapter also provides an overview of optimisation fields including the terminology
of optimisation. The three components of the optimisation are detailed. The
optimisation hierarchy tree is also introduced.
Classical optimisation is the formal title given to the branch of computational science
that seeks to answer the question ^What is best?' for problems in which the quality of
any answer can be expressed as a numerical value. Such problems arise in all areas of
mathematics, physical, chemical and biological sciences, engineering, architecture,
economics, and management, and the range of techniques available to solve them is
wide.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 37
Chapter 3
Principles of Evolutionary Optimisation and Its
Applications in Engineering
Natural evolution is a population-based optimisation process. When trying to find new
methods and problem-solving strategies for their research, scientists often turn to nature
for inspiration. An excellent example of this is the application of Darwin's Theory of
Evolution, particularly the notion of the "survival of the fittest", in computer programs
designed to search for optimal solutions to many kinds of problems. Simulating the
natural evolution process on a computer results in stochastic optimisation techniques
that can often out-perform classical methods of optimisation when applied to difficult
real-world problems. Evolutionary algorithms (EAs) are simply search methods that
take their inspiration from natural selection and survival of the fittest in the biological
world [SNL, 1997]. These "evolutionary algorithms" start from a population of possible
solutions to a given problem and, by applying evolutionary principles, evolve successive
generations with improved characteristics until an optimal, or near-optimal, solution is
obtained.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 38
This chapter first gives an introduction of evolutionary optimisation, followed by
comparison between the classical optimisation and the evolutionary optimisation, then
reviews the industrial applications of evolutionary optimisation, especially in the steel
industry.
3.1 INTRODUCTION
The term evolutionary algorithm (EA) stands for a class of stochastic optimisation
methods that simulate the process of natural evolution. The origins of EAs can be traced
back to the late 1950s, and since the 1970s several evolutionary methodologies have
been proposed, mainly genetic algorithms, evolutionary programming, evolutionary
strategy and genetic programming. All of these approaches operate on a set of candidate
solutions. Using strong simplifications, this set is subsequently modified by the two
basic principles of evolution: selection and variation. Selection represents the
competition for resources among living beings. Some are better than others and more
likely to survive and to reproduce their genetic information. Each iteration of an EA
involves a competitive selection that weeds out poor solutions. The solutions with high
"fitness" are "recombined" with other solutions by swapping parts of a solution with
another. Solutions are also "mutated" by making a small change to a single element of
the solution. Recombination and mutation are used to generate new solutions that are
biased towards regions of the space for which good solutions have already been seen.
To compare the natural evolution and the simulated evolution (evolutionary algorithms),
some terminology that has been derived from the natural evolution has been listed in the
following table.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 39
In evolutionary algorithms, natural selection is simulated by a stochastic selection
process. Each solution is given a chance to reproduce a certain number of times,
dependent on their quality. Here, quality is assessed by evaluating the individuals and
assigning them scalar fitness values. The other principle, variation, imitates natural
capability of creating "new" living beings by means of recombination and mutation.
Table 3.1 Terminology comparison between the natural evolution and evolution
algorithms
Natural Evolution
Chromosome Gene
Allele
Locus
Genotype
Phenotype
Epistasis
Evolutionary Algorithms
String Feature/Character
Feature Value
Position on String
Coded Form of Parameters
Actual Parameter Set
Nonlinear
Although the underlying principles are simple, these algorithms have proven themselves
as a general, robust and powerful search mechanism. In particular, they possess several
characteristics that are desirable for problems involving:
(1) multiple conflicting objectives, and
(2) intractably large and highly complex search spaces.
This section provides a background in history and application of evolutionary
computation methods with particular emphasis on Genetic Algorithms, which is the core
of the optimisation approaches employed in the thesis.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 40
3.1.1 Composition of Evolutionary Optimisation
Evolutionary Algorithms [Fogel, 1995] include the following domains:
(1) Genetic Algorithms, which are intended to optimize discrete structures with a
special attention to general combinatorial problems;
(2) Evolutionary Programming, which focuses on optimizing continuous functions
without recombination;
(3) Evolutionary Strategies, which focus on optimizing continuous functions with
recombination, and
(4) Genetic Programming, which is well suited for problems that require the
determination of a function, this is called evolution program.
As Genetic Algorithm is suitable for general combinatorial problems, it is broadly
applied in engineering area. In the following sections, the origin, principles and
mathematical foundations of Genetic Algorithms will be addressed.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 41
3.1.2 Genetic Algorithms
3.1.2.1 Concept and Origin of Genetic Algorithms
Genetic Algorithms (GA) are search algorithms based on the mechanics of natural
selection and natural genetics [Goldberg, 1989]. They combine survival of the fittest
among string structures with a structured yet randomized information exchange to form
a search algorithm with some of the innovative flair of human search.
In 1966, L.J. Fogel, A.J. Owens, and M.J. Walsh proposed and investigated the
evolution of simple automata, which was intended to predict the symbols in digital
sequences [Fogel, 1966]. In 1975, J.H. Holland, in his book "Adaptation in Artificial
and Natural Systems", proposed the scheme of the genetic algorithms [Holland, 1975].
Approximately at the same time, I. Rechenberg and H.-P. Schwefel developed the
evolutionary strategies [Rechenberg, 1973; Schwefel, 1977]. These works formulated
the main directions of the developments of evolutionary algorithms.
John Holland was one of the first developers of genetic algorithm schemes. A good
introduction to the topic can be found in his contributions [Holland, 1969; Holland,
1975]. Goldberg and Forrest reviewed Genetic Algorithms in 1989 [Goldberg, 1989]
and 1993 [Forrest, 1993], respectively. In general, GA operates over a population of
individuals and aims at possible solutions to the problem being solved using a set of
genetic operators.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 42
3.1.2.2 Genetic Operators
The operators of a simple GA include: reproduction (selection), crossover, and
mutation [Goldberg, 1989; Holland, 1992]. Each iteration is known as a generation.
Pseudo-code for a basic genetic algorithm is shown as follows:
Generate initial population
Evaluate the population
While termination conditions not satisfied
{
Select the fittest individuals
Apply genetic operators to generate new solutions
Evaluate the solutions in the population
Insert new individuals to the population
}
For example, if the problem is to maximize the function f(x) = x2 on the integer
interval [0, 31]. Suppose the population size is assigned to four, and four individuals are
randomly created as 13, 24, 8 and 19. They can be coded as binary strings as illustrated
in Table 3.2.
The first generic operator is reproduction. It is a process in which individual strings are
copied according to their objective function values (the fitness function f(x)). The
reproduction operator identifies the fittest individuals from the population using the
value returned by an evaluation function that depends on the specific problem being
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 43
solved. In the example, reproduction means copying strings according to their fitness
values. The strings with a higher value have a higher probability of contributing one or
more offspring in the next generation. The reproduction operator may be implemented
in algorithmic form in a number of ways. The easiest one is to create a biased roulette
wheel where each current string in the population has a roulette wheel slot sized in
proportion to its fitness. The corresponding weighted roulette wheel for this
generation's reproduction is shown in Figure 3.1.
Table 3.2 Four initial individuals in the first generation
No. Individuals
1 13 2 3 4
24 8 19
Total
Binary Strings
01101 11000 01000 10011
Fitness f(x)
169 576 64 361 1170
Percentage of Total
14.4% 49.2% 5.5% 30.9% 100.0%
Figure 3.1 The roulette-wheel selection
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 44
To reproduce, simply spin the weighted roulette wheel four times. For the example
problem, the No. 1 string has a fitness value of 169 representing 14.4% of the total
fitness. As a result, the string is given 14.4% of the biased roulette wheel, and each spin
turns up the No. 1 string with a probability of 0.144. Thus an offspring can be
reproduced by a simple spin of the roulette wheel, and it is an exact replica of the string.
In this way, better fit strings which have higher fitness values will have a higher number
of offspring in the succeeding generation. The produced offspring is then entered into a
tentative new population for further genetic operator actions.
The individuals in the tentative population are then combined between themselves using
the crossover operator. Firstly, individuals of the newly reproduced offspring in the
tentative population are mated at random. Secondly, each pair of strings undergoes
crossover by selecting one crossover position along the string at random between 1 and
the string length, then swapping all characters on the right side of the crossover position.
For example, assume the crossover position is 4 for the following two strings:
4 = 0 1 1 | 0 1
4 = 1 1 0 | 0 0
After the crossover operation, two new strings are yielded by swapping all characters on
the right side of position 3:
4=01100
4=11001
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 45
The procedure of crossover can be summarized as: (1) random number generation to
determine the crossover position, and (2) exchange all characters on the right of the
crossover position. It should be noted that even though reproduction and crossover
effectively search and recombine genes, they may occasionally lose some potentially
useful genetic material (1 's and O's at particular locations). Thus, mutation is introduced
in Genetic Algorithms to overcome this shortcoming.
The mutation operator is applied eventually to some individuals to randomly change a
part of their chromosome, from 0 to 1 or from 1 to 0, and to introduce new genetic
material to the population.
3.1.2.3 Mathematical Foundation of Genetic Algorithms
(1) Representation of Parameters
In practical problem solving, parameters involved are usually coded as binary strings. If
a parameter falls into the interval [ux, u2 ], the parameter can be mapped to an integer
space [0, 2" ], where n is the coding length of the parameter. The length depends on the
precision of the solution requirements on the parameter, which is also named as
"granularity" of the solution. For example, if the parameter takes on a value from [-5.12,
5.11], when the granularity is 0.1, the length of the coded string will be 7; however, if
the granularity is 0.01, the length will be 10. The granularity is usually determined by
the precision requirement of the solution. Thus the resolution of the parameter can be
controlled by defining granularity, and the precision of the solution is defined as:
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 46
§ = u2 -Mj
2"-l (3-1)
If two or more variables are involved in the problem solving, each parameter should be
coded individually as binary string. The binary string for each of the parameters is then
linked together into a single chain as illustrated in Figure 3.2. The genetic operations are
applied to the chain as a single variable.
Single parameter x is coded (n = 4 ) as:
0 0 00- umm
1 1 u -> « M
Multiple parameters (« = 10) are coded as:
0001
«1
0101
u2
1 100
M9
1111
«10
Figure 3.2 A n illustration often encoded variables
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 47
(2) Fitness Function
Fitness is simply a figure of merit to measure how good an individual of a population is.
It must be non-negative. In optimisation problems, the objective is usually stated as the
minimization of some cost function g(x) rather than the maximization of some profit
function u(x) [Goldberg, 1989]. Even if the problem is stated in maximum form, this
does not guarantee that the profit function is positive for all x. Thus, it is often
necessary to map the underlying natural objective function to a fitness function form
through one or more mappings.
If the objective function formulation is a cost function, the fitness function should be
transformed. In operation research, to transform a minimization problem to a maximum
problem, the cost function is usually multiplied by a minus one. However, this operation
can not guarantee that the fitness obtained is non-negative in all instances. Thus, an
input coefficient Cmax is introduced, and it may be taken as the largest value of g(x).
The following cost-to-fitness transformation is commonly employed:
Cmax - gO) when g(x) < Craax
0 otherwise
When the objective function is a profit function, the fitness should simply take the
formulation of the objective function while introducing an input coefficient Cmin to
guarantee f(x) is non-negative. The coefficient Cmin may be assigned to the absolute
value of the worst u{x).
f(x) =
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 48
f,x) = J<x) + C-" when u{x) + Cmin > 0
0 otherwise
Besides the above two situations, transformation is needed in the following cases:
a) If the optimisation problem is not to find the largest value of its objective, and
b) In the early stage of evolution, the early domination of extraordinary individuals
occurs, and therefore leads to premature convergence.
Three commonly used procedures to transform the objective function to the fitness
function for the above cases are [Bagley, 1967; Rosenberg, 1967; Forrest, 1985; Gillies,
1985]:
a) Linear scaling such as f(x) -ax + b. The coefficients a and b may be chosen in a
number of ways. However, in all cases, the average scaled fitness should be equal to
the average raw fitness, or the maximum fitness after transformation is the product
of the average raw fitness and the number of expected copies desired for the best
population member. This is because subsequent use of the selection procedure will
insure that each average population member contributes one expected offspring to
the next generation [Goldberg, 1989].
b) Power law scaling such as f(x) = xa. In the power law form of scaling, the scaled
fitness is taken as some specified power of the raw fitness. The coefficient a is
problem-dependent and may require changes during a run to stretch and shrink the
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 49
range as needed. The fitness after transformation is the power of the raw fitness to
the scaling coefficient a.
c) Exponential scaling such as f(x) = e(_ax). The exponential scaling is based on the
idea of Simulated Annealing. The Simulated Annealing is a heuristic optimisation
algorithm based on the idea of annealing in physics [Laarhoven and Aarts, 1987;
Aarts and Lenstra, 1997]. This technique allows physicists to create configurations
in materials, preferably with low energetic values. It can also be used to solve
combinatorial optimisation problems, especially to avoid local minima that cause
problems when using simpler local search methods.
The purpose of all the above transformations is to guarantee that the fitness is
nonnegative. Meanwhile, the transformations make the fitness stretched to a certain
range, therefore, guarantee the best individual has the best chance to be reproduced
while preventing the early domination of extraordinary individuals, and encourage a
healthy competition among near equals.
(3) Operation Procedure of Generic Algorithms
The operation of genetic algorithms is remarkably straightforward. The procedure of
solving a minimization optimisation problem is as follows:
(a) Binary coding of the parameters involved in the problem to be solved. There are two
basic principles for choosing a GA coding. The principle of meaningful building
blocks and the principle of minimal alphabets [Goldberg, 1989].
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 50
(b) Determine the population size and randomly initialize the first generation
population.
(c) Determine the fitness function.
(d) Determine the selection strategy such as roulette wheel.
(e) Conducting genetic operations such as reproduction, crossover and mutation.
(f) Evaluate the individuals created in the generic operations and select individuals for
the next generation population.
(g) Check the convergence criteria, if it is satisfied, stop the evolution process;
otherwise, repeat steps (e), (f) and (g).
(4) Mathematical Description of Genetic Operations
For a GA based optimisation problem, assume the fitness function is f(x), the
population size is m{m is an even number). Each of individuals in the population is
coded as binary string with the length of n. Suppose the /"* individual in the gth
generation is coded as:
xg-'=(xf'',xZ't...,x?f) (3-4)
where g = 0,1,..., and / = 1,2,..., m. Suppose the gth generation population has been
created, then the principles to create the (g + X)th generation population are:
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 51
(a) Reproduction. Randomly choose one of the individuals in the g generation, the
probability for an individual to be chosen is:
p{x^)=f{xg'i)lfjf{xg") (3-5)
1=1
where / = 1,2, ...,m, f(xsJ) is the fitness of x8'' and f is the fitness function.
The new individuals created are named temporary individuals, and designated as
x'gJ.
(b) Crossover. Assume the crossover rate is pc for two individuals selected to conduct
crossover operation, and xg+u , xg+h2, ... , xg+u (0<i<m) denote the individuals
in the (g +1)'* generation population to be created. If the following two temporary
individuals generated in the reproduction operation are selected to conduct crossover
operation, and the crossover position is s(l<s<ri).
x'g'h = (*;*•*,xg'h,...,x*-h,xi/1,...,xg'h ) (3-6)
x'g'h = (x*h,x''h >...>x'f'htx&h >-'x'n'h ) (3~7)
After the crossover operation, the new individuals xg+l'h and xg+l'h
(0 < /,, ;2 < m) for the (g +1)* generation are:
x^h = (X;..A t x'*.h ;...5 x'«h, x'«h v..? x>*.k) (3.8)
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 52
xg+u> = (*;*'>,*?•'',.,xg'\xg^,...,xHg'h ) (3-9)
(c) Mutation. Mutation is the random alternation of a single position of an individual.
Suppose the mutation rate for a gene is pm, which means each of the genes of the
/* individual of the (g + V)'h generation population has a probability of pm to
change to its counterpart gene (changed from 0 to 1 or 1 to 0). After the mutation
operation, there is:
x*+u=JC-*+u (3_10)
where xrg+1'; is the counterpart of the gene xf+L/.
(5) Fundamental Theorem of Genetic Algorithms
In Genetic Algorithms, binary strings are directly involved in the genetic operation. The
most important concept relevant to the binary string is "schema". This section will
describe the concept of the "schema", and its theorem.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 53̂
(a) Concept of Schema
Schema is a similarity template describing a subset of strings with similarities at certain
string positions [Holland, 1968 and 1975]. For example, 5* = *0000 is a schema of
length 5, where * denotes a gene which is not determined yet. It matches two strings
{10000, 00000}. The schema *111* represents a subset with four members {OHIO,
11110, 01111, 11111}. In a schema, the number of fixed positions presented in the
template is defined as the order of the schema. So the order of the schema *111* is 3.
The defining length of a schema is the distance between the first and last specific string
position. For example, the defining length of the schema 011*1** is 4 because the first
specific position is 1 and the last specific position is 5, so the distance is 5-1=4. For the
schema 1*****, the defining length is 0 because the first and last specific positions are
the same. For a binary coded string of length n, there are 2" schemata. For example, an
individual 1100 represents the following 16 schemata:
1100, 110*, 11*0, 11**, 1*00, 1*0*, 1**0, 1***
*100 *10* *1*0 *1** **oo, **0*, ***0, ****
For a population with m individuals, there are totally Ns schemata, where
2"<Ns<m2n (3-11)
In fact, the total number of schemata in a population is dependent on the difference
among these schemata. If a population has ten exactly the same individuals, then each of
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 54
them represents the same set of schemata. The total number of the schemata is the same
as that represented by each of the individuals, which is 2". Thus, the population with
totally different individuals represents a larger number of schemata than those
populations if some of their individuals are the same.
(b) Schema Theorem
Schemata and their properties provide the basic means for analyzing the effect of
generic operations. This section discusses the individual and combined effect of
reproduction, crossover and mutation on schemata contained within a population of
strings.
Suppose a schema S has n(S,t) individuals at time t. The probability and expectation
for one of the individuals to be reproduced to the next generation are f(St) / Fs and
f(St)/ f(p), respectively. The reproductive schema grows and the number of
individuals at time (t +1) is as follows:
n(S,t + l) = n(S,t)f(S)/f(p) (3-12)
where
f(S{) is the fitness of the /'* individual
Fs is the summation of fitness values of all individuals of the schema S
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 55_
f(S) is the average fitness of all individuals of the schema S
f(p) is the average fitness of all individuals in the population
If f(S) = (1 + k) f(p), then equation (3-12) can be rewritten as follows:
n{S,t + \) = n(S,t)[{l + k)ap)Vf(p)^n(S,t)(\ + k) (3-13)
Thus, after n times reproduction, the number of individuals that the schema S
represents is:
n(S,t + l) = n(S,t)(\ + k)n+l (3-14)
From the above equation, if the average fitness f(S) of all individuals of the schema S
is greater than the average fitness f(p) of all individuals in the population, then the
number of individuals of the schema S in the next generation will be increased
exponentially. Otherwise, it will be decreased exponentially.
Although reproduction is an efficient mean to increase the number of individuals with
merits, it can not create new individuals. New individuals can only be created by
crossover and mutation operations. Table 3.3 shows the individual A of the schemata
Sl and S2, individual B, and the new individuals A' and B' created during crossover
of the individuals A and B.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 56
Table 3.3 Individuals, schema and crossover
£ _ * * Q * J * * * |
£ _****JQ|**
4=010010101
5=101011000
A' =01 001 1000
B' = 101010101
Although A is the individual of the schemata Sx and S2, A' is not. However, B' is an
individual of the schema S2. Thus, after the crossover, the number of schemata is
reduced. Before the crossover, there is an individual A for the schema 5,, however, it is
lost after the crossover. The schema still has the individual A. This is because the
defining length of the schema Sl is longer than that of the schema S2. Thus, the
probability of a schema losing its individuals can be formulated as follows:
Pl=8{S)l{l-l) (3-15)
where, 5(5') denotes the defining length of the schema S, / is the length of the schema
S. If Pr is the probability for the schema 5 to keep its individuals, then
Pr=\-Pl=l-8(S)/(l-l) (3-16)
If the crossover rate is P , then
P=\-Pc8{S)l{l-\) (3-17)
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 57
Based on Equations (3-12) and (3-17), after the crossover, the number of individuals of
the schema S is:
n(S,t + l)>[n(S,t)f(S)/f(p)][l-Pc8(S)/(l-l)] (3-18)
The above equation denotes that the number of individuals of a schema depends on its
defining length and the fitness of other individuals in the population.
Any one of genes of an individual in a population could be changed to its counterpart,
from 1 to 0 or from 0 to 1. This is implemented by mutation operation. Whether
mutation occurs on a gene of an individual is independent of other genes of the
individual. Assume Pm is the mutation rate, and A is one of the individuals of the
schema S with r genes, then the probability of the gene not being mutated is (1 - Pm )r,
i.e., after mutation, the probability PAS of the individual being one of the individuals of
the schema is (1 - Pm)r. As Pm is a very small value, (1 - Pm )
r can be rewritten as:
PA,s=l-rPm (3-19)
Combine Equations (3-12), (3-18) and (3-19), the number of individuals of the schema
5* after reproduction, crossover and mutation operations is:
n(S,t + \)>[n(S,t)f(S)/f(pW-Pc8(S)/(l-\)-rPm] (3-20)
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 58
The above equation is so called the schema theorem. It indicates that the number of
individuals of the schema with large fitness, small defining length and small schema order
will be increased exponentially during evolution.
3.2 COMPARISON OF EVOLUTIONARY AND CONVENTIONAL
OPTIMISATION
Literatures show there are three main types of conventional search methods. They are
calculus-based, enumerative, and random. They are useful but not robust [Goldberg,
1989]. Other methods are necessary to attack more complex problems. Figure 3.3
illustrates the efficiency of three search schemes: a specialized scheme (such as gradient
technique and hill climbing), an enumerative scheme, and an idealized robust scheme
such as GA [Goldberg, 1989].
combinatorial unimodal multimodal
Figure 3.3 Comparison of searching efficiency of three types of schemes
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 59
The above figure shows the specialized scheme, such as gradient searching, performs
well in its narrow problem domain, however, it becomes highly inefficient elsewhere.
This is because the specialized scheme is designed for specific problems solving only.
The vertical coordinate, Efficiency, in Figure 3.3 is defined in terms of time cost when a
search method is applied to find a solution to the problem being solved. More details
can be found in Goldberg's book [Goldberg, 1989]. The enumerative scheme performs
with egalitarian inefficiency across the spectrum of problems in comparison with the
robust scheme. However, the most desirable one should combine the best of local search
method with the more general robust scheme. The Genetic Algorithms can be used to
fulfill this requirement [Goldberg, 1989].
GA is often viewed as a global optimisation method although convergence to a global
optimum is only guaranteed in a weak probabilistic sense. However, GA differs from
more traditional optimisation techniques in that:
(1) GA works with the coding of a parameter set, not the parameters themselves. GA
requires the natural parameter set of the optimisation problem to be coded as a
finite-length string over some finite alphabet [Goldberg, 1989]. In the example
given in Section 3.1.2.2, to maximize the function f(x) = x2 on the integer interval
[0, 31]. With traditional methods, the parameter x is directly operated. However,
with GA, the parameter x is coded into a finite-length string first, the genetic
operations are conducted on the coded string, not the parameter x.
(2) As compared to usual optimisation methods, the GA has the following specific
particularities: parallel search, random variations, and recombination of already
found solutions, i.e. GA involves a search from a "population" of solutions, not
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 60
from a single point. In most of traditional optimisation, searching is moved from a
single point in the decision space to the next using some transition rules to
determine the next point. This point-to-point method could lead to a false peak in
multi-modal (many peaks) search spaces. However, GA works from a pool of
candidate points (a population of strings) simultaneously, climbing many peaks in
parallel. By working from a population of well-adapted diversity instead of a single
point, the GA adheres to the old adage that there is safety in numbers. This makes
the GA more robust in searching for a solution.
(3) GA is well suited for a wide range of combinatorial and continuous problems,
though the different variations are tailored towards specific domains.
Combinatorial optimisation problem is involved in many domains in which
assignment or partition is involved, i.e. allocate a certain number of units to a
limited capacity container respecting certain constraints of the problems. The
recombination operation used by GA requires that the problem can be represented
in a manner that makes combinations of two solutions likely to generate interesting
solutions. Consequently selecting an appropriate representation is a challenging
aspect of applying these methods.
(4) GA uses objective function information instead of derivatives or other auxiliary
knowledge. Many search techniques require much auxiliary information in order to
work properly. For instance, gradient techniques require derivatives in order to be
able to climb the current peak. However, GA requires objective function values
only associated with individual strings.
(5) In contrast to classical methods, Genetic Algorithms based technique of optimum
search allows for incorporation of knowledge of the environment to be embedded
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 61
in the algorithm structure, and enables the search to continue (no breaks) in case of
environment changes. There are several ways to combine problem-specific
information with GA such as hybrid techniques, knowledge-directed operators and
approximate function evaluation methods. This makes GA a heuristic method of
global optimum search. It performs well on "noisy" and discontinuous functions
where there may be multiple local optima. GA tends not to get "stuck" on a local
minimum and can often find a globally optimal solution. Although optimal
solutions cannot be guaranteed all the time, on the other hand many reasonable sub-
optimal solutions could be reached.
(6) GA uses probabilistic transition rules instead of deterministic rules to guide search.
Unlike many traditional searching methods, GA uses random choice as a tool to
direct a search toward regions of the search space.
All of the above features of GA contribute to GA's robustness searching and advantages
over other commonly used conventional techniques.
3.3 GA'S APPLICATIONS IN STEEL INDUSTRY
GAs have been successfully applied to a variety of optimisation problems such as wire
routing, scheduling, traveling salesman, image processing, engineering design,
parameter fitting, computer game playing, and transportation problems. There are
numerous GA research papers published in the last two decades, and some of them are
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 62
engineering applications oriented. Figure 3.4 shows the numbers of published papers
regarding GA's engineering applications, which were collected in Inspec Database.
G A has been applied in the steel industry in the areas of rolling scheduling, production
planning and optimisation, rolling mill optimisation and steel product design. The
following are some examples of GA's applications in the steel industry.
Papers on Genetic Algorithms Collected in Inspec Database
1600 -i
(0 i_
0)
o. re 0-o a> JQ
E 3 z
1400 --
1200-
1000 --
800 --
600 -
400 -
200 --
0 + T • T f i i 1 1 1 1 1 1 1 1
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Year
Figure 3.4 Papers collected in Inspec Database relevant to engineering applications of
Genetic Algorithms
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 63
3.3.1 Rolling Scheduling
The operation of hot strip mill rolling scheduling (HSMRS) at China Steel Corporation
is an extremely difficult and time-consuming process due to the complexity of the
problem. Fang and Tsai explored how this problem could be solved through the use of a
genetic algorithm in 1998 [Fang and Tsai, 1998]. One of the key aspects of this
approach is the use of specially designed representations for such scheduling problems.
The representations explicitly encode a schedule by encoding information for building
cycles. The authors have found that this representation cooperates with a stochastic
violation directed mutation operator and suitable fitness function and can quickly
produce results comparable to human scheduling. The efficient and flexible GA
approach presented is potentially useful in other similar rolling cycle scheduling
applications in large steel companies.
Scheduling for tandem cold mills refers to the determination of inter-stand gauges,
tensions and speeds of a specified product. Optimal schedules should result in
maximized throughput and minimized operating cost. Wang and Tieu presented a
genetic algorithm based optimisation procedure for the scheduling of tandem cold
rolling mills in 1998 and 2000 [Wang, Tieu et al., 1998; Wang, Tieu et al., 2000]. The
proposed optimisation procedure initiates searching from a logical starting point, an
empirical rolling schedule, and ends with an optimum cost. Cost functions were
constructed to heuristically direct the genetic algorithm's searching, based on the
consideration of power distribution, tension, strip flatness and rolling constraints.
Numerical experiments have shown that the proposed method is more promising than
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 64
those based on semi-empirical formulae. The results generated from a case study show
that the proposed approach could significantly improve empirically derived settings for
the tandem cold rolling mills.
3.3.2 Production Planning and Optimisation
Production planning problem in steel industry is very complex. However, researchers
and engineers have been making their efforts to deal with the problem. Following are
some successful examples.
In 1989, Fogarty introduced a rule-based system for optimising combustion in a
multiple burner furnace and its manual adaption for use on another installation [Fogarty,
1989a; Fogarty, 1989b]. Genetic algorithm was employed to adapt the rule base. The
performance of the expert system on the 108 burner furnace of a continuous annealing
line for rolled steel was critically examined. The results of experiments to compare the
performance of the two systems in dealing with "noise" data were reported. The results
show that the genetic algorithm can be used to learn specific actions for particular
situations in this domain. In 1990, Fogarty published another paper presenting a rule-
based system for optimizing combustion in multiple-burner installations, which was
built and tested on the furnace of a continuous annealing line for rolled steel [Fogarty,
1990]. The furnace has only one firing level, and the rules were elicited from energy
experts with this problem in mind. The system was then installed on a multiple-burner
boiler in the steel industry, but it did not respond very fast to changes in the firing level
of the boiler. One particular solution to this problem is to enter new rules into the rule
base to deal with the changed situation. A more general solution is to build into the rule
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 65^
base a learning component to help it to cope with new situations. Promising results have
been obtained with the genetic algorithm, which has proved to be robust in this noisy
domain and suitable for learning control rules that give performance comparable to that
of rules elicited from the experts. Experiments carried out on multiple-burner
installations with two firing levels, using the genetic algorithm to learn the best actions
for given situations, are described, and the results are discussed.
In 1993, Takahashi and Konishi et al. developed a scheduling method for steelmaking
process, in which the problem was formulated as a minimum cost flow [Takahashi and
Konishi et al., 1993]. Sequential search algorithm is one of the well-known optimisation
methods, but it often results in local optimisation. Above all, for an integer problem
such as scheduling, which has a complex discrete structure, a sequential search approach
is not available in many cases. To cope with this difficulty, genetic algorithm has been
attempted, but its processing time is too great for actual application systems. As a result
of this shortcoming, it is often effective to utilize heuristics to find a feasible semi-
optimal solution first, then employ GA to find the optimal solution. The developed
method was applied to the scheduling system in a steel plant, and the effectiveness of
this approach was revealed in the actual application.
In 1995, Hamada and Baba et al. proposed an approach to divide the production
planning problem of a steelworks into sub-problems, and combine procedural methods,
a rule-based expert system, and a genetic algorithm [Hamada & Baba et al., 1995]. This
strategy produces time-saving, cost-effective schedules.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 66
In 1996, Iwaya and Ohtsuka et al. presented a successful application of Genetic
Algorithms in the distribution field where the Genetic Algorithms were effectively
applied to achieve excellent results from a logistic point of view [Iwaya and Ohtsuka et
al., 1996]. They described a storage system for steel sheets at Nippon Steel's Kimitsu
Works introduced as an application example of GA, and a vehicle allocation planning
system implemented at another company as a test application of GA.
In 1997, Wan, Chen and Xu established a new clustering model for rolling planning in
the iron steelmaking industry using artificial intelligence and genetic algorithm [Wan,
Chen and Xu, 1997]. They developed a two-phase improvement algorithm to resolve the
planning problem, and the results obtained showed GA's superiority in the quality of
solution over other heuristics.
In 1997, Paul and Chanev also presented a simple real-coded genetic algorithm which
was employed to optimize management parameters of plant, demonstrating the
versatility genetic algorithms' offer in solving difficult inverse problems [Paul and
Chanev, 1997]. A steelworks model was selected as representative of the stochastic and
unpredictable behaviour of a complex discrete event model. The steelworks has a
number of different entity or object types. Using the number of each entity type as
parameters, it is possible to find better and worse combinations of parameters for
various management objectives.
In the order allocation to in-stock slabs problem, it is necessary to optimize the selection
of both thick and flat pieces of steel plate for each order and the order arrangement in
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 67
hot rolling plates. The optimum solution in this problem is to produce as many high
priority orders as possible while using the slabs as efficiently as possible, i.e. reducing
waste material as much as possible. Matsuda and Yoshida et al. approached this in-stock
allocation problem as a combinatorial optimisation problem and successfully applied a
combinatorial optimisation method known as the genetic algorithm (GA) to achieve an
optimal allocation and arrangement in 1997 [Matsuda and Yoshida et al., 1997]. As the
in-stock allocation problem became more complicated (number of constraints included,
number of orders, number of slabs, etc.), the calculation time began to exceed the
allocated production planning time. In this system they divided the problem into sub
problems while maintaining solution quality. By allocating the given calculation time
among the small problems they were able to obtain a good solution in a feasible
calculation time. In general, it is difficult to solve the problem that includes a vast of
array of solution constraints and restrictions using GA. The order allocation problem has
many constraints such as order/slab material composition restriction, machine dictated
order arrangement constraints and so on. In the proposed system, by introducing a new
coding and decoding method that considers these constraints, it was possible to
implement the application of GA. In this problem, two objectives including order
priority and yield rate have to be satisfied. By the construction of the evaluation
function and the tuning of the weighting coefficients, the authors were able to develop a
system that satisfied the above two objectives.
In the same year, a production ordering problem for the acid rinsing process of a
steelworks was presented by Sannomiya, Kako, and Kobayashi [Sannomiya, Kako, and
Kobayashi, 1997]. The problem is to arrange hot strip coils in an optimal order subject
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 68^
to complicated constraints. The strip coils are classified into several groups on the basis
of their specifications, and the coils belonging to a same group must be arranged
successively. Other constraints are precedence relationship between coils and bounded
difference in size between two adjoining coils. The genetic algorithm was applied to
obtain a sub-optimal arrangement of strip coils. The individual description and the
corresponding decoding rule were also proposed for this problem.
In 1998, Barker, Price and Evans described the validation of a dynamic model of
moving steel strip [Barker, Price and Evans, 1998]. The model is based on a physical
model that accurately replicates the dynamic behaviour of moving steel strip. A pseudo
random signal was used as a perturbation signal to persistently excite each model over
the whole of its dynamic range. The resulting output signals were compared and a
genetic algorithm was used to optimize the model parameters by minimising the root-
mean-square of the error. After optimisation, there is close agreement between both the
time and frequency responses of the models. It is therefore concluded that the proposed
model is a valid representation of the dynamics of the physical model and of moving
steel strip.
Filipic and Sarler also presented a computational approach to process parameter setting
in continuous casting of steel in 1998 [Filipic and Sarler, 1998]. The objective was to
determine parameter values such that the highest possible quality of the cast steel could
be achieved. A numerical model of the casting process has been developed and
integrated with a genetic algorithm to explore the space of parameter settings with
respect to predefined metallurgical criteria. The results obtained at a continuous casting
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 69
unit of a steel plant indicate a potential improvement of product quality and
productivity.
Deb and Chakraborti also demonstrated a model of preheating of blooms in a fuel fired
furnace using a combined GA and heat transfer formulation [Deb and Chakraborti,
1998]. A re-heating furnace containing three asymmetrically placed burners were
considered, which is a typical configuration used in many integrated steel plants. This
study shows that the GA is an ideally suited tool for studying such complex problems.
In 1998, Lima, Sannomiya and Wakasugi also proposed a method for solving
production scheduling problem with many complicated constraints [Lima, Sannomiya
and Wakasugi, 1998]. They dealt with the production-planning problem in the acid
rinsing process of a steel-making plant. Strong requirements assigned to the constraints
usually deteriorate the objective value, and in many cases it is impossible to satisfy all
the constraints completely. Therefore, fundamental constraints of the problem are
treated as hard constraints, the other constraints as soft constraints. The genetic
algorithm was applied to the problem. A two-phase method was proposed in such a way
that: in Phase 1, soft constraints conflicting strongly with the objective function were
found; and in Phase 2, the optimal solutions for representing the tradeoff between the
objective value and the relaxation of the constraints were obtained. The method is
effective because the computation load for tuning penalty parameters is decreased and
the method has given a better result than the real operation data.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 70
In 1999, Li and W a n g presented an approach to production lot planning of steel pipe in
iron and steel industries from the point of view of cutting crude pipe [Li and Wang et
al., 1999]. Two objectives were considered. The first was to minimize leftovers and the
second to minimize all over-grade. Since this problem is generally a large scale one, the
traditional search methods have lost their power. The authors applied a genetic
algorithm scheme to obtain the near-optimal solution. The experiments show that the
genetic algorithm is efficient for the planning problem.
In 2000, Tang and Liu et al. presented a system developed and implemented for hot
rolling production scheduling [Tang and Liu et al., 2000]. The project was part of a
large-scale effort to upgrade production and operations management systems of major
iron and steel companies in China. Hot rolling production involves sequence dependent
setup costs. Traditionally the production is scheduled using a conventional method and
the setup cost is very high. They proposed a parallel strategy to model the scheduling
problem and solve it using a modified genetic algorithm. Combining the model and a
man-machine interactive method, a scheduling system was developed. The result from
one year's running in Shanghai Baoshan Iron and Steel Complex shows 20%
improvement over the previous manual-based system. As the company is one of the
largest steel companies in China, the successful application of the scheduling system in
the company sets an example for other steel companies which have potential for
improvement.
Zang and Wang et al. proposed an integer programming model for production
scheduling, using the genetic algorithm based on a repeatable natural scale code and 3-
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 71
mutation operator in 2000 [Zang and W a n g et al., 2000]. To test this model, the
production scheduling of a hot-rolling plant was considered. The numerical analysis has
shown that the model corresponds well to the production process, the solutions thus
obtained are superior to those obtained by the human-machine system, and the proposed
model has proved to be effective.
3.3.3 Rolling MiU Optimisation
In 1997, Nolle and Armstrong et al. introduced their work in the optimisation of the
profile of work rolls in the finishing stands of a hot rolling mill [Nolle and Armstrong et
al., 1997]. Computational intelligence including neural networks and genetic algorithms
were used in the optimisation. The results show that the approach could be used in a
wide range of combinatorial optimisation problems of technical systems. In 1999, they
modeled the finishing train of a hot strip mill by using a constant volume element theory
[Nolle, Armstrong and Hopgood, 1999]. A non-linear rank based genetic algorithm has
been developed for the optimisation of the work roll profiles in the finishing stands of
the hot strip mill. It has been compared with eight other experimental optimisation
algorithms: random walk, hill climbing, and simulated annealing (SA) etc. The work
roll profiles have been optimised by using the proposed approach and the quality of the
strip from the mill has been significantly improved accordingly.
In 2000, Collins and Zhao et al. employed genetic algorithms to tune finite word length
(FWL) PID controllers for fixed-point or floating-point implementation that optimizes a
cost function [Collins and Zhao et al., 2000]. The formulation enables the consideration
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 72
of all sources of F W L errors, i.e., coefficients, input and output quantization effects, and
the round-off error of the arithmetic operations. The results illustrated the effectiveness
of the methodology in developing a FWL PID control law for a steel rolling mill.
3.3.4 Steel Products Design
In 1996, Hirayama and Kajihara et al. presented a new approach to solving the problem
of slab design in the iron and steel industries [Hirayama and Kajihara et al., 1996]. The
problem is a kind of bin-packing problem, which is known as NP-complete problem. To
obtain the optimal solution for slab design problem is very difficult, as it involves many
restrictions. The authors proposed an approach applying a hybrid genetic algorithm
(GA) which was combined with heuristics in order to carry out an effective search and
obtain a near-optimal solution. The main feature of the approach is a coding scheme
designed to combine ordered plates by using heuristics. The results demonstrated that
the hybrid genetic algorithm within certain restrictions was a valuable tool for solving
combinatorial optimisation problems.
3.3.5 Other Fields in Steel Industry
In 1999, Kim and Lee et al. proposed a method to recognize the various defect patterns
of cold rolled strip using a binary decision tree [Kim and Lee et al., 1999]. In classifying
complex patterns with high similarity like the defect patterns of the cold rolled strip, the
selection of an optimal feature set and an appropriate recognizer is important to achieve
high recognition rate. In this literature the GA and K-means algorithm was used to
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 73
select a subset of the suitable features at each node in the binary decision tree. The
feature subset with maximum fitness was chosen and the patterns were classified into
two classes using a linear decision function. This process was repeated at each node
until all the patterns were classified into individual classes. In this way, the classifier
using the binary decision tree was constructed automatically. After constructing the
binary decision tree, the final recognizer was accomplished by a neural network learning
sets of standard patterns at each node. The classifier using the binary decision tree was
applied to the recognition of defect patterns of the cold rolled strip, and the experimental
results were given to demonstrate the usefulness of the proposed scheme.
In 1998, Bureerat and Cooperm reviewed the applications of evolutionary methods for
the engineering systems [Bureerat and Cooperm 1998]. A number of the most common
methods were compared in terms of their philosophical basis and implementation. The
applications of these approaches, including genetic algorithms, evolutionary
programming, evolutionary strategies, simulated annealing and population-based
incremental learning, were illustrated. As a case study, the shape optimisation problem
of a steel plate with buckling and stress constraints was investigated.
There are also numerous applications of GA in other industries such as manufacturing
industry, mining industry, power industry, construction industry, and food industry.
They are reviewed and addressed in Appendix B.
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 74
3.4 SUMMARY AND CONCLUDING REMARKS
This chapter addresses evolutionary optimisation with emphasis placed on Genetic
Algorithms, their mathematical foundation and their engineering applications.
Evolutionary Algorithms include a family of Genetic Algorithms, Evolutionary
Programming, Evolutionary Strategies and Genetic Programming. The background of
Evolutionary Algorithms is a general understanding of Darwinian evolution features.
Evolutionary Algorithms are rather simple heuristic optimisation techniques, which are
well suited for the optimisation of a wide variety of multi-dimensional, ill-defined
functions. Genetic Algorithms are being widely applied in many fields of engineering,
they are well suited for the optimisation of general combinatorial problems. In GA's
evolutionary process, a group of individuals compete with each other to survive and
reproduce. Fitness of an individual controls chance of its sexual reproduction and
chance of selection to the next generation of its offspring. Briefly, the evolution is an
iterated process of reproduction and selection. In the practical applications of GA, a
group of solutions from the domain of the problem is represented by a group of
individuals. A new group of offspring individuals is generated with the search operators,
and then the next generation of solution is selected. In fact, it is the same iterated
process of reproduction and selection, as in natural evolution.
Comparing with the classical optimisation, Genetic Algorithms based optimisation is
often viewed as a global optimisation method especially in the following aspects:
Chapter 3 Principles of Evolutionary Optimisation and Its Applications in Engineering 75
(1) GA works with a coding of the parameter set, not the parameters themselves. The
genetic operations are conducted on the coded string instead of the parameters.
(2) GA has the following specific particularities: parallel search, random variations, and
recombination of already found solutions.
(3) GA is well suited for a wide range of combinatorial and continuous problems.
(4) GA uses objective function information instead of derivatives or other auxiliary
knowledge.
(5) GA based techniques for optimum search allow for incorporation of knowledge of
the environment to be embedded in the algorithm structure, and enable the search to
continue (no breaks) in case of environment changes.
(6) GA uses probabilistic transition rules instead of deterministic rules to guide search.
This chapter also provides a broad overview of GA's applications in steel industry.
There are many successful examples of engineering applications of Genetic Algorithms
in industries. This indicates that GA has a potential in robust searching of an
engineering optimal solution.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 76
Chapter 4
Evolutionary Optimization of Rolling Scheduling for
Tandem Cold Rolling Mills
Scheduling for tandem cold mills refers to the determination of inter-stand gauges,
tensions and speeds for a specified product to be rolled. Optimal schedules should result
in maximized throughput and minimized operating cost. This chapter presents a Genetic
Algorithm (GA) based optimization procedure for the scheduling of tandem cold rolling
mills. The optimization procedure initiates searching from a logical staring point - an
empirical rolling schedule - and ends with an optimum cost. Cost functions are
constructed to heuristically direct the GA's searching based on the consideration of
power distribution, tension, strip shape and rolling constraints. Numerical experiments
have shown that the proposed GA based approach is more promising than those based
on semi-empirical formula. The results generated from a case study show that the
proposed approach could significantly improve empirically derived settings for the
tandem cold rolling mills. This chapter has been published on International Journal of
IF AC, Engineering Applications of Artificial Intelligence, Vol. 13, No. 4, July 2000.
The contents of this chapter are organized as follows: the background and scenario of
the rolling scheduling for tandem cold rolling mills is addressed in the first section. The
basic principles of tandem cold rolling mill scheduling are described in Section 4.2; the
GA based optimization for scheduling is detailed in Section 4.3, followed by numerical
experiments and conclusions.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 77
4.1 INTRODUCTION
Automation systems for the tandem cold rolling mills are continuously being improved
due to today's high throughput, quality and low scrap loss requirements of products. To
consolidate competitive strengths in the global market, many steel companies engage in
maximizing the reduction and minimizing the cost of manufacture [Bryant, 1973; Yuen
and Nguyen, 1996; and Ozsoy et al., 1992]. Rolling scheduling is an important aspect in
the operation of tandem cold rolling mills. It defines stand reductions, tensions, rolling
forces, roll torque, mill maximum speeds, and threading adjustments. Optimized
scheduling should lead to improved thickness and shape performance of the products.
In the last two decades, only a few papers have addressed the rolling scheduling
problem, especially for tandem cold rolling. An early work by Bryant [Bryant, 1973]
has led to the development of mill scheduling systems which achieve correct output and
satisfactory shape in a tandem cold mill. The scheduling is described as a constrained
two-point boundary-value problem that is solved using conjugate gradient and
projection techniques. Cost functions defined for the optimization problem include strip
shape cost, tension cost and thermal-crown cost. Although the results generated from the
optimized schedules are better than that from the original empirical schedule, and the
uniform power distribution is desirable for the tandem cold rolling mills, power cost has
not been considered by other researchers [Bryant, 1973; Dixit and Dixit, 2000; Reddy
and Suryanarayana, 2001]. Moreover, the calculation of the costs mostly relies on some
linear equations of rolling parameters. Although the conjugate gradient methods are
frequently used in practice even when the cost function for the optimization problem is
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 78
not convex, there are reasons to believe that such a use leads to the computation of a
local minimum [Polak, 1971; Luenberger, 1984; and Nash and Sofer, 1996]. The
computing equipment at that time also limited the calculation. Ozsoy, Ruddle and
Crawley [Ozsoy et al., 1992] employed a nonlinear programming method called hill-
climb algorithm to optimize rolling schedules for a hot rolling process. The results show
that although the optimization problem cannot be solved in a closed form because of the
non-linearity of the defining equations and the amplitude constraints on the system
variables, it can be solved numerically on a digital computer by nonlinear programming.
However, the convergence behavior of the nonlinear programming method employed in
the reference [Ozsoy et al., 1992] is directly affected by the initial searching point used.
Some other nonlinear programming methods such as sequential quadratic programming
also have a number of disadvantages besides their added complexity (derivative
calculations etc.), such as the local minimum problem, non-guaranteed convergence,
and expensive calculation cost [Nash and Sofer, 1996]. As an intelligent searching
mechanism, Genetic Algorithms (GA) seem to be flexible and have shown significant
potential in overcoming the above mentioned disadvantages and have been applied in
other fields as well [Holland, 1969; Holland, 1975; Goldberg, 1989; Davis, 1991; Fogel,
1995; Wang and Xu, 1996; Wang and Tieu et al., 1998; Wang and Tieu et al., 2000].
This chapter investigates an optimal scheduling for tandem cold rolling mills. It is the
first time that GA is applied to the optimisation of rolling scheduling for tandem cold
rolling. The scheduling process includes the following steps:
1) A rolling model is set up to establish the relationships among the rolling parameters.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 79^
2) The constraints in the GA based optimization, such as cost functions and mill
capacity constraints, are defined and formulated. The cost functions include the
power distribution cost function, the tension cost function and the optimum flatness
condition. The constraint checks include the roll force and torque limitations, work
roll speed references, strip exit thickness, threading conditions, and tension
limitations. Once the effective cost functions are properly constructed, the optimum
scheduling problem is transformed into a nonlinear optimization problem.
3) The Genetic Algorithms (GA) are used to optimize the schedule. A total cost
function is employed as the fitness function of the GA during the searching for
optimized rolling parameters.
4) The optimal rolling parameters generated from the GA optimization procedure are
further checked against the practical rolling constraints to ensure that the optimal
rolling parameters would not result in unrealistic settings.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 80
4.2 BASIC SCHEDULING FOR TANDEM COLD ROLLING MILLS
The basic procedure for the scheduling of the tandem cold rolling mills is usually based
on experiences, trials or on rules of thumb [Bryant, 1973]. A typical scheduling
procedure for the setup of tandem cold rolling mills is illustrated in Figure 4.1.
Coil & Mill data Material grade Dw,N W,H,h
Normalised Stand reduction patterns
_L" Yes
Entry & Exit gauges Hi, hi
X Inter-stand tension stresses te & tw
X Roll forces Fi
J: Forward slip fi
_c Roll torque Gi
Yes
JL
i<N> 'No
Roll speed cot Yes
£ Power Pi
Yes
i<N> No
^<4<N?. No
Parameters validity check
Figure 4.1 A typical rolling scheduling procedure for the setup of a tandem cold rolling
mill
As an illustration, the following describes the existing semi-empirical scheduling
procedure:
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 81
1) Based on coil data and semi-empirical stand thickness reduction patterns i?,'
(1 < i < N, N is the number of stands), a reduction at each stand is allocated. The
coil data includes strip material grade, strip width W, strip entry thickness H and
strip exit gauge h . The stand thickness reduction patterns are empirically generated
on the basis of providing uniform power distribution and consistent flatness at each
stand. The patterns vary with total mill reduction and last stand conditions, namely,
shot-blast or bright work rolls.
2) According to empirical equations, inter-stand tension (including front tension tf
and back tension tb) are determined. These tensions depend on the strip width,
nominal yield stress deviation of the coil material and the mill exit thickness at each
stand generated in Step 1).
3) Based on the Bland-Ford-Hill force formula, the roll force Pt (1 < i < N) at each
stand is calculated. The formula is a function of variables including the deformed
work roll radius Rw, average yield stress k , the stress state coefficient Q , the front
and back tensions determined in Step 2), the strip width, strip entry and exit gauges
at each stand, and the friction coefficient at each stand.
4) The forward slip ratio ft (\<i<N)at each stand is calculated from the Bland-Ford
forward slip formula, hence yielding the location of the neutral point of roll bite.
5) Based on the torque formula derived by Bryant [Bryant, 1973], the roll torque Gt
(l<i<N) for each stand is determined. The torque formula is a function of the
work roll diameter Dw, angle of contact, the yield stresses at the roll bite entry
k(H) and exit k(h) and the front and back tensions.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 82^
6) According to the constant mass flow principle, the roll speed a)(. (1 < / < N) at each
stand is calculated. The forward slip factor and the motor droop are considered in
the calculation.
7) The power PW1. (1 < i < N) at each stand is estimated based on the results of the roll
speed and the roll torque in Steps 5) and 6).
8) The constraints of the rolling parameters are checked to ensure that all rolling
parameters have not exceeded the mill capability such as the limitations of physical
and design capability of the rolling mill and electrical requirements.
After the schedule is determined, the exit strip thickness and speed of each stand, inter-
stand tension etc. are employed for further calculation of corresponding actuator
references. On the basis of these references, the rolling mill can then be preset to roll the
required product.
4.3 OPTIMIZED SCHEDULING
A schedule is a sequence of elements whose every element is considered as the
aggregate of all important system parameters described as the combination of their
respective cost functions. An optimal sequence is arrived wherever there exists a
minimum mismatch between succeeding sequence elements. Optimum rolling
scheduling deals with the selection of rolling mills' schedule parameters such as the
reductions and motor speeds etc. which lead to the desired thickness, surface finish and
material properties, and shape for a given product at a minimum cost. In fact, these key
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 83
schedule parameters fall in individual intervals of their data space as described in 1) of
Section 4.3.3.1. The objective of the optimal scheduling is to find the optimal presetting
parameter combination. The optimization is essentially a non-linear data sampling
process. Usually the effective cost functions are first defined, then optimization
techniques are employed to search for the optimal solution. An optimization
methodology based on Genetic Algorithms (GAs) is employed here to solve the mill
schedule problem.
4.3.1 Cost Functions
During optimization, one of the constraints should be the "Constant Mass Flow" at each
of the stands. Meanwhile, the optimization should meet the requirements of all
constraints and ntinimize the cost of rolling operation. The cost functions include power
distribution cost function, tension cost function and the perfect shape condition .
1) Power Distribution Cost
One of the most important objectives in the scheduling is to provide uniform power
distribution and consistent flatness at each stand. Hence, the power distribution is a
suitable measurement of how the current schedule meets this requirement. The power
distribution cost function is defined as follows:
Cost _Power = W,YJfj(Pwi - /to)2 (4-1)
/=1 /=2
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 84
where Wx is weighting constant, N is the total number of stands. The power at each
stand is calculated according to the following equation:
Ki^^Gj (4-2)
where co* and G. are the work roll rotational speed and torque at Stand i, respectively.
According to the constant mass flow guideline, Whivl = WHjVt = constant, where W is
the strip width; H. and ht are input and output strip thickness, and Vi and v{ are input
and output strip velocities, respectively. With the forward slip factor (Equation (4-3))
and the motor droop (Equation (4-4)), the work roll speed can be described as Equation
(4-5):
f^Vj/VjRj (4-3)
fi>;=0>,(l +43/(7^) (4-4)
a>lh^R^(l + djGi/Gbasei) (4.5) h, ftRj
where Gbasei is the motor base torque at Stand i, di is the motor droop parameter
representing the fractional change in motor speed per unit base torque change, h5 and
©5 are the strip thickness and work roll speed at stand 5, R5 and f5 are the work roll
radius and the forward slip at Stand 5, respectively. The roll torque formula can be
derived as follows [Bryant, 1973]:
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 85
G(=RPCp +R(tfH-tbh)/2.0 (4-6)
where
L[\(Kh)-tb) + \(k(H)-tf)}
<to=— (4-7) [k(H)-tf] + [k(h)-tb]
K }
L = ylR'(H-hm) (4-8)
hm=h{l-(l-v2)(k(h)-tf)/E] (4-9)
where tf and tb are the front and back tension at the stand, respectively; k(H) and
k(h) are the yield stresses at entry and exit points of the roll bite, respectively; v is the
Poisson's ratio of the strip; E is Young's modulus of the strip; R' is the deformed
work roll radius which is described by Equation (4-15) according to Hitchcock's
equation [Hitchcock, 1935]; and P is the roll force which can be obtained based on
Bland-Ford-Hill force calculation formula [Bland and Ford, 1948; and Bland and Ford,
1952], namely
\IC_
H P=Wk{r\\—Qpnt (4-10)
where k{r) is the average yield stress evaluated at the mean reduction r defined as
Equation (4-12) [BHP, 1979]. Although this formula is proposed 50 years ago, if is still
widely used by steel industry and universities as described in Section 2.1.3.1. For cold
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 86
rolling, the yield stress is largely independent of strain rate and temperature, and a
simple work hardening law relating yield stress to accumulated strain is used. Based on
Equation (4-12), when the strip is rolled at the last stand, the average yield stress is
much greater than that when it is rolled at the first stand, as both the entry and exit
thickness at the last stand are much thinner than that at the first stand.
k{r) = ax{r+a2)ai (4-11)
f = 1 - 0.4/7 / H0 - 0.6/z / H0 (4-12)
where ax and a2 are constants, H0 is called hot band thickness, or the annealed strip
thickness. According to Hill's results,
H-h JR' H-h Qp = 1.08 + 1 . 7 9 M — ^ — ^ — - 1 - 0 2 - ^ - (4-13)
k(r)
cP R' = R{\+ } (4-15)
1• W(H-hy }
c = 16(l-v2roll)/(7rEroll) (4-16)
where A is a constant; vroll is the Poisson's ratio of the work roll; Eroll is the Young's
modulus of the work roll; tf and tb axe the front and back tension of the strip,
respectively; R the work roll radius, and u the friction coefficient.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 87
2) Tension Cost Function
Tension between stands should be kept midway between its lower limit t'fmiD and upper
limit t'fmax [Bryant, 1973]. The lower limit is specified as the maximum value of the
measured noise under mill operating condition; while the upper limit is assigned based
on the consideration of strip skidding and tearing situations. Normally the tension cost
function is described as follows [Bryant, 1973]:
Cost_Tension = W2Y}tn -(fmin +'fmax )]2 (4-17)
i=l 2
where tfi is the tension between two adjacent stands (the i'h and (i + \)th stands), W2 is
the weighting constant of the tension cost function, N is the total number of the stands.
When N = 5, the tension tf5 means the tension between the fifth stand and the coiler of
the mill. For example, the minimum and maximum desired tension forces for stands 1 to
5 of a five stand tandem cold mill are as follows:
Stand 1: minimum 45KN, maximum 450KN
Stand 2: minimum 30KN, maximum 350KN
Stand 3: minimum 25KN, maximum 300KN
Stand 4: minimum 20KN, maximum 250KN
Stand 5: minimum 20KN, maximum 40KN
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 88_
3) Perfect Shape Condition
Good rolling scheduling should lead to uniformity of the tension distribution in the strip
width direction or minimal possibility of strip buckling during rolling. The optimum
shape condition is obtained when the strip is uniformly rolled across the strip width
(neglecting the insignificant lateral spread in cold rolling). This means that the deformed
roll profile should geometrically and perfectly match the incoming transverse strip
thickness profile [Sabatini and Yeomans, 1968; Yuen and Wechner, 1990; and
O'connor and Weinstein, 1972]. For example, the input profile of the incoming strip
may be described as follows [Sabatini and Yeomans, 1968]:
H(x) = Hc+Pxx2+Qxx
4 (4-18)
The reason to choose the model with x2 and x4 in the equation (4-18) is that the strip
to be rolled and after rolled is usually symmetrical about its centerline. So neither x nor
x3 is considered in the model. The parameters Hc, Px, and Qx are determined by
measured strip thickness profile. Where H(x) is the entry thickness of the strip at a
distance x from the strip centerline, Hc is the entry thickness of the strip at the strip
centerline, and Px and Qx are constants. The output profile is shown in Figure 4.2 and
described as follows:
h(x) = hc+P2x2+Q2x
4 (4-19)
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 89
where h(x) is the exit thickness of the rolled strip at a distance x away from the strip
centerline, hc is the exit thickness of the strip at its centerline, P2 and Q2 are constants.
t
hc >
>
Y
< . ^ _ ^ _ — — — •
V
h(x) , X
i
W/2 * >
X
Figure 4.2 Transverse profile of rolled strip
It can be shown that the condition for perfect shape requires [Sabatini and Yeomans,
1968]:
P2=Px(l-r)
Qi=QxQ-r)
(4-20)
(4-21)
where r is the thickness reduction.
Thus, if the profile of the work rolls exactly matches the incoming transverse gauge
profile of the strip geometrically, perfect strip shape will be obtained. So the cost
function for perfect shape can be described as follows:
Cost_Shape =W,Y?\ [yJL^-C^xj-Hxifdx i=\
(4-22)
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 90
where Wi is weighting constant; yw{x) is the deformed roll profile, which is the
function of roll force and roll bending force if used. It is calculated by summing the
deflections due to bending, shear and work roll flattening. Cw (x) is the total work roll
crown including machined crown, thermal crown and roll wear. For mills with roll
bending jacks, the deformed roll profile could be geometrically adjusted by changing
the jack force, which would be reflected in the term yw(x). The deformed work roll
profile is obtained by calculating the roll deflections due to bending, shear, effect of
Poisson's ratio, bending moment, interference between work and backup rolls, and
interference between work roll and strip, which are described in the following
paragraphs.
(a) Roll deflection due to bending
Beam theory for the bending and shear components has been widely employed to
calculate the roll deflections [Bryant, 1973; Yuen and Wechner, 1990; O'connor and
Weinstein, 1972; and Cresdee et al, 1991]. A typical roll deflection model under the
effect of point loads is shown in Figure 4.3.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 91
2L
q(z)
P(z)
x
y i Figure 4.3 Deflection of the neutral axis due to point loads
The roll deflection of the beam under the effect of bending at a position x can be
described as follows:
y(x) = [q(z)- p(z)](L-z)2[3(L-x)-L + z]l(6EI) x<z
[q(z) - p(z)][(x - z)3 - (L - yf (3x - y - 2L)]/(6FJ) x > z (4-23)
where E is the Young's modulus, / the second moment of area, p(z) and q(z) point
loads.
(b) Roll deflection due to shear
According to O'connor [O'connor and Weinstein, 1972], the deflection of the neutral
axis for short stubby beams due to shear is given by:
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 92
'4(L-z)[q(z)-p(z)]/(3AJ) x<z y(x) = < (4-24)
\4(L-x)[q(z)-p(z)]/(3AJ) x>z
where A is the cross-sectional area and J the shear modulus of the beam.
(c) Roll deflection due to a bending moment
If there is a bending moment, the deflection of the neutral axis can be illustrated in
Figure 4.4 and expressed as follows:
y(x) = M[(L - x)2 + vR(x)2 ] l(2EI) (4-25)
where v is the Poisson's ratio, and M the bending moment.
(d) Roll deflection due to the effect of Poisson's ratio
The deflection of the roll neutral axis due to the effect of Poisson's ratio on the
movement of the surfaces is given by [O'connor and Weinstein, 1972:
v(x) = l ° X~Z (4-26) \vR
2[q{z)-p{z)]{x-z)l{2EI) x> z
where R is the work roll radius.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 93
2L — >
M
1)-* Figure 4.4 Deflection due to a bending moment
(e) Interference between work and backup rolls
Based on the assumption of two infinitely long elastic cylinders in contact, the
interference under inter-roll pressure q(x) can be described as follows [Bryant, 1973;
Yuen and Wechner, 1990; and Cresdee et al., 1991]:
y„b(x) = q(x)(Cw +CB)\n{e2,3(Dw + DB)/[2q(x)(Cw +CB)]} (4-27)
where ywb(x) is the interference between work and backup rolls; subscripts w and B
refer to the work roll and backup roll, respectively; Cw and CB are determined by the
following equation:
M
y
C„=C.=(l-v2 )I(KE) (4-28)
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 94
where v is the Poisson's ratio.
Based on the contours of the rolls, the interference between the work roll and backup
roll can be calculated as follows [Bryant, 1973; and Cresdee et al., 1991]:
ywb (*) = y wb (o) + yb (*) - y w (x) -cB-cw (4-29)
where ywb{o) is the total roll interference at the strip center, yb{x) the total backup
roll-axis deflection, and CB(x) the total backup roll crown including ground crown,
thermal crown and roll wear, yw (x) is the total work roll-axis deflection, and Cw (x) the
total work roll crown including ground crown, thermal crown and roll wear.
(f) Work roll flattening
Work roll flattening at the contact area of work roll and strip can be described as follows
[Bryant, 1973]:
yws (*) = ft + h p(x)]yH (x) 0<x<W/2 (4-30)
where W is the strip width, and yH (x) is given by
yH (x) = 2p(x)cw ln{e2,3Dw l{2Cwp{x))} (4-31)
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 95
where p(x) is the roll gap pressure, bx and b2 are constants determined by experiments
[Bryant, 1973]. For mild steel (0.1 - 0.25% C), bx and b2 are estimated as
32.92mm2 /kN and 0.86mm2 /kN, respectively. When rolling tinplate, the strip can
be very thin and the deflection of work rolls can be sufficient to result in work roll
contact beyond the edges of the strip. Nowadays the roll bending systems are employed
to separate work rolls from touching each other beyond the edges of the strip. The
interference between work rolls, y w (x) , can then be calculated as follows:
ymv(x) = yws(o)-h(o) + 2[yw(x)-Cw(x)] WI2<x<LI2 (4-32)
where L is the work roll width, h(o) is the exit strip thickness at the strip center.
F/2 F/2
> f
/ iir: q(x)
"~-r~-_
im 0.
k*=£E pi(x)
'ttttt P(x)
n o tti pr(x)
'
X
Figure 4.5 Interference between work rolls beyond the edges of the strip
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 96
In the above figure, the left and right hand sides beyond the edges of the strip being
rolled are named roll edge contact region. p,(x) and pr(x) are contact pressures
between work rolls at the left and right hand contact regions, respectively.
(g) Total roll deformation
The total roll deformation can be obtained by summing the deflections given by
Equations (4-23), (4-24), (4-26), (4-29), (4-30), and (4-25) if a bending moment exists.
4) Total Cost Function
In the above cost functions (Equations (4-1), (4-17) and (4-22)), the power is a function
of the roll torque and speed. The roll torque is related to roll force, which is in turn a
function of the entry- and exit-strip tension, and input- and output strip thickness. So the
optimization problem addressed here involves combinations of numerous variables. A
method named Totally Ordered Set (TOS) is employed to determine the final format of
total cost function. TOS simply means that there is a full list of the preferences of all
costs and/or constraints. The priority of each constraint is measured by a weight.
Individual costs are normalized to a value between 0 and 1 before the total cost is
calculated. In this study, the complete cost function is given as follows:
cost = wxYJ(Pw-PWJ)2 + w22[^-(-^V^
)]2 +w^2l iyM-cM-Kxtfdx
(4-33)
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 97
where Wx is the weighting constant for power distribution cost defined in Equation (4-
1), W2 the weighting constant for tension cost described in Equation (4-17), and W3 the
weighting constant for strip shape cost in Equation (4-22). Wx, W2 and W3 are
determined based on the significance of individual rolling conditions. A large value of
Wx means that dominant consideration is given to uniform power distributions for
maximizing the throughput of a rolling mill. If a thin strip is rolled, W2 should be
assigned a large value to make sure neither skidding nor tearing occurs. Strip shape is
always an important factor to be considered for final product quality. So W3 should be
assigned to an as large as possible value compared with Wx and W2 such that the
condition Wx + W2+W3=\ satisfies. As an example, W3=0A, Wx=0.3 and W2=0.3.
4.3.2 Constraint Checks
1) Roll Force and Roll Torque
The total roll force and roll torque are limited to the corresponding maximum values
due to the mechanical design limit with the manufacturer of the rolling mill and
electrical drive motors. This constraint can be described as follows:
P. < P (4-34) i max v '
G. < G (4-35) w i max v '
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 98
2) Strip Exit Thickness
The exit strip thickness (h., 1 < i < 5) at the exit of Stand i is kept within its upper and
lower limits as follows:
h . <h<h (4-36) mm — i — max
The bounds hmin and hmax are determined based on the constraint check requirements of
the rolling mill. They are physically determined by the mechanical design limit of the
mill, as screw positions on adjacent stands should not differ by more than a certain
amount (for example, 1.5 mm for a local tin mill) and should lie between a range (for
example, 0.5 to 9.5 mm for the local tin mill). This constraint check is used to ensure
that operator adjustments do not result in an outrageous set of screw positions on
adjacent stands.
3) Tension
The tension at each stand should not exceed its upper and lower limits.
4mi„ * '/» * 'U (4"37)
The t'r is determined based on a lower limit on the tension set by measurement noise j mm
so that the strip does not loop between two neighboring stands. The t'fmax is set by the
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 99_
strip tearing and skidding considerations. The t'fmax is usually assigned to a value of
about one-third of the yield stress of the strip.
After the construction of the cost functions and constraints, the schedule optimization
problem may be described as: starting from an initial searching point {hx,h2,h3,h4,h5} ,
find out a satisfactory combination of the strip exit gauges at minimum cost without
violating the constraint limitations.
4) Mill Shudder
Nowadays, significant advances in technology have allowed rolling mills to be
scheduled to operate at relatively high speeds to increase productivity and efficiency.
However, the speed increases may be accompanied by mechanical vibrations, generally
referred to as chatter or shudder. This causes periodic, transverse, bands of light and
dark appearance across the strip. In some cases, a matching thickness variation is
associated with the "chatter bands". This phenomenon is highly undesirable. Not only is
the affected product unacceptable, but strip breaks may lead to damage in the mill
equipment. Usually it has been found that as mill speeds are increased the vibrations
become more severe. Thus, at present, the only reliable remedy for chatter is to reduce
the operating speed of the rolling mill which adversely affects mill productivity
[Meehan et al., 1998].
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 100
To stabilize the rolling operation, rolling schedules should meet the criterion for
avoidance of sensitive and critical rolling speeds of a tandem mill and prevent the
occurrence of self-excited chatter.
4.3.3 Genetic Algorithm (GA) Based Optimization
4.3.3.1 Genetic Encoding and Operation for Rolling Scheduling Problem
The Genetic Algorithm was firstly proposed by Holland as a heuristic search mechanism
for intelligent systems. Although it has been growing since the early 1970s [Holland,
1969; and Holland, 1975], only recently has its commercial potential been demonstrated
[Goldberg, 1989; Davis, 1991; and Fogel, 1995]. The main operations in GA include
parent selection, reproduction, crossover, and mutation. The aim of these operations is
to generate meaningful offspring. A typical step by step procedure for a GA is described
as follows [Wang and Xu, 1996]:
(a) Define fitness function for optimization problem, e.g. Equation (4-33).
(b) Encode variables into binary codes, e.g. hx, h2, h3, h4, then combine individual
binary code for each variable together into a binary chain as a combined single
variable.
(c) Initialize population as the population for the first generation.
(d) Evaluate each individual in population.
(e) Genetic operations: reproduction, crossover, and mutate.
(f) Rank the population.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 10J_
(g) Delete lowest-ranked genomes in the population and keep highly ranked individuals.
(h) Repeat (e) until the evolutionary termination criterion is satisfied.
1) Genetic Encoding of Rolling Scheduling Problem
In the optimization of the rolling schedule, as the exit thickness at the last stand is fixed,
for a 5 stand tandem rolling mill, only the exit gauges at the first four stands hx,h2,h3,h4
need to be encoded. For such a multiple variable encoding problem, each variable
should be encoded first and then linked together as a chain. For example, suppose a steel
strip with an initial gauge of 2.5 mm is to be rolled to 0.5mm, with the following exit
gauges based on the empirical reduction patterns:
&, =1.874, h2=l.2l4, h3 =0.853, h4 =0.623, A5=0.50 (4-38)
If the average of two adjacent exit gauges is taken as the boundary between the possible
value fields of these two exit gauges. For example, 1.54 is the average value of
scheduled exit thickness at Stands 1 and 2, which is equal to rounded value of
(1.874+1.214)/2. The possible value fields for the exit gauges are as follows:
1.54< hx <2.5, 1.03 < h2 < 1.54, 0.74 < h3 <1.03, 0.56 < h4 <0.74, /z5=0.50 (4-39)
If the granularity for the exit gauge at each stand is 0.01, the total number of binary bits
representing hx should be 7, since 26 < (2.5-1.54)/0.01 < 27; 6 bits for representing h2
since 25 < (1.54-1.03)70.01 < 26; 5 bits for representing h3 since 24<(1.03-
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 102
0.74)/0.0T < 25; and 5 bits for representing h4 since 24 <(0.74-0.56)/0.01< 25. The
binary bits representing each exit gauge are finally linked together as a binary chain. So
the total bit number of the final chain is 23.
The initial population is usually chosen at random or by a heuristic approach. In order to
get the best solution, as many points in the search space as possible should be searched.
Hence some variations are introduced into the new population by genetic operations,
which include reproduction, crossover, and mutation.
2) Genetic Algorithms Operation
To maximize the possibility of an improvement achieved by a new population,
reproduction is employed to select some parent binary strings (also called
chromosomes) with high fitness values. This means that if a binary structure represents
a point of a high-performance area of the search space, it may lead to further exploration
in this area of the search space. The selected individuals are then duplicated and go to
the next generation. The most important genetic operation is crossover. By the crossover
operation, two binary structures in the new population exchange portions of their binary
bits. This is implemented by randomly choosing a crossover point, and exchanging the
segments to the right of the crossover point. Also, multiple crossover points could be
randomly assigned to get higher evolution efficiency. To avoid convergence to a local
minimum area during searching, mutation is employed to make the search jump outside
the local minimum area. This is implemented by changing the binary value 1 to 0 or 0 to
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 103
1 at a randomly selected bit of a binary representation. The whole genetic operation is a
systematic recombination process of the binary structure.
3) Evaluation
A fitness function is used to measure the performance of each binary string. How close a
binary string is to the final optimization destination is judged by its corresponding
fitness value and whether the rolling parameters resulted from it violate the constraints
described in Section 4.3.2. The total cost function (Equation (4-33)) is employed as the
fitness function. All binary strings are ranked based on how small their fitness values
are. The higher it is ranked, the smaller its fitness value is. The top 20 binary strings will
be selected as the candidates for the next generation. The whole evolution process
ceases when the termination criterion is met. The termination criterion in this
optimization is 5000, the total number of trials. If the trial number is too small, the
evolution process may cease before an optimal result is achieved. However, if it is too
large, the evolution process will continue even the optimal result has already obtained.
This will waste computing time without any further improvement on the result.
Numerical experiments [Wang and Xu, 1996; Wang and Tieu et al., 1998; Wang and
Tieu et al., 2000] show this proposed termination criterion is suitable for the addressed
optimization.
The general genetic model has been constructed in C/C++. Numerical calculations were
carried out on Pentium 200, and Pentium II 266 and 400 PCs. The whole optimization
process took 33 seconds on the Pentium II 400MHZ PC, 43 seconds on the Pentium II
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 104
266MHZ PC and 99 seconds on Pentium 200MHZ PC. The final optimization result is
presented in the following section.
4.3.3.2 Optimization Parameters and Results
1) Optimization Parameters
The evolution parameters are determined based on previous experience and experiments
considering the searching efficiency and precision of the solution. As described in
Chapter 3, if the number of trials is too small, it may result in a premature termination of
the optimization process. However, it is too large, the optimization process will continue
until the number of trials is reached even if the optimal result has already been achieved.
The length of binary structure is determined by the precision requirement of the
solution. In this study, the total number of trials is 5000 for this optimization procedure.
The initial population size is assigned to 20, which means the number of initial
randomly generated binary strings is 20. The binary structure length is 23. The
crossover and mutation rates are set to 0.65 and 0.005, respectively.
To demonstrate the different rolling schedules for different considerations on the
significance of power, tension and shape, different weighting constants are assigned to
power, tension and shape costs. A greater weighting constant means more significance
is considered. If the equal weighting is applied to power, tension and shape costs, the
same significance is considered on the costs. For example, if the tension and shape are
more important than power distribution, then the weighting coefficients may be
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 105
assigned as follows: Wx =0.1 (power distribution), W2 = 0.45 (tension), and W3 = 0.45
(shape). The numerical simulation is conducted under the following three conditions:
(a) The coefficients of the cost function are assigned to 0.4 for Wx, 0.2 for W2, and 0.4
for W3,
(b) The coefficients of the cost function are assigned to 0.35 for Wx, 0.3 for W2, and
0.35 for ^3;and
(c) The coefficients of the cost function are assigned to 0.3333 for Wx, 0.3333 for W2,
and 0.3333 for W3.
In the numerical calculation, the assumed rolling condition is that: the work roll
diameter is 500mm, backup roll diameter 1300mm, roll face width 1426mm, strip width
960mm, strip entry gauge 2.5mm, and the final strip exit gauge 0.5mm.
2) Optimization Results
The optimized results are listed in Table 4.1 in comparison with the empirical method
based rolling schedule. The distributions of roll force, power, work roll speed, tension
force and torque at each of all 5 stands are shown in Figures 4.6 to 4.10 for the
comparison purpose between the schedules based on the empirical method and GA
based optimization approach. The data describing the costs of power, tension and shape,
and total cost in Table 4.1 has been normalized based on the power, tension and shape
costs generated from the empirical schedule. The smaller the cost value, the better has
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 106
the rolling schedule been obtained in terms of that cost. For example, in the optimized
schedule 1, the cost of power is 0.6673, which is the smallest among all schedules listed
in Table 4.1. This means the optimized schedule 1 is the best solution in terms of
uniform power distribution. The cost of tension is 1.0516, which is the largest among
the costs of tension for all schedules. This indicates the tension cost generated from the
optimized schedule 1 is greater than that from the empirical schedule.
Table 4.1 Comparison of empirical rolling schedule and optimized schedules
Rolling Parameters
Exit
gauge
(mm)
Cost oi
Std. 1
Std. 2
Std. 3
Std. 4
Std. 5
"power
Cost of tension
Cost of shape
Total cost
Empirical schedule
1.874
1.214
0.853 0.623
0.500
1.0 1.0 1.0 1.0
Optimized schedule 1
WX=0A,W2=0.2,
W3=0A
1.715
1.223
0.894
0.644
0.500
0.6673
1.0516
0.9963
0.8758
Optimized schedule 2
^=0.35,1^2=0.3,
W3=035
1.778
1.248
0.915
0.662
0.500
0.8980
0.6880
0.9981
0.8700
Optimized schedule 3
JVX = W2 = W3=0.3333
1.792
1.257
0.925
0.668
0.500 0.9714
0.6069
0.9986
0.8590
10 Roll force (MN)
1 2 3 The semi-empirical schedule
The optimised schedules
Stand No. (1-5)
Figure 4.6 Roll forces for different schedules
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 107
2000
1500
1000
500
Power (KW)
1 2 3N
The semi-empirical schedule
r- -J^,-
The optimised schedules
2 3 4
Stand No. (1-5)
Figure 4.7 Power distributions for different schedules
800
600
400
200
Work roll speed (RPM)
Stand No. (1-5)
Figure 4.8 W o r k roll speeds for different schedules
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 108
250 Tension force (KN)
200'-
150
100
Stand No. (1-5)
Figure 4.9 Tension forces for different schedules
60 Torque (KNM)
The semi-empirical schedule
Figure 4.10 Torque at each stand for different schedules
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 109
As defined in Equation (4-2), power is the product of the work roll rotational speed and
torque. Figure 4.8 shows the work roll speeds for all five stands under different
schedules including the semi-empirical schedule, and the three optimized schedules.
Figures 4.7 and 10 demonstrate the power distribution and torque for each stand under
the different schedules. From Figure 4.7, conclusions can be drawn that the power cost
under the optimized schedule 1 is the lowest among four schedules including the semi-
empirical schedule. This is also reflected in the cost of power shown in Table 4.1. The
roll force shown in Figure 4.6 is an important factor during rolling in addition to the
results for the power, the tension and the strip flatness. Figure 4.9 shows that the tension
distributions under the three optimized schedules are likely to be kept midway between
the lower and upper limits for each neighboring pair of the mill stands, which is also
numerically demonstrated in Table 4.1.
When the parameters for the optimized schedule 3 are applied, the deflections of
work/backup rolls along the roll barrel against roll force are shown in Figures 4.11 and
4.12, respectively. Figures 4.13 and 4.14 show the deflections of backup roll along the
backup roll barrel due to bending and shear, respectively. Figure 4.15 shows the work
roll/backup roll flattening, and Figure 4.16 illustrates the deflection of backup roll due
to the effect of Poisson's ratio. The deflections of work roll due to bending and shear are
shown in Figures 4.17 and 4.18, respectively. Figure 4.19 demonstrates the work
roll/strip flattening, and Figure 4.20 shows the deflection of work roll due to the effect
of Poisson's ratio along the work roll barrel.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 110
Total Deflection of Work Roll
60 70
Roll Force, 0.5 x M N 0 0 Coordinate Across the Strip, 10 x m m
Figure 4.11 Relationship between deflection of work roll and roll force
Total Deflection of Backup Roll
Roll Force, 0.5 x M N 0 0 Coordinate Across the Strip, 10 x m m
Figure 4.12 Relationship between deflection of backup roll and roll force
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 111
Deflection of Backup Roll Due to Bending
0 0 Roll Force, 0.5 x M N Coordinate Across the Strip, 10 x m m
Figure 4.13 Deflection of backup roll due to bending
Deflection of Backup Roll Due to shear
0 0 Roll Force, 0.5 x M N Coordinate Across the Strip, 10 x m m
Figure 4.14 Deflection of backup roll due to shear
Chapter 4 Evolutionary Optimization of Rolling Scheduling for T a n d e m Cold Rolling Mills 112
Work Roll/Back-up Roll Flattening
0 0 Roll Force, 0.5 x M N Coordinate Across the Strip, 10 x m m
Figure 4.15 Work roll/backup roll flattening
Deflection of Backup Roll Due to Effect of Poisson Ratio
Roll Force, 0.5 x M N 0 0
Coordinate Across the Strip, 10 x m m
Figure 4.16 Deflection of backup roll due to the effect of Poisson's ratio
Chapter 4 Evolutionaiy Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 113
Deflection of Work Roll Due to Bending
Force, 0.5 x M N 0 0 Coordinate Across the Strip, 10 x m m
Figure 4.17 Deflection of work roll due to bending
Deflection of Work Roll Due to shear
Coordinate Across the Strip, 10 x m m
Figure 4.18 Deflection of work roll due to shear
Chapter 4 Evolutionaiy Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 114
Flattening at Work Roll/Strip
Roll Force, 0.5 x M N 0 0 Coordinate Across the Strip, 10 x m m
Figure 4.19 Flattening at work roll/strip
Roll Force, 0.5 x MN Coordinate Across the Strip, 10 x m m
Figure 4.20 Deflection of work roll due to the effect of Poisson's ratio
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills U 5
4.3.3.3 Discussion of the Results
As illustrated in Table 4.1, optimized schedules result in lower cost of power, more
uniform distributions of roll forces at the different stands and better shape. The tensions
are more likely to be kept midway between upper and lower limits. For example, the
power distribution cost value generated from the GA approach is reduced by about
33.4% in comparison with the schedule based on semi-empirical formula when the
weight coefficient (Wx) of power cost is assigned to a value of 0.4. The cost of power is
increased from 0.6673 to 0.9714 when the weight coefficient is assigned to a smaller
value, from 0.4 to 0.3333. The cost of tension is significantly reduced when the weight
coefficient {W2) of the tension cost function is changed from 0.2 to 0.3 and then 0.3333.
The cost of tension in the optimized schedule 1 is slightly greater than that in the
empirical schedule. This is the result of a small weight coefficient of the tension cost
function of 0.2, which means less consideration is paid on the tension distribution. With
the change of the weight coefficient of the shape cost function from 0.3333 to 0.4, the
shape cost is slightly decreased from 0.9986 to 0.9963. Although the cost is slightly
reduced, this is an important move towards a better shape. The optimized schedules
shown in Table 4.1 demonstrate that the weight coefficient can be assigned to a certain
value based on a different consideration of the significance of the shape, power and
tension distributions. Generally, the weight coefficients of the power, tension and shape
costs should be determined based on the significance consideration given to the shape,
power and tension distributions.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 116
Figure 4.6 shows that the roll force distributions under three optimized rolling schedules
are more uniform than that from the empirical schedule. Roll force under the empirical
schedule at Stand 2 is much higher than the roll force under the optimized schedules.
For example, it is about 9.46% higher than the roll force under the optimized schedule
1. A uniform roll force distribution is desirable for a tandem cold mill [BHP, 1979].
This will significantly increase the roll bearing life and reduce the roll wear at Stand 2,
and, therefore, benefit both rolls and roll bearings. Table 4.1 and Figure 4.7 show that
the overall power cost under three optimized schedules is lower than that under the
empirical schedule. This indicates that lower power consumption has been achieved and
lower production cost has been obtained accordingly. Figure 4.10 shows that the torque
at Stand 2 under the empirical schedule is much higher than that under the optimized
schedules. For example, it is 31.6% higher than that under the optimized schedule 3. A
lower torque will be beneficial to the components of mechanical devices and reduce the
chances of malfunctions of the components accordingly.
Figure 4.11 shows the relationship between total deflection of work roll and roll force.
Along the width direction of the work roll, the deflection is the parabolic function of the
distance from the left edge of the strip. The total deflection is the sum of the deflections
due to bending shown in Figure 4.17, shear shown in Figure 4.18, flattening shown in
Figure 4.19 and the effect of Poisson's ratio shown in Figure 4.20. The total deflection
of backup roll is shown in Figure 4.12. It is obtained by adding the deflections due to
bending shown in Figure 4.13, shear in Figure 4.14, roll flattening in Figure 4.15 and
the effect of Poisson's ratio in Figure 4.16. As shown in Figures 4.11 to 4.12, the roll
deflections are the functions of both roll force and coordinate across the strip, i.e.
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 117
distance from the left edge of strip. Based on the perfect shape condition, only when the
roll profile exactly matches with the strip profile, good shape can be obtained. So a
proper roll force should be determined and exerted so as to form the work roll profile
most likely matching with the strip entry profile. The shape cost function defined in
Section 4.3.1 is exactly based on this principle. It should be pointed out that strip
profile, work roll machined crown, wear and thermal crown should all be taken into
account when coil to coil rolling scheduling is conducted as they are directly relevant to
the strip shape.
The results shown in Table 4.1 demonstrate that the optimized schedule 3 in Table 4.1 is
the best among all four schedules including the empirical schedule. The total cost is the
lowest (only 85.9% of the cost of the empirical schedule) when equal weight
coefficients WX=W2= W3 is considered. In fact, three weight coefficients can be
assigned differently based on specific requirements in industry. For example, in a cold
mill, an appropriate set of weight coefficients may be 0.15 for power, 0.45 for tension
and 0.4 for shape.
The empirical rolling reduction schedule is used as a starting point of GA's searching.
This potentially reduces the solution searching space and time. The optimized results
show that the empirical schedule is a reliable starting point for GA's heuristic searching.
To verify the searching capability of Genetic Algorithms in the rolling schedule
optimizations, optimizations have also been conducted based on a perturbed empirical
schedule which is obtained by changing the exit gauges for Stand 1, 2, 3 and 4 by 10%
of their original values, respectively. The perturbed empirical schedule is as follows:
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 118
A, =2.061, A2 =1.335, h3 =0.938, h4 =0.685, h5=0.50 (4-40)
As described in Section 4.3.3.1, taking the average of two adjacent exit gauges as the
boundary between the possible value fields of these two exit gauges, the possible value
fields for the exit gauges for Stands 1, 2, 3, 4 and 5 are as follows:
1.70< hx <2.5, 1.14< h2 < 1.70, 0.81 < h3 <1.14, 0.59< h4 <0.81, A5=0.50 (4-41)
The optimization results are demonstrated in Table 4.2. Comparing with the optimized
rolling schedules based on the original empirical schedule, the variations of the
optimized results based on the perturbed empirical schedule are shown in Table 4.3. The
maximum variation of the exit thickness is 0.008, and the maximum variation of the
cost is 0.0502. The optimization results show that the GA employed has a good
performance and is robust in searching optimized rolling schedules.
Table 4.2 Optimized schedules based on the perturbed empirical schedule (4-40)
Rolling Parameters
Exit
gauge
(mm)
Cost ol
Std. 1
Std. 2
Std. 3
Std. 4
Std. 5
'power
Cost of tension
Cost of shape
Total cost
Empirical Schedule
1.874
1.214
0.853
0.623
0.500
1.0 1.0 1.0 1.0
Optimized schedule 1
WX=0A,W2=0.2,
W3=0A
1.708
1.223
0.894
0.644
0.500
0.6537
1.0791
0.9961
0.8758
Optimized schedule 2
^=0.35,^=0.3,
W3=0.35
1.782
1.256
0.92
0.668
0.5 0.9424
0.6378
0.9983
0.8706
Optimized schedule 3
WX = W2=W3 =0.3333
1.794
1.256
0.925
0.668
0.5 0.9759
0.6023
0.9986
0.8589
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 119
Table 4.3 Variations of the optimized schedules based on the perturbed empirical
schedule (4-40)
Rolling Parameters
Exit
gauge
(mm)
Cost o:
Std. 1
Std. 2
Std. 3
Std. 4
Std. 5
"power
Cost of tension
Cost of shape
Total cost
Variations for optimized schedule 1
^= 0 . 4 , ^ = 0 . 2 ,
W3 =0.4
-0.007
0 0 0 0
-0.0136
0.0275
-0.002
0
Variations for optimized schedule 2
^=0.35,^2=0.3,
W3=035
0.004
0.008
0.005
0.006
0 0.0444
-0.0502
0.0002
0.0006
Variations for optimized schedule 3
WA = W2 = W3=03333
0.002
-0.001
0 0 0
0.0045
-0.0046
0 -0.001
The approaches proposed are still applicable if some rolling conditions are changed. For
example, different initial thickness of strip, different exit thickness at the last stand of
the mill, different material grades, and different friction conditions due to different
lubrications. However, for very thin strip rolling, for example, the final strip thickness is
less than 0.2mm, some formulae employed in this chapter may need to be revised to suit
the thin strip rolling conditions. Please refer to [Fleck et al., 1987 and 1992] and Section
7.2 of Chapter 7 for more details. Foil rolling is beyond the scope of this thesis.
4.4 SUMMARY AND CONCLUDING REMARKS
A n optimal rolling scheduling method is introduced which aims at achieving uniform
power distribution, maximum safe level of strip tension and good flatness. The results
generated from the case study show that the proposed GA based optimization approach
Chapter 4 Evolutionary Optimization of Rolling Scheduling for Tandem Cold Rolling Mills 120
has the potential to significantly improve the empirical schedules for tandem cold
rolling mills. With the GA based optimization procedure, a considerable cost reduction
can be obtained. The numerical evaluation also shows that the GA based evolutionary
searching is efficient and with extension potential for on-line scheduling purpose. The
power distribution of the tandem cold rolling mill is more uniform than those generated
from the experience-based scheduling, and the overall power cost is also reduced under
the optimized schedules. Moreover, the shape for optimized schedules is likely to be
improved. The tension under the optimized schedules 2 and 3 in Table 4.1 is more likely
to be kept midway between the upper and lower limits. Numerical experimental results
show that the proposed approach is efficient and effective for solving rolling scheduling
problems.
The optimized results show that the empirical schedule is a reliable starting point for
GA's heuristic searching in terms of reducing the solution space and searching time.
This is applicable to other engineering problems where empirical engineering solutions
can be used as reliable heuristic knowledge to guide GA's searching for optimal
solutions in neighborhood spaces of the empirical solutions. Thus, the original solutions
can be potentially improved.
Future work will focus on the on-line adaptation capability due to disturbances in rolling
conditions such as threading, tailing out, and the passing of a weld through the mill, and
of strip properties such as entry thickness and hardness.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 121
Chapter 5
Shape Analysis and Optimization in Tandem Cold
Rolling
Chapter 4 addresses the optimization of a rolling schedule of a tandem cold mill. This
chapter investigates a theoretical predictive model for analysis of strip shape during cold
rolling. The model can be used to deal with the set up of both rolling force and roll
bending force. To obtain the best shape, the bending forces and/or rolling forces are
optimized. The models are flexible to deal with rolling mills such as those with work
roll side-shifting systems and variable crown rolls. A numerical simulation has been
conducted based on typical plant data. The proposed model is also used to optimize
shape problem during threading by using optimal roll ground crowns or an optimum roll
bending force to minimize threading delays caused by poor shape.
5.1 INTRODUCTION
Tighter tolerance requirements for strip products with uniform thickness and good shape
have placed increased research emphasis on the development of effective systems for
both strip gauge and flatness control in the steel industry. Shape is an important aspect
of the dimensional accuracy of strip transverse thickness (crown) [Shohet and
Townsend, 1971; Guo, 1990; Cresdee et al., 1991; and Yuen and Miller, 1993] to which
flatness is directly related. Typical shape defects include edge buckle (left and right
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 122
slopes) and center buckle. The edge buckle is caused by over-rolling the strip edges as
shown in Figure 5.1. Conversely, as a result of over-rolling the center of the strip, the
center buckle will be produced as illustrated in Figure 5.2. Bad shapes may damage rolls
and cause safety problems. It also results in threading delay and negatively affects
production. For example, in a local plant, the average number of coil rolled a day is
about 80. Average threading delay per coil is about 0.6-0.8 minutes. If it takes 12-15
minutes to roll one coil, four more coils could be rolled per day if there is no threading
delay. When the threading delay occurs, temporary rectifying procedures include (1)
stop the rolling mill, (2) cut the buckled nose of strip, and then (3) re-thread the strip
until the strip is successfully threaded through the last stand of the mill.
Usually there are several means to control shape, the conventional shape controls
include:
(1) Asymmetric change in the screw positions on both sides of rolls.
(2) Adjustment of roll bending pressure through roll bending systems as shown in
Figure 5.3, where F, is the roll bending (jack) force.
(3) Change of roll coolant distribution (segmented headers) on the last stand.
(4) Continuously Variable Crown (CVC) Roll (side shifting rolls as illustrated in
Figure 5.4).
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 123
Figure 5.1 Edge buckle (long-edge shape)
Figure 5.2 Center buckle (long-middle shape)
F, " " - "
" Fj >'
Figure 5.3 Roll bending systems
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 124
^^^^^^=5 (a)
(c)
Figure 5.4 CVC rolls with the roll shifting (a) neutral crown; (b) positive crown; and
(c) negative crown
To determine the most effective way for the shape control, efficient models for the
analysis and prediction of shape are needed. For the last few decades, many efforts have
been made to model, predict and control strip shape. In general, previous methods can
be classified into four categories [Guo, 1990] including:
• Elastic Foundation Method (EFM) [Hetenyi, 1947; Tong, 1963; Stone and Gray,
1965; Gohar, 1974; Grimble, 1976; Lin and Lee, 1997; and Senalp, Gokler, and
Akkok, 1997];
• Influence Coefficient Method (ICM) [Shohet and Townsend, 1968; Kuhn and
Weinstein, 1970; Tozawa, 1970; Tozawa, 1971; Shohet and Townsend, 1971;
O'connor and Weinstein, 1971; Berger et al., 1976; Kizaki and Sato, 1983; Pawelski
et al., 1984; Pawelski et al., 1985; Pawelski et al., 1987; Lawrence et al., 1988;
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 125
Yamada and Sekiguchi et al, 1989; Shigaki and Herman, 1999; and Gonzalez and
Abascal, 2000];
• Finite Element Method (FEM) [Kihara et al., 1987; Chen and Zou, 1987; and
Lawrence et al., 1988; Gratacos, Montmitonnet, Fromholz, and Chenot, 1992; Lin
and Lin, 1995; Fukumura and Fujikake et al, 1998; Iguchi, Owen and Liu, 1998;
and Kiuchi and Matsumoto, 1998]; and
• Spring and Beam Method (SBM) [Poplawski et al., 1980; Guo, 1986; Guo, 1990;
Wang, 1990; Yuen and Wechner, 1990; Yuen and Miller, 1993; and Yuen, Duval
and Wechner, 1998].
EFM provides a simple and convenient way for engineers and researchers to roughly
calculate and estimate the forces and profile of the gap between loaded work rolls.
However, it is not accurate enough to be used for mill setup and on-line flatness control
[Guo, 1990]. Moreover, it is not suitable for calculating roll deformation and strip
thickness profile when both roll force and roll bending force exist. So it is only used for
approximate estimation at the mill design stage.
ICM was first developed by Shohet and Townsend [Shohet and Townsend, 1968; and
Shohet and Townsend, 1971] and further investigated by Bryant [Bryant, 1973] and
Cresdee et al. [Cresdee et al., 1991], Tozawa [Tozawa, 1970; Tozawa, 1971], Berger
[Berger et al., 1976], Pawelski et al. [Pawelski et al., 1984, 1985, 1987], Lawrence
[Lawrence et al., 1988], Yamada and Sekiguchi [Yamada and Sekiguchi et al., 1989],
Shigaki and Helman [Shigaki and Helrnan, 1999], Gonzalez and Abascal et al.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 126
[Gonzalez and Abascal, 2000]. The basic principle of I C M is based on the following
two aspects:
1) the deformation compatibility conditions between work roll and backup roll, and
work roll and the rolled strip, and
2) the static equilibrium of vertical forces and bending moments for both work roll and
backup roll.
ICM has been proven an efficient and powerful approach dealing with the predictions of
roll deformation and strip thickness distribution since the late 1960's. The advantage of
ICM is that the mill is considered as a whole system in approaching a better solution for
both the roll force and strip thickness distributions. However, the mill system is
nonlinear with respect to the strip rolling schedules, control devices, and the mill system
configurations such as the arrangements of the mill stand, rolls and bearings. Therefore,
the control models for the mill system based on ICM usually cannot be readily
implemented with on-line control systems due to the complexity and excessive
computation of nested iterative loops. Moreover, the rigid body motion of the backup
roll is not considered in Shohet and Townsend's ICM [Shohet and Townsend, 1968; and
Shohet and Townsend, 1971], and thus is not realistic.
FEM has so far become a powerful tool to analyze detailed mill performance. This is
indicated by several successful commercial FEM software packages such as ANSYS,
ABAQUS etc. FEM can provide not only detailed deformation performance of both
strip and rolls, but also the dynamic analysis of the rolls and thermal crown estimation.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 127
The disadvantages of F E M include that of a large mesh being required to model the
deformation correctly, and hence large computation time is required. So it is not suitable
for on-line shape control in production runs.
SBM considers the behavior of both backup roll and strip as a linear spring, the inter-
roll specific force and specific roll force are proportional to the indentation between the
backup and work rolls, and the relative motion between the work roll and strip,
respectively. This simplifies the modeling of the interference between the work roll and
backup roll and the interference between the work roll and strip, as the strip and backup
roll are treated as springs and the work roll as a beam.
The research in this chapter focuses on the shape optimization for the threading process
in tandem cold rolling [Wang and Tieu, 1999]. A shape predictive model is proposed in
combination of the strengths of both ICM and SBM. The approach for the shape
analysis consists of deriving the physical equations in a form suitable for efficient
computer solution. This makes the developed models more flexible, accurate and
convenient for practical use. The research also extends the models to the setting up of
the bending forces for those mill stands with roll bending systems, and to optimize
reduction schedule for those without roll bending systems with an intelligent
optimization scheme.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 128
5.2 ROLL PRESSURE AND PROFILE MODEL
To successfully predict final strip thickness distribution along its width, a good
understanding of the mechanical deflection of the rolls and strip, inter-roll force and roll
force distributions is needed.
5.2.1 Roll Pressure Distribution
A four-high tandem cold rolling mill usually consists of three to five stands. Each stand
contains two work rolls supported between two larger backup rolls. To control the strip
shape, roll-bending systems are usually installed between the work rolls, the backup
rolls, or between a work roll and a backup roll. The bending system can effectively
influence the strip shape by exerting bending forces. In Figure 5.5, JL and JR are roll
bending forces exerted between two work rolls; JL stands for the left hand side roll
bending force (operator side) and JR is the right hand side roll bending force (drive
side). Lx is the distance between the centerlines of two backup roll bearings; L2 is the
distance between two jacks of the work roll; L3 is the distance between the centerline of
the backup roll bearing and roll barrel; L4 is the distance of the work roll bearing
centerline from roll barrel; F is the total load exerted on the top backup bearing chocks;
p(x) is the specific roll force between work roll and strip at position x; q(x) is the
inter-roll specific force between backup roll and work roll at position x measured from
the vertical centerline of the strip, and W the strip width.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 129
L3 LI
F/2 q(x) F/2
2 ^a T—I fTrTTTTTTTrt i ?
•%< ̂ >i—i^^^^^^"™"™1—i |JR
r^1 . w . n—^ L2
Figure 5.5 Force distribution model for rolling mills with bending systems
The total load exerted on top backup roll bearing chocks can be described as follows:
F-K/BL+YBR ASL+ASR) (5-1)
where Kx is the combined stand flexibility including mill modulus, bearing and screw
elastic coefficients, it is mill specific; YBL and YBR are the rigid body motions of the
backup roll on the left and right sides, respectively; ASL and ASR are the relative
changes of screw down on the left and right sides, respectively.
5.2.2 Roll Pressure Analysis
Both inter-roll force and roll force are parabolic-shaped [Yuen and Wechner, 1990; and
Yuen and Miller, 1993]. Based on Shohet and Townsend's influence coefficient method,
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 130
the roll barrel length of backup roll can be divided into k equal portions (elements) and
m equal portions for work roll, and the strip can be split to n equal portions. The length
of each portion is Ax, which should be made small enough to ensure elastic continuity.
From Figure 5.5, the following equation can be obtained:
Ax^LB_=L]L:=W (5_2) k m n
where Lw and LB are the length of work roll barrel and the length of backup roll barrel,
respectively.
The inter-roll force between the work roll and the backup roll can then be replaced by k
concentrated loads Q( /) (1 < / < it), and the roll force between the work roll and the
strip can be represented by m concentrated loads P{ /) (1 < / < m). Q{ /) and P( /)
can be described as follows:
Q(j) = q(i)Ax (!</<£) (5-3)
P(f) =
m-n . m + n p(f)Ax ~^2T~]~ 2
0 . .m-n , m + n u 1 < / < and < / < m
(5-4)
where
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 131
p( j) is the unit contact load between the work roll and strip, and
q( /*) is the unit contact load between the work roll and backup roll.
5.3. ROLL DEFORMATION THEORY AND DERIVATION OF EQUATIONS
5.3.1 Initial Roll and Strip Profile
5.3.1.1 Incoming Strip Profile
If the initial strip profile is parabolic in shape, the thickness at a position x in Figure 4.2
of Chapter 4 is:
H(x) = Hc [1 - ^-] + 4He -£- (5-5) v c yy2 e w2
where H is the thickness at the strip edges (both the left and right sides).
5.3.1.2 Work/Backup Roll Crown
A crowned roll can simply be illustrated as shown in Figure 5.6.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 132
Crown
Figure 5.6 Crowned roll
The crowns of work roll Cw (x) and backup roll CB (x) at the position x include
thermal, machined crowns and wear, they are:
W \X) — ^W ^^ffr\X L/fp I £*) ' Li)y (5-6)
2 IT 2 CB(x) = CB-4CB(x-LB/2Y/L (5-7)
where Cw is the work roll crown at its vertical centerline, and CB is the backup roll 'W
crown at its vertical centerline.
5.3.2 Static Equilibrium of the Work/Backup Rolls
5.3.2.1 Static Force Equilibrium
For the work roll, if a roll bending system is installed in the mill as shown in Figure 5.5,
the following equation for equilibrium can be written:
(5-8)
/=! H
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 133
If there is no roll bending system, JL and JR should be assigned to 0. For the backup
roll, the static equilibrium is:
fiq{f)Ax = F (5-9) /=i
Substitute Equation (5-1) in Equation (5-9):
Xq(/)Ax - Kl(YBL+ YBR} = -Kx(ASL +ASR)/2 (5-10) j=\ 2
5.3.2.2 Static Bending Moment Equilibrium
For the work roll, the static bending moment equilibrium is:
k m L -W £ ? ( ;)Ax[L4 +AZ + U- 0-5)Ax] - £ p( j)Ax[-±—- + (/ - 0.5)Ax] = JRL2 (5-11)
where AZ is the side-shifting value of work roll, the sign of AZ is positive when the
work roll is left side-shifted, otherwise, it is negative. For the backup roll, the moment
equilibrium is as follows:
5>(/)A*[A +(j-0.5)Ax] + KxYBRLx =KXLXASR (5-12)
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 134
5.3.3 Derivation of Roll Deformation Compatibility Equations
The roll deformation on the surface includes three parts:
1) deflection of the roll axis due to bending and shear,
2) rigid-body motion of the roll axis, and
3) local surface deformation.
The deformation of the roll axis can be calculated based on Shohet's influence
coefficient theory [Shotet and Townsend, 1968 and 1971]. The rigid-body motion of the
roll axis is assumed to be proportional to the relative motion of the roll left and right
bearings. The backup/work roll flattening at each portion of the rolls is assumed to be
proportional to the corresponding inter-roll Q( j) (1 < j < k). The work roll/strip
flattening at each portion is assumed to be proportional to the corresponding roll force
P(/)(l</<m).
5.3.3.1 Calculation of Influence Coefficients
The influence coefficients for both work roll and backup roll are calculated based on
bending and shear theory. The reference location for influence coefficient is the central
point of the left roll bearing. The influence coefficient calculation model is shown as
Figure 5.7.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 135
1
T
d i
P <
>
•n
L
/i\ /-̂ 0)
<
D
• — > '
> •
UJ
\
\
ID
Id
Figure 5.7 Influence coefficient calculation model
The influence coefficient a(i, j) is defined as the deflection of the i element due to
the influence of the bending and shear of the unit load at the /'* element. In Figure 5.7,
the load distribution is assumed to be symmetrical. Therefore, when / > i, there is the
following equation:
L-rj,P 16 , / p-LL-T} a{i,j) = --—L^ + c p+C2)+ — (—r +
J1-r-) — ' LEIn 6 x 2 3nG d2 D2 L
(5-13)
when / < i, the influence coefficient is calculated as follows:
r -\ ^ r(L~P)3 , nn K\±rM. 16 u l ^-KL-n {p-n)n
(5-14)
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 136
where E is Young's modulus of the roll, ID is the moment of inertia of the roll barrel
about the horizontal axis, and Id is the moment of inertia of the roll neck about the
horizontal axis. CX,C2,C3,C4 are constants and can be determined from:
C 2 = C 4 = ^-(1-^) (5-15) 3 /,
Cx=^{n-2L) + ̂ ~^-C2 (5-16) 6 (77 - L)L
C3=^-^- + {--hc2 (5-17) 6 77 L
5.3.3.2 Deformation Compatibility Condition for Contact of Backup and Work
Rolls
The work roll and backup roll are initially machined with a crown. During rolling the
total crown includes not only machined crown, but also thermal camber and wear.
Without loading, the backup and work rolls are actually in line contact. When the strip
is rolled, the backup and work rolls are deformed and in contact with each other along
the whole roll barrel. The deformation compatibility is described as follows [Shohet and
Townsend, 1968]:
YB(i)-Yw{i)+^- = -[Cw-Cw(i) + CB-CBii)} (5-18) K2
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 137
where YB(i) and Yw(i) are the deflections of the backup and work rolls at the i'h
element due to bending and shear plus the roll axis rigid body motion at the position,
respectively (which are considered positive in sign for upwards motion); CB(i) and
Cw (/) are the crowns at the i'h element of the backup and work rolls, respectively.
yB^hu^B(u) + ̂ -Y-)[Y(i-°-5)AX]^L (5-19)
v ,-A V M A r ^ V M A r ^. (YWR-YWL)[L4+(i-0.5) Ax] Yw{i) = 2^p{l)Axaw{i,])-2^q{i)Axaw(i,i)+ + YWL
j=\ j=\ L2
(5-20)
where aB(i, j) and aw(i, j) are influence coefficients for the backup and work rolls,
respectively, determined by Equation (5-14); YBL is the rigid body motion of the backup
roll on the left side, YWL and YWR are the rigid body motions of the work roll on the left
and right sides, respectively.
Substitute Equations (5-19) and (5-20) to Equation (5-18):
Yjq{j)AxaB(i, j) + J > ( j)Axaw (i, /) + q(i) / K 2 - ^ p ( j)Axaw (i, /) + YBL - YWL y=l y=l /='
, (YBR -YBL)[L3 +(/-0.5)Ax] (Ym -YWL)[L4 +(/-0.5)Ax] _ ^ _ ^ m + r _c m i
Lx E2
(5-21)
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 138
where K2 is the elastic contact coefficient between the backup and work rolls [Loo,
1958; Stone and Gray, 1965], and
q(i)/K2=ZBW(i) + ZWB(i) (5-22)
where ZBW(i) and ZWB(i) are local contact deformations at the center of the /'
element of the backup roll and the work roll, respectively.
5.3.3.3 Deformation Compatibility Equation for Contact of Work RoE and Strip
The deformation situation between the work roll and the strip is shown in Figure 5.8
based on Shohet and Townsend [Shohet and Townsend, 1968]:
In Figure 5.8, H(i)/2 and h(i)/2 are half entry and exit thickness of the strip at the
position of the ith element, respectively. C(i) is the semi-height of roll gap at the same
position.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 139
The center of the ith element of work roll
h(i)/2-Z sw __
_ _ A . C(i) 1
Strip profile before rolled
- Loaded roll gap profile
~~ Unloaded work roll profile
Figure 5.8 Deformation compatibility condition of the work roll and strip
The deformation compatibility equation for contact of work roll and strip is described as
follows:
h(i) -Zsw(i)-C(i) = Yw(i) + Zws(i) (5-23)
where Zsw (i) and Zws (i) are the local contact deformations of the strip and the work
roll at the location of the ith element, respectively. Hence, for the center point of the
strip,
h -°--7 -C =Y +Z
(5-24)
where h0 is the exit thickness at the center of the strip; Zsw0 and Zws0 are the local
contact deformations at the center of the strip and the work roll, respectively. If the
plastic modulus of the strip is M [Shiozaki, 1968], the following equation applies:
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 140
h{i) = H(i)-E^>+2Zsw(i) (5-25)
At the center point of the strip,
h°=H°~^[ + 2Zs"o (5-26)
where p0 is the specific roll force at the center point of the strip. W h e n Equations (5-
25) and (5-26) are substituted into (5-23) and (5-24), respectively, and then subtract (5-
24) from (5-23):
2M[YW {i) - Ywo ] + (1 + 2¥-)p(j) = M[H(i) - H 0 ] - 2 M R C W (i) + (1 + ^ - ) P o (5-27) A 3 K3
where K3 is the combined elastic deformation coefficient of work roll surface and strip
[Shiozaki, 1968]. Edwards et al. [Bryant, 1973] considered the correlation between the
total roll flattening for both work rolls and the Hertzian flattening which occurs between
two infinitely long, elastic cylinders [Hertz, 1896; Loo et al., 1958]. However, this
introduces complexity of logarithm calculation in solving equation set. Shiozaki
[Shiozaki, 1968] proposed the following equation to resolve K3:
— = — + 4r (5-28) K3 M K\
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 141
where K*3 is the elastic deformation coefficient of the work roll surface, it is calculated
as follows:
1_ = £(ln 23L cIo } (5_29)
K3 8 H0-h0+ cp0 H0-h0+ cp0
where Dw is the work roll diameter. In Equations (5-27), Rcw(i) can be described as
follows:
Rcw (i) = C{i) -C0=CW-CW (0 (5-30)
ZmV)=lf- (5-3D
Z = — (5-32) K3
In Equation (5-32), p0 can be obtained based on Bland-Ford-Hill force calculation
formulae [Bland and Ford, 1948; and Bland and Ford, 1952], which has been widely
used in cold rolling [Hikino et al., 1977; Jimba et al., 1990; Hwang et al, 1996;
Sekiguchi et al., 1996; Cho et al., 1997; Geddes et al., 1998; Guo, 2000]. Namely
p0=mJ^-QPnt (5-33) tin
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 142
where R' is the deformed work roll radius at the central point of the strip, k(r) is the
average yield stress evaluated at the mean reduction r as defined in Section 4.3.1 of
Chapter 4.
5.3.4 Objective Function of Strip Shape Optimization
The strip shape is determined by the deformed work roll profile. Bad shape is caused by
mismatch of the roll gap and initial strip profile. The exit thickness distribution can be
predicted as follows:
h(i) = h0+ 2[C(i) -C0+Yw (0 + Zws (0 + Zsw (/) - Ywo - Zwso - Zswo ] (5-34)
The objective function of shape optimization can be described by a shape optimization
parameter y , which is defined as the summation of differences between actual thickness
and theoretical thickness when the best shape is obtained at each portion of the strip,
namely:
Y=Jt[h(i)-H(i)(l-r)]2 (5-35)
where r is the thickness reduction, the theoretical thickness H(i)(\-r) is derived
based on the perfect shape condition as described in Chapter 4.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 143
The flatness in I Units due to a transverse variation in the stand reduction is given below
[Spooner, 1994]:
Flatness I(i) = \n(2^) x 105 (5-36) hH(i)
where H and h are the average entry and exit thickness of the strip, respectively.
Equation (5-36) is derived based on the stand exit tension distribution:
tJi) = -Es(e(i)-s) (5-37)
where tsh(i) is the exit tension at the i* portion of the strip; £(i) is the natural strain
distribution due to the stand reduction at the ith portion of the strip; and s is the mean
value for the natural strain.
e(0 = ln(^) (5-38) h(i)
e = ln(i) (5-39) h
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 144
5.4 EQUATION SET SOLVING
In the equation set containing Equations (5-8), (5-10), (5-11), (5-12), (5-18) and (5-27),
there are totally k + n + 4 unknown variables, which are q{ /) (1 < / < k),
p( /) (—~— ^ / ^ ) and YBL, YBR, YWL, YWR. And there are also k + n + 4 equations
including k equations contained in Equation (5-18), n-\ equations contained in
Equation (5-27) (except the element at the vertical centerline of the work roll), one
equation from the Bland-Ford-Hill force calculation formula (5-33) for solving p0, and
four equations (5-8), (5-10), (5-11), and (5-12). Therefore, the equation set is solvable.
If there is no work roll side-shifting, and the rolling mill is considered as symmetric to
the vertical centerline of the backup roll, then the rigid motions of the backup roll at the
left and right sides should be identical, which is:
YBL=YBR (5-40)
The rigid motions of the work roll at both the left and right sides should also be
identical, namely:
YWL=YWR (5-41)
For this case, there are totally k + n + 2 unknown variables, the equation set includes
k + n + 2 equations contained in Equation (5-8) (one equation), Equation (5-10) (one
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 145
equation), Equation (5-18) (k equations), Equation (5-27) ( « - l equations), and one
equation from Formula (5-33). Thus a solution can be obtained for the set of equations.
The basic method employed to solve the above equation set is Gauss Elimination
Algorithms [Engeln-Miillges and Uhlig, 1996] which has been found to be efficient and
convenient for on-line use purpose, details about the calculation time to solve the
equation set will be discussed in Section 5.6.
5.5 FLOWCHART FOR SHAPE ANALYSIS
The flowchart for shape analysis is demonstrated in Figure 5.9. Based on the results
from the equation set, the tension stress distribution along the strip width is analyzed,
followed by the iteration convergence test. The convergence condition is pre-defined as
follows:
e>±J-t(C(i)-t^))2 (5-42)
where £ is the convergence tolerance with a value of 0.001 in the reference book
[Bryant, 1973], fs is the mean tension stress of the strip, tkJl(i) and t]h(i) are the
tension stresses at the ith element of the strip in the (k + l)"1 and k,h iterations,
respectively. It should be noted that, in the iteration, the step towards convergence is
named "convergence parameter". The greater the convergence parameter, the faster does
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 146
the iteration go towards achieving the tolerance of convergence. However, if it is too
large, the iteration may miss its target and goes towards a wrong direction.
Initial Data
l(Lp,, LI, L2, L3, U, W, DB, D W , C W , H, h, Kj)
Calculate Various Parameters fR.jK a»(i.i).qw(ij) et al)
J Construct Equation Set
Determine Coefficient Matrix of Equation Set
I Solving Equation Set Obtain q(i) et al
Yes
Calculate Yffl and W i )
Calculate Strip Thickness
I Calculate Tension
Calculate
Rolling Pressure No
Test Convergence, Yes
q(i)=0
Save Results
Figure 5.9 Flow chart for strip shape analysis
5.6 MODEL VALIDATION AND NUMERICAL EXPERIMENTAL RESULTS
The strip shape analysis program has been implemented in C/C++ for execution on a
Pentium II 400 PC. A case study for a tandem cold rolling mill has been conducted to
verify the proposed model. The numerical experiment conditions are listed in Table 5.1:
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 147
Table 5.1 Numerical experiment conditions
Mill Parameters
Work Roll Face Length (mm)
Work Roll Diameters (mm)
Combined Work Roll Crown (mm)
Young's Modulus (GPa)
Backup Roll Face Length (mm)
Backup Roll Diameters (mm)
Strip Width (mm)
Full Running Speed (m/min)
Scheduled Total Roll Force (MN)
Yield Stress (Mpa)
Exit Tension (Mpa)
Centerline Input Thickness (mm)
Centerline Output Thickness (mm) Coefficient of Friction
Standi
1169.2
574.8
0.055
207 1169.2
1324.9
775 150.4
6.4 496 179 1.98
1.387 0.0797
Stand 2
1169.2
582.1
0.04
207 1169.2 1291.4
775 304.9
7.6 669 192 1.387 1.094
0.0449
Stand 3
1169.2
569.2
0.03
207 1169.2
1343.2
775 490.0
7.7 740 201 1.094
0.458
0.0392
Stand 4
1169.2
562.9 0.04
207 1169.2
1273.2
775 744.5
6.0 770 172 0.458 0.302
0.0312
Stand 5 1169.2
578.5 0.04
207 1169.2
1256.4
775 1058.8
9.0 883 136 j 0.302
0.212 0.0244
In the above table, the combined work roll crown includes machined crown, thermal
crown and wear. When the rolls are divided into elements with different lengths,
solution time is shown in Figure 5.10. The larger the element length Ax, the shorter the
solution time is. However, the element length Ax should be made small enough to
ensure elastic continuity [Shohet and Townsend, 1968]. The length of each of the
elements into which the strip, backup and work rolls are divided is Ax = 6mm in this
simulation. When the convergence tolerance is assigned to 0.001 and the convergence
parameter is taken the value of 0.45, the convergence performance of the simulation is
shown in Figure 5.11.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 148
1000 Solution Time (s)
1 2 3 4 5 6 7 8 9 10 11 12 15 20
Length of Elements (mm)
Figure 5.10 Solution time for different element lengths
0.25
0)
u c l_ 0 >
c o u
0.2
0.15
0.1
0.05
0
Convergence Toierance=0.001
Convergence Parameter=0.45
2 — i 1 1 —
3 4 5
Number of Iterations
Figure 5.11 Convergence performance (convergence tolerance = 0.001)
A measured work roll surface temperature distribution at Stand 1 from a tandem cold
rolling mill is demonstrated in Figure 5.12. The coordinate 0 in Figure 5.12 is the
operation side of the work roll, and the coordinate 1169.2 corresponds to the drive side
of the work roll. The corresponding thermal crown of the work roll is shown in Figure
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 149
5.13. Besides the thermal crown, the work roll wear is another important factor to be
taken into account. Figure 5.14 shows the real profile of the work roll at Stand 1 taking
into account combined machined and thermal crowns, and wear which is demonstrated
in Figure 5.15.
The comparison of the strip thickness distribution generated from the proposed model
and the actual strip thickness profile which is measured from off-cuts of head end strip
exiting Stands 1 and 2 is illustrated in Figures 5.16 and 5.18, respectively. Figures 5.17
shows the comparison of the actual and predicted flatness for Stand 1. Figure 5.16
indicates that a reasonably good agreement between the predicted and the real strip
thickness distribution at Stand 1 has been achieved, thus providing a validation of the
model. Figure 5.18 demonstrates that the predicted strip thickness on the left hand side
exiting Stand 2 is thinner than the actual strip thickness. Investigation results show that
this is resulted from the misalignment of the strip from the mill centerline. Thus, the
proposed model is considered reliable in the prediction and analysis of the strip shape in
cold rolling.
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 150
0 146.15 292.3 438.45 584.6 730.75
Work Roll Wdth (mm)
876.9 1023.05 1169.2
Figure 5.12 Measured work roll surface temperature distribution (Degree C) at Stand 1
0.016
0 146.15 292.3 438.45 584.6 730.75 876.9 1023.05 1169.2
Work Roll Width (mm)
Figure 5.13 Thermal crown of the work roll at Stand 1
Chapter 5 Shape Analysis and Optimization in Tandem Cold Rolling 151
The real work roll profile at Stand 1
0 150 300 450 600 750 900 1050 Work roll width (mm)
Figure 5.14 Real profile of the work roll at Stand 1
The actual wear profile of the work roll at Stand 1
0 100 200 300 400 500 600 700 800 900 1000 1100
Work roll width (mm)
Figure 5.15 Wear profile of the work roll at Stand 1
Chapter 5 Computational Modeling for Shape Analysis and Optimization in Tandem Cold Rolling 152
3
(A
5 _ w E d> c C —" J*
o a.
1.4
1.39
1.38
1.37
1.36
1.35
1.34
1.33 4
Actual Strip Thickness
Calculated Strip Thickness
100 200 300 400 500
Strip Width (mm)
600 700 800
Figure 5.16 Comparison of calculated and actual strip thickness at Stand 1 exit
(Coordinate 0 - operation side)
Figure 5.17 Comparison of actual and predicted strip flatness at Stand 1 exit
Chapter 5 Computational Modeling for Shape Analysis and Optimization in Tandem Cold Rolling 153
c o 3 -C i-
V)
a V) V)
c Jt o £ 1-Q. w +* (0
E E
1 11
1.1 1.U9
1.08
1.07
1.06
105 1.04
Actual Strip Thickness
200 400
Strip Width (mm)
600 800
Figure 5.18 Comparison of calculated and actual strip thickness at Stand 2 exit
5.7 O P T T M I Z A T T O N O F S H A P E U N D E R T H R E A D I N G C O N D I T I O N
Special consideration for shape analysis under threading condition has been given in this
research, and possible ways to alleviate threading delays have been suggested. During
threading, there is no front tension when the strip is heading towards the next stand. If
the threading force is too high, wavy edge will occur. However, if the threading force is
inadequate, center buckle will appear. As introduced in Section 5.1, when these shape
problems occur during threading, delays could be caused. Normally, there are three
means to change the strip shape:
(1) schedule dependent adjustments to be applied to the screw position references, thus,
rolling reductions at individual stands, for the short duration of the threading
sequence,
(2) adjusting roll bending forces if available, and
Chapter 5 Computational Modeling for Shape Analysis and Optimization in Tandem Cold Rolling 154
(3) altering the total roll crown combining machined crown, thermal crown and roll
wear.
In the following paragraphs, an optimized procedure is introduced to optimize rolling
reduction patterns for a tandem cold mill.
Simulations are conducted to analyze the influence of different work-roll mechanical
crowns. Simulation results show that the same level of flatness can be achieved under
different rolling conditions. As discussed earlier, Figure 5.17 shows the flatness of the
exit strip corresponding to the thickness distribution demonstrated in Figure 5.16 at the
first stand under the following rolling conditions:
(1) entry thickness = 1.980mm, work roll crown =0.055mm, Exit Thickness =1.387mm
and
(2) entry thickness = 1.980mm, work roll crown=0.02mm, Exit Thickness =1.520mm.
This means that reasonably good shape at Stand 1 can be obtained by using an optimum
work-roll ground crown and less reduction, while leaving more room for the following
stands to adjust their reductions. To determine an optimum ground crown for the work
roll, the exit thickness against different work roll crown to maintain the same level of
flatness shown in Figure 5.17 is demonstrated in Figure 5.19.
Possible ways proposed to deal with shape problems during threading include not only
using work rolls with optimum ground crown especially for the threading consideration,
Chapter 5 Computational Modeling for Shape Analysis and Optimization in Tandem Cold Rolling 155
but also installation of roll bending system at the first stand. Meanwhile, the installation
of the roll bending system at Stand 1 will leave more room for the following stands to
adjust their reductions. The numerical simulation has been conducted under the rolling
conditions listed in Table 5.1. To determine the roll bending force, a heuristic searching
based scheme is proposed and implemented. Genetic algorithms [Holland, 1969;
Holland, 1975; Goldberg, 1989; Davis, 1991; and Fogel, 1995] are employed as the core
of the scheme. During searching, the bending force is firstly self-locked in an interval.
The fitness function is defined as the shape optimization parameter y defined in Section
5.3.4.
The evolutionary parameters employed in the GA for determination of an optimum
bending force are as follows: the total number of trials is 1000; the initial population
size is 20; the length of binary chains describing bending force is 11; the crossover rate
is 0.6; and mutation rate is 0.001. The solution time for one stand executed on a
Pentium II 400 PC is about 26 seconds. The flatness of exit strip at Stand 1 under
different roll bending forces for the conventional work roll crown is shown in Figure
5.20. A roll bending force of 400kN produces a perfect shape in this case. As discussed
in Chapter 4, the evolutionary parameters are determined based on tests, inappropriate
evolutionary parameters may lead to premature termination of the evolutionary process
or excessive computing time [Wang and Xu, 1996; Wang and Tieu et al., 1998; Wang
and Tieu et al., 2000].
Chapter 5 Computational Modeling for Shape Analysis and Optimization in Tandem Cold Rolling 156
1.6
1.55
(0 m « c o sz 1-
"5 LU
15
1.45
1.4
1.35
1.3 0.01 0.02 0.03 0.04
Work Roll Crown (mm)
0.05 0.06
Figure 5.19 Setup values for exit thickness to maintain the same flatness level with
different work roll crowns at Stand 1
80
-40
800
Strip Width (mm)
Figure 5.20 Flatness of exit strip under different roll bending forces (KN)
Chapter 5 Computational Modeling for Shape Analysis and Optimization in Tandem Cold Rolling 157
5.8 SUMMARY AND CONCLUDING REMARKS
In this study, a hybrid model of ICM and SBM is proposed and verified. Numerical
experiments have been conducted to predict strip shape and the results show that it is
efficient and effective. The following conclusions may be drawn:
(1) A theoretical predictive model has been presented to analyze strip shape during
cold rolling. To obtain the best strip shape, the bending forces and/or rolling forces
are optimized. A case study of strip shape for threading has been conducted.
(2) A shape prediction model including an optimized roll based bending force set up
procedure is proposed.
(3) Proposals to improve shape during threading, which include smaller work roll
ground crowns and adjustment of screw position references (rolling reductions) and
roll bending forces at individual stands, have been presented.
It should be noted that the proposed models are still valid if some rolling conditions are
changed such as different initial and exit gauges and different material grades. As no
further plant data is available at the time of thesis writing, more validation could not be
done with different gauge and width. As the variations of strip entry thickness along its
width was not available due to the fact that there is no shape-meter at the hot strip mill,
the assumed parabolic strip profile is employed in the shape prediction model. Future
work in the above aspects is recommended.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 158
Chapter 6
Numerical Simulation and Analysis of Unsteady
Processes in Tandem Cold Rolling
This chapter addresses the mechanism of how strip thickness deviation is produced
during the threading, acceleration, deceleration and tailing out phases in cold rolling,
and then details the simulation results of these processes. Motor speed droop during
threading, speed dependent parameters during acceleration such as friction coefficient
between the rolls and strip, and oil film thickness in hydrodynamic roll neck bearings
are taken into account in the simulation. The interactions of tension, roll force and exit
thickness fluctuations during unsteady rolling are demonstrated and investigated. The
effect on the unsteady rolling processes of friction, strip properties and work roll profile
is also discussed.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 159
6.1 INTRODUCTION
The control of head and tail off-gauge is becoming an urgent need in the steel industry
to reduce crop loss and produce superior-quality steel products. As there are various
kinds of fluctuations in unsteady states such as threading, acceleration, deceleration, and
tailing out processes, these fluctuations are always accompanied with non-linearity and
hysteresis, it is difficult to accurately model and control these processes. In the steel
rolling industry, to meet the requirements of thickness accuracy, reduce off gauge length
and hence minimize wastage during the unsteady-states is more difficult than that
during steady-state (constant speed rolling).
Although a few technical papers suggest several possible approaches for the
improvement of off-gauge with little details, technical articles addressing a substantial
and detailed solution for the problem are relatively sparse. One United States Patent
suggests a method for making delivery gauge corrections at low mill speed during
threading and tailing out by varying the inter-stand tension [Peterson and Larsen, 1991].
The delivery AGC by tension is selectively energized so that it is only operational when
the delivery AGC by speed system is not functional. The speed of each stand is
controlled by both tension controller and delivery AGC by speed controller. However,
the method is inadequate for those control systems already existing in some five stand
tandem cold rolling mills shown in Figure 2.5, where Stand 3 is usually a pivot, and
cannot be changed to suit the patent. James M. Hennes reported the modernization of a
1970 tandem cold rolling mill [Hennes, 1997]. The modernization has turned the facility
to world class status with significant quality and productivity improvements. The major
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 160
components of the modernization include automatic mill setup, conversion to fully
hydraulic mill, hydraulic cylinder control, hydraulic power unit control, digital drives,
upgrade of roll bending, and a new man-machine interface. The improvements
addressed above are palliative rather than remedial for the head and tail off-gauge
problem. Schipper addressed a new approach to using and developing a preset model for
the modernization and automation of a cold rolling mill in 1998 [Schipper, 1998]. The
results show the proposed preset model is more flexible than old statistical-based
models, and leads to excellent thickness performance, about 97% of the coils are seen to
show a thickness deviation within ± 1%. Siemens AG introduced a new patent named
"Extended Mass Flow Control" [Aeberli, 1995] which relies on the well-known
constant mass flow principle [Fainberg, 1963]. The object of this patent is to extend the
constant mass flow principle to Stand 0 (uncoiler). The principle has been used only
after the first stand of a tandem rolling mill. The basic idea of the patent is that,
assuming the thickness control at the first stand fully compensates the incoming errors,
a rigid control of the speeds of the subsequent stands will ensure the correct reduction
and accuracy because of constant volume. Based on the consideration above, the
measured entry speed is now used as the actual value in a closed uncoiler speed control
loop. Thus thickness errors measured at the entry to the mill can be compensated by
changing the uncoiler speed and the roll gap of Stand 1, as the thickness errors enter the
first stand. Therefore, there is no need to calculate the roll gap, which is the bottleneck
as the parameters concerned (e.g. friction, yield stress and strip stiffness) cannot be
exactly determined, since the entry tension controller automatically sets the correct gap
by keeping the strip tension constant. This method seems to be a promising one that
eliminates disturbances during threading especially those from strip properties. In 1993,
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 161
the extended mass flow gauge control was successfully applied to a two-stand
aluminum cold rolling mill in Japan and it proved to be a milestone in the production of
cold-rolled aluminum sheet for cans [Aeberli, 1995]. A notable feature of this mill was
that the very tight gauge tolerance required by the canning industry (less than 0.6%) had
been achieved.
Literature shows the various ways in which the mill speed may influence the specific
roll force and strip thickness in cold rolling are considered. Sims and Arthur reported
their experimental work for a range of mill speeds in 1952 [Sims and Arthur, 1952]. The
results show that the variation of the coefficient of friction between the rolls and strip is
the principal cause of roll force and thickness variations. Starchenko presented a
relationship between the deviations of the strip thickness from the nominal as a function
of rolling speed in 1966 [Starchenko, 1966]. The author suggested periodical use of an
efficient lubricant during acceleration of a rolling mill to decrease the thickness
deviations. Faizullin investigated the relationship between rolling speed and the
thickness of the lubricating film in the backup bearings and the relationship between the
rolling speed and the friction coefficient between work roll and strip in 1967 [Faizullin,
1967]. Ogan addressed the speed influence on the output of continuous strip mills in
1969 [Ogan, 1969]. Yuen and Matthews presented a quantitative formula showing the
variations of the friction coefficient with rolling speed in 1998 [Yuen and Matthews,
1998]. None of the papers describe in details a full simulation of the acceleration
process.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 162
This chapter addresses an endeavor to dynamically model, analyze and numerically
simulate the threading and acceleration processes of a tandem cold rolling mill so as to
investigate the correlation of unsteady rolling parameters [Wang and Tieu, 2001a; Wang
and Tieu, 2001b]. In the following sections, a theoretical analysis of the threading and
acceleration processes is addressed first, then the modeling of the processes and
simulation results are detailed, followed by discussions of the effect on the unsteady
rolling processes of some rolling parameters.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 163
6.2 THREADING AND ACCELERATION PROCESSES
In the cold rolling process, a gauge deviation is produced between uniform (body)
section and the end (head and tail) sections due to the change of rolling conditions. The
whole rolling process for a coil of strip includes threading, acceleration, constant speed
rolling, deceleration, and tailing out phases. Before a process is controlled, it is
important to simulate the process and then investigate its performance. Moreover,
various sources of disturbances encountered should be included in the modeling process.
This section focuses on the theoretical analysis of the threading and acceleration
processes together with accompanied disturbances.
6.2.1 Threading Process
The threading process begins when the head of the strip enters the first stand of a
tandem rolling mill and ends when the rolling mill accelerates. To guarantee no strip
rupture, the threading speed is usually maintained at a low value between 10-130m/min.
Due to the lack of front tension plus strip entry thickness and hardness deviation,
disturbances can occur during the threading process. When the strip enters the first stand
of a five-stand rolling mill shown in Figure 2.5, the back-tension is first established. The
tension between the first stand and the second stand does not exist until the strip head
enters the second stand. The process continues until the strip head enters the fifth stand.
The tension between the coiler and the fifth stand is finally established after the strip
head goes into the coiler. The whole threading process ends when the rolling mill begins
to accelerate. During threading, the deviation of strip thickness is caused by several
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 164
influence factors from the rolling stand and strip. The exit strip thickness is calculated
follows:
as
h = S0+--O, (6-1)
In Equation (6-1), S0 is the no-load roll gap, K is the rolling mill elastic modulus, and
Of is the backup bearing oil film thickness. The above roll gap model can be
graphically illustrated as the following diagram, where the slope of line A is mill elastic
modulus K (N/m), and the slope of line B is plastic modulus M (N/m) of the material
being rolled. The lines A and B intersect at point O, which determines the value of the
roll force P and exit thickness h.
Force
• Roll Gap
Figure 6.1 Roll gap model diagram
Equations (4-10) and (4-14) show that the back and front tensions are directly related to
the roll force, which, in turn, influence the strip exit thickness as described in Equation
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 165
(6-1). The strip properties such as hardness and entry thickness variations directly affect
the average yield stress. Disturbances to the threading process originates in the stands
and input strip, however, vary significantly from one mill to another and from one strip
to another.
Previous experience has shown that on thin strip satisfactory threading shape cannot be
achieved with full speed actuator references, even after correcting for the reduced
backup roll bearing speed effect [BHP, 1979]. In practice, to avoid coil collapse,
threading delays and roll damage, it is necessary to sacrifice a controlled length of out-
of-tolerance material so as to meet shape requirements for the threading. For example,
when rolling thin strip in a cold mill, if the final strip gauge is less than 0.4 mm, during
threading the exit thickness at the last stand is normally set to 0.4 mm so as to prevent
coil collapse. The off-gauge strip is rolled until 2-3 wraps of strip is on the coiler, then
the gauge is set to desired thickness.
6.2.2 Speed Dependent Variables Analysis during Acceleration
Acceleration phase begins when the strip head enters the coiler, and ends when rolling
speed reaches its preset value for the mill to be at full running speed. With the
advancements in mechanical, electrical, hydraulic and computer systems, especially
advanced control techniques, acceleration can usually achieve a value of more than
2m/s2 [Yang, 1988].
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 166
During the acceleration phase, the strip thickness decreases as the mill is accelerated
[McDonald, 1993]. Experimental results show that the principal causes leading to the
thickness deviation may be as follows:
(1) Oil film thickness of hydro-dynamically lubricated backup bearings
The oil film separating the roll neck and the chock increases with mill speed. This
results in the change of roll gap position and exit thickness accordingly. Based on
Polukhin's result [Polukhin, 1975], the oil film thickness O f can be determined by
the following formula:
Of= 2DbxV(\ --cos<p) (6-2)
a
a = 1 + (0.0432£>6 + 0.0056) f cr\ \0-85
{PW¥: (6-3)
<p = 0.23 llog(a-l) +0.6 (6-4)
where Db is the bearing diameter of the backup roll; *¥ is the bearing radial
clearance; a and (p can be calculated from Equations (6-3) and (6-4), respectively;
vB is the peripheral speed of the bearing, and r\ is the bearing oil viscosity. From the
above equations, the oil film thickness increases with acceleration, and the exit
thickness decreases accordingly.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 167
(2) Friction coefficient between rolls and strip
Friction is affected by the rolling speed, arising from the increased hydrodynamic
effect in the roll bite as the rolling speed increases. As expected, the friction
coefficient decreases with increasing rolling speed, and the trend is expressed by
the following equation [Yuen and Matthews, 1998]:
u = 0.0288 exp(-0.00452F) + 0.0268 (6-5)
In the above equation, u is the mean friction coefficient between the work rolls and
the strip, and V the rolling speed (m/s). On closer examination of the correlation of
the friction with rolling speed, Yuen and Matthews also found that although there
were significant scatters in their experimental data due to the wide variations in mill
condition, the friction trend followed that of the average variation closely [Yuen
and Matthews, 1998]. Hence start-up calculations may be performed using
adaptation values (carried out at full speed), with the speed effect taken into
consideration using the average trend expressed by Equation (6-5) [Yuen and
Matthews, 1998]. Based on the above equation, the friction coefficient decreases
with rolling speed, thus, resulting in a decrease in rolling force and the exit
thickness. The speed dependent friction coefficient is an important influence factor
for the rolling force. The variation of rolling force with friction is significant,
especially in the later stands, in which the friction hill effect is prominent [Yuen,
Popelianski andProuten, 1996].
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 168
(3) The tensions applied to the rolled strip at entry and exit fluctuate
As the tensions from inter-stand speed differential cannot be returned to preset
values as quickly as the hydraulic roll gap control due to motor and roll inertia, this
results in a strip thickness deviation.
(4) Yield stress of the strip
In the roll force equation (Equation (4-10)), the roll force is proportional to the mean
yield strength of the material, which in turn is a function of the instantaneous values of
the yield strength along the arc of contact. However, experimental results suggest that
the yield stress has no significant influence on the change of thickness with speed [Sims
and Arthur, 1952].
6.3 MODELING OF DYNAMIC THREADING AND ACCELERATION
PROCESSES
6.3.1 Modeling of Threading Process
6.3.1.1 Motor Speed Droop Consideration
One commonly known motor characteristic is the inverse proportional relationship
between torque and speed. While the strip head is being threaded into the mill, roll force
is applied and work roll motor torque is increased. As a result of the increasing torque,
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 169^
the motor speed is dropping [Kondo and Tajima, 1985]. To keep the motor speed
constant, the close-loop motor speed control system is designed to reduce the impact of
the increasing load as shown in Figure 6.2. The relationship between the final work roll
rotational speed co* at the ith stand and the preset speed coi is empirically described as
Equation (4-4). The motor speed droop is taken into account when constructing
threading simulation models as seen in Equation (6-9).
6)
fi).
Figure 6.2 Motor speed droop and control
6.3.1.2 Threading Process Modeling
The laws governing threading process include "constant mass flow" and "tension
variation principle" according to Fainberg's result [Fainberg, 1963]. The constant head
end speed may also be applied, which means, whenever the strip head end is threaded
into a stand, the rolling speed at that stand shall be adjusted to a preset constant
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 170
threading speed. The threading starts from the moment the strip head enters the first
stand. The tension between the first and second stand is established after the strip head
enters the second stand. As described in Equation (6-6), the exit tension t"+] at time step
(n +1) at Stand i is equal to the summation of tension t" at time step (n) and product
rttn
of time increment At and rate of tension change —-. The rate of tension change is dt
described by Equation (6-7), where Ls is the distance between two neighboring stands,
VH"+l is the strip velocity at the entry of the (i + \)'h stand of the mill, and Vh" is the
strip velocity at the exit of the ith stand.
tr=t?+^At (6-6)
dtl = ~(VHlx-Vh^) (6-7) dt L s
According to the constant mass flow principle, VH"+X can be derived from equation (6-
8):
^ r ^ v ^ l + f " , ) (6-8)
Vh? =0*111,(1 +C) (6-9)
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 171
where h"+x is the strip thickness at the exit of the (i + \),h stand, H"+x stands for the strip
thickness at the entry of the (i + l)'h stand, v"+l denotes the strip velocity at the exit of
the (i +l)'h stand, and f"+x is the forward slip ratio at the {i +V)th stand. Based on the
definition of the forward slip ratio (Equation (4-3)), Vh" can be derived from Equation
(6-9), where co* and f" are the work roll rotational speed as described in Equation (4-
4) and the forward slip ratio at the ith mill stand, respectively.
During steady state rolling, the strip thickness at the entry of the (i + \)'h stand is equal
to the strip thickness at the exit of the ith stand. However, during threading, the strip
thickness at a certain point X of the exit of the ith stand is equal to the strip thickness at
the entry of the (i + \)th stand only when the same element of strip comes to the entry of
the (i + \)th stand as described in Equation (6-11). Therefore, a thickness time delay
occurs, and the time delay ATY between.the ith and {i+\)th stands is equal to the
quotient of the inter-stand distance Ls/ and the strip velocity Vh" at the exit of the i'h
stand as Equation (6-10). If the speed of a certain point on the strip at time x" is Vh",
the time r"+1 can be described as the summation of the time delay AT,, and the time T"
at the nth moment. At time r"+1, the strip thickness at the exit of the ith stand is
changed to h"+l and the exit speed is changed to Vh"+1. After a time period At, the
distance from the point X to the i* stand is calculated by Equation (6-13), which is the
summation of the distance that the strip traveled in the time period Ar and the distance
from the point X to the ilh stand at the time t".
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 172
to,=La/Vh? (6-10)
^,=*rAT| (6-11)
z"+l=T"+At (6-12)
V£=fyh!(At + T") (6-13) /=«
If L"Sl. >L5/, the point X is going to enter the (i + \)th stand of the mill. The above
procedure will be repeated between the (i + \)th and (/ + 2)'h stands of the mill and until
the strip head enters the coiler. In this simulation, a short time period A? =0.005 seconds
is employed as the time step.
During threading, whether threading tension control between Stand / and Stand i + \ is
activated or not is mill specific and subject to some conditions, for example [BHP,
1979], mill master speed > 1%, total roll force > 2MN on Stand i and Stand i + l,
tension measurement available, and strip in Stand i and Stand i + l.
There are also some constraints such as tracking requirements to be considered during
threading. Some mill automation functions require a reliable indication of where the
strip is in the mill. This can be achieved by detecting a change in roll force on each
particular stand as the strip either enters or leaves the stand. Alternatively, the strip tail
or front-end position can be derived from the knowledge of where tension is established.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 173
Usually roll force and tension detection is used jointly to determine if the strip is in a
stand. As the roll force at a stand is not necessarily zero before the strip is threaded into
the stand, absolute roll force detection cannot be used, and a roll force change is
detected instead. The minimum roll force and tension changes required are mill specific.
For example, an increase in roll force by more than 10% of the maximum roll force is
required for Stand 1 of a tandem mill [BHP, 1979], and 4% for other stands. This means
that the minimum threading force should be the summation of the required minimum
roll force change and no-load roll force. Similarly, the minimum tension during
threading for each stand is 5% of its maximum tension. These constraints must be
satisfied when determining threading parameters.
6.3.2 Modeling of Acceleration Process
6.3.2.1 AGC Control during Acceleration
As the actual rolling mill control model is confidential to the company and is not
available to the public, a typical AGC control loop is employed instead.
The Automatic Gauge Control (AGC) loop is operational before the acceleration starts
and after the threading process ends [McDonald, 1993]. During the rolling process, the
rolling force P and the no-load roll gap S0 may be measured at any time (refer to
Figure 6.1). Therefore, the exit thickness calculated from the gauge-meter Equation (6-
1) can be used to replace the delayed gauge signal by measurement [Wang, 1999]. This
gauge-meter control was developed jointly by BISRA and Davy-United [Dendle, 1978],
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 174
as illustrated in Figure 6.3. It compensates for the mill elastic deformation, so the exit
thickness will not vary with a variation in entry thickness or hardness, or some other
external disturbance. However, in order to achieve an optimum gauge and strip flatness
characteristics, a roll gap compensation coefficient C is introduced to the gauge-meter
control model as shown in Figure 6.4 [Wood, 1995]. The relationship between the exit
thickness error Ah and no-load gap adjustment AS0 can be represented by Equation (6-
14), where K is rolling mill elastic modulus and M is plastic modulus of the material
being rolled. When C = 1, the mill elastic deformation is completely compensated for,
the effective mill elastic modulus becomes infinite, and the mill stand seems infinitely
stiff. When C = 0, the no-load gap adjustment AS0 is equal to the exit thickness error
Ah, partial compensation is obtained, the mill stand becomes less rigid. In a tandem
mill, the gauge-meter control provides high mill stiffness for earlier mill stands and
reduces stiffness for subsequent mill stands.
M 1 + —
A50= ^-Ah (6-14)
i + d-Q-
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 175
External disturbance
href + ^ A/l
i I l+M/K AS;
•
'
Rolling Mill
f '—•
i •
c JB
So+f/K
h
Figure 6.3 B 1 S R A and Davy-United gauge-meter control model
External disturbance
Kef + ^ Ah
T-
l+M/K \+(\-Q(M/K)
Figure 6.4 A practical gauge-meter control model
6.3.2.2 Modeling of Acceleration Process
The algorithms for acceleration simulation are similar to that used in the threading
simulation. The differences are that:
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 176
(1) The AGC control loop is operational during acceleration, but not during threading.
The control scheme shown in Figure 6.4 is considered in this simulation in
dynamically adjusting roll gap.
(2) The speed dependent parameters addressed in Section 6.2.2 are taken into account.
6.4 NUMERICAL SIMULATION OF THREADING PROCESS
6.4.1 Flow Chart of Threading Simulation
The threading process is successfully simulated on a Pentium III 500 computer based on
the model detailed in the above section. The flow chart is outlined in Figure 6.5. Details
is described as follows:
(1) Preset threading parameters such as reduction and speed patterns.
(2) Initialize simulation parameters including the time t = 0 when the strip head enters
the first stand, time step number n (since the strip head enters the first stand) is
initialized as zero. The tensions between all two neighboring stands are initially
assigned to zero at the moment as follows: t"=0 (between Stands 1 and 2), t2=0
(between Stands 2 and 3), t"=0 (between Stands 3 and 4), t4=0 (between Stands 4
and 5), t" =0 (between Stand 5 and coiler). Stand number / is initialized as 1, total
number of stand N is assigned 5.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 177
(3) Calculate the roll force Ft", strip forward slip ft", strip exit thickness h" and
velocity Vh" at Stand i.
(4) Calculate the distance L" between the strip head and Stand 1, if it is equal or
greater than the stand spacing LX2 between Stands 1 and 2, then go to step (5).
Otherwise, n=n+l, go to the step (3).
(5) Calculate the roll force F"+x, strip forward slip f"+x, strip exit thickness h"+x and
velocity Vh*M at Stand i +1.
(6) Calculate the strip velocity VH"+X at the entry of Stand i + 1 based on Equation (6-
8).
rtt"
(7) Calculate the rate of change of tension — - based on Equation (6-7). dt
(8) Calculate the strip tension t"+1 at time step number n +1 based on Equation (6-6).
(9) The same procedure from Steps (4) to (8) applies for the remaining stands and so
on until the strip head reaches the coiler after Stand 5.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 178̂
Calculate AS„
F " f h" Vh" r i'+1 ' / i +1 » " i+ 1 ' r " i + 1
At" VVt " ' tn + l T n ' ^ j t l i - i 11 •> A-' i+l
at
Preset Threading Parameters
3L Initialize parameters:
t = 0,n = 0,t" = 0,/2"
t'3 = 0,r4" = Oto5" = 0
N = S,i = 1
= 0,
JSkL
Calculate Ff" , f^h^Vh^L* ^
<^
No
No
> n = « + 1
> n - n + I
^
^
Calculate AS 0
Fi+4, fi+4,hi + 4,Vhi + 4
dtU VHlt,
dt to+1 V £ / + 3 i J-'i + t
<r
Yes
Yes V
No
-> n = « + 1 ^
Stop simulation, and print out the simulation results
Figure 6.5 Flow chart for threading process simulation of a tandem cold mill
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 179
6.4.2 Simulation Condition and Results
6.4.2.1 Threading Simulation Parameters
The following simulation parameters are employed for a tandem cold rolling mill:
Strip dimensions: entry thickness 2.50mm, final exit thickness 0.555mm, and width
930mm. The Mill Modulus is 4MN/mm. Other parameters are illustrated in Table 6.1.
Table 6.1 Threading simulation parameters
Scheduled Reduction Rates (%)
Scheduled Exit Gauges (mm)
Work Roll Diameter (mm)
Standi
28.4
1.789
594.66
Stand 2
30.1
1.250
578.82
Stand 3
28.3
0.896
556.0
Stand 4
22.3
0.696
591.0
Stand 5
20.3
0.555
590.64
Stand spacing between Stands 1 and 2, Stands 2 and 3, and Stands 3 and 4 is 3.96m. The
stand spacing between Stand 4 and Stand 5 is 4.26m. The distance between uncoiler and
the first stand is 6.0m and the distance between the last stand and coiler is 6.5m.
6.4.2.2 Threading Simulation Results
During threading, the constant head end speed is applied. This means, whenever the
strip head end is threaded into a stand, the rolling speed at that stand shall be adjusted to
a preset threading speed. For example, if the preset threading speed is 20mpm (meter
per minute), when the strip is threaded into Stand 1, the rolling speed at this stand will
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 180
be adjusted to 20mpm; when the strip head comes into Stand 2, the rolling speed at
Stand 2 will be adjusted to 20mpm while the rolling speed at Stand 1 is decreased to a
value determined based on "Constant Mass Flow" principle; and so on for other stands.
Thus, this kind of threading is named "Constant Head End Speed Threading".
Generally, the threading speed is low and falls in an interval within which the strip can
be threaded into individual stands of a tandem mill without violating threading
conditions. If a strip is required to be rolled to less than or equal to 4 mm, the exit
thickness at last stand is normally set to 4 mm. Two to three wraps of 0.4 mm thick strip
should be rolled in order to avoid coil collapse. Simulations have been conducted with a
constant head end speed of 20 mpm, and the simulation results are presented in the
following figures. For an academic exercise, threading simulation with different
threading speeds (non constant head end speed threading) has also been conducted, the
effect of threading speed on the exit thickness has been investigated and the results are
demonstrated in Appendix C.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 181
Comparison of roll forces at different stands during threading
Stand No. (1-5)
Figure 6.6 Comparison of threading forces from the simulation with actual measurements
(Constant head end speed = 20 mpm)
The threading force generated from the simulation is compared with the actual
measurement from a steel plant and shown in Figure 6.6. The threading force at each
stand corresponds well to the measurement. Because of mill stretch, the scheduled roll
gap may be slightly different from the actual one at each stand, and this leads to the
prediction error.
The tension is gradually established during threading. W h e n the strip head enters Stand 2
to 5, an overview of the whole process of tension establishment is shown in Figure 6.7.
Four lines in Figure 6.7 from left to right are for the tensions between Stands 1 and 2,
Stands 2 and 3, Stands 3 and 4, and Stands 4 and 5. The individual dynamical processes
of the tension establishment after the strip head enters Stands 2 to 5, are all demonstrated
in Figures 6.8, 6.9, 6.10 and 6.11, respectively. It should be noted that when the strip
head enters a stand, the tensions between all neighboring stands before this stand are
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 182
affected. As the effect is small, it is not obviously clear at points a, b and c as shown in
Figure 6.7.
Tension establishment process during threading
300
250 4
200
| 150
-• 100
50
0
---
n a b c
Enter Stand 2 Enter Stand 3 > /
/ , / ,
Enter Stand 4
/ > .
Enter Stand 5
k / '
10 15 20 25 30 35 40 45 50 55 60 65 Time (s)
Figure 6.7 Tension establishment process during threading
Tension establishment process after the strip head enters Stand 2
u •o
c
S £ £ w
% -g •° § c o '5 c
350
300
250
200
150
100
50
11.16 11.21 11.26 11.31 11.36 11.41 11.46 11.51 11.56
Time (s)
Figure 6.8 Dynamic process of tension establishment after the strip head enters the
second stand
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling
Tension establishment process after the strip head enters Stand 3
CN « •a
c n *-•
u * a> w
» 7 •° is c 0 10 c O H
300 j
250 -
200-
150
100 -
50 -
o -22.8 22.85 22.9 22.95 23 23.05 23.1 23.15 23.2
Time (s)
Figure 6.9 Dynamic process of tension establishment after the strip head enters the third
stand
Tension establishment process after the strip head enters Stand 4
« 160 -| 140 -
between Sta
and 4 (KN)
ro »
o to
o o o o
o 40 J
I 20-•- n U i
^ — « ^ _ _ _-_.
^ ^ - — • - — ' ' "
^ ^
y^ s s /
7 /
7 34.36 34.41 34.46 34.51 34.56 34.61
Time (s)
34.66
Figure 6.10 Dynamic process of tension establishment after the strip head enters the
fourth stand
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 184
Figure 6.11 Dynamic process of tension establishment after the strip head enters the
fifth stand
The threading forces at Stands 2 to 5 are gradually decreasing after the strip head enters
these stands, with the tension gradually establishing. It should be noted that the
threading forces at both stands 1 and 2 are affected when the strip head enters the
second stand. In turn, each time when the strip head enters the next stand, the threading
forces at that stand and its immediate previous stand will be affected. The threading
force fluctuations at each of all five stands during threading are shown in Figures 6.12 to
6.19.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 185
Stand 1 roll force variation after the strip head enters Stand 2
1 11.23 11.28 11.33 11.38 11.43 11.48 11.53
Time (s)
Figure 6.12 Stand 1 threading force variation after the strip head enters Stand 2
Stand 2 roll force variation after the strip head enters Stand 2
Z.
1 1.95 -o " 19-o la
u_
1 1-85 1.8
i i i i i
11.23 11.28 11.33 11.38 11.43 11.48
Time (s)
11.53
Figure 6.13 Stand 2 threading force variation after the strip head enters Stand 2
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 186
Stand 2 roll force variation after the strip head enters Stand 3
22.9 22.95 23 23.05 23.1
Time (s)
23.15 23.2
Figure 6.14 Stand 2 threading force variation after the strip head enters Stand 3
Stand 3 roll force variation after the strip head enters Stand 3
2.9 22.9 22.95 23 23.05 23.1
Time (s)
23.15 23.2
Figure 6.15 Stand 3 threading force variation after the strip head enters Stand 3
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 187
**
2 2.8-
g 2.6-
i£ 2.4-
o tt 2.2
9 J
Stand 3 roll force variation after the strip head enters Stand 4
£ . i i i | i
34.425 34.475 34.525 34.575 34.625 34.675
Time (s)
Figure 6.16 Stand 3 threading force variation after the strip head enters Stand 4
Stand 4 roll force variation after the strip head enters Stand 4
3.75
3.65 -
o L.
o u. o
3.55-
3.45-
34.425 34.475 34.525 34.575 34.625 34.675
Time (s)
Figure 6.17 Stand 4 threading force variation after the strip head enters Stand 4
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 188
or , O.O
| 3 . 4 -
8 3.3
£ 3.2-
S. 3.1 -
Stand 4 roll force variation after the strip head enters Stand 5
O I I I !
46.915 46.965 47.015 47.065 47.115
Time(s)
Figure 6.18 Stand 4 threading force variation after the strip head enters Stand 5
Stand 5 roll force variation after the strip head enters Stand 5
46.915 46.965 47.015
Time (s)
47.065 47.115
Figure 6.19 Stand 5 threading force variation after the strip head enters Stand 5
The strip thickness at the exit of each of the five stands is dropping after the strip head
enters Stands 2 to 5. This is resulted from the gradual tension establishment, which in
turn leads to the decrease of threading force. According to Equation (6-1), the strip
thickness will be dropping when the threading force decreases due to the establishment
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 189
of inter-stand tension. A n overview of the strip thickness variations at each of all five
stands during threading is shown in Figure 6.20. Individual strip thickness variation at the
exit of each stand is also illustrated in Figures 6.21 to 6.28. Each time, when the strip
head enters a stand, the exit thickness at all other stands before this stand will be
affected. Similarly for tension and roll force, as this effect is small, it is not obviously
clear, as shown in Figure 6.20 (refer to the points a, b and c in Figure 6.7).
Exit thickness variation during threading
1.85
: 1.65
1.25
E
& ' </> tfl CU
c i 1.05 s 0.85
w 0.65 4
0.45
• ^
\
Enter Standi
i ^
-~ Enter Stand 2
~\
Enter Stand 3---^.
Enter Sta TY\ A h. *™ * m.
bnter otana o r • r i i i i
0 10 20 30 40 Time (s)
50 60
Figure 6.20 Overview of the strip exit thickness variation at each of all five stands
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 190
Stand 1 exit thickness variation after the strip head enters Stand 2
. - 1.8 E E, 1.78 g 1.76 4
JE 1-74 4 u j= 1.72 4 •S 1-7 LU
1.68
11.235 11.285 11.335 11.385 11.435
Time(s)
11.485
Figure 6.21 Exit thickness variation at Stand 1 after the strip head enters Stand 2
Stand 2 exit thickness variation after the strip head enters Stand 2
1.21 11.235 11.285 11.335 11.385
Time(s)
11.435 11.485
Figure 6.22 Exit thickness variation at Stand 2 after the strip head enters Stand 2
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 191
Stand 2 exit thickness variation after the strip head enters Stand 3
22.905 22.955 23.005 23.055
Time (s)
23.105 23.155
Figure 6.23 Exit thickness variation at Stand 2 after the strip head enters Stand 3
Stand 3 exit thickness variation after the strip head enters Stand 3
| 0.86
£ 0.84 JZ
~ 0.82 x LU 0.8
22.905 22.955 23.005 23.055
Time (s)
23.105 23.155
Figure 6.24 Exit thickness variation at Stand 3 after the strip head enters Stand 3
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 192
Stand 3 exit thickness variation after the strip head enters Stand 4
34.43
Enter Stand 4
34.48 34.53 34.58
Time (s)
34.63 34.68
Figure 6.25 Exit thickness variation at Stand 3 after the strip head enters Stand 4
Stand 4 exit thickness variation after the strip head enters Stand 4
0.63
34.43 34.48 34.53 34.58
Time(s)
34.63 34.68
Figure 6.26 Exit thickness variation at Stand 4 after the strip head enters Stand 4
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 193
Stand 4 exit thickness variation after the strip head enters Stand 5
46.92 46.97 47.02
Time(s)
47.07 47.12
Figure 6.27 Exit thickness variation at Stand 4 after the strip head enters Stand 5
Stand 5 exit thickness variation after the strip head enters Stand 5
0.56
0.55
0.53 -
E
— C 54 ifl 0)
c S 0.52 4 x: t 0.51 x w 0.5 0.49
46.92
Enter Stand 5
46.97 47.02
Time(s)
47.07 47.12
Figure 6.28 Exit thickness variation at Stand 5 after the strip head enters Stand 5
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 194
6.5 NUMERICAL SIMULATION OF ACCELERATION PROCESS
6.5.1 Flow Chart of Acceleration Simulation
The acceleration simulation is conducted on a Pentium III 500, and the flow chart is
illustrated in Figure 6.29.
Preset Acceleration Parameters
-&-
Initialize parameters: i=\,n = 0,N = 5 , t = 0
Initialize the tension for all stands as the tension value at the last moment of threading process
-ikl
Calculate v? ,U? ,0? , F," ,t? , h? <-
_^_
Calculate h" - ht
Reset roll gap Sx :> n = n + 1
^L i = i + l
^L Calculate v5 , w5 , U 5 , b 5 ,t5, hs
Stop simulation, and print out the simulation results
Yes
<-
No ^L
Calculate ^s" ~ h5
Reset roll gap S5 ^ n = n + 1
->
->
Figure 6.29 Flow chart for the acceleration process simulation of a tandem cold mill
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 195
6.5.2 Acceleration Simulation Parameters
The strip and mill parameters used in the simulation are as follows:
The strip entry thickness 2.50mm, final exit thickness 0.555mm, and width 930mm;
the mill modulus is 4MN/mm.
The constant head end threading speed is 20m/min, the maximum rolling speed is
19.2m/s, and other parameters employed are listed in Table 6.2.
Table 6.2 Acceleration simulation parameters
Mill Parameters Work Roll Diameter (mm) Backup Roll Diameter (mm)
Backup Bearing Diameter (mm) Backup Bearing Width (mm)
Bearing Radial Clearance (mm) Bearing Oil Viscosity (N s/mm2)
Standi 594.66 1324.9 920 750 0.65 0.539
Stand 2 578.82 1291.4 920 750 0.65 0.539
Stand 3 556.0 1343.2 920 750 0.65 0.539
Stand4 591.0
I 1273.2 920 750 0.65 0.539
Stand 5 590.64 1256.4 920 750 0.65 0.539
6.5.3 Acceleration Simulation Results
The simulation is conducted based on an acceleration rate of 2.0m/s2 at Stand 5. The
acceleration rates for the other four stands are determined using the following formula
[Yang, 1988]:
at=a5^- (6-16) V05
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 196
where ai and a5 are the acceleration rates at the /* and 5th stands, v0i and v05 are the
work roll surface speed at the i'h and 5rt stands (z = l,2, 3, 4) during threading,
respectively. The rolling speed variations at Stand 5 during acceleration are illustrated in
Figure 6.30, and the friction coefficient between the work roll and strip is changed with
rolling speed accordingly, as shown in Figure 6.31.
To verify the simulation results, the simulated roll force at each stand is compared with
the actual measurement. The comparison results demonstrated in Figure 6.32 show that
they match well. The roll force deviations at all 5 stands are also shown in Figure 6.33.
It can be seen that with increasing rolling speed, the friction coefficient is dropping and
accordingly the roll force is decreasing and directly causes the exit thickness variation.
With the AGC feedback control loop illustrated in Figure 6.4, the exit thickness
variation is mostly corrected, and the exit thickness at the exit of Stands 1 and 5 are
shown in Figures 6.34 and 6.35, respectively. As shown in Figure 6.35, the simulated
maximum percentage error of the strip exit thickness at Stand 5 is about 5.77% when
the AGC control is operational. Figures 6.34 and 6.35 also show that the exit thickness
will keep dropping without AGC control.
The periodical fluctuations of exit thickness at both Stands 1 and 5 illustrated in Figures
6.34 and 6.35 are caused by the time delay between the roll force measurement and
feedback control action. It should be pointed out that the results shown in Figures 6.34
and 6.35 are based on the assumption of time delay 0.05 seconds. The time delay
depends on the speed of computer used for the control, and the response time of the
screw-down drive system. If the time delay is 0.1 seconds, the exit thickness variations
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 197
at Stands 1 and 5 will be as shown in Figures 6.36 and 6.37. It can be seen the periodical
fluctuations are less frequent than that for the time delay 0.05 seconds.
Figures 6.34, 6.35, 6.36 and 6.37 show that the simulated gauges at Stands 1 and 5 with
AGC control are close to their desired exit thickness, although they are slightly below
the desired thickness lines. This is because:
a) The acceleration process follows the threading process. When the mill is accelerated,
the strip entry thickness is less than the desired one due to the off-gauge strip
produced during threading.
b) When the mill speed increases, speed-dependent rolling parameters keep changing
as addressed in Section 6.2.2. Namely, variable friction between work roll and strip,
inter-stand tension, roll force and backup roll bearing oil film thickness lead to the
exit thickness deviations.
As addressed in Section 6.3.2.1, the control compensation coefficient C in Equation (6-
14) is usually empirically determined. When it is assigned to 1, the mill elastic
deformation is completely compensated for, the mill stand seems infinitely stiff. When
it is assigned to 0, partial compensation is obtained, the mill stand becomes less rigid.
Based on experiences, the control compensation coefficient is assigned to a greater
value for earlier mill stands and smaller value for subsequent stands. The compensation
coefficient is also product and mill dependent. More discussions on the control
compensation coefficient C can be found in Appendix C.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 198
25
Rolling speed variation at Stand 5 during acceleration
Full speed rolling phase
4 6 8 Time (s)
10 12
Figure 6.30 Rolling speed variations at Stand 5 during acceleration
0.08
_ 0.07 c 0 » 0.06 it a> a 0.05 c o 1 0.04 EL
0.03 0.02
7 Friction variation during acceleration
Stand 1 Stand 2 Stand 3
Stand 4
4 6 Time (s)
Stand 5
z 8
Figure 6,31 Variation of mean friction coefficient during acceleration at all 5 stands
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 199
Comparison of roll forces at different stands during acceleration
Actual measurements
Simulation results
Stand No. (1-5)
Figure 6.32 Comparison of roll forces from the simulation with actual measurements
10
9
I 8
o £ 7
Roll force variation during acceleration
Stand ̂
Stand 2
k Stand 3
k Stand 4 to^ Stand 5
0 1 4 5 Time (s)
~i r
Figure 6.33 Roll force variation during acceleration
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 200
1.84
182
6 F Ul 40
C Jt (1
18
178
1./B
X 174 LU
17
Comparison of exit thickness at Stand 1 during acceleration
Desired thickness
± With AGC control
Without AGC control
1.72 4-
3 4 5 6
Time (s)
Figure 6.34 Exit thickness variation at Stand 1 during acceleration (time delay = 0.05
seconds)
0.64 0.62
0.6 +
E
£ in
X HI
0.46 0.44 +
0.42
0.4
Comparison of exit thickness at Stand 5 during acceleration
With A G C control Without A G C control
4 5
Time (s)
Desired exit thickness
Figure 6.35 Exit thickness variation at Stand 5 during acceleration (time delay = 0.05
seconds)
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 201
184
182
E E t8
in
m 178
% 176
X 174 LU
172
17
Comparison of exit thickness at Stand 1 during acceleration
Desired thickness
JJUli 1 m mmm i
Wth AGC control -A Without AGC control
3 4 5
Time (s) 8 9
Figure 6.36 Exit thickness variation at Stand 1 during acceleration (time delay = 0.1
seconds)
Comparison of exit thickness at Stand 5 during acceleration
With AGC control Without AGC control
0 1 2 3 4 5 6 7 8 9
Time (s)
Figure 6.37 Exit thickness variation at Stand 5 during acceleration (time delay = 0.1
seconds)
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 202
6.6 OTHER RELEVANT ISSUES
6.6.1 Deceleration and Tailing out processes
Deceleration and tailing out processes are similar to acceleration and threading
processes, respectively, in the modeling and simulation. The difference between
threading and tailing out processes is that there is no front tension during threading,
while there is no back tension during tailing out. The distinction between acceleration
and deceleration processes is that rolling speed is increasing during acceleration, while
decreasing during deceleration. The proposed models for threading and acceleration can
be applied to deceleration and tailing out with a little modification.
6.6.1.1 Deceleration Process
At the end of the rolling for a coil, to prepare for the strip tailing out, the rolling mill is
decelerated so as to achieve the tailing out speed. It is an opposite process to the
acceleration. During deceleration, the strip thickness increases [Sims and Arthur, 1952].
The exit thickness variations are mainly resulted from the following causes:
(1) as a result of decreasing rolling speed, friction coefficient rises and, thus, roll force
increases as shown in Figure 6.38, and
(2) decrease of the thickness of the oil film separating the roll neck and the chock as the
mill speed decreases.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 203
These directly lead to the change of roll gap position and accordingly produce an
increase in the strip exit thickness as shown in Figure 6.39. During deceleration, AGC
control loop is still operational, thus the exit thickness is still being controlled. As shown
in Figure 6.39, the simulated maximum percentage error of the strip exit thickness at
Stand 5 is about 5.59%.
10 -
—. 9 -E E. H in
g 8-ic
o 3E I-
* 7 LU '
6
Roll force variation during deceleration
t Stand 1
Stand 2 Stand 3
ir ,..,. ,
- ' ' ' " A
T" 0 Stand 4 Stand 5
0 1 2 3 4 5 6 7 8 9
Time (s)
Figure 6.38 Roll force variation during deceleration
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 204
Beit thickness variation during deceleration
E E 10
v
1.6
1.2
I 0.8
x "" 0.4
to* : t o t o ~ ~ Standi Stano>2
StancKJ
4 ~~~~~~~~~~™ / ... *
Stand4 Stands
2 3 4 5 6 7 8 9
Time (s)
Figure 6.39 Exit thickness variation during deceleration
6.6.1.2 Tailing Out Process
Tailing out is also named de-threading. It is the last phase of the rolling of a coil. The
tailing out process begins when the strip tail being rolled leaves the uncoiler and ends
when the strip is completely wrapped onto the coiler. Normally, the tailing out speed is
as low as threading speed. When the strip tail leaves the uncoiler, the back tension of
Stand 1 disappears. The tension between Stands 1 and 2 is removed when the strip tail
exits from Stand 1, and so on for other stands. The process continues until the strip tail is
coiled onto the coiler. As there is no back tension, the tailing out process is accompanied
with tension, roll force and exit thickness fluctuations.
Similar to the threading process, the deviation of strip thickness is resulted from several
influence factors such as the lack of the back tensions and other disturbances from stand
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 205
and strip. Figure 6.40 shows the roll force variations during tailing out. It should be
noted that the roll force at Stand 2 increases when the strip tail goes out from Stand 1.
The roll forces at other stands are also affected when the strip is tailing out from Stand 1,
however, as the effect is small, it is not obviously clear, as shown in Figure 6.40.
Similarly, for the roll forces at Stands 3, 4 and 5, they increase whenever the strip tail
goes off from their immediate previous stands. Accordingly, the exit thickness at
individual stands rises when the strip tails out from their immediate previous stands as
shown in Figure 6.41. The exit thickness is increased by 11.1% at Stand 2 when tailing
out from Stand 1, by 11.0% at Stand 3 when tailing out from Stand 2, by 12.1% at Stand
4 when tailing out from Stand 3, and by 11.3% at Stand 5 when tailing out from Stand 4.
It should be pointed out that the AGC control loop is not operational during tailing out.
Roll force variation during tailing out
10
IT 7H
6
4) P 5
Tailing out from Stand 1
4 Roll force at Stand 4
Roll force at Stand 2
3
2 •--
Tailing oul^fromStand 4
Tailing outfromStand 2
Roll force at Stand 3
. .T^i'Qd.ouLflPJTL _ Stand 3
Roll force at Stand 5
15
Time (s)
20 25 30
Figure 6.40 Roll force variation during tailing out
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 206
r Beit thickness variation during tailing out
1.6
1.4 -I
1.2 --
in in
<u c •x
u
0.8
Z 0.6 -
Tailing out from Stand 2
Tailing out from Stand 3
J Beit thickness at Stand 5
Exit thickness at Stand 2
Exit thickness at Stand 3
1 Exit thickness at Stand 4
Tailing out from Stand 4 "^ ioa
% Tailing out from Stand 5
10 15
Time (s)
20 25 30
Figure 6.41 Exit thickness variation during tailing out
6.6.2 Effect of Friction on Unsteady Rolling Processes
In Section 6.5, unsteady rolling processes are simulated, and the interactions among
rolling parameters including tension, roll force and exit thickness are graphically
illustrated. This section focuses on the effect of some rolling parameters such as friction
coefficients on unsteady rolling processes. For example, during threading process, if the
rolling schedule (the stand reduction and rolling speed) is kept as listed in Table 6.1 and
apply the following friction conditions on Stands 1 and 5, respectively, threading force
varies significantly and, hence, the exit thickness is affected:
(1) apply 3 0 % less friction,
(2) keep the normal friction condition, and
(3) apply 30% more friction.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 207
The variation of roll force with friction is significant. Figures 6.42 and 6.44 illustrates
the threading force variations at Stands 1 and 5 with the above friction conditions
applied, and the exit thickness at Stands 1 and 5 are demonstrated in Figures 6.43 and
6.45, respectively. It should be noted that the initial roll gap setup positions at Stands 1
and 5 are adjusted to keep the scheduled reduction at the same stand when different
friction conditions are applied. Figure 6.42 shows that roll force falls by about 15.19%
when 30% less friction is applied on Stand 1, and the roll force rises by 16.54% when
30% more friction is applied. Figure 6.44 shows that the roll force falls by about
19.23% when 30% less friction is applied on Stand 5, and roll force rises by 26.97%
when 30% more friction is applied. To produce strip of constant thickness, the initial
roll gap is set to a certain position so as to compensate the roll force change due to
different friction conditions applied, mill stretch, roll eccentricity etc. Thus, when the
strip head is just threaded through Stand 5, the initial exit thickness at Stand 5 is kept
constant under the different friction conditions. After the strip head is threaded through
Stand 5 and before it reaches the coiler of the mill, the strip will be rolled under a
relatively stable condition. As shown in Figure 6.45, the exit thickness at Stand 5 is
increased by 2.2% if 30% less friction is applied, comparing with the exit thickness
when the normal friction condition is applied, and it is dropped by 1.8% if 30% more
friction is applied. This is because that the roll gap at Stand 5 is set to a smaller value
when 30% more friction is applied than when 30% less friction applied. Thus, the strip
off-gauge situation can be alleviated a little by applying appropriate friction conditions
during threading.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 208
When the friction changes, as a result, the forward slip varies. It will rise when the
friction coefficient increases and vice versa when the friction coefficient decreases.
Hence, when 30% more friction is applied to Stand 1, the exit strip speed is faster than
that for the cases when normal friction and 30% less friction condition apply. This leads
to less time for the strip head to travel from Stand 1 to Stand 2. Refer to Figures 6.42
and 6.43, among the above three friction conditions, it takes the longest time for the
strip to travel from Stand 1 to Stand 2 when 30% less friction is applied. From this point
of view, greater friction coefficient may help to reduce threading time and, hence, strip
off-gauge length. However, the friction coefficients required for threading are primarily
governed by the threading conditions such as roll wear, roll gap setup, which, in turn, is
influenced by the mill stretch effect. The roll wear is relevant to both strip and roll
roughness. Thus, the friction condition should be determined based on overall
consideration on its effect on threading time, in turn, off-gauge length, and exit
thickness error as addressed in the last paragraph.
There are many factors affecting friction between the work roll and strip. They are strip
surface condition such as pickling oil, surface roughness of the rolls and the strip,
surface hardness of the rolls and the strip, coolant lubrication conditions and flow rates,
the roll and strip temperature, and rolling speed which is discussed in Section 6.5. The
application of the coolant flow, rolling speed and work roll surface roughness has an
enormous effect on the friction coefficient. Details of how to control the friction
coefficient is out of the scope of this thesis.
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in T a n d e m Cold Rolling 209
Roll force variation at Stand 1 during threading with different friction
conditions applied
0) p
o 1.1 -
0.9
W h e n threading into Stand 2
W h e n 3 0 % m o r e friction
applied, toi StandII
£ When 3 0 % less When normal
friction apllFedtb "" Thctfori condition
Stand 1 applied to Stand 1
11.12 11.22 11.32 11.42
Time(s)
11.52 11.62
Figure 6.42 Roll force variation at Stand 1 during threading with different friction
conditions applied
£ £
c -X
o
1.85
1.8
1.75
Exit thickness variation at Stand 1 during threading when different friction conditions applied
S 1.7 x OJ
1.65
When threading into Stand 2
When 30% more friction applied to Stand 1
11.12 11.22
When 3 0 % less friction applied to Stand 1
11.32 „ , ,11.42 Time (s)
When normal friction condition applied to 15fand "T
11.52 11.62
Figure 6.43 Exit thickness variation at Stand 1 during threading with different friction
conditions applied
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 210
Roll force variation at Stand 5 during threading when different friction
applied
- 5.5
£ 5 <D O
o 4.5 u. =5 4 3.5
W h e n 3 0 % more friction applied to Stand 5
W h e n normal friction condition applied to Stand 5
W h e n 3 0 % less friction applied to Stand 5
46.92 46.97 47.02 47.07
Time(s)
47.12 47.17
Figure 6.44 Roll force variation at Stand 5 during threading with different friction
conditions applied
Exit thickness variation at Stand 5 during threading when different
friction applied
0.48
46.92
W h e n 3 0 % less friction applied to Stand 5
W h e n normal friction condition applied to Stand 5
W h e n 3 0 % more friction applied to Stand 5
46.97 47.02 47.07
Time (s)
47.12 47.17
Figure 6.45 Exit thickness variation at Stand 5 during threading with different friction
conditions applied
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 211
6.6.3 Effect of Strip Hardness and Entry Thickness
Strip hardness and entry thickness also have significant effect on unsteady rolling
processes.
Different grades of steel, from the softest to hardest, lead to different rolling conditions
[D'Alessio, 1992]. The harder the steel strip, the greater is the yield stress. In turn,
greater yield stress results in greater roll force and exit thickness. On the other hand, the
increase in strip hardness also results in more of the asperity of the strip and work rolls
riding on each other, rather than on meshing into the valleys, thus, making the roll bite
more "slippery" [Yuen, Popelianski and Prouten, 1996], namely, less friction.
If the strip has been annealed, the hardness will usually be more or less uniform when it
is threaded at Stand 1. However, the strip off-gauge in unsteady rolling processes may
be associated with a change in hardness when it is rolled in the successive stands of a
tandem cold mill. Strip yield stress during threading may be greater than that during full
speed rolling due to the lower strip temperatures [BHP, 1979]. Taking the change in
hardness into account will improve the simulations of unsteady rolling processes. The
proposed modes can be extended to include the hardness change.
Apart from the entry thickness change at Stands 2 to 5 due to strip off-gauge at Stand 1
exit, the initial thickness at the entry of Stand 1 may be slightly variable, which in turn
results in changes in roll force and exit thickness. In tracking the strip thickness change
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 212
during unsteady rolling, the initial entry thickness variation should also be taken into
account in the simulation models if the entry thickness measurement is available.
6.6.4 Effect of Work Roll Profile
The thermal camber and wear profile of work roll is required to be calculated
continuously in order to correctly compensate for the screw positions during unsteady
rolling. The presence of cold rolls after a roll change should also be taken into account.
6.6.5 Strip Shape Consideration
During unsteady rolling, strip shape requirements should be met in order to avoid delays
and roll damage. In general, this requires schedule dependent adjustments to be applied
to the roll speed and screw position references. The dynamic control systems reset the
screw position by feedback control action at an appropriate time. As a result it is
necessary to sacrifice a certain length of off-gauge strip to achieve a satisfactory shape.
6.7 SUMMARY AND CONCLUDING REMARKS
The threading process of a tandem cold rolling mill is dynamically simulated and the
result from the simulation is well matched with the actual measurements from the plant.
The gradual tension establishment process, threading force variation and the strip
thickness variation are demonstrated when the strip head threads through the whole mill
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 213
with a time step of 0.005 seconds. This provides the basis for theoretical analysis of the
interactions of rolling parameters during threading.
The acceleration process of tandem cold rolling is also modeled and simulated, taking
into account the speed dependent parameters including backup bearing oil film
thickness and friction coefficient. The relationships between the final exit thickness and
speed are presented graphically with and without AGC control. Methods of controlling
the strip gauge during acceleration are addressed, and the exit thickness deviations with
and without AGC control are compared. The simulation results show the AGC control
provides an efficient correction of the speed effect on strip gauge and roll force. The
simulated roll force at Stands 1 to 5 matches the actual plant data well. The simulated
results show that the simulated maximum percentage error of exit thickness at Stand 5 is
5.77% when the AGC feedback control loop is operational.
The deceleration and tailing out processes are also briefly addressed, as they are similar
to the acceleration and threading processes, respectively. During deceleration, the
friction coefficients at all 5 stands increase when the rolling mill speed is decreasing,
thus, the roll forces at all 5 stands are increasing accordingly. This results in increasing
exit thickness at the exit of all five stands. During the deceleration, the AGC is still
applied to the rolling mill. The simulated maximum percentage error of exit thickness at
Stand 5 is 5.59%. Thus, the AGC is still one of the most important means to reduce the
off-gauge length of the strip during deceleration. During the tailing out, the strip tail end
gradually goes out from the uncoiler, Stand 1, Stand 2, Stand 3, Stand 4 and Stand 5,
and then is wrapped onto the coiler. When the strip end tails out from a stand, the back
Chapter 6 Numerical Simulation and Analysis of Unsteady Processes in Tandem Cold Rolling 214
tension between the stand and its immediate next stand is removed. This results in
increased roll force at the immediate next stand, thus, increased exit thickness.
The effect of friction on roll force and exit thickness during threading is also
investigated. When the friction increases, the forward slip will rise, and vice versa when
the friction decreases. The roll force and exit thickness variations at Stands 1 and 5 are
demonstrated when 30% more friction, normal friction, and 30% less friction conditions
are applied to both Stands 1 and 5. The simulated results show that applying appropriate
friction condition by a combination of roll surface roughness and lubrication can lead to
a little improvement of the off-gauge situation on account of exit thickness variation and
threading time.
Chapter 7 Conclusions and Recommendations 215
Chapter 7
Conclusions and Recommendations
7.1 SUMMARY OF MAIN CONCLUSIONS
The main conclusions drawn from this research are as follows:
(1) The Generic Algorithms based searching procedure is reliable for rolling schedule
optimization for tandem cold rolling and can solve complex engineering
optimization problems solving, where the best combination of multiple objectives
and/or multiple parameters is the optimization objective instead of individual
objectives and/or parameters.
(2) Heuristic knowledge, such as a starting point of searching, is important in directing
GA's searching in terms of reducing the solution space and searching time.
Empirical engineering solution is reliable heuristic knowledge. When it is used to
guide GA's searching for an optimal solution in a neighborhood space of the
empirical solution, the original empirical solution can be potentially improved.
(3) The proposed approach for rolling scheduling optimization is robust, effective and
efficient for tandem cold rolling and the approach could be employed to
significantly improve empirically derived settings for tandem cold rolling mills.
Moreover, the approach is generic and can be extended to tandem hot rolling. The
Chapter 7 Conclusions and Recommendations 216
results generated from the proposed approach show that uniform power distribution
and good flatness are achieved - shoot two birds with one arrow - in terms of
energy saving and product quality. It has the potential to be applied in steel
industry.
(4) The proposed theoretical predictive model combining the strength of both Influence
Coefficient Method (ICM) and Spring and Beam Method (SBM) is promising in
the analysis of strip shape during tandem cold rolling. The model is flexible to deal
with various rolling mills such as those with roll side-shifting systems, variable
crown rolls etc. For on-line efficient use in industry, the model is simplified and a
numerical simulation has been conducted based on typical plant data.
(5) The proposed roll profile model in combination with GA based optimization can be
used to deal with the presetting of reductions, roll force and/or roll bending force,
especially for the threading process of tandem cold rolling. Based on perfect shape
condition, more accurate threading parameters can be determined. Shape
optimization parameter is defined in terms of accumulated strip reduction error at
each point across the strip width to assist the searching of optimal threading
parameters in their solution space.
(6) During threading process of tandem cold rolling, smaller work roll crowns and
optimal presetting of roll bending force are two effective proposals to achieve
quality shape of thin strip such as tinplate and in turn minimizing threading delay.
Chapter 7 Conclusions and Recommendations 217
(7) The proposed models for threading and acceleration processes of tandem cold
rolling are valuable and innovative for dynamically simulating the unsteady rolling
process and understanding the relationships between interactive rolling parameters
and disturbances.
(8) The effect of friction, strip hardness and entry thickness on unsteady rolling
processes is analyzed. Simulation results demonstrate that the exit thickness at the
last stand during threading is increased by 2.2% if 30% less friction is applied,
comparing with the exit thickness when the normal friction condition is applied,
and it is dropped by 1.8% if 30% more friction is applied. Preliminary research
results show that applying appropriate friction condition between the work roll and
the strip may decrease off-gauge portion of downgraded strip on account of
threading time and exit thickness variation.
7.2 RECOMMENDATION FOR FUTURE RESEARCH
Tandem cold rolling process is nonlinear and time varying. The task to improve the
production performance of tandem cold rolling mills is endless. In the last decade, more
attentions have been paid on the improvement of rolling models and applications of
computational intelligence. The research presented in this thesis is only a small part, and
future work is recommended for further investigation in the area.
Chapter 7 Conclusions and Recommendations 218
The optimum rolling scheduling reported in Chapter 4 can be extended for on-line
adaption capable of dealing with changes both in plant conditions and strip properties
such as entry thickness and hardness.
In analyzing the strip deformation, the deformed work roll profile is calculated based on
the assumption that the work rolls were deformed into a circular arc. This is applicable
in normal cold rolling conditions where the strip thickness is not very thin. However,
when rolling very thin strip, for example, the strip thickness is less than 0.2mm, the
work rolls deform substantially, and numerical instability could occur [Yuen, 1995]
with significant roll flattening. The elastic recovery of the strip has not been taken into
consideration in the plastic strip deformation formulation. Fleck et al. [Fleck et al., 1987
and 1992] proposed a new theory of cold rolling thin strip. Therefore, the new theory
should be applied when the proposed models in this thesis are to be employed to deal
with thin strip.
The modeling and simulation of deceleration and tailing out processes are not repeated
in details, as they are similar to acceleration and threading processes, respectively.
Friction coefficient is a crucial rolling parameter for unsteady tandem cold rolling
processes. Preliminary research into its influence on strip thickness irregularities during
threading and acceleration has been conducted, further investigation is recommended
regarding its effect on other unsteady rolling processes, and how to control it.
Opportunities may exist to further alleviate head end off-gauge situation by improving
AGC performance during acceleration. Assuming the acceleration process can be
Chapter 7 Conclusions and Recommendations 219
divided into n portions, the screw positions for each portion may be optimized using
the optimization procedure proposed in this thesis instead of using feedback exit
thickness error. Another way to obtain the optimized screw positions for the individual
portions may be "coil-to-coil" learning. After rolling several coils with the same strip
dimensions and reduction requirements, the screw position curves controlled by AGC
may be kept in long term memory of the mill control system. When rolling a coil,
simply recall the screw position memory learned from the previous rolled coils with the
same dimensions and reduction requirements. The "coil-to-coil" learning may be
applicable to deceleration process as well.
To obtain further data for transitional phenomena such as tensions, roll forces, shape
and strip gauges may be beneficial to further investigation into the unsteady rolling
processes. In this thesis, some results such as strip exit thickness, shape and roll force
generated from the simulations have been compared with that obtained from a 5-stand
tandem cold mill to verify the proposed models. However, the transitional tension
measurements are not available at the time of thesis writing. It will be useful for further
improvement of the proposed models to do more experiments to get a complete set of
data for all transitional phenomena from all unsteady processes including threading,
acceleration, deceleration and tailing out.
The proposed models have a potential to be applied in steel industry. Although the
results generated from the models and simulations have been compared with the real
data from steelworks, they should be further tested on tandem cold rolling mills.
References 220
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Appendix A Classical Optimisation Hierarchy 251
Appendix A
Classical Optimisation Hierarchy
The hierarchy of classical optimisation is illustrated in Figure 2.9 of Chapter 2. The
optimisation is classified into continuous optimisation, and discrete optimisation
[NEOS, 1996]. The discrete optimisation is composed of integer programming and
stochastic programming. The continuous optimisation includes Unconstrained and
Constrained Optimisation. The Unconstrained Optimisation comprises of Nonlinear
Equations, Nonlinear Least Squares, Global Optimisation, and Non-differentiable
Optimisation. The Constrained Optimisation consists of Linear Programming, Semi-
definite Programming, Nonlinearly Constrained, Bound Constrained, Quadratic
Programming, Network Programming, and Stochastic Programming.
A. DISCRETE OPTIMISATION
The discrete optimisation includes integer programming and stochastic programming.
They are outlined in the following paragraphs.
(a) Integer Programming
The book by [Nemhauser and Wolsey, 1988] provides a comprehensive guide to integer
programming, while their review paper, [Nemhauser et al., 1989], is a more concise
Appendix A Classical Optimisation Hierarchy 252
overview. Schrijver's book [Schrijver, 1986] also covers a survey of integer
programming.
Integer programming problems, such as the fixed-charge network flow problem and the
famous traveling salesman problem, are often expressed in terms of binary variables. In
many applications, the solution of an optimisation problem makes sense only if a certain
number of the unknowns are integers. Integer linear programming problems have the
general form
mm{cTx:Ax = b,x>0,xeZ"} (A-l)
where cTx is the direct cost, Z" is the set of n -dimensional integer vectors, A and b
are constants. In mixed-integer linear programming, some components of x are allowed
to be real.
(b) Stochastic Programming
Most of the model formulations in the optimisation tree have assumed that the data for
the given problem are known accurately. However, for many problems, the solution
data cannot be known accurately for a variety of reasons. The first reason is due to
simple measurement error. The second and more fundamental reason is that some data
represents information about the future (e.g., predicted steel strip thickness and flatness
for a future time period) and simply cannot be known with certainty. The fundamental
idea behind stochastic linear programming is the concept of recourse [Abel, 1993].
Appendix A Classical Optimisation Hierarchy 253
Recourse is the ability to take corrective action after a random event has taken place
[Ahmed et al., 2000]. A simple example of two-stage recourse is the following:
(1) Choose some variables, for example, x, to control what happens today.
(2) Overnight, a random event happens.
(3) Tomorrow, take some recourse action, y, to correct what may have been affected
by the random event.
Such kind of optimisation problems can be formulated to choose x and y in an optimal
way. In this example, there are two periods; the data for the first period is known with
certainty and some data for the future periods is stochastic, that is, random. Stochastic
programming enables the modeler to create a solution that is optimal over a set of
scenarios [Abdreatta, 1995].
B. CONTINUOUS OPTIMISATION
Continuous optimisation is composed of Unconstrained and Constrained Optimisations.
The Unconstrained Optimisation comprises of Nonlinear Equations, Nonlinear Least
Squares, Global Optimisation, and Non-differentiable Optimisation. The Constrained
Optimisation consists of Linear Programming, Semi-definite Programming, Nonlinearly
Constrained, Bound Constrained, Quadratic Programming, Network Programming, and
Stochastic Programming.
Appendix A Classical Optimisation Hierarchy 254
(A) Unconstrained Optimisation
The unconstrained optimisation problem [Kowalik and Osborne, 1968; and Powell,
1970] is central to the optimization problem solving. Constrained optimisation
algorithms are often extensions of unconstrained algorithms, while nonlinear least
squares and nonlinear equation algorithms tend to be specialization.
In the unconstrained optimisation problem, the optimisation objective is to seek a local
minimum of a real-valued function, f(x), where x is a vector of n real variables. In
other words, a vector, x*, is the solution, such that f(x* ) < f(x) for all x close to x .
Global optimisation algorithms try to find a vector, x*, that minimizes f(x* ) over all
possible vectors. This is a hard problem to solve, and so far, no efficient algorithm is
known for performing this task. For many applications, local minima are good enough,
particularly when the user can draw on his/her own experience and provides a good
starting point for the algorithm. There are four areas in the Unconstrained Optimisation,
they are: Nonlinear Equations, Nonlinear Least Squares, Global Optimisation and Non-
differentiable Optimisation.
(a) Nonlinear Equations
The interest in solving systems of nonlinear equations originated in the work on
numerical integration. The construction of integration rules forced us to study systems
of polynomial equations. Nowadays this is an independent research area. The research
concentrates on methods to find all solutions of systems on nonlinear equations in a
Appendix A Classical Optimisation Hierarchy 255
given region. Nonlinear systems of equations are the basis for many of the models of
phenomena in science and engineering, and their efficient numerical solution is critical
to progress in these areas. Systems of nonlinear equations arise as constraints in
optimisation problems [Ortega and Rheinboldt, 1970], but also arise, for example, when
differential and integral equations are discretized [Dennis and Schnabel, 1996], In
solving a system of nonlinear equations, a vector is searched such that f(x) = 0 where
xisan -dimensional vector with n variables.
(b) Nonlinear Least Squares
Nonlinear Least Square algorithms are discussed in the books of Bates and Watts [Bates
and Watts, 1988]; Dennis and Schnabel [Dennis and Schnabel, 1983]; Fletcher
[Fletcher, 1987]; Gill, Murray, and Wright [Gill et al., 1981]; and Seber and Wild
[Seber and Wild, 1989]. The books by Bates and Watts and by Seber and Wild
introduce Nonlinear Least Square from a statistical point of view. Bates and Watts
emphasize applications, while Seber and Wild concentrate on computational methods.
Bjorck [Bjorck, 1990] discusses algorithms for linear least square problems in a
comprehensive survey that covers, in particular, sparse least square problems and
nonlinear least squares.
The nonlinear least square problem has the general form
min{r(x): x e IR" } (A-2)
Appendix A Classical Optimisation Hierarchy 256
where r(x) is the function defined by r(x) = -\f(x^2 for some vector-valued
function f(x) that maps R" to Rm.
Least square problems often arise in data-fitting applications. Suppose that some
physical process is modeled by a nonlinear function <j>(x,t,) that depends on a
parameter vector x and time tt. If bt is the actual output of the system at time tt, then
the residual (j)(x, r.) measures the discrepancy between the predicted and observed
outputs of the system at time f.. A reasonable estimate for the parameter x may be
obtained by defining the i"1 component of f.(x) by f.(x) = <j>(x,ti )-bn and solving
the least square problem with this definition of f(x).
(c) Global Optimisation
Global optimisation is the task of finding the absolutely best set of admissible
conditions to achieve optimisation objective, formulated in mathematical terms [Pinter,
1995]. In general, there can be solutions that are locally optimal but not globally
optimal. Consequently, global optimisation problems are typically quite difficult to
solve exactly. Global optimisation problems often fall within the broader class of
nonlinear programming (NLP). A basic reference on most aspects of global optimisation
is the book edited by R. Horst and P.M. Pardalos [Horst and Pardalos, 1995]. Global
optimisation includes the following fields, such as global optimality conditions,
complexity issues, concave minimization, dc methods, indefinite quadratic
Appendix A Classical Optimisation Hierarchy
programming, fractional programming, network problems, continuation methods,
interval methods, and stochastic methods (including simulated annealing).
Methods for global optimisation problems can be categorised based on the properties of
the problem that are used and the types of guarantees that the methods provide for the
final solution. A NLP has the form
minimize F(x)
subject to g(x) = 0 for i = 1,..., mx where m, > 0
i++
h(x) > 0 for /' = mx +1,..., m where m > m,
/++
where F(x) is a function of a vector of real variables x, which is subject to equality
and inequality constraints. Some of the most important classes of global optimisation
problems are differential convex optimisation, complementary problems, minimax
problems, bi-linear and bi-convex programming, continuous global optimisation and
quadratic programming.
(d) Non-differentiable Optimisation
In some well-known instance, the problem functions themselves are not smooth, but
rather are composites of smooth functions [Gill, Murray, and Wright, 1981]. The
optimisation with this kind of problem functions is defined as Non-differentiable
Appendix A Classical Optimisation Hierarchy 258
Optimisation which is often required to obtain good lower bounds and approximate
solutions in the combinatorial problems solving. In this case, one method of solution is
to transform the unconstrained composite non-differentiable problem into a smooth, but
nonlinearly constrained, problem. This problem, although more complex than the
original, can then be solved using a method for nonlinear constraints, which takes
advantages of the structure of the problem. The transformation would normally be
performed implicitly by software designed to solve the original non-smooth problem.
(B) Constrained Optimisation
Many applications of optimisation to manufacturing process are naturally expressed as
"minimize an objective while meeting a set of requirements." For example, the objective
might be power consumption; the requirements might be maximum allowable force and
torque. In order to solve this type of problem, a constrained optimisation algorithm is
required.
(a) Linear Programming
The basic problem of linear programming is to minimize a linear objective function of
continuous real variables, subject to linear constraints. For purposes of describing and
analyzing algorithms, the problem is often stated in the standard form:
min{cTx: Ax = b,x>0) (A-3)
Appendix A Classical Optimisation Hierarchy 259
where xeIR" is the vector of unknowns, ceR" is the cost vector, and AeRmxn is the
constraint matrix. The feasible region described by the constraints is a polytrope, or
simplex, and at least one member of the solution set lies at a vertex of this polytrope.
The simple algorithms [Murtagh, 1981; Chvatal, 1983; Gill, Murray and Wright, 1991]
and interior-point algorithms [Wright, 1996] form the basis for a vast majority of the
methods for linear programming.
(b) Semi-definite Programming
The Semi-definite Programming problem (SDP) is essentially an ordinary linear
program where the non-negativity constraint is replaced by a semi-definite constraint on
matrix variables [Vandenberghe, 1996; Alizadeh, 1991]. The standard form for the
primal problem is
min(C*X) subject to Ak • X = bk (k = l,...,m);X>0 (A-4)
where C, Ak and X are all symmetric nxn matrices, bk is a scalar, and the constraint
X > 0 means that X, the unknown matrix, must lie in the closed, convex, and positive
semi-definite cone. Here, • refers to the standard inner product on the space of
symmetric matrices, i.e. for symmetric matrices A and B, A»B = trace(AB).
Semi-definite Programming reduces to Linear Programming when all the matrices are
diagonal. Semi-definite Programming is a special instance of a more general problem
Appendix A Classical Optimisation Hierarchy -^n
class called conic linear programs, where a linear objective function is minimized
subject to linear constraints and a cone constraint.
(c) Nonlinearly Constrained Optimisation
The general constrained optimisation problem is to minimize a nonlinear function
subject to nonlinear constraints [Gill et al., 1981; Fletcher, 1987; Gill et al., 1989]. Two
equivalent formulations of this problem can be used to describe algorithms. They are the
formulae (A-5) and (A-6).
min{ f(x): c; (x) < 0, i e I, ct (x) = 0, i e e} (A-5)
where each ci is a mapping from R" to R , I and £ are index sets for inequality and
equality constraints, respectively; and
min {f(x): c(x) = 0, / < x < u) (A-6)
where c maps R" to R, the lower- and upper-bound vectors, / and u, may contain
some infinite components.
The main techniques that have been proposed for solving constrained optimisation
problems are reduced-gradient methods, sequential linear and quadratic programming
methods, and methods based on augmented Lagrangians and exact penalty functions.
Appendix A Classical Optimisation Hierarchy 261
(d) Bound Constrained Optimisation
Bound-constrained optimisation problems play an important role in the general
constrained problem solving as many constrained optimisation approaches reduce the
solution of the general problem to that of a sequence of bound-constrained problems
[Fletcher, 1987; Gill et al., 1981; Conn, Gould, and Toint, 1992]. A typical bound
constrained problem can be described as
min { f(x) :l<x<u) (A-7)
where xeIR" is the vector of unknowns, / and u are the lower- and upper-bound
vectors, respectively.
Algorithms for the solution of bound-constrained problems focus on a local minimum
x* of f. The standard first-order necessary condition for a local minimum x* can be
expressed in terms of the binding set
B(x) = {i:xi=Li,dif(x)>Q}vj{i:xi=ui,dif(x)<Q} (A-8)
by requiring that dt f(x*) = 0,i£B(x*). There are some other ways to express this
condition, but this form brings out the importance of the binding constraints.
Appendix A Classical Optimisation Hierarchy 262
(e) Quadratic Programming
The quadratic programming problem involves minimization of a quadratic function
subject to linear constraints [Fletcher, 1987; Gill et al, 1981; Lawson and Hanson,
1974; Golub and Loan, 1989]. It can be described using the following formula:
min{— xTQx + cT x: af x < bt,i e I, afx = bt,i e e} (A-9)
where QeR"xn is symmetric, the index sets I and e specify the inequality and equality
constraints, respectively.
The difficulty of solving the quadratic programming problem depends largely on the
nature of the matrix Q. In convex quadratic programs, which are relatively easy to
solve, the matrix Q is positive semi-definite. Otherwise, the objective function may
have more than one local minimum.
(f) Network Programming
Network problems arise, as the name indicates, in applications that can be represented
as the flow of a commodity in a network [Kennington and Helgason, 1980; Ahuja et al.,
1993]. The resulting programs can be linear or non-linear. Network problems cover a
large number of applications [Rockafellar, 1984; Murty, 1992], some of them are:
Appendix A Classical Optimisation Hierarchy 263
(1) Transportation Problem. Suppose that we have a commodity that can be produced
in different locations and needs to be shipped to different distribution centers.
Given the cost of shipping a unit of commodity between two points, the capacity of
each production center, and the demand at each distribution center, find the
minimum-cost shipping plan. Note that the example that we described above is a
transportation model.
(2) Assignment Problem. There are n individuals that need to be assigned to n
different tasks. Each individual is assigned to one job only and each job is
performed by one person. Given the cost that each individual charges for
performing each of the n jobs, find a minimal cost assignment. Clearly, this
problem is a special case of the transportation problem. Many techniques are
developed to solve this type of problems since it is often encountered in
applications.
(3) Maximum Value Flow. Given a directed network of roads that connects two cities
and the capacities of these roads, find the maximum number of units (cars) that can
be routed from one city to another. Here, the constraints are the equilibrium or
balance equations at each node (or road intersection); i.e., flow of the cars into a
node is equal to the flow of the cars out of that node.
(4) Shortest Path Problem. Given a directed network and the length of each arc in this
network, find the shortest between two given nodes.
(5) Minimum Cost Flow Problem. Given a directed network with upper and lower
capacities on each of its arcs, and given a set of external flows (positive or
negative) that need to be routed through this network, find the minimal cost routing
Appendix A Classical Optimisation Hierarchy 264
of the given flows through this network. Here, the cost per unit of flow on each arc
is assumed to be known.
(g) Stochastic Programming
Stochastic programming has been introduced in Subsection (b) of Section A. 1 as it is a
component of both Discrete and Continuous Optimisation.
Appendix B Overview of GA's Applications in Other Industries 265
Appendix B
Overview of GA's Applications in Other Industries
GAs have been successfully applied to a variety of industries. In Chapter 3, the
overview of GA's applications in steel industry has been conducted. This appendix will
address its applications in other industries.
A. PROCESS CONTROL IN INDUSTRY
In 1983, Goldberg applied Genetic Algorithms to optimisation and machine learning
problems in natural gas pipeline control [Goldberg, 1983]. The problems are governed
by nonlinear state transition equations that dictate the pressure drop through the
pipelines and pressure rise across compressor. The optimisation result was compared
with that obtained by Wong and Larson [Wong & Larson, 1968] using dynamic
programming, and the comparison indicated that near optimal result was obtained by
using GA.
In 1998, Shin and Park presented the control problem of nonlinear processes handled by
using a GA-based optimisation technique. Details can be found in [Shin & park, 1998].
Although fuzzy control has been applied to various industrial processes; however, its
control rules and membership functions are usually obtained by trial and error. In 2000,
Appendix B Overview of GA's Applications in
Zhou and Lai proposed an optimal design for membership functions and control rules
simultaneously by using a genetic algorithm (GA) [Zhou and Lai, 2000].
B. IMAGE PROCESSING
In 1984, Fitzpatrick et al. successfully applied GA to a medical imaging system, in
which there was no closed-form mathematical representation of its fitness function
[Fitzpatrick et al., 1984]. The system was used to deal with the medical image
registration for the calculation of the different images. The GA was employed to search
for coefficients that minimized the difference between two medical images. Numerical
experiments with both artificial images and real x-rays were successful.
In 1997, Krishna and Ramakrishnan et al. addressed the problem of designing a globally
optimum code book using genetic algorithms (GAs). A hybrid GA called the genetic K-
means algorithm (GKA) that combines the advantages of gradient descent algorithms
and GA have been proposed and applied to image compression [Krishna and
Ramakrishnan et al., 1997].
C. POWER INDUSTRY
In 1995, Li and Morgan et al. proposed a novel constraint handling technique to
improve the efficiency with Genetic Algorithms, which could effectively attack the
Appendix B Overview of GA's Applications in Other Industries 267
problem of economic-environmental power dispatch (EED) under the constraint of
generation rate [Li and Morgan et al., 1995]. EED is a sophisticated and difficult task
because of the conflicting requirements of minimising generation cost and reducing
environmental pollution. The modified GA was applied to the highly constrained EED
problem on a four-generator system. The study showed that the proposed distance based
constraint handling performed better than the conventional equation based method, and
provided easy and efficient means to attack the EED problem.
In the same year, Yang et al. proposed an innovative genetic algorithm (GA) approach
to solve the thermal unit commitment (UC) problem in power generation industry
through a constraint satisfaction technique [Yang et al., 1995]. To effectively deal with
the constraints of the problem and greatly reduce the search space of the GA, the
minimum up- and down-time constraints were embedded in the binary strings that were
coded to represent the on-off states of the generating units. The violations of the other
constraints were handled by integrating penalty factors into the cost function. The
results from Taiwan Power (Taipower) system of 38 thermal units over a 24-hour period
showed that the features of easy implementation, fast convergence, and highly near-
optimal solution in solving the UC problem could be achieved by the proposed GA
approach.
In 2000, Borges et al. reported their work on developments of GA based power analysis
system [Borges et al, 2000]. All applications were developed based on modeling
techniques currently adopted in the power system industry and tested with realistic size
Appendix B Overview of GA's Applications in Other Industries 268
power system models. The results indicated that this low cost computer system had a
great potential for practical use in the power utility industry.
D. FOOD INDUSTRY
Genetic algorithms and fuzzy logic have been proven to be effective in a variety of
control applications. In 1995, Freisleben and Strelen presented an approach for
automatically determining the control parameters of a machine used in a manufacturing
environment for producing cookies and other pastry [Freisleben and Strelen, 1995]. The
proposed solution was based on augmenting a genetic algorithm with elements of fuzzy
logic, since either of them alone could not produce satisfactory results. The feasibility of
the approach was demonstrated by presenting experimental results obtained in a series
of test productions. The tests showed that high quality cookies could be produced in a
very short time.
In 2000, Shaw & Lee et al. demonstrated the difficulties of attempting to schedule the
combination of discrete tasks of varying cycle times with continuous elements [Shaw &
Lee et al., 2000]. Two implementations of a heuristic genetic algorithm (GA) approach
were demonstrated on a processing problem that has similar characteristics to a sugar
mill. The implementations included problem-specific representations, single and multi-
objective approaches to handle the problem cost, and various use of penalty functions to
avoid constraint violations. Highly tailored problem-specific operators allowed the GA
Appendix B Overview of GA's Applications in Other Industries 269
to match its behavior to the critical elements in the problem definition, specifically those
relating to a highly constrained shared storage facility.
E. CONSTRUCTION INDUSTRY
In 1986, Goldberg achieved a great success in the application of GA in structural
optimisation of plane trusses [Goldberg & Samtani, 1986]. The objective of the
optimisation is to minimize the weight of the structure subject to maximum and
minimum stress constraints on each of trusses. The performance of Genetic Algorithms
in the optimisation has been compared with other methods.
In 1997, Corno and Prinetto described their application of a genetic algorithm for the
minimization of glass loss in cutting large sheets into several pieces [Corno & Prinetto
et al., 1997]. The algorithm took into account several industrial constraints, stemming
both from the glass cutting technology and from the requirement that the optimisation
must run in real-time, concurrently with the cutting operation. The algorithm has
delivered comparable or better results than optimisation procedures embedded in
comprehensive commercial software systems, and is now distributed with all flat glass
cutting machines sold by Bottero SPA.
Appendix B Overview of GA's Applications in Other Industries 270
F. MINING INDUSTRY
In 1996, Karr and Week developed a technique in which linguistic models could be
developed exclusively from engineering data for mineral processing equipment. This
technique combines traditional fuzzy, rule-based systems and genetic algorithms. The
efficacy of such an approach has been demonstrated with the development of linguistic
models for both a grinding circuit and a hydrocyclone separator [Karr and Week, 1996].
G. MANUFACTURING INDUSTRY
In 1993, a scheme for the scheduling of flexible manufacturing systems (FMS) was
developed by Rabelo and Jones [Rabelo and Jones, 1993]. It integrates neural networks,
parallel Monte-Carlo simulation, genetic algorithms, and machine learning. Modular
neural networks were used to generate a small set of attractive plans and schedules from
a list of such plans and schedules. Genetic algorithms were utilized to combine
attractive alternatives into a single best decision. Induction mechanisms were used for
learning and simplifying the decision process for future performance. A scheduling
example has been used to illustrate the scheme capabilities including speed,
adaptability, and computational efficiency.
In 1994, Morikawa et al. addressed a basic concept of evolution of computer integrated
manufacturing (CIM) with genetic algorithms (GA) [Morikawa, et al., 1994]. A multi-
agent model of the CIM system was presented. Each agent solves its own manufacturing
Appendix B Overview of GA's Applications in Other Industries 271
problem independently using the G A and has a function of strategic search through
interactions with the environment. The results showed that the model could produce
better solutions and could be applied to job shop scheduling problems and transportation
problems.
Logan and Rhomas also addressed an evolutionary algorithm implemented in an
electronic card production line in the same year [Logan and Rhomas, 1994]. The
program partitioned a week worth of jobs, awaiting assembly, into groups, with each
group corresponding to a single day's production. The objective was to partition the
groups such that every day's workload mix was basically identical. Operator
effectiveness and results were presented.
Grouping machines into manufacturing cells is now a commonly used technique for
improving the flow of products in some manufacturing systems. The machines are
grouped into cells to minimize the traffic of parts between the cells under certain
constraints. This problem was addressed using a genetic algorithm by Pierreval and
Plaquin [Pierreval and Plaquin, 1994]. Two concrete cases of manufacturing job shops,
one under design and one in rearrangement have been used to verify the approach.
Remote diagnosis of industrial and manufacturing facilities constitutes a feasible
alternative at high-risk or remote sites where unmanned operation is preferred.
Computer-aided diagnostic tools can reduce downtime by providing support to remote
monitoring centers and on-site plant operators. In 1994, Rojas-Guzman and Kramer
described a novel technique to perform on-line remote monitoring and diagnosis of
Appendix B Overview of GA's Applications in Other Industries 272
industrial and manufacturing systems based on Bayesian belief networks and genetic
algorithms [Rojas-Guzman and Kramer, 1994]. An implementation of the methodology
in chemical process industry was presented and potential applications for different types
of industrial systems were discussed.
In 1995, Lu and Morton et al. presented an approach using a genetic algorithm
technique to carry out the inspection path planning for a multi-component inspection
application. Details have been reported by Lu and Morton [Lu and Morton et al., 1995].
Porter, Mak and Wong also presented a genetic approach to determining the optimal
number of machines required in a manufacturing system for meeting a specified
production schedule [Porter, Mak and Wong, 1995]. The application of genetic
algorithms was illustrated by solving a typical machine-requirement-planning problem.
Comparison of the respective results obtained by using the proposed approach and a
standard mixed-integer programming package showed that the proposed approach could
be used effectively for optimal manufacturing systems design.
In 1995, Fujimoto et al. also reported their work on applications of GA based intelligent
approaches to a production scheduling problem in Flexible Manufacturing Systems
(FMS) [Fujimoto et al., 1995].
In 1996, Morad and Zalzala employed the genetic algorithm (GA) technique to handle
two problems in manufacturing systems: (1) the formation of manufacturing cells in
cellular manufacturing and (2) batch scheduling [Morad and Zalzala, 1996]. Unlike
Appendix B Overview of GA's Applications in Other Industries 273
conventional methods, which merely rearrange the part-incidence matrix, this algorithm
incorporates other parameters, such as the processing time of each part and the number
of parts required. The batch-scheduling problem described in this paper was the problem
of scheduling a single machine with jobs of different due dates and arrival times. The
effectiveness of two types of crossover and mutation operators, the position-based and
order-based operators, was evaluated. Two different objective functions were used to
minimise the completion time and the total tardiness, respectively.
In 1999, Patel presented a scheduling approach, based on genetic algorithms (GA),
developed to address the scheduling problem in manufacturing systems constrained by
both machines and workers [Patel et al., 1999]. A new chromosome representation in
the GA was proposed, which took into account machine and worker assignments to
jobs. A study was conducted, using the proposed scheduling method to compare the
performance of six dispatching rules with respect to eight performance measures for two
different shop characteristics: (1) dual-resources (machines and workers) constrained
shop, and (2) single-resource constrained shop (machines only).
Selection of a production plan is a crucial decision making problem in manufacturing
systems due to the presence of alternative plans arising from the availability of several
machines, tools, fixtures, etc. Selecting an optimal set of plans for a given set of parts
becomes a NP complete problem under multi-objective and fairly restrictive conditions.
Tiwari [Tiwari et al., 1999] presented a genetic algorithm (GA) based approach used to
obtain a set of feasible plans, for given part types and production volume, to minimize
Appendix B Overview of GA's Applications in Other Industries 274
the processing time, setup time and materials handling time constrained by not
overloading the machines.
Heuristic procedures based on priority rules are quite frequently used to solve the
multiple resource-constrained project-scheduling problem (RCPSP) such as task
programming with limited resources. The rules are based on the problem knowledge.
Different local search procedures have been proposed in order to look for acceptable
solutions in scheduling problems. Bautista proposed a local search procedure [Bautista
et al, 1999] that defines the solution neighborhood based on heuristics, to assign
assembly operations to a fixed number of robots in a manufacturing cell. A genetic
algorithm was used to generate the solution.
In 1999, Bautista et al. also reported the application of GA in the optimisation of
designing manufacturing systems [Bautista et al., 1999]. The pre-manufacture
simulation is often used in designing manufacturing systems. It is helpful in finding
potential problems such as bottlenecks and gridlock, and in determining the necessary
rates of machines or the capacity of conveyor systems. The manufacturing systems
usually have a mixture of continuous and discrete parameters with highly nonlinear
responses. Most nonlinear programming packages are not built to handle discrete
variables and also take a long time to converge when a simulation must be run for each
function evaluation. The genetic algorithm was used for optimizing the net present value
of a mechanized assembly system.
Appendix B Overview of GA's Applications in Other Industries 275
In 2000, Tan et al. also developed a novel genetic algorithm (GA) based methodology
for optimal tuning of a reported fuzzy dispatching system for a fleet of automated
guided vehicles in a flexible manufacturing environment [Tan et al., 2000]. The reported
dispatching rules were transformed into a continuously adaptive procedure to capitalize
the on-line information available from a shop floor at all times. The results obtained
showed that the GA was very powerful and effective to achieve optimal fuzzy
dispatching rules for higher shop floor productivity and operational efficiency.
H. COMPONENTS AND PACKAGING
Surface mount technology (SMT) is a robust methodology that has been widely used in
the past decade to produce circuit boards. Analyses of the SMT assembly line have
shown that the automated placement machine is often the bottleneck, regardless of the
arrangement of these machines (parallel or sequential) in the assembly line. Wang and
Nelson et al. presented experimental results using genetic algorithms to optimise the
feeder slot assignment problem for a high-speed parallel, multi-station SMT placement
machine [Wang and Nelson et al, 1999]. Four crossover operators, four selection
methods, and two probability settings were used in their experiments. A penalty
function was used to handle constraints. A comparison of genetic algorithms with
several other optimisation methods (human experts, vendor-supplied software, expert
systems, and local search) was presented, which supported the use of genetic algorithms
for this problem.
Appendix B Overview of GA's Applications in Other Industries 276
In 2000, Woonghwan et al. reported their work on the high-frequency SPICE model of
the ACF flip-chip interconnections up to 13 GHz [Woonghwan et al., 2000]. The
extraction process is based on a GA optimisation procedure, which is known as a robust
optimisation tool. The proposed equivalent circuit model of the ACF interconnection
can readily be used in SPICE circuit simulations for signal integrity analysis of high-
frequency packages.
A significant amount of risk is involved in the wafer fabrication due to huge investment
costs, long production cycle time, and short production life cycle. A genetic algorithm
(GA) embedded search strategy over a hybrid color-timed Petri-net (HCTPN) for wafer
fabrication was proposed by Chen and Fu in 2000 [Chen and Fu et al., 2000]. Through
the HCTPN model, all possible behaviors of the wafer manufacturing systems such as
WIP status and machine status could be completely tracked down by the reachability
graph of the net. The chromosome representation of the search nodes in GA was
constructed directly from the HCTPN model, recording the information about the
appropriate scheduling policy for each workstation in the fabrication. The results
showed that a near-optimal schedule could be generated.
Appendix C Additional Results of Threading and Acceleration Simulations 277
Appendix C
Additional Results of Threading and Acceleration
Simulations
In Chapter 6, the simulation results of unsteady rolling processes have been presented.
As an academic exercise, additional simulations for threading and acceleration have
been conducted. The results generated from these simulations are not included in
Chapter 6 due to the limited space. This appendix covers the additional results of
threading and discussions on the control compensation coefficient in acceleration.
A. SIMULATION OF NON-CONSTANT HEAD END SPEED THREADING
The simulation for constant head end speed threading has been conducted and presented
in Chapter 6. As part of the research conducted, non-constant head end speed threading
has also been investigated. This section focuses on the non-constant head end speed
threading and discusses the effect of threading speed on exit thickness.
(a) Threading Simulation Conditions
The following parameters are employed for the non-constant head end speed threading
simulation:
Appendix C Additional Results of Threading and Acceleration Simulations 278
Strip dimensions: entry thickness 2.20mm, final exit thickness 0.273mm, and width
965mm. The Mill Modulus is 6MN/mm. Other parameters are illustrated in Table C.l.
Table C. 1 Parameters for non-constant head end threading simulation
Scheduled Reduction Rates (%)
Scheduled Exit Gauges (mm)
Work Roll Diameter (mm)
Stand 1
28.1
1.582
594.66
Stand 2
42.7
0.906
578.82
Stand 3
36.9
0.572
556.0
Stand 4
31.3
0.393
591.0
Stand 5
30.6
0.273
590.64
Stand spacing between Stands 1 and 2, Stands 2 and 3, and Stands 3 and 4 is 3.96m. The
stand spacing between Stand 4 and Stand 5 is 4.26mm. The distance between uncoiler
and the first stand is 6.0m and the distance between the last stand and coiler is 6.5m.
With the non-constant head end speed threading, the work roll surface speed of last
stand of a tandem cold mill is usually determined first, then other stands based on the
constant mass flow principle. For example, in this simulation, the work roll surface
speed of 0.966m/s is selected as the threading speed for the fifth stand (which is higher
than current practice), then the threading speeds for other four stands (from the first to
the fourth) are: 0.167m/s, 0.291m/s, 0.461m/s and 0.671m/s.
Appendix C Additional Results of Threading and Acceleration Simulations 279
(b) Threading Simulation Results
This section presents the simulation results especially the fluctuations of the tension,
roll force and exit thickness during threading.
During threading, the tension is gradually established. Similar to Figure 6.7, when the
strip head enters Stand 2 to 5, an overview of the whole process of tension
establishment is shown in Figure C.l. Four lines in the figure from top to bottom are for
the tensions between Stands 1 and 2, Stands 2 and 3, Stands 3 and 4, and Stands 4 and
5. The individual dynamical processes of the tension establishment after the strip head
enters Stands 2 to 5, are shown in Figures C.2, C.3, C.4 and C.5, respectively.
180
_ 150 -
1 120
.2 90 (0
§ 60 l-
30
20
Tension establishment process during threading
Enter Stand 2 Enter Stand 3
-̂ -----NH — — i 1 — i
Enter Stand 4
- - N ^
_ EnterStand 5
^
25 30 35 40 45
Time (s)
50 55 60
Figure C.l Tension establishment process during threading
Appendix C Additional Results of Threading and Acceleration Simulations 280
Tension establishment process after the strip head enters Stand 2
23.1 23.15 23.2 23.25 23.3 23.35 23.4 23.45
Time (s)
Figure C.2 Dynamic process of tension establishment after the strip head enters the
second stand
Tension establishment process after the strip head enters Stand 3
5 *
CO
150
100
50 +
36.2 36.25 36.3
Time (s)
36.35 36.4
Figure C.3 Dynamic process of tension establishment after the strip head enters the third
stand
Appendix C Additional Results of Threading and Acceleration Simulations 281
Tension establishment process after the strip head enters Stand 4
44.18 44.185 44.19 44.195 44.2 44.205 44.21 44.215 44.22
Time (s)
Figure C.4 Dynamic process of tension establishment after the strip head enters the
fourth stand
Tension establishment process after the strip head enters Stand 5
si 80
60 4
40
20 -
50.059 50.061 50.063 50.065 50.067 50.069 50.071 50.073 50.075
Time (s)
Figure C.5 Dynamic process of tension establishment after the strip head enters the fifth
stand
Appendix C Additional Results of Threading and Acceleration Simulations 282
The threading forces at Stands 2 to 5 are gradually decreasing after the strip head enters
these stands, with the tension gradually establishing. The threading force fluctuations at
each of all five stands during threading are shown in Figures C.6 to C.l 3.
1 R r -
z 1.5 -2 «" 1.4 -, o £ 1.3-~** o 12-0£ •••*
1 1 -
^N
Stand 1 roll force variation after the strip head enters Stand 2
23.12
i i i i i • " i " "
23.17 23.22 23.27 23.32 23.37 23.42
Time (s)
Figure C.6 Stand 1 threading force variation after the strip head enters Stand 2
a u o LL O DC
Stand 2 roll force variation after the strip head enters Stand 2
23.125 23.175 23.225 23.275
Time (s)
23.325 23.375
Figure C.7 Stand 2 threading force variation after the strip head enters Stand 2
Appendix C Additional Results of Threading and Acceleration Simulations 283
Stand 2 roll force variation after the strip head enters Stand 3
2 S a> u o u. o DC
2.8 j
2.7 --
2.6 -
2 5
2.4-
2.3 -
2 ? -
36.245 36.265 36.285 36.305 36.325 36.345 36.365 36.385
Time (s)
Figure C.8 Stand 2 threading force variation after the strip head enters Stand 3
Stand 3 roll force variation after the strip head enters Stand 3
4.8 | —• ——
^ 4 . 7 -
!• 4.6 -V o 4 5 X o ^^ £ 4.4 ^ ^ ^ o —^>_ 4.2 -I —T- — ' '
36.245 36.295 36.345 36.395
Time (s)
Figure C.9 Stand 3 threading force variation after the strip head enters Stand 3
Appendix C Additional Results of Threading and Acceleration Simulations 284
z S 0 u i_
0 u. 0 cr
4 3 -
4.2
4.1
4 -
3.9
3.8
37 -44
Stand 3 roll force variation after the strip head enters Stand 4
19 44.195 44.2 44.205 44.21 44.215 44.22 44.225 44.23
Time (s)
44.235
Figure C I O Stand 3 threading force variation after the strip head enters Stand 4
5.1 ^ 5.05 4.
2 ni u w. LL.
"5 DC
5 4.95
4.9 4.85
4.8
4.75
4.7
Stand 4 roll force variation after the strip head enters Stand 4
44.19 44.195 44.2 44.205 44.21 44.215 44.22 44.225 44.23
Time (s)
Figure C.l 1 Stand 4 threading force variation after the strip head enters Stand 4
Appendix C Additional Results of Threading and Acceleration Simulations 285
d ft
- 4 . 7 . 2 4.6 -• 4 5 -
£ 4.4-
DC A O
4 1 -
Stand 4 roll force variation after the strip head enters Stand 5
50.065
i i i i i - -I -
50.067 50.069 50.071 50.073 50.075 50.077
Time (s)
'•'• i
50.079
Figure C.l2 Stand 4 threading force variation after the strip head enters Stand 5
Stand 5 roll force variation after the strip head enters Stand 5
5.6
50.0665 50.0685 50.0705 50.0725 50.0745
Time (s)
50.0765 50.0785
Figure C.l3 Stand 5 threading force variation after the strip head enters Stand 5
Appendix C Additional Results of Threading and Acceleration Simulations 286
The strip thickness at the exit of each of the five stands is dropping after the strip head
enters Stands 2 to 5. An overview of the strip thickness fluctuations at each of all five
stands during threading is shown in Figure C.14. Individual strip thickness variation at
the exit of each stand is also illustrated in Figures C.15 to C.22.
E (A (A 4) C
1.5
1.25
0.75
Z 0.5 x UJ 0.25
Exit thickness variation during threading
Enter Stand 1 Enter Stand 2
Enter Stand 3
Enter Stand 4-
Enter Stand 5-
10 20 30 Time (s)
40 50
Figure C.14 Overview of the strip exit thickness variation at each of all five stands
Stand 1 exit thickness variation after the strip head enters Stand 2
E E, M <A CD
1.59
1.58
1.57
1.56
1.55
o a 1.54
1.53
1.52
Enter Stand 2
23.105 23.155 23.205 23.255 23.305
Time (s)
23.355 23.405
Figure C.15 Exit thickness variation at Stand 1 after the strip head enters Stand 2
Appendix C Additional Results of Threading and Acceleration Simulations 287
Stand 2 exit thickness variation after the strip head enters Stand 2
0 91 -i
"p 0 9-E T 0 89 -CO 0)
j= 0.88 -o H 0.87 -
0 RR -
\
^ v >>. ^ \
^^^ ^ ^ - ^
—"*-»-. " - ~ - — — _
23.128 23.178 23.228 23.278 23.328 23.378
Time (s)
23.428
Figure C.l6 Exit thickness variation at Stand 2 after the strip head enters Stand 2
Stand 2 exit thickness variation after the strip head enters Stand 3
0.88
E" 0.86 --
0.84
0.82 4
0.8
0.78
^ — — Enter Stand 3
X~ \ ^ _ _ ^ ^
i r i 1 T - — — i 1 i
36.23 36.25 36.27 36.29 36.31 36.33 36.35 36.37 36.39 36.41
Time (s)
Figure C.l7 Exit thickness variation at Stand 2 after the strip head enters Stand 3
Appendix C Additional Results of Threading and Acceleration Simulations 288
0 58 T
£ 0.56 -E ^ 0.54 -w CO
= 0.52 -o
J5 0.5 -0 48 -
Stand 3 exit thickness variation after the strip head enters Stand 3
36.25 36.3 36.35 36.4 36.45
Time (s)
36.5
Figure C. 18 Exit thickness variation at Stand 3 after the strip head enters Stand 3
Stand 3 exit thickness variation after the strip head enters Stand 4
44.185 44.195 44.205
Time (s)
44.215 44.225 44.235
Figure C.l9 Exit thickness variation at Stand 3 after the strip head enters Stand 4
Appendix C Additional Results of Threading and Acceleration Simulations 289
Stand 4 exit thickness variation after the strip head enters Stand 4
0.396
~ 0.386 -
E, 0.376 -
8 0.366 -I c •g 0.356 -
44.195 44.205 44.215 44.225 44.235 44.245
Time (s)
Figure C.20 Exit thickness variation at Stand 4 after the strip head enters Stand 4
Stand 4 exit thickness variation after the strip head enters Stand 5
0.05 0
50.0515 50.0615 50.0715 50.0815 50.0915 50.1015
Time (s)
Figure C.21 Exit thickness variation at Stand 4 after the strip head enters Stand 5
Appendix C Additional Results of Threading and Acceleration Simulations 290
Stand 5 exit thickness variation after the strip head enters Stand 5
0.3 -. _ 1
0.25 - ^ O T ^ ^ - — , ^ _ ^ _ ^ ^
E, 0.2 ^^^~:~~r^ CO
» 0.15 c o 0.1 j *- 0.05
0 -j , , , ,
50.0715 50.0765 50.0815 50.0865 50.0915
Time (s)
Figure C.22 Exit thickness variation at Stand 5 after the strip head enters Stand 5
(c) Effect of Threading Speed on Exit Thickness
Threading speed is one of important rolling parameters during threading. As an
academic exercise, the effect of threading speed on the exit thickness has been studied.
Simulations have been conducted with different threading speeds while keeping the
preset reduction patterns and satisfying the threading conditions.
Usually the threading parameters are determined based on previous experiences. The
reduction patterns for threading are preset; the threading speed for each stand is
determined according to the constant mass flow principle and threading conditions. An
effort has been made here to investigate the effect of different threading speeds on the
exit thickness deviation. The smaller the deviation, the better is the threading speed in
terms of alleviating off-gauge situation. In Equation (C-l), ah stands for the standard
Appendix C Additional Results of Threading and Acceleration Simulations 291
thickness deviation, h5 is the desired thickness at the exit of the fifth stand, N' is the
total sample number, and h, stands for the thickness at the f sample point.
li>5-A/)2 CT*=vj^—^— (c_1)
Figure C.23 demonstrates the exit thickness with different threading speeds (0.966m/s
and 0.233m/s at Stand 5). It should be noted that the results shown is based on the
assumption of fixed threading reduction patterns at individual stands then varying
threading speed while satisfying threading constraints. It can be seen that the exit
thickness off-gauge situation is alleviated by 15.46% with the threading speed of
0.233m/s. GA based optimization of threading speed can be conducted based on the
objective function (Equation (C-l)), which is the standard deviation of the strip
thickness at the exit of the last stand. The optimization is actually a process searching a
threading speed at the last stand leading to a minimized standard deviation of the strip
thickness. During threading, AGC control loops are not activated, there is no front
tension, the back tension is not stable when the strip head is being threading, thus the
improvement of off-gauge may be limited.
Appendix C Additional Results of Threading and Acceleration Simulations 292
Comparison of exit thickness after the strip head enters Stand 5
0.28
0.2 0.02 0.04 0.06 0.08 0.1 0.12
Time since the strip head enters Stand 5 (s)
0.14
Figure C.23 Comparison of exit thickness fluctuations at Stand 5 with different threading
speed
B. DISCUSSION OF CONTROL COMPENSATION COEFFICIENT
The control compensation coefficient C in Equation (6-14) is a critical parameter in
acceleration. It is product and mill dependent as it is related to both the plastic modulus
(M) of the strip in the roll gap and the elastic modulus (K) of the mill stand.
In Section 6.3.2.1, Equation (6-14) describes the relationship between the no-load gap
adjustment AS0 and the exit thickness error Ah. The compensation coefficient C is
usually determined by experience. As discussed earlier, when C = 1, the mill elastic
deformation is completely compensated for, the effective mill elastic modulus becomes
infinite, and the mill stand seems infinite stiff. When C = 0, the no-load gap adjustment
AS0 is equal to the exit thickness error Ah, a partial compensation is obtained, and the
Appendix C Additional Results of Threading and Acceleration Simulations 293
mill stand becomes less rigid. So the compensation coefficient C is ideally a value
between 0 and 1. It is usually assigned to a greater value for earlier mill stands and a
smaller value for subsequent mill stands. Wood plotted family C-curves in both dh/dH
vs. M/K and dh/dS vs. MIK charts [Wood, 1995], where dh is an increment of change in
outgoing strip thickness; dH is an increment of change in incoming strip thickness; and
dS is an increment of change in screw-down position. It shows that dh/dH is dropping
and dh/dS is rising while C increases. This means, when C is assigned to a greater value,
a change in the incoming strip thickness will result in a less change in the outgoing strip
thickness, and a change in screw-down position will result in a greater change in the
outgoing strip thickness. It also demonstrates that when MIK increases, a change in the
incoming strip thickness will result in a greater change in the outgoing strip thickness,
and a change in the screw position will result in a less change in the outgoing strip
thickness.