j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 3592–3596
journal homepage: www.elsevier.com/locate/jmatprotec
Adaptation for tandem cold mill models
Carlos Thadeu de Ávila Pires a , Henrique Cezar Ferreira b,∗ , Roberto Moura Sales b
a
Companhia Siderúrgica Paulista (Cosipa), Estrada de Piaçagüera, km. 6, CEP 11573-900, Cubatão SP, Brazil
University of São Paulo, Department of Telecommunication and Control Engineering, Av. Prof. Luciano Gualberto, trv. 3, n. 158, CEP
05508-900, São Paulo, SP, Brazil
b
a r t i c l e
i n f o
a b s t r a c t
Article history:
The ideal conditions for the operation of tandem cold mills are connected to a set of refer-
Received 18 March 2008
ences generated by models and used by dynamic regulators. Aiming at the optimization of
Received in revised form
the friction and yield stress coefficients an adaptation algorithm is proposed in this paper.
31 July 2008
Experimental results obtained from an industrial cold rolling mill are presented.
Accepted 25 August 2008
© 2008 Elsevier B.V. All rights reserved.
Keywords:
Cold rolling
Set-up model
Adaptation
Optimization
1.
Introduction
Physical models for rolling mills have been intensively developed in the last years, hoping to increase quality of steel
strip and productivity of rolling processes. Nevertheless, these
two characteristics are only achieved using models sufficiently
accurate. The availability of accurate models is associated to
set-up optimization and adaptation methods, and, in function of this, has been intensely explored in the literature.
Among optimization works, Pires et al. (2006) report good
results of quality and productivity after having applied a nonlinear simplex optimization method to set-up an industrial
cold mill; Wang et al. (2000) present the results of an investigation into an optimal scheduling for tandem cold mills based on
genetic algorithm; Sekiguchi et al. (1996) analyze set-up accuracy of low thickness reduction when using rougher rolls in an
industrial mill and Fiebig and Zander (1982) present interesting concepts of productivity applied to set-up of cold rolling
mills. At the same time, models will only be precise enough
∗
Corresponding author. Tel.: +55 11 3091 5273; fax: +55 11 3091 5718.
E-mail address: henrique@lac.usp.br (H.C. Ferreira).
0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmatprotec.2008.08.020
if equated with parameters representing the present status
of the rolling process. Among these parameters are the friction , between the strip and the rolls, and the yield stress
k of the strip. These coefficients cannot be measured with
the instrumentation available today and their effect on the
process are quite similar, making the question of accuracy
even more complex. In order to face this problem, different
adaptation schemes have been proposed in the literature.
Originally, Bryant (1973) pointed in his book, Automation of
Tandem Mills, the importance of such adaptation. Recently,
it has been the object of several works. Atack and Robinson
(1996) demonstrate how process models implemented with
adaptation schemes can predict, with accuracy, the load and
torque of flat wide strip rolling; Randall et al. (1997) describe
how a set-up model with adaptive scheme improved the head
and tail off-gage in a hot finishing mill; Nishino et al. (2000),
based on a physical model of a plate rolling process, present an
empirical and adaptive approach to improve the accuracy of
a rolling load prediction model; and finally, Wang et al. (2005)
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 3592–3596
show how the accuracy of rolling force calculation in a tandem cold mill can be improved through the adaptation of strip
deformation resistance model.
In the present work, an adaptation scheme is proposed and
applied to a four stand tandem cold mill at Cosipa plant, Brazil.
Such scheme is based on the minimization of a cost function
which takes into consideration the contribution of the friction
coefficient and the yield stress coefficient.
For the minimization of the cost function, Nelder and Mead
(1965) simplex algorithm was employed. An objective function
is defined such that the main variable force is taken into consideration. Although more sophisticated model has recently
been proposed, for example, by Pawelski (2003), force, torque,
slip and power are calculated using a classical cold rolling
model developed by Bland and Ford (1948), because this is the
model employed in the automation system of the cold mill.
This paper is organized as follows: in Section 2, the
mechanical and electrical characteristics of the tandem cold
mill are introduced and the automation and control system architecture is described; in Section 3, the mathematical
model of the cold rolling mill process is presented and some
advantages of its use are summarized; in Section 4 details of
the adaptation algorithm and the cost function are presented;
simulation and experimental results are explained in Section
5; and finally, Section 6 presents the main conclusions.
2.
Cosipa four stand tandem cold mill
Cosipa cold mill is a coil to coil four high, four stand mill, in
which each stand is composed by two backup rolls and two
work rolls, the later coupled to dc motors controlled by digital speed regulators, totaling 16 MW of nominal power. Two
hydraulics actuators, installed at the top of the stands, complete the set of reduction of each stand. Table 1 presents the
main electrical and mechanical characteristics of the tandem
cold mill and the entry and exit dimensions of the processed
material, which consists of low, medium and high carbon steel
sheet.
Table 1 – Electrical and mechanical characteristics
Annual production (ton)
Maximum speed (m/min)
Work rolls diameter (mm)
Back up rolls diameter (mm)
1,248,000
1080
490–575
1270–1422
Motors
Stand
Power (kW)
Speed (rpm)
Voltage (V)
1
2 × 1800
433–1046
900
2
2 × 1800
433–1046
900
3
2 × 1800
433–1046
900
4
2 × 1482
200–485
700
Material
Carbon steel
Entry thickness (mm)
Exit thickness (mm)
Coil width (mm)
Coil internal diameter (mm)
Coil external diameter (mm)
2.00–4.75
0.38–3.00
650–1575
610
1930
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The automation architecture of Cosipa four stand tandem
cold mill includes the following four levels, as described by
Bolon (1996).
• Level 3 (Production planning level): This level is responsible
to decide which product will be produced and according to
which specification.
• Level 2 (Process optimization level): From entry and exit coil
data specifications, this level is responsible for finding the
best mill set-up in order to ensure high quality and productivity. Based on static models this level includes a set-up
optimization procedure (Pires et al., 2006) and an adaptive
loop which improves the set-up specification for every new
coil.
• Level 1 (Process dynamic control): According to the reference signals from level 2 and measured process signals,
suitable control signals are generated in this level for the
actuators. This level includes the dynamic model and the
mill master logic. In addition, it records the process variables necessary to the adaptive functions of level 2.
• Level 0 (Actuators and sensors): This level includes sensors,
motor drives and hydraulic actuators for gap control.
The present paper refers to the adaptation procedure of
level 2. More specifically, the coefficients and k are adapted
in order to optimize the predicted forces to be applied by the
mill stands.
3.
The rolling model
Bland and Ford (1948) cold mill model was chosen as the mathematical process model used in this paper to calculate the
rolling loads of the tandem cold mill. According to Bland and
Ford theory, the strip is subjected to three different zones in
the arc of contact between the strip and the work rolls. In the
first zone, located at the entry of this region, the strip is elastically compressed until the yield stress condition is achieved.
In the second zone, the strip is plastically deformed until a
minimum thickness while in the third and last zone, it suffers elastic recover. The mathematical development to find
the final expressions of force, torque and deformed roll radius
can be followed in Bland and Ford (1948). Hereafter, only the
final expressions of these dimensions will be shown.
The rolling force by unit width P is a nonlinear function
of the entry thickness hin , the exit thickness hout , the entry
tension in , the exit tension out , the entry yield stress kin ,
the exit yield stress kout , the coefficient of friction and the
deformed work roll radius R ,
P = fP (hin , hout , in , out , kin , kout , , R ).
(1)
The deformed roll radius is a function of the elastic and
plastic forces and can be calculated in the following manner,
R = R 1 +
16P(1 − R2 )
ER (hin − hout )
,
(2)
where R and ER are the Poisson’s ratio and the Young’s modulus for the work roll, respectively.
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Finally, the yield stress is expressed by the following equation
kin(out) = k · [A + B εin(out) ] · {1 − C exp[−Dεin(out) ]},
h 0
εin = ln
,
hinh 0
εout = ln
,
hout
(3)
where A, B, C and D are material dependent constants and h0
is the strip thickness at entry of the mill, assumed to be the
thickness of annealed strip. Both of Eq. (1) and k of Eq. (3) are
initialized and adjusted in the adaptation phase.
4.
Adaptation procedure
This section is focused on the adaptation of the friction coefficients and yield stress k parameter, which are central for the
Bland and Ford rolling model. More specifically, before rolling
the first coil of a batch, estimates for these parameters are
adopted for the calculation of the reduction for each stand.
This calculation is performed through an optimization procedure, which is described in detail in Pires et al. (2006). Besides
the reduction for the stands, the optimization procedure generates predicted values for the rolling forces. After the rolling
of the first coil, these predicted values are compared to the
measured values and the parameters and k are then adapted,
aiming at improving the predictions of force for the second
coil. This section presents the proposed adaptation scheme,
which is the main contribution of this paper. In Section 5 some
experimental results are also presented.
The proposed adaptation scheme consists of two main
phases as shown in Fig. 1: in the first phase, using Bland and
Ford model and an initial guess for and k, predicted values
of forces and their corresponding reduction for each stand
are calculated. These calculated forces are compared to the
measured forces and, in the second phase, through an optimization algorithm, the parameters and k are adjusted in
order to minimize an objective function. The simplex nonlin-
Table 2 – Parameters of the adaptation cost function
Stand #
Kf
Nf
1
2
3
4
10
2
10
2
10
2
10
2
ear method, initially proposed in Nelder and Mead (1965) is
used in the optimization phase.
Some comments on the Bland and Ford model have already
been presented in Section 3. In what follows, a brief description of the Nelder and Mead simplex algorithm is presented.
Among the various existing optimization methods, some
recent applications related to cold mill set-up optimization include nonlinear programming (Ozsoy et al., 1992),
genetic algorithms (Wang et al., 2000), and more specifically
the (Nelder and Mead, 1965) simplex method, which was
employed in Fiebig and Zander (1982) and also by Cosipa cold
mill automation system (Pires et al., 2006). A detailed description of the simplex method can be found in Walters et al.
(1991).
Like for every optimization algorithm, the definition of the
objective function plays a central role for the Nelder and Mead
algorithm. In the present case, the adopted objective function
is given by
J=
4
i=1
(i)
Kf ·
(i)
(i)
Fcalc − Fmeas
(i)
Fmeas
Nf(i)
(4)
where the index i = 1, 2, 3 and 4 refers to each stand of the
(i)
(i)
mill; Fcalc and Fmeas are the calculated and measured forces,
respectively and Kf and Nf are constants adjusted according to
the values of Table 2.
The search for the parameters and k that minimize the
objective function is implemented in the Nelder and Mead
algorithm through the following steps:
(i) Set initial values for and k.
(ii) Introduce disturbances in and k and, for each new point,
calculate the corresponding objective function value.
Three main operations may be accomplished from step
(ii): reflection, contraction and expansion, as illustrated in
Fig. 2, for an example of two optimization variables. Yet for the
two dimensions case, the algorithm proceeds in the following
steps.
(1) Sorting: The iterative process is initiated sorting the points
xW , xN and xB for which the function has its maximum
value JW , the second maximum value JN , and the minimum
value JB , respectively.
(2) Reflection: The average point or centroid xc is determined
finding the average of all points xi , except xW (see Fig. 2).
From equation:
Fig. 1 – Adaptation scheme.
x = xC + b · (xC − xW ),
(5)
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Table 3 – Force error for the actual and proposed
adaptation
Stand #
1
2
3
4
k
Fcalc
Fmeas
Error %
0.0283
1.34
1100
1112.7
−1.14
0.0288
1.34
972
977.3
−0.58
0.0241
1.34
841
832.1
1.07
0.0965
1.34
953
982.1
−2.96
k
F calc
Error %
0.0387
1.30
1113.2
0.05
0.0290
1.30
976.7
−0.60
0.0173
1.30
833.4
0.156
0.1104
1.30
982.5
0.041
Fig. 2 – Simplex algorithm steps.
and assuming the minimization step b = 1, it results x = xR ,
known as reflection of xW with respect to xC . If JB < JR < JN ,
then xW is replaced by xR and the process is restarted from
step 5.
(3) Expansion: If JR < JB < JN , then set b = 2 and get x = xE , known
as expansion of xR with respect to xC . If JE < JB , xW is
replaced by xE and a new process is restarted from step
5.
(4) Contraction: If JN < JR < JW , a contraction is made, generating
a vertex x = xU for which b = 1/2. If JB < JU < JN , xW is replaced
by xU and a new process is restarted from step 5; If JW < JR ,
a contraction with change in direction must be done, generating a vertex x = xT for which b = −1/2. If JT < JW , xW is
replaced by xT and a new process is restarted from step 5.
(5) Stop condition: Sort the points of the new simplex as xW , xN
and xB for which the function has its maximum value JW ,
the second maximum value JN , and the minimum value JB ,
respectively. If J(xW) − J(xB) < ε, then stop; otherwise, go to
step 2. The value of the first stop criterion ε was selected as
0.001 and proved sufficient to get accurate results with a
Fig. 3 – Results of the present rolling mill model.
Fig. 4 – Convergence of the adaptation cost function.
not too big number of iterations. As a second stop criterion,
the maximum number of iterations was adjusted to 200.
Finally, as a last guarantee of halting the whole iteration
process, a maximum computation time may be chosen.
5.
Main results
Set-up accuracy of the actual rolling mill model, working with
the original adaptation technic are shown in Fig. 3 for 300
coils processed in sequence. Intending to assess these results,
simulations using the proposed adaptation procedure were
performed and compared to the currently operating method
for a batch of coils.
The upper part of Table 3 shows the accuracy of the predicted rolling load for the adaptation scheme currently in
use in the tandem cold mill, while the lower part presents
the same characteristic for the proposed adaptation procedure. For comparison purpose, it was considered a batch of
10 dimensionally similar coils, not considering the first coil.
It can be noted a slight improvement in the mean force error
when the friction and yield stress coefficients, for the next coil,
are calculated using the proposed adaptation technic.
The adopted strategy was to adjust the step of the yield
stress coefficient greater than that of the friction coefficients,
taking into consideration that most part of the force error is
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references
Fig. 5 – Comparison of force error.
certainly due to the steel strip hardness changes. To evaluate the convergence capacity of the method, the number of
steps of one of those adaptation set-up simulation is shown
in Fig. 4.
Considering the two methods, a comparison of the mean
error and error distribution, for each one of the mill stands,
can be shown in Fig. 5. In this figure, the first four distributions correspond to the present adaptation while the next
four to the simulated method. It can be noted a significant
improvement in the model accuracy when using the proposed
method.
6.
Conclusions
This work presents an adaptation procedure for the set-up
generation of a four stand tandem cold rolling mill, located
at Cosipa plant, Brazil. The proposed algorithm consists of
an objective function, which takes into consideration friction and yield stress coefficients and is minimized using
Nelder and Mead (1965) simplex method. Simulation results
show that the methods adopted is robust, very fast and
can be tuned very easily leading to results according to the
expected for the industrial tandem cold mill into consideration.
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