Application of artificial neural networks for the prediction of roll force and roll torque in hot...

Applied Mathematical Modelling 37 (2013) 4593–4607

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Applied Mathematical Modelling

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Application of artificial neural networks for the prediction of roll forceand roll torque in hot strip rolling process

Mahdi Bagheripoor ⇑, Hosein BisadiDepartment of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

a r t i c l e i n f o

Article history:Received 27 February 2012Received in revised form 23 August 2012Accepted 25 September 2012Available online 4 October 2012

Keywords:Artificial neural networkFinite element simulationHot strip millRolling forceRolling torque

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.09.070

⇑ Corresponding author. Tel.: +98 21 77240540; fE-mail address: [email protected]

a b s t r a c t

This paper introduces an artificial neural network (ANN) application to a hot strip mill toimprove the model’s prediction ability for rolling force and rolling torque, as a function ofvarious process parameters. To obtain a data basis for training and validation of the neuralnetwork, numerous three dimensional finite element simulations were carried out for dif-ferent sets of process variables. Experimental data were compared with the finite elementpredictions to verify the model accuracy. The input variables are selected to be rollingspeed, percentage of thickness reduction, initial temperature of the strip and friction coef-ficient in the contact area. A comprehensive analysis of the prediction errors of roll forceand roll torque made by the ANN is presented. Model responses analysis is also conductedto enhance the understanding of the behavior of the NN model. The resulted ANN model isfeasible for on-line control and rolling schedule optimization, and can be easily extended tocover different aluminum grades and strip sizes in a straight-forward way by generatingthe corresponding training data from a FE model.

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1. Introduction

Roll force and roll torque are critical in aluminum rolling processes. They affect the mill set-up schedule, the thickness ofthe rolled metal, and the final product profile. Nowadays, maintaining product consistency and quality in the manufacturingprocess has become a widespread concern as a result of increasing competition in the world markets. Increasing demands onthe quality of rolling mill products have led to great efforts to improve the control and automation systems of the rollingprocess and modern roll gap set-up and profile control systems are highly interested in accurate predictions of the roll forceand torque under various rolling conditions. However, the non-linear nature of the hot rolling process, the close couplingbetween the thermal, mechanical and material phenomena, and the multistep nature of optimization make the process dif-ficult to formulate and solve. To overcome this limitation, there have been several attempts to develop analytical and numer-ical solutions in order to calculate the relationships between rolling loads and the degree of deformation.

The analysis of rolling is dated back to the pioneering work of Orowan [1] who developed a comprehensive theory basedon an extension of the slab method by introducing non-homogeneity of plastic deformation of the sheet and elastic defor-mation of the rolls. Sims [2] developed analytical expressions of pressure distribution, roll force and roll torque by avoidingmost of the numerical integration in Orowan’s theory. Ford and Alexander [3] modified the Orowan’s model by using thetechnique of limit analysis, and compared their results with Sim’s results for nonferrous metal. Green and Wallace [4] devel-oped an upper-bound approach for hot rolling. It was assumed that the material in the roll gap is perfectly rigid-plastic andthe shear stress along the arc of contact attains the value of the shear yield stress in plane strain. Freshwater [5] simplified

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4594 M. Bagheripoor, H. Bisadi / Applied Mathematical Modelling 37 (2013) 4593–4607

the equations for roll force, roll torque and roll pressure with considering the inhomogeneity of the rolling process. Said et al.[6] determined the roll separating forces and torques via low carbon steel rolling as a function of the area reduction at dif-ferent entry temperatures. They examined several empirical, mathematical models of the process for their ability to predictthe roll separating forces. However, due to the complexity of the modern rolling process, pure analytical models often re-quire considerable simplification and assumptions, which inevitably lead to model errors when these assumptions arenot satisfied.

One solution to the analytical model is to use the finite element method (FEM), which is able to perform complicated cal-culations under realistic process constraints and various deformation conditions. A penalty rigid-viscoplastic finite elementformulation was presented by Hwang and Joun [7] for investigation of hot strip rolling. The effect of the friction coefficient onthe rolling force, roll pressure and tangential stress distribution is examined in their investigation. Shangwu et al. [8] carriedout the 3D modeling of hot rolling process of flat strips by combining the finite element and boundary element methods.They predicted rolling force, rolling torque and contact pressure on the roll for both rigid and flexible roll cases. Kwaket al. [9] developed an approximate model predicting roll force and power applicable to finishing stand of tandem hot stripmill using rigid viscoplastic FEM and concept of hypothetical deformation mode of material in roll gap. Duan and Sheppard[10] simulated the hot flat rolling of aluminum alloy 3003 by the commercial FEM program FORGE3. They concluded thatviscoplastic friction law is slightly better than the Coulomb and Tresca friction laws in terms of predicting rolling loadand torque. A large-deformation constitutive model applicable to the calculation of roll force and torque in heavy-reductionrolling was presented by Byon et al. [11]. Three-dimensional finite element analysis coupled with the proposed constitutivemodels has been carried out to calculate workpiece deformation and the rolling force in their investigation. Moon and Lee[12] proposed an approximate model for predicting roll force and torque in plate rolling process. Peening effect whichunavoidably occurs in plate rolling due to a small ratio of work roll radius over mean thickness of slab being deformedhas been considered in their model. Wang et al. [13] investigated comprehensive influences of normal stress and shear stressin longitudinal, transverse direction and altitude to improve the accuracy of the calculation of rolling force. In their study, therolling force was solved in three-dimensional finite differential method. A new model for the prediction of the roll force andtension profiles was presented by Kim et al. [14]. Their approach was based on an approximate 3-D theory of plasticity, andconsidered the effect of the pre-deformation of the strip. They used a rigid-plastic finite element model to examine the pre-diction accuracy of the proposed model. Zhang and Cui [15] established a 3-D thermo-mechanical coupled elasto–plastic FEmodel to consider the influence of roll force on the plate shape and its final profile. Although the finite element method is avery powerful tool for simulation of the engineering problems, the FE simulation of nonlinear problems is a time-consumingprocedure and the accurate setting of the various aspects of the deformation conditions is difficult.

Another approach that can be used to predict the relevant relationships among the parameters involves neural networkmodels. There has been considerable focus on neural networks in the recent past as it is widely applicable and easy to usefor problems with highly non-linearity and complex data. Portmann et al. [16] introduced a neural network learning schemein a rolling mill control system by combining a neural network and a classical physical-based mathematical model. Picanet al. [17] presented an artificial neural network for presetting the rolling force in a temper mill. They tried to solve thenetwork’s performance degradation in aberrant points by using multiple networks per domain. Chun et al. [18] studiedthe ability of an artificial neural network model, using a back-propagation learning algorithm, to predict the flow stress, rollforce and roll torque obtained during the hot rolling of aluminum alloys. Jeon and Kim [19] designed an algorithm of neuralnetworks by which, before the strip reaches the rolls, the proper rolling force and torque are quickly and exactly calculatedand executed to the rolling system to optimize the strip manufacturing process. Lee and Lee [20] proposed a long-termlearning method using neural-network to improve the accuracy of rolling-force prediction in hot-rolling mill. They com-bined neural-network method with the conventional learning algorithm in the pre-calculation stage to reduce the thicknesserror at the head–end part of the strip. Yang et al. [21] described a neural network modeling approach for the roll force andtorque predictions, without the requirement of a physically-based or an empirical model. Ensemble modeling techniqueshave been adopted by them to improve the model prediction. Lee and Choi [22] proposed an on-line adaptable networkfor the rolling force set-up and discussed such important matters as neural network structure, input selections, debugging,development environments and test results. A neural network model for roll load prediction was developed by Yang et al.[23], based on rolling data generated via an FE model. As they observed, the roll load prediction will be quite robust againstthe errors introduced by setting of the heat transfer and friction coefficient values. Son et al. [24] presented an on-line learn-ing neural network for both long-term learning and short-term learning in order to improve the prediction of rolling force inhot-rolling mill. Moussaoui et al. [25] discussed the combination of an artificial neural network with analytical models toimprove the performance of the prediction model of finishing rolling force in hot strip rolling mill process. Park and Hwang[26] proposed a new width control system composed of a RF (roll force) AWC (automatic width control) in the roughing milland a FVM (finishing vertical mill) AWC in the finishing mill in order to obtaining the desirable width margin in the hotstrip mill.

The main objective of the present research is to develop a neural network model for a hot strip mill which is capable ofpredicting accurate rolling force and torque under various rolling conditions. A three-dimensional finite element model hasbeen developed to provide a data basis for training and validation of the neural network. A three level, full factorial analysishas been carried out for studying the influence of input parameters. The results show that the ANN model is very successfulin the prediction of roll force and torque and quite useful instead of time consuming experimental trials.

M. Bagheripoor, H. Bisadi / Applied Mathematical Modelling 37 (2013) 4593–4607 4595

2. Finite element modeling

A 3D thermo-mechanical finite element model has been developed for data generation, with careful validation under var-ious rolling conditions. Arbitrary Lagrangian–Eulerian (ALE) method has been employed as a formulation technique throughthe use of the commercial general-purpose FE program ABAQUS™. In the model, the interaction of the thermal and mechan-ical phenomena is considered. The flow stress is coupled to the rolling temperature, strain rate, and strain. At the same time,the friction and the heat of deformation cause an increase in the temperature of the metal and the work-roll surface. Hence,thermal and mechanical models should be coupled to include the effects of the above mentioned factor simultaneously.

2.1. Metal flow model

According to optimization theory, kinematically acceptable velocity field can be obtained by minimizing total energy con-sumption rate, then deformation and mechanics parameters can be solved. The basic idea in this approach is utilizing an en-ergy function, p, which satisfies the boundary condition, compatibility equations and the volume constancy situations.Therefore, the following energy functional [27] should be minimized as:

dp ¼Z

V

�rd _�edV þZ

VK _evd _evdV �

ZSF

Fiduids ¼ 0: ð1Þ

Here, �r is the effective stress which is a function of strain, strain rate and temperature, _�e the effective strain rate, Fi the sur-face tractions, ui the velocity field, V the volume, SF the boundary surface of sub-domain, K the penalty constant and _ev is thevolumetric stain rate. Since the terms of the above equation cannot be determined analytically, the numerical evaluationshould be performed. It has been carried out by means of the finite element method where the working domain is discretizedinto small regions called element and at the same time the governing equations are applied.

Commercial pure aluminum alloy (AA1100) is studied in this work due to its wide industrial applications. The strip is as-sumed to behave as a thermo-viscoplastic material with temperature dependent elastic modulus and Poisson’s ratio. Forinvestigating the metal plastic deformation behavior, it is appropriate to consider uniform or homogeneous deformationconditions. The range of temperatures and strain rates experienced by the material during the rolling process is large, henceit is necessary to define the strip’s plastic behavior as a function of both temperature and strain rate. This was done using ahyperbolic sine equation, developed by Brown et al. [28] from hot compression tests, as follows:

_e ¼ A exp�Q def

Rh

� �sinh n

~rs

� �� 1m

; ð2Þ

where A, n and m are material constants, _e is the mean equivalent plastic tensile strain rate, ~r the equivalent tensile stress,Qdef the activation energy for deformation, R the universal gas constant, h the absolute temperature and s is a scalar internalvariable with dimensions of stress, called the deformation resistance. The following flow stress parameters have been uti-lized in the constitutive equation of AA1100, A = 1.91E7, Qdef = 175.3 kJ/mole, n = 7 and m = 0.23348 [28]. In order to imple-ment this material behavior in the model, data tables of flow stress at specified strain rates and temperatures were generatedand input to ABAQUS™. The values of the elastic constants and the initial values for the deformation resistance, s0, are listedin Table 1, as a function of temperature.

On the contact surface, frictional stress between the work-roll and the strip must be considered. Interfacial friction for thecontact area is proportional to the normal force as shown in the following equation:

scrit ¼ lp; ð3Þ

where scrit is the critical shear stress, l the coefficient of friction and P is the contact pressure. The friction coefficient l variesin wide ranges due to rolling conditions and modeling assumptions of the authors. In the present paper, different frictioncoefficients are used according to the experimental rolling conditions (Section 2.3).

Table 1Elastic modulus, poisson’s ratio and initial values of internal variables for AA1100 [28].

Temperature (�C) E(GPa) m So(MPa)

300 59.4 0.356 36.6350 57.3 0.359 34.4400 55.1 0.362 29.7450 52.9 0.365 29.5500 50.7 0.368 27.6600 46.4 0.337 21.6


2.2. Heat transfer model

During the rolling process, the temperature distribution in the strip and the work-roll can be calculated using the gov-erning partial differential equation shown in the following expression [29]:

@

@xk@T@x

� �þ @

@yk@T@y

� �þ @

@zk@T@z

� �þ _Q ¼ qc

@T@t; ð4Þ

where q is the density of the rolled strip, c the specific heat of the rolled metal, k is the thermal conductivity of the rolledstrip and _Q represents the volumetric rate of heat generation arising from the deformation. The heat generation term _Q iscalculated using Eq. (5):

_Q ¼ gr _e; ð5Þ

where g is the efficiency of conversion of plastic work into heat. The latter is assumed to be about 0.95, which is consideredreasonable for aluminum alloys [30]. The shear friction model is used to model the friction phenomenon during the rollingprocess. The friction heat q between rolled piece and rollers is written as follows [31]:

qfric ¼ jstj; ð6Þ

where s is the shear stress and t is the sliding velocity. At the contact interface between the strip and the work roll, an inter-facial heat transfer coefficient is assumed. The boundary conditions in deformation zone are as below [32]:

�k@T@n¼ hconðT � TrÞ � qfric; ð7Þ

�k@T@n¼ h1ðT � T1Þ þ re T4 � T4

1

� �; ð8Þ

where Tr is the work-roll surface temperature, T1 the surrounding temperature, hcon the interface heat transfer coefficient,which is determined to be about 40 kW m�2 K�1 based on the experimental work that has been conducted by Chun and Le-nard [33] and h1, the heat transfer coefficient between the strip and the air, is set to 0.01 kW m�2 K�1 [31]. The thermo phys-ical properties for the strip and steel work roll are listed in Tables 2 and 3, respectively [34,31].

The work roll geometry is limited to a 90� section with a thickness of 5 mm. In both the strip and work roll, eight-nodeisoparametric brick elements were employed. The chosen elements were suitable for coupled thermal-stress analysis. Thesteelwork roll was defined as an elastic material with a Young’s modulus of 200 GPa. The large differences in elastic modulusbetween the work roll and the strip causes the work roll to behave as a virtually rigid material. Employing symmetry alongthe center line, a quarter of the strip is considered in the model. The geometry of the sheet, and work roll is shown in Fig. 1.

2.3. Model verification

The FEM model developed in this investigation is validated by comparing the model predictions of rolling force and roll-ing torque with experimental results of Hum et al. [35]. The material is commercial purity aluminum (AA1100) and the strip

Table 2Thermo physical properties used for AA1100 [34].

Temperature(�C)

Thermal conductivity(Wm�1 K�1)

Heat capacity(J kg�1 K�1)

37.8 162 94593.3 177 978

148.9 184 1004204.4 192 1028260 201 1052315.6 207 1078371.1 217 1104426.7 223 1133

Table 3Thermo physical properties used for the steel work roll [31].

Thermal conductivity(Wm�1 K�1)

Heat capacity(J kg�1 K�1)

Density(kg m�3)

14 460 7876

Table 4Conditions of hot rolling experiments [35] employed for the simulation.

Sample no. Initial thick. (mm) Thick. red. (%) Roll speed (rpm) Friction coef.

1 6.32 21.36 80 0.08442 6.29 30.68 20 0.11413 6.28 30.41 60 0.19234 6.28 30.40 80 0.11805 6.29 31.48 100 0.12536 6.30 31.27 140 0.07347 6.30 39.05 80 0.1805

Fig. 1. Finite element analysis model.


is 6.32 mm in thickness and 50 mm in width. As described in previous section, only a quarter of strip is modeled due to thesymmetric nature of flat rolling. The strip length is taken to be 40 mm in simulation, for purpose of reducing analysis time.The rolling was performed at 500 �C in a single stand mill with roll diameters of 250 mm. Table 4 summarizes the employedexperimental rolling conditions [35] used in the hot rolling simulations.

The computed time behavior of rolling force and torque during the process, as the material traverses the roll gap, is shownin Fig. 2 for the rolling conditions of pass no. 4. As expected the rolling force and torque increase gradually when the stripfeeds into the roll gap and reach relatively steady values when the deformation is in the steady state region. Since only aquarter of the strip is modeled, the actual rolling force and torque are calculated as twice of the average values in the steadystate intervals shown in Fig. 2(a) and (b).

A comparison between the FEM predicted values of roll separating force and roll torque with experimental data is pre-sented in Fig. 3. As observed, the prediction relative error has a mean of 11.4% for rolling force and 9.8% for rolling torque.In fact Fig. 3, indicates a good correlation between the numerical and experimental results. So, the FE model is able to pro-duce accurate roll force and torque predictions for the rolling process under investigation, and can confidently be used as asurrogate resource to generate the training data for NN modeling.

3. Artificial neural network model development

3.1. Brief description of back-propagation neural networks

The theory of neural networks is inspired by the structure of the brain and how it processes a huge amount of informa-tion. Artificial neural networks are adaptive models that can learn from the data and generalize the things learned. They canbe used to build mappings from inputs to outputs, giving information about how the phenomenon behaves in practice [36].

Fig. 2. Variation of: (a) roll force, and (b) roll torque during the time for the complete rolling program of pass 4.


Multi-layer neural network consists of an input layer, one or more hidden layers, and an output layer. The input layer isfirst layer and accepts symptoms, signs, and experimental data. The layers which are placed between the input and outputlayer called hidden layers. The hidden layer processes the data it receives from the input layer, and sends a response to theoutput layer. The output layer accepts all responses from the hidden layer and produces an output vector.

Each layer has a certain number of processing elements (neurons) which are connected by connection links with adjust-able weights. These weights are adapted during the training process, most commonly through the back-propagation algo-rithm, by presenting the neural network with examples of input–output pairs exhibiting the relationship the network isattempting to learn. The output of each neuron is calculated by multiplying its inputs by a weight vector, summing the re-sults, and applying an activation function to the sum [37]:

y ¼ fXn

k¼1

xkwk þ bk

" #; ð9Þ

where n is the number of inputs, bk the bias of the neuron, xk the input value received from the preceding layer neuron, wk

the corresponding weight of each connection and f is the activation function that is used for limiting the amplitude of theoutput of a neuron.

During training, Q sets of input and output data are given to the neural network. An iterative algorithm adjusts theweights so that the outputs (ok) according to the input patterns will be as close as possible to their respective desired outputpatterns (dk). Considering a neural network with K which is the total number of outputs, the mean square error (MSE) func-tion is to be minimized [38]:

MSE ¼ 1Q � K

�XQ

q¼1

Xs

k¼1

K dkðqÞ � okðqÞ½ �2: ð10Þ

The back-propagation algorithm is most widely used to minimize MSE by adjusting the weights of connection links [39].Learning in this method includes three stages:

In the first stage, input data are entered to the network. After that based on the weights and biases which are chosen ran-domly, also on the function of each neuron, the outputs of each neural cell are calculated until the final outputs are obtained.

(a)

(b)

Fig. 3. Comparison between the experimental data and FEM predicted values of the (a) roll force at different reductions and (b) roll torque at differentrolling speeds.


This stage is called feedforward. In the second stage the error of the final output is calculated and then using this error, theerrors of the outputs of all neurons are calculated subsequently from front to the back. This stage is called back-propagation.In the third stage according to the results of the second stage and using a special algorithm, the weights and biases are chan-ged until the least error is reached. Each running of the above is called a cycle. The cycles are repeated until one of the stop-ping conditions is reached [31].

3.2. Input and output parameters

The aim of the present work is to use artificial neural network modeling technique to construct a quantitative model foraluminum alloys, which can predict the roll force and torque of hot strip mill stands when the initial thickness of strip isknown. Omitting the parameters which are not important benefits the development of the model, the independent inputparameters are selected to be as initial temperature, thickness reduction, rolling speed and interfacial friction coefficientin the contact area. The patterns of the neural network models for rolling force and torque prediction according to the inputparameters are shown in Fig. 4.

3.3. Network training and testing dataset

To develop a robust NN model, a sufficiently large amount of data is required for training. The actual amount of data re-quired will depend on the complexity of the process to be modeled, the number of the input variables, and the quality of theprocess data. Typically, hundreds or even thousands of data will be required for training a NN model. A FE model usually

Fig. 4. Schematic illustration of the neural network structure.


involves heavy computation and is a very slow process. However, we have the advantage of full flexibility to choose the lev-els of all the individual inputs, in contrast to traditional data collection from a real rolling mill where the range of excitationsis often heavily constrained by the process conditions. Design for input excitation becomes necessary in order to minimizethe number of FE runs for generating the training data, while still providing the necessary information required for training aNN model.

The required data set in this study was created with the help of statistical design of experiments method (DOE). Rollingforce and torque values corresponding to training data are calculated for parameter values generated by full factorial design,which is one of the most common experimental designs. In a factorial design, variable space is divided into levels between thelowest and the highest values. A three-level full factorial design creates 3n training data, where n is the number of variables(n = 4 in this study). The constructed levels and factors are presented in Table 5. Hence, 34 = 81 data samples were generatedfrom the validated FE model that was developed in Section 2. Commercial pure aluminum (AA1100) strips with dimension of6.32 mm in thickness and 50 mm in width were rolled with a 250 mm diameter work roll in all samples. Some of the exper-imental results of Hum et al. [35] (nine samples in which process parameters are in the range of Table 5) are also employed asthe required data sets of ANN model. The 90 data sets for roll force and torque are divided into two parts; 14 data are ran-domly selected to test the reliability of the ANN model as a test set, and remaining set was used for training the network.

3.4. Selection of the best network topology

In this study, a computer program was devised using MATLAB based on the back-propagation algorithm (BPA) for thedeveloped ANN model. Back-propagation networks are the most commonly used NN for modeling because of their powerfulmodeling capability and the existence of many well established back-propagation training algorithms, such as the gradientdescent algorithm, quasi-Newton optimization, conjugate gradient algorithm, stochastic approximation, Levenberg–Marqu-ardt optimization, etc. [23]. The tangential sigmoid transfer function is used in the hidden and output layers and the Leven-verg–Marquardt back-propagation algorithm is selected as training function because it is the fastest method for trainingmoderate-sized feedforward neural networks [40].

To reduce the effect of overfitting, the early-stopping method is applied during the training. In this method, the trainingdata are divided into two subsets. The first subset is the training set which is used for adjusting the network weights andbiases. The second subset is the validation subset. During the training process by the first subset, the error on the validationset is concurrently monitored. Normally, the validation error decreases during the training, as does the training set error.However, when the network begins to overfit the data, the error on the validation set will typically begin to rise. Whenthe validation error increases for a specified number of iterations, the training is stopped, and the weights and biases atthe minimum of the validation error are returned.

Table 5Level designation of different process variables.

Factors Levels

1 2 3

Roll speed (RPM) 20 60 100Reduction (%) 20 30 40Temperature (�C) 450 500 550Friction coef. 0.1 0.23 0.35


Before presenting the patterns to the BP network, it is usually necessary to normalize the input and target data so thatthey can fall into a specified range. All the input and target data were scaled in the range between [0.1,0.9] using the follow-ing normalization equation:

Table 6Tried N

Netw

123456789

101112131415161718192021222324252627282930

Table 7The AN

Netw

TrainLearTranPerfoNumNumNumNumNum

Xi ¼ 0:1þ 0:8� X � Xmin

Xmax � Xmin

� �; ð11Þ

where X is the original data, Xmin the minimum value of X, Xmax the maximum value of X, and Xi is the unified data of thecorresponding X.

One of the most important tasks in ANN studies is to determine the optimal network architecture which is related to thenumber of hidden layers and neurons in it. Generally, the trial and error approach is used. In this study, the best architectureof the network was obtained by trying different number of hidden layers and neurons. The trial started one hidden layer withthree neurons, and the performance of each network was checked by correlation coefficient (R). The goal is to maximize cor-relation coefficient to obtain a network with the best generalization. Many different network models were tried and their Rvalues are calculated. Some of the best obtained R values for different trials are presented in Table 6. As it is seen in Table 6,the highest correlation coefficient for both roll force and roll torque prediction was obtained at a network, which has two

N models and their correlation coefficients (R) for defining the optimal network architecture.

ork no. Number of neurons in hidden layer 1 Number of neurons in hidden layer 2 R values of roll forceprediction

R values of roll torqueprediction

Training Test Training Test

3 – 0.9786 0.9752 0.9542 0.96524 – 0.9774 0.9823 0.9911 0.99175 – 0.9713 0.9799 0.9633 0.96596 – 0.9939 0.9872 0.9960 0.98977 – 0.9545 0.9887 0.9870 0.9833

11 – 0.9924 0.9852 0.9950 0.980312 – 0.9845 0.9745 0.9961 0.978813 – 0.9892 0.9686 0.9840 0.982314 – 0.9938 0.9537 0.9967 0.987915 – 0.9865 0.9438 0.9791 0.9535

3 3 0.9789 0.9847 0.9541 0.95044 3 0.9723 0.9827 0.9837 0.97815 5 0.9736 0.9484 0.9839 0.91775 4 0.9434 0.9153 0.9659 0.96155 3 0.9898 0.9764 0.9940 0.98386 6 0.9809 0.9796 0.9618 0.95436 5 0.9928 0.9740 0.9838 0.89156 3 0.9830 0.9613 0.9909 0.98517 6 0.9923 0.9899 0.9916 0.99447 5 0.9829 0.9765 0.9876 0.98107 4 0.9864 0.9840 0.9943 0.99239 6 0.9837 0.9683 0.9703 0.94669 5 0.9649 0.9627 0.9768 0.9576

10 10 0.9801 0.8853 0.9946 0.985310 9 0.9794 0.9665 0.9900 0.978110 8 0.9653 0.9669 0.9757 0.944510 5 0.9779 0.9336 0.9912 0.994211 7 0.9859 0.9283 0.9890 0.964611 5 0.9841 0.9379 0.9913 0.973912 6 0.9877 0.9145 0.9948 0.9905

N architecture and functions.

ork Feed-forward backpropagation network

ing function Levenberg–Marquardtning function Gradient descent with momentum weight & bias learning functionsfer function Tan sigmoid functionrmance function Mean squared errorber of input layer unit 4ber of hidden layers 2ber of first hidden layer units 7ber of second hidden layer units 6ber of output layer units 2


hidden layers with seven and six neurons, respectively. Based on this analysis, the optimal architecture of the ANN was con-structed as 4–7–6–2, representing the number of inputs, neurons in hidden layers, and outputs, respectively. The artificialneural network architecture and functions used in the resulted model are summarized in Table 7.

4. Model analysis and discussion

The ANN model developed in this study is used to predict the roll force and roll torque of the 90 hot strip rolling processdesign data. The performance of the proposed ANN model was plotted in Figs. 5 and 6 for both roll force and roll torque, ontraining and testing data respectively. It was observed that a high prediction capability was achieved for both training andtesting data sets of roll force and roll torque even though the latter was not used for the training of the ANN. Therefore, theANN appears to have a high generalization capability.

The overall performances of both sets for roll force and roll torque were evaluated via mean absolute percentage error(MAPE), maximum percentage error, sum squared error (SSE), the root mean-square error (RMSE), the correlation coefficient(R) and the coefficient of variation in percent (COV). As seen in Table 8, a high coefficient of correlation and a low mean abso-

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

R=0.99232

Target Value

AN

N P

red

icti

on

Data Point

Best Linear FitA = T

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

R=0.99162

Target Value

AN

N P

red

icti

on

Data Point


(a)

(b)

Fig. 5. Training performance of proposed ANN model in prediction of (a) roll force and (b) roll torque.

Table 8Statistical analysis of the performance of ANN model for training and testing predictions.

Training Testing

Roll force Roll torque Roll force Roll torque

MAPE (%) 5.014625 5.583072 8.007738 5.210827Maximum percentage error (%) 17.413783 20.384114 18.873612 15.710502SSE 0.048838 0.058761 0.011543 0.008715RMSE 0.025350 0.027806 0.028714 0.024949R 0.992321 0.991619 0.989901 0.994372COV (%) 6.640889 6.236267 7.674229 5.641910

0.1 0.2 0.3 0.4 0.5 0.60.1

0.2

0.3

0.4

0.5

0.6

R=0.9899

Target Value

AN

N P

red

ictio

n

Data Point

Best Linear Fit

A = T

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R=0.99437

Target Value

AN

N P

red

ictio

n

Data Point


(a)

(b)

Fig. 6. Test performance of proposed ANN model in prediction of (a) roll force and (b) roll torque.


lute percentage error were obtained for the training and testing data sets for both roll force and roll torque. The proposedANN model for roll force and roll torque prediction had correlation coefficients of 0.9923 and 0.9916, respectively, for

(a)

(b)

Fig. 7. Percentage error deviations of predicted roll force and roll torque from actual target results (a) training set and (b) test set.

(a)

(b)

Fig. 8. Model capability of predicting accurate rolling force and torque for practical unforeseen conditions.



training data sets, 0.9899 and 0.9944, respectively, for the testing data sets. Moreover, the mean absolute percentage error ofthe roll force prediction was about 5.01% and 8.01% for the training and testing set, respectively. Similarly, the former of5.58% and the latter of 5.21% were obtained for the model developed for roll torque prediction. As it is seen these mean abso-lute percentage errors are fairly reasonable. Furthermore, the models provided highly reasonable SSE of as low as 0.0115 and0.0087 for the testing sets of roll force and roll torque, respectively.

Fig. 7 demonstrates the percentage error deviations of predicted roll forces and roll torques using ANN from actual targetresults (Denormalized data). Comparison of the ANN predicted roll forces and roll torques with respect to actual target re-sults of test data set is given in Fig. 8. As it is observed from Figs. 7 and 8, the ANN was capable of generalizing between inputvariables and the output reasonably well. The predictions are good for not only the training set, but also for the unseen test-ing set, with all predictions of the testing data fall inside the 10.5% boundaries. It is sufficient to use the developed ensembleNN models for the roll force and torque prediction, instead of using the time consuming FE model. This quick response is adesirable property for online control and optimization, such as mill setup, dynamic adjustment and control. Hence, the re-sulted model is quite suitable for online control of hot rolling mills to predict appropriate rolling conditions due to inputparameters.

The neural network model is essentially a complicated nonlinear model, containing numerous activation functions andinterconnecting weights. Model response surfaces which can be obtained via varying two selected inputs, while all theremaining inputs are used at their nominal intermediate values, are a good way to check the compatibility between thedeveloped model and the existing physical understandings, thus providing some indications on whether an adequate modelhas been developed from the training data. Some of the typical response surfaces of the resulted NN model, which show therolling force and torque trends against two selected inputs, are shown in Figs. 9 and 10. These model response surfaces can

Fig. 9. Evolution of roll force vs. (a) rolling speed and thickness reduction, and (b) friction coefficient and thickness reduction.

Fig. 10. Evolution of roll torque vs. (a) initial temperature and thickness reduction, and (b) initial temperature and friction coefficient.


serve the purpose of checking the validity of the developed models against existing process knowledge. Fig. 9 indicates thatreduction and friction coefficient are the dominant parameters for the rolling force, and the roll force rises as the reduction orfiction coefficient increases. However, the rolling speed has no clear effect on rolling force and its influence differs in variousreductions. As it can be observed from Fig. 10, the factors of thickness reduction and friction coefficient have more influentialeffects on rolling torque. Initial temperature has a negative correlation and rolling torque decreases as the initial tempera-ture increases. The behavior of the neural network thus reflects the existing knowledge about the rolling process and thecharacteristics of the data with which it is trained.

5. Conclusions

This paper focuses on developing an ANN model to address the need for accurate prediction of the roll separating forceand roll torque in a hot strip mill which aims to meet the quality standards of the final products in aluminum manufacturing.A three-dimensional finite element model was developed and verified to provide a data basis for training and validation ofthe neural network. The well-known Levenberg–Marquardt feed-forward network trained with error BP algorithm was useddue to its successful applications in various practical problems and its simplicity in implementation. The optimum networkarchitecture was obtained by evaluating the performance of ANNs with different number of hidden layers and neurons. TheANN predictions are found to be in extremely good agreement with the simulation results which indicates the capability ofthe model as a tool for accurate estimation of roll force and torque in different hot rolling conditions. The model is capable ofconsidering the effects of important parameters such as strip temperature, percentage of thickness reduction, rolling speedand friction coefficient in the contact area. The resulting ANN model is feasible for on-line control and rolling schedule opti-


mization, and can be easily extended to cover different aluminum grades by generating the corresponding training data froma FE model.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm.2012.09.070.

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