Regular Article

Prediction of Mean Flow Stress during Hot Strip Rolling Using Genetic Algorithms

2014 Volume 54 Issue 1 Pages 171-178

Details

Abstract

In order to satisfy the demands for high accuracy and efficient rolling, it is necessary to establish favourable mathematical model of roll force calculation, which is one of the most important terms for process control. For this purpose it is necessary to predict the Mean Flow Stress (MFS) with good accuracy, since it is the predominant factor of the roll force model. In this paper this problem is dealt on data coming from a real industrial plant and hot compression tests. Various steels have been tested; these can be divided into 2 principal groups: Niobium/Titanium microalloyed, and plain Carbon-Manganese.

Particularly, MFS has been found out by measurements taken in the industrial strip rolling mill converting log data in MFS using the Sims approach. Moreover, in order to evaluate the dispersion of MFS measurements, thermomechanical deformation tests have been conducted by a Gleeble 3800 thermomechanical simulator simulating all the seven passes of the studied finishing stand.

The results have been analysed and compared to the predictions of some mathematical models developed in literature and it is shown how inadequate well known literature models are. Alternative models have been then proposed by improving existing formulae by means of genetic algorithms based optimization. The performance of the proposed methods have been compared. Moreover, their prediction abilities have been evaluated using the MFS dispersion data measured experimentally. The satisfactory results obtained by optimized based models put into evidence the advantages of the use of artificial intelligence techniques in the industrial framework.

1. Introduction

Computer control of rolling process has been eagerly developed in modern rolling mills. There have been strong demands for increasing the efficiency in the process control and for improving the accuracy in the gage and shape control of rolled products. In order to satisfy the demands for high accuracy and high efficient rolling, it is necessary to establish favourable mathematical model of roll force calculation, which is one of the most important terms for process control.^{1)} For this purpose it is necessary to predict the Mean Flow Stress with good accuracy, since it is the predominant factor of the roll force model.

The present work aims at modelling the MFS for carbon-manganese and microalloyed steels at the hot strip mill logs in Taranto (Italy) by measurements taken in the industrial plant. This work is firstly focused on the empirical models as this class of approaches is widely used in the industrial fields as it combines the theoretical knowledge of the interaction between input and output variables with the exploitation of real plant data to obtain performing models in terms of predictions. For this reason, some of the literature models (for carbon and microalloyed steels) have been firstly tested and subsequently optimised by means of Genetic Algorithms (GAs), which tuned some of their parameters in order to best fit the available data. Their prediction abilities have been evaluated using the MFS dispersion data measured experimentally by a Gleeble 3800 thermomechanical simulator simulating all the seven passes of the studied rolling mill finishing stand.

The aim of the work is, in fact, to provide a model which can predict rolling forces which are sufficiently fast and accurate. Before starting rolling, in fact, a drafting schedule is generated. This drafting schedule specifies the reduction rate at each pass. The reduction rate and the rolling speed affect the material characteristics of the rolled plate as well as the productivity. Thus, the optimization of a drafting schedule is an important problem. A rolling load prediction model plays the central role to calculate the drafting schedule, as it describes what load causes what deformation of a strip.

This paper is organized as follows: in section 2 it is reported the definition of MFS and the method used to estimate it experimentally both by plant mill data and by hot compression tests. In section 3 a short review of the state of the art in the modelling of MFS is provided, focusing on most widely used empirical predictive models, and then new approaches based on different artificial intelligence methods are presented. Section 4 illustrates the experimental works used for the construction of the dataset necessary for the implementation of the mathematical models for the prediction of MFS by plant mill data. The average standard deviation of MFS measurements by hot compression tests has been estimated. Section 5 depicts how the literature empirical models are used to forecast the MFS related to the strip rolling in the considered hot rolling plant and their performances are evaluated. Subsequently, in order to overcome the criticalities encountered by standard models, they have been optimized by means of GAs and the main achievements are discussed. Finally, in section 6, conclusions are drawn and the future perspectives of the proposed approaches are outlined.

2. MFS Definition

In mill plant the conventional way of estimating the rolling force is via mathematical model, which is mostly based on thermomechanical theory (Orowan;^{2)} Sims;^{3)} Ford and Alexander^{4)}). The rolling load per unit width is given generally by the following equation:

(1) |

With Sims calculations *Q _{P}* is calculated from a complex function of roll gap geometry, while Ekelund

In laboratory, hot deformation resistance of metallic materials can be usually determined by using methods, such as high-temperature tension, compression and torsion testing. Based on measurement of forces in the laboratory rolling of flat samples graded in thickness, the effective methods of description of the mean flow stress (MFS) values were developed and applied to many steel grades, some iron aluminides, magnesium alloy *etc.*^{6,7,8)} Particularly, the Mean flow stress (MFS) is defined as the area under a given stress-strain curve for the strain interval selected.^{9)} The MFS is calculated as follows:

(2) |

3. Current Approaches on MFS Estimation

A considerable literature exists on the modelling of the rolling process using data recorded in experimental or industrial rolling operations. These models generally establish a relationship between the rolling force/torque and the deformation resistance of the workpiece in the roll gap.

Dimatteo *et al.*^{10)} have carried out an excellent analysis of the different types of mathematical models and have supplied a list of the advantages associated with the application of a particular model to a given practical situation. According to Hodgson,^{11)} the main types of models fall into the following categories:

A) Phenomenological: These describe the actual physical processes that occur. However, the prediction ability with only a thermo-mechanical-based mathematical model is limited because the model cannot describe all the physical phenomena perfectly due to the very nonlinear, complex, and immeasurable data such as friction, yield stress, and system disturbances.^{12,13)}

B) Empirical: These are characterized by the empirical analysis of data. The aim of this approach is to provide relationships between the process variables and parameters of interest. While simple regressions have been the most applied technique, there are increasing numbers of sophisticated methods available for constructing empirical models.^{14)}

C) Semi-Empirical: These combine certain features of the above two methods. An example of this type is the modeling of the high temperature recrystallization behaviour of austenite. Mathematical models for the recrystallization kinetics use an Arrhenius temperature term combined with expressions relating the changes in the driving and drag forces (strain, dislocation density and particle/solute drag), and the surface area per unit volume (grain size and morphology). Such relationships have a fair degree of universality, although they are essentially empirical in nature.^{15,16,17)}

D) Heuristic: These include the rule-based models that are now gaining widespread acceptance within the steel industry. These models do not contain mathematical representations of the physical processes, but predict the outcome of a series of events on the basis of previous experience. The application of neural networks to predict the rolling force is a current example of this type of models.^{18,19,20)}

Empirical and semi-empirical relationships obtained from experimental data are widely used in deformation models because they are easier to develop and complex internal material variables need not be considered. The developed relationships often used present flow strength in terms of either strain and strain rate or strain rate and temperature; the empirical constants are defined for different process conditions. The most famous are *Bruna*, *Kirihata*, *Misaka*, *Poliak*.^{14)} The *Shida*^{21)} model also considers the softening that occurs in steel near the A_{r3} temperature.

In **Table 1**, some of the most used formulae for C–Mn and microalloyed steels are reported, where [X] is the concentration of the chemical element X in weight percent, T is the temperature in Kelvin, *ε* is the applied strain and *ε*_{s} is the strain rate:

Table 1. MFS literature models. MFS is expressed in units of MPa.

ID | Formula | Parameters |
---|---|---|

Misaka |
| a_{m}=0.126 b_{m}=1.75c _{m}=0.594 d_{m}=2851f _{m}=2968 g_{m}=1120α=0.21 β=0.13 |

Bruna |
| a_{b}=3126 b_{b}=68c _{b}=2117 d_{d}=54e _{b}=152δ=0.21 χ=0.13 |

Siciliano |
| a_{s}=2704 b_{s}=3345c _{s}=220θ=0.21 φ=0.13 |

Poliak |
| a_{p}=1.09 b_{p}=0.056c _{p}=4.54 d_{p}=1.21e _{p}=0.056 f_{p}=0.1 |

Shida |
| a_{h}=0.28 b_{h}=5c _{h}=0.01 d_{h}=0.05e _{h}=1.3 f_{h}=0.2g _{h}=0.3 h_{h}=0.2i _{h}=0.41 l_{h}=0.07m _{h}=10n _{h}=0.019 p_{h}=0.126q _{h}=0.076 r_{h}=0.05 |

All equations reported in Table 1 assume that full recrystallization takes place between passes. However, in Nb-containing steels, recrystallization is frequently incomplete when the interpass times are short, particularly during the later passes of strip rolling, when the temperature is relatively low. This leads to strain accumulation, which may in turn cause the initiation of dynamic recrystallization followed by metadynamic recrystallization. These three microstructural processes can be expected to contribute to the departures from the Misaka and other equations.

The biggest limitation of this approach is that formulae are tuned by exploiting a particular set of data, representing specific conditions, whose generalization can lead to criticalities in the estimation of the desired output when the actual conditions are sensibly different from those used for the tuning of the model.

3.2. MFS Prediction Using Genetic AlgorithmsGAs can be applied for modelling hot strength of steels as they offer a relatively simple and fast way to determine the main rolling process parameters. In particular, GAs are especially useful to accurately tune existing models with a minimum of experimentation, given a procedure indicating how good any candidate the solution (*i.e.* the parameters of the model to be tuned) is. The stochastic nature and the robustness of this procedure allow to determine the most optimal values of parameters with a minimum of experimentation. GAs have already proven to be very efficient to this purpose (see an example in Colla *et al.*^{22)}). Moreover, Yang and Linkens^{23)} have suggested that the hybrid neural network model with GA-based optimization can predict the complex flow stress behaviour provided that a suitable training data set is available, and they have demonstrated that genetic algorithms are highly effective in finding the optimal parameters for the constitutive equations. Gama and Mahfouf^{24)} have employed GAs to optimise the parameters of the constitutive equations demonstrating that the hybrid neural network modelling with GA-based optimization has obvious advantages over the corresponding white, black or grey (without GAs) modelling schemes for stress–strain predictions.

One of the most important advantages of GA-based parameters tuning lies in the fact that the mechanism of search for the optimal solution is linked to the model to optimise by the coding mechanism, *i.e.* the way in which a set of parameters is “transformed” into a so-called *chromosome*. The different chromosomes follow an evolution procedure which depends on the selected evolutionary strategy and not on the model or system to optimize, *i.e.* does not require the formulation of derivatives of the analytical model to tune: only an objective (or some objectives, in case of multi-objective optimization^{25,26)}) function need to be formulated, which quantify the goodness of the performance of the model when a particular set of parameters is adopted. This feature of GAs provide a great flexibility with respect to the different models which can be adopted to predict a variable (such as the MFS, in the present case): one can adopt the same search algorithm to tune different models, once a coding procedure and a performance function are selected, and choose the best one for a selected HSM and products range. Moreover one can even add further models with minimal variations of the prediction software and no alteration of the search procedure, by thus expanding the potential capabilities of the overall prediction.

4. Materials and Methods

4.1. Industrial Data
Hot strip mill logs from Taranto steel plant are the basic source of data for this work. The analysis has involved data relating both to microalloyed steels and to carbon steels for a total of 1620 coils.

**Figure 1** shows a scheme of the considered rolling plant. It is composed, from right to left of the scheme, of four reheating furnaces, a press for the sizing of the slabs (SP), a vertical scale breaker that removes the oxide at the exit of the furnaces (VSB), six stands of the roughing mill (R1–R6), seven stands of the finishing (F1–F7), the run-out-table (ROT), and, finally, the three coilers (C1–C3).

Fig. 1.

Strip rolling plant scheme. (Online version in color.)

The examined data include chemical composition, final mechanical properties of the strip, size of the strip sheet (thickness, length, width), rolling parameters both of the roughing and finishing stands (thicknesses, rolling temperature, forces rolling, interpass time, reduction rate, strain rate, diameter of the rolling cylinders), parameters of heating furnace (temperature evolution), parameters of coiler (strip sheet rate and coiler temperature).

The MFS value for each pass has been calculated using the approach of Sims.^{10)}

Real coded genetic algorithms have been used to tune the parameters of the literature formulae introduced in section 3.1. More in detail, GAs perform a smart search over the space of possible parameters combinations in order to minimize the discrepancy between the MFS predicted through the candidate optimized formula and the measured one. The fitness function utilized by GAs is the Mean Square Error. At each generation of GAs, 20% individuals of the current generation survive with a probability proportional to their fitness. The remaining 80% of the new population is formed by crossing over individuals of the previous generation (probabilistically selected according to their fitness). The crossing over operator creates a new candidate solution by picking randomly the *genes* (the parameters values, in this case) from the two parents of the new individual. Subsequently, 5% of the so-created population is modified by the mutation operator which changes the values of two genes of the selected individuals in the range +/– 10%.

The MFS is influenced by the complicated microstructural changes associated happening at high temperature deformation processing, which include strain hardening, recovery, static and dynamic recrystallization, precipitation all of which operate simultaneously at different rates under different conditions of strain, strain rate and temperature. For this reason MFS is strongly affected by small variations of process parameters, such as strain, strain rate and temperature. In order to evaluate the standard deviation of MFS measurements, hot compression tests has been performed both on Carbon-Manganese steels and Microalloyed steels. In order to have enough data for statistics, for each chemical composition, the MFS values have been calculated for six different tests.

4.2.1. Materials and MethodsLaboratory cast and forged HSLA steels of type C–Mn, C–Mn–Nb–Ti, C–Mn–B served as the experimental material. The steel compositions are listed in **Table 2**.

Table 2. Chemical compositions of steels tested by hot compression tests.

steel | C | Mn | Si | V | Al | Ti | B | Nb | T_{nr} (°C) |
---|---|---|---|---|---|---|---|---|---|

B040 | 0.032 | 0.239 | 0.002 | 0.001 | 0.040 | 0.001 | – | – | 901 |

PH20 | 0.005 | 0.175 | 0.002 | 0.002 | 0.037 | 0.022 | – | 0.029 | 977 |

RS75 | 0.072 | 0.873 | 0.016 | 0.004 | 0.039 | 0.023 | – | 0.039 | 1019 |

CH3N | 0.031 | 0.233 | 0.007 | 0.002 | 0.034 | 0.006 | 0.0021 | – | 897 |

The first steel in Table 2 was selected as a reference steel.

Deformation tests were conducted by a Gleeble 3800 thermomechanical simulator. Rectangular bar test specimens (of 10 mm length and 20×15 mm basis) have been heated, in a vacuum chamber, up to 1250°C with a heating rate of 3.3°C/s, soaked for 300 s, and then cooled down with a cooling rate of 3°C/s to the first deformation temperature. Subsequent compressions between tungsten carbide anvils have been performed. The deformation schedule is reported in **Table 3**. Finally, specimens have been controlled cooled at cooling rates of 10°C/s. The deformation schedule has been carried out in order to exactly reproduce the nominal schedule used in the Taranto strip log mill for the data collected in subsection 4.1.

Table 3. Hot compression test schedule.

Pass | Strain rate (1/s) | reduction | Temperature (°C) | Interpass time (s) |
---|---|---|---|---|

1 | 7 | 0.40 | 980 | 3.9 |

2 | 14 | 0.50 | 965 | 2.6 |

3 | 28 | 0.42 | 950 | 2 |

4 | 42 | 0.35 | 942 | 1.6 |

5 | 60 | 0.30 | 933 | 1.3 |

6 | 81 | 0.27 | 925 | 1.1 |

7 | 87 | 0.20 | 910 | 1.1 |

For each chemical composition at least six analogous hot compression tests have been conducted. Tantalum foils with a thickness of 0.1 mm have been used to prevent the specimen sticking to the anvils, and graphite foils for lubrication. Tantalum foil can prevent not only interface sticking but also helps reducing the specimen bulging produced by the interfacial friction and thus ensure a homogenous deformation of the rectangular bar specimen. The temperature has been measured using a Pt/PtRh thermocouple welded to the surface of the specimen. The axial and radial displacement and the axial compressive force have been measured continuously during the experiment.

For the chemical compositions of the steels tested in the present work, the reheating temperature of 1250°C has been chosen. In this way for the C–Mn–Nb–Ti steels, all the titanium is precipitated as TiN by avoiding austenite grain size coarsening, while all the niobium is dissolved and can precipitate in the last stage of rolling to pancake austenite. Generally speaking, in fact these steel compositions are optimised for size control by TiN; thus, it is obvious that no titanium is available to provide precipitation strengthening. In fact, TiN is extremely stable and can withstand dissolution at high reheating temperatures prior to hot rolling inhibiting the austenite grain coarsening during reheating and controlling the austenite grain size during hot rolling. In order to design high-strength ferrite–pearlite steel and to capitalise on any available excess nitrogen, titanium is complemented with niobium. Niobium carbide or carbonitride, in fact, has a relatively low solubility and may precipitate in the later stages of rolling.

The temperature of non-recrystallization temperature, Tnr, has been calculated according the formula used by Zaki^{10)} where the chemical elements’ amount regard their amount dissolved in austenite. The measured values are reported in Table 3. As control of the efficiency of alloying design, the dissolved niobium and titanium have been calculated according to the approach reported of Gorni.^{27)}

The Mean flow stress (MFS) has been measured according to formula (2) from stress-strain curves. It is reported in **Fig. 2** as function of the inverse of temperature for the studied steels.

Fig. 2.

MFS calculation for RS75, CH3N, B040, PH10 steels. (Online version in color.)

Generally speaking, for all the tested steels the MFS naturally increases as the temperature decreases. In particular, the carbon steel exhibits the lowest MFS, while the RS75 microalloyed exhibits the highest ones. The differences in the MFS values between the microalloyed RS75 and PH10 steels are imputable to their different content of carbon: higher is the carbon content higher is the MFS.

Particular attention must be paid on B040. As expected, in fact, the MFS of this steel is higher than that of CH3N steels being the boron content of B040 steel higher. This behaviour is confirmed by the actual literature,^{28)} where higher flow stresses were found for Boron steel than those found in B-free steel were due to solute B at grain boundaries, which delayed dynamic recrystallisation and enhanced the solute drag effect. Boron segregation to austenite grain boundaries, in fact, retards their mobility and hence, the recrystallisation kinetics during hot working.^{29)} It has been suggested that when austenite grain boundaries move slowly enough (*i.e.* comparable to the diffusion rate of B at low austenite temperatures), solute B atoms are “swept up” leading to an increased flow stress.

For each chemical composition the MFS values (calculated for the six different tests) are reported in **Fig. 3**.

Fig. 3.

(a) MFS calculation for RS75, CH3N, B040, PH10 steels. (b) Standard deviation for RS75, CH3N, B040, PH10 steels. (Online version in color.)

As evident from Fig. 3(a), some differences are present in the MFS values belonging to the same chemical composition. In Fig. 3(b) the standard deviation is reported for all the four examined chemistries as function of the inverse of temperature. The standard deviation has resulted to be higher for the microalloyed steels and when the temperature decreases. The reason of this discrepancy/dispersion has to be found in the major sensitivity of microalloyed steels to changes in annealing time/temperature and on the strain/strain rate.

The standard deviations increases also with the number of passes. This is because, pass after pass, the microstructural differences sums giving up higher discrepancies/dispersion.

5. Numerical Results

5.1. MFS Prediction Using Current Literature Formulae
The results obtained by comparing the MFS predicted by means of the formulae reported in Table 1 are depicted in **Tables 4** and **5** for the 7 stands composing the hot strip mill. The results, which refer to the two whole available datasets, are reported in terms of average percentage error for each formula and for each pass. The formula used for the calculation of the average prediction error is reported in the following:

(3) |

Table 4. Average prediction error for the MFS (expressed in MPa) at each pass for Carbon-Manganese steels, which is obtained through equations reported in Table 2 by using the literature values for all the parameters; in the last column, the global average error in the prediction of the MFS is reported.

formula | Pass 1 | Pass 2 | Pass 3 | Pass 4 | Pass 5 | Pass 6 | Pass 7 | average |
---|---|---|---|---|---|---|---|---|

Misaka | 70.8 | 57.3 | 81.5 | 139.9 | 157.4 | 184.5 | 194.9 | 126.6 |

Shida | 43.2 | 51.4 | 78.8 | 98.3 | 85.2 | 90.1 | 112 | 79.9 |

Poliak | 56.6 | 41.2 | 63.1 | 119.8 | 136 | 163 | 176.6 | 108.0 |

It is worth noting that the whole dataset has be divided in two subgroups: carbon-manganese steels (1480) and microalloyed steels (140) for which different the formulae have been tested. Particularly for C–Mn steels (results in Table 4) the Misaka, Poliak and Shida formulae, while for microalloyed steels (results in Table 5) the formulae of Bruna, Siciliano, Shida and Poliak have been tested. It worth noting that when the Poliak formula has been used to predict the MFS of Carbon-Manganese steels, the Niobium and the Titanium terms have been considered equal to zero.

Table 5. Average prediction error for the MFS (expressed in MPa) at each pass for microallayed steels, which is obtained through equations reported in Table 2 by using the literature values for all the parameters; in the last column, the global average error in the prediction of the MFS is reported.

formula | Pass 1 | Pass 2 | Pass 3 | Pass 4 | Pass 5 | Pass 6 | Pass 7 | average |
---|---|---|---|---|---|---|---|---|

Bruna | 56.3 | 69 | 95.6 | 169.9 | 188 | 188 | 237.5 | 136.1 |

Siciliano | 84.6 | 102.1 | 131 | 206.8 | 226.1 | 226.1 | 273.5 | 170.7 |

Poliak | 35.9 | 46.1 | 71.6 | 145.2 | 163 | 163 | 214.1 | 112.7 |

Shida | 26.7 | 40.1 | 73.8 | 155 | 164.9 | 164.9 | 188.3 | 108.1 |

As it clearly stands out from the results reported in Tables 4 and 5, the predictive performance of the standard formulae when coping with the proposed plant data are not satisfactory, as the average error is higher with respect to the industrial requirements. Moreover, it is worth noting that, both for microalloyed and Carbon-Manganese steels, the performance of the tested models worsens in the last passes.

For the microalloyed steels, among the tested approaches, the performance of Shida and Poliak formulae are comparable and definitely low. The worst performance is that of the Siciliano formula that takes into account the lowest number of microalloying/alloying elements. Also for the Carbon-Manganese steels, the performance of the Shida model is the best. As expected, the performance of the Poliak formula is better than the Misaka one. This is because the Poliak formula takes into account also the effect of the alloying elements (Mn, Mo, Al).

For both Carbon-Manganese and microalloyed steels, the good prediction ability of the Shida models is probably due to the dependence of the exponent of strain rate from temperature. This is evident considering that the increment of average error with the temperature is lower for the Shida formula than for all the other ones.

The poor results achieved by literature formulae are probably due to their empirical nature: the presented models, in facts, are, on one hand, theoretically driven as far as the choice of involved variables concerned. On the other hand, the coefficients defining the equations are determined by means of statistical and mathematical techniques, aiming at the minimization of the prediction error with respect to the data used for the tuning. This latter process can possibly lead - as in this case - to the over-fitting of the developed model to training data. The over-fitted model will not be able to generalize when coping with different data, achieving a worse performance than expected.

In order to overcome the criticalities encountered, in the next section new methods for the prediction of MFS will be described and assessed. The main aim of the proposed approaches is the improvement of the existing models as well as the development of new ones for obtaining a better predictive performance which allows models use on plant.

5.2. Tuning of Sensitive Parameters of Standard Formulae by Means of Genetic AlgorithmsIn this section the models introduced in Table 2 are tuned in order to improve their predictive performance with respect to the available set of training data. More in detail, some of the coefficients of the handled formulae are modified in order to reduce the error in MFS prediction through the 7 stands of the rolling mill. This approach has the advantage of exploiting an existing and coherent model and slightly modify it in order to obtain better performance. The obtained updated formulae maintain the theoretical relations among inputs and outputs and their interpretability from the operator point of view.

For the Carbon-Manganese steels, the exponents associated to strain and strain rate respectively take part to the tuning process both in the Misaka and in the Poliak formulae.

For microalloyed steels, as far as the Poliak formula is concerned, in a first attempt only a subset of the parameters take part to the optimization, in order to reduce the search space dimensionality. In this case, the selected parameters are the coefficients associated to the micro-alloying elements [Nb] and [Ti] and the exponents associated to strain and strain rate values. In a second attempt one more parameter - the coefficient associated to [Mn] - has been involved into the optimization, trying to further improve the quality of the provided solution. This choice is driven by theoretical considerations, as the selected parameters are those ones which are associated to the variables which mostly influence MFS values.

Regarding the Bruna formula, the exponents associated to strain and strain rate respectively take part to the tuning process. Whereas, for the Siciliano formula, the tuning process has involved not only the exponents associated to strain and strain rate values, but also the coefficients associated to the alloying elements [Nb], [Ti], and [Mn].

Both for microalloyed and Carbon-Manganese steels, regarding the Shida formula, two different attempts have been tested, changing the parameters taking part to the optimization process: in the first attempt all the parameters contributing to the determination of the exponent of the strain rate are considered; in the second one, in order to broaden the search space and not to limit the search to a restricted and possibly not promising area of the search space, all the parameters of the formula are tuned. This latter optimization process, due to the high dimensionality of the problem, took several minutes for reaching the depicted optimal solution. By the way, as shown in **Fig. 4**, the GAs search converges quickly to an optimal solution. This latter figure assesses the effectiveness of the tested GAs approach.

Fig. 4.

Convergence behaviour of the objective function. (Online version in color.)

The results in terms of average error (calculated using(1)) per stand for each of the tested optimized formulae are depicted in **Tables 6** (for C–Mn steels) and **7** (for microalloyed steels). In **Table 8** the values of the tuned parameters are reported.

Table 6. Average prediction error for the MFS (expressed in MPa) at each pass for the Carbon-Manganese steels, which is obtained through equations reported in Table 2 after optimization; in the last column, the global average error in the prediction of the MFS is reported.

formula | Pass 1 | Pass 2 | Pass 3 | Pass 4 | Pass 5 | Pass 6 | Pass 7 | average |
---|---|---|---|---|---|---|---|---|

Misaka | 21.1 | 15.4 | 16.7 | 15.4 | 12.6 | 22.9 | 38.4 | 20.4 |

Poliak | 15.1 | 23.1 | 22.5 | 13.6 | 12.6 | 23.6 | 39.3 | 21.4 |

Shida (M coeff.) | 45.6 | 10.8 | 11.8 | 17.6 | 18.5 | 23.5 | 52.9 | 25.8 |

Shida (full opt.) | 28.2 | 11.8 | 15.9 | 13.1 | 13.3 | 22.4 | 39.7 | 20.6 |

Table 7. Average prediction error for the MFS (expressed in MPa) at each pass for the microalloyed steels, which is obtained through equations reported in Table 2 after optimization; in the last column, the global average error in the prediction of the MFS is reported.

formula | Pass 1 | Pass 2 | Pass 3 | Pass 4 | Pass 5 | Pass 6 | Pass 7 | average |
---|---|---|---|---|---|---|---|---|

Bruna | 15.8 | 25.6 | 28.8 | 20.7 | 25.4 | 41.1 | 41.1 | 26.2 |

Siciliano | 15.6 | 18.1 | 23.2 | 21 | 25.2 | 42.1 | 42.1 | 24.2 |

Poliak | 21.9 | 32.8 | 32.9 | 19.6 | 24.8 | 43.7 | 43.7 | 29.3 |

PoliakPlus | 13.6 | 19.9 | 25.2 | 23.3 | 27.1 | 41.5 | 41.5 | 25.1 |

Shida (M coeff.) | 53.2 | 24.5 | 19.4 | 31.5 | 46.7 | 65.2 | 65.2 | 40.1 |

Shida (full opt.) | 15.4 | 14 | 19.6 | 20.1 | 20.2 | 30.3 | 30.3 | 19.9 |

Table 8. Optimized MFS models. MFS is expressed in units of MPa.

formula | Parameters C–Mn steels | Parameters microalloyed steels |
---|---|---|

Misaka | α_{m}=0.0489 β_{m}=0.2274 | |

Poliak | α_{m}=0.0746 β_{m}=0.2095 | c_{p}=1.6104 d_{p}=4.1135α_{m}=0.01 β_{m}=0.1958 |

PoliakPlus | a_{p}=0.6718 b_{p}=0.0692c _{p}=4.8507 d_{p}=6.7067α_{m}=0.01 β_{m}=0.2472 | |

Bruna | δ=0.0010 χ=0.2149 | |

Siciliano | a_{s}=2883.7 b_{s}=2246.3c _{s}=88.9 θ=0.04 φ=0.3 | |

Shida (M coeff.) | n_{h}=0.2367 p_{h}=0.067q _{h}=0.2149 r_{h}=0.3328 | n_{h}=0.1738 p_{h}=0.1322q _{h}=1.3709 r_{h}=0.1455 |

Shida (full opt.) | a_{h}=0.4956 b_{h}=3.8596 c_{h}=0.0465d _{h}=0.1767 e_{h}=2.0925 f_{h}=0.2034g _{h}=0.625 m_{h}=10.5029 i_{h}=0.1044h _{h}=0.0802 n_{h}=0.0748 p_{h}=0.2827q _{h}=0.8462 r_{h}=0.1521 | a_{h}=0.618 b_{h}=7.135 c_{h}=0.1945d _{h}=0.1811 e_{h}=0.268 f_{h}=1.6177g _{h}=0.216 m_{h}=2.0614 i_{h}=0.0667h _{h}=0.0771 n_{h}=0.155 p_{h}=0.8774q _{h}=0.0542 r_{h}=0.8653 |

As it can be seen from Tables 6 and 7, both for microalloyed and Carbon-Manganese steels, the average prediction error is about 20 MPa. In particular, the results obtained by all the tuned formulae are very good as the prediction error has almost be reduced by five times.

As example, for the full optimazed Shida formula, the predicted MFS versus measured MFS is shown in **Fig. 5** both for Carbon-Manganes and microalloyed steels for the sixth pass. As it can be seen, the dispersion around the measured value is independent from the measured MFS value.

Fig. 5.

Predicted MFS versus measured MFS using the full optimized Shida formula for C–Mn and microalloyed steels for the sixth pass. (Online version in color.)

In order to evaluate the goodness of the prediction ability of the optimized models, their average prediction errors have been compared with the dispersion of experimental data measured in subsection 4.2.3.

This analysis gives an idea of the error in the prediction of MFS stress that is industrially acceptable. Particularly, the average of unavoidable error is 10 MPa for Carbon-Manganese steels, 14 MPa for microalloyed steels. These values are consistent with the average error found for the optimized models that is around 20 MPa.

6. Conclusion

The paper presents a deep investigation on the prediction of the mean flow stress during strip rolling of microalloyed and Carbon-Manganese steels.

(1) MFS has been found out by both thermomechanical tests simulating the seven compression stages of the strip rolling process and measurements taken in the industrial mill.

(2) The dispersion of MFS measurements has been calculated using the Gleeble data. The average standard deviation has resulted to be 15 MPA for microalloyed steels, 10 MPa for Carbon-Manganese steels.

(3) Several literature formulas for the prediction of MFS have been tested on industrial data without achieving satisfactory results. It is probably due to the empirical nature of these models which, even though valid and efficient, are tuned on specific data and can be prone to the lack of generalization capabilities.

(4) For this reason, within this work, the literature models have been tuned by means of genetic algorithms in order to produce optimized formulae which minimize the prediction error on the available data. This process led to optimized models whose performance are by far better with respect to original literature formulae. The obtained average prediction error is about 20 MPa both for Carbon-Manganese and Microalloyed steels.

(5) In order to have an idea if the obtained error in the prediction of MFS is industrially acceptable, it has been compared by the average standard deviation measured by Gleeble measurements (point 2). These values are consistent with the average error found for the optimized models that are so satisfactory with respect to the industrial standard and encourages the use of GAs optimization of existing literature formulae in the industrial framework.

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