Model Based Optimization and Control of the Rolling Process for Microstructure Design

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An integrated approach to carry-out advanced optimisation and control of the hot-rolling of steel for microstructure design
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    MODEL-BASED OPTIMISATION AND CONTROL OF THE HOT-ROLLING PROCESS FOR THEDESIGN OF STEEL MICROSTRUCTUREMiguel Angel Gama and Mahdi Mahfouf   IMMPETUS (Institute for Microstructural and Mechanical Process Engineering: The University of Sheffield) Department of Automatic Control and Systems Engineering The University of Sheffield, Sheffield S1 3JD, UK    Abstract: A new integrated approach to carry-out advanced optimisation and control of the hot-rolling of steel is presented. The proposed design includes two stages: (1)OPTIMISATION: Genetic Algorithms and physically-based models describing thekinetics of dynamic microstructural behavior are used, along with an optimisationcriterion, to set-up the optimal rolling schedule; (2) CONTROL: advanced controlstrategies, including Model-based Predictive Control, are used to guarantee the optimal performance of the mill during the rolling process. The overarching aim of this researchwork is to integrate knowledge about both the stock and the rolling process to findoptimal hot-deformation profiles and achieve the desired final steel microstructure and properties. Copyright © 2007 IFAC   Keywords: Steel Manufacture and Industry, Predictive Control, Genetic Algorithms,Optimisation.   1.   INTRODUCTIONThe development of optimal design and controlmethods for steel-making processes is needed for improving metal quality, reducing costs and producing specified properties on a repeatable basis.The quality of steel depends on the propertiesassociated with its microstructure, that is, on thearrangements, volume fraction, sizes andmorphologies of the various phases of transformationwith a given composition in a given processedcondition (Krauss, 2004). Each type of microstructure and product is developed tocharacterize property ranges by specific processingroutes that control and exploit microstructuralchanges. Thus, processing technologies not onlydepend on microstructure but are also used to tailor final microstructures (Brandon, 1999). Existingdesign methods are generally ad hoc and lack adequate capabilities for finding effective process parameters such as rolling speed, temperature,amount of deformation, number of rolling passes,etc. Usually, the selection of the mill set-up is ademanding task that requires not only theunderstanding of the material behavior but also theknowledge about the mill’s limitations during rolling.On the other hand, it is known that there exist astrong correlation between the hot-deformation of steel and its microstructure and final properties(Honeycombe, 1995).In a model-based design, the model is at the centre of the development process, from analysis and design toimplementation and testing. There have been a fewattempts to optimise steel microstructures usingcontrol theory principles and expert systems(Venugopal, et al  ., 1997; Dixit and Dixit, 2000). Inthis work, a new model-based approach to optimisethe scheduling and control of the hot-rolling processis proposed. It is carried-out through two mainstages. The first stage consists of optimising therolling schedule according to the desired finalmicrostructure and properties. The optimisation usesthree basic components for defining and approachingthe problem: (1) the stock model; (2) the optimalitycriterion; and (3) the process constraints. GeneticAlgorithms (GA) are used to solve the optimisation problem.   ( a ) ( b ) ( c )Fig. 1. ( a ) Hot-rolling mill at The University of Sheffield; ( b ) Hot-rolling of steel; ( c ) Model-based integratedsystem for microstructure optimisation and control.In the second stage, the mill carries-out the rollingschedule using advanced control strategies such asModel-based Predictive Control (MPC) to guaranteeoptimal process performance when applying thedeformation profile. Thus, the knowledge integrationof both the stock and the rolling mill allows one toachieve optimal scheduling and control of the whole process on a repeatable basis for different steelmicrostructures and properties.This paper is organized as follows: Section 2 presents the general view of the proposed integratedsystem describing the optimisation methodology, the physically-based model of the C-Mn steel alloy and adescription of the self-tuning predictive controller implemented in the mill. Sections 3 present theresults from a real-time hot-rolling experiment.Finally, concluding remarks and further work relating to this overall study are presented in Section4.2.   THE INTEGRATED SYSTEM FOR OPTIMISATION AND CONTROLAn experimental laboratory-scale hot-rolling mill(Fig. 1), located in the Department of EngineeringMaterials at Sheffield University, is used as part of the research undertaken by IMMPETUS 1 , which fallsinto the investigation of steel and aluminiummicrostructure evolution in hot-rolling. Fig. 1( c)  depicts the block diagram of the model-basedintegrated system for microstructure optimisationand control of the hot-rolling process. Under thisframework, one first specifies the desiredmicrostructure and the initial conditions of rolling;then, by using thermomechanical and physically- based models, knowledge integration of both therolled stock and the mill is used (1) to perform a GA- based optimal search to find the rolling schedule for a particular microstructure and, (2) to control therolling process by using model-based predictive 1   Institute for Microstructural and Mechanical ProcessEngineering: The University of Sheffield. control. As a result, a “right-first-time” design and production of different alloy microstructures can beachieved. Also with this approach, production timesand costs are reduced compared to those obtained bycurrent methodologies which lack adequatecapabilities for finding effective rolling schedules.As shown in Fig.1( c) , the rolling schedule is set-up based on microstructure models of the rolled materialand not via empirical assumptions.It is worth noting that only quantitative parameters of the microstructure are considered in the optimisation.These parameters, for example  grain size or  dislocation density , are critically related to the alloymechanical properties. Indeed, the refinement of thegrain size provides one of the most importantstrengthening routes in steels; the finer the grain size,the higher the resulting yield stress and, as a result,increased strength is obtained (Brandon, 1999) .   Theunique feature of grain size strengthening is that it isthe only strengthening mechanism which alsoincreases toughness (Honeycombe, 1995). For theabove reasons, special attention to the refinement of the grain size as a main parameter of themicrostructure has been given in this work. 2.1.   The Optimisation of the Rolling Schedule The scheduling problem is treated here as anoptimisation problem which minimizes the error  between the desired microstructure and the finalmicrostructure in terms of its quantitativecharacteristics. The design approach requires three basic components: the stock model, the process or  physical constraints present both in the stock and inthe mill, and the optimality criterion represented byan “objective” function. The stock model describesthe microstructure evolution of the material duringhot-rolling. Constraints include the limitations of theforming process and the hot workability of the stock.Optimality criteria are related to achieving a particular final microstructure in terms of quantitative elements within “feasible” processingwindows for hot deformation. The optimisation    problem is solved by using Genetic Algorithmswhich are known to be a powerful evolutionarymethodology to solve a variety of optimisation problems, including those in which the objectivefunction is discontinuous, nondifferentiable,stochastic, or highly nonlinear (Kalyanmoy, 2001).   The Stock Model. Although this strategy attempts to be applied to a wide range of steel alloys, consider the case of the physically-based model of the C-Mnalloy, for which the microstructural state is definedmainly in terms of the grain size and the percentageof recrystallised material (Sellars, 1980). Thestatically recrystallised grain size (drex) can beexpressed using the following system equations: 150301670 0 108150 ... .. −− ×==  Z d d d  rexrex ε    ( )( ) ** ε ε ε ε  ≥< (1)where 150670 04 1082 ..* .  Z d  − ×= ε  , d  0 is the initialgrain size ε  is strain, ε  & is strain rate and  Z  is theZener-Hollomon parameter given by the followingexpression: ( ) T  Z  31.8/312000exp ε  & = (2)where T  is the deformation temperature. The volumefraction recrystallised (  χ  ) is calculated according tothe following system equations: ( ) [ ] ( ) [ ] 05.205 /639.0exp1 /639.0exp1 t t t t  −−=−−=  χ  χ    ( )( ) cc ε ε ε ε  ≥< (3)where 15050 04 10923 .. .  Z d  c − ×= ε  , t  is time inseconds and t  50 is time for 50% staticrecrystallisation.When recrystallisation is complete, further graingrowth may take place even in the relatively shorttime available between passes. The time dependenceof grain growth may be represented by the followingequation: ( ) T t d d  rex gg  31.8/567800exp1019.1 391010 −×+= (4)The above equations clearly show a high degree of sensitivity of microstructure to the operatingconditions during rolling. Therefore, the determiningvariables during the whole process are mainlycharacterized by the deformation temperature ( T  ),strain ( ε  ) and strain rate ( ε  & ), that is the deformation profile. However, for the vast majority of steels, theevolution of the grain size through the different phases defines the final microstructure andmechanical properties. In this case, ferrite grain sizeis a very important factor when defining mechanical properties such as Lower Yield Strength (LYS) andUltimate Tensile Strength (UTS). The transformationfrom austenite to ferrite can be expressed by thefollowing formula (Pickering, 1978): ( ) [ ] ( ) ε  γ α  45.01015.0exp1 5.0 −⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−−+⎟ ⎠ ⎞⎜⎝ ⎛ += − d cdt dT bad  (5)where d  α and d  γ are the grain size in the ferrite andaustenite phase respectively, dT/dt  is the cooling rateduring transformation, ε is the strain in austenite, a , b , and c are material constants for the C-Mn steelalloy. Although the above models were determinedon the basis of very few data, they have proven to berobust and form the basis of many such models thathave been since developed. The Optimality Criteria. To find the most appropriatedesign solution, the optimality criterion is formulatedas a series of “objective” functions to be minimizedin order to obtain the specified microstructuralfeatures (Venugopal, et al. , 1997). The objectivefunctions are lumped together into a single scalar optimality criterion (  J  M  ) in the following form:  F  N  F  F M   J  J  J  J  +++= L 21 (6)where, 2 )( d i F i  x x J  −= β     N i K ,2,1 = (7)where the superscript  F  refers to the requirements onthe desired final states of the microstructure.  β  i is aweight factor that scales various terms of   J  M  toexpress priorities in the overall criterion. In this casea quadratic cost-function is used when it is desirablethat a microstructure feature  x achieves a value  x d  atthe termination of the deformation process. In thiscase, the optimisation problem can be defined asfollows:Minimize 2 )( d rexrexM  d d  J  −= (8)Subject to( i ) maxmin ω ω ω  ≤≤ i   ni ,,2,1 K =  ( ii ) maxmin %%% r r r  i ≤≤   ni ,,2,1 K =  ( iii ) max  P  P  i ≤   ni ,,2,1 K =  where ω , % r  and  P  is the rolling speed, percentage of reduction, and rolling torque respectively; n is thetotal number of passes. These process parameters arevery important because they define the boundaries of the feasible solution space for the optimisation problem. It is clear that other constraints can beconsidered such as rolling power or exit thickness.The rolling speed, reduction, and rolling torque arecalculated by the following equations: r r   RV  π ω  260 = (9)   10023exp1% ×⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ −= ε  r  (10)and),,,%,( kF kH r T  f  P  ω  = (11)In equation (9) ε ε  / h RV  r r  ∆= & , and represents the peripheral roll speed;  R r  is the roll radius, and ∆ h thedecrement in thickness. In regards to the rollingtorque, it is represented by a nonlinear relationship of the operating conditions, the heat transfer coefficient( kH  ), and the friction coefficient ( kF  ) between theroll and the stock. In this work an artificial neuralnetwork is used to model such relationship (seeYang, et al  ., 2003). GA Optimal Search. GA begins its search with arandom set of solutions representing the deformation profile, i.e. values for  T  , ε  and ε  & . Each solution isevaluated using the stock model (equations 1-4); thenconstraint violations are obtained by calculating the process parameters (equations 9-11); finally theobjective function value (i.e. assigning a fitnessvalue to the solution) is calculated using equation(8). This process is carried-out until GA finds theminimum of the objective function. Fig. 2 shows aflow chart indicating the process of searching for theoptimal rolling schedule.As shown in Fig. 2, the initial assumption is that thedesired grain size ( d  desired  ) can be achieved only inone pass. However, for most experiments this is notthe case. For instance, for certain deformationconditions the smallest grain size achievable in one pass might be much larger than the desired grainsize. In that case, although GA has minimized  J  M  , thedeformation profile given will not lead to the desiredmicrostructure in one pass so that a new grain sizetarget ( d  target  ) must be provided by multiplying the previous target by a scaling factor ( α ). Thisautomatically leads to the search and optimisation for subsequent passes until the desired grain size isreached.   It is important to mention that in order to obtain thedesired ferrite grain size at the end of rolling, carefulanalysis of the phase transformation, cooling rate,chemical composition and any accumulated strains prior transformation must be carried-out. In thiswork, inverse modeling of equation (5) to obtain theaustenite grain size ( d  desired  in Fig.2) from the ferritegrain size was used. 2.2.   The Rolling Mill Control  The mill shown in Fig. 1 is a one-stand 50-tons hot-rolling mill with a maximum torque of 3467 Nmavailable in two main work rolls which have directcontact with the rolled product producing asequential reduction in thickness. Beginpass = 1Calculate processparameters per passRollingSchedulestopgen = 1Model evaluation& constraintsAssign fitness  J   M  min?ReproductionCrossoverMutationgen = gen +1NoYes GeneticAlgorithm Initializepopulation ε ε  & ,, T  Set initial conditions forthe rolling pass:Stock dimensionsInitial TemperatureInitial grain size d  target  = d  desired  Set Grain Sizetarget d  target  Grain Sizetarget achieved d  rex  ≈  d  target  ?Scale Grain Sizetarget d  target  = α · d  target  desired GrainSize achieved d  rex  ≈  d  desired  ?YesYesNoNopass = pass + 1  Fig. 2. GA-based schedule optimisationIn this case, Model-based Predictive Control (MPC)is used to guarantee optimal performance at highrolling speed and during changes in the rollingdirection. It has been shown that the use of MPC inthis application eliminates the problems caused byconventional PID controllers such as very activecontrol signal and shaft chattering when working athigh speeds. For a comprehensive description aboutthe successful real-time implementation of the MPCalgorithm in the rolling mill the reader is referred toGama and Mahfouf (2006), and Mahfouf andLinkens (1998).At the centre of the mill control is the process modelwhich is built upon a CARIMA (ControlledAutoregressive Integrated Moving Average) modelwhich solves offset problems inherently and becauseof its adaptive capabilities, the mill can be used in awide range of operating conditions.
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