Dynamic optimization of a copolymerization reactor using tabu search

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  • Dynamic optimization of a copolymerization reactor using tabu search

    P. Anand a, M. Bhagvanth Rao b, Ch. Venkateswarlu c,n

    a Chemical Engineering Sciences Division Indian Institute of Chemical Technology Hyderabad 500 007, Indiab Anurag group of Institutions Hyderabad 501301, Indiac Chemical Engineering Department Padmasri Dr BV Raju Institute of Technology Narsapur 502313, India

    a r t i c l e i n f o

    Article history:Received 18 July 2013Received in revised form12 July 2014Accepted 18 July 2014This paper was recommended forpublication by Dr. A.B. Rad

    Keywords:Dynamic optimizationPolymerizationSimulationTabu searchIterative dynamic programming

    a b s t r a c t

    A novel multistage dynamic optimization strategy based on meta-heuristic tabu search (TS) is proposedand evaluated through sequential and simultaneous implementation procedures by applying it to asemi-batch styreneacrylonitrile (SAN) copolymerization reactor. The adaptive memory and responsiveexploration features of TS are exploited to design the dynamic optimization strategy and compute theoptimal control policies for temperature and monomer addition rate so as to achieve the desired productquality parameters expressed in terms of single and multiple objectives. The dynamic optimizationresults of TS sequential and TS simultaneous implementation strategies are analyzed and compared withthose of a conventional optimization technique based on iterative dynamic programming (IDP). Thesimulation results demonstrate the usefulness of TS for optimal control of transient dynamic systems.

    & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

    1. Introduction

    Batch and semi-batch reactors are well suited for the production ofpolymers of varying grades whose quality is assessed in terms ofstrength, stiffness and processability. The properties of the polymersproduced in these reactors are closely related to the operatingconditions specified for the reactors. Ensuring product quality is amajor issue in a polymerization process since the molecular ormorphological properties of a polymer product strongly affects itsphysical, chemical, thermal, rheological and mechanical properties aswell as polymer applications [1]. In free radical copolymerization,different reactivities of comonomers can cause composition driftsunless a more reactive monomer is added to the reactor continuouslyto maintain a constant mole ratio. It is well known that two styreneacrylonitrile (SAN) copolymers differing more than 4% in averageacrylonitrile content are incompatible and result in products with poorphysical and mechanical properties [2]. Significant variations in batchto batch copolymerization can also result in off-specification products.The copolymer composition influences the end use properties of thefinal product with respect to its flexibility, strength and glass transitiontemperature, where as the molecular weight (MW) and molecularweight distribution (MWD) affects the important end use propertiessuch as viscosity, elasticity, strength, toughness and solvent resistance.

    The determination of the open-loop time varying control policiesthat maximizes or minimizes a given performance index is referred to

    as optimal control/dynamic optimization. These optimal controlpolicies that ensure the satisfaction of the product property require-ments and the operational constraints can be calculated off-line, andare implemented on-line such that the system is operated in accor-dance with these control policies. There has been tremendous intereston the optimal control/dynamic optimization of polymerization reac-tors. Most of the studies on optimal control of polymerization reactorsare based on classical methods of solution. When the process model iscomplex and its dimension is large, it becomes tedious to compute theoptimal control policies by conventional optimal control techniques.For problems, where the Hamiltonian is a linear function in control,the optimal control problem becomes singular and the classicalanalytical methods are unable to provide a complete solution [3].Pontryagin's maximum principle is a classical control technique thathas been widely used to solve the optimal control problems ofpolymerization reactors [46]. However, this method requires goodinitial guess for the control inputs and the rate of convergence of thesolution is very sensitive to this guess. In case of complex systems, thesearch space quite often becomes very narrow and choosing the initialguess for the search region requires tedious effort. Chang et al. [7] useda two step method to compute the reactor temperature and timeprofiles to obtain a polymer with a prescribed molecular weightdistribution in a free radical polymerization reactor. The parametersearch of this method requires complicated numerical calculations tosolve the nonlinear algebraic equations. Salhi et al. [8] employedcontrol vector parameterization method to find the optimal jackettemperature profile to maximize the productivity in a batch emulsioncopolymerization reactor of styrene and -methyl styrene. The control

    Contents lists available at ScienceDirect

    journal homepage: www.elsevier.com/locate/isatrans

    ISA Transactions

    http://dx.doi.org/10.1016/j.isatra.2014.07.0140019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

    n Corresponding author. Tel.: 91 8458 2222000; fax: 91 8458 222002.E-mail address: chvenkat.iict@gmail.com (Ch. Venkateswarlu).

    Please cite this article as: Anand P, et al. Dynamic optimization of a copolymerization reactor using tabu search. ISA Transactions (2014),http://dx.doi.org/10.1016/j.isatra.2014.07.014i

    ISA Transactions ()

    www.sciencedirect.com/science/journal/00190578www.elsevier.com/locate/isatranshttp://dx.doi.org/10.1016/j.isatra.2014.07.014http://dx.doi.org/10.1016/j.isatra.2014.07.014http://dx.doi.org/10.1016/j.isatra.2014.07.014mailto:chvenkat.iict@gmail.comhttp://dx.doi.org/10.1016/j.isatra.2014.07.014http://dx.doi.org/10.1016/j.isatra.2014.07.014http://dx.doi.org/10.1016/j.isatra.2014.07.014http://dx.doi.org/10.1016/j.isatra.2014.07.014

  • vector parameterization technique requires a set of trial functions withweights to pre-specify the functional form of the control and a deepphysical insight into the process behavior. Among the derivative freemethods, iterative dynamic programming (IDP) is a popular methodthat has been applied to solve different optimal control problems [912]. Vicente et al. [13] employed iterative dynamic programming tocompute the time optimal monomer and chain transfer agent feedprofiles to attain the desired copolymer composition and molecularweight distribution in a semi-batch methylmethacrylate reactor.Though this method is reliable in finding the global optimum andcan be applied to problems where the objective function is non-differentiable, it has low efficiency and computationally time consum-ing. Of recent is the class of stochastic optimization methods derivedfrom the principles of natural phenomena that are becoming increas-ingly popular to solve complex optimization problems due to theirflexibility and ease of operation. Various stochastic search andoptimization algorithms such as differential evolution (DE) [1416],genetic algorithm(GA) [1720], simulated annealing(SA) [21,22] andant colony optimization (CO) [23] have been successfully employed toachieve the global optimum of complex engineering problems.

    Tabu search (TS) is a meta-heuristic approach and it is one of themost efficient optimization techniques that incorporates adaptivememory and responsive exploration to overcome the limitations oflocal optimality. The use of adaptive memory enables TS to learn andcreates a more flexible and effective search strategy than memoryless methods such as SA and GA. TS differs from other stochasticoptimization techniques by maintaining lists of previous solutions byusing a memory that helps to guide the search process. It exploits thememory to generate a sequence of progressively improving solutionsthrough repetitive modification of the current solution. TS uses aneighborhood search approach to explore the search space thatfacilitates escaping the solutions from local optima. The memory inTS allows to drive the method forward to discover regions that harborone or more solutions better than the current best. A set ofcoordinated strategies such as intensification and diversificationemployed in TS allow to explore the search space more thoroughly,thus helping to avoid becoming stuck in local optima. TS originallydeveloped by Glover and Laguna [24] has now become an establishedsearch procedure and has been successfully applied to solve a widespectrum of optimization problems [2527]. The theoretical aspects ofTS has been investigated by various researches [28,29]. TS has beenrecently applied to solve design and synthesis problems in chemicalengineering [30,31]. However, the potentiality of TS has not beenexplored to solve dynamic optimization problems in chemical/poly-merization reactors.

    This study explores the use of tabu search (TS) as a tool fordynamic optimization of a polymerization reactor. A multistagedynamic optimization methodology based on TS is presented andevaluated by applying it to a semi-batch reactor producing styreneacrylonitrile (SAN) copolymer. In SAN copolymerization, styrene andacrylonitrile are the monomers, and xylene and azobisisobutyroni-trile (AIBN) are the solvent and initiator, respectively. Due to thedifferent reactivities of the comonomers, composition drift occursunless a more reactive monomer is added to the reactor in order tomaintain a constant mole ratio. It has been reported that the SANcopolymers differing by more than 4% in the average acrylonitrilecontent are incompatible resulting in poor physical and mechanicalproperties. The control objective is to maintain the copolymercomposition and the molecular weight as constants during thecourse of polymerization by manipulating the reactor variables suchas the monomer addition rate and reactor temperature. The SANcopolymerization is commercially important and optimal control isnecessary to maintain the copolymer composition and molecularweight distribution at their target values. Thus this system forms anideal test bed for design and implementation of the proposedmultivariable open loop optimal control strategy. In SAN polymer-ization, the copolymer composition, MW, MWD and PD are theproduct quality parameters. The objective is to find the optimalcontrol policies for monomer addition rate and reactor temperatureto produce a polymer with the desired copolymer composition andmolecular weight distribution. When a polymer quality parameter iscontrolled by one manipulated variable, the uncontrolled productquality parameter may deviate from its desired specification and thisrequires the need for simultaneous optimization of more than oneobjective. In this work, TS is designed and implemented to deter-mine the optimal control policies that satisfy the individual andmultiple objectives of SAN copolymerization reactor. Multistagedynamic optimization by TS is carried out by using sequential andsimultaneous implementation strategies. The results of TS strategiesare compared with those of another widely used discrete multistagedynamic optimization strategy, namely, iterative dynamic program-ming (IDP). The multistage dynamic optimization strategies basedon TS and IDP to compute the optimal control policies of our workare different than those reported in recent literature [32,33]. In thework of Pahija [32] batch cycle time has been discretized into stagesand optimal control policies are determined by using dynamicoptimization methods based on optimization package solvers. Pisti-kopolous [33] presented a parametric programming approach fordynamic optimization where present states and future decisionvariables are considered as parameters and the present decisions

    Nomenclature

    If initiator concentration in feedkd initiator decomposition rate constantkfij chain transfer rate constant of species i, jkpij propagation rate constant species i, jktcij combination termination rate constant of species i, jktdij disproportionation termination rate constant of

    species i, jMif monomer concentration in reaction in feed of species iP total growing polymer concentration of type-1 [mol/l]Pi moment of total number MWD of radicals of type-1Q total growing polymer concentration of typeQi moment of total number MWD of radicals of species irij monomer reactivity ratioui manipulated variable of species iui manipulated variable of species iV reactor volume

    wi molecular weight of monomer of speciesL length of time stepN number of grid pointsP number of time stagesq pass numberr region size used for allowable controlR number of allowable values for control used at eachtf residence time or batch timexi state variablex state vector region contraction factor region restoration factor molar ratio of monomerf monomer mol ration in feed streams desired value of molar ratio of monomer in reaction

    mixturet cross termination factor

    P. Anand et al. / ISA Transactions () 2

    Please cite this article as: Anand P, et al. Dynamic optimization of a copolymerization reactor using tabu search. ISA Transactions (2014),http://dx.doi.org/10.1016/j.isatra.2014.07.014i

    http://dx.doi.org/10.1016/j.isatra.2014.07.014http://dx.doi.org/10.1016/j.isatra.2014.07.014http://dx.doi.org/10.1016/j.isatra.2014.07.014

  • are used to represent the optimization variables. Before implement-ing the TS for dynamic optimization of copolymerization reactor, itsefficacy is tested by applying it to five test functions of differentcomplexities. In all these cases, the method has achieved globalminimum in respective objective functions while determining thefunction variables exactly. The implementation results of TS for thetest problems are given in Appendix B.

    2. Multistage dynamic optimization

    Dynamic optimization is increasingly used in batch and semi-batch process operations, in which the process variables undergosignificant changes. The major objective in these operations is todetermine the time varying open-loop control policies that max-imize or minimize the objective function specifying the systemperformance as a function of process variables and their changes.Optimizing the objective function means, for example, eitherachieving a desired product quality or maximizing the productyield of a batch process. The general open-loop optimal controlproblem with fixed terminal time, considering a lumped para-meter batch/semi-batch process can be stated as follows.

    Find a control vector u(t) over tf [to, tf] to maximize (minimize)a performance index J(x,u):

    J x;u Z tft0

    xt;ut; t dt 1

    Subject to

    _xt f xt;ut; t; xt0 x0 2

    h xt;ut 0 3

    g xt;ut r0 4

    xLrxtrxU 5

    uLrutruU 6

    where J is the performance index, x is the vector of state variables, u isthe vector of control variables. Eq. (2) is the system of ordinarydifferential equations with their initial conditions, Eqs. (3) and (4) arethe equali...

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