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Journal of Process Control 24 (2014) 399–414

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Journal of Process Control

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

Realization of online optimizing control in an industrial semi-batchpolymerization

Tiago F. Finklera,b,∗, Michael Kawohl b, Uwe Piechottkab, Sebastian Engell a

a Process Dynamics and OperationsGroup, Technical University of Dortmund, Germanyb Evonik Industries AG, Germany

a r t i c l e i n f o

Article history:

Received 16 January 2013Received in revised form 26 May 2013Accepted 10 September 2013Available online 25 December 2013

Keywords:

Model based controlNonlinear controlOptimizing controlPolymerization controlSemi-batch processes

a b s t r a c t

In this work, the realization of an online optimizing control scheme for an industrial semi-batch polymer-ization reactor is discussed in detail. The goal of the work is the automatic minimization of the durationof the batch without violating the tight constraints for the product specification which translate intostringent temperature control requirements for a highly exothermic reaction. Crucial factors for a suc-cessful industrial implementation of the control scheme are the development and the validation of aprocess model that is suitable for process optimization purposes andthe estimation of unmeasured pro-cess states and the online compensation of model uncertainties. Two implementations are proposed, adirect online optimizing control scheme and a simplified scheme that combines a model-predictive tem-perature controller and a monomer feed controller that steers the cooling power to a predefined valuein a cascaded fashion. We show by simulation results with a validated process model that both schemesachieve the goals of tight temperature control and reduction of the batch time. The performance of theNMPC controller is superior, on the other hand the cascaded scheme could be directly implemented intothe DCS of the plant and is in daily operation while the online optimizing scheme requires an additionalcomputer and is currently in the test phase.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Linear MPC (Model Predictive Control) was first proposed andimplemented in the 1970s by Cutler and Richalet. Since then, ithas emerged as the standard solution for multivariate high perfor-mance control problems in the chemical industry, especially in thepetro-chemical sector. This technique was successfully applied inreal production units long before it was well understood from thetheoretical point of view. By now linear MPC is quite mature andit is routinely applied in industry [1]. However, as in the chemicalindustry most of the process dynamics are inherently nonlinear,linear models are often only capable to describe the behavior of the processes accurately near a fixed operating point. Especially

in time-varying processes as batch and semi-batch processes withtight operating windows, linear MPC may fail to deal provide suf-ficient control performance. In such cases, the use of nonlinearmodels may improve the operation of the process, because thesafety margins due to model inaccuracies can be reduced. There-fore, the use of NMPC (Nonlinear Model Predictive Control) hasbeen motivated by the growing demands on process economics

∗ Corresponding author at: EvonikIndustries AG, Germany.Tel.: +49 6181592087.E-mail addresses: [email protected], [email protected]

(T.F. Finkler).

under tighter product quality specifications and environmentalregulations. While the structure of MPC enables a straightforwardextension by employing nonlinear models in the optimization overa finite prediction horizon, in contrast to the linear case, the result-ing problems are usually non-convex and numerically demanding.A solid theoretical background has also been developed for NMPCoverthelastdecades [2,3] andseveralsuccessfulapplicationsinrealsystems have been reported [4,5]. But, despite of the effort that hasbeen made, there are still several obstacles to be overcome for theimplementation of NMPC controllers in real production units, inparticular the need for accurate models or online compensation of model inaccuracies and the robustness of the online optimizationunder significant uncertainties. Therefore the utilization of NMPC

in industry is still quite limited.During the last years, it has been realized that the potential

of MPC can go far beyond tracking references and rejecting dis-turbances. Instead, economic cost functions that are usually onlyconsidered in an upper steady state real time optimization layercan be optimized within the MPC formulation. This enables thepossibility of updating the operation point or the trajectories toexternal factors (like changes in the availability of utilities or inthe cost of the raw materials) online during the process operation,resulting in what is called DRTO (Dynamic Real Time Optimiza-tion)or online optimizingcontrol [6]. Dynamic online optimizationcan also be realized by using economics-motivated cost functions

0959-1524/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.

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400 T.F. Finkler et al. / Journal of Process Control 24 (2014) 399–414

Fig. 1. Scheme of the industrial polymerization process.

or by tracking necessary conditions of optimality [7]. Our NMPC

controller follows this approach by maximizing the monomer feedonline over a finite prediction horizon.

Inthiswork,whichisanextendedversionof [8], thechallengeof implementing an optimizing NMPC controller in a real productionunit is addressed using an industrial semi-batch polymerizationprocess as a case study. An online optimizing NMPC scheme thatsimultaneously optimizes the cooling power and the monomerdosage is proposed and thoroughly tested in simulations. Becauseof a simpler implementation and good robustness properties thathave been shown in [9,10], a simplified batch time minimizationscheme that combines a scalar MPC temperature controller and amonomer feed ratecontroller is also investigated. The effectivenessof both control schemes is illustrated by simulations and by resultstaken from the application at the real plant. Crucial factors for a

successful industrial implementation of the NMPC, as the develop-ment and validation of a process model that is suitable for onlineoptimization purposes, the estimation of unmeasured states andthe online compensation of uncertainties, are discussed in detail.

After this introductory section, the remainder of the paper isorganized as follows. In Section 2, the industrial polymerizationprocess is described, different control schemes that have been pro-posed in the literature are briefly discussed and the need for anadvanced control solution is exposed by means of historical pro-cess data. In Sections 3 and 4 issues regarding the development,calibration and validation of a semi-rigorous process model arediscussed. Aiming at using the model for online optimization of the batch operation, the issues of state estimation and online com-pensation of model errors are dealt with in Section 5 and a NMPC

scheme for the online optimization of the process is proposed andinvestigated in the Section 6. In Section 7, the improvement of thecurrentlyusedcontrolsolutionisrevisitedonthebasisofthedevel-oped process model and the alternative optimizing scheme thatcan be easily implemented in the DCS is introduced andreal resultsfrom the plant are shown. Conclusions are drawn in Section 8.

2. Process description andmotivation

The unit investigated in this work is an industrial semi-batchpolymerization reactor that produces a liquid polymer with a rela-tively low molecular weightin a catalyzed solution polymerization.The reactor system, which is schematically shown in Fig. 1, iscomposed of the reactor vessel where a solution polymerization

reaction takes place, a mechanical stirrer that keeps the reactant

mixture homogeneous and a cooling circuit that removes the heatof reaction. The reactor is equipped with four inlet ports throughwhich solvent, monomer, and two catalyst components can beindependently dosed into the reactor.

As it is usual in the chemical industry, this reactor is used toproduce different commercial polymer grades which, in this case,differ from each other by the polymer viscosity. Since this propertyis mainly determined by the temperature at which the polymer-ization takes place, a precise temperature control is required toensure that the end-product will have an acceptable quality. Cur-rently, the product quality is checked by off-line measurements of the viscosity of the product. The different products are obtained bydifferent recipes that can be generally described by the three fol-lowing steps. Firstly, during the pre-reaction step, the two catalystcompounds and a certain amount of solvent are inserted into thereactor. During the feeding or reaction step, monomer and solventare continuouslyfed into the reactor, the polymerization startsandthe reactor temperature is raised to the desired value. Finally, dur-ing the holding step, the monomer feed is stopped andthe reactantmixture is kept inside the reactor for a pre-established period suchthat a high monomer conversion (low residual monomer content)is achieved.

The standard operation strategy for this process consists in run-

ning the batch with constant monomer feed during the wholereaction period while a cascade of PID controllers, which is illus-trated in Fig.2, takes care of controlling thereactiontemperaturetothe desired value. Although this operation strategy has been suc-cessfully applied during the last years, several difficulties relatedto the control of the reaction temperature (especially at the begin-ning of the reaction period) have been reported by the operators.In Fig. 3, some real process trends are presented in order to illus-trate potential for improving the operation and control of theprocess.1

These process trends show that, for both products, a relativelylarge temperature peak (which has a negative impact on the prod-uct quality) is observed at the beginning of the reaction period.For Product B, the system can be operated with the maximum

monomer feed rate during the whole batch and the cooling con-straint is not active (the opening of the cooling valve is below50% during almost the whole reaction period). There is a relativelylarge temperature peak at the beginning of the reaction period,but the controller can then bring the reaction temperature backto the setpoint without intervention. For Product A, the coolingconstraint becomes active at the beginning of the feeding period.When the temperature peak occurs, the maximum cooling poweris reached and the operators are forced to reduce the monomerfeed to bring the system under control. After the intervention bythe operator, the system is operated close to the limit of the cool-ing capacity and during most of the reaction period the quality of the temperature control is not good. An alternative for eliminat-ing the temperature peak is to feed the monomer more slowly into

the reactor. This could improve the product quality but would alsoincrease the batch duration, which is not desired. The problem of finding the right compromise between the quality of the temper-ature control and the batch duration, i.e. to choose the velocityat which the reactants are fed into the reactor, is a well-knownproblem in the operation of semi-batch reactors. Several possiblecontrol schemes to handle this challenge can be found in the lit-erature. For example, [8,11,12] investigate the idea of maximizingthe monomer conversionby tracking optimal trajectories thatwerecomputed offline by manipulating the reactant feed and the jackettemperature. In semi-batch emulsion polymerizations, usually a

1

Note that, for confidentiality reasons, all the values are scaled.


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to check the robustness of the developed NMPC controller againstuncertainties in the model and uncertainties in the operation sce-narios and to predict the viscosity of the final product. The reducedmodel was developed for the online optimization of the processoperation. It canbe simulatedmuchfaster than thecompletemodeland contains correction terms that are updated online in order tocompensatemodeluncertaintiesusingtheavailablemeasurementsin an extended Kalman filter scheme.

3.1. Complete model

The polymerization investigated here is a live polymerizationreaction where there is no termination reaction andthe only chain-breaking reaction is the chain-transfer to monomer. The reactionmechanism is based on the assumption that two different types of active complexes, I and II, are involved in the reaction. The activecomplex of type I is formed when the catalyst complex is first com-bined with monomer. It is highly reactive but not stable and it isgradually converted into the active complex of type II that is sta-ble but less reactive. Both the active complexes of types I and IIreact with monomer to form live polymer molecules that grow bythe propagation reaction. By letting the variablesM , CAT A and CAT B,

P 1,0* and P 2,0* , P 1,i* and P 2,i* and P i denotethe monomer, thecatalystcompounds of types A and B, the active sites of types I and II, thelive polymer of type I and II molecules with chain length equal toi and the dead polymer molecules with chain length equal to i, thepolymerization mechanism is described by Eqs. (1)–(6):

CAT A +M + CAT Bki−→P ∗1,0 (1)

P ∗1,ikr −→P ∗2,i, i = 1, 2, . . .,∞ (2)

P ∗1,i +M k p1−→P ∗1,i+1, i = 1, 2, . . .,∞ (3)

P ∗

2,i +M

k p2

−→P ∗

2,i+1, i = 1, 2, . . .,∞ (4)

P ∗1,i +M k fm−→P i + P

∗

1,0, i = 1, 2, . . .,∞ (5)

P ∗2,i +M k fm−→P i + P

∗

2,0, i = 1, 2, . . . ,∞ (6)

Based on this reaction mechanism, the complete process modelwas developed by setting up appropriate material and energy bal-ances around the reactor and the cooling circuit. In the completemodel, the polymer molecules of different sizes are independentlybalanced.2 The model is composed of Eqs. (7)–(18):

dM M dt = F M − r i −

N

i=1

(r p1,i + r p2,i + r fm1,i + r fm2,i ) (7)

dM sdt = F s (8)

dM CAT Adt

= F CAT A − r i (9)

2 Theoretically thisleads to an infinitely largenumberof balance equations.How-ever, forpracticalpurposes, as theaverage degreeof polymerization of this productis small, the amount polymer molecules with more than one thousand monomer

units can be neglected.

dM CAT Bdt

= F CAT B − r i (10)

dM P ∗1,0

dt = r i − r p1,0 − r r,0 +

N i=1

r fm1,i (11)

dM P ∗2,0

dt = r r,0 − r p

2,0

+

N

i=1

r fm2,i

(12)

dM P ∗1,i

dt = r p1,i−1 − r p1,i − r r,i − r fm1,i , i = 1, 2, . . . ,N (13)

dM P ∗2,i

dt = r p2,i−1 − r p2,i + r r,i − r fm2,i , i = 1, 2, . . . ,N (14)

dM P d,idt

=

N i=1

r fm1,i +

N i=1

r fm2,i , i = 1, 2, . . . ,N (15)

dT Rdt = KA(T

J − T R)+

i=S,M F iCpi(T i − T R)+N

i=1(r p1,i + r p2,i )H RCpRM R

(16)

dT J out dt =

KA(T R − T J )+ F J Cp J (T J in − T J out )

Cp J M J (17)

dT J indt =

F cool()(T cool − T J out )+ F J (T J out − T J in )

M mix(18)

where M M and M S are the monomer holdup and solvent holdup,M CAT A , and M CAT B are the holdups of the two catalyst compoundsA and B, M

P ∗

1,0and M

P ∗

2,0are the holdups of the free active sites

of types I and II, M P ∗1,i

and M P ∗2,i

are the holdups of living polymer

molecules of types I and II with chain length equal to i, M P d,i is theholdups of dead polymer molecules with chain length equal to i, T R,T cool, T J in , T J out , T J , T M and T S are the inner reactor temperature, thebrinetemperatureat thecooling circuit inlet, thebrinetemperatureat the jacket inlet, the brine temperature at the jacket outlet, theaverage jacket temperature between jacket inlet and outlet, themonomer inlet temperature and the solvent inlet temperature, r iis the rate of the initiation reaction, r r ,i is the rate of the conversionof active species of type I into active species of type II for the livepolymer molecules with length equal to i, r p1,i and r p2,i are theratesof the propagation reaction for the live polymer molecules of typesI and II and length equal to i, r fm1,i and r fm2,i are the rates of the

chain transfer to monomer reaction for the live polymer moleculesof types I and II and length equal to i, N is the maximum degree of polymerization that is considered,K is the heat transfer coefficient, A is the heat transfer area, V is the volume of the reactive mixtureinside the reactor, F M , F S , F CAT A and F CAT B are the inlet flow rates of monomer, solvent and catalyst components A and B, respectively,F J is the constant flow rate of cooling brine that crosses the jacket,F cool is the flow rate of cooling brine that enters the cooling system,which depends on the position of the cooling valve , CpR is thethermal capacity of the reactant mixture, M R is the overall massof the reactant mixture, Cp J is the thermal capacity of the coolingbrine;M J is themass ofcooling brine insidethe jacket,M mix isatimeconstant that describes the dynamics of the jacket back-mixing,CpM , CpP , and CpW are the thermal capacities of monomer, polymer

and cooling brine, respectively, H R is the heat of reaction.


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T.F. Finkler et al. / Journal of Process Control 24 (2014) 399–414 403

3.2. Reduced model

Thereduced model hasbeen developedfor predicting thefuturebehavior of the process during the online optimization. As it willbe discussed in Section 6, the objectives of the optimization are themaximizationofthemonomerfeedandthebestpossibletrackingof the desired reaction temperature. The description of the completemolecular weight distribution is not relevant for these purposes.Therefore, instead of the complete reaction mechanism describedby Eqs. (2)–(6), only a lumped polymerization reaction is consid-ered and, instead of solving separate balance equations for thepolymer molecules of different sizes, only the bulk polymer massis balanced, what reduces the model size considerably. Nonethe-less, aiming at quality monitoring, it is desired that the operatorsreceive some information about the average molecular during theonline optimization. Thus, the number and the weight averages of the molecular weight distribution are predicted along the batchfrom the full model using the method of moments technique.3 Thereduced model is composed of Eqs. (19)–(33):

dM M dt = F M − r

∗ p (19)

dM sdt = F s (20)

dM CAT Adt

= F CAT A − r i (21)

dM CAT Bdt

= F CAT B − r i (22)

dM P dt = r ∗ p (23)

dM C ∗

dt = r i (24)

dT Rdt =

KA∗

(T J − T R)+

i=S,M F iCpi(T i − T R)+ r ∗ p H R

CpRM R(25)

dT J out dt =

KA∗(T R − T J )+ F J Cp J (T J in − T J out )

Cp J M J (26)

dT J indt =

F ∗cool

()(T cool − T J out )+ F J (T J out − T J in )

M mix(27)

d(0V )dt

= k0C CAT AC CAT BC M (28)

d(1V )dt

= k0C CAT AC CAT BC M + k pC M 0 + k fmC M (0 − 1) (29)

d(2V )dt

= k0C CAT AC CAT BC M + k pC M (21 − 0)+ k fmC M (0 − 2)

(30)

3 It is well known that the number-based average molecular weight (Mn) andthe mass-based average molecular weight (Mw) canbe directly computed from therelevant moments of the molecular weight distribution. The ODEs for computingthe variation of these moments along the batch can be easily obtained by combin-ing the definition of each moment with the kinetic expressions that describe the

polymerization mechanism. For brevity this derivation is not reproduced here.

d(0V )dt

= k fmC M 0 (31)

d(1V )dt

= k fmC M 1 (32)

d(2V )dt

= k fmC M 2 (33)

where C M is the concentration of monomer, C CAT A and C CAT B arethe concentrations of catalyst compounds of types A and B, M C*is the overall mass of active species inside the reactor, M P is theholdup of the lumped polymer mass, i and i are moments of order i of the molecular weight distribution of live and dead poly-mer chains. The variables r ∗ p , KA* and F

∗

cool, which represent the

corrected polymerization rate, the corrected heat transfer coeffi-cient and the corrected flow rate of cooling brine, are computedaccording to Eqs. (34)–(36):

r ∗ p =r pr p (34)

KA∗ =KA KA (35)

F ∗cool = (KV )F cool (36)

where r p , KA and KV are correction factors that are updatedonline so that the uncertainties in the model computations for thereaction rate r p, for the heat transfer coefficient KA and for the flowrate of cooling brine F cool are compensated.

4. Model calibration and validation

In order to solve the mass and energy balances of both models,several parameters, input variables and algebraic relations have tobe known. All the flow rates and temperatures of the inlet streamsare measured and the values of the constants F J ,M J , M mix, H R andthethermal capacities of thedifferent substanceswere provided bytheprocess engineering team of theindustrial unit. Butno informa-tion was available regarding the unknown vales KA, F cool as well asthe constants of all the reactions that compose the polymerizationmechanism.In this section, algebraicrelations that enablethe com-putation of these quantities as a function of thestates of the systemandoftheinputsthatareappliedtothesystemaredeterminedwiththe help of historical process data.

4.1. Computation of the heat transfer coefficient

The energy flow that is established between the reactor and the

jacket depends on the product between the heat transfer coeffi-cient K and the heat transfer area A. At the beginning of the batch,while the reactant mixture is being prepared, the product KA isexpected to increase continuously due to the addition of mate-rial that leads to an increase of the heat transfer area A. In thelater stages of the batch, on the other hand, the product KA isexpected to decrease due to the formationof polymer. In this work,an empirical correlation for the computation of KA as a functionof total mass in the reactor and the monomer mass that reflectsthis dependence is proposed. This empirical correlation is given by(37):

KA = C 1(M M + M S +M P )+ C 2M P . (37)


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Fig. 4. Estimation of theheat transfer coefficient forProduct A.

Fig. 5. Estimation of theheat transfer coefficient for Product B.

By combining (37) with (17) it is easy to show that, aftersomealgebraicmanipulation,the linearexpression (38) thatrelatesthe constants C 1 and C 2 with the process measurements can beobtained.4

Cp J M J dT J out dt − F J Cp J (T J in − T J out ) = (M M +M S +M P )(T R − T J )C 1

+M P (T R − T J )C 2 (38)

This equation canthen be used to estimate these constantsfromhistorical process data by linear regression. In Figs. 4 and 5, twohistograms with the results of independent estimations for C 1 andC 2 based on historical process data of around 130 batches are pre-sented for both products.5 The averages of these estimatesare then

taken as the nominal values of C 1 and C 2 for the proposed correla-tion.

In order to quantitatively check the predictions provided by theproposed correlation,one caninvertthe energybalance aroundthe

4 Note that the term (M M +M S +M P ) that multiplies C 1 corresponds to the over-all mass of content inside the reactor, which is continuously measured during theoperation. The polymer massM P that multiplies the C 2 is notmeasured online. ButvaluefromM P duringthebatchcanbe well approximatedby a straight line that goesfrom zero to the total mass of polymer at the end of the batch, which is measured.As it can be seen from the results that will be presented later in Fig. 10, the errorassociated to this approximation is very small.

5 As the viscosity of both products is different, the parameters C 1 and C 2 wereestimatedseparatelyfor eachof theproducts. Forsake ofbrevity,only theestimation

results forProduct B are shown here.

jacket (17) and obtain an explicit formula so that the product KAcan directly be computed from the process measurements along abatch. In Figs. 4 and 5 the result from (37) is compared with thevalue of KA computed along several batches as well. One can seethat, after the initial period where the inversion of (17) is known tobe inaccurate,6 the predictions of the proposed correlation matchthe process measurements quite well.

This correlation is then employed in both the complete andthe reduced model. When the complete model is used to playthe role of the real processes in simulation studies, the uncertain-ties on the heat transfer efficiency can be simulated by letting C 1and C 2 vary within the range shown in Figs. 4 and 5. When thereduced model is used for online optimization purposes, the cor-rection factor KA is updated to compensate such uncertainties,as it will be discussed later. As it can be seen from the resultsin Figs. 4 and 5, one can expect KA to vary within the range[0.75–1.25].

4.2. Computation of the flow rate of the cooling brine

In order to solve the energy balance at the jacket inlet (18), therelationship between the position of the cooling valve and the flowrate of fresh cooling brine that is fed at the jacket inlet must be

6 The computation of KA is expected to be quite inaccurate at the beginning of the batch, when the reactant mixture is being prepared and heated up from roomto reaction temperature. During this initial period, the reaction activity is low andthe difference between jacket temperature and reactor temperature is very small,

what makes theinversion of (17) error-prone.


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Fig. 6. Characteristic curve of thecooling valve.

known. Therefore, the characteristic curve of this valve was deter-mined experimentally with help of an external flowmeter.In Fig.6,theexperimentalmeasurements of the flow rateare plotted against

the valve position and this curve is approximated by a forth orderpolynomial, which is taken as the nominal characteristic curve of the cooling valve.

The nominal characteristic curve is used in both, the completeand the reduced model. When the complete model is used to sim-ulate the real process, the coefficients of the polynomial can bechanged in order to represent the effect of disturbances that affectthecooling supply, e.g. pressure oscillationsat theinletof thevalve.When the reduced model is used for the online optimization of theprocess operation, the correction factor KV is updated to compen-sate such disturbances. Such compensation is important,especiallybecause the reactor investigated in this work shares the coolingsupply with other production units. Therefore relatively large vari-ations of the correction factor may occur in order to provide an

accurate calculation of the available cooling power, depending onwhat is going on in theotherplants. Based on operationexperience,KV is expected to vary within the range [0.25–1.0].

4.3. Kinetic relations

The rates of the different reactions involved in the polymeriza-tion mechanism are given by Eqs. (39)–(44):

r i = kiC CATAC CATBC M V (39)

r r,i = kiC P ∗1,iV, i = 1, 2, . . .,N (40)

r p1,i

= k p1

expEa1RT RC

P ∗

1,i−1C M V, i = 1, 2, . . .,N (41)

r p2,i = k p2 expEa2RT R

C P ∗

2,i−1C M V, i = 1, 2, . . . ,N (42)

r fm1,i = k fmC P ∗1,iC M V, i = 1, 2, . . . ,N (43)

r fm2,i = k fmC P ∗2,iC M V, i = 1, 2, . . . ,N (44)

where ki, kr , k p1 , k p2 and k fm are unknown reaction constants thathave to be estimated and C P ∗

1,iand C P ∗

2,iare the concentrations of

polymer molecules of types I and II with length equal to i. As theinitiation step is very fast,it canbe assumed as being instantaneous,i.e. ki is set to infinity. The reaction constants for the propagation

steps and for the formation of active species of types I and II were

computed based on the results of a calorimetric analysis of the his-torical process data, which are presented in Fig. 7. This data wasgenerated fromthe explicit formula for computingthe reaction ratefrom theprocess measurementsalonga batch that is obtained frominverting the energy balance (16) of the reactor. The trajectoriesof the reaction rate along more than 200 batches are presented inFig.7. Theaccuracyofthecalorimetriccomputationsisalsocheckedin the histogram where the monomer conversion computed fromthecalorimetricanalysisiscomparedtothelaboratoryanalysisthatis performed at the end of each batch. As it can be seen from thishistogram, the cumulative error along one trajectory is in the orderof 5%.

The fitting of the propagation constants was divided into twosteps. In the first step k p2 and Ea2 were estimated using data fromthe end of the batches, where the reactor temperature is practi-cally constant and only reactant species of type II are expected tobe present. Note that in this region all the trajectories for the reac-tion rate are almost identical. Moreover, as data of two differentproducts whichare produced at different temperatures is available,k p2 and Ea2 can be independently and precisely estimated in thisfirst fitting step. The second step of the fitting procedure was basedon average trajectories for the reaction rate. The parameters kr , k p1 ,and Ea1 were adjusted so that thekinetic model fitsthe average tra-

jectories of both products as well as possible. Here it is importantto remark that the estimation of these constants is highly depend-ent on the data at the startup of the reaction, where the reactantspecies of type I are the dominant ones. In this region, the trajec-tories for the reaction rate are spread over a relatively large region.Therefore it is must be taken into account that the estimations of kr , k p1 , and Ea1 are significantly less accurate than the estimates of k p2 and Ea2.

The reaction constants that lead to the trajectories shown inFig. 7 are taken as the nominal reaction constants. When the com-plete model is used to represent the real process, these nominalvalues can be modified to test the robustness of the controller withrespect to modeling inaccuracies. The lumped reaction rate of thereduced model r p, which is given byEq. (45), depends only on k p2 ,

Ea2, T R, C M andthe overall concentration of activespecies insidethereactorC C ∗ .Whenthereducedmodelisusedintheonlineoptimiza-tion, the reaction rate computed from Eq. (45) is multiplied by thecorrectionfactorr p that compensates themodel errors. Note that,in the absence of a correction, i.e. for r p equal to one, the reducedmodel behaves as if only active species of type II were present. Thisis a convenient choice because, as explained before, this reactionkinetics can be modeled very well. The correction factorr p is thenused to compensate any uncertainties that cause deviations fromthis behavior, e.g. the presence of active species of type I. As it canbe seen from the results in Fig. 7, when the rate of polymerizationcomputed by the reduced model is corrected one may expect r ptovary upto a value of 5.0.

r p = k p2 exp Ea2RT RC C ∗C M V (45)

Finally, the reaction constant k fm that determines the speed of the chain transfer to monomer was estimated based on the labora-tory measurements of the polymer average molecular weights forboth products.

4.4. Model validation

The full model was intensively checked with respect to its abil-ity to predict the dynamic behavior of the process. By comparingsimulations of batch runs using the cascaded temperature controlstructure with the same controller tuning parameters as in the realplant with historical data of more than 200 batches, it was verified

that the model represents the dynamics of the real plant faithfully.


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Fig. 7. Calorimetric estimation of the reaction kinetics.

Fig. 8. Comparison of the model predictions (continuous blue line) with real process measurements (dotted blue line) for typical batch runs. (For interpretation of thereferences to color in this figure legend, thereader is referred to theweb version of thearticle.)

In order to illustrate the prediction capability of the model, theoperation scenarios of the real batches that were previously pre-sented in Fig. 3 are reproduced in the closed-loop simulationsshowed in Fig.8. In these simulations,the cascade of PIDcontrollersis tuned using exactly the same parameters as in the real process.Apart from the flow rate of cooling brine, which is computed fromthe closed-loop simulations, all the other model inputs, e.g. flowrates and temperatures of the inlet feeds, are taken directly fromthe process measurements. As it can be seen from the plots, themodel is able to reproduce the dynamics of the plant very well.

In Fig. 9, the molecular weight distribution of the final productis presented andthe average molecularweightdistribution is com-pared with thetarget value of theproduct specification. As it canbeseen from thefigure, themodelprediction matches thetarget value

almost exactly. The variation of the average molecular weights andthe variation of the polydispersity along the batch are also shownin Fig. 9.

5. State estimation and compensation ofmodel

uncertainties

Sincenotallthestatesofthereducedprocessmodelcanbemea-sured online, the unmeasured states have to be computed in orderto initialize the NMPC controller at every sampling period along abatch. In this work, this issue is addressed by an EKF – extendedKalman filter [25] – that uses the temperature measurements T R,T J out and T J in and the energy balances ((25)–(27)) to estimate r

∗ p , KA*

and F ∗

cool online during a batch.

Following the methodology described in [20], one can invertthe energy balances ((25)–(27)) to build the observability mapdescribed by ((46)–(48)).

r ∗ p =(dT R/dt )CpRM R − KA∗(T J − T R)−

i=S,M

F iCpi(T i − T R)

H R(46)

KA∗ =(dT J out /dt )Cp J M J − F J Cp J (T J in − T J out )

T R − T J (47)

F ∗cool =M mix(dT J in/dt )− F J (T J out − T J in )

T cool − T J out (48)

Eqs. (46)–(48) are always defined, except for T R /= T J ,T cool /= T J out . Moreover, apart from KA*, all the variables on the righthand side of (46)–(48) are either known or measured. Therefore,according to this observability map, KA* is observable if T R /= T J ,F ∗cool

is observable if T cool /= T J out and r ∗ p is observable if KA* is known.

By analyzing the historical data of the process, one verifies that,apart from some short periods at the very beginning of the batchwhen the difference between T R and T J is small, these observabilityconditions are always fulfilled. Therefore, it is possible to estimatethecurrentvalueofthevariablesr ∗ p ,KA* and F

∗

cool duringa batch and

use these estimates to compute the monomer holdup, the overallpolymer holdup and the moments of the molecular weight dis-tribution by integrating the corresponding Eqs. (19)–(33), what isrealized by setting the corresponding elements on the diagonal of

the covariance matrixQ to zero.


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Fig. 9. Molecular weight distribution and its evolution fora typical batch run.

Considering that the goal here is to use the reduced model topredict the future behavior of the process in the online optimiza-tion, besides of estimating the values of r ∗ p, KA* and F

∗

cool at a given

time instant, it is important to know how these values evolve overthe consider prediction horizon in the all possible operation sce-narios. In order to adapt the predictions of the nominal model to

the different uncertainties (e.g. reactivity, heat transfer efficiencyand cooling capacity) the computation of the correction factors ishere mixed within the EKF computations. This is realized by insert-ing((34)–(36)) into the energy balances ((25)–(27)) and extendingthe reduced model with the pseudo states as described by Eqs.(49)–(51):

dr pdt = 0 (49)

dKAdt = 0 (50)

dKV

dt = 0 (51)

In this representation,the outcome of theEKF provides thecom-putations for the correction factors directly. The estimations for r ∗ p ,KA* and F ∗

cool can then be obtained based on the computed correc-

tion factors with help from Eqs. (34)–(36).The EKFwas tuned withinthe simulation environment that uses

the complete model to represent the real process. A simple tuningstrategy was employed to determine the diagonal elements of thecovariance matrix Q . The elements corresponding to T R, T J in andT J out are set to one or two orders of magnitude smaller than thevariance of the temperature measurements7 and the elements cor-responding to the correction coefficients were adjusted so that thesimulated reaction rate,heat transfer coefficient and cooling power

are well tracked without excessive oscillations.8

The state vectorof the EKF is given in Eq. (52) and the diagonal elements of thecovariance matrix are presented in Eq. (53).9

ˆ x = [M M ,M S,M CAT 1,M CAT 2,M P , T R, T J out , T J in ,r p,KA,

KV,0,1,2, 0, 1, 2] (52)

7 Thismeans thatthe measurements areconsidered morereliablethanthe model.8 The oscillations in the EKF predictions are not desired because they are trans-

mitted tothe plant whenthe loop of the NMPC controller is closed.9

Note that theelements corresponding to thesimulated states are set to zero.

diag (Q ) = [0,0,0,0,0,2.5× 10−5,2.5× 10−2,2.5× 10−2,

10−4,10−4,10−6,0,0,0,0,0,0] (53)

The extended Kalman filter provides very good estimates of the system state over the full range of uncertainties exposed in

Figs. 4, 5 and 7. In order to illustrate the effectiveness of the EKF,the estimates for r ∗ p , KA* and F

∗

cool, as well as the predictions of

monomer consumption, polymer formationand average molecularweight are presented in Fig. 10. The robustness of the EKF can alsobe seen from therobustnessanalysisthatis discussedin Section 6.2.After several simulation tests, the EKF was integrated with the DCSof the real process so that online inferences of the non-measuredstates became available.

Note that the computation of the correction factors depends onthe simulated process states, e.g. M M and M P , and that the simula-tion error is integrated within their computations. In practice it isnot an issue because, as it is shown in Fig. 10, it is possible to tunethe EKF such that the simulation error is almost negligible. Fur-thermore, when compared to the usual representation that uses r p

and KA as pseudo-states [20] the approach of computing the cor-rection factors turned out to be advantageous because of its betterprediction qualities.

6. Online optimizing NMPC

In an NMPC scheme, the closed loop control inputs are com-puted based on the repeated solution of an open-loop optimalcontrol problem in which the future behavior of the process is pre-dicted based on a process model. At every sampling period, thecontroller is initialized based on the current information on thestate of the process and the discrete control movements that min-imize an objective function over a given prediction horizon are

determined. The first control movement from this optimal discretesequence is then applied to the plant and the controls are heldconstant until the next measurement becomes available. The con-troller is reinitialized using the newest process information andthe whole procedure is repeated. It is not the target of this paperto provide a deep description of the MPC principles. The inter-ested reader can find several good descriptions in the literature[21,22].

Thereduced process model is used to predict theprocess behav-ior in the NMPC computations. As it is not possible to directlymeasure all the relevant system states (only reactor and jackettemperature measurements are available), the controller initial-ization is done at every sampling period based on the informationprovided by the EKF. The correction factors in the model are also

updated every sampling period and they are held constant in the


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Fig. 10. EKF estimates fora simulation of a typical batch run. (a) Thedotted blue lines and thesolid redlines denotethe simulated trajectories and theEKF estimates for r p,KA and F cool. The solid green lines show the respective correction factors, i.e. r p , KA and KV . (b) The dotted blue lines and the solid red lines denote the simulated valuesand theEKF estimates forM M , M P and M w . (For interpretation of the references to color in this figure legend, thereader is referred to theweb version of thearticle.)

optimal control computations over the prediction horizon. Twodifferent implementations of the NMPC were tested, a sequen-

tial implementation and a simultaneous implementation [23,24].Both implementations use the solver SNOPT from the Tomlab®

optimization suite. In the sequential implementation, the model isembedded in the objective function and it is repeatedly integratedover thepredictionhorizon ineveryiteration of theoptimization byan explicit first order methodwith constant integration step. In thesimultaneous implementation, the dynamic optimization problemis formulated so that the system states and the control movementsat each sampling period along the prediction horizon are degreesof freedom for the optimizer, and, in order to guarantee that thesolution of the optimization respects the process model, the modelequations are incorporated to the optimization problem as addi-tional equality constraints, which are integrated by an implicit firstorder method with constant integration step. Both implementa-

tions are capable of solving the problem considerably faster thanreal time and, for the prediction horizons considered in this work,the results provided by both methods are very similar. For safetyreasons, the implementation of the controller at the real industrialprocess must consider that eventually theoptimization might haveto be terminated before convergence. In case of premature termi-nation, there is no guaranty that the suboptimal solution providedby the simultaneous implementation will make sense physically(because if the constraints are not fulfilled the model equationsmight not be satisfied). The sequential approach on the other handwill alwaysprovide a suboptimal solution that respects theprocessmodel. Therefore the sequential implementation was preferred forthe industrial application.

In the optimization setup that is proposed here, the monomer

dosage and thecooling powerhave to be optimizedsimultaneously.This is here realized by minimizing the objective function given by(54), which is composed of the weighted summation of the tem-perature tracking residual errors, a dominant term that maximizesthemonomer feed anda penalty term that penalizes control move-ments.

J =

NP i=1

wT i (T Ri − T ref Ri

)2+

NP i=1

wF M i(F M i − F

maxM i

)2

+

NC i=1

wi (i − i−1)2 (54)

The optimization is constrained by pre-defined bounds for thereaction temperature, by the valve opening limits and by the max-

imal monomer feed rate, as described by Eqs. (55)–(57).

T minR < T Ri < T maxR (55)

0 < i < 100% (56)

0 < F M i < F maxM (57)

Note that, in the absence of plant-model mismatch, the satis-faction of the hard constraint (55) suffices to keep the reactiontemperature within the specification bounds even in the absenceof the quadratic cost for the tracking of the reference temperaturein (54). In our implementation, the quadratic cost is kept becauseit brings the reaction temperature the center of the specificationrange, providing more margin for the controller to react to fast

disturbances.When the optimal control problem is solved, the control move-ments over the control horizon as well as the evaluation of theobjective function and constraints over the prediction horizon arediscretized considering a discretization period of one minute, withNC= 2 and NP= 5. These relatively short prediction horizons werechosen based upon simulations, picking the shortest horizons thatgave a good performance in all situations. As the computation of the solution of the open-loop optimal control problem takes onlyaroundfiveseconds,theNMPCcomputationandtheEKFestimationareruneverytenseconds.Thisshortsamplingperiodisveryhelpfulbecause it provides the controller with more accurate informationabout the state of the real process, what is important for updatingthe correction factors in order to compensate model uncertainties.

The NMPC cost parameters were tuned by extensive tests withinthesimulation environment that uses thecompletemodelto repre-sent the plant. After several simulation trials the weights wTRi , wF M iand wi were set to 400, 100 and 400, respectively.

10 This tuningprovides a good compromise between the main control objectives,i.e. temperature tracking and feed maximization, without excessiveoscillation of the position of the cooling valve. The nonlinear con-troller was checked with respect to robustness and performanceby simulating several scenarios with plant-model mismatch whichwere realized by varying key parameters of the complete model as

10 Note however that the weights depend directly on the units of the reactortemperature and themonomer feed, which arehere arbitrarily scaled due to confi-

dentiality.


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Fig. 13. Simulation of the NMPC for Product A (scenario with plant-model mismatch and cooling failure after half of the feeding period). (a) The solid blue lines are thesimulatedprocess measurements. Thedotted green line is thesetpointfor T R . Thered dottedlinesare thespecificationbounds (±1K). (b) The dottedbluelines and the solidred lines denote the simulated trajectories and the EKF estimates for r p, KA and F cool. The solid green lines show the respective correction factors, i.e. r p , KA and KV . (Forinterpretation of thereferences to color in this figure legend, the readeris referred to theweb version of the article.)

Fig. 14. Simulation of the NMPC for Product B (scenario with plant-model mismatch and cooling failure after half of the feeding period). (a) The solid blue lines are thesimulatedprocess measurements. Thedotted green line is thesetpointfor T R . Thered dottedlinesare thespecificationbounds (±1K). (b) The dottedbluelines and the solidred lines denote the simulated trajectories and the EKF estimates for r p, KA and F cool. The solid green lines show the respective correction factors, i.e. r p , KA and KV . (Forinterpretation of thereferences to color in this figure legend, the readeris referred to theweb version of the article.)

6.2. Robustness analysis

In order to illustrate therobustness of theNMPC controller, sim-ulations with considerable plant-model mismatch are presentedhere. The model errors are simulated by changing the key param-eters of the complete model so that the trajectories of the reaction

rate andthe heat transfer coefficient vary by±25%when comparedto nominal ones (corresponding to the nominal values of the keyparameters). Moreover, in order to simulate a strong cooling fail-ure, the cooling capacity is reduced by 50% during the second half of the feeding period. The results show that the NMPC controller isrobust over the whole uncertainty range.

The simulation of the worst case scenario where the reactivityis 25% bigger and the efficiency of the heat transfer is 25% smallerasin the nominal case are presented in Fig. 13 f ora batch where theProductAisproduced.Inthisscenario,thecoolingpowerconstraintbecomes active at the early stages of the feeding period and theNMPC adjusts the monomer feed so that the system is operatedon the limit of its cooling capacity. When the cooling failure takesplace, the NMPC reacts quickly by adjusting the monomer feed and

the disturbance is rejected and the temperature of the reactor is

kept around the target value during the whole batch. In Fig. 14,the same worst case scenario is simulated for a batch where theProduct B is produced. In this case, the NMPC starts driving thesystemalong themaximal feed bound.Whenthe failure eventtakesplace, the cooling constraintis activatedand thecontrollerkeeps ondriving the system along the maximal cooling bound by adjusting

the monomer dosage.

7. Alternative optimizing control structure

A disadvantage of the optimizing nonlinear model-predictivecontroller is that it cannot be run on the DCS by which the plant iscontrolled but requires an additional computer and software envi-ronment for its implementation. This is technically possible, butnot very much liked in industry. DCS are robust systems with errorchecking mechanisms, redundancy etc., while additional hard- andsoftware imply an additional effort for installation and in particularfor long-term maintenance. The two systems have to be synchro-nized and additional visualizations have to be built. Moreover,running a nonlinear optimization online inevitably raises concerns

about its stability and robustness. Therefore, and also to have a


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Fig. 15. Block diagram of thealternative control scheme.

“quick win” strategy to convince the plant management and plantoperators, an alternative scheme was investigated as well. Thisscheme is based on the simple idea that in order to minimize thebatch time, the cooling capacity should be fully used, with somesafetymarginto reactto fast disturbances. Sothe cooling valve posi-tion is controlled by the monomer feed in an outer loop while the

temperature controller acts as the inner loop. The scheme is illus-trated in Fig. 15. This combination results in an optimizing controlscheme that automatically balances the amount of heat producedby the reaction with the available cooling capacity of the system.The main motivation for using this alternative control structure isto take advantage of the fact that, independent of any external dis-turbancesthat may affectthesystem,e.g.fluctuationsinthecoolingwater or in the monomer temperatures, the position of the coolingvalvewillalwaysprovideaquantitativemeasureofhowmuchcool-ing power is still available. As a result, operation conditions whichare close to optimality are tracked by the additional controller andthe system is operated almost on the limit of its cooling capacity,significantly reducing the batch duration.

In [9,10] it has been shown for the well-known benchmarkexample of the Chylla–Haase polymerization reactor [26] thatevenwith simple PID controllers this scheme realizes most of the poten-tialof a full NMPCscheme and is even more robustfor model errorsthat are not compensated by model adaptation. The same schemewas demonstrated for an emulsion polymerization in [27].

This optimizing scheme is here applied to a real industrialprocess. Aiming at improving the control performance by takingadvantage of the information provided by the process model, thecascade of PID controllers for the control of the reactor tempera-ture is replaced by a cascade of predictive functional control (PFC)modelpredictive controllers. The combination of the cascade of PFCcontrollers with an additional PID controller that manipulates themonomer feed is illustrated in Fig. 15.

In this work, the alternative optimizing scheme is implementedin a discrete-time fashion. At the beginning of every sampling

period, the next position of the cooling valve is computed by thecascade of PFC controllers and the monomer feed rate is computedby the PID controller. The inputs are then applied to the processand held constant during the next sampling period. In the fol-lowing the cascade of PFC controllers is described first. Then theoptimizing scheme is investigated in simulations so that it can becompared with the full NMPC controller. Finally the effectivenessof the scheme is illustrated by results from the implementation inthe real plant.

7.1. Cascade of PFC controllers

The predictive functional control (PFC) method has been

pioneered by Richalet [19]. The PFC algorithm is an effective

model-predictive controller that makes use of a reference trajec-tory to specify thedesired behavior of theclosed loop system. In thePFC algorithm, as in the standard MPC approach, a nonlinear pro-cess model is used tosimulate theprocess over a prediction horizonand discrete control movements are computed every samplingperiod. The control action is computed by determining the input

change that makes the process model match the reference trajec-tory at given coincidence points over the prediction horizon. Forthe special case where only one coincidence point is used and afirst order reference trajectory is specified, the control movementcan be determined by solving an algebraic system of equations,i.e. one can compute the input movement that makes the modelmatch the coincidence point at the end of the prediction horizonby inverting the model equations and there is no need to solve anoptimizationproblem. This feature makes it easy to implement PFCcontrollers directly within most of the commercial digital controlsystems. Therefore PFC has become popular way of realizing MPCregulators in industry [19,28].

In this work, the internal model of the master PFC controller isgiven by Eq. (58), which represents a modified energy balance forthe reactor where the terms accounting forthe heat transfer coeffi-cient and for the reaction rate are replaced by the approximationsKAPFC and r PFC

P . The value of the reaction rate within the computa-

tions of themaster PFCcontroller is approximatedas being equal tothe value of the current monomer feed rate that enters the reactor,i.e. r PFC P is assumed to be equal to F M . Note that this correspondsto a pseudo-stationary assumption with respect to the amount of monomer inside the reactor and, as the additional PID controllerin fact uses the monomer feed to control the reactivity of the sys-tem, F M provides the PFC controller with valuable information withrespect to thevariations of thereactionrate. The heat transfer coef-ficient KAPFC is approximated by a constant value taken from theplateaus that can be seen in Figs. 4 and 5. At every sampling period,the master PFC controller tracks the desired reaction temperatureby applying a step change in the setpoint of the jacket temperature

while the slave PFC controller tracks the desired jacket tempera-ture by applying a step change in the position of the cooling valve.The step change in the jacket temperature is computed so that themodel prediction forthe reactor temperature, which is obtained byintegrating (58), matches a reference trajectory at the end of theprediction horizon. All the terms on the right hand side of Eq. (58)are kept constant during a sampling period.

dT Rdt =

KAPFC (T J − T R)+

i=S,M F iCpi(T i − T R)+ r PFC P H R

CpRM R(58)

The internal model of the slave PFC controller is described by Eq.(59), which represents a modified energy balance for the jacket.

In the convective term of the modified energy balance (59), the


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Fig. 16. Simulations of thealternative control structurefor ProductA. (a)Nominal scenario: thesolid bluelines arethe simulated process measurements. The dotted greenlines are the setpoints for T R and and the red dotted lines are the specification bounds (±1K). (b) Worst case scenario: The solid blue lines are the simulated processmeasurements. Thedotted greenlines arethe setpointsfor T R and andthe reddotted lines arethe specificationbounds (±1 K).(Forinterpretationof thereferencesto colorin this figure legend, thereader is referred to theweb version of thearticle.)

temperature of the cooling brine entering the jacket is computedfrom a static energy balance performed at the back mixing pointwhere the fresh coolant entering the system is mixed with the hot

coolant stream thatleaves the jacket. Within the PFC computations,the characteristic curve of the cooling valve is represented by thefunction (), which is a linear fit to the experimental data pre-sented in Fig. 6. At every sampling period, the slave PFC controllercomputesthestepchangeinthepositionofthecoolingvalvesothatthe model prediction for the jacket temperature, which is obtainedby integrating (59), matches a reference trajectory at the end theprediction horizon. All the terms on the right hand side of (59) arekept constant in the computations within a sampling period.

dT J out dt =

KAPFC (T R − T J )+ Cp J ()(T cool − T J out )

Cp J M J (59)

7.2. Simulation and robustness analysis

The alternative optimizing scheme was tuned based on simula-tions where the complete model represented the real process. Thedesired tracking speed of the PFC loops was regulated by choosinga proper timeconstant for the first-orderreferencetrajectories thatareusedinthecomputationofthecontrolmovements(asdescribedpreviously). The reference trajectories of the master PFC controllerand of the slave PFC controller have time constants in the order of a few minutes. The PID controller that manipulates the monomer

feed was tuned in such a way that the variations in the monomerflow rate enter as a slow disturbance into the cascaded PFC loops.The setpoint for the valve position set is a tuning parameter that

determines the desired cooling power. In order to provide somesafety margin for the cascaded PFC controllers to reject fast distur-bances, e.g. sharp decreases in the cooling supply, set was set to75%.

In Fig. 16, the alternative control scheme is simulated for theProduct A considering the same nominal and worst case scenariosthat were used for evaluating the NMPC controller in Section 6. Thefeeding period is started with maximal monomer feed and after-wards the monomer feed controller is activated. By comparing thesimulations shown in Fig. 16 with the simulation in Figs. 11and12it can be concluded that the quality of the simplified temperaturecontrolschemeisnotasgoodasthatoftheNMPCcontrollerbutstillsatisfactory. The batch time with the simplified scheme is longerwhich is due to the use of only 75% of the cooling water valve

opening, on the other hand, this provides a safety margin and in areal implementation the bound for the NMPC controller might alsobe reduced using a soft-constraint for this reason. In the nominalscenario, the system is driven with maximal monomer feed untilthe cooling power reaches the desired level, which is specified bythesetpointforthepositionofthecoolingvalve.Thenthemonomerfeed is adjusted during the remainder of the feeding period sothat the position of the cooling valve remains fine around its set-point. Note that, as r PFC P is approximated by the measurement F M , a

Fig. 17. Simulations of thealternative control structure for Product B. (a) Nominal scenario: the solid bluelines are thesimulated process measurements. The dotted greenlines are the setpoints for T R and and the red dotted lines are the specification bounds (±1K). (b) Worst case scenario: the solid blue lines are the simulated processmeasurements. Thedotted greenlines arethe setpointsfor T R and andthe reddotted lines arethe specificationbounds (±1 K).(Forinterpretationof thereferencesto color

in this figure legend, thereader is referred to theweb version of thearticle.)


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Fig. 18. Tests of the alternative optimizing scheme at the real plant rate. (a) Test for Product A: the solid purple line, the solid gray line and the dashed gray line denote thereactor temperature,the position of thecooling valve andthe monomer feed. (b) Test forProduct B: thesolid purple line, thesolid gray line and thedashed gray line denotethe reactor temperature,the position of thecooling valve and themonomer feed. (For interpretation of thereferences to color in this figure legend, thereader is referred tothe web version of thearticle.)

relatively large error is introduced in the energy balance (58) whenthe monomer feed drops abruptly to zero at the end of the feedingperiod.Asaresult,atemperaturepeakisobservedatthisregion.Forthe worst case scenario, the maximum cooling power is reached atthe beginning of the feeding period. When the cooling constraint isreached, the feed controller reduced the monomer feed and bringsthe system under control. A slight temperature peak is observedat the beginning of the feeding period, but the reaction temper-ature is always kept within the ±1K bounds. When the coolingfailure event takes place (at t = 0.5), the monomer feed is automat-ically adjusted so that the position of the cooling valve remainsat 75%. The differences in the reaction to the disturbance at the

middle of the batch in the worst-case scenario are surprisinglysmall.

In Fig. 17, the alternative control scheme is simulated for theProduct B. As before, the simulations consider the same nominaland worst case scenarios that were used for evaluating the NMPCcontroller in Section 6 and the feeding period is started with maxi-mal monomer feed. By comparingthe simulations from Fig.17 withthe ones from Figs. 13 and 14, similar conclusions can be drawn asfor Product A. The control performance is not as good as for theNMPC controller, but still satisfactory.

7.3. Implementation at the real plant

The alternative optimizing structure was implemented within

the DCS of the real process. In Fig. 18a, the performance of thealternative optimizing scheme is illustrated by trend lines of a realbatchwheretheProductAisproduced.Whencomparedtotheorig-inal cascade control structure, the alternative optimizing schemeimproves the quality of the temperature control and reduces theduration of the feeding period considerably, by around 25%. InFig. 18b, results of a real batch where the Product B is producedare shown. In this case, the cooling constraint does not becomeactive and the monomer is inserted into the reactor with maximalflow rate during the whole feeding period. As it can be seen fromthe plot, the temperature peak at the beginning of the batch wasconsiderably reduced (from 3 K to round 0.5 K).

Thesimplifiedcontrolschemeachievesasignificantreductioninthebatch time by reducing the feeding periodby 25%for Product A,

andfor bothproducts it significantly improves temperature control.

Manual reductions of the feed rate in order to bring the processback to the operating window are no longer required. The schemecould be easily implemented in the DCS of the plant and is in dailyoperation. A further advantage is that the monomer feed controllercan be replaced by manual operationwhile leaving the temperaturecontrolschemeunaltered.Itisplannedtocomparetheperformanceof the two schemes, the simplified one and the full NMPC schemealso at the real plant, but implementation issues so far delayed thetest of the NMPC controller.

8. Conclusion

In this work, the operation of an industrial polymerization reac-tor is optimized online using two different model-based controlschemes. A process model that describes the dynamics of the rele-vantsystemstatesandtheMWDofthepolymeralongthebatchwasdeveloped and validated using historical process data. After inten-sive investigations in a simulation environment, important issuesrelated to the practical implementation of the NMPC scheme, e.g.state estimation and real-time solution of the optimization prob-lem were addressed. In addition, a simplified control scheme withmodel-predictive temperature control and a master controller forthe monomer feed was developed that could be integrated into theDCS of the real process and is in daily operation at the real plant.This scheme improves the quality of the temperature control and

reduces the duration as well as the variation of the batches alreadysignificantly. For the case where the cooling constraint is not activeand the monomer can be inserted into the reactor with maximalflow rate during the whole feeding period, the results provided bythe PFC temperature controller are as good as the ones provided bythe generic NMPC scheme. For the case where the monomer feedrate and the cooling power have to be simultaneously optimized,the NMPC scheme provides a superior performance in simulations.This is due to the fact that the NMPC controller takes the monomerdosage and the cooling power simultaneously into account in theoptimal control computations while the alternative scheme opti-mizesthe cooling power in an inner loop and the monomer dosagein an outer loop. While better results can be expected from the useofthefullNMPCschemeattherealplant,itisanopenissuewhether

the gains over the simplified scheme will justify the additional


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effort for the maintenance of a dedicated solution implementedon special soft- and hardware.

The main engineering effort for the development of the solu-tions presented here went into the setup, parameter estimationand validation of the model that is used in the controller and in thedevelopment of a robust and reliable state and parameter estima-tor. What made the situation favorable for this endeavor was thefact that thestructure of a first-principles model of the polymeriza-tion was known and that from large amounts of historical data, themissing parameters could be determined and the reliability of themodel could be judged. This model also predicts the influence of the operating parameters on the product quality sufficiently wellwhich is a prerequisite for an optimization of recipe parameters.

It can be concluded that online optimizing NMPC provides aversatile tool to simultaneously achieve very good control andperformance of polymerization processes. The available methodsand optimization algorithms are sufficient for the implementa-tion at real processes of medium complexity as the one consideredhere. However, the solutions must be engineered carefully whichinvolves in particular a significant effort for process modeling andparameter tuning and simulation studies. Simpler schemes whichapproximately give the same performance are attractive becauseof easier implementation, easier understanding by the operators,

and easier switching between manual and automatic operation.The cascaded scheme proposed here has the structural advantageto rely only on measured information and not on a plant model,reducing the dependency on model quality and the computationaleffort at the price of a reduced performance. Both schemes handlethe trade-off between accurate temperature control and maximiz-ing the feed flow rate automatically and also improve the safetyof process operation, as the efficiency of the cooling system (theavailable cooling power) is automatically considered during oper-ation whereas in a manual operation of the feed the operators haveto observe the temperature control loop and to reduce the feed if they observe that the control valve saturates and the temperaturecannot be kept at the set-point any more. Of course there may beother situations where thebenefitof model-based control schemes

is smaller. The engineering of simple near-optimal schemes is anopen field for research, as well as the systematic robustificationof model-based schemes for which we propose the approach of scenario-based multi-stage optimization [29,30].

References

[1] S.J. Qin, T.A. Badgwell, An Overview of NMPC Applications, Birkhäuser, Basel,2000.

[2] D.Q. Mayne,J. Rawlings, C. Rao, P. Scokaert,Constrainedmodelpredictive con-trol: stability and optimality, Automatica 36 (6) (2000) 798–814.

[3] H. Chen, F . Allgöwer, A quasi- infinite horizon nonlinear model predic-tive control scheme with guaranteed stability, Automatica 34 (2000)1205–1217.

[4] Z.K.Nagy, B. Mahn,R. Franke, F. Allgöwer,Nonlinear model predictivecontrolof batchprocesses: anindustrialcase study,in: 16thIFAC World Congress,Prague,

July 4–8, 2005.[5] A. Küpper, S. Engell, Optimizing control of theHashimoto SMB process: exper-imental application, in: Preprints of the 8th International IFAC Symposium onDynamics andControl of Process Systems, 2007.

[6] S. Engell, Feedback control foroptimal process operation, J. Process Control 17(2007) 203–219.

[7] D. Bonvin, B. Srinivasan, Optimal operationof batch processes via thetrackingof active constraints, ISA Trans. 42 (2003) 123–134.

[8] T.F. Finkler, M. Kawohl, U. Piechottka, S. Engell, Realization of optimizingcontrol in an industrial polymerization reactor, in: Preprints of the 8th IFACSymposiumon Advanced Control of Chemical Processes,Singapore, July10–13,2012.

[9] T.F. Finkler, M.B. Dogru, S. Engell, Optimizing control of semi-batch polymer-ization reactors, in: Preprints of the 2nd Conference on Advances on Controland Optimization of Dynamics Systems, Bangalore, India, February 16–18,

2012.[10] T.F. Finkler, S. Lucia, M.B. Dogru, S. Engell, A simple scheme for the time-

optimal operation of semi-batch polymerizations, Ind. Eng. Chem. Res. 52(2013) 5906–5920.

[11] J.S. Chang, W.Y. Hsieih, Optimization and control of semibatch reactors, Ind.Eng. Chem. Res. 34 (1995) 545–556.

[12] B. Srinivasan, D. Bonvin, E. Visser, S. Palanki, Dynamic optimization of batchprocesses:Part 2, Comput. Chem. Eng. 27 (1) (2003) 27–44.

[13] M. Vicente, S.B. Amor, L.M. Gugliotta, J.R. Leiza, J.M. Asua, Control of molec-ular weight distribution in emulsion polymerization using on-line reactioncalorimetry, Ind. Eng. Chem. Res. 40 (2001) 218–227.

[14] C. Kravaris, R.A. Wright, J.F. Carrier, Nonlinearcontrollers for trajectorytracingin batch processes,Comput. Chem. Eng. 13 (1989) 73–82.

[15] C. Gentric, F. Pla, M.A. Latifi, J.P. Corriou, Optimization and nonlinear con-trol of a batch emulsion polymerization reactor , Chem. Eng. J. 75 (1999)31–46.

[16] R. Gesthuisen, S. Krämer, S. Engell, Hierarchical control scheme for time-optimal operationof semibatchpolymerizations, Ind.Eng. Chem.Res. 43 (2004)7410–7427.

[17] W. Mauntz, A contribution to observation and time-optimal control of emul-sion co-polymerization reactions, Dortmund Technical University,2010 (Ph.D.Thesis).

[18] V. Hagenmeyer, M. Nohr, Flatness-based two-degree-of-freedom control of industrial semi-batch reactors using a newobservationmodelfor an extendedKalman filter approach,Int. J. Control 81 (2008) 428–438.

[19] J. Richalet, D. O’Donovan, K.E. Åström, Predictive Functional Control: Principlesand Industrial Applications (Advances in Industrial Control), Springer, London,2009.

[20] S. Krämer, Heat balance calorimetry and multirate state estimation applied tosemi-batch emulsion copolymerization to achieve optimal coltrol, DortmundTechnical University, 2006 (Ph.D. Thesis).

[21] L. Magni, D.M. Raimondo, F. Allgöwer, Nonlinear Model Predictive Control:Towards New Challenging Applications, Springer, Berlin, 2009.

[22] P.S.Agachi, Z.K.Nagy, M.V.Cristea,A. Imre-Lucaci, ModelBased Control, Wiley-VCH, Weinheim (Germany), 2006.

[23] M. Diehl,S. Findeisen, S.Schwarzkopf, I.Uslu,F. Allgöwer,H.G.Bock,E.D.Gilles, J.P. Schlöder, An efficient algorithm for nonlinear predictive control of large-

scale systems, Part I: Description of the method, Automatisierungstechnik 50(2002) 557–567.[24] C. Kirches, L. Wirsching,H.G. Bock,J.P. Schlöder,Efficientdirect multiple shoot-

ing fornonlinear model predictive control on long horizons, J. Process Control22 (2012) 1832–1843.

[25] R.E.Kalman, R. Bucy, Newresults in linearfilteringand prediction, J. Basic Eng.(ASME) 83D (1961) 98–108.

[26] R. Chylla, D. Haase, Temperature control ofsemi-batchpolymerizationreactors,Comput. Chem. Eng. 17 (1993) 257–264.

[27] K. Pelz, H. Brandt, T.F. Finkler, S. Engell, Scheme for time-optimal operation of semi-batchemulsionpolymerizationreactors,in: Preprintsof theInternationalSymposium on Advanced on Control of Chemical Processes, Singapore, July10–13, 2012.

[28] H. Bouhenchir, M. Cabassud, M.V. Le Lann, Predictive functional control forthetemperature control of a chemical batch reactor, Comput. Chem. Eng. 30 (6/7)(2006) 1141–1154.

[29] S. Lucia, T.F. Finkler, X. Basak, S. Engell, A new robust NMPC scheme and itsapplication toa semi-batch reactor example,in: Preprintsof the8th IFAC Sym-posium on Advanced Control of Chemical Processes, Singapore, July 10–13,

2012.[30] S. Lucia, T.F. Finkler, S. Engell, Multi-stage nonlinear model predictive control

applied to a semi-batch polymerization reactor under uncertainty, J. ProcessControl 23 (9) (2013) 1306–1319.
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om/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0145http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref0140http://refhub.elsevier.com/S0959-1524(13)00197-2/sbref014

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