Temperature Control of Ethylene to Butene-1 Dimerization Reactor

Emad Ali* and Khalid Al-humaizi

Chemical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

Dimerization reactions of ethylene to butene-1 which take place in a continuously stirred tankreactor with recycle suffer from serious temperature runaway. Therefore, it is essential to controlthe dimerization reactor temperature to secure safe plant operation and to deliver good qualityproduct. It is also desirable to maintain optimal operation of high ethylene conversion and desiredbutene-1 yield in the face of plant upsets. The objective of this paper is, thus, to study thesimulated implementation of standard proportional-integral (PI) control and linear modelpredictive control (MPC) for temperature stabilization of such a reactor to maintain favorableethylene conversion and butene-1 yield conditions. The mathematical model used in thesimulations is developed by the author’s research group (Trabzoni, 1998). The simulation resultsrevealed the possibility of stabilizing such a reactor by both control algorithms. However, carefultuning of the parameters of these controllers is necessary, without which very aggressive oreven unstable closed-loop performance will be obtained.

Introduction

The catalytic dimerization of ethylene is consideredto be one of the most promising methods for producingbutene-1, the first member of the even-numbered linear1-alkenes which have diversified applications. Thisprocess uses a homogeneous titanium-based catalystwhich demonstrates high dimerization activity coupledwith excellent selectivity to butene-1 at moderate pres-sure (20-30 psia) and temperature (50-60 °C). Thedimerization reaction is regarded as a degenerate eth-ylene polymerization reaction, and therefore the forma-tion of heavier oligomers is expected. An industrialethylene dimerization reactor operates in liquid phaseat bubble point conditions. Fresh ethylene and homo-geneous catalyst are fed continuously to the reactorwhere the exothermic reaction is removed by means ofan external loop equipped with a cooler.

The ethylene dimerization process, which is stronglynonlinear, is very sensitive to external disturbances. Themain reaction is highly exothermic, and its rate in-creases rapidly with temperature, incurring thermalrunaway risks. The side reactions produce heavieroligmers such as hexene and octene that might causecontinuous fouling of the overall system of the reactor,recycle loop, and heat exchangers. Industrial experienceshows that the ethylene dimerization reactor demandsextremely regular monitoring of the process variables.A mathematical model for this unit has been developedby Trabzoni (1998) to understand the dynamic behaviorof this reactor. The steady-state analysis of the modelshows that the system can exhibit unique steady stateand multiplicity in the form of an S shape (hysteresis).Generally, exothermic polymerization reactions withback mixing are well-known for reactor temperatureinstability (Dadebo et al., 1997; Luyben, 1998). In fact,the transient behavior of the dimerization reactortemperature exhibits phases of variation such as per-sistent oscillation or runaway (Braunschweig et al.,1992; Trabzoni, 1998). In fact, our steady-state bifurca-tion and dynamic analysis confirmed that the desired

operation condition of maximum butene-1 yield occursat an open-loop periodically unstable point. For thisreason, control of the reactor temperature is essentialfor stable operations. In addition, good temperaturecontrol is important to maintain the process around theoptimum conditions of maximum yield and to minimizelarge duration of off-spec products. The objective of thispaper is, thus, to design and test a good temperaturecontroller. Specifically, conventional PI and linear (modelpredictive control) MPC algorithms will be investigated.

Because the process is highly unstable, achieving goodcontroller performance or even stable feedback responseis a challenging task. Careful control design and tuningis inevitable in this case to attain such control objectives.For the PI algorithm, the conventional tuning methodsare the reaction curve method (Cohen and Coon, 1953)and the continuous cycling method (Ziegler and Nichols,1942) (ZN). In this paper, the reaction curve method isnot applicable because the process is open-loop-unstable.The continuous cycling method is applicable but isextremely difficult to implement. For this purpose,beside the ZN method, other existing model-basedtuning approaches designed particularly for unstableprocesses are tested in this paper (Rotstein and Lewin,1991; Poulin and Pomerleau, 1996; Venkatashankar andChidambaram, 1994; DePaor and O’Malley, 1989).

In addition, model predictive control that utilizes stepresponse models in output predictions will be tested. Thealgorithm is originally proposed by Cutler and Ramaker(1980) and was further modified, tested, and analyzedby many researchers. It became one of the most popularcontrol algorithms used in the chemical process indus-try. Good review of such algorithm can be foundelsewhere (Ogunnaike, 1986; Qin and Badgwell, 1997).Testing this specific algorithm does not necessarilyimply total superiority over the classical PID algorithm.The reason is rather based on past comparison studies,which had shown that MPC provides better performancein many cases (Shridhar and Cooper, 1997). Neverthe-less, application of MPC to open-loop-unstable processeswas frequently avoided since it requires a stable linearmodel for prediction. Alternatively, a nonlinear versionof MPC is used to stabilize unstable systems (Ali et al.,

* Fax: ++(9661) 467-8770. Telephone: ++(9661)467-6871.E-mail: amkamal@ksu.edu.sa.

1320 Ind. Eng. Chem. Res. 2000, 39, 1320-1329

10.1021/ie9903846 CCC: $19.00 © 2000 American Chemical SocietyPublished on Web 03/25/2000

1998). Ali (1998), however, managed to stabilize anopen-loop-unstable fluidized bed reactor using linearMPC through utilization of a proposed disturbanceestimation technique. In this paper, we try to achievesimilar successful implementation of MPC throughproper tuning. This will not be an easy task since MPCis commonly tuned by trail-and-error and the existingguidelines are applicable for stable unconstrained sys-tems. For unstable ill-conditioned systems corruptedwith model uncertainty, the tuning task becomes sig-nificantly problematic.

The paper is organized as follows. The followingsection presents the derivation of the mathematicalmodel for the dimerization reactor and its coolingsystem, followed by a section devoted to the open-loopanalysis of the model. The fourth section discusses thecontrol objectives of the dimerization reactor. The fifthsection discusses the control structure design using thesteady-state disturbance analysis. The sixth sectiongives an overview of the various PI tuning proceduresused and lists the obtained numerical values of the PIsettings. Section seven outlines the linear MPC formu-lation used in this work, and the eighth section isdevoted to illustrative closed-loop simulations. The lastsection offers the final conclusions.

Reactor Model

The dimerization reactor considered in this study isassumed to be a liquid-phase, perfectly mixed reactor;i.e., no mass transfer limitation is considered in thissystem. A schematic of the process is depicted in Figure1. The liquid is homogenized by a high recirculation ratearound the reactor through a heat exchange used toremove the high exothermic heat of reaction. The modeluses the homo- and copolymerization mechanisms sug-gested by Galtier et al. (1988). The reaction stages,initiation, propagation, and termination, are of first-order kinetics with respect to each reactant. The eth-ylene dimerization reaction is composed of the following:Initiation and propagation stages:

The rate of initiation and propagation, which alsorepresents the rate of disappearance of C2n in thesestages, has the following rate equation:

Termination stage:

The chain termination reactions, which are assumed tooccur in parallel with the growth reactions have thefollowing rate equation:

This rate expression defines the rate of disappearanceof C2n in the termination stage. For all stages, thereaction rate constant dependence on temperature fol-lows the Arrhenius formula. Under these assumptions,the mathematical model for the dimerization reactorconsists of the following: material balance equations forthe ethylene monomer (C2), the butene-1 and heavieroligomers (C2n, n g 2), the catalyst (K), and theintermediate catalytic activators (KC2n). In addition, themodel contains two energy balance equations for thereactor temperature and the recycle temperature.

By assuming the physical properties of the reactionmixture to be constant for all system streams, the twoenergy balance equations can be represented in thefollowing form:

K + C2n f KC2n n g 1

KC2m + C2n f KC2(n+m) n, m g 1

Figure 1. Schematic of the dimerization reaction process.

ra2n) a2n[C2n] ∑

m)0

∞

[KC2m]

KC2m + C2n f K + C2(n+m) n, m g 1

rb2n) b2n[C2n] ∑

m)1

∞

[KC2m]

Vd[C2]

dt) Fe[C2f] - Qâ[C2] - V(ra2

+ rb2)

Vd[C2n]

dt) -Qâ[C2n] - V(ra2n

+ rb2n) +

V ∑m)1

∞

b2m[C2m][KC2(n-m)] n > 1

Vd[K]

dt) Fe[Ak] - Qâ[K] - V[K]∑

n)1

∞

a2n[C2n] + V∑n)1

∞

rb2n

Vd[KC2n]

dt) -Qâ[KC2n] - V[KC2n] ∑

m)1

∞

(a2m[C2m] +

b2m[C2m]) + V ∑m)1

∞

a2m[C2m][KC2(n-m)] n g 1

VFCpdT

dt) FeFCp(Tf - Tr) + Q(1 - â)FCp(TR -

Tr) - QFCp(T - Tr) + V∑n)1

∞

ra2n(-∆Hra2n)

VcFCpdTR

dt) Q(1 - â)FCp(T - TR) -

UaA( T + TR

2- Tcav)

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1321

where

The dynamic of the coolant fluid is not included, andthe coolant flow rate is obtained by solving the steady-state equation:

To simplify the model, we neglect the formation ofoligomer compounds with higher than eight carbonatoms (n > 4). Under this assumption, the modelconsists of eight mass balance equations and two energybalance equation. In addition, the rates of disappearanceof C6 and C8 in the dimerization reaction stages are notconsidered, which means that a2n and b2n equal zero forall n g 3. The kinetic parameters used in this study arebased on the rate constants obtained by Galtier et al.(1988) and Woo et al. (1991) and are given as follows:

Table 1 shows the design parameters for the dimer-ization reactor system. Detailed description and deriva-tion of the model can be found elsewhere (Trabzoni,1998). The complete model and its Fortran coding isavailable upon request from the authors.

Open-Loop Analysis

The steady-state behavior of the model is traced inthe bifurcation diagram (Figure 2). The figure demon-strates the dependence of the ethylene conversion,butene-1 yield, selectivity, and reactor temperature onthe fresh ethylene flow rate. The other process opera-tional parameters such as Ak, â, Tf, and Wc are fixed attheir given values in Table 2. The system shows uniquesteady state for all Fe possible range of values. Theportions of the solution branches drawn with solid linesrepresent regions in which the reactor is stable and thedotted lines indicate unstable steady states. The systemloses and gains back its stability through Hopf bifurca-tion points. The simulations around the unstable regionlead to sustained oscillations around the unstablepoints, which might lead to a thermal runaway situa-tion. It can be seen that as Fe increases especially above0.02, the ethylene conversion and reactor temperatureincrease while the butene-1 yield and selectivity de-creases. Thus, a tradeoff between conversion and selec-tivity exists. For this reason, it is desired to operate theplant around suboptimal or a favorable point of Fe ) 4× 10-3 m3/s, which corresponds to 95.7% conversion and69.6% yield. This point also corresponds to a practicaltemperature operation which has to be around theheavy mixture bubble point. Table 2 lists the processparameters at this favorable plant operating point. Thevalues of the other operational parameters, i.e., Ak, â,Tf, and Wc, listed in Table 2 are determined by similaropen-loop bifurcation analysis which is discussed else-

where (Trabzoni, 1998). Nevertheless, as Figure 2indicated, the desired operating point is unstable. Thestable regions for this process are economically unac-ceptable. For example, a stable region exists at highthroughput, i.e., high Fe, but, at the same time, at lowyield and selectivity. Another stable region (not shown)is located at very low Fe, which corresponds to highselectivity but low conversion and production rate.Therefore, there is a potential for utilizing a good controldesign to stabilize the reactor around the desired open-loop-unstable point.

Control Objectives

The main control objective of such a process is thestabilization of the reactor temperature. This is es-sential to secure safe plant operation and to deliver agood quality product. It is also desirable to maintainoptimal operation of high ethylene conversion anddesired butene-1 yield as given in Table 2 in the face ofplant upsets. In practice, the coolant feed temperature,Tc, is one possible source of disturbances to the processwhich may cause thermal runaway due to temperatureinstability and consequently loss of conversion or yield.Figure 3 depicts clearly this effect. For this reason, ourclosed-loop simulations focus on temperature stabiliza-tion and maintaining desired yield in the face of upsetsin Tc. In this case, the controlled variable would be thereactor temperature, T, and the butene concentrationat the outlet stream, C4. In due course, there are fivepossible manipulated variables (MV), namely, the cool-ant flow rate, Wc, the feed flow rate, Fe, the catalystconcentration in the feed, Ak, the feed temperature, Tf,and the recycle ratio, â. Attaining the above controlobjective will be attempted through utilization of SISO/MIMO PI and MPC algorithms as discussed in thefollowing section.

Control Structure Design

There are different ways of designing a specific controlstructure (Ali et al., 1998). In this paper, we will mainlyuse the information deducted from the steady statedisturbance analysis (SSDA) (Yi and Luyben, 1995) plussimple reasoning for the control system design. In the

TCav )Tc + Tco

2

WcCpw(Tco - Tc) ) UaA(T + TR

2- Tcav)

a2 ) a20e-6000(1/T-1/Tr) b2 ) b20e

-3000(1/T-1/Tr)

a4 ) a40e-6000(1/T-1/Tr) b4 ) b40e

-3000(1/T-1/Tr)

b20

a20) 24

a40

b20) 0.0015

b40

a20) 0.05

Figure 2. Steady-state bifurcation diagram.

Table 1. Dimerization Reactor Design Parameters

V ) 500 m3 C2f ) 25 000 mol/ m3 UaA ) 27 500 cal/mol sVc ) 50 m3 F ) 500 kg/ m3 Cp ) 0.55 cal/g °CTr ) 25 °C ∆Hra2 ) ∆Hra4 )

∆Hra6 ) 25 000 cal/molCpw ) 1.0 cal/g °C

1322 Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

following sections, the part of the design associated withselecting the appropriate manipulated variables is ap-plicable to both PI and MPC algorithms. On the otherhand, the input-output pairing design is only applicableto the PI algorithm.

SISO Control Structure. First, we will try to controlonly the reactor temperature, T, hoping that this willalso keep the ethylene conversion and butene-1 yieldat their corresponding desired values. Selecting theappropriate MV can be determined, in general, as theone with the largest singular value (Cao and Biss, 1996)or equivalently static gain. Table 3 lists the static gainbetween T and the five available inputs. However, forconstrained inputs, the most effective input is the onehaving the largest reproducible output range (Cao etal., 1996) as listed in Table 3. The static gain iscomputed by exact linearization of the model. Thelargest output range is computed as the product of theinput static gain with its admissible range. Accordingto this criterion, the best MV is Fe followed by Wc andthen â. However, this method is based on the linearizedmodel and does not reflect how the MV actually doesrespond to disturbances. An alternative way for select-ing the suitable manipulated variable can be inferredfrom the SSDA (Yi and Luyben, 1995) shown in Figure4. Specifically, Figure 4a illustrates how much thereactor temperature changes at steady state for variousdifferent values of Tc when all the manipulated variableare fixed at their corresponding steady-state values. Theother parts of Figure 4 demonstrate how much a specificinput should change to keep T at its set point at steadystate for various values of Tc while the other inputs arefixed. For example, Figure 4f shows how Wc shouldchange, while other inputs are fixed at their nominalvalues, to keep T at 67 °C when Tc is stepped from itsnominal value by the amount shown in the x-axis. TheSSDA is obtained by solving the static version of themodel for different values of Tc with T and all the inputs,except the specified one, are fixed. The numericalsolution is carried out using ZSPOW subroutine of theIMSL library.

As Figure 4 demonstrated, Tf, Ak, and Wc may not besuitable as MVs. Tf has a very limited span; therefore,only a small range of upsets in Tc can be handled. Akand Wc have partly limited span and, moreover, respondnonlinearly to upsets in Tc, which may result in their

quick saturation if used as MVs. On the other hand, Feand â are regarded as reasonable MVs since they reactlinearly over the entire range of disturbances. Moreover,Fe is superior since it can handle wider range of upsetswithout violating its limits. Nevertheless, Wc can stillbe selected as the MV for two reasons. First, the coolantis considered, from the operational cost point of view,less demanding than the feed raw materials. Second,rejection of positive upsets in Tc, which are more serioussince they lead to temperature runaway, requires lowvalues for Fe and â. This means low throughput and,thus production rate. It can be argued that Ak can beas good as Wc. However, as shown by Figure 4, sup-pression of positive upsets in Tc requires small valuesfor Ak, which indicates lower reaction activity andtherefore ethylene conversion. In addition, from atechnical point of view, flow rate can be manipulatedeasier and more reliably than can concentration, i.e.,Ak.

MISO Control Structure. As will be demonstratedlater by closed-loop simulations, the SISO controlscheme cannot stabilize the reactor temperature at largeupsets that lead to high positive temperature excursion.The control failure is due to the input saturation, whichwas expected as anticipated from the steady-statedisturbance analysis shown in Figure 4. One way toovercome such a problem is to increase the degrees offreedom by utilizing an additional manipulated variablein a split-range framework. In this case, the feed flow(Fe) is used as the additional degree of freedom. It isdesirable to limit the use of the feed flow since it affectsthe production rate directly. For this reason, the split-range scheme (Figure 5) will be implemented such that90% of the control signal is used to actuate Wc, whilethe rest of the signal is used to actuate Fe.

MIMO Control Structure. As will be demonstratedlater, the MISO control scheme will manage to stabilizethe reactor temperature. However, the use of Fe as a

Table 2. Favorable Operating Condition

variable Wc (kg/s) Fe (m3/s) â Tf (°C) Ak (mol/m3) T, TR (°C) C2 (mol/m3) C4 (mol/m3)

value 500 4.0e-3 0.02 30.0 1.25 67, 43 1065 8700

Table 3. Static Gain for the Reactor Temperature

variable Wc (kg/s) Fe (m3/s) â Ak (mol/m3) Tf (°C)

static gain -0.025 16 253 854 5.2 0.082input range 100-1300 2-6e-3 0.01-0.04 0.1-2.5 20-90reproducible output range 35 65 25.6 12.5 5.7

Figure 3. Open-loop simulation of the reactor temperature fortwo values of Tc.

Figure 4. Steady-state disturbance analysis for controlled reactortemperature. Dashed lines represent the available range for MVs.

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1323

manipulated variable will drift the ethylene conversionand butene-1 yield away from their favorable conditions.Thus, it is desirable to maintain both the reactortemperature and the yield under control through MIMOcontrol scheme. In this case, the control problem consistsof two controlled variables, namely, the reactor tem-perature and the butene concentration. According to thesteady-state disturbance analysis shown in Figures 4and 6, there exist three possible manipulated variables,namely, Wc, Fe, and Ak. There are different rigorousmethods for selecting the best control structure (Ali etal., 1998). Without getting involved into the screeningof different control schemes, the control structure designwill be handled by simple reasoning. Apparently, Wc willbe used to regulate the reactor temperature. This isbecause it is the least expensive variable and it wasfound to be effective, to some extent, in regulating thereactor temperature. Therefore, Fe and Ak are left forcontrolling the butene concentration. To determinewhich input should be used beside Wc, the SSDA isreconstructed for the 2 × 2 system as discussed in thenext section.

It is thus advisable to inspect the SSDA when eitherinputs Wc and Fe (case 1) or Wc and Ak (case 2) areallowed to vary and T and C4 states are fixed. The otherstates are free and the other inputs are fixed. The resultof the analysis is shown in Figure 7. The available rangefor Fe and Ak are not shown in Figure 7 since they areout of the plotting scale. It is clear that Fe and Ak areslightly affected. However, In both cases, Wc saturatesdue to temperature runaway. Since Wc saturation isworse for case 2, Fe is considered more favorable thanAk as MV for the second loop.

PI Controller Tuning

Generally, conventional PI controller can be tuned bytwo popular methods; the reaction curve approach(Cohen and Coon, 1953) and the continuous cyclingapproach (Ziegler and Nichols, 1942). Recently, model-

based approaches designed particularly for unstableprocesses are reported (Rotstein and Lewin, 1991;Poulin and Pomerleau, 1996; Venkatashankar andChidambaram, 1994; DePaor and O’Malley, 1989). Thereaction curve method requires open-loop tests to builda small-order linear transfer function from which thePI settings are inferred. Thus, it cannot be used in ourcase since the process is open-loop-unstable. The con-tinuous cycling approach is based on determining theultimate gain and ultimate period of oscillation beyondwhich the closed-loop response becomes unstable. Theultimate values are determined either by a relayfeedback approach (Astrom and Hagglund, 1984) or bytrial-and-error closed-loop tests (Ziegler and Nichols,1942). The model-based schemes are associated withidentifying an unstable linear model represented by afirst-order plus dead time transfer function. The iden-tification is carried out via closed-loop experimentsusing a proportional controller only. In this work, onlythe continuous cycling and model-based tuning proce-dures will be tested.

SISO Case. Generally, the continuous cycling meth-odology may not be easy for open-loop-unstable pro-cesses since instability occurs at both high and lowcontroller gains (Yuwana and Seborg, 1982). Howeverby careful and iterative closed-loop procedure, thefollowing ultimate controller gains and periods of oscil-lation for the (T f Wc loop) are obtained:

The upper limits can be used to estimate the PIparameters as listed in Table 4. The first column liststhe regular Ziegler-Nichols (RZN) tuning parameters,while the second column lists the modified ZN (MZN)tuning parameters (Seborg et al., 1989). The modifiedZN parameters are more conservative and thus moredesirable for such cases since the regular ones mayexhibit serious oscillation.

According to Luyben (1998) the above lower and upperlimits can also be used to identify a first-order modelwith time delay. Implementation of such an approachyields the following:

Figure 5. Schematic of the split-range scheme.

Figure 6. Steady-state disturbance analysis for controlled C2(ethylene concentration). Dashed lines represent the availablerange for MVs.

Figure 7. Steady-state disturbance analysis for 2-input/2-outputsystem for various changes in Tc. Dashed lines represent theavailable range for MVs.

Table 4. PI Tuning Parameters for the SISO Case (T fWc Loop)

param. reg. ZN mod. ZNIMC

(τc ) 0.5, 1)PP

(Mr ) 15 dB) VC DO

kc -240 -120 -529, -285 -455 -186 -84τI 10 6 1, 2.3 34 85 25

kumin ) -58 Pu

min ) 8 min

kcmax ) -600 Pu

max ) 12 min

1324 Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

where kp is the process static gain, τ is the process timeconstant, and θ is the time delay. The obtained linearmodel can be used to determine the PI parameters forunstable process using the model-based methods as willbe discussed later. For comparison purposes, the linearmodel can also be identified via closed-loop simulations(Venkatashankar and Chidambaram, 1996). The methodassumes the process to be characterized by a first-orderplus time delay (FOPDT) model. By simple set pointchange using a proportional controller, the closed-loopsystem can be made to have an underdamped response.The closed-loop response can then be used to determinethe parameters of a second-order system from which theparameters of the original FOPDT can be evaluated.Implementation of such procedure for a unit set pointchange, the FOPDT parameters are found to be

which are almost consistent with those found fromultimate gain information. Having identified the FOP-DT system, the model-based tuning methods proposedfor unstable processes can be now tested.

(1) IMC-Based. According to Rotstein and Lewin(1991), the PI parameters can be estimated from thefollowing relations:

where τc is the IMC filter constant used as adjustableparameter for robustness. For two values of τc and byusing the obtained values for the model parameters, thePI settings were found and are given in Table 4.

(2) Poulin and Pomerleau (PP). Poulin and Pomerleau(1996) proposed a tuning procedure for unstable systemsbased on a desired maximum closed-loop log moduluspeak, Mr. Our investigation found that values greaterthan 11 for Mr result in positive values of reset time.By use of a value of +15 dB for Mr, the PI settings areobtained as in Table 4.

(3) DePaor and O’Malley (DO). DePaor and O’Malley(1989) proposed a tuning scheme based on the ratio ofthe dead time to the time constant. Implementation ofthe proposed scheme with the obtained values for τ andθ suggests the PI settings given in Table 4.

(4) Venkatashankar and Chidambaram (VC). Ven-katashankar and Chidambaram (1996) proposed a tun-ing scheme that is based on the values of kc

min, kcmax, τ,

and θ. Implementation of such a method suggests thePI settings given in Table 4.

MISO Case. The additional T f Fe control loop istuned by the continuous cycling method. By carefulclosed-loop trial-and-error simulations and by use of themodified ZN criteria, the PI settings are determinedsuch that kc ) 4.5 × 10-4 and τI ) 6 min.

MIMO Case. The PI settings for the first loop (T fWc) will remain the same as those of the MZN used inthe SISO case. The PI settings for the second loop, i.e.,(C4 f Fe), are then determined by the continuous cyclingmethod. The model-based tuning methods are found tobe difficult to apply for this case because they requireidentification of a FODPT model for the C4 f Fe loop.Our experience revealed that identification of theFOPDT model for the second loop by using the same

approach as that used for the SISO case is extremelydifficult. The reason is attributed to the internal insta-bility of the process and to high interaction between thetwo control loops, as indicated by the relative gain array.The RGA for the 2-input/2-output system is listed inTable 5. It should be noted that the RGA was not usedfor designing the MIMO control structure. The controlpairing according to the RGA is not reliable due to thehigh interaction indicated by the negative sign of theoff-diagonal elements. It should be noted that tuning ofthe second loop via continuous cycling method cannotbe handled easily without closing the first loop, i.e.,setting the reactor temperature under control. Never-theless, by extensive closed-loop simulations, the ulti-mate dynamic properties of the C4 f Fe loop is deter-mined, from which PI settings for that loop are foundto be

Linear MPC Algorithm

A usual MPC formulation (notation follows that inLee et al., 1994) solves the following on-line optimizationfunction:

subject to

The symbol ||‚‚‚|| denotes the Euclidean vector norm andk denotes the current sampling point. Γ and Λ arediagonal weight matrices. R(k + 1) ) [r(k + 1)...r(k +P)]T contains the desired output trajectories over horizonP. ∆U(k) ) [∆u(k)...∆u(k + M - 1)]T is a vector of Mfuture changes of the manipulated variable vector u thatare to be determined by the on-line optimization. YP(k+ 1) ) [y(k + 1)...y(k + P)]T includes the predictedoutputs over the future horizon P, where y is the outputvector, assuming ∆u(k + i) ) 0; i g M. y(k) ) [y(k)-...y(k + n - 1)]T includes the predicted outputs over thetruncation horizon n (length of FIR) based on ∆u(k + i)) 0; i g 0. Equation 2 represents the output predictionbased on the process model. A disturbance estimatedenoted d should also be added in eq 2, or alternatively,it can be absorbed in R(k + 1). The latter is assumedfor simplicity. In the implementation of standard MPC,the disturbance is assumed constant over the predictionhorizon and is set equal to the difference between plantand model outputs at present time k. The matrices Mp

and SPm are defined as in Lee et al. (1994). Mp is a

constant matrix consisting of ones and zeros and SPm is

what is usually referred to as the dynamic matrix ofstep response coefficients. The above objective function(1) is minimized subject to constraints on the manipu-

Table 5. Relative Gains Array for Three Possible ControlStructures

S1 S2 S3

Wc Fe Fe Ak Wc Ak

T 1.6255 -0.6255 -0.1054 1.1054 -0.3293 1.3293C4 -0.6255 1.6255 1.1054 -0.1054 1.3293 -0.3293

kc ) -1.0 × 10-7 τI ) 10 min

min∆u(k1),...,∆u(k+M-1)

||Γ(YP(k + 1) - r(k + 1)||2 +

||ΛU(k)||2 (1)

YP(k + 1) ) MPy(k) + SPm∆U(k) (2)

FT∆U(k) e b (3)

kp ) (kcmin)-1 ) -0.017 τ ) 5.8θ ) 2.4

kp ) -0.022 τ ) 5.6θ ) 2.7

kc ) -τγτc

2kp

τI ) γ γ ) τc(τc

τ+ 2)

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1325

lated variables and on the change of manipulatedvariables represented by eq 3.

To implement the above control algorithm, a stablestep response model should be developed for the process.This issue is discussed in the simulation section. Thecontroller implementation also requires fixing the val-ues of its tuning parameters, namely, the diagonalelements of the input weight (move suppression coef-ficients), Λ, and of the output weight, Γ, the controlhorizon, M, and the prediction horizon, P. Generally,adjustment of these parameters are conducted throughsystematic trial-and-error procedure. However, generalguidelines are available (Shridhar and Cooper, 1998).Basically, for stable processes, fine-tuning of MPC isneeded for robustness and good performance require-ments. However, for unstable processes such as the oneconsidered in this paper, fine-tuning is a must andwithout it controller failure is inevitable. This issue isaddressed in the simulation section.

Closed-Loop Simulations

PI Controller. Simulation of the SISO control prob-lem for +4 °C step change in Tc using the PI settingsobtained from the ZN method is shown in Figure 8. Asampling time of 0.1τ ) 0.5 h is used. The regular ZNmethod provided feedback performance superior to thatof the modified ZN. The figure also includes the responseof the C2 and C4 concentrations and coolant flow rate.It is obvious that by regulating the reactor temperature,the reaction conversion and yield were also kept at theirdesired optimum set points of 95.7% and 69.6%, respec-tively. Figure 9 demonstrates the simulation of the samecontrol problem using the model-based tuned PI algo-rithms. As demonstrated by the figure, the DO methoddelivered very oscillatory performance, whereas the VCmethod provided some oscillation initially but managedfinally to bring the reactor temperature back to its setpoint. The PP method delivered the best performance.Likewise, the IMC-tuned controller had an excellentperformance for τc ) 0.5. However, as τc increases, theclosed-loop performance worsen and even becomesunstable for large values of τc. This is in fact contradic-tory to the role of τc, because increasing it should providerobustness. The simulations shown in Figures 8 and 9revealed that large values for kc provide improvedfeedback performance.

However, at an upset of magnitude of +6 °C in Tc, alarger amount of coolant is needed to remove the heatof reaction leading to saturation of the coolant flow rate(Figure 10). This degrades the closed-loop performance.

The PI controller used in this simulation is fine-tunedstarting from the modified ZN parameters by stepwisetrial-and-error procedure. The specific PI settings usedare kc ) -400 and τI ) 5. As a remedy, Fe can be usedas a second manipulated variable in a split-rangescheme, which was discussed in the MISO structuresection. Improved feedback performance is obtained asshown in Figure 11. However, the tight control of Tcaused reduction in the feed flow rate Fe and thus inthe production rate, a situation which also led to someoffset in the ethylene and butene-1 concentrations.Specifically, 96% conversion and 69.3% yield are ob-served. This confirms that utilization of Fe for reductionof the temperature runaway is not favorable. It couldbe argued though, that the offset in C2 is favorable sinceit increases the conversion. However, the increase isminute and, as expected, is associated with reductionin the butene-1 yield.

Figure 8. Closed-loop response to +4 °C step change in Tc usingPI algorithm with (solid) regular ZN and (dashed) modified ZNtuning methods.

Figure 9. Closed-loop response to +4 °C step change in Tc usingPI algorithm with (a) IMC, (b) DO, (c) PP, and (d) VC tuningmethods.

Figure 10. Closed-loop response to +6 °C step change in Tc usingPI algorithm.

Figure 11. Closed-loop response to +6 °C step change in Tc usingPI algorithm with split-range scheme. Solid line: set point.

1326 Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

Another way to deal with the case of rejecting +6 °Cchange in Tc is to use a MIMO control structure asdiscussed earlier. Figure 12 illustrates the effect ofemploying the MIMO control scheme to handle suchchallenging problem. In this case, the MIMO schememanaged to keep both controlled variables, T and C4,within their corresponding set points. However, this isachieved at very low rate since more conservative PIsettings than what have been determined in the previ-ous section were used. Specifically, kc ) [-300, -1 ×10-7] and τI ) [50, 10 000] are employed in the simula-tion shown in Figure 12. Apparently, using the rawvalues for the PI settings led to very oscillatory responsedue to high loop interaction, bearing in mind that thefirst loop was tuned while the second loop was open.Our investigations observed that any attempt to speedup the feedback performance would create persistentoscillatory response.

Linear MPC. The above SISO and MIMO controlproblems are retested using the MPC algorithm. To usethe MPC, a linear stable step response model for the 2× 2 system is required by the controller. A linear state-space model is obtained by linearizing the nonlinearmodel around a stable operating condition. The linear-ized model is converted into a step response model usingMATLAB software. The stable operating point usedcorresponds to that at which the heat transfer coef-ficient, Ua, is 20% less than the nominal value. Theobtained model captures all of the dynamics features,i.e., static gain and time constant, of the true process(nonlinear model). Another stable operating point canbe obtained at very low feed rate, but in that case, thedynamic features of the obtained model are far fromthose of the true process. It should be noted that theobtained model is highly ill conditioned. The conditionnumber is on the order of 107. A sampling time of 0.5 hand a truncation number of 60 are used during con-struction of the step response model. The truncationnumber used was found to be enough to ensure that themodel correctly predicts the steady state. In the entiresimulations, the MPC is implemented on the fullnonlinear plant model. This makes the control problemmore challenging due to modeling error introduced fromusing the linearized model in the controller. Selectionof the appropriate MPC tuning parameters is anotherimplementation issue that is addressed in the nextsection.

Regarding the SISO case, the PI simulations indicatedthat aggressive tuning parameters should be used.Therefore, M ) P ) 1 and Λ ) 0 were used. However,this set of tuning parameter values delivered unstable

performance. Any further attempts to stabilize theclosed-loop response through increasing P or Λ was alsofound to be useless. Perfect feedback performance isobtained for the case of ∆Tc ) 0 °C (nominal case) onlyat a sampling rate of 0.1 h, as shown in Figure 13a.However, when upsets are introduced in Tc, a samplingrate of 0.05 h must be used to retain stability of thecontroller (Figure 13b,c). In these cases, the stepresponse model is reconstructed using the specificsampling rate. The poor performance for the case of ∆Tc) 6 °C is attributed to the input saturation as shownin Figure 13d and not improper tuning. It should benoted that, although the feedback response for the caseof ∆Tc ) 6 °C is oscillatory, the oscillation is stable andoutperformed that for the open-loop case shown inFigure 3. Although, Λ is used by many researchers totune the MPC (Shridhar and Cooper, 1997), our simula-tions revealed that it not only does not help but mayworsen the performance.

Building upon the tuning difficulties faced in theSISO case and knowing that the system is unstable, ithas significant cross-loop interactions, and moreover theprocess model is ill-conditioned, we expected the tuningof the MIMO system to be even more complicated. Infact, our simulations supported this opinion. The fol-lowing is an elaboration on how the MPC parametersare selected. Generally, a large prediction horizon (P)is commonly suggested to improve nominal stability(Rawlings and Muske, 1993); however, it is found hereto have an adverse nonmonotonic effect on the controllerperformance. For this reason, M ) P ) 1 is fixed in thesimulations and found to simplify the selection of theother parameters. The move suppression coefficient, Λ,is believed to serve dual functions of suppressing theaggressive control actions and condition the modeldynamic matrix (Shridhar and Cooper, 1998). Thisguideline was found helpful as Λ had to be adjustedcarefully to remove the ill conditioning and stabilize theMIMO control system. Moreover, because the outputsare of different order of magnitude, it was important toadjust the controlled variable weight (Γ) to properlyscale the outputs and consequently improve the perfor-mance. The following simulations illustrate our finaltuning results.

Figure 14 depicts the closed-loop response to +6 °Cupset in the inlet coolant temperature (Tc) using Γ )[100, 1] and Λ ) [0.5, 5 × 105]. As the figure demon-strated, a moderate initial oscillation is observed in the

Figure 12. Closed-loop response to +6 °C step change in Tc usingPI algorithm with MIMO scheme. Solid line: set point.

Figure 13. Closed-loop response using MPC. ∆Tc ) (a) 0, (b) +4,(c) +6, and (d) (dashed line) +6, (solid line) +4, and (dashed/dottedline) 0 °C.

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1327

reactor temperature response followed by settling aroundits desired value. However, after a long time, theresponse of butene-1 concentration (C4) drops sharply,creating large offset and persistence stable oscillationof small amplitude. By careful adjustment of λ2 (to reacha value of 2e+6), tight control of the reactor temperature(T), i.e., elimination of final oscillation, can be obtained(Figure 15a,b). However, the C4 offset remains. Furtherstepwise adjustment of both λ2 and γ1 leads to tightcontrol of both T and C4 as shown in Figure 15c,d.Specifically, Γ ) [500, 1] and Λ ) [0.5, 8 × 106] are used.Apparently, the initial oscillation in T and sluggishtracking of C4 set point seem to be inevitable in orderstabilize the process. A sampling rate of 0.5 h is usedin the simulations. Further improvement of the control-ler performance was found to be difficult due to thecontroller sensitivity to its tuning parameters. Forexample, slight changes in sampling rate, M, or P willsubstantially deteriorate the performance in an unpre-dictable manner.

Conclusions

A previously developed first-principle model for theethylene to butene-1 dimerization reactor is used forsteady-state bifurcation analysis. The analysis indicatedthe existence of optimal plant operation of maximumethylene conversion and butene-1 yield. However, thefavorable operation condition is located at a highly

unstable and periodic state, which is common to manypolymerization reactions. For this purpose, classical PIand MPC algorithms were tested for possible stabiliza-tion of such a reactor. Application of SISO PI algorithmrevealed the ability to stabilize the reactor at low upsetsin the coolant temperature. At high upsets, saturationof the coolant flow rate occurs to degrade the controllerperformance. Similarly, implementation of the SISOMPC algorithm also observed this finding. For a situ-ation that calls for better control structure such as theMIMO control system, simulation of the MIMO controlsystem demonstrated successful control operation andexistence of tradeoff between stability and good perfor-mance. This observation was confirmed by both algo-rithms, i.e., PI and MPC. It should be pointed out herethat careful tuning of both algorithms is a must toachieve successful implementation. The PI algorithmwas tuned mainly by the ZN method and other existingtechniques particular for unstable systems in additionto trial-and-error. On the other hand, the MPC wastuned by adjusting the sampling rate, Λ, and Γ througha trial-and-error procedure.

Nomenclature

a2j ) rate constant for consumption of component C2j inthe initiation and propagation stages (j g 1) (m3/mol s)

A ) cooler area for heat transfer (m2)Ak ) catalyst concentration at the fresh feed (mol/m3)b ) vector of upper and lower bounds for the linear

constraints of MPCb2j ) rate constant for consumption of component C2j in

the termination stage (j g 1) (m3/mol s)C2n ) reactant and products based on number of carbon

atoms (n ) 1 for ethylene, n ) 2 for butene-1, n ) 3 forC6, and n ) 4 for C8)

[C2], [C4] ) concentration of ethylene and butene-1 in thereactor, respectively (mol/m3)

[C2n] ) concentration of component C2n (mol/m3)[C2f] ) ethylene feed concentration of component C2n (mol/

m3)Cp, Cpw ) heat capacity of reactor mixture and water

respectively (cal/g °C)d ) disturbance estimates for MPC algorithmF ) constant matrix for the linear constraints of MPCFe ) reactor fresh feed (m3/s)[K] ) catalyst concentration (mol/m3)[KC2n] ) intermediate catalyst activator concentration

(mol/m3)k ) sampling time (h)kc, ku ) controller gain and its ultimate value, respectivelykp ) process static gainM ) control horizonMp ) constant matrix for the MPC algorithmMr ) maximum closed-loop log modulusP ) prediction horizonPu ) ultimate periodQ ) reactor product volumetric flow rate (m3/s)r ) vector of set pointra2n ) rates of chain initiation and propagation (n g 1) (mol/

m3 s)rb2n ) rates of chain termination (n g 1) (mol/m3 s)R ) vector of P-future set pointsSp

m ) step response matrixT, Tc, Tf ) reactor, coolant, and feed temperatures,

respectively (°C)TR, Tr ) recycle flow and reference temperatures, respec-

tively (°C)u ) vector of manipulated variables

Figure 14. Closed-loop response to step change of +6 °C in Tcusing MPC for the MIMO case. Γ ) diag[100, 1], and Λ ) diag-[0.5, 5 × 105].

Figure 15. Closed-loop response for step change of +6 °C in Tcusing MPC for the MIMO case. (a) and (b) Γ ) diag[100, 1] and Λ) diag[0.5, 2 × 106]. (c) and (d) Γ ) diag[500, 1] and Λ ) diag[0.5,8 × 106].

1328 Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

U ) vector of M-future manipulated variablesUa ) heat transfer coefficient (cal/m2 s °C)V ) reactor volume (m3)Vc ) cooler volume (m3)y ) vector of controlled variablesyp, YP ) vector of P-future predicted outputsW1 Wc ) coolant flow rate (kg/s)

Greek Letters

∆Hr ) heat of initiation and propagation reactions (cal/mol)

∆u ) vector of manipulated variable moves∆U ) vector of M-future manipulated movesâ ) recycle ratioΛ ) diagonal matrix of input weightsΓ ) diagonal matrix of output weightsλi, γi ) ith diagonal element of Λ and Γ, respectivelyF ) mixture density (kg/m3)θ ) time delayτ, τI, τc ) time constant, reset time, and IMC filter time

constant, respectively

Literature Cited

Ali, E. Robust Model-Based Control of Open-loop Unstable Pro-cesses, the Arabian J. Sci. Eng. 1999, 24, 33-48.

Ali, E.; Abasaeed, A.; Al-zahrani, S. Optimization and Control ofIndustrial Gas-Phase Ethylene Polymerization Reactors. Ind.Eng. Chem. Res. 1998, 37, 3414-3423.

Astrom, K. J.; Hagglund, T. Automatic Tuning of Simple Regula-tors with Specifications of Phase and Amplitude Margins.Automatica 1984, 20 (5), 645-651.

Braunschweig, S.; Galtier, P.; Glaize, Y. ALEXIP. Revue DeEinstitut Francais Du Petorle 1990, 47, 375-383.

Cao, Y.; Biss, D. An Extension of Singular Value Decompositionfor Assessing Manipulated Variable Constraints. J. Proc. Cont.1996, 6, 36.

Cao, Y.; Biss, D.; Perkins, J. Assessment of Input-Output Control-lability in the Presence of Control Constraints. Comput. Chem.Eng. 1996, 20, 337-346.

Cohen, G. H.; Coon, G. A. Theoretical Investigations of RetardedControl. Trans. ASME 1953, 75, 827-834.

Cutler, C. R.; Ramaker, D. L. Dynamic Matrix Control-A Com-puter Control Algorithm. Proc. Joint Auto. Cont. Conf.; SanFrancisco, CA, 1980.

Dadebo, S.; Bell, M.; Mclellan, P.; McAuley, K. TemperatureControl of Industrial Gas-Phase Polyethylene Reactors. J. Proc.Cont. 1997, 7, 83.

DePaor, A.; O’Malley, M. Controllers of Ziegler-Nichols Type forUnstable Process with Time Delay. Int. J. Control 1989, 49,1273-1284.

Galtier, P. A.; Forestiere, A. A.; Glaize, Y. H.; Wauquire, J. P.Mathematical Modeling of Ethylene Oligomerization. Chem.Eng. Sci. 1988, 43, 1855-1860.

Lee, J. H.; Morari, M.; Garcia, E. State Space Interpretation ofModel Predictive Control. Automatica 1994, 30, 707-717.

Luyben, W. Tuning Temperature Controllers on Open-loop Un-stable Reactors. Ind. Eng. Chem. Res. 1998, 37, 4322-4331.

Ogunnaike, B. A. Dynamic Matrix Control: A NonstochasticIndustrial Process Control Technique with Parallel AppliedStatistics. Ind. Eng. Chem. Fundam. 1986, 25, 712.

Poulin, E.; Pomerleau, A. PID Tuning for Integrating and UnstableProcesses. IEE Proc. Cont. Theory. Appl. 1996, 143, 429-435.

Qin, J. S.; Badgwell, T. A. An Overview of Industrial ModelPredictive Control Technology. Proc. Chem. Process Cont. Conf.CPC V; Kantor, J. C., Garcia, C. E., Carnahan, B., Eds.; AIChESymp. Ser. 1997, 316 (93), 201.

Rawlings, J.; Muske, K. The Stability of Receding Horizon Control.IEEE Trans. Auto. Cont. 1993, 38, 1512-1516.

Rotstein, G.; Lewin, D. Simple PI and PID Tuning for Open-loopUnstable Systems. Ind. Eng. Chem. Res. 1991, 30, 1609-1615.

Seborg, D.; Edgar, T.; Mellichamp, D. Process Dynamics andControl; Wiley and Sons: New York, 1989.

Shridhar, R.; Cooper, D. J. A Tuning Strategy for UnconstrainedSISO Model Predictive Control, Ind. Eng. Chem. Res. 1997, 36,729-746.

Shridhar, R.; Cooper, D. J. A Tuning Strategy for UnconstrainedMultivariable Model Predictive Control. Ind. Eng. Chem. Res.1998, 37, 4003-4016.

Trabzoni, F. Modeling and Simulation of an Ethylene DimerizationReactor Dynamics. M. Sc. Thesis, Chemical Engineering De-partment, King Saud University, Riyadh, Saudi Arabia, 1998.

Venkatashankar, V.; Chidambaram, M. Design of P and PIControllers for Unstable First-Order Plus Time Delay Systems.Int. J. Control 1994, 60, 137-144.

Venkatashankar, V.; Chidambaram, M. On-line Controller Tuningfor Unstable Systems. Comput. Chem. Eng. 1996, 20 (3), 301-305.

Woo, T. W.; Woo, S. I. Kinetics Study of Ethylene DimerizationCatalyzed over Tu(O-nC4H9)4/AlEt3. J. Catal. 1991, 132, 68-78.

Yi, C.; Luyben, W. Evaluation of Plant-Wide Control Structuresby Steady-State Disturbance Sensitivity Analysis. Ind. Eng.Chem. Res. 1995, 34, 2393-2405.

Yuwana, M.; Seborg, D. A New Method for On-line ControllerTuning. AIChE 1982, 28 (3), 434-439.

Ziegler, J.; Nichols, N. Optimum Settings for Automatic Control-lers. Trans. ASME 1942, 64, 759-768.

Received for review June 1, 1999Revised manuscript received December 6, 1999

Accepted January 11, 2000

IE9903846

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