Optimization and nonlinear model predictive control of batch polymerization systems

European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.

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Optimization and Nonlinear Model Predictive Control of Batch Polymerization Systems Dulce C.M. Silva and Nuno M.C. Oliveira a aDepartamento de Engenharia Qufmica, Universidade de Coimbra P61o II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. Tel: +351-239-798700. Fax: +351-239-798703. E-mail: {dulce,nuno}@eq.uc.pt.

This paper addresses the optimization of batch polymerization systems, using a feasible path approach, with roots on Model Predictive Control (MPC) theory. The approach allows the reuse of many concepts previously developed for nonlinear MPC of continuous plants. It also provides an efficient and well integrated methodology for the optimal supervision of discontinuous chemical processes. The application of this technique in the optimization of the batch suspension polymerization of vinyl chloride is also described.

1. INTRODUCTION Due to their non-stationary nature, batch processes present interesting challenges in their control and online optimization, and create unique opportunities for the development of advanced supervision strategies. In the case of batch polymerization processes, optimal operation involves computing and accurately maintaining the optimal temperature and initiator (or coreactants) addition policies that can lead to a product with desired properties (such as molecular weight average, polidispersivity, chain length distribution) and final conversion, while minimizing the total operation time. In many cases, batch operations are still carried according to recipes based on heuristics and past experience. However, the recent availability of detailed mechanistic models, experimentally validated, provide a significant incentive for a wider use of newly developed optimization algorithms. Previous research on the determination of optimal policies for batch processes concentrated on techniques for the solution of optimization problems subject to algebraic and differential constraints. To avoid the numerical difficulties associated with the solution of a nonlinear twopoint boundary value problem, various methods based on the discretization of the differential equations have been proposed, using a simultaneous solution and optimization approach [ 1]. On the other hand, nonlinear MPC algorithms using a feasible path approach have been tested with success on continuous processes, in the presence of general constraints [2,3]. While considering different objectives, these algorithms constitute in fact general NLP optimizers, and are able to deal efficiently with differential constraints. Their use for general optimization of discontinuous processes is therefore investigated in this paper.

750 2. PROBLEM FORMULATION AND SOLUTION STRATEGY The dynamic optimization problems to be solved can be formulated as min

W(x(t),u(t))

u(t)E Y{ik

s.t.

:~ = fp (x, u, d; O)

y - gp(x; O)

(1)

Ul ~ U ~_~ Uu XI ~ _ X ~ Xu

Yl <_ Y < Yu, where fp and gp are usually assumed differentiable and continuous, except perhaps at a finite number of switching points. Depending on the task at hand, different forms of the objective function can be considered. For instance, the Newton Control formulation for nonlinear MPC uses a quadratic objective of the form

w(,) -- f~+,o~(y-Ysp)TQy(t)(y-Ysp)+(U-ur)TQu(t)(U-Ur)dZ,

(2)

9It k

where Ur and Ysp are reference trajectories for both the inputs and outputs. Other common objectives can be related to the properties of the final product, and be formulated as soft constraints of the form W(o) -- (y(tF) --Ysp)TQ1 (y(tF) -- Ysp), (3) where Ysp represents the desired final value. Important product specifications might require their formulation as hard equality or inequality constraints. In this case the objective can coincide directly with the minimization of the operation time

V(o) =tv, subject to several product constraints, which can be expressed in terms of restrictions involving the u, x and y variables of formulation (1). Piecewise constant input profiles are assumed. The resulting nonlinear programming problem can be solved using a successive quadratic programming (SQP) approach, requiting the linear and quadratic terms of the constraints and objective. To simplify the notation, augmented vectors U, X and Y are defined, containing all values of the corresponding variables inside an operating horizon. An exact linearization of the model around a nominal trajectory can be written as

~Y

AU

- - Y -Jr-S m A U ,

where Sm represents the dynamic matrix of the model, containing the first order information for the system relative to the input variables. This matrix can be efficiently computed from the original differential model through the use of appropriate sensitivity equations. When the objective has the form of (2) or (3) the algorithms described in [2,3] can be directly used. However, some modifications are required in this formulation to treat minimum-time problems. In these problems, the final time is usually defined by a certain output variable, which

751 reaches a predefined value YF at the end of the operation. We assume that this happens during the nth discretization interval, from tn to tn+l, inside a larger horizon defined as a maximum bound on tv. Given the previous assumptions about the model, it is possible to write tF as an implicit function of the initial condition and input variables during this interval

(4)

tF -- h(xn, Un).

The first order information for tF can then be obtained by writing a Taylor series in this interval: tF - - t-F +

The derivatives

x=x

and

~tF

~tF x=~ 9

. u=~

UUn u=~

9 (btn--Un ) .

are, in some cases, difficult to obtain directly, by integration

of the sensitivity coefficients, since (4) is usually not available in explicit form. However, since these coefficients are only needed in the last time interval, they can also be approximated by finite differences, without a great penalty. Applying the previous concepts, the linearization of tF with respect to the input variables can be written as tF -- ~ + S*AU.

This allows formulation of the optimization problem as the SQP iteration of minJ2 - t~ + S*AU + A U T H A U AU s.t. UId <_ AU <_ Uud Yld <_ Sm AU <_ Yud Sm.,jAU - AyF,

where AyF = Ysp -- y(tF), and H represents an approximation of the Hessian of the Lagrangian of (1). This formulation is closely similar to the one used in the nonlinear Newton control law, making the algorithms developed for its solution applicable for minimum-time problems as well. 3. APPLICATION TO THE PVC SUSPENSION POLYMERIZATION

In this section we consider the application of the previous strategy to the optimization and nonlinear control of the batch suspension polymerization of vinyl chloride (VCM). This system involves four phases (monomer, polymer, aqueous and gas), and an heterogeneous reaction. Various kinetic models have been proposed to describe the process, with significant differences at the level of complexity and detail given to various chemical and physical phenomena taking place. In order to compare the optimization results, and to better assess their sensitivity, two mechanistic models for this process were built, based on the kinetics information provided by Xie and co-workers [4-6] and Kiparissides and co-workers [7]. Both of these models consider diffusion controlled reactions. The monomer distribution in the different phases is computed as a function of the conversion and reactor operation conditions. In the Xie model the rate constants are

752 modelled using the free volume theory, while in Kiparissides's model the termination and propagation rates are expressed in terms of reaction limiting and diffusion limiting terms; the later term depends on an effective reaction radius and on the diffusion coefficients, calculated from the extended free volume theory [7]. The state variables in the Xie's model are the monomer conversion, the first and second moments of the dead polymer distribution in the polymer phase, number and weight accumulated molecular weight average and quantities of the initiator. In the Kiparissides's model the state variables are: conversion, zeroth, first and second moments of the dead polymer distribution, and quantity of initiators. The predictions from both models are compared in Figure 1, using a constant polymerization temperature of 55~ As can be observed the conversion profiles are close until a conversion of 70% is reached. Their divergence after this point can be attributed to the fact that in Kiparissides's model the initiation efficiency after the critical conversion is not considered to be diffusionaly controlled, as in the Xie's model. With respect to average molecular weights, we can observe in Figure 2 that for the same temperature the models predict polymers with slightly different properties at the end of the operation. These differences can be due to the values of the kinetic parameters used in each model, especially the chain transfer to monomer that controls the molecular weight of the polymer, as well as other considerations made in their development.

k

200

1 ..........................

. . . . . . . . . . . . . .

175 0.8

~. 150

0.6
0.4

~

0.2

~

0

0

I00

200 300 t (rain)

400

500

Fig. 1. Conversion profile for isothermal operation ( - Xie's model; - 99 Kiparissides's model).

PMw

125

.

I00

I~....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 : < PMn 5o ...................................................... PMn 25 0

0.2

0.4

0.6

0.8

X

Fig. 2. Conversion dependence of number and weight average molecular weights ( Xie's model . . . . Kiparissides's model).

4. OPTIMIZATION RESULTS The main variables available for manipulation in this example are the reaction temperature and the initiator (or co-reactants) concentration. We studied the independent effect of each of these variables separately.

4.1. Effect of the reaction temperature: Figure 3 shows the results obtained for the optimal temperature profile that minimizes the total batch time. This optimal profile was obtained subject to restrictions in the polidispersivity and weight average molecular weight that guaranteed that the same product was obtained. Also, due to operational constraints, bounds of i 5 ~ relatively to the nominal temperature were imposed. When compared with isothermal operation (the current industrial practice), the profiles

753 obtained with Kiparissides's and Xie's models are able to achieve reductions of 8.7% and 23% in the reaction time, respectively. Xie's model shows more sensitivity to the reactor temperature and therefore has lower deviations relatively to the nominal trajectory.

1.03

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.

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.

.

1.02 ..... ~--~..~ ..~

1.01

i

"'~'-%

1

"'%..

I

1:

~..~.. -~___X .......

0.99 0.98

0

I00

200 300 t (min)

400

500

Fig. 3. Normalized optimal reactor temperature policy ( - Xie's model; - . model).

Kiparissides's

The on-line implementation of the optimal trajectory obtained in the Kiparissides's model was considered, using the nonlinear Newton-type model predictive control formulation. This is illustrated in Figures 4 and 5, where these results are also compared with a linear control strategy. For the predictive controller the tuning parameters used w e r e Qa = I, Q2 = 10-31, with a sampling time of 200 seconds. The discrete PI controller used a sampling time of 100 seconds. Since the process was found to be open-loop unstable, the linear controller was tuned using the Ziegler-Nichols criteria, and the resulting parameters were subsequently fine-tuned, in order to avoid the appearance of oscillation in some operating zones; the final settings obtained were kc - 20 and I:/-- 1500 seconds. Besides being easier to tune, smaller amplitude changes are observed in the MPC profile, with a closer tracking of the optimal trajectory.

1.03 1

9

. . . . . . . . . . . . . . . . . . . . . . . . .

02

0.99

0.98

1.35 1.3

5,

1.25

ii!iiiii

~

1.2

1.15 I.i 1.05

6 . . . . 160. . . . 260. . . . 360. . . . 460. . . .

s~0

t (min)

Fig. 4. Output profile for the optimal policy of the reactor temperature (... PI; - MPC).

. . . .

1+o . . . .

2+o . . . . t (rain)

3~o . . . .

4+o

Fig. 5. Normalized inlet temperature jacket profile for the optimal profile of the reactor temperature (... PI; - MPC).

754 4.2. Effect of the initiator quantities: The composition of the initiator, and the best amount that should be added at the beginning of the operation, in order to produce a polymer with desired properties in minimum-time, were also the subject of optimization. The results are described in Table 1. In this case, both models also allow for important reductions in the cycle time, with a similar optimal initiator composition. The reduction obtained with the Xie model is smaller, because the initiation efficiency becomes diffusionaly controlled after the critical conversion. Significantly, no important reductions in the reaction time were found when the continuous addition of initiators was considered with this system. Additional optimization results, such as optimal profiles for polymers with improved properties, that cannot be formed with simple isothermal operation, can be found in [8].

Table 1 Optimization of the initial quantities of the initiators. Isothermal case Xie's model Total amount (mol) 18.0 19.4 Initiator A (%) 50 53 Initiator B (%) 50 47 Reduction in the cycle time (%) 7.2

Kiparissides's model 22.1 53 47 14.4

5. CONCLUSIONS The application of a nonlinear feasible path optimization strategy for the determination of optimal policies for batch processes was considered. The results obtained with a batch suspension polymerization reactor clearly illustrates the advantages and possible improvements in the operation of discontinuous processes, associated with a more generalized use of theses methodologies. REFERENCES

1. J.E. Cuthrell, L.T. Biegler, AIChE Journal, 33(8) (1987) 1257. 2. N.M.C. Oliveira, L.T. Biegler, Automatica, 31 (1995) 281. 3. L.O. Santos, N.M.C. Oliveira, L.T. Biegler, Proc. Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes (DYCORD+'95), (1995) 33. 4. T.Y. Xie, A.E. Hamielec, EE. Wood, D.R. Woods, J. Applied Polymer Science, 34 (1987) 1749. 5. T.Y. Xie, A.E. Hamielec, EE. Wood, D.R. Woods, Polymer, 32 (1991) 537. 6. T.Y. Xie, A.E. Hamielec, EE. Wood, D.R. Woods, Polymer, 32 (1991) 1098. 7. C. Kiparissides, G. Daskalatis, D.S. Achilias, E. Sidiropoulou, Ind. Eng. Chem. Res., 36(4) (1997) 1253. 8. D.C.M. Silva, N.M.C Oliveira, Proc. Controlo'2000, 4th Portuguese Conference on Automatic Control, (2000) 49.