Abstract
A model-based predictive control system is designed for a copolymerization reactor. These processes typically have such a high nonlinear dynamic behavior to make practically ineffective the conventional control techniques, still so widespread in process and polymer industries. A predictive controller is adopted in this work, given the success this family of controllers is having in many chemical processes and oil refineries, especially due to their possibility of including bounds on both manipulated and controlled variables. The solution copolymerization of methyl methacrylate with vinyl acetate in a continuous stirred tank reactor is considered as an industrial case study for the analysis of the predictive control robustness in the field of petrochemical and polymer production. Both regulatory and servo problems scenarios are considered to check tangible benefits deriving from model-based predictive controller implementation.
1. Introduction
Operations management of polymerization plants is quite complex, since such processes are characterized by strong nonlinearities, intense state variable interactions, and wide variations in operating conditions. Beyond these issues that are strictly related to the physical process, it is worth underlining that the field of polymer production, and generally speaking petrochemical plants, is undergoing more and more reduced net profit margins and frequent market dynamics, both making the operations management of production sites a problematic issue. Controlling polymer reactors and related operations has always been a challenging task even accounting for the fact that operators rely on conventional methods to do it, rather than model-based methodologies. This is mainly attributed to the lack for rigorous/detailed process knowledge, reliable kinetic models, and online measures of some specific properties of the final product.
Thanks to the increased computing power and the spread of detailed process modeling and programming tools, the use of rigorous models for generating accurate system predictions and for making use of predictive control methodologies is nowadays feasible for many processes. It is not a coincidence that many techniques based on rigorous models have emerged. The one that has become very popular in recent years is model predictive control (MPC).
MPC has been the most successful advanced control technique applied in the process industries. Its formulation naturally handles timedelays, multivariable interactions, and constraints [2–5]. The name MPC arises from use an explicit prediction model of the process to be controlled so as to foresee its future behavior. Such a capacity of predicting process behavior can be exploited to get the so-called online optimal control, where tracking error, that is, the difference between the predicted output and the desired reference trajectory, is minimized over a future horizon. For this reason, MPC strategy is also known as moving horizon control or receding horizon control, where the horizon which the output variable is predicted on is moved in a sampling instant in the direction of the future, each sampling instant.
MPC benefits have been validated on polymerization reactors [6–9] by providing good results (i.e., satisfactory tracking performance, robustness, ability to suppress disturbances, and accuracy under real-time constraints) without significantly increasing computational complexity and effort [10–15]. Some recent works also deal with stability and fault-tolerant design issues [16–18].
One of the well-known MPC algorithms is dynamic matrix control (DMC), originally developed by Cutler and Ramaker [19] and rigorously derived for linear systems [20]. A DMC procedure uses the system information in the context of an optimizer that solves the control problem for the trajectory of the manipulated variable over a future time horizon based on a dynamic model of the process. DMC basic ideas are [21] the following:(1)use a linear model to predict future deviations from a set point over a prediction horizon forming a quadratic objective function to be minimized,(2)adjust a control horizon of manipulated input moves,(3)implement the first move and measure the resulting output at the next sample time,(4)update the model and loop to (1).
The condition of the DMC in employing a linear model and a quadratic objective function results in a convex optimization problem easily solved by means of quadratic programming. There is a good deal of literature focusing on the application of DMC in polymerization processes. Peterson et al. [22] developed a control algorithm that uses an explicit nonlinear process model and the basic elements of the classical DMC and applied it to a semibatch polymerization reactor. Gobin et al. [23] applied a DMC algorithm to the polymerization of styrene in the presence of a binary initiator mixture, involving a cascade of two continuous stirred-tank reactors (CSTRs). Meziou et al. [24] have performed a simulation study to assess the performance of DMC for an ethylene-propylene-diene polymerization reactor. Yüce et al. [25] have investigated experimentally and by simulation the dynamic matrix control of a batch solution polymerization reactor. Lima [26] developed and implemented a DMC algorithm for a copolymerization process in a well-mixed jacketed tank reactor.
In this study, the application of DMC is investigated for a copolymerization reactor in solution to control the polymer production rate, the copolymer composition, the molecular weight, and the reactor temperature. Four monovariable control loops are designed and analyzed separately; multivariable strategies will be accounted for in future developments. The copolymerization of methyl methacrylate with vinyl acetate is considered as a case study. A nonlinear dynamic model of the system is used to simulate both regulatory and servo responses of DMC.
2. Dynamic Matrix Control
Dynamic matrix control (DMC) was developed at Shell Oil Company in 1979 [19]. The basic idea is to use a time-domain step-response model (called convolution model) of the process to calculate the future changes in the manipulated variable that will minimize a given performance index. In the DMC approach, we would like to have the PH (prediction horizon) future output responses matching some “optimum” trajectory by finding the “best” values of the CH (control horizon) future changes in the manipulated variables. This is exactly the concept of a least-squares problem of fitting the PH data points with an equation having CH coefficients. This is a valid least-squares problem as long as the PH is greater than the CH.
The aim of a predictive control law is to drive future outputs close to the reference trajectory. The computation sequence is first to calculate the reference trajectory and estimate the output predictions using the convolution model. Then, the errors between predicted and reference trajectories are calculated [27]. The next step is to estimate the sequence of the future controls by the minimization of an appropriate quadratic objective function . However, only the first predicted control action is really implemented. At this point, the data vectors are shifted so that the calculations can be repeated at the next sample instant. This function is defined by: where and are the time interval, is the output variable (controlled variable), is the input variable (manipulated variable), with , and is the suppression factor for the movements of the manipulated variable. This control parameter assures that no drastic control action is calculated. A too small results in large control actions, which can result in an instable response, while a too large results in a slow response. Thus, is even a weighting factor for the movements of the manipulated variable.
In the original form of DMC strategy, the term is the set point. In the present work, to prevent drastic control actions, a term is introduced based on Model Algorithmic Control strategy [28]. The desired output is calculated through an optimal trajectory defined by a first-class filter: where is the vector of the current measured value of the controlled variable at the sampling time , is the vector of the set point of the controlled variable at the sampling time , and is the reference trajectory parameter, with .
Predicted values in (1) can be obtained directly from a model of the process. However, since this model is never perfect because of unavoidably ideal assumptions and simplifications, the controller will be not sufficiently robust [8, 9]. Therefore, the following correction is applied: where is defined by a convolution model. It is considered that the difference between the predicted and actual values in the previous instant is valid for the current instant. Thus, the system reaches the desired value after successive corrections. Details on DMC and on obtaining the convolution model are given by Luyben [29].
The controller tuning is carried out through the Integral of the Absolute value of the Error (IAE), defined by (4), looking up the best combination of parameters (PH, CH, , ) that minimizes this performance criterion. Therefore, the following optimization problem is solved during tuning procedures:
In (4), and are the initial and final times, respectively, of the evaluation period.
3. Case Study: Copolymerization of Methyl Methacrylate with Vinyl Acetate
The process considered in this paper is the solution copolymerization of methyl methacrylate with vinyl acetate within a continuous stirred tank reactor [1]. Figure 1 is a qualitative process flow diagram of a copolymerization reactor including the recycle loop (streams 6, 8, and 2). Monomer A is methyl methacrylate, monomer B is vinyl acetate, the solvent is benzene, the initiator is azobisisobutyronitrile (AIBN), and the chain transfer agent is acetaldehyde. The monomer stream may also contain inhibitors such as m-dinitrobenzene (m-DNB).
Monomers A and B are continuously added with initiator, solvent, and chain transfer agent. In addition, an inhibitor may enter with the fresh feeds as an impurity. Feed streams (stream 1) are mixed to the recycle stream (stream 2) to give the reactor inlet flowrate (stream 3). The reactor is assumed to be a jacketed well-mixed tank. A coolant flows through the jacket to remove heat generated during the copolymerization process. Polymer, solvent, unreacted monomers, initiator, and chain transfer agent flow out of the reactor (stream 4) to enter the separator. Here, polymer is ideally separated (stream 5). Residual initiator and chain transfer agent are also removed in this step. In a real process, the separation process is particularly complex, as it often involves a series of steps, which may include dryers and distillation columns. We assume also unreacted monomers and solvent (stream 6) are recycled upstream, accounting for purge line only (stream 7), which represents vent and other losses. Purge line is required to prevent any accumulation of inerts within the system. After the purge, the monomers and solvent (stream 8) are stored in the recycle hold tank, which acts as a surge capacity to smooth out variations in the recycle flow and composition. The effluent (stream 2) recycle is then added to the fresh feeds.
The important reactor output variables to control polymer quality are the polymer production rate (), the mole fraction of monomer A in the copolymer (), the weight average molecular weight (), and the reactor temperature (). Inputs are the reactor flows of monomer A (), monomer B (), initiator (), chain transfer agent (), solvent (), inhibitor (), the reactor jacket temperature (), and the reactor feed temperature (). Reactor, separator, and hold tank contain pure solvent preheated to 353.15 K at startup.
The steady-state operating point is reported in Table 1. Under these conditions, the reactor residence time is h, and the overall reactor monomer conversion is 20%. These operating conditions ensure that the viscosity of the reaction medium remains moderate. Table 1 also indicates that the temperature of the reactor feed is practically equal to the reactor temperature , because in this work, we have chosen to simulate reactor operation with a preheated feed, where the source of heat removal is through the jacket.
4. Feedforward Control of Recycle
The presence of the recycle stream introduces disturbances in the reactor feed which affect the polymer properties. Congalidis et al. [1] implemented a feedforward controller in the process to compensate for these disturbances by manipulating the fresh feeds to maintain a constant feed composition and flow to the reactor. Feedforward control of the recycle stream enabled the designer to separate the control of the reactor from the rest of the process. Thus, the feedforward control is also applied in this work and the reactor can be analyzed separately.
The feedforward control equations were obtained by writing component balances around the recycle addition point [1, 9, 30]. For example, the mole balance for monomer A is:
Equation (5) is then solved for the fresh feed of monomer A since it is desired to keep to goal flow of monomer A to the reactor () constant:
Since only monomers A and B and solvent are present in the recycle, only these three components have feedforward control equations. The corresponding equations for fresh feeds of monomer B and solvent are:
If any feedforward control equation causes a fresh feed to go negative, the value of that fresh feed is set to zero.
5. Deterministic Model
This case study is described by a nonlinear deterministic mathematical model. This model consists of a set of algebraic and ordinary differential equations which formally replace the real plant for the controller implementation. The deterministic model, as well as the kinetic mechanism and initial concentrations, is explained in detail in Appendix. More information on the nonlinear model is given in Congalidis et al. [1] and Maner and Doyle [30].
6. Open-Loop Behavior and Selection of the Control Loops
This system consists of six inputs (, , , , , and ) and four outputs. The temperature of the reactor feed () is assumed to be constant and the purge ratio () is regulated by the feedforward controller. As pointed out in Table 1, the inhibitor feed rate is normally no flow. With these considerations, the algebraic equations in the deterministic model were solved and the ordinary differential equations were numerically integrated by means of Runge-Kutta algorithm type implemented in Fortran 90 language.
The following four output variables of the process are analyzed separately: copolymer production rate (), mole fraction of A in copolymer (), weight average molecular weight (), and reactor temperature ().
The dynamic behavior of these four output variables is reported in Figure 2, where the system undergoes an inhibitor disturbance of 4 parts per 1000 (mole basis) in the fresh feed. This disturbance is the same as that considered by Maner and Doyle [30]. The fresh feed corresponds to the addition of the molar outflows of monomers, initiator, solvent, and chain transfer agent in the system entrance.
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Figure 3 presents the dynamic trend of the weight average molecular weight () for symmetric step disturbances of in the reactor jacket temperature (), by highlighting process nonlinearities.
Lima et al. [31] developed a factorial planning using Statistica Version 7.0 Software to discriminate the higher impact variables on the process performance. According to this sensitivity analysis, the four selected SISO control loops are shown in Table 2. Figure 4 presents the location of each monovariable DMC loop and the feedforward controller. For the DMC control, the controlled variables are displayed in blue color, and the manipulated variables are displayed in red color.
7. Performance of the Dynamic Matrix Control
An algorithm for the proposed predictive controller was developed in Fortran 90 and further inserted in the simulation program. Both regulatory and servo mechanism problems are taken into account to check DMC performances. Each SISO control loop was tuned separately.
7.1. Regulatory Problem
In this problem, an inhibitor disturbance of 4 parts per 1000 (mole basis) in the fresh feed was considered. Table 3 shows the parameters used for DMC in each control loop together with the control errors for this specific configuration. A comparison between DMC and open-loop responses is given in Figure 5; the corresponding manipulated variables for the closed-loop simulation are shown in Figure 6. The computational effort is in the order of 3 s for each control loop run.
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As can be observed in Table 3 and Figure 5 the DMC structure performs satisfactorily, without abrupt changes and with small overshoots. Furthermore, Figure 6 shows that the manipulated variable behavior is quite acceptable presenting small oscillations, which is important for a stable operation.
7.2. Servo Problem
According to the regulatory problem, the parameters used for the DMC in each control loop and the control errors are given in Table 4. Figure 7 shows closed-loop performances of output variables, while the system undergoes a series of set point changes. The corresponding manipulated variables are given by Figure 8. The computational effort is in the order of 2 s for each control loop run.
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Table 4 and Figure 7 show that the performance of the DMC is fairly good with the polymer properties, since it is able to approach quickly the new set point. However, the shows a more complex dynamic response, which is characteristic of the highly interactive dynamics of solution polymerization reactors. Moreover, Figure 8 shows that the manipulated variable behavior is quite acceptable without drastic changes in reaching the control objective illustrated in Figure 7.
8. Effect of Control Parameters on Controller Performances
Prediction horizon (PH) and suppression factor () are the control parameters of greatest impact on the four output variables of the process, thus a sensitivity analysis of the control structures has been carried out for them.
Dealing with the regulatory problem for an inhibitor disturbance of 4 parts per 1000 (mole basis) in the fresh feed, Figure 9 and Figure 10 both show the closed-loop performance of the four controlled variables for changes in PH and in , respectively. The values of the other control parameters are listed in Table 5. Figures 9 and 10 point out that the increase in PH or in takes to a larger control error; however, random values of the IAE are observed for changes in the PH for the loop number 3. It is also shown that the increase in provides a larger damping of the system, as somehow expected. This can be explained from the analysis of the control objective function (1).
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For example, by analyzing the servo problem for the weight average molecular weight, Figure 11 shows the closed-loop performance of this output variable for changes in PH and in f. The values of the other control parameters are presented in Table 6. Figure 11 shows that the increase in PH provides a smaller control error, whereas the opposite effect occurs for changes in f. Moreover, the increase in f provides a larger damping of the system.
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9. Conclusions
The servo and regulatory performance of DMC, applied to a solution copolymerization jacketed reactor, has been analyzed. The simulation case-study was based on a nonlinear mathematical model that describes the liquid-full reactor. Closed-loop computer simulation results showed the successful behavior and the potential of the DMC methodology to reduce off-specifications during changes in copolymer production rate, mole fraction of monomer A in the copolymer, weight average molecular weight, and reactor temperature. From this perspective, performances of DMC methodology give the opportunity to move towards the so-called demand-driven production and, hence, to increase net operating margins of polymer plants by forcing the production to fast follow, when possible, the more and more frequent market dynamics and price/cost volatilities.
The DMC strategy showed robustness, having stable behavior for the four control loops even when large changes due to the optimization convergence are imposed on the system. Another important breakthrough of the analyzed control strategy is its capacity to deal with nonstationary and nonlinear features, which are typical of polymerization systems. This means that the whole procedure proposed in this work has significant potential for application in several industrial processes similar to the type considered here.
Appendix
A. Nonlinear Mathematical Model for the Copolymerization Reactor
A.1. Kinetic Mechanism
The free radical kinetic mechanism shown in Table 7 is postulated for the polymerization of monomers A and B in the presence of initiator (), solvent (), chain transfer agent (), and inhibitor () [1]. In this mechanism, and symbolize growing polymer chains containing units of monomer A and units of monomer B that terminate in A and B, respectively. represents a “dead” polymer chain containing units of monomer A and units of monomer B. In the calculation of the cross-termination rate constants, it has been assumed that and .
Each of the kinetic rate constants shown in Table 7 is computed by an Arrhenius form such as:
Values for the Arrhenius factor and the activation energy are given in Table 8 [30].
A.2. Material and Energy Balances
Assuming that the reaction occurs in a CSTR with no volume change in the reacting mixture, the following mole balances can be written for the monomers, the initiator, the solvent, the chain transfer agent, and the inhibitor:
The reactor feed volumetric flow rate, concentrations, and reactor residence time are calculated by:
Using the quasi steady-state assumption, the following expressions can be derived for the total reactor concentrations of the free radicals terminating in A or B:
These equations are coupled with the following reactor energy balance:
The instantaneous polymerization rate is:
A.3. Dead Copolymer Composition
The following mole balances can be written for the calculation of the molar concentrations of the monomers in the dead polymer:
The molar fraction of monomer A in dead polymer is calculated as follows:
A.4. Moments of the Molecular Weight Distribution of the Dead Copolymer
Assuming that the reaction occurs in a CSTR, the following expressions can be derived:
The number and weight average molecular weights of the dead copolymer are then computed by the following relationships:
A.5. Moments of the Molecular Weight Distribution of the Live Copolymer
These moments are the same for all reactors and depend only on the local reaction environment: where and denotes and , respectivly.
A.6. Separator and Hold Tank Balances
These pieces of equipment are modeled as first-order lags on the species concentrations with constant level:
Notation
| A: | Monomer A |
| : | Arrhenius pre-exponential factor |
| B: | Monomer B |
| : | Concentration, kmol/m³ |
| : | Heat capacity, kJ/kg/K |
| CH: | Control horizon |
| CSTR: | Continuous stirred tank reactor |
| DMC: | Dynamic matrix controller |
| : | Activation energy, kJ/kmol |
| : | Molar flow rate, kmol/s |
| : | Suppression factor |
| : | Mass flow rate, kg/h |
| : | Enthalpy, kJ/kmol |
| : | Initiator |
| IAE: | Integral of the absolute value of the error |
| : | Kinetic rate constant |
| : | Intermediate variable in molecular weight distribution calculations |
| : | Molecular weight, kg/kmol |
| MPC: | Model predictive control |
| : | Dead polymer |
| PH: | Prediction horizon |
| : | Volumetric flow rate, m³/s |
| : | Gas constant, kJ/kmol/K; reaction rate, kmol/m³/s; ratio |
| : | Reactivity ratio |
| : | Surface area, m²; solvent |
| : | Temperature, K; chain transfer agent |
| : | Time |
| : | Overall heat transfer coefficient, kJ/m²/s/K |
| : | Input variable of the process |
| : | Volume, m³ |
| : | Output variable of the process, mole fraction |
| : | Mole fraction |
| : | Intermediate variable in molecular weight distribution calculations |
| : | Inhibitor. |
| : | Reference trajectory parameter, intermediate variable in molecular weight distribution calculations |
| : | Intermediate variable in molecular weight distribution calculations |
| : | Intermediate variable in molecular weight distribution calculations |
| : | Initiator efficiency |
| : | Residence time |
| : | Molar concentration of monomer in polymer macromolecules |
| : | Molar purge fraction |
| : | Density, kg/m³ |
| : | Moment of molecular weight distribution |
| : | Monomer A |
| : | Monomer B |
| : | Termination by coupling |
| CL: | Closed-loop |
| : | Termination by disproportionation |
| : | Feed to the reactor, final time of the evaluation period |
| : | Hold tank |
| : | Initiator |
| : | Cooling jacket |
| : | Time instant |
| : | Number of B units in polymer chain |
| : | Number of A units in polymer chain |
| : | Initial value |
| : | Dead polymer, propagation |
| : | Number of B units in polymer chain |
| : | Reactor, number of A units in polymer chain |
| : | Solvent, steady-state value, separator |
| : | Chain transfer agent |
| : | Weight (average polymer property) |
| : | Inhibitor |
| 0: | Initial time of the evaluation period |
| (): | Free radical. |
| actual: | Actual value |
| : | Desired output value |
| Future: | Future value |
| Pred: | Predicted value |
| Set: | Set point |
Acknowledgments
The authors acknowledge FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for financial support.