^{1}

^{1}

^{2}

^{1}

^{3}

^{1}

^{1}

^{2}

^{3}

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.

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 [

MPC benefits have been validated on polymerization reactors [

One of the well-known MPC algorithms is dynamic matrix control (DMC), originally developed by Cutler and Ramaker [

use a linear model to predict future deviations from a set point over a prediction horizon forming a quadratic objective function to be minimized,

adjust a control horizon of manipulated input moves,

implement the first move and measure the resulting output at the next sample time,

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. [

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.

Dynamic matrix control (DMC) was developed at Shell Oil Company in 1979 [

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 [

In the original form of DMC strategy, the term

Predicted values

The controller tuning is carried out through the Integral of the Absolute value of the Error (IAE), defined by (

In (

The process considered in this paper is the solution copolymerization of methyl methacrylate with vinyl acetate within a continuous stirred tank reactor [

Basic process configuration [

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 steady-state operating point is reported in Table

Steady-state operating conditions.

Inputs | |

Monomer A (MMA) feed rate | |

Monomer B (VAc) feed rate | |

Initiator (AIBN) feed rate | |

Solvent (benzene) feed rate | |

Chain transfer (acetaldehyde) feed rate | |

Inhibitor (m-DNB) feed rate | |

Reactor jacket temperature | |

Reactor feed temperature | |

Purge ratio | |

Reactor parameters | |

Reactor volume | |

Reactor heat transfer area | |

Outputs | |

Copolymer production rate | |

Mole fraction of A in copolymer | |

Average molecular weight | |

Reactor temperature |

The presence of the recycle stream introduces disturbances in the reactor feed which affect the polymer properties. Congalidis et al. [

The feedforward control equations were obtained by writing component balances around the recycle addition point [

Equation (

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.

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. [

This system consists of six inputs (

The following four output variables of the process are analyzed separately: copolymer production rate (

The dynamic behavior of these four output variables is reported in Figure

Open-loop simulation to inhibitor disturbance.

Figure

Nonlinear behavior of the weight average molecular weight (

Lima et al. [

SISO control loops.

Loop | Manipulated variable | Controlled variable |
---|---|---|

1 | Reactor jacket temperature ( | Copolymer production rate ( |

2 | Monomers A/B feed rate ( | Mole fraction of A in copolymer ( |

3 | Reactor jacket temperature ( | Weight average molecular weight ( |

4 | Reactor jacket temperature ( | Reactor temperature ( |

Schematic representation of the controllers evaluated.

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.

In this problem, an inhibitor disturbance of 4 parts per 1000 (mole basis) in the fresh feed was considered. Table

Parameters of the DMC to the regulatory problem.

Parameters | Loop 1 | Loop 2 | Loop 3 | Loop 4 |
---|---|---|---|---|

Convolution horizon | 48 | 111 | 26 | 4 |

Prediction horizon (PH) | 2 | 5 | 9 | 2 |

Control horizon (CH) | 1 | 1 | 1 | 1 |

Suppression factor ( | 0.001 | 0.003 | 1.500 | 0.001 |

Reference trajectory parameter ( | 0.002 | 0.001 | 0.010 | 0.001 |

Sampling time ( | 0.25 | 0.25 | 0.25 | 0.25 |

IAE* | 6.946 | 0.121 | 23,803 | 1.744 |

*IAE units: loop 1 (kg/h), loop 2 (−), loop 3 (kg/kmol), and loop 4 (K).

Closed-loop and open-loop simulations for an inhibitor disturbance of 4 parts per 1000 (mole basis).

Manipulated variables profile for an inhibitor disturbance of 4 parts per 1000 (mole basis).

As can be observed in Table

According to the regulatory problem, the parameters used for the DMC in each control loop and the control errors are given in Table

Parameters of the DMC to the servo problem.

Parameters | Loop 1 | Loop 2 | Loop 3 | Loop 4 |
---|---|---|---|---|

Convolution horizon | 48 | 111 | 26 | 4 |

Prediction horizon (PH) | 2 | 5 | 9 | 2 |

Control horizon (CH) | 1 | 1 | 1 | 1 |

Suppression factor ( | 1.500 | 2.500 | 4.000 | 2.000 |

Reference trajectory parameter ( | 0.002 | 0.001 | 0.010 | 0.001 |

Sampling time ( | 0.25 | 0.25 | 0.25 | 0.25 |

IAE* | 99.245 | 11.976 | 209,180 | 365.420 |

*IAE units: loop 1 (kg/h), loop 2 (−), loop 3 (kg/kmol), loop 4 (K).

Closed-loop simulations for changes in set point.

Manipulated variables profile for changes in set point.

Table

Prediction horizon (PH) and suppression factor (

Dealing with the regulatory problem for an inhibitor disturbance of 4 parts per 1000 (mole basis) in the fresh feed, Figure

Sensitivity analysis parameters of DMC for regulatory problem.

Parameters | Loop 1 | Loop 2 | Loop 3 | Loop 4 |
---|---|---|---|---|

Convolution horizon | 48 | 111 | 26 | 8 |

Prediction horizon (PH)* | 2 | 5 | 9 | 2 |

Control horizon (CH) | 1 | 1 | 1 | 1 |

Suppression factor ( | 0.001 | 0.003 | 1.500 | 0.001 |

Reference trajectory parameter ( | 0.002 | 0.001 | 0.010 | 0.001 |

Sampling time (h) | 0.25 | 0.25 | 0.25 | 0.25 |

*Value for changes on the

**Value for changes on the PH.

PH sensitivity simulations for an inhibitor disturbance of 4 parts per 1000 (mole basis).

For example, by analyzing the servo problem for the weight average molecular weight, Figure

Sensitivity analysis parameters of DMC for servo problem.

Parameters | Loop 3 |
---|---|

Convolution horizon | 26 |

Prediction horizon (PH) | 9* |

Control horizon (CH) | 1 |

Suppression factor ( | 4.000** |

Reference trajectory parameter ( | 0.001 |

Sampling time (h) | 0.25 |

*Value for changes on the

**Value for changes on the PH.

Closed-loop simulations for the servo problem for sensitivity analysis on PH and

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.

The free radical kinetic mechanism shown in Table

Kinetic mechanism for deterministic model.

Initiation | Propagation |
---|---|

Termination by coupling | Termination by disproportionation | Inhibition |
---|---|---|

Chain transfer to monomer | Chain transfer to solvent | Chain transfer to agent |

Each of the kinetic rate constants shown in Table

Values for the Arrhenius factor

Kinetic and thermodynamic parameters for deterministic model.

Kinetic parameters |
---|

Thermodynamic parameters |

^{3}, ^{2} |

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:

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:

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:

These moments are the same for all reactors and depend only on the local reaction environment:

These pieces of equipment are modeled as first-order lags on the species concentrations with constant level:

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

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

Initial time of the evaluation period

Free radical.

Actual value

Desired output value

Future value

Predicted value

Set point

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.