Dynamic Real-Time Optimization of Industrial Polymerization

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Article

Dynamic Real-Time Optimization of Industrial Polymerization Processes with Fast Dynamics Karen Valverde Pontes, Inga Wolf, Marcelo Embiruçu, and Wolfgang Marquardt Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b00909 • Publication Date (Web): 28 Oct 2015 Downloaded from http://pubs.acs.org on November 4, 2015

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Industrial & Engineering Chemistry Research

Dynamic Real-Time Optimization of Industrial Polymerization Processes with Fast Dynamics Karen V. Pontes1,2*, Inga J. Wolf2, Marcelo Embiruçu1 and Wolfgang Marquardt2 1

Industrial Engineering Graduate Program - Federal University of Bahia, Brazil

2

AVT – Process Systems Engineering, RWTH Aachen, Germany

KEYWORDS dynamic real-time optimization, economic model-predictive control, polymerization reactor, quality control, grade transition

ABSTRACT This paper addresses real-time optimization strategies which can be readily implemented in industrial polymerization processes, even in case they show very fast dynamics. At the upper-layer dynamic and steady-state real-time optimization (D-RTO and RTO) are suggested and compared. A novel multi-stage formulation for the real-time dynamic optimization problem is introduced. It relies on a purely economic objective without additional stabilizing terms and facilitates an integrated treatment of a sequence of alternating dynamic and stationary operational stages. A case study shows that closed-loop D-RTO allows reducing off-spec material as well as exploiting or rejecting disturbances to maximize overall profit. The computational delay not only determines closed-loop performance but significantly impacts the profit margin. The results indicate that even 1 ACS Paragon Plus Environment


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the simplest variant of the investigated strategies can significantly improve economic performance, since the transitions can be completed much faster than in current industrial practice.

1. Introduction The highly competitive market demands just-in-time production of tailor-made products in agile plants to promptly satisfy customer needs. In industrial practice, however, product quality is controlled by tracking controllers which follow a constant reference derived from process design studies and operational experience. The polymer quality control problem has recently been approached by nonlinear model-predictive controllers (NMPC) with a tracking objective1-5. Process profitability, however, depends not only on minimum deviation from the target quality, but rather on the economic performance during phases of steady-state and transient operation. For strongly nonlinear processes with frequent product change-overs, such as polymerization processes, best economic performance while maintaining product quality cannot be realized by operational experience only, but requires the use of model-based operational strategies. In order to account for economic performance, the regulatory objective of NMPC can be replaced by an economic objective. This strategy has been termed dynamic real-time optimization6 or economic model-predictive control7-8. In the context of polymerization processes Zavala and Biegler9, for example, employ a dynamic real-time optimization (D-RTO) approach to a lowdensity polyethylene tubular reactor. Due to persistent, but slow dynamic disturbances, such as fouling, optimal steady-state targets are not suitable. The authors formulate a nonlinear constrained dynamic optimization (DO) problem with an economic objective and employ a full-discretization approach with a sparsity-exploiting interior point method for its solution. They assume that the optimization can be obtained at every controller sampling instant without computational delay. However, the DO problems encountered in industrial practice are usually too complex to be converged reliably and efficiently even within the controller sampling time. Hence, computational delay is always an issue regarding stability and performance10. Despite the significant progress 2 ACS Paragon Plus Environment

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made on fast real-time optimization algorithms (cf.

9,11-12

, for recent contributions, and

13-14

, for

comprehensive reviews), the D-RTO approach is rarely reported in industrial applications15 because of the high engineering effort, its complexity, and a lack of transparency for industrial operators. The D-RTO approach has been extended to a two-layer architecture (Figure 1) which separates the optimization and the control objectives into two time scales6,16. This is motivated by established two-layer real-time optimization (RTO) architectures, which combine an upper-layer economic optimizer and a lower-layer controller: the solution of an economic optimization problem subject to a rigorous steady-state model on the upper layer provides the set-points for an underlying (modelpredictive) controller on the lower layer, which adjusts the manipulated variables to track the setpoints in the presence of process disturbances and model mismatch (Figure 1- left). The economic optimization is triggered periodically if the plant is in (or rather sufficiently close to) steady-state, after the model has been updated by parameter estimation. Such RTO architectures are wellestablished for continuous processes operating at steady-state1,17-18. D-RTO explicitly accounts for transient behavior and hence requires the solution of a dynamic rather than a steady-state optimization problem based on a rigorous dynamic model on the upper-layer (Figure 1 - right). This two-layer architecture avoids the solution of the dynamic optimization (DO) problem during each sampling time of the controller, but rather solves it during the sampling time ∆ on the upper

layer, which is a multiple of the MPC sampling time ∆ on the lower layer.

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Objectives

Steady-state optimization

Measurements

Constraints

Set-points for MVs and CVs

MPC (tracking) CV measurements

Dynamic optimization

Measurements

Constraints

Reference Trajectories for MVs and CVs

MPC (tracking)

MVs and CVs

Process

Objectives

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Disturbances

CV measurements

MVs and CVs

Process

Disturbances

Figure 1: RTO (left) and D-RTO (right) architectures.

Though RTO has been successfully applied to many large-scale and complex plants in the chemical and petrochemical industries17,19, polymerization processes have been largely neglected, because they undergo frequent product change-overs. D-RTO strategies have recently been explored in process control research to address this issue. Wang et al.20 and Dünnebier et al. 21, for example, employ a hierarchical two-layer approach to the grade transition of industrial slurry polymerization reactors. The optimal reference trajectories, however, have been computed off-line and may become sub-optimal or even infeasible in the presence of disturbances or model uncertainty. Bath et al.22 and Sangwai et al.23 implement an on-line optimizing control coupled with periodic adjustment of the kinetic model of the laboratory bulk methyl methacrylate (MMA) polymerization reactor. Every 6 minutes, the model is adjusted emplyoing soft sensor predictions (Bath et al.24) before a new optimization run is trigged to suppress disturbances. The optimization problems, which minimizes a weighted sum of batch time and tracking terms, are solved by means of a genetic algorithm within a few minutes, which is possible because of the small size of the model (9 differential and 12 algebraic equations). Maneti and Rovaglio25 approach optimal operation of a large-scale polyethylene terephthalate (PET) plant, represented by a mathematical model with 1500 algebraic and differential equations, maximizing productivity and considering tracking terms as stabilizing penalty. The solution of the economic optimization problem takes about 50 minutes. It is computed 4 ACS Paragon Plus Environment

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on the upper-layer every 2 hours. Würth et al.

15

address grade transitioning for an industrial

polymerization reactor. On the upper-layer, a rigorous dynamic optimization problem is solved at a slow time-scale to capture time-varying model mismatch and process disturbances requiring 10 to 20 minutes of computational time. The objective combines tracking and productivity terms, while polymer properties are targeted by inequality constraints. On the lower-layer, a fast neighboringextremal controller is used for trajectory tracking and disturbance rejection. The DO problem is triggered only if deviations from the optimal trajectories are detected. Hartwich26 reported the implementation of a two-layer architecture on a pilot semi-batch polymerization reactor. At the lower layer, a linear time-varying MPC is used, and, at the upper layer, a DO problem with economic objective is solved using an adaptive control vector parameterization and switching structure detection method. In order to ensure stationary operation for polymer production with the purely economic objective function, the steady-state optimum is computed separately. Since the solution of the optimization problem took much longer (about 650 s) than the sampling time (20 s), the feasible-path optimization was stopped when the available time expired. Hence, a sub-optimal but feasible result could be provided. For all these contributions, the bottleneck is the computation of the DO problem at the upper-layer. Würth et al.15 address this problem and propose a more realistic two-layer architecture, where the computational delay of the rigorous optimization problem is accounted for on the upper layer. The objective of the present paper is to propose strategies for maximizing economic performance while maintaining product quality within specification for large-scale multi-grade polymerization processes with frequent grade transition and fast dynamics. Since implementation in an industrial environment is a key requirement for any of the strategies to be presented, only mature technologies which can easily be communicated to operators should be considered. The two-layer architectures depicted in Figure 1 constitute a promising starting point, because they rely on time-scale separation and they reflect current industrial practice of hierarchical process control systems27. At the upper



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layer, two real-time algorithms are considered to handle computational delay which cannot be neglected due to fast process dynamics and high computational complexity. The first algorithm considers the computational delay prior to implementing the new optimal trajectory as suggested by Würth et al.15. The influence of the computational delay on closed-loop performance is investigated (Figure 1, right) to assess requirements for fast algorithms relying a.o. on warm-start strategies. The second algorithm employs steady-state rather than dynamic optimization on the upper layer (Figure 1, left). Unlike in conventional RTO17, the steady-state optimization is triggered periodically and not only after steady-state has been detected, thus neglecting deliberately the impact of transient modes possibly present on the optimization result. This real-time strategy will be termed adapted RTO (A-RTO) to indicate the similarity to conventional RTO. The contribution of this paper in contrast to previous studies is the application of a fully nonlinear dynamic real-time optimization algorithm to a challenging problem, i.e., to the optimal operation of a solution polymerization plant with very small dominant time constant. Further, alternative approaches are suggested and investigated to deal with undesirable effects resulting from computational delay in D-RTO implementation in industrial practice. In particular, a novel multistage formulation for the real-time DO problem is introduced. In contrast to existing literature15,2526,28-33

, it casts all operational stages into a single optimization problem with a purely economic

objective, i.e., without additional penalty terms for stabilization. This approach facilitates an integrated treatment of a sequence of alternating dynamic and stationary operational stages. The paper is organized as follows. The process is presented first in the following Section 2. Section 3 formulates the dynamic optimization problem and discusses some benefits of the suggested approach compared to other strategies in literature. Section 4 presents the investigated two-layer DRTO architectures in detail, followed by some results, which also investigate the computational delay, since the case study is a large-scale process with fast dynamics. Finally, conclusions are discussed in Section 5, stressing the benefits of each strategy investigated. 6 ACS Paragon Plus Environment

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2. Process Description and Hierarchical Control Architecture The process considered in this work is the polymerization of ethylene in solution with a ZieglerNatta catalyst. The reaction takes place in a non-ideal continuous stirred tank reactor (CSTR) in series with a plug-flow reactor (PFR) as sketched in the flowsheet of Figure 2, which further illustrates the distributed control strategy proposed. The feed is a mixture of a chain transfer agent, CTA ( ), the monomer ( ) and a solvent. The total feed is split in two streams, the main feed is introduced at the bottom and a side feed is fed at the top of the reactor in order to enhance mixing. ), according to The melt index (MI) is a measure of the polymer average molecular weight (

, where and are parameters estimated from industrial data. A detailed

description of the process model and the operation of the reactor has been presented by Pontes et al. 34

and Embiruçu et al.35 and is – for the sake of brevity – not repeated here.

Figure 2: Flowsheet of industrial polymerization process with the proposed control strategy.



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The target controlled variable for polymer property control is the melt index (MI), which is difficult to measure reliably online at high frequency. In current industrial practice, the reactor temperature is therefore used as an inferential control variable due to its significant correlation with MI. All states are assumed to be accessible, either by measurements or state estimation. The temperature is constrained to ensure safe operation and existence of a single phase in both reactors. Based on process knowledge, sensitivity analysis36 and steady-state optimization34,37, the concentrations of CTA ( ) and monomer ( ) in the feed are used as set-points of the manipulated variables for underlying PID ratio controllers to comply with industrial practice. Despite its negligible effect on reaction temperature and profit, CTA has a great impact on the molecular weight and consequently on the MI of the polymer, which is the only quality target considered in this case study. Monomer not only influences MI but also temperature and profit; therefore, it has to be manipulated to follow the optimal trajectories to implement profitable process operation. The process is represented by a dynamic model involving 148 differential and 2435 algebraic equations and variables to describe the coordination polymerization in a series of reactors. Figure 3 illustrates a step response when the monomer concentration in the feed is increased by 20% at tstep. Despite the observed delay time which is due to transport delay and reactor holdup, a typical grade transition only takes about 2 to 5 minutes in total. This fast dynamics presents a challenge for the implementation of any D-RTO scheme. The steady-state process model employed in A-RTO has been taken from Pontes et al.34. It comprises a set of differential-algebraic equations with the axial coordinate along the tubular reactor as the independent variable with 12 differential and 1265 algebraic equations and variables.


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Figure 3: Open-loop step response to a 20 % increase of the monomer concentration in the feed: MI (left); temperature (right).

3. Off-line dynamic optimization The models sketched in the previous section are used to compute economically optimal operating conditions. The dynamic optimization problem is based on a multi-stage formulation with an economic objective, which covers phases (or stages) of transient and stationary operation. Figure 4 illustrates a typical production sequence of three polymer grades ∈ , , . For each

product, there is a transition (Tj) and a production (Pj) stage. The production schedule as well as the demand of each produced grade is assumed to be known a priori. If we assume that the production stages Pj are long enough, each one can be subdivided into three intervals, where the produced polymer satisfies all the quality constraints: the first interval ∆tA,j denotes the optimal path to the steady state (SSj), the second interval ∆tSS,j corresponds to the steady state and the third interval ∆tB,j denotes the optimal trajectory to the successive transition stage. The optimal steady state during the second interval SSII is independent of the grade specifications of the prior and the following production stages. Consequently, the intervals ∆tSS,j with steady-state operation decouple the time intervals with transient operation. In particular, the optimal trajectory of transition stage TI is decoupled from the optimal trajectory of transition stage TII by the steady-state stage SSII. To reduce computational complexity, a receding horizon with exactly one transition stage and at most two neighboring production stages is thus considered in the implementation of the upper-layer 9 ACS Paragon Plus Environment


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optimization of the D-RTO architecture. More specifically, the horizon starting at a production stage Pj, ∈ , , has to cover three stages: the production stage Pj, the transition stage Tj and the

production stage Pj+1. If the optimization is trigged during a transition Tj, the optimization horizon has to cover two stages: the transition stage Tj and the production stage Pj+1. If the optimization is trigged during the last production stage, a single-stage problem results.

ΔtA,I

ΔtSS,I

ΔtA,II

ΔtB,I

ΔtSS,II

ΔtB,II ΔtA,III

ΔtSS,III ΔtB,III

MI

SSI

1

PI

SSII

TI

PII

=

TII

]

SSII

PIII

^ t

Figure 4: Typical production sequence with three grades and two transitions

The multi-stage dynamic optimization (DO) problem is then formulated in case of three stages as

s.t.

min

, ,! , , ,!

Φ#

$% # &'$ # , # , (# , )*, + ∈ 1,2,3, /# 0'$ # , # , (# , )*, + ∈ 1,2,3, $ 1 $ 1 , $ # # − $ #3 # 0 + ∈ 1,2, $ #,56 ≤ $ # ≤ $ #,6 ∈ 8#9 , # : + ∈ 1,2,3 ; ; ; $ #,56 ≤ $ # ≤ $ #,6 ∈ 8#9 , # : + ∈ 1,3, ; ; ; ; ; $ =,56 ≤ $ ≤ $ =,56 + ? if $ =,56 > $,6 , < ; ; ; $ =,6 − ? ≤ $ ≤ $ =,6 otherwise IJ;

# # # + 1,3, 56 ≤ # ≤ 6 ∈ K#9 , # L, + ∈ 1,2,3,

( 1) (1a) (1b) (1c) (1d) (1e) (1f) (1g) (1h) (1i)

where $ M ∈ X ⊆ ℝQR are state variables, /M ∈ Y ⊆ ℝQT are algebraic output variables,

U# ∈ U ⊆ ℝQW are degrees of freedom, X# ∈ D ⊆ ℝQZ are disturbances, ) ∈ P ⊆ ℝQ\ are

parameters, + is the stage index and ? is a tolerance. While Eqs. (1a) and (1b) represent the 10 ACS Paragon Plus Environment

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nonlinear system, Eq. (1d) maps the final states of stage + to the initial condition of the states of the next stage + + 1, where $ 1 is the initial conditions of the states in the first stage (1c). The subscripts

_` and ` represent the lower and upper bounds, respectively. The superscripts a and b indicate

process (e.g. temperature) and quality (e.g. MI) variables, respectively. Eq. (1e) denotes path constraints (for reactor temperature, for example) to ensure safe operation. Eqs. (1f) refers to path constraints on quality variables. Eq. (1g) is an end-point constraint targeting the quality variables of the second grade within a given tolerance. Eq. (1h) fixes polymer produced during production stages + ∈ 1,3, where # # is the accumulated amount of product at time # and #

IJ;

is the

required amount of product. At the economic optimum, the amount of off-spec material produced during the transition stage ( + 2 ) will always be minimal. Additional constraints on the

manipulated variables such as feed concentrations of CTA and monomer (cf. Figure 2) are given by Eqs. (1i). The solution to this problem results in the optimal trajectories of the manipulated variables # in stages + ∈ 1,2,3 as well as in the optimal switching times, , and = .

The objective function is chosen to be consistent with the one of the steady-state optimization problem formulated previously34,37, i.e., m

=

Qk

Φ c de f# ∙ h# − e `k ∙ k l , n

#i #j

ki

( 2)

where f# is the polyethylene sales price (€/kg), h# is the polymer production rate (kg/s), `k is the

cost of raw-material o (€/kg), po is the number of raw-materials and k is the mass flow rate of

component o (kg/s). In particular, the costs of monomer, catalyst, co-catalyst, solvent and hydrogen are considered. The sales price of polyethylene is a binary function, i.e., on-spec polymer sells at

1.2€/kg, while off-spec polymer has to be reprocessed or disposed, such that f −0.8.

Therefore, the amount of off-spec material will always be minimal as stated above. Note, that this objective is different from the one used by Prata et al.38, who suggested a similar multi-stage



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problem but chose an objective comprising a weighted sum of transition time, mass of off-spec material and raw-material consumption, because less raw-material consumption does not necessarily mean higher profitability34. The solution polymerization takes place in an adiabatic reactor where the inlet temperature does not show great influence on MI and outlet temperature11; therefore, energy and utility costs have been neglected. In contrast to the tracking objective, the economic objective ensures that all available dynamic degrees of freedom are used to maximize the profit of the plant on a given time horizon6,39. However, some authors point out that a purely economic objective on a finite horizon may lead to ill-posed dynamic optimization problems, since the uniqueness of the solution cannot be guaranteed9,40. Uniqueness and stationary operation at the end of a transition can be ensured by either (i) including a tracking term in the objective function9,25,41-43 or by (ii) introducing a terminal constraint for a specific steady-state26,32,43, which might be computed previously by a decoupled steady-state optimization problem26,42,44. The multi-stage formulation addressed here, on the other hand, solves a purely economic objective function, casting stationary and transient operation into a single optimization problem45. There is no need to set terminal constraints, neither to include tracking terms into the objective function. The resulting problem is solved by DyOS46, a software tool developed at AVT – Process Systems Engineering, RWTH-Aachen University, which is based on single-shooting with adaptive control vector parameterization47-48. The availability of such a sophisticated dynamic optimization tool facilitates the application of the suggested multi-stage approach. The differential-algebraic process model (Eqs. (1a) and (1b)) is implemented in gPROMS49, which is interfaced to DyOS. The controls # are discretized by piecewise-constant basis functions on a not necessarily equidistant

grid. The results in the following Section 3.1 compare and discuss different discretizations. The

path constraints are relaxed, such that they only need to be satisfied at discretization grid points47.


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Constraint violations may occur between grid points, which can only be reduced by a higher resolution of the path constraint grid. 3.1 Results The solution to the multi-stage DO problem (1) for different numbers of discretization intervals ps# for the control variables on every stage + is presented in the following. Three runs with a

fixed number of discretization intervals are compared: t0 82, 2, 2:, t0 810, 2, 10: and t0

816, 2, 16: are compared to an adaptive choice of grid points48. These four cases are denoted by A, B, C and D, respectively. Special attention is given to computational effort which determines the delay in real-time implementation. The resulting optimal trajectories are compared to step responses to the optimal steady-state (OS for short) after setting the controls to the values of the optimal steady-state of the following production stage. Figure 5 illustrates the set-point trajectories IJv to the underlying controller, i.e., the concentrations

of monomer and CTA, when t0 82, 2, 2:, as well as the flow rates w delivered by the PID

ratio controllers at the regulatory level (cf. Figure 2). The horizontal lines refer to the bounds according to Eqs. (1e), (1f) and (1g). Since the switching times , and = are degrees of freedom,

the required amount of polymer at each production stage, and = = , determine the

optimal duration of the corresponding stages, represented by vertical dotted lines in Figure 5. The transition occurs as fast as possible since off-spec polymer ( ) reduces profit. The finer

discretization used in case C significantly reduces the transition time and the amount of off-spec

polymer, but at the expense of a highly oscillatory behavior shown in Figure 6, which may not be realizable. This undesired behavior can be accommodated by some kind of regularization of the optimal solution, i.e. by imposing a penalty such as constraints on the rate of change of the manipulated variables, or an appropriate choice of the discretization. We decided for the latter option and employed an adaptive choice of the grid points47 in Case D, which offers effective



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regularization and a small optimality gap at the same time. This approach relies on a repetitive solution of increasingly refined finite-dimensional optimization problems. The grid points of each variable are inserted or deleted based on a-posteriori wavelet analysis47. The adaptation strategy

used in case D determines the discretization t0 814, 12, 12: for CTA and t0 812, 8, 12: for

monomer. The results are similar to case C and are not shown for the sake of brevity. As Table 1 shows, there is a tradeoff between computational load and profit. Pragmatically, the discretization scheme t0 82, 2, 2: is chosen for on-line implementation. Note that the computational time can

be significantly reduced in case of on-line implementation if warm-start strategies are used. Hence, we will later conservatively assume that the computational time can be reduced to 80 s for this discretization scheme in on-line implementation. The optimization results evidence the good performance of the DO problem formulation, which accounts for both, dynamic and steady-state operational phases simultaneously in a comprehensive approach.

Figure 5: Open-loop trajectories for set-point trajectories of CTA and monomer concentrations, respectively. Multi-stage approach – case A, where the discretization grid is t0 82, 2, 2:. Vertical dashed lines indicate the optimal duration of each stage. Horizontal lines indicate bounds.


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Figure 6: Open-loop trajectories for set-point trajectories of CTA and monomer concentrations, respectively. Multi-stage approach – case C, where the discretization grid is t0 816, 2, 16:. Vertical dashed lines indicate the optimal duration of each stage. Horizontal lines indicate bounds.

Table 1: Computational performance of off-line multi-stage dynamic optimization for various grids (cases A to D) compared to OS approach.

Case A B C D OS

ps(PC) 2+2+2 10+2+10(5[3]) 16+2+16(2[3]) adaptation ---

ΔΦv --12.73% 13.85% 13.85% -55.73%

[1]

[2]

CPU (s) 127.6 1228.7 2879.6 6209.9 4.49

Stage 1 Stage 2 ∆

∆

440.0 3.001 86.8 0.603 441.8 3.001 36.4 0.252 441.2 3.001 31.2 0.221 440.4 3.001 31.3 0.222 405.0 2.999 305.5 2.141

Stage 3 ∆=

= 'v * 438.4 3.001 428.5 3.001 430.9 3.001 434.7 3.001 443.0 3.002

[1]

Gain in profit based on case A. All computations are carried out on a computer with Intel Xenon 2 Core (3.47GHz, 3.46GHz) and 96Gb RAM. [3] Grid number for path constraints. [2]

The step response to the optimal steady-state (OS) is compared with the results of the multi-stage DO (case A) in Figure 7. The step to the optimal steady-state is triggered when the desired amount of polymer has been produced, i.e., when

IJ;

3.00, which results in 405 s. The

vertical dotted lines delimit the production and the transition stages of OS: the transition starts as 15 ACS Paragon Plus Environment


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soon as the desired amount of polymer,

IJ;

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, has been produced; the transition ends when the MI

of the following grade is reached. As expected, the DO approach not only reaches the target MI range faster than the OS approach, but also shows an accumulated profit which is 55.73 % higher (Table 1). However, the computational time of the off-line OS approach is just 4.49 CPU seconds, while the one for the off-line DO approach is 127.6 CPU seconds.

Figure 7: Open-loop trajectories: dynamic optimization (DO) versus optimal step (OS).

4. Real-time Optimization Strategies for Closed-Loop Process Control The control strategy proposed for the ethylene polymerization reactor (Figure 2) is presented in Figure 8 employing a two-layer architecture according to Figure 1. Figure 8 indicates that a decision maker sets the production schedule as well as the desired amounts of each grade produced. When the desired amount of the current grade has been achieved, the economical optimization (problem (1)) is solved at the upper-layer to provide reference trajectories of the manipulated (IJv ) and controlled variables ( / IJv ) to the underlying controller for the next grade to be produced. The

objective function for on-line optimization is the same as in problem (1), the decision variables


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becomes the manipulated variables and the outlet temperature and MI are the controlled variables. Note that feedback is introduced on both layers, because the initial conditions (or other uncertain parameters) are adjusted in every sampling interval. While this grade is produced, the necessary conditions of optimality (NCO) are checked in order to detect at every controller sampling instant, if the manipulated and controlled variables deviate from the reference trajectories due to slowly varying disturbances. If the NCO are violated more than a user-specified tolerance, the optimization is triggered on the upper-layer in order to update the reference trajectories. The controller on the lower layer should be set up in a manner which is consistent with the upper-layer optimization50. Note, that the architecture sketched in Figure 8 neglects updates of the model parameters and state estimation for the sake of simplicity.

Decision maker

False

1

NCO Trigger

Production schedule Desired amount of polymer production

/, $, (

/, $, (

True

Optimizer

Optimal trajectories IJv , / IJv , $ IJv

1

Controller /, $

/~

Process

(

Figure 8: Control architecture proposed for the polymerization reactor (assuming perfect models and measurements of all states and disturbances).

For the A-RTO architecture, the trigger cannot be considered to initiate an economic optimization on the upper layer, since the necessary conditions of optimality are not available in this setting. Rather, a steady-state optimization is only carried out to prepare for a scheduled grade transition after the desired amount of the currently produced polymer has been achieved. In particular, the optimal set-points U{ , |{ , }{ for the following production stage are computed rather than the 17 ACS Paragon Plus Environment


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optimal trajectories IJv , / , $ for the transition and the following production stages. The reference trajectories for transitioning are then constructed as a ramp linking the process variables at the current time 1 (U , | , } ) to the optimal stationary process variables of the following production

stage ({ , / { , $ { ):

IJv −

− ∙ − 1 , ∈ 81 , 1 + Δ :, Δ

IJv , > 1 + Δ .

( 3) ( 4)

Δ is the estimated transition time for OS, which is estimated from the off-line results shown in

Table 1.

4.1 D-RTO at the upper layer According to Figure 8, the DO problem is triggered at either a scheduled grade change (A-RTO and D-RTO) or when the NCO are not fulfilled (D-RTO only). We will distinguish two cases in the following regarding the computational delay introduced by the upper-layer optimization. Negligible computational delay If a scheduled transition has to be computed at time 1 , a three-stage dynamic optimization problem

(Section 3.1) is solved starting from a production stage, say PI, resulting in optimal trajectories schematically shown in Figure 9a. If another grade change will be scheduled, a new three-stage

problem must be formulated and solved, say in the production stage PII for ∈ 8 , = : at , as illustrated in Figure 9b. Given the choice of the horizon only covering a sequence of a first

production stage, a subsequent transition stage and a second production stage, the optimal

trajectories in the time interval Δ, computed at 1 are typically different from those computed at

, whereas the first part of the trajectories in the first production and the following transition stages

computed at 1 remain (largely) the same. Although it would be sufficient to compute the trajectories for the second production stage only at = − Δ, , since Δ, is unknown, this 18 ACS Paragon Plus Environment

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strategy is infeasible. If no further grade change were scheduled, i.e., the plant is supposed to be operated continuously producing the current grade at steady state for ≥ ] , a steady-state

optimization problem could be solved at ] to fix the operating conditions for the production

stage corresponding to ≥ ] . a) 1

b)

PI IJv ,

1

Δ,

PII

Δ,

PI

IJv ,

1

Δ,

= PIII

PII

Δ, =]

^

Figure 9: a) Optimal trajectories computed at t 1 ; b) Optimal trajectories computed at t .

Deviation of states and disturbances from their reference values results in parametric uncertainty of the optimization problem, which can be monitored by a criterion which measaures the violation of the NCO computed at every controller sampling instant using the current (measured or estimated) values of the uncertain parameters15.

More precisely, an optimization run is triggered if the

sensitivities of the Lagrangian (optimality condition) or the nonlinear constraint (feasibility condition) with respect to the uncertain parameters are violated more than a user-specified tolerance. For a complete presentation of this strategy the reader should refer to the original work of Würth et al.15. 19 ACS Paragon Plus Environment


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This way, the optimal trajectories are regularly updated in the presence of disturbances with a sampling frequency (much) lower than that of the lower-layer controller. Fast disturbances are

rejected by the lower-layer controller, executed at a smaller sampling period ∆ , which is chosen to

guarantee desired control performance for given disturbance characteristics and process dynamics. Note that any slow disturbances (like trends due to fouling of a heat exchanger, for example) or sudden major events (like the loss of cooling capacity, for example) which influence the economic objective significantly, are taken into account on the upper-layer. Non-negligible computational delay Despite the use of fast optimization algorithms tailored to D-RTO or economic NMPC14, the computational time may not be neglected since the large-scale polymerization process shown in Figure 2 presents very fast dynamics. Even though fast (suboptimal) algorithms including warmstart strategies are considered to reduce the computational time, the optimization might not be solved within the controller sampling time period. Therefore, the computation of the new trajectories should be trigged some sampling times in advance for a scheduled grade transition. If Δ is the maximal expected computational time, the optimization should start at { −

Δ as sketched in Figure 10a. Since the reference trajectories have to be implemented at , the state of the system $ at has to be predicted. To this end, a forward simulation from { to has

to be carried out (as also suggested in fast NMPC algorithms, e.g., by 51), maintaining the controls at their previously computed values to ensure stability in closed-loop despite the computational delay10. Figure 10a illustrates the new reference trajectories by a dashed line, which is only available at although its calculation started at { . Likewise, if the upper-layer controller is

triggered at ∈ 81 , due to disturbances, a forward simulation has to be done first to predict

∗ the state of the process $ at

+ Δ , as illustrated by Figure 10b. The gray dotted

line indicates the measured output, which deviates from the reference trajectory due to disturbances.


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∗ although its As before, the optimal trajectory, represented by the dashed line, is only available at

calculation started at . During the computational delay no new optimization should start, even if a disturbance occurs within this period. If this happens, the reference trajectory available at or

∗ would become sub-optimal and, if the NCO are violated significantly, a new optimization run is

trigged considering the current states and disturbances. For non-persistent disturbances, therefore, the computational delay might deteriorate the closed-loop performance, as the results in Section 4.4 illustrate. An approach based on stationary real-time optimization, such as the A-RTO approach of this work, might then be considered as an alternative.

a)

1

Δ

IJv

t0 b)

}

{

t3

Δ

IJv

t0

t4

t5

t

}

∗

t

Figure 10: Delayed optimal trajectories trigged a) by decision maker, b) by optimality criteria. Black lines denote reference trajectories while gray lines measured outputs.

4.2 Lower-layer controller At the lower-layer, the controller tracks the reference trajectory with a higher sampling rate at the shorter sampling time ∆ . In general, this controller can vary from a simple PID to a linear, time-

varying linear or even a non-linear MPC. Control performance can be significantly improved if an 21 ACS Paragon Plus Environment


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optimal controller is used on the lower-layer, the formulation of which is consistent with the upperlayer controller. A neighboring-extremal control strategy15,39,52 would fulfill this requirement appropriately. Since the focus of the present paper is on established methods which can be readily used in industry, a suboptimal regulatory MPC is employed which catches process non-linearities5354

rather than a neighboring extremal controller which would be consistent50. On each control

horizon, the non-linear model is linearized on a pre-specified grid along the reference trajectories to obtain a linear discrete-time model describing the effect of the manipulated variables on the controlled variables. The suboptimal regulatory MPC then minimizes the quadratic deviation from the reference trajectories ( , / ) subject to the linear discrete-time model to determine the movements of the manipulated variables.

4.3 Implementation of the two-layer D-RTO algorithm The implementation of the suggested two-layer D-RTO algorithms is based on the software tool OptoEcon-Toolbox, which has been developed at AVT – Process Systems Engineering, RWTHAachen University, to realize multi-layer optimization and control architectures55. It is based on Matlab, which communicates with other software tools via OPC. The upper layer optimization is also carried out by DyOS, which has already been used for off-line optimization in Section 3. The underlying controller is implemented in Matlab. The plant replacement model and the model used for upper layer optimization are the same; this large-scale DAE system is implemented in the commercial simulator gPROMS. 4.4 Results and discussion This section presents and discusses some results obtained with the two-layer architecture, either implementing D-RTO or A-RTO, when controlling polymer quality of the large-scale polymerization process with fast dynamics introduced in Section 2. A production schedule of three grades, I → II → III, is assumed with desired amounts of production of 1.000 kg, 3.500 kg and


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2.000 kg, respectively. Since computational delay has to be considered for any on-line implementation, the closed-loop performance is investigated under different delay times, which depend on the type NMPC algorithm14 used in an actual implementation. In this context, four scenarios are considered: 1. D-RTO without computational delay (Case 1); 2. D-RTO with computational delay of 80 s, using warm-start strategies (Case 2); 3. D-RTO with computational delay of 120 s, using warm-start strategies (Case 3); 4. A-RTO (Case 4). Figure 11 illustrates the profiles of the manipulated and controlled variables for Case 1 for the disturbances acting on the process, as displayed in Figure 12a (continuous line), and the estimate of the disturbance used by the optimization (dashed line). All values are normalized due to confidentiality reasons. Vertical lines indicate the time points, where an optimization is triggered, either by a scheduled grade transition or by the trigger criteria based on the NCO. According to the discussion in Section 4, an optimization has to be carried out at which corresponds to

220 and 830 for Case 1. The D-RTO strategy satisfactorily compensates mild disturbances and, at 350, when there is a sudden disturbance on catalyst concentration, the

NCO triggers a new optimization. The optimization, then, updates the reference trajectory immediately based on the current catalyst concentration towards profit maximization.



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Figure 11: Profiles of controlled (top) and manipulated (bottom) variables for Case 1, D-RTO with negligible computational delay. Vertical dashed lines indicate time point when the optimization is triggered.

(b) (a) Figure 12: Disturbance profiles for Case 1 (a) and Case 2 (b). Vertical dashed lines indicate time point when the optimization is triggered. When computational delay is considered, the optimization has to start Δ earlier than where

the transition is triggered by the decision maker (Figure 10). If the NCO are violated, the

optimization promptly starts at . The state of the system has to be propagated forward by simulation for Δ based on the current state and reference trajectory. For Case 2, when 24 ACS Paragon Plus Environment

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Δ 80 , the optimization starts at 140 to be available at 220 , where the grade

transition is triggered. Since a disturbance occurs after 140 while the optimization is already

progressing (see Figure 12b), there is a mismatch in the predicted state, deteriorating the optimality of the new reference trajectory when it becomes available at 220 (Figure 13). Consequently, the NCO-based trigger immediately starts a new optimization. As the lower-layer controller was following a sub-optimal reference, the NCO are again violated at 320, such that another

optimization has to be triggered. Due to a sudden change in the catalyst inlet concentration just after

the optimization is started (see Figure 12b), the predicted state deviates from the actual state at

400 and the NCO-based trigger starts again a new optimization. The reference trajectory is then available at 480 when the controlled variables can follow reasonably well the reference

trajectory until the next scheduled grade change at 800. The oscillatory behavior illustrated in

Figure 13 might be reduced by a different controller tuning. For the sake of comparison between Case 1 and Case 2, though, the same tuning was employed.

Figure 13: Profiles of controlled (top) and manipulated (bottom) variables for Case 2, D-RTO with computational delay of 80 s. Vertical dashed lines indicate time point when the optimization is triggered. 25 ACS Paragon Plus Environment


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The trigger, therefore, may improve the closed-loop performance for D-RTO, especially with computational delay, since it detects slowly-varying uncertainties and performs an optimization towards profit maximization, thus rejecting or taking advantage of disturbances. Depending on the computational delay, though, due to the presence of disturbances, the process deviates from the predicted state, and the reference trajectory, when it has been computed, may become sub-optimal or infeasible, compromising the closed-loop performance.

The results obtained for A-RTO are shown in Figure 14. Much smoother profiles compared to the ones obtained for D-RTO with computational delay (Figure 13) are obtained, but the transition takes much longer to be completed.

Figure 14: Profiles of manipulated and controlled variables for Case D (A-RTO). Vertical dashed lines indicate time point when the optimization is triggered.

Table 2 summarizes the results for each case investigated, where the polymer amount is computed based on the time length of the production stage. The higher the computational delay, the higher is


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the off-spec amount produced, which is due to a slower transition. The deviations from the reference trajectories are indicated by the ISE (Integral Square Error) and ITAE (Integral Time Absolute Error) indexes. As expected, the profit is highest for D-RTO with negligible computational delay. The large profit losses occurring for the even small computational delay of 80 s clearly underlines the importance of the recent research efforts targeted at fast algorithms or online economic optimization reviewed by14. The negative profit in case of A-RTO is due to the short production stage (as a result of the small desired amount of polymer) compared to the duration of the grade transition, a case, which may not happen in practice. The performance loss in case of A-RTO is also caused by the lack of counteracting disturbances between grade changes, because the NCO-based trigger is not available.

It is important to point out that a typical grade transition in current industrial practice takes from 45 min to 1h, while, with the aid of the very simple A-RTO strategy, it can be reduced to less than 10 minutes, resulting in significant additional profit margins. The D-RTO can reduce the transition time even further to less than 2 minutes, highlighting the significant economic potential of D-RTO (or economic NMPC) strategies with fast algorithms, which allow for feedback with negligible computational delay.

Table 2: Comparison of the different real-time optimization approaches.

Off-spec (%) ISE ITAE (ton) (ton) (ton) (s) Cost (dim.) Revenue (dim.) Profit (dim.)

Case 1 17.5 0.028 1282 1.013 3.596 2.010 1160 5879 7750 1871

Case 2 29.6 134.3 148768 1.013 4.073 2.042 1190 6030 6739 709

Case 3 35.2 95.15 130442 1.013 5.580 2.061 1140 5796 6046 250

Case 4 49.1 3.752 3998 1.013 3.469 2.001 2020 10596 8056 -2539



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5. Conclusions This paper addresses the quality control of large-scale industrial polymerization processes which have to be operated in an economically optimal way during frequent grade transitions and at steadystate, towards excellence in operation in a competitive market. Therefore, the focus of this work is on state-of-the-art approaches, well-known in industrial practice, in order to assess suitable solutions which can be readily implemented on industrial (polymerization) plants with fast dynamics. Two approaches based on a two-layer architecture for real-time optimization are proposed, differing on how the optimal profiles provided to the underlying regulatory controller are computed, either by dynamic optimization (D-RTO) or by stationary optimization (A-RTO). This paper applies a rigorous D-RTO strategy to a challenging process with very fast dynamics for the first time and compares this strategy to a straightforward stationary real-time optimization approach in the context of computational delay. The novel formulation of the dynamic optimization problem presented in this work to implement the D-RTO strategy can be efficiently solved within few seconds for the large-scale model of the polymerization plant considered. The formulation of a multi-stage dynamic optimization problem is suggested to account for a production sequence targeting at different grades with frequent changes to implement an agile plant operation. The multi-stage problem minimizes a purely economic objective without the need of additional stabilizing terms to overcome ill-posedness during the whole production cycle, maintains the polymer properties via path constraints and accounts for additional path and point constraints to assure safe operation. Though the real-time optimization strategies have been illustrated and demonstrated for particular industrial polymerization process, it can be extended to any other polymerization or other chemical process system. The closed-loop results obtained for the case study illustrate the efficiency of the strategies proposed, even if computation delay may not be neglected. The simple A-RTO strategy significantly reduces the transition time, and hence improves profitability, compared to industrial 28 ACS Paragon Plus Environment

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practice. An even better improvement can be reached by D-RTO because it is based on dynamic optimization and thus can account for disturbances between grade changes. The case study nicely demonstrates that even the A-RTO strategy constitutes a promising and readily available solution to implement model-based optimization on multi-grade polymerization processes in case of fast transitions and in the presence of unpredictable disturbances.

6. Acknowledgments The authors acknowledge CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), FAPESB (Fundação

de

Amparo

à

Pesquisa

do

Estado

da

Bahia)

and

DAAD

(Deutscher

AkademischerAustauschdienst) for financial support.

7. Notation f

polyethylene sales price (€/ton)

CTA

chain transfer agent

`

∆

Δ

Φ

+

MI ps )

cost of raw-material (€/ton)

controller sampling time (lower-layer) optimization sampling time (upper-layer) objective function stage flow rate polymer melt index number of discretization points vector of parameters



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

P

production stage

SS

steady state of production

T

time transition stage

{

time to trigger scheduled transition

vector of controls (manipulated variables)

∗

time to trigger transition when the NCO are not fulfilled

polymer accumulated production

/

vector of algebraic output variables

$

Page 30 of 37

vector of state variables

Subscripts ref

reference trajectory for manipulated or controlled variables

`

upper bound

_`

a

b

lower bound

production variable quality variable

8. Literature Cited (1) BenAmor, S.; Doyle, F. J.; McFarlane, R.. Polymer grade transition control using advanced real-time optimization software. J. Proc. Cont. 2004, 14, 349; (2) Bindlish, R.; Rawlings, J. B. Target Linearization and Model Predictive Control of Polymerization Processes. AIChE J. 2003, 49, 2885; 30 ACS Paragon Plus Environment

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(3) Embiruçu, M.; Fontes, C. Multirate multivariable generalized predictive control and its application to a slurry reactor for ethylene polymerization. Chem. Eng. Sci. 2006, 61, 5754 (4) Ali, M. A.; Ali, E. M. Control of Polyethylene Properties Using nonlinear Model Predictive Control. In: Proceedings of UKACC 2008, Manchester, UK , 2008; (5) Ali, E. M.; Ali, M. A. Control of Molecular Weight Distribution and Density of Polyethylene in Gas Phase Reactors. J. Chem. Eng. Jap. 2010, 43, 880; (6) Helbig, A.; Abel, O.; Marquardt, W. Structural concepts for optimization based control of transient Processes. Proceedings of the Nonlinear Model Predictive Control Workshop Assessment and Future Directions, Ascona, Switzerland, 2000, 26, 295; (7) Zanin, A. C.; Tvrzska de Gouvea, M.; Odloak, D. Industrial implementation of a real-time optimization strategy for maximizing production of LPG in a FCC unit. Comp. Chem. Eng. 2000, 24, 525; (8) Zanin, A. C.; Tvrzska de Gouvea M.; Odloak, D. Integrating realtime optimization into the model predictive controller of the FCC system. Cont. Eng. Pract. 2002, 10, 819; (9) Zavala, V. M.; Biegler, L. T. Optimization-based strategies for the operation of low-density polyethylene tubular reactors: nonlinear model predictive control. Comp. Chem. Eng. 2009, 33, 1735; (10) Findeisen, R.; Allgöwer, F. Computational delay in nonlinear model predictive control. In: Proceedings of the Int. Symp. Adv. Control of Chemical Processes, ADCHEM'03, HongKong, 2004; (11) Diehl, M.; Bock, H.; Schlöder, J.; Findeisen, R.; Nagy, Z.; Allgöwer, F. Real-Time Optimization and Nonlinear Model Predictive Control of Processes Governed by DifferentialAlgebraic Equations. J. Proc. Cont. 2002, 12, 577; (12) Würth, L.; Hannemann, R.; Marquardt, W. Neighboring-extremal updates for nonlinear modelpredictive control and dynamic real-time optimization. J. Proc. Cont. 2009, 19, 1277;



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(13) Ellis, M.; Durand, M.; Christofides, P. D. A tutorial review of economic model predictive control methods. J. Proc. Cont. 2014, 24, 1156; (14) Wolf, I. J.; Marquardt, W. Fast NMPC schemes for regulatory and economic NMPC - a review. To be submitted to Journal of Process Control, 2015; (15) Würth, L.; Hannemann, R.; Marquardt, W. A two-layer architecture for economically optimal process control and operation. J. Proc. Cont. 2011, 21, 311; (16) Backx, T.; Bosgra, O.; Marquardt, W. Integration of model predictive control and optimization of processes. In: Proceedings of the IFAC ADCHEM, Pisa, Italy , 249, 2000; (17) Marlin, T. E.; Hrymak, A. N. Real-Time Operations Optimization of Continuous Processes. In: Chemical Process Control-V Conference, Tahoe City, Nevada , 1996; (18) Engell, S. Feedback control for optimal process operation. J. Proc. Cont. 2007, 17, 203; (19) Qin, S. J.; Badgwell, T. A. A survey of industrial model predictive control technology. Cont. Eng. Pract. 2003, 11, 733; (20) Wang, Y.; Seki, H.; Ohyama, S.; Akamatsu, K.; Ogawa, M. Optimal grade transition control for polymerization reactors. Comput. Chem. Eng. 2000, 24, 1555; (21) Dünnebier, G.; van Hessem, D.; Kadam, J.; Klatt, K. U.; Schlegel, M. Optimization and control of polymerization processes. Chem. Eng. Tech. 2005, 28, 575; (22) Bhat, S. A.; Saraf, D. N.; Gupta, S.; Gupta, S. K. On-Line Optimizing Control of Bulk Free Radical Polymerization Reactors under Temporary Loss of Temperature Regulation: Experimental Study on a 1-L Batch Reactor. Ind. Eng. Chem. Res. 2006, 45, 7530; (23) Sangwai, J. S. S.; Bhat, S. A.; Saraf, D. N.; Gupta, S. K. An experimental study on on-line optimizing control of free radical bulk polymerization in a rheometer–reactor assembly under conditions of power failure. Chem. Eng. Sci. 2007, 62, 2790; (24) Bhat, S. A.; Saraf, D. N.; Gupta, S.; Gupta, S. K. Use of Agitator Power as a Soft Sensor for Bulk Free-Radical Polymerization of Methyl Methacrylate in Batch Reactors. Ind. Eng. Chem. Res. 2006, 45, 4243; 32 ACS Paragon Plus Environment

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(25) Manenti, F.; Rovaglio, M. Integrated Multilevel Optimization in Large-Scale Poly(Ethylene Terephthalate) Plants. Ind. Eng. Chem. Res. 2008, 47, 92; (26) Hartwich, A. Adaptive Numerical Methods for Dynamic Real-Time Optimization of Chemical Processes. Fortschritt-Berichte VDI, Reihe 3, Nr. 935, VDI-Verlag, Düsseldorf German, 2013; (27) Tatjewski, P. Advanced control and on-line process optimization in multilayer structures. Annu. Rev. Control, 2008, 32, 71; (28) Cervantes, A. M.; Tonelli, S.; Brandolin, A.; Bamdoni, J. A.; Biegler, L. T. Large-scale dynamic optimization for grade transitions in a low density polyethylene plant. Comput. Chem. Eng. 2002, 26, 227; (29) Kiparissides, C.; Seferlis, P.; Mourikas, G.; Morris, A. J. Online Optimizing Control of Molecular Weight Properties in Batch Free-Radical Polymerization Reactors. Ind. Eng. Chem. Res. 2002, 20, 6120; (30) Zhang, J. A Reliable Neural Network Model Based Optimal Control Strategy for a Batch Polymerization Reactor. Ind. Eng. Chem. Res. 2004, 43, 1030. (31) Saliakas, V.; Chatzidoukas, C.; Krallis, A.; Meimaroglou, D.; Kiparissides, C. Dynamic Optimization of Molecular Weight Distribution Using Orthogonal Collocation on Finite Elements and Fixed Pivot Methods: An Experimental and Theoretical Investigation. Macromol. React. Eng. 2007, 1, 119; (32) Terrazas-Moreno, S.; Flores-Tlacuahuac, A.; Grossmann, I. E. Simultaneous Cyclic Scheduling and Optimal Control of Polymerization Reactors. AICHE J. 2007, 53, 2301; (33) Chen, X.; Chen, L.; Feng, J.; Yao, Z.; Qian, J. Weights and Polydispersity Index in Batch Free Radical Polymerization. Ind. Eng. Chem. Res. 2009, 48, 6739; (34) Pontes, K. V.; Maciel, R.; Embiruçu, M.; Hartwich, A.; Marquardt, W. Optimal Operating Policies for Tailored Linear Polyethylene Resins Production. AIChE J. 2008, 54, 2346; (35) Embiruçu, M.; Lima, E. L.; Pinto, J. C. Continuous Soluble Ziegler-Natta Ethylene Polymerizatons in Reactor Trains. I. Mathematical Modeling. J. App. Pol. Sci. 2000, 77, 1574; 33 ACS Paragon Plus Environment


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