- Email: [email protected]
Integration of Scheduling and Control Using Internal Coupling Models
Jiří Jaromír Klemeš, Petar Sabev Varbanov and Peng Yen Liew (Editors) Proceedings of the 24th European Symposium on Computer Aided Process Engineering – ESCAPE 24 June 15-18, 2014, Budapest, Hungary. Copyright © 2014 Elsevier B.V. All rights reserved.
Integration of Scheduling and Control Using Internal Coupling Models Jungup Parka, Juan Dua, Iiro Harjunkoskib, Michael Baldeaa,* a
McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 East Dean Keeton St., Stop C0400, Austin, TX 78712, USA b ABB AG Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany [email protected]
Abstract In this paper, we present a novel framework for the integration of scheduling and control of process systems. We introduce internal coupling models (ICMs), defined as (low-order) representations of the closed-loop input-output behaviour of the process under supervisory control. We explore the derivation of ICMs for a specific class of input-output linearizing nonlinear controllers. Then, we formulate the scheduling problem as a mixed-integer dynamic optimization under the constraints imposed by the ICM, aimed at finding the optimal setpoint trajectory for the supervisory controller. We illustrate these concepts with a case study, demonstrating that ICM-based scheduling has comparable performance to other scheduling approaches in the absence of plantmodel mismatch, and vastly outperforms them when mismatch is present. Keywords: Short-term Scheduling, Control, Integrated Scheduling and Control
1. Introduction Scheduling and control are essential components of the operation of chemical process systems, and it is anticipated that a closer interaction and integration of these operational layers can lead to important economic benefits (Harjunkoski et al., 2009). As the scheduling and control problems are closely related at the fundamental mathematical level, one would expect this integration process to be seamless, and to be supported by recent advances in communication technology, which simplify enterprisewide sharing of the relevant data and information. Yet, and in spite of industrial efforts and academic interest, the development of a comprehensive and rational framework for integrating scheduling and control continues to be confronted several theoretical (Harjunkoski et al., 2009) and organizational challenges (Shobrys and White, 2002), and remains an important open problem in process systems engineering. In this paper, we propose the use of an explicit low-order model of the closed-loop input-output behaviour of a process under supervisory control as the internal coupling model for integrating short-term scheduling and supervisory control. Within this novel framework, the scheduling layer relies on the closed-loop process model to perform dynamic scheduling in a computationally efficient fashion. The result of the scheduling calculation is a sequence of setpoint changes, which are then imposed in the process by the supervisory controller, with the latter being designed to produce a well-defined closed-loop behaviour which is captured by the low-order model. This is possible owing to, i) the time scale separation between supervisory control and distributed control in a process, (i.e., the supervisory controller acts over a longer time horizon), as well as, ii) the proximity in time scale between supervisory control and short term scheduling (see,
530
J. Park et al.
e.g., Baldea and Daoutidis, 2012). Our approach allows scheduling and control calculations to be performed transparently and independently using a decompositionbased strategy, increasing solution speed as well as the robustness and resilience of the integrated system. The proposed concepts are illustrated using a case study concerning a four-product reactor with highly nonlinear behaviour.
2. Internal Coupling Models for Integration of Scheduling and Control 2.1. Scheduling Problem Formulation We rely on the developments in (Flores-Tlacuahuac and Grossmann, 2006) to formulate the cyclical process scheduling problem as described below. The objective is to maximize overall production profit (Eq.(1)).
§
NP
NP
max ¨ ¦ CiPWi / Tc ¦ ®Ciinv Gi Wi TC
©i1
i 1
¯
NS
¦ tp
i ,k
k 1
½· / 2¾ ¸ ¿¹
(1)
where the first term represents profit and the second term stands for inventory cost. Index ݅ represents the product number and index ݇ represents the time slots for the production cycle. ܰ is the total number of products and ܥ is the product price. The decision variables are the total production time (ܶ ), the production time for product ݅ in time slot ݇ (ݐǡ ) and the optimal production sequence (ݖǡ ). Eq.(2) enforces that only one product can be manufactured in each time slot. Eq.(3) states that within one production cycle, the product can be manufactured only once.ݖǡ is a binary variable that determines the production of product ݅ in time slot ݇. ܰௌ represents the number of time slots, which is the same as the total number of products. NP
¦z
i ,k
1, k
(2)
1, i
(3)
i 1 NS
¦z
i ,k
k 1
Eq.(4) defines the amount of the product ݅ (ܹ ) produced as a product of the production time (σ ݐǡ ) and the production rate (ܩ ). Eq.(5) enforces that the manufactured amount is greater than the demand rate of the product (ܦ ) times the total production time (ܶ ).
Wi
NS
Gi ¦ tpi , k , i
(4)
k 1
Wi t DiTC , i
(5)
The timing relation in Eq.(6) computes the time at the end of slot ݇ (ݐ ) as the sum of the start time (ݐ௦ ), the processing time (σ ݐǡ ) and the transition time (߬ ). Eq.(7) states that the start time of each slot is the end time of the previous slot. Eq.(8) enforces that the end time of each slot is not greater than the total production time. tke
Np
tks W k ¦ tpi ,k , k
(6)
i 1
tks
tke1 , k z 1
(7)
Integration of Scheduling and Control Using Internal Coupling Models
tke d Tc ,
k
531
(8)
2.2 Internal coupling model (ICM) We note that the formulation of the scheduling problem above depends on the dynamics of the process through the transition times ߬ . Intuitively, there are two ways to determine ߬ : i) via process experiments, after which the transition times become part of a transition table, which makes the problem static MI(N)LP and, ii) via a rigorous dynamic simulation of the process. Here, the full dynamic model of the process must be included as a set of model constraints (Flores-Tlacuahuac and Grossmann, 2011) and the scheduling problem becomes a mixed-integer dynamic optimization (MIDO). Both approaches have advantages and drawbacks. The static scheduling formulation is easier to solve but is agnostic to process dynamics and control performance. On the other hand, the second approach, which we will refer to as “full dynamic scheduling” presents the advantage of rigorously accounting for the process dynamics in calculating the transition times and the corresponding control moves. On the downside, it is computationally intensive. Further, computing the control moves offline for the entire makespan leaves the system vulnerable to disturbances and plant-model mismatch. In order to mitigate the aforementioned challenges, we propose a novel approach based on an internal coupling model (ICM), which captures the closed-loop, input-output behaviour of the process. The process is now assumed to be under a supervisory control system (potentially part of a hierarchical control structure including a distributed/base layer of control). We define the ICM as the explicit function relating the setpoints of the supervisory controller to the measured process outputs that are of interest to scheduling (e.g., product grade, production rate), of the form:
f ysp t
y t
(9)
Previous research (Baldea and Daoutidis, 2012) suggests that this overall input-output behaviour of process systems evolves over a longer time horizon, which is in the order of magnitude of the scheduling time horizon. It is thus this behaviour that should be accounted for in scheduling calculations. ICM-based scheduling therefore proceeds similar to full dynamic scheduling, with the exception that the full process model is replaced with the ICM, and the decision variables that are related to process ௦ manipulated variables are replaced by the setpointsݕ of the supervisory controller. The solution thus consists of the optimal setpoint sequence that imposes, via the supervisory controller, the optimal production sequence for the original problem, i.e., Np
y
sp k
¦y
ss i i ,k ,
z
k
(10)
i 1
௦
whereݕ୩ represents the setpoint to be tracked by the state variable ݕin each time slot ݇. ݕ୧௦௦ is the operating condition for producing product ݅Ǥ Using the ICM presents several important advantages. First, scheduling becomes aware of the process dynamics. Second, the process operates in closed loop and is thus robust to disturbances and plant-model mismatch. Third, the scheduling and control calculations are performed independently and using a low-order ICM can significantly
532
J. Park et al.
ease the computational burden of scheduling. Evidently, the above are dependent on the availability of a closed-form expression for the closed-loop input-output behaviour of the process. In the present work, we rely on using input-output linearizing controllers with integral action (see, e.g., Daoutidis and Kravaris, 1994) for imposing a wellr r defined (multivariable) linear closed loop behaviour of the form ¦W d y y (where r sp dt t i 1 is the relative order and ߬ are tunable time constants), which will be used as the ICM for scheduling. In what follows, we illustrate these concepts with a case study.
3. Case Study We consider a non-isothermal multiproduct CSTR with four products manufactured at different operating conditions. Our goal is to maximize overall production profit while meeting the manufacturing demand for each product. The scheduling problem thus consists of determining the total production time, the optimal production sequence and the processing times for each product. We solve this problem following the three approaches described above, i.e., static scheduling, full dynamic scheduling, and dynamic scheduling using an internal coupling model. The full dynamic scheduling problem for this system has been formulated and solved by Flores-Tlacuahuac and Grossmann (2006) and we follow closely their developments in that direction, as well as using the same model parameters as in their paper. The static scheduling problem is solved assuming a constant transition time ߬ = 10 h for all transitions. Let us now focus on the development of the ICM for this process. The process model is given in Eq.(11) and (12), where ݕଵ is the dimensionless concentration, and ݕଶ stands for the dimensionless temperature. dy1 dt
dy2 dt
1 y1
W
y f y2
W
(11)
k10e N y2 y1
k10e N y2 y1 D u y2 yc
(12)
In Eq.(12),߬represents reactor residence time, ݇ଵ is the pre-exponential factor, and ܰ is the activation energy. In Eq.(13), ݕ denotes the dimensionless feed temperature, ݕ is the dimensionless coolant temperature, ߙ is the dimensionless heat transfer area. The coolant flow rate ݑis the manipulated variable available for changing the output of interest, which in this case is the composition ݕଵ . The relative order of this system is = 2. We design a nonlinear input-output linearizing controller with integral action to impose a critically damped second-order input-output behaviour: 2 W CM
d 2 y1 dy 2W CM 1 y1 2 dt dt
y1sp
(13)
with ߬ெ = 2 h (note that this value was chosen so that the step response of the closedloop system reaches steady state in about 10 h, which is the same as the transition time used for static scheduling. ICM-based scheduling is thus formulated using the above equation as a constraint, aiming to determine the optimal setpoint profile in Eq.(10).
Integration of Scheduling and Control Using Internal Coupling Models
533
4. Results and Discussion 4.1. Optimal Solutions to Scheduling Problem Formulations The static, full dynamic and ICM-based scheduling problems were solved using GAMS/CPLEX. The dynamic optimization problems were reformulated as MINLPs using a full-discretization approach as described in (Flores-Tlacuahuac and Grossman, 2006). The optimal solutions are shown in Tables 1–4. The optimal solution for ICMbased scheduling was validated via simulation on the closed-loop system using the derived input-output linearizing controller. 4.2. Performance in the Presence of Model Uncertainty The aforementioned results were obtained assuming no plant-model mismatch, i.e., that the dynamic model used is perfect. Here, we assume that the reaction rate constant has been overestimated in the model by 10 % compared to the plant. The control actions computed using full dynamic scheduling were imposed on the mismatched plant. Clearly, in the absence of feedback control, most of the products are off-spec. Then, we imposed the setpoint sequence computed from IBO-based scheduling on the closed-loop Table 1. Optimal Solutions to Three Different Scheduling Formulation Problems
Case Proposed Method Static Full dynamic
Profit 36.615 36.615 34.218
Production Sequence 4→2→3→1 2→3→4→1 2→1→3→4
Total Production Time 119.585 h 119.585 h 124.016 h
Table 2. Optimal Solution for IBO-based Scheduling
݇ (Time Slot) ߬ ݅ (Product) ܹ ݐǡ
1 10 1 35.100 (i = 4, k = 1) 18.35
2 10 2 21.525 (i = 2, k = 2) 16.064
3 10 3 20.330 (i = 3, k = 3) 15.171
4 10 4 27.526 (i = 1, k = 4) 30
3 10 3 20.330 (i = 4, k = 3) 18.350
4 10 4 27.526 (i = 1, k = 4) 30
3 12.291 3 21.083 (i = 3, k = 3) 15.733
4 10.937 4 27.594 (i = 4, k = 4) 18.396
Table 3. Optimal Solution for Static Scheduling
݇ (Time Slot) ߬ ݅ (Product) ܹ ݐǡ
1 10 1 35.100 (i = 2, k = 1) 16.064
2 10 2 21.525 (i = 3, k = 2) 15.171
Table 4.Optimal Solution for Full Dynamic Scheduling
݇ (Time Slot) ߬ ݅ (Product) ܹ ݐǡ
1 10 1 35.100 (i = 2, k = 1) 16.659
2 10 2 22.323 (i = 1, k = 2) 30
y1
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J. Park et al.
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1
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Figure 1. Process response using full dynamic (left) and IBO-based (right) scheduling in the presence of plant-model mismatch. Dashed lines represent the target values of the variables.
system using the input-output linearizing controller. While each of the products is initially off-spec, feedback control with integral action compensates for plant-model mismatch and helps recover product purity (Figure 1, right).
5. Conclusions In this paper, we proposed a novel path for integrating short term scheduling and supervisory process control. We defined the internal coupling model (ICM) as a loworder model that describes the closed-loop input-output behaviour of the process, and derived ICMs for a specific class of supervisory controllers. We then formulated the scheduling problem as a MIDO, in terms of identifying the optimal sequence of setpoints for the supervisory controller, subject to the closed loop dynamics described by the ICM. Using a case study, we showed that ICM-Based scheduling is comparable to full dynamic scheduling in the nominal cases, and outperforms it in terms of on-spec production when plant-model mismatches are present.
Acknowledgements Financial support from ABB Corporate Research is gratefully acknowledged.
References M.Baldea, P. Daoutidis, 2012, Dynamics and Nonlinear Control of Integrated Process Systems, Cambridge University Press, Cambridge, UK. P. Daoutidis, K. Kravaris, 1994, Dynamic output feedback control of minimum-phase multivariable nonlinear processes, Chem. Eng. Sci., 49, 433–447. A. Flores-Tlacuahuac, I. Grossmann, 2006, Simultaneous cyclic scheduling and control of a multiproduct CSTR, Ind. Eng. Chem. Res., 45, 6698–6712. A. Flores-Tlacuahuac, I. Grossmann, 2011, Simultaneous cyclic scheduling and control of tubular reactors: Parallel production lines, Ind. Eng. Chem. Res., 50, 8086–8096. I. Grossmann, 2005, Enterprise-wide optimization: A new frontier in process systems engineering, AIChE J., 51, 7, 1846–1857. I. Harjunkoski, R. Nystrom, A. Horch, 2009, Integration of scheduling and control – Theory or practice?, Comput. Chem. Eng., 33, 12, 1909–1918. D. Shobrys, D. White, 2002, Planning, scheduling and control system: why cannot they work together, Comput. Chem. Eng., 26, 2, 149–160.
Integration of Scheduling and Control Using Internal Coupling Models Jungup Parka, Juan Dua, Iiro Harjunkoskib, Michael Baldeaa,* a
McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 East Dean Keeton St., Stop C0400, Austin, TX 78712, USA b ABB AG Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany [email protected]
Abstract In this paper, we present a novel framework for the integration of scheduling and control of process systems. We introduce internal coupling models (ICMs), defined as (low-order) representations of the closed-loop input-output behaviour of the process under supervisory control. We explore the derivation of ICMs for a specific class of input-output linearizing nonlinear controllers. Then, we formulate the scheduling problem as a mixed-integer dynamic optimization under the constraints imposed by the ICM, aimed at finding the optimal setpoint trajectory for the supervisory controller. We illustrate these concepts with a case study, demonstrating that ICM-based scheduling has comparable performance to other scheduling approaches in the absence of plantmodel mismatch, and vastly outperforms them when mismatch is present. Keywords: Short-term Scheduling, Control, Integrated Scheduling and Control
1. Introduction Scheduling and control are essential components of the operation of chemical process systems, and it is anticipated that a closer interaction and integration of these operational layers can lead to important economic benefits (Harjunkoski et al., 2009). As the scheduling and control problems are closely related at the fundamental mathematical level, one would expect this integration process to be seamless, and to be supported by recent advances in communication technology, which simplify enterprisewide sharing of the relevant data and information. Yet, and in spite of industrial efforts and academic interest, the development of a comprehensive and rational framework for integrating scheduling and control continues to be confronted several theoretical (Harjunkoski et al., 2009) and organizational challenges (Shobrys and White, 2002), and remains an important open problem in process systems engineering. In this paper, we propose the use of an explicit low-order model of the closed-loop input-output behaviour of a process under supervisory control as the internal coupling model for integrating short-term scheduling and supervisory control. Within this novel framework, the scheduling layer relies on the closed-loop process model to perform dynamic scheduling in a computationally efficient fashion. The result of the scheduling calculation is a sequence of setpoint changes, which are then imposed in the process by the supervisory controller, with the latter being designed to produce a well-defined closed-loop behaviour which is captured by the low-order model. This is possible owing to, i) the time scale separation between supervisory control and distributed control in a process, (i.e., the supervisory controller acts over a longer time horizon), as well as, ii) the proximity in time scale between supervisory control and short term scheduling (see,
530
J. Park et al.
e.g., Baldea and Daoutidis, 2012). Our approach allows scheduling and control calculations to be performed transparently and independently using a decompositionbased strategy, increasing solution speed as well as the robustness and resilience of the integrated system. The proposed concepts are illustrated using a case study concerning a four-product reactor with highly nonlinear behaviour.
2. Internal Coupling Models for Integration of Scheduling and Control 2.1. Scheduling Problem Formulation We rely on the developments in (Flores-Tlacuahuac and Grossmann, 2006) to formulate the cyclical process scheduling problem as described below. The objective is to maximize overall production profit (Eq.(1)).
§
NP
NP
max ¨ ¦ CiPWi / Tc ¦ ®Ciinv Gi Wi TC
©i1
i 1
¯
NS
¦ tp
i ,k
k 1
½· / 2¾ ¸ ¿¹
(1)
where the first term represents profit and the second term stands for inventory cost. Index ݅ represents the product number and index ݇ represents the time slots for the production cycle. ܰ is the total number of products and ܥ is the product price. The decision variables are the total production time (ܶ ), the production time for product ݅ in time slot ݇ (ݐǡ ) and the optimal production sequence (ݖǡ ). Eq.(2) enforces that only one product can be manufactured in each time slot. Eq.(3) states that within one production cycle, the product can be manufactured only once.ݖǡ is a binary variable that determines the production of product ݅ in time slot ݇. ܰௌ represents the number of time slots, which is the same as the total number of products. NP
¦z
i ,k
1, k
(2)
1, i
(3)
i 1 NS
¦z
i ,k
k 1
Eq.(4) defines the amount of the product ݅ (ܹ ) produced as a product of the production time (σ ݐǡ ) and the production rate (ܩ ). Eq.(5) enforces that the manufactured amount is greater than the demand rate of the product (ܦ ) times the total production time (ܶ ).
Wi
NS
Gi ¦ tpi , k , i
(4)
k 1
Wi t DiTC , i
(5)
The timing relation in Eq.(6) computes the time at the end of slot ݇ (ݐ ) as the sum of the start time (ݐ௦ ), the processing time (σ ݐǡ ) and the transition time (߬ ). Eq.(7) states that the start time of each slot is the end time of the previous slot. Eq.(8) enforces that the end time of each slot is not greater than the total production time. tke
Np
tks W k ¦ tpi ,k , k
(6)
i 1
tks
tke1 , k z 1
(7)
Integration of Scheduling and Control Using Internal Coupling Models
tke d Tc ,
k
531
(8)
2.2 Internal coupling model (ICM) We note that the formulation of the scheduling problem above depends on the dynamics of the process through the transition times ߬ . Intuitively, there are two ways to determine ߬ : i) via process experiments, after which the transition times become part of a transition table, which makes the problem static MI(N)LP and, ii) via a rigorous dynamic simulation of the process. Here, the full dynamic model of the process must be included as a set of model constraints (Flores-Tlacuahuac and Grossmann, 2011) and the scheduling problem becomes a mixed-integer dynamic optimization (MIDO). Both approaches have advantages and drawbacks. The static scheduling formulation is easier to solve but is agnostic to process dynamics and control performance. On the other hand, the second approach, which we will refer to as “full dynamic scheduling” presents the advantage of rigorously accounting for the process dynamics in calculating the transition times and the corresponding control moves. On the downside, it is computationally intensive. Further, computing the control moves offline for the entire makespan leaves the system vulnerable to disturbances and plant-model mismatch. In order to mitigate the aforementioned challenges, we propose a novel approach based on an internal coupling model (ICM), which captures the closed-loop, input-output behaviour of the process. The process is now assumed to be under a supervisory control system (potentially part of a hierarchical control structure including a distributed/base layer of control). We define the ICM as the explicit function relating the setpoints of the supervisory controller to the measured process outputs that are of interest to scheduling (e.g., product grade, production rate), of the form:
f ysp t
y t
(9)
Previous research (Baldea and Daoutidis, 2012) suggests that this overall input-output behaviour of process systems evolves over a longer time horizon, which is in the order of magnitude of the scheduling time horizon. It is thus this behaviour that should be accounted for in scheduling calculations. ICM-based scheduling therefore proceeds similar to full dynamic scheduling, with the exception that the full process model is replaced with the ICM, and the decision variables that are related to process ௦ manipulated variables are replaced by the setpointsݕ of the supervisory controller. The solution thus consists of the optimal setpoint sequence that imposes, via the supervisory controller, the optimal production sequence for the original problem, i.e., Np
y
sp k
¦y
ss i i ,k ,
z
k
(10)
i 1
௦
whereݕ୩ represents the setpoint to be tracked by the state variable ݕin each time slot ݇. ݕ୧௦௦ is the operating condition for producing product ݅Ǥ Using the ICM presents several important advantages. First, scheduling becomes aware of the process dynamics. Second, the process operates in closed loop and is thus robust to disturbances and plant-model mismatch. Third, the scheduling and control calculations are performed independently and using a low-order ICM can significantly
532
J. Park et al.
ease the computational burden of scheduling. Evidently, the above are dependent on the availability of a closed-form expression for the closed-loop input-output behaviour of the process. In the present work, we rely on using input-output linearizing controllers with integral action (see, e.g., Daoutidis and Kravaris, 1994) for imposing a wellr r defined (multivariable) linear closed loop behaviour of the form ¦W d y y (where r sp dt t i 1 is the relative order and ߬ are tunable time constants), which will be used as the ICM for scheduling. In what follows, we illustrate these concepts with a case study.
3. Case Study We consider a non-isothermal multiproduct CSTR with four products manufactured at different operating conditions. Our goal is to maximize overall production profit while meeting the manufacturing demand for each product. The scheduling problem thus consists of determining the total production time, the optimal production sequence and the processing times for each product. We solve this problem following the three approaches described above, i.e., static scheduling, full dynamic scheduling, and dynamic scheduling using an internal coupling model. The full dynamic scheduling problem for this system has been formulated and solved by Flores-Tlacuahuac and Grossmann (2006) and we follow closely their developments in that direction, as well as using the same model parameters as in their paper. The static scheduling problem is solved assuming a constant transition time ߬ = 10 h for all transitions. Let us now focus on the development of the ICM for this process. The process model is given in Eq.(11) and (12), where ݕଵ is the dimensionless concentration, and ݕଶ stands for the dimensionless temperature. dy1 dt
dy2 dt
1 y1
W
y f y2
W
(11)
k10e N y2 y1
k10e N y2 y1 D u y2 yc
(12)
In Eq.(12),߬represents reactor residence time, ݇ଵ is the pre-exponential factor, and ܰ is the activation energy. In Eq.(13), ݕ denotes the dimensionless feed temperature, ݕ is the dimensionless coolant temperature, ߙ is the dimensionless heat transfer area. The coolant flow rate ݑis the manipulated variable available for changing the output of interest, which in this case is the composition ݕଵ . The relative order of this system is = 2. We design a nonlinear input-output linearizing controller with integral action to impose a critically damped second-order input-output behaviour: 2 W CM
d 2 y1 dy 2W CM 1 y1 2 dt dt
y1sp
(13)
with ߬ெ = 2 h (note that this value was chosen so that the step response of the closedloop system reaches steady state in about 10 h, which is the same as the transition time used for static scheduling. ICM-based scheduling is thus formulated using the above equation as a constraint, aiming to determine the optimal setpoint profile in Eq.(10).
Integration of Scheduling and Control Using Internal Coupling Models
533
4. Results and Discussion 4.1. Optimal Solutions to Scheduling Problem Formulations The static, full dynamic and ICM-based scheduling problems were solved using GAMS/CPLEX. The dynamic optimization problems were reformulated as MINLPs using a full-discretization approach as described in (Flores-Tlacuahuac and Grossman, 2006). The optimal solutions are shown in Tables 1–4. The optimal solution for ICMbased scheduling was validated via simulation on the closed-loop system using the derived input-output linearizing controller. 4.2. Performance in the Presence of Model Uncertainty The aforementioned results were obtained assuming no plant-model mismatch, i.e., that the dynamic model used is perfect. Here, we assume that the reaction rate constant has been overestimated in the model by 10 % compared to the plant. The control actions computed using full dynamic scheduling were imposed on the mismatched plant. Clearly, in the absence of feedback control, most of the products are off-spec. Then, we imposed the setpoint sequence computed from IBO-based scheduling on the closed-loop Table 1. Optimal Solutions to Three Different Scheduling Formulation Problems
Case Proposed Method Static Full dynamic
Profit 36.615 36.615 34.218
Production Sequence 4→2→3→1 2→3→4→1 2→1→3→4
Total Production Time 119.585 h 119.585 h 124.016 h
Table 2. Optimal Solution for IBO-based Scheduling
݇ (Time Slot) ߬ ݅ (Product) ܹ ݐǡ
1 10 1 35.100 (i = 4, k = 1) 18.35
2 10 2 21.525 (i = 2, k = 2) 16.064
3 10 3 20.330 (i = 3, k = 3) 15.171
4 10 4 27.526 (i = 1, k = 4) 30
3 10 3 20.330 (i = 4, k = 3) 18.350
4 10 4 27.526 (i = 1, k = 4) 30
3 12.291 3 21.083 (i = 3, k = 3) 15.733
4 10.937 4 27.594 (i = 4, k = 4) 18.396
Table 3. Optimal Solution for Static Scheduling
݇ (Time Slot) ߬ ݅ (Product) ܹ ݐǡ
1 10 1 35.100 (i = 2, k = 1) 16.064
2 10 2 21.525 (i = 3, k = 2) 15.171
Table 4.Optimal Solution for Full Dynamic Scheduling
݇ (Time Slot) ߬ ݅ (Product) ܹ ݐǡ
1 10 1 35.100 (i = 2, k = 1) 16.659
2 10 2 22.323 (i = 1, k = 2) 30
y1
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J. Park et al.
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1
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60 80 time (hour)
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20
40
60 80 time (hour)
100
120
20
40
60 80 time (hour)
100
120
Figure 1. Process response using full dynamic (left) and IBO-based (right) scheduling in the presence of plant-model mismatch. Dashed lines represent the target values of the variables.
system using the input-output linearizing controller. While each of the products is initially off-spec, feedback control with integral action compensates for plant-model mismatch and helps recover product purity (Figure 1, right).
5. Conclusions In this paper, we proposed a novel path for integrating short term scheduling and supervisory process control. We defined the internal coupling model (ICM) as a loworder model that describes the closed-loop input-output behaviour of the process, and derived ICMs for a specific class of supervisory controllers. We then formulated the scheduling problem as a MIDO, in terms of identifying the optimal sequence of setpoints for the supervisory controller, subject to the closed loop dynamics described by the ICM. Using a case study, we showed that ICM-Based scheduling is comparable to full dynamic scheduling in the nominal cases, and outperforms it in terms of on-spec production when plant-model mismatches are present.
Acknowledgements Financial support from ABB Corporate Research is gratefully acknowledged.
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