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An Industrial Case Study in Simultaneous Design and Control using Mixed-Integer Dynamic Optimization
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
163
An Industrial Case Study in Simultaneous Design and Control using Mixed-Integer Dynamic Optimization V. BansaP'*, R. Ross''\ J.D. Perkins\ E.N. Pistikopoulos^'* and S. de Wolf ^ Process Systems Enterprise Ltd., 107a H'smith Bridge Rd., London W6 9DA, U.K. ^ Dept. of Chemical Engineering, Imperial College, London SW7 2BY, U.K. "" Shell International Chemicals B.V., Badhuisweg 3, 1031 CM Amsterdam, The Netherlands.
Abstract This paper describes an industrial case study concerned with the simultaneous optimization of the design of a high-purity, azeotropic distillation system and its control scheme. The resulting mathematical problem is a large-scale, mixed-integer dynamic optimization (MIDO) one, where the integer variables correspond to discrete decisions such as the determination of the optimal feed tray location. The optimal solution found has a total annualised cost that is 18% less than that of the existing design, which was obtained using a sequential design and control approach.
1. Introduction This paper is concerned with the process design and control of a two-column distillation system that forms part of a Shell iso-propyl alchohol (IPA) separation train (see Figure 1). This system was previously studied by Ross et al. (1999, 2001). In that work, the design of the process equipment and control system's tuning parameters were simultaneously determined by applying a methodology developed by Pistikopoulos, Perkins and co-workers (see Mohideen et al, 1996; Bansal et al, 2000a). The study showed that if such a simultaneous optimization approach had been used when the distillation system was originally designed, then the operability difficulties that had been experienced during its operation would have been avoided. Furthermore, this approach led to a system with a substantially lower total annualised cost than the existing design. The problem studied by Ross et al. is one of the largest dynamic optimization problems whose solution has been reported in the open literature. However, it only considered continuous design decisions such as determination of the optimal column diameter and Pl-controller gain. The objective of this study is to demonstrate how recently developed, state-of-the-art algorithms for solving mixed-integer dynamic optimization (MIDO)
* Current address: Orbis Investment Advisory Limited, 5 Mansfield St., London, WIG 9NG, U.K. ^ Current address: United Technologies Research Center, East Hartford, CT 06108, U.S.A. * Corresponding author. Tel: +44 (0)20 7594 6620. Fax: +44 (0)20 7594 6606. E-mail: e . p i s t i k o p o u l o s ( i i c .ac .uk.
164 problems can be utilised to also make discrete process and control design decisions such as where to optimally locate the feed tray in the IPA column.
2. Problem Statement The problem to be solved can be stated in the following terms. The objective is to design the distillation system and its required control system at minimum total annualised cost, capable of feasible operation over the whole of a given time horizon in the face of disturbances in the feed flow rate and composition. As in the study of Ross et a/., feasibility is defined through the satisfaction of the following constraints: (i) composition specifications on the NPA and water in the IP A/water azeotrope product from the IPA column; (ii) a composition constraint on the IPA in the bottoms stream from the NPA column; (iii) minimum column diameters to avoid flooding in each column; and (iv) fractional entrainment limits in each column. Solution of the problem thus requires the determination of: 1.
The optimal process design, in terms of the locations of the feed and draw-off trays in the IPA column (discrete decisions), and the columns' diameters, reboilers' heat duties and flow rates of the draw-off stream to the NPA column and the return stream back to the IPA column (continuous decisions); and
2.
The optimal control design, in terms of the tuning parameters of the control loops used. For this study, the control loops under consideration are the same as those used by Ross et al and the tuning parameters to be determined correspond to the gain of the ratio controller in the feedforward loop and the gain and reset fime of the Pl-controller in the feedback loop. In the feedback loop, the controlled variable is the average of the temperatures on the trays two and four above the IPA feed tray.
3. Modelling Aspects The high-fidelity, dynamic model of Ross et al. was also used in this study but with the introduction of binary variables yfk and ydk, /:=!,...,30, in order to account for the locations of the feed and draw-off trays, respectively, within the bottom thirty trays of the IPA column. Thus, yfk = 1 if the feed enters tray k and is 0 otherwise; similarly ydk = 1 if liquid is drawn-off from tray k to the NPA column and is 0 otherwise. The following new equations and constraints apply: •
Only one tray receives the feed and only tray is used as a draw-off: 30
30
k=l
k=\
•
The draw-off tray cannot be located below the feed tray:
165 30
yf,-'Zyd.<0,
^ = l,...,30.
(2)
it=i
•
The feed and draw-off trays must be even-numbered due to the arrangement of "inner" and "outer" trays within the column:
yf2H-yd2j-i=0, •
7 = 1,...,15.
The feed flow rate to tray k:
Feed^ = Feed - yf,,, •
(3)
k = 1,...,30.
(4)
The draw-off flow rate from tray k:
Draw^ = Draw • yd^,
A: = 1,... ,30.
(5)
In keeping with the original model given by Shell, the free area of the trays below the feed is different than the free area of the feed tray and above: 30
Ajree., =0.019-DJ,,
• ^
/
30
} / , +0.1 1 5 - D , ^ , • 1 "
k=k+\
J^jf^
^
^ = 1,...,29,
k =k^\
(6)
A,,,,,, =0.115-D,^,,
^=30,...,70.
(7)
4. Problem Formulation In this study, the simultaneous process and control design optimization problem has the following mathematical form: Cost = mm ( Q ^ + C „ J
f s.t. h . XA(t),x^(t),x^(t),i(t),d,y,t
= 0,
h,(xa(O,x^(0,i(0,
Xd(0),x^(0),x,(0),i (0),d,y,/ = 0 = 0,
g<
XA(t.),x^{t.),x^(t
),iit
),d,y,t
= 0,
g.(y)=o. In (8), Ccap and Cop are the total annualised capital and operating costs, respectively; hd=0 and ha=0 represent the differential and algebraic equations, respectively; ho=0 is the set of initial conditions; and Xd and Xa are the differential and algebraic variables.
166 ge< 0 denote end-point constraints which result once the original path constraints listed in Section 2 have been converted using the method described in Section 7.2.10.2 of Bansal (2000). gy < 0 is the set of pure integer constraints such as (1) to (3). The two disturbances in this study, v(t), are a step change in the feed flow rate from 115 to 135 t/hr. and a sinusoidal variation in the IPA feed mass fraction (nominal 0.12, ±5% amplitude, 3 hr. period), d is the set of six continuous process design variables and three controllers' tuning parameters described in Section 2; and y is the set of binary variables yfkdindydk, /:=1,...,30.
5. Solution and Results (8) corresponds to a MIDO problem with 509 differential equations, 8295 algebraic equations (not including the 20,000 or so physical properties equations), 475 end-point inequalities, 32 pure binary constraints, 9 continuous search variables and 60 binary search variables. This formidably-sized problem was solved using the MIDO algorithm described in Chapter 6 of Bansal (2000). This algorithm has already been used to solve industrial-type problems (Chapter 7 of Bansal, 2000 and Bansal et al, 2000b). In this study, gPROMS vl.8.4 (Process Systems Enterprise Ltd, 2000) was used to solve the primal problems and GAMS v2.5/CPLEX (Brooke et a/., 1992) was used to solve the master problems. Using a termination tolerance of e = 0, the MIDO algorithm converged in just 8 iterations and was able to find three structures that are cheaper than the original structure considered in the study of Ross et al (1999,2001). The optimal solution was found on the fourth primal solution, and as shown in Table 1, its total annualised cost is 18% less than the existing Shell design. These savings are achieved by removing more NPA from the IPA column (increased draw-off rate) so that the DPA column capacity can be significantly reduced (smaller diameter and reboiler duty). Although the capacity and hence cost of the NPA column increase, the savings in the cost of IPA column easily outweigh this. Figure 2 plots the mass fraction of NPA in the IP A/water azeotrope product for the optimally designed system. It can be seen that the constraint that the mass fraction is less than 800 ppm is satisfied at all times during operation. Other plots like this one are automafically given as part of the MIDO solution. Note that Figure 2 shows the dynamic operation over one week, which is more than ten times longer than the 15 hr. time horizon that was used for solving the primal dynamic optimization problems. The cyclical behaviour and the fact that the constraint is never violated thus suggest that the optimally designed process and control system is also stable.
6. Concluding Remarks The previous study by Ross et al (1999, 2001) illustrated the economic and operability benefits of simultaneously considering process design and control decisions within an optimization framework, compared to traditional approaches where process design and process control are considered sequentially. This study has demonstrated that further
167 benefits are possible by also considering discrete decisions within the optimization framework with the result that a design has been found for the LPA/NPA system that is 18% cheaper than the existing design. In order to achieve this, very difficult, mixedinteger dynamic optimization problems need to be solved. This study has also shown that the state-of-the-art algorithms for solving such problems have progressed to a point where complex, industrial problems can be readily tackled.
References Bansal, V, 2000, Analysis, Design and Control Optimization of Process Systems under Uncertainty. PhD Thesis, University of London. Bansal, V., R. Ross, J.D. Perkins and E.N. Pistikopoulos, 2000a, The Interactions of Design and Control: Double-Effect Distillation, J. Proc. Control 10, 219. Bansal, V., R. Ross, J.D. Perkins, E.N. Pistikopoulos and J.M.G. van Schijndel, 2000b, Simultaneous Design and Control Optimisation under Uncertainty, Comput. Chem. Eng. 24,261. Brooke, A., D. Kendrick and A. Meeraus, 1992, GAMS Release 2.25: A User's Guide. The Scientific Press, San Francisco. Mohideen, M.J., J.D. Perkins and E.N. Pistikopoulos, 1996, Optimal Design of Dynamic Systems under Uncertainty, AIChE J. 42, 2251. Process Systems Enterprise Ltd., 2000, gPROMS Advanced User Guide, http://www.psenterprise.com. Ross, R., V. Bansal, J.D. Perkins, E.N. Pistikopoulos, G.L.M. Koot and J.M.G. van Schijndel, 1999, Optimal Design and Control of a High-Purity Industrial Distillation System, Comput. Chem. Eng. 23, S875. Ross, R., J.D. Perkins, E.N. Pistikopoulos, G.L.M. Koot and J.M.G. van Schijndel, 2001, Optimal Design and Control of an Industrial Distillation System, Comput. Chem. Eng. 25, 141. Table 1: Comparison of the Design obtained via MIDO with the Existing
Design Variable Draw-off Location Feed Location Qreb (IPA) (MW) D,,KIPA)(m.) Draw (t/hr) Return {t/hr) Qreb (NPA) (MW) D,,KNPA)(m.) Capital Cost ($M yf^) Operating Cost (SMyr"^) Ljotal Cost
Optimal Design using MIDO 18 8 14.67 2.58 4.20 3.43 2.45 1.33 0.56 3.56 4.12
Design.
Existing Design 22 1 14 19.54 3.17 1.69 1.20 0.87 0.87 0.63 4.37 5.00
168 IPAAVater/DMK P = 2.8 bar
IPAAVater Azeotrope Product Water content < 13.07% NPA content < 800 ppm
70 Trays
P = 1.5 bar
48 Trays
12.0 wt% Iso-Propanol (IPA) 87.9 wt% Water 0.1 wt%Propanol(NPA) trace Acetone (DMK)
Water/NPA/MIBC
Water
trace Methyl Isobutylcarbinol (MIBC) Figure 1: Schematic of the IPA/NPA Distillation System.
E
900
1 fiiiiliifiiiiiii Ullllgl^lPPlllllgPlllllj www www
r1
-S 500
I 400 300
9
•4
200,
i
0
10
20
30
40
50
60
70
80
90
100
110
120
Time (hr)
Actual
Maximum Allowable
Figure 2: Mass Fraction of NPA in the Azeotrope Product.
130
140
150
160
170
163
An Industrial Case Study in Simultaneous Design and Control using Mixed-Integer Dynamic Optimization V. BansaP'*, R. Ross''\ J.D. Perkins\ E.N. Pistikopoulos^'* and S. de Wolf ^ Process Systems Enterprise Ltd., 107a H'smith Bridge Rd., London W6 9DA, U.K. ^ Dept. of Chemical Engineering, Imperial College, London SW7 2BY, U.K. "" Shell International Chemicals B.V., Badhuisweg 3, 1031 CM Amsterdam, The Netherlands.
Abstract This paper describes an industrial case study concerned with the simultaneous optimization of the design of a high-purity, azeotropic distillation system and its control scheme. The resulting mathematical problem is a large-scale, mixed-integer dynamic optimization (MIDO) one, where the integer variables correspond to discrete decisions such as the determination of the optimal feed tray location. The optimal solution found has a total annualised cost that is 18% less than that of the existing design, which was obtained using a sequential design and control approach.
1. Introduction This paper is concerned with the process design and control of a two-column distillation system that forms part of a Shell iso-propyl alchohol (IPA) separation train (see Figure 1). This system was previously studied by Ross et al. (1999, 2001). In that work, the design of the process equipment and control system's tuning parameters were simultaneously determined by applying a methodology developed by Pistikopoulos, Perkins and co-workers (see Mohideen et al, 1996; Bansal et al, 2000a). The study showed that if such a simultaneous optimization approach had been used when the distillation system was originally designed, then the operability difficulties that had been experienced during its operation would have been avoided. Furthermore, this approach led to a system with a substantially lower total annualised cost than the existing design. The problem studied by Ross et al. is one of the largest dynamic optimization problems whose solution has been reported in the open literature. However, it only considered continuous design decisions such as determination of the optimal column diameter and Pl-controller gain. The objective of this study is to demonstrate how recently developed, state-of-the-art algorithms for solving mixed-integer dynamic optimization (MIDO)
* Current address: Orbis Investment Advisory Limited, 5 Mansfield St., London, WIG 9NG, U.K. ^ Current address: United Technologies Research Center, East Hartford, CT 06108, U.S.A. * Corresponding author. Tel: +44 (0)20 7594 6620. Fax: +44 (0)20 7594 6606. E-mail: e . p i s t i k o p o u l o s ( i i c .ac .uk.
164 problems can be utilised to also make discrete process and control design decisions such as where to optimally locate the feed tray in the IPA column.
2. Problem Statement The problem to be solved can be stated in the following terms. The objective is to design the distillation system and its required control system at minimum total annualised cost, capable of feasible operation over the whole of a given time horizon in the face of disturbances in the feed flow rate and composition. As in the study of Ross et a/., feasibility is defined through the satisfaction of the following constraints: (i) composition specifications on the NPA and water in the IP A/water azeotrope product from the IPA column; (ii) a composition constraint on the IPA in the bottoms stream from the NPA column; (iii) minimum column diameters to avoid flooding in each column; and (iv) fractional entrainment limits in each column. Solution of the problem thus requires the determination of: 1.
The optimal process design, in terms of the locations of the feed and draw-off trays in the IPA column (discrete decisions), and the columns' diameters, reboilers' heat duties and flow rates of the draw-off stream to the NPA column and the return stream back to the IPA column (continuous decisions); and
2.
The optimal control design, in terms of the tuning parameters of the control loops used. For this study, the control loops under consideration are the same as those used by Ross et al and the tuning parameters to be determined correspond to the gain of the ratio controller in the feedforward loop and the gain and reset fime of the Pl-controller in the feedback loop. In the feedback loop, the controlled variable is the average of the temperatures on the trays two and four above the IPA feed tray.
3. Modelling Aspects The high-fidelity, dynamic model of Ross et al. was also used in this study but with the introduction of binary variables yfk and ydk, /:=!,...,30, in order to account for the locations of the feed and draw-off trays, respectively, within the bottom thirty trays of the IPA column. Thus, yfk = 1 if the feed enters tray k and is 0 otherwise; similarly ydk = 1 if liquid is drawn-off from tray k to the NPA column and is 0 otherwise. The following new equations and constraints apply: •
Only one tray receives the feed and only tray is used as a draw-off: 30
30
k=l
k=\
•
The draw-off tray cannot be located below the feed tray:
165 30
yf,-'Zyd.<0,
^ = l,...,30.
(2)
it=i
•
The feed and draw-off trays must be even-numbered due to the arrangement of "inner" and "outer" trays within the column:
yf2H-yd2j-i=0, •
7 = 1,...,15.
The feed flow rate to tray k:
Feed^ = Feed - yf,,, •
(3)
k = 1,...,30.
(4)
The draw-off flow rate from tray k:
Draw^ = Draw • yd^,
A: = 1,... ,30.
(5)
In keeping with the original model given by Shell, the free area of the trays below the feed is different than the free area of the feed tray and above: 30
Ajree., =0.019-DJ,,
• ^
/
30
} / , +0.1 1 5 - D , ^ , • 1 "
k=k+\
J^jf^
^
^ = 1,...,29,
k =k^\
(6)
A,,,,,, =0.115-D,^,,
^=30,...,70.
(7)
4. Problem Formulation In this study, the simultaneous process and control design optimization problem has the following mathematical form: Cost = mm ( Q ^ + C „ J
f s.t. h . XA(t),x^(t),x^(t),i(t),d,y,t
= 0,
h,(xa(O,x^(0,i(0,
Xd(0),x^(0),x,(0),i (0),d,y,/ = 0 = 0,
g<
XA(t.),x^{t.),x^(t
),iit
),d,y,t
= 0,
g.(y)=o. In (8), Ccap and Cop are the total annualised capital and operating costs, respectively; hd=0 and ha=0 represent the differential and algebraic equations, respectively; ho=0 is the set of initial conditions; and Xd and Xa are the differential and algebraic variables.
166 ge< 0 denote end-point constraints which result once the original path constraints listed in Section 2 have been converted using the method described in Section 7.2.10.2 of Bansal (2000). gy < 0 is the set of pure integer constraints such as (1) to (3). The two disturbances in this study, v(t), are a step change in the feed flow rate from 115 to 135 t/hr. and a sinusoidal variation in the IPA feed mass fraction (nominal 0.12, ±5% amplitude, 3 hr. period), d is the set of six continuous process design variables and three controllers' tuning parameters described in Section 2; and y is the set of binary variables yfkdindydk, /:=1,...,30.
5. Solution and Results (8) corresponds to a MIDO problem with 509 differential equations, 8295 algebraic equations (not including the 20,000 or so physical properties equations), 475 end-point inequalities, 32 pure binary constraints, 9 continuous search variables and 60 binary search variables. This formidably-sized problem was solved using the MIDO algorithm described in Chapter 6 of Bansal (2000). This algorithm has already been used to solve industrial-type problems (Chapter 7 of Bansal, 2000 and Bansal et al, 2000b). In this study, gPROMS vl.8.4 (Process Systems Enterprise Ltd, 2000) was used to solve the primal problems and GAMS v2.5/CPLEX (Brooke et a/., 1992) was used to solve the master problems. Using a termination tolerance of e = 0, the MIDO algorithm converged in just 8 iterations and was able to find three structures that are cheaper than the original structure considered in the study of Ross et al (1999,2001). The optimal solution was found on the fourth primal solution, and as shown in Table 1, its total annualised cost is 18% less than the existing Shell design. These savings are achieved by removing more NPA from the IPA column (increased draw-off rate) so that the DPA column capacity can be significantly reduced (smaller diameter and reboiler duty). Although the capacity and hence cost of the NPA column increase, the savings in the cost of IPA column easily outweigh this. Figure 2 plots the mass fraction of NPA in the IP A/water azeotrope product for the optimally designed system. It can be seen that the constraint that the mass fraction is less than 800 ppm is satisfied at all times during operation. Other plots like this one are automafically given as part of the MIDO solution. Note that Figure 2 shows the dynamic operation over one week, which is more than ten times longer than the 15 hr. time horizon that was used for solving the primal dynamic optimization problems. The cyclical behaviour and the fact that the constraint is never violated thus suggest that the optimally designed process and control system is also stable.
6. Concluding Remarks The previous study by Ross et al (1999, 2001) illustrated the economic and operability benefits of simultaneously considering process design and control decisions within an optimization framework, compared to traditional approaches where process design and process control are considered sequentially. This study has demonstrated that further
167 benefits are possible by also considering discrete decisions within the optimization framework with the result that a design has been found for the LPA/NPA system that is 18% cheaper than the existing design. In order to achieve this, very difficult, mixedinteger dynamic optimization problems need to be solved. This study has also shown that the state-of-the-art algorithms for solving such problems have progressed to a point where complex, industrial problems can be readily tackled.
References Bansal, V, 2000, Analysis, Design and Control Optimization of Process Systems under Uncertainty. PhD Thesis, University of London. Bansal, V., R. Ross, J.D. Perkins and E.N. Pistikopoulos, 2000a, The Interactions of Design and Control: Double-Effect Distillation, J. Proc. Control 10, 219. Bansal, V., R. Ross, J.D. Perkins, E.N. Pistikopoulos and J.M.G. van Schijndel, 2000b, Simultaneous Design and Control Optimisation under Uncertainty, Comput. Chem. Eng. 24,261. Brooke, A., D. Kendrick and A. Meeraus, 1992, GAMS Release 2.25: A User's Guide. The Scientific Press, San Francisco. Mohideen, M.J., J.D. Perkins and E.N. Pistikopoulos, 1996, Optimal Design of Dynamic Systems under Uncertainty, AIChE J. 42, 2251. Process Systems Enterprise Ltd., 2000, gPROMS Advanced User Guide, http://www.psenterprise.com. Ross, R., V. Bansal, J.D. Perkins, E.N. Pistikopoulos, G.L.M. Koot and J.M.G. van Schijndel, 1999, Optimal Design and Control of a High-Purity Industrial Distillation System, Comput. Chem. Eng. 23, S875. Ross, R., J.D. Perkins, E.N. Pistikopoulos, G.L.M. Koot and J.M.G. van Schijndel, 2001, Optimal Design and Control of an Industrial Distillation System, Comput. Chem. Eng. 25, 141. Table 1: Comparison of the Design obtained via MIDO with the Existing
Design Variable Draw-off Location Feed Location Qreb (IPA) (MW) D,,KIPA)(m.) Draw (t/hr) Return {t/hr) Qreb (NPA) (MW) D,,KNPA)(m.) Capital Cost ($M yf^) Operating Cost (SMyr"^) Ljotal Cost
Optimal Design using MIDO 18 8 14.67 2.58 4.20 3.43 2.45 1.33 0.56 3.56 4.12
Design.
Existing Design 22 1 14 19.54 3.17 1.69 1.20 0.87 0.87 0.63 4.37 5.00
168 IPAAVater/DMK P = 2.8 bar
IPAAVater Azeotrope Product Water content < 13.07% NPA content < 800 ppm
70 Trays
P = 1.5 bar
48 Trays
12.0 wt% Iso-Propanol (IPA) 87.9 wt% Water 0.1 wt%Propanol(NPA) trace Acetone (DMK)
Water/NPA/MIBC
Water
trace Methyl Isobutylcarbinol (MIBC) Figure 1: Schematic of the IPA/NPA Distillation System.
E
900
1 fiiiiliifiiiiiii Ullllgl^lPPlllllgPlllllj www www
r1
-S 500
I 400 300
9
•4
200,
i
0
10
20
30
40
50
60
70
80
90
100
110
120
Time (hr)
Actual
Maximum Allowable
Figure 2: Mass Fraction of NPA in the Azeotrope Product.
130
140
150
160
170