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Automated Design of Batch Water-Allocation Network
10th International Symposium on Process Systems Engineering - PSE2009 Rita Maria de Brito Alves, Claudio Augusto Oller do Nascimento and Evaristo Chalbaud Biscaia Jr. (Editors) © 2009 Elsevier B.V. All rights reserved.
843
Automated Design of Batch Water-Allocation Network Hong-Guang Dong,a Li-Juan Li,a Rui-Jie Zhoub a
School of Chemical Engineering, Dalian University of Technology, Dalian 116012, PRC b Department of Economics, Dalian University of Technology, Dalian 116024, PRC
Abstract The mathematical technique presented in this work deals with one step design of batch water-allocation network (WAN), where batch schedules, water-reuse subsystems, and wastewater-treatment subsystems are all taken into account. In the first place, a flexible schedule model is introduced to represent the precedence order of all operations. Then, two novel superstructures incorporating all basic elements (i.e. states, tasks, equipment and time) are adopted to capture all production schemes and batch WAN configurations. Specifically, by adding novel components in the original superstructure, a series of optimal network structures with multistage splitting and mixing options which have never been contained within previous superstructure can be easily generated. Finally, a reliable optimization strategy, where deterministic and stochastic searching techniques are combined, is suggested to deal with the resulting mixed integer non-linear programming (MINLP) model. Two illustrative examples are presented to demonstrate the effectiveness of the proposed approach. Keywords: superstructure, batch schedules, batch water-allocation network.
1. Introduction In the past, the tasks of optimizing batch schedules, water reuse and wastewater treatment subsystems were performed individually. Cheng and Chang (2007) first developed an effective procedure to incorporate these three components into a single comprehensive model. However, in their study, the discrete-time model embedded was inflexible. In addition, not all possible network structures were included in their superstructure and this superstructure may fail to wholly reflect the essential relationship between units and corresponding operations. Finally, interactions between process schedules and batch WAN were not clearly addressed mathematically, which might result in solving the proposed model on a sequential basis. To circumvent these problems, Zhou et al. (2008) integrated batch schedules and WAN into a simultaneous optimization model based on continuous time representation and the modified state space concept. Although the relationship between operations and units was addressed by the STN (Kondili, 1993) and SEN (Smith, 1996) extensions of the original state space framework (Bagajewicz and Manousiouthakis, 1992), yet assigning a fixed mixer and splitter to each unit during the whole time horizon inevitability leads to the preclusion of a class of optimal network structures, where the best cost-optimal scheme may actually lies. Finally, the continuous time formulation is unable to express the sequence of all operations and the optimization strategy has to be executed interactively between different components. Given these shortages, there is therefore a need to develop a more comprehensive design method for optimizing the batch WANs.
844
D. Hong-Guang et al.
2. Superstructures 2.1. Superstructure for batch production A novel superstructure has been introduced in this work to represent batch productions. (see Figure 1). In the first place, all material states embedded in distribution network (DN block) are classified into three groups-raw materials, intermediate products and final products, all of which are further divided into two parts: material states to be consumed/sold and produced/bought. Then, through the process operators (OP block), each water-using unit is illustrated as a block with many sub-blocks which correspond to operations performed at certain event points. Finally, all material states and units are connected through streams to reflect the material flows and a reference axis of time is set to identify the starting and ending points of each operation.
i2 i3 i 4
N
N
N
! ! ! !
! ! ! ! N
n i2 i3 i4 j2 j2 j2
i5
i2 i3 i 4
n1
n1 i2n1 i3n1 i4n1 j3 j3 j3
n1
!
N
N
!
!
! !
n1 i1 j1
n i1j1
n1
n1 i2 i3 i4 j2 j2 j2
n
n i nN i n N i N j j j N
n1 i4j 4
! !
n1
n1
!
i1
nN
n i4 j4 N
Figure.1. Superstructure for batch production. 2.2. Superstructure for WAN In the previous superstructure (Zhou et al. 2008), the splitting and mixing can only take place in DN where a one-to-one correspondence between mixers/splitters and water users is ensured, and the possibilities that streams bypass water users without mass exchange were completely ruled out. As a result, streams can only mix once before entering the water user and in the same way they are only allowed to split once after leaving the water user. To overcome these deficiencies, we have developed a comprehensive STS superstructure for optimizing batch WAN. The improved state-space superstructure for batch WAN is also viewed as two interconnected blocks (see Figure 2). One is referred to as the distribution network (DN), in which all connections between units and junctions are embedded. The other is the socalled process operators (OP), which can be further divided into two sub-blocks, i.e., units and junctions. Detailed explanations of our superstructure are described in the sequel. In one time interval, while all input streams to DN are allowed to connect all exits leading to OP block or the environment, only inputs from junctions are allowed to split into several branches. Likewise, more than one input is allowed to mix before all junctions. Finally, in any time interval, splitting of inputs from units and mixing of streams before units are forbidden. As for OP, each unit is illustrated as a block with many sub-blocks which correspond to operations performed at certain event points. Furthermore, the optimal number and the corresponding assignment of units to junctions are not fixed a priori, thus being subjected to optimization process. It is worthy of note, that junctions can provide increased flexibility in formulating network
845
Automated Design of Batch Water-Allocation Network
structures with multistage mixing/splitting options, which can never be captured by other frameworks. Specifically, all mixed inputs to DN block have further opportunities to match with other possible streams at all junctions and, if necessary, such looping procedure within junctions can be carried out for arbitrary times.
!
!
#
!
!
#
!
!
!
!
#
#
#
#
#
#
Figure.2. Superstructure for WAN.
3. Mathematical Model Due to the length of this paper, only parts of the constraints are presented as follows. 3.1. Models for Batch schedules 3.1.1. Material balances STs , n
STsin ¦ U sc,i ¦ Bi , j , n rs ,n d s ,n iI s
STs ,n
s S , n N ( n
STs ,n 1 ¦ U sc,i ¦ Bi , j ,n ¦ U sp,i ¦ Bi , j ,n rs ,n d s ,n iI s
STsin
STs , N
n1 )
(1)
jJ i
jJ i
iI s
s S
s S , n N (n ! n1 )
( 2)
jJ i
(3)
Constraint 3 expresses that the amount of each state at the starting and ending times of any operation cycle should be kept identical. 3.1.2. Duration constraints Ti f Ti s D i , j wv(i, n) E i , j bi i I , j J i , n N (4) n j
n j
n j
Bi n Bi , j d bi n Bi n Bi , j j
j
j
Bi , j
i I , j J i , n N
Ti ns =F i n 't
i I , j J i , n N
(6)
Ti n =Mi n 't
i I , j J i , n N
(7)
j
j
f
j
j
(5)
where D i , j and Ei , j denote respectively the constant and each variable term of processing time; Bi , j and b are assumed, respectively, to be the amount processed at i n j
each variable term and the number of variable terms needed; 't represents the length of each variable interval. Constraints 4-7 show that the processing time of each operation is made up of two parts: fixed term and variable terms.
846
D. Hong-Guang et al.
3.2. Model for WAN 3.2.1. Mass and flow balances of each operation fuin,t fuin,t
¦ fs c ¦ fs u c , u ,t
u cU in u , k ,t
u U , t T
u c ,u ,t
csu c, k ,t
u cU
fuout ,t
¦
csu ,k ,t
u U , k K , t T
u ,u c ,t
u U , k K , t T
fs jun ,u ,t
(9)
junJUN
¦ fs
u cU
cuout,k ,t
(8)
¦
fsu , jun ,t
u U , t T
(10)
junJUN
(11)
3.2.2. Mass and flow balances of each unit f jin,t ¦ fi in,t j J , t T (12) i nj I nj
f jout ,t
¦
i nj I nj
c inj ,k ,t
n j
finout,t j
¦c
in i nj , k ,t
i nj I nj
c out j , k ,t
¦c
i nj I nj
out i nj , k ,t
j J , t T
(13)
j J , k K , t T j J , k K , t T
(14) (15)
3.3. Objective function The criterion of this study is the minimizing of cost, which can be expressed as follows: Obj = (total income - purchasing cost) - (cost of fresh water + cost of water treatment + cost of buffer tanks cost of junctions) (16)
4. Application examples 4.1. Example 1 The first example presented was originally solved simultaneously based on discretetime representation by Cheng and Chang (2007). All the data used can be found in the original work. In a latter study, when network complexity was taken into account, Zhou et al. (2008) obtained a batch WAH design, of which the profit of batch production and cost of network were found to be 1366.7 and 388.31 units respectively. In the present study, On the basis of a time of horizon of 4 hr, the most appropriate production scheme and network configuration are presented in Figures 3. It can be found that while the schedule scheme is the same as the former study, the network configuration has subjected to a major transformation, as the multistage mixing/splitting stream from Jun1-Jun2 can be clearly identified. The corresponding cost of network in a production cycle can be reduced to 377.01 units, which represents a 3% improvement. This can be attributed to the fact that by allowing multistage mixing/splitting, the economic tradeoff issues in WAH design can be carried out effectively. Specifically, the costs of freshwater supply, wastewater treatment, buffer tanks, junctions and pipelines were found to be 123.33, 201.68, 0 and 52 units respectively. As evident from the network structure, buffer tank is not embedded in the optimal network configuration and this can explain that buffer tanks are not necessary in some batch WAN if optimal production sequences and operating policies of water flows are adopted. Finally, it can be observed that the introduction of junctions allows a higher integration of the overall batch WAN, since only 2 junctions and 14 pipelines are needed.
847
Automated Design of Batch Water-Allocation Network
i1 46.67
n1
i3 50.00
i5
i2
i4
#
n1
50.00
n2
44.66
n3
70.00
n1
72.00
n3
100.00
n1
#
#
#
Figure.3. Optimal production scheme and network configuration for Case I 4.2. Example 2 In this example, multi-contaminant system design options are incorporated. Consider the process presented by Zhou et al. In this study, the removal ratios of tr1 and tr2 are chosen to be 0.9, 0.85, 0.9 and 0.95, 0.9, 0.95 respectively; the corresponding cost is set to 0.75 and 1 respectively. The resulting schedule results and its corresponding network are shown in Figure 4 and 5. Notice that the rewards of production and cost of network can be found to be 2108.57 and 333.95 units respectively. The freshwater and wastewater treatment costs in this case are 121.71 and 78.03 units respectively, while the costs of buffer tanks, junctions and pipelines are 67.21 and 67 units respectively. It can be concluded from this example that multistage mixing/splitting is indeed a desirable fashion and such methods enable cost-optimal designs of complex and large scale multi-contaminant systems.
Figure.4. Optimal production scheme for Case II
848
D. Hong-Guang et al.
tr1
#
jun1
#
jun2
#
#
b1
Figure.5. Optimal network configuration for Case II
5. Conclusions Two superstructures have been presented in this study to formulate a MINLP model for one step design of batch WAN. The advantage of this approach is that not only all possible alternative network topologies can be captured, but the sequential order of all operations as well, can also be properly addressed. To ensure the quality and efficiency of the solutions, a hybrid optimization strategy integrating DICOPT and EA search techniques is developed. Finally, two examples dealt with single- and multicontaminant WAN are presented to demonstrate the feasibility and effectiveness of the proposed automated design method. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant No.20876020.
References Bagajewicz, M., & Manousiouthakis, V. (1992). On the mass/heat exchanger network representations of distillation networks. AIChE Journal, 38, 1769. Cheng, K. F., & Chang, C. T. (2007). Integrated Water Network Designs for Batch Processes. Ind. Eng. Chem. Res., 46, 1241. Kondili, E., Pantelides, C. C., & Sargent, R. W. H. (1993). A general algorithm for short-term scheduling of batch operations—I. MILP formulation. Comput. Chem. Eng., 17, 211. Smith, E. M. (1996). On the optimal design of continuous processes, Ph.D. Dissertation, under supervision of C. Pantelides. Imperial College of Science, Technology and Medicine, London, UK. Zhou, R.J., Li, L.J.,Xiao, W. & Dong H.G. (2008) Simultaneous optimization of batch process schedules and water-allocation network. Comput. Chem. Eng. (In press)
843
Automated Design of Batch Water-Allocation Network Hong-Guang Dong,a Li-Juan Li,a Rui-Jie Zhoub a
School of Chemical Engineering, Dalian University of Technology, Dalian 116012, PRC b Department of Economics, Dalian University of Technology, Dalian 116024, PRC
Abstract The mathematical technique presented in this work deals with one step design of batch water-allocation network (WAN), where batch schedules, water-reuse subsystems, and wastewater-treatment subsystems are all taken into account. In the first place, a flexible schedule model is introduced to represent the precedence order of all operations. Then, two novel superstructures incorporating all basic elements (i.e. states, tasks, equipment and time) are adopted to capture all production schemes and batch WAN configurations. Specifically, by adding novel components in the original superstructure, a series of optimal network structures with multistage splitting and mixing options which have never been contained within previous superstructure can be easily generated. Finally, a reliable optimization strategy, where deterministic and stochastic searching techniques are combined, is suggested to deal with the resulting mixed integer non-linear programming (MINLP) model. Two illustrative examples are presented to demonstrate the effectiveness of the proposed approach. Keywords: superstructure, batch schedules, batch water-allocation network.
1. Introduction In the past, the tasks of optimizing batch schedules, water reuse and wastewater treatment subsystems were performed individually. Cheng and Chang (2007) first developed an effective procedure to incorporate these three components into a single comprehensive model. However, in their study, the discrete-time model embedded was inflexible. In addition, not all possible network structures were included in their superstructure and this superstructure may fail to wholly reflect the essential relationship between units and corresponding operations. Finally, interactions between process schedules and batch WAN were not clearly addressed mathematically, which might result in solving the proposed model on a sequential basis. To circumvent these problems, Zhou et al. (2008) integrated batch schedules and WAN into a simultaneous optimization model based on continuous time representation and the modified state space concept. Although the relationship between operations and units was addressed by the STN (Kondili, 1993) and SEN (Smith, 1996) extensions of the original state space framework (Bagajewicz and Manousiouthakis, 1992), yet assigning a fixed mixer and splitter to each unit during the whole time horizon inevitability leads to the preclusion of a class of optimal network structures, where the best cost-optimal scheme may actually lies. Finally, the continuous time formulation is unable to express the sequence of all operations and the optimization strategy has to be executed interactively between different components. Given these shortages, there is therefore a need to develop a more comprehensive design method for optimizing the batch WANs.
844
D. Hong-Guang et al.
2. Superstructures 2.1. Superstructure for batch production A novel superstructure has been introduced in this work to represent batch productions. (see Figure 1). In the first place, all material states embedded in distribution network (DN block) are classified into three groups-raw materials, intermediate products and final products, all of which are further divided into two parts: material states to be consumed/sold and produced/bought. Then, through the process operators (OP block), each water-using unit is illustrated as a block with many sub-blocks which correspond to operations performed at certain event points. Finally, all material states and units are connected through streams to reflect the material flows and a reference axis of time is set to identify the starting and ending points of each operation.
i2 i3 i 4
N
N
N
! ! ! !
! ! ! ! N
n i2 i3 i4 j2 j2 j2
i5
i2 i3 i 4
n1
n1 i2n1 i3n1 i4n1 j3 j3 j3
n1
!
N
N
!
!
! !
n1 i1 j1
n i1j1
n1
n1 i2 i3 i4 j2 j2 j2
n
n i nN i n N i N j j j N
n1 i4j 4
! !
n1
n1
!
i1
nN
n i4 j4 N
Figure.1. Superstructure for batch production. 2.2. Superstructure for WAN In the previous superstructure (Zhou et al. 2008), the splitting and mixing can only take place in DN where a one-to-one correspondence between mixers/splitters and water users is ensured, and the possibilities that streams bypass water users without mass exchange were completely ruled out. As a result, streams can only mix once before entering the water user and in the same way they are only allowed to split once after leaving the water user. To overcome these deficiencies, we have developed a comprehensive STS superstructure for optimizing batch WAN. The improved state-space superstructure for batch WAN is also viewed as two interconnected blocks (see Figure 2). One is referred to as the distribution network (DN), in which all connections between units and junctions are embedded. The other is the socalled process operators (OP), which can be further divided into two sub-blocks, i.e., units and junctions. Detailed explanations of our superstructure are described in the sequel. In one time interval, while all input streams to DN are allowed to connect all exits leading to OP block or the environment, only inputs from junctions are allowed to split into several branches. Likewise, more than one input is allowed to mix before all junctions. Finally, in any time interval, splitting of inputs from units and mixing of streams before units are forbidden. As for OP, each unit is illustrated as a block with many sub-blocks which correspond to operations performed at certain event points. Furthermore, the optimal number and the corresponding assignment of units to junctions are not fixed a priori, thus being subjected to optimization process. It is worthy of note, that junctions can provide increased flexibility in formulating network
845
Automated Design of Batch Water-Allocation Network
structures with multistage mixing/splitting options, which can never be captured by other frameworks. Specifically, all mixed inputs to DN block have further opportunities to match with other possible streams at all junctions and, if necessary, such looping procedure within junctions can be carried out for arbitrary times.
!
!
#
!
!
#
!
!
!
!
#
#
#
#
#
#
Figure.2. Superstructure for WAN.
3. Mathematical Model Due to the length of this paper, only parts of the constraints are presented as follows. 3.1. Models for Batch schedules 3.1.1. Material balances STs , n
STsin ¦ U sc,i ¦ Bi , j , n rs ,n d s ,n iI s
STs ,n
s S , n N ( n
STs ,n 1 ¦ U sc,i ¦ Bi , j ,n ¦ U sp,i ¦ Bi , j ,n rs ,n d s ,n iI s
STsin
STs , N
n1 )
(1)
jJ i
jJ i
iI s
s S
s S , n N (n ! n1 )
( 2)
jJ i
(3)
Constraint 3 expresses that the amount of each state at the starting and ending times of any operation cycle should be kept identical. 3.1.2. Duration constraints Ti f Ti s D i , j wv(i, n) E i , j bi i I , j J i , n N (4) n j
n j
n j
Bi n Bi , j d bi n Bi n Bi , j j
j
j
Bi , j
i I , j J i , n N
Ti ns =F i n 't
i I , j J i , n N
(6)
Ti n =Mi n 't
i I , j J i , n N
(7)
j
j
f
j
j
(5)
where D i , j and Ei , j denote respectively the constant and each variable term of processing time; Bi , j and b are assumed, respectively, to be the amount processed at i n j
each variable term and the number of variable terms needed; 't represents the length of each variable interval. Constraints 4-7 show that the processing time of each operation is made up of two parts: fixed term and variable terms.
846
D. Hong-Guang et al.
3.2. Model for WAN 3.2.1. Mass and flow balances of each operation fuin,t fuin,t
¦ fs c ¦ fs u c , u ,t
u cU in u , k ,t
u U , t T
u c ,u ,t
csu c, k ,t
u cU
fuout ,t
¦
csu ,k ,t
u U , k K , t T
u ,u c ,t
u U , k K , t T
fs jun ,u ,t
(9)
junJUN
¦ fs
u cU
cuout,k ,t
(8)
¦
fsu , jun ,t
u U , t T
(10)
junJUN
(11)
3.2.2. Mass and flow balances of each unit f jin,t ¦ fi in,t j J , t T (12) i nj I nj
f jout ,t
¦
i nj I nj
c inj ,k ,t
n j
finout,t j
¦c
in i nj , k ,t
i nj I nj
c out j , k ,t
¦c
i nj I nj
out i nj , k ,t
j J , t T
(13)
j J , k K , t T j J , k K , t T
(14) (15)
3.3. Objective function The criterion of this study is the minimizing of cost, which can be expressed as follows: Obj = (total income - purchasing cost) - (cost of fresh water + cost of water treatment + cost of buffer tanks cost of junctions) (16)
4. Application examples 4.1. Example 1 The first example presented was originally solved simultaneously based on discretetime representation by Cheng and Chang (2007). All the data used can be found in the original work. In a latter study, when network complexity was taken into account, Zhou et al. (2008) obtained a batch WAH design, of which the profit of batch production and cost of network were found to be 1366.7 and 388.31 units respectively. In the present study, On the basis of a time of horizon of 4 hr, the most appropriate production scheme and network configuration are presented in Figures 3. It can be found that while the schedule scheme is the same as the former study, the network configuration has subjected to a major transformation, as the multistage mixing/splitting stream from Jun1-Jun2 can be clearly identified. The corresponding cost of network in a production cycle can be reduced to 377.01 units, which represents a 3% improvement. This can be attributed to the fact that by allowing multistage mixing/splitting, the economic tradeoff issues in WAH design can be carried out effectively. Specifically, the costs of freshwater supply, wastewater treatment, buffer tanks, junctions and pipelines were found to be 123.33, 201.68, 0 and 52 units respectively. As evident from the network structure, buffer tank is not embedded in the optimal network configuration and this can explain that buffer tanks are not necessary in some batch WAN if optimal production sequences and operating policies of water flows are adopted. Finally, it can be observed that the introduction of junctions allows a higher integration of the overall batch WAN, since only 2 junctions and 14 pipelines are needed.
847
Automated Design of Batch Water-Allocation Network
i1 46.67
n1
i3 50.00
i5
i2
i4
#
n1
50.00
n2
44.66
n3
70.00
n1
72.00
n3
100.00
n1
#
#
#
Figure.3. Optimal production scheme and network configuration for Case I 4.2. Example 2 In this example, multi-contaminant system design options are incorporated. Consider the process presented by Zhou et al. In this study, the removal ratios of tr1 and tr2 are chosen to be 0.9, 0.85, 0.9 and 0.95, 0.9, 0.95 respectively; the corresponding cost is set to 0.75 and 1 respectively. The resulting schedule results and its corresponding network are shown in Figure 4 and 5. Notice that the rewards of production and cost of network can be found to be 2108.57 and 333.95 units respectively. The freshwater and wastewater treatment costs in this case are 121.71 and 78.03 units respectively, while the costs of buffer tanks, junctions and pipelines are 67.21 and 67 units respectively. It can be concluded from this example that multistage mixing/splitting is indeed a desirable fashion and such methods enable cost-optimal designs of complex and large scale multi-contaminant systems.
Figure.4. Optimal production scheme for Case II
848
D. Hong-Guang et al.
tr1
#
jun1
#
jun2
#
#
b1
Figure.5. Optimal network configuration for Case II
5. Conclusions Two superstructures have been presented in this study to formulate a MINLP model for one step design of batch WAN. The advantage of this approach is that not only all possible alternative network topologies can be captured, but the sequential order of all operations as well, can also be properly addressed. To ensure the quality and efficiency of the solutions, a hybrid optimization strategy integrating DICOPT and EA search techniques is developed. Finally, two examples dealt with single- and multicontaminant WAN are presented to demonstrate the feasibility and effectiveness of the proposed automated design method. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant No.20876020.
References Bagajewicz, M., & Manousiouthakis, V. (1992). On the mass/heat exchanger network representations of distillation networks. AIChE Journal, 38, 1769. Cheng, K. F., & Chang, C. T. (2007). Integrated Water Network Designs for Batch Processes. Ind. Eng. Chem. Res., 46, 1241. Kondili, E., Pantelides, C. C., & Sargent, R. W. H. (1993). A general algorithm for short-term scheduling of batch operations—I. MILP formulation. Comput. Chem. Eng., 17, 211. Smith, E. M. (1996). On the optimal design of continuous processes, Ph.D. Dissertation, under supervision of C. Pantelides. Imperial College of Science, Technology and Medicine, London, UK. Zhou, R.J., Li, L.J.,Xiao, W. & Dong H.G. (2008) Simultaneous optimization of batch process schedules and water-allocation network. Comput. Chem. Eng. (In press)