StateâTimeâSpace Superstructure-Based MINLP Formulation for

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Ind. Eng. Chem. Res. 2010, 49, 236–251

State-Time-Space Superstructure-Based MINLP Formulation for Batch Water-Allocation Network Design Li-Juan Li,† Rui-Jie Zhou,‡ and Hong-Guang Dong*,† School of Chemical Engineering, Dalian UniVersity of Technology, Dalian, 116012, PRC, Department of Economics, Dalian UniVersity of Technology, Dalian, 116024, PRC

The mathematical technique presented in this work deals with one step design of single- and multicontaminant batch water-allocation network (WAN), where batch production, water-reuse subsystems, and wastewatertreatment subsystems are all taken into account. In the first place, a flexible schedule model combining the merits of discrete and continuous time formulations is introduced to integrate batch production and WAN. Then, two novel state-time-space (STS) 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 nonlinear programming (MINLP) model. Two illustrative examples are presented to demonstrate the effectiveness of the proposed approach. 1. Introduction In recent literature, many studies on the designs of WAN in chemical plants were concerned with batch processes. It has been well-recognized that batch processes are suitable for producing multiple products in small quantities and the process configuration of a batch plant can be easily adjusted to meet the market demand. However, the water network designs for batch processes are obviously more complicated than the continuous ones due to the inherent time dimension. The techniques to synthesis subsystems of water network for batch processes can be broadly classified into two general categories: the insight-based graphical pinch analysis and the mathematicalbased optimization techniques. Wang and Smith1 presented a modified version of pinch method to target the minimum wastewater for a given set of batch water-using processes, in which time is treated as the primary constraint. Foo et al.2 proposed a two-stage water cascade analysis for the synthesis of maximum water recovery network in batch processes, including both mass transfer-based and nonmass transfer-based water-using processes. Majozi et al.3 developed a technique for freshwater and wastewater minimization in completely batch operations, where the intrinsic two-dimensionally constrained nature of batch processes is taken into consideration. Subsequent works on graphical techniques were respectively presented by Liu et al.4 and Chen and Lee.5 Apart from the insight-based approach, mathematical optimization techniques have been developed to handle more complex cases, such as multicontaminant systems and pipeline connections. Almato and his co-workers6-8 first proposed the systematic rationalizing of water reuse in batch process and developed a nonlinear programming (NLP) model to optimize a water reuse network based on the proposed superstructure. In a later study, Kim and Smith9 proposed a new design method for discontinuous water systems considering time constraints and the network designs in which the resulting optimization solved is a MINLP model. * To whom correspondence should be addressed. E-mail: hgdong@ dlut.edu.cn. Tel.: +(86) 159 4265 1993. Address: School of Chemical Engineering, Dalian University of Technology, Dalian, PRC 116012. † School of Chemical Engineering. ‡ Department of Economics.

Majozi10 presented a mathematical formulation for freshwater and wastewater minimization in a given schedule by fixing the outlet concentration of stream at its maximum. Li and Chang11 proposed a general mathematical programming model for the design of discontinuous water-reuse systems. The wastewater equalization options are also incorporated into the system design to address the practical needs. In a recent study, Chen and his co-workers12,13 introduced a model to synthesize applicable water-using networks with the minimum freshwater consumption and analyzed the impact of central storage facilities on freshwater reduction. Shoaib et al.14 introduced a three stage hierarchical approach for the synthesis of cost-effective batch water networks, where all water reuse/recycle between water sources and sinks is conducted between two consecutive batches of operation via water storage to avoid scheduling problems. Notice that in foregoing studies, it is assumed that the production sequence is already known or bypassed for batch production campaign. However, it is really an oversimplification to assume that the production schedule of an overall plant is known a priori. Thus, it is necessary to incorporate the scheduling framework into previous procedures. The key issue pertaining to batch schedules is concerned with time representation. Generally speaking, there are two types of time representations to be employed: discrete time representation and continuous time representation. Kondili et al.15 presented a discretetime-based model by dividing the entire time horizon into a finite number of time intervals with constant duration in which all variables are kept identical. Pinto and Grossmann16 developed a continuous slot-based model to eliminate the binary variables, which further reduces computational requirements. Ierapetritou and Floudas17 proposed a novel continuous-time formulation which is marked by the concept of event points. Later, Floudas and Lin18 presented an overview of all existing approaches in the scheduling of batch processes and found that both kinds of time models have their unique advantages in different scenarios. In the past, the tasks of optimizing batch schedules and subsystems of water network were performed individually. Majozi19 and Gouws et al.20 introduced a framework to embed wastewater minimization within an established scheduling framework in which starting and finishing times become

10.1021/ie900427b CCC: $40.75  2010 American Chemical Society Published on Web 11/23/2009

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010

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Figure 1. STS superstructure for batch production.

optimization variables. Later, Gouws and Majozi21,22 presented an MINLP model to minimize multiple contaminant wastewaters where there are multiple storage vessels. The technique is extended to take into consideration operations where wastewater is reused as part of the product, thereby allowing an almost zero-effluent fashion. Rabie and Halwagi23 developed a sequential strategy for design and scheduling of batch recycle networks with a source-tank-sink representation for the potential water network configurations. However, the common drawback of these works19-23 is that the overall batch production scheme was not fully incorporated. Recently, Cheng and Chang24 developed an effective procedure to incorporate batch production, water reuse, and wastewater treatment subsystems into a single comprehensive model. However, in their study, not all possible network configurations can be generated by their superstructure and this superstructure fails to reflect the essential relationship between units and corresponding operations. Furthermore, the discrete-time model embedded cannot address the interactions between batch production and WAN mathematically, and this may result in solving the proposed model in a sequential manner. To circumvent these problems, Zhou et al.25 integrated batch production and multicontaminant WAN into a simultaneous optimization model based on continuous time representation and the modified state space concept.26,27 Although the relationship between operations and units was addressed by the state task network (STN)15 and state equipment network (SEN)28 extensions of the state space framework, such a relationship was not fully demonstrated in batch production. More importantly, 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 lie. Finally, the continuous time formulation is unable to mark the intervals in which all operating status are kept constant and the optimization strategy has to be executed interactively between different components. Given these shortcomings, there is therefore a need to develop a more comprehensive design method for optimizing the batch WANs. To illustrate the batch WAN design method developed in this work, the rest of this paper is organized as follows. Design

specifications of the superstructures are formally illustrated in section 2 and issues pertaining to the mathematical model are given in section 3. A one step hybrid optimization strategy has been developed to solve the proposed model, and an outline of the solution algorithm can be found in the following section. Two demonstrative examples are then presented in section 5 and the conclusions of this research are provided in the last section. 2. STS Superstructures Oftentimes the structure of the problem is not only as important as the solution, but often determines the solution. Traditionally, the superstructure-based process synthesis makes use of simplifying assumptions, which allow the problem to be computationally solvable at a cost of not considering all possible alternatives. However, the state space approach constitutes a significant departure from the previous superstructure-based model in that it does not contain simplifying assumptions.27 That is the best solution is merely the best of those contained within the problem superstructure. Furthermore, when it comes to batch process synthesis, the basic elements of superstructures are identified as states, tasks, equipments, and time. In previous works, no general framework clearly illustrates the basic elements simultaneously and these existing superstructures fail to capture a whole family of alternative structures. To overcome these problems, the concepts of STN, SEN, and Gantt chart are incorporated into the modified state space notion, and in our work we term this new framework as STS superstructure. The predominant superiority of this superstructure over previous ones is that all batch process can be projected onto the time and space dimension, thereby facilitating simultaneous optimization of batch WANs in both dimensions. 2.1. STS Superstructure for Batch Production. The STS 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.

238


Figure 2. STS superstructure for batch WAN.

Then, through the process operators (OP block), each waterusing unit is illustrated as a block with many sub-blocks which correspond to operations performed at certain event points. More specifically, suppose there are M kinds of operations which can be performed in certain unit and N event points are selected in the overall time horizon; then this unit can be viewed as a set of M × N subgrids. 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. 2.2. STS Superstructure for Batch WAN. To develop a systematic design framework, it is necessary to build a superstructure to allow a common basis for all physical configurations to be included. Zhou et al.14 successfully applied the state space concept to synthesize the batch WAN, while taking account of network structures in considerable generality. However, in this superstructure, 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. Specifically, the overall STS framework for WAN design is viewed as a system of two interconnected blocks (see Figure. 2). One is referred to as the DN block, in which the external inputs, system outputs, and all connections among WAN are embedded. The other block is the so-called OP block, which can be further divided into two sub-blocks, that is, OP_UNIT and OP_JUN. While all water users, potential wastewater treatment units, and buffer tanks are placed in the former subblock, all junctions are set in the latter one. Detailed explanations of our superstructure are described in the sequel. 2.2.1. Distribution Network. In DN block, the freshwater streams and the recycle streams from the OP block are considered, respectively, as the external and internal inputs to DN block. In one time interval, while all input streams are allowed to connect all exits leading to OP block or the environment, only internal inputs from OP_JUN are allowed to split. Likewise, more than one input is allowed to mix before all junctions in OP_JUN. Finally, in any time interval, splitting from and mixing before units in OP_UNIT are forbidden.

2.2.2. Process Operators. 1. OP_UNIT Sub-block. Generally speaking, the presence of every water-using unit is dictated by process requirements. On the other hand, the wastewater treatment units and buffer tanks used in the STS model can be viewed as off-line equipments available for possible installation. In our work, two buffer tanks and treatment units are initially embedded in the superstructure. However, the number of such units may be less than that in the superstructure and it may not even be necessary to use every type of unit in the optimum solution. Another practical issue to be addressed is the time representation. Here, like the superstructure in batch production, each unit is illustrated as a block with many sub-blocks which correspond to operations performed at certain event points. Notice that, for buffer tanks and wastewater treatment units, operations and equipments are identical conceptually. 2. OP_JUN Sub-block. Junctions are designed for mixing/ splitting purposes and can significantly improve splitting and mixing opportunities. Specifically, all mixed inputs from OP_ JUN to DN block have further opportunities to split and match with other possible streams at OP_JUN and, if necessary, such multistage splitting and mixing policies, characterized as looping procedures within OP_JUN in the STS superstructure, can be carried out for arbitrary times. Here, it is worthy of note that such flow policies, which have never been explored before, provide plenty opportunities for water with various grades to split and mix, so that the corresponding mass exchange process can be implemented under a low-level mass transfer driving force. Furthermore, since all junctions can be shared by different units during the time horizon, the trade-offs between various costs of the WAN can also be better balanced and a higher integrated WAN can be achieved. Finally, we must note that it is imperative to place sufficient junctions to provide enough opportunities to match all streams with different concentration grades. In this study, the initial number of junctions is chosen heuristically as the total number of units and the actual number rests on the optimization process. 3. Mathematical Model The overall integrated mathematical model is made up of two modules. One of the modules focuses on batch schedules and the other on WAN. For the sake of clearness and simplicity, these two parts will be introduced relatively separately.


3.1. Model for Batch Process Schedules. Our scheduling model is adapted from the continuous time formulation proposed by Ierapetritou and Floudas.17 Combining both discrete and continuous time formulation, the main characteristics are that (I) the starting and ending points of operations were only allowed at interval boundaries and (II) the operating time of batch operations takes the form of a step function. The main advantages of our combined schedule framework are the necessities of integrated modeling between batch production and WAN and flexibilities. Specifically, the discrete features are used to identify the time intervals in which the water using conditions of all operations are kept identical, while the event points are adopted to specify the operations performed in the units. On the other hand, this model allows us to adjust the relationship between the operating time and the amount processed to reflect the scenarios of different plants. All constraints of our model are summarized as follows: 3.1.1. Distribution Network. 1. Material Balances. STs,n ) STsin -

∑F ∑B c s,i

i∈Is

j∈Ji

ijn

∑F ∑B c s,i

i∈Is

ijn

+

j∈Ji

STs,N ) STsin

∀i ∈ I, j ∈ Ji, n ∈ N

∑F ∑B p s,i

i∈Is

ijn-1

Tifjn ) Tisjn + Ri,j · wV(i, n) + βi,j · bijn ∀i ∈ I, j ∈ Ji, n ∈ N Bijn Bi,j

Bijn + Bi,j

e bijn
n1)

STsin

2. Capacity Constraints.

+ rs,n - ds,n ∀s ∈ S, n ∈ N(n ) n1)

STs,n ) STs,n-1 -

239

∀j ∈ J, n ∈ N

(8)

i∈Ij

where wV(i,n) represents if operation i is performed at event point n, and yV(j,n) denotes if unit j is utilized at event point n. Constraint 8 ensures that only one operation can be performed in unit j at event point n.

s Tinj

Tisjn ) χijn · ∆t

∀i ∈ I, j ∈ Ji, n ∈ N

(12)

Tifjn ) φijn · ∆t

∀i ∈ I, j ∈ Ji, n ∈ N

(13)

f Tinj

and represent, respectively, the times that operation where inj starts and finishes; Ri,j and βi,j denote, respectively, the constant and each variable term of processing time; Bi,j and bijn are assumed, respectively, to be the amount processed at each variable term and the number of variable terms needed; ∆t represents the length of each variable interval; χijn and φijn are positive integers marking the number of intervals of the starting and ending points of inj . Assuming a variation of 33% around the mean value of the average processing time ˜ti,j, Ri,j takes the value 2/3˜ti,j which corresponds to the minimum processing time ti,jmin; and βi,j )

min tmax i,j - ti,j

· Bi.j ∀i ∈ I, j ∈ Ji (14) min Bmax i,j - Bi,j max ) 4/3˜ti,j. Constraint 10 shows the processing time is where ti,j dependent on the amount of state being processed and constraints 12 and 13 ensure that the starting and ending time of any operations are only allowed at interval boundaries. 4. Time Sequence Constraints for the Same Task in the Same Units. Tisjn+1 g Tifjn - H · [2 - wV(i, n) - yV(j, n)] ∀i ∈ I, j ∈ Ji, n ∈ N(n * N)

(15)

Tisjn+1 g Tisjn

∀i ∈ I, j ∈ Ji, n ∈ N(n * N)

(16)

Tifjn+1 g Tifjn

∀i ∈ I, j ∈ Ji, n ∈ N(n * N)

(17)

where H is the time horizon of interest. Constraints 15-17 show that task i starting at event point n + 1 should start after the end of the same task performed in the same unit j, which has already started at event point n. 5. Time Sequence Constraints for Different Tasks in the Same Units. Tisjn+1 g Ti'f jn - H · [2 - wV(i', n) - yV(j, n)] ∀j ∈ J, i ∈ Ij, i' ∈ Ij, i * i', n ∈ N(n * N)

(18)

Constraint 18 shows that when task i and i′ are performed at the same unit, task i should be performed after the end of task i′, which has already started at event point n.

240


6. Time Sequence Constraints for Completion of Previous Tasks. Tisjn+1 g

∑∑

n'en i'∈Ij

(Ti'f jn' - Ti's jn')

∀j ∈ J, i ∈ Ij, n ∈ N, n' ∈ N, n * N

(19)

Constraint 19 represents that task i can only start after the completion of all tasks performed in previous event points at the same unit j. 7. Time Horizon Constraints. Tisjn e H

∀i ∈ Ij, j ∈ J, n ∈ N

(20)

Tifjn e H

∀i ∈ Ij, j ∈ J, n ∈ N

(21)

Constraints 20 and 21 show that every task i should start and end within the time horizon H. To integrate the batch schedules and WAN in one step, we should further specify the status of all batch operations in each interval (∆t). Therefore the following two groups of constraints are imposed. i. Operations with Identical Charging and Discharging. For operations with identical charging and discharging, binary variable w(inj ,t) is introduced to specify if inj is performed in interval t. Their values can be expressed mathematically as follows: 1 · (∆t · t - Tisjn - ε) · (Tifjn - ∆t · t) < w(inj , t) H2 i ∈ SB ∪ UB ∪ OB, j ∈ Ji, n ∈ N, t ∈ T

(22)

1 < (∆t · t - Tisjn) + 1 H i ∈ SB ∪ UB ∪ OB, j ∈ Ji, n ∈ N, t ∈ T

(23)

1 f (T n - ∆t · t) + 1 H ij i ∈ SB ∪ UB ∪ OB, j ∈ Ji, n ∈ N, t ∈ T

(24)

w(inj , t)

w(inj , t) e

where ε is a sufficient small positive number. Constraint 22 ensures that w(inj ,t) is 1 if t locates between, not including, the starting and ending points. If t is at the starting point, constraint 23 will enforce w(inj ,t) to zero. On the other hand, if t is at the ending point, constraint 23 will enforce w(inj ,t) to 1. For other cases, w(inj ,t) will be set to zero by these three constraints. ii. Operations with Nonidentical Charging and Discharging. For operation with nonidentical charging and discharging (operation ua), another variable T imjn should be introduced to specify the time boundaries of charging and discharging. Such boundary points can be determined as follows: Timjn ) γijn · ∆t

∀i ∈ I, j ∈ Ji, n ∈ N

1 1 s (T n + Tifjn) e Timjn < (Tisjn + Tifjn + ∆t) 2 ij 2 ∀i ∈ I, j ∈ Ji, n ∈ N win(inj ,t)

wout(inj ,t),

(25)

(29)

1 · (∆t · t - Timjn - ε) · (Tifjn - ∆t · t) < wout(inj , t) H2 i ∈ UA, j ∈ Ji, n ∈ N, t ∈ T

(30)

1 (∆t · t - Timjn ) + 1 H i ∈ UA, j ∈ Ji, n ∈ N, t ∈ T

(31)

1 f (T n - ∆t · t) + 1 H ij i ∈ UA, j ∈ Ji, n ∈ N, t ∈ T

(32)

wout(inj , t)
0). The existence of all other pipelines can be identified through constraints 95-97. 3.3. Objective Function. The objective function in this scheme is to maximize the overall profit of a production cycle, taking into account the profit of production and costs associated with freshwater supply, wastewater treatment, buffer tanks as well as piping and junctions to link the entire network. To facilitate a clear-cut description, the objective function is decoupled into several terms and they will be quantified separately as follows. 1. Net Income in Production. The net profit in a production cycle is expressed as follows: sp

2 b

jun∈JUN

t∈T

∑ ∑ pr

· ∆t

t∈T

(102)

∑ nfs(jun', jun, t) e M·ne(jun', jun)

sp∈Sp n∈N

in tr,t

tr

1 b

(96)

net profit in production )

∑ pr ∑ f

∑ [pr ·nb(b) + pr

∀e, e' ∈ E

jun, jun' ∈ JUN

(100)

(101)

t∈T

ne(jun', jun) e

· ∆t

t∈T

tr∈Tr

t∈T

ne(e, e') e

in sa,t

sa

sa

∑ nfs(jun, e, t) e M · ne(jun, e) ∀e ∈ EQ, jun ∈ JUN

∑ pr ∑ f

(93)

∑ nfs(e, jun, t) e M · ne(e, jun) ∀e ∈ EQ, jun ∈ JUN

where prsm and prsp represent the cost coefficients of raw materials (sm) and products (sp) respectively; dsp,n denotes the amount of product sp sold at event point n; rsm,n denotes the amount of raw material sm purchased at event point n. 2. Overall Cost of Network. The overall cost of networks is written mathematically as

b∈B

where Fsmin and Fsmax represent, respectively, the lower and upper bounds of the flow rate in the system. 5. Number of Pipelines. To wholly reflect the overall cost and complexity of the network, it is also desirable to incorporate the cost of pipelines. So it is necessary to determine the actual number of pipelines, and we have ne(e, jun) e

243

(98) Figure 3. Solution strategy.

244


Figure 4. Flowsheet for examples 1 and 2. Table 1. Process Data (1) of Examples 1 and 2

units

capacity (kg)

suitability

average processing time (h)

heater (j1) reactor 1 (j2) reactor 2 (j3) separator (j4)

100 50 80 200

heating (ub) reaction 1, 2, 3 (ua1, sb1, ob1) reaction 1, 2, 3 (ua2, sb2, ob2) separation (ua3)

1.0 2.0,2.0,1.0 2.0,2.0,1.0 2.0

Table 2. Process Data (2) of Examples 1 and 2

states

storage capacity in example 1 (kg)

storage capacity in example 2 (kg)

feed A, B, C (s1, s2, s3) hot A (s4), impure E (s7) int AB (s5) int BC (s6) product 1, 2 (s8, s9)

unlimited 1000 500 0 unlimited

unlimited 100, 200 200 150 unlimited

price per unit 10

20

3. Objective Function. The overall objective function can be built as follows: OBJ ) net profit in production - overall cost of network (104) 4. Solution Strategy Given all aforementioned features one may want to consider in the one-step design, the number of variables and constraints in this superstructure-based optimization process may overwhelm the capabilities of any available solvers. Therefore, the present model still calls for the development of a dedicated solution strategy owing to its complexity. In the present study, by incorporating both deterministic and stochastic components,29 a hybrid optimization strategy is developed with iterative procedures to enhance the solution quality. Specifically, the proposed solution procedure can be broadly divided into two stages. The first stage is to decompose the original MINLP scheme into a MILP-MINLP formulation that is solved in a sequential manner to provide feasible solutions as the initial starting points for the next stage. These solutions from this initialization stage are then refined in the original MINLP scheme to get a final optimal solution. In the whole process, the solvers employed are DICOPT30 for MINLP model as well as CPLEX and CONOPT for MILP and NLP problems, respectively. Furthermore, in stage two DICOPT is interfaced with MATLAB31 for controlling and executing evolutionary programming (EP) automatically. The complete solution strategy is illustrated in Figure 3.

In stage one, first consider the MILP problem which consists of constraints 1-32. The objective function in this scheme is to maximize the net profit, consists of linear equations, and assumes the form of eq 98. After generating the fixed initial s guesses of Tijn, this formulation is solved to obtain schedule schemes and then all relevant results are passed to the following MINLP model as parameters. The succeed MINLP which is made up of constraints 33-97 is focused on WAN synthesis, and with the initial guesses of a set of specified variables, it is solved against an objective function with the form of eq 99 that minimizes the overall cost of network. However, if the solution process of the MILP or MINLP model is not convergent, the search procedure should be restarted. The procedure is repeated until a predetermined number of feasible solutions can be obtained, and these feasible solutions are caught in MATLAB as the initial population in EA. In stage two, EA is introduced to provide an evolutionary region of initial values, which facilitate the identification by the DICOPT solver of the final optimum in the original MINLP scheme. The initial population generated in stage one consists of kpop individuals, each of which is represented by three vectors, namely Xl, Yl, and σl (l ∈ 1,2,...,kpop). Xl and Yl altogether constitute a point in the search space which has real and binary components, respectively: in out in in out in out X ) [Tisjn, Bijn, fiout n , fin,t, cin,k,t, cin,k,t, fjun,t, fjun,t, cjun,k,t, cjun,k,t] j ,t j j j

Y ) [wV(i, n), yV(j, n), n(b), n(tr), n(jun), ne(eq, eq')] and each σl is the coordinate deviation for an individual, which can be expressed as σl ) √OBJ(Xl, Yl) To facilitate the mathematical expression of the mutation operator, the components of Xl are written as a series, which is made up of all aforementioned components in a sequential order. Now, Xl can be expressed as follows: Xl ) [xl(1), xl(2), ..., xl(m)]

l ∈ 1, 2, ..., kpop

Then, according to Gaussian mutation operator, each individual l produces an offspring l′ as follows: σl'(q) ) σl(q) exp{τaN(0, 1) + τbNj(0, 1)} xl'(q) ) xl(q) + σl'(q)Nj(0, 1)

q ∈ 1, 2, ..., m

q ∈ 1, 2, ..., m

where N(0,1) denotes a standard Gaussian random variable (fixed for a given l), Nj(0,1) stands for an independent Gaussian random variable generated for each component q.


245

Table 3. Process Data (3) of Examples 1 and 2

equipments j1 (heater) j2 (reactor 1)

j3 (reactor 2) j4 (separator) sa b1 b2 tr1 tr2 oa

operations

inlet/outlet flow rate (m3/h)

inlet/outlet concn in example 1 (ppm)

inlet/outlet concn (k1, k2, k3) in example 2 (ppm)

ub ua1 sb1 ob1 ua2 sb2 ob2 ua3 sa b1 b2 tr1 tr2 oa

0-80 0-80 0-60 0-60 0-80 0-60 0-60 0-80 0-200 0-200 0-200 0-200 0-200 0-200

0/0-5 0-6/0-14 -/10 0-7/0-6/0-14 -/10 0-7/0-10/0-15 -/0 0-15/0-15 0-15/0-15 7-15/0-6 5-10/3-5 0-2.5/-

0, 0, 0/0-5, 0-4, 0-3 0-5, 0-4, 0-2/0-15, 0-11, 0-10 -,-,-/8,3,5 0-7, 0-5, 0-8/-,-,0-5, 0-4, 0-2/0-15, 0-11, 0-10 -,-,-/8,3,5 0-7, 0-5, 0-8/-,-,0-5, 0-5, 0-3/0-12, 0-8, 0-13 -,-,-/0, 0, 0 0-15, 0-12, 0-13/0-15, 0-12, 0-13 0-15, 0-12, 0-13/0-15, 0-12, 0-13 7-15,6-11,8-14/0-6, 0-5, 0-7 3-9,3-8,2-6/0-4, 0-3, 0-3 0-4, 0-2, 0-3/-,-,-

The control parameters, τa and τb, are chosen to be 1/(2kpop)1/2 and 1/(2(kpop)1/2)1/2, respectively. On the other hand, for Yl, each time one of the binary variables in each group (six groups in total) is selected randomly. Its value is then changed from 0 to 1 or vice versa. Finally, the fitness evaluation function assumes the form F(Xl, Yl) )

{

1 OBJ(Xl, Yl) MINLP feasible 1 MINLP infeasible

Here, OBJ(Xl,Yl) denotes the objective values obtained based on the initial values of Xl and Yl. It should be noted that the

Table 4. Design Specifications of the Network in Case I of Example 1 pipeline

flow rate (m3/h)

concentration (ppm)

time period (h)

sa-jun2 sa-jun2 sa-jun2 j1-jun1 j2-jun1 j3-jun1 j3-jun1 j4-jun1 tr1-jun2 tr1-jun2 jun1-j2 jun1-tr1 jun1-jun2 jun1-tr1 jun2-j3 jun2-j4 jun2-j1 jun2-j2 jun2-oa

86.67 13.33 11.67 50 22.33 20 36 16.67 17.79 58.33 50 17.79 18.88 58.33 20 16.67 50 50 70

0 0 0 4.67 10 14 10 15 4.34 3 4.67 14.46 14.46 10 0 0 0 7 2.5

0-1 1-2 2-4 0-1 2-4 1-2 2-4 1-2 1-2 2-4 0-1 1-2 1-2 2-4 0-1 0-1 0-1 1-2 2-4

Table 5. Comparisons between Case I in Example 1 and the Original Study

Figure 5. Optimal production scheme for case I in example 1.

overall cost (cost unit) freshwater cost (cost unit) treatment cost (cost unit) buffer tank cost (cost unit) network cost (cost unit) number of buffer tanks number of treatment units number of pipelines number of junctions/splitters and mixers

the present case

the original study

377.01 123.33 201.68 0 52 0 1 14 2

388.31 86.16 185.28 51.87 65 1 1 12 7

Table 6. Parameters for Batch Schedules in Case II of Examples 1 and 2a

i1 i2 i3 i4 i5 i6 i7 i8 Figure 6. Optimal WAN configuration for case I in example 1.

a

amount processed at each variable term

max number of variable terms

50 12.5 20 12.5 20 25 40 50

2 4 4 4 4 2 2 4

∆t is assumed to be 1/3 h in this study.

246


Figure 7. Optimal production scheme for case II in example 1. Table 7. Design Specifications of the Network in Case II of Example 1

Figure 8. Optimal WAN configuration for case II in example 1.

MINLP model may not be feasible based on certain pair of Xl and Yl, and for such case, 1 will be assigned to the fitness value. In the selection process, based on the q-Tournament approach and traditional winning function, kpop individuals from parent and offspring population with highest winning values are selected for the next generation. All the steps of EA procedure are repeated for improved solution until the iteration steps hit the upper limit. 5. Application Example To illustrate the relative merits of the proposed formulation, we considered the following examples. Example 1 was originally solved by Cheng and Chang24 with single contaminant based on discrete-time representation and the mixer-unit-splitter superstructure.32 Example 2 was tailored for a multicontaminant

pipeline

flow rate (m3/h)

concentration (ppm)

time period (h)

sa-jun1 j1-jun2 j2-jun1 j2-jun1 j2-jun2 j3-jun1 j3-jun1 j4-jun1 tr1-jun2 tr1-jun2 tr1-jun2 b1-jun1 b1-jun1 jun1-j1 jun1-j4 jun1-tr1 jun1-tr1 jun1-tr1 jun1-b1 jun1-b1 jun1-jun2 jun1-jun2 jun1-jun2 jun2-j2 jun2-j3 jun2-j3 jun2-j3 jun2-oa

45.97 45.97 11.4 18.73 18.73 30 34.57 69.6 26.08 27.33 27.24 23.64 41.29 45.97 69.6 26.08 27.33 27.24 32.61 16.04 10.91 13.96 5.45 11.4 34.57 55.72 60.02 32.69

0 4.7 14 10 10 10 8.68 15 1.5 1.4 1 10 13.95 0 10 15 13.95 10 15 10 15 13.95 10 4.7 4.7 7 7 2.5

0-1.33 0-1.33 1.33-2.67 5.33-8 2.67-5.33 5.33-8 1.33-2.67 2.67-4 2.67-4 4-5.33 5.33-8 1.33-2.67 4-5.33 1-1.33 1.33-2.67 2.67-4 4-5.33 5.33-8 2.67-4 5.33-8 2.67-4 4-5.33 5.33-8 0-1.33 0-1.33 2.67-4 4-5.33 5.33-8

WAN and studied by Zhou et al.25 based on the original state space approach. A set of common backgrounds and model parameters are used in both examples. Specifically, a common flowsheet is adopted (see Figure 4); the minimum market demand for products and min and the maximum supply for materials at any event point (Ds,n max Rs,n ) are designed to be 0 and 120 kg, respectively; the cost of fresh water (prsa) is 1 unit/m3; the installation cost model for a


247

Table 8. Design Specifications of the Network in Case I of Example 2

Figure 9. Batch production scheme for case I in example 2.

Figure 10. Optimal WAN configuration for case I in example 2.

conventional buffer tank is taken to be 4.8 + 6(Vbmax)0.6 units; the costs of each junction and pipeline are assumed to be 5 and 3 units, respectively; all relevant water loss and water loss rates are set to 0; the lower and upper bounds of the flow rate in the system were chosen to be 3 and 100 m3/h, respectively; the minimum and maximum of the throughput of all junctions are selected to be 0 and 200 m3/h, respectively, and the minimum and maximum number of streams connected to junctions min max ,Njun ) are assigned to be 3 and 20, respectively. Finally, (Njun in the first case of example one, all proportional parameters here are adjusted so as to make the mass loads and flow rates keep in the same level with the original study. For other cases, proportional parameters are adopted according to the previous work;21 specifically, λsb and λob are assigned to 1; λua,k and λub,k are set to 3, 2.8, 2.5 and 4, 3.2, 2.4, respectively, on the basis of the three different kinds of pollutants. 5.1. Example 1. Consider the process data (see Tables 1-3) first presented by Cheng and Chang. In a latter study, when network complexity was taken into account, Zhou et al. obtained a batch WAN design, of which the profit of batch production and cost of network were 1366.7 and 388.31 units, respectively. In the present example, two cases are provided. Case I is intended to compare with the original computational results; while case II is studied to demonstrate the validity and advantages of our novel schedule scheme. In case I, to be able to compare different strategies on the same basis, the following parameters and equations should be manipulated. Specifically, constraints 11-14 are removed; ∆t

pipeline

flow rate (m3/h)

concentration k1, k2, k3 (ppm)

time period (h)

sa-jun1 sa-jun1 sa-jun1 sa-jun1 sa-jun2 sa-jun2 sa-jun2 j1-jun2 j2-jun1 j2-jun2 j2-jun2 j3-jun1 j3-jun2 j4-jun1 b1-jun1 b1-jun2 b1-jun2 b1-jun2 b1-jun2 b1-jun2 jun1-j1 jun1-j2 jun1-j3 jun1-b1 jun1-b1 jun1-jun2 jun1-jun2 jun2-j3 jun2-j3 jun2-j4 jun2-oa jun2-oa jun2-oa

100 36.78 73.13 3.69 52.02 14.54 15.22 70 15.54 18.8 15.54 11.33 21.55 51.46 13.19 9.63 7.64 26.78 29.26 5.48 70 28.2 22.66 51.46 100 30 29.66 60.02 60.02 51.46 98.52 100 35.14

0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 2.9, 2.32, 1.74 8,3,5 8,3,5 8,3,5 12.59, 10.43, 10 8, 3, 5 10.11, 8, 7.64 2.77, 1.74, 1.99 2.77, 1.74, 1.99 10.11, 8, 7.64 10.11, 8, 7.64 10.11, 8, 7.64 10.11, 8, 7.64 0, 0, 0 6.36, 5, 4.95 2.38, 0.89, 1.49 10.11, 8, 7.64 2.67, 1.65, 1.91 0, 0, 0 2.38, 0.89, 1.49 7, 4.5, 4.96 7, 4.68, 5.02 3.68, 2, 2.28 3.68, 2, 2.28 4, 1.82, 2.6 3.58, 2, 2.45

0-1.026 3.387-4.69 4.69-4.927 4.927-6 1.026-2.052 2.052-2.66 2.66-3.387 0-1.026 3.387-4.927 0-2.66 2.66-3.387 4.69-6 0-2.052 1.026-2.052 4.927-6 0-1.026 1.026-2.052 2.052-2.66 2.66-3.387 3.387-4.69 0-1.026 4.927-6 3.387-4.69 1.026-2.052 4.69-4.927 0-1.026 3.387-4.69 2.052-2.66 2.66-3.837 0-1.026 0-1.026 1.026-2.052 3.387-4.69

is assumed to be 1; the removal ratios of tr1 and tr2 are 0.7, and the corresponding cost is set to 1.5 and 2 units/m3, respectively; all constant terms of processing time (Ri,j) are set to the average processing time while the variable terms (βi,j) are taken to be zero. The typical MINLP model for case I consists of 1820 variables (with 585 binary variables) and 2318 constraints. Solving this MINLP model with a time of horizon of 4 h needs three major iterations and yields the most appropriate schedule, as well as the network configuration (see Figures 5 and 6, respectively). It can be found that while the schedule scheme (see Figure 5) is the same as the former study, the network configuration (see Figure 6) is subjected to major changes, as the multistage mixing and splitting stream from jun1-jun2 can be clearly identified. Detailed network designs and comparisons of key features between the present case and the original one are summarized in Tables 4 and 5. It can be concluded from the results that our proposed strategy is clearly superior to the former one. This can be attributed to the fact that by allowing multistage mixing and splitting, the economic tradeoff issues in WAN design can be carried out effectively. More specifically, although the freshwater and treatment costs are larger than those in the former study, the cost of buffer tanks is drastically reduced to 0 unit, so as to lower the overall cost in a production cycle. Furthermore, as evident from the network structure, the buffer tank is not embedded in the optimal network configuration, and this can demonstrate that buffer tanks are not necessary in some batch WAN if optimal production sequences and operating policies of water flows are adopted. Finally, compared with the original study, 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.

248


Figure 11. Optimal production scheme for case II in example 2. Table 9. Design Specifications of the Network in Case II of Example 2

Figure 12. Optimal WAN configuration for case II in example 2.

Case II is studied to show the advantages of our schedule scheme. Notice in this example, the removal ratios of tr1 and tr2 are set to 0.9 and 0.95, respectively; the corresponding cost is assumed to 0.75 and 1 units/m3, respectively; and all relevant parameters concerned with schedules are listed in Table 6. With a time horizon of 8 h, this model has 6824 continuous and 3181 binary variables, as well as 14132 constraints. Four major iterations between NLP subproblem and MILP master problem were needed to get to the optimal solution. The optimal production schedule, as well as the network structure and its detailed design, is illustrated, respectively, in Figures 7-8 and Table 7. Note that this network is assembled with 2 junctions and 16 pipelines. The profit of production and cost of network were estimated to be 2108.57 and 298.34 units, respectively. Specifically, the costs of freshwater supply, wastewater treat-

pipeline

flow rate (m3/h)

concentration k1, k2, k3 (ppm)

time period (h)

sa-jun1 sa-jun1 sa-jun2 sa-jun2 j1-jun2 j2-jun2 j2-jun2 j2-jun2 j3-jun2 j3-jun2 j4-jun1 tr1-jun1 tr1-jun2 b1-jun2 b1-jun2 jun1-j1 jun1-j4 jun1-tr1 jun1-oa jun2-j2 jun2-j3 jun2-j3 jun2-j3 jun2-tr1 jun2-b1 jun2-b1 jun2-b1 jun2-jun1 jun2-jun1 jun2-oa

43.21 20.96 20.67 6.68 43.21 18.73 18.73 7.313 13.2 30 52.39 20.52 57.55 3 36.99 43.21 52.39 57.55 20.52 7.37 60.02 55.72 13.3 20.52 4.92 17.3 18.05 31.43 5.16 89.75

0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 5, 4, 3 8, 3, 5 8, 3, 5 11.79, 11, 9.82 11.79, 11, 9.82 8, 3, 5 11.44, 8, 8.54 1.18, 1.65, 0.98 1.08, 1.12, 0.8 5.95, 2.57, 3.74 5.95, 2.57, 3.74 0, 0, 0 4.8, 1.8,3 10.78, 7.45, 8 1.18, 1.65, 0.98 0, 0, 0 5.91, 3.65, 3.61 6.64, 2.71, 4.16 0, 0, 0 11.79, 11, 9.82 5.91, 3.65, 3.61 8, 3, 5 4, 1.86, 2.56 8, 3, 5 4, 1.86, 2.56 4, 1.86, 2.56

0-1.33 5.33-6.67 2.67-4 6.67-8 0-1.33 0-2.67 5.33-8 4-5.33 4-5.33 5.33-8 6.67-8 4-5.33 6.67-8 0-1.33 1.33-2.67 0-1.33 5.33-6.67 6.67-8 4-5.33 2.67-4 0-1.33 1.33-2.67 2.67-4 4-5.33 0-1.33 5.33-6.67 6.67-8 5.33-6.67 6.67-8 6.67-8

ment, buffer tanks, junctions, and pipelines are found to be 61.33, 108.04, 71.17, and 58 units, respectively. It is worth mentioning that our schedule strategy has its unique advantages. First, it contains the discrete time representation as a special case and it is much more flexible as operating times are dependent on the amount processed. Then, unlike continuous time formulation, it can provide unified time durations (∆t in


our example) to identify the overall number of time intervals in which operating conditions of the batch production and WAN are constant. On the other hand, it is obvious that the state time space framework for batch production is more effective than traditional approaches, as it illustrates all basic elements (states, tasks, equipment, and time) in a single and comprehensive representation. 5.2. Example 2. In this example, multicontaminant system design options are incorporated (also see the process data in Tables 1-3). Case I is first introduced to explore the optimal batch WAN design under a predefined schedule, while case II is performed for the purpose of exhibiting the capability of our model when all design options are considered. Let us first consider the predefined schedule (as shown in Figure 9) for case I. Suppose the removal ratios of tr1 and tr2 for the three pollutants are 0.7, 0.5, 0.6 and 0.7, 0.6, 0.8, respectively, and the corresponding costs are 1.5 and 2 units/ m3, respectively. When our proposed methods are applied, a more cost-effective network configuration can be easily identified (see Figure 10) with three major iterations. This network presented is assembled with 2 junctions and 18 pipelines. Its corresponding cost in a production cycle can be reduced to 372.76 units, which represents a 4% improvement. Specifically, the costs of freshwater supply, wastewater treatment, buffer tanks, junctions, and pipelines were 245.09, 0, 63.67, and 64 units, respectively. Detailed operating policies of water flows in the network can be found in Table 8. Note that wastewater treatment units are not selected in the optimal network structure, and this is because using treated wastewater might not be desirable when treatment costs are set against freshwater cost. In the last case study (case II), the removal ratios of tr1 and tr2 for the three pollutants are chosen to be 0.9, 0.85, 0.9 and 0.95, 0.9, 0.95, respectively; the corresponding costs are set to 0.75 and 0.95 units/m3, respectively; the parameters concerned with schedules are the same as those in case II of the above example. Under this condition, the corresponding model with an 8 h time horizon entails 19004 constraints, as well as 11472 continuous and 3181 binary variables. Five major iterations were required to find the optimal solution. The schedule results and its corresponding network are shown in Figure 11 and 12. Design specifications for this network are itemized in Table 9. 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. On the basis of the designs produced above, it can be concluded that the STS superstructure for WAN, which enables multistage mixing and splitting, is more effective than previous superstructures and that our method is indeed suitable for cost-optimal designs of complex and large scale multicontaminant systems. At the end of our examples, we have to mention that the global optimal solutions of all cases cannot be guaranteed, because of the nonlinearity and nonconvexity of the proposed mathematical model. 6. Conclusions Two state-time-space superstructures are introduced to provide a complete, integrated, and interconnected framework for modeling, optimization, and illustration of batch WAN in both time and space dimensions. The advantage of this approach is that not only all possible alternative network topologies can

249

be captured but the operating conditions of the overall system in any time can also be properly described. To ensure the quality and efficiency of the solutions, a hybrid optimization strategy integrating DICOPT and EA search techniques is developed. Finally, two examples dealing with single- and multicontaminant WAN are presented to demonstrate the feasibility and effectiveness of the proposed automated design method. Acknowledgment The authors would like to acknowledge the financial support provided by the National Natural Science Foundation of China, under Grant No. 20876020. Nomenclature Sets S ) set of all involved material states Sm ) set of all raw materials Sp ) set of all final products SB ) set of operations which generate wastewater in batch production UA ) set of operations in batch production with nonidentical charging and discharging time intervals UB ) set of operations in batch production with identical charging and discharging time intervals OB ) set of operations in batch production which only consume water I ) set of all operations in batch production, I ) SB∪UA∪UB∪OB J ) set of all involved units in batch production SA ) set of freshwater sources B ) set of buffer tanks Tr ) set of wastewater treatment units OA ) set of water sinks U ) set of all units not involved in batch production, U ) SA∪TR∪B∪OA JUN ) set off junctions N ) set of all event points within the time horizon Ijn ) set of operation i performed in j(j ∈ Ji) at event point n E ) set of all units, E ) U∪J EQ ) set of all units and junctions, EQ ) SA∪TR∪B∪OA∪J∪JUN T ) set of all intervals of equal durations, T ) {t|t ) 1,2,3,...,T(H/ ∆t)} Is ) set of operations which either produce or consume state s Ji ) set of units where operation i can be performed Ij ) set of operations which can be performed in unit j K ) set of all pollutants in water-allocation network Parameters min max , Bi,j ) the minimum and maximum capacities of unit j when Bi,j used for performing operation i c p Fs,i , Fs,i ) the proportions of state s consumed, produced from operation i, respectively STsmax ) the maximum storage capacity for state s min Ds,n ) the minimum market demand for state s at event point n max Rs,n ) the maximum supply of state s at event point n Ri,j, βi,j ) the constant and variable terms of processing time of operation i at unit j Bi,j ) the amount processed at each variable term of operation i at unit j bijn ) number of variable term needed for operation ijn H ) the fixed time horizon prsm,prsp ) the cost coefficients of raw materials and products, respectively

250


Fin,max , Fout,max ) the upper bounds of the inlet and outlet flow rates e e of e out ) the concentrations of pollutant k in water generated from Csa,k sa in,max out,max , Cu,k ) the upper bounds of the inlet and outlet Cu,k concentrations of pollutant k of u Fiout,max, Fiout,min ) the upper and lower bounds of water generating rate of operation i Fiin,max, Fiin,min ) the upper and lower bounds of water consuming rate of operation i out,max out,min , Ci,k ) the upper and lower bounds of the concentration Ci,k of pollutant k in water generated from operation i in,max in,min , Ci,k ) the upper and lower bounds of the concentration Ci.k of pollutant k in water consumed by operation i in,max in,min , Fjun ) the upper and lower bounds of the inlet flow rate Fjun of jun max min , Njun ) the upper and lower bounds of the number of streams Njun connected to jun Fsmax, Fsmin ) the upper and lower bounds of the flow rates in the system Rtr,k ) the removal ratio of pollutant k in tr loss loss , Cua,k ) the fixed amount of water loss and relevant DVua concentrations in ua loss ) the fixed water loss rate and relevant concentrations in ub dVub loss loss , ctr,k,t ) the fixed water loss rate and relevant concentrations in ftr,t tr Vbmax, Vbmin ) the upper and lower bounds of the designing volumes of b prsa, prtr ) the cost coefficients of freshwater and wastewater treatment prjun, prpipe ) the cost coefficients of junctions and pipelines prb1, prb2, prb3 ) the cost coefficients of buffer tanks ∆T ) the length of each variable interval λi,k ) the proportional coefficient correlating mass loads/flow rate and amount processed of operation i Continuous Variables Bijn ) amount of material undertaking operation ijn STsin ) initial amount of state s STs,n ) amount of state s at event point n rs,n, ds,n ) amount of state s purchased and sold at event point n, respectively s f Tijn, Tijn ) time that operation ijn starts and finishes m Tijn ) time boundaries of charging and discharging for ijn Muajn,k,t ) the accumulated mass load of pollutant k in uajn µubjn,k ) the nominal instantaneous mass load of pollutant k in ubjn mubjn,k,t ) the instantaneous mass load of pollutant k in ubjn in interval t in out ,feq,t ) the inlet and outlet flow rates of eq in interval t feq,t in out , ceq,k,t ) the concentrations of pollutant k in waters consumed ceq,k,t and generated by eq in interval t Vb,t ) the water volume of b in interval t in out fijn,t, fijn,t ) the water consuming and generating rates of ijn in interval t ) the nominal inlet and outlet flow rates of ijn Fiinjn , Fiout n j in out cijn,k,t, cijn,k,t ) the concentrations of pollutant k in the input and output streams of ijn in interval t fseq,eq′,t ) the flow rate in stream from eq to eq′ in interval t Binary Variables wV(i,n) ) binary variable used to signify if operation i is performed at event point n yV(j,n) ) binary variable used to signify if unit j is utilized at event point n

win(inj ,t), wout(inj ,t) ) binary variables used to denote if inj is consuming or generating water in interval t w(ijn,t) ) binary variable used to denote if ijn is being performed in interval t n(b),n(tr),n(jun) ) binary variables used to denote if b, tr or jun is selected in the optimal network configuration w(jun,t) ) binary variable used to denote if jun is occupied in interval t nfs(eq,eq′,t) ) binary variable used to denote if actual stream exists between eqand eq′ in interval t ne(eq,eq′) ) binary variable used to denote if pipeline between eq and eq′ is selected in optimal network configuration

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ReceiVed for reView March 16, 2009 ReVised manuscript receiVed October 16, 2009 Accepted October 23, 2009 IE900427B

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