Unified Approach for the Optimization of Energy and Water in

Article pubs.acs.org/IECR

Unified Approach for the Optimization of Energy and Water in Multipurpose Batch Plants Using a Flexible Scheduling Framework Omobolanle Adekola,† Jane D. Stamp,† Thokozani Majozi,*,†,‡ Anurag Garg,§ and Santanu Bandyopadhyay⊥ †

Department of Chemical Engineering, University of Pretoria, Lynnwood Road, Pretoria, 0002, South Africa Modelling and Digital Science, CSIR, Meiring Naudé Road, Pretoria, 0002, South Africa § Centre for Environmental Science and Engineering and ⊥Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India ‡

ABSTRACT: A unified framework providing for the simultaneous reduction of energy and wastewater in multipurpose batch plants is presented in this paper. Unlike many current methods, the schedule is not predefined. Time is treated as a variable leading to the optimization of the schedule together with water and energy usage simultaneously. Since a flexible process schedule is employed, an improved result in the form of a better overall production schedule compared to schedules obtained from optimizing water and energy separately is achievable. The ability of the current method to handle industrial scale problems is also highlighted using a complex case study. Opportunities for direct water reuse and indirect water reuse, using wastewater storage, are explored as well as direct and indirect heat integration for reducing external utilities. The objective is to improve the profitability of the plant by minimizing wastewater generation and utility usage. The results from three examples are presented.

1. INTRODUCTION Utility and water requirements in many batch plants, such as in the food industry, breweries, dairies, biochemical plants, and agrochemical facilities, contribute largely to their overall running costs.1−4 Heating and cooling are unavoidable aspects of many chemical processing facilities, with operations where heat is generated and others where heat is required. It is because of this occurrence that heat integration becomes a possibility. Process equipment cleaning is usually associated with large amounts of water, due to the inherent sharing of equipment by different tasks. Hence, efficient use of water for process equipment cleaning is pursued to ensure that the amount of freshwater consumed as well as the amount of wastewater generated are minimized. Minimizing energy use and water consumption is influenced by the need to comply with stricter environmental regulations, counter high energy prices, and conserve scarce environmental resources. Published literature exists for the independent consideration of wastewater minimization5−7 and heat integration8,9 in batch plants. The simultaneous consideration of both wastewater minimization and heat integration has been largely ignored, due to the potential complexities of such a problem. This problem has been addressed in continuous processes using graphical methods10,11 and mathematical optimization.12−15 Halim and Srinivasan16 developed a framework that integrated scheduling, direct heat integration, and direct water reuse optimization based on a sequential technique. First, the process schedule was optimized to meet an economic objective. This was followed by the generation of alternate schedules through a stochastic search-based integer cut procedure. These alternate schedules were used to optimize energy and water consumption with the aim of retaining the optimality of the scheduling solution. Next, heat integration analysis and water reuse synthesis were each © XXXX American Chemical Society

performed to optimize water and energy requirements on each schedule. In this paper, a unified framework to optimize both water and energy usage within the scheduling problem is developed. Due to the fact that a flexible process schedule is employed, an improved result in the form of a better overall production schedule compared to the schedules obtained from optimizing water and energy separately is achievable. With respect to wastewater minimization, opportunities for direct water reuse and indirect water reuse utilizing wastewater storage are explored. With respect to energy optimization, both direct and indirect heat integration are explored for reducing external utility requirements. The objective is to improve the profitability of the plant by minimizing wastewater generation and utility usage.

2. PROBLEM STATEMENT The problem addressed in this paper involves the optimization of water and energy within a production scheduling framework. In production scheduling, the objective is either profit maximization or minimization of makespan. During production, certain tasks require cooling or heating, for instance, exothermic reactions that require cooling or endothermic reactions that require heating. The requirements for heating or cooling afford opportunities for heat integration. In multipurpose batch plants, particularly at the completion of a processing task in a unit, the equipment unit is washed before performing subsequent tasks. This is to ensure product integrity through prevention of contamination. The washing operations present Received: October 4, 2012 Revised: May 13, 2013 Accepted: May 24, 2013

A

dx.doi.org/10.1021/ie302500t | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Figure 1. Superstructure for mathematical formulation: (a) when units perform heating/cooling tasks; (b) when units perform washing tasks.

Determine an optimal production schedule that achieves a maximum profit or minimum makespan, requiring the minimum amount of external utilities and freshwater use.

the opportunity for wastewater minimization. It is assumed that no heat or water losses occur. The problem can be stated as follows: Given: Scheduling data (i) production recipe for each product (ii) available units and their capacities (iii) maximum storage capacity for each material (iv) task durations (v) time horizon of interest or predetermined production (vi) costs of raw materials (vii) selling price of final products Heat integration data (i) hot duties for tasks requiring heating and cold duties for tasks that require cooling (ii) operating temperatures of heat sources and heat sinks (iii) minimum allowable temperature differences (iv) heat capacities of materials (v) costs of hot and cold utilities (vi) design limits on heat storage Wastewater minimization data (i) washing time for each unit (ii) mass load of contaminants in each unit (iii) maximum inlet and outlet concentrations of contaminants for each unit (iv) maximum storage available for water reuse (v) cost of freshwater (vi) cost of effluent treatment

3. MATHEMATICAL FORMULATION The mathematical formulation consists of the three modules given in the problem statement, i.e. production scheduling, heat integration, and wastewater minimization modules. The mathematical formulation comprises the following sets, variables, parameters, and constraints: Sets. C {c|c = contaminant} J {j|j = processing unit} Jc {jc|jc = processing unit which may conduct tasks requiring heating} ⊆ J Jh {jh|jh = processing unit which may conduct tasks requiring cooling} ⊆ J Jw {jw|jw = processing unit which requires washing after conducting a task} ⊆ J P {p|p = time point} S {s|s = any state} Sin {sin|sin = input state into any unit} Sin,j {sin,j|sin,j = input state to a processing unit} ⊆ S Sout {sout|sout = output state from any unit} Sout,j {sout,j|sout,j = output state from a processing unit} ⊆ S Sout,j,j′ {sout,j,j′|sout,j,j′ = recycled state from unit j to unit j′} ⊆ S U {u|u = heat storage unit} B



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Continuous Variables. batch size, either fixed or variable B(sin,j, p) cin(sin,j, c, p) inlet concentration of contaminant c, to unit j ∈ Jw at time point p cout(sout,j, c, p) outlet concentration of contaminant c, from unit j ∈ Jw at time point p cooling load for hot state CL(sin,jh, p) CL̇ (s cooling load per time, for hot state in,jh, p) csin(c, p) inlet concentration of contaminant c, to storage at time point p csout(c, p) outlet concentration of contaminant c, from storage at time point p external cooling required by unit jh cw(sin,jh, p) conducting the task corresponding to state sin,jh at time point p d(s, p) amount of state delivered to customers at time point p dur(sin,j, p) duration of task, dependent on batch size H time horizon of interest, optimization variable for makespan minimization problem heating load for cold state HL(sin,jc, p) heating load per time, for cold state HL̇ (sin,jc, p) MB(sin,j, c, p) mass load of contaminant c in unit j ∈ Jw at time point p after processing sin,j that is added to the water stream mp(sout,j, p) amount of state sout,j produced at time point p mu(sin,j, p) amount of state sin,j consumed at time point p msin(j, p) mass of water transferred from unit j ∈ Jw to storage at time point p msout(j, p) mass of water transferred from storage to unit j ∈ Jw at time point p mwe(sout,j, p) mass of effluent water from unit j ∈ Jw at time point p mwf(sin,j, p) mass of freshwater into unit j ∈ Jw at time point p mwin(sin,j, p) mass of water into unit j ∈ Jw for washing at time point p mwout(sout,j, p) mass of water exiting unit j ∈ Jw at time point p after washing mwr(sout,j,j′, p) mass of water recycled/reused from j to unit j′ (j,j′ ∈ Jw) at time point p Q(sin,j, u, p) heat exchanged with heat storage unit u at time point p q(sin,jh, sin,jc, p) amount of heat exchanged during direct heat integration qs(s, p) amount of state s stored at time point p qws(p) amount of water stored in storage at time point p external heating required by unit jc st(sin,jc, p) conducting the task corresponding to state sin,jc at time point p T0(u, p) initial temperature in heat storage unit u at time point p Tf(u, p) final temperature in heat storage unit u at time point p t0(sin,j, u, p) time at which heat storage unit commences activity tf(sin,j, u, p) time at which heat storage unit ends activity

tout(sout,j, p) tu(sin,j, p) tsin(j, p) tsout(j, p) twin(sin,j, p) twout(sout,j, p) twr(sout,j,j′, p) W(u) Γ(sin,j, u, p) Ψ(sin,j, u, p)

time at which a state is produced from unit j at time point p time at which a state is used in unit j time at which water is transferred from unit j ∈ Jw to storage at time point p time at which water is transferred from storage to unit j ∈ Jw at time point p time at which water enters unit j ∈ Jw at time point p time at which water exits unit j ∈ Jw at time point p time at which water is recycled from unit j to unit j′(j,j′ ∈ Jw) at time point p capacity of heat storage unit u Glover transformation variable reformulation−linearization variable

Binary Variables. x(sin,jc, sin,jh, p) binary variable associated with heat integration between unit jc conducting the task corresponding to state sin,jc and unit jh conducting the task corresponding to state sin,jh, at time point p y(sin,j, p) binary variable associated with usage of state s in unit j for production at time point p ysin(j, p) binary variable showing transfer of water from unit j ∈ Jw to storage at time point p ysout(j, p) binary variable showing transfer of water from storage to unit j ∈ Jw at time point p yw(sin,j, p) binary variable showing usage of water in unit j ∈ Jw at time point p ywr(sout,j,j′, p) binary variable showing reuse/recycle of water from unit j to unit j′ (j,j′ ∈ Jw)at time point p z(sin,j, u, p) binary variable associated with heat integration between unit j conducting the task corresponding to state sin,j with heat storage unit u at time point p Parameters. A(c) contaminant loading CUin(sin,j, c) maximum inlet concentration of contaminant c in unit j ∈ Jw CUout(sout,j, c) maximum outlet concentration of contaminant c from unit j ∈ Jw CE cost of effluent water treatment CF cost of freshwater fixed cooling load for hot state CL(sin,jh) Cost_cw cost of cooling water Cost_st cost of steam Cpfluid specific heat capacity of heat storage fluid Cpstate(sin,j) specific heat capacity of state CP(s) selling price of product s, s = product CSoout(c) initial concentration of contaminant in storage H time horizon of interest, a parameter for profit maximization problem fixed heating load for cold state HL(sin,jc) M(sin,j, c) mass load of contaminant c in unit j ∈ Jw after processing sin,j that is added to the water stream MM any large number MUw (sin,j) maximum inlet water mass of unit j ∈ Jw C



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Qmax(sin,j)

Constraints 1 and 2 are active simultaneously and ensure that one hot unit will be integrated with one cold unit when direct heat integration takes place, in order to simplify operation of the process. Also, if two units are to be heat integrated at a given time point, they must both be active at that time point. However, if a unit is active, it may operate in either integrated or standalone mode.

maximum possible heating load or cooling load for a cold state or hot state, respectively Qos (s) initial amount of state s in storage QUs (s) maximum capacity of storage for state s Qwos initial amount of water in storage QwUs maximum capacity of storage for water T(sin,j) operating temperature for processing unit j conducting the task corresponding to state sin,j, for constant temperature processes TL lower bound for heat storage temperature TU upper bound for heat storage temperature Tin(sin,j) inlet temperature for state sin,j Tout(sin,j) outlet temperature for state sin,j ΔTmin minimum allowable thermal driving force minimum capacity of a unit Vmin j Vmax maximum capacity of a unit j WL lower bound for heat storage capacity WU upper bound for heat storage capacity α constant coefficient of processing time β variable coefficient of processing time τ(sin,j) duration of the task corresponding to state sin,j conducted in unit j τw(sout,j) duration of washing for unit j ∈ Jw The necessary heat integration and wastewater minimization constraints are embedded in the scheduling framework. The model uses the scheduling framework of Majozi and Zhu,17 which features the state sequence network (SSN) recipe representation and an uneven discretization of the time horizon. It is a continuous-time unit specific event based formulation, where the time associated to events can be different across the units. This scheduling framework has proven to result in improved mathematical models compared to formulations based on other representations. The necessary scheduling constraints have been included in the Appendix. Recent state of the art scheduling techniques include those of Shaik and Floudas,18 Susarla et al.,19 and Seid and Majozi.20 Reviews on other scheduling methods may be obtained from the literature.21 The heat integration constraints of Stamp and Majozi9 are presented first, followed by the necessary wastewater minimization constraints of Adekola and Majozi.7 The mathematical model is based on the superstructures in Figure 1. Figure 1a is the superstructure for heat integration while Figure 1b is the superstructure for water optimization. Heat Integration Constraints. The heat integration constraints are based on the superstructure in Figure 1a. In Figure 1a, each processing unit may operate using either direct or indirect heat integration. Direct heat integration refers to the use of heat generated from a processing unit to supply a processing unit requiring heat without use of storage. Indirect heat integration refers to the use of heat previously stored in a heat storage vessel to supply a processing unit requiring heat. The heat is stored through a heat transfer fluid medium, e.g. water. Processing units may also operate in standalone mode, using only external utilities. This may be required for control reasons or when thermal driving forces or time do not allow for heat integration. If either direct or indirect heat integration is not sufficient to satisfy the required duty, external utilities may make up for any deficit. Constraints 1−39 constitute the heat integration model, useful for multipurpose batch processes. The formulation is based on the model by Stamp and Majozi9 and includes direct and indirect heat integration.

∑ x(sin,j , sin,j , p) ≤ y(sin,j , p), c

h

h

∀ p ∈ P , s in, j ∈ Sin, j h

s in, j

c

(1)

∑ x(sin,j , sin,j , p) ≤ y(sin,j , p), c

s in, j

h

c

∀ p ∈ P , s in, j ∈ Sin, j c

h

(2)

Constraints 3 and 4 ensure that heat integration between a unit and heat storage may occur only if the unit is active at that time point. However, if a unit is active, it will not necessarily integrate with heat storage. z(sin, j , u , p) ≤ y(sin, j , p), c

∀ p ∈ P , sin, j ∈ Sin, j , u ∈ U

c

c

(3) z(sin, j , u , p) ≤ y(sin, j , p), h

∀ p ∈ P , sin, j ∈ Sin, j , u ∈ U

h

h

(4)

Constraint 5 ensures that heat storage is heat integrated with either one hot unit or one cold unit at any point in time. This is to simplify and improve operational efficiency in the plant.

∑ z(sin,j , u , p) + ∑ z(sin,j , u , p) ≤ 1, c

s in, j

s in, j

c

h

h

∀ p ∈ P, u ∈ U

(5)

Constraints 6 and 7 ensure that a unit cannot simultaneously undergo direct and indirect heat integration. This condition simplifies the operation of the process.

∑ x(sin,j , sin,j , p) + z(sin,j , u , p) ≤ 1, s in, j

c

h

c

h

∀ p ∈ P , s in, j ∈ Sin, j , u ∈ U c

(6)

∑ x(sin,j , sin,j , p) + z(sin,j , u , p) ≤ 1, c

s in, j

h

h

c

∀ p ∈ P , s in, j ∈ Sin, j , u ∈ U h

(7)

Constraints 8 and 9 quantify the amount of heat received from or transferred to the heat storage unit, respectively. There will be no heat received or transferred if the binary variable signifying use of the heat storage vessel, z(sin,j, u, p), is zero. These constraints are active over the entire time horizon, where p is the current time point and p − 1 is the previous time point. Q (s in, j , u , p − 1) = W (u)Cpfluid (T0(u , p − 1) c

− Tf (u , p))z(s in, j , u , p − 1), c

∀ p ∈ P , p > p0 , s in, j ∈ Sin, j , u ∈ U c

D

(8)



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minimal thermal driving forces between the two integrated tasks need also to be enforced.

Q (s in, j , u , p − 1) = W (u)Cpfluid (Tf (u , p) h

− T0(u , p − 1))z(s in, j , u , p − 1),

Tf (u , p) − T (s in, j ) ≥ ΔT min

h

c

∀ p ∈ P , p > p0 , s in, j ∈ Sin, j , u ∈ U

(9)

h

− MM(1 − z(s in, j , u , p − 1)), c

Constraint 10 quantifies the heat transferred to the heat storage vessel at the beginning of the time horizon. The initial temperature of the heat storage fluid is T0(u, p0).

∀ p ∈ P , p > p0 , s in, j ∈ Sin, j , u ∈ U

(15)

c

T (s in, j ) − Tf (u , p) ≥ ΔT min

Q (s in, j , u , p0 ) = W (u)Cpfluid (Tf (u , p1 ) − T0(u , p0 ))z

h

h

(s in, j , u , p0 ),

∀ s in, j ∈ Sin, j , u ∈ U

h

h

− MM(1 − z(s in, j , u , p − 1)), h

(10)

∀ p ∈ P , p > p0 , s in, j ∈ Sin, j , u ∈ U

Constraint 11 ensures that the final temperature of the heat storage fluid at any time point becomes the initial temperature of the heat storage fluid at the next time point. This condition will hold regardless of whether or not there was heat integration at the previous time point. T0(u , p) = Tf (u , p − 1),

∀ p ∈ P, u ∈ U

Constraints 17 and 18 give the heating load for a cold state and cooling load for a hot state, respectively, for variable batch size and changing temperature. HL(sin, j , p) = B(s in, j , p)Cpstate(sin, j )(Tout(s in, j ) − Tin(s in, j )),

(11)

c

s in, j

h

c

(17)

h

− Tout(s in, j )),

h

h

∀ p ∈ P , s in, j ∈ Sin, j

h

(18)

h

Constraint 19 ensures that the heating of a cold state will be satisfied by either direct or indirect heat integration as well as external utility if required. HL(sin, j , p) = Q (s in, j , u , p) + st(sin, j , p) c

c

+

c

c

∑ q(sin,j , sin,j , p), s in, j

∀ p ∈ P , p > p0 , u ∈ U

h

c

CL(s in, j , p) = B(s in, j , p)Cpstate(s in, j )(Tin(s in, j )

c

∑ z(sin,j , u , p − 1)),

c

c

T0(u , p − 1) ≤ Tf (u , p) + MM(∑ z(s in, j , u , p − 1) +

c

∀ p ∈ P , sin, j ∈ Sin, j

Constraints 12 and 13 ensure that temperature of heat storage does not change if there is no heat integration with the heat storage unit, unless there is heat loss from the heat storage unit. MM is any large number, thereby resulting in an overall “big M” formulation. If either z(sin,jc, u, p − 1) or z(sin,jh, u, p − 1) is equal to one, constraints 12 and 13 will be redundant. However, if these two binary variables are both zero, the initial temperature at the previous time point will be equal to the final temperature at the current time. s in, j

(16)

h

c

∀ p ∈ P , sin, j ∈ Sin, j , u ∈ U

h

c

h

(19)

h

(12)

Constraint 20 states that the cooling of a hot state will be satisfied by either direct or indirect heat integration as well as external utility if required.

T0(u , p − 1) ≥ Tf (u , p) − MM(∑ z(s in, j , u , p − 1) s in, j

c

c

+

∑ z(sin,j , u , p − 1)), s in, j

CL(sin, j , p) = Q (sin, j , u , p) + cw(sin, j , p)

∀ p ∈ P , p > p0 , u ∈ U

h

h

+

h

h

The upper bounds of the heating load of a cold state, the cooling load of a hot state, and the amount of heat exchanged during direct integration are given in constraints 21−23. HL(s in, j , p) ≤ y(s in, j , p)Q max(s in, j ), c

min

c

c

∀ p ∈ P , s in, j ∈ Sin, j

(21)

c

− MM(1 − x(s in, j , s in, j , p − 1)),

CL(s in, j , p) ≤ y(s in, j , p)Q max(s in, j ),

h

∀ p ∈ P , p > p0 , s in, j , s in, j ∈ Sin, j c

h

(20)

c

c

h

∀ p ∈ P , sin, j ∈ Sin, j , u ∈ U

c

Constraint 14 ensures that minimum thermal driving forces are obeyed when there is direct heat integration between a hot and a cold unit. This constraint holds when both hot and cold units operate at constant temperature, which is commonly encountered in practice. An example is when there is heat integration between an exothermic and an endothermic reaction. h

c

s in, j

(13)

T (s in, j ) − T (s in, j ) ≥ ΔT

h

∑ q(sin,j , sin,j , p),

h

h

(14)

h

h

∀ p ∈ P , s in, j ∈ Sin, j

(22)

h

Constraints 15 and 16 ensure that minimum thermal driving forces are obeyed when there is heat integration with the heat storage unit. Constraint 15 applies for heat integration between heat storage and a heat sink, while constraint 16 applies for heat integration between heat storage and a heat source. In constraints 15 and 16, the units operate at fixed temperatures. For units not operating at fixed temperatures, both inlet and outlet

q(sin, j , s in, j , p) ≤ x(sin, j , s in, j , p)min{Q max(sin, j ), Q max(sin, j )} c

h

c

∀ p ∈ P , sin, j , s in, j ∈ Sin, j c

h

h

c

h

(23)

For the specific case where the heating and cooling loads are fixed, constraints 24 and 25 are used instead of 19 and 20. E



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̇ s in, j , p)dur(s in, j , p), q(s in, j , s in, j , p) ≤ HL(

HL(s in, j )y(s in, j , p) = Q (s in, j , u , p) + st(s in, j , p) c

c

c

c

c

+ x(s in, j , s in, j , p) ∑ min {HL(s in, j ), CL(s in, j )}, c

h

s in, j

s in, j , s in, j c

h

c

h

c

c

∀ p ∈ P , s in, j , s in, j ∈ Sin, j

h

c

(30)

h

h

∀ p ∈ P , s in, j ∈ Sin, j , u ∈ U

̇ s in, j , p)dur(s in, j , p), q(s in, j , s in, j , p) ≤ HL( c

(24)

c

h

c

h

∀ p ∈ P , s in, j , s in, j ∈ Sin, j c

CL(s in, j )y(s in, j , p) = Q (s in, j , u , p) + cw(s in, j , p) h

h

h

h

̇ s in, j , p)dur(s in, j , p), q(s in, j , s in, j , p) ≤ CL(

+ x(s in, j , s in, j , p) ∑ min {HL(s in, j ), CL(s in, j )}, c

h

s in, j , s in, j

s in, j

c

c

c

c

h

(25)

c

c

h


(26)

h

h

c

(33)

h

In Constraints 28−33, the duration is a function of batch size. If the duration is fixed, τ(sin,j) is used and these constraints are then linear. Constraints 34 and 35 ensure that the times at which units are active are synchronized when direct heat integration takes place. Starting times for the tasks in the integrated units are the same. This constraint may be relaxed for operations requiring preheating or precooling and is dependent on the process.

c

c

h

c

(Tout(s in, j ) − Tin(s in, j ))x(s in, j , s in, j , p), c

(32)

h

∀ p ∈ P , s in, j , s in, j ∈ Sin, j

q(s in, j , s in, j , p) ≤ B(s in, j , p)Cpstate(s in, j ) c

h

̇ s in, j , p)dur(s in, j , p), q(s in, j , s in, j , p) ≤ CL(

The amount of heat transferred through direct heat integration will be limited by the smaller heating or cooling requirement of the heat integrated tasks. Constraints 26 and 27 express this. Constraint 26 calculates the heat load of the cold task, while constraint 27 calculates the cooling load of the hot task. h

h

c

h

c

h

∀ p ∈ P , s in, j , s in, j ∈ Sin, j

h

∀ p ∈ P , s in, j ∈ Sin, j , u ∈ U

(31)

h

tu(s in, j , p) ≥ tu(s in, j , p) − MM(1 − x(s in, j , s in, j , p)) h

q(s in, j , s in, j , p) ≤ B(s in, j , p)Cpstate(s in, j ) c

h

h

h

c

c

h

h

(27)

h

c

HL(s in, j , p) c

dur(s in, j , p)

,

h

h

dur(s in, j , p)

,

c

h

h

(35)

Constraints 36 and 37 ensure that if indirect heat integration takes place, the time at which a heat storage unit starts either to transfer or receive heat will be equal to the time a unit is active. tu(s in, j , p) ≥ t 0(s in, j , u , p) − MM(y(s in, j , p) − z(s in, j , u , p)) ∀ p ∈ P , u ∈ U , s in, j ∈ Sin, j

(36)

tu(s in, j , p) ≤ t 0(s in, j , u , p) + MM(y(s in, j , p) − z(s in, j , u , p)) ∀ p ∈ P , u ∈ U , s in, j ∈ Sin, jz

(37)

Constraints 38 and 39 state that the time when heat transfer to or from a heat storage unit is finished will coincide with the time the task transferring or receiving heat has finished processing.

∀ p ∈ P , s in, j ∈ Sin, j c

(28)

CL(s in, j , p)

c

c

c

̇ s in, j , p) = CL(

(34)

h

∀ p ∈ P , s in, j , s in, j ∈ Sin, j

Furthermore, it is possible that a given pair of tasks cannot be heat integrated or that a possible ΔTmin violation may occur. In this work, the possibility of heat integration between pairs of tasks as well as possible ΔTmin violations was investigated for each pair of hot and cold tasks beforehand. If ΔTmin violations occur, the temperatures in constraints 26 and 27 should be adjusted for this. The amount of heat transferred through direct heat integration can also be limited by the duration of the shorter task if the tasks have different durations. Constraints 28−33 capture this. Constraints 28 and 29 calculate the heating load per time and cooling load per time, of the cold task and hot task, respectively. ̇ s in, j , p) = HL(

h

tu(s in, j , p) ≤ tu(s in, j , p) + MM(1 − x(s in, j , s in, j , p))


c

∀ p ∈ P , s in, j , s in, j ∈ Sin, j

h

(Tin(s in, j ) − Tout(s in, j ))x(s in, j , s in, j , p), h

c

tu(s in, j , p − 1) + τ(s in, j)y(s in, j , p − 1) ≥ t f (s in, j , u , p) − MM(y(s in, j , p − 1) − z(s in, j , u , p − 1))

∀ p ∈ P , s in, j ∈ Sin, j h

∀ p ∈ P , p > p0 , u ∈ U , s in, j ∈ Sin, j

h

(29)

(38)

tu(s in, j , p − 1) + τ(s in, j)y(s in, j , p − 1) ≤ t f (s in, j , u , p)

Constraint 30 calculates the heat load of the cold task based on the duration of the same cold task. Constraint 31 calculates the heat load of the cold task based on the duration of the hot task. Constraint 32 calculates the cooling load of the hot task based on the duration of the same hot task. Constraint 33 calculates the cooling load of the hot task based on the duration of the cold task. The amount of heat integrated directly will effectively be the minimum of these four quantities.

+ MM(y(s in, j , p − 1) − z(s in, j , u , p − 1)) ∀ p ∈ P , p > p0 , u ∈ U , s in, j ∈ Sin, j

(39)

Wastewater Minimization Constraints. The wastewater minimization constraints are based on the superstructure in Figure 1b. The task being performed in unit j ∈ Jw is a washing F



Article

product of the contaminant loading and the batch size of material processed.

operation in which the water used could consist of freshwater, stored water, or recycled/reused water. Water from unit j can be recycled into the same unit, reused by other units or sent to storage. Direct water reuse refers to the use of an outlet wastewater stream from one unit in another unit, while indirect water reuse refers to the use of previously stored wastewater in a unit. Constraints 40−90 constitute the wastewater minimization model useful for multipurpose batch processes, which involve multiple contaminants. The formulation is based on the model by Adekola and Majozi7 and includes both direct water reuse and indirect water reuse due to the presence of a central storage vessel. Mass balances around each processing unit and the central storage vessel are formulated as follows. Mass Balance around a Unit. Constraint 40 is the water balance over the inlet to a unit. Water entering the unit is a combination of reuse/recycle streams from other units, j′, freshwater, and water from storage. Constraint 41 states that the water leaving a unit could be recycled/reused, sent to storage, or discarded as effluent. Constraint 42 states that the amount of water exiting a unit must equal the amount of water entering the unit at the previous time point. This constraint captures the fact that water is neither produced nor lost in the unit during the washing operation.

∑

mwin(s in, j , p) =

sout, j

+ msout(j , p) mwout(sout, j , p) =

+ mwin(sin, j , p − 1)c in(sin, j , c , p − 1) ∀ j ∈ Jw , sin, j ∈ Sin, j , sout, j ∈ Sout, j , p ∈ P , p > p1 , c ∈ C

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P , c ∈ C

∑

c in(s in, j , c , p) ≤ C inU(s in, j , c)yw(s in, j , p) ∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P , c ∈ C

∀ j ∈ Jw , s in, j ∈ Sin, j , sout, j ∈ Sout, j , p ∈ P , p > p1 , c ∈ C

(40)

(48)

mwin(s in, j , p) ≤ Mw U(s in, j)yw(s in, j , p)

′

∀ j , j′ ∈ Jw , sout, j ∈ Sout, j , p ∈ P

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P

(41)

mwr(sout, j , j , p) ≤ Mw U(s in, j)ywr(sout, j , j , p) ′ ′ ∀ j , j′ ∈ Jw , s in, j ∈ Sin, j , sout, j , j ∈ Sout, j , j , p ∈ P ′ ′

mwin(s in, j , p − 1) = mwout(sout, j , p) (42)

Constraint 43 represents the inlet contaminant mass balance. The contaminant mass load in the inlet stream is the sum of the contaminant mass load in recycle/reuse water and that in water from storage. Constraint 44 represents the outlet contaminant mass as the sum of the mass of contaminant that entered the unit at the previous time point and the mass load of contaminant picked up in the unit during washing. In constraint 44, the mass load of contaminant in a unit is a given parameter. mwin(s in, j , p)c in(sin, j , c , p) =

∑ sout, j

∀ j ∈ Jw , s in, j ∈ Sin, j , sout, j ∈ Sout, j , p ∈ P

mwr(sout, j , j , p)cout(sout, j , c , p) ′ ′

∈ Sout, j , j , p ∈ P , c ∈ C ′

(50)

(51)

The maximum quantity of water into a unit is calculated using eq 52. It is important to note that for multicontaminant wastewater the outlet concentration of the individual components cannot all be set to the maximum, since the contaminants are not limiting simultaneously. The limiting contaminant(s) will always be at the maximum outlet concentration and the nonlimiting contaminants will be below their respective maximum outlet concentrations.

′,j

′,j

(49)

msout(sout, j , p) ≤ Mw U(s in, j)ysout(sout, j , p)

+ msout(j , p)csout(c , p) ∀ j , j′ ∈ Jw , s in, j ∈ Sin, j , sout, j

(47)

U cout(sout, j , c , p) ≤ Cout (sout, j , c)yw(s in, j , p − 1)

mwr(sout, j , j , p)+mwe(sout, j , p) ′

∀ j ∈ Jw , s in, j ∈ Sin, j , sout, j ∈ Sout, j , p ∈ P , p > p0

(46)

Constraints 47 and 48 ensure that the inlet and outlet contaminant concentrations do not exceed the allowed maximum. Similarly, the maximum allowable water in a unit must not be exceeded. This is governed by constraint 49. Constraint 50 restricts the mass of water entering the unit from recycle/reuse, to the maximum allowable for the unit. Likewise, constraint 51 restricts the mass of water entering the unit from storage, to the maximum allowable for the unit.

′,j

∀ j , j′ ∈ Jw , s in, j ∈ Sin, j , p ∈ P

(45)

MB(s in, j , c , p) = A(c)B(s in, j , p)

mwr(sout, j , j , p)+mwf (s in, j , p) ′

sout, j , j

+ ms in(j , p)

mwout(sout, j , p)cout(sout, j , c , p) = MB(s in, j , c , p − 1)yw(s in, j , p − 1)

⎧ ⎫ M(s in, j , c) ⎪ ⎪ ⎬ Mw U(s in, j) = max⎨ U U ⎪ c ∈ C ⎩ Cout(sout, j , c) − C in (s in, j , c) ⎪ ⎭

(43)

mwout(sout, j , p)cout(sout, j , c , p) = M(s in, j , c)yw(sout, j , p − 1)

∀ j ∈ Jw , s in, j ∈ Sin, j , sout, j ∈ Sout, j , c ∈ C

+ mwin(s in, j , p − 1)c in(s in, j , c , p − 1)

(52)

Mass Balance around Central Storage. Constraint 53 is the water mass balance around the storage tank. The amount of water stored in the storage tank consists of water stored at the previous time point and the difference between water entering the storage tank from a unit and water leaving the storage tank to a unit. Constraint 54 defines the initial amount of water in the tank.

∀ j ∈ Jw , s in, j ∈ Sin, j , sout, j ∈ Sout, j , p ∈ P , p > p1 , c ∈ C (44)

In the case where the mass load of contaminant in a unit is defined as a function of batch size of material processed in the unit, constraint 45 represents the outlet contaminant mass. Constraint 46 defines the variable contaminant mass load as the G


Industrial & Engineering Chemistry Research qws(p) = qws(p − 1) +

Article

∑ msin(j , p) − ∑ msout(j , p) j ∈ Jw

tout(sout, j , p) = tu(s in, j , p − 1) + τ(s in, j)y(s in, j , p − 1)

j ∈ Jw

∀ p ∈ P , p > p0


(60)

(53)

yw(s in, j , p) = y(s in, j , p − 1) qws(p0 ) = Qwso −

∑

msout(j , p0 )

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P , p > p0

(54)

j ∈ Jw

twin(s in, j , p) ≥ tout(sout, j , p) − MM(1 − yw(s in, j , p))

The definition of the inlet contaminant concentration to the storage tank is given in constraint 55. The concentration of water exiting the storage tank is assumed to be equal to the concentration of water in the tank as given in constraint 56. This condition is true in the case of perfect mixing. The initial concentration in the storage tank is expressed in constraint 57. Constraint 58 ensures that the maximum capacity of the tank is not exceeded. cs in(c , p) =



+ τw(s in, j)yw(s in, j , p − 1)

∑j ∈ J ms in(j , p)


w

(55)

+ [ ∑ ms in(j , p)]cs in(c , p)} j ∈ Jw

∑ msin(j , p)] j ∈ Jw

o csout(c , p0 ) = CSout (c )

qws(p) ≤ QwsU

(56)

∀c∈C

∀p∈P

(57) (58)

ywr(sout, j , j , p) ≤ yw(s in, j , p) ′ ′ ∀ j , j′ ∈ Jw , s in, j ∈ Sin, j , sout, j , j ∈ Sout, j , j , p ∈ P ′ ′

Constraint 59 ensures that no water is stored in the storage vessel at the end of the time horizon in order to give a true optimum. Otherwise the resulting minimum effluent could be misleading. qws(p) = 0

∀ p = |P |

(64)

Recycle/Reuse Scheduling. Wastewater can only be recycled/reused directly, if the unit producing wastewater and the unit receiving wastewater finish operating and begin operating at the same time, respectively. Constraint 65 describes the relationship between usage of water in a unit and the opportunity for recycle and reuse. The constraint states that for a unit j to transfer water to unit j′, unit j′ should require water at that time point. It does not, however, mean that unit j′ must use water from unit j, it could still obtain water from other sources. Constraints 66 and 67 state that the time at which water recycle/reuse takes place coincides with the time at which the water is produced. Constraints 68 and 69 ensure that the time at which water recycle/reuse takes place coincides with the starting time of the unit receiving the water.

csout(c , p) = {qws(p − 1)csout(c , p − 1)

∀ p ∈ P , p ≥ p0 , c ∈ C

(63)

twout(sout, j , p) = twin(s in, j , p − 1)

w

/[qws(p − 1) +

(62)

twin(s in, j , p) ≤ tout(sout, j , p) + MM(1 − yw(s in, j , p))

∑j ∈ J (ms in(j , p)cout(s in, j , c , p))

∀ s in, j ∈ Sin, j , p ∈ P , c ∈ C

(61)

(65)

twr(sout, j , j , p) ≤ twout(sout, j , p) − MM(1 − ywr(sout, j , j , p)) ′ ′ ∀ j , j′ ∈ Jw , sout, j ∈ Sout, j , sout, j , j ∈ Sout, j , j , p ∈ P (66) ′ ′

(59)

The scheduling constraints for the wastewater minimization model are as follows. Task Scheduling Constraints. Unlike heating or cooling that can occur during a production task, a unit can only perform either a production task or a washing task at any given point in time. The task scheduling constraints ensure the proper sequencing of equipment washing and production tasks. Constraint 60 expresses the duration of the production task.17 Constraint 61 stipulates that washing can only commence at time point p if the unit was active at the previous time point performing a production task, i.e., y(sin,j, p − 1) has a value of 1. Constraints 62 and 63 together ensure that unit j is washed immediately after a production task, which produced sout,j, has ended in the unit. If washing is being performed in the unit, yw(sin,j, p) has a value of 1 causing constraints 62 and 63 to become active and the start time of washing is forced to coincide with the end time of production. Otherwise, when water is not used in the unit, i.e., yw(sin,j, p) has a value of zero, the two constraints become relaxed. Constraint 64 represents the duration of the washing task performed in unit j.

twr(sout, j , j , p) ≥ twout(sout, j , p) + MM(1 − ywr(sout, j , j , p)) ′ ′ ∀ j , j′ ∈ Jw , sout, j ∈ Sout, j , sout, j , j ∈ Sout, j , j , p ∈ P (67) ′ ′

twr(sout, j , j , p) ≤ twin(s in, j , p) − MM(1 − ywr(sout, j , j , p)) ′ ′ ′ ∀ j , j′ ∈ Jw , s in, j ∈ Sin, j , sout, j , j ∈ Sout, j , j , p ∈ P (68) ′ ′ twr(sout, j , j , p) ≥ twin(s in, j , p) + MM(1 − ywr(sout, j , j , p)) ′ ′ ′ ∀ j , j′ ∈ Jw , s in, j ∈ Sin, j , sout, j , j ∈ Sout, j , j , p ∈ P (69) ′ ′ Central Storage Scheduling. Constraint 70 relates water usage in a unit and water transfer from storage. It states that water can only be transferred to a unit if it uses water at the same time point. However, it is not a prerequisite for the unit to use stored water, the water could be provided from other sources. Constraints 71 and 72 ensure that the time at which water is sent from storage to a unit coincides with the start time of washing of the unit. H


Industrial & Engineering Chemistry Research ysout(j , p) ≤ yw(s in, j , p)

Article

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P

tsout(j , p) ≥ tsout(j′, p) − MM(2 − ysout(j , p)

(70)

− ysout(j′, p))

∀ j , j′ ∈ Jw , p ∈ P

(80)

tsout(j , p) ≥ twin(s in, j , p) − MM(2 − ysout(j , p) − yw(s in, j , p))

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P

tsout(j , p) ≤ tsout(j′, p) + MM(2 − ysout(j , p)

(71)

− ysout(j′, p))

tsout(j , p) ≤ twin(s in, j , p) + MM(2 − ysout(j , p) − yw(s in, j , p))

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P

ts in(j , p) ≥ tsout(j′, p) − MM(2 − ys in(j , p) − ysout(j′, p)) ∀ j , j′ ∈ Jw , p ∈ P

∀ j , j′ ∈ Jw , p ∈ P

− yw(s in, j , p − 1))

tsout(j , p) ≥ ts in(j′, p′) − MM(2 − ysout(j , p)


− ys in(j′, p′))

(74)

− yw(s in, j , p − 1)) ∀ j ∈ Jw , s in, j ∈ Sin, j , sout, j ∈ Sout, j , p ∈ P , p > p1

(84)

(75)

ywr(sout, j , j , p) + ywr(sout, j , j , p) ≤ 1 ′ ′ ∀ j , j′ ∈ Jw , sout, j , j , sout, j , j ∈ Sout, j , j , p ∈ P ′ ′ ′

If water is transferred from storage to a unit at a later time point, the time at which this happens must correspond to a later time in the time horizon. This is specified in constraint 76. Constraint 77 ensures that if water is transferred from a unit to storage at a later time point, the time at which this happens corresponds to a later time in the time horizon.

(85)

Constraints 86−90 ensure that each event occurs within the time horizon of interest. twin(s in, j , p) ≤ H

tsout(j , p) ≥ tsout(j′, p′) − MM(2 − ysout(j , p) ∀ j , j′ ∈ Jw , p , p′ ∈ P , p ≥ p′

twout(sout, j , p) ≤ H

(76)

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P

(86)

∀ j ∈ Jw , sout, j ∈ Sout, j , p ∈ P (87)

ts in(j , p) ≥ ts in(j′, p′) − MM(2 − ys in(j , p) − ys in(j′ , p′)) ∀ j , j′ ∈ Jw , p , p′ ∈ P , p ≥ p′

∀ j , j′ ∈ Jw , p , p′ ∈ P , p ≥ p′

The following feasibility and time horizon constraints also hold. Constraint 85 ensures that if a processing unit j is reusing water from unit j′ at time point p, then unit j′ cannot reuse water from unit j at the same time point.

ts in(j , p) ≤ twout(sout, j , p) + MM(2 − ys in(j , p)

twr(sout, j , j , p) ≤ H ′

(77)

Constraints 78 and 79 state that if water is transferred to storage from more than one unit at the same time point, the time at which they do so must coincide. Constraints 80 and 81 state that if water is discharged from storage to more than one unit at the same time point, the time at which the water is discharged must coincide.

ts in(s in, j , p) ≤ H tsout(sout, j , p) ≤ H

∀ j ∈ Jw , sout, j , j ∈ Sout, j , j , p ∈ P ′ ′

(88)

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P

(89)

∀ j ∈ Jw , sout, j ∈ Sout, j , p ∈ P (90)

The objective of the formulation is either the maximization of profit or the minimization of makespan. Constraint 91 expresses the profit as the difference between the product revenue and the sum of freshwater, effluent treatment, cooling water, and steam costs. When the objective is the maximization of profit, constraint 91 is maximized. As a result, the amount of external utilities as well as freshwater consumption is minimized.

ts in(j , p) ≥ ts in(j′, p) − MM(2 − ys in(j , p) − ys in(j′, p)) (78)

ts in(j , p) ≤ ts in(j′, p) + MM(2 − ys in(j , p) − ys in(j′, p)) ∀ j , j′ ∈ Jw , p ∈ P

(83)

Constraint 84 ensures that if water leaves storage at a later time point compared to water entering the storage, the time at which water leaves the storage must correspond to a later time in the time horizon.

(73)

ts in(j , p) ≥ twout(sout, j , p) − MM(2 − ys in(j , p)

∀ j , j′ ∈ Jw , p ∈ P

(82)

ts in(j , p) ≤ tsout(j′, p) + MM(2 − ys in(j , p) − ysout(j′, p))

ys in(j , p) ≤ yw(s in, j , p − 1)

− ysout(j′, p′))

(81)

If water is simultaneously being transferred to and discharged from storage, the time at which this happens should coincide. This is given in constraints 82 and 83

(72)

Constraint 73 relates water usage in a unit and water transfer to storage. It states that water can only be transferred from a unit to storage if the unit used water at the previous time point. However, washing can take place in the unit without discharging water to the storage tank. The water could be discharged to other sinks. Constraints 74 and 75 ensure that the time at which water is sent to storage from a unit must coincide with the finishing time of washing of the unit.

∀ j ∈ Jw , s in, j ∈ Sin, j , p ∈ P , p > p1

∀ j , j′ ∈ Jw , p ∈ P

(79) I



Article

Figure 2. Recipe representation of the simple sequential process for case study I.

Table 1. Data Pertaining to Production, for Case Study I task (i) task 1 task 2 task 3

unit (j) unit unit unit unit unit

profit =

max batch size (kg)

processing time (h)

washing time (h)

material state (m)

initial inventory (kg)

max storage (kg)

revenue/cost ($/kg or $/MJ)

100 150 200 100 150

1.25 1.70 1.50 0.75 1.20

0.25 0.30 0 0.25 0.30

A B C D washwater wastewater cooling water steam

1000 0 0 0

1000 200 250 1000

0 0 0 5 0.1 0.05 0.02 1

1 2 3 4 5

28−33, 43−45, 55 and 56 have bilinear terms where two continuous variables are multiplied. This results in a nonconvex MINLP formulation. The bilinearity resulting from the multiplication of a continuous variable with a binary variable as found in Constraints 26 and 27 may be handled effectively with the Glover transformation.23 This is an exact linearization technique and as such will not compromise the accuracy of the model. The procedure is demonstrated for constraint 9. Let

∑ ∑ CP(s)d(s , p) s

p

− CF ∑ ∑ mwf (sin, j , p)−CE ∑ ∑ mwe(sout, j , p) sin, j

p

sout, j

p

− cost_cw ∑ ∑ cw(sin, jh , p)−cost_st ∑ ∑ st(sin, jc , p) sin, jh

p

s in, jc

p

(91)

Constraint 92 is the objective function for makespan minimization. In constraint 92, the quotient of profit and time horizon is maximized. As a result, the makespan is minimized and the amount of external utilities as well as freshwater consumed are also minimized using the same reasoning as above. profit max H

Tf (u , p)z(s in, j , u , p − 1) = Γ1(s in, j , u , p) h

(93)

h

with lower and upper temperature bounds known T L ≤ Tf (u , p) ≤ T U

(94)

Then

(92)

Γ1(s in, j , u , p) ≥ Tf (u , p) − T U(1 − z(s in, j , u , p − 1)) h

h

(95)

4. SOLUTION PROCEDURE The overall model, whether for a profit maximization problem or a makespan minimization problem is an mixed-integer nonlinear program (MINLP). When solving a profit maximization problem, the model is linearized and solved as an MILP, the solution of which is then used as a starting point for the exact MINLP model. If the solutions from the two models are equal, the solution is globally optimal, as global optimality can be proven for MILP problems. If the solutions differ, the MINLP solution is locally optimal. The possibility also exists that no feasible starting point is found. This solution procedure was used by Gouws et al.,22 Adekola and Majozi,7 and Stamp and Majozi.9 When solving a makespan minimization problem, the MINLP cannot be linearized completely to an MILP due to the nonlinear objective function, constraint 92. However, the MINLP was linearized to a relaxed MINLP problem which provides a starting point for the exact MINLP problem. However, the resulting solution to the exact MINLP cannot be guaranteed to be a global optimum. Constraints 8, 9, and 10 have trilinear terms where a binary variable and two continuous variables are multiplied. Constraints

L

Γ1(s in, j , u , p) ≤ Tf (u , p) + T (1 − z(s in, j , u , p − 1)) h

h

(96)

Γ1(s in, j , u , p) ≥ z(s in, j , u , p − 1)T L

(97)

Γ1(s in, j , u , p) ≤ z(s in, j , u , p − 1)T U

(98)

h

h

h

h

The result from the Glover transformation for constraint 9 is seen in constraint 99 and includes the addition of one new continuous variable and four new continuous constraints. Q (s in, j , u , p − 1) = W (u)cp(Γ1(s in, j , u , p) h

h

− Γ2(s in, j , u , p − 1)), h

∀ p ∈ P , p > p0 , s in, j ∈ Sin, j , u ∈ U h

(99)

The heat storage capacity, W(u), is also a continuous variable and is multiplied with the continuous Glover transformation variable. This results in another type of bilinearity, which results in a nonconvex model. A method to handle this is a reformulation− J



Article

linearization technique24 as discussed by Quesada and Grossmann.25 This is demonstrated for constraint 99, resulting in constraints 100−107. Let

Table 2. Data Pertaining to Energy Requirements, for Case Study I Tin (°C)

Tout (°C)

task 1

140

60

task 2 task 3

60 40

40 80

20 170

30 160

task (i)

cooling water steam


Cp (kJ/(kg °C))

1 2 3 4 5

4.0 4.0 3.5 3.0 3.0

W (u)Γ1(s in, j , u , p) = Ψ1(s in, j , u , p) h

with lower and upper heat storage capacity and temperature bounds known

task (i) task 1 task 2 task 3


1 2 3 4 5

max outlet concentration (ppm)

500 50

1000 100

150 300

300 2000

(101)

T L ≤ Γ1(s in, j , u , p) ≤ T U

(102)

Then Ψ1(s in, j , u , p) ≥ W L Γ1(s in, j , u , p) + T LW (u) − W LT L h

contaminant loading (g contaminant/kg batch)

Ψ1(s in, j , u , p) ≥ W U Γ1(s in, j , u , p) + T UW (u) − W UT U

0.2 0.2 0.2 0.2 0.2

Halim and Srinivasan

16

h

(103) h

h

(104) U

L

h

h

(105)

Ψ1(s in, j , u , p) ≤ W L Γ1(s in, j , u , p) + T UW (u) − W LT U h

h

(106)

this formulation

4,764.1 43.9 313.9 1238.37 5,000 43.9 6.28 123.8 61.9 7 N/A not reported 1500 not reported

U L

Ψ1(s in, j , u , p) ≤ W Γ1(s in, j , u , p) + T W (u) − W T

Table 4. Comparison of Results for Case Study I profit ($) steam (MJ) cooling water (MJ) total freshwater (kg) revenue from product ($) cost of steam ($) cost of cooling water ($) cost of freshwater ($) cost of wastewater ($) number of slots number of time points number of binary variables number of iterations CPU time (s)

W L ≤ W (u) ≤ W U h

Table 3. Data Pertaining to Water Requirements, for Case Study I max inlet concentration (ppm)

(100)

h

This is an inexact linearization technique and increases the size of the model by an additional type of continuous variable and four types of continuous constraints. The final completely linearized form of constraint 9 can be seen in constraint 107.

4,775.28 66 336 1013.33 5,000 66 6.72 101.3 50.7 N/A 15 385 N/A 28797

Q (s in, j , u , p − 1) = cp(Ψ1(s in, j , u , p) h

h

− Ψ2(s in, j , u , p − 1)) h

∀ p ∈ P , p > p0 , s in, j ∈ Sin, j , u ∈ U h

(107)

For the profit maximization problem, the linearization procedure is carried out for each of the nonlinear terms in constraints 8−10, 26−33, 43−45, 55 and 56 to produce a MILP

Figure 3. Resulting production schedule for case study I. K



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Figure 4. STN representation of the complex production facility for case study II.

Table 5. Data Pertaining to Production, for Case Study II task (i) heating (H) reaction 1 (R1) reaction 2 (R2) reaction 3 (R3) separation (S)

unit (j)

max batch size (kg)

αij (h)

βij (h)

washing time (h)

HR RR1 RR2 RR1 RR2 RR1 RR2 SR

100 50 80 50 80 50 80 200

0.667 1.084 1.034 1.09 1.034 0.417 0.367 1.334

0.007 0.027 0.017 0.027 0.017 0.013 0.008 0.007

0 0.25 0.3 0.25 0.3 0.25 0.3 0

material state feed A (s1) feed B (s2) feed C (s3/s4) hot A (s5) IntAB (s8) IntBC (s6) impure E (s9) product 1 (s7) product 2 (s10) washwater wastewater cooling water steam

problem, which provides a starting point for the exact MINLP problem. The objective function for the makespan minimization problem was also nonlinear, however, only constraints 8−10, 26−33, 43−45, 55 and 56 were linearized to produce a relaxed MINLP problem, the solution of which provides a starting point for the exact MINLP problem.

initial inventory (kg)

max storage (kg)

revenue/cost ($/kg or $/MJ)

1000 1000 1000 0 0 0 0 0 0

1000 1000 1000 100 200 150 200 1000 1000

10 10 10 0 0 0 0 20 20 0.1 0.05 0.02 1

3. A single contaminant is present 4. The energy requirements of the operations are a function of the batch size and the given initial and final temperatures. Due to this fact, careful consideration must be given to ensure that whenever heat integration occurs between two operations, the minimum temperature difference for heat transfer, 10 °C is not violated. The first case study was solved with the proposed formulation, using constraints 1, 2, 17−23, 26−35, 40−43, and 45−50, 52, 60−69, and 85−88. The objective function was the maximization of constraint 91. It is important to note that in constraints 40, 41, 43 and 45, variables associated with wastewater storage are not included. Due to the fact that a single contaminant was present, the formulation for this case study could be reduced to a MILP. The outlet contaminant concentration was fixed to the maximum, constraint 43 was substituted into constraint 45 and a Glover transformation was performed on the resulting equation.26 The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using CPLEX as the MIP solver.

5. CASE STUDY I16 This case study is a simple sequential process. A single product is produced via three tasks in five units. The case study has been adapted to include water and energy using operations. Figure 2 shows the recipe representation of the process. Steam is available for heating, cooling water is available for cooling, and freshwater is available for washing. The objective is to maximize profit over a 12 h time horizon, while minimizing freshwater usage and utility requirements. Data required for case study I may be obtained from Tables 1−3. The following should be noted: 1. The processing duration of a batch is fixed. 2. The mass load of contaminants is dependent on the batch size. L



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The results from the first case study as compared to the results achieved by Halim and Srinivasan16 may be obtained from Table 4. As can be observed from Table 4, a better profit overall was obtained with the proposed formulation. Although the amounts for steam and cooling water were lower for the model by Halim and Srinivasan,16 the amount of freshwater required was lower in the proposed formulation, which also results in a lower effluent production. This is a consequence of differences in the schedule obtained by Halim and Srinivasan16 compared to the current formulation. The advantage of the current work is that the variable nature of time ensures that the schedule is flexible and results that could have been overlooked when time is fixed are available during the optimization search. The CPU time required to solve the problem was approximately 8 h. This highlights the complexity introduced by solving scheduling, wastewater minimiza-

tion, and heat integration simultaneously, whereas the standalone scheduling problem resulted in an objective of $5000 obtained in 0.097 s. Similar to the method by Halim and Srinivasan,16 the washing of a unit may not necessarily occur immediately after the end of a processing task, but can be delayed to improve reuse opportunities with other units. To cater for this, constraint 63 was removed from the formulation while constraint 62 remained. Figure 3 shows the Gantt chart with the resulting production schedule for the first case study. The clear horizontal blocks represent the tasks associated with production processes. The amount of material processed in each unit is labeled within the clear blocks. The dark horizontal blocks represent the washing of a unit. The amount of water associated with washing is labeled underneath the relevant dark blocks. The double sided arrows (up−down arrows) represent direct heat integration between two tasks. The numbers in round brackets represent the amount of energy associated with heat integration. The numbers in curly brackets represent steam duties while the numbers in square brackets represent cooling water duties.

Table 6. Data Pertaining to Energy Requirements, for Case Study II task (i)

Tin (°C)

Tout (°C)

unit (j)

Cp (kJ/(kg °C))

heating (H) reaction 1 (R1)

50 100

70 70

reaction 2 (R2)

70

100

reaction 3 (R3)

100

130

separation (S) cooling water steam

130 20 170

100 30 160


2.5 3.5 3.5 3.2 3.2 2.6 2.6 2.8

6. CASE STUDY II This case study comprises of the popular literature example by Kondili et al.27 Halim and Srinivasan16 adapted this complex production process to include energy and water using aspects, and it is described here. For the production process, two products, product 1 and 2 are to be produced from three raw materials feed A, B, and C. A heater, HR is used to heat feed A. Two reactors, RR1 and RR2, are available to perform three different chemical reactions, reactions 1, 2, and 3. Finally, a

Table 7. Data Pertaining to Water Requirements, for Case Study II max inlet concentration (ppm)

max outlet concentration (ppm)

task (i)

unit (j)

ar

br

cp

dw

ar

br

cp

dw

contaminant loading (g contaminant/kg batch)

reaction 1 (R1)

RR1 RR2 RR1 RR2 RR1 RR2

300 300 700 700 500 500

500 500 600 600 200 200

800 800 300 300 400 400

400 400 400 400 300 300

700 700 1200 1200 800 800

800 800 1000 1000 500 500

1200 1200 600 600 700 700

900 900 800 800 900 900

0.2 0.2 0.2 0.2 0.2 0.2

reaction 2 (R2) reaction 3 (R3)

Figure 5. Resulting production schedule for case study II. M



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separator exists to purify impure E. Figure 4 shows the STN representation of the problem. Steam is available for heating, cooling water is available for cooling, and freshwater is available for washing. The production recipe is as follows: 1. Heating: Feed A is heated from 50 to 70 °C inside HR. Steam is required as a heating medium for this reaction. 2. Reaction 1: A mixture of 50% feed B and 50% feed C, on a mass basis, is reacted in either unit RR1 or RR2. The product of this reaction is IntBC. The reaction requires cooling from 100 to 70 °C. 3. Reaction 2: A mixture of 40% hot A and 60% IntBC is reacted to form product 1 (40%) and IntAB (60%). This reaction can be performed in either unit RR1 or RR2 and requires heating from 70 to 100 °C. 4. Reaction 3: A mixture of 20% feed C and 80% IntAB is reacted in either units RR1 and RR2. The reaction produces impure E and requires heating from 100 to 130 °C. 5. Separation: In SR, impure E is purified to produce product 2 (90%) and intermediate IntAB (10%). The separation requires cooling from 130 to 100 °C. Water is required for washing RR1 and RR2 at the end of any reaction. In this case study, four contaminants, ar, br, cp, and dw are present. Table 5 shows information regarding the

production process. Data pertaining to heating and cooling are given in Table 6, while data pertaining to the washing of RR1 and RR2 are given in Table 7. The objective in this case study was to determine the minimum makespan required to produce 200 kg each of products 1 and 2, while also minimizing external utility and freshwater consumption. It is important to note the following from the data in Tables 5−7: 1. The processing duration of a batch is dependent on the batch size. 2. The mass load of contaminants is not fixed, but dependent on the batch size. 3. The energy requirements of the operations are a function of the batch size and the given initial and final temperatures. Due to this fact, careful consideration must be given to ensure that whenever heat integration occurs between two operations, the minimum temperature difference for heat transfer, 10 °C, is not violated. The second case study was solved using the proposed formulation, using constraints 1, 2, 17−23, 26−35, 40−43, 45−50, 52, 60−69, and 85−88. The objective function was the minimization of makespan, constraint 92. It is important to note that in constraints 40, 41, 43, and 45, variables associated with wastewater storage are not included. This is in order to make a direct comparison with the results obtained by Halim and Srinivasan.16 The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using DICOPT for the MINLP with CPLEX as the MIP solver and MINOS as the NLP solver. Figure 5 shows the Gantt chart with the resulting production schedule for the case study. The clear horizontal blocks represent the tasks associated with production processes. The amount of material processed in each unit is labeled within the clear blocks. The dark horizontal blocks represent the washing of a unit. The amount of water associated with washing is labeled underneath the relevant dark blocks. The double sided arrows (up−down arrows) represent direct heat integration between two tasks. The numbers in round brackets represent the amount of energy associated with heat integration. The numbers in curly brackets represent steam duties while the numbers in square brackets represent cooling water duties.

Table 8. Comparison of Results for Case Study II Halim and Srinivasan16

this formulation

19.96 61.36 35.39 275.09 8000 5,604.4 61.36 0.7078 27.509 13.75 2,292.3 8 N/A 3500 not reported not reported

19.93 44.88 19.72 341.2 8000 5,444.4 44.88 0.3994 34.12 17.06 2,459.1 N/A 17 N/A 24532 836 9

makespan (h) steam (MJ) cooling water (MJ) total freshwater (kg) revenue from products ($) cost of raw materials ($) cost of steam ($) cost of cooling water ($) cost of freshwater ($) cost of wastewater ($) profit ($) number of slots number of time points number of iterations CPU time (s) number of binary variables major iterations

Table 9. Production Data for Case Study III task (i) heating (H) reaction 1 (R1) reaction 2 (R2) reaction 3 (R3) separation (S)

unit (j)

max batch size (kg)

mean processing time (h)

washing time (h)


100 50 80 50 80 50 80 200

1 2 2 2 2 1 1 1 for Prod 2 2 for IntAB

0 0.25 0.3 0.5 0.25 0.25 0.25 0

material state feed A (s1) feed B (s2) feed C (s3/s4) hot A (s5) IntAB (s8) IntBC (s6) impure E (s9) product 1 (s7) product 2 (s10) washwater wastewater cooling water steam N

initial inventory max storage (kg) (kg) unlimited unlimited unlimited 0 0 0 0 0 0

unlimited unlimited unlimited 0 0 0 0 0 0

revenue/cost (cost units/kg or cost units/kJ) 0 0 0 0 0 0 0 100 100 2 3 2 10



Article

In this case study, storage facilities for heat storage and water reuse are available. Different scenarios of the case study were

A comparison between the results of this case study obtained by Halim and Srinivasan16 and the proposed formulation is described in Table 8. As can be observed from Table 8, improved objectives were obtained using the proposed formulation with the exception of the amount of freshwater used for washing. This was probably due to the fact that the cost associated with using steam was more significant than the cost associated with using freshwater. Hence, to minimize overall costs, promoting opportunities for heat integration took precedence over promoting opportunities for water reuse. This can be observed from the production schedule in Figure 5. With the operations aligned to promote heat integration, no opportunities for water reuse exist at all. The MINLP model was solved by the aforementioned initialization procedure. The objective value from the relaxed MINLP model was 123.543 $/h and the objective value from the exact MINLP model was 123.369 $/h. As both the relaxed model and exact models were MINLP, due to the nonlinear objective function, the global optimality of the solution could not be guaranteed.

Table 11. Heat Integration Data for Case Study III reaction RX1 RX2 RX3

reaction 2 (RR1) reaction 3 (RR1) reaction 1 (RR2) reaction 2 (RR2) reaction 3 (RR2)

reaction 1 reaction 2 reaction 3

max max max max max max max max max max max max

reactor reactor reactor reactor reactor reactor

1 2 1 2 1 2

inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet inlet outlet

0.5 1 0.01 0.2 0.15 0.3 0.05 0.1 0.03 0.075 0.3 2 contaminant

profit (cu) amount of product 1 (kg) amount of product 2 (kg) cooling water (kWh) steam (kWh) freshwater (kg) time points CPU time (s) binary variables initial storage temperature (°C) heat storage capacity (ton) major iterations

0.5 2.3 0.9 3 0.05 0.3 0.1 1.2 0.2 0.35 1 1.2 0.2 0.05 1 12 0.1 0.2 0.2 1 0.6 1.5 1.5 2.5 mass load (g)

C2

C3

4 15 28.5 9 15 22.5

80 24 7.5 2 80 45

10 358 135 16 85 36.5

4.2 10 20 180 1 3

18537 96

19836 116

22235 116

162

156.4

188

390

250

190

240 816 11 3.1 128

180 1 020 11 14.8 508

10 896 13 42 275 954 82.9 2.024

3

3

4

solved to demonstrate the capabilities of the model. These scenarios are as follows: • Scenario 1: During production, only freshwater is available for washing. Heating and cooling are provided exclusively by steam and cooling water. • Scenario 2: In addition to freshwater for washing, steam and cooling water for heating and cooling respectively, opportunities for direct water reuse and direct heat integration are explored. • Scenario 3: The effect of the inclusion of storage facilities for heat and wastewater (indirect water reuse and indirect heat integration) to scenario 2 is explored. Constraints 1−16, 24−25, 34−39, 40−43, 44, and 47−91 were used to solve this case study. The bilinear terms present in the model were linearized and the solution procedure as described above was used. The capacity of the storage vessel for water was 200 kg. The objective of this case study was to maximize profit while minimizing wastewater production and energy consumption, within a time horizon of 12 h. The storage capacity and the initial storage temperature of the storage vessel were variables to be optimized.9 Heat losses were not considered.

C3

C1

100 60 140 values

freshwater direct water reuse direct/indirect water and and direct heat reuse and direct/indirect utilities integration heat integration

maximum contaminant concentration (g contaminant/kg water) reaction 1 (RR1)

operating temperature (°C)

Table 12. Comparison between Different Scenarios for Case Study III

Table 10. Wastewater Minimization Data for Case Study III

C2

exothermic 60 (cooling) endothermic 80 (heating) exothermic 70 (cooling) heat storage parameters Cpfluid (kJ/(kg °C)) ΔTmin (°C) TL (°C) TU (°C) WL (ton) WU (ton)

7. CASE STUDY III This case study was obtained from Majozi and Gouws28 and was extended to include heat integration opportunities. The multipurpose batch facility investigated is similar to that discussed in case study II. The heating and separation tasks performed in HR and SR, respectively, are not to be heat integrated with any other units. Heat integration can only occur between RR1 and RR2 depending on the tasks they perform. Similarly, water is required for washing RR1 and RR2 at the end of any reaction. In this case study, three contaminants, C1, C2, and C3, are present. Table 9 shows information regarding the production process, while data pertaining to the washing of RR1 and RR2, obtained from the work of Majozi and Gouws,28 are given in Table 10. The heat integration data is provided in Table 11.

C1

heating/cooling requirement (kW h)

type

O



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Figure 6. Resulting process schedule when only direct heat integration and direct water reuse are possible.

Figure 7. Resulting process schedule when both direct/indirect heat integration and wastewater minimization are possible.

of freshwater. The unit operations are aligned in such a way as to promote as much heat integration as possible. In so doing, opportunities for direct water reuse are lost. In scenario 3, storage for water is available and hence a decrease is observed in the amount of freshwater used. The resulting process schedule for the results of scenario 2 is illustrated in Figure 6, while the corresponding schedule for scenario 3 is illustrated in Figure 7. The clear blocks represent the production operation in the unit while the dark blocks represent the washing operations which take place after the reactions are completed. The numbers above the clear blocks represent the amount of material processed in the unit during production and the numbers below the washing operations represent freshwater. Water transfer to and from storage has been clearly labeled. The up−down arrows represent direct heat integration, while the bent arrows represent indirect heat integration to or from the heat storage unit. The results of scenario 1 are globally optimal with the objective function of the linearized MILP and exact MINLP being 18 537 c.u.

The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using DICOPT for the MINLP with CPLEX as the MIP solver and CONOPT as the NLP solver. A comparison between the three scenarios is provided in Table 12. The results of scenarios 1, 2, and 3 are contained in columns 2, 3, and 4, respectively, of Table 12. From Table 12 it can be observed that profit increases from scenario 1 to 3, with a decrease in steam and cooling water requirements. However, while cooling water and steam decreased in scenario 2 compared to scenario 1, the amount of freshwater increased. The total amount of product in scenario 1 is 258 kg, while the total amount of product in scenario 2 is 272.4 kg. This is as a result of additional unit operations being performed in scenario 2 than are performed in scenario 1. These additional unit operations contribute to the increased amount of washing water required. Furthermore, no opportunities for direct water reuse were realized due to the cost of steam relative to the cost P



Article

water and s′ as effluent. This implies that there has to be a unit operation between these two due to the change in states. Figure A.1b shows the mixing of different states to yield a new state, e.g. raw materials entering a reactor to yield a product. Figure A.1c illustrates a splitting/separation unit with state s as the input and states s′ and s″ as outputs. Capacity Constraint

Constraint A.1 states that the total amount of all the states consumed at time point p is limited by the capacity of the unit which consumes the states. V jminy(s in, j , p) ≤

∑ mu(s , p) ≤ V jmaxy(sin,j , p), s in, j

∀ p ∈ P, j ∈ J

(A.1)

Material Balances

Constraint A.2 is the material balance around a particular unit j. It implies that the sum of the masses for all the input states used at time point p − 1 should be equal to the sum of the masses for all the output states produced at time point p.

∑

mu(s , p − 1) =

s = s in, j

∑

mp(s , p),

s = sout, j

∀ p ∈ P , p > p1 , j ∈ J

Constraint A.3 states that the amount of state s stored at the first time point, is the difference between the amount stored before the beginning of the process and that being utilized at the first time point.

Figure A.1. Building blocks of state sequence network (SSN).

Similarly, the results of scenario 2 are globally optimal, with the objective function of the linearized MILP and exact MINLP being 19 836 c.u. In scenario 3, the objective function of the linearized MILP was 25 270 c.u, while the objective function of the exact MINLP was 22 235 c.u. Hence, the results for scenario 3 are locally optimal.

qs(s , p1 ) = Q s0(s) − mu(s in, j , p1 ), ∀ s in, j ∈ Sin, j , s ≠ product

qs(s , p) = qs(s , p − 1) − mu(s in, j , p), ∀ p ∈ P , p > p1 , s in, j ∈ Sin, j , s = feed

(A.4)

Constraint A.5 only applies to intermediates, since they are both produced and used in the process. qs(s , p) = qs(s , p − 1) + mp(sout, j , p) − mu(s in, j , p), ∀ p ∈ P , p > p1 , s in, j ∈ Sin, j , sout, j ∈ Sout, j , s ≠ feed, product

(A.5)

Constraints A.6 and A.7 only apply to products and byproducts, since they are the only states that have to be taken out of the process as shown by the terms d(s, p). Constraint A.6 is used at the first time point, and constraint A.7 is used for the remainder of the time horizon.

APPENDIX

Scheduling Model of Majozi and Zhu

(A.3)

Constraint A.4 is the storage balance applicable to feed states, as they are only used in the process.

8. CONCLUSIONS A simultaneous method for the optimization of energy and water embedded within a scheduling framework has been developed. Furthermore, opportunities for direct and indirect heat integration as well as direct and indirect water reuse have been explored. The mathematical formulation led to an MINLP problem for which an initialization procedure was employed. The applicability of the method has been demonstrated with three case studies. The developed formulation has proved to effectively solve a complex makespan minimization problem in which duration is a function of batch size and which included multiple contaminants. The advantage of the developed model is that the variable nature of time ensures that the schedule is flexible and results that could have been overlooked when time is fixed are available during the optimization search, leading to better overall objectives. In the future, the problem can be extended to include other water using operations other than equipment washing.

■

(A.2)

qs(s , p1 ) = Q s0(s) − d(s , p1 ),

17

The state sequence network (SSN) representation is based on states only. Tasks and units are implicitly incorporated and this reduces the number of binary variables required for solution. The only binary variable involved is y(sin,j, p), which equals 1 if state s is used at time point p and is zero otherwise. The building blocks of the SSN are shown in Figure A.1. Figure A.1a shows the transition from state s to state s′. For an example, this could be a washing operation with s as washing

s = product

(A.6)

qs(s , p) = qs(s , p − 1) + mp(sout, j , p) − d(s , p), ∀ p ∈ P , p > p1 , sout, j ∈ Sout, j , s = product, byproduct (A.7)

Duration Constraints

Constraint A.8 shows the duration of a task as a function of the batch size, while constraint A.9 is for a fixed task duration. Q



■

tout(sout, j , p) = tu(s in, j , p − 1) + α(s in, j)y(s in, j , p − 1)

*E-mail: [email protected].

s in, j

Notes

∀ p ∈ P , p > p1 , s in, j ∈ Sin, j , sout, j ∈ Sout, j , j ∈ J

The authors declare no competing financial interest.

■

(A.8)

ACKNOWLEDGMENTS The authors gratefully acknowledge support from the South African National Energy Research Institute and the National Research Foundation, South Africa.

tout(sout, j , p) = tu(s in, j , p − 1) + τ(s in, j)y(s in, j , p − 1), ∀ p ∈ P , p > p1 , s in, j ∈ Sin, j , sout, j ∈ Sout, j , j ∈ J

(A.9)

■

Sequence Constraints

Constraints A.10 and A.11 imply that state s can only be used in a particular unit, at any time point, after all the previous states have been processed. Constraint A.10 is only relevant in situations where more than one task can be conducted in one unit; otherwise, it is redundant in the presence of constraints A.11 and A.12.

∑∑ ∑

tout(sout, j , p′) − tu(s in, j , p′ − 1),

∀ p , p′ ∈ P , p > p1 , s in, j ∈ Sin, j , sout, j ∈ Sout, j , j ∈ J (A.10)

tu(s in, j , p) ≥ tout(sout, j , p), ∀ p ∈ P , s in, j ∈ Sin, j , sout, j ∈ Sout, j , j ∈ J

(A.11)

Constraint A.12 stipulates that a state can only be processed at a particular time point p in a particular unit j after it has been produced from another unit j′. In case of a recycle, j is the same as j′. It is worthy of note that constraints A.11 and A.12 are only applicable to intermediates, since they are the only states that are both produced and used. tu(s in, j , p) ≥ tout(sout, j ′ , p), ∀ p ∈ P , sout, j ′ = s in, j , j , j′ ∈ J

(A.12)

Assignment Constraint

Constraint A.13 is the assignment constraint which is aimed at ensuring that only one task is conducted in a unit at any time point. It is therefore apparent that the assignment is only necessary if more than one task can be performed in a given unit. Otherwise, it is also redundant. y(s in, j , p) ≤ 1,

∀ p ∈ P, j ∈ J

s in, j ∈ Sin, j

(A.13)

Time Horizon Constraints

Constraints A.14 and A.15 respectively stipulate that the usage or production of a state should be within the time horizon of interest. tu(s in, j , p) ≤ H , tout(sout, j , p) ≤ H ,

∀ p ∈ P , s in, j ∈ Sin, j , j ∈ J

(A.14)

∀ p ∈ P , sout, j ∈ Sout, j , j ∈ J (A.15)

Storage Constraint

Constraint A.16 states that the amount of state s stored at each time point cannot exceed the maximum allowed. qs(s , p) ≤ Q sU (s),

∀ p ∈ P, s ∈ S

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s in, j sout, j p ′≤ p

∑

AUTHOR INFORMATION

Corresponding Author

+ β(s in, j) ∑ mu(s , p − 1),

tu(s in, j , p) ≥

Article

(A.16) R



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Unified Approach for the Optimization of Energy and Water in

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