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Computers and Chemical Engineering 38 (2012) 171– 184

Contents lists available at SciVerse ScienceDirect

Computers and Chemical Engineering

jo u rn al hom epa ge : www.elsev ier .com/ locate /compchemeng

ptimal production planning under time-sensitive electricity prices forontinuous power-intensive processes

umit Mitraa, Ignacio E. Grossmanna,∗, Jose M. Pintob, Nikhil Arorac

Center for Advanced Process Decision-making, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, United StatesPraxair Inc., Danbury, CT 06810, United StatesPraxair Inc., Tonawanda, NY 14150, United States

r t i c l e i n f o

rticle history:eceived 2 July 2011ccepted 29 September 2011vailable online 26 October 2011

a b s t r a c t

Power-intensive processes can lower operating expenses when adjusting production planning accordingto time-dependent electricity pricing schemes. In this paper, we describe a discrete-time, deterministicMILP model that allows optimal production planning for continuous power-intensive processes. Weemphasize the systematic modeling of operational transitions, that result from switching the operatingmodes of the plant equipment, with logic constraints. We prove properties on the tightness of several logic

eywords:roduction planningower-intensive systemsILP models

ime-varying electricity pricesir separation plants

constraints. For the time horizon of 1 week and hourly changing electricity prices, we solve an industrialcase study on air separation plants, where transitional modes help us capture ramping behavior. Wealso solve problem instances on cement plants where we show that the appropriate choice of operatingmodes allows us to obtain practical schedules, while limiting the number of changeovers. Despite thelarge size of the MILPs, the required solution times are small due to the explicit modeling of transitions.

ement plants

. Introduction

The profitability of industrial power-intensive processes isffected by the availability and pricing of electricity supply.owadays, two major trends increase the complexity of manag-

ng power-intensive processes. First, deregulation in the 1990sntroduced hourly as well as seasonal variations. Second, the envi-onmental pressure to reduce CO2 emissions and diminishingatural resources lead to an increasing share of renewable ener-ies, which intensifies the aforementioned problem. These trendsave added a considerable amount of uncertainty and variability inhe daily operating expenses of power-intensive industries, whichn turn affect their competitiveness.

One important component of the current and future power sys-em is the concept of Demand Side Management (DSM), consistingf Energy Efficiency (EE) and Demand Response (DR). A reporteleased by The World Bank (Charles River Associates, 2005) definesSM as the “systematic utility and government activities designed

o change the amount and/or timing of the customer’s use of elec-ricity for the collective benefit of the society, the utility and itsustomers.” While EE aims for permanently reducing demand for

nergy, DR focuses on the operational level (Voytas et al., 2007).he official classification of DR by the North American Electric

∗ Corresponding author. Tel.: +1 412 268 3642; fax: +1 412 268 7139.

098-1354/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2011.09.019

© 2011 Elsevier Ltd. All rights reserved.

Reliability Corporation (NERC) distinguishes between dispatchableand non-dispatchable programs (see Fig. 1).

Dispatchable DR programs include any kind of demand responsethat is according to instructions from a grid operator’s control cen-ter. They are divided into capacity services, such as load controland interruptible demand, and ancillary services, such as spin-ning and nonspinning reserves as well as regulation. The controlactions, which balance the electricity supply and demand, differon the time scale and usually range from a few seconds to onehour. Hence, participation in one of these dispatchable DR programsrequires the process to be highly flexible, while process feasibilityand safety have to be maintained. Nowadays, chemical companies,which operate flexible processes like chlor-alkali synthesis, mar-ket already a few percent of their total load as operative capacityreserve (e.g. in Germany; Paulus & Borggrefe, 2011). The poten-tial of ancillary services for aluminum production was recentlyevaluated in a case study by ALCOA (Todd et al., 2009). However,both processes, chlor-alkali synthesis and aluminum production,are examples of capital-intensive processes that are operated at ahigh level of capacity utilization. Thus, these processes usually onlyshift production on a minute level around a predefined setpoint.

In contrast to dispatchable DR programs, non-dispatchable DRprograms do not involve instructions from a control center. Instead,

the electricity consumption of industrial customers is influenced bythe market price of electricity. Typical examples of time-sensitiveelectricity prices are time-of-use (TOU) rates and real-time prices(RTP). While TOU rates are usually specified in terms of on-peak,

172 S. Mitra et al. / Computers and Chemical Engineering 38 (2012) 171– 184

Nomenclature

SetsP (index p) the set of plants or production linesM(p) (index m), abbreviated as M the set of modes, depend-

ing on plant pI(p, m) (index i), abbreviated as I the set of extreme points

that relate to mode m of plant pG (index g) the set of products. For air separation plants it is{

LO2, LN2, LAr, GO2, GN2}

.Storable ⊆ G the subset of products that are storable. For air

separation plants it is{

LO2, LN2, LAr}

Nonstorable ⊆ G the subset of products that are not storable.For air separation plants it is

{GO2, GN2

}ST the set of shared storage tanksH (index h) the set of hours of a week in the operational

modelHx ⊆ H subset of hours, where x can stand for, e.g. a certain

dayMinStay(m, m ′) a placeholder for the sets UT, DT and TTUT(p), abbreviated as UT the set of hours that a plant p has to

stay online, once it was startedDT(p), abbreviated as DT the set of hours that a plant p has to

stay offline, once it was shut downTT(p), abbreviated as TT the set of hours that a plant p has to

stay in a transitional modeTrans(p, m, m ′ , m ′′) the set of possible transitions for plant

p from mode m to a production mode m′′ with thetransitional mode m′ in between

AL(p, m, m ′) the set of allowed transitions for plant p frommode m to another mode m′

DAL(p, m, m ′) the set of disallowed transitions from mode mto mode m′ of plant p

Binary variablesXh

st,g indicates which product g is stored in storage st athour h (for shared storage)

yhp,m determines whether plant p operates in mode m in

hour hZh

p,m,m′ indicates whether there is a transition from mode mto mode m′ at plant p from hour h − 1 to h

Continuous variablesPr

hp,m,g production amount of product g in mode m at plant

p in hour hPrh

p,g total production of product g at plant p in hour h

�hp,m,i

variable for the convex combination of slates i todescribe the feasible region of the plant p of modem in hour h

INVhp,g inventory level of product g at plant p in hour h

Shp,g sales of product g from plant p in hour h

Storehp,st,g keeps track how much of product g is transferred

from production line p to storage tank st in hour h(for shared storage)

OBJ objective function variable

Parametersˆ̨ LN2, ˆ̨ LO2, ˆ̨ LAR conversion parameters for equivalent liquid

rate˛p,m, ˇp,m, �p,m fitting parameters for mode m of plant p in

[power/volume]

ıp,g cost coefficient for inventory of product g at plant pin [$/volume]

�p,m,m′ cost coefficient for transitions from mode m to m′ atplant p in [$]

ehp electricity prices for plant p in hour h

xp,m,i,g extreme points i of the convex hull of mode m ofplant p in terms of the products g

M̄p,m,g BigM constant for bounds on production for plant p(i.e. max. production of product g in mode m)

Kminm,m′ number of hours the plant has to stay in mode m′

after a transition from mode mKmax

m,m′ number of hours the plant can stay at most in modem′ after a transition from mode m

rp,m,g maximum rate of change for product g at plant p inmode m

ddailyp,g , dweekly

p,g daily/weekly demand for the products g ofplant p. For air separation plants, it is the demandfor the liquid products.

dh,hourlyp,g hourly demand for the products g of plant p in hour

h. For air separation plants, it is the demand for thegaseous or the liquid products.

INVUp,st,g tank capacity of tank st for product g at plant p

PWhmax maximum power consumption at hour h (under

restricted power availability)

mid-peak and off-peak hours, real-time prices vary every hourand are quoted either on a day-ahead or hourly basis. Other pric-ing models exist but strongly depend on the characteristics of theregional market (NERC study; (Voytas et al., 2007)).

Non-dispatchable DR programs allow industrial customers toperform production planning based on predefined hourly prices.At first glance it may seem that production planning due to pricefluctuations is only attractive for processes that are operatedsignificantly below the process capacity, and therefore have oper-ational flexibility. However, major demand drops due to economicchanges, such as the 2008 recession, can lead to over-capacities,which in turn make a systematic production planning more attrac-tive. Promising examples can be found in the industrial gases sector(cryogenic air separation plants) and in the cement industry.

The purpose of this paper is to describe a general model thathelps decision-makers for power-intensive production processesto optimize their production schedules with respect to operatingcosts that are due to fluctuations in electricity prices, which are inturn caused by non-dispatchable DR programs.

2. Literature review

The economic potential for DSM of industrial processes in devel-oped countries has been recognized by numerous institutions andauthors (World Bank report: (Charles River Associates, 2005); NERCstudy (Voytas et al., 2007); Klobasa, 2007; Gutschi & Stigler, 2008;Paulus & Borggrefe, 2011). In these publications, we can find alist of various chemical processes where the consumption of elec-tricity is due to different unit operations: grinding (cement, paperpulp production), compression (air separation), electrolysis (chlor-alkali, aluminum) and drying (paper production). Fig. 1 illustratespossible DR applications for these processes. In chemical engineer-ing, two different lines of research address optimizing operations

of these processes according to time-sensitive pricing.

The first line of research proposes a control approach. For theeconomic optimization of air separation plants, Zhu, Legg, andLaird (2011) developed a model based on heat and mass balances,

S. Mitra et al. / Computers and Chemical Engineering 38 (2012) 171– 184 173

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Fig. 1. Illustration of DR applications for chemical processes according to t

ncluding nonlinear thermodynamics. Huang (2010) proposed thepplication of economically-oriented nonlinear model predictiveontrol. While these approaches have the advantage of providingn accurate process representation, they cannot readily incorpo-ate discrete decisions such as equipment shutdowns and startupsnd corresponding operational constraints such as enforcing a min-mum up- or downtime. Furthermore, the resulting NLP problemsend to be large, and therefore problems of only 24–48 h time hori-on can be efficiently solved. The problems become even hardero solve when parallel production lines or whole process networksre included.

The second line of research comes from the area of planningnd scheduling. An initial approach by Daryanian, Boln, and Tabors1989) shows the economic benefit of demand response for aireparation plants, but does not consider discrete operating deci-ions such as equipment shutdowns and startups. Ierapetritou, Wu,in, Sweeny, and Chigirinskiy (2002) as well as Karwan and Keblis

2007) extend the methodology to incorporate discrete operat-ng decisions. However, their models do not capture the transientehavior between different operating modes. Moreover, some ofheir logic constraints show room for improvement with respect tohe tightness of the linear relaxation.

For continuous paper pulp production, Pulkkinen and Ritala2008) develop a scheduling model to determine the number ofperating units and production levels for each time period. They usetochastic constraints to handle uncertainty in model parameters,ntroducing nonlinearities. However, their optimization procedures not rigorous in the sense that heuristic methods such as simulatednnealing and genetic algorithms are employed. Furthermore, theodel does not include a detailed description of plant dynamics in

erms of transitions and logic constraints.For cement plants, Castro, Harjunkoski, and Grossmann (2009,

011) extend the MILP formulation of the Resource Task NetworkRTN) to incorporate hourly changing electricity prices. In theiraper from 2009, Castro and co-workers propose a discrete timeormulation (DT) that cannot be solved to optimality within a rea-onable amount of time for some case studies, although the finalap is small. Additionally, the obtained schedules may contain aignificant number of changeovers. Castro and co-workers alsohow a continuous time formulation (CT), which is computation-

lly intractable. In their paper from 2011, Castro and co-workersevelop an aggregate formulation (AG) and a rolling horizon for-ulation (RH). While AG and RH can be solved considerably faster

nd yield schedules with fewer transitions than DT, the obtained

sification of DR programs by NERC (Voytas et al., 2007); reduced diagram.

schedules may be suboptimal under power restrictions. Further-more, AG seems to only adequately work in a setting withtime-of-use (TOU) prices, since it aggregates periods with the sameprice level.

In this paper, we first introduce the generic problem statementfor a continuous power-intensive process in Section 3. In Section 4,we generalize the approaches proposed by Ierapetritou et al. (2002)and Karwan and Keblis (2007). We discuss the concept of transi-tional modes to allow for a more detailed modeling of transitionalbehavior that is due to plant dynamics. Furthermore, we enhancethe computational efficiency of their formulations by improvingthe formulation of logic constraints. In Section 5, we apply the pro-posed methodology to a variety of power-intensive processes. Wesolve an industrial case study on air separation plants where transi-tional modes help us capture ramping behavior, as well as probleminstances on cement plants that are reported in Castro et al. (2009,2011). We show that the appropriate choice of operating modesallows us to solve the cement problem instances with a discretetime model to optimality within a reasonable amount of time, whilelimiting the number of changeovers.

The focus of this paper is to obtain an efficient deterministic for-mulation that can handle transient plant behavior. Therefore, we donot consider uncertainty in electricity prices explicitly. However, itshould be mentioned that our formulation can be used as a basisto also incorporate uncertainty in electricity prices, e.g. in a rollinghorizon fashion (e.g. Karwan and Keblis, 2007) or with a stochasticprogramming approach (e.g. Ierapetritou et al., 2002). Furthermore,note that seasonal electricity tariffs, e.g. time-of-use (TOU) con-tracts, contain no uncertainty for the timespan of the operationalplanning (usually a few days or a week). Hence, the deterministicoptimization model allows to obtain the optimal production planfor seasonal electricity tariffs.

3. Generic problem statement

Given is a set of products g ∈ G that can be produced in differentcontinuously operated plants or production lines p ∈ P. While someproducts can be stored on-site (g ∈ Storable), other products mustbe directly delivered to customers (g ∈ Nonstorable). The plantshave to satisfy demand for the products that can be specified on

a weekly, daily or hourly basis. The costs of production vary everyhour h ∈ H and are related to electricity costs, which also undergoseasonal changes. We assume that a seasonal electricity price fore-cast for a typical week is available on an hourly basis and that our

1 emical Engineering 38 (2012) 171– 184

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74 S. Mitra et al. / Computers and Ch

roduction decisions do not influence the electricity prices. Theroblem is to determine production levels, modes of operationincluding start-up and shut-down of equipment), inventory lev-ls and product sales on an hourly basis, so that the given demands met. The objective is to minimize the operating costs that are

ostly due to energy expenses.

. Model formulation

The traditional way of modeling a process involves heat andass balances for each individual equipment, which requires the

etailed description of the system’s performance (e.g. thermody-amics, kinetics). The disadvantage of this approach is that theodel can become prohibitively hard to solve due to its nonlin-

arities and its size for longer time horizons.An alternative approach is to model the plant in a reduced space,

.g. the product space. To determine the feasible region of the plantn the product space, either a sequence of steady-state simula-ions or the insights of flexibility theory by Swaney and Grossmann1985), Grossmann and Floudas (1987) and Ierapetritou (2001) cane used. While it is still useful to have a detailed plant model basedn heat and mass balances, we shift the computational burden ofvaluating the feasible region to offline computations to obtain aurrogate model. A similar idea is employed by Sung and Maravelias2007, 2009) for batch planning and scheduling.

In absence of a detailed plant model, the surrogate model canlso be built from historic plant data. The evaluation of the produc-ion levels for the feasible region must be performed according tohe chosen time discretization �t, which is usually on an hourlyasis due to the hourly price changes of non-dispatchable DR pro-rams.

The model can be classified into three sets of constraints. Therst set deals with the previously mentioned surrogate description

or the feasible region in the product space. The second set con-ains constraints for transitions between different operating pointsnd modes. The third and last set describes the mass balances thatapture inventory relationships and demand.

.1. Production modes and feasible region

To describe the feasible region of a power-intensive plant, thenit operations that consume electricity have to be first identified.he equipment operation involves discrete as well as continuousperating decisions. A discrete operating decision refers to the statef an equipment, e.g. “off”, “production mode” or “ramp-up transi-ion”. A continuous operating decision refers to production levels,.e. flowrates of material.

To distinguish discrete operating decisions of a plant, we for-ally introduce the concept of a mode.A mode is a set of operating points for which the same discrete

perating decisions are active (i.e. selection of running equipment).he operating points vary in terms of the continuous variables in theroduct space. The feasible region of each mode is approximated by

convex region consisting of known operating points. In addition,he energy consumption is approximated by a correlation for thentire feasible region of a mode. In any given time period only oneode can be active, in other words the modes are disjoint. However,odes may overlap in the product space while usually exhibiting

ifferent correlations. In Fig. 2, a plant can be seen that produceswo products (A and B) having three distinct production modes.

We assume that offline computations or plant measurements

rovide the data that describes the disjunctive set of modes for theeasible region. We can formulate constraint (1) to describe the dis-unction between the modes m ∈ M for each plant p and every hour hsee nomenclature section). Each term of the disjunction is defined

Fig. 2. Feasible region with distinct operating modes.

in terms of the binary variable yhp,m, which when true (yh

p,m = 1)defines the term that applies in the disjunction. The data for allmodes is represented as a collection of operating points (slates).xp,m,i,g are the extreme points i of mode m of plant p in terms of theproducts g. These extreme points have to be determined a-prioriby using an appropriate tool such as MATLAB (The MathWorks Inc.,2010) or PORTA (Christof & Lobel, 1997). The convex combinationof the extreme points, with weight factors �h

p,m,i, determines the

production Prhp,g at hour h for each plant p and product g.

∨m∈M

⎛⎜⎜⎜⎜⎜⎝

∑i∈I

�hp,m,ixp,m,i,g = Prh

p,g ∀g∑i∈I

�hp,m,i = 1

0 ≤ �hp,m,i

≤ 1yh

p,m = 1

⎞⎟⎟⎟⎟⎟⎠ ∀p ∈ P, h ∈ H (1)

The disjunction (1) is reformulated using the convex hull (Balas,1985) to allow for solving the program with an MILP solver. Theconvex hull reformulation can be written with constraints (2)–(7).Note that M̄p,m,g is the maximum production of product g in modem.∑i∈I

�hp,m,ixp,m,i,g = Pr

hp,m,g ∀p ∈ P, m ∈ M, g ∈ G, h ∈ H (2)

∑i∈I

�hp,m,i = yh

p,m ∀p ∈ P, m ∈ M, h ∈ H (3)

0 ≤ �hp,m,i ≤ 1 ∀p ∈ P, m ∈ M, i ∈ I, h ∈ H (4)

Prhp,g =

∑m∈M

Prhp,m,g ∀p ∈ P, g ∈ G, h ∈ H (5)

Prhp,m,g ≤ M̄p,m,gyh

p,m ∀p ∈ P, m ∈ M, g ∈ G, h ∈ H (6)∑m∈M

yhp,m = 1 ∀p ∈ P, h ∈ H (7)

4.2. Transitions

Due to the hourly changing electricity prices, it might be eco-nomically desirable to run the plant at different time-dependent

operating points. If the operating point is changed, a transitionoccurs. Transitions deserve special attention because they areclosely related to the system dynamics. There are two major typesof transitions:

S. Mitra et al. / Computers and Chemica

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h + K . The corresponding logic relationship is:

ig. 3. State graph of a plant with one production and one transitional ramp-upode. The edges indicate the direction of the allowed transitions and operational

onstraints of the system.

. Transitions between operating points of different modes – thesetransitions are described first and are the major focus of thissection.

. Transitions between operating points that are both part of thesame mode – these transitions are described at the end of thissection (rate of change constraints).

.2.1. Transitions between operating points of different modesA typical case of a transition between operating points of differ-

nt modes is the start-up of a plant. During the first few hours thelant might produce a specified amount of off-spec product until

steady-state is reached. Furthermore, the plant cannot be turnedff immediately after. The dynamics of the plant in terms of switch-ng behavior can be visualized with a state graph as shown in Fig. 3.ach node represents the state of the system. The edges representhe direction of the allowed transitions and show the operationalonstraints.

We can model the ramp-up characteristics using three modes:off”, “ramp-up transition” and “production mode”. In the givenxample, the plant can switch from “off” to “ramp-up transition”mmediately, but has to stay in the “ramp-up transition” mode for

h. Then, a transition occurs to the mode “production mode”, inhich the plant has to stay at least 24 h. The plant can switch from

production mode” to mode “off” immediately, but if the plant ishut down, the plant has to stay in the mode “off” at least 48 h. Allransitional constraints faced by the plant operator, who decidesn a sequence of states as a control input, have to be expressed inerms of logic constraints.

.2.1.1. Switch variables constraints. To model the transitional con-traints efficiently, we introduce the binary transitional variablehp,m,m′, which is true if and only if a transition from state m totate m′ at plant p occurs from time step h − 1 to h. Hence, the logicelationship to express is the following:

yh−1p,m ∧ yh

p,m′) ⇔ zhp,m,m′ ∀p ∈ P, m ∈ M, m′ ∈ M, h ∈ H (8)

Using propositional logic (Raman & Grossmann, 1993), the fol-owing constraints can be derived for the “⇐” direction:

h−1p,m ≥ zh

p,m,m′ ∀p ∈ P, m ∈ M, m′ ∈ M, h ∈ H (9)

hp,m′ ≥ zh

p,m,m′ ∀p ∈ P, m ∈ M, m′ ∈ M, h ∈ H (10)

In a similar fashion, the constraint for the “⇒” direction can beerived:

h−1p,m + yh

p,m′ − 1 ≤ zhp,m,m′ ∀p ∈ P, m ∈ M, m′ ∈ M, h ∈ H (11)

Note that the constraints (9)–(11) are also used in Ierapetritout al. (2002) and Karwan and Keblis (2007) to model an equiva-ent relationship for air separation plants. However, as shown byahinidis and Grossmann (1991) as well as by Erdirik-Dogan androssmann (2008), the logic relationship (8) can be more effectively

odeled by the following set of constraints:∑

′∈M

zhp,m′,m = yh

p,m ∀p ∈ P, h ∈ H, m ∈ M (12)

l Engineering 38 (2012) 171– 184 175

∑m′∈M

zhp,m,m′ = yh−1

p,m ∀p ∈ P, h ∈ H, m ∈ M (13)

The reformulation (12)–(13) requires the same number ofvariables but fewer constraints ((2 · |M|) · |P| · |H| vs 3|M|2 · |P| · |H|).Furthermore, its relaxation is tighter. Note that the logic isexpressed differently, for more details see the aforementionedreferences, where the modes are equivalent to tasks in batchscheduling.

4.2.1.2. Minimum stay constraint. Some modes have a minimumtime that the plant has to stay in that particular mode, dependingon the previous mode that was active before the plant switched.An example can be found in Fig. 3, where the plant has to stayfor 6 h in mode “ramp-up transition” after it switched from mode“off”. Formally, this behavior can be expressed by the followinglogic relationship:

zhp,m,m′ ⇒ yh+�

p,m′ ∀(p, m, m′) ∈ AL, ∀h ∈ H, � ∈ MinStay(m, m′)Let AL be the set of allowed transitions of plant p from mode m

to mode m′. Some of the allowed transitions require a minimumstay constraint.

For these transitions, let MinStay(m, m ′) be the associatedplaceholder, which describes a set that contains the elements{0, 1, . . . , Kmin

m,m′ − 1} to model the duration of the minimum stayconstraint. Kmin

m,m′ is the minimum amount of time the plant has tostay in mode m′ after a transition from mode m. Examples for substi-tutes of MinStay(m, m ′) include the set TT (to model the duration ofa transition), the set UT (minimum uptime for a production mode)and the set DT (minimum downtime after a shut-down). Note thatMinStay(m, m ′) is the empty set for transitions that do not requirea minimum stay constraint.

The previous logic statement can be converted into the followingconstraint:

yh+�p,m′ ≥ zh

p,m,m′ ∀(p, m, m′) ∈ AL, ∀h ∈ H, � ∈ MinStay(m, m′) (14)

Note that Ierapetritou et al. (2002) (constraints (8) and (9) intheir paper) and Karwan and Keblis (2007) (constraint (11) in theirpaper) used a different constraint for the same relationship.

With our nomenclature their constraint can be written as:

Kminm,m′ − 1∑

�=0

∑m′′∈M,m′′ /= m′

yh+�p,m′′ ≤ Kmin

m,m′(1 − zhp,m,m′) ∀(p, m, m′) ∈ AL, ∀h ∈ H

(15)

We show in Appendix I that despite fewer number of con-straints, formulation (15) is weaker in the LP relaxation comparedto that of constraint (14). Note that both properties (number ofconstraints and tightness of LP relaxation) can impact the compu-tational performance and depend on the specific application.

4.2.1.3. Maximum stay constraint. If the duration that a mode canbe operated is restricted, one has to express a maximum stay con-straint. An example can be found for an auxiliary equipment thatcannot be operated longer than a specified amount of time. Once atransition occurs from mode m to m′, the plant cannot stay in modem′ after Kmax

m,m′ number of hours.However, the maximum stay constraint only applies if no tran-

sition zhp,m,m′ from mode m to m′ is active between h + Kmin

m,m′ andmax

m,m′

(zhp,m,m′ ∧ ¬z

h+Kminm,m′

p,m′′,m′ ∧ · · · ∧ ¬zh+Kmax

m,m′p,m′′,m′ ) ⇒ ¬y

h+Kmaxm,m′+1

p,m′∀(p, m, m′) ∈ AL, m′′ ∈ M, m′′ /= m′, ∀h ∈ H

176 S. Mitra et al. / Computers and Chemica

Fig. 4. Illustration of transitions involving transitional mode m′ and minimum staycKm

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onstraint for mode m′ . Following the terminology of Fig. 3, the plant has to stayminm,m′ (6) hours in mode m′ (ramp-up transition) and switches thereafter in mode′′ (production mode).

This logic statement can be reformulated to the following con-traint:

− yh+Kmax

m,m′+1p,m′ ≥ zh

p,m,m′ −Kmax

m,m′∑�=Kmin

m,m′

∑m′′,m′′ /= m′

zh+�p,m′′,m′

∀(p, m, m′) ∈ AL, ∀h ∈ H (16)

.2.1.4. Transitional mode constraints. As seen in the state graph inig. 3, the transition from “off” to “ramp-up transition” is coupledith the transition to “production mode” after 6 h. More formally,

et (p, m, m ′ , m ′′) ∈ Trans be the set of transitions from mode mo mode m′′, which require the plant to stay in the transitional

ode m′ for a certain specified time Kminm,m′, as defined before. Aside

rom a minimum stay constraint for the transitional mode m′, theransition from m to m′ and the transition from m′ to m′′ have toe coupled, which is conceptually visualized in Fig. 4. In terms ofropositional logic, we can express it as follows:

h−Kminm,m′

p,m,m′ ⇔ zhp,m′,m′′ ∀(p, m, m′, m′′) ∈ Trans, ∀h ∈ H

This can be written as:

h−Kminm,m′

p,m,m′ − zhp,m′,m′′ = 0 ∀(p, m, m′, m′′) ∈ Trans, ∀h ∈ H (17)

.2.1.5. Forbidden transitions. Since the logic relationship (8)mplies ¬(yh−1

p,m ∧ yhp,m′) ⇔ ¬zh

p,m,m′ ∀p, m, m′, h, it is sufficient to fixhp,m,m′ to zero for transitions that are not allowed (denoted by theet DAL):

hp,m,m′ = 0 ∀(p, m, m′) ∈ DAL, ∀h ∈ H (18)

The concept of the zhp,m,m′ variables is very useful to describe

ystems with slow dynamics, i.e. where the impact of transitionsn later time periods has to be modeled. However, if the systemsynamics are fast and there are no minimum stay restrictions, thenhe zh

p,m,m′ variables can be avoided. Thus, forbidden transitionsave to be modeled in a slightly different way.

Let us denote the set of all combinations for possible transi-ions of plant p from mode m to mode m′ by (p, m, m ′) ∈ PT. We canartition PT into two disjoint subsets: The subset of allowed tran-itions AL and the subset of disallowed transitions DAL. Note thatL∩ DAL = ∅.

To model forbidden transitions, one of the following logic rela-ionships, which are equivalent since AL and DAL are disjoint setsnd can be translated by negating, has to be satisfied.

h∨

h−1

p,m′ ⇒

m:(p,m,m′)∈ALyp,m ∀p ∈ P, m′ ∈ M, ∀h ∈ H (19)

hp,m′ ⇒

∧m:(p,m,m′)∈DAL

¬yh−1p,m ∀p ∈ P, m′ ∈ M, ∀h ∈ H (20)

l Engineering 38 (2012) 171– 184

These logic statements (19) and (20) can be expressed by the cor-responding constraints (21) and (22):∑m:(p,m,m′)∈AL

yh−1p,m ≥ yh

p,m′ ∀p ∈ P, m′ ∈ M, ∀h ∈ H (21)

yh−1p,m + yh

p,m′ ≤ 1 ∀(p, m, m′) ∈ DAL, ∀h ∈ H (22)

Constraint (21) has the advantage that fewer constraints in totalare needed (|M| · |P| · |H| instead of |M| · |M ′|· |P| · |H| (M′ = disallowedm′)) and it is furthermore tighter than constraint (22). The proof canbe found in Appendix II.

Note that Ierapetritou et al. (2002) do not consider the case offorbidden transitions. Karwan and Keblis (2007) model forbiddentransitions using constraint (22).

4.2.2. Transitions between operating points that are both part ofthe same mode4.2.2.1. Rate of change constraints. From a control perspective, wehave assumed so far that for a given discrete operating decision,each point of the feasible region is reachable within the given timediscretization. However, for transitions between operating pointsthat are both part of the same mode, the rate of change might berestricted by a parameter rp,m,g. To model these restrictions in terms

of the production variables Prhp,m,g , the following constraints on the

production levels from hour h to hour h + 1 can be formulated:

|Prh+1p,m,g − Pr

hp,m,g | ≤ rp,m,g ∀p ∈ P, m ∈ M, g ∈ G, h ∈ H (23)

Note that rp,m,g is specified in [mass/�t], where �t is the chosentime discretization (e.g. 1 h).

4.3. Mass balances

A mass balance has to be enforced to describe the relationshipbetween production levels, inventory levels and satisfied demand.The demand can be specified on an hourly, daily or weekly basis. Theproducts G can be partitioned into two sets: the storable productsStorable and the non-storable products Nonstorable.

The amount of each product g that is stored at each plant p foreach hour h is denoted by INVh

p,g and the hourly sales for each prod-uct are described by Sh

p,g . The mass balance for the storable productsis:

INVhp,g + Prh

p,g = INVh+1p,g + Sh

p,g ∀p ∈ P, g ∈ Storable, h ∈ H (24)

The inventory levels INVhp,g are constrained by the outage levels

(INVLp,g , lower bound) and by the storage capacities (INVU

p,g , upperbound):

INVLp,g ≤ INVh

p,g ≤ INVUp,g ∀p ∈ P, g ∈ Storable, h ∈ H (25)

If a product g cannot be stored, the inventory level is simplyINVh

p,g = 0, ∀h and the mass balance becomes:

Prhp,g = Sh

p,g ∀p ∈ P, g ∈ Nonstorable, h ∈ H (26)

We have to satisfy demand that is usually specified on an hourlybasis:

Shp,g ≥ dh,hourly

p,g ∀p ∈ P, g ∈ G, h ∈ H (27)

However, it is also possible to specify demand for the storableproducts on a daily or weekly basis (dweekly

p,g , ddailyp,g ), e.g. if the exact

sales points for the products are unknown:∑h∈Hweek

Shp,g ≥ dweekly

p,g ∀p ∈ P, g ∈ Storable (28)

emica

h

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4

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S. Mitra et al. / Computers and Ch

∑∈Hday

Shp,g ≥ ddaily

p,g ∀day, p ∈ P, g ∈ Storable (29)

Note that stockouts are not allowed, but the extension can easilye made by introducing slack variables in the demand constraintsith corresponding penalty terms in the objective function. Fur-

hermore, the notation used above assumes that every plant p hasts own demand. However, it is also possible to aggregate demandf the customers for multiple plants. Due to space limitation theorresponding equations are omitted.

.4. Objective function

The objective function minimizes the total cost that consists ofroduction cost, inventory cost and transition cost for every hour. Hence, we need a correlation between the power consumption

nd the production levels Prhp,m,g . We denote by �p,m,g the coeffi-

ients that have to be determined by a multivariate regression (inpower/volume]). The inventory cost coefficients are denoted byp,g (in [$/volume]) and the cost coefficients related to transitionost are described by �p,m,m′ (in [$]). Hence, the objective functionan be written as follows:

BJ =∑p,h

ehp

(∑m,g

�p,m,gPrhp,m,g

)+∑p,g

ıp,g

∑h

INVhp,g

+∑

p,m,m′�p,m,m′

∑h

zhp,m,m′ (30)

.5. Power availability restrictions

If the available power is restricted, e.g. due to a contracted powerurve, the previously defined correlation between the power con-

umption and the production levels Prhp,m,g can be used to model

he following constraint, where PWhmax is the maximum power con-

umption at hour h:∑,m,g

�p,m,gPrhp,m,g ≤ PWh

max ∀h ∈ H (31)

. Case studies

.1. Air separation plants

The first case study is on air separation plants, which produceiquid oxygen (LO2), nitrogen (LN2) and argon (LAr) as well asaseous oxygen (GO2) and nitrogen (GN2). All liquid products cane stored on-site in storage tanks. In contrast to that, the gaseousroducts must be provided to industrial on-site customers accord-

ng to previously negotiated contracts on an hourly basis and cannote stored if the air separation plant is in proximity to the consumer.herefore, plants that are connected to a gas pipeline have feweregrees of freedom in their daily operation.

In our case study, we optimize two different air separationlants that supply the liquid merchant market without on-site cus-omers. Thus, the power consumption for cryogenic air separations mostly due to the compressors that are required to provide cool-ng for liquefaction by exploiting the Joule-Thompson effect. Hence,

ach operating mode can be related to a liquefier in the air separa-ion plant. If the plant ramps up production, a transitional mode isctive in which a specified amount of off-spec product is generatednd vented. If the plant has a second liquefier installed and switches

l Engineering 38 (2012) 171– 184 177

its operation from one to two liquefiers, the plant also undergoes atransitional mode for a few hours.

5.1.1. Model formulationThe feasible region for each mode is modeled with constraints

(2)–(7). The switch variables constraints (12) and (13) are used tolink mode variables yh

p,m and transitional variables zhp,m,m′. Mini-

mum stay constraint (14) is used to enforce a minimum uptimeonce the plant is switched on, a minimum downtime once theplant is switched off and transition times for transitional modes.Constraints (17) and (18) are used to express logic relationshipsbetween different modes. The mass balance for the storable liq-uid products (LO2, LN2, LAr) is expressed by constraints (24) and(25), the mass balance for the non-storable gaseous products (GO2,GN2) is described by constraint (26). The demand for all productsis specified on an hourly basis using constraint (27).

5.1.2. Objective functionFor air separation plants, Karwan and Keblis (2007) as well as

Ierapetritou et al. (2002) suggest linear objective functions. Thestructure of our linear correlation is the same as the one reportedby Ierapetritou et al. (2002), namely between the equivalent liq-uid rate for liquid production, the gaseous oxygen production, thegaseous nitrogen production and the power consumption. Startupcosts are described with respect to the energy consumption of thecorresponding transitional modes.

The equivalent liquid rate, Prhp,LIQ , is defined as follows, with

conversion parameters ˆ̨ LN2, ˆ̨ LO2, ˆ̨ LAR:

Prhp,m,LIQ = ˆ̨ LN2Pr

hp,m,LN2 + ˆ̨ LO2Pr

hp,m,LO2 + ˆ̨ LARPr

hp,m,LAR (32)

A multivariate regression, based on a given set of operatingpoints, was applied to obtain the model parameters for each oper-ating mode of each plant. Note that the statistical significance ofthis cost model can be measured by several metrics. Student’s t-test for a 95% confidence interval reported t-values greater than 2for all parameters of every plant configuration. Also, the R2 value,the standard deviation in the residual and the overall F-test wereanalyzed. With fitting parameters ˛p,m, ˇp,m, �p,m and the hourlyelectricity prices at each plant p (eh

p), the expenses for electricitycan be formulated based on (30) as:

OBJ =

⎛⎝∑

p,m,h

ehp(˛p,mPr

hp,m,LIQ + ˇp,mPr

hp,m,GO2 + �p,mPr

hp,m,GN2)

⎞⎠(33)

Note that highly insulated tanks allow to keep the liquid prod-ucts without additional cooling and negligible losses. Hence, weneglect inventory cost. The transitional cost term of (30) is removedbecause transitional cost are implicitly modeled by a higher energyconsumption in transitional modes, which reflects the efficiencylosses during mode transitions. Therefore, Eq. (33) is the objectivefunction of our optimization model, which we intend to minimize.

5.1.3. ResultsBased on real industrial data, we modeled two different air sep-

aration plants: a plant with one liquefier (P1) and a plant with twoliquefiers (P2). We tested the computational performance of ourmodel for five different cases (A–E) that differ in the demand thatis specified for the liquid products on a 6-h basis (which is similarto assume that a certain number of tanker trucks arrive within that

time window) for a time horizon of 1 week (168 h). The electric-ity price forecast of the day-ahead prices (see Fig. 5) was given onan hourly basis. Due to confidentiality issues we cannot disclose

178 S. Mitra et al. / Computers and Chemica

Fig. 5. Top: Electricity prices and energy consumption profiles for optimal cyclicsp

tso

iusumsena

TSd

chedules of plants P1 and P2 (test case B). Middle: Optimal LO2 and LN2 inventoryrofiles for plant P1. Bottom: Optimal LO2 and LN2 inventory profiles for plant P2.

he actual LO2 and LN2 demands and production levels. Therefore,cales are omitted, but the total demand is reported as a functionf the plants’ production capacity in Table 1.

For the same demand data, we solved the problem using timentervals of 1 h for two different setups: for a non-cyclic sched-le and a cyclic schedule. An optimal non-cyclic schedule mightuggest to shut down the plant at the end of the week. This is anndesirable situation if the decision maker can assume that thearket conditions will be similar in the following week, because a

hut-down hinders successful operations in the following week,

.g. due to minimum downtime restrictions. In contrast to theon-cyclic schedule, the cyclic schedule also satisfies all mass bal-nces and transitional constraints in a cyclic manner throughout

able 1avings of the optimal noncyclic schedules for all test cases as function of theemand/capacity ratio.

Case Demand/capacity Savings

P1A 82% 4.58%P1B 74% 12.02%P2E 95% 3.76%P2D 85% 4.90%P2C 72% 7.44%P2B 51% 13.78%

l Engineering 38 (2012) 171– 184

the week. We use the wrap-around operator (Shah, Pantelides, &Sargent, 1993) to modify constraints (12), (13), (14), (17) and (24).

To assess the economic impact of our optimization model, wecompare our solutions for the noncyclic schedules with a heuris-tic: we assume constant operation throughout the week and finda set point that locally minimizes the energy consumption whilesatisfying the demand constraints.

In Table 1 we can see that for both plants the potential savingsincrease with decreasing demand that is reported in percent of thetotal production capacity. For test cases A and B the realized savingsare mostly due to plant shutdowns, which is illustrated at the topof Fig. 5 for the electricity consumption profiles of plants P1 andP2 for test case B. We can also see that the electricity consumptionis reduced during hours of high electricity prices – due to higheroperational flexibility plant P2 can produce 5.2% cheaper than plantP1. The corresponding inventory profiles for plant P1 are reportedat the middle of Fig. 5 and the inventory profiles for plant P2 areshown at the bottom of Fig. 5. The lower bound corresponds to theoutage level and the upper bound is the storage tank capacity. Wecan clearly observe how the demand is served from the storage tankduring plant shutdowns.

For test cases C, D and E no shutdowns are reported since thestorage tank capacity is not large enough to store the productsrequired for the minimum downtime of the plant. Nevertheless,savings are observed that are due to the presence of plant P2 sec-ond liquefier that is turned on during hours of low electricity pricesand turned off during hours of peak prices.

In Table 2 the problem sizes are reported. The problem sizesdiffer significantly for the two plants since plant P1 has only threeoperating modes, in contrast to plant P2, which has six operat-ing modes (including transitional modes). However, the problemsizes for the cyclic and noncyclic schedules differ only slightly,because the implementation for the noncyclic schedule includesa few redundant constraints that could be filtered with conditionalstatements. However, the pre-solve operations of the MILP solvereliminate those constraints reliably.

The computational results can be found in Table 3. We solved alltest cases with the commercial solvers CPLEX 12.2.0.1 and XPRESS21.01 (both with default settings, no parallel computing featureswere used) in GAMS 23.6.2 (Brooke, Kendrick, & Meeraus, 2010)on a Intel i7 (2.93 GHz) machine with 4 GB RAM, using a termina-tion criterion of 0.1% optimality gap. Except for test case P2B, alltest cases can be solved within less than 30 s. CPLEX and XPRESSperform similarly, except for test cases A and B with advantageson both solvers. We can observe that with increasing demandthe problems become easier to solve since operational flexibilitydecreases. This can be seen in terms of CPU time and also in termsof the tightness of the LP relaxation (RMIP). For case P2E withthe highest demand the initial gap is in fact 0 %. Furthermore, thecyclic schedule is harder to obtain when compared to the non-cyclicschedule.

5.2. Cement plant

Castro et al. (2009, 2011) study the case of cement plants,where the final processing stage is a power-intensive grindingoperation. Based on the resource task network (RTN) formulation,

Table 2Problem sizes for two air separation plants.

Plant Type of schedule # constraints # variables # binary

P1 Noncyclic 22,508 16,129 1512P1 Cyclic 22,513 16,129 1512P2 Noncyclic 44,173 29,401 3528P2 Cyclic 44,185 29,401 3528

S. Mitra et al. / Computers and Chemical Engineering 38 (2012) 171– 184 179

Table 3Computational results for test cases A–E, plants P1 and P2. cyc: cyclic schedule, ncyc: noncyclic schedule, RMIP gap% shows the tightness of the linear relaxation.

Case cyc/ncyc RMIP MIP CPU Times (s) RMIP gap% Final Gap%CPLEX XPRESS CPLEX XPRESS CPLEX XPRESS

P1A ncyc 88,416 90,129 90,129 2 3 1.9006% 0.0000% 0.0000%P1A cyc 89,197 90,448 90,448 3 26 1.3831% 0.0000% 0.0000%P1B ncyc 72,806 74,932 74,932 4 5 2.8372% 0.0000% 0.0000%P1B cyclic 73,305 76,011 76,011 15 27 3.5600% 0.0588% 0.0000%P2B ncyc 67,659 70,017 70,017 135 47 3.3678% 0.0995% 0.0118%P2B cyc 67,173 72,027 72,027 844 462 6.7391% 0.0977% 0.0936%P2C ncyc 98,645 100,796 100,837 7 10 2.1340% 0.0955% 0.0740%P2C cyc 99,726 101,280 101,242 6 8 1.4974% 0.0684% 0.0085%P2D ncyc 113,521 114,715 114,715 4 4 1.0408% 0.0076% 0.0000%P2D cyc 115,173 116,125 116,122 5 6 0.8172% 0.0079% 0.0016%P2E ncyc 128,019 128,054 128,054 4 4 0.0273% 0.0000% 0.0000%

4

CtOttsndusiCtc

utw

5

eaIopsp

5tuFPipDtm|

5piPppft

2. Model TM+RT, which restricts the total number of transitions

P2E cyc 130,186 130,186 130,186

astro and co-workers present solution methods that most impor-antly differ in their time representation (see literature review).ne of the time representations they use is a discrete formula-

ion with a 1-h time grid. They outline three major issues withhis approach. First, due to the time discretization the obtainedchedules might be slightly suboptimal because changeovers can-ot take place between two time points. Second, due to solutionegeneracy the obtained schedules might have a large number ofndesirable changeovers. Third and lastly, the presence of sharedtorage adds even more symmetry to the problem. Altogether thesessues lead to the observation that the discrete time formulation ofastro et al. cannot be solved to optimality within a reasonableime. However, they report only a small final gap in all their testases.

In the following we will describe how our framework can besed to model the cement plant with a discrete time formulationhat can be solved to optimality within a reasonable amount of time,hile limiting the number of changeovers.

.2.1. Model formulationWe make the same basic assumptions that are reported in Castro

t al. (2009) and use the same model parameters. We model the par-llel lines as individual manufacturing facilities denoted by index p.n all cases, we use constraints (2)–(7) to define the feasible regionf the operating modes. However, the main challenge is the appro-riate definition of the operating modes. This “procedure” can beeen as choosing how and which of the capacity constraints of plant

are projected in the product space for each mode.

.2.1.1. Single product modes (model SPM). The most intuitive wayo define the operating modes is to have one mode for each prod-ct. The concept is illustrated for two products in the left part ofig. 6, where each node represents a single product mode (“Mode1” and “Mode P2”). The feasible region of each mode m is thenterval [0, �max

p,g ], where �maxp,g is the maximum processing rate of

roduction line p of product g. This approach is very similar to theT formulation of Castro et al. since we only allow for transitions at

he predefined event points (every hour). Obviously, the number ofodes in model SPM scales linearly with the number of products

G|.

.2.1.2. Convex hull mode (model CHM). The feasible region for eachroduction line p can also be described with only one mode, as

llustrated in the middle part of Fig. 6 by “Mode P1, P2”. “Mode P1,2” is the convex hull of all maximum production levels for eachroduct and the origin. In other words, all capacity constraints of

lant p are projected in the product space for one single mode. Theeasible region of model CHM for two products is also shown inhe middle part of Fig. 6. Note that this definition of the feasible

4 0.0000% 0.0000% 0.0000%

region allows for product transitions at any point in time and thereare no restrictions on the batch processing time. Hence, we canobtain the lowest possible production costs. The drawback of thisfeasible region is that we cannot limit the number of transitionsthat are occurring since the transitions are not linked to any binaryvariables.

5.2.1.3. Transitional modes (model TM and extensions). If we wantto combine SPM with the ability to allow for transitions at any pointin time, we need to introduce ideas from model CHM, e.g. transi-tional modes that account for changeovers between two products.In the right part of Fig. 6, model TM is illustrated with the stategraph for two products. As one can see in the upper right cornerof Fig. 6, model CHM is used for each pair of products to derivethe corresponding feasible region of the transitional mode (“ModeP1|P2”). More formally, the feasible region of the transitional modem of production line p that accounts for the transition from prod-uct g to g′ is the convex combination of the origin, �max

p,g and �maxp,g′ .

We modify the feasible region of the single product modes (“ModeP1” and “Mode P2”) as well. In order to avoid batch interruptionsthat may result from a definition of the feasible region as in SPM,we define the feasible region for each product g as a single point,namely [�max

p,g ]. Therefore, we have to introduce a “Mode 0” cor-responding to a shutdown of the plant, which only contains [0] inthe feasible region. We also define transitional modes (“Mode 0|P1”and “Mode 0|P2”) to allow for production ramp-ups at any point intime. The feasible region of these transitional modes is the interval[0, �max

p,g ].We use the switch variables constraints (12) and (13) to link

mode variables yhp,m and transitional variables zh

p,m,m′. Forbiddentransitions are implemented with constraint (18). To ensure thecoupling of transitions between transitional and single productmodes as well as shutdown, transitional and production modes,constraint (17) is applied. In Fig. 6, for each transitional mode thetwo transitions with dashed lines as well as the two transitions withsolid lines are coupled as previously described in Section 4.2.4.

With the setup of modes in model TM, we can limit the numberof transitions in different ways:

1. Model TM+MS, where we utilize the minimum stay constraint(14) to enforce a minimum uptime once the plant is switched onand a minimum downtime once the plant is switched off (e.g. atleast two subsequent time periods).

that occur.3. Model TM+MSRT, which is the combination of both aforemen-

tioned methods.

180 S. Mitra et al. / Computers and Chemical Engineering 38 (2012) 171– 184

F a cemw odes

n

dt(nad

5

w

cklob

P

I∑

I

i(a

5

msspbs

ig. 6. Different choices of defining modes and corresponding feasible regions for

ith convex hull (one mode), TM: model with transitional modes, single product m

Alternatively, changeover costs can be introduced, which we doot further explore in this paper.

In Fig. 7, we illustrate how the number of modes increasesepending on the number of products |G| with the case ofhree products. We observe that despite model TM growing|G|2 − |G|)/2 + 2|G| + 1 in the number of modes as a function of theumber of products |G|, the computational results show favor-ble CPU times. Note that the restricted power availability can beescribed in terms of Eq. (31).

.2.2. Mass balances for shared storage constraintsTo model the shared storage and the associated mass balances

e need additional constraints. Xhst,g is a binary variable that indi-

ates which product g is stored in storage st at hour h. Storehp,st,g

eeps track how much of product g is transferred from productionine p to storage tank st in hour h. Therefore, independent of theperating modes, the material transfers are tracked on an hourlyasis. The shared storage constraints can be written as:

rhp,g =

∑st∈ST

Storehp,st,g ∀p ∈ P, g ∈ G, h ∈ H (34)

NVhst,g + Sh

st,g ≤ INVUstX

hst,g ∀st ∈ ST, g ∈ G, h ∈ H (35)

g

Xhst,g = 1 ∀st ∈ ST, h ∈ H (36)

NVh−1st,g +

∑p

Storehp,st,g = INVh

st,g + Shst,g ∀st ∈ ST, g ∈ G, h ∈ H (37)

In addition, we use constraint (27) to enforce the hourly spec-fied demand for each product. The objective function is based on30). The terms for inventory cost and transition cost are neglected,nalogous to the papers by Castro and co-workers (2009, 2011).

.2.3. ResultsWe compare the different model choices for the operating

odes with respect to the known issues of the discrete timecheduling model of Castro et al. (2009): potentially suboptimal

olutions, including a small final gap since optimality cannot beroven within a reasonable amount of time, as well as large num-er of changeovers. All test cases were solved with the commercialolver GUROBI 4.0.0 (with default settings, no parallel computing

ent plant with two products. SPM: model with single product modes, CHM: modeland zero production mode.

features were used) in GAMS 23.6.2 (Brooke et al., 2010) on a Intel i7(2.93G Hz) machine with 4 GB RAM, using a termination criterion of10−6 optimality gap (as in Castro et al., 2009). GUROBI was foundto be more efficient than CPLEX and XPRESS on these problems.Our computational results for test cases Ex5-Ex10 (the “harder”problem instances) are reported in Table 4.

5.2.3.1. Single product modes (model SPM). The first observation isthat model SPM yields exactly the same suboptimal objective val-ues as the discrete time formulation of Castro et al. (2009), exceptfor Ex8. Moreover, the number of binary variables, which indicatethe status of the equipment and the storage tanks, is the same.Despite the newer hard- and software, we cannot solve the testcases to optimality within 2 h neither (except Ex7).

5.2.3.2. Convex hull mode (model CHM). On the contrary, modelCHM yields the same optimal objective values as the ones reportedin Castro et al. (2011) for all test cases. Since each productionline has only one operating mode, the problem size is signifi-cantly smaller and CHM can be solved in less than 20 s for alltest cases. Unfortunately, the number of transitions cannot be con-trolled with model CHM, which leads to schedules that are hard toimplement.

5.2.3.3. Transitional modes (model TM and extensions). The compu-tational results show that the ability to obtain the optimal solutiondue to the introduction of transitional modes in conjunction withthe tightness of the linear relaxation are key to the success of modelTM. This observation is somewhat counter-intuitive since the prob-lem size of model TM is roughly one order of magnitude largercompared to model SPM.

Test cases Ex5–7 and Ex9 can be solved to optimality in less than30 s, yielding the same objective value as CHM. For test case Ex8,TM yields a slightly suboptimal solution within 420 s. This is dueto the fact that TM implicitly assumes a batch processing time ofat least 1 h, but Ex8 (restricted power availability) requires a fewbatches with batch processing times less than 1 h. If we use the

convex hull mode instead of the previously described transitionalmode for production ramp-ups, we can allow for batch processingtime less than 1 h and we obtain the optimal objective value within36 s (model TMb).

S. Mitra et al. / Computers and Chemical Engineering 38 (2012) 171– 184 181

Fig. 7. Scalability of different choices for operating modes for cement plants as a function of the number of products |G|, here illustrated for three products. In the lower partof the figure, the number of modes and transitional variables per production line is reported. SPM: model with single product modes, CHM: model with convex hull (onemode), TM: model with transitional modes, single product modes and zero production mode.

Table 4Computational results for cement production. SPM: single product modes, CHM: convex hull mode, TM: model with transitional modes, single product modes and zeroproduction mode, TMS: TM strengthened with bound from CHM. Extensions: +MS: minimum stay constraints on up- and downtime, +RT: restricted number of transitions,+MSRT: both aforementioned restrictions.

Case Model constr. vars. bin. RMIP MIP CPUs Gap%

Ex5 Castro DT 6739 10,953 2016 26,738 26,780 7200a 0.04SPM 12,433 12,097 2016 26,738 26,780 7200b 0.02CHM 6385 8401 1008 26,738 26,758 1 0TM 58,389 59,473 13,776 26,738 26,758 5 0TM+MS 62,421 59,473 13,776 26,738 26,791 15 0TM+RT 58,390 59,473 13,776 26,738 26,758 5 0TM+MSRT 62,422 59,473 13,776 26,738 26,791 19 0

Ex6 Castro DT 8423 14,155 2520 43,250 43,259 7200a 0.02SPM 13,609 14,617 2520 43,250 43,259 7200b 0.02CHM 7561 10,921 1512 43,250 43,250 1 0TM 59,565 61,993 14,280 43,250 43,250 4 0TM+MS 63,597 61,993 14,280 43,250 43,250 8 0TM+RT 59,566 61,993 14,280 43,250 43,250 16 0TM+MSRT 63,598 61,993 14,280 43,250 43,250 44 0

Ex7 Castro DT 10,780 18,534 3528 68,282 68,282 19.9a 0SPM 19,489 22,681 3528 68,282 68,282 1 0CHM 10,417 17,137 2016 68,282 68,282 1 0TM 88,423 93,745 21,168 68,282 68,282 8 0TM+MS 94,471 93,745 21,168 68,282 68,282 8 0TM+RT 88,424 93,745 21,168 68,282 68,282 12 0TM+MSRT 94,472 93,745 21,168 68,282 68,282 11 0

Ex8 Castro DT 12,464 21,736 4032 101,139 104,622 7200a 0.22SPM 20,665 25,705 4032 104,375 104,607 7200b 0.09CHM 11,593 20,161 2520 104,375 104,375 5 0TM 89,599 96,769 21,672 104,375 104,502 420 0TMb 89,599 99,793 21,672 104,375 104,375 36 0TMb+MS 95,647 99,793 21,672 104,570 104,891 350 0TMb+RT 89,600 99,793 21,672 104,375 104,375 115 0TMb+MSRT 95,648 99,793 21,672 104,570 104,891 852 0

Ex9 Castro DT 13,810 24,092 4704 87,817 87,868 7200a 0.06SPM 29,569 32,257 4704 87,817 87,868 7200b 0.06CHM 13,441 22,681 2688 87,817 87,817 2 0TM 149,845 184,969 32,928 87,817 87,817 23 0TM+MS 159,925 184,969 32,928 87,817 87,838 32 0TM+RT 149,846 184,969 32,928 87,817 87,817 32 0TM+MSRT 159,926 184,969 32,928 87,817 87,838 101 0

Ex10 Castro DT 16,840 29,650 5880 86,505 86,582 7200a 0.09SPM 41,665 42,841 5880 86,505 86,582 7200b 0.04CHM 16,465 28,225 3360 86,505 86,550 17 0TM 232,924 331,633 47,208 86,505 86,550 4079 0TMS 232,924 331,633 47,208 86,550 86,550 118c 0TMS+MS 248,044 331,633 47,208 86,550 86,550 401c 0TMS+RT 232,925 331,633 47,208 86,550 86,550 734c 0TMS+MSRT 248,045 331,633 47,208 86,550 86,550 870c 0

a Castro et al. (2011), Intel Core2 Duo T9300 (2.5 GHz), 4 GB RAM, Windows Vista Enterprise, CPLEX 11.1 with default options.b Maximum computational time.c 17 s required to solve CHM already included.

182 S. Mitra et al. / Computers and Chemica

Table 5Parameter values for the maximum number of allowed transitions for models TM+RTand TM+MSRT that we used in our computational experiments as a function of thenumber of orders.

Case Ex5 Ex6 Ex7 Ex8 Ex9 Ex10

# of orders 9 9 17 18 21 22# of allowed transitions 70 70 120 140 150 150

Wamb

pstttt

btvi

Fidpp

For test case Ex10, the optimal solution of TM requires 4079 s.e can speed up the solution process if we strengthen the relax-

tion of TM by adding the objective value that we computed withodel CHM as a lower bound. The corresponding model TMS can

e solved within 118 s (including 17 s from CHM).We use model TM+MS to enforce a minimum uptime once the

lant is switched on and a minimum downtime once the plant iswitched off, both for at least two subsequent time periods. Despitehe additional constraints, we can obtain the same optimal objec-ive value as CHM for Ex6, Ex7 and Ex10. For Ex5, Ex8 and Ex9he objective is slightly larger. In all cases, the optimization runserminate in less than 6 min.

As previously mentioned, another possibility to limit the num-er of occurring transitions is restricting the total number of

ransitions (model TM+RT). Thus, we have to find an appropriatealue for the total number of allowed transitions. One possibil-ty is counting the number of orders and estimate the number

ig. 8. Cement production: optimal schedule and storage profiles for problemnstance Ex10 using model TMS+MSRT. The optimal schedule avoids productionuring hours of peak prices (brown vertical blocks). M1–M3 are the three parallelroduction lines, S1–S4 are the four storage tanks, P1–P5 indicate the five differentroducts.

l Engineering 38 (2012) 171– 184

of batches required to fulfill each order. Ramping up produc-tion, production of a single batch and shutting down productionrequires 4 transitions. If we assume that each order needs between1 and 2 batches, reasonable values for the total number of transi-tions can be found in the range [4 × no. orders, 8 × no. orders].In Table 5, we report the parameter values for the number ofallowed transitions that we used for our computational experi-ments.

The computational results for model TM+RT show that we canobtain the optimal solution with the same objective as CHM in lessthan 2 min for all test cases except Ex10, which takes less than 8 minto solve.

Combining the minimum up- and downtime constraints withthe restriction of the total number of transitions leads to modelTM+MSRT, which can produce even more practical schedules. Forthe number of allowed transitions, we use again the values reportedin Table 5. The obtained objective values are the same as for modelTM+MS, the solution times are longer than the ones reported formodel TM+MS but smaller than 15 min. In Fig. 8 we show a practicalschedule obtained with model TM+MSRT for Ex10. One can see thatthe schedule avoids production during hours of peak prices and hasa comparable number of batches to the schedule obtained by therolling horizon approach in Castro et al. (2011).

6. Conclusions

In this paper, we have presented a model for the optimaloperational production planning for continuous power-intensiveprocesses that participate in non-dispatchable demand responseprograms. We described a deterministic MILP model that allowsan accurate and efficient modeling of transitions between operat-ing modes using a discrete time representation. Properties on thetightness of several logic constraints were proved. We successfullyapplied the model to two different real-world air separation plantsthat supply to the liquid merchant market, as well as cement plants.

In the air separation case study, the operational optimizationmodel produced savings of more than 10% when compared to a sim-ple heuristic. We also learned that operational flexibility, in termsof production and storage capacity, is the key to lower operatingexpenses. For the same demand profile, a plant with two liquefierswas able to save 5% on costs when compared with a single liquefierplant.

We observed in the case of the cement plant that the intro-duction of transitional modes resulted in a large-scale model.Nevertheless, the model was superior compared to a smaller modelthat only had single product modes, and thus lacked the capabilityof limiting the number of occurring transitions, and could not besolved to optimality. Despite the large size of the MILP model withtransitional modes, the required solution times to obtain the opti-mal solution were small for all test cases. Furthermore, the obtainedschedules were practical to implement because we were able tolimit the number of occurring transitions.

Future work will examine the impact of long-term capitalinvestments in additional production or storage capacity thatincrease process flexibility and potentially facilitate savings on theoperational level. We also intend to model the effect of uncertaintyin the problem data.

Acknowledgements

We thank the Center for Advanced Process Decision-making(CAPD) at Carnegie Mellon University and Praxair for the financialsupport. We are also grateful to Michael Baldea and Larry Meganfor insightful feedback on the air separation case study.

emica

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S. Mitra et al. / Computers and Ch

ppendices for proofs

. On the tightness of constraint (14) for the minimum stayelationship

We prove that constraint (14) is tighter in the LP relaxationhan constraint (15). If we add over index � all Kmin

m,m′ constraintsf constraint (14) we get the following inequality:

minm,m′−1∑�=0

yh+�p,m′ ≥ Kmin

m,m′zhp,m,m′ ∀(p, m, m′) ∈ AL, h ∈ H (38)

We can substitute each yh+�p,m′ using the equality constraint (7),

y yh+�p,m′ = 1 −

∑m′′∈M,m′′ /= m′y

h+�p,m′′ and get:

minm,m′ −

Km,m′−1∑�=0

∑m′′∈M,m′′ /= m′

yh+�p,m′′ ≥ Kmin

m,m′zhp,m,m′

∀(p, m, m′) ∈ AL, ∀h ∈ H (39)

Note that there are exactly Kminm,m′ different yh+�

p,m′ variablesnvolved. By rearranging the equation we get the surrogate con-traint:

minm,m′−1∑�=0

∑m′′∈M,m′′ /= m′

yh+�p,m′′ ≤ Kmin

m,m′(1 − zhp,m,m′)

∀(p, m, m′) ∈ AL, ∀h ∈ H (40)

hich is exactly the constraint proposed by Ierapetritou et al.2002) and Karwan and Keblis (2007). Therefore, the set of con-traints (14) implies constraint (15). Furthermore, let us considerhe following non-integral point: assume one of the yh+�

p,m′′ in con-

traint (15) is 1. zhp,m,m′ = (1 − (1/Kmin

m,m′)) is feasible for constraint15), but it is infeasible in constraint (14) if MinStay(m, m ′) isonempty. Hence, constraint (14) is tighter. �

I. On the tightness of constraint (21) for forbidden transitions

We prove that constraint (21), which we rewrite here as

hp,m′ −

∑m:(p,m,m′)∈AL

yh−1p,m ≤ 0 ∀p ∈ P, m′ ∈ M, ∀h ∈ H (41)

s tighter in the LP relaxation than constraint (22),

h−1p,m + yh

p,m′ ≤ 1 ∀(p, m, m′) ∈ DAL, ∀h ∈ H

ote that both constraints are equivalent for integer values since ALnd DAL are disjoint sets. The key to the proof is equality constraint7), which results from the disjunction of modes. We can rewrite7) for a given m′ in the following way:

∈M

yhp,m =

∑m∈AL

yhp,m +

∑m∈DAL

yhp,m = 1 ∀p ∈ P, h ∈ H (42)

If we add over index m all K constraints (K = number of for-idden transitions for m′) of constraint (22) we get the following

nequality:∑∈DAL

yh−1p,m + Kyh

p,m′ ≤ K ∀p ∈ P, m′ ∈ M, ∀h ∈ H (43)

l Engineering 38 (2012) 171– 184 183

Using equality (42) we can rewrite (43) as

∑m∈DAL

yh−1p,m + Kyh

p,m′ ≤ K

(∑m∈AL

yh−1p,m +

∑m∈DAL

yh−1p,m

)

∀p ∈ P, m′ ∈ M, h ∈ H (44)

⇔ yhp,m′ −

∑m:(p,m,m′)∈AL

yh−1p,m − K − 1

K

∑m∈DAL

yh−1p,m ≤ 0

∀p ∈ P, m′ ∈ M, h ∈ H (45)

If we compare the resulting inequality (45) with constraint (41),we can see that the additional term −(K − 1)/K

∑m∈DALyh−1

p,m makes

(45) weaker compared to (41) because yh−1p,m are binary (nonnega-

tive). �

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