Model reformulations for Work and Heat Exchange Network (WHEN) synthesis problems

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Computers and Chemical Engineering

journalhomepage:www.elsevier.com/locate/compchemeng

Model reformulations for Work and Heat Exchange Network (WHEN) synthesis problems

Haoshui Yu

^a

, Matias Vikse

^a

, Rahul Anantharaman

^b

, Truls Gundersen

^a,∗

aDepartment of Energy and Process Engineering, Norwegian University of Science and Technology, Kolbjoern Hejes v. 1A, NO-7491 Trondheim, Norway

bSINTEF Energy Research, Kolbjoern Hejes v. 1A, NO-7491 Trondheim, Norway

a rt i c l e i n f o

Article history:

Received 15 December 2018 Revised 21 February 2019 Accepted 28 February 2019 Available online 1 March 2019 Keywords:

Work and heat exchange networks Duran-Grossmann model Reformulations disjunctive programming MINLP

a b s t r a c t

The Duran-Grossmannmodel can deal with heatintegrationproblems withvariable processstreams.

Work and Heat ExchangeNetworks (WHENs) represent an extension of Heat Exchange Networks. In WHEN problems, the identities of streams (hot/cold) are regarded as variables. The original Duran- Grossmannmodelhasbeenextendedand appliedtoWHENswithoutknowingtheidentityofstreams apriori.IntheoriginalDuran-Grossmannmodel,themaxoperatorisachallengeforsolvingthemodel.

ThispaperanalyzesfourwaystoreformulatetheDuran-Grossmannmodel.SmoothApproximation,Ex- plicit Disjunctions, DirectDisjunctions and Intermediate Temperaturestrategy arereviewedand com- pared.TheExtendedDuran-GrossmannmodelforWHENproblemsconsistsofbothbinaryvariablesand non-smoothfunctions.TheExtendedDuran-Grossmannmodelcan bereformulatedinsimilar ways.In thisstudy, the performance of different reformulationsof the Extended Duran-Grossmann model for WHENproblemsarecomparedbasedonasmallcasestudyinthispaper.

© 2019TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Heat integration has been widely used to save hot/cold utili- ties because thermalenergy contributes significantly to the total cost ofa process(HuangandKarimi,2013). Theclassicalheat in- tegration techniques, such aspinch technology (Klemeš and Kra- vanja, 2013), can only deal with the heat integration problem withknownstreamdata.Ifheatintegrationandprocessoptimiza- tion are performed simultaneously, i.e.heat integration consider- ing variable process streams,more benefits canbe achieved.Du- ran andGrossmann proposed a mathematical model forsimulta- neousprocessoptimizationandheatintegration(DuranandGross- mann, 1986). The Duran-Grossmann model is a powerful tool to solve theheatintegration problemwithvariableprocess streams.

Thispaperhasbeencitedmorethan350timesbytheendof2018.

TheirmodelhasbeensuccessfullyappliedtoorganicRankinecycle systemsrecovering low-temperaturewaste heat(Yuetal., 2017a), processes for liquefaction of naturalgas (Wechsung et al., 2011), optimalreactornetworksynthesis(LakshmananandBiegler,1996), andfuelcellsystems (Marechaletal.,2005).Toimprovetheper- formanceofthemodel,severalreformulationsareproposedinpre- viousstudies,whichwillbereviewedandcomparedinthisstudy.

∗ Corresponding author.

E-mail address: [email protected] (T. Gundersen).

The new topic(see e.g. Yu et al., 2018a) referred to as Work andHeat ExchangeNetworks (WHENs) arise ifpressure manipu- lations are considered while designing Heat Exchanger Networks (HENs). There are many potential applications of WHENs the- ory, such as a novel process for offshore liquefaction of natu- ral gas (Aspelund and Gundersen, 2009), effluent gas recovery (Liaoetal., 2017),process integrationincarboncaptureprocesses (Fu and Gundersen, 2016), and optimal distillation column inte- gration (Nair etal., 2018). More applicationscan be foundin the literature (Yu et al., 2018b). More generally speaking, not only temperatures but also pressures have to meet some specifica- tionsinasystem.Pressurespecificationsforprocessstreamsmake theproblemmorechallengingcomparedwithconventionalHENs.

Holiastos and Manousiouthakis (2002) proposed a mathemati- cal model minimizing hot/cold/work utility cost for HENs. Here

“work utility” refers to the generation or consumption of work.

Aspelundetal.(2007)proposedamanualmethodologyreferredto asExtendedPinchAnalysisandDesign(ExPAnD),wheretraditional Pinch Analysis is extended with pressure considerations andEx- ergy Analysis. Marmolejo-Correa and Gundersen (2012) proposed amethodology combiningExergy andPinch Analysesto designa ReverseBraytoncyclefortheliquefactionofnaturalgas.Basedon thisstudy, Marmolejo-Correa and Gundersen (2013) developed a novel diagram for exergy andenergy targeting for a heat recov- erysystemsubject tochanges in bothtemperature andpressure.

Thismethod is particularlysuitable forlow temperature systems https://doi.org/10.1016/j.compchemeng.2019.02.018

0098-1354/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

Nomenclature

CC CompositeCurve C ColdStreamSet cu ColdUtility def Heatdeficit

DG Duran-GrossmannModel

EDG ExtendedDuran-GrossmannModel ExPAnD ExtendedPinchAnalysisandDesign FCp HeatCapacityFlowrate

GAMS GeneralAlgebraicModelingSystem GCC GrandCompositeCurve

H Enthalpy/HotStreamSet HEN HeatExchangeNetwork

HRAT HeatRecoveryApproachTemperature hu Hotutility

i HotStreams

IT IntermediateTemperature j ColdStreams

LogMIP Logic-baseddisjunctivemodelsolver MINLP MixedIntegerNon-LinearProgramming NLP Non-LinearProgramming

ORC OrganicRankineCycle

P Pressure

p Pinchcandidate PC Pinchcandidateset PSE ProcessSystemEngineering

QSOA Heatloadofhotstreamsabovepinchcandidates QSIA Heatloadofcoldstreamsabovepinchcandidates R Alarge numberused intheexplicitdisjunction re-

formulation S Entropy/Streamset SA SmoothApproximation sup Supplystate

tar Targetstate T Temperature

WHEN WorkandHeatExchangeNetwork WSK Worksinkstream

WSR Worksourcestream Y Booleanvariables y Binaryvariables

γ

^{Specificheatcapacityratio}

ω

^{Processvariables}

suchasLNGprocesses.FuandGundersen(2015a)presentedasys- tematicgraphicaldesignprocedurefortheintegrationofcompres- sors inHENs above ambienttemperature. Similarly, Fuand Gun- dersen (2015b) integrated compressors into heat exchanger net- worksbelowambienttemperature. Fourtheoremswere proposed andused as the basis forthe design methodology. Fu and Gun- dersen also integrated expanders into heat exchanger networks above (Fu andGundersen, 2015c) and below(Fuand Gundersen, 2015d) ambient temperature. Wechsung et al. (2011) combined Pinch Analysis, Exergy Analysis, and Mathematical Programming tosynthesize HENsbelowambienttemperaturewithcompression andexpansionofprocessstreams.

The WHENs problem involvesboth heat integration andwork integration.TheDuran-Grossmannmodelcanbeextendedtosolve WHENproblems.Sincethe thermodynamicpath andtheidentity (hot/cold) of process streams are unknown in WHENs, classical heatintegrationmethodscannotbeapplied.Inaddition,theiden- tityofstreamscanalsotemporarilychangeinWHENs.Thispaper isamorecomprehensiveversionofYuetal.(2018c)thatextended theDuran-GrossmannmodeltoWHENproblems,wheretheiden-

titiesofstreamsareunknownapriori.Thepresentpaperaddsyet anotherreformulation basedona so-calledIntermediate Temper- ature strategy. The various reformulations ofthe original Duran- Grossmann modelare applied to theExtended Duran-Grossmann modelforWHEN synthesisproblems.There are fourdifferentre- formulationsfortheExtendedDuran-Grossmannmodelpresented in the literature. This study investigates the different reformula- tionsandtheircomputationalexpenses.

2. OriginalDuran-Grossmannmodelandreformulations

The original Duran-Grossmann model can consider the utility cost andother economicindicators simultaneously,andit can be writteninacompactwayasfollows:

min ob j=F

( ω

,x

)

+ChuQhu+CcuQcu

s.t.

(

^DG.1

)

^h

( ω

,x

)

⁼⁰

(

^DG.2

)

^g

( ω

^,x

)

^≤0

(

^DG.3

)

^T_i^p=T_iⁱⁿ

∀

^i∈H

(

^DG.4

)

^T_j^p=T_jⁱⁿ+HRAT

∀

^j∈C

(

^DG.5

)

^QSOA

(

^x

)

^p=

i∈H

FCpi

max

0,T_iⁱⁿ−T^p

−max

0,T_i^out−T^p

∀

^p∈PC

(

^DG.6

)

^QSIA

(

^x

)

^p=

j∈C

FCp_j

max

0,T_j^out−

(

^Tp−HRAT

)

−max

0,T_jⁱⁿ−

(

^Tp−HRAT

)

∀

^p∈PC

(

^DG.7

)

^Z_{de f}^p

(

^x

)

=QSIA

(

^x

)

^p−QSOA

(

^x

)

^p

(

^DG.8

)

^Z_{de f}^p

(

^x

)

≤Q_hu

(

^DG.9

)(

^x

)

+Qhu−Qcu=0

(

^DG.10

)(

^x

)

=

i∈H

FCpi

(

^Tiⁱⁿ−T_i^out

)

−

j∈C

FCpj

(

^Tj^out−T_jⁱⁿ

)

(

^DG.11

)

^Qhu≥0,Qcu≥0

DG.1 andDG.2 are the equality andinequality constraints for the industrial process. The vector

ω

^{denotes process parameters}

such aspressure,temperature, orparameters incostcorrelations.

DG.3 andDG.4 are used to assign the inlet temperature of each streamtopotentialpinchcandidates.Itshouldbenoticedthatonly cold stream inlet temperatures are modified to consider the ef- fectoftheHeatRecoveryApproachTemperature(HRAT).DG.5and DG.6denotethe totalhotstream heatloadandtotal coldstream heatloadaboveeachpinchcandidatetemperature.DG.7andDG.8 aimatidentifyingthecorrectpinchpoint,whichfeaturesthemax- imumheatdeficitamongallthepinchcandidates.DG.9andDG.10 areenergybalancesforthesystem.

The Duran-Grossmann model incorporates max operators, which result in non-differentiabilities at T^p. Max operators are challengingfordeterministic solversandhave tobe removedbe- foresolving themodel.The originalDuran-Grossmann modelhas proven to be powerful in process design. Thus, interest has in- creasedintheProcessSystemEngineering(PSE)fieldtofindways toreformulatethemodel.Fourdifferentreformulationshavebeen found in the literature, andtheseare presented andused in the ExtendedDuran-Grossmann model forWHENsynthesis. The four reformulations are the following: (1) Smooth Approximation, (2) Explicit Disjunction, (3)Direct Disjunction, and(4)an Intermedi- ateTemperaturestrategy.

2.1. Smoothapproximationfortheheatintegrationmodel

The max operator in the Duran-Grossmann model was reformulated by using smooth approximations proposed by

Balakrishna and Biegler (1992). This reformulation has been ap- pliedtoheatintegrationproblemsconsideringorganicRankinecy- cles (Yu et al., 2017b) and carbon capture processes considering waste heat recovery (Yu et al., 2018d). The max operator in the originalDuran-Grossmannmodelcanbereformulatedbyusingthe equationshowninEq.(1)tomodifyDG.5andDG.6.

max

{

^0,x

}

^∼=1 2

x+

x²+

ε

⁽¹⁾

Here,ɛisasmallconstant,typicallybetween10⁻³ and10⁻⁶. However, this reformulation may encounter problems when dealing withisothermal streams. In addition,the performance of the approximation depends on the value of the small constant, which may cause numerical conditioning problem if chosen im- properly (Grossmann et al., 1998). The small parameter is close to zero, and the Smooth Approximation can sometimes be ill- conditioned.

2.2. Explicitdisjunctionfortheheatintegrationmodel

Toremove themax operatorin theoriginal Duran-Grossmann model,Grossmannetal.(1998)proposed adisjunctive reformula- tion. Thisreformulationcan even handleisothermal streams ina system. Thekeyideaofthedisjunctive formulationistheexplicit treatmentofthreepossibilitiesforprocessstreamtemperatures:a process stream istotally above,totally beloworacross thepinch candidate temperature, asshownin Fig.1.When a stream isto- tally above the pinch candidate temperature, both the inlet and outlettemperaturesaregreaterthanthepinchcandidatetempera- ture. Whenastreamistotallybelowthepinchcandidatetemper- ature,boththeinletandoutlettemperaturesare belowthepinch candidatetemperature. These twostatementsare validregardless ofthestreamsbeinghotorcold.However,ifthestreamisacrossa pinchcandidate temperature,the constraintsare differentforhot andcoldstreams.Forhotstreams,theinlettemperatureisgreater thanthepinchcandidatetemperature,andtheoutlettemperature islessthanthepinchcandidatetemperature. Incontrast,different constraintsapply tocoldstreams.Toavoidtheuseofmaxopera- tors,intermediatevariablesareintroducedtocalculatethecorrect heat loadof hot and cold streams respectively, asshown in the Eq.(2).

Then the max operators in the original Duran-Grossmann model can be replaced by the disjunctions shown in Eq. (2). In

ourstudy,werefertothisdisjunctivereformulationasExplicitDis- junction.

⎡

⎢ ⎢

⎢ ⎣

Y1_i^p T_iⁱⁿ≥T^p T_i^out≥T^p T_i^in,p=T_iⁱⁿ−T^p T_i^out,p=T_i^out−T^p

⎤

⎥ ⎥

⎥ ⎦

^∨

⎡

⎢ ⎢

⎢ ⎣

Y2^p_i T_iⁱⁿ≥T^p T_i^out≤T^p T_i^in,p=T_iⁱⁿ−T^p

T_i^out,p=0

⎤

⎥ ⎥

⎥ ⎦

^∨

⎡

⎢ ⎢

⎢ ⎣

Y3^p_i T_sⁱⁿ≤T^p T_s^out≤T^p T_s^in,p=0 T_s^out,p=0

⎤

⎥ ⎥

⎥ ⎦

⎡

⎢ ⎢

⎢ ⎢

⎣

Y1^p_j T_jⁱⁿ≥T^p T_j^out≥T^p T_j^in,p=T_jⁱⁿ−T^p T_j^out,p=T_j^out−T^p

⎤

⎥ ⎥

⎥ ⎥

⎦

^∨

⎡

⎢ ⎢

⎢ ⎢

⎣

Y2^p_j T_jⁱⁿ≤T^p T_j^out≥T^p T_j^in,p=0 T_j^out,p=T_j^out−T^p

⎤

⎥ ⎥

⎥ ⎥

⎦

^∨

⎡

⎢ ⎢

⎢ ⎢

⎣

Y3^p_j T_jⁱⁿ≤T^p T_j^out ≤T^p T_j^in,p=0 T_j^out,p=0

⎤

⎥ ⎥

⎥ ⎥

⎦

QSOA

(

^x

)

^p=

i∈H

FCps

(

^Ti^in,p−T_i^out,p

)

QSIA

(

^x

)

^p=

j∈C

−FCp_j

(

^Tj^in,p−T_j^out,p

)

⁽²⁾

2.3.Directdisjunctionfortheheatintegrationmodel

Recently,Quiranteetal.(2017)proposedanothernovelandro- bustdisjunctivereformulation.Thismethodreformulatesthemax operator from a pure mathematical point of view without any physicalinsightregardingtheheatintegrationbackground.Were- fer to thisreformulationas Direct Disjunction in thisstudy. This reformulationhasfewerBoolean variablescomparedwiththeEx- plicitDisjunction (Grossmann etal., 1998), thus showsbetter re- laxationgapsandreducednumberofequations.

Themaxoperatorisexpressedasfollows:

φ

^=max

(

⁰,c^Tx

)

⁽³⁾

Basedonmathematicalanalysis,themaxoperatorcanbeeither 0orapositivenumber.Therefore,adirectdisjunctionisproposed asshowninEq.(4).

Y ¬Y

c^Tx≥0

φ

=c^Tx

∨

c^Tx≤0

φ

=0

Y∈

{

^True,False

}

⁽⁴⁾

Fig. 1. Relationship between pinch candidate temperature and process streams ( Yu et al., 2018e ).

Usingthisformulation,themaxoperatorinEqs.DG.4andDG.5 canbereplacedbythedisjunctionsasshowninEq.(5).

_Yin

T_iⁱⁿ−Ti^p≥0

φ

ⁱⁿi =T_iⁱⁿ−T^p

∨

_¬Yin

T_iⁱⁿ−Ti^p≤0

φ

iⁱⁿ=0

_Yout

T_i^out−iT^p≥0

φ

^outi =T_i^out−T^p

∨

_¬Yout

T_i^out−iT^p≤0

φ

i^out=0

⎡

⎣

^Y

in j

T_jⁱⁿ+HRAT−T^p≥0

φ

ⁱⁿj =T_jⁱⁿ+HRAT−T^p

⎤

⎦

∨

⎡

⎣

^¬Y

in j

T_jⁱⁿ+HRAT−T^p≤0

φ

ⁱⁿj =0

⎤

⎦

⎡

⎣

^Y

out j

T_j^out+HRAT−T^p≥0

φ

^outj =T_j^out+HRAT−T^p

⎤

⎦

∨

⎡

⎣

^¬Y

out j

T_j^out+HRAT−T^p≤0

φ

^outj =0

⎤

⎦

QSOA

(

^x

)

^p=

i∈H

FCp_i

( φ

ⁱⁿi −

φ

i^out

)

QSIA

(

^x

)

^p=

j∈C

−FCpj

( φ

ⁱⁿj −

φ

^outj

)

⁽⁵⁾

2.4.IntermediatetemperaturestrategyfortheHeatIntegrationModel

Anantharaman et al. (2014) revisited the Duran-Grossmann model to improve the solution of the formulation. They pointed outthattheExplicitDisjunctionreformulationhasthedrawbackof introducingalarge numberofbinaryvariables.The novelidea in thisstudyistointroduceanewvariablenamedintermediatetem- perature,torepresentthepinchcandidatetemperatureandavoid using max operators. We refer to this reformulation as the In- termediate Temperature (IT) strategy in thisstudy. The key idea ofthe three reformulations discussed in Sections 2.1–2.3is how toreformulatethemaxoperatorsintheDuran-Grossmannmodel.

TheIntermediateTemperature strategy,however,isdifferentfrom thethree previous reformulations. Eqs. DG.5-DG.8 inthe original Duran-Grossmann model can be written in one single compact equationasshowninEq.(6).

Q_hu≥

j∈C

(

^Tj^out−t^M_j,p

)

·FCpj−

i∈H

(

^Tiⁱⁿ−t_i^M_,p

)

·FCpi (6)

In Eq. (6), intermediate temperatures (t_i,p^M andt^M_j,p) are intro- duced. Hotutility consumption isdetermined by the heat deficit between hot and cold streams above each potential pinch tem- perature.Todetermine thecorrect intermediate temperaturecor- responding to the correct pinch temperature, more constraints are incorporated in the model. More detailed and updated in- formation about this model can be found in the updated notes (Anantharaman, 2018). In this reformulation, max operators are avoided but binary variables are introduced. The reformulated Duran-GrossmannmodelwiththeITstrategyisasfollows:

min ob j=F

( ω

^,x

)

^+ChuQ_hu+CcuQcu

s.t.

(

^IT.1

)

^h

( ω

^,x

)

⁼⁰

(

^IT.2

)

^g

( ω

^,x

)

^≤0

(

^IT.3

)

^Ti^p=T_iⁱⁿ

∀

ⁱ∈H

(

^IT.4

)

^Tj^p=T_jⁱⁿ+HRAT

∀

^j∈C

(

^IT.5

)

^Qhu≥

j∈C

FCp_j

T_j^out−T_j^M_,p

−

i∈H

FCp_i

T_iⁱⁿ−T_i^M_,p

∀

^p∈PC

(

^IT.6

)

^Ti^M,p≥T_i^out

∀

ⁱ∈H,p∈PC

(

^IT.7

)

^Ti^M,p≥T^p−M_i,p·y_i,p

∀

ⁱ∈H,p∈PC

(

^IT.8

)

^Ti^M,p≥T^p−Ui,p·

1−yi,p

∀

ⁱ∈H,p∈PC

(

^IT.9

)

^Tj^M,p≤T_i^out

∀

^j∈C,p∈PC

(

^IT.10

)

^Tj^M,p≤T^p+M_j,p·

1−y_j,p

∀

^j∈C,p∈PC

(

^IT.11

)

^Tj^M,p≤T_jⁱⁿ+U_j,p·y_j,p

∀

^j∈C,p∈PC

(

^IT.12

)(

^x

)

^+Qhu−Qcu=0

(

^IT.13

)(

^x

)

⁼

i∈H

FCp_i

(

^Tiⁱⁿ−T_i^out

)

⁻

j∈C

FCp_j

(

^Tj^out−T_jⁱⁿ

)

(

^IT.14

)

^Qhu≥0,Qcu≥0

Here, y_i,p and y_j,p are binary variables indicating whether a stream is above or below a pinch candidate. The case where a stream is across the pinch candidate temperature is not treated separately.Forhotstreams,y_i,p=1correspondstothecasewhere streamiisbelowthepinchcandidatetemperatureandy_i,p=0cor- respondstothe casewherestream iisabove oracrossthepinch candidate temperature. For cold streams, y_j,p=1 corresponds to the case where stream j is across or below the pinch candidate temperatureandy_j,p=0 corresponds tothe casewherestream j isabovethepinchcandidatetemperature.MandUarevalidupper boundsassociatedwithbinaryvariablesy_i,pandy_j,p.

2.5. Modelcomplexity

The four reformulations are proposed in the following chronological order: Smooth Approximation (Balakrishna and Biegler,1992), ExplicitDisjunction(Grossmannetal.,1998),Inter- mediateTemperaturestrategy(Anantharamanetal.,2014)andDi- rectDisjunction(Quiranteetal.,2017).SmoothApproximationhas the following advantages:no binary variables are needed, and it iscomputationally efficient.However, the reformulationhasdiffi- cultywhenhandlingisothermalstreamsandintermediateutilities.

Inaddition,theSmoothApproximationparameterhastobechosen properly,otherwisenumericalissuescouldarise.Toovercomethe limitations of Smooth Approximation, Explicit Disjunction, which is capable of handling isothermal streams and multiple utilities, isproposed.However,3Booleanvariablesareintroducedforeach pairofstreamsandpinchcandidates.The numberofbinariesare increasing rapidlywiththescaleoftheproblem. Therefore,itbe- comeschallengingtosolvethemodeliftheproblemsizeislarge.

Motivatedby thischallenge,Direct Disjunction,whichonlyneeds 2Booleanvariablesforeachpairofstreamsandpinchcandidates, provides a better reformulation of theoriginal Duran-Grossmann model. Direct Disjunction should perform much better than Ex- plicit Disjunction, especially formedium orlarge-scale problems.

The Intermediate Temperature strategy introduces a new contin- uous variable to avoid using max operators. One binary variable to activate/deactivatethe corresponding constraints hasto be in- troduced.The differentreformulationsare subjecttothetrade-off betweencontinuousvariablesandbinaryvariables.

3. ExtendedDuran-Grossmannmodelforworkandheat integration

In this study, we mainly focus on the application of the Duran-Grossmann model forWork and Heat Exchange Networks (WHENs).The Duran-Grossmann modelhasbeensuccessfullyex- tended toWHENproblems(Yuetal.,2018e).Abriefintroduction totheWHENsproblemispresentedhere.TheWHENsproblemcan be statedas follows:Given a setof process streams withsupply andtarget state (temperature, pressure), aswell ashot, coldand powerutilities; theobjectiveis todesigna networkconsistingof

Fig. 2. Superstructure for streams belonging to WSK ( Yu et al., 2018e ).

heattransferequipmentsuchasheatexchangers,heatersandcool- ers,andpressuremanipulationequipmentsuchasexpanders,com- pressors,pumpsandvalveswithminimumExergyConsumptionor minimumTotalAnnualizedCost.

IntheWHENsproblem,aprocessstreamwhosetargetpressure is greater than the supply pressure is calleda work sinkstream (WSK). Opposite, a work source stream(WSR) can be defined as a process stream whose target pressure is less than the supply pressure.Anyprocess streamcanbeheated, cooledorsimplynot changedbeforepressuremanipulation.Fig.2illustratesthesuper- structureofastreaminthecategoryofWSK.Detailedinformation aboutthesuperstructureisavailableinYuetal.(2018e).

SincetheidentityofstreamsintheWHENisunknownapriori, the Duran-Grossmann model cannot be applied directly andhas to be extended to anew modelusing binaryvariables todenote the identity ofstreams.In theExtended Duran-Grossmann (EDG) model, separate sets of hot andcold streams do no longerexist.

Binaryvariablesareusedtoautomaticallydistinguishthehot and coldstreamsinthemodel.TheExtendedDuran-Grossmannmodel canbeformulatedasfollows:

Min ob j=ExergyConsumption s.t.

(

^EDG.1

)

^h

( ω

,x

)

⁼⁰

(

^EDG.2

)

^g

( ω

^,x

)

^≤0

(

^EDG.3

)

^Ts^p=T_sⁱⁿ+ys·HRAT

∀

^s∈S

(

^EDG.4

)

^QSOA

(

^x

)

^p=

s∈S

(

¹−ys

)

^FCps

max

0,T_sⁱⁿ +ys·HRAT−T^p

}

^−max

0,T_s^out +ys·HRAT−T^p

}

^]

∀

^p∈PC

(

^EDG.5

)

^QSIA

(

^x

)

^p=

s∈S

ys·FCps

max

0,T_s^out +ys·HRAT−T^p

}

^−max

0,T_sⁱⁿ +ys·HRAT−T^p

}

^]

∀

^p∈PC

(

^EDG.6

)

^Zde f^p

(

^x

)

=QSIA

(

^x

)

^p−QSOA

(

^x

)

^p

∀

^p∈PC

(

^EDG.7

)

^Zde f^p

(

^x

)

≤Q_hu

∀

^p∈PC

(

^EDG.8

) (

^x

)

+Q_hu−Qcu=0

(

^EDG.9

) (

^x

)

=

s∈S

(

¹−ys

)

^FCps

(

^Tsⁱⁿ−T_s^out

)

−

s∈S

ys·FCps

(

^Ts^out−T_sⁱⁿ

)

(

^EDG.10

)

^Qhu≥0,Qcu≥0

Here, x represents the flow rates and temperatures of the streams involved in heat integration.

ω

^{represents all the other}

processvariables.Eqs.EDG.1andEDG.2 denotetheprocessequal-

ity and inequality constraints as those in the original Duran- Grossmann model. ys is a binary variable to denote the identity ofaprocessstream.Inthisstudy,y_s=1meansstreamsisacold stream.QSOAandQSIAdenotethetotalheatloadofhotandcold streams above each pinch candidatep∈PC. Z_{de f}^p (^x) ^{is heat deficit} above each pinch candidate.^{(x) is the heat loaddifference be-} tween hotandcold streams. HRATdenotesthe heatrecovery ap- proachtemperature. Theobjective functionis minimizing theex- ergy consumption of the system, which is related to the use of thermalutilities and shaft work consumed in thesystem. In the nextsections,thepreviouslyreviewedreformulationsfortheorig- inalDuran-Grossmannmodelare appliedtothe ExtendedDuran- Grossmannmodel.Sincetheidentityofthestreamsareunknown apriori, thesereformulationshaveto be revisedaccordingly.The reformulationsfortheextendedDuran-Grossmannmodelarepre- sentedasfollows.

3.1. Smoothapproximationfortheworkandheatintegrationmodel

FortheExtended Duran-Grossmannmodel,the maxoperators can be replaced by Smooth Approximations as well. It issimilar to the reformulationfor the original Duran-Grossmann model as discussedinSection2.1.However,binaryvariablesareinvolvedin theSmoothApproximationreformulationinthiscase.Thedetailed modelisomittedinthissectionsinceitisstraightforward.

3.2.Explicitdisjunctionfortheworkandheatintegrationmodel For the explicit disjunction reformulation, it is not neces- sary to distinguish between hot and cold streams in the Ex- tended Duran-Grossmannmodel.In contrasttothe reformulation inSection2.2,onlythreedisjunctionsareneededintheExtended Duran-Grossmannmodel.However,moreconstraintsareneededto taketheidentityofstreamsintoaccountinthedisjunction.Espe- ciallyforthecasewhereastream operatesacrossthe pinchcan- didatetemperature, theconstraints aredifferent forhotandcold streams.Therefore,thereare3moreconstraintsintheseconddis- junctionasshowninEq.(7)comparedwiththe ExplicitDisjunc- tionreformulationfortheoriginalDuran-Grossmannmodel.

⎡

⎢⎢

⎢⎣

Y1_s^p T_sⁱⁿ+ys·HRAT≥T^p T_s^out+ys·HRAT≥T^p T_s^in,p=T_sⁱⁿ+ys·HRAT−T^p T_s^out,p=T_s^out+y_s·HRAT−T^p

⎤

⎥⎥

⎥⎦^∨

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

Y2^p_s

T_sⁱⁿ+y_s·HRAT≥T^p−y_s·R T_s^out+ys·HRAT≥T^p−(¹−ys)·R

T_sⁱⁿ+ys·HRAT≤T^p+(¹−ys)·R T_s^out+ys·HRAT≤T^p+ys·R T_s^in,p=(¹−ys)^Tsⁱⁿ−(¹−ys)^Tp T_s^out,p=ys·T_s^out+ys·HRAT−ys·T^p

⎤

⎥⎥

⎥⎥

⎥⎥

⎦

∨

⎡

⎢⎢

⎢⎣

Y3_s^p T_sⁱⁿ+ys·HRAT≤T^p T_s^out+ys·HRAT≤T^p

T_s^in,p=0 T_s^out,p=0

⎤

⎥⎥

⎥⎦

QSOA(^x)^p=

s∈S

(¹−y_s)^FCps(^Ts^in,p−T_s^out,p)

QSIA(x)^p=

s∈S

−ys·FCps(T_s^in,p−T_s^out,p) (7)

Here, Risavalidupperbound torelaxthe constraintsforthe binaryvariablesdenotingstreamidentities.The valueofRcanbe estimatedbasedontemperaturesoftheprocessstreams.

After thereformulation,theExtendedDuran-Grossmannmodel becomes a disjunctive model, which can be transformed into a Mixed Integer Non-Linear Programming (MINLP) problem by the Big-M method or the convex hull method (Türkay and Grossmann, 1996). In this study, LogMIP (Vecchietti and Gross- mann, 2004), a specially designed program for disjunctive pro- gramming,is adopted asthe solver. Users can freely choose the Big-Mmethod orconvexhullmethod intheGAMS environment, whichfacilitiesthemodelingandsolutionsubstantially.

3.3.Directdisjunctionfortheworkandheatintegrationmodel

Recently, Quirante et al. (2018) proposed a disjunctive model considering unclassified streams and area estimation. In their study, the stream identity is expressed as a disjunction. This is in contrast to our study, where the stream identity is denoted by using binary variables. Based on the reformulation presented in Section 2.4, the direct disjunction reformulation can be ap- pliedto the ExtendedDuran-Grossmann model in a similar way.

However,only two disjunctionsare necessarysince the Extended Duran-Grossmannmodelonlyhasonecommonsetfortheprocess streams anddoesnot distinguishbetween hot andcold streams.

ThedirectdisjunctioncanreplacethemaxoperatorinEqs.EDG.4 and5.Theresultingdisjunctions areshowninEq.(8).Intermedi- atevariables

φ

in and

φ

^{out areintroduced inthedirect}disjunction reformulation.

_Y

in

T_sⁱⁿ+ys·HRAT−T^p≥0

φ

in=T_sⁱⁿ+ys·HRAT−T^p

∨

_¬Y

in

T_sⁱⁿ+ys·HRAT−T^p≤0

φ

in=0

_Y

out

T_s^out+ys·HRAT−T^p≥0

φ

^out=T_s^out+ys·HRAT−T^p

∨

_¬Y

out

T_s^out+ys·HRAT−T^p≤0

φ

^out=0

QSOA

(

^x

)

^p=

s∈S

(

¹−ys

)

^FCps

( φ

in−

φ

^out

)

QSIA

(

^x

)

^p=

s∈S

−ys·FCps

( φ

in−

φ

^out

)

⁽⁸⁾

ComparedwithExplicitDisjunction,onlytwoBooleanvariables are needed foreach pair of streams and pinch candidates. With the above disjunctions, the model can easily be implemented in theGAMSenvironment along withother equationsrelatedtothe process.

3.4.Intermediatetemperaturestrategyfortheworkandheat integrationmodel

FortheIntermediateTemperaturestrategy,thereformulationis verydifferentfromthat fortheoriginalDuran-Grossmann model.

IntheExtendedDuran-Grossmannmodel,thereisonlyonesetin- cludingbothhotandcoldstreams.Therefore,alltheequationsare definedbasedonasinglestreamset.Toactivatethecorresponding constraintsforastreamchangingfromhotstreamtocoldstream, abig-M relaxationstrategy isadopted.The Intermediate Temper- aturereformulation for work and heat integration model can be expressedasfollows:

min ob j=F

( ω

,x

)

^{+ChuQhu+CcuQcu} s.t.

(

^EDG−IT.1

)

^h

( ω

^,x

)

⁼⁰

(

^EDG−IT.2

)

^g

( ω

,x

)

^≤0

(

^EDG−IT.3

)

^Ts^p=T_sⁱⁿ+ys·HRAT

∀

^s∈S

(

^EDG−IT.4

)

^Qhu≥

s∈S

ys·FCps

T_s^out−T_s^M_,p

−

s∈S

(

^1−ys

)

^FCps

T_sⁱⁿ−T_s,p^M

∀

^p∈PC

(

^EDG−IT.5

)

^Ts^M,p≥T_s^out−Ms·ys

∀

^s∈S,p∈PC

(

^EDG−IT.6

)

^Ts^M,p≥T^p −Ms,p·ys,p

−Ms·ys

∀

^s∈S,p∈PC

(

^EDG−IT.7

)

^Ts^M,p≥T^p −U_s,p·

(

^1−ys,p

)

−Ms·ys

∀

^s∈S,p∈PC

(

^EDG−IT.8

)

^Ts,p^M≤T_s^out+Ms·

(

^1−ys

) ∀

^s∈S,p∈PC

(

^EDG−IT.9

)

^Ts,p^M≤T^p +Ms,p·

(

^1−ys,p

)

+Ms·

(

¹−ys

) ∀

^s∈S,p∈PC

(

^EDG−IT.10

)

^Ts^M,p≤T_sⁱⁿ+U_s,p·y_s,p

+Ms·

(

^1−ys

) ∀

^s∈S,p∈PC

(

^EDG−IT.11

)(

^x

)

+Q_hu−Qcu=0

(

^EDG−IT.12

)(

^x

)

=

i∈H

FCp_i

(

^Tiⁱⁿ−T_i^out

)

−

j∈C

FCpj

(

^Tj^out−T_jⁱⁿ

)

(

^EDG−IT.13

)

^Qhu≥0,Qcu≥0

Itshould be noticedthat ys is abinaryvariableto denotethe streamidentity,whileys,pisabinaryvariabletodenotetherela- tionshipbetweentheintermediatetemperatureandthepinchcan- didatetemperature.Ifthestreamidentityisahotstream(i.e.ys= 0),thenconstraintsEDG-IT.5-7areactiveandconstraintsEDG-IT.8- 10arerelaxed. Ifthestreamidentityisacoldstream(i.e.ys=1), thenconstraintsEDG-IT.8-10areactiveandconstraintsEDG-IT.5-7 are relaxed. Ms arevalid upperboundsfortemperatures to relax the constraints relatedto binaryvariables y_s. Similarly, M_s,p and Us,p arevalidupperboundsassociatedwithbinaryvariablesys,p. It should be notedthat the value ofthese parameters willaffect thecomputationaltimeofthemodel.

4. Casestudy

This case study is taken from the study by Fu and Gunder- sen(2015a).ThestreamdataarelistedinTable1.Thereare4pro- cessstreams. Stream C1is subjectto pressure changeand needs tobecompressedfrom100kPato300kPa.Thehot andcoldutili- tiesaresuppliedat400°Cand15°Crespectively.Theproblemisto determineifstreamC1needstobe splitintosub-streams,andto findthe optimalinlettemperature(s)forthecompressor(s). Since thisisasmall-scaleproblem,C1issplitonlyintotwosub-streams in the superstructure to reduce the modelsize. The HRAT is set tobe20°C.Theambienttemperatureisassumedtobe15°C,thus theexergyofcoldutility(at15°C)iszerointhiscase.Thefluidsto

Table 1

Stream data for the case study.

Stream T ^sup (^◦C ) ^T

tar

(^◦C ) ^F (kW ^{C p} / ^◦C ) H (kW ) ^P

sup

(kPa ) ^P

tar

(kPa )

H1 300 50 4 1000 – –

H2 120 40 4 320 – –

C1 70 380 3 930 100 300

C2 30 180 3 450 – –

Hot utility 400 400 – – – –

Cold utility 15 15 – – – –

Fig. 3. The optimized superstructure of C1.

Table 2

Optimal stream data for C1 for all reformulations.

Stream T ^sup( ^◦C) T ^tar( ^◦C) FCp (kW/ ^◦C) H (kW) P ^sup(kPa) P ^tar(kPa)

C1_S1 70 280 1.47 308.7 100 100

C1_S2 484 380 1.47 152.9 300 300

C1_S3 70 35 1.53 53.6 100 100

C1_S4 148.6 380 1.53 354 300 300

becompressedareassumedtobehavelikeidealgaswithconstant specificheatcapacityratio

γ

=1.4.

The ExtendedDuran-Grossmann modelcandetermine theop- timal split ratio of stream C2 and the optimal temperature(s)of stream C1 before compression. For this casestudy, all the refor- mulations are able to find the global optimum. The detailed re- sultsconcerning streamC1 are listedinTable 2.Without consid- ering pressure manipulation, the original pinch temperatures are 120/100 °C forhot andcold streams respectively. In the optimal configuration, stream C1 is split into two sub-streamswith heat capacityflowratesbeing1.47kW/°Cand1.53kW/°Crespectively.A newpinchiscreatedandlocatedat300/280°C.Itcanbeseenthat partofC1(sub-streamC1_S1)isheatedtothenewpinchtemper- ature 280 °C before compression. The other part is cooled down toambienttemperaturebeforecompression.Theoptimizedsuper- structureofC1 isshowninFig.3.Theresultsareconsistent with thestudybyFuandGundersen(2015a).

The overall systemperformance underoptimal conditions are summarizedinTable3.TheCompositeCurvesandtheGrandCom- positeCurveareshowninFig.4.Thehotutilitydemandiszerobe- causethecompressionheatofC1_S2 canbefullyutilizedtoheat cold streams in the systemabove the pinch. Stream C1 is com- pressedatthenewpinchtemperatureandatambienttemperature.

In contrast,compressionattheoriginal pinchtemperatureorthe supplytemperature arenot good optionsfromthe perspectiveof exergyconsumption.ThecompressionofC1_S1hasthesimilaref- fectasaheatpump.After compression,thesub-streamC1_S2be- comesahotstreaminthesystemthatwillreduce hotutility.The hotandcoldCompositeCurvesare closertoeachother andthere are two pinch pointsinthe GCC.This demonstrates that an effi-

Table 3

System performance under the optimal configuration.

Items Value

Hot utility (kW) 0

Cold utility (kW) 413.9

Pinch temperature ( °C) 290

Compression work (kW) 473.8

Exergy consumption (kW) 473.8

Original Pinch compression flowrate (kW/ °C) 0 New Pinch compression flowrate (kW/ °C) 1.47 Ambient compression flowrate (kW/ °C) 1.53 Compression at T ^supflowrate (kW/ °C) 0

cient heatexchanger network canbe derived withthe optimized superstructureforstreamC1.

Eventhough all thereformulations canreach the sameglobal optimumasdiscussedabove,thecomputationalexpenseshowsbig differences for the different reformulations. For the Explicit Dis- junction and the Direct Disjunction reformulations, the disjunc- tiveprogrammingmodelscanbereformulatedintoMINLPmodels by theBig-M orconvexhull methodswithLogMIP asthe solver.

In essence, the LogMIP solver calls other MINLP algorithms to solvethe disjunctivemodel.Forthissmall-sized problem,BARON (TawarmalaniandSahinidis,2004)isadoptedastheMINLPsolver.

Table 4 shows the computational performance of each re- formulation. It is clear that Smooth Approximation performs much better than the other reformulations for this case study.

The Smooth Approximation reformulation has fewer continuous variablesandsignificantlyfewerbinaryvariables.Thecomputation time is also considerably less than for the other three reformu- lations. The Direct Disjunction model has more disjunctions and continuousvariablesbutfewerbinaryvariablescomparedwiththe Explicit Disjunction model. The advantage of the Direct Disjunc- tionreformulationisthat itcan easily beextended tocaseswith isothermalstreamsandmultipleutilities.However, intheWHENs problem,phasechangeprocessstreamsaredifficulttohandleina generalway.Suchstreams need specialattentioninWHENprob- lems.Theconvexhullreformulationperformsslightlybetterthan the Big-M method for both Explicit and Direct Disjunction. It is clearthat theintermediate temperature strategyhas muchfewer

Table 4

Computational results for the case study.

Items SA IT Explicit disjunction Direct disjunction strategy Big-M Convex hull Big-M Convex hull

Disjunctions – – 49 49 98 98

Continuous variables 161 146 411 908 462 994

Binary variables 4 53 151 151 102 102

Equations 171 339 762 1448 467 663

CPU time (s) 17.5 4000 207.3 196.3 76.6 61.2

Objective function (kW) 473.8 473.8 ^a 473.8 473.8 473.8 473.8

aUpper bound obtained with the maximum computational time being 40 0 0 s.