Dynamic hybrid simulation of batch processes driven by a scheduling module

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DOI:10.1016/j.compchemeng.2011.04.007

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Fabre , Florian and Hétreux, Gilles and Le Lann, Jean Marc and Zaraté, Pascale

Dynamic hybrid simulation of batch processes driven by a scheduling module.

(2011) Computers & Chemical Engineering, vol. 35 (n° 10). pp. 2098-2112.

ISSN 0098-1354

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Dynamic

hybrid

simulation

of

batch

processes

driven

by

a

scheduling

module

Florian

Fabre

a

_{, Gilles}

_Hétreux

a,∗

_{, Jean-Marc}

_Le

_Lann

a

_{, Pascale}

_Zaraté

b

a_{LaboratoiredeGénieChimique(PSI–GI),UMR-CNRS5503/INPT-ENSIACET,4,alléeEmileMonso,31030Toulouse,Cedex4,France} b_{IRIT/INPT-ENSIACET,118RoutedeNarbonne,31062Toulouse,Cedex9,France}

Keywords:

DynamichybridsimulationEnvironment Batchprocesses

Scheduling Petrinets

MixedIntegerLinearProgramming Objectorientedmodeling

a

b

s

t

r

a

c

t

SimulationisnowaCAPEtoolwidelyusedbypracticingengineersforprocessdesignandcontrol.In particular,itallowsvariousofflineanalysestoimprovesystemperformancesuchasproductivity,energy efficiency,wastereduction,etc.Inthisframework,wehavedevelopedthedynamichybridsimulation environmentPrODHySwhoseparticularityistoprovidegeneralandreusableobject-orientedcomponents dedicatedtothemodelingofdevicesandoperationsfoundinchemicalprocesses.Unlikecontinuous processes,thedynamicsimulationofbatchprocessesrequirestheexecutionofcontrolrecipestoachieve asetofproductionorders.Forthesereasons,PrODHySiscoupledtoaschedulingmodule(ProSched)based onaMILPmathematicalmodelinordertoinitializevariousoperationalparametersandtoensureaproper completionofthesimulation.Thispaperfocusesontheprocedureusedtogeneratethesimulationmodel correspondingtotherealizationofascenariodescribedthroughaparticularscheduling.

1. Introduction

For severaldecades,processingand recoveryof raw materi-alshascauseda tremendousexpansionofindustrial chemistry. Iftheunitsinthissectortraditionallyoperatecontinuously,food, biotechnology,pharmaceuticalorelectronicsindustriesare func-tioningpreferentiallyinabatchmode.Indeed,locatedonmarkets subject tohigh turnover of productsand fluctuating or

unpre-dictable demand, batch processes are characterized by these

qualitiesofflexibility.Generallyusedtomanufacturehighadded valueproducts,profitsmadesofarweresuchthatitseemed some-whatinterestingtodeveloptoolsandmethodologiestoimprove theperformanceoftheseunits.Buttheinternationalizationof mar-ketsandthegrowingneedsofsocietyhaveledtonewindustrial strategies.Locatedinhighlycompetitivemarkets,thefunctionof theprocessisthencomplicatedbyadesiretoconsolidate produc-tionfacilitiesandreducecosts.Thesenewconstraintsarereflected todaybyanundeniableinterestofindustrialandscientific commu-nitytobetterdesignandmoreimportantly,tobetterexploitthese batchprocesses.

AmongtheavailableCAPEtools(ComputerAidedProcess Engi-neering), process engineers are showing a growing interest in

dynamicsimulationforitsabilitytocarryoutvariousanalyses (con-figurations,operatingpolicies,etc.)ona“virtual”plant,extremely usefultoprocessengineersin theirdailyworktoimprove

sys-∗ Correspondingauthor.Tel.:+33562883660.

E-mailaddress:gilles.hetreux@ensiacet.fr(G.Hétreux).

temperformance(productivity,energyefficiency,wastereduction, etc.).Duringthedevelopmentofanewprocess,massandenergy balances,equipmentssizing,utilitiesneedsassessment,estimation oftimecycleorcostanalysisaregenerallyperformedandthese toolscansignificantlyreducedesignvarianceandthelaboratory workonpilotwhichisoftencostlyandtimeconsuming.In oper-ation,havingareliablesimulationmodelimprovesunderstanding ofthewholeprocessbytheoperatorsandfacilitates communica-tion.Productionengineerscanassessinafewminutestheimpact ofcriticalparametersonkeyindicatorssuchasproductioncosts, timecycle,energyefficiencyorproductivity.Thesimulationalso providesthemeanstomonitortheoccupancyofalltanksduring

acampaignandverifythattheminimumandmaximumloadsare

alwaysmetinallpartsoftheprocess.Itcanvalidateoperating con-ditionsofeachtaskandsetsthecontrolloopsrequiredtomaintain theseoperatingconditions.Finally,inasafetypointofview,the impactofdefaultsintheoperativeorcommandpartcanbequickly estimatedbysimulationandcorrectiveactionscanbetested.

Inthiscontext,theunificationofresearchinmodelingand sim-ulationofprocessescarriedoutformanyyearsintheLGChasledto thedevelopmentofPrODHyS(Fabre,2009;Hétreux,Théry,Perret, Lelann,&Joulia,2002;Jourda,Joulia,&Koehret,1996;Moyse,2000; Olivier-Maget, 2007; Perret,2003; Sargousse,1999), a dynamic

hybridsimulationenvironmentdedicatedtochemicalprocesses

(Fig.1).

Basedonobjectconcepts,thisenvironmentoffersextensibleand reusablesoftwarecomponentsallowingarigorousandsystematic modelingofthetopologyandthebehaviorofprocesses.Thehybrid featureismanagedwiththeObjectDifferentialPetriNets(ODPN)

Nomenclature

Indices

i processingtasks

st storagetasks

j units

n eventpointsrepresentingthebeginningofatask

s states

Sets

I setofprocessingtasksi

ST setofstoragetasksst

STs setofstoragetasksstforstates

Ij tasksithatcanbeperformedinunitj

Is setoftasksithatusestates

Isp setoftasksithatproducestates

Isc setoftasksithatconsumestates

J setofunitsj

Ji setofunitsjthataresuitableforperformingtaski

N setofeventpointsnwithinthetimehorizon

S setofstatess

Sf _{stateswithfiniteintermediatestorage}

Sn _{stateswithnointermediatestorage}

Sp _{statesthatarefinalproducts}

Sr _{statesthatarerawmaterials}

Sz _{stateswithzero-waitpolicy}

Parameters

Vimin,Vimax minimumandmaximumcapacityfortaski

Csmax maximumamountofstatesthatcanbestored

p

i,s/ci,s proportionofstatesproducedorconsumedbytask

i

pfi fixedpartoftheprocessingtimeoftaski

pvi variablepartoftheprocessingtimeoftaski

H maximumdurationofthecampaign→timehorizon

hs storagecostofstates

Ds amountofstatesdeliveredattheendofthe

cam-paign

Variables

Bi,n amountofmaterialundertakingbytaskiatevent

pointn

Bsi,n amountofmaterialstartingprocessingbytaskiat

eventpointn

Bfi,n amountofmaterialfinishingprocessingbytaskiat

eventpointn

Bstst,n amountofmaterialstoredbystoragetaskstatevent

pointn

Ss,n amountofstatesateventpointn

SFs finalamountofstatesattheendofthetimehorizon

S0s initialamountofstatesatthebeginningofthetime

horizon

Tsi,n timeatwhichtaskistartsateventpointn

Tfi,n timeatwhichtaskifinishesateventpointn

pti,n processingtimeoftaskiateventpointn

Tsstst,n timeatwhichstoragetasksststartsateventpointn

Tfstst,n timeatwitchstoragetasksstfinishesateventpoint

n

Wi,n 1iftaskiisactivatedateventpointn,else0

Wsi,n 1ifthetaskibeginsateventpointnelse0(binary

variable)

Wfi,n 1ifthetaskiendsateventpointnelse0(binary

variable)

Plan durationoftheproductionscheduling

Table1

Batchmanagementwithoptimizationordynamicsimulationapproach. Optimisationmethod Dynamicsimulation Advantages Exhaustiveexplorationof

candidatesolutions

Morerealisticmodelingof processes

Globalconsiderationofall theconstraints

Processingtimes determinedby

phenomenologicalmodels Efficientsolvingmethod

Drawbacks Modelingoftenbasedon simplifyingassumptions, whichdonotpermitto exploittheentireflexibility oftheprocess

Evaluationofacandidate solution simulationof theprocessforagiven sequenceandagivenbatch sizes limitedexploration ofcandidatesolutions Fixedandoften

overestimatedprocessing times

Myopicview difficulties totakeintoaccounttime constraints(no-wait,

conditioningcalendar, cleaningpolicy)

formalism.Itcombines inthesamestructure, asetof

differen-tialand algebraic equations (DAE) systems which describe the

continuousevolutionofthesystem(primarilybasedonthe

ther-modynamicandphysicochemicallaws)andhighlevelPetrinets

whichdefinethelegalcommutationsequencesbetweenstates(i.e. oneofthepossibleconfigurationsofDAEsystems).

Nevertheless,inoppositetocontinuousprocesses,studieson batchunitsoftennecessitatetotakeintoaccountboththe physic-ochemicalphenomenathattakeplaceineachdevice(localvision)

andthemanagementofbatches(nature,size,numberandstarting date)passingthroughtheunit(globalvision).Obviously,thesetwo featureshaveasignificantimpactontheperformancesandinduce thatthesystemhastobetackledasawholetoestablishaconsistent analysis.Inthiscontext,thesimulatormustbeabletoruna sce-nariodescribedbyaproductionplanincludingproductionorders (PO),eachPOindicatingamongotherthings,thetypeofproduct, thequantitytobeproducedandtheperiodofexecution(starting andendingdateofthejobs).

Toachievetheproductionplanandmeetthevariousconstraints, atemporalandquantitativesynchronizationmustbeensured.But,

themanagementofbatchesonlybysimulationdoesnotalways

givesatisfactoryresultsandmayevenleadtoabortanexecution. First,inordertotakeintoaccountthecapacityofequipment,itis oftennecessarytosplitproductionordersintoseveralbatches.The numberandsizeofthesesbatcheshavetobecalculated.In addi-tion,themyopicviewofthesimulationpreventsaproperhandling of time constraints(delivery dates, zero-wait policy,maximum delay,etc.),resourceallocationconstraintsorcleaningconstraints. Inmanydynamicsimulators,thesecalculationsareeitherassumed bytheuser,eitherbasedonheuristicsorsimplepriorityrules.So, inordertotacklerigorouslyeachpartoftheproblemandimprove thesolutions,thestrategyadoptedinPrODHySconsistsindriving thesimulationbyfollowingaproductionscenarioobtainedfrom

aschedulingmodulebasedonoptimizationtechniques.Table1

summarizedthemaincharacteristicsofthesetwokindsoftool. Thepurposeofthispaperistopresentthetoolsand method-ologiesusedtoimplementtheinterfacebetweenthisscheduling moduleandthesimulationmodel.Therestofthepaperisorganized asfollow.InSection2,theproblemstatementandtheprinciple oftheproposedapproacharedescribed.Eachmoduleofthistool isthen describedin thefollowing order.In Section3,theERTN

graphicalformalismisbrieflypresentedandillustrated.Section4

describestheimplementedmathematicalformulation.Section5

presentsthemajorconceptsonwhichthedynamichybrid

simu-latorisbased.Finally,Section6dealswiththecommandlevelof thesimulationmodelandithighlightsthenecessityofcoupling

Fig.1.PackagesofthedynamichybridsimulationenvironmentPrODHyS.

thesimulatorwitha schedulingmodule.Theinterfacewiththe simulatoristhenpresentedandsomeremarksarediscussed. 2. Mainstepsoftherecipe-drivendynamichybrid simulation

2.1. Generalfeaturesoftheconsideredbatchprocesses

The addressed processes are multi-purpose batch or

semi-continuousplant.Inthiskindofunit,eachproductfollowsaspecific sequenceofoperationsandisproducedusingsharedprocessing equipment.Thesegeneralnetworkprocessescorrespondtothemore generalcaseinwhichbatchescanbemergedand/orsplit.This fea-tureinducesthatmaterial balancesmustbetakenintoaccount explicitly (inopposite tosequential processes that areorder- or batch-orientedanddonotrequiretheconsiderationofmass bal-ances).Consequently,thecorrespondingsimulationmodelshave toincorporateseveralgeneralcharacteristicsthatinclude: •disjunctive(devices,operators,etc.)andcumulative(materials,

utilities,etc.)resourcesconstraints,

•variousstorageandtransferpolicies(UIS:UnlimitedIntermediate Storage,FIS:FiniteIntermediateStorage,NIS:NoIntermediate Storage,ZW:Zero-Wait,etc.)

•fixedand/ordependentprocessingtimes(dependingonbatch

size),

•mixingandsplittingofbatches,inducingvariablebatchsizealong theproduction

2.2. Modelingofrecipes

Recipeisanentitythatdescribestheformulation(setof chem-ical substances and proportions),the procedure(set of physical stepsrequiredtomaketheproduct)andtherequiredequipment.

Totacklecomplexprocesses,thestandardISA/SP88(www.isa.org) hasspecifiedahierarchicalmodelincluding4levels(Fig.2),each oneprovidinginformationinanappropriategranularity:

•Generic(orgeneral)recipespecifiesthemanufacturingmethodof eachfinishedproduct.Itcontainsinformationaboutthe

mate-Fig.2.Hierarchicalmodelingoftherecipe.

rials(rawmaterialsandintermediates),proportions,operating parameters,but,nodataabouttheequipmentoftheproduction systemisprovided.

•Siterecipeisaninstantiationofthegenericrecipeinwhichthe detailsabouttheproductionsiteareincorporated.Thisinvolves thegeneraltopologyoftheprocessandcleardefinitionsofthe characteristics of theprocessing equipment(capacity, energy consumption,etc.).

•Masterrecipeisaninstantiationofthesiterecipethatsetsthetype andamountoffinishedproduct(s)tobeproducedinagiven oper-ationalhorizon.Itthereforeclarifiestheproductionorderstobe achieved.Thislevelofrecipemakesuseofscheduling,which cal-culatesthenumberandsizeofeachbatchaswellasthesequence ofthesebatchesonequipment.

•Controlrecipeisappliedtoaparticularbatchorlotanddescribes theexecutionofeachtaskindetail.Itisimplementedata super-visionlevel.

2.3. Graphicalframeworkforthemodelingofrecipes

Tofacilitatethemodelingphasebynon-expertusersin

opti-mization and simulation, a way is to build optimization and

simulationmodelswhicharestructurallygenericandconfigurable

withparameters entered through a well-defined graphical

for-malism.Indeed,theimplementationandthetuningofaMILPor

simulation model can becomerather technical and complex in

somecases.Thus,thesupportofavisualrepresentationcanbevery helpfulforthemodelingoftheproductionsystem.Providedthat thesemanticofthisgraphicalformalismissufficientlygeneral,it allowstheusertodescribeaprobleminasimpleandintuitiveway whileignoringthemathematicalsupportusefultoitsresolution. Anotheradvantageofsuchformalismistheabilityto unambigu-ouslymodela problembyaddingspecific constructionrules.It reduces(butitdoesnotavoid)potentialmodelingmistakesand

userscanspendmoretime inanalysingthesystemratherthan

developingthemodel.

Inthisframework,theExtendedResourceTaskNetwork(ERTN) formalismhasbeendevelopedforthemodelingofrecipes.Basedon

thewell-knownResourceTaskNetwork(RTN)formalismproposed

in(Agha,2009;Fabre,2009;Pantelides,1994)haveintroducednew semanticelements(seeSection3)notablyamongothers,inorderto handleexplicitlycumulativeresources(suchasutilitiesfor exam-ple)andmulti-modalresources.Intheseworks,theERTNformalism ismorepreciselyusedtomodeltheprocedurepartofthesiterecipe.

Itisconstructedfromtheprocedureofthegenericrecipeandthe topologyoftheunitchosentoexecutethisrecipe.

2.4. Mainstepsofthedynamicsimulationprocedure

Fig.3showsaschematicdiagramoftheprocedureimplemented torunadynamicsimulationofacompleteprocessforagiven pro-ductioncampaigninPrODHyS.

Giventhegenericrecipeofthemanufacturedproductsandthe topologyoftheunit,theprocedureofthesiterecipeismodeledin ourtoolusingtheERTN(ExtendedResourceTaskNetwork)graphical formalism.

Tomanageoverallflows passingthroughtheunit, a

“simpli-fied”butstructurallygenericschedulingmodelbasedona MILP

formulationissetandinstantiatedwithdataprovidedthroughthe

ERTNview(setofestimatedparametersforduration,capacityof devicesaccordingtothestoredmaterial,etc.).Thus,givenatime horizonanda productionplan(obtainedbyaMRPprocedurefor example),thepackageProSchedcalculatesaschedulingbycalling thecommercialsolverXPRESS-MPforagivencomputationeffort. Thistreatmentgivesrisetothemasterrecipeandtheresultinglist oftaskscanbedepictedonaGanttchart.Datacharacterizingeach taskaretransmittedviaafiletothedynamicsimulatorPrODHySin ordertoparameterizethecommandlevelofthesimulationmodel (i.e.thecontrolrecipe),previouslyconstructedinaccordancetothe

ERTNviewbyassemblingpredefinedoperationobjects.Theprocess levelofthesimulationmodelisbuiltaccordingtothetopologyof theunitwithdeviceorcompositedeviceobjects.Thesimulationof this“detailed”modelisthenexecuteduntilthecompletionofthe productionplan.

Anormallyendedsimulationindicatesthatallcapacityandtime constraintsaremet.Theproductionplanisvalidatedandthe analy-sisoftheoperationalandphysicochemicalpropertiescanbemade. Ifasimulationfailsthenitmeansthatconstraintsareviolatedand theuserhastoanalyzethesimulationresults(viatheevaluationof variousindicators)toundertakecorrectiveactions.Nevertheless, accordingtotheobjectiveofthestudy,someparametersofthe (simplifiedand/ordetailed)modelmaybemodifiedorrefinedby exploitingthesimulationofthepreviousiteration.Thisprocedure canberestarteduntiltheuserfindssatisfactoryresults.

Insummary,themainideaofthiscombinedapproachistotake advantageofthestrengthsofdynamicsimulationand mathemati-calprogrammingtoachieveaconsistentbatchmanagementinthe

workshopandthus,toenhancetheachievementofthedynamic

simulation.

3. TheERTNgraphicalformalism

3.1. BriefdescriptionoftheERTNformalism

Theexpressivequalityofformalismisjudgedbyitsaptitudeto summarizeonasinglegraphtheinformationnecessaryto repre-sentaprocess.Inthiscontext,StateTaskNetwork(STN)proposed by(Kondili,Pantelides,&etSargent,1993)hasbeenafirststep towarddeveloping a universalrepresentationfor abatchplant. Later,(Pantelides,1994)hasproposedtheResourceTaskNetwork

(RTN)formalism,anextensionoftheSTNthatcontainsmore infor-mationaboutprocessingequipmentandtheirconnectivity.Based uponthemajorconceptsofthewell-knowRTNformalism(Agha, 2009;Fabre,2009)haveintroducednewsemanticelementsand theresultingframeworkiscalledExtendedResourceTaskNetwork

(ERTN). Thus, this graph represents the main features

encoun-teredinbatchprocesses.Theunderlyingsemanticelementsare listedinFig.4.Accompaniedbywell-establishedconstructionrules,

it clearlyand unambiguouslyrepresents production procedures

(precedenceconstraints),materialsandenergyflows(ratioofinlet andoutletflows,freeflows,recycling,separation andmixingof batches) andresource constraints(topology ofunit, capacity of devices,fixedordependentoperatingtime,sharedandmultimodal devices,etc.).ThegenericnatureoftheERTNformalismoffers a directcorrespondencebetweenthegraphicalelementsand mathe-maticalconstraints.So,severalformulationscanbeassociatedwith

theERTNframework.

3.2. Exampleofbatchprocessmodeling

Toillustratea subpartoftheERTNsemantic, atypicalbatch processispresented.Inthisexample,theproductionoftwofinal productsisconsidered.ProductP1necessitatesthreesuccessive operations:a preheatingofreactantA, nexta reaction(reaction

1:A+B→IntAB)andfinally,adistillationtoseparatefinal

prod-uct P1 and residue P2. If we suppose that intermediate IntAB

alreadyexists,therecipeofproductP3iscomposedoftwo oper-ations:apreheatingofreactantC,followedbyareaction(reaction

2:C+IntAB→P3).

Thetopology of theunit is shown in Fig.5.It consists of a

preheater/mixer,tworeactors(calledREACTOR1andREACTOR2),

acolumn(ensurestheseparationofreactionproducts)and sev-eralstoragetanks(forrawmaterialsA,B,C,intermediateproduct IntAB,residueproductP2andfinalproductsP1,P3).Tocontrol thesedevices,theunitisequippedwithseveralactuators(pumps Pi,valvesVi,heatingsystemsQi,electricmotorsMi)andsensors (retentionUi, temperatureTi, composition XPi, flow Fi). REAC-TION1canbeperformedindifferentlyinthetwo reactorswhile

Fig.3.GeneralprocedureofadynamicsimulationinPrODHyS.

Fig.5.Topologyoftheunit.

regardingtheenergypointofview,reactorsandheaterconsume electricitytomaintaintheoperatingconditionsandthecolumn requireshighpressuresteam(HP)atboilerandacoolant(CW)at condenser.

In these conditions, theprocedure of the site recipe is then

modeled by the ERTN shown in Fig. 6. The capacity of tanks

is given in kg and the processing times include a fixed and a

variable (dependent of batch size) part. The absence of

stor-age tankbetween thepreheater and reactors induces that the

stateHotAhasa“capacity”equaltozeroandazero-wait trans-ferpolicy.Moreover,notethatdevices aredisjunctiveresources whilethedifferentkindsofenergyareconsideredascumulative resources.

4. MILPformulationoftheschedulingproblem

4.1. Keyfeaturesoftheconsideredschedulingmodel

Differentmethodsareproposedintheliteraturetosolvethese

problemsrecognizedasNP-complex.Givenourgoal,anymethod

allowingthesimultaneousdeterminationofthestartingdateand thebatch-sizeofeachtaskisacandidate,providingthatagood solutionisobtainedinareasonabletime.So,thequalityoftheplan andtheprovidedcomputationaleffortareparametersforwhich acompromisemustbefound.Severalexcellentreviews(Burkard &Hatzl,2005;Floudas&Lin,2004;Kallrath,2002;Méndez,Cerdá, Grossmann,Harjunkoski,&Fahl,2006)clearlypointoutthatMixed IntegerLinearProgramming(MILP)hasbeenwidelyusedfor

solv-ing the batch process scheduling problem. In this framework,

variousformulationsoftheproblemareproposedinthe litera-ture(Kondilietal.,1993;Maravelias&Grossmann,2003).Globally, wecandistinguishMILPmodelsbasedondiscretetimeformulation (suchasGlobaltimeintervals)orbasedoncontinuoustime formu-lation(suchasGlobaltimepoints,Unit-specifictimeevent,Time slots,Unit-specificimmediateprecedence,Immediateprecedence, Generalprecedence, etc.).In our study,the bestsuited models

regarding the combination of optimization and simulation are

thosebasedonacontinuous-timeformulation.Adetailed compar-isonofthesecontinuous-timemodelscanbefoundin(Shaiketal., 2005).Notably,itpresentsthroughseveralexamplesofbenchmark thegoodcompromiseoftheUnitSpecificEventformulationinterm ofresolutiontimeandrobustness.

4.2. Descriptionoftheoptimizationmodel

Originally,theUnitSpecificEventformulationhasbeen devel-oped by (Ierapetritou & Floudas, 1998). This continuous-time formulationforshort-termschedulingintroducestheoriginal con-ceptofeventpoints,whichareasequenceoftimeinstanceslocated alongthetimeaxisofaunit,eachrepresentingthebeginningofa taskortheutilizationoftheunit.Thelocationoftheeventpoints isdifferentforeachunit,allowingdifferenttaskstostartat dif-ferenttimesin eachunitfor thesameeventpoint. Thetimings oftaskarethen accountedforthrough specialsequencing con-straints.Becauseoftheheterogeneouslocationsoftheeventpoints fordifferentunits,aswellasthedefinitionofaneventasonlythe startingofatask,forthesameschedulingproblem,thenumberof eventpointsrequiredissmallerthanotherscontinuous-time for-mulationsandsubsequently,reducesnotablythenumberofbinary variables.

ThemodelcurrentlyimplementedinProSchedcorrespondsto theformulationfoundin(Janak,Lin,&Floudas,2004)witha lim-iteduseof“BigM”constraintsandtheaggregationof sequence constraints.Moreover,thecapacitylimitsofstoragetankaretaken

into accountpartially by the mathematicalmodel described in

(Ierapetritou&Floudas,1998).Indeed,thematerialbalancesare calculatedonlyatthebeginningoftasks.Insomecases,thiscan locallyleadtooverflowthecapacityofstoragetanks(Fig.7a). How-ever,thatis unacceptablein termsof simulationsince physical constraintsareviolated.

Thus,additionalconstraintshavebeenimplementedby(Janak etal.,2004)totacklethisfeature.Forthis,storagetasksaredefined.

Thesequence andtimingofthesenewtasksandtheprocessing

tasksarethenrelatedsothattheamountsofstateswillbe consis-tentandspecifiedlimitscanbeenforced(Fig.7b).

Notehoweverthatinourformulation,thevariablesareonly

indexedby a eventnumber n and a taskithat correspondsto

a couple (operation, EquipementUnit) and not by anoperation

i,a devicejand eventnasin (Janak etal.,2004).Thisreduces thenumberof variablesand itis consistentwiththeERTN for-malism.AsdescribingindetailthewholeMILPmodelisnotthe aimofthispaper,only thefundamental equationsarereported belowgroupedbyfunctions(forexample,utilityconstraintsarenot givenherealthoughtheyareincludedinourmodel).An exhaus-tivedescriptionoftheseconstraintscanbefoundin(Janaketal., 2004).

A (4000,∞) B (4000,∞) HotA (0,0) ZW P2 (0,∞) Preheater/ Mixer T1 –Preheating1 (0, 100, 0, 0.032) T2 –Reaction1 (0, 100, 6, 0) T4 -Separation (0, 100, 0, 0.04) Reactor1 Column 0.625 0.375 IntAB (0,600) P1 (0,∞) 0.2 0.8 T3 –Reaction1 (0, 50, 6, 0) 0.625 0.375 C (4000,∞) HotC (0,0) ZW T5 –Preheating2 (0, 100, 0, 0.04) T6 –Reaction2 (0, 50, 4, 0) 0.5 Reactor2 P3 (0,∞) 0.5 HP (∞,∞) CW (∞,∞) 4.5,0 2.3,0 Elec (∞,∞) 3,0 3.2,0 0.2,0.03 0.2,0.015 0.1,0.05 A (4000,∞) A (4000,∞) B (4000,∞) B (4000,∞) HotA (0,0) ZW HotA (0,0) ZW P2 (0,∞) P2 (0,∞) Preheater/ Mixer Preheater/ Mixer T1 –Preheating1 (0, 100, 0, 0.032) T1 –Preheating1 (0, 100, 0, 0.032) T2 –Reaction1 (0, 100, 6, 0) T2 –Reaction1 (0, 100, 6, 0) T4 -Separation (0, 100, 0, 0.04) Reactor1 Reactor1 Column Column 0.625 0.375 IntAB (0,600) IntAB (0,600) P1 (0,∞) P1 (0,∞) 0.2 0.8 T3 –Reaction1 (0, 50, 6, 0) T3 –Reaction1 (0, 50, 6, 0) 0.625 0.375 C (4000,∞) C (4000,∞) HotC (0,0) ZW HotC (0,0) ZW T5 –Preheating2 (0, 100, 0, 0.04) T5 –Preheating2 (0, 100, 0, 0.04) T6 –Reaction2 (0, 50, 4, 0) T6 –Reaction2 (0, 50, 4, 0) 0.5 Reactor2 Reactor2 P3 (0,∞) P3 (0,∞) 0.5 HP (∞,∞) HP (∞,∞) CW (∞,∞) CW (∞,∞) 4.5,0 2.3,0 Elec (∞,∞) Elec (∞,∞) 3,0 3.2,0 0.2,0.03 0.2,0.015 0.1,0.05

Fig.6.ERTNviewofthesiterecipeoftheprocess.

Fig.7.Refinedstoragetankcapacityconstraints.

Fig.8. Allocationconstraints(disjunctiveresources).

Thenomenclatureassociatedwiththis modelisgiveninthe appendixattheendofthearticle.Onthebasisofthisnotation,the mathematicalmodelinvolvesthefollowingconstraints:

Allocationconstraints(cf.Fig.8):

X

i∈Ij Wi,n≤1

∀

j∈J,

∀

n∈N (A.1) Wi,n=

X

n′_≤n Wsi,n′−

X

n′_<n Wfi,n′

∀

_i∈I,

∀

n∈N (A.2)

X

n∈N Wsi,n=

X

n∈N Wfi,n

∀

i∈I (A.3) Wsi,n≤1−

X

n′_<n Wsi,n′+

X

n′_<n Wfi,n′

∀

_i∈I,

∀

n∈N (A.4) Wfi,n≤

X

n′_≤n Wsi,n′−

X

n′_<n Wfi,n′

∀

_i∈I,

∀

n∈N (A.5)

Fig.9.Capacityconstraintsofprocessingtasks.

CapacityconstraintsandBatch-sizematchingconstraintsof Pro-cessingTasks(cf.Fig.9)

Vmin

i Wi,n≤Bi,n≤VimaxWi,n

∀

i∈I,

∀

n∈N (A.6)

Bi,n≤Bi,n−1+Vimax(1−Wi,n−1+Wfi,n−1)

∀

i∈I,

∀

n∈N|n>1(A.7)

Bi,n≥Bi,n−1−Vimax(1−Wi,n−1+Wfi,n−1)

∀

i∈I,

∀

n∈N|n>1(A.8)

Bsi,n≤Bi,n

∀

i∈I,

∀

n∈N (A.9)

Bsi,n≤VimaxWsi,n

∀

i∈I,

∀

n∈N (A.10)

Bsi,n≥Bi,n−Vimax(1−Wsi,n)

∀

i∈I,

∀

n∈N (A.11)

Bfi,n≤Bi,n

∀

i∈I,

∀

n∈N (A.12)

Bfi,n≤VimaxWfi,n

∀

i∈I,

∀

n∈N (A.13)

Bfi,n≥Bi,n−Vimax(1−Wfi,n)

∀

i∈I,

∀

n∈N (A.14)

Capacityconstraintsofstoragetasks

Fig.10.Materialbalancesonstate.

Fig.11.Durationofprocessingtasks.

MaterialBalancesincludingStorageTasks(cf.Fig.10) Ss,n=Ss,n−1+

X

i∈Is p_i,sBfi,n−1−

X

i∈Is c i,sBsi,n+

X

st∈STs Bstst,n−1 −

X

st∈STs Bstst,n

∀

s∈S,

∀

n∈N|n>1 (A.16) Ss,1=S0s−

X

i∈Is c i,sBsi,1−

X

st∈STs Bstst,1

∀

s∈S (A.17) SFs=Ss,N−Ds+

X

i∈Is p_i,sBfi,N+

X

st∈STs Bstst,N

∀

s∈S, (A.18) Ss,n≤Cs

∀

s∈S,

∀

n∈N (A.19) Ss,n=0

∀

s∈Se_∪Sf_∪Sn_,

_∀

n∈N (A.20)

DurationconstraintsofProcessingtasks(cf.Fig.11)

Tfi,n≥Tsi,n

∀

i∈I,

∀

n∈N (A.21)

Tfi,n≤Tsi,n+MWi,n

∀

i∈I,

∀

n∈N (A.22)

Tsi,n≤Tfi,n−1+M(1−Wi,n−1+Wfi,n−1)

∀

i∈I,

∀

n∈N|n>1(A.23)

pti,n=pfiWsi,n+p

v

_iBs_i,n

∀

i∈I,

∀

n∈N (A.24)

Tfi,n′ −Ts_i,n≥pt_i,n−M(1−Ws_i,n)−M(1−Wf_i,n′)

−M

X

n≤n′′_<n′

Wfi,n′′

∀

i∈I,

∀

n∈N,

∀

n′∈N,n≤n′ (A.25)

Tfi,n′ −Ts_i,n≤pt_i,n+M(1−Ws_i,n)+M(1−Wf_i,n′)

+M

X

n≤n′′_<n′

Wfi,n′′

∀

i∈/Ips,

∀

n∈N,

∀

n′∈N, n≤n′ (A.26)

DurationConstraintsofStorageTasks

Tfstst,n≥Tsstst,n

∀

st∈ST,

∀

n∈N (A.27)

Sequenceconstraintsofprocessingtasks:sametaskinthesame unit

Tsi,n≥Tfi,n−1

∀

i∈I,

∀

n∈N|n>1 (A.28)

Sequence constraintsof processingtasks: different tasksin the

sameunit

Tsi,n≥Tfi′_,n−1+_i′_,i−M(1−Wf_i′_,n−1−Ws_i,n)

∀

j∈J,

∀

i∈Ij,

∀

i′∈Ij|i=/i′,

∀

n∈N|n>1 (A.29)

Sequenceconstraintsofprocessingtasks:differenttasksindifferent units

Tsi,n≥Tfi′_,n−1−M(1−Wf_i′_,n−1)

∀

s∈S,

∀

i∈Isc,

∀

i′∈Isp,

∀

j∈Ji,

∀

j′∈Ji′|j=/j′,

∀

n∈N|n>1 (A.30)

Sequenceconstraintsofprocessingtasks:no-waitcondition(ZW transferpolicy)

Tsi,n≤Tfi′_,n−1+M(2−Wf_i′_,n−1−Ws_i,n)

∀

s∈Sz∪Sn∪Sf,

∀

i∈Isc,

∀

i′∈Ips,

∀

j∈Ji,

∀

j′∈Ji′|j=/j′,

∀

n∈N|n>1

(A.31) Sequenceconstraintsofstoragetasks

Tsi,n≥Tfstst,n−1

∀

s∈S,

∀

i∈Izc,

∀

st∈STs,n∈N|n>1 (A.32) Tsi,n≤Tfstst,n−1+M(1−Wsi,n)

∀

s∈Sf,

∀

i∈Icz,

∀

st∈STs, n∈N|n>1 (A.33) Tsstst,n≥Tfi′_,n−1−M(1−Wf_i′_,n−1)

∀

s∈S,

∀

i′∈I_sp,

∀

st∈STs, n∈N|n>1 (A.34) Tsstst,n≤Tfi′_,n−1+M(1−Wf_i′_,n−1)

∀

s∈Sf,

∀

i′∈Ip_s,

∀

st∈ST_s, n∈N|n>1 (A.35) Tsstst,n=Tfstst,n−1

∀

st∈STs, n∈N|n>1 (A.36) Boundconstraints Tfi,n≤H

∀

i∈I,

∀

n∈N (A.37a) Tsi,n≤H

∀

i∈I,

∀

n∈N (A.37b) 0<Wi,n<1

∀

i∈I,

∀

n∈N (A.38)

Plandurationconstraint:

Tfi,n≤Plan

∀

i∈I,

∀

n∈N (A.39)

Objectivefunction min a. Plan+

X

s∈S hsSFs+

X

s∈S

X

n∈N hsSs,n

!

(A.40)

4.3. ComplementarytoolsdevelopedforProSchedmodule

Asmentioned previously,anysemanticelementofERTN for-malismdescribedinSection3hasadirecttranslationwithsets ofconstraintsofthemathematicalmodel.Thisgivesthegeneric natureofthismodelsinceeachprobleminstanceissimplydefined throughadatafile(Fig.12).

Inordertofacilitateparametersentry,a“draganddrop”tool hasbeendeveloped.TheusercancreateitsERTNgraphicallyand choosealltheparameters(units,tasks,durations,sequences...) ofitsmodel.Afteranautomaticverificationofthevalidityofthe

ERTN,theprogramcreatestheinitializationfilecompatiblewith themodelofoptimizationimplementedinXpressMP.

Aftertheschedulingphase,theusercanproceeddirectlytoa firstanalysisbasedonGanttdiagram.Asecondtoolcaninterpret directlythedataprovidedbytheoptimizeranddisplayitasaGantt chartwiththeevolutionofamountofstatesandbatchesonthetime horizon(seeFig.13).

Fig.12.Toolsassociatedtothefirststepoftheprocedureofdynamicsimulation.

Fig.13. ProSchedGeneratorandGanttChartManager.

5. ThedynamichybridsimulatorPrODHyS

This section presents briefly themain characteristicsof the dynamicsimulatorPrODHySandthen,describesthestructureof thesimulationmodel,andespeciallythecommandlevel.

5.1. TheODPNformalism(ObjectDifferentialPetriNet)

Batchprocessesaregenerallyclassifiedasdynamichybrid sys-tems(Zaytoon,2001).Thiskindofsystemrequiresspecificdynamic simulatorsabletohandlerigorouslyboththecontinuousevolution ofthestatevariables(temperatureincrease,chemicalkinetics,etc.) andthediscretebehavior(on/offpump,open/closevalve,etc.).In thisframework,theplatformPrODHySusestheObjectDifferential PetriNetformalism(ODPN)tomodelthehybridbehaviorofboth devicesandmaterialitcontains.Fig.14recallsthesemantic ele-mentsofODPN.Theformaldefinitionandevolutionrulesofthis formalismanditsimplementationwithinPrODHySaredescribedin detailin(Perret,2003)andpresentedin(Hétreux,Perret,&LeLann,

2003;Hétreux,Thery,Olivier,&LeLann,2007;Perret,Hétreux,& LeLann,2004;Hétreux,Perret,&LeLann,2004).

5.2. Generalstructureofthesimulationmodel

To make the simulation of a discontinuous process, it is

necessary to model both the control part (the supervisor) and

the operative part (the process). In PrODHyS, the simulation

model located at the command level (presumably specific to

the recipe, the topology of the considered process and the

production plan to achieve) is completely separated from the

simulation models of devices. Indeed, models of devices must

be reusable regardless of the context (concept of component).

Thus, differentrecipescanbe implementedand testedwithout

changingthemodelsassociatedwiththedevices(i.e.theprocess

level).

ThemodelofthecommandlevelisthemasterODPN(calledrecipe PetriNet)whoseevolutioncauseschangesintheODPNofthe enti-tieslocatedattheprocesslevel.ThisODPNcorrespondssomehowto

Fig.15. SubpartoftheprocessshowninFig.4andGRAFCETsequence.

thecontrolrecipeorGRAFCETprogramexecutedbyaProgrammable LogicController(seeexampleinFig.15).

The signals exchanged between the command part and the

process section correspondeither to transmit a command (sig-nalOpen V2),ortoreceiptaninformationfromasensor(signal SignalUR2).A signalismodeled bya place(called respectively, commandorinformationplace)anditsstatusisassociatedwiththe markingofthisplace.Theseplacesaretheuniquelinkbetween

processandcommandlevels.

Regardedasblack boxes, thereare twotypes ofdevice (see

Fig.16):

•activedevices:objectswhosePetrinethasoneormorecommand and/orinformationplacessuchasactuators(cf.VALVEV2)and

sensors(seeCAPTORUR2)

•passivedevices:objectswhosePetrinethasnodirectlinkwith therecipePetrinetsuchastanks,reactors(seeREACTOR2)or material.

Themarkingofacommandplaceofanactivedeviceinduces gener-allychangesinitsownPetrinet,itselfcausingevolutionincascade inpassivedevicesthroughthenetworkformedbytheconnectionof thevariousmaterialorenergyports.Inconsequence,theevolution ofODPNmodelsisconditionedbytwodistincttypesofevent: •first,theexternaleventsthatcausecontrolledswitching.These

eventsare issuedfromtherecipePetrinettodrive theactive

devicesoritistheoccurrenceofastateevent(threshold)ora tem-poralevent.Specifiedbytheuser,theseeventsappearexplicitly ontherecipePetrinet.

•secondly,theintrinsiceventswhoseoccurrencedependsonlyon

thespontaneousevolution ofthe process.These autonomous

switchesareforexample,achangeinmaterialstate(the transi-tionfromliquidtoliquid/vaporwhentheboilingpointisreached)

or a commutationin a passivedevice. Theytherefore do not

appearexplicitlyintherecipePetrinet(theuserdoesnothave tospecifythem)andaretreatedsolelywithinthemodelofthe entity.

InteractionsbetweenrecipePetrinetandprocessPetrinetare illus-tratedinFig.16.Thisisthetranslationofthesystemshown in

Fig.15anditrepresentsanoperatingsequenceinwhichareactor isfeduntilafixedvolumeisreached.Thefillingoperationis con-trolledbytherecipePetrinetbyplacingatokenonthecommand

placeofthevalveobject(placeOpenV2).Thefeedofreagentis keptopenwhilethiscommandplaceismarked.Todetectthe end-ingtimeofthetransfer,aleveldetectorobjectisused.Themarking oftheinformationplace(placeSignalD1)oftheleveldetectorobject indicatesthatthevolumeofreagenthasreachedthetargetvalue. Thetransitionisfired.Theabsenceoftokenonthecommandplace thencausestheclosureofthevalveobject.Inthefollowing,only

recipePetrinetisshown(sequenceofoperations)andequipment areseenonlythroughtheirsignalplaces.

6. Drivingadynamichybridsimulation

6.1. Hierarchicalmodelingoftherecipe

TherecipePetrinetisthelinkbetweentheoptimizationmodel

andthesimulationmodeloftheoperativepart.However,when

theproductionsystemincludesextensivefacilitiesortheproduct developmentrequiresmanyoperations,thesizeofODPN associ-atedwiththecommandlevelcangrownquickly.Inthiscase,itis necessarytostructurethecontrolrecipeinsuccessiverefinements.

6.1.1. Notionofparameterisedmacro-place

Based on the decomposition advocated by the ISA-SP88

(Fig.17a),thecontrolrecipehasahierarchyonseverallevels

(pro-cedure,operation,phase,step,instruction,etc.).Toimplementthis structureintheODPNofthecommandlevel,thenotionof macro-placesisexploited.Itreplacesasequenceofplacesandtransitions relativetoanoperationoraphasebya singlemacro-place.This sequenceisthendelimitedbytwospecialplacesEandSbetween whichalltypesofplacesdefinedintheODPNformalismmayarise, includingothermacro-places(Fig.17b).

Atthehighestlevelofthehierarchy(theprocedure),a macro-placerepresentstheexecutionofaunitoperation.However,some operationsmaytakeplaceindifferentdevices.Forexample,inthe processshowninFig.5,theoperationcalledREACTION1canrunin theREACTOR1and/orREACTOR2.Forthisreason,anymacro-place canbesetwithanEquipmentUnitobject.Thisobjectrepresentsthe maindevice(forexample,thevesselofthereactor1)andallthe

actuatorsandcontrolequipment(here,thevalveV1,pumpsP2

andP4,heatingsystemQ2,engineM2andcaptorsUR1,TR1and

XPR1).Aninstanceofthisobjectdefinesaunitofequipmentand attributesofthisinstanceisthenusedtodefinethecommands orsignalsrequiredforeachsequence(Fig.17b).Thishierarchical structurefacilitatesthespecificationofthecontrolrecipeandthe setting-upofthesimulationmodelbytheuseofreusablesequences storedinmacro-place(seeFeedphaseinFig.17b).Anexampleof

recipecontrolisgivenin(Hétreux,Théry,&LeLann,2006).This functionalityisratherimportantsincethemodelofthecommand

leveliscompletelydisconnectedfromthemodelsoftheprocess

level.

6.1.2. Notionoftasktoken

The macro-place operation are parameterized by the used

devices(seeSection6.1.1)butalsobythecharacteristicsoftasksto perform.Forexample,areactorinwhichseveralreactionscantake placerequiresineachcasedifferentoperatingconditions (temper-ature,pressure,composition,etc.).Itisthecaseofthereactorcalled REACTOR2intheprocessshownonFig.5.Similarly,twotasks per-formingthesameoperationinthesameunitmaystillhavedifferent settings,especiallywhentheydependonthebatchsize.Toaddress theseissuesanddefinemoregenericoperationobjects,ataskobject hasbeenintroduced.Theattributesofataskobjectinclude,among othersthings:

•theearlieststartingdateofthetask, •thebatchsize,

•areferencetotheEquipmentUnitobjectallocatedtothistask, •areferencetotheoperationobjecttoperform,includingall

oper-ational parameters (temperature, pressure, composition, etc.) necessarytodefinetheconditionsandactionsofthesequence (stateeventsassociated withcontinuousvariables oftheDAE

Fig.16.Interactionbetweenprocessandcommandlevel.

Fig.17. Hierarchicalstructureofthecontrolrecipeusingmacro-places.

Fig.19.Simulationresultsoftask<REACTION1,REACTOR1,700mol>.

Ataskobjectthusdefinesthetriplet<Operation,EquipmentUnit, BatchSize>.However,thisinformationshouldbedisseminatedto instantiatetheconditionsandactionsofthetransitionsdispatched ontheODPNofanoperation.Forthisreason,thetaskobjectis asso-ciatedwithatokenobjectoftypeTaskToken(noted<T>).Whenan instanceofatoken<T>sensitizesatransition,thentheformal vari-ablesusedtodefinetheconditionsoractionsarereplacedbyits attributes(Fig.18).Finally,notethatthistokendoesnot material-izealotofmaterial,butaninformationalentityusedtolauncha task.Itcanthereforebeassimilatedtoanexecutionorder.

6.1.3. Simulationofaunitoperation

Toillustratetheabovediscussion,asimulationlimitedtothe task<REACTION1,REACTOR1,700mol>isexecuted.TheODPN

associatedwiththisoperationisshowninFig.18.Forthelaunchof asinglebatch(700molofIntAB),Fig.19aandbshowsrespectively theevolutionofthecompositioninREACTOR1andtheretention inthevariousconcerneddevices.

6.2. StructureoftheODPNofthecontrolrecipewithinthe procedurelevel

Foreachunitoperationopoftheprocedurecarriedoutonthe equipmentunitres(calledcouple<operation,EquipmentUnit>), a structurecalled“decisioncenter” isimplemented asshown in

Fig.18.Furthermore,aninstanceofTaskTokenobject<T>is cre-atedforeachtaskcorrespondingtothetriplet(op,res,size).This

ODPNmanageboththetemporalandtheresourceavailability:

•thetemporalaspectissupportedbyatimedplace(place Starting-DateinFig.18)formanagingthelaunchofeachtask.Thedelay

parameteroftheplaceisequaltothestartingdateofthetask carriedbythetoken<T>(P(<T>).delay←<T>.StartingDate).When thestartingdatehasexpired,thetokenisreleasedandmarksthe placededicatedtothemanagementofaqueue(placequeuein

Fig.18)whennecessary.

•amutexplaceisassociatedwitheachdisjunctiveresource(shared devicesbetweenoperationsornot)andmanageitsavailability (placeResAvailableinFig.18).Whenthisplaceisnotmarked,this indicatesthattheresourceisalreadyrequisitionedbyanother taskandpreventsthecrossingofthetransitioncalledstart.So,it avoidsthestartingofanewtaskbeforetheendoftheprevious one.

•ataskcanbestartedonlyafterensuringtheavailabilityof materi-als.Indeed,atthesimulationlevel,therealdurationofoperations canbeshorterorlongerthanthemeandelaytakenintoaccount attheschedulinglevel.Forthis,aconditionplacedonthe tran-sitionlocatedbeforetheoperationmacro-placeverifiesthatthe amountofmaterialsareequaltoorgreaterthantheproportion requiredforthebatchsizecarriedbythetoken.

Notealsothat:

•alltasksthatdonotsharethesameequipmentunitcan poten-tiallybeperformedinparallel,manytasksassociatedwiththe same couple <Operation, EquipmentUnit> can exist. In this context,thetimedplaceStartingDateissimplymarkedwitha numberoftoken<T>equaltothecorrespondingnumberoftasks.

Fig.20.ODPNofthecontrolrecipe(procedurelevel).

•ataskTokenobject<T>representsthedatarelativetoasingletask. Itbecomesobsoletewhenthetaskiscompleted.Inotherwords, thesametokencannotbeusedfortwosuccessiveoperations (eveniftheywereidentical).Asaresult,noprecedence relation-shipappearsexplicitlyintheODPNofthecontrolrecipeatthe

procedurelevel.

6.3. Applicationontheprocessexample

BasedontheERTNshowninFig.6,theoptimizationmodule establishesaschedulingwiththeMILPmodelsolvedwith XPRESS-MP.Theparametersofthemathematicalmodelareinitializedwith estimatedaveragedurationsandlinearizedparameters.Giventhe characteristicsoftheprocessinFig.5,theschedulingofasingle productionorderequalto100kgofP1isshowninFig.20.

Aftertheschedulingstep,thesequenceoneachprocessingunit, thestartingdatesaswellasthenumberandthebatchsizesare transmittedtothesimulator.Eachtaskisinstantiatedand associ-atedwithataskTokenobject<T>.Fig.20showstheODPNofthe

controlrecipeattheprocedurelevelcorrespondingtotheERTNin

Fig.5instantiatedwiththeaforementionedscheduling.

The ODPNof thecontrol recipeis built byassembling a set ofdecisioncenter,eachoneassociatedwithacouple<Operation, EquipmentUnit>.Thus,operationscarriedoutbyseveral process-ingunitsmustbeduplicatedasitisdoneintheERTNformalism. ThiscaseconcernstheoperationREACTION1performedeitherin REACTOR1orREACTOR2.Inaddition,ifthesameresourceresis usedbyseveraloperationsopitheneachdecisioncenter

associ-atedwithacouple<opi,res>sharesthesamemutexplace(named

ResAvailable)which modelstheavailabilityoftheresourceres.

ThiscaseconcernsforexampleREACTOR2whichperformsboth

REACTION1andREACTION2.

Thesimulationisthenperformedbyfollowingtheproduction plansodefined.Performanceindicatorscanbecalculatedinorder toevaluatethequalityofthesolution.Fig.20showsthe succes-siveexecutionoftwobatchesofidenticalsizeinthesamedevice

(here, REACTOR1).Thecurves show thatthe durationsof each

batchare different(change infeedratedue toa gravity trans-fer).Thisexamplehighlightsthemodelinggap(modelsusedare differentbynature)existingbetweenthetwomodules (optimiza-tion/simulation)andtheneedtoprovidedecisionalautonomyto thesimulatorfor thestarting(ornot)ofproduction tasks.Asa result,schedules obtainedby simulationand thoseobtainedby optimizationarenotdirectlycomparable.

Severalcaseshavebeensolvedandgenerally,thesimulations

have been correctly completed. Nevertheless, some time

con-straints maynotbecompletelyfulfilled due inmostcases toa inaccurateestimationof theprocessingtimesatthescheduling level.Indeed,ifthedurationtakenintoaccountinthe optimiza-tionmodelisunderestimated,thesimulatorstartsthetaskatthe earliestwhentheallocatedresourceandtherequiredamountof materialareavailable.Nevertheless,futuretimeconstraintscould notbemet.Intheopposite,ifthedurationtakenintoaccountin theoptimizationmodelisoverestimated,thesimulatorisforcedto waittheexpiryofthescheduledstartingdate.Hereagain,future timeconstraintscannotbeguarantied.Fig.21illustratesthiscase. Infact,asestablishedin(Méndezetal.,2006),agapalwaysexists betweentheoryandpracticalduetothesimplifyingassumptions sometimesintroducedtomaketheproblemtractable.Thisisthe reasonwhythemodeliscalled“simplified”fortheschedulingpart, inoppositiontothe“detailed”modelforthesimulationpartwhich

Fig.21.Simulationresultsfor3tasks<REACTION1,REACTOR1>withdifferentbatchsizes.

describesthephysicochemicalphenomenonbydifferential alge-braicequationssystems.Moreover,thisinducesthatthesearchof amathematicaloptimumofsimplifiedmodelscanseemuseless,in practice.Forvariousreasons,theimplementationofsuch schedul-ingisoftenlimitedwhenitisconfrontedwiththesimulationmodel oftheprocess.Inparticular,optimizationmodelareoften

estab-lishedundertheassumptionofconstant andknownprocessing

times.However,thisrepresentsasevererestrictiontowardthe sen-sitivityofcertainoperationstotheadjustmentsoftheoperating conditions.Thebatchcolumnisanexamplewheretheprocessing timedependsonseveralparameters:thequalityoftheinitialload, theheatingpolicyoftheboiler,therefluxpolicy,therackingside flows,thethermallosses,etc.Inaddition,thedurationofataskcan alsodependonthestateofthesystematagiventime.Forexample, thedurationofatransferbygravityisdependentontheretentionin thesourcetank.Inthesameway,theheatingdurationofaproduct dependsontheinitialtemperature,itselfbeingabletodependon thewaitingdurationoftheproductintheupstreamstoragetankif thermallossesexist.Finally,criterionisoftenreducedtoasubpart oftheoverallobjectivesconsideredbyend-users.So,ifa schedul-ingisonlya“good”solutionoftheproblem,itisnotadrawback andtheusercanadjustsomeparametersatthesimulationlevel. Forthesereasons,inthisprocedure,theoptimizationcalculations areoftenstoppedwhenafixedtimedelayoranintegralitygapis met.

Inordertorefinetheresults,theabovesimulationresultscanbe usedtoresetthedataofthemathematicalmodelandthusimprove theproductionplansobtainedthroughaniterativeprocedure.An anotherstrategyisthesimulationofeachoperationindependently forasetofparametersinordertoobtainedaccurateinitialdata fortheschedulingmodule.Nevertheless,itseemslikelythatthe simulatedplansaremoreeasilyexploitablebecausetheyarebased onamoreaccuraterepresentationoftherealphenomenaandcan providereferencepoints(temperature,pressure,composition,etc.) duringtheprogressionoftheinsituoperations.

7. Conclusion

Basedonobjectconcepts,PrODHySprovidessoftware compo-nentsforthemodelingandthedynamicsimulationofindustrial processes(Hétreuxetal.,2002;Hétreuxetal.,2003;Perretetal., 2004).Theimplementationofahighlevelformalism(Object Differ-entialPetriNet)associatedwithefficientnumericalmethods(Gear, 1971)hasledtothedevelopmentofahybriddynamicsimulator numericallyrobust.Inordertodealefficientlywiththesimulation ofbatchprocess,thispaperpresentsapackagewhoseroleistobuild automaticallyoptimizedproductionscenariosthatshouldrunthe simulator.Forthis,severalkeyissueshavebeenaddressed.First,it hasbeenintroducedtheERTNgraphicalformalismthatmodelsthe maincharacteristicsofaprocess.Thisformalismisusedinthe

soft-wareProSchedGeneratordesignedtogeneratetheinputparameters oftheschedulingmodel.Thisgenericmathematicalmodel(MILP) isbasedonacontinuoustimeformulationcalledUnit-SpecificTime Event.Thismodulecalculatesallinputdatausefultothesimulation model.Secondly,theinterfacebetweentheoptimizationmodeland thesimulationmodelhasbeenestablished.

For this, the ODPN of the control recipe is structured into

several levels by using parameterized macro-places.Moreover,

informationassociated witheach taskisdistributed throughout thenetworkthankstotasktokenobject.

Currently,theeffectivenessofthisframeworkhasbeenproved andseveralstudiesonbatchprocesseshavebeenconductedwith success.Nevertheless,itmight beinterestingtotestother opti-mizationmodelstoimprovethequalityoftheschedulingobtained inthefirststepoftheprocedure.Especially,manyrobust optimiza-tiontechniquescanbeappliedinordertoexplicitlymodelsystem uncertaintyandgenerateaschedulewhichisnotonlyfeasiblefor thenominalsystemconditionsbutalsorobustwhenconsidering thedistributionoftheunknownsystemparameters(Lin,Janak,& Floudas,2004;Janak,Lin,&Floudas,2007;Shaik&Floudas,2009). To conclude, note that this procedure is included as a part

ofa moregeneralmethoddedicatedtotheschedulingofbatch

processes.Thefundamentalprincipleistosupposethatan “approx-imate”solution(intermofbehavior)providedbyanoptimization modelwithareducedcomputationaleffort,iscompensatedbya finermodelingoftheprocesscarriedoutatthesimulationlevel. Thisapproachshouldmakemorerobusttheproductionplansand facilitatesthephysicochemicalanalysisofphenomena.However, inordertovalidatethisapproachandevaluatequantitativelyits effectiveness,severalmodulesarecurrentlyindevelopment.

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