Uncertainty quantification and global sensitivity analysis of complex chemical process using a generalized polynomial chaos approach

ComputersandChemicalEngineering90(2016)23–30

Contents lists available atScienceDirect

Computers

and

Chemical

Engineering

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / c o m p c h e m e n g

Uncertainty

quantification

and

global

sensitivity

analysis

of

complex

chemical

process

using

a

generalized

polynomial

chaos

approach

Pham

Luu

Trung

Duong

a

_,

_Wahid

_Ali

b

_,

_Ezra

_Kwok

c

_,

_Moonyong

_Lee

b,∗

a_{LuxembourgCentreforSystemsBiomedicine,SystemsControlGroup,UniversityofLuxembourg,Luxembourg}

b_{SchoolofChemicalEngineering,YeungnamUniversity,Gyeongsan,RepublicofKorea}

c_{Chemical&BiologicalEngineering,UniversityofBritishColumbia,Vancouver,Canada}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received6December2015

Receivedinrevisedform18February2016

Accepted22March2016

Availableonline29March2016

Keywords:

Generalizedpolynomialchaos

Uncertaintyquantification

Processuncertainty

Sensitivityanalysis

Monte-Carloapproach

a

b

s

t

r

a

c

t

Uncertaintiesareubiquitousandunavoidableinprocessdesignandmodeling.Becausetheycan signifi-cantlyaffectthesafety,reliabilityandeconomicdecisions,itisimportanttoquantifytheseuncertainties andreflecttheirpropagationeffecttoprocessdesign.Thispaperproposestheapplicationofgeneralized polynomialchaos(gPC)-basedapproachforuncertaintyquantificationandsensitivityanalysisofcomplex chemicalprocesses.ThegPCapproachapproximatesthedependenceofaprocessstateoroutputonthe processinputsandparametersthroughexpansiononanorthogonalpolynomialbasis.Allstatistical infor-mationoftheinterestedquantity(output)canbeobtainedfromthesurrogategPCmodel.Theproposed methodologywascomparedwiththetraditionalMonte-CarloandQuasiMonte-Carlosampling-based approachestoillustrateitsadvantagesintermsofthecomputationalefficiency.Theresultshowedthat thegPCmethodreducescomputationaleffortforuncertaintyquantificationofcomplexchemical pro-cesseswithanacceptableaccuracy.Furthermore,Sobol’ssensitivityindicestoidentifyinfluentialrandom inputscanbeobtaineddirectlyfromthesurrogatedgPCmodel,whichinturnfurtherreducestherequired simulationsremarkably.Theframeworkdevelopedinthisstudycanbeusefullyappliedtotherobust designofcomplexprocessesunderuncertainties.

©2016ElsevierLtd.Allrightsreserved.

1. Introduction

Mostrigorousprocessdesignproblemsarecarriedoutunder adeterministicsettingwithfixedspecifications.Inreality, how-ever,theprocessinputsandparametersexhibitsomerandomness asdepictedinFig.1,whichcanhave asignificanteffectonthe safety,reliabilityandeconomicdecisions.Therefore,itisimportant toexaminetheeffectsoftheseuncertaintiesandanalyzethe sen-sitivityoftheprocessmodelwithrespecttotheseuncertaintiesin thedesignstage.Monte-Carlo(MC)andQuasiMonte-Carlo(QMC) methodsarerepresentativeprobabilisticapproachesforthe propa-gationofuncertaintiesinthemodelinputtoitsoutput(Niederreiter etal.,1996;Liu,2001;Kroeseetal.,2011;Abubakaretal.,2015).The brute-forceimplementationofthesemodelsfirstinvolvesthe gen-erationofanensembleofrandomrealizationswitheachparameter drawnrandomlyfromitsuncertaintydistribution.Deterministic solversarethenappliedtoeachmembertoobtainanensembleof results.Theensembleofresultsisthenpost-processedtoestimate

∗ Correspondingauthor.Fax:+82538113262.

E-mailaddress:mynlee@yu.ac.kr(M.Lee).

therelevantstatisticalproperties,suchasthemean,standard devi-ationandquantile.Despitethis,estimationsofthemeanconverge withtheinversesquarerootofthenumberofruns,making MC-andQMC-basedapproachescomputationallyexpensiveandeven infeasibleforcomplexchemicalprocessproblems.

Recently,uncertaintyanalysisusingageneralizedpolynomial chaos(gPC)expansionwasstudiedinvariousapplications includ-ingmodeling,control,robustoptimaldesign,andfaultdetection problems.NagyandBraatz(2007)consideredthegPCapproachfor uncertaintyquantificationandrobustdesignofbatchcrystalized process.Intheirwork,itwasshownthatthegPCapproachtobe morecomputationallyefficientthantheMC/QMCmethodsfora systemwithamoderatenumberofrandominputs.DuongandLee (2012,2014)appliedthegPCmethodtothePIDcontrollerdesign for fractionalorder andinteger ordersystems. Duet al.(2015) consideredthefaultdetectionproblembycombiningmaximum likelihood withthegPC framework.ThegPC methodoriginated fromWienerchaos(Wiener,1938).Thismethodisaspectral repre-sentationofarandomprocessbytheorthonormalpolynomialsofa randomvariable.ExponentialconvergenceisexpectedforthegPC expansionofinfinitelysmoothfunctions(i.e.,analyticandinfinitely differentiable).GhanemandSpanos(1991)reportedthatthegPC http://dx.doi.org/10.1016/j.compchemeng.2016.03.020

Fig1.Uncertaintypropagationandquantificationinchemicalprocesses.

isaneffectivecomputationaltool forengineeringpurposes.Xiu andKarniadakis(2002)furthergeneralizedPCforusewith non-standarddistributions.

Thisworkdemonstratesandvalidatestheapplicabilityof poly-nomialchaostheoryforuncertaintyquantificationandsensitivity analysisforcomplexchemicalprocessessuchasnaturalgasand syngasproduction.TheproposedgPCbasedmethodcanreduce sig-nificantlythecomputationalcost(simulationtime)foruncertainty quantificationovertraditionalapproaches, suchastheMC/QMC methods.Moreover,Sobol’ssensitivityindices(Sobol,2001)can alsobedirectlyobtainedfromthegPCsurrogateanalyticalmodel (Crestauxetal.,2009;Sandovaletal.,2012),whichcaninturnbe usedtodetecttheinfluentialinputsinthepropagationofprocess uncertainty.

2. Uncertaintyquantificationusingpolynomialchaos theory

Considerasteady-stateprocessthatisdescribedwithasetof followingnonlinearequations:

F(y,)= 0 (1)

where=(1,2,...N)isaprocessinputvariablevectorexpressed

byarandomvectorofmutuallyindependentrandomcomponents withprobabilitydensityfunctionsofi(i):i→R+;andydenotes

aprocessstateandoutputvariablevector.

Thejointprobabilitydensity oftherandom vector,, is=

N



i=1

_i,andthesupportofis _≡ N



i=1

_i∈RN_{.Theuncertainties}

intheprocessinputsarethenpropagatedthroughtheentire pro-cess,asshowninFig.1.Thesetofone-dimensionalorthonormal polynomials,{i(i)dmi=0},canbedefinedinfinitedimensionspace,

i,withrespecttotheweight,i(i).Basedonaone-dimensional

setofpolynomials,anN-variateorthonormalsetcanbeconstructed withPtotaldegreesinspace,,usingthetensorproductofthe

one-dimensionalpolynomials,thebasisfunctionofwhichsatisfiesthe following:



 ˚m()˚n()()d=



1,m=n 0,m=/n (2)

Considera response functionf(y())for a processstate vari-able,y, withthestatistics (e.g.,mean,variance)of interest,the N-variatePth_{orderapproximationoftheresponsefunctioncanbe}

constructedasfollows: fNP(y())= M



i=1 ˆfi˚i(); M+1=



N+P N



=(N+P)! N!P! (3)

wherePistheorderofpolynomialchaos,and ˆfmisthecoefficient

ofgPCexpansionthatsatisfiesEq.(2)asfollows: ˆfi=E[if(y)]=





f(y)i()()d (4)

whereE[·]denotestheexpectationoperator.

ThecoefficientsofthegPCexpansionfromEq.(4)arenormally obtainednumericallyusingthefollowingprocedure(Xiu,2010): • Choosea N-dimensionalintegrationrule(cubaturenodes and

weights) Q[g]=(Qq1 (1)_⊗...⊗Q qN (N)_{) [g]=} q1



j1=1 ... qN



jN=1 g((j1) j ,..., (jN) j ) (w1(j1)...w1(jN)) ∼=



 g()()d (5)

where⊗denotesthetensorproduct,andQ[·]denotesthe multi-variatecubatureapproximation.

P.L.T.Duongetal./ComputersandChemicalEngineering90(2016)23–30 25 ˜ˆfi=Q[f(y,)()]= Q



m=1 f((m))j( (m) )w(m) for j=1,...,M (6) where˜ˆfrepresentsthenumericalapproximationof ˆfandf(␰)j(␰)

playstheroleofg(␰)inEq.(5);Qisthenumberofnodesincubature approximation.Thenumberofcubaturenodes(simulations)rises exponentiallyandhencethetensorcubatureshouldbeusedonly forsmallnumberofparameters.However,formoderatenumber ofparameters(5–10parameters),asparsegridquadraturecanbe usedforcalculatingthegPCinEq.(6)(Crestauxetal.,2009;Xiu, 2010).

• Construct an N-variate Pth _{order gPC approximation of the}

responsefunctionintheform:

˜fP N= M



j=1 ˜ˆf_jj() (7)

OnceallthegPCcoefficientsandthesurrogatemodelinEq.(7) havebeenobtained,apost-processingprocedureisthencarriedout toobtainthestatisticalpropertiesoftheresponsefunctionf(y()).

Thefirstexpansioncoefficientisthemeanvalueasfollows:

E[˜f_NP]= f=



 ˜fP N()d=





⎡

⎣



M j=1 ˜ˆf_j()˚j()

⎤

⎦

()d=˜ˆf1 (8)

Thevarianceoftheresponsefunctionf(y())canbeevaluated asfollows: Df =2f =E[(f− f)2] =



 ( M



j=1 ˜ˆf_j()j()−˜ˆfj)( M



j=1 ˜ˆf_j()˚j()−˜ˆf1)()d= M



j=2 ˜ˆf_j2 (9)

Eqs.(8)and(9)employthepropertythatthepolynomialset beginswith1()=1.Theweightfunctionofthepolynomialisthe

probabilitydensityfunction.

Thedistributionfunctionoftheresponsefunctionisobtained bysamplingthesurrogatemodelinEq.(7).

Ifaresponsefunctionf(y)ischosenforasstatevariable,y,the meanandvarianceofthesystem’sstatesaregivenapproximately byEqs.(8)and(9),respectively.DuongandLee(2012)andthe referencesthereinpresentdifferentmethodsforconstructingaset oforthonormalpolynomialswithrespecttoagivendistribution. Remark. IntheMC/QMC methods, it isnecessary tosimulate theprocessnumeroustimesatthesamplingpointstoobtainthe requiredprobability. On theotherhand,the gPCrequires solv-ingEq.(1)onlyatthecubaturenodesforobtainingtheanalytical representationinEq.(7).TheanalyticalmodelbyEq.(7)canbe sam-pledeasilywithlittlesimulationeffort.Thestatisticalmomentsare availabledirectlyfromthesurrogatemodel,asexpressedinEqs.(8) and(9).

3. Variancebasesensitivityanalysis

3.1. Brieftheoryofvariancebasesensitivityanalysis

Inadditiontotheuncertaintyquantification,globalsensitivity analysisisanessentialsteptoidentifytheparametersthatare rel-ativelymoreimportantthantheothersovertheentireparameter

spaceofthemodelsinthescienceandengineeringfields.One pop-ularapproachforsensitivityanalysisisvariancebased;itstheory isgivenbrieflyinthissection.

ConsiderthesystemdescribedinFig.1andEq.(1).Themean andvarianceoftheresponsefunctionf(y())canbeexpressedas

fy=



1 ...



N f(y(1,...,N)) N



i=1 i(i)di (10) Dfy=



1 ...



N



f(y(1,...,N))− fy

2 N



i=1 i(i)di. (11)

Theresponsefunctionofthesystemoutputcanbedecomposed intosumoftermswithincreasingdimensions(Satellietal.,2004):

f(y())=f0+ N



i=1 fi(i)+ N−1



i=1 N



j>i fij(i,j)+...+fi1...iN(i,...,N), (12) wheref0= fy.

ThetermsinEq.(12)arefunctionsofthefactorsinitsindexand canbeexpressedas fi(i)=E



f(y())|i

− f fij(i,j)=E



f(y())|i,j

−fi−fj− f ... , (13)

where_E



f(y())_|i

(resp._E



f(y())_|i_j

)istheconditional expec-tation of f(y()) when i is set (resp. i and j are set). The

decompositioninEqs.(12)and(13)isuniqueprovidedthatthe randomfactorsareindependent.

Therefore,thevarianceoftheoutputfunctioncanbe decom-posedasfollows: Dfy= N



i=1 Di+ N−1



i=1 N



j>i Dij+...+D1,2,...,N (14) where Di=var



E



f(y())|i



Dij=var



E



f(y())|i,j



−Di−Dj ... D1,2,...,N=Dfy− N



i=1 Di−...−



1≤i1<···<in−1≤N Di1...iN−1 (15)

Notethatinvar



E



f(y())|i,j



,theinnerexpectationisover allfactorsexceptfori,j,andtheoutervarianceisoveri,j.

Thefirstordersensitivityindex(function)isdefinedasfollows: Si=

Di

Dfy

(16) Thefirstorderindexrepresentsthemaineffectcontributionofeach randominputfactortothevarianceoftheoutputalone.Siliesin

[0,1].Thesumofthefirstorderfunctionsequals1fortheadditive models.

Similarly, let the sensitivity functions of a higher order be definedas

Si1,...,ik= Di1,...,ik

Dfy

Fig.2. Syngasproductionprocess(Example1).

Thesensitivityfunctionofahigherorderisasensitivitymeasure thatdescribeswhatpartofthetotalvarianceisduetothe uncer-taintiesinthesetofparameters,{i1,...,is}.

Thetotaleffectfunctionforthefactor,i,canbewrittenas

Ti=1−

var(E(fy()|_∼i))

Dfy(t)

. (18)

Thistotaleffectindexmeasuresthecontributiontotheoutput varianceofi,includingallvariancescausedbyitsinteractions,of

anyorder,withanyotherinputvariables.Thefunction,Ti,isthe

educatedanswertothequestion,“Whichfactorcanbefixed any-whereoveritsrangeofvariabilitywithoutaffectingtheoutput?” (Satellietal.,2004,2008).

3.2. VariancebasedsensitivityanalysisusinggPC

ThegPCexpansionsinEq.(7)canbereorderedtoseparatethe singleandcollectivecontributionofeachparameterasfollows.

Definethesetofmulti-indicesIk1,...,kssuchthat(Sandovaletal., 2012): Ik1,...,ks=



(k1,k2,...,ks) :0≤␥j_k≤P,␥j_k=0,k



1,...,n



\



k1,...,ks



(19) where_kjistheone-dimensionalpolynomialdegree.Usingthis notation,thefirstordersensitivityfunctioncanbeexpressedas

Si=



j∈Ii ˆf2 j D_f . (20)

Theestimatedsensitivityfunctionofahigherordercanbeobtained inthesamemannerasfollows:

S_i1,...,is=



j∈I i1,...,is ˆfj2 Df . (21)

Remark. Normally,forestimationoffirstorderandtotal sensi-tivityindicesthecomputationalcomplexityis(N+2)×M,whereN isthenumberofvariables,andMisthenumberoftheMC/QMC samples(Satellietal.,2004,2008).NotethatonlyMsamplesare neededforuncertaintyquantificationproblem.

4. Chemicalprocessexamples

Inthissection,theproposedgPCbasedmethodwasappliedto uncertaintyquantificationandsensitivityanalysisoftwochemical processexamplesstudiedin(Abubakaretal.,2015),suchasa syn-gasproductionprocessandasalesgasprocess,andvalidatedusing standardMC/QMCmethods.

4.1. Example1.a:syngasproductionprocesswithGaussian uncertainties

Fig.2showsaprocessflowdiagramofasynthesisgas(or alterna-tivelyknownassyngas)productionprocess.Inthisprocess,natural gas(NG)richinmethaneentersthereformer,wherealargeamount ofmethanereactswithsteamandproducessyngaswithCO2.The

effluentfromthereformeristhensenttothecombustorwithanair stream.Steamisalsofedintothecombustortomaintainthe reac-tortemperatureandensurethecompletecombustionofmethane. Theproductfromthecombustorpassesthroughtheshiftreactors, wherethewatergasshiftreactionsarecarriedout.Thesynthesis gasisfinallytakenasthefeedstockforanammoniaproduction plant.Tofacilitatetheproduction ofammonia,thehydrogento nitrogenmolarratio(HNr)inthesyngasneedstobecontrolled preciselyat3:1,whichrepresentsthestoichiometricamountsof thereactantsintheammoniaprocess.

Inthisexample,theNGfeedtemperature,flowrate,andthe reformer steam flow rate were assumed to be uncertain and normally distributed independently with a mean and standard deviation of {370◦_{C, 100kgmol/h, 240kgmol/h} and {55.5}◦_C,

15kgmol/h, 36kgmol/h}, respectively. Allother process inputs (suchasairstreamandcombustionsteamtothecombustor)and processparameters(suchasthevesselvolume,heatdutyand pres-sure)wereassumedtobedeterministic.Acodetogeneratethe cubaturenodesandweightsinthegPCmethodwasdevelopedin aMatlabTM _{environmentandconnectedtoHYSYS}TM_,wherethe

syngasprocess in Fig. 2was modeled rigorously. Thecubature nodesfromgPCcodearepassedtoHYSYSTM_{,andtheimpactof}

P.L.T.Duongetal./ComputersandChemicalEngineering90(2016)23–30 27 MC/QMCmethodscannotofferanaccurateresultwithalow

num-berofsamples.Sincetrueestimatesoftheoutputforthestudied processarenotavailable,theresultfromtheproposedmethodwas comparedwiththosefromtheMC/QMCmethodswithasufficiently largenumberofsamples.Table1liststhestatisticalpropertiesof HNrobtainedfromthegPCmethodandtheconventionalMC/QMC methods.Fig.3comparesthedensityfunctionsofHNrobtained fromthegPC/MC/QMCmethods.TheresultsfromtheproposedgPC methodmatchedthosefromthetraditionalMCandQMCmethods. NotethatthenumberofsimulationsrequiredbytheproposedgPC methodwassignificantlylowerthanthosebytheothertwo meth-ods.AsshowninFig.3,thetraditionalMCand/orQMCmethods requireahugenumberofsamplestogetsimilarresults.Itisnoted thattheperformanceofthegPCmethodmaydegradeforhighorder statisticalmomentsofnon-smoothquantityofinterest(output).

AnotheruniqueadvantageoftheproposedgPCmethodisthat Sobol’ssensitivityindicestoidentifytheinfluentialinputsinthe propagationofprocessuncertaintycanbeobtaineddirectlyfrom thesurrogateanalyticalmodel,whichcanbeusedtoreducethe numberofsimulationsrequiredfrom10–100times,andthe com-putationaleffortsneededfortheuncertaintyquantification.Inthis example,thesensitivityindicesfortheprocessvariableHNrwith respecttothreeuncertaininputs(theNGfeedtemperatureand flowrate,thereformersteamflowrate)canbeobtainedfromthe surrogategPCmodel.Table2liststhesensitivityindicesobtained fromthesurrogategPCmodel.Theonlyinputthatmattersisthe secondone,i.e.,theNGfeedflowrate.Thenaturalgas tempera-tureandthereformersteamflowarenon-influentialandcanbe excludedfromtheanalysisofuncertaintypropagation.Fig.4 com-paresdensitydistributionspredictedbythegPCandQMCmethods withonerandomparameter(i.e.,theNGfeedflowrate)andthatby theQMCmethod(with10,000simulationsfromHaltonsequence) withallthree randomparameters.Note thatinthecase ofone randominput,onlyfivesimulationswereusedfortheproposed gPCmethod,whereas10,000simulationsfortheQMCmethod.The densityfunctionsfrombothmethodswithonerandomvariable showedaclosematchwiththatusingallthreerandomvariables. The number of samples for the MC/QMC methods was chosen accordingtoChernoffbound(Tempoetal.,2013)foraccurate esti-mationoftheprobability.

4.2. Example1.b:syngasproductionprocesswithuniform uncertainties

In this example, all the three process inputs of the syngas productionprocessinFig.2wereassumedtohaveindependent uncertainties distributed uniformly in intervals of {370±50◦C, 100±20kgmol/h,240±54kgmol/h}.TheGaussLegendre cuba-ture nodes were used to obtain the surrogate gPC model, as describedinSection2.TheresultfromtheproposedgPCmethod wasthencompared withthetraditionalMCandQMCmethods. Tables1 and2listthestatisticalpropertiesof theHNrandthe sensitivityindices,respectively.

Fig.5comparesthedensityfunctionspredictedusingthe pro-posedgPCmethod(7thordergPCwith343simulations),theQMC methodwith16,000simulationsandtheMCmethodwith1000 simulations.TheMCmethodwithalownumberofsamples can-notprovideanaccurateresult,eventhoughthenumberofsamples wasmuchlargerthanthatusedinthegPCmethod.Theresultby thegPCmethodshowedaclosematchwiththatusingtheQMC methodwithalargenumberofsamples.Sobol’sindicesinTable2 alsoshowedthattheimportantrandomparameterinfluencingthe uncertaintypropagationwasonlyaNGfeedflowrateandtwoother randominputscanbeexcludedfromtheanalysis.Fig.6compares thedensityfunctionofHNrwiththreerandominputsandthose

withonerandominput(i.e.,theNGfeedflow),whereastheother Table

Table2

Sobol’ssensitivityindicesfromthesurrogatedgPCmodelforExamples1.a,1.b,and2.

Sobol’ssensitivityindices(Sij ... n)

Example1.a S1 S2 S3 S12 S13 S23 S123

2.272E-05 0.9806 0.0137 1.646E-05 0.0020 0.0022 0.0014

Example1.b S1 S2 S3 S12 S13 S23 S123

4.940E-05 0.9479 0.0062 5.220E-04 8.840E-04 0.0371 0.0074

Example2 S1 S2 S3 S4 S12 S13 S14 S23

0.9036 0.0795 0.0167 9.7715E-06 6.7285E-05 2.5839E-06 1.8018E-12 6.4715E-05

S24 S34 S123 S124 S134 S234 S1234

3.2377E-09 1.8833E-09 5.4235E-08 3.9457E-14 7.5100E-14 5.8284E-12 4.6192E-14

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Hydrogen-Nit

rogen rati

o in syng

as (HNr

)

Dens

ity f

u

nc

tion

f(

HN

r)

MC with 1000 simulations MC with 10000 simulations QMC with 10000 simulations gPC with 125 simulations

Fig.3.Densitydistributionsofthehydrogen-nitrogenratioinsyngaswith3

Gaus-sianrandominputs(Example1.a).

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Hydrogen-Nitrogen ratio in syngas (HNr)

De ns ity f un ction f(H N r) QMC: 3 random inputs QMC: 1 random input gPC: 1 random input

Fig.4. Densitydistributionsofthehydrogen-nitrogenratioinsyngaswith3and1

Gaussianrandominputs(Example1.a).

twoinputsareassumed tobeconstantattheirmeanvalues.As showninthefigure,theresultsusingtheimportantrandominput onlywereidenticaltothoseusingallthree randominputs.The proposedgPCmethodwithonerandominputrequiredonlyseven simulations,whereas theQMCmethodused16,000simulations toobtainanaccurateresult.Notethatforcalculatingsensitivity indicesusingtheQMCmethod,onemayneed16,000×5 simula-tions.Henceduetothelargecomputationalburdenforcalculating sensitivityindicesbytheQMCmethod,wewillnotdotheQMC approachforvalidationofthegPCmethod.However,Fig.6shows thatthegPCapproachcorrectlydetectsnon-influentialinput.

2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Hydrogen-Nitrogen ratio in

the synga

s (HNr)

De

ns

ity

fun

ct

ion

f(

HN

r)

MC with1000 simulations QMC with 16000 simulations gPC with 343 simulations

Fig.5. Densitydistributionsofthehydrogen-nitrogenratioinsyngaswith3uniform

randominputs(Example1.b).

2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Hydrogen-Nitrogen ratio in the syngas (HNr)

De ns ity f unction f( HN r)

QMC: 3 random inputs

QMC: 1 random input

gPC: 1 random input

Fig.6. Densitydistributionsofthehydrogen-nitrogenratioinsyngaswith3and1

uniformrandominputs(Example1.b).

P.L.T.Duongetal./ComputersandChemicalEngineering90(2016)23–30 29

Fig.7.Salesnaturalgasprocess(Example2).

1900 200 210 220 230 240 250 260 270 280 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Lean

na

tural ga

s production rate G (kg

mol/h

r)

Dens

ity

fun

ction f

(G)

MC with 10000 simulations QMC with 10000 simulations gPC with 625 simulations

Fig.8. Densitydistributionsofleangasproductionratewith4Gaussianrandom

inputs(Example2).

rate,pressure,temperature,and thechiller duty wereassumed tobevariedindependentlywithameanandstandarddeviation of{300kgmol/h,4150kPa,15◦C,2×105_{kJ/h}and{10kgmol/h,}

150kPa,3◦C,500kJ/h},respectively.Fig.8showsthedensity func-tionsforleandrygasproductionpredictedfromthegPC/MC/QMC methods.TheresultfromthegPCmethod(5thordergPCwith625 simulations)closelymatchedtheresultfromtheMC/QMCmethods with10,000simulations.Table1liststhestatisticalpropertiesof leangasproductionandsimulationparametersfromtheproposed gPC and MC/QMC methods. Table2 lists thesensitivity indices obtainedfromthesurrogategPCmodel.Thesensitivityindices indi-catethatthecrudeNGfeedflowrateandpressureareinfluentialfor uncertaintypropagationwhilethecrudeNGfeedtemperatureand thechillerdutyarenon-influential.Fig.9showsthedensity

func-1900 200 210 220 230 240 250 260 270 280 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Lean natu

ral gas producti

on rate G (kgm

ol/h

r)

De

ns

ity

function

f(

G)

QMC: 4 random inputs QMC: 2 random inputs gPC: 2 random inputs

Fig.9. Densitydistributionsofleangasproductionratewith4and2Gaussian

randominputs(Example2).

tionsofleangasproductionwithtwoinfluentialrandominputs bythegPCmethod(with25simulations)andbytheQMCmethod (with10,000simulations),andcompareswiththatusingallfour randomparametersbytheQMCmethod(with10,000simulations). ThesensitivityindicesfromthegPCmethodcorrectlyidentifythe influentialinputs.

5. Conclusions

A stochastic spectral approach based methodwas proposed tomodeltheuncertaintypropagationandsensitivityincomplex chemicalprocesses.ThegPCmethodwasappliedtodeterminethe solutiontotheseproblems.AninterfacebetweenMatlabTM _and

Theresultsshowedpreciseagreementwiththoseofthe conven-tionalapproachessuchastheMC/QMCmethods,whichmightbe beyondthecomputationalcapabilityforlargescalecomplex chem-icalprocessproblems.ThegPCapproachhasadvantagesoverthe popularMC/QMCapproaches,mainlyintermsofthecomputational cost:therequirednumberofsamplesfortheuncertainty predic-tioncanbereducedsignificantlybythegPCmethod.Moreover, Sobol’ssensitivityindicestoidentifytheinfluentialrandominputs foruncertaintypropagationcanbeobtaineddirectlyfromthe sur-rogatedgPCmodel,whichinturnreducestherequiredsamples remarkablyand allowsa focusontheimportant inputs forthe robustdesignofcomplexprocessesunderarangeofuncertainties. Acknowledgments

ThisstudywassupportedbytheBasicScienceResearch Pro-gramthroughtheNationalResearch FoundationofKorea (NRF) fundedbytheMinistryofEducation(2015R1D1A3A01015621),and alsobythePriorityResearchCentersProgramthroughtheNational ResearchFoundationofKorea(NRF)fundedbytheMinistryof Edu-cation(2014R1A6A1031189).

References

Abubakar,U.,SrirnivasSriramula,S.,Renton,C.,2015.Reliabilityanalysisof

complexchemicalengineeringprocesses.Comput.Chem.Eng.74,1–14.

Crestaux,T.,LeMaitre,O.,Martinez,J.M.,2009.Polynomialchaosforsensitivity

analysis.Reliab.Eng.Syst.Saf.94(7),1161–1172.

Du,Y.,Duever,T.A.,Budman,H.,2015.Faultdetectionanddiagnosiswith

parametricuncertaintyusinggeneralizedpolynomialchaos.Comput.Chem. Eng.76,63–75.

Duong,P.L.T.,Lee,M.,2012.RobustPIDcontrollerdesignforprocesswith

stochasticparametricuncertainties.J.ProcessControl22,1559–1566.

Duong,P.L.T.,Lee,M.,2014.Probabilisticanalysisandcontrolofsystemswith

uncertainparametersovernon-hypercubedomain.J.ProcessControl24, 358–367.

Ghanem,R.G.,Spanos,P.D.,1991.StochasticFiniteElements:ASpectralApproach.

DoverPublications,NewYork.

Kroese,D.P.,Taimre,T.,Botev,Z.I.,2011.HandbookofMonteCarloMethods.John

WileyandSons,Hoboken,NewJersey.

Liu,J.S.,2001.MonteCarloStrategiesinScientificComputing.Springer-Verlag,

NewYork.

Nagy,Z.K.,Braatz,2007.Distributionaluncertaintyanalysisusingpowerseriesand

polynomialchaosexpansions.J.ProcessControl17,229–240,SpecialIssueon AdvancedControlofChemicalProcesses.

Niederreiter,H.,Hellekalek,P.,Larcher,G.,Zinterhof,P.,1996.MonteCarloand

Quasi-MonteCarloMethods.Springer-Verlag,Berlin.

Sandoval,E.,Collin,F.A.,Basset,M.,2012.Sensitivitystudyofdynamicsystem

usingpolynomialchaos.Reliab.Eng.Sys.Saf.104(8),15–26.

Satelli,A.,Tarantola,S.,Campolongo,F.,Ratto,M.,2004.SensitivityAnalysisin

Practice:AGuidetoAssessingScientificsModels.JohnWiley&Sons, Chichester.

Satelli,A.,Ratto,M.,Andres,T.,Campolongo,F.,Cariboni,J.,Gatelli,D.,Saisana,M.,

Tarantola,S.,2008.GlobalSensitivityAnalysis:ThePrimer.JohnWiley&Sons,

Chichester.

Sobol,I.M.,2001.Globalsensitivityindicesfornonlinearmathematicalmodelsand

theirMonteCarloestimates.Math.Comput.Simul.55,271–280.

Tempo,R.,Calafiore,G.,Dabbene,F.,2013.RandomizedAlgorithmforAnalysisand

ControlofUncertainSystems:WithApplications,2ndedition.Springer-Verlag, London.

Xiu,D.,Karniadakis,G.E.,2002.TheWiener-Askeypolynomialchaosforstochastic

differentialequations.SIAMJ.Sci.Comput.24,619–644.

Xiu,D.,2010.NumericalMethodforStochasticComputations:ASpectralMethod

Approach.PrincetonUniversityPress.