steady statedetermination for hemial proess plants
Kavouras,A 1
,Kevrekidis,IG 1;
, Georgakis,C 2
,Kelley,CT 3
orrespondingauthor
Email: yannisarnold.prineton.edu
1
Department of Chemial Engineering, Prineton University, Prineton, NJ 08544,
USA
2
Department ofChemialand BiologialEngineering,Tufts University, Sieneand
TehnologyCenter,Medford,MA02155,USA
3
DepartmentofMathematis,Center forResearhin SientiComputation,North
CarolinaStateUniversity,Raleigh,NC27695-8205,USA
Abstrat: Given a legay dynami modelof anopen-loopunstable hemial
proess plant weonstruta omputational\wrapper" that extratsitssteady
states(bothstableandunstable)andtheirdependeneoninputparameters. We
applythissteadystatedeterminationapproahtoahallengeproblemforproess
ontrol presentedby DownsandVogel(1993), whoalso providedaFORTRAN
proessmodel. UsingtheFORTRANmodelasalegaydynamisimulator,we
enable it to systematially omputesteadystates and their parametri
depen-dene; our so-alled \equation-free" approah is basedon matrix-freeiterative
linear algebramethods. The existene of inventories and their interplay with
the problemformulation and the solutionproedure is laried. Separation of
timesales,whenpresent,istakenadvantageofineetivelyreduingthemodel
orderandenhaningomputationaleÆieny. Thisapproahenableslegay
dy-namisimulators toalulate amultitudeof linearized dynamiharateristis
thatouldbeusedforontrollerdesignoroptimization,forwhihtheyhavenot
beenintentionallydesigned.
Topialarea: proesssystemsengineering
Keywords: dynami model, unstableproesses, GMRES, inventory ontrol,steady
state
1 Introdution
Frequentlyinthehemialproessindustriessimulatorsaredevelopedthatprovidea
dynamirepresentationofsingleunitsorintegratedproesses. Gooddynami
simula-torsleadtoabetterunderstandingoftheproess,allowtheexplorationofasestudies
vari-aurate,itan beusedasatoolforoptimization,orforontrollerdesign.
Thesoure odes ofsuhdynami models areoftennotaessible, henetheterm
legay ode; even if they are, their size (and possibly lak of doumentation) may
be daunting when urrent users onsider adapting them to new purposes. When
alternative tasks suh as optimization or ontroller design are onsidered, one may
end up oding themodelagain from srath forthe newpurpose, insteadof reusing
the available legay dynami simulator. Computational enabling tehnologies exist
(seethedisussioninSiettosetal.(2003))thatanllthisgapbywrappingamaster
ode around the legay dynami simulator (the \time-stepper", as we will refer to
it through this paper). The master ode uses the time-stepper as a omputational
experiment that providesoutput information for presribed inputs; the master ode
presribes initial onditions and parameter values and alls the time-stepper for a
short timeas ablakbox. Theproessstatesafterthespeiedintegrationtime are
returned,andproessedbythemasterodetowardsitsultimatetask. Theequations
\hidden"intheblakboxodeneednotbeknown,noraretheyeverextratedinlosed
form;importantquantities(residuals,ationsofJaobians,derivativeswithrespetto
parameters) are estimated from the time-stepper output ondemand; for this reason
wehavetermedtheapproah\equation-free". Ineet,wesolvetheequationsoded
in the blak boxdynami simulator withoutever obtainingthem in losed form. It
is worthnoting that for ourTennesee-Eastman(TE) illustrativeexample thelosed
form equations have been extrated from the original soure ode (Riker and Lee,
1995a) to be used for model preditive ontrol purposes and dynami optimization
(Albuquerque etal.,1999).
A fundamental omputational taskin proessmodelingis thesystemati
determi-nation of steady states. Computing the proess steady states, as well as evaluating
Jaobiansandderivativeswithrespettoparametersatasteadystateunderpinsboth
ontrollerdesignandoptimizationomputations. Whenadynamimodelisavailable,
a simple,diret wayto nd stable steadystates isto integrate in time until nothing
hanges; alternatively, when the steady stateequations are available, xed point
al-gorithms like the Newton-Raphson method anbe implemented using the Jaobian
of thesteadystateequations andomputationallinearalgebra. Thesteady statesof
the TEproblem are, however,not soeasy to ndeither way; the\referene"steady
state given by Downs and Vogel (1993) is dynamially unstable, so that transient
integration of the open-loop system moves away from it. Furthermore, the \legay
ode"doesnotproduederivativesofthesteadystateequationsasanoutput. Thus,
performingNewton-Raphsontond steadystatesisnotimmediatelyimplementable.
Finally (and more importantly), beause of the existene of two so-alled
invento-ries (seee.g. MAvoyandYe(1994),LymanandGeorgakis(1995))thesteadystate
problem requiresarein itsdenition. Inventoriesareommonly presentinhemial
proessplants. Generallyspeakinganinventoryisavessel,inwhihtheleveldoesnot
aettheothersteadystatevariables. Thusproessmodelswithinventoriesallowfor
rametervaluestherearenosolutions,andforother parametervaluesinnitelymany
solutionsexist.
These ompliations havebeentakledin dierent ways bymany researhersover
thelasttenyears,espeiallyinlowdimensionalproblemsandinproblemsinwhihthe
equations areexpliitlyavailableratherthanbeinghiddeninsidelegayodes.
Feed-bakontroltehniqueshavebeenusedtostabilizetheopen-loopunstableTEproess
and toobtainsteadystates. Timeintegrationofthelosed-loopdynami systemwill
then omputationally loatetheopen-loop steadystates; andby hangingsetpoints,
theopenaswellaslosed-loopsteadystatedependene onoperatingparametersan
be traked. Inventories, however, ompliate the losed-loop ontroller design, and
the assoiated zero eigenvalues also interfere with steady state approximationusing
Singular-Value-Deomposition and approximate inverses (Arkun and Downs, 1990;
MAvoy,1998).
Inthis paperweimplementomputationalsuperstruturesaroundthelegayode
that enablethesystemati omputation of steadystates (stableas well asunstable)
and their dependene on parameterswithout the aid of feedbakontrol. However,
throughaugmentationwithanappropriatenumberofso-alled\pinningonditions",
well dened algebrai problems with isolated solutions an be formulated; we will
disussthree dierentformsofsuh wellposedaugmentedproblems.
TheTElegayodeanbealled byouromputationalsuperstruturein two
dis-tintformulations. Intherstformulationthelegayodereturnsthetimederivatives
ofthemodelvariablesatapresribedstate;werefertothisastheright-hand-side
for-mulation. Intheseondformulationaalltothelegayodepresribesinitial
ondi-tionsforthestate,performstimeintegrationforasettimehorizonstartingfromthese
initial onditions,andreturnsthesystemstateattheend ofthisperiod;weallthis
thetime-stepperformulation. Beauseofthelegaynatureoftheproblem,whihdoes
notexpliitlyprovidederivativesofthestateequations,weusematrix-freenumerial
xedpointandontinuationmethods. Newton-GMRESisourmethodofhoiehere,
and with thetime-stepperbased formulationofthesteady stateonditions,time
in-tegration of thelegay dynami equations an leadto a signiantredution of the
number of overall iterations; the existene of a separation of time sales in the TE
problem ombines withshort time integrationto what anbethoughtof aseetive
\on-linemodelredution". AnearlyreferenetotheexploitationofGMRESformodel
redutionisgiveninWigtonetal.(1985).
The paper is organizedasfollows: In Setion 2westart with abriefpresentation
of the TE problem and some urrent approahes to the omputation of its steady
states. In Setion 3 we disuss the interplay of inventories with xed point
ompu-tation and ontinuationand wederivethree well posed augmentationsof thesteady
state equations. We desribe both the right-hand-side and the time-steppersteady
state formulations. We reall a few basi features of Newton-GMRES and present
our omputational results. In Setion 4 we make omparisons to other steady state
astheloaldynamisystemharateristis.
2 The TE proess and its time integration to steady
state
A owsheet that gives anoverview of the units that onstitute the TE proess and
illustratesthestronginterlinkingbetweenthemisgiveninFig.1. Themodel
represen-tationoftheproess(DownsandVogel,1993)onsistsof50statevariablesxR 50
,12
manipulatedvariablesuR 12
and41measurementsyR 41
. TheTEodereturnsthe
timederivativesf(x;u)ofthestates-theright-hand-sidesofthetransientdierential
equations (1) at the urrent statesx and manipulated variables u. In addition the
measurementsy ,whiharedependentontheurrentxalone,arereturnedbytheall
of theTEode.
dx
dt
=f(x;u) (1)
y=g (x) (2)
A randomnumbergeneratorthat -ifwanted- distortsthemeasurementswithnoise
is also supplied with the routine; we didnot use this feature. Basease valuesof a
referenesteadystatex
0 andu
0
arealsoprovided.
At the beginning of our study welinked the original Fortran TE ode to Matlab
asa mexle. Wealulated theJaobianf=x at thereferenesteady stateusing
numerialderivativesandobtaineditseigenvaluesthroughMatlab. Asiswellknown,
theeigenvalueanalysisoftheJaobianatthebaseasevaluesyieldsseveralverylarge
negativeeigenvalues,twozero-eigenvalues(omputationally10 6
)and twopairsof
omplexonjugateeigenvalueswithpositiverealparts. Thelargenegativeeigenvalues
representfaststabledynamis,whilethepositiverealparteigenvaluesindiatethatthe
referenesteadystateisopen-loopunstable. Asaonsequene,whentimeintegration
is performed todynamiallyonvergetothereferenesteadystate, theproessveers
awayfromitandshutsdown(violatesoperatingonstraints)withinashorttime. Fig.2
showstheeigenvaluesofthereferene steadystateJaobianf=x. Theeigenvalues
with realparts largerthan-10(ofwhihthere are19)aregiveninthe upperpartof
theplot;thelowerpartoftheplotshowsalleigenvalues.
Forstable systemsit isommon pratieto determinesteady statesthroughtime
integration: when slightly perturbed from anisolated stable steady state, the
simu-lation will asymptotiallyreturn to it. Depending on operatingparameters, the TE
proessanalsoexhibitopen-loopstablesteadystatesandthetransientopen-loop
re-sponseoftheproessafteraperturbationform onesuh stablesteadystate(givenin
orlossof theA feed streamwasonesenariofordisturbane rejetionstipulated by
Downs and Vogel (1993) andis onsidered oneof the mosthallengingdisturbanes
for ontrol strutures (Rikerand Lee,1995b; Lymanand Georgakis, 1995; Prieet
al.,1994). Forourrepresentativeopen-loopstablesteadystate,theapplied
perturba-tionisspontaneouslydampedandtheoriginalsteadystateisasymptotiallyreovered
after approximately150 h proess time. Time integration washere performed by a
lassialexpliitRungeKutta4 th
ordershemewithindividual timesteps1.5s.
Atthis representativestable steadystategivenin Tab.1tworealparteigenvalues
are again zero; but nowthe largestnon-zeroreal parteigenvalue is-0.053, implying
anoverallneutrallystablesystem. Aninterestingfeatureofthedynamisimulationis,
that theinventory levelsofprodutseparatorand stripperdonotregaintheirinitial
valuesafter the otherproess measurementshaveequilibrated. It is wellunderstood
that this type of dynami observation is harateristi of systems with inventories;
the two inventories in the TE proess are embodied in the two zero eigenvalues of
the (singular)Jaobian. The seond importantfeature isthat these innitiesdo not
ourateveryoperatingparametersetting-formostoperatingparametersettingsthe
open-loopproblemhasnosteadystates;astheoperatingparametersvary,theseentire
familiesof steadystatesappearonly forspeialparameterombinations. Atagiven
set ofoperatingparameters,allstatevaluesremainonstantwhen theholdupin the
inventoriesvaries; the nonzero eigenvaluesof the Jaobianf=x do, however,vary
with theholdup,and theeetonthelargestrealparteigenvalueanbesigniant.
For an inrease of the levels of the produt separator and the stripper at the TE
proessbasease,thelargestrealparteigenvaluedereases,thus makingtheproess
less unstable. Thereverse is observed when the levels of separatorand stripper are
dereased. Theleveloftheprodutseparatorhasamuhstrongerinuenethanthat
of thestripper-inreasingthe leveloftheprodutseparatorto 100%whilekeeping
thestripperlevelat50 %dereasestheleadingeigenvaluereal partto 2.528whereas
at the base ase it is 3.065. These observations are easily rationalized, onsidering
that thetwovessels serveto dampen utuations ofthe onentrations ofthe liquid
streamsexitingfromthem. Thiseetofinventoriesonproessdynamisanbetaken
advantageof indesigningeetivedynami ontrollersforaproess; forthenominal
TE proess in partiular, dynami operation is favorably aeted by a large liquid
holdupin theseparatorratherthanthestripper.
In engineering pratie,unstable steadystates are stabilizedthroughthe addition
of ontrolloops. These giverisetolosed-loop dynamisystemspossessingthe same
steady statesastheopen-loopones,but withdierentstabilityharateristis.
Con-sider -as an illustration-theveloityformulationof aSISOPIontrollerwith gain
G, time onstant and setpoint of ameasurementy
set
, whih anbe be viewed as
atransientdierentialequation: u
t+t =u
t
+G[(y
t y
t t )+(y
t y
set
)= t℄.
In a ontrol ontext oneaims at transforming an open-loopunstable system like(1,
2) into a stabilized system through addition of an appropriate set of suh feedbak
ontrollers. Ifthe losed-loopsystemdynamisonvergeto thedesiredsetpoint y
set t
u=u
set
. This approahhas been used forthe numerial omputation ofTE model
steady states(e.g. Subramanian andGeorgakis (2004)),in thesame way onewould
doitforphysialexperiments.
3 Steady state determination by ontinuation methods
Ingeneral,thesolutionofsquaresystemsofalgebraiequationsoftheformf(x;u)=0
withf R n
forthestatesxR n
withpresribedparametersuR m
isperformedusing
ontinuationmethodsandontrationmappingsliketheNewton-Raphsonalgorithm.
When the Jaobian is nonsingular, suh algorithms trae both stable and unstable
steady statesfamilies asthe parameters(forourpurposes, manipulatedvariables) u
arevaried;thereisnoneedforlosingontrolloopstostabilizethesystemso thatits
steadystatesanbefoundbyintegration.
Oneiterativelysolvesthelinearsetofequations
f(x;u)+
f(x;u)
x
Æx=0 (3)
for Æx and adding it to the urrent iterate x until onvergene. Inversion of the
Jaobian f=x through the standard Gauss algorithm requires, that the Jaobian
hasfullranki.e. is nonsingular.
Wealreadyknow(fromeigenvalueanalysis,butalsofromphysialreasoningabout
inventories)that the linearization of the TE equations is singular, and theirsteady
states - if they exist - are notisolated. Sine the Jaobianof the TE model is
sin-gular,standardNewton-Raphsonfails; alreadyMAvoy(1998)appliedsingularvalue
deompositiontotheJaobianf=x,andreportedtheexisteneoftwosingular
val-uesequaltozeroatthebase ase. Theiranalysis wasarriedoutin theframeworkof
Arkun and Downs (1990)'s method to identify gain matriesfor proesses that
on-tain integrating tanks. MAvoy (1998)used aSVD based pseudo inverse to further
onvergethe steadystate ofthe base asegiven byDowns andVogel (1993)(as the
steady statetheyreported is notwell onverged)and then derivethegainmatrix of
theproessthroughnumerialderivatives.
Whilef(x;u)isofdimension50,f=xisrank48(hastwonullvetors). Ingeneral,
ahangeinonemanipulatedvariableu
i
resultsinahangef=u
i
off,whihisnotin
therangeoftheJaobianmatrixf=x. Aswedisussed,thefull nonlinearproblem
haseithernosteadystatesorinnitelymany. Inordertoobtainawell-denedproblem,
oneaugmentstheset ofequationsf(x;u)=0from(1) withtwoadditionalalgebrai
equations(theso-alled\pinningonditions");moregenerallyonewouldneedasmany
pinningonditionsasindependentinventoriesintheproblem. Thepinningonditions
selettwooutofthetwo-parameterinnityofsteadystatesolutions. Theaugmented
system has n+2 equations, and n+2 variables - the old n variables plus two more;
typiallythetwoadditionalunknownswillbetwoofthemanipulatedvariables. Inthis
approahthefatthatnotall12manipulatedvariablesanbeseletedatwillislearly
tenproess measurements;thevaluesofalltwelvemanipulatedvariablesaswellasa
singlesteadystate(outofthetwo-parameterinnity)willresultfromanappropriately
augmentedalgebraisystem. Oneasinglerepresentativesolutionhasbeenfound,the
entiretwo-parameter familyan be routinelyreonstruted using the nullvetors of
the steady stateJaobian. Clearly, while theinventoriesare vitalfor dynami plant
modeling,theyareeetivelysuperuousforthedeterminationofsteadystates. Note
that in onventionalPIontrol shemes eah inventoryis stabilizedbymanipulating
oneproessvariablein alosed-loop. An analogythereforeexistsbetweenwell-posed
xedpoint/ontinuationproblems(twoadditionalpinningonditions,usedtondtwo
ofthemanipulatedvariables)andthelosed-loopontrolapproah(twoontrolloops
linking twomanipulatedvariableswiththetwoinventoryholdups).
Augmented SteadyState Problems
We wanttond steadystatesof theproblem (1),(2). Intheright-hand-side
formu-lation, wesolve forthevetorof timederivativesof thesystemstate variablesto be
zero. Inthetime-stepperformulation,wesolvefor(4)tobezero,where(x
t
0 ;u
t
0 ;)
istheresultofintegratingthesystemequationsforatimehorizon. Clearly,asteady
statex(asolutionof(1)equaltozerof(x,u) =0)alsosatises(5)forall.
x
t0+ =
Z
t0+
t=t0 f(x
t0 ;u
t0
)dt(x
t0 ;u
t0
;) (4)
Thetime-stepperbasedequivalenttof(x;u)=0forthedeterminationofsteadystates
basedonthexed pointequation(4)is(5):
^
f(x;u;)=x (x;u;)=0 (5)
Thefollowingdisussionappliesthereforetobothformulations.
As disussed, the TE proess Jaobian (in both formulations!) has a rank
de-ieny oftwo,beauseofthe twoinventories. Forparametervaluesforwhihsteady
statesexist,theyappearasatwo-parameterinnityofsolutions,parametrizedbythe
inventorylevels;theremainingsteadystatevariablesremainonstantoverthefamily.
We haveimplementedthree dierentwaysof obtainingnon-degenerate formulations
ofthesteadystateproblemWerefertothesetofequationsbyFandtotheirvariables
byX.
Augmentation1a: Thisformulationonsistsoftheaugmentedsetofequations
F
1a =
f(x;u)
g
l (x) y
lset !
=0 (6)
where f(x;u) is the vetor of 50 time derivatives of state variables of the system
and g
l
\pinning" onstraints,whihin this aseare y
12 =g
12
(x) andy
15 =g
15
(x), seleta
single member of the two-parameter innity of solutions; for the TE problem these
orrespondto\pinning"measurements12and15i.e. produtseparatorandstripper
levels. Wesolvefor thesteadystate with50 % levelin thetwoinventories (produt
separatorandstripper);wehave52equations,andanthereforesolvefor52unknowns,
whihwean arrangeintheolumnvetorX
1a . X 1a = x u d ! (7)
The olumnsubvetoru
d
is ofdimension2and ontainstwodependent manipulated
variables(theonesthatwewillhoosetosolveforafterpresribingtheremainingten).
OurNewtonstepisthen:
f(x;u)
g l (x) y lset ! + " f(x;u) x f(x;u) ud g l (x) x 0 # Æx Æu d !
=0 (8)
TheNewtononvergeswhentheaugmentedsystemabovehasanonsingularJaobian;
ourhoieoftwodependentmanipulated variablesisguidedbythisrequirement. For
this spei proess any ombination of two from the 12 manipulated variables of
the TEproblemwouldgiveanonsingularJaobian;ertainhoies,however,leadto
betteronditionedproblems,orproblemsthatdonoteasilyhitsaturationboundaries
during parameter-spaeontinuation. Fromthe proess sideonemightselet asthe
new dependentvariablestwoproessowsthatare losestin physial proximityand
largestineet onthetwomeasurements,whosevaluesarespeied.
Augmentation 1b: A simplevariation of the rst formulationpins twostates,
in-volved in the equation(s) desribing the rate of hange of the inventory variables,
instead of twomeasuredvariables. This formulationpiksa solutionout of thetwo
parameterinnitybysolvingtheset ofequations
F
1b =
f(x;u)
x l x l set !
=0 (9)
where x
l
ontains two elementsof the omplete state variable vetorx. These two
variablesx
l
musthavenon-zeroontributionstothenull-eigenvetorsoff(x;u)=x,
so that the twoparameterinnity anbe pinned down to a uniquesolution (again,
anyombinationoftwomanipulatedvariablesastheadditionalunknownsinthe
f(x;u) x l x lset ! + 2 6 4 f(x;u) x1 ::: f(x;u) xn 1 f(x;u) xn f(x;u) u d
0 ::: 1 ::: 0 0 0
0 ::: 1 ::: 0 0
3 7 5 Æx Æu d !
= 0 (10)
Augmentation 2: Here we substitute the last two lines of (9) i.e. the ondition
x
l =x
l
set
diretlyinto f(x;u) =0,so that the equations forthis augmentation are
F
2
f(x;u) = 0. Instead of pinning at speied positions (like augmentations 1a
and 1b)augmentation2 removestwostatevariables, whih representinventories, as
unknownsfromtheproblemandreplaesthem(asunknowns)byanequalnumberof
parameters (manipulated variables). The values of these manipulated variables will
nowbefoundasafuntionoftheremainingparameters.
The vetorof 50 unkonwsnowonsists of the48-longsubvetorx
f
, that ontains
allbut twostatevariables, augmentedbythetwoadditionalu
d asabove. X 2 = x f u d ! (11)
This leadstotheNewton-Raphsonsheme:
f(x;u)+ h f(x;u) xf f(x;u) ud i Æx f Æu d !
=0 (12)
Werefertothesetsofsteadystateequationsofformulations1a,1band2,thatuse
thetime-stepperformulation ^
f(x;u;t)insteadofthetimederivativesf(x;u)by ^
F.
Clearly,theresultsofaugmentation2willbethesameastheresultsofaugmentation
1b(theyusethesamepinning);augmentation1ausesadierentpinning,andsoitwill
nd dierent representativesofthe familyof solutions. Upon onvergene,weobtain
the values of the two dependent manipulated variables for whih a two-parameter
innity of solutions exists, along with asingle representative of the family(the one
seleted by our pinning). The rest of the steady state family for the manipulated
variable valuesin question(thetenwepresribedandthetwowefound)isroutinely
obtainedbyusingthenullvetorsofthedegenerateproblemJaobian.
This disussion is independent of whether the ode is available in a legay form
or expliitly(i.e., independently of whether we anevaluatethe Jaobians from the
equations, orwe needto estimate them from the ode). As asanityhek, weused
the three above augmentations for the right-hand-side of the TE equations, using
diretlinearalgebra(LUdeomposition)onaugmentedJaobianswhoseelementswere
estimated throughnite dierenes from the TE ode. We reprodued the optimal
operatingpointsgivenbyRiker(1995)(in\optimizationmodes"1and4ofhispaper).
of Tab. 1forwhih wepresentedtransientsimulationsin Figs.3and4. Figs. 5and
6showthevariation, alongthis branh oftheomputed steadystatemeasurements.
The numbersonthe absissaorrespondto thestepalongthis branh; # 1refersto
thebaseaseand# 11isthestable steadystateofTab.1.
At the stable steady state of Tab. 1 the proess operating onstraints stay well
within thebounds forsafeoperation. AsRikerandLee(1995b) desribed,arisein
thereatortemperatureausesenhanedformationofbyprodutF.Thisisapparently
the ase for the steady state of Tab. 1 as an be seen in a plot of the reator feed
onentration along the branh in Fig. 7. The produt onentration at the stable
steady state (not shown) undergoes only minor hanges and the produt stream of
valuable omponents G+H even inreases slightly when ompared to the base ase,
whiletheprodutionostdereasesslightly.
Using time-steppersto ompute steady states
In priniple steady states an be omputed as xed points of equations formulated
with the time-stepper for any time reporting horizon . A number of papersin the
reentnumerialliterature(seetheworkofJaraushandMakens(1987),Shroand
Keller (1993), Lust et al.(1998) aswellaswork in our group: Siettos et al. (2003),
Kevrekidis et al. (2003,2004), Kelleyet al. (2004))disuss theenablingof temporal
simulationodestoonvergeonsteadystatesolutions. FortheTEproblem,thexed
point equation f(x;u) = 0i.e. the steady state onditionof (1) also givesrise to a
rank-deientJaobian,andtheaugmentationsdisussedabovewillworkequallywell
for thetime-stepperformulationastheydofortheright-hand-sideformulation. The
important element, however, in the time-stepper formulation, is the omputation of
theaugmentedJaobianforNewton-Raphson. Inpriniple,ifequationsareavailable,
integration of variational and sensitivity equations (with odes like ODESSA, Leis
and Kramer (1988)) will provide the derivatives with respet to initial onditions
and parametervalues that onstitute Jaobian elements. The solution of the linear
equations that onstitute a Newton step, however, need not be arried out through
diretlinearalgebra(LUdeomposition);theyanbearriedoutiteratively(through
methodslikeGMRES:Saad(1981,1984),Kelley(1995),Brown(1994)).SineGMRES
omputations atuallyrequirematrix-vetormultipliationsof the(Jaobian)matrix
with a sequene of omputed vetors, GMRES an be performed in a matrix-free
fashion,usingonlyallstotheintegrator,aswewillillustratebelow. Partialderivative
omputationsan thusbeirumvented.
OurmethodofhoieforthelinearsetsofequationsthatarisefromNewton-Raphson
augmentedsteadystateproblems(8),(10)and(12),bothfortheright-hand-sideand
for the time-stepper basedformulations, was theiterativeGMRES solver, whih we
now briey review. GMRES is a subspaemethod for the iterativesolution of sets
of linear equations Ax+b = 0. Starting from the initial residual vetor r
0 =
Ax
0
+baKrylovsubspae(i.e. thespanofasequeneof vetorsarisingfrom the
matrix-vetorproduts hr
0 ;Ar
0 ;A
2
r
0 ;A
3
r
0
diretionalderivatives. CalulatingasequeneoforthonormalizedKrylovbasesleads
to the equality AZ
k = Z
k +1 H
k +1
, where Z
k
onsists of the orthonormal basis
vetorsandH
k +1
isaHessenbergmatrix.
A( x
0 +Z
k
)+b=Z
k +1 H
k +1 +r
0
=r (13)
Atthek+1stepthelinearleastsquaresproblem(14)oforderk+1issolvedto nd
thesolutionthatminimizestheresidualoverthesubspae.
kH
k +1
+kr
0 ke
1
k!min (14)
An extensivesurveyofthis familyofNewton-Krylovmethods anbe foundin Knoll
and Keyes(2004). Forour alulationswithGMRES weused theodepresentedin
Kelley(1995).
One Newton-Raphson step of (8), (10) and (12) is termed an \outer iteration"
whereastheiterationstosolveonesetoflinearequationsarealled\inneriterations".
Ifthetime-stepper(5)i.e. ^
f(x;u)=0isusedinsteadoff(x;u)=0,theiterative
GM-RES solvertakespartiular advantageof theeigenvaluestruture of(x;u;)=x.
Newton-GMRES solvers perform favourably with lustered eigenvalues of the
Jao-bian, whiharises whenusing time-stepperstosolve ^
f(x;u)=0; undersuh
irum-stanesgoodsolutionsareoftenfoundafterarelativelysmallnumberofiterations(in
low-dimensionalKrylovsubspaes).
Whatmakesthismatrix-freeomputationalapproahaninterestingonetoombine
withtime-steppers,isthatitaneasilyexploittime-saleseparationwhenitispresent
in theproblem ofinterest -and this isindeed the asein theTE proess. Probably
therstarhivalobservationofthiseet anbefoundin Wigtonetal.(1985). Itis
obviousfromtheeigenvaluestrutureof theJaobianf=x givenin Fig.2that the
dynamisoftheTEproblemevolveondrastiallydierenttimesales. Integratingthe
dierentialequations withatime-stepperoverlargetime horizonseetivelyredues
theproblemsize(inalinearproblemiteliminatestheomponentsalongeigenvetors
assoiatedwiththefaststabledynamis fromtheoutputx
t
0 +
). Considerthe
eigen-valuesoftheJaobian(x;u;)=x(atsteadystateandforperfetintegrationthese
areexp() ,where isthereportinghorizon andistherespetiveeigenvaluesof
the steadystateJaobian. As thereporting horizonof thetime-stepperisinreased,
thefasteigenmodesofthetime-stepperlinearizationmovetowardszero,whiletheslow
(unstable)onesstayaround(outside)theunitirle. Fortimesteppingweusedbotha
lassialRungeKuttaexpliitmethodof4 th
orderandanimpliitEulermethod. Due
tothestinessofthesystemweemployedindividualtimestepsof1swiththeexpliit
method, whileweoveredtheompletereportinghorizoninonesingletimestepwith
theimpliitEulermethod. Whenweevaluatedtheeigenvaluesof(x;u;)=xwith
the Runge Kutta time-stepper at the base ase, the number of eigenvalues lose to
zero (absolute valuebelow 10 4
) inreasedfrom 0at reporting horizon 10 s to 8at
thetime-stepperevaluation forvarious reporting horizonsat thebase aseoperating
point;onemore,forperfetintegrationtheseoughttobeexp() ,whilefor
numer-ial integration theyare approximationsof exp()that depend on thepartiular
method.
Tohighlight theadvantages of theNewton-GMRES approah in the time-stepper
formulation, wealulatedasteady stateof the TE proess usingall three
augmen-tations 1a,1band2. Thesteady statetobedeterminedwasdesignatedbyahange
of the ratio of the D and E feed mass streams o the base ase (derease of u
1 by
0.25,inreaseofu
2
by0.17391). Forallthreeaugmentationsu
7 andu
8
werehosenas
thedependent manipulatedvariablesinu
d
. Thisorrespondstothetypialinventory
ontrol sheme, where the underowsof theinventoriesare manipulated in order to
ontrol the vessel holdups. In augmentation 1b the twostate variables x
13 and x
22
werearrangedinx
l
tobepinnedbytheadditionalequations. Inaugmentation2these
two state variables were kept at the values of the TE base ase, suh that x
f
on-tainedallstatevariablesbutx
13 andx
22
. ConvergeneoftheouterNewton-Raphson
iterationwasdelared,whenthequantity ^
F T
^
Ffell below10 10
.
The GMRES onvergene is ontrolled by two tolerane parameters. The rst (a
so-alled \foring term", ) is the ratio, by whih the residual of the linear set of
equations must be reduedbefore asequeneof inner GMRES iterationsisdelared
suessful,andtheapproximatesolutionofthelinearsetofequations(8),(10)or(12)
is aepted. The seond parameterlimits the maximum number of inner iterations
performedinoneouter(Newton)iteration. Anof0.1performedsatisfatorilyforall
omputationsreportedhere. Wedidnotlimitthemaximumnumberofinneriterations
to asmaller numberthanthe problem size(i.e. we allowedfor afull solutionof the
set of linear equations by the GMRES method if neessary; this never ourred in
the time-stepperformulation). RestartedGMRES (seee.g. Kelley (1995))isalso an
optionforkeepingthesizeofthesystemsolvedsmall;wedidnotuseitinthispaper.
Convergene results of the three augmentations for time-stepper based
Newton-GMRESaregiveninTabs.2and3forbothaRungeKuttaandanimpliitEulertime
stepper. InpartiularTab.2givesthetotalnumberofallstotime-stepper(x;u;t)
(for one reporting horizon) during all outer iterations until onvergene. Note that
eah diretional derivative, by whih a diretion is added to the Krylov subspae,
involves onefuntion all i.e. a all to the time-stepper overthe reporting horizon.
Furthermore, the number of outer Newton-Raphson iterations until onvergene is
displayedinTab.2. IntegrationoveratimehorizonrequiresrepeatedallstotheTE
ode. Thetotalnumberof suh alls of theTE ode (righthand side evaluations of
(1)inurredforafullsteadystateomputationandthetotalCPUtimeforitisgiven
in Tab.3. OurMatlab programsandtheTEode,that wasonnetedto Matlabas
amex le, were runonMatlabRelease13on aPentiumIV,3GHzomputerunder
RedHat Linux.
Theonvergeneresultsarestrikinglydierentforthethreeproblemaugmentations.
time-stepperallssigniantly. Thesametendenyisobservedwithaugmentation1b,
but notwithaugmentation 2..
Fig.9showsthefullonvergenehistoryofaugmentation1awiththeexpliitRunge
Kutta time-stepper anddierent reporting horizons. On theordinate isthe residual
^
F T
^
F, on the absissa is the umulative number of time-stepper or funtion alls,
whih is equalto the umulativenumber of inner GMRES iterations. Eah marker
in this plot representsthe residual at an outer iteration, whih was evaluated after
ompletion of a sequene of inner GMRES iterations. Due to the inexat solution
of thelinear set of equationsin the Newton-Raphsonaugmentation(8) byGMRES,
the ontration of the residual with the Newton-Raphson iterations is slower than
quadrati, even in the last few iterations. With inreasing reporting horizon of the
time-stepper,thedelineof theresidualversusthenumberof funtionalls beomes
steeper. Infat,withaugmentations1aand1bittakesaboutthesamenumberofouter
Newton-Raphsoniterationstoonverge,butthedimensionoftheproblem(thenumber
of innerGMRESiterationsforoneapproximate linearsolve) redues. Thereasonfor
this is that time integration, by eliminating the fast dynami modes of the system,
atseetivelyasapreonditioner;theapproximatelinearsolutionsareobtainedinan
eetivelysmallersubspaethantheoriginalproblemspae. Foraugmentation1aand
the RungeKutta time-stepperwithareporting horizonof 2000sapproximatelinear
solvesanbeobtainedin,roughly,a10-dimensionalsubspae(downfromtheoriginal
full spaesize of52). As disussedin Kelleyet al.(2004)theorrespondingproblem
beomes a ompat perturbation of the identity, the eigenvalues of its linearization
luster,andbetterperformaneisexpetedfrom GMRES.
For a stable proess, of ourse, a all with a relatively long time horizon would
requirenoiterations;yetitwouldtakealargeomputationaleort. Wetrytoexploit
separationoftimesales,sothattheeetivepreonditioningbenetfromarelatively
short integration-ombinedwiththeNewton-GMRESalgorithm-outperformsboth
diretintegrationandNewton-GMRESonaright-hand-sideformulation. Ofoursein
manyproblems (e.g. splitstepdynami simulators,orin aseswheretheinner
time-stepperinvolvesastohastiormirosopiode,Kevrekidisetal.(2003),Kevrekidis
et al.(2004))the right-hand-side formulationis notan option. Due to the unstable
nature of theTE proess, thereporting horizonannot be inreasedat will. Even if
theinputx
t ,u
t
tothetime-stepperisthesteadystateitself(toomputerroundo),
itwill veerawayfromthesteadystateiftheopen-loopunstableproessis integrated
overtoolongreportinghorizons. Arule ofthumbfor theseletionofanappropriate
timehorizonwouldput itsinverseinthegapbetweenthestrongstableeigenvaluesof
thesystemlinearizationandtheabsolutevaluesorrespondingtotheslow(bothstable
and unstable) eigendiretions; so,fast omponents die (getslaved to slowones), yet
wedonotintegratelongerthanneessary, andtheunstablemodesdonot\explode"
awayfromthesteadystateofinterest.
Augmentation 1bshowsthe samequalitativebehaviorasin Fig.9with bothtime
for onvergene(not shown). This suggeststhat thelinearization ofaugmentation2
doesnotexhibiteigenvaluelustering as1aand1bdo. Plotsoftheeigenvaluesofthe
Jaobiand ^
F=dXofaugmentations1a,1band2withtheRungeKuttatimeintegration
atthebaseasevaluesaregiveninFigs.10,11and12. Augmentation2withtheRunge
Kuttatime-stepperproduesonenegativerealeigenvaluewithlargeabsolutevalueat
reportinghorizons 500and 1000, notshownin Fig.12. An important fatorfor the
GMRES method to perform wellona linearset ofequations is thelustering of the
eigenvaluesoftheproblemmatrix(Kelley, 1995). Welearlyseethattheeigenvalues
ofaugmentations1aand1bindeedform lustersasthereportinghorizonofthe
time-stepper is inreased and the GMRES-time-stepper approah eetively redues the
problem size. It appears that these advantageous lustering harateristis do not
arise in augmentation 2,and webelievethat this underlies theobservation that the
problem sizedoesnoteetivelyreduewithinreasingtime-reportinghorizon.
Itmightbeargued,thattheonvergeneriterionusedforomparisonbetweenthe
three formulationsisnot fair, beause thenorm ^
F T
^
F ofthe vetorfuntion sales
dierentlywith dierentproblem augmentationsand reporting horizons ofthe
time-stepper. Wethusalsoused-asaonvergeneriterion-theresidualf T
f oftheright
hand side of (1), whih orrespondsto theset of equations that shall ultimately be
solved;thisriterionisthesameforallproblemaugmentationsandreportinghorizons.
Yet,thetrendspresentedabovedidnothange.
InearlierworkKelleyet al.(2004)usedGMREStoperformpseudo-arlength
on-tinuationforsteadystatesofdynamialsystemsusingtime-steppers. Theyfound,that
the augmentationof thesystemequationswithoneadditionalequation (the
pseudo-arlengthequationforontinuation)doesnotdestroytheeigenvaluelustering ofthe
augmentedsystem.
The ruial question for the viability of the legay time-stepper based
Newton-GMRESapproahtosteadystateomputationis,ofourse,underwhihirumstanes
it performs better than diret time integration. Clearly, if one needs to ompute
unstablebranhesofsteadystatesormarginallystablesolutions,integrationissimply
not anoption. Aswesawfor theTEproess,diret open-loop integrationalso does
not onstitute an option for problems with inventories, whether they be stable or
unstable. If wedonot knowthe right ombination ofmanipulated variables,steady
states simply do not exist. For problems with asymptotially stable steady states,
performanedependsdrastiallyonthedistributionoftheeigenvaluesoftheJaobian
f=x of the systemdierentialequations (1). Time-stepper Newton-GMRES is in
priniplebestsuitedforlargesystems,haraterizedbyafewslow(possiblyunstable)
eigenmodesandalargemajorityoffast,stableeigenmodes.
Onethingshouldbemadelearatthispoint. Usingseveraltimestepsofanimpliit
integrator, and performing the assoiated nonlinearsolves, is learlynot aneÆient
wayofsolvingaxedpointproblem(asinglenonlinearsolve). Theintegratorisused
hereasalegayode,aodethatinprinipleoneannotmodify. Itisin thisontext
MonteCarlo).
In our disussion of time-sale separationand the orresponding gap between the
system eigenvalues, we assumed for simpliity that the short numerial integration
was perfetly aurate; when numerial integrators are used, the eigenvaluesof the
linearizedtime-stepperarenotanymoreexponentialsofthesystemeigenvalues
multi-pliedbythereportinghorizon. Whatintegrationmethodisusedwillnothangethe
xed point, but may drastially hange theeigenvalues ofthe timestepper
lineariza-tion(largeexpliitstepsmayevenrenderastablesteadystatenumeriallyunstable).
These linearized numerial timestepper eigenvalues aet GMRES performane, as
anbeseeninTab.3. Inourexampleweusedbothanexpliitandanimpliit
time-stepper and found - for our omputational protools - dierent trends of the CPU
timeforsteadystatedeterminationwithGMRES,whenthereportinghorizonwas
in-reased. FortheimpliitEulertime-stepper,forexample,wehoseasingletime-step
equal tothe entirereporting horizon. With reporting horizons largerthan100s the
timestepsizewasunreasonableintermsofauray,yettheresultingdampeningwas
beneial forsteadystatedetermination. Infat theredutionof theCPU timeand
number ofrighthand side(1) evaluations forinreasing reporting horizons with the
impliit Euler and augmentations1a and 1b (as shown in Tab. 3) is entirelydue to
theGMRES\eetivemodelredution"andtheresultingredutionofrequired
time-stepperalls. ThesavingsintheGMRESmatrixinversionduetoasmallernumberof
basisdiretions isnotsignianthere. Inonlusion,aninreasingreportinghorizon
inreases the omputational expense for time integration, while possibly leading to
inreasedGMRESperformane;optimalperformanewillresultfrombalaningthese
twotrends.
4 Further system augmentation
Duringontinuationwithhemialproessmodels,operatinglimitsoftheproessneed
to beobservedand failure to dosoresultsin proess shutdown. Interms ofthe TE
proess, these shutdown limits are the pressure and temperature high limits in the
reatorandthehighorlowlimitofthereatorliquidlevel.
Intheaugmentationsdisussedaboveweusedtheminimumnumberofpinning
on-ditionsthatwouldgiveawell-posedproblem-wepresribed10manipulatedvariables
and allowed two to be omputed from the model. In priniplewean now perform
pseudo-arlength ontinuationalong anyone-parameter pathin ten-dimensional
ma-nipulatedvariablespae. Tomakesure,however,thatwedonotfrequentlyenounter
operatinglimits,wemaywanttoexplorethisten-dimensionalspaebyaddingfurther
onstraintsthat keepkeysystemmeasurementsat xedlevelsawayfromoperability
boundaries. Additional suh pinnings result in redutionof the numberof
indepen-dent manipulated variables (and orresponding growth of the number of dependent
variables) - one additional dependent variable for eah additional pinned
and thustheequationsforonstrainedsteadystatedetermination (16)arederived.
g
(x) y
=0 (15)
0
B
f(x;u)
g l (x) y lset g (x) y 1 C A
=0 (16)
TheassoiatedNewton-Raphsonshemeis(17),thevariablesare thestatesx, the
dependentmanipulatedvariablesu
d
andthemanipulatedvariablesu
.
0
B
f(x;u)
g l (x) y l set g (x) y 1 C A + 2 6 4 f(x;u) x f(x;u) u d f(x;u) u gl(x) x 0 0 g (x) x 0 0 3 7 5 0 B Æx Æu d Æu 1 C A
=0 (17)
Inthesamemannerformulations1band2anbeaugmentedtosolveforonstrained
steadystates.
Suh additionalaugmentations nd theirounterpart in the losed-loop approah
tosteadystatestabilization/omputation-additionalsetpointsrequirefurtherontrol
loops and \slave" additional manipulated variables. Considerthe set of ontrol and
outputvariablesin ontrolstruture2inLymanand Georgakis(1995): There,single
SISOloopstieallmanipulated variables(exeptfortheagitatorspeed) toontrolled
variables in the proess. The ontrolled variables are the reator temperature, the
reyle ow rate and the stripper underow (measurements 9, 5, 17), the levels in
the stripper, the separator and the reator (measurements 8, 12, 15), and besides
onentrationsin thereatorfeed, thepurgeand theprodutstream (measurements
23,26,27,30,38). Notethatreatorpressureisnotontrolledbythisontrolsheme.
Inoursteadystateaugmentationformulation,ifweimposethesamehangeof
rea-tor temperaturefrom120.4to 121.6and seletthesamesetsofadditionalpresribed
measurementsanddependentmanipulatedvariablesasLymanandGeorgakis(1995),
the values of the manipulated variables we nd are given in Tab. 4. Also given in
Tab. 4 is an example, where the produt ow rate is inreased by 3.5 % with the
remainingtenpinnedmeasurementsremainingatthebaseaseTEproessvalues.
Both ases of setpoint hange show, that with the ontrol struture proposed in
LymanandGeorgakis(1995),themanipulatedvariableu
9
(strippersteam)undergoes
extremehangesensuingfromthegenerisetpointhangesofreatortemperatureand
produtionrate;theontrollooplinkingu
9
withtheonentrationofomponentEin
the produtstreammayeasilysaturate. RikerandLee(1995a)alsoreportthat the
strippersteamratehasnegligibleeetontheproessbehavior. Removingthestripper
steam rate (manipulated variable u
9
38
of theontrol struture yieldsaset of 62equationswith 2independentmanipulated
variables. Keepingnowu
9
atitsbaseasevalueandontinuinginreatortemperature
- withoutattempting to piny
38
, whihoats -wend that thereatortemperature
anbeinreasedto y
9
=128:2 beforeoneofthedependentmanipulated variables
hitsitsbound. Inthispartiularasetheompressorreylevalvepositionu
5 hitsits
lowerbound. Continuationin theprodutowrateshowsthatitanbeinreasedby
13.1 % (y
17
=25:95m 3
=h)before thereator pressurehitsthe high limitfor normal
operationof2895kPa.
Havinganalgorithmthateasilyomputesandparametriallyontinuessteadystates
subjetto various sets of onstraints (keepingagivenset ofmeasurementsonstant
throughvaryingagivensetofmanipulatedvariables)allowsasystematiexplorationof
thesystemoperability. Inpartiular,oneanexplorewhethergenerisetpointhanges
anbein prinipleperformedusing presribed setsof manipulated variables tokeep
all ontrolled measurements at their setpoints without ontrol saturation. One an
evenenvisionpinningonditionsthataskforamanipulated variableorstatevariable
to marginally attain saturation as a omputational approah for traing operability
boundaries.
5 Disussion and Conlusions
Steady states of a dynami proess simulator were determined through matrix-free
tehniques(Newton-GMRES)thatemploysequenesofinput-outputallstothelegay
ode. Two formulations (a dierential one, where the legay ode returns
time-derivatives of the state variables, and an integral one, where it returns the result
of integrationoveratimereportinghorizon) anbeused forthis purpose. This
om-putationalapproahdoesnotrequirepriorstabilizationofunstableproessesthrough
losing ontrolloops.
Theadaptationofthisapproahtoproblemswith inventorieswasexplored. When
workingwithdynamimodelsofhemialproessplants,inventoriesgiverisetoertain
omputational problems. Inventoriesturn thesteady state problem into anonlinear
eigenproblemwitheitherzeroorinnitefamiliesofsolutions. Theirsteadystatelevels
havenoinueneontheotherproessstatessuhasstreams,onentrations,pressures
and temperaturesnoron theinoworoutow ofthe inventoryitself. Theyare also
assoiated with singular Jaobians(Arkun and Downs (1990)) and the steady state
equationsmustbeappropriatelyaugmentedtoyieldwell-posedproblems. Thenumber
of freely adjustable manipulated variables among thetotal of manipulated variables
redues byoneforeahinventory. Forasteadystatetoexist,dependentmanipulated
variablesneedto bedened,whosevaluesaredeterminedbythesolutionproedure.
Whileusingatime-stepperbasedNewton-GMREStosolvetheseappropriately
aug-mentedproblems,weexploredtheeetoftimesaleseparationandthetime-stepper
reportinghorizononGMRESperformane. Inreasingthetime-stepperreporting
presentedresultspertainingtotheeetsofdierentaugmentationsaswellasdierent
numerialintegrationmethodsforthetime-stepper.
Asystematisteadystateomputation/ontinuationroutineanbeavaluabletool
forvariouspurposeslikestabilityanalysis,ontrollerdesignandoptimization.
Deter-mination ofsteadystategains isvaluabletowardsontrollerdesign;yetforproblems
with inventoriesand/orunstablesetpointsthesegains maybediÆultto obtain(for
the TE problem see (MAvoy and Ye, 1994)). Systemati onstrained ontinuation
(implemented through further augmentations, as just disussed) an provide useful
diagnostis in strongly stronglynonlinear problems forgainsign hange and ontrol
saturation.
For optimization, where the steady state equations appear as onstraints, a
fea-sible approah an take advantage of a fast, systemati steady state solverto turn
the problem into an unonstrained one of redued dimension. If the steady state
solverunderlyingafeasibleoptimizationodeisbasedonontrolloopsandintegration
(Subramanian and Georgakis,2004;MAvoyandYe,1994), thelosed-loopstability
requirementwilllimitthesteady statesoneannd,andthuspossiblyaetthe
op-timization. Notethat,thankstotheopenstrutureoftheoriginalTEFortransoure
ode, steady state optimization has been aomplishedby linking theTE ode to a
standardoptimizationroutine(Riker,1995).
It isinteresting,espeiallyforlegaynumerialapproahes(inludingtime-stepper
based ones) where derivatives arenot available, to explore the appliability of more
diretsearhtehniques(suhasNelder-MeadorHooke-Jeevesalgorithms)whihhave
beenexperieningarevivalinbothliteratureandpratie(Koldaetal.,2003). \Blak
box" steady state solvers, turning the problem into a feasible one, may be an asset
in suh omputations. Wean provide somesupporting evidene forthis statement:
loating anopen-loop stable steady state for the TE proess (the onewepresented
in Tab.1) was theresult ofaNelder-Meadoptimization, where ourobjetivewasto
optimize steady state stability. In partiular, we tried to minimize the largestreal
part of the eigenvaluesof the TE linearization at steady state. Wesolvedthis asa
feasibleproblem; oursteadystatesolverunderlyingtheobjetivefuntion evaluation
was a Newton-GMRES time-stepper based one, and the Nelder-Mead diret searh
algoritm was the one of Kelley (1999). The optimization was performed over four
manipulatedvariablesthatwerehosenas\designparameters"(numbers5,6,10and
11);thetwodependentmanipulatedvariablesin ouraugmentationwere7and8,and
theremainingones werekeptatthebase asevalues. Inaforthomingpaperweuse
this methodologyforTEproessoptimization.
Aknowledgements: ThisworkwassupportedbyaresearhsholarshipoftheMax
KadefoundationforAndreasKavouras,whihisgratefullyaknowledged. I.G.K.also
aknowledegesthe partial support of AFOSR (Dynamis and Control) and an NSF
A oeÆient/massmatrixofalinearset ofequations
b oeÆientvetorof alinearsetofequations
statevetorxprojetedontheKrylowsubspae
f funtionofthederivativesofthestatestowardstime
intheTEproess
f time-stepperbasedset ofequationsforsteadystate
determinationin theTEproess
F non-degeneratereplaementforsteadystateequations
f,formulations1a,1band2
g orrelationbetweenstatevetorandmeasurements
H Hessenbergmatrixin GMRESmethod
t time
r residualvetorof alinearsetofequations
u vetorofmanipulatedvariables
u
d
vetorofdependentmanipulated variables
x statevetoroftheTEproess,solutionvetorofaset
oflinearequations
x
f
reduedstatevetoroftheTEproess
X variablesofthenon-degeneratesteadystateequations
F=0,formulations1a,1band2
y measurements
Z
k
orthonormalizedKrylowsupspaewithkbasisvetors
oftheJaobianin aNewton-GMRESsolver
Greek letters:
eigenvalue
righthandsideofatime-stepperformulationofthe
dierentialevolutionequation
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Figure 1:FlowsheetoftheTEproess
−10
−8
−6
−4
−2
0
2
4
−10
−5
0
5
10
Eigenvalues of df/dx
−2000
−1500
−1000
−500
0
500
−20
−10
0
10
20
0
50
100
150
0
500
1000
1500
2000
2500
3000
time [h]
measurements [−]
A feed*10000
Recy. flow*100
Re. feed*10
Re. pressure
Re. level*10
Re. temp.*10
Purge rate*1000
Sep. temp.*10
Sep. level*10
Figure 3:Proess measurements1-stablesteadystate
0
50
100
150
0
500
1000
1500
2000
2500
3000
3500
time [h]
measurements [−]
Sep. pressure
Sep. underflow*100
Strip. level*10
Strip. pressure
Strip. underflow*100
Strip. temp.*10
Strip. steam flow*10
Compr. work*10
Re. c. w. temp.*10
Cond. c. w. temp.*10
Figure 4:Pituremeasurements2-stable steadystate
1
2
3
4
5
6
7
8
9
10
11
0
1000
2000
3000
4000
5000
continuation point [−]
measurements [−]
Recy. flow*100
Re. feed*100
Re. pressure
Re. level*10
Re. temp.*10
Purge rate*10000
Sep. temp.*10
1
2
3
4
5
6
7
8
9
10
11
500
1000
1500
2000
2500
3000
3500
continuation point [−]
measurements [−]
Sep. pressure
Sep. underflow*100
Strip. pressure
Strip. underflow*100
Strip. temp.*10
Strip. steam flow*10
Compr. work*10
Re. c. w. temp.*10
Cond. c. w. temp.*10
Figure 6:ContinuationtostablesteadystateofTab.1-measurements 2
1
2
3
4
5
6
7
8
9
10
11
0
5
10
15
20
25
30
35
continuation point [−]
concentrations [mol%]
A
B
C
D
E
F
Figure 7:ContinuationtostablesteadystateofTab.1-reatorfeedonentrations
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
1000 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
500 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
250 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
100 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
50 s
Figure 8:PlotofeigenvaluesoftheJaobian=xoftheRungeKuttatime-stepperatthe
0
50
100
150
200
250
10
−14
10
−12
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
10
2
function evaluations [−]
residual [−]
50 s
100 s
250 s
500 s
1000 s
Figure 9:Convergene historyofformulation1awiththetime-stepper
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
0
2
1000 s
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
0
2
500 s
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
0
2
250 s
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
0
2
100 s
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
0
2
50 s
Figure 10:PlotofeigenvaluesofJaobiand ^
F
1a =dX
1a
atbasease(formulation1awiththe
RungeKutta timestepper)atvarious reportinghorizons t
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
1000 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
500 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
250 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
100 s
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−2
0
2
50 s
Figure 11:Plotofeigenvalues ofJaobian d ^
F
1b =dX
1b
atbasease(formulation1b withthe
−20
−15
−10
−5
0
5
10
15
20
−2
0
2
1000 s
−20
−15
−10
−5
0
5
10
15
20
−2
0
2
500 s
−20
−15
−10
−5
0
5
10
15
20
−2
0
2
250 s
−20
−15
−10
−5
0
5
10
15
20
−2
0
2
100 s
−20
−15
−10
−5
0
5
10
15
20
−2
0
2
50 s
Figure 12:Plot of eigenvalues of Jaobian d ^
F2=dX2 at base ase (formulation 2 with the
Table 1:Manipulatedvariablesandkeyproess measurementsatanopen-loopstable
operat-ing pointoftheTEproess
Manipulatedvariables[%℄
Dfeedowu
1
63.053
Efeedowu
2
53.980
Afeedowu
3
24.644
A+Cfeedowu
4
61.302
Compressorreyleu
5
22.881
Purgevalveu
6
42.322
Separatorunderowu
7
38.256
Produtowu
8
46.421
Strippersteamu
9
47.446
Reatoroolingwateru
10
35.620
Condenseroolingwateru
11
19.620
Agitatorspeedu
12
50.000
Measurements
Reatorpressurey
7
[mbar℄ 2678
Reatorlevely
8
[%℄ 55.9
Reatortemperaturey
9
[℄ 130.7
Stripperunderowy
17 [m
3
tinuationstepwithformulations1a,1band2asafuntionofthereportinghorizon
t
t 1a 1b 2 1a 1b 2
RungeKutta TSalls NR iters.
10s 201 315 406 8 14 11
50s 236 252 408 10 10 10
100s 224 208 367 10 9 9
250s 132 185 365 8 10 9
500s 130 164 361 9 11 9
1000s 107 96 384 10 8 12
2000s 96 102 399 12 14 13
3000s 96 99 416 12 12 15
ImpliitEuler
50s 245 251 321 10 10 8
100s 226 247 407 11 10 10
500s 111 164 374 9 11 10
1000s 98 113 338 9 8 10
5000s 62 85 7 8
Table 3:Totalnumberofallsoftherighthandsideof(1)andCPUtimefortheontinuation
step withformulations1a,1band2asafuntionofthereportinghorizont
t 1a 1b 2 1a 1b 2
RungeKutta 10 3
RHSalls CPUtime
10s 8.7 13.7 17.1 0.6 1.0 1.2
50s 51.2 54.4 85.6 2.4 2.6 4.0
100s 97.6 90.4 154 4.3 4.0 6.7
250s 148 205 383 6.2 8.7 16.0
500s 296 372 758 12.3 15.4 31.3
1000s 508 448 1632 20.9 18.5 67.1
2000s 960 1040 3400 39.3 42.5 139.6
3000s 1440 1476 5352 58.9 60.8 218.1
ImpliitEuler
50s 44.5 44.7 61.8 3.4 3.4 4.8
100s 42.0 49.4 77.5 3.2 3.9 6.0
500s 23.8 34.2 84.8 1.9 2.7 6.8
1000s 22.5 24.2 75.6 1.8 1.9 6.8
121:6 andataprodut owrate thatisinreased by3.5% (y
17
=23:752m 3
=h)
with the set of ontrol andoutput variables used in ontrol struture 2 in Lyman
andGeorgakis(1995)
MV basease y
9
=121:6 y
17
=23:752m 3
=h
1 63.0526 63.6584 64.1466
2 53.9797 54.2347 56.1574
3 24.6436 25.5426 25.0296
4 61.3019 61.6400 63.0798
5 22.2100 18.9699 22.7734
6 40.0637 41.6996 40.2187
7 38.1003 37.6623 39.8558
8 46.5342 46.8157 47.8766
9 47.4457 4.4434 95.6305
10 41.1058 40.8412 42.4343
