Economicplantwidecontrol:Automatedcontrolledvariableselectionforareactor-separator-recycleprocessVladimirosMinasidis,JohannesJaschkeandSigurdSkogestadDepartmentofChemical Engineering,NorwegianUniversityofScienceandTechnology(NTNU),Trondheim,NorwayAbstract:A systematic plantwide control design procedure was proposed in [Skogestad, 2000].The maingoal of this procedure is, to designanoptimal control structure for a completechemical plant based on steady state plant economics, also known as economic plantwide control.Inthiswork, weautomatedakeystepof thisprocedure, whichistheselectionof controlledvariables, basedonquantitative local methods. We appliedthe economic plantwide controldesignproceduretoatypical chemical plantprocess, whichconsistsof areactor, aseparatoranda recycle streamwithpurge. We evaluatedthe economic performance of the designedcontrol structures for various disturbances andfoundthat, althoughtheautomaticselectionofthecontrolledvariableswasbasedonlocalmethods,thecontrolstructuresperformedquitewell,evenforlargedisturbances.Keywords:Economicplantwidecontrol,Exactlocalmethod,Nullspacemethod,self-optimizingcontrol,controlstructureselection1. INTRODUCTIONProduction plants today are facing dicult challenges im-posedbythemodernglobalizedmarkets. Global compe-titiondemandscheaperandmoreexibleproduction, inordertoretainprotabilityandcompetitiveness.Mostofthe industrial process control strategies are designedtoworkatsomenominal operatingconditions, andusually,are not designedtooptimallyhandle frequent (dailyorweekly) changes in market conditions and prices, and thus,to retainanoptimal economic operation. Engell [2007]statesthat, inordertominimizetheoperationcost, inasuch demanding environment of frequent market conditionchanges,anewlookonintegrationofprocesscontrolandprocess operations is needed. Inthesamepaper, Engellprovidesareviewofstateoftheartinintegratedprocessoptimization and control of continuous processes. The twostateof theartapproachesforimplementinganoptimalplant operation presented in detail are: the self-optimizingcontrol [Skogestad, 2000] and direct online optimizing con-trol[MarlinandHrymak,1997].Forthedirectonlineoptimizingcontrol themainideais,tocalculatetheoptimal inputtrajectoryoversomecon-trol horizonbyoptimizingarigorousnonlineardynamicmodel of aplant over somepredictionhorizon. For self-optimizing control the idea is to turnthe optimizationintoasetpointcontrol problemandthetricktodothatistondsomeself-optimizingvariablestocontrol.Thesemagic variables are dened by Skogestad [2004b] as: thecontrolledvariables,that,whenkeptconstantatnominal

This work was supportedby the NorwegianResearchCouncil.Correspondingauthor:SigurdSkogestadskoge@.ntnu.nooptimal values, using the available degrees of freedom,indirectly result in a close-to-optimal operation despite theoccurrenceofdisturbances.Since the direct online optimization approach guaranties anear-optimal operation, it has gain a lot of attention fromprocessindustry, buthasfailedtobeadaptedwidelybytheindustry,becauseitistoocomplicatedandexpensiveinmanycases.DownsandSkogestad[2011]mentionthattheusualindustrialpracticeistofocusonunitoperationcontrol, mainlybecausethis is asimplestrategythat iseasilyunderstoodbytheoperators andengineers. Thusaplantwide control designprocedure has tohave threeessential characteristics in order to be applied by theprocess industry engineers: (1) it has to be simple (withoutthe need for complex control technology, like real-timeoptimization), (2) it has to be able to achieve near-optimaloperation, and(3) it shouldnot require the care andfeedingofexperts.Theeconomicplantwidecontrol designprocedure, asde-scribedinSkogestad[2012],canbeusedtodesigncontrolstructuresthathavethersttwoofthekeycharacteristicmentioned above and the third characteristic could berealizedbyautomatingthis procedure. The hierarchicaldecompositionof theeconomicplantwidecontrol designinto a stepwise top-down and bottom up procedures couldprovide the basic framework for automating the entiredesignprocedure, whichis the ulterior goal behindtheworkpresentedinthispaper. Akeystepinautomatingtheentireprocedureis toautomatetheselectionof thecontrolled variables for the economic layer. Our mainfocus here, is to showthat this can done by selectingPreprints of the 10th IFAC International Symposium on Dynamics and Control of Process SystemsThe International Federation of Automatic ControlDecember 18-20, 2013. Mumbai, IndiaCopyright 2013 IFAC 87OptimizerFeedbackProcess(G, Gd)Measurementcsudynyymncc +nH(RTO)controller combinationFig. 1. Feedback implementation of optimal operation withseparatelayersforoptimization(RTO)andcontrol.[Alstadetal.,2009]self-optimizingcontrolledvariablesbasedthequantitativelocalmethodsdescribedin[Alstadetal.,2009].In this work we apply the economic plantwide controldesignproceduretoatypicalchemicalplant,emphasizingthe automatic selection of the self-optimizing control struc-ture, usingquantitativelocal methods. Weevaluatetheeconomic performance of the selectedcontrol structuresand compare them against a simple strategy of keeping theunused degrees of freedom at their nominal optimal values.Asatypicalchemicalplant,weconsideraprocess,whichconsists of a reactor, a separator and a recycle stream withpurge.This paper is structured as follows: Section 2 presentsabrief summaryof the top-downpart of the economicplantwide control design procedure. In Section 3, wepresentamathematical formulationofoptimal operationandthe mainideas behindthe derivationof Nullspacemethod and the Exact local method. We describe thereactor-separator-recycle (RSR) process model and thesoftwareusedforthisworkinSection4. Astepbystepapplication of the top-down part of the economic plantwidecontrol procedurefortheRSRprocesswithemphasisonthe automatic selection of self-optimizing CVs is presentedinSection5, andinSection6weevaluatetheeconomicperformance of these self-optimizing control structures,discusstheresultsandconcludethepaper.2. ECONOMICPLANTWIDECONTROLDESIGNPROCEDUREThissectionpresentsbriefsummaryofthetop-downpartof the economic plantwide control design procedure asdescribedinSkogestad[2004a]andSkogestad[2012].Thenomenclaturefortheinputs, disturbances, measurementsandnoise,usedinthissection,isdepictedinFig.1.Step 1: Dene the operational objectives (eco-nomics)andconstraints.First, operational objectives aredenedas ascalar costfunctionJandtheoperational constraintsareidentiedandformulated.Step2: Determinethesteady-stateoptimal opera-tionInthisstep, rst anoperational modehastobechosenbeforeproceedingwiththeanalysis: Mode1: Given throughput (maximize the eciency).This mode corresponds to some tradeo betweenvaluable product recovery and minimal energy usage. Mode2: Maximumthroughput (maximizethepro-duction). When the product prices are high comparedto energy andrawproduct prices it is optimal toincreasetheproductiontomaximum.(a) IdentifythesteadystatedegreesoffreedomIdentify all the dynamic DOFs uD. They corre-spondtotheactualmanipulatedvariables.Then,ex-cludeall theDOFs, thateitherdonthaveanyeectonthecostfunctionJorareusedtocontroloutputsthathavenoeectonthecost.TheremainingDOFsarethesteadystateDOFsuSS.(b) IdentifytheimportantdisturbancesandtheirexpectedrangeHere, the important disturbances are identied. Theimportanceof adisturbanceisproportional tothesensitivityof the cost functiontothat disturbance.Typically the important disturbances are: the feedows,feedcompositionandback-ofrom constraints.(c) IdentifytheactiveconstraintsregionsHere, the active constraints regions for the expecteddisturbance range are identied. Astraightforwardapproach, for mapping these regions, is to optimize theprocess over anegridof points inthedisturbancespace, thusdeterminingwhichconstraintsareactiveat everypoint. Amore resourceful approach, whichtrackstheactiveconstraintboundariesisreportedbyJacobsenandSkogestad[2011].Step3: Selectprimary(economic)controlledvari-ables.EverysteadystateDOFidentiedinthestep2(a)needstobepairedwithaprimarycontrolledvariable.First,pairthemwiththeactiveconstraints,whichcanbeconsideredastheobviousself-optimizingvariables, sincekeepingthematthenominaloptimalvalues,resultsinanoptimaloperation.Theactiveconstraintscouldbeinputsuoroutputsy.While,theimplementationfortheactiveinputs constraints is trivial (e.g. valve fully open or closed),for output constraints that cannot be violated, specialcare,intermsofsafetymargins(back-o),isrequired.Then, select self-optimizing controlledvariables for theremainingDOFs.Themainstepsinvolvingtheselectionofthosevariablesare:(a) IdentifythecandidatemeasurementsIdentify all the candidate measurements yand esti-matetheexpectedstaticmeasurementerrorny.Themeasurements should include the inputs too u(e.g. theowratesmeasurements).(b) Selecttheprimary(economic)controlledvari-ables(CVs)fortheremainingDOFsAsprimaryCVscwemayselectasingleorcombi-nation of measurements, based on the structure of theHmatrix, thatis: c=Hy. Theselectionisbasedon:IFAC DYCOPS 2013December 18-20, 2013. Mumbai, India88(I) Qualitative approach (based on the followingSkogestadsheuristicrules)(1) The optimal value of a CV should be insen-sitivetodisturbances.(2) The CVshould be easy to measure andcontrol.(3) The CV should be sensitive to manipulatedvariable(MV)variations.(4) For cases with two or more CVs, theyshouldbenotcloselycorrelated.(II) Quantitativeapproach(1) BruteforceapproachGridtheexpecteddisturbancespaceandevaluate the cost function Jat each point ofthegrid, includingestimatedmeasurementnoise,whilekeepingasubset(ofnusize)ofcandidateCVssetattheirnominalvalues.Chooseforpairingthesubsetof CVsthatgivesthelowestcost.(2) LocalapproachesLocal approachesarebasedontheTay-lor series expansion of the cost functionaroundthe optimal nominal point. Theseare: the Maximum gain rule [Skogestad andPostlethwaite, 2005], the Nullspace methodandtheExactlocal method[Alstadetal.,2009].Thelasttwolocal methodsaredescribedinthesection3.Step4: Selectthelocationofthroughputmanipu-lator(TPM)The TPM or the process gas pedal is usually a ow in theprocessthatisset. TherearetwokeyconcernsregardingthelocationoftheTPM:(a) EconomicsThe locationof the TPMis going to aect howtightsomeactiveconstraintscanbecontrolled, thusaectingtheeconomicloss.(b) StructureoftheregulatorycontrolsystemThe location of the TPM has a signicant impact onthe regulatory control structure, because of the radia-tionrule[PriceandGeorgakis,1993],whichlinksthetop-downandthebottom-uppartsofthisprocedure.We do not consider the bottom-up part of these procedure,becauseitisoutof thescopeof thispaper. Wereferthereaderto[Skogestad,2012].3. LOCALQUANTITATIVEMETHODSHere, we focus onstep3(b) of the procedure describedin the previous section. We present the mathematicalformulation of the optimal operation and the lay down theideas behind the two local methods, the Nullspace methodandtheExactlocal method. Theworkpresentedhereisfollows[Alstadetal., 2009], unlessotherwisestated. Thenomenclature used in this section, if not explicitly dened,isdepictedintheFig.1.Optimal operation of the plant, with respect to the steadystateDOFsu,canbedenedas:minimizeuJ(u, d)subjectto ci(u, d) 0.(1)where: the ci(u, d) are the operational constraints. Weassume that the u here includes only the remaining steadystateDOFsandthatthedincludestheparametervaria-tionstoo.For a given d, the solution of the problem 1 gives the opti-mal value for the cost function Jopt(d) , the optimal inputvaluesuopt(d)andtheoptimaloutputvaluesyopt(d).We dene the loss as the dierence between the cost usingnonoptimalinputsuandtheoptimalcostJopt(d):L = J(u, d) Jopt(d) (2)The linearized (local) model in terms of deviation variablesisformulatedas:y=Gyu +Gydd (3)c = HGyu +HGydd (4)Wedenethescaleddisturbancesd

andscaledmeasure-mentsnoiseny

as:d = Wdd

(5)ny= Wnyny

(6)Theself-optimizingcontrol canbedescribed, intermsofthe variables denedinthis section, as the selectionofoptimalHinc = Hy,suchthat,ifckeptatitsnominaloptimalvaluescs(constantsetpointpolicy),itresultsinminimal or acceptable loss [Yelchuru and Skogestad, 2011].NullspacemethodThe derivationof this methodis quite straightforward,details canbe foundin[Alstadet al., 2009]. Assumingthat thereis nomeasurement noiseny=0, anoptimalcombinationofmeasurementscanbeformulatedas:copt= Hyopt(7)yoptcanbewrittenas:yopt= Fd (8)Combiningtheequations(7)and(8)results:copt= HFd (9)where: F=yoptd= (GyJ1uuJudGyd)istheoptimalsensitivity matrix. For practical purposes its easier tocalculate the F byreoptimizingnon-linear steadystateplantmodelforsmalldisturbancevariations.Anoptimaloperationbasedonconstantsetpolicymeanscopt= 0foranyd = 0.Theorem1.(Nullspacemethod). [Alstad et al., 2009] Ifthenumber of measurements nyis equal or larger thanthenumberofinputsnuplusthenumberofdisturbancesnd, that is ny nu+ndand Fis evaluated with constantactive constraint set, then it is possible to select the matrixHasabasisforthenull spaceof F, H N(FT), suchthat,HF= 0WhichmeansthatanyHsuchthat, HF=0resultsinanoptimaloperation.IFAC DYCOPS 2013December 18-20, 2013. Mumbai, India89Exactlocal methodTo minimize the average and the worst case loss for the ex-pectednoiseanddisturbances,Alstadetal.[2009]formu-lated the the problem for nding an optimal measurementcombinationasfollows:H= arg minH

J12uu(HGy)1HY

(10)where: Y =[FWdWn], the denotes 2-normfor theworst case scenarioandFrobenius normfor minimizingtheaverageloss.Inthesamepaper theanalytical solutionfor Frobeniusnormcaseoftheproblem(10)isderived:HT= (Y YT)1Gy(GyT(Y YT)1Gy)1J1/2uu(11)Kariwalaet al. [2008] showedthat the HobtainedforFrobenius normis insomesensesuper optimal. It min-imizes both worst case loss and the average case loss.Yelchuru and Skogestad [2011] based on the analyticalsolution(11)derivedandprovedthefollowingtheorem:Theorem2.(Simpliedanalyticalsolution). [Yelchuru andSkogestad, 2011] Another analytical solution to problem in(10)isHT= (Y YT)1GyQ (12)where:Qisanynon-singularmatrixofncncThis method is recommended when the measurment noiseisnotnegligible.4. PROCESSDESCRIPTIONIn this section we describe the details of the reactor-separator-recycle process used as a case study in this work.4.1 Processmodel descriptionAmodel of agenericchemical plant, whichconsistsof areactor, aseparator andarecyclestreamwithpurge, isusedasacasestudy.Thespecicprocesswaschosenbe-cause it incorporates the basic structure of many chemicalplantsandbecauseithasbeenstudiedextensivelyintheprocesscontrol literature[Larssonetal., 2003],[JacobsenandSkogestad, 2011],[WuandYu, 1996]. Inthis work,weusedthesameprocessparametersandreactionsetas[JacobsenandSkogestad,2011],sothereadercanrefertothatpaperforanydetailsomittedhere.TheprocessowdiagramisillustratedinFig.2.AfreshfeedF0of rawproduct AandrecyclestreamRarefedintoacontinuouslystirredtankreactor(CSTR).TwoparallelreactionstakeplaceintheCSTR:A B (13a)A 2 C (13b)where: Bisthedesiredproductand Cabyproduct. Thereaction rates are modelled as rst-order kinetics. Thereactionratemodelisgivenby:ri= Ai exp

E,iRT

(14)where: E,iis the activation energy, R is the gas constantandT is the temperature. The reactionparameters aregiveninTable1.Table1.ReactionkineticsparametersReaction, Reactionrateconstant, Activationenergy,i Ai,[units1] E,i, [J/mol]A B 1 1056 104A 2 C 5 1068 104MDMBMRTRSFITIFCPCLCLCLCCCPICIF0VLBDXDXBXFFRPLCFFig. 2. Process owdiagramfor the reactor-separator-recycleprocessTable2.DistillationcolumnparametersParameter Value Parameter ValueAC0.70 numberofstages 30BC0.60 feedstagelocation 15The euent F of the reactor is sent tothe distillationcolumn, wheretheproduct Bis separatedas abottomproductandtheunreactedAandCaredistilled.Partofthe distillate Dis removed as Ppurge, while the rest of itRisrecycledbacktotheCSTR.The column model used here is an ideal multicompo-nentmodel basedontheColumnAin[SkogestadandPostlethwaite, 2005]. SomeofthecolumnparametersaregiveninTable2.4.2 ProcessoptimizationandsimulationForthisinvestigationwedevelopedasteadystateMAT-LABmodel and a dynamic SIMULINKmodel of thedescribed above process. For the optimization needsthebuiltinMATLABR non-linear optimizationfunction(fmincon) was used. For simulationpurposes we usedSIMULINKR withode15ssolver.5. CASESTUDYThissectionpresentstheimplementationofthetop-downpartof theeconomicplantwidecontrol designprocedureasdescribedin[Skogestad, 2004a] and[Skogestad, 2012].ThenomenclatureusedinthissectionisdepictedinFig.2.IFAC DYCOPS 2013December 18-20, 2013. Mumbai, India90Step 1: Dene the operational objectives (eco-nomics)andconstraints.Cost function: The cost function for this process isdenedasfollows:J= pFF0 +pVV pPP pB Bwhere: F0isthereactorfeedowrate, V isthecolumnboilup, Bisthebottomproduct owrateandPisthepurge ow rate. The pF, pV , pBand pPare their respectiveprices.Operational constraints: The operational constraintsare the following: the product specications andequip-mentlimitations.xB,B 0.9 TR 390 [K]MR 11000 [mol] V 30 [mol/s]R 0 [mol/s] P 0 [mol/s]where: xB,Bis the composition of B in the bottom productstream B, TR is the reactor temperature, MR is the reactorholdup,Rrecyclestream,Ppurgestream.Step2: Determinethesteady-stateoptimal opera-tionForthisprocessweassumedthatthefeedisgiven, thustheoperationalmodeisMode1:Giventhroughput.(a) Identify the optimizationdegrees of freedomanalysisDynamicdegreesfreedom:uD= [L, V, D, B, F, R, TR]Steadystatedegreesfreedom:uSS= [L, V, F, R, TR]DandBareusedtocontrol thelevelsMDandMBrespectively,whichhavenosteadystateeectonthecost but have to be controlled in order to stabilize theplant.(b) IdentifytheimportantdisturbancesandtheirexpectedrangeWeconsiderasmaindisturbanceforthiscasethefeedow,F0andtheenergyprice,pVuctuations:d = [F0, pV ](c) IdentifytheactiveconstraintsregionsTomapthe active constraints regions the distur-bance space was griddedinto200 200points andforeachpointoptimalsolutionwasfound.TheactiveconstraintsregionsareillustratedinFig.3.Step3: Selectprimary(economic)controlledvari-ables.We assume that the process operates in the rst region (I).Thenominaloperationalpointis:F0,nom= 0.8 [kmol/s], pV,nom= 0.06 [$/mol]Wechoosethefollowingpairings (Input,Output) for theactive constraints basedonSkogestads rule [Skogestad,2012]pairclose.(V, xB,B) (S, TR) (F, MR) (R, Closed valve)The control structure of the selected pairings are depictedinFig2.(a) IdentifythecandidatemeasurementsFig. 3. Active constraints regions for the reactor-separator-recycleprocessWe assume that the following measurements areavailable and estimate their expected noise magni-tudes:columnandreactortemperatures(noise 1 [K])T1, T5, T9, T13,T17, T21, T25, T30, TR,theinputows (noise 10 %)L, V, D, B, F, F0, Preactorlevel (noise 100 [mol])MRcompositions (noise 0.01)xB,D, xB,B, xB,F.(b) SelecttheprimarycontrolledvariablesfortheremainingDOFsAfterpairingtheactiveconstraintsandlevels,onlyone steady-state DOFis available, andthat is thereuxow, L. Weselectaself-optimizingcontrolledvariable to pair with L, based on the two quantitativemethods,describedinsection3.First, we calculate the optimal sensitivities F byperturbing the disturbances and re-optimizing theprocessfortheperturbedvalues:F=yoptdThen,wecalculatethegainmatrixGybyperturbingthe inputs andthensimulatingthe process until itreachesthenewsteadystate:Gy=yuTo select an H,based on the Nullspace method, weselectabasisfromtheleftnullspaceofFT.For the selectionof Hbasedonthe Exact localmethod we use the equation (12), where we set Q = I.The selection of the self-optimizing control structureforanynominal pointhasbeenautomatedbasedontheproceduredescribedinthisstep.Step4: Selectthelocationofthroughputmanipu-latorForagivenfeedtheTPMisalreadysetbythefeedowF0. Thelevel control isselectedtobeinthedirectionoftheowdownstreamofthelocationofthefeed.Formoredetailswereferthereaderto[AskeandSkogestad,2009]IFAC DYCOPS 2013December 18-20, 2013. Mumbai, India9120 0 20 4001020304050(a)Feedowratechange,F0[%]Relativeloss,LRel[%]LossLkeptconstantNullspacemethodCVsExactlocalmethodCVsLkeptconstant+noiseNullspacemethodCVs+noiseExactlocalmethodCVs+noise20 0 20 4001020304050(b)Energypriceandfeedowchange,[%]Relativeloss,LRel[%]Fig. 4. Evaluationoftheperformanceofdierentcontrolstructuresforvariousdisturbance6. RESULTSANDCONCLUSIONSHere we evaluate the economic performance of the self-optimizing control structure against the simplest strategy,that is, tokeepthe unconstraineddegree of freedomLatitsnominaloptimalvalue.Wedisturbtheprocessandevaluate the relative loss LRelfor various disturbances.WheretheLRelisdenedasfollows:LRel=J(u, d) Jopt(d)Jopt(d)(15)Theeconomicperformances of all thecontrol structuresfordierentdisturbancevaluesaredepictedinFig.4.First, we evaluate the relative loss for feed ow changes F0forthecontrolstructureswithandwithoutmeasurementnoise(see Fig. 4(a)). Then, we evaluate the relative lossforsimultaneouspVandF0changesforthesamecontrolstructures(seeFig.4(b)).It is clear that keepingtheselectedself-optimizingcon-trolledvariableconstantoutperformskeepingtheLcon-stant inbothof thecases. Thereforethesystematicse-lectionof the controlledvariables basedonexact localmethodscouldbeconsideredasasuccessful approachtoautomate this essential stepof the economic plantwidecontrolprocedureTheautomationoftheeconomicplantwidedesignproce-dure,especiallythe integrationofthe automatic designinthe major process simulators, couldpotentiallyimprovetheoptimalityoftheproductionplantsonaglobalscale,thus having a large eect onthe productioncosts andenvironmentalfootprint.REFERENCESVidar Alstad, Sigurd Skogestad, and Eduardo S. Hori.Optimal measurementcombinationsascontrolledvari-ables. Journal of ProcessControl, 19(1):138148, Jan-uary2009. ISSN09591524.ElviraMarieB. AskeandSigurdSkogestad. ConsistentInventoryControl. 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