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Robust control solutions for stabilizing flow from the reservoir: S-Riser experiments
Mahnaz Esmaeilpour Abardeh
Chemical Engineering
Supervisor: Sigurd Skogestad, IKPCo-supervisor: Ole Jørgen Nydal, EPT
Esmaeil Jahanshahi, IKP
Department of Chemical Engineering
Submission date: June 2013
Norwegian University of Science and Technology
Robustcontrolsolutionsforstabilizingflowfromthereservoir:S‐Riserexperimentsandsimulations
MahnazEsmaeilpourAbardeh
June26,2013
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Preface
ThisthesisiswrittenasthefinalpartofmyMasterdegreeinChemicalEngineeringattheNorwegianUniversityofScienceandTechnology(NTNU),classof2013.
I would like to express my greatest gratitude to my highly knowledgeablesupervisor, professor Sigurd Skogestad, for all his helps, his good guidance andencouragements. I am also grateful tomy co‐supervisors, professor Ole JorgenNydaland PhD student Esmaeil Jahanshahi, who helped and supportedme throughout mythesis. It has been a great opportunity and honor forme to be part of your team ingeneratingnewideasandIamconfidentwhatIhavelearnedthroughthisthesiswillbesurelyusedinpracticeinmyprofessionalcareer.
DeclarationofCompliance
I,MahnazEsmaeilpourAbardeh,herebydeclare that this is an independentworkaccording to the exam regulations of the Norwegian University of Science andTechnology(NTNU).
Dateandsignature:
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Abstract
One of the best suggested solutions for prevention of severe‐slugging flowconditionsatoffshoreoilfieldsistheactivecontroloftheproductionchokevalve.Thisthesis is a studyof robust control solutions for stabilizingmultiphase flow inside therisersystems;throughS‐riserexperimentsandOLGAsimulations.“Nonlinearity”astheimportantcharacteristicofsluggingsystemposessomechallengesforcontrol.Focusofthisthesis isononlinetuningrulesthat takeintoaccountnonlinearityofthesluggingsystem.Themainobjectivehasbeentoincreasethestabilityofrisersystemsathigherlevelsofvalveopeningswithmoreproductionrates.
Similarresearchhasbeendonepreviously,butisrepeatedinthisthesisusingnewsystematic tuningmethods.Threedifferent tuningmethodshavebeen applied in thisthesis. One is Shams’s set‐point overshoot method developed by Shamsozzhoha(ShamsuzzohaandSkogestad2010).TheotherisIMC‐(InternalModelControl)basedtuningmethodwithrespecttotheidentifiedmodelofthesystemfromclosed‐loopsteptest.ThelasttuningmethodissimplePItuningruleswithgainschedulingforthewholeoperatingrangeof thesystemconsidering thenonlinearityof thestaticgain.The twolatter methods have been developed very recently by Jahanshahi and Skogestad(JahanshahiandSkogestad2013).
Twoseriesofexperimentshavebeencarriedoutusingamedium‐scaletwo‐phaseflow S‐riser loop. A single loop control scheme with riser‐base pressure as themeasurementwasused.Therobustnessofdifferenttuningmethodswascomparedbyslowlydecreasingtheset‐pointoftheclosed‐loopsystem,whichwastheinletpressure,until instability was reached. The choke valve opening was increasing gradually bydecreasing the set‐point.A controlwith a robust tuningmethod canmaintain systemstability in a large range of conditions. The choke valve was then replaced with aquicker valve after the first set of experiments. The sameexperimentswere repeatedandtheeffectofcontrolvalvedynamicswasinvestigatedthereafter.
The experiments were simulated in OLGA and the same control tests wereperformed.TheOLGAcasewasconstructedbasedonthefirstseriesoftestswithvalve1andthedesignedcontrollerswithdifferenttuningstrategieswereapplied.Resultsoftheexperimentsverifiedthoseofthesimulations.
The tuning method with the highest robustness was thus the one which couldstabilizethesystematthelargestchokevalveopening(thelowestinletpressure).Thebesttuningmethod,withrespecttorobustnessisthesimplePItuningruleswithgainscheduling for the whole operating range of the system. With this method, it waspossibletostabilizetheexperimentalrisersystemuptoachokevalveopeningof37%from an open‐loop stability of 16%. It was also able to stabilize the simulated risersystemuntilachokevalveopeningof75%fromanopen‐loopstabilityof26%.
Top side measurements were in general difficult to use in anti‐slug control.Measurement of the topside density using a conductance probe installation was notsuccessful.Therefore,nocascadeanti‐slugcontrolschemescouldbetested.
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iv
Contents
Preface.....................................................................................................................................................i
Abstract.................................................................................................................................................ii
1 Introduction................................................................................................................................1
1.1 Scopeofthethesis...........................................................................................................................2
2 Background.................................................................................................................................3
2.1 Multiphasetransport.....................................................................................................................3
2.2 Slugflow..............................................................................................................................................5
2.3 Riserscontainingmultiphaseflow...........................................................................................6
2.4 Riserslugging....................................................................................................................................7
2.5 Anti‐slugoperations.....................................................................................................................11
2.5.1 Choking.....................................................................................................................................11
2.5.2 Gaslift........................................................................................................................................11
2.5.3 Slugcatchers...........................................................................................................................12
2.5.4 Activecontrol.........................................................................................................................12
2.6 Modelingofrisersystems..........................................................................................................13
2.7 Bifurcationdiagrams....................................................................................................................13
2.8 PIDandPIcontrollers..................................................................................................................14
2.9 TuningofPIDandPIcontrollers.............................................................................................15
2.9.1 Method1:Shams’sset‐pointovershootmethodforclosed‐loopsystems..15
2.9.2 Method2:TuningbasedonIMCdesign......................................................................18
2.9.3 Method3:SimpleonlinePItuningmethodwithgainscheduling...................22
3 Experimentalwork................................................................................................................26
3.1 SetupDescription..........................................................................................................................27
3.2 Equipment........................................................................................................................................30
3.2.1 Mainwaterstoragetank....................................................................................................30
3.2.2 Airreservoirtank.................................................................................................................31
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3.2.3 Airbuffertank........................................................................................................................32
3.2.4 Overflowtank.........................................................................................................................33
3.2.5 Pressuretransmitters.........................................................................................................33
3.2.6 Smallseparator......................................................................................................................34
3.2.7 Centrifugalwaterpump.....................................................................................................34
3.2.8 Airflowmeter........................................................................................................................35
3.2.9 Waterflowmeter..................................................................................................................35
3.2.10 Chokevalves.......................................................................................................................37
3.2.11 Conductanceprobe(C)..................................................................................................38
3.2.12 LabVIEW..............................................................................................................................39
4 Simulationofexperimentalcases....................................................................................41
4.1 OLGA®,multiphasesimulationtool.......................................................................................41
4.2 Constructionofthecase.............................................................................................................41
4.2.1 Flowpathgeometry.............................................................................................................42
4.2.2 Fluidproperties.....................................................................................................................43
4.2.3 Boundaryandinitialconditions.....................................................................................43
4.2.4 Numericalsetting.................................................................................................................44
4.3 ImplementingPIDcontrollerinOLGA.................................................................................44
5 Resultsanddiscussion.........................................................................................................46
5.1 Experimentalresults....................................................................................................................46
5.1.1 Seriesofexperimentswithvalve1(slowchokevalve)........................................47
5.1.2 Seriesofexperimentswithvalve2(fastchokevalve).........................................57
5.1.3 CascadeControlusingtoppressurecombinedwithdensity............................68
5.2 ComparisonofSlowvalveandFastvalve...........................................................................70
5.3 SimulatedresultsfromOLGAmodel.....................................................................................74
5.3.1 Open‐loopsimulations.......................................................................................................74
5.3.2 Controlbytrialanderror..................................................................................................75
5.3.3 Tuningthecontroller..........................................................................................................79
5.4 Comparisonofexperimentalandsimulatedresults......................................................95
5.4.1 Open‐loopbifurcationdiagrams....................................................................................95
5.4.2 ComparisonofcontrolresultsfromIMC‐basedtuningmethod......................96
5.4.3 ComparisonofcontrolresultsfromSimpleonlinetuningmethod................96
5.4.4 Comparisonoftuningmethods......................................................................................97
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6 Discussionandfurtherworks........................................................................................100
6.1 Tuningmethods..........................................................................................................................100
6.2 Controlstructures......................................................................................................................101
6.3 Discussableissuesrelatedtoexperimentalactivities................................................102
6.3.1 Oscillationsinflowrates................................................................................................102
6.3.2 Waterflowbackintothebuffertank........................................................................102
6.3.3 Leakageinsteelconnection..........................................................................................102
7 Conclusion..............................................................................................................................104
7.1 Stabilizingcontrolexperimentsusingbottompressure...........................................104
7.2 TestingonlinetuningrulesonS‐riserexperiments....................................................104
7.3 Controlusingtoppressurecombinedwithdensity....................................................105
7.4 Investigatingeffectofcontrolvalvedynamics..............................................................105
7.5 ControlsimulationsusingOLGA..........................................................................................106
8 References..............................................................................................................................107
A.LowpassfilterinLabVIEW..............................................................................................109
B. SimulatedresultstogetthebeststeptestsforShams’smethod.....................110
C. SomeexamplesofMATLABscripts..............................................................................111
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1
1 Introduction
MultiphasepipelinesareacommonfeatureofoffshoreproductionintheNorthSea.Theyconnectsubseawellstothetopsideprocessingfacilitiesortheplatforms.Inmanypointsof transportation, thesepipelinesget theshapeofL‐shapedorS‐shapedrisers.The stability of multiphase flow inside these pipeline‐riser systems is of greatimportance andmany efforts have been spent on this issue so far. In low reservoirpressuresorlowflowrateconditionstheliquidphasestendtoaccumulateinlowpointsand form liquid slugs. This leads to the pipeline or riser blockage and can be moredangerous when the length of slugs is comparable to the length of the riser. Thisphenomenon iscalledSevereslugging(alsoTerrainsluggingorRiserslugging)and ischaracterizedbylargeoscillatoryvariationsinpressureandflowrates(Storkaas2005).These large variations lead to a poor separation, unwanted flaring and even a plantshutdownintheworstcase.
Reducingopeningofthetopsidechokevalvehasbeenatraditionalwaytosuppresssevere slugging. However, this increases the valve back pressure and thereforedecreasestheproductionratefromthewell.
Activefeedbackcontrolofthetopsidechokevalvecanmakeitpossibletostabilizetheflowattheconditionswherenormallyseveresluggingispredicted.Thisreducestheneed for additional topside equipment and allows a higher rate of oil recovery. Thecontrol system is called anti‐slug control and its main objective is to keep themultiphaseflowasstableaspossiblebymanipulatingthetopsidechokevalveusingtheparameterssuchaspressureordensityasthecontrolvariables.
In the way of developing new technologies for stabilizing control of severeslugginginrisersystemsmanyresearcheshavebeendoneattheNorwegianUniversityofScienceandTechnology.TheworkhasbeenguidedbySkogestad(Skogestad2003;Storkaas 2005; Shamsuzzoha and Skogestad 2010; Jahanshahi and Skogestad 2011;SkogestadandGrimholt2011; Jahanshahi andSkogestad2013)andperformedat thedepartment of Chemical Engineering. Storkaas (Storkaas 2005), Sivertsen (Sivertsen2008), Jahanshahi (Jahanshahi and Skogestad 2011) and numerous master studentshaveworkedonmodelingandcontrollingofrisersystems.
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Companies like ABB (Havre, Stornes et al. 2000), Statoil and Total have allresearched prevention of slugging and built installations at offshore locations. Statoilcompleted in 2001 their first slug control installation at the Heidrun oil platform.Siemens is also involved in slugging research and funds a PhD program, which thisthesisisconnectto.
Intheanti‐slugcontrolsystem,it isveryimportantthatthecontrollersarefinetuned. Otherwise, the control system is not robust in practice and the closed‐loopsystembecomesunstableafteraplantchange.Thesluggingsystemishighlynonlinearsincethegainchangesatdifferentoperatingpoints.Forsuchasystemthecontrollersneedtoberetunedateachoperatingpoint.
1.1 Scopeofthethesis
Inthisthesisthreedifferenttuningmethodswillbetestedwithexperimentsandsimulations to find the most robust solution for anti‐slug control system. Highrobustnesswillbeobtainedifthesystemcanmaintainstabilityatlargedeviationsfromopenloopconditions.Thismeanslargechokevalveopenings.Thetuningmethodsaresystematicandhavebeendevelopedveryrecently(ShamsuzzohaandSkogestad2010;JahanshahiandSkogestad2013).
TheexperimentsofthisthesiswillbecarriedoutatthedepartmentofEnergyandProcessEngineering.Twoseriesofexperimentswillberunusingamedium‐scaletwo‐phase flow S‐riser loop. The difference between the two series is the type of chokevalve.Theaimistoinvestigatetheeffectofcontrolvalvedynamicsonperformanceofthe control system in addition to robustness of the tuning methods. Possibility ofdifferentcontrolstructureswillbealsoinvestigated.
The experimentswill be simulated inmultiphase flow simulator,OLGA, and thesame control testswill be performed. Finally the simulated and experimental resultswillbecompared.
3
2 Background
2.1 Multiphasetransport
Whenitcomestooffshoreproductionofoilandgas,longtransportofmultiphaseflowhasrecentlybecomeofgreatattention.Manypipelinesandrisersarecarryingthecombination of natural gas, condensate, oil and water from the North Sea to shore.Previously, large production platforms equippedwith process facilitieswere built onthe sea floor with the aim of separating gas, oil and water. Today this can be tooexpensive and multiphase transportation can save billions of dollars for the oilcompaniesinstead.
Design and operation of multiphase transportation systems raise many newchallenges. These challenges could be either related to the flow, fluid or the pipeintegrity. Pressure drop/ boosting, Slugging, liquid emulsion, temperature change,scaling, hydrate and wax formation can be examples of them. Overcoming thesechallengesandhavingasafeanduninterruptedmultiphaseflowreferstotheterm“flowassurance”.This termwas firstusedbyPetrobras in theearly1990sand itoriginallyreferred to only thermal hydraulics and production chemistry issues encounteredduringoilandgasproduction(Fabre,Peressonetal.1990).
One important issue in flowassurance is stabilizing themultiphase flow insidethe pipeline‐riser systems. From a control engineering point of view, this can bereferred as control of the disturbances in the multiphase flow as the feed to theseparationprocess.Avoidingvariationsintheflowenteringtheprocessingunit,attheoutlet of themultiphasepipelines is the issueof interest for control (Bratland2010).The ability of predicting the flow patterns and reserving a stable flow is of greatimportance, which is the objective of the thesis. Figures 2.1 and 2.2, adapted fromBratland (Bratland 2010), describe possible flow patterns inside the horizontal andverticalpipelines.
4
Figure2.1:Gas‐liquidflowregimesinhorizontalpipes.
Figure2.2:Gas‐liquidflowregimesinverticalpipes.Slugflowisthepointofinterestinthethesis.
Changingmultiphaseflowbetweendifferentflowregimescanbedescribedbyatypical flow regimemap shown in figure 2.3, adapted from Taitel (Taitel 1986). Theboundariesbetweenstableandunstableregionsareclearlyshownin the flowregimemap.Withapplying feedbackcontrol theseboundariescanbemovedand thereby thestableregioncanbeincreased.
Figureclearly
2.2
AdeliverSlugfloflowing“slugs”differenor pipecommoshownslug floproducinhomoproducequiliboscillat
2.3: Stabilshownint
Slu
Among thrability hasowisoneoginthepipofliquid,tntvelocitieeline geomon in risersinfigure2ow is its iction facilitogeneous dction as larium prodtions can
lity map fothemap.
ugflow
he flow ass receivedmoftheflowppes.Inthistravelfromesofgasanmetrywhichs and itsm2.4,adaptedinstabilityties. The pdistributionarge as 50duction andamage th
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ssurance cmuch interpatternschtypeofflo
moneendondliquidph is referremain reasondfrom(Yanthat has aperiodic osn cause o0%. The and this gihe pipe an
5
hase flow (
concerns,rest in receharacterizeowregime,ofthepipethasewhiched as terran is the grnandChe2a negativecillations oscillatory pverage ofives the pnd the se
(Taitel 198
managemeent years (edbyalternelongatedtotheothehisreferreain inducedavity.A sc2011).Theimpact onof liquid apressuresthese oscproductionparation p
86). Stabilit
ent of slu(Godhavn,natingslugbubblesoferend.Itcaedashydrod slugging.hematicmmasterunn the operand gas phand decrecillations isn losses. Mprocess. Fo
ty boundar
ugging inFard et al.sofgasanfgasseparanbeeitherodynamics The latter
mapof slugfavorableeration of ohases due teases the ls lower thMore overor these r
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system. 2005).dliquidratedbyrduetosluggingr one isflow iseffectofoffshoreto theirlevel ofhan ther thesereasons,
6
suppressingtheslugflowisofdominantimportance.Ahomogeneoussteadyflowwithvery small bubbles of gas well distributed in the continuous liquid phase is mostdesired.Insuchdesiredsituation,thepressureremainsconstantovertime.
Figure 2.4:Schematicmap of slug flow in a vertical pipe in a slug unit (Yan and Che2011)
2.3 Riserscontainingmultiphaseflow
Risers are a special type of pipeline developed for vertical transportation ofmaterialsfromseafloortoproductionanddrillingfacilitiesonthewater'ssurface.Theycanbeintypesofrigidrisers,flexiblerisersandhybridrisersthatisacombinationofthe rigid and flexible. Risers canhavemanydifferent configurations.However in thisthesisalltheS‐shapedtypesarethepointofinterestregardlessoftheirdifferences.Thefunctionalsuitabilityandlongtermintegrityoftherisersystemaffectstheselectionofriserconfiguration(Bai2001).Figure2.5showsprevalentriserconfigurations.
7
2.4 Riserslugging
Riser slugging (also called severe slugging/ terrain induced slugging) is thetoughest type of slugging happening in a pipeline‐riser system where a downwardinclinedpipelineisconnectingintoanupwardriser.Storkaas(Storkaas2005)explainsthe cyclic behavior of riser slugging illustrated schematically in figure 2.6. It can bebroken down into four steps. Step 1: Slug formation: gravity causes the liquid toaccumulateinthelowpointandaprerequisiteforseveresluggingtooccuristhatthegas and liquid velocity is low enough to allow for this accumulation. Step 2: Slugproduction:Theliquidblocksthegasflow,andacontinuousliquidslugisformedintheriser.Aslongasthehydrostaticheadoftheliquidintheriserincreasesfasterthanthepressuredropovertheriser,theslugwillcontinuetogrow.Step3:Blowout:Whenthepressuredropover theriserovercomes thehydrostaticheadof the liquid in theslug,theslugwillbepushedoutofthesystemandthegaswillstartpenetratingtheliquidinthe riser. Since this is accompanied with a pressure drop, the gas will expand andfurtherincreasethevelocitiesintheriser.Step4:Liquidfallback:Afterthemajorityoftheliquidandthegashaslefttheriser,thevelocityofthegasisnolongerhighenoughto pull the liquid upwards. The liquidwill start flowing back down the riser and theaccumulationofliquidstartsagain.
Figure2.5:Commonriserconfigurationsappliedintheoilandgasindustry(Bai2001)
Figureshownbackin
followeLengthhighlyuthesepflaring,field(Jacompremarginfromth
slugginbend apressur
2.6:Graphinpart1.
npart4.
Severesluged by veryofliquidsundesirablparator.Flu,andthehansen,Shoession unitnstohandleheoptimal
Severeslugng the liquiand gas pareandacce
hical illustrSlugprodu
ggingcausy high liquislugscanble.Thelarguctuationsighpressuohametal.ts. The redethelargeroperationp
ggingcanoid fully bloasses throueleratedblo
rationofauction is sh
esperiodsid and gasbeseveraltgeliquidpringasprodrefluctuat1996). Itcduced capardisturbanpoint,andt
occurintwock the benugh. The towout.In
8
slugcyclehown inpa
ofnoliquis rates, whtimestheleroductionmductionmigtionsmightcanreduceacity is caunces.Largerthusreduc
wodifferenndwhile intype I is cfactthepre
(Yanandart2,Blow
idorgasphen the liqengthofthmightcauseghtcauseotreducetheoperatingused by thrdisturbaningthethr
ntmodesofn type II thcharacterizessureosci
Che2011)wout inpar
roductionquid slug isheriser.Theoverflowoperationaheproductigcapacity fhe need ofncesrequiroughput(S
fIandII. Ihere is a pzed by larillationsre
).Slug formrt3and liq
intotheses being prhisphenomandshutdlproblemsoncapacityforseparatf larger oprealargerbStorkaas20
IntypeIofpartial blocrge oscillatflectstatic
mation isquid fall
eparatoroduced.
menonisdownofsduringyofthetionandperatingback‐off005).
fsevereckage attions inheadof
9
theriser.TherearesmallpressureoscillationsintheseveresluggingoftypeIIandthesluglengthisshorterthantheheightoftheriser.Butflowoscillationscanbelarge.TypeIIsluggingisnotusuallycriticalforastableoperation.Figures2.7and2.8,adaptedfromMalekzadeh (Malekzadeh, Henkes et al. 2012), illustrate SS1 and SS2 respectively.
Figure 2.7 is based on a measured cycle of the riser P for SS1 corresponding to10.20SLU ms and 11.00SGOU ms .Figure2.8isbasedontheexperimentalcycleforthe
riser P ofSS2correspondingto 10.10SLU ms and 12.00SGOU ms .
Figure 2.7: Stages for SS1 (a) a graphical illustartion (b) marked on a cycle of an
experimental riser P trace ( 10.20SLU ms and 11.00SGOU ms ) (Malekzadeh,Henkes et
al.2012)
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Figure 2.8: Stages for SS2 (a) a graphical illustartion (b) marked on a cycle of an
experimentalriser P trace( 10.10SLU ms and 12.00SGOU ms )(Malekzadeh,Henkesetal.
2012)
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2.5 Anti‐slugoperations
Asthefieldsbecomemorematurethemoreadvancedtechnologyisdemanded.Thereasonisthattheenergyofreservoirdecreasesduetoitsaging.Thisleadstolowerpressureandtemperaturesinreservoir.Thelowerpressureofreservoircauseslimiteddrivingforcetotheflowandthereby lowerphasevelocities inresultandfinallymoreprobable riser slugging formation. Low temperatures also increase the probability ofsolidformation.Changingthedesignofpipe‐risersystemtoavoidsluggingcannotbeeconomicallyfeasible.Themostcommonmethodsforavoidingsluggingarepresentedbelow.
2.5.1 Choking
Schmidtetal. (1979) first suggestedchoking(decreasing theopeningZ)of thevalveattherisertopasaneliminationwayofsevereslugging.Thetheorybehindthissuggestionisthatthesteadyflowisgainediftheaccelerationofthegasabovetheriseris stabilized before reaching the choke valve (Jansen, Shoham et al. 1996). Thisincreasesthebackpressureandthevelocityatthechokethereupon.Themechanismisexplainedasapositiveperturbationintheliquidholdupinapipeline‐risersystemwithastableflowwillincreaseweightandwillcausetheliquidto“falldown”.Theresultofthis is an increased pressure drop over the riser. The increased pressure drop willincreasethegasflowandpushtheliquidbackuptheriser,resultinginmoreliquidatthetopof theriser thanprior totheperturbation.Withavalveopening larger thanacertaincriticalvalue(Zcrit)toomuchliquidwillleavethesystem,resultinginanegativedeviation in the liquid holdup that is larger than the original positive perturbation.Thus,wehaveanunstablesituationwheretheoscillationsgrow,resultinginslugflow.ForavalveopeninglessthanthecriticalvalueZcrit,theresultingdecreaseintheliquidholdupissmallerthantheoriginalperturbation,andwehaveastablesystemthatwillreturntoitsoriginal,non‐sluggingstate(Storkaas2005).
2.5.2 Gaslift
Gaslifthasbeensuggestedasanothermethodofeliminatingsevereslugging.Inthismethodthehydrostaticheadoftheriserisreducedwithgasinjectionandthusthepipelinepressurewillbereduced.Theinjectedgasliftstheliquidtowardsuptheriser.If sufficientgas is injectedthe liquidwillbecontinuously liftedandasteady flowwilloccur.Thedrawbackofgasliftisthelargegasvolumesneededtoobtainasatisfactorystabilityoftheflowintheriserandthisistooexpensive(Storkaas2005).
12
2.5.3 Slugcatchers
Oneotherwaytoaccommodatesluggingiscommonto installa largeseparatorasaslugcatcherattheexitofthepipeline.Theslugcatcheristhefirstelement intheprocessing facilityanddetermining itsproper size is vital to theoptimaloperationoftheentirefacility.Thefundamentalpurposeofslugcatcheristoremovefreegasfromthe liquid phase and to deliver a relatively even supply of liquid to the rest of theproduction facility.Anadvantageof thisset‐up is that inspectionandmaintenanceonthe slug catcher can be done without interrupting the normal operation. There aremainlytwotypesofslugcatchers,thevesselandthemultiple‐pipetypesandtheuseofeach type depends on the type of flow stream. Multiple‐pipe separators have beenwidelyappliedingas‐condensateprocessingfacilities(Miyoshi,Dotyetal.1988).
Installingslugcatchershasseveraldrawbacks;itputsalowerboundontheoperatingpressureofthepipe,whichagainlimitstheflowfromthereservoir.Italsoincreasesthemechanicalwear of thepipelinedue to largeoscillations inpressure.The capital andmaintenancecostsofaslugcatcherarerelativelylarge(Olsen2006).
2.5.4 Activecontrol
Risersluggingcanbepreventedusingstabilizingfeedbackcontrol.AnapproachbasedonfeedbackcontrolwasfirstproposedbyShmidt(Shmidtet.al.1978).Theideaofpaperwastosuppressterrainsluggingbyusingthetop‐sidechokevalveandasimplefeedback loop, measuring pressure at the inlet and upstream the riser or the toppressurebefore the chokevalveas inputs. With feedbackcontrol, the stabilityof theflowregimescanbechangedtoenhanceoperation.Infacttheboundariescanbemovedvia feedback control, thereby stabilizing adesirable flow regimewhere riser slugging“naturally”occurs(Storkaas2005).Anti‐slugcontrolcanmovetheboundaries in flowregime map resulting in increased stable region. It sounds to be one of the bestsolutions for prevention of severe‐slugging. Several models have been suggested byresearchers to describe the system dynamics and several controllers have beendesigned.Themodelsaremeanttoaidtuningofcontrollerswhichusetheproductionchokevalveastheactuatorandtrytostabilizethesystemwithamoreproductionratein a higher valve opening. The objective could be defined as obtaining the mostrobustness for the system against large inflow disturbances. “Nonlinearity” as theimportant characteristic of slugging system provides some challenges for control.However, a good control systemusingamodel that ismost consistentwith theplantcouldhavegoodresultsinachievingdesiredstableflowregimes.
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2.6 Modelingofrisersystems
Themainobjectivesofmodelingofproductionflowinpipelinesandrisersaretopredict the pressure drop, the phase distributions, the potential for unsteady phasedelivery(slugging)andthe thermalcharacteristicsof thesystem(Pickering,Hewittetal. 2001). The reliability of these simplified models is however questionable. Theanalysisandmodelingofmultiphaseflowsreliesheavilyonempiricalcorrelationsandthepredictionsforthemodelsareonlyasreliableastheempiricaldataonwhichtheyarebased.Thereforeitcanbequestionedwhetherthemodelswouldbevalidifappliedtorealsystems.Theyaretestedbytheuseofsmalldiametersexperimentalrisersandmaybemorethangoodenoughforsuchsystems,buttheystillmaybeinvalidforuseinlargersystems(Pickering,Hewittetal.2001).
The tuning methods used in this work are provided via linear and nonlinearmultiphase flowmodelsbasedonthemassbalancesover thedifferentsectionsof thepipeline‐risersystem.Thesimplifiedfour‐statemechanisticmodelmadebyJahanshahiandskogestad(JahanshahiandSkogestad2011)usessimplerelationshipstocalculatethephasedistributionsoverthedifferentsectionsofsystem.Themodelhasbeenthenlinearizedaroundanunstableoperatingpointandafourth‐orderlinearmodelwithtwounstable poles, two stable poles and two zeros is produced. Since amodel with twounstable poles is enough for control design, the model order is reduced by usingbalancedmodeltruncationviasquarerootmethod.Thisidentifiedmodelofthesystemis then used for an IMC (Internal Model Control) design and finding new IMC‐basedtuningrules.(JahanshahiandSkogestad2013).Moreover,asimplemodelforthestaticnonlinearity of the system is proposedby Jahanshahi andbased on this staticmodel,simplePItuningrulesconsideringnonlinearityofthesystemaregiven(JahanshahiandSkogestad2013).Thesetuningruleshavebeenusedinthesimulationsandexperimentsofthisthesisandaclearcomparisonoftheresultshavebeenpresented.
2.7 Bifurcationdiagrams
Bifurcation diagrams have been used in this thesis in order to plot the values ofpressureversusthevaluesofvalveopeningforthesluggingsystemeitherinopen‐looppositionor inclosed‐looppositionwithdifferentcontrollers.Bifurcationdiagramsarethesimplestwayto illustratethestabilityofthesystem.Inthestableregionstheplotconsistsofaunitcurveshowingtheexactvalueofthepressure(insimulations)ortheaverage of very small pressure oscillations (in experiments) while in the unstableregions theplot consists of three curves, one for steady state conditions and the twoothers showing the maximum and minimum of oscillations at each value of valveopeningovertheworkrangeofchokevalve.
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2.8 PIDandPIcontrollers
PI(proportional‐integral)andPID(Proportional‐integral‐derivative)controlareoftheearliercontrolstrategies.ThePIDcontrollerincludestheproportionalaction(P),integralaction(I),andderivativeaction(D).Thecontrollerusestheerrorsignal ( )e t to
generatetheproportional, integral,andderivativeactions.AmathematicaldescriptionofthePIDcontrolleris:
0
1 ( )( ) [ ( ) ( ) ( ) ]
t
p di
de tu t K e t e d T
T dt
Equation2.1
Where ( )u t istheinputsignaltotheplantmodel.Theerrorsignal ( )e t isdefined
as ( ) ( ) ( )e t r t y t and ( )r t is thereferenceinputsignal(Fabre,Peressonetal.1990).
AfteraLaplaceTransformthecontrollercanbeshownas:
1
(1 )c dI
c K ss
Equation2.2
Where cK , I and d are the respective tuning parameters for the P, I and D
actions. PI and PID controllers are themost widely used controllers in the industry.However,theyneedtobetunedappropriatelyforrobustnessagainstplantchangesandlarge inflow disturbances. (Jahanshahi and Skogestad 2013) Thus finding the mostappropriateamountsof cK , I and d couldbeextremelyrequired.Atypicalstructureof
aPIDcontrolsystemisshowninFigure2.9.
Figure2.9:AtypicalPIDcontrolstructure
15
2.9 TuningofPIDandPIcontrollers
Many tuning methods for different systems have been introduced so far byresearchersandengineers.Dependingonthecharacteristicsofthesystem(plant), forinstancenonlinearityandstability,differentlevelsofrobustnessisachievedbydifferenttuningmethods.Threedifferenttuningmethodshavebeenappliedinthisthesis.Twoofthem are quite new and have been recently developed (Jahanshahi and Skogestad2013).Theyarespecifiedforthesluggingsystem.Infact,thisthesisisaverificationofthesenewmethods.
2.9.1 Method1:Shams’sset‐pointovershootmethodfor
closed‐loopsystems
Some systems like slugging system are originally unstable in open‐loop. Forthese systemsmodel fromclosed‐loop responsewithP‐controller canbeused to findthe appropriate tuning parameters. A method called “Shams’s set‐point overshootmethod” was first constructed by Shamsuzzoha et al. (Shamsuzzoha and Skogestad2010).Skogestadetal. (SkogestadandGrimholt2011)developedthismethodfurtherintoatwo‐stepclosed‐loopprocedure.Astepbystepdescriptionofthetwostepclosed‐loopShams’smethodispresentedbelow.
Theclosed‐loopsystemwithP‐controllershouldbeatsteady‐stateinitially,thatis,beforetheset‐pointchangeisapplied.Then,aset‐pointchange, sy ,isapplied.The
step change and the P controller gain ( 0cK ) should be adjusted in a way that the
overshoot (D) is approximately 30%. Figure 2.10 shows a graphical illustration andequation2.4findstheovershoot.
FigureGrimho
Extract
2.10: closolt2011)
tinformatio
Timetofir
Maximum
Relativest
Alternative
atfirstund
Overshoot
Steadystat
sed‐loop se
onfromthe
rstpeak:
outputcha
teadystate
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et‐point re
egraphical
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ange: py
outputcha
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16
esponse w
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ange:
matedfrom
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py yD
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sy y
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y
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)uy
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2.6,usingth
r (Skogest
heoutputch
Equati
Equat
Equat
tad and
hange
ion2.3
tion2.4
tion2.5
17
Theparameter(A):
21.152 1.607 1A D D
Equation2.6
Theparameter(r):
2A
rB
Equation2.7
Thefirstorderplusdelaymodelparameters:
Steadystategain:
0
1
c
kK B
Equation2.8
Delay:
0.61(0.309 0.209 )rp et
Equation2.9
Timeconstant:
1 r
Equation2.10
Nowafirstorderplusdelaymodelisfoundandwithrespecttothismodel,thetuningparametersare:
1 1
.cc
Kk
Equation2.11
1min ,4I c Equation2.12
InthepaperbySkogestad(Skogestad2003),itwasrecommendedtouse c
asagoodcompromisebetweenperformanceandrobustness.
18
2.9.2 Method2:TuningbasedonIMCdesign
The Internal Model Control (IMC) method was developed by Morari. et.al.(Morari and Zafiriou 1989) The method supposes a model, states desirable controlobjectives,and,fromthese,proceedsinadirectmannertoobtainboththeappropriatecontrollerstructureandparameters.Fortheobjectivesandsimplemodelscommontochemical process control, the IMC design procedure leads naturally to PID‐typecontrollers,occasionallyaugmentedbyafirst‐orderlag.(Rivera,Morarietal.1986)
Consider theblockdiagram for the IMC structure (See figure2.11).Here, g ismodelof theplant that ingeneralhassomemismatchwiththeplant. cg is inverseof
minimumphasepartof g andf(s)isalow‐passfilterforrobustnessoftheclosed‐loopsystem.
Thegoalofcontrolsystemdesignisfastandaccurateset‐pointtracking:
sy y t , d
Equation2.13
Efficientdisturbancerejection:
sy y d t , d
Equation2.14
andinsensitivitytomodelingerror.
Figure2.11:Theinternalmodelcontrol(IMC)structure
instead
Figure
TheydusetheA summpresent
2.9.2.1
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20
Thefourparameters( 2K , z , and )areestimatedbyusingsixdata( py , uy ,
y , sy , pt , and t ) observed from the closed‐loop response (see figure2.10). Then,
they use a systematic procedure to back‐calculate the parameters of the open‐loopunstablemodelinequation2.16(JahanshahiandSkogestad2013).
2.9.2.2 IMCdesignforunstablesystems
TodesigntheIMCcontroller(C),theidentifiedmodel( g )isusedastheplantmodel.
1 0
21 0 1 2
( )k sb s b
g ss a s a s s
Equation2.18
1 21 /( )c
k s sg s
s
Equation2.19
Theyalsodesignthefilter ( )f s forrobustnessofthesystemasexplainedby
Morari.et.al.(MorariandZafiriou1989).Thefilterisinthefollowingform:
22 1 1
( )1
n
s sf s
s
Equation2.20
λisanadjustablefiltertime‐constant.Thecoefficients 1 and 2 arecalculated
bysolvingthefollowingsystemoflinearequations:
32111 1
2 322 2 2
1 1
1 1
Equation2.21
Filteronlyactstothederivativeaction.
FinallytheresultingIMCcontrollerisfoundasthefollowing:
22 13
11
( )s s
kC s
s s
Equation2.22
21
2.9.2.3 PIDandPItuningbasedonIMCcontroller
Jahanshahi writes the IMC controller of equation 2.22 in form of a PIDFcontroller andpropose the tuningparametersbasedon that. PIDF is aPID controllerwhichalow‐passfilterhasbeenappliedonitsderivativeaction.
( )1
i dPID p
f
K K sK s K
s s
Equation2.23
Wherethetuningparametersare:
1 /f
Equation2.24
3
fiK
k
Equation2.25
1p i i fK K K
Equation2.26
2d i p fK K K
Equation2.27
AnimportantpointtobeconsideredintuningofPI/PIDcontrollersbasedonIMCdesign is choosing an appropriate . It must be chosen in a way that the requiredfollowingconditionsaresatisfied:
0pK
Equation2.28
0dK Equation2.29
22
APIcontrollerhasbeenalsoobtainedbyreducingtheorderofIMCcontrollerto1.
1
( ) (1 )PI cI
K s Ks
Equation2.30
Andthesuggestedtuningrulesare:
23cK
k
Equation2.31
2I
Equation2.32
2.9.3 Method3:SimpleonlinePItuningmethodwith
gainscheduling
One main part of the thesis is tuning the controller by a new method called“SimpleonlinePItuningrules”proposedbyJahanshahiandSkogestad(JahanshahiandSkogestad 2013). One advantage of this method is that Nonlinearity of the sluggingsystem has been considered when providing the tuning rules. Gain of the sluggingsystem changes drastically for different operating conditions and as the source ofnonlinearity,makescontrolofthesystemdifficult.Themethodconsiststwoparts:
First,asimpleMATLABstaticmodelforthestaticnonlineargainisidentifiedateachoperatingpoint(valveopening).
Then,theidentifiedmodelateachoperatingpoint isusedandsimplePItuningrulesbasedonsinglesteptestbutwithgaincorrectiontocounteractnonlinearityofthesystemareproposedasfunctionsofvalveopening.
In thismethodof tuning, Jahanshahi andSkogestadhaveusedgain‐schedulingwithmultiplecontrollersbasedonmultipleidentifiedmodels.TheMATLABmodelandtheobtainedPItuningrulesforeachcontrollerwillbeexplainedbelow.
23
2.9.3.1 SimpleMATLABstaticmodel
ThesimplemodelforanL‐shapedriserconsideringstaticnonlinearitywasmadeby Jahanshahi (Jahanshahi and Skogestad 2013). The model is based on the massbalancesanditcalculatesthephasedistributionsoverthedifferentsections.ThismodelneededtobemodifiedforanS‐shapedrisertobeusedinthethesis.
A good assumption of valve equation is very important in using the simplemodel.Thereasonisthattheslugginggainofthesystemasafunctionofvalveopening,is derived based on this equation. Jahanshahi assumes the valve equation as thefollowing:
( )pcw K f z p
Equation2.33
Wherewistheinletmassflowratetotheriser, pcK isthevalveconstantand
( )f z isthecharacteristicsofthevalvewhichisdefinedasthefollowingforthelinearvalveusedinexperiments:
( )f z z Equation2.34
andasfollowsfortheOLGAvalvemodelinsimulations:
2 2( )
.
1 .f z
z cd
z cd
Equation2.35
p isthepressuredropoverthevalveandasitisclearinthevalveequation,it’safunctionofvalveopeningthatcanbewritteninthefollowingform:
2
1
. ( )pc
wp
K f z
Equation2.36
Thenthesimplemodelfortheinletpressureis:
in foP p P
Equation2.37
24
foP istheinletpressureatfullyopenpositionofthevalveandhasbeencalculated
fromthebelowequation:
* *foP P p
Equation2.38
*P isalargeenoughinletpressuretoovercometheriserslugging:
*,min. .L r s vP g L P P
Equation2.39
Here L isthedensityofliquidwhichiswaterinoursystem. g isthegravityand
rL is the length of riser. sP is the separator pressure in downstream and ,minvP is the
minimum pressured drop over the valve and has been considered zero in thesimulations.
*p is the pressure drop over the valve at the critical valve opening of the
system(bifurcationpoint).
Then based on the above equations, the static gain of the slugging system isderivedasafunctionofvalveopeningbydifferentiating inP withrespecttoZ.Finallythe
simplemodelforthestaticgainofthesystemis:
( ) inPk z
z
Equation2.40
2.9.3.2 SimplePItuningrulesbasedonidentifiedMATLABmodel
Jahanshahi and Skogestad (Jahanshahi and Skogestad 2013) then perform aclosed‐loop step test with a P‐only controller at the initial valve position of 0Z . The
parameter ( ) is then calculatedby using data ( py , uy , y , pt , and t ) observed
fromtheclosed‐loopresponse(see figure2.10)andthestaticmodelgiveninequation2.40.
0 0ln ( )
2 4
u pc
p
p
y y y yK k z
y y y
t t
Equation2.41
25
Where 0cK istheproportionalgainusedforthesteptest.ThesuggestedPItuning
parametersasfunctionsofvalveopeningaregivenasthefollowing:
0 *0 0
( )( ) /
oscc
TK z
k z z z
Equation2.42
*0 03 ( / )I oscz T z z
Equation2.43
oscT istheperiodofsluggingoscillationswhenthesystemisinopen‐loopposition
and *z isthecriticalvalveopeningoftheopen‐loopsystem(wheresluggingstarts).
26
3 Experimentalwork
ControlofSevereSluggingandcreatingastableflowregimebyapplyingcontrolusing new online tuning methods has been verified in this thesis. Air‐water Seversluggingcontrolexperiments inS‐shapedriserhasbeenoneof themainpartsof thisthesisinadditiontomodelingandsimulations.Aseriesoftestshavebeenconductedatamedium scale setup located in NTNUmultiphase flow laboratory at department ofEnergy and Process Engineering (See figure 3.1). It has been tried to evaluate theapplicability of three tuning methods explained previously in different conditions.Experimentsinthisissueandcomparingthemwithsimulatedresultsarealsovaluableinthewayofapprovingpredictionofsimulations.
Theexperimentalworkincludetryingtwodifferentchokevalveswithdifferentdynamics as the actuator and running series of control experiments for each valveseparately. Series of control experiments have been in the following order: First theopen‐loop experiments have been run in order to make the open‐loop bifurcationdiagramofthesystem.ThenaP‐onlycontrollerhasbeenusedtoclosetheloopandtheset‐point step change test has been run with the aim of finding appropriate tuningparameters. Finally, after calculating different tuning rules based on the data of stepchange test, closed‐loop experiments were run and the closed‐loop responses ofdifferentcontrollerstunedwithdifferentmethodswereevaluated.Buffertankpressure(riserinletpressureintherealsystems)hasbeenselectedasthecontrolvariable(CV)inseriesofcontrolexperiments.
Moreover cascade control experiment using topside pressure combined withoutflowdensityasthecontrolvariableshasbeentried.
27
Figure3.1:Mediumscaleexperimental setupofmultiphase flow laboratory locatedatdepartmentofEnergyandProcessEngineeringofNTNU
3.1 SetupDescription
Thethree‐dimensionaloverviewofthemultiphaseflowrigusedtoperformtheseriesofexperimentsinthisthesisisshowninfigure3.2.Theflowloopwasconsistingofwaterandcompressedairsupply.
details.air tancontinuriserwconnecflow wmixingshould
Athe S‐rpressurmanualandPTplacesi
Amovedseparatlargest
scaleis
Figur
Figure 3.3.Thewholnks (T1 anuedtolab‐lwereplacedctedtoamwas forcedpoint to ibelargeen
Asoneoftiser. Itwareandoutlllywhile rT2)andtheinthesetup
After the Sintoasmated there.toragetank
Thedimensgiveninm
re3.2:Mult
shows aesystemisnd T2) andlevelandadatthis levixer/inletsup the S‐rncrease thnoughtofo
themostimsused as tletflowdenunning theeconductanp,andwere
S‐riser, airallseparatoThewaterkinthebas
nsionsofthmeters.
tiphaseTes
schematicsplacedatd water puallflowmetvel.Theflosectionconriser. The ahe air volumorcetheliqu
mportanteqthe contronsityastheesystem innceprobeaeusedtoco
andwateror(T5)thror is then resement.The
eexperime
28
stRigLayou
overviewttwolevelsump (P1)ters,controowlineof tntainingthair bufferme and emuidupthe
quipment,l actuatorecontrolvanopen‐loopasthedensonstructa
rwere entoughalargeturned froeairisven
entalsetup
ut,NTNU(
of the exps.Largestowere placolvalves,htestwith ineair/watetank (T3)mulate a loriserandc
chokevalvfor controariables.Itpposition.sitymeternumberof
tered intogeflexiblepom the testedoutwit
pareillustr
Lilleby200
perimentaloragewateed at basehorizontaltnnerdiameersupplyanwas instalng pipelinausesluggi
ve(V)waslling the inwasalsopPressure(C)wereindifferentc
an overflowpipemadeost section bthoutfurth
atedinfigu
03)
setup witerandpresement. Flotestsectioneterof50mndthemullled upstree. The airingtooccu
mountedanletpressuossibletoatransmittenstalledatcontrolstru
w tank (T4ofhoses,anback to thehertreatme
ure3.4.The
th moressurizedw linesnandS‐mmwasltiphaseeam thevolumeur.
attopofure/ topadjustitrs (PT1variousuctures.
4), thenndweree waterent.
elength
29
Figure3.3:MediumscaleTestRigLayoutwithmoredetails,NTNU
Figure3.4:ConfigurationoftheS‐shapedrisertestsection(Lilleby2003)
30
3.2 Equipment
Inthissectionpropertiesandpurposeofthemainequipmentaregiven.Allthepipes,bendsandotherconnectionsaremadeofacid‐proofsteel,AISI316L.Thisisthecasefortheentirepipinguptothetestsections.Thevalvesaremadeoftreatedbrass,andarequiteresistanttocorrosion.
3.2.1 Mainwaterstoragetank
Water is filled in a separator (T1). It is a 3 3m acid proof tank placed in thebasement.Fromtheseparator,waterispumpedthroughtheinfrastructure,intothetestsectionandreturnedtotheseparatoragain.
Figure3.5:Mainwaterstoragetanklocatedinbasement
31
3.2.2 Airreservoirtank
Theairsupply(T2)isconnectedtothecentralhigh‐pressuresupply.Thissupplyis a pressure vesselmade byNessco and gives a pressure of 6‐7 bars, which is thenreduced through a pressure reduction valve to the operational pressure of (usually)approximately3bars.
Figure3.6:Airreservoirtanklocatedinbasement
32
3.2.3 Airbuffertank
Theairbuffertank(T3)withavolumeof200litersandthetype“DN50flange”has beenmade by the company “Laguna”. It is installed before the mixing point. Tomake slugging possible, a large pipe volume for pressure buildup is necessary. Thebuffertankisusedtoemulatethislargepipevolume.Themaximumpressurethebuffertankcanwithstandislimited.Forsafety,thetankhasbeenequippedwithasafetyvalve,toensurethatthepressurenotwillexceed3Bars.
Figure3.7:Airbuffertank
33
3.2.4 Overflowtank
An overflow system ismade to achievepressure dependent liquid flow. It is aventedsteeltank(T4)filledwithwater.Flexiblepipesconnectthetanktotheseparator.Abypassflowwillflowintothetankandbacktotheseparatorandmaintainaconstantliquidlevelinsidethetank.Thepressureattheoverflowtankwillbeconstantequaltothe hydrostatic pressure of the liquid column from the tank. This will simulate aconstantreservoirpressureandmakethe inflowto the testsectiondependentontheinletpressure.Thesupplypipes for theplasticoverflowtankaresmall,so itwillonlyworkproperlyiftheflowthroughitisverylow.
Figure3.8:Overflowtankattopofriser
3.2.5 Pressuretransmitters
Pressure transducers (PT1 and PT2) made by Siemens were installed on thebuffer tank and riser to measure the buffer pressure and top pressure respectively.Theyhaveaworkingrangeof0‐4bars.
34
3.2.6 Smallseparator
The flow from the overflow tank (T5) ismoved into a small separator locateddownthehosespipe.Apictureoftheseparatorisshownunderneathinfigure3.9.Theairfromtheriserisreleasedfromthetopoutlet.Thebottomoutletisusedforthewaterrecycleandreturnsthewatertothewaterstoragetank.
Figure3.9:Smallseparator
3.2.7 Centrifugalwaterpump
A large centrifugal water pump (P1) of the type DN100 flange made byWiloNorgeASwasused topushthewater into thesystem. Inorder topreventwater flowoscillationsthecentrifugalwaterpumpwasruninaveryhighlevelofpower(80%ofthemaximum).Howevertogetthedesiredflowrateofwaterwhichwasnothigh(0.39kg/sec)thewatercontrolvalvewasopeninsmallvalues,instead.
35
Figure3.10:Centrifugalwaterpump
3.2.8 Airflowmeter
Thevortex flowmeterof typeDN40wafermanufacturedby JF Industrisensorerwasusedtomeasuretheairflowrate(FIT1.01).ThenumberthatitgavewasintheunitofKg/hourandneeded tobe converted into thedesiredunit (kg/sec). Itwas locatedupstreamtheairbuffertank.Theworkingrangeoftheairflowmeterwas5‐2180kg/h.
3.2.9 Waterflowmeter
TheElectro‐magneticwater flowmeterof type1/2''union,manufacturedby JFIndustrisensorerwas locatedupstreamof themixingpoint (FIT2.01). Ithasaworkingrangeof0.19‐6.4 3m /h.
36
Figure3.11:Airflowmeter
Figure3.12:Waterflowmeter
37
3.2.10 Chokevalves
Two different choke valves (V) have been used in this thesis and the series ofexperimentshavebeenrunwithboth.Firstaslowvalvewasusedastheactuatortorunthecontrolexperimentsand then itwasreplacedwitha fastvalve.Theeffectof theirdynamicswas then investigated. They are angle seat valves located on the topof theriserupstreamoftheseparator.Thechokevalveisoperatedbypressurizedair(4bars)supplied from the pressurized air system in the laboratory, through the valvepositioner. The specifications of the old slow valve were not available, while thespecificationsofthefastvalveareasfollows:Manufacturer:ASCO Diameter:2inchMaterial:StainlessSteel Operation:NC(NormallyClosed)PilotPressure:4‐10bar MaximumWorkingPressure:6barOperatorDiameter:90mm Signal:4‐20mAmpOpeningTime:2sec ClosingTime:2.5sec
Figure3.13:Chokevalves;left:Fastvalve,Right:Slowvalveanditspositioner
38
3.2.11 Conductanceprobe(C)
In the second series of experiments with the fast valve a cascade controlstructurewasusedwithoutflowdensityandthetoppressureasthecontrolvariables.Conductanceprobewasapplied tomeasure thedensityof theoutflow from the riser.The probe has been calibrated byKazemihatami (Kazemihatami 2012) very recently.Theoutputoftheprobewasintheformofvoltage.ThecalibrationcurvepresentedbyKazemihatamiwasusedtofindtherelationbetweenvoltageandholdup.Equation3.1showsthisrelation.HmeansholdupandVmeansvoltage.
0.9857H V
Equation3.1
Thedensityofmixedflowisfoundfromtheequation3.2:
. . 1m water airH H
Equation3.2
Afterinsertingtherelatedvaluesintheaboveequation,thedensityofmixedflowisfoundasafunctionofvoltage:
984.513 1.204m V
Equation3.3
Figure3.14:Theconductanceprobe
39
3.2.12 LabVIEW
The Laboratory Virtual Instrumentation Engineering Workbench (LabVIEW)softwaredevelopedbyNationalInstrumentswasusedforinstrumentationcontrolanddata logging.Theuser interface is illustrated in figure3.15. Thepressures, flow ratesand valve position could bemonitored directly from the interface. In addition itwaspossible to run the loop manually by manipulating choke valve opening, orautomatically by setting tuning parameters for PID/PI/P controllers. Somemodificationswereappliedincaseofcontrol.Twomodesofcontrolwereimplementedin the program; a single mode and a cascade mode. The single mode used bufferpressureascontrolvariableandthecascademodewasusingtoppressureandoutflowdensityascontrolvariables.Aschematicviewofcontrolmodesarepresentedinfigure3.16.
Figure3.15:LabVIEWuserinterface
40
Figure3.16:ImplementedcontrolmodesinLabVIEW
41
4 Simulationofexperimentalcases
4.1 OLGA®,multiphasesimulationtool
OLGA® (OiL and GAs simulator) is a commercial multiphase flow simulatorwidelyusedintheoilandgasindustry.Itsolvesmanynumericalequationstosimulatetheflowbyconsideringthesystemdynamicsandoffersheatandmasstransfermodels.
TheexperimentalcasewasconstructedinOLGA.Thedesignedcontrollerswithdifferenttuningstrategieswereusedandtheresultswerecompared.InordertofittheOLGAmodelwiththeMATLABmodelsandexperimentssomeoftheparametersweremanipulatedwithinlimitedranges.OLGA®version7.1wasusedforthesimulations.
In this chapter the case construction with implementing the S‐shaped risergeometry, fluid properties, numerical settings and boundary conditions is explainedstepwise.
4.2 Constructionofthecase
Establishment of a good case with appropriate particular items such as fluidproperties,numericalsettings,initialandboundaryconditionsandflowpathgeometry,wastheinitialstepforsimulationprocess.The“S‐risersimple”casemadebyJahanshahi(Jahanshahi and Skogestad 2011) was basically used for the open‐loop simulations.Some improvements and modifications were applied after the file was received. Foropen‐loopsimulations themodificationswere in termsofnumericand for theclosed‐loopsimulationstheywererelatedtoimplementingthePIDcontrollerintothecase.IntermsofnumericsomeIntegrationparametersweremanipulatedinPropertieswindowoftheprogram.
42
4.2.1 Flowpathgeometry
The “S‐riser simplecase”withageometrybasedon theexperimental set‐upattheDepartmentofEnergyandProcessEngineeringwasused.The reason tousesuchgeometry is that the simulation results are to be compared with the experimentalresultsinthethesis.Theexactgeometryispresentedintable4.1.
The X‐Y coordinates have been calculated with respect to table 4.1and theresultinggeometryhasanoverviewofthefigure4.1.
Accordingtotheexperimentalsetupinmultiphaseflowlaboratory, thesourcesofairandwaterareplacedinthebeginningandtheendofthebuffertankrespectively.
Table4.1:ThegeometryoftheS‐riserexperimentalset‐up
Pipe L[m] D[m] [ ] out in
1 8.125 0.20 ‐45.0
2 3.000 0.05 ‐10
3 6.050 0.05 ‐4.0
4 1.200 0.05 ‐1.8
5 1.106 0.05 ‐1.8 ‐61.8
6 4.110 0.05 61.8
7 0.709 0.05 61.8 ‐32.0
8 2.160 0.05 ‐32
9 1.716 0.05 ‐32 79.0
10 1.820 0.05 79.0
11 1.150 0.05 90
43
Figure4.1:GeometryofS‐riserinOLGA
4.2.2 Fluidproperties
AllfluidpropertieshadbeenwritteninPVTfileby(JahanshahiandNilsen2012).Itisatableofphasecompositionsatdifferenttemperaturesandpressuresandismadeby aprogram calledPVT‐Sim. By specifying temperature andpressure limits and thecompositions of the fluids involved, the program calculates the values for the phasecompositions.Heattransferandtemperaturechangewerenotimportantinsimulationsdue to experimental condition. Water was assumed as an incompressible flow. Heattransfer and temperature related properties such as enthalpy or entropywere filledwithdummynumbers.
4.2.3 Boundaryandinitialconditions
Thetypesoftheairandwatersourcesasinletnodsweredefinedasinletmassflow. The flow rates were fixed for all simulations. The volume fractions wereestablishedto1forbothnodessinceonlywaterorairwasinjectingthroughthenode.Theoutletnod typewasselectedtopressure typeand ithasbeenset toatmosphericpressure.
‐5 0 5 10 15
‐1
0
1
2
3
4
5
6
7
x[m]
y[m]
S‐risergeometry
44
4.2.4 Numericalsetting
Thenumericalsettingspecificationssuchassimulationtimeandtimestepwereadjusted indifferentnumbers fromcase to case.This isdue to thediversityofphasevelocityindifferentcases.
4.3 ImplementingPIDcontrollerinOLGA
InordertoimplementaPIDcontrollerinOLGAfirstapositivecheckvalvewasplacedrightafter thewatersource inpipe2,section1of thecase.Thereasonwastomake sure that the flow will move only in the defined direction. Then a pressuretransmitterwaslocatedinpipe2,section2thatistheinletoftheriser,rightafterthebuffertank.Itwasaimedtomeasurethebufferpressureandsendthepressuresignalinto the PID controller. The PID controller was used in a way that it received themeasurementsignal fromthepressuretransmitterandsenttheoutputsignal intothechokevalvelocatedattopoftheriser(Pipe8,section3).ChokevalvescanbesimulatedbyselectingtheHydrovalveforthevalvemodelinOLGA.
Figure 4.2: OLGA case with PID controller. The controller receives themeasurementsignal fromthepressure transmitterandsends theoutputsignal into thechokevalvelocatedattopoftheriser.
When applying a PID controller in OLGA several specifications need to beestablished by user, depending on the desired conditions and results. The moreimportantspecificationsthathavebeenmanipulatedmanytimesduringsimulationsarethe PID parameters and the time varying specifications. When it comes to PID
45
parametersinpropertywindowofthesimulator,AMPLIFICATIONreferstothegainofthecontroller;BIASisthedesiredinitialoutputvalue(itwasusedasthedesiredvalveopeninginoursimulations);DERIVATIVECONSTisthetimeconstantforthederivativeaction and INTEGRALCONST is the time constant for the integral action. As the timevarying specifications the MODE was set to AUTOMATIC and the SET‐POINT valueswerechangedfromonesimulationtoanother.
46
5 Resultsanddiscussion
The purpose of this chapter is to present the results from experiments andsimulationsandaclearcomparisonofthem.Theexperimentalresultsfromtwoseriesofexperimentsusingaslowandafastchokevalvewillbepresentedinsection5.1.Theeffort of cascade control experiment using top pressure combined with density asmeasurements and the faced issues has been also mentioned there. Section 5.2evaluatestheeffectofcontrolvalvedynamicsthroughcomparingresultsofslowvalvewiththoseoffastvalve.Thesimulatedresultswillbeexplainedinsection5.3.Insection5.4 the experimental results are comparedwith simulated results. In section 5.5 thethreedifferentusedtuningmethodshavebeencomparedandthebest tuningmethodhasbeeninvestigated.
5.1 Experimentalresults
Theoperatingproceduresand theresults fromexperimentalactivitiesdoneatNTNUmultiphaseflowlaboratoryarediscussedinthissection.
Foreachseriesofexperimentswithvalve1(slowvalve)orvalve2(fastvalve)the open‐loop system with basic conditions would be explained first. Then theprocedureofimplementingclosed‐loopsteptestandcalculatingthetuningparametersbyusingdifferenttuningmethodswillbediscussed.Theresultsoftuningintheformoftuningrulesareexplainedthereafter.Finallytheclosed‐loopresponsesusingcalculatedtuningparameterswillbepresentedasthemainresultsoftheexperimentalwork.
47
5.1.1 Seriesofexperimentswithvalve1(slowchoke
valve)
Theexperimentalworkinthisthesisstartedwithusingslowchokevalveastheactuator.Thegoalwas torepeatthesameseriesof testswithaslowandafastchokevalveandthenevaluatetheeffectofcontroldynamicsonthefinalresults.
5.1.1.1 Open‐loopexperiments
Thestartingpointintheexperimentswasrunningtheloopinmanualmode.Thetestswererunindifferentvalveopeningswithfixedliquidandgasflowrateswhilenocontrollerwasimplementedinthesystem.Itwasaimedtopresentthesystembehaviorinnaturalconditionswithoutcontrol.Theinflowconditionsandtherelatedbifurcationdiagramarepresentedbelow.
5.1.1.1.1 Inflowconditions
The applied fixed flow rates have been 0.3927lw [ / ]kg sec for water and
0.0024gw [ / ]kg sec for air (See figure 5.1.) These flow rates correspond to 0.2slU
[ / ]m sec and 1Usg [ / ]m sec astheliquidandgassuperficialvelocities.Thewaterflow
rate couldbe set in lab viewby adjusting thepump frequency and the control valve,whiletheairflowrateneededtobesetwithamanualvalveinthepathoftheflow.Thereasonwasthatthecontrolvalvefortheairwasbroken.Themanualvalvewasfarfromthescreenandthismadeitdifficulttoobtaintheexactflowrate.
Thewater flow ratewas not also easy to set. Large variations in the flow ratewereeliminatedby running thepumpwithahigh frequencyandopening the controlvalve ina smallvalue.Themoreopening thechokevalve, themoreslugging the flowregimeand themoreunstable the flowrateswere resulted. In the following seriesofexperiments,aconstantflowrateofairandwaterwasused.Asaresult,thewaterandair flow rates needed to be readjusted when the valve opening in open‐loop waschanged.However,whenusingacontrollerinclosed‐loopmode,itwasconsiderednotto be reasonable to readjust the inflow conditions. Figure 5.1 compares variations offlowratesandpressureintwodifferentvalveopenings.
48
Figure5.1:Illustrationofbasisopen‐loopconditionsincaseofflowratesandpressure.The left series of plots are illustrating the system with valve opening Z=0.2 that isrelated to the stable regionwhile the right side plots present the systemwith valveopeningZ=1thatisrelatedtotheunstableregion.LargeoscillationsareclearsignsofinstabilityatZ=1.
5.1.1.1.2 Bifurcationdiagram
Theexperimentswere startedwith thevalveopeningof Z=0.2.Then thevalvewasopenstepwiseuntil itwas fullyopen.Theresultsofbufferpressurewere loggedand the related bifurcation diagramwasplotted, presented in Figure 5.2. The criticalstabilitypoint(thebifurcationpoint) isthemaximumchokevalveopeningthesystemcanhavewhilebeingstable.Inthepresentedbifurcationdiagram,thetoplinetracksthemaximum values of pressure at each operating point, the bottom line presents theminimumvaluesofpressureandthemiddlelineshowstheaveragevaluesofthebufferpressureatdifferentvalveopenings.Asclearinthefigurethecriticalstabilitypointwasfoundtobeatapproximately26%chokevalveopening(Z=0.26).
0 100 200 300 4002
2.2
2.4
2.6x10
‐3AirFlowrate[kg/sec]
Basisconditionwith20%valveopening
0 100 200 300 4000.32
0.34
0.36
0.38
WaterFlowrate[kg/sec]
0 100 200 300 400190
192
194
196
198
200
time[sec]
BufferPressure[kPa]
0 50 100 150 200 250 300 3502
2.2
2.4
2.6x10
‐3
AirFlowrate[kg/sec]
Basisconditionwith100%valveopening
0 50 100 150 200 250 300 350
0.35
0.4
0.45
0.5
WaterFlowrate[kg/sec]
0 50 100 150 200 250 300 350100
120
140
160
180
200
time[sec]
BufferPressure[kPa]
49
Figure5.2:Open‐loopbifurcationdiagramfromtheslowchokevalveexperiments.ThebifurcationpointoccursatvalveopeningofZ=0.26.Thetopandbottomline illustratethemaximumandminimumvaluesofoscillationsforinletpressurerespectivelyateachoperatingpoint.Themiddlelineshowstheaveragevaluesofpressure.
5.1.1.2 Closed‐loopsteptest
Inordertoapplyeachoftuningmethodstogetanappropriatecontrollerforthesluggingsystemaclosed‐loopsteptestisrequiredwithastepchangeinset‐point(thebufferpressure).Todothisitwastriedtocontrolthesystembytrialanderror.AP‐onlycontrollerwasselectedandastheinitialguessforthegain,abigvalueof100wastried.Thereasonwasthattheset‐pointvaluewasasmallnumber(pressureinbars)andthegainhadtobeselectedinawaythatitcouldchangetheoutput(Z)inalargerangeaftera small change in set‐point. Increasing the gain resulted in a more stable flow withsmaller pressure variations or smaller amplitude of slugs. Finally a high value of
0 220cK wasselectedtoperformthesteptest.Set‐pointwasmanipulatedtogetthe
averagevalveopeninghigherthan0.26andtheobtainedvalueof0.29wassatisfying.Itwas aimed to do the test in a region that is unstable in open‐loopposition. After thesystem was stabilized, four step tests were implemented and data were logged. Therelatedspecificationsarepresentedintable5.1andtherelateddiagramsareshowninfigure5.3.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1110
120
130
140
150
160
170
180
190
200
210
Z
InletPressure[Kpa]
OpenloopBifurcationDiagram‐Valve1
50
Table5.1:Closed‐loopsteptestspecificationsrunwithslowchokevalve
0cK I Initialset‐point Finalset‐point
Test_1
220
1.52 1.72
Test_2 1.73 1.54
Test_3 1.54 1.73
Test_4 1.49 1.70
Figure 5.3: Presentation of different tests of set‐point step change for a closed‐loopfeedback experimentwith a P_only controller using inlet (buffer) pressure as controlvariable.Test‐4showsthebestcharacteristicsincaseofdesiredovershootandsteadystategainrequiredfortuningthecontroller.
Afterevaluatingdatafromsteptestsitseemedthatthelastone(test_4)hasbettercharacteristicscomparedtotheotherswithrespecttothepointthataunitsteptestwas going to be used for all tuningmethods. Itwas decided to use test_4 in thetuningof controllerbydifferentmethods. Some important considerations in selectingthebeststeptestwere:
1. ForthesteptesttobeusedinShams’smethodtherecommended0.3overshootwasdesired.
0 100 200 3001.5
1.55
1.6
1.65
1.7
1.75
Inletpressure[bar]
Test‐1
SetpointData
0 200 400 600
1.4
1.6
1.8
2
Inletpressure[bar]
Test‐2
SetpointData
0 200 400 6001.5
1.55
1.6
1.65
1.7
1.75
time[sec]
Inletpressure[bar]
Test‐3
SetpointData
0 200 400 600 800
1.5
1.6
1.7
1.8
time[sec]
Inletpressure[bar]
Test‐4
SetpointData
51
2. The steady state gain of the systemmust be smaller than one ( 1s
y
y
) to be
usedinIMC‐basedtuningmethod.
Since the response was noisy, a low‐pass filter in MATLAB from the type of Simpleinfiniteimpulseresponsefilterwasusedtoreducethenoiseeffect.Asmoothingfactorof 0.001 was used to smooth the signal as well as required ( 1 means nofiltering).Figure5.4illustratesthestepresponseusedinthetuningmethods.
Figure5.4:Set‐pointstepchangeforaclosed‐loopfeedbackexperimentwithaP_onlycontroller using inlet (buffer) pressure as control variable. A low pass filterwith asmoothingfactorof 0.001 wasusedtoremovethenoiseeffectfromtheresponse.
5.1.1.3 Tuningthecontroller
The tuning methods explained in section 2.9 have been used to tune thecontrollerusingbuffer(inlet)pressureasthecontrolvariableandslowchokevalveasthe actuator. The tuning procedure and the related results are explained in thefollowing.
0 100 200 300 400 500 600 700 800 900145
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175
time(sec)
InletPressure[Kpa]
SetpointDataFiltered
52
5.1.1.3.1 TuningbyShams’sclosed‐loopmethod
The first method to be used for tuning of the controller was Shams’s methoddeveloped by Shamsuzzoha (Shamsuzzoha and Skogestad 2010). In order to tune byShams’smethod,explainedinsection2.9.1theinformationfromthesteptestexplainedinprevioussection(Seefigure5.4)wereused.Then,theovershootwascalculatedandthe appropriate tuning parameters were found. Table 5.2 shows the resulted tuningparameters by Shams’smethod. 0cK is the initial gain used in the step test, cK is the
calculatedproportional gain, and I is the integral tuningparameter. The systemhas
beenconsideredasafirstorderplusdelaymodel.
Table5.2:TuningparametersfromSham’smethodforthesluggingsystem
0cK aveZ Overshoot Offset cK I
220 0.29 0.3846 0.6501 121.5189 224.3679
Itwastriedtocontrolthesystembytherelatedtuningparametersseenintable5.2.Yet,thementionedtuningparameterscouldn’twork;meaningthatthePIcontrollerwiththeseparameterswasnotabletostabilizethesystemandseveresluggingwasnoteliminated. WemaysaythattheSham’stuningmethodisnotasuitableapproachforthesluggingsystem.
5.1.1.3.2 TuningbasedonIMCdesign
Nextmethodapplied in tuningof controller inexperimentswas the IMC‐basedtuning described in section 2.9.2. To do this, it was tried to identify the closed‐loopstable system with respect to the data from step test and according to the methodproposedbyJahanshahi(JahanshahiandSkogestad2013)explainedinsection2.9.2.1.Theidentifiedmodelofclosed‐loopsystemwasintheformof:
2
11.74 S + 0.6
96.38 10.88( )
1
06cl S
GS
s
Equation5.1
Theidentifiedclosed‐looptransferfunctionisshownbytheblacklineinfigure5.5.
53
Figure5.5:Presentationof identifiedclosed‐loopstepresponse.Thedashedblacklineshowstheidentifiedclosed‐looptransferfunctionobtainedfromIMCdesign.
Then, the open‐loop unstable system has been back calculated by using theprocedure proposed by Jahanshahi (Jahanshahi and Skogestad 2013). The open‐loopunstablesystemhastheformof:
2
-0.0005538 S - 2.858
0.00898
e-
4
05
0.004(
0 8)
8SP s
S
Equation5.2
ThentheIMCcontroller(C)isdesignedbyusingthemethodexplainedinsection2.9.2.2. The time constant of the closed‐loop system is an important manipulatedparameterandhasbeenselectedas 20 .Thisnumberwasobtainedbytrialanderrorandexperiencingdifferentresults.ThedesignedIMCcontrolleris:
2287.0673( 0.02146 0.00078
( ) S(S+0.051
62)
61)
S SC s
Equation5.3
The IMCcontroller isa secondorder transfer functionwhichcanbewritten inform of a PIDF controller. PIDF is a PID controller which a low‐pass filter has beenappliedonitsderivativeaction.ItwillbementionedasPIDcontroller.
250 300 350 400 450 500 550 600 650 700145
150
155
160
165
170
175
time(sec)
Inletpressure[kPa]
Closed‐loopstepresponse
set‐pointExp.dataFilteredidentified
54
APIcontrollerwasalsoobtainedbyreducingtheorderofIMCcontrollerto1.
Therelatedtuningparametershavebeenobtainedandareshownintable5.3.
Table5.3:IMC‐basedPIDandPItuningparameter
0cK cK I D F
PID 220 34.6387 7.92 141.2113 19.3773
PI 220 287.0673 65.6371 _ _
TheapproachofimplementingthelowpassfilterintheexperimentsisdescribedinappendixA.
To find the control results all related tuning parameterswere implemented inLabVIEWand the loopwas run in the stable regionwithanaveragevalveopeningofZ=25%.Thenitwastriedtodecreasetheset‐pointvalueinastepwisemanner.Ateachstepitwaswaiteduntilthesteadystatewasreachedandthenanewstepofreductionwasdone. Figures5.6and5.7describe the results of controlusing the IMC‐basedPIDandPIcontrollersrespectively.Theexperimentalsluggingsystemcouldbestabilizedupto Z= 40% with IMC‐based PID controller and up to Z= 38.4% with IMC‐based PIcontrollereventhoughthecontrollershavebeendesignedatvalveopeningofZ=28%.
55
Figure5.6:ResultofcontrolusingtheIMC‐basedPIDcontroller.ThecontrollerhasbeenabletomovethebifurcationpointfromZ=26%uptoZ=40.2%.
Figure5.7:ResultofcontrolusingtheIMC‐basedPIcontroller.ThecontrollerhasbeenabletomovethebifurcationpointfromZ=26%uptoZ=38.39%.
0 200 400 600 800 1000 1200 1400140
150
160
170
180
190
InletPressure[Kpa]
IMCbasedPIDController
SetpointMeasurement
0 200 400 600 800 1000 1200 14000
10
20
30
40
50 X:1363Y:40.2
time[sec]
Z[%]
0 200 400 600 800 1000 1200140
150
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190
InletPressure[Kpa]
IMCbasedPIController
SetpointMeasurement
0 200 400 600 800 1000 1200 14000
10
20
30
40
50
X:1266Y:38.39
time[sec]
Z[%]
56
5.1.1.4 Inconclusiveeffortsandtherelatedpracticalissues
Whenworkingwiththefirstvalve,someeffortswereinconclusiveandnoresultswereproduced.Belowsomeexplanationsaregiven.
5.1.1.4.1 TuningthecontrollerbySimpleonlinemethodbasedonidentified
MATLABmodelofthesystem
AsthelastmethodoftuningitwastriedtousesimplePItuningrulesdescribedin section 2.9.3. The method has been proposed by Jahanshahi (Jahanshahi andSkogestad2013)andisbasedontheidentifiedMATLABstaticmodelofthesystem.Toimplement this method, first the simple static MATLAB model of the system whichtuningrulesarebasedonneededtobemodifiedandfittotheexperimentalsteadystatemodel. For a reasonable result, it was required to have an accurate model of theexperiments. Though, right in that time the lab technician replaced the current valvewiththefastvalvesincehewasgoingtovacationandthiscouldn’tbedonefora longtime.Thereforethistuningmethodwastriedonlybythesecondvalve.
5.1.1.4.2 Applyingtimedelayinthecontroller
Oneimportantissueregardingtheslowvalveteststhatneedstobementionedisaboutapplying timedelay. Itwasaimed tochecktherobustnessofcontrol systembyimplementing delay on measurement. In order to do this, an algorithm wasimplementedinLabVIEWbyoneofthelabtechnicians.Itwasadigitaldelaylinewhichdelayed thesamplesof themeasureddatabyadesiredgiven time.Thedesireddelaytimecouldbesetfromthefrontpanel.Yetthedelaysettingcouldn’tworkinadesiredway,meaning that even for very small values of delay the systemswitched to severesluggingandthecontrolwasimpossible.Itwasclearthatforsuchalongpipelinerisersystemsmallvaluesofdelayintherangeofmillisecondscouldn’tcrashthecontrolandthereasonofinconveniencymaybefromLabVIEW.Itmightbebecauseofmistakesinthealgorithmor in theconnections insideLabVIEW.Since thesystemwas inmediumscaleandnooneelseexceptforthelabtechnicianswasabletodomodificationsinthesystemorLabVIEWandalsoduetotimeissuesitwasdecidedtoignoreimplementingtimedelayaftercounselingwithmysupervisor.Itwasanextraworktobedoneinthethesiswhilethenextrequiredexperimentswerenotstartedyetatthattime.
57
5.1.2 Seriesofexperimentswithvalve2(fastchoke
valve)
Thenextseriesofexperimentalworkinthisthesiswasrepeatingthefirstseriesof testswithanew fastvalveas theactuator.Anewmethodof tuninghasbeenusedhereinadditiontothetuningmethodsofprevioussection.
5.1.2.1 Open‐loopexperiments
The loopwasrun inmanualmodewithfixedflowratesof 0.3927lw [ / ]kg sec
forwaterand 0.0024gw [ / ]kg sec forair.Theseflowratesarethesamevaluesusedfor
the slow valve. The related inflow conditions have been fully described in section5.1.1.1. The testswere run in different valve openingswith fixed liquid and gas flowrates without applying control. The system behavior in natural conditions was thenpresentedwiththerelatedbifurcationdiagramasseeninfigure5.8.
5.1.2.1.1 Bifurcationdiagram
The starting point was the valve opening of Z=0.1. Then the valve was openstepwiseuntil itwasfullyopen.Theresultsofbuffer(inlet)pressurewereloggedandtherelatedbifurcationdiagramwasplotted.Thecriticalstabilitypoint(thebifurcationpoint)isthemaximumchokevalveopeningthesystemcanhavewhilebeingstableandis locatedatZ=0.16forthesystemwithvalve2. Inthepresentedbifurcationdiagram,thetoplinetracksthemaximumvaluesofpressureateachoperatingpoint,thebottomlinepresents theminimumvaluesofpressureand themiddle line shows theaveragevalues of the buffer pressure at different valve openings. Small pressure oscillationsbefore the bifurcation point are due to hydrodynamic slugs and are not the signs ofinstabilities.
58
Figure5.8:Open‐loopbifurcationdiagramfromthefastchokevalveexperiments.ThebifurcationpointoccursatvalveopeningofZ=0.16.Thetopandbottomline illustratethe maximum and minimum values of inlet pressure respectively at each operatingpoint.Themiddlelineshowstheaveragevaluesofpressure.
5.1.2.2 Closed‐loopsteptest
Justliketheexperimentserieswithvalve1,thefirststeptotunethecontrollerwithanytuningmethodwasaclosed‐loopsteptestwithastepchangeinset‐point(thebuffer pressure). The loop was closed with a P‐only controller with a gain value of
0 250cK .Thestepchangewasdoneinaregionthatisunstableinopen‐loopposition.
The average of valve openingwas 0.18Z . The related plot is shown in figure5.10.Since the response was noisy, a low‐pass filter in MATLAB from the type of Simpleinfiniteimpulseresponsefilterwasusedtoreducethenoiseeffect.Asmoothingfactorof 0.25 wasusedtosmooththesignalaswellasrequired( 1 meansnofiltering).
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9120
140
160
180
200
220
Z
InletPressure[Kpa]
OpenloopBifurcationDiagram‐Valve2
59
Figure5.9:Set‐pointstepchangeforaclosed‐loopfeedbackexperimentwithaP_onlycontroller using inlet (buffer) pressure as control variable. A low pass filterwith asmoothingfactorof 0.25 wasusedtoremovethenoiseeffectfromtheresponse.
5.1.2.3 Tuningthecontroller
Three different methods explained in section 2.9 have been used to tune thecontrollerusingbuffer(inlet)pressureas thecontrolvariableand fastchokevalveastheactuator.Thetuningprocedureandtherelatedresultswillbepresentedbelow.
5.1.2.3.1 TuningbyShams’sclosed‐loopmethod
Shams’s tuning method developed by Shamsuzzoha (Shamsuzzoha andSkogestad 2010) was used as the first tuning method. Table 5.4 shows the resultedtuningparametersbyShams’smethod.Thesystemhasbeenconsideredasafirstorderplusdelaymodel.The information from the closed‐loop step test (See figure5.9)wasusedtofindthetuningparameters.
Table5.4:TuningparametersfromSham’smethodforthesluggingsystem
0cK aveZ Overshoot Offset cK I
250 0.18 1.5738 1.3823 331.0775 246.7640
0 100 200 300 400 500 600148
150
152
154
156
158
160
162
164
166
time(sec)
InletPressure[Kpa]
SetpointMeasurementFiltered
60
Asexpected,accordingtoresultsofvalve1, thePIcontrollerwiththesetuningparameterscouldn’tstabilizethesystem,meaningthatShams’smethodisnotasuitablemethodtotunethesluggingsystemcontroller.
5.1.2.3.2 TuningbasedonIMCdesign
IMC‐based tuning method described in section 2.9.2 was applied as the nextmethodtotunethesystemwithfastvalve.Datafromsteptest(Seefigure5.9)wereusedandThemodelofclosed‐loopsystemwasidentifiedasexplainedinsection2.9.2.1.
2
9.076 S + 0.7406( )
64.76 4.63 15clG sS S
Equation5.4
Theidentifiedclosed‐looptransferfunctionisshownbytheblacklineinfigure5.10.
Figure5.10:Presentationofidentifiedclosed‐loopstepresponse.Thedashedblacklineshowstheidentifiedclosed‐looptransferfunctionobtainedfromIMCdesign.
Theopen‐loopunstablesystemwasthencalculatedastheformofequation5.5byusingtheprocedureproposedbyJahanshahi(JahanshahiandSkogestad2013).
0 100 200 300 400 500 600148
150
152
154
156
158
160
162
164
166
time(sec)
InletPressure[Kpa]
Closed‐loopstepresponse
set‐pointMeasurementFilteredidentified
61
2
-0.0005606 S - 4.574e-05( )
0.06858 0.004006SP s
S
Equation5.5
TheIMCcontroller(C)wasobtainedthenastheequation5.6(Seesection2.9.2.2).Avalueof 24.5 wasusedforthetimeconstantoftheclosed‐loopsystem.Thisvaluewasmanipulatedbytrialanderroruntilasatisfyinggain,phaseanddelaymarginwasobtainedforthecontroller.
2340.7491( 0.005194 0.000356)
( 0.0816)
S SC
S S
Equation5.6
TheIMCcontrollerasasecondordertransferfunctionwasthenwritteninformofaPIDcontrollerwithalow‐passfilterappliedonitsderivativeaction(WemaysayaPIDFcontroller).
APIcontrollerwasalsoobtainedbyreducingtheorderofIMCcontrollerto1.
Therelatedtuningrulesareshownintable5.5.
Table5.5:IMC‐basedPIDandPItuningparameters
0cK cK I D F
PIDF 250 3.4736 2.3368 1189.9378 12.2552
PI 250 340.7491 229.2276 _ _
Thefunction“PIDAdvancedVI”fromLabVIEWwasusedtoimplementthelow‐passfilterintheexperiments(seeappendixA).
ThePIDtuningparameterswereimplementedinLabVIEW.Firstthesystemwasruninopen‐loopmannerwithamanualvalveopeningofZ=0.2anddatawerelogged.Then the loopwasclosedwithaset‐pointP=170kPa that results inanaveragevalveopeningofZ=0.16.Aftercoupleofminutesitwastriedtodecreasetheset‐pointvalueinastepwisemanner.Ateachstep itwaswaiteduntil thesteadystatewasreachedandthen a new step of reduction was applied. The same was done with PI tuningparameters.Figures5.11and5.12describetheresultsofcontrolusingthe IMC‐basedPID and PI controllers respectively. The experimental slugging system could bestabilizeduptoZ=0.30withIMC‐basedPIDcontrolleranduptoZ=0.29withIMC‐basedPIcontroller.
62
Figure 5.11: Result of control using the IMC‐based PID controller. The controller hasbeenabletomovethebifurcationpointfromZ=16%uptoZ=30.19%(aboutthedoublevalue).
Figure5.12:ResultofcontrolusingtheIMC‐basedPIcontroller.ThecontrollerhasbeenabletomovethebifurcationpointfromZ=16%uptoZ=29.35%.
0 200 400 600 800 1000 1200 1400 1600 1800 2000120
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220
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IMCbasedPIDController
SetpointMeasurement
0 200 400 600 800 1000 1200 1400 1600 1800 20000
50
100
X:1810Y:30.19
time[sec]
Z[%
]
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140
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IMCbasedPIController
SetpointMeasurement
0 200 400 600 800 1000 1200 1400 1600 18000
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X:1526Y:29.35
time[sec]
Z[%
]
63
5.1.2.3.3 SimpleonlinePItuningbasedonMATLABmodelwithgainscheduling
SimplePItuningrulesdescribedinsection2.9.3wasusedasthelastmethodoftuningthecontroller.SincethemethodisbasedonthesimplestaticMATLABmodelofthe system, the MATLAB model needed to be identified and fit to the experimentalsteadystatemodel.
For a reasonable result, itwas first required to have an accuratemodel of theexperiments.TofindthemodeltheloopwasclosedwithaPIcontrollerandwasruninthe region after stability point in open‐loop bifurcation diagram. Thiswas done suchthatset‐pointwassettoavaluelowerthanthecorrespondingvalueofthebifurcationpoint, then it was waited until steady state was reached and data were logged. Theaverage of valve opening was found from logged data and the obtained point waslocatedon thesteadystateexperimentalmodel.By repeating this for someotherset‐pointvaluesthesteadystatelineofexperimentalmodelwasfound.Itisshowninfigure5.13withtheblackmidline.
Next step was to modify the MATLAB static model and fit that with theexperimentalmodel.Asdescribedinsection2.9.3.1theMATLABmodelisderivedbasedonthevalveequationandisafunctionofvalveopening.Thereforeagoodassumptionofvalveequationisveryimportantinusingthesimplemodel.Thevalveequationisastheform:
( )pcw K f z p Equation5.7
Thevalveislinearanditscharacteristicisdefinedas:
( )f z Z Equation5.8
Thesimplemodelfortheinletpressureisasdefinedinsection2.9.3.1andthestaticgainofthesystembecomesinformof:
2
3 2
2( )
. . pc
wk z
z K
Equation5.9
Since the tuning parameters are found based on this MATLAB model, a goodmatchbetweenthismodelandtheexperimentalmodelisveryimportantmeaningthatthevaluesofinletpressureandthestaticgainobtainedbythemodelneededtobetruevalues.Theparameters rL (lengthof riser), min_P V (minimumPressuredropover the
valve) and pcK (the valve constant) were manipulated many times until the desired
match with the experimental model was reached. Below is a discussion of theseparameters.
64
Lengthofriser
InMATLABmodellengthofriserisdirectlyusedtocalculatethestaticpressureoftheriserwhenitisfilledwithliquidandthereafterthisstaticpressureisusedtofindtheinletpressureatanylevelofvalveopening.Thereforemanipulatingofthatcouldbeveryhelpfulinproducingdesiredresults.Theexactlengthofriserintheexperimentalsetuphasbeen6.433m.Though,itwaschangedto6.7minmodeltoprovidethebestresults.
Minimumpressuredropoverthevalve
Thisparameterisusedinseveralcalculationsinthemodel.Themostimportantoneisthevalueofinletpressureinthefullyopenpositionofthevalvethatuses min_P V
directly(Seesection2.9.3.1).Levelof thecurve in the inletpressureplotof themodelwas quite affected by inlet pressure at fully open position of the valve. A value of
min_ 3P V kPa wasusedtogetthebestfitnessofthemodels.
Valveconstant
Thevalveconstant pcK hasamajoreffecton theslopeof thecurve in the inlet
pressureplotofthemodel.Avalueof 31.6 10pcK wasusedintheMATLABmodel.
Figure5.13comparessimple staticMATLABmodel to theexperimentalmodel.As clear in the figure there is a good match between the twomodels. The MATLABmodelisattachedinAppendixC.5.TheblackmidlineinthefigurepresentsthesteadystatevaluesoftheinletpressureandtheredmidlineisthevaluesofinletpressurefromtheMATLABmodel.Thetopandbottomblacklinesshowthemaximumandminimumvaluesofpressureoscillationsateachoperatingpointintheopen‐loopsystem.
65
Figure 5.13: Simple static MATLABmodel compared to the experimentalmodel. TheblackmidlineinthefigurepresentsthesteadystatevaluesoftheinletpressureandtheredmidlineisthevaluesofinletpressurefromtheMATLABmodel.Thetopandbottomblack lines show themaximum andminimum values of pressure oscillations at eachoperatingpointintheopen‐loopsystem.
In order to find the appropriate tuning parameters based on the identifiedMATLABmodelaclosed‐looptestwithstepchangeofset‐pointisrequired(Seesection2.9.3.2).Datafromthesteptestexplainedinsection5.1.2.2wasused(Seefigure5.9)andtheparameter wasfoundfromtheequation2.41as =0.038.Theperiodofslugging
oscillationsinopen‐loopexperimentshasbeen oscT =90Sec.Whenrunningthemodelat
aspecialoperatingpointtheparameters ( )cK z and ( )I z werefoundforthespecified
operatingpointasfunctionsofvalveopening(Z)bytheequations2.42and2.43.
ByrunningthemodelwithaloopforawiderangeofZvalues,itwaspossibletofind multiple tuning parameters each for a controller at a specified operating point.Thengain‐schedulingwithmultiplecontrollerswasusedtostabilizethesystem.Todothis in the experiments, five PI controllers were used with the related found tuningparameters.Table5.6 shows the resulted controller for eachoperatingpointof valveopening.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9120
140
160
180
200
220
Z
InletPressure[kPa]
SimplestaticMATLABmodelExperimental
66
In order to perform the gain scheduling between the controllers thecorrespondingvalueof inletpressuretothatspecificoperatingpointofvalveopeningwasdeterminedfromthemodelandthenthispressurevalueastheset‐pointtogetherwiththerelatedtuningvalueswereenteredinLabVIEW.Theclosed‐loopwasrunanditwas waited until the systemwas in steady state. Then the next pressure value (set‐point) corresponding to the next valve openingwas tried and the new tuning valueswereenteredinLabVIEWveryfast(Iwasworkingasthecontrolwoman!).Thisactionwasrepeateduntiltheinstabilitywasappeared.
Figure 5.14 shows the results of controlwith gain scheduling tunedby simpleonlinetuningmethod.ThecontrollerscouldstabilizetheflowuptoavalveopeningofZ=0.35. Changing bifurcation point from Z=16% into Z=35% could be a very goodresult.
67
Table5.6:PItuningvaluesandthecorrespondingoperatingpointsfromsimpleonlinetuningmethod based onMATLABmodel. These five controllers were connected andperformedgainschedulingwithmultiplecontrollersforthenonlinearsluggingsystem.
cK I Set‐point Valveopening
360.6481 320.625 166 0.19
511.9335 354.375 165 0.21
816.9311 405 161 0.24
2000.8383 523.125 157 0.31
4080.2676 641.25 156 0.38
Figure5.14:ResultsofgainschedulingusingfourPIcontrollers.Whenthesystemwasswitched into the fifth controller the instability was appeared; meaning that themaximum level of stabilitywas reachedwith four controllers tuned by simple onlinetuningrules.ThecontrollershavebeenabletomovethebifurcationpointfromZ=16%uptoZ=35%.
0 200 400 600 800 1000 1200 1400 1600 1800
140
160
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200
InletPressure[Kpa]
Multiplecontrollerswithgainscheduling
SetpointMeasurement
0 200 400 600 800 1000 1200 1400 1600 18000
50
100
X:1604Y:34.63
time[sec]
Z[%
]
5.1.3
controlvariablanalyze
5.1.3.1
pressurof densstructuThe deexplain
Figurereceivepoint sconduc
Cas
density
Oneofthel structurees.Theintethecontro
1 The
Theoutflorewasselesity couldure.Figureevice usednedinsecti
5.15:Asches signal frsignal forctanceprob
scadeC
y
etasks inthe with seletentionsweollabilitych
etestpra
wdensityectedasthebe very im5.15illustrd for meason3.2.11.
hematicvierom top prthe innerbeinstalled
ontrolu
hethesishecting toperetuningharacterist
acticaliss
wasusedecontrolvamportant iratesaschesuring the
ewofcascaressure senloop. The
dintheoutl
68
usingto
hasbeenrupressure athiscontroticsincomp
sue–test
as theconariableforin controllematicovere outflow
adecontrolnsor as thee inner loletoftheri
ppressu
unningtheand densitolstructureparisonwit
tincompl
trolvariabtheouterling slugginrviewofcadensity w
lstructureemeasuremop uses thiser.
urecom
closed‐looy of outfloebytrialaththesingl
lete
bleof innerloop.Havinng systemascadestruwas the co
inLabVIEWment andhe density
mbinedw
opusingaow as thenderroraleloopstru
r loopandngarightmwith thectureinLaonductance
W.Theoutproduces ty signal fr
with
cascadecontrolndthenucture.
the topmeasurecascadeabVIEW.e probe
terloopthe set‐om the
69
However itseemedthat theconductanceprobe isnotagoodmeasuringdeviceforthedensity.Aftersomeunsuccessfultriestocontrolthesystembytuningtheloopswithtrialanderror,itwasdecidedtoperformasteptestinopen‐loopsituationofthesystem and evaluate the open‐loop step response of the conductance probe. It wasaimed to check the applicability of probe as an appropriate sensor to measure thedensity.Todothis,theloopwasruninmanualmodeatthestableregionwithavalveopening of Z=7%. Data from density meter (Conductance probe) and top pressuresensorwas logged.Aftersomeminutes thevalvewaschanged toZ=12%while itwastried tokeep thesystem in thestableregion.Figure5.16presents theopen‐loopstepresponseoftheprobeasthedensitymeter.
Figure 5.16: Open‐loop step response of conductance probe (density meter) in thestableregion.Thefirstplot(Z)showsthestepchangeinvalveopening,thesecondplotpresentsthestepresponseof toppressure inKiloPascalandthethirdplot illustratesthestepresponseoftheoutflowdensityinkg/sec.
As seen in the figure, thedensity signaldoesnot showa clear response to thestep changeand isverynoisy.This signal couldn’tbea suitablemeasurement for thecontrol targets. Inorder tohaveanefficientcascadecontrolwithdensityas the innerloopcontrolvariable,moreaccuratesignalsofdensityarerequired.
Tryingeach loopseparately toevaluate theirresponse independently,couldbeconsidered as an alternativework. But thiswasnotpractical since thebackupof theLabVIEW file was lost and the compiled file couldn’t be manipulated or modified.Makinganewfilewasnotpossibleduetotimeissues.
0 100 200 300 400 500 6000.06
0.08
0.1
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Openloopresponse
0 100 200 300 400 500 6000
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0 100 200 300 400 500 60015
20
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Density
70
5.2 ComparisonofSlowvalveandFastvalve
In this section ithasbeen tried to compare thedynamicsof the applied controlvalves(slowvalveandfastvalve)byinvestigatingtheirrelatedresults.Forthistargettheopen‐loopandclosed‐loopresultsofthetwovalveswerecompared.Ourcriteriontoevaluate control loop is the stability. For a fast stability the dynamic response of thevalveisimportant.Itmeansasmalldeadtimeforthevalve.
The criterion for evaluating the stability of the slugging control loop has beenusually the level of valve opening (Z). However, this criterion couldn’t be useful forcomparingcontrolvalveswitheachother.Thereasongetsback to thevalve inherentcharacteristicsthatwillbeexplainedinthefollowing.
Therelationbetweentheflowrateandthelevelofvalveopeningisaninherentcharacteristicofthevalvethathasbeendefinedasthevalveequation:
( )mix pcmix
Pq K f z
Equation5.10
Here mixq isthevolumetricflowrate, pcK isthevalveconstant, P isthepressure
dropoverthevalveand mix isthemixeddensityofoutflow.
Bothvalveshavebeenconsideredlinearwith ( )f z z .Butinrealityvalve2(fast
valve)couldbenonlineartosomeextent,meaningthatitproducesthesameflowrateas theslowvalveevenwith lower levelsofvalveopenings.Figure5.17describes thisconceptmoreclearlyby illustrating thecharacteristiccurves for the twovalves. Ifwespecifyalevelofflowrateandtrytofindthecorrespondinglevelsofvalveopeningforeach of the valves, it will be seen that the slow valvemay give higher level of valveopeningforthesameflowrate.
The main desired result that can be affected by the valve dynamics is theminimum inlet pressure the system could obtain. For the open‐loop system, this isdefinedas theminimum inletpressure at fullyopenpositionof thevalve and for theclosed‐loop system it will be defined as the minimum set‐point the controller canstabilizethesystem.Figure5.18comparestheopen‐loopbehaviorofthesystemfortheslowvalvewiththatoffastvalve.Asclearinthefigure,theslowvalvegiveslowerinletpressuresatmostoperatingpointsofvalveopeningincludingthefullyopenpositionofthevalve(Z=1).
71
Figure 5.17: Characteristic curves for slow (linear) and fast (quick opening) valves.There is a lower level of valve opening for the fast valve at a specific flow rate (forinstance50),meaning that the fast valve canproduce the same flow rateas the slowvalveevenatlowerlevelsofvalveopening.
Figure5.18:Comparisonofinletpressurebetweentheslowvalveandthefastvalveattheir different operating points for the open‐loop system. At a certain level of valveopening,theslowvalvegivesalowerinletpressure.
0.2 0.4 0.6 0.8 1100
120
140
160
180
200
220
Valveopening(Z)
InletPressure[Kpa]
OpenloopBifurcationDiagrams
SlowvalveFastvalve
72
Based on the previous descriptions, it was decided to compare the minimumachievable set‐points in the closed‐loop responses. Figure 5.19 present the controlresults with IMC‐based PI and IMC‐based PID controllers. The valve opening is alsopresented,justincase,andisnotapointofinteresttocomparetheresults.
Fromthefiguresitcanbesaidthattheslowvalvehashadabetterperformancecomparedtothefastvalve.Thismeansthattheslowvalvehasbeenalreadyfastenoughforourcontroltargetsandtherehasbeennoneedtovalve2(fastercontrolvalve).Inother words the stability of the slugging system is more affected by the tuningparametersforthecontrollerinsteadofcontrolvalvedynamics.
73
Figure5.19:ComparisonofIMC‐basedcontrolresultsfromslowvalvewiththoseoffastvalve.Withtheslowvalve ithasbeenabletodecreaseset‐point inawiderrangetoalower level.Withthefastvalve, theopen‐loopsystemswitchedtosluggingatP=170kpa while with the slow valve the instability started at P=180 kpa in the open‐loopsystem. These are the initial points, respectively, to start control. The minimumachievableset‐pointhasbeenP=154kpafortheslowvalveandP=158kpaforthefastvalve.Thismeansthattheslowvalvehasshownabetterperformanceforthesluggingsystemandhasbeenalreadyfastenoughaswell.
0 200 400 600 800 1000 1200 1400140
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InletPressure[Kpa]
IMCbasedPIController
SetpointfromSLOWvalveMeasurementfromSLOWvalveSetpointfromFASTvalveMeasurementfromFASTvalve
0 200 400 600 800 1000 1200 14000
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ValveopeningfromSLOWvalveValveopeningfromFASTvalve
0 500 1000 1500140
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IMCbasedPIDController
SetpointfromSLOWvalveMeasurementfromSLOWvalveSetpointfromFASTvalveMeasurementfromFASTvalve
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74
5.3 SimulatedresultsfromOLGAmodel
InthischaptersimulationoftheexperimentalcasesinOLGA®arepresented.Thesimulationshavebeenmatchedwiththeexperimentalmodelsfromvalve1(Slowvalve).The open‐loop simulations are discussed in section 5.3.1. In section 5.3.2 results ofcontrol simulations by trial and error would be explained. Section 5.3.3 deals withfinding the appropriate tuning rules based on the methods explained in section 2.9.Results of control by applying the calculated tuningparameters are alsodiscussed inthissection.
5.3.1 Open‐loopsimulations
The first step before implementing the controller is running simulations fordifferent valveopeningswith fixed liquid and gas flow rates. Thevalues of Zand therelatedflowregimetypesarepresentedbytable5.7.
Table5.7:Differentvaluesofvalveopening(Z)usedinopen‐loopsimulations
Z Flowregimestability 0.20 Stable 0.25 Stable 0.26 Stable 0.27 Unstable 0.28 Unstable 0.30 Unstable 0.40 Unstable 0.50 Unstable 0.60 Unstable 0.70 Unstable 0.80 Unstable 0.90 Unstable 1.00 Unstable
Figure5.20describes theopen‐loopbifurcationdiagram fromsimulations.Thediagram shows the maximum, minimum, average and steady state values of bufferpressureversusthevalveopenings.Theappliedfixedflowrateshavebeen 0.3927lw
[ / ]kg sec forwater and 0.0024gw [ / ]kg sec for air (The sameas experiments). These
flow rates correspond to 0.2slU [ / ]m sec and 1Usg [ / ]m sec as the liquid and gas
superficialvelocities.Thecriticalstabilitypoint(thebifurcationpoint)isthemaximumchokevalveopeningthesystemcanhavewhilebeingstable. Inabifurcationdiagram,thecritical stabilitypoint iswhere themaximumandminimumpressuresapproachafinitevalue. In thepresentedbifurcationdiagram, the red line shows thesteady state
75
valuesofthebufferpressureatdifferentvalveopeningsandtheaveragevaluesofthepressure are on the mid black line that is higher than the steady state line. The
coefficientofdischargewaschangedtocd=0.34inordertomanipulatetheplacementofthe critical valve opening (the bifurcation point) based on the experimental result ofvalve 1 and also fit the steady state OLGA values with the steady state values fromexperimentsandmodels.Thediagramcomparingsteadystatevalues fromOLGAwiththat of themodelwill be presented in section5.3.3. As clear in the figure the criticalstabilitypointwasfoundtobeatapproximatelychokevalveopeningofZ=26%.
Figure 5.20: Open‐loop bifurcation diagram from OLGA simulations. The bifurcationpoint occurs at valve opening of Z=0.26. The top and bottom line illustrate themaximum andminimum values of oscillations for inlet pressure respectively at eachoperatingpoint.Themid‐blacklineistheshowstheaveragevaluesofpressure.
5.3.2 Controlbytrialanderror
As the first work after implementing PID controller in OLGA, it was tried tostabilizetheflowbytrialanderror.TwotypesofcontrollerincludingP‐onlycontrollerandPIcontrollerweretriedtobetunedbytryingmanydifferentvaluesastherelatedtuningparameters.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100
120
140
160
180
200
220
Z
InletPressure[Kpa]
OpenloopBifurcationDiagram‐OLGAmodel
Steadystate
76
5.3.2.1 P‐onlycontroller
As the first try a P‐only controller was used to stabilize the system. P‐onlycontrollerhasbeendesignedby inserting 0D and 1010I .Apoint inunstable
regionwith 0.3Z was selectedanddifferent valuesof gainparameterwere tried tocheckwhich gain can create stabilitywith the highest level of valve opening (Z). Foreachgainvalue itwas tried to findtheminimumamountofbuffer(inlet)pressureasSet‐point or in other words the maximum level of valve opening as manipulatedvariablebystepwisereductionoftheSet‐point.Table5.8showsdifferentvaluesofgainthathavebeentriedandthecorrespondingminimumvalueofSet‐pointandmaximumvalueofZ.
Table5.8:differenttriedP‐onlycontrollers
initialvalveopening
cK MinimumSet‐pointvalue
(P)
MaximumManipulatedvariable
(Z)
0.3
0.01* _ _0.05 141 0.36860.1 138 0.44970.5 135.6 0.64541 136 0.63052 _ _5 _ _10 _ _
ThebestcontrollerthatgivesstabilitywiththehighestlevelofZandthelowestlevel of achievable set‐point is the one with 0.5cK . With this controller, the
bifurcationpointwasmovedfromZ=26%intoZ=65%.Figure5.21showstheresultofcontrolbyP_Onlycontrollerfor 0.5cK .Forthecontrollerwith 0.01cK ,specifiedby
thestarinthetable,thesimulatorcouldconvergeinsomevaluesofSet‐point.However,theresultwasnotgoodandthereweremanyoscillationsinpressureandvalveopening.ItwasalmostimpossibletomakeareductionintheSet‐point.
For the controllerwith 1cK , controlwasdifficult and the Set‐point reduction
was challenging. Figure 5.22 shows the result of control by for 1cK . The steps of
reduction had to be selected very small and the simulator could not convergewith alargerstepthanitisobservedinthefigure.ForthevaluesfilledwithdashthesimulatorcouldnotconvergeforanyvaluesofSet‐point,meaningthatitwasimpossibletocontrolthesystemwiththegainvalueshigherthan1.AP‐onlycontrollerwith 0.05 1cK can
stabilizethesystem.
77
Figure 5.21: Simulation result of control by P‐Only controller for 0.5cK withOLGA.
Thishasbeenthebestresultfromtrialanderrorduetothelowestachievableset‐pointorinotherwordsthehighestlevelofvalveopening.
Figure5.22:SimulationresultofcontrolbyP‐Onlycontrollerfor 1cK
0 500 1000 1500 2000130
140
150P[Kpa]
InletPressure(Controlledvariable)
Set‐pointMeasurement
0 500 1000 1500 20000
0.5
1
X:2179Y:0.6436
time[sec]
Z
Valveposition(Manipulatedvariable)
0 500 1000 1500 2000 2500130
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InletPressure(Controlledvariable)
Set‐pointMeasurement
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Valveposition(Manipulatedvariable)
78
5.3.2.2 PIcontroller
PIcontrollerwasusedtostabilizethesysteminthesecondseriesofsimulationsbytrialanderror.Thecontrollerwasdesignedbyinserting 0D andtryingdifferent
valuesfor cK and I .Theaimasthepreviouspartwastotunethecontrollertocreate
stable flowwith the highest production rate (the highest level of valve opening (Z)).StepwisereductionoftheSet‐pointwas implementedastheone forP‐onlycontroller.Table 5.9 shows different values of tuning parameters that have been tried and thecorrespondingminimumvalueofSet‐pointandmaximumvalueof Z .
Table5.9:SimulationResultsofdifferenttriedtuningparametersforPIcontroller
cK
I
0.01 0.05 0.1 0.5 1
Min.P
Max.Z
Min.P
Max.Z
Min.P
Max.Z
Min.P
Max.Z
Min.P
Max.Z
80 _ _ 143.2 0.368 140 0.416 136.5 0.621 _ _
130 _ _ 141.9 0.379 140 0.434 136.5 0.627 _ _
180 _ _ 141.7 0.388 139.6 0.457 136.2 0.639 _ _
300 _ _ 141.5 0.395 139.6 0.460 136.2 0.644 _ _
800 _ _ 141.3 0.397 139.4 0.463 136.2 0.646 _ _
As it is observed in the table, the best tuning parameters are 0.5cK and
800I .Highervaluesof800werealsotriedfortheparameter I andnodifference
wasmadeinresult.ResultofcontrolbyPIcontrollerwiththebesttuningparametersispresentedinfigure5.23.Increasingtheparameter I decreasedthesystemoscillations
verywellandeveneliminateditinsomecases.However,itcausedalongertimetoberequiredfortheoutputtotracktheSet‐point ineachstepofSet‐pointreduction.Thisimportant effect of applying integral time constant could be verified by comparingfigures5.23and5.22.Asitisobserved,alessoscillatorysystemwithlongersimulationtimeistheresultofPIcontrollercomparedwithP‐onlycontroller.
Thesameas theone forP‐onlycontrollerhappened for thePIcontrollerswiththegainvaluesof 0.01cK and 1cK .Thesimulatorcouldnotconvergeforanyvalues
of Set‐point, meaning that it was impossible to control the system with the tuningparametersfilledwithdash.
79
Figure5.23:simulationresultsofcontrolbyPIcontrollerfor 0.5cK and 800I
5.3.3 Tuningthecontroller
Threetuningmethodsexplainedinsection2.9wereusedinsimulations.Theaimhasbeentocomparethetuningmethodsbasedonsimulationsaswellasexperiments.
5.3.3.1 TuningusingShams’sclosed‐loopmethod
InordertotunethecontrollerwithShams’smethodaclosed‐loopsteptestwasrequired. As explained in section 2.9.1 a P‐only control is required to determine theoptimal tuning parameters. The P‐only controllers described in section 5.3.2.1 wereusedtoachieveastepresponseclosetotherecommend0.3overshoot.Firstthesystemwassetwith thechokeopeningatZ=30%where it isunstable inopen‐looppositionandstableinclosed‐loopposition.Itwasatsteady‐stateinitiallyandthenastepchangewas applied in set‐point. Different values of step change were tried to get a stepresponse close to the recommend 0.3 overshoot. Then the system was set with thechokeopeningatZ=40%andthesametrieswereimplemented.Valueoftheresultingovershootwashighlydependedon the initial gain and the amount of step change. Insome cases with the same initial gain several tests with different amounts of stepchangewereruninordertogetthedesired0.3overshoot.Allsimulationsruntogetthe
0 500 1000 1500 2000 2500 3000120
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InletPressure(Controlledvariable)
Set‐pointMeasurement
0 500 1000 1500 2000 2500 30000
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Valveposition(Manipulatedvariable)
80
desired overshoot at different basis conditions of the controller are presented inAppendixB.
When thedesiredovershootwasachieved theShams’smethod for closed‐loopsystemsexplainedinsection2.9.1wasusedtofindtheappropriatetuningparameters.Table5.10showstheresultingtuningparametersbyShams’smethodatdifferentinitialpositionsof chokevalve. 0cK is the initial gainused in the tuning simulation, cK is the
calculatedproportionalgain,and I istheintegraltuningparameter.
Table5.10:TuningparametersfromSIMCmethodforthesluggingsystem
Initialvalveposition 0cK Overshoot Offset cK I
0.3 0.1 0.3085 0.0787 0.0614 34.5702
0.4 0.15 0.3210 2.1132 0.0904 3.1150
UsingPIcontrollerswiththeparametersfoundintable5.10,thesystembecameunstableatachokevalveopeningofapproximateZ=38.84%withthecontrollertunedattheinitialpositionof30%andatachokevalveopeningofapproximateZ=39.45%withthecontrollertunedattheinitialpositionof40%.
Figures5.24and5.25showthesteptestusinginitialchokevalveopeningof30%asthebasisinflowconditionandtheresultofcontrolbyShams’smethodrespectively.
Figures5.26and5.27arepresentedfortheinitialchokevalveopeningof40%.
Twoinitialpointswereusedfortuningtoimprovetheresults.Howeverasit isclear from the figures, no notable change is observed in the results of control.Decreasingset‐pointevenforaverysmallvaluemorethanthefinalvalueshowninthefigurescausedsystemtobecomeunstable.Severesluggingoccurredandthesimulatorcouldnotconverge.
Asseenintheresults,thesecondcontrollertunedattheinitialpointofZ=40%hasn’tbeenable to stabilize thesystem forany furthervalveopenings. Ithasn’tbeenableeventoachievethepointthathasbeentunedfor.ThismaynotbestrangesincetheShams’smethodhasbeendesignedforthestablesystemswhilethesluggingsystemisunstable.
81
Figure 5.24: Set‐point step change using initial choke valve opening of 30% and theinitialgainof 0 0.1cK .AnovershootofD=0.3,astherecommendedvaluebyShamshas
beenachieved.
Figure5.25:SimulationresultofcontrolbyShams’sclosed‐loopmethodforthe initialchoke valve position of 30%. The values of Z=0.389 and P=142 kpa have been themaximumachievedvalveopeningandtheminimumachievedset‐point,respectively.
0 100 200 300 400 500 600149
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P[Kpa]
SetpointMeasurement
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P[Kpa]
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X:3452Y:0.3884
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time(sec)
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Figure5.26: Set‐pointchangeusinginitialchokevalveopeningof40%andtheinitialgainof 0 0.15cK .AnovershootofD=0.32,neartotherecommendedvaluebyShams
hasbeenachieved.
Figure5.27:SimulationresultofcontrolbyShams’smethodfortheinitialchokevalveposition of 40%. The values of Z=0.395 and P=142 kpa have been the maximumachievedvalveopeningandtheminimumachievedset‐point,respectively.
0 100 200 300 400 500 600
142
144
146Inletpressure(controlledvariable)
P[Kpa]
SetpointMeasurement
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time(sec)
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P[Kpa]
SetpointMeasuremet
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X:1952Y:0.3945
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time(sec)
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83
5.3.3.2 TuningbasedonIMCdesign
Next method used in tuning of controller in simulations was the IMC‐basedtuningdescribedinsection2.9.2.Asexplainedbefore,theopen‐loopsystemswitchestosluggingflowatvalveopeningofZ=26%anditisunstableatZ=30%or40%.Tuningbythis method was done for two different operating points of the system; Z=30% andZ=40%.Bothsimulationsaswellastheirresultsarepresentedinthissection.
5.3.3.2.1 IMC‐basedtuningatZ=30%astheinitialvalveposition
TheloopwasclosedbyaP‐onlycontrollerwithaninitialgain 0 0.1cK andset‐
pointwaschangedby2kPa,atZ=30%.Thenwithrespecttothedatafromsteptestandaccording to the method proposed by Jahanshahi (Jahanshahi and Skogestad 2013)explainedinsection2.9.2.1,closed‐loopstablesystemwasidentifiedasthefollowing:
2
8.105 S + 0.9
17.73 3.765( )
1
19cl S
GS
s
Equation5.11
Figure 5.28 illustrates the implemented step change and the identified closed‐looptransferfunctionshownbytheblackline.
Then, the open‐loop unstable system has been back calculated by using theprocedureproposedbyJahanshahi.Theopen‐loopunstablesystemhastheformof:
2
-4.572 s - 0.51
0.2448
8
0.00)
45
4(
7s sP s
Equation5.12
Then the IMCcontroller(C) is thendesignedbyusing themethodexplained insection2.9.2.2.Thetimeconstantoftheclosed‐loopsystemhasbeenselectedas 10 .This numberwas obtained by trial and error and experiencing different results. ThedesignedIMCcontrolleris:
20.11916( 0.04668 0.0018
( )35
S (S+0.
)
1134)
S SC s
Equation5.13
84
Figure 5.28: Closed‐loop response of step change at initial valve opening Z=0.3. ThedashedblacklineshowsthetransferfunctionoftheIMC‐basedidentifiedmodel.
TheIMCcontrollerisasecondordertransferfunctionandcanbewritteninformofaPIDFcontroller.PIDFisaPIDcontrollerwhichalow‐passfilterhasbeenappliedonitsderivativeaction.ItwillbementionedasPIDcontroller.
APIcontrollerhasbeenalsoobtainedbyreducingtheorderofIMCcontrollertoone. The related PID and PI tuning parameters have been calculated as described insection2.9.2.3andareshownintable5.11.
Table 5.11: IMC‐based PID and PI tuning parameters tuned at the initial choke valvepositionof30%
0cK cK I D F
PIDF 0.1 0.03204 16.6113 23.9802 8.8191
PI 0.1 0.11916 61.7797 _ _
Implementing lowpass filterwas not possible inOLGA. Therefore, despite thefact that the filter time constantwas an important part of tuning parameters, it wasneglectedinsimulationsandaPIDcontrollerwasusinginstead.
Figures5.29and5.30describetheresultsofcontrolusingtheIMC‐basedPIDandPIcontrollersrespectively.ThecontrollersweretunedforvalveopeningofZ=30%.But,theycanstabilizethesystemuptoverylargervalveopenings.ThePIDcontrollercouldstabilizetheflowwithamaximumof50.27%valveopeningandthePIcontrollercouldstabilize the system up to valve opening of Z=47%. The PID controller has shown abetterperformance compared to thePI.A lower set‐pointaswell asahigher levelofvalveopeninghasbeenachievedwithPIDcontroller. Inaddition, theoutput fromthePIDcontrollerislessoscillatory.
180 200 220 240 260 280 300147.5
148
148.5
149
149.5
150
150.5
151
151.5
time(sec)
Inletpressure[kPa]
Closed‐loopstepresponsefromOLGAsimulations
SetpointOLGAmeasurementIdentifiedmodel
85
Figure5.29:SimulationresultofcontrolusingtheIMC‐basedPIDcontrollertunedattheinitialchokevalvepositionof30%.
Figure5.30:SimulationresultofcontrolusingtheIMC‐basedPIcontrollertunedattheinitialchokevalvepositionof30%.
0 500 1000 1500 2000 2500
140
145
150Inletpressure(controlledvariable)
P[Kpa]
SetpointMeasurement
0 500 1000 1500 2000 25000
0.5
1
X:2440Y:0.5027
Valveposition(manipulatedvariable)
time(sec)
Z
0 500 1000 1500 2000 2500
140
145
150Inletpressure(controlledvariable)
P[Kpa]
SetpointMeasurement
0 500 1000 1500 2000 25000.2
0.4
0.6
X:2417Y:0.4707
Valveposition(manipulatedvariable)
time(sec)
Z
86
5.3.3.2.2 IMC‐basedtuningatZ=40%astheinitialvalveposition
ThesamesimulationastheoneexplainedinprevioussectionwasrunatZ=40%.TheloophasbeenclosedbyaP‐onlycontrollerwithaninitialgain 0 0.15cK andset‐
pointhasbeenchangedby1kPa,atZ=40%.ThesameprocedureandcalculationsasdescribedinprevioussectionwasusedtofindIMC‐basedPIDandPItuningparameters.
Theclosed‐loopstablesystemwasidentifiedasthefollowing:
2
7.011 S + 0.
23.64 2.27
805( )
1cl S SG s
Equation5.14
The implemented step change and the identified closed‐loop transfer functionareillustratedinfigure5.31.
Figure 5.31: Closed‐loop response of step change at initial valve opening Z=0.4. ThedashedblacklineshowsthetransferfunctionoftheIMC‐basedidentifiedmodel.
Theopen‐loopunstablesystemhastheformof:
2
-2.966 S - 0.34
0.2006
0
0.00)
82
5(
5SP s
S
Equation5.15
250 300 350 400 450 500 550 600 650 700141
141.5
142
142.5
143
time(sec)
InletPressure[kPa]
Closedloopstepresponse
SetpointOLGAMeasurementIdentifiedmodel
87
ThedesignedIMCcontrolleris:
20.16877( 0.04345 0.0019
( )98
S (S+0.
)
1148)
S SC s
Equation5.16
And finally the related PID and PI tuning parameters have been calculated asshownintable5.12.
Table 5.12: IMC‐based PID and PI tuning parameters tuned at the initial choke valvepositionof40%
0cK cK I D F
PIDF 0.15 0.038293 13.0406 29.6774 8.7097
PI 0.15 0.16877 57.4753 _ _
Just like the previous part, the filter time constant was neglected due toimpossibilityofapplyinglow‐passfilterinOLGAandaPIDcontrollerwasusedinstead.
Figures5.32and5.33describetheresultsofcontrolusingtheIMC‐basedPIDandPI controllers respectively, tuned for Z=40%. The controllers were tuned for valveopeningofZ=40%.ThePIDcontrollercouldstabilizethesystemuptoZ=54.61%.Infactwiththiscontroller,thebifurcationpointhasbeenmovedfromZ=26%intoZ=54.61%.ThePIcontrollercouldstabilizethesystemuptoZ=51%.
Aswell as the result for the initialpointofZ=30%, thePIDcontroller showsabetterperformancewithlessoscillationsinoutputandahigherlevelofvalveopeninghasbeenachieved.
88
Figure5.32:SimulationresultofcontrolusingtheIMC‐basedPIDcontrollertunedattheinitialchokevalvepositionofZ=40%.
Figure5.33:SimulationresultofcontrolusingtheIMC‐basedPIcontrollertunedattheinitialchokevalvepositionofZ=40%.
0 200 400 600 800 1000 1200 1400 1600 1800136
138
140
142
144
146Inletpressure(controlledvariable)
P[Kpa]
SetpointMeasurement
0 200 400 600 800 1000 1200 1400 1600 18000
0.5
1
X:1742Y:0.5461
Valveposition(manipulatedvariable)
time(sec)
Z
0 500 1000 1500 2000136
138
140
142
144
146Inletpressure(controlledvariable)
P[Kpa]
SetpointMeasurement
0 500 1000 1500 20000
0.2
0.4
0.6
X:1929Y:0.5101
Valveposition(manipulatedvariable)
time(sec)
Z
89
5.3.3.3 Tuningusingsimpleonlinemethodwithgainscheduling
Simple PI tuning rules based on identified MATLAB static model of nonlinearpartofthesystemwasusedasthelasttuningmethodinthesimulations.Themethodhas beenproposed by Jahanshahi (Jahanshahi and Skogestad 2013) and described insection2.9.3.
5.3.3.3.1 ModifyingMATLABmodel
AsthefirststepinimplementingthismethodthesimplestaticMATLABmodelofthesystemwhichtuningrulesarebasedon,neededtobemodifiedtobesimilartotheOLGAcaseusedinthesimulationsofthethesis.Asexplainedbefore,theOLGAcasewasthepipeline‐S‐shapedrisersetuplocatedatmultiphaselaboratoryofNTNU.
Asdescribedinsection2.9.3,thesimplemodelisbasedonthevalveequation:
( )pcw K f z p Equation5.17
ForthevalveusedinOLGAsimulationsthevalvecharacteristicisdefinedas:
2 2( )
.
1 .f z
z cd
z cd
Equation5.18
Here cd is the discharge coefficient of the valve and had an important role infittingtheMATLABmodeltothesimulations.
pcK wasconsideredas:
2pc AK Equation5.19
Aisthecrosssectionalareaofthepipeandfinallythemodelwasfoundasthefollowingforthesimulations:
2
3 2 2
2( )
. . . pc
wk z
z cd K
Equation5.20
Themodelisafunctionofvalveopeningandthereforthevalueofinletpressureandthestaticgainachievedataspecifiedoperatingpoint(valveopening)wasdifferentfromtheoneinanotheroperatingpoint.Sincethetuningparametersarefoundbasedonthismodel,itisveryimportantthatthemodeltoberealistic,meaningthatthevaluesofinletpressureandthestaticgainobtainedbythemodelneededtobetruevalues.InordertomakeagoodmatchbetweenthemodelandtheOLGAcasethegeometrywaschangedtosuit theexperimentalsetup.However it soonbecameclear that themodelneeded to be manipulated to achieve the desired results. As the manipulated
90
parameters, length of riser and the discharge coefficient of the valve were quiteeffective.Adescriptionregardingthisissuewillbepresentedbelow.
Lengthofriserasthefirstmanipulatedparameter
InMATLABmodellengthofriserisdirectlyusedtocalculatethestaticpressureoftheriserwhenitisfilledwithliquidandthereafterthestaticpressureoftheriserisusedtofindtheinletpressureatanylevelofvalveopening.Thereforemanipulatingthatcouldbe veryhelpful inproducingdesired results. Theexact lengthof riser thatwasusedinsimulationsis7.7054m.Though,itwaschangedto5.15minmodeltoprovidethebestresults.
Dischargecoefficientofchokevalve(cd)asthesecondmanipulatedparameter
Thecoefficientofdischargeinthevalveequationisaconstantwhichdependsonthepressuredropoverthevalve.InordertofitthesimplestaticMATLABmodeltotheOLGA case this parameter was manipulated. Decreasing the value of cd caused themodeltohaveabettermatchwiththesimulations.TheparametercdusedinOLGAcasewas0.34whileavalueof0.31wasimplementedinMATLABmode.
Thesimplestwaytocheckifthemodeliscorrectiscomparingthevaluesofinletpressure and static gain from the MATLAB model by the same values from OLGAsimulations.Todothis,thesteadystatevaluesofinletpressurefromOLGAsimulationswereused.Thesimulatorgivesthesteadystatevaluesastheinitialvalueofanyvariableincluding inlet pressure in the simulations. Therefor the initial value of the inletpressure at each open‐loop simulation for a specified valve opening was used to becomparedwith thoseofobtained from themodel. Figure5.34compares simple staticmodeltotheOLGAcase.AsclearinthefigurethereisquiteagoodmatchbetweenthemodelandOLGA.TheMATLABmodelisattachedinAppendixC.7.
91
Figure 5.34: Simple static MATLAB model compared to the OLGA model. The bluemidline inthe figurepresents thesteadystatevaluesofthe inletpressure fromOLGAsimulationsandtheredmidlineisthevaluesofinletpressurefromtheMATLABmodel.The top and bottom blue lines show themaximum andminimum values of pressureoscillationsateachoperatingpointintheopen‐loopsystem.
5.3.3.3.2 CalculatingTuningParametersbasedonMATLABmodel
In order to find tuning parameters based on the identified MATLAB model aclosed‐looptestwithstepchangeofset‐pointwasrequired.ThesteptestwasdonebyaP‐only controlleras itwasproposedby Jahanshahi (JahanshahiandSkogestad2013).The samestep testsapplied in section5.3.3.2wereusedhere too.Twodifferent steptests,onewiththegainvalueof 0 0.1cK attheinitialvalvepositionof 0 0.3Z andthe
otherwiththegainvalueof 0 0.15cK attheinitialvalvepositionof 0 0.4Z wereused
tofindtwosetsoftuningparameters.Themethodofhowtofindthetuningparametershasbeendescribedinsection2.9.3.2.
5.3.3.3.3 ResultsoftuningusinginitialvalvepositionofZ0=0.3
With respect to the informationextracted from the step test, theparameter hasbeenfoundfromtheequation2.41as =0.2848.Theperiodofsluggingoscillations
in open‐loop simulations have been oscT = 140 Sec. Themodel has been run for each
operatingpointseparately,meaningthattheparameterZhasbeenchangedaftereach
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100
120
140
160
180
200
220
Z
InletPressure[kPa]
SimplestaticMATLABmodelOLGAcase
92
running of theMATLABmodel. The parameters ( )cK z and ( )I z have been found as
functionsofvalveopening(Z)bytheequations2.42and2.43andarepresentedintable5.13.
Table5.13:PItuningvaluesinOLGAsimulationswithinitialchokevalvepositionof30%
cK I Set‐point
(Inletpressure)[kpa]
Valveopening
0.0499 484.6154 148.5 0.3000
0.0774 549.2308 145.5 0.3244
0.1142 613.8462 143 0.3616
0.1622 678.4615 141 0.4033
0.2229 743.0769 140 0.4307
0.2985 807.6923 138.5 0.4846
0.3908 872.3077 138 0.5084
0.5650 969.2308 137 0.5673
0.7477 1050 136.5 0.6061
0.9691 1130.8 136 0.6539
1.2338 1211.5 135.5 0.7145
1.5465 1292.3 135.3 0.7562
Then gain‐scheduling with multiple controllers based on multiple identifiedmodelswasusedtostabilizethesystem.Todothisinthesimulations,12PIcontrollerswereimplementedinOLGAwiththerelatedfoundtuningparameters.Thecontrollerscouldstabilizetheflowupto75.5%ofvalveopening.ChangingbifurcationpointfromZ=26%intoZ=75.5%couldbeaverygoodresult.Figure5.35illustratestheresultofcontrolusinggain schedulingbetweenPI controllers tuned for the initial chokevalvepositionofZ=30%.
93
Figure5.35:SimulationresultofcontrolusinggainschedulingbetweenPI controllerstunedfortheinitialchokevalvepositionofZ=30%.
5.3.3.3.4 ResultsoftuningusinginitialvalvepositionofZ0=0.4
EverythinghasbeendoneinthesamewayasexplainedinprevioussectionforZ0=0.3 except for the step test that has been run for Z0=0.4. With respect to theinformation extracted from the step test, the parameter has been found as: =
0.8183.Inordertodogain‐schedulingwithmultiplecontrollersinthesimulations,eightPI controllerswere implemented in OLGAwith the related found tuning parameters.Thecontrollers couldstabilize the flowup toZ=66.34%ofvalveopening.Table5.14and Figure 5.36 describe the result of tuning and control using control using gainschedulingbetweeneightPIcontrollerstunedfortheinitialchokevalvepositionofZ=40%.
0 500 1000 1500 2000 2500 3000 3500 4000135
140
145
150Inletpressure(controlledvariable)
P[Kpa]
SetpointOLGAMeasurement
0 500 1000 1500 2000 2500 3000 3500 40000.2
0.4
0.6
0.8
X:3881Y:0.7554
Valveposition(manipulatedvariable)
time(sec)
Z
94
Table5.14PItuningvaluesinOLGAsimulationwithinitialchokevalvepositionofZ=40%
cK I Set‐point Valveopening
0.3927 646.1538 141 0.4024
0.5482 710.7692 140 0.4323
0.7434 775.3846 138.5 0.4871
0.9838 840 138 0.5111
1.2751 904.6154 137 0.5713
1.8207 1001.5 136.5 0.6111
2.2661 1066.2 136.2 0.6398
2.7843 1130.8 136 0.6633
Figure5.36:SimulationresultofcontrolusinggainschedulingbetweenPI controllerstunedfortheinitialchokevalvepositionofZ=40%.
0 500 1000 1500 2000 2500134
136
138
140
142
144Inletpressure(controlledvariable)
P[Kpa]
SetpointOLGAMeasurement
0 500 1000 1500 2000 25000
0.5
1
X:2476Y:0.6634
Valveposition(manipulatedvariable)
time(sec)
Z
95
5.4 Comparisonofexperimentalandsimulated
results
Thesimulations in this thesishavebeenmatchedwith theexperimentalmodelsfromvalve1(Slowvalve).Thereforeincaseofnumericalcomparison,itisreasonabletocomparesimulatedresultswithexperimentalresultsfromvalve1.Butgenerallyincaseofcomparisonofdifferenttuningmethodsandfindingthebesttuningapproachforthesluggingsystemthesimulatedresultsdoagreewiththeexperimentalresultsfromthebothvalves.Inthissectioneachtuningmethodwouldbediscussedseparatelyandtheresult of control from simulations and experiments will be compared. Finally acomparisonofalltuningmethods,usedinthethesis,basedonthesimulationsandbothvalvesexperimentswillbepresented.
5.4.1 Open‐loopbifurcationdiagrams
A comparison of the simulated open‐loop results from the OLGA case and theexperimental results fromvalve1 is shown in figure5.37. It canbe seen in the figurethat thebifurcationpoint is fairly thesameforthebothmodels. Itoccursat thesamevalveopeningofZ=0.26forbothmodelsbutatahigherpressurefortheexperiments.Modelsareslightlydeviatedfromeachother.FortheOLGAsimulationsthemaximumofinletpressureoscillationsarelocatedathighervalues.
Figure 5.37: simulated results from the OLGA case compared with the experimentalresultsfromvalve1.Thebifurcationpointisfairlythesameforbothmodels.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100
120
140
160
180
200
220
Valveopening[Z]
InletPressure[Kpa]
OpenloopBifurcationDiagrams
OLGAsimulationsExperiments
96
5.4.2 ComparisonofcontrolresultsfromIMC‐based
tuningmethod
A comparison of simulated and experimental closed‐loop responses fromcontrollerstunedwithIMC‐basedmethodispresentedintable5.15. MaxZshowsthemaximumvalveopeningachievedwiththatcontrollerandMinPpresentstheminimumvalue of set‐point that is inlet pressure in kilo Pascal. The numbers are the roundedvalues.ThecontrollershavebeentunedattheinitialvalvepositionofZ=0.30.
Table5.15:ComparisonofsimulatedandexperimentalresultsfromcontrollerstunedwithIMC‐basedmethod.ZisthelevelofvalveopeningandPistheinletpressureinKPa.
Stabilitybefore
control
StabilityaftercontrolwithIMC‐basedPI
controller
StabilityaftercontrolwithIMC‐basedPID
controllerMaxZ MinP MaxZ MinP MaxZ MinP
Experiments 0.26 177.8 0.38 154.5 0.40 154
Simulations 0.26 153 0.46 139.5 0.50 138.5
Althoughthesimulatedclosed‐loopresultsshowahigherlevelofvalveopenings,still the amount of set‐point reduction is larger for the experiments. Both modelsconfirmthat the IMC‐basedtuningmethod isa fineapproach for thesluggingsystem.Moreover they agree that the IMC‐based PID controller has a better performancecomparedtothePI.
5.4.3 ComparisonofcontrolresultsfromSimpleonline
tuningmethod
SimpleonlinetuningmethodbasedonMATLABmodelwasnottriedintheseriesof experimentswith valve1 (See section5.1.1.4 formore explanations). Instead itwastriedwithvalve2.Thereforetheresultscan’tbenumericallycomparedsincetheOLGAsimulationsarebasedontheexperimentswithvalve1.However,theexperimentalandsimulatedresultsdoagreeonconfirmationofthismethodasthebestmethodoftuningwiththehighestlevelofstabilityforthesluggingsystem.Thiswillbeseenmoreclearlyinthenextsection.
97
5.4.4 Comparisonoftuningmethods
Anoverviewof all experimentaland simulated results from theapplied tuningmethods is presented in table 5.16 in numeric form. The maximum valve openingachieved as well as the minimum obtained set‐point for each closed‐loop test orsimulationisillustrated.
ItcanbesaidthatthebesttuningmethodforthesluggingsystemisthesimpleonlinePItuningruleswithgainschedulingforthewholeoperatingrangeofthesystembased onMATLABmodel. Tuning based on IMC design also works very well for theslugging system. These tuning methods are able to move the critical stability pointsignificantlyandconsiderablyincreasetheproductionrateasaresult.
Itwasalsotriedtomakeaclearcomparisonbetweentheappliedtuningmethodsby using figures. To do this the open‐loop and the closed‐loop bifurcation diagramswereplottedforthesimulationsandeachseriesofexperiments.Figure5.38comparestheresultsofstabilizingcontrolsimulationsbydifferenttuningmethods.Figures5.39and5.40dothesamefortheresultsofcontrolexperiments.
Thebifurcationpointasasignofstabilitylevelisshownbeforeandaftercontrolwitheachtuningmethod.Therightmostbifurcationpointisrelatedtothebesttuningmethodthatprovidedthemoststabilityineachseries.
ItshouldbenotedthatShams’smethoddidn’tworkintheexperiments.Thisisnotsurprisingsinceithasbeendevelopedforthesystemsthatarestableinopen‐loopwhile the slugging system is highly unstable. Simple online tuningmethod based onMATLAB model was not tried in the series of experiments with valve1 (See section5.1.1.4formoreexplanations).
98
Table5.16:Comparisonofsimulatedandexperimentalresultsfromalltuningmethods.
Z Max is the maximum level of valve opening and P Min is the minimum set‐pointachievedthatistheriserinletpressureinKiloPascal.
Open‐loop
stabilitylimit
Shams’sset‐pointovershootmethod
IMC‐basedPItuningmethod
IMC‐basedPIDtuningmethod
simplePItuningwithgainscheduling
Set1ofExperimentswithslowvalve
PMin 177.8 ‐ 154.5 154NOT
PerformedZMax 0.26 ‐ 0.38 0.40
Set2ofExperimentswithfastvalve
PMin 170.8 ‐ 158.5 157.5 156
ZMax 0.16 ‐ 0.29 0.30 0.35
OLGAsimulations
PMin 153 142 139.5 138.5 135.5
ZMax 0.26 0.38 0.46 0.50 0.75
Figure 5.38: Comparison of stabilizing control results from different tuningmethodsappliedinthesimulations.Itcanbesaidthatsimpleonlinemethodwithgainschedulingis the most stabilizing and the IMC‐based designed method is the second best assystematicmannerstotunethecontrollers.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100
120
140
160
180
200
220
Valveopening[Z]
InletPressure[Kpa]
Comparisonofdifferenttuningmethodsfromsimulations
Openloop
Shams
IMCbased
Trialanderror
Simpleonline
99
Figure5.39:ComparisonbetweenthestabilizingcontrolresultsfromIMC‐basedtuningmethodandtheopen‐loopsystemfortheexperimentswithvalve1.
Figure 5.40: Comparison of stabilizing control results from different tuningmethodsapplied in the experimentswith valve 2. Simple onlinemethodwith gain schedulingshowsthebestperformance.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1110
120
130
140
150
160
170
180
190
200
210
Valveopening[Z]
InletPressure[Kpa]
Comparisonofclosedloopandopenloopstabilityfromexperimentswithvalve1
OpenloopIMCbased
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9120
140
160
180
200
220
Valveopening[Z]
InletPressure[Kpa]
Comparisonofdifferenttuningmethodsfromexperimentswithvalve2
OpenloopIMCbasedSimpleonline
100
6 Discussionandfurtherworks
6.1 Tuningmethods
Themainobjective inthis thesiswastoverify theveryrecentlydevelopedtuningmethods (Jahanshahi and Skogestad 2013) by medium scale experiments and OLGAsimulationsandidentifythemostrobusttuningmethodforthesluggingsystem.Fromtheresults itcanbeseenthatthehighest levelofstability isrelatedtothecontrollerstunedbysimpleonlinemethodbasedonMATLABmodel.Itshouldbenotedthatinthisthesis wherever simple online method has been applied gain scheduling betweenmultiple controllers has been also performed. Since the slugging system is nonlinearandthegainofsystemchangesdrasticallywithchanginglevelofvalveopening,tuningthe controllers at eachoperatingpoint and then connecting themvia gain schedulinghas a huge effect on the control performance. It can be said that applying gainschedulingbetweentheIMC‐basedcontrollersmayalsoleadtoahigherlevelofstabilitycompared to applying a single IMC‐based PID/PI controller. This can be tried in thefuture.
Generally, both IMC‐based method and simple online method are very usefulsystematicapproachestotunethecontrollersforthesluggingsystem.Previously,trialanderrorwasmostlyusedfortuning thecontrollers in thesluggingsystem. Itcanbesaidthatthetuningrulesusedinthisthesisarefromthefirstsystematicrulesforanti‐slugcontrolandgiveveryclearfineresults.
Infutureworkstimedelaycanbeaddedtothemeasurementsintheexperimentsinordertohaveabetterinvestigationofthesystemrobustness.Actuallythiswastriedinthisthesisasaninconclusiveeffort(Seesection5.1.1.4.2).
One importantpointneeds tobementioned inrelationwith tuningbasedon IMCdesign.TheIMC‐basedPIDtuningrulesincludeafiltertimeconstantthatmeansanIMCfilter must be implemented on the derivative action of the PID controller. This wasimpossibleinOLGAandthereforehadtobeneglected.Althoughthesimulationresultsdo agreewith the experimental results it can’t be denied that neglecting filter actiondeviatesthesimulatedresultsfromthereality.Thismaybepossibleinfutureversionsofthesimulator.
101
Aboutsimpleonlinemethod,theMATLABmodelisdiscussable.FromtheresultsitcanbeobservedthattheMATLABmodeldidfittothesimulatedresultsfromOLGAandalsotheexperimentalresults.However,themanipulationsdonetofitthemodeltothesimulations and experiments may lead to the inaccuracy of the results. Also, in theMATLABmodelthevalveisassumedtohavealinearcharacteristic;howeverthismaynotbethecaseforthevalveintheexperiment(Seesection5.2).
Shams’stuningmethodhasbeendesignedforthestablesystemswhilethesluggingsystem is unstable. Therefore itmay not be far from the expectation that it couldn’twork for the slugging system. This method didn’t work in any of the experimentalseries. Though it worked in simulations but didn’t give very good results. This evensmall stability found with this method in OLGA simulations may deviate from thereality. This deviationmay be due to inappropriate assumptions or inaccurate initialand boundary conditions in OLGAmodel. The overall result can be that thismethodcan’t be a suitable one to tune the controllers in the slugging system. Instead, therecentlydevelopedIMC‐basedandsimpleonlinemethodsperformmuchbetter.
6.2 Controlstructures
The control structure used in the series of experiments and simulations was aSISOcontrolwithbufferpressureasthecontrolvariable.Thismeasurementistheriserinletpressureinrealsubseasystemsandmaynotbeveryeasytomeasure.Howeverithasbeenprovedpreviously that it’s thebest controlvariable for theactivecontrolofsevere slugging (Jahanshahi, Skogestadet al. 2012) (Meland2011) . In thearticlebyJahanshahi (Jahanshahi, Skogestad et al. 2012) one pressure measurement from thepipelinecombinedwithchokeflowratehasbeensuggestedasthebestmeasurementsfor a multivariable structure. At the beginning of the thesis it was decided to try asimilar structure with top pressure combined with riser outflow density as themeasurements. But this didn’t become practical during the thesis due to theinconvenientdensitysensor(Seesection5.1.3).Thenewtuningmethodsappliedinthisthesiscanbetriedbyothermeasurementsandcontrolstructuresinthefuture.Asthefirst step an accurate density sensor shall be used to give correct measurements ofdensities.Thenitcanbeusedinthenewcontrolstructures.
102
6.3 Discussableissuesrelatedtoexperimental
activities
6.3.1 Oscillationsinflowrates
Inordertohaveafixed slU and sgU ineachtestitwasimportanttohaveconstant
andconsistentflowrates.Theairandwaterflowrateshadmanyoscillationsanditwasvery difficult to set the exact required flow rates. Specially, for the case of air thisproblemwasmore challenging.Thereasonwas that the controlvalve for theairwasbrokenandtheairflowratehadtobesetwithamanualvalvefarfromthescreen.Themanualvalvemadeabigchangeinairflowrateevenwhenitwastriedtoopenorcloseitverylittle.Itwasnecessarytogoandcomemanytimestomakeaflowrateclosetothe desired value. Forwater flow rate the centrifugalwater pumpwas the reason ofoscillations.However,itwastriedtodealwiththisissuethroughrunningthepumpinaveryhighlevelofpower(80%ofthemaximum)andopeningthewatercontrolvalveinsmallvalues,instead.
6.3.2 Waterflowbackintothebuffertank
Whenthebufferpressurebecamelowerthanthepressureinsidepipeline,waterdidflowback intothebuffertank.Thisreducedthevolumeofbuffertankandcausedthebufferpressuredeviates from the real values.Thisdiscrepancy coulddistract thecontrollerperformanceandtherefore itwasveryimportanttoremembertodrainthebuffer tankbetween the experiments. Installing an automatic sensor toquickly sensethewaterinthebuffertankcouldbeveryhelpfultoovercomethisissue.
6.3.3 Leakageinsteelconnection
Thelaboratoryfacilitywasinawaythatasinglepipelineneededtobeconnectedtoanyoftherisers(SteelS‐riser,HoseL‐riserorHorizontalpipeline).Ontheotherhandseveralpeoplewereworkinginthelabandonthedifferentsetupsduringthesemester.Thiscausedthepipeline‐risersconnectionsneededtobechangedseveraltimesaweek.Thiswasnotaveryeasyjobandsometimestheconnectioncouldn’tbefittedquitewelleven with trying many different sealing rubber O‐rings, screws and nuts. Thereforetherewassomeflowleakagefromtheconnectionduringthework.Thiscouldaffecttheaccuracyoftheexperimentssincetheflowmeterswerelocatedbeforethisconnection.
103
However, the flowmeters themselveswerenotof thebestqualityand theirnumbersmay be also inaccurate. One way to overcome this occasional leakage is to make amultipleconnectionbetweenthepipelineandallriserswiththemanualvalvesforeachconnection.Thenthevalvescanbemanipulatedtochangeflowdirectionsinsteadofthetimeconsumingchangeoftheconnectionsbymechanicalwork.
104
7 Conclusion
Thischapterisorganizedbasedonthetasksdefinedinthethesisdescription.Thesetaskshavebeenfollowedandthedesiredresultshavebeenobtainedmostly.
7.1 Stabilizingcontrolexperimentsusingbottom
pressure
Stabilization control experiments using the medium scale S‐riser setup provedthattheseveresluggingphenomenacanbedelayedtoalargeextentbyactivecontrolofproduction choke valve and using the bottom pressure (buffer tank pressure) as thecontrolvariable.Twosetsofexperimentswithtwodifferentchokevalvesshowedthatthe anti‐slug control structure using bottom pressure as measurement and a goodtuningmethodaswell,thestabilityregioncouldbeextendedwidely.
7.2 TestingonlinetuningrulesonS‐riser
experiments
Threedifferenttuningmethodsforanti‐slugcontrolweretestedonlineandtheirrobustnesswascomparedwith respect to thestability limits theyprovided(See table5.16andalsofigures5.39and5.40).
The Shams’s set‐point overshoot method (Shamsuzzoha and Skogestad 2010)failed to stabilize the system in both sets of experiments. This was not far from theexpectation,sinceShams’smethodhasbeendevelopedforthestablesystemswhilethesluggingsystemishighlyunstable.
For implementing the IMC (Internal Model Control) based tuning method(JahanshahiandSkogestad2013)themodelofthesystemwasidentifiedfromaclosed‐
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loopstep test.The identifiedmodelwasused foran IMCdesign,and thenPIDandPItunings were obtained from the resulted IMC controller. The IMC‐based PID tuningrulescouldincreasethestabilitylimitfrom26%to40%ofchokevalveopeninginthefirst set of experiments using the slow valve and from 16% to 30% of choke valveopeninginthesecondsetofexperimentsusingthefastvalve.
ThesimplePItuningruleswithgainschedulingforthewholeoperatingrangeofthe system, was used as the last tuning method and proved to be the best tuningapproach for the slugging system. To implement thismethod, aMATLABmodelwasmodified and fitted to the steady state model of experiments. Then based on thisMATLABmodelandalsoasingleclosed‐loopsteptest,thesimplePItuningruleswerefound.Thistuningmethodcouldincreasethestabilitylimitfrom16%to35%ofchokevalveopeninginthesecondsetofexperimentsusingthefastvalve.
7.3 Controlusingtoppressurecombinedwith
density
Measurement of the topsidedensity using a conductanceprobe installationwasnot successful. The open‐loop step test proved that the probe is not applicable as anappropriatesensortomeasuretheflowdensity.Theprobesignalcouldn’tshowaclearresponse to the step change and therefore was not a suitable measurement for thecontrol targets (See figure 5.16). In order to have an efficient cascade control withdensityastheinnerloopcontrolvariable,moreaccuratesignalsofdensityarerequired.Thereforenocascadeanti‐slugcontrolschemescouldbetested.
7.4 Investigatingeffectofcontrolvalvedynamics
Thecriteriontoevaluatethesluggingcontrol loop is thestabilityandsincethevalves’inherentcharacteristicsaredifferent,thelevelofvalveopeningcan’tbeusedtocomparethevalves’performanceinthecontrolloop.Instead,theminimumachievableset‐points in the closed‐loop responses and also the achieved range of set‐pointreductionwereusedtocomparethevalvebehaviors.Fromtheclosedloopresponses,itwasproved that the slowvalvehas abetterperformance compared to the fast valve.Thismeansthattheslowvalvehasbeenalreadyfastenoughforourcontroltargetsandtherehasbeennoneedtovalve2(fastercontrolvalve).Inotherwordsthestabilityofthesluggingsystemismoreaffectedbythetuningparametersforthecontrollerinsteadofcontrolvalvedynamics.Figure5.19comparestheclosedloopresponseofIMC‐based
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controller for the twovalves.With the slowvalve, the IMC‐based controllerhasbeenabletodecreaseset‐pointinawiderrange,downtoalowerlevel.
7.5 ControlsimulationsusingOLGA
TheOLGAmodelwasdevelopedbasedonthefirstseriesofexperimentswithvalve1 and the implemented PID controller was fine‐tuned using the different tuningstrategies.Resultsoftheexperimentsverifiedthoseofthesimulations.
Inopen‐loopconditiontherewasagoodmatchbetweentheOLGAmodelandtheexperimentalmodelofvalve1(seefigure5.37).
The same as the experimental results, the simulated ones proved that simple PItuning rules with gain scheduling for the whole operating range of the system(Jahanshahi and Skogestad 2013) is the best tuning method providing the largeststabilityregion for thesluggingsystem.ThePIcontroller in thesimulations, tunedbythismethod,couldincreasethestabilitylimituptothevalveopeningofZ=75%fromtheopenloopstabilityofZ=26%(seetable5.16orfigure5.38).
From the simulation results it can be said that the IMC‐based tuning method(Jahanshahi and Skogestad 2013) is the second best systematic manner to tune thecontrollersforthesluggingsystem.ThePIDcontrollertunedbythismethod,increasedthestabilitylimitfrom26%to50%ofchokevalveopening.
TheShams’sset‐pointovershootmethod(ShamsuzzohaandSkogestad2010)wasused to tune the PI controller in two initial points of Z=30% and Z=40% in thesimulations.TheonetunedattheinitialpointofZ=30%couldsurprisinglystabilizethesystemuptothevalveopeningofZ=38%.However,theotheronetunedattheinitialpoint of Z=40%wasn’t able even to achieve the stability for the point that has beentunedfor.
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8 References
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http://www.drbratland.com/PipeFlow2/chapter1.html.Fabre,J.,L.Peresson,etal.(1990)."Severeslugginginpipeline/risersystems."SPE
ProductionEngineering5(3):299‐305.Godhavn,J.‐M.,M.P.Fard,etal.(2005)."Newslugcontrolstrategies,tuningrulesand
experimentalresults."Journalofprocesscontrol15(5):547‐557.Havre,K.,K.O.Stornes,etal.(2000)."Tamingslugflowinpipelines."ABBreview4:55‐
63.Jahanshahi,E.andS.Skogestad(2011).Simplifieddynamicalmodelsforcontrolof
severeslugginginmultiphaserisers.WorldCongress.Jahanshahi,E.andS.Skogestad(2013).Closed‐loopmodelidentificationandPID/PI
tuningforrobustanti‐slugcontrol.Mumbai,India,10thIFACInternationalSymposiumonDynamicsandControlofProcessSystems.
Jahanshahi,E.,S.Skogestad,etal.(2012).Controllabilityanalysisofsevereslugginginwell‐pipeline‐risersystems.IFACWorkshop‐AutomaticControlinOshoreOilandGasProduction,Trondheim,Norway.
Jansen,F.,O.Shoham,etal.(1996)."Theeliminationofsevereslugging—experimentsandmodeling."Internationaljournalofmultiphaseflow22(6):1055‐1072.
Kazemihatami,M.(2012).ExperimentsonLiquidFlushinginPipes.MasterProject.Trondheim,NorwegianUniversityofScienceandTechnology.
Lilleby,K.(2003).User'sManualforMultiphaseFlowLoop.Trondheim,Norwegianuniversityofscienceandtechnology.
Malekzadeh,R.,R.Henkes,etal.(2012)."SevereSlugginginaLongPipeline‐RiserSystem:ExperimentsandPredictions."Internationaljournalofmultiphaseflow.
Meland,K.O.(2011).Stabilizationoftwo‐phaseflowinrisersfromreservoirs.ChemicalEngineering.Trondheim,NorwegianUniversityofScienceandTechnology.Master.
Miyoshi,M.,D.Doty,etal.(1988)."Slug‐catcherdesignfordynamicslugginginanoffshoreproductionfacility."SPEProductionEngineering3(4):563‐573.
Morari,M.andE.Zafiriou(1989).Robustprocesscontrol,Morari.Olsen,H.(2006).Anti‐slugcontrolandtopsidemeasurementsforpipeline‐risersystem,
Master’sthesis,NorwegianUniversityofScienceandTechnology.Pickering,P.,G.Hewitt,etal.(2001).Thepredictionofflowsinproductionrisers‐truth
&myth.IIRConference.
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Rivera,D.E.,M.Morari,etal.(1986)."Internalmodelcontrol:PIDcontrollerdesign."Industrial&engineeringchemistryprocessdesignanddevelopment25(1):252‐265.
Shamsuzzoha,M.andS.Skogestad(2010)."Thesetpointovershootmethod:Asimpleandfastclosed‐loopapproachforPIDtuning."Journalofprocesscontrol20(10):1220‐1234.
Sivertsen,H.(2008).Stabilizationofdesiredflowregimes.DepartmentofChemicalEngineeringNorwegianUniversityofScienceandTechnology.Master.
Skogestad,S.(2003)."SimpleanalyticrulesformodelreductionandPIDcontrollertuning."Journalofprocesscontrol13(4):291‐309.
Skogestad,S.andC.Grimholt(2011).TheSIMCmethodforsmoothPIDcontrollertuning.PIDControlintheThirdMillennium,Springer:147‐175.
Storkaas,E.(2005).Controlsolutionstoavoidslugflowinpipeline‐risersystems.ChemicalEngineering.Trondheim,NorwegianUniversityofScienceandTechnology.PhD.
Storkaas,E.(2005).Stabilizingcontrolandcontrollability.Controlsolutionstoavoidslugflowinpipeline‐risersystems,NorwegianUniversityofScienceandTechnology.
Taitel,Y.(1986)."Stabilityofsevereslugging."Internationaljournalofmultiphaseflow12(2):203‐217.
Yan,K.andD.Che(2011)."Hydrodynamicandmasstransfercharacteristicsofslugflowinaverticalpipewithandwithoutdispersedsmallbubbles."Internationaljournalofmultiphaseflow37(4):299‐325.
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A.LowpassfilterinLabVIEW
In order to implement the low‐pass filter in the experiments the function “PIDAdvancedVI”fromLabVIEWwasused.ThefunctionimplementsaPIDcontrollerusingaPIDalgorithmwithadvancedoptional features.Figure5.6,adapted from theNationalInstruments’website,showstheblockdiagramofrelatedthefunction.
FigureA.1:PIDAdvanced(DBL)
Inthepresentedfigurealphaspecifiesthederivativefiltertimeconstantandcanbeavaluebetween0and1.ThedefaultisNaN,whichspecifiesthatnoderivativefilterisapplied.Therelationbetween F fromthemethodand fromLabVIEWisasfollows:
f
D
EquationA.1
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B. Simulated results to get the best step tests forShams’smethod
In this appendix all simulations run to get the desired overshoot at differentbasisconditionsofthecontrollerarepresented.ThesimulationsarerelatedtotuningofthecontrollerbyShams’smethod(Seesection5.3.3.1).TableB.1presentstheinitialandfinalvaluesofbuffer(inlet)pressureusedascontrolvariableinsimulationsbeforeandafter stepchangeand the resultingovershoot.Theunits are inkiloPascal. 0cK is the
initialgainusedinthetuningsimulations.Thevaluesspecifiedbytheredcolorarethebestresultsthosewereusedtofindtuningparameters.
TableB.1:Resultingovershootstothedifferentsteptestsatdifferentinitialpositionsofchokevalve
Bias 0cK Initialset‐pointvalue
Finalset‐pointvalue Overshoot
0.30
0.5142 143 3.4058142 144 1.7426145 147 2.1049
0.1
142 143 0.5741142 144 0.6318142 145 1.2219143 144 0.5402143 145 0.5609144 145 0.4894149 144 0.3108149 150 0.3308150 151 0.3085
0.400.1
141 142 0.8839145 142 0.4471145 146 0.5028148 149 0.3535
0.15 145 142 0.3210
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C. SomeexamplesofMATLABscripts
C.1 TuningbyShams’smethod
clc clear all load Data Kc0 = -0.1; dy_s = 1; t_init = 200; dt = 0.1; t = Data(:,1); r = Data(:,2); y = Data(:,3); u = Data(:,4); %% figure(1) clf subplot(2,1,1) plot(t,r,'r',t,y,'k','Linewidth',1.5) xlim([0,1000]) title('Inlet pressure(controlled variable)') ylabel('P [Kpa]') legend('Setpoint','Data',2) grid on hold on subplot(2,1,2) plot(t,u,'k','Linewidth',1.5) xlim([0,1000]) xlabel('time(sec)') ylabel('Z') title('Valve position (manipulated variable)') grid on %% i_init = find(t==t_init); y_plant = y(i_init:end); t_plant = t(i_init:end); u_plant = u(i_init:end); y_init = y(i_init-200); u_init = u(i_init-200); yp = max(y_plant); dy_p = abs(yp - y_init); i_yp = find(y_plant==yp); t_p1 = mean(t_plant(i_yp)); yu = min(y_plant(i_yp:10*i_yp)); dy_u = abs(yu - y_init); i_yu = find(y_plant==yu); t_u = mean(t_plant(i_yu)); y_inf = y_plant(end); dy_inf = abs(y_inf - y_init); tp = t_p1 - t_init;
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Overshoot = abs((dy_p-dy_inf)/dy_inf); D = Overshoot Offset = abs((dy_s-dy_inf)/dy_inf) B = Offset; A = 1.152*D^2 - 1.607*D +1; r = 2*A/B; K = 1/(Kc0 * B); Tetha = tp*(0.309 + 0.209*exp(-0.61*r)); tau1 = r*Tetha; Kc = tau1/(K*2*Tetha) tauI = min (tau1, 8*Tetha) Tau_c=Tetha
C.2 ModelidentificationbasedonIMC‐design
clc clear all close all load z30_148_150 Kc0 = -0.1; dy_s = 2; t_init = 200; dt = 0.1; t = z30_148_150(:,1); y = z30_148_150(:,2); r = z30_148_150(:,3); u = z30_148_150(:,4); %% figure(1) plot(t,r,'--r','LineWidth',2.25); hold on plot(t,y,'b','LineWidth',2.25); xlabel('time(sec)'); ylabel('Inlet pressure [kPa]'); xlim([170 300]); title('Closed-loop step response from OLGA simulations'); grid on hold on %% i_init = find(t==t_init); y_plant = y(i_init:end); t_plant = t(i_init:end); u_plant = u(i_init:end); y_init = y(i_init-100); u_init = u(i_init-100); yp1 = max(y_plant); dy_p1 = abs(yp1 - y_init); i_yp1 = find(y_plant==yp1); t_p1 = mean(t_plant(i_yp1)); yu = min(y_plant(i_yp1:10*i_yp1)); dy_u = abs(yu - y_init); i_yu = find(y_plant==yu); t_u = mean(t_plant(i_yu));
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yp2 = max(y_plant(i_yu:2*i_yu)); dy_p2 = abs(yp2 - y_init); y_inf = y_plant(end); dy_inf = abs(y_inf - y_init); D0 = (dy_p1 - dy_inf)/dy_inf; deltaT = t_u - t_p1; v1 = (dy_inf - dy_u)/(dy_p1 - dy_inf); z = -log(v1)/sqrt(pi^2+(log(v1))^2); Tau = (deltaT/pi)*sqrt(1-z^2); K = dy_inf/(dy_inf-dy_s); K2 = K/(K-1); alpha = (K+1)/(K-1); Tau1 = 2*z*Tau*(K-1)+sqrt(4*z^2*Tau^2*(K-1)^2+(K+1)*(K-1)*Tau^2); tp = t_p1 - t_init; Phi = atan((1-z^2)/z)-tp*sqrt(1-z^2)/Tau; E = sqrt(1-z^2)/Tau; D1 = D0/(exp(-z*(tp)/Tau)*sin(E*(tp)+Phi)); Tauz = z*Tau + sqrt(z^2*Tau^2-Tau^2*(1-D1^2*(1-z^2))); s=tf('s'); disp('The identified closed loop model:') G2 = K2*(1+Tauz*s)*exp(-0*s)/(Tau^2*s^2 + 2*z*Tau*s + 1) u = [zeros(1,round(t_init/dt)) dy_s*ones(1,round((3600-t_init)/dt)+1)]; t = 0:dt:3600; y1 = lsim(G2,u,t); plot(t,y1+y_init,'--k','LineWidth',2.25); legend('Setpoint','OLGA measurement','Identified model',3); %% %BACK CALCULATION OF THE OPEN LOOP UNSTABLE SYSTEM%% A0 = 1/Tau^2; A1 = 2*z/Tau; B0 = K2/Tau^2; B1 = K2*Tauz/Tau^2; Kp = dy_inf/(Kc0*abs(dy_s-dy_inf)); a0 = A0/(1+Kc0*Kp); b0 = -Kp*a0; b1 = -B1/Kc0; a1 = -A1-Kc0*b1; s = tf('s'); disp('Identified model:') Ge = (-b1*s-b0)/(s^2-a1*s+a0) gcl = feedback(Kc0*Ge,1);
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C.3 DesignofInternalModelController
%% Internal Model Controller (IMC) % Plant Information [Zero,Pole,Gain,Ts] = zpkdata(Ge,'v'); indRHPzero = (real(Zero)>0); % indices of open RHP zeros indRHPpole = (real(Pole)>0); % indices of open RHP poles RHPpoles = Pole(indRHPpole); % RHP poles NumRHPzeros = sum(indRHPzero); % number of open RHP zeros NumRHPpoles = sum(indRHPpole); % number of open RHP poles Tauc = 10; % Tuning parameter: time constant of the closed-loop system % for MP systems q_tilde = zpk(Pole,Zero,1/Gain); k = NumRHPpoles+1; % since Vm always contains an pole at origin for step input m = max(length(zero(q_tilde))-length(pole(q_tilde)),1); % make sure q=q_tilde*f is proper filterOrder = m+k-1; % 3. calculate filter as sum(a(k)s^k)/(tau*s+1)^filterOrder coefficients = ones(1,k); if NumRHPpoles>0 A = zeros(NumRHPpoles,NumRHPpoles); for ctRHPpole = 1:length(RHPpoles) A(ctRHPpole,:) = RHPpoles(ctRHPpole).^(1:NumRHPpoles); end b = (Tauc*RHPpoles+1).^filterOrder-coefficients(1); coefficients(2:end) = (real(A\b))'; end % computing f num = fliplr(coefficients); den = fliplr(poly(repmat(-Tauc,1,filterOrder))); f = tf(num,den); q = minreal(q_tilde*f); C = feedback(q,Ge,+1); disp('The IMC controller:') C = minreal(C) L1 = C*Ge; allmargin(L1)
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C.4 FindingPID/PItuningrulesbasedonIMC‐design
disp('IMC based PID tuning:') [Kc_PID,Ki_PID,Kd_PID,Tf_PID] = piddata(C) Ti_PID = Kc_PID/Ki_PID; Td_PID = Kd_PID/Kc_PID; disp(['Kp = ' num2str(Kc_PID)]) disp(['Ti = ' num2str(Ti_PID)]) disp(['Td = ' num2str(Td_PID)]) disp(['Tf = ' num2str(Tf_PID)]) disp('FEED TO OLGA AND CLOSE THE LOOP!') C2 = Kc_PID*( 1 + 1/(Ti_PID*s) + Td_PID*s/(Tf_PID*s+1)); L2 = C2*Ge; allmargin(L2) %% %Reduce to PI Controller C3 = balancmr(C,1); [Kc_PI,Ki_PI] = piddata(C3); Ti_PI = Kc_PI/Ki_PI; disp('IMC based PI tuning:') disp(['Kp = ' num2str(Kc_PI)]) disp(['Ti = ' num2str(Ti_PI)]) disp('FEED TO OLGA AND CLOSE THE LOOP!') C3 = Kc_PI*(1+1/(s*Ti_PI)); L3 = C3*Ge; allmargin(L3)
C.5 Simplestaticmodelfittedtoexperiments
%%%%Simple Static Model%%% clc clear all g = 9.81; %Gravity (m/s2) Wg_in=0.0024; %Inlet mass flow rate of gas (Kg/sec) Wl_in=0.39298; %Inlet mass flow rate of liquid(Kg/sec) W = Wg_in+Wl_in; %Inlet mass flow rate (Kg/sec) R = 8314; %Gas constant (J/(K.Kmol)) M_g = 29; %Molecular weight of Gas (kg/kmol) p_s=101325; %Separator pressure (pa) p_vmin = 0; %minimum Pressure drop over the valve (Pa) T=15+273.15; %Riser temperature (K) par.r2 = 0.025; %Radius of riser (m) par.A2 = pi*par.r2^2; %Cross section area of riser (m2) rho_g= (p_s+p_vmin)*M_g/(R*T); %Average gas density at the outlet(Kg/m3)
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rho_l=1000; %Liquid density (Kg/m3) alpha_g = Wg_in/(Wg_in+Wl_in); %Average gas mass fraction alpha_l=(1-alpha_g)*rho_g/((1-alpha_g)*rho_g+alpha_g*rho_l); %liquid volume fraction rho = alpha_l*rho_l+(1-alpha_l)*rho_g; %Average mixture density L_r = 5.15; %Length of riser z_star= 0.26; %Bifurcation point cd = 0.31; %Discharge coefficient of valve K_pc = sqrt(2)* par.A2; %Valve constant (m2) a = (1/rho)*((W/K_pc)^2); %Constant parameter p_star= (rho_l*g*L_r)+p_s+ p_vmin; fz_star = z_star*cd/sqrt(1-z_star^2*cd^2); delta_p_star = a/fz_star^2; p_fo = p_star-delta_p_star; %inlet pressure at fully open position of the valve(at z=1) (pa) z_t = 0.2:0.001:1; %Different valve openings n = length(z_t); Pin = zeros(1,n); %Inlet Pressure K_z_t = zeros(1,n); %Static gain of the system (pa) for i = 1:n fz = z_t(i)*cd/sqrt(1-z_t(i)^2*cd^2); Pin(i) =(a/fz^2 + p_fo)/1000; K_z_t(i) = (-2*a/z_t(i)^3*cd^2)/1000; end figure(1) clf subplot(2,1,1); plot(z_t,Pin,'k','LineWidth',2); xlabel('Z'); ylabel('Inlet Pressure [kPa]'); hold on grid on subplot(2,1,2); plot(z_t,K_z_t,'k','LineWidth',2); xlabel('Z'); ylabel('K(z)'); hold on grid on
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C.6 Onlinetuningbasedonsimplestaticmodelandaclosedloopsteptest
clc clear all load z40_141_142 z0 = 0.4; %Initial valve position in step-test Kc0= -0.15; %gain used for the step test dy_s = 1; t_init = 300; dt = 0.1; t = z40_141_142(:,1); y = z40_141_142(:,2); r = z40_141_142(:,3); u = z40_141_142(:,4); i_init = find(t==t_init); y_plant = y(i_init:end); t_plant = t(i_init:end); u_plant = u(i_init:end); y_init = y(i_init-10); u_init = u(i_init-10); yp = max(y_plant); %Step change is positive dy_p = abs(yp - y_init); i_yp = find(y_plant==yp); t_p1 = mean(t_plant(i_yp)); yu = min(y_plant(i_yp:10*i_yp)); dy_u = abs(yu - y_init); i_yu = find(y_plant==yu); t_u = mean(t_plant(i_yu)); y_inf = y_plant(end); dy_inf = abs(y_inf - y_init); tp = t_p1 - t_init; deltat = t_u - t_p1; %% %%%%%%MODEL%%%%%% z = 0.3 %The operating point g = 9.81; %Gravity (m/s2) Wg_in=0.0024; %Inlet mass flow rate of gas (Kg/sec) Wl_in=0.39298; %Inlet mass flow rate of liquid (Kg/sec) W = Wg_in+Wl_in; %Inlet mass flow rate (Kg/sec) R = 8314; %Gas constant (J/(K.Kmol)) M_g = 29; %Molecular weight of Gas (kg/kmol) p_s=101325; %Seperator pressure (pa) p_vmin = 0; %minimum Pressure drop over the valve (Pa) T=15+273.15; %Riser temprature (K) par.r2 = 0.025; %Radius of riser (m) par.A2 = pi*par.r2^2; %Cross section area of riser (m2)
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rho_g= (p_s+p_vmin)*M_g/(R*T) %Average gas density at the outlet(Kg/m3) rho_l=1000; %Liquid density (Kg/m3) alpha_g = Wg_in/(Wg_in+Wl_in) %Average gas mass fraction alpha_l=(1-alpha_g)*rho_g/((1-alpha_g)*rho_g+alpha_g*rho_l) %liquid volume fraction rho = alpha_l*rho_l+(1-alpha_l)*rho_g %Average mixture density L_r = 5.30; %Length of riser z_star= 0.26; %Bifurcation point cd = 0.31; %Discharge coefficient of valve K_pc = sqrt(2)* par.A2 %Valve constant (m2) a = (1/rho)*((W/K_pc)^2) %Constant parameter p_star= (rho_l*g*L_r)+p_s+ p_vmin fz_star = z_star*cd/sqrt(1-z_star^2*cd^2) delta_p_star = a/fz_star^2 p_fo = p_star-delta_p_star %inlet pressure at fully open position of the valve(at z=1) (pa) fz = z*cd/sqrt(1-z^2*cd^2) K_z = -2*a/z^3*cd^2 %static gain of the system (pa) K_z0 = -2*a/z0^3*cd^2 %% %%%PI tuning parameters%% T_osc= 140 %period of slugging oscillations in sec in the open loop system Betha=-log((dy_inf-dy_u)/(dy_p-dy_inf))/(2*deltat)+ Kc0*K_z0 *((dy_p-dy_inf)/dy_inf)^2/(4*tp) Kc=Betha*T_osc/K_z*sqrt(z/z_star) tauI_z=3*T_osc*(z/z_star) disp('FEED TO OLGA AND FIND THE MAXIMUM STABILITY!') %% %%%%%%%%%%Plot%%%%%%% %MODEL z_t = 0.2:0.001:1; n = length(z_t); Pin = zeros(1,n); K_z_t = zeros(1,n); for i = 1:n fz = z_t(i)*cd/sqrt(1-z_t(i)^2*cd^2); Pin(i) =(a/fz^2 + p_fo)/1000; K_z_t(i) = (-2*a/z_t(i)^3*cd^2)/1000; end figure(1) clf plot(z_t,Pin,'r','LineWidth',2.5); xlabel('Z'); ylabel('Inlet Pressure [kPa]'); hold on grid on %%
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%%%%%%OLGA MODEL load Openloop z_olga = Openloop (:,1); P_max = Openloop (:,2); P_min = Openloop (:,3); P_ss = Openloop (:,4); figure (1) plot (z_olga,P_ss,'b','LineWidth',2.5); hold on legend('Simple static MATLAB model','OLGA case',2) plot (z_olga,P_max,'b','LineWidth',2.5); hold on plot (z_olga,P_min,'b','LineWidth',2.5); grid on
C.7 SimplestaticmodelfittedtotheOLGAsimulatedmodel
clc clear all %% %%%%%%%STEP TEST INFORMATION%%%%%%%% % load z30_148_150 % z0 = 0.3; %Initial valve position in step-test % Kc0= -0.1; %gain used for the step test % dy_s = 2; % t_init = 200; % dt = 0.1; % % t = z30_148_150(:,1); % y = z30_148_150(:,2); % r = z30_148_150(:,3); % u = z30_148_150(:,4); load z40_141_142 z0 = 0.4; %Initial valve position in step-test Kc0= -0.15; %gain used for the step test dy_s = 1; t_init = 300; dt = 0.1; t = z40_141_142(:,1); y = z40_141_142(:,2); r = z40_141_142(:,3); u = z40_141_142(:,4); i_init = find(t==t_init); y_plant = y(i_init:end); t_plant = t(i_init:end); u_plant = u(i_init:end);
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y_init = y(i_init-10); u_init = u(i_init-10); yp = max(y_plant); %Step change is positive dy_p = abs(yp - y_init); i_yp = find(y_plant==yp); t_p1 = mean(t_plant(i_yp)); yu = min(y_plant(i_yp:10*i_yp)); dy_u = abs(yu - y_init); i_yu = find(y_plant==yu); t_u = mean(t_plant(i_yu)); y_inf = y_plant(end); dy_inf = abs(y_inf - y_init); tp = t_p1 - t_init; deltat = t_u - t_p1; %% %%%%%%MODEL%%%%%% z = 0.3 %The operating point g = 9.81; %Gravity (m/s2) Wg_in=0.0024; %Inlet mass flow rate of gas (Kg/sec) Wl_in=0.39298; %Inlet mass flow rate of liquid (Kg/sec) W = Wg_in+Wl_in; %Inlet mass flow rate (Kg/sec) R = 8314; %Gas constant (J/(K.Kmol)) M_g = 29; %Molecular weight of Gas (kg/kmol) p_s=101325; %Seperator pressure (pa) p_vmin = 0; %minimum Pressure drop over the valve (Pa) T=15+273.15; %Riser temprature (K) par.r2 = 0.025; %Radius of riser (m) par.A2 = pi*par.r2^2; %Cross section area of riser (m2) rho_g= (p_s+p_vmin)*M_g/(R*T) %Average gas density at the outlet(Kg/m3) rho_l=1000; %Liquid density (Kg/m3) alpha_g = Wg_in/(Wg_in+Wl_in) %Average gas mass fraction alpha_l=(1-alpha_g)*rho_g/((1-alpha_g)*rho_g+alpha_g*rho_l) %liquid volume fraction rho = alpha_l*rho_l+(1-alpha_l)*rho_g %Average mixture density L_r = 5.30 %Length of riser z_star= 0.26; %Bifurcation point cd = 0.31; %Discharge coefficient of valve K_pc = sqrt(2)* par.A2 %Valve constant (m2) a = (1/rho)*((W/K_pc)^2) %Constant parameter p_star= (rho_l*g*L_r)+p_s+ p_vmin fz_star = z_star*cd/sqrt(1-z_star^2*cd^2) delta_p_star = a/fz_star^2 p_fo = p_star-delta_p_star %inlet pressure at fully open position of the valve(at z=1) (pa) fz = z*cd/sqrt(1-z^2*cd^2) K_z = -2*a/z^3*cd^2 %static gain of the system (pa) K_z0 = -2*a/z0^3*cd^2 %%
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%%%PI tuning parameters%% T_osc= 140 %period of slugging oscillations in sec in the open loop system Betha=-log((dy_inf-dy_u)/(dy_p-dy_inf))/(2*deltat)+ Kc0*K_z0 *((dy_p-dy_inf)/dy_inf)^2/(4*tp) Kc=Betha*T_osc/K_z*sqrt(z/z_star) tauI_z=3*T_osc*(z/z_star) disp('FEED TO OLGA AND FIND THE MAXIMUM STABILITY!') %% %%%%%%%%%%Plot%%%%%%% %MODEL z_t = 0.2:0.001:1; n = length(z_t); Pin = zeros(1,n); K_z_t = zeros(1,n); for i = 1:n fz = z_t(i)*cd/sqrt(1-z_t(i)^2*cd^2); Pin(i) =(a/fz^2 + p_fo)/1000; K_z_t(i) = (-2*a/z_t(i)^3*cd^2)/1000; end figure(1) clf plot(z_t,Pin,'r','LineWidth',2.5); xlabel('Z'); ylabel('Inlet Pressure [kPa]'); hold on grid on %% %%%%%%Openloop and Steady-state load Openloop z_olga = Openloop (:,1); P_max = Openloop (:,2); P_min = Openloop (:,3); P_ss = Openloop (:,4); figure (1) plot (z_olga,P_ss,'b','LineWidth',2.5); hold on legend('Simple static MATLAB model','OLGA case',2) plot (z_olga,P_max,'b','LineWidth',2.5); hold on plot (z_olga,P_min,'b','LineWidth',2.5); grid on
