Model predictive control for combined cycles integrated with CO2 capture plants

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Computers and Chemical Engineering

journalhomepage:www.elsevier.com/locate/compchemeng

Model predictive control for combined cycles integrated with CO ₂ capture plants

Jairo Rúa

^a,∗

, Magne Hillestad

^b

, Lars O. Nord

^a

aDepartment of Energy and Process Engineering, Norwegian University of Science and Technology, Norway

bDepartment of Chemical Engineering, Norwegian University of Science and Technology, Norway

a rt i c l e i nf o

Article history:

Received 14 September 2020 Revised 23 December 2020 Accepted 28 December 2020 Available online 31 December 2020 Keywords:

Gas turbine combined cycle Amine absorption process Monoethanolamine (MEA) Dynamic modelling and simulation Advanced control strategy Post-combustion CO 2capture

a b s t r a c t

FlexiblethermalpowerplantsintegratedwithCO₂capturesystemscanbalancetheintermittentpower generationofrenewableenergysourceswithlow-carbonelectricity.Amongthesepowersystems,natu- ralgascombinedcycleswillplayafundamentalrolebecauseoftheirfasteroperationandhighereffi- ciency.Optimisation-basedcontrol strategiescanenhancetheflexiblepowerdispatchofthesesystems andimprovetheirperformanceduringtransientoperation.Thisworkproposesamodelpredictivecon- trol(MPC)strategy tostabilisethese power plantswithpost-combustion CO₂ capture basedontem- peratureswingchemicalabsorptionandprovideoffset-freereferencetracking.Adelta-inputformulation withdisturbancemodellingisproposed,asitprovidesmoreefficientcomputationwithoffset-freecon- trol.Data-basedmodelsweredevelopedtoreplicatetheperformance oftheactualpowerand capture plants.Predictionofnonlinearbehaviourwasaccomplishedbycreatinganetworkoflocallinearmodels, whichallowedtheformulationofthedynamicoptimisationprogram intheMPCstrategyas aconvex quadraticprogrammingproblem.Acase studydemonstratedthe effectivenessoftheproposed MPCto balancedrasticchangesonpowerdemandandkeepspecifiedcapture ratios.Furthermore,thereduced deviationsachievedinthereboilertemperaturesuggestthatthenominalvalueofthisparametercould beincreasedtoimprovethedesorptionprocesswithoutrisksofreachingtemperatureswherethesolvent woulddegradate.

© 2020 The Author(s). Published by Elsevier Ltd.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Climate change mitigation requires a profound reduction of greenhousegasemissions(IPCC,2014;2018).Bysector,powergen- eration is the main contributor to global CO₂ emissionsbecause ofits relianceonfossilfuels(IEA,2019).Deploymentofintermit- tentrenewableenergysources,mainlywindandsolar,hasconcen- trated mostof the efforts to decarbonise thissector (IEA, 2019).

However, abroaderportfoliooftechnologiesisnecessarytomeet theincreasing powerdemandwhilst ensuringasafe,efficientand sustainableelectricmarket.Inthiscontext,theintegrationofflex- ible carboncapture andstorage(CCS)withthermalpowerplants isexpectedtoplayafundamentalroleinthereductionoftheCO₂ emissionsassociatedwiththepowersector(IPCC,2005;2014).

Thermal power plants, especially natural gas combinedcycles (NGCC),arerecognisedasaviabletechnologytoaccommodatethe intermittentpowergenerationfromrenewableenergysourcesand

∗Corresponding author.

E-mail addresses: [email protected] (J. Rúa), [email protected] (L.O. Nord).

balancetheelectricgrid(KondziellaandBruckner,2016;Eseretal., 2017;González-Salazaretal.,2017).FlexibleCCSmayenhancethis dispatchablenatureofflexiblethermalpowerplantsby providing lowcarbonelectricityinacosteffectivemanner(Montañésetal., 2016;Heubergeretal., 2016;2017a;2017b).Post-combustionCO₂ capture(PCC)basedonliquid-absorbentsisarguablythemostma- tureCCStechnology, withtwo commercial-scalecapture facilities integratedwith coal powerplants in operation (Bui etal., 2018).

Nevertheless,thedeploymentofthistechnologyinpowermarkets dominatedby intermittentrenewableenergysources requiresthe demonstration that integration of CCS andthermal power plants doesnotinhibitflexibleandefficientpowergeneration,andstable CO₂ capture.

The dominant dynamics that govern the transient opera- tion of thermal power plants, CO₂ capture plants and systems integrated by both technologies were extensively analysed by Rúaet al.(2020b). Twodifferentdynamicbehaviour define tran- sientoperationofthesetechnologies. Thermalpowerplantsoper- ateinshorttime-scalesandare limitedbythelarge heatcapaci- tanceof thesteam generator, whereas post-combustionCO₂ cap-

https://doi.org/10.1016/j.compchemeng.2020.107217

0098-1354/© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Nomenclature

LatinSymbols ˆ

A Stateestimation A Delta-inputstatematrix A Statematrix

A(^q−1) ^{PolynomialARXmodel} A_a Augmentedstatematrix a Coefficientssimplifiedmodels B Delta-inputinputmatrix B Inputmatrix

B(^q−1) ^{PolynomialARXmodel} Ba Augmentedinputmatrix B_d Disturbanceinputmatrix b Coefficientssimplifiedmodels C Delta-inputoutputmatrix C Outputmatrix

c Centrevalidityfunction Ca Augmentedoutputmatrix C_d Disturbanceoutputmatrix ^u Delta-inputcontrolvector

δ

^u Delta-inputcontrolaction d Disturbancevector

F MIMOdelta-inputpenaltyvector f Delta-inputpenaltyvector

G MIMOdelta-inputinequalitymatrix g Delta-inputinequalitymatrix H Delta-inputmatrixoutputequation I Identitymatrix

J Objectivefunction K Observergainmatrix K_f Kalmanfilter

M numberlocalARXmodels N Timehorizon

P MIMOdelta-inputinequalityvector p Delta-inputinequalityvector Q Weightmatrix

q⁻¹ Backwardsshiftoperator Qp Processnoisecovariance R Penaltyvector

R² Coefficientofdetermination Rm Measurementnoisecovariance

t Time(s)

u Manipulatedvariable w Widthvalidityfunction

x Delta-inputstatevector x Statevector

xa Augmentedinputstatevector y Predictedvariable,outputvector Z Estimatorcovariancematrix GreekSymbols

Delta-inputweightmatrix

γ

^{Localoperatingpoint}

λ

^{Weightsobjectivefunction MIMO}delta-inputweightmatrix ^{Unitlowertriangularmatrix}

ξ

^{Localvalidityfunction}

σ

^{2 Covariance}

ε

^{Stochasticerror}

Subscripts

0 Initialconditions d Disturbance

nu OrderARXinput ny OrderARXoutput

pow Power

ramp Rampingrate ref Referencetrajectory

u Inputs

x States

Superscripts

- Previousestimation low Lowerbound up Upperbound

ture plants are characterised by slow responses and long time- scales owing to the large volumes of stored solvent, the impact oflarge vessels onresidencetime, andthetransport delayintro- ducedbysomeequipment.Thisdifferenttransientbehaviourdoes not limit power generation since variable steam extraction from theintermediateandlowpressurecross-overofthesteamturbine does not significantly affect the steam cycleof the power plant, albeit it has an impact on process variables of the CO2 capture plant (Rúa et al., 2020b). Thus, control strategies must consider the different dynamicnature of thermal power plants andpost- combustionCO2 captureplantstoadequately stabilisetheprocess variableofeachplantwithintheiroperationtime-scales.

Control of traditional thermal power plants refers to match- ing the power generation to the demand and the stabilisation of the steam cycle. Natural gas combined cycles utilise the gas turbine to control power generation owing to their fast dynam- ics(Kehlhoferetal., 2009). Coal andbiomass powerplants must adaptthefuelandairinjectedintheboilerandthrottlethesuper- heatedandreheatedsteamflow attheinletofthesteamturbine (Alobaidetal.,2017).Powergenerationcontrolincoalandbiomass power plantsis hence dominatedby the heat capacitance of the boiler. Therefore,the fasttransient operation ofgas turbinesand their capability to adapt the power output within seconds make NGCCsmoresuitable forflexible operation andgridbalance than coalandbiomasspowerplants(Hentscheletal.,2016;Eseretal., 2017). Furthermore,NGCCs can under-and over-shootthe power generatedby the gasturbine tocompensate the slowertransient ofthesteamcycle,enhancingtheflexibilitythatthistypeofpower plantsprovidetothegrid(Rúaetal.,2020a;RúaandNord,2020).

Steamcyclecontrol includestheregulationofthe fluidinven- toryinthesteamdrums, deaerators,condensers,andstorageves- sels;pressurecontrolofthelow-,medium-andhigh-pressuresec- tions of the steam cycle; and temperature limitation of the su- perheated and reheated steam to avoid damaging the pipe sys- tem and the steam turbine. Inventory control refers to the sta- bilisation of the mass flows so the steady-state mass balances foreach ofthe components andtheoverall power plantare sat- isfied (Aske and Skogestad, 2009). Proportional-integral (PI) con- trollers are normally used for control of water levels since the main objective of this control layer is to stabilise power plant operation, although three-element controllers where the drum level,feedwater flow andlive-steamfloware embeddedina PID (proportional-integral-derivative) cascade controller are tradition- allyimplementedin thermalpowerplants(Mansouretal., 2003;

Kehlhoferetal.,2009).Thesecontrollersadjustthefeedwatermass flow by changing the speed ofthe pumps orthe openingof the control valves, depending on the type and design of the power plant.Modelpredictivecontrol(MPC)strategiesleadtofurtherim- provementsintheinventorycontroloftraditionalpowerplantsbe- cause of the dynamic optimisation carried out to determine the most suitable control action (Lu and Hogg, 1997; Prasad et al., 2000).

Pressure control is achieved by adjusting the feedwater mass flow rateand by valve throttling, specially in the lower-pressure sectionsofNGCCswherethe pressureinthedrum anddeaerator may be controlled (Casella and Pretolani, 2006; Montañés et al., 2017c). Inthehigh-pressuresection ofthesteam cycle,strategies such partial arc andsliding pressure control lead to better part- load performance (Kehlhofer etal., 2009; Jonshagen andGenrup, 2010).Partialarccontrol regulates thesteam admittanceintothe steamturbinewithseveralvalvesinthestatorofthefirststage.In contrast, thesevalvesare close to fully-openduringsliding pres- sure operation to allow the variation of the high pressure and keep almost constant volumetric flow in the turbine, which re- sultsinhigherpart-loadisentropicefficiency(JonshagenandGen- rup, 2010). Ifthehighpressureof thesteamcycleisnot allowed tofluctuate,optimisation-basedstrategiesleadtoimprovedcontrol ofthispressureastheyreducethedeviationfromitsset-point(Lu andHogg,1997;Prasadetal.,1998;2000;Pengetal.,2009)

Thetemperatureinthehotsectionsofthesteamcycle,i.e.the outletofthesupeheaterandreheater,mustbecontrolledtoavoid damagingthematerials.Spraycoolingishencenecessarytoinject pressurised waterin the steam flow andreduce its temperature.

Theopeningoftheattemperatorvalvesregulatingtheflowofpres- surised watermay be defined by PID controllers (Alobaid et al., 2008;Kehlhoferetal.,2009;Montañésetal.,2017c;Garðarsdóttir et al., 2017), adaptative controllers (Matsumura et al., 1998), or optimisation-based controllers (Peng et al., 2009; Prasad et al., 1998, 2000; Rúa etal., 2020a; Rúa andNord, 2020). Among the differentalternativestoregulatethemaximumtemperatureinthe steam cycle, model predictive control showsthe minimum offset from the set-point and the fastest stabilisation time (Rúa et al., 2020a;RúaandNord,2020).

Incontrasttothermalpowerplants,controlofpost-combustion CO₂ capture plants is not a mature field and most of the avail- able knowledge comes fromdynamicstudies andtest campaigns in pilot plants. Basic control of PCC plants reduces to stabilise liquid levels insumps ofabsorber and strippercolumns,reboiler andcondenser;regulatethetemperatureofleansolventandcon- denser; adaptthe pressure of the reboiler and CO₂ product, and maintain a constant solvent composition (Panahi and Skogestad, 2011; Schach et al., 2013; Flø et al., 2015; 2016; Walters et al., 2016;Montañésetal.,2017a;2018;Wuetal.,2020).Temperature control is achievedbyheat exchangers wherethe massflow rate of cooling water is the manipulated variable, whereas inventory controlrequiresseveralpumpstostabiliseliquidlevelsindifferent equipment,althoughvalvesmayalsobeused.Throttlingregulates thepressureofproductofCO₂andthemassflowrateofmake-up solvent,orwater,neededforaconstantcomposition.Controlofall theseprocessvariablesmayleadtoover-constrainedsystems,and some mightbeleftuncontrolled.Forinstance,thelevelinthere- boileriscontrolledandthesumplevelinthestrippervariesfreely in theBrindisipilot plant(Flø et al., 2016), whereas theopposite inventory control approachis implemented at Technology Centre Mongstad(TCM)(Montañésetal.,2017a;2018).

Thisbasiccontrolstrategyaimsatstabilisingthemainprocess variablesandensuringsafeoperationofPCCplants.Therefore,PID controllers are normallyimplemented. This control layer is simi- laramongdifferentpilotplantsanddynamicprocess models(see e.g. the reviewsby Salvinderet al.(2019) andWu et al.(2020)).

The maindifference incontrolstrategies andperformance ofPCC plants lies on the pairings and methods used to control perfor- mance indicators,i.e. capturerate orCO₂ product, liquidsolvent to gas (L/G) ratios, energy performance ratios, and reboiler per- formance, where the lattermay refer to outletsolvent tempera- ture,outletleanloadingorheatduty.Themajorityofpairingsbe- tweencontrolledandmanipulatedvariablesoriginatefrominsights obtained duringprocess dynamicsimulations, albeit relative gain

array (RGA) analyses and self-optimisation procedures have been proposed (Panahi andSkogestad,2011; 2012;Schach etal., 2013;

Nittayaetal.,2014;SahraeiandRicardez-Sandoval,2014;Luuetal., 2015;Manafetal.,2016;Gasparetal.,2016).Differentcontrolde- sign strategiesmay lead to distinct pairings with various perfor- mance,butnoneofthedesignmethodshaveprovedsystematically superior.

Traditional PID controllers are able to reject disturbances and trackreferencesofCO₂ captureratesby modifyingthemass flow rateoflean/richsolventattheinlet/outletoftheabsorbercolumn (Lawaletal., 2010;Nittayaetal.,2014;Garðarsdóttiretal.,2015;

Luuetal.,2015;Manafetal.,2016;Gasparetal., 2016;Montañés etal.,2017a),orthesteamflow inthereboiler,i.e.theheat duty (PanahiandSkogestad,2011;Nittayaetal., 2014;Montañésetal., 2017a). Similarly, PIDs can achieve closeto contant reboilertem- perature(Lawaletal.,2010;PanahiandSkogestad,2011;2012;Nit- tayaetal.,2014;Waltersetal.,2016;Montañésetal.,2017a;2018), L/Gratios(Garðarsdóttiretal.,2015;Montañésetal.,2017a;2018), leansolventloading(Garðarsdóttiretal.,2015;Gasparetal.,2016) or energyperformance indicators (Luu et al., 2015; Manaf et al., 2016)bymanipulatingthemassflowrateofsolventorthereboiler heatduty.Thesestudies demonstratePID controllerscan stabilise PCC plantssubjected to large disturbances within reasonablepe- riodsoftime, albeitthe lackofagreement onthe mostadequate pairingforkeyprocessvariables.

Nevertheless,PIDcontrollers maynot be ableto stabilisepro- cessvariableswithindesirableboundsandcanrequireexcessively long settling times if the tuning is not adequate or the distur- bance toodrastic (Luu etal., 2015).Model predictive control can addressthesechallengesbycomputingthecontrolinputthrougha dynamicoptimisationproblemwhereconstraintsinthecontrolled andmanipulatedvariables ensurethatprocessparameters remain within acceptable limits. MPC also originates less oscillations of smaller amplitudethan PIDs for a givendisturbance (Arceet al., 2012;SahraeiandRicardez-Sandoval,2014;Luuetal.,2015;Zhang etal.,2016;Heetal.,2018;Lietal.,2018;Wuetal.,2018a;2019a).

This behaviour is due to the optimisation of predicted trajecto- riesoveratimehorizon,whichleadstoshortersettlingtimesand tighter control of PCC plants. Hauger etal. (2019) demonstrated thetightcontrolachievedbyMPCstrategiesindifferenttestsper- formedintwopilotfacilities(TillerandTCM).

Furthermore,economiccriteriasuchasmarketpricesorenergy cost maybe includedinMPC formulationsto reduce thepenalty ofCCSsystemswhilekeepingPCCplantsstable(Arceetal.,2012;

Decardi-Nelsonetal.,2018).Thiseases theintegrationofschedul- ingandcontrolstrategiessincetheoutputsoftheschedulingpro- cess may modify, in addition to the set-points of the controlled variables,tuningparameters intheoptimisationproblemincluded intheMPC(Heetal.,2016).

Whilstthereareseveralstudiesanalysingcontrolstrategiesfor thesedifferenttechnologiesoperatingindependently,therearerel- ativefewstudiesconsidering thecontrolofthermalpowerplants integratedwithpost-combustionCO₂ captureplants(Lawaletal., 2012; Mechleri et al., 2017; Garðarsdóttir et al., 2017; Montañés etal.,2017c;Marx-SchubachandSchmitz,2019;Wuetal.,2019b;

2019c).DecentralisedPIDcontrollerscanstabilisetheseintegrated systemswithintheirdifferenttime-scales,wherethedominantdy- namicsofeach plantdictatethesettlingtime. However,the inte- gration of CO₂ captureplants increases the settling time of pro- cess variables (e.g.steam pressure) in coal andnatural gas ther- malpowerplantsbecauseofthelongstabilisationperiods ofCO₂ capture systems (Lawal et al., 2012; Garðarsdóttir et al., 2017;

Montañésetal.,2017c; Mechlerietal.,2017). Similarlyto control strategiesin individual PCC plants,pairing ofcontrolled and ma- nipulated variables affects notably the performance of thesede- centralised controllers, as it influences the amplitude of fluctua-

tionandsettlingtimeofdifferentprocess variablesinbothplants (Garðarsdóttiretal., 2017;Montañésetal.,2017c;Mechlerietal., 2017).Moreover, PID controllers can alsoregulate the start-up of integrated systems and achieve desirable CO₂ capture rates and powergeneration(Marx-SchubachandSchmitz,2019).

Model predictive control can improve the control of thermal power plants integrated with CO₂ capture systems and reduce the settling time ofkey performance variables(Wu etal., 2019b;

2019c). MPC also enables the definition of different operation modes, whichallowsprioritisingpowergeneration,gridbalancing or CO₂ captureaccordingtomarket conditionsand currentregu- lations(Wuetal.,2019b;2019c).However,powergenerationfrom coal-fired powerplants isstill limitedby the heat capacitance of the steam generator, and MPC strategies can only enhance their flexible operation by reducingthe steamextractionfromthe CO₂ capture plant, which leads to momentarily decreases of carbon capture(Wuetal.,2019b;2019c).Naturalgascombinedcyclesreg- ulatetheir powergeneration throughthegasturbine, anddonot needtomodifythesteamextractionfromthecaptureplanttobal- ance the grid. Therefore, applicationof MPC strategies to NGCCs integratedwithPCC plantscanfurther enhancetheflexible oper- ation ofbothsystemswhiletakingadvantageofthefasttransient operationofNGCCstobalancepowergenerationanddemand.

This work demonstrates the application of model predictive control strategies to full-scale natural gas combined cycles inte- grated withpost-combustion CO2 capture plantswith the objec- tive of minimising the deviation of key process variables from their set-points.Section2describesthedynamic,full-scaleNGCC- PCC model andthe simplified models used in the MPC strategy, while Section 3 discusses how to achieve offset-free MPC with thesesimplified modelsanddetails itsmathematical formulation.

Section 4demonstratesthefastcontrol achievedby theproposed MPC strategy through a casestudy where the integrated system needs tobalancea decreaseinpowerdemand.Finalremarksand conclusionsareincludedinSection5.

2. Modelling

Thissectionincludesthedifferentmodelsdevelopedtodemon- strate the application of model predictive control strategies to natural gas combined cycles integrated with capture plants.

Section 2.1describesthehigh-fidelitymodelusedto replicatethe behaviour ofthe NGCC-PCCsystem, whereas Section2.2 presents thesimplifiedmodelsincludedinthedynamicoptimisationprob- lemtopredictthefuturebehaviouroftheactualsystem.

2.1. DynamicmodellingofaNGCC-PCCsystem

Natural gas combined cycles are expected to balance the in- termittent power generation associated with renewable energy sources because of their fast and flexible operation. Moreover, triple-pressure NGCCs with reheating are the most efficient and lesspollutingfossil-fuelledthermalpowerplants(Kehlhoferetal., 2009; Alobaid et al., 2017). This study considers a full-scale 615 MWe NGCC with this configuration. The design was carried out withGTPRO (Thermoflow, 2014)becauseit providesdetailedde- scriptions of the geometry of the equipment, off-design perfor- mance, andoperationmapsofpumpsandgasturbines.Thisdata was implementedin a high-fidelitydynamicmodel developedin Modelica (Modelica Association, 2019; Dassault Systemes, 2016) with thespecialized TPL library (Modelon, 2015), which isbased on conservation equations, detailed heat transfer and pressure dropcorrelations, andmapsofperformancefortheturbomachin- erycomponents.

This thermal power plant wasintegrated with a full-scale 30 wt% MEA-based post-combustion capture process, as this is the

mostmatureCCStechnologyavailable.Systemintegrationoccurred betweentheintermediate-andlow-pressuresteamturbinesofthe NGCC and the reboiler of the PCC plant, where steam extracted fromthesteamcycleprovidestheenergytoregeneratethesolvent inthecaptureplant.Thedesignofthelow-pressuresectionofthe steam turbine was adapted to nominal operating conditions, i.e.

steamisextractedto achievea 90%capturerateat100%gastur- bineload(Jordaletal.,2012;Rezazadehetal.,2015).Furthermore, the designof the PCCplant considered the nominalCO₂ capture rate, the exhaust gas CO₂ concentration and conditions (i.e. flow rate, temperature, pressure), the allowable pressure drops in the absorber andstrippercolumns,columnflooding limitsanda rea- sonablebalancebetweencapitalandoperationalcosts(Jordaletal., 2012;Duttaetal.,2017).BecauseofthesizeoftheNGCC andthe amountoffluegasgenerated,theserequirementsweremetwitha parallelconfigurationwithtwoabsorbercolumnsandonestripper (Montañés etal., 2017c; Dutta etal., 2017). A detailedmodelling descriptionandthoroughvalidationresultsofthesedynamicmod- els can be found in the work by Montañés et al.(2017c). Fig. 1 representsthelayoutoftheNGCC-PCCsystem.

Theseplantsexhibitdifferentdynamicbehaviour.Loadchanges inthegasturbineleadtoimmediatevariationsintheexhaustgas conditions.However,thesechanges affectprogressivelythe steam cycle.Thus,theheatcapacitanceoftheHRSGdominatesthetran- sientperformance of theNGCC. Forthermalpower plantsofthis type and size, step changes in the exhaust gas conditions show dominant dynamics of approximately 10 min, with stabilisation timesof20–25min(Hentscheletal.,2016;Montañésetal.,2017c).

PCCplantshaveslowertransientperformancebecauseofthelong residence time of the solvent, the transport delay introduced by heat exchangers, andthe large amount ofsolvent stored in ves- selsandliquidhold-ups(Rúaetal.,2020b).Similarly,stepchanges in the exhaustgas conditions show that the dominant dynamics of PCC plants of this size occur in approximately 60 min with stabilisation timesof severalhours (Lawal etal., 2010; 2012;Flø etal.,2015;2016;Garðarsdóttiretal.,2015;Montañésetal.,2017c;

2017b).

2.2. SystemIdentification

Thecomputationalcostofsimulatingthehigh-fidelitydynamic modeloftheNGCC-PCCsystemdescribedinSection2.1inhibitsits utilisationinoptimisation-basedcontrolstrategies.Therefore,sim- plified models that replicate the behaviour of specific thermody- namicvariables (e.g.reboiler temperature, capturerate, mechani- cal powergeneration)are requiredto predict theperformance of theintegratedsysteminthemodelpredictivecontrolstrategypro- posedinthiswork.

Systemidentification refers to the developmentof data-based dynamicmodels (Ljung, 1987), andwasutilised to develop auto- regressivemodelswithexogenousvariables(ARX)thatpredictthe dynamicbehaviour ofvariables ofinterest. Eq.(1)represents the generalstructureofanARXmodel:

A

(

^q−1

)

^y

(

^t

)

=B

(

^q−1

)

^u

(

^t

)

+

ε (

^t

)

⁽¹⁾

wherey isthepredictedandcontrolled variable,uisthe manip- ulatedvariableassociatedwithit,AandBarepolynomialsinthe backwardsshiftoperatorq⁻¹ ofordernyandnu,respectively,and

ε

∈N(⁰,

σ

²)^.

A

(

^q−1

)

^=1+a1q⁻¹+a₂q⁻²+· · · +anyq^−ny B

(

^q−1

)

=b1q⁻¹+b2q⁻²+· · · +bnuq^−nu

Table 1 summarises the set of input-output pairs, i.e. the con- trolled variable and its associated manipulated variable, consid- eredin thiswork to control the operationof the NGCC-PCCsys- tem. These input-output pairs present nonlinear behaviour and

Fig. 1. Process diagram of the natural gas combined cycle integrated with the post-combustion capture plant.The nomenclature is as follows. E: Economiser, B: Boiler, S:

Superheater, R: Reheater P: Pressure, L: Low, I: Intermediate, H: High, FWC: Feed-water cooling, RS: Reheated steam, SS: Superheated steam, SE: steam extraction, DCC:

Direct contact cooler, c.w.: cooling water.

singleARXmodelscannotpredictthemaccuratelyinbroadopera- tionrangesbecauseoftheirlinearity.Localmodelnetworksoflin- ear ARXmodels canovercomethislimitation(JohansenandFoss, 1993;Wuetal.,2018b;Jungetal.,2020).Thismodellingapproach relies onthedevelopmentofseverallinearARX modelsatdiffer- ent operation pointsfor each input-output pair. The overall pre- diction ofalocalmodelnetworkistheresultofinterpolatingthe individual predictions of the local ARX models accordingto cur- rentoperationpoint(JohansenandFoss,1993).Thus,localmodels neighbouringthisoperationconditioncontributemoretotheover- allpredictionthanlocalsmodelsofregimesfarfromtheoperation point.Theoutputofalocalmodelnetworkis:

y

(

^t

)

= M

i=1

y_i

(

^t

) ξ

i

( γ )

⁽²⁾

whereMisthenumberoflocalmodelsforeachinput-outputpair, y_i(^t)^{representstheoutputs ofthelocalARX models,}

ξ

^isthelo-

cal validity function that weights the contribution of each local ARXmodel,and

γ

^{istheparameterdefiningthecurrentoperating}

point.Thisisequivalenttofirstinterpolatingtheparameters(^a,b) of the local ARX models using the local validity function

ξ

^and

then computing the output ofthe overall ARX modelwith these parameters.

This work considered a Gaussian validity function because it satisfies a necessary conditionto achieve arbitrarily good predic- tionswithlocalmodelnetworks(JohansenandFoss,1993):

ξ

i

( γ )

^{= exp}

−¹₂[

( γ

^−ci

)

^/wi]2

_M

j=1exp

−¹₂

γ

^−cj

/wj

2

⁽³⁾

wherec_iandw_iare,respectively,thecentresandwidthsofthelo- calGaussianinterpolationfunctions.TableA.4includesthenumber

oflocalmodelsforeachinput-outputpair,theparametersofeach localARXmodel,andthevariablesdefiningtheirvalidityfunctions.

Datatogeneratethesemodelswasobtainedfromexcitationof thehigh-fidelitymodeldescribedinSection 2.1atdifferentoper- ation conditions. Therefore, each set of data was used to gener- ateasinglelocalARX modelforevery input-outputpair.Random gaussian signals (RGS) were superimposed on the controllers of theNGCC-PCCsysteminclosed-loopsincethisapproachenhances theidentificationofARXmodels(GeversandLjung,1986;Forssell andLjung,1999;Gevers,2005;Geversetal.,2006;Miškovi´cetal., 2008). In addition, an unique validation set of data covering the entireoperationrangeoftheNGCC-PCCsystemwasgeneratedfol- lowingthesameapproach.

Table1summarisesthepredictionaccuracyofthelocalmodel networkforeachinput-outputpairmeasuredbythecoefficientof determinationR².Thelow R² ofthesimplifiedmodelsforthesu- perheating and reheating temperature originate from the nature ofthevalidationdata.TheRGSsignalssuperimposed onthecon- trollers to generate the identification data fluctuated faster than the dominant dynamics of the steam cycle, which lead to dras- tic andfast changes in the controlled and manipulated variables ofthe NGCC. This created a challengingset of datathat allowed testingwhetherthe localmodel network could predictlarge and frequentfluctuations.Incontrast,thePCCdatadoesnotshowthis behaviour because of theslower dominant dynamicsof the cap- ture plant and its buffering effect, mainly through solvent ves- selsandliquidhold-ups (Rúaetal.,2020b).Thistransientperfor- manceresultsinsmootherandslowervariationseasiertopredict thatlead tohigherR² values.Fig.B.4illustrates thisdifferentbe- haviourbetween theNGCC andPCC plantsfora smallset ofthe validationdata,andhowtheARXmodelsoftheNGCC adequately predictthetrajectoryoftheoutputvariablesdespitethelowerR² values.

Table 1

Input-output pairs with model order and coefficient of determination.

Plant Input-output pair Order Nominal

R ²[%]

Controlled variable ( y ) Manipulated variable ( u ) n y n u n y n u

NGCC Power generation Gas turbine load 99.95

Superheated steam temperature Opening attemperator valve 1 2 2 592.7 ^◦C 0.02655 69.59 Reheated steam temperature Opening attemperator valve 2 2 2 592.5 ^◦C 0.07882 74.37

PCC Capture rate Mass flow lean solvent 1 1 90 % 614 98.40

Reboiler temperature Opening steam extraction valve 1 1 119.22 ^◦C 0.69 99.09

Incontrasttotheothersimplifiedmodels,thepowergeneration of the NGCC waspredictedusing an unique polynomial overthe entiresetofoperatingconditions.Asimplerepresentationforthis variable is possible owing to thelinear relationship betweenthe powergenerationoftheNGCCandtheloadofthegasturbineover abroadoperatingregion.Thestructureofthismodelis:

y

(

^t

)

=a+bu

(

^t

)

⁽⁴⁾

ARX models are suitable for system identification procedures because the computation of their coefficients becomes a sim- ple least-squareproblemora convexoptimisation,whereas other structuresmayinvolvemorecomplex,possiblynon-convex,identi- ficationproblems(Huusometal.,2010).However,foranalysispur- poses,state-spaceformsofARXmodelsarepreferred.Therealisa- tioninobservableformoftheARXmodelinEq.(1)is:

xk+1=Axk+Buk (5a)

y_k=Cx_k (5b)

with

A=

⎡

⎢ ⎢

⎢ ⎢

⎣

−a1 1 0 · · · 0

−a2 0 1 · · · 0 ..

. ... ... ... ...

−any−1 0 0 · · · 1

−any 0 0 · · · 0

⎤

⎥ ⎥

⎥ ⎥

⎦

^B=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

0 .. . b1

.. . bnu

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦

C=[10· · ·0]

where B has ny−nu zeros, and x∈R^ny, u,y∈R, A∈R^nyxny, B∈ R^nyx1,andC∈R^1xny.ThisrealisationisvalidwhentheARXmodel leadstoproperrationaltransferfunctions,i.e.n_y≥n_u.Thestochas- ticerrorterminEq.(1)isnotincludedbecauseofthedeterminis- ticdatausedduringsystemidentification.

3. Modelpredictivecontrol

ControlstrategiesbasedonMPCformulationsrequirethedevel- opmentof differentmodels andoptimisation problemsto ensure optimalcomputationofcontrol inputs,offset-freetrackingofcon- trolled variables and adequate estimation of states. Fig. 2 shows a diagram of theMPC strategy proposed in thiswork. The high- fidelity dynamic model of the NGCC-PCC system described in Section 2.1 replicates the behaviour of a real power plant with post-combustionCO₂ capture. Measurementsfromthis modelal- lowtheestimationofthestatesinthesystem.Thisestimatoruses aKalmanfiltertoupdate thestate estimationsandcorrectpossi- blemismatches between thepredictions of the responses by the simplifiedmodelsandthemeasurements fromthedynamicsimu- lationof theNGCC-PCCplant. Theseestimates define thecurrent state, i.e.the initial conditions, fromwhere the dynamicoptimi- sationproblemintheMPCstrategystartstocomputetheoptimal sequence of control inputs. The first element ofthis sequence is the control action imposed in the actual system. This process is repeatedperiodically, with a frequency dictated by the sampling time, to stabilise the operation of the NGCC integrated with the

Fig. 2. Diagram of the proposed MPC strategy with a Kalman filter. Expressions within the diagram are developed throughout Section 3 , while the dynamic model of the NGCC-PCC system is described in Section 2.1 .

PCC plant. This MPC strategy includes all simplified models in a singlecontrollerasshowninFig.2.

Thissectiondescribesdifferentmodelsandformulationsofthe MPCstrategy,anddetailshowtheyarecombinedintheintegrated control structure represented in Fig. 2. Section 3.1 discusses ref- erencetrackingandoffset-freeMPC,anddescribestheformulation ofthisoptimisationproblem,whereasSection3.2buildsuponthis formulationanddefinesasimplerdynamicoptimisationproblem, called delta-input formulation, that only depends on the manip- ulated variables.Section 3.3describesthe estimatorthat predicts the statesontheactualNGCC-PCCsystemsandpresentsan algo- rithmtosolvetheMPCcontrolproblem.

3.1. Referencetrackingandoffset-freeMPC

Referencetrackingisoneofthemainapplicationsofmodelpre- dictive control.Thiscontrol strategy minimisesthedifference be- tweenoutputsofasystemandreferencetrajectoriesbycomputing controlinputsthroughdynamicoptimisationproblemsandimple- menting thefirst elementofthecalculatedcontrol sequence.The generalformulationoflinearMPCproblemsforreferencetracking is:

minx,u N−1

k=0

1

2

^Q

(

^yk−y_ref

)

⁺

^R

(

^uk−u_k−1

)

^(6a)

s.t.

x_k+1=Ax_k+Bu_k (6b)

y_k=Cx_k (6c)

y^low≤ yk≤y^up (6d)

u^low≤u_k≤u^up (6e)

where

·

^{representsthetwo-normthatleadstoaquadraticpro-}

gramming(QP)optimisationproblem.Eq.(6b)and(6c)ensurethat the state-space realisation ofthe identified ARX models is satis- fied. Eqs.(6d)and(6e) limit theminimumandmaximum values ofthe controlledandmanipulated variables,respectively.The ob- jectivefunctioninEq.(6a)minimisesthedifferencebetweencon- trolledvariablesandtheirreferencesy_refandimposesapenaltyin excessiveutilisationofcontrolinputs.

Nevertheless,referencetrackingformulationsofMPCstrategies asinEq.(6)canleadto offsetsinthecontrolledvariables dueto unmeasured disturbances and plant-model mismatches. To over- come this limitationandensure zero offset,models representing actualsystemscanbeaugmentedwithadisturbancemodel,which acts as an integrator driving the tracking error to zero. This al- lowsfindingthecontrolinputsthatminimiseboththeeffectofthe disturbance on the controlled variables and differences between model and system (Pannocchia and Rawlings, 2003; Borrelli and Morari, 2007;Pannocchia, 2015;Rawlingsetal., 2017).The state- spacemodelinEq.(5)becomes:

x_a,k+1=Aax_a,k+Bau_k (7a)

y_k=Cax_a,k (7b)

wherevectorsandmatricesare:

x_k+1 d_k+1

=

A B_d

0 I

x_k d_k

+

B 0

u_k

yk=

C Cd

x_k dk

Thisaugmented modelachieves offset-freetracking ifthesystem isstabilisable,thepair(^A,C) ^isobservable,thenumberofdistur- bancesn_d:

n_d=p=1

andthefollowingconditionholds(PannocchiaandRawlings,2003;

Borrelli and Morari, 2007; Pannocchia, 2015; Rawlings et al., 2017):

rank

A−I B_d C C_d

=ny+n_d

Since the disturbancematrices B_d∈R^nyxnd andC_d∈R^1xnd can be chosenfreely,thelastconditionholdsif(^A,C)^isobservable.Inthis work,thestate-spacerealisationoftheidentifiedARXmodelswas expressedinobservable form,andhencethepair(^A,C)^isalways observable (Chen,2013).Therefore,offset-freetracking reducesto theadequateselectionofdisturbancematricesB_dandC_d.

TheMPCformulationinEq.(6)forthesystemaugmentedwith adisturbancemodelbecomes:

minx,u N−1

k=0

1

2

^Q

(

^yk−y_ref

)

⁺

^R

(

^uk−u_k−1

)

^(8a)

s.t.

x_a,k+1=Aax_a,k+Bau_k (8b)

y_k=Cax_a,k (8c)

y^low≤yk≤y^up (8d)

u^low≤uk≤u^up (8e)

3.2. Delta-inputformulation

Delta-input formulations of the MPC describedin Eq. (8) are moresuitableforreferencetrackingproblems,astheypenalisedi- rectlytherateofchangeofthemanipulatedvariables(Borrelliand Morari, 2007). Furthermore, it reduces the number of optimisa- tionvariables andthe computationalcost ofthe dynamicoptimi- sation. Section 3.2.1 describes the delta-input formulation of the MPCprobleminEq.(8),whereasSection3.2.2discusseshowsev- eralstate-spacemodelscanbemergedintoacommonMPCprob- lem.

3.2.1. Delta-inputformulationforSISOsystems

Definethe delta-input control actionthat determines therate ofchangeofamanipulatedvariable:

δ

^uk:=uk−uk−1 (9)

and augment the state-space equation in Eq. (7) with this new stateandcontrolinput:

x_a,k+1 u_k

=

Aa Ba

0 I

x_a,k u_k−1

+

Ba

I

δ

^uk (10a)

y_k=[Ca 0]

xa,k

u_k−1

(10b)

whichcanbewritten:

x_k+1=Ax_k+B

δ

^uk

yk=Cxk

Definethe vectorsofcontrolled andmanipulated variablesovera timehorizonN:

δ

^u=[

δ

^u0

δ

^u1 ...

δ

^uN−1]^T y=[y1 y2 ... yN]^T

andeliminatethestatesinEq.(10).Theoutputequation,overthe timehorizonN,becomes:

y=H

δ

^{u+A0x˜0 (11)}

where

H=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

H1 0 · · · 0 H₂ H₁ 0 · · · 0

..

. ... ... ... ... ..

. H2 H1 0

HN · · · H2 H1

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦

A₀=

⎡

⎢ ⎢

⎢ ⎢

⎣

CA CA² CA³ .. . CA^N

⎤

⎥ ⎥

⎥ ⎥

⎦

with

Hi=CAⁱ⁻¹B i∈

{

^1,2,...,N

}

x0=x[0]

Withthisreducedoutputequation, Eq.(11),andthedefinitionof thedeltacontrolinputinEq.(9),theinequalityconstraintsinthe standardMPCformulation,Eq.(8d)andEq.(8e),canbewrittenas:

⎡

⎢ ⎣

−H H

−

⎤

⎥ ⎦ δ

^u≤

⎡

⎢ ⎣

−

(

^ylow−A0x0

)

y^up−A₀x₀

−

(

^ulow−u₋1

)

u^up−u₋₁

⎤

⎥ ⎦

⁽¹²⁾

where u₋₁ wasthecontrol action intheprevious samplingtime, and^{isanunitlowertriangularmatrix:}

=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

1 0 · · · 0 1 1 ... ...

..

. ... ... ... ... ..

. ... 1 0

1 · · · 1 1

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

Following the sameapproach, the objectivefunction Eq.(8a) be- comes:

J= 1

2

(

^Q

(

^y−yref

)

⁺

^R

δ

^u

)

= 1

2

(

^Q

(

^H

δ

^u+A0x˜₀−y_ref

)

⁺

^R

δ

^u

)

= 1 2

δ

^uT

H^TQH+R

δ

^u

+2

(

^{A0x˜0−yref}

)

^QH

δ

^u

+

(

^A0x˜0−yref

)

^TQ

(

^A0x˜0−yref

)

⁽¹³⁾

wherethelasttermmaybedroppedsinceisconstant.

Therefore,the MPCstrategy canbe expressedastheQP prob- lem:

δmin^u∈R^N

1

2

δ

^uT

δ

^u+fT

δ

^{u (14a)}

s.t.

g

δ

^u≤p (14b)

withthematrixandvectorinEq.(14b)definedinEq.(12),and:

=H^TQH+R f =

(

^A0x˜0−yref

)

^QH

ThedevelopmentoftheMPCdelta-inputformulationforthepoly- nomialmodelinEq.(4)followsthesameapproachandissumma- rizedinAppendixC.

3.2.2. Delta-inputformulationforMIMOsystems

Systems generally require the control of severalprocess vari- ables. Thus, the delta-input formulation of the MPC problem in Eq.(14)isexpandedtoconsidermulti-inputmulti-output(MIMO) systems.Considermsingle-inputsingle-output(SISO)modelswith manipulated variables definedas delta-input control actions and groupedinavectoras:

^u:=[

δ

^u1

δ

^{u2 ...}

δ

^{um]T (15)}

whereeachcomponentisasequenceofcontrolactionsoveratime horizonNforagivenmanipulatedvariable:

δ

^uj=[

δ

^uj,1 ...

δ

^uj,N]^T j∈

{

¹,...,m

}

TheMPCdelta-inputformulationcanbeextendedas:

u∈minR^(Nxm)x1

1

2

^{uT u}+F^T

^{u (16a)}

s.t.

G

^u≤P (16b)

where

=

⎡

⎢ ⎢

⎢ ⎣

1 0 · · · 0

0

2

... ... ..

. ... ... 0 0 · · · 0

m

⎤

⎥ ⎥

⎥ ⎦

^F=

⎡

⎢ ⎢

⎣

f₁ f2

.. . fm

⎤

⎥ ⎥

⎦

G=

⎡

⎢ ⎢

⎢ ⎣

g1 0 · · · 0 0 g2

... ... ..

. ... ... 0 0 · · · 0 gm

⎤

⎥ ⎥

⎥ ⎦

^P=

⎡

⎢ ⎢

⎣

p1

p₂ .. . pm

⎤

⎥ ⎥

⎦

3.3. Estimator

States anddisturbances need to be estimated from the mea- surementsoftheactualsystemateachsamplingtimetoobtainthe current state of the NGCC-PCC plant. The estimator, or observer, computestheaugmentedstateateachdiscretetimekasacombi- nationofthecurrent,orapriori,statepredictionandacorrection basedonthemeasuredoutputy_k:

xˆ_k=Axˆ_k−1+B

δ

^uk−1+K

(

^yk−C

(

^Aˆxk−1+B

δ

^uk−1

))

⁽¹⁷⁾

whereˆ· indicatesestimatedvariables, andK∈R^(ny+nd+1)x1 isthe observergain:

K:=

_K

x

Kd

Ku

in whichK_x,K_d, K_u are the observergains forthe states,distur- bancesandcontrolinput,respectively.ThisobservergainKischo- sen so the observer is stable, i.e.the eigenvalues of the system (^A−KCA)^{lieinsidetheunitcircle.}

Pole placement routines compute observer gain matricesthat fix the eigenvalues of a matrix pair in specific coordinates and make the estimator stable (see, e.g. Pannocchia, 2015). How- ever,thiswork considers a Kalman filterasobservergain matrix (Kalman, 1960). Calculation ofthe Kalman filter matrix gain is a two-stepprocess.First,thea prioristatexˆ⁻_k−1andcovariancema- trixZ_k⁻arecomputedfrompreviousestimations:

xˆ⁻_k =Aˆxk−1+B

δ

^uk−1 (18a)

Z⁻_k =AZ_k−1A^T+Qp (18b)

Algorithm1 MPCforNGCC-PCCsystems

Require: coefficients(^a,b) ^{inTable~A.4, B}d,C_d,Qp,Rm,Q,R, y_ref, y^low,y^up,u^low,u^up,GTramp,^,N

Require: ˆx_k−1,^uk−1,y_k,m˙_exhaust,Z_k−1

Compute:interpolatedcoefficients(^a,b)^withEqs.~2,3 Compute:A,B,CinEq.~10

Compute:H,A₀inEq.~11 Compute:xˆ_k,Z_kinEq.~18 Set:x₀:=xˆ_k

Compute:g,pinEq.~12 Compute:^{, finEq.~14a} Compute:G,P,^,FinEq.~16 Solve:

^u∈minR^(Nxm)x1

1

2

^{uT u+FT}

^u

s.t.

G

^u≤P

return^uk,xˆ_k,P_k

with Qp representing the covariance of the process noise w∈ N(⁰,Qp)^{.Then,theseaprioriestimatesareupdatedbasedoncur-} rentmeasurements:

K_f = Z⁻_kC˜^T C˜Z⁻_kC˜^T+Rm

(18c)

ˆ

x_k=xˆ⁻_k +K_f

(

^yk−Cxˆ⁻_k

)

^(18d)

Z_k=

(

^I−KfC

)

^Z−k (18e)

where Rm isthe covarianceassociatedto themeasurement noise

v

∈N(⁰,Rm),andK_f istheKalmanfilterusedtoestimatethecur- rentstateˆx_kandthecovariancematrixZ_kthatwillbeusedatthe nextsamplingtime.

Algorithm1summarisesthesequenceofcomputationsneeded to implementthe MPC strategy at each sampling time. The first require condition refers to the parameters, matrices and vectors providedoff-line,whilstthesecondrequireconditionindicatesthe parameters that areupdated every samplingtime.The massflow rate ofexhaustgas m˙_exhaust belongsto thissecond group asit is the parameter neededto interpolate the coefficients ofthe local ARX modelsforthecaptureratioandreboilersteam temperature (seeTableA.4).Moreover,notethatthefirstelementofeachinput controlsequencemustbeselectedfrom^uk.

4. DynamicoperationofNGCC-PCCintegratedsystems

A casestudywhere theNGCC-PCCsystemneeds toreduce its power generation tobalance the griddemonstrates the effective- ness of the proposed MPC strategy to respond to fast changes in power demand and stabilise the operation of the integrated plants. The dominant dynamics of the NGCC and PCC described inSection 2.1occurwithin10 and60min,respectively.Thus,the MPCstrategy consideredasamplingtimeof30sinordertocap- turethetransientbehaviourintheshortesttime-scale,i.e.thedy- namic operation of the NGCC. Atime horizonN=20was hence selected to consider the entire period of dominant dynamics in the NGCC. Table 2 includes the bounds for the controlled and manipulatedvariablesconsideredduringthedynamicsimulations.

Table 3 summarises the matrices and vectors to create the aug- mented models,theestimatorbasedontheKalmanfilter,andthe weightsintheobjectivefunctionforeachinput-outputpair.These

Table 2

Lower and upper bounds of the controlled and manipu- lated variables.

Variable Lower Upper

Power [MW] 450 615

Gas turbine load [%] 60 100

Superheating temperature [ ^◦C] 587.7 597.7 Attemperator valve 1 [-] 0.01 1 Reheating temperature [ ^◦C] 587.5 597.5 Attemperator valve 2 [-] 0.01 1

Capture ratio [-] 0.85 0.95

Mass flow lean solvent [kg/s] 300 800 Reboiler temperature [ ^◦C] 115.22 120.22 Steam extraction valve [-] 0.01 1

Table 3

Matrices and vectors defining the disturbance ( B d, C d) and noise ( Q p, R m) models;

and weights for controlled variables ( λQ) and penalties in movement of manipu- lated variables ( λR).

Variable B d C d Q p R m λQ λR

Power - - - - 1 1

Superheating temperature

⎡

⎣ ⁰ 0 0 . 01

⎤

⎦ 0 I 4x4 0.01 10 0.01

Reheating temperature

⎡

⎣ ⁰ 0 0 . 01

⎤

⎦ 0 I 4x4 0.01 10 0.01 Capture ratio

0 . 1 0 . 1

0 I 3x3 0.1 50000 0.001 Reboiler temperature

0 . 01 0 . 01

0 I 3x3 0.1 100 10

weightsaimed atcompensatingthedifferentordersofmagnitude betweencontrolled and manipulatedvariables and atprioritising thetrackingof theprocess variables,albeittheir tuning wasout- sideofthescopeofthiswork.

A step change in the power demand drives the transient op- eration ofthe powerplant, which adaptsthe gasturbine loadto adjustthenetpower output.Similarly,the changeinexhaustgas conditionsdisturbstheoperationofthecaptureplant.Fig.3shows keyprocessvariablesintheNGCC-PCCsystemduringdynamicop- eration and demonstrates the effectiveness ofthe proposed MPC strategytoachieveoptimaloffset-freecontrol.

ProcessvariablesfromtheNGCCreachtheirset-pointfasterbe- causeoftheshorterdominantdynamicsofthepowerplantcom- paredto the capturesystem. Net power generation is thefastest variableto meetits target owing tothe fastdynamicsof thegas turbine,whichcontrolstheoverallpoweroutputoftheNGCCand compensatestheslowdynamicsofthesteam cycle.Consequently, powerdemandandsupplyarebalanced withinthe dominantdy- namicsoftheNGCC.Temperature controlinthesuperheatingand reheatingsections ofthe HRSG requires more time. Heatcapaci- tance inthe HRSG slows down the transientperformance of the steam cyclecomparedto the changein gasturbineload. Theat- temperatorvalvesneedtocompensateandanticipatethesevaria- tionsintheoperatingconditionsforalongerperiodoftime.Nev- ertheless,the proposedMPCstrategy limitedtheoffsetanddrove bothtemperaturestotheirset-point.

DynamicsinthePCCplantarenotablyslowerthaninanytype ofthermal powerplant(Rúaetal., 2020a). However, Fig.3 illus- trates how the MPC strategy controlled the capture ratioalmost simultaneouslytothetemperatureinthesteamcycleoftheNGCC.

Thisbehaviouroriginatesfromtheuseofoptimisation-basedcon- trolstrategies. MPC considers the dynamicoperation of the cap- tureplantandcomputesoptimalcontrolactionsthatachievedbet- ter andfaster offset free in key process variables. Fig. 3 also il-

Fig. 3. Dynamic behaviour of process variables from the NGCC-PCC system with the proposed MPC strategy during a power demand reduction of 70 MW.

lustrates how traditional PID controllers require more time and lead to larger offsets than MPC, albeit offset-free control is also achievedbecauseoftheirintegralaction(Montañésetal.,2017c).

Incontrast,thereboilertemperatureneededmoretimetoreach its set-point.Controlactionsinthemassflowrateoflean solvent to stabilise the capture ratio modify the operation of the desor- ber,whichalsoaffectstheleanloadingofthesolventattheoutlet ofthiscolumnandthetemperatureinthereboiler.Theseprocess changes arecharacterisedby slowdynamics becauseoftheinter- action betweentheabsorber andstrippercolumns,large volumes ofsolventanddelaysfrompipingandheatexchangers(Rúaetal., 2020b). Therefore, the MPC needs to adapt the steam extraction from the NGCC to anticipate the interaction between both ab- sorption anddesorption sectionsandcompensatetheseoperation changes.Thisleadstothesaturationofthesteamextractionvalve in the first 20 min of transientperformance of the CO₂ capture plant, whichresultsfromthecombinedeffectofchangingloading in thesolvent,theMPC strategytrying toanticipate thedynamic behaviour ofthe reboiler temperature andthe slow dynamics of

thedesorptionsectionofthePCCplant.Thesteamcycleandcap- ture stabilise completely duringthis time and reduce hence the variationsinsteam availabilityandfluctuationintherich loading ofthesolvent.Thesesteadierconditionseasethecontrolofthere- boilertemperatureandallow amorestableandprolonged move- mentsofthesteamextractionvalveafterthisstabilisationperiod.

Despitethe saturation of the steam extractionvalve, the pro- posed MPC strategy obtained smaller offsets than 0.15^◦C and achievedoffsetfreeinan hour,which isbetterperformance than usingPIDcontrollers (Montañésetal.,2017c).Thisreducedoffset achievedbytheMPCstrategyduringdrasticchangesofloadisspe- ciallyimportantinthe reboilertemperature, asitcould allowin- creasingits set-point, andhencethe strippingefficiency, without reachingtemperaturesthatleadtosolventdegradationduringthe regenerationprocess.

Tuning of the MPC was not the main objective ofthis study.

Improved performance might be achieved with adequate weight valuesinthe objectivefunction,

λ

Q and

λ

R,disturbancematrices andvectors(Pannocchia, 2003), B_d andC_d,and noise modelsfor