Hierarchical nonlinear optimization-based controller of a continuous strip annealing furnace

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This document contains a post-print version of the paper
Hierarchical nonlinear optimization-based controller of a continuous strip
annealing furnace
authored by S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi
and published in Control Engineering Practice.
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Cite this article as:
S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based controller of
a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.
conengprac.2017.12.005
BibTex entry:
@Article{acinpaper,
author = {S. Strommer and M. Niederer and A. Steinboeck and A. Kugi},
title = {Hierarchical nonlinear optimization-based controller of a continuous strip annealing furnace},
journal = {Control Engineering Practice},
year = {2018},
volume = {73},
pages = {40--55},
doi = {10.1016/j.conengprac.2017.12.005},
}
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acontinuousstripannealingfurnace(cid:17),Control Engineering Practice,vol.73,pp.40(cid:21)55,2018.doi: 10.1016/j.conengprac.2017.12.005
Hierarchical nonlinear optimization-based controller of a continuous strip annealing
furnace
S. Strommera, , M. Niederera, A. Steinboeckb, A. Kugia,b
∗
aAITAustrianInstituteofTechnology,CenterforVision,Automation andControl, Argentinierstrasse2/4,1040 Vienna,Austria
bAutomation andControl Institute,TechnischeUniversita¨tWien,Gusshausstrasse 27–29, 1040 Vienna,Austria
Abstract
Continuous strip annealing furnaces are complex multi-input multi-output nonlinear distributed-parameter systems.
They are used in industry for heat treatment of steel strips. The product portfolio and different materials to be heat-
treatedissteadilyincreasingandthedemandsonhighthroughput,minimumenergyconsumption,andminimumwaste
have gained importance over the last years. Designing a furnace control concept that ensures accurate temperature
trackingunderconsiderationofallinputandstateconstraintsintransientoperationsisachallengingtask,inparticular
in view of the large thermal inertia of the furnace compared to the strip. The control problem at hand becomes even
more complicated because the burners in the different heating zones of the considered furnace can be individually
switched on and off. In this paper, a real-time capable optimization-based hierarchical control concept is developed,
which consists of a static optimization for the selection of an operating point for each strip, a trajectory generator for
the strip velocity, a dynamic optimization routine using a long prediction horizon to plan reference trajectories for the
strip temperature as well as switching times for heating zones, and a nonlinear model predictive controller with a short
prediction horizon for temperature tracking. The mass flows of fuel and the strip velocity are the basic control inputs.
The underlying optimization problems are transformed to unconstrained problems and solved by the Gauss-Newton
method. The performance of the proposed control concept is demonstrated by an experimentally validated simulation
model of a continuous strip annealing furnace at voestalpine Stahl GmbH, Linz, Austria.
Keywords: Nonlinear model predictive control, direct- and indirect-fired strip annealing furnace, reheating and heat
treatment of metal strips, nonlinear MIMO system, unconstrainedoptimization, Gauss-Newtonmethod
1. Introduction Since the strip temperature is a distributed process vari-
able, which can only be measured at a very few discrete
Continuousannealing processesareused for heattreat- points, the control task is further complicated. Moreover,
ment of steel strips. Controllers should ensure that the thethermalinertiaofthefurnaceisratherhighcompared
strip temperature follows a set-point temperature trajec- tothatofthestrip. Thus,theresponsetimeofthefurnace
tory as closely as possible. The set-point trajectories de- is also high comparedto the processing time of a strip.
pend on metallurgical requirements and may vary from The CSAF constitutes a multiple-input multiple-output
strip to strip. Typically, a diverse portfolio of prod- nonlineardistributed-parametersystem. Themaincontrol
uctsisheat-treatedincontinuousstripannealingfurnaces inputsarethefuelsuppliesoftheheatingzones. Theycan
(CSAF). Therefore, a variety of different CSAF can be beindividuallyswitchedon/offdependingontherequired
found in industry (Mullinger and Jenkins, 2014; Imose, heatinput,whichmakesthetaskoffindingoptimalcontrol
1985). The CSAF considered in this paper is part of a inputsamixed-integerprogrammingproblem(Grossmann
hot-dip galvanizing line of voestalpineStahl GmbH, Linz, and Kravanja, 1997). The strip velocity serves as an ad-
Austria. ditional control input. It is subject to several restrictions
The accurate temperature control of a CSAF is essential which are mainly defined by downstream process steps.
toensureahighproductquality. Thisisinparticularchal- All control inputs are bounded from above and below. In
lenging in transient operational situations when a welded thiswork,anonlinearoptimization-basedhierarchicalcon-
joint moves through the furnace. In this case, the strip trolstrategyfortheconsideredCSAFofvoestalpineStahl
dimensions (thickness, width), the steel grade, the set- GmbH is presented.
pointstriptemperature,andthestripvelocitymaychange.
1.1. Continuous strip annealing furnace
∗Corresponding author. Tel.: [phone], fax: [phone] Figure 1 shows a schematic of the considered CSAF,
2813.
Email address: [email protected] (S.Strommer) which consists of a direct- and an indirect-fired furnace
Preprint submittedtoControl EngineeringPractice November9,2017
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’s(cid:28)nallayoutorcopyediting.
Nomenclature W weighting matrix
x algebraicand state variables
y system outputs
Latin symbols
z spatial coordinate
b width
c specific heat capacity
Greek symbols
= hza,hzb,hzc,hzd set of abbreviations for the
D { } ε emissivity
heating zones a–d
Γ system dynamics
d thickness
Λ Lagrangemultiplier
d searchdirection
λ air-fuel equivalence ratio
= dff,rth,rts set of abbreviations for furnace sec-
F { } ϕ nonlinear transformation
tions
ρ mass density
g gradient
H˙ enthalpy flow τ switching time
υ unconstrainedoptimization variable
H Hessian
χ stoichiometriccoefficient
h specific enthalpy
i index
Subscripts
J objective function
flue gas
k index with respect to time g
roll
L vector-valuedLagrangefunction h
radianttube
l index r
M˙ mass flow s strip
M¯ molar mass w wall
m vector of the mass flows of fuel
Superscripts
m˙ slope of the mass flows of fuel
N number of discretized elements + upper bound
Q˙ heat flow − lower bound
q˙ heat flux ad adiabatic flame
R objective function a air
S surface cb combustion
s switching state d set point
T temperature f fuel
t time in incoming
U system input out outgoing
u optimization variables r reference
v strip velocity sp nitrogenflushing
s
v˙ slope of the strip velocity ˆ observer
s
separated by an air lock. In the considered CSAF, stan- post combustion chamber (PCC) by adding fresh air via
dardsteelfortheautomotivearea(bodywork)isproduced. an air intake. The flue gas leaving the PCC contains ex-
Thestripwidthvariesfrom0.8mupto1.8m,whereasthe cess oxygen and streams into the preheater, where it is
strip thickness typically varies from 0.4mm up to 1.2mm used to preheat the incoming strip. The DFF is a coun-
duetotheproductionneedsbutcanacceptmaterialshav- terflowheatexchangerbecausethefluegasstreamsinthe
ing lower thickness. The steel strip, which is conveyed opposite direction of the strip motion.
through the furnace by rolls, couples both parts. This In the heating zone a and b, an array of burners is used,
furnace type is designed for heat treatment of steel strips whereadefined numberofburnerscanbedeactivatedde-
intermsofthroughput,energyconsumption,andproduct pending on the width of the strip (narrow,middle, wide).
quality (Imose, 1985). Theseheatingzonesareresponsibleforthebaseload. Us-
In the direct-fired furnace (DFF), the strip is heated by ingshut-offvalves,thefuelsupplyoftheheatingzonesa–d
means of hot flue gas, which comes from the combustion can be individually switched on/off. Deactivated burners
offuel. Thefuelisburntfuelrichinthefourheatingzones are flushed with cold nitrogen to protect the burner noz-
(hz a–d) to avoid scale formation of the strip. Thus, the zles from thermal damage (Strommer et al., 2014b). The
flue gas contains unburnt products, which are burnt in a media supplies of air and fuel are coupled by the air-fuel
2
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’s(cid:28)nallayoutorcopyediting.
Direct-firedfurnace Indirect-firedfurnace
Postcombustionchamber PyrometerPrth zrth
rth1 rth2 rts
Inertgasatmosphere
Fuel-rich
atmosphere Radianttube
aa
Airintake aa Burner
Burner Fuel aa
aa
aa
Gasflow aa
Heatingzoned aa
Air aa
aa
Fluegasaa Recuperator
Heatingzonec
Preheater
Stripmotion
Heatingzoneb
Hotstrip
zrts PyrometerPrts
Strip Heatingzonea
PyrometerPdff
z=0
Wall Roll
Coldstrip
EnvironmentwiththeambienttemperatureT
Fuel-leanatmosphere ∞
Airlock zdff
Figure1: Combineddirect-andindirect-firedstripannealingfurnace.
equivalence ratio, which is controlled by a cross-limiting this controller were determined by an optimization prob-
controller. lem. In (Kelly et al., 1988), a nonlinear model of an IFF
The indirect-fired furnace (IFF) is separated into three was introduced. Moreover, a Kalman filter and a linear
sections, the radiant tube heating sections 1 and 2 (rth 1 quadratic controller were designed based on a linear sys-
and 2) and the radiant tube soaking section (rts). Each tem model.
ofthesesectionsisequippedwithW-shapedradianttubes To support the operators, mathematical models are em-
and can be separately controlled. The tubes are perma- ployed to calculate optimal process parameters, e.g.,
nentlysuppliedwithfuelandairtoavoidflameextinction. the strip velocity and the target strip temperature,
Inside the IFF, an inert gas atmosphere is established to see (de Pis´on et al., 2011; Yahiro et al., 1993). The op-
prevent scale formation of the strip. Due to a controlled eratorsuse this information to adjust the process, in par-
pressure gradient, a gas flow in the direction of the DFF ticular when a welded joint moves through the furnace.
is alwaysassured. In (Bitschnau et al., 2010), a nonlinear model of an IFF
The strip temperature is measured by three pyrometers wasderivedwith a controllerconsisting ofa feedbackand
(P ,P , and P ), see Fig. 1. Additionally, thermo- a feedforwardpart.
dff rth rts
couples,whichmeasureseverallocalfluegastemperatures, In (Norberg, 1997), the challenges of controlling a CSAF
walltemperatures,andradianttubetemperatures,arein- are discussed in more detail, in particular the problems
stalled for safety reasons. andtherequirementswithrespecttothestripvelocity,the
combustion process, the gas atmosphere, and the scaling
1.2. Existing solutions process are considered. As a conclusion, model predictive
Controlconceptsthatcanbefoundintheliteraturesig- controlis highly recommended to tackle all these issues.
nificantly differ in their complexity and application. In To control the strip temperature in an IFF, real-time
CSAFs, simple PID controllers are widespread to control implementations of linear model predictive controllers
the heating of the strip. In (Dunoyer et al., 1998; Mar- (MPC) are given in (Bitschnau and Kozek, 2009; Lewis
tineau et al., 2004), PID controllersbased on mathemati- et al., 1994; Wu et al., 2014). For all these approaches,
cal models of the furnace are applied. The PID controller onlythemassflowsoffuelserveascontrolinputs,whereas
proposedbyLietal.(2004)isusedtocontrolthetemper- thestripvelocityandotherprocessparametersaredefined
ature of a workpiece inside a furnace. The parameters of by the operator.
3
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’s(cid:28)nallayoutorcopyediting.
In(Guoetal.,2009),aparticleswarmoptimizationispro- early stage. Pure feedback control does not appear to be
posedtoensureanaccurateheatingofthestripaccording suitablefor the consideredcontroltaskbecauseofrapidly
to a desired trajectory. Here, a neural network was used changing operating conditions and the high thermal iner-
for modeling the system. tia of the CSAF.
In (Niederer et al., 2016), a nonlinear model predictive All these aspects motivate the development of a new fur-
controller for the considered strip annealing furnace was nacetemperaturecontroller,whichutilizestheadvantages
designed, where the strip velocity can be arbitrarily var- of MPC. Moreover, a control structure with several mod-
iedwithinapermissiblerange. Thecontrolinputsarethe ules seems to be useful to tackle the challengesassociated
massflowsoffuelandthestripvelocitytorealizeanopti- withmixed-integerprogramming,theselectionofthestrip
mal tracking controlof the strip temperature. velocity,thehighthermalinertiaoftheCSAF,thecompu-
Infurnacecontrol,hierarchicalcontrolstructuresarequite tationaleffort,andthereal-timerequirements(Mingetal.,
common. In (Ming et al., 2008), a control structure with 2008;Ueda et al., 1991;Yoshitani and Hasegawa,1998).
three layers is used. One layer adapts the weakly known Theproposedfurnacetemperaturecontrollerconsistsofa
parametersofthemathematicalmodelbasedonmeasure- nonlinear temperature regulator (TR), an optimization-
ments. The model is used in a second layer to determine based trajectory planner (OTP), a static optimization
a target trajectory for the strip temperature. Finally, in module, and a trajectory generator for the strip veloc-
a third layer,a nonlinear MPC usesthis targettrajectory ity. The static optimization choosesan optimal operating
to calculate the optimal fuel supply. Based on a nonlin- point characterized by the strip velocity and the switch-
ear model of an IFF, Ueda et al. (1991) solved an opti- ing state of the heating zones (on or off) for each strip.
mal control problem. The control inputs are the adjust- The optimal strip velocities are used by the trajectory
ment times of the strip velocity and the furnace temper- generator for the strip velocity to design a desired trajec-
ature. A hierarchical structure is also used by Yoshitani tory, which is then utilized by the TR and the OTP. The
and Hasegawa (1998). First, a reference speed is deter- OTP calculates an optimized reference trajectory of the
mined based on a static model. Then, the optimal time striptemperatureandoptimizedswitchingtimesforturn-
of the speed change is calculated. Furthermore, a target ing on/off the heating zones. The OTP uses a long time
trajectory of the strip temperature is specified. Finally, a horizon(15min)totakeintoaccountthehighthermalin-
tracking controller (MPC) is used to determine the mass ertiaoftheCSAF.TheOTPisrequiredtoplanaheadfor
flows of fuel. long-term transient changes of operating situations, e.g.,
when a new strip with other properties than the preced-
1.3. Motivation and objectives ing strip enters the furnace. The TR is based on MPC
Usually, CSAFs are not equipped with a DFF. How- technology, uses the fuel supplies as control inputs, and
ever, the key advantage of a DFF compared to an IFF performs tracking control for the strip temperatures de-
is its fast response characteristic. With a DFF, a nearly pendingontheiroptimizedreferencetrajectoriesobtained
instantaneousheatingofthestripcanbeachieved(Delau- fromtheOTP.TheTRusesashorttimehorizon(3min).
nay, 2007; Imose, 1985; Mould, 1982). Most of the exist- Adetaileddescriptionofthechoiceofthetimehorizonsis
ing controlstrategiesonly concern IFFs and are based on presented in Sec. 4.6.
simple furnace models that capture the heaters and the The main control objectivesare:
strip. Althoughsuchmodelsmaysimplifytheoverallcon-
trol design, this approach often restricts the capabilities Accurate heating of the strip
ˆ
and accuracyof the controlled furnace. Minimum scrap of material
ˆ
ACSAFisarathercomplexdynamicalsystem,wheretyp- Maximum throughput of strip
ˆ
ically a diverse portfolio of products having a notable va- Minimum energy consumption and CO2 emissions
ˆ
riety of steel grades is processed. Moreover, the require-
Clearly, by minimizing the energy consumption also the
ments in termsofproductqualitymayvaryfromapplica-
CO emissionsarereduced. Inthecourseofthecontroller
tiontoapplication. Thisiswhyatailoredcontrolconcept 2
design, a number of constraints have to be considered in
has to be developed to fully utilize the potential of the
terms of a reliable and safe operation of a hot-dip galva-
CSAF under consideration.
nizing line:
In this paper, the strip velocity is not allowed to vary in
anarbitrarywayduetodownstreamprocesssteps. More- The temperatures of the gas, the radiant tubes, and
over, the heating zones can be switched on/off. This is ˆ the walls inside the furnace are limited (damage and
a substantial extension to the control strategy proposed wear prevention).
by Niederer et al. (2016). Limitations of controlinputs have to be respected.
A control strategy based on predictive control seems ˆ The strip temperature is bounded.
promising for the considered task because the strips are ˆ
known in advance (Norberg, 1997). Thus, the require- The strip velocity has a significant influence on the evo-
ments and the parameters of these strips can be incor- lution of the strip temperature. A change of the speed
porated into the design of optimized control inputs at an causes a much quicker response of the strip temperature
4
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’s(cid:28)nallayoutorcopyediting.
compared to a change of the fuel supply (Yoshitani and objectives. Section4isdevotedtotheformulationandthe
Hasegawa,1998). However,thespeedissubjecttoseveral numericalsolutionoftheassociatedunconstrainednonlin-
restrictions: ear optimization problem. Finally in Section 5, the per-
formanceoftheproposedcontrolconceptisdemonstrated
A change of the strip velocity is programmed only
by simulation on an industrially validated model.
ˆ
in case a new strip enters the furnace. However, cer-
tainproductionexceptionsmightrequiresuddenvari-
ations of the strip velocity, which may be considered 2. Mathematical model
as unforseen disturbances to be compensated by the
The mathematical modeling of the overall process is a
temperature controlloop.
complex task due to the underlying nonlinear physical ef-
A change of the strip velocity should always be
ˆ fects and the variety of products and materials yielding
monotonous, i.e., the target acceleration should not
to a large range of operating conditions. Nonlinear mod-
change its sign during coil transition.
els of the DFF and IFF were developed and validated by
The heat-treatment time of a strip section has to be
ˆ measurements in Strommer et al. (2014a) and Niederer
limited due to annealing and material aging, which
etal.(2014),respectively. Moreover,acombinedmodelof
may reduce the product quality.
the DFF and IFF is presented in (Niederer et al., 2015).
There is a maximum and minimum speed, which are
ˆ It serves as a simulation model, however, it is not suit-
caused by metallurgical requirements and motor siz-
able for control design due to its high system dimension
ing.
and complexity. Therefore, a reduced model in terms of
Speed limitations demanded by the operator or due
ˆ complexity, dimension, and computational effort will be
to downstreamprocess steps have to be respected.
usedfor controlpurposes. Thismodelconsistsofthe sub-
Abigdifferencebetweentherollandthestriptemperature systems flue gas, radiant tube, roll, strip, and wall. The
may cause mechanical tensions in the strip, which result following reduction steps are performed:
in a distortion of the strip. Thus, the product quality
Coarserspatial discretization of the furnace
suffersandintheworstcase,scrapisproduced. Thisphe- ˆ
Simplified calculationof the heat transfer coefficient
nomenon is called heat buckling (Paulus and Laval,1985; ˆ
Simplified combustionin the DFF
Sasakietal.,1984). Suchatemperaturedifferencefollows ˆ
Simplified model of the radianttubes
fromarapidchangeofthestripvelocityaswellasachange ˆ
ofthestripwidth. Aheatingzonemaybeswitchedon/off Thecoarserspatialdiscretizationsignificantlyreducesthe
depending on the required heat input. Since the switch- system dimension from 850 for the full simulation model
ing of the burners may cause several disadvantages like
to 150 for the reduced controller design model. The cal-
reducedenergyefficiencyandahightemperaturegradient culationof the heat transfercoefficient isbasedon empir-
inside the furnace, the switching should be restricted: ical relations, which depend on the properties of the flue
gas(BaehrandStephan,2006;Incroperaetal.,2007). To
Ensureaminimumtimebetweentwoswitchingcycles.
ˆ determine these properties for a gaseousmixture, the for-
Switching ofburnersisonlyallowedin thevicinityof
ˆ mula suggested by Buddenberg and Wilke (1949) is typi-
strip transitions.
callyapplied. Thisformulaiscomplexandcausesasignifi-
There exist further control objectives and requirements cantcomputationaleffort. Therefore,thegaseousmixture
which should be incorporated into an advanced model- is only represented by nitrogen for calculating the heat
based furnace controlstrategy: transfer coefficient and hence, the complexity can be re-
duced substantially.
Consideration of recuperators, heat recircula-
ˆ
tion (Katsuki and Hasegawa,1998). 2.1. Fuel supplies
The mathematical model aims at predicting the pro-
ˆ Generally,the fuel supply of the four heating zonesa–d
cess behavior in all operating conditions and for all
of the DFF can be switched off using shut-off valves and
product types.
thus, 24 =16 different possibilities have to be considered.
The control concept has to realize a fuel-rich gas at-
ˆ However, switching off the heating zones a and b is not
mospheretoensurethatthefluegasdoesnotcontain
useful because they cover the base load. Therefore, only
oxygen,whichmaycauseundesirablescaleformation
the heating zones c and d are switched depending on the
(product quality).
required heat input. Based on a thorough energy analy-
sis (Strommer et al., 2013) and the fact that the furnace
1.4. Contents
operatesasacounterflowheatexchanger(Incroperaetal.,
Thisworkhasthefollowingstructure: InSection2,the 2007),it can be shown that it is not useful to fire heating
mathematical model of the CSAF is briefly summarized. zone d if heating zone c is off. Thus, only three different
The controller is based on a hierarchical structure, which casescanoccur,whichareindicatedbytheswitchingstate
isoutlinedinSection3togetherwiththecontroltasksand s 1,2,3 , see Tab. 1. In case 1 (s = 1), all heating
∈ { }
5
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’s(cid:28)nallayoutorcopyediting.
s M˙ f M˙ f oxygenO2, and nitrogen N2. These reactionproducts are
hzc hzd
summarized in the set = CO ,CO,H O,H ,O ,N .
2 2 2 2 2
1 M˙hfz,mcin,M˙hfz,mcax M˙hfz,mdin,M˙hfz,mdax The parameters χνi denGote th{e stoichiometric coefficient}s
2 hM˙hfz,mcin,M˙hfz,mcaxi h {0} i olefntcheercaotmiopλon.eInnttνhe∈hGeaatnindgdzeopneensdoofnththeeDaFirF-f,uaelfueeqlu-rivicah-
i
3 h 0 i 0 combustionisrealized,whichimpliesλ <1 andχO2 =0.
{ } { } i i
The remaining parameters χν, ν O , are deter-
i ∈ G \{ 2}
mined by mole balancesand the equilibrium equation
Table1: Differentcasesandallowedrangesofthemassflowsoffuel
totheheatingzonescandd. χCO2χH2
i i =K (Tad) (2)
χCOχH2O c g
s M˙sp M˙sp i i
hzc hzd
of the water-gas-shiftreaction (Moe, 1962)with the equi-
1 0 0
librium constant K (Tad). In contrast to Strommer et al.
2 0 M˙sp c g
hzd (2014a), the adiabatic flame temperature Tad = const.
3 M˙hszpc M˙hszpd is used in (2). This simplification is justifiged because
K (Tad) K (T ). Thus, the stoichiometric coefficients
c g ≈ c g,i
χν, ν , can be determined from mole balances, (1),
Table 2: Different cases and the corresponding mass flows due to i ∈ G
and (2) independently of the flue gastemperature.
nitrogenflushingtotheheatingzonescandd.
ThemassflowM˙cb,ν ofacombustionproductν which
i ∈G
enters the furnace zone i can be calculated by
zones of the DFF are active, in case 2 (s = 2), heating
zone d is switched off, and in case 3 (s = 3), the heating M˙ cb,ν = M¯ν χνM˙ f,
zones c and d areswitched off. The switching state s also i M¯CH4 i i
defines the mass flows M˙f of fuel to the heating zones c
α with the mass flow M˙ f of fuel supplied to this zone and
and d, α hzc,hzd , respectively. If a heating zone is i
switched o∈ff,{it is flush}ed with a constant amount M˙αsp of tThheemmoalassrmfloawssoMf¯cνomofbtuhseticoonmapironteontthνe∈zoBne=iGis∪g{ivCeHn4b}y.
nitrogen, see Tab. 2. In the IFF, the radiant tubes are
M˙ a =M˙a,O2 +M˙ a,N2 with
permanentlysuppliedbyfuelandair. Thus,themassflow i i i
M˙αf of fuel to an active heating zone of the DFF and to M¯κ
the IFF can vary in the range M˙αf ∈ M˙αf,min,M˙αf,max , M˙ia,κ = M¯CH4χai,κM˙if,
α rth1,rth2,rts , with = hhza,hzb,hzc,hzdi
Ma˙nα∈df,mtDhaxe∪mo{finfiumelu,mreaspnedctm}ivaexlyim. uTmDhme abs{osuflnodws MM˙˙ααff,,mmiinn aanndd} mκa∈ssAfl=ow{OM˙21,a2N2=},Mχ˙ai1a,2,OO22=+2Mλ˙1ia,2,Na2ndofχaaii,Nr2su=pp7l.i5e2dλit.oTthhee
M˙f,max,α hza,hzb ,dependonthewidthofthestrip. PCC reads as
α ∈{ }
M¯κ 5
2.2. Flue gas in the DFF M˙ a,κ= (χa,κ χa,κ)M˙f,
12 M¯CH4 12 − i i
The DFF is discretized into Ng = 13 volume zones, Xi=2
where each zone is considered as a well-stirred reactor with κ , χa,O2 = 2λ , χa,N2 = 7.52λ , and the air-
with a uniform flue gas temperature T , i = 1,...,N . ∈ A 12 12 12 12
g,i g fuel equivalence ratio λ in the PCC.
12
The volume zones 2–5 correspond to the heating zones
Remark 1. To ensure that the flue gas which leaves the
a–d and zone 12 representsthe PCC. The flue gas is con-
furnacecontainsadesiredamountofexcessoxygen,λ is
sidered quasi-stationary due to its fast response charac- 12
feedback-controlled. This ensures a fuel-lean gasmixture,
teristiccomparedtotheremainingsubsystems(Strommer
i.e., λ >1 (Strommer et al., 2017).
et al., 2014a). It is assumed that the combustion occurs 12
right at the burner nozzle and no further chemical reac- The stationary mass balance of each component ν
tions are considered within a volume zone. Furthermore, canbeutilizedtodeterminetheoutgoingmassflowM˙o∈ut,Gν
i
it is assumed that natural gas consists only of methane from an individual volume zone i (Baehr and Stephan,
CH . The stationarycombustion reactionin eachheating 2006;Incropera et al., 2007)
4
zone readsas (Turns, 2006)
M˙out,ν =M˙ in,ν +M˙ cb,ν+M˙sp,ν, (3)
i i i i
CH +2λ (O +3.76N ) χCO2CO +χCOCO
4 i 2 2 −→ i 2 i (1) whereM˙ in,ν istheincomingmassflowfromtheupstream
+χH2OH O+χH2H +χO2O +χN2N i
i 2 i 2 i 2 i 2 zone, i.e., M˙ in,ν = M˙ out,ν. M˙ sp,ν is the mass flow due
i i 1 i
iwdiethCiO∈,{c2a,r3b,o4n,5m}.onMoextihdaenCeOis,owxaidteizreHdiOnt,ohcyadrrboognendiHox-, Mto˙ snp,iNtr2o=geMn˙ sflpu,ssheiengT,abi.−.e.2,.M˙isp,ν = 0 for ν 6= N2 and
2 2 2 i i
6
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The flue gas temperature T of volume zone i is calcu- with i = 1,...,N and the radiative heat flux q˙ from
g,i r r,i,1
latedbasedonthestationaryenthalpybalance(Baehrand thefurnacechambertothefirstpipe. Theheatfluxq˙ =
c,i,1
Stephan, 2006;Incropera et al., 2007) wQ˙ into the inner surface of the tube is assumed to be
c,i
uniform, with a weighting factor w (0,1), see (Niederer
0=H˙iin+H˙ia+H˙if +H˙isp−H˙iout+Q˙g,i. et al., 2014)for more details. ∈
Here, H˙iin, H˙ia, H˙if, H˙isp, and H˙iout correspondto the en- 2.4. Furnace wall
thalpy flows of the incoming bulk flow, combustion air,
ThelayeredfurnacewallisdiscretizedintoN wallseg-
w
fuel, nitrogen flushing, and the outgoing flue gas stream,
ments. The temperature of the outer wall surface is as-
respectively. Q˙ isthenetheatflowintothefluegasand
g,i sumed to be equal to the ambient temperature T . On
includes convection and thermal radiation. An enthalpy ∞
the inner surface, the boundary condition is defined by
flow can be determined by H˙ = M˙ νhν(T) with the
specific enthalpy hν(T) and the maνs∈sBflow M˙ ν of compo- thheeathceoantdfluuctxioqn˙wthdruoeugthotchoenvweacltliocnanabnedcrhaadriaacttieorni.zedTbhye
P
nentν (Turns,2006). Thisyieldsanonlinearequation theone-dimensionalheatconductionequation(Baehrand
∈B
Stephan, 2006; Incropera et al., 2007). To obtain a low-
0= M˙iou1t,νhν(Tg,i 1)+ M˙ia,κhκ(Tia) dimensionalandcomputationallyundemandingmodel,the
− −
νX∈G κX∈A Galerkin weighted residual method is applied (Fletcher,
+M˙ fhCH4(Tf)+M˙ sphN2(Tsp) (4) 1984). Here, the stationary solution of the heat conduc-
i i i i
tion equation serves as a trial function. Finally, the dy-
− M˙iout,νhν(Tg,i)+Q˙g,i, namicbehaviorofthewalltemperaturecanbedefined by
ν a lumped-parameter model
X∈G
where Tia, Tif, and Tisp are the temperatures of combus- dT = q˙w,i + K2,i (T T ) (6)
tion air, fuel, and nitrogen, respectively. For determining dt w,i K1,i K1,i ∞− w,i
the fluegastemperaturesT bymeansof (4), theoutgo-
g,i
representingtheinnersurfacetemperatureT ofthewall
ingmassflowsM˙out,ν arerequired. Sincethesemassflows w,i
i segment i with i = 1,...,N . The parameters K and
onlydepend onthe massflowsoffuel suppliedto thefour w 1,i
K aredefined in (Niederer et al., 2015).
heating zones mD = M˙αf α ∈ R4, the air-fuel equiva- 2,i
lence ratios λ = [λα]αh∈D∪i{pc∈cD} ∈ R5, and the mass flows 2.5T.hReoslltraipndesnttreirps the furnace at z = 0 (cf. Fig. 1)
of nitrogen flushing mN = M˙αsp α ∈R4, they directly withambienttemperatureT andmovesthroughthefur-
follow from (3). h i ∈D nace with the velocity v . ∞The strip is characterized by
s
the thickness d , the width b , the mass density ρ , and
s s s
2.3. W-shaped radiant tube the temperature-dependent specific heat capacity c . The
s
The N W-shaped radiant tubes are equipped with dynamic behavior of the strip temperature T of a dis-
r s,i
gas-fired burners and local recuperators, cf. Fig 1. In cretized section i is given in the form (Strommer et al.,
the radiant tubes, the combustion is fuel lean, i.e., λ 2014a)
≥
1. In (Niederer et al., 2014), a semi-empirical nonlin-
d 2q˙ T T
eaMr˙rftmh1a,pMp˙irfnthg2,ΨMi˙rftdsepTen∈dinRg3 oonf futheel ims asussggflesotwesd mtoIca=l- dtTs,i = ρscs(Tss,,ii)ds,i −vs s,i−∆zs,i−1, (7)
with the boundary condition T (t) = T at z = 0 and
chulate the heat inpuit due to the combustionof fuel in the s,0
∞
form Q˙ =Ψ (m ), i=1,...,N . Note, the effect of the thelocalheatfluxtothestripq˙s,i(t), whichcomprisesthe
c,i i I r
convective,conductive,andradiativeheattransfer. In(7),
recuperator is incorporatedinto the mapping Ψ .
i
the backward finite difference formula is used to approxi-
In (Niederer et al., 2014), a radiant tube is approximated
mate the transport term. The strip is discretized into N
by four straight pipes each with a thickness d , a mass s
r
sections with equidistant length ∆z and locally uniform
density ρ , anda temperature-dependentspecificheat ca-
r
temperature T , i=1,...,N .
pacityc . Inthecurrentwork,onlythetemperatureT s,i s
r r,i,1
Thermal conduction occurs if the strip touches one of the
ofthefirstpipeoftheradianttubei iscalculatedandthe
N rolls. The temperature T of a single roll i can be
remaining temperatures are chosen proportional to T . h h,i
r,i,1
determined based on the heat balance, which gives
Starting from the first pipe, a nearly linear temperature
drop over the pipes can be observed(Imose, 1985). d 1 Sc Sc
The temperature of the first pipe of the radiant tube i is dtTh,i = ρ c (T )d 1− Sh q˙h,i+ Shq˙hc,i . (8)
based on the heat balance, resulting in h h h,i h (cid:18)(cid:18) h(cid:19) h (cid:19)
Here, d is the wall thickness of the roll, ρ is its mass
h h
d q˙ +q˙
dtTr,i,1= ρrcc,ir,1(Tr,i,1r,)id,1r, (5) dpeancistiyt,y,Schhiisstithsetteomtaplersautrufarec-edeopfetnhdeenrotlls,paecnidficShhceaisttchae-
7
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
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contact area between the strip and the roll. Moreover, gas temperatures are assembled in the vector x . Equa-
2
q˙ captures the heat flux to the roll by convection and tion (10) is nonlinear and time variant. The steady state
h,i
radiation and q˙c denotes the conductive heat flux. of (10) can be computed from
h,i
2.6. Measurement of the strip surface temperature f(t,x1,x2,m,vs)
0=Π(t,x,U)= , (11)
Itiswellknownthatnon-contacttemperaturemeasure- "a(t,x1,x2,m,mN)#
ment is a sophisticated task, in particular when the mea-
suredobjectmoves. Typically,apyrometerisusedinsuch with the augmented state vector xT = xT1,xT2 ∈ R150
cases to measure the intensity. In the considered furnace, and the system input UT = mTD,mTI,(cid:2)mTN,vs(cid:3) ∈ R12.
three pyrometers are available to measure the intensities For computer implementation, (10) has to be integrated
(cid:2) (cid:3)
I with α = dff,rth,rts (cf. Fig. 1). in time. In (Niederer et al., 2014, 2015; Strommer et al.,
α
∈F { } 2014a), Euler’s explicit method is used with a small sam-
Remark 2. Because the material quality depends on the
pling time ∆t . To decreasecomputationalcosts,a larger
k
temperature but not on the intensity and because it is
sampling time is preferable, which can, however, jeopar-
difficult for most furnace operators to interpret intensity
dizetheaccuracyorevencausenumericalinstability. Nu-
values,furnacecontrolsystemscommonlyusethetemper-
merical methods of higher order may help to keep the ac-
ature rather than the intensity.
curacy high even for larger sampling times. In (7), the
Based on I =σε T4 , the correct strip temperature Courant-Friedrichs-Lewy (CFL) condition has to be met
α s,α s,α
is given in the form (Iuchi et al., 2010; Michalski et al., for numerical stability (Strikwerda, 2004), i.e., ∆tkvs
≤
2001) ∆z with the sampling time ∆tk, the spatialdiscretization
∆z of the strip, and the strip velocity v . Hence, a larger
s
Ts,α = σεIα 1/4, (9) ∆sazm.pAlipnpglytiimnget∆hetkserceoqnudir-eosrdaenrahpaplfr-oexpprilaictietdRisucnrgeet-izKautitotna
(cid:18) s,α(cid:19) method(AscherandPetzold,1998),thediscrete-timesys-
withtheStefan-Boltzmannconstantσ andthe(unknown) tem can be expressed by a predictor step
real strip emissivity ε at the corresponding pyrometer
s,α
positionzα. Thestripemissivityεs,α canbeestimatedby X1,k+1 =x1,k+∆tkfk(x1,k,x2,k,mk,vs,k) (12a)
means of a state estimator (Strommer et al., 2016). 0=a (X ,X ,m ,m ) (12b)
k 1,k+1 2,k+1 k N,k
and a correctorstep
2.7. State-space model and discrete-time representation
Intheprevioussections,theindividualsubsystemswere x =x + ∆tk f (x ,x ,m ,v )
presented. Now, these subsystems are assembled in a re- 1,k+1 1,k 2 k 1,k 2,k k s,k (13a)
(cid:16)
duced mathematical model of the considered CSAF. The +f (X ,X ,m ,v )
k+1 1,k+1 2,k+1 k+1 s,k+1
subsystems(4)–(8)areinterconnectedviatheheattransfer
mechanisms (Niederer et al., 2014,2015;Strommer et al., 0=ak+1(x1,k+1,x2,k+1,mk+1,mN,k+1),(cid:17) (13b)
2014a). The reduced model can be written in state-space
form whereX1,k+1 andX2,k+1 areintermediatevaluesandx1,k
and x are the values of x and x at the grid points
2,k 1 2
ddtx1 =f(t,x1,x2,m,vs) (10a) ttkim=ets0ys+temki=ca1n∆btie.wInristetretninags(12) into (13), the discrete-
0=a(t,x ,x ,m,m ) (10b) P
1 2 N
y=bTx1, (10c) 0=Γk(xk,xk+1,Uk,Uk+1), (14)
with the state vector x1 R137, the algebraic variables with the initial state xT0 = xT1,0,xT2,0 . Here, the ini-
∈
x2 ∈R13, the vector of mass flows mT = mTD,mTI ∈R7 tial state x2,0 follows from sol(cid:2)ving (10b)(cid:3)with x1,0, m0 =
offuel,themassflowsmN ofnitrogen,andtheinitialcon- m(t=t0), and mN,0=mN(t=t0).
(cid:2) (cid:3)
dition x (t=t )=x . The mass flows of fuel, the strip
1 0 1,0 Remark 3. In the dynamic optimization (cf. Sec. 4), the
velocity,andthemassflowsofnitrogenarethesystemin-
system (14) is evaluated recurrently during a particular
puts,wherethemassflowsoffuelandnitrogendependon time interval ti,ti , i.e., 0 = Γ (x ,x ,U ,U )
the switching state s, see Tabs. 1 and 2. Equation (10c) b e k−1 k−1 k k−1 k
with the initial condition x , k K, and the set K =
definesthesystemoutputywhichcorrespondstothestrip (cid:2) (cid:3) k0 ∈
k +1,...,k ,seeFig.2. Theparametersk ,...,k are
0 1 0 1
ytem=p[eTratu]res .atTthheelpinyeraormmetaeprppinogsitinion(1s0zcα),isαd∈efiFne,di.be.y, s{ampling point}s which correspond to tib = tk0,...,tie =
thevecst,oαrαb∈.FThestatevectorx1 summarizesthetemper- ttk11=. tAt=thte baengdintnhienginiotfiatlhceonfidristtioonpxtimi=zatxio.n horizon,
atures of radiant tubes, rolls, strip, and walls. The flue b k0 0 k0 0
8
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
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4 Variable Description
Optimization 3
horizon 2 Tl,d Set-point strip temperature at z
1 s,α α
Tl, Lower bound on strip temperature at z
t1 t2 t3t1 t4t2 t3 t4 t s,α− α
b b b e b e e e
Tl,+ Upper bound on strip temperature at z
s,α α
Optimalsolutionappliedtothefurnace
tl Time when a strip transition takesplace
Optimalsolutionusedasinitialguessinthenexthorizon
tl Beginning of a changeof the strip velocity
Figure2: Recedinghorizonapproach. s
λl Set-point air-fuel equivalence ratio in hz a–d
β
λl Set-point air-fuel equivalence ratio in PCC
pcc
3. Furnace control system
ρl Mass density of the strip material
s
Themassflowoffuelintoeachheatingzone(hza,hzb, cl Specific heat capacityof the strip material
s
hzc,hzd,rth1,rth2,andrts),thecorrespondingair-fuel dl Strip thickness
s
equivalence ratios, the switching state, the strip velocity, bl Strip width
s
and the strip properties (geometry and steel grade) con-
Ll Length of strip
stitute theprimaryinputsoftheCSAF.Next, thecontrol s
structure is briefly discussed.
The primary control objective is that the strip tempera-
Table3: Processdataofthestripldefinedbythesupervisoryfurnace
controllerwithα andβ .
∈F ∈D
Supervisory
Processdata
furnace
controller Temperature vs cVoenltorcoiltlyerSpeed
controller m MassflowAfuierl& Furnace Is gradeandmetallurgicalrequirements. Thescalartemper-
controller ature values Tsl,,αd and Tsl,,α±, α ∈ F, are defined at the
positions z , i.e., the positions of the pyrometersP . De-
α α
xˆ,εˆs State pending onthesescalarvaluesandthepositiontrajectory
observer
of the strip, the supervisory furnace controller generates
piecewise-constant trajectories Td (t) and T (t). In the
s,α s±,α
Figure 3: Hierarchical control structure of the furnace control sys- same way, the piecewise-constant trajectory λ(t) of the
tem. air-fuel equivalence ratiosis designed based on λl.
The second layer(temperature controller)consists of four
turefollowsaset-pointtrajectoryatthepositionsz and
rth modules,seeSec.4andFig.5. Inthislayer,optimization-
z ,wherethepyrometersP andP areinstalled,see
rts rth rts based methods are applied to determine the mass flows
Fig. 1. These strip temperatures are the most important
of fuel and the strip velocity so that a desired heating of
temperatures in terms of material properties and control.
the strip is achieved. This layer is the centerpiece of this
Maximization of throughput and minimization of energy
paper andthecorrespondingcontroltasksarespecifiedin
consumptionaresecondarygoals. Tocaterforallthesere-
Sec. 3.2.
quirements and constraints, a control structure with sev-
Since the CSAF is only equipped with few measurement
eral hierarchical layers seems appropriate. Additionally,
devices, a state observer is employed to provide the tem-
different response times of certain parts of the plant mo-
perature controller with estimated system states. In
tivate the use of a cascaded structure. For instance, the
fact, the observer estimates the augmented state xˆ and
responsetimeofthecontrolvalvesofthefuelandairsup-
the badly known strip emissivity εˆ (Strommer et al.,
s
plyistypicallymuchsmallerthantheresponsetimeofthe
2016). The observer is an adaptive estimator, which uses
temperaturesofthefurnacewallandtherolls,seeSec.1.3.
a copy of the system (11) and a heuristic update law
ˆε˙ =Wo(I ˆI ). Wo is a positive-definite diagonal ma-
3.1. Hierarchical control layers s s− s
trix, I are the measured intensities of the strip, see Fig.
s
Figure3showsthehierarchicalcontrolsystemconsisting 3, and ˆI are the estimated intensities of the strip at the
s
ofthreelayers,i.e.,thesupervisoryfurnacecontroller,the pyrometer positions z , z , and z . The estimated
dff rth rts
temperaturecontroller,andthesubordinatecontrollersfor
intensities can be computed byˆI = σεˆ Tˆ4 with
the mass flowand the strip velocity. The supervisoryfur- s s,α s,α
α
nacecontrollerhasthehighestauthorityandstipulatesall the estimated strip emissivity εˆ anhd temperaitu∈rFe Tˆ ,
s,α s,α
production steps of the hot-dip galvanizing line. It de- see Sec. 2.6.
fines the sequence of strips, their set-point temperatures, The third layer performs subordinate tracking control
andthecorrespondingconstraints. Thisinformationisre- tasks for the mass flows of fuel and air and the strip
ferredtoasprocessdata. Table3showssomeprocessdata velocity. Generally, this layer uses decentralized SISO
specified by the supervisory furnace controller for each PI-controllers. Moreover, cross-limiting controllers en-
individual strip l. The set-point values of the air-fuel sure that the desired air-fuel equivalence ratios are real-
equivalence ratios λl are chosen depending on the steel ized(Froehlichetal.,2016;Strommeretal.,2014b,2017).
9
Post-print version of the article: S. Strommer, M. Niederer, A. Steinboeck, and A. Kugi, (cid:16)Hierarchical nonlinear optimization-based
controller of a continuous strip annealing furnace(cid:17), Control Engineering Practice, vol. 73, pp. 40(cid:21)55, 2018. doi: 10.1016/j.conengprac.
2017.12.005
Thecontentofthispost-printversionisidenticaltothepublishedpaperbutwithoutthepublisher’s(cid:28)nallayoutorcopyediting.