speciÞc heat capacity (J Keywords - mie.uth.gr · IdentiÞcation of catalytic converter kinetic model using a genetic algorithm approach ... instead of elementary employing the downhill

1455

Identification of catalytic converter kinetic model usinga genetic algorithm approach

G N Pontikakis and A M Stamatelos*Mechanical and Industrial Engineering Department, University of Thessaly, Volos, Greece

Abstract: The need to deliver fast-in-market and right-first-time design for ultra-low-emissionvehicles at a reasonable cost is driving the automotive industries to invest significant manpower incomputer-aided design and optimization of exhaust after-treatment systems. To serve the abovegoals, an already developed engineering model for the three-way catalytic converter kinetic behaviouris linked with a genetic algorithm optimization procedure, for fast and accurate estimation of theset of tuneable kinetic parameters that describe the chemical behaviour of each specific washcoatformulation. The genetic-algorithm-based optimization procedure utilizes a purpose-designed per-formance measure that allows an objective assessment of model prediction accuracy against a set ofexperimental data that represent the behaviour of the specific washcoat formulation over a typicallegislated test procedure.

The identification methodology is tested on a demanding case study, and the best-fit parametersdemonstrate high accuracy in matching the measured test data. The results are definitely more accuratethan those usually obtained by manual or gradient-based tuning of the parameters. Moreover, the setof parameters identified by the genetic algorithm methodology is proven to describe in a valid waythe chemical kinetic behaviour of the specific catalyst, and this is tested by the successful predictionof the performance of a smaller-size converter.

The parameter estimation methodology developed fits in an integrated computer-aided engineeringmethodology assisting the design optimization of catalytic exhaust systems that extends all the waythrough from model development to parameter estimation, and quality assurance of test data.

Keywords: catalytic converter kinetic model, genetic algorithm, catalytic exhaust systems

DH molar heat of reaction (J/mol )NOTATIONh convection coefficient (W/m2 s)k thermal conductivity (W/m K)aj,k stoichiometric coefficient of species j in

reaction k km

mass transfer coefficient (m/s)K inhibition term (Table 2) (—)A pre-exponential factor of reaction rate

expression (mol K/m3 s) m exhaust gas mass flowrate (kg/s)M molecular mass (kg/mol )c species concentration (—)

cp

specific heat capacity (J/kg K) N number of time intervals for evaluation ofperformance function=t/Dte error between computerization and

experiment (—) Qamb heat transferred between converter andambient air (J/m3 s)E activation energy of reaction rate expression

(J ); conversion efficiency (—) r rate of reaction (mol/m3 s)Rg universal gas constant=8.314 J/mol Kf performance function (—)

F performance measure (—) R rate of species production/depletion per unitreactor volume (mol/m3 s)G inhibition term (Table 2) ( K)

S geometric surface area per unit reactorvolume (m2 /m3)

t time (s)The MS was received on 20 August 2003 and was accepted after revisionfor publication on 9 July 2004. T temperature ( K)* Corresponding author: Laboratory of Thermodynamics & Thermal

uz

exhaust gas velocity (m/s)Engines, Mechanical & Industrial Engineering Department, Universityof Thessaly, 383 34 Volos, Greece. email: [email protected] z distance from the monolith inlet (m)

D18903 © IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part D: J. Automobile Engineering

1456 G N PONTIKAKIS AND A M STAMATELOS

c catalytic surface area per unit washcoat follow the Langmuir–Hinselwood formalism, modifiedby empirical terms. Generally, the form of the ratevolume (m2 /m3 washcoat)

d washcoat thickness (m) expressions of such models for the reaction between twospecies a and b ise emissivity factor (radiation) (m−1)

q tuneable parameters vectorr density (kg/m3)

r=A e−E/RT cacb

G(ca , cb , … ; K1, K2, …)

(1)s Stefan–Boltzmann constant (W/m2 T4)t duration of an experiment (s)

Thus, the Langmuir–Hinselwood rate expressions deter-y fractional extent of the oxygen storagemine an exponential (Arrhenius-type) dependence oncomponent (—)temperature while G is an inhibition term, a functionYcap washcoat capacity of the oxygen storageof temperature and concentrations c of various speciescomponent (mol/m3)that may inhibit the reaction.

In the above expression, factors A and E (the pre-Subscripts exponential factor or frequency factor and the activation

energy) as well as factors K included in the inhibitionamb ambientterm G are considered as fitting (tuneable) parameters.g gasThe effects of all phenomena not included explicitly intoi parameter indexthe model are lumped in these terms. Therefore, theirj species indexvalues are dependent on the chemical composition of thek reaction indexcatalyst’s washcoat and must be estimated by fitting themon monolithmodel to a set of experimental data, which representn time indexthe behaviour of the catalyst in typical operating cycles.in inlet

The identification process of the model’s tuneables solid or solid–gas interfaceparameters is commonly referred to as model tuning.z axial directionApplicability of the lumped-parameter models is signifi-cantly affected by the successful identification of theirtuneable parameters. Once accurate parameter identifi-1 INTRODUCTIONcation is successful, the model may be used subsequentlyfor the prediction of the catalytic converter efficiency forCatalytic converters have been in use for the past 30 yearsdifferent geometrical and design characteristics or underas an efficient and economic solution for the reductiondifferent operation conditions.of pollutants emitted by passenger car engines. Ever-

Traditionally, fitting of lumped-parameter catalyticdecreasing legislated emission levels trigger the develop-converter models was accomplished manually, a processment of high-efficiency exhaust after-treatment systems,which is highly empirical, requires experience and doeswhich involve optimization of the engine, catalytic con-not guarantee the success of the undertaking over averter, exhaust piping and control for each application.wide range of operating conditions (i.e. full legislatedCatalytic converter modelling tools significantly assist thiscycles with significant variation of reactor conditions).development process. Modern modelling methodologiesTo circumvent these drawbacks, several efforts havehave demonstrated their capacity to be successfullybeen made towards a systematic methodology for modelincorporated in the process of exhaust after-treatmenttuning. All of these are based on the transformation ofsystems design [1–5]. Among the plethora of catalytic con-the tuning problem into an optimization problem, whereverter models that have been published in the literature,a quantity that indicates goodness of fit is optimized forkinetic models with reduced reaction schemes and semi-the tuneable parameters of the model. The goodness-of-empirical rate expressions appear to be better suitedfit quantity may be viewed as a performance measureto the requirements and constraints of the automotiveof the model, since it indicates the performance of theengineer [6 ]. They have been proven able to match themodel compared with the experimental data.accuracy levels required in the prediction of catalyst

Montreuil et al. [7] were the first to present a systematicperformance in legislated driving cycle tests and thus toattempt to tune their steady state three-way catalytic con-provide the engineer with reliable, fast and versatile toolsverter model, using a conjugate gradients optimizationthat may significantly decrease the cost and developmentprocedure. Dubien and Schweich [8] presented a con-time of new exhaust lines.ceptually similar methodology to determine the kineticsReduced reaction scheme models employ a limitedof simple rate expressions from light-off experiments,number of phenomenological reactions that contain onlyemploying the downhill simplex method. Pontikakis andinitial reactants and final products instead of elementaryStamatelos [9] used the conjugate-gradients techniquereactions on the catalyst active sites. The complexity andto determine kinetic parameters of a transient three-details of the reaction path are lumped into the kinetic

rate expressions of these models. Rate expressions usually way catalytic converter model from driving cycle tests.

D18903 © IMechE 2004Proc. Instn Mech. Engrs Vol. 218 Part D: J. Automobile Engineering

1457IDENTIFICATION OF CATALYTIC CONVERTER KINETIC MODEL

Glielmo and Santini [10] presented a simplified control- optimization algorithm of the procedure are updated, inan attempt to approach the problem of computer-aidedoriented three-way catalytic converter model that was

tuned using a genetic algorithm. All the above efforts identification of the kinetic model more systematically.Specifically, a performance measure is first formulatedused performance measures based on the least-squares

error [11] between measured and computed results. The that is suited to the problem of catalytic converter modeltuning in driving cycle tests. Then, a purpose-designedwork of Glielmo and Santini must be distinguished

because of its optimization aspects, as it uses a multi- genetic algorithm is used to extract a set of tuneableparameters that optimizes the performance measure toobjective optimization procedure for the identification

of the model. The genetic algorithm has a potential to obtain a good fit of the model to the experimental data.avoid local optima in the optimization space and thusto fit the model to the experimental data with higheraccuracy. 2 MODEL DESCRIPTION

The combination of a catalytic converter modelwith an optimization procedure for the identification The catalytic converter model used in this study is briefly

described below. The model’s underlying concept is theof the model’s parameters is only a first step towardsa complete computer-aided methodology for catalytic minimization of degrees of freedom and the elimination

of any superfluous complexity in general. A more detailedconverter design and optimization, which is under con-tinuous development at the authors’ laboratory during description of the model and its design concept has been

given in references [6 ] and [15].the last decade. The complete methodology is basedon the following fourfold framework: The prevailing physical phenomena that occur in the

catalytic converter are heat and mass transfer in both(a) catalytic converter model and software based on

gaseous and solid phases. They are described by a systemtuneable Langmuir–Hinselwood kinetics approach;

of balance equations, which is summarized in Table 1.(b) kinetic parameter estimation software based on a

The model has the following features:properly adapted optimization procedure;

(c) emissions measurements quality assurance method- (a) transient, one-dimensional heat transfer calculationsfor the solid phase of the converter;ology and software;

(d) design and implementation of critical experiments (b) quasi-steady one-dimensional calculations of tem-perature and concentration axial distributions forto improve understanding and modelling of catalytic

converters. the gaseous phase;(c) simplified reaction scheme featuring a minimum set

This work is a continuation of the work presented inof Langmuir–Hinselwood-type reduction–oxidation

reference [9] and addresses the interaction of the first(redox) reactions and an oxygen storage submodel

two of the above issues. It is based on the CATRANfor three-way catalytic converter washcoats.

three-way catalytic converter model, which is con-tinuously developed and validated against real-world Heat transfer in the solid phase involves a fully

transient calculation. Nevertheless, quasi-steady heat andcase studies [12–15]. The performance measure and the

Table 1 Model equations and tuneable parameters

Model equation Tuneable parameters

Mass balances

Gas phase (channel ) rguzqcj(z)

qz=rgkm,jS[cj(z)−cs,j(z)] —

Solid phase (washcoat)rgMg

km,jS(cj−cj,s)=Rj —

Heat balances

Gas phase (channel ) rscpuzqTg(z)

qz=hS[Ts(z)−Tg(z)] —

Solid phase (washcoat) rscp,sqTsqt=ks,z

q2Tsqz2+hS(Tg−Ts)+ ∑

NR

k=1(−DH

k)rk+Qamb —

Reaction rates Rj=dcS ∑NR

k=1(aj,k r

k) (rk

is defined for each reaction in Table 2) Ak, k=1, … , 9

Boundary conditions Qamb=hamb(Ts−Tamb)+es(T4s−T4amb) —cj(t, z=0)=cj,in(t)Tg(t, z=0)=Tg,in(t)m(t, z=0)=min(t)j=CO, O2, H2, HCfast, HCslow, NO

x, N2



mass balances are employed for the gas phase, since the with a rate rk, the rate of consumption or production of

a species j isheat and mass accumulation terms in the gas phase areneglected, which is a realistic assumption [16, 17].

Rrea,j=dcS ∑NR

k=1(aj,krk) (3)

2.1 Washcoat level modellingwhere aj,k is the stoichiometric coefficient of species j in

A ‘film model’ approach (see, for example, references reaction k, d is the washcoat thickness and c is the[12] and [18]) approximates the washcoat (usually about specific catalyst area, i.e. catalytically active area per50 mm thickness) with a solid–gas interface, where it is washcoat volume. Thus, any washcoat diffusion effectsassumed that all reactions occur. This approximation are lumped in the kinetics expressions.essentially neglects diffusion effects in the pore system The two primary categories of heterogeneous catalyticof the washcoat and assumes that all catalytically active reactions occurring in the washcoat are redox reactionssites are directly available to gaseous-phase species at and oxygen storage reactions (see reaction scheme inthis solid–gas interface [18, 19]: Table 2). Oxygen storage phenomena play a principal

role in the efficiency of the modern three-way catalyticrgMg

km,jS(cj−cj,s)=Rj (2) converter, as they supply the lack of oxygen under

reducing environment. Oxygen storage occurs mainly onthe ceria surface. Ceria (together with zirconia whichOn the right-hand side of equation (2), the rate Rj refers

to the production or consumption of each species at the has analogous activity) are contained in large quantitiesin the catalyst’s washcoat (order of 30 wt %). Under netsolid–gas interface. For NR reactions, each taking place

Table 2 Reaction schemes and rate expressions of the model

Reactionnumber Reaction Rate expression

Oxidation reactions

1 CO+0.5O2�CO2 r1=

A1

e−E1/RgT cCOcO

2G

2 H2+0.5O2�H2O r2=

A2

e−E2/RgT cH2

cO2

G

3 CH1.8(fast)+1.45O2�CO2+0.9H2O r3=

A3

e−E3/RgT cHC

fast

cO2

G

4 CH1.8(slow)+1.45O2�CO2+0.9H2O r4=

A4

e−E4/RgT cHC

slow

cO2

G

NO reduction5 2CO+2NO� 2CO2+N2 r

5=A5

e−E5/RgT cCOcNO

Oxygen storage6 Ce2O3+0.5O2� 2CeO2 r

6=A6

e−E6/RgT cO2

(1−y)7 Ce2O3+NO� 2CeO2+0.5N2 r

7=A7

e−E7/RgT cNO(1−y)

8 2CeO2+CO�Ce2O3+CO2 r8=A8

e−E8/RgT cCOy9 CH1.8(fast)+3.8CeO2� 1.9Ce2O3+CO2+0.9H2O r

9=A9

e−E9/RgT (cHC

fast

+cHCslow

)y10 CH1.8(slow)+3.8CeO2� 1.9Ce2O3+CO2+0.9H2O r

10=A10

e−E10/RgT (cHC

fast

+cHCslow

)y

The inhibition term is

G=T (1+K1cCO+K

2cTHC)2(1+K

3c2COc2THC)(1+K

4c0.7NO)

where

Ki=Aie−Ei/RgT, i=1, … , 4,

with

A1=65.5, A

2=2080, A

3=3.98, A

4=4.79×105

E1=−7990, E

2=−3000, E

3=−96 534, E

4=31 036

and where

cTHC=cHCfast

+cHCslow

The auxiliary quantities are

y=2×mol CeO

22×mol CeO

2+mol Ce

2O3

dy

dt=

r6+r7

Ycap−

r8+r9+r10

Ycap



oxidizing conditions, three-valent cerium oxide (Ce2O3) solid–gas interface. The parameter h is the heat transfercoefficient and is calculated as a function of the Nusseltmay react with O2 or NO and oxidize to its four-valent

state (CeO2). Under net reducing conditions, CeO2 may number. Finally, the boundary conditions for the tem-perature, mass flowrate and concentrations are givenfunction as an oxidizing agent for CO, hydrocarbons

(HC) and H2 . from measurement at the converter’s inlet.The model uses the auxiliary quantity y to express

the fractional extent of oxidation of the oxygen storage2.3 Reactor-level modellingcomponent [12]. Specifically, the oxidation rate of the

oxygen storage component is assumed proportional to The reactor model presented in this work is a one-the active sites of Ce2O3 , i.e. to Ycap(1−y). On the dimensional heat transfer model for the transient heatother hand, the oxidation rate of CO and HC by CeO2 conduction in the monolith. Heat losses to the environ-is assumed proportional to Ycapy. Moreover, the rates ment via convection and radiation are also taken intoof these reactions should be linearly dependent on the account. The model’s primary assumptions are thelocal concentration of the corresponding gaseous-phase following.reactant. The extent of oxidation, y, is continuously The temperature field in the converter is describedchanging during transient converter operation. Its value by the equation of transient heat conduction in oneis affected by the relative reaction rates of reactions 6 dimension, with heat sources being convection from theto 10 in Table 2. The rates of reactions are expected to exhaust gas, the enthalpy released from the reactions andbe linear functions of y for CeO2 reduction and 1−y convection to ambient air:for Ce2O3 oxidation. The rate of variation in y is thedifference between the rate that Ce2O3 is oxidized and rscp,s

qTsqt=ks,z

q2Tsqz2+hS(Tg−Ts)reduced:

+ ∑NR

k=1(−DH

k)rk+Qamb (7)dy

dt=

r6+r7

Ycap−

r8+r9+r10

Ycap(4)

Finally, the boundary condition needed for the solutionThe above equation is solved analytically for y at eachof the heat conduction equation refers to the heat lossesnode (typically 15–30 nodes are employed to modelto ambient air:the full channel length), along the catalyst channels. The

quantity Ycap is the total oxygen storage capacity and Qamb=Smon [hamb(Ts−Tamb)+es(T4s−T4amb)] (8)its value may be estimated by the content of ceria in the

The one-dimensional approximation of the converterwashcoat.neglects any non-uniformity of inlet flow profiles [20].Two- or three-dimensional model versions exist [12]; how-2.2 Channel-level modellingever, it would be unrealistic to require such huge amount

For the formulation of the channel-level model, two of input data (time series of inlet flow profiles) in orderusual simplifications are employed: firstly, the axial to predict the performance of a catalytic converter ondiffusion of mass and heat in the gas phase is negligible a car.and, secondly, the mass and heat accumulation in the For the formulation of the reaction scheme, two typesgas phase is negligible. (This includes the assumption of heterogeneous catalytic reaction are considered: redoxfor the quasi-steady state nature of the problem.) reactions and oxygen storage reactions. The complete

Using the quasi-steady state approximation and reaction scheme of the model, together with the rateneglecting diffusion and accumulation terms, the mass expression for each reaction, is summarized in Table 2.balance for the gas phase becomes The constants for the inhibition term are taken from

references [19] and [21]. Below, the features of therguzqcj(z)

qz=rgkm,jS[cj(z)−cs,j(z)] (5) reaction scheme are examined in some more detail.

Redox reactions take place on the precious metal load-where cj is a mean bulk value employed for the gas-phase ing of the washcoat (a combination of platinum (Pt)concentration of each species and cs,j the corresponding palladium (Pd) and rhodium (Rh) depending on thevalue at the solid–gas interface. Energy is transferred to formulation) and involve oxidation of CO, H2 and aand from the exhaust gas only due to convection with complex mixture of HC compounds in the exhaust gas,the channel walls (gas-phase energy balance): as well as reduction of nitrous oxides (NO

x) to N2 .

Oxygen storage reactions proceed on the ceria com-rscpuz

qTg(z)

qz=hS[Ts(z)−Tg(z)] (6) ponent of the washcoat, where three-valent cerium oxide

(Ce2O3) is oxidized by O2 and also by NO (see alsoreferences [22] and [23]) to its four-valent counterpartSimilarly to the above, a mean bulk value Tg is used

for the exhaust gas temperature, and a solid-phase (CeO2). In turn, CeO2 is reduced by CO and HC toCe2O3 .temperature Ts is introduced for the monolith and the



In the present model, the oxidation reactions rates here apparent kinetics and not real kinetics are beingof CO and HC are based on the expressions obtained considered. Owing to the complexity of the reactionby Voltz et al. [24], which were originally developed for scheme and the characteristic range of conditions ofa Pt oxidation catalyst but, interestingly enough, they the specific reactor, differences in apparent activationproved successful, with certain improvements made by energies between different washcoats can be usuallyother researchers [19, 21], in describing the performance accommodated by the tuning of the pre-exponentialof Pt : Rh, Pd, Pd : Rh and even trimetal catalyst wash- factors reported above. Furthermore, the Voltz inhibitioncoats. Of course, a search for further improvements in factor without modification in its term has been foundthe inhibition expressions and parameters can be under- to give consistently satisfactory results for a wide rangetaken in the future with the assistance of the genetic of washcoats. Of course, there exists the possibility toalgorithm optimization tool described here. However, this tune activation energies together with the pre-exponentialwill require dedicated high-accuracy experiments, and factors. This alternative has also been tested and thethe improved expressions will be valid for the specific genetic algorithm usually resulted in values that are closewashcoats. The philosophy here is to fit the needs of the to those originally known.automotive manufacturer to the models of global validity Additionally, the kinetic constants of H2 oxidationand minimized requirement in specialized experiments. are also not tuned in this work. The H2 content of the

Another simplification is made regarding the modelling exhaust gas is typically of the order of one-third of theof the HC reactions. In practice, flame ionization detector CO content (due to the thermodynamics of the water–gasanalysers measure only the total hydrocarbon content of shift reaction in the exhaust gas [25]). It is not routinelythe exhaust gas and make no distinction of the separate measured in exhaust emissions testing. In this work theHC species. Therefore, for modelling purposes, the total kinetics of H2 oxidation are assumed to be equal to thoseHC content of the exhaust gas is divided into two broad of CO oxidation. A justification for this assumption iscategories: easily oxidized HC (‘fast’ HC), and less that in presence of CO the oxidation rate of hydrogeneasily oxidized HC (‘slow’ HC). Throughout this work, is inhibited by CO to approximately the same extent asit is assumed that the exhaust hydrocarbon consisted of the oxidation of CO itself [26 ]. Also, this practice was85 per cent fast HC and 15 per cent slow HC. This is suggested from experience with computer-aided para-a rough approximation introduced because of lack of meter estimation, where it was noticed that the geneticmore accurate data. According to our experience, it gives algorithm does not converge for the specific reaction.satisfactory results [9, 12]. The same practice has been Thus, there are nine tuneable parameters in total, oneemployed by other researchers in the field. Both fast HC for each reaction except for the reaction of H2 .and slow HC are represented as CH1.8 , since the average Since the problem of model tuning is a parameter-fittingratio of hydrogen to carbon atoms in the exhaust gas is problem, it may be tackled as an optimization problem.1.8. Thus, the two HC compounds are distinguished in This involves the development of two components:the model only by the difference in the kinetic parameters.

(a) a performance measure, which qualitatively assessesthe goodness of fit of the model for each possible

4 TUNING PROCEDURE set of parameter values;(b) an optimization procedure, which finds a set of

4.1 General comments tuneable parameters giving an optimum value forthe performance measure, i.e. yields in modellingThe reaction rate expressions introduce into the catalyticresults that are as close to the measured results asconverter model a set of parameters that has to bepossible.estimated with reference to a set of experimental data.

In the present model, the set of tuneable parametersThe most usual performance measure used is basedis formed by the pre-exponential factors A

kthat are

on the least-squares error between measured and com-included in the reaction rates rk. Our objective is to fit

puted instantaneous concentrations of pollutants at thethem against experimental data from a routine drivingconverter’s outlet. Here, a new performance measurecycle test.is modified that is more beneficial for optimizationIn concept, the activation energy E

kof each reaction

purposes and may also be used independently as anand the set of terms Ki, included in the Voltz inhibition

objective generic measure to compare the performanceterm G, may also be considered as tuneable parameters,of different models.but no attempts are made to tune them for simplicity

The performance measure was optimized using aand better control of the set of tuneable parameters.genetic algorithm, because previous experience has shownIndicative activation energy values for each reactionthat the problem of model parameter estimation is multi-are known from previous experience and publishedmodal and the genetic algorithm is a powerful techniquework [19, 21, 24]. Their variation over different wash-

coat formulations is not very significant. Of course, for multi-modal optimization. The genetic algorithm was



properly adapted to the problem at hand. The details of time period of interest:performance measure and genetic algorithm formulationare presented below. F(q)=

1

N∑N

n=0f (tn; q)=

1

N∑N

n=0

|e(tn; q) |

emax(tn)(14)

The performance measure defined in equation (13) is4.2 Formulation of the performance measure used for the assessment of the performance of each of

the three pollutants CO, HC and NOx. The total per-The performance measure that is formulated below

formance measure is computed as the mean of theseexploits information on the measurement of species con-three values:centrations at the inlet and the outlet of the catalytic

converter. Specifically, it is based on the conversionF=

FCO+FHC+FNOx

3(15)efficiency Ej for a pollutant j. Herein, the three legislated

pollutants are taken into account; thus j=CO, HCThe above performance measure presents advantageousor NO

x.

features compared with the classical least-squaresTo account for the goodness of computation resultsperformance measure:compared with a measurement that spans over a certain

time horizon t, an error e for each time instance 1. It ranges between two, previously known finite extrememust be defined. The latter should give the deviation values. Extremes correspond to zero and maximumbetween computation and measurement for the con- deviation between calculation and experiment.version efficiency E. Summation over time should then 2. The extrema of the performance measure are thebe performed to calculate an overall error value for the same for all physical quantities that may be used andwhole extent of the measurement. Here, the error is all different measurements where the performancedefined as measure may be applied; i.e. the performance measure

is normalized so that its extrema do not depend on|e|=|E−E|. (9)either the measured quantities or the experimental

Absolute values are taken to ensure error positiveness. protocol.This error definition also ensures that 0∏|e|∏1, since

It should be noted here that, because of the aboveit is based on conversion efficiency.properties, this performance measure may be used as aThe error between computation and measurement isgeneral measure to compare the model’s performancea function of time and the tuneable parameter vector:under different case studies or to compare alternativee=e(t; q), where q is formed by the pre-exponentialmodels for a single case study; i.e. it is a genericfactor of each reaction of the model:quantitative measure to assess the model’s performance.This should be contrasted with the usual practice forq= [A

1, A2, … , A

NP

]T (10)model assessment, which is simply based on inspection.

The performance function f (t; q)= f (e(t; q)) is a function Although a visualization procedure is necessary to gainof the error e, which is subsequently summed over some insight to the model’s results, it is a subjective criterion.time horizon t to give the performance measure F. Here, A least-squares performance measure, on the otherthe performance function is defined as hand, depends on the measurement at hand and is not

helpful for comparison purposes. A normalized per-f (tn; q)=

|e(tn; q) |

emax(tn)(11) formance measure such as that defined above eliminates

this problem and should provide more insight to modelassessment.Time t take discrete values, t

n=n Dt, with Dt being the

From the optimization point of view, normalizationdiscretization interval which corresponds to the frequencyof the performance measure is required because thethat data is measured. The quantity emax is the maximumtotal performance measure F is computed as the meanerror between computation and measurement, and it isof FCO , FHC and FNO

x

. If each of the individual per-defined asformance measures was not normalized by definition, they

emax(tn)=max [E(tn), 1−E(t

n)] (12) would take values of different points of magnitude. Then,

arbitrary scaling factors (weights) would be necessaryThe performance measure can be subsequently formedbefore taking the average to compute F. With the currentusing some function of the sum of the performanceperformance measure definition, this is avoided.function over time:

4.3 Optimization procedureF(q)=FA ∑Nn=0

f (tn; q)B , N=

t

Dt(13)

Having defined the performance measure for the model,the problem of tuneable parameter estimation reduces inIn this work, the performance measure F is defined as

the mean value of the performance function over the finding a tuneable parameter vector q that minimizes F.



Owing to the multimodal character of the problem, algorithm. The present implementation is summarizedin Table 3. The genetic algorithm is a real-coded genetica genetic algorithm has been employed for the task.

Since genetic algorithms are maximization procedures, algorithm and uses the SBX [29] for the mating andrecombination of individuals.the problem is converted into a maximization problem

for F ∞, defined as F ∞=1−F. The SBX operator works directly on the real-parametervector that represents each individual, thus eliminatingSummarizing the above, the mission of the genetic

algorithm is to solve the following problem: the need for a real-to-binary encoding–decoding requiredin binary encoded genetic algorithm. The SBX operatorMaximize F ∞(q)=1−F(q)also works on arbitrary precision, which should becontrasted with the finite precision of binary encodings.

=1

3N∑

j=CO,HC,NOx

∑N

n=0

|ej(tn ; q) |ej,max(tn)

(16) The randomized nature of the genetic algorithmenables it to avoid local extrema of the parameter

This is a constraint maximization problem, since the space and converge towards the optimum or a near-components of the vector q are allowed to vary between optimum solution. It should be noted, however, thattwo extreme values, i.e. q

i,min∏qi∏qi,max . this feature does not guarantee convergence to theThe genetic algorithm is a kind of artificial evolution, global optimum. This behaviour is common to all multi-

where a population of solutions evolves similarly modal optimization techniques and not a specific geneticto nature’s paradigm: individual solutions are born, algorithm characteristic.reproduce, are mutated and die in a stochastic fashion The above-described genetic algorithm optimizationthat is nevertheless biased in favour of the fittest procedure forms the basis of software with the generalindividuals [27]. The implementation of the algorithm that flow chart in Fig. 1.has been developed in this work takes the following steps.

Step 1: initialization. A set of points in the optimizationspace is chosen at random. This is the initial population

5 APPLICATION CASE STUDYof the genetic algorithm, with each point (each vector oftuneable parameters) corresponding to an individual

The validity of the approach that is described above isof the population.assessed in a real-world application case. Manual tuningStep 2: fitness calculation. The fitness of each individualof the model is originally performed, and the results arein the population is computed using equation (16). Itsubsequently compared with the identification resultsshould be noted that the fitness calculation requiresproduced by the genetic algorithm. It is found that thethat the model be called for each individual, i.e. asgenetic algorithm manages to find a set of tuneable para-many times as the population size.meters that fits the experimental data with much higher

Step 3: selection. Random pairs of individuals are accuracy than the manual efforts. In order to check thesubject to tournament, i.e. mutual comparison of their useability of the genetic-algorithm-tuned model, it isfitnesses [28]. Tournament winners are promoted for applied to a second set of driving cycle data, obtainedrecombination. with a catalyst with significantly reduced size. It is found

that the model is able to predict the efficiency of theStep 4: recombination (mating). The simulated binarysecond catalyst successfully.crossover (SBX ) operator [29] is applied to the couples

Specifically, in this application example, a set ofof individuals that are selected for recombinationmeasurements of emissions upstream and downstream(parents). The resulting chromosomes are inserted inof a Pt : Rh (5 : 1) catalyst installed on a 1.8 l gasolinethe children population.engine, tested according to a simulated new European

Step 5: mutation. A small part of the population is driving cycle (NEDC) test on the computer-controlledrandomly mutated; i.e. random parameters of the engine bench, is employed. The washcoat thickness ischromosomes change value in a random fashion [30]. about 50 mm. The precious metal loading is 50 g/ft3. The

Step 6: The original parent population is discarded; thechildren population becomes the parent population.

Table 3 Parameters of the genetic algorithmStep 7: Steps 3 to 6 are repeated for a fixed number of

Encoding type Real parameter encodinggenerations or until an acceptably fit individual hasAlgorithm type Generationalbeen produced. Selection scheme Deterministic tournamentCrossover operator (SBX )Genetic algorithms are not black-box optimizationMutation operator Random mutation

techniques. On the contrary, a genetic algorithm should Population size 100Crossover probability 0.6be adapted by the user to the target problem [28]. ThereMutation probability 0.02are a number of design decisions and parameters thatParameter range 105<A<1025

influence the operation, efficiency and speed of the genetic



Fig. 1 Flow chart of genetic algorithm (GA) optimization methodology (DARWIN code), which employsrepeated application of the CATRAN catalytic converter simulation code to produce, after theprescribed number of generations, a set of kinetics parameters with the best fit to the test data

catalytic converter has a circular cross-section of 127 mm light-off point of the catalyst. Increased accuracy isdemanded during the whole extent of the cycle test. Thediameter and it consists of two beds with a total length

of 203 mm. CO, HC NOx, O2 and CO2 analysers measure low levels of concentrations at the catalyst exit, com-

pared with the corresponding concentrations at the inlet,the exhaust gas content upstream and downstream ofthe catalyst. Data acquisition is made at 1 Hz frequency further complicate the undertaking. Its success is thus

heavily dependent on both the careful model formulationand the analysers’ response time constants range from0.1 to 0.5 s. Figure 2 presents an overview of the emissions and the accurate tuneable parameter identification.

Proceeding to the model identification, Table 4 presentsmeasurements set-up.Figure 3 presents a summary of the measured results the set of kinetics parameters of the model and their

values tuned manually and by the genetic algorithm. Inafter preprocessing with the data consistence and errorchecking routines [31]. Evidently, the catalytic converter principle, the kinetic parameters are 21 in total: one

parameter for the oxygen storage capacity, and ten pairslight-off occurs at about 50 s after the beginning of themeasurement. After light-off, and up to about 800 s, of parameters A and E for the 10 reactions incorporated

in the reaction scheme.emissions are almost zeroed. The first 800 s correspond tothe urban phase of the driving cycle. During this phase, As previously discussed, not all kinetic parameters are

tuned. The activation energies are more or less knownonly few emission breakthroughs occur. Comparativelymore pollutants are emitted in the period from 800 to from previous experience [21]. They could be varied a

little, but this is not necessary since the rate depends on1180 s (which corresponds to the extra-urban phase ofthe NEDC), because of the higher space velocity of the both A and E and any small difference can be com-

pensated by respective modification of A. The oxygenexhaust gas.It is important to note that, because of the very low storage capacity is also not tuned, since its approximate

magnitude is estimated on the basis of the washcoatemission standards, it does not suffice to predict thecomposition (Ce and Zr) [32] and is also checked by itsemptying behaviour at characteristic parts of the cycle.Finally, the H2 oxidation kinetics are assumed to beapproximately equal to those of CO oxidation.

Thus, nine pre-exponential factors are left to be tuned:four reactions of gaseous phase species on the Pt surface,and another five reactions on the ceria–zirconia com-ponents of the washcoat. The manual tuning that wasinitially performed gives the results that are illustratedin Figs 4 and 5. Manual tuning was performed followinga trial-and-error procedure and was mainly aided byprevious experience with similar catalysts. Figure 4 givesthe cumulative emissions for all pollutants. Although thecomputed total mass of pollutants matches the measure-ments, at least regarding CO and NO, the computationFig. 2 Emissions measurements set-up. The legislated cycleloses accuracy in several parts of the cycle. For example,for the respective vehicle is simulated by the computer-

controlled engine dyno controller the error in the prediction of cumulative CO emissions



Fig. 3 Measured instantaneous CO, HC and NOx

emissions at converter inlet and exit, over the 1180 sduration of the cycle: 1.8 l-engined passenger car equipped with a 2.4 l underfloor converter with50 g/ft3 Pt:Rh catalyst

Table 4 Values of activation energy, tuneable pre-exponential factors (manual estimates versus values determinedby the genetic algorithm) and oxygen storage capacity value inserted in the model

A

Reaction E Manual Genetic Ycapnumber Reaction (fixed value) tuning tuning (fixed value)

1 CO+0.5O2�CO2 90 000 1×1019 4.89×10202 H2+0.5O2�H2O 90 000 1×1019 4.89×10203 CH1.8(fast)+1.45O2�CO2+0.9H2O 95 000 2×1019 3.61×10204 CH1.8(slow)+1.45O2�CO2+0.9H2O 120 000 5×1019 1.83×10175 2CO+2NO� 2CO2+N2 90 000 4×1014 1.54×10116 Ce2O3+0.5O2� 2CeO2 90 000 2×1010 2.94×1009 6007 Ce2O3+NO� 2CeO2+0.5N2 90 000 3×109 4.68×1010 6008 2CeO2+CO�Ce2O3+CO2 85 000 2×109 7.85×109 6009 CH1.8(fast)+3.8CeO2� 1.9Ce2O3+CO2+0.9H2O 85 000 9×1010 1.35×1010 600

10 CH1.8(slow)+3.8CeO2� 1.9Ce2O3+CO2+0.9H2O 85 000 1×1010 2.43×1013 600

in Fig. 4 approaches 20 per cent at about 800 s from of 5 per cent, which indicates that also the instantaneousemissions of the model are fitted with good accuracy.start. Much higher errors are reported in the prediction

of instantaneous emissions. For example, error in the This becomes obvious in the comparison of computedand measured instantaneous emissions for CO, HC andprediction of instantaneous CO emissions (Fig. 5) some-

times exceeds 500 per cent. The situation presented in NOx

in Figs 7, 8 and 9 respectively. Here it could bealso said that there is a good quantitative fit of theFig. 5 can be described as some kind of qualitative fit of

the computed instantaneous CO concentrations at the instantaneous emissions in a real cycle. Maximum errorcan reach even 100 per cent in certain cases (e.g. HCconverter’s outlet.

The model’s accuracy concerning instantaneous emissions between 800 and 1050 s in Fig. 8), but theseare exceptions that hint at specific problems (possiblyemissions is significantly improved when the model

is fitted using the genetic algorithm. The comparison with the storage submodel and the data quality), to beinvestigated in the future.of computed versus measured cumulative emissions is

illustrated in Fig. 6. The form of the curves for all three It must be noted that the fit of the model is moresuccessful for the CO and HC curves than for the NO

xpollutants matches the measured data much more closely(maximum error in cumulative emissions is of the order curve. This is mainly attributed to the Voltz inhibition



Fig. 4 Manual tuning: comparison of computed versus measured cumulative emissions for CO, HC andNOx

(full-size converter)

Fig. 5 Manual tuning: comparison of computed versus measured CO instantaneous emissions (full-sizeconverter)

term for the CO and HC oxidation reactions. On the con- The evolution of the genetic algorithm populationof solutions is indicative of the problem difficulty andtrary, no appropriate inhibition term has been extracted

for the reactions that involve NOx. Furthermore, com- explains the limited success of manual tuning or tuning

that uses gradient-based methods. To illustrate theparing the fits for CO and HC curves, it may be readilyfound that the HC fit is inferior. This is expected since evolution process, a graph of the evolution of maximum

and average fitness of the population is presented inthe complicated mixture of HC types contained in theexhaust gas is approximated by only two components: Fig. 10. The genetic algorithm quickly improves the maxi-

mum performance measure solution at the beginning offast HC and slow HC. Bearing in mind that this is avery gross approximation, the model may be considered the run. Then, evolution is slower and, after some point, it

completely stalls. This indicates that the genetic algorithmfairly satisfactory.



Fig. 6 Computer-aided tuning: comparison of computed versus measured cumulative emissions for CO, HCand NO

x(full-size converter)

Fig. 7 Computer-aided tuning: comparison of computed versus measured instantaneous CO emissions(full-size converter)

population has converged to a specific attraction basin of The spreads of individuals in the 20th, the 45th andthe last (135th) generation are given in Figs 11, 12 andthe optimization space and not much improvement may

be achieved. At this point, the algorithm is stopped. The 13 respectively. The individuals are sorted in descendingorder according to their performance measure.specific computation required about 72 h on a 2.4 GHz

Pentium 4 computer. Figure 11 visualizes the spread of the kinetic para-meters in the population of the genetic algorithm nearIt may be noted that the absolute value of the per-

formance measure does not vary much during the genetic the beginning of the procedure. The kinetic parametersare allowed to vary in certain intervals that are inducedalgorithm run. This is a property of the performance

measure formulation but also indicates the multi- on the basis of previous experience and are consistentwith their physical role in the respective reactions. Themodality of the problem, since it appears that many

combinations of kinetic parameters lead to the same different kinetic parameters pertaining to reactions thatoccur on the three distinct catalytic components of theoverall performance of the model.



Fig. 8 Computer-aided tuning: comparison of computed versus measured instantaneous HC emissions(full-size converter)

Fig. 9 Computer-aided tuning: comparison of computed versus measured instantaneous NOx

emissions(full-size converter)

washcoat (in our example, Pt, Rh and Ce) fall in three oxygen do not converge, whereas the rest of the para-meters show clear signs of convergence. This could bedistinct intervals.

Figure 12 gives the spread of individual solutions attributed to the fact that the slow HC are only 15 percent of the total HC content and thus influence thein the 45th generation of the population. Apparently,

the population has started to converge for the pre- total HC efficiency of the catalyst much less than thefast HC. The same absence of convergence is noticed forexponential factors of some reactions. This indicates that

the kinetics of these reactions influence the quality of the kinetics of CO+NO reaction, whereas the com-plementary reaction of ceria+NO shows clear signs ofthe model fit (and thus the performance measure value)

much more significantly than the rest of the reactions. convergence. This fact suggests a lack of sensitivity ofthe model regarding the above three reactions. It shouldFigure 13 presents the last population of the genetic

algorithm run. It is evident that the parameters for the not be deduced at this early investigation point that thesereactions are less important than the rest to the model’soxidation of slow HC with oxygen on Pt or with stored



Fig. 10 Evolution of the genetic algorithm: maximum and average population fitness during the first 135generations

Fig. 11 Spread of genetic algorithm population at the 20th generation (HCf, HCfast ; HCs, HCslow)

accuracy and predictive ability. Experience shows that by the genetic algorithm for the full-scale converter areapplied in the prediction of the behaviour of a reducedfurther reduction in the number of reactions leads to an

observable deterioration of the model-fitting ability. size converter with the same washcoat formulation andloading. This is a demanding exercise for any model ofFor comparison purposes, the best set of kinetic

parameters values derived in the three characteristic this type. The model’s prediction is checked againstexperimental results obtained with a cylindrical con-generations of the genetic algorithm evolution is

presented, together with the manually derived set, in verter of 120 mm diameter and 60 mm length. The resultsare presented in Fig. 15 in the form of cumulative CO,Fig. 14.

The above discussion should make apparent that the HC and NOx

emissions.The results, which are considered good with theparameter identification methodology developed gives

significant feedback also to the reaction modelling. This exception of HC emissions, are indicative of the pre-dictive ability of the one-dimensional model for typicalis a subject of continuing investigation.

As a next step in the evaluation of the parameter quality test data. To check further the model’s per-formance regarding HC, the computed instantaneousidentification methodology, the kinetic parameters derived



Fig. 12 Spread of genetic algorithm population at the 45th generation (HCf, HCfast ; HCs, HCslow)

Fig. 13 Spread of genetic algorithm population at the last (135th) generation (HCf, HCfast ; HCs, HCslow)

HC emissions are compared with the experimental viously developed three-way catalytic converter model.This is a one-dimensional model for the heat and massvalues in Fig. 16. A certain degree of discrepancy is

observed, which could be attributed to the enhanced transfer in the catalytic converter that features areduced kinetics scheme.three-dimensional effects introduced by the very small

length-to-diameter ratio of the reduced-size catalytic 2. This scheme involves rate expressions which containa limited number of apparent kinetic parameters thatconverter. Overall, it should be stressed that the model

prediction continues to stay close to the experimental may be viewed as fitting parameters. Their values areidentified in order to fit a set of experimental datadata, both qualitatively and quantitatively, which further

supports the validity of the kinetic model approach. that represents the behaviour of the specific washcoatformulation over a typical test procedure.

3. In this paper, a complete identification methodology6 CONCLUSIONS for the above problem is formulated in two steps.

Firstly, a performance measure is defined that is suit-1. A genetic algorithm methodology was developed for able for the assessment of the model’s performance

in fitting the data. Secondly, a genetic algorithm isthe identification of the kinetic submodel of a pre-



Fig. 14 Comparison of manually derived kinetics and kinetics identified at the 20th, 45th and 135thgenerations of the genetic algorithm run

Fig. 15 Application of the kinetic parameters identified by the genetic algorithm for the full-size converter,to predict the behaviour of the reduced-size converter: comparison of computed versus measuredcumulative emissions for CO, HC and NO

x

employed that uses the performance measure as an the search space is highly multi-modal, which causesnon-stochastic search procedures to become trappedobjective function. The genetic algorithm searches the

parameter space to find the optimal set of parameters to local optima.5. Moreover, the set of parameters identified by theproducing the best fit to the data.

4. The identification methodology is tested on a charac- genetic algorithm methodology is found to describein a valid way the chemical kinetic behaviour ofteristic case study, and the best-fit parameters pro-

duced demonstrate a high accuracy in matching the the specific catalyst. This is confirmed by applyingthe specific set of kinetic parameters to predict thetest data describing the behaviour of a specific catalyst

installed on a 1.8 l passenger car engine tested accord- behaviour of a reduced-size converter with the samecatalyst formulation. The prediction accuracy is remark-ing to the NEDC procedure. The results are far more

accurate than those that may be obtained by manual able if the statistical variation in the performance ofsuch a complex system is taken into account.or gradient-based tuning of the parameters, because



Fig. 16 Application of the kinetic parameters identified by the genetic algorithm for the full-size converter,to predict the behaviour of the reduced-size converter: comparison of computed versus measuredinstantaneous HC emissions

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speciÞc heat capacity (J Keywords - mie.uth.gr · IdentiÞcation of catalytic converter kinetic model using a genetic algorithm approach ... instead of elementary employing the downhill

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