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2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.Peer review under responsibility of International Federation of Automatic Control.10.1016/j.ifacol.2015.10.062

Stefano Sabatini et al. / IFAC-PapersOnLine 48-15 (2015) 434–440

© 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

A New Semi-Empirical Temperature Modelfor the Three Way Catalytic Converter

Stefano Sabatini∗, Irfan Kil∗∗, Joseph Dekar ∗∗∗,Travis Hamilton ∗∗∗, Jeff Wuttke ∗∗∗, Michael A Smith ∗∗∗,

Mark A Hoffman∗, Simona Onori∗,∗∗

∗ Department of Automotive Engineering- Clemson University,Clemson, SC 29607 USA

∗∗ Department of Electrical and Computer Engineering - ClemsonUniversity, Clemson, SC 29634 USA

e-mail: (ssabati, ikil, mhoffm4, [email protected])∗∗∗ FCA US LLC - Auburn Hills, MI 48326 USAe-mail: (joseph.dekar, travis.hamilton, jeff.wuttke,

[email protected])

Abstract: This paper proposes a semi-empirical model to predict the temperature of the three-way catalytic converter. The model is derived from the partial differential equations describingthe energy balance in the catalyst and it is simplified by lumping the heat produced by theexothermic reactions occurring in the catalyst into a single parameter. This simplificationallows the model to be run real time in production vehicles, since no information about theconcentration of the chemical species in the exhaust gas is requested as input to the model. Theparameters are identified using a particle swarm optimization algorithm. It is shown that themodel accurately predicts the system behavior.

1. INTRODUCTION

Future tightening of emissions standards and the 54.5 mpgfleet wide average fuel efficiency target on production ve-hicles by 2025 have spurred great interest from automotivecompanies toward the modeling, control and optimizationof engine and aftertreatment systems (EPA, 2011). TheThree Way Catalyst (TWC) has been extensively usedto reduce oxides of nitrogen (NOx), and to oxidize hy-drocarbons (HC) and carbon monoxide (CO) in currentproduction vehicles.

The conversion efficiency of the TWC is highly dependenton temperature, as shown in Figure 1. The ”light-offtemperature” is defined as the temperature at which thedesired reduction efficiency reaches 50% (Brandt et al.,2000). The time it takes the catalyst temperature to reachlight-off temperature should be minimized for improvedTWC functionality. Also, knowing and monitoring thecatalyst temperature is crucial to: 1) protect the TWCfrom excessive temperatures leading to premature TWCfailure and 2) estimate the temperature dependent oxygenstorage component (OSC) of the catalyst. Models can beused to estimate the catalyst temperature in that they area more cost effective alternative to temperature sensors.

A significant amount of research has been dedicated tomodeling the TWC. Two modeling approaches are pre-dominant in literature: physics-based modeling and empir-ical modeling. Physics-based models of the TWC are de-veloped in Auckenthaler (2005), Montenegro and Onorati(2009), Kang et al. (2014),Kumar et al. (2012) and Depcikand Assanis (2005). In those works, converter operationwas described using energy and mass balance equations

Fig. 1. The effect of temperature on a TWC conversionefficiency (jointly for CO, HC and NOx) (reproducedfrom Bresch-Pietri et al. (2013)).

with the purpose of predicting conversion of undesiredengine emission species within the catalyst. Since the per-formance of the catalyst strongly depends on temperature,these models estimate catalyst temperature by solvinga system of partial differential equations describing theenergy balance coupled with a detailed chemical kineticsmodel. The estimation of the kinetic parameters makesthe identification of these models quite challenging. Inaddition, due to their computational complexity, thesemodels are not suitable for real time catalyst temperatureestimation and OSC control.

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1. INTRODUCTION














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1. INTRODUCTION














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1. INTRODUCTION








Empirical models, like those developed by Shafai et al.(1996), Ammann et al. (2000), Brandt et al. (2000), Ro-duner et al. (1997), Cioffi et al. (2001) and Tomforde et al.(2013) are more computationally suitable for real-time cat-alyst monitoring and control. The objective of these mod-els is to estimate the oxygen storage dynamic, which highlyinfluences the exhaust gas emissions during transients. Themodels presented in Brandt et al. (2000) and Cioffi et al.(2001) also include catalyst temperature estimation. InBrandt et al. (2000) the temperature estimation is used asan input to a static catalyst conversion efficiency map inorder to predict CO, NO and HC emissions. In Cioffi et al.(2001), the adsorption rate of an empirical oxygen storagemodel depends on temperature estimation. In both cases,the models consist of a first order differential equationwhose parameters are empirically identified. Because oftheir simplicity, these models lack accuracy and are notreliable over wide ranging operating conditions, especiallyduring the warm-up phase.

In Bresch-Pietri et al. (2013) a control-oriented TWCtemperature model is obtained by simplifying the partialdifferential equations describing the energy balance into atime-varying input delay model. In that work TWC inletconcentrations of CO, NO and HC are model inputs forcharacterization of reaction heat. While complex modelscan approximate reaction heat through species estimationfrom lambda, actual species measurement are not avail-able in production vehicles. Models this complex are notdirectly suitable for real time operation.

This paper presents a new semi-empirical temperaturemodel suitable for real time vehicle operation. The model,suitable for control purposes, simulates thermal transientsinside the catalyst. The heat produced by the chemicalreactions is lumped into a single term so that the speciesconcentrations are not necessary inputs. Experimental re-sults, obtained under real driving conditions, demonstratethe accuracy of the model.

This paper is organized as follows: Section 2 providesinformation about the TWC system. The temperaturemodel development is described in Section 3. In Section4 the experimental setup used for model identificationand validation is presented. Section 5 focuses on theparameter identification procedure and, in Section 6, theexperimental results are presented and discussed.

2. TWC SYSTEM

Modern TWC converters are capable of conversion effi-ciencies approaching 100% when the catalyst is properlyheated and the air fuel ratio is controlled in a narrowband around the stoichiometric value. However, conver-sion efficiency describes only the steady state behavior ofthe TWC while tailpipe emissions are highly affected bytransient variations of the pre-catalyst air fuel ratio. Thedynamic behavior of the TWC is dominated by its abilityto store and release oxygen. For this reason, a considerablenumber of empirical oxygen storage models have been de-veloped (Shafai et al., 1996; Ammann et al., 2000; Brandtet al., 2000; Roduner et al., 1997; Tomforde et al., 2013).Most oxygen storage models neglect the dependence ofoxygen storage dynamics on catalyst temperature. In fact,catalyst temperature affects the reaction rates occurring

in the TWC and consequently the oxygen adsorption andrelease rates.

0 1 2 3 4 5

lam

bda

[-]

0.95

1

1.05200 °C

lambda prelambda post

0 1 2 3 4 5

lam

bda

[-]

0.95

1

1.05400 °C

Time [s]0 1 2 3 4 5

lam

bda

[-]

0.95

1

1.05600 °C

Fig. 2. Effect of temperature on the oxygen storage dy-namics (reproduced from Brinkmeier (2006)).

In Brinkmeier (2006), the temperature dependence of theoxygen storage dynamics is proven empirically by theresults shown in figure 2. In this experiment an oscillatingair-fuel ratio profile is imposed upstream of the catalystand the air fuel ratio response downstream the catalyst isevaluated for different catalyst temperatures. The higherthe temperature, the more the oxygen storage influencesthe post catalyst lambda response. At low temperatures,besides the transport delay, the post catalyst lambdareplicates the pre catalyst lambda. At 600◦C the air fuelratio oscillations are completely absent in the post TWClambda due to the ability of the catalyst to store andrelease oxygen. It is indeed important to estimate thecatalyst temperature and include its effect in the OSCmodeling.

The proposed TWC control oriented model can be repre-sented by the structure of figure 3. The catalyst thermalmodel is designed as a function of exhaust gas temperature(Texh)and exhaust mass flow rate ( mexh) measured up-stream of the catalyst. Inputs to the oxygen storage modelare: 1) pre-catalyst lambda (measured), λpre, 2) exhaustmass flow (measured), mexh and 3) catalyst temperature

(estimated by the proposed thermal model) Tcat. Themodel inputs λpre, mexh and Texh can be either mea-sured by commercially available sensors (oxygen sensors,air mass flow meters, thermocouples) or calculated by theECU. The overall outputs of the TWC are: oxygen storage(state estimated), Φ, catalyst temperature (estimated),

Tcat, and post-catalyst lambda (measured), λpost.This paper focuses on the development, identification andvalidation of the catalyst thermal model.

IFAC E-COSM 2015August 23-26, 2015. Columbus, OH, USA

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436 Stefano Sabatini et al. / IFAC-PapersOnLine 48-15 (2015) 434–440

Fig. 3. Proposed structure of TWC.

3. TWC TEMPERATURE MODEL

Using a one dimensional approach, the TWC can bedescribed as a single channel consisting of a gas phase anda solid phase. The gas phase represents the exhaust gasflowing into the converter and the solid phase representsthe catalyst washcoat and substrate. The temperaturedynamics of the TWC can be described by performing theenergy balance on the gas phase and the energy balanceon the solid phase.

The energy balance of the gas phase can be written as:

ρgεcpg

∂Tg

∂t= ελg

∂2Tg

∂z2− mexh

Acscpg

∂Tg

∂z+ hAgeo(Tcat − Tg)

(1)

where the first term on the right hand side accountsfor the heat conduction and the second term for theconvective heat transport in the axial direction. The lastterm describes the heat exchange between the gas and thesolid phase. In a similar way the energy balance for thesolid phase is performed resulting as:

ρs(1− ε)cs∂Tcat

∂t= (1− ε)λs

∂2Tcat

∂z2− hAgeo(Tcat − Tg)

+Acat

∑j

(−∆HjRj)−Aout

Vcathout(Tcat − Tamb)

(2)

It is assumed that no transport phenomena are present inthe solid phase. The term Acat

∑j(−∆HjRj) accounts for

the heat produced by the exothermic reactions occurringin the catalyst where Rj stands for the net reaction rate ofeach single chemical reaction occurring in the converter. Inorder to estimate the reactions rates, an accurate kineticmodel should be developed and another system of PDEsshould be added to the energy balance to account forthe chemical species mass balance. Developing a kineticmodel requires an extensive experimental campaign anda time consuming parameter calibration, in addition themodel complexity would increase dramatically. Moreover,accounting for reaction kinetics, would require measure-ments of the concentrations of chemical species in theexhaust gas at the inlet of the converter.

For these reasons, simplifications will be made to themodel in order to obtain a model suitable for real timeapplications.

Since gas phase conduction has a minor influence oncatalyst temperature it can be neglected (Depcik andAssanis, 2005). In addition, given that the dynamics ofthe gas phase are much faster than the dynamics of thecatalyst temperature, the storage term in equation (1) canbe neglected, leaving Tg at steady state. In Montenegroand Onorati (2009) the authors showed the influence ofeach term in equation 2 on the catalyst temperature. Fromthat study it is clear that the heat exchanged with the gasand the heat coming from the exothermic reactions arethe dominant factors during normal catalyst operation.Conduction along the substrate is less important due tothe low conductivity of the substrate material (cordierite)and therefore can be neglected. In TWC converters witha metallic substrate the conduction effect may not benegligible. The heat lost to the ambient becomes relevantat high temperature and is kept in the energy balance.As already mentioned, the development of a kinetic modelis not suitable for control purposes. Therefore the heatproduced by the chemical reactions is lumped into a singleterm Qreac. This term is related to the catalyst efficiencyand dominates the energy balance after light-off while itis close to zero for temperatures below the catalyst light-off. Furthermore Qreac is considered proportional to theexhaust mass flow rate mexh and no dependence from theinlet species concentration is considered.

Qreac = Kreacmexhη(Tcat) (3)

This assumption is accurate as long as the catalyst oper-ates around stoichiometric conditions where the majorityof the hydrocarbons and carbon-monoxide are reducedby the incoming oxygen. This is assured by today’s air-fuel ratio controls that have the goal to keep the lambdaupstream of the catalyst very close to 1, allowing only fastdeviations from the stoichiometric value. If the catalystis operated for a long period in severe lean or severerich conditions, the modeling of the heat produced bythe chemical reactions is not expected to be accurate.The TWC efficiency has been chosen to be an hyperbolicfunction of the catalyst temperature trying to reproducethe s-shaped efficiency behavior presented in literature(e.g. Kang et al. (2014)):

η(Tcat) = 0.5tanh(s(Tcat − Tlight−off )) + 0.5 (4)

where s is a parameter describing the slope of the efficiencycurve while Tlight−off represents the catalyst light offtemperature (figure 4). In conclusion the PDEs 1 and 2are simplified as:

mexh

Acscpg

∂Tg

∂z= hAgeo(Tcat − Tg) (5)


∂t= −hAgeo(Tcat − Tg)

+Qreac −4

Dcathout(Tcat − Tamb)

(6)

To enable simulation, PDEs 5 and 6 are discretized in100 computational nodes using an upwind finite difference


436


Fig. 3. Proposed structure of TWC.

3. TWC TEMPERATURE MODEL

Using a one dimensional approach, the TWC can bedescribed as a single channel consisting of a gas phase anda solid phase. The gas phase represents the exhaust gasflowing into the converter and the solid phase representsthe catalyst washcoat and substrate. The temperaturedynamics of the TWC can be described by performing theenergy balance on the gas phase and the energy balanceon the solid phase.

The energy balance of the gas phase can be written as:

ρgεcpg

∂Tg

∂t= ελg

∂2Tg

∂z2− mexh

Acscpg

∂Tg

∂z+ hAgeo(Tcat − Tg)

(1)

where the first term on the right hand side accountsfor the heat conduction and the second term for theconvective heat transport in the axial direction. The lastterm describes the heat exchange between the gas and thesolid phase. In a similar way the energy balance for thesolid phase is performed resulting as:


∂t= (1− ε)λs

∂2Tcat

∂z2− hAgeo(Tcat − Tg)

+Acat

∑j

(−∆HjRj)−Aout

Vcathout(Tcat − Tamb)

(2)

It is assumed that no transport phenomena are present inthe solid phase. The term Acat

∑j(−∆HjRj) accounts for

the heat produced by the exothermic reactions occurringin the catalyst where Rj stands for the net reaction rate ofeach single chemical reaction occurring in the converter. Inorder to estimate the reactions rates, an accurate kineticmodel should be developed and another system of PDEsshould be added to the energy balance to account forthe chemical species mass balance. Developing a kineticmodel requires an extensive experimental campaign anda time consuming parameter calibration, in addition themodel complexity would increase dramatically. Moreover,accounting for reaction kinetics, would require measure-ments of the concentrations of chemical species in theexhaust gas at the inlet of the converter.

For these reasons, simplifications will be made to themodel in order to obtain a model suitable for real timeapplications.

Since gas phase conduction has a minor influence oncatalyst temperature it can be neglected (Depcik andAssanis, 2005). In addition, given that the dynamics ofthe gas phase are much faster than the dynamics of thecatalyst temperature, the storage term in equation (1) canbe neglected, leaving Tg at steady state. In Montenegroand Onorati (2009) the authors showed the influence ofeach term in equation 2 on the catalyst temperature. Fromthat study it is clear that the heat exchanged with the gasand the heat coming from the exothermic reactions arethe dominant factors during normal catalyst operation.Conduction along the substrate is less important due tothe low conductivity of the substrate material (cordierite)and therefore can be neglected. In TWC converters witha metallic substrate the conduction effect may not benegligible. The heat lost to the ambient becomes relevantat high temperature and is kept in the energy balance.As already mentioned, the development of a kinetic modelis not suitable for control purposes. Therefore the heatproduced by the chemical reactions is lumped into a singleterm Qreac. This term is related to the catalyst efficiencyand dominates the energy balance after light-off while itis close to zero for temperatures below the catalyst light-off. Furthermore Qreac is considered proportional to theexhaust mass flow rate mexh and no dependence from theinlet species concentration is considered.

Qreac = Kreacmexhη(Tcat) (3)

This assumption is accurate as long as the catalyst oper-ates around stoichiometric conditions where the majorityof the hydrocarbons and carbon-monoxide are reducedby the incoming oxygen. This is assured by today’s air-fuel ratio controls that have the goal to keep the lambdaupstream of the catalyst very close to 1, allowing only fastdeviations from the stoichiometric value. If the catalystis operated for a long period in severe lean or severerich conditions, the modeling of the heat produced bythe chemical reactions is not expected to be accurate.The TWC efficiency has been chosen to be an hyperbolicfunction of the catalyst temperature trying to reproducethe s-shaped efficiency behavior presented in literature(e.g. Kang et al. (2014)):

η(Tcat) = 0.5tanh(s(Tcat − Tlight−off )) + 0.5 (4)

where s is a parameter describing the slope of the efficiencycurve while Tlight−off represents the catalyst light offtemperature (figure 4). In conclusion the PDEs 1 and 2are simplified as:

mexh

Acscpg

∂Tg

∂z= hAgeo(Tcat − Tg) (5)


∂t= −hAgeo(Tcat − Tg)

+Qreac −4

Dcathout(Tcat − Tamb)

(6)

To enable simulation, PDEs 5 and 6 are discretized in100 computational nodes using an upwind finite difference


436

Temperature [°C]

0 100 200 300 400 500 600

η

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S

Tlight-off

Fig. 4. Model of the TWC efficiency.

scheme. Since the real time capability of the model hasto be preserved, the explicit Euler forward method isemployed to integrate the equations because of its lowcomputational demand. A sensitivity study with respect tothe number of computational nodes is discussed in section7. The boundary and initial conditions are:

Tg(t)

∣∣∣∣x=0

= Texh(t)

Tcat(x)

∣∣∣∣t=0

= Tcat0(x)

(7)

4. EXPERIMENTAL SETUP

All the experiments for model identification and validationwere performed on a FCA production vehicle in a chassisdyno test-cell. The catalyst was instrumented with onethermocouple at the inlet of the TWC measuring theexhaust gas temperature and one thermocouple locatedinside the catalyst brick to measure the substrate catalysttemperature. This second thermocouple is instrumented atthe brick mid-length location as shown in figure 5. Eventhough no lambda measurements are needed as input tothe proposed thermal model, a wide band lambda sensorplaced at the TWC inlet is used for the validation anddiscussion in section 6.

Fig. 5. Scheme of the sensors configuration

5. PARAMETERS IDENTIFICATION

The parameters to be identified are summed in table 1.

Parameters

Symbol Description

V HC=ρscs Volumetric heat capacity of the catalysth Convective heat transfer coefficienthout Heat transfer coefficient with ambientKreac Scaling factor for Qreacs Slope for the TWC efficency functionTlight−off Light-off temperature

Table 1. Parameters to be identified.

The volumetric heat capacity V HC and the heat transfercoefficients h and hout are related to physical proprietiesof the catalyst such as material and geometry. Kreac, sand Tlight−off are related to the effects of the chemicalreactions occurring in the TWC. Since the only way to testthe catalyst was during vehicle operation in the chassisdyno cell, isolating the effect of each parameter on thecatalyst temperature response was impractical. In fact,during normal vehicle operation the light-off temperatureis reached very quickly after the engine is started and itis not possible to clearly distinguish between the thermaldynamics of the heat exchanged with the exhaust gas andthe heat produced by chemical reactions. For this reasonall the parameters have been identified simultaneouslyfrom the same data set.

The identification procedure finds the optimal set of pa-rameters that minimize a function of the error betweenthe simulated catalyst temperature Tcat−sim at the sensorlocation and the actual measured catalyst temperatureTcat−meas. The data set used for identification is thecombination of a cold start test (figure 6) and an urbandriving cycle (figure 7). Including the cold-start test in theidentification data set turned out to be necessary sinceparameters like Tligth−off and s play an important roleonly during the catalyst warm-up but do not influence thetemperature response thereafter.

The cost function J to be minimized during the optimiza-tion is the summation of the error computed during thecold start and over a driving cycle:

J = RMS(e)cold−start +RMS(e)driving−cycle (8)

where:

RMS(e) =

√√√√ 1

Nsample

Nsample∑k=1

(Tcat−sim(k)− Tcat−meas(k))2 (9)

Furthermore, in order to guarantee that the light-off periodhas a significant effect on the cost function, the catalystlight-off period was extended as much as possible. Thiswas done by advancing the spark timing with respectto the baseline calibration strategy in order to keep theexhaust temperature relatively low during the first secondsof engine operations.


437


The simultaneous identification of six parameters repre-sents an optimization problem with a very wide searchspace. A Particle Swarm Optimization (PSO) algorithmwas used to find the parameters because it showed goodperformance in optimization problems with a wide searcharea despite its simple structure (Ebbesen et al., 2012).The Matlab implementation of the PSO algorithm pre-sented in Ebbesen et al. (2012) is used in this work.The performance of the PSO algorithm can depend onthe initial parameter estimates. The catalyst specificationsand geometries are used to compute an initial staring pointfor physics related parameters while for Tlight−off and s aninitial state is derived from experimental results presentedin literature (Kang et al., 2014). Figures 6 and 7 show theidentifications results on the cold start and driving cyclerespectively.

Time [s]0 50 100 150 200 250 300 350 400 450

Tem

pera

ture

[°C

]

0

50

100

150

200

250

300

350

400

450

500

T exhaustT catalyst measT catalyst sim

Fig. 6. Identification results on the cold-start test

Time [s]0 100 200 300 400 500 600

Tem

pera

ture

[° C

]

250

300

350

400

450

500

550

600

650


Fig. 7. Identification results on an urban driving cycle test

6. MODEL VALIDATION AND DISCUSSION

The model has been validated with a different data setfrom the one used for identification. Figure 8 shows theresponse of the model simulated on an highway drivingcycle. The simulation results exhibit very good agreementwith the experimental data.

There are some situations in which the model cannot accu-rately estimate the catalyst temperature. These situationsare more prevalent in the urban driving cycle of figure7 and they are highlighted in figure 9 and 10 together

Time [s]0 100 200 300 400 500 600 700 800

Tem

pera

ture

[°C

]

250

300

350

400

450

500

550


Fig. 8. Model Validation on an highway driving cycle test

with the lambda measurements. From seconds 40 to 80,the model overestimates the catalyst temperature. Duringthis time period the experimental lambda measured atthe inlet of the catalyst deviates significantly from thestoichiometric value (see figure 9). Since the model’s heatof reactions does not depend on lambda, the catalyst tem-perature estimation is expected to lack accuracy when theair fuel ratio is far from the stoichiometric value. Specifi-cally, if the engine is running very rich, there is not enoughoxygen in the exhaust to convert all the hydrocarbons andcarbon monoxide, reducing the heat from reactions. Onthe other hand, if the engine is running very lean, there isan excess of oxygen in the exhaust and a lack of HC andCO, reducing the heat produced from the most exothermicreactions in the catalyst. Only more accurate modeling ofthe heat produced by the chemical reactions can improvecatalyst temperature accuracy during these conditions.

Time [s]40 45 50 55 60 65 70 75 80 85 90

Tem

pera

ture

[°C

]

400

500

600T exhaustT catalyst measT catalyst sim

Time [s]40 45 50 55 60 65 70 75 80 85 90

lam

bda

[-]

0.8

1

1.2

Fig. 9. Overestimation of the temperature when thelambda significantly deviates from stoichiometric

From seconds 120 to 160, the catalyst is in a coolingcondition (figure 10). A fuel cut-off occurs (the two lambdasensors saturate at the highest measurable lean value at120 and 130 seconds). The model is not able to reflectthe thermal inertia of the catalyst in this sudden coolingcondition. A better characterization of the catalyst solid


438


The simultaneous identification of six parameters repre-sents an optimization problem with a very wide searchspace. A Particle Swarm Optimization (PSO) algorithmwas used to find the parameters because it showed goodperformance in optimization problems with a wide searcharea despite its simple structure (Ebbesen et al., 2012).The Matlab implementation of the PSO algorithm pre-sented in Ebbesen et al. (2012) is used in this work.The performance of the PSO algorithm can depend onthe initial parameter estimates. The catalyst specificationsand geometries are used to compute an initial staring pointfor physics related parameters while for Tlight−off and s aninitial state is derived from experimental results presentedin literature (Kang et al., 2014). Figures 6 and 7 show theidentifications results on the cold start and driving cyclerespectively.

Time [s]0 50 100 150 200 250 300 350 400 450

Tem

pera

ture

[°C

]

0

50

100

150

200

250

300

350

400

450

500


Fig. 6. Identification results on the cold-start test

Time [s]0 100 200 300 400 500 600

Tem

pera

ture

[° C

]

250

300

350

400

450

500

550

600

650


Fig. 7. Identification results on an urban driving cycle test

6. MODEL VALIDATION AND DISCUSSION

The model has been validated with a different data setfrom the one used for identification. Figure 8 shows theresponse of the model simulated on an highway drivingcycle. The simulation results exhibit very good agreementwith the experimental data.

There are some situations in which the model cannot accu-rately estimate the catalyst temperature. These situationsare more prevalent in the urban driving cycle of figure7 and they are highlighted in figure 9 and 10 together

Time [s]0 100 200 300 400 500 600 700 800

Tem

pera

ture

[°C

]

250

300

350

400

450

500

550


Fig. 8. Model Validation on an highway driving cycle test

with the lambda measurements. From seconds 40 to 80,the model overestimates the catalyst temperature. Duringthis time period the experimental lambda measured atthe inlet of the catalyst deviates significantly from thestoichiometric value (see figure 9). Since the model’s heatof reactions does not depend on lambda, the catalyst tem-perature estimation is expected to lack accuracy when theair fuel ratio is far from the stoichiometric value. Specifi-cally, if the engine is running very rich, there is not enoughoxygen in the exhaust to convert all the hydrocarbons andcarbon monoxide, reducing the heat from reactions. Onthe other hand, if the engine is running very lean, there isan excess of oxygen in the exhaust and a lack of HC andCO, reducing the heat produced from the most exothermicreactions in the catalyst. Only more accurate modeling ofthe heat produced by the chemical reactions can improvecatalyst temperature accuracy during these conditions.

Time [s]40 45 50 55 60 65 70 75 80 85 90

Tem

pera

ture

[°C

]

400

500

600T exhaustT catalyst measT catalyst sim

Time [s]40 45 50 55 60 65 70 75 80 85 90

lam

bda

[-]

0.8

1

1.2

Fig. 9. Overestimation of the temperature when thelambda significantly deviates from stoichiometric

From seconds 120 to 160, the catalyst is in a coolingcondition (figure 10). A fuel cut-off occurs (the two lambdasensors saturate at the highest measurable lean value at120 and 130 seconds). The model is not able to reflectthe thermal inertia of the catalyst in this sudden coolingcondition. A better characterization of the catalyst solid


438

phase may improve the catalyst cool-down estimation. Forexample, considering the external metallic layer and theinsulating material as shown in Depcik and Assanis (2005).However, this is not of interest for a control-orientedmodel.

Fig. 10. Catalyst cooling down during fuel cut-off

7. NUMBER OF COMPUTATIONAL NODES

Since the objective is to develop a control-oriented modelsuitable for real time applications, the complexity of themodel must be minimized. For this reason a sensitivitystudy with respect to the number of computational nodeshas been performed with the purpose of analyzing themodel’s performance when less than 100 computationalnodes are considered. Some of the parameters found withthe identification procedure described in section 5 (espe-cially Kreac and s) have no well defined physical meaning,thus they can be considered lumped parameters repre-sentative of complex chemical phenomena. Therefore, thevalues of these parameters may depend on the numberof computational nodes utilized during the identificationprocedure. In order to perform a meaningful and fair com-parison, a new identification of parameters is performedevery time the numbers of computational nodes is varied.In figure 11 the cost function value J defined in equa-tions (8) and representative of the model’s agreement tothe experimental data is shown for different number ofcomputational nodes considered. The results presented infigure 11 show that using more than 10 computationalnodes does not lead to any significant improvement inthe model accuracy. A number of computational nodesbetween five and ten can be chosen depending on theaccuracy requested by the application.

8. CONCLUSIONS

In this paper a simplified control oriented model of theTWC temperature is derived from PDEs equations de-scribing the catalyst energy balance. The model has beenvalidated with experimental tests reflecting real vehicleoperations. The heat produced by the chemical reactionsis lumped into a single term so that no concentrationinformation about the species upstream the catalyst is

Number of computational nodes [-]0 10 20 30 40 50

J=Jn/J1

0.5

0.6

0.7

0.8

0.9

1

Fig. 11. Model error for different numbers of computa-tional nodes considered. J is normalized with respectto the value computed with a single computationalnode.

Nomenclature

TWC Three Way CatalystOSC Oxygen Storage ComponentPSO Particle Swarm OptimizationPDE Partial Differential EquationTg gas temperature [K]Tcat TWC solid phase temperature [K]mexh exhaust gas mass flow rate [Kg/s]ρg exhaust gas density [Kg/m3]ρs TWC solid phase density [Kg/m3]cpg

specific heat of the exhaust gas [J/(KgK)]cs TWC solid phase specific heat [J/(KgK)]λg exhaust gas conductivity [W/(mK)]λs TWC solid phase conductivity [W/(mK)]h convective heat transfer coefficient [W/(m2K)]ε TWC open cross sectional area [0− 1]

Acs TWC cross sectional area [m2]Ageo TWC specific geometric area [m2/m3]Aout TWC external surface [m2]Acat TWC specific active surface [m2/m3]Vcat TWC volume [m3]Dcat TWC diameter [m]Rj reaction rate [mol/(m2s]∆Hj reaction enthalpy difference [J/mol]Qreac heat produced by reactions [W/(m3)]

required. Thanks to this approximation, the model is suit-able for real time applications like estimation of the light-off temperature during cold start or as support to an OSCmodel for emission control applications.

REFERENCES

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Auckenthaler, T.S. (2005). Modelling and controlof three-way catalytic converters. Diss., TechnischeWissenschaften, Eidgenossische Technische HochschuleETH Zurich, Nr. 16018, 2005.

Brandt, E.P., Wang, Y., and Grizzle, J.W. (2000). Dy-namic modeling of a three-way catalyst for si engineexhaust emission control. Control Systems Technology,IEEE Transactions on, 8(5), 767–776.


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Ebbesen, S., Kiwitz, P., and Guzzella, L. (2012). Ageneric particle swarm optimization matlab function. InAmerican Control Conference (ACC), 2012, 1519–1524.IEEE.

EPA (2011). http://yosemite.epa.gov/opa/admpress.nsf.Kang, S.B., Han, S.J., Nam, I.S., Cho, B.K., Kim, C.H.,and Oh, S.H. (2014). Detailed reaction kinetics fordouble-layered pd/rh bimetallic twc monolith catalyst.Chemical Engineering Journal, 241, 273–287.

Kumar, P., Makki, I., Kerns, J., Grigoriadis, K., Franchek,M., and Balakotaiah, V. (2012). A low-dimensionalmodel for describing the oxygen storage capacity andtransient behavior of a three-way catalytic converter.Chemical Engineering Science, 73, 373–387.

Montenegro, G. and Onorati, A. (2009). 1d thermo-fluiddynamic modeling of reacting flows inside three-waycatalytic converters. Technical report, SAE TechnicalPaper.

Roduner, C., Onder, C., and Geering, H. (1997). Auto-mated design of an air/fuel controller for an si engineconsidering the three-way catalytic converter in the happroach. In Proceedings of the fifth IEEE Mediter-ranean Conference on Control and Systems, Paphos,Cyprus.

Shafai, E., Roduner, C., and Geering, H.P. (1996). Indirectadaptive control of a three-way catalyst.

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