AVEC ‘06

Proceedings of AVEC ‘06 The 8th International Symposium on Advanced Vehicle Control, August 20-24, 2006, Taipei, Taiwan AVEC060229

A Simplified Catalytic Converter Model for Automotive Coldstart Applications with Adaptive Parameter Fitting Pannag R Sanketi*, J. Carlos Zavala*, J. K. Hedrick*, M. Wilcutts#, T. Kaga# * University of California Berkeley, # Toyota Motor Engineering and Manufacturing, North America 2162 Etcheverry Hall, University of California, Berkeley, CA- 94720, USA Phone: +15106426933 Fax: +15106426163 E-mail: [email protected]

It is well known that a major portion of the unburned hydrocarbon (HC) emissions in a typical drive cycle of an automotive engine is produced in the initial 1-2 minutes of operation, commonly called as the “coldstart" period. Catalyst light-off is essential for reducing these emissions. Model-based paradigm jis used to develop a control-oriented, thermodynamics based simple catalyst model for coldstart analysis. The catalyst thermal submodel is modeled as a hybrid system consisting of three discrete states and one continuous state. The discrete states are "initial warm-up", "evaporation of condensed gas" and "light-off". The continuous dynamics consists of the catalyst temperature. In each of the discrete states, energy balance of a control mass is used to model the catalyst temperature. Parameter adaptation algorithm (PAA) is used to identify the parameters in each of the discrete states. Effectiveness of the average values of the adjusted parameters is also given. Wiebe profiles are adopted to empirically model the HC emissions conversion properties of the catalyst as a function of the catalyst temperature and the air-fuel ratio. The static efficiency maps are further extended to include the effects of spatial velocity of the feedgas. Experimental results indicate good agreement with the model estimates for the catalyst warm-up. Topics / Powertrain & Drivetrain Control, Modeling & Simulation Technology, Vehicle Diagnostics 1. INTRODUCTION During the first minute of engine operation, the cold engine walls make the flame unstable due to the heat transfer from the gas to the walls; the catalyst is not active at low temperatures; and the oxygen sensor does not reach its operating temperature. Due to this, as much as 70-80% of the hydrocarbon (HC) emissions in a typical drive cycle are emitted during this period, commonly called as "coldstart". A lot of interest has been shown in modeling and analysis of catalysts for coldstart analysis. The models vary over a broad spectrum ranging from some that are based on a very detailed physics-based approach to some that rely on black-box modeling techniques. In Shen et al. [10], a very detailed physical catalyst model involving 13-step kinetics and 9-step oxygen storage mechanisms is described. In Chan et al. [2], the chemical conversion of carbon monoxide and unburned hydrocarbons in the oxidation process is considered with detailed heat transfer modeling, which includes the effects of water vapor condensation. Further, numerical algorithms to predict catalyst characteristics including detailed oxygen storage effects are given in Ohsawa et al. [8]. Being complicated, these are not suitable for real-time control purposes. Semi-empirical models that are less complicated and more suitable for real time control and on-board diagnostics have also been studied. For instance, oxygen storage dominated models (simplified storage and conversion modeling) are presented in Jones et

al. [5] and Jones et al. [6]. In Fiengo et al. [3], use of a control oriented model in which genetic algorithm is applied for identifying the model parameters is suggested. Kinetics based thermal response of the catalyst and the flow distribution in the catalyst is studied in Koltsakis et al. [7]. It is shown that flow maldistribution does not affect the light-off behaviour of the catalyst. In Soumelidis et al. [11], four nonlinear (NARMAX) dynamic models are used to predict the catalyst transient response, but it is mainly focused on the oxygen storage phenomenon and does not address the coldstart issue. In Gonatas et al. [4], a black-box method is proposed to find the emissions and oxygen storage. The problem with this method is that it requires extensive experimentation each time. A phenomenological control-oriented model using least squares for identifying model parameters is developed in Brandt et al. [1]. However, the thermal submodel in the paper is first order dynamics cascaded with an algebraic function determined by regression. It is not a physics-based model, presenting which is one of the objectives of this paper. Also, the validation results are given only for the warmed-up condition. A simplified control-oriented thermodynamic model of the catalytic converter is developed in Shaw et al. [9]. Though this model is suitable for designing controllers, it is assumed that the brick temperature and the feedgas temperature are same. It is also assumed that the engine exhaust and the post-catalyst temperatures are same, which is oversimplified.

AVEC ‘06 In most of the cases where a physics-based modeling approach is followed, the model is found to be very complicated and not suitable for use in real-time. The goal of this work is to develop a simplified yet realistic thermodynamic model of a catalyst which offers a broader scope for coldstart analysis and controller development. A single set of continuous dynamics parameters is not enough to represent the actual dynamics of the catalyst as it undergoes various phases. The proposed model consists of 3 discrete states and one continuous state. The discrete phases correspond to the changes in the modes of heat transfer in the catalyst whereas the continuous state is the catalyst temperature, . Energy balance of a control mass is used to model . In each of the discrete states, parameter adaptation algorithm (PAA) is used to identify the parameters in the continuous dynamics. Further, a nominal model is presented and the effectiveness of the average values of the adjusted parameters is tested. The hypothesis proposed by Shaw et al. [9] that oxygen storage phenomenon is not prevalent during the warm-up is adhered to here. It is found that the conversion efficiency of the catalytic converter is primarily a function of and air-fuel ratio ( ), and is modeled using Wiebe profiles. Furthermore, it is shown that the efficiency of the catalyst can be more accurately represented by including the effect of the spatial velocity of the feedgas. A factor to that effect is added to the static efficiency map. Experimental results indicate that and the catalyst HC conversion efficiency are predicted with good accuracy.

(1)

‘ Fig. 1 Thermal submodel hybrid characteristics

2. MODEL DESCRIPTION The basic structure of a catalytic converter model as proposed in Brandt et al. [1] consists of three submodels: the oxygen storage submodel, thermal submodel and static efficiency curves submodel. Oxygen storage is not important during the warm-up of the engine as described in Shaw et al. [9]. In this paper, we concentrate on the thermal and the efficiency submodels. In this paper, the dynamics of the catalyst is divided into three discrete phases. This according to the authors, is an important aspect in rendering the catalyst model more realistic and accurate, still simple enough for control purposes. 2.1 Thermal Submodel Fig. 1 shows the catalyst thermal submodel proposed using hybrid automata modeling paradigm. Here, is the saturation temperature of the vapor condensed inside the catalyst. The reset maps indicate that does not change when the system enters one discrete phase from another. Each of the discrete states corresponds to a physical process that the catalyst goes through and its effect on the continuous dynamics is represented by functions through in the figure. More details about the transitions from one state to another will be given further in this section. In each of the states, the catalyst temperature depends on the heat obtained from the feedgas flowing through the catalyst , the amount of heat generated due to oxidation of pollutants from the feedgas and the heat transfer to the surroundings . These are given by,

(2)

(3) where, is the conversion efficiency of the catalyst, is the mass flow rate of engine out raw pollutant, is the heat generated by the conversion of the pollutant inside the catalytic converter, and are respectively the inner and outer effective heat transfer coefficients of the catalyst, and are respectively the inner and outer surface areas of the catalyst, is the temperature of the exhaust gas flowing through the catalyst. In addition to these heat transfers, heat is lost due to evaporation in the "evaporation" state. The dynamics are much faster than that of the brick temperature and calculated using the exhaust and the post-catalyst temperatures. The state equation for in each of the discrete states is briefly described next. 2.1.1 Warm-up State The first phase is constituted by the initial few seconds of engine operation during which the catalyst warm-up begins and is low. For how long the system remains in this phase is determined by the catalyst temperature. Typically, this constitutes the initial 8-10s of the engine operation. The dynamics of in this state is given by,

AVEC ‘06

where, is the catalyst mass and is the catalyst specific heat. In this state, is low; hence is low. The efficiency of the catalyst is also low. As a result, is the dominating factor. Also, due to the cold catalyst, the water vapor in the exhaust gas condenses on the inner surface of the catalyst (see Chan et al. [2]). This adds to the vapor that had condensed after the engine shutdown. When the temperature reaches the saturation temperature of the condensed vapor, the vapor starts evaporating. The saturation temperature was found constant in all the experiments done. 2.1.2 Evaporation State The transition from the warm-up state to this state occurs when the condensed exhaust gas starts evaporating at its saturation temperature. The dynamics of in this state is given by

(5)

(6) where, is the heat lost due to evaporation, is the mass rate of evaporation and is the enthalpy of evaporation at the corresponding saturation temperature. Due to the heavy heat loss in evaporation, the catalyst temperature remains almost constant or increases at a very low rate. Once the condensed gas is evaporated, which is determined on the basis of , the system enters the next discrete state, i.e. the light-off state.

change of catalyst, etc. are compensated for in the PAA. However, to model the evaporation will necessitate the PAA to be very complex, since the evaporation occurs in certain time interval. Hence, the effective heat transfer coefficients and are adapted to compensate for the heat loss due to evaporation. The calculation of the evaporation heat loss will be explained in Sec. 2.2.1. The parameter values at the end of one discrete state become the initial conditions for the PAA in the next state. PAA picks up the transition in the discrete state by itself. In summary, PAA allows the model to be simplified, and still be able to capture complex physical characteristics, which otherwise will require a very complex model. 2.2.1 Calculation of Heat Loss Due to Evaporation The PAA gives a model in which the is not modeled explicitly, however, can be calculated from . The profiles of and as given by the PAA for three different experiments are shown in Fig. 2. The experiments differ in the value of maintained constant throughout value is the experiment. As seen from the figure, the almost constant for the whole time of interest and the PAA converges to the value quickly. The value changes according to the state the system is in and the PAA is able to catch it efficiently. As expected, the value is very low during the warm-up phase, rises to a very high value during the evaporation phase and finally settles to its value in the light-off state. The bump in the middle part of the value represents the heat loss due to evaporation. Internal heat transfer coefficient through PAA 2000 1500 hin

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2.1.3 Light-off State In this state, the temperature of the catalyst increases rapidly and the light-off is achieved. The dynamics of in this state is given by

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(7) Fig. 2 Heat transfer coefficients given by PAA As compared to the warm-up state, the heat transfer to the surroundings is much higher here. So the heat transfer coefficients are different than those in the warm-up state. 2.2 Use of Parameter Adaptation Algorithm The parameters to be identified are the heat transfer coefficients and the evaporation rate of the condensed vapor. Though the parameters in each discrete state are constant, they might vary a bit from experiment to experiment. Hence, a parameter adaptation algorithm (PAA) is used to identify the parameters in each of the discrete states. Another important advantage of using the PAA is that it can find the parameters accurately in each of the cases. Additionally, any physical changes to the system, for example catalyst ageing,

To calculate , it is assumed that the value is same in the evaporation and the light-off states. Hence, the difference in the as calculated by this assumption and the as given by the PAA gives the . This can be summarized by the following set of equations.

(8)

(9)

AVEC ‘06

where, the subscripts ‘ ’ and ‘ ’ respectively represent the evaporation state and the light-off state. Also, the subscripts ‘ ’ and ‘ ’ stand respectively for the actual value of the parameter and its value as given by the PAA. Fig. 3 below shows calculated as described previously using the PAA algorithm for one of the experiments. Fig. 4 shows the profiles of calculated for the experiments corresponding to those shown in Fig. 2. This curve profile makes sense because the condensed vapor evaporation rate increases steadily as the temperature increases and then drops off until the condensed gas evaporates off completely.

PAA for the three sets of experiments corresponding to those shown in Fig. 2 and Fig.4. The model estimates match the experimental data very closely in each of the cases. The temperature profiles are very similar in experiments 1 and 3 500 450 400 350

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2.2.3 Results of Tcat model using constant set of parameters Further, average values of parameters over a set of experiments were calculated so that a nominal model of the system can be represented. Fig. 7 shows the performance of such a model. The model is able to predict the catalyst temperature reasonably well is most of the cases. In particular, it is able to model the evaporation effect on the catalyst temperature. However, as expected, the model estimates are not as accurate as the ones using PAA since a constant set of parameters is being used for all the cases. Hence, if the catalyst temperature can be measured or observed, PAA will be more effective in modeling the catalyst temperature.

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Fig. 4 Calculating the evaporation rate using PAA

2.2.2 Results of Tcat model using PAA PAA is very effective in capturing the condensed vapor evaporation part, and hence in obtaining an accurate model. Fig. 5 shows that the model predictions are not accurate if the evaporation is neglected. Clearly such models cannot model the leveled part in the catalyst temperature profile due to the evaporation. Fig. 6 shows that the model estimates of using

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Fig. 6 Catalyst temperature model considering the effect of water condensation using PAA

AVEC ‘06 2.3 Static Efficiency Curves Submodel The conversion efficiencies are generally measured over a range of temperatures and values, requiring extensive data fitting and lookup tables. However, the conversion efficiency can be described by a -shaped Wiebe function as proposed in Shaw et al. [9]. However, it was found that just and were not sufficient to represent the HC conversion efficiency accurately. Here, it is proposed that the spatial velocity of the feedgas is also important in determining the efficiency of the catalyst.

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Fig. 8 shows the efficiency prediction model when the exhaust gas flow rate into the catalyst is not accounted for. Such a model is not able to predict the efficiency precisely since there is a level part in the model corresponding to that in the catalyst temperature. It can be seen from Fig. 9 and Fig. 10 that this flaw can be removed from the model estimates by including the effect of feedgas flow. The efficiency estimate can be further improved if a more complex base function is chosen. However, for control purposes, the current accuracy level is acceptable.

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3. CONCLUSIONS A simplified control-oriented model of a catalytic converter for automotive coldstart analysis consisting of thermal dynamics and static efficiency maps is proposed. The thermal subsystem is modeled as a hybrid system consisting of 3 discrete states and one continuous state. The discrete phases correspond to changing physical processes in the catalytic converter resulting in different continuous dynamics. In each of the discrete states, energy balance of a closed system is used to model the catalyst temperature. First, a

AVEC ‘06 parameter adaptation algorithm (PAA) is used to identify the parameters. It is shown to accurately capture the complex physical characteristics through its adapted parameters. Further, a nominal model of the catalyst temperature based upon the average values of parameters found through PAA is presented. This model, though not as accurate as that using PAA, is able to capture the catalyst temperature dynamics well, especially the level part of the temperature profile due to the evaporation of the condensed vapor inside the catalyst. Wiebe profiles are used to model the static conversion properties of the catalyst dominated by its thermal behavior and air-fuel ratio. The static efficiency maps are further extended to include the effects of spatial velocity of the feedgas. The model estimates for the catalyst temperature and the HC conversion efficiency during the warm-up agree well with the experimental results. REFERENCES [1]. Brandt, E., Wang, Y., Grizzle, J.W.: “Dynamic modeling of three-way catalyst for si engine exhaust emission control”, IEEE Transactions on Control Systems Technology, vol. 8, no. 5, pp. 767-776 (2000). [2]. Chan, S.H., Hoang, D.L.: “Modeling of catalytic conversion of co/ch in gasoline exhaust at engine cold-start”, SAE Technical Paper 1999-01-0452 (1999). [3]. Fiengo, G. et al.: “Control oriented models for twc-equipped spark ignition engines during the warm-up phase”, Proceedings of the American Control Conference, pp. 1761-1766, (2002). [4]. Gonatas, E., Stobart, R.: “Prediction of gas concentrations in a three-way catalyst for on-board diagnostic applications”, SAE Technical Paper 2005-01-0054, (2005). [5]. Jones, J. P. et al.: “Modeling the transient characteristics of a three way catalyst”, SAE Technical Paper 1999-01-0460, (1999). [6]. Jones, J. P., Roberts, J., Bernard, P.: “A simplified model for the dynamics of a three-way catalytic converter”, SAE Technical Paper 2000-01-0652, (2000). [7]. Koltsakis, G. C., Tsinoglou, D. N.: “Thermal response of close-coupled catalysts during light-off”, SAE Technical Paper 2003-01-1876, (2003). [8]. Ohsawa, K., Baba, N., Kojima, S.: “Numerical prediction of transient conversion characteristics in a three-way catalytic converter”, SAE Technical Paper 982556, (1998). [9]. Shaw, B., Fischer, G. D., Hedrick, J. K.: “A simplified coldstart catalyst thermal model to reduce hydrocarbon emissions”, Proceedings of 15th Triennial World Congress of the International Federation of Automatic Control, (2002). [10]. Shen, H., Shamim, T., Sengupta, S.: “An investigation of catalytic converter performances during cold starts”, SAE Technical Paper 1999-01-3473, (1999). [11]. Soumelidis, M. I. et al.: “A nonlinear dynamic model for three-way catalyst control and diagnosis”, SAE Technical Paper 2004-01-1831, (2004).

ACKNOWLEDGEMENTS This work was supported in part by the Center for Hybrid and Embedded Software Systems (CHESS) at UC Berkeley, which receives support from the National Science Foundation (NSF award #CCR-0225610), the State of California Micro Program, and the following companies: Agilent, DGIST, General Motors, Hewlett Packard, Infineon, Microsoft, and Toyota. Authors also acknowledge the support provided by Toyota and CONACYT (Consejo Nacional de Ciencia y Technología de Mexico). We would also like to thank the reviewers of this paper for their useful suggestions and comments.

Proceedings of AVEC ‘06 The 8th International Symposium on Advanced Vehicle Control, August 20-24, 2006, Taipei, Taiwan AVEC060229

A Simplified Catalytic Converter Model for Automotive Coldstart Applications with Adaptive Parameter Fitting Pannag R Sanketi*, J. Carlos Zavala*, J. K. Hedrick*, M. Wilcutts#, T. Kaga# * University of California Berkeley, # Toyota Motor Engineering and Manufacturing, North America 2162 Etcheverry Hall, University of California, Berkeley, CA- 94720, USA Phone: +15106426933 Fax: +15106426163 E-mail: [email protected]

It is well known that a major portion of the unburned hydrocarbon (HC) emissions in a typical drive cycle of an automotive engine is produced in the initial 1-2 minutes of operation, commonly called as the “coldstart" period. Catalyst light-off is essential for reducing these emissions. Model-based paradigm jis used to develop a control-oriented, thermodynamics based simple catalyst model for coldstart analysis. The catalyst thermal submodel is modeled as a hybrid system consisting of three discrete states and one continuous state. The discrete states are "initial warm-up", "evaporation of condensed gas" and "light-off". The continuous dynamics consists of the catalyst temperature. In each of the discrete states, energy balance of a control mass is used to model the catalyst temperature. Parameter adaptation algorithm (PAA) is used to identify the parameters in each of the discrete states. Effectiveness of the average values of the adjusted parameters is also given. Wiebe profiles are adopted to empirically model the HC emissions conversion properties of the catalyst as a function of the catalyst temperature and the air-fuel ratio. The static efficiency maps are further extended to include the effects of spatial velocity of the feedgas. Experimental results indicate good agreement with the model estimates for the catalyst warm-up. Topics / Powertrain & Drivetrain Control, Modeling & Simulation Technology, Vehicle Diagnostics 1. INTRODUCTION During the first minute of engine operation, the cold engine walls make the flame unstable due to the heat transfer from the gas to the walls; the catalyst is not active at low temperatures; and the oxygen sensor does not reach its operating temperature. Due to this, as much as 70-80% of the hydrocarbon (HC) emissions in a typical drive cycle are emitted during this period, commonly called as "coldstart". A lot of interest has been shown in modeling and analysis of catalysts for coldstart analysis. The models vary over a broad spectrum ranging from some that are based on a very detailed physics-based approach to some that rely on black-box modeling techniques. In Shen et al. [10], a very detailed physical catalyst model involving 13-step kinetics and 9-step oxygen storage mechanisms is described. In Chan et al. [2], the chemical conversion of carbon monoxide and unburned hydrocarbons in the oxidation process is considered with detailed heat transfer modeling, which includes the effects of water vapor condensation. Further, numerical algorithms to predict catalyst characteristics including detailed oxygen storage effects are given in Ohsawa et al. [8]. Being complicated, these are not suitable for real-time control purposes. Semi-empirical models that are less complicated and more suitable for real time control and on-board diagnostics have also been studied. For instance, oxygen storage dominated models (simplified storage and conversion modeling) are presented in Jones et

al. [5] and Jones et al. [6]. In Fiengo et al. [3], use of a control oriented model in which genetic algorithm is applied for identifying the model parameters is suggested. Kinetics based thermal response of the catalyst and the flow distribution in the catalyst is studied in Koltsakis et al. [7]. It is shown that flow maldistribution does not affect the light-off behaviour of the catalyst. In Soumelidis et al. [11], four nonlinear (NARMAX) dynamic models are used to predict the catalyst transient response, but it is mainly focused on the oxygen storage phenomenon and does not address the coldstart issue. In Gonatas et al. [4], a black-box method is proposed to find the emissions and oxygen storage. The problem with this method is that it requires extensive experimentation each time. A phenomenological control-oriented model using least squares for identifying model parameters is developed in Brandt et al. [1]. However, the thermal submodel in the paper is first order dynamics cascaded with an algebraic function determined by regression. It is not a physics-based model, presenting which is one of the objectives of this paper. Also, the validation results are given only for the warmed-up condition. A simplified control-oriented thermodynamic model of the catalytic converter is developed in Shaw et al. [9]. Though this model is suitable for designing controllers, it is assumed that the brick temperature and the feedgas temperature are same. It is also assumed that the engine exhaust and the post-catalyst temperatures are same, which is oversimplified.

AVEC ‘06 In most of the cases where a physics-based modeling approach is followed, the model is found to be very complicated and not suitable for use in real-time. The goal of this work is to develop a simplified yet realistic thermodynamic model of a catalyst which offers a broader scope for coldstart analysis and controller development. A single set of continuous dynamics parameters is not enough to represent the actual dynamics of the catalyst as it undergoes various phases. The proposed model consists of 3 discrete states and one continuous state. The discrete phases correspond to the changes in the modes of heat transfer in the catalyst whereas the continuous state is the catalyst temperature, . Energy balance of a control mass is used to model . In each of the discrete states, parameter adaptation algorithm (PAA) is used to identify the parameters in the continuous dynamics. Further, a nominal model is presented and the effectiveness of the average values of the adjusted parameters is tested. The hypothesis proposed by Shaw et al. [9] that oxygen storage phenomenon is not prevalent during the warm-up is adhered to here. It is found that the conversion efficiency of the catalytic converter is primarily a function of and air-fuel ratio ( ), and is modeled using Wiebe profiles. Furthermore, it is shown that the efficiency of the catalyst can be more accurately represented by including the effect of the spatial velocity of the feedgas. A factor to that effect is added to the static efficiency map. Experimental results indicate that and the catalyst HC conversion efficiency are predicted with good accuracy.

(1)

‘ Fig. 1 Thermal submodel hybrid characteristics

2. MODEL DESCRIPTION The basic structure of a catalytic converter model as proposed in Brandt et al. [1] consists of three submodels: the oxygen storage submodel, thermal submodel and static efficiency curves submodel. Oxygen storage is not important during the warm-up of the engine as described in Shaw et al. [9]. In this paper, we concentrate on the thermal and the efficiency submodels. In this paper, the dynamics of the catalyst is divided into three discrete phases. This according to the authors, is an important aspect in rendering the catalyst model more realistic and accurate, still simple enough for control purposes. 2.1 Thermal Submodel Fig. 1 shows the catalyst thermal submodel proposed using hybrid automata modeling paradigm. Here, is the saturation temperature of the vapor condensed inside the catalyst. The reset maps indicate that does not change when the system enters one discrete phase from another. Each of the discrete states corresponds to a physical process that the catalyst goes through and its effect on the continuous dynamics is represented by functions through in the figure. More details about the transitions from one state to another will be given further in this section. In each of the states, the catalyst temperature depends on the heat obtained from the feedgas flowing through the catalyst , the amount of heat generated due to oxidation of pollutants from the feedgas and the heat transfer to the surroundings . These are given by,

(2)

(3) where, is the conversion efficiency of the catalyst, is the mass flow rate of engine out raw pollutant, is the heat generated by the conversion of the pollutant inside the catalytic converter, and are respectively the inner and outer effective heat transfer coefficients of the catalyst, and are respectively the inner and outer surface areas of the catalyst, is the temperature of the exhaust gas flowing through the catalyst. In addition to these heat transfers, heat is lost due to evaporation in the "evaporation" state. The dynamics are much faster than that of the brick temperature and calculated using the exhaust and the post-catalyst temperatures. The state equation for in each of the discrete states is briefly described next. 2.1.1 Warm-up State The first phase is constituted by the initial few seconds of engine operation during which the catalyst warm-up begins and is low. For how long the system remains in this phase is determined by the catalyst temperature. Typically, this constitutes the initial 8-10s of the engine operation. The dynamics of in this state is given by,

AVEC ‘06

where, is the catalyst mass and is the catalyst specific heat. In this state, is low; hence is low. The efficiency of the catalyst is also low. As a result, is the dominating factor. Also, due to the cold catalyst, the water vapor in the exhaust gas condenses on the inner surface of the catalyst (see Chan et al. [2]). This adds to the vapor that had condensed after the engine shutdown. When the temperature reaches the saturation temperature of the condensed vapor, the vapor starts evaporating. The saturation temperature was found constant in all the experiments done. 2.1.2 Evaporation State The transition from the warm-up state to this state occurs when the condensed exhaust gas starts evaporating at its saturation temperature. The dynamics of in this state is given by

(5)

(6) where, is the heat lost due to evaporation, is the mass rate of evaporation and is the enthalpy of evaporation at the corresponding saturation temperature. Due to the heavy heat loss in evaporation, the catalyst temperature remains almost constant or increases at a very low rate. Once the condensed gas is evaporated, which is determined on the basis of , the system enters the next discrete state, i.e. the light-off state.

change of catalyst, etc. are compensated for in the PAA. However, to model the evaporation will necessitate the PAA to be very complex, since the evaporation occurs in certain time interval. Hence, the effective heat transfer coefficients and are adapted to compensate for the heat loss due to evaporation. The calculation of the evaporation heat loss will be explained in Sec. 2.2.1. The parameter values at the end of one discrete state become the initial conditions for the PAA in the next state. PAA picks up the transition in the discrete state by itself. In summary, PAA allows the model to be simplified, and still be able to capture complex physical characteristics, which otherwise will require a very complex model. 2.2.1 Calculation of Heat Loss Due to Evaporation The PAA gives a model in which the is not modeled explicitly, however, can be calculated from . The profiles of and as given by the PAA for three different experiments are shown in Fig. 2. The experiments differ in the value of maintained constant throughout value is the experiment. As seen from the figure, the almost constant for the whole time of interest and the PAA converges to the value quickly. The value changes according to the state the system is in and the PAA is able to catch it efficiently. As expected, the value is very low during the warm-up phase, rises to a very high value during the evaporation phase and finally settles to its value in the light-off state. The bump in the middle part of the value represents the heat loss due to evaporation. Internal heat transfer coefficient through PAA 2000 1500 hin

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2.1.3 Light-off State In this state, the temperature of the catalyst increases rapidly and the light-off is achieved. The dynamics of in this state is given by

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(7) Fig. 2 Heat transfer coefficients given by PAA As compared to the warm-up state, the heat transfer to the surroundings is much higher here. So the heat transfer coefficients are different than those in the warm-up state. 2.2 Use of Parameter Adaptation Algorithm The parameters to be identified are the heat transfer coefficients and the evaporation rate of the condensed vapor. Though the parameters in each discrete state are constant, they might vary a bit from experiment to experiment. Hence, a parameter adaptation algorithm (PAA) is used to identify the parameters in each of the discrete states. Another important advantage of using the PAA is that it can find the parameters accurately in each of the cases. Additionally, any physical changes to the system, for example catalyst ageing,

To calculate , it is assumed that the value is same in the evaporation and the light-off states. Hence, the difference in the as calculated by this assumption and the as given by the PAA gives the . This can be summarized by the following set of equations.

(8)

(9)

AVEC ‘06

where, the subscripts ‘ ’ and ‘ ’ respectively represent the evaporation state and the light-off state. Also, the subscripts ‘ ’ and ‘ ’ stand respectively for the actual value of the parameter and its value as given by the PAA. Fig. 3 below shows calculated as described previously using the PAA algorithm for one of the experiments. Fig. 4 shows the profiles of calculated for the experiments corresponding to those shown in Fig. 2. This curve profile makes sense because the condensed vapor evaporation rate increases steadily as the temperature increases and then drops off until the condensed gas evaporates off completely.

PAA for the three sets of experiments corresponding to those shown in Fig. 2 and Fig.4. The model estimates match the experimental data very closely in each of the cases. The temperature profiles are very similar in experiments 1 and 3 500 450 400 350

Cat Temp (deg C)

(10)

300 250 200 150 100 Model1 Experiment Model2

50

dQin

0

8000

0

10

20

30

40

50 Time(s)

60

70

80

90

100

J/s

6000

Fig. 5 Catalyst temperature model without the effect of water condensation

4000 2000 0

0

10

20

30

40

50

60

70

dQout 10000

dQPAA dQavg dQevap

J/s

5000

whereas the horizontal leveled part is longer in the case of experiment 2. This agrees with the results in Fig. 4 where the is higher in experiment 2 than that in experiments 1 and 3. Hence, more heat is lost due to the evaporation of the condensed vapor in this case.

0

-5000

0

10

20

30

40

50

60

70

Time(s)

Fig. 3 Calculating heat of evaporation using PAA

3.5 m

evap, Exp1

mevap, Exp2

3

m

evap, Exp3

2.5 2 1.5

2.2.3 Results of Tcat model using constant set of parameters Further, average values of parameters over a set of experiments were calculated so that a nominal model of the system can be represented. Fig. 7 shows the performance of such a model. The model is able to predict the catalyst temperature reasonably well is most of the cases. In particular, it is able to model the evaporation effect on the catalyst temperature. However, as expected, the model estimates are not as accurate as the ones using PAA since a constant set of parameters is being used for all the cases. Hence, if the catalyst temperature can be measured or observed, PAA will be more effective in modeling the catalyst temperature.

1

Catalyst Temperature Model Using PAA

0.5

250

−0.5

0

10

20

30

40

50

60

70

Fig. 4 Calculating the evaporation rate using PAA

2.2.2 Results of Tcat model using PAA PAA is very effective in capturing the condensed vapor evaporation part, and hence in obtaining an accurate model. Fig. 5 shows that the model predictions are not accurate if the evaporation is neglected. Clearly such models cannot model the leveled part in the catalyst temperature profile due to the evaporation. Fig. 6 shows that the model estimates of using

Cat Brick Temp (deg C)

0

200

150

Model Exp1 Exp 1 Model Exp2 Exp 2 Model Exp3 Exp3

100

50

0

10

20

30 Time(s)

40

50

60

Fig. 6 Catalyst temperature model considering the effect of water condensation using PAA

AVEC ‘06 2.3 Static Efficiency Curves Submodel The conversion efficiencies are generally measured over a range of temperatures and values, requiring extensive data fitting and lookup tables. However, the conversion efficiency can be described by a -shaped Wiebe function as proposed in Shaw et al. [9]. However, it was found that just and were not sufficient to represent the HC conversion efficiency accurately. Here, it is proposed that the spatial velocity of the feedgas is also important in determining the efficiency of the catalyst.

300

250

250

200 150

1.4 Experimental Model 1.2

200 150

1

100

50

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Model Exp 0

20

40 Time(s)

60

0

80

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Cat Temp Avg Model Exp C

20

40 Time(s)

Model Exp 60 80

ηconv, HC

100

0

HC Conversion Efficiency

Cat Temp Avg Model Exp B

300

Tcat(C)

Tcat(C)

Cat Temp Avg Model Exp A

Fig. 8 shows the efficiency prediction model when the exhaust gas flow rate into the catalyst is not accounted for. Such a model is not able to predict the efficiency precisely since there is a level part in the model corresponding to that in the catalyst temperature. It can be seen from Fig. 9 and Fig. 10 that this flaw can be removed from the model estimates by including the effect of feedgas flow. The efficiency estimate can be further improved if a more complex base function is chosen. However, for control purposes, the current accuracy level is acceptable.

Cat Temp Avg Model Exp D

400

300

0.8

0.6

250

Tcat(C)

Tcat(C)

300 200

0.4

200 150

0.2

100 100 0

0

20

40 Time(s)

Model Exp 60 80

50 0

0

20

40 Time(s)

Model Exp 60 80

0

Fig. 7 Catalyst temperature model considering the effect of water condensation using nominal parameter values

0

10

20

30

40 50 Time(s)

60

70

80

90

Fig. 9 Catalyst HC conversion efficiency considering feedgas rate- Exp1 1.4

The catalyst efficiency map is given as follows.

1.2

(11) where, given by

= exhaust mass air flow rate influence factor is

ηconv, HC

1

0.8

0.6

0.4

(12) 0.2

and

Exp Model

being the fitting parameters. 0

0

10

20

HC Conversion Efficiency

30

40 50 Time(s)

60

70

80

90

1.5

η

conv, HC

Experimental Model

Fig. 10 Catalyst HC conversion efficiency considering feedgas rate- Exp2

1

0.5

0

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40 50 60 Time(s) Catalyst Temperature

70

80

90

400

T

cat

300 200 100 0

0

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40 50 Time(s)

60

70

80

90

Fig. 8 Catalyst HC conversion efficiency without feedgas rate

3. CONCLUSIONS A simplified control-oriented model of a catalytic converter for automotive coldstart analysis consisting of thermal dynamics and static efficiency maps is proposed. The thermal subsystem is modeled as a hybrid system consisting of 3 discrete states and one continuous state. The discrete phases correspond to changing physical processes in the catalytic converter resulting in different continuous dynamics. In each of the discrete states, energy balance of a closed system is used to model the catalyst temperature. First, a

AVEC ‘06 parameter adaptation algorithm (PAA) is used to identify the parameters. It is shown to accurately capture the complex physical characteristics through its adapted parameters. Further, a nominal model of the catalyst temperature based upon the average values of parameters found through PAA is presented. This model, though not as accurate as that using PAA, is able to capture the catalyst temperature dynamics well, especially the level part of the temperature profile due to the evaporation of the condensed vapor inside the catalyst. Wiebe profiles are used to model the static conversion properties of the catalyst dominated by its thermal behavior and air-fuel ratio. The static efficiency maps are further extended to include the effects of spatial velocity of the feedgas. The model estimates for the catalyst temperature and the HC conversion efficiency during the warm-up agree well with the experimental results. REFERENCES [1]. Brandt, E., Wang, Y., Grizzle, J.W.: “Dynamic modeling of three-way catalyst for si engine exhaust emission control”, IEEE Transactions on Control Systems Technology, vol. 8, no. 5, pp. 767-776 (2000). [2]. Chan, S.H., Hoang, D.L.: “Modeling of catalytic conversion of co/ch in gasoline exhaust at engine cold-start”, SAE Technical Paper 1999-01-0452 (1999). [3]. Fiengo, G. et al.: “Control oriented models for twc-equipped spark ignition engines during the warm-up phase”, Proceedings of the American Control Conference, pp. 1761-1766, (2002). [4]. Gonatas, E., Stobart, R.: “Prediction of gas concentrations in a three-way catalyst for on-board diagnostic applications”, SAE Technical Paper 2005-01-0054, (2005). [5]. Jones, J. P. et al.: “Modeling the transient characteristics of a three way catalyst”, SAE Technical Paper 1999-01-0460, (1999). [6]. Jones, J. P., Roberts, J., Bernard, P.: “A simplified model for the dynamics of a three-way catalytic converter”, SAE Technical Paper 2000-01-0652, (2000). [7]. Koltsakis, G. C., Tsinoglou, D. N.: “Thermal response of close-coupled catalysts during light-off”, SAE Technical Paper 2003-01-1876, (2003). [8]. Ohsawa, K., Baba, N., Kojima, S.: “Numerical prediction of transient conversion characteristics in a three-way catalytic converter”, SAE Technical Paper 982556, (1998). [9]. Shaw, B., Fischer, G. D., Hedrick, J. K.: “A simplified coldstart catalyst thermal model to reduce hydrocarbon emissions”, Proceedings of 15th Triennial World Congress of the International Federation of Automatic Control, (2002). [10]. Shen, H., Shamim, T., Sengupta, S.: “An investigation of catalytic converter performances during cold starts”, SAE Technical Paper 1999-01-3473, (1999). [11]. Soumelidis, M. I. et al.: “A nonlinear dynamic model for three-way catalyst control and diagnosis”, SAE Technical Paper 2004-01-1831, (2004).

ACKNOWLEDGEMENTS This work was supported in part by the Center for Hybrid and Embedded Software Systems (CHESS) at UC Berkeley, which receives support from the National Science Foundation (NSF award #CCR-0225610), the State of California Micro Program, and the following companies: Agilent, DGIST, General Motors, Hewlett Packard, Infineon, Microsoft, and Toyota. Authors also acknowledge the support provided by Toyota and CONACYT (Consejo Nacional de Ciencia y Technología de Mexico). We would also like to thank the reviewers of this paper for their useful suggestions and comments.