Model Based Catalyst Control
Irman Svraka and Linus Österdahl Wetterhag
Model Based Catalyst Control is a project done in the course Examensarbete in Electrical En-gineering, TQET33 30.0 ECTS perfomed at Volvo Cars in Gothenburg for Department of ISY, Vehicular Systems, Linköping University.
Supervisors: Anton Ajne Volvo Cars, Johan Tubbero Volvo Cars and Olov Holmer, Department of Vehicular Systems, ISY, Linköping University
“A catalyst is that couch in the middle of the room during a party, where you sit down and happen to meet someone you fall in love with. Then you leave as one.”
A one dimensional discretized model of a two brick three way catalyst (TWC) system was de-veloped and implemented in MATLAB, Simulink and TargetLink in collaboration with Volvo Cars and Linköpings Universitet - ISY. The purpose of this thesis was to increase system un-derstanding and create a model based TWC control for further development at Volvo Cars. A total of 50 states were modelled, including emission concentrations (O2, CO, C3H6, C3H8, H2,
N Ox, CO2, H2O), temperature and oxygen buffer level (OBL). A model based control structure
was implemented in the form of five separate PID-controllers enabling possibilities to control the OBL of each separate slice of each brick individually and through simple reference handling. The control structures includes anti-windup, feedforward control and feedback safety for model reset during sensor indication of leakage. Specific equipment and software used included MAT-LAB, Simulink, TargetLink, Volvo SULEV30 TWC and testing rigs. Overall increase in system understanding was achieved in comparison with contemporary TWC modelling and control, as well as sufficient system performance in regard to estimate emissions, simulation duration and pedagogical value. Concluding thoughts of the thesis revolve the complexity of the actual TWC modelling, parameter estimation as well as control. The model presented in this thesis has great potential of describing TWC systems but with great effort during parameter estimation. With ECU performance available in temporary vehicle production year 2019, a complex model may be combined with a simple control strategy whilst a simple model may be combined with a complex control strategy.
lades samt implementerades i MATLAB, Simulink och TargetLink i samarbete med Volvo Cars och Linköpings Universitet - ISY. Syftet bakom arbetet var att utöka systemförståelsen samt skapa en modellbaserad regulator för en trevägskatalysator för vidareutveckling på Volvo Cars. Totalt modellerades 50 tillstånd inkluderande emissionskoncentrationer (O2, CO, C3H6, C3H8,
H2, N Ox, CO2, H2O), temperatur och syrebuffertnivå. En modellbaserad regulatorstruktur
im-plementerades i form av fem separate PID-regulatorer tillåtandes möjligheten att reglera syrebuf-fertnivån i vardera skiva individuellt och genom simpel referenshantering. Regulatorstrukturen inkluderar anti-integratoruppvridning, framkopplingsreglering och återkopplingssäkerhet resul-terande i modellåterställning vid sensorindikering av läckage. Specifik utrustning och mjukvara som användes inkluderade MATLAB, Simulink, TargetLink, Volvo SULEV30 trevägskatalys-ator och testrig. Generell ökad systemförståelse uppnåddes i jämförelse med samtida model-lering och styrning av trevägskatalysatorer, samt tillfredsställande systemprestanda angående simulerade emissioner, simuleringstider samt pedagogiskt värde. Sammanfattande tankar an-gående arbetet rör komplexiteten av faktiskt modellering, parameter estimering samt reglering av en trevägskatalysator. Modellen som presentera i arbetet har stor potential att beskriva trevägskatalysator-systemet men med ansenlig börda i form av parametersättning. Dagens ECU-prestanda i produktion år 2019 medför möjlighet att applicera en komplex modell i kombination med en simpel regulatorstruktur medan en simpel modell kan kombineras med en komplex reg-ulatorstruktur.
“Jag drog upp den på 6000 varv, sen bara dog den”
We would like to thank our supervisors Anton Ajne and Johan Tubbero at Volvo Cars for all the help with the work done in this thesis. We would also like to thank our supervisor from Linköping University Olov Holmer for assistance during the work. Additionally we would like to give an extra thanks to Fredrik Wemmert for the expertise, support and impulsion during the work. Other people at Volvo Cars we would like to thank for their support are Olaf Eickel, Damien Eymeric, Anders Johnsson, Victor Gylling, Martin Larsson, Anders Botéus and Ulf Nordgren. We would also like to thank Lars Eriksson at Linköping university for giving us the opportunity to do our master thesis at Volvo Cars in the first place. We would also like to thank our friends and colleagues Robert Widén and Jonas Vedin for our time together during our education. Last but not least we would also like to thank our respective families, friends and loved ones for their continuous support.
Irman Svraka, Linus Österdahl Wetterhag Linköping, Sweden,
1 Introduction 1
1.1 Background . . . 1
1.2 Problem Formulation . . . 1
1.3 Purpose and Goals . . . 1
1.4 Approach . . . 2 1.5 Expected Results . . . 2 2 Theory 3 2.1 Three-Way Catalyst . . . 3 2.1.1 Chemical Reactions . . . 4 2.2 Lambda Sensors . . . 6
2.2.1 Linear Lambda Sensor . . . 6
2.2.2 Binary Lambda Sensor . . . 6
2.2.3 Lambda Sensor Modelling . . . 6
2.3 Oxygen Storage . . . 7
2.3.1 Water-Gas Shift Reaction . . . 7
2.4 Earlier Work . . . 8
2.4.1 Physical Modelling . . . 8
2.4.2 Black-Box Models . . . 10
2.4.3 Oxygen Storage Models . . . 11
2.5 Controller . . . 15
2.5.1 Bang-Bang Control . . . 15
2.5.2 Model Predictive Control . . . 15
2.5.3 Linear-Quadratic Regulator . . . 15
2.5.4 Earlier Control Strategies for TWC Control . . . 16
3 Modelling 17 3.1 Model Description . . . 17
3.1.1 Langmuir-Hinshelwood Reaction Rates . . . 18
3.1.2 Arrhenius Reaction Rates . . . 19
3.1.3 Oxygen Storage Modelling . . . 20
3.1.4 Temperature Model . . . 20
3.1.5 Engine Out Model . . . 21
3.1.6 Species Balance Solver . . . 22
3.1.7 Rate Limiter . . . 22
3.1.8 Brettschneider Lambda . . . 23
3.1.9 Model Reset . . . 23
3.2 Matlab and Simulink Implementation . . . 23
4 Controller 26 4.1 PID Controller . . . 26
5 Results 29 5.1 Simulation Results for Model Validation . . . 29
5.2 Controller Evaluation Simulation . . . 39
5.2.1 PID Controller Simulation Results . . . 39
5.2.2 Model Reset Function . . . 41
6 Discussion 43 7 Conclusion 46 Bibliography 47 Appendix A Parameters A.1 Parameters used for simulation . . . 49
Appendix B Rate Limiter B.1 Code used in the rate limiter for simulations . . . 51
Appendix C Controller code C.1 Code used for the controller in simulations . . . 54
Appendix D Result plots D.1 Richlean transients at 1250 rpm . . . 58
D.2 Richlean transients at 2000 rpm . . . 63
D.3 Bang-Bang control results . . . 67
Abbreviations
AF R Air-Fuel-Ratio
OBL Oxygen Buffer Level
P P M Parts per million
SI Spark Ignition
T OSC Total Oxygen Storage Capacity
T W C Three Way Catalyst
Variables ˙
V Volume flow m3/s
λ AFR proportion −
Θ Oxygen buffert level −
a Area of contact m2
Ci Concentration of species i mol/m3
cp Specific heat capacity J/kgK
Ej Activation Energy J /mol
G Inhibition factor −
Hj Exothermic heat value J /mol
hs Heat transfer coefficient W /m2K
kj Kinetic parameter −
Kwgs Water gas shift reaction factor −
m Mass kg
p Pressure P a
Q Energy J
R Gas constant J /Kmol
Rj Reaction Rate mol/s
V Volume m3
vj Stoichiometric coefficient −
Introduction
1.1
Background
With tougher legislation regarding emissions from spark ignition (SI) engines Volvo Cars are looking for new possibilities to decrease emissions. Currently a bang-bang control strategy for lambda control is utilized. To increase the understanding regarding the three-way catalyst (TWC), this thesis aim is to investigate the possibilities to construct a model for a TWC. The model will thereafter be used to construct a model based controller to reduce tailpipe emissions from SI engines. In this thesis earlier attempts to construct models of the TWC will be presented and investigated and used as the base material for the constructed model. Just as important as legislation, minimizing emissions entails less environmental impact from combustion engine propelled vehicles along with a reduction of general human health impact.
1.2
Problem Formulation
To improve the understanding of what goes on in the TWC, chemically and physically, Volvo Cars wants to expand its knowledge around what actually goes on inside the catalyst. To do so this thesis was proposed with the purpose of building a model of the TWC and use it to increase TWC performance. The problem with modelling of a TWC is the great complexity of chemical reactions occurring in the catalyst and the lack of available methods to measure chemical reac-tions. The biggest problem with catalyst modelling is the vast amount of parameters that has to be estimated and the difficulty in this comes from the large amount of reactions occurring at the same time in the catalyst. Another big problem with modelling is to actually be able to implement the model in a vehicle since the models usually are complex. The ECU’s used today are often not powerful enough to handle such a complex model. Therefore a good compromise of model complexity and usability is needed if the model is to be used for control purposes.
1.3
Purpose and Goals
The goal of this entire project is to explore alternative approaches to controlling a TWC, com-pared to conventional control strategies. Also, the development of a model increases system understanding in regard to chemical reactions, temperature developments and oxygen buffer levels. It is desirable from Volvo Cars to establish a specific TWC model that will increase system understanding out of a pedagogical stand point. Acquiring a model of this sort may assist in future control development, component explanation and general testing.
1.4
Approach
The work will start with a literature study of earlier work done on TWC modelling and control. These attempts will be investigated and summarized and with the knowledge gained an approach for the model to be constructed will be chosen. Then the model chosen will be built in Simulink for easy simulation and validation. The model will then be evaluated against measurement data using software such as INCA and MDA. Thereafter the model will used to choose and evaluate different control strategies. At last the model will be built with TargetLink blocks in Simulink to conform with Volvo Cars Standard, allowing easy code generation for testing the controller in a test rig or vehicle.
1.5
Expected Results
This thesis should result in a TWC model allowing for greater system understanding regarding concentrations, temperatures, theoretical lambda values as well as oxygen buffer level. The mentioned model shall be used as underlying for a model based catalyst control approach. Due to the complexity of the model and hypothetically high computational burden, the control strategy applied will likely consist of simple controllers such as PID-controllers. The model in combination with it’s controllers performance should be equal to or better than conventional control methods used currently by Volvo Cars, meaning minimizing leakage of legislated emissions out of the exhaust system tailpipe.
Theory
2.1
Three-Way Catalyst
In a spark ignition (SI) engine exhaust gases contain nitrogen (N2), carbon dioxide (CO2),
water (H2O), oxygen (O2), carbon monoxide (CO), nitrogen oxides (N Ox) and hydrocarbons
(HC). Of these gases carbon monoxide (CO), hydrocarbons (HC) and nitrogen oxides (N Ox) are considered emissions and are regulated by law [1]. The concentration of the different gases vary with different operating conditions, however the amount of these gases allowed as tailpipe emissions are regulated. To reduce these emissions a TWC converter is needed in an SI engine exhaust system. To reduce all these emissions effectively the engine has to be operated with the correct air to fuel ratio (AFR). This means the stoichiometric AFR. A typical stoichiometric AFR value for gasoline is 14.7. The air to fuel ratio is defined as λ seen in (2.1) and for the TWC to have high conversion efficiency the engine has to be operated at λ ≈ 1. The TWC conversion efficiency as a function of lambda can be seen in Figure 2.1, reproduced with permission [2].
λ = AF Ractual AF Rstoichiometric
(2.1)
To maintain the engine running at correct air fuel mixture, a lambda feedback control is needed to ensure that the air fuel mixture is kept at stoichiometry. This control typically contains of a oxygen sensor in front of the first brick and a sensor after the first brick. These sensors are used in the feedback control and ensure that the right amount of fuel is injected to the engine [3]. Performance of a TWC is highly unique from TWC to TWC and strongly dependant of specifications including number of cells per square inch (CPSI), materials, TWC volume and material composition.
The TWC consists of a composition of different metals. Mainly P latinum (P t), P alladium (P d), Rhodium (Rh), Cerium (CeO2), Zirconia (ZrO2), LanthanumOxide (LaO2) , StrontiumOxide
(SrO) where P t, P d and Rh are the main part of the catalytic components coating. The con-struction of the catalyst is a honeycomb structure to make the surface area as large as possible to allow increased possibility of reactions to occur. The ceramic brick used in this thesis can be seen in Figure 2.2. In Figure 2.3 a cut through of a catalytic converter with 3 bricks can be seen. The ground layer of the catalyst is coated with two layers of washcoat, being thin layers of alloy. The inner layer of the washcoat often consists of aluminiumoxide (Al2O3) and P d and
its main purpose is to oxidize HC. The outer layer of the washcoat consists of Al2O3, CeO2,
Figure 2.1: The percent of emissions as a function of lambda after the TWC. Figure from [2] (p. 139) reproduced with permission.
2.1.1
Chemical Reactions
A number of chemical reactions occur in the TWC. The reactions presented in Ramanathan and Sharma are the ones modelled in this thesis with modifications to reaction 13 and 14 [4]. In (2.2) to (2.16) 15 of these reactions can be seen. All of these reactions can be divided in to subcategories. Reactions (2.2) to (2.5) may be classified as oxidation reactions using O2.
Reactions (2.6) to (2.8) are reduction reactions for N Ox. Reactions (2.9) and (2.10) models
the water gas shift reaction which will be explained further in 2.3.1. Oxygen storage reactions may be described as reactions (2.11) and (2.12), whilst oxygen release reactions are taken in to account by (2.13) to (2.16). CO + 0.5O2−→ CO2 (2.2) C3H6+ 4.5O2−→ 3CO2+ 3H2O (2.3) C3H8+ 5O2−→ 3CO2+ 4H2O (2.4) H2+ 0.5O2−→ H2O (2.5) CO + N O −→ CO2+ 0.5N2 (2.6) C3H6+ 9N O −→ 3CO2+ 3H2O + 4.5N2 (2.7) H2+ N O −→ H2O + 0.5N2 (2.8) CO + H2O −→ CO2+ H2 (2.9) C3H6+ 3H2O −→ 3CO + 6H2 (2.10) 2Ce2O3+ O2−→ 4CeO2 (2.11) Ce2O3+ N O −→ 2CeO2+ 0.5N2 (2.12) CO + 2CeO2 −→ Ce2O3+ CO2 (2.13) C3H6+ 18CeO2 −→ 9Ce2O3+ 3CO2+ 3H2O (2.14) C3H8+ 20CeO2 −→ 10Ce2O3+ 3CO2+ 4H2O (2.15) H2+ 2CeO2 −→ Ce2O3+ H2O (2.16)
Figure 2.2: Picture of the ceramic in the TWC used in this project. The honeycomb pattern for increased surface area can be seen.
Figure 2.3: Picture of a cut through catalytic converter with three bricks. Photo by Linus Österdahl Wetterhag at Volvo Museum, Gothenburg.
2.2
Lambda Sensors
Varying setups regarding both types and placement are common within the automotive industry regarding lambda sensors. Typically a linear lambda sensor is placed upstream before the first brick of the TWC followed by a switch type binary lambda sensor after the first brick. Lambda sensors measure the excess (λ > 1) or lack (λ < 1) of oxygen in exhaust gases in regards to stoichiometry. Lambda itself is defined as a fraction of the actual- and stoichiometric Air-Fuel-Ratio (AFR), see (2.20), the stoichiometric AFR is fuel dependant [5]. For the lambda sensor to give a correct signal, often the sensor has to reach a certain temperature before it starts to display the correct value. This means that during a cold start the sensor is not used for control purposes and the ECU uses an open-loop controller during cold start.
2.2.1
Linear Lambda Sensor
The linear lambda sensor provides a wider spectrum of lambda value measurements whilst the binary lambda sensor has greater measurement precision for lambda values close to one. Linear Lambda sensors are often used in front of brick one for the inner control loop for the lambda feedback. Using the linear lambda sensor in combination with relevant chemical reactions, the concentrations of O2, CO, H2 and H2O in to the TWC may be modelled [1] [6] . The O2term
being important during lean operations whilst CO, H2 and H2O dominate rich conditions. The
H2O term being key for the water-gas shift reaction. Concentrations of the gases mentioned
previously may be modeled using look up tables which is sufficient in some cases, an actual concentration model leading to the TWC model allows greater accuracy[1][6].
2.2.2
Binary Lambda Sensor
A binary (switch type) lambda sensor output voltage is directly affected by varying concentra-tions of the exhaust gases. Switch type lambda sensors consist of multiple layers of protection, cathodes, electrolytes, and anodes. Diffusion occurs from the exhaust gases in to the sensors own reference gas (being ambient air) and thereafter λ is determined by the electrochemical process within the sensor giving a output in voltage. Previously, the sensor-layers have been divided in various modules and thereafter detailed models have been developed with regards to diffusion, adsorption, desorption and more [7]. Exhaust temperatures have been shown to have severe impact upon binary lambda sensor voltage outputs, most notably during rich operation (λ < 1). The water-gas shift reaction’s H2-generation may result in a voltage-increase for the
downstream binary lambda sensor [7]. Binary lambda sensors are typically better at determining if the lambda value is rich or lean with great precision around the value of lambda close to 1, whilst they are more poor at determining wider spectrum of lambda values compared to linear lambda sensors.
2.2.3
Lambda Sensor Modelling
To model the sensor dynamics can have an impact on the model and propsed lambda models are presented below.
2.2.3.1 Linear lambda model
Though some applications assume look up tables are satisfactory regarding linear lambda out-puts (upstream), their accuracy may be of poor quality [6] [1]. A two-step calculation may be implemented for modelling, the primary one regarding the concentration being excessively rich or lean (λ-dependent) along with the secondary regarding temperature (unburned components).
All oxidizing agents are referred to as O2 occurring during lean lambda values and CO as well
as H2whilst the lambda values display rich combustion.
yO2= 1 − 1λ α+β2 (α+β4)λ+ (1 − 1 λ) + 79 21 (2.17) yCO = 2 1+φH2/CO( 1 λ− 1) α+β2 (α+β/4)λ+ 79 21 (2.18) yH2= 2θH2/CO 1+φH2/CO( 1 λ− 1) α+β 2 (α+β/4)λ+ 79 21 (2.19)
2.2.3.2 Binary Lambda Model
Voltage values displayed by the binary lambda model are influenced by reducing species such as hydrogen, which should be taken in to account when modelling [7]. Reactions occurring in the lambda sensor are separated in to two: electrode-electrode transitions and solely electrolyte transitions (third model). Though this model gives lesser accuracy during rich inputs than lean inputs, hypothetical reasons may be decoupling of electrodes and electrolytes [7]. Another hypothetical reason (though examined on a linear lambda sensor, might be the case for binary as well) may be the concentration of H2 post catalyst, being affected by catalyst deactivation
[8]. Discrepancies in the measurements of the down stream lambda sensor may be used as a measurement of H2 from the water-gas shift reaction [8].
2.3
Oxygen Storage
CeO2 and ZrO2 have the ability to store and release oxygen in the catalyst [9]. This process
works as a buffer for when the engine switches between lean and rich air fuel ratio. This property of the catalytic converter makes it possible to switch between lean and rich λ during stationary conditions and maintaining a good conversion rate of pollutants. One big challenge with oxygen storage capacity occurs when for instance a fuel cut is performed, then air is pumped through the engine into the exhaust filling the oxygen storage to it’s max capacity. This means that the TWC has to be neutralized after the fuel cut to resume it’s optimal conversion efficiency once again. Since the oxygen buffer level (OBL) is immeasurable, a model is required to estimate the current state of the OBL.
The catalyst is also temperature dependent and is not effectively converting emissions until the temperature is high enough. Therefore a critical part of the emissions SI engines release are emitted during the cold start period before the catalyst is above the critical light off temperat-ure. The light off temperature is the critical temperature for the catalyst work properly.
During a fuel cut the engine is pumping fresh air directly through the engine out to the ex-haust system. This means that the OBL will be filled to its maximum level. After a fuel cut the catalyst has to be neutralized, meaning emptied of excess oxygen to work properly and have a good conversion efficiency. This is most commonly done by a short period of extremely rich air fuel mixture to use up the excessive oxygen stored in the ceria.
2.3.1
Water-Gas Shift Reaction
Whilst the available concentration of O2 stored in the TWC during rich operation remains at
[5]. When these conditions are not met, what is know as the water gas shift reactions can occur.
2CO + O2−→ 2CO2 (2.20)
What happens during the water gas shift reactions is that CO is reduced using water molecules to form CO2and H2 as in reaction (2.21) [5].
CO + H2O CO2+ H2 (2.21)
Impact of the gas shift reaction is of various importance depending on source. The water-gas shift reaction may result in inhibition on the lambda sensor and thereby alter the downstream λ-values, the altering of λ is related to the generation of H2 in the water-gas shift reaction [8].
In some cases, impact of the water-gas shift reaction is neglected due to CO and H2 both being
seen as resulting in the same overall concentration when interchanged [10].
2.4
Earlier Work
Several different methods to model the TWC have previously been utilized. Ranging from simple black-box models to advanced 3D CFD models. Most common amongst TWC-modeling is the modeling of the catalysts oxygen storage. Models regarding oxygen storage are popular due to their simplicity compared to direct emission modelling. Below, different approaches will be investigated and evaluated based on advantages and disadvantages. When selecting model type, factors such as simplicity, computational effort for calculations and validity are very important. An advanced and precise model may be too cumbersome for an ECU’s computational potential and therefore eliminates the chance of vehicular implementation. Factors influencing the possib-ility to implement the model to the ECU are sampling time, number of states, solver, amongst others.
2.4.1
Physical Modelling
In Otsuka et. al., three different models were implied for the chemical reactions namely the Arrhenius model, competitive adsorption model (Langmuir-Hinshelwood) as well as an adsorp-tion model (representing the rate of gas per unit of area at the TWC) [11]. All tests were performed at a constant space velocity, unlike previously mentioned papers [11] [12]. Tests were performed at five various temperatures ranging from 373 K to 773 K, to test conversion rate contra temperature. The adsorption and desorption characteristics, the TWC was filled with O2
(oxidizing) and CO (reducing) gas alternating each other and whilst measuring concentrations upstream and downstream of the TWC. The physical model used was a one-dimensional model (discretized), allowing information regarding rate, concentrations and temperatures throughout the axial direction of the TWC. Physical properties modelled included diffusion rates (Fick’s law) along with four different heat transfers (convection, conduction, reaction and release) within the TWC.
F. Aimard et al. presented a mathematical approach to model the TWC [13]. In the pa-per two models were presented, one advanced model consisting of partial differential equations (PDE) who’s main purpose was to perform accurate simulation of the behaviour of the catalyst for validation or design purposes. The other model presented was a simplified model consisting only of ordinary differential equations (ODE). Both models describe the conversion efficiency’s of HC, CO and N Oxand the oxygen storage capacity. The model was evaluated during an
model in F. Aimard et al. should therefore be investigated further[13]. The model proposed was not tested in control applications but only as a simulation tool, however given the satisfact-ory results it would be interesting to investigate the capabilities of the model in control purposes.
A detailed model for the temperature dynamics in the TWC was presented in Onori and Godi, where a set of PDEs were used to describe the temperature dynamics of the TWC [14]. To re-duce the system further and make the model more control-oriented a POD-Galerkin projection was used to reduce the PDE to a simplified ODE system. Both the full model and the reduced model presented satisfying results during the FTP drive cycle. The model presented in Onori and Godi could be used as a base for temperature modelling of the TWC [14] .
Modelling the energy balance within the exhaust system assumptions such as that the thermal dynamic effect of conduction is minimal in comparison to convection may be applied [4] [15] . The change in gas temperature Tg in the TWC axial direction is assumed to be the difference
in gas temperature in (Tg,in) and gas temperature out (Tg,out) across a substrate length (L) as
can be seen in (2.22).
∂Tg
∂x =
Tg,out− Tg,in
L (2.22)
However, (2.23) does contain a term regarding conduction in the axial direction. More over, no exo- nor endothermic reactions were taken in to account and no dynamics were acknow-ledged regarding the temperatures between the TWC and the exhaust gases, however these were taken in to account in Ramanthan and Sharma as term two in (2.23) [4] [15] .
(fsbρsbcp,sb+ fwcρwccp,wc)) ∂Ts ∂t = fsb ∂ ∂z(λsb ∂Ts ∂z ) − nrxn X j=1 aj(z)(∆H)jRj+ hS(Tg− Ts) (2.23)
The solid- and gas-phase energy balance may be lumped together (2.24) or alternatively de-scribed separate as (2.23), (2.25) [4] [15]. Overall the energy phase balance dede-scribed by Ram-anthan and Sharma appeared more detailed and more relevant for a TWC instead of an SCR (being Selective Catalytic Reduction, generally implemented in diesel powered vehicles) when compared to Holmer and Eriksson [4] [15] .
dTg,out dt = − 1 (1 − )ρsCp,s (νρgCp,g Tg,out− Tg,in
L + hs↔aas↔a(Tg,out− Ta)) (2.24) w
Acp,g ∂Tg
δz = hS(Ts− Tg) (2.25)
Mass balance regarding gas and the TWC surface dependant of species may be modelled as in (2.26) (where xg and xs are gas- and solid phase mole fractions), along with its boundary
condition (2.27) [4].
w A
∂xg,i
∂z = −km,iS(xg,i− xs,i) =
nrxn
X
j=1
aj(z)Rjsi,j (2.26)
Determining the amount of, as well as which chemical reactions shall be taken in to account greatly affects the model’s complexity and implementation effort. Selection of chemical reac-tions also affects the choice of reaction rate calculareac-tions (being Arrhenius equation, Langmuir-Hinshelwood and more) which largely determine overall model precision [16]. Simplification of chemical reactions within the TWC and the actual feed gas reduce implementation time and further effort. Lumping oxidating reactions (O2, N O) and reducing reactions (CxHy, CO) along
with hydrogen reactions being independent (H2) severely diminishes computational burden
ne-cessary for the model and results in only three to six chemical reactions related to the TWC’s cerium [10] [17]. Alternatively, specific reactions may be assumed dominating due to dynamic rates for various working conditions (i.e. lean or rich), though these assumptions significantly affect model accuracy [5]. Higher number of chemical reactions ensues greater computational burden in vehicle applications and far more tunable parameters, hence substantial modelling efforts but also increased precision [4] [16]. An increased number of chemical reactions entails an enhanced number of tunable parameters, tuning a great number of parameters manually requires vast understanding of the system and is a tedious, iterative and time consuming process. Para-meters may be optimized for models resulting in desired behaviour by optimization procedures such as conjugate gradients (multi-dimensional) and genetic algorithms [18] [19].
2.4.2
Black-Box Models
In Stobart et. al. a library of Nonlinear AutoRegressive Moving Average with eXogenous in-put (NARMAX) models were used to predict the TWC transient response [5]. The model was aimed for control purposes and aimed to predict the post TWC lambda. The model was divided into four different operating regions, lean, rich, oxygen release and oxygen storage. To identify which operating condition the model operated in, a simplified chemical model was utilized. The simplified model was fundamentally an oxygen storage model. The model assumed that under lean conditions, the reaction seen in (2.28) is the main reaction occurring.
CeO + 1
2O2−→ CeO2 (2.28)
And under rich conditions (2.29) was the main reaction taking place. Where OST R was the
amount of oxygen stored in the catalyst.
CO + OST R−→ CO2 (2.29)
This model also considered the reduction of CO by water using the water-gas shift reaction seen in (2.21). The quantity Θ was defined as the normalized stored oxygen (2.30) where OSC was the total oxygen storage capacity.
Θ = [O2]
OSC (2.30)
When the OSC is full, Θ takes the value 1, and when it is empty the value 0. The derivat-ive of Θ during oxygen storage was defined in (2.31) where k1 is the rate constant of the ceria
oxidation reaction and pO2 is the partial pressure of O2. The derivative of the oxygen storage
during oxygen release can be seen in (2.32) where k2is the rate constant and pCO is the partial
pressure of CO.
dΘ
dt = k1(1 − Θ)p
0.5
dω
dt = −k2ωpCO, λ < 1 (2.32)
The chemical model was then used to choose which operating condition was active and from there the correct NARMAX model was chosen to predict the AFR post TWC. The models were estimated using only the ∆AF Rpre as input and ∆AF Rpost as output. The model predicted
∆AF Rpost with good accuracy. However the model was only validated in one operating region
and therefore it was not able to handle many different operating conditions. The model could however be expanded to be valid in more operating conditions. It is mostly the chemical model that sets the limitations for the proposed model structure. More cases for the model could be added and more accuracy could hypothetically be obtained but all of this requires a lot of work.
The model presented in Stobart et. al. seemed to be usable for lambda control [5]. How-ever the shortcoming of the model was the expansion to handle different operating conditions. One weakness with the model was that it also did not give a deeper understanding of what happens chemically in the catalyst due to the fact that it is a black-box model. Also the para-meters k1and k2 simplified the tuning process of the OBL greatly, however once again it lacked
complexity and explanation of the true system reactions.
In Gonatas and Stobart another black box model was proposed [20]. This model aimed to predict ∆AF Rpost (y(t)) lambda using only measurements of ∆AF Rpre (u(t)). It also used
the ∆AF Rpre together with the predicted ∆AF Rpost to estimate output gases after the TWC.
The model used can be seen in (2.33) and a recursive-least square (RLS) algorithm was used to estimate the parameters. Where Θ was the parameter vector that minimizes (2.34) where f f is defined as a forgetting factor defined as (2.35) which was used to discard earlier information. Too small value of f f however can decrease the performance and make the prediction unstable. The term e was the error between measured output and model output which was to be minimized.
y(t) = UT(t)Θ + e(t) (2.33) Vt(Θ) = N X t=1 (f f )e2(t − 1) (2.34) f f = samples − 1 samples (2.35)
The model predicts the ∆AF Rpost with good accuracy during stationary conditions, however
during transients the model predictions are poor. In the article it is explained as a factor of non-linear dynamics in the system that the model cannot capture. The model also tries to predict N Ox and CO emissions out. For this, different regressors are evaluated and it does to
some extent predict the output emissions with satisfactory results. However the model does not predict HC and CO2 emissions. It is mentioned in [20] that the model can be extended to
capture the non-linear dynamics. This extension would probably lead to a similar approach as shown in Stobart et. al. [5]. This method rapidly becomes very complex with different models for different operating conditions.
2.4.3
Oxygen Storage Models
In Kerns et. al. a low dimensional TWC-model based upon OBL and total oxygen storage capacity (TOSC) was obtained [10]. Reductions of a three-dimensional system was achieved by assuming symmetry, axial averaging and mass transfer coefficient concepts resulting in a zero-dimensional model. This zero-dimensional model was in the form of a few ODEs based
on energy balances and species. All reducing agents A were lumped together for simplification reasons as seen in (2.36). The same reasoning was used for Ox. All oxidizing agents in equation
(2.37) consisting of O2 and N O [10]. Equation (2.38) displays the definition of total oxidation
products. [A] = (2 +y 2)[CHy] + [CO] + [H2] + 3 2[N H3] (2.36) [Ox] = [O2] + 1 2[N O] (2.37) [AO] = [CO2] + [H2O] (2.38)
It is noteworthy that in this model, the water-gas shift reaction and steam reforming is neglected. The global kinetic reactions are described by (2.39), (2.40) and (2.41):
A +1 2O2−→ AO (2.39) Ce2O3+ 1 2O2−→ Ce2O4 (2.40) A + Ce2O4−→ Ce2O3+ AO (2.41)
Each of the reactions in (2.39), (2.40) and (2.41) are described by a corresponding reaction rate, namely R1, R2 and R3. The OBL is defined using a fraction based on the total concentra-tion on the ceria, see (2.42). Another important factor is the changing rate of OBL (θ) which is defined as in (2.43). θ = Ce2O4 Ce2O4+ Ce2O3 (2.42) dθ dt = 1
2T OSC(Rstore− Rrelease) = 1
2T OSC(R2− R3) (2.43)
Kumar and Makki is an extension of Kerns et. al. in which H2 was introduced as a third
agent along with the previous lumped oxidants and reductants [10] [17]. Additionally, three supplementary kinetic reactions were implemented to previously mentioned (2.39), (2.40) and (2.41). The three new reactions were displayed in (2.44), (2.45) and (2.46). Once again, each of the reactions included entailed a corresponding rate constant R4, R5and R6. The rate of which
OBL is changing with was now extended to contain the rate of (2.46) in the form of an release factor.
H2+
1
2O2−→ H2O (2.44)
H2+ Ce2O4↔ H2O + Ce2O3 (2.46)
In Onder et. al., similarly to Kerns et al. a control oriented model was based upon a num-ber of chemical reactions in the TWC as well as containing models for lambda sensors [6] [10]. Unlike Kers et. al. this paper did take the water-gas shift reaction in to account [10]. The TWC model was also split in to three slices along the axial direction, meaning that all six partial differential equations ((2.47)) - (2.54)) were solved three times resulting in 18 state vari-ables. Reaction rates for each chemical reaction taking place in the model were calculated using Langmuir-Hinshelwood, Eley-Riddeal or Arrhenius.
1 2O2,g+ ∗ Ce−→ OCe (2.47) CO,g+ OCe−→ CO2,g+ ∗Ce (2.48) H2,g+ OCe−→ OH2Ce (2.49) H2O,g+ ∗Ce−→ OH2Ce (2.50) 1 2O2,g+ OH Ce 2 −→ O Ce+ H 2O,g (2.51) OH2Ce−→ OCe+ H 2,g (2.52) OCe−→ ∗Ce+1 2O2,g (2.53) H2,g+ OCe−→ ∗Ce+ H2O,g (2.54)
The reactions were used for calculating concentrations of CO, O2 and H2along with occupancy
of oxygen on the catalytic surface (ΨO) and occupancy of hydroxyl on the catalytic surface
(ΨOH) and the temperature T .
The quantities of O2, H2, CO and H2O used as input to the TWC model were the output
of the linear lambda sensor. These were provided from a look up table which is deemed suffi-cient regarding precision [6]. However, other sources do claim accuracy must be high with the use of look up tables and simple models are of preference [1]. Modelling of the binary lambda sensor requires inputs in the form of concentrations of O2, CO and H2 as well as post-catalyst
tem-perature Tout provided by the TWC-model. Voltage of the binary sensor (2.55) is the modelled
parameter.
Vλout = A + E(T )f (cCO, cH2) − F g(cO2) (2.55)
In Wang et. al. the oxygen storage dynamics was considered fast, and the temperature dy-namics slow [21]. A static map was then used to obtain the conversion efficiency for the different pollutants. The proposed model can be seen in (2.56), where C was the total oxygen storage capacity of the catalyst as a function of mass air flow (MAF). The term ˙Θ is the derivative of
the oxygen storage capacity and ρ is the exchange of oxygen between the exhaust gas and the catalyst surface as a function of λpreand Θ.
˙ Θ = ( 1 C(M AF )· ρ(λpre, Θ) · 0.21 · M AF · (1 − 1 λpre), 0 ≤ ˙θ ≤ 1 0, otherwise (2.56)
The lambda value after the catalyst is then calculated assuming the conservation of mass. This can be seen in (2.57). The model predicts the post λ sufficiently.
λpost= λpre− ρ(λpre, Θ) · (λpre− 1) (2.57)
The model in Wang et. al. also consisted of a temperature model which both takes into ac-count the warm up phase of the catalyst (namely the light off process) and then changes model when the temperature is high enough [21]. The temperature model was used as an input para-meter for the static map since conversion efficiency is dependent on temperature as well as air fuel ratio. This article thoroughly modelled the conversion efficiency’s mapping the efficiency’s during steady state measurements and then used the data collected to fit non-linear functions representing the static efficiency’s curves. The results show that the model predicted N Ox and
CO conversion efficiently but underestimated the HC conversion during rich operation. The authors concluded that there might be some other dynamics in the TWC besides oxygen storage since the HC conversion was poor during rich conditions. This indicates that a more chemically detailed model could give better results.
In Peyton and Muske, a model to estimate the current state of the oxygen storage level was proposed [22]. The model was initially based on the oxygen storage model presented in James et. al. [12]. However this model implemented a sensor distortion model which improved per-formance of the model. It predicted λpostwith satisfactory results. The model did however not
estimate emissions out, it was strictly a model to estimate oxygen storage level. However, using the estimated oxygen storage level in combination with reaction rates and mapped engine out emissions leading in to the TWC, the emissions may be estimated.
Oxygen storage based models may be further elaborated by the addition of space velocity [12]. The oxygen storage itself was based upon lambda values pre- and post-catalyst being supplied by the lambda sensors and described as (2.59) where K1and K2 are parameters of proportionality.
∆λ = λ − 1 = actual AFR stoichiometric AFR− 1 (2.58) ∆λpost = ( ∆λpre(1 − K1K2N (θ)) 0 ≤ θ ≤ 1 ∆λpre, otherwise (2.59) ˙ θ = ( K3∆λpre (∆λpre< 0), (θ > 0) K3(∆λpre− N (θ)) otherwise (2.60)
In (2.60) the parameter K3 is regarded as an engine condition-constant and N (θ) is potential
feedgas supply which itself is nonlinear. The nonlinearity of N (θ) is explained by the increas-ing difficulty of oxygen storage and release the further θ is from zero (θ = 0 corresponds to equilibrium) and implemented in form of a look up table. Space velocities were assumed to be proportionally related to the rate of storage and release, normalization of the model with space velocity change factor gave decent results when the same engine was used (but not necessar-ily for a new engine). An interesting process explained in the paper was the generation of an automated estimation algorithm for parameter estimation. Manual adjustments and tuning of models are tedious and time consuming, therefore an algorithm is to be preferred. A recursive nonlinear least squares estimator was developed but further work would be needed due to its heavy computational requirement.
2.5
Controller
A multitude of control strategies are used in current production vehicle’s depending on circum-stances of temperature, fuel injection and more. During vehicle start up, TWC temperature is far too low to enable full potential of reaction acceleration or promotion. Instead of using conventional bang-bang or feedback control (see section Bang-Bang Control), an open loop con-trol strategy based on feed-forward concon-trol is used until the lambda sensors is warm enough to output a correct signal. Thereafter the system uses a cascade control structure consisting of a faster dynamic inner control loop in combination with a slower dynamic control loop outputting reference lambda values.
2.5.1
Bang-Bang Control
Today a bang-bang control scheme is used to control lambda. The control scheme is based upon setting the target lambda based on the voltage of the binary lambda sensor. First a period of rich air fuel mixture is used until the binary sensor outputs a mapped voltage then the target lambda switches to lean until the sensors voltage again tells it to switch to rich, on so on the control scheme goes on. The control strategy aims to keep the average lambda close to one for optimal catalyst conversion efficiency. The bang-bang control is very effective and does provide sufficient control of the TWC, however it could be improved, especially after fuel cuts or after enrichment for engine protection.
2.5.2
Model Predictive Control
Control of the TWC may benefit from use of more complex controllers, such as an Model Predictive Controller (MPC). This has been attempted earlier by in [23] with satisfying results. The model used in [23] is simple and therefore an MPC controller can be used. However, the considerable number of states in each slice of the discrete model (10) and the adoption of multiple slices entails notable computatinal power necessary for reasonable simulation duration. An MPC would require information from each state of all slices during all time samples, thereafter it must calculate optimal control outputs based upon the inputs and the chosen prediction horizon. The prediction horizon is chosen by the control engineer and could hypothetically be multiple seconds, which in an ECU sampling each 10 ms would require thousands of calculations. Using performance available from an ECU in the year 2019, the time required for these predictions and calculations would be far too lengthy and annihilate the plausibility of vehicle-applications if the model is complex [24].
2.5.3
Linear-Quadratic Regulator
Linearizing the TWC-model significantly simplifies controllability. A Linear-Quadratic Regu-lator (LQR) in combination with a linear model can be tuned with various weight matrices Q (may require multiple iterations by the engineer) in order to control the system optimally to desired references at a minimum cost. Implementing an LQR will display an optimized control of the system whilst allowing tuning possibilities for the control engineer. LQR is a simpler alternative to an MPC and its control behavior may be used as an inspiration for implementing simpler and fundamental PID-controllers which may resemble the control of the optimal control LQR. Compared to an MPC, the controller is inadequate due to its lack of control signal limits and lack of prediction horizon. Linearizing the model at an operating point (ex. λ=1) may be done using MATLAB-functions linmod, this function creates a continous-time linear state space model. Due to in-vehicle applications must be done in discrete-time, the model provided by linmod may simply be transformed to an discrete-time linear state space model using the MATLAB-function c2d [25]. The use and further implementation and evaluation of an LQR controller for the catalyst model is out of the scope for this thesis and will be left as future work.
2.5.4
Earlier Control Strategies for TWC Control
In Balenovic and Edwards an oxygen storage level model is applied in a control oriented manner [26]. Here the model is implemented as an observer since the OBL can’t be measured. The es-timated oxygen storage level is then subtracted from the desired oxygen storage level and which then is used to set the desired target lambda value. The controllers performance is then com-pared to a conventional lambda controller using the sensor between brick one and brick two to update the lambda reference and a sensor in front of brick one for a fast feedback control of the fuel injection. The standard controllers purpose is to keep the engine running at stoichiometry, very similar to the controller Volvo Cars uses today. The results in Balenovic and Edwards show that the controller reduces emissions of N Oxand HC compared to the older strategy [26]. This
shows that it is a promising control strategy which could reduce emissions further, especially during transients. The authors also point out that the performance could be increased further if a third sensor was added after the second brick. Though suggestions such as these should be taken lightly in productional implementations due to effects regarding cost. Application of third lambda sensors do occur but sparsely and only in cars sold in geographical locations with extremely strict emission legislation.
Simple control strategies and tuning parameters may result in satisfying control of oxygen stor-age within the TWC. Controlling the outer loop of a cascade control structure using a simple PI-controller is simple and an elemental set of tuning parameters can be easily calculated using Ziegler-Nichols method for deciding the P − and I−gains. The application of a PI-controller entails a necessity for an anti windup loop in order to ensure stability due to the integrator part of the controller. This controller outputs the reference λref to the inner control loop [17].
Modelling
In this chapter the model developed and implemented will be explained in detail. The choice of which of model approach that would be used was consisting of multiple conditions. One condition for the model to be developed was that it would be implementable in an ECU, for this some of the easier models proposed in 2.4.3 would have been preferable, however at the same time a deeper understanding of what physically happens in the catalyst was of interest. Therefore a compromise of the two were chosen and the model is mainly based on the work presented in Ramanthan and Sharma, with modifications to reaction 13 and reaction 14 [4]. The final model will contain states in the form of the gasses O2, CO, C3H6, C3H8, H2, N Ox,
CO2and H2O. The OBL Θ will also be modelled as a state in each slice. A dynamically varying
temperature is also modelled throughout the TWC. A theoretical Brettschneider-lambda-value will be implemented. In Figure 3.1 a principal description of the whole layout of controllers, model and configuration.
Figure 3.1: Schematic figure of the model and controller layout modelled and built in this thesis.
3.1
Model Description
The experimental setup for the thesis will be a SULEV30 (Super ultra low emissions vehicle) catalyst from Volvo Cars with a linear lambda sensor before the first brick and a binary sensor between brick one and brick two in the catalyst. The model will be based on this setup but will be easy to modify if it is to be used in another application.
encompass characteristics differing along the TWC’s axial length. Main variables to be mon-itored will be OBL along the length of the TWC, as well as the temperature, Brettschneider lambda and concentrations. The focus on correct emissions output from the model has been of lower priority due to the fact that it is the OBL that gives most flexibility in terms of control. Therefore the main focus has been on tuning the model to work properly regarding OBL.
The model is mostly based on Arrhenius expressions and Langmuir-Hinshelwood expressions to describe reaction rates in the catalyst. A total of 15 chemical reactions are modelled and a total of 10 states exists in each slice of the final model. These states include concentration of species in the exhaust gas, OBL and temperature.
Excluded factors from the TWC model will be aging, which has direct impact upon the total oxygen storage capacity (TOSC). Lacking controllability of the catalyst during low temperatures means that the light off time will also be neglected from the TWC-model, meaning that cold starts will be excluded from the scope of this thesis. Other factors that have been excluded are engine out model and temperature model which have been provided by Volvo Cars and the weight on explaining these models profoundly has been neglected.
The model is one dimensional model split into five slices in axial direction as can be seen in 3.2. For the purpose of later implementation the model represents both bricks in the catalytic converter. The three first slices represents the first brick and the second brick is represented by the two last slices. In each slice, the concentrations of O2, CO, C3H6, C3H8, H2, N Ox, CO2,
H2O and OBL (Θ) are modelled as states. All states are modelled using the species balance
seen in equation (3.1). Where Cinis the concentration of species in to the slice and Cout is the
concentration exiting. Rj is the rate for the j’th reaction and vj is the stoichometric coefficient
for the reaction, ˙V is the volume flow and V is the catalyst segment volume.
Figure 3.2: Schematic figure of the Three-way catalyst setup modelled in this thesis. dCn
dt = (Cin− Cout) ˙ V
V + ΣRjvj (3.1)
Fifteen reactions are modelled in this work and can be seen in equations (2.2)-(2.16), as described in the theory-chapter. These include oxidation reactions with O2, reduction reactions
with N Ox, water gas shift reactions and ceria reactions for the oxygen storage.
3.1.1
Langmuir-Hinshelwood Reaction Rates
The rates for reactions (2.2)-(2.10) are modelled using Langmuir-Hinshelwood expressions with an inhibition factor G. The expressions to calculate the rates for each individual reaction can be seen in (3.2)-(3.10) where Ej is the activation energy, kj is a kinetic parameter, Tg is the
temperature of the gas and R is the gas constant.
R1=
k1CCOCO2exp(−RE1
gTg)
R2= k2CC3H6CO2exp(− E2 RgTg) G (3.3) R3= k3CC3H8CO2exp(− E3 RgTg) G (3.4) R4= k4CH2CO2exp(− E4 RgTg) G (3.5) R5= k5CCOCN Oexp(−REgT5g) G (3.6) R6= k6CC3H6CN Oexp(− E6 RgTg) G (3.7) R7= k7CH2CN Oexp(− E7 RgTg) G (3.8) R8= k8 Gexp(− E8 RgTg ))(CCOCH2O− CH2CCO2 KW GS ) (3.9) R9= k9 Gexp(− E9 RgTg ))CC3H6CH2O (3.10)
Where KW GS can be seen in (3.11) where ∆GW GS is the reaction energy of the water gas
shift reaction from Ramanthan and Sharma [4]. The inhibition factor G is expressed as seen in (3.12) where Ki,G is expressed as seen in (3.13) where ki,G and Ei,G are kinetic parameters
which may be found in appendix A.
KW GS = exp( −∆GW GS RgTg ) (3.11) G = (1 + K11CCO+ K2CC3H6) 2(1 + K 3CCO2 C 2 C3H6)(1 + K4CN O) (3.12) Ki= ki,Gexp(− Ei,g RgTg ), i = 1, ..., 4 (3.13)
3.1.2
Arrhenius Reaction Rates
For the reactions involving ceria, Arrhenius expressions are used to model the rates due to lack of inhibition and can be seen in (2.11)-(2.16) where Θ represents the OBL.
R10= k10exp(− E10 RgTg )CO2(1 − Θ) (3.14) R11= k11exp(− E11 RgTg )CN O(1 − Θ) (3.15) R12= k12exp(− E12 RgTg )CCOΘ (3.16) R13= k13exp(− E13 RgTg )CC3H6Θ (3.17) R14= k14exp(− E14 RgTg )CC3H8Θ (3.18)
R15= k15exp(−
E15
RgTg
)CH2Θ (3.19)
In the implementation all rates are gained with the parameter ai where i is the index for the
i’th reaction. This parameter is seen as a tuning parameter which can be used to tune the rates for each individual reaction. In general the kinetic parameters ki and Eiare the ones tuning the
rates. Since these parameters were not estimated in this thesis, the tuning parameter ai were
introduced to be able to tune the rates without changing the kinetic parameters. The unit of ai
is [molm3] and represents the amount of active sites in the TWC participating in the reactions.
3.1.3
Oxygen Storage Modelling
The oxygen storage property of the TWC is very important, especially for control purposes. Therefore it is important that the model of this attribute is modelled correctly. Since the model is discrete in axial-direction, an assumption made is that the oxygen storage capacity in the first slice has to be close to 1 before there can occur any leakage to the next slice. This assumption is based on the knowledge collected at Volvo Cars during different measurements monitoring the exothermic reactions in the catalyst. This property has been pursued and the model has been tuned to satisfy this assumption. Doubling the number of slices along the models axial-direction would automatically result in slices of half their original size, meaning double the rate of OBL increase and decrease per slice for identical circumstances.
The change in OBL (Θ) can be expressed as the sum of rates R10-R15 seen in (3.20), which is integrated to obtain the oxygen storage level. Note that (3.20) is a modified version of the oxygen storage model used in Ramanthan and Sharma due to modifications of (3.17) and (3.18) [4]. Since there is a maximum oxygen storage capacity the sum of the integral is divided by to total active mole sites in the catalyst per m3. Θ is then saturated between extreme values of
zero and one. The parameter aj is the total mole sites per m3 in the catalyst and is a tuning
parameter that is individually unique for each catalyst. To obtain the current active sites in the catalyst, aj has to be multiplied with the catalyst volume. This parameter is denoted Θden.
dΘ dt =
V Θden
((4R10a10+ 2R11a11) − (2R12a12+ 18R13a13+ 20R14a14+ 2R15a15)) (3.20)
All constants for the rate expressions, the kinetic parameters, has to be estimated which is a tedious work in itself. Therefore in this master thesis, the parameters estimated in [4] were used and can be seen in appendix A in table A.1 and A.2. Compared to [4] reactions (3.17) and (3.18) have been modified increasing the amount of HC consumed and the production of CO2
instead of CO due to problems with CO being produced in slice one which reduced the OBL in slice 2. This caused problems for control and the reactions were modified. This might not be chemically correct in terms of reactions occurring, but since the desired behaviour was that the slices should be completely leakage free these reactions were modified.
3.1.4
Temperature Model
Temperature development throughout the one-dimensional model is simulated using a modified pre-existing temperature model provided by Volvo Cars with some slight modifications added to it, which is also modelled as a state. Addition of the exothermic heat reactions is simple by multiplying the reaction rates with a corresponding heat of reaction constants (H). The temperature is modelled as temperature in the gas and in the solid. The main equations can be seen in (3.21) and (3.22). dQgas dt = dQgasf low dt + dQconv,solid−gas dt (3.21)
dQsolid dt = dQconv,solid−gas dt + dQconv,solid−ambient dt + dQenthaply dt (3.22)
For the gas temperature model the expressions in (3.21) can be seen in (3.23) - (3.25), where mg is the gas mass, cp,s and cp,g is the specific heat capacity for the solid and gas, hsg and
hsa is the heat coefficient between the solid and gas and solid and ambient respectively. The
heat-technical area between solid, gas and ambient are asg and asa
dQgas dt = mgcp,gdTgas dt (3.23) dQgasf low dt = ˙mcp,g(Tin− Tout) (3.24) dQconv,solid−gas dt = hsgasg(Tgas− Tsolid) (3.25) For the solid temperature model seen in (3.22) the expressions can be seen in (3.26) - (3.28). where ∆Hj is the heat value for the exotermic energy from reaction j.
dQsolid dt = mscp,sdTsolid dt (3.26) dQconv,solid−ambient dt = hsaasa(Tsolid− Tamb) (3.27) dQenthalpy dt = −Σ∆Hjvj (3.28)
The temperature model was not tuned to fit the catalyst material properties and this might effect the results from both the temperature model and the catalyst model. The parameters used were the ones that are used for the GPF (Gasoline particulate filter) at Volvo Cars. The paramters used in the model can be found in Appendix A.3.
3.1.5
Engine Out Model
The engine out model used in this work was developed by our supervisors Anton Ajne and Fre-drik Wemmert at Volvo cars and uses measured lambda from a linear sond in front of the first brick and gives the output of O2, CO, HC and N Ox. The outputs are based on maps obtained
from experiments measuring the engine out emissions. Also the hydrocarbons are measured in carbon atoms. Since the model requires C3H6and C3H8 the output from the model is divided
by 3 and then the remaining output is split into 2 3C3H6
1
3C3H8of the total output. This might
not be the correct ratios but to continue the work it was decided to use this ratio. The amount of H2was also missing in the engine out block which made it necessary to model this separately.
To model the fraction of H2 a model proposed in Auckenthaler which can be seen in (2.19)
where φH2/COis the ration between H2and CO which is assumed to be constant set to 0.35 [1].
Fuel constants α and β are obtained from data sheets and the parameters used can be found in appendix A.4.
CO2 was modelled creating a simple map as a function of lambda input. The data for the
map was retrieved from measurements at Volvo cars. H2O was modelled as constants into the
first brick due to lack of measurements of this species. To improve the model, a map of H2O
could be added. The effects of this would not be significant.
requires the output to be converted to fractions. To convert the fractions to concentrations the equation seen in (3.29) is used where pexh is the exhaust gas pressure, Rg is the ideal gas
constant, Tg is the temperature in the gas and x is the fraction of the species that is to be
converted. C is the converted species in [mol m3].
C = pexh RgTg
x (3.29)
3.1.6
Species Balance Solver
Calculation of (3.1) in the discrete system entails a number of complications regarding order of calculations. The equation is therefore split, first the preliminary concentrations are calculated and thereafter the rates are calculated. Using the calculated rates the new species concentrations is obtained by addition of the chemical reactions net production. Splitting (3.1) allows for faster simulation time and increased model stability when compared to both calculations running in parallel. Cn∗= ( Cn+ (Cin− Cn) ˙ V VTs, ˙ V VTs< 1 Cin, otherwise (3.30) Cn+1= Cn∗+ ΣRj(Cn∗)vjTs (3.31)
Since there is a problem with high mass flows and slow sampling time, an assumption is made that if VV˙ Ts> 1 the volume of the slice is passed through faster than we can sample, the output
of the slice is equal to the input subtracted with the reaction rates.
A modified semi-implicit solver was also tested to increase protection against instability when increasing the sample time. The solver can be seen in (3.32) to (3.34). A dilemma stemming from this semi-implicit solver, is the dependency of the flow rate ˙V and the sampling time Ts.
Due to the nature of the fraction, changes in sampling time and flow rate will severely impact the model state outputs. The implicit solver underestimated the states so that it was chosen to go back to the explicit solver seen in (3.30).
Cn+1∗ = ( Cold+ (Cin− Cn+1) ˙ V VTs, ˙ V VTs< 1 Cin, otherwise (3.32) Cn+1∗ = cold+ cin ˙ V VTs 1 +VV˙Ts (3.33) Cn+1= Cn+1∗ + σR(Cn+1∗ )vjTs (3.34)
3.1.7
Rate Limiter
Complete stability may not be guaranteed through only the solver for (3.30) and (3.31) whilst implementing a solution in the form of a implicit solver for (3.1) is tedious due to cross correlation between species balances due to net production in (3.31). A simple alternative that increases model stability is a rate limiter which hinders excessive consumption of each state and thereby hindering instability. The implemented rate limiter calculates if reaction rates (3.2) - (3.19) for each sample is greater than the actual concentration available of each state. Analysis of each states total consumption through reaction rates is done as following:
Ca,lim=
Ca
ΣRa,jva,j
In case of excessive total state consumption due to high reaction rates, the rates are modified to consume the maximum amount of the corresponding state available.
3.1.8
Brettschneider Lambda
Being able to calculate a theoretical lambda-value throughout the model further enhances the understanding of the TWC. Due to the model being divided in axial direction, Brettschneider lambda-values may allow for predictions of varying OBL (Θ) and may benefit when implementing a control strategy. The Brettschneider lambda is only included for theoretical and pedagogical purposes, though it may be used for control strategies as well and can be seen in (3.36).
λ = CO2+CO2 + O2+N O2 + ((HCV4 3.5+3.5CO CO2 ) −OCV 2 ) (1 +HCV 4 − OCV 2 )(CO2+ CO + (nHC)) (3.36)
3.1.9
Model Reset
To protect from incorrect model estimation of the OBL, a safeguard in the OBL-block was ad-ded. This function consists of a switch in Simulink that if the binary sensor detects either too lean or too rich mixture mid brick, the oxygen storage state is set to respective state. So for instance if the binary sensor detects that the mixture is lean but the model says that we should not leak, the OBL state in slice one to three will be set to one. This prevents us from estimating wrong OBL in the first brick and makes sure that we won’t leak unnecessary emissions even if the model estimates wrong. The same method applies for the rich case, but then the OBL is set to zero if the binary sensor detects too rich mixture. This function does of course need to be cal-ibrated to get it working properly but if the model is to be used in the future it has been prepared.
One drawback is that there is no way to reset the second brick since there in this case is no sensor after brick two. This could cause problems in real applications if the model estimates the states completely wrong. A cure to fix this issue would of course be to add a secondary sensor behind brick two. Then the same functionality could easily be implemented on brick two as well. This drawback is however present in contemporary modelling and control of a TWC.
3.2
Matlab and Simulink Implementation
The model is implemented in Simulink both for simulation purposes and for code generation for easy implementation and testing. There exists two versions of the model that has been de-veloped, one model consisting only of Simulink compatible blocks and one model that consists of TargetLink blocks for easy code generation. The purpose of the Simulink model is for easy val-idation and simulation, and the TargetLink model is focused on code generation at Volvo Cars. The output of the models are exactly the same, yet they are constructed using different blocks due to limitations of code generation in TargetLink. Here only the Simulink model will be ex-plained. Due to the massive amount of subsystems in the model, not all blocks will be exex-plained.
In Figure 3.3 the top view of the implemented Simulink model can be seen. The model takes lambda from the linear sensor in front of the catalyst, the engine speed [rpm], the engine load [mgstk], the mass flow [kgs] and the exhaust temperature [K] after the turbocharger. The model outputs the OBL in each slice, engine out emissions, tailpipe emissions, calculated Brettschneider lambda in slice three of brick one and the temperature in each brick. The model is set up for the catalyst seen in table 5.2 with three slices in the first brick and two slices in the second brick. The controller takes a reference, the oxygen storage state of each slice and the binary sensor as input and outputs a target lambda. The target lambda should not enter the model directly as it is implemented in Figure 3.3, but instead be the requested target lambda for the ECU in a real vehicle application. However for simulation purposes we can in the ideal case set the target
lambda as the input to the model.
Figure 3.3: The top view of the Simulink model of the Three-way Catalyst
The sub level of the TWC block can be seen in 3.4 and consist of many subsystems. This level consists of 9 subsystems which will be explained further. In Figure 3.5 the first three blocks of the TWC model can be seen. The first block contains the engine out model which mostly consists of mapped values from emissions measurements. The next block is a simple conversion from fractions to [molm3]. Note that the exhaust pressure is assumed constant in the model. Slice
one is where the states, rates and temperature is calculated. The subsystems slice one to slice five are the same inside except constants for the volume and oxygen storage capacity due to the fact that the model represents two bricks.
Figure 3.4: The view inside the TWC block.
Figure 3.5: The three first subsystems inside the TWC model subsystem.
The last block in the TWC model is a model of the binary lambda sensor which only is a lookup table taking the calculated brettschneider lambda from slice three in brick one as input and
Figure 3.6: Lambda sensor model and concentration to fractions conversion blocks in the TWC model subsystem.
In Figure 3.7 the view inside of a slice subsystem can be seen. Here all the concentration states, OBL, temperature and rates are calculated. The two Matlab functions in the block is the rate limiter mentioned earlier and the Brettschneider lambda calculation. The code for the rate lim-iter can be seen in appendix B. Due to the massive amount of subsystems any lower level than this will not be explained in detail. All calculations described in the model description in 3.1 are calculated in this block.
Figure 3.7: Lambda sensor model and concentration to fractions conversion blocks in the TWC model subsystem.
Controller
The main purpose during control of an automotive exhaust system would be avoiding leakage in the form of tailpipe emissions. Multiple control strategies may be used to obtain the same result ranging from commonly used bang-bang control to model based control strategies. Common amongst all strategies is keeping lambda values near stoichiometric if possible regarding driving conditions and varying lean and rich AFR to obtain cleaner emissions. The control signal of the controller is a reference lambda value and the reference values used is desired OBL. The feedback signal from sensors are the linear lambda sensor and the binary sensor before and after the first brick respectively.
4.1
PID Controller
Controlling the TWC-model is not completely straight forward and depends on what behaviour is desired. In this thesis a series of five PID-controllers are implemented, one for each slice of the model. These controllers are in turn switched on and off through a number of IF-statements deciding if the error is small enough. Due to physical limitations and the propagating unidirec-tional behaviour of the exhaust gas throughout the TWC, the last slice must be allowed to reach it’s reference point primarily before the previous slice may be controlled. When all slices are sufficiently near their corresponding reference points, disturbances may occur due to emission slips between the various slices. Decisively concluding a reference point for each of the slices is not straight forward and may vary. As a result of varying engine out emissions in the form of different engine speeds, mass flows and loads, one single tune for a PID-controller is not enough. This may be solved by using a varying controller gain which takes variations in engine speed, mass flow and load in to account before controlling the oxygen storage level of each slice.
The control structure used is a PID controller with adjustment to the integral of the controller found in [27]. The control structure consists of a serial PID control structure which can be seen in (4.1). Where index k stands for value in current sample and index k − 1 stands for value from previous sample. In (4.2) the adjustment factor to the integral part is calculated. Note that the calculation in (4.2) is done in the previous sample. As can be seen the sampling time is a part of this control structure due to the fact that the controller is implemented in discrete time. All five controllers have separate Kp-, Ti- and Td-parameters although the integrator part (4.2)
accumulates total value from previously active controllers to the current active controller.
vk = Kpek+ Ik−1+ Kp T s T iek+ Kp Td Ts (ek− ek−1) (4.1) Ik−1= Ik+ Ts Tt (uk− vk) (4.2)
