Reduced Order Model Design for Three Way Catalytic Converter Temperature Dynamics

Transcription

1 2017 IEEE Conference on Control Technology and Application (CCTA) Augut 27-30, Kohala Coat, Hawai'i, USA Reduced Order Model Deign for Three Way Catalytic Converter Temperature Dynamic Romit Godi 1, Student Member, IEEE and Simona Onori 2, Senior Member, IEEE Abtract Thi paper preent a reduced order model for three way catalyt (TWC) temperature dynamic developed uing Proper Orthogonal Decompoition (POD) and Galerkin projection method. The TWC thermal dynamic are decribed uing a et of coupled, non-linear partial differential equation (PDE). The Galerkin projection method i applied to the PDEmodel to contruct a et of ordinary differential equation (ODE) to decribe the dynamic of the ytem. POD extract a low dimenional bai function from a high dimenional data et obtained by numerical imulation of the ytem. Thi i achieved by performing ingular value decompoition (SVD) on the data et and electing the bai function correponding to the mot dominant eigenvalue of the ytem. Uing the POD bai function along with Galerkin projection lead to a low order ODE ytem that capture the ytem dynamic decribed by the phyic baed model. The reult obtained indicate that the a three tate reduced-order model obtained uing POD-Galerkin i able to capture the thermal dynamic with atifactory accuracy. The development of the newly propoed control oriented model for TWC thermal dynamic i carried out uing experimental data collected from a TWC running on the Federal Tet Protocol (FTP) driving cycle. NOMENCLATURE TWC Three Way Catalyt ODE Ordinary Differential Equation PDE Partial Differential Equation OSC Oxygen Storage Capacity λ Normalized air fuel ratio [ mol m 3 t Time [ z Spacial coordinate [m T g Ga temperature [K T cat TWC olid phae temperature [K T exh Exhaut ga temperature [K T cat0 Initial temperature [ [K Exhaut ga ma flow rate kg ṁ exh ρ g Exhaut ga denity [ kg m 3 L TWC length [m A out TWC external urface area [m 2 V cat TWC volume [m 3 Q react η Heat produced by reaction TWC efficiency [ W m 3 1 R. Godi i with the Department of Electrical and Computer Engineering, Clemon Univerity, SC, 29634, USA rgodi@clemon.edu 2 S. Onori i with the Department of Automotive Engineering, Clemon Univerity, Greenville, SC, 29607, USA onori@clemon.edu I. INTRODUCTION Automotive indutry ha been triving to improve vehicle efficiency and meet the alway more tringent emiion tandard requiring to governing air pollutant releaed into the atmophere [1, [2. In order to meet the trict emiion tandard, advance have been made toward both developing advanced engine technologie a well a through powertrain electrification [2. A the deign complexity of automotive engine increae, automaker have been challenged to develop efficient after-treatment ytem to control and reduce harmful exhaut gae and particulate matter. Common and harmful exhaut gae under regulation include carbon monoxide (CO), nitrogen oxide (NO x ), and hydrocarbon (HC). A three way catalyt i the mot widepread emiion control device ued with internal combution engine in modern vehicle [3. It reduce imultaneouly (NO x ) and oxidize HC and CO with an efficiency cloe to 99% when the device operate under predefined condition, namely, engine operation near toichiometry and ufficiently high temperature [4. Developing phyically-meaningful control oriented model of uch device i the key enabler for improving the device efficiency through the development of robut control and diagnotic trategie. The focu of thi paper i the deign of a reduced (i.e., control-oriented) model to predict temperature dynamic inide the TWC. Fig. 1: Schematic view of a TWC (reproduced from [5). A three way catalytic converter i redox (reductionoxidation) reaction-baed device in which the harmful pollutant are converted to le toxic emiion. The efficiency with which the redox reaction occur highly depend on a quantity called air-fuel ratio (AFR). Thi i the ratio between the ma of air to fuel during combution proce. The AFR i uually denoted in term of it normalized /17/$ IEEE 886

2 value given a: where λ = = AFR (actual) AFR (toichiometric) m air /m fuel AFR (toichiometric) m air = ma of air m fuel = ma of fuel { 14.7 for gaoline AFR (toichiometric) = 9 for ethanol If the AFR i within a narrow region around the toichiometric value, the chemical reaction occur with higher efficiency, i.e. the ucce of complete reduction and oxidation reaction i the highet. The efficiency decreae when the AFR fall out of thi range. The region outide of the toichiometric value are claified a lean and rich condition. When the AFR i le than the toichiometric value, the combution occur in rich condition (λ < 1). In thi cae the oxygen i utilized during fuel combution and there in't ufficient amount left to carry out the oxidation reaction inide the catalyt. When the AFR i greater than the toichiometric value, the engine operate under lean condition (λ > 1), i.e there i an exce of oxygen available inide the catalyt, hence the reduction reaction doe not occur completely. The catalytic converion efficiency alo depend on the temperature inide the TWC. Only when the catalyt temperature ha reached a threhold level, called light-off temperature, the chemical converion take place with an efficiency cloe to 50%. Therefore the catalyt temperature dynamic, play an imperative role in the control of harmful emiion. The preent technology doe not allow for uch kind of meaurement in real-time application. Hence the ability to monitor and predict the catalyt temperature along the flow direction i key to deign an efficient exhaut ytem control trategy. The common approach to model development to capture the TWC dynamic relie on either phyic baed or empirical model a in [6, [7, [8 and [9. The drawback of developing a phyic baed model i in that the reulting Partial Differential Equation (PDE)-baed repreentation i computationally unfeaible to etimate the real time behavior of the tate variable. When it come to empirical model, a for example the one propoed in [10, the dependence of the converion efficiency on the catalyt temperature i neglected, which make their accuracy a not guaranteed acro driving cycle and engine condition. Thi paper propoe a reduced order control-oriented model which not only capture the TWC dynamic efficiently but i alo computationally feaible for online implementation. The POD baed Galerkin' projection follow a bai function repreentation of the ytem obtained by performing ingular value decompoition (SVD) on the matrix containing ytem generated output data. The reulting model (1) (2) imulate the tate variable of the TWC ytem acro variou driving cycle. To the bet of our knowledge, a rigorou TWC control-oriented model deign uing formal reduction method applied to an experimentally validated phyic-baed ytem ha never been invetigated. The ret of the paper i organized a follwow: Section II dicue the phyic-baed PDE model of the TWC. In Section III, the POD-Galerkin' projection method for reduced order modeling (ROM) i preented. In Section IV, the imulation reult of the reduced model are preented. Section V give the concluion and cope for future work. II. THREE WAY CATALYST MODEL The dynamic of the TWC are expreed a a et of coupled nonlinear PDE [7 decribing the temperature profile of the exhaut ga and the catalyt ubtrate. A. Thermal Model The thermal behavior of the TWC follow a one dimenional approach, and can be decribed by carrying out energy balance on the exhaut ga and on the olid phae. The ga phae comprie of the exhaut ga flow coming from the engine and flowing into the catalyt and the olid phae i repreented by the ubtrate and the wahcoat lumped together [11. The partial differential equation decribing thee dynamic are given a follow: Energy balance of the ga phae The PDE model decribing the energy balance of the ga phae i given by [9: ρ g ɛ c pg T g = ṁexh c pg T g A c z + h A geo (T cat T g ) where Tg repreent the time dynamic of the ga phae, ṁ exh T A c c g pg z, decribe the convective heat tranport in the axial dimenion and ha geo (T cat T g ) i the heat exchange between the ga and the olid phae. Conidering the ga phae dynamic to be changing rapidly in comparion to the olid phae, we aume the ga temperature T g to be T at teady tate. Therefore, the term ρ g ɛc g pg i neglected. The implified equation i: ṁ exh A c (3) c pg T g z = h A geo (T cat T g ) (4) The boundary condition for the above equation i: T g (z = 0, t) = T exh (t) (5) where the exhaut temperature T exh, meaured at the input of the catalyt, i conidered to be the initial ga temperature. 887

3 Energy balance of the olid phae The energy balance of the olid phae i decribed by: ρ (1 ɛ) c T cat where the term Tcat olid phae, (1 ɛ)λ 2 T cat = (1 ɛ) λ 2 T cat z 2 h A geo (T cat T g ) + Q react A out V cat h out (T cat T amb ), repreent the time dynamic of the z, account for the conduction 2 loe in the ubtrate, ha geo (T cat T g ) repreent the heat exchange between the ga and the olid phae, Q react i the heat produced by the exothermic reaction in the wahcoat. Finally the term Aout V cat h out (T cat T amb ), account for the radial loe in the ambient phae. The term Q react decribe the heat dynamic due to the chemical reaction occurring in the catalyt and i a function of the converion efficiency. It dependency on the exhaut ma flow rate i valid under toichiometric condition [9. It can be expreed a follow: (6) Q react = K react ṁ exh η(t cat ) (7) The contribution of the efficiency term in (7) i ignificant only above a certain threhold temperature value called T lightoff and i generally expreed through a hyperbolic function of the catalyt temperature a [12: η(t cat ) = 0.5 tanh((t cat T lightoff )) (8) where i the lope of the efficiency curve. The term A out and V cat in (6) which repreent the TWC external urface area and volume, repectively, are expreed in term of it diameter by replacing their ration with the term 4 D cat. The olid phae equation become: ρ (1 ɛ) c T cat The initial condition for (6) i given a: B. Implementation = (1 ɛ) λ 2 T cat z 2 h A geo (T cat T g ) + Q react 4 D cat h out (T cat T amb ) (9) T cat (z, t = 0) = T cat0 (z) (10) The TWC phyic-baed thermal model i imulated uing a the Finite Difference Method (FDM). Although uing FDM a a tandalone modeling technique doe allow for temperature monitoring, the mall tep ize for pacial and time domain reult in a large ytem of ODE which i not deirable for real time implementation. Decreaing the tep ize reduce the number of ODE, but reult in large error in comparion to meaurement value. Equation (4) and (9) are dicretized with uing 40 cell dicretization tep ize over the pacial coordinate and Parameter c pg ɛ λ ρ c h h out K reac Decription Specific heat [ of the exhaut ga J (kg K) TWC open cro ectional area [ m 2 TWC olid [ phae conductivity W (m K) Volumetric heat capacity of the catalyt [ J m 3 K Convective heat tranfer coefficient [ W m 2 K Heat tranfer coefficient with ambient [ W m 2 K Scaling factor for Q reac term Slope of the TWC efficiency curve function T lightoff Light-off temperature [K T amb Ambient temperature [K A c TWC cro ectional area [ m 2 A geo D cat L cat TWC pecific geometric area [ m 1 TWC diameter [m TWC length [m TABLE I: Lit of parameter of the TWC model [9, [13. econd a time tep. The input ignal to the TWC are the exhaut temperature T exh, exhaut ga ma flow rate ṁ exh, and AFR λ pre meaured at the inlet of the catalyt uing the enor layout depicted in Figure 2. The input ignal are hown in Figure 3 over the FTP driving cycle. The dicretized equation are written a: dt j cat dt T j g = = λ ρ c T j1 g ( T j+1 cat + h Ageo Ac z ṁ exh c pg 1 + h Ageo Ac z ṁ exh c pg T j cat 2 T j cat + T j1 cat z 2 ) Q react (11a) h A geo (T j cat Tg j ) + ρ (1 ɛ) c ρ (1 ɛ) c 4 h out (T j cat T amb ) D cat ρ (1 ɛ) c (11b) Figure 4 how the olid phae temperature variation imulated uing FDM over the FTP driving cycle. In order to olve the ytem PDE, the parameter in it have been previouly identified uing a Particle Swarm Optimization algorithm [15. The validated model i found in [9. 888

4 0 z Fig. 2: TWC enor placement (reproduced from [14) Time [ Fig. 3: Input to the TWC model over FTP cycle (reproduced from [13). III. REDUCED MODEL USING POD-GALERKIN In thi ection, we develop a reduced order model of the thermal dynamic of a TWC ytem uing POD-Galerkin projection approach. POD, which i alo referred to a Karhunen-Loève decompoition (KLD) or principal component analyi (PCA) [16, i a numerical technique that generate bai function from ytem output data [17. The input to the POD method i a data matrix of the thermal model, in time and pacial domain, computed through imulation over a certain time period and the output i a et of time independent function that repreent the mot energetic mode of the original ytem. The bai function are generated by performing SVD on the data matrix over which the minimum number of mode required to recontruct the ytem i elected. A. Singular Value Decompoition The data matrix generated from the FDM imulation of Figure 4 i the naphot matrix S nap, containing value from the ytem output in time and pace domain. SVD i performed on thi matrix reulting in the following matrix decompoition, SV D(S nap ) = UΣV T (12) Fig. 4: Simulation reult uing FDM over FTP driving cycle for T cat where (a) U = [ ψ 1 ψ 2... ψ z R zxz V = [ v 1 v 2... v t R txt σ Σ = 0 σ Rzxt σ z (13) U and V are orthonormal matrice, and Σ i a diagonal matrix containing real poitive ingular value σ of S nap matrix arranged in decreaing order σ 1 σ 2... σ z 0. The matrix dimenion z i the number of dicretized cell along the length of the catalyt, which in thi cae i 40. The dimenion t i the number of ample in the driving cycle which in thi cae i 3,00,000, conidering a FTP driving cycle of 1500 econd ampled at econd time tep. The bai function are contained in the matrix U, and atify the propertie of orthogonality and orthonormality defined a: { ψi T 1 if i=j ψ j = (14) 0 if i j where ψ i (z) = pace dependent bai function. B. Formulating POD bai vector The number of mode in the bai function define the order of the reduced model. Since the ingular value in the Σ matrix are in decreaing order, the firt few value are more dominant in the ene that they capture mot of the ytem dynamic. Hence, a heuritic criterion i applied for electing the optimal number of bai mode uch that the error between the actual dynamic (from the PDE model) and the reduced one i being minimized. A common approach of electing the number of mode i by defining an energy criterion a in [18: O n = n i=1 σ2 i N i=1 σ2 i (15) 889

5 where O n = truncation degree N = total number of mode n = reduced number of mode (16) Thi ratio i ued in thi work to determine the order of the reduced ytem model. O n need to be cloe to 1 for the bai function to recontruct the approximate olution accurately. Thi condition i applied to the ingular value in the Σ matrix. In order to etup a reduced order model, we et the energy criterion uch that the number of mode choen hould capture 99% of the ytem dynamic. From the plot we conclude that for the temperature dynamic n=3, number of mode atify thi criteria for FTP the driving cycle Number of mode Fig. 5: Truncation degree plot for T cat over FTP driving cycle. C. Applying Galerkin' projection The Galerkin' projection approach look toward developing an approximate olution of a PDE. An exact olution of PDE ytem i complex due to non-linearity and nonhomogeneou boundary condition. Hence finding an approximate olution with guaranteed accuracy turn out to be a very ueful approach. A bai repreentation of the ytem PDE i found by plitting a continuou function (in pace and time) into a dicrete ummation of the two quantitie eparated. To find the approximate olution, the inner product of the PDE and a weight function i computed over the pacial domain and equated to zero. Thi enure that the approximate olution ha the leat error. The reduced form of the TWC ytem i derived by applying the Galerkin' projection on the POD bai derived in the previou ection. The ytem variable under conideration in the thermal model i the catalyt temperature. Let u define an approximate olution for T cat a [19: n T cat (z, t) = x i (t) ψ i (z) (17) i=0 = ψ n x T (t) where x T (t) = [ x 1 x 2... x t R nxt ψ 1 ψ 2 ψ(z) =. (18) Rzxn ψ z x T (t) are the time dependent fourier coefficient [20 and ψ i (z) are the pace dependent bai function. The next tep i to define the reidual function R, a: R( T cat, T cat) = d T cat λ d2 Tcat dt ρ c dz 2 h A geo + ( ρ (1 ɛ) c T cat T g) Kreact ṁ exh (0.5 tanh(( T cat T lightoff )) ) ρ (1 ɛ) c 4 h out + ( D cat ρ (1 ɛ) c T cat T amb ) (19) To olve for the approximate olution, we take the inner product of the reidual and weight function. The weight function i choen imilar to the bai function a: R( T cat, T cat ), ψ j (z) = 0 (20) Subtituting the approximate olution of T cat, we can implify the above expreion a: ψn ẋ T (t), ψ j (z) = f( T cat, u cat, t), ψ j (z) (21) where, ( ) f( T cat, u cat, t) = λ d 2 ψ n ρ c dz 2 xt h A geo (ψ n x T T g) ρ (1 ɛ) c + Kreact ṁ exh (0.5 tanh((ψ n x T T lightoff )) ) ρ (1 ɛ) c 4 h out (ψ n x T T amb ) D cat ρ (1 ɛ) c u cat = [ T exh ṁ exh (22) contain the non-linear term of (19). Uing the property of orthogonality and orthonormality from (14), and olving for the inner product, the reduced order non-linear model can be written a: ẋ T = ψ T f( T cat, u cat, t) (23) We can expre the above equation in term of the catalyt temperature by multiplying both ide with ψ n a follow: T cat = f( T cat, u cat, t) (24) IV. SIMULATION RESULTS The control-oriented model i compared againt the temperature dynamic imulated from the center location of the TWC over a FTP driving cycle. The olid phae temperature T cat imulated by the control oriented model developed uing POD-Galerkin uing three tate i compared to the high 890

6 dimenional PDE-baed ytem in Figure 6. The reduced model i able to capture the thermal tranient with atifactory accuracy. Tcat [K Meaured Simulated Simulated Galerkin 600 Time [ Fig. 6: T cat plot for meaured v/ FDM imulated (top) and FDM imulated v/ Galerkin (bottom) value at the center of the catalyt brick. V. CONCLUSION In thi paper, a control oriented non-linear model i developed for the TWC ytem with reduced number of mode uing the POD-Galerkin projection approach. The bai function for POD are obtained from a FDM imulated model of the original PDE ytem. Galerkin projection i then applied to arrive at a control oriented model equation. The truncation degree i plotted to elect the order of the control model developed by conidering the mot dominant mode of the ytem. From the imulation reult, the control-oriented model capture the thermal dynamic with atifactory accuracy. [9 S. Sabatini, I. Kil, J. Dekar, T. Hamilton, J. Wuttke, M. A. Smith, M. A. Hoffman, and S. Onori, A New Semi-Empirical Temperature Model for the Three Way Catalytic Converter, IFAC Workhop on Engine and Powertrain Control, Simulation and Modeling E-COSM, vol. 48, no. 15, pp , [10 M. Soumelidi, R. Stobart, and R. Jackon, A Nonlinear Dynamic Model for Three-Way Catalyt Control and Diagnoi, SAE Technical Paper Serie, [11 R. M. Heck, R. J. Farrauto, and S. T. Gulati, Catalytic air pollution control: commercial technology. John Wiley & Son, [12 S. B. Kang, S. J. Han, I. C. Nam, B. K. Cho, C. H. Kim, and S. H. Oh, Detailed reaction kinetic for double-layered Pd/Rh bimetallic TWC monolith catalyt, Chemical Engineering Journal, [13 S. Sabatini, S. Gelmini, M. A. Hoffman, and S. Onori, Deign and experimental validation of a phyic-baed oxygen torage - thermal model for three way catalyt including aging, in Control Engineering Practice, [14 S. Gelmini, S. Sabatini, M. A. Hoffman, and S. Onori, Development and experimental validation of a dual extended kalman filter for three way catalytic converter, in American Control Conference, [15 S. Ebbeen, P. Kiwitz, and L. Guzzella, A generic particle warm optimization Matlab function, Proceeding of the American Control Conference, [16 R. Pinnau, Model reduction via proper orthogonal decompoition, vol. 13, pp , [17 W. H. Schilder, H. A. van der Vort, and J. E. Romme, Model Order Reduction: Theory, Reearch Apect and Application. Springer, [18 A. Marquez, J. Oviedo, and D. Odloak, Model reduction uing proper orthogonal decompoition and predictive control of ditributed reactor ytem, Journal of Control Science and Engineering, [19 S. Oh and J. Cavendih, Tranient of monolithic catalytic converter: Repone to tep change in feedtream temperature a related to controlling automobile emiion, Indutrial & Engineering Chemitry Product Reearch and Development, no. 21, pp , [20 D. Luchtenburg, B. Noack, and M. Schlegel, An introduction to the POD Galerkin method for fluid flow with analytical example and MATLAB ource code, in Technical Report, Berlin Intitute of Technology, REFERENCES [1 M. Muratori, H. Khehgi, B. Mignone, L. Clarke, H. McJeon, and J. Edmond, Carbon capture and torage acro fuel and ector in energy ytem tranformation pathway, International Journal of Greenhoue Ga Control, vol. 57, pp , [2 EPA. air-pollution-current-and-future-challenge. [3 E. Brandt, Y. Wang, and J. Grizzle, Dynamic modeling of a three-way catalyt for i engine exhaut emiion control, IEEE Tranaction on Control Sytem Technology, pp , [4 A. Faiz, C. S. Weaver, and M. P. Walh, Air pollution from motor vehicle: Standard and technologie for controlling emiion, World Bank Publication, [5 PaionInChemitry/Automobile/. [6 P. Kiwitz, C. Onder, and L. Guzzella, Control-oriented modeling of a three-way catalytic converter with obervation of the relative oxygen level profile, Journal of Proce Control, vol. 22, no. 6, pp , [7 T. Shamim, H. Shen, S. Sengupta, S. Son, and A. Adamczyk, A comprehenive model to predict three-way catalytic converter performance, Journal of engineering for ga turbine and power, vol. 124, no. 2, pp , [8 C. Ngo, D. Koenig, O. Sename, and H. Béchart, A reduced model of Three Way Catalyt converter and tored oxygen rate etimation uing witched oberver, 12th biannual European Control Conference,