A control-oriented model for cold startoperation of spark ignition engines

Chris Manzie ∗ Farzad Keynejad ∗ Denis I. Andrianov ∗Robert Dingli ∗ Glen Voice ∗∗

∗Department of Mechanical Engineering, The University ofMelbourne, VIC 3010, Australia (e-mail: manziec@unimelb.edu.au,

keynejad@unimelb.edu.au, denisa@pgrad.unimelb.edu.au,rdingli@unimelb.edu.au ).

∗∗ Ford Motor Company of Australia, (e-mail: gvoice1@ford.com)

Abstract: The use of mean value models in replicating the key characteristics of automotivepowertrains has been well established. There has been considerable success in the applicationof these models to controller design, with improved emissions and performance the primarybenefits. However, these low order models typically must make certain assumptions about theengine - with constant temperature operation a standard approach. As economy and emissionsat every point in the drive cycle become scrutinised, the cold start operation of the enginebecomes more critical and the constant temperature assumption is limiting. This paper seeksto develop an engine model framework for capturing the temperature transients and gaseousconcentrations throughout the engine. It is intended for use in controller design and optimisationstudies that incorporate the engine warm up period.

Keywords: Engine modelling, engine systems, reduced-order models, automotive control,engine emissions

1. INTRODUCTION

The characteristics of automotive engines following a coldstart make this region of engine operation one of themost difficult. Fuel used to overcome engine friction atidle is typically two to three times higher relative to afully warm engine, while emissions will be high until thethree way catalyst has lit off requiring high exhaust gastemperatures among other considerations. As emissionsand fuel consumption requirements become increasinglystrict, it is apparent there is a need to model enginecharacteristics during a cold start transient.

In general, the approach favoured by many control andoptimisation researchers is to use low order models thatsufficiently capture the basic physics to reduce the needfor significant storage of black box relationships. Thedevelopment of low order models for automotive enginepowertrains began in the 1970s with the work of Hazelland Flower (1971), but it was not until the 1980s thatthe potential to use these models as an effective simula-tion and control tool became apparent in work such asthat conducted by Dobner (1980), Moskwa and Hedrick(1992), and Cho and Hedrick (1989). In the early 1990s,these types of models were termed ’mean value models’by Hendricks and Sorenson (1990) and typically based onsimple physical models with approximations that includedtreating the pulsating flows through the engine as contin-uous flows averaged over an entire engine cycle and theassumption of constant temperature operation.

The success of these models in dealing with various power-train control strategies is well established with a multitude

of examples in the literature. However, there are severallimitations that preclude the use of existing models incold start studies. Most critically, of course, is the typ-ical absence of temperature as an engine variable. Theremoval of a constant temperature assumption requiresthe addition of several new states within the model, butis necessary in order to capture thermal transients. Thepioneering work described in Kaplan and Heywood (1991)and Sandoval and Heywood (2003) form an excellent basisin this domain.

Furthermore, as pre-catalyst emissions become an impor-tant consideration for catalyst light off and modelling, aframework to incorporate species concentrations withinthe low order model is required along with the exhaustgas temperature. Previous attempts in this regard havetypically been of high order or are phenomenologicallybased, e.g. Fiengo et al. (2003).

The aim of this work is to extend the mean value modelsthrough the inclusion of thermal characteristics, as well asproposing a framework for modelling gas concentrationsthrough the engine. The model developed is intendedto serve as a validated starting point for further modelreduction to be used in controller design and optimisationstudies. To ensure portability, there is a high emphasis onphysics-based aspects in the model.

2. ENGINE MODEL

The automotive engine can be split into three intercon-nected physical domains - the mechanical domain wherethe dynamics of engine speed are considered, the thermal

domain describing the temperature dynamics of the fluidsflowing throughout the engine system and the thermody-namic domain containing both the combustion process andthe gaseous pressure and temperature dynamics within thevarious control volumes in the engine. While the main con-tribution of this work is incorporating the thermal domain,for completeness and to clearly demonstrate the inter-domain linkages, the complete model is described in thissection. The overall system being modelled is representedin the block diagram of Figure 1.

Fig. 1. Overall system model

2.1 Mechanical Domain

The crankshaft velocity, ωcrank, is dictated by Newton’ssecond law, with the torques supplied by the individualcombustion processes in each cylinder and load torquesemanating from mechanical friction and brake torque

Jtotalωcrank =Ncyl∑i=1

τcrank,i − τfric − τbrake (1)

Assumption 1. The inertia of the crankshaft, Jtotal, issufficiently large to ensure that the speed fluctuationsarising from individual cylinder combustion events aresmall in comparison to the mean engine speed over thecycle.Remark 1. A standard result of Assumption 1, there is noneed for consideration of individual cylinder combustionevents, and thus a single approximation of indicated torquemay be utilised. Consequently (1) may be reduced to:

Jtotalωcrank = τcrank − τfric − τbrake (2)

The mean torque generated by combustion over an en-gine cycle, τcrank is an output from the thermodynamicdomain, and will be calculated in Section 2.3.

The engine mechanical friction is described in terms ofa mean effective pressure, Pfme, which varies with oiltemperature, Toil, and crankshaft velocity. It is used tospecify the frictional torque acting on the engine andclearly scales with the number of cylinders,Ncyl, and sweptvolume, Vs, of the engine:

τfric =NcylVs

4πPfme(Toil, ωcrank) (3)

The friction mean effective pressure, Pfme, has four sourceterms arising from rubbing losses from the crankshaft,reciprocating components, the valve train and other auxil-iary devices such as the water and oil pumps. Temperaturedependence arises through the viscosity, µ, being a func-tion of oil temperature. A structure for these equations

was developed and validated by Sandoval and Heywood(2003) on multiple engines and is the model adopted here.For convenience, the equations are repeated below:

Pcfme = c1 + c2

√µ(Toil)µ0

ωcrank + c3ω2crank (4)

Prfme = c4 + c51

ωcrank+ c6

√µ(Toil)µ0

+ c7ωcrank

√µ(Toil)µ0

(5)

Pvfme = c8 + c91

ωcrank+ c10

√µ(Toil)µ0

+ c11

√µ(Toil)ωcrank

µ0(6)

Pafme = a0 + a1ωcrank + a2ω2crank + a3ω

3crank (7)

Pfme = Pcfme + Prfme + Pvfme + Pafme (8)

The parameters c1 to c11 are known functions of the enginegeometry (validated for different engines in Sandoval andHeywood (2003)), while a0 to a3 are obtained as part of themodel calibration process for the specific engine of interest.

The brake torque, τbrake is the amount of torque the engineperforms to drive the vehicle through a drive cycle. In ourexperiments, this is simulated by a brake dynamometer,which can be run either in a torque-tracking or a speed-tracking mode and the throttle angle α adjusted throughclosed loop control, i.e. if the dynamometer is run in torquetracking mode with a desired drive cycle torque, τdrive:

α =(Kp +KI

∫+Kd

d

dt

)(τbrake − τdrive) (9)

A similar equation to (9) penalising speed deviations maybe used for the throttle if the dynamometer is run in aspeed-tracking mode.

2.2 Thermal domain

This domain is used to describe the various heat flowsin the engine, with source terms being the heat gener-ated by mechanical friction (arising from the Mechanicaldomain) and the heat generated by combustion (arisingfrom the Thermodynamic domain). Ambient conditionsact as a heat sink. The temperature of each of the enginecomponents and fluids must be modelled in the thermaldomain.Assumption 2. The head, block, piston and liner of theengine can be treated as lumped masses.Remark 2. The actual masses experience temperature gra-dients as a result of proximity to the heat source (frictionor combustion). The lumped mass temperature will there-fore not necessarily be indicative of the mean temperatureacross the entire body, but the mean temperature of thebody at the point of contact with the coolant or oil. Thissubtle difference is taken into account in the procedure forcalibrating the heat transfer coefficients in Section 3.

Assumption 3. The coolant and oil are treated as lumpedfluids existing in the block and head (coolant) and thehead, block and sump (oil).Remark 3. As with Assumption 2, there will still be tem-perature gradients within the fluid in practice. The pos-sibility of local boiling is not considered under this as-sumption, as it is assumed the flow velocity of the fluids,which is set by the pump characteristic and typically afunction of engine speed, is sufficient to avoid this effectoccurring. Despite the lumped fluid assumption there isstill enthalpy flow as will be described in the system’sdynamic equations. The temperature of the fluid is thesame as the outlet temperature for each of the volumes.Assumption 4. The thermostat remains closed during en-gine warmup.Remark 4. Since the thermostat is closed there is noneed to model the entire coolant circuit during thermaltransients, and thus only the head, block, piston, linerand the fluid flows within them are considered relevant.Once the engine operating temperature reaches steadycondition, the whole thermal behaviour of the engine canbe neglected with the exception of dynamics associatedwith the opening and closing of the thermostat valve - i.e.a constant temperature model such as the one described inCho and Hedrick (1989), becomes a valid approximation.

From Assumptions 2 to 4 there are nine states in the enginecorresponding to the temperatures of the four lumpedmasses (head, Th, block, Tb, piston, Tp and liner, Tl);coolant flowing through the block, Tc,b and the head Tc,h;and oil flowing through the block, To,b the head, To,h, andremaining in the sump, To,s.

Each of the nine temperature states in the thermal modelhas dynamics dictated by the standard heat capacitancerelationship:

CiTi = Qi,in − Qi,out (10)where the total heat capacitance of each lumped elementis the product of its mass and the specific heat capacity,i.e. Ci = micp,i.

Naturally, the heat flowing out of one state flows intoanother state (or exits the system to the environment).The thermal flow diagram presented in Figure 2 depictsthe interconnections between the lumped parameters inthe thermal domain.

The values of heat transfer, Qi,j , between lumped massesi and j is calculated from the relation

Qi,j = Gi,j(Ti − Tj) (11)

When forced convection takes place between a fluid andsolid, the thermal conductance, Gi,j of (11) is given by theproduct of the contact surface area Ai,j and the convectiveheat transfer coefficient, hi,j :

Gi,j = hi,jAi,j (12)

Furthermore, the heat transfer coefficient is well describedby the Nusselt correlation (see e.g. Bohac et al. (1996))in terms of kinematic fluid viscosity, ν, mean velocity, v,of the fluid, and the characteristic diameter, D and fluidthermal conductivity, k, as:

hi,j = ack

D

(vD

ν

)bc

(13)

Head

Combus,onOilinblock

Coolantinblock

Coolantinhead

Liner

Piston

Block

Oilinhead

Oilsump

Ambient

Ambient

Ambient

Auxiliaryfric,on

Recip.fric,on

Cranksha@fric,on

Valvetrainfric,on

Fig. 2. Flows in the thermal domain of the cold start enginemodel. The rounded boxes represent lumped masseselements, the square boxes represent fluids and thetrapezoids represent heat sources.

Parameters ac and bc must be identified by experiment.Since the kinematic fluid viscosity is a function of tem-perature, the heat transfer coefficient also maintains sometemperature dependance. Note that for in-cylinder heattransfer, the characteristic diameter is the cylinder bore,and the mean velocity is calculated from the mean massflow through all cylinders.

If the heat flows to the environment, then a radiation termwith conductance coefficient, Gr = Ai,ambεσ, is added sothat the heat flow becomes:Qi,amb = hi,ambAi,amb(Ti − Tamb) +Gr(T 4

i − T 4amb) (14)

Note that while the Stefan-Boltzman constant, σ, is well-defined, there may be some difficulty in specifying theemissivity, ε, precisely.

The heat flow rate of the circulating fluids (i.e. oil andcoolant) is calculated from the net change of enthalpy dueto mass replacement. For a mechanically-driven system,the mass replacement is a function of engine speed accord-ing to the pump’s characteristic, so that this heat flow ratemay be expressed as:

Qfluid = mfluid(ωcrank)cp,fluid [Tinlet − Toutlet] (15)

As an example using Figure 2, applying (15) to the oil inthe block results in:

Qo,b = moil(ωcrank)cp,oil [To,h − To,b] (16)

The losses in mechanical energy due to the friction torquewere described in (3), and represent the first source of heat.This effective power is given by:

Qfric = τfricωcrank

=(NcylVs

4π

)Pfme(Toil, ωcrank)ωcrank (17)

Assumption 5. The heat flow from the cylinder may betreated as a continuous process (i.e. averaged over thefour engine strokes) and furthermore the heat transfercoefficients between the combustion by-products in thecylinder and each of the cylinder walls is the same. Alsothe gas temperatures in the exhaust port and the cylindermay be considered equal.

Remark 5. Given the time scales of the combustion pro-cess relative to the mean value time scales considered in theremainder of the model this approximation is reasonable.

The heat source due to the combustion results in heattransfer from within the cylinder to the surroundinglumped masses, Qcyl, as well as from the exhaust port tothe head, Qport. Using Assumption 5 and the convectionequation (11), the following relationships hold:

Qport(Tcyl, Th) = hexh,hAport(Tcyl − Th) (18)

Qcyl(Tcyl, Th, Tp, Tl) = hcyl,walls(Ap(Tcyl − Tp)+Ah(Tcyl − Th) + Al(Tcyl − Tl)) (19)

The heat transfer coefficient between the gas in the ex-haust port and the head, hexh,h is given in Bohac et al.(1996). Averaged over all four engine strokes, the meanheat transfer coefficient to the cylinder walls, hcyl,walls, isgiven by equation (13), where the characteristic diameteris taken to be the cylinder bore, and the characteristicvelocity is calculated from the mean value mass flow ratethrough the cylinders, mcyl.

The cross sectional areas used in (18) - (19) are takenfor all N cylinders, while the area of the liner, Al, is theaverage area over the combustion stroke.

The derivation of the gas temperature in the cylinder, Tcyl,will be detailed more fully in the next section dealing withthe thermodynamic domain.

2.3 Thermodynamic domain

The thermodynamic domain describes the combustionprocess occurring in the cylinders resulting in the produc-tion of heat and mechanical work required for the thermaland mechanical domains. To model the process completely,it is necessary to include the temperature, pressure andspecies concentrations in the intake and exhaust manifoldsas states of the domain, and the combustion process itself.In modelling this domain it is useful to make some simpli-fying assumptions and define some terminology.Assumption 6. The temperatures and pressures through-out each manifold are constant throughout the volumes.Remark 6. This is a standard assumption in mean valuemodels, and is equivalent to Assumptions 2 and 3 in thethermal domain.Assumption 7. The dynamics in the intake and exhaustmanifolds induced by the in-cylinder events including valveopening and heat release are of a different time scaleto the mean flows. Consequently their effects on enginetemperature, manifold pressures are assumed negligible.Remark 7. Again, this is a standard assumption in meanvalue models and can be rigorously justified using singularperturbation techniques, e.g. Sharma et al. (In press).The significance of the mean value approach is that eachcontrol volume can be modelled using three internal stateequations for pressure, temperature and composition.Assumption 8. The closed loop control of air fuel ratio issufficiently accurate that the engine controller is able todeliver the requested air fuel ratio.Remark 8. This simplifying assumption negates the needfor fuel pooling dynamics to be considered in the model,

and requires the closed loop lambda controller to eitherhave fuel puddle information or a reasonable online esti-mate, see e.g. Choi and Hedrick (1998). The main purposeof this assumption is to allow air fuel ratio to be usedas a control input. Relaxation of this assumption wouldrequire an additional fuel pooling state equation(s), whichare typically a variant of the form originally presentedin Aquino (1981), as well as the fuel injection controllerdynamics to be included in the model.Definition 1. Define xi as an n-vector whose elementsrepresent the relative composition in the volume i of eachof the n-gaseous species considered in the model. The massflow into the volume, mi is also an n-vector representingthe mass flow of each considered gaseous species fromcontrol volume i.Definition 2. Define cP (T ), cv(T ) and R as n-vectorswhose elements represent the specific heat and gas con-stants for each element of xi. Further, let the ratio of dotproducts, γ(x, T ) = cp(T )·xi

cv(T )·xi, represent the ’mean’ ratio of

specific heats in the volume i with temperature, T .Definition 3. For notational convenience, let ||.|| representthe 1-norm of the n-vector.

To begin, the intake manifold state equations can be foundby considering the flows into the manifold (consisting offlow through the injectors, minj , fresh charge, mfc, andexhaust gas recirculation megr) as well as the mass flowleaving the manifold (and entering the cylinder mcyl).Only the element of the minj vector corresponding to fuelwill be non-zero, and the flow rate is specified by the enginecontroller in order to deliver the requested air fuel ratio byAssumption 8.

The mass flow into the manifold from ambient, mfc, dueto internal exhaust gas recirculation, megr, and out of thetailpipe, mtp, are static functions of the throttle angle, α,and pressure differences:

mfc =Pamb√RTamb

Aeff (α)Ψ1(Pamb, Pim) (20)

megr = Ψ2(Pem, Pim, θvo) (21)mtp = Ψ3(Pamb, Pem, Tem) (22)

mfuel =mfc

λ AFRs(23)

The nonlinear functions Ψ1(),Ψ2(),Ψ3() have been previ-ously identified in Heywood (1988), Fox et al. (1993) andShayler et al. (1993) respectively.

The total mass flow from the intake manifold into thecylinders is represented by the well-known speed-densityrelationship, which uses a static mapping of volumetricefficiency ηv(·) to give:

mcyl =(NcylVs

4π

)1

RTimηv(·)Pimωcrank (24)

There are enthalpy and mass flows into the intake manifoldfrom the fresh charge (Eh,fc = mfccP (x, Tim)Tamb andmfc), fuel (Eh,fuel and mfuel) and exhaust gas recircula-tion (Eh,egr and megr). The flow out of the intake manifoldis into the cylinders, Eh,cyl and mcyl, resulting in netenthalpy and net mass flows into the intake manifold givenby:

Eim = Eh,fc + Eh,fuel + Eh,egr − Eh,cyl (25)

Mim = mfc + mfuel + megr − mcyl (26)

In Appendix A of Keynejad (2009) it is shown from thefirst law of thermodynamics, mass conservation and theideal gas law that the following dynamic equations hold inthe intake manifold:

Pim =(γ(x, Tim)− 1) Eim

Vim

+(R · xim

R · xim− cv(Tim) · xim

cv(Tim) · xim

)Pim (27)

Tim =(γ(x, Tim)− 1)Tim

PimVim

×(Eim −

(cv(Tim) · Mim

)Tim

)(28)

xim =RTim

PimVim

(Mim − xim||Mim||

)(29)

The state equations for the exhaust manifold can bedefined similarly, but must include terms reflecting thelosses from the exhaust manifold to the environment,Qem,amb, and the net flows into the exhaust manifoldincludes gain from the combustion process occurring inthe cylinders and losses due to exhaust gas recirculationand the tailpipe:

Eem = Eh,cyl − Eh,egr − Eh,tp (30)

Mem = mcyl − megr − mtp (31)

Consequently, the state equations for the exhaust manifoldare:

Pem =(γ(x, Tem)− 1)

Vem

(Eem − Qem,amb

)+(R · xem

R · xem− cv(Tem) · xem

cv(Tem) · xem

)Pem (32)

Tem =(γ(x, Tem)− 1)Tem

PemVem

×(Eem −

(cv(Tem) · Mem

)Tem − Qem,amb

)(33)

xem =RTem

PemVem

(Mem − xem||Mem||

)(34)

Assumption 9. The rate of change of gas composition, x,in each manifold is assumed negligible.Remark 9. With this assumption, equations (29) and (34)become redundant and the second terms on the righthand side of (27) and (32) disappear. Consequently, themanifolds can be modelled in the thermodynamic domainusing only equations (28), (33), and the following twosimplified versions of (27), (32):

Pim =(γ(x, Tim)− 1)

VimEim (35)

Pem =(γ(x, Tem)− 1)

Vem

(Eem − Qem,amb

)(36)

With the intake and exhaust manifold dynamics modelled,the in-cylinder behaviour can now be considered.Assumption 10. The in-cylinder dynamics can be replacedby static maps.

Remark 10. Torque and heat production as well as compo-sition change all take place within the cylinder, primarilydictated by chemical reaction rates in the order of millisec-onds. Meanwhile, the inertia of the engine is sufficient thatthe mechanical domain is characterised by time constantsof the order of seconds, while the thermal domain has evenslower time constants. The other dynamic equations inthe thermodynamic domain, (27) - (34), are mean valueapproximations and by Assumption 7 will be of the orderof several engine cycles. With this degree of time scaleseparation, a static map approximation to the dynamicequations in the cylinder is well justified.

The control signals for the static maps are valve overlap,θov, and spark advance, θsa, while the influence of theconditions in the bounding volumes must also be reflected,so that the static maps resulting from Assumption 10 havethe form:

τcrank = f1(Pim, ωcrank, xim, θov, θsa) (37)Tcyl = f2(Pim, Tim, xim, θov, θsa) (38)xcyl = f3(xim, Tim) (39)

While (37)-(39) are static maps whose outputs feed intothe other domains, it is useful to impose a physics-basedstructure to these mappings to simplify the calibrationprocess. They can also be easily related to other requiredquantities (such as the temperature and enthalpy of theexhaust gas) using these relations.

A chemical balance can be used to relate the powerentering the cylinder (given as the mass of fuel multipliedby its lower heating value, QLHV ) to the power deliveredas mechanical power (including the power delivered to thecrankshaft and that required to overcome pumping losses,Wpump) and that released as heat. The indicated efficiency,ηi(ωcrank, Pim, x, θsa), is the fraction of mechanical powerto the chemical energy entering the cylinder, and is acalibrated quantity.

ηimfuelQLHV = τcrankωcrank + Wpump (40)

The pumping losses can be written in terms of the pump-ing mean effective pressure, Ppme(Pamb, Pim) defined inSandoval and Heywood (2003) to give:

Wpump =NcylVs

4πPpme(Pamb, Pim)ωcrank (41)

Equations (40) and (41) can be combined to model thetorque delivered to the crankshaft as:

τcrank =ηimfuelQLHV

ωcrank− NcylVs

4πPpme(Pamb, Pim) (42)

To calculate the cylinder temperature, the other side of thechemical balance equation is used, where the heat flows tobe considered are the heat transfer from the cylinder tothe piston, liner, head and exhaust port.

(1−ηi)mfuelQLHV =Qcyl(Tcyl)+Qport(Tcyl)+Eh,cyl (43)

Thus, defining the enthalpy flow Eh,cyl = mcylcPTcyl, itfollows from Assumptions 6-10, (19), (24) and (43) thatthe cylinder temperature is given by:

Tcyl =1

hcyl,walls

∑iAi + hexh,hAport + mcylcP

×(

(1− ηi)mfuelQLHV + hexh,hAportTh

+ hcyl,walls

∑i

AiTi

)(44)

It is worth noting that the mass of fuel entering thecylinder, mfuel, may be calculated using the total massflow rate into the cylinder, mcyl, and the composition, xim.

The final static map representing the effect of combustionon composition, (39), can only be specified when theelements in the vector x are explicitly specified. Fordemonstration purposes, the following assumption allowsa static map for xcyl to be postulated.Assumption 11. Only three gaseous species are assumedpresent throughout the engine and these are regarded asfuel (C8.26H15.5), air (STP concentrations of nitrogen andoxygen) and exhaust (consisting of only CO2 and H2O atstoichiometry). Further, the gaseous species in the cylinderfollowing combustion are assumed functions of air-to-fuelratio only.Remark 11. Assumption 11, along with the usual defini-tion of λ as the ratio of actual and stoichiometric air-to-fuel ratios, allows the following structure for (39) to beproposed:

xcyl =

[xfuel

xair

xexh

]=

1−min(λ, 1) 0 00 1−min(λ−1, 1) 0

min(λ, 1) min(λ−1, 1) 1

xim (45)

2.4 Sensor domain

While not strictly necessary to model the dynamics insidethe engine during warm up, a model for the exhaustgas thermocouple mounted at the exhaust gas port ispresented here as the thermocouple response is typicallyquite slow and hides some of the faster dynamics presentedby the model.

As the sensor will be subject to both convection from theexhaust gas and radiation to the head, the structure ofequation is:CtcTtc = htc,exhAtc(Tcyl − Ttc)−Atcεσ(T 4

tc − T 4h ) (46)

The parameter htc,exh in (46) can be determined usinga Nusselt number correlation for a cylindrical object ina fluid stream as described in Churchill and Bernstein(1977), while ε was taken to be 0.04 (corresponding topolished aluminium).

3. ENGINE TESTBED AND MODEL CALIBRATION

The test engine used was as a 4.0 litre, Ford Falcon BFinline 6 cylinder engine with 10.3:1 compression ratio, andwas mounted on a 460kW transient AC dynamometer.Aside from the standard engine sensors and dynamometerload feedback, a thermocouple was mounted at the ex-haust port to measure Tcyl, and in-cylinder pressure wasrecorded to measure indicated mean effective pressure.

From manufacturer data and CAD drawings, many of theengine parameters could be either directly measured or

inferred. Consequently there are only eleven heat transfercoefficients parameters and three surfaces (Pfme, ηv andηi) in the model described in Section 2 that require calibra-tion across the three domains using data measurements.

In the mechanical domain, the coefficients of (7), describ-ing the auxiliary friction mean effective pressure polyno-mial, require calibration. Since this quantity is not di-rectly measurable, the total friction mean effective pres-sure was estimated using measurements of indicated (fromin-cylinder pressure measurements) and braking mean ef-fective pressure (from measurements of brake torque andengine speed), i.e.

Pfme = Pimep − Pbme

= Pcfme + Prfme + Pvfme + Pafme + e (47)

The estimates of Pcfme, Prfme and Pvfme were calculatedbased on Sandoval and Heywood (2003). Although theseestimates will have some error, the approach suggestedhere is to associate all estimation error, e, with theestimate of Pafme. Consequently, the best estimate of thecoefficients of the Pafme polynomial is obtained by solvingthe following optimisation problem

[a∗0, . . . , a∗3] = arg min

a0,...,a3

∑e2 (48)

The data set used in solving (48) was obtained usinga constant set temperature of 373K, with engine speedvaried in 50 rpm increments from 1000rpm to 3000rpm.The resulting parameters following the implementation ofan SQP algorithm were obtained as:

a0 = 6.01× 10−3; a1 = 2.91× 10−3

a2 = −4.74× 10−6; a3 = 7.5× 10−9

The thermal domain has eleven parameters to be cali-brated, consisting of three heat transfer coefficients fromthe head, block and oil sump to ambient (hh,amb, hb,amb

and hs,amb); the heat transfer coefficients from the coolantto the block and head and from inside the cylinder to thecylinder walls (specified by two sets of coefficients ac andbc); the heat transfer from the piston to oil sump, and thehead and block to the oil (hp,o, hh,o and hb,o); and the heattransfer coefficient from the piston to the liner (Gp,l).

Existing thermocouples in the coolant path in the headand the oil path in the block were used to pose thefollowing optimisation problem:

J =∫ tf

t0

(To,b − To,b

)2

+(Tc,h − Tc,h

)2

dt (49)

[h∗h,amb, . . . , a∗c,(c→b), . . . , G

∗p,l] = arg min

hh,amb,...,Gp,l

J (50)

A steady speed-load warm up test was conducted at 1000rpm and 50 Nm with initial conditions of Tc,h(0) = 296K,To,b(0) = 289K. Ambient temperature was recorded asTamb(t) = 288K. The constant heat transfer coefficientsresulting from the optimisation were:

hh,amb = 25W/(m2K); hb,amb = 15W/(m2K)hs,amb = 200W/(m2K); hh,o = 80W/(m2K)hb,o = 130W/(m2K); hp,o = 400W/(m2K)Gp,l = 30W/K

The heat transfer for the coolant flow to the head andblock, and for the cylinder gas to the cylinder walls aredescribed by (13) and the parameters of this equation werefound to be:

ac,(c→b) = ac,(c→h) = 0.46; ac,(cyl→w) = 0.11bc,(c→b) = bc,(c→h) = 0.56; bc,(cyl→w) = 0.74

Simulating the model over the NEDC cycle, this leads tothe engine speed dependant heat transfer coefficients tofall within the range hc,b ∈ [300, 550] W/(m2K), hc,h ∈[280, 480] W/(m2K) and hcyl,w ∈ [8, 30] W/(m2K). Notethat the different heat transfer coefficients in the head andblock arise from the different diameter of the flow passages,and hence the different fluid velocities, while heat transfercoefficient from the cylinder to walls is effectively averagedover the four engine cycles.

Finally, the thermodynamic domain requires calibrationof the maps for volumetric, ηv(Pim, ωcrank, λ, θvo) andindicated efficiency, ηi(Pim, ωcrank, θsa, λ). Static sweepsof spark timing, valve timing, engine speed, air fuel ratioand engine load were conducted over a wide range of enginesteady state operation and curve fits were produced.

Assumption 11 and (45) were used in the modelling withfuel, air and exhaust assumed the only species present.

4. VALIDATION

The model from Section 2 with parameters identified inSection 3 was then produced in the Modelica simulationenvironment. It’s accuracy was validated by running theengine over constant speed-load and NEDC drive cycles.Modelled and experimental results are compared in thefollowing subsections.

4.1 Validation of mechanical domain model

The default calibration strategy of the OEM was usedduring all testing to set the engine input variables suchas air-fuel ratio and spark timing. With the engine at-tached to the dynamometer, the brake torque was spec-ified explicitly and the engine speed was controlled bythe throttle in closed loop to match the specified drivecycle. Consequently, the friction mean effective pressureis considered as a useful quantity to validate the proposedmodel. The friction mean effective pressure cannot be mea-sured directly, but is implied from the difference betweenthe indicated mean effective pressure and the losses due topumping and brake mean effective pressures.

The comparison of the modelled and experimental frictionmean effective pressure is presented in Figure 3. Goodagreement between the modelled and experimental datais observed across both engine speed and oil temperatureranges. An exception to the high accuracy is the spike inPfme seen at 150C for both constant engine speeds whenthe engine starts (possibly due to the initial oil distributionbefore the oil pump is operational not completely coatingthe surfaces).

4.2 Validation of thermal domain model

The thermal domain was validated by placing thermocou-ples in the oil and coolant flow pathways, and the engine

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was run over the NEDC cycle until the thermostat opened.The left plot in Figure 4 illustrates very good correlationbetween the predicted and experimental temperatures inboth the coolant and oil.

As further validation, the cumulative fuel use was bothlogged from the engine computer and measured using anexternal fuel flow meter, and compared to the actual fuelused. The requested air-fuel ratio was maintained betweenthe modelled and simulation results. This comparison, seenin the right hand side of Figure 4, is a good indicator thatthe friction model and static maps in the engine modelreflect accurately the real situation. The cumulative errorsin oil, coolant and fuel use can be quantitatively measuredas 0.24%, 0.54% and 1.6% respectively.

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4.3 Validation of thermodynamic domain model

The key aspect of the thermodynamic model presentedhere is the inclusion of the gas composition coupling,and the mean value cylinder model represented as aset of static maps. The composition of the gases in thevarious reservoirs could not be measured due to a lack ofequipment, however its effect can be implicitly ascertainedthrough the functional dependance of γ on x.

The manifold pressure dynamics have been well estab-lished in the literature previously, so their validation isnot repeated here. Instead, the focus is given to the tem-perature of the gas in the cylinder and how well the modelpredicts the actual value as measured at the exhaust port.Figure 5 shows the relationship between the temperaturepredicted by the model and the experimental data. Thereare significant high frequency dynamics predicted by the

model that are not reflected in the experimental data,however when the thermocouple model is included thereis very good agreement between the two over both drivecycle and steady state operation.

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4.4 Sensitivity analysis of the model

As a basic sensitivity analysis, the parameters of the modelidentified in Section 3 were randomly perturbed over alarge number of trials by up to 20%. The effects on thestates and cumulative fuel consumption is summarisedin Figure 6, where it is observed that a wide operatingenvelope exists. On the surface, this indicates that thecalibration process is effective, and the sensitivity of themodel to changes in the parameters is significant in someinstances, although further study is required to makequantifiable statements regarding the critical parametersand states of the model.

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5. CONCLUSIONS

The model presented in this paper is able to accuratelyreproduce the thermal, mechanical and thermodynamiccharacteristics of a spark ignition engine during the warmup region of operation. This model makes use of severalsimplifying assumptions, most notably the elimination ofthe in-cylinder dynamics (the fastest dynamics presentin the engine) and their replacement with a static map,as well as the development of previous models from theliterature.

The model also proposes a structure for considering thecomposition of the gas species in different volumes in

the engine, and discusses how these are coupled to thedynamics of pressure and temperature in the intake andexhaust manifolds.

While the accuracy of the multi-domain model is sufficientto faithfully reproduce the engine dynamics during warmup, the complexity of the model is low enough to lend itselfto offline optimisation studies (simulating 500 seconds ofthe warm up duration using the model implemented inModelica on a Pentium 4 processor takes approximately17 seconds). For closed loop model-based control purposesthis is still too complex, but the model provides an ex-cellent basis for further simplification using engine modelreduction techniques such as proposed in Sharma et al. (Inpress).

ACKNOWLEDGEMENTS

The authors would like to acknowledge the ARC fundingof Linkage Project LP0453768 and the in-kind supportprovided by the Ford Motor Company of Australia. Fur-thermore, the resources within the ACART research centreenabled the experimental work to be conducted.

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