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A modeling approach for engine dynamicsbased on electrical analogy
Antonio Palma ∗∗, Angelo Palladino ∗, Giovanni Fiengo ∗,Ferdinando De Cristofaro ∗∗, Fabio Garofalo ∗∗ and
Luigi Glielmo ∗
∗Dipartimento di Ingegneria, Universita degli Studi del Sannio,Piazza Roma 21, 82100 Benevento, Italy.
E-mail: [email protected], [email protected],[email protected]
∗∗ Systems and Controls, Gasoline EMS, PWT dpt., Elasis S.C.p.A.Via ex Aeroporto snc, 80038, Pomigliano d’Arco, Napoli, ITALY.
E-mail: [email protected],[email protected],
Abstract: In this paper, a multi-cylinder, internal combustion engine model is presented. Thepaper is focused on a simple modular, physically based and lumped parameter (zero dimensional)approach, leading to a complete and coherent model structure. The model is conceived to beused with general purpose simulation software, as MATLAB/Simulink. The mean value outputsof the model (pressure, temperature, mass flow, torque) are compared with experimental data,collected by a Fiat 1.8 liter engine, in order to perform a standard identification and validationprocedure.
Keywords: Modeling; Automotive system; SI engine model.
GLOSSARY
SI Spark IgnitionDI Direct InjectionCVCP Continuously Varying Cam PhasersWOT Widely Open ThrottleV V T Variable Valve TimingHCCI Homogeneous Charge Compression Ignitionp gas pressure [N/m2]T gas temperature [K]Tsl temperature of lateral surface of cylinder [K]Tsb temperature of basic surface of cylinder [K]m air mass flow rate [kg/s]Qext heat flow rate [kJ/s]V volume [m3]ϑ crank angle [deg]θ spark advance [deg]cx specific heat capacities (x= p or v)[J/(kgK)]γ ratio of specific heats , cp/cv [−]R gas constant [kJ/kgK]hi enthalpy [kJ/kg]mfuel fuel mass [kg]Qhv fuel lower heating value [J/kg]ηcb combustion efficiency [−]λ air/fuel ratio [−]S heat release [W/kg]mc choked air mass flow rate [kg/s]mnc not choked air mass flow rate [kg/s]CD discharge coefficient [−]AT effective flow area [m2]α throttle angle [deg]
1. INTRODUCTION
The availability of complete software packages for thesimulation of the internal combustion engine behavior isan useful tool for engine designers in order to have, inadvance, a good insight of what their design will be able (ornot) to perform. The complexity of the models, containedinside these software packages, involves the physical andchemical aspects of the combustion process (mono-zoneor multi-zone), the air flow dynamics (wave effects) orthe liquid deposition and vaporization dynamics (wallwetting).
Widely accepted method for modeling Naturally Aspi-rated (NA) SI engines is the Mean Value Engine Model(MVEM) [Muller et al., 1998] thanks to its great flexi-bility, good performance and low power computation. Itdescribes dynamically the development of engine physicalvariables over time periods which are long compared to thedominant time constants of the engine, see among others[Hendricks, 1989], [Hendricks, 2001] and [Deur et al., 2003].Mean value engine models are aimed at describing theaverage dynamic of key internal engine variables, such ascrank shaft speed, manifold pressure and air/fuel ratio, orvariables that cannot be measured directly, as thermal andvolumetric efficiencies [Aquino, 1981], [Powell and Cook,1987]. Nevertheless, MVEM does not include explicitly adescription of the intake, exhaust or combustion processes,but simply represents the overall result.
In literature, particular interest is dedicated to enginemodeling, both for analysis and control purpose. As an
Proceedings of the 17th World CongressThe International Federation of Automatic ControlSeoul, Korea, July 6-11, 2008
978-1-1234-7890-2/08/$20.00 © 2008 IFAC 2037 10.3182/20080706-5-KR-1001.2601
example, in [Karlsson and Fredriksson, 1999] a cylinder-by-cylinder model of experimental variable valve timing 4-cylinders engine has been developed. The model includesthe cylinder and manifold air mass flow, temperature,pressure and valve actuator dynamics, but it doesn’t de-scribe the combustion effects and burned gas residual.In [Bengtsson et al., 2004] and [Rausen et al., 2004]mathematical engine models are developed to study thedynamics of the HCCI engines. It is an alternative piston-engine combustion process that could potentially reachhigher efficiencies than direct-injection. The charge is opti-mally mixed resulting in a minimization of emissions. Theproposed model structures consist of a zero-dimensionalcylinder models combined with a reduced chemical systemsto describe the ignition phase. In [Cook and Powell, 1988]and [Fiengo et al., 2002] are presented engine models aimedto control designs. The main features of these models arethe simplicity (typically linear models are adopted) andthe accuracy necessary to reach the control goals.
The focus of this paper is to introduce a simple approachto model all the components of the internal combustion en-gine. The authors, starting from an analogy with electricalsystems, have tried to simplify the approach eliminatingthe space dynamics (multi-zone combustion and wave ef-fects), while preserving the time dynamics (resonance andtuning phenomena). In this way they have obtained anengine description similar to an electrical (although notlinear) circuit, with all the useful consequences in term ofexistence and numerical availability of the solution. Theadvantages are in the specific comparison that is foundbetween the engine components and variables (as throttlevalve, cylinder, inertial flows), with electrical counterparts(current, voltage, resistance).
The main benefits achievable with this methodology isthe simplicity to compose the whole engine model andcustomize it including all the latest devices. So it ispossible to easily implement both a baseline engine anda high complex automotive system. Moreover, in order todesign a new engine model adopting this approach, it isnot necessary to have a deep mechanical and electronicalknowledge, but to possess a familiarity with the moresimple electric circuits.
The application domain of the engine model, that ispossible to obtain with this methodology, is the analysis ofdynamical behavior of engine, as an example scavengingor exhaust gas recirculation, allowing to test the enginedynamics under different operative conditions with a goodlevel of reliability and accuracy. The approach has beenhere tested on SI engine, but it is remarked that only fewchanges are necessary to extend the model to other kindsof engines, as Diesel, TurboCompressor, Tri-fuel and HCCIengines.
It is highlighted that the proposed modeling methodologyis thought to be independent from the mathematicalenvironment used for the simulation. In particular, in thiswork Matlab/Simulink platform has been used, but othersimilar software packages could be adopted, as Modelicaand GT-Power [Schweiger et al., 2005], [Tiller, 2001],[Batteh et al., 2003], [Eriksson, 2003].
Table 1. Variables
element q i v
electric charge current voltageq [C] i [A] v [V ]
hydraulic volume flow rate pressureV m3 m [kg/sec] p [N/m2]
mechanic angle speed torqueθ [rad] ω [rad/s] T [Nm]
Table 2. Parameters
element parameters
electric resistance inductance capacity
R = vi
L = vdidt
C = idvdt
hydraulic pneumatic pneumaticresistance capacity
R = pm
C = mdpdt
mechanical friction inertia elasticity
B = Tω
J = Tdωdt
1K
= ωdTdt
2. SIMPLE APPROACH TO ENGINE MODEL
A physical system can be decomposed in elementarysubsystems or elements. For each elementary subsystemis possible to define three variables named [Balestrino andCelentano, 1997]:
q quantity, i.e. variable of the elementi flow, equal to dq/dt, i.e. the variable that flows
across the elementv forcing, i.e. the variable acting at the extremes
of the element
As an example, table 1 reports these variables for theelectric, hydraulic and mechanical elements. The mainfeatures is that one variable can be considered constantdespite the others or, alternatively, can be consideredconstant the following ratios
v
i,
vdidt
,i
dvdt
(1)
as shown in table 2. Moreover, it is possible to demonstratethat, for a network of elements and in particular condi-tions, the variables are related thought Kirchhoff laws, asfollows [Feynman and Leighton, 2007]:
• for each node of the network, the algebraic sum of theflow variables is zero ∑
h
ih = 0, (2)
where ih is the flow across the h-th element.• for each mesh of the network, the algebraic sum of
the forcing variables is zero∑h
vh = 0, (3)
where vh is the forcing acting on the h-th element.
Now, considering the engine formed by mechanical com-ponents, as throttle valve, manifolds, cylinders and crankshaft, crossed by a gas, the approach proposed in this workis based on the analogy among the electric, simpler tomodel, and the mechanic and hydraulic elements.
Regarding the mechanic systems, an analogy can be foundamong speed and torque respectively with electric currentand voltage, named ”Maxwell’s analogy” [Feynman and
17th IFAC World Congress (IFAC'08)Seoul, Korea, July 6-11, 2008
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Leighton, 2007]. It results in considering the mechanicalfriction B as an electric resistance R and, similarly, theinertia J as an electric inductance L and the inverse ofthe elasticity K as a capacitor C. The same considerationscan be done for the hydraulic elements. Here the analogiesare between the gas flow rate m with the current i, thepressure p with the voltage v and the volume V with thecharge q. Then, the electrical resistance corresponds to thepneumatic resistance, that is the resistance of gas flowingacross an orifice, and the volume to an electric capacitor.
In this scenario, all the parts composing the engine canfind an equivalent electrical circuit or element as detaileddescribed in the following sections. Moreover, in order toexactly describe the operation of electrical circuit, it isnecessary to use the Maxwell equations. These can captureboth the dynamics of the electrical quantities, such ascurrents and voltages, and the related electromagneticalphenomena, as transmission and radiation. Fortunately,if the size of the circuit is small compared to the wave-length of the electrical variables (i.e. the ratio betweenthe light speed and the frequency of the pulsating events),these electromagnetical phenomena can be neglected. Asa consequence, the partial differential relationships of theMaxwell equations can be simplified to the widely usedelectrotechnic equations, that are the Kirchhoff laws andthe current/voltage relationships of circuit components.Similarly, the same approach can be extended to the in-ternal combustion engine, providing that the wavelength(in this case the ratio between the sound speed and thefrequency of its pulsating events) is large enough comparedto the length size of the engine. As an example, a fourcylinder four stroke engine, running at 3000 rpm, generatesintake pulses at 100 Hz, resulting in a of 3.4 meters.Considering the engine size of approximately 1 meter, thelumped parameter approach seems to be reasonable.
To conclude, the analogy between the mechanical andelectrical components can be extended to the relation-ship governing the relative quantities. In fact, the firstKirchhoff equation has a specific counterpart in the massconservation law and, similarly, the Bernoulli equation cancorrectly replace the second Kirchhoff equation.
3. COMPONENTS
The engine is seen as an array of cylinders, having commonconnections with an intake and an exhaust manifold. Theconnections are regulated by valves opening. According tothe previous section, it is possible to distinguish separatesubsystems interconnected each others, such as the intakemanifold equipped with throttle valve, the exhaust man-ifold and cylinders. From the phenomenological point ofview, the elements composing the engine can be classifiedin the following categories: volumes, orifices, inertial effectsand combustion. In the following, each category is intro-duced and the relationship among the interested variablesare reported.
3.1 Volumes
Here are grouped the intake and exhaust manifold andcylinders, respectively as constant and variable volumes.The electric counterpart is the quantity of charge stored
Fig. 1. Volume equivalent circuit: a) constant capacity; b)variable capacity.
Fig. 2. Orifice equivalent circuit.
in a capacitor, as shown in Figure 1 1, reporting the corre-sponding circuit.
Applying the corresponding current/voltage relationshipand considering the analogies with pressure and tempera-ture inside the volume, it is possible to obtain the classicalequations [Muller et al., 1998]. Starting from ideal gasequations
pV = mRT (4)where R is the specific gas constant and m is the mass ofgas.
p =Rγ
V
∑i
miTi − T∑
j
mj +γ − 1Rγ
Qext −pV
R
(5)
T =RγT
pV
∑i
miTi(1−T
γTi)− T
∑j
mj(1−1
γT)
+γ − 1Rγ
Qext −pV
R(1− 1
γT)
] (6)
where i represents the entering mass flow and j theoutgoing mass flow. For sake of brevity, the details on howto obtain equations 5 and 6 are omitted.
It is remarked that, regarding the intake and exhaustmanifolds, since the volume V is constant, the derivativeterms in the equation disappear.
3.2 Orifices
The orifices are responsible of the pressure drops alongthe gas path. They are modeled as variable resistancescausing equivalent voltage drops, as illustrated in Figure2. The size of the orifice is variable and regulated by valveopening, as throttle valve, air bypass, intake and exhaustvalves [Heywood, 1988].
The electrical resistance is governed by a static relation-ship between voltage and current, corresponding to a staticrelationship between the analogue variables, i.e. pressureand flow rate, in according to the well known equations[Hendricks, 1989].
1 The circuits have been drawn using Modelica.
17th IFAC World Congress (IFAC'08)Seoul, Korea, July 6-11, 2008
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Fig. 3. Inertial effects equivalent circuit.
Fig. 4. Combustion equivalent circuit. mc = CDAT p0√RT
γ 12( 2
γ+1 )γ+1
2(γ−1) pT
p0> 1
mnc = CDAT p0√RT
(pT
p0)
1γ { 2γ
γ−1 [1− (pT
p0)
γ−1γ ]} pT
p0< 1
(7)where pT and p0 are respectively the pressure upstreamand downstream the orifice.
3.3 Inertial effects
The inertial phenomena can be considered as minor effortsbut not complectly negligible. They describe the reductionor the increase of the pressure upstream the valve of aquantity proportional to the derivative of the mass flowthrough the same valve. Here, they are modeled as anlinear inductance, see Figure 3, regulated by a differentialrelationship between voltage and current, correspondingto the following equation
pcorr = p− km (8)where pcorr is the manifold pressure and k is a parameterto be set. It is remarked that this kind of relationship isnot present in literature. In order to justify the adoptedchoice, both the analogy with the electrical circuit andthe simulation results illustrated later on the paper can beadduced.
3.4 Combustion description
The combustion process constitutes the most meaningfuland complex phenomenon occurring into the engine. Inorder to model the in-cylinder cycle pressure, an equivalentelectric circuit has been adopted, as shown in Figure 4. Thecircuit is formed by a variable condensator, representingthe cylinder volume according to section 3.1, equipped byan impulsive voltage generator. This causes an impulsiveincrease of the voltage at the condensator extremitiesand, consequently, generates a current flow thought thecapacitor.
This phenomenon corresponds to the well known combus-tion process, i.e. an impulsive increase of the in-cylinderpressure caused by the combustion resulting in a torquegeneration and in mass flow through the exhaust valves.The equation regarding this process is described by thefollowing relationship
Qext =hiAl(θ)(Tsl − T ) + h2Ab(Tsb − T )
+ mfuelηcbηburn(λ, p)S(θ, p)QHV
(9)
where h1 and h2 are parameters to be set and Al andAb the lateral and base area respectively. It is remarkedthat equation 9 represents the heat power generated bythe combustion affecting the pressure 5 and temperature6. The cyclic variation has been implemented as a functionof the operating conditions (ηburn in 9) of engine and,partially, equipped with a random behavior needed torepresent the combustion cycle-by-cycle and cylinder-by-cylinder irregularity [Hendricks, 2001] and [Deur et al.,2003].
4. ENGINE MODEL
4.1 Equivalent circuit
Based on the analogies depicted in the previous section,the entire engine can be represented by the circuit shownin Figure 5.
The model starts describing the dynamic of the air crossingthe intake manifold, i.e. driven by the ambient pressure(a current generator), the air mass passes the filter (aresistance) and the throttle body (a variable resistance)and arrives into the cylinder through the intake valves (anew variable resistance). The cylinder are described bya parallel of ”n” combustion equivalent circuit (see Figure1), with ”n” the number of cylinders composing the engine.Finally, the gas mixture is discharged into the exhaustmanifold through the exhaust valves (a variable resistance)and ends into the ambient crossing the muffler (a resis-tance). For sake of completeness, it is possible to introducethe inductors that naturally represent the inertial effects ofthe current and are able to describe the analogous effectsof the fluid columns. The validity of the modeling choicesadopted in this work has been tested with experimentaldata of Fiat engine 1.8 liters with 8 valves and with aCVCP.
4.2 Simulation results
Two experiments are illustrated, the first at 1500 rpm andthe second at 5500 rpm. In particular for the last exper-iment, the condition necessary to simplify the Maxwellequations, i.e. the wavelength of the electric variables mustbe much larger than the size of the circuit, is still validbut the approximation error is higher. In fact, a fourcylinder pulses at 183 Hz, resulting in a wavelength atthe fundamental frequency of 1.8 meter.
Figures 6 to 8 report the first experiment. In particular,Figures 6 and 7 show the simulated intake and exhaustmass flows for each cylinder and the pressure cycle andmass flows through the intake and exhaust valves, re-spectively. Figure 8 completes the experiment comparingthe simulated in-cylinder pressure cycle with experimentaldata highlighting the good performance. The intake andexhaust valve lift ends the Figure 8.
Similarly, Figures 9 to 11 report the second experimentat 5500 rpm. Again, firstly are illustrated the simulatedmass flows and pressure and finally the comparison withexperimental data is shown in Figure 11. It is remarkedthat the performance are still high despite the loweraccuracy of the condition at the base of the proposedapproach.
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Fig. 5. Internal combustion engine equivalent circuit.
Fig. 6. Experiment 1 at 1500 rpm and WOT. Intake andexhaust mass flows for the four cylinders.
Fig. 7. Experiment 1 at 1500 rpm and WOT. The first plotdepicts the pressure inside intake manifold (solid-blueline), exhaust manifold (dotted-red line) and cylinder(dashed-magenta line). The second plot describes thevalves lift, showing the exhaust mass flow (solid-redline) and intake mass flow (dotted-blue line).
Fig. 8. Experiment 1 at 1500 rpm and WOT. The firstplot compares experimental data (dotted-black line)of the pressure inside cylinder and simulated results(solid-magenta line); the second plot reproduces theintake and exhaust valves lift (experimental data).
Fig. 9. Experiment 2 at 5500 rpm and WOT. Intake andexhaust mass flows for the four cylinders.
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Fig. 10. Experiment 2 at 5500 rpm and WOT. Thefirst plot depicts the pressure inside intake manifold(solid-blue line), exhaust manifold (dotted-red line)and cylinder (dashed-magenta line). The second plotdescribes the valves lift, showing the exhaust massflow (solid-red line) and intake mass flow (dotted-blueline).
Fig. 11. Experiment 2 at 5500 rpm and WOT. The firstplot compares experimental data (dotted-black line)of the pressure inside cylinder and simulated results(solid-magenta line); the second plot reproduces theintake and exhaust valves lift (experimental data).
5. CONCLUSION
A simple modeling approach for SI engine based on theequivalence with electric circuit has been introduced. Theresulting mean value model has been tested by comparingwith experimental data, showing a good level of reliabilityand accuracy.
Two points are been remarked, the calibration phase andthe simulation complexity. The first aspect required theidentification of the characteristic parameters. In particu-lar, the model is based on known geometric constants andonly few parameters need to be set. It results in a simpleand quick calibration phase.
Conversely, the drawback is the high computational com-plexity necessary to simulate the model, causing the im-possibility to run it for long test.
ACKNOWLEDGEMENTS
The authors would like to thanks Luigi Maresca and ElenaGiugliano for their contribution to this work and Elasis-FPT for providing the experimental data.
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