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SAE TECHNICALPAPER SERIES 1999-01-0908

Turbocharger Modeling for AutomotiveControl Applications

Paul MoraalFord Forschungszentrum Aachen

Ilya KolmanovskyFord Research Laboratories

Reprinted From: SI Engine Modeling(SP-1451)

International Congress and ExpositionDetroit, Michigan

March 1-4, 1999

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1

1999-01-0908

Turbocharger Modeling for Automotive Control Applications

Paul MoraalFord Forschungszentrum Aachen

Ilya KolmanovskyFord Research Laboratories

Copyright © 1999 Society of Automotive Engineers, Inc.

ABSTRACT

Dynamic simulation models of turbocharged Diesel andgasoline engines are increasingly being used for designand initial testing of engine control strategies. The turbo-charger submodel is a critical part of the overall model,but its control-oriented modeling has received limitedattention thus far. Turbocharger performance maps aretypically supplied in table form, however, for inclusion intoengine simulation models this form is not well suited.Standard table interpolation routines are not continuouslydifferentiable, extrapolation is unreliable and the tablerepresentation is not compact. This paper presents anoverview of curve fitting methods for compressor and tur-bine characteristics overcoming these problems. Weinclude some background on compressor and turbinemodeling, limitations to experimental mapping of turbo-chargers, as well as the implications of the compressormodel choice on the overall engine model stiffness andsimulation times.

The emphasis in this paper is on compressor flow ratemodeling, since this is both a very challenging problemas well as a crucial part of the overall engine model. Forthe compressor, four different methods, including neuralnetworks, are presented and tested on three differentcompressors in terms of curve fitting accuracy, modelcomplexity, genericity and extrapolation capabilities.Curve fitting methods for turbine characteristics are pre-sented for both a wastegated and a variable geometryturbine.

INTRODUCTION AND BACKGROUND

Dynamic simulation models of turbocharged Diesel andgasoline engines are increasingly being used for designand initial testing of engine control strategies. The turbo-charger submodel is a critical part of the overall model,but its control-oriented modeling has received limitedattention thus far. A standard approach still appears to beto include the turbocharger performance data in the formof lookup tables directly into the model [6], [12]. However,this form is not ideally suited for use in control-oriented

engine models because the standard linear interpolationroutine is not continuously differentiable, sometimes lead-ing to apparent discontinuities in simulations. Further-more, and more seriously, this type of model does notadequately handle operating conditions outside of themapped data range, for example at very low turbochargerrotational speeds.

While engine mapping usually covers the entire operatingrange, the situation for the turbocharger unit is different.Generally, it is possible to obtain the performance charac-teristics from the supplier. However, the turbochargercharacteristics are typically only mapped for higher turbospeeds (typically 90000 RPM and up) and pressureratios, whereas the operating range on the engine ex-

Figure 1. Typical compressor map. Usually, such a compressor map shows constant speedlines and constant isentropic efficiency lines. Here, we have omitted the efficiency lines and instead superimposed the compressor flow rate determined from engine mapping data. It is readily apparent that the compressor mapping data does not cover the operating range of the compressor on the engine. In particular, low speed and low pressure ratio data are lacking.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.351

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tends down to 10000 RPM shaft speed and pressureratios close to unity. This is illustrated in Figure 1. As aresult, standard interpolation methods (polynomialregression, look-up tables) generally fail to produce rea-sonable results outside of the region where experimentaldata is available. In fact, because of the nonlinear natureof the compressor and turbine characteristics even inter-polation through lookup tables has been found to causeunacceptable performance in simulations.

In this paper we will present a number of different curvefitting methods for turbochargers which allow a morecompact implementation of the turbocharger submodel.Special emphasis will be on the abilities of the differentmethods to extrapolate the performance characteristicsinto regions which are usually not mapped, but which areencountered during normal operation of the engine. Firstwe'll provide some insight into the limitations of experi-mental turbocharger mapping.

LIMITATIONS OF EXPERIMENTAL TURBOCHARGER MAPPING

Whenever the choice is given, one should obviously aimfor extending the range of available experimental data,rather than trying to predict/extrapolate the behavior out-side of the given range (unless good models of theunderlying physics are available, in which case a fewexperimentally measured data points may suffice to char-acterize a large operating region and experimental devel-opment time can thus be reduced). With regard toextending the range of experimental mapping of turbo-charger units on a flow bench (in particular, lower turbinespeeds and lower mass flow rates), two problems arise,which would require significant effort to work around. Thefollowing is an explanation provided by David Flaxington[2], at the time working at Allied Signal - Garrett.

FLOW SENSOR ACCURACY

The flow through the compressor (the same argumentshold for the turbine) is usually measured by determiningthe differential pressure across a properly sized orificeplaced in the flow path, and using Bernoulli's law, assum-ing incompressible flow (constant density), to determinethe flow rate [3]. This gives a smooth variation of com-pressor characteristics as the flow rate is varied. One canmap at lower speeds if smaller orifice plates and nozzlesare used to measure the lower flows, but this produces aproblem. As the flow rate is reduced, the accuracy of thereadings with any one measuring device reduces.Changing at some point to another measuring devicesized for lower flow rates then causes a step change inthe mass flow readings as the accuracy of the measure-ment is again improved.

With a vast set of orifice plates and nozzles this problemcould be circumvented. However, this is not standardpractice and improvements are largely dependant uponthe goodwill of the supplier.

HEAT TRANSFER EFFECTS

A second problem associated with the low speed regionof the maps is caused by the heat conduction from thehot lubricating oil to the cool compressor end. At higherspeeds, the conduction can reverse and go from the hotair to the lubricating oil. This heat added to the air is mea-sured by the temperature sensors measuring the temper-ature rise of the air and used to calculate the work inputand hence the efficiency of the compressor and the effi-ciency of the turbine. It makes the compressor look worsebecause apparently more turbine work is required than isreally used. The same effect makes the turbine look bet-ter. This problem is worst at low speeds because of thehigher temperature differences. With relatively largeareas, higher temperature drops and low flow rates, thebackplate and compressor housing are relatively efficientheat exchangers. This effect can be seen on the com-pressor maps which all show efficiency collapsing at lowpressure ratios. This is not true aero performance. On theturbine maps, the opposite effect of the turbine lookingbetter than it really is is apparent. Again this is not trueaerodynamic performance. Lower speeds would showthe heat transfer effect as more extreme. This means thatthe predicted transient performance would be in error ifthe maps were used without correction because the tur-bine work to drive the compressor is overestimated andmore work is actually available to accelerate the rotor.The heat transfer problem is not easily overcome withoutseparate turbine, bearing and compressor dynamome-ters. When these are in use, the compressor mappingmethod could be adjusted to control the temperature dif-ferences driving the heat transfer so that it could belargely eliminated. However, different oil temperaturesmay require different bearings to be used and either ofthese two changes would affect the turbine maps. This isa problem the turbocharger manufacturers are well awareof, but simple solutions presently don't appear to be avail-able.

COMPRESSOR

BACKGROUND – The rotating impeller of a centrifugalcompressor imparts a high velocity to the air. The air isthen decelerated in a diffuser with a consequent rise instatic pressure. Neglecting heat losses, the power, P,required to drive the compressor can be related to themass flow rate through the compressor, W, and the totalenthalpy change across the compressor from the first lawof thermodynamics as

(1)

Assuming constant specific heat coefficients, the poweris given by

(2)

The subscripts 01 and 02 refer to the stagnation condi-tions at the compressor inlet and outlet, respectively. If cp

3

is specified in kJ/kg/K, W in kg/sec and T01,T02 are in Kthen units of P are kW. For an isentropic process the tem-perature ratio can be related to the pressure ratio usingthe relation,

(3)

To account for the fact that the compression process isnot isentropic, the compressor isentropic efficiency, 0 ≤ηc,is ≤ 1, is introduced and defined as the ratio of theoret-ical (isentropic) temperature rise and actual temperaturerise:

(4)

Combining equations (3) and (4),

(5)

hence,

(6)

An insight into various terms comprising equation (6) canbe obtained by considering the enthalpy change acrossthe compressor, ∆h. From equations (1) and (6), weobtain the following relation between enthalpy changeand pressure risse across the compressor:

(7)

Assuming that there are no losses (i.e. in the ideal case),∆h can be estimated from Euler's equation for turboma-chinery. Consider a compressor with a radial vanedimpeller, no backsweep and no inlet pre-whirl. Then [13],

(8)

where U2 is the velocity of the impeller at the impeller tip,Cθ2 is the tangential component of the air velocity leavingthe impeller, while U1, is the velocity of the impeller at theimpeller entry (where air enters the impeller), and Cθ1the tangential component of the air velocity entering theimpeller (see Figure 2 for a velocity diagram). The no inletpre-whirl assumption implies that Cθ2 = 0 and

(9)

The ratio

(10)

is known as the slip factor and depends, for example, onblade spacing, backsweep angle, and mass flow rate

through the compressor [13]. Thus, from equation (10)we obtain

(11)

Figure 2. Generic compressor flow velocity diagram at impeller tip, indicating blade tip velocity U2, air flow absolute velocity C2 , and the tangential component of the air leaving the compressor, Cθ2.

If the dependence of the slip factor on the mass flow rateis neglected, equations (7) and (11) suggest that, ideally(i.e. if ∆h = ∆hideal), the pressure increase across thecompressor should only be a function of the turbochargerspeed Ntc. However, this is not the case because of thelosses that do depend on the mass flow rate, W. Theactual enthalpy change across the compressor is largerthan ∆hideal and the ratio is precisely the compressor effi-ciency:

(12)

There are several sources of losses that can be broadlycategorized as (1) incidence losses in the impeller anddiffuser caused by flow instantaneously changing direc-tion to comply with geometry, (2) friction losses in impel-ler, diffuser and collector (e.g. due to viscous drag on thewalls), (3) clearance losses in the impeller, (4) losses inthe impeller due to the backflow and several others, see[13]. Accounting for these losses individually with phys-ics-based sub-models may be very difficult. Further-more, it is not clear how to validate the individual sub-models from the experimental data. Consequently, onecan view their use as simply a way of introducing a partic-ular parametrization with multiple parameters to beregressed from compressor performance maps. Simplerexpressions can potentially be generated by only partiallyrestricting ourselves to the structure suggested by thephysics based models. This is, basically, the curve fittingapproach pursued in this paper.

The turbocharger manufacturer specifies the perfor-mance characteristics in terms of the mass flow rate andisentropic efficiency for varying compressor speeds andpressure ratios on a flow stand and supplies the informa-tion as performance maps in table form. It is conven-tional to specify the performance maps in terms of scaledmass flow rate parameter, φ, and compressor rotationalspeed parameter, , that are defined as

C2

U2Cθ2

ω

4

(13)

where Ntc denotes the compressor rotational speed(rpm). The use of the scaled parameters eliminates thedependence of the performance maps on inlet condi-tions (Tin and pin). For the remainder of this section, wewill assume that the inlet and outlet velocities for thecompressor are small enough to ignore the differencebetween static and stagnation pressure and temperature.

The dependencies are

(14)

(15)

Figure 1 shows several speedlines (lines of stable operat-ing points of pressure ratio versus mass flow parameterfor constant speeds) for a typical compressor. For eachspeedline, there are two limits to the flow range. Theupper limit is due to choking, when the flow reaches thevelocity of sound at some cross-section. In this regime nofurther flow increase can be obtained by reducing thecompressor outlet pressure and the speedline slopebecomes infinite. The lower limit is due to a dangerousinstability known as surge [13]. During surging a noisyand often violent flow process can occur causing cycleperiods of backflow through the whole compressor andthe installation downstream the compressor. The specificvalue of φ at which surge occurs depends not only on thecompressor characteristics but also on the properties ofthe installation downstream of the compressor. Typically,this value is where the slope of the speedline is zero orslightly positive. The left-hand extremities of the speed-lines may be joined up to form what is known as thesurge line. When the turbocharger compressor is con-nected to an engine intake manifold, the volume of themanifold is often not sufficient to damp out the pressurefluctuations arising from periodic suction strokes of thepistons. As a result, even though the mean value of φmay lie to the right of a surge line obtained in a steady-state flow stand, the minimum mass flow rate (at the peakof the pulse) may cause surge to develop. Other instabili-ties that can develop during operation of a centrifugalcompressor include stalls. See [13] for a discussion ofinducer stalls, impeller stalls and rotating stalls.

CURVE FITTING OF COMPRESSOR MASS FLOW

For the mathematical representation of the compressorflow characteristics, there are two options. The flowthrough the compressor can be expressed as a functionof pressure ratio pout/pin and turbine shaft speed Ntc

(16)

or the pressure ratio across the compressor can be mod-eled as a function of compressor flow Wc and turbinespeed Ntc:

(17)

When viewing the compressor model in isolation, ModelII appears to be the better choice:

• In the areas where the speedlines are almost hori-zontal,coinciding with a large part of the engine oper-ating regime, as illustrated in Figure 1, this model isless sensitive to input or modeling errors. In Model I,lines of constant pressure ratio intersect the speed-lines with a very small angle, resulting in high sensi-tivity of Wc for small changes in pressure ratio pout/pin.

• Unlike Model II, Model I cannot be easily incorpo-rated into a dynamic model that exhibits surgebehavior (as will be made clear below).

However, when the compressor model is implemented aspart of an engine model, other aspects need to be takeninto account.

SENSITIVITY – When the compressor model is con-nected to the engine model, the equilibrium compressorflow and boost pressure levels are determined from anequilibrium of the engine pumping rate and the compres-sor flow map. The engine pumping is given by the follow-ing equation:

(18)

where, for fixed fueling rate, the volumetric efficiency ηvoland engine speed N are only weak functions of boostpressure pi and compressor flow Wc. The subscript irefers to intake manifold conditions. For fixed enginespeed, the term ηvol Vd N/(120 R Ti) is sometimesreferred to as engine pumping constant, and the pumpingmap is a linear function of intake manifold pressure.Superimposing the engine pumping map for various val-ues of the engine pumping constant onto the compressormap (Figure 3) illustrates that the sensitivity issue haschanged significantly: the engine operating points aredetermined by the intersection of the two sets of lines.The angle of intersection is never close to zero, but ratheraround 60-90 degrees, indicating that the sensitivity tocompressor modeling error is almost the same for Mod-els I and II. Of course, this analysis is only correct forsteady-state operating points. During transients theengine operating trajectory behaves differently withrespect to the compressor speed lines due to the fact thatthe pressure ratio changes much faster than the com-pressor flow rate.

5

Figure 3. Generic compressor speed lines and engine pumping map (for fixed engine speeds). The intersections of the two sets of lines determine engine operating equilibria

MODEL STIFFNESS – Model I can easily be incorpo-rated into a mean value engine model with ambient pres-sure pamb as (constant) input and intake manifoldpressure pi as state variable. The intake manifold dynam-ics are modeled by

(19)

where the implicit assumptions of constant temperatureand absence of EGR are without loss of generality. UsingModel I, the compressor flow Wc is simply given by

(20)

Using Model II, an additional state variable must be intro-duced based on the momentum equation for the air massmc in the tube connecting the compressor outlet and theintake manifold. With pc denoting the compressor outletpressure, and A and l representing the cross sectionalarea and length respectively of the connecting tube, themomentum equation is given by

(21)

hence

(22)

Depending on the geometry of the intake assembly, theaddition of this state variable can increase the model stiff-ness considerably. Model stiffness refers to the ratio ofsmallest to largest eigenvalue of the linearization arounda given operating point, or, equivalently, to the differenceof time scales in the system’s dynamics. Dynamic modelsfor the complete engine dynamics can be found in, e.g.,[5]-[8]. Using typical values for A and l, say A=50cm2 andl=20cm, at a nominal operating point (fuel=1.5 kg/hr,load=40Nm, no EGR), the range of eigenvalues of the lin-

earized system was found to be between -0.5 and -280.The additional state introduced an eigenvalue of -4100 tothe system. The resulting increase in simulation time by afactor of 3 - 4 may be unacceptable. However, if the con-necting tube between compressor and intake manifold islonger (this may be required for packaging of the inter-cooler, for example), say, l=100cm, the additional stateintroduces an eigenvalue of -450 to the system. In thislatter case, the simulation time will not be significantlyaffected.

In the remainder of the section we'll describe four differ-ent curve fitting techniques for the flow characteristics ofa radial compressor. Alternative approaches have beenreported in [10].

Jensen & Kristensen Method (5) – In [5], Jensen & Kris-tensen present a simplification of a model due to Winkler[14]. The model uses the dimensionless head parameterΨ, equivalent to the slip factor defined in equation [10]:

(23)

where Uc is the compressor blade tip speed

(24)

The normalized compressor flow rate Φ is defined by

(25)

and the inlet Mach number M introduced by

(26)

The head parameter Ψ and compressor efficiency ηc arethen expressed as functions of Φ and M in the followingway:

(27)

(28)

The coefficients k and a are determined through a leastsquares fit on experimental data. Since equation (27) isinvertible, this compressor model is also capable ofdescribing the compressor flow as a function of pressureratio (Model I) by

6

(29)

Depending on the model chosen, the output is then givenby

(30)

or by

(31)

Due to the particular choice of basis functions, namelyrational polynomials, this method is effective in describingboth the flat speedlines at low flow rates, as well as thealmost vertical speedlines at high compressor speeds.The functional form (27) has a singularity at Φ=k3. Onlythe curve to the left of this asymptote is used. During sim-ulations, care must be taken to avoid crossing this singu-larity during the numerical integration.

Even though the model is not entirely physics-based,equation (31) shows that extrapolation to low compressorspeeds and low mass flow rates causes the resultingpressure ratio to decrease continuously to unity for zeromass flow, which is exactly what one would expect tohappen physically. The only constraint on the curve fittingparameters is that k3>0 for all compressor speeds.

Mueller method – In his Master's Thesis [8], MartinMueller from DTU derives a compressor model from firstprinciples, incorporating the underlying physical princi-ples as well as the compressor assembly geometry. Themodels thus derived can predict the compressor charac-teristics surprisingly well. However, once experimentaldata is available, still better accuracy can always beachieved by appropriate curve fitting.

Based on physical considerations, Mueller proposes tomodel Ψ as a quadratic function in Φ:

(32)

This model is claimed to be generic, however, the param-eters A, B, and C are known to be speed dependent andthe way in which this speed dependence is modeled isagain a design choice. The obvious choice is to model A,B, and C as linear or quadratic in Uc. However, Muellerobserves that this choice does not lead to a genericmodel because it fails to give satisfactory results forsome of the compressor types under investigation. Itturns out that, rather than fitting A, B, and C indepen-dently, it is advantageous to exploit one more observa-tion, namely that the curve connecting the maximummass flows Wc,top on each speedline is typically a qua-dratic function in Uc. This then leads to the followingparametrization:

(33)

This model is expressed in the form of Model II and can-not easily be inverted to a Model I form. Whatever finiteperiod of time is needed for determining the sign of thesquare root term in the inversion (during which the signmay be incorrect, and if a globally stable sign detectionalgorithm can be found at all) will more than likely beunacceptable in dynamic simulations.

Zero Slope Line method (ZSLM) – Another model for thecompressor flow map was developed in an internal publi-cation [7]. It describes the compressor flow parameter, φ,as a function of pressure ratio, r, and speed parameter,

(Model I). First, the curve connecting the maximummass flows on each speedline (also referred to as thezero-slope line) is characterized by a quadratic in :

(34)

where rp,top are the values of the pressure ratio corre-sponding to φtop. Then, in order to capture the steepslope of the speedlines near the choke limit, the speed-lines to the right of the zero-slope line are modeled asexponentials:

(35)

for rp,< rp,top and are linearly extended to the left of thezero-slope line:

(36)

where the parameter α is modeled as a constant, or as afunction of :

(37)

Neural networks – Neural networks are becomingincreasingly popular for a wide range of applications,including curve fitting, system identification, etc. The useof neural networks for representing turbocharger charac-teristics is reported in [4], among others. However, eventhough neural networks are claimed to be universal func-tion approximators, which they are in even a quite gen-eral sense, finding the right structure and coefficientsrequires some amount of trial and error.

7

In [4], a network with one hidden layer and five neurons isused to represent a compressor flow map in the form ofmodel II, i.e., pressure ratio is fitted as a function of massflow and compressor rotational speed. A network with nuinputs (nu=2 in this case), one hidden layer containing nnneurons and ny outputs (ny=1 for now) would mathemat-ically be represented by

(38)

where b and W are coefficient vectors and matricesrespectively, and the function f is the neuron transferfunction (basis function) typically of the form

(39)

In general, such a network requires a total of(nu+ny+1)*nn+ ny coefficients to be fitted. A network with2 inputs, 5 hidden neurons and 1 output would thereforehave 21 parameters to be fitted, the same network withtwo outputs would have 27 coefficients. Considering thefact that a typical compressor map is specified by 25-40points, this is a large number of coefficients. Of course, ifone given network is used to fit a family of compressormaps, the ratio of data points to coefficients improvessubstantially.

Here we have used a smaller network to fit the compres-sor maps; in fact, both mass flow map and efficiency arerepresented in one single network with two inputs andtwo outputs. For a compressor map in the form of modelII, i.e., pressure ratio is fitted as a function of mass flowand compressor rotational speed, 3 and 4-node net-works with 17 and 22 coefficients respectively gaveexcellent results for the compressor pressure ratio (seeFigure 7). The problem of fitting a compressor model oftype I, i.e., compressor mass flow as a function of pres-sure ratio and compressor speed, proved much more dif-ficult - practically impossible actually. Any combination of3-5 neurons and one or two hidden layers was tried up to10 times, with initial coefficients generated in a partiallyrandom way without success. The problem here is thatthe fit through the mapped data points may have to beless than optimal in order to get sensible extrapolationresults. This type of behavior is difficult to enforce in ageneric structure such as a neural network. Even theaddition of "artificial" mapping points in the surge regionand at low compressor speeds, or the deletion of the pos-itive slope speedline segments was without success.

Of course, with an increasing number of neurons and lay-ers, the optimization problem to determine the best coef-ficients increases in dimensionality and number of localextrema. This means that the optimization process willtake longer, and at the same time, it may become moredifficult to find a sufficiently good initialization because ofthe larger number of local minima. Therefore, the optimi-

zation method (also referred to as network training)needs to be computationally very efficient and have built-in mechanisms for escaping from local minima. Over thepast couple of years, the issue of computational speedhas improved significantly compared to the initial back-propagation algorithms, which are a type of gradientdescent method. Commercially available software, suchas The Mathworks' Neural Network Toolbox [1], imple-ment a number of different second order methods, suchas Levenberg-Marquardt, which are orders of magnitudefaster than gradient descent based backpropagationmethods.

The issue of global convergence on the other hand, hasnot been solved with the same level of success. Thecrude method is simply to start the optimization manytimes from different, semi-randomly generated initial con-ditions (knowledge about the output range allows one torestrict some coefficients to a sensible range). However,this can be quite tedious. A large number of trials with 4and 5-node networks for the model I compressor maprevealed that about one out of ten times the resulting net-work starts to look acceptable, but a sufficiently accuratefit was not obtained. A more systematic way of dealingwith local minima was developed by Puskorias and Feld-kamp [9] and, undoubtedly, more exist; however, areview of the neural network literature is beyond thescope of this paper.

The conclusion is that for a generic compressor charac-teristics curve fitting method, neural networks are verywell suited for models of type II, but appear not to besuited for models of type I (or, at least, it is not straightfor-ward to find the right structure and training procedure,and significant manual modifications of the mapping dataare required).

CURVE FITTING RESULTS FOR THREE DIFFERENT COMPRESSORS

In this section we present the curve fitting results onthree compressors from two different manufacturers forthe methods discussed sofar. The compressors are iden-tified here only as compressors 1,2, and 3. In Figures 4-7, the curve fitting methods are applied to the compres-sor flow map for three compressors used for engine dis-placements in the range from 1.1 to 2.4 liter. The datasupplied by the manufacturer is indicated by the solidlines - the speedlines cover speeds in the range of 90 -230 kRPM. Superimposed onto the manufacturer's dataare the results of the curve fits (dashed lines), extendeddown to very low turbine speeds (10kRPM, 30kRPM, and60kRPM) and pressure ratios or mass flows. The exten-sion of the maps gives an indication as to whether thecurve fits produce sensible results in extrapolation. Espe-cially for lower turbine speeds, this is important, becausethey represent operating conditions which are frequentlyencountered.

8

It can be noted that all methods yield quite similar resultsfor the lower speed lines of the supplied compressormaps. Also, except for the neural network approach, eachmethod has difficulties describing the highest speed linesfor at least one of the given compressors. The extensionsinto the surge region, i.e., extending the speed lines tothe left, yield very different results depending on thecurve fitting method chosen. This was to be expectedsince the methods of Jensen & Kristensen and ZSLMrequire the speed lines to be strictly monotonic. However,for use in mean value control oriented engine models,any of the proposed extensions will work since they allprovide bounded and continuously differentiable exten-sions, and the surge region is expected to be enteredonly for very short periods during transients, if at all.Finally, Mueller's method applied to compressors 2 and 3reveals some slight difficulties with the extension to verylow speed lines: the extrapolated speed lines for 30kRPM in the second and third fit in Figure 5 are obviouslyincorrect.

In order to validate the extrapolation results to the lowerspeed lines, all the compressor fits were repeated andsupplied with the manufacturer's data excluding the low-est speed line. All methods showed acceptable results forthe extrapolated lowest speedline. In fact, the methods byJensen & Kristensen and Mueller yielded almost identicalresults compared to the fits where all the original datawere used for curve fitting.

CURVE FITTING OF COMPRESSOR EFFICIENCY

One method for describing the compressor isentropicefficiency was already given in equation (28). It modelsthe efficiency as a quadratic function of the compressorflow rate, with coefficients depending on the speedparameter. For compressor 1, the corresponding effi-ciency fit is shown in Figure (8)

Alternatively, if the compressor flow is represented usinga neural network, the network can be augmented with asecond output (and additional nodes if necessary) andbe retrained to provide an approximation for the isen-tropic efficiency as well as for the mass flow rate. Thiswas actually already done for the neural network curvefitsshown in Figure (7). For compressor 1, the correspond-ing efficiency fit is shown in Figure (9)

Figure 4. Curve fits for three different compressors using method proposed by Jensen & Kristensen [5].

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mean eror -0.217840%, std 0.094753

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mean eror 1.664988%, std 0.097928

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Figure 5. Curve fits for three different compressors using curve fitting method proposed by Mueller [8]

Figure 6. Curve fits for three different comprressors compressors using curve fitting method proposed in [7].

0 0.05 0.1 0.15 0.2 0.25 0.31

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

90000

120000

140000

160000

180000

scaled mass flow parameter

pres

sure

rat

io

Compressor 1

13

6

9

12

14

16

18

compressor datacurve fit

mean eror 0.053687%, std 0.058212

0 0.05 0.1 0.15 0.2 0.25 0.31

1.5

2

2.5

3

100000

120000

140000

160000

180000

200000

scaled mass flow parameter

pres

sure

rat

io

Compressor 2

36

1012

1416

18

20

compressor datacurve fit

mean eror 0.041969%, std 0.060226

0 0.04 0.08 0.12 0.16 0.21

1.5

2

2.5

3

3.5

90000110000

130000150000

170000

190000

210000

230000

scaled mass flow parameter

pres

sure

rat

io

Compressor 3

36

911

1315

17

19

21

23

compressor datacurve fit

mean eror 0.189370%, std 0.055547

0 0.05 0.1 0.15 0.2 0.25 0.31

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

90000

120000

140000

160000

180000

scaled mass flow parameter

pres

sure

rat

io

Compressor 1

1 3 6 9 12 14 16 18

compressor datacurve fitzero slope line

mean error = -4.73764 % , std = 0.0023737

0 0.05 0.1 0.15 0.2 0.25 0.31

1.5

2

2.5

3

100000

120000

140000

160000

180000

200000

scaled mass flow parameter

pres

sure

rat

io

Compressor 2

1 3 6 10 12 14 16 18 20

compressor datacurve fitzero slope line

mean error = -5.47799 % , std = 0.0031010

0 0.05 0.1 0.15 0.2 0.251

1.5

2

2.5

3

3.5

90000110000

130000150000

170000

190000

210000

230000

scaled mass flow parameter

pres

sure

rat

io

Compressor 3

1 3 6 9 11 13 15 17192123

compressor datacurve fitzero slope line

mean error = 0.114804 % , std = 0.00070898

10

Figure 7. Curve fits for three different compressors using a neural network with 3 neurons in one hidden layer.

Figure 8. Compressor efficienc curve fit for compressor 1 using curve fitting method proposed by Jensen & Kristensen [5].

Figure 9. Compressor efficiency curve fit for compressor 1 using the same 3 node neural network as was used for the compressor flow rate in the top graph of Figure (7).

TURBINE

BACKGROUND – The turbine is powered by the energyof the exhaust gas. The power input to the turbine, P,can be obtained from the first law of thermodynamics,neglecting heat transfer, as

(40)

where W is the mass flow rate of the exhaust gasthrough the turbine. As previously, the subscripts 01 and02 refer to stagnation conditions at the turbine inlet andthe turbine outlet, respectively. Treating the exhaust gasas an ideal gas, we obtain

(41)

0 0.05 0.1 0.15 0.2 0.25 0.31

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

90000

120000

140000

160000

180000

scaled mass flow parameter

pres

sure

rat

io

Compressor 1

136

9

12

14

16

18

compressor datacurve fit

mean eror 0.002116%, std 0.016118

0 0.005 0.01 0.015 0.02 0.025 0.031

1.5

2

2.5

3

100000

120000

140000

160000

180000

200000

scaled mass flow parameter

pres

sure

rat

io

Compressor 2

136

10

12

14

16

18

20

compressor datacurve fit

mean eror -0.004643%, std 0.014558

0 0.04 0.08 0.12 0.161

1.5

2

2.5

3

3.5

90000110000

130000150000

170000

190000

210000

230000

scaled mass flow parameter

pres

sure

rat

io

Compressor 3

136911

13

15

17

19

21

23

compressor datacurve fit

mean eror 0.312992%, std 0.02571

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

90000

120000 140000160000

180000

Scaled mass flow parameter

Com

pres

sor

effic

ienc

y

Compressor 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

90000

120000 140000160000180000

Scaled mass flow parameter

Com

pres

sor

effic

ienc

y

Compressor 1

11

If W is specified in kg/sec, cp in kJ/kg/K, and T in K thenP is in kW. For a given pressure ratio across the turbine,the outlet temperature can be computed assuming isen-tropic expansion,

(42)

In order to account for the fact that the expansion throughthe turbine is not isentropic, the turbine (total-to-static)isentropic efficiency is introduced and defined as

(43)

where T2,is is the temperature of the exhaust gas leavingthe turbine if the expansion were isentropic. Note that theturbine outlet temperature is evaluated as static, becauseno use can be made of the kinetic energy left in theexhaust gas at the turbine outlet.

Using equation (42) and the above defined isentropic effi-ciency, we obtain the following expression for the turbinepower:

(44)

The turbine outlet temperature T02 is given by

(45)

Similar to the analysis done for the compressor, it is pos-sible to obtain more insight into various terms that com-prise the above equation. From Euler's equation for theturbine rotor we obtain

(46)

where ∆hideal is the ideal (or isentropic) enthalpy dropacross the turbine, Cθ1 is the tangential velocity compo-nent of the flow at the entry to the rotor and U1 is thevelocity of the turbine rotor at the point where the flowenters. Assuming that there is no swirl at the turbine out-let we obtain Cθ2 = 0, and ∆hideal = U1Cθ1. Similar tothe compressor slip factor, the ratio Cθ1/U1 is a functionof several variables, e.g. the number of rotor blades. Con-sequently, in the ideal case

(47)

is proportional to the square of the turbo speed. Ofcourse, various sources of energy losses (accounted forby the turbine efficiency) do introduce the dependence onthe turbine mass flow rate. See [13] for more details.

Typically, the turbine model is used during simulations tocalculate turbine power and mass flow rate given inletand outlet pressure values and turbocharger speed. Theturbocharger speed and inlet pressure are usually statevariables whose behavior is determined by differentialequations based on compressor-turbine power balanceand ideal gas law respectively. The turbine outlet pres-sure is a function of the flow restriction of the exhaustsystem assembly downstream of the turbine. Ideally, it isequal to atmospheric. As a consequence of this use ofthe model, it is convenient to express the turbine charac-teristics in the same form as the compressor Model I, i.e.,the turbine mass flow is expressed as a function of tur-bine pressure ratio and rotational speed.

The turbocharger manufacturer characterizes the massflow rate and isentropic efficiency over a certain operat-ing range (typically for turbine speeds between 100kRPM and 180 kRPM and pressure ratio between 0.3and 0.8) on a flow stand and supplies the information intable form. For turbines with variable inlet geometry(generically abbreviated as VGT - variable geometry tur-bochargers), an additional input for these maps is theinlet geometry setting ϑvgt

Again, we'll use the scaled mass flow parameter, φ, andturbine speed parameter, , and neglect differencesbetween static and stagnation pressures and tempera-tures:

(48)

The use of these parameters eliminates the dependenceof the performance maps on inlet conditions (Tin and pin).These maps are now only a function of speed parameter,pressure ratio across the turbine, and, if applicable, VGTsetting ϑvgt:

(49)

and

(50)

where U/C is the blade-speed ratio [13], defined as

(51)

and D denotes the turbine blade diameter. Note that theblade speed ratio is a function of pressure ratio andspeed parameter, and hence no new independent vari-ables are introduced.

12

It should be noted that the manufacturer supplied data onturbine characteristics is not as representative of actualturbine behavior on the engine as is the case for com-pressor data. The turbine is characterized on a flowbench with steady flow, whereas the turbine on theengine experiences strong pressure pulsations, whichwill influence its performance. For that reason, it may bepreferable to model the turbine using data measureddirectly on the engine. However, this does require the tur-bine speed measurement, which may not always beavailable. Figure (10) illustrates this to some extent: theflow measured on the engine is slightly higher than themapped flow for a given mean value of the pressure ratioacross the turbine. The figure also clearly illustrates theeffect of the wastegate: for expansion ratios close to 2and higher, an increasing portion of the exhaust flowbypasses the turbine through the wastegate. As a result,the total flow through the exhaust increases without build-ing up additional boost pressure or exhaust manifoldpressure. A final observation on Figure (10) is that here,again, we see that the turbine map does not cover theentire region of turbine operation on the engine. Just aswith the compressor, lower flows and lower expansionratios were not mapped. In contrast to the compressor,however, the turbine characteristics can be extrapolatedinto those regions without real difficulty.

Figure 10. Comparison of turbine flow characteristics provided by the turbine manufacturer and measured on the engine. At higher boost pressures, the wastegate opens and part of the exhaust flow bypasses the turbine. Since the wastegate flow was not measured separately, the turbine flow parameter includes the wastegate flow.

CURVE FITTING OF TURBINE MASS FLOW

FIXED GEOMETRY TURBINE – The mass flow ratethrough the turbine can be modeled as an adiabatic noz-zle flow, where the effective flow area is a function of theturbine speed parameter, and pressure ratio. In [11] it ispointed out that the standard orifice flow equations arederived assuming isentropic expansion. In the case of the

turbine flow, not only do we know that the expansion isnot isentropic, but we also know the isentropic efficiency.Hence, we could use a more general form of the equa-tions presented here, taking into account the isentropicefficiency. However, at least for the turbines consideredhere, this modification did not result in noticeably betterfits, hence we'll stick with the standard equations. There-fore, turbine flow equations are given by

(52)

where the effective turbine area At . is modeled as afunction of turbine pressure ratio and speed parameter.For a fixed geometry turbocharger, a simple fit for theeffective turbine flow area At was proposed by Jensen &Kristensen:

(53)

where the parameters kti are functions of the speedparameter:

This representation is only approximate because thepressure ratio across the turbine at which choked flowoccurs is actually lower than the expected value ofapproximately 0.55 (depending on exhaust gas composi-tion and temperature). The reason is that the turbineeffectively behaves as a series of two nozzles (inlet noz-zle vanes and rotor passages), which each individuallyexperience higher pressure ratios than the total acrossthe turbine. However, by including the pressure ratio as aparameter in the effective area fit (53), the model can stillrepresent an increasing mass flow parameter at pressureratios beyond the critical pressure ratio. A second restric-tion on the validity of the adiabatic nozzle flow model isthe fact that, in the limit, the turbine can actually act as acompressor with flow against a positive pressure gradi-ent, whereas in an adiabatic nozzle flow reversal wouldoccur under those conditions.

Figure (11) illustrates the results of the turbine flow curvefit. The accuracy is more than adequate.

VARIABLE GEOMETRY TURBINE – For variable geom-etry turbochargers, the turbine area is modified by chang-ing the inlet geometry. For example, in the case ofvariable nozzle geometry, the throat area of the nozzlesis modified, resulting in a change in expansion ratio forthe same mass flow rate. One way to model the turbineflow in the presence of a variable inlet geometry is to useequation (51) where the effective area is also a functionof the (normalized) geometry setting. For turbine 2 a rea-sonably good fit could be obtained this way without a tur-bine speed dependence, i.e., the mass flow parameter is

1 1.5 2 2.5 3 3.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18Turbine flow data

Turbine expansion ratio

Tur

bine

mas

s flo

w p

aram

eter

φ

Turbine mapMeasured engine data

13

expressed as a function only of the pressure ratio and theinlet geometry setting. A slight variation of equation (51)is given by

where 1-g is the theoretical zero flow pressure ratio (theintersection of the curves with the abscissa in Figure 12),and g is fitted as a quadratic function in vane position.The result of this fit is shown in Figure 12.

Figure 11. Curve fit of turbine flow for turbine 1 for different values of the speed parameter.

An alternative is to use a neural network with all inputs(speed parameter, pressure ratio, and vane setting) andone hidden layer with three neurons. The result of that fitis shown in Figure 13. An interesting note on extrapola-tion: the asterisks in Figure 13 mark the extrapolated tur-bine flows at 10kRPM. It is apparent that theseextrapolation results do not make physical sense. Hence,in order to get sensible extrapolation results for low tur-bine speeds, the neural network needs to be suppliedwith additional, artificial mapping points forcing the net-work to provide much lower turbine flows at these lowspeeds. For the previous fit based on the orifice flowequations the extrapolation did give sensible results.

CURVE FITTING OF TURBINE EFFICIENCY

For fixed turbine speed, the turbine efficiency typicallyhas the shape of an inverted parabola, and can usuallybe modelled by a quadratic or cubic polynomial in bladespeed ratio, with coefficients depending on the speedparameter [5], [7], [8]. In [10], a correction factor at lowexpansion ratios is used to account for the observedaccelerated decrease in efficiency at those lower expan-

sion ratios. However, in light of the comments on turbo-charger mapping limitations described earlier, theobserved decreased efficiency at lower expansion ratiosmay be an artifact of heat transfer effects occurring dur-ing the experimental procedure.

Figure 12. Curve fit of turbine flow for turbine 2, using modified version of adiabatic nozzle flow. VGT setting, normalized between 0 and 1 is indicated in the graph.

Figure 13. Curve fit of turbine flow for turbine 2, using 3-node neural network. VGT setting, normalized between 0 and 1 is indicated in the graph. The asterisks near the y-axis are the result of extrapolation of the turbine flow parameter to 10kRPM turbine speed for the given VGT settings. This network would have to be retrained and supplied with artificial mapping points at low turbine speeds to force it to give more sensible low speed turbine flows.

The curve fit illustrated in Figure 14 is a quadratic polyno-mial in blade speed ratio with coefficients linearly depen-dant on the speed parameter:

(54)

10

3045

4061

4738

5415

6092

Turbine expansion ratio

Sca

led

mas

s flo

w p

aram

eter

φ

Turbine data for turbine 1

1.5 2.0 2.5 3.0

0.08

0.16

0.24

10

00.167

0.333

0.5

0.667

1

Turbine expansion ratio

Sca

led

turb

ine

mas

s flo

w p

aram

eter

φ

Turbine 2

0.2

0.4

1.5 2.0 2.5 3.0

0

00.167

0.333

0.5

0.667

1

Turbine expansion ratio

Turbine data for turbine 2S

cale

d tu

rbin

e m

ass

flow

par

amet

er

φ

0.2

0.4

1.5 2.0 2.5 3.0

14

SUMMARY AND CONCLUSIONS

In this paper we have presented an overview of curve fit-ting techniques for automotive turbochargers, motivatedby two facts: Turbocharger mapping data usually do notspan the operating range experienced on the engine,hence a need for reliable extrapolation, and, even thoughit still appears to be commonly used, the representationin lookup tables is not well-suited for implementation indynamic engine simulation models. .

Figure 14. Curve fit of turbine efficiency turbine 1 for different values of the speed parameter

The emphasis was on compressor flow rate curve fittingfor two reasons: engine models are generally more sensi-tive to errors in mass flow than to errors in temperatures(within reason, of course), and the compressor character-istics are more difficult to capture based on first principlesmodels than the turbine characteristics. In other words,the compressor flow rate representation is both the morecritical and the more challenging task. Four differenttechniques were described and illustrated on three differ-ent compressors. Based on the choice of input and out-put, the methods can be classified into two categories:Model I uses pressure ratio and speed to compute massflow, Model II uses mass flow and speed to computepressure ratio. The particular choice has interesting impli-cations on the overall engine model. For a compressorcharacterisation in the form of Model II, a neural networkappeared to be the most accurate technique for all threecompressors. It is also the most flexible in that its com-plexity is easily changed by changing the number of neu-rons or hidden layers. Furthermore, by combiningcompressor pressure ratio and isentropic efficiency inone network, the total number of coefficients comparesnot unfavorably to the other techniques described. For acompressor characterisation in the form of Model I how-ever (typically used in control oriented mean valueengine models), a neural network was not found to givean acceptable representation. Sensible extrapolation ofthe model to lower turbocharger speeds and compressorflow rates could not be obtained, even after manual mod-ifications in the compressor mapping data. Instead, an

empirical model proposed by Jensen & Kristensenappeared to be best suited.

For the representation of the turbine characteristics wemade a similar experience as for the compressor Model Irepresentation: a neural network can give an accurate fitof the mapping points, but a sensible extrapolation tolower expansion ratios and turbine speeds can only beobtained by augmenting the experimental mapping datawith a suitable number of articial mapping points in thoseareas. On the other hand, a more physically based modelusing adiabatic nozzle flow equations extrapolated verywell without manual intervention in the mapping data.

ACKNOWLEDGEMENT

Thanks to Michiel van Nieuwstadt from Ford ResearchLaboratories and an anonymous reviewer for carefulreading of the draft paper. Thanks also to David Flaxing-ton from Allied Signal/Garrett for a number of fruitful dis-cussions.

REFERENCES

1. Demuth, H., Beale, M., "Neural Network Toolbox, version3.0", The Mathworks, Inc., 1988.

2. Flaxington, D., Allied Signal/Garrett. Personal communica-tion. July1996.

3. Fraden, J., "AIP Handbook of modern sensors", AIP Press,1993.

4. Nelson, S.A., Filipi, Z.S., Assanis, D.N., "The use of neuralnetworks for matching compressors with diesel engines,"Spring Technical Conference, volum ICE-26-3, pages 35-42, 1996.

5. Jensen, J.P, Kristensen, A.F., Sorenson, S.C., Houbak, N. ,Hendricks, E., "Mean value modeling of a small turbo-charged diesel engine," SAE 910070.

6. Kao, M., Moskwa, J.J., "Turbocharged diesel engine model-ing for nonlinear engine control and estimation", ASMEJournal of Dynamic Systems, Measurement and Control,Vol 117, 1995.

7. Kolmanovsky, I.V., Moraal, P.E., van Nieuwstadt, M.J., Crid-dle, M., Wood, P., "Modeling and identification of a 2.0 l tur-bocharged DI diesel engine". Ford internal technical reportSR-97-039, 1997.

8. Mueller, M., "Mean value modeling of turbocharged sparkignition engines", Master’s thesis, DTU, Denmark, 1997.

9. Puskorius, G.V., Feldkamp, L.A., "Decoupled extended Kal-man filter training of feedforward layered networks", Pro-ceedings IJCNN, 1991.

10. Sher, E., Rakib, S., Luria, D., "A practical model for the per-formance simulation of an automotive turbocharger", SAE870295.

11. Sokolov, A.A., Glad, S.T., "Identifiability of turbocharged ICengine models", SAE 1999.

0.55 0.6 0.65 0.7 0.75 0.80.66

0.67

0.68

0.69

0.7

0.71

0.72

0.73

0.74

0.75

3046

4061

4738

5415

6092

Blade speed ratio U/C

Tur

bine

effi

cien

cy

Turbine data for turbine 1

15

12. Watson, N., "Dynamic turbocharged diesel engine simula-tor for electronic control system development", Journal ofDynamic Systems, Measurement, and Control 106, pp.27-45, 1984.

13. Watson, N., Janota, M.S., "Turbocharging the internal com-bustion engine", John Wiley & Sons, 1982.

14. Winkler, G., "Steady state and dynamic modeling of engineturbomachinery systems", PhD Thesis, University of Bath,1977.