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Modeling and Estimation of Unmeasured Variables in a Wastegate Operated

Turbocharger

Article  in  Journal of Engineering for Gas Turbines and Power · May 2014

DOI: 10.1115/1.4025498

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G.R. Vossoughi

Sharif University of Technology

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Modelling and Estimation of unmeasuredVariables in a Wastegate Operated Turbocharger

Rasoul Salehi

School of Mechanical EngineeringSharif University of Technology

Azadi Ave., Tehran,1458889694,IranEmail: r salehi@mech.sharif.ir

Gholamreza Vossoughi∗Professor

School of Mechanical EngineeringSharif University of Technology

Azadi Ave., Tehran,1458889694,IranEmail: vossough@sharif.edu

Aria AlastyProfessor

School of Mechanical EngineeringSharif University of Technology

Azadi Ave., Tehran,1458889694,IranEmail: aalasti@sharif.edu

AbstractABSTRACT

Estimation of relevant turbocharger variables is crucial for proper operation and monitoring of turbocharged(TC) engines which are important in improving fuel economy of vehicles. This paper presents mean-value modelsdeveloped for estimating gas flow over the turbine and the wastegate (WG), the wastegate position and the com-pressor speed in a TC gasoline engine. The turbine is modelled by an isentropic nozzle with a constant area andan effective pressure ratio calculated from the turbine upstream and downstream pressures. Another physicallysensible model is developed for estimating either the WG flow or position. Provided the WG position is available,the WG flow is estimated using the orifice model for compressible fluids. The WG position is predicted consideringforces from the WG passing flow and actuator. Moreover, a model for estimating the compressor speed in low andmedium compressor pressure ratios is proposed, using the compressor head and efficiency modified by the turbineeffective pressure ratio. The estimates of the turbocharger variables match well with the experimentally measureddata. The three proposed models are simple in structure, accurate enough to be utilized for engine modelling, andsuitable to be validated and calibrated on an internal combustion engine in a test cell.

NomenclatureA Area (m2)Cd Discharge coefficient (−)Cp Specific heat at constant pressure (kJ/kg.K)d Diameter (m)F Force (N)K Spring stiffness (N/m)ṁ Mass flow (kg/s)P Pressure (kPa)Pr Pressure ratio (−)R Gas constant (kJ/kg.K)T Temperature (K)

∗Address all correspondence to this author.

U Speed (m/s)X Displacement (m)γ Ratio of specific heats (−)ρ Density (kg/m3)ω Rotational speed (rad/s)

Subscritsa airamb ambientb boostc compressorcorr correctedcyl cylinderds downstreamem exhaust manifoldeff effectiveeng enginef fuelK kineticP pressureref referencesp springt turbineWG wastegateus upstream

1 IntroductionEngine downsizing using turbochargers in gasoline engines is a common solution to reducing vehicles’ fuel consumption

and emissions for automotive manufacturers [1]. In fact, displacement reduced turbocharged engines regenerate thermalpower from exhaust gas either by a wastegate controlled or a variable geometry turbine. The turbochargers controlled by thewastegate (WG) benefit from low cost and high durability in the harsh thermal environment of the exhaust manifold in thecurrent automotive market [2].

When a turbocharger is added to a naturally aspirated engine, different strategies should be included in the engine controlunit (ECU). These strategies are either to control the turbocharger [3, 4] or to monitor its performance for proper and safeoperation of both the engine and turbocharger [5, 6]. Among the strategies, gas flow estimation is imperative for the ECUto control the gasoline engine torque and emission. The ECU requires either measuring or estimating the air mass flowas it enters the intake path from the air filter and continues monitoring the flow streamline until it exits from the tailpipe.However, measuring the turbine and the WG flows is not possible due to harsh exhaust environment. Moreover, measuringthe compressor variables such as its rotational speed and efficiency requires additional sensors to be implemented in theengine which raises cost concerns.

The compressor speed and efficiency can be predicted using its characteristic map [7]. But this introduces large errorsdue to linear interpolation and extrapolation [8,9]. Another problem associated with using maps provided by the compressorsupplier is limited information at low compressor pressure ratios [9]. On the other side of the engine, there exist challengeswith estimation of gas flow over the turbine and the WG when the WG is not closed. The mean-value modelling (MVM),termed also as Zero-Dimensional (0-D) modelling in some literatures, is the main approach to get reliable estimation ofthe turbocharger variables in the ECU. Simple structure of MVM is shown to be accurate enough even for transient enginecontrol applications [10, 11]. The turbine flow is estimated using MVM techniques such as empirical modelling [12, 13] orsemi physical modelling [14]. In semi physical modelling the main idea is using a model of an adiabatic flow through onenozzle with a variable area or two successive nozzles with constant areas [15]. For the WG, the gas flow is estimated utilizingan orifice model with a variable discharge area [16, 17].

This paper presents MVM formulations to estimate the wastegate and turbine flows along with the compressor speed andefficiency. The turbine flow is estimated from an isentropic compressible orifice flow model with a constant discharge areaand an effective pressure ratio calculated from the turbine’s actual upstream and downstream pressures. Then, the turbineflow model is used to estimate parameters required to develop a physically sensible model for the wastegate position andflow. As a new method, the coupling between the compressor and the turbine is used to predict the compressor speed andefficiency which can be easily calibrated on an engine test bench. The compressor speed model can replace existing models

Table 1. Engine specifications

Description Value Unit

Displacement Volume 1.7 Litre

Compression ratio 9.9 -

No. of cylinders 4 -

Bore × Stroke 78.6 × 85 mm

Max. torque 215@2200-4800RPM N.m

Max. power 110@5500RPM kW

Max. WG Disp. 12 mm

to improve accuracy of estimation algorithms presented in [7, 18]. The proposed models for the turbocharger main variablesare easy to be calibrated on a new engine for possible control and monitoring applications.

After the introduction, the experimental setup used for validation in this paper is described in section 2. Section 3addresses modelling of the turbine gas flow. The WG model to estimate its flow and position over different engine operatingpoints is explained in section 4. Finally, the model for detection of the compressor variables is outlined in section 5.

2 Experimental setupA 4-cylinder port fuel injected gasoline turbocharged engine is used for validation of all models in this work. As shown

in Fig. 1, the engine is equipped with a turbocharger with a radial turbine and a centrifugal compressor. The turbine iscontrolled by a wastegate which allows bypassing the exhaust gas from the turbine. Opening the WG reduces exhaust backpressure (which improves the fuel economy) in operation points where the engine does not require high boost pressure.The WG position is controlled by a pneumatic actuator. Conventionally, the ECU regulates the pressure inside the actuatorcylinder (Pcyl) by controlling an electronic solenoid valve (solenoid valve1 in Fig. 1) fed by pressures from downstream andupstream of the compressor. To additionally make it possible to adjust the WG position independent of the ECU command,a general control unit (GCU) is set up and used in this work. The GCU regulates the pressure inside the WG actuator bycontrolling solenoid valve2 (Fig. 1). Therefore, using the setup shown in Fig. 1, two modes are available to control the WGactuator:

Control mode-I: ECU-controlled operation. In this mode the ECU regulates Pcyl (via the dashed line from solenoidvalve1 to the WG actuator) utilizing solenoid valve1.

Control mode-II: GCU-controlled operation. In this mode the solenoid valve1 is disconnected from the WG actuatorand Pcyl is regulated using solenoid valve2 controlled by the GCU. Connected to a pressure reservoir, solenoid valve2 canincrease or decrease the WG opening compared to control mode-I.

In Fig. 1, Pcyl is measured using two pressure sensors PS1 and PS2 in the two control modes and PS3 and PS4 areused to measure the upstream and downstream turbine pressures. Figure 1 also illustrates the engine instrumentation. Aproximity sensor is used to measure the compressor speed with ±500(rpm) resolution. Temperatures are measured usingK-type thermocouples with an accuracy of ±2◦C and pressures are measured using piezoresistive pressure transmitters with±0.1kPa accuracy. The fuel mass flow is measured using a temperature-controlled fuel mass flow meter with a measurementaccuracy of 0.12% and maximal measuring frequency of 20HZ. The measured fuel in the test cell is used along with themeasured engine airflow to compute the exhaust gas flow. A hall-effect sensor with −40 to +150◦C working temperature isintegrated with the engine setup for measuring the WG displacement. The hall sensor is connected to the WG actuator usinga thermal isolating pad. Further engine characteristics are presented in Table. 1.

3 Turbine flow modellingTurbomachine performance characteristics such as mass flow and efficiency are specified in terms of pressure ratio and

rotating speed. For these machines, the effect of inlet conditions are cancelled by defining corrected mass flow and correctedspeed. Therefore, for a machine of a specified size which handles a single gas, the dimensional analysis for compressiblefluids suggests the following [19]:

Fig. 1. Schematic of the engine test bed and the GCU to adjust the WG position independent of the engine operation

ṁcorr = f(Pr, ωcorr) (1)

where ωcorr = ω/√Tus is the corrected rotating speed of the machine shaft and the corrected mass flow is defined as

ṁcorr = ṁ√Tus/Pus for a turbine and ṁcorr = ṁ

√Tus/Tref/(Pus/Pref ) for a compressor. The pressure ratio, Pr, is

either the upstream to downstream turbomachine pressure ratio or the reverse.Radial turbines have rather weak dependencyupon the corrected speed [19], and the pressure ratio is the most significant variable in Eqn. 1. Figure 2 compares themeasured corrected flow over the radial turbine with that of a conventional nozzle. The test procedure for measuring thepresented data includes a large region of the TC engine map. The flow over the turbine is measured by closing the wastegatecompletely thus the entire engine flow passes through the turbine. As plotted in Fig. 2, the turbine flow resembles a nozzlemodel but it does not reach the choking point at a pressure ratio about 0.53 as the nozzle model predicts [15, 20]. Therefore,in this work the turbine flow is estimated using a single nozzle model with a constant area but with an ”effective” pressureratio (Fig. 3) to avoid early choking prediction. The effective pressure ratio is calculated from the turbine actual pressureratio using; Prt,eff = (Pds,t/Pus,t)α , thus the turbine flow is estimated by the following:

ṁt =Pus,t√RemTus,t

At,eff

√2γemγem−1 (Pr

∗t

2γem − Pr∗t

(γem+1)γem )

Pr∗t = max(Prt,eff ,2

γem+1

γem/(γem−1))

Prt,eff =Pds,tPus,eff

= (Pds,tPus,t

)α

(2)

where Aeff,t is the effective area of the turbine, α is a constant to include the turbine effective pressure ratio in theturbine flow model instead of its real pressure ratio, Tus,t is the turbine upstream temperature, Rem is the gas constant forthe exhaust manifold and, Pus,t and Pds,t are the turbine’s upstream and downstream pressures . More details on how tooptimally estimate α and to calculate Aeff,t are presented in [21].

The turbine flow modelled by Eqn.( 2) with α = 0.39 and At,eff = 2.85 × 10−4 (m2) is compared to the measuredengine flow in Fig. 4. As shown, predicted flow for the turbine agrees well with measured gas flow. The estimation error ofthe results in Fig. 4 has an average of 1.4% with a standard deviation, σ, of 1.5%.

Fig. 2. Comparison between the turbine corrected flow and a conventional orifice corrected flow

Fig. 3. The turbine schematic and 0-D model

4 WG flow and displacement modellingThe WG is an orifice with a variable area. Therefore the flow over the wastegate is modeled using the orifice isentropic

flow equation,

ṁWG = ACdPus,t√RemTus,t

f(Pds,tPus,t

) (3)

where the discharge factor is a function of the WG actuator position, ACd = g(XWG) . The WG flow in Eqn. (3) can alsobe calculated from the difference between the engine flow and the turbine flow,

ṁWG = (ṁeng + ṁf )− ṁt (4)

Fig. 4. Modeled turbine flow compared to measured exhaust flow with closed WG

where ṁeng is the engine air mass flow, ṁf is the injected fuel mass flow and ṁt is the turbine mass flow from Eqn. (2).When the WG flow is known from Eqn. (4), the discharge factor can be computed using Eqn. (3). Fig. 5 shows ACdat different measured WG positions. As shown, the discharge factor has a linear relation to the WG displacement for adisplacement range of [0,5] (mm). The linear relation between the WG displacement and the discharge factor makes it easyto estimate the WG flow using Eqn. (3). However, the WG position is not always measured in production TC engines and itneeds to be estimated.

Fig. 5. WG discharge factor at different WG positions

The WG position is predicted based on forces applied to its actuator and the WG flapper. Fig. 6 shows the schematic ofthe wastegate and its force diagram. As illustrated, there are three main sources of force which affect the WG position. Thefirst force is from the pressure difference across the actuator cylinder (Fcyl) which opens the WG. This pressure difference(Pcyl−Pamb) is controlled by the WG electronic solenoid valve which incorporates pressures from downstream and upstreamthe compressor. The second source of the force is the WG spring force (Fsp) preloaded by an initial displacement, X0. Thespring is to close the WG allowing more flow over the turbine. The third force is applied by the exhaust gas flow inside the

Fig. 6. Wastegate schematic and its force diagram

exhaust manifold, Fflow. Therefore, when the WG is open, XWG is estimated using the torque balance at point A, shown inFig. 6, as:

[Fsp − Fcyl].r2.cos(θ)− Fflow.r1 =[Ksp(XWG +X0)− (Pcyl − Pamb).Acyl].

r2.cos(θ)− Fflow.r1 = 0(5)

in which, Acyl is the area of the piston fitted inside the WG actuator cylinder, r1 and r2 are the length of the connecting rods,Pamb and Pcyl are the ambient and the WG cylinder pressures, and Ksp is the spring stiffness. Table 2 in the Appendix liststhe measured values of parameters used in Eqn. (5).

When the exhaust flow hits the WG flapper, a force is applied to the flapper due to change in the flow momentum. Thisforce is calculated by applying the conservation of momentum for a control volume bounded by a surface S around the WGflapper (Fig. 6) as:

Fflow +∫∫S

FsurfdS +∫∫∫∫CV

βρemdv =∫∫S

U(ρemUdS) +∂∂t

∫∫∫∫CV

U(ρemdv)(6)

where Fsurf is the force acting on the control surface S inN/m2, β is the body force inN/kg, v is the volume of an elementinside the control volume in m3 , ρem is the exhaust gas density in kg/m3 and U is the fluid velocity at the control surfaceS in m/sec. The following are assumptions used to calculate Fflow from Eqn. (6):

1. Assumption I: Since we present models to predict steady-state flow, the time dependency term is eliminated as follows.

∂

∂t

∫∫∫∫CV

U(ρemdv) = 0. (7)

2. Assumption II: The effect of body forces exerted by the surroundings are ignored (β = 0). The body force is mainlyexerted by the gravity and the weight of elements inside the control volume is negligible.

3. Assumption III: The pressure in front of the flapper surface is Pus,t and in the back is Pds,t when the WG opens. Theassumption is considered for small displacement of the WG, where XWG affects ṁWG noticeably. Therefore,∫∫

S

FsurfdS = (−Pus,t + Pds,t)Aflap. (8)

4. Assumption IV: The gas flow hitting the flapper surface leaves it parallel to the surface. This results the following:

∫∫S

U(ρemUdS) = ṁWGUflow. (9)

(a)

(b)

Fig. 7. a) Schematic of the geometry used for the WG 3-D numerical simulation; b) Normalized pressure difference across the WG flapperarea at different radial positions.

A closer look into Assumption III is made by a 3-D numerical simulation of the pressure distribution on the flappersurface in different WG positions. The simulation is done by numerical calculation of the fluid dynamics through a simplifiedgeometry which resembles the WG in the exhaust path (Fig. 7-a). The simulated pressure difference across the WG flapperis shown in Fig. 7-b. As plotted, in 1(mm) WG displacement, the normalized pressure difference on most of the WG flappersurface is one. Moreover, when the WG displacement increases to 5(mm), the normalized pressure difference is still closeto unity in most area of the flapper surface. The normalized pressure difference in Fig. 7-b is calculated from the following:

Norm. ∆PWG =Pflapper − Pds,tPus,t − Pds,t

(10)

where Pflapper is the pressure on the flapper surface. Using the 4 assumptions above, the flow force from Eqn. (6) is:

Fflow = [ṁWGUflow + (Pus,t − Pds,t)Aflap] =[Rus,tTus,tṁ

2WG

Pus,tAWG+ (Pus,t − Pds,t)Aflap]cos(θ)

(11)

in which Aflap and AWG are the areas of the flapper and the WG hole respectively (Fig. 6). In Eqn. (11), ṁWGUflow iscalled the force from the gas kinetic energy (Fflow,K) and (Pus,t−Pds,t)Aflap is called the force from the exhaust pressure,

(Fflow,P ). Plugging Eqn. (11) into Eqn. (5), the WG displacement from its equilibrium point is estimated as the following:

XWG =Tt

ksp.r2−X0 if Tt > Fs,0.r2

XWG = 0 if Tt < Fs,0.r2(12)

where Tt = Fcyl.r2 + Fflow.r1 and Fs,0 is the WG initial load from the spring. As Eqn. (12) proposes, the WG actuatorand the flow forces should be large enough to cancel the WG spring force otherwise the WG stays in its initial position(XWG = 0). Figure 8 depicts results of the WG modelling. In Fig. 8-a, the modelled WG flow from Eqn. (3) with aknown XWG is compared to the WG flow computed from Eqn. (4). In Fig. 8-b, the WG position calculated from Eqn. (12)is compared to the measured WG position. Results of both the WG flow and displacement models reveal good agreementbetween measured and estimated data.

(a)

(b)

Fig. 8. Results of the wastegate model; a) wastegate flow; b) wastegate displacement.

Each force term in Eqn. (5) has a different contribution to the final WG displacement. To check the contribution of theforces in the total displacement of the WG, the contribution factor is defined as:

Contribution factor =Fx

Fcyl + Fflow,K + Fflow,P∗ 100 (13)

where Fx is either Fcyl, Fflow,K or Fflow,P . Figure 9 shows how much each force in Eqn. (5) has contributed to openthe WG. As expected the cylinder force has the most significant and the kinetic force has the least effect on the wastegateposition. The pressure force, Fflow,P , is also considerable especially when the downstream and upstream turbine pressuredifference increases. This happens when the WG area is small and most of the exhaust flow passes over the turbine.

Fig. 9. Contribution factor of forces applied to the WG actuator.

5 Compressor rotational speed estimationThe rotational speed of a centrifugal compressor can be predicted using its characteristic map once the compressor

pressure ratio and corrected airflow, ṁc,corr , are known. Figure10-a shows the compressor map used in this work alongwith the compressor operating points in three engine tests; a) full load test, b) a test with the WG kept closed and, c) atest with the WG left open (i.e. the WG operates normally in the control mode-I as described in section 2). As shown, thecharacteristic map represents the compressor flow and pressure ratio at specific compressor rotational speeds. Therefore,at a speed not included in the map, the compressor speed can be predicted by interpolating the map. However, linearinterpolation in the characteristic map introduces large errors in speed prediction [8]. Moreover, the map is not suitable foranalytical works like stability investigation for an estimation algorithm such as presented in [18]. Dimensional analysis isapplied, to the compressor map for estimating the compressor speed, reducing the interpolation error. Two non-dimensionalvariables known as the head parameter, Ψ and the normalized compressor flow rate, Φ are defined as follow [22]:

Ψ = [Cp,aTus,c((Pds,c/Pus,c)(γa−1)/γa − 1)]/0.5U2c

Uc = π/60.dc.ωcΦ = ṁc/(ρus,c.d

3c .ωc)

(14)

where dc is the compressor impeller diameter, Uc is the impeller tip velocity, ρus,c is the air density upstream the compressor,ωc is the compressor rotor speed and, Pds,c and Pus,c are the compressor upstream and downstream pressures. Results ofthis transformation is shown in Fig. 10-b. As observed, the wide and scattered operating points in a compressor characteristicmap are transformed into a band by the Φ−Ψ transformation. The band is represented by a curve fitted to the transformationresults (Fig. 10-b). This curve can be used to estimate the compressor variables.

To investigate application of the non-dimensional variables, Φ − Ψ, for estimating the compressor performance inlow and medium pressure ratios, two extremely conditioned tests are discussed. In the first extreme test, the turbochargerWG is closed completely. The WG-closed test shows the maxima of operation variables (speed and pressure ratio) for thecompressor when operating on an engine. Compared to an engine full load test in Fig. 10-a, it shows that from low to mediumcompressor pressure ratios, the WG-closed condition also occurs during the engine full load operation. In the second test, asanother possible extreme condition for the turbocharger on an engine, the WG is opened as much as possible by the engineboost pressure. In this case the entire boost pressure is applied to the WG actuator and the WG operates based on its appliedforces. The WG-open test shows the minima of operating points for the compressor operating on an engine. The compressor

(a)

(b)

Fig. 10. a) The compressor characteristic map and its trajectory during the engine test points. b) Transformation results from the compressormap and the engine test points.

trajectories for the WG-open and WG-closed tests are plotted in Fig. 10-a. As shown, in both tests the compressor has thesame trajectories for pressure ratios below that of point “A”, but the two tests can be distinguished at higher pressure ratios.The reason is the fact that the boost pressure is not high enough to open the WG until reaching the point ”A”.

Transformation results of the two extreme tests are shown in Fig. 10-b. As observed, when the compressor is coupledto the engine, the Φ − Ψ relation, observable from transforming the whole map of the compressor, is confined to a specificregion with weak dependency of Φ on Ψ. The weak dependency of Φ on Ψ and the width of transformed compressor mapare two sources of error when the fitted curve is used for the compressor flow estimation.

In a turbocharged engine, the coupling between the compressor and the turbine created by the engine limits the rangeof the compressor operation in its characteristic map (Fig. 10-a). To use the effect of this coupling for estimation of thecompressor operation point, the compressor speed coefficient, CFc, is proposed as:

CFc =ηc

Ψ.P r2t,eff(15)

where ηc is the compressor isentropic efficiency from the following:

ηc =Tds,c|s−Tus,cTds,c|a−Tus,c =

Tus,c((Pds,c/Pus,c)(γa−1)/γa−1)

Tds,c|a−Tus,c(16)

in which Tus,c is the compressor upstream temperature and, Tds,c|s and Tds,c|a are the isentropic and actual compressordownstream temperatures. As Eqn. (15) suggests, the CFc uses the exhaust manifold information from either the estimated[5] or measured turbine effective pressure ratio. To explore more conditions possible for the turbocharged engine, two newtests are designed that resemble compressor operation under some deviations of the engine from the normal operation. Thefirst test is with a WG that is stuck open and cannot be closed. This test extends the compressor trajectory to lower pointsthan those of the WG-open test. The test is carried out using the GCU which controls the WG position utilizing an externalpressure source (Fig. 1). The second test is a test with a gas leakage in the exhaust manifold.

The compressor corrected speed (ωc/√Tus,c) during the four described tests is shown in Fig. 11-a. As plotted, there is

no unique dependence of the compressor speed with its mass flow in the four tests. If one plots ηc/Ψ at different compressormass flow rates for the four tests, on the other hand, the test points are merged towards a curve with higher correlation factoras shown in Fig. 11-b . Although ηc/Ψ is correlated to ṁc,corr, but this correlation reduces as ṁc,corr increases. Thiscreates an estimation error if the fitted curve is used to estimate ηc/Ψ, specifically at high compressor mass flow rates. Byusing CFc as defined in Eqn. (15), correlation with the compressor corrected mass flow is further improved as shown inFig. 11-c (compare the correlation coefficientR2 in Figs. 11-b and c). Moreover, as Fig. 12 shows, CFc has better sensitivityto ṁc,corr since the fitted curve in Fig. 11-c has higher slops (i.e. higher gradients) compared with the curve in Fig. 11-bspecifically at high compressor flow rates.

5.1 Modelling the compressor downstream temperatureThe proposed strategy to estimate the compressor rotational speed in Eqn. (15) requires the compressor isentropic

efficiency, ηc, to be known a priori. From Eqn. (16) ηc is a function of the compressor pressure ratio, Prc, Tus,c andTds,c. The compressor pressure ratio can be estimated using the conventionally measured boost pressure, Pb; compensatingthe pressure drop due to the airflow through the engine intercooler. However, Tds,c is not usually measured in productionengines. To reduce the cost of integrating an additional temperature sensor downstream of the compressor, the temperatureis estimated using Prc, Tus,t, Tus,c and ṁc . The idea of using the turbine upstream temperature is due to the effect ofthe exhaust temperature on the compressor downstream temperature and isentropic efficiency particularly at low compressorspeeds and flows [23]. Moreover, Tus,t is already required in this work to estimate the WG and turbine flows.

The compressor downstream temperature is modelled using a simple structured neural network, NN. A two-layer feed-forward network with 9 sigmoid neurons in the hidden layer and a linear output neuron is trained to estimate Tds,c (Fig. 13).The error back propagation with steepest descent is used as the training algorithm. For training the NN network, 60% ran-domly selected data from all the four tests in Fig. 11 are used to train the NN network. The NN performance is demonstratedin Fig. 14. As shown, the NN can effectively estimate Tds,c for all the four engine test conditions described in section 5.

5.2 Results of the compressor speed estimationFinally, the compressor speed is estimated using the following algorithm:

1. Estimate CFc at each compressor corrected flow using a curve fitted to the results of Fig. 11-c.2. Estimate the compressor downstream temperature utilizing the developed NN, and then use it to compute ηc.3. Compute Ψ from Eqn. (15), and then invert it to get ωc using Eqn. (14).

The above algorithm is schemed in Fig. 13. Results of the estimated compressor speed are shown in Fig. 15. As shownthere is good agreement between the estimated and measured compressor speed for the four test conditions. The averageestimation error is ē = 430(RPM) with a standard deviation of σ = 2100(RPM). This proves the suitability of theproposed model to estimate the compressor speed at low and medium pressure ratios and airflows.

6 ConclusionSteady state models suitable for mean-value computations in an engine control unit are presented to estimate a tur-

bocharger working conditions. The turbine flow is estimated using the model of an isentropic nozzle with an effectivepressure ratio. The use of the effective pressure ratio circumvents wrong prediction of early choking for the turbine. TheWG flow is estimated using the orifice model for compressible fluids, if the WG position is measured. For engines that theWG position is not measured, its position is estimated using a physically sensible model of the WG and its applied forces.Experimentally validated WG displacement model takes into account forces from the WG passing flow, cylinder pressure,and spring. It is shown that about 85% of the WG displacement is due to the WG cylinder force and 15% is from the flow

(a)

(b)

(c)

Fig. 11. a,b) The compressor corrected speed and ηc/Ψ at different compressor corrected flows. c) Alignment of the test points to a singlecurve using the defined speed coefficient.

Fig. 12. Sensitivity (∂/∂ṁc,corr) of CFc and ηc/Ψ to the compressor corrected flow

Fig. 13. Overall structure of the compressor speed model.

force which comprises the force from the flow kinetic energy and the exhaust manifold pressure. Another model for esti-mation of the compressor rotational speed is proposed, which utilizes the coupling between the compressor and the turbine.The new compressor model uses the turbine effective pressure ratio to modify the compressor head parameter utilized tocalculate the compressor speed. The estimated compressor speed by the new model matches well with the experimentallymeasured speed in an engine test cell.

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20 40 60 80 100 120 14020

40

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Mod

eled

T ds,

c (o C

)

Measured Tds,c

(oC)

WG−closedWG−openWG−offsetExhaust leak

ē = 0.2 oCσ = 1.5 oC

Fig. 14. Results of modelling the compressor downstream temperature.

Fig. 15. Estimation results for the compressor rotational speed, ωc

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Appendix : Turbocharger parametersThis appendix lists values for the constant parameters used in modelling the turbocharger.

Table 2. Turbocharger parameters

Par. Value Unit Par. Value Unit

Acyl 5.9× 10−4 m2 r1 19× 10−3 m

Aflap 5.3× 10−4 m2 r2 30× 10−3 m

AWG 2.8× 10−4 m2 Ksp 12.4 kN/m

X0 4.9× 10−3 m dc 8× 10−2 m

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