control-oriented turbine power model for a variable …...with the turbine modeling issues addressed...

Original article

Proc IMechE Part DJ Automobile Engineering2018 Vol 232(4) 466ndash481 IMechE 2017Reprints and permissionssagepubcoukjournalsPermissionsnavDOI 1011770954407017702996journalssagepubcomhomepid

Control-oriented turbine power modelfor a variable-geometry turbocharger

Tao Zeng and Guoming G Zhu

AbstractA control-oriented model for the variable-geometry turbocharger is critical for model-based variable-geometry turbo-charger control design Typically the variable-geometry turbocharger turbine power is modeled with a fixed mechanicalefficiency of the turbocharger on the assumption of an isentropic process The fixed-efficiency approach is an oversimpli-fication and may lead to modeling errors because of an overpredicted or underpredicted compressor power This leadsto the use of lookup-table-based approaches for defining the mechanical efficiency of the turbocharger Unfortunatelysince the vane position of a variable-geometry turbocharger introduces a third dimension into these maps real-timeimplementation requires three-dimensional interpolations with increased complexity Map-based approaches offergreater fidelity in comparison with the fixed-efficiency approach but may introduce additional errors due to interpolationbetween the maps and extrapolation to extend the operational range outside the map Interpolation errors can be man-aged by using dense maps with extensive flow bench testing smooth extrapolation is necessary when the turbine isoperated outside the mapped region eg in low-flow and low-speed conditions Extending the map to this regionrequires very precise flow control and measurement using a motor-driven compressor which currently is not a standardtest procedure In this paper a physics-based control-oriented model of the turbine power and the associated powerloss is proposed and developed where the turbine efficiency is modeled as a function of both the vane position of thevariable-geometry turbocharger and the speed of the turbine shaft As a result the proposed model eliminates the inter-polation errors with smooth extension to operational conditions outside typically mapped regions

KeywordsVariable-geometry turbine internal-combustion engine control-oriented model

Date received 2 November 2016 accepted 3 February 2017

Introduction

The variable-geometry turbocharger (VGT) is ubiqui-tous in modern diesel engines The benefits of using theVGT over the traditional fixed-geometry turbocharger(FGT) have long been established1ndash4 Federally man-dated emission standards on nitrogen oxides (NOx)have forced the use of exhaust gas recirculation (EGR)in diesel engines This introduces a coupling betweenthe VGT and the EGR loops in the air path Thedynamics and control of this system have been topics ofactive research within the engine control community5ndash8

for the last two decades It is common for real-timeVGTndashEGR control systems to use supplier-providedturbocharger efficiency maps for given VGT vanepositions

Unfortunately supplier-provided turbocharger per-formance maps often do not cover the entire engineoperating range In Figure 1 the turbine pressure ratioand the mass flow rate regions are shown for three

maps constructed on the basis of the data from stan-dard hot-gas turbocharger flow bench tests steady-state engine dynamometer tests and transient vehicletests for the same VGT and the same engine It shouldbe noted that the steady-state engine dynamometertests cover the entire engine operational conditionsDuring transient operations the operational range ofthe turbocharger covers a much larger area than thesteady-state engine tests do in particular the supplier-provided map obtained from hot-gas flow bench testdata does not cover either the steady-state engine mapor the transient map (see Figure 1(a)) Although mostof the mass flow rate range for steady-state engine

Michigan State University East Lansing Michigan USA

Corresponding author

Guoming G Zhu Michigan State University 1497 Engineering Research

Court Room E148 East Lansing MI 48824 USA

Email zhugegrmsuedu

dynamometer tests remains inside the region for hot-gas flow bench tests a significant number of the transi-ent test data are outside the region for the hot-gas flowbench test data as shown in Figure 1(b) It is clear thatthe flow bench map does not provide full coverage Asa result operating the turbocharger outside its perfor-mance map requires extrapolation This has led to sev-eral investigations into extrapolation methods8ndash12

Typical extrapolation schemes rely on the second-orderor third-order polynomials as a function of the bladespeed ratio (BSR) This can lead to physically impossi-ble values and extremely large modeling errors Thetypical performance map or the data set which is usedto calibrate and validate the model is provided by theturbocharger manufacturer and is based on the dataobtained from hot-gas flow bench tests Since thesetests are performed in steady-state flow conditions themapped data cannot match the pulsating-flow opera-tional conditions when the turbocharger is coupled toan internal-combustion engine121314

First the turbine efficiency maps provided by tur-bine manufacturers combine the turbine efficiency withthe mechanical loss It is difficult to measure themechanical loss15ndash17 and it is not a required measure-ment in the standard flow bench tests (see the test codedescribed in SAE J182618) As a result the change inthe enthalpy across the compressor is calculated on thebasis of the turbine output power This makes the cal-culated turbine efficiency depend on the compressorcharacteristics and the measured turbine efficiency isnot the actual turbine efficiency since it is a combina-tion of both the turbine efficiency and the mechanicalefficiency As a result the turbine performance mapobtained from flow bench tests is different from the tur-bine characteristics when it is coupled to the engine asshown in Figure 1 In automotive turbocharger systems

the turbine extracts energy from the engine exhaust todrive the compressor for an increased boost pressureIn this application the available turbine power due tothe exhaust gas expansion process needs to overcomeboth the mechanical bearing loss and the heat transferloss to drive the compressor (Figure 2) Friction loss isdue to both the journal and the thrust bearings in theradial direction and the axial direction heat transferloss also drives the turbocharger operation away fromadiabatic behavior It should be noted that both lossesaffect the turbine efficiency However in transientengine operations the friction loss dominates the turbo-charger behavior in particular in fast transient opera-tions the thrust friction loss is determined by thebalanced thrust force between the turbine and the com-pressor and this is not available from turbochargermanufacturers Hence the turbine efficiency map pro-vided by the manufacturer is not suitable for transientoperations Most existing control-oriented turbochar-ger models assume a constant mechanical loss efficiencyfor simplicity257ndash10 This is because the axial frictiondue to the thrust bearing load and the radial frictiondue to the journal bearing cannot be directly measuredon the hot-gas flow bench which makes it challengingto model the mechanical loss directly

Second the required interpolation and extrapolationof the manufacture turbocharger performance maps toobtain the turbine efficiency in the current operationalconditions lead to multiple-dimensional lookup tableswith at least three inputs (the upstream and down-stream pressure ratio the reduced mass flow rate andthe VGT vane position) This can introduce an addi-tional computational burden for real-time controlFurthermore the map-based traditional isentropicmodel needs both the upstream condistions and thedownstream conditions (the pressure or the

Figure 1 Turbine operational ranges for hot-gas flow bench tests steady-state tests and transient operational conditions (a)turbine pressure ratio range (b) turbine mass flow rate rangeTC turbocharger dyno dynamometer

Zeng and Zhu 467

temperature) as the inputs which adds extra dynamicsstates for the engine air path model and extra sensorsfor real-time control

Finally the map-based model is not applicable tothe assisted and regenerative turbocharger since theoperational range of the associated turbine and com-pressor is quite different from that of a turbochargerwithout assisted power With the assisted power on theturbocharger shaft the turbine can operate in condi-tions with a much lower pressure ratio and a highershaft speed or in other words can operate the turbineoutside its traditional operational range Also thethrust friction torque can change dramatically since thethrust force direction can vary during transient opera-tions This is because the thrust force balance betweenthe turbine and the compressor is a function of theassisted or regenerative load on the turbocharger shaftThis results in a different thrust force for the thrustbearing and therefore the mechanical loss is differentfrom that of the non-assisted turbocharger

To summarize the turbocharger modeling for theturbine the compressor and the mechanical loss needsto be separated on the basis of their own dynamics Inthe literature some researchers proposed a fluid-dynamics-based approach to model the turbine powerrather using the supplier map19 The advantage of thismodeling approach is its compactness since it needsonly the upstream conditions However the associatedmodel needs the rotor inlet conditions (after the vanenozzle) which are difficult to measure and the mechani-cal loss model to calculate the mechanical efficiencyUntil now the turbine downstream conditions havenot been used for modeling the turbine Since the tur-bine downstream conditions are shared with both theturbine and the after-treatment system it is a good

candidate for turbine modeling as well as for model-based after-treatment control

With the turbine modeling issues addressed abovethis paper proposes to develop a physics-based control-oriented model of the turbine power based on the tur-bine downstream conditions and a general approach toidentifying the mechanical loss of the turbocharger foraccurately modeling the turbine power in both steady-state conditions and transient conditions The proposedmodeling approach can be directly applied to power-assisted turbochargers The fundamental Euler turbineequation is used to model the turbine power1242021 byincorporating both the flow dynamics and the rotordynamics Four different shaft mechanical loss (frictionloss only) models are studied In order to improve thetransient dynamic model of the shaft speed themechanical loss of the shaft is modeled on the basis ofboth the thrust torque and the axial friction torqueThe turbine power model and the mechanical lossmodel are validated for the given compressor powerwhich is calculated on the basis of the standard isentro-pic compression assumption with the measured com-pressor upstream and downstream conditions (thetemperature the pressure and the compressor massflow rate) The turbocharger rotor dynamic equation isused to couple the compressor the turbine and themechanical loss powers during both steady-state opera-tions and transient operations The results of a prelimi-nary study have been introduced by Zeng et al22 andin this paper detailed investigation results for the pro-posed modeling approach are presented with threeexisting friction loss models and one newly proposedmodel

The main contributions of this paper are twofold anew control-oriented turbine power model as a

Figure 2 Architecture of the turbocharger system

468 Proc IMechE Part D J Automobile Engineering 232(4)

function of the turbine vane position and a generalizedapproach of identifying the friction loss for the pro-posed turbine model The rest of this paper is organizedas follows In the second section the turbine modelbased on the Euler equation with four mechanical losscandidates is discussed the third section provides boththe steady-state results and the transient validationresults The last section adds some conclusions A listof notation is available in Appendix 1

VGT turbine power model using the Eulerturbine equation

The Euler turbine equation

The Euler equation for turbomachinery provides a cou-pling between the flow power removed from the turbine(or added to the compressor) and the characteristics ofthe rotating blades122324 it is easily established byconservation of the angular momentum and the energyFor completeness the derivation of the Euler equa-tion24 is reproduced below The turbine torque t isdefined as the rate of change in the angular momentumaccording to

t = _m vinrin voutrouteth THORN eth1THORN

where vin is the inlet turbine flow velocity vout is theoutlet turbine flow velocity rin is the radius of the flowinlet with respect to the rotation axis rout is the radiusof the flow outlet with respect to the rotation axis and_m is the mass flow rate The work _W per unit time (orpower) can then be defined as

_W=vt

=v _m vinrin voutrouteth THORNeth2THORN

where v is the turbine speed Now we consider thesteady-state flow energy equation

_Q _W= _m DhT eth3THORN

where DhT is the enthalpy change across the turbineand _Q is the turbine heat transfer rate Assuming anadiabatic process ( _Q=0) for the gas expansion processinside the turbine24 leads to

_W= _m hin houteth THORN eth4THORN

When ideal-gas assumptions with a constant-pressurespecific heat value25cp are considered the enthalpychange model (4) can be equated to the momentumchange model (2) according to

_W= _mcp TT1 TT2eth THORN= _mv vinrin voutrouteth THORN

eth5THORN

Equation (5) is the well-known Euler turbine equationwhich couples the temperature ratio (and hence thepressure ratio) across a turbine to the rotational speedand the change in the momentum per unit mass

VGT turbine power model as a function of the VGTvane angle

In order to deal with the flow velocities the turbinevelocity triangles must be used Figure 3 shows the geo-metric configuration of a VGT turbine and the associ-ated velocity triangles at the rotor inlet and outletstations Details of the velocity triangles can be foundin the work by Zinner1 and by Watson and Janota23

In the rest of this paper the inlet station is referencedby a subscript 1 or in and the outlet station is refer-enced by a subscript 2 or out If the linear velocity atsome radial distance k is defined as uk = vrk the tur-bine power in equation (5) can be redefined as

_W=vt

=v _m vinrin voutrouteth THORN= _m u1vin u2vouteth THORN

eth6THORN

On the assumption of full energy recovery at the tur-bine exit this allows zero exit swirl23 and hence vout =0 which leads to the power equation

_W= _mu1vin eth7THORN

Standard turbine designs allow the radial velocity cr1 atthe rotor inlet to be the same as the radial velocity cr2 atthe rotor exit ie cr1 = cr2 = c1 cos a1 This assump-tion leads to the velocity relationship

vin = c1 sina1

= cr2sina1

cosa1

= cr2 tana1

eth8THORN

where a1 is the gas entry angle to the rotor and is deter-mined by the angular position of the vane guide bladecontrolled by the VGT position actuator As a resulta1 = f(uvgt) where uvgt is the VGT position controlduty cycle The radial gas velocity at the turbine outletcan be expressed in terms of the mass flow rate and theexit area A2 as

cr2 =_m

A2r

=_m

A2

ltTT2

PT2

eth9THORN

where TT2 is the turbine outlet temperature and PT2 isthe turbine outlet pressure (see Figure 3) respectivelyThe turbine outlet area A2 is defined using the turbineoutlet and the nut diameters Dt2 and Dtn as

A2 =p

4D2

t2 D2tn

eth10THORN

The blade-tip linear velocity u1 in equation (8) is calcu-lated using the turbine wheel diameter Dt1 as

u1 =vDt1

2eth11THORN

Zeng and Zhu 469

When equations (7) and (11) are combined the turbinepower can be defined by

_WT = _mvDt1

2

_m

(p=4) D2t2 D2

tn

ltTT2

PT2tana1

=2v _m2 Dt1

p D2t2 D2

tn

ltTT2

PT2tana1

eth12THORN

The angular velocity of the turbocharger shaft isexpressed in terms of the rotational speed of the turbo-charger in revolutions per minute v = 2pNTC60rmin If _m= _mex the power _WT and the torque tTproduced by the turbine can be expressed as

_WT =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth13THORN

tT =2

p_m2ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth14THORN

A similar power model was proposed by Isermann19

according to

_WT =1

60NTC _m2

ex

1

B

ltTP

tana1 eth15THORN

where B is the turbine inlet clearance as shown in Figure3 T is the temperature at the rotor inlet and P is thepressure at the rotor inlet (see location 1 in Figure 3)These inputs are difficult to measure even in an experi-mental bench set-up The model proposed in equation(13) is in contrast practically realizable In both modelversions however the turbine power is directly influ-enced by the VGT vane position The direct couplingbetween the vane position and the turbine power allowsdirect VGT vane position regulation and eliminates theneed to obtain the inverse model for obtaining the targetvane position

Turbocharger mechanical loss model andidentification method

Power transmission between the turbine and the com-pressor is influenced by its mechanical loss In thispaper only the friction loss is considered The two pri-mary sources of turbocharger friction are related toradial or journal bearing friction and axial or thrustbearing friction The bearing system and the forcesapplied on the rotor are shown in Figure 2 The

Figure 3 VGT layout and its velocity triangles123


associated mechanical loss _Wloss (also called the shaftloss) is expressed as the sum of these two loss termsaccording to

_Wloss = _Wjournal + _Wthrust eth16THORN

where _Wjournal is the loss associated with the journaland _Wthrust is the loss associated with the thrust bear-ings The journal bearing friction depends on the angu-lar speed of the shaft the oil-film thickness and the oilviscosity the thrust bearing friction depends on theaxial thrust and the angular speed of the shaft Theaxial thrust is a result of the pressure difference betweenthe compressor inlet pressure PC1 and the turbine outletpressure PT2 (see Figure 2) It should be noted that thethrust contribution from the axial flows in the compres-sor and turbine is typically ignored for the axial thrustload The compressor wheel action force FC due to itsinlet pressure PC1 and the turbine wheel action force FT

due to PT2 are used to determine the thrust bearingload Three existing models from the literature areinvestigated and shown in Table 1 where model 1 andmodel 3 define the lumped thrust and the journal bear-ing friction loss using a polynomial function of thespeed of the turbocharger shaft model 2 separates thethrust friction loss and the journal bearing friction lossby considering the turbine upstream pressure PT1 andthe compressor downstream pressure PC2 In order toobtain the transient behavior of the turbochargermodel 2 is modified by a transient compensation termc3 _NTC

which is referred to as model 4 For steady-state operations ( _NTC

=0) model 2 and model 4 areidentical All the four friction models are investigatedin the next section for both steady-state operations andtransient operations For simplicity all friction modelcoefficients are assumed to be constant and lossesrelated to the heat transfer the bearing oil-film thick-ness the viscosity of the lubrication oil as well as thevariations in the oil temperature are not considered inthis paper

The power balance between the turbocharger tur-bine and the compressor is typically determined usingthe inertial dynamics of the turbocharger defined bythe governing equation

vJTCdv

dt=hm

_WT _WC eth17THORN

This representation implies that the mechanical loss_Wloss = 1 hmeth THORN _WT is a fraction of the turbine powerleading to

vJTCdv

dt= _WT _Wloss _WC eth18THORN

In steady-state operations the turbocharger is at a con-stant speed (dvdt = 0) and the turbine and compres-sor are operated in power equilibrium according to

_WC = _WT _Wloss

=hm_WT

eth19THORN

In this paper the compressor power is obtained fromthe isentropic power corrected by the compressor isen-tropic efficiency and is given by

_WC =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

eth20THORN

Substituting equations (13) and (19) into equation (20)results in an explicit expression for hm tan a1 given by

hm tana1 =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

G1

eth21THORN

where

G =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

From equation (21) it is clear that for a given vaneposition (fixed a1) hm varies with the operating condi-tions of the turbocharger This is verified using theexperimental data from steady-state tests performed ona heavy-duty turbocharged diesel engine2627 The entireengine operational range is mapped via 195 steady-statespeed and load points The vane position is based onthe turbocharger control signal uvgt and varies from thefully closed position to the fully open position Sincetypical data sets provide only the vane position(0ndash100) the angular position a1 corresponding touvgt must be determined on the basis of the known link-age kinematic relationship Hence for convenience thevariation in hm tan a1 calculated using the right-handside of equation (21) with uvgt is shown in Figure 4(a)

Table 1 Friction models

Model

Model 12 _Wloss = c1N2TC + c2NTC + c3

Model 228 _Wloss = _Wjournal + _Wthrust

where _Wjournal = c1N2TCand _Wthrust = c2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPT1 PC2

3p

N2TC

Model 319 _Wloss = c1N2TC

Model 4 (this work) _Wloss = c1N2TC + c2


3p

N2TC + c3

_NTC

Zeng and Zhu 471

Each point on this plot represents a unique steady-stateoperational condition for the power-balanced turbo-charger Since tan a1 is a constant for a given vaneposition the variations observed in Figure 4(a) must beattributed to hm This is the motivation for obtaining amechanical efficiency model which varies with the oper-ating conditions of the turbocharger

It is clear that for the vane position between about60 open (40 closed) and fully open (0 closed)the variation in hm is small This is an important obser-vation since it indicates that for large vane openings(with unrestricted flow across the turbine) the steady-state mechanical (or the overall) efficiency of the turbo-charger can be approximated as a constant which issimilar to the approach employed in early work257910

However when the vane position is between 0 and60 opening because of the flow restrictions themechanical efficiency of the turbocharger varies signifi-cantly and the constant-efficiency assumption is nolonger valid The impact of the turbocharger speed onhm tan a1 is also investigated using this data set (seeFigure 4(b)) It is clear that hm tan a1 varies with boththe turbocharger speed and the vane position depend-ing on the loss models in Table 1

Model validation and mechanical lossidentification

Model identification and mechanical loss estimationusing steady-state engine test data

The coefficients in the turbine power model and thefriction model are determined by minimizing a costfunction J defined as the squared summation of errorsbetween the measured values and the model predictedvalues according to

Jk =Xni=1

_WTcalci _WCi + _Wlosski 2 eth22THORN

where _WTcalciis the calculated turbine power from

equation (13) _WCi is the calculated compressor power

from equation (20) and _Wlosski is the power loss

model k (= 1 2 3 4) defined in Table 1 for i = 1 2 n Since the steady-state test data are used the erroris the deviation of the turbocharger power balancefrom zero A least-squares minimization scheme isused and the model coefficients (see Table 2 later) areidentified In practice the exact relationship betweenthe vane angle a1 and the VGT actuator control dutycycle uvgt (Figure 5 shows a typical actuation mechan-ism of the VGT) is unknown and for this study thevane angle a1 is estimated on the basis of the VGT con-trol duty cycle uvgt It should be noted that the controlduty cycle uvgt is a one-to-one mapping to the vaneangle a1 The polynomial

a1 = f uvgt

= c4u3vgt+ c5u

2vgt+ c6uvgt + c7

eth23THORN

is used to define the relationship between a1 and uvgtAs a result this increases the number of parameters

to be determined to a maximum of seven depending onthe loss models in Table 2 It is seen that the parametersc4 to c7 always converge to similar values providingconfidence in the parameter identification The vaneangles corresponding to the minimum (0) and themaximum (100) vane positions are reproduced usingequation (23) and the identified coefficients in Table 2The range of VGT angles in absolute geometric degrees

Figure 4 Variation in hm tan a1 as a function of the VGT vaneposition and the turbocharger speedVGT variable-geometry turbocharger RPM rmin

Figure 5 Actuation mechanism of the VGTVGT variable-geometry turbocharger


varies between 32 for a 0 vane position (uvgt = 0)and 80 for a 100 vane position (uvgt = 08) It can beseen that a non-zero geometric angular position for a0 vane position is in line with practical design consid-erations which guarantee a minimum leak path andhence a minimum flow for the exhaust gas The identi-fied coefficients for the different loss models are shownin Table 2 Identification using the steady-state datamakes model 2 and model 4 the same as discussed ear-lier The modeling error is defined by

Error=Xni1

_WTcalci _Wlosski

_WCi

Xni=1

_WCi

1

eth24THORN

The identification results are presented in Figure 6The vane angle models for the three different fits returnvery similar values The mechanical loss of the turbo-charger is seen to increase with increasing turbochargerspeed with a maximum loss of about 85 kW at the peakturbocharger speed This result matches the test datadiscussed by Serrano et al16 and Francisco et al17 Thevane opening angle is around 52 which agrees with thevane design range The deviation between model 1 andmodel 3 is not significant for steady-state operationssince the quadratic term in both models dominates thefriction loss From Table 2 it is seen that the term N2

TC

is dominantModel 2 and model 4 deviate from model 1 and

model 3 possibly because of the thrust impact Sincemodel 2 and model 4 account for the pressure differencebetween the turbine and the compressor wheel the var-iation in the friction power between 40000 rmin and100000 rmin is due to the change in the thrust loadThe variation in the friction occurs only for steady-stateoperations for this validation For transient operationsthe thrust friction increases owing to the unbalancedthrust force from both the turbine and the compressor

As shown in Figure 1 the maximum turbine pressureratio for steady-state engine operations is much smallerthan the maximum for transient operations Thus

during transient operations the thrust force is highbecause of the increased pressure difference between theturbine and the compressor Thus model 2 and model4 have higher thrust friction values owing to the higherpressure differences across the turbocharger Thedynamics compensation term in model 4 is furtherinvestigated using the transient test data in the subsec-tion on model validation using the transient test data ofthe vehicle

An estimate of the steady-state compressor power isderived from the difference between the predicted tur-bine power and the predicted mechanical loss This esti-mation is compared with the measured compressorpower in Figure 6(c) The prediction error defined inequation (24) yields an error margin of 64Admittedly there are certain modeling errors due tothe unknown relationship between the VGT positionand the vane angle as well as unmodeled physics dueto simplification From Table 2 it shows that the aver-aged errors for the proposed models are about 4 forsteady-state operations

The results indicate that the investigated four powerloss models all perform equivalently Although model 2and model 4 inherently account for more physics byincluding the trust load impact model 1 and model 3have simpler structures and rely on fewer inputsHowever using the proposed identification process itis apparent that for turbochargers with a low loss itmay be possible to model the loss as a function of theturbocharger speed alone

Turbine model identification using the standardhot-gas flow bench test data

Next the test data from the standard hot-gas flowbench of a different turbocharger called turbocharger2 are used to validate the turbine model In this casethe turbine inlet has a steady-state flow which is quitedifferent from the engine dynamometer test data forthe pulsating flow The detailed test setup has beendescribed in SAE J182618 The test points over theVGT operational range are shown in Figure 7 The

Table 2 Model validation results using the steady-state engine test data for turbocharger 1

Value for the following models

Model 1 Model 2 and model 4 Model 3

Friction model coefficients c1 6957 3 10210 4507 3 10210 6958 3 10210

c2 2357 3 10214 5909 3 10211 mdashc3 75023 3 10212 000 mdash

Vane angle coefficients c4 1011 0961 1003c5 0171 0206 0175c6 0349 0344 0349c7 0572 0572 0556

Vane angle (equation (23)) (rad (deg)) Minimum 0572 (3277) 0572 (3277) 0556 (3186)Maximum 1478 (8468) 1471 (8428) 1460 (8365)

Model error () mdash 438 423 438

Zeng and Zhu 473

speed range for turbocharger 2 is between 5000 rminand 150000 rmin and the pressure is between 11 and50 bar The turbine model described in the subsectionon model identification and mechanical loss estimationusing steady-state engine test data is used Frictionmodels 1 to 3 were chosen for this study

It should be emphasized that in this study the vaneangle equation (23) is used as a function of the VGTcontrol duty cycle in percentages The optimizedmechanical loss and vane angle coefficients are shown

in Table 3 for four models and the predicted mechani-cal loss and the predicted vane angle as functions of theturbine speed and the VGT position are shown inFigure 8 Since this turbocharger has a maximumoperational speed of 150000 rmin a high friction loss(peaked at 16 kW) is expected The vane angle openingrange is around 41ndash43 which is smaller than that ofturbocharger 1 because of the difference in the vanedesign The error function is defined in equation (24)The results are comparable with those presented in the

Figure 6 (a) Predicted vane angle (b) predicted mechanical loss and (c) predicted compressor power in the steady state forturbocharger 1VGT variable-geometry turbocharger TC turbocharger rpm rmin


previous subsection It shows that the proposed turbinemodel is also suitable for the data set obtained fromsteady-state flow bench tests and can be used to extra-polate the turbine efficiency based on the given turbineefficiency map

Turbine power model identification using theGT-Power transient simulation data

In order to understand the model characteristics duringtransient operations the proposed model is calibratedusing the transient responses obtained from the GT-Power model in this subsection The GT-Power modelis developed for the engine and turbocharger 1described in the subsection on model identification andmechanical loss estimation using steady-state enginetest data The detailed GT-Power model and its simula-tion setup have been decribed by Sun et al27 In thetransient simulations the engine follows the desiredtorque based on the acceleration pedal position TheVGT feedback control is used to track the target cali-brated boost pressure The purpose of this study is tostudy the model behavior in transient operationalconditions

First the developed turbine model is calibratedusing the Federal Test Procedure 75 (FTP 75) simula-tion data to obtain the vane angle model with the VGTcontrol duty cycle as the input and then the calibratedmodel is used to predict the turbine power for theSupplemental Federal Test Procedure (US 06) drivingcycle Since the turbine map used in GT-Power simula-tions is based on the steady-state flow bench map thefriction loss is not modeled in the GT-Power modeland the turbine power is an output in the GT-Powersimulations by interpolating or extrapolating themanufacturer-provided efficiency map28 The turbinevane angle can be calculated using

tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN

where _WsimGTis the GT-Power simulated turbine power

The relationship between the vane angle and the VGTcontrol duty cycle is shown in Figure 9 and the vaneangle fitting is obtained as

a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN

where uvgt is the VGT turbine control duty cycle Usingthe proposed turbine model equation (13) and the fittedvane relationship equation (26) the predicted turbinepower can be obtained

In order to show the consistency of the calibratedmodel the model calibrated using the FTP 75 cycle isused to simulate the turbine power using the US 06driving cycle for the same engine The simulation resultsand the associated errors are shown in Figure 10 andTable 4 where the power model error is defined as

Error=Xni1

_WsimGTi _Wpredictedi

Xni=1

_WsimGTi

1

eth27THORN

It should be noted that the error may be due to the mapextrapolation inaccuracy based on the empirical equa-tion used in the GT-Power model the unmodeled fric-tion loss in the GT-Power model may also contribute to

Table 3 Model validation results using the steady-state flow bench test data for turbocharger 2



Friction model coefficients c1 6988 3 10210 6513 3 10210 mdashc2 1800 3 10210 ndash48812 3 10214 mdashc3 1000 3 10210 000 mdash

Vane angle coefficients c4 08765 1107 1085c5 0002 476 ndash02495 ndash02191c6 01581 02169 00121c7 02541 02479 02508

Vane angle (rad (deg)) Minimum 02541 (145) 02479 (142) 02508 (143)Maximum 0972 (556) 0986 (554) 0995 (570)

Figure 7 Test range of turbocharger 2VGT variable-geometry turbocharger rpm rmin

Zeng and Zhu 475

the error The above study shows that the proposed tur-bine power model can be used for modeling the transi-ent operations of the turbine

Model validation using the transient test data of thevehicle

Validation of the fitted model is carried out using thetransient test data from a vehicle equipped with

turbocharger 1 that was identified earlier in the sub-section on model identification and mechanical lossestimation using steady-state engine test data The tur-bine power is calculated on the basis of equation (13)and the mechanical loss and the vane angle models arefrom equation (23) and Table 2 For transient opera-tions the turbocharger energy balance is defined inequation (18) and is reorganized in terms of the tur-bine power as



_WT =vJTcdv

dt+ _WC + _Wloss eth28THORN

The turbine power representation in equation (28) iscompared with the calculated values using the model inequation (13) the measured inputs TT2 PT2 NTC and_mex and the geometric parameters of the turbine Thecompressor power _WC in equation (28) can be calcu-lated on the basis of the change in the flow enthalpy(see equation (20)) which is also calculated directly frommeasurements The four mechanical loss _Wloss modelsdefined in Table 1 are used with the calibrations shownin Table 2 where the transient compensation coefficientc3 in model 4 is set to 0001 to improve the transientoperation behavior For comparison purposes the tur-bine power was reproduced from the supplier-basedmaps using23

_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN

where htm is the combined turbine and mechanical effi-ciency and is a parabolic function of the turbine BSRwhich is given by

htm= f BSReth THORN

= f Rv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cexp TT1 1 PT2

PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN

The efficiency htm in equation (30) is identified fromthe same data as in the subsection on model identifica-tion and mechanical loss estimation using steady-stateengine test data by utilizing the method from the workby Jankovic and Kolmanovsky7 Other model inputsare based on the data measured directly in this study Itshould be noted that the efficiency in equation (30) isnot a function of the VGT position Good agreementswere obtained between the turbine power calculatedfrom the physics-based model (13) and the turbinepowers calculated using equation (28) based on theexperimental data for all four friction models in con-trast the turbine power from the map-based model(29) deviates significantly from that calculated fromequation (28) This indicates that the proposed turbinepower model and the proposed turbocharger mechani-cal loss model are adequate The top diagram inFigure 11 clearly shows that the proposed physics-based turbine power model represents the turbochargertransient operations better than the map-based modeldoes and also that the map-based model is reasonablewhen the turbine operates within the map provided bythe flow bench test data and the transient operationdata shown in Figure 1 However the range for turbinetransient operations is quite different from the flowbench test range Since the mapped data are not avail-able at high turbocharger speeds with a low pressureratio and at low turbocharger speeds with a light loada large extrapolation error over both regions leads to afairly large modeling error (see the top diagram ofFigure 11) As a result the map-based turbine effi-ciency is significantly lower than the actual value Thisstudy also shows that the proposed model is able torepresent the fast transient characteristics during tip-out because the proposed friction loss models dependon both the turbocharger speed and the pressure differ-ence across the turbocharger When the VGT is fullyopened during tip-out initially the turbine mass flow

Figure 10 Turbine power between the proposed model andthe GT-Power modelFTP 75 Federal Test Procedure 75 US 06 Supplemental Federal Test

Procedure

Figure 9 Vane angle and VGT control input with GT-PowersimulationsVGT variable-geometry turbocharger

Table 4 Averaged turbine power error between the proposedmodel and the GT-Power model

Value for the following driving cycles

Federal TestProcedure 75

US 06 SupplementalFederal Test Procedure

Error () 66 82

Zeng and Zhu 477

rate increases and the turbine upstream pressuredecreases This leads to an increased turbine powerwith a higher turbine mass flow rate at the beginning oftip-out and as a result to an increased compressordownstream pressure with a higher turbine power Thisnon-minimum phase behavior has been explained wellby Kolmanovsky et al3 Hence with increased com-pressor downstream pressure and decreased turbineupstream pressure the thrust load increases during tip-out The thrust load direction is from the compressorwheel side to the turbine wheel side All these lead toan increased friction power at the beginning of tip-out

which makes the proposed model different from themap-based method

From the lowest diagram in Figure 11 the results alsoagree with the previous discussion that the thrust frictionincreases as the pressure difference across the turbine andthe compressor increases The modeling errors of the fourfriction models are similar when the turbocharger oper-ates within the envelope of engine steady-state operations(see the middle two diagrams of Figure 11 for the speedand the pressure ratio) the thrust friction increases signif-icantly with increasing pressure difference across the tur-bine and the compressor during transient operations

Figure 11 Model validation with transient vehicle test dataTC turbocharger


which leads to a mechanical efficiency of about 10Also the mechanical efficiency decreases similarly duringthe transient tip-outs leading to an increased compressormodeling error (see Table 5) The error for the physics-based model is defined by

Error=Xni=1

_WTi _Wlossi _Wkinetic

_WC(i)

Xni=1

_WC(i) 1

eth31THORN

The error for the map-based model is the same as thatin equation (31) by setting _WLossi =0 since themapped-based model lumps the friction loss _WLossi

together with the turbine power _WTi It should benoted that the large error in the map-based model inTable 5 is due to the map extrapolation error Althoughthe physics-based model is able to represent the turbo-charger dynamics over the entire operating range a cer-tain error exists that is due to the measurement errors(eg the temperature sensor time constant limits thetransient measurement response) and to the unmodeledphysics such as the heat transfer and the bearing fric-tion loss due to the variation in the oil viscosity

From Table 5 it can be observed that for the thrustload model (friction model 2 and friction model 4) theturbine power accuracy can be improved by 5 and6 for model 2 and model 4 respectively in the transi-ent operations It should be noted that the steady-stateoperation study results in Table 2 and Table 3 show nosignificant error difference This shows the importanceof including the thrust friction in the turbine powermodel during transient operations Even though theproposed turbine model is derived on the basis of apower-balanced (steady-state) turbocharger with thehelp of the friction model (particularly model 2 andmodel 4) it can also be used in transient operationssince the modeling error is reasonable (10) in transi-ent operations In conclusion the proposed physics-based model with friction model 4 reduces the modelerror in both steady-state operations and transientoperations in comparison with the conventional map-based model

Conclusion

A physics-based turbine power model of a VGT is pro-posed in this paper together with a thrust friction

model The turbine power model is derived on the basisof the Euler turbine equation with the VGT vane posi-tion as the control parameter Three existing frictionloss models and one newly proposed friction model arealso investigated All four friction models have thepotential of including the oil viscosity and the heattransfer effects in the model The proposed turbinepower model together with the friction models areinvestigated for two steady-state data sets (enginedynamometer tests and flow bench tests) and two tran-sient data sets (one-dimensional GT-Power transientsimulation results and vehicle transient test data) Thesteady-state data study shows that the proposed modelis fairly accurate (with less than 45 modeling error)and the four friction models provide a similar modelingaccuracy for the transient data investigation the pro-posed turbine power model and the acceleration-basedthrust friction model are able to reduce the transientmodeling error from 228 (conventional map-basedmodel) to 101 This indicates that an accurate thrustfriction model is a key to having an accurate transientmodel It should be noted that the proposed turbinepower model and its mechanical efficiency modelare suitable for model-based VGT control because ofits analytical nature as a function of the VGT vaneangle

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) received no financial support for theresearch authorship andor publication of this article

References

1 Zinner KA Supercharging of internal combustion engines

additional Berlin Springer 20122 Watson N and Banisoleiman K A variable-geometry

turbocharger control system for high output diesel

engines SAE paper 880118 19883 Kolmanovsky I Morall P Van Nieuwstadt MI and Ste-

fanopoulou A Issues in modelling and control of intake

flow in variable geometry turbocharged engines In Polis

MP Dontchev AL Kall P et al A (eds) System modeling

and optimization Chapman amp HallCRC Research Notes

Table 5 Average errors for different models


Map-basedturbine model

Physics-basedturbine model+ friction model 1




Transient error () 228 159 111 163 101

Zeng and Zhu 479

in Mathematics Boca Raton Florida London Chapmanamp HallCRC 1999 pp 436ndash445

4 Konishi K and Yoshiki H Characteristics of radial

inward turbines for exhaust gas turbochargers undernonsteady flow conditions effects of waveforms on pre-

diction of turbine performance JSME Int J Ser 2 Fluids

Engng Heat Transfer Power Combust Thermophys

Properties 1992 35(2) 228ndash2375 Upadhyay D Utkin VI and Rizzoni G Multivariable

control design for intake flow regulation of a dieselengine using sliding mode In 15th triennial IFAC world

congress on transportation and vehicles control of

advanced engines Barcelona Spain 21ndash26 July 2002 6pp New York IFAC (IFAC Proc Vol 2002 35(1)

277ndash282)6 Zhu G Wang J Sun Z and Chen X Tutorial of model-

based powertrain and aftertreatment system controldesign and implementation In 2015 American control

conference Chicago Illinois USA 1ndash3 July 2015 pp2093ndash2110 New York IEEE

7 Jankovic M and Kolmanovsky I Robust nonlinear con-

troller for turbocharged diesel engines In 1998 American

control conference Philadelphia Pennsylvania USA 26June 1998 Vol 3 pp 1389ndash1394 New York IEEE

8 Wahlstrom J and Eriksson L Modelling diesel engines

with a variable-geometry turbocharger and exhaust gasrecirculation by optimization of model parameters for

capturing non-linear system dynamics Proc IMechE Part

D J Automobile Engineering 2011 225(7) 960ndash9869 Moraal P and Kolmanovsky I Turbocharger modeling

for automotive control applications SAE paper 1999-01-

0908 199910 Jensen JP Kristensen AF Sorenson SC et al Mean

value modeling of a small turbocharged diesel engine

SAE paper 910070 199111 Stricker K Kocher L Koebelein E et al Turbocharger

map reduction for control-oriented modeling In ASME

2011 dynamic systems and control conference and Bath

ASME symposium on fluid power and motion control Vol

2 Arlington Virginia USA 31 Octoberndash2 November

2011 paper DSCC2011-5992 pp 619ndash626 New YorkASME

12 El Hadef J Colin G Chamaillard Y and Talon V Physi-cal-based algorithms for interpolation and extrapolation

of turbocharger data maps SAE paper 2012-01-04342012

13 Schorn N The radial turbine for small turbocharger

applications evolution and analytical methods for twin-entry turbine turbochargers SAE paper 2014-01-16472014

14 Rajoo S and Martinez-Botas R Variable geometry mixed

flow turbine for turbochargers an experimental study IntJ Fluid Mach Systems 2008 1(1) 155ndash168

15 Deligant M Podevin P and Descombes G Experimental

identification of turbocharger mechanical friction lossesEnergy 2012 39(1) 388ndash394

16 Serrano JR Pablo O Andres T et al Theoretical andexperimental study of mechanical losses in automotive

turbochargers Energy 2013 55 888ndash89817 Payri F Serrano JR Olmeda P et al Experimental meth-

odology to characterize mechanical losses in small turbo-

chargers In ASME Turbo Expo 2010 power for land

sea and air Vol 5 industrial and cogeneration microtur-

bines and small turbomachinery oil and gas applications

wind turbine technology Glasgow UK 14ndash18 June 2010paper GT2010-22815 pp 413ndash423 New York ASME

18 SAE J1826 Turbocharger gas stand test code WarrendalePennsylvania SAE International 1995

19 Isermann R Engine modeling and control BerlinSpringer 2014

20 Saravanamuttoo HIH Rogers GFC and Cohen H Gasturbine theory Upper Saddle River New Jersey PearsonEducation 2001

21 Dale A and Watson N Vaneless diffuser turbocharger

turbine performance In 3rd international conference on

turbocharging and turbochargers IMechE Conference

Publications 1986-4 London UK 6ndash8 May 1986 paperC11086 pp 65ndash76 London Mechanical EngineeringPublications

22 Zeng T Upadhyay D Sun H and Zhu GG Physics-based turbine power models for a variable geometry tur-bocharger In 2016 American control conference BostonMassachusetts USA 6ndash8 July 2016 pp 5099ndash5104 NewYork IEEE

23 Watson N and Janota MS Turbocharging the internal

combustion engine London Macmillan 198224 Dixon S and Hall C Fluid mechanics and thermodynamics

of turbomachinery Oxford ButterworthndashHeinemann2013

25 Cengel YA and Boles MA Thermodynamics an engineer-

ing approach 7th edition New York McGraw-Hill2010

26 Deraad S Fulton B Gryglak A et al The new Ford 67L V-8 turbocharged diesel engine SAE paper 2010-01-1101 2010

27 Sun H Hanna D Niessen P et al Experimental evalua-tion of advanced turbocharger performance on a lightduty diesel engine SAE paper 2013-01-0920 2013

28 GT-Power userrsquos manual version 75 Westmont IllinoisGamma Technologies 2015

Appendix 1

Notation

A geometric area (m2)cairp specific heat of air at a constant pressure

(J kg K)cexp specific heat of the exhaust gas at a

constant pressure (Jkg K)h specific flow enthalpy (J)JTC inertia of the turbocharger shaft (kgm2)_m mass flow rate (kgs)NTC rotational speed of the turbocharger

(rmin)P pressure (Nm2)lt universal gas constantT temperature (K)W work (J)_W power (kW)

a vane angle of the variable-geometryturbocharger (rad)


gair isentropic index for airgex isentropic index for exhaust gash efficiencyhm mechanical efficiencyhis isentropic efficiencyr density of the gas(kgm3)t torque (N m)v angular velocity of the turbocharger

(rads)

Subscripts

C compressorin inlet or upstreamout outlet or downstreamT turbine1 inlet or upstream2 outlet or downstream

Zeng and Zhu 481

dynamometer tests remains inside the region for hot-gas flow bench tests a significant number of the transi-ent test data are outside the region for the hot-gas flowbench test data as shown in Figure 1(b) It is clear thatthe flow bench map does not provide full coverage Asa result operating the turbocharger outside its perfor-mance map requires extrapolation This has led to sev-eral investigations into extrapolation methods8ndash12

Typical extrapolation schemes rely on the second-orderor third-order polynomials as a function of the bladespeed ratio (BSR) This can lead to physically impossi-ble values and extremely large modeling errors Thetypical performance map or the data set which is usedto calibrate and validate the model is provided by theturbocharger manufacturer and is based on the dataobtained from hot-gas flow bench tests Since thesetests are performed in steady-state flow conditions themapped data cannot match the pulsating-flow opera-tional conditions when the turbocharger is coupled toan internal-combustion engine121314

First the turbine efficiency maps provided by tur-bine manufacturers combine the turbine efficiency withthe mechanical loss It is difficult to measure themechanical loss15ndash17 and it is not a required measure-ment in the standard flow bench tests (see the test codedescribed in SAE J182618) As a result the change inthe enthalpy across the compressor is calculated on thebasis of the turbine output power This makes the cal-culated turbine efficiency depend on the compressorcharacteristics and the measured turbine efficiency isnot the actual turbine efficiency since it is a combina-tion of both the turbine efficiency and the mechanicalefficiency As a result the turbine performance mapobtained from flow bench tests is different from the tur-bine characteristics when it is coupled to the engine asshown in Figure 1 In automotive turbocharger systems

the turbine extracts energy from the engine exhaust todrive the compressor for an increased boost pressureIn this application the available turbine power due tothe exhaust gas expansion process needs to overcomeboth the mechanical bearing loss and the heat transferloss to drive the compressor (Figure 2) Friction loss isdue to both the journal and the thrust bearings in theradial direction and the axial direction heat transferloss also drives the turbocharger operation away fromadiabatic behavior It should be noted that both lossesaffect the turbine efficiency However in transientengine operations the friction loss dominates the turbo-charger behavior in particular in fast transient opera-tions the thrust friction loss is determined by thebalanced thrust force between the turbine and the com-pressor and this is not available from turbochargermanufacturers Hence the turbine efficiency map pro-vided by the manufacturer is not suitable for transientoperations Most existing control-oriented turbochar-ger models assume a constant mechanical loss efficiencyfor simplicity257ndash10 This is because the axial frictiondue to the thrust bearing load and the radial frictiondue to the journal bearing cannot be directly measuredon the hot-gas flow bench which makes it challengingto model the mechanical loss directly

Second the required interpolation and extrapolationof the manufacture turbocharger performance maps toobtain the turbine efficiency in the current operationalconditions lead to multiple-dimensional lookup tableswith at least three inputs (the upstream and down-stream pressure ratio the reduced mass flow rate andthe VGT vane position) This can introduce an addi-tional computational burden for real-time controlFurthermore the map-based traditional isentropicmodel needs both the upstream condistions and thedownstream conditions (the pressure or the

Figure 1 Turbine operational ranges for hot-gas flow bench tests steady-state tests and transient operational conditions (a)turbine pressure ratio range (b) turbine mass flow rate rangeTC turbocharger dyno dynamometer

Zeng and Zhu 467















_W=vt








eth5THORN





_W=vt


eth6THORN




vin = c1 sina1

= cr2sina1

cosa1

= cr2 tana1

eth8THORN


cr2 =_m

A2r

=_m

A2

ltTT2

PT2

eth9THORN


A2 =p

4D2

t2 D2tn

eth10THORN


u1 =vDt1

2eth11THORN

Zeng and Zhu 469


_WT = _mvDt1

2

_m

(p=4) D2t2 D2

tn

ltTT2

PT2tana1

=2v _m2 Dt1

p D2t2 D2

tn

ltTT2

PT2tana1

eth12THORN


_WT =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth13THORN

tT =2

p_m2ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth14THORN


according to

_WT =1

60NTC _m2

ex

1

B

ltTP

tana1 eth15THORN













vJTCdv

dt=hm

_WT _WC eth17THORN


vJTCdv



_WC = _WT _Wloss

=hm_WT

eth19THORN


_WC =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

eth20THORN


hm tana1 =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

G1

eth21THORN

where

G =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2



Model





3p

N2TC




3p

N2TC + c3

_NTC

Zeng and Zhu 471







Jk =Xni=1






a1 = f uvgt

= c4u3vgt+ c5u

2vgt+ c6uvgt + c7

eth23THORN







Error=Xni1

_WTcalci _Wlosski

_WCi

Xni=1

_WCi

1

eth24THORN


TC













c2 2357 3 10214 5909 3 10211 mdashc3 75023 3 10212 000 mdash




Zeng and Zhu 473










tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN



a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN



Error=Xni1


Xni=1

_WsimGTi

1

eth27THORN









Zeng and Zhu 475







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481















_W=vt








eth5THORN





_W=vt


eth6THORN




vin = c1 sina1

= cr2sina1

cosa1

= cr2 tana1

eth8THORN


cr2 =_m

A2r

=_m

A2

ltTT2

PT2

eth9THORN


A2 =p

4D2

t2 D2tn

eth10THORN


u1 =vDt1

2eth11THORN

Zeng and Zhu 469


_WT = _mvDt1

2

_m

(p=4) D2t2 D2

tn

ltTT2

PT2tana1

=2v _m2 Dt1

p D2t2 D2

tn

ltTT2

PT2tana1

eth12THORN


_WT =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth13THORN

tT =2

p_m2ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth14THORN


according to

_WT =1

60NTC _m2

ex

1

B

ltTP

tana1 eth15THORN













vJTCdv

dt=hm

_WT _WC eth17THORN


vJTCdv



_WC = _WT _Wloss

=hm_WT

eth19THORN


_WC =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

eth20THORN


hm tana1 =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

G1

eth21THORN

where

G =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2



Model





3p

N2TC




3p

N2TC + c3

_NTC

Zeng and Zhu 471







Jk =Xni=1






a1 = f uvgt

= c4u3vgt+ c5u

2vgt+ c6uvgt + c7

eth23THORN







Error=Xni1

_WTcalci _Wlosski

_WCi

Xni=1

_WCi

1

eth24THORN


TC













c2 2357 3 10214 5909 3 10211 mdashc3 75023 3 10212 000 mdash




Zeng and Zhu 473










tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN



a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN



Error=Xni1


Xni=1

_WsimGTi

1

eth27THORN









Zeng and Zhu 475







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481


_WT = _mvDt1

2

_m

(p=4) D2t2 D2

tn

ltTT2

PT2tana1

=2v _m2 Dt1

p D2t2 D2

tn

ltTT2

PT2tana1

eth12THORN


_WT =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth13THORN

tT =2

p_m2ex

Dt1

D2t2 D2

tn

ltTT2

PT2tana1 eth14THORN


according to

_WT =1

60NTC _m2

ex

1

B

ltTP

tana1 eth15THORN













vJTCdv

dt=hm

_WT _WC eth17THORN


vJTCdv



_WC = _WT _Wloss

=hm_WT

eth19THORN


_WC =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

eth20THORN


hm tana1 =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

G1

eth21THORN

where

G =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2



Model





3p

N2TC




3p

N2TC + c3

_NTC

Zeng and Zhu 471







Jk =Xni=1






a1 = f uvgt

= c4u3vgt+ c5u

2vgt+ c6uvgt + c7

eth23THORN







Error=Xni1

_WTcalci _Wlosski

_WCi

Xni=1

_WCi

1

eth24THORN


TC













c2 2357 3 10214 5909 3 10211 mdashc3 75023 3 10212 000 mdash




Zeng and Zhu 473










tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN



a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN



Error=Xni1


Xni=1

_WsimGTi

1

eth27THORN









Zeng and Zhu 475







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481








vJTCdv

dt=hm

_WT _WC eth17THORN


vJTCdv



_WC = _WT _Wloss

=hm_WT

eth19THORN


_WC =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

eth20THORN


hm tana1 =1

hisC

_mairTC1cairp

PC2

PC1

gair1eth THORN=gair

1

G1

eth21THORN

where

G =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2



Model





3p

N2TC




3p

N2TC + c3

_NTC

Zeng and Zhu 471







Jk =Xni=1






a1 = f uvgt

= c4u3vgt+ c5u

2vgt+ c6uvgt + c7

eth23THORN







Error=Xni1

_WTcalci _Wlosski

_WCi

Xni=1

_WCi

1

eth24THORN


TC













c2 2357 3 10214 5909 3 10211 mdashc3 75023 3 10212 000 mdash




Zeng and Zhu 473










tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN



a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN



Error=Xni1


Xni=1

_WsimGTi

1

eth27THORN









Zeng and Zhu 475







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481







Jk =Xni=1






a1 = f uvgt

= c4u3vgt+ c5u

2vgt+ c6uvgt + c7

eth23THORN







Error=Xni1

_WTcalci _Wlosski

_WCi

Xni=1

_WCi

1

eth24THORN


TC













c2 2357 3 10214 5909 3 10211 mdashc3 75023 3 10212 000 mdash




Zeng and Zhu 473










tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN



a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN



Error=Xni1


Xni=1

_WsimGTi

1

eth27THORN









Zeng and Zhu 475







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481


Error=Xni1

_WTcalci _Wlosski

_WCi

Xni=1

_WCi

1

eth24THORN


TC













c2 2357 3 10214 5909 3 10211 mdashc3 75023 3 10212 000 mdash




Zeng and Zhu 473










tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN



a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN



Error=Xni1


Xni=1

_WsimGTi

1

eth27THORN









Zeng and Zhu 475







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481










tana1 =1

15NTC _m2

ex

Dt1

D2t2 D2

tn

ltTT2

PT2

_W1simGT

eth25THORN



a1 = 3358u3vgt+5881u2vgt 1594uvgt+058

eth26THORN



Error=Xni1


Xni=1

_WsimGTi

1

eth27THORN









Zeng and Zhu 475







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481







_WT =vJTcdv



_WTmap=htm _mexc

exp TT1 1 PT2

PT1

11=gex

eth29THORN


htm= f BSReth THORN

= f Rv


PT1

11=gex

vuut8lt

9=10

B1CAeth30THORN



Procedure






Error () 66 82

Zeng and Zhu 477







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481







Error=Xni=1


_WC(i)

Xni=1

_WC(i) 1

eth31THORN




Conclusion





Funding


References
















Transient error () 228 159 111 163 101

Zeng and Zhu 479




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481




























































Appendix 1

Notation








(rads)

Subscripts


Zeng and Zhu 481


(rads)

Subscripts


Zeng and Zhu 481

control-oriented turbine power model for a variable …...with the turbine modeling issues addressed...

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