A Generalized Compressor Power Model for Turbocharged ... Articles/A... · W fdC wt w( outout inin) ( mvrvr mUC UC T T 2 2 1 1 ) (2) For the centrifugal impeller, it is assumed that

Tao Zeng Mechanical Engineering

Michigan State University East Lansing, Michigan 48824

Email: [email protected]

Devesh Upadhyay and Harold Sun Ford Motor Company

Dearborn, Michigan 48124 Email: [email protected]

[email protected]

Guoming G. Zhu Mechanical Engineering

Michigan State University East Lansing, Michigan 48824

Email: [email protected]

ABSTRACT Control-oriented models of automotive turbocharger

compressors typically describe the compressor power assumming

an isentropic thermodynamic process with a fixed isentropic

efficiency and a fixed mechanical efficiency for power

transmission between the turbine and compressor. Although these

simplifications make the control model tractable, they also

introduce additional errors due to unmodeled dynamics,

especially when the turbocharger is operated outside its normal

operational region. This is also true for map-based approaches,

since these supplier-provided maps tend to be sparse or

incomplete at the boundary operational regions and often ad-hoc

extrapolation is required, leading to large modeling error.

Furthermore, these compressor maps are obtained from the

steady flow bench tests, which introduce additional errors under

pulsating flow conditions in the context of internal combustion

engines. In this paper, a physics-based model of compressor

power is developed using Euler equations for turbomachinery,

where the mass flow rate and compressor rotational speed are

used as model inputs. Two new coefficients, speed and power

coefficients are defined. This makes it possible to directly estimate

the compressor power over the entire compressor operating range

based on a single analytic relationship. The proposed modeling

approach is validated against test data from standard

turbocharger flow bench, steady state engine dynamometer as

well as transient simulation tests. The validation results show that

the proposed model has adequate accuracy for model-based

control design and also reduces the dimension of the parameter

space typically needed to model the compressor dynamics.

NOMENCLATURE P : Pressure [N/m

2]

T : Temperature [K]

W : Work [J]

m : Mass flow rate [kg/sec]

: Torque [Nm]

: Compressor angular velocity [rad/sec]

: Gas density[kg/m3]

: Universal gas constant[J kg−1

K−1

]

A: Geometric area [m2]

: Isentropic index

Subscripts:

1, in: represent the compressor upstream or inlet condition.

2, out: represent the compressor downstream or outlet

condition.

Symbols not in this list are defined locally in the text.

INTRODUCTION It is common for plant models, used in the air path control of

turbocharged (TC) diesel engines, to assume that the power

consumed by the compressor is an isentropic thermodynamic

process. The actual power is then derived from either a map-based

compressor isentropic efficiency [1]-[3] or an empirically fitted

isentropic efficiency [6],[8],[9]. A common alternative approach

is to define the compressor power as a first order dynamic with an

ad-hoc time constant with the turbine power as the source term [4],

[5]. Map-based approaches of compressor power, on the other

hand, rely on the overall TC system efficiency available as a series

(for various vane positions) of supplier provided maps [7] applied

to the calculated turbine power. Alternately empirically fitted

compressor efficiencies are typically regressions as a 2nd

or 3rd

order polynomial in the Blade Speed Ratio (BSR). The

polynomial coefficients are often dependent on the shaft speed

[8]-[11] and are identified against the populated regions of the

manufacturer supplied maps. These polynomial models are,

A GENERALIZED COMPRESSOR POWER MODEL FOR TURBOCHARGED INTERNAL COMBUSTION ENGINES WITH REDUCED COMPLEXITY

Proceedings of the ASME 2016 Dynamic Systems and Control Conference DSCC2016

October 12-14, 2016, Minneapolis, Minnesota, USA

DSCC2016-9792

1 Copyright © 2016 by ASME

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therefore, also subject to extrapolation for operating conditions

beyond the manufacturer supplied test points. Note that the typical

manufacturer provided compressor performance map is based on

hot gas flow bench test data. These tests are performed under

steady flow conditions, hence, practical applications could deviate

from these maps under pulsating flow when coupled to an Internal

Combustion (IC) engine [1], [2], [13]. Another issue, often

encountered when operating at light load conditions, is the

sparsity of manufacturer provided compressor performance maps.

This is as indicated in Figure 1, for a sample turbocharged engine.

As a result, operating the compressor outside the available map

requires extrapolation under the assumption of a convex hull

extension of the supplied map. Several investigations into

extrapolation methods, based on these assumptions are reported in

open literature as in [14], [15].

The traditional compressor power is computed from the

standard isentropic efficiency equation in (1) based on the inputs

as indicated in Figure 2.

Figure 1. Operating range deficit between mapped and desired engine operational

ranges

11

1

air

air

in

outairpinis

C

CP

PcTmW

(1)

Figure 2. Traditional approach for deriving compressor power

In (1), the compressor isentropic efficiency must be defined.

In a review of existing literature, compressor isentropic efficiency

is commonly modeled using empirical fits based on manufacturer

provided compressor maps. In [9], the efficiency is expressed as a

polynomial function of the non-dimensional mass flow rate. The

polynomial coefficients are three individual functions of Mach

number. In [26] the efficiency is modeled using an elliptical fit

based on mass flow rate and pressure ratio. This model depends on

the maximum efficiency and the location in terms of corrected

mass flow rate and compressor pressure ratio. In [23], the

efficiency model form [26] is further modified for pressure ratio

variation. In [25] the corrected compressor work is defined as a

polynomial function of corrected mass flow rate and the model is

further fitted with experimental data. TABLE 1 summarizes these

approaches and our assessment of these models for performance

in the interior as well as the exterior of the supplier maps.

TABLE 1. Comparison among different modelling approaches

Efficiency prediction

method

Performance

within the

mapped area

Performance

outside the

mapped area

Mapping Maps provided by

manufacturer Good No data

Compressor

Models

based on

empirical

fitting

Jensen et al. [9] Good

Subject to

extrapolation

methods and

base map

granularity.

General

performance

varied form

average to poor.

Guzzella-Amstutz [27] Good

Kolmanovsky [10] Good

Andersson [23] Good

Canova et al. [24],[25] Good

Eriksson et al. [26] Good

Sieros et al.: Simple

linear [28] Good

Sieros et al.: Generalized

linear 1 [28] Good

Sieros et al.: Generalized

linear 2 [28] Good

In this paper, the compressor power model is developed based

on the Euler equations for turbomachinery. The model uses the

mass flow rate and compressor speed as inputs. It is found that

with the proposed approach a quadratic analytic function of two

parameters, the speed and power coefficients, is adequate to

characetrize the compressor performance over the entire operating

range. The model is validated using hot gas flow bench test data,

steady-state engine dynamometer test data, and transient

simulation data. Results indicate that the proposed reduced order

model is suitable for both steady state, pulsation flow conditions

as well as transient operation.

The paper is organized as follows. Section II discusses the

generalized compressor model based on the Euler equations with

different slip factors and Section III provides validation results

from turbocharger flow bench, turbocharged engine dynamometer

and transient simulation data. The last section adds some

conclusions.

GENERALIZED COMPRESSOR POWER MODELING

Compressor power model based on turbomachinery Euler equation

A typical centrifugal compressor geometry layout is shown in

Figure 3, and the velocity triangles of the gas flow at the

compressor impeller inlet and outlet are shown in Figure 4, and

are adapted from [1], [17]. Using the Euler equations the

compressor power can be expressed as:



1122 CUCUmrvrvmW ininoutoutC (2)

For the centrifugal impeller, it is assumed that the air enters

the impeller eye in the axial direction so that the initial angular

momentum of the air at the inlet of the compressor can be assumed

to be zero ( 011 CU ) [1], [17]. The ideal compressor power

equation can then be reduced to:

22_ CUmW idealC (3)

or equivalently:

22_ CRmW idealC (4)

Ideally, the flow exits the blade with a back sweep angle 2

with some slip. The theoretical absolute flow velocity can be

expressed as

222,2 tan rCUC (5)

where, 2rC is impeller outlet flow radial velocity. Neglecting slip

loss for the radial direction, the ratio between 2C and 2C in a

impeller is defined as slip factor as shown in (6); The slip factor

depends on a number of factors such as the number of impeller

blades, the passage geometry, the impeller eye tip to impeller exit

diameter ratio, the mass flow rate as well as compressor speed

[17].

2

2

C

C

(6)

With the slip factor the compressor power equation (4) becomes:

222222 tan_

rCURmCRmWidealC

(7)

For simplicity, 12 rr CC is assumed, where 1rC is impeller inlet

radial velocity. This leads to (constant inlet density assumption)

112212

A

m

A

mCC rr

(8)

Substituting (8) into (7), the compressor power can be formulated

as

2

11

22222

tan_

mA

RmRW

idealC

(9)

As a result of flow and ‘windage’ losses, the actual required

input work is greater than the theoretical value necessary to

achieve the target flow rate [33]. To account for this, power loss

factor and friction loss term fk are introduced to (10):

fC WWWidealC

_

(10)

In this paper, is assumed to be constant. Based on the study

from [17] and [34], the friction loss for impeller and diffuser are:

33 mkkmkW fdfiff (11)

where fik and fdk are friction constants for impeller and diffuser,

respectively. Including the friction losses, the actual desired

compressor power can now be written as:

32

11

22222

tanmkm

A

RmRW fC

(12)

Now, define two new parameters: the power coefficient

PowerC and speed coefficient SpeedC as below:

3m

WC C

Power

(13)

m

CSpeed

(14)

From (12), (13) and (14), the new compressor power can be

expressed in terms of these two coefficients as a quadratic

function:

fSpeedSpeedPower kCA

RCRC

11

22222

tan

(15)

Figure 3. Centrifugal compressor layout

Figure 4. Velocity triangles of a centrifugal compressor at the rotor inlet and outlet

Figure 5. Expected relationship between the speed and power coefficients from the

equation (14), n is different compressor operating point



Expected variation of Cpower with Cspeed based on the

relationship in (15) is shown in Figure 5. The range of this

variation will be influenced by variation in one or more of the

operating/design parameters as indicated in Figure 5. Different

compressor designs will result in different impeller wheel

diameters, power loss factor, slip factor, and exit area. For a given

compressor design and inlet condition, the inlet gas density 1 as

well as the parameters and fk may be assumed to be constant.

The power coefficient, as in (15), can then be expressed compactly

as a function of the speed coefficient and the slip factor for a given

operating condition, i.

, n,, i CfC iiSpeediPower 21 , ,__ (16)

Equation (16) indicates that the power coefficient is a function of

speed coefficient and the slip factor for a given compressor

design. One of the goals of this work was to investigate the

prediction capabilities of the power and speed coefficients in

conjunction with an appropriate slip factor model such that the

dimension of the parameter space in (16) may be reduced from

two to one is indicated in (17).

SpeedPower CfC (17)

Equation (18) shows the compressor power models in terms

of the expanded forms of Cpower and Cspeed. It is clear that in order

to achieve a form as in (17) the slip factor must be considered

either as a parameter fixed by design or defined in terms of the

speed and/or power coefficient to maintain the homogeneity of the

compressor power model with respect to Cpower and Cspeed. The

effectiveness of such approximations will therefore need to be

evaluated.

fC k

mA

R

mR

m

W

11

222

223

tan (18)

Slip factor investigation

Several slip factors models are readily available in open

literature [17, 22, 29-32]. The slip factor typically depends on the

compressor design parameters as well as the operating conditions,

such as: compressor rotational speed and mass flow rate. This led

to the development of several empirical slip factor models as in

[17, 22, 29-32]. In order to maintain the impact of flow conditions

on slip, non-constant slip factor candidate models, as proposed in

[22], [29], and [32] are investigated in this work:

Slip 1: 10 e-1*0.5-tan

12cos

-2

211

2

Z

RA

m (Reffstrup) (19)

Slip 2:

10

tan

cos/1

211

2

22

A

mR

RZ

(Stodola)

(20)

Slip 3: 10132

R

ma

(Stahler) (21)

where a is a design parameter in (21) related to the flow exit

angle and is a constant for a given impeller design. Z is the number

of impeller blades. Integrating these three slip models into the

proposed compressor power model (15), a generalized compressor

power model can be established as follows:

322

1 SpeedSpeedPower CCC (22)

where coefficients 1 , 2 , and 3 depend on the selected slip

factor model and are defined in TABLE 2. The model (22) is

referred to as the generalized compressor power model in the rest

of this paper. Note that the three coefficients in (22) take constant

values and are fixed by compressor design under the assumption

of constant or slowly varying inlet conditions. Using the

definitions of speed and power coefficients, the proposed

compressor power model (22) becomes:

32

2

13

mmm

WC

(23)

Equations (22) and (23) are the proposed analytical models

for the compressor power. The proposed model shows that the

compressor operation can be represented as an analytic quadratic

function rather than a two dimensional map. The proposed model

has only two inputs: mass flow rate and TC rotational speed. For

practical applications, these two parameters are available as

measurements. Alternately the speed can also be obtained by

solving turbocharger rotor dynamic differential equations as in

[3]-[5], and the compressor mass flow rate could be modeled with

shaft speed and pressure ratio as inputs. In this paper, the

compressor power model is validated under the former

assumption of the two parameters are available as measurements.

TABLE 2. Model coefficients for various slip models

Sli

p m

odel

1

1

2cos-2

22 0.5e.50

ZR

2

5.00.5etan 2cos

-2

11

22

Z

A

R

3

2

11

2tan

Ak f

Sli

p m

odel

2

1 )cos/1( 222 ZR

2 11

22 tan

A

R

3 fk

Sli

p m

odel

3 1 2

2R

2

11

22

2

tan

A

R

R

a

3 fkRA

a

211

2tan

MODEL VALIDATION



Model validation using standard flow bench test data

Data from standard hot gas flow bench tests for three different

production compressors was used to verify the proposed model

and the relationship between Cpower and Cspeed The test setup for

generating these data sets can be found in [12]. The inlet

conditions of compressors are considered as fixed [21], which

agrees with the constant inlet condition assumption used here. The

test matrix for the three different compressors is shown in Figure

6. The minimum test speed for Compressor-1 and Compressor-2

are 30,000 RPM, while the lowest speed for Compressor-3 is

46,000 RPM.

Power and speed coefficients are calculated based on (13) and

(14) for the test data points as in Figure 6. The values are shown in

Fig 7 (labeled as “compressor-i test data”) for the three different

compressor designs considered in this work. The Cpower vs Cspeed

characteristic curves for each compressor can be characterized as

a unique quadratic function. This representation is an important

and significant order reduction when considering the large

two-dimensional efficiency maps typically used for defining

compressor power [8],[11],[23],[25]-[27]. In order to identify the

model (22), a fitting cost function is defined as follows:

n

i

iWiWJ

1

2

measmodel

(24)

where n is the number of steady state compressor operating points

from hot gas flow bench test results; modelW is the calculated

compressor power using (23); and measW is the standard

compressor power calculated using (25) with measured inputs:

inoutpmeas TTcmW (25)

Figure 6. Turbo flow bench test ranges for 3 different compressors

The design parameters for Compressor-1 were available from

the supplier. The first step was to identify the power loss

coefficient and friction loss power for each slip model candidate

for Compressor-1. A Least-Squares optimization was used to

identify , fk and a such that the cost function (24) is

minimized.

The identified parameters are shown in TABLE 3. The

identified power loss parameters ( , fk and a ) are in a

reasonable range, indicating that compressor input power

necessary to achieve the desired flow rate must be larger than the

ideal power calculated in (4). The identified parameter a in Slip

Model-3 agrees with the result from [29]. The maximum friction

loss power identified was around 2.38kW over the entire

compressor operating range. Based on the fitting results in

TABLE 3. , the Slip Model-3 offers the best fit (lest error). The

identified model for Compressor-1 using the Slip-Model 3 is

shown in Figure 7. Fitting results confirm that the model structure

proposed in (22) can be generalized for modeling centrifugal

compressor power.

TABLE 3. Identified model coefficient for Compressor-1with different slip factor

models

fk a R2

Slip model 1 1.295 19040 - 0.968

Slip model 2 1.323 6374 - 0.968

Slip model 3 1.351 11390 1.334 0.989

Since compressor design parameters were not available for

the other two compressors (Compressor-2 and Compressor-3), we

used the generalized model structure (22) to identify the 3

parameters 1 , 2 and 3 directly and the identified values are

shown in TABLE 4. It is interesting to note from Figure 7 that the

three different compressor designs lead to three distinct

characteristic curves. This is expected and reflects the design

differences between the different compressors. In fact this

representation also allows a quick and easy method to compare

different compressor designs. As an example, it is clear that for a

given mass flow rate and TC shaft speed the compressor power

requirement follows the trend, Power1 > Power2 > Power3,

indicating that Design-1 may perhaps have a larger loss relative to

the other designs. Also note that it appears that Compressor-3 may

offer the widest operating range of the three designs considered

here.

TABLE 4. Identified model coefficient for compressor-2 and compressor-3 with

model structure (22)

1 2 3 R2

Compressor 2 0.001762 10.15 15600 0.995

Compressor 3 0.001179 0.01179 13900 0.996

Logarithmic scale plots offer new insights. The Log scale

plots for the three compressor designs are shown in Figure 7, and

are represented by three corresponding straight lines. This follows

directly from the Log scale representation of the generalized

model as (26)

qCpC SpeedPower logloglog (26)

In (26), p is the slope of each characteristic line and q is the y-

intercept for each design. Equation (26) can also be formulated as:

pSpeed

qPower CC 10 (27)

or

2 4 6 8 10 12 14 16

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TC Speed [RPM]

Com

pre

ssor

ma

ss f

low

ra

te[k

g/s

]

Compressor-1

Compressor-2

Compressor-3



pqC

mm

W

10

3 (28)

Using the same fitting method as described in the previous

section, identified p and q and the associated results are shown in

TABLE 5. One important advantage of the straight line model is

that (27) can, ideally, be obtained from only two compressor

operating conditions (one in high load, one in low load). This

method significantly reduces the experimental burden during the

early stages of compressor development.

Figure 7. Model validation for three different compressors

TABLE 5. Identified model coefficient for three compressor with (27)

p q R2

Compressor 1 2.51 5.795*10^(-6) 0.994

Compressor 2 2.50 4.769*10^(-6) 0.989

Compressor 3 2.36 1.375*10^(-5) 0.996

In summary, model validation results confirm the

effectiveness of generalized compressor power structure proposed

in this work. These results also demonstrate that for a centrifugal

compressor the operating characteristics can be reduced to a single

analytic quadratic function in linear scale and a straight line in

logarithmic scale.

Model validation based on steady state engine dynamometer test data

In this section, Compressor-1, identified earlier, was tested on

the engine dynamometer with a heavy duty turbocharged diesel

engine. Detailed engine and test setup information can be found in

[19] and [20]. The steady state test covers the whole engine

operating range (185 testing points). The engine speed changes

from 500 to 3,500 RPM. The engine brake torque varies from 10

to 1200 Nm. The compressor operating range (corresponding to

each engine operating point) results in a mass flow rate between

0.01 and 0.45 kg/s and a compressor speed between 5.9k and 109k

RPM. The compressor operating range (above 30,000 RPM) from

the engine dynamometer tests is a subset of the standard

turbocharger flow bench test data for the same operating range as

shown in Figure 8.

Figure 8. Test ranges of TC flow bench and engine dyno tests

Figure 9. Comparison of the standard TC flow bench and engine steady state

dynamometer test data for Compressor-1

The test bench data set was limited to 30kRPM TC speed at

the low end. Validation results in Figure 9 show the Cpower vs Cspeed

curves for the engine data, the flow bench test data and the fitted

model (based on the test bench data). Since the engine data set

extended to lower load conditions (TC RPM < 30k), the

0 0.5 1 1.5 2 2.5

x 106

0

1

2

3

4

5

6

7x 10

4

Speed coeff

Pow

er

co

eff

Compressor-1 test data

Compressor-1 model


Compressor-2 model


Compressor-3 model

106

102

103

104

105

Speed coeff

Pow

er

co

eff


Compressor-1 model


Compressor-2 model


Compressor-3 model

0 2 4 6 8 10 12 14

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TC speed [RPM]

Com

pre

ssor

ma

ss f

low

ra

te [

kg/s

]

TC flow bench test

Engine steady state dyno test

0 2 4 6 8 10 12

x 105

0

1

2

3

4

5

6

7x 10

4

Speed coeff

Pow

er

co

eff

TC Flow bench test

Engine steady state dyno(above 30000 RPM)

Engine steady state dyno(below 30000 RPM)

Model

106

102

103

104

105

Speed coeff

Pow

er

co

eff

TC Flow bench test

Engine steady state dyno(above 30000 RPM)

Engine steady state dyno(below 30000 RPM)

Model



characteristic curves for the engine data are plotted in two sets to

cover both operating regimes around the 30k RPM TC speed. It is

clear from Figure 9 that while the model adequately reproduces

the engine performance for the operating condition with TC speed

> 30K RPM, there is significant error for operating conditions

with TC speed < 30kRPM. These operating conditions are typified

by low mass flow rates and investigations reveal that the

measurements available from the engine at these low load

conditions suffer from severe variability and are therefore quite

suspect. An additional source of error could be related to the heat

transfer effect from the hot end (turbine) adding a positive bias to

the compressor outlet temperature. This effect of heat transfer on

compressor efficiency under light load operating conditions was

verified by experimentally and reported in [35], [36]. The higher

compressor outlet temperature leads to a higher calculated

compressor power (based on measurements), which makes the

measured power coefficient (below 30,000RPM) to be higher than

the model predicted value, as is observed in Figure 9. This is also

confirmed when considering transient test data from a GT-Power

based engine simulation as discussed below.

. Figure 10. Transient simulation validation

Transient Model Validation through GT-Power simulations

In order to verify the behavior of the proposed reduced order

model, transient engine simulation data is used in this section.

The main reason for using data from engine simulation was to

avoid measurement errors and biases, as discussed earlier.

GT-power engine simulation results used in this section are based

on the same engine with Compressor-1 connected to a variable

geometry turbocharger (VGT). The GT-power simulation setup

and model validation can be found in [20]. In this transient

simulation, the engine follows the target torque based on the pedal

position. The VGT vane feedback control is used to track the

target boost pressure generated from calibrated maps. A US06

driving cycle was studied for this case as shown in Figure 10.

Compressor speed varies from 14,000 to 105,000 RPM for this

transient simulation. The relationship between the power and

speed coefficients shows reasonable agreement between data and

model prediction for Compressor-1 with Slip Model 3.

CONCLUSION A compressor power model, based on the Euler

turbomachinery equations and realistic assumptions was

developed. Two new performance coefficients, the power and

speed coefficients were proposed as an alternative to multiple

performance maps. The proposed correlation between Cpower and

Cspeed is especially useful in defining the compressor power

necessary for regulating a desired compressor mass flow rate.

This can then be translated into a VGT vane position or the assist

demand in assisted boosting systems. This relationship can also be

easily used to compare compressor design variants and

compressor power requirements. The model is validated against

data sets from standard turbocharger flow bench tests, steady-state

engine dynamometer tests as well as transient engine simulations.

Validation results indicate that the proposed model provides

accurate compressor power prediction over a broad range of

compressor operating conditions and provides for an easy and

reliable extrapolation for operating conditions outside the

standard mapping domain. Further, the proposed model reduces

the dimensionality of the parameter space typically necessary for

such applications. The reduced order, reduced complexity model

is especially useful for the control applications. Future work will

focus on improving prediction accuracy in the face of

measurement noise as previously discussed.

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0 100 200 300 400 500 6000

2

4

6

8

10

12x 10

4

Time [s]

Turb

och

arg

er

Sp

eed

[R

PM

]

Turbocharger speed

2 3 4 5 6 7 8 9

x 104

0

5000

10000

15000

Speed coeff

Pow

er

co

eff

GT simulation

Model



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