3. enhance dcm

Attempts to enhance the differential coefficientsmethod for reconstruction of transient emissionsfrom heavy-duty vehiclesM R Madireddy and N N Clark*

Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia, USA

The manuscript was accepted after revision for publication on 29 October 2008.

DOI: 10.1243/14680874JER01808

Abstract: Engine tailpipe emissions as measured by a sampling system and analyser do notclearly reflect the actual transient emissions from the engine at the tailpipe of the vehicleexhaust. With increasing demand for accurate emissions measurements, several researchefforts are being made to compensate for the measurement distortions of the analyser system.The differential coefficients method is one such approach which reconstructs the input signalby approximating the analyser input as a linear combination of the output and the first twoderivatives of the output. While the results with this approach were found to be acceptable, thispaper focuses on the possibilities of improving the accuracy of the reconstruction by alteringthe ways in which the numerical derivatives are computed and by considering higher orderderivatives in the linear combination. It was found that the use of backward differences incomputing the numerical derivatives proved more effective than forward differences. Usinghigher order derivatives has shown improvement of about 10 per cent over using just the firsttwo derivatives, but this margin may be important for model accuracy or for assessing pass/failemissions criteria.

Keywords: differential coefficients method, time delay, time alignment, cross-correlation,instantaneous input, dispersion, reconstructed input, dispersed output, central/ forward/backward differences

1 INTRODUCTION

‘Instantaneous’ emissions are the actual emissions

produced by the engine (or at the tailpipe) due to

combustion of the fuel. They may be modified by

after-treatment systems before they leave the engine

stack or tailpipe. A measurement system can be used

to record these instantaneous emissions, but the

system will distort the signal that corresponds to the

instantaneous emissions and the system will produce

an output signal, which represents the ‘measured’

emissions. For an ideal analyser system, the measured

emissions will be the same as the instantaneous

emissions. However, any real analyser system will

report a distorted signal during the process of

measurement. For example, the emissions reported

by the analyser may be delayed and dispersed relative

to the instantaneous emissions [1–5]. Knowledge of

the instantaneous emissions may play an important

role in developing both inventory models and superior

engine control strategies, and the prediction of

instantaneous emissions from measured emissions.

This paper presents conclusions related to a single gas

species, nitric oxide (NO), but the results are equally

applicable to all species encountered in engine

exhaust.

2 TIME DELAY AND COMPENSATION FOR THETIME DELAY

When the steady state condition of the engine

changes, the corresponding engine-out emissions

*Corresponding author: Center for Alternative Fuels, Engines,

and Emissions, Department of Mechanical and Aerospace

Engineering, West Virginia University, Morgantown, West Virgi-

nia, USA. email: [email protected]

65

JER01808 F IMechE 2009 Int. J. Engine Res. Vol. 10

levels also change. There is a time delay between the

point when the engine/vehicle experiences an

operating condition and the point when the corre-

sponding analyser measures the emissions related to

that operating condition. The time delay is a

combination of transport time taken by the exhaust

to reach the appropriate gas analyser and the

response time of the analyser. The sum total of

these two measurement delay times should account

for the time shift between the instantaneous (actual)

emissions at the tailpipe and measured emissions.

Time delay was addressed in detail by Messer et al.

[3, 4]. In order to compensate for the delay, the

emissions data should be shifted back in time to

align with the engine operating variables such as

power and speed. The importance of time alignment

was addressed in detail by Hawley et al. [5]. There

are several ways in which the emissions data can be

time aligned. Some researchers prefer visual time

alignment, which is usually implemented by match-

ing the crests and troughs of the continuous

emissions dataset with that of an engine or vehicle

operating parameter such as power. The most widely

followed procedure for time alignment is cross-

correlation, where two datasets are compared

against a common variable and the time shift that

best matches the two sets is calculated. It is based on

the assumption that a correlation exists between the

two datasets, but this may fail for the case of

emissions species that do not correlate well with

engine power. For example, emissions of carbon

monoxide or particulate matter from diesel engines

may depend both on power and the rate of change of

power with respect to time. Cross-correlation be-

tween the time derivatives of power and emissions

mass flowrate has also been used.

3 TIME DISPERSION AND ESTIMATION OF TIMEDISPERSION

The measured response from a transient (oscillating)

instantaneous emissions output experiences ampli-

tude reduction in addition to the time delay. The

amplitude of a peak or a trough in the measured

response is smaller than the one actually experi-

enced by the engine. Hence, there is a need to

compensate for the time dispersion of the data by

the analyser system. For this purpose, estimating the

transient response of the measurement system is

required [6, 7].

The current researchers examined the response

characteristics of two analysers (with the sampling

system) used to collect the data that were processed

in this study. NO was measured using a Rosemount

955 analyser and CO2 was measured using a Horiba

AIA 210 analyser. Data analysed were measured by

only these analysers connected to a 0.45 m diameter

dilution tunnel. More detail is given in reference [8].

The response of the Rosemount 955 NO analyser

(manufactured by Emerson Electric Co., St Louis,

MD, USA) to an instantaneous pulse input of NO is

shown in Fig. 1(a). The response was obtained

through the following experiment conducted in the

engine test cell at West Virginia University. A balloon

was filled with approximately 1 litre of NO with a

concentration of 1000 p/million. and inserted in the

dilution tunnel. The balloon was burst to simulate an

instantaneous pulse. The pulse travelled via the

sampling lines, and the output of the Rosemount 955

was collected. The time delay due to transport of the

pulse through the tunnel depends on the length of

the tunnel. The time delay shown in Fig. 1(a) was a

function of the length of the sampling lines and the

Fig. 1 Response of (a) the Rosemount 955 NO analyser and (b) the Horiba AIA 210 analyser to aninstantaneous input of NO at time t 5 0

66 M R Madireddy and N N Clark

Int. J. Engine Res. Vol. 10 JER01808 F IMechE 2009

speed of the exhaust gas travel through the lines, as

well as a contribution from the analyser. Only the

dispersion of the instantaneous pulse is of interest in

this analysis. The response was found to be

dispersed over a period of 6 s. The probability

density function (pdf) is shown in Fig. 1 as a set of

probabilities with 0.2 s increments. If the fractions

(probabilities) of the response were less than 0.05

per cent, all such fractions were considered to be

insignificant and were added as one fraction on

either side of the response. In other words, if there

was a negligible response from t 5 0 to t 5 7 as in

Fig. 1(a), the sum of all those small responses was

shown in t 5 7. In Fig. 1(a), the function is normal-

ized such that the area under the curve is unity,

which is a definitive characteristic of a pdf. A similar

response for the Horiba AIA 210 analyser (manufac-

tured by Horiba Automotive Test Systems, Ann

Arbor, MI, USA) for CO2 is shown in Fig. 1(b). A

comprehensive description of these types of analyser

and their operation is provided in reference [9].

4 DATA USED FOR ANALYSIS

The data used for testing these techniques were

collected from the Federal Test Procedure (FTP)

cycle on a 1992 Detroit Diesel Corporation (DDC)

Series 60 engine and a 2004 Cummins ISM 370

engine and the testing was conducted on a dynam-

ometer in a transient cell. Further details of the test

cell and typical operation can be obtained elsewhere

[10]. The NO data were collected using a fast NO

analyser, which was manufactured by Cambustion

Ltd, Cambridge, UK [11, 12]. Since the response

from the fast NO analyser was rapid and had low

diffusion (T10–90% 5 12 ms), the data were treated as

instantaneous data without dispersion of the type

shown in Fig. 1(a). No NO2 to NO converter was

used with the Cambustion fast NO analyser. With

early model year engines, the NO is a reasonable

approximation of NOx because the NO2 to NO ratio

is generally small [13]. This paper demonstrates the

methodology for NO, but the methodology may be

applied to NOx or to any other emission species that

is measured continuously.

The dispersion of the instantaneous emissions to

yield the measured emissions in this study includes

dispersion in the exhaust system, dispersion in the

tunnel, dispersion in the sample lines, and disper-

sion due to the nature of the analyser function. It is

true that each of these could be studied separately

using the fast NO analyser, but from a practical

perspective they are all used in concert and the

system needs to be considered as a whole for real-

world data. However, the methods used in this paper

could also be applied to each separate dispersion

system using an appropriate dispersion function if

desired. The dispersion function shown in Fig. 1(a)

was used to diffuse the instantaneous (fast NO) data

to obtain the diffused NO data, so that they

simulated the data that would be measured by the

Rosemount 955 NO analyser. This analyser was set in

NO mode without an NO2 to NO converter. In this

way the authors had companion sets of both

instantaneous and (simulated) measured emissions

to test instantaneous emissions reconstruction algo-

rithms.

5 REVISITING THE DIFFERENTIALCOEFFICIENTS METHOD (DCM)

Ajtay and Weilenmann [1] and Madireddy and Clark

[2] have discussed a mathematical approach to

reconstruct the true emission signals from the

measured output of the analyser. The DCM ap-

proach of Ajtay and Weilenmann [1] defines real

input as a linear combination of the output and its

time derivatives. Let U(t) be the value measured by

the fast NO analyser, which was considered to yield

instantaneous data. Now consider that U(t) is the

input to the conventional analyser, and let Y(t) be

the output (measured) value from the conventional

analyser. The output was simulated by mathemati-

cally dispersing or convoluting the input U(t) with

the pdf (i.e. dispersion function), which was shown

in Fig. 1. Y9(t) and Y0(t) are the first and second

derivatives of the simulated measured emissions

(simulating the conventional analyser output). The

method assumes that the input can be expressed as

the sum of the output and some linear combinations

of the first and second derivatives of the output. The

input U(t) and output Y(t) and its derivatives are

related by

U tð Þ~Y tð Þza1Y ’ tð Þza2Y ’’ tð Þ ð1Þ

In order to obtain the values of the coefficients, the

known response of the analyser, which was the

response to an instantaneous pulse, was used in

equation (1). The dispersion function, which was

experimentally estimated, was considered as the

output Y(t) to a unit input U(t) and the output Y(t)

was differentiated numerically to obtain Y9(t) and

Y0(t) over a period of the dispersion. The numerical

derivatives for this study were computed using

central, forward, and backward differences in 1 s

Attempts to enhance the ‘differential coefficients method’ 67


time intervals. Then the derivatives were mapped

with the unit impulse input and the time sequence

was fitted over the dispersion period and the error

was then computed at each second as the absolute

value of [U(t) 2 Y(t) 2 a1Y9(t) 2 a2Y0(t)]. The least

squares error was computed as the sum of the

squares of the computed errors at all points and this

was minimized for the best fit that generated the

values of a1 and a2, the coefficients of the derivatives

of the output. Values of a1 and a2 depended on the

actual analyser response. The values of a1 and a2

were used to obtain the input U of the analyser from

any measured output Y of the analyser. The inherent

assumption in the method was that the analyser was

consistent, and linear in its dispersion behaviour

with respect to time and rate of change of input.

6 EFFECT OF FORWARD, CENTRAL, ANDBACKWARD DIFFERENCES

The DCM suggested by Ajtay and Weilenmann [1]

involved the first two differentials numerically

computed using backward differences. It was of

interest to compare the results when different

methods of computing the derivatives were used.

Continuous data from a heavy-duty engine FTP cycle

were measured by the fast NO analyser from the

engine tests conducted on two different engines

(DDC Series 60 and Cummins ISM 370). Each of

these cases was examined to understand the effect of

the method by which numerical derivatives were

computed. The forward, central, and backward

differences were used independently while comput-

ing the differentials. All the differences were com-

puted numerically with 1 Hz frequency for 1200 data

points. A forward difference at time t 5 n was given

by Dx(t 5 n) 5 x(t 5 n + 1) 2 x(t 5 n). Similarly, a

backward difference at time t 5 n was given by

Dx(t 5 n) 5 x(t 5 n) 2 x(t 5 n 2 1), and a central dif-

ference at time t 5 n was given by Dx(t 5 n) 5 1/

2[x(t 5 n + 1) 2 x(t 5 n 2 1)].

The DCM was used to reconstruct the original data

from the dispersed data and in each of the cases, the

percentage error was computed as follows. If U(t) is

the original input and Y(t) is dispersed output, and if

U*(t) represents the reconstructed input, then the

percentage error is computed according to

Percentage error~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPU tð Þ{U� tð Þ½ �2

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPU tð Þ½ �

p 2ð2Þ

In each of the cases, percentage error (equation (2))

was computed and the results when more deriva-

tives were used are shown in Table 1. Cases 1 to 3

represent three separate runs for the DDC engine

and cases 4 to 6 represent three separate runs for the

Cummins engine. The backward differences showed

better results than the forward differences. This

could be attributed to the fact that the reconstruc-

tion at any point in time depends more on the data

points that precede than the data points that follow

as a result of the asymmetry of the diffusion

function. In addition, typical engine transients may

contribute to different behaviour under increasing

and decreasing emissions scenarios, but this is still

speculative. In all cases the backward difference

approach yielded fewer errors than the forward

difference approach. For the DDC engine, the

central difference approach was slightly superior to

the backward difference approach, but overall for

the dataset, the backward difference approach is

recommended.

7 EFFECT OF MULTIPLE DERIVATIVES ON DCM

Instead of using just two derivatives as in equation

(1) for the DCM, multiple derivatives were also used,

as follows

U tð Þ~Y tð Þza1Y 0 tð Þza2Y 00 tð Þza3Y 3 tð Þ

za4Y 4 tð Þza5Y 5 tð Þza6Y 6 tð Þ ð3Þ

Table 1 Errors in DCM with different numerical methods for computing the derivatives for theDDC Series 60 and Cummins ISM engines

Numerical methodBackward differences methodof Ajtay and Weilenmann [1] Forward differences Central differences

Case 1 2.03% 2.45% 1.91%Case 2 2.04% 2.65% 1.93%Case 3 2.07% 2.42% 1.96%Case 4 3.55% 4.03% 4.38%Case 5 3.58% 4.16% 4.32%Case 6 3.95% 4.39% 4.25%

Cases 1 to 3 are the results for the DCC engine and cases 4 to 6 are for the Cummins engine.



Clearly more fitting parameters were needed.

Several cases were examined for reconstruction

using DCM with more derivatives. Equation (3)

holds good for the case in which six derivatives

were used for computation. The same equation can

be used for different cases. For example, in the case

that uses only the first three derivatives, only the

first four terms exist on the right-hand side. Back-

ward differences (which were comparatively better

than forward differences from the results shown in

Table 1) were used for each of these cases and the

percentage error was again computed according to

equation (2), the results are shown in Table 2.

Employing higher order derivatives improved the

accuracy of reconstruction, but the computation

time and the complexity of the procedure increased

with the addition of higher order derivatives. Noise

levels during data measurement could affect

the accuracy of reconstruction. This effect was

discussed in detail in reference [2]. In the case

of heavy-duty-engine NO emissions reconstruc-

tion, it can be concluded that the use of more

derivatives was advantageous offering in the order

of a 10 per cent improvement. For researchers

seeking simplicity, only the first two derivatives

were recommended for reconstruction, as origin-

ally suggested by Ajtay and Weilenmann [1], but for

more accurate modelling, additional derivatives

proved desirable.

8 CONCLUSION

The original DCM method [1] was shown again [2]

by the authors to recreate instantaneous data from

dispersed data successfully. The original method was

extended to use more than two derivatives and the

improvement over the use of two differentials

offered at best a 10 per cent advantage in reducing

the root mean square error. For further application

of this procedure in emissions reconstruction, the

use of higher order derivatives can improve the

accuracy of results, given a way in which computa-

tional (rounding) errors can be minimized. This 10

per cent improvement may be important for model

accuracy or for assessing pass/fail emissions criteria.

ACKNOWLEDGEMENTS

The authors appreciate the research efforts of the WestVirginia University engine test cell for providing thedata analysed in this study. The authors also thankBradley Ralston and David McKain for assisting inobtaining the transient response shown in Fig. 1.

REFERENCES

1 Ajtay, D. and Weilenmann, M. Compensation ofthe exhaust gas transport dynamics for accurateinstantaneous emission measurements. Environ.Sci. and Technol., 2004, 38, 5141–5148.

2 Madireddy, R. M. and Clark, N. N. Sequentialinversion technique and differential coefficientsapproach for accurate instantaneous measure-ment. Int. J. Engine Res., 2006, 7(JER6), 437–446.

3 Messer, J. T. Measurement delays and modalanalysis for two heavy-duty transportable emissionstesting laboratories and a stationary engine emis-sion testing laboratory. M.Sc. Thesis, Department ofMechanical and Aerospace Engineering, WestVirginia University, 1995.

4 Messer, J. T., Clark, N. N., and Lyons, D. W.Measurement delays and modal analysis for aheavy-duty transportable emissions testing labora-tory. SAE technical paper 950218, 1995.

5 Hawley, J. G., Brace, C. J., Cox, A., Ketcher, D., andStark, R. Influence of time-alignment on thecalculation of mass emissions on a chassis dynam-ometer. SAE technical paper 2003-01-0395, 2003.

6 Ganesan, B. and Clark, N. N. Relationship betweeninstantaneous and measured emissions in heavy-duty applications. SAE technical paper 2001-01-3536, 2001.

7 Weilenmann, M., Soltic, P., and Ajtay, D. Describ-ing and compensating gas transport dynamics for

Table 2 Percentage errors in DCM with multiple derivatives for the DDC Series 60 and CumminsISM engines

Derivatives used

First two derivatives‘suggested by Ajtay andWeilenmann [1]’

First threederivatives

First fourderivatives

First fivederivatives

First sixderivatives

Case 1 2.03 1.83 1.83 1.78 1.78Case 2 2.04 1.81 1.81 1.75 1.75Case 3 2.07 1.83 1.82 1.77 1.77Case 4 3.55 3.48 3.29 3.29 3.28Case 5 3.58 3.55 3.42 3.42 3.41Case 6 3.95 3.82 3.29 3.60 3.57

Cases 1 to 3 are the results for the DDC engine and cases 4 to 6 are for the Cummins engine

Attempts to enhance the ‘differential coefficients method’ 69


accurate instantaneous emission measurements.Atmos. Environ, 2003, 37, 5137–5145.

8 Madireddy, R. M. Methods for reconstruction oftransient emissions from heavy-duty vehicles. Ph.D.Thesis, Department of Mechanical and AerospaceEngineering, West Virginia University, 2008.

9 Nayak, G. A. Development of a test method tomeasure in-use emissions from stationary andportable diesel sources. M.Sc. Thesis, Departmentof Mechanical and Aerospace Engineering, WestVirginia University, 2004.

10 Aravelli, A. Real time measurement of oxides ofnitrogen from heavy-duty diesel engines. M.Sc.Thesis, Department of Mechanical and AerospaceEngineering, West Virginia University, 2003.

11 Cambustion, Fast Gas Analysers. Information re-trieved from www.cambustion.com on 29 June 2006.

12 Cambustion Limited. fNOx 400, Users manual,2006 (Cambustion Limited, Cambridge, UK).

13 Lavoie, G. A., Heywood, J. B., and Keck, J. C.Experimental and theoretical study of nitric oxideformation in internal combustion engines. Sci.Technol., 1970, 1, 313–326.

APPENDIX

Notation

U(t) original input to the analyser

U*(t) reconstructed input

Y(t) dispersed analyser input



3. enhance dcm

Documents