FALCON S^iSM
DENVER BALTIMORE ALBUQUERQUE BUFFALO
ONE AMERICAN DRIVE
BUFFALO. NEW YORK 14225
TELEPHONE: (716) 632-4932
EMISSION CORRECTION FACTORS:
A STUDY TO DEVELOP AN IMPROVED MATHEMATICAL APPROACH
Prepared for:
Environmental Protection Agency
Office of Air and Waste Management
Mobile Source Air Pollution Control
Under:
Purchase Order No. CD-8-0145-A
Prepared by: H. T. McAdams Approved by: A. Stein
June 1978
A SUBSIDIARY O F i 11 "O
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TABLE OF CONTENTS
Section Title
1 INTRODUCTION
2 SUMMARY
3 RECOMMENDATIONS
4 WORKING PAPERS
4.1 Working Paper No. 1 -
A Factor-Analytic Approach to Emission
Correction Factors
4.2 Working Paper No. 2 -
Speed-Temperature-Hot/Cold Correction
Factors: A Critique and Prospectus
4.3 Working Paper No. 3 -
Hot/Cold/Stabilized Vehicle Operation:
A Critique and Candidate Approach
i
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I. INTRODUCTION
This report is submitted as deliverable item in fulfill-
ment of work performed under Purchase Order No. CD-8-0145-A
for the Environmental Protection Agency (EPA), Office of Air
and Waste Management (OAWM), Mobile Source Air Pollution
Control (MSAPC). The objective of the program was to examine
the mathematical formulation of existing emission correction factors
and to suggest a new mathematical approach which is simple
to use, simple to update when new data are available, and
which considers the dependencies among various factors.
The scope of work was aimed at isolating a set of pro-
spective statistical techniques and discussing the advantages/
disadvantages of each technique. Evaluation criteria were
to include but not be limited to, the following:
• Ease in application (manually and computer).
• Ease in updating when additional data are available.
• Required input data sample.
• Ability to relate to engineering concepts.
• Ability to assess and express correction factor
uncertainty.
• Required test procedure changes to obtain needed
data.
This report summarizes findings and offers recommendations
for future correction factor development. Because of the
broad scope of the effort, these recommendations are neces-
sarily exemplary rather than definitive. Their purpose is
to serve as an outline or prospectus for further development
of correction factor methodology.
1
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II. SUMMARY
This report is developed as a series of three working
papers developed in chronological sequence. Each of these
papers touches on one or more issues pertinent to the formu-
lation of emission correction factors.
The formulation and application of emission correction
factors represents an attempt to deal with the multiplicity
of variables which influence vehicle emissions. Mathematically
the problem can be represented as a function
Yp = f(xlf x2, ..., xk) (1)
where yp represents emission of a particular pollutant in
grams per mile and x^, X2, . are variables such as
speed, ambient temperature and vehicle variables known to
affect emissions. Equation (1) is often referred to as a
response relation and can be represented as a hypersurface
(response surface) in (k + 1)-dimensional space. The sub-
space consisting of the variables xj, X2, ..., x^ is often
referred to as the "treatment space" or the "x-space."*
To explore the emission response space in sufficient
detail to allow formulation of the response relation (.1) is
the crux of the correction-factor problem. The standard
emission test as formulated in the Federal Test Procedure
(FTP) constitutes only a single point in the treatment space
and consequently provides no information pertinent to emis-
sions under any conditions other than those specified in
the FTP. Moreover, because of the large number of variables
which can affect emissions and the prospect of interaction
among these variables, it is quite costly to explore the
x-space with a sufficient number of treatments and emission
tests to represent all possible driving and use scenarios.
The term "treatment" is a vestige of the fact that the
science of experiment design originated in agricultural
research, in which various fertilizer treatments were
applied to plots of land to determine the effect of these
treatments on crop yield.
2
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Efforts of the EPA to provide correction factors applicable
to a wide range of scenarios provide an approximation to the
ideal response-surface approach. Because of cost constraints
and other considerations, data-acquisition methods were some-
times less than optimum and it was necessary to draw heavily
on engineering knowledge and judgment in order to fill in
data lacunae. For example, it is well known that if two
variables and X2 are subject to interaction in the
sense of experiment design, then it is necessary to explore
the (xj, X2)-plane with a set of treatments which are distri-
buted areally in that plane. Under realistic circumstances,
however, it may be possible to sample x^ only at some fixed
value of X2r and to sample X2 only at some fixed value of
x^. Such a sampling design can not provide the information
needed to assess the interaction between x^ and X2/ and
it thus becomes necessary to supplement such data with engi-
neering judgment.
The working papers which constitute the bulk of this
report provide a critique of past methodology and offer
suggestions for future correction factor development. They
address certain general considerations in the philosophy
of correction-factor formulation as well as certain specific
issues involved in implementing that philosophy.
Working Paper No. 1 well exemplifies these two aspects
of the present study. Though the paper is concerned primarily
with the representation of preliminary speed correction factors,
it also raises such basic and general issues as the degree of
commonality of correction-factor functions for various groups
of vehicles and the most parsimonious representation of
functional relations. In this connection it was shown that
principal component analysis provides a means for structuring
the information content of correction factors in a concise
way, as well as a means for identifying areas of commonality
among various makes and models or other homogeneous groups
of vehicles. Specifically, it was shown that the functional
relation between correction factor and average speed tends to
have a common shape for various groups of vehicles and that
among-group variation can be accommodated by one or two
group-specific parameters rather than the four or five param-
eters assumed in the exponential and polynomial regression
relations.
3
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Working Paper No. 2 extends the analysis of preliminary
speed correction factors to their incorporation in the more
general "R-factors," which take into account, in addition to
average speed, a number of other factors affecting emissions.
These include mileage accumulation, the prevailing ambient
temperature, and the fraction of vehicle operation performed
in the cold transient, hot transient, and stabilized modes.
In addition, the paper makes generalizations pertaining to
the identification and definition of variables affecting
emissions and the important role played by nondimensionali-
zation in the formulation of correction factors.
One of the difficulties encountered in the use and
interpretation of the R-factors arises from the notion that
the fraction of miles driven in the cold transient, stabilized
and hot transient conditions can be driven at arbitrary
speeds, whereas the FTP definition of these conditions imply
associated speeds of 26, 16 and 26 mph, respectively. When
average speed and mode of operation are incorporated in a
common correction factor, therefore, this combination
impacts on trip lengths and the proportion of trips origi-
nated in the cold-start mode. To reconcile all the con-
straints it is necessary to invoke certain assumptions about
either the lengths or time durations of trips and the
portions thereof spent in a Bag 1, Bag 2 or Bag 3 condition
of operation. Otherwise, the implied warm-up times may
exceed the time allowable under the FTP test. Reformulation
of the R-factors, therefore, is indicated to be a fruitful
area for refinement of correction factors.
In the general area of methodology, Working Paper No. 2
provides a good example of how correction factors may be
simplified by an adroit definition of variables affecting
emissions. In particular, it is shown that it may be
advantageous to represent the effect of speed on emissions
in terms of grams per unit time rather than in terms of
grams per unit distance. This observation arises from a
dimensional analysis type of argument and is shown empiri-
cally to lead to considerable simplification of the speed
correction factor function. It is further shown th^t diffi-
culties arising from numerical constraints imposed by choice
of units can often be avoided by nondimensionalization of
the independent as well as the dependent variables in a
functional expression.
4
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Working Paper No. 3 delves further into the problem of
transient versus stabilized operation. In particular/ it
suggests that the thermal state of operation of a vehicle
comprises a continuara and that it may be more advantageous
to view the operating history of a vehicle in this light
rather than in the light of discrete cold transient, hot
transient and stabilized states. This type of treatment
would circumvent the need to define and estimate "warm-up
times" under various ambient and use conditions. An accretion-
depletion model of thermal transient effects is outlined and
it is proposed that any state of operation of a vehicle can
be represented as a linearly weighted combination of initial
and final emission values as represented by cold transient
and stabilized operation, respectively.
In summary, several directions for possible refinement
of correction factors have been identified. In most instances,
these directions have been made explicit by an example based
on actual data.
5
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3. RECOMMENDATIONS
Recommendations aimed at an optimal approach for future
correction factor development are evolved primarily in Working
Paper No. 2. These recommendations are based on the incontro-
vertible fact that correction factors represent an attempt
to express the functional dependence of emissions on a host
of variables and that a designed experiment is best suited
for defining this function. The experiment should be based
on the best available engineering estimates and experience
pertinent to the complexity of the functional relation, the
interaction of variables, and the relative importance of
these variables in the emission-generation process. The
thrust of the emission-factor formulation should be to separate
vehicle or vehicle-class dependent aspects of emissions from
incremental effects more or less common to all vehicles.
Specific recommendations, therefore, are as follows;
o Systematic analysis of the degree of commonality of
the effects of emission-related variables should
precede attempts to develop a correction-factor
response function.
o Correction factors are best determined through a
designed experiment in which the allocation of
"treatments" (that is, combinations of levels of
the variables , X2, Xp) is made according
to the anticipated degree of nonlinearity within
variables and degree of interaction among variables.
o Mathematical representation of correction factors
should be approached with due regard to the magni-
tude and engineering importance of variables
perceived to be important. The prospect of
combined or derived variables should not be over-
looked nor should the fact that variables originally
identified may be highly covariant.
o Difficulties arising from numerical constraints
imposed by choice of units can often be avoided by
nondimensionalization of independent or predictor
variables as well as the dependent variable.
6
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It is recognized that any proposed program aimed at
refinement of correction factors must be balanced against
associated costs. Though such considerations are beyond
the scope of the current effort, a cost-benefit examination
of correction-factor refinement is considered to be an
essential prelude to future correction factor development.
7
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4. WORKING PAPERS
8
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4.1 WORKING PAPER NO. 1:
A FACTOR-ANALYTICAL APPROACH TO EMISSION
CORRECTION FACTORS
9
-------
FALCON
DENVER BALTIMORE ALBUQUERQUE BUFFALO
ONE AMERICAN DRIVE
BUFFALO. NEW YORK 14225
TELEPHONE: <716> 632-4932
Working Paper No. 1
Project No. 8411
Environmental Protection Agency
April 5, 197 8
H. T. McAdams
A FACTOR-ANALYTIC APPROACH TO
EMISSION CORRECTION FACTORS
1. INTRODUCTION
Vehicle exhaust emissions are functions of a large
number of variables. Some of these variables are vehicle
related, others pertain to the operating environment, and
still others are the consequence of use priorities dictated
by the needs of a mobile society. Consequently, one can
envision an infinite variety of emission scenarios, each
having its own peculiar impact on air quality.
Because emissions are either continuous or discrete
functions of a large number of variables it is impractical
if not impossible to measure emissions for every scenario
of interest. The only alternative is to attempt to derive
a modeling or scaling procedure by means of which a limited
number of measurements can be employed to predict emissions
over the entire domain of the multivariate emission function.
Perhaps the most "vehicle-related" variable is the vehicle
itself. Different vehicles exhibit different emission charac-
teristics, and these differences, of course, stem from many
design variables which differentiate one vehicle from another.
If all these variables are pooled, however, one can think of
a "vehicle variable" which changes value discretely as one
goes from one make or model year to another. In short,
vehicles can be considered as comprising a state space.
Viewed as a process, pollutant generation assumes succes-
sively. different states as attention is directed succes-
sively to different automobiles c$r different classes of
A SUBSIDIARY OF W.Htaker
-------
automobiles. Within a given state (i.e., a particular auto-
mobile or automobile group) the emission process is responsive
to other, continuous, process variables, such as operating
speed, ambient temperature, and the like.
It is important to distinguish between data analysis
applicable to a given state of the process and that applicable
to some aggregation of states. In short, it is important
to know to what extent the effect of the process variables
are common to all states and to what extent these effects
must be particularized to a given state. Factor analytic
methods--in particular, principal component analysis—provide
an approach to this problem and is indicated to have important
bearing on the parsimonious formulation of emission correction
factors. The method is demonstrated in relation to speed
correction factors but is evidently applicable to other
variables in different contexts.
2. SPEED CORRECTION FACTORS
Speed correction factors for 18 groups of vehicles are
given in the attached Tables 11.12, 11.13, 11.14 from
"Supplement 8 Light Duty Vehicle Correction Factors," Memo
Janet Becker to J. Hidirtger/J. Horowitz (3/7/77). The vehicle
groups play the role of states in the process, whereas speed
plays the role of a continuous variable affecting emissions
within a given state. Consider, for example, the formulation
of CO correction factors as
in CF = A. + A. s + A„ s2 + A_ s3 + A. s4 + Ac s5 ...
U x j 4 3 l-LJ
or
CF = exp (A + A, s + A„ s* + A0 s3 + A„ s* + At ss)
U 1 Z J 4 o {£}
The correction factors are normalized to yield CF = 1.0 at
s = 19.6 mi/hr. Note that separate coefficients (and separate
correction factors as a function of speed) are given for each
of the 18 vehicle groups. One might well ask if some
"commonality" might not exist among the groups so that a
single functional form might unify all groups except for a
state "scaling factor" peculiar to each vehicle group.
Falcon R&D
Working Paper No. 1
Page 2
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Tabic XI-12
Croup Definitions
Group l
-------
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o sn xr.ble II.i",
k- q Speed Correction factors
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Table 11.14
Selected Speed Correction "actors - Warm Operation
'vdrocarbon
Average Speed
y.
3.107
3.297
3.033
3 .4*7 0
3.419
3,123
3.160
2.700
2 ,Q03
3.039
2.793
2.92S
2.705
3.276
2.315
2.763
3.963
3.194
000
10.000
1,679
1.7^9
1.703
i.800
1.773
1.695.
1.70?
1,543
1.60 0
1 ,64215
, 16 0
, 170
. 190
r 1 £>6
, 1 B'i-
,165.
,216
,172
. lfli
, 283
.2?. 3
20.000
0.987
0.956
0.936-
0 i 934
0,985
0. 987
0 .986
0 ,990
0 i 990
0. 986
0,9-39
0.9BS
0.939 '
0.986
0.989
0.937
0.981
0.954 .
0 .891
0,367
0.377
0.371
0.871
0 .544
0,876
0.843
0.734
0,803
25.COO
0.358 0
0.S44 0
0.341 0
0,321 0
0.834 C
0.353 0
0.845 0
0.SS9 0
0
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«761
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.733
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35.000
0. 6 ft 4
0»659
0,650
0»506
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0.709
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0,650
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0.540
n
40.000
.629 (
0.600 <
0.592 <
0.538
0 .565
0 .622
0 .600
0. 703
0.720
0 . 649
0.667
0.659
0.640
0.589
C . 669
0.573
0 .446
0 .463
45.000
.597
.565
. 556
.497
.526
.591
.567
.601
.699
.619
.644
.632
.616
.554
,646
.543
.401
.432
50.000
0,535
0.547
0.534
0,472
0,5 04
0.576
0 .551
0.678
0,694
0.609
0,638
0.624
0.61.0
0.53A
0.64]
0,531
0.373
0.414
- 55.000
0,571 0
0 .530
0,503
0,445
0 .479
0.557
0.529
0.672
0.691
0.596
o,M7
Q-.612
0.589
0,519
0,623
0.497
0.337
P r 373
.60.000
.516
.432
.429
.331
.414
. f>95
,(.60
,616
.64 9
,538
.525
.553
.493
,^56
,553
1 383
,253
,260
-------
-•COO ii>.uuu ci. v j.' >vjv 35.000 40.0 00 4d,0 00 ^0,0 00 5d«COO 60<000
§¦ pf ] ?, 3 iJ 9 1.463 1.142 0.091 0,3?]? 0.r-C3 0,733 0,636 0.665 0.663 0,447 0.55")
h m j 3,319 1.751 1.22S 0,986 C.?41 0,73'+ 0,650 0.592 0,557 '0.539 0.513 0.454
E O 3 3.656 1.856 1.2:51 0.935 0,637 0.7JFI q,663 0,608 0- 57? 0.556 0,542 0,515
3 3 3.021 1.B44 1,253 O.^o'f 0»323 a. 707 0.619 0,556 Q .517. 0.493 0.465 0. 399
W 5 4.554 2,120 1,32" 0.979 . 0.731 0.644 0.543 0,469 0.41B 0.334 0,358 0,323
y g 6 6,511 2.103 1 .326 0.979 0.773 0.637 0.533 0.457 0.407 0,374 0,345 0.29?
•D 7 4,17 4 2.R03 -1,299 0,975 0.775 0.633 0. 526 0,453 0,406 C.375 0.341 0.273
2 8 2,345 1,415 1,121 0.992. 0.905 0.527 C. 760 0,717 0.703 0.711 0 . 70 1 0,600
9 2.277 1,395 1.113 0,993 0,913 0.SM 0.760 0.74.1 0,737 0,755 0,756 0.662
g 10 2.541 ,1.483 1, 149 0.990 0.673 0.770 0.6S4 0.629 0.60S 0,503 0 .539 0,47c.
¦ 5 1 ft 1.47/* 1.14/3 Q.969 9.B63 0,746 0.651 0.591 0 .563 0.566 0.534 0.397
0,729 0.677 0.657 0.658 0.643 0.53 0
0,496- 0.42 0 0.37? 0,355 0,324 0.246
0,577 0.510 0.470 0,451 0,436 0.391
0 ,739 0.691 0.675 0.681 0 .661 0 .530
0.527 0.450 0.427 0,410 0.360 0.227
0.525 0,454 0.415 0,395 0.364 0.275
0,557 0.493 0.475 0.47/3 0.434 0,265
(Table II.
14 con
'0
I-
Ave.
CiVJV?
5 . C 0 0 i C . 1
300 15.000
20,000
25.0 00 30•0 DC
]
?.3i:9
1,463
1 . 142
0.091
0 . 33 9
0,r-C3 !
•s
c
3.319
1.751-
1 .225
0.986
0 , 34 1
0.734 !
3.656
1.856
1 .251
0.935
0.637
0,73 3 (
'*
3.021
1.344
1.253
0.934
0.323
0.707 !
5
4,554
2.120
1.329
0.979 .
0.731
0 . 644
6
4,511
2.103
1 .326
0.979
0.773
0.637
7
4, 174
2. 003
-1.299
0.975
0.775
0.633
5
2.345
1,415
1.121
0.992
0.905
0.527
9
2.277
1.395
1.113
0.993
0.913
0,34 1
10
2.541
.1.485
1,149
0 .990
0.673
0.770
1!
2.516
1.474
1.140
0.969
0.B63
0.746
12
2.685
1.54?
1,149
0.991
0.^91
0.604
13
3,791
1.916
1.291
0, 930
0.771
0,612
14
4,056
1.950
1.281
0.962
0.604
0.675
15
2.599
1.459
1. 127
0.992
0.90 0
0.314
16
3,384
1,744
1.237
0.9S3
0.795
0.641
17
4,239
1.980
1.293
0.961
0,762
0.634
1*
2.983
1,530
1.153
0.986-
0.321
0.671
Nitric
Oxide
CTvO'J?
hVCXCLZZ StJCCCl
5.000 10.
000 15.000 20 .0C 0 25.03
0 30.000
1
1.505.
1,060
C.941
1.010
1.161
1.319
2
1.242
1.031
C.974
1. 034
1.074
1.14 5
3
0,990
0,945
0.960
1.004
1 .036
3 ,109
4
1.063
0.^92
0 .900
1.002
1.03 '3
1.075
c
0.978
0,970
c ,9a l
1 i 002
1 .026
1.049
6
0.927
0.924
0,956
1. 004
1.0 56
1.102
7
1,003
0.949
0,960
1 .004
1.059
1.110
8
) ,2i34
1.006
0,944
1 . 006
1.123
1.255
9
1. ] 43
0.966
0 ,944
1. V 0 7
1 .ICS
I .200
10
1 ,324
0.997
0,930
1 . C 1 o.
1.152
3.297
11
1.181
0,931
0,946
1.007
1.109
1,214-
12
1.014
0.360
o. r>p. 7
1.012
1. 174
1.330
13
0.^9
0.306
0,934
1.0 04
1.043
1,070
14
0.999
0.903
0.924
1. 003
1.112
3.208
15
1 .082
0.907
0.909
l.oio
1.14 3
1.230
16
0. 6 5 5
0,943
0.9rt6
1. 001
1. 002
3.000
17
0.803
0.364
0.934
1.006
1,069 .
1 .'121
10
O, K1 6
0,619
0.097
1.009
1.124
1.222 '
-35,000 40,COO 45,000 50.000 55,000 60.000
1,440 1.511 1 .551 1 ,606 1,754 2.129
1,203 1.239 1,265 1.306 1.404 1.615
1,150 1.152 1.213 2.258 1,340 1.4S?
1,105 1,129 1.152 1.1£9 1,257' 1,334
1,070 1 .091 1 . 1 15 1 . 151 1.2C0 1.2
-------
Evidence of undue complexity in the formulation of speed
correction factors is found in the fact that the coefficients
in (1) or (2) appear to be highly correlated, as shown in
Figure 1, in which Aq has been plotted against for CO
for the 18 groups. The correlation coefficient for the plot
is 0.69. Similarly, one can compute correlations for all
pairs of coefficients to obtain a 6x6 correlation matrix,
as shown in Table I. Note that many of the correlation
coefficients approach either +1 or -1. The result is that
the correlation matrix is effectively of less than full rank.
Indeed, if one computes the eigenvalues of the matrix only
two are found to be of appreciable magnitude. This fact suggests
that the coefficients do not vary independently from group
to group but are highly covariant. Consequently, a simpler
expression with fewer terras should suffice to represent the
speed vs. emission relations for the various vehicle groups.
Similar conclusions can be drawn from Table II for hydrocarbon
emissions (HC) arid from Table III for nitrogen oxide emissions
(NOx) . Note, particularly, that the correlation matrix for
NOx has only one eigenvalue of appreciable magnitude and is
consequently of rank 1.
Falcon R&D
Working Paper No. 1
•Page 9
-------
s:
o a
n ¦<->
X o
(-• 0
3 3
i£
- W
*3 a-
O
13
fD
Hf
Z
o
&
U3
rt>
M
o
0.25 0.26 0.27 0.2B 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40
-Al"
FIGURE 1 - Relation Between Coefficients in Speed Correction Factor Equation
-------
TABLE I
CORRELATION MATRIX FOR COEFFICIENTS
IN CORRECTION FACTOR vs SPEED EQUATION
FOR CARBON MONOXIDE (CO)
*2
A,
1.0000 -0.6871 0.0933 0.0330 -0.0705 0.0851
-0.6871 1.0000 -0.7825 0.6860 -0.6489 0.6309
0.0933 -0.7825 1.0000 -0.9868 0.9741 -0.9655
0.0330 0.6860 -0.9868 1.0000 -0.9977 0.9943
-0.0705 -0.6489 0.9741 -0.9977 1.0000 -0.9992
0.0851
0.6309 -0.9655 0.9943 -0.9992 1.0000
EIGENVALUES:
4.5115
1.4534
0.0350
0.0002
0.0000
0.0000
TRACE = 6.0001 = SUM OF EIGENVALUES
Falcon R&D
Working Paper No. 1
Page
-------
TABLE II
CORRELATION MATRIX FOR COEFFICIENTS
IN CORRECTION FACTOR vs SPEED EQUATION
FOR HYDROCARBONS (KC)
A0 A1 A2 a3 A4 a5
1.0000 -0.8320 0.4854 -0.3922 0.3568 -0.3375
-0.8320 1.0000 -0.8821 -0.8148 -0.7823 -0.7622
0.4854 -0.8821 1.0000 -0.9883 0.9752 -0.9650
-0.3922 0.8143 -0.9383 1.0000 -0.9975 0.9935
0.3568 -0.7823 0.9752 -0.9975 1.0000 -0.9991
-0.3375 0.7622 -0.9650 0.9935 -0.9991 1.0000
EIGENVALUES: 4.958 9
1.0004
0.04 05
0.0001
0.0000
0.0000
TRACE = 5.9999 = SUM OF EIGENVALUES
Falcon R&D
Working Paper No. 1
Page 12
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TABLE III
CORRELATION MATRIX FOR COEFFICIENTS
IN CORRECTION FACTOR vs SPEED EQUATION
FOR NITROGEN OXIDES (NOx)
Aq a^j^ a2 a3 a4
Aq 1.0000 -0.9868 0.9677 -0.9584 0.9525
h1 -0.9868 1.0000 -0.9957 0.9918 -0.9886
A2 0.9677 -0.9957 1.0000 -0.9993 0.9979
A3 -0.9584 0.9918 -0.9993 1.0000 -0.9996
A4 0.9525 -0.9886 0.9979 -0.9996 1.0000
EIGENVALUES: 4.93 56
0.0637
0.0008
0.0000
0.0000
TRACE = 5.0001 = SUM OF EIGENVALUES
Falcon R&D
Working Paper No, 1
Page
-------
The correlations among coefficients of the emission versus
speed equations suggest that attention be directed to Table II.4,
in which the equations have been used to compute emissions for
5-mph increments of speed. One can compute the covariance (or
correlation) between correction factors for 5 mph and 10 mph,
5 mph and 15 mph, . .., 55 mph and 60 mph to produce a 12 x 12
matrix of covariances or correlations for each of the three
pollutants. For the purpose of this analysis the covariance
matrix is preferred, because it retains the scale aspect of
the relation between correction factors for various speeds.
It can be shown that the number of linearly independent
functions needed in a regression equation to represent emissions
as a function of speed can be deduced from the eigenvalues
of the covariance matrix and that the form of these "basis
functions" can be deduced from the corresponding eigenvectors.1
First, let us direct attention to the eigenvalues of the
12 x 12 covariance matrices (see Table 4). Only eigenvalues
TABLE 4
EIGENVALUES OF COVARIANCE MATRICES
FOR SPEED CORRECTION FACTORS
EVALUATED AT 5-mph SPEED INTERVALS
CO
0.7126
0.0250
0.0009
HC
0.1364
0.0141
0.0002
0.0001
NOx
0.2499
0.0298
0.0005
significant to the fourth decimal place are tabulated. Note
that the first eigenvalue constitutes 96.5% of the trace for
CO, and 90.6% and 89.2% of the trace for HC and N0X respectively.
Thus it appears that "most" of the particularization for vehicle
group could be achieved by a single basis function. If
further refinement is required, a second basis function could
be used, but this second function would serve only to "trim"
the effect of the first function and provide a second-order
refinement.
H. T. McAdams, "A Factor Analytic Approach to the Identification of
Manufacturing Systems," Proc. of the CIRP Seminars on Manufacturing
Systems, Vol, 1, No. 2 (1972), pp. 79-97
Falcon R&D
Working Paper No. 1
Page 14
-------
The eigenvectors resulting from the covariance analysis
consist of 12 components corresponding to 5 mph, 10 mph,
60 mph. These components are plotted versus speed in Figure 2.
for the eigenvectors corresponding to the eigenvalues of
greatest magnitude. Note that the eigenvectors for CO and
HC are essentially monotonic for increasing speed, whereas the
eigenvector for N0X peaks at about 15 mph. The eigenvectors
corresponding to the second largest eigenvalues are plotted
in Figure 3. Note that certain similarities are apparent in
the first eigenvectors for CO and HC and the second eigenvector
for NOx. It is known that factors affecting CO and HC emissions
often tend to affect NOx emissions inversely, and it is hypo-
thesized that this tendency is reflected in the emission
eigenvectors.
It is informative to compare the eigenvectors in Figure 2
with the plots in Figure 4, which represents the actual relations
between correction factors and speed for Group 2. Except for
sign (sign is arbitrary!) corresponding curves in Figure 2 and
Figure 4 have quite similar shapes.
The implications of the analysis can now be expressed as
follows. Let f(x) denote the correction factor for a given
pollutant at speed x when averaged over all 18 groups. For
the present it is assumed that the domain of the function f(x)
is
but the matter of extension to a continuous domain will be
considered later (see Section 3, Summary and Conclusions.)
Let v^(x) denote the first eigenvector and let V2(x) denote
the second eigenvector for that pollutant. Considering v^(x)
and v2(x) as functions defined on the same domain as f(x),
one can then write
where and b2i are regression coefficients for the ith
group and (x) is an approximation to the observed correction
factor vs speed relation for the ith group. The coefficients
b^£ and b2^ can be determined very simply by virtue of the
fact that the vectors v^(x) and V2(x) are orthogonal.
Falcon R&D
Working Paper No. 1 Page 15
• • • f
gi (x) = f(x) + bji VjJx) + b2i v2(x)
-------
. 0
. 9
. 8
.7
. 6
. 5
.4
. 3
.2
.1
0
. 1
.2
.3
.4
. 5
. 6
.7
.8
.9
.0
FIGURE 2
PLOT OF FIRST EIGENVECTORS
I
\
I
^ CO
\
0
\
I
5
in A o'n 'ok ' 1 1 I J 1 '
10 15 20 25 30 35 40 45 50 55 60
MPII
Page 16
-------
. 0
.9
. 8
.7
.6
. 5
.4
. 3
. 2
.1
0
.1
. 2
.3
.4
. 5
. 6
.7
.8
.9
.0
FIGURE 3
SECOND LARGEST EIGENVALUE
-------
4
2
0
8
6
.4
. 2
.0
8
6
4
2
0
8
6
1.4
) • 2
FIGURE 4
PLOT OF GROUP 2 CORRECTION FACTORS
I
4
5
10 15
25 30 35 40 45 50 55 60
MPH
Page 3 8
-------
Also, by virtue of orthogonality, the last term in the
equation can be deleted if it makes only a small contribution
to g^{x) without the need to recompute the coefficient b^-
An illustration is informative. Consider the correction
factors for CO as displayed in Table 11.14. The mean computed
across all 18 groups gives rise to the following "correction-
factor vector" or "correction-factor function" (see Table 5).
TABLE 5
MEAN CO CORRECTION FACTOR FOR
18 VEHICLE GROUPS AT 5 mph INCREMENTS
Speed (mph) Mean CO Correction Factor
5 3.325
10 1.727
15 1.217
20 0.985
25 0.836
30 0.718
35 0.627
40 0.567
45 0.536
50 0.525
55 0.501
60 0.413
Each of the vehicle groups deviates to some extent from
these correction factors, and it is the intent of the principal
component analysis to allow for this correction in the most
parsimonious way.
Consider Group 2, for example. Its deviations from the
mean correction-factor relation are tabulated in Table 6.
Falcon R&D
Working Paper No. 1
Page 19
-------
TABLE 6
GROUP 2 DEVIATIONS FROM MEAN CO
CORRECTION FACTORS AT 5 mph INCREMENTS
SPEED DEVIATION
(mph) FROM MEAN CORRECTION FACTOR
5
-0.0057
10
0.0214
15
0.0074
20
0.0006
25
0.0053
30
0.0156
35
0.0227
40
0.0251
45
0.0206
50
0.0140
55
0.0170
60
0.0411
The object is to express these deviations as a linear combination
of the significant eigenvectors vj(x) and V2(x) for CO as
derived from the principal-component analysis. In short,
we want to solve the following equation for and b2
0.9015
0. 2881
0.0867
,0059
0541
,0793
0937
1059
1206
1374
¦0.1436
¦0.1159
•0,
¦0,
•0
¦0,
¦0
¦0,
•0
+ b.
U
-0. 2909
-0.0923
0.0149
-0.0067
-0.1061
-0.2179
-0.2942
-0.3246
-0.3239
-0.3263
-0.3887
•0. 5417
-0.0057
JLm
e2
0.0241
e3
0.0074
e4
0.0006
e5
0.0053
i
e6
0.0156
X
e7
e8
e9
e10
ell
e12
0.0227
0.0251
0.0206
0.0140
0.0170
0.0411
(3)
Falcon R&D
Working Paper No. 1
Page 20
-------
in such a way as to minimize the sum of squared errors
12
>
i=l
, ei
Equation (3) thus falls in the framework of a linear model
X b + e
(4)
and one can solve for b by means of the least-squares normal
equations
X'X b = X* y
(5)
from which
(X 'X) ""1 X ' £
(6)
It should be noted, however, that the column vectors of the
matrix X are orthogonal, so that (6) becomes simply
b = X' y.
(7)
Solution of (7) for the specific case of (3) gives
-0.0154784
-0.0593595
(8)
With the computed values of bj and b2 one can then
compute an estimated correction factor vector for Group 2
and can compare this vector with the observed correction
Falcon R&D
Working Paper No. 1
Page 21
-------
factors. Thus,
-0.0155
0.9015
-0.2902
0.0033
-0.0057
0.2881
-0.0923
0.0010
0.0214
0.0867
0.0149
-0.0022
0.0074
-0.0059
-0.0067
0 . 0005
0.0006
-0.0541
-0.1061
0.0071
0.0053
-0.0793
-0.0594
-0.2179
0.0142
^ w
0.0156
-0.0937
-0.2942
0.0189
0.0227
-0.1059
-0.3246
0.0209
0.0251
-0.1206
-0.3239
0.0211
0.0206
-0.1374
-0.3263
0.0215
0.0140
-0.1436
-0.3887
0.0253
0.0170
-0.1159
-0.5417
0.0340
0.0411
Computed Deviations from Mean-
J
Observed Deviations from Mean
The error vector can now be computed as the difference between
the observed and the computed vectors according to
as shown below.
£ = Z ~ *
b
.ow.
-0.0057
0.0033
-0.0090
0.0214
0.0010
0.0204
0.0074
-0.0022
0.0096
0.0006
0.0005
0.0001
0.0053
0.0071
-0.0018
0.0156
0.0142
0.0014
0.0227
0.0189
0.0038
0.0251
0.0209
0.0042
0.0206
0.0211
-0.0005
0.0140
0.0215
-0.0075
0.0170
0.0253
-0.0083
0.0411
0.0340
0.0072
Note that the maximum error is 0.02, at 10 rnph, where the
actual value of the correction factor as tabulated in Table II.
is 1.7 51. Thus the maximum error is only slightly over 1%.
Falcon R&D
Working Paper No. 1
Page 22
-------
Similar analysis can be performed for other vehicle
groups and for other pollutants, but such analysis is not
within the scope of this paper, which is intended to be
exemplary only. The inputs required for additional analysis
are, however provided in the Appendix.
3. SUMMARY AND CONCLUSIONS
It has been demonstrated, in connection with emission
speed correction factors, that principal component analysis
provides a useful tool for consolidating correction factor
data. By virtue of the fact that the speed correction factor
curves for various groups of vehicles tends to have a common
'shape," this common (mean) curve can be expressed as the
basic input to speed correction. Refinement to this curve
can be achieved by means of a linear model having no more
than two correction terms, the coefficients of which are
vehicle-group specific. Thus, if g^ (x) is the mean speed-
correction function for the ith vehicle group and f(x) is
the mean speed-correction function for all groups, then
gi(x) = f (x) + b1:L v-^x) + b2i v2 (x)
(10)
where v^(x) and v2(x) are determined from principal
component analysis. Thus to characterize the data of
Table 11.14 it is necessary to know only:
f(x), the mean curve
v1(x), the first eigenvector
v2 (x), the second eigenvector
b ; 1 1, 2 | 10
'2i
i = 1, 2,
18
(12 values)
(12.values)
(12 values)
(18 values)
(18 values)
Thus a total of 72 (and possibly as few as 42) key parameters
retains the correction factor information for each pollutant,
as opposed to 18 x 12 = 216 in the complete tabulation.
If viewed in the form of Table 11.13, employing logarithmically
transformed exponential functions, the correction-factor
milieu required 18 x 5 = 90 coefficients for CO and HC
and 18 x 4 = 72 coefficients for N0X, in addition to the
assumption of specific forms of equations for the correction
factor vs speed relations.
Falcon R&D
Working Paper No. 1
Page 23
-------
Though it is recognized that in (10) each of the functions
gj[(x), f (x) , v^(x) and V2 (>0 are considered as being defined
on a discrete domain of vehicle speeds, extension to a
continuous interval of speeds is straightforward. Indeed,
if f(x), vj(x) and V2(x) are expressed as functional forms,
each of these functions can be represented by much fewer than
the 12 values which emerge as components of the eigenvectors.
For example, each of these characteristic functions could be
expressed as an approximating polynomial of relatively small
degree. Note that the coefficients in these polynomials
need be defined only once, not for each vehicle group, since
the group-specific part of the representation is contained
entirely in the coefficients b^ and b2i>
In summary, principal component analysis provides a means
for structuring the information content of correction factors
in a concise way, as well as identifying the areas of commonality
among various vehicle makes and models or other homogeneous
groups. The approach is applicable to other multivariate
aspects of correction factors.
Falcon R&D
Working Paper No. 1
Page 24
-------
APPENDIX
-------
TABLE A-l
CO CORRECTION FACTORS AT 5 mph INCREMENTS
Speed (mph)
5
10
15
20
25
30
35
40
45
50
55
60
Raw Data
Column Means
3.3247
1.7269
1.2176
.9854
.8357
.7184
.6273
.5669
.5364
.5250
.5010
.4129
Raw Data Column Sigmas
.7624
.2447
.0739
.0053
.0489
.0755
.0919
.1032
.1141
.1272
.1361
.1308
Transformed Data (Deviation Scores)
- .9357
-.2639
-.0756
.0056
.0533
.0846
.1057
.1191
.1286
.1380
.1460
.1451
- .0057
.0241
.0074
.0006
.0053
.0156
.0227
.0251
.0206
.0140
. .0170
.0411
.3313
.1291
.0334
-.0004
.0013
.0196
.0357
.0411
.0356
.0290
.0410
.1021
.2963
.1171
.0354
-.0014
-.0127
-.0114
-.0083
-.0109
-.0194
.-.0320
-.0360
-.0139
1.2293
.3931
.1114
-.0064
-.0547
-.0744
-.0843
-.0979
-.1184
-.1410
-.1430
-.0899
1.1863
.3761
.1084
-.0064
-.0577
-.0814
-.0943
-.1099
-.1294
-.1510
-.1560
-.1139
.8493
.2761
.0814
-.0104
-.0597
-.0854
-.1013
-.1139
-.1304
-.1500
-.1600
-.1399
- .9797
-.3089
-.0966
.0066
.0693
.1086
.1327
.1501
.1666
.1860
.2000
.1871
-1.0477
-.3319
-.1046
.0076
.0773
.1226
.1527
.1761
.2006
.2300
. .2550
.2491
- .7837
-.2389
-.0686
.0046
.0373
.0516
.0567
.0621
'.0716
.0830
.0880
.0651
- .8087
-.2529
-.0696
.0036
.0273
.0276
.0237
.0241
.0316
.0410
.0330
-.0159
- .4397
-.1869
-.0686
.0056
.0553
.0856
.1017
.1101
.1206
.1330
.1420
.1251
.4663
.1891
.0734
-.0054
-.0647
-.1064
-.1313
-.1469
-.1584
-.1700
-.1770
-.1669
.7313
.2231
.0634
-.0034
-.0317
-.0434
-.0503
-.0569
-.0664
-.0740
-.0650
-.0219
- .7257
-.2679
-.0906
.0066
.0643
.0956
.1117
.1241
.1386
.1560
.1600
.1171
.0593
.0171
.0194
-.0024
-.0407
-.0774
-.1003
-.1089
-.1054
-.1150
-.1410
-.1859
.9143
.2531
.0754
-.0044
-.0537
-.0844
-.1023
-.1129
-.1214
-.1300
-.1370
-.1349
-.3367
-.1469
-.0346
.0006
-.0147
-.0474
-.0703
-.0739
-.0614
-.0470
-.0670
-.1479
A-l-a
-------
Covariance Matrix
.5813
.1856
.0555
-.0038
-.0339
.1856
.0599
.0180
-.0012
-.0109
.0555
.0180
.0055
-.0004
-.0034
-.0038
-.0012
-.0004
.0000
.0002
-.0339
-.0109
-.0034
.0002
.0024
-.0494
-.0158
-.0050
.0004
.0037
-.0580
-.0185
-.0059
.0004
.0044
-.0657
-.0210
-.0067
.0005
.0050
-.0751
-.0241
-.0076
.0006
.0055
-.0858
-.0276
-.0087
.0006
.0062
-.0894
-.0287
-.0091
.0007
.0066
-.0706
-.0223
-.0072
.0006
.0058
Eigenvalues
.7126
.0250
.0009
.0002
.0000
Eiaenvectors
.9015
-.2902
-.2991
.0498
.2881
-.0923
-.7316
-.2374
.0867
.0149
-.3209
.0096
-.0059
-.0067
.0183
.0038
-.0541
-.1061
.1256
-.1809
-.0793
-.2179
.0798
-.3750
-.0937
-.2942
.0132
-,4617
-.1059
-.3246
.0126
-.3906
-.1206
-.3239
.1046
-.1901
-.1394
-.3263
.2200
.1212
-.1436
-.3887
.1410
.3908
-.1159
-.5417
-.4127
. .*453
TABLE A-l (Continued)
-.0494
-.0580
-.0657
1
o
<_n
-.0858
-.0894
-.0706
-.0158
-.0185
-.0210
-.0241
-.0276
-.0287
-.0223
-.0050
-.0059
-.0067
-.0076
-.0087
-.0091
-.0072
.0004
.0004
.0005
.0006
.0006
.0007
.0006
.0037
.0044
.0050
.0055
.0062
.0066
.0058
.0057
.0069
.0078
.0086
.0096
.0102
.0095
.0069
.0084
.0095
.0104
.0116
.0124
.0117
.0078
,0095
.0106
.0117
.0130
.0140
.0131
.0086
.0104
.0117
.0130
.0145
.0155
.0143
.0096
.0116
.0130
.0145
.0162
.0173
.0157
.0102
.0124
.0140
.0155
.0173
.0185
.0171
.0095
.0117
.0131
.0143
.0157
.0171
.0171
.0000
.0000
.0000
.0000
.0000
.0000
.0000
(Other eigenvectors make insignificant contributions)
A-l-b
-------
TABLE A-l (Continued)
Transformed Data (Standard Scores)
-1.2273
-1.0785
-1.0230
1.0536
1.0904
1.1204
1,1497
1.1544
1,1275
1,0852
1.0730
1.1092
- .0075
.0985
.1000
.1054
.1080
.2067
,2466
.2434
,1807
.1101
,1249
.3139
.4345
.5277
.4518
- .0843
.0262
.2597
.3881
.3984
.3122
.2280
.3013
.7804
.3886
.4786
.4788
.2739
- .2604
- .1508
- .0907
- .1055
- .1700
- .2516
- .2646
- .1066
7.6123
1.6067
1.5071
-1.2222
-1.1200
- .9851
- .9176
- .9487
-1.0378
-1.1088
-1.0509
- .6877
1.5559
1.5372
1.4665
-1.2222
-1.181*
-1.0778
-1.026*
-1.0650
-1.1343
-1.1874
-1.1464
- .8713
1.1139
1.1285
1.1012
-1.9808
-1.2233
-1.1307
-1.1025
-1.1037
-1.1430
-1.1795
-1.1758
- .0701
-1.2650
-1.2624
-1.3071
1.2433
1.4179
1.4382
1.4435
1.4548
1.4606
1.4626
1.4698
1.4303
-1.3742
-1.3564
-1.4154
1.4329
1.5816
1.6236
1.6611
1.7068
1.7586
1.8086
1.2740
1.9044
-1.0279
- .9764
- .9283
.8640
.7629
.6834
.6165
.6019
.6278
.6527
.6467
.4974
-1.0607
-1.0336
- .0^18
.6743
..5583
.3656
.2575
.2337
.2771
.3224
.2425
- .1219
- .5767
- .7638
- .9233
1.0536
1.1318
1.1337
1.1062
1.0671
1.0573
1.0459
1.0436
.9562
.6116
.7729
.9929
-1.0326
-1.3246
-1.4088
-1.4289
-1.4236
-1.3885
-1.3368
-1.3008
-1.2765
.9591
.9119
.6577
- .6532
- .8492
- .5746
- .5476
- .5513
- .5820
- .5819
- .4777
- .1678
- .9518
-1.0949
-1.2260
1.2433
1.3155
1.2661
1.2150
1.2028
1.2151
1.2267
1.1753
.8951
.0777
.0699
.2623
- .4636
- .8334
-1.0248
-1.0917
-1.0553
- .9589
- .9043
-1.0362
-1.4218
1.1991
1.0345
1.0200
- .8429
-1.0995
-1.1175
-1.1134
-1.0941
-1.0641
-1.0223
-1.0868
-1.0318
- .4416
- .6003
- .4683
.1054
- .3013
- .6275
- .7652
- .7161
- .5382
- .3696
- .4524
-1.1312
Correlation Matrix
1.0000
.9947
.9853
- .9349
- .9108
- .8573
- .8280
- .8345
- .8637
- .8854
- .8616
- .7085
.9947
1.0000
.9947
- .9422
- .9146
- .8565
- .8245
- .8311
- .8628
- .8874
- .8619
- .6968
.9853
.9947
1.0000
- .9590
- .9469
- .8977
- .8691
- .8744
- .9021
- .9232
- .9010
- .7499
- .9349
- .9*22
- .9590
1.0000
.9683
.9380
.9191
.9224
.9408
.9538
.9408
.8335
- .9108
- .9146
- .9469
.9683
1.0000
.9912
.9807
.9820
.9906
.9945
.9881
.9150
- .8593
- .8565
- .8977
.9380
.9912
1.0000
.9979
.9979
.9982
.9946
.9949
.9572
- .8280
-.8245
- .8691
.9191
.9807
.9979
1.0000
.9999
.9963
.9893
.9928
.9723
- .8345
- .8311
- .8744
.9224
.9820
.9979
.9997
1.0000
.9978
.9917
.9948
.9716
- .8637
- .8628
- .9021
.9408
.9906
.9982
.9963
.9978
1.0000
.9980
.9982
.9583
- .8854
- .8874
- .9232
.9538
.9945
.9946
.9893
.9917
.9980
1.0000
.9981
.9466
- .8616
- .8619
- .9010
.9408
.9881
.9949
.9988
.9948
.9982
.9981
1.0000
.9609
- .7085
- -6968
- .7499
.8335
.9150
.9572
.9723
.9716
.9583
.9436
.9609
1.0000
A-l-c
-------
Eigenvalues
11.2318 .6865
Eigenvectors
-.2738
.4599
-.2738
.4772
-.2826
.3798
.2892
-.1802
.2979
.0020
.2954
.1574
.2926
.2294
.2933
.2172
.2960
.1469
.2973
.0851
.2957
.1484
.2747
.4567
.0225 .0110
-.5005
.3833
-.0164
-.1524
.3291
-.3008
.2482
-.0855
-.3409
-.1294
-.2652
-.3200
-.1612
-.3767
-.0971
-.2364
-.0873
.0471
-.0709
.3772
.0751
.4964
.5823
.1554
0456
3528
1535
0941
8977
0685
0220
0597
0791
0532
0197
0310
1341
TABLE A-1 (Continued)
.0023 .0002 .0000 .0000 .0000 .0000 .0000
.0979 -.3596
.0182 .3743
.2482 .3324
.0428 -.0516
.4432 .4267
2544 0195
" CCf '10,00 (Other eigenvectors make insignificant
.Ubbb -.luoo , • i T• >
.3444 -.2477 contribution)
.4713 -.1036
.3753 .2402
.0308 .4392
.4240 -.3187
A-1-d
-------
TABLE A-2
HC CORRECTION FACTORS AT 5 mph INCREMENTS
Speed (rrph)
5
10
15
20
25
30
35
40
45
50
55
60
Raw Data (Column Means)
3.0969
1.6768
1.2035
.9868
.8509
.7464
.6644
.6072
.5759
.5622
.5414
.4701
Raw Data (Column
Siqmas)
.3110
.0987
.0321
.0023
.0276
.0475
.0615
.0704
.0765
.0829
.0918
.1022
Transformed Data (Deviation
Scores)
.0102
.0022
-.0025
.0002
.0071
.0146
.0196
.0218
.0211
.0223
.0296
.0459
.2002
.0722
.0205
-.0008
-.0069
-.0064
-.0054
-.0072
-.0109
-.0152
-.0144
.0119
-.0138
.0312
.0155
-.0008
-.0099
-.0134
-.0144
-.0152
-.0199
-.0282
-.0384
-.0411
.3732
.1312
.0425
-.0028
-.0299
-.0464
-.0584
-.0692
-.0789
-.0902
-.0964
-.0891
.3222
.0962
.0275
-.0018
-.0169
-.0264
-.0344
-.0422
-.0499
-.0582
-.0624
-.0561
.0262
.0162
.0045
.0002
.0021
.0076
.0126
.0148
.0151
.0138
.0156
.0248
.0632
.0322
.0115
-.0008
-.0059
-.0064
-.0064
-.0072
-.0089
-.0122
-.0124
-.0101
-.3968
-.1288
-.0435
.0032
.0381
.0646
.0836
.0958
.1051
.1158
.1306
.1459
-.1938
-.0768
-.0335
.0032
.0401
.0726
.0976
.1128
.1231
.1318
.1496
.1789
-.0578
-.0278
-.0135
.0012
.0161
.0286
.0376
.0418
.0431
.0468
.0546
.0679
-.2988
-.1058
-.0355
.0022
.0261
.0416
.0516
.0598
.0631
.075"
.0756
.0549
-.1688
-.0538
-.0195
.0012
.0201
.0346
.0466
.0518
.0561
.0618
.0706
.0829
-.3918
-.1288
-.0385
..0022
.0201
.0266
.0296
.0328
.0401
.0478
.0476
.0229
.1792
.0492
.0125
-.0008
-.0069
-.0104
-.0144
-.0182
-.0219
-.0242
-.0224
-.0141
-.2818
-.0948
-.0315
.0022
.0251
.0416
.0526
.0618
.0701
.0788
.0865
.0829
-.3338
-.1018
-.0225
.0002
-.0029
-.0174
-.0294
-.0342
-.0329
-.0312
-.0444
-.0871
.8662
.2542
.0815
-.0058
-.0669
-.1114
-.1414
-.1612
-. 1749
-.1892
-.2044
-.2121
.0972
.0312
.0245
-.0028
-.0479
-.0934
-.1244
-.1392
-.1439
-.1482
-.1684
-.2101
A-2-a
-------
Covariance Matrix
.0967
.0304
.0097
-.0006
-.0073
.0304
.0097
.0031
-.0002
-.0023
.0097
.0031
.0010
-.0001
-.0008
.000$
-.0002
-.0001
.0000
.0001
.007?
-.0023
-.0008
.0001
.0008
.0117
-.0038
-.0014
.0001
.0013
.0147
-.0047
-.0017
.0001
.0017
.0169
-.0055
-.0020
.0002
.0019
.0188
-.0061
-.0022
.0002
.0021
.0209
-.0068
-.0024
.0002
.0023
.0223
-.0072
-.0026
.0002
.0025
.0214
-.0069
-.0026
.0002
.0027
Eigenvalues
.1364 .0141 .0002 .0001 .0000
Eigenvectors
-.8282
.4684
.2934
-.0457
-.2627
.1224
-.8847
-.0793
-.0862
-.0038
-.3290
.0286
.0060
.0043
.0111
-.0014
.0694
.0843
.0698
-.1288
.1141
.1822
-.0305
-.2645
.1450
.2538
-.1212
-.3175
.1666
.2869
-.1314
-.3262
.1840
.2937
-.0201
-.3124
.2024
.3005
.1488
-.2027
.2191
.3630
.2013
.0860
.2196
.5199
-.1568
.7423
TABLE A-2 (Continued)
1
O
_ j
^1
-.0147
-.0169
o
J
00
CO
-.0209
-.0223
-.0214
-.0038
-.0047
-.0055
-.0061
-.0068
-.0072
-.0069
-.0014
-.0017
-.0020
-.0022
-.0024
-.0026
-.0026
.0001
.0001
.0002
.0002
.0002
.0002
.0002
.0013
.0017
.0019
.0021
.0023
.0025
.0027
.0023
.0029
.0033
.0036
.0039
.0043
.0047
.0029
.0038
.0043
.0047
.0051
.0056
.0062
.0033
.0043
.0050
.0054
.0058
.0064
.0071
.0036
.0047
.0054
.0058
.0063
.0070
.0077
.0039
.0051
.0058
.0063
.0069
.0076
.0083
.0043
.0056
.0064
.0070
.0076
.0084
.0092
.0047
.0062
.0071
.0077
.0083
.0092
.0104
.0000
.0000
.0000
.0000
.0000
.0000
.0000
(Other eigenvectors make insignificant contribution)
A-2-b
-------
TABLE A-2 (Continued)
Transformed Data (Standard Scores)
.0327
.0225
- .0778
.0977
.2359
,3067
,3177
,3100
,2761
,2754
,3227
,4497
.6437
.7318
.6378
- .3420
- .2519
- ,1358
- ,0885
- ,1018
- ,1424
- ,1829
- ,1241
,1169
- .0445
.3164
.4822
- .3420
- .3607
- .2832
- .2347
- .2154
2602
- ,3397
- .4184
- .4019
1.2001
1.3296
1.3222
-1.2213
-1.0862
- .9785
- .9496
- .9822
-1.0319
-1.0875
-1.0604
- .8718
1.0061
.9750
.8555
- .7817
- .6147
- .5571
-.5597
- .5988
- .6526
- .7016
- .6799
- .5487
.0841
.1846
.1400
.0977
.0746
.1592
.2040
.2106
.1977
.1668
.1701
.2442
.2031
.3265
.3572
- .3420
- .2156
- .1358
- .1047
- .1018
- .1163
- .1347
- .1350
- .0934
-1.2762
-1.3048
-1.3533
1.4167
1.3805
1.3600
1.3577
1.3609
1.3749
1.3971
1.4234
1.4286
- .6233
- .7779
-1.0422
1.4167
1.4530
1.5286
1.5851
1.6023
1.6103
1.5901
1.6365
1.7517
- .1860
- .2815
- .4200
.5374
.5824
.6016
.6102
.5940
.5639
.5649
.5952
.6651
- .9610
-1.0718
-1.1044
.9771
.9452
.8755
.8377
.8497
.8909
.9146
.8240
.5378
- .5430
- .5449
- .6066
.5374
.7275
.7280
.7240
.7360
.7339
.7458
.7695
.8119
-1.2601
-1.3048
-1.1977
.9771
.7275
.5595
.4802
.4662
.5247
.5769
.5189
.2246
.5762
.4987
.3889
- .3420
- .2519
- .2200
- .2347
- .2580
- .2863
- .2915
- .2440
- .1376
- .9063
- .9603
- .9800
.9771
.9089
.8755
.8539
.8781
.9171
.9508
.9439
.6119
-1.0736
-1.0313
- .7000
.0977
- .1068
- .3675
- .4784
- .4852
- .4302
- .3759
- .4837
- .6522
2.7855
2.5759
2.5354
-2.5404
-2.4284
-2.3479
-2.2983
-2.2886
-2.2876
-2.2816
-2.2274
-2.0758
.3125
.3164
.7622
-1.2212
-1.7392
-1.9687
-2.0220
-1.9762
-1.8821
-1.7871
-1.8351
-2.0562
Correlation Matrix
1.0000
.9920
.9702
- .9132
- .8465
- .7918
- .7681
- .7730
- .7926
- .8094
- .7828
- .6728
.9920
1.0000
.9863
- .9274
- .8634
- .8045
- .7797
- .7845
- .8057
- .8250
- .8005
- .6892
.9702
.9863
1.0000
- .9727
- .9313
- .8860
- .8658
- .8692
- .8862
- .9018
- .8835
- .7919
- .9132
- .9274
- .9727
1.0000
.9851
.9632
.9525
.9548
.9648
.9727
.9625
.9044
- .8485
- .8634
- .9313
.9851
1.0000
.9939
.9881
.9886
.9926
.9952
.9908
.9557
- .7918
- .8045
- .8860
.9632
.9939
1.0000
.9989
.9987
.9987
.9973
.9969
.9795
- .7681
- ,7797
- .8658
.9525
.9881
.9989
1.0000
.9998
.9984
.9955
.9967
.9862
- .7730
- .7845
- .8692
.9548
.9886
.9987
.9998
1.0000
.9991
.9966
.9974
.9857
- .7926
- .8057
- .8862
.9648
.9926
.9987
.9984
.9991
1.0000
.9991
.9986
.9806
- .8094
- .8250
- .9018
.9727
.9952
.9973
.9955
.9966
.9991
1.0000
.9987
.9752
- .7828
- .8005
- .8835
.9625
.9908
.9969
.9967
.9974
.9986
.9987
1.0000
.9845
- .6728
- .6892
-..7919
.9044
.9557
.9795
.9862
.9857
.9806
.9752
.9845
1.0000
A-2-c
-------
Eigenvalues
11.1529 .8122
Eigenvectors
-.2607 -.5349 -.
-.2645 -.5191
-.2827 -.3571
.2970 .1215 -.
.2989 -.0435 -.
.2960 -.1608
.2940 -.2053
.2945 -.1974
.2962 -.1603
.2974 -.1250 -.
.2956 -.1719 -.
.2834 -.3498 -.
.0085 .0043
-.2229 -.0409
-.0848 -.1730
.1653 -.2503
.1419 -.8186
-.3910 .2240
-.3741 .1280
-.2514 -.0158
-.1358 -.0942
-.0511 -.1808
.0539 -.0866
.2499 .0968
.6733 .3323
0189
6983
1984
4992
8421
0768
1306
2159
2163
0964
0745
1810
0453
TABLE A-2 (Continued)
.0030 .0001 .0000 .0000 .0000 .0000 .0000
¦.0537 -.2138
.1608 .3254
-.0351 .0334
.3742 .0678
.3041 .0276
2465 3547 ,
*... 'n77 (Other eigenvectors make insignificant
• iili * i I * / / , • ¦ , • \
-.1446 -.4720 contribution)
-.3862 -.2388
-.5113 .1332
-.2972 .5941
.3802 -.2276
A-2-d
-------
TABLE A-3
N0X CORRECTION FACTORS AT 5 mph INCREMENTS
Speed (mph)
5
10
15
20
25
30
35
40
45
50
55
60
Raw Data (Column
Means)
T.0477
.9402
.9437
1.0062
1.0910
1.1726
1.2375
1.2837
1.3208
1.3702
1.4651
1.6499
Raw Data (Column
Sigmas)
.2100
.0683
.0274
.0032
.0492
.0965
.1325
.1528
.1618
.1712
.2021
.2868
Transformed Data (Deviation
Scores)
.4603
.1198
-.0027
.0038
.0700
.1464
.2025
.2273
.2302
.2378
.2989
.4791
.1973
.0908
.0303
-.0022
-.0170
-.0266
-.0345
-.0447
-.0558
-.0642
-.0611
-.0349
-.0547
.0058
.0163
-.0022
-.0330
-.0636
-.0875
-.1017
-.1078
-.1122
-.1251
-.1609
.0183
.0518
.0363
-.0042
-.0530
-.0976
-.1325
-.1547
-.1688
-.1812
-.2081
-.2659
-.0667
.0298
.0373
-.0042
-.0650
-.1236
-.1675
-.1927
-.2058
-.2192
-.2571
-.3519
-.1177 -
.0162
.0123
-.0022
-.0350
-.0706
-.0965
-.1107
-.1148
-.1202
-.1421
-.2049
-.0417
.0088
.0163
-.0022
-.0320
-.0626
-.0875
-.1037
-.1138
-.1202
-.1431
-.1669
.2393
.0658
.0003
.0018
.0370
.0824
.1215
.1453
.1562
.1608
.1759
.2271
.0983
.0258
.0003
.0008
.0140
.0274
.0375
.0393
.0372
.0358
.0459
.0831
.2793
.0568
-.0137
.0038
.0610
.1244
.1725
.1963
.2032
.2118
.2559
.3811
.1363
.0408
.0023
.0008
.0180
.0414
.0625
.0773
.0862
.0918
.1049
.1361
-.0307 -
.0802
-.0567
.0058
.0830
.1574
.2165
.2583
.2852
.3078
.3349
.4201
-.4557 -
.1342
-.0097
-.0022
-.0480
-.1026
-.1405
-.1517
-.1458
-.1492
-.2071
-.3819
-.1457 -
.0372
-.0197
.0018
.0210
.0354
.0455
.0483
.0482
.0508
.0609
.0861
.0373 -
.0332
-.0347
.0038
.0570
.1074
.1455
.1683
.1802
.1938
.2259
.3051
-.1887
.0028
.0423
-.0052
-.0890
-.1726
-.2355
-.2697
-.2848
-.3032
-.3621
-.5139
-.2367 -
.0762
-.0097
-.0002
-.0220
-.0516
-.0765
-.0907
-.0948
-.0962
-.1121
-.1639
-.2287 -
.1212
-.0467
.0028
.0330
.0494
.0565
.0603
.0652
.0758
.0949
.1271
A-3-a
-------
Covariance Matrix
.0441
.0123
.0004
.0003
.0058
.0123
.0047
.0011
-.0000
.0002
.0004
.0011
.0007
-.0001
-.0011
.0003
-.0000
-.0001
.0000
.0002
.0058
.0002
-.0011
.0002
.0024
.0124
.0008
-.0020
.0003
.0047
.0175
.0013
-.0025
.0004
.0065
.0198
.0014
-.0031
.0005
.0075
.0202
.0012
-.0033
.0005
.0079
.0207
.0010
-.0035
.0005
.0084
.0257
.0017
-.0041
.0006
.0099
.0408
.0042
-.0053
.0009
.0139
Eigenvalues
.2499 .0298 .0005 .0000 .0000
Eigenvectors
-.2964
-.8616
-.0563
-.0341
-.3819
-.1699
.0358
-.1188
-.1082
-.0059
.0063
.0071
-.0966
.0531
.0779
-.1915
.0686
.0515
-.2636
.0796
-.0682
-.3033
.1037
-.2478
-.3195
.1372
-.4121
-.0371
.1654
-.4179
-.4007
.1541
-.1155
-.5709
.0561
.7236
TABLE A-3 (Continued)
.0124
.0175
CO
o
.0202
.0207
.0257
.0408
.0008
.0013
.0014
.0012
.0010
.0017
.0042
-.0020
-.0026
-.0031
-.0033
-.0036
-.0041
-.0053
.0003
.0004
.0005
.0005
.0005
.0006
.0009
.0047
.0065
.0075
.0079
.,0084
.0099
.0139
.0093
.0128
.0147
.0156
.0165
.0195
.0275
.0128
.0176
.0202
.0214
.0226
.0268
.03 77
.0147
.0202
.0233
.0247
.0261
.0309
.0434
.0156
.0214
.0247
.0262
.0277
.0326
.0457
.0165'
.0226
.0261
.0277
.0293
.0345
.0482
.0195
.0268
.0309
.0326
.0345
.0408
.0574
.0275
.0377
.0434
.0457
.0482
.0574
.0818
.0000 .0000 .0000 .0000 .0000 .0000 .0000
(Other eigenvectors make insignificant contribution)
A-3-b
-------
TABLE A-3 (Continued)
Transformed Data (Standard Scores)
2.1922
1.7548
- .0995
1.1976
1.4231
1.5171
1.5277
1.4875
1.4233
1.3889
1.4793
1.6747
.9397
1.3300
1.1066
- .7045
- .3456
- .2758
- .2603
- .2927
- .3448
- .3751
- .3021
- .1220
- .2603
.0846
.5949
- .7045
- .6709
- .6592
- .6601
- .6657
- .6663
- .6555
- .6168
- .5624
.0872
.7586
1.3259
-1.3385
-1.0775
-1.0116
- .9996
-1.0126
-1.0^34
-1.0586
-1.0295
- .9294
- .3175
.4363
1.3624
-1.3385
-1.3215
-1.2810
-1.2637
-1.2613
-1.2722
-1.2805
-1.2720
-1.2300
- .5603
- .2377
.4487
- .7045
- .7116
- .7318
- .7280
- .7246
- .7096
- .7023
- .7029
- .7162
- .1984
.1286
.5949
- .7045
- .6506
- .6489
- .6601
- .6788
- .7034
- .7023
- .6634
- .5884
1.1397
.9637
.0102
. 5636
.7522
.8538
.9166
.9508
.9658
.9392
.8706
.7939
.4683
.3777
.0102
.2466
.2846
.2838
.2829
.2571
.2201
.2090
.2273
.2905
1.3302
.8318
- .5015
1.1976
1.2402
1.2891
1.3014
1.2846
1.2564
1.2371
1.2665
1.3322
.0492
.5974
.0832
.2466
.3660
.4289
.4715
. 5058
. 5331
.5361
.5193
.4785
- .1460
-1.1753
-2.0731
1.8316
1.6874
1.6311
1.6334
1.6904
1.7633
1.7978
1.7069
1.4685
-2.1700
-1.9665
- .3553
- .7045
- .9759
-1.0634
-1.0600
- .9930
- .9012
- .8717
-1.0246
-1.3249
- .2175
- .5453
- .7208
.5626
.4269
.3667
.3357
.3160
.2981
.2966
.3016
.3010
.1778
- .4867
-1.2690
1.1976
1.1588
1.1129
1.0902
1.1013
1.1142
1.1319
1.1180
1.0665
- .8985
.0407
1.5452
-1.6555
-1.8094
-1.7888
-1.7767
-1.7653
-1.7606
-1.7712
-1.7916
-1.7963
-1.1270
-1.1167
- .3553
- .0704
- .4473
.5349
- .5771
- .5937
- .5859
- .5621
- .5545
- .5789
-1.0890
-1.7760
-1.7076
.0806
.6709
.5118
.4263
.3945
.4032
.4426
.4698
.4448
Correlation Matrix
1.0000
.8615
.0700
.4144
.5603
.6130
.6283
.6174
.5933
.5767
.6061
.6796
.8615
1.0000
.5668
- .1027
.0627
.1284
.1488
.1369
.1087
.0883
.1216
.2135
.0700
.5668
1.0000
- .8740
- .7862
- .7431
- .7280
- .7342
- .7506
- .7638
- .7457
- .6834
.4144
- .1027
- .8740
1.0000
.9825
.9691
.9632
.9642
.9682
.9720
.9684
.9449
.5603
.0627
- .7862
.9825
1.0000
.9976
.9953
.9946
.9943
.9949
.9965
.9375
.6130
.1284
- .7431
.9691
.9976
1.0000
.9995
.9987
.9970
.9962
.9991
.9947
.6283
.1488
- .7280
.9632
.9953
.9995
1.0000
.9996
.9976
.9967
.9992
. 99^9
.6174
.1369
- .7342
.9642
.9946
.9987
.9996
1.0000
.9998
. 9984
.9994
. 9919
.5933
.1087
- .7506
.9682
.9943
.9970
.9978
.9993
1.0000
.9998
.9988
.9868
. 5767
.0383
- .7538
.9720
. 3949
.9952
.9957
.5984
. 9998
i,o:oo
.9984
. 9845
. 6G61
.1216
- .7457
.9634
.9965
.9991
.9992
.9994
. 9988
.9984
1.0000
. 9926
.5796
.2135
- .5834
.9449
.9875
.99^7
.9949
.3919
.9358
.9545
.9925
i.o:oo
A-3-c
-------
TABLE A-3 (Continued)
Eigenvalues
9.8535
2.1174
.0215
.0064
.0011
.0001
Eigenvectors
-.1911
.5490
.2833
-.0780
.1775
.0326
-.0359
.6828
.0335
.0793
.0297
.4294
.2399
.4507
-.3083
.3065
-.2173
-.6857
-.3099
-.1472
.3962
.8396
-.1400
.0279
-.3130
-.0344
.2010
-.1497
.3286
-.5231
-.3185
.0109
.0626
-.0923
. .2R05
i—
CO
o
1
-.3183
.0252
-.0909
-.0187
.2733
-.0906
-.3182
.0170
-.2737
.0439
.1861
.0385
-.3179
-.0023
-.4386
.0923
.0301
-.0820
-.3179
-.0164
-.4449
.0516
-.1790
.1721
-.3184
.0064
-.1582
-.1150
-.4631
.0261
-.3163
.0704
.3518
-.3672
-.5978
-.1531
.0000
.0000 .0000 .0000 .0000
.0000
(Other eigenvectors make insignificant
contribution)
A-3-d
-------
TABLE A-4
REGRESSION COEFFICIENTS FOR CO CORRECTION FACTORS
PRINCIPAL COMPONENTS PROGRAM
Covariance Matrix
.11180E +00
.78435E -
02
.98686E
_
04
.15818E
_
05
-.64443E
_
07
.52888E -
09
.11655E -
02
-.84526E
-
04
.33546E
-
05
-.60594E
-
07
.40040E -
09
.10011E
-
04
-.44718E
-
06
.84291E
-
08
-.56788E -
10
. 20514E
-
07
-.39083E
-
09
.26477E -
11
.74804E
-
n
-.50808E -
13
.34561E -
15
Eigenvalues
.11235E + 00
.62202E -
03
.13417E
-
06
.46185E
-
12
.10209E
-
12
.30625E -
15
Eigenvectors
•99752E + 00
.11940E -
01
.79114E
_
02
-.26439E
_
08
.82568E
_
05
-.12681E -
07
-.70371E - 01
.98962E +
00
.12518E
+
00
.50308E
-
02
-.16204E
-
03
-.23231E -
06
.92923E - 03
.12541E +
00
.98851E
+
00
-.04350E
-
01
-. 31904E
-
02
-.22265E -
05
.11940E - 04
.56053E -
01
—.843 51E
-
CI
.99453E
+
00
-.61425E
-
01
. 13586E -
03
—.53415E - 06
.10535E -
03
.2011OE
-
02
.61471E
-
01
-.99800E
+
00
.14408E -
01
.44445E - 08
.70796E -
06
-.15285E
-
04
-.75079E
-
03
•14389E
-
01
.99990E +
00
A-4
-------
TABLE A-5
REGRESSION COEFFICIENTS FOR HC CORRECTION FACTORS
PRINCIPAL COMPONENTS PROGRAM
Covariance Matrix
.23837E - 01
.23725E
02
.12123E -
03
-.45466E
_
05
.80177E
_
07
-.51954E - 09
.34111E
-
03
-.26353E -
04
.11299E
-
05
—.21030E
-
07
. 14038E - 09
.26167E -
05
-.12003E
-
06
..22962E
-
08
15567E - 10
.56372E
-
08
—.10901E
-
09
.74382E - 12
..21187E
-
11
-. 14501E - 13
.99441E - 16
Eigenvalues'
.24074E - 01
.10587E
-
03
.55098E -
07
.58084E
-
13
. 11181E
-
17
.24840E - 21
Eigenvectors
.99503E +00
.99251E
01
.83972E -
02
-.26351E
_
03
--95207E
_
05
.12889E - 06
-.99474E - 01
.98583E
+
00
. 13497E +
00
50465E
-
02
-.18636E
-
03
•25240E - 05
.51201E - 02
.13508E
+
00
.98722E +
00
-.84279E
-
01
-.36026E
-
02
.51042E - 04
-.19261E - 03
.64121E
-
02
-.34239E -
01
-.99421E
+
00
-.66324E
-
01
..11165E - 02
.34012E - 05
.12360E
-
03
. 20101E -
02
.66396E
-
01
-.99736E
+
00
.29268E - 01
-.22057E - 07
.83998E
-
06
-.15353E -
04
-.82734E
-
03
.29278E
-
01
.99957E + 00
A-5
-------
TABLE A-6
REGRESSION COEFFICIENTS FOR NOX CORRECTION FACTORS
PRINCIPAL COMPONENTS PROGRAM
Covariance Matrix
.26107E + 00
.40772E -
01
.21252E
_
02
-.42892E
_
04
.30230E
_
06
.65390E -
02
-.34608E
-
03
.70242E
-
05
49658E
-
07
.18475E
-
04
-.37619E
-
06
.26644E
-
08
.76712E
-
08
54385E
-
10
.38584E
-
12
Eigenvalues
.26746E + 00
.16858E -
03
. 13350E
-
08
.82688E
-
15
.25399E
-
18
Eigenvectors
-.98798E +00
.15453E +
00
.47 653E
_
02
.18743E
_
03
.40029E
05
.15439E +00
.98456E +
00
.82412E
-
01
.36301E
-
02
•79084E
-
04
-.80505E - 02
.82215E +
00
.99434E
+
00
•66750E
-
01
. 16147 E
-
02
.16251E - 03
.18897E -
02
-.66836E
-
01
•99710E
+
00
.36236E
-
01
-.11454E - 05
.14214E -
04
.81028E
-
03
-.36263E
-
01
.99934E
+
00
A-6
-------
4.2
WORKING PAPER NO. 2:
SPEED-TEMPERATURE-HOT/COLD CORRECTION FACTORS:
A CRITIQUE AND PROSPECTUS
-------
FA f PON Rl S1 AR("
F/\JL^ wl\ ni \ i.i oiwn.vi
DENVER BALTIMORE ALBUQUERQUE BUFFALO
ONE AMERICAN DRIVE
BUFFALO. NEW YORK 14225
TELEPHONE: (716) 632-4932
Working Paper No. 2
Project No. 8411
Environmental Protection Agency
May 12, 1978
H. T. McAdams
SPEED-TEMPERATURE-HOT/COLD CORRECTION FACTORS:
A CRITIQUE AND PROSPECTUS
1. INTRODUCTION
One of the most important components of the corrections
applied to emission factors is the correction factor denoted
Ripstv/x anc^ sometimes referred to as the speed-temperature-
hot/cold correction factor. To understand the omnibus
effect of this factor one must examine the influence of
each of the variables which enter into its makeup. The
R-factor is, of course, specific for vehicle group i and
pollutant p, but the items of interest here are the average
speed s, the ambient temperature and the hot/cold
partitioning specified by the quantities w and x.
2. AVERAGE SPEED CONSIDERATIONS
Speed correction has been previously discussed in
Working Paper No. 1. ^ The speed correction factors treated
in that document, however, are only the "preliminary" inputs
to the R-factors. It is appropriate, therefore, to examine
the assumption whereby these preliminary speed-correction
factors were incorporated into the more inclusive R-factors.
^ H. T. McAdams, "A Factor-Analytic Approach to Emission Correction
Factors," Working Paper No. 1, Project 8411, Falcon Research and
Development Company, Buffalo, N. Y. (April 5, 1978).
A SUBSIDIARY OF Wwttau er
-------
The preliminary speed-correction factors were generated
by application of the Modal Emission Model (MEM). It should
be noted, however, that this model applies only to warmed-up
vehicles and to vehicles operating in a standard ambient of
75°F. A further complication is introduced by the fact that
the MEM takes into consideration the actual speed vs time
profile prevailing in a driving sequence, whereas any correction
factor based on average speed does not. In reality there
is an unlimited number of speed-time profiles which could
map into the same average speed, but a concession made in
the R-factor is that this many-to-one mapping is permissible.
The preliminary speed correction factors, as derived by
application of the MEM, were found to be nonlinear functions
of average speed. Whether the complexity of the functional
forms assumed for these relations is justifiable is subject
to question, however. For example, the correction factor
vs speed relations for HC and CO show steep gradients only
at low speed, and it is possible that a simpler functional
form would be capable of retaining "most" of the information
in the exponential fifth-order polynomials. Some further
observations pertinent to this point are given in Appendix I.
Whatever the form of the preliminary speed correction
factors, one needs now a way to adjust these correction
factors for the operating condition of the engine, whether
cold start, hot start or stabilized. In other words one
must make the speed correction factors for warmed-up engines
bag specific.
The method employed for this purpose, as described by
Becker z, is based on two factors which can affect bag emissions.
One is that the averaqe speeds for the several bags are
different and different from the average speed of 19.6 mpg
over the FTP driving cycle. The other stems from the state
of warm-up of the vehicle as that state affects the actual
generation of pollutants. In other words, even if the average
speed in Bag 1 (cold start) were the same as for the total
FTP driving cycle, there would still be a difference
between cold-start emissions and emissions from vehicles in
the warmed-up state.
^ "Supplement 8 Light Duty Vehicle Correction Factors," Memo Janet Becker
to J. Hidinger/J. Horowitz (3/7/77).
2
-------
It is at this point that difficulties in terminology
seem to arise as well as difficulties in assumptions. To
speak of measuring emissions over the FTP driving cycle
at an arbitrary speed seems contradictory, because the FTP
in a strict sense implies an average speed, namely 19.6 mph.
Similarly, to speak of measuring Bag i emissions at an
arbitrary speed also seems incorrect, since with each bag
there is an associated specific average speed. Further
challenging of these concepts will be held in abeyance
for the present, however, in order to set forth what was
actually done in the generation of the bag-specific speed
correction factors.
First consider Bag 1. This is the cold-start bag and
has an associated average speed of 26 mph. For Bag 1, the
final speed correction factor for a specific pollutant is
given as
V2' S1
Vg' S1 = V2, 26
where the subscript g refers to vehicle group, and the
subscript 2 identifies Group 2, low-altitude pre-controlled
vehicles. This definition is based on the assumption that
"emission dependency on speed during cold operation is ...
similar for all model year vehicles and ... is equal to the
dependency of pre-controlled vehicles during warmed-up
operation."^ In short, it is assumed that it is necessary
only to normalize the Group 2 correction factor at an arbitrary
speed by dividing by the Group 2 correction factor at the
bag speed of 26 mph (see Table 1). This adjustment has the effect
of re-referencing the correction factor curves to the Bag 1 average
speed of 26 mph rather than to the FTP average speed of 19.6 mph.
Adjustment of Bag 2 and Bag 3 preliminary correction
factors were performed in a somewhat different manner. Here
the re-referencing for bag average speed is done on a group-
wise basis.
^ Becker, op. cit.
3
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TABLE 1
DERIVATION OF BAG 1 SPEED CORRECTION FACTORS
| Speed (mph)
i-
5
10
15
20
25
C 26J 30
35
40
45
50
55
60
\
Hydrocarbons
Preliminary Correction
Factor
3.297
1.749
1.224
.986
.844
[J.82l] .740
.659
.600
.565
.547
.530
.482
Final Correction
Factor
4.016
2.130
1.491
1.201
1.028
1.000 .901
.803
.731
.688
.666
.646
.587
Carbon
Monoxide
Preliminary Correction
Factor
3.319
1.751
1.225
.986
.841
P.817J .734
.650
.592
.557
.539
.518
.454
Final Correction
Factor
4.062
2.143
1.499
1.207
1.029
1.000 .898
.796
.725
.682
.660
.634
I
'
.556 ,
i
Nitrogen Oxides
j
j
i
Preliminary Correction
Factor
1.242
1.031
.974
1.004
1.074
[1.0891 1.146
1.203
1.239
1.265
1.306
1.404
1
1.615
Final Correction
Factor
1.140
.947
.894
.922
.986
1.000 1.052
1.105
1.138
1.162
1.199
1.289
1.483 j
-------
Thus,
V
g- s.
9' S2 Vg, 16
for Bag 2
and
V
g, s.
g, s.
g, 26
for Bag 3
The assumption in all cases is that "emissions as measured
over the cold (hot) (sic) bag driving cycles and over the
stabilized driving cycle at a given average speed are egual
to emissions as measured over the FTP driving cycle at that
speed." W
Though understandable because of data limitations, the
assumptions on which the speed correction factors are based
seem tenuous, and certain aspects of the procedure raise
questions. For example, it is asserted that in the Monte
Carlo implementation of the Modal Emission Model "all cycles
were transient; no steady state cycles were generated." ^
Yet to obtain an average speed of--say--60 mph, much of the
time would have to be spent, it would seem, in a high-speed
cruise mode. A second consideration applies to the bag-
specific average speeds. It might be considered to be
"very unlikely" that a vehicle operating in the cold-start
mode would do so at a high average speed; in short, there
is an upper bound of speed beyond which the correction factor
for cold start is essentially irrelevant. Finally, there
is an anomaly in the notion of "FTP driving cycle" or "Bag
i driving cycle" at anything other than the design average
speeds.
CS O
Becker, op. cit
5
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3. TEMPERATURE-DETUNING-AGE DETERIORATION CONSIDERATIONS
According to Becker (op. cit.) ambient temperature was
found to affect only Bag 1 emissions for only HC and CO.
Results obtained for 1975 vehicles showed an exponential
relation between these emissions and ambient temperature.
Pre-1975 vehicles were assigned the relations:
. 5.6548 - .015965 t
Bag 1 CO =
i /^~\2.9310 - . 014779 t
Bag 1 HC = £3
where the response is in units of gms/mi.
When t = 75°, one has:
*—v 4 4 ^74 ?
Bag 1 CO = = 86.266 gms/mi.
Bag 1 HC = £2^1.822575 _ ^ gms/mi.
It might seem reasonable to divide the general Bag 1
expressions by their corresponding standard emissions at
75° to obtain a "Bag 1 temperature correction factor." In
the case of carbon monoxide, this would have led to
= e 5-6548 e-015965 t
286.015 -.015965 t
86.266 ^
= 3.318 0"-°1596St
Note that when t = 75°,
Q -.015965 t = 0-1.197375 = 3Q2
and the correction factor is (3. 318) (.302) = 1.00.
6
-------
Further insight would be gained by differentiating the correc-
tion factor with respect to temperature to obtain
Thus a 1° change from a specified temperature would change
the correction factor by about 1^% of its value at the initial
temperature. A similar analysis for HC would indicate the
range of the correction factor and its rate of change at
any given temperature.
Though the above type of approach sheds light on the
specific effect of ambient temperature on Bag 1 emissions,
this was not the approach used in developing the current
correction factors. Instead, terms were added to account
for the "detuning" of vehicles in use and for mileage-
accumulation "deterioration." In these contributions
distinction is made between pre-1968 vehicles and 1968-74
vehicles, whereas no distinction is made in the exponential
parts of the Bag 1 expressions (see Table 2).
d (CO Bag 1 CF)
dt
-.015965 (CO Bag 1 CF)
Table 2
DETUNING AND DETERIORATION CONTRIBUTIONS TO
CORRECTION FACTOR EXPRESSIONS (A = age in years - 1)
Detuning Deterioration
HYDROCARBONS
Pre-1968
0.673
0.569A
1968-74
-2.410
0.863A
CARBON MONOXIDE
Pre-1968
14.74
9.62A
1968-74
33.89
9.77A
7
-------
Moreover, the correction factors incorporating temperature,
age and detuning effects are normalized in terms of the overall
FTP emissions rather than in terms of the Bag 1 emissions.
Let us examine the implications of this approach for
HC. As given in Table II.1 of Becker (op. cit.), the
relevant part of the general expression for the R-facto, is:
02.9310 .014779 t + _6?3 + >569A
______ for pre-1958 vehicles
and
^ 2.9310 - .014779 t
£3 ~ 2.41 + 863A
for 1978-74 vehicles
2.8 + .64A
When t = 7 5° and A = 0, these expressions reduce to:
6.188 + .673 6.861 n c inco u • i
~—r=—- - 7:—- 1.21 for pre-1968 vehicles
5.67 5.67
6.188 - 2.41 3.778 , „ r ^ u . ,
—— = = 1.35 for 1968-74 vehicles
• o ^ • o
These results imply that hydrocarbon emissions in Bag 1 are
21% higher than emissions over the FTP for pre-1968 vehicles
and 35% higher for 1968-74 vehicles. A much stronger conclusion
than this, however, can be made: these factors are independent
of age so long as temperature is held at 7 5°. For:
6.861 + .569A _ ^ ^
5.67 + .47A
and
3.778 + .863A _ _
2.8 + .64A "
8
-------
Similar conclusions can be drawn for Bag 2 and Bag 3
for pre-1968 and for 1968-74 vehicles. For, in Table II.1
one has for HC:
1.002 for pre-1968 vehicles
0.932 for 1968-74 vehicles
0.837 for pre-1968 vehicles
0.867 for 1968-74 vehicles
Thus for pre-1975 vehicles the age-effect term is a "fictional"
or "dummy" correction. Note, however, that this is not the
case for 1975 vehicles.
In summing up the considerations of this section several
points can be made. In Table 2, the effects of detuning are
tabulated. In three of the four entries the detuning terms
are negative. This fact suggests that as-received, in-use
vehicles give lower emissions than tuned-up vehicles. Though
it is not inconceivable that such could be the case, it
seems more likely that the results may be an artifact of
the sampling involved in obtaining the data on which the
corrections are based. This possibility further suggests
that a program aimed at assessing deteriorations in use
should exercise as much control as possible and practical
over the sampling procedure. Finally,-there may be advantage
in normalizing emission correction factors on a bag-by-bag
basis rather than on the FTP basis and to report emission
factors bagwise rather than (or in addition to) FTP-wise.
More on this point will follow in the next section, which
deals with the hot start/cold start aspect of the R-factor.
For Bag 2
For Bag 3
f5.69 + .471A _
5. 67
+
. 4 7A
2. 61
+
. 597A
2.8
+
. 64A
4.75
+
. 3 9 3A
5. 67
+
. 4 7A
2.43
+
. 55 5A
2.8 + .64A
9
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4. HOT START/COLD START CONSIDERATIONS
The three bags in the FTP test are combined to give the
overall emission factor. The weightings, under standard
circumstances, are as follows:
w = cold start fraction = 20.58% (Bag 1)
1-w-x - stabilized fraction = 52.13% (Bag 2)
x = hot start fraction = 27.28% (Bag 3)
The average speeds for these bags under standard conditions
are respectively 26 mph, 16 mph and 26 mph.
In the overall expressions (Tables II.1, II.2, II.3 of
Becker, oj3. cit.) the factors w, 1-w-x, and x are used as
simple weighting factors. When these factors assume their
standard values and when both temperatures and speeds are at
their standard values, then the R-factor should compute to
unity. That this is true can be noted by returning to the
previous section and considering weighted combinations of
the bag-specific correction factors for HC for pre-1968
vehicles and for 1968-1974 vehicles. For pre-1968 vehicles,
one has
1.21 (.2058) + 1.002 (.5213) + 0.837 (.2728) = 1.00
(Bag 1) (Bag 2) (Bag 3)
and, for 1968-74 vehicles,
1.35 (.2058) + 0.932 (.5213) + 0.867 (.2728) = 1.00
It is when the values of w and x depart from their
standard values of w = 0.2058 and = 0.2728 that the
correction factors become useful. However, that is also when
any difficulties or errors in the correction-factor expressions
will be manifested. A particularly troublesome point, it
would seem, arises when both the fraction of occupancy time
as well as the prevailing speeds in the three bags depart from
their nominal values. Indeed, as was mentioned earlier, such
departures are "coincidentia oppositorum" if the usual
concepts of "Bag i" emissions are to hold in a strict sense.
What is really implied, when one postulates a Bag 1 speed of--
say--10 mph, is a driving cycle quite different from the normal
Bag 1 cycle, but still one in which the vehicle is operating
in the cold-start condition. Apparently the assumption relied
upon to resolve this apparent contradiction is that any scaling
of emissions for speed effects, whether in Bags 1, 2 or 3, can
be performed "as if" the vehicle were operating over some
portion of the FTP consistent with that average speed.
-------
In applying the weighting fractions w, x and 1-w-x
to the three bags certain questions of a sample-space nature
arise. According to definition, these weighting factors
relate to the fraction of total miles driven in the cold
start, hot start and stabilized conditions, respectively.
Note, however, that the fraction of miles driven in a given
mode is not, in general, the same as the fraction of the
number of vehicles operating in that mode at a given point
in time nor to the fraction of time during which a vehicle
operates in that mode. Indeed, when corrections are made
simultaneously for mode weight fraction and for average
speed in mode, the prospect of interaction of the two should
not be overlooked.
To investigate this possibility let us return to the
basis by which the standard weightings
w = 20.58%
1-w-x = 52.13%
x = 27.28%
arise. Consider the following.
Bag 1. First 505 seconds (0.1403 hr. )
Mileage = 3.59 miles
Average speed = 25.6 mph
Bag 2. Next 870 seconds (0.2417 hr.)
Mileage = 3.91 miles
Average speed = 16 mph
Bag 3. Repeat first 505 seconds
Mileage = 3.59 miles
Average speed = 2 5.6 mph.
If it is assumed that 43% of the vehicle trips begin in
the cold start mode and 57% in the hot start mode, then a
"typical" or "expected" vehicle trip would be represented as
FTP GMS/MI = [o-43 Bag 1 Gms. + Bag 2 GMS.
+ 0.57 Bag 3 Gms. ]/ 7.5 mi.
11
-------
and
FTP mi. = 0.4 3 Bag 1 mi. + Bag 2 mi. + 0.57 Bag 3 mi.
= (0.43) (3.59) + 3.91 + (0.57) (3.59)
= 1.5437 + 3.91 + 2.0463
= 7.5 mi.
Then
1.5437
7.5
3.91
7.5
= .2058 = fraction of miles in cold start mode
= .5213 = fraction of miles in stabilized mode
and
2^0463
7.5
.2728 = fraction of miles in hot start mode
Now it is presumed that the FTP weightings were devised
with a view toward "representative" operation of vehicles
in cold-start, stabilized and hot-start modes. The gms/mi.
emissions are derived on the basis that the driving cycle
consists of a transient part 3.59 miles long and a stabilized
part 3.91 miles long and that the total trip length is 7.5 miles,
When average speeds in the three stages of operation are taken
into account, the assumptions of the FTP translate into certain
fractions of "vehicle-hours" spent in cold-start, stabilized
and hot-start modes, given a trip length of 7.5 miles.
(°'o\3)g(3'59/umi>) = 0.0603 hrs. (3.6 min.) in "Bag 1"
25.6 mi/hr
¦^ * 91 ¦;")*'— = 0.2444 hrs. (14.7 min) in "Bag 2"
16 mi/hr
) = 0.0799 hrs. (4.8 min) in "Bag 3"
25.6 ini/hr
12
-------
Thus the total "trip time" is
0.0603 + 0,2444 + 0.0799 = 0.3846 hrs. = 23 minutes
and the fractions of time in each "bag" are:
~ 0. 1568 for cold start (Bag 1)
» J U T D
0.2444
b:334g = 0.6355 for stabilized (Bag 2)
0.0799
0.3846
0.2077 for hot start (Bag 3).
The point to be made here is that the FTP driving sequence
taken as the reference to which other driving scenarios are
compared, implies not only certain average speeds and fractions
of miles in each mode but also a "typical" or average trip
length and certain fractions of total vehicle operating times
in each of the three modes of operation. It is to be under-
stood, of course, that either the vehicle executes the trip
from a cold start or from a hot start. Thus the time for
"warm-up" in the cold start, when it occurs, is (3.59 mi.)/
(25.6 mi. hr.) = 0.14 hr. = 8.4 minutes, not the "expected"
value of 3.6 minutes as calculated above.
In view of the above considerations it seems logical to
consider their implications when one envisions a scenario
departing from the reference conditions. For example, consider
one of the cases given in Table II.5 of Becker (op. cit.):
% cold, % stable, % hot start = 40, 30, 30
Ave. speed Bag 1, 2, 3 = 10, 10, 10
The fraction of trips originating "cold" is given by
.40
= 0.5714 (i.e. , 57%)
.40 + .30
13
-------
and the fraction of trips originating "hot" is given by
.30
40 + .30
0.30
0.40
0.4286 (i.e., 43%)
To reconcile all the constraints one must make certain assump-
tions about either trip lengths or trip times and the portions
thereof spent in each bag.
For example, suppose that the trip length is assumed to
be 7.5 miles, as in the FTP, and that the disposition of this
length is 3.59 miles in either cold or hot transient and 3.91
miles in stabilized operation. Then
3.59 miles „ _
—77: r-yr— = 0.359 hr. = 21.5 mm.
10 mi/hr
which is too long for the vehicle to operate without being in
stabilized mode, whether the start is cold or hot. Suppose,
instead, that one assumes the warm-up time of 8.4 minutes
(0.14 hr.) as in the FTP. The corresponding distance at
10 mi/hr. is (0.14 hr.) x (10 mi/hr.) = 1.4 mi. If a trip
length of 7.5 miles is assumed, then 7.5 - 1.4 = 6.1 miles
would have to be in stabilized operation. Note that
(.57) (1.4) + 6.1 + (.43) (1.4) = 7.5
satisfies the trip length requirement but violates the original
assumption of 40%/30%/30% mileage split in Bag 1/Bag 2/Bag 3.
For,
(0.57)(1.4 mi.)
7.5 mi.
6.1 mi.
7.5 mi.
(0.43)(1.4 mi.)
7.5 mi.
= .1064 ^ 0.40
= .8133 ? 0.30
= .0803 f 0.30
14
-------
Evidently, then, trip length must be much shorter than in the
FTP, a fact made evident by the consideration that average
speed over every segment is no greater than about half the
average speed of the FTP (19.6 mph). A consistent solution,
based on adjusted trip length, would be found by solving
x + 1.4
0.3
where x is the distance covered during the stabilized
portion of the trip. The solution is x = 0.6 miles, for a
total trip length of 1.4 +0.6 =2 miles. Then
(0-57)(1-4) + 0.6 + (0.43) (1.4) = 2 miles
and
(0.57)(1.4)
0.4
0 . 6
0.3
(0.43) (1.4)
= 0.3
and it is seen that the desired split of cold start, stabilized
and hot start driving is preserved butvonly if an average
trip length of 2 miles is postulated.
To summarize, cold/stabilized/hot fractions, average
speeds, trip lengths and fractions of trips initiated in
cold start or hot start—all are interrelated. Correction
factors designed to accomodate both average speed and mode
of operation must, therefore, be developed and used with care.
Specific points to consider are the fact that the fraction
of miles driven in cold start is different from the fraction
°f time spent in the cold start mode and is also different from
the fraction of trips which originated from a cold start.
15
-------
Moreover, for a given fraction of miles driven in cold start
conditions, the average cold-start speed acts as a constraint
on trip length. Though rate of warm-up may vary with driving
speed, it is likely that vehicles tend to stabilize after some
fixed length of time rather than after some fixed distance
driven. Further study to define warm-up time as a function of
speed, ambient temperature and other factors would help
resolve this question. Any survey to define local use patterns
for the purpose of defining the impact of vehicle emissions
on air quality should aim at consistency among the various
elements involved.
5. SUMMARY AND PROSPECTUS
Correction factors for ambient temperature, average
speed, percent hot start/percent cold start operation and
other factors are incorporated in a quantity Ripstwx*
Assumptions on which this factor is based have been critically
examined, certain difficulties and possible anomalies
indicated, and some suggestions made for simplification and/or
revision. It is realized that the approach taken in the
formulation of the R-factor is to a certain extent expedient
in that definitive data pertinent to variables affecting the
correction factor were not always available. This fact
necessitated the use of cogent estimation based on engineering
judgment and reasonable assumptions. Without calling into
question the efficacy of such an approach, which certainly
was reasonable under the circumstances, it is nonetheless
appropriate to consider alternatives, given the resources to
supplement data sources. This prerogative is exercised in
response to the charge given in the Scope of Work to offer
a recommendation "as to the optimal approach for future
correction factor development." A comprehensive pronouncement
on this point is premature at this time, but certain pertinent
observations deriving from the examination of the R-factor
will be advanced. These observations deal generally with the
assumptions and the data on which the R-factor is based, as
well as the mathematical form into which the factor is cast.
At the outset it is observed that correction factors
represent an attempt to express the functional dependence
of emissions on a host of variables known to influence them.
Mathematically, therefore, one hopes for an expression or
response relation of the form
16
-------
Emissions (gms./mi.) = f (x^ , x^, ... x ) (1)
where x^, X2 , •••, Xp refer to such quantities as ambient
temperature,vehicle average speed, and the like. Even to
identify the appropriate "variables" is problematical, as
evidenced by concern previously expressed with regard to the
adequacy of the concept of average speed. Moreover, the
equation (1 ) is vehicle specific, and to develop such an
equation for every vehicle or cLass of vehicle would clearly
entail exorbitant effort. The hope of the correction factor
concept is that the vehicle or vehicle-class dependent aspect
of equation (1 ) can be extracted as a scalar multiplier and
that the equation can thereby be reduced to nondiraensional form.
In short, it is assumed that the relative effect on emissions
of incremental changes Ax^, Ax2/ Axp is unaffected by
the absolute level of emissions factored out of the response
relation.
Viewed in the above light, equation (1) becomes
Emissions (gms/mi.) = Emission factor (gins/mi.) * CF(x^, X2, • Xp)
(2 )
where the first quantity on the right-hand side is vehicle
or group dependent and the second quantity is a non-dimensional
function of the variables xj_, X2, . • . Xp and is considered
to be common to all vehicles in the class of interest. Thus
the very existence of CF(xj_, X2/ xp) rests on an implicit
assumption which can be either too strong or too weak, depending
on the complexity of the response relation and the degree of
commonality of that function as one goes from one vehicle or
class of vehicles to another. Means for analytically evalu-
ating the degree of commonality reside in factor analytic
methods such as principal component analysis, as illustrated
in Working Paper No. 1. The correction factor function,
viewed as a response "surface" (hypersurface) does not
necessarily have the same "shape" for all cases of interest
and may need to be resolved into two or more component
surfaces, each of which has to be scaled by a multiplier
much as in the original assumption of single separable scalar
and non-dimensional functional parts.
17
-------
A first recommendation, therefore, is that:
o Systematic analysis of the degree of commonality
of the effects of emission-related variables
should precede attempts to develop a correction-
factor response function.
It is believed that such an approach might better structure
the correction process, which should be viewed not necessarily
as a factor but as a general mathematical transformation until
its form has been delineated by systematic analysis.
Once the form of CF(x^, X2/ •••, xp) has been postulated,
either by mathematical or engineering analysis, data require-
ments for its definition can be specified. The function can
be structured as a "linear model"
CF (x^ , X2 f » • • , Xp) ^ ' x2 » * * * / ^p) •••
+ bkfk (x;l , x2 , . • • , xp)
in which f-^, f2/ ••• f^ are linearly independent "basis
functions" of arbitrary (and often nonlinear) form. It
follows, therefore, that the effects of x^ , X2, . Xp
on the correction factor may be both nonlinear and interactive--
that is, the effect of x^ on emissions may depend on the
level at which Xj is set (i j). Accordingly, an experiment
designed to evaluate the correction-factor surface should take
such nonlinearities and interactions into account. On the
other hand the experiment should not be "overstructured."
For example, if the effect of speed on emissions were thought
to be quadratic, it would be wasteful to structure an experi-
ment at—say—ten speed levels.
A second recommendation thus arises:
o Correction factors are best determined through a
designed experiment in which the allocation of
"treatments" (that is, combinations of levels of
the variables x-^, X2, •••, Xp) is made according
to the anticipated degree of nonlinearity within
variables and degree of interaction among variables.
18
-------
One notes that the above recommendations are idealized
and break with historical precedent, in which it is often
necessary to gain information by a "piggy-back" process. In
other words, data made available from an experiment designed
for a purpose other than correction factor development must
be adapted to that purpose and, in the process, becomes
subject to uncertainties over which the investigator has little
control.
So much for the assumptions regarding emission factor
correction and the nature of the data-acquisition process. Let
us now move to the question of mathematical representation
of the correction-factor response relation.
There is evidence that, in some instances, the dependence
of emissions on certain variables—e.g., average speed—is
less complicated than is implied by available expressions
in current use. Excess complication in functional representation
can arise from: (1) incorporation of refinements having only
minor engineering significance; (2) less than optimum choice
of independent variable; (3) covariance of variables influencing
emissions; and (4) inopportune choice of units.
Appreciation of the engineering importance of variables
influencing emissions can be obtained by evaluating the
differential effects of variables. For example, it would be
informative to know the incremental change in grams per mile
for a 1-mile-per-hour increment of average speed or a 1°F
change in ambient temperature. It is realized, of course,
that if interaction is present these incremental changes are
not constant, but the lack of constancy can be evaluated
by means of mixed partial derivatives. further, if the response
of emissions to a given variable is substantially linear, then
incorporation of additional terms may add little to the
precision of estimation of emissions even though the added
terms can be shown to be statistically significant. In such
cases it is possible that the simplification of representation
more than offsets the small gain in precision, expecially
in view of the uncertainties that may exist in the estimation
of levels of the independent variable. For example, if average
speed must be estimated by a rough sampling process, it is
doubtful that a fourth or fifth order polynomial or exponential
is justified in representing the relation between speed and
emissions.
19
-------
What is referred to as less than optimum choice of inde-
pendent variable is exemplified by the observation that
emissions expressed in qrams per unit time may plot as a
simpler function of average speed than when expressed in
grams per unit distance. In short, the effect of speed tends
to be "unified" by the distance-to-time transformation. The
fact that certain variables are perceived as having important
effects on emissions is to a certain extent an accident of
human perception--that is, we are accustomed to think in terms
of speed, temperature and the like when other, derived or
even "contrived" variables may lead to simpler relations or
scaling laws.
One of the ways to originate variables which seem artificial
but which are in reality quite meaningful is to note the
covariance or. interdependence of two or more variables. Such
covariance can be imposed by real physical constraints, such
as those influencing temperature, absolute humidity and
relative humidity. A cogent example, also, is afforded by
the fact that high speeds are not likely to be associated
with short trip lengths or with cold transient operation.
The mutual interdependence of cold start/hot start ratio,
average speeds and trip length is another case in point.
If added complexity of representation serves no other purpose
than to extend the domain of functional representation
to such unlikely or impossible combinations, then it is
clearly not justified. In the event that two or more variables
are so interrelated that this interrelation can be expressed
mathematically, then it may be possible to combine several
variables into a single, "combined" variable having the
same effect on emissions as the original variables within their
allowable ranges of variation under the constraints.
Another recommendation, therefore, naturally arises:
o Mathematical representation of correction factors
should be approached with due regard to the magni-
tude and engineering importance of variables
perceived to be important. The prospect of
combined or derived variables should not be over-
looked nor should the fact that variables originally
identified may be highly covariant.
20
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Finally, a word is in order regarding choice of units.
The issue can be illustrated by currently used expressions to
define emissions as a function of average speed. For example,
N0X emission correction factors are related to average speed
through a fourth-degree polynomial:
CF = aQ + a^ s + a2 sz + a3 s3 + a4 s"
Since speed s is in miles per hour, s2 has units of (mph)2,
s3 has units of (mph) 3 and s4 has units of (mph)4. In reality
however, each term in the sum must be dimensionless. Therefore,
a^ must units which are the reciprocal of mph, a2 must have
units of (mph)~2, and so on up to a^ , which must have units of
(mph)-4. The result is that when s = 60 mph, s4 = 12,960,000 mi.
hr4. To compensate for such a numerically large quantity in
the fourth-degree term, the magnitude of a.% must be very
small. Under such conditions computational precision becomes
critical and it is difficult to appreciate the actual importance
of the higher powers of speed. It would be preferable to
express speed in nondimensional form before developing the
regression relation. Such nondimensionalization can be achieved
by expressing speed as a dimensionless multiple of some
nominal or "standard" speed, such as the 19.6 mph average
speed of the FTP. In the case of exponential expressions
such a transformation facilitates a Taylor-series expansion
which may be capable of capturing "most" ofvthe effect of the
variable in a linear or quadratic expression.
Consideration of units and dimensional homogeneity
therefore suggests that:
o Difficulties arising from numerical constraints
imposed by choice of units can often be avoided
by nondimensionalization of independent or
predictor variables as well as the dependent
variable.
In conclusion, examination of the R-factor and its basis
of derivation suggests the desirability of an experimental
program specifically aimed at developing a data base for
the formulation of improved correction factors.
21
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APPENDIX I
SOME CONSIDERATIONS PERTINENT TO SPEED CORRECTION FACTORS
The speed correction factors for CO and HC, as developed
in Mobile Source Emission Factors, Final Document (January 1978)
and "Supplement 8 Light Duty Vehicle Correction Factors"*
take the functional form
F = pA0 + A1 S + A2 + A3 s' + A4 s" + A5
(1-1)
Differentiating (1-1) with respect to s one obtains
• dF
F = — = (A^ + 2A^ s + 3A^ sz + 4 A^ s3 + 5A^ s**) F
(1-2)
Thus the fractional change in correction factor per mph is
F
—= A + 2A s + 3A_ s2 + 4A, s3 + 5A_ s1*
* I 1 3 4 5
(1-3)
Similarly, for NO the correction factor relation takes the
form
F = A + A s + A~ s2 + A_ s3 + A. s*1
0 12 3 4
-------
and
dF 9 , i
F = -r— = An + 2A_ s + 3A s + 4A. ss
ds 1 2 3 4
or
• A, + 2A„ s + 3A s2 + 4A. s3
F _ 1 2 3 4
F AQ + A^ s + A2 s2 + s3 4 A4
An appreciation of these relations can be obtained by
comparing correction factors computed at 1-mph increments
in Table II.le of Becker (op. cit.). In Table 1-1 below
these incremental changes in correction factors are tabulated
at 5 mph to 6 mph, 10 mph to 11 mph, ..., 60 mph to 61 mph
for HC, CO and NOx for Group 2 vehicles. Values are expressed
both as incremental changes in correction factors and as
fractions of the correction factors prevailing at 5, 10, ...,
60 mph respectively. It is noted that although the correction
factors for CO and HC range from 3.3 to less than 0.5, most
of the sensitivity to speed is in the range below the average
speed of 19.6 mph prevailing in the FTP. This fact coulu
possibly be of value in attempts to simplify the correction
factor vs speed relations.
The shape of the correction factor curves for CO and HC
suggest a hyperbola of the form xy = constant in cartesian
coordinates. Quite clearly it would be fortuitous if such
a simple function applied. Note, however, that dimensionally
this notion has an interesting aspect. Recalling that the
correction factors are simply multiples of grams/mile
emissions at standard conditions (19.6 mp^), one has
gms/mi • mi/hr. = gms/hr
or emissions per unit time rather than emissions per unit
distance traveled. It is quite possible that this change of
basis, though not reducing to a constant, could give a much
simpler relation and one having a certain amount of engi-
neering credibility.
To examine this suggestion, consider Table 1-2. In this
table the correction factors for Group 2 hydrocarbons are
tabulated at 5 mph increments together with the product of
correction factor and speed. In the last column there is
(1-5)
(1-6)
A2
-------
Table 1-1
DIFFERENTIAL ANALYSIS OF CORRECTION FACTORS
Group 2
SPEED (mph) 5-6 10-11 15-16 20-21 25-26 30-31 35-36 40-41 45-46 50-51 55-56 60-61
HYDROCARBONS
Delta
A/Initial
Delta
A/Initial
Delta
A/Initial
3.2 97 1.749 1.224 .986 .844 .740 .659 .600 .565 .547 .530 .482
2.816 1.601 1.163 .953 .821 .722 .645 .591 .560 .544 .525 .464
-.481 -.148 -.061 -.033 -.023 -.018 -.014 -.009 -.005 -.003 -.005 -.018
-.146 -.085 -.049 -.033 -.027 -.024 -.021 -.015 -.009 -.005 -.009 -.037
CARBON MONOXIDE
3.319 1.751 1.225 .986 .841 .734 .650 .592 .557 .539 .518 .454
2.829 1.602 1.164 .952 .817 .715 .637 .583 .552 .536 .510 .433
-.490 -.149 -.061 -.034 -.024 -.017 -.013 -.009 -.005 -.003 -.008 -.021
-.148 -.085 -.050 -.034 -.029 -.023 -.020 -.015 -.009 -.006 -.015 -.046
NITROGEN OXIDES
1.242 1.031 .974 1.004 1.074 1.146 1.203 1.239 1.265 1.306 1.404 1.615
1.184 1.010 .975 1.017 1.089 1.159 1.211 1.244 1.271 1.320 1.436 1.677
-.058 -.021 .001 .013 .015 .013 .008 .005 .006 .014 .032 .062
-.047 -.020 .001 .013 .014 .Oil . .007 .004 .005 .011 .023 .038
-------
s
5
10
15
20
25
30
35
40
45
50
55
60
TABLE 1-2
GROUP 2 CORRECTION FACTOR ANALYSIS
HYDROCARBONS
CORRECTION FACTOR PRODUCT SUCCESSIVE
F s_F RATIOS
3.297 16.485
1.749 17.490 1.061
1.224 18.360 1.050
.986 19.720 1.074
.844 21.100 1.070
.740 22.200 1.052
.659 23.065 1.034
.600 24.000 1.041
.565 25.425 1.059
.547 27.350 1.076
.530 29.150 1.066
.482 28.920 0.992
A4
-------
> (1
s
5
10
15
20
25
30
35
40
45
50
55
60
TABLE 1-3
GROUP 2 CORRECTION FACTOR ANALYSIS
NITROGEN OXIDES
CORRECTION FACTOR PRODUCT SUCCESSIVE
F S F RATIOS
1.242 6.210
1.031 10.310 1.660
.974 14.610 1.417
1.004 20.080 1.374
1.074 26.850 1.337
1.146 34.380 1.415
1.203 42.105 1.225
1.239 49.560 1.177
1.265 59.925 1.209
1.306 65.300 1.090
1.404 77.220 1.182
1.615 96.900 1.255
A5
-------
tabulated the ratio of successive products. The fact that
these ratios are relatively constant suggests that an exponential
relation applies:
s~\ a + b s
s F = g
or
f = i/s ea + b s
where a and b are constants. Further,
log (s F) = a + b s
and the product s F should plot as a linear function of
speed on semilog coordinates. The validity of this hypothesis
is shown in Figure 1-1. It is further evident that CO
would exhibit similar behavior because the correction factors
for CO and HC are closely related.
Let us now examine the multiplicative relation for N0X,
as shown in Table 1-3. Though a simple exponential relation
is not evident, the product s F is a monotonic increasing
function of speed, and when plotted on semilog coordinates
appears capable of being represented by perhaps a quadratic
function of s (see Figure 1-2) .
In conclusion, it appears that it may be advantageous to
represent the effect of speed on emissions in terms of grams
per unit time rather than in terms of grams per unit distance.
A.S
-------
s, speed (mph)
FIGURE 1-1 TRANSFORMED PLOT OF CORRECTION FACTOR AS A FUNCTION OF.SPEED
FOR GROUP 2 HYDROCARBONS
-------
0 5 10 15 20 25 30 35 40 45 50 55 60
s, speed (mph)
FIGURE 1-2
TRANSFORMED PLOT OF CORRECTION FACTOR AS A FUNCTION OF SPEED
FOR GROUP 2 OXIDES OF NITROGEN
-------
4.3
WORKING PAPER NO. 3:
H 0*?/C CL D/5"AE ILI ZED VEHICLE OPERATION:
A CRITIQUE AND CANDIDATE
APPROACH
-------
FALCON
DENVER BALTIMORE ALBUQUERQUE BUFFALO
ONE AMERICAN DRIVE
BUFFALO. NEW YORK 14225
TELEPHONE: (716) 632-4932
Working Paper No. 3
Project 8411
Environmental Protection Agency
May 25, 1978
H. T. McAdams
HOT/COLD/STABILIZED VEHICLE OPERATION:
A CRITIQUE AND CANDIDATE APPROACH
1. INTRODUCTION
The concepts of cold-start, stabilized and hot-start
operation of automotive vehicles represents an attempt to
address the effects of "state of warm-up" on emissions and
fuel economy and were developed in recognition of the importance
of correcting for these effects. Though one is tempted to refer
to "vehicle temperature," it is recognized that to do so would be
simplistic, because (1) temperature takes on different meanings
according to where in the vehicle it is measured, and (2)
the mechanism by which such temperatures translate into
emissions is complicated and not well understood.
Presently used correction factors based on fraction of
miles driven in the cold-start, stabilized and hot-start
modes represent a compromise with reality. The conditions
under which a vehicle operates vary continuously rather than
discretely, and it is safe to say that it is seldom, if
ever, that a vehicle can be said to be ideally in the cold-
start, stabilized or hot-start state as defined in the FTP.
For example, few indeed must be the times when a vehicle is
restarted exactly 10 minutes after completing a trip which
is the exact equivalent of the FTP driving cycle. Conse-
quently, some protocol must be adopted by which operation
can be classified into one of the three states on the basis
that a given set of circumstances is sufficiently "like" one
of the states to justify its inclusion in that state. As
A SUBSIDIARY Of
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will be suggested later, however, there may be an advantage
in abandoning such a compartmentalized approach in favor of
a methodology which views heating and cooling of a vehicle
as a continuous process and modifies emission rates accordingly.
In the continuum, warm-up is considered as a matter of degree
rather than as a "go" or "no-go" affair.
2. PRESENT PROCEDURES: A REVIEW
The present method for dealing with hot and cold starts
is to view them as processes which are switched on or off ac-
cording to cold soak time (that is, how long the vehicle has
been standing unused) and length of time the vehicle has been
running since start-up. It is recognized, of course, that the
nature and severity of the driving cycle can modify the ef-
fects of soak times and run times on emissions, as can also
ambient conditions.
According to an EPA "Emission Factor User Information Sheet"
(see Appendix I), the break between hot-start and cold-start
operation can be defined in terms of a threshold for engine-off
time. For example, in the case of a catalyst-equipped vehicle,
the threshold is based on the maximum engine-off time "that can
occur without causing the catalyst to cool down sufficiently"
so that upon engine restart the catalyst is still operational.
Any such threshold time will, as noted above, be affected by
ambient temperature. The definition given in the cited docu-
ment is:
"Following an engine-off period, vehicle
operation is said to be hot transient (hot
start) if the engine-off time is less than
30 minutes and the temperature is 75° F or
greater. If the temperature is 20° F or less,
the allowable engine-off time drops to 10
minutes. Interpolation can be used between
the two temperature levels."
In the previous EPA emission factor document, AP-4 2, Supplement 5,
cold operation was defined as 505 seconds of operation following
a 4 hour engine-off period for noncatalyst vehicles and a 1 hour
engine-off period for catalyst vehicles.
Falcon Research & Development Co.
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Even though engine-off time can be used, albeit arbitrarily,
to differentiate between hot transient and cold transient opera-
tion, it is clear that the ensuing transient period "remembers"
the past operating history of the vehicle. It seems reasonable
to believe that a vehicle restarted after only a 2-hour soak
would exhibit a different transient response than one not re-
started until after a 12-hour soak. Thus a complete model of
transient phenomena should take into account both heating and
cooling cycles.
In the "Emission Factor User Information Sheet" of Appen-
dix I, it is proposed that the time to reach stabilized emissions
can be defined by an equation of the type
t - 2.51 s°*35 (1)
where t is the time, in minutes, required to reach stabilized
emissions, and s is the soak time in hours. Evaluation of this
formula for various soak times is shown in Table 1.
TABLE 1
Relation Between Stabilization and Soak Times
t = 2.51 s0-36, Temperature = 75° F
s t
soak time stabilization time
(hrs) (min.)
0 0
1 2.51
2 3.22
4 4.14
16 6.81
32 9.09
64 11.22
Now it is clear that, according to the table, the time required
for stabilization after a 16-hour soak is only 6.8 1 minutes
whereas the transient portion of the FTP is based on 505 seconds
(8.4 minutes) of operation.
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The equation can be adjusted so as to constrain the time
to a value of 8.4 minutes at a soak time of 16 hours. The re-
vised equation is
t - 3.11 s0,36 (2)
and is evaluated for various soak times in Table 2.
TABLE 2
Constrained Relation Between Stabilization and Soak Times
t = 3.11 s^-3^, Temperature = 75° F
s
soak time
(hrs)
0
1
2
4
16
32
64
t
stabilization time
(min.)
0
3.11
3.99
5.12
8.44
11. 26
13. 90
An interpretation of Table 2 is that it gives the times
reguired to bring the vehicle to a state of operation comparable
to that which occurs at the end of the Bag 1 sequence in a stan-
dard FTP test (following the prescribed 16-hour soak). Inasmuch
as the vehicle stabilized, according to Equation (1), in less
than the allotted 8.4 minutes, however, it might be said to
have spent 8.4 - 6.8 = 1.6 minutes in stable operation. This
time increment, expressed as a percent of the actual warm-up
time of 6.8 minutes, is
x 100% = 24%
D • O
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Note that by expressing the ratio in terms of the actual times,
as predicted by Equation (1), one is able to apply the same
arqument to soak times other than the standard 16-hour period.
For example, when s = 4 hours, one has
5.12 - 4.14
4.14
x 100% =
0.98
4 .14
x 108% = 24%
Thus the shift is a proportional one and is shown graphically
in Figure 1.
A similar approach applied to tests conducted at 20° F
suggested (see Appendix I) that a constant difference of 1.27
minutes between 20° F ambient and 75° F ambient applies regard-
less of engine-off time. Thus the time to reach stabilized
emissions becomes
2.51 s0'36 + 1.27
(3)
or, upon renormalization,
t = 2.61 s°'36 + 1.32 (4)
and it is proposed that linear interpolation/extrapolation be
used between 0° F and 100° F.
\
Several comments regarding the suggested approach can now
be made. First, the notion of a power law to represent the
relation between stabilization times and soak times does not
seem compatible with physical reality. According to Equations
(1) - (4), the time required for stabilization continues to
increase as a monotonic function of soak time, rather than
approaching some limiting value. Second, units seem difficult
to reconcile. Of course it can always be argued that the
equations are only for descriptive purposes and are not to be
applied outside a specified range of soak times. On the other
hand,their use is not in keeping with criteria of the Purchase
Order, one of which is "ability to relate to engineering
concepts."
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10
1 1
>
TZ
Jir
i
t = 3.
Tffi
i rr
im
¦ r - H r _r
li
rt-
r,
t+
Hb
Tffi
P
tiii
1 s
0.36
TF
.ii
n-ii
m
m
i
¦-Hri
I
HI
tri't
m
rfrl
Hi!
HI
m
-Ftet
rrb:
. _ 0 r, 0.36":
t — 2.51 s -ft
5 6 7 8 9 10
i rt'-i
ill
1
tHi
Mil
m
!tH
>
HV
-ilj
10
8 9 10
] 00
s, Soak time (hours)
FIGURE 1
Time to Stabilize as a Function of Soak time
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Perhaps the most serious drawback of an approach based
on definition of a specific time required for stabilization
is the difficulty in selecting a criterion for defining when
vehicle operation is "warm or stabilized." Temperature
levels, such as catalyst temperature, oil temperature, water
temperature or air-intake temperature can be considered as
bases, but these do not necessarily track each other over
time and are not readily translated into effects on emissions.
If emission levels are taken as indicators, it may well be
that each pollutant has its own "stabilization time" so that
it would not be possible to specify a unique time required
to reach a stabilized state.
3. AN ACCRETION-DEPLETION VIEW OF THERMAL TRANSIENT EFFECTS
Insight pertinent to the development of a predictive
equation for engine warm-up is afforded by an examination of
the physical processes involved. An internal combustion
engine is both a heat source and a heat reservoir. Much of
the heat generated is converted to mechanical energy to
drive the vehicle, but a certain amount goes to raise the
temperature of elements of the engine itself, and excess
heat is dissipated by the cooling system. When the engine
is turned off, the heat stored in the engine and associated
elements such as the catalytic converter is lost to the
surrounding atmosphere. It is conjectured that heat is
transfered mostly through a slow process of radiation. Thus
it might be expected that heat is lost by the engine much
more slowly than it is gained, and that the effect of the
ambient temperature would be to increase or decrease the
temperature differential between engine and atmosphere and
hence influence the rate of heat loss accordingly. Also for
any given ambient temperature, it is reasonable to expect
that as the engine cools and approaches ambient, the rate of
heat dissipation decreases so that the cooling cycle might
be expected to be essentially an exponential process. By
similar reasoning it might be conjectured that heat accretion
is also exponential, but with a much shorter time constant
caused by the much larger temperature differential driving
the process.
In short, it appears that the accretion and depletion
of heat might be analogous to the charging and discharging
of an electrical condenser in a simple resistance-capacitance
circuit, as developed in Appendix II. Because some of the
processes involved (such as the action of the choke and the
activation of the analytic converter) are discontinuous or
only quasi-continuous, the simple exponential process postu-
lated may be subject to step-function perturbations which
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may need to be modeled by non-linear circuit elements. In
any event, however,, such a view has the conceptual advantage
that the "heat budget" of the vehicle, or its effects on
emissions, can be visualized as a continuous function of time.
A schematic view of how the accretion and depletion of
heat might be manifested is shown in Figure 2. At what point
Time
FIGURE 2
in the vehicle this temperature is measured is not particularly
germane to the argument, but it is assumed to be related in
some way to emission performance. When the vehicle is first
started, designated by the notation "ON" at Time = 0, the
temperature rises, presumably exponentially, as the vehicle
accumulates heat from the combustion process. Because of the
cooling system, however, a maximum operating temperature is
approached as time continues. As the difference is narrowed
between the limiting temperature and the temperature at time t,
the rate of temperature rise tends to become smaller. When the
vehicle is stopped, designated by the notation "OFF" at
Time = a, the temperature falls, again presumably exponentially,
as temperature is lost to the surround. Again, as the difference
is narrowed between ambient temperature and vehicle temperature,
the rate of cooling tends to decrease with time.
Figure 2 does not suggest any mechanism by which vehicle
temperature is translated into emissions. Let us suppose,
however, that there exists a critical, minimum temperature level
requisite to "stable operation"—for example, the operating
temperature of the catalytic converter. As shown in Figure 3,
this critical level would be reached sooner after a relatively
short "OFF" time than after a quasi-infinite time {compare
and t2), as was assumed to be the case at Time = 0. Because
of the mechanism by which temperature is translated into
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FIGURE 3
emissions, emission rates as functions of time are not
necessarily exponential though driven by an exponential
thermal process. On the other hand, an exponential decay of the
choke effect has been postulated as part of a computer simu-
lation of emissions and fuel economy.
In the event that emissions are directly expressible as
an exponential function of time, the accretion-depletion model
could be implemented by determining the time constants of the
heating and cooling cycles. An attempt to evaluate the required
time constants was undertaken, using data from an EPA-funded
program on vehicle soak and run times and their effects on
emissions (see Appendix III). For reasons cited above, the
attempt had only limited success, but it is suggestive for
further, less simplistic analysis. The implications of the
approach are also exploited in the discussion of the following
section, in which transient emission rates are regarded as
linear combinations of two limiting emission rates associated,
respectively, with cold transient and hot transient operation.
W. K. Juneja, w. J. Kelly and R. W. Valentine, "Computer
Simulations of Emissions and Fuel Economy," SAE Paper
No. 780287, Society of Automotive Engineers {1978)
^ R. L. Srubar, "Emission and Fuel Economy Sensitivity to
Changes in Light Duty Vehicle Test Procedures," Final
Report of Task No. 10, EPA Contract 68-03-2196, Southwest
Research Institute (May 1977)
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4. A CONTINUUM APPROACH TO TRANSIENT OPERATION
The 197 5 Federal Test Procedure employs three types of
driving: a cold transient phase {representing vehicle start-up
after a long engine-off period); a hot transient phase (repre-
senting vehicle start-up after a short engine-off period); and a
stabilized phase (representing warmed-up vehicle operation).
Emissions measured during these three phases are combined as
a weighted sum, the weighting factors being 0.20, 0.27, and
0.53, respectively. It is presumed that the FTP is run at a
standard temperature of 75° F, but a range from 68 to 86° is
allowed.
The basis for quoting FTP emission results suggest that
the measure is a sort of "hybrid" quantity partaking of the
properties of both cold-start and hot-start operation. By an
extension of this argument it is not a great step to propose
that a vehicle restarted after some period of soak time can
similarly be characterized as a "hybrid" in the sense that
it represents neither a cold-start nor a hot-start situation,
but some weighted combination of the two. Viewed in this way,
adjustment of emissions for various combinations of soak and
run times can be achieved by the device of variable weighting
factors rather than by the device of predicting warm-up times.
An example of how this approach might be used is shown in
Table 3 below. The data are taken from R. L. Srubar (op. cit.)
for a 1976 Chevrolet Impala. Emission tests were run after
TABLE 3
Soak
Time
HC (gms/mi)
First 505 sec. of FTP
HC
Percent Cold Start
10 min
20 min
30 min
1 hr
2 hr
4 hr
8 hr
16 hr
36 hr
0.39*
0.48
0.53
0.69
0.64
0.85
1.03
1.19*
1.27
0,
11.
17,
37 ,
31.
57,
80.0
100. 0
110.0
0
2
5
5
2
. 5
Reference values (see text)
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various soak times as indicated. The starred values are of
especial significance. The results shown for a 10-minute soak
are essentially Bag 3 results, that is, emissions representing
hot transient conditions. The results shown for a 16-hour
soak are essentially Bag 1 results, that is, emissions repre-
senting cold transient conditions.
A procedure is now proposed whereby results for soak times
other than 10 minutes and 16 hours can be interpreted as linear
combinations of hot transient and cold transient contributions.
For example, consider the emissions for the 1-hour soak as
consisting of a fraction P of cold-start emissions and a
fraction 1-P of hot-start emissions. Then
P(1.19 gms/mi) + (1-P)(0.39 gms/mi) = 0.69 gms/mi
Solving this equation gives P = 0.375. Thus the 1-hour test
acts "as if" it were a composite of 37.5% cold transient and
62.5% hot transient operation. The weightings should not be
construed as fractions of either the total time to execute the
test (8.4 minutes) or the total distance covered (3.59 miles).
Rather, their only purpose is to generate, from available
Bag 1 and Bag 3 emissions, a quantity which is equivalent to
the results observed for any given soak time. Note that this
approach would have the advantage that the required computation
could be performed in terms of emission results readily available
as components of the standard FTP test, provided that the
fraction P is known as a function of soak time and run time.
Tables 3 through 7 provide further analysis for the five
vehicles studied by Srubar (op. cit.). Results are given for
HC and CO emissions and for fuel economy. No attempt was
made to analyze the results for NOx, since that pollutant
does not seem to be very sensitive to the cold-start phenomenon.
Two anomalies may be noted in these tables. One is that
negative values of P sometimes occur. The other is that
emissions after a 36-hour soak may be greater than emissions
after a 16-hour soak. This fact implies values of P greater
than 1.0 and negative values of 1-P.
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TABLE 4
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 1976 Chevrolet Impala
Over the First 505 Seconds of the FTP
(Data from Table A-2, Srubar, op. cit.)
Soak
HC
PHC
CO
PCO
Fuel Economy (F.E.)
PF.E.
Time
(girts/mi)
(%)
(grr.s/mi)
(1)
tmi/gal)
10 min
0.53
0
7.13
0
13.47
0
j
j 20 min
0.61
6.45
6.08
-4.3
13.20
12.05
30 min
0.88
28.20
6.87
-1.1
13. 08
17.41
1 hr
1.09
45.20
7.95
3.3
12.61
38.39
2 hr
1.38
68.5
10.10
12.0
12.14
59.38
4 hr
1.34
65.3
14.18
28.5
11.88
.70.98
B hr
1.27
59.7
16.52
38.0
11.28
97.77
16 hr
1.77
100.0
31.86
100.0
11.23
100.00
36 hr
1.56
83.0
28.70
87.2
11.06 1
[
107.6
-------
TABLE 5
*3
o
t—¦
o Percent Cold Start Operation as a Function of Soak Time
3
for HC, CO and Fuel Economy for a 1977 Ford LTD
O
Over the First 505 Seconds of the FTP
(Data from Table B-2, Srubar, op. cit.)
Soak
Time
HC
(gms/mi)
PHC
(%)
CO
(gms/mi)
pCO
(%)
Fuel Economy (F.E.)
(mi/gal)
PF.E.
10 min
0.39
0.0
3.91
0.0
15.98
0.0
20 rain
0.72
37.50
2.51
-5.49
15.10
21.10
30 min
0.93
61.36
3.67
-0.94
15.52
11.03
1 hr
1.03
72.73
3.38 | -2.08
15.29
16. 54
2 hr
1.08
78.41
1
3.56 j -1.37
j
14.84
27.34
4 hr
0.78
44.32
3.96 S 0.20
I
J
14.15
43.88
8 hr
1.23
95.45
12.48
33.63
12.40
85.85
16 hr
1.27
100.00
29.39
100.00
11.81
100.00
36 hr
2.06
189.80
37.63
132.33
12.46
84.41
—
-------
TABLE 6
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 1976 Plymouth Fury
Over the First 505 Seconds of the FTP
{Data from Table C-2, Srubar, op. cit.)
Soak
Time
HC
(gms/mi)
PHC
(%J
CO
(gms/mi)
PC0
(%)
Fuel Economy (F.E.)
(mi/gal)
PF.E.
10 min
0.52
0.0
1.71
0.0
16.87
0.0
20 min
0.70
13.7
1.88
0.8
15.82
26.05
30 min
0.72
15.3
2.70
4.4
15.43
35.73
1 hr
0.87
26.7
3.18
6.5
15.42
35.98
2 hr
1.40
67.2
12.91
49.7
14.67
54.59
4 hr
1.42
68.7
16.95 | 67.6
i
13.88
74.19
8 hr
1.64
85.5
\
20.84 ! 92.4
i
13.18
91.56
16 hr
1.83
100.0
24.24 1 100.0
5
12.84
100.00
36 hr
1.89
104.6
28.37 S 118.3 1
t »
12.18
116.40
-------
TABLE 7
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 1976 Chevrolet Vega
Over the First 505 Seconds of the FTP
(Data from Table D-2, Srubar, op. cit.)
Soak
HC
PHC
CO
PC0
Fuel Economy (F.E.)
PF.E.
Time
(gms/mi)
(%)
(gms/mi)
(%)
(mi/gal)
10 min
0.39
0.00
1.01
0.00
22.71
0.00
20 min
0.48
11.25
2.09
8.05
22.84
-3.05
30 min
0.53
17. j50
1.89
6.56
23.20
-11.50
1 hr
0.69
37.50
2.24
9.16
22.19
12.21
2 hr
0.64
31.25
3.68
19.89
20.97
40.84
4 hr
0.85
57.50
5.36
32.41
19.70
70.66
8 hr
1.03
80.00
14.63
101.49
19.41
77.46
16 hr
1.19
100.00
14.43
100.00
18.45
100.00
36 hr
1.27
110.00
19.90
140.80
18.45
100.00
-------
TABLE 8
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 197 6 Honda Civic CVCC
Over the First 505 Seconds of the FTP
{Data from Table E-2, Srubar, op. ext.)
Soak
Time
HC
(gras/mi)
PHC
(%)
CO
(gras/rai)
PC0
(%)
Fuel Economy (F.E.)
(mi/gal)
PF.E.
10 rain
0.85
0.00
4.83
0.00
31.53
0.00
20 rain
0.97
8.39
4.39
-14.33
31.08
9.49
30 rain
1.00
10.49
4.17
-21.50
31.58
-1.05
1 hr
0.80
-3.50
4.38
-14.66
30.75
16.45
2 hr
1.00
10.49
4.89
1.95
28.72
59.28
4 hr
1.24
27.27
6.32
48.53
27.84
77.85
8 hr
2.03
82.52
7.82
97.39
25.03
137.13
16 hr
2.28
100.00
7.90
100.00
26.79
100.00
36 hr
2.65
125.90
9.48
151.46
24.13
i
156.10
-------
These anomalies, however, are not considered serious nor
obstructive to the formulation of a weighting-factor approach,
Part of the difficulty, it is believed, arises from errors of
emission measurement and the manner in which these errors
propagate in the computation of P. In general,
p Et - Ehs
" Ecs - ehs
where Et = emissions after soak time t, E^s = ho,t start
emissions, and Ecs = cold start emissions. Thus errors in
the determination of Efjs an<3 Ecs could induce appreciable
error in P. In general application of the methodr however,
it is likely that P would be determined on an aggregated
basis (i.e., for groups of vehicles) and errors would tend
to be averaged out. It is also possible, of course, that
soak times somewhat greater than 10 minutes could produce
emissions slightly lower than those associated with the
classical hot transient by virtue of evaporative losses and
other minor effects of relatively short duration. It is
believed, however, that such phenomena would have such a
small effect that they could be ignored. Finally, the
tendency for the 36-hour soak to exhibit higher emissions
than the 16-hour soak suggests that a reference other than
the 16-hour soak might be appropriate. On the other hand,
such long soak times would rarely occur in practice and for
that reason could likewise be ignored.
Implementation of the variable-weighting approach requires
further development in order to be practical. For example,
it would be necessary to explore the question of vehicle-to-
vehicle commonality of P, expressed as a function of soak
and run times, and to develop the necessary functional
representations. Moreover, it would be necessary to develop
a strategy by which a local or area survey of vehicle use
patterns could be converted into either a distribution of P
values or into some form of appropriate average P value.
Finally, it might well be that cold-start and hot-start
transient emissions do not represent the best choice of
limiting emissions on which to base the continuum approach;
perhaps cold-start and stabilized emission levels might be
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better (see Appendix 4). At any rate it is suggested that
two emission levels, representing respectively the most
favorable and least favorable operating conditions could
form the basis for a weighted combination reflecting different
scenarios of soak and run times. The proposed approach is
believed to offer a viable alternative to the determination
of time to stabilize and is considered worthy of further
evaluation.
5. SUMMARY AND PROSPECTUS
One of the most important scenario variables affecting
emissions is the "thermal operating history" of the vehicle.
The term in quotation marks, admittedly a coined one, is
meant to signify the state of warm-up of the vehicle as a
function of time. It suggests that the vehicle can, at
certain times in its operating history, be in a fully
warmed up condition but at other times can fall short of
this condition to varying degrees.
The present approach to the thermal history problem is
to postulate three "states" of operation: cold transient,
stabilized, hot transient. Because these three states do
not represent the whole spectrum of degrees of "warmed-up-
ness," it becomes necessary to implement a methodology by
which an arbitrary driving scenario can be decomposed into
the three discrete states. An important aspect of this
methodology is to develop a criterion for determining the
length of time required in order to declare a vehicle
"stabilized."
Equations previously developed by EPA for this purpose
are of such a nature that they predict stabilization times
which increase without limit as soak times increase. It
seems more reasonable to believe, however, that there exists
an upper limit for stabilization time and that this limit
would be approached asymtotically with increasing length of
soak time. The lower bound for stabilization time is, of
course, zero and occurs at zero soak time. Thus stabilization
times have "floor" and "ceiling values," though it is understood
that the range between these two values can vary with ambient
conditions and with vehicle characteristics and the nature of
the emission-control system.
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A hypothesis advanced in this working paper is that emission
rates, expressed in terms of grams/mile, also have "floor"
and "ceiling" values. The floor value, mathematically
speaking, must be zero, but in practice the emission rates will
tend to be of some finite value even when the vehicle is
operating under the most ideal conditions. As a practical
assumption, these "most ideal conditions" can be taken as
"stabilized operation." The ceiling value is not so clearly
limited, but it would have to be of finite, albeit very large,
magnitude even if all the fuel burned were converted to
pollutant. In practice, one can visualize a "worst possible"
condition which is seldom if ever exceeded. For the purposes
of the arguments in this paper it is not particularly damaging
that the proposed ceiling is occasionally exceeded if these
exceedances are very rare or tend to occur under use scenarios
(e.g., very long soak times) which practically never occur.
It is now proposed that the determination of realistic
floor and ceiling emission rates may suffice to define
emissions over the entire spectrum of thermal operating
histories, once the envelope of histories has been "calibrated"
for various combinations of soak and run times. Note that
this approach eschews completely the notion of "warm-up time"
or "time to stabilization." Under the assumption that
stabilization is approached asymtotically, time to stabili-
zation can be defined only arbitrarily. Moreover, if a means
is provided for computing emissions for all scenarios of
interest, the question of stabilization time is irrelevant.
Data required to implement the proposed approach are
acquired readily with little, if any, modification of present
testing practice. If cold-start emissions (Bag 1) are taken
as the ceiling and stabilized emissions (Bag 2) are taken
as the floor, the spectrum of all values between these limits
can be represented as weighted sums of the two. The presently
used hot-transient phase (Bag 3) could provide a check point
for such weighting and could be augmented by one or more
additional check points representing soak times intermediate
between 10 minutes and 16 hours. In this way the standard
method of reporting emissions according to the FTP would not
need to be disturbed, but the information provided by the FTP
could be augmented and used in a more effective way. One diffi-
culty, however, does remain. Since cold-start (Bag 1) emissions
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Page 19
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and stabilized emissions (Bag 2) are measured over different
driving cycles, it might be preferable to normalize the two
results to the same average speed. The necessary calibration
for this purpose could be provided by a series of tests in
which the first 505 seconds of the FTP is followed by a
repetition of the first 505 seconds without any vehicle
off time. In this way the Bag 2 results could be referenced
to a "Bag 1 result obtained under-stabilized conditions."
It is true, of course, that to implement the methodology
for purposes of regional air-quality assessment one must have
available a distribution of soak and run times. It is
believed, however, that the required data could be obtained
by sample survey methods and that average or "effective"
soak and run times characteristic of the scenario can be
derived. These effective times would have to be defined in
such a way that they reflect the impact of the joint distri-
bution of soak and run times, rather than as a simple average,
but statistical approaches to such problems are well known.
In summary, an alternative to the three-bag method of
adjusting emissions for cold-start/stabilized/hot-start
fractions is proposed. It is believed that the method could
be implemented with only minor modification of present test
procedures and that it has the advantage of circumventing the
need to determine effective warm-up times for various use
scenarios. Further evaluation of the approach is recommended.
An essential part of this evaluation would be the development
of a data base to define how the weighting factors vary as a
function of soak and run times. As noted above, the required
data base could be developed as an addendum to existing
emission-testing programs, such as certification and in-use
surveillance.
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APPENDIX I
EMISSION FACTOR USER INFORMATION SHEET:
HOT/COLD/ STABILIZED OPERATION
a 1-1
-------
Emission Factor User Information Sheet:
Hot/Cold/Stabilized Operation
Problem Identification:
The recent Revised Emission Factors Document contains a single correction
factor for hot/cold weighting, average speed, and ambient temperature.
In the previous EPA emission factor document, AP-42, Supp 5, cold
operation was defined as 505 seconds of operation following a 4
hour engine-off period for non-catalyst vehicles and a 1 hour
engine-off period for catalyst vehicles. The recent emission
factor document did not provide a definition of cold or hot transient
operation. This information guide will provide users with the
appropriate definitions and some methodologies which can be used to
collect the necessary input data.
Definitional Constraints:
A correction factor for vehicle temperature would ideally relate
emissions to a series of variables including time since vehicle
start-up, time vehicle was turnecl-off prior to start-up, severity
of cycle over which vehicle has been driven since start-up, vehicle
identifying information and ambient conditions. This relationship
would be a predictive regression relationship and would be normalized
to equal one if the vehicle were completely warmed-up. This relationship
would then be applied on a second by second basis to the EPA modal
emission model. Since vehicle temperature changes on a second by
second basis, it would not be absolutely correct to apply a single
correction factor to an entire cycle.
The data are not available to develop a functional relationship of
the type just describedj the development of such a relationship
would require extensive amounts of second by second emission data
as a function of all of the variables of interest. An equally
important limitation is that Federal Test Procedure emission data
collected in the large studies of in-use vehicles have divided the
data into three distinct operational categories. A single emission
value is available for the first 505 seconds of operation following
a 16 hour engine off period. A second emission value is available
for an 870 second period of stabilized (warmed-up) operation.
Fina.lly, a third emission value is available for 505 seconds of
operation following a ten minute engine off period. These three
pieces of data can be used to develop average correction factors
for cold start, stabilized, and hot start operation. In each case,
the correction factor is an average of the effect of vehicle temperature
oyer a fairly long'time period during which, vehicle temperature
and vehicle emissions are changing.
A 1-2
-------
-2-
The cold start correction factor presented in AP-42 will under-
estimate the emissions during the first minute after a 16 hour soak
and overestimate the emissions during the seventh minute after a 16
hour soak. Therefore, the correct application of the factor
requires a knowledge of the number of vehicles which are operating
within a 505 second period after a 16 hour soak. J^t jLs assumed
that the distribution of vehicles is equally distributed throughout
the 50j5 second period. For example, if 16 percent of the vehicles
are operating in the first 505 seconds since start-up, it is assumed
that 2 percent are operating in the first minute, 2 percent in the
second minute, ... and 2 percent in the eighth minute.
From an emission standpoint, cold operation can occur when a vehicle
soaks (engine off condition) for less than 16 hours. Limited data
are available to determine the length of time it takes before
emissions stabilize as a function of engine-off time and time since
engine start-up. If emissions averaged over start-up periods which
are less than 505 seconds following a less than 16 hour engine off
period can be shown to be equivalent to emissions averaged over 505
seconds following a 16 hour engine off period, then all such
equivalent operation should be defined as cold start operation and
the AP-42 correction factor should be applicable. Again, the
assumption of equal vehicle distribution throughout the time period
is required.
The differentiation between cold start operation and hot start
operation is strictly dependant upon the length of the engine off
period. A hot start condition attempts to simulate a case where
the length of engine off time is sufficiently short so that vehicle
engine temperatures/ emission control systems do not cool down
significantly; for example,a hot-start situation would not activate
the vehicle choke. Emissions are increased during a hot start
condition due to the dumping of excess fuel which is stored in the
carburetor, evaporative canisters, etc. Thus, emissions following
a hot start quickly return to their normal stabilized levels.
Again, as in the case of the cold start, the AP-42 correction
factor averages these emissions over 505 seconds and assumes that
vehicles are equally distributed throughout the time period.
Thus, the break between hot start and cold start operation is
dependant upon engine-off time. It is that engine-off time that
differentiates between a vehicle where the engine/emission control
system is still in a warmed-up condition and a vehicle where the
engine/emission control system has cooled down. In the case of a
catalyst vehicle, it is the engine-off time that can occur without
causing the catalyst to cool down sufficiently so that during a hot
start the catalyst is still operational. EPA does not have data to
define this point. Clearly, the time period could be expected to
be a function of ambient temperature. At this time, the recommended
definition is given below.
A 1-3
-------
-3-
Following an engine-off period, vehicle operation is said to
be hot transient (hot start) if the engine-off time is less
than 30 minutes and the temperature is 75° F or greater,
if the temperature is 20° F or less, the allowable engine-off
time drops to 10 minutes. Interpolation can be used between
the two temperature levels.
Once engine-off time is used to differentiate between hot transient
and cold transient operation, a definition Is needed for transient
operation as a function of ertgine-off time and ambient temperature.
Several studies have been performed by EPA to address the transient
operation definition. These studies are.
1. Bureau of Mines, "Ambient Temperature and Vehicle Emissions",
EPA 460/3/74-028, December 1974; 26 vehicles, 4 different
ambient temperatures, one soak condition, emission readings at
2, 5.5, 8.4, 15.6, and 22.9 minutes.
2. In house work on five vehichles, January, 1977 (unpublished);
2 different ambient conditions, five different soak conditions,
emission readings at 2, 5.5, 7.1, 8.4, 12.8, 17.1, and 22.9
minutes.
3. Ongoing contract work on five vehicles; one ambient condition,
nine different soak conditions, emission readings at 1, 2, 3,
4, 5, 6, 7, and 8 minutes.
At the present time, detailed analyses have not been performed.
However, the necessary stages of detailed analysis can be outlined.
First, it is necessary to define when operation is "warm or stabilized".
Two criteria are possible for this assessment; emission levels or
temperature levels. Temperature levels can be catalyst temperature,
oil temperature, water temperature, or intake air temperature.
While emission levels have been recorded in discrete bag samples,
temperature levels are normally recorded continuously. Given the
difference in measurement/recording techniques, there may be less
variability in using a temperature definition. However, since the
bottom line item is emissions, extra variability is introduced with
the undefined link between temperature and emissions. Future
analyses will pursue both approaches.
The first stage of analysis will be to develop a curve of time to
reach stabilized emissions as a function of engine-off time for
ambient temperatures in the range of the FTP. Data sources 2 and 3
can be used for this assessment. If possible, separate curves
should be developed for pre-1975 models, 1975 and later catalyst
equipped models, and 1975 and later non-catalyst modes.
h 1-4
-------
-4-
The seond stage of analysis is to factor in ambient temperature
effects. Data sources 1 and 2 can be applied and two approaches
can be used. The approach used with data source 2 would be a
straightforward graphical application of the data. Data source 1
has the largest amount of information regarding ambient temperature
effects. However, all data were collected at one soak time, an
overnight soak taken to be 16 hours. By defining the 16 hour point
and making the assumption that temperature and soak time effects
are independent, a parallel set of curves can be drawn for a range
of ambient temperatures.
Finally, the definition of transient operation must be related to
the definition used in the development of emission correction
factors. Iri the most recent AP-42 work, cold and hot transient
emissions were defined as emissions over the first 505 seconds
where the first 505 seconds contain some stabilized operation. In
order to appropriately apply the AP-42 factors, the warm-up tiine/engine-
off time curves need to be shifted. The cold correction factors
assume a sixteen hour soak period and 505 seconds (8.4 minutes) of
transient operation while the hot transient correction factors
assume a 10 minute engine-off period and 505 seconds of transient
operation. Thus, the curves should be shifted so that the 16 hour
and 10 minute soak periods respectively are equivalent to 8.4
minutes of operation. The shift is a percentage shift. That is,
if the time for the 16 hour soak period is shifted up by 25%, the
time for the 2 hour soak period is also increased by 25%. This
approach allows the same percentage mix of cold/stabilized or
hot/stabilized operational time in the estimate of the cold or hot
transient correction factors.
Definitions:
The data sources referenced above were analyzed on a preliminary
basis. At 75° F, the time to reach stabilized emission operations
can be defined as
t = 2.51 s*36 , r = .86
where t is the time to reach stabilized emissions (in minutes) and
s is the engine-off time (in hours).
To get the AP-42 definition for cold operation, the equation must
be adjusted so that a 16 hour engine-off period results in 505
seconds of cold operation. This is accomplished by including 24%
stabilized operation in the definition. Thus,
A 1-5
-------
-5-
A vehicle is operating in a cold transient condition if:
1) The ambient temperature is 75°F
2) The engine-of£ period is 30 minutes or greater
3) The vehicle has b|gn operating for t minutes or less
vhere t = 3.11 s" and s is the engine-off time in hours.
To get the AP-42 definition of hot operation, the equation must be
adjusted so that a 10 minute engine-off period results in 505
seconds of hot operation. This is accomplished by including 538%
stabilized operation in the definition. Thus,
A vehicle is operating in a hot transient condition if:
1) The ambient temperature is 75°F
2) Hie engine-off period is 30 minutes or less
3) The vehicle been operating for t minutes or less where
t = 16.01 s" and s is the engine-off time in hours.
Based on very limited data, it appears that a fixed difference in
time to reach stabilized emissions exists between t at 75°F and t
at 20°F, regardless of engine-off time. The fixed difference was
obtained from two catalyst vehicles in data source 2; the fixed
difference is 1.27 minutes. Thus, at 20°F, the time to reach
stabilized emissions is defined as
t = 2.51 s*36 + 1.27
where t and s are given earlier. Using the same normalization
schemes discussed above, the following definitions hold
A vehicle is operating in a cold transient condition if:
1) The ambient temperature is 20°F
2) The engine-off period is 10 minutes or longer
3) The vehicle has been operating for t minutes or less
where t = 2.61 s* + 1.32.
At temperatures different from 20°F or 75°F, linear interpolation/
extrapolation can be used between 0°F and 100°F by computing a t/°F
rate and then normalizing. The t/°F rate is .023 minutes/°F.
Thus, at 30°F, the time constant is 1.04 rather than 1.27, the
normalizing multiplier is 1.07, and the final equation would be t =
2.69 s" + 1.11.
A vehicle is operating in a hot transient condition if:
1) The ambient temperature is 20°F
2) The engine-off period is 10 minutes or less
3) The vehicle-has been operating for t minutes or less where
t = 8.15 s + 4.12
A 1-6
-------
-6-
At temperatures different from 20°F or 75°F, linear interpolation/
extrapolation can be used between 0°F and 100°F by computing a t/°F
rate and then normalizing. The t/°F rate is .023 minutes/°F.
Thus, at 30°F, the time constant is 1.04 rather than 1.27, the
normalizing multiplier is 3.56, and the final equation would be t =
8.94 s + 3.70.
Sources of Data:
Various methods are available to obtain data on the percentage of
vehicles which are operating in cold transient, hot transient, or
stabilized emission scenarios. These are discussed below.
1. Origin - Destination Studies - On a regional basis, these data
banks can be analyzed to determine the number of trips which
begin after various engine-off times. By adding information
on average trip length and average trip speed, the percentage
of miles of each type of operation can be readily determined.
On a local basis, 0-D studies can be used to determine the
distribution of vehicles at a given location with respect to
operating time and engine-off time prior to start-up. Average
link speed may have to be added to the data base to perform
the analysis. This methodology was recently used by GCA under
contract to EPA. The report is titled Characterization of
Cold Mode Operation and is available from Mr. Jim Wilson,
EPA, Research Triangle Park, NC 27711.
2. Survey data - Clearly, only two pieces of information are
needed in order to determine whether a vehicle is in a transient
or a stabilized emission condition. A quick roadside survey
can be designed to ask motorists how long ago they started
their engine (time and/or miles) and how long the engine was
off prior to start-up. In tnany cases, a rough estimate is
entirely adequate. Depending upon the design of the survey,
localized or regional percentage values can be determined.
3. Direct Measurement Data - Direct measurement techniques exist
vhich can determine whether a vehicle is in a stabilized
emission configuration. These techniques can be difficult to
implement due to the need to perform extensive calibration.
Some measurement methods require the vehicles to stop for
several minutes while other techniques require only a 30
second stop or no stop. Broadly, the measurement techniques
can be divided into two types; those which use a thermocouple
measurement technique and those which use an infrared measurement
technique. Temperature measurements from a number of areas of
A 1-7
-------
-7-
the vehicle can be fairly well correlated with emissions in a
gross sense (that is, emissions stabilized or emissions not
stabilized). These areas include: inlet to vehicle radiator,
oil pan, catalyst skin, exhaust gas, and difference between
vehicle hood and vehicle trunk. The EPA is in the process of
implementing this type of study. Results and documentation of
the methodology should be available by July, 1978.
A 1-8
-------
APPENDIX II
A CAPACITIVE MODEL OF SOAK AND RUN TIME EFFECTS
ON VEHICLE EMISSIONS AND FUEL ECONOMY
Experimental data collected on vehicles during varying
periods of run and soak . (vehicle on and off) indicate that
temperature, as well as emissions and fuel economy, may
respond in time as voltage on a capacitor responds in time
to periods of charge and discharge. The accumulation of
electric charge in the capacitor is analogous to the accumu-
lation of heat in the vehicle when it is in the "run" condition.
The discharge of the capacitor is analogous to the cooling
phase when the vehicle is in the "soak" condition.
Consider an electric circuit represented by Figure II-l
below:
R1
-/WW
—©
V
R-
in
1 Vout
SC
FIGURE II-l
In this figure, R2 >;> R1 an<3 niost of the current passes
through Rt when switch is closed. This is equivalent
to the "on cycle when the car is started. The rate at which
the capacitor is charged (or the car heats up) is a function
h II-l
-------
of the time constant,
C.
R2 represents a condition in the circuit allowing some leakage
from the capacitor during charging, and is analogous to the
cooling effect of the ambient conditions which act in opposition
to the accumulation of heat when the vehicle is started. For
hot ambient, R2 is much larger than and the time constant
is approximately R^C. For cold ambient, R2 is of smaller
magnitude than for hot ambient and can provide an appreciable
leakage for the capacitor. In short, a longer time would be
required for the capacitor to charge, just as a longer time
would be required for the vehicle to heat up. When switch
is opened, the battery and R^ are eliminated from the circuit
and the capacitor is discharged through the resistance R2
(the engine is cooled by heat transfer to ambient with a time
constant R2C).
When charging the capacitor the voltage time response is
However, when discharging the capacitor, the voltage response is
where R is
where
Vq is the voltage attained during charging for a time of
t = a. Thus
for t > a
A II-2
-------
If a is "long enough" to approximately charge the capacitor,
then the magnitude of Vout(t) is approximately
0-t/R2C
By means of Laplace transform, methods, Vout(S) may be
described in terms of ^n(S) the frequency domain (S-space)
by the relation:
V . (S)
out
and
vin:> R1 ^°r char9in9
Vcut
-------
The output may be represented by Figure II-3 which follows:
Time
FIGURE II-3
V (S) _2
The transform of - °.ur.—_ is represented bv L
V. (S)
xn
and
V
(sf
out
V.
(S)
m
V ^ - e
r / -a/Rc\ Q-b/» C, -C/RC\ "I
[(i- e J e 2 (i- e J J
v.
times
e"t/v
for t > c
b < t < c
This treatment of a capacitive model in frequency space
is easily generalized to include inputs which may be character-
ized by functions other then step functions.
A 11-4
-------
APPENDIX III
TIME-CONSTANT CONSIDERATIONS FOR A CAPACITIVE MODEL
OF EMISSIONS AND FUEL ECONOMY
For the capacitive model to be applicable to emission and
fuel economy of automobiles, it is necessary to be able to
evaluate the time constants corresponding to the heating and
cooling cycles of a vehicle. These cycles correspond to vehicle
on (run time) and vehicle off (soak time), respectively.
Data relevant to time-constant evaluation are afforded
by R. L. Srubar in a report on soak-time and run-time effects.*
Five vehicles were tested for emissions and fuel economy after
various soak times. The only way in which the effects of run
time were monitored, however, was by bagging emissions for
various periods of "time into the driving cycle." Though the
bagged results do, indeed, reflect warm-up effects, these
effects are confounded with changes in the driving cycle from
one bag to another. Accordingly, an estimation of time constants
must be by an indirect process, but this fact does not preclude
the possibility of designing an experiment specifically for the
purpose of time constant estimation.
Consider the data tabulated in Table III-l for a 1977
Ford LTD. Each of the bags corresponds to a 63-second period
of operation, but the distances covered, average speeds and
driving sequence severity are different for the various bags.
If it is assumed that the 10-minute soak represents essentially
stabilized conditions (actually it probably exhibits a short
transient), then one can compute for each bag a "correction
factor" reflecting only the effects of bag differences in
driving sequence.
The average fuel consumption for the 505-second run is
15.35 £/100 km. By dividing the fuel consumption in the first
bag by this number one obtains 24.99/15.34 = 1.63. Thus one
can assume that the first 63 seconds of operation is 1.63
R. L. Srubar, Emission and Fuel Economy Sensitivity to
Changesin Light Duty Vehicle Test Procedures, Final Report
of Task No. 10, Contract 68-03-2196, Southwest Research
Institute (May 1977).
A, IT1-1
-------
TABLE III-l
FUEL CONSUMPTION FOR A 1977 FORD LTD AS A FUNCTION OF RUN TIME
Bag
NO.
Time
(sec.)
Distance
km.
Fuel Consumption
10-minute
Soak* (A)
U/100 km)
16-hour
Soak** (B)
Ratio B/A
1
0-63
0.35
24.99 (1.63)***
59.22
2.37
2
64-126
0.72
11.90 (0.78)
16.37
1.38
3
127-189
0.24
32.31 (2.11)
37.63
1.16
4
190-252
1.37
15.39 (1.00)
19.22
1.25
5
253-315
1.40
9.20 (0.60)
11.14
1.21
6
316-378
0.55
20.00 (1.30)
19.87
0.99
7
379-441
0.42
19.02 (1.24)
20.59
1.08
8
442-505
0.73
14.60 (0.95)
16.18
1.11
Combined
5.78
15.38
*
**
***
Average of two tests
Average of three tests
Numbers in parentheses are fuel consumptions divided by overall fuel
consumption of 15.34 Jl/100 km
-------
times "as severe" as the composite 505 seconds of operation.
One can now adjust the fuel consumption figures for the 16-hour
soak by dividing each of the tabulated fuel consumption values
by the corresponding values in parentheses. These results are
the same as are obtained by dividing the 16-hour soak results
by the 10-minute soak results on a bag-by-bag basis.
As shown in Figure III-l, the ratios fall off rapidly with
increasing time into cycle and tend to approach as asymtote of
unity. It appears that about 6 minutes of running time from
a cold start is sufficient to reduce fuel consumption to
within about 10% of the quasi-stabilized values represented
by the hot start condition.
Whereas the duration of the warm-up cycle is of the
order of minutes, the duration of the cool-down cycle is of
the order of several hours. This effect can be. seen in
Table III-2, in which average fuel consumption rates are
tabulated for the 1977 Ford LTD as determined for results
bagged over the first 505 seconds of the FTP driving cycle.
TABLE III-
-2
FUEL
CONSUMPTION FOR A
1977
FORD LTD
AS A FUNCTION OF !
SOAK
TIME
Soak
length
Fuel Consumption
Jl/100 km
Ratio
10
min
14.72
1.00
20
min
15. 58
1.04
30
min
15.16
1.03
1
hr
15. 39
1.05
2
hr
15.85
1.08
4
hr
16. 62
1.13
8
hr
18.97
1.29
16
hr
19.92
1.35
36
hr
18.89
1.28
A HI-3
-------
Taking the 10-mirmte soak time as reference, one sees that it
takes approximately 2 hours of cooling to degrade fuel economy
to the extent that about 10% more fuel is consumed over the
505-second cycle. This time is compared wihtout about 6 minutes
in the warm-up cycle.
The "time constants" estimated by the above procedures
can not be interpreted in the strict exponential sense because
of the mechanisms involved in translating temperature into
fuel economy (or emissions). In reality, fuel economy is a
composite function
F.E. = g |"f (t , t ,)~|
3 [_ run soak J .
Although f (trun, tSoak) can be regarded as temperature and
although temperature may vary exponentially as the run and
soak times trun and tsoa^, fuel economy may take a somewhat
different form from exponential. Similar comments can be made
for HC, CO and NOx emissions. Nevertheless, the prospect
of a "heat budget" or time-history approach to emissions as a
function of soak and run times is attractive from the standpoint
of physical understanding of the process.
A II1-4
-------
o
•H
¦p
-------
APPENDIX IV
WEIGHTING FACTORS FOR ADJUSTING EMISSIONS
FOR VEHICLE THERMAL OPERATING HISTORY
An approach is proposed for adjusting emissions according
to the thermal operating history of a vehicle, as expressed
by its profile of soak and run times. The approach assumes
a "floor" value representing the lowest level of emissions
which might ever be expected and a "ceiling" value representing
the highest level anticipated. It is then postulated that
emission levels for all combinations of soak and run times can
be expressed as weighted combinations of these two extreme
values.
Reasonable definitions of the floor and ceiling values
might be based on the hot stabilized (HST) and cold transient
(CTR) values of the FTP respectively. Then one can assume
that
E,
P E + (1-P) E
CTR v ' HST
(IV-1)
or
E.
E
HST
E - E
CTR HST
(IV-2)
E^. denotes emissions during the first 505 seconds of
where
the FTP after a soak time
t.
Values of P computed by equation (IV-2) are given in
Tables IV-1 through IV-5. The data are taken from Srubar
(op. cit.) and represent five vehicles which were subjected
to tests after various soak times. One notes that the 10
minute soak times exhibit P values which are generally of
the order of 10% to 20%. Since the 10-minute soak times
represent conditions which are essentially hot transient,
the results suggest that this condition is only slightly
worse than stabilized operation but is "slightly contaminated"
with cold-start behavior.
A IV-1
-------
A word of caution is in order concerning literal inter-
pretation of the results in the tables, however. The first
505 seconds of the FTP and the next 870 seconds represent
different driving sequences, different average speeds, and
hence different levels of severity as far as both emissions
and fuel economy are concerned. It is for this reason that
the fuel-economy calculations show a rather severe anomaly
of negative P values for short soak times. Clearly an
improved basis for the calculation of weighting factors would
be obtained if either (1) driving cycles for the end-points
were the same, or (2) adjustments were made for speed
differences or differences in severity for the two driving
sequences. In that sense the results presented here are only
suggestive of further refinements of approach.
A IV-2
-------
TABLE IV-1
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 197 6 Chevrolet Impala
Over the First 505 seconds of the FTP
(Data from Table A-2, Srubar, op. cit.)
Soak
Time
HC
(gms/mi)
PHC
(%)
CO
(gms/mi)
pco
(%}
Fuel
(gal/mi)
PF.E.
10 min
0.53
22.98
7.13
16.14
13.47
-128.57
20 min
0.61
27.95
6. 08
12.58
13.20
-101.02
30 min
0.88
44.70
6.87
15.26
13.08
-88.77
1 hr
1.09
57.76
7.95
18.92
12.61
-40.82
2 hr
1.38
75.78
10.10
26.21
12.14
7.14
4. hr
1.34
73.29
14.18
40.04
11. 88
33.67
3 hr
1.27
68.94
16.52
47.93
11.28
9 4.89
16 hr
1. 77
100.00
31.86
100.00
11.23
100.00
3 6 hr
1.56
86.96
28.70
f
89.28 | 11.06
117.3
Hot Stabilized
0.16
2.37
12.21
[
-------
TABLE IV- 2
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 1977 Ford LTD
Over the First 505 seconds of the FTP
(Data from Table B-2, Srubar, 0£. cit.)
Soak
Time
HC
(gms/mi)
PHC
(%)
CO
(gms/mi)
pco
(%)
Fuel
(gal/mi)
PF.E.
10 min
0.39
10.20
3.91
12.53
15.98
-66.89
20 min
0.72
43.88
2. 51
7.72
15.10
-28.51
30 min
0.93
65.31
3.67
11.71
15.52
-44.92
1 hr
1.03
75.51
3.38
10.71
15.29
-35.94
2 hr
1.08
80.61
3.56
11.33
14.84
-18.36
4 hr
0.78
50.00
3.96
12.70
14.15
8.59
8 hr
1.23
95.92
12.48
41.95
12.40
76.95
16 hr
1.27
100.00
29.39
100.00
11.81
100.00
36 hr
2.06
180.60
37. 63
128.30
12.46
74.61
¦
Hot Stabilized
0.29
| 0.26
14.37
-------
TABLE IV- 3
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 197 6 Plymouth Fury
Over the First 505 seconds of the FTP
(Data from Table C-2, Srubar, ojd. cit.)
Soak
HC
p
HC
CO
pco
Fuel
^F E
Time
(gms/mi)
(%)
(gms/mi)
(%)
(gal/mi)
r • •
10 min
0.52
21.56
1.71
5.53
16.87
-110.00
20 min
0.70
32.33
1.88
6.25
15.82
-56.02
30 min
0.72
33.53
2.70
9.68
15.43
-35.60
1 hr
0.87
42.51
3.18
11.70
15.42
-35.08
2 hr
1.40
74.25
12.91
52.49
14.67
4.19
4 hr
1.42
75.45
16.95
69.43
13.88
45. 55
8 hr
1.64
88.62
20.84
85. 57
13.18
82.20
16 hr
1.83
100.00
24.24
100.00
12.84
100.00
36 hr
1.89
103.60
28.37
117.30
12.18
134.50
Hot Stabilized
0.16
0.39
14.75
-------
TABLE IV- 4
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 1976 Chevrolet Vega
Over the First 505 seconds of the FTP
{Data from Table D-2, Srubar, oj>. cit.)
Soak
Time
HC
(gms/mi)
PHC
(%)
CO
(gms/mi)
pco
(%)
Fuel
(gal/mi)
PF.E.
10 min
0.39
26. 60
1. 01
-0.90
22.71
-8.67
20 min
0.48
34.86
2.09
7.22
22.84
-11.99
30 rain
0.53
39.45
1.89
5.71
23.20
-21.17
1 hr
0.69
54.13
2.24
8.34
22.19
4.59
2 hr
0. 64
49.54
3. 68
19.17
20.97
35.71
4 hr
0. 85
68.81 5.36
31.80
19.70
68.11
8 hr
1.03
85.32 14.63
101.50
19.41
75. 51
16 hr
1.19
100.00 14.43
100.00
18.45
100.00
36 hr
1.27
107.3 j 19.90
141.13
18.45
100.00
Hot Stabilized
0.10
1.13
22.37
-1
-------
TABLE IV-5
Percent Cold Start Operation as a Function of Soak Time
for HC, CO and Fuel Economy for a 1976 Honda Civic CVCC
Over the First 505 seconds of the FTP
(Data from Table E-2, Srubar, 0£. cit.)
Soak
Time
HC
(gms/mi)
?HC
<%)
CO
(gms/mi)
pco
(%)
Fuel
(gal/mi)
?F.E.
10 min
0.85
16.86
4. 83
3.15
31.53
-138.20
20 min
0.97
23.84
4.39
-10.72
31.08
¦ -115.60
30 min
1.00
25.58
4.17
-17.66
31.58
-140.70
1 hr
0.80
13.95
4.38
-11.04
30.75
-99.00
2 hr
1.00
25. 58
4.89
5. 05
28.72
3.01
4 hr
1.24
39.53
6.32
50.16
27. 84
47.24
8 hr 2.03
85.46
7.82
97.48
25.03
188.40
16 hr | 2.28
. ... i
100.00
7.90
100.00
26.79
100.00
36 hr
2.65
121.50
9.48
149.80
24.13
233.60
Hot Stabilized
" • ' ¦ -
0.56
1
• 4.73
28.78
------- |