TEMPERATURE CORRECTION FORMULAS
FOR ADJUSTING ESTIMATES OF AUTOMOBILE
FUEL CONSUMPTION
by
Norman Morse
Report 3520-1/BUF-35
May 1980
RESEARCH AND
DEVELOPMENT
-------
-------
TABLE OF CONTENTS
Section
1.0
2.0
Title
3.0
4.0
Appendix
Introduction
Description of the Study
2.1 Data Base
2.2 Model-Year/Standard Groups of Vehicles
2.3 Analytical Method
2.4 Temperature Correction Coefficients Obtained
Graphical and Tabular Results
Limitations of Present Analysis and Recommendations
for Further Study
Applying Constrained Least Squares to the
Y-vs-T Data
1
2
2
3
3
5
7
38
40
-------
1.0 INTRODUCTION
This report describes an analysis of test data leading to formulas
reflecting temperature effects on automobile fuel consumption. The
analysis was conducted by Falcon Research and Development Company as a
task under Contract No. 68-03-2835 for the Environmental Protection
Agency. The purpose of the task was to provide factors which, when
used to multiply fuel consumption estimates for vehicle operation at
standard FTP temperatures, would yield "corrected" estimates of fuel
consumption for operation outside the FTP ambient temperature range.
The report is divided into three additional sections, supported by
an appendix. Section 2 summarizes the work and the outputs of the study.
Section 3 presents the derived correction formulas in both graphical and
tabular forms. In the graphs, the plotted formulas are superimposed on
scatter diagrams of the input data. The output tables are series of
temperature correction factors calculated at 'intervals of 5°F. In the
final section some observations are offered on the results of the study
and on the analytical method employed.
-------
2.0 DESCRIPTION OF THE STUDY
This study was performed according to a task order generated at
EPA in which the analysis method was prescribed. The method itself is
one which had previously been used by Parrel I1 in deriving temperature
correction formulas for automobile emissions. The important features
of the study are as follows.
2.1 Data Base
The data base for the study was the same emissions vs. temperature
data base2 used by Farrell in deriving the temperature correction
formulas for emissions. Data from several test series are contained
in this data base:
Bureau of Mines, 1974 (BM1)
Bureau of Mines, 1975 (BM2)
Bureau of Mines, 1977 (BM3)
Canadian Data, 1975 (CA1)
Canadian Data, 1978 (CA2)
Gulf Data, 1977-79 (GUI)
Chicago Cold Test, 1978 (CHI)
The references contain full descriptions of these test series. In total,
after appropriate editing, the input contained data from 143 vehicles.
There were 854 individual test points pertinent to this analysis.
Each test point consisted of a run of a test vehicle through the
FTP cycle at a given ambient temperature. The standard FTP terminology,
in which "bag number" refers to one of the three major regimes within
the driving cycle, is used herein. Of the total of 854 tests, 499 were
at temperatures below the FTP range, 315 were within the FTP range, and
90 were at higher temperatures. Each test yielded CO, HC, and C02
emissions for each bag number. This allowed composite fuel consumption
to be calculated by the carbon balance method, and by weighting Bag 1,
Bag 2, and Bag 3 fuel consumptions in the ratio 21:52:27. In some of
the older test series, the C02 values were not available, but composite
fuel economy was provided as an explicit input datum.
Robert L. Farrell, "Temperature Correction Formulae for Adjusting
Estimates of Emissions from Automobiles," Vector Research, Inc.,
Report No. VRI-EPA-6 (Draft), September 1979
6. Miller, and K. Wilkinson, "Data Base for the Development of
Improved temperature Correction Factors for Emissions," Vector
Research, Inc., Report No. VRI-EPA-5, July 27, 1979
-------
Considerable effort was expended checking the validity of individual
data points before performing the analysis. Because the input data varied
with respect to availability of fuel economy and/or C02 data, the
checking process differed from test series to test series. Where fuel
economy (mpg) data were provided as inputs, these were converted to fuel
consumption (gpm) data before analysis began.
2.2 Model-Year/Standard Groups of Vehicles
Fifteen groups or subpopulations of automobiles had been defined for
the previous work on temperature correction factors for emissions, and
the task work statement prescribed use of these same groups. These were
termed model-year/standard (MYST) groups, which were as follows:
MYST = 1 67 FED
2 69 FED
3 70 FED
4 71 FED
5 72 FED
6 73 FED
7 74 FED
8 75 CAL
9 75 FED
10 76 FED
11 77 CAL
12 77 FED
13 78 CAL
14 78 FED
15 80 FED
The work statement required that individual correction formulas and sets
of correction factors be derived for each MYST group.
2.3 Analytical Method
Each temperature datum was classified as being in the COLD range
(T 1 670F), in the HOT range (T;>87°F), or in the FTP range
(68°F l T 1 86°F). Each vehicle was tested at one or more COLD or HOT
temperatures and at one or more FTP temperatures. This allowed each
fuel consumption (FC) figure obtained in the COLD or HOT ranges to be
expressed as a ratio relative to FC at FTP temperatures for the same
vehicle.
-------
Explicitly, let C be an individual FC value for a_vehicle,
obtained at temperature T not in the FTP range. Let Cpjp be the
geometric mean of all the FC's obtained for the same vehicle at FTP
temperatures. Define
u
U will then provide an estimate of the ratio by which FTP FC estimates
must be multiplied in order to produce an FC estimate for temperature T.
Suppose T is in the HOT range. It is conjectured that, except
for random errors, the correction required is one at the boundary
between the FTP and HOT regions, and changes exponentially with distance
from that boundary. The input temperature data had been rounded to
the nearest integer. For analysis purposes, the boundary is given the
idealized location T = 86.5°. Thus the correction formula is of the
form
'U = exp [>(T - 86.5)3 (T = 86.5°)
where b is appropriately chosen. The constant b is estimated
by providing that value which "best fits" the HOT temperature data
from all vehicles in the given MYST group. To determine that value,
let Y = In U. Then
Y = b (T - 86.5) (T = 86.5°)
and note that Y = In 1 = 0 when T = 86.5°. If there are n HOT
data points in the given MYST group, then each is represented by a
pair of values (T-j , Y-j). The constant b can then be obtained by
the method of least squares.
The most common form of linear regression, which allows for a
Y-intercept to be estimated, is not appropriate in the present case.
Here, since the fitting equation is constrained to go through the point
(T,Y) = (86.5,0), only the slope b has to be estimated. Equations
suitable for this constrained regression analysis are given in the
Appendix.
For COLD temperatures, the same basic approach is followed, which
results only in the change of one or two details. The boundary between
COLD and FTP temperature ranges is idealized as T = 67.5° for analytical
purposes. The model becomes
U = exp [> (67.5 - TO (T = 67.5°)
-------
which goes into the form
Y - b (67.5 - T) (T < 67.5)
after taking natural logarithms. The fitting equation is then constrained
to go through the point (T,Y) = (67.5,0).
Where sufficient data were available, two fitting equations were
obtained for each MYST group: one for COLD and one for HOT temperatures.
In one case (75 CAL) insufficient data were available to allow the analysis
for HOT to be performed.
2.4 Temperature Correction Coefficients Obtained
Table 2.1 contains the constants b obtained for the COLD and HOT
ranges for the various MYST groups. These constants may be used to
"correct" FC estimates based on FTP temperatures for temperature effects
outside the normal range. Specifically, if b is the coefficient
obtained from the table, then
FC at temp. T r, ,,-. c TN-I
Jl = exp Lb (67.5 - T)J
FC at FTP *-
for COLD temperatures, and
lL at temP. I f i /-r rt/- r\ I
JL = exp |_b (T - 86.5)_|
for HOT temperatures.
-------
Table 2.1
COMPOSITE FUEL CONSUMPTION TEMPERATURE EFFECTS COEFFICIENTS
Model Year/Std.
67 FED
69 FED
70 FED
71 FED
72 FED
73 FED
74 FED
75 CAL
75 FED
76 FED
77 CAL
77 FED
78 CAL
78 FED
80 FED
Low Temperatures
.002037
.002682
.001697
.002261
.002555
.001775
.003021
.003203
.002941
.002310
.001521
.002608
.002600
.002982
.002958
High Temperatures
.000161
-.000048
-.002261
-.000933
-.000733
-.000305
-.000627
.000000
-.002192
.000000
.000304
-.000593
-.000483
.002810
-.002456
-------
3.0 GRAPHICAL AND TABULAR RESULTS
On the following pages, Tables 3.1 through 3.15 and Figures 3.1
through 3.15 give the results of the study in graphical and tabular
form. Each figure contains the input data, displayed by means of a
scatter diagram, for the COLD and HOT temperature correction
formulas for the given MYST group. The corresponding correction
equations are also depicted. Note that the vertical, or U axis,
has a logarithmic scale, so that the fitting equations appear as
straight lines.
Each figure also conveys percentages labelled STD ERROR. One is
given for each of the b coefficients, COLD and HOT. The figure
is the estimate of the standard error of b, s^, expressed as a per-
cent of b itself, i.e.,
STD ERROR = 100x(sb/b) %
The formula for s^ is given in the Appendix. Note that large per-
centages are cases with large variability in the estimated values of b,
and vice versa. Thus only where small percentages appear could the
estimated b be significantly different from zero. Explicit significance
tests were not performed because of concerns with the form of the distri-
bution of deviations from the regression lines.
Each of the Tables 3.1 through 3.15 contains the correction factors
obtained by substituting various values of T into the fitted equations.
-------
1 . 953 _,
G 7 F E D
1.0-
. o 0 Li -
0.5] 2
0 . 0 0
MIN TEMP = 20.0
M H :' T E M F = 110. 0
N (LOW TEMP) 6
N i.FTP TEMPJ 3
N iHIGHTEMP) 3
X
FTP fMHNC-E
40.0 0 G 0 . U 0 :3 i"i. n fi
TEMPERRTURE
100. 0 0 120. 00
EPFECT fLOW TEMP) =EXP CO. 002037 (67. 5-TL-i
EPFECT CHIGHTEMP) =EXP (0. 000161 t.TEMP-66.:
TD ERROR (LOW TEMP)
.TD ERROR I.HIGHTEMPJ
15.H5X
428. is
FIGURE 3.1
Temperature Effects on Fuel Consumption, Model-Year/Standard = 67 FED
8
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Table 3.1
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 67 FED
TEMP (F) CORRECTION FACTOR
0. 0 1. 1474
5. 0 1. 1358
10. 0 1. 1243
15. 0 1. 1129
20. 0 1. 1016
25. O 1. 0904
30. 0 1. 0794
35. 0 1. 0684
40. 0 1. 0576
45. 0 1. O469
50. 0 1. 0363
55. 0 1. 0258
60. 0 1. 0154
65. 0 1. 0051
70. 0 1. OOOO
75. 0 1. OOOO
BO. 0 1. OOOO
85. 0 1. OOOO
90. 0 1- 0006
95. 0 1- 0014
100. 0 1. 0022
105. 0 1. 003O
1 10. 0 1- 0038
-------
6TIFED
1.0-
. oUU -
0.512
0.00
20.00
X
X
FTP RflNGE
T
40.00 60.00 SO
TEMPERflTURE
00
100.00 120.00
HIM TEMP = 20.0
MH:< TEMP= 110.0
N (LOW TEMP) S
N (FTP TEMP) 4
N (HIGMTEMP) 4
EFFECT (LOW TEMP) =EXP (0.002682 (67.5-TEMP))
EFFECT (HIGHTEMP) =EXP (-0. 000048 (TEMP-86.5)
5TD ERROR(LOW TEMP)
STD ERROR (HIGHTEMP)
13.41X
251 1. 172
FIGURE 3.2 .
Temperature Effects on Fuel Consumption, Model-Year/Standard = 69 FED
10
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Table 3.2
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 69 FED
TEMP (F) CORRECTION FACTOR
0. 0 1. 1985
5. 0 1. 1825
10. 0 1. 1667
15. 0 1. 1512
20. 0 1. 1359
25. 0 1. 1207
30. 0 , 1. 105B
35.0 .1.0911
40. 0 1. 0765
45. 0 1. 0622
50. 0 1. 0481
55. 0 1. 0341
60. 0 1. 0203
65. 0 1. 0067
70. 0 1. 0000
75. 0 1. OOOO
80. 0 1. OOOO
85. 0 1. OOOO
90. 0 0. 9998
95. 0 0. 9996
100. 0 O. 9994
105. 0 0. 9991
1 10. 0 0. 9989
11
-------
70FED
1 .
1.0-
. .son
0.51
0.00
20. 00
HIN TEHP= 20.0
M H >: T E M F = 110.0
N i LOW TEMPI 4
N iFTF TEMF) 2
N(HIGHTEMP) 2
FTP F.ftNGE
140.00 60.00 80.00
TEMPERRTUP.E
100.00
120.00
EFFECT (LOW TEMP) = EXP (0. u"01697 (67. 5-TEMPJJ
EFFECT (HIGHTEMP) =EXP (-0. 002261 (TEKP-86.5J
STO ERROR (LOW TEMF) 21.18'X
STD ERROR (HIGHTEMP) 49.34X
FIGURE 3.3
Temperature Effects on Fuel Consumption, Model-Year/Standard = 70 FED
12
-------
Table 3.3
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 70 FED
TEMP (F) CORRECTION FACTOR
0. 0 1. 1214
5. 0 1. 1119
10. O 1. 1025
15. 0 1. 0932
20. 0 1. 0839
25. 0 1. 0748
30. 0 1. 0657
35. 0 1. 0567
40. 0 1. 0478
45. 0 1. 0389
50. 0 1. 0301
55. 0 1. 0214
60. 0 1. 0128
65. 0 1. 0043
70. 0 1. 0000
75. 0 1. 0000
SO. 0 1. 0000
85. 0 1. OOOO
90. 0 0. 9921
95. 0 O. 981O
100. 0 0. 9699
105. 0 0. 9590
1 10. 0 0. 9483
13
-------
11 f: t LI
1.95::
1 . 25 -
1.0-
. SOO -
0.512
X
FTP RANGE
0.00 20.00 40.00 60.00 80.00
TEMPERRTURE
100.00 120.00
MIN TEMF= 20.0
MHX TEMP = 110.0
N (LOW TEMP) 10
N IFTP TEMP) 5
N (HIGHTEMP) 5
EFFECT (LOW TEMP) =EXP (0 . 00226 1 (67.5-TEMP)
EFFECT (HIGHTEMP) =EXP (-0. 000933 (TEMP-86.5
STD ERROR (LOW TEMP) 1U.152
STD ERROR (HIGHTEMP) 89.87X
FIGURE 3.4
Temperature Effects on Fuel Consumption, Model-Year/Standard = 71 FED
14
-------
Table 3.4
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 71 FED
TEMP. (F) CORRECTION FACTOR
0- 0 1. 1649
50 1. 1518
1O- 0 1. 1388
15. O 1. 1260
20. 0 1. 1134
25. O 1. 1009
30. 0 1. 0885
35. 0 1. O762
40. 0 1. 0642
45. 0 1. O522
50. 0 1. 0404
55. 0 1. 0287
60. 0 1. 0171
65. 0 1. O057
70. 0 1. 0000
75. O 1. OOOO
80. 0 1. 0000
85. 0 1. OOOO
90. 0 0. 9967
95. O O. 9921
100. O 0. 9875
1O5. O O. 9829
110. 0 0. 9783
15
-------
t <-
2FED
1.0-
. o 0 u -
0 . 5 1 2
FTP F.HNGE
0.00 20.00 40.00 60.00 80.00
TEMPERRTURE
100.00 1
20.0fll
HIM TEMP = 0.0
Mfl:-: TEMP= 110.0
rULCH TEMP) 16
H (F T P TEMP) 12
N fMIOHTEMP) 5
EFFECT (LOW TEMP) =EXP (0. 002555 (67. 5-TEMPJ
EFFECT (HIGHTEMP) =EXP (-0. 000733 (TEMP-86.^
STD ERROR (LOW TEMP) 12.52X
STD ERROR (HIGHTEMP) 105. 12X
FIGURE 3.5
Temperature Effects on Fuel Consumption, Model-Year/Standard = 72 FED
16
-------
Table 3.5
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 72 FED
TEMP (F) CORRECTION FACTOR
0. 0 1. 1882
5. 0 1. 1731
10. 0 1. 1583
15. 0 1. 143
-------
7'JFED
1.9'
1 . 0
300
0.51 2
o. o u
x
x
FTP F;HNGE
10.00 40.00 60.00 30.00
TEMFERRTLIRE
100.00 120.00
MIN TEMP= £0.0
M H :: T E M P = 110. 0
N i.LOH TEMP) 12
H i.FTP TEMP) 12
N fHlGHTEMP) 4
EFFECT (LOH TEMP) =EXP (0.001775 167.5-TEMP) ) (
EFFECT (HIGHTEMP) =EXP (-0. 000305 (TEMP-36. 5))'
STO ERROR (LOH TEMP) 30.307.
STD ERROR (HIGHTEMP) 281.43X
FIGURE 3.6
Temperature Effects on Fuel Consumption, Model-Year/Standard = 73 FED
18
-------
Table 3.6
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 73 FED
TEMP (F) CORRECTION FACTOR
0. 0 1. 1273
5. 0 1. 1173
10. 0 1. 1075
15. 0 1. 0977
20. 0 1. OB80
25. 0 1. 0784
30. 0 1. 0688
35. 0 1. 0594
40. 0 1. 0500
45. 0 1. 0407
50. 0 1. 0315
55. 0 1. 0224
60. 0 1. 0134
65. 0 1. O044
70. 0 1. 0000 '
75. O 1. OOOO
80. 0 1. 0000
85. 0 1. OOOO
90. 0 0. 9989
95. 0 0. 9974
100. 0 0. 9959
105. 0 0. 9944
1 10. 0 0. 9929
19
-------
74FED
1.25-
1.0-
. oDO -
0.51 2
0. 00
FTP RftNGE
20.00 40.00 60.00 80.00
TEMPERRTURE
X
X
X
100.00 120.00
HIM TEMP= 0.0
MHX TEMF= 110.0
N i LOW TEMP) 22
N IFTP TEMP) 20
N f H I C- H T E M P j 12
EFFECT (LOU TEMP) =EXP (0 . 00302 1 (67.5-TEMP)
EFFECT fHIGHTEMP) =EXP (-0. 000627 (TEMP-86.5
STD ERROR (LOW TEMPJ
STD ERROR (HIGHTEMP)
133. 78X
FIGURE 3.7
Temperature Effects on Fuel Consumption, Model-Year/Standard = 74 FED
20
-------
Teible 3.7
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 74 FED
TEMP (F) CORRECTION FACTOR
0. O 1. 2262
5. 0 1. 2078
10. 0 1. 1897
15. 0 1. 1719
20. O 1. 1543
25. O 1. 1370
30. 0 1. 1200
35. O 1. 1032
40. 0 1. 0866
45. 0 1. 0703
50. 0 1. 0543
55. 0 1. 0385
60. 0 1. 0229
65. O 1. 0076
70. 0 1. 0000
75. 0 1. 0000
80. 0 1. 0000
85. O 1. OOOO
90. 0 0. 9978
95. O 0. 9947
100. 0 0. 9916
105. 0 0. 9885
1 10. 0 0. 9854
21
-------
7f.CHL
1 . 9 5 3 .,
1 . 25 -
1 . 0
. 800 -
0.512
X
X
X
XX
X
X
XX
X
FTP RANGE
0.00 20.00 40.00 60.00 30.00
TEMPERRTURE
100.00 120.00
MIN TEMP= -22.0
MflX TEMP= 89.0
n (LOU TEMP) 93
N (FTP TEMP) 53
N (HIGHTEMP) 1
EFFECT (LOW TEMP) =EXP (0. 003203 (67. 5-TEMP)!
STD ERROR (LOW TEMP) 7.912
* The data set on which the fitting equation was based contained an additional
15 points at temperatures below 0 F, and one point in the FTP range which fell
below the lower boundary of the graph.
FIGURE 3.8
Temperature Effects on Fuel Consumption, Model-Year/Standard = 75 CAL
22
-------
Table 3.8
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 75 CAL
TEMP
0.
5.
10.
15.
20.
25.
30.
35.
40.
45.
50.
55.
60.
65.
70.
75.
80.
85.
90.
95.
100.
105.
110.
(F)
0
O
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
O
0
0
O
0
0
CORRECTI
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
ON FA(
.2414
. 2216
. 2022
. 1831
. 1643
. 1458
. 1276
. 1097
. 0921
. 0747
. 0577
. 0408
. 0243
. OOSO
. OOOO
. OOOO
. OOOO
. OOOO
. OOOO
. OOOO
. OOOO
. OOOO
. OOOO
23
-------
75FED
1 .953 -,
1.25-
1.0-
. 800 -
0.512
X
X
xxxx
FTP
1 1 1 1
0.00 20.00 40.00 60.00 80.00
TEMPERRTURE
100.00 120.00
HIM TEMF= -S.5
HflX TEHP= 100.0
\i (LOU TEMP) 71
U fFTP TEMP) 81
II (HIGHTEMF) 1 1
EFFECT (LOW TEMP) =EXP (0. 002941 (67. 5-TEMPJ
EFFECT (HI GHTEMP) =EXP (-0.002192 (TEMP-36.'
STD EFiPiOR (LOW TEMP) 6.63X
STD ERROR (HIGHTEMP) 25.777.
* The data set on which the fitting equation was based contained an additional
10 points at temperatures below 0°F.
FIGURE 3.9
Temperature Effects on Fuel Consumption, Model-Year/Standard = 75 FED
24
-------
Table 3.9
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 75 FED
TEMP
0.
5.
10.
15.
20.
25
30.
35.
40.
45.
50.
55.
60.
65.
70.
75.
80.
85.
90.
95.
100.
105.
110.
(F)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
CORRECTION FA<
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
0.
0.
0.
0.
0.
2196
2018
1842
1670
1499
1331
1166
1003
0842
0684
0528
O374
0223
0074
OOOO
0000
0000
0000
9924
9815
9708
9603
9498
25
-------
71... FED
3 -> -< -I
1 . Hi
. 3 0 0 -
0.51
0. 00
X
FTP FiflNGE
10.00 40.00 60.00 80.00
TEMPERH"lUKt~
100.00 120.00
MIN TEMF = 22.0
MRX TEMP= 78.7
N(LOW TEMP) 19
N (FTP TEMP) 19
N (H I G H T E M P) 0
EFFECT (LOW TEMP) = EXP (0.002310 (67.5-TEMP))
EFFECT (HIGHTEMP) =EXP (0.000000 (TEMP-86.5))
'TO ERROR (LOU TEMP)
13.022
FIGURE 3.10
Temperature Effects on Fuel Consumption, Model-Year/Standard
= 76 FED
26
-------
Table 3.10
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 76 FED
TEMP.
0.
5.
10.
15.
20.
25.
30.
35.
40.
45.
50.
55.
60.
65.
70.
75.
80.
85.
90.
95.
100.
105.
110.
(F)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
CORRECTION FACTOR
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1687
1553
1421
1289
1160
1032
0905
0780
0656
0533
0413
0293
0175
0058
0000
OOOO
0000
OOOO
0000
OOOO
OOOO
OOOO
OOOO
27
-------
77CRL
1 . 953-,
1.0-
. SOD -
0.512
0. 00
X
X
FTP RANGE
20.00 40.00 60.00 80.00
TEMPERRTURE
100.00 120.00
HIM TEMP = 0.0
HPU TEMF= 110.0
N iLOH TEMP) 6
N fFTP TEMP) 4
N (HIGHTEMP) 4
EFFECT (LOU TEMP) =EXP (0. 001521 (67. 5-TEMP)
EFFECT (HIGHTEMP) =EXP (0. 000304 (TEMP-86.5)
STD ERROR (LOW TEMP) 8.582
STD ERROR (HIGHTEMP) 120. IS'/.
FIGURE 3.11
Temperature Effects on Fuel Consumption, Model-Year/Standard = 77 CAL
28
-------
Table 3.11
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 77 CAL
TEMP. (F) CORRECTION FACTOR
0. 0 1. 1081
5. 0 1 O997
10. O 1. 0914
15. 0 1. O831
20. 0 x 1. 0749
25. 0 1. 0668
30. 0 1. 0587
35. 0 1. 0507
40. 0 1. 0427
45. 0 1. 0348
50. 0 1. O27O
55. 0 1. 0192
60. 0 1. O115
65. 0 1. O038
70. 0 1. 0000
75. O 1. 0000
80. 0 1. 0000
85. 0 1. 0000
9O. O 1. 0011
95. O 1. O026
100. O 1. OO41
105. 0 1- OO56
110. 0 1. 0072
29
-------
77FED
X
X
1.0-
. 300 -
0.512
H
0. 00
X
FTP RflNGE
20.00 140.00 60.00 80.00
TEMPERRTURE
100.00
120.00
MIN TEHP= 0.0
MFT-: TEHF= 110.0
N (LOW TEMP)
N iFTP TEMP)
N IHIGHTEMP)
65
48
12
EFFECT (LOW TEMP) =EXP (0. 002608 (67. 5-TEMP))
EFFECT (HIGHTEMP) =EXP (-0.000593 (TEMP-S6.5)
STD ERROR (LOW TEMP) 8.1422
STD ERROR (HIGHTEMP) 110.57X
FIGURE 3.12
Temperature Effects on Fuel Consumption, Model-Year/Standard = 77 FED
30
-------
Table 3.12
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 77 FED
TEMP. (F) CORRECTION FACTOR
0. 0 1. 1925
5. O 1. 1770
10. 0 1. 1618
15. 0 1. 1467
20. 0 1. 1319
25. 0 1. 1172
30. 0 1. 1027
35. O 1. O885
40. 0 ,1. 0744
45. 0 1. 0604
50. 0 1. 0467
55. 0 1. 0331
60. 0 1. 0198
65. 0 1. 0065
70. 0 1. 0000
75. 0 1. 0000
80. 0 1. 0000
85. 0 1. OOOO
90. 0 O. 9979
95. 0 0. 995O
100. 0 0. 9920
1O5. 0 0. 9891
110. 0 O. 9862
31
-------
76CRL
1 . 9!
1 . 25 :
1.0-
. sou -
0 . 5 1 2
X
X
X
X
0
-------
Table 3.13
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 78 CAL
TEMP (F) CORRECTION FACTOR
0. 0 1. 1918
5. 0 1. 1764
10. 0 1. 1613
15. 0 1. 1463
20. 0 1. 1314
25. 0 1'. 1168
30. 0 1. 1024
35. 0 1. O882
40. O 1. 0741
45. 0 1. 0602
50. 0 1. 0466
55. 0 1. 033O
6O. 0 1. 0197
65. 0 1. O065
70. 0 1. 0000
75. 0 1. 0000
80. 0 1. 0000
85. 0 1. OOOO
90. 0 0. 9983
95. O 0. 9959
100. 0 0. 9935
105. 0 O. 9911
1 1O. 0 0. 9887
33
-------
78FED
1 . 25 -
1.0-
. 800 .
0.512
X
>, X
-
x
X
X
X
X
X
X
X
X
X
FTP
0.00 20.00 140.00 60.00 30.00
TENFERRTURE
100.00 120.00
MIN TEMP= 0.0
MR.X TEMF= 110.0
N (LOU TEMP) 41
N (FTP TEMP) 1 1
N (HIGHTEMP) 4
EFFECT (LOW TEMP) =EXF (0.002932 (67.5-TEMFJ
EFFECT (HIGHTEMP) =EXP (0.002310 (TEMP-S6.5)
STD ERROR(LOW TEMP) 13.832
STD ERROR (HIGHTEMP) 75.55X '
FIGURE 3.14
Temperature Effects on Fuel Consumption, Model-Year/Standard = 78 FED
34 '
-------
Table 3.14
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 78 FED
TEMP. CF) CORRECTION FACTOR
0.0 1. 2230
?. 0 1. 2049
10. 0 1. 1870
15. 0 1. 1695
20. 0 1. 1522
25. 0 1. 1351
30. 0 1. 1183
35. 0 1. 1018
40. 0 1. 0855
45. 0 1. 0694
50. 0 1. O536
55. 0 1. O3BO
60. 0 1. 0226
65. 0 1. O075
70. 0 1. 0000
75. 0 1. OOOO
80. 0 1. 0000
85. 0 1. OOOO
90. 0 1. 0099
95. 0 1. 0242
100. 0 1. 0387
105. 0 1. 0534
1 10. 0 ' 1. O683
35
-------
80FED
1 .953
1.0,
. 800
0. 5
0. LI 0
X
FTP' FiflNGE
!0.00 40.00 60.00 60.00
TEMFERfiTURE
100-00
120.00
MIN TEMP= 0.0
MHX TEMF= 110.0
N (L 0 M TEMP) 17
N iFTF TEMP) 12
fMHI OHTEMP) 10
EFFECT (LOW TEMP) =EXP (0.002958 (67.5-TEMP))
EFFECT (HIGHTEMF) = EXP (-0.002456 (TEMP-86.5) )
STD ERROR (LOW TEMP) 12.617.
STD ERROR (HIGHTEMP) 27.1 IX
FIGURE 3.15
Temperature Effects on Fuel Consumption, Model-Year/Standard = 80 FED
36
-------
Table 3.15
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 80 FED
TEMP (F) CORRECTION FACTOR
0. 0 1. 2210
5. 0 1. 2031
10. 0 1. 1854
15. 0 1. 168O
20. 0 1. 1509
25. 0 1. 1340
30. O 1. 1173
35. 0 1. 1009
40. 0 1. 0847
45. O 1. 0688
50. 0 1. 0531
55. 0 1. 0377
60. 0 1. 0224
65. O 1. 0074
70. 0 1. 0000
75. 0 1. OOOO
80. 0 1. 0000
85. 0 1. 0000
90. 0 O. 9914
95. O O. 9793
100. O 0. 9674
105. 0 O. 9556
110. 0 0. 9439
37
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4.0 LIMITATIONS OF PRESENT ANALYSIS AND
RECOMMENDATIONS FOR FURTHER STUDY
The results of this report were produced by a "first-step" analysis
which is subject to a number of refinements. It is appropriate to comment
on the limitations of the analysis in its current form, and on related
questions that are subject to further investigation.
Users of the correction factors should be aware that the derived
equations are only appropriate for vehicle operations reproducing the
FTP cycle, and not other mixes of operating regimes. It might be
useful to derive sets of correction factors for the individual bag
numbers, which could then be combined appropriately for given operating
cycles. Using the existing data base and little additional effort,
temperature correction factors for fuel consumption could be produced for
individual bag numbers.
j
The assumptions of additive errors whose variance is constant with
respect to the independent variable (temperature) are implicit in
applying linear regression methods to the transformed variables. With
respect to the untransformed data (the U's) these assumptions imply a
model with multiplicative errors with constant variance. There is
implied a tendency for sampling errors to be proportional to U, and
thus to increase with distance from the FTP range. This is an aspect of
adoption of the exponential model used on this study which should be
considered in judging its validity.
Users should also be aware that the temperature "effect" implied
by the derived fitting equation might, in some cases, be estimated with
wide variability due to large sampling errors. Some of the estimates
have been based on extremely small samples. The given standard error
values should be used as a guide to detect cases where the fitted
equation should be used with caution.
Furthermore, note that the constrained regression method can yield
an estimate of the slope even where only one value of the independent
variable is represented. This is the case for two of the HOT tempera-
ture analyses: only one temperature was represented. The result of
any such analysis depends even more heavily than usual on the linearity
assumption, since the sample itself provides no information with which
to check the form of the assumed relationship. Depending on the number
of observations concentrated at that single value, it may even happen
that the standard error is small in such cases.
38
-------
Examination of the coefficients and correction factors could lead
one to question whether there is a HOT temperature effect at all, or
whether the coefficients obtained for the different MYST groups are
in fact randomly distributed estimates of the same zero coefficients.
Similarly, although there appear to be significant COLD temperature
effects, it is questionable that there are fifteen individual effects
rather than some smaller set, or indeed a single one. These questions
suggest areas for further investigation. In a further study, it would
be useful to consider whether or not there are, in fact, significant
HOT temperature effects, and whether or not there is a smaller set of
COLD temperature correction equations applicable over broader classes
of vehicles.
Finally, some users may object to the fact that the correction
relationship for a given MYST group, viewed as a function over the
entire temperature range, has discontinuous slopes at T = 67.5 and
T = 86.5. It would be easy to "smooth" the function in the neighbor-
hoods of these values by appropriate weighting of the adjacent relation-
ships. This might also be considered in further investigations.
39
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APPENDIX
APPLYING CONSTRAINED LEAST SQUARES
TO THE Y-vs-T DATA
For HOT temperature cases, the fitting equation is of the
form
Y = b(T - 86.5)
Let X = T - 86.5. Then Y = bX. Thus the relationship is constrained
to go through the origin, just as the Y-vs-T relationship was constrained
to go through (86.5, 0). For COLD temperature cases, the fitting equation
is of the form
Y = b(67.5 - T)
Letting X = 67.5 - T, one again obtains the constrained relationship
Y = BX.
Except for the constraint, the assumptions and the approach are as
with ordinary (i.e., including a Y-intercept) linear regression. The
model is
Y. = bX. + e. (i = 1, ...., n)
i 11
where E(e.j) = 0, Var(e.j) = a2 for all i, and where the e-j's are
independent of one another. There is only one "normal equation" in this
case, namely
ZX.Y. = blX2.
i i i
ys.
Thus the fitting equation is Y = bX, where
b = ZX.Y./IX2
An estimate of a2 is given by
40
-------
It can be shown that E(a2) = a2. It can also be shown that
a2; = Var(b) = a2/ZX*
a. = a/ /ZX?
b i
Thus an estimate of a, is provided by
= a/ ZX?
It is clear from the above that E(s2) = a2
D D
41
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