TEMPERATURE CORRECTION FORMULAS
FOR ADJUSTING ESTIMATES OF AUTOMOBILE
FUEL CONSUMPTION
by
Norman Morse
Report 3520-1/BUF-35
May 1980
RESEARCH AND
DEVELOPMENT
-------
FALCON RESEARCH
Falcon Research & [Development Co.
A Subsidiary of Whittaker Corporation
One American Drive
Buffalo, New York 14225
716/632-4932
WhfttakgR
TEMPERATURE CORRECTION FORMULAS
FOR ADJUSTING ESTIMATES OF AUTOMOBILE
FUEL CONSUMPTION
by
Norman Morse
Report 3520-1/BUF-35
May 1980
Prepared under Contract 68-03-2835
Task Order 1
(Final Report)
Prepared for
ENVIRONMENTAL PROTECTION AGENCY
Ann Arbor, MI 48105
Approved by:
A. -Stfetn"for H. T. Me Adams,
Program Manager
-------
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
Page
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.
Sectton 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!1 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 COg
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 C0£ values were not available, but composite
fuel economy was provided as an explicit input datum.
Robert L. Parrel!, "Temperature Correction Formulae for Adjusting
Estimates of Emissions from Automobiles," Vector Research, Inc.,
Report No. VRI-EPA-6 (Draft), September 1979
G. 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
(Tl 67°F), in the HOT range (Tl;>87°F), or in the FTP range
(68°F l T * 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 PC's obtained for the same vehicle at FTP
temperatures. Defi ne
u • c/cFTp.
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 - 86.50 (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, Yj). 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°)
4
-------
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-. ,„ r
FC at FTP = 6XP tb (67'5 -
for COLD temperatures, and
FC at temp. T „ r-. /T oc
FC at FTP exp L> (T ' 86'
for HOT temperatures.
-------
Table 2.1
COMPOSITE FUEL CONSUMPTION TEMPERATURE EFFECTS COEFFICIENTS
Model Year/Std.
Low Temperatures
High Temperatures
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
.002037
.002682
.001697
.002261
.002555
.001775
.003021
.003203
.002941
.002310
.001521
.002608
.002600
.002982
.002958
.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, 55, expressed as a per-
cent of b itself, i.e.,
STD ERROR = 100x(sb/b) %
The formula for 55 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.
-------
67FED
1 . 953-,
1.25-
1.0.
800
0.512
X
FTP FiftNGE
0.00 20.00 40.00 60.00 30.00
TEMPERflTURE
100.00 120.0
MIN TEMF= 20.0
MflX TEMP = 110.0
r-4 (LOW TEMP) 6
N (FTP TEMP) 3
N(HIGHTEMP) 3
EFFECT (LOW TEMP) =EXF (0.002037 (67.5-TEM
EFFECT (HIGHTEMP) =EXP (0.000161 (TEMP-86-
3TD ERROR (LOW TEMP) 15.45%
STD ERROR (HIGHTEMP) 428. IQ7.
FIGURE 3.1
Temperature Effects on Fuel Consumption, Model-Year/Standard = 67 FED
8
-------
Table 3.1
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 67 FED
TEMP. (F) CORRECTION FACTOR
O. 0 1. 1474
5. O 1, 1358
10. O 1. 1243
15. O 1. 1129
20. 0 1. 1016
25.0 1.0904
30. O 1.0794
35. 0 1. 0684
40.0 1.0576
45.0 1.O469
50. O 1. 0363
55.0 1,0258
60.0 1.0154
65.0 1.OO51
70.0 1.0000
75.0 l.OOOO
80.0 1.0000
85. 0 1. OOOO
90.0 1.0006
95.0 1.0O14
1OO. O 1.OO22
105.0 1.O03O
110.0 1.0038
-------
1 . 953 -,
1.25-
1.0-
800-
0.512
69FED
X
X
X
FTP RfiNGE
0.00 20.00 40.00 60.00 80.00
TEMPERRTURE
100.00 120.00
MIN TEMP = 20.0
MflX TEMP= 110.0
N (LOW TEMP) 8
N (FTP TEMP) 14
H(HIGHTEMP) 4
EFFECT (LOW TEMP)=EXP(0.002682 (67. 5-TEMP)
EFFECT (HIGHTEMP)=EXP(-0.000018(TEMP-86.5
STD ERROR (LOW TEMP)
STO ERROR (HIGHTEMP)
13.U12
2511. 177.
FIGURE 3.2
Temperature Effects on Fuel Consumption, Model-Year/Standard = 69 FED
10
-------
Table 3.2
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 69 FED
TEMP. (F) CORRECTION FACTOR
0. 0 1. 1985
5. O 1. 1825
10. O 1. 1667
15. 0 1. 1512
20. 0 1. 1359
25. 0 1. 12O7
30. Q 1. 1058
35. O 1.O911
40.0 1.O765
45.0 1.O622
50.0 1.0481
55.0 1.0341
60. O 1.O203
65.0 1.O067
70. O 1. OOOO
75.0 l.OOOO
80. 0 1. OOOO
85.0 l.OOOO
9O. O O. 9998
95. 0 O. 9996
1OO..O O. 9994
1O5. O O. 9991
110. O O. 9989
11
-------
70FED
1. 953-,
1 . 25 -
1.0.
800-
0.512
FTP RfiNGE
0.00 20.00 UO.OO 60.00 - 80.00
TEMPERRTURE
^ 100.00 120.01
MIN TEMP = 20.0
MflX TEMP= 110.0
N (LOW TEMP) 4
N (FTP TEMP) 2
N(HIGHTEMP) 2
EFFECT (LOW TEMP)= EXP (0.001697 (67.5-TEMP
EFFECT(HIGHTEMP) =EXP (-0.^002261 (TEMP-86.
STD ERROR (LOW TEMP) 21.182
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. O l. 1119
1O. 0 1. 1O25
15.0 1.0932
20.0 1.O839
25.0 1.0748
30.0 1.0657
35.0 1.0567
40.0 1.0478
45.0 1. O3S9
50.0 1.0301
55.0 1.0214
60.0 1.0128
65. 0 1. O043
70. 0 1. 0000
75.0 l.OOOO
80. 0 1. OOOO
85. 0 1. OOOO
9O. 0 0. 9921
95. 0 O. 9810
100. 0 0. 9699
105. 0 O. 9590
110. 0 0. 9483
13
-------
1 .953-,
1.25-
1.0-
800
0.512
X
71FEO
X
FTP RANGE
0.00 20.00 UO.OO 60.OQ 80.00
TEMPERRTURE
100.00 120.00
MIN TEMP= 20.0
MflX TEMP= 110.0
N (LOW TEMP) 10
N (FTP TEMP) 5
N(HIGHTEMP) 5
EFFECT (LOW TEMP) =EXP (O.O02261 (67.5-TEMPi
EFFECT (HIGHTEMP) =EXP (-0. 000933 (TEMP-86.
STD ERROR (LOW TEMP) 11.
STD ERROR (HIGHTEMP) 89.872
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: CORRECTION FACTOR
O. 0 1. 1649
5. 0 1. 1518
1O. 0 1. 1388
15. 0 1. 1260
20. 0 1. 1134
25. 0 1. 1009
30.0 1.O885
35.0 1.0762
40. O 1. O642
45.0 1.O522
50. O 1.O404
55.0 1.O287
60. O 1.O171
65.0 1.0057
7O. O 1. 0000
75. 0 l. OOOO
8O. 0 1. OOOO
85. O 1. OOOO
90. O 0. 9967
95. O 0. 9921
100. O 0. 9875
1O5. O O. 9829
110. 0 O. 9783
15
-------
1 . 953-,
1. 25 .
1.0-
. 800 -
0.512
X
FTP FiftNGE
0.00 20.00 40.00 60.00 80.00
TEMPERflTURE
72FED
100.00 - 1
20.00!
HIM TEMP= 0.0
MflX TEMP= 110.0
N (LOW TEMP) 16
N (FTP TEMP) 12
NWIGHTEMP) 5
EFFECT CLOW TEMP)=EXP (0.002555(67.5-TEMF
EFFECT (HIGHTEMP) = EXP(-0.000733(TEMP-86.
STD ERROR (LOW TEMP) 12.522
STD ERROR (HIGHTEMP) 105.12%
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. 18B2
5. 0 1. 1731
10. O 1. 1583
15. 0 1. 1436
20.0 1.129O
25. 0, 1. 1147
30. 0 1. 1006
35.0 1.0866
40.0 1.0728
45. 0 1. 0592
50.0 1.0457
55. 0 1. 0325
60.0 1.0193
65. 0 1. 0064
70. 0 1. OOOO
75.0 l.OOOO
80.0 1.0000
85. 0 1. 0000
90. 0 O. 9974
95. 0 0. 9938
100. 0 0. 99O2
1O5. 0 0. 9865
110.0 O. 9829
17
-------
II™.
. b
1.
1.0
300
0.512
73FED
X
X
X
X
FTP RflNGE
0.00 20.00 40.00 60.00 80.00
TEMPERRTURE
100.00 120.00
MIN TEMP= 20.0
MflX TEMF= 110.0
N (LOW TEMP) 12
N (FTP TEMP) 12
N(HIGHTEMP) 4
EFFECT (LOH TEMP) =EXP (0. 001775 (67. 5-TEMP)!
EFFECT (HIGHTEMP) =EXP (-0.000305 CTEMP-36.51
STD ERROR'(LOW TEMP)
STD ERROR (HIGHTEMP)
30. SOX
231.
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
O. 0 1. 1273
5. 0 1. 1173
10. O 1. 1075
15. O 1.O977
20. O 1. 0880
25.0 1.O784
30.0 1.06BB
35. O 1.0594
40.0 1.0500
45.0 1.O407
50.0 1.0315
55.0 1.0224
6O. O 1.0134
65. O 1. O044
7O. 0 1. OOOO
75.0 l.OOOO
80. O 1. 0000
85. 0 1. OOOO
9O. O 0. 9989
95. O O. 9974
10O. O O. 9959
105. 0 O. 9944
HO. O 0, 9929
19
-------
1.953-,
1.25-
1.0-
800
0.512
X
FTP RflNGE
0.00 20.00 40.00 60.00 80.00
TEMPERRTURE
7UFED
X
100.00 120.0
HIM TEMP= .0.0
MftX TEMP= 110..0
N (LOW TEMP) 22
N (FTP TEMP) 20
H(HIGHTEMP) 12
EFFECT (LOW TEMP) =EXP (0. 003021 (67. 5-TEMIJ
EFFECT (HIGHTEMP) =EXP (-0.000627 (TEMP-86.
STD ERROR (LOW TEMP)
STD ERROR (HIGHTEMP)
33. 78*
FIQURE 3,7
Temperature Effects on Fuel Consumption, Model-Year/Standard = 74 FED
20
-------
Table 3.7
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 74 FED
TEMP. (F) CORRECTION FACTOR
0.0 1.2262
5.0 1.2078
10. O 1. 1897
15. O 1. 1719
20. 0 1. 1543
25. O 1. 1370
30. 0 1. 12OO
35. O 1. 1032
40. O 1. 0866
45.0 1.O7O3
50. O 1.O543
55. O 1. O385
60.0 1.0229
65. O 1. O076
70.0 1.0000
75. O 1. OOOO
80. O 1. OOOO
85. 0 1. OOOO
90. 0 0. 9978
95. O O. 9947
100. O O. 9916
105. O O. 9885'
110.0 0.9854
21
-------
1.953-,
1.25-
1.0.
800.
0.512
X X
X
x xx
X
x
xx
X
X
X
XX
X
X
RANGE
0.00 20.00 40.00 60.00
TEMPERRTURE
80.00
75CfiL
100.00 120.00
MIN TEMP= -22.0
MflX TEMP= 89.0
H (LOW TEMP) 93
N (FTP TEMP) 53
N (HIGHTEMP) 1
EFFECT (LOW TEMP) =EXP (0.003203 (67.5-TEMPl
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. (F)
0. 0
5. 0
10. 0
15. 0
20. O
25. 0
30. 0
35. 0
40. 0
45. 0
50. 0
55. 0
60. 0
65. 0
70. 0
75. O
80. 0
85. O
90. 0
95, 0
100. O
105. 0
110. 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.
2414
2216
2022
1831
1643
1458
1276
1097
0921
0747
0577
0408
0243
OO80
OOOO
OOOO
OOOO
OOOO
OOOO
OOOO
OOOO
OOOO
OOOO
23
-------
1.953-,
1.25-
1.0-
.800
0.512
X
X
x
X
xxxx
FTP RANGE
0.00 20.00 40.00 60.00 80.00
TEHPERRTURE
75FED
100.00 120.01
MIN TEMP-= -8.5
MflX TEMP = 100.6
N (LOW TEMP) 71
N(FTP TEMP) 81
N(HIGHTEMP) 11
EFFECT(LOW TEMP)=EXP (0.002911 (67.5-TEMP
EFFECT (HIGHTEMP)=EXP (-0.002192 CTEMP-86.
STD ERROR (LOW TEMP) 6.632
STD ERROR (HIGHTEMP) 25.77X
* 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. (F) CORRECTION FACTOR
O. O 1.2196
5.0 1.2018
10. 0 1. 1842
15. O 1. 1670
2O. 0 1. 1499
25. 0 1. 1331
30. 0 1. 1166
35. O 1, 1003
40.0 1.O842
45. 0 1. 0684
5O. O 1.O528
55. O 1.O374
6O. 0 1.0223
65.0 1.0074
7O. 0 1. 0000
75. 0 l'. OOOO
8O.0 1. OOOO
85. 0 1. OOOO
9O. 0 O. 9924
95. 0 O. 9815
1OO. 0 0. 97O8
105. O 0. 9603
HO. 0 O. 9498
25
-------
1.953^
1.25-
1.0
.800
0.512
X .
7GFED
FTP RRNGE
0.00 20.00 40.00 60.00 80.00
TEMPERHTUHt"
100.00 120.00
MIN TEHP= 22.0
MflX TEMP= 78.7
N (LOW TEMP) 19
N (FTP TEMP) 19
N (HIGHTEMP) 0
EFFECT(LOW TEMP)=EXP(0.002310(67.5-TEMP)
EFFECT (HIGHTEMP)=EXP (0.000000 (TEMP-86.5)
STD ERROR (LOW TEMP)
13.02X
FIGURE 3.10 .
Temperature Effects on Fuel Consumption, Model-Year/Standard = 76 FED
26 •
-------
Table 3.11
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 77 CAL
TEMP. (F) CORRECTION FACTOR
0. O 1. 1081
5. O 1.O997
1O. 0 1.O914
15. 0 1. 0831
20.0 1.O749
25! 0 1. O668
30. O 1.0587
35.0 1.0507
40. 0 1. 0427
45.0 1.0348
50.0 1.O270
55.0 1.O192
60.0 1.O115
65. 0 1. O03S
70. 0 1. OOOO
75. O 1. OOOO
80. O 1. OOOO
85. O 1. OOOO
90. O 1. O011
95. O 1. OO26
1OO. 0 1. OO41
105. 0 1. OO56
110.0 1.O072
29
-------
1 .953 „
1.25
1.0-
.300
0.512
X
X
X
FTP RfiNGE
0.00 20.00 40.00 60.00 80.00
TEMPERflTURE
77FED
X
X
X
100.00 120.00
MIN TEMP= 0.0
MflX TEMP= 110.0
N (LOW TEMP) . 65
N (FTP TEMP) U8
N(HIGHTEMP) 12
EFFECT (LOW TEMP) =EXP (0.002608 (67.5-TEMP)
EFFECT (HIGHTEMP)=EXP (-0.000593 (TEMP-86.5
STD ERROR (LOW TEMP) 8.U2X
STD ERROR (HIGHTEMP) 110.57X
FIQURE 3.12
Temperature Effects on Fuel Consumption, Model-Year/Standard = 77 FED
30
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Table 3.12
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 77 FED
TEMP. (F) CORRECTION FACTOR
0. 0 1. 1925
5. 0 1. 177O
10. O 1. 1618
15. 0 1. 1467
20. 0 1. 1319
25. O 1. 1172
30. 0 1. 1O27
35. O 1. O885
4O. 0 1.O744
45. O 1.O604
50.0 1.0467
55.0 1.0331
6O. 0 1 O198
65. 0 1. O065
70. 0 1. OOOO
75. 0 1. OOOO
BO. O l.OOOO
85.0 l.OOOO
9O. 0 O. 9979
95. 0 O. 995O
100. O O. 9920
105. O O. 9891
HO. 0 O. 9862
31
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1.953-,
1.25
1.0-
.800
0.512
X
X
X
X
X
FTP RftNGE
X
0.00 20.00 UO.OO 60.00 80.00
TEMPERRTURE
78CRL
100.00 120.00
M1N TEMP= -3.8
MflX TEMP= 110.0
N (LOH TEMP) 57
N (FTP TEMP) 29
H(HIGHTEMP) 13
EFFECT (LOH TEMP) =EXP (0. 002600 (67- 5-TEMP)
EFFECT (HIGHTEMF) =EXP(^0.000483(TEMP-86.5
STD ERROR CLOW TEMP) 8.2«45i
STD ERROR (HIGHTEMP) 104.95>i
* The data set on which the fitting equation was based contained an additional
2 points at temperatures below 0°F.
FIGURE 3.13
Temperature Effects on Fuel Consumption, Model-Year/Standard = 78 CAL
32
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Table 3.13
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 78 CAL
TEMP. (F) CORRECTION FACTOR
O. 0 1. 1918
5. O 1. 1764
10. 0 1. 1613
15. 0 1. 1463
20. O 1. 1314
25. 0 1. 1168
30. O 1. 1O24
35. O 1.0882
40. O 1.0741
45. O 1.O602
50.0 1.O466
55. 0 1. 033O
60.0 1.O197
65. O 1.OO65
70. O 1. OOOO
75. O 1. OOOO
80. 0 1. OOOO
85.0 l.OOOO
90. O O. 9983
95. O O. 9959
100. O 0. 9935
105.0 O. 9911
110.0 0.9887
33
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1 .953 -,
1.0-
800
0.512
X
X
X
X
X
X
X
X
FTP RANGE
0.00 20.00 140.00 60.00 80.00
TEMPERflTURE
78FED
X
X
100.00 120.00
MIN TEMP= 0.0
MflX TEMF= 110.0
N CLOW TEMP) 141
N (FTP TEMP) 11
N(HIGHTEMP) 4
EFFECT (LOW TEMP) = EXP (0. 002982 (67. 5-TEMF)
EFFECT (HIGHTEMP) =EXP (0.002310 CTEHP-86.5)
STD ERROR (LOW TEMP)
STD ERROR (HIGHTEMP)
13.832
75.55X •
FIGURE 3,14
Temperature Effects on Fuel Consumption, Model-Year/Standard = 78 FED
34
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Table 3.14
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 78 FED
TEMP. CORRECTION FACTOR
0.0 1.2230
5. O 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. 1O18
40.0 1.O855
45.0 1.O694
50. O 1. O536
55.0 1.0380
60. O 1.O226
65.0 1.O075
70. O i. OOOO
75. O 1. OOOO
80.0 1.0000
85.0 l.OOOO
90. O 1.O099
95.0 1.O242
100. O 1.O387
105. O 1.O534
110. O 1. O683
35
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1.953^
1.25-
1.05
.800
0.512
X
FTP FiRNGE
0.00 20.00 40.00 60.00 80.00
TEMPERRTURE
80FED
X
100.00 120.00
MIN TEMP= 0.0
MRX TEMP= 110.0
H (LOW TEMP) 17
N (FTP TEMP) 12
H (HIGHTEMP) 10
EFFECT(LOW TEMP)= EXP(0.002958(67.5-TEMP))
EFFECT (HIGHTEMP)=EXP (-0.002U56(TEMP-S6.5)
STD ERROR (LOW TEMP) 12.61X
STD ERROR (HIGHTEMP) 27-112
FIQURE 3.15
Temperature Effects on Fuel Consumption, Model-Year/Standard = 80 FED
36
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Table 3.15
TABLE OF ESTIMATED TEMPERATURE EFFECTS
FUEL CONSUMPTION 80 FED
TEMP. (F) CORRECTION FACTOR
O. 0 1.2210
5.0 1.2031
1O. 0 1. 1854
15. O 1. 1680
20. 0 1. 1509
25.0 1.1340
30. 0 1. 1173
35. 0 1. 1O09
4O. O 1.O847
45.0 1.0688
50.0 1.O531
55. 0 1. O377
60. 0, 1. O224
65. O 1. OO74
7O.0 1. OOOO
75. O l.OOOO
80.0 l.OOOO
85.0 l.OOOO
9O. 0 0. 9914
95. O 0. 9793
100. O 0. 9674
105. 0 O. 9556
,I1O. 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.
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 1n such cases.
38
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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
Yi = 5X1 + ei (i = 1 ...... n)
where E(ej) = o, Var(e-j) = a2 for all i, and where the e^'s are
independent of one another. There is only one "normal equation" in this
case, namely
ZX1Y1 = bZX2
Sl
Thus the fitting equation is Y = bX, where
An estimate of a2 is given by
52
40
-------
It can be shown that E(a2) = a2. It can also be shown that
= Var(b) = a2/ZX2
a. = a/ /ZX?
b i
Thus an estimate of a. is provided by
s. = a/ /ZX?
b i
It is clear from the above that E(sf) = a2.
b b
41
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