TEMPERATURE CORRECTION  FORMULAS
       FOR ADJUSTING ESTIMATES  OF AUTOMOBILE
               FUEL  CONSUMPTION

                     by

                Norman  Morse
Report  3520-1/BUF-35
May 1980
                                 RESEARCH AND
                                 DEVELOPMENT

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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

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                            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

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                           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.

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                     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

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     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.

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                            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

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                   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.

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                                                             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

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                  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

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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

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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. 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

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                                                       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

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                 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

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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

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                 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

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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

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                 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

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  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

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                 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

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                 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

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               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

-------
                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

-------
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

-------
                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

-------
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

-------
                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

-------
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

-------
                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

-------
                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

-------
     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

-------
                                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

-------