United States        Air and Radiation        EPA420-P-02-017
           Environmental Protection                  June 2002
           Agency                         NR-007a
&EPA    Calculation of Age
           Distributions in the
           Nonroad Model:
           Growth and Scrappage
                                     > Printed on Recycled Paper

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                                                          EPA420-P-02-017
                                                                 June 2002
          Calculation of Age Distributions in the
         Nonroad Model: Growth and Scrappage
                              NR-007a
                     Assessment and Standards Division
                    Office of Transportation and Air Quality
                    U.S. Environmental Protection Agency
                                NOTICE

  This technical report does not necessarily represent final EPA decisions or positions.
It is intended to present technical analysis of issues using data that are currently available.
       The purpose in the release of such reports is to facilitate the exchange of
     technical information and to inform the public of technical developments which
      may form the basis for a final EPA decision, position, or regulatory action.

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                 Calculation of Age Distributions in the Nonroad Model:
                                 Growth and Scrappage

                                  Report No. NR-007a
                                     June 10, 2002

                            Assessment and Standards Division
                       EPA, Office of Transportation and Air Quality
Introduction

       The Draft NONROAD2002 version of the US EPA nonroad engine emission inventory
model ("NONROAD") calculates nonroad equipment populations by age (i.e., an age
distribution) for given equipment types and scenario years. This calculation is necessary for the
model to account for factors which affect nonroad emissions over time as the in-use fleet ages
and turns over to newer equipment, including emissions deterioration, new emissions standards,
technology changes, and changes in equipment population resulting from sales growth trends.

       The NONROAD model calculates equipment age distributions for each horsepower range
of each equipment type based on the median life (hours at full load), activity (hours of use per
year), load factor, and current growth rate for that equipment, combined with the model's
generalized scrappage function.  Age distribution refers to the fractions of a given equipment's
population that are one, two, three, etc., years old in a given target year. This age distribution is
treated as constant regardless of the target calendar year being evaluated, as described in more
detail below. This methodology differs from the previous versions of NONROAD, which
attempted to calculate age distributions for future years by stepping through each year between
the base year and future evaluation year, calculating equipment populations and scrappage for
each model year of equipment still  in service, and the necessary equipment sales to meet the
projected population (see NR-007,  1998) [1].

       Following are descriptions of the NONROAD model calculation methodology  for
population growth and equipment age distribution, followed by detailed descriptions of the
relevant model inputs and a glossary that defines the key terms used. Terms contained in the
glossary are italicized the first time they appear in this report subsequent to this introduction.
Methodology

       The NONROAD model calculates equipment populations and the age distribution of
those populations. The methods used by the model to perform these calculations are described
below. Basically, the model uses the population growth rate to project equipment populations
from a base year to the desired evaluation year.  Then it applies the age distribution to that
equipment population to estimate the number of units of each age (model year) back to two times
the median life in years, up to a maximum of 50 years.

                                           1

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       Population Growth: Growth Rates and Projected Populations

       The NONROAD model projects equipment populations in past and future years (before
and after the input population base year) by applying a growth rate to the base year equipment
population. This growth rate is calculated by determining the linear annual growth rate in
equipment population between  1989 and 1996, as reported by Power Systems Research (PSR).
This approach is discussed more fully in EPA technical report "Nonroad Engine Growth
Estimates" (NR-008b) [2]. Other than certain exceptions, such as All Terrain Vehicles (ATV's),
the model assumes that the population growth rate remains constant for all years before and after
the base year.

       The growth rates are used to generate the growth index values that are contained in the
growth input data file (nation.grw), which is more completely described in the Inputs section
below. The model uses linear interpolation between these growth indexes to calculate any index
for target years between the input growth index years. For target years prior to the  earliest
growth index year or after the latest growth index year the model uses linear extrapolation from
the two closest growth index inputs. The target year total equipment population for each
horsepower range of each equipment type is then calculated by multiplying the base year
population by the ratio of the target year growth index to the base year growth index.

       Age Distribution: Scrappage Curve Adjusted for Growth

       The NONROAD model assumes a constant age distribution for each power range  of each
equipment type. E.g., according to the model the percentage of 50-100  hp diesel agricultural
tractors that are ten years old in the year 2000 is the same as the percentage that are ten years old
in 2020. This age distribution algorithm is essentially the same as what was used in previous
versions of the NONROAD model for the base year and earlier years and for negative growth in
future years. But the model now also uses that  same methodology in cases  of positive growth in
future years, rather than using a more complex method that stepped through each future year's
scrappage and calculated the sales that would be required to achieve the predicted population.
Also, instead of directly using the input scrappage curve values (percent scrapped versus
percentage of median life used) from the growth data file as an age distribution curve, the model
now uses a modified version of those values that accounts for growth.  Thus, an equipment type
that has zero growth would have an age distribution that exactly matches the input  scrappage
curve (see Figure 1).  But if there is positive growth, the age distribution will be skewed more
toward newer engines (Figures 2, 3, and 4).

       The methodology can be summarized as follows:  For each SCC use the linear growth
from the first period supplied in the growth input file (e.g., 1996 - 1998) plus the input scrappage
curve to build a modified "scrappage curve" that is actually a customized age distribution  curve
for that particular growth rate. Apply this age distribution curve to all calendar years including
the base year, earlier calendar years, and all future calendar years.

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       Following is a more detailed step-by-step technical description of the methodology used
inNONROAD:

1.  For each SCC, read the first two growth entries in the growth input data file and calculate the
linear annual growth rate relative to the first (earlier) year's value.  For most equipment types,
these are currently the entries for calendar years 1996 and 1998, where the 1996 entry is always
arbitrarily set to 1,000, and the year 1998 entry is  1,000 plus whatever relative growth is assumed
for that category.  E.g., for diesel farm equipment, the year 1998 entry is currently 1,063,
indicating a linear growth of (1,063-1,000)71,000/2 yrs = 3.15% per year relative to 1996.

2.  For each SCC, create a temporary array the size of MXSCRP, which is currently a 50 element
array, and initialize it with a contrived "sales ratio history" using the growth rate from Step 1.
I.e., for a 3.0% linear growth rate the first (earliest) entry would be 1.00, the second would be
1.03, the third would be  1.06, etc., increasing by 0.03 each time. An example of the results of
this and subsequent calculation steps, using 3.0%  linear growth, is provided in the appendix of
this report. This ratio sales history is labeled as "Orig. Sales Ratio" in the appendix.

3.  Multiply each value in this temporary array by the corresponding survival fraction ( = 1 -
percent scrapped) from the scrappage array (see Table 1 below) to generate a value for the
relative number of units  surviving in the target year.  This is labeled as "Surviving Pop Ratio" in
the appendix. Replace the original temporary array value with this new value.

4.  Divide each value in the temporary array by the sum of all of the array values, effectively
turning it into an age distribution.  Replace each original temporary array value with the new
value.  This is labeled as "Age Distrib." in the appendix.

5.  Normalize the values in the temporary array to represent a new version of percent scrapped
that includes growth. An example of this calculation is shown below, and the results appear  as
the column labeled "Normalized as % Scrapped" in the appendix.

       Percent "Scrapped" = 100 * (1 - (MYfrac/MaxMYfrac))

       where:
       Percent "Scrapped" = the percent reduction from the initial population (sales) for each
                           model year. "Scrapped" is in quotes because in this use it doesn't
                           actually mean percent scrapped, but rather a relative measurement
                           of scrappage with respect to the other model years.
       MYfrac =
       MaxMYfrac =
the surviving population of the model year of interest divided by
the sum of all model years' surviving populations in the calendar
year being evaluated.

the maximum MYfrac of all the model years for the equipment
type being considered.

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6. Use this new temporary array in place of the scrappage curve values to generate populations
for all past model years in whatever calendar year is being modeled (past, base, or future years).

                  Figure 1 Input Scrappage Curve (no growth)
                                    Default Scrappage Curve
                          0.00       0.50       1.00        1.50
                                          Age / Median Life
2.00
                  Figure 2 Age Distribution (with growth)
                       o.oo
                                       Age Distribution
                                     (with 3% linear growth)
                           0.00       0.50       1.00       1.50
                                          Age / Median Life
2.00

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                 Figure 3 Survival Fraction (with growth)
                       1.00
                       0.80 -
                       0.60 -
                       0.40 -
                       0.20 -
                       0.00
                              Re-Normalized as Fraction Surviving
                                    (with 3% linear growth)
                          0.00
0.50       1.00       1.50
     Age / Median Life
2.00
                  Figure 4 Age Distribution Example (with growth)
                               NONROAD Pop by Age 50-1 OOhp
                                     Diesel Ag Tractors
                       35,000
                                             15    20
                                            Age (years)
Limitations of this Methodology:

       As in previous versions of NONROAD, any type of equipment that is relatively new to
the world, or has had substantial increases in sales in recent years, will not have a real world age
distribution based simply on median life and scrappage parameters, which is how it is modeled.
I.e., the model will show more older units than are actually out there, since they haven't been on
the market long enough to fill out their full scrappage curve, or because the population is
dominated by recent year sales.
                                             5

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       The calculation of age distributions requires the following data to be supplied to the
NONROAD model:

      Equipment Population and Median Life at full load
      Growth Indicators
      Scrappage Curve
      Activity and Load Factor

       These data are input to the model using data files.  The format of the data files is
described below, examples of data are given, and the location of the data is specified. For more
information on equipment activity, load factors, median life, and populations see the EPA
technical reports on these subjects, NR-005b [3] and NR-006b [4].
       Equipment Populations and Median Life at Full Load

       Equipment population data for each state and nationwide are provided in the directory
"data/pop." If estimates are provided for more than one year, the NONRO AD model will use the
closest year which comes before the target year.  Since median life depends on equipment type
and horsepower range, the median life at full load parameter is also included in the population
data file.

       The format of the data in the /POPULATION/ packet is as follows:

Columns     Description
 1 - 5       FIPS code
 7-11      subregion code (used for subcounty estimates)
 13- 16      year of population  estimates
 18-27      SCC code (no globals accepted)
 29 - 68      equipment description (ignored)
 70 - 74      minimum HP range
 76 - 80      maximum HP range (ranges must match those internal to model)
 82- 86      average HP in range (if blank model uses midpoint)
 88- 92      median expected life (in hours of use at full load)
 93-102     flag for scrappage distribution curve (DEFAULT = standard curve)
106-122     population estimate

       An example of data from a /POPULATION/ packet is given below (note that lines are
wrapped to fit the page).

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00000

00000

00000

1996 2265004010 4-
47.9
DEFAULT
636892
1996 2265004010 4-
47.9
DEFAULT
31652672
1996 2265004010 4-
400
DEFAULT
10436
Str

Str

Str

Lawn

Lawn

Lawn

mowers

mowers

mowers

(res)

(res)

(res)

1

3

6

3

6

11

2

4

6

55

1

24

       Growth Indicators

       The default data are based on national growth estimates for the various source category
groups. Data are located in the "data/growth" directory in a single file that contains both data
packets:

       /INDICATORS/      Cross reference between SCC code and growth indicator code.
       /GROWTH/         Numerical values for different growth indicator codes.

       An indicator code is an alphanumeric code used to identify which set of growth index
inputs to use for a given type of equipment. The growth indicator codes used in the
/INDICATORS/ packet must match one of the codes provided in the /GROWTH/ packet. Cross
referencing between the /INDICATORS/ and /GROWTH/ packets is based on FIPS code, SCC
code, horsepower (HP) range, and technology type. The model uses the best match to the codes
provided, falling back on global values if a unique match is not found.

       The format of the data in the /INDICATORS/ packet is as follows:

Columns     Description
1-5          FIPS code     (00000 = applies to entire nation)
                          (ssOOO = applies to all of state ss)
7-10         Indicator code (arbitrary alphanumeric code)
12-21        SCC code     (2260004000 = applies to all 2-stroke lawn and garden)
                          (2600000000 = applies to all 2-stroke)
23-27        Beginning of HP range
28-32        Ending of HP range
34-43        Technology type (ALL = applies to all tech types)

       An example data record from the /INDICATORS/ packet is given below.
 00000  052  2265004000     0 9999  ALL         4-Stroke Lawn  &  Garden Equip.
       The format of the data in the /GROWTH/ packet is as follows:

1-5          FIPS code     (00000 = applies to entire nation)
                          (ssOOO = applies to all of state ss)
                                         7

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6-10         Subregion code (blank = applies to all subregions)
11-15        Year of estimate (4-digit year)
17-20        Indicator code (arbitrary alphanumeric code)
26-45        Indicator value

       Example of a few data records from the /GROWTH/ packet:
00000
00000
00000
00000
00000
00000
00000
00000
1996
1998
2000
2005
2010
2015
2025
2045
052
052
052
052
052
052
052
052
1000
1045
1090
1210
1329
1449
1688
2166
       Scrappage Curve
       A single default scrappage curve is used by the NONROAD model.  The scrappage curve
is read in from the same data file as the growth data in the "data/growth" directory.

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             Table 1. Default Scrappage Curve
               Age/Median    Percent
                            Scrapped
                                o
 Life*
0.0000
0.0588
0.1694
0.2710
0.3639
0.4486
0.5254
0.5948
0.6570
0.7125
0.7617
0.8049
0.8425
0.8750
0.9027
0.9259
0.9451
0.9607
0.9730
0.9824
0.9894
0.9942
0.9973
0.9990
1.0000
                                1
                                3
                                5
                                7
                                9
                               11
                               13
                               15
                               17
                               19
                               21
                               23
                               25
                               27
                               29
                               31
                               33
                               35
                               37
                               39
                               41
                               43
                               45
                               50
Age/Median Percent
Life Scrapped
continued. . .
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
.0010
.0027
.0058
.0106
.0176
.0270
.0393
.0549
.0741
.0973
.1250
.1575
.1951
.2383
.2875
.3430
.4052
.4746
.5514
.6361
.7290
.8306
.9412
.0000
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
95
97
99
100
             * Age in hours/load factor;
               Median Life  in  hours at full load.
      Activity and Load Factor

      The /ACTIVITY/ packet defines how often a piece of equipment is used in a year. The
file also contains other information about the equipment, most notably the average load factor.

      The format of the /ACTIVITY/ packet is as follows:

Columns     Description
1-10        SCC code
12-51       Equipment description (not used)
52-56       Region code
57-66       Technology type
67-71       Minimum HP
72-76       Maximum HP

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77-81        Load factor
82-86        (not used)
87-96        Activity level units
97-106       Activity level
107-116      Identifier for age adjustment curve (DEFAULT = no adjustment)

      An example data record from the /ACTIVITY/ packet is (note that lines are wrapped to fit
on page):
     2265004010 4-Stroke Lawn mowers  (Residential)                   ALL
       0 9999 0.33         Hrs/Yr          25  DEFAULT
                                         10

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                                   Glossary of Terms
Term
Accumulated scrappage



Activity level


Age distribution



Annual Growth Rate



Annualized scrappage curve


Average lifetime


Base year


In-service fraction



Load factor


Median life at full load
Definition
The total amount of scrappage that has occurred for
equipment of a given model year since its introduction into
service

The number of hours per year that the equipment in
question operates

The function that describes the proportion of in-service
equipment by age; consists of the full set of model year
fractions for a given year

The linear rate, relative to 1996, at which the equipment
population is projected to grow each year in order to reach a
specified level

The result of scaling the scrappage curve by the average
lifetime of the equipment in question

The age in years at which half of the equipment will have
been scrapped (removed from service)

The year for which the population of a given type of
equipment is specified in the population input file

The fraction of the engines originally sold in a given year
which are still in service; the inverse of accumulated
scrappage

The average power level at which the engine operates
divided by the maximum available power

The number of hours that a given type of equipment is
expected to survive, if it were operated at full load
                                           11

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                               Glossary of Terms, cont'd
Term
Definition
Model year
Model year fraction


Population growth rate


Scrappage curve

Scrappage function
Scrappage rate
Target year
Total equipment population
Refers to the year in which equipment was produced.
Equipment of the same model year was produced in the
same year. To clarify the relationship between age and
model year, consider the following example:  1990-model
year equipment is (on average) six years old in 1996 and
ten years old in 2000.

The fraction of the total equipment population represented
by a given model year at a given point in time

The rate at which the equipment population increases each
year

A graphical representation of the scrappage function

The relationship between equipment age (expressed in
terms of the fraction of average lifetime) and the proportion
of equipment that has been removed from service, i.e.,
scrapped

The percentage of equipment of a given age removed from
service in a given year

The calendar year for which the NONROAD model's user
wishes to estimate emissions and other quantities

The total number of pieces of equipment in service at a
given point in time; the sum of the populations of each
model year still in service
                                          12

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References

[1] "Calculation of Age Distributions in the Nonroad Model: Growth and Scrappage," NR-007,
U.S. Environmental Protection Agency, Office of Transportation and Air Quality, February 18,
1998.

[2] "Nonroad Engine Growth Estimates," NR-008b, U.S. Environmental Protection Agency,
Office of Transportation and Air Quality, May 2002.

[3] "Median Life, Annual Activity, and Load Factor Values for Nonroad Engine Emissions
Modeling," NR-005b, U.S. Environmental Protection Agency, Office of Transportation and Air
Quality, May 2002.

[4] "Nonroad Engine Population Estimates," NR-006b, U.S. Environmental Protection Agency,
Office of Transportation and Air Quality, May 2002.
                                           13

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Appendix:
Frac Med
Life Used
0.0000
0.0588
0.1694
0.2710
0.3639
0.4486
0.5254
0.5948
0.6570
0.7125
0.7617
0.8049
0.8425
0.8750
0.9027
0.9259
0.9451
0.9607
0.9730
0.9824
0.9894
0.9942
0.9973
0.9990
1.0000
1.0010
1.0027
1.0058
1.0106
1.0176
1.0270
1.0393
1.0549
1.0741
1.0973
1.1250
1.1575
1.1951
1.2383
1.2875
1.3430
1 .4052
1 .4746
1.5514
1.6361
1.7290
1.8306
1.9412
2.0000
Sum:
Example
Percent
Scrapped
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
50
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
95
97
99
100

of Scrappage
Fraction
Surviving
1
0.99
0.97
0.95
0.93
0.91
0.89
0.87
0.85
0.83
0.81
0.79
0.77
0.75
0.73
0.71
0.69
0.67
0.65
0.63
0.61
0.59
0.57
0.55
0.5
0.45
0.43
0.41
0.39
0.37
0.35
0.33
0.31
0.29
0.27
0.25
0.23
0.21
0.19
0.17
0.15
0.13
0.11
0.09
0.07
0.05
0.03
0.01
0

Age
Years
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49

Curve Modification for
Orig. Sales
Ratio
2.4400
2.4100
2.3800
2.3500
2.3200
2.2900
2.2600
2.2300
2.2000
2.1700
2.1400
2.1100
2.0800
2.0500
2.0200
1.9900
1 .9600
1.9300
1.9000
1.8700
1 .8400
1.8100
1.7800
1.7500
1 .7200
1 .6900
1 .6600
1 .6300
1 .6000
1.5700
1 .5400
1.5100
1 .4800
1 .4500
1 .4200
1.3900
1 .3600
1.3300
1.3000
1 .2700
1 .2400
1.2100
1.1800
1.1500
1.1200
1.0900
1 .0600
1.0300
1.0000

Surviving
Pop Ratio
2.4400
2.3859
2.3086
2.2325
2.1576
2.0839
2.0114
1.9401
1.8700
1.8011
1.7334
1 .6669
1.6016
1.5375
1 .4746
1.4129
1 .3524
1.2931
1.2350
1.1781
1.1224
1.0679
1.0146
0.9625
0.8600
0.7605
0.7138
0.6683
0.6240
0.5809
0.5390
0.4983
0.4588
0.4205
0.3834
0.3475
0.3128
0.2793
0.2470
0.2159
0.1860
0.1573
0.1298
0.1035
0.0784
0.0545
0.0318
0.0103
0.0000
48.5456
Growth
Age
Distrib.
0.05026
0.04915
0.04756
0.04599
0.04444
0.04293
0.04143
0.03996
0.03852
0.03710
0.03571
0.03434
0.03299
0.03167
0.03038
0.02910
0.02786
0.02664
0.02544
0.02427
0.02312
0.02200
0.02090
0.01983
0.01772
0.01567
0.01470
0.01377
0.01285
0.01197
0.01110
0.01026
0.00945
0.00866
0.00790
0.00716
0.00644
0.00575
0.00509
0.00445
0.00383
0.00324
0.00267
0.00213
0.00161
0.00112
0.00066
0.00021
0.00000
1.00000
(3% linear)
Normalized as
%"Scrapped"
0.00
2.22
5.39
8.50
11.57
14.59
17.57
20.49
23.36
26.18
28.96
31.68
34.36
36.99
39.57
42.09
44.57
47.00
49.39
51.72
54.00
56.23
58.42
60.55
64.75
68.83
70.75
72.61
74.43
76.19
77.91
79.58
81.20
82.77
84.29
85.76
87.18
88.55
89.88
91.15
92.38
93.55
94.68
95.76
96.79
97.77
98.70
99.58
100.00

14

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