Air and Radiation EPA420-R-05-018
December 2005
United NR-007c
Environmental Protection
Agency
Calculation of Age
Distributions in the
Nonroad Model:
Growth and Scrappage
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EPA420-R-05-018
December 2005
in the
NR-OO/c
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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Introduction
The NONROAD2005 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 the base year (as given in the
equipment population input file) based on estimated engine populations for that year combined
with the scrappage function and growth inputs. The model calculates age distributions for future
years by stepping through each year between the base and future years; for each year, the model
projects equipment populations, scrappage for each model year of equipment still in service, and
equipment sales needed to attain the projected population.
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. The NONROAD model calculates base year
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. The term
"scrappage" as used here means the final scrapping of equipment (permanently retiring it from
service), such that it no longer contributes to the emissions or fuel consumption of the fleet
This methodology is an enhancement of the original methodology (see NR-007, 1998)1,
which also used an iterative step-wise approach but failed to properly account for prior growth in
the base year age distribution. The new methodology also differs from the most recent model
version (Draft NONROAD2004, as documented in NR-007b, 2004)2, which used a static age
distribution for all target years.
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 estimated equipment populations, the age distribution
of those populations, annual equipment sales, and equipment scrappage. These calculations are
performed for the base year., and then for every year from the base year up to, and including, the
specified target year for the model run. The model uses the population growth rate to project
equipment populations from the base year, as provided in the population input file, through the
target evaluation year. For each projection year the model back-calculates expected equipment
sales for each year as the difference between projected total equipment population and
population remaining from each prior model year after that projection year's scrappage has been
applied.
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The scrappage calculation uses a scrappage curve to determine the proportion of
equipment of a given age (model year) that has been removed from service. This proportion is
then multiplied by the initial population of that age (model year) to determine the accumulated
scrappage for equipment of that age. Subtracting the accumulated scrappage from initial sales
yields the population of equipment still in service. With this information, the age distribution of
the population is calculated by dividing the population at each age by the total population. The
details of each of these calculations are discussed more fully below, and an example calculation
is provided in the Appendix.
Base Year Equipment Population
Estimates of the base year equipment populations (fleet totals, by SCC and power range,
but not separated by model year) are contained in the NONROAD model's input files.
NONROAD2005 uses 2000 as the base year for most diesel equipment, and 1998 for most spark-
ignition equipment. The basis for these estimates is discussed at length in technical report NR-
006d, "Nonroad Engine Population Estimates".3
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 for each equipment type (SCC) and power range. The growth rates for most
equipment types are calculated as linear annual growth rates in equipment populations, as
reported by Power Systems Research (PSR). This approach is discussed more fully in EPA
technical report "Nonroad Engine Growth Estimates" (NR-008c).4 With certain exceptions, such
as All Terrain Vehicles (ATV's), the model assumes that the population growth rate, in terms of
number of units per year, 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.
Scrappage
The term "scrappage" as used here means the final scrapping of equipment (permanently
retiring it from service), such that it no longer contributes to the emissions or fuel consumption
of the fleet. This may be due to aging of the engine or to some other part of the equipment
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breaking. The determination of the equipment age at the time of scrappage is covered more fully
in technical report NR-005c, "Median Life, Annual Activity, and Load Factor Values for
Nonroad Engine Emissions Modeling."5
The NONROAD model uses a scrappage curve to determine the proportion of equipment
that has been scrapped as a function of equipment age. The scrappage curve is read in from the
same data file as the growth data in the "data\growth" directory. The default scrappage curve as
shown in Figure 1, and Table 1 comes from the Power Systems Research (PSR) Partslink
database.6 The curve is scaled to the average lifetime of the equipment, such that half of the
units sold in a given year will be scrapped by the time those units reach the average expected
lifetime, and all units will be scrapped by twice the average lifetime. The average lifetime (in
years) is calculated as the median life (in hours at full load) divided by the activity level
(hours/year) and the load factor.
Average Lifetime (years) = Average Life (hrs) / [Activity (hrs/yr) * Load Factor]
Figure 1 Input Scrappage Curve
Default Scrappage Curve
0.00 0.50 1.00 1.50
Age / Median Life
2.00
Note that in Figure 1, the proportion of equipment that has been scrapped (one minus the
fraction surviving) represents the accumulated scrappage from the year that the equipment was
placed in service.
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Table 1. Default Scrappage Curve
Age/Median
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
Cumulative
Percent
Scrapped
0
i
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
Life
Cumulative
Percent
Scrapped
continued. . .
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
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.
The default scrappage curve used in the NONROAD model is based on a normal
distribution of accumulated scrappage versus age. Other distributions could be used; for
example, EPA's original proposed 1996 standards for small spark-ignited engines used a Weibull
distribution to project fleet turnover. That distribution yields reduced scrappage in the first
(newest) few years and increased scrappage in later years than is the case for a normal
distribution. The NONROAD model allows a user-specified curve to be substituted for the
default curve for any or all equipment types. A user-specified curve can vary the rate of
scrappage with age (the shape of the curve) but must conform to the assumption that all units are
scrapped within twice the average lifetime.
The scrappage rate is defined as the percentage of equipment of a given age removed
from service in a given year. The scrappage rate can be derived from the scrappage curve by
determining the slope of the scrappage curve at the age in question. Note that for the default
scrappage curve, the scrappage rate changes as the equipment age changes: the scrappage rate is
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low when units are new, reaches a maximum when unit age is equal to the average lifetime, and
then declines again for units that are older than the average lifetime.
Age Distribution in the Base Year
To determine the equipment population's actual age distribution as of the base year, one
would need to know past equipment sales and scrappage. However, estimates of past equipment
sales suffer from missing or incomplete data, are of poor quality, or are based on estimation
methods that are incompatible with the methods used to estimate the base year equipment
populations used in the NONROAD model. Given these uncertainties, the age distribution in the
base year is determined using the relationship between sales growth and population growth,
where this relation is a function of average lifetime and the scrappage curve. The methodology
involves building up a simulated fleet for each SCC and power range using an arbitrary starting
sales value (1000 units/year) and projecting sales out into subsequent years for twice the median
life of the engine, taking into account the scrappage that would occur each year. These are just
projections of relative sales, not estimates of actual sales. For purposes of this methodology, the
last year of this projection is the same as the base population year.
The model growth inputs are given in terms of population growth, rather than sales growth, so
the assumed sales growth needs to be consistent with the input population growth at the time of
the base year. By performing the sales/scrappage simulation described in the Appendix for a
range of sales growth and median life inputs, EPA was able to determine the following
relationship between sales growth and population growth:
SalesGrw = PopGrw / { [ (-1.4306 x PopGrw) x MedLifeYrs ] + (-0.24 x PopGrw) + 1.0 }
where
SalesGrw = Growth in engine sales as a percent of the initial year sales. Thus, the
same value (percent and absolute number) is applied each year.
PopGrw = Growth in fleet population for a given SCC and power range as
determined from the Growth input file (nation.grw), as a percent of the
base year population.
MedLifeYrs = Median Expected Life in Years.
Thus, using the model inputs for population growth, median life, and scrappage curve, an initial
base year age distribution is generated by calculating the number of engines sold into the fleet in
each year minus the number scrapped in each year of that model year's life. Figure 2 shows an
example of this initial age distribution for 75 - 100 hp diesel agricultural tractors, which use
model inputs of 16.7 years median life (4,667 hrs median life at full load, 475 hrs/year, 0.59 load
factor) and 3.0% growth (relative to 1996).
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Figure 2 Age Distribution Example (with growth)
Age Distribution
(with 3% linear growth)
0.00 0.25 0.50
0.75 1.00 1.25
Age / Median Life
1.50 1.75 2.00
Age Distribution in Years Prior to the Base Year
For evaluation years prior to the base population year the model uses the same age
distribution as for the base year. The algorithm that is used to project growth and scrappage into
future years does not function properly for backcasting, and use of a static age distribution for
backcasting is consistent with the way the model has been operating for all prior verions. Thus,
this is not considered a significant issue.
Annual Sales and Age Distribution in Future Years
As the model steps through each year from the base year to the target year, it estimates
the sales expected to take place during each year. As discussed above, annual sales are
determined as the increase in equipment population projected for that year plus the number of
units projected to be scrapped that year. I.e., new sales would replace all the equipment lost to
scrappage that year, and it would also add enough to increase the equipment population by the
amount of growth expected that year.
Sales(Y) = Population(Y) - Population(Y-l) + Scrappage(Y)
where
Sales(Y)
Population(Y)
Population(Y-l)
Scrappage(Y)
= Equipment sales in given year
= Equipment population at the end of given year
= Equipment population at the end of prior year
= Equipment scrapped during given year from all model years
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No attempt is made to calculate partial-year scrappage or sales for monthly or seasonal
model runs prior to the end of the year; the same population and age distribution are used for
any month within the same calendar year.
The age distribution in any given evaluation year for each SCC and power range is
simply the number of units of equipment remaining in service from each model year (after
applying the sales and scrappage described above) divided by the total remaining from all model
years.
The calculation of age distributions requires the following data to be supplied to the
NONROAD model as input data files. The input filenames are shown in parentheses along with
a citation for the NONROAD technical report documenting the data.
Base Year Equipment Population (??.POP, NR-006d)3
Median Life at full load (??.POP, NR-005c)5
Growth Index inputs (NATION.GRW, NR-008c)4
Scrappage Curve (NATION.GRW, this report NR-007c)
Activity (hours/year), (ACTIVITY.DAT, NR-005c)5
Load Factor (ACTIVITY.DAT, NR-005c)5
Limitations of Methodology
As in previous versions of NONROAD, any type of equipment that is relatively new to
the world, or which has experienced 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
have not been on the market long enough to fill out their full scrappage curve, or because the
population is dominated by recent model year sales.
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Glossary of Terms
Term
Accumulated scrappage
Activity level
Age distribution
Annual Growth Rate
Average lifetime
Base year
Fraction Surviving
(In-service fraction)
Cumulative percent scrapped
Load factor
Median life at full load
Model year
Definition
The total amount of scrappage that has occurred for
equipment of a given model year since its introduction into
service (cumulative percent scrapped)
The number of hours per year that the equipment in
question operates
The function that describes the proportion of surviving (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 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 (not yet scrapped); the inverse of
accumulated scrappage
See 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
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.
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Glossary of Terms, cont'd
Term
Model year fraction
Population growth rate
Sales growth
Scrappage
Scrappage curve
Scrappage function
Scrappage rate
Target year
Total equipment population
Definition
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, in number of units per year
The increase (or decrease) in equipment sales each year
that is needed to yield the target population growth rate,
given specific assumptions of median life, activity, and
scrappage.
The final scrapping of equipment (permanently retiring it
from service), such that it no longer contributes to the
emissions or fuel consumption of the fleet
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
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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 "Calculation of Age Distributions in the Nonroad Model — Growth and Scrappage," NR-007b
(EPA420-P-04-007), U.S. Environmental Protection Agency, Office of Transportation and Air
Quality, April 2004.
3 "Nonroad Engine Population Estimates," NR-006d, U.S. Environmental Protection Agency,
Office of Transportation and Air Quality, December 2005.
4 "Nonroad Engine Growth Estimates," NR-008c, U.S. Environmental Protection Agency, Office
of Transportation and Air Quality, April 2004.
5 "Median Life, Annual Activity, and Load Factor Values for Nonroad Engine Emissions
Modeling," NR-005c, U.S. Environmental Protection Agency, Office of Transportation and Air
Quality, April 2004.
6 Power Systems Research, Inc. "Reference Guide, U.S. PartsLink." Edition 6.2 St. Paul, MN.
10
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APPENDIX
Numeric Example of Projected Population Growth and Scrappage
As an example of the growth/scrappage calculations we will consider residential lawn mowers.
The basic parameters that the model uses in these calculations are:
Base US Population in 1996:
Median Life
Activity
Load Factor
Growth (linear from 1996)
Scrappage Curve
31,652,672 (3-6 hp bin, from US.POP)
47.9 hours at full load (from US.POP)
25 hours/year (from ACTIVITY.DAT)
0.33 (from ACTIVITY.DAT)
2.25%/yr (1045-1000)7(1998-1996) (fromNATION.GRW)
(from ACTIVITY.DAT)
Table Al. Scrappage as Function of Agea
Mower
Age (yrs)
1
2
3
4
5
6
7
8
9
10
11
12
13
Fraction of
MedLife used
0.000
0.172
0.344
0.517
0.689
0.861
1.033
1.206
1.378
1.550
1.722
1.895
2.067
Cumulative
Percent
scrapped
0.0
3.0
6.5
10.5
16.0
24.0
66.0
79.5
86.0
90.5
94.5
98.0
100.0
1 -Year Fraction
of Original Sales
Scrappedb
0.0
0.030
0.035
0.040
0.055
0.080
0.420
0.135
0.065
0.045
0.040
0.035
0.020
1-Year Fraction
of Remains
Scrapped0
0.0
0.0300
0.0361
0.0428
0.0615
0.0952
0.5526
0.3971
0.3171
0.3214
0.4211
0.6364
1.0000
3 The values in this table are based on the scrappage curve values of Table 1 ,
but are different points on that curve representing one year increments for an
engine with a median life of a residential lawnmower per NONROAD inputs.
b The portion of the original equipment population that is scrapped in the
given year of its life. The sum of these values equals 1.0 (100% of the
original equipment is eventually scrapped).
c The portion of equipment population surviving at the given age that is
scrapped in that year of its life. In the final year of life 100% of the surviving
equipment is scrapped, which equals 2.0% of the original population.
Al
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Using the methodology described in this report, the age distribution in the base year is calculated
as shown in Table A2, using an arbitrary initial "sales" of 1,000 and applying the scrappage from
Table Al along with the sales growth (derived from the input population growth).
In the first year 1,000 units are sold. In the second year 1,028 units are sold, and 30 engines
(3.0%) of the first year's sales are scrapped. In the third year, sales grow to 1,056 units, while
3.0% of the second year's sales are scrapped and an additional 35 units (3.5%) of the first year's
sales are scrapped. This pattern is followed for as many years as it takes to reach a steady-state
point — i.e., where all of the first year's sales have been scrapped. In this particular case using
the median life and activity for lawnmowers that takes 13 years. The first "Pop" column of
Table A2 shows the sum of units sold that year plus those remaining in use that year (not yet
scrapped) from prior years' sales. The row of most interest here is the final (Year 13) row in
which the scrappage curve has been completed and steady-state sales/scrappage has been
reached. The age distribution in this year, for the given set of inputs, represents a starting point
from which future fleet growth (with scrappage) can be calculated. Thus the ratio of each value
in the last row of columns 2 through 13 to the total population (7,819) in that year represents the
in-use model year fraction for a model year that many years old. These fractions are shown in
the first column of Table A3, which represents the base year age distribution that will serve as
the starting point for the growth calculations.
Table A3 shows how the age distribution is calculated for future projection years. Starting from
the base year, which was calculated per the prior paragraph, the entries for the next year
(Base+1) are calculated as follows. For Mower Age 2, multiply the prior year (Base Year) entry
of 0.1706 by the "1-Year Fraction of Remains Scrapped" entry from Table Al for Mower Age
of 2 years (0.03), which equals 0.0051. This is the portion of those engines scrapped that year,
so the remaining portion would be 0.1706 - 0.0051 = 0.1655, which shows up as the Base+1
entry for Mower Age 2. This same procedure is followed for all the older mower ages (3
through 12) in the Base+1 column. Finally, the Mower Age 1 entry is computed as the
difference between the total fleet population (Base grown by one year) and the sum of all the
older model years. I.e., Mower Age 1 fraction = 1.0225 - Sum(0.1655+0.1562+... +0.0027) =
0.1744.
Subsequent growth years (Base+2, Base+3, etc.) are computed the same way, applying the "1-
Year Fraction of Remains Scrapped" values from Table Al to the age distribution fractions of
one year to determine the age distribution in the subsequent year.
A2
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Table A2 Build-up of Base Year Population by Age from Sales and Scrappage
Pop
1028
2053
3069
4070
5039
5949
6449
6820
7129
7396
7625
7819
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
6255
Remains
1000
970
935
895
840
760
340
205
140
95
55
20
0
2
6429
1027.9
997.0
961.0
919.9
863.4
781.2
349.5
210.7
143.9
97.6
56.5
20.6
3
6603
0
1055.7
1024.0
987.1
944.9
886.8
802.3
358.9
216.4
147.8
100.3
58.1
4
6778
0
0
1083.6
1051.1
1013.1
969.8
910.2
823.5
368.4
222.1
151.7
102.9
5
6952
0
0
0
1 1 1 1 .4
1078.1
1039.2
994.7
933.6
844.7
377.9
227.8
155.6
6
7126
0
0
0
0
1139.3
1105.1
1065.2
1019.7
957.0
865.9
387.4
233.6
7
7300
0
0
0
0
0
1167.1
1132.1
1091.3
1044.6
980.4
887.0
396.8
8
7475
0
0
0
0
0
0
1195.0
1159.1
1117.3
1069.5
1003.8
908.2
9
7649
0
0
0
0
0
0
0
1222.9
1186.2
1143.4
1094.5
1027.2
10
7823
0
0
0
0
0
0
0
0
1250.7
1213.2
1169.4
1119.4
11
7997
0
0
0
0
0
0
0
0
0
1278.6
1240.2
1195.5
12
8172
0
0
0
0
0
0
0
0
0
0
1306.4
1267.2
13
8346
0
0
0
0
0
0
0
0
0
0
0
1334.3
A3
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Table A3 Age Distribution with Growth
Mower
Age (years)
1
2
3
4
5
6
7
8
9
10
11
12
13
Total
Fraction of Fleet
Base Year
0.1706
0.1621
0.1529
0.1432
0.1314
0.1161
0.0507
0.0299
0.0199
0.0132
0.0074
0.0026
0.0000
1.0000
Base+1
0.1744
0.1655
0.1562
0.1463
0.1344
0.1189
0.0520
0.0306
0.0204
0.0135
0.0076
0.0027
0.0000
1 .0225
Base+2
0.1780
0.1692
0.1595
0.1495
0.1374
0.1216
0.0532
0.0313
0.0209
0.0138
0.0078
0.0028
0.0000
1.0450
Base+3
0.1816
0.1726
0.1631
0.1527
0.1403
0.1243
0.0544
0.0321
0.0214
0.0142
0.0080
0.0028
0.0000
1.0675
A4
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