Air and Radiation EPA420-R-05-018 

December 2005 

NR-007c 

Calculation of Age 

Distributions in the 

Nonroad Model: 

Growth and Scrappage 
EPA420-R-05-018 

December 2005 

Calculation of Age Distributions in the Nonroad Model:

Growth and Scrappage

NR-007c

Assessment and Standards Division 

Office of Transportation and Air Quality 

U.S. Environmental Protection Agency 

NOTICE

This technical report does not necessarily represent fi nal 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 fi nal EPA decision, position, or regulatory action.
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. 

1 
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 sparkignition equipment. The basis for these estimates is discussed at length in technical report NR006d, "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 

2 
Fraction Surviving

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 

0.2 

0.4 

0.6 

0.8 

1 

0.00 0.50 1.00 1.50 2.00 

Age / Median Life 

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. 

3 
Table 1. Default Scrappage Curve 

Age/Median 

Life* 

Cumulative 

Percent 

Scrapped 

Age/Median 

Life 

Cumulative 

Percent 

Scrapped 

0.0000 0 continued... 

0.0588 1 1.0010 55 

0.1694 3 1.0027 57 

0.2710 5 1.0058 59 

0.3639 7 1.0106 61 

0.4486 9 1.0176 63 

0.5254 11 1.0270 65 

0.5948 13 1.0393 67 

0.6570 15 1.0549 69 

0.7125 17 1.0741 71 

0.7617 19 1.0973 73 

0.8049 21 1.1250 75 

0.8425 23 1.1575 77 

0.8750 25 1.1951 79 

0.9027 27 1.2383 81 

0.9259 29 1.2875 83 

0.9451 31 1.3430 85 

0.9607 33 1.4052 87 

0.9730 35 1.4746 89 

0.9824 37 1.5514 91 

0.9894 39 1.6361 93 

0.9942 41 1.7290 95 

0.9973 43 1.8306 97 

0.9990 45 1.9412 99 

1.0000 50 2.0000 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 

4 
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). 

5 
Figure 2 Age Distribution Example (with growth) 

Age Distribution 

(with 3% linear growth) 

0.00 

0.01 

0.02 

0.03 

0.04 

0.05 

0.06 

0.07 

0.08 

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 

Age / Median Life 

P

op Fractio

n i

n Base Year 

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-1) + Scrappage(Y) 

where 

Sales(Y) = Equipment sales in given year 

Population(Y) = Equipment population at the end of given year 

Population(Y-1) = Equipment population at the end of prior year 

Scrappage(Y) = Equipment scrapped during given year from all model years 

6 
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. 

Inputs 

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. 

7 
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 

Glossary of Terms 

 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 (inservice) 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. 

8 
Glossary of Terms, cont'd 

Term

Model year fraction 

Definition 

The fraction of the total equipment population represented 

by a given model year at a given point in time 

Population growth rate The rate at which the equipment population increases each 

year, in number of units per year 

Sales growth 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. 

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 

Scrappage curve A graphical representation of the scrappage function 

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 

Scrappage rate The percentage of equipment of a given age removed from 

service in a given year 

Target year The calendar year for which the NONROAD model's user 

wishes to estimate emissions and other quantities 

Total equipment population 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 

9 
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 
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: 31,652,672 (3-6 hp bin, from US.POP) 

Median Life 47.9 hours at full load (from US.POP) 

Activity 25 hours/year (from ACTIVITY.DAT) 

Load Factor 0.33 (from ACTIVITY.DAT) 

Growth (linear from 1996) 2.25%/yr (1045-1000)/(1998-1996) (from NATION.GRW)

Scrappage Curve (from ACTIVITY.DAT) 

Table A1. Scrappage as Function of Agea 

Mower 

Age (yrs) 

Fraction of 

MedLife used 

Cumulative 

Percent 

scrapped 

1-Year Fraction 

of Original Sales 

Scrappedb 

1-Year Fraction 

of Remains 

Scrappedc 

1 0.000 0.0 0.0 0.0 

2 0.172 3.0 0.030 0.0300 

3 0.344 6.5 0.035 0.0361 

4 0.517 10.5 0.040 0.0428 

5 0.689 16.0 0.055 0.0615 

6 0.861 24.0 0.080 0.0952 

7 1.033 66.0 0.420 0.5526 

8 1.206 79.5 0.135 0.3971 

9 1.378 86.0 0.065 0.3171 

10 1.550 90.5 0.045 0.3214 

11 1.722 94.5 0.040 0.4211 

12 1.895 98.0 0.035 0.6364 

13 2.067 100.0 0.020 1.0000 a

 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. 

A1 
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 A1 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 A1 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 A1 to the age distribution fractions of 

one year to determine the age distribution in the subsequent year. 

A2 
Table A2 Build-up of Base Year Population by Age from S

ales and Scrappage 

6255 2 3 4 5 6 7 8 9 10 

Pop Year Remains 6429 6603 6778 6952 7126 7300 7475 7649 7823 

1 1000 

1028 2 970 1027.9 0 0 0 0 0 0 0 0 2053 3 935 997.0 1055.7 0 0 0 0 0 0 0 3069 4 895 961.0 1024.0 1083.6 0 0 0 0 0 0 4070 5 840 919.9 987.1 1051.1 1111.4 0 0 0 0 0 5039 6 760 863.4 944.9 1013.1 1078.1 1139.3 0 0 0 0 5949 7 340 781.2 886.8 969.8 1039.2 1105.1 1167.1 0 0 0 6449 8 205 349.5 802.3 910.2 994.7 1065.2 1132.1 1195.0 0 0 6820 9 140 210.7 358.9 823.5 933.6 1019.7 1091.3 1159.1 1222.9 0 7129 10 95 143.9 216.4 368.4 844.7 957.0 1044.6 1117.3 1186.2 1250.7 7396 11 55 97.6 147.8 222.1 377.9 865.9 980.4 1069.5 1143.4 1213.2 17625 12 20 56.5 100.3 151.7 227.8 387.4 887.0 1003.8 1094.5 1169.4 17819 13 0 20.6 58.1 102.9 155.6 233.6 396.8 908.2 1027.2 1119.4 1

A3 
Table A3 Age Distribution with Growth 

Mower 

Age (years) 

Fraction of Fleet 

Base Year Base+1 Base+2 Base+3 

1 0.1706 0.1744 0.1780 0.1816 

2 0.1621 0.1655 0.1692 0.1726 

3 0.1529 0.1562 0.1595 0.1631 

4 0.1432 0.1463 0.1495 0.1527 

5 0.1314 0.1344 0.1374 0.1403 

6 0.1161 0.1189 0.1216 0.1243 

7 0.0507 0.0520 0.0532 0.0544 

8 0.0299 0.0306 0.0313 0.0321 

9 0.0199 0.0204 0.0209 0.0214 

10 0.0132 0.0135 0.0138 0.0142 

11 0.0074 0.0076 0.0078 0.0080 

12 0.0026 0.0027 0.0028 0.0028 

13 0.0000 0.0000 0.0000 0.0000 

Total 1.0000 1.0225 1.0450 1.0675 

A4