This chapter discusses the way HERS models discrete processes: pavement wear, vehicle operating costs, crash costs, traffic forecasts, speed calculation, travel time costs, demand elasticity, agency costs, and external costs. All of these processes are affected by the implementation of an improvement or the passage of time on an unimproved section.
HERS distinguishes the following components of user costs: travel time costs, vehicle operating costs, and safety costs, which includes both property damage and personal injury. Within the context of the demand elasticity model, these costs make up the user price. User benefits are simply the difference in costs between two predicted future states of the section under consideration: typically, an improvement will lower user costs, producing a benefit. User costs are calculated per vehicle mile of travel; total user costs are a product of user costs per vehicle mile times section length times AADT.
HERS models pavement wear as a function of traffic and environment. First, HERS calculates the effects of vehicular traffic on a section's PSR. Then, HERS figures both a minimum and a maximum rate of deterioration. The minimum rate is designed to reflect the effects of weather. The maximum rate of deterioration is designed to limit deterioration on sections with low structural numbers^{1}. HERS applies these limits to the PSR value (which reflects pavement wear due to traffic) to arrive at a forecast pavement condition. HERS does not deteriorate unpaved sections, and roads without reported truck traffic are deteriorated at the minimum rate.
Except for roads with relatively light traffic volumes, the rate of pavement deterioration is dependent primarily on the number of 18,000 pound (18 kip) equivalent single-axle loads (ESALs). For any time period, ESALs on the most heavily traveled lane of each sample section are estimated using
1. Structural numbers (SN), which range from 1.0 to 6.0, indicate the strength of pavement. Sections whose SN is in the range from 1.0 through 3.0 are considered "light".
The 18-kip equivalent load factor is a function of pavement type, functional class, and truck type; values for this factor are given in Table 5-1. The lane load adjustment factor provides an estimate of the percentage of trucks that travel in the lane most heavily used by trucks as a function of the number of lanes in one direction; these values follow the AASHTO Pavement Design Guide^{2} and are given in Table 5-2.
Single Unit Trucks | Combination Trucks | |||
---|---|---|---|---|
Flexible Pavement | Rigid Pavement | Flexible Pavement | Rigid Pavement | |
Rural: | ||||
Interstate | 0.2898 | 0.4056 | 1.0504 | 1.6278 |
Other Principal Arterials | 0.3141 | 0.4230 | 1.1034 | 1.7651 |
Minor Arterials | 0.2291 | 0.3139 | 1.0205 | 1.0819 |
Collectors | 0.2535 | 0.3485 | 0.7922 | 1.3265 |
Urban: | ||||
Interstate and Other Freeways and Expressways | 0.6047 | 0.8543 | 2.3517 | 3.7146 |
Other Principal Arterials | 0.5726 | 0.8123 | 0.8584 | 1.3047 |
Minor Arterials | 0.3344 | 0.4109 | 1.0433 | 1.5276 |
Collectors | 0.8126 | 1.1595 | 0.6417 | 0.9968 |
Number of Lanes (One Direction) | Lane Factor |
---|---|
1 | 1.0 |
2 | 0.9 |
3 | 0.7 |
4 or more | 0.6 |
HERS estimates pavement deterioration using Percent Average Daily Single Unit Commercial Vehicles and Percent Average Daily Combination Commercial Vehicles. HERS allows the user to specify a set of annual growth factors to be applied to each section's percent truck values. (See section 2.11 "The Fleet Composition Model" on page 2-15.)
For any time period beginning at t_{0} and ending at t_{f}, HERS first calculates the total traffic:
2. American Association of State Highway and Transportation Officials, AASHTO Guide for Design of Pavement Structures, Washington, D.C., 1986.
TOTRAF = | (AADT_{t0} + AADT_{t1}) | × 365 × (t_{f}∠t_{0}) |
2 |
where (t_{f} ∠ t_{0}) represents the length of the period in years.
HERS then calculates ESALs for the time period:
ESALS = (TOTRAF × PCAVSU × ELF_{SU} × LF) + (TOTRAF × PCAVCM × ELF_{CM} × LF)
where:
ESALS | = | ESALs accumulated during the time period; |
---|---|---|
PCAVSU | = | average percentage of single-unit trucks during the time period; |
ELF_{SU} | = | equivalent load factor for single unit trucks for this pavement type and functional class (from Table 5-1); |
PCAVCM | = | average percentage of combination trucks during the time period; |
ELF_{CM} | = | equivalent load factor for combination trucks for this pavement type and functional class (from Table 5-1); and |
LF | = | lane load distribution factor (from Table 5-2). |
HERS uses one-half the length of a funding period as the time period for calculating total traffic and incremental ESALs in order to capture changes in both AADT and average percentages of trucks. Therefore, when estimating the number of ESALs which will accumulate during a funding period, it utilizes Equations 5.1 through 5.2 twice, once for the first half and once for the second half of the funding period.
HERS determines present and future pavement condition using the AASHTO 1993 guidelines. The first step is to obtain the number of ESALs that would have resulted in causing PSR to decline from 5.0 to its base-year value. The number of ESALs applied during any subsequent period is then estimated and added to the previous ESAL value. This result is then used to estimate PSR at the end of this period.
For flexible pavement, the HPMS database contains either the structural number (SN) or pavement weight (light, medium or heavy); for rigid pavement it contains either thickness (D) or pavement weight. If any of the optional information is not provided for a section, HERS uses the default values shown in Table 5-3 to obtain values describing the initial pavement. When the pavement is improved, procedures described in section 4.2.4 "Effects of HERS Improvements" on page 4-20 are used to obtain the thickness of the overlays or of the new pavement and, for flexible pavements, a new value of SN.
For flexible pavements, the number of ESALs that would cause PSR to decline from 5.0 to its base-year value is obtained using the equation:
ESAL = 10^{LOGELA} |
where:
LOGELA = XA + XG/XB + X0 + XM |
XA = 9.36 × log(SNA) ∠ 0.2 |
XB = 0.4 + 1094/SNA^{5.19} |
XG = log((5 ∠ PSRI)/3.5) |
SNA = SN + (6/SN) 5.8 |
X0 = ZR × S0 |
XM = 2.32 × log(MR) ∠ 8.07 |
ZR = ANORIN(REL) |
and
PSRI | = | PSR at the beginning of the base year; |
---|---|---|
ANORIN | = | name of the function that evaluates the inverse of the standard normal (Gaussian) distribution function; |
S0, REL, MR | = | input parameters taken from Table 5-4; |
and all logarithms are taken to the base ten.
Pavement Section | |||
---|---|---|---|
Heavy | Medium | Light | |
SN (Flexible Pavement) | 5.3 | 3.8 | 2.3 |
D (Rigid Pavement) | 10.0 | 8.0 | 6.5 |
Input Parameter | Description | Varies By | Default Values | |
---|---|---|---|---|
S0 | Prediction error | Pavement type | 0.49 | Flexible |
0.39 | Rigid | |||
REL | Reliability Factor | Functional Class | 90% | Interstates |
85% | Other Arterials | |||
80% | Collectors | |||
MR | Modulus of Resistance | Functional Class | 4000 for all FC's |
The PSR at the end of any subsequent time period, PSRF, is then obtained by adding the number of ESALs incurred during that time period to the initial value of ESALs, substituting PSRF for PSRI in Equation 5.7, solving the above system of equations for PSRF, and performing the indicated computations. Solving Equation 5.7 for PSRF produces:
PSRF = 5 ∠ 3.5 × PDRAF_{pt} × 10^{XG} |
where:
PDRAF_{pt} | = | A user-specified pavement deterioration rate adjustment factor for pavement type pt, normally set to one^{3}; |
---|
and solving Equation 5.3 and Equation 5.4 for XG produces:
XG = XB × (log(ESALTF) ∠ XA ∠ X0 ∠ XM) |
where ESALTF is the total number of ESALs accumulated at the time of interest.
The procedure for obtaining the pavement condition of rigid pavements follows the same approach as that for flexible pavements: it also is based upon Equation 5.3. But for rigid pavements, LOGELA is defined:
LOGELA = X0 + XA + XG/XB + XN × XC |
where:
XA = 7.35 × log(D + 1) ∠ 0.06 |
XB = 1 + 16.24 × 10^{6}/(D + 1)^{8.46} |
XN = 4.22 ∠ 0.32 × PT |
and
X0 | = | as per flexible pavement in Equation 5.9; |
---|---|---|
XG | = | as per flexible pavement in Equation 5.7; |
D | = | pavement thickness; |
and PT, SCP, CD, J, EC, and K are input parameters as shown in Table 5-5.
For rigid pavement, the solution for XG (the analogue to Equation 5.13) becomes:
XG = XB × (log(ESALTF) ∠ XA ∠ X0 ∠ XN × XC) |
and HERS uses Equation 5.12 to solve for PSRF, the PSR at the time of interest.
3. If HERS is being used to analyze data for a single state, PDRAF_{pt} can be used to reflect the effects of the state's environment and materials used in that state. Separate values of PDRAF_{pt} can be specified for flexible and rigid pavement types.
Input Parameter | Description | Default Value |
---|---|---|
PT | Design terminal serviceability index | 2.5 |
SCP | Modulus of rupture | 600 |
CD | Load transfer coefficient | 1 |
J | Drainage coefficient | 3.0 |
EC | Modulus of elasticity | 3.5 x 10^{6} |
K | Modulus of subgrade reaction | 200 |
For both flexible and rigid pavements, minimum deterioration rates are used to reflect pavement deterioration due to environmental conditions. HERS uses the following equation to calculate an appropriate minimum deterioration rate:
PSRMAX_{t} = PSR_{t0} × ((PDL)/(NPSRAI))^{((t ∠ t0)/ML)} |
where:
t | = | any time of interest; |
---|---|---|
PSRMAX_{t} | = | upper limit on the PSR of a given section at time t; |
t_{0} | = | time at which the section was last improved or, if not known, six months before the beginning of the HERS run; |
PDL | = | pavement deficiency level; |
NPSRAI | = | "normal" PSR after improvement; and |
ML | = | maximum life of the section in years from Table 5-6. |
The use of Equation 5.20 requires knowing the time that each section was last improved (t_{0}) and the PSR immediately after the improvement (PSR_{t0}). For all improvements analyzed or selected by HERS, this information is readily available. For improvements that occurred prior to the start of a HERS run, the preprocessor uses the time of last improvement specified in the HPMS dataset, if available, or the middle of the year preceding the start of the HERS run. If the preprocessor finds that the section's initial PSR is greater than the maximum PSR on that section after resurfacing, the time of improvement t_{0} is set to six months before the beginning of the analysis period and the initial PSR is used as the PSR after improvement NPSRAI. Otherwise, the preprocessor uses the maximum permissible PSR after resurfacing.
The maximum pavement life values for rigid and flexible pavements for three types of pavement section (light, medium and heavy) are shown in Table 5-6.
Surface Type | Pavement Section | ||
---|---|---|---|
Heavy | Medium | Light | |
Flexible | 25 | 20 | 15 |
Rigid | 30 | 25 | 20 |
The HERS model then enforces the minimum deterioration rate:
PSRMX_{t} = the lesser of | PSRMAX_{t} | |
PSR_{tESALS} |
where:
PSRMAX_{t} | = | upper limit on PSR at time t from Equation 5.20; |
---|---|---|
PSR_{tESALS} | = | PSR at time t as a function of ESALs (PSRF from Equation 5.12); and |
PSRMX_{t} | = | PSR at time t after enforcement of the minimum deterioration rate. |
A user-specified maximum PSR deterioration rate is used to limit pavement deterioration on sections with low values of SN. The default value for this maximum rate of deterioration is 0.3 per year. This maximum rate is applied after the enforcement of the minimum deterioration rate:
where:
PSR_{t} = the larger of | PSR_{t0} ∠ MAXPDR × (t ∠ t_{0}) | |
PSRMX_{t} |
t | = | any time of interest; |
---|---|---|
PSR_{t} | = | PSR at the time t after enforcement of both the maximum and minimum deterioration rates; |
t_{0} | = | time at which the section was last improved or, if not known, six months before the beginning of the HERS run; |
PSRMX_{t} | = | PSR at time t after enforcement of the minimum deterioration rate from Equation 5.21; and |
MAXPDR | = | maximum PSR deterioration rate per year. |
Having forecast the future PSR based upon ESALs, and then applied limitations based upon minimum and maximum pavement deterioration rates, HERS applies a minimum PSR level below which no section is permitted to deteriorate. The minimum level is defined by PSRUPS, the PSR value for unpaved sections. The user specifies PSRUPS as a control input to HERSPP (see paragraph 3.2.1, "The Preprocessor Control Inputs"). The supplied PSRUPS value is 1.0.
The cost of operating a vehicle on a given section is a function of costs for fuel, oil, tires, maintenance and repair, and mileage-related depreciation. This section discusses the method by which HERS estimates operating costs. These estimates exclude the effect of taxes.^{4}
HERS treats operating costs as having three sources, and derives its estimates using a three-step procedure:
Figure 5-1 provides an overview of the operating cost calculations.
The operating cost calculation process, as outlined above and detailed in the paragraphs below, is performed for each of the seven vehicle types. For the two truck categories the process is performed once for each direction unless free-flow speed and uphill free-flow speed are the same (see section 5.4.1 "free-Flow Speed and the APLVM" on page 5-33). The process is performed only once for four-wheel vehicles, as HERS assumes that grades do not affect free-flow speed for these vehicles.
HERS recognizes five components of operating costs:
All five components are included in the calculation of constant-speed costs and excess costs due to speed change cycles: for excess costs due to curves, only fuel, tire wear, and maintenance and repair are included.
Table 5-7 shows estimates of component prices in 1997 dollars for use in estimating operating costs. The sources of these estimates are described below.
Fuel prices for two-axle vehicles were derived by subtracting federal and state gasoline taxes^{5} from the 1997 retail price of gasoline, and fuel prices for larger vehicles were derived by subtracting taxes on diesel fuel from the average 1997 retail price of highway diesel fuel.^{6}
4. From the standpoint of the user, taxes are part of user costs. However, from the standpoint of the overall economy, taxes are transfer payments that entail no resource costs.
5. U. S. Department of Transportation, Federal Highway Administration, Highway Statistics, 1997, Washington, D.C., 1998, Table MF-121T.
6. U.S. Department of Energy, Energy Information Administration, "On-Highway Diesel Fuel Price Survey," Form EIA-888, 1995.
Figure 5-1. Operating Cost Calculation Flow
Vehicle Type | Fuel ($/gallon) | Oil ($/quart)^{a} | Tires ($/tire) | Maintenance and Repair ($/1,000 miles) | Depreciable Value ($/vehicle) |
---|---|---|---|---|---|
a. The unit cost for oil includes the labor charge for changing the oil. | |||||
Automobiles | |||||
Small | $0.871 | $3.573 | $45.2 | $84.1 | $18,117 |
Medium/Large | $0.871 | $3.573 | $71.5 | $102.1 | $21,369 |
Trucks | |||||
Single Units | |||||
4 Tires | $0.871 | $3.573 | $78.8 | $129.8 | $23,028 |
6 Tires | $0.871 | $1.429 | $190.1 | $242.9 | $34,410 |
3+ Axles | $0.762 | $1.429 | $470.7 | $343.5 | $75,702 |
Combination | |||||
3-4 Axles | $0.762 | $1.429 | $470.7 | $355.8 | $87,690 |
5+ Axles | $0.762 | $1.429 | $470.7 | $355.8 | $95,349 |
Values for the cost of oil and tires were obtained by applying appropriate price indexes to the 1995 estimates previously developed^{7} from the original Zaniewski estimates^{8}. The price index used for oil is the consumer price index (CPI)^{9} for motor oil, coolant, and fluids (SS47021). Tire costs were indexed using the CPI for tires (SETC01). The tire-cost index reflects the effects of improvements in quality (as downward adjustments in the index) - improvements that generally decrease the rate of tire wear. Maintenance and repair costs were indexed using the CPI for motor vehicle maintenance and repair (SETD).
For medium and heavy trucks, following Zaniewski, depreciable value was obtained by subtracting tire costs from the vehicle's retail price and then subtracting ten percent salvage value. For the three heaviest vehicles, the vehicle prices were those used by the recent Federal Highway Cost Allocation Study^{10} (for three-axle dump trucks and for combinations with a tandem-axle van semi-trailer). The retail price of a 1995 28,000 pound gross vehicle weight six-tire truck was obtained from the Truck Blue Book^{11} and adjusted to include a van body.
For the two classes of automobiles, 1995 depreciable value was obtained by adjusting the 1993 values^{12} for changes in the average price paid for a new car.^{13} For four-tire trucks, 1995 depreciable value was obtained judgmentally from the 1995 value for medium/large automobiles by comparing the range of list prices of minivans and sport-utility vehicles to the range for medium and large automobiles.^{14} For all vehicle classes, 1997 depreciable value was then obtained by applying the change in the average price of a new car between 1995 and 1997.
7. Cambridge Systematics, Inc., Revisions to HERS, prepared for the Federal Highway Administration, December 1997, Chapter 7.
8. J.P. Zaniewski, et.al., Vehicle Operating Costs, Fuel Consumption, and Pavement Type and Condition Factors, Texas Research and Development Foundation, prepared for U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., June 1982, Table 2, p. 7.
9. U.S. Department of Labor, Bureau of Labor Statistics, Consumer Price Index Database.
10. U. S. Department of Transportation, FHWA, 1997 Federal Highway Cost Allocation Study, August 1997.
11. Maclean Hunter Market Reports, Truck Blue Book, Chicago, January 1995.
12. Cambridge Systematics, Inc., op. cit. The 1993 values were derived from the original Zaniewski values using the same procedure.
13. U.S. Department of Commerce, Bureau of Economic Analysis, "Average Transaction Price of a New Car," quoted in American Automobile Manufacturers Association, Motor Vehicle Facts and Figures, 1996, Detroit, 1996, page 60. This source provides a better indication of changes in vehicle prices than the appropriate components of the CPI and PPI, because the latter indexes are adjusted (downward) to exclude the effect on prices of improvements in the quality of new vehicles. On the other hand, none of the adjustments reflect the effects that some of these improvements have had on servicing requirements or depreciation rates (which, ideally, should be handled by modifying the operating cost equations for maintenance and repair).
14. The alternative approach of adjusting the original Zaniewski values using data on changes in the price of a new car was rejected because it does not adequately reflect the increase in the quality of appointments of four-tire trucks that has occurred during the last several years. The rejected procedure produces a 1995 value of only $17,002 (instead of the $20,742 value actually used).
The parameters used by the operating cost equations have been indexed to reflect reductions in fuel and oil consumption rates and depreciation rates that occurred between 1980 and 2000. Increases in tire durability are reflected in the consumer and producer price indexes (which have increased by only a few percent since 1980), so separate adjustments are not needed for changes in the rate of tire wear. Similarly, no adjustments were made in the amount of maintenance required; reductions in the requirements for routine maintenance are reflected in the data used for adjusting maintenance costs per mile through 1995^{15} (but not in the BLS data used for the 1995-2000 adjustment).
The adjustments for changes in fuel efficiency, oil consumption, and vehicle depreciation are discussed below.
The fuel efficiency adjustment factors for automobiles and four-tire trucks were obtained by dividing on-road fuel efficiency for the 2000 fleet of automobiles and light trucks by corresponding 1980 values. The 1980 values were obtained from Energy and Environmental Analysis.^{16} The 2000 values were developed using the following data from the Transportation Energy Data Book published annually by Oak Ridge National Laboratory (ORNL):^{17}
The surviving fleets of pre-1970 light trucks and pre-1976 automobiles were assumed to be three times the number of surviving 1970 light trucks and 1976 automobiles, respectively. Fuel efficiencies of pre-1976 automobiles were estimated by extrapolation, while fuel efficiencies of pre-1976 light trucks were assumed to be the same as those of 1976 light trucks (which are 11 percent below those of 1977 light trucks). All averaging was performed using fuel consumption rates (gallons per mile); and in-use fuel efficiency was assumed to be 15 percent below the EPA value.
15. Runzheimer International, quoted in American Automobile Manufacturers Association, Motor Vehicle Facts and Figures, 1996, Detroit, 1996, p. 58.
16. Energy and Environmental Analysis, Inc., The Motor Fuel Consumption Model: Thirteenth Periodical Report, prepared for the U.S. Department of Energy, Washington, D.C., January 1988, page B-1.
17. Oak Ridge National Laboratory, Transportation Energy Data Book, various editions.
A single 20-year fuel-efficiency adjustment factor for the three classes of heavy trucks was developed by obtaining the ratio of the 1997 and 1977 fuel-efficiency estimates for in use Class 8 trucks developed by ORNL using data from the Truck Inventory and Use Survey (TIUS). The relatively modest increase in fuel efficiency (39.6 percent over 20 years) is due, in part, to increases in vehicle weights.
The fuel efficiency adjustment factor for six-tire trucks was similarly developed from TIUS data using a weighted average of fuel efficiency estimates for Class 6 trucks (19,500 to 26,000 pound gross vehicle weight). Class 6 is the largest of the five truck classes (Classes 3-7) that consist primarily of six-tire trucks. Use of data for a single truck class minimizes the effect of changes in the mix of six-tire vehicles occurring over the period.
The resulting fuel efficiency adjustment factors are shown in Table 5-8.
Vehicle Type | Factor |
---|---|
Small Automobiles | 1.550 |
Medium/Large Automobiles | 1.550 |
4-Tire Trucks | 1.666 |
6-Tire Trucks | 1.344 |
3+ Axle Trucks | 1.396 |
3-4 Axle Combinations | 1.396 |
5+ Axle Combinations | 1.396 |
The most common recommended oil-change interval for new automobiles was 7,500 miles in both 1980 and 2000. However, for various reasons, some slight reduction in oil consumption between these two years was likely. (These reasons include a reduction in the number of older cars with shorter oil-change intervals and reduced burning of oil.) Accordingly, an oil-consumption reduction factor of 1.05 was assumed for all vehicle classes.
The average age of the automobile fleet increased from 6.6 years in 1980 to 8.6 years in 1996.^{18} Extrapolating to 2000, we estimate average age to be 9.1 years, suggesting a 38 percent increase in longevity (or a decline in the average rate of depreciation of about 28 percent). The same increase in average longevity was assumed for trucks.
18. R. L. Polk and Company, as quoted in American Automobile Manufacturers Association, Motor Vehicle Facts and Figures, 1996, Detroit, 1996, p. 39.
For each vehicle type (vt), constant-speed operating cost per thousand vehicle-miles (CSOPCST) is estimated as the sum of five cost components representing costs for fuel, oil, tires, maintenance and repair, and vehicle depreciation. The overall equation for combining these components is:
CSOPCST_{vt} = | CSFC × PCAFFC × COSTF_{vt}/FEAF_{vt} |
+ CSOC × PCAFOC × COSTO_{vt}/OCAF_{vt} | |
+ 0.01 × CSTW × PCAFTW × COSTT_{vt}/TWAF_{vt} | |
+ 0.01 × CSMR × PCAFMR × COSTMR_{vt} /MRAF_{vt} | |
+ 0.01 × CSVD × PCAFVD × COSTV_{vt}/VDAF_{vt} |
where:
CSOPCST_{vt} | = | constant speed operating cost for vehicle type vt; |
---|---|---|
CSFC | = | constant speed fuel consumption rate (gallons/1000 miles); |
CSOC | = | constant speed oil consumption rate (quarts/1000 miles); |
CSTW | = | constant speed tire wear rate (% worn/1000 miles); |
CSMR | = | constant speed maintenance and repair rate (% of average cost/1000 miles); |
CSVD | = | constant speed depreciation rate (% of new price/1000 miles); |
PCAFFC | = | pavement condition adjustment factor for fuel consumption; |
PCAFOC | = | pavement condition adjustment factor for oil consumption; |
PCAFTW | = | pavement condition adjustment factor for tire wear; |
PCAFMR | = | pavement condition adjustment factor for maintenance and repair; |
PCAFVD | = | pavement condition adjustment factor for depreciation expenses; |
COSTF_{vt} | = | unit cost of fuel for vehicle type vt; |
COSTO_{vt} | = | unit cost of oil for vehicle type vt; |
COSTT_{vt} | = | unit cost of tires for vehicle type vt; |
COSTMR_{vt} | = | unit cost of maintenance and repair for vehicle type vt; |
COSTV_{vt} | = | depreciable value for vehicle type vt; |
FEAF_{vt} | = | fuel efficiency adjustment factor for vehicle type vt; |
OCAF_{vt} | = | oil consumption adjustment factor for vehicle type vt; |
TWAF_{vt} | = | tire wear adjustment factor for vehicle type vt; |
MRAF_{vt} | = | maintenance and repair adjustment factor for vehicle type vt; and |
VDAF_{vt} | = | depreciation adjustment factor for vehicle type vt. |
Equations for estimating constant-speed consumption rates for fuel, oil, tires, maintenance and repair, and vehicle depreciation are shown in Appendix E, "Operating Cost Equations." In these equations, AES is average effective speed in miles per hour (an output of the speed model), and GR is grade (in percent). The equations were estimated by applying ordinary least squares regression to the consumption tables presented in Zaniewski,^{19} and have been modified to handle the higher speeds that HERS will encounter as a result of the recent increase in speed limits.
The Zaniewski tables represent estimated consumption rates for equipment in use in 1980 on roads with PSR = 3.5. Table e-16 on page e-11 presents equations for estimating pavement-condition adjustment factors for oil consumption, tire wear, maintenance and repair, and vehicle depreciation. These equations also were estimated by applying ordinary least squares regression to the adjustment factors presented in Zaniewski.^{20} Zaniewski does not provide pavement-condition adjustment factors for fuel consumption.
Accordingly, the corresponding adjustment factor used for HERS is set to one. However, the factor has been included in the code for symmetry and to allow development of such a factor in the future.
HERS calculates excess operating costs due to speed-change cycles (or speed variability) for sections which have stop signs or traffic signals. The overall formula for calculating these costs is similar to that for constant speed operating costs (see Equation 5.23) with two exceptions: the consumption rates are derived from a different set of equations, and no pavement condition adjustment factors are used. For each vehicle type (vt), excess operating costs per thousand vehicle-miles due to speed variability (VSOPCST) is estimated:
VSOPCST_{vt} = | VSFC × COSTF_{vt}/FEAF_{vt} |
+ VSOC × COSTO_{vt}/OCAF_{vt} | |
+ VSTW × COSTT_{vt}/TWAF_{vt} | |
+ VSMR × COSTMR_{vt}/MRAF_{vt} | |
+ VSVD × COSTV_{vt}/VDAF_{vt} |
where:
VSOPCST_{vt} | = | excess operating cost due to speed variability for vehicle type vt; |
---|---|---|
VSFC | = | excess fuel consumption rate due to speed variability (gallons/1000 miles); |
VSOC | = | excess oil consumption rate due to speed variability (quarts/1000 miles); |
VSTW | = | excess speed tire wear rate due to speed variability (% worn/1000 miles); |
VSMR | = | excess speed maintenance and repair rate due to speed variability (% of average cost/1000 miles); |
VSVD | = | excess depreciation rate due to speed variability (% of new price/1000 miles); |
COSTF_{vt} | = | unit cost of fuel for vehicle type vt; |
COSTO_{vt} | = | unit cost of oil for vehicle type vt; |
COSTT_{vt} | = | unit cost of tires for vehicle type vt; |
COSTMR_{vt} | = | unit cost of maintenance and repair for vehicle type vt; |
COSTV_{vt} | = | depreciable value for vehicle type vt; |
FEAF_{vt} | = | fuel efficiency adjustment factor for vehicle type vt; |
OCAF_{vt} | = | oil efficiency adjustment factor for vehicle type vt; |
TWAF_{vt} | = | tire wear efficiency adjustment factor for vehicle type vt; |
MRAF_{vt} | = | maintenance and repair efficiency adjustment factor for vehicle type vt; and |
VDAF_{vt} | = | depreciation adjustment factor for vehicle type vt. |
20. Ibid., Figure 5 and Tables 12, 15, and 19.
These equations were also derived from Zaniewski, and are only applied to sections with stop signs or traffic signals. The equations are shown in Tables e-28 through e-91.
Signals and stop signs (as a group) are assumed to be uniformly spaced on each section. (This assumption is also used in the speed model.) Sections with both signals and stop signs are treated as having all signals at one end of the section and all stop signs at the other end. The two portions of the sections are analyzed separately, producing separate estimates of excess costs per 1000 cycles for the stop-sign and traffic signal portions of the section.
For each section, the estimates of excess costs per 1000 cycles are converted to excess costs per 1000 miles by dividing by the average distance between stops for stop signs and traffic signals. For traffic signals, this denominator reflects an adjustment for the probability of actually being stopped at a traffic signal. If both stop signs and traffic signals exist on the section, the sum of the excess costs for the two parts of the section is used.
HERS uses the original Zaniewski tables^{21} and equations derived from those tables for estimating excess operating costs due to curves. two-dimensional linear interpolation of table values is used for sections with average effective speed below 55 m.p.h., and equations fit to the tables are used for sections with average effective speed above 55 m.p.h. On sections with zero degrees of curvature, excess costs are set to zero.
For medium and high speeds (generally above 40 m.p.h.), the Zaniewski values for excess costs due to curves with one degree of curvature are higher (and sometimes substantially higher) than those due to curves with two degrees of curvature. The values for one degree of curvature were deemed to be excessive and were ignored in estimating the equations for average effective speeds above 55 m.p.h. Similarly, the questionably high values for one degree of curvature were modified to more reasonable values in the tables used for sections with average effective speeds below 55 m.p.h.
21. Ibid., Appendix A, Tables A.73-A.80.
HERS uses the individual Zaniewski tables for the effects of curves on fuel consumption, tire wear, and maintenance and repair. (The effects of curves on vehicle depreciation and oil consumption were assumed to be negligible by Zaniewski.) During program initialization, the values in these tables are:
The result is a single table of excess costs due to curves for each vehicle type (in dollars per 1000 vehicle miles) as a function of curvature and speed (up to 55 m.p.h.). For individual sections, excess costs due to curves for each vehicle type are estimated using average effective speed and curvature on the sections and using two-dimensional linear interpolation between entries in the table.
For sections with average effective speeds equal to or greater than 55 m.p.h., HERS uses equations fit to the Zaniewski values given for speeds of 55-70 m.p.h. and two degrees of curvature or more. Equations for use with sections having two or less degrees of curvature were devised to match the modified table values. Similar to the overall formula for constant-speed operating costs, HERS calculates the excess cost due to curves (COPCST) for each vehicle type on sections with average effective speed greater than 55 m.p.h.:
COPCST_{vt} = | CFC × COSTF_{vt}/FEAF_{vt} |
+ 0.01 × CTW × COSTT_{vt}/TWAF_{vt} | |
+ 0.01 × CMR × COSTMR_{ct}/MRAF_{vt} |
where:
COPCST_{vt} | = | excess operating cost due to curves for vehicle class vt; |
---|---|---|
CFC | = | excess fuel consumption rate due to curves (gallons/1000 miles); |
CTW | = | excess tire wear rate due to curves (% worn/1000 miles); |
CMR | = | excess maintenance and repair rate due to curves (% of average cost/1000 miles); |
COSTF_{vt} | = | unit cost of fuel for vehicle type vt; |
COSTT_{vt} | = | unit cost of tires for vehicle type vt; |
COSTMR_{vt} | = | unit cost of maintenance and repair for vehicle type vt; |
FEAF_{vt} | = | fuel efficiency adjustment factor for vehicle type vt; |
TWAF_{vt} | = | tire wear adjustment factor for vehicle type vt; and |
MRAF_{vt} | = | maintenance and repair adjustment factor for vehicle type vt. |
The equations used to produce CFC, CTW, and CMR are shown in Tables e-40 through e-53.
The HERS operating cost process is implemented as two nested loops. The outer loop propels the model through each vehicle type in turn. The inner loop is executed twice, once in each direction for each vehicle type. The calculation of the three categories of operating costs is performed within this inner loop. When operating costs in both directions for all vehicle types have been calculated, the model weights the costs using the procedures in section 2.11 "The Fleet Composition Model" on page 2-15 to arrive at the total operating cost per vehicle mile over the section.
The HERS safety analysis is a three-step procedure:
The procedures for estimating the number of crashes are described and documented in the next paragraph. The subsequent section, 5.3.2, "Fatalities and Injuries," presents the injury/crash ratios and fatality/crash ratios. Section 5.3.3, "Secular Trends," develops estimates, for use by HERS, of the extent to which recent secular declines in crash rates, fatality/crash ratios, and injury/crash ratios are due to factors not analyzed by HERS. In section 5.3.4, "Costs of Crashes," data from a recent report by the National Highway Traffic Safety Administration (NHTSA) is used to update HERS' estimates of costs per injury and property damage costs per crash, as well as to provide estimates of the cost of travel-time delay per crash.
HERS estimates the numbers of crashes and crash rates using separate procedures for three types of rural facility and three types of urban facility. The facility types distinguished are:
The freeway procedures are used for all divided roads^{22} with four or more lanes and full access control, and also for all one-way roads with two or more lanes and full access control. These procedures are used for these roads regardless of functional system. For all other facilities with four or more lanes and all other one-way facilities with two or three lanes, the "multi-lane" procedures are used, again regardless of functional system. Finally, the "two-lane" procedures are used for all two-way facilities with fewer than four lanes and for all one-lane facilities.
Five of the procedures are slightly modified versions of procedures recommended by Richard Margiotta based on an extensive review of the literature.^{23} The sixth procedure is derived from the results of an analysis by Vogt and Bared^{24} that was performed after the completion of Margiotta's work. All procedures were modified to produce estimates of crash rates per 100 million vehicle-miles of travel (VMT) and calibrated to crash-rate data for 1995. The six procedures (after calibration) are described in sections 5.3.1.1 through 5.3.1.6, and the calibration is described in section 5.3.1.7.
The procedure for estimating crashes on rural two-lane roads develops separate estimates of crashes within 250 feet of an intersection and crashes on segments between intersections. Both sets of estimates are developed using equations based on those developed by Vogt and Bared.^{25} These estimates are then combined:
CRASH = 1.056 × (CNINT + CINT) |
where
CRASH | = | total number of crashes on the section per 100 million VMT; |
---|---|---|
CNINT | = | non-intersection crashes per 100 million VMT; |
CINT | = | crashes occurring within 250 feet of an intersection, per 100 million VMT on the section; |
and 1.056 is the calibration factor developed in section section 5.3.1.7 "Calibration" on page 5-26. The procedures for estimating CNINT and CINT are presented below.
The equation for estimating non-intersection crashes is based on an equation developed by Vogt and Bared using Highway Safety Information System (HSIS) data for Minnesota and Washington.^{26} The HERS equation is:
22. For the purpose of the safety analysis, a divided road has a positive barrier median or a median width of at least four feet. This definition is slightly narrower than the HPMS definition of a divided (FHWA, Highway Performance Monitoring System Field Manual, Appendix I, pp. I-1 and I-8, January 1998). The HPMS definition, which is used by the HERS capacity procedures, also classifies roads with curbed medians of any width as being divided. Because narrow curbed medians provide relatively limited protection from median crossing, our safety procedure treats roads with curbed medians and median widths of less than four feet as being undivided.
23. Richard Margiotta, Incorporating Traffic Crash and Incident Information into the Highway Performance Monitoring System Analytical Process, prepared by COMSIS Corporation and Science Applications International Corporation for FHWA, September 1996, Chapter 2.
24. Andrew Vogt and Joe Bared, "Accident Models for two-Lane Rural Segments and Intersections," presented at the TRB Annual Meeting, January 1998.
26. Ibid., p. 6. Vogt and Bared also use the data from Minnesota and Washington separately to develop two additional equations for non-intersection crashes (p. 5).
where
SLEN | = | section length (in miles); |
---|---|---|
ADJSL | = | section length adjusted to exclude segments within 250 feet of an intersection; |
LW | = | lane width (in feet); |
SHW | = | shoulder width (in feet); |
RHR | = | roadside hazard rating (3.0); |
DD | = | driveway density (per mile) (3.7 for rural type of development, 50 for dense development); |
CURV_{i} | = | average degrees of curvature in HPMS curve class i; |
LCURV_{i} | = | total length (in miles) of all curves in curve class i; |
GRD_{i} | = | average percent grade in HPMS grade class i; |
LGRD_{i} | = | total length (in miles) of all grades in grade class i; and |
CCGR | = | crest curve grade rate in percent per hundred feet (zero for flat terrain, 0.03 for hilly terrain, and for mountainous terrain). |
In Equation 5.27, HERS uses the factor of 100 to convert the estimate of crashes from being expressed per million VMT (as in Vogt and Bared) to being expressed per 100 million VMT in HERS.
The ADJSL/SLEN factor adjusts the estimate of non-intersection crashes to reflect only travel that occurs more than 250 feet from an intersection. (The procedure treats crashes occurring within 250 feet of an intersection as intersection crashes.) For this purpose, ADJSL is obtained from SLEN by subtracting 500/5,280 times the number of intersections; if the result is negative, ADJSL is set to zero. This adjustment enables the HERS procedure to avoid producing unreasonably high estimates of total crashes for sections with moderate to high numbers of intersections per mile.^{27}
Vogt and Bared use a dummy variable, STATE, to distinguish between Minnesota and Washington data. In Equation 5.27, STATE has been set to 0.5 (effectively weighting data from both states equally), and the term corresponding to STATE has been combined with the constant term to produce a modified constant term (0.72).
The value used in Equation 5.27 for the roadside hazard rating (RHR) is 3.0, which approximates the average value for all sections used in the Vogt and Bared analysis.
The values used for driveway density (DD) are assumed to vary by type of development. For rural development, driveway density should be somewhat below the median for rural types of development, and for dense development it should be appreciably higher. The median values in the data used by Vogt and Bared are 3.73 for Minnesota and 6.12 for Washington; the means are 6.58 and 10.12, respectively; and the maxima are 85.1 and 100, respectively. This data suggests that it is appropriate to set DD to 3.7 where development is rural and to set it to 50 where development is dense. When these values are used with 1995 HPMS data, the VMT-weighted average value of DD is 8.29, just slightly below 8.35, the unweighted average of the means for Minnesota and Washington. (The value of 50 for dense development implies an average of 211 feet between driveways.)
The values for crest curve grade rate (CCGR) were also based on a judgmental review of Vogt and Bared data. In their data, the median values for this variable were 0.024 in Washington and 0.037 in Minnesota, suggesting that 0.03 is a reasonably typical value for hilly terrain. Similarly, the maximum value was 2.0 in Washington (and 0.89 in Minnesota). Since most crests can be assumed to have CCGR values that are appreciably below the maximum, a typical value of 0.4 was assumed for mountainous terrain. Finally, for flat terrain, CCGR values are likely to be zero or close to zero; accordingly, a value of zero was used.
Finally, if necessary, the HPMS-coded lengths of curves (LCURV_{i}) are scaled so that their sum equals the coded section length (SLEN); and, if necessary, a corresponding adjustment is made to the lengths of the grades (LGRD_{i}).^{28}
Vogt and Bared used HSIS data for Minnesota to develop separate equations for estimating crashes at three-legged intersections and crashes at four-legged intersections. The HPMS database does not distinguish between three and four-legged intersections, but it does distinguish between:
To avoid double-counting, HERS assigns all crashes at the second type of intersection to the intersecting road and all crashes at the third type to the sample section. A portion of crashes at intersections with traffic signals are assigned to the sample section using assumptions presented subsequently.
27. Vogt and Bared did not incorporate an ADJSL/SLEN adjustment in their analysis. The inclusion of this adjustment in the HERS equation would, by itself, reduce HERS' estimate of total crashes on rural two-lane roads. However, since the HERS estimates are calibrated to 1995 data, the actual effect of the adjustment is to increase the size of the calibration factor (1.056 in Equation 5.26), leaving the estimate of total crashes unchanged but shifting some crashes from sections with high numbers of intersections per mile to sections with lower numbers of intersections per mile.
28. The adjustments are made in the HERS preprocessor and are applied to all sections for which curves or grades are coded. (Previously, adjustments were made when the total length of curves (or grades) exceeded coded section length but not when they were less than coded section length.)
HERS treats all signalized intersections as four-legged intersections. "Other" intersections (i.e., Type 3 intersections) are treated as a mix of three- and four-legged intersections by using a weighted average of the two Vogt and Bared equations. The weights used are 0.55 for three-legged intersections and 0.45 for four-legged intersections.^{29} Since all crashes at intersections with stop signs are assigned to the intersecting road, crashes at these intersections are not estimated.
With the above assumptions, estimates of the number of crashes at intersections are obtained from the following equations:
CINT = | 10^{8} | × (CSINT + COINT4 + COINT3) |
VMT |
CSINT = | 0.2 × NSIG × FSICAS |
× exp(∠ 7.74 + 0.64 ln(ADT1) 0.58 × ln(ADT2) | |
0.33 × CCGR ∠ 0.053 × ADJIA + 0.11 × ND) |
COINT4 = | 0.2 × 0.45 × NOINT |
× exp(∠ 7.74 + 0.64 × ln(AADT) + 0.58 × ln(ADT2) | |
+ 0.33 × CCGR ∠ 0.053 × ADJIA + 0.11 × ND) |
COINT3 = | 0.2 × 0.55 × NOINT |
× exp(∠ 11.48 + 0.82 × ln(AADT) + 0.51 × ln(ADT2) | |
+ 0.26 × CCGR + 0.036 × DC + 0.027 × SPDLIM | |
+ 0.18 × RHR3LI + 0.24 × PRTL) |
where CINT and CCGR^{30} have been defined above and the other variables are:
VMT | = | vehicle miles traveled on the section in one year; |
---|---|---|
CSINT | = | annual crashes at signalized intersections; |
COINT4 | = | annual crashes at "other" four-legged intersections (i.e., intersections with neither signals nor stop signs on the sample section); |
COINT3 | = | annual crashes at "other" three-legged intersections; |
NSIG | = | number of signalized intersections; |
FSICAS | = | AADT/(ADT1+ADT2) = fraction of total AADT on the inventoried section; |
ADT1 | = | at signalized intersections, AADT on the road with the higher traffic volume; |
ADT2 | = | at any intersection, AADT on the road with the lower traffic volume; |
ADJIA | = | "adjusted intersection angle" (2.0); |
NOINT | = | number of "other" intersections; |
AADT | = | Annual Average Daily Traffic; |
ND | = | number of driveways within 250 feet of a given intersection = (500/5,280) x DD; |
DC | = | average degrees of curvature on the section;^{31} |
SPDLIM | = | speed limit (mph); |
RHR3LI | = | roadside hazard rating for three-legged intersections (2.1); and |
PRTL | = | probability that a three-legged intersection has a right-turn lane (0.42). |
29. The database used by Vogt and Bared contained data for 389 three-legged intersections and 327 four-legged intersections. The weights used were obtained by reducing the latter figure by an estimate of the number of signalized intersections (based on 1995 HPMS data), all of which are treated as four-legged intersections by the HERS procedure.
30. We considered the possibility that crest curve grade rate and average curvature would have lower values in the vicinity of an intersection than they would have for the section as a whole. However, this hypothesis was not supported by the data in Vogt and Bared.
In the case of signalized intersections, AADT on intersecting roads is assumed to vary with the functional class of the road section being analyzed. In the case of principal arterials, the intersecting road is assumed to carry less traffic than the section in question, so ADT1 is set to AADT and ADT2 is assumed equal to one-half AADT. In the case of major collectors, the reverse is assumed, so ADT2 is set to AADT and ADT1 is assumed equal to twice AADT. For minor arterials, traffic volumes on both roads are assumed (on average) to be equal, so ADT1 and ADT2 are both set to AADT.
Crashes at signalized intersections are allocated to the inventoried section and to the intersecting roads in proportion to their relative traffic volumes. Thus, FSICAS (fraction of signalized-intersection crashes attributed to the inventoried section) is two-thirds for principal arterials, one-half for minor arterials, and one-third for major collectors.
"Other" intersections are unsignalized intersections which do not have stop signs on the inventoried section. These sections are assumed to have stop signs on the intersecting roads and relatively low volumes on these roads. For these intersections ADT2 is assumed to be the lesser of 500 and one-half AADT. All crashes at these intersections are allocated to the inventoried section (and all crashes at intersections with stop signs on the inventoried section are allocated to the intersecting roads).
If the total number of intersections (signalized, stop sign, and "other") exceeds 20 per mile, the number of each type of intersection is scaled so that the total is reduced to 20 per mile and the scaled-down numbers are used for NSIG and NOINT in the above equations (and also for deriving ADJSL for use in Equation 5.27).
Vogt and Bared define "adjusted intersection angle" (ADJIA) to be (α - 15)^{2}/100, where α is the departure of the intersection angle from 90°, measured in degrees. This variable equals 2.25 when α = 0° or 30°, equals zero when α = 15°, is below 2.0 when α is between 1° and 29°, and exceeds 2.25 when α > 30°. The Vogt and Bared data for α suggests that 2.0 is a reasonable average value for ADJIA.
Vogt and Bared assigned roadside hazard ratings of one to seven for conditions on the main road within 250 feet of an intersection. The average value of these ratings for three-legged intersections, 2.1, has been adopted as the default value for the corresponding variable (RHR3LI) in Equation 5.31. Similarly, the default value of 0.42 assumed for the probability that a three-legged intersection has a right-turn lane (PRTL) represents the fraction of such intersections with right-turn lanes in the data used by Vogt and Bared.
The factor 0.2 in Equations 5.29 through 5.31 is used to transform the corresponding Vogt and Bared equations, which estimate intersection crashes over a five-year period, into equations that produce estimates of the expected numbers of annual crashes.
The equation for estimating the number of crashes per 100 million vehicle-miles on rural multilane roads is:
CRASH = | 132.2 × AADT^{0.073} |
× exp(0.131 × RHRRML ∠ 0.151 × AC + 0.034 × DDRML | |
+ 0.078 × INTSPM ∠ 0.572 × RPA + 0.0082 × (12 ∠ LW) | |
∠ 0.094 × SHLDW ∠ 0.003 × MEDW + 0.429 (DEVEL ∠ 1)) |
where:
RHRRML | = | roadside hazard rating for rural multilane roads (2.45); |
---|---|---|
AC | = | 1 for sections with (full or partial) access control, = 0 for other sections; |
DDRML | = | driveway density (per mile) for rural multilane roads (0.41 for rural type of development, 5.6 for dense development); |
INTSPM | = | intersections per mile (maximum =10); |
RPA | = | 1 for rural principal arterials and rural Interstate, = 0 for lower functional systems; |
LW | = | lane width, in feet (between 8 and 13 feet); |
SHLDW | = | right shoulder width, in feet (maximum = 12 feet); |
MEDW | = | 50 if positive barrier median, = median width, in feet, otherwise (maximum = 50); and |
DEVEL | = | type of development (1 for rural, 2 for dense). |
Equation 5.32 is a modified version of an equation for estimating crashes on rural highways that was fit to Minnesota HSIS data for rural four-lane roads by Wang, Hughes and Stewart.^{32} The following modifications were made to the estimated equation:
The equation for estimating the number of crashes per 100 million vehicle-miles on rural freeways is:
CRASH = 17.64 × AADT^{0.155} × exp(0.0082 × (12 ∠ LW)) |
This equation incorporates a lane-width factor into an equation originally developed by Persaud^{38} for four-lane freeways and calibrated to HPMS data for all rural free-ways. The effect of lane width (LW) is assumed to be the same as that assumed for other rural multilane roads. The variables used by this equation have been defined previously.
32. Jun Wang, Warren Hughes and Richard Stewart, Safety Effects of Cross-Section Design of Rural four-Lane Highways, FHWA Report FHWA-RD-98-071, May 1998, Equation 6.
33. The coefficient (0.078) of the combined term was obtained as a weighted average of the original coefficients, using data from Wang, Hughes and Stewart to obtain weights representing the approximate number of intersections with turn lanes per mile (0.22) and the approximate number of intersections without turn lanes per mile (0.74).
34. The variable actually used by Wang, Hughes and Stewart was width of road surface.
36. Wang, Hughes and Stewart, op. cit., Table 2.
37. The current HPMS database has a moderate number of sections with 20 or more intersections per mile.
38. B. N. Persaud, Roadway Safety: A Review of the Ontario Experience and of Relevant Work Elsewhere, prepared for the Ministry of Transportation, Ontario, 1992.
Persaud's equation actually estimates crashes per mile per year. Equation 5.33 was derived from Persaud's equation by multiplying by section length, dividing by annual VMT (equal to 365×AADT×section length), multiplying by 108, and multiplying by the estimated calibration factor (0.8442).
Persaud also derived a separate equation for rural freeways with more than four lanes. However, for any value of AADT, the equation for freeways with more than four lanes produces appreciably higher crash rates than the equation for freeways with four lanes, implying that crash rates vary inversely with congestion. This result is inconsistent with those of most other analyses.^{39}, ^{40}
The equation for estimating the number of crashes per 100 million vehicle-miles on urban freeways is:
CRASH = | (154.0 ∠ 1.203 × ACR + 0.258 × ACR^{2} ∠ 0.00000524 × ACR^{5}) |
× exp(0.0082 × (12 ∠ LW)) |
where ACR = AADT divided by two-way hourly capacity and the other variables have been defined previously. This equation incorporates a lane-width factor into an equation developed by Margiotta^{41} using results from Tedesco, et. al.,^{42} and Margiotta and Cohen.^{43} The effect of lane width is assumed to be the same as that assumed for rural multilane roads.
The equation for estimating the number of crashes on urban multilane surface streets and on urban expressways lacking full access control is:
CRASH = A × AADT^{B} × NSIGPM^{C} |
where:
A, B, and C | = | values from Table 5-9; and |
---|---|---|
NSIGPM | = | number of signals per mile. |
The value of NSIGPM, the number of signals per mile, has a minimum value of 0.1 and a maximum value of eight. This equation was derived from an equation for estimating annual crashes per mile that was estimated by Margiotta^{44} using data from Bowman and Vecellio.^{45} The derivation involved multiplying Margiotta's equations by section length, dividing by annual VMT, multiplying by 108, and incorporating calibration factors. The upper and lower limits on the number of signals per mile were recommended by Margiotta.
39. See B. Persaud and L. Dzbik, "Accident Prediction Models for Freeways," Transportation Research Record 1401, 1993.
40. For low values of AADT (less than 24,000), crash rates were also found to be higher on freeways with more than four lanes in FHWA's Highway User Investment Study (as reported in HPMS Analytical Process, Volume II, Technical Manual, 1987, page IV-41 and Appendix J). However, this result is of little significance since very few freeways with such low values of AADT have more than four lanes. To enable HERS and the AP to avoid assuming that adding lanes to four-lane freeways will increase crashes, Margiotta recommended that only Persaud's equation for four-lane freeways be used.
41. Richard Margiotta, op. cit., pp. 15-19, 25 and 28.
42. Shelby A. Tedesco, et. al., "Development of a Safety Model to Assess the Impact of Implementing IVHS User Services," Proceedings of the IVHS America 1994 Annual Meeting, April 1994.
43. Richard Margiotta and Harry Cohen, Roadway Usage Patterns: Urban Case Studies, prepared by Science Applications International Corporation and Cambridge Systematics for Volpe National Transportation Systems Center, June 1994.
44. Op. cit., pp. 19-22, 25 and 29.
45. Brian L. Bowman and Robert L. Vecellio, "Effect of Urban and Suburban Median Types on Both Vehicular and Pedestrian Safety," Transportation Research Record 1445, 1994, pp. 169-179. This source actually provided data only for roads with raised medians, roads with two-way left-turn lanes, and other undivided roads. (Roads with two-way left-turn lanes are considered to be divided roads in the safety literature and are treated as such in this report; however, they are classified as undivided in the HERS and HPMS capacity analyses.)
Type of Section | A | B | C |
---|---|---|---|
two-Way with Left-Turn Lane | 95.1 | 0.1498 | 0.4011 |
one-Way, or two-Way with a median: | |||
1) wider than 4 feet, or | 82.6 | 0.1749 | 0.2515 |
2) curbed, or | |||
3) a "positive barrier" | |||
Otherwise | 115.8 | 0.1749 | 0.2515 |
For two- and three-lane urban streets, crashes are estimated using the equation:
CRASH = ∠19.6 × ln(AADT) + 7.93 × (ln(AADT))^{2} |
This equation was developed by using ordinary least squares regression to fit a function of this form to the data shown in Table 5-10 and multiplying by the calibration factor in Table 5-11. The r^{2} for this regression is 0.99.
AADT Range | Mean Value of AADT Within Range^{a} | Crashes per 100 Million VMT^{b} |
---|---|---|
a. Weighted average mean value obtained from the 1995 HPMS database for streets to which the "two-lane urban streets" procedure is applied (i.e., all one-lane urban streets and two-way urban streets with two or three lanes). | ||
b. Crash rates used in original HERS safety procedure. | ||
< 4,000 | 1,978 | 345 |
4,000 - 7,999 | 5,739 | 490 |
8,000 - 15,999 | 11,101 | 590 |
> 15,999 | 20,417 | 660 |
The crash-rate equations were calibrated in two steps.
In the first step, the equations used by the six procedures presented above were calibrated separately to crash rates for the corresponding highway types. These rates were obtained by Margiotta^{46} as a VMT-weighted average of rates developed by Zegeer and Williams^{47} using data for four states from the HSIS. The rates used are shown in the first column of Table 5-11.
Facility | Crashes per 100 million VMT | Calibration Factor |
---|---|---|
a. Derived from rates developed using Highway Safety Information System data for Illinois, Maine, Minnesota and Utah. Separate rates were developed for each state by C.V. Zegeer and C. Williams (Calculation of Accident Rates by Roadway Class for HSIS Status, University of North Carolina Highway Research Center, June 1994). The above rates are VMT-weighted averages of these rates developed (using 1994 HPMS data) by Richard Margiotta (Incorporating Traffic Crash and Incident Information into the Highway Performance Monitoring System Analytical Process, prepared by COMSIS Corporation and Science Applications International Corporation for FHWA, September 1996, Table 2.3). | ||
Rural | ||
Freeway | 68.0 | 0.8842 |
Multilane | 146.6 | 2.4123 |
Two Lane | 163.8 | 1.0557 |
Urban | ||
Freeway | 131.0 | 1.1453 |
Multilane | ||
Divided | 439.1 | |
Median | 0.9367 | |
Two-Way Left-Turn Lane | 0.7494 | |
Undivided | 554.8 | 1.3131 |
Two-Lane | 378.7 | 0.8743 |
In the second step, the six calibration factors were scaled uniformly to produce an overall rate of 309.7 crashes per 100 million VMT. This rate was obtained by multiplying an estimate of total crashes in 1995 from the National Highway Traffic Safety Administration's General Estimates System (GES)^{48} by an undercapture correction factor of 1.12^{49} and dividing by national VMT in that year.^{50} ^{51} The calibration factors produced by this two-step procedure are shown in the right column of Table 5-11. These factors are included in Equations 5.26 through 5.36 as presented in sections 5.3.1.1 through 5.3.1.6.
46. Margiotta, op. cit., pp. 7-8.
47. C.V. Zegeer and C. Williams, Calculation of Accident Rates by Roadway Class for HSIS States, University of North Carolina Highway Research Center, June 1994. The four states used were Illinois, Maine, Minnesota and Utah. This source contains crash rates for five states. However, rates for the fifth state, Michigan, are appreciably higher than those for the other four states (for rural areas, they are, on average, twice as high). Accordingly, Michigan data were excluded from our calibration.
48. National Highway Traffic Safety Administration, Traffic Safety Facts - 1996, Table 1.
49. Lawrence J. Blincoe and Barbara M. Faigin, The Economic Cost of Motor Vehicle Crashes, 1990, NHTSA, Report DOT HS 807 876, 1992, as quoted in Ted R. Miller, Diane C. Lestina and Rebecca S. Spicer, "Highway Crash Costs in the United States by Driver, Age, Blood Alcohol Level, Victim Age, and Restraint Use," Accident Analysis and Prevention, Vol. 30, No. 2, 1998.
50. FHWA, Highway Statistics - 1995, Table VM-1.
51. No adjustment was made for differences in overall crash rates between the nine functional systems covered by HPMS sample-section data and the three systems (rural minor arterials and the two local systems) that are not covered. To the extent that crash rates on the latter systems may be lower than average, our calibration procedure may result in a slight upward bias in the HERS estimates of crashes on the nine systems covered by HERS. Since the other three systems account for only 15 percent of national VMT, the effect of this bias should be fairly small.
The calibration factor for urban two-lane streets results from the replacement of the step function previously used by a continuous function and from a decline in crash rates between 1988 and 1995. (The step function was last calibrated using 1988 data.) The calibration factors for urban multilane streets incorporate separate calibration factors developed by Margiotta^{52} for the three types of multilane streets distinguished.
The other calibration factors generally represent differences, that are not explained by any of the independent variables, between the crash rates observed in the data used in developing the original equations and the HSIS and national crash rates used in the calibration process. The high calibration factor for rural multilane roads is due to very low average crash rates in the HSIS data used by Wang, Hughes and Stewart in their analysis of crash rates on rural multilane roads. The factor for rural two-lane roads incorporates an upward adjustment to counter the effect of the ADJSL/SLEN factor that we added to Equation 5.27 (see section 5.3.1.1.1 "Non-intersection Crashes" on page 5-18).
The HERS safety procedure estimates fatalities and nonfatal injuries as being directly proportional to the number of crashes, with separate ratios used for each functional system. The ratios were obtained by:
The resulting ratios are shown in Table 5-12.
53. FHWA, Highway Statistics, 1995, Table FI-1.
54. An alternate approach for obtaining ratios by highway type (instead of by functional system) was also investigated. This approach used a calibration process that was more complicated than the one finally adopted, along with HSIS data on numbers of crashes, fatalities and injuries by highway type in six states, and corresponding estimates of 1995 VMT obtained from HPMS sample-section data. However, the HSIS estimates of the numbers of crashes on rural multilane roads (and, in particular, undivided rural multilane roads) in these states were found to be inconsistent with the corresponding HPMS estimates of VMT in these states, making it impractical to calibrate the equations appropriately.
Functional System | Fatalities per Crash | Injuries per Crash |
---|---|---|
Rural | ||
Rural Interstate | 0.01408 | 0.4546 |
Other Principal Arterial | 0.01685 | 0.6317 |
Minor Arterial | 0.01362 | 0.5610 |
Major Collector | 0.01370 | 0.6261 |
Urban | ||
Urban Interstate | 0.00382 | 0.4908 |
Other Freeway or Expressway | 0.00396 | 0.3640 |
Other Principal Arterial | 0.00273 | 0.4113 |
Minor Arterial | 0.00237 | 0.3401 |
Collector | 0.00257 | 0.3496 |
Over time, the rates of injuries and fatalities in highway crashes have shown steady declines. In the past twenty years, fatalities per 100 million vehicle-miles have declined at an average annual rate of about four percent, and nonfatal injuries have declined at an average rate of about 2.25 percent.^{55} Although highway improvements have contributed to this decline, several other factors have been major contributors. These include improvements in: vehicle designs; emergency medical care; and driver behavior (including reductions in drunk driving).
In order to allow HERS to incorporate the effects of these secular trends into its forecasts of crashes and crash costs, the safety model allows the user to specify annual percentage declines in:
Estimates of annual crashes for 1988 and subsequent years are available from NHTSA's General Estimates System (GES).^{56} These values, when combined with FHWA's estimates of annual fatalities and nonfatal injuries,^{57} indicate that, over the 1988-1995 period, the average annual rates of decline have been 1.0 percent for the ratio of (nonfatal) injuries to crashes and 1.3 percent for the ratio of fatalities to crashes, and the rates of decline since 1990 have been appreciably higher. Since the ratios developed in section 5.3.2 are for 1995, the decline in these ratios is assumed to begin in 1996. The year of the data used for calibrating the fatality and injury ratios (currently 1995) is provided to HERS as a parameter and should be changed by the user whenever these ratios are changed.
55. Derived from FHWA, Highway Statistics Summary to 1995, July 1997, Tables FI-210 and FI-220.
56. National Highway Traffic Safety Administration, Traffic Safety Facts - 1996, Table 1.
57. FHWA, op. cit. Estimates of 1996 fatalities and injuries (from Highway Statistics, Table FI-1) were excluded from this analysis because of inconsistencies between the FI-1 data and the FI-210 and FI-220 data. The latter data is currently being revised.
Obtaining a forecast rate of decline to be applied to crash rates presents a somewhat greater problem. Combining the GES estimates of annual crashes since 1988 with FHWA estimates of annual VMT^{58} produces estimates of crashes per 100 million vehicle-miles that drop from 340 in 1988 to 266 in 1993, rise more slowly to 277 in 1995, and decline very slightly to 276 in 1996. The average annual rate of decline between 1988 and 1995 (the time period used for estimating the decline rates for the fatality and injury ratios) is 2.6 percent, but a focus on more recent data would produce an appreciably lower annual rate of decline (and an increase if only data since 1993 is used^{59}).
As the above discussion implies, data for the last few years suggests that there may be some weakening in the long-term trends toward reductions in crash, fatality and injury rates. It is not yet clear whether this weakening represents a temporary or permanent change in the secular rates of decline in the crash, fatality and injury rates. In preparing data for the 1999 Conditions and Performance Report, the rate of decline was set to zero (no decline).
The HERS safety model estimates crash costs as the sum of the value of lives lost and the costs of injuries, property damage, and delay to other highway users. The value of lives lost is estimated by multiplying fatalities by the U.S. Department of Transportation's estimate of the value of life (currently $2.7 million). Unit costs for estimating the three other components of crash costs have been derived in large part from information contained in a recent National Highway Traffic Safety Administration (NHTSA) study of crash costs in 1994.^{60}
HERS' estimates of injury costs are derived from estimates of comprehensive costs per injury developed by Ted Miller in 1991^{61} and updated to 1994 dollars by NHTSA.^{62} These estimates, which are based on the willingness-to-pay concept used by HERS, are provided by the Maximum Abbreviated Injury Scale (MAIS).^{63} They range from $10,840 for MAIS Level 1 to $2,509,310 for MAIS Level 5 (and $2,854,500 for fatal injuries). Weighting the estimates for nonfatal injuries by the relative frequency of injuries of each severity^{64} produces an overall estimate of $47,657 per police-reported injury.
58. Ibid., Table FI-200; and FHWA, Highway Statistics, 1996, Table VM-1.
59. In another test, the new HERS crash-estimation procedures were applied without a temporal adjustment to 1993 and 1996 HPMS data for 42 states. The procedures indicate a 0.6 percent increase in crash rates over this three-year period, apparently because of increased congestion. (All six procedures produce estimated crash rates that vary with AADT or AADT per lane.) However, the increase produced by the HERS procedures is appreciably lower than the 3.7 percent increase indicated for this three-year period by the GES and FHWA data.
60. Lawrence J. Blincoe, The Economic Cost of Motor Vehicle Crashes, 1994, NHTSA, 1996.
61. Ted R. Miller, et. al., The Costs of Highway Crashes, The Urban Institute, 1991.
62. Blincoe, op. cit., Table A-1. These estimates of comprehensive costs, based on willingness to pay, are, on average, roughly three times the NHTSA estimates of economic or "human capital" costs summarized in Table 1 of the Blincoe report.
63. An alternative to using costs by MAIS is the use of costs by police-reported "KABCO" code (killed; A, B or C injury; property damage only). Estimates of comprehensive costs by KABCO code are available in Ted R. Miller, Diane C. Lestia, and Rebecca S. Spicer, "Highway Crash Costs in the United States by Driver Age, Blood Alcohol Level, Victim Age, and Restraint Use," Accident Analysis and Prevention, Vol. 30, No. 2, 1998. However, these estimates probably should not be used with crash data from small numbers of states, since the source observes that KABCO coding varies appreciably across states.
64. Blincoe, op. cit., Table 3.
Corresponding estimates of property-damage costs per crash and travel-delay costs per crash were obtained by dividing the NHTSA estimates of total 1994 costs of these two types^{65} by an estimate of crashes in 1994 that is consistent with the 1995 estimate used to calibrate the HERS crash-estimation procedures.^{66} This step produced overall costs-per-crash estimates of $7,164 for property damage and $605 for travel delay.
The next series of steps in the development of unit cost factors for use by the HERS safety model involved using the above overall estimates of unit costs for injuries and property damage to develop estimates by functional system. This was accomplished by:
This last step produced scale factors of 3.1062 for injury costs and 1.2532 for property damage costs. These scale factors were then applied to the 1988 HERS estimates of unit costs by functional system to produce a revised set of unit costs for injuries and property damage. The revised unit costs are shown in Table 5-13.
Delay costs vary with AADT per lane. HERS uses the equation:
DELCC = | 0.0886 × AADT | × CRASH |
LANES |
where
DELCC | = | cost of delay due to crashes (per 100 million VMT); |
---|---|---|
CRASH | = | crash rate on the section (per 100 million VMT); and |
LANES | = | number of lanes. |
The coefficient (0.0886) was set so that, when applied to 1994 data, HERS would produce an overall average cost of delay that matches the above estimate of $605 per crash.
The assumption that the delay cost of crashes is linear with a simple measure of traffic volume (AADT per lane) undoubtedly understates the complexity of this relationship. Hence, this simple procedure is likely to underestimate the delay cost of crashes on congested roads and to overestimate this cost on uncongested roads.
66. The resulting estimate of 1994 crashes, 7.28 million, was obtained by applying an undercapture correction factor of 1.12 (see section 5.3.1.7) to the NHTSA GES estimate of crashes in 1994.
Functional System | Injury Cost per Injury (1994 dollars) | Property-Damage Costs per Crash (1994 dollars) |
---|---|---|
Rural | ||
Interstate | $52,800 | $5,000 |
Other Principal Arterial | 68,300 | 6,300 |
Minor Arterial | 55,900 | 6,300 |
Major Collector | 77,650 | 6,300 |
Urban | ||
Interstate | 55,900 | 6,300 |
Other Freeway or Expressway | 46,600 | 7,500 |
Other Principal Arterial | 49,700 | 7,500 |
Minor Arterial | 40,400 | 7,500 |
Collector | 31,100 | 6,300 |
For sections with stop signs and for free-flow sections with only one lane in one or both directions, the delay cost of crashes is multiplied by the average value of an hour of travel time to estimate hours of delay due to crashes. For these sections, this value is used as an estimate of incident delay. For other sections, DELCC is no longer used by HERS.
HERS allows the costs of property damage, delay and injuries to be indexed from 1994 dollars to dollars of a subsequent year using separate, user-supplied index values. For property-damage costs, an appropriate index to use is the Consumer Price Index (CPI) component for automobile body work.^{67} Using 1994 as a base year, the 1997 value of this index is 1.126, so property-damage costs per crash in 1997 are estimated as being 12.6 percent higher than in 1994.
For travel delay, the index used should be the same one as is used for the value of time, but the base year for the travel delay index would be 1994 (instead of 1995, the year currently used for value of time). The index currently being used for this purpose is the U.S. Bureau of Labor Statistics (BLS) Employment Cost Index (which reflects total compensation of all civilian workers).^{68} Using 1994 as a base year, the 1997 value of this index is 1.089.
Injury costs have previously been indexed by HERS using the CPI component for medical care. However, since the HERS estimates of the comprehensive costs of injuries are based on willingness to pay, rather than on economic costs (which include medical costs), the cost of medical care may not be the most appropriate basis for indexing injury costs. A measure of perceived wealth or earnings ability is probably a better indicator of changes in willingness to pay. For simplicity, HERS uses the BLS Employment Cost Index for this purpose (as well as for indexing the cost of delay); therefore the 1997 value of this index is also 1.089.
67. U.S. Bureau of Labor Statistics, Consumer Price Index - All Urban Consumers, Series ID CUUR0000SETD01, Motor Vehicle Body Work (U.S. city average, not seasonally adjusted).
68. U.S. Bureau of Labor Statistics, Employment Cost Index, Series ID: ECS10001I, Total Compensation of All Civilian Workers (seasonally adjusted).
HERS uses computed vehicle speed for three purposes: calculation of travel time costs; calculation of external costs due to vehicular emissions; and calculation of vehicle operating costs^{69}. Average effective speed (AES) across the section is used in the first two calculations above and for most of the operating cost calculations. To calculate excess operating costs due to speed change cycles induced by traffic signals and/or stop signs, HERS uses distance travelled between traffic control devices and the average travel speed over the portions of the section which contain signals and stop signs.
Previously, the HERS speed model was based on the Texas Research and Development Foundation (TRDF) adaptation^{70} of the "Aggregate Probabilistic Limiting Velocity Model" (APLVM), one of four related procedures originally developed by the World Bank.^{71} HERS now uses a simplified version of the APLVM procedures to calculate "free-flow" speed (FFS). It then applies algorithms developed by Science Applications International Corporation (SAIC) and Cambridge Systematics, Inc.^{72} (CSI) for FHWA to incorporate the effects vehicle speed of grades (free-flow speed uphill, or FFSUP), traffic-control devices, and congestion on speed.
For each section, HERS models speed for each of the seven vehicle types (except for autos and pickup trucks) in each direction of travel. Overall average speed per section is aggregated from the speeds of the individual vehicle types. HERS uses vehicle speed data in calculating operating costs and travel time costs.
The HERS version of APLVM involves a four-step procedure; the first three of which involve the computation of three "limiting velocities." These limiting velocities represent the approximate speeds that would be obtained should a single factor (e.g., pavement condition) limit speed to a value much lower than would otherwise be the case. The three limiting velocities^{73} are:
VCURVE | = | maximum allowable speed on a curve; |
---|---|---|
VROUGH | = | maximum allowable ride-severity speed; and |
VSPLIM | = | maximum speed resulting from speed limits. |
The fourth step is to combine the three limiting velocities, using the APLVM, to determine the free-flow speed.
69. See Figure 2-3 "Prediction and calculation model linkages." on page 2-3.
70. G.C. Elkins, et al., Estimating Vehicle Performance Measures, Texas Research and Development Foundation, prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., July 1987, pp. 128-177.
71. Thawat Watanatada, Ashok M. Dhareshwar and Paulo Roberto S. Rezende Lima, Vehicle Speeds and Operating Costs, The World Bank, Johns Hopkins University Press, Baltimore, 1987.
72. Cambridge Systematics, Inc., Harry Cohen, and Science Applications International Corp., Sketch Methods for Estimating Incident-Related Impacts, prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., December 1998, Section 2.3.; Science Applications International Corporation and Cambridge Systematics, Inc., Roadway Usage Patterns: Urban Case Studies, prepared for Volpe National Transportations Systems Center and the Federal Highway Administration, June 1994, Appendix A; Science Applications International Corporation, et al., Speed Determination Models for the Highway Performance Monitoring System, prepared for the U. S. Department of Transportation, Federal Highway Administration, Washington, D.C, October 31, 1993; Cambridge Systematics, Inc., Revisions to the HERS Speed and Operating-Cost Procedures, prepared for the U. S. Department of Transportation, Federal Highway Administration, Washington, D.C, January 25, 1996, Section 2; Cambridge Systematics, Inc., 2000 Revisions to HERS, prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., August 2002.
73. HERS 2.0 used four limiting velocities: VCURVE, VROUGH, VDRIVE, and VMISC. VDRIVE was a function of vehicle characteristics and average grade, which are handled in HERS through calculation of uphill free-flow speed. The fourth limiting velocity, VMISC, was a replacement for a factor called "desired speed" (VDESIR) by the World Bank and TRDF, and reflected the effects of speed limits, safety concerns, and congestion. In HERS, these factors are represented by VSPLIM and the congestion delay algorithms applied to free-flow speed.
In APLVM, the dominant role in the determination of free-flow speed^{74} is played by the smallest of the limiting velocities.^{75} Each of the other limiting velocities are assumed to play some probabilistic role in influencing the speed of some drivers, but, except when they have values close to that of the lowest velocity, their influence on average free-flow speed tends to be negligible.
The following subsections describe the four steps of the HERS version of APLVM.
In the World Bank procedure, the effect on speed attributed to the presence of curves is represented by VCURVE. The World Bank estimates VCURVE, in meters per second:
VCURVE = ((FRATIO + SP) × g × RC) |
where:
RC | = | radius of curvature (meters); |
---|---|---|
SP | = | superelevation; and |
g | = | the force of gravity = 9.81 m/sec^{2}. |
The remaining variable in Equation 5.38, FRATIO, known as the maximum perceived friction ratio, is the ratio of the lateral force on a horizontal curve to the normal force. TRDF derived values for FRATIO of 0.103 for combination trucks and 0.155 for automobiles; and they suggest the use of the 0.155 figure for single-unit trucks as well. HERS uses these values.
Replacing radius of curvature in Equation 5.38 by degrees of curvature (DC) and converting the equation to estimate VCURVE in miles per hour produces:
VCURVE = 292.5 × (FRATIO + SP)/(DC) |
74. The World Bank and TRDF use the term "steady-state speed," or Vss, to refer to free-flow speed.
75. The original version of APLVM also uses a fifth limiting velocity, maximum allowable braking speed on downhill sections (VBRAKE). This limitation affects only heavy trucks and only on long, steep downhill sections (e.g., a five percent grade more than 10.5 miles long, or an eight percent grade more than three miles long). Since most HPMS sections are less than five miles long, there are likely to be only a handful of sections for which one would find braking speed to be a limiting factor (though, undoubtedly, some additional HPMS sections are part of longer descents for which braking speed is indeed a factor). Accordingly, TRDF's recommendation to exclude VBRAKE from the procedure has been adopted.
For all arterial sections except urban minor arterials, a weighted average value of degrees of curvature can be obtained from detailed data on curves by class contained in the HPMS database. For all collectors and for urban minor arterials, typical values of degrees of curvature are produced by existing HPMS software from horizontal alignment adequacy and type of terrain. (The HPMS submittal software is used in preparation of the HPMS database prior to HERS' processing the data.)
Although data on superelevation are not contained in the HPMS database, typical superelevation can be estimated from degrees of curvature using the equation:
SP = | 0.0 | for DC ≤ 1 | |
0.1 | for DC ≥ 10, otherwise: | ||
0.0318 + 0.0972 + ln(DC) ∠ 0.0317 × DC + 0.007 × DC × ln(DC) | |||
This equation was derived by regression from a table presented by Zaniewski^{76}, but fits so well (R2 = 0.9999) as to suggest that it may be the equation that was used to generate the values in the table.
The effect of pavement roughness on speed is represented by VROUGH. HERS uses PSR to measure pavement roughness. Descriptions of pavement characteristics corresponding to the various PSRs are presented in Table 3-20 on page 3-33.
A review of these descriptions indicates that pavement condition begins to become a limiting factor on high speed roads at approximately the boundary between the Good (3.0 to 4.0) and Fair (2.0 to 3.0) ratings, suggesting that VROUGH should play a minimal role in limiting speed when PSR greater than or equal to 3.0. Similarly, the descriptions suggest 52.5 mph as an appropriate value for VROUGH when PSR equals 2.0.
In order to avoid a speed of zero when PSR drops to zero (which can occur in HERS when funds are short), and to allow additional user control over the function used for VROUGH, HERS allows the user to specify VROUGH as a pair of line segments with different slopes meeting at a user-specified breakpoint, PSRB. These parameters are specified in the parameter file PARAMS.DAT.
Specifically, HERS uses the function below when the section's PSR is greater than the PSR breakpoint PSRB:
VROUGH = VR2 + VRSLOP × (PSR ∠ PSRB) |
HERS uses the following equation when PSR is less than or equal to the breakpoint PSRB:
VROUGH = VR1 + (VR2 ∠ VR1) × | PSR |
PSRB |
where:
VR1 | = | value of VROUGH when PSR is zero; |
---|---|---|
VR2 | = | value of VROUGH when PSR = PSRB (the breakpoint); and |
VRSLOP | = | slope of the function when PSR > PSRB |
76. J.P. Zaniewski, et al., Vehicle Operating Costs, Fuel Consumption, and Pavement Type and Condition Factors, Texas Research and Development Foundation, prepared for U.S. Department of Transportation, Washington, D.C., March 1982, p. e-7.
The default values of the above parameters are:
PSRB | = | 1.0 |
---|---|---|
VR1 | = | 5 mph |
VR2 | = | 20 mph |
VRSLOP | = | 32.5^{77} |
For PSR below 2.5, the default values produce intentionally lower estimates of speed than either the current AP procedure or that proposed by TRDF. For PSR = 1.5, they produce VROUGH = 36 mph, while TRDF's formula would produce values of 48 mph for automobiles and 44 mph for large trucks,^{78} and the AP procedure would permit speeds of 49 mph. On the basis of the preceding discussion, 36 mph appears to be a more appropriate speed; however, users can choose different values for the four parameters if higher values of VROUGH are desired.
For purposes of deriving VROUGH for unpaved sections, HERS treats these sections as having a PSR of 1.0 (i.e., when VR1, VR2 and VRSLOP are set to their default values, VROUGH for unpaved sections is 20 mph); the user may change this PSR value if desired.
For HERS, the same formula for VROUGH is used for all vehicle classes. Using Brazilian data, the World Bank study^{79} obtained results that imply a very significant difference (about 30 mph) between the effects of roughness on automobiles and on combination trucks; and TRDF has proposed formulas that produce a much more modest difference (two to four mph). However, TRDF did not provide any recommended formulas for use with single-unit trucks.
HERS represents the effect of speed limits on speed with VSPLIM. VSPLIM is assumed to be 10 or 15 kilometers per hour greater than the speed limit. Fifteen kmph is used for urban freeways by design and rural multilane roads with partial or full access control and a median which is either a positive barrier or has a width of at least 4 feet. Ten kmph is used for all other sections. These values correspond to 6.215 and 9.323 mph.
The general formula for estimating free-flow speed, FFS, is:
where σ^{2} and Β are parameters discussed below.
77. This represents a change from HERS v2, where the default value for VRSLOP was set at 20. The previous value produced a limiting velocity of 60 mph for a pavement of PSR 3.0 - This change produces a limiting velocity of 85 mph when PSR is 3.0.
VROUGH = | 1 | for automobiles |
0.025 ∠ 0.00275 PSR |
VROUGH = | 0.9 | for large trucks |
0.0255 ∠ 0.00333 PSR |
79. Elkins, et al., op. cit., pp. 144-149.
In the above equation, Β is a parameter that may vary with vehicle class and reflects the standard deviation of the sensitivity of drivers of vehicles in that class to the different conditions reflected in the equation. For the moment, ignore the effect of σ^{2} (i.e., assume σ^{2} = 0). In this case, when two factors produce very similar limiting velocities, the variation in sensitivities results in some vehicles being limited more by one factor while some vehicles are limited more by the other, with an overall average speed somewhat lower than either of the limiting velocities. The smaller the value chosen for Β, the more this average speed approaches the lower of the two limiting velocities.
The World Bank^{80} used Brazilian data to estimate Β for six vehicle classes, deriving values of 0.24 to 0.31. After comparing the effects of values of 0.01, 0.1 and 0.3 on the behavior of the FFS equation, TRDF recommended a value of 0.1 for all vehicle classes.^{81}
In Equation 5.43, σ^{2} is described by the World Bank as the variance of the logarithm of section-specific errors of observed speed. The World Bank's estimates^{82} for σ^{2} are between 0.007 and 0.036; and TRDF^{83} suggests using 0.01. The effect of these values for σ^{2} is a small upward adjustment in FFS (of about 0.5 percent σ^{2} using = 0.01, and about 1.8 percent σ^{2} using = 0.036). For simplicity, the effect of σ^{2} has been omitted from the HERS equation.
Setting Β = 0.1 and = σ^{2} = 0, Equation 5.43 becomes:
FFS = ((1/VCURVE)^{10} + (1/VROUGH)^{10} + (1/VSPLIM)^{10})^{∠ 0.1} |
where VCURVE is given by Equation 5.39, VROUGH by Equations 5.41 and 5.42, and VSPLIM is derived from the section's speed limit as described above.
Equation 5.44 produces estimates of free-flow speed that are always below the lowest of the limiting velocities in the equation, but are exceedingly close to that velocity whenever that velocity is appreciably smaller than the other limiting velocities.
Using an SAIC algorithm^{84}, HERS next calculates free-flow speed in the uphill direction (FFSUP) for trucks^{85}. (For "personal vehicles" - automobiles and pickup trucks - HERS assumes that grades have no effect on free-flow speed.) First, crawl speed for the section is estimated as follows:
CRAWLS = 1/(j + k × GRADE) |
where:
CRAWLS | = | Crawl speed in miles per hour; |
---|---|---|
j, k | = | constants which depend upon vehicle characteristics; and |
GRADE | = | the average grade of the section (expressed as a fraction). |
80. Watanatada et al., op. cit., Table 4.3(a), p. 85.
81. Elkins, et al., op. cit., p. 156.
82. Watanatada et al., Table 4.3(c), p. 86. The reference uses σ_{e}^{2} to represent our σ^{2}.
83. Elkins, et al., op. cit., p. 156.
84. Science Applications International Corporation, et al., Speed Determination Models for the Highway Performance Monitoring System, pp 78-79.
85. Since the HPMS database does not contain any information on the direction of grades, one-way facilities are treated in the same way as two-way facilities; i.e., as if traffic may be moving either uphill or downhill.
The values used for constants j and k are shown in Table 5-14^{86}.
Vehicle Type | j | k |
---|---|---|
6-Tire Truck | 0.0090 | 0.0815 |
3-4 Axle Truck | 0.0090 | 0.2755 |
4-Axle Combination | 0.0090 | 0.2755 |
5-Axle Combination | 0.0090 | 0.2755 |
HERS then calculates the delay due to grades:
DGRADE = | a(1 ∠ exp(b/a)) + b | if CRAWLS < FFS |
0 | otherwise |
where:
b = SLEN(1/CRAWLS ∠ 1/FFS) |
a = ∠ 0.05(1/CRAWLS ∠ 1/FFS)^{0.6} |
DGRADE | = | delay in hours; and |
---|---|---|
SLEN | = | length of the section |
The delay due to grades is then combined with free-flow speed to yield free-flow speed uphill, FFSUP:
FFSUP = 1/(1/FFS + DGRADE/(SLEN)) |
86. Science Applications International Corporation, et al., Speed Determination Models for the Highway Performance Monitoring System, Table 3-3, and Herbert Weinblatt, "The Effects of Grades on Truck Speed," memorandum, Cambridge Systematics, Inc., Feb. 25, 1998.
The CSI/SAIC algorithms address four types of highway conditions based upon number of lanes and the presence of traffic control devices. HERS identifies two additional conditions, defining a total of six highway classifications for use within the speed model. Table 5-15 lists the salient characteristics of each of the six classifications, and indicates which of the equations are used for each classification. Note that the number of lanes is not a factor when either signals or stop signs are present on the section.
Total Lanes in Both Directions | Stop Signs | Traffic Signals | HERS Classification | Algorithms Used |
---|---|---|---|---|
N/A | Yes | No | Sections with Stop Signs | Urban Arterials with Unsignalized Intersections |
N/A | No | Yes | Sections with Traffic Signals | Urban Arterials with Signalized Intersections |
N/A | Yes | Yes | Sections with Stop Signs and Traffic Signals | Both: Urban Arterials with Unsignalized Intersections and Urban Arterials with Signalized Intersections |
2 | No | No | Free-Flow Sections, One Lane per Direction | Two-Lane Rural Sections |
3 | No | No | Free-Flow Sections, three-Lane two-way | Two-lane Rural Sections and modified Freeways and Multilane Rural Highways |
4 or more | No | No | Free-Flow Sections, Two or More Lanes per Direction | Freeways and Multilane Rural Highways |
Each of the implemented algorithms consists of two or more equations. Selection of the appropriate equation hinges upon the ratio of the section's Annual Average Daily Traffic (AADT) to the section's two-way peak hour capacity^{87}. This AADT/Capacity ratio is referred to as the ACR. Each of the routines below yields delay in hours per 1000 vehicle miles. HERS converts this to average effective speed using the equation:
AES = 1/(1/FFS + D/1000) |
where:
AES | = | Average Effective Speed; |
---|---|---|
FFS | = | Free Flow Speed (or FFS Uphill), as calculated above; and |
D | = | average delay in hours per 1000 vehicle miles, with delay due to incidents, other congestion, and/or traffic control devices. |
87. Other than in the calculation of the AADT/Capacity ratio (used in the speed calculations and the elasticity calculations), capacity generally means one-way capacity (except for two-lane rural roads) as reported in the HPMS data records.
The procedures used for estimating delay on multi-lane free-flow sections and on sections with traffic signals develop separate estimates of delay due to incidents, due to other congestion, and due to traffic signals (where present). For other types of sections, the procedures used do not estimate incident delay. For these sections, estimates of delay due to crashes are inferred from HERS' estimates of the delay cost of crashes (see section 5.3.4.1), and these estimates are used (without further adjustment) as estimates of delay due to incidents.
For roads with stop signs, HERS selects an equation based upon both the AADT/Capacity ratio and the number of stop signs per mile. The equations are presented in Table 5-16.
AADT/C Range | Stop Signs/mile | |
---|---|---|
<6 | D_{ss} = N_{sspm} × (1.9 + 0.067 × FFS + 0.103 × ACR + 0.0145 × ACR^{2}) | |
>6 and <15 | <10 | D_{ss} = N_{sspm} (3.04 + 0.067 × FFS ∠ 0.029 × (ACR ∠ 6)^{2}) + 0.354 × (ACR ∠ 6)^{2} |
>10 | D_{ss} = N_{sspm} × (3.04 + 0.067 × FFS) + 0.064 × (ACR ∠ 6)^{2} | |
>15 | <10 | D_{ss} = N_{sspm} × (0.691 + 0.067 × FFS) + 0.354 × (ACR ∠ 6)^{2} |
>10 | D_{ss} = N_{sspm} × (3.04 + 0.067 × FFS) + 0.354 × (ACR ∠ 6)^{2} ∠ 23.49 |
where:
D_{ss} | = | Delay due to stop signs and non-incident congestion in hours per 1000 vehicle miles; |
---|---|---|
N_{sspm} | = | Number of stop signs per mile (average); |
FFS | = | Free flow speed (or free flow speed uphill); and |
ACR | = | the AADT/Capacity ratio for the section. |
Total delay on these roads is estimated as the sum of the above delay plus delay due to crashes (per 1,000 vehicle-miles). The latter value is generated by HERS' safety analysis (see 5.3.4.1).
For sections with traffic signals, HERS uses modified versions of the CSI/SAIC equations^{88} in a multi-step process to estimate traffic volume and delay in the peak period (separately by direction) and in the offpeak period. The variables below are used in the equations contained in Table 5-17 through Table 5-20 (for sections with traffic signals) and also Table 5-22 through Table 5-25 (for free-flow sections).
88. Cambridge Systematics, Inc., Harry Cohen, and Science Applications International Corp., Sketch Methods for Estimating Incident-Related Impacts, prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., December 1998, Section 2.3; and Cambridge Systematics, Inc., 2000 Revisions to HERS, prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., August 2002.
ACR | = | ratio of AADT to two-way capacity during the off-peak period |
---|---|---|
BPM | = | bottlenecks per mile (set to 0.083 for multi-lane free-flow sections and 0.5 for sections with traffic signals) |
CPCAP | = | capacity during peak periods in the counter-peak direction (one way) |
CPLANES | = | through lanes in peak period, counter-peak direction |
DFAC | = | directional factor |
FFS | = | free-flow speed (mph) |
NSIG | = | number of signalized intersections per mile (average) |
OP2WCAP | = | two-way capacity during off-peak period |
PKCAP | = | capacity during peak period in peak direction (one way) |
PLANES | = | through lanes in peak period, peak direction |
Index K, K = 1, 2, or 3 | ||
1 | = | peak period, peak direction |
2 | = | peak period, counter-peak direction |
3 | = | off-peak period |
DINC(K) | = | incident-related delay (hours per million vehicle miles) in period/direction K (K = 1, 2, 3) |
NITR(K) | = | travel rate (hours per vehicle-mile) without incidents in period/direction K (K = 1, 2, 3) |
SHFAC | = | shoulder factor (based on left and right shoulder widths, see Table 5-26) |
TRAVF(K) | = | fraction of annual travel in period/direction K (K = 1, 2, 3) |
VCRPP(K) | = | peak-period volume/capacity ratio - average hourly volume in the indicated direction during a three-hour peak period divided by capacity (K = 1 for peak direction, 2 for counter-peak direction) |
ZVDSIG | = | zero-volume delay due to traffic signals (hours per vehicle-mile |
The steps of this process are:
ZVDSIG = 0.0687 × (1 ∠ e^{∠NSIG/24.4}) |
AES = 1/(NITR + D_{inc}) |
where:
AES | = | Average Effective Speed; |
---|---|---|
NITR | = | Non-Incident Travel Rate (hours per vehicle mile); and |
D_{inc} | = | Delay due to incidents (hours per vehicle-mile). |
This procedure produces estimates of overall average speed, average speed in the peak and offpeak periods, and three components of delay (relative to free-flow speed): zero-volume delay, incident delay, and other congestion delay.
Peak/Off-Peak | Condition | Fraction of Travel |
---|---|---|
Off-Peak | ACR ≤ 7 | TRAVF(3) = 0.7106 |
ACR > 7 and ACR ≤ 9 | TRAVF(3) = 0.7106 − 0.00160 (ACR-7) + 0.00240 (ACR-7)^{1.48} e^{−0.0431(ACR-7)} | |
ACR > 9 and ACR ≤ 12 | TRAVF(3) = 0.7074 − 0.00227 (ACR-9) + 0.00240 (ACR-7)^{1.48} e^{−0.0431(ACR-7)} | |
ACR > 12 | TRAVF(3) = 0.7006 + 0.00373 (ACR-12) + 0.00240 (ACR-7)^{1.48} e^{−0.0431(ACR-7)} | |
Peak | Peak Direction | TRAVF(1) = DFAC × ( 1 − TRAVF(3) ) |
Counter-peak Direction | TRAVF(2) = ( 1 − DFAC ) × ( 1 − TRAVF(3) ) |
Direction | Volume-to-Capacity Ratio |
---|---|
Peak Direction | VCRPP(1) = 0.243 × ACR × TRAVF(1) × OP2WCAP/PKCAP |
Counter-peak Direction | VCRPP(2) = 0.243 × ACR × TRAVF(2) × OP2WCAP/CPCAP |
Peak/Off-Peak | Condition | Travel Rate (hours per vehicle-mile) |
---|---|---|
Off-Peak Period | ACR ≤ 7 | NITR(3) = (1/FFS+ZVDSIG) (1 + 0.0213 ACR^{1.05} ) |
ACR > 7 and ACR ≤ 11 | NITR(3) = (1/FFS+ZVDSIG)(1 + 0.0213 ACR^{1.05} ) + 4.56E−08 BPM (ACR-7) ^{8.25} e ^{−0.561 (ACR-7)} | |
ACR > 11 | NITR(3) = (1/FFS+ZVDSIG) (1.05 + 0.0247 ACR − 0.000504 ACR^{2} + 2.68E−06 ACR^{3}) + 4.56E−08 × BPM × (ACR-7) ^{8.25} e ^{−0.561 (ACR-7)} | |
Peak Period | VCRPP(K) ≤ 0.5767 | NITR(K) = (1/FFS+ZVDSIG) (1 + 0.455 VCRPP(K)^{1.02} ) |
VCRPP(K) > 0.5767 | NITR(K) = (1/FFS + ZVDSIG)(0.889 + 0.680 × VCRPP(K) + 0.0423 VCRPP(K)^{2} − 0.182 VCRPP(K)^{3}) + 0.228 BPM (VCRPP(K) − 0.5767) ^{2.66} e ^{3.61 (VCRPP(K)−0.5767} |
Peak/Off-Peak | Condition | Incident Delay (hours per vehicle-mile) |
---|---|---|
Off-Peak Period | ACR ≤ 11 | DINC(3) = 7.52E−06 (ACR)^{1.11} e^{0.132 ACR} |
ACR > 11 | DINC(3) = 7.74E−09 (ACR)^{5.20} e ^{−0.135 ACR} | |
Peak Period | VCRPP(K) ≤ 0.5767 | DINC(K) = 0.000111 (VCRPP(K)) ^{−0.828} e ^{2.83 VCRPP(K)} |
VCRPP(K) > 0.5767 | DINC(K)= 1.34E−06 (VCRPP(K)) ^{−2.05} e ^{7.74 VCRPP(K)} |
To calculate the average effective speed for sections with both types of traffic control devices, HERS calculates two speeds over the section: one, as if all the devices were stop signs, and two, as if all the devices were signals. HERS then averages these speeds together, weighted by the ratio of traffic signals to stop signs.
The equation selection for two-lane roads depends only upon the AADT/Capacity ratio. The equations are presented in Table 5-21.
AADT/C Range | |
---|---|
< 10 | D_{cong1} = 0.432 × ACR |
> 10 | D_{cong1} = 9.953 ∠ 1.66 × ACR + 0.109 × ACR^{2} |
where:
D_{cong1} | = | average delay due to non-incident congestion in hours per 1000 vehicle miles; and |
---|---|---|
ACR | = | the AADT/Capacity ratio for the section. |
Total delay on these roads is estimated as the sum of the above delay plus delay due to crashes (per 1,000 vehicle-miles). The latter value is generated by HERS' safety analysis (see 5.3.4.1).
For three-lane, two-way roads without traffic control devices, HERS assumes that the volume is split evenly between the two directions, and that capacity is split 7:5 in favor of the two-lane direction. This is implemented by multiplying the section's AADT/Capacity ratio by 0.857 to derive the AADT/Capacity ratio in the two-lane direction, and by 1.2 to derive the AADT/Capacity ratio in the one-lane direction. These modified AADT/Capacity ratios are then used in the respective delay calculations. HERS calculates the delay in the single-lane direction using the equations for two-lane rural roads, and in the two-lane direction using the multilane equations. HERS then figures total delay as the average of the two.
For sections with at least two lanes in each direction and no traffic signals or stop signs, HERS uses modified versions of the CSI/SAIC equations^{89} in a multi-step process to estimate traffic volume and delay in the peak period (separately by direction) and in the offpeak period. The procedure produces estimates of overall average speed, average speed in the peak and offpeak periods, and two components of delay: incident delay and other congestion delay.
The steps of this process are:
Peak/Off-Peak | Condition | Fraction of Travel |
---|---|---|
Off-Peak Period | ACR ≤ 7 | TRAVF(3) = 0.6970 |
ACR > 7 and ACR ≤ 9 | TRAVF(3) = 0.6970 ∠ 0.00085(ACR ∠ 7) + 0.00212(ACR ∠ 7)^{1.39} × e^{∠0.00798(ACR ∠ 7)} | |
ACR > 9 and ACR ≤ 12 | TRAVF(3) = 0.6953 ∠ 0.00187(ACR ∠ 9) + 0.00212(ACR ∠ 7)^{1.39} × e^{∠0.00798(ACR ∠ 7) ×} | |
ACR > 12 | TRAVF(3) = 0.6897 ∠ 0.00408(ACR ∠ 12) + 0.00212(ACR ∠ 7)^{1.39} × e^{∠0.00798(ACR ∠ 7)} | |
Peak Period | Peak Direction | TRAVF(1) = DFAC × (1 ∠ TRAVF(3)) |
Counter-peak Direction | TRAVF(2) = (1 ∠ DFAC) × (1 ∠ TRAVF(3)) |
Direction | Volume-to-Capacity Ratio |
---|---|
Peak Direction | VCRPP(1) = 0.243 × ACR × TRAVF(1) × OP2WCAP/PKCAP |
Counter-peak Direction | VCRPP(2) = 0.243 × ACR × TRAVF(2) × OP2WCAP/CPCAP |
Peak/Off-Peak | Condition | Travel Rate (hours per vehicle-mile) |
---|---|---|
Off-Peak Period | ACR ≤ 7 | NITR(3) = (1/FFS) × ( 1 + 9.19 × 10^{∠11} × ACR^{7.71}) |
ACR > 7 and ACR ≤ 11 | NITR(3) = (1/FFS) × (1 + 9.19 × 10^{∠11} × ACR^{7.71}) + 0.00000133 × BPM × (ACR ∠ 7)^{6.97} × e^{∠0.356 × (ACR ∠ 7)} | |
ACR > 11 | NITR(3) = (1/FFS) × (1.0367294 ∠ 0.0169 × ACR + 0.00177 × ACR^{2} ∠ 0.0000407 × ACR^{3}) + 0.00000133 × BPM × (ACR ∠ 7)^{6.97} × e^{∠0.356 × (ACR ∠ 7)} | |
Peak Period | VCRPP(K) ≤ 0.5995 | NITR(K) = (1/FFS) × (1 + 0.388 × VCRPP(K)^{7.27}) |
VCRPP(K) > 0.5995 | NITR(K) = (1/FFS) × 1.4060195 ∠ 1.84 × VCRPP(K) + 2.54 × VCRPP(K)^{2} ∠ 0.985 × VCRPP(K)^{3}) + 0.00000133 × BPM × (VCRPP(K) ∠ 0.5995)^{2.54} × e^{1.94 × (VCRPP(K) ∠ 5.995)} |
Peak/Off-Peak | Number of Lanes per Direction | Condition | Incident Delay (hours per vehicle-mile) |
---|---|---|---|
Off-Peak Period | 2 | ACR ≤ 11 | DINC(3) = 4.05 × ACR^{∠0.251} × e^{(0.603 × ACR)} × SHFAC |
ACR > 11 | DINC(3) = 3.55826 × 10^{∠11} × ACR^{17.1} × e^{∠0.865 × ACR} × SHFAC | ||
3 | ACR ≤ 11 | DINC(3) = 0.789 × ACR^{∠0.834} × e^{0.854 × ACR} × SHFAC | |
ACR > 11 | DINC(3) = 3.41342 × 10^{∠11} × ACR^{16.9} × e^{∠0.847 × ACR} × SHFAC | ||
4 or more | ACR ≤ 11 | DINC(3) = 0.153 × ACR^{∠0.881} × e^{1.01 × ACR} × SHFAC | |
ACR > 11 | DINC(3) = 3.2672 × 10^{∠11} × ACR^{16.8} × e^{∠0.82 × ACR} × SHFAC | ||
Peak Period | 2 | VCRPP(K) ≤ 0.5995 | DINC(K) = 4.6 × VCRPP(K)^{∠0.247} × e^{9.2 × VCRPP(K)} × SHFAC |
VCRPP(K) > 0.5995 | DINC(K) = 29173.1 × VCRPP(K)^{5.16} × e^{∠0.789 × VCRPP(K)} × SHFAC | ||
3 | VCRPP(K) ≤ 0.5995 | DINC(K) = 0.011 × VCRPP(K)^{∠2.3} × e^{16.9 × VCRPP(K)} × SHFAC | |
VCRPP(K) > 0.5995 | DINC(K) = 560567 × VCRPP(K)^{7.99} × e^{∠3.92 × VCRPP(K)} × SHFAC | ||
4 or more | VCRPP(K) ≤ 0.5995 | DINC(K) = 0.00035 × VCRPP(K)^{3.19} × e^{21.8 × VCRPP(K)} × SHFAC | |
VCRPP(K) > 0.5995 | DINC(K) = 5841010 × VCRPP(K)^{10.1} × e^{∠6.12 × VCRPP(K)} × SHFAC |
Number of Lanes per Direction | Number of Six Foot (or wider) Shoulders in Each Direction | ||
---|---|---|---|
0 | 1 | 2 | |
2 | 5.22 | 3.04 | 1.00 |
3 | 4.77 | 2.83 | 1.00 |
4 or more | 4.45 | 2.68 | 1.00 |
89. Cambridge Systematics, Inc., Harry Cohen, and Science Applications International Corp., Sketch Methods for Estimating Incident-Related Impacts, prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., December 1998, Section 2.3; and Cambridge Systematics, Inc., 2000 Revisions to HERS, prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., August 2002.
90. These equations produce values of DINC in hours per vehicle-mile for consistency with the corresponding equations for sections with traffic signals (in Section I.3). The equations in the HERS code for multi-lane free-flow sections (but not those for sections with traffic signals) actually produce values in hours per million vehicle-miles that are subsequently converted to hours per thousand vehicle-miles and hours per vehicle-mile as needed.
HERS uses the distance travelled between stops when traversing a section as an input to the operating cost model. HERS assumes that drivers stop at all stop signs, and at traffic signals when they are red. For sections with stop signs, this distance is simply the length of the section divided by the number of traffic control devices. For sections with traffic signals, HERS allows for signals which might be green and not require a stop.
HERS incorporates U.S. Department of Transportation values of time per person for personal travel and for business travel.^{91} Table 5-27, "Value of One Hour of Travel Time (1995 Dollars)," presents a summary of the major components of the revised HERS estimates of the 1995 value of travel time, by vehicle type. The values used for each of the components are documented below.
Small Auto | Med. Auto | 4-Tire Truck | 6-Tire Truck | 3-4 Axle Truck | 4-Axle Comb. | 5-Axle Comb. | |
---|---|---|---|---|---|---|---|
Avg Value per Vehicle | $ 15.71 | $ 15.75 | $ 17.84 | $ 19.98 | $ 23.66 | $ 25.49 | $ 25.24 |
Business Travel | |||||||
Value per Person | $ 18.80 | $ 18.80 | $ 18.80 | $ 16.50 | $ 16.50 | $ 16.50 | $ 16.50 |
Avg. Occupancy | 1.43 | 1.43 | 1.43 | 1.05 | 1.0 | 1.12 | 1.12 |
Vehicle | $ 1.09 | $ 1.45 | $ 1.90 | $ 2.65 | $ 7.16 | $ 6.41 | $ 6.16 |
Inventory | - | - | - | - | - | $ 0.60 | $ 0.60 |
Personal Travel | |||||||
Value per Person | $ 8.50 | $ 8.50 | $ 8.50 | - | - | - | - |
Avg. Occupancy | 1.67 | 1.67 | 1.67 | - | - | - | - |
Percent Personal | 89% | 89% | 75% | - | - | - | - |
For the purpose of indexing the value of time from 1995 dollars to dollars of a subsequent year, HERS allows separate indexing of the value of time per person, the vehicle cost, and inventory-cost components. The indexes currently used for the three components are, respectively: The U.S. Bureau of Labor Statistics (BLS) Employment Cost Index for total compensation of all civilian workers; U.S. Department of Commerce Bureau of Economic Analysis (BEA) data on average expenditures per car; and the implicit gross domestic product (GDP) price deflator, also obtained from BEA. The index values used to convert these components to 1997 dollars (for the 1999 C&P Report) are, respectively: 1.059, 1.110, and 1.038.
HERS obtains the value of time to vehicle occupants from the U.S. Department of Transportation (USDOT) Departmental Guidance.^{92} The values used are the values (in 1995 dollars) for travel via surface modes. For on-the-clock travel for all occupants of four-tire vehicles, HERS uses the recommended value for "business travel" ($18.80 per person-hour), while the value used for all occupants of larger vehicles is the slightly lower recommended value for truck drivers ($16.50 per person-hour). For personal travel, HERS uses the recommended value for personal local travel ($8.50 per person hour).^{93}
HERS derives values for average vehicle occupancy (AVO) of four-tire vehicles from 1995 National Personal Travel Survey (NPTS)^{94} estimates of VMT and person-miles of travel by trip type. The NPTS data indicates that AVO for "work-related business" (exclusive of commuting) is 1.43, while AVO for all other purposes is 1.67.
For combination trucks, AVO was set to 1.12 on the basis of Hertz' analysis of the frequency of the use of two-driver teams in crash-involved trucks.^{95} Six-tire vehicles, which include pick-up-and-delivery vehicles that sometimes carry a helper, were assumed to have an average occupancy of 1.05, while heavier single-unit trucks were assumed to have only one occupant.
Approximately 4.7 percent of automobiles are estimated to be in commercial fleets of four or more vehicles, excluding fleet vehicles that are individually leased or used for daily rental;^{96} and 6.7 percent of the VMT of the remaining automobiles is for work-related business.^{97} These figures indicate that just under 89 percent of automobile VMT represents personal travel (including commuting), while the remainder represents business travel.
For four-tire trucks, the percentage of VMT that was not for personal use was 31 percent in 1992;^{98} however, this percentage has undoubtedly dropped in the last several years as small truck-based vehicles have become increasingly popular as personal vehicles. Accordingly, HERS assumes that personal use accounts for 75 percent of the VMT of four-tire trucks and business use accounts for 25 percent of this VMT.
91. U. S. Department of Transportation, "The Value of Saving Travel Time: Departmental Guidance for Conducting Economic Evaluations," April 1997, Table 4.
93. The Departmental Guidance recommends using a higher value ($11.90 per person-hour) for personal intercity travel, implying that, at least in rural areas, an average value for personal travel that is slightly higher than $8.50 might be appropriate. (HERS does not accept separate values of personal travel time and vehicle occupancy for business and personal travel. The input values of personal travel time and vehicle occupancy used by HERS 3.2 for four-tire vehicles are set so that, when combined with weighted averages of the average vehicle occupancy values in Table 5-27, they will produce the overall average values of time shown at the bottom of the table.)
94. Oak Ridge National Laboratories, 1995 National Personal Travel Survey, Table NPTS-1, October 1997.
95. Robin P. Hertz, "Sleeper Berth Use as a Risk Factor for Tractor Trailer Driver Fatality," 31^{st} Annual Proceedings, American Association for Automotive Medicine, September 1987, pp. 215-227.
96. American Automobile Manufacturers Association, Motor Vehicle Facts and Figures, 1995, Detroit, 1995, pp. 39 and 43.
97. Oak Ridge National Laboratories, op. cit.
98. U. S. Bureau of the Census, 1992 Truck Inventory and Use Survey, May 1995.
Vehicles depreciate as a result of their use and as a result of aging that is independent of vehicle use. The former type of depreciation is estimated by HERS' vehicle operating-cost procedure, while the latter type is a time-related cost incurred by all vehicle owners and included as a component of travel-time cost of commercial vehicle operators. For HERS 3.2, time-related depreciation was estimated by:
The estimation process is described below.
For autos in commercial motor pools and four-tire trucks, total depreciation per hour was computed as the average vehicle cost per year (assuming a five-year life, with a 15 percent salvage value at the end, with initial cost from the American Automobile Manufacturers Association^{100}) divided by 2,000 hours per year of sign-out time (essentially the day shift or other shift when maximal vehicle use occurs). For heavier trucks, total depreciation per hour was computed as the average vehicle cost per year^{101} divided by the number of hours in service per year. Six-tire trucks and four-axle combination trucks were assumed to be in service 2,000 hours per year; and five-axle combinations were assumed to be in service 2,200 hours per year. Because three- and four-axle single-unit trucks include many dump trucks that have down time between jobs, especially during cold periods of the winter, they were assumed to be used only 1,600 hours per year.
The resulting estimates of total depreciation per hour of operation are shown in the first column of Table 5-28. The relatively high value shown for three- and four-axle single-unit trucks is the result of the low number of hours per year that they are used and relatively small differences between the initial costs of these vehicles and those of tractor-trailer combinations.
The second column of Table 5-28 shows estimates of average annual mileage for the seven vehicle types distinguished by HERS. Annual mileage for automobiles is from Highway Statistics;^{102} and annual mileage for the five categories of trucks is from the 1992 Truck Inventory and Use Survey.^{103}
99. In earlier versions of HERS, Steps 2 and 3 were not performed. Thus, usage-related depreciation was included in HERS estimates of travel-time costs as well as HERS estimates of operating costs. The new procedure is designed to eliminate this double counting.
100. American Automobile Manufacturers Association, Motor Vehicle Facts and Figures, 1996, Detroit, 1996, p. 60.
101. Estimates of average vehicle cost per year are those used in the 1997 Federal Highway Cost Allocation Study (U.S. Department of Transportation, July 1997). Sources used in developing these estimates were: Jack Faucett Associates, "The Effect of Size and Weight Limits on Truck Costs," prepared for the U.S. Department of Transportation, Federal Highway Administration, Washington, D.C., 1990; Maclean Hunter Market Reports, The Truck Blue Book, January 1995, Chicago (sales prices for tractors and chassis); U.S. Bureau of the Census, Current Industrial Reports, Truck Trailers, summaries for various years (price adjustments for trailers); and a survey of truck dealers (prices for single-unit trucks).
102. Federal Highway Administration, Highway Statistics 1997, November 1999, Table VM-1.
Vehicle Type | Total Depreciation ($/hr.) | Miles per Year^{a} | Mileage-Related Depreciation | Time-Related Depreciation ($/hr.) | |
---|---|---|---|---|---|
($/mile) | ($/hr.) | ||||
a. For automobiles, from Federal Highway Administration, Highway Statistics 1997, November 1999, Table VM-1; for trucks, from U. S. Bureau of the Census, 1992 Truck Inventory and Use Survey, May 1995, Table 2a. | |||||
Small Autos | $ 1.72 | 11,575 | $0.109 | $ 0.63 | $ 1.09 |
Medium/Large Autos | 2.02 | 11,575 | 0.098 | 0.57 | 1.45 |
four-Tire Trucks | 2.18 | 12,371 | 0.045 | 0.28 | 1.90 |
Six-Tire Trucks | 3.08 | 10,952 | 0.079 | 0.43 | 2.65 |
3+ Axle Trucks | 8.80 | 15,025 | 0.175 | 1.64 | 7.16 |
3-4 Axle Combinations | 7.42 | 35,274 | 0.057 | 1.01 | 6.41 |
5+ Axle Combinations | 7.98 | 66,710 | 0.060 | 1.82 | 6.16 |
The third column of Table 5-28 shows the estimates of mileage-related depreciation, in cents per mile. The estimates of annual hours of operation presented above and those of annual miles per year shown in the second column of the table were then used to convert the estimates of mileage-related depreciation to dollars per hour (as shown in the fourth column); and this result was subtracted from total depreciation to produce the estimates of time-related depreciation that are shown in the last column of Table 5-28, and also in Row 3 of Table 5-27.
The estimates of time-related depreciation and mileage-related depreciation shown in the fourth and fifth columns of Table 5-28 are internally consistent in that, for each vehicle type, the two values add up to the estimate of total depreciation (in the first column). These two sets of estimates are thus appropriate for use by HERS (or by any similar system making joint use of both sets of estimates). However, some of the individual values in the last three columns do raise questions. In particular, the values for mileage-related depreciation for trucks appear to be low relative to the corresponding values for automobiles.^{104} A brief investigation into the causes of this result suggests that it probably is due to differences between the procedures used for automobiles and those used for trucks in the original estimation of mileage-related depreciation.^{105}
104.A comparison of Columns 4 and 1 indicates that mileage-related depreciation accounts for 28 percent of total depreciation of medium/large automobiles and 37 percent of depreciation for small automobiles. The corresponding figures for the three types of single-unit trucks are only about half as large (13 to 19 percent). Even for five-axle combinations, which have average annual mileages that are five times those of automobiles, mileage-related depreciation represents only 23 percent of total depreciation. Observing that styling obsolescence is a significant contributor to time-related depreciation for automobiles but not for trucks, this suggests that, for vehicles with comparable annual mileages, mileage-related depreciation probably should be smaller for automobiles than for trucks.
105.J. P. Zaniewski, et. al., Vehicle Operating Costs, Fuel Consumption, and Pavement Type and Condition Factors, Texas Research and Development Foundation, prepared for FHWA, June 1982, pp. 60-67.
To compute the inventory costs for five-axle combination trucks, an hourly discount rate was computed and multiplied by the value of a composite average shipment. The discount rate selected was 9.8 percent, equal to the average prime bank lending rate in 1995 plus one percent. Dividing this rate by the number of hours in a year produces an hourly discount rate is 0.0033 percent. The average payload of a five-axle combination is about 35,000 pounds. In 1993, the average value of commodities shipped by truck was $1.35 per pound (on a ton-mile weighted basis).17 Inflating to 1995 dollars using the GDP deflator and multiplying by the average payload produces an average payload value of roughly $50,000. The resulting time value of the average payload is approximately $0.60 per hour (ignoring any costs for spoilage and depreciation over time).
Payload for four-axle combination trucks is lower than for five-axle combination trucks, but the value of the cargo probably is higher. Consequently, the value per shipment was assumed to be the same for both types of trucks.
For each vehicle type, these values are used by HERS to develop estimates of travel time costs on each section from the equation:
TTCST_{vt} = | 1000 | × TTVAL_{vt} |
AES_{vt} |
where:
TTCST_{vt} | = | average travel-time cost (in 1995 dollars per thousand vehicle-miles) for vehicles of type vt; |
---|---|---|
AES_{vt} | = | average effective speed of vehicles of type vt on the highway section being analyzed; and |
TTVAL_{vt} | = | average value of time (in 1995 dollars) for occupants and cargo of vehicles of type vt (as shown on the bottom line of Table 5-27). |
For each section, the average travel-time cost (per thousand vehicle-miles) is obtained by taking a weighted average of the corresponding costs for each vehicle type. In HERS the weights are obtained by using section-specific HPMS data on the percentages of four-tire vehicles, single-unit trucks, and combination trucks, and then allocating these percentages to the seven vehicle types using distributions (by functional system) obtained from the HPMS Vehicle Classification Study^{106} (see section 2.11 "The Fleet Composition Model" on page 2-15).
The 1999 version of HERS introduced travel demand elasticity to the travel forecast model. See Appendix B, "Induced Traffic and Induced Demand" for a discussion of the concepts guiding these modifications to HERS. See Appendix C, "Demand Elasticities for Highway Travel" for a discussion of appropriate elasticity values for use in HERS. Appendix D, "Basic Theory of Highway Project Evaluation" presents the principles that apply generally to evaluating highway improvements.
106. U. S. Department of Transportation, Federal Highway Administration, Highway Performance Monitoring System Analytical Process Technical Manual, Version 2.1, December 1987, Table IV-20.
This section first addresses the tasks performed by HERS during initialization:
It then discusses the specific steps utilized by the model in:
The section input data includes AADT for the data year and also for a future data year, typically 20 years beyond the data year. HERS generally assumes that the future volume forecast is based upon a continuation of the initial level of service, as defined by volume-to-capacity (V/C) ratio and PSR. The exception is congested sections, in which case HERS assumes the forecast includes an improvement to increase capacity. HERS calculates the initial user price as the sum of the operating, travel time, and safety costs at the beginning of the analysis period and saves the initial V/C ratio as the baseline V/C level. The baseline price is set equal to the initial price. However, if the section's initial V/C ratio is equal to or greater than one, then HERS sets the baseline V/C level to one, and calculates the baseline price at a volume consistent with a V/C of one and a minimum PSR of two.
HERS next calculates an initial adjusted volume for the section at the beginning of the analysis period. The adjusted volume is used as the "departure point" for the calculation of future baseline traffic volumes. During initialization, HERS calculates adjusted volume:
VADJ = (AADT/(INPRI^{SRE})) × BASPRI^{SRE} |
where:
VADJ | = | adjusted volume at beginning of analysis period; |
---|---|---|
AADT | = | reported volume - AADT at beginning of analysis period; |
INPRI | = | initial price to user at beginning of analysis period; |
BASPRI | = | baseline price; and |
SRE | = | short run elasticity. |
Note that, for sections with initial V/C less than one, the adjusted volume will equal the reported volume because the initial price is equal to the baseline price. For sections with initial V/C greater than one, the re-calculated baseline price is likely to be lower than the initial price (the lower level of congestion should lower travel time costs). As a result, the adjusted volume should be higher than the initial volume, and reflects backing out the effects of short run elasticity.
To calculate future traffic volume, HERS performs the following steps:
The model applies elasticity separately to each funding period. When the traffic prediction model calculates the volume after a time span of more than one funding period, it calculates the volume for each funding period successively. The calculation period for volume prediction is from the midpoint of one funding period to the midpoint of the following funding period, a schedule which coincides with the implementation of improvements (at the middle of the funding period) and the benefit-cost analysis period.
The baseline projection begins with the adjusted volume from the "current" funding period (that is, the last period for which volume data is known). The adjusted volume represents the volume on the section before the application of within period short run elasticity. As shown in Equation 5.55, any of the baseline forecast options discussed in section 3.3.3.7 "Traffic Growth Rates" on page 3-24 may be used in the calculation of baseline traffic volume at time t_{1} from previous long run adjusted volume (at time t_{0}):
VBASE_{t1} = | VADJ_{t0} + (AAGRSL × LFP) | |
VADJ_{t0} × AADTGR^{LFP} | ||
2 × (VADJ_{t0} + (AAGRSL × LFP))∠VADJ_{t0} × AADTGR^{LFP} |
where:
VBASE_{t1} | = | baseline traffic volume at the midpoint of funding period t_{1} |
---|---|---|
VADJ_{t0} | = | adjusted traffic volume at the midpoint of the previous funding period t_{0}; |
AAGRSL | = | linear growth rate (see section 5.6.2); |
AADTGR | = | constant growth rate (see section 5.6.2); and |
LFP | = | the length of a funding period. |
HERS next applies the long run share of elasticity to get adjusted volume at time t_{1}:
VADJ_{t1} = VBASE_{t1} × ( 1 + LRS × (FINPRI_{t0} ∠ BASPRI)/(BASPRI)) |
where:
VADJ_{t1} | = | the adjusted volume at time t_{1}; |
---|---|---|
VBASE_{t1} | = | the baseline volume at time t_{1} (from Equation 5.55); |
LRS | = | the long run share; |
FINPRI_{t0} | = | the final user price at time t_{0}, based upon the AADT at time t_{0}; and |
BASPRI | = | the baseline price. |
Short run elasticity is applied to the adjusted volume to estimate an initial volume:
VINIT_{t1} = ALPHA × FINPRI_{t0}^{SRE} |
where:
VINIT_{t1} | = | the initial volume at time t_{1}; |
---|---|---|
ALPHA | = | VADJ_{t1}/BASPRI^{SRE}; and |
SRE | = | short run elasticity. |
To locate a point on the demand curve, knowing the price is sufficient to determine volume. If the price were constant with respect to volume, there would be a simple functional relationship with a single argument. However, as both demand and price vary with volume, HERS must find a simultaneous resolution of the supply and demand functions. The equilibrium is the intersection of the supply and demand.
For most of the components of price to the highway user, price does not vary with volume (that is, the rate of flow). Pavement condition is related to cumulative usage (not immediate volume), and the effects of volume-to-capacity (V/C) on accident costs is not well understood, so these are treated as unit costs invariant with flow volume. The exception is congestion, which is clearly related to V/C, although the relationship is not precisely known.
Starting with a price (p0) that includes all components other than delay, and a demand curve, the volume (v0) is determined from the price, as shown in Figure 5-2. To this price, adding the additional cost for delay, measured off the curve marked "price of delay," generates the upper curve "price with delay." At the price with delay corresponding to the initial volume, pd, demand would be reduced to some point to the left of the vertical axis (this axis is not at zero volume); delay, however, would be largely eliminated, so the price would no longer apply at this volume. The correct solution is the circled point "equilibrium," which balances the increase in price with the reduction in congestion.
Ideally, this equilibrium point could be found by solving for the intersection of the two functions. While the demand curve is either a straight line or a constant elasticity curve, either of which is a simple single-valued function with two parameters, the delay curve is more complex. The delay curve differs with each of the six road types, and for three of these has different equations for different volume levels. Because of the variety and complexity of these equations, closed-form solutions to the supply-demand intersection are not feasible. Numerical solutions could be obtained to any precision desired, but convergence under all possible conditions would be difficult to ensure, and computational effort might be excessive.
Figure 5-2. Adjustment of Calculated Delay for Congestion Reduction
HERS instead uses the alternative approach of a numerical approximation, whose properties are:
This strategy is acceptable because a high degree of numerical precision is not required; the only purpose is to adjust the volume of traffic to a reasonable level given congestion and other generalized price factors.
The approximation strategy uses the slopes of the two curves to estimate the intersection as the apex of a triangle (as if the curves were straight lines), and uses the resulting volume adjustment to re-estimate the slopes as the average of two slopes. In Figure 5-3, the first iteration is shown in heavy solid lines, and the second iteration in heavy dashed lines. The first iteration uses the tangents at the initial volume, VINIT, shown as sdlay_{i} and sdem_{i} (for delay and demand, respectively), to yield the volume RVOL and the price Price_{RVOL}. Averaging the demand slope at RVOL with that for VINIT (sdem_{i}) gives the arc slope of the demand curve (shown as sdem_{r}) between those two points, which is a much closer approximation of the slope between the equilibrium and the initial volumes than is the tangent at VINIT. Doing the same for the delay function gives the revised delay slope sdlay_{r}, and applying the two revised slopes to VINIT produces the second-iteration result of volume V_{elas} and price Price_{Velas}. This is still not the true equilibrium point, but it is close enough.
Figure 5-3. Details of Successive Approximation
The lower the AADT/Capacity ratio, the more accurate this procedure becomes. The situation displayed in Figure 5-3 is an extreme scenario, in that VINIT represents an AADT/Capacity ratio of 19.5, which means the facility is operating at capacity for 15-19 hours per day. For AADT/C under 12, the second iteration is almost indistinguishable from the first, and for AADT/C under 8 the first iteration is indistinguishable from the equilibrium.
HERS performs a sequence of steps preparatory to the simultaneous solution of the demand and supply functions. HERS first calls upon the pavement model to determine the condition of the pavement at time t_{1} based upon the initial volume estimate. Using this provisional PSR and the initial volume, HERS:
HERS first calculates the slope of the initial demand curve:
SDEM_{i} = | PWOD^{(1 ∠ SRE)} |
ALPHA × SRE |
where:
SDEM_{i} | initial slope of demand curve; and |
---|---|
PWOD | price without delay. |
HERS then calculates an initial estimate of the amount of delay at the volume VINIT at time t_{1}, (EDLAY_{i}) and calculates the slope of the delay function (SDLAY_{i}) at that volume. The specific algorithms are based upon the SAIC/CS equations used in the speed model, and like them are dependent upon the road type, the AADT/Capacity ratio, and the number of traffic signals and stop signs per mile, if any. The equations for specific road types are in Table 5-29 and Table 5-30 in section 5.6.5, "Delay Equations by Road Type." The examples below (in Equations 5.59 and 5.60) are for two-lane, two-way roads without traffic control devices where the AADT/Capacity ratio (figured using the initial volume as AADT) is less than 10. The initial estimate of delay is determined:
EDLAY_{i} = 0.432 × | VINIT | × VOT/1000 |
Capacity |
where:
EDLAY_{i} | = | estimate of delay at the initial volume; |
---|---|---|
VINIT | = | initial volume estimate (from Equation 5.57); and |
VOT | = | value of an hour of travel time. |
The slope of the delay equation at the initial volume is:
SDLAY_{i} = 0.432 × VOT/(Capacity × 1000) |
where:
SDLAY_{i} | = | slope of the delay curve at the initial volume. |
---|
The first approximation for a revised volume is the height of a triangle (laying on its side) whose base is the initial estimate of delay and whose sides slope at SDEM_{i} and SDLAY_{i} (as shown in Figure 5-3):
RVOL = VINIT + | EDLAY_{i} |
SDEM_{i} ∠ SDLAY_{i} |
where:
RVOL | = | revised volume estimate |
---|---|---|
EDLAY_{i} | = | initial estimate of delay (from road type specific equation) |
SDLAY_{i} | = | initial slope of delay function (from road type specific equation) |
The model substitutes this revised volume for the initial volume in Equation 5.59 to yield a revised estimate of delay (EDLAY_{r}). (Note that the revised volume is also substituted for the initial volume in determining the AADT/Capacity ratio used to select the specific form of the equation.) HERS next calculates the price associated with the initial volume:
Price_{VINIT} = (VINIT/ALPHA)^{(1/(SRE))} |
and the price associated with the revised volume:
Price_{RVOL} = (RVOL/ALPHA)^{(1/(SRE))} |
The next step is the calculation of revised slopes for the demand and delay functions. The demand slope is taken as the difference between the initial and revised prices over the difference between their associated volumes:
SDEM_{r} | (Price_{RVOL} ∠ Price_{VINIT}) |
(RVOL ∠ VINIT) |
where:
SDEM_{r} | = | revised demand slope |
---|
The delay slope is taken as the difference between the revised and initial delay estimates divided by the difference in the associated volumes:
SDLAY_{r} = | (EDLAY_{r} ∠ EDLAY_{i}) |
(RVOL ∠ VINIT) |
where:
SDLAY_{r} | = | revised delay slope |
---|
HERS calculates the price of delay at the intersection of the revised slopes:
Price_{Delay} = EDLAY_{i} × | SDEM_{r} |
(SDEM_{r} ∠ SDLAY_{r}) |
and uses it to estimate the final, elasticized volume, V_{elas}:
V_{elas} = ALPHA × (PWOD + Price_{Delay})^{SRE} |
The process used to determine delay for sections with signals and multi-lane freeflow sections (for which see paragraphs 5.4.3.2 and 5.4.3.6) is not conducive to the development of equations (such as those in Tables 5-29 and 5-30 below) useful in the procedure described above. For such sections, the elasticity procedure makes successive calls to the congestion/delay routines (described in section 5.4.3) with slightly different input volumes in order to estimate the slope of the delay curve.
These tables contain the delay equations used during the simultaneous solution for two-lane, two-way roads without traffic control devices, and for the two-lane direction of three-lane, two-way roads without traffic control devices. (The calculations for the single lane direction of the three-lane road employ the equations used for the two-lane, two-way road.) Within each road type, equations are selected based upon the AADT/Capacity ratio and the number of stop signs per mile. The equations are based upon the SAIC^{107} and CSI^{108} equations implemented in the speed model. The equations yielding EDLAY (the estimated delay) would replace Equation 5.59 in the computations detailed above. The equations yielding SDLAY (the slope of the delay function) would replace Equation 5.60 as used above.
The "wheres" below apply to the delay equations in Tables 5-29 through 5-30:
EDLAY | = | estimate of delay (equation substitutes for Equation 5.59) |
---|---|---|
SDLAY | = | delay slope (equation substitutes for Equation 5.60) |
ACR | = | AADT/Capacity Ratio |
VOT | = | value of an hour of travel time for the section |
NSS | = | the average number of stop signs per mile |
NTS | = | the average number of traffic signals per mile |
FFS | = | free flow speed for the section |
COMPF | = | the factor 1 ∠ exp(∠NTS/24.4) |
The two hybrid road types are treated in the same manner as in the speed model. On sections with both stop signs and traffic signals, the final elasticized volume is the average of the volumes on the two portions of the section, weighted by the relative numbers of stop signs and signals. On three-lane sections in two directions, volume is split equally between the two directions, and capacity is split 7:5 in favor of the two-lane direction. Elasticity is applied using the equations for rural multilane roads (in the two-lane direction) and two-lane roads (in the one-lane direction). The sum of the elasticized volumes is taken as the final elasticized volume for the section.
For agencies in charge of building and maintaining highways, HERS recognizes two potentially accruing benefits resulting from improving a highway section:
107. Science Applications International Corporation and Cambridge Systematics, Inc., Roadway Usage Patterns: Urban Case Studies, prepared for Volpe National Transportations Systems Center and the Federal Highway Administration, June 1994, Appendix A; Science Applications International Corporation, et al., Speed Determination Models for the Highway Performance Monitoring System, prepared for the U. S. Department of Transportation, Federal Highway Administration, Washington, D.C, October 31, 1993.
108. Cambridge Systematics, Inc., Revisions to the HERS Speed and Operating-Cost Procedures, prepared for the U. S. Department of Transportation, Federal Highway Administration, Washington, D.C, January 25, 1996, Section 2.
The second type of benefit is referred to as the "residual value" of the improvement. The estimation of residual value is discussed at some length in conjunction with the presentation of the HERS benefit-cost analysis procedure under section 7.7 "Residual Value" on page 7-5. The HERS procedure for estimating the other type of agency benefit, reductions in maintenance costs, is presented below. These benefits take their place in the numerator of the benefit-cost equation.
In HERS, all improvements are analyzed over a benefit-cost analysis (BCA) period that begins at the midpoint of one funding period and ends at the midpoint of some subsequent funding period. To simplify the analysis of maintenance expenditures, a "maintenance cost (MC) period" is defined as a period beginning at the midpoint of a funding period and ending at the midpoint of the next funding period. Estimates of pavement maintenance expenditures over each MC period are then derived from PSR estimates for the beginning and end of each period.
AADT/C Range | |
---|---|
<8 | EDLAY = (0.0797 × ACR + 0.00385 × ACR^{2}) × VOT/1000 |
SDLAY = (0.0797 + 0.00385 × 2 × ACR) × VOT/(Capacity × 1000) | |
>8 and <12 | EDLAY = (12.1 ∠ 2.95 × ACR + 0.193 × ACR^{2}) × VOT/1000 |
SDLAY = (∠ 2.95 + 0.193 × 2 × ACR) × VOT/(Capacity × 1000) | |
>12 | EDLAY = (19.6 ∠ 5.36 × ACR + 0.342 × ACR^{2}) × VOT/1000 |
SDLAY = (∠ 5.36 + 0.342 × 2 × ACR) × VOT/(Capacity × 1000) |
AADT/C Range | |
---|---|
< 10 | EDLAY = 0.432 × ACR × VOT/1000 |
SDLAY = 0.432 × VOT/1000 | |
> 10 | EDLAY = (9.953 ∠ 1.66 × ACR + 0.109 × ACR^{2}) × VOT/1000 |
SDLAY = (∠ 1.66 + 0.218 × ACR) × VOT/(Capacity × 1000) |
Estimates of maintenance costs per lane-mile for flexible pavements have been developed by Witczak and Rada^{109} as a function of PSR and structural number (SN). Their results are presented in Table 5-31. The middle column of this table presents estimates of maintenance costs (in 1984 dollars) incurred per lane-mile during periods when PSR (PSI in the exhibit) drops from 4.5 to 4.0, from 4.0 to 3.5, etc. The last column shows estimates of cumulative maintenance costs per lane-mile for a section that starts with a PSR of 4.5 and has various indicated terminal PSRs ranging from 4.0 to 1.5. These estimates are independent of the time required for the deterioration to occur.
109.Matthew W. Witczak and Gonzalo R. Rada, Microcomputer Solution of the Project Level PMS Life Cycle Cost Model, University of Maryland, Department of Civil Engineering, prepared for Maryland Department of Transportation, State Highway Administration, Baltimore, Md., December 1984, Chapter 4.
Final PSI | Maintenance Cost Between PSI Levels ($/lane mile) | Cumulative Cost ($/lane mile) |
---|---|---|
a. Source: Matthew W. Witczak and Gonzalo R. Rada, Microcomputer Solution of the Project Level PMS Life Cycle Cost Model, University of Maryland, Department of Civil Engineering, prepared for Maryland Department of Transportation, State Highway Administration, Baltimore, MD., December 1984, p. 132 | ||
Low SN/traffic: (SN = 2.16) | ||
4.0 | 221.57 | 221.57 |
3.5 | 767.03 | 988.60 |
3.0 | 1,314.95 | 2,302.55 |
2.5 | 1,859.47 | 4,163.02 |
2.0 | 2,413.74 | 6,576.76 |
1.5 | 2,957.34 | 9,534.10 |
Medium SN/traffic: (SN = 3.60) | ||
4.0 | 339.10 | 339.10 |
3.5 | 1,174.05 | 1,513.15 |
3.0 | 2,012.72 | 3,525.87 |
2.5 | 2,845.76 | 6,371.63 |
2.0 | 3,604.98 | 10,066.61 |
1.5 | 4,526.45 | 14,593.06 |
High SN/traffic: (SN = 5.04) | ||
4.0 | 456.63 | 456.63 |
3.5 | 1,581.05 | 2,037.38 |
3.0 | 2,710.50 | 4,748.18 |
2.5 | 3,832.04 | 8,580.22 |
2.0 | 4,976.21 | 13,556.43 |
1.5 | 6,095.55 | 19,651.98 |
Regressing the values for cumulative maintenance costs shown in Table 5-31 against the values for PSR (or PSI) and SN yields the following equation:
COST = | 4427.24 ∠ 1989.7 × PSR + 223.57 |
× PSR^{2} + 7996.11 × SN ∠ 3594.56 | |
× SN × PSR + 403.99 × SN × PSR^{2} |
where:
PSR | = | terminal PSR; |
---|---|---|
SN | = | structural number; and |
COST | = | where cost is cumulative maintenance cost per lane-mile, in 1984 dollars, for the time over which the pavement is deteriorating from an initial PSR of 4.5 to the terminal PSR. |
The R^{2} for the above equation exceeds 0.9999.
Equation 5.68 can be modified to produce cost estimates in 1988 dollars by multiplying all coefficients by 1.2118, the ratio of the 1988 and 1984 values of FHWA's Cost Index for Highway Maintenance and Operation.^{110}
To estimate maintenance costs per lane-mile on any section during a period beginning at time i and ending at time f, Equation 5.68 is evaluated using the section's PSR at times i and f, and the difference between the two results is obtained. The general form of the HERS equation to provide this result, MCOST, in 1988 dollars, is:
MCOST = | ∠ (2411 + 4355 × SN) × (PSR_{f} ∠ PSR_{i} |
+ (270.9 + 489.6 × SN) × (PSR_{f}^{2} ∠ PSR_{i}^{2}) |
This equation (Equation 5.69) would produce negative values of MCOST whenever PSR_{i} > PSR_{f} ≥ 4.5. To avoid this undesirable effect, 4.5 is substituted for any PSR values above 4.5. The resulting costs can be adjusted to dollars of another year. To index the 1988 dollars to 1997 dollars for use in the 1999 Conditions and Performance Report, a factor of 1.231 was used for rural sections and 1.242 for urban sections.^{111}
110.U.S. Department of Transportation, Federal Highway Administration, Highway Statistics: 1988, U.S. Government Printing Office, Washington, D.C., 1989, Table Pt-5.
111. Indexing since 1988: Office of Program Administration, Price Trends for Federal-Aid Highway Construction, U.S. Department of Transportation, Federal Highway Administration, Washingon, D.C., quarterly.
In the absence of readily available information about maintenance costs for rigid pavements, HERS assumes these costs are identical to those for flexible pavements with a structural number (SN) of 5.625. This is the SN of flexible pavements with a thickness of 5.5 inches, the thickest flexible pavement considered by HERS.
The HERS model includes estimates of the costs of damages from vehicular emissions of air pollutants in its calculation of benefits and disbenefits resulting from the implementation of an improvement. HERS employs a set of tables that specify the average cost of air pollutant emissions generated per vehicle-mile by the three different HERS vehicle classes operating at various speeds on each of the nine HERS roadway functional classes. HERS uses the projected mix of vehicle classes and the average speed of travel on each sample section to determine the average cost of emissions per vehicle-mile, and multiplies this value by its forecast of total vehicle-miles to calculate the total cost of air pollutant emissions generated by travel on the section.
The effect of a proposed improvement to a sample section on air pollution costs is measured by the difference between total pollution costs generated by the forecast volumes of travel on the section under baseline and improved conditions. Because the cost of air pollutant emissions per vehicle-mile varies both by travel speed and among vehicle classes, this effect can be negative (a benefit) or positive (a disbenefit) depending on how a proposed improvement affects forecast travel volumes, the mix of vehicle types, and travel speeds on a sample section.
HERS' estimates of the average cost of air pollutant emissions per vehicle-mile for each of its three vehicle classes differ among the nine HERS roadway functional classes for several reasons:
The average cost of air pollutant emissions per vehicle-mile for each HERS vehicle class and roadway functional class also declines gradually over future years. The decline in air pollution costs reflects projected reductions in the rates at which all types of vehicles emit various air pollutants, as well as projected changes in the composition of the U.S. vehicle fleet and the resulting mix of vehicle classes operating on each roadway functional class.
Air pollution costs during future years for each HERS vehicle class and roadway functional class are estimated by applying an annual rate of decline to the appropriate values for the year 2000. The average annual rate of decline in air pollution costs for each vehicle and roadway functional class was calculated by fitting an exponential function (which represents a constant annual percentage rate of decline) using detailed estimates of air pollution costs prepared for the years 2000 and 2020.
Table 5-32 below provides an example of HERS' estimates of air pollution damage costs. It shows average air pollution costs per vehicle-mile of travel at selected speeds, for each of HERS' three vehicle classes operating on Urban Arterial sections during the year 2000. Table 5-32 reveals the typical patterns of variation in air pollution costs: at each speed, costs are higher for single-unit trucks than for four-tire vehicles (automobiles and light-duty trucks), and highest for combination trucks, while costs for each vehicle class decline significantly from very low speeds through moderate speeds (30-40 mph), after which they increase gradually. Appendix G of this report includes similar tables for each of the nine roadway functional classes employed by HERS, including a complete discussion of the derivation of these costs.
Average Speed (mph) | Emission Damage Cost per vehicle-Mile of Travel ($2000) | ||
---|---|---|---|
Four-Tire Vehicles | Single-Unit Trucks | Combination Trucks | |
r* | 6.06% | 7.79% | 11.30% |
5 | 0.0358419 | 0.0560533 | 0.1275028 |
6 | 0.0316863 | 0.0514504 | 0.1211734 |
7 | 0.0287181 | 0.0481626 | 0.1166524 |
8 | 0.0264919 | 0.0456968 | 0.1132616 |
9 | 0.0247604 | 0.0437789 | 0.1106243 |
10 | 0.0233752 | 0.0422446 | 0.1085145 |
… | |||
30 | 0.0153292 | 0.0296437 | 0.0786490 |
31 | 0.0152326 | 0.0295601 | 0.0784381 |
32 | 0.0151421 | 0.0294816 | 0.0782404 |
33 | 0.0150570 | 0.0294080 | 0.0780546 |
34 | 0.0149769 | 0.0293386 | 0.0778798 |
35 | 0.0149014 | 0.0292732 | 0.0777150 |
36 | 0.0148982 | 0.0292916 | 0.0786260 |
37 | 0.0148952 | 0.0293089 | 0.0794878 |
38 | 0.0148923 | 0.0293254 | 0.0803043 |
39 | 0.0148896 | 0.0293410 | 0.0810789 |
40 | 0.0148870 | 0.0293558 | 0.0818147 |
41 | 0.0148946 | 0.0294674 | 0.0818474 |
42 | 0.0149019 | 0.0295736 | 0.0818786 |
43 | 0.0149088 | 0.0296749 | 0.0819083 |
44 | 0.0149155 | 0.0297716 | 0.0819367 |
45 | 0.0149218 | 0.0298640 | 0.0819638 |
46 | 0.0149325 | 0.0300613 | 0.0831937 |
47 | 0.0149427 | 0.0302503 | 0.0843713 |
48 | 0.0149524 | 0.0304313 | 0.0854998 |
49 | 0.0149618 | 0.0306050 | 0.0865823 |
50 | 0.0149708 | 0.0307717 | 0.0876214 |
… | |||
60 | 0.0151521 | 0.0341296 | 0.1095182 |
61 | 0.0151613 | 0.0347254 | 0.1136012 |
62 | 0.0151702 | 0.0353021 | 0.1175524 |
63 | 0.0151789 | 0.0358604 | 0.1213782 |
64 | 0.0151872 | 0.0364013 | 0.1250845 |
65 | 0.0151953 | 0.0369255 | 0.1286767 |
66 | 0.0152034 | 0.0374580 | 0.1323721 |
67 | 0.0152116 | 0.0379988 | 0.1361736 |
68 | 0.0152198 | 0.0385481 | 0.1400843 |
69 | 0.0152280 | 0.0391061 | 0.1441073 |
70 | 0.0152362 | 0.0396728 | 0.1482459 |