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Time-of-Day Modeling Procedures: State-of-the-Practice, State-of-the-Art

3.0 Innovative Approaches

As peak hour congestion increases on urban highways, drivers wishing to avoid the added delay have a number of options, including:

Existing travel demand models can predict the extent to which some of these options (rerouting, mode shifts, destination shifts) will be chosen, but not the complete set of possible responses. Several methodologies are available for assessing the travelers' temporal response to congestion. This section describes innovative methods used by MPOs or state agencies that go beyond the relatively simple factoring methods described in the previous section.

Three approaches to improving the time-of-day modeling process are being addressed in this section. These "Peak Spreading" methodologies work within the confines of the current "four-step" modeling process. The peak spreading process addresses the problem that projected demand exceeds capacity in certain corridors during the peak period and that failing to account for the excess demand results in a flawed assessment of travel conditions in the future. The three general approaches to implementing peak spreading analysis discussed here include:

3.1 Link-Based Peak Spreading

Description and Applicability

The effect of traffic congestion on route choice has led to the development of equilibrium-based traffic assignment techniques. However, as congestion levels have increased, the limitations of the equilibrium traffic assignments based on static, regional diurnal factoring of trip tables has also become apparent, and revised approaches for time-of-day modeling have attempted to improve the modeling process. One such approach is to account for congestion at the link level and divert trips to the "shoulder" hours on either side of the peak.

Experience with urban traffic suggests that peaking is sensitive to congestion. One of the most well known examples of a peak spreading method was developed for Phoenix, Arizona (Loudon et al, 1988). The objective of this method was to provide an estimate of the net effect of traffic congestion without identifying the magnitude of each type of behavioral response. The result was a set of significantly more realistic estimates of future traffic volumes and speeds on congested highways, as well as more realistic estimates of regional measures such as vehicle-miles of travel (VMT).

The Phoenix study was based on data collected from 49 corridors in Arizona, California, and Texas. These data provided relationships between peak hour and peak period volume as a function of facility type and volume/capacity ratio in the peak period. The procedure hinges on the assignment of peak period, not peak hour, trips. The Phoenix study was based on a three-hour peak period with the total number of trips within that peak period based on a fixed percentage of total daily trips by trip purpose and direction (to or from home) using split factors developed for the region (these TOD factors are applied between trip generation and trip distribution - see Section 2.4). Typically, in peak period assignments, link-specific, hourly capacities are related to period-specific trips through a peak hour factor. This factor is typically applied at a regional level and, in effect, relates the percent of the period's trips that take place in the most congested one-hour time period.

In the Phoenix study, the peak hour factor was allowed to vary for each link based on link congestion levels as measured by volume/capacity ratios. The modeling process which was implemented in Phoenix is illustrated in Figure 3.1. The first step in the process is to produce separate trip tables for each of the three time periods: a three-hour a.m. peak, a three-hour p.m. peak, and an off-peak which includes all other times. Peak spreading and computation of traffic volumes and speeds were applied to each link each time link speed updating is required using the following steps:

  1. Compute the ratio of the current assigned three-hour volume to the three-hour link capacity
  2. Apply the peak-spreading model to calculate a peaking factor (the ratio of one-hour volume to three-hour volume);

    The functional form chosen for the peak spreading model was:



    P = the ratio of peak hour volume to peak period (three-hour) volume,
    V/C = the volume/capacity ratio for the three-hour period, and
    a, b = model parameters.

  3. Determine the revised peak hour volume as the product of the peaking factor and the assigned volume;
  4. Compute link-level peak hour volume/capacity ratios;
  5. Apply a peak hour speed model to estimate revised link speeds; and
  6. Continue this link volume updating process throughout the iterative equilibrium procedure.

Flow chart as described in the text.

The peak spreading procedure was applied as part of a peak period (typically three hours) equilibrium assignment. As each link is considered, in turn, during the equilibrium assignment's travel time updating, peaking factors representing the ratio of peak hour volume to peak period volume are computed using a decreasing function of the link three-hour volume-to-capacity ratio. The peaking factor function was estimated with time series and/or cross-sectional vehicle count data. The peak hour volume corresponding to this peaking function was used to estimate revised travel times during each iteration of the equilibrium assignment procedure.

When applied in the Phoenix area, this technique was found to improve the estimates of average speed and VMT. The root mean squared error (RMSE) of speeds on links was reduced from 56 percent (24-hour trips - no peak spreading) to 46 percent (24-hour trips - 24-hour peak spreading) to 36.6 percent (3-hour trips - 3-hour peak spreading). Also, the percent VMT error declined from 16.4 percent to 3.2 percent as compared to observed VMT estimates computed from regional traffic counts.

The study noted that there were some limitations with the procedure:

3.2 Trip-Based Peak Spreading

In the Phoenix link-based peak spreading approach, an underlying assumption was that all the trips would occur in the three-hour period under consideration, although the percentage of trips occurring in the peak hour within that three-hour timeframe could spread as congestion increased. An alternative to the link-based peak spreading approach is a trip-based approach that spreads the number of trips for an origin-destination interchange that occur in the peak period or peak hour.

Three examples of the trip-based peak spreading are available from the literature. These include:

Tri-Valley Model Peak Spreading

One trip-based peak spreading scheme has been applied for the Tri-Valley Subarea Model in Alameda and Contra Costa Counties, California. For consistency with the regional San Francisco Bay Area model, the Tri-Valley model was designed as a focused subarea model. The region outside of the subarea was included in the model, albeit at a very aggregate level. The Tri-Valley trip-reduction approach recognizes the overall constraint of the future highway network system capacity (by time-of-day) by limiting the assignment of trips to that network based on the overall capacity of the future network at selected gateways.

The Tri-Valley subarea is transected by two major freeways (I-580 and I-680) that define four major gateways for access into, out of, and through the study area. These gateways were identified as the key capacity constraint locations. Without the trip-based peak spreading process, peak hour traffic assignments through the gateways overwhelmed the subarea network leaving little additional capacity for subarea trips. The regional trips resulted in peak hour volume/capacity ratios at the gateways in excess of 1.0.

For reasonableness, peak hour traffic assignments were constrained at the gateways so that the volume/capacity ratios were equal to 1.0 (or slightly higher). An approach to adjust origin-destination (O-D) matrices to better fit observed or estimated link volumes in a network was used. The structure of the Tri-Valley peak spreading methodology is shown in Figure 3.2. The following steps are used for the trip table reduction and assignment process:

  1. Peak hour volumes were assigned to the highway network and V/C ratios calculated;
  2. For gateways with V/C ratios in excess of 1.0, target volumes were estimated so that the V/C ratio would be 1.0 (or slightly higher);
  3. A mathematical approach1 for adjusting trip tables was used to reduce the interchange volumes on the O-D pairs using the over-assigned gateways;
  4. The revised trip table was assigned and the gateway V/C ratios were checked for reasonableness; and
  5. The process was repeated if a close match between the assigned and desired link volumes was not obtained for the gateway links.

In the link-based peak spreading approach the trips were all assumed to occur within a three-hour timeframe regardless of capacity. In the trip reduction approach there is no explicit treatment of the trips being reduced. This trip table reduction process does not assume that the excess trips on each congested interchange are not made. Rather, it is assumed that these trips cannot be completed in the peak hour (used for planning and design purposes) and, thus, have been forced to travel outside of the peak hour. In addition, the trip reduction approach does not account for changes in traveler behavior due to congestion.

Peak Spreading in the Central Artery/Tunnel Project

When forecast year peak hour vehicle trip tables are assigned to highway networks which are at capacity or congested in the base year, the resulting forecast year traffic volume estimates can exceed capacities by unrealistic amounts. This is because, typically, growth rates are applied during the trip generation phase of the modeling system, without consideration for traffic conditions. Trip distribution models and mode choice models reflect the highway capacity constraints by shortening trip lengths and increasing HOV and transit shares, but the effect of peak spreading (where tripmakers who would previously prefer to travel during peak hours make their trips earlier or later to avoid congestion) is not captured in peak hour analyses.

Figure 3.2 - Trip-based peak spreading

To combat this problem, a technique was developed to reduce a trip table selectively. This technique is described in (Rossi et al, 1990). Figure 3.2 shows the structure of this approach. In this procedure, individual origin-destination cells of the trip table are reduced according to congestion levels in the corridor corresponding to the origin-destination pair. This approach was implemented in the Boston area for the Central Artery/Tunnel Project. This is an iterative-factoring procedure applied only to highway trips. In this study, the motivating factor was that base-year peak hour traffic volumes were already at or over capacity throughout downtown Boston. Since daily travel was projected to grow, in the absence of transportation improvements or vehicle travel reduction measures, the use of time-of-day factors based on existing conditions resulted in impossibly high peak hour travel estimates for the future.

The Boston area approach uses a trip reduction process that consists of five iterative steps:

1. Perform Unconstrained Trip Assignment. Here, initial peak hour trip tables are developed by factoring daily vehicle trip tables based on land use at the origin and destination level;

2. Select Congested Links to be Examined. These are links where congestion is likely to necessitate lower demand for peak hour auto travel. Key links are then examined to determine whether assigned unconstrained volumes are above the estimated maximum volumes.

3. Sequentially Adjust Traffic Volumes for Origin-Destination Pairs in the Selected Link Trip Tables of Congested Links. The peak reduction process decreases trips for individual origin-destination pairs according to the congestion level of the corridor in which the trips would be made. The process of making adjustments to cells in the overall trip matrix is similar to the Fratar process of matrix adjustment. Using this process the trip table is adjusted to produce the desired row and column totals in the selected link trip tables and alternates among all of the selected links until the trip table converges to the desired totals.

4. Reassign Using Adjusted Trip Tables. This step is necessary to reflect reroutings which are likely to occur as trip table reductions are made. Typically, trips will shift to the selected links from parallel facilities under these conditions, necessitating additional iterations to ensure that the final trip table reductions represent corresponding network routing patterns.

5. Compare Final Link Volumes with Link Capacities. If the assigned link volumes have not been sufficiently reduced so that target capacities are not exceeded and if significant improvements in volumes since the last assignment are indicated, the process is repeated using the new selected link trip tables. Trips in selected link trip tables for links that have met their capacity targets are not adjusted. Origin-destination pairs in the overall trip table corresponding to non-zero cells in these selected link trip tables also remain unadjusted. This limits the number of cells in other selected link trip tables that can be used in the reduction process. The process is considered complete when the overall assignment has converged with the study area network capacity.

To make the process more practical for large models, compressed overall and selected link trip tables can be used to calculate the reductions although the original zone system should be used for traffic assignment. In a compressed trip table reduction, the same adjustments are applied to all zone-to-zone pairs comprising a district-to-district pair. The districts should be chosen so that easily identifiable corridors of travel using the selected links can be identified.

This selective reduction, which is accomplished using "selected link analysis," is superior to global reduction (which implies a general decrease in trip generation rates) because predicted traffic volumes in uncongested corridors are not changed by unrealistic amounts.

The creation of the matrix of factors provides an important analysis capability. Specifically, conservation of the total amount of daily trips can be assured by modifying time-of-day factor matrices for the other time periods. In this way, daily assigned volumes can be obtained (by adding results of multiple time-of-day assignments).

There are two major differences between the Tri-Valley and the Boston approaches to trip table reduction to account for peak spreading:

Like the link-based peak spreading approach, a major limitation to the trip reduction approach is the treatment of the trips being reduced. In the link-based peak spreading approach the trips were all assumed to occur within a three-hour timeframe regardless of capacity. In the trip reduction approach there is no explicit treatment of the trips being reduced. It is left up to the individual analyst implementing the approach as to how and when the trips being reduced show up on the transportation system. In addition, neither approach accounts for changes in traveler behavior due to congestion.

Washington D.C. Peak Spreading Model

Another vehicle trip-based peak spreading procedure was developed for the Washington, D.C. area. This research was conducted as part of a larger project to develop a complete set of travel demand models for the Washington D.C. region for travel analysis in the Dulles airport corridor. This technique is described in (Allen et al, 1996). As in the link-based procedure used in Phoenix, the Washington procedure assumes that a three-hour peak period has a fixed travel demand and that trips will spread throughout the peak period.

The Washington peak spreading model was developed as a post-mode choice procedure, to be applied to a.m. peak period auto driver trips. The model was calibrated using household travel survey data and uses a stratification of data by trip purpose. This stratification includes home-based work (HBW), home-based university (HBU), and three non-home based trip purposes. The prevailing assumption is that the non-work trip purposes would have flatter peaking than the work and university trip purposes. Based on the survey data, home-based work and university trips had 40 percent or more of their a.m. peak period vehicle-hours of travel (VHT) occurring in the a.m. peak hour. For the non-work trip purposes, this share was 34 percent or less.

The procedure estimates the percentage of peak period travel at the vehicle trip interchange level that occurs during the peak hour based on two variables:

Essentially, a set of curves relating this percentage to the travel time difference for each trip purpose was estimated from the survey data. Each curve represented a trip distance range. Examples of these curves are shown in Figure 3.3. As the travel time difference grows, more traffic can be expected to shift from the peak hour to the shoulders of the peak period.

The final model is specified in terms of a maximum share (the leftmost part of the curve), a slope (the drop in peak hour share per minute of congested time difference), a "limit," which is the point (congested time difference at which the line begins to slope downward), and the minimum share (the rightmost part of the curve). These parameters vary by distance range and trip purpose. Thus, the mode's function is:

Shared = MAX([maxshared + sloped * MAX(timediff - limitd,0)], minshared)


d = distance range
timediff = congested time-free flow time, minutes

Figure 3.3 - Washington DC peak-spreaking model

Table 3.1 lists the final peak hour share model parameters.

The number of trip distance ranges varied by trip purpose from one (for home-based university trips) to five (for home-based work trips), depending on the amount of data available for estimation. One of the key findings was that trip distance strongly influences the peak hour percentage for work trips. Longer trips tend to have less peaking, while short trips tend to occur mainly during the peak hour. Short trips (less than five miles) have over 45 percent of the peak period trips in the peak hour, while long trips (over 35 miles) have less than 30 percent of trips in the peak hour. Trips with minimal congestion (less than five minutes) have almost 45 percent of their trips in the peak hour while trips with major congestion (greater than 25 minutes) have 30 percent of their trips in the peak hour.

Table 3.1 AM Peak Hour Model Parameter
Purpose Distance Range (miles) Maximum Share Slope Limit Minimum Share

A FORTRAN program was written to apply the peak spreading model. This program reads an a.m. peak period auto driver trip table for a specific trip purpose, a matrix file containing congested travel time and distances, and a matrix file with off-peak travel times. It then applies the peak spreading model to each cell in the input trip matrix and outputs a matrix of a.m. peak hour auto driver trips.

The Washington procedure appears to be transferable to other areas. The data required for model estimation can all be obtained from a traditional household travel survey and travel model system.

The assumption of a constant three hour peak period is a limitation of the model. While data from various regions imply that the three hour peak, as a percentage of daily trips, is fairly stable, this may simply reflect a lack of in-depth analysis of this type of information. A more rigorous peak spreading model will need to take into account trip chaining and trip tours.

3.3 Systemwide Peak Spreading

This third method includes a systemwide peak spreading approach that has been implemented by the Volpe National Transportation System Center (VNTSC) within a modeling framework applied in evaluating Intelligent Transportation Systems (ITS). The peak spreading module included in this approach stands on its own and is not required to be used in conjunction with the full ITS Benefits Assessment Framework. It can be used with traditional travel demand models and model systems.

This peak spreading approach considers the systemwide excess travel demand and delay and distributes excess travel demand between the individual travel hours that comprise the peak period. This approach is neither link-specific nor trip-specific; because it was designed to model the travel impacts of ITS deployment, it assumes that a significant amount of travel information is available to travelers and thus the traveler's temporal response to congestion can be modeled on a systemwide basis rather than on a trip-specific or link-specific basis.

The overall VNTSC model framework - shown in Figure 3.4 - is an analytical tool that improves the sensitivity and capability of currently available transportation software to assess the impacts of implementing ITS user services. The model framework is comprised of a set of transportation and impact assessment models linked together by interface software facilitating the transfer of data between the models. In an iterative process estimates of mode split and assigned traffic volumes produced by a travel demand model are input to a peak spreading module and subsequently to two macroscopic simulation models (one for freeway analysis and one for arterial analysis) via the interfaces to produce revised speeds for freeways and signalized arterials. The revised speeds are then re-input to the travel demand model. The process is iterated until the travel speeds and volumes converge, at which point the impact assessment models are used to estimate emissions, fuel, and safety impacts.

Figure 3.4 - ITS impact assessment framework - Model system structure

The peak spreading interface module enables the model framework to estimate congestion-dependent travel distribution within the peak period and to distribute excess travel demand among competing times of travel. An increase in traffic congestion (due to historical, recurrent, or incident-related reasons), for instance, may prompt commuters to change their departure times and travel in a different time than initially intended. In the current version of the peak spreading interface the peak period consists of three peak hours that are analyzed separately; however the methodology can be expanded to include any number of peak travel hours that comprise the peak period.

There are several analytical components in the peak spreading methodology:

Figure 3.5 - Systemwide peak-spreading module

The peak spreading module is operationalized using code written in C++ and instructions for its use are included in the VNTSC report "Program Reference Guide - IVHS Benefits Assessment Framework - 1994". In this peak spreading module the user is given the option to:

The peak spreading module, however, stands on its own and does not require the previous use of traffic simulation models. It can be used in conjunction with the traditional travel demand models run using aggregate performance measures (VHT, VMT, VHD) across facility types (freeways and arterials) produced by all travel models.

The systemwide peak spreading approach was applied in a study examining the impacts of ITS user services on the I-880 corridor in Alameda County, California. I-880 is the major north-south route serving the east San Francisco Bay Area extending from San Jose to Oakland, a distance of approximately 50 miles. This section of I-880 offers continuous alternative arterial routes located within one mile of either side of the freeway. Combinations of five types of traffic management services were evaluated using the systemwide peak spreading approach, including:

A limitation of the systemwide peak spreading approach is that it is not sensitive to different trip purposes. For instance, work trips may be less flexible to temporal distribution than shopping or other home-based travel. It is likely that the majority of temporal shifts is associated with non-work trips.

Another limitation of the systemwide peak spreading approach relates to not being sensitive to traffic congestion on specific links or specific origin-destination flows. However, the basic premise of this approach is that significant amounts of travel information are available to travelers and thus the traveler's temporal response to congestion can be modeled on a uniform, systemwide basis rather than on a trip-specific or link-specific basis.


1. INRO Consultants, EMME/2 User's Manual, Software Release 7, May 24, 1994.

Updated: 3/25/2014
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