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Day-long origin-destination (OD) demand estimation for transportation forecasting is advantageous in terms of accuracy and reliability because it is not affected by hourly variations in the OD distribution. In this paper, we propose a method to estimate the time coefficient of day-long OD demand to estimate hourly OD demand and to predict hourly traffic for urban transportation planning of a large-scale road network that lacks discrete-time rich traffic data. The model proposed estimates the time coefficients from observed link flows given a proven day-long OD demand based on a bilevel formulation of the generalized least square and semidynamic traffic assignment (OD-modification approach). The OD-modification approach is formulated as a static user-equilibrium assignment with elastic demand, based on the residual demand at the end of each period. Our model does not require setting many parameters regarding the OD demand matrices and the discrete-time dynamic traffic assignments. Applying the model to large-scale road network demonstrates that it efficiently improves estimation accuracy because the 24-hour time coefficients of survey data are slightly biased and may be modified properly. In addition, the methods that partially relax the assumption of OD-modification approach and transform the estimated demand into demand based on departure time are examined.

The four-step prediction technique on a one-day unit to predict day-long origin-destination (OD) demand and link flow is generally used worldwide for urban and transportation planning. For one-day activity, a normal pattern exists, in which people commute and shop in the morning and return home in the evening every day on weekdays. Therefore, the estimated day-long OD demand has the advantage of accuracy and reliability, because it is not affected by hourly variations in OD distribution.

However, hourly traffic prediction is important for transportation analysis. Thus, a simple estimate of hourly link flow or OD demand which multiplies the average measured time coefficients by the day-long values of link flow or OD demand is sometimes adopted for practical use in a large-scale road network that lacks real-time data. In this paper, the time coefficients of OD demand are calculated as follows:

Such estimated hourly values cannot ensure user equilibrium on route-choice behavior and prediction accuracy. Accurately estimating hourly link flow and OD demand of a large-scale road network by harnessing the reliable day-long OD demand is highly desired. Additionally, the input data of road network for our study can now benefit from a nationwide survey that includes an OD survey with questionnaire and link flow observations with manual counting; however, it does not provide real-time and discrete-time rich traffic data.

We therefore propose a time coefficient estimation (TCoE) model to obtain the hourly OD demand and to analyze hourly traffic predictions and measures for urban transportation planning of a large-scale road network that lacks discrete-time rich traffic data. The proposed TCoE model estimates the time coefficients from observed link flows given a proven day-long OD demand, which in turn is based on a bilevel formulation of the generalized least square and semidynamic traffic assignment. In the model, hourly OD demands are deduced from both the time coefficient and day-long demand. The semidynamic traffic assignment is a static user-equilibrium assignment with elastic demand and accounts for residual demand at the end of study period, as discussed later.

Previous work has developed several approaches of frameworks to estimate OD matrices from observed link flows, for example, entropy maximization [

Recent years have seen the development of a bilevel formulation for OD demand estimation from observed link flow [

Other works have developed dynamic OD demand estimates based on the time-dependent proportional and dynamic assignments operated over continuous discrete periods (i.e., several minutes) [

Zhou et al. [

When we consider uncertain demand information, Tsekeris and Stathopoulos [

Note that such OD demand matrix estimation problems generally require many parameters corresponding to the number of centroids of origin and destinations. This number becomes multiples of the number of study periods. The greater this number is, the shorter the study period is. Such a detailed model may be difficult to apply to large-scale road networks including arterial roads and expressways over 24-hour periods. If the study network for estimating time-varying OD demand is large and has many routes for many OD pairs but cannot provide sufficient data, for example, provided by on-line detection equipment, a different approach may be adopted.

The TCoE model proposed herein uses a bilevel formulation, in which the upper problem is based on the generalized least square to estimate 24-hour time coefficients given the day-long OD demand matrix and observed link flows. Thus, the TCoE model does not require many parameters to be set (e.g., for the origins and destinations of the OD demand matrices) so that the hourly OD demand matrices can be calculated by multiplying the given day-long OD demand for the 24-hour time coefficients estimated by the model. The TCoE model thus efficiently improves estimation accuracy because the 24-hour time coefficients aggregated from OD surveys are somewhat biased, thereby reducing accuracy. Therefore, the results show that the TCoE model reduces the number of parameters and computational cost but retains high accuracy for the 24-hour model when applied to a large-scale network that lacks rich real-time data. The characteristics of the time coefficients of the OD survey data are discussed in Section

Fujita et al. [

The semidynamic UE assignment formulation, which considers the residual traffic at the end of each period, has been proposed in three approaches: the OD demand-modification approach [

The TCoE model uses a static UE assignment to obtain link flow proportions of hourly OD demand in the lower problem. The assignment adopted is the same model as the TUE with toll road [

In this paper, we first review the semidynamic concept for the OD-modification approach by comparing it with the OD modification in the TCoE model. In addition, a partial relaxation of the assumption about the length of the study period in the OD-modification approach is developed for practical use. Next, we develop the TCoE model from observed traffic flow by using the generalized least square under a given day-long OD demand and the OD-modification approach. Third, before application to the study network, we clarify that a slight bias exists in the time coefficients of OD demand aggregated from survey. Finally, we demonstrate that the TCoE model can improve the accuracy of estimates of hourly link flow compared with the traffic assignments adopting the initial hourly OD demand aggregated from the survey. In addition, the method transforming the estimated hourly OD demand into OD demand based on departure time is also examined.

The TCoE is a model that justifies the 24-hour time coefficients under a given day-long OD demand by minimizing the least square error between hourly observed link flow and estimated link flow. Conventional estimates of the OD demand matrix based on observed link flows require link flow proportions estimated by traffic assignments. The TCoE model also uses the semidynamic OD-modification approach to obtain link flow proportions for each hour of OD demand. The residual demand for the semidynamic approach in the TCoE model is not operated in the lower problem of traffic assignment but rather the upper problem of the TCoE model.

This section reviews the semidynamic concept in the OD-modification approach of Fujita et al. [

When an hour is set for a study time period, the OD-modification approach semidynamically assigns hourly OD demand for a day by each hour based on the UE principle that drivers select a route with minimum time. Let

The OD-modification approach assumes that the maximum travel time between OD pair

Figure

Modification image of residual demand at the end of period in OD modification approach.

Even though many paths exist between the OD pair

Here, part of the demand of

At this time, link flows, which presume observed link flows, at several points along a path between the OD pair are expressed along the downward-slopping solid line in Figure

Therefore, to minimize the error between observed and assigned link flows midway along the path, the OD-modification approach modifies

We average the burden of a residual demand in both the current and next periods by using the parameter of 1/2 in

By considering continuous periods, the residual OD demand of the previous period also flows in the current network, so it must be added to the current OD demand. Therefore, the following equation expresses the hourly OD demand based on the midway (

In the above equation,

When we set

Note that, after reformulating the Lagrange function above, we can obtain the optimality conditions. The user-equilibrium conditions can be obtained by applying Karush-Kuhn-Tucker conditions with regard to path flows to the Lagrange function. By applying the Karush-Kuhn-Tucker conditions with regard to

References [

As mentioned earlier, the OD modification is the semidynamic assignment method that modifies the hourly OD-dep (

The OD-modification approach assumes that the length of the period must be set longer than the maximum travel time. However, the model may be hard to treat if we cannot set the period length sufficiently long in practical use, which may force us to give up the application, change the strategy of not satisfying the assumption half way through calculation, or recalculate after resetting the length of the period. Therefore, provided we keep the accuracy for practice use, we consider a partial relaxation of the assumption for trips longer than the length of the period. This is done as follows.

From (

To keep the current theoretical OD demand

We now present the formulation and solution algorithm for the TCoE model proposed herein. Additionally, we propose a calculation method of the hourly OD demand based on departure time (hourly OD-dep) from the hourly OD demand based on midway (hourly OD-mid) estimated by the TCoE model.

The TCoE model justifies the time coefficients given a day-long OD demand by minimizing the least square error between estimated link flow and observed link flow. Generally, estimates of the OD demand matrix based on observed link flow require the link flow proportion for each OD pair, which is estimated by traffic assignment.

In this study, we use a set of 24-hour time coefficients in a day for a pair of departure and arrival areas as a pattern of the time coefficients. The TCoE model can significantly reduce the number of operation variables regarding time coefficients from a pattern to a few dozen patterns of time coefficients with high accuracy, as shown in the result of the application in Section

We set a departure subarea to

This model is a bilevel problem in which the upper problem is the above minimization and the lower problem is the TUE-f assignment. Therefore, the upper problem estimates the time coefficients under the given link flow proportions and the lower problem estimates link flow proportions by using the TUE-f assignment with the hourly OD demand, which is calculated by multiplying the given day-long OD demand by the time coefficients of the upper problem. As mentioned in Section

The time coefficients of the solution that minimizes the optimum function

Additionally, we calculate the following iterative steps under the nonnegative constraint condition

Obtain the optimum if all

Exclude

Keep excluding

Otherwise, add

Solve the simultaneous equation (

Set

The optimum time coefficients obtained are multiplied by day-long OD demand to estimate the hourly OD demand which is used for TUE-f assignment in the next step. The link flow proportions given from the result of TUE-f assignment are applied to obtain the new time coefficients for the upper problem of the TCoE model. The solution of time coefficients and hourly OD demand to minimize the square errors of link flows can be obtained by converging the values of time coefficients through these calculations.

Conversely, when setting only a pattern of time coefficients in the study area, the optimum time coefficients can be calculated by using only link flows from the TUE-f assignment without link flow proportion for each OD pair as follows.

When

By substituting above

We get a pattern of time coefficients when (

The optimum solution for the time coefficient can be obtained by the convergence of

Hourly OD demand obtained by TCoE is the hourly OD-mid that minimizes the error of estimated and observed link flows midway along paths between OD pairs. Therefore, the hourly OD demand by the TCoE model differs a little from OD demand aggregated based on departure time (hourly OD-dep) from survey data. When the hourly OD-dep is required for practical use, we explain the method to calculate the hourly OD-dep from the TCoE result.

When the hourly OD demand by TCoE is assumed to be the hourly OD-mid

By using the above equation, hourly OD-dep can be calculated from hourly OD-mid of the TCoE model. In Section

The road traffic census is one of the most important national traffic surveys in Japan. This survey includes observations of hourly link flow on arterial road and expressway and questionnaire surveys of origin destination for each automobile trip nationwide. In this section, we examine the hourly variation pattern of OD demand and compare it with the hourly variation pattern of link flows observed and assigned by TUE.

Figure

Hourly variation patterns by survey type and vehicle type.

Figure

Note that the OD survey gives larger values in the daytime, but the link flow survey gives larger values in the nighttime. The reason for these differences is that the OD variation pattern tends to be underestimated at nighttime and overestimated at peak hours because data is missing from questionnaires especially at nighttime, although OD variation pattern is naturally a little different from link flow variation due to the different survey type. This bias can be seen notably in trucks because trucks usually make more trips at nighttime than other vehicles. We examine this bias of OD variation pattern in view of the results of traffic assignment for a real network in the next section.

The hourly OD demand based on departure time aggregated by the road traffic census 2010 is hereafter called “initial OD.” We examine the characteristics of the hourly variation pattern of the initial OD by applying it to the TUE, TUE-f, and TCoE mentioned in Sections

The study network is composed of 484 zones, 6683 links, and 4468 nodes, which is the Chukyo metropolitan network based on the road traffic census 2010, as shown in Figure

Study network.

The study network (and study area) to which TUE is applied is a large road network within the Chukyo metropolitan area that is also generally used for day-long traffic assignments for practical use. This network connects several cities inside the study area and also simply connects with the outside network including the main cities throughout Japan outside the study area. A cordon line defines the boundary of the study area. When the TUE assignment is applied to such a large network connected with the outside network, new centroids must be set along the cordon line as origins of inflow traffic into the study network. In addition, the periods during which outside traffic departs from its initial origins should be adjusted to the periods that outside traffic enters the study network from new centroids along the cordon line. However, because adjusting the origin and periods for outside trips requires significant manpower, we adjust the initial departure time of outside trips to the periods departing from the cordon line according to the travel time between the initial origin and the cordon line. By using the adjusted departure time for outside trips, hourly OD demand is aggregated from survey data. In the application of TUE, TUE-f, and TCoE, the hourly OD demand from the outside network is assigned as a fixed OD demand separately without residual demand in each hour. In the next chapter, computational time and the PC used for all calculations are described.

Figure

Result of day-long UE assignment.

Figure

Result of hourly UE assignment with initial OD at 7:00.

TUE-f (without residual demand)

TUE (with residual demand)

Result of hourly UE assignment with initial OD at 22:00.

TUE-f (without residual demand)

TUE (with residual demand)

Therefore, from the comparison of hourly variation patterns in Section

In this section, we examine the validity of the TCoE model by applying it to the same road network as in Section

The convergence criterion for bilevel problems of TCoE is set as follows: the difference of the totals of current and previous steps of RMSEs is less than or equal to 0.002. This criterion is applied separately to two vehicle types. The convergence of the TCoE model is judged when the criteria for the two vehicle types are satisfied. The total computational time for above convergence is about 240 minutes with a personal computer (Intel(R) Core(TM) 4.00 GHz processor with 64 GB RAM), at which the iteration number for bilevel problem is four times. The iterative algorithms of the upper problem and the lower problem (TUE) are implemented in FORTRUN and MATLAB, respectively. The average computational time for TUE in a peak hour is approximately 10 minutes in the condition that the iteration number for UE traffic assignment is limited up to 20 times. The average travel time for all OD pairs within study region is about 35 minutes in peak hours. The proportion of traffic which cannot reach its destination within a period to all OD demand in the study region is about 3% and the proportion of residual demand to all OD demand in each hour is about 25–35%.

Table

Comparison of RMS error for link flows for all vehicles

7:00 | 8:00 | 9:00 | 22:00 | Total in a day | |
---|---|---|---|---|---|

TUE-f (with initial OD) | 936 | 646 | 470 | 340 | 10177 |

TUE (with initial OD) | 706 | 766 | 534 | 276 | 9450 |

TCoE | 571 | 532 | 434 | 264 | 8695 |

Comparison of RMS error for link flows for the vehicles on expressway

7:00 | 8:00 | 9:00 | 22:00 | Total in a day | |
---|---|---|---|---|---|

TUE-f (with initial OD) | 1297 | 839 | 640 | 340 | 12145 |

TUE (with initial OD) | 881 | 1020 | 698 | 346 | 10177 |

TCoE | 732 | 749 | 597 | 302 | 11423 |

Result of TCoE at 7:00.

Result of TCoE at 22:00.

In Section

Figure

Comparison of hourly variation patterns.

Figure

Comparison between results by the TCoE and the TUE with OD-dep from TCoE.

Result at 7:00

Result at 22:00

Additionally, from Table

Now the TCoE model is applied to several subareas and dual directions into which the study area is divided. In the study area, Chukyo metropolitan area contains most of Aichi prefecture (including Nagoya city) and part of Mie and Gifu prefectures, which are the commuting areas for Nagoya city. For an analysis that increases the number of time coefficients in the TCoE, we set three subareas and directions (Aichi area to Aichi area, Aichi area to outside of Aichi area, outside of Aichi area to Aichi area, and outside of Aichi area to outside of Aichi area). Therefore, we apply the TCoE under the condition that the time coefficients are set to the above three subareas (four patterns) for two vehicle types, with the other conditions being the same as the application in Section

From the RMS errors of estimated link flows in Table

RMS errors for TCoE with basic model and 3-subarea model.

Period | 7:00 | 8:00 | 9:00 | 22:00 | Total in a day |
---|---|---|---|---|---|

TCoE (3 subareas) | 567 | 526 | 431 | 252 | 8605 |

Difference (TCoE with 3 subareas-basic model) | −4 | −6 | −3 | −12 | −90 |

Hourly variation patterns for TCoE result of three subareas.

The day-long OD demand for transportation forecasting has advantages of accuracy and reliability because it is not affected by hourly variation of OD distribution. In this paper, we proposed the time coefficient estimation (TCoE) model to obtain the hourly OD demand from observed link flows given a proven day-long OD demand. It was constructed based on a bilevel formulation of the generalized least square and the semidynamic traffic assignment (OD-modification approach). Since the hourly OD demand matrices can be calculated by multiplying the given day-long OD demand for the 24-hour time coefficients estimated by the TCoE, TCoE is not needed to set many parameters regarding origins and destinations of the OD demand matrices. TCoE could significantly improve estimation accuracy because the initial OD demand by survey had some bias due to many data missing at nighttime, whereas the TCoE could cancel the bias with a few parameters. From the result of assignments, the accuracy at 7:00 for the TCoE reduced by about 40% the RMS errors in comparison with the TUE-f with initial OD demand. Additionally, we adopted the generalized least square formulation for TCoE to improve the accuracy of hourly OD demand because the maximum-entropy formulation requires a prior hourly OD demand and the prior hourly OD demand in our study network has some bias and is not a reliable demand.

We have reviewed the semidynamic concept for the OD-modification approach (TUE) and compared it with the OD modification in the TCoE model and newly proposed a partial relaxation method of the assumption about the study period length in the OD-modification approach. The TUE is formulated as a static user-equilibrium traffic assignment with elastic demand which modifies the OD demand in the current period to consider the residual traffic volume at the end of each period in a congested network. That is, the residual traffic of TUE is semidynamically subtracted from the demand in the current period and added to the demand in the next period corresponding to the degree of congestion in the study network in which the original hourly OD demand is preserved in both the current and next periods. The treatment of residual demand in the TCoE is almost the same as in the TUE because the OD modification is considered in the upper problem in the TCoE but in the lower problem of traffic assignment.

Hourly OD demand obtained by the TCoE is the hourly OD-mid that minimizes the error of estimated and observed link flows midway along paths between OD pairs. Therefore, the hourly OD demand by the TCoE is a little different from OD demand aggregated based on departure time (hourly OD-dep) from survey data. In case the hourly OD-dep is required for practice use, we also explored and examined the method to calculate the hourly OD-dep from the TCoE result.

The OD-modification approach (TUE) assumes that the period length must be set longer than the maximum travel time. Although a partial relaxation of the assumption was proposed, it is difficult to apply TUE into the network that many OD pairs have more travel times than the period length, such as the network with 15 minutes of period length and 30 minutes of average travel time, because in this case almost the traffic cannot reach its destination within the current period and the treatment of residual flows in Figure

However, since TUE is a user-equilibrium traffic assignment with the elastic demand, it can integrate the logit model that expresses travel behaviors such as a route choice between normal roads and toll roads of expressway. Thus, TUE can be properly applied to the route-choice analysis to predict the change of hourly traffic volume after the construction of new bypass road and expressways or after the congestion charging in peak hours. If there are a day-long OD demand for future transportation planning and a traffic assignment system in large-scale road network with the observed link flows (that may be not autocounted data) which have already been prepared, TCoE is able to be applied to the same network only by changing several parameters such as the hourly traffic capacity and simultaneously estimate the hourly OD demands and link flows during 24-hour periods. Since TUE can use a simple expression of intersections in network as a static traffic assignment, TUE has a characteristic to reduce the maintenance cost of the network with high accuracy.

Future research should examine how to set subareas and durations in a study area for good accuracy and efficiency. This paper executed the TCoE model by using as many observed link flows as possible. We could not clarify the relationship between the estimation accuracy and the number of observed links and sizes in study network. The relationship between estimation accuracy and location of observed links should also be analyzed in the future.

The authors declare that they have no conflicts of interest.

This work was supported by JSPS KAKENHI (Grant no. JP16K06534).