Vehicle trajectory changes on a closed road network due to major renewal works

To predict the impact of traffic regulations using a theoretical approach, it is common to verify changes using a traffic equilibrium model in a scenario where the road network is disrupted. These are mathematical model developments with many assumptions, such as complete information that does not validate the actual vehicle movements. There are also user equilibria and dynamic user equilibrium based on bounded rationality that relax perfect rationality to make the traveler’s behavior more realistic (Simon 1987; Szeto and Lo 2006). Some studies have applied and extended the proposed model to actual road-network degradation. Di and Liu (2016) showed that an analysis based on a boundedly rational user equilibrium is effective for differences in route-choice behavior for planned and unexpected road closures. The motivation for introducing user equilibrium in response to a network collapse was that user equilibrium could not predict the return of users when the Mississippi River Bridge was restored. Several mathematical model analyses were conducted for the Mississippi River Bridge collapse in 2007. Guo and Liu (2011) developed a model for irreversible network change based on a boundedly rational user equilibrium and reproduced the phenomenon of irreversible change in the Mississippi River Bridge collapse. He and Liu (2012) presented a day-to-day model that represents the equilibrium process of disruption after collapse. Di et al. (2017) proposed a boundedly rational route switching model assuming that travelers will not switch to a new route unless travel time savings exceed a threshold. The parameters were estimated based on a behavioral survey using a GPS. Mathematical models have been refined to bridge the gap between the models and real phenomena, and surveys and GPS data have been used as complementary tools. Nevertheless, some behavioral assumptions remain.

Examples of the use of real data to validate the differences between the prediction model and reality are also presented. Watling et al. (2012) compared the predictions and empirical observations of changes in route-choice patterns due to the closure of the Lendal Bridge and Fishergate capacity reduction on York's Inner Ring Road. A traffic assignment simulation based on the concept of an asymmetric static Wardrop equilibrium built into the SATURN suite (Van Vliet 1982) was used as a predictive model. The comparisons show that the calibration of the parameter controlling the relative valuation of travel distance to time in the generalized cost equation is important for a reasonable estimation. In addition, the empirical data show some features that cannot be accommodated in the Wardrop equilibrium. The authors suggest that a more detailed empirical analysis can be achieved by utilizing data, such as mobile communication traces, to precisely link spatial data and trajectories followed by individuals on different days. Theoretical analyses using traffic assignment models have been studied historically, and examples of actual road closures are often used to reproduce the proposed model and parameter settings. In addition to setting the parameters, real data were used for calibration by comparing predictions and experience. However, because traffic data does not track the movement of each vehicle, they are used only as supplemental information. It is suggested that the detailed movement of each vehicle with respect to the date and time will be useful for understanding the phenomenon of road-network disruptions.

Many analyses have been based on observational surveys and questionnaires to understand traffic conditions before and after the actual traffic control period. Compared with trajectory data, questionnaires are more likely to ask about intentions and reasons for actions; however, they are also more dependent on respondents’ memories and less likely to be accurate. Hunt et al. (2002) studied the impact of the closure of Centre Street Bridge in Calgary, Canada. This was a 14-months closure of cars and trucks owing to major road repairs that started in August 1999. Observational data before and during the closure, as well as driver interviews, revealed that only a few drivers changed their traffic mode or canceled their trips. The main responses were to find another route to the same destination and to shift the departure time to avoid traffic congestion. Zhu et al. (2010) studied the traffic behavior in response to the collapse of the Mississippi River Bridge using loop detectors, bus users, and survey data during the weeks leading up to equilibration. The traveler’s typical reaction to a bridge collapse was to change routes and departure times, and few changed their traffic mode to public transportation. Tennøy and Hagen (2021) investigated user adaptations to a 14-months capacity reduction in a tunnel with 70,000 vehicles per day in Oslo, Norway. Traffic volume and speed data based on surveys conducted every two weeks and user interviews were analyzed. In response to long-term capacity reduction, users showed adaptive behaviors such as changing traffic modes to public transportation and bicycles. Such behavioral changes were not observed in the analyses of unexpected short-term network disruptions. The survey revealed long-term behavioral changes.

Although interviews with users before and after network degradation are effective, traffic-flow surveys are essential. This section describes an analysis of the data obtained from these devices. Fixed-point observations have long been the mainstream method for understanding traffic flow. However, data analysis using mobile devices such as GPS is increasing every year. Geroliminis and Daganzo (2008) presented a Microscopic Fundamental Diagram using loop detectors and vehicle GPS data. Regarding the possibility of using mobile-device data on expressways, Sohn and Hwang (2008) compared vehicle speeds obtained from aerial photographs and estimated passing speeds obtained from mobile phones, and showed that speed can be estimated more stably in sections where the speed is faster. The effectiveness of data from many types of mobility devices has been debated, and in recent years, analysis of movement trajectory data has become commonplace.

An example of data analysis for various road networks experiencing failures was presented. Several studies have been conducted on various types of disruption, including special events and disasters. Ferreira et al. (2013) handled over 520 million trip data from taxi sensors in New York City in 2009, 2011, and 2012, and developed a technical tool that can visualize them in space and time and search for attributes. The developed tool examined the difference in taxi usage trends when Hurricanes Sandy and Irene occurred and found that Irene had a greater impact on taxi trips in Manhattan than in Sandy. Donovan and Work (2017) detected abnormal situations in transport infrastructure using taxi GPS data and measured their impact in terms of travel time. The proposed method analyzes Hurricane Sandy in New York City in detail and shows that the largest increase in travel time during the hurricane occurred two days after the storm hit. This shows that traffic management is important not only during but also after a disaster and that the resilience of infrastructure can be measured using GPS data. The emergency traffic situation was verified using spatially aggregated data based on GPS data rather than the trajectory of each vehicle’s movement.

In addition to the traffic volume and speed obtained through conventional fixed-point observations, capturing the detailed movements of each vehicle using a mobile device is effective in understanding traffic situations during road restrictions and closures. These include the possibility of simultaneously performing a survey by identifying the past behavior of each vehicle. Another advantage of following each vehicle is the ability to identify specific routes. As described above, route choice (or route choice and departure time, if dynamic) is an important factor in traveler behavior modeling. Recent empirical studies have shown that perfect rationality cannot fully explain route choice. As a comparison between a theoretical model and a real phenomenon, Xie et al. (2017) used trajectory data to verify whether the proportionality condition for obtaining a unique path flow in user equilibrium matches the actual traffic behavior. They identified ODs from taxi trajectory data and showed that the proportionality condition satisfied the statistical significance of the routes between the identified ODs. For descriptive route-choice studies, González Ramírez et al. (2021) conducted a computerized route-choice experiment in which they provided an experimenter with estimated travel times and map information for each route candidate. The results of the verification of which route to choose revealed that travelers evaluated the relative difference in travel time, not the absolute difference, and that the directness of the route had a significant impact on choice, in addition to travel time information. Deng et al. (2023) pointed out that there is an insufficient understanding of individual route-choice behavior, which requires movement data for each vehicle, and clarified the factors that contribute to route stability by analyzing the trajectory data of 5641 vehicles. Luong et al. (2018) applied six metrics, including the degree of overlap between routes and standardized Shannon entropy, to GPS truck data to measure route diversity. They indicated that short-distance and urban OD pairs have a greater diversity in route choices. Many analyses of route choices using vehicle-trajectory data have been studied (e.g. Ciscal-Terry et al. 2016; Hess et al. 2015; Mepparambath et al. 2023; Yang et al. 2017), and their necessity and usefulness have been indicated. Nevertheless, there has been no verification of the effects of planned roadwork using detailed data for each vehicle. It would be valuable to empirically and quantitatively evaluate how route-choice changes under unusual conditions owing to traffic regulations using tracking data.

Even in Japan, where road infrastructure renewal is urgent and projects with planned traffic restrictions are relatively common, empirical studies focusing on specific cases remain scarce. Sakurai et al. (2023) analyzed lane restrictions on an intercity expressway between Tokyo and Nagoya using probe data in addition to traffic counter data. The actual traffic capacity by lane and type of regulation was determined from the traffic-flow rates. Katai and Shimamoto (2022) compared route traffic volumes obtained from probe data by using an inverse estimation model for bus lane restrictions in Miyazaki City. They showed that an increase in the number of vehicles passing toll roads farther away from the restricted section observed in the probe data did not appear in the results based on an inverse estimation model. These analyses revealed the mechanism of congestion generation by traffic regulation type and showed that differences between the model and the phenomenon exist but were insufficient in analyzing how the behavior changed. This study seeks to capture, visualize, and quantify changes in traffic behavior that result from the planned unavailability of road infrastructure by tracking individual vehicles. The literature review introduces both theoretical and empirical studies that identify or reproduce behavioral changes, and it also covers recent research that employs state-of-the-art tracking data to analyze traffic volumes, route choices, and travel times. Nevertheless, for predictable disruptions that are large in scale and long in duration, and that affect many vehicles, only a few studies have examined origin and destination impacts or tendencies to choose alternative routes using individual vehicle trajectories. The novelty of the present research lies in quantifying the effects of such drastic changes in infrastructure conditions by directly observing the routes and behaviors of individual vehicles, thereby informing existing theory-based predictions and simulation-based analyses through the identification of gaps and the calibration of parameters.

Several examples of network disruptions and their analyses have been described. There are two types of disruptions: unexpected disruptions, such as bridge collapses and disasters; and expected disruptions, such as road construction. Disruptions in expectations are likely to increase ATM effectiveness. Given the background of the need for infrastructure management, it can also be said that expected disruptions can occur to prevent unexpected disruptions in advance. Expected network disruptions are considered to have a smaller magnitude of driver avoidance behavior than unexpected disruptions, and daily equilibration is less clear (Danczyk et al. 2017). This may be because there is room for planned changes in the traffic modes and departure times. However, expected disruptions are more amenable to preventive measures, because understanding the scenario in advance (location, period, and magnitude of capacity reduction or road closure) is a significant advantage. Therefore, accurate prediction is very important, and understanding the empirical situation can greatly improve the prediction accuracy by setting parameters for theoretical models and simulations.

This study examines the impact of renewal/improvement work implemented on the Hanshin Expressway in Japan, as an example of planned long-term construction on an expressway with heavy traffic. This construction was conducted on the No. 4 Wangan Line of the Hanshin Expressway in 2019, as part of a longevity renewal project. This line is a major artery used by 100,000 of the 720,000 average daily users of the Hanshin Expressway, and has a high utilization rate for large vehicles.

One-lane restriction (daytime) and two-lane restriction (nighttime) in the 4.7 km section of the No. 4 Wangan Line operated for 14 d from November 6–20, 2019. The road closures in the 8.8 km section of No. 4 Wangan Line and 1.4 km section of No. 6 Yamatogawa Line operated for 10 d from November 21–30, 2019. Figure 1 shows the lane restrictions and road closures. The shaded areas in Fig. 1 are under construction and impassable to all vehicles. Lane restrictions close either the outer or inner lanes of each three-lane carriageway, so traffic is confined to the remaining lanes. Although the closure pattern differs between outer and inner lanes, capacity is reduced by about two thirds during the day and by one third at night. During the full-closure period, all lanes are shut and the road is completely impassable, as shown on the right-hand side of Fig. 1. The Hanshin Expressway is a toll road. During the road-closure period, a system that provides a detour route and does not charge a new toll for getting on and off within four hours was introduced.

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The vehicle-trajectory data used in this study were provided by Transtron Inc. The data is collected from network digital tachographs with the permission of freight carriers for use only in road maintenance and safety improvement. In addition, the data is anonymized and does not include business or corporate information. All target vehicles are recorded at 1-s intervals. The point sequence data collected every second were absorbed into the link on the digital road map by map-matching processing. The target of these data was commercial vehicle trucks, and the data were collected for approximately 22% of the commercial vehicle trucks.

This study uses the data contained in vehicle-trajectory data: vehicle ID, trip ID, link ID on the corresponding road network, time stamp, direction. The vehicle ID did not change, even if the trip or date changed. This is useful for understanding the characteristics of usage. The routes and frequency of use of each vehicle were also identified. The vehicle was uniquely identified using the combined data of vehicle ID and trip ID. Because the vehicle-trajectory data are adsorbed on the road network by a digital road map, all of these data are linked to the road network. The average speed was calculated from the time required to pass through the link and the length of the link. In addition, the digital road map contains available data on the road type, width, number of lanes, and administrators. These on the road network can be connected to vehicle trajectory data based on link IDs. Although the study area is extensive, the digital road map used in the analysis retains sufficient resolution, including city streets. Thus, the breadth of the study area does not compromise the validity or applicability of the findings.

The Hanshin Expressway No. 4 Wangan Line, where the renewal project was conducted, is located in Osaka and Sakai City, Japan. As shown in Fig. 2, the analysis in this study covered an area of approximately 20 × 30 km, including the regulated area. The black and orange dashed lines represent regulated roads that are expressways. Other roads indicate their type based on their thickness. Some roads in the target area were named to explain the results of the analysis found in this study. The construction project involves a major expressway with high traffic volumes and is implemented over an extended period, not just a few hours. For such large-scale construction projects with traffic restrictions, it is essential to evaluate not only the localized impacts (e.g., congestion near the work zone) but also the broader network-wide effects. This allows us to capture potential widespread detour behaviors. While expanding the target area inevitably introduces the influence of other factors, it allows for insights that would not be attainable through an analysis limited solely to the vicinity of the construction zone. The vehicle-trajectory data for all trips through the links in the target area were used for the analysis.

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A normal week without restrictions, a week with lane restrictions, and a week with road closures before and after the construction period in 2019 were set for analysis as follows:

The number of days was the same for the comparison. During the designated three-week period, a total of 867,330 trip IDs and 222,851 vehicle IDs were recorded. Trends in the number of trips and trip lengths were examined for all trips, those using the Hanshin Expressway, and those using the construction section. The results showed that the number of trips decreased on weekends in all groups. The median trip length using the construction section exhibited a slightly different trend from that of the other groups. On some days, the trip length was longer during the normal period than that during the restricted period. However, we found no significant change in the number of trips and median trip length, regardless of the implementation of lane restrictions and road closures. Therefore, a few vehicle users canceled their trips or changed their destinations, and many vehicle users may have changed their route or the time of using the route. (Detailed numbers are shown in Figs. 11 and 12 in the Appendix.) The trend in the number of trips remained unchanged, regardless of serious lane restrictions and road closures. To summarize the changes in traffic situations due to lane restrictions and closures, we show the results of the change for each road section (link) on the road network. As mentioned in Sect. “Vehicle-trajectory data and road networks”, it was assumed that the trajectory data used in this study had already been matched to the links in the road network. Before describing the change analysis, situations on a normal day without restrictions can be observed from the trajectory data within the target area. On Thursday, October 24, 2019 (the first day of the analysis period), the number of links with one or more vehicles observed on that day was 31,782, more than half of which had daily traffic volumes of 50 vehicles or less. The number of links with traffic volumes below 50 (vehicles/d) is particularly high. A histogram of the link traffic volume for that day is included in the Appendix. Because commercial vehicle probe data with this observed situation were used in this study, weekly traffic volumes (vehicles/week) were analyzed to verify the impact of lane restrictions or road closures. In addition, to set the target links where the traffic volume was above a certain level, links with an average daily traffic volume of 50 vehicles or more (350 vehicles/week) in any week during the normal, lane restriction, and road-closure periods were included in the analysis. The number of links analyzed in the road network was 11,360 (expressways: 2.3%; urban expressways: 6.9%; national roads: 28.9%; main prefectural roads: 30.2%; main city roads: 9.1%; prefectural roads: 6.3%; city roads: 1.9%; others: 14.3%).

To understand the spatial impact of road closures, the difference and rate of change in traffic volume (vehicles/week) on each target link were identified. Owing to the characteristics of commercial vehicle-trajectory data, the recorded vehicles were sampled. Therefore, the difference and rate of change were used instead of the traffic volume to compare before and after the road closures. The traffic volume difference and change rate for link \(a\) are expressed by Eqs. (1) and (2), respectively:where.

\({v}_{a}^{normal}\): total traffic volume (vehicles/week) during the normal period.

\({v}_{a}^{stop}\): Total traffic volume (vehicles/week) during the road closure period.

Figure 3 shows the \({D}_{a}\) and \({R}_{a}\) on the road network, with the black dashed line indicating the construction section. To improve visibility in understanding the spatial characteristics, links with particularly significant changes of more than 500 or less than − 500 in the difference and more than 20% or less than − 20% in the rate of change were colored. The Hanshin Expressway No. 4 Wangan Line (H-Exp No. 4) around the closed section had a large decrease in both \({D}_{a}\) and \({R}_{a}\), indicating the direct impact of the closures. In terms of \({D}_{a}\), there were no links where there was a reduction of more than 500 vehicles, except in H-Exp No. 4. Only lines containing construction sections were directly affected by reduced use.

Conversely, the expressway with the highest increase in traffic volume was H-Exp No. 15, which also had the highest rate of increase of 2.014. H-Exp No. 15 would have been the major candidate for detours for the users of H-Exp No. 4. Traffic volume also increased in the E26Hanwa and H-Exp No. 14, which were distant from the construction section. The rate of change for these lines was also high, at 1.221 and 1.265 times, respectively. These lines do not belong to the neighborhood of the construction sections but are assumed to have been used as detour roads. These results are a part of the widespread effects of road closures. With respect to \({R}_{a}\), the small roads around the construction sections had high values. Although ordinary roads around construction sections do not have a high traffic volume, the traffic volume increased significantly during the road-closure period compared with that during the normal period. Road closures on expressways had an impact on ordinary roads, which originally only had a few users. The most heavily affected ordinary road was L29, which was parallel to H-Exp No. 4, and both \({D}_{a}\) and \({R}_{a}\) were large. However, most of the impacts on ordinary roads are limited to the neighborhoods of construction sections. Thus, the extent and magnitude of the impact of road closures can be identified by comparing the differences in traffic volume and rate of change between the normal and closure periods. Spatial visualization of the impact is important because detour behavior can affect traffic conditions far from a closed section.

The relationship between \({R}_{a}\) and road type was verified. Figure 4 presents a boxplot of \({R}_{a}\) for each road type. The quartile range shows that the range is very narrow, except for the urban expressway (Hanshin Expressway in the target area), and most links have a change rate close to 1. Very few links within each road type exhibited high rates of change. In particular, expressways and city roads do not have links that show large rates of change, in terms of either increase or decrease. In contrast, the urban expressways had the widest range of maximum and minimum values, indicating that the fluctuation was the largest among the road types. The impact of traffic reduction caused by road closures is not significantly different for all types, except for urban expressways, which include closed sections. However, there are links that have been significantly affected by increased traffic volumes on national and prefectural roads, other than urban expressways. As with spatial considerations, the impact on the increased side of the traffic volume affects a wide variety of road types, whereas the impact on the decreased side is confined to a limited road type. The verification of these relationships shows that the impact of road closures varies according to road type.

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To ensure a sufficient number of trips for the analysis, representative gate–OD pairs were established instead of OD pairs. These representative gates, which are virtual ODs, are comprised of several links in a road network. The difference from the usual trips between the OD pairs is that trips that start from other locations and pass through a representation gate or pass through a representation gate and then reach other destinations are also included. In other words, a trip is considered a target trip if it passes between representation gates, regardless of the OD. Whether a vehicle terminates its trip at the virtual destination is irrelevant; only the act of passing through the gate is considered. The same rule applies at the origin. A vehicle that passes through only the virtual departure gate or only the virtual destination gate is not regarded as a trip between the virtual gates and is therefore excluded from the analysis of gate OD pairs.

The concept of gate OD pairs is illustrated in Fig. 5. Three trips whose original destinations and origins are far apart are treated as the same-gate OD pair, because they pass through a set gate origin and destination. This is because the travel-time changes and the variety of route choices generated by the renewal works are not limited to vehicles that begin at a fixed origin and end at a fixed destination. Using gate-based OD pairs allows trips that differ greatly in length, road type, and other attributes to be grouped for behavioral analysis. Moreover, this identification method is applicable to map-matched trajectory data collected at very short sampling intervals, such as the one-second data used here.

Several representative zones consisting of spatial areas are set up and the algorithm for extracting the trips that passed through them is described below. The zones defined for this analysis should encompass all potential virtual origins and destinations relevant to the study objectives. Although there is no strict size requirement, zones that are too small may allow some vehicles to pass through without being recorded. The zone size should therefore correspond to the temporal resolution of the data. When the goal is to examine broad spatial or directional trends, larger zones are suitable because they capture overarching movement patterns without requiring fine spatial detail.

Thus, we must distinguish the route of the trip between gate–OD pairs. The route enumeration method between the gate–OD pairs, based on the observed data, adopts the same approach as the trip extraction method. Multiple intermediate passage places (links) are established between the gate and OD pairs, and the route of a vehicle is specified according to whether the vehicle passes through these places. Links that were set as intermediate passages were selected from those with large observed traffic volumes. Because the links selected for passage determination were partial, not all the routes were covered. The coverage rate of vehicles by the enumerated routes is calculated, and intermediate passage places are iteratively added until the vehicle coverage reaches the desired threshold. Even if minute differences exist in the links used, they can be identified using the same route. This method was used in this study to analyze the route choice for detours on a large spatial scale, rather than a detailed route choice. This enables the identification of routes based on commercial vehicle probe data, which does not contain a large number of observation vehicles. Hereafter, the virtual gate ODs defined above are referred to as the ODs.

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A method is proposed to evaluate the impact of road closures on route-choice behavior among the virtual gate ODs defined above. Quantifying the degree of variation in route choice examines the differential impact of road closures according to the gate OD. The most problematic aspect of analyzing the trend in route choice in situations where roads become unusable is that the set of routes that can be chosen before and after closure is different. Several studies have been conducted on the various route choices (e.g. Li et al. 2013; Liu et al. 2011; Luong et al 2018). However, there is no established method for measuring route-choice diversity that considers the difference in the number of possible routes to choose depending on the situation.

A variety of route choices were verified using a measure of variability based on the entropy theory. Before referring to the number of route choices, we note that the routes enumerated by the method described in the previous section do not cover all travel demands between gate OD pairs. Therefore, Eq. (3) was used to normalize the observed route-choice rate. Only the users who used the enumerated routes were included in the analysis. \({\sum }_{k}^{N}{\theta }_{k}^{rs}\) in Eq. (3) shows the number of users who use the enumerated routes.

Next, a method that considers the number of enumerated routes, which varies according to traffic regulations, is presented. Road closures inevitably lead to the loss of frequently used alternatives. When using entropy indices to measure the variety of route choices, changes in the number of available routes pose a methodological issue. Even if vehicles are distributed uniformly across all routes, the resulting entropy value depends on how many routes exist. The entropy index, as a measure of uniformity, reaches its maximum when all options are equally likely, representing a state of high uncertainty in choice. Therefore, as the number of available paths increases, even if the states in the variety of "uniform probability" are equal, the overall uncertainty also increases, leading to a higher entropy value. Thus, the theory of normalized entropy (Abramson 1963) was used to understand the diversity of route choices, considering different numbers of route alternatives. Equation (4) shows the entropy index for the variability of route choices, normalized by the number of routes.where

N^rs: The number of routes between rs

θ_k^rs: The number of vehicles that used route k between virtual gate pair rs

The simple example shown in Fig. 6 explains the variety of route choices using normalized entropy theory. The normalized entropy index defined by Eq. (4) for the heterogeneous and homogeneous route selection states is shown when the number of routes that can be selected (\({N}^{rs}\)) is different: three in normal situations and two under construction. In the case of perfectly homogeneous route choice, the \({E}_{rs}(\mu )\) remains unchanged at 1 in both normal and under construction situations. In the heterogeneous example in Fig. 6, if all users with \({N}^{rs}=3\) of the closed route (Route B) select the same alternative route, the demand would split between 0.9 and 0.1. The heterogeneity increases further when all users use the same alternative route (Route A). Entropy values decreased, indicating a more heterogeneous route choice. For \({N}^{rs}=2\) to have the same value of \({E}_{rs}(\mu )\)=0.582 as example of the heterogeneous state with \({N}^{rs}=3\), the demand is divided into 0.861 and 0.139 for each route. This means that the users of Route B are split to 0.761 (95.1%) and 0.039 (4.89%) out of 0.8 (100%). Here, we briefly examine how the evaluation changes when using the general (unnormalized) entropy index. The normal entropy corresponds to the case where the denominator in Eq. (4), \(\text{ln}{N}^{rs}\), equals 1. In this case, for the homogeneous examples with \({N}^{rs}=3\) and \({N}^{rs}=2\) in Fig. 6, the entropy values are 1.099 and 0.693, respectively. As these considerations show, entropy varies with the number of available routes. To enable consistent comparisons across differing route sets caused by road closures, the study therefore employs the normalized entropy index.

Normalized entropy makes it possible to evaluate route choice homogeneity or heterogeneity even when the observed choice proportions differ. The numerical examples confirm that normalized entropy reflects the degree of uniformity within any given set of routes, independent of the value of \({N}^{rs}\). Thus, \({E}_{rs}(\mu )\) evaluates the variety of route choices under conditions where the number of possible choices differs between \(r\) and \(s\).

The average number of daily trips between gate ODs for the four zones and the rate of change from the normal period during the lane restriction and road-closure periods are listed in Table 1. The vertical axis represents the gate origin zone, and the horizontal axis represents the gate destination zone. These results reveal that some gate–OD pairs increase, and others decrease the average number of trips, owing to lane restrictions and road closures. The gate–OD pair with a particularly significant reduction was C:Sukematsu JCT—D:Nakashima IC, where the average number of trips in both directions decreased by more than 75% during the closure period. This means that there is a gate–OD pair where the impact of road closure is very significant. Both the Sukematsu JCT and Nakashima IC were located in H-Exp No. 4, and most vehicles passing through both zones would have used the closed section during the normal period. As these vehicles detoured, the average number of trips between C and D decreased significantly.

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Conversely, the gate–OD pair with the largest rate of increase in trips was B:Miharaminami IC–D:Nakashima IC. Although this gate pair had a low daily average number of trips during the normal period, the number of trips increased more than twice during the closed period. This means that more than half of the trips during the road-closure period were vehicles traveling on detours owing to road closure. The gate–OD pairs A:Kadoma JCT—B:Miharaminami IC and B:Miharaminami IC—C:Sukematsu JCT, which have relatively high average daily trip numbers, also show increases of approximately 20–30% regardless of direction during the road-closure period. These gate pairs, as well as B—D, can be interpreted as gate pairs in which the number of trips increased owing to detours caused by road closures. The difference in the increase or decrease compared with the normal period, owing to the direction between the gates, occurred in B—C during the lane-restriction period. This gate pair also had a high average number of daily trips. The reason the direction from C to B is affected more by the reduction owing to lane restrictions than the opposite direction is assumed to be a spatial characteristic of the road network. Starting from C, E90 is likely to be selected on highways, where lane-restricted sections can be avoided. However, vehicles that plan to use the lane-restricted section starting from B are more likely to use N26 or H-Exp No. 15 instead of E90. This is probably due to the fact that the direction of the expressway connecting E90 and N26 is not close to a straight line, and the direction changes dramatically along the way shown in Fig. 2. There are examples in which the same road has different impacts in both directions, depending on the alignment of the road, which is likely to be used as a detour.

The effect of lane restrictions showed a high rate of increase between the gate–OD pairs A:Kadoma JCT and D:Nakashima IC. For this gate pair, the impact of lane restrictions was greater than that of road closure. The change in the rate of the daily average number of trips between other gate pairs during the lane-restriction period was smaller than that during the road-closure period. Even for gate pairs C and D, which exhibited a significant rate of decrease during the closure period, the rate of decrease during the lane-restriction period was small. Based on the results for gate–OD pairs, where both the change rate and number of trips are large, the impact of road closures is greater than that of lane restrictions.

The effects of road regulations and closures on users were shown to have different characteristics, depending on the gate–OD. In addition to the changes in the number of trips, this section analyzes how the travel time between gate ODs has changed. Statistical tests were used to examine whether lane restrictions and road closures caused significant differences in travel times between gate pairs. Two types of significance tests were applied to the travel time data for each period: the first was for normal and regulation periods, and the second was for normal and closure periods. The time zone was divided into morning (6:00–10:00 a.m.), daytime (10:00–4:00 p.m.), and evening (4:00–8:00 p.m.). Because the travel time data were assumed to have no correspondence, and exhibit unequal variances, and differ in size, Welch’s t-test was applied. The null hypothesis stats that there is no difference between the means of the data groups. Here, three pairs, gate pairs diagonal to the target area (A and C, B and D), and a gate pair including the construction section (C and D) were analyzed.

Table 2 presents the t-test results for the travel time data. For the road-closure condition, the null hypothesis was significantly rejected in fifteen of eighteen tests, covering six gate pairs across three time periods. By contrast, only four rejections were observed under lane restrictions. These findings indicate that road closures have a substantially greater effect on travel time than lane restrictions. Moreover, every gate–period combination that showed a significant difference under lane restrictions also displayed a significant difference during road closures.

First, we focused on the differences in the results depending on the gate–OD pair. For gate pair C–D, travel times during the road closure period differed from normal conditions at the 0.1 percent significance level in every time zone. This pair was the only one to show significant differences in both directions across all time zones during the closure period. The analysis in the previous sections identified C and D as gate pairs with a significant decrease in the average daily number of trips during the road-closure period. However, the average travel time increased, particularly in the morning. It is understandable that despite the fact that there has been a decrease in the number of users traveling between these gate ODs, a significant increase in travel time is inevitable. During the traffic regulation period, differences at the 0.1 percent significance level also appear in both directions during the morning period and in one direction during the evening period. These results indicate that gate pair C-D is more affected in terms of travel time than any other pair.

Gate pair A–C presents an interesting case. During the daytime period, travel times from A to C are shorter under both the regulation and closure conditions; these rare positive t-values indicate a significant decrease in average travel time despite increased traffic volume. In the morning and evening time zones, by contrast, average travel times rise, although only the evening closure shows a significant increase. The route-choice analysis and Appendix Table 5 reveal that the principal route from A to C follows the E26 Kinki and E26 Hanwa expressways, which do not traverse the construction zone, and that both traffic regulations and full closures further concentrate traffic on this main route. Because routes that traverse the construction zone are unavailable, vehicles shift to the most efficient main route, which shortens travel times. Evidently, only the daytime period can absorb the higher traffic volume and the concentration of vehicles without causing delays. This pattern does not appear in the reverse direction from C to A. In that direction, even when the road is closed, some drivers proceed toward the edge of the construction zone before beginning a detour. That behavior does not occur from A to C. In other words, once a detour begins, drivers maintain it until the construction zone ends, whereas in the opposite direction some drivers reach the closure point before diverting. Consequently, travel times from C to A do not decrease because traffic is not concentrated to the same extent on the most efficient route.

An analysis of time-of-day effects shows that the evening time zone is the only time interval in which average travel time increases significantly across all gate pairs during the road closure phase. For gate pair C-D, the t values indicate a consistently large effect in every period. For gate pairs A-C and B-D, the t values reveal that the impact is more pronounced in the evening than in the morning or daytime. Although average travel times in the morning are greater under normal, regulated, and closed conditions, the relative increase caused by the closure is smaller in the morning than in the evening.

The impact of traffic restrictions on the number of trips, rate of change, and travel time differed significantly between gate and OD pairs. This suggests that behavioral changes would occur in an even wider range than that of the gate set in this study. The fact that the impact was the greatest for gate pairs located across the construction section is a result that is close to intuitive as a direct effect of construction. However, the impact of lane restrictions is greater for some gate–OD pairs in the diagonal directions that do not include the construction section. Although vehicles change their behavior during road closures, they do not change their route during lane restrictions, which may result in a larger impact during lane restrictions than during closures. The result that the number of vehicles using diagonally located gate pairs increases in road closures supports the fact that the change appears not only in the neighborhood of the closed section, but also over a wider area. Previous studies have shown that there is already a discrepancy between theoretical models and actual phenomena during traffic restrictions (e.g. Di and Liu 2016; Watling et al. 2012). Hence, a detail understanding of the actual phenomena has been sought to realize the gap. The results of these studies show the existence of ODs where demand itself fell owing to traffic restrictions and that, despite this, travel times increased significantly. The clarification of these changes in travel time by type of traffic regulation is useful knowledge for calibration processes that aim to match theoretical models with actual phenomena.

The spatial spread of detour behavior from C: Sukematsu to D: Nakashima, which has the greatest impact on decreasing the number of trips and increasing travel time, was visualized. Figure 8 shows the link traffic volume aggregated for trips between gates C and D during the normal and closed periods. First, the “spread” of the links used during the closure period was visualized on the road network. Comparing the periods, the links used were finely distributed near H-Exp. No. 4 with closed sections, whereas the number of links used was limited to detours far from the closed sections. As large spatial detours use high-standard roads, such as expressways, the number of links used is limited. Figure 8 shows that the distribution of the links used differed depending on the distance from the road-closure section.

Because the spatial characteristics indicated that a wide range of alternative routes was used, the relationship between the selected route and trip length was also analyzed. When detours occur over a wide spatial area, the distances of the routes are not similar, which means that there are vehicles that will detour, even if it is a long distance away. We hypothesized that the length of the original trip may be effective, particularly for detour routes with longer distances. In other words, while trips that are originally long in distance can easily accept wide detours, trips that are short in distance, such as those that fall within the target area of this study, may resist long detours. To test this hypothesis, Table 3 shows the percentage of routes selected, median travel distance, travel time between gate–OD pairs, and median trip lengths during the closure period. The trip length is not the travel distance between the gate–OD pairs, but the distance traveled from the origin point to the destination point of a trip (referring to the distance between the OD of the trips in Fig. 5). In contrast, the travel distance between the gate–OD pairs is the median relative to the sum of the link distances used to travel from C to D, that is, the gate–OD pair. Route CD9 was excluded from the comparison because the number of vehicles selected was less than 10. As shown in Fig. 1, Routes CD3 and CD8 using E26Hanwa, E90, H-Exp No. 13, and No. 14 were regarded as spatially extensive detours. The median total link distance used between gate ODs was also significantly larger for Routes CD3 and CD8 than for the others, indicating that the distances were also longer. In contrast, the median of the original trip length exceeded 80 km for Routes CD7 and CD8. The median trip length of 73.8 km for Route CD3 was close to that of the other routes, allowing for detours with a large spatial spread, even for trips that were not particularly long. In addition, the median travel time between gate–OD pairs for Routes CD3 and CD8, which have longer detour distances, was not significantly different from the others.

The hypothesis that trips with a longer distance from the origin point to the destination point are less resistant to spatially extensive detours and more likely to be selected was rejected by the observed data analysis. The common factors of Routes CD7 and CD8, which have larger median trip lengths, are that they minimize the use of ordinary roads. For the road names that constitute the routes in Table 5, italicized text indicates ordinary roads. Route CD7 replaces only the closed section with an ordinary road and immediately returns to the expressway. Conversely, Routes CD4 and CD6, which have short medians of the original trip lengths, are routes where ordinary roads are used for a large proportion of travel between gate ODs. These results are independent of the distance required for the detour because vehicles with longer original trip lengths are more likely to select routes with less use of ordinary roads, and vehicles with shorter original trip lengths are more likely to substitute routes that use ordinary roads. This characteristic was also true for trips in the opposite direction (D to C). As shown in the route details in the Appendix, the route using the ordinary road only in the closed section and the route using E26Hanwa and E26Kinki H-Exp Nos. 13 and 16 correspond to Routes DC7 and DC8, respectively, for route choices D to C. The median trip lengths for Routes DC7 and DC8 are the top two (77.1 km, 82.3 km) of the six routes available during the road closures. Vehicles selecting a route that avoided the use of ordinary roads as much as possible also exhibited longer original trip lengths from D to C. The differences between expressways and ordinary roads include maintaining the speed of vehicles, ease of driving, and the number of traffic signals. These results suggest that these factors are more important for route choice than the spatial extent of detours.

The median travel time from C to D varied by route. The difference between the minimum and maximum median travel times among the seven routes, excluding Routes CD1 and CD9, was 21.2 min. This is a large difference, based on the fact that the average travel time in the daytime from C to D during the normal period is 22.34 min shown in Table 1. There was no obvious relationship between the route-choice rate and median travel time. For example, the median travel time between gate pairs differs by 12.9 min for Routes CD2 and CD5, where the rate of route choice differed by only 0.203%. It should be noted that accurate travel times between gate pairs cannot be determined prior to route choices. In particular, unlike travel times from origin to destination, the willingness to search for gate-to-gate travel times in advance may be small. Further analysis of the trajectory data is required to investigate the extent to which travel time histories and forecast information contribute to route choices.

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As shown in Fig. 8, the route choices during road closures are widely distributed. To confirm whether the travel between gate–OD pairs has a similar phenomenon, each gate pair was evaluated using a variety of route choices. This section analyzes the characteristics of each gate–OD with respect to changes in the variety of route choices when road closures occur. For the trips between gate and OD pairs extracted in Sect. “Methodology”, both directions of the three gate pairs, A and C, B and D, and C and D, are covered. Before normalizing the observed route-choice rates, it is necessary to indicate the percentage of vehicles included in the routes enumerated by the method described in Sect. “Trip identification between gate–OD pairs”, which identifies a route using several intermediate links specified between the gate ODs. The average user coverage of the enumerated routes between gate ODs during normal, regulation, and closure periods was 94.62%. The maximum coverage was 99.50% from D: Nakashima to C: Sukematsu during the normal period and the minimum coverage was 81.51% from C: Sukematsu to D: Nakashima during the closure period. Hence, the enumerated route choices include most users. These were normalized using Eq. (3) as the route-choice rate per gate–OD among vehicles using the enumerated routes.

The normalized entropy model shown in Eq. (4) calculates the variation in route choice according to the route alternatives. The route choice rates by direction in each gate OD are indicated in the Appendix, whereas the number of routes \({N}^{rs}\) that could be used varied with road closure as follows. Between A and C, and B and D, it is reduced by one, and between C and D, it is reduced by two. The number of available routes includes routes that can be selected, but not used. Figure 9 shows the normalized entropy index based on the data for one week in each period. It is interesting to note that although gate pairs A and C had a greater diversity of route choices than the other gate pairs during the normal period, their diversity decreased during the closed period. This is because there is another route, E26Hanwa-H26Kinki, which is highly selective, other than H-Exp No. 4 during the normal period, and road closures have further concentrated on that route. According to the percentages reported in Appendix Tables 5 and 6, the share of trips choosing the E26 Hanwa-E26 Kinki route is highest during the closure period. This result aligns with the interpretation that a decrease in entropy reflects a stronger concentration on specific routes and hence greater predictability in route choice. The observed data are therefore consistent with the theoretical expectation. Conversely, gate pairs B, D, C, and D have greater diversity, owing to lane restrictions and road closures. For B and D, the variety of route choices increases with lane restrictions, and further increases with road closures. Although H-Exp No. 4 had a high choice rate under normal and lane-restriction periods, the results showed that the rate of using other routes increased slightly owing to lane restrictions.

Another feature of Sites B and D is that they differ in the variety of route choices in each direction. B to D were more diverse than D to B in all the periods. This seems to be due to the spatial characteristics of the roads. Examples C and D are extreme examples. Table 4 presents the percentage of route choices from C to D during each period. A hyphen indicates an unusable route and distinguishes it from routes with 0% utilization. There was no change in the variety of route choices owing to lane restrictions in either direction, and the route choice was nearly uniform when the roads were closed. These results indicate that variations in route choices during lane restrictions are not directly related to route-choice behavior during road closures. The following section summarizes the implications of the analysis of the three bidirectional gate pairs. Gate pairs with a high percentage of sections to be closed during the normal period have various route options when the road is closed and the normalized entropy increases. In contrast, for gate pairs with a main route that does not include the section to be closed during the normal period, the route choices are concentrated when the road is closed, and the normalized entropy decreases.

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Focusing on gate–OD pairs C: Sukematsu and D: Nakashima, where the variety of route choices during the road-closure period was the largest and spatially spread, the results of the distribution of travel time for each vehicle by selected routes are presented.  Figure 10 shows the distribution of travel time by time of day for each period in both directions. Dots for each trip, with the color and shape of the dots indicating the route adopted for that trip. In the normal period, a peak time zone with increased travel times appeared in the morning hours for C to D, and in the evening hours for D to C. When the road regulations started, the travel time from C to D changed significantly compared to that from D to C, which did not change significantly. During the travel times from C to D, two-tiered mountains of morning peak hour times were observed, with many trips being twice as long as those at the same time of day during the normal period. Interestingly, there are two levels of increase in travel times on the same route and at the same time of day. Evening peak hours that did not appear during the normal period were also visible. The differences in the amount of change in each direction could be influenced by the time of day when they were most frequently used. The direction from C to D, where the travel time in the morning increased significantly from the normal period, was used more often in the morning, and it may have been more difficult to change the time compared with the evening. Most of the routes used remained in H-Exp. No. 4; however, in terms of travel times, the changes were particularly in directions C–D.

During closed road periods, the distribution of travel times by time of day changed dramatically. In both directions, the travel time increased in all time zones irrespective of the peak hour. Many types of routes exist; however, they do not avoid increased travel times at all times of the day. However, the route-specific travel-time distributions in Fig. 10 for the closure period differ noticeably. Routes CD2 and CD3 occupy positions with longer travel times. Routes CD4, CD5, CD6 and CD7, plotted in green and red, show similarly high values during peak hours, yet observations in the early morning and late evening appear to reduce their median travel times relative to CD2 and CD3, as indicated in Table 3.

Levene’s test was applied to the travel-time data for each route. The assumption of equal variances held for routes from C to D, whereas heteroscedasticity was detected for routes from D to C. Normality was present in only three bidirectional routes, so the non-parametric Kruskal–Wallis test, which does not require normality or homoscedasticity, was used to evaluate median differences. The null hypothesis that all route medians are equal was rejected in both directions for the C–D gate pair (from C to D: p-value 5.07e-12, H-stat 67.3; from D to C: p-value 0.000174, H-stat 24.5). Thus, travel times differed across routes, with some showing significant discrepancies. Post hoc comparisons revealed significant median differences in 12 of the 28 route pairs from C to D, and in 2 of the 15 route pairs from D to C. Both significant pairs from D to C involved Route DC7, the most frequently used route (Appendix Table 7), indicating that this heavily selected path differed markedly from others. To assess deviations from traffic equilibrium, in which all vehicles experience equal travel times and no route change reduces travel time, observed behavior should be compared with an equilibrium assignment. The trajectory analysis, which demonstrates unequal travel times across routes, suggests that traffic conditions during the road closures were far from equilibrium.

Consistent with the trends observed during the normal and regulation periods, the closure period also exhibited a morning peak in the C to D direction and an evening peak in the D to C direction. Additionally. The distribution of travel times between the other gate pairs is shown in the appendix. For gate pairs A and C, the most commonly used routes remained the same; however, the peak hour in the travel time distribution was more pronounced at road closures. For gate pairs B and D (particularly D to B), the routes used have changed; however, it can be seen that few vehicles have a significant increase in travel time at road closures.

The relationship between the original travel distance and route choice for the virtual ODs indicates that vehicles with longer travel distances prefer routes with a higher proportion of expressways. However, as the original distance traveled increases, the sensitivity may change because the ratio of the distance between the virtual ODs decreases as the original travel distance increases. By creating it between virtual gate pairs, it is possible to specify the gates to be verified. However, the relationship with the original OD is an issue that needs to be addressed.

Trips between C and D, which diverge into many routes in closures, do not avoid an increase in travel time regardless of the route selected, a result indicative of the chaotic state of affairs. Even during off-peak hours, confusion continued throughout the day, indicating that this was not a typical traffic situation. Even in the case of gate pairs that did not change during the lane-restriction period, many vehicles were affected by the distribution of travel times, in terms of route variety. For these gate pairs, the increase in the travel time during lane restrictions can be reduced by encouraging effective route changes.

Employing an entropy-based index to track changes in route-choice variety provides a quantitative means of showing how lane restrictions and full closures affect behavior. The fact that there is a relationship between the characteristics of route choice under normal conditions and the reactions that occur during road closures has led to the recognition of gate pairs that are particularly susceptible to confusion caused by road closures regarding the routes used. This narrows down the target for which alternative routes should be presented, based on the relationship between the route-choice rate under normal conditions and the links containing construction sections. The finding that the characteristics of impact due to traffic regulations vary depending on the usual route choice behavior for each OD pair is likely to serve as a generalizable observation.

There were cases in which the characteristics of the gate–OD pairs differed according to the direction. The reason the characteristics of the same-gate OD pair differ depending on the direction is difficult to consider using the proposed variety index. By classifying routes according to the type of road and directness of the route, the proposed index can be improved to evaluate its variety. The ability to reflect the characteristics of a route other than the distance is an advantage of route-by-route analysis. The difference between ODs with various and uniform routes can only be understood via a route-based analysis, not an OD-based analysis. Previous vehicle trajectory data analyses of events including disasters visualized changes in the time-spatial characteristics and travel times of each OD. The results of this study show that previously unobserved changes in route choice tendencies are also present in the confusion and recovery processes associated with disasters, as indicated in previous studies (Ferreira et al. 2013; Donovan and Work 2017).

In addition, it is interesting to note that there is no clear relationship between travel time and route choice; therefore, factors other than time and distance, such as ease of driving and previous driving history, could be further analyzed as factors leading to route choice. The ability to link historical data is a good feature of the vehicle trajectories used in this study, and can complement interview surveys and questionnaires.