Statistical distribution for inter platoon gaps, intra-platoon headways ...

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University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. Phone:(217) ... inter-platoon gap for platoon (i) is the time needed for the leader of platoon (i) to travel the distance X as ..... More than 90 percent of the data have an absolute.
Statistical distribution for inter platoon gaps, intra-platoon headways and platoon size using field data from highway bottlenecks Hani Ramezani Graduate Research Assistant B106 Newmark Civil Engineering Laboratory University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA Phone:(217) 333-5967 [email protected]

Rahim F. Benekohal Professor of Civil and Environmental Engineering 1213 Newmark Civil Engineering Laboratory University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA Phone:(217) 244-6288 [email protected]

Kıvanç A. Avrenli Graduate Research Assistant B106 Newmark Civil Engineering Laboratory University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA Phone:(217) 333-5967 [email protected]

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Introduction In this paper, field data are used to develop models for inter-platoon gap, average intra-platoon headway, and platoon size distributions in highway bottlenecks created by roadway construction activities. Inter-platoon gap is defined as the time elapsed since the rear bumper of the last vehicle in a platoon passes a particular reference line until the front bumper of the leader of the next platoon crosses the same reference line. In other words, the inter-platoon gap for platoon (i) is the time needed for the leader of platoon (i) to travel the distance X as shown in Figure 1. Intra-platoon headway is the headway between vehicles within a platoon, excluding the headway of the platoon leader. Headway for a given vehicle is equal to the time interval between the moment that the front bumper of the lead vehicle crosses the reference line and the moment the front bumper of the given vehicle passes the line. Platoon size is defined as the number of vehicles in a platoon, including the leader of the platoon.

Figure 1: Schematic for showing inter-platoon ap The three distributions are needed in generating real-world traffic conditions in microscopic or mesoscopic simulation models. Traffic stream and roadway could be modeled as a queuing system with batch arrival where an event is defined as the arrival of each platoon (i.e. arrival of the leader of the platoon). By analogy with a system of queue which has batch arrival, the batch size could be either one, in which case we have a single free flowing vehicle, or more than one, which is equal to the platoon size. The inter-arrival headway could be transformed to inter-platoon gap. The processing time is the interval that a platoon occupies a particular point of the road and it is a function of platoon size and average headway of in-platoon vehicles. The above-mentioned system could be a method for analyzing highway bottlenecks as an alternative to well-known “inflow-outflow” analysis, especially when traffic volume is close to capacity. In such a condition, traffic condition could be switching from undersaturated condition to oversaturated condition and vice versa, and hence intermittent queue is observed in the field. In intermittent queue conditions, traffic volume is lower than capacity and inflow-outflow analysis does not detect the impact of the queuing condition. The main reason that inflow-outflow analysis does not detect intermittent queue is that the assumed

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arrival pattern is uniform. Instead, one can observe arrival of a group of in-platoon vehicles with headways of few seconds followed by a large gap between platoons. When a platoon of vehicles arrives at a bottleneck, the traffic volume could exceed capacity and as a result, in-platoon vehicles may experience queuing condition and delay. Depending on the time gap between platoons, the next platoon may encounter the residual queue. If time gap is long enough during which the queue vanishes, the next platoon will not encounter residual queue. The above-mentioned mechanism of intermittent queue formation implies that the sequence of headway occurrences is important in modeling the real world traffic condition. In close-to-capacity condition, after the occurrence of a particular number of short or medium headways that equals the platoon size, there is a time gap between platoons. Short and medium headways can be modeled by average intra-platoon headway whereas the interplatoon gaps distribution can be used for modeling the time gap between platoons. Hence, the platoon characteristics defined in this paper can potentially describe the real world conditions and sequence of headway occurrences. This paper is the first building block of the idea of batch arrival for traffic stream and aims to find statistical distributions for platoon characteristics. There are a few studies which investigated some of the platoon characteristics. Sadeghhosseini and Benekohal (1995) investigated the headway distribution on freeways and effects of traffic volume on the parameters of headway distribution and percent of in-platoon vehicles. Sun and Benekohal (2005) suggested shifted negative exponential distribution for platoon size and Weibull distribution for time gap distribution using work zone data. The above works have not studied inter-platoon gaps and intra-platoon headways, hence new field data are analyzed to study these platoon characteristics. Also platoon size distribution is studied to see if a similar trend as the one, mentioned by Sun and Benekohal (2005) will be found for the new data set. In this paper, first field data used for this study are explained. Then, different statistical distributions are examined in terms of how well they fit the platoon size, average intra-platoon headway and inter-platoon gap data. Finally, a discussion is made on effects of traffic conditions on the distributions.

Data sets Data were collected form a work zone located over a bridge on, I74EB, a U.S. Interstate Highway. Construction activity was mainly going on under the bridge and practically did not have any influence on traffic. The interstate highway has normally two open lanes per direction, but the right lane was closed within the work zone. The length of the work zone was about 0.5 mile and concrete barriers separated work area from travel lane. Speed limit was 55 mph and no treatments like flagger, police car and ITS were present at the site. Data were collected on the 21st of August, 2008, during a weekday. Two data sets, called “AM data” and “PM data” are available for this site. The AM data were collected from 9:00 AM to 9:45 AM whereas the PM data were collected from 4:15 PM to 5:00 PM. For each individual vehicle, headway, speed and vehicle type are recorded. Table 1 illustrates traffic composition data. Traffic volume during the 45 minutes of video was 555 and 599 vehicles for the AM and PM data, respectively. The percentage of heavy vehicles for the AM data is 21% while it is 28% for the PM data.

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In terms of car following conditions, traffic is divided into three groups, namely Single Free Flowing (SFF) vehicles, platoon leaders and platoon followers. A vehicle is considered to be a follower if its time headway is less than 4.0 seconds or its distance headway is less than 250 ft. Platoon leaders and SFF vehicles are not followers, which means that both have time headways of greater than or equal to 4.0 seconds and distance headways of greater than or equal to 250 ft. However, a group of in-platoon vehicles were following each platoon leader whereas SFF vehicles were not immediately followed by in-platoon vehicles. Table 1: Traffic composition data Data set

Volume (veh)

AM PM

555 599

HVs 21 28

Percentage of SFF vehicles Platoon Leaders 13 23 13 21

Followers 64 66

Thirteen percent of the vehicles in both data sets consist of SFF vehicles. On the other hand, 23 and 21 percent of the observed vehicles are platoon leaders in the AM and PM data, respectively. The percentage of vehicles that are followers in the AM and PM data is 64 and 66 percent, respectively, which constitute a major proportion of the traffic in both data sets.

Platoon size distribution The number of vehicles in each platoon was treated as a random variable. The descriptive statistics are tabulated as below. Table 2: Descriptive statistics for platoon size data Data AM PM

No. 130 129

Mean 3.72 4.05

Std. dev. 2.01 2.69

Minimum Maximum 2 10 2 13

Since each platoon is composed of one leader and at least one follower, the minimum platoon size that could be observed in the field data is two. The sample size includes all vehicles but single free flowing (SFF) vehicles. As mentioned in the previous section, the proportion of SFF vehicles is 13% in both data sets. Figures 2- 3 show the observed platoon distributions vs. the theoretical distribution that may fit to the observed AM and PM data. Considering the fact that all observed distributions are skewed to the right and the minimum platoon size is two, shifted negative exponential, shifted lognormal and shifted Weibull distributions are examined. Since SAS 9.2 was used for the statistical analyses, equations and parameters are according to SAS users’ guide and are shown in Table 3.

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Table 3: Probability density functions Distribution

Equation

1        exp   0  1 log   exp         √2  2  0 ! "  # $%&  # $  '     #       0  ! # 1  (   '   ∞ ∞ 2  √2

Negative Exponential Lognormal

Weibull Normal

Parameters Location Scale Shape 





(

#

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Plsize Curves:

Lognormal(Theta=1.49 Shape=0.96 Scale=0.37) Weibull(Theta=1.49 Shape=1.14 Scale=2.35)

Exponential(Theta=1.49 Scale=2.23)

Figure 2: Platoon size distribution for the AM data

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Plsize Curves:

Lognormal(Theta=1.49 Shape=1 Scale=0.45) Weibull(Theta=1.49 Shape=1.04 Scale=2.61)

Exponential(Theta=1.49 Scale=2.56)

Figure 3: Platoon size distribution for the PM data The location parameter is selected considering the minimum observation and the rest of the parameters are estimated using the maximum likelihood method. If the location parameter is equal to the minimum observation, the occurrence probability of the minimum observation computed from the theoretical distribution will be zero when the random variable is continuous. By choosing a value slightly less than the minimum observation, a small probability is associated with the minimum value. Thus, a value slightly less than the lower bound of the first bin is considered as the location parameter. Since the lower bound of the first bin is 1.5, it was decided to select 1.49 as the location parameter for all distributions. The maximum likelihood estimates of the scale and shape parameters are tabulated as below: Table 4: Statistics of the fitted distributions to the platoon size data Data set AM

PM

Distribution Parameters Location Scale Exponential 1.49 2.23 Lognormal 1.49 0.37 Weibull 1.49 2.35 Exponential 1.49 2.56 Lognormal 1.49 0.45 Weibull 1.49 2.61

Shape 0.96 1.14 1 1.04

P-value for chisquare test 0.16 0.004 0.054 0.54 0.31 0.38

Chi-square test was used to evaluate the goodness of fit for each distribution. The pvalues are given in Table 4. The shifted exponential distributions fit both data sets while the 6

shifted Weibull and lognormal distributions do not fit the AM data at a 0.10-probability of Type I error. Thus, it was concluded that the platoon sizes for both the AM and PM data sets are exponentially distributed. If the Weibull and lognormal distributions had fit to the data sets, the exponential distribution would still be preferred over them because it needs fewer parameters to estimate. The findings for platoon size distribution are consistent with those mentioned by Sun and Benekohal (2005). The scale parameter for the exponential distribution that fit to the AM data is close to that of the PM data (2.23 versus 2.56) which could be due to the similarity of traffic conditions between the both data sets. It is recommended to use more data sets to investigate the effects of various traffic conditions such as traffic volume, percentage of heavy vehicles and work zone length, on platoon formation and the parameters of the distributions. The percentage of heavy vehicles changed from 21% and 28%, but that seems not enough to create a change in shape of distribution.

Average intra-platoon headway Average intra-platoon headway is the average headway for all followers within a platoon. Thus, each platoon has one average intra-platoon headway. The descriptive statistics of average intraplatoon headways are shown in Table 5.

Table 5: Descriptive statistics for average intra-platoon headways Data sets AM PM

No. 130 129

Mean (sec) 2.08 2.27

Std. dev. (sec) 0.57 0.60

Minimum (sec) 0.5 0.93

Maximum (sec) 3.77 3.97

Knowing that each follower has a headway of less than 4.0 seconds, the appropriate distribution is expected to have a maximum value less than 4.0 seconds as shown in Table 5. The mean is approximately located in the middle of the range of data and the standard deviation is close to a quarter of the mean. Based on the central limit theorem, the distribution of the average of a random variable is close to the normal distribution, so it is first investigated if normal distribution fits the data. The location and scale parameters of the normal distribution are defined according to Table 3. The maximum likelihood estimates of these parameters and p-value for chi-square test are displayed in Table 6. As expected, the maximum likelihood estimates of the location and scale parameters are equal to the mean and standard deviation of the observed data, respectively. The p-values show that the observed distribution of average intra-platoon headways is not significantly different than the fitted normal distribution for the data sets.

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index Curv e:

Normal(Mu=2.0757 Sigma=0.5707)

Figure 4: Average intra-platoon headway distribution for the AM data

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Normal(Mu=2.2748 Sigma=0.6036)

Figure 5: Average intra-platoon headway distribution for the PM data

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Table 6: Statistics of the distribution, fitted to the average intra-platoon data Data set

Distribution

AM PM

Normal Normal

Parameters Location Scale 2.08 0.57 2.27 0.60

P-value for chisquare test 0.79 0.56

Inter-platoon gap If one compares the definition of time headway for each vehicle with the definition of interplatoon gap mentioned in the introduction, it is seen that the inter-platoon gap is equal to the headway of either a SFF vehicle or a platoon leader minus the time needed for the preceding vehicle to pass the reference line. Thus, knowing headway, speed and length of each vehicle, the following equation can be used to compute inter-platoon gaps. 3456 *+,-./  011 2 7 (1) 456

where *+,-./ =Inter-platoon gap (sec), 011 =Headway (sec) of either a single free flowing (SFF) vehicle or a platoon leader, 2=Conversion factor from mph to ft/sec (1.47), 89/. =Speed (mph) of the preceding vehicle, :9/. =Length (ft) of the preceding vehicle. The length was determined to be 7 ft for motorcycles, 14 ft for passenger cars, 32 ft for large single unit trucks, and 64 ft for semi trailers. In the appendix, the data analyses that returned these vehicle lengths are explained. As shown in Table 6, the sample size for inter-platoon gap is equal to the number of SFF vehicles and platoon leaders. The minimum inter-platoon gap is slightly less than 4.0 seconds, and this is expected because the 4 sec threshold for platooning vehicles. .

Table 7: Descriptive statistics for inter-platoon gaps Data sets AM PM

No.

Mean (sec) Std. dev. (sec) Minimum (sec) Maximum (sec) 201 9.87 6.49 3.25 38.0 205 8.40 4.86 3.24 25.3

As displayed in Figures 6 and 7, the observed inter-platoon distributions are skewed to the right Similar to the platoon size distribution, exponential, lognormal and Weibull distributions are examined for their goodness of fit. The equations and definition of parameters for these distributions were introduced in Table 3. Similar to the platoon size distributions, a value slightly less than the lower bound of the first bin is selected for the location parameter. The maximum likelihood estimate of the other parameters and p-value for the chi-square tests are reported in Table 8. Based on the p-values for both data sets, the three candidate distributions are not significantly different than the observed inter-platoon distributions. However, the authors recommend the shifted exponential distributions for inter-platoon gap distribution since it needs only one parameter to estimate. Moreover, it is known that the inter-arrival headway for 9

independent arrivals is exponentially distributed. Since platoon leaders are not under the influence of lead vehicles, it can be assumed that the arrival of platoon leaders is independent of the lead vehicles and inter-platoon headways have exponential distribution. Thus, the distribution of inter-platoon gaps is expected to be exponential.

Discussion and conclusion In this paper, theoretical distributions were fit to the platoon sizes, inter-platoon gaps and average intra-platoon headways obtained from field data. Two data sets from the same location, one for AM and the other one for PM time, were used in this study. Based on the data analyses, shifted negative exponential distribution was recommended for both platoon size and inter-platoon gap distributions. Normal distribution was fitted to the average intra-platoon distribution. Overall traffic conditions are the same for the two data sets except that the PM data set had about 10% more volume and 7% more heavy vehicles. Data showed that theoretical distributions of the same type were fitted to the AM data and PM data for platoon characteristics. Hence, the two data sets are supporting each other and the trend may be reproducible at different times at the same location with similar traffic conditions. Since the traffic conditions of the two data sets used in this study were similar enough, they did not show effects of volume and percent heavy vehicles in platoon characteristics. However it is expected that traffic conditions have some effects on platoon characteristics. For instance, if one considers the traffic in oversaturated condition, it is expected that as volume increases, the number of platoons decreases while average platoon size goes up. Moreover, as the percentage of heavy vehicles in platoon increases, the average intra-platoon headway is also expected to increase. However, these trends should be supported and quantified using field data. Thus, it is recommended to use field data from different locations and under different traffic conditions to investigate the effects of traffic condition on platoon characteristics.

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index Lognormal(Theta=3.19 Shape=1.15 Scale=1.39) Weibull(Theta=3.19 Shape=1.05 Scale=6.82)

Curves:

Exponential(Theta=3.19 Scale=6.68)

Figure 6: Inter-platoon gap distribution for the AM data

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index Curves:

Lognormal(Theta=3.19 Shape=0.99 Scale=1.23) Weibull(Theta=3.19 Shape=1.15 Scale=5.49)

Exponential(Theta=3.19 Scale=5.21)

Figure 7: Inter-platoon gap distribution for the PM data

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Table 8: Statistics of the distributions, fitted to the inter-platoon data Data set AM

PM

Distribution Parameters Location Scale Exponential 3.19 6.68 Lognormal 3.19 1.39 Weibull 3.19 1.05 Exponential 3.19 5.21 Lognormal 3.19 0.99 Weibull 3.19 1.15

P-value for chiShape square test 0.99 1.15 0.56 6.82 0.99 0.38 1.23 0.98 5.49 0.26

Appendix The average lengths of different vehicle types used in Equation 1 are based on a study which was specifically done for this purpose. A data set from an interstate highway with 662 vehicles of different type was used. The data include the time that the front and rear bumper of each vehicle pass a marker as well as individual vehicle speeds. Using these data, length of individual vehicles was computed and thereby, average length of different vehicle types was obtained. For the purpose of validation the time that each vehicle passes the marker was computed using two methods: 1) The exact method in which the difference between the moments that the front and rear bumper of each vehicle pass the marker was computed, 2) The approximate method which computes average vehicle length divided by the speed of individual vehicles. Error was defined as the value obtained by the exact method minus the approximate one. The error distribution is tabulated as below: Table 9: Error distribution for the inter-platoon gaps Upper bound of ≤-0.3 the bin 2 Frequency Percentage of 0.3 frequency

-0.2

-0.1

0

0.1

0.2

0.2>

7

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304

24

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1.1

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46.2

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0.5

The upper bounds of the bins were shown in Table 9. Each bin includes the upper bound. For example, bin 0 includes all errors which are greater than -0.1 and less than or equal to zero. The error values range from -0.31 to 0.28 seconds. More than 90 percent of the data have an absolute error less than or equal to 0.1 seconds and about 0.3 percent of the data have an absolute error greater than 0.3 seconds. The authors believe that using Equation 1 to estimate inter-platoon gaps returns reliable data for the purpose of fitting a distribution, especially when one considers that the bin width of observed inter-platoon data is 2 seconds. Moreover, since vehicles with a time headway of more than or equal to 4.0 seconds and distance headway of more than or equal to 250 ft are considered as free-flowing vehicles, most inter-platoon gaps are greater than 4.0 12

seconds. Thus, the percentage of error for most of the data is expected to be within ±2.5%, which is negligible for fitting distributions.

References 1) Sadeghhosseini, S. and Benekohal, R.F., Space Headway and Safety of Platooning Highway Traffic, 2005, Traffic Congestion and Traffic Safety in the 21st Century, Chicago, Illinois, pp. 472-478. 2) Sun, D. and Benekohal, R.F., Analysis of Work Zone Gaps and Rear-End Collision Probability, 2005, Journal of Transportation and Statistics, Volume 8 Number 2.

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