Hysteresis in traffic flow revisited: an improved measurement method

Transcription

1 Hysteresis in traffic flow revisited: an improved measurement method Jorge A. Laval a, a School of Civil and Environmental Engineering, Georgia Institute of Technology Abstract This paper presents a method for measuring traffic flow variables from trajectory data consistently with kinematic wave theory. This method unveils new insights on the hysteresis phenomenon in freeway traffic flow. It is found that that theories of hysteresis based on different congested branches for acceleration and deceleration are incomplete since a key component appears to be aggressive and timid driver behavior. The method is also compared with other methods used in the literature, explaining the scatter usually observed in empirical fundamental diagrams. Keywords: Hysteresis, traffic flow, kinematic wave theory Introduction Typically, traffic flow data are analyzed by aggregating the raw data within predefined regions in the time-space plane. For example, loop-detector data is commonly aggregated over consecutive time intervals of fixed duration, which defines a succession of rectangles in time-space. The problem of this approach is twofold: (i) inside a given rectangle traffic may not be in stationary conditions, and (ii) different lanes may exhibit different traffic states. Problem (i) was already pointed out in Cassidy (1998), who showed that carefully chosen regions in time-space where traffic conditions are stationary lead to well-defined and scatter-free empirical fundamental diagrams. Laval (200) also showed that lane-changes are the main cause of problem Tel. : +1 (0) ; Fax :+1 (0) address: jorge.laval@ce.gatech.edu (Jorge A. Laval ) Preprint submitted to Transportation Research Part B February 11, 2010

2 (ii) as they induce voids in the traffic stream that, when aggregated across lanes, produce significant scatter. An additional source of error comes from the method of aggregation within the given region. It is now commonly accepted that Edie s generalized definitions of flow, speed and density (Edie, 1961) should be used whenever possible (Cassidy and Coifman, 1997). This reference also shows that other aggregation methods may lead to misleading relationships among traffic variables. In this paper, several trajectory data sets under congested conditions are analyzed with Edie s definitions inside areas that maximize the chances of containing stationary conditions. Our results indicate that previous theories of hysteresis loops in the fundamental diagram explain the phenomenon only partially. The first theory for explaining hysteresis is due to Newell (1962). He conjectured the existence of two different congested branches in the fundamental diagram. He argued that in the flow-density diagram, the deceleration branch should be above the acceleration branch. Zhang (1999) postulated that these branches may intersect and reverse the relative position of acceleration and deceleration branches, possibly describing multiple loops. Recently, Yeo and Skabardonis (2009) have analyzed the I-80 NGSIM trajectory data (NGSIM, 2006) and validated Newell s conjectures. They pointed out that the cause for traffic instability might be human error, i.e. anticipation and overreaction. Interestingly, we found that it is indeed common to observe the acceleration branch above the deceleration branch (we did not find multiple loops, however), and this may be explained by human error called aggressive and timid driver behavior in this paper. This paper is organized as follows. Section 2 presents the measurement methodology applied to the trajectory datasets described in section 3. The results are included in section, the implications of these results to hysteresis in section, and a discussion in section Methodology This paper proposes a measurement method based on Edie s definitions inside parallelograms in the time-space plane slanted along the backwards wave speed, w; see Fig. 1a. In this way, one maximizes the chances of having stationary conditions inside the area. This is true because in congested conditions changes in traffic variables propagate upstream at a nearly constant 2

3 wave speed of w 20 km/hr. For an n-vehicle platoon inside an arbitrary region A in time-space, Edie s generalized definitions read as follows: k = q = n t i / A, (1a) i=1 n x i / A, i=1 v = q/k = n n t i / x i, i=1 i=1 (1b) (1c) where the symbols k, q and v stand for density, flow and speed in A, respectively, A is the area of A, and t i, x i are the ith vehicle travel time and distance traveled inside A, respectively; see Fig. 1b. In this paper, A is a parallelogram containing n 1 vehicles, with two opposite sides with slopes w and the other two with a slope comparable to vehicles speed; Fig. 1b. The definition of these areas was performed manually over a digital image of the trajectory dataset. Software was specially designed to allow the user to input n, calculate w directly from the image and draw A accordingly. The software then computes traffic variables according to (1). With this, the experiment consisted in following a platoon free of lanechanging activity as it traverses a traffic disturbance, by computing traffic variables inside successive areas along the platoon s trajectory. In this way, we can readily compare our results with the famous experiment in Treiterer and Myers (197). The main differences between the experiment proposed here and the one in Treiterer and Myers (197) are that (i) instead of an area A (as defined above) they used vertical lines, and (ii) instead of using (1) they calculated average speed and average density; the flow was computed as the product of these two variables. 3. The datasets In this paper we analyzed the following trajectory datasets: 1. Treiterer and Myers (197). Data gathered by aerial photographs taken in July 2, 1967 from an helicopter over Interstate 71 from the downtown area of Columbus, Ohio, north to Route 61. Only the median lane (lane 1) was considered here; see Fig. 2a. 3

4 Treiterer et al. (1969). Same as the above but for lane Coifman (1996). A collection of 12 time-space diagrams for twelve shock waves, video taped from southbound I-680, near Walnut Creek, California. Individual sequences were digitized at one frame per second and vehicle positions were manually recorded in each frame.. Sugiyama et al. (200). A controlled experiment on a 230 m-perimeter single-lane circular road on flat terrain. A stream of 22 vehicles were set up along the circular road, which corresponds to near critical density. A 360-degree video camera was set at the center to record vehicle trajectories.. Xing and Koshi (199). Data gathered on a Japanese sag bottleneck north of Tokyo using a video recorder mounted on a zeppelin. 6. NGSIM (2006). This is a 60 m segment of U.S. Highway 101 in Los Angeles, California. The data were collected on June 1th, 200 and extracted from video data. Here, we used data from lanes 1, 2 and 3 between 7:0 am and 8:0 am. It is worth noting that the digital images for each data set were obtained by converting the relevant figure from each publication to a bitmap file. The only exception is NGSIM (2006) where the bitmap file was created directly with the raw database provided in that reference. These files were then analyzed with a specially designed software.. Results This section presents the results of the experiment defined in section 2 applied to the datasets shown above. A total of platoons were analyzed. The comparison with Treiterer and Myers (197) is interesting because it allows us to contrast different measurement methods and its influence on the conclusions one may draw regarding the hysteresis phenomenon. Fig. 2a shows the famous platoon in Treiterer and Myers (197) and the original flowdensity measurments in part b of the figure. There can be seen two hysteresis loops, labeled A and B in this figure. Loop A corresponds to the process of decelerating to approach the disturbance and accelerating away from it, while loop B can be interpreted as the return to the initial equilibrium state. Hereafter, we focus on type-a loops, which result in different acceleration and deceleration branches; see part b of the figure. Parts c and d of Fig. 2 show our definitions of areas A for this platoon, and the corresponding flow-density plot, respectively. The differences are clear.

5 Mainly, our method shows that (i) the magnitude of the hysteresis loop is much smaller than originally reported; (ii) the shape of the loop is different: larger flow differences are observed at lower densities with our method; (iii) our method captures the jam state (point 10 in Fig. 2d), which is absent in the original measurements. Based on our discussion in the introduction section, it is clear that the bias in the method used by Treiterer and Myers (197) lies in the aggregation of very different traffic states. For example, if one draws a vertical line within the platoon around t = 90 in Fig. 2a, it becomes clear that one would be aggregating a state in complete jam with other states where vehicles are moving. To see if the hysteresis phenomenon is an artifact of the measurement method, we assessed the level of hysteresis obtained for all platoons in the sample: Strong level, if at any given density one observes a difference in flow greater than 300 veh/hr. Weak level, if at any given density one observes a difference in flow less than 300 veh/hr but greater than than 0 veh/hr. Negligible level, if flow differences are less than 0 veh/hr. Negative level, if the hysteresis loop is reversed; i.e., the acceleration branch on top of the deceleration branch. This assessment is clearly arbitrary; but it is useful to identify the main trends. A sample of experiments are reported in Fig. 3. In all, we have obtained a total of 19 (3%) strong-level cases, 10 (23%) weak-level cases, 9 (20%) negligible-level cases, and 6 (1%) negative-level cases; see Fig... Implications for hysteresis The results in the previous section indicates that hysteresis exhibits features that were previously unknown. In particular, (i) hysteresis does not occur every time a platoon traverses a disturbance; and (ii) negative hysteresis. This implies that theories based on the existence of an upper deceleration branch and a lower acceleration branch reproduce hysteresis only partially. A more general explanation for hysteresis seems to be related to aggressive and timid driver behavior. To see this, Fig. analyzes two pairs of

6 typical vehicle trajectories from the NGSIM US-101 freeway dataset, passing through a disturbance. We focus our attention on the deviations of these trajectories with respect to the prediction of the kinematic wave theory with triangular fundamental diagram (Newell, 2002), which does not predict hysteresis loops. These predictions or Newell trajectories correspond to the trajectory of the leader shifted along the wave speed for a distance of approximately equal to the jam spacing. Fig. b of the figure presents a detailed view of the two trajectories in part a, which result in a marked negative hysteresis loop. The solid curves in part b are actual trajectories while the dotted curve represents Newell s trajectory. It can be seen that before the passage of the disturbance, trajectories obey the kinematic wave model very accurately. As soon as the acceleration wave is felt by the follower he decides to stay above Newell s trajectory but to converge to it shortly afterwards. But in doing this aggressive maneuver, the resulting traffic states appear above the deceleration branch. Conversely, parts c and d of Fig. show a timid maneuver, where the driver stays below Newell s trajectory and thus the acceleration branch appears below the deceleration branch. According to this theory, platoons of more than two vehicles will exhibit positive or negative hysteresis loops depending on the relative proportion of aggressive and timid drivers and their respective deviations from Newell s trajectories. 6. Discussion and outlook Using Eddie s generalized definitions of traffic characteristics measured within carefully defined time-space regions, this paper showed that the hysteresis phenomena first shown in Treiterer and Myers (197) exhibits previously unknown properties. We conclude that hysteresis is better explained in terms of aggressive and timid driver behavior rather than acceleration and deceleration phases as previously thought (Newell, 1962). We have confirmed the existence of these two branches, but their relative position in the fundamental diagram depends on the prevailing driver behavior inside the platoon. This result may be exploited to complement the mathematical theory proposed by Zhang (1999). From a modeling perspective, it might not be necessary to explain the cause for aggressive and timid driver behavior, which could be quite challenging if at all possible. Alternatively, one could observe the proportion of 6

7 aggressive and timid drivers and model the deviations from Newell s trajectories explicitly. We are currently building the case for such a theory using the framework in Laval and Leclercq (2008). References Cassidy, M., Coifman, B., The relation between average speed, flow and occupancy and the analogous relation between density and occupancy. Transportation Research Record (191), 1 6. Cassidy, M. J., Bivariate relations in highway traffic. Transportation Research Part B 32 (1), 9 9. Coifman, B., Time space diagrams for thirteen shock waves. Tech. rep., University of California - PATH. URL coifman/documents/tsw.pdf Edie, L. C., Car following and steady-state theory for non-congested traffic. Operations Research 9, Laval, J. A., 200. Linking synchronized flow and kinematic wave theory. In: Schadschneider, A.and Poschel, T., Kuhne, R., Schreckenberg, M., Wolf, D. (Eds.), Traffic and Granular Flow 0. Springer, pp Laval, J. A., Leclercq, L., Microscopic modeling of the relaxation phenomenon using a macroscopic lane-changing model. Transportation Research Part B 2 (6), URL Newell, G. F., Theories of instability in dense highway traffic. J. Operations Research Japan 1 (), 9. Newell, G. F., A simplified car-following theory : a lower order model. Transportation Research Part B 36 (3), NGSIM, Next generation simulation. URL Sugiyama, Y., Fukui, M., Kikuchi, M., Hasebe, K., Nakayama, A., Nishinari, K., Tadaki, S., Yukawa, S., 200. Traffic jams without bottlenecksexperimental evidence for the physical mechanism of the formation of a jam. New Journal of Physics 10 (033001). 7

8 Treiterer, J., Clear, D., Tolle, J., Lee, J., Erion, J., Austin, B., Nemeth, Z., Myers, J., Investigation of traffic dynamics by aerial photogrammetry techniques (interim report 2). Tech. Rep. EES 278-2, The Ohio Dept. of Highways, U.S. Bureau of Public Roads. URL Treiterer, J., Myers, J. A., 197. The hysteresis phenomenon in traffic flow. In: Buckley, D. J. (Ed.), 6th Int. Symp. on Transportation and Traffic Theory. A.H. and A.W. Reed, London,, pp Xing, J., Koshi, M., 199. A study on the bottlenecked phenomena and carfollowing behavior on sags of motorways. In: Japanese Society of Civil Engineers. Vol. IV-26. pp.. Yeo, H., Skabardonis, A., Understanding stop-and-go traffic in view of asymmetric traffic theory. In: Lam, W., Wong, S. C., Lo, H. (Eds.), 18th International Symposium of Traffic Theory and Transportation. Hong- Kong, China, pp Zhang, H. M., A mathematical theory of traffic hysteresis. Transportation Research Part B: Methodological 33 (1), characteristic - w a - w trajectories space space Region A A - w b i=1 i=2 i=3 i= i=n x n time t n time Figure 1: Basic definitions: (a) trapezoidal regions used in this paper; (b) time and distance traveled by vehicles inside a time-space region. 8

9 a 10 b space, 100-ft B acceleration branch deceleration branch A time, min c space km/hr (strong) 11 km/hr d time 10 (veh/km) (veh/mi) Figure 2: Platoons and corresponding flow-density plot for the trajectory data in Treiterer and Myers (197). Top row: original measurements; bottom row: measurement method proposed in this paper. 9

10 a (weak) x t (negative) b c (strong) d (strong) 6 (negative) e Figure 3: A sample of experiments from the remaining trajectory data sets used in this paper: (a) Treiterer and Myers (197); (b) Coifman (1996); (c) Sugiyama et al. (200); (d) Xing and Koshi (199); (e) NGSIM (2006).

11 NGSIM (2006) Xing and Koshi (199) Sugiyama et al. (200) Coifman (1996) Treiterer et al. (1969) Treiterer and Myers (197) 0 strong weak negligible negative Figure : Summary of results. 11

12 a b acceleration deceleration Newell trajectories x -w acceleration wave deceleration wave t c d acceleration deceleration x t Figure : Empirical trajectory data showing how aggressive and timid driver behavior leads to negative (a-b) and positive (c-d) hysteresis loops. 12