The comfortable driving model revisited: Traffic phases and phase transitions
TThe comfortable driving model revisited: Trafficphases and phase transitions
Florian Knorr and Michael Schreckenberg
Fakultät für Physik, Universität Duisburg-Essen, 47048 Duisburg, GermanyE-mail: [email protected] , [email protected] Abstract.
We study the spatiotemporal patterns resulting from different boundaryconditions for a microscopic traffic model and contrast them with empirical results. Byevaluating the time series of local measurements, the local traffic states are assigned tothe different traffic phases of Kerner’s three-phase traffic theory. For this classificationwe use the rule-based FOTO-method, which provides ‘hard’ rules for this assignment.Using this approach, our analysis shows that the model is indeed able to reproducethree qualitatively different traffic phases: free flow (F), synchronized traffic (S), andwide moving jams (J). In addition, we investigate the likelihood of transitions betweenthe three traffic phases. We show that a transition from free flow (F) to a wide movingjam (J) often involves an intermediate transition; first from free flow F to synchronizedflow S and then from synchronized flow to a wide moving jam. This is supported bythe fact that the so-called F → S transition (from free flow to synchronized traffic) ismuch more likely than a direct F → J transition.The model under consideration has a functional relationship between traffic flow andtraffic density. The fundamental hypothesis of the three-phase traffic theory, however,postulates that the steady states of synchronized flow occupy a two-dimensional region inthe flow-density plane. Due to the obvious discrepancy between the model investigatedhere and the postulate of the three-phase traffic theory, the good agreement that wefound could not be expected. For a more detailed analysis, we also studied vehicledynamics at a microscopic level and provide a comparison of real detector data withsimulated data of the identical highway segment.PACS numbers: 45.70.Vn, 64.60.De, 89.40.Bb a r X i v : . [ phy s i c s . s o c - ph ] J u l he comfortable driving model revisited
1. Introduction
Assessing a model’s quality is probably best done by comparing its results with empiricalfindings. For microscopic traffic models, this approach is particularly interesting becausedifferent scales have to be considered. On the one hand, microscopic traffic models arebased on the assumption that a vehicle’s motion is governed by the next (or the nexttwo) vehicles ahead. Hence, from a single vehicle’s perspective it makes no differencein these models whether there are only two or two thousand vehicles on the road(for a review of microscopic traffic models from a physical point of view, see [1, 2]).Many empirical studies of traffic, however, focus on macroscopic phenomena. Emergingbehaviors (e.g., traffic breakdown, jam formation), for instance, require the coordinatedmotion of hundreds of vehicles (see the recent review article [3], which also discussescurrent challenges on traffic modeling).To characterize traffic flow, the distinction between freely flowing and congestedtraffic is obvious, but also quite coarse-grained. A more detailed analysis of trafficreveals a rich variety of spatiotemporal patterns in congested traffic: the meticulousstudy of empirical traffic data has led to the development of the three-phase theoryof traffic (exhaustively presented in the books by Kerner [4, 5]). According to thistheory, congested traffic subdivides into two phases: ‘synchronized traffic’ (S) and ‘widemoving jams’ (J). Low average velocities, low vehicle flow rates, and a downstream frontthat propagates with a constant velocity against the vehicles’ direction of travel arecharacteristic features of the latter. In contrast, the downstream front of synchronizedtraffic is often located at a bottleneck and, although average velocities are considerablybelow the velocities of free flow, traffic flow is higher than in jammed traffic—sometimesclose to the rates observed in free flow.Kerner’s detailed analysis has led to several conclusions about the characteristicsof traffic flow. One of these is the fundamental hypothesis of three-phase traffic theory .It states that “steady states of the synchronized phase cover a two-dimensional regionin the flow density-plane” [5, p. 46]. From this follows that models with a functionalrelationship between vehicle flow and density are not able to adequately reproduce thephases of congested traffic. Recently, this conclusion was discussed controversially though[6, 7, 8], and an alternative explanation for the two-dimensional region of steady stateswas given [9].The distinction between these ‘traffic phases’ gets even more complicated for severalreasons: (i) Synchronized traffic itself subdivides into various classes with differentspatiotemporal characteristics. (ii) Although some of these sub-classes seem to beidentical to the ones found by Schönhof and Helbing [6, 7], Helbing et al [10], and Treiber et al [8], the just mentioned authors and Kerner [4, 5] use a different vocabulary toclassify these patterns. (iii) In addition, the identification of distinct traffic phases isdifficult, if not impossible, based on macroscopic traffic data [11].In this context, it has to be noted that the term ‘traffic phase’ does not (necessarily)correspond to a phase in the physical sense. Although traffic flow can be interpreted as he comfortable driving model revisited et al [16, 17], to distinguish the three phases of traffic. Despite some imperfections, theFOTO-method allows an objective classification of traffic states that may not only helpto compare empirical with simulated data, but can also be applied to compare trafficmodels with each other.
2. Simulations
In the following, we study the dynamics of the comfortable driving model [14, 15], towhich we will refer as CDM from now on, with open boundary conditions. The CDM isa cellular automaton model with extensions for anticipatory driving behavior. Earlierinvestigations showed a good agreement of the model with empirical traffic data ona microscopic level (e.g., headway distributions) [15, 18]. As is common for cellularautomata, space is discretized in sites (of length 1.5 m) and time is discretized in (timeor update) steps (of duration 1 s).
As we use a similar simulation setup to Barlovic et al [19], we give only a brief summaryof the simulation method. (Minor modifications were necessary because each vehicleoccupies l veh > sites.) We consider a one-lane road of N sites, on which vehiclesmove from left to right. The left boundary or entrance section consists of the leftmost v max + l veh + 1 sites, where v max denotes the vehicles’ maximum velocity. Vehicles, whichenter the road from the left boundary with probability α , are inserted with velocity v max at position x insert = min( v max + l veh + 1 , x last − v max ) , where x last is the rear positionof the vehicle closest to the entrance section. Then, all vehicles, including the newlyinserted one, move according to the CDM’s rules of motion [14]. If a vehicle is not ableto leave the entrance section, it is removed afterwards. This insertion strategy allows forhigher inflow rates compared with the obvious insertion strategy, which places a vehiclein the leftmost site with probability α if this site is empty [19].The right boundary or exit section is modeled as follows: before other vehicles move he comfortable driving model revisited β , and it is cleared after thevehicles have moved. Moreover, a vehicle is removed from the road if it will reach therightmost site or beyond during the next update step by maintaining its current velocity.One can interpret the probability α as the inflow of vehicles to the road segment,and the probability β determines the strength of local perturbations caused at thedownstream boundary. These perturbation may, for instance, result from vehicles thatenter the road via an on-ramp and occur randomly in front of vehicles on the main road.Consequently, high values of β result in a low exit probability of the system.The following results were obtained on a road consisting of N = 5001 sites. Thefirst × from a total of . × time steps were discarded in each simulation run toavoid transient behavior (e.g., in some cases, it took several thousand time steps until ajam formed at the exit boundary reached the entrance boundary). Vehicles could move v max = 22 sites per time step at most and had a length of l veh = 5 sites. We have analyzed the vehicle dynamics for all combinations of entrance and exit boundaryconditions resulting from a step size of . (i.e., α, β ∈ [0 . , . , . . . , . ). Thephysical phase diagram resulting from these measurements in the bulk of the system,which excludes the first and the last N/3 sites of the road, is depicted in figure 1. v e l o c i t y [ k m / h ] αβ v e l o c i t y [ k m / h ] (a)
0 0.2 0.4 0.6 0.8 1 α β v e l o c i t y [ k m / h ] F C (b)
Figure 1. (a) The rotated 3D representation of the phase diagram illustrates verywell that the transition from free flow to congested traffic is discontinuous. For betterreadability, we have added straight lines to the equivalent 2D representation in (b),which represents a stylized phase diagram. (For the units of the z-axis we have usedthe standard conversion, where a site measures . m and an update step correspondsto 1 s [14].) The lines separating free flow (F) and congested traffic (C) in figure 1(b) weredetermined from the average velocity in the bulk: if the average velocity was at least99.5% of the vehicles’ maximum velocity v max , free flow was assumed. (The value for the he comfortable driving model revisited p d (see Appendix B), with which vehicles randomlyreduce their velocity in free flow.) This classification is probably more intuitive thanthe application of the extremal principle [20], which was used by Appert and Santen[21] or Barlovic et al [19] for simpler models. Since the application of the extremalprinciple requires the knowledge of the vehicle densities induced at the boundaries, thisapproach relies on extensive simulations as well. (Only for very simple models such as theTASEP [2], are the boundary densities identical to the entrance and exit probabilities α and β .) We admit, however, that our very simplistic approach of constructing the phasediagram might not be suited for models with a more complex fundamental diagram (e.g.,with local maxima of the flow [20]).As one can see, especially from figure 1(a), free flow (F) and congested traffic (C)are separated by a very sharp, discontinuous line for β (cid:38) . . In free flow (F) practicallyall vehicles move at their maximum velocity. The averaging process, which led to theabove figures, hides spatial and temporal information. Kerner, however, defines trafficphases as “a set of traffic states considered in space and time that exhibit some specificspatiotemporal features” [4]. Therefore, it is necessary to investigate the microscopicdynamics of vehicles in more detail.Figure 2 shows four spatiotemporal plots of the entire road during a one-hour interval(i.e., 3600 consecutive update steps) for four distinct combinations of inflow and outflowprobabilities. All configurations were taken from the region C in figure 1(b). The patternof figure 2(a) is taken from the bottom right corner of figure 1(b). In this area high inflowrates coincide with a low probability of the exit section being blocked. The figure showstwo phenomena which are characteristic for this combination of boundary conditions: (i)Due to the high inflow rates, random velocity fluctuations are likely to occur close tothe left (upstream) boundary. These fluctuations cause congested traffic propagatingbackwards and, thereby, reduce the effective inflow probability. Consequently, the flowof the remaining vehicles corresponds to the outflow of congested traffic, where no otherjams occur (see figure 2(a) during the first 1000 time steps). (ii) Jams, which often, butnot necessarily, occur at the right boundary (at t ≈ in figure 2(a)), have a sharpupstream front. This is again a consequence of the high inflow rates, because any localperturbation immediately affects the following vehicle and propagates upstream.Figure 2(b) depicts several waves of congested traffic traveling upstream at a nearlyconstant velocity. These stop-and-go waves are known even from the most simplistic trafficcellular automata (e.g., the Nagel-Schreckenberg model [22] or the VDR model [23]).A localized congested pattern is presented in figure 2(c). Relatively low exitprobabilities (i.e., high values of β ) constantly provoke jams at the right boundary. Theinflow probability, however, does not suffice to supply enough vehicles for the jams topropagate to the left boundary. Hence, the jams get saturated and end close to the exitsection.Quite interesting is the pattern of figure 2(d). Here, nearly the entire road is coveredby a state of intermediate velocities (30–70 km/h). At the same time, relatively highflows, ranging from 1020 vehicles/h to 1440 vehicles/h (see figure 3(d)), predominate. he comfortable driving model revisited (a) (b)(c) (d) Figure 2.
Examples of different spatiotemporal patterns resulting from differentboundary conditions: (a) α = 0 . , β = 0 . , (b) α = 0 . , β = 0 . , (c) α = 0 . , β = 0 . , and (d) α = 0 . , β = 0 . . (The entrance section is located at position x = 0 and the exit section at x = 7 . km.) Such a state has not been observed for simpler models [19] and has led to differentinterpretations [24, 25] for the CDM and a subsequent model [26] (see section 4). Thequestion arises: to which traffic phase(s) should one assign this spatiotemporal pattern?And more general: how can such an assignment be done for any of the above trafficpatterns?
3. Classification of Traffic Phases
Kerner et al [17] have presented a method called “FOTO” (Forecasting of Traffic Objects)that can be used to identify traffic states (see Appendix A). The method uses 1-min-aggregated data from a local detector (i.e., velocity and traffic flow). Based on a set ofrules, it decides whether the local traffic state is ‘free flow’ (F), ‘synchronized traffic’ (S),or ‘wide moving jam’ (J). The underlying set of rules can be summarized as follows: (i) ifthe average velocity is high, free flow predominates, (ii) if both the average velocity andthe flow are low, a wide moving jam passes the detector, and (iii) if at medium velocitiesthe flow is still high, then the corresponding traffic phase is “synchronized flow”. (It isimportant to note that the classification of traffic phases results from the simultaneous he comfortable driving model revisited fl o w [ v e h i c l e s / h / l a n e ] t [min] (a) fl o w [ v e h i c l e s / h / l a n e ] t [min] (b) fl o w [ v e h i c l e s / h / l a n e ] t [min] (c) fl o w [ v e h i c l e s / h / l a n e ] t [min] (d) Figure 3.
The time series of the flow rate from local measurements for the patternsdepicted in figure 2. Data was collected by a detector positioned in the middle ofthe road at kilometer . . Figures (a)–(d) correspond to the spatiotemporal plots offigure 2: (a) α = 0 . , β = 0 . , (b) α = 0 . , β = 0 . , (c) α = 0 . , β = 0 . , and (d) α = 0 . , β = 0 . .States of free flow are depicted with a white background color and triangles ( (cid:52) )as data points. Synchronized traffic is shown with a light gray background color andrectangular symbols ( (cid:3) ). A dark gray background color and circular symbols ( ◦ )indicate wide moving jams.From figures (a) and (b) it becomes evident that there is no unique flow rate abovewhich the traffic flow breaks down in the CDM. In (a) the flow rate reaches valuesconsiderably above 2500 vehicles/h/lane before congestion sets in, whereas it barelyexceeds 2000 vehicles/h/lane in (b) before a breakdown occurs. Also note that figures (c)and (d) are assigned to different traffic phases, even though the flow rates are at thesame level. (The corresponding velocity time series is given in figure 4.) analysis of both the flow rate and the average velocity because the analysis of only onevariable usually does not suffice to identify the traffic phase.) In combination with amethod called ‘ASDA’ (Automatische Stau-Dynamik Analyse; Automatic Tracking ofMoving Jams), it is even possible to track the propagation of traffic phases detectedby FOTO [4] between detectors. An advantage of both FOTO and ASDA is that they„perform without any validation of model parameters in different environmental andtraffic conditions” [17]. Hence, we could apply the FOTO-method to our simulationresults without modifications. he comfortable driving model revisited v e l o c i t y [ k m / h ] t [min] (a) v e l o c i t y [ k m / h ] t [min] (b) v e l o c i t y [ k m / h ] t [min] (c) v e l o c i t y [ k m / h ] t [min] (d) Figure 4.
The velocity time series from local measurements for patterns depictedin figure 2. Data was collected by a detector positioned in the middle of the road atkilometer . . (The corresponding values of α and β are: (a) α = 0 . , β = 0 . , (b) α = 0 . , β = 0 . , (c) α = 0 . , β = 0 . , and (d) α = 0 . , β = 0 . .)Based on the local measurements of traffic flow and vehicle velocity, a classificationof the current traffic state was performed according to the FOTO-method. Statesof free flow are depicted with a white background and triangles ( (cid:52) ) as data points.Synchronized traffic is shown with a light gray background color and rectangularsymbols ( (cid:3) ). A dark gray background color and circular symbols ( ◦ ) indicate widemoving jams. For better readability, dashed horizontal lines indicate the values of km/h and km/h, respectively. (Note that these lines are not associated with theclassification of traffic states.) average velocity in figure 2(a) manifests as a wide moving jam, which lasts for eightminutes. Note that the detector data do not show an abrupt transition from free flowto a wide moving jam ( F (cid:54)→ J ). First, we can observe a transition from free flow to asynchronized phase ( F → S ) before a wide moving jam is detected ( S → J ). Similarly,the recovery to free flow is achieved by a sequence of two transitions ( J → S and S → F ). he comfortable driving model revisited F → S → J → S → F .As the congested states of figure 2(c) are located close to the exit boundary, thedetector in the middle of the road measures only free flow (see figure 4(c)).More interesting is the time series of figure 4(d), which belongs to the spatiotemporalpattern of figure 2(d): We have already seen (figures 3(c) and 3(d)) that the flow ratesmeasured at the detector are approximately the same for the patterns of figures 2(c) and2(d). The measurements of figure 4(d) are assigned to the synchronized phase S, whereasthe measurements of figure 2(c) belong to free flow. The difference between the twomeasurements becomes evident in the velocity time series. In figure 4(d), all detectedvelocities are between 30 km/h and 70 km/h, and thus they are distinctly lower than theaverage velocities of figure 4(c). The maximum change during subsequent measurementsis slightly below 29 km/h (from t = 43 min to t = 44 min). The absolute value ofthese changes in the average velocity are relatively high compared with real traffic flow.Kerner [27, p. 6], for example, reports fluctuations in the range of ± at averagespeeds of 65 km/h. However, the FOTO-method classified all 1-min-measurements ofvelocity and flow as synchronized traffic (S). This observation is in agreement withKerner [11], who attributes a self-sustaining character to synchronized traffic. The trafficpatterns at the detector’s position cover the entire road, as a comparison with figure2(d) shows. Consequently, we also expect the results of figure 4(d) to be representativefor the entire road—independent of the detector’s position.To get a more quantitative result on the likelihood of the transitions between trafficphases, we have analyzed the time series of all simulations that are not labeled ‘F’ infigure 1(b). The exclusion of the free flow states ‘F’ has two reasons: (i) In the free flowregime, where vehicles can move without hindrance, we do not expect any transitions tooccur. (ii) In very dilute traffic (i.e., a small value of α ), the application of FOTO is likelyto produce erroneous results due to very low vehicle flows. As very few vehicles enterthe road, a detector will detect no vehicle most of the time. However, if it does detecta vehicle, there is a high probability that no vehicle was detected during the previoussampling interval. This in turn, leads to a measurement of high velocity (the singlevehicle travels at maximum velocity) following a measurement with very low velocity(no vehicle), which will be interpreted as a J → F transition by the FOTO-method.In addition to the detector in the middle of the road, which we have used in theanalysis of figure 4, we have added another detector close to the right boundary atposition x = 7 . km (site 4800), where more congestion is expected due to its proximityto the exit section. We analyzed the time series of both detectors and classified theobserved traffic states. The classification was performed both with the standard set ofrules of FOTO, which we have used up to now and which comprises four distinct rules,and with an extended set of rules comprising 13 different rules [17], which offers a betterdistinction between the states J and S.Table 1 shows the resulting probabilities of observing a given transition. As we have he comfortable driving model revisited Table 1.
Probabilities for transitions from one traffic phase to another. We use boththe standard and the extended rule set of FOTO. (As a consequence of rounding, theprobabilities do not necessarily add up to 100%.) probability [%] for the detector at x = 3 . km x = 7 . kmtransition standard extended standard extended J → F J → S S → F S → J F → S F → J J → F ) are veryunlikely ( ≤ . ). Similarly, F → J transitions occur with probability below 2%. Moreimportantly, it has to be noted that transitions from free flow to synchronized traffic( F → S ) are more than two to three times more likely than F → J transitions. At thedetector at position x = 7 . km, they are even more than eight times more likely.This is in good agreement with empirical data. According to Kerner, spontaneous F → J transitions cannot be observed in real traffic, but wide moving jams J alwaysemerge from synchronized flow [4, 5, 28].Concerning the different sets of rules, we can say that they led to slightly differentquantitative results, but they did not change the qualitative character of our results. The findings of the previous section might lead to the conclusion that the CDM can,indeed, reproduce all the traffic phases proposed by Kerner. This finding appears as aclear contradiction to the three-phase traffic theory, as the model under considerationviolates the fundamental hypothesis of this theory. One might object that the usage ofdata aggregated over intervals of 1 min masks to some extent the inter-vehicle dynamics.Therefore, we felt it necessary to inspect the single vehicle data, too.Figure 5 shows a snapshot of the vehicles’ headways (i.e., bumper-to-bumper distance)and velocities at a fixed time t = 30 min.As in figures 2(a) and 4(a), the massive jam surrounded by free flow is clearly visiblein figure 5(a). Within the jam, both the vehicles’ headway (i.e., the bumper-to-bumperdistance) and velocity are equal or close to zero. A similar observation can be made infigure 5(b), where two jam waves can be identified. Although, in the second jam wave, at he comfortable driving model revisited d i s t a n ce [ m ] position [km] 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 v e l o c i t y [ k m / h ] position [km] (a) d i s t a n ce [ m ] position [km] 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 v e l o c i t y [ k m / h ] position [km] (b) d i s t a n ce [ m ] position [km] 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 v e l o c i t y [ k m / h ] position [km] (c) d i s t a n ce [ m ] position [km] 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 v e l o c i t y [ k m / h ] position [km] (d) Figure 5.
Snapshots of the vehicles’ spatial headways (left) and their current velocity(right) at a fixed time t . The figures (a)–(d) correspond to the the simulations depictedin figures 2 and 4. The snapshot was taken at t = 30 min. Note that figure (c) usesa different scale for the distance-axis. (The corresponding values of α and β are: (a) α = 0 . , β = 0 . , (b) α = 0 . , β = 0 . , (c) α = 0 . , β = 0 . , and (d) α = 0 . , β = 0 . .) he comfortable driving model revisited (cid:98) = 5 . / h ) at time t = 30min .)Moreover, the distance between the first and last stopped vehicle in figure 5(d) was132 m. Similar to the total number of stopped vehicles, we consider the spatial extentas small. Therefore, we also determined the time the 11 vehicles had to wait until theycould move again. The average waiting time was 14 s, and the longest waiting time was25 s.Studying the vehicles’ time headways allows for a more objective analysis: Kerner et al [29] reported that one finds regions of interrupted flow within wide moving jams.These flow interruptions are characterized by maximum time headways of t h , max ≥ sbetween two vehicles. So, for each stopped vehicle, we recorded the time headways that alocal detector at the corresponding vehicle’s position would have measured. The averagetime headway of the next ten following vehicles was 4 s. Yet, once, a time headway of33 s was observed. All other time headways were below 8 s. Consequently, based onKerner’s microscopic criterion t h , max ≥ s , we would have to reject the classification ofthe FOTO-method for some time intervals. Moreover, we would have to conclude thatthe pattern of figure 2(d) is not entirely jam-free, even if no wide moving jam could beidentified via FOTO.At this point, it has to be mentioned that the three-phase traffic theory also treatsso-called narrow moving jams [5, p. 259], which, in contrast to wide moving jams,consist merely of an upstream and downstream front. These narrow moving jams caneither grow into wide moving jams or disappear completely, and they are associatedwith the synchronized flow phase. Hence, if the aforementioned sequence of stoppedvehicles represents a narrow moving jam, which its length of 132 m suggests, then theFOTO-method’s classification as synchronized flow is still correct. he comfortable driving model revisited In contrast to the simplistic setup of section 2.1, we finally want to provide somemore realistic results. Therefore, we apply the FOTO-method to both empirical andsimulated traffic data. In a recent article [30], the authors investigated the ability ofthree microscopic traffic models—including the CDM—to reproduce a traffic breakdown,i.e., the abrupt and spontaneous transition from free flow to congested traffic. Thisinvestigation was based on empirical data collected by 10 detectors on the GermanAutobahn A44, as sketched in figure 6(a). In great detail we have simulated the depictedtwo-lane Autobahn-segment with two types of vehicles (i.e., cars and trucks). The flowrates for both cars and trucks follow the rates measured by the detector D1. From thedetectors’ time series of November 4, 2010, shown in figure 6(b), this breakdown caneasily be identified in the morning peak hour between 7 a.m. and 9 a.m.In the following, we will contrast these empirical measurements with data obtainedfrom simulations with the CDM. The computer simulations emulated the highwaysegment of figure 6(a), and the inflow and outflow via the boundaries (including theramps) followed the empirical data. (For a detailed description of the simulation setup,see [30].)To apply the FOTO-method, we have averaged the vehicle flow across lanes, andwe have calculated the average velocity by a weighted average of the velocity of trucksand vehicles over both lanes.The resulting classification of traffic states is given in figures 7(a), 7(b), and 7(c)for the detector cross-sections D4, D5, and D6, respectively (see figure 6(a)). The plotson the left show the empirical data, and the plots on the right show the detectors’ timeseries from simulations with the CDM.The comparison between real and simulated data confirms some earlier findings [30]:the CDM overestimates the temporal extent of congested traffic during the morning peakhour. At detectors D5 and D6, an uninterrupted sequence of congested traffic couldbe found between 7:20 a.m. and 8:55 a.m. for the CDM. The real time series at theselocations show free flow during a 10 min-interval at approximately 8:10 a.m.Table 2, which lists the number of intervals assigned to the traffic phase J, F, and S,confirms this observation as well, but allows for a quantitative characterization. On the he comfortable driving model revisited D10.377 D22.131 D33.663 D45.018 D56.809 D67.875 D79.019 D89.934 D911.334D1011.916 [km] (a) D3 D4 detector data D5 D6 v e l o c i t y [ k m / h ] time (b) Figure 6.
Empirical data used for a comparison with simulated data were takenfrom several detectors on German Autobahn A44. The Autobahn-section considered isdepicted in (a), where the detector cross-sections are labeled D1,. . . ,D10. Time series ofthe average velocity for detectors D3–D6 are shown in (b). A solid black line representsthe time series of cars, whereas the time series of trucks is shown with a solid gray line. one hand, the simulation results overestimate the occurrence of congested traffic, butthey underestimate the number of intervals with wide moving jams (J).
Table 2.
Overview of the traffic phases found in the time from 6:00 a.m. to 10:00 a.m.at the detectors downstream of the off-ramp. → S and the subsequent S → J transition were only one or twominutes apart. Hence, one might assume that the intermediate step S is a mere artifact he comfortable driving model revisited v e l o c i t y [ k m / h ] timeD4 (empirical data) 0 20 40 60 80 100 120 07:00 08:00 09:00 v e l o c i t y [ k m / h ] timeD4 (CDM) (a) v e l o c i t y [ k m / h ] timeD5 (empirical data) 0 20 40 60 80 100 120 07:00 08:00 09:00 v e l o c i t y [ k m / h ] timeD5 (CDM) (b) v e l o c i t y [ k m / h ] timeD6 (empirical data) 0 20 40 60 80 100 120 07:00 08:00 09:00 v e l o c i t y [ k m / h ] timeD6 (CDM) (c) Figure 7.
A comparison of the traffic state classification according to the FOTO-method for both the empirical (left) and the simulated (right) time series.States of free flow are depicted with a white background color and triangles ( (cid:52) ) asdata points. Synchronized traffic is shown with a light gray background and rectangularsymbols ( (cid:3) ). A dark gray background color and circular symbols ( ◦ ) indicate widemoving jams. he comfortable driving model revisited → J follows. On the otherhand, we observe such short-lasting S-states in the time series of the real detectors infigures 7(a)–7(c). This proves that the CDM is also able to exhibit long-lasting intervalsof synchronized flow that precede a wide moving jam.It should be noted that a few F → J transitions could be found in the empirical dataas well. To ensure that this effect did not result from the chosen averaging process, wealso applied the FOTO-method to the raw data. The analysis of the raw data confirmedthe existence of F → J transitions. This observation, which contradicts three-phasetheory, is rather an evidence of an imperfection of the FOTO-method than of the theoryitself. Remember that the traffic phases are spatiotemporal phenomena, whereas theFOTO-method relies on local information only.
4. Discussion
In this article, we studied the spatiotemporal dynamics of the comfortable driving model(CDM). We felt such an analysis was necessary, as many newly presented traffic models(e.g., [25, 31]) claim to reproduce the synchronized phase of Kerner’s three-phase traffictheory simply by presenting space-time plots such as the ones shown in figure 2. Hence,we used the rule-based FOTO-method which, although it certainly has some limitationsin assessing the spatiotemporal dynamics as it is based on local measurements only, stillprovides hard and objective criteria for this purpose.In the article by Kerner et al [24], for instance, in which also the CDM wasinvestigated, the spatiotemporal pattern of figure 2(d) was classified as an ‘oscillatingmoving jam’. It appears, however, that the same pattern was classified as synchronizedflow by Jiang and Wu [25, Figs. 3+4], who analyzed an extension of the CDM [26].(Their extended model incorporates some findings of the three-phase traffic theory.)This observation only confirms the difficulty of classifying traffic phases which we havementioned in the introductory section. Therefore, a ruled-based method such as theFOTO-method is preferable, as it promises an objective classification and facilitates thecomparability of our results with both empirical data (figure 7) and other traffic models.By applying the FOTO-method to the CDM, we found that the obtained resultsare in both good quantitative and qualitative agreement with findings of the three-phasetraffic theory. In particular, we have demonstrated that the CDM exhibits three clearlydistinguishable traffic states or “phases”. Notwithstanding these results, it is still notclear whether the CDM reproduces the synchronized phase in the sense of Kerner’sthree-phase traffic theory. For, according to Kerner, any traffic model with a functionalrelationship between traffic flow and density, such as the CDM, cannot adequatelydescribe synchronized flow [24], where such a relationship does not exist. At this point,it has to be mentioned that traffic states where traffic density and flow are practicallyuncorrelated exist in the CDM as well [18]. This uncorrelated behavior is a consequenceof strong fluctuations of flow, velocity, and density, and does not contradict the existence he comfortable driving model revisited → S transition as found by Kerner [5, ch. 3]),although it correctly reproduces many aspects of the three-phase theory. Therefore, itsapplication to large scale traffic networks is still justified [32].
Acknowledgments
FK thanks A. Schadschneider for fruitful discussions and the German ResearchFoundation (DFG) for funding under grant number SCHR 527/5-1. Both authorsthank the anonymous reviewers for their helpful comments.
Appendix A. The FOTO-method
Here, we briefly review the FOTO-method which allows a classification of locally measuredtraffic data according to the three-phase traffic theory. The classification is based on aset of rules. It uses the aggregated data provided by a local detector (i.e., velocity andflow) and decides to which traffic state the measured combination of observables mostlikely belongs. As the traffic phases S and J both denote phases of congested traffic, thedistinction is not always obvious. Therefore, the set of rules employs a fuzzification ofthe input parameters as shown in figure A1.The fuzzification process transforms the measured value into fuzzy values whichdenote the degree of membership to the given classes. Hence, one value may belong tomore than one class. This is illustrated in figure 1(b), where fuzzificating the velocity of75 km/h shows that this velocity is both a “medium” and a “high” velocity. Yet the degreeof membership to the class “high” is larger (0.75) than to the class “medium” (0.25).Based on this fuzzification of flow and velocity, the classification of traffic states viathe FOTO-method works as follows [4, 17]: F1 If the measured average velocity is classified as “high”, then the associated trafficphase is free flow (F)—independent of the flow rate’s value. he comfortable driving model revisited d e g r ee o f m e m b e r s h i p flow [vehicles/h/lane]low high (a) d e g r ee o f m e m b e r s h i p velocity [km/h]low medium high (b) Figure A1.
Illustration of the fuzzification process. For each value of (a) flow and(b) velocity one can read off the associated degree of membership to the classes “low”,“high”, and, in the case of the velocity, “medium”. The dashed lines in (b) illustrate thatone value (75 km/h) can be member of more than one class. F2 If the measured average velocity is a member of the class “medium”, the associatedtraffic phase corresponds to synchronized traffic (S). F3 If the measured average velocity is “low” but the flow is “high”, the associated trafficphase is synchronized traffic (S). F4 If both the measured average velocity and the flow are “low”, the associated trafficphase should be classified as a wide moving jam (J).Due to non-exclusive memberships, a pair of flow and average velocity may match morethan one of the above criteria. In this case, one has to chose the traffic state with thehighest degree of membership of both velocity and flow.
Appendix B. The comfortable driving model (CDM)
The CDM, originally called brake-light model , is an advancement of the Nagel-Schreckenberg (NSM) cellular automaton with extensions for anticipatory driving.Thereby, the CDM enables a vehicle to react more carefully to the preceding one.(In the NSM any preceding vehicle is ignored, unless a collision is imminent.) The modelincludes anticipatory effects by considering the status of the preceding vehicle’s brakelight b n +1 , which may be on (i.e., b n +1 ( t ) = 1 ) or off (i.e., b n +1 ( t ) = 0 ), by anticipatingits own velocity v anti ( t ) = min( v n +1 ( t ) , d n +1 ( t )) , and by calculating the effective distance d eff n ( t ) as d eff n ( t ) = d n ( t ) + max ( v anti ( t ) − d safe , (B.1)where the parameter d safe governs the effectiveness of the anticipation.Vehicle motion results from the simultaneous application of several rules that areexplained in the following:(i) Acceleration : In the first step, a vehicle tries to accelerate to its maximumvelocity. To avoid unnecessary acceleration, it checks the status of its own and the he comfortable driving model revisited t h ( t ) = d n ( t ) /v n ( t ) to a velocity-dependent interaction horizon t s ( t ) = min ( v n ( t ) , h ) . v n ( t + 1) ← (cid:26) min ( v max n , v n ( t ) + 1) , if b n ( t ) = b n +1 ( t ) = 0 or t h ( t ) ≥ t s ( t ) , v n ( t ) , otherwise. (B.2)(ii) Braking : Here, the vehicle checks whether it actually has to brake and updatesthe status of its brake light. The function Θ( · ) denotes the Heaviside step function. v n ( t + 1) ← min (cid:0) d eff n ( t ) , v n ( t + 1) (cid:1) (B.3) b n ( t + 1) ← − Θ ( v n ( t + 1) − v n ( t )) (B.4)(iii) Determination of randomization parameter p : p ← p b , if b n +1 ( t ) = 1 and t h ( t ) < t s ( t ) , p , if v n = 0 , (B.5) p d , otherwise.(iv) Dawdling : In the CDM, this step influences both the vehicle’s velocity and thestate of its brake light. Let ξ be a (pseudo-)random number, uniformly distributedin [0 , : v n ( t + 1) ← (cid:26) max ( v n ( t + 1) − , , if ξ < p , v n ( t + 1) , otherwise. (B.6) b n ( t + 1) ← (cid:26) , if ξ < p and p = p b , b n ( t + 1) , otherwise. (B.7)(v) Vehicle motion : x n ( t + 1) ← x n ( t ) + v n ( t + 1) (B.8)The variables are set to p d = 0 . , p b = 0 . , p = 0 . , h = 6 , and d safe = 7 .From the definition of the anticipated velocity v anti ( t ) follows that in the CDM avehicle’s motion does not only dependent on the velocity of and the distance to theleading vehicle but also on the distance of the latter to its own predecessor. Thisallows vehicles to accept time headways of below 1 s when driving at high velocities (cf.[18, 33]). References [1] Helbing D 2001
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