Capacity funnel explained using the human-kinetic traffic flow model

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1 Capacity funnel explained using the human-kinetic traffic flow model Chris Tampère 1,3, Serge P. Hoogendoorn 2, and Bart van Arem 1 1 TNO Inro PO box JA Delft The Netherlands b.vanarem@inro.tno.nl 2 Delft University of Technology PO box GA Delft The Netherlands s.hoogendoorn@citg.tudelft.nl 3 Katholieke Universiteit Leuven Kasteelpark Arenberg Heverlee Belgium chris.tampere@bwk.kuleuven.ac.be Summary. We present a macroscopic traffic flow model that explicitly builds on continuous individual driving behaviour. Not only do we start from classical car-following rules (like the kind that are used in microscopic simulators), the model also explicitly accounts for the finite reaction times of drivers, anticipation behaviour, anisotropy in driver responses and the finite space requirement of drivers in the stream. Moreover we allow variations in driver psychology, which lets drivers adopt different driving styles dependent on traffic conditions, like the presence of a merging zone. We illustrate the potential of such model by simulating a busy highway with an on-ramp. Plausible assumptions about driver psychology allow us to reproduce the so-called capacity funnel, i.e. the onset of congestion typically occurs some distance downstream of the merge area. Keywords: traffic flow model, gas-kinetic traffic flow model, driver behaviour, capacity funnel, micro-macro link 1 Introduction: individual driver behaviour in macroscopic traffic flow models In this article we present some recent developments of traffic flow models in which special attention is given to the representation of individual driver behaviour. The motivation for this work was raised by criticism heard in our daily co-operation with human factors experts on the role of individual driver behaviour in traffic flow models (e.g.[1]) and by the desire to explore the impact of (semi-) automated driving on traffic dynamics. The latter would require a macroscopic traffic flow model that does not build on any empirical macroscopic equilibrium or fundamental relationship. In the remainder of this chapter we summarise our findings with respect to the state-of-the-art on this topic. In the next chapters we show how we combine various elements that we found in the literature into a new traffic flow model, based on the gas-kinetic approach but with explicit human

2 behaviour of the vehicles in the flow. For this reason we refer to our model as the human kinetic traffic flow model. A common component of all macroscopic traffic flow models is the equation for the density dynamics. With k for the density, q for the flow and V for the speed, it writes: k( t, x) q( t, x) (1) + = 0 with q ( t, x) = k( t, x) V ( t, x) This equation is a purely physical law of conservation of vehicles. All reference to individual driver behaviour is therefore found in the equations accompanying this density equation. Traditionally a dominant role is played by the so-called equilibrium speed or fundamental relationship V e ( k( t, x) ) between the macroscopic speed and the density. This holds equally for the first order model in which V is set equal to V e at all times and for higher order models in which the speed V is allowed to vary dynamically around V e. The equilibrium speed implicitly contains all information on individual driver behaviour, since it is the result of individual actions and interactions between drivers, which is either obtained empirically or (theoretically) through calculations in the gas-kinetic theory. We found four approaches in literature that aim at refining the representation of individual driver behaviour in macroscopic traffic flow models mostly through the equilibrium speed: Extensions in analogy to hydrodynamics: Some authors propose extra terms in the speed dynamics equation (of second or higher order models) to represent anticipation of drivers or the tendency to go along with the flow (viscosity term); these extensions are seldom built on individual driver behaviour specifications, but are inspired by qualitative behavioural interpretations of terms that are introduced to obtain better dynamic behaviour or numerical performance of the model. Contributions of this kind are numerous, among others: [6], [8], [9], [11], [12], [20]. Direct micro-macro links: A microscopic car-following rule is directly scaled up to the macroscopic level by a Taylor expansion (in analogy to [16], e.g. [21]) or by an appropriate transition from discrete to continuous variables ([7]). Interestingly the latter author adds non-local behaviour to this model, i.e. it is numerically reflected that drivers respond to conditions some distance downstream. Direct refinement of the equilibrium speed: Inspired by behavioural considerations or by macroscopic empirical findings, the equilibrium relationship is refined, for instance with different branches for accelerating or decelerating traffic ([15], [22]) or different branches for distinct classes of drivers (e.g. the rabbits and slugs in [4] and [5]). Interestingly, the equilibrium speed of the rabbits in the latter example has a special reversed-lambda shape that is explained by assuming a collective loss of motivation of drivers to follow their predecessor closely. (Gas-) kinetic approaches: This theory accounts for the fact that the flow consists of individual vehicles, but their behaviour is often modelled only coarsely, e.g. with

3 instant deceleration. This theory provides an analytical foundation of the equilibrium speed as a macroscopic balance between the tendency of the driver population to relax to its desired speed and the gross effect of interactions with other vehicles. The basic assumptions ([17] and [18]) were explicitly refined with respect to driver behaviour modelling in [13], [14], [6] and [10]. Still, interactions between vehicles in these models have a discrete nature and are little refined. In our human-kinetic traffic flow model we use the mathematical framework of the gas-kinetic theory to scale up microscopic car-following relationships (chapter 2), we use non-local and even non-temporal interactions inspired by the idea of Hennecke [7] (chapter 3) and we allow changes in driver psychology as an extension to what is suggested by Daganzo [4] (chapter 4). 2 Human-kinetic traffic flow model with smooth accelerations and decelerations balanced on the microscopic level Let us summarise our application of the gas-kinetic theory as follows (more details can be found in [19]). We interpret V and k as the moments of a probability distribution of individual vehicle-driver combinations in a road section around x. The density k is the inverse of the average of the probability distribution of the rear-bumper to rear-bumper headway s; V is the average of the individual speeds v. The combined (mesoscopic) generalized density function ρ(t,x,v,s) is interpreted as the probability distribution function describing the probability of finding at time t at location x a vehicle with speed v and distance s to its predecessor. The predecessor has an individual speed w drawn from a probability distribution with average V(t, x+s). For each of these predecessor-follower pairs we can apply a microscopic longitudinal or car-following model acc = CF(v,s,w) to calculate the individual acceleration or deceleration of the follower. After averaging this individual acceleration i.e. multiplying it with the probability of occurrence of the vehicle pair and integration of all possible vehicle pairs whose follower is located at x we find the expected acceleration or macroscopic acceleration at x. This procedure is illustrated in figure 1. acc=cf(v,s,w) v w s Fig. 1 calculation of the expected macroscopic acceleration by integration of the car-following relation CF weighed with the probabilities of its inputs v, s and w

4 Note how we balance the individual decisions to either accelerate or decelerate (dependent on the outcome of the CF function) on the microscopic scale so that a positive or negative macro acceleration can occur. The gas-kinetic theory now states that we can transform a generalized law for the conservation of probability: ρ ( t, S ) ds ( ) ( t, S ) dρ( t S ) + S ρ t, S =, dt dt with S = s,..., and S the state vector (e.g. S = (x,v)) by applying the 1 sn so-called method of moments. This method turns eq. 2 into dynamic equations for the density dynamics (eq. 3) and for the speed dynamics (eq. 4, in which we have used the notation dv as the outcome of the integration dt v described above and in [19] to obtain the macro acceleration and θ for the speed variance): k kv dk + = dt kv + ( kv + kθ ) v 2 dv dkv = k + (4) dt dt With eq. 3 and 4 we now have a macroscopic traffic flow model that is directly derived from a microscopic car-following model. Since the carfollowing model typically aims at a velocity dependent following distance and we only look to cars in front our model implicitly obeys the finite space requirement (no point-sized vehicles) and is anisotropic (not sensitive to traffic conditions upstream but only downstream). 3 Refinements to the basic human-kinetic traffic flow model We now introduce finite reaction times and anticipation behaviour as typical behavioural aspects and show how they can be accounted for in the humankinetic model. 3.1 Finite reaction times We know that traffic flows are not always stable in the sense that local small perturbations of the speed or density can grow in amplitude under certain conditions. From asymptotic stability analysis in microscopic models we know that the reaction time of drivers is an important contributor to instability that should therefore also be part of our human-kinetic model. We implement a finite reaction time T explicitly in the numerical scheme a non-temporal term analogously to the non-local term in [7]. For this purpose we calculate the expected acceleration at time t and location x as described in the previous chapter, but only implement it at time t+t, when traffic is at location x(t+t) = x(t)+t V(t). It turns out that this short delay has (2) (3)

5 little effect in smoothly changing traffic conditions; but can cause instabilities in dense and rapidly changing traffic conditions, as is the case in real traffic. 3.2 Anticipation It is known that finite reaction times cannot result in collision free traffic flows unless it is compensated by some sort of anticipation. In the model of Payne ([16]) for instance an anticipation term is present in the speed equation, and stability can be mathematically proven to depend on the ratio of the anticipation coefficient and the reaction time. We model anticipation by taking a subjective or perceived speed w of the predecessor in the integration procedure for the expected acceleration of chapter 2. That is: instead of the actual average speed V(t,x+s) we take the speed of the predecessor as if it came from a distribution with perceived average V perc (t,x+s). Although other definitions are equally possible, we define the perceived speed here as follows. We assume that the driver scans some range (0, x anticip ) for the steepest speed drop that can be perceived in perceived this interval: dv ; instead of the actual average speed V(t,x+s), we dx take V perc perceived (t,x+s) = V(t,x)+s dv. dx Note that this specification models anticipation of drivers with an asymmetric sensitivity for accelerating or decelerating spatial speed profiles. It models anticipation, since the driver already accounts for the fact that soon his predecessors will have to adapt their speeds to the decreasing speeds further ahead. The model is asymmetrically sensitive for deceleration or acceleration because only the steepest (lowest) speed gradient is taken into account. If speeds are lower at the maximum anticipation range than immediately in front then the driver will anticipate to this trend. If speeds far ahead are higher but the immediate predecessors have not yet accelerated, then neither will do the driver since the minimum speed gradient is then the one over the nearby range x << x anticip. Indeed, it would not be realistic in this case to assume long-range anticipation, because nearby traffic restricts the acceleration possibility of the driver. 4 Modelling (temporal) driving style variations Just like other macro or micro models we have assumed so far that driver behaviour is rigid, i.e. drivers respond in an identical manner to the same actual traffic conditions, no matter what has happened before. In reality the driving style of an individual might change over time, for instance due to increased motivation, awareness or attention level. This might be modelled through an adapted reaction time, shorter headways, longer anticipation range etc. We now illustrate how attention level dynamics can be modelled in the human-kinetic traffic flow model and apply this to the case of a freeway onramp in the next chapter. As an extension to chapter 2 and 3, we characterise an individual driver s state not only by the individual speed v (macro equivalent V) and the distance s tot the predecessor (macro eq. k), but also by the attention level a of the

6 driver (macro eq. A). Before we can simulate we have to mathematically describe: how the macro property A flows with traffic; (see next sections) how the attention level of an individual driver affects the driving behaviour as specified so far; (see next sections) how the attention level varies as a result of autonomous processes of the driver and of (changes in) traffic conditions (see next chapter). We mentioned in chapter 2 that the speed dynamic equation 4 is obtained directly from the basic eq. 2 by applying the method of moments. We do the same here but with respect to the attention level instead of the speed. For that purpose we multiply both sides of eq. 2 by the attention level a and integrate over a and v. One can verify that this operation yields eq. 5 as a complement to eq. 3 and 4: A A + V = da dt v, a 1 + k a v dρ a dt s A dk dv da k dt In this equation, the first term on the right hand side stands for the gross effect of autonomous (driver induced) changes of A, the second term is the gross effect due to s in the flow and the last term redistributes the total attention level A over the new population k in case the density does not remain constant (nett in- or outflow dk of traffic). dt Next we specify the effect of the attention level on the individual driver behaviour. Among many possible specifications we choose for illustrative purposes the simple assumption that the space required by a driver for carfollowing at a certain speed decreases with increasing attention level. Macroscopically speaking this means that an increase in the density cannot only be accommodated by a decrease of the average speed (as always in traffic flow models) but possibly also by an increased attention level, so that less space for the same driving speed suffices and speeds can (temporarily) remain unaffected. 5 The attention level based human-kinetic model explaining the capacity funnel In this chapter we illustrate the potential of the newly developed traffic flow model by means of a case study of a bottleneck due to merging traffic. It is well-known that near merges the so-called capacity funnel phenomenon can occur. This means that the on-set of congestion is located some distance after the actual merging zone. This phenomenon was first described in [2]. In [3] a well-documented example is presented of a capacity funnel in Toronto where congestion sets in 1000 m or more after the end of the slip lane. 5.1 Specification of the attention level dynamics near a merge zone A final step in the specification is to describe how the attention level changes in time. We assume (a) that the of other vehicles merging in front of a driver is primarily handled by raising the attention level to a main (and not by immediate braking due to the suddenly reduced headway), (b) that merging drivers from an on-ramp enter with the same high attention level a merge, which (5)

7 enables them to merge into relatively small gaps, and (c) in absence of any of the previous causes for a raised attention level the drivers tend towards a comfortable low attention level a min. Without going into the details of the specification and the mathematical derivation we here immediately present the rewritten equation 5 which together with eq. 3 and 4 forms our human-kinetic traffic flow model: A A amin A dk amerge A dk amain A + V = + + τ merge merge a dt k dt k ( c) relaxation ( b) nett effect of inflow ( a) interaction 5.2 Effect of attention level variations on traffic flow near a merge: the capacity funnel Let us now investigate the effect of attention level dynamics on traffic flows near a merge zone. As a case study we look at a circular road of length 3000 m. We start with homogeneously distributed speeds and density. Starting from x 0 =1375 m a 250m long merging zone is located. The inflow increases linearly from 0 at t=0 to 100 veh/h at t=100s and remains constant thereafter. Moreover we assume that the speed of merging traffic is always equal to the current average speed on the main lanes adjacent to the middle of the merging zone. In figure 2 we show the solution of three experiments for t [0,500]. (a) (5 ) (b) (c) Fig. 2: Effect of the attention level dynamics near a merge (at x=1500m). From left to right: density, speed and the attention level (if applicable). In (a)

8 no attention level variation is modelled and congestion sets in at the location of the merge. In (b) and (c) the merging traffic causes an increase of the attention level that relaxes with time constant τ a =12 in (b) and τ a =18 in (c). Congestion sets in further after the actual merge as τ a increases. 6 Conclusions Challenged by human factors experts and inspired by 50 years of macroscopic traffic flow modeling and the creative solutions proposed by various authors to represent valid driver behaviour in these models, we developed the human-kinetic traffic flow model. The model borrows the mathematical framework from the gas-kinetic traffic flow modeling to build macroscopic equations based on a microscopic car-following rule. The model is refined with finite reaction times and a specification of anticipation behaviour and accounts for (temporal) variations of the driving style by taking the attention level as a dynamically varying index for the driving style. The result is a flexible traffic flow simulation the parameters of which all have a specific microscopic meaning directly related to individual driver behaviour. The potential of such a model is illustrated by simulating delayed onset of congestion near a merge (so-called capacity funnel) by purely making assumptions on the microscopic level. Acknowledgement The authors wish to thank the TNO-TRAIL programming committee T3 for its financial support of this research. References 1. Boer, E.R. (1999), Car following from the driver s perspective, Tr. Res. F Vol. 2, No. 4, pp Buckley, D.J. & S. Yagar (1974), Capacity funnels near on-ramps, Proceedings of the 6th International Symposium on Transportation and Traffic Theory, Sydney, Australia. 3. Cassidy, M.J. & R.L. Bertini, (1999), Observations at a Freeway Bottleneck, Proc. of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem 1999, pp Daganzo, C.F. (2002a), A behavioral theory of multi-lane traffic flow. Part I: Long homogeneous freeway sections, Transportation Research B vol 36B, issue 2, pp (available at 5. Daganzo, C.F. (2002b), A behavioral theory of multi-lane traffic flow. Part II: Merges and the onset of congestion, Transportation Research B vol 36B, issue 2, pp (available at 6. Helbing, D., (1997), Verkehrsdynamik, Neue physikalische Modellierungskonzepte, Springer Verlag, Berlin. 7. Hennecke, A., M. Treiber & D. Helbing, (2000), Macroscopic Simulation of Open Systems and Micro-Macro Link, Traffic and Granular Flow '99, edited by: Helbing, D., H.J. Herrmann, M. Schreckenberg & D.E. Wolf (available at: 8. Hoogendoorn, S.P., (1999), Multiclass Continuum Modelling of Multilane Traffic Flow, Dissertatoin thesis, Delft University of Technology, Faculty of Civil Engineering and Geosciences. 9. Kerner, B.S. & P. Konhäuser, (1993), Cluster effect in initially homogeneous traffic flow, Phys. Rev. E 48(4), pp

9 10. Klar, A. & R. Wegener (1999), A hierarchy of models for multilane vehicular traffic I: modeling, SIAM Journal of Applied Mathematics, Vol. 59, No. 3, pp (available through: Kühne, R.D., (1984), Macroscopic freeway model for dense traffic stop-start waves and incident detection, Proc. of the 9th International Symposium of Transportation and Traffic Theory, pp.21-42, ed. by: I. Volmuller & R. Hamerslag. 12. Liu, G., A.S. Lyrintzis & P.G. Michalopoulos (1998), Improved higher-order model for freeway traffic flow. Tr. Res. Rec. 1644, pp Nelson, P. (1995), A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, Transport Theory and Statistical Physics, Vol. 24(1-3), pp Nelson, P., D.D. Bui & A. Sopasakis (1997), A novel traffic stream model deriving from a bimodal kinetic equilibrium, Proceedings of the 1997 IFAC meeting, Chania, Greece, pp Newell, G.F. (1965), Instability in dense highway traffic, a review, In: J. Almond, ed., Proceedings of the second International Symposium on the Theory of Traffic Flow, pp Payne, H.J., (1971), Models of freeway traffic and control, In: Mathematical models of public systems, volume 1 of simulation councils proc. ser., ed. by: Bekey, G.A., pp, Prigogine, I. & R. Herman (1971), Kinetic Theory of Vehicular Traffic, American Elsevier, New York. 18. Prigogine, I. (1961), Boltzmann-like approach to the statistical theory of traffic flow, In: Theory of Traffic Flow (Ed.: R. Hermand). 19. Tampère, C.M.J., S.P. Hoogendoorn & B. Van Arem (2003), Gas kinetic traffic flow modelling including continuous driver behavior model, Preprints of the 82nd Annual Meeting of the Transportation Research Board. 20. Zhang, H.M. (1998), A theory of nonequilibrium traffic flow, Tr. Res. B, Vol. 32B, No. 7, pp Zhang, H.M. (2003), Driver memory, traffic viscosity and a general viscous traffic flow model, Tr. Res. B, Vol. 37, No. 1, pp Zhang, H.M., (1999), A mathematical theory of traffic hysteresis, Tr. Res. B 33. Pp.1-23

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