A Within-day Dynamic Traffic Assignment Logit

A Within-day Dynamic Traffic Assignment Logit Model to Multimodal Urban Networks Guido Gentile, Lorenzo Meschini, Natale Papola Dipartimento di Idraulica, Trasporti e Strade, Università di Roma “La Sapienza”

1.

ABSTRACT

In this paper, with reference to congested multimodal urban networks, within-day dynamic traffic assignment is formalized as a fixed-point problem in terms of arc flow and transit frequency temporal profiles. The main innovation is to represent transit supply using a frequency approach, instead of a run approach, thus not requiring a diachronic graph. By so doing, intra and inter modal congestion effects can be easily reproduced. For this purpose, a new dynamic transit line performance model is introduced.

2.

INTRODUCTION

The quantitative analysis of urban network traffic is usually performed through multimodal static assignment models, yielding the transport demand-supply equilibrium under the assumption of within-day stationariety. This implies that the relevant variables of the system are assumed to be constant within the reference period. The static assignment models satisfactorily reproduce the congestion effects on the traffic flow pattern; on the other hand they are not able to reproduce some important dynamic phenomena, such as the formation and dispersion of vehicle queues along road arcs, and of user queues at transit stops, due respectively to the temporary over-saturation of road intersections and of transit stops. The within-day Dynamic Traffic Assignment (DTA) models are conceived to overcome this limit. With reference to road networks, several theoretical approaches and analytical formulation have been proposed in the literature (e.g., refer to Cascetta, 2001, chapter 6). With reference to transit networks, so far the within-day DTA has been formalized and solved by modeling supply through a diachronic graph (Nuzzolo e Russo, 1997), both for systems with high regularity and low frequency, and for systems with low regularity and high frequency. The introduction of a diachronic graph in the assignment model requires to take into account each run that constitutes the transit service and the corresponding arrival times at stops. Referring to high frequency systems, this implies some significant drawbacks: on the supply side, diachronic graphs are not able to represent congestion effects on travel times (unless the graph structure itself depend on the flow pattern); on the demand side, since in urban transit networks with high frequency users perceive lines as unitary supply facilities, the single runs could be not explicitly introduced, as usually done in the static case (De Cea and Fernandez, 1993; Wu et al., 1994; Nguyen et al., 1998); on the algorithm side, the complexity of the assignment problem increases more than linearly with transit line frequencies, because this implies the growth of graph’s dimension. In this work we achieve avoiding to introduce in the assignment model a diachronic graph, by adopting a line frequency approach within the representation of the transit mode. Our approach is to regard within-day DTA as a demand-supply equilibrium problem, as in Cascetta (2001). Specifically, we extend to the multimodal case the road network assignment model proposed in Bellei, Gentile, Papola (2001), where it is shown how to formulate the dynamic user equilibrium within an implicit path enumeration approach (Figure 1), by combining the arc performance functions with the Network Loading Map (NLM) into a fixed-point problem. To this 1

end, the flows of transit line vehicles must be considered together with the flows of private cars within the NLM, as proposed for the static case in Bellei, Gentile, Papola (2000). o-d demand flows

network loading map

mode choice model

calculation of arc conditional probabilities

node satisfactions

o-d modal flows

arc conditional probabilities

calculation of node satisfactions

network flow propagation model arc flows

arc performances arc performance model

arc performance functions

Figure 1 – Implicit path DTA model

On the supply side, the flows of private cars and transit vehicles are represented through a partially compressible mono-dimensional fluid model, which ensures that the FIFO rule is satisfied. It is worth noting that, in this framework, line frequencies are represented as continuous flows of transit vehicles. As a result, the detailed description of waiting times at every transit stop, available when the single runs of each line are represented explicitly, is here replaced by a temporal profile of the average line access time. The arc performance function takes into account the effects of both intra modal (vehicle queues at road intersections, user queues at transit stops, effect of the boarding and alighting user flows on the time spent by transit vehicles at stops, effect of vehicle occupancy on boarding and alighting rate) and inter modal (mutual conditioning among private and transit vehicles) congestion phenomena. The demand model, based on random utility theory, is a Nested Logit with two levels: mode choice and route choice. Users’ decisions are assumed to be completely preventive, leaving the extension to hyper paths to further investigations. The fixed-point problem can be efficiently solved through the Bather’s method (Bottom, Chabini, 2001). The complexity of the resulting dynamic equilibrium algorithm is equal to the one resulting in the static case multiplied by the number of time intervals in which the period of analysis is divided.

3.

TRANSIT SUPPLY MODEL

The performance functions presented in this section are aimed at reproducing the effects of transit service intrinsic discontinuity within a supply model where the lines are represented as a continuous flow of vehicles. In the following, the temporal profile of vehicles’ departure frequency from each line terminal is assumed to be known.

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The characteristic element of the transit supply model is the stop model, depicted in Figure 2. line node LN(a) stop arc SA(a) line arc a∈AL(l)

previous arc A(a)∈AL(l)

boarding arc BA(a) waiting arc WA(a) alighting arc AA(a) queue arc QA(a) stop node stop access arc walking arc

walking node

Figure 2 – Transit stop model

The stop access arc, having zero travel time and cost, allows to distinguish different stop nodes corresponding to the same walking node. The boarding arc and the stop arc, both having zero travel time and cost, permit to express, respectively, the time spent at the stop by the transit vehicle and the available line capacity as a function of the arc entering flows v(τ), thus avoiding the dependency from the arc exiting flows e(τ), as it will be clearer further on. The walking arc and the alighting arc are assumed to be uncongested. In order to take into account the terminal time without affecting the travel times of users boarding or alighting at the corresponding stop, a terminal arc may be introduced, as depicted in Figure 3. line arc

line node line arc stop arc

terminal arc

boarding arc waiting arc alighting arc

queue arc stop node stop access arc

walking arc

walking node

walking arc

Figure 3 – Transit terminal model

The travel time sa(τ΄) of the generic line arc a∈AL(l), where AL(l) is the set of line arcs of line l, for users entering it at time τ΄ is assumed to be the sum of three terms: the vehicle dwelling time at the stop DTa; the uncongensted running time URTa; the extra time due to road congestion CDa, calculated at time τ΄+DTa(τ΄) when the vehicle leaves the stop. Thus we have: sa(τ΄) = DTa(τ΄) +URTa +CDa(τ΄ +DTa(τ΄))

(1)

The waiting arc represents the average time spent to access the line. Its travel time sWA(a)(τ) is assumed to be proportional to the inverse of the line frequency temporal profile φa(τ) calculated at boarding time; thus we have:

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tWA(a)-1(τ΄) = τ΄ -αl /φa(τ΄) ,

(2)

where tWA(a)-1(τ) is the temporal profile of the entering times (obtained as the inverse of the exit time temporal profile), and αl takes account of vehicle irregularity. The queue arc represents the time spent due to the line over saturation, that is the wait time that the next vehicle approaching the stop becomes available. In fact, if the flow of users overwhelms the line capacity at the stop, a residual queue arises, which is to be summed up to the queue formed by the users that will board the next arriving vehicle. The queue arc is modelled as a bottleneck with a time-variable capacity. Its temporal profile coincides with the profile of the available capacity perceived at the head of arc QA(a), denoted AKQA(a)(τ), which is related to the available capacity profile AKa(τ) at the node LN(a). Referring to the generic instant τ΄, AKa(τ) is equal to: AKa(τ΄) = VKl ⋅φa(τ΄) -vSA(a)(τ΄)

(3)

In fact, as the stop arc travel time is zero, we have: eSA(a)(τ) = vSA(a)(τ). For the vehicle conservation law, the profile AKQA(a)(τ) is obtained by propagating profile AKa(τ) backwards from the node LN(a) to the head of arc QA(a). Referring to the generic instant τ΄, and recalling that the travel time of the boarding arc is zero, we have: AKQA(a)(tWA(a)-1(τ΄)) = AKa(τ΄) /∂(tWA(a)-1(τ΄))/∂τ

(4)

Let’s analyse now the exit time of the queue arc tQA(a)(τ΄). Two cases are possible: a) eQA(a)(τ΄) < AKQA(a)(τ΄), the flow of users doesn’t activate the capacity constraint. In this case we have: tQA(a)(τ΄) = τ΄ , eQA(a)(τ΄) = vQA(a)(τ΄)

(5a)

b) eQA(a)(τ΄) = AKQA(a)(τ΄), the flow of users activates the capacity constraint. In this case, denoted τ ∗ ≤ τ΄ the instant when the constraint becomes active, based on the vehicle conservation law we have: τ'

∫ vQA(a ) (τ )dτ =

τ*

tQA ( a ) (τ ' )

∫τ AK

QA( a )

(τ )dτ

,

(5b)

*

where clearly tQA(a)(τ ∗) = τ ∗. Equation (5b) expresses implicitly the exit time tQA(a)(τ΄) as an extreme of integration, and can be solved numerically. The temporal profiles of the arrival times from the terminal to each stop of line l, denoted Ta(τ), are determined recursively through the following relations: TFA(l)(τ΄) = t FA(l)(τ΄) , Ta(τ΄) = ta(TPA(a)(τ΄)) ∀a∈[AL(l) -{FA(l)}] ,

(6)

where FA(l) is the first arc of line l and PA(a) is the arc preceding a. Contrary to the static case, the line frequency temporal profiles are generally not constant along the line. On one side, in fact, there is a translation of the terminal frequency in space and time; on the other side, the variation in time of the line travel times makes vehicles spread and jam (see for example Figure 4). Recalling that frequency is a flow of transit vehicles, for the vehicle conservation law the frequency at the stop of arc a∈AL(l) at instant τ΄ is given by any one of the following two equations:

φa(τ΄) = φFA(l)(Ta-1(τ΄)) ⋅(∂Ta-1(τ΄) /∂τ) ,

(7a)

φa(Τa(τ΄)) = φFA(l)(τ΄) /(∂Ta(τ΄)/∂τ)

(7b)

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TPA(a)(τ΄) TPA(a)(τ΄΄)

a∈AL(l)

φa(τ)

space

PA(a)

FA(l) Terminal

φ FA(l)(τ)

time

τ΄

τ΄΄

Figure 4 – Frequency temporal profiles along the transit line

4.

REFERENCES

Cascetta E. (2001). Transportation systems engineering: theory and methods, Kluwer Academic Publisher Bellei G., Gentile G., Papola N. (2000). Transit assignment with variable frequencies and congestion effects. Proceedings of the 8th Meeting of the Euro Working Group Transportation EWGT, ed.s M. Bielli, P. Carotenuto, Roma, Italia Bellei G., Gentile G., Papola N. (2001). Un modello logit di assegnazione dinamica intraperiodale alle reti stradali urbane. In Metodi e tecnologie dell’ingegneria dei trasporti – Seminario 2000, a cura di G. E. Cantarella e F. Russo, Franco Angeli, Milano Bellei G., Gentile G., Papola N. (2002). A within-day dynamic traffic assignment Logit model for urban road networks. Internal Report Bottom J., Chabini I. (2001). Accelerated Averaging Methods for Fixed Point Problems in Transportation Analysis and Planning. Preprints of Tristan IV, pp. 69-79, Azores, Portugal De Cea J. , Fernandez E. (1993). Transit Assignment for Congested Public Transport Systems: An Equilibrium Model. Transpn. Sci. 27, pp. 133-147 Nguyen S. , Pallottino S. , Gendreau M. (1998). Implicit Enumeration of Hyperpaths in a Logit Model for Transit Networks. Transpn. Sci. 32, pp. 54-64 Nuzzolo A., Russo F. (1997). Modelli per l’analisi e la simulazione dei sistemi di trasporto collettivo, Franco Angeli, Milano Wu J. H. , Florian M. , Marcotte P. (1994). Transit Equilibrium Assignment: a Model and Solution Algorithms. Transpn. Sci. 28, pp. 193-203

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