A New Framework for Dynamic Traffic Assignment

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(1)Title. A New Framework for Dynamic Traffic Assignment. Author(s). YING, Jiang Qian. Citation. [岐阜大学地域科学部研究報告] vol.[46] p.[55]-[62]. Issue Date. 2020. Rights. Version. Department of Regional Policy, Faculty of Regional Studies, Gifu University. URL. http://hdl.handle.net/20.500.12099/79155. ※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。.

(2) 55 岐阜大学地域科学部研究報告第 46 号：55 – 62. (2020). A New Framework for Dynamic Traffic Assignment Jiang Qian YING Department of Regional Policy, Faculty of Regional Studies, Gifu University ying@gifu-u.ac.jp (Received January 9, 2020). 1. Introduction Traffic flow estimation is an indispensable step for the evaluation and optimization of various policies in transportation planning and traffic management, and related urban and regional policies. In this paper we consider the problem of urban travels and assume that travelers finish their trips within a day. The core task of traffic estimation is to calculate the flows on the links of the transportation network with given travel demands with various originations and destinations across an urban area. This task is called traffic assignment [1, 2]. There are basically two kinds of models for traffic assignment: static and dynamic [1,2,3,4]. In a static model, travel demands are assumed to last stationarily during a certain time period. Static traffic assignment can be applied with high accuracy if such a time period is long enough, such that the yielding traffic flows on the transportation network remain approximately stationary. In practical static models, the time period can be several hours, even the whole day in some applications. Of course, in reality travel demands may vary largely within a day. In a dynamic traffic assignment model, travel demands are faithfully assumed to be time varying. In theory, travel demands can have a continuously time varying profile. In practical application, of course, discretization which divide time into sufficiently small time periods is necessary. In any case, dynamic traffic model is distinguished from a static model with the feature that the time varying microscopic movement and interactions of travelers or vehicles on the network has to be addressed. In a static model, such movement and interaction are expressed by a macroscopic flow-travel time function, possibly with link capacity constraints. In this paper a new framework is proposed for dynamic traffic assignment which inherits the desirable features of conventional static models and has high flexibility to cope with the nonstationary characteristics of traffic flows..

(3) 56 應江黔. 2. The model 2.1 Basic Model with One Arrival Time. The road network consists of a set of links A Let R. N. and D. a, b,. and a set of nodes N. i, j ,. .. N denote the set of origins and destinations, respectively. Travel. demands are represented by platoons. qrs. which are to arrive at the destinations at a. r R, s D. same average time T. Suppose that travel time is the only cost that matters to the travelers. The following is a list of notations and equations describing the road network traffic system. r: origin; s: destination a: a link, sometimes expressed as the tail-head pair a=ij. cas : cost incurred by the platoons of travelers on link a going to s Ckas : cost that a traveler incurs on a route k from (the head of ) link a to s Ckas = as k ,b :. as k ,b. s as b b k ,b. c. as k ,b. 1 if link b is on route k;. 0 otherwise. Pk js : probability that a traveler departing from node j to s uses route k Pkas. Pk js probability that a traveler departing from link a to s uses route k (a=ij) (this notation is redundant, but convenient). exp( Ckas ) , exp( Ckas ) k. as k. P. the summation in the denominator ranges over all possible routes : a dispersion parameter in the logit type route choice model adopted in this paper. Pbas : probability that a traveler departing from (the head of) link a to s passes link b as b. P. k. as as k k ,b. P. k. exp( k. exp(. Ckas ). as k ,b. Ckas ). (1).

(4) 57. A New Framework for Dynamic Traffic Assignment. xas : the flow on link a going to destination s; also referred to as a platoon. xas. r. qrs Pars. (2). as. C : the average time cost that travelers incur en route from link a to destination s C. as. Pkas Ckas. k. C. as. k. Pkas. s as b b k ,b. c. b. Pbas cbs. (3). depends on the destination, not directly on the origins.. as. T as T C : the time that the platoon xas leaves link a xas , T as. s. : the traffic profile for a link a, composing of the destination specific flows and the time that they are on the link. The travel time needed for the travelers going to s to pass through a link a can be written as. cas. s a. xas , T as. (4). s. cas. At an equilibrium state, the destination specific link costs. as. satisfy the following system of. equations s a. If. cas. as. cas. s a. xas , T as. are determined, then. Ckas. s D. as ,k. ,. 0, a. A,s. Pbas. ,. as ,b. As will be seen later, the calculation of the path costs. xas. Ckas. D.. as. (5). are all determined subsequently.. as ,k. are not necessary in solving the. model. 2.2 General Model In the above we assumed that all travel demands are to arrive at their destinations at a same m. time T. Here we extend the model to the general case that there is plural arrival time. Let qrs be the travel demand to arrive at s at time Tm , m=1,…,M, T1. TM . Let T. TM . Let.

(5) 58 應江黔. s1 , , sM. 1. be M-1 imaginary nodes connected by M-1 imaginary links as1 ,. , asM. 1. to the real. node s, respectively. Let T Tm be the constant travel time cost of link asm , m=1,…,M-1. Let. D be the union set of D and the imaginal new nodes corresponding to distinct arrival time. m M Rewrite qrs as qrsm for m=1,…,M-1 and qrs as qrs . We have C. asm. C. as. T Tm .. By working on the extended network with single arrival time T , we can obtain the solution for the original problem with plural arrival time of OD demands. The new system of equilibrium equations becomes s a. cas. s a. xas , T as. s D. 0, a. A,s. D.. (6). Note that the link time costs of the imaginary links are constants in the equilibrium equations.. 3. Link Cost Function In this section we examine some basic properties of the link travel time cost functions. The link. xas , T as. travel time cost is determined by the profile s. The general function form ca. s a. xas , T as. s. s. of flows on the link (see Figure 1).. may be constructed so as to describe the precise. physical interaction between platoons on the links. However, a precise description of the interactions may be too much complicated to be useful in practical application. Here we propose a simplified formulation that may capture the main features of such interactions that are essential in calculating average travel time cost.. ( xas1 , T as1 ). ( xas2 , T as2 ). ( xas , T as ). ( xas3 , T as3 ). ( xas4 , T as4 ). Figure 1. The profile of traffic flows on a link.. The link travel time cas of the platoon with destination s should increase with the volume xas of the platoon itself. The problem is how to calculate the influences of other platoons on cas . We.

(6) 59. A New Framework for Dynamic Traffic Assignment. approximately take account of these influences by aggregating them into a single quantity, with weights calculated from the time T as . That is. X as. s D. X as. xas ws ,. cas. a. ws. ws T as T as .. and. (7). ,. (8). where. a. ws. X as. (9). is a smooth increasing function in X as ; the shape of the weighting function. ws T as T as. is illustrated in Figure 2. When T as is much larger or much less than. T as , meaning that the platoon xas is on the link a at a much later or much earlier time than xas , the influence of this platoon on cas is negligible. Suppose that Ta 0 is the time difference such that xas has the largest weight when T as. T as = Ta 0 . Plausibly, Ta 0 is 0 or a small positive. number, that is, xas has the strongest influence on cas if it reaches the link a slightly earlier than xas . The influence of platoons arriving on link a later than xas fade rapidly away when the time difference becomes large.. ws. Ta 0. T as T as. Figure 2. The weighting function that depends on the difference of time of platoons on the link..

(7) 60 應江黔. 4. Solution method. The equilibrium equations for the independent state variables s a. cas. s a. xas , T as. s D. 0, a. A,s. D. cas. as. (5’). can be solved by Newton method, with the Jacobian matrix of the equations being computed in a link based manner.. s a s b. c. s,s a ,b. 1 if s. xas cbs. xas cbs. x. s. s a s a. s,s a ,b. s,s a ,b. s a s a. xas cbs. x. T. T. s a as. s a as. T as cbs. T as cbs. (10). s and a b ; 0 otherwise.. Pars cbs. q r rs. T as cbs Pbas. C as cbs. r. as b. P. a. cas Paas Pbas. a. Pars Pbrs. qrs. s a. c. Paas,b. Pars,b. (11). Paas cbs (12). rs. The terms Paas and Pa ,b can be efficiently computed by sensitivity analysis algorithms [5,6].. Therefore, once. s a s a. x. and. T. s a as. are computed, existing computational methods that have. been developed for analyzing static traffic models can be applied for solving the dynamic model formulated in this paper. 5. Discussion and Conclusion In this paper a new model for dynamic road traffic assignment is formulated, and a gradient.

(8) 61. A New Framework for Dynamic Traffic Assignment. based method is provided for solving the associated equilibrium equations. In the model, the time varying link travel time cost is formulated as a function of the profiles of platoons on the link. This formulation embodies a new framework for dynamic traffic assignment in the sense that it inherits the desirable features of conventional static models and has high flexibility to cope with the nonstationary characteristics of traffic flows. A plausible instance of macroscopic link cost function is presented in the paper for illustrating the possible simplification of models for dealing with time varying interactions between traffic flows. Of course, extensive empirical studies are necessary for identifying satisfactory link cost functions of traffic profiles in practice. In our model, “average time” and “average arrival time” are involved without much explanation about their physical meaning. A thorough examination of the relationship of physical traffic movements and these concepts is an interesting topic for future research. Here we suffice to comment that any practical models, so long as they only deal with average aggregated traffic behaviors, but not the probabilistic distributions, essentially involve only some kind of average time. Note that in the literature there is a kind of eclectic semi-dynamic models for traffic assignment which divide a day into multiple time periods within each period travel demands are approximately stationary [7,8,9,10]. In these models, the traffic flows during a given period are assumed stationary such that the static type models can be applied to compute the flows in that period, although various methods have been proposed to deal with the demands that cannot pass through the network to their destination in a single period. In the model proposed in this paper, stationarity requirement is week because the macroscopic link cost functions can be elaborated to cope with the nonstationarities. Note that in [10], sensitivity analysis method is also used to calculate the remaining flows in a model with the period-wise stationary characteristics, which is different from the model formulated here. Acknowledgments This research is funded by the Grants-in-Aid for Scientific Research [16K01238] and [17H03323] from the Japan Society for the Promotion of Science. References [1] Sheffi, Y. (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods, Prentice-Hall, Englewood Cliffs, N.J. [2] 土木計画学研究委員会「交通ネットワーク」出版小委員会 (1998) 交通ネットワー.

(9) 62 應江黔. クの均衡分析-最新の理論と解法(Committee of Infrastructure Planning and Management: Equilibrium Analysis of Traffic Network – State-of-the-Art Theories and Solution Methods) [3] 赤松隆, 和田健太郎 (2014) 動的な交通ネットワーク流問題, Proceedings of the Twenty-Sixth RAMP Symposium Hosei University, Tokyo, October, 2014. (Akamatsu T. Wada, K. Dynamic Network Traffic Flow Problems) [4 ] Wang W. Y., Szeto B. C., Han K., Friesz Terry L. (2018) Dynamic traffic assignment: A review of the methodological advances for environmentally sustainable road transportation applications, Transportation Research Part B 111, pp. 370–394 [5] Ying, J. Q. and Miyagi, T. (2001) Sensitivity Analysis for Stochastic User Equilibrium Network Flows-- A Dual Approach, Transportation Science 35(2) pp.124-133. [6] Ying, J. Q. (2015) Optimization for multiclass residential location models with congestible transportation networks. Transportation Science, 49(3) pp. 452-471. [7] 藤田素弘, 松井寛溝上章志 (1988) 時間帯別交通量配分モデルの開発と実用化に関する研究土木学会論文集 No.389/IV-8，pp.111-119，1988． (Fujita, M, Matsui, H., Mizokami, S: Modeling of the Time-of-Day Traffic Assignment over a Traffic Network，Journal of Japan Society of Civil Engineers) [8] 宮城俊彦，牧村和彦：時間帯別交通配分手法に関する研究，交通工学 Vol. 26，No. 2， pp. 17-28，1991． (Miyagi, T., K., Makimura: Study on Time-of-Day Traffic Assignment Methods, Japan Society of Traffic Engineers) [9] 菊地志郎赤松隆 (2007) リンクの流入・流出交通量を内生化した時間帯別交通均衡配分に関する基礎的研究土木計画学研究・論文集 24, pp. 577-585. (Kikuchi, S., Akamatsu T.: A Semi-Dynamic Traffic Equilibrium Assignment Model with Link Arrival and Departure Rates, Infrastructure Planning Review) [10] 板垣雄哉, 中山晶一朗, 高山純一 (2014)感度分析を用いた交通混雑内生型時間帯別配分モデル, 土木学会論文集 D3 Vol.70 (5) pp. 569-577 (Itagaki, Y., Nakayama I., J-I., Takayama: A Semi‐dynamic traffic assignment model with endogenous traffic congestion using the sensitivity analysis, Journal of Japan Society of Civil Engineers, Ser. D).

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