a dynamic model for traffic network flow

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Physica A 387 (2008) 2603–2610www.elsevier.com/locate/physa

A dynamic model for traffic network flow

T.Q. Tanga,b,∗, H.J. Huangb, C.Q. Meib,c, S.G. Zhaod

a Department of Automobile Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, Chinab School of Management, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

c School of Statistics, Capital University of Economics and Business, Beijing 100070, Chinad Solid Mechanics Center of Beijing, University of Aeronautics and Astronautics, Beijing 10083, China

Received 4 November 2007; received in revised form 10 December 2007Available online 9 January 2008

Abstract

Concerning the link properties in traffic networks, we introduce a dynamic equation of road flow into each link, and therebypropose a dynamic model for network flow. Using this model, we investigate the evolutions of inflow, outflow and flow on each linkcaused by a small perturbation of the network inflow under different route choice rules. Numerical results show that the dynamicmodel can reasonably capture the basic characteristics of network flow.c© 2008 Elsevier B.V. All rights reserved.

Keywords: Dynamic model; Network traffic flow; Inflow; Outflow

1. Introduction

Since the 1950s, many models have been developed to study complex traffic phenomena. Chowdhury [1] in detailintroduced the existing traffic flow models (including kinematics models, dynamics models, gas-based kinematicsmodels, car-following models and cellular automaton models), which can perfectly reproduce stop-and-go traffic,phase transitions, local clusters and various traffic waves. In addition, Gipps [2] presented the Gipps model, andSpyropoulou [3] recently investigated its mathematical properties. However, these models cannot directly be usedto study network flow. On the other hand, the network models mainly concentrate on some macro issues (e.g. routechoice) [4] but cannot depict the evolution of the flow and the density of each link. Therefore, developing a modelwhich can deal with the macro properties of network flow and the drivers’ micro behaviors of each link simultaneouslyhas become an important topic in the field of traffic science.

Recently, some scholars adopted a cellular automaton model and a differential equation model to study networkflow, and they obtained some important results [5–12]. Recently, some network flow simulation programs such asAIMSUN and DRACULA have been developed. In this paper, we develop a dynamic model for traffic network flow,and numerically test the model in a network with one OD pair connected by two routes. A small perturbation of thenetwork inflow is set and the resultant evolutions of inflow, outflow and flow on each link are obtained. It is found thatthe dynamic model can reasonably capture the basic characteristics of network flow.

∗ Corresponding author at: Department of Automobile Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China.E-mail address: [email protected] (T.Q. Tang).

0378-4371/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.01.020


2604 T.Q. Tang et al. / Physica A 387 (2008) 2603–2610

2. Model

Lighthill and Whitham [13] and Richard [14] independently developed a first-order continuum traffic model (LWRmodel for short), which can be written as follows:

∂ρ

∂t+

∂(ρve)

∂x= s(x, t), (1)

where ρ is the traffic density, ve the equilibrium velocity, s(x, t) the traffic generation rate, and x and t the space andtime variables respectively. If the freeway has no ramp, then s(x, t) = 0, and s(x, t) 6= 0 otherwise. In addition, theequilibrium velocity ve(ρ) is a decrease function of the density ρ and the equilibrium flow ρve(ρ) is a concave functionof the density ρ. The LWR model can perfectly reveal the formation, propagation and evolution of shocks [15], butdoes not faithfully describe the non-equilibrium traffic flow because the velocity cannot deviate from the equilibriumvelocity ve(ρ). For these reasons, many scholars have later developed various high-order models, which can beclassified into the density-gradient (DG) model and speed-gradient (SG) model. The DG model can be written asfollows [16–20]:

∂v

∂t+ v

∂v

∂x=

ve − v

τ−

c2(ρ)

ρ

∂ρ

∂x+ µ

∂2v

∂x2 , (2)

where τ is the reactive time, c(ρ) > 0 the sonic speed, and µ the viscous coefficient. The DG model can capture thebasic properties of non-equilibrium flow, but may produce backward motion under some given condition [21]. Jianget al. [22,23] recently presented an SG model:

∂v

∂t+ v

∂v

∂x=

ve − v

τ+ c0

∂v

∂x, (3)

where the constant c0 > 0 is the propagating speed of small perturbation. They proved that this model never givesbackward motion and can reproduce such non-equilibrium phenomena as stop-and-go, local clusters and ghost jams.However, all the above models can only be used to describe road flow, rather than network flow.

In fact, it is very easy to extend the classical LWR model to the case of network flow. The extended LWR modelcan be written as follows:

∂ρk

∂t+

∂(ρkvke)

∂xk= sk(xk, t), k = 1, 2, . . . , N , (4)

where ρk and vke are respectively the traffic density and the equilibrium velocity of link k, sk(xk, t) is the flowgeneration rate of link k, and N is the number of links in the network. In addition, the equilibrium velocity vke(ρk)

is a decrease function of the density ρk , and the equilibrium flow ρkvke(ρk) is a concave function of the density ρk .Note that sk(xk, t) in Eq. (4) is different from s(x, t) in Eq. (1). It has the following form:

sk(xk, t) =

the inflow of the link k, if xk = 00, if 0 < xk < Lkthe outflow of the link k, if xk = Lk,

(5)

where Lk is the length of link k.From Fig. 1, we can see that link k is independent of its adjacent links when 0 < xk < Lk , so we can directly

introduce the full velocity difference (FVD) equation [22,23] into Eq. (4) and obtain a dynamic model for trafficnetwork flow, i.e.

∂ρk

∂t+

∂(ρkvk)

∂xk= sk(xk, t)

∂vk

∂t+ vk

∂vk

∂xk=

vke − vk

τk+ ck

∂vk

∂xk

, k = 1, 2, . . . , L , (6)

where τk and ck are the reaction time and the propagation speed of a small perturbation on link k, respectively. Ingeneral, τk and ck are functions of density ρk . For simplicity, we here let them be constant, i.e., τk = τ and ck = c0.


T.Q. Tang et al. / Physica A 387 (2008) 2603–2610 2605

Fig. 1. The scheme of network flow.

In order to better understand the interaction of each link and each node in a traffic network, we should here definesk(0, t) and sk(Lk, t) again, i.e.

si, jk (0, t) = q i, j,k

inflow and si, jk (Lk, t) = q i, j,k

outflow, (7)

where si, jk (0, t) and si, j

k (Lk, t) are respectively the inflow and the outflow of the link k which connects the origin node

i with the destination node j . Since the vehicles should be conservative at any node, si, jk (0, t) and si, j

k (Lk, t) shouldsatisfy the following condition:∑

i,k

s j,ik (0, t) =

∑i,k

si, jk (Lk, t), (8)

where∑

i,k s j,ik (0, t) is the total inflow of node j , and

∑i,k si, j

k (Lk, t) the total outflow of node j . Eq. (8) shows thatthe total inflow is equal to the total outflow at node j .

Note: Eq. (6) is similar to the two-lane traffic flow model [24,25]. However, their differences are as follows: thesum of sk(x, t) will be equal to zero in the traffic flow model on two lanes without ramp, and sk(xk, t) in Eq. (6)should satisfy Eq. (8).

3. Numerical simulation

We first rewrite Eq. (6) in a conservative form, i.e.

(uk)t + f (uk)xk = Sk(uk), (9)

where uk =

{ρkvk

}, f (uk) =

{ρkvk

0.5v2k − c0vk

}, Sk(uk) =

{sk (xk , t)

(vke(ρk ) − vk )/τ

}. We then adopt the conservative scheme [26] to

discretize Eq. (9):

un+1k,i = un

k,i −1t

1xk( f n

k,i+1/2 − f nk,i−1/2) + Sk(u

nk,i )1t, (10)

where n represents the n-th time step, i the i-th space step of link k, 1t the length of the time step, and 1xk the lengthof the space step of link k. f n

k,i+1/2 = f (unk,i , un

k,i+1) is defined as

f (unk,i , un

k,i+1) = 0.5( f (unk,i ) + f (un

k,i+1) − αk(unk,i+1 − un

k,i )). (11)

For simplicity, we here set αk = vk f , where vk f is the free flow velocity of link k.Next, we test the proposed model in a network with one OD pair connected by two routes. In the simulation, we

set initial conditions as follows:

ρ1(x, 0) = ρ10 = 0.02, ρ2(y, 0) = ρ20 = 0.06. (12)

We considering the fact that the routes of most vehicles are fixed and only few vehicles have the chance to updateroutes. Hence, we let the network inflow be

qinflow(t) = q10 + q20(1 + sin(π t/30)), (13)

where q10 is the inflow of fixed route vehicles which enter the link i (=1, 2) with probability ri , and q20(1 +

sin(π t/30)) is the inflow of time-varying route vehicles, representing a small perturbation. A linear equilibrium



Fig. 2. Evolution of inflows on two routes (links) by Principle A.

velocity vie = vi f (1 − ρi/ρi j ) is adopted in the simulation,1 where vi f and ρi j are the free-flow velocity and thejam density of link i , respectively.

In fact, all the parameters of a traffic network need to be calibrated by the use of many observed data, and theparameters of different network are often different. For simplicity, in this paper we set the other parameters as follows:1x1 = 1x2 = 100 m, 1t = 1 s, ρ1 j = ρ2 j = 0.2 vel/m, c0 = 11 m/s, τ = 10 s, q10 = 0.5, q20 = 0.05, r1 = 0.4,r2 = 0.6, v1 f = 30 m/s, v2 f = 26 m/s, L1 = 6 km, and L2 = 4 km.

In the simulation, the following two route choice principles are adopted to allocate the perturbation flow.Principle A: for time interval t , if L1

v1(0,t) <L2

v2(0,t) , the vehicle chooses link 1, otherwise link 2. This principleassumes that at any time interval t , drivers probably compute the route travel times required according to the velocitiesat the entrances of routes, and then choose the route with smaller travel time.

Principle B: for time interval j , if∑ j

t=0L1

v1(0,t) <∑ j

t=0L2

v2(0,t) , the vehicle chooses link 1, otherwise link 2. Thisprinciple assumes that, for each route, drivers compute the sum of route travel times during the past j time intervals,and then choose the route having the smaller value just computed.

The main numerical results are shown in Figs. 2–7. From these figures, we can conclude the following results:(i) It can be seen from Figs. 2 and 3 that significant oscillation exists in the route inflows by both principles.

Certainly, oscillation is only subject to the vehicles which have the chance to update routes. There is a longer cycle foroscillation, which is caused by the sine function in Eq. (13). Meanwhile, the amplified part shown in the right-handside of Fig. 3 indicates that in each longer cycle, the inflow is valued in a bang–bang manner, which is clearly becauseof the bang–bang route choice principle. Therefore, the model proposed in this paper is able to reasonably describethe route choice behavior in a network. Comparing Figs. 2 and 3, it can be found that the bang–bang route choiceappears later in Principle B than in Principle A because the route choice decision in Principle B is based on all historicinformation.2

(ii) The outflows of both links finally become stable; see Figs. 4 and 5. Thus, a small perturbation of the networkinflow has little effect on link outflows regardless of the route choice principle adopted.3

1 We can obtain some similar results if we adopt another perturbation function or another equilibrium velocity.2 As for other more complex networks, the oscillations might be relatively small, and might even be dissipated under some given conditions,

which proves that we should use the dynamics model to further study the complex phenomena of a complex traffic network.3 The outflow of each link might be unstable in other more complex traffic networks because the interactions of links and nodes are often

relatively complex. Generally speaking, the more complex a traffic network is, the more complex the evolution of the outflow of each link will be.Therefore, our on-going research is to further study the evolution of the density, the inflow and the outflow of each link in more complex trafficnetworks.



Fig. 3. Evolution of inflows on two links by Principle B. The right-hand side is the amplified one in part.

Fig. 4. Evolution of outflows on two links by Principle A.

(iii) Figs. 6 and 7 show that the flows of both links easily evolve to stable states and the traffic waves propagatebackward. That is, our model can smooth a small perturbation of the network inflow no matter which route choiceprinciple is used.

The above results verify that the network flow dynamic model proposed in this paper can explore the microproperties of link flows (e.g., their spatial–time evolutions), describe such macro behavior as route choice and analyzethe interactive impacts between them. It should be noted that a simple network is adopted in the simulation for clearlydemonstrating the model’s ability in depicting macro/micro behavior. If the model is used in such general networks asthat shown in Fig. 1, we will find the interaction among the nodes and links. In Fig. 1, the travel time of route 1 relieson the times of links 1, 2, 4 and 6; the travel time of route 2 is dependent upon the times of links 1, 3, 5 and 6. Thenode for route choice decision making is at the end of link 1. We can employ a method similar to Principle A or B, toassign the outflow of link 1 onto links 2 and 3. The outflow of link 2 becomes the inflow of link 4, and the sum of theoutflows of links 4 and 5 is the inflow of link 6. The central problem is how to exactly estimate the travel times of the



Fig. 5. Evolution of outflows on two links by Principle B.

Fig. 6. Spatial–time evolution of flows on two links by Principle A.

two routes starting from the decision making node (i.e., the end of link 1). A route’s travel time consists of the traveltimes on all links covered by it, while the travel time of a link depends on the time by which a vehicle enters this linkand the time-varying traffic conditions encountered after that time. It is impossible for a driver to know the exact linktravel time before he or she exits from the link. The two principles of governing route choice presented in this paperare based on the estimated route times which are determined by the velocities at route entrances. Other route choiceprinciples can be used, for example, the real travel time strategy and the on-line average velocity strategy [4]. The realtravel time of a vehicle is the departure time from a route minus the entering time; obviously, the route guidance forfollow-up vehicles is postponed.

It should be explained that the small difference between Figs. 4 and 5 (also Figs. 6 and 7) is due to the use of asimple network in the simulation, that is, both route choice principles are not very sensitive to a small perturbation ofnetwork inflow. It is expected that distinct differences caused by different route choice principles will be observed inlarge and complex networks.



Fig. 7. Spatial–time evolution of flows on two links Principle B.

4. Conclusions

Existing road traffic flow models and network models are independent of each other and few of them can be used instudying the combined problem. In this paper, we have developed a dynamic model for network flow by introducingthe inflow and outflow variables between associated links and applying the LWR model of road flow together witha route choice principle. Numerical results show that the model can reproduce the route choice behavior in a simplenetwork and the micro properties of each link. Recently, scholars have found that bifurcation and chaos often appearin road traffic flow models [27–30]. Our on-going research is to test the model in more complex networks and use themodel to further study the bifurcation and chaos of network flow.

Acknowledgements

This study has been jointly supported by the National Natural Science Foundation of China (70701002, 70521001)and the National Basic Research Program of China (2006CB705503).

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a dynamic model for traffic network flow

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