Physics Letters A 365 (2007) 6–9

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Spatiotemporal congested traffic patterns in macroscopic versionof the Kerner–Klenov speed adaptation model

Rui Jiang a,∗, Mao-Bin Hu a, Bin Jia b, Ruili Wang c, Qing-Song Wu a

a School of Engineering Science, University of Science and Technology of China, Hefei 230026, PR Chinab School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, PR China

c Institute of Information Sciences and Technology, Massey University, New Zealand

Received 26 September 2006; received in revised form 28 November 2006; accepted 20 December 2006

Available online 29 December 2006

Communicated by C.R. Doering

Abstract

In this Letter, a macroscopic version of the microscopic three phase speed adaptation model proposed by Kerner and Klenov [B.S. Kerner,S.L. Klenov, J. Phys. A 39 (2006) 1775], which employing the idea that steady states of free flow are separated from that of synchronized flow,is presented through the micro–macro link. The simulations of the macroscopic model with an on-ramp show that the various congested patternsfound in empirical observations are reproduced. We have compared the results in microscopic speed adaptation model and in the macroscopicversion. The existing problems in the model are pointed out.© 2006 Elsevier B.V. All rights reserved.

Keywords: Traffic flow; Three phase traffic theory; Macroscopic model

Recently the traffic flow problems has attracted the interestsof many physicists and engineers [1–6]. There are two differentconceptual frameworks for modeling traffic: the macroscopicmodels and microscopic models. In the macroscopic fluid-dynamical description, the traffic is viewed as a compressiblefluid formed by the vehicles but these individual vehicles donot appear explicitly in the theory. In contrast, in the so-called“microscopic” models of vehicular traffic attention is explicitlyfocused on individual vehicles; the nature of the interactionsamong these vehicles is determined by the way the vehicles in-fluence each others’ movement.

Based on the empirical observations, Kerner [2] classifiedtraffic flow into three states: free flow, synchronized flow andwide moving jam. He further pointed out the existing mod-els, which are based on the fundamental diagram approach, failto reproduce the empirical spatiotemporal patterns (the readersmay refer to the book of Kerner [2] for the details of problemsof the fundamental diagram approach).

* Corresponding author.E-mail address: rjiang@ustc.edu.cn (R. Jiang).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.12.058

In 2002, Kerner et al. proposed a first microscopic modelbased on the three-phase traffic theory [7]. The model can de-scribe the empirical phase diagram quite satisfactorily. Lateron, several other microscopic models based on the three-phasetraffic theory were presented [8–12]. Nevertheless, there is nomacroscopic models based on three phase traffic theory untilnow.

In a recent paper, Kerner and Klenov proposed a determin-istic speed adaptation microscopic model [12], where a sepa-ration of steady states of free flow and synchronized flow isassumed. In this Letter, we present a macroscopic version ofthe microscopic speed adaptation model. It is shown that themacroscopic model can qualitatively exhibit the empirical fea-tures of the F(free flow) → S(synchronized flow) → J (jams)transitions and the spatiotemporal congested traffic patterns.

The macroscopic model is presented through the followingmicro–macro links [13]. (i) The microscopic vehicle speed v

of vehicle n is replaced by the macroscopic vehicle speed u:vn(t) → u(x, t); (ii) the distance between a vehicle n and thepreceding vehicle n− 1 is expressed in term of the vehicle den-sity ρ: �x = xn−1(t) − xn(t) → 1

ρ(x,t), therefore, V (�x) →

R. Jiang et al. / Physics Letters A 365 (2007) 6–9 7

Fig. 1. The steady states of the free flow, synchronized flow and jams.

ue(ρ); (iii) the speed difference between a vehicle and the pre-ceding vehicle is replaced by the macroscopic speed gradient:vn−1(t) − vn(t) → ∂u

∂x; (iv) the microscopic parameters τ , κ

are replaced by macroscopic ones: τ → T , κ → c0. Here V isan optimal velocity function in microscopic model, ue, T , c0are equilibrium speed density relationship, relaxation time, andpropagation speed of small disturbance in macroscopic model.

Applying the above replacements to the microscopic speedadaptation model, we obtained the equations of the macro-scopic model, which read as follows:

du

dt= ∂u

∂t+ u

∂u

∂x

(1)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

ue,f (ρ)−u

Tf+ c0,f

∂u∂x

at u � uf

min and ρ < ρj

min,

ue,s (ρ)−u

Ts+ c0,s

∂u∂x

at u < uf

min and ρ < ρj

min,

− uTj

at ρ � ρj

min.

Eq. (1) together with the conservation equation

(2)∂ρ

∂t+ ∂ρu

∂x= 0

constitute the macroscopic version of the speed adaptationmodel. The steady states of the model are shown in Fig. 1.Steady states of free flow are related to a curve q = ρue,f (ρ)

(the curve F in Fig. 1) associated with the condition

(3)u = ue,f (ρ) at u � uf

min.

Steady states of synchronized flow are related to a curve q =ρue,s(ρ) (the curve S in Fig. 1) associated with the condition

(4)u = ue,s(ρ) at u < uf

min and ρ < ρj

min.

Steady states for a wide moving jam are given by a horizontalline q = 0 and ρ

j

min � ρ � ρmax.As in Ref. [12], in the model, an F → S transition is mod-

eled through discontinuity of steady speed solutions. Movingjam emergence is simulated through an instability of some ofthe synchronized flow model steady states.

Next we carry out the simulation of the macroscopic modelat an on-ramp. Here the on-ramp is included into the modelthrough the continuum equation as follows:

(5)∂ρ

∂t+ ∂(ρu)

∂x= qon

Lramp,

where qon is on-ramp flow rate and Lramp is length of on-rampmerge region.

Eqs. (1), (2) and (5) are discretized as in Ref. [13]. For thecontinuum Eqs. (2) and (5), we have

(6)ρn+1i = ρn

i + �t

�x

(ρn

i−1uni − ρn

i uni+1

),

(7)ρn+1i = ρn

i + �t

�x

(ρn

i−1uni − ρn

i uni+1

) + �tqon

Lramp.

For Eq. (1), we have:

• When ρni < ρ

j

min– When un

i > c0,phase(ρni )

un+1i = un

i − (un

i − c0,phase) �t

�x

(un

i − uni−1

)

(8)+ �tue,phase − un

i

Tphase.

– When uni � c0,phase(ρ

ni )

un+1i = un

i − (un

i − c0,phase) �t

�x

(un

i+1 − uni

)

(9)+ �tue,phase − un

i

Tphase.

• When ρni � ρ

j

min

(10)un+1i = max

(0, un

i − uni

�t

�x

(un

i − uni−1

) − �tun

i

Tj

).

Here phase denotes f (free flow) or s (synchronized flow), in-dex i represents the road section and index n represents time,�t is time step and �x is the length of a mesh. The para-meters are �t = 1 s, �x = 50 m, Lramp = 50 m (i.e., one

mesh is classified as merge region), uf

min = 22.22 m/s, ρj

min =0.15 vehicle/m, c0,f = 10 m/s, Tf = 2 s, c0,s = 0.5ue,s , Ts =e20ρ , Tj = 1 s,

ue,f = uf tanh

(1/ρ − 5

uf τ

)

with uf = 33.33 m/s, τ = 0.85 s,

ue,s = (1/ρ − 1/ρj

min)

τs

with τs = 1.2 s.Fig. 2 shows the phase diagram of the model at an on-ramp.

Here the length of the main road is 50 km, the on-ramp is at37.5 km. To consider the randomness of real traffic, we assumethat the on-ramp flow is a random number uniformly distributedbetween qon − 18 and qon + 18 vehicles/h.

8 R. Jiang et al. / Physics Letters A 365 (2007) 6–9

In the phase diagram, the moving synchronized pattern(MSP), the widening synchronized pattern (WSP), the generalpattern (GP) and the localized synchronized pattern (LSP) areidentified. This is qualitatively consistent with the empiricalobservations (Fig. 3). Comparing the LSP with that from mi-croscopic simulations (Fig. 8(e) in Ref. [12]), one finds that inthe LSP from macroscopic simulations, synchronized flow isusually observed as in empirical LSP (Fig. 4).

In Fig. 2, the dashed line separates the space into two re-gions. In the right region, congested pattern occurs even if theon-ramp flow qon remains constant. In the left region, the freeflow will not transit into congested flow if the on-ramp flowremains constant. However, in the region between the dashedline and line FC, if a perturbation large enough is exerted, the

Fig. 2. Phase diagram of the model with an on-ramp.

congested patterns will be induced (Fig. 3(c)). In other words,the states between the dashed line and line FC are metastablestates. Once the congested patterns (except LSP) are induced,the capacity will drop.

In Fig. 5, the density and speed profiles are shown wherethe main road flow qin is given and qon increases gradually.Compared with that from microscopic simulations (Figs. 6(a)and (b) in Ref. [12]), the deterministic perturbation does notappear from macroscopic simulations. Therefore, the F → S

transition occurs when qin + qon > qfmax. In other words, the

dashed line is determined by qin + qon = qfmax.

Compared with the phase diagrams of other macroscopictraffic flow models with an on-ramp, such as those shown in[14,15], Fig. 2 is qualitatively different because it is based onthe three phase traffic flow theory while others are based onfundamental diagram approach.

Nevertheless, there are several issues need discussion in thephase diagram. Firstly, compared with the empirical observa-tions, the on-ramp flow rate corresponding to boundary betweenGP and WSP in Fig. 2 is much larger. This is, maybe, due to thenumerical viscosity in the numerical scheme, which enhancesthe stability of traffic flow. Therefore, one of our next goals isto seek more suitable numerical scheme.

Secondly, we find that the outflows from the jams are alwayssynchronized flow (Fig. 6), which is in opposite to the micro-scopic results, where only free flow can be formed betweenwide moving jams within the GP (Fig. 10(d) in Ref. [12]). Thisis, maybe, due to that the slow-to-start rule is not taken intoaccount in the macroscopic model. In the microscopic modelin Ref. [12], the slow-to-start rule is considered by choosinga smaller value of K(acc) than K(dec). Therefore, in our fu-

(a) (b)

(c) (d)

Fig. 3. Typical congested patterns at an on-ramp. (a) MSP, qin = 2725 vehicles/h, qon = 180 vehicles/h. (b) WSP, qin = 2725 vehicles/h, qon = 1080 vehicles/h.(c) LSP qin = 1195 vehicles/h, qon = 1800 vehicles/h between t = 1000 s and t = 1200 s and qon = 1440 vehicles/h during other times. (d) GP,qin = 2200 vehicles/h, qon = 1800 vehicles/h.

R. Jiang et al. / Physics Letters A 365 (2007) 6–9 9

Fig. 4. The snapshot of the density profile of LSP in Fig. 3(c). Here the widthof the LSP is about 0.35 km, much larger than the length of the merge region(50 m).

Fig. 5. The density profiles where main road flow qin is given and qon increasesgradually. qin = 2200 vehicles/h, for curve I qon = 540 vehicles/h, for curve II

qon = 648 vehicles/h. Note qfmax = 2880 vehicles/h.

Fig. 6. The snapshot of the density profile of GP in Fig. 3(d).

ture work, the slow-to-start rule needs to be incorporated intomacroscopic model.

Thirdly, we find that GP always appears providing qon islarge enough, even if qin is quite small. This is in contradictto empirical observations. In real traffic, the on-ramp will becongested itself when qon is large. In this case, further increas-ing qon will not increase the number of vehicles merging intomain road but increase the congestion propagation speed on on-ramp. This feature is not modeled in macroscopic model whensimply regarding on-ramp as a source as in Eq. (4). Therefore,more realistic modeling of on-ramp effect is needed.

To summarize, in this Letter, we have presented a macro-scopic version of the microscopic speed adaptation modelthrough micro–macro links. Our simulations show that, themodel with an on-ramp can reproduce qualitatively the empir-ical phase diagram. The MSP, WSP, LSP, GP are reproduced.Nevertheless, several problems exist in the present model in-cluding the numerical scheme issue, the modeling of on-ramp,and the slow-to-start rule issue. These problems need furtherinvestigations.

Acknowledgements

We thank an anonymous referee for his suggestions to im-prove the Letter. We acknowledge the support of NationalBasic Research Program of China (No. 2006CB705500), theNational Natural Science Foundation of China (NNSFC) un-der Key Project No. 10532060 and Project Nos. 10404025,70501004, 70601026, 10672160, and the CAS special Founda-tion. R. Wang acknowledges the support of the ASIA:NZ Foun-dation Higher Education Exchange Program (2005), MasseyResearch Fund (2005), and International Visitor Research Fund(2006).

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