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Zhongmin Huang, M.N. Smirnova, N.N. Smirnov, Zuojin Zhu. 2024: Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model. Journal of Traffic and Transportation Engineering (English Edition), 11(4): 733-746. DOI: 10.1016/j.jtte.2022.08.005
Citation: Zhongmin Huang, M.N. Smirnova, N.N. Smirnov, Zuojin Zhu. 2024: Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model. Journal of Traffic and Transportation Engineering (English Edition), 11(4): 733-746. DOI: 10.1016/j.jtte.2022.08.005

Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model

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  • Author Bio:

    Zhongmin Huang is now a PhD candidate in Institute of Plasma Physics, Chinese Academy of Sciences and University of Science and Technology of China. His primary research interest is in the direction of Plasma control system. But recently, he has also taken part in the project NSFC 11972341 and carried out some studies on traffic flow modeling. E-mail address: zhongmin.huang@ipp.ac.cn

    M.N. Smirnova is an associate professor in Faculty of Mechanics and Mathematics, Lomonosov Moscow State University (LMSU). She received her PhD degree in physics and mathematics from LMSU in the year 2013. Currently, she has done some studies in fluid mechanics and transportation engineering. Now she has the interest in mathematical modelling multi-phase media and wave processes in natural, technical and social systems. E-mail address: wonrims@inbox.ru

    N.N. Smirnov is a professor in Faculty of Mechanics and Mathematics, Lomonosov Moscow State University (LMSU). He received his MS and PhD degrees in physics and mathematics from LMSU in the years 1976 and 1980 respectively, habilitated doctor degree in 1990. He has been an academician of Russian Academy of Natural Sciences since 2007. E-mail address: ebifsun1@mech.math.msu.su

    Zuojin Zhu is an associate professor in Faculty of Engineering Science, University of Science and Technology of China. He obtained his PhD degree in 1990 at Shanghai Jiao Tong University, and worked as a research fellow at the Hong Kong Polytechnic University in 2008. His current research interest is mainly in thermal fluid flow and transportation engineering fields. E-mail address: zuojin@ustc.edu.cn

  • Corresponding author:

    M.N. Smirnova

    Zuojin Zhu, Faculty of Engineering Science, University of Science and Technology of China, Hefei 230026, China

  • Peer review under responsibility of Periodical Offices of Chang'an University.

  • Received Date: December 07, 2021
  • Revised Date: July 23, 2022
  • Accepted Date: August 14, 2022
  • Published Date: August 01, 2024
  • Fluid flow throttling is common in industrial and building services engineering. Similar tunnel throttling of vehicular flow is caused by the abrupt number reduction of roadway lane, as the tunnel has a lower lane number than in the roadway normal segment. To predict the effects of tunnel throttling of annular freeway vehicular flow, a three-lane continuum model is developed. Lane Ⅲ of the tunnel is completely blocked due to the need of tunnel rehabilitation, etc. There exists mandatory net lane-changing rate from lane Ⅲ to lane Ⅱ just upstream of the tunnel entrance, which is described by a model of random number generated through a golden section analysis. The net-changing rate between adjacent lanes is modeled using a lane-changing time expressed explicitly in algebraic form. This paper assumes that the annular freeway has a total length of 100 km, a two-lane tunnel of length 2 km with a speed limit of 80 km/h. The free flow speeds on lanes Ⅰ, Ⅱ and Ⅲ are assumed to be 110, 100 and 90 km/h respectively. Based on the three-lane continuum model, numerical simulations of vehicular flows on the annular freeway with such a tunnel are conducted with a reliable numerical method of 3rd-order accuracy. Numerical results reveal that the vehicular flow has a smaller threshold of traffic jam formation in comparison with the case without tunnel throttling. Vehicle fuel consumption can be estimated by interpolation with time averaged grid traffic speed and an assumed curve of vehicle performance. The vehicle fuel consumption is lane number dependent, distributes with initial density concavely, ranging from 5.56 to 8.00 L. Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.

    HIGHLIGHTS
    ● The effects of tunnel throttling of annular freeway vehicular flow is analyzed.
    ● Golden section-based analysis is used to model the mandatory net lane-changing rate just upstream the tunnel entrance.
    ● A three-lane continuum model is proposed and used to code a simulation platform.
    ● Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.
  • equals to uc2/{1sech[Λlln(ρc2l/ρm)]}
    traffic sound speed on lane , m/s
    equilibrium speed at saturation point on segment , m/s
    CFL Courant number
    vehicle grid fuel consumption on lane , mL
    vehicle fuel consumption on lane , L
    length scale, m
    average length of cars, m
    total length of ring road, km
    tunnel length, km
    traffic pressure on lane , veh·m/s2
    mandatory net lane-changing rate from lane Ⅲ to lane Ⅱ, veh/h
    product of and traffic flow acceleration on lane , veh/s2
    traffic speed on lane , m/s
    equilibrium speed on segment , m/s
    second critical speed, m/s
    speed scale, m/s
    free flow speed on segment , m/s
    braking distance on segment , m
    kth ramp intersection
    position of tunnel entrance, km
    position of tunnel exit, km
    equals to , parameter used to define traffic pressure
    ratio of lane-changing time to lane averaged relaxation time
    random number used to define the mandatory net lane-changing rate
    equals to cτl/uc2
    lane averaged traffic density, veh/km
    traffic density on lane , veh/km
    traffic jam density, veh/km
    random variable to describe ramp flow
    relaxation time on segment , s

    Fluid flow throttling is common in industrial and building service engineering. In transportation science and engineering for vehicular flow, it is renamed as bottleneck. A review of mathematical modeling of vehicular flows indicates that many models have been developed, as reported by Zhang et al. (2018, 2023).

    After considerations for macroscopic modeling of multi-lane vehicular flow reported by Michalopoulos et al. (1984), Helbing and Greiner (1997) put forward a macroscopic model of multi-lane traffic flow from a Boltzmann-like gas-kinetic approach. Daganzo (2002) developed a macroscopic behavior theory of traffic dynamics for homogeneous multi-lane freeways. Focusing on the onset of congestion, Pestov et al. (2019) examined the performance of multi-lane freeway traffic on entering road crossings. Smirnova et al. (2014) proposed a multi-lane continuum model, in which an expression of the parameter to describe the acceleration component in the perpendicular direction in relation to lane-changing was derived.

    For the studies on bottleneck effects, Goñi-Ros et al. (2016) found that to make reduction of congestion at sags, a highly effective and innovative way is to use cooperative adaptive cruise control systems. Jin (2018) developed a behavioral kinematic wave model for better design and control strategies to enhance the performance of tunnel bottlenecks.

    Recently, Li et al. (2023) have investigated freeway tunnel effects on ring road traffic flow by proposing a two-lane traffic model. Huang et al. (2023) have explored the effects on vehicular fuel consumption of a freeway work zone which has a length of 100 m and occupies lanes Ⅲ and Ⅳ by a four-lane model.

    The aim of this paper is to predict the effects of tunnel throttling of annular freeway vehicular flow by developing a three-lane continuum model, where the net lane-changing rate between adjacent lanes is described by a vehicular lane-changing time model. The mandatory lane-changing rate from lane Ⅲ to Ⅰ just upstream of the tunnel entrance is described be a random number model, with the random number generated in a small range around a median based on golden section analysis. For vehicular flows on an annular freeway with a tunnel having a length of 2 km, but with just two lanes thus inducing the throttling of vehicular flow. Numerical simulations with a reliable numerical method of 3rd-order accuracy are conducted to provide spatiotemporal evolution of traffic density, and grid traffic speed. The grid traffic speed on different lanes is used to calculate the mean travel time and its fluctuation through each lane. With an artificially-assumed vehicle performance curve relating to vehicle fuel consumption on vehicular speed, both the vehicle fuel consumption through each lane of the annular freeway and the additional vehicle fuel consumption in comparison with the case of lane blockage vanished (LBV) (i.e., the absence of lane blockage or without tunnel throttling) are predicted.

    Numerical results reveal that tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling. Under the conditions in this paper, the density threshold of traffic jam formation normalized by jam density is around 0.19, while for the case without tunnel throttling the threshold is 0.22. The ramp effect slightly delays the occurrence of traffic jam, making the threshold become 0.20. The mean travel time for the case without tunnel throttling is slightly longer than that for the case with tunnel throttling. As soon as the tunnel has originated a traffic jam at the tunnel entrance, the tunnel mean travel time is approximately equal to 0.062 h, suggesting that the time averaged traffic speed through the tunnel is around 32.26 km/h, which agrees well with existing data, suggesting that the three-lane model has the application potential for analyzing the effect of tunnel throttling of vehicular flow and understanding the tunnel bottleneck phenomenon.

    In this paper, model equations are presented in Section 2, the numerical method is described in Section 3, numerical results of vehicular flows on freeway with tunnel throttling are discussed in Section 4, and some conclusions are provided in Section 5.

    In the three lanes Ⅰ, Ⅱ and Ⅲ, for lane number l{1,2,3}, traffic densities and traffic speeds are labeled by ρl and ul respectively, as shown in Fig. 1. Generally, vehicles on lane Ⅰ have the highest free flow speed, and lowest on lane Ⅲ, with a value between the two for vehicles on lane Ⅱ. However, in the tunnel, due to the existence of tunnel throttling, lane Ⅲ is blocked completely and traffic density is equal to jam density, a two-lane continuum model is suitable for vehicular flow on lane Ⅰ and Ⅱ. The road has one initial jam at xA, the tunnel with a length of Lt has an entrance and exit are at xt1 and xt2 respectively. The intersections are at xR1 and xR2, which are connected with lane Ⅲ. In the normal segment, any vehicle attempting to leave the annular road initially makes a lane change and shifts onto lane Ⅲ, except lane-changing happened in the tunnel. It is assumed that lane-changing arises spontaneously to keep the local homogeneity in the perpendicular direction, regardless of whether lane-changing is mandatory or discretionary, except for lane-changing at the point just upstream of the tunnel entrance.

    Figure  1.  Illustration of a three-lane vehicular flow with freeway tunnel throttling, as lane Ⅲ in the tunnel is blocked completely (tunnel length Lt=xt2xt1).

    In the three-lane model, traffic densities ρl and flow rates ql=ρlul are taken as the main variables. Let the lane average density be ρ=(ρ1+ρ2+ρ3)/3, for vehicles on lane Ⅰ attempting to shift onto lane Ⅱ and Ⅲ, the density is ρ1ρ in unit of km. Putting the traffic relaxation time on lane l as τ, labeling the average relaxation time as τ=(τ1+τ2+τ3)/3, if the vehicular lane-changing time is τβ*, the net lane-changing rate for the governing equation of ρ1 can be written as (ρ1ρ)/(τβ*). Using random ramp parameter σ generated in a small range Δσ with a median, the equations of the three-lane traffic model may be written as Eq. (1).

    {ρlt+qlx=(ρlρ)/(τβ*)+σq3δl3/l0+(δl3/l0+δl2/l0)qt1xt1wult+ululx=Rl/ρl (1)

    where l0 is the length scale of traffic flow, xt1w=xt1l0 denotes the location of the net lane-changing rate qt1, subscript t represents the tunnel, subscript w denotes the west side of the position xt1, subscript t represents the time derivative, subscript x denotes the spatial derivative of some variables, such as ρl and ql.

    However, for traffic flow in the tunnel, the lane average density is ρ=(ρ1+ρ2)/2, for l{1,2}, the relevant equations are as shown in Eq. (2).

    {ρlt+qlx=(ρlρ)/(τβ*)ult+ululx=Rl/ρl (2)

    where δl2 and δl3 are respectively the Kronecker deltas.

    Eqs. (1), (2) indicate that there is a mass conservation of vehicular motion between lanes Ⅰ, Ⅱ and Ⅲ. Similarly, Rl can be expressed as follows (Bogdanova et al., 2015; Ma et al., 2018; Zhu and Yang, 2013).

    Rl=(qelql)/τlplx+(ρlνlulx)x (3)

    where qel and Rl/ρl are equilibrium flow rate and acceleration on lane l{1,2,3} respectively, νl and τl are traffic kinematic viscosity and relaxation time, pl is traffic pressure which equals to ρlcl2, and cl is traffic sound speed.

    Adopting τβ* to represent lane-changing time, its ratio to relaxation time is given by Eq. (4).

    β*={|ρ1ρ2|or|ρ2ρ3|<1veh/km(e1)ρ/(ρmρ)otherwise (4)

    It shows that if the absolute value of traffic densities between the two lanes is less than 1 veh/km, drivers have not attempted to carry out lane-changing so that the ratio β*=; otherwise it is inversely proportional to vehicular spatial headway, equally to unity when traffic flow is at saturation with ρs=ρm/e. When traffic flow is completely jammed, β*=, indicating that lane-changing is hard to happen for congested flow. For unsaturated traffic flow, β*<1. For instance, at the first critical point on lane Ⅱ, ρ*2=0.0819 (Table 1), the ratio β*=0.15328, the lane-changing time used at the critical point is about 15% of relaxation time, implying that a driver in free flow needs 1.401 s to change lanes once if the average relaxation time is 9.142 s.

    Table  1.  Parameters of vehicular flow on annular freeway.
    Parameter Value Parameter Value
    vf1 (km/h) 110 ρ*1 0.0736
    vf2 (km/h) 100 ρ*2 0.0819
    vf3 (km/h) 90 ρ*3 0.0909
    vf4 (km/h) 80 ρ*4 0.1021
    Xbr1 (m) 73 ρc21 0.6525
    Xbr2 (m) 65 ρc22 0.6374
    Xbr3 (m) 58 ρc23 0.6190
    Xbr4 (m) 51 ρc24 0.5985
    cτ1 5.147 xR1 40
    cτ2 4.879 xR2 60
    cτ3 4.582 xA 20
    cτ4 4.280 xt1 50
    τ1 (s) 7.762 xt2 52
    τ2 (s) 9.007 Lt (km) 2
    τ3 (s) 10.657 lveh (m) 5.8
    τ4 (s) 12.834 uc2 (km/h) 18
    ρm (veh/km) 172 Imax 1001
    l0 (m) 100 γ 0.382
    v0 = ρ*2vf2 (m/s) 2.2756 Δγ 0.05
    t0 (s) 43.945 g0 = 1−γ 0.618
    L (km) 100
    Note: ρ * j and ρ c 2 j are normalized by ρ m , tunnel length L t = x t 2 x t 1 , cτ1, cτ2, cτ3, and cτ4 are measured by v0.
     | Show Table
    DownLoad: CSV

    In comparison with existing modules of lane-changing, the present model is simpler than the behavioral theory reported by Daganzo (2002), which has been used to make predictions for separate groups of lanes while recognizing that the traffic stream is usually composed of aggressive and timid drivers. The simplest version of the theory is found to be qualitatively consistent with experimental observations. However, eight observable parameters are used to specify the simple model.

    In the normal road segment excluding the tunnel, lane numbers are l{1,2,3}, vehicles have different free flow speed and braking distance on different lanes, indicating that equilibrium traffic flow rate is lane-dependent, as shown in Fig. 2. In Fig. 2, ρ is measured by jam density ρm, traffic flow rate has the unit of q0=ρmv0, and qesj=cτj/e(ρ*jvfj/ρ*2vf2),j{1,2,3,4}. Let jam density be ρm, in normal road segment, equilibrium traffic flow rate can be written as proposed by Zhang et al. (2018).

    qel={ρlvflρlρ*lcτlρlln(ρl/ρm)ρ*l<ρlρc2lBlρl{1sech[Λlln(ρl/ρm)]}ρc2l<ρlρm (5)
    Figure  2.  Fundamental diagrams (FD) for a three-lane vehicular flow.

    In the tunnel, the lane number is limited to l{1,2}, as the free flow speed is just the same as tunnel speed limit vf4, equilibrium traffic flow rate is shown in Eq. (6)

    qel={ρlvf4ρlρ*lcτ4ρlln(ρl/ρm)ρ*l<ρlρc2lB4ρl{1sech[Λ4ln(ρl/ρm)]}ρc2l<ρlρm (6)

    For j{1,2,3,4}, in the definition of the equilibrium traffic speed at saturation state cτj, the average length of vehicles lveh has to be used. In the definition of Bj, the ratio of cτj to the second critical speed uc2 on lane l, i.e., Λl, should be used, as proposed by Zhang et al. (2021).

    In particular, qt1 at xt1w is the mandatory net lane-changing rate from lane Ⅲ to lane Ⅱ due to the existence of tunnel throttling, the rate is approximated by the random number model based on golden section analysis: the ratio of the mandatory net lane-changing rates to q3 is approximately equal to γ=1g0, here g0=0.618 is the golden section. Hence, let γ be a random number with the median γ=0.382 and the number range Δγ=0.05, the net changing rate at xt1w can be written as Eq. (7).

    qt1=γq3(t,xt1w) (7)

    As an example, when t(0.2,0.27) h, the evolution curve of qt1 for initial density normalized by jam density ρ0=0.29 can be seen in Fig. 3(a), with Fig. 3(b) showing the random numbers σ1 and σ2 for assigning ramp flows at the intersections xR1 and xR2 (Table 1).

    Figure  3.  Temporal evolution of qt1, σ1 and σ2. (a) Mandatory lane-changing rate qt1 in the unit of ρmv0 for ρ0=0.29 when t(0.2, 0.27) h in the case of RF1. (b) Random numbers for on and off ramp flows at intersections xR1 and xR2.

    Details of calculation of the numerical flux with the WENO5 scheme (Henrick et al., 2005; Jiang and Shu, 1996) and time derivative term with the RK3 (Shu, 1988; Shu and Osher, 1988) are reported by Zhang et al. (2021). The numerical method WENO5-RK3 is explicit. The vector form of the equations of the continuum model and the condition of numerical stability are presented as below.

    In the normal road segment, the pressure gradient plx is related to sound speed as in Eq. (8).

    plx=cl2ρlx (8)

    Using

    \begin{aligned} & {R}_{{l}^*}={R}_{{l}}+p_{{lx}}=u_l[ \\ & \left.\quad-\left(\rho_{{l}}-\rho\right) /\left(\tau \beta_*\right)+\sigma {q}_3 \delta_{{l3}} / {l}_0+\left(-\delta_{{l3}} / l_0+\delta_{{l} 2} / l_0\right) q_{\mathrm{t} 1} x_{\mathrm{t} 1 \mathrm{w}}\right] \end{aligned} (9)

    Instead of Rl for l{1,2,3}, the governing Eq. (1) can be written as Eq. (10).

    Ut+F(U)x=S (10)

    where U=(ρ1,q1,ρ2,q2,ρ3,q3)T, F(U)=(q1,q12/ρ1+p1,q2,q22/ρ2+p2,q3,q32/ρ3+p3)T, and

    S=[(ρ1ρ)/(τβ*)R1*(ρ2ρ)/(τβ*)+qt1xt1w/l0R2*(ρ3ρ)/(τβ*)qt1xt1w/l0+σq3/l0R3*] (11)

    In the tunnel, x[xt1,xt2], for l{1,2}, according to Eq. (2),

    Rl*=Rl+plx (12)

    Taking U=(ρ1,q1,ρ2,q2)T, F(U)=(q1,q12/ρ1+p1,q2,q22/ρ2+p2)T, the source term becomes Eq. (13).

    S=[(ρ1ρ)/(τβ*)R1*(ρ2ρ)/(τβ*)R2*] (13)

    The WENO5-RK3 satisfies the Courant-Friedrichs-Lewy (CFL) condition, the Courant number is given by Eq. (14).

    CFL=ΔtΔxmax|ak,i|<1i=0,1,,Imax1 (14)

    where ak,i is the kth eigenvalue of A at xi, Δx is the uniform grid length, Imax is the maximum of grid number.

    The Courant number CFL (Shui, 1998) is fixed at 0.3 to ensure numerical stability. The main reason for choosing WENO5 rather than other Riemann solvers is that essentially non-oscillatory (ENO) reconstruction (Shu and Osher, 1988) is based on adaptive stencils, such that the optimal stencil is chosen. This provides high-order accuracy and essentially non-oscillatory behavior. WENO reconstruction (Shu, 1988) consists of a convex combination of all the candidate stencils, and constitutes an improvement on ENO schemes on many levels, as reported by Johnsen and Colonius (2006).

    To predict the tunnel throttling effect of annular freeway vehicular flow, numerical simulations are conducted with parameters shown in Table 1. It can be seen that the free flow speeds on lanes Ⅰ, Ⅱ and Ⅲ are vf1=110, vf2=100, and vf3=90 km/h respectively, with the tunnel speed limit vf4 being equal to 80 km/h. With the second critical speed uc2=18 km/h, and the corresponding braking distances Xbr for j{1,2,3,4}, the fundamental diagrams can be determined, as shown by Fig. 2. The second column of Table 1 shows the first and second critical densities ρ*j and ρc2j, and positions of the first and second ramp intersections xR1, xR2, the position of initial jam at xA, the tunnel entrance at xt1=50 km, tunnel exit at xt2=52 km, and tunnel length of Lt=2 km. The column also shows the average length of cars of lveh=5.8 m, and the second critical speed of uc2=18 km/h.

    The third column of Table 1 shows the jam density ρm, the scales of length, speed and time l0, v0 and t0, the maximum number of grid Imax = 1001, total length of the annular freeway L=(Imax1)l0=100 km. The median of the random parameter from a golden section analysis γ is 0.382, and the relevant range of Δγ=0.05. In the numerical tests, Reynolds number is given by Re=l0v0/ν=128. In Table 2, the medians for random numbers, i.e., σ1av and σ2av, and the relevant range Δσ are shown for the cases LBV, RF0 and RF1, here LBV is the abbreviation of lane blockage vanished. The off and on ramp intersections located respectively at xR1 and xR2.

    Table  2.  Parameters of random number generator for ramp flows.
    Case σ1av σ2av Δσ
    LBV 0.000 0.000 0.00
    RF0 0.000 0.000 0.00
    RF1 −0.191 −0.191 0.05
     | Show Table
    DownLoad: CSV

    In particular, it is noted that the average length of cars lveh=5.8 m is just suitable for passenger car, as reported by McShane et al. (1998), but for single unit truck and single unit bus, the length can be changed to 9.1 m and 12.1 m respectively. Indeed, the choice of lveh could have some influences on simulation results, as traffic pressure pl and traffic sound speed cl depend on the parameter α=lvehρm, as described by Zhang et al. (2018).

    In the numerical simulations, in the normal segment excluding the tunnel, x[xt1,xt2], the initial density is assumed to be as Eq. (15).

    ρl(0,x)={1x[(xA1)/2,(xA+1)/2]ρ0(1+0.05δl2)otherwise (15)

    While in the tunnel, x[xt1,xt2], it is assumed to be just 38.2% of ρ0, such that

    ρl(0,x)=ρ0(1g0) (16)

    with ql(0,x)=qel(0,x).

    To examine the macroscopic model characterized by the ratio of lane-changing time to relaxation time β*, a numerical study has been carried out by Li et al. (2022) to seek the effects of tunnel speed limit on traffic flow, where the observed speed data (Koshi et al., 1992) and the calculated speed on the basis of a behavioral kinematic wave model developed by Jin (2018) were used to make a comparison, which has shown that the predicted speeds on the basis of a three lane traffic model agree well with published data.

    As the average vehicle length lveh is 5.8 m, we choose the lowest contour level as 7 m to determine whether vehicular flow jam has formed. In Fig. 4, spatiotemporal evolution of spatial headway hd2 is shown, from which one can see that there exists a larger hd2 segment that is close to the tunnel end, the length of the segment grows with the increase of time; a lower hd2 segment near the tunnel entrance. Also the initial jam propagates in the downstream direction, and the initial jam has induced several spontaneously jams propagating forward.

    Figure  4.  Spatiotemporal evolution of spatial headway hd2=(1−ρ2)/(ρmρ2). (a) RF0, ρ0=0.18. (b) RF0, ρ0=0.19. (c) RF1, ρ0=0.19. (d) RF1, ρ0=0.20.

    Comparing Fig. 4(a) and (b) and Fig. 4(c) and (d), the density threshold of traffic jam formation ρth=0.19 or 0.20 for RF0 and RF1 (Table 2) are obtained, indicating that the off ramp flow at the ramp intersection at xR1=40 km and the on ramp flow at the intersection at xR2=60 km delay the occurrence of traffic jam slightly.

    For ρ0=0.29, the instantaneous distributions of traffic density ρl and traffic speed ul, l{1,2,3} and density fraction s1=ρ1/ρl at the time of t=0.5 h are shown in Fig. 5. Caused by traffic jam at xt1=50 km, in the segment near the entrance, traffic density ρl is higher, with lower traffic speed ul. There is a peak of ρl and a corresponding trough of ul at a position upstream the initial jam point xA=20 km, suggesting that the tunnel triggered traffic jam propagates in the upstream direction. In the segment near the tunnel exit, ρl dropped abruptly to a level of 0.1, with small magnitude oscillation at this level, correspondingly ul rises suddenly to a level of about 0.9, and then oscillates with a small magnitude, caused by the propagation and interaction of traffic waves. When traffic density ρl is equal to 0.29, the traffic speed ul has a value around 0.5.

    Figure  5.  Variable distributions at t = 0.5 h on the ring road for ρ0 = 0.29 in the case of RF1. (a) Traffic density. (b) Traffic speed. (c) Density fraction s1=ρ1/ρl.

    In the tunnel, the density fraction s1=ρ1/(ρ1+ρ2) has a value of around 0.5. However, in the freeway normal segment, s1=ρ1/(ρ1+ρ2+ρ3) has a value around 1/3, just because of the crucial role played by local vehicular homogeneity.

    Correspondingly, Table 3 shows the data of ρl, ul and s1 near the tunnel. It can be seen that in the traffic jam triggered by the tunnel, ρl is generally over 0.5, excluding the grid on lane Ⅲ at x=49.90 km which is nearest to the tunnel entrance at xt1=50 km. In particular, on lanes Ⅰ and Ⅱ, the jam terminates at x=50.4 km, as there is a sudden drop of traffic density (ρ1 or ρ2) when x approaches to the adjacent grid at x=50.5 km. Downstream in the tunnel segment when the jam terminates, traffic density decreases gradually from a value of around 0.38 to about 0.34, as can be seen in the 2nd and 4th columns of Table 3.

    Table  3.  Predicted (ρl,ul) for l{1,2,3} and s1 near the tunnel at t=0.5 h for RF1 and ρ0=0.29.
    x ρ1 u1 ρ2 u2 ρ3 u3 s1
    49.5 0.5474 0.2509 0.5717 0.2140 0.5391 0.2359 0.3301
    49.6 0.5476 0.2509 0.5792 0.2055 0.5333 0.2432 0.3298
    49.7 0.5476 0.2512 0.5889 0.1941 0.5238 0.2544 0.3298
    49.8 0.5466 0.2529 0.6009 0.1800 0.5098 0.2699 0.3298
    49.9 0.5418 0.2529 0.6319 0.1852 0.2734 0.4889 0.3744
    50.0 0.5634 0.2281 0.5835 0.2204 1.0000 0.0000 0.4912
    50.1 0.5520 0.2330 0.5446 0.2362 1.0000 0.0000 0.5034
    50.2 0.5242 0.2454 0.5126 0.2509 1.0000 0.0000 0.5056
    50.3 0.4957 0.2595 0.4899 0.2626 1.0000 0.0000 0.5030
    50.4 0.4809 0.2675 0.4789 0.2679 1.0000 0.0000 0.5010
    50.5 0.3846 0.3344 0.3731 0.3448 1.0000 0.0000 0.5076
    50.6 0.3717 0.3459 0.3722 0.3456 1.0000 0.0000 0.4997
    50.7 0.3704 0.3471 0.3714 0.3463 1.0000 0.0000 0.4993
    50.8 0.3691 0.3483 0.3697 0.3478 1.0000 0.0000 0.4996
    50.9 0.3676 0.3497 0.3675 0.3498 1.0000 0.0000 0.5001
    51.0 0.3655 0.3517 0.3651 0.3520 1.0000 0.0000 0.5002
    51.1 0.3632 0.3538 0.3628 0.3541 1.0000 0.0000 0.5003
    51.2 0.3610 0.3559 0.3606 0.3562 1.0000 0.0000 0.5002
    51.3 0.3588 0.3579 0.3585 0.3582 1.0000 0.0000 0.5002
    51.4 0.3567 0.3599 0.3565 0.3601 1.0000 0.0000 0.5002
    51.5 0.3547 0.3618 0.3544 0.3621 1.0000 0.0000 0.5002
    51.6 0.3527 0.3638 0.3523 0.3641 1.0000 0.0000 0.5003
    51.7 0.3507 0.3657 0.3502 0.3662 1.0000 0.0000 0.5004
    51.8 0.3485 0.3679 0.3479 0.3685 1.0000 0.0000 0.5004
    51.9 0.3463 0.3700 0.3456 0.3706 1.0000 0.0000 0.5005
    52.0 0.3438 0.3725 0.3431 0.3732 1.0000 0.0000 0.5005
    52.1 0.2350 0.4611 0.2375 0.4549 0.0684 0.5235 0.4345
    52.2 0.1773 0.5494 0.1806 0.5280 0.1214 0.5781 0.3699
    52.3 0.1584 0.6095 0.1587 0.5860 0.1413 0.5775 0.3456
    52.4 0.1368 0.6724 0.1362 0.6461 0.1278 0.6388 0.3413
    52.5 0.1196 0.7320 0.1204 0.7006 0.1147 0.6768 0.3372
    52.6 0.1108 0.7769 0.1140 0.7361 0.1077 0.7115 0.3333
    Note: ul (l = 1, 2, 3) is normalized by vf2.
     | Show Table
    DownLoad: CSV

    Downstream of the tunnel end xt2=52 km, with x increasing from 52.1 to 52.6 km, ρ1 drops from 0.2350 to 0.1108, ρ2 drops from 0.2375 to 0.1140; while ρ3 increases from 0.0684 to 0.1413 at x=52.3 km at first and then decreases gradually to 0.1077.

    In this study, travel time is predicted using grid traffic speed as previously reported by Zhang et al. (2018), which is different from the work of Chang and Mahmassani (1988) and Wang et al. (2016). It is noted that in the tunnel lane Ⅲ x[xt1,xt2], traffic speed is zero, which is not appropriate for the prediction of travel time through lane Ⅲ using grid traffic speed. Alternatively, the density-averaged grid traffic speed, i.e., (q1+q2)/(ρ1+ρ2) is used, as the tunnel length is just 2% of the total road length.

    In this paper, it is found that in the case of LBV (or the case of three lane tunnel), as shown by the coarse solid purple curve in Fig. 6(a), the mean travel time through lane Ⅱ Tt,2av is slightly higher than that predicted for the cases RF0 and RF1, implying that tunnel throttling only has a slight influence on the mean travel time.

    Figure  6.  Distributions of travel time with ρ0. (a) Mean travel time Tt,2av. (b) Tunnel mean travel time Ttu,2av.

    Tt,2av for the case RF0 is illustrated by the blue solid curve with circles, as shown in Fig. 6(a), with the corresponding data given in the 3rd column of Table 4. Tt,2av grows with the increase of initial density ρ0. When ρ0=0.07, Tt,2av=1.01 h, which is almost equal to the mean travel time calculated directly with the free flow speed vf2. When ρ0 increases from the second threshold of jam formation 0.19 to the saturation density of vehicular flow ρs=1/e0.368, the mean travel time Tt,2av increases from 1.482 to 2.496 h. For ρ0=0.37, at over saturation, the mean travel time Tt,2av is 2.507 h. The green deltas represent the mean travel time Tt,2av for the case of RF1, which almost overlap the circles for the case of RF0, indicating that the ramp flow influences on the mean travel time is very small.

    Table  4.  Distributions of Tt, lav, Ttu, lav and Tt,l for RF0.
    ρ0 Tt,1av (h) Tt,2av (h) Tt,3av (h) Ttu,1av (h) Ttu,2av (h) Ttu,3av (h) Tt,1 (h) Tt,2 (h) Tt,3 (h)
    0.070 0.924 1.010 1.120 0.024 0.025 0.025 0.003 0.002 0.002
    0.150 1.233 1.305 1.372 0.044 0.045 0.044 0.022 0.030 0.022
    0.190 1.399 1.482 1.550 0.060 0.060 0.060 0.034 0.044 0.037
    0.200 1.439 1.524 1.593 0.061 0.061 0.061 0.038 0.047 0.041
    0.210 1.484 1.574 1.641 0.062 0.062 0.062 0.041 0.050 0.045
    0.368 2.344 2.496 2.595 0.063 0.062 0.062 0.030 0.036 0.034
    0.370 2.354 2.507 2.607 0.063 0.062 0.062 0.031 0.038 0.035
     | Show Table
    DownLoad: CSV

    In Fig. 6(b), the tunnel mean travel time through lane Ⅱ Ttu,2av is plotted as a function of initial density ρ0, where the blue solid cure with circles show the tunnel mean travel time through lane Ⅱ Ttu,2av, which is consistent with the data given by the 6th column of Table 4, the green deltas represent Ttu,2av predicted for the case of RF1 with ramp flow effect, while the coarse solid purple curve shows Ttu,2av for the case of LBV. For the cases of RF0, when ρ0 passes through the turning point at ρth=0.19, the mean travel time exhibits a slowly growing plateau that ends at the point ρ0=0.37. As shown in Fig. 6(b), when ρ0 is lower than the threshold ρth=0.19, Ttu,2av for the case of RF1 is almost the same as that calculated for the case RF0. As in the case of RF1, the threshold of traffic jam formation is slightly delayed to 0.20, as shown in Fig. 4, indicating that the ramp flow effect on Ttu,2av is also very small. However, for the case of LBV, the turning point occurs at 0.22, indicating that tunnel throttling results in a smaller density threshold of traffic jam formation.

    As shown in the 6th column of Table 4, the platform height is about 0.062 h, implying that as soon as a traffic jam appears and propagates in the upstream direction, the time-averaged traffic speed through the tunnel (Lt/Ttu,2av) is around 32.26 km/h, which agrees well with the time averaged speed of 28.9 km/h from observed speed data (Koshi et al., 1992) and the time averaged speed of 33.6 km/h calculated on the basis of a behavioral kinematic wave model (Jin, 2018).

    In Fig. 7, the mean travel time Tt,lav, root mean square (RMS) of travel time Tt,l, tunnel mean travel time Ttu,lav, and its root mean square Ttu,l are plotted as functions of initial density ρ0.

    Figure  7.  Distributions of travel time and root mean square with ρ0. (a) Mean travel time Tt,lav. (b) RMS of travel time Tt,l. (c) Tunnel mean travel time Ttu,lav. (d) RMS of tunnel travel time Ttu,l.

    For the case of RF0, Tt,lav and Ttu,lav for l=1,2,3 are illustrated by the black deltas, green circles, and blue gradients respectively. While for the case of RF1, the ramp parameter has a median of 0.191 (Table 2) with a range of 0.05, Tt,l and Ttu,l for l=1,2,3 are shown by black, green and blue curves respectively. It can be seen that ramp flow has brought a very small influence on Tt,lav, but an observable small influence on Tt,l, usually causing a smaller value of Tt,l for a given initial density, such an effect becoming more acute when the initial density is beyond the threshold ρth.

    As the free flow speed vfl determines equilibrium traffic speed on lane l, i.e., qel/ρl, with the value of vfl given in the first column of Table 1, for a given initial density ρ0, it is reasonable that Tt,3av takes a value equal to the largest among the three mean travel times, as shown in Fig. 7(a). However, those values are all lower than that predicted for the case of LBV, as shown by the coarse solid purple curve.

    In Fig. 7(b)(d), it can be seen that the coarse solid purple curve denoting the corresponding variable for l=3 in the case of LBV, deviates clearly from the gradient symbols, suggesting that tunnel throttling has brought a significant effect on the RMS of travel time, tunnel mean travel time as well as the relevant RMS value when the initial density is below the threshold.

    When traffic jam originated by the tunnel appears at the tunnel entrance, there is a rarefaction wave propagating in the downstream direction, which decreases the responsiveness of grid traffic speed in the tunnel to the variation of initial density ρ0. As a result, the fluctuation of travel time has a peak near the threshold ρth (Fig. 7(b)), the peak value is usually below 0.05 h; the tunnel mean travel time has a plateau with a height of about 0.062 h (Fig. 7(c)). For the case of LBV, the fluctuation of tunnel travel time also occurs a peak at the density threshold of 0.22, the peak is around 0.008 h, as can be seen in Fig. 7(d).

    To predict vehicle fuel consumption by interpolation, time averaged grid traffic speed is used with a vehicular performance curve as shown in Fig. 8. The performance curves for lane Ⅰ, Ⅱ or Ⅲ have steps, reflecting the dependence of vehicle fuel consumption fc on vehicular speed u.

    Figure  8.  Distribution of fuel consumption fc as a function of vehicular speed (the volume unit 1 L = 1000 mL, and the vehicular speed is in the unit of vf2).

    According to the three vehicle performance curves, with the time averaged grid traffic speed u¯l, the vehicle fuel consumption through the grid length Δxi=l0 for i[1,Imax], i.e., fcl can be estimated by linear interpolation. As l0 is assigned as 100 m, the unit of fc is L/100 km. For the convenience of illustration, mL is selected as the unit of the vehicle grid fuel consumption.

    For ρ0=0.190,0.250and0.368, the distributions of u¯l and vehicle grid fuel consumption fcl are shown in Fig. 9. It can be seen that the time averaged grid traffic speed u¯l has a trough at the tunnel entrance at xt1=50 km. As illustrated by the purple solid curve, the off-ramp flow at xR1=40 km results in an abrupt rise of u¯l for l=1,2 to a peak then decreases gradually in the downstream segment near xR1, where u¯3 has an obvious peak for ρ0=0.368. Due to the on-ramp flow at xR2=60 km, u¯3 also has a trough in the downstream segment near xR2.

    Figure  9.  Distributions of time averaged grid traffic speed u¯1,u¯2,u¯3 and fuel consumption through a grid length fc1, fc2, fc3. (a) u¯1 at ρ0=0.19. (b) fc1 at ρ0=0.19. (c) u¯2 at ρ0=0.25. (d) fc2 at ρ0=0.25. (e) u¯3 at ρ0=0.368. (f) fc3 at ρ0=0.368.

    The right-hand side of Fig. 9 indicates that there are corresponding peaks for the troughs of u¯l. In particular, for ρ0=0.368, being relevant to the effect of initial jam, a peak of fc3 appears at xA, the black solid curve for RF1 deviates from the green circles for RF0 in the downstream segment near xR1.

    Taking the sum of fcl(xi), the total vehicle fuel consumption through the annular freeway can be obtained as Eq. (17).

    \mathrm{Fu}_l=10^{-3} \sum\limits_{i=1}^{I_{\max }} \mathrm{fc}_l\left({x}_i\right) (17)

    where the factor 103 is used to change the unit from mL to L.

    Being consistent with the data given in Table 5, the vehicle fuel consumption through the freeway ring Ful is plotted as a function of initial density ρ0, as shown in Fig. 10(a), where coarse solid curves are obtained in the case of LBV, with circles and squares corresponding to the cases of RF0 and RF1 respectively. The coarse solid curves are shown in order to demonstrate the tunnel throttling of vehicular flow more clearly. Ful is lane dependent, and distributed in the range of ρ0[0.07,0.37] concavely, implying that there exists a local minimum. Fig. 10(a) shows that in the range of ρ0[0.07,0.37], Ful is below 8.0 L.

    Table  5.  Distributions of Ful for LBV, RF0 and RF1.
    ρ0 LBV RF0 RF1
    Fu1(L) Fu2(L) Fu3(L) Fu1(L) Fu2(L) Fu3(L) Fu1(L) Fu2(L) Fu3(L)
    0.070 5.965 6.970 7.959 5.974 6.987 7.968 5.975 6.987 7.968
    0.080 5.911 6.908 7.930 5.937 6.950 7.952 5.938 6.952 7.953
    0.090 5.848 6.829 7.897 5.895 6.880 7.937 5.895 6.881 7.938
    0.190 5.572 6.397 7.189 5.656 6.505 7.316 5.644 6.490 7.306
    0.200 5.557 6.392 7.138 5.642 6.499 7.274 5.633 6.484 7.265
    0.210 5.567 6.401 7.106 5.638 6.488 7.236 5.627 6.480 7.230
    0.260 5.724 6.410 7.061 5.765 6.471 7.166 5.746 6.460 7.152
    0.270 6.209 6.420 7.062 6.181 6.479 7.160 6.133 6.468 7.145
    0.280 6.264 6.439 7.070 6.214 6.491 7.163 6.168 6.479 7.148
    0.300 6.327 6.481 7.086 6.264 6.521 7.166 6.246 6.507 7.156
    0.330 6.433 6.573 7.153 6.377 6.613 7.228 6.385 6.588 7.206
    0.360 6.560 7.067 7.260 6.500 7.043 7.326 6.490 7.001 7.280
    0.368 6.596 7.098 7.291 6.535 7.081 7.354 6.516 7.027 7.291
    0.370 6.607 7.106 7.292 6.531 7.081 7.347 6.510 7.030 7.283
     | Show Table
    DownLoad: CSV
    Figure  10.  Initial density dependents of the variables. (a) The fuel consumption Ful. (b) The additional fuel consumption δFul.

    Comparing the fuel consumption illustrated by circles and squares for the cases RF0 and RF1, it can be seen that ramp flow merely results in an observable small deviation of Ful when ρ0>0.27.

    To show the tunnel throttling effect on fuel consumption through the annular freeway more clearly, the difference of fuel consumption to that consumed in the case of LBV called as the additional fuel consumption is calculated by Eq. (18).

    δFul=Ful(Ful)LBV (18)

    As shown in Fig. 10(b), the difference δFul is positive for l=3, but negative only when initial density ρ0 is above 0.35 for l=2, or above 0.26 for l=1.

    Examining the distributions illustrated in Fig. 10(b), in the cases of RF0 and RF1, if ρ0 is not over 0.368, δFu3 is positive, with a local maximum around 0.13 L δFu2 changes from positive to negative at ρ0=0.35 in the case of RF1 as shown by the blue gradients, but the change happens at ρ0=0.345 in the case of RF1 as illustrated by the green filled circles. While both δFu1 change to negative at ρ0=0.27 as shown by the red deltas. In the case of RF0, δFu3 is above 0.05 L for the range of ρ0[0.1,0.36], but δFu2 is above 0.05 L merely for a narrowed range of ρ0[0.09,0.28].

    A three-lane continuum model is developed to predict the effect of tunnel throttling on annular freeway vehicular flow. Based on the continuum model, numerical simulations with a reliable method of 3rd-order accuracy are conducted, with the following conclusions.

    (1) The three-lane continuum model is characterized by the vehicular lane-changing time model to describe net lane-changing rate between adjacent lanes, the model of random number generated in a small range around a median from a golden section analysis, is proposed to describe the mandatory lane-changing rate just upstream of the tunnel having two lanes. The three-lane model has the application potential for analyzing the effect of tunnel throttling of vehicular flow and understanding the tunnel bottleneck phenomenon.

    (2) Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling. Under the conditions adopted in this study, the density threshold of traffic jam formation in unit of jam density is around 0.19, while for the case without tunnel throttling the threshold is 0.22. The ramp effect slightly delays the occurrence of traffic jam, making the threshold become 0.20.

    (3) It is found the mean travel time for the case without tunnel throttling is slightly longer than that for the case with tunnel throttling. If the tunnel has originated a traffic jam at the tunnel entrance, the tunnel mean travel time is approximately equal to 0.062 h, suggesting that the time averaged traffic speed through the tunnel is around 32.26 m/h, which agrees well with existing data.

    (4) The time averaged traffic speed can be used to estimate travel time and vehicle fuel consumption through every lane. The vehicle fuel consumption through each lane is lane dependent, can be estimated by interpolation with vehicle performance curve. The vehicle fuel consumption varies concavely with initial density. Under the conditions of in this paper, the vehicle fuel consumption is in the range of 5.56–8.00 L. As a counterpart of comparison, for the case of lane blockage vanished without ramp flow effect, the additional fuel consumption for vehicles through lane Ⅲ is above 0.05 L, when initial density is in the range from 0.10 to 0.36; but for vehicles through lane Ⅱ, the additional fuel consumption is above 0.05 L only when initial density is in the range of 0.09–0.28.

    This study is supported by the project of National Natural Science Foundation of China "exploring the road condition effect of travel time using emergency mitigation traffic flow models" (grant 11972341) and fundamental research project of Lomonosov Moscow State University "mathematical models for multi-phase media and wave processes in natural, technical and social systems". We thank Prof. X.Y. Yin and Prof. C.K. Chan respectively at USTC and The Hong Kong Polytechnic University and Dr. Yinglin Li at Peking University for some useful private communications.

    Conflict of interest

    The authors do not have any conflict of interest with other entities or researchers.

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