1 Introduction
2 Macroscopic traffic flow modelling
3 Model design
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Taking into account “multiphasic” hybrid aspect of traffic flow
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Choosing the pressure expression according to density
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Elasticity principle is applicable
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There is an internal energy “potential”.
3.1 Elasticity principle
3.2 Internal energy density and pressure
3.2.1 Pressure expression
3.2.2 Multiphasic traffic flow concept
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ρ ≤ ρ cThere is no constraint on the traffic flow here, so the internal energy vanishes. The expression of the density of the total energy is reduced to , ie we have only one form of energy storage and, thus, one state variable. Then, ρ and ν are linked algebraically.
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ρ c < ρ ≤ ρmaxIn this case, we should consider the total energy density given by Eq. 12. This corresponds to the presence of both forms of energy, which implies the presence of two state variables.
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Free flow (ρ ≤ ρ c )
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Congested flow (ρ c < ρ < ρmax)
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Congested and saturated flow (ρ = ρmax).
4 Proposed expression
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The model depends explicitly on characteristic parameters of the considered road (ρ c , ν c and ρmax).
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The corresponding eigenvalues of this matrix are .
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As the wave velocities do not exceed the traffic flow speed.
5 Various phases of the traffic flow
5.1 Free flow phase (ρ ≤ ρ c )
5.2 Congested phase and forced regime (ρ c < ρ < ρmax)
5.2.1 Speed deterioration effects
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Evolution at upstream of congestionThis case corresponds to the presence of a cluster forced to drive at a speed determined by a given vehicle, a leader. For instance, the presence of a truck, travelling at a speed lower than the critical one, compels the following vehicles to drive at its speed. Then, a cluster is formed which moves at a speed ν l (t, x) < ν c . We have, therefore, the appearance of an internal energy which compensates for the reduction of the kinetic one. Note that the speed of the leader represents exogenous distributed action. This corresponds to a forced regime for a dynamic system and to take it into account, we add a second member to the second equation of the system (20):where ν l (x, t) and τ l represent, respectively, the leader speed and a time constant.(23)
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Evolution inside congestionWhen the situation persists, the traffic flow tends towards a metastable state, mentioned by Kerner [15]. In the particular case when the leader speed is constant and the situation continues, a stationary regime establishes at ν = ν l . Then a cluster is formed, moving with constant speed. Note that this case represents the so-called moving jam phenomenon for low ν l (x, t) values.To get the corresponding particular model, we rewrite the system (23) with ν = ν l . We, therefore, obtain a single equation that characterizes the cluster that moves at a constant speed:In other words,we have a convective equation with constant speed fixed by the leader. In contrast to the free flow case, we do not have here an algebraic relationship between ρ and ν l .(24)
5.2.2 Density deterioration effects (bottleneck)
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Evolution inside congestionThe equations to be used in this case have to reflect the temporary characteristics of the concerned zone. Indeed, the maximum density is reduced, the reduction ratio depending on the narrowing rate of the road. We should also change the values of the critical density and critical speed that must be adapted into the area. The model then takes the formwhere .(25)Note that the zone concerned can move (case a) or be fixed (cases b and c).
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Evolution at upstream of congestionUpstream of the congestion, vehicles must slow down to adapt their speed to that imposed by the density of jam zone. This corresponds to the case of the “follow-the-leader” seen above, but, here, the leader is fictitious.The leader speed νlbo ≤ νcbo and it depends on density values within, and upstream of, the congestion.(26)
5.3 Congested and saturated phase (ρ = ρmax)
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an accident or incident that jams the whole road
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traffic lights that regulate highway access at the ramps.
5.4 Congested phase and free regime (ρ c < ρ < ρmax)
6 Numerical scheme and simulation
6.1 Numerical scheme
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we have N=2 real eigenvalues
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∀ u l ,u r ∈ Ω ,(32)
6.2 Simulation examples
6.2.1 Example 1
ρ
c
= 40 vehicles/km | ρmax = 100 vehicles/km |
ν
c
= 30 km/h | νmax = 50 km/h |
time r-l = 6.5 s | time g-l = 41.5 s |
6.2.2 Example 2
Time t | Behaviour | Systems of equations involved in same time |
---|---|---|
Before time-in | free flow | Eq. 22 |
At time-in | cluster forming | |
From time-out | Progressive return to the free flow |
ρ
c
= 40 veh./km | ρmax = 100 veh./km |
ν
c
= 90 km/h | νmax = 130 km/h |
ν
l
= 40 km/h | position-in= 20 km |
time-in = 40 s | time-out= 78 mn |