Step III: Speed limitation for stabilization

The vehicles following the most upstream vehicle that is slowed down to resolve the jam, or the most downstream vehicle in the stabilization area when the jam has re-solved, should realize the target densityρ^[4] (veh/km) in addition to the target speed v^eff. In microscopic terms this means that the following distance d^headway should be

d^headway = 1/ρ^[4], (2.14)

on the average. The density is a tuning variable and is chosen such that it corresponds to stable traffic.

This density is realized by properly slowing down the vehicles that are in free flow upstream of already speed-limited vehicles. At each time t^rs = k^rsT^rs a reference vehiclej^ref is determined upstream of which the densityρ^[4]should be realized. In the case that there is a jam, this vehicle is the vehiclej that should be speed-limited to resolve the jam according to (2.11). In the case that the jam has resolved, it is the first

Time Position Reference vehicle xrefxtarget ixi dheadway

v^eff

vi

t^rs

∆xi

t^rs+ ∆t^free_i + ∆ti

t^rs+ ∆t^free_i

Figure 2.4: Vehiclei will reach the target trajectory which lies a distance of d^headwaykm upstream of the reference trajectoryx^refby starting to decelerate at timet^rs+ ∆t^freei .

vehicle that is upstream of the S-head line – i.e., the line describing the downstream end of the stabilization area – that will be defined in Section 2.3.4.

Note that the reference vehiclej^ref is at locationx^ref_j (t^rs) (km) and is traveling with speed v_j^ref(t^rs) (km/h). The vehicle is speed-limited and in the case that it has not reached the effective speed v^eff yet it will do so after time∆t^ref_j (t^rs) (h) and distance

∆x^ref_j (t^rs) (km) given as:

∆t^ref_j (t^rs) = max{v^eff, v^ref_j (t^rs)} − v^eff

a^T (2.15)

∆x^ref_j (t^rs) = 1

2(max{v^eff, v_j^ref(t^rs)} + v^eff)∆t^ref_j (t^rs) . (2.16) The reference trajectory that this vehicle determines is then defined by the speed v^eff and positionx^ref(t^rs) (km) given as:

x^ref(t^rs) = x^ref_j (t^rs) + ∆x^ref_j (t^rs) − v^eff∆t^ref_j (t^rs) , (2.17) that together define the following reference line:

x^ref(t) = x^ref(t^rs) + (t − t^rs)v^eff. (2.18) The idea is that every vehicle i upstream of vehicle j^ref should reach its own target trajectory line defined by the positionx^target_i (t^rs) given as:

x^target_i (t^rs) = x^ref(t^rs) − N_j^ref−id^headway, (2.19) whereN_j^ref−i (veh) is the number of vehicles between vehiclej^ref and vehiclei as is illustrated in Figure 2.4.

This is realized by checking whether a vehicle will need to start decelerating during the current time step. This is done iteratively in the upstream direction starting from the vehicle upstream of the vehiclei^us(-). Vehiclei^usis the last vehicle in the ‘platoon’

directly upstream ofj^ref that is traveling with a speedvi^us(t^rs) ≤ v^eff+ ǫ^veff, whereǫ^veff (km/h) is a threshold. If no such vehicle exists, then it is equal to the reference vehicle j^ref.

It might be the case that the vehiclei^ushas decelerated too fast and is traveling too far – i.e., more than a threshold γd^headway (km) withγ (-) a tuning parameter represent-ing a fraction of the followrepresent-ing distance – upstream of its target trajectory line. This may prevent the next upstream vehicle from reaching its target trajectory line (simply because the vehicle is blocked), and so on. In the worst case, this process may con-tinue for several vehicles, leading to a local accumulation of vehicles, and to a possible breakdown. Therefore, the reference linex^ref(t^rs) and index j^ref are reset to the vehicle i^usif this happens:

x^ref(t^rs) =

(x_i^us(t^rs) ifx_i^us(t^rs) ≤ x^target_ius (t^rs) − γd^headway

x^ref(t^rs) otherwise, (2.20)

j^ref =

(i^us ifxi^us(t^rs) ≤ x^target_i^us (t^rs) − γd^headway

j^ref otherwise. (2.21)

Now, for the vehicles upstream ofi^usthe time∆t^free_i (t^rs) (h) after which they will need to start decelerating can be calculated. Note that it will take a vehicle a time∆ti(t^rs) (h) and distance ∆xi(t^rs) (km) to reach the effective speed. A vehicle that has not reached the effective speed yet will travel first with its free speedvi(t^rs) (km/h) for a time∆t^free_i (t^rs) after which it has to start to decelerate – see Figure 2.4 for a graphical representation of these variables. It will then reach the point xi(t^rs + ∆x^free_i (t^rs) +

∆ti(t^rs)) given by:

xi(t^rs+ ∆x^free_i (t^rs) + ∆ti(t^rs)) = xi(t^rs) + vi(t^rs)∆t^free_i + ∆xi(t^rs) . (2.22) The target trajectory line is then located at:

x^target_i (t^rs+ ∆t^free_i (t^rs) + ∆ti(t^rs)) = x^target_i (t^rs) + v^eff(∆t^free_i (t^rs) + ∆ti(t^rs)) . (2.23) Solving these two equations for∆t^free_i (t^rs) gives:

∆t^free_i (t^rs) = x_i(t^rs) + ∆x_i(t^rs) − x^target_i (t^rs) − v^eff∆t_i(t^rs)

v^eff − vi(t^rs) . (2.24) During this time, the vehicle has traveled∆x^free_i (t^rs) (km) given by:

∆x^free_i (t^rs) = vi(t^rs)∆t^free_i (t^rs) . (2.25) Vehicles for which it holds that ∆t^free_i (t^rs) ≤ T^rs will have to start decelerating for stabilization during the current sampling time step.

In a real-world implementation it is not necessary (and may not be possible) to test all vehicles upstream of a speed limited vehicle whether it needs to slow down during the current time step as well. However, it is not always sufficient to find the first upstream vehicle i^last that does not have to slow down because there may be another vehicle upstream of it that is traveling faster and needs to slow down earlier than or within the same time step as vehiclei^last.

Consider the worst case where a vehicle is traveling directly upstream of vehiclei with the maximum speedv^max (km/h). The maximum timet^maxthat this vehicle will have to decelerate to reach the effective speed is

t^max= v^eff − v^max

a^T , (2.26)

and the distance traveled during deceleration is

d^max = t^max(v^eff + v^max)

2 . (2.27)

Given this, the time∆t^free,min_i (t^rs) (h) when this vehicle should start to decelerate is given by:

∆t^free,min_i (t^rs) = xi(t^rs) + ∆d^max− x^target_i (t^rs) + d^headway− v^eff∆t^max

v^eff− v^max . (2.28)

If it is the case that ∆t^free,min_i (t^rs) < T^rs the possibility exists that there is a vehi-cle upstream that needs to decelerate earlier than vehivehi-cle i. However, if it holds that

∆t^free_i (t^rs) > T^rs, and∆t^free,min_i (t^rs) > T^rs, this is the last vehiclei^lastthat should be speed-limited for stabilization.

The speed-limits are communicated to the vehicles using the S-tail line L^S−tail(k^rs).

The S-tail line is defined as the set of lines connecting the points defined by the times t^S_i(t^rs) = t^rs+ ∆t^free_i (t^rs) when and locations x^S_i(t^rs) = xi(t^rs) + ∆x^free_i (t^rs) where the vehicles have to start decelerating for alli^us ≤ i ≤ i^last:

L^S−tail(k^rs) = {(t^S_i(t^rs), x^S_i(t^rs))} ∀ i^us ≤ i ≤ i^last. (2.29)