Traffic flow dynamics

This section details the description of the traffic flow dynamics. The main elements of the model used in this paper are links, origins, and nodes as illustrated in Figure 4.1.

The approach also includes the possibility to impose restrictions on the outflow at exits of the network, see e.g. link 2 in Figure 4.1. The description of the link dynamics fol-lows the LTM of Yperman [2007] which is briefly introduced in Section 4.2.2. In con-trast to that model a node model to connect the links is not explicitly included, since, the connection between links is an outcome of the optimization problem. Section 4.2.2 details the formulation of the LTM using linear state equations and constraints.

Brief introduction to the LTM

The LTM describes the link dynamics using two traffic states, namely, the cumulative inflowN_i^L,in(k^c) (veh) and outflow N_i^L,out(k^c) (veh) of every link i (-) in the network.

This is an advantage when compared to approaches that divide a link into segments which require much more traffic states to describe the link dynamics. More important may actually be that the numerical stability of these segment-based schemes requires a small time step due to the CFL condition. Since some segments are small, the simula-tion often needs to run with a small time step. Another advantage of the LTM is that it is capable of modelling all traffic regimes and specifically considers downstream and upstream propagating waves.

Figure 4.2 illustrates the description of the traffic dynamics in the LTM. It is assumed that the free-flow speed v_i^free (km/h) is known and constant, and that a vehicle cannot exit the link before the timet^free_i (h) that it requires to travel through the link with the free-flow speed as illustrated by the trajectory of vehicle 6. Thus, the maximum link outflow depends on the link inflow in the past. In saturated regimes, the link outflow is equal to the saturation rate q^sat_i (veh/h). Finally, in oversaturated regimes the wave

speed v_i^shock (km/h) is included as illustrated by the wave that starts when vehicle 8 exits the link. Note that it takes a time t^shock_i (h) for the upstream propagating wave caused by spill back to travel through the link. Due to this, vehicle 14 can only enter the link a timet^shock_i (h) after vehicle 8 has exited the link. The implies that the maximum link inflow depends on the outflow of the link in the past.

Thus, in order to model the traffic dynamics using the LTM it is required to know the cumulative inflow and outflow in the past. For instance, free-flow dynamics can be modeled by assuring that the cumulative outflow at time t is not larger than the cumulative inflow at time t − t^free_i in the past as illustrated in Figure 4.2. The next subsection will formally describe the traffic flow modelling.

Formulation of the LTM using linear state equations and constraints

The above mentioned dynamics are modelled using linear state update equations of the cumulative curves and linear constraints. In order to realize this, the control variables used are the effective fractions of green timeb^L,eff_i (k^c) (-) used by the links, and the effective fractions of green timeb^O,eff_j (k^c) (-) used by the origin queues. For a link, this is defined as the realized link outflowq^realized_i (k^c) (veh/h) divided by the link saturation flow:

b^L,eff_i (k^c) = q^realized_i (k^c)

q^sat_i . (4.1)

By using the effective fractions it is possible to model the link dynamics using linear equations and include free flow travel times and upstream and downstream propagating waves by adding linear inequality constraints as detailed below. The optimization will take care of matching the outflows and inflows of links that are connected to each other so that the optimization problem is essentially serving as the node model.

The cumulative flow out of linki is updated as follows:

N_i^L,out(k^c+ 1) = N_i^L,out(k^c) + b^L,eff_i (k^c)q_i^satT^c. (4.2) Note that this equation assumes that the link outflow is equal to the saturation rate.

However, the effective fraction of green timeb^L,eff_i (k^c) used enables to limit the outflow when there is no queue. In this way, free-flow dynamics can be modelled using the following constraint:

N_i^L,out(k^c+ 1) ≤ γ_i^c,freeN_i^L,in(k^c− k^c,free_i + 2) + (1 − γ_i^c,free)N_i^L,in(k^c− k_i^c,free+ 1) , (4.3) where the number of time steps k_i^c,free = ⌈t^free_i /T^c⌉ (-), and the fraction γ_i^c,free = k^c,free_i − t^free_i /T^c (-) the residual of a sampling time step that the free-flow travel time t^free_i (h) is exceeded byk_i^c,free. The mathematical operator⌈·⌉ rounds the argument of the function to the nearest integer that it higher than the argument of the function. The

interpretation of this constraint is that the cumulative outflow curve should always lie below the cumulative inflow curve shifted with the free-flow travel time as illustrated by the dashed line named ‘Maximum Outflow’ in Figure 4.2. Note that k^c,free_i ≥ 2 to guarantee CFL conditions. In the case that link i is at an exit of the network, a constraint is introduced to limit the maximum outflowq^out,max_i (k^c) (veh/h) out of that link:

b^L,eff_i (k^c)q^sat_i ≤ q_i^out,max(k^c) ∀i ∈ I^Exit, (4.4) where the set I^Exit is the set of exit links. This maximum outflow is modeled as an external disturbance to the process so that, for instance, the impact of a (temporal) bottleneck on the traffic flow at an exit of the network can be included.

The cumulative inflow to linki is updated using:

N_i^L,in(k^c+ 1) = N_i^L,in(k^c)+ X

i^us∈I_iⁱⁿ



η_i^us_,i(k^c)b^L,eff_i (k^c)q_i^satT^c



+ . . . (4.5)

X

j∈J_iⁱⁿ



ηj,i(k^c)b^O,eff_j (k^c)q_j^capT^c

 ,

where the setI_iⁱⁿis the set of links directly upstream of linki and the set J_iⁱⁿis the set of origins directly upstream of linki. The fraction ηi^us,i(k^c) indicates the turn fraction form linki^usto linki, and the fraction ηj,i(k^c) (-) indicates the turn fraction form origin j to link i. In order to model upstream propagating waves, the following constraint is used:

N_i^L,in(k^c+ 1) ≤ γ_i^c,shockN_i^L,out(k^c− k^shock_i + 2) + . . . (4.6) (1 − γ_i^c,shock)N_i^L,out(k^c− k^shock_i + 1) + N_i^max

with the number of vehiclesN_i^max(veh) the maximum number of vehicles that can fit in a link – i.e., the link length multiplied with the jam density – the number of time stepsk^c,shock_i = ⌈t^shock_i /T^c⌉ (-) , and the fraction γ_i^c,shock = k^c,shock_i − t^shock_i /T^c(-) the residual of a sampling time-step that the upstream propagating wave travel timet^shock_i (h) is exceeded by k^c,shock_i . It should hold thatk^c,shock_i ≥ 2 in order to guarantee CFL conditions. This constraint limits the inflow as indicated with the dashed line named

‘Maximum Inflow’ in Figure 4.2.

Origins are modelled as vertical queues as illustrated in Figure 4.1. The cumulative inflowN_j^O,in(k^c) to origin j is updated as follows:

N_j^O,in(k^c+ 1) = N_j^O,in(k^c) + q_jⁱⁿ(k^c)T^c. (4.7) The cumulative outflowN_j^O,out(k^c+ 1) out of origin j is updated using:

N_j^O,out(k^c+ 1) = N_j^O,out(k^c) + b^O,eff_j (k^c)q_j^capT^c, (4.8)

which should satisfy:

N_j^O,out(k^c+ 1) ≤ N_j^O,in(k^c+ 1) . (4.9)

Apart from these dynamical update equations and constraints, constraints on the con-trol signals should be added:

0 ≤ b^L,eff_i (k^c) ≤ 1 , (4.10) 0 ≤ b^O,eff_j (k^c) ≤ 1 , (4.11)

X

i∈I_y^conflict

b^L,eff_i (k^c) ≤ 1 , (4.12)

where the setI_y^conflictis the sety of signals which are in conflict with each other. The first two constraints make sure that the effective fractions are bounded between0 and 1 while the third constraint makes sure that the green time is distributed over conflicting links. Note that clearance times between conflicts may be modeled by limiting the sum of the green-fractions of the conflicting links to be less than 1. These constraints essentially serve as the node model. To see this, note that in the original LTM model of Yperman [2007] the green times that are given may result in a violation of constraints (4.3) and (4.6). Hence, a node model is required to determine the transition flows from one link to another so that the constraints are not violated. In the approach proposed in this paper, the model is re-written as an optimization model where the effective fractions of green time are optimized so that the constraints (4.3) and (4.6) cannot be violated. Note that this can be seen as a modification of the generic class of first order node models proposed by Tamp`ere et al. [2011] where instead of maximizing the node outflows, the total network outflows are maximized subject to supply and demand constraints of the nodes.

The statex^c,L_i (k^c) ∈ R^{nL,si ,1}of linki is given as:

x^c,L_i (k^c) = . . . (4.13)

h

N_i^L,out(k^c) . . . N_i^L,out(k^c− k_i^c,shock) N_i^L,in(k^c) . . . N_i^L,in(k^c− k_i^c,free)i^⊤ where n^L,s_i = k^c,shock_i + k_i^c,free + 2 is the length of the vector. Similarly, the state x^c,O_j (k^c) ∈ R^{nO,sj ,1}of an origin has the following structure:

x^c,O_j (k^c) =N_j^O,out(k^c) N_j^O,in(k^c)^⊤

, (4.14)

wheren^O,s_j = 2 is the length of the vector.