Network coordination layer: LML-U approach

The task of the network coordination layer – i.e., the top layer of the proposed frame-work – is to determine the reference outflows that optimize the netframe-work throughput.

Recall that the coordination layer sampling timeT^ref (s) is in the range of one to sev-eral minutes. Hence, in order to satisfy real-time feasibility, the coordination layer has to be able to compute the reference outflow trajectories within one to several minutes.

To this end, the recently developed linear model predictive control strategy using the link transmission model for urban traffic networks (LML-U) is chosen in the coordina-tion layer [van de Weg et al., 2016]. This approach has the advantage that it considers all relevant first-order traffic dynamics – i.e., upstream and downstream propagating waves – using only two traffic states. Compared to segment-based models, such as the CTM, this is more efficient from a computational point of view. The approach requires a prediction of the traffic demand, turn-fractions, and maximum network outflows. Its

output consists of the optimized fractions of green time used by the traffic streams in the network. The remainder of this section first discusses the prediction model used in more detail, next the optimization problem is introduced, and finally the approach to compute the reference outflow trajectories from the optimization output is presented.

The prediction model

The prediction model used in the LML-U control strategy is the LTM. The main ele-ments used here are links – indicated with indexi^L (-) – and origins – indicated with indexi^O (-). The traffic dynamics of origins and links are updated using two traffic states; the cumulative link inflowN_iⁱⁿL(k^c) (veh) and outflow N_i^outL (k^c) (veh), and the cumulative origin inflowN_i^O,inO (k^c) (veh) origin outflow N_i^O,inO (k^c) (veh). Every out-flow is controlled using a control parameterb^eff_iL(k^c) for links and b^eff,O_iO (k^c) for origins that expresses the effective fraction of green time used during the time stepk^c. Note that this optimization approach is presented in more detail in van de Weg et al. [2016].

The interested reader is referred to [Yperman, 2007] for a more detailed description of the LTM.

The cumulative link outflow is updated using the following equation:

N_i^outL (k^c+ 1) = N_i^outL (k^c) + q_i^satL T^cb^eff_iL(k^c) , (6.1) whereq_i^satL (veh/h) is the saturation rate. The cumulative link inflow is modeled as the sum of the outflows of upstream linksj^L ∈ I_i^L,usL and originsi^O ∈ I_i^O,usL multiplied with the turn-fractionsη_j^L_,i^L(k) given as:

N_iⁱⁿL(k^c+ 1) = N_iⁱⁿL(k^c)+ X

j^L∈I^L,us

iL



η_j^L_,i^L(k^c)b^eff_iL(k^c)q^sat_iL T^c



+ . . . (6.2)

X

i^O∈I^O,us

iL



η_i^O_,i^L(k^c)b^eff,O_iO (k^c)q_i^capO T^c

 ,

where the set I_i^L,usL is the set of links directly upstream of link i^L and the set I_i^O,usL is the set of origins directly upstream of linki^L. The fractionηj^L,i^L(k^c) indicates the turn fraction form linkj^Lto linki^L, and the fractionη_i^O_,i^L(k^c) (-) indicates the turn fraction form origini^O to linki^L.

In order to model free-flow dynamics, the cumulative link outflow is bound from above, so that vehicles cannot travel through the link faster than the free flow travel timet^free_iL

(s). This can be written as a constraint on the cumulative outflow given as:

N_i^outL (k^c+ 1) ≤ γ^c,free_iL N_iⁱⁿL(k^c− k^c,free_iL + 2) + (1 − γ_i^c,freeL )N_iⁱⁿL(k^c− k_i^c,freeL + 1) . (6.3) In (6.3) the number of time steps k^c,free_iL = ⌈t^free_iL /T^c⌉ (-), and the fraction γ_i^c,freeL = k^c,free_iL − t^free_iL /T^c(-) are used to linearly interpolate the cumulative curve as detailed in

[van de Weg et al., 2016]. The mathematical operator⌈·⌉ rounds the argument of the function to the nearest integer that it higher than the argument of the function. In order to satisfy CFL conditions it should hold thatk^c,free_iL ≥ 2.

Similarly, upstream propagating waves caused by spillback are included by bounding the cumulative link inflow from above so that a vehicle can only enter a linkt^shock_iL (s) seconds after the vehiclen^max_iL (veh) has exited the link given as:

N_i^inL(k^c+ 1) ≤ γ^c,shock_iL N_i^outL (k^c− k_i^c,shockL + 2) + . . . (6.4) (1 − γ_i^c,shockL )N_i^outL (k^c− k^c,shock_iL + 1) + n^max_i^L ,

with the number of time steps k_i^c,shockL = ⌈t^shock_iL /T^c⌉ (-), and the fraction γ_i^c,shockL = k_i^c,shockL − t^shock_iL /T^c (-). It should hold that k_i^c,shockL ≥ 2 in order to guarantee CFL conditions.

Outflow limitations at the network are modeled as external disturbances – i.e., inputs that cannot be affected by the control signal. So, when a link is at an exit of the network, an extra constraint is added:

N_i^outL (k^c+ 1) ≤ N_i^outL (k^c) + q_i^out,maxL (k^c)T^c, (6.5) whereq_i^out,maxL (k^c) (veh/h) is the maximum outflow that can exit the link at time step k^c.

Origins are modeled as vertical queues via the following state update equations and constraints:

N_i^O,inO (k^c+ 1) = N_i^O,inO (k^c) + dⁱⁿ_iO(k^c)T^c, (6.6) N_i^O,outO (k^c+ 1) = N_i^O,outO (k^c) + q^cap_iO T^cb^eff,O_iO (k^c) , (6.7)

N_i^O,outO (k^c+ 1) ≤ N_i^O,inO (k^c+ 1) . (6.8)

withq_i^capO (veh/h) the origin capacity.

The final constraints concern the effective fractionsb^eff_iL(k^c) and b^eff,O_iO (k^c) of green-time which should be between 0 and 1. Additionally, if there is a conflict between links at an intersection – i.e., {j^L, i^L} ∈ I_i^conflictcon – the sum of the effective green fractions b^eff_iL(k^c) + b^eff_jL(k^c) should be less than 1 − θi^con. The tuning parameterθi^con (-) is used to prevent infeasible reference outflows which can occur when a clearance time has to be respected when switching linki^L toj^L. This results in the following constraints:

0 ≤ b^eff_iL(k^c) ≤ 1 , (6.9)

0 ≤ b^eff,O_iO (k^c) ≤ 1 , (6.10)

0 ≤ b^eff_iL(k^c) + b^eff_jL(k^c) ≤ 1 − θi^con. (6.11)

The optimization problem

The objective of the linear optimization problem is to minimize the total time spent (TTS)J^TTS (veh·h) used by all the vehicles in the network over a prediction horizon

N^p (-) subject to the linear model and constraints presented in the previous section.

The TTS can be expressed as the total number of vehicles in the network at every time step k^c multiplied with the sampling time T^c and summed over the time steps k^c = (k^ref − 1)ǫ^c,ref+ 1, . . . , (k^ref − 1)ǫ^c,ref + N^p+ 1 given as:

Here,I^L(-) represents the set of all links andI^O(-) represents the set of all origins.

As in [van de Weg et al., 2016], minimizing the TTS can be written as the following linear optimization problem:

¯min

u(k^ref)Z ˜B ¯u(k^ref) + Z( ˜Ax(k^ref) + ˜C ¯d(k^ref)) , (6.13) Subject toM^inequ(k¯ ^ref) ≤ V^ineq,

Here, the matrices ˜A, ˜B, and ˜C as detailed in [van de Weg et al., 2016] describe the traffic dynamics, so that a prediction of the traffic statex(k¯ ^ref), as defined by equations 6.1, 6.2, 6.6, and 6.7, can be computed by multiplication of the control vectoru(k¯ ^ref) by ˜B, the initial traffic state x(k^ref) by ˜A, and a prediction of the disturbances ¯d(k^ref) – i.e., inputs that cannot be controlled – by ˜C. The matrix M^ineq and vectorV^ineq as detailed in [van de Weg et al., 2016] contain the inequality constraints of equations 6.3, 6.4, 6.5, 6.8, 6.9, 6.10, and 6.11. Multiplication of the vectorZ by the predicted state gives the TTS.

The disturbance vector ¯d(k^ref) contains the traffic demands d(k^c) at time steps k^c = (k^ref − 1)ǫ^c,ref+ 1, . . . , (k^ref − 1)ǫ^c,ref+ N^p:

The control vectoru(k^c) and disturbance vector d(k^c) at a time step k^care defined as follows:

wheren^L(-) indicates the number of links andn^O(-) the number of origins.

The reference trajectory

The outcome of the optimization problem (6.13) is the vector u¯^∗(k^ref) (-). As noted before, this signal cannot be directly applied to the local intersection controllers due to the aggregated nature of the traffic flow model that is used to formulate the linear optimization problem. Instead, a reference trajectory is derived from the optimized signalu¯^∗(k^ref).

A prediction of the traffic statesx(k¯ ^ref) can be obtained as follows:

¯

x(k^ref) = ˜Ax(k^ref) + ˜B ¯u^∗(k^ref) + ˜C ¯d(k^ref) . (6.18) The prediction of the state x(k¯ ^ref) consists of the traffic states x(k^c) at time steps k^c= (k^ref − 1)ǫ^c,ref+ 2, . . . , (k^ref − 1)ǫ^c,ref + N^p is given as:

Now, a reference cumulative outflow trajectoryN_i^out,refL (k^c), defined as:

N_i^out,refL (k^ref) =

can be derived fromx(k¯ ^c) for every link i^L ∈ I^controlled for all the time steps where k^c= (k^ref − 1)ǫ^c,ref+ 1, . . . , (k^ref − 1)ǫ^c,ref + N^p.

Since the sampling time of the prediction model is a multiple of the measurements sampling time – i.e T^c = ǫ^cT –, the signal N_i^out,refL (k^ref) has to be resampled. The