Freeway traffic control considering capacity drop phenomena: comparison of different MPC schemes

Freeway traffic control considering capacity drop phenomena: comparison of different MPC schemes L. Maggi, S. Sacone, S. Siri Abstract— This paper proposes different MPC-based traffic controllers in order to reduce congestion in freeway systems via ramp metering. These controllers differ for the adopted prediction model and for the considered cost function to be minimized. In particular, both a standard CTM and a CTM modified version representing the capacity drop phenomenon are used, while the two different cost functions considered penalize congested states in different ways. These MPC controllers are compared via simulation, both evaluating the performances of the controlled freeway system in the different cases, and from a computational point of view.

I. I NTRODUCTION Ramp metering is one of the most widespread and successful traffic control strategies adopted to reduce recurrent and nonrecurrent congestion phenomena in freeways. Since the first control strategies developed in the Nineties, e.g. the local feedback controller ALINEA [1], researchers have studied and developed more and more sophisticated control schemes, often based on Model Predictive Control (MPC) techniques. As known, the application of MPC presents many advantages, such as the prediction capability, the use of optimization, and the compliance with different constraints, but it entails the important disadvantage related to the computational load. These aspects directly affect the decision about the traffic model to be used for the prediction in the Finite-Horizon Optimal Control Problem (FHOCP) to be solved. Indeed, the higher is the accuracy of the model, the better is the prediction, but the more difficult is the FHOCP to be solved. In the literature, some MPC schemes for freeway traffic control are based on the second-order traffic flow model METANET (see e.g. [2], [3]). In these cases, the FHOCP is nonlinear, computationally intensive and the solutions obtained are generally local optima. In other cases, the simpler first-order Cell Transmission Model (CTM) is used, and the resulting FHOCP is either properly relaxed [4] or rewritten in a mixed-integer linear form [5], [6], so that efficient linear solvers can be used. One of the major drawbacks associated with the use of the CTM for prediction in MPC schemes is that this model is not always accurate in representing the dynamic behavior of freeway systems. In particular, as widely discussed in the literature, the standard CTM, developed in [7], is not able to incorporate capacity drop. Capacity drop phenomena are directly related to bottlenecks. A bottleneck is commonly defined as a point in a freeway stretch which presents a reduction in capacity. The authors are with the Department of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Italy.

According to [8], bottlenecks have been classified into different categories (i.e. weaving sections, bridges or tunnels, horizontal curves, uphill grade stretches and lane drops) and may show two different states, active or inactive. An active bottleneck has free-flow conditions downstream and congested conditions upstream and its performance is not affected by any bottlenecks occurring downstream [9], [10], [11]. When a bottleneck activates, a capacity drop occurs: even though the demand upstream the bottleneck is higher than the capacity, the flow exiting the bottleneck is not the maximum supported capacity but is lower. In the literature, different methodologies to estimate the capacity drop have been developed (see e.g. [8], [9], [12], [13]. In this paper, an MPC scheme to control freeway systems via ramp metering is proposed, using the CTM for the prediction. As aforementioned, the standard CTM does not model capacity drop phenomena, but some efforts have been made to include these phenomena in first-order traffic models (see e.g. [14], [15], [16], [9], [17]). In particular, in this paper the extended version of the CTM proposed in [9] is used as simulation model. In this model the fundamental diagram is represented as a 5-step piecewise linear function, in which two values of capacity are explicitly considered. In the present paper, different FHOCPs are proposed and compared, using for the prediction either the standard CTM or the CTM including the capacity drop proposed in [17]. In this latter model, the capacity drop is included by defining slight changes compared with the original model. Specifically, when the density is over the critical value, the maximum flow requiring to exit a cell decreases proportionally with the density. The choice of using, in the FHOCP, the CTM modified according to [17], instead of the one proposed in [9], is motivated by computational issues. As it will be better clarified in the next sections, in the FHOCP the CTM is rewritten in mixed-integer linear form, and the CTM version provided by [9] is too complicated and would generate a very high number of binary variables, hence decreasing the computational efficiency of the MPC scheme. The idea of adopting the CTM in MPC schemes taking into account capacity drops has been also exploited in [18], where, differently from the present paper, the Link-Node CTM is considered and the resulting FHOCP is non-convex. This paper is organized as follows. In Section II the simulation model used to represent the dynamic evolution of the freeway is described. The MPC scheme adopted to control the freeway via ramp metering is described in Section III, by presenting four different FHOCPs to be compared via simulation. The simulation analysis is reported in Section IV,

whereas some conclusions are drawn in Section V. II. T HE SIMULATION

MODEL

In order to represent the dynamic evolution of the freeway system taking into account also capacity drop phenomena, the modified version of the Cell Transmission Model proposed in [9] is adopted. In this model the fundamental diagram in a given location, i.e. the steady-state relation between traffic flow q and density ρ, is assumed to be a 5-step piecewise linear function, which can be written as follows  vρ if 0 ≤ ρ ≤ ρa ∧ σ = 0    ′   κ + v ρ if ρa ≤ ρ ≤ ρb ∧ σ = 0 if ρb ≤ ρ ≤ ρc ∧ σ = 0 (1) q = Q(ρ) = F H  L b d  F if ρ ≤ ρ ≤ ρ ∧ σ = 1    w(¯ ρ − ρ) if ρd ≤ ρ ≤ ρ¯ ∧ σ = 1 A representation of the piecewise linear relation (1) is given in Fig. 1. Each block of this function is defined by the density boundaries ρa , ρb , ρc , ρd , the jam density ρ¯ and the congestion state σ ∈ {0, 1}. This latter is a binary quantity, equal to 0 when the state is uncongested and equal to 1 when it is congested. Note that the density boundaries must verify 0 < ρa < ρb < ρc < ρd < ρ¯. q FH FL

ρa ρb

ρc ρd

ρ ρ¯

which it is assumed that a different fundamental diagram is defined for each cell of the freeway stretch. Let N be the number of cells, with Li indicating the length of cell i = 1, . . . , N , K the number of time steps, with T representing the sample time. For each cell i = 1, . . . , N , and for each time step k = 0, . . . K − 1, the following quantities are considered: • ρi (k) traffic density of cell i [veh/km]; • li (k) queue length in the on-ramp of cell i [veh]; • φi (k) mainstream flow entering cell i from cell i − 1 [veh/h]; • ri (k) flow entering cell i from the on-ramp [veh/h]; • si (k) flow exiting cell i through the off-ramp [veh/h]; • di (k) on-ramp demand referred to cell i [veh/h]; • Di (k) demand of cell i [veh/h]; • Si (k) supply of cell i [veh/h]; • σi (k) ∈ {0, 1} congestion state of cell i. The model parameters, referred to cell i, i = 1, . . . , N , are the split ratio βi ∈ [0, 1), the priority of on-ramp flow with respect to mainstream flow pi ∈ [0, 1], and all the parameters of the 5-step piecewise linear fundamental diagram, i.e. the free-flow speed vi [km/h], the under-saturated speed vi′ [km/h], the constant κi [veh/h], the high and low capacity values FiH and FiL [veh/h], the congestion wave speed wi [km/h], the jam density ρ¯i [veh/km], the density boundaries ρai , ρbi , ρci , ρdi [veh/km]. The dynamic model is given by the following state equations for the traffic density and the queue length:   T ρi (k+1) = ρi (k)+ φi (k)+ri (k)−φi+1 (k)−si (k) (2) Li   li (k + 1) = li (k) + T di (k) − ri (k)) (3)

where the flow exiting through the off-ramp is computed as βi si (k) = 1−β φi+1 (k), i = 1, . . . , N , k = 0, . . . , K − 1. i Fig. 1: 5-step piecewise linear fundamental diagram. Taking into account (1) and Fig. 1, it is possible to split the graph into two parts: the left part of the graph (from The first two blocks represent the uncongested phase of the first to the third block) is related to the demand function, traffic. The first block is for light conditions in which vehicles while the right part (forth and fifth blocks) is associated with move at free-flow speed v. The second block represents the the supply function. Specifically, referring to time step k = under-saturated state of traffic, in which the interactions 0, . . . K − 1, the demand of cell i − 1 and the supply of cell among vehicles decrease the average speed (that is, v ′ < v). i, i = 1, . . . , N , are respectively defined as The third and fourth blocks represent, respectively, the pre congestion and post-congestion situations. Indeed, there is a Di−1 (k) = min (1 − βi−1 )vi−1 ρi−1 (k),   H time interval in which, despite the high density, the freeway ′ (4) (1 − βi−1 ) κi−1 + vi−1 ρi−1 (k) , Fi−1 works at the maximum capacity F H , but after that time a   breakdown occurs and the capacity decreases to a lower value min wi (¯ ρi − ρi (k)), FiH if σi (k − 1) = 0 (5) Si (k) = F L . Finally, the fifth block represents the behavior in the min wi (¯ ρi − ρi (k)), FiL if σi (k − 1) = 1 congested phase, therefore the slope is assumed to be equal The congested state variable σi (k) indicates if the state to the congestion wave speed with a change in the sign, i.e. of cell i = 1, . . . , N at time step k = 0, . . . , K − 1 is −w. Moreover, if the density value is equal to the maximum uncongested or congested and is given by: value ρ¯, the flow is equal to zero.

 According to this 5-step piecewise linear fundamental 1 if ρi (k) ≥ ρci ∨ ρi (k) ≥ ρbi ∧ σi (k − 1) = 1 σ (k) = i diagram, the standard CTM has been modified, by changing 0 otherwise (6) the demand and supply functions, as well as by introducing a relation to update the value of the congestion state variable According to [9], in the 5-step piecewise linear fundamen[9] . Let us describe in the following this modified CTM in tal diagram, there is not one value for the critical density,

but two values, i.e. ρbi and ρci . Following the standard CTM merge connection model, the mainstream and on-ramp flows are obtained considering two alternative cases. The first case regards a situation in which there is enough space for vehicles that want to enter cell i: li (k) If Di−1 (k) + di (k) + ≤ Si (k) T (7) li (k) then φi (k) = Di−1 (k) ri (k) = di (k) + T The second case is the opposite situation in which not all the vehicles that want to enter cell i can enter: li (k) ≥ Si (k) If Di−1 (k) + di (k) + T then  li (k) , (1 − pi )Si (k) φi (k) = mid Di−1 (k), Si (k) − di (k) − T  li (k) ri (k) = mid di (k) + , Si (k) − Di−1 (k), pi Si (k) T (8) III. T HE MPC SCHEME To control the freeway with ramp metering, an MPC scheme is proposed in four different versions. These versions differ for the considered FHOCPs in which two different prediction models and two different cost functions are adopted. A. The considered prediction models The first prediction model considered in the FHOCP is the standard CTM. In this model, referring to time step k = 0, . . . K − 1, the demand of cell i − 1 and the supply of cell i, i = 1, . . . , N , are defined as  Di−1 (k) = min (1 − βi−1 )vi−1 ρi−1 (k), Fi−1 (9)  (10) Si (k) = min wi (¯ ρi − ρi (k)), Fi

where only one value for the capacity is considered, i.e. Fi , i = 1, . . . , N . Note that (9) and (10) do not model the capacity drop. When the density is higher than a given value, the demand function is assumed to be constant. Such value of density is the critical density ρcr i . The standard CTM is then given by relations (2)-(3), (7)-(8), (9)-(10). The second prediction model to be adopted is the CTM including capacity drop phenomena proposed in [17], in the following referred to as modified CTM. In this model, the drop is modeled by simply modifying the demand function, so that in case of congestion the demand function is linearly decreasing. More specifically, referring to time step k = 0, . . . K − 1, the demand of cell i − 1 and the supply of cell i, i = 1, . . . , N , are given by  Di−1 (k) = min (1 − βi−1 )vi−1 ρi−1 (k), ′ Fi−1 + wi−1 (ρcr (11) i−1 − ρi−1 (k))  Si (k) = min wi (¯ ρi − ρi (k)), Fi (12)

where wi′ is the decreasing capacity rate due to the capacity drop phenomenon referred to cell i, while ρcr i is the critical

density causing a breakdown in capacity. The modified CTM is then given by relations (2)-(3), (7)-(8), (11)-(12). The two prediction models are nonlinear due to the logic relations and the mid function in (7) and (8), as well as to the min functions in the demand and supply definitions. In order to avoid these nonlinearities, both models are properly rewritten in Mixed Logical Dynamical (MLD) form [19], by introducing some sets of inequalities and auxiliary variables. For space limitations, we do not report all the inequalities and the auxiliary variables of the MLD formulation (for more details on the CTM in MLD form refer to [5]). B. The considered cost functions The first cost function to be used in the FHOCP solved at time step k over a prediction horizon Kp is J1 (k) =

k+Kp −1 N  X X h=k

γδ (1 −

i=1

δiM (h))

 + γl li (h)

(13)

where γδ and γl are weighing coefficients and δiM (h), i = 1, . . . , N , h = k, . . . , k + Kp − 1, are auxiliary binary defined in the MLD formulation of the CTM. In particular, these variables have been introduced with the following meaning: [δiM (k) = 1] iff [Di−1 (k) + ri (k) ≤ Si (k)]. Then, cost function J1 (k) penalizes queue lengths and the cases in which δiM (h) = 0, i.e. when Di−1 (k) + ri (k) > Si (k), which correspond to congested situations. The second cost function, again to be adopted in the FHOCP solved at time step k over a prediction horizon Kp , is given by J2 (k) =

k+Kp −1 N  X X h=k

i=1

 γρ ρ˜i (h) + γl li (h)

(14)

where γρ and γl are weighing coefficients and ρ˜i (h), i = 1, . . . , N , h = k, . . . , k+Kp −1, is a set of auxiliary variables defined as follows:  ρi (h) − ρ∗i if ρi (h) ≥ ρ∗i ρ˜i (h) = (15) 0 otherwise where ρ∗i is a set-point value for the density in cell i = 1, . . . , N . Cost function J2 (k) penalizes queue lengths, analogously to J1 (k), and a term which is equal to 0, if ρi (h) < ρ∗i , and is equal to the difference between the actual density and the set-point value, otherwise. In this way, differently from J1 (k), densities much higher than the setpoint are penalized more than densities slightly higher than the set-point. In other words, the seriousness of congestion situations is here explicitly taken into account, not only their occurrence. C. The FHOCP The four different FHOCPs are generated by the combination of the two aforementioned models and the two cost functions. In general, the FHOCP problem to be solved at time step k can be stated as the following mixed-integer linear mathematical programming model.

Problem 1: Given the initial conditions on the density and the queue length ρi (k) and li (k), i = 1, . . . , N , the demand of the cell before the first one D0 (h), h = k, . . . , k + Kp − 1, the supply of the cell after the last one SN +1 (h), h = k, . . . , k+Kp −1, the on-ramp demands di (h), i = 1, . . . , N , h = k, . . . , k + Kp − 1, find the optimal control variables ri (h), i = 1, . . . , N , h = k, . . . , k + Kp − 1, minimizing

density evolution in the no-control case for Dataset 1.2 (a similar behavior is obtained with Dataset 1.1, not reported in the paper for space limitations). The queues at the onramps are null in the no-control case for both datasets. No-control case

8

250

7 200

J2 (k)

(16)

6

Cells

J1 (k) or

subject to the CTM (standard or modified) in MLD form.

150

5

4

IV. S IMULATION

AND COMPUTATIONAL ANALYSIS

100

3 50

This section presents both a performance analysis and a computational evaluation of the proposed MPC schemes.

2

1

0

20

40

60

80

100

120

140

160

180

0

Time steps

Fig. 2: Density evolution in the no-control case (Dataset 1.2).

Standard CTM-J1 (k)

8

Standard CTM-J2 (k)

8

250

7

250

7 200

200

6 150

5 4

Cells

Cells

6

100

3

150

5 4

100

3 50

50

2 1

2

0

30

60

90

120

Time steps

150

180

Modified CTM-J1 (k)

8

1

0

0

30

60

90

120

Time steps

151

0

180

Modified CTM-J2 (k)

8

250

7

250

7 200

200

6 150

5 4

Cells

Cells

6

100

3

150

5 4

100

3 50

50

2 1

2

0

30

60

90

120

Time steps

150

180

1

0

0

30

60

90

120

Time steps

150

0

180

Fig. 3: Density in the controlled cases (Dataset 1.2).

On-ramp 3

200

Queue length [veh]

The proposed MPC schemes have been tested trough simulation in order to evaluate their effectiveness in comparison with the no-control case. To do that, some performance indexes need to be introduced. Let us denote with Jinc , i = 1, 2, the cost computed in the no-control case using the FHOCP cost function Ji (k) (i.e. (13) and (14)) over the whole simulation period. Analogously, let Jic , i = 1, 2, indicate the cost computed in the controlled cases using the FHOCP cost function Ji (k) over the whole simulation period. The first index to be adopted is the cost reduction of the controlled case compared with the J nc −J c no-control case, i.e. ∆Ji = i J nc i , i = 1, 2. i Analogously, it is possible to consider, as commonly done in the literature, the Total Time Spent (TTS) by vehicles in the system accounting for both travel time and time in the on-ramp queues. The TTS is computed both when no control is applied (denoted as T T S nc) and in the controlled case (denoted as T T S c). The second performance index is the TTS reduction of the controlled system compared with nc T Sc the no-control case, i.e. ∆T T S = T T ST T −T . nc S The three-lane freeway stretch considered for the simulation tests is characterized by N = 8 cells and two on-ramps in cells i = 3, 6. The following parameters have been used: K = 180, T = 20/3600 [h], Li = 0.7 [km], ρ¯i = 400 [veh/km], vi = 105 [km/h], wi = 35 [km/h], Fi = 8000 [veh/h], pi = 0.4, βi = 0.05, wi′ = 5 [km/h], vi′ = 65 [km/h], κ = 2500 [veh/h], FiH = 8000 [veh/h], FiL = 7000 [veh/h], Fi = 8000 [veh/h], i = 1, . . . , 8. The initial conditions have been set equal to 80 [veh/km] for the traffic density in all the cells and equal to 0 [veh] for all queue lengths. The boundary conditions have been fixed as follows: D0 (k) = 5000 [veh/h] and S9 (k) = 8000 [veh/h], k = 0, . . . , 179. Two different datasets have been chosen, which differ for the demands at the on-ramps (Dataset 1.1 creates lighter conditions of traffic than Dataset 1.2), i.e. • Dataset 1.1: di (k) = 1800 [veh/h], k = 0, . . . , 120 and di (k) = 700 [veh/h], k = 121, . . . , 179, i = 3, 6; • Dataset 1.2: di (k) = 2000 [veh/h], k = 0, . . . , 107 and di (k) = 700 [veh/h], k = 108, . . . , 179, i = 3, 6. In case no control is applied, cell 6 quickly becomes an active bottleneck causing heavy congestion. Fig. 2 shows the

Standard CTM-J1 (k) Standard CTM-J2 (k)

150

Modified CTM-J1 (k) Modified CTM-J2 (k)

100

50

0

0

20

40

60

80

100

120

140

160

180

Time steps

On-ramp 6

200

Queue length [veh]

A. Performance analysis

Standard CTM-J1 (k) Standard CTM-J2 (k) Modified CTM-J1 (k) Modified CTM-J2 (k)

150

100

50

0

0

20

40

60

80

100

120

140

160

180

Time steps

Fig. 4: On-ramp queue in the controlled cases (Dataset 1.2). The MPC schemes have been applied considering the following parameters: γδ = 50, γρ = 1, γl = 1 and ρ∗i = 95 [veh/km], i = 1, . . . , 8, Kp = 10. Analyzing the evolution of the traffic densities and the queue lengths in the four controlled cases (reported respectively in Fig. 3 and in

TTS

∆T T S

13641.00 10283.18 14206.00 10729.08

16.76% 45.73% 13.31% 43.38%

450.70 440.46 458.23 443.39

4.80% 6.96% 3.21% 6.34%

24679.00 25436.83 25355.00 25805.36

4.49% 34.21% 1.87% 33.26%

525.10 528.99 529.04 530.66

9.52% 8.85% 8.84% 8.56%

TABLE I: Controlled cases - Performance indexes Table I shows the performance indicators introduced before referred to the four controlled cases. Note that, in the columns labeled with Jic and ∆Ji , J1c and ∆J1 are reported in case J1 (k) is considered, whereas J2c and ∆J2 are reported otherwise. The results are generally satisfying. All the analyzed controlled cases show an improvement in the performance compared with the no-control case. While comparing the different versions of MPC schemes proposed, it seems that using the modified CTM does not imply any advantages in performance compared with the use of the standard CTM. Moreover, the use of J1 (k) or J2 (k) provides different behaviours of the controlled system. In particular, referring to Dataset 1.1, ∆T T S is always remarkably higher in case J2 (k) is chosen. When considering Dataset 1.2, the differences between the use of J1 (k) and J2 (k) are slighter, but with an opposite trend, i.e. higher Total Time Spent reduction in case J1 (k) is used. B. Computational analysis Due to the real-time application logic of MPC schemes, each FHOCP has to be solved in a very short time. From this point of view, an analysis devoted to investigate the computational times of the four FHOCPs in different situations has been conducted. Specifically, the computational load is investigated by varying the size of the problem to be solved and the traffic congestion level considered. First of all, 9 groups of instances are introduced, by varying the finite horizon length Kp from 10 to 20 time steps, and the number of cells N from 8 to 12, as indicated in Table II. Group Kp N

1 10 8

2 10 10

3 10 12

4 15 8

5 15 10

6 15 12

7 20 8

8 20 10

9 20 12

TABLE II: The 9 selected groups of instances Then, 3 different datasets are considered corresponding to three different levels of congestion in the freeway (Dataset

Dataset 2.1 2.2 2.3

ρ0 (h) [75-95] [85-105] [95-115]

D0 (h) [4000-5000] [5000-6000] [6000-7000]

SN + 1(h) [7000-8000] [7000-8000] [7000-8000]

di (h) [1000-2000] [2000-3000] [3000-4000]

TABLE III: The 3 sets of data Five instances have been randomly generated for each group and for each dataset. Then, for each random instance the four FHOCPs have been solved with Cplex 12.5 solver on a pc Intel(R) Core(TM) i5 CPU M460 @ 2.53 GHz with an installed RAM of 4.00 GB. A time limit of 60 seconds have been set to the solver. 60 50

Standard CTM-J1 (k) N =8 N =10 N =12

40 30 20 10 0

60 50

10

15

Kp

Modified CTM-J1 (k) N =8 N =10 N =12

30 20 10

10

15

Kp

20

50

Standard CTM-J2 (k) N =8 N =10 N =12

40 30 20 10 0

20

40

0

60

Average CPU time [s]

∆Ji

60

Average CPU time [s]

Jic

Average CPU time [s]

Dataset 1.1 Standard CTM-J1 (k) Standard CTM-J2 (k) Modified CTM-J1 (k) Modified CTM-J2 (k) Dataset 1.2 Standard CTM-J1 (k) Standard CTM-J2 (k) Modified CTM-J1 (k) Modified CTM-J2 (k)

2.1 shows the lightest condition of traffic, while Dataset 2.3 is the most congested one). In particular, the values of the initial density ρi (0), i = 1, . . . , 8, the demand from the cell before the first one D0 (k), k = 0, 179, the supply in the cell after the last one SN +1 (k), k = 0, 179, and the on-ramp demands di (k), i = 3, 6, k = 0, 179, have been generated from the uniform distribution in the intervals defined in Table III. All the other parameters have the same values used for the performance analysis described in Section IV-A.

Average CPU time [s]

Fig. 4 for Dataset 1.2), it can be noted that the adoption of cost function J1 (k) or J2 (k) in the FHOCP, almost independently of the prediction model used, implies rather different behaviors of the controlled system. In particular: when using J1 (k), the congestion is only partially avoided, so that it does not spread upstream of the bottleneck; when using J2 (k), congestions are almost completely avoided in the mainstream and longer queues at the on-ramps are created.

50

10

15

Kp

20

Modified CTM-J2 (k) N =8 N =10 N =12

40 30 20 10 0

10

15

Kp

20

Fig. 5: Average computational times. Fig. 5 shows the average computational times for solving each FHOCP, by varying N and Kp (these times have been computed as average on the three datasets and on the five random instances). It can be easily noted that the solving time is highly influenced by Kp . When Kp = 20 the solver is stopped by the time limit and the optimal solution is not found. Note that the optimality gaps indicated by the solver when stopped by the time limit are high in some cases, in the most difficult instances exceeding 50%. Analyzing again Fig. 5, it can be argued that the CPU times required to solve the FHOCPs are not influenced by the number of cells N . In some cases, an increase in the number of cells even corresponds to a decrease of the average solving time. Fig. 6 and Fig. 7 show, respectively for the FHOCPs adopting the standard and the modified CTM, the average CPU times for the different groups of instances and the different

Average CPU time [s]

Standard CTM 60 J1 (k) J2 (k)

50 40 30 20 10 0

1-2.1 1-2.2 1-2.3 2-2.1 2-2.2 2-2.3 3-2.1 3-2.2 3-2.3

Average CPU time [s]

Standard CTM 60 J1 (k) J2 (k)

50 40 30 20 10 0

4-2.1 4-2.2 4-2.3 5-2.1 5-2.2 5-2.3 6-2.1 6-2.2 6-2.3

Average CPU time [s]

Standard CTM 60 J1 (k) J2 (k)

50 40 30

R EFERENCES

20 10 0

7-2.1 7-2.2 7-2.3 8-2.1 8-2.2 8-2.3 9-2.1 9-2.2 9-2.3

Fig. 6: Average computational times - standard CTM.

Average CPU time [s]

Modified CTM 60 J1 (k) J2 (k)

50 40 30 20 10 0

1-2.1 1-2.2 1-2.3 2-2.1 2-2.2 2-2.3 3-2.1 3-2.2 3-2.3

Average CPU time [s]

Modified CTM 60 J1 (k) J2 (k)

50 40 30 20 10 0

4-2.1 4-2.2 4-2.3 5-2.1 5-2.2 5-2.3 6-2.1 6-2.2 6-2.3

Modified CTM Average CPU time [s]

prediction model (taking into account the capacity drop phenomenon or not) and for the considered cost function (which penalizes congestion states in different ways). As for the cost functions, it has been shown that using J2 (k) is better than choosing J1 (k), since it allows to better solve the congestion in the freeway and implies lower computational times for the solver. The comparison between the use of the standard CTM and the modified version which accounts for capacity drop leads to less clear conclusions. With these preliminary tests it seems that using the modified CTM does not provide big advantages. Future works (possibly using traffic microsimulators) will be devoted to further compare the two prediction models and their ability to predict future traffic states in the controller.

60 J1 (k) J2 (k)

50 40 30 20 10 0

7-2.1 7-2.2 7-2.3 8-2.1 8-2.2 8-2.3 9-2.1 9-2.2 9-2.3

Fig. 7: Average computational times - modified CTM.

datasets. Specifically, on the x-axis of the bar graphs, groups and datasets are indicated (for example 1-2.1 indicates Group 1, i.e. with Kp = 10 and N = 8, and Dataset 2.1). When comparing the FHOCPs adopting J1 (k) with those using J2 (k), it can be observed that the use of J2 (k) generally leads to shorter CPU times. This happens even though the use of J2 (k) implies a larger number of variables and constraints in the FHOCP (due to the additional auxiliary variables introduced). Finally, the comparison between the standard and the modified CTM shows that, when the modified CTM is included in the FHOCP, the computational times needed to solve the problem are higher. V. C ONCLUSION In this paper, different MPC schemes for traffic control have been proposed and compared. The main focus has been related to the different FHOCPs, which differ for the adopted

[1] M. Papageorgiou, H. Hadj-Salem, J.-M. Blosseville, ALINEA: A local feedback control law for on-ramp metering, Transportation Research Record, vol. 1320, 1991, pp 58-64. [2] A. Hegyi, B. De Schutter, H. Hellendoorn, Model predictive control for optimal coordination of ramp metering and variable speed limits, Transportation Research C, vol. 13, 2005, pp 185-209. [3] S. Sacone, S. Siri, A control scheme for freeway traffic systems based on hybrid automata, Discrete Event Dynamic Systems: Theory and Applications, vol. 22, 2012, pp 3-25. [4] A. Muralidharan, R. Horowitz, Optimal control of freeway networks based on the Link Node Cell transmission model, Proc. of American Control Conference, 2012, pp 5769-5774. [5] A. Ferrara, S. Sacone, S. Siri, Event-triggered model predictive schemes for freeway traffic control, Transportation Research C, published on line, doi:10.1016/j.trc.2015.01.020, 2015. [6] A. Ferrara, A. Nai Oleari, S. Sacone, S. Siri, Freeways as Systems of Systems: A Distributed Model Predictive Control Scheme, IEEE Systems Journal, vol. 9, 2015, pp. 312-323. [7] C.F. Daganzo, The cell transmission model, part II: Network Traffic, Transportation Research part B, Vol 29, 1995, pp. 79-93. [8] L. Zhang, D. Levinson, Ramp metering and freeway bottleneck capacity, Transportation Research Part A, vol. 44, 2010, pp. 218-235. [9] A. Srivastava, N. Geroliminis, Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model, Transportation Research C, vol. 30, 2013, pp. 161-177. [10] C.F. Daganzo, Fundamentals of Transportation and Traffic Operations, Elsevier Science, Inc., New York, 1997. [11] R.L. Bertini, M.T. Leal, Empirical study of traffic features at a freeway lane drop, Journal of Transportation Enginerring, Vol. 131/6, 2005, pp. 397-407. [12] M.J. Cassidy, R.L. Bertini, Some traffic features at freeway bottlenecks, Transportation Research Part B, vol. 33, 1999, pp. 25-42. [13] K. Chung, J. Rudjanakanoknad, M.J. Cassidy, Relation between traffic density and capacity drop at three freeway bottlenecks, Trasportation Research Part B, vol. 41, 2007, pp. 82-95. [14] J. Lebacque, Two-phase bounded-acceleration traffic flow model: analytical solutions and applications, Transportation Research Record, vol. 1852, 2003, pp. 220-230. [15] J.A. Laval, C.F. Daganzo, Lane-changing in traffic streams, Transportation Research B, vol. 40, 2006, pp. 251-264. [16] L. Leclercq, J.A. Laval, N. Chiabaut, Capacity drops at merges: an endogenous model, Procedia - Social and Behavioral Sciences, vol. 17, 2011, pp. 12-26. [17] C. Roncoli, M. Papageorgiou, I. Papamichail, Optimal control for multi-lane motorways in presence of vehicle automation and communication systems, Proc. of the 19th IFAC World Congress, 2014, pp. 4178-4183. [18] A. Muralidharan, R. Horowitz, P. Varaiya, Model Predictive Control of a freeway network with capacity drops, Proc. of ASME 2012 5th Annual Dynamic Systems and Control Conference joint with the JSME 2012 11th Motion and Vibration Conference, 2012, pp. 303-312. [19] A. Bemporad, M. Morari, Control of systems integrating logic, dynamics, and constraints, Automatica, vol. 35, 1999, pp. 407-427.