Controlling a discrete model of two cascading intersections Jia Wu, Abdeljalil Abbas-Turki, Aur´elien Corr´e¨ıa and Abdellah El Moudni SeT, UTBM, 90010 Belfort Cedex

Abstract— In this paper, we propose a new control of traffic lights of two cascading intersections, taking into account vehicle behaviour, integral red and orange phases. We consider the intersection as a resource shared between vehicles of two roads. Hence, the control of traffic light signal is performed to take into account each vehicle arrival individually. On the one hand, this allows us to distinguish different kinds of vehicles such as public transport vehicles and emergency vehicles. On the other hand, since the main objective is the minimization of the evacuation time, then the waiting time of each vehicle is dramatically reduced. Moreover, such a control can lead to autonomous vehicles, which would not need traffic lights anymore.

I. I NTRODUCTION As the rapid growth of urban traffic, the demand of transport becomes hard to meet. Furthermore, the transport is generally the principal source of air pollution in big cities. Several politicians stake on the reduction of the use of individual vehicles. Thus, they have to face the dilemma of the choice between pollution and the economic expansion of the city. This situation leads several researchers to look for new solutions for controlling the traffic in order to minimize stop times, queue lengths and situations of acceleration and deceleration. Traffic controls are mainly based on the estimation of the flow rate of vehicle arrivals. This is the case of both major kinds of techniques of the traffic light control, fixed cycle time and adaptive signal control. In the fixed signal control, historical data is used to set up control strategies. The fixed signal control is simple and does not require sensors to obtain information about real-time traffic condition. It is just based on the average flow rate of roads. The drawback of the fixed signal control is obvious: It can not respond to any change in traffic condition. The most widely known and used system is TRANSYT [1] which was developed originally at Transport and Road Research Laboratory (TRRL). In order to improve such a control, adaptive control strategies are proposed. They receive real-time data through sensors and create an optimal timing plan. Several approaches are proposed to develop successfully adaptive control at a single intersection, such as the ones based on fuzzy logic [2], [3], continuous or hybrid Petri Nets [4], [5], and Markov chains [6]. The basic principle of fuzzy signal control is to model the control based on human expert knowledge. The basic advantage of fuzzy control is that it fires many soft rules simultaneously and makes a decision which offers the compromise. The Petri Nets are widely used in traffic control. The safety requirements such as avoiding conflicting movements at the intersection can be easily ensured by PN model. Besides, the assumption of a continuous flow can be

modelled by continuous places and transitions. The Markov model provide a way of computing the queue dynamics and the probability of a queue to be observed at a certain time. Right now, various responsive systems of traffic control have been established. For example, SCOOT [7], one of the first major real-time traffic control system, based on feedforward strategy, makes continual incremental adjustments in real-time of cycle lengths, splits, and offsets. SCAT [8] uses voting algorithms to find the best signal plan. OPAC [9], PRODYN [10], operate in a different manner. OPAC uses a rolling horizon approach to operate a dynamic optimisation. PRODYN uses automatic control methods like: Bayesian estimation, Dynamic Programming, Decentralized methods. One can note that there are several approaches for the traffic light control. However, we draw the reader attention to the fact that all these approaches are mainly based on the estimation of the flow rate for minimizing vertical queue length. It is widely known that the estimated flow rate is not an effective measure for controlling traffic lights because it leads to a confusion about which actual traffic situation occurs. Moreover, there is a gap between the estimated traffic flow of the next time and the actual one. Hence, such systems are still requiring a human intervention when the traffic fluctuates dramatically. Another limitation of such systems is that their operation rules cannot be used for new concepts of traffic control such as the one of autonomous vehicles. Indeed, flow rate cannot be applied to vehicles that communicate their situation to each other in order to define their priority for passing through intersection. Such a new concept requires taking into consideration each vehicle individually. Besides, flow rate estimation can not make use of each second of green time efficiently. Because the portion of the signal timing cycle is allocated to each phase, there is the possibility that when the green light is on, no vehicle gets through the intersection. The intersection as a resource is not utilized in the maximum degree. Since approaches we find in the literature meet the limitation of efficiency and of flexibility for facing new technologies of communication, naturally, we are forced to look for new approaches. In this case, it is interesting (i) to consider only present vehicles according to their availability for passing through intersection and (ii) to try to vacate them as soon as possible. If we could reduce the evacuation time, the crossroad would absorb efficiently the new vehicle arrivals without the need of estimation of the future flow rate. Since we treat individually each vehicle arrival, we need a discrete event model. We use the model of street proposed

in [11]. The model is based on Timed Event Graphs (TEG) and allows us to evaluate microscopic and macroscopic performances. These evaluations meet the fundamental diagram properties. Besides, we can compute sojourn time of each vehicle in order to evaluate the time at which it will be ready to pass through intersection. By this way, we present in this paper an approach that attempt to expend each second efficiently. The remain of this paper is organized as follows. Section II presents the model that our control strategy is based on. Section III briefly describes traffic light control algorithm that we use. The simulation results are given in Section IV. II. S TUDIED SYSTEM We consider two cascading three-phase intersections with controllable traffic lights on each corner, see Figure 1 for description of a simple intersection. It is supposed that there are two traffic flows of vehicles to be served in the intersection. Each road has one lane. We assume that there is a minimum waiting time s between two successive vehicles coming from different roads. This corresponds to the integral red time plus the wasted time for acceleration. Likewise, two successive vehicles from the same road are spaced at least by a time space p. Obviously, p < s.

Traffic Management

response

control

Traffic Simulator Fig. 2.

Simulator’ is to represent the real world. It reflects the influence of the traffic control strategy. ‘Traffic Management’ generates the traffic control strategy. 1) Traffic Simulator: This part simulates the movements of vehicles by car-following models. The car-following model captures acceleration behaviour in the car-following regime. In this regime, the drivers are close to their leaders and follow their leaders. For simplicity, the linear model is used: ′′

′

′

dn (t) = L + hxn (t)

Road 2 3 one segment

2

of Road 2

one segment

1

of Road 1

Road 1 5

4

2

3

′

xn (t + T ) = c1 [(xn−1 (t))−xn (t)]+c2 [xn−1 (t)−xn (t)−dn t]

4

sensor

structure of program

1

c1 = 0.4 m/s c2 = 0.14 m/s2 T =1s L=5m h = 1.8 s The program simulates two successive intersections and the roads connected to them. Vehicles from Roads 1 and 2 probably go straight or turn to right or left. Vehicles from Roads 3 and 4 just go straight ahead.

shared surface Road 2

sensor

detector

a vehicle

Fig. 1.

Road 3

detector

λ2

λ3

a simple intersection λ1

For each movement of traffic flow, the signals are given in the following order: Red, Green and Yellow. A green interval is followed by a yellow change interval indicating that a vehicle must stop if it can ensure safety. One can note that in the described system, both movements of the traffic flow obey the same rule. Therefore, we are interested only on the modelling of one movement of flow. A. Simulation program structure The structure of the simulation program is given by figure 2. ‘Traffic Simulator’ simulates the movements of individual vehicles in microscopic simulation. The role of ’Traffic

Road 1

Road 4

Fig. 3.

Two successive intersections

The inputs of simulator are flow rates of Road 1 (λ1 ), Road 2 (λ2 ) and Road 3 (λ3 ). 2) Traffic Management(Green Waves): The ‘Optimal’ cycle for the first intersection is given by: c0 =

1.5T + 5 1−Y

T = 2s + 2P = 2 × 3 + 2 × 2 = 10s y1 =

λ1 × 3600 2000

y2 =

λ2 × 3600 2000

Road

2

1

3

2

1

1

4

3

2

5

4

t (s) 0

Y = y1 + y2

2

4

6

8

Fig. 4.

The green light time for Road 1 is given by:

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the arrival time of vehicles

Road 1

v1 = y1 × (c0 − T )/Y

2

1

3

4

5

The green light time for Road 2 is given by: v2 = y2 × (c0 − T )/Y The ‘Optimal’ cycle for the second intersection is given by: 1.5T + 5 c0 = 1−Y T = 2s + 2P = 2 × 3 + 2 × 2 = 10s y3 = y4 =

λ3 × 3600 2000

(λ1 + λ2 ) × 3600 2 × 2000

1

Road 2

Fig. 5.

2

3

the optimal strategy for traffic lights control

A. Problem Specification Before introducing the control strategy, we need data presented in table I that are derived from the intrinsic infrastructure parameters such as the length, the maximum allowed speed and flow rate of each road. Besides, we need, at each second, information about traffic situation like the number of vehicles at each road, the arrival time of vehicles and so on.

Y = y3 + y4

TABLE I ROAD PARAMETERS

The green light time for Road 3 is given by: v3 = y3 × (c0 − T )/Y The green light time for Road 4 is given by:

4

variable g ng

definition the index of road, g = 1, 2 the number of vehicles in road g

v4 = y4 × (c0 − T )/Y

(qg , g)

the qg th vehicle in road g, 1 ≤ qg ≤ ng

III. T RAFFIC C ONTROL S TRATEGY

r(qg ,g)

the arrival time of vehicle (qg , g)

p(qg ,g)

the time which vehicle (qg , g) need to cross the shared surface

sg′ ,g

the integral red time plus the wasted time for acceleration when the traffic light changes between road g ′ and g, g ′ 6= g, g ′ = 1, 2.

G (q1 , q2 , g)

by traffic control strategy, the minimum total travelling time for vehicles (1, 1), . . . , (q1 , 1) and (1, 2), . . . , (q2 , 2). The last vehicle (qg , g)th in this strategy is from road g

As we have mentioned, we consider the intersection as a resource shared between vehicles of two roads. Hence, the control strategy is proposed to allocate the authority of using the intersection to different roads according to the arrival time of vehicles. The objective is to free the resource as soon as possible. This means that we aim at vacating both roads and intersection as earlier as we can. For example, see figure 1, there are two directions of traffic flow. In each direction, the arrival times are the following: 5 vehicles in road 1, the arrival times are 0s, 4s, 10s, 15s, 21s; 4 vehicles in road 2, the arrival times are 2s, 8s, 15s, 22s, see figure 4. For simplicity reasons, we suppose vehicles of two roads use identical time units p1 = p2 = 2s when crossing the shared surface. We assume that the integral red time plus the wasted time for acceleration are s1−2 = s2−1 = 5s. The optimal solution in this case is known and is given in figure 5. The minimum evacuation time of all vehicles in this example is 33s. For sake of generality, the question is how can we allocate the resource of intersection to each vehicle to carry out our objective? The Petri net model informs us about vehicle arrivals and conflicting situations. However, a control strategy is still required.

We assume that vehicles of each road leave the intersection in a stringent order according to their arrival time. This means that we vacate each lane according to FIFO rule. Indeed, for sake of correctness, we can not admit vehicles to pass through intersection before a previous vehicle of the same road. Hence, we have: (1, g) ≺ (2, g) ≺ . . . ≺ (ng , g) Since our objective is to evacuate the vehicles in the road as quickly as possible, we have to reduce the travelling duration of each vehicle in the road. We have defined G (q1 , q2 , g) as the total travelling time of vehicle (1, 1), . . . , (q1 , 1) and (1, 2), . . . , (q2 , 2) (see Table I). It can be comprehended

as the duration of the last vehicle departure time (time when it leaves the intersection) minus the arrival time of the first vehicle (when the first vehicle is detected). Hence, the optimal strategy is the arrangement of traffic light that minimizes this objective function: min {G(n1 , n2 , 1), G(n1 , n2 , 2)} For example, see figure 1, the objective is to find an optimal strategy that satisfies: min {G(5, 4, 1), G(5, 4, 2)} B. Forward Dynamic Programming One can note from the problem definition and the PN model Pthat the problem can be considered as 1|rj , pj , sP Cj [12] where rj is r(qg ,g) , pj is p(qg ,g) , sf is f| sg′ ,g and Cj is G (q1 , q2 , g). In [12], it proposes a forward dynamic programming for solving the problem in order to find the optimal resolution. Since the completion time of this algorithm is polynomial (O(n2 )), it can be implemented for a real traffic situation. Thus, right now, we use the forward dynamic programming that is described in the following. Now, we give the formula of recursion. G (q1 , q2 , g) = min {G(q1′ , q2′ , g ′ )+sg,g′ +p(qg ,g) +τ } (1) ′ g =1,2

Where qf′ = qf for f 6= g, otherwise qf′ = qf − 1 for f = g, f = 1, 2. If the last vehicle of the previous strategy is also from road g, we have then sg′ ,g = 0, where g ′ = g, that means the traffic light does not change. Otherwise, if the last vehicle of the previous strategy is from another road g ′ , where g ′ 6= g, then the traffic light change time is involved. At this stage, we can give the formula for computing the waiting time τ of vehicle (qg , g). We have: τ = max



0, r(qg ,g) − G(q1′ , q2′ , g ′ ) 0, r(qg ,g) − G(q1′ , q2′ , g ′ ) − sg′ ,g

g′ = g g ′ 6= g

(2)

The formula of recursion selects the previous optimal strategy by which vehicle (qg , g) is the last vehicle to pass through the intersection. The recursion keeps on until it reaches the initialization condition. To illustrate formula 1, we cite again the example of figure 1 and figure 4, G(5, 4, 1) = min{G(4, 4, 1) + p(5,1) + τ1 , G(4, 4, 2) + p(5,1) + s1,2 + τ2 } τ1 = max{0, r(5,1) − G(4, 4, 1)} τ2 = max{0, r(5,1) − G(4, 4, 2) − s1,2 } And r(5,1) = 21 s.

C. Initialization We are now ready to give initialization for forward dynamic programming. The recursion stops when there is one vehicle in the road. For our problem, there are four cases that can be considered as initializations: G(1, 0, 1), G(0, 1, 2), G(1, 0, 2), G(0, 1, 1). s0,g denotes the traffic light change time before the first vehicle (1, g) in road g arrives.  G(1, 0, 1) =    + G(0, 1, 2) =    +

s0,1 + max(r(1,1) − s0,1 , 0) p(1,1) s0,2 + max(r(1,2) − s0,2 , 0) p(1,2)

(3)

If s0,g = 0, the initialization can be simplified as follows:  G(1, 0, 1) = r(1,1) + p(1,1) (4) G(0, 1, 2) = r(1,2) + p(1,2) Indeed, the other two cases, which are G(0, 1, 1) and G(1, 0, 2), can not be considered. Because they are contradictory of reality. Taking G(1, 0, 2) as example, on the one hand it denotes that there is no vehicle on the road 2, but on the other hand it denotes at the same time that there is one vehicle on this road that pass through intersection. G(0, 1, 1) in the same way can be considered as impossible. We denote these two impossible cases as:  G(0, 1, 1) = ∞ (5) G(1, 0, 2) = ∞ D. Time Complexity As widely known, dynamic programming has three properties: overlapping sub problems, optimal substructure, and memorization. The overlapping means that the sub problems are solved by dividing into similar sub-sub problems until we reach the initialization condition. During the process, the same sub problems maybe appear several times. Hence, we do not need to compute them every time. Instead, we save the solutions to sub problems we have already solved and use them to solve many different larger problems. Then, if we need to solve the same problem later, we can retrieve and reuse our already-computed solution. This is called memorization. So, in our method, the time complexity of the algorithm is O(n2 ), where n is the number of vehicles (proof can be found in [13]). E. Extension of the traffic control strategy We can consider other elements in real traffic situation. For example, there are some vehicles which have urgency, like the police, the ambulance. The traffic control system should give the high authority to these important vehicles, and minimize their waiting times. We make a change of the dynamic programming to solve this problem. Suppose that every vehicle has a weight for evaluating its degree of urgency. To reduce the waiting time of the high urgency vehicles, we should minimize their weighted total travelling time.

9

18

8

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14

the number of vehicles

the number of vehicles

20

12 10 8 6

6 5 4 3

4

2

2

1

0

0

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150 time[s]

200

250

0

300

Fig. 6. The waiting time of vehicles in the second intersection: λ1 = 0.1 veh/s, λ2 = 0.1 veh/s, λ3 = 0.1 veh/s

fixed cycle time forward dynamic programming

0

50

100

150 time[s]

200

250

300

Fig. 7. The number of vehicles in the second intersection: λ1 = 0.1 veh/s, λ2 = 0.1 veh/s, λ3 = 0.1 veh/s

w(pg ,g) denotes the weight of vehicle (qg , g), qg = 1, . . . , nf , g = 1, 2. Redefine G(q1 , q2 , g) as the weighted total travelling time of vehicles (1, 1), . . . , (q1 , 1) and (1, 2), . . . , (q2 , 2). Hence, we have:

fixed cycle time forward dynamic programming

350

300

g =1,2

+w(qg ,g) (sg,g′ + p(qg ,g) + τ )}

(6)

Where the last vehicle (qg , g) in this strategy is from road g. Then qf′ = qf for f 6= g, otherwise qf′ = qf −1 for f = g. And sg′ ,g = 0 if g = g ′ , otherwise traffic light change time is involved. Definitions of other variables are the same as the precedent assumption, i.e. without given weights. The initializations for the extended traffic control strategy are given in the following:

waiting time[s]

250

G (q1 , q2 , g) = min {G(q1′ , q2′ , g ′ ) ′

200

150

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0

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150 time[s]

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Fig. 8. The waiting time of vehicles in the second intersection: λ1 = 0.1 veh/s, λ2 = 0.1 veh/s, λ3 = 0.3 veh/s

G(1, 0, 1) = w(1,1) (s0,1 +max(r(1,1) −s0,1 , 0)+p(1,1) ) (7) 14 fixed cycle time forward dynamic programming

G(0, 1, 2) = w(1,2) (s0,2 +max(r(1,2) −s0,2 , 0)+p(1,2) ) (8)

(9)

The other two impossible cases, G(0, 1, 1), G(1, 0, 2), are considered equal to infinity as it is shown in (5). IV. S IMULATION R ESULTS In this section, we evaluate our approach regarding a classical, fixed-cycle green wave approach. Simulation results are presented through figures 6 to 11. As it is shown in simulation, the denser the traffic, the best our approach towards classical green waves approach.

the number of vehicles

If the traffic light change time s0,g = 0, then:  G(1, 0, 1) = w(1,1) (r(1,1) + p(1,1) ) G(0, 1, 2) = w(1,2) (r(1,2) + p(1,2) )

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150 time[s]

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Fig. 9. The number of vehicles in the second intersection: λ1 = 0.1 veh/s, λ2 = 0.1 veh/s, λ3 = 0.3 veh/s

1500 fixed cycle time forward dynamic programming

waiting time[s]

1000

500

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150 time[s]

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250

300

Fig. 10. The waiting time of vehicles in the second intersection: λ1 = 0.3 veh/s, λ2 = 0.3 veh/s, λ3 = 0.2 veh/s

20

fixed cycle time forward dynamic programming

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the number of vehicles

16 14 12 10 8 6 4 2 0

0

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150 time[s]

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Fig. 11. The waiting time of vehicles in the second intersection: λ1 = 0.3 veh/s, λ2 = 0.3 veh/s, λ3 = 0.2 veh/s

V. C ONCLUSION In this paper, a new control for traffic signal has been presented. This strategy is based on the model [14] which treats the vehicle individually. Thus, the arrival time of vehicle is obtained easily. We use forward dynamic programming to find the optimal traffic control strategy for vacating all vehicles in the roads as soon as possible. Then, we extend the control strategy to minimize the waiting time of vehicles which have urgency. The weight is added to each vehicle for evaluating the urgency of each vehicle. Vehicles behaviours are simulated by using VISSIM. By comparing the fixed cycle time and the adaptive signal control method, our method proves its ability of efficiently reducing the waiting time of vehicles and the number of vehicles in real traffic situation. R EFERENCES [1] C. B.M. and L. C.J., “Transyt-the latest developments,” Traffic engineering and control, vol. 28, pp. 387–390, 1987.

[2] J. Niittymaki and M. Pursula, “Signal control using fuzzy logic,” Fuzzy sets and systems, vol. 116, pp. 11–22, 2000. [3] J.Lin and K.Y.Kuo, “Application of fuzzy set theory on the change intervals at a signalized intersection,” Applied Soft Computing, vol. 1, pp. 161–177, 2001. [4] G. F. List and M. Cetin, “Modeling traffic signal control using petri nets,” IEEE Transacations on Intelligent Transportation Systems, vol. 5, no. 3, pp. 177–187, September 2004. [5] A. D. Febbraro, D. Giglio, and N. Sacco, “Urban traffic control structure based on hybrid petri nets,” IEEE Transactions on Intelligent Transportation Systems, vol. 5, no. 4, December 2004. [6] L. Qiaoru, W. Lianyu, and M. Shoufeng, “The model analysis of vehicles situation and distribution in intersections based on markov process,” Intelligent Transportation Systems, 2003. Proceedings. 2003 IEEE, vol. 2, pp. 1076–1080, October 2003. [7] D. I.Robertson and R. Bretherton, “Optimizing networks of traffic signals in real time-the scoot method,” IEEE Transacations on Vehicular Technology, vol. 4, no. 1, pp. 11–15, February 1991. [8] A.G.Sims and K.W.DOBINSON, “The sydney coordinated adaptive traffic(scat) system philosophy and benefits,” IEEE Transactions on vehicular technology, vol. VT-29, pp. 130–137, May 1980. [9] ——, “Implementation of the opac adaptive control strategy in a traffic signal network,” IEEE Intelligent Transactions Systems Conference Proceedings, pp. 25–29, August 2001. [10] J.J.Henry, J.L.Farges, and J.Tuffal, “The prodyn real time traffic algorithm,” Proc. 4th IFAC/IFORS Conf. Control Transportation System, pp. 305–309, 1983. [11] A. Corr´e¨ıa, A. Abbas Turki, R. Bouyekhf, and A. El Moudni, “A (max,+)-linear model for the analysis of urban traffic networks,” in 10th IEEE international conference on Emerging Technologies and Factory Automation (ETFA), Catania, Italy, Sept, 2005. [12] C. N.Potts and M. Y.Kovalyov, “Scheduling with batching: A review,” European Journal of Operational Research, vol. 120, pp. 228–249, 2000. [13] B. Peter, Scheduling Algorithms. Springer, 2004. [14] A. Corr´e¨ıa, A. Abbas Turki, R. Bouyekhf, and A. El Moudni, “Modelling urban intersections in dioid algebra,” WSEAS transactions on systems, vol. 5, pp. 1223–1228, May 2006.