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Transportation Research Procedía 3 (2014) 992 - 1001

Transportation

Procedía

www.elsevier.com/locate/procedia

17th Meeting of the EURO Working Group on Transportation, EWGT2014, 2-4 July 2014,

Sevilla, Spain

Decentralized spatial decomposition for traffic signal

synchronization

Ludovica Adacher*a, Adriano Gemmaa, Gabriele Olivab

aEngineering Department, University Roma Tre, Via della Vasca Navale 79, 00146 Roma, Italy b University Campus Bio-Medico of Rome, via A. del Portillo 21, 00128 Rome, Italy.

Abstract

The Traffic Signal Synchronization is a traffic engineering technique of matching the green light times for a series of intersections to enable the maximum number of vehicles to pass through, thereby reducing stops and delays experienced by motorists. Synchronizing Traffic Signals ensures a better flow of traffic and minimizes gas consumption and pollutant emissions. In this paper we provide a solution to the the traffic signals problem via simulation. The objective function used in this work is a weighted sum of the delays caused by the signalized intersections, and it is calculated by platoon model. Urban signal timing is a non-convex problem and finding an optimal solution for not very small and simple networks may take long time, wherever possible. For this reason we proposed a spatial decomposition of the network, it is obtained by the distributed consensus algorithms. In this paper we provide a distributed communication architecture for a network of smart traffic lights. Each semaphore shares information in a fully distributed way, only with its neighbors according to the topology of the communication network, hence avoiding to resort to a central authority. Given the subnetwork a surrogate method is applied to solve the Traffic Signal Synchronization problem.

© 2014TheAuthors. Publishedby ElsevierB.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Scientific Committee of EWGT2014 Keywords: Traffic Signal Synchronization, Distributed Systems

* Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: adacher@dia.uniroma3.it

2352-1465 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Scientific Committee of EWGT2014 doi: 10.1016/j.trpro.2014.10.079

1. Introduction

Traffic signal control can bring substantial reduction to traffic congestion, leading to improved conditions both for the drivers (better travel times, safety and convenience) and the environment (reduced air pollution and energy consumption). Urban signal timing is a non-convex problem, hence finding an optimal solution for not very small and simple networks may take long time, wherever possible. Furthermore, recent advancements in electronics, sensing, and ICT (information and communication technology) allow the real-time collection and processing of traffic data, as well as the deployment of intelligent controllers for the efficient operation of a transportation system. Nevertheless, controlling the traffic signals of a transportation network constitutes a significant challenge due to the large-scale nature and complexity of the problem, the uncertain and dynamic behavior of the network (e.g. weather, accidents, events) and the patterns of different driver behaviors. For this reason, several different approaches have been proposed; many strategies apply to single intersections, others use historical data to determine fixed plans, while a family of strategies attempt to dynamically decide on the traffic signal timing plans in a distributed and online manner. Traffic signal variables typically controlled are the cycle length, split plan, and offset. Cycle length is the time required for a complete sequence of signal indications. The split plan refers to the time assigned to different phases (simultaneous movement combinations that have the right-of-way) during a signal cycle. Finally, the offset is used to coordinate phases of adjacent intersections to reduce vehicle stops. The majority of techniques considers the single intersection traffic signal control problem, neglecting interrelation effects with other intersections. Nonetheless, by considering intersections atomically, the offset between intersections is not optimized leading to frequent vehicle stops. Also measures-of-interest are optimized locally instead of globally and may lead to poor global performance (Adacher and Cipriani, 2010; Adacher, 2012; Cantarella et al., 2012; Cascetta et al., 2006; Fusco et al., 2004). The majority of techniques consider the multiple intersections traffic signal control problem. Fixed or pre-timed signal control strategies optimize offline the signal timing plans based on historical data so that fixed signal programs are applied for different periods of the day. Fixed- time for multiple intersections methods either attempt to adjust the offset between adjacent intersections so as to maximize progression along multiple corridors using MILP methods, e.g. in MULTIBAND (Lo and Chow, 2004), and global optimization techniques (Li, 2010), or optimize split plans and cycle according to some measure of effectiveness that combines different traffic metrics such as delay, minimum number of stops and throughput, e.g. TRANSYT (El-Tantawy and Abdulhai, 2012). To account for stochastic variations of traffic flows, several online adaptive traffic signal control (ATSC) systems have been developed. These approaches collect information from different sources on-demand, and use them to adaptively optimize traffic signal plan parameters such as splits, offsets and cycle, e.g. SCOOT (Robertson and Bretherton, 1991) or MOTION (Bielefeldt and Busch, 1994). The centralized solution of the TSS problem may provide better performance if it can be derived, but it has several shortcomings as a solution strategy. Firstly, solving this problem is usually intractable and hence not suitable for online decision making because the problem is complex (NP-hard) and of large-scale (especially when the problem involves a large time horizon and several intersections). Secondly, centralized solutions require global information about the status of the network and hence may be prone to communication related failures. Thirdly, solving the problem centrally is not robust, as failure of the central unit will result in complete failure of the system. On the other hand, distributed strategies can be more robust to failures. In this paper we propose a new distributed procedure to obtain a spatial decomposition to reduce the complexity of the problem. Given the subnetwork a surrogate method is applied to solve the Traffic Signal Synchronization problem.

The remainder of the paper is as follows: Section 2 discusses the traffic signal problem and describes a way to solve it; Section 3 deals with a distributed way to cluster a network of elements, based on distributed consensus algorithms; Section 4 provides some simulation results and Section 5 draws some conclusions and future work directions.

2. Traffic Signal Problem

Delay is the one of the most important measures of effectiveness at a signalized intersection because "it relates to the amount of lost travel time, fuel consumption, and the frustration and discomfort of drivers" (Akgungor and Bullen, 1999). Delay can be seen also as an effective way to evaluate and compare different control schemas. Several, often uncontrollable factors, however, typically affect the estimate of such a parameter: among others, the

presence of random traffic flows. Notice that a common approximation (Gentile and Tiddi, 2009) of user total travel time is the sum of the free flow travel time over the non-signalized network, which is often assumed to be constant, and of a delay due to traffic signals, congestions, etc. As a consequence of this choice, minimizing the total delay implies minimizing total travel time.

As for urban networks, it is fair to assume that a relevant fraction of the total delay time is spent on main arterial roads, as these roads have larger flows and higher congestion. A possible control strategy, therefore, is to prioritize a high flow along these corridors. To do this an optimization problem has to be solved, although in many cases a closed-form objective function may not exist or may be hard to identify: a possible solution is to resort to simulation. Simulation approaches can be adopted to estimate the value of the objective function, and can be combined with algorithms aimed at finding the best solution.

In this work we resort to the approach proposed in (Adacher and Cipriani, 2010; Fusco et al., 2013), where the adopted objective function is a weighted sum of the delays caused by the signalized intersections and is calculated via a platoon model (Gaur and Mirchandani, 2001; Jiang et al., 2006), and the algorithm adopted is the surrogate method (Gokbayrak and Cassandras, 2002). We will review the aspects related to the objective function, then we will discuss the surrogate method. According to Newell the delay caused by a signalized intersection can be defined as the difference in the road section travel time in the presence of traffic lights compared to the travel time of the same section for a vehicle with constant speed vt , which is referred to as the speed of synchronization (Newell, 1989).

In Figure 1 an example of delay calculated following the approach in (Newell, 1989) is reported. We used the platoon based delay model, which allows to deal with even non stationary traffic demand and non synchronized signal settings. The model is rather similar the well- established TRANSYT solving procedure. With respect to TRANSYT, however, the platoon model is more flexible as it is aimed at improving the algorithm efficiency. Several different hypotheses about drivers' behavior can be considered. If there is no information about the synchronization speed, it is usual assuming that the vehicle speed on each link depends on the average link traffic density. However, if one envisions that an information system advises drivers about the synchronization speed, it is possible to consider several alternative hypotheses, depending if we allow drivers accelerating to gain possible available space produced by exiting and entering maneuvers or not.

In each hypothesis we assume to include the transient phase of motion into the effective red time. Three reasonable hypotheses are: 1) all drivers travel at the synchronization speed vt ; 2) if there are no empty spaces inside of the platoon, all vehicles travel at speed vt ; otherwise, if vehicles exiting the artery leave available some spaces inside of the platoon, following vehicles accelerate in order to refill the empty space, compressing then the platoon; 3) the first vehicle passed during the green time travels at the synchronization speed vs, while all following vehicles along the artery, including the leading vehicles of successive platoons, tend to travel at least at vt . As for the objective function, it can be defined as a linear combination of the total delay on each direction of the artery, i.e.,

Fig. 1. Example of delay calculation according to (Newell, 1983).

/ = (1 - wt) (Wl £ ^ D? + (1 - Wl) £ ^ Dfa J + wt £ ^ Dfh = /CD) V ¿=i ¿=i ' ¿=i

where:

• Dia is the total delay at node i in one direction of the artery a

• Di-a is the total delay at node i in the opposite way of the artery a

• Diht is the total delay at node i of queue h in lateral approach t

• wa is the weight of delay in direction a

• wt is the weight of the delay at lateral approaches

• roi is the weight of node i

Given the complexity of finding a closed form for the above objective function, we choose to evaluate its value via simulation. For a road artery, the minimum travel time (or minimum delay) problem for traffic signals synchronization (DTSS) can be expressed as follows:

min J = min f (C, g,6 , L,s,X ,Q)

subject to:

• 0 ^ 0i < Ci

• C ^ C ^ C

^min ^i ^max

• max{yia Ci} ^ gi ^ Ci - Li - (max{yi,t Ci)} where:

•Ci is the traffic light cycle for the intersection i. The traffic light cycle is defined as any complete sequence of switch on (and off) of traffic lights at the end of which returns the same configuration of the lights existing at the beginning of the sequence.

• 0i is the offsets for the intersection i.

• gi is the effective green time of node i;

• Li is the time loss at the node i. That is the time in which the intersection is not completely used. The time lost is mainly due to three contributions: transient state of vehicles in the queue at the beginning of the green phase; transient state of exiting vehicles at the end of the green phase and during the yellow phase; the time between the end of yellow and the beginning of green of the next phase. The lost times at the beginning and at the end of green are used to determinate the duration of effective green.

• yi,a, Yi,t are the saturation degree of the approach a along the artery and of the transversal approach t of the i node. The saturation degree is the ratio between traffic flow and the saturation flow. This quantity is an indicator of the level of congestion.

• s is the saturation flow vector for each arc. The saturation flow is the maximum number of vehicles that can cross a stop intersection line per unit time, in the presence of continuous queue. The saturation flow depends on the geometric characteristics of the intersection, on flow composition and on the control traffic lights.

• X is the urban artery geometry.

• Q is the demand level or vehicular flow. It defines the flow of a current the average number of vehicles passing through a section in unit time.

The travel time of a road section is closely linked to the geometry of the road itself and to the configuration of traffic light plans. Saturation flow and geometry are studied and designed in earlier phases and often they cannot be changed a posteriori. In this work we take these factor into account by means of suitable parameters (see [1], [5]) for details. In addition, to maximize the intersection performance, time loss are already designed to be both minimum and ensure the safety levels required. Among the variables that have been used to express the system are only three: cycles, green splits and offsets are our state variables.

2.1. The Surrogate Method (SM)

The main idea is transforming a discrete optimization problem into a "surrogate" continuous optimization problem which is not only easier, but also much faster to solve using standard gradient-based approaches. The two key issues related to this approach are (a) obtaining the actual solution of the original problem from the surrogate one, and (b) using this approach on- line, i.e., making sure that at every step of the iterative solution process a feasible discrete state is defined from an (infeasible) surrogate state. This has two advantages:

• First, the cost of the original system is continuously adjusted (in contrast to an adjustment that would only be possible at the end of the surrogate minimization process);

• and Second, it allows us to make use of information typically employed to obtain cost sensitivities from the actual operating system at every step of the process.

This scheme is intended to combine the advantages of a stochastic approximation type of algorithm with the ability to obtain sensitivity estimates with respect to discrete decision variables. This method work well in different areas, see (Adacher and Cipriani, 2010; Adacher et al., 2014). For a fixed value of the cycle, we iteratively adjust the (integer) green split ratios gj, . . . , gn and the offsets . . . , 9n for n intersection links based on data directly observed and aiming at minimizing the global performances of the network, the overall mean delay (or delay) of cars, denoted by J(D) = J(C, gj, . . . , gn, 9i, . . . , 9n). The cycle is analyzed by a binary search, for a given fixed value of the cycle Ci, g and 9 are evaluated on the bases of the surrogate approach, we denote with x the state variable of the surrogate method. The surrogate method is applied two times in sequence, first x=9 and when an optimal value of the offset is defined x=g.

For a fixed Ci, the DTSS problem can be formulated as mi n(g.e )eAd Jd (Cj, g, 0 ), where Ci is the cycle fixed by the binary research, g is an n-dimensional decision vector with gi E Z+ denoting the green time ratio for intersection link i and 9 is an n-dimensional decision vector with 0 i E Z+ denoting the offset for intersection link i. The capacity constraint is given by:

Ad = {g := [g1, -gn], gmin < gi <gmax, gi 6 Z+; 9 := [91, ■■■ 9n], 0 < 0i < 1; , 0i 6 Z+}

and J(C,g,9) =J(D), that is the total travel time on the network when the variables (green split vector, offsets and cycle) are fixed.

3. Distributed Algorithm for Spatial Decomposition

Spatial problem decomposition refers to the process of partitioning the geographical area over which the problem is optimized into small regions. A good policy towards this direction, is to divide the considered area in regions of some intersections, so that each area controller decides its own schedule; solutions towards the global optimum can be attained through collaboration with neighboring area controllers, see (Adacher et al, 2014; Adacher and Meloni, 2005).

3.1. Definitions

Let G = {V , E , W } be a weighted graph, where V is a set of n vertices vj, . . . vn and E is the set of links of edges (vi, vj ). W is the set of weights wij associated to each edge (vi, vj). A graph is said to be undirected if (vi, vj) E E whenever (vj, vi) E E, and is said to be directed otherwise. A graph G is connected if for any vi, vj E V there is a path whose endpoints are in vi and vj, without necessarily respecting the orientation of edges. A graph G is strongly connected if for any vi, vj E V there is a path whose endpoints are in vi and vj, respecting the orientation of edges. A graph G is balanced if for each node vi E V

j=l j=l

i.e., the sum of the weights of incoming and outgoing edges coincide. Let the neighborhood Ni of a vertex vi be the set of vertices {vj: (vj,vi) EE}.

3.2. Distributed Max-Consensus Algorithm

Let us describe an algorithm (Olfati-Saber and Murray, 2004; Olfati-Saber et al., 2007) to allow a set of distributed agents, each provided with a vectorial initial state, to reach an agreement on the component-wise maximum of the states among the agents in the network. Let a set of n agents, each described by the discrete time dynamic equation Zi(t+l)=Zi(t)+Ui(t), zi(0)=zi0 where zi(t),zi0 E Rn . The problem, for a connected graph (for undirected graphs) or strongly connected and balanced graph (for directed graphs), is known to have a solution in finite time (Olfati-Saber and Murray, 2004; Olfati-Saber et al., 2007) if the following distributed control law is (chosen:

ui(Ni, t) = rnaxz,- (t)

With the above control law, the problem is solved in m ^ n iterations, where m is the length of the diameter of the graph. However, the nodes do not know m nor n, hence a number of iterations tmax > n has to be selected.

3.3. Distributed Average-Consensus Algorithm

In the average-consensus problem the nodes are required to converge to the component-wise average of their initial conditions. If the graph G is connected (for undirected graphs) or strongly connected and balanced (for directed graphs), then the problem admits an asymptotic solution (Olfati-Saber and Murray, 2004; Olfati-Saber et al., 2007), if the following control law is chosen:

where parameter T is assumed to be

^ —-™-

max £y=1 wu

i=l...n '

Note that T plays the role of a sampling time and is a global quantity. To solve the average-consensus problem the nodes need to use the same T. For instance if wjj G {0,1} then 1 is a n possible upper bound for t; however, this would require each node to know n. Such an issue is generally overcome by choosing a considerably small value for T.

Algorithm 1: Distributed Network Size Computation: i'-th node

Data: tmaXi^i

Result: Network Size NS;

/* Initialization */

tID, = i\ cID; = i; NS,- = 0;

while tIDi > 0 do

/* Execute a max consensus on the tIDj */

cID; = max-consensusftlD,',^,*^!!;);

/* If i coincides with the result of the max consensus set

tID; to 0 */

if cID, == i then I tID, = 0; end

/* The node adds 1 to the network size counter */

NS; = NSj + 1;

return NS,;

Fig. 2 Distributed network size computation algorithm

3.4. Distributed Network Size Computation

Let us discuss a distributed algorithm based on max-consensus to estimate the number of agents in a network (Oliva and Setola, 2013). The pseudo code of the algorithm is reported in Algorithm 1 . Supposing that each agent has a unique identifier i; the agents are counted by iterating a max-consensus procedure with initial conditions equal to the identifiers: at each iteration the node whose identifier coincides with the current maximum sets its identifier to 0, while each node increases the node counter. Using Algorithm 1 to compute the network size, it is possible to set tmax = NSi for the max-consensus algorithms, hence saving computational time.

3.5. K-means Algorithm

Consider a set of n observation xj,...,xn, where each observation xi is a vector in Rd. Suppose we want to partition the n observations into k (k ^ n) sets or clusters Sj,...,Sk. Specifically, we want to find a set of centroids cj,...,ck, each associated to a cluster and we want to solve the following optimization problem:

V" Vk 9

minD= > > rij\\xi — cj\r lj=l !_/ = 1

subject to

Y nj = l Vi = l,...,n [r0- e {0,1} Vt = 1,...,n,Vj = 1,...,k

where rij =1 if observation xi is assigned to the set Sj and rij =0 otherwise, and cj ERd is the centroid of the observations within the set Sj. The problem (9) is hard to solve, and in the literature several iterative algorithms have been proposed. Among the others, the k-means algorithm (MacQueen et al, 1967) proved its effectiveness. Specifically, starting with a random set of k centroids cj(0),...,ck(0), the algorithm alternates for each step an assignment and a refinement phase. During the assignment phase, each observation xi is assigned to the set characterized by the nearest centroid, i.e.,:

r_ = (lifh = argmin.j\\xi - Cj(T)\\ LJ l 0 else

During the refinement phase each centroid Cj is updated as the centroid of the observations is associated to Sj(T),

Cj(T + 1) =:

2?=1r0CO

The two steps are iterated until convergence or up to a maximum of M iterations. The k-means algorithm is granted to converge to a local optimum value, while there is no guarantee to converge to the global optimum (MacQueen et al., 1967; Brucker, 1978).Since there is a strong dependency on the initial choice of the centroids, a common practice is to execute the algorithm several times and select the best solution.

3.6. Distributed K-means Algorithm

Let us now describe the distributed k-means algorithm proposed in (Oliva and Setola, 2013). In the following we assume that the algorithm is executed synchronously by each node and that the nodes exchange the necessary information with their neighbors or with a subset of the nodes in their neighborhood. The initialization phase of the algorithm is as follows. In order to choose the initial random centroids, a leader is elected and is in charge to choose the centroids. Supposing that each node has a unique identifier i, the nodes challenge on their identifiers via max-consensus. The node i* whose identifier is the greatest is elected as leader and chooses the centroids Ci*h (0) E Rd at random (within the range of values of interest for each component) for h = 1,...,k. The other nodes choose for all h = 1,...,k

cih(.°) = Vi * i*

Step 1

StepIO

• CO ' . ' " * « oo * * * « oo . /,4- * * * » oo

0.5 0.5 0.5 0.5

TM + + x ° ° + + x ° ° K ° O K ° O

Fig. 3. Example of distributed k-means clustering in 2D for n=30, k=3.

After the initialization phase, the main cycle is executed M times; at every step T it is composed of the following

phases:

1) Centroid Propagation: each node selects a vector c0i (T ) E Rkd as follows:

0m_i [%(0)'.....Cik(0)'] ifT = 1

Ci (№cr - l)®qk.,(r)cr) ifT> 1

where cik.,(r)(T) £ Rdis the centroid chosen by node i at step T and ^¿(T)E Rk is a vector representing the choice of a centroid at step T. An expression for ^i(T) is given below. Vectors c0i (T ) are structured so that, if used as initial conditions for a vectorial max-consensus procedure, the resulting vector is the stack vector [ci1(T) ' ,...,cik(T)' ]' containing all the k centroids. In fact at step T = 1 the only vector with components greater than is Zf-(T) and the result of the max-consensus is zt*(T) for all nodes. For T > 1 each node choses a vector c0i (T ) that has non-zero components only in correspondance of the chosen centroid, hence a max-consensus results in a stack vector containing all the centroids

2) Nearest Centroid Choice: each node chooses the index ki*(T) of the nearest among the current centroids, i.e.,

k*(T) = argminh=1.....n\cih(T)

and updates ^¡(T) as follows

3) Nearest Centroid Choice Broadcast: each node provides its choice of ki*(T) to the nodes in its neighborhood Ni.

4) Cluster Neighborhood Choice: each node selects, among its neighbors in Ni, those nodes j that share the same choice of ki*(T ). As a result a new neighborhood Nic(T ) G Ni is is obtained. The new neighborhood Nic(T ) is then used to update the centroids.

5) Centroid Refinement: The new neighborhood represents a subgraph Gc(T ) of G , where each node is connected only to nodes with the same centroid choice. During this phase an average consensus on the nodes'observations xi is performed over Gc(T) in order to obtain Qk.„cr)(T).

4. Simulation Results

Preliminary numerical experiments have been conducted on different subnetworks considering different dimensions and characteristics. The subnetworks are given by the decentralized spatial decomposition and different values of the flow are tested. The dimension of the subnetwork starts from a minimum of 4 signalized intersections to a maximum of 20 signalized intersections. Different demand scenarios have been considered to verify the robustness of the synchronization solution with respect to possible demand fluctuations. Starting from the actual average demand, two other scenarios, high and low, have been obtained by increasing and reducing the average demand level as +15 per cent and -15 per cent, respectively. The simulation results highlight that the optimizing procedure improves the average unitary delay at the nodes from 50 per cent to 15 per cent. A discontinuous, noisy and high-dimensional objective function characterized by the presence of many local minima makes the solution searching process a difficult optimization problem. The analysis of the shape of the objective function shows the existence of many local minima so highlighting the importance of convergence capabilities of the solving algorithm. These preliminary results point out the capabilities of the SM approach to jump out of local minima. It is important also notice that the quality of the initial population does not affect significantly the SM convergence, effectively different trials are tested for each test networks and the algorithm given practically the same area of convergence, the perturbation is attested around 3%. The performance of the our proposed approach (SM) has also been compared for the different topologies of intersections against the performance of well known Genetic Algorithm (GA) (Gentile and Tiddi, 2009). In Figure 4 an example of results on a networks of 20 signalized intersections is reported, during the peak our. The total travel time of the two different heuristic schemes is reported, and the independency from the start points of SM is highlighted.

424000 423000 422000 421000

"D 420000

q 419000

418000 417000 416000

415000

414000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-•-GA 418623 413232 417878 419980 419403 423131 418877 419967 422870 419064 421524 420241 422268 419518 419967 422570 419064 421524 420241 422268

-■-SM 417516 417447 417532 418229 418552 418376 418184 418215 417611 418143 418318 417612 418060 418011 416707 418030 418249 418012 417322 417534

Different start points

Fig. 4. Comparison of Surrogate Method (SM) with a Genetic Algorithm (GA) for a networks of 20 signalized intersections.

5. Conclusions

Urban signal timing is a non-convex problem and finding an optimal solution for not very small and simple networks may take long time, wherever possible. For this reason we propose a spatial decomposition of the network, it is obtained by the distributed consensus algorithms. We provide a distributed communication architecture for a network of smart traffic lights. Each semaphore shares information in a fully distributed way, only with its neighbors according to the topology of the communication network, hence avoiding to resort to a central authority. Given the subnetwork a Surrogate Method is applied to solve the Traffic Signal Synchronization problem. This decomposition gives comforting results, and other indexes for the composition are under study. This work is at the beginning and we are studying the possibility to introduce particular characteristics of the subgraph in the decomposition phase. We are testing the possibility to dynamically chance the network clustering depending on the traffic conditions and to select a desired number of traffic lights for each subnetwork.

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