This paper suggests the adoption of a spatial decomposition method to solve the signal synchronization problem. A good signal setting maximizes the number of vehicles passing through intersections, while minimizing gas emissions and possible delays experienced by drivers. The signals synchronization issue can be defined as the problem of finding the offsets, the green timings, and the cycle length for a series of controlled intersections, minimizing the total delay of the network subject to admissibility constraints. In this paper, the authors optimized the signal setting through a new Surrogate Method calculating the objective function via the

Traffic congestion can be reduced through a traffic signal control. A traffic signal control gives an improvement both for the drivers (minimizing travel time and delay) and for the environment (reducing both the energy consumption and gas emission). Nevertheless, controlling the traffic signals of a transportation network is a significant challenge due to its large-scale and complexity. Traffic congestion on roads is a serious problem, especially for big cities in the world.

The traffic signal synchronization is a nonconvex problem and sometimes finding a quick and optimal solution even for small networks can be difficult [

The signal synchronization problem consists in the simultaneous optimization: the offsets, the green timings and the cycle length at each junction (computed by delay minimization) for a series of junctions see [

According to the classification presented in [

Centralized approach: The majority of signal timing optimization algorithms use a centralized formulation and architecture. At the same time, for all intersections, they optimize various signal timing parameters (i.e., cycle length, green times, and offsets). However, network signal timing optimization is an NP-hard problem and a central optimization technique will not be scalable and applicable to large transportation networks [

Hierarchical or distributed approaches decompose the network optimization problem into a multilevel control problem with distinct objectives at each level. The underlying concept of most hierarchical approaches is to make network level decisions at the upper (or central) level and the real-time, small-area computations in the lower (or intersection) level. The exchange of information is a crucial aspect [

Decentralized approaches decompose the network into regions with varying number of intersections. As the result of the decentralization, these approaches are scalable and can be real-time; however, rather than global optimization, they mostly locally control signals and may find a suboptimal solution [

It is evident that a centralized system will theoretically be able to find optimal solutions, though with a higher amount of information on the system than its distributed counterpart. However, the computation time of a centralized control increases exponentially together with the size of the urban network thus preventing its processing. It is also expected that a centralized system may theoretically provide a more effective control policy than its decentralized counterparts with a better coordination among network components. As a result of the decentralization, these approaches are scalable and can be on real-time basis; however, they control local signals and often find suboptimal solutions. The first developed commercial software is based on a centralized control system, where one computing unit decides for all intersections (e.g., SCOOT and SCATES) or hierarchical architectures where one part of the decision is centralized whereas the other is local (e.g., MOTION and RHODES).

Often the literature suggests a decentralized approach which optimizes each intersection separately and the information used is not sufficient to optimize the offset among intersections, finding suboptimal solutions. Several authors tend to suggest a decomposition based on a single intersection either without or a limited exchange of information. The adaptive traffic signal control systems are used to accommodate real-time traffic conditions. In fact recent development of artificial intelligence, especially the success of deep learning, gives the possibility to use information of individual vehicles to control traffic signals. However, those studies are limited to isolated intersections and their effectiveness was only evaluated in ideal simulated traffic conditions by hypothetical benchmarks (see [

Despite the decentralized approaches presented in literature, there are only a few studies comparing the performance of the distributed system to its centralized version. The authors of [

The objective function is the delay minimization. The delay is considered as the number of vehicles obstructed in the road section. Consequentially, the time saving due to signals synchronization reduces pollution produced by traffic and fuel loss due to low running speed.

The decision variables for the delay minimization are: (i) the green time, (ii) the offset, and (iii) the cycle for each controlled intersection (rather than only the green time).

The network is decomposed in sub-networks (rather than in individual intersections); network, rather than the neighborhood concept.

The proposed clustering is based on the physical topology and on the features of the urban road network (rather than the neighborhood concept).

The analysis is performed on a real-world network with 56 signalized intersections and 39 intersections under control (rather than on small networks or corridors).

In our previous studies, we have demonstrated the effectiveness of the Surrogate Method to solve the traffic signal synchronization problem proposing a centralized approach. However, the computing time grows with the size of decision vector (i.e., the urban network that needs to be optimized). The Surrogate Method is applied to single sub networks, minimizing the total delay and optimizing the cycle, the offset, and the green time ratio. Different levels of exchange of information between subnetworks are considered. In this paper, a decentralized approach is presented, and a comparison with a centralized approach is reported. Generally, a decentralization approach is created as a way to improve efficiency and take advantage of potential economies of scale. In fact, decentralization looks to improve the speed and flexibility. Our aim is to find a decentralized approach with a good trade off between optimality and computational time.

As shown previously, signal timing optimization in an urban network is an NP problem and a central approach is not able to find the optimal solution in a reasonable amount of time. The hierarchical approaches can find solutions faster but require significant investment in infrastructure to provide communications between a central unit and each local optimizer. The existing decentralized approaches find suboptimal signal timing parameters. Given the interactions among travellers and between travellers and the network in the transportation system, it is difficult to formulate pure mathematical models to evaluate performances. Simulation has been more widely used as the tool to evaluate transportation system performance under different policies. When the system operates in a stochastic environment and no closed form expression for objective function is available, the problem is further complicated by the need to estimate the function. In this case, traditional optimization methods based on derivatives cannot be applied. Most known approaches are based on some form of random search, or ordinal optimization approach. In addition, also being the simulation computationally expensive, the extensive exploration of the entire solution domain would imply unacceptable calculation time [

Different decomposition of the network is proposed

Different decentralized approaches based on different levels of cooperation are suggested

Some simplifications to reduce the calculation time of

A comparison between centralized and decentralized approach is presented

Given the characteristics of the signal setting problem, in this paper, some simplifications for the

The signal setting improves driver safety, but it also provokes delays. For this reason, many researches try to minimize total delay, being the sum of all vehicles delays. This objective function is often calculated through simulation approaches. The

The approach is described in Figure

Scheme of our approach.

The remainder of the paper is organized as follows: The problem description is given in

Over the past years, the well-known Cell Transmission Model (CTM by Daganzo [

The

The

Given the urban network

The delay is considered as the number of vehicles obstructed in the road section (as in the paper of Lo see [

The delay of cell

The problem is formulated as follows:

where

The

The analysis of the objective function shows that this function is convex with respect to the cycle; instead it presents many local minimum with regard to the green split vector and offsets. Since the Surrogate Method calculation time is strictly dependent on the size of the problem, the cycle is not optimized by Surrogate Method. The Cycle is the same for all junctions and fixed by the binary search, and only after it has been found its optimum value the Surrogate Method is applied. The shape of the objective function looks convex with respect to the cycle variations and it is quasiconvexity respect when the green ratio changes and non convex with many peaks, with respect to the offset variations. From this analysis, it seems that the hardest task for signal synchronization problem is the offset setting. Figure

Shape of the objective function.

Spatial problem decomposition is the process that divides in small areas the space used for the optimization. The optimization of the decomposition is to identify some subsets of traffic lights that can be representative for the entire network [

Based on the physical topology model of the urban road traffic network, urban network

A classification for the nodes of the urban network is introduced to define a new clustering method. A Node Priority considering three different parameters is here introduced:

The degree of the node

Betweenness

Given

Flow

The Node Priority

Given the network

The node priority is calculated by the sum of three normalized elements, this implies that each element varies from 0 to 1. For a node, the normalized degree increases with the number of the incident edges on it, and the betweenness increases with the number of the shortest paths traversed it and the normalized flow increases with the flow directed into the node. It is evident that the first two terms can change only if the urban network changes. While considering these priorities it is possible to understand the importance of each node with respect to the network. The Node Priority is also used to determine the Priority of each subnetwork. The Priority of the sub-network is given by the sum of the Priority of each node in the subnetwork. Given the partition, for each node and for each subnetwork, the Priority is calculated.

Given the procedure to calculate the Priority, the proposed algorithm to clustering the network is introduced.

The Hybrid algorithm suggests a clustering of the network considering the Priority method. The Hybrid algorithm adds to the clustering one subnetwork

In Figure

Given a clustering of the entire urban network, we proposed different decentralized approaches based on different levels of cooperation between the individuated subnetworks.

Step 0 Given:

the network

Step 1 Calculate

Step 2 WHILE

Step 3 WHILE

Step 4 RETURN

Example of Hybrid clustering based on Newman partition. (a) Newman clustering. (b) Hybrid clustering.

Before introducing the different approaches, the concept of Traffic Signal Control is introduced.

Traffic Signal Control

Given the network

The

In practice, the Traffic Signal Control, based on the

Coopnet

The Coopnet is a function that, given the network

This process utilizes the same criteria of Hybrid algorithm without limiting the dimension of the subnetwork.

COOP means that

The level of cooperation is minimal, and the Surrogate Method is applied only on

Apply

SM Optimizes

Fix randomly

COOP is the method that optimizes only the subnetwork

COOP1 means that

Apply

SM Optimizes

Update

Apply

COOP2 means that

Apply

SM Optimizes

Update

UNTIL

{

SM Optimizes

COOP3 means that the

Apply

SM Optimize

Update

UNTIL

{

SM Optimizes

A scheme of the Network Clustering method and the cooperation methods is reported in Figure

Considering the different levels of cooperation, the signal setting parameters for a subset nodes (just the node in

Procedure providing the clustering and the signal setting parameters of the urban network.

The Surrogate Method (

The steps sequence of the algorithm is reported in Algorithm

0 Initialize

For any iteration

1 Determine the selection set

Initialize

Repeat the following steps Until

∗

∗

∗

∗

2 Select a

(i)

3 Evaluate the gradient estimation

(i)

(ii) using the following relationship

(iii)

(iv)

4 Update state:

5 If some stopping condition is not satisfied, repeat steps for

Vector

The basic idea of this method is to solve a continuous optimization problem by stochastic approximation methods and establish the fact that when (and if) a solution of the relaxed problem

Note, however, that the sequence

This has two advantages:

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 optimization process)

It allows us to make use of information typically employed to obtain cost sensitivities from the

Note that there is an additional operation: the

For each updating step of the Surrogate state (Step 4), it is computed (

A different green time ratio vector is able to provide the same value of

if

else

end if

for

if

else

end if

end for

For every gradient estimation, the SM requires

The green splits constraints imply a simplification of the decision vector.

For every intersection, there is one independent variable only; as a matter of fact, the sum of green for every links of an intersection is equal to the cycle time. One decision variable for each intersection is taken into consideration (i.e., if the cycle is 100, and the intersection is composed by three intersection links, given the green times vector

The results of the optimization approach applied on the entire network (centralized control) are compared to the results obtained while considering different network partitions and different levels of cooperation. The goal is to analyze whether the optimization must be applied on the entire network or obtain satisfying results just applying the Surrogate Method on subnetworks.

The aim of this paper is to evaluate if the decentralized control can be used to solve the synchronization signal problem. For this reason the evaluation of the trade-off between efficiency (goodness of the solution) and efficacy (computational time) is fundamental. Each method is compared in terms of efficacy (

The Hybrid Algorithm is applied to Newman and K-means clustering methods. The Newman clustering is based on betweenness, considering in part our Priority Method. K-means takes in consideration the neighborhood concept, often used in the literature when a set of nodes are detected to be optimize. The results put in evidence that the best results are given by the Hybrid Method considering the Newman clustering, which provides the same solution of the centralized method. This highlights the efficiency and efficacy of our Priority Method that finds a subset of nodes that represent the whole network.

In this section, to better explain the Network Clustering Method, it is carried out a small case study, and some results are presented. The small network is the same presented in [

Small case study. The centroids are red, signalized intersections are red, and links are green.

The

An inflow demand of 800 (veic/h) to exam different levels of congestion is also performed. The delay is calculated in seconds. All experiments are performed on a desktop computer with an Intel i5-3470 processor (3.2 GHz) with 8 GB of DDR3 RAM running 64-bit Windows 7. Even if each iteration takes 100 msec to complete, plus the communication overhead to calculate the objective function by

It is here first reported a comparison for the extensions on the

The extensions of Surrogate Method are compared in terms of efficiency (number of times that the

Efficiency of extensions.

SM | 7906 | 291 | |

7906 | 106 | ||

7743 | 271 | ||

7743 | 208 |

Despite the simplifications on the

An application of the clustering method is given in Figure

Clustering individuated by 4-means partition. (a) 4-means clustering. (b) Priority process.

Given the small dimension of the test network the dimension of the sub network introduced by Hybrid method is fixed to 5. The Hybrid clustering adds a cluster formed by the cluster with high Priority (1,2) together with the node with high Priority. One is for the others clusters, obtaining the clustering:

In

Table

These preliminary results stress that it is possible to find a good compromise between subnetwork and efficiency. The comparison is performed between the centralized approach (

Decentralized vs centralized with Newman decomposition.

Subnetworks | CT | ||||
---|---|---|---|---|---|

Newman | [1,2,3,5,6],[4,7],[8,9] | ||||

Tot net | [1,2,3,4,5,6,7,8,9] | 5997 | 35 | ||

Main net | [1,2,3,5,6] | 6330 | |||

COOP | [1,2,3,4,5,6,8] | 6318 | 23 | ||

COOP1 | [1,2,3,4,5,6,8],[4,7],[8,9] | 6214 | 5,6 | ||

COOP2 | [1,2,3,4,5,6,8],[4,7],[8,9] | 6195 | 6 | ||

COOP3 | [1,2,3,4,5,6,8],[4,7],[8,9] | 5999 | 11 |

Decentralized vs centralized with 4-means inflow decomposition.

Sub-networks | CT | ||||
---|---|---|---|---|---|

4-means | [1,2],[3,5,6],[7,8,9],[4] | ||||

Tot net | [1,2,3,4,5,6,7,8,9] | 5997 | 35 | ||

Main net | [1,2] | 6489 | 0.7 | ||

COOP | [1,2,4,6,8] | 6411 | 6 | ||

COOP1 | [1,2,4,6,8],[5,6,9],[7,8,9],[4] | 6411 | 6 | ||

COOP2 | [1,2,4,6,8],[5,6,9],[7,8,9],[4] | 6382 | 7 | ||

COOP3 | [1,2,4,6,8],[5,6,9],[7,8,9],[4] | 6017 | 10.5 |

Decentralized vs centralized with 3-means adjacent decomposition.

Subnetworks | CT | ||||
---|---|---|---|---|---|

3-means | [5,7,8,9],[2,3,6],[4,1] | ||||

Tot net | [1,2,3,4,5,6,7,8,9] | 5997 | 35 | ||

Main net | [5,7,8,9] | 6411 | 6 | ||

COOP | [2,4,5,7,8,9] | 6175 | 17 | ||

COOP1 | [[2,4,5,7,8,9],[2,3,6],[4,1] | 6173 | 17 | ||

COOP2 | [[2,4,5,7,8,9],[2,3,6],[4,1] | 6102 | 19 | ||

COOP3 | [[2,4,5,7,8,9],[2,3,6],[4,1] | 6093 | 18 |

Comparison between Newman clustering and Hybrid clustering (the delay is given in seconds).

The real case study has been conducted on a large-size network located in Rome, the area of Eur. Figure

Real case study.

The Newman algorithm gives 4 clusters, and this is why K is set to 4 and 5. In Figure

Trade-off efficiency-efficacy.

It is important to notice that the decomposition approaches provide optimal results, and the worsening of the objective function can reach a maximum of

The best results are given by the Hybrid Method considering the Newman clustering, which provides the same solution of the centralized method. The results of Hybrid decomposition based on Newman clustering are explicitly reported in Table

Decentralized vs centralized with Hybrid decomposition based on Newman clustering.

Sub-networks | CT | ||||
---|---|---|---|---|---|

Hybrid | [4 7 9 14 16 18 35 41 45 48], | ||||

[1 2 3 5 6 8], [31 32 33 38 39 40 46 47], | |||||

[20 21 36 37 42 43 44],[22 23 24 25 26 27 28 29 30] | |||||

Tot net | 15146 | 222 | |||

COOP | [4 6 7 9 14 16 18 19 20 28 31 35 41 45 48] | 15177 | 33 | ||

COOP1 | 15275 | 51 | |||

COOP2 | 15159 | 55 | |||

COOP3 | 15159 | 106 |

Decentralized vs centralized with Hybrid decomposition based on 4-means.

Sub-networks | CT | ||||
---|---|---|---|---|---|

Hybrid | [1,2,4,5,8],[6,3],[7,9] | ||||

Tot net | [1,2,3,4,5,6,7,8,9] | 5997 | 35 | ||

Main net | [1,2,4,5,8] | 6415 | 3 | ||

COOP | [1,2,4,5,6,7,8] | 6081 | 17 | ||

COOP1 | [1,2,4,5,6,7,8],[6,3],[7,9] | 6051 | 18 | ||

COOP2 | [1,2,4,5,6,7,8],[6,3],[7,9] | 6051 | 18 | ||

COOP3 | [1,2,4,5,6,7,8],[6,3],[7,9] | 5999 | 19 |

The signal setting problem is a nonconvex problem; usually to find an optimal solution for simple networks may take long time, when it is possible. It is here suggested a decentralized control, considering different clustering approaches for the network, together with a procedure to classify the nodes. The Surrogate Method is applied to solve the Traffic Signal Synchronization problem, for each identified subnetwork. This decomposition provides comforting results for other indexes concerning the composition is currently under study. The reduced computation time is not sufficient to run online; however, it is possible to run the decentralized approach more times in a day during principal time slices, while obtaining optimal results. An improvement of calculation time can give the possibility to apply the

In this section, the principal characteristics of

Representation of the node.

The flow conservation equation used for

The number of vehicles present in each cell

Inflow of the cells belongs to the

When

It permits to maximize the demand of upstream lane

Because of conflict between turning vehicles and ahead vehicles, the total inflow of channelized zone can be formulated as follows:

Inflow of each direction can be calculated as

To access the channelized zone, the vehicles directed to different turns may obstruct each other. For this reason, in oversaturated conditions, their behavior could block different movements. The following simple case considers only the interactions between left-turn (

It considers the maximum flow given by (

Channelized zone, for

For each

The

This model is based on the same method used by Flotterod to define the demand constraint function (gap acceptance method). The

In order to have a realistic capture of the potential capacity of minor flow

This equation can be applied on the node model to evaluate the max capacity of minor flow. The capacity determination of minor streams of the

K-means is one of the simplest unsupervised learning algorithms able to solve the well known clustering problem; for more details, see [

Steps of K-means algorithm.

Obviously, the algorithm is also highly sensitive to the choice of initial centers. The K-means algorithm can be run multiple times to reduce this effect.

In Girvan-Newman algorithm the number of clusters is not fixed a priori, and it detects the clusters by removing edges from the original network; for more details, see [

Steps of Newman algorithm.

The data used in the study are available on request.

The authors declare that they have no conflicts of interest.