A new mobility field and gradient-based traffic signal control approach applicable to large-scale road networks

Abstract

Previous traffic control models were usually developed on small or medium sized road networks. The traffic control models applicable to large-scale road networks have received growing interest. In this study, we develop a new mobility field and gradient-based traffic signal control approach applicable to large road networks. First, we introduce an emerging analytical technique, the mobility field approach, to generate the mobility field of urban travels and measure the gradients of the mobility field. Next, a gradient-based approach is proposed to identify the signalized intersections for implementing traffic control. Finally, a gradient-based traffic control model is developed to alleviate traffic congestion during mass events. A new solution algorithm, termed DBSCAN-FW-GA, is proposed to solve the developed traffic control model. The developed mobility field and gradient-based traffic signal control approach is validated using actual road network data and travel demand data. Results indicate that the proposed new traffic control approach can reduce by 17.97% the travel time compared with the widely used perimeter control approach.

1. Introduction

Mass events held in big cities (e.g. sporting events, festival activities, commercial activities) usually generate extraordinarily large volumes of travel demand, posing tremendous pressure to urban transportation infrastructures. Indeed, it is extremely challenging to effectively alleviate traffic congestion in the surrounding area of the event location, first because of the large volume of traffic flow and second because of the generally long duration of mass events [1]. The severe traffic congestion caused by mass events increases fuel consumption and CO₂ emissions [2], and can lead to a dramatic increase in travel time and accidental risk [3]. However, the adaptive traffic signal control approaches applicable to anomalous travel demand situations are still lacking. This may be caused first by the lack of traffic signal control approaches applicable to large road networks (mass events attract participants all around the city) and second by the lack of mobility data documenting the anomalous growth of travel demand during mass events.

Here, we aim to fill the research gap by developing a traffic signal control approach applicable to both large road networks and anomalous travel demand situations. Here, we first review previous studies in this area.

Traffic signal control approaches can be classified into fixed-time signal control approaches and adaptive signal control approaches. Fixed-time signal control approaches aim to maintain stable performance under daily traffic conditions [4–6]. Nonetheless, fixed-time approaches are not capable of dealing with the travel demand uncertainties caused by accidents or mass events, which usually result in serious traffic jams. In the past few decades, adaptive signal control approaches have been developed and implemented to cope with dynamic traffic conditions. The well-known adaptive signal control systems include SCOOT (Split, Cycle and Offset Optimization Technique) [7] and SCATS (Sydney Coordinated Adaptive Traffic System) [8]. These two adaptive signal control systems have been widely used in practice; however, these systems may encounter difficulties when they are applied in oversaturated or congested road networks [9].

In order to cope with traffic flow fluctuations, researchers have made great efforts in the field of adaptive traffic signal control. Addressing the uncertain travel demand at signalized intersections is one important research direction. Researchers have developed various traffic signal optimization models for coordinated traffic control [10–14] and traffic blockage prevention [15, 16]. These models, in general, have improved the traffic control performance by simultaneously controlling the signalized intersections of a single road segment. Another research direction focuses on improving the overall network efficiency [17, 18]. A series of adaptive traffic signal control approaches have been proposed, such as the traffic-responsive hybrid signal control approach [19], the iterative learning approach [20] and the two-stage stochastic model [21]. A common feature of previous adaptive traffic signal control approaches is that they were in general developed on small or medium sized road networks. In another words, these approaches may not be applicable to large road networks due to the high computational cost.

Different from the aforementioned traffic signal control approaches, the perimeter control approach can be applied in large road networks. When implementing perimeter control, the spatial partitioning methods are first used to partition the studied region into several homogenous clusters, in which road segments are characterized with similar density and speed properties [22, 23]. Consequently, the intersections connecting different clusters are identified to control the transfer flows between clusters. There are a number of spatial partitioning methods proposed, of which the Normalized-cut (N-cut) method proposed by Ji and Geroliminis [24] and the Symmetric Non-negative Matrix Factorization (SNMF) framework proposed by Saeedmanesh and Geroliminis [25] are the two most widely used ones. Previous studies have demonstrated that the perimeter control approach could well maintain the overall performance of a large road network. Yet, it is still not clear if the perimeter control approach is capable of dealing with the local, severe and long-duration traffic congestion caused by mass events.

In this study, we aim to fill the research gap by introducing an emerging analytical technique, the mobility field approach [26, 27], to develop a new mobility field and gradient-based traffic signal control model applicable to large road networks and anomalous travel demand situations. A new solution algorithm, termed DBSCAN-FW-GA (combining Density-Based Spatial Clustering of Applications with Noise, Frank-Wolfe Algorithm, and Genetic Algorithm), is proposed to solve the developed traffic control model. The actual large-scale road network data and licence plate recognition (LPR) data of Changsha, a major southern city of China, are used to validate the proposed mobility field and gradient-based approach. We have also compared the proposed approach with the widely used perimeter control approach.

The main contributions of the paper are as follows:

1. We introduce the mobility field approach to analyze the anomalous travel demand in an urban road network.

2. The gradient measure is proposed to identify the key intersections for implementing traffic control during a mass event.

3. We propose a gradient-based two-stage traffic signal control model, which can effectively alleviate traffic congestion in the peripheral area of the mass event.

The remainder of this paper is organized as follows. The road network data and travel demand data are introduced in Section 2. In Section 3, we describe the methods for generating the mobility field and calculating the gradient. Section 4 formulates the mobility field and gradient-based traffic signal control approach. Section 5 presents the results of the proposed approach compared with other benchmark models. We summarize the conclusions and future works in Section 6.

2. Data

2.1. Road network data

The road network data of Changsha recorded the coordinates of road segments and road intersections, the number of lanes and the length of each road segment. The road network is composed of 5882 road segments and 2329 road intersections, covering the major urban area of Changsha (Fig. 1). The number of signalized intersections connecting four road segments is 365. For simplicity, in this paper we denote road segments as links. According to the code for the design of urban road engineering [28], we set the speed limit of each link to 60 km/h and the lane capacity to 1800 veh/h/lane. The free-flow travel time of each link is then calculated. The saturation flow of each link is calculated by multiplying the lane capacity by the number of lanes. The geographic area of Changsha is partitioned into 1088 |$1\!\times\!1$| km² grids for further analysis.

2.2. Licence plate recognition data

The LPR data were collected from 2 to 8 November 2017. We find that the LPR stations are widely distributed in Changsha (Fig. 1). Each time a vehicle passed by an LPR station, the time, licence plate number and station ID were recorded (licence plate number was anonymized). In the original LPR data, a total of 3,948,734 vehicles were identified, and 120,450,624 records were collected by 540 LPR stations.

To extract vehicle trips from the LPR data, we first filter out 391,961 records with information missing and 49,249 records with anomalous licence plate numbers. In addition, a total of 3,255,220 records with licence plate numbers appearing only once are filtered out (vehicle trips cannot be extracted from these records). Next, for each vehicle we sort its LPR records according to the data collection time. When the time difference between two consecutive records is larger than 30 minutes, the two records are assigned to different trips [29]. Besides, a total of 63,818 (0.414%) trips that started and ended at the same LPR station and took less than 1 minute are filtered out. Finally, a total of 15,312,076 trips are extracted during the data collection period.

The first record and the last record of each vehicle trip are denoted as the origin LPR station and the destination LPR station of the trip. After obtaining the trips between each pair of LPR stations, we can further calculate the number of trips between each pair of |$1\!\times\!1$| km² grids. In this study, a day is divided into 96 15-min time windows. The number of trips generated in each day is shown in Fig. 2(a), and the number of trips generated during each time window is shown in Fig. 2(b).

2.3. Simulating the anomalous travel demand during a mass event

There was no mass event during the data collection period, hence we simulate a mass event occurring on a weekday, which is described in the following.

• Step 1: Generate the average number of daily trips. We calculate the mean |$\mu _{mn}^h$| and the standard deviation |$\sigma _{mn}^h$| of the number of trips between LPR stations |$m$| and |$n$| during time window |$h$|⁠. The average number of daily trips |$\bar{T}_{mn}^h$| is generated using the normal distribution:

  $$\begin{eqnarray} \bar{T}_{mn}^h = \left\{ {\begin{array}{@{}*{1}{c}@{}} {N\left( {\mu _{mn}^h,{{{\left( {\sigma _{mn}^h} \right)}}^2}} \right),\ {\rm{if}}\ N\left( {\mu _{mn}^h,{{{\left( {\sigma _{mn}^h} \right)}}^2}} \right) > 0}\\ {0,\ {\rm{if}}\ N\left( {\mu _{mn}^h,{{{\left( {\sigma _{mn}^h} \right)}}^2}} \right) \le 0} \end{array}} \right. \end{eqnarray}$$

  (1)

  where |$\bar{T}_{mn}^h$| follows a normal distribution with mean |$\mu _{mn}^h$| and variance |${{( {\sigma _{mn}^h} )}^2}$| (i.e. |${\rm{N}}( {\mu _{mn}^h,{{{( {\sigma _{mn}^h} )}}^2}} )$|⁠). If |${\rm{N}}( {\mu _{mn}^h,{{{( {\sigma _{mn}^h} )}}^2}} )$| equals or is smaller than 0, |$T_{mn}^h$| is 0.

• Step 2: Estimate the number of additional trips. The LPR station within the grid of the event location is denoted as |$c$|⁠. We calculate the average number of trips from LPR station |$m$| to LPR station |$c$|⁠, |$\mu _{mc}^h$|⁠. An expansion coefficient |$\lambda $| is proposed to quantify the ratio between the number of additional trips on the event day and the average number of daily trips. We assume that |$\lambda ( h )$| follows a Gaussian distribution (Equation (2)), where |$h$| represents the index of time window (i.e. |$h \in \{ {0,{\rm{\ }}1,{\rm{\ }}\ldots,{\rm{\ }}95} \}$|⁠), and |${{\lambda }_{\rm max}}$|⁠, |${{\mu }_\lambda }$| and |${{\sigma }_\lambda }$| represent the maximum, mean and standard deviation of |$\lambda $|⁠, respectively [30]. The number of additional trips attracted by the mass event |$\hat{T}_{mc}^h$| is estimated by multiplying |$\mu _{mc}^h$| by |$\lambda $|⁠.

  $$\begin{eqnarray} \lambda \left( h \right) = {\rm{\ }}{{\lambda }_{\rm max}} \times {{\rm e}^{ - \frac{{{{{\left( {h - {{\mu }_\lambda }} \right)}}^2}}}{{2{{\sigma }_\lambda }^2}}}} \end{eqnarray}$$

  (2)

   

  $$\begin{eqnarray} \hat{T}_{mc}^h = \mu _{mc}^h \times \lambda \left( h \right) \end{eqnarray}$$

  (3)
• Step 3: Estimate the number of trips in an event day. The number of trips in an event day |$T_{mn}^h$| is the sum of the average number of daily trips |$\bar{T}_{mn}^h$| and the number of additional trips |$\hat{T}_{mc}^h$|⁠:

  $$\begin{eqnarray} T_{mn}^h = \left\{ {\begin{array}{@{}*{1}{c}@{}} {\bar{T}_{mn}^h,\ {\rm if}\ n \not= c}\\ {\bar{T}_{mn}^h + \hat{T}_{mc}^h,\ {\rm if}\ n = c} \end{array}} \right. \end{eqnarray}$$

  (4)

Finally, we map each LPR station to the grid in which it is located. The mean and standard deviation of the number of trips between grids |$i$| and |$j$| during time window |$h$| are denoted as |$\mu _{ij}^h$| and |$\sigma _{ij}^h$|⁠. The number of trips between grids |$i$| and |$j$| in an event day is denoted as |$T_{ij}^h$|⁠.

3. Mobility field and gradient measure

We employ the emerging mobility field approach to identify the signalized intersections for implementing traffic control.

3.1. Pre-processing of the trips in a mass event

We follow a method proposed in Yang et al. [31] to process the trips in a mass event.

First, the z-score method is used as an outlier detection method to identify the origin-destination (OD) pairs with trips much larger than the average number of daily trips. According to the z-score method presented in Equation (5), the degree |$z_{ij}^h$| between grids |$i$| and |$j$| is calculated by dividing (⁠|$T_{ij}^h - \mu _{ij}^h$|⁠) by |$\sigma _{ij}^h$|⁠. If |$z_{ij}^h$| is greater than 3 (which is a commonly used threshold value), the OD pair (⁠|$i$|⁠, |$j$|⁠) and its additional trips |$\hat{T}_{ij}^h$| (Equation (6)) are kept for further analysis.

Second, we employ the K-MWO algorithm to select the OD pairs characterized with large volumes of additional trips. Note that the OD pairs with small volumes of additional trips are filtered out since they play negligible roles in the traffic congestion during mass events. The K-MWO algorithm is a swarm intelligence-based clustering method which can achieve global optimization by taking advantage of the swarm intelligence. The K-MWO algorithm is introduced in detail in Kang et al. [32]. Two important parameters of the K-MWO algorithm are the feature |$F$| and the number of clusters |$K$|⁠. In this study, feature |$F$| is the number of additional trips. The number of clusters |$K$| is determined using the Silhouette coefficient [33]. The value of |$K$| is tested from 2 to 10 (i.e. |$K = {\rm{2,\ 3,\ 4,\ \ldots ,\ 10}}$|⁠), and the optimal value of |$K$| is obtained when the highest Silhouette coefficient is reached. After implementing the K-MWO algorithm, the cluster of OD pairs characterized by small volumes of additional trips is filtered out; the additional trips in the remaining |$K\!-\!1$| clusters of OD pairs are kept for further calculation.

3.2. The mobility field, potential and gradient

The processed additional trips are used to generate the mobility field. For each grid |$i$|⁠, we introduce a vector |$\vec{W}_i^h$| to capture the cumulative moving direction of the additional trips originated from grid |$i$| in time window |$h$|⁠. Vector |$\vec{W}_i^h$| is calculated using Equations (7) and (8), where |$| {W_i^h} |$|⁠, the module of |$\vec{W}_i^h$|⁠, ranges between 0 and 1.

where |$\vec{W}_i^h$| excludes the internal trips of grid |$i$|⁠. The unit vector |${{\vec{u}}_{ij}}$| directs from grid |$i$| to grid |$j$|⁠. The resultant vector of grid |$i$|⁠, |$\vec{T}_i^h$|⁠, is calculated by vectorially summing the vector |$\hat{T}_{ij}^h{{\vec{u}}_{ij}}$| pointing to all destinations |$j$| (⁠|$\vec{T}_i^h = \mathop \sum \limits_j \hat{T}_{ij}^h{{\vec{u}}_{ij}}$|⁠). A local mass |$m_i^h$| represents the volume of additional trips originated from |$i$|⁠, including the internal trips within |$i$|⁠.

The module |$| {W_i^h} |$| is a relative value quantifying the concentration of the directions of additional trips originated from grid |$i$|⁠. The module |$| {W_i^h} |$| is small if the directions of additional trips are diverse, whereas |$| {W_i^h} |$| is large if the directions of additional trips are concentrated. In extreme cases, |$| {W_i^h} | = 0$| when the additional trips originated from |$i$| are balanced in opposite directions, and |$| {W_i^h} |$| equals to 1 when the additional trips originated from |$i$| have the identical direction. After calculating the vector |$\vec{W}_i^h$| of each grid, the mobility field of the additional trips is generated.

In order to compute the potentials of the grids, we set |$V = 0$| at the corner grid of the city and calculate the potential of other grids using Equations (9) and (10).

where |$\alpha $| and |$\beta $| represent that grid |$i$| is at |$\alpha $| row and |$\beta $| column of a two-dimensional plane. |$V_{( {\alpha + 1,i} )}^h$| represents the potential of the grid located on the east of grid |$i$|⁠, and |$V_{( {\beta + 1,i} )}^h$| represents the potential of the grid located on the north of grid |$i$|⁠. |$WI_i^h$| and |$WJ_i^h$| are the x-direction and y-direction components of |$\vec{W}_i^h$|⁠. |$\Delta x$| and |$\Delta y$| are the side length of each grid, which are both 1 km.

For example, we first calculate the potential of the grid located on the east of the corner grid using Equation (9). Next, we calculate the potential of the grid located on the north of the corner grid using Equation (10). The potential of other grids can also be iteratively calculated. The zero potential |$V = 0$| is also set to the other three corner grids of the city to calculate the potentials of the grids. The obtained potentials are then averaged to decrease noise.

In this study, potential is used to quantify the energy of each grid. Given that energy always flows from the high potential area to the low potential area, we can identify the direction and the intensity of additional trips by analysing the potential difference between grids. This is achieved by calculating the gradient of each grid. A second-order finite difference method [34] is adopted to calculate the gradient |$G_i^h$| using Equation (11).

For a region with trips occupying diverse directions, the modules |$| {W_i^h} |$| of the grids within the region are relatively small. Consequently, the potential differences of neighbouring grids within the region are small (Equations (9) and (10)). This will consequently result in small grid gradients (Equation (11)). On the contrary, if there is a large volume of trips sharing similar directions within a region, the modules |$| {W_i^h} |$| of the grids within the region are large, which will consequently generate large potential differences between neighbouring grids and large grid gradients. Hence, the large gradient of a grid indicates that there is a large volume of trips sharing similar directions passing through it, hinting that a larger gradient may incur a higher probability of traffic congestion.

4. The mobility field and gradient-based traffic signal control approach

We propose a gradient-based two-stage model to address the traffic signal control problem during mass events. A solution algorithm, termed DBSCAN-FW-GA, is developed.

4.1. Notations

Notations used in this section are summarized in Table 1.

4.2. Model formulation

We propose a gradient-based two-stage model to solve the optimal traffic signal green splits and minimize the total travel time within the grid of the event location. Fig. 3 shows the model structure diagram. In Stage 1, we establish the P1 model to minimize the total travel time in the road network with respect to a fixed green split (Equation (12)). The three kinds of constraints in the P1 model are those of user equilibrium, of flow conservation, and of green split and travel time. The P1 model outputs the fixed green split and equilibrium routes of drivers on ordinary days. In Stage 2, we establish the P2 model to minimize the total travel time within the grid of the event location with respect to adaptive green splits (Equation (23)). The five kinds of constraints in the P2 model are those of user equilibrium, of flow conservation, of mobility field, potential and gradient, of gradient-based perimeter control, and of green split and travel time.

This paper only considers phase splits of signalized intersections; cycle length and offset are predefined and fixed [35]. The two-stage model for traffic signal control is formulated as follows:

1) P1 model.

subject to:

Constraints of user equilibrium:

Constraints of flow conservation:

Constraints of green split and travel time:

Equations (13) to (16) are the constraints of user equilibrium for ordinary days. Complementary conditions: Equation (13) indicates that if the average daily flow |$f$| on route |$r$| of OD pair |$w$| is greater than 0, the travel time on route |$r$| is the shortest travel time between OD pair |$w$| (i.e. |$\bar{b}_r^w - {{\bar{\pi }}_w} = 0$|⁠); otherwise, if the average daily flow |$f$| on route |$r$| of OD pair |$w$| equals to 0, the travel time on route |$r$| is greater than the shortest travel time between OD pair |$w$| (i.e. |$\bar{b}_r^w - {{\bar{\pi }}_w} > 0$|⁠). Equations (14) and (15) are non-negative constraints. Equation (16) specifies that the sum of |$\bar{f}_r^w$| equals to the average number of daily trips between OD pair |$w$| (⁠|${{\bar{d}}_w}$|⁠).

Equations (17) to (19) are the constraints of flow conservation. Equation (17) indicates that trips |${{\bar{d}}_w}$| of an ordinary day include the trips |${{\bar{d}}_{w1}}$| made by the drivers who change their routes on the event day, and the trips |${{\bar{d}}_{w2}}$| made by the drivers who do not change their routes on the event day. The route flows are composed of the flows generated by these two types of drivers, as specified in Equation (18). Link flows are calculated using the route flows in Equation (19).

Equations (20) to (22) are the constraints of green split and travel time. Equation (20) specifies that for a link |$a$| where vehicles are moving east to west at a controlled intersection |$c$|⁠, the fixed green split |${{u}_a}$| equals to |${{u}_c}$|⁠, and for a link |$a$| where vehicles are moving south to north at |$c$|⁠, |${{u}_a}$| equals to (⁠|$1 - {{u}_c}$|⁠). The lower bound and the upper bound of |${{u}_c}$| are defined in Equation (21). The minimum and maximum green time split values are respectively set to 0.45 and 0.55. The link travel time is calculated using Equation (22). We use the Bureau of Public Roads (BPR) function to estimate the link travel time, where |$\alpha $| and |$\beta $| are set to 0.15 and 4, respectively.

2) P2 model.

Subject to:

Constraints of user equilibrium:

Constraints of flow conservation:

Constraints of mobility field, potential and gradient: Eqs. (7) to (11).

Constraints of gradient-based perimeter control:

Constraints of green split and travel time:

Equations (24) to (27) are the constraints of user equilibrium for the event day. Complementary conditions: Equation (24) indicates that if the flow |$f$| on route |$r$| of OD pair |$w$| on the event day is greater than 0, the travel time on route |$r$| is the shortest travel time between OD pair |$w$| (i.e. |$b_r^w - {{\pi }_w} = 0$|⁠); otherwise, if the flow |$f$| on route |$r$| of OD pair |$w$| on the event day equals to 0, the travel time on route |$r$| is greater than the shortest travel time between OD pair |$w$| (i.e. |$b_r^w - {{\pi }_w} > 0$|⁠). Equations (25) and (26) are non-negative constraints. Equation (27) specifies that the sum of |$f_r^w$| equals to the first part of daily trips (⁠|${{\bar{d}}_{w1}}$|⁠) and the additional trips on the event day (⁠|${{\hat{d}}_w}$|⁠).

Equations (28) and (29) are the constraints of flow conservation. Equation (28) specifies that for each OD pair |$w$|⁠, the number of trips on the event day |${{d}_w}$| is composed of the average number of daily trips |${{\bar{d}}_w}$| and the number of additional trips |${{\hat{d}}_w}$|⁠. Equation (29) calculates the link flows based on the route flows. The route flows consist of two parts: the route flows |$\bar{f}_r^{w2}$| generated by the drivers who follow the routes used on ordinary days, and the route flows |$f_r^w$| generated by the drivers who follow the equilibrium routes on the event day.

Equations (7) to (11) are the constraints of mobility field, potential and gradient. Equations (7) and (8) are used to generate a mobility field in time window |$h$| of the event day. Equations (9) and (10) calculate the potentials of grids in the mobility field. Equation (11) calculates the gradient of each grid according to the obtained potential distribution of the mobility field.

Equations (30) to (32) are the constraints of gradient-based perimeter control. Equation (30) calculates the mean gradient of each grid |$i$| during |$H$| 15-min time windows. Equation (31) specifies that the signalized intersection |$c$| within the grid |$i$| has the same mean gradient. Equation (32) indicates that if the mean gradient of intersection |$c$| is larger than the gradient threshold |${{\delta }_G}$|⁠, intersection |$c$| should be adaptively controlled (i.e. |${{e}_c} = 1$|⁠); otherwise, intersection |$c$| uses the fixed green split (i.e. |${{e}_c} = 0$|⁠).

Equations (33) to (35) are the constraints of green split and travel time. In this study, only the signalized intersections connecting four links are selected for implementing traffic control. We assume a two-phase signal control scheme for each signalized intersection [35]. Equation (33) specifies that for a link |$a$| where vehicles are moving east to west at an adaptively controlled intersection |$c$|⁠, the adaptive green split |${{g}_a} = {{g}_c}$|⁠, and for a link |$a$| where vehicles are moving south to north at an adaptively controlled intersection |$c$|⁠, |${{g}_a} = ( {1 - {{g}_c}} )$|⁠. Equation (34) defines the lower bound and the upper bound of the adaptive green splits (i.e. 0.05 and 0.95) [36]. Equation (35) indicates that if the signalized intersection |$c$| is adaptively controlled (i.e. |${{e}_c} = 1$|⁠), the green split at |$c$| is |${{g}_a}$|⁠; otherwise, a fixed green split |${{u}_a}$| is used.

4.3. Solution algorithm

To solve the gradient-based two-stage model, we combine the DBSCAN (Density-Based Spatial Clustering of Applications with Noise) algorithm, the Frank-Wolfe algorithm and the genetic algorithm (DBSCAN-FW-GA) to search the intersections for implementing traffic control and the optimal adaptive green splits |${{g}_c}$|⁠. Specifically, the DBSCAN algorithm [37] is used to determine the gradient threshold and identify the intersections greatly influencing the total travel time within the grid of event location. The Frank-Wolfe algorithm [38] assigns trips to the road network. The genetic algorithm [39] generates different schemes with regard to |${{g}_c}$| and obtains the optimal solution for minimizing the total travel time within the grid of event location.

1. Gradient clustering

    We use the density-based spatial clustering of applications with noise (DBSCAN) algorithm to determine the gradient threshold and identify the set of signalized intersections for implementing traffic control. First, we divide the mass event period into |$H$| 15-min time windows. The gradient of each grid (i.e. |${{G}_i}$|⁠) during the mass event period is represented as |$\{ {G_i^1,G_i^2, \cdots ,G_i^h, \cdots ,G_i^H} \}$|⁠. Second, we calculate the mean of the gradients of each grid, denoted as |${{\mu }_{G( i )}}$|⁠. Next, the mean gradient of grid |$i$| (⁠|${{\mu }_{G( i )}}$|⁠), the grid horizontal index (⁠|$\alpha $|⁠) and the vertical index (⁠|$\beta $|⁠) are used as features of the DBSCAN algorithm. The maximum and minimum normalization is applied to each of these three features.

    The Euclidean distance is used to measure the |$\varepsilon $| neighbourhood of each grid, and a cluster partition requires to include |$minpts$| minimum number of grids, where |$minpts$| is set to 4. The value of |$\varepsilon $| is determined using a widely accepted method, which includes two steps. (1) Calculate the distance |${{d}_4}$| from each grid to its fourth-nearest neighbour; (2) sort the distance |${{d}_4}$| in a descending order, find the inflection point of the |${{d}_4}$| curve and use the value of |${{d}_4}$| at the inflection point as the value of |$\varepsilon $|⁠. Using the DBSCAN algorithm, grids are partitioned into different clusters. The gradient threshold |${{\delta }_G}$| equals to the maximum mean gradient in the cluster with the smallest average |${{\mu }_{G( i )}}$|⁠. Hence, the cluster of grids with the smallest average |${{\mu }_{G( i )}}$| is filtered out. The signalized intersections within the grids in the remaining clusters are identified as the intersections for implementing traffic control (i.e. |${{e}_c} = 1$|⁠).

2. DBSCAN-FW-GA algorithm

   The detailed process of the DBSCAN-FW-GA algorithm is elaborated as follows.

   • Step 1: Solve the P1 model using different fixed green splits |${{u}_c}$| ranging from 0.45 to 0.55 with an interval of 0.01. Under different |${{u}_c}$|⁠, use the Frank-Wolfe algorithm to assign the average number of daily trips |${{\bar{d}}_w}$| to the road network. Calculate the link flow |${{x}_a}$| and obtain the total travel time in the road network. The |${{u}_c}$| that minimizes the total travel time in the road network is used as the fixed green split on ordinary days.

   • Step 2: We determine the gradient threshold |${{\delta }_G}$| and identify the intersections for implementing traffic control using the DBSCAN algorithm.

   • Step 3: For each controlled intersection, initialize the green split |${{g}_c}$|⁠. A traffic control scheme is composed of the adaptive green split of each controlled intersection. Pseudorandom numbers are used to generate the chromosomes for each traffic control scheme in the population. The population size is set to |$p$|⁠. Set |$k$| as the iteration indicator, |$\tau $| as the crossover rate and |$M$| as the mutation rate.

   • Step 4: Solve the P2 model using the Frank-Wolfe algorithm. The variables |$f_r^w$|⁠, |${{\pi }_w}$| and |${{v}_a}$| are calculated. The total travel time within the grid of event location |$TTEL$| of each chromosome in the population is calculated.

   • Step 5: The |$TTEL$| is used as the result of the fitness function. The |$TTEL$| of all chromosomes is analysed to find the optimal solution of this population. Use the tournament method to select the offspring. For the existing chromosomes, two chromosomes are randomly selected with a probability |$\tau $| to carry out the crossover operator. Set a uniformly distributed random number |$R$| between |$[ {0,1} ]$| for each gene in the existing chromosomes. Change the value of gene (i.e. from 0 to 1, or from 1 to 0) if |$R < M$| and a new chromosome is generated.

   • Step 6: Stop the algorithm if |$k > {{k}_{\rm max}}$|⁠, where |${{k}_{\rm max}}$| is the maximum number of iterations. Output the minimal |$TTEL$| and the corresponding green splits |${{g}_c}$|⁠. If |$k \le {{k}_{\rm max}}$|⁠, set |$k = k + 1$| and go to Step 3.

5. Results

5.1. The mobility field and gradient

We assume that a mass event occurs at Wuyi Square of Changsha on a weekday. Wuyi Square is located in the centre of Changsha. Expansion coefficient |$\lambda ( h )$| in Equation (2) quantifies the number of additional trips attracted by the mass event. The mass event is assumed to attract participants from 3 pm to 10 pm. Therefore, the 15-min time window |$h$| ranges from 60 to 88. The parameters |${{\lambda }_{\rm max}}$|⁠, |${{\mu }_\lambda }$| and |${{\sigma }_\lambda }$| in Equation (2) are respectively set to 2, 15 and 4.5. The number of trips on the event day changes with the expansion coefficient |$\lambda ( h )$| and the average number of daily trips (Fig. 4). The additional trips between each pair of grids and the additional trips between each pair of LPR stations can be calculated using the methods proposed in Section 2.3.

Taking 7 pm to 7:15 pm as an example, the number of trips on ordinary days is 36,819, the expansion coefficient |$\lambda $| is 2 and the total number of additional trips is 879. Trips between each pair of grids are preprocessed using the z-score method and the K-MWO algorithm introduced in Section 3.1. We use the remaining additional trips to generate the mobility field (Fig. 5(a)) and calculate the potential distribution (Fig. 5(b)). The equipotential lines in Fig. 5(b) indicate that different regions have heterogenous attraction strengths to urban travel demand.

Next, we use Equation (11) to calculate the gradient of each grid during |$H$| 15-min time windows from 3 pm to 10 pm. Fig. 5(c) shows that the mean and standard deviation of the gradients of each grid are positively correlated. The physical meaning of mean gradient can be explained as follows. First, a grid with a large mean gradient indicates that a large volume of additional trips sharing similar directions passes through the grid (see the analysis in Section 3.2). Second, a grid with a large mean gradient is also featured with a large gradient standard deviation, suggesting that the grid experiences large gradient changes during the mass event; in another words the gradient of the grid may be influenced by the mass event. Therefore, trips attracted by the mass event are more likely to pass through the grids with a large mean gradient. This inspires us to select the grids with a large mean gradient as the candidate regions for implementing traffic control schemes. In this study, the signalized intersections within the same grid are defined to have the same gradient, and the mean gradient of each signalized intersection is shown in Fig. 5(d). We find that the signalized intersections near Wuyi Square are characterized by large mean gradients.

5.2. Gradient-based two-stage model

We first determine the intersections for implementing traffic control using the DBSCAN algorithm. The |$minpts$| (i.e. minimum number of points) is set to 4, and the value of |$\varepsilon $| is set to 0.095 (the inflection point of the |${{d}_4}$| value curve) (Fig. 6(a)). The grids are partitioned into two clusters and a cluster of noise (Fig. 6(b)). The cluster of noise represents the grid with an anomalously large mean gradient. The cluster depicted in yellow has the smallest mean gradient, with a maximum mean gradient of 0.245. Therefore, the gradient threshold |${{\delta }_G}$| equals to 0.245. The signalized intersections with a mean gradient larger than the gradient threshold |${{\delta }_G}$| are adaptively controlled to alleviate congestion caused by the mass event. Fig. 6(c) shows that the controlled intersections are located near Wuyi Square.

We focus on the time period of 5:30 pm to 7:30 pm when heavy traffic emerges. The experiment is carried out under 10 randomizations. In each randomization, we randomly select 80% of the trips to follow the equilibrium routes on the event day [40], and the remaining 20% of the trips still follow the routes used on ordinary days (see Section 4.2). Taking one randomization (denoted as |${{R}_1}$|⁠) as an example, we solve the gradient-based two-stage model using the proposed DBSCAN-FW-GA algorithm, in which the population size |$p$| is set to 50, the iteration indicator |$K$| is set to 50, the crossover rate |$\tau $| is set to 0.8 and mutation rate |$M$| is set to 0.02 [41]. Fig. 7 shows the convergency of |$TTEL$|⁠, and the optimal |$TTEL$| is 349.87 (veh.h). The fixed green split |${{u}_c} = 0.45$|⁠, and the adaptive green splits at 42 controlled intersections are presented in Table 2. The optimal traffic signal control schemes are considerably different from |${{u}_c}$|⁠. Some signalized intersections are strictly controlled, with the green split set to 0.05 or 0.95. Experiments on other randomizations receive similar results, as shown in Section 5.3.

5.3. Comparison of three traffic signal control models

We use three benchmark models to validate the effectiveness of the proposed mobility field and gradient-based traffic signal control model (denoted as the GBC model). The first benchmark model employs the fixed green split scheme |${{u}_c} = 0.45$| (denoted as the FC model). The second benchmark model selects intersections for implementing traffic control using a well-established spatial partitioning method (denoted as the SPC model). The third benchmark model selects all intersections for implementing traffic control schemes during the mass event (denoted as the AC model).

The SPC model is built upon the N-cut algorithm [42] improved by Ji and Geroliminis [24]. The SPC model first regards each road segment as a node and generates edges between the nodes. The weight of a node |$i$| is the volume over capacity (VOC) of the road segment |${{d}_i}$|⁠, where the traffic volume of the road segment is calculated using the FC model. The edge connecting node |$i$| and node |$j$| has a weight |$w( {i,j} )$|⁠. If node |$i$| and node |$j$| are spatially connected in the undirected form of the road network, |$w( {i,j} )$| is set to |${{\rm e}^{( { - {{{( {{{d}_i} - {{d}_j}} )}}^2}} )}}$|⁠; otherwise, |$w( {i,j} ) = 0$|⁠. Next, the SPC model partitions the road network into multiple clusters by finding an optimal number of clusters minimizing the evaluation metrics |$NS$|⁠. Readers can refer to Ji and Geroliminis [24] for detailed descriptions of partition criterion and evaluation metrics. For randomization |${{R}_1}$|⁠, we partition the road network into two clusters (Fig. 8(a)) and three clusters (Fig. 8(b)). Figs. 8(a) and (b) show that the optimal |$NS$| is obtained when the number of clusters is set to two. The perimeter of these two clusters and the identified 41 intersections for implementing traffic control are shown in Fig. 8(c).

We compare the total travel time within the grid of event location (TTEL) and the number of controlled signalized intersections (NCI) when applying the GBC model and the three benchmark models (Table 3). Using the SPC model the TTEL is reduced by 11.55%, whereas using the GBC model the TTEL is decreased by 29.52%, indicating that the GBC model can mitigate traffic congestion within the grid of event location in a more efficient manner. The TTEL decreases by 32.42% when using the AC model, however all intersections have to be controlled and a large number of ordinary travellers far from the mass event are influenced. Hence, the proposed GBC model can effectively identify the key intersections for implementing traffic control and alleviate traffic congestion in the peripheral area of the mass event.

The traffic flow distributions under the scenarios of implementing the FC model, the SPC model and the GBC model are illustrated in Fig. 9. We find that Wuyi Square is congested with a VOC larger than 0.6 when the fixed green split scheme (i.e. the FC model) is adopted (Fig. 9(a)). The traffic control scheme generated by the SPC model could not alleviate the congestion at Wuyi Square either (Fig. 9(b)). However, under the scenario of the GBC model, the VOC distribution in Fig. 9(c) shows that the road network around the event location becomes much less congested.

Moreover, we compare the traffic flow differences when implementing the FC model, the SPC model and the GBC model. Fig. 9(d) shows the traffic flow distribution when implementing the FC model, and Fig. 9(e) shows the traffic flow differences when implementing the SPC model and the FC model. We find that the SPC model only affects the traffic flows of a small number of road segments, resulting in the sight decrease in total travel time within the grid of the event location. Under the scenario of the SPC model, drivers may keep travelling through the grid of the event location after passing the controlled intersections on the perimeter. Fig. 9(f) shows the traffic flow differences when implementing the GBC model and the FC model. We find that traffic flows of the road segments within the grid of the event location considerably decrease and the traffic flows of the road segments in the periphery area of the event location increase when implementing the GBC model. An explanation is that the drivers whose destinations are not within the grid of the event location change their routes under the GBC model.

6. Conclusions

In summary, we employ an emerging mobility field analytical technique to develop a new traffic signal control approach applicable to both large road networks and anomalous travel demand situations, which we think can fill the research gaps in the traffic signal control research area. In particular, we generate the mobility field of urban travels and measure the gradients of the mobility field. Consequently, a gradient-based two-stage model is proposed to generate the optimal traffic control schemes for minimizing the travel time in the vicinity of the event location. The two-stage model is solved by developing the DBSCAN-FW-GA algorithm. Our results indicate that the proposed mobility field and gradient-based traffic signal control approach outperforms the widely used perimeter control approach on alleviating the local, severe and long-duration traffic congestion caused by mass events.

Different from previous models and approaches focusing on traffic signal control in a small-scale network, the present study provides an effective method to address the traffic signal control problem in large road networks under anomalous travel demand conditions. The gradient of the generated mobility field serves as a useful tool to identify the signalized intersections for implementing traffic control. Controlling the signalized intersections characterized with a large mean gradient can alleviate the traffic congestion caused by a mass event in a more efficient manner. In future studies, actual traffic data (e.g. LPR data and GPS data) during mass events can be collected to accurately calculate the gradient of the road network. The phase sequences and cycle lengths at each signalized intersection are worthy of further refinement. Traffic flow can be simulated using dynamic traffic assignment approaches and simulation software. In addition, the present study provides a generic framework for traffic signal optimization in large road networks. The gradient-based two-stage model can be reformulated to alleviate congestion during peak hours. More complicated applications can be explored, such as the implementation of deep-learning methods [43] at controlled intersections.

Acknowledgements

This research was supported by the Hunan Provincial Natural Science Fund for Distinguished Young Scholars (Grant No. 2022JJ10077), the National Natural Science Foundation of China (Grant No. 71871224) and the 2021 Science and Technology Progress and Innovation Plan of Department of Transportation of Hunan Province (Grant No. 202102).

Conflict of interest statement

None declared.

References