Traffic Flow Model for Coordinated Traffic Light Systems ^†

Abstract

Traffic in large cities is increasing due to continuous urbanization, the construction of new housing complexes and the accompanying new street network. The growth of cities creates prerequisites for increasing the intensity of transport, pedestrian, and bicycle flows, especially during peak periods. To improve the conditions in which traffic flows, it is necessary to introduce an effective method for reducing delays that arise at intersections, especially those regulated by traffic light systems. One of the possible approaches to this is to coordinate the operation of traffic light systems. The main thing in this is to determine relatively accurate times for the movement of individual flows, for which adequate traffic models are needed. This article presents a model of the movement of transport flows when starting from the first intersection in a coordinated mode of operation of traffic light systems. This is of particular importance when determining the times of individual signals and, above all, has an impact on the moment for switching on the permitting signal at the next intersection. The presented model aims to provide an opportunity to determine accurate times of passage of vehicles through consecutive intersections that operate in a coordinated mode of traffic light systems.

1. Introduction

Coordination of the operation of traffic light systems at consecutive intersections requires in-depth research and the development of adequate models for the movement of individual groups of traffic participants—transport, pedestrian, and bicycle flows. A key point in building a comprehensive model of the movement of transport flows through different intersections is the behavior of individual vehicles when they start moving from the first of the coordinated intersections. This largely determines the set of decisions that will be made and applied to the remaining intersections. Some of these decisions are related to the duration of individual signals, the type of phases, the duration of traffic light cycles, and others. A basic solution is the application of justified intervals for shifting the moment of switching on the green signals at consecutive synchronized intersections. This must also be justified by the nature of the traffic, taking into account the intensity and saturation flow of individual intersections.

In some of the developments in this area, the different speed at which vehicles cross the stop line when entering the intersection has been studied and taken into account [1]. In this regard, the authors indicate a model for crossing the stop line—a multinomial logit, which they consider as a polynomial variable. Shepelev et al. in [2], on the other hand, pays attention to the dynamics of vehicles and especially to the acceleration when leaving an intersection. In addition, the authors show that other side factors can also influence the model by referring to the condition of the road surface. On the other hand, Kim et al. in [3] indicate that the speed of the vehicle, in addition to being influenced by the speed of the preceding vehicle, is also influenced by its own speed. This is precisely why a thorough study of the processes of starting and moving vehicles is necessary to build the algorithms necessary for modeling these processes. Research on modern living conditions, determined by the technological progress of the population, can be supported by a number of innovative technologies used in [3]. Similar innovative research approaches are also presented by the authors in [4,5]. In turn, Nuli and Mathew in [6] show the application of such an approach directly in the operation of coordinated traffic light systems.

The presented developments are actually part of the global goal, which requires their implementation in intelligent transport systems. Some of those interested in this area present the advantages of using such systems and how much they contribute to reducing travel time [7,8], the impact on movement on motorways [9], and reducing transport delays that occur under different conditions [10]. Others are specific by showing the importance of using these systems to improve the movement of emergency vehicles [11] or their use to improve the organization and movement in other conditions [12]. The development of these systems requires taking into account the specific features of different types of intersections and, in particular roundabouts, for which some authors propose appropriate methods for research and application of methods for improving the passage through them [13,14]. This also places additional requirements for creating a comprehensive model of the movement of transport flows in a coordinated mode of operation of traffic light systems, in which the flow passes through a roundabout with different options for its regulation. When developing these options, the model presented in this publication is a possible initial version of such models. The overall performance of these systems requires continuous monitoring, which can be implemented with the progress of modern technologies, as presented by Miletiev et al. in [15]. In this case, it is recommended to use advanced technologies used in other areas of scientific research development, presented by the authors in [16,17]. In this way, the presented model can be applied to achieve the benefits that are set as goals of such systems, since it allows for its integration into the overall monitoring algorithm.

The crossing of pedestrians during the operation of traffic light systems in a synchronized mode is an issue that poses the mandatory measures that need to be carried out for the compilation of such models to be included in intelligent transport traffic management systems. In this case, some authors have already developed such models of intersections [18], and others provide a basis for their development with research conducted for other purposes [19,20]. In these cases, the presented model can also be used to improve the models that relate to the crossing of pedestrians and determine the various possible options when accidents occur with them and vice versa—these developments can add requirements for the subsequent construction of the overall model, including the passage of conflicting traffic flows.

2. Materials and Methods

The development of a model of the movement of the traffic flow when starting from the first intersection in the mode of coordinated operation of consecutive traffic light systems refers to the micro models of the transport flows, which consider the movement of individual units in the transport flow. In doing so, some assumptions are made, and certain considerations are taken into account, which aim to achieve maximum compliance of the results with the actual traffic conditions and vehicle movement.

The model is developed using the scheme shown in Figure 1. STOP Line 1 is the stop line at the first intersection (Intersection 1) in synchronized operation of consecutive intersections, from which cars depart. STOP Line 2 is the stop line at the second intersection (Intersection 2), which is reached by those departing from their respective place in the queue in front of STOP Line 1.

The model assumes that the cars at the first intersection are stopped at a red signal waiting for a green signal to allow them to pass. This is the starting point in the calculations. The calculations for the time for the first and each subsequent car to reach STOP Line 2 are presented sequentially. The fact that the first car starts a few moments after the green signal is turned on is taken into account. Also, the fact that each subsequent car does not start simultaneously with the car in front of it, but a few moments after, is taken into account.

The authors in [1] define this start-up delay as 1 s. The same has been proven by a team from the Technical University of Sofia, some of which are the authors of this publication. The conducted study proved that the average time for the first car to start after the green signal is turned on is 1 s, which is identical to the average time for the next car in the queue to start after the car in front of it starts moving [21].

3. Results

The results take into account two components of the movement of each of the stopped cars when starting from their respective place in the queue in front of STOP Line 1 at Intersection 1 until reaching STOP Line 2. One component determines the time ($t_{ua}$) from the moment the car starts from a stopped position with an initial speed ($V_0$) equal to 0 m/s, until reaching the maximum speed for the road section ($V_{max}$) with uniform acceleration (Formula (1)). The second component determines the time of the same car ($t_u$) to reach STOP Line 2 after reaching the maximum speed for the respective road section (Formula (2)). When determining the second component, it is assumed that the movement is uniform with the maximum speed for the respective road section.

$$t_{ua}={\displaystyle \frac{V_{max}-V_0}{a}},s$$

(1)

where $a$ is the acceleration of the vehicle, which for mass-produced car brands and models can be assumed to be between $2.5÷3.0$ m/s² (data from an unpublished study conducted by the authors of the publication).

$$t_u={\displaystyle \frac{S_u}{V_{max}}},s$$

(2)

where $S_u$ is the distance traveled by the car with $V_{max}$, m.

The distance traveled by each car to reach its maximum speed ($S_{ua}$) is formulated in (3):

$$S_{ua}={\displaystyle \frac{a\ast t_{ua}^2}{2}},m$$

(3)

The distance each car travels at maximum speed is the difference between the distances between the stop lines ($S_{stop}$) and what is needed to reach the maximum speed for the road section ($S_{ua}$):

$$S_{V_{max}}=S_{stop}-S_{ua},m$$

(4)

To calculate distance $S_{V_{max}}$ for each car, it is necessary to add the distance it travels to reach STOP Line 1 ($S_{STOPLine1}$). When determining these distances separately for each car, it is assumed that the first car is at a certain distance from STOP Line 1 ($\Delta S$), and each subsequent one is at a distance with the average length of the cars in front of it added ($\overline{l_a}$). The distance determined in this way is assumed to be traveled at the maximum speed for the section.

Following the presented considerations, the time to reach STOP Line 2 is expressed as the result of three summing components—the time to reach the maximum speed for the section, the time from the moment the green signal turns on until the respective vehicle starts moving, and the time to travel at the maximum speed for the road section. For the first three vehicles in the queue in front of STOP Line 1, the calculations are expressed in Formulas (5)–(7).

$$t_{STOPLine2}^{C1}=t_{ua}+1+{\displaystyle \frac{S_{V_{max}}+\Delta S}{V_{max}}},s$$

(5)

where $t_{STOPLine2}^{C1}$ is the time to reach STOP Line 2 from the first car departing from STOP Line 1.

$$t_{STOPLine2}^{C2}=t_{ua}+2\ast 1+{\displaystyle \frac{S_{V_{max}}+\Delta S+\overline{l_a}}{V_{max}}},s$$

(6)

where $t_{STOPLine2}^{C2}$ is the time to reach STOP Line 2 by the second car that left STOP Line 1.

$$t_{STOPLine2}^{C3}=t_{ua}+3\ast 1+{\displaystyle \frac{S_{V_{max}}+\Delta S+2\ast \overline{l_a}}{V_{max}}},s$$

(7)

where $t_{STOPLine2}^{C3}$ is the time to reach STOP Line 2 by the third car starting from STOP Line 1.

The presented sequence allows us to reach the final equation, which allows us to calculate the time to reach STOP Line 2 for each of the cars departing from Intersection 1:

$$t_{STOPLine2}^{Cn}=t_{ua}+n+{\displaystyle \frac{S_{V_{max}}+\Delta S+(n-1)\ast \overline{l_a}}{V_{max}}},s$$

(8)

where $n$ is the sequence number of each car waiting in line in front of STOP Line 1.

4. Discussion

The presented model of the movement of cars starting from the first intersection in the coordinated operation of consecutive traffic light systems allows for the determination of one of the main parameters in the application of such approaches to traffic regulation. The time for covering the distance between the two intersections and the moment of the arrival of cars from the previous intersection to the stop line of the next one is a determining factor for synchronizing the operation of traffic light systems and predicting unnecessary shifts in the time of switching on the permitting signals at successive intersections. This largely determines the number of possible cars passed, the number of cars that we will expect to accumulate in a queue for each of the intersections, and the subsequent determination of the exact moment for the start of the green signals for each of the intersections in a coordinated operation mode.

A key point in the presented model is the ability to determine the times for reaching the stop line at the next intersection even in cases where the distance between the stop lines ($S_{stop}$) is less than the distance required to reach the maximum speed for the road section ($S_{ua}$). In addition, the model allows for determining the time for reaching the stop line at each subsequent intersection even in situations where no acceleration is required to reach the maximum speed for the section. The statements made are based on a numerical experiment carried out in laboratory conditions.

The developed model allows for its application in heterogeneous traffic flows. It is necessary to take into account the equalization coefficient of each vehicle and present the results for passenger car unit (PCU).

5. Conclusions and Future Work

The developed model of traffic flow when starting from the first intersection in the mode of coordinated operation of consecutive traffic light systems describes well enough the real process of movement of cars from one intersection to another. The model was developed for application in the conditions for which it was set, but its capabilities are not limited to this. It offers options for determining the times for reaching the stop line at the next intersection and in conditions of facts not accounted for by the model, which it interprets with sufficient accuracy to achieve adequate results. This applies to conditions in which the distance between the intersections is less than the distance required to reach the maximum speed for the respective road section and cases in which the times for reaching the next intersection can be calculated in the absence of uniformly accelerated traffic.

The future work of the model developers is to verify and prove its adequacy through experiments in real traffic conditions, on a street with consecutive intersections regulated by traffic lights in the city of Sofia. The subsequent work also includes the development of a model for determining the number of cars that will manage to pass through the intersection (first or subsequent) in the coordinated traffic light regime, depending on the duration of the green signal and the shift in the green signals for the remaining intersections after the first.