Abstract
Modernized trams usually run on exclusive rail lanes along urban streets, but they share the right of way with general vehicles at intersections and often get interrupted by traffic signals. We developed a mixed integer model to resynchronize traffic signal timings to favor tram movements. The objective is to balance the operational needs between minimizing bidirectional tram travel times and reducing the likelihood of activating the green extensions. The model depicts both tram and vehicle progressions in one signal timing plan, making it possible to control the impact of signal timing resynchronization through traffic. Trams following the tram bands produced by the proposed model are prevented from being stopped by red phases at signalized intersections. The applicability and effectiveness of the proposed model were demonstrated in a real-world case study. Compared with the state-of-the-art practice approach, the developed model reduced tram travel time by 10% with lower negative impacts on traffic on side streets. The reduction in tram travel time was obtained without sacrificing the mobility of through traffic.
1. Introduction
Modernized tram systems have been adopted as an environment-friendly transit mode in many cities. Compared with buses, modernized trams possess larger capacity, improved level of comfort, and a higher degree of right of way. Modernized trams, especially in Asian counties, run on exclusive rail lanes along urban streets, but they share the right of way with general vehicles at intersections [1]. As a result, tram operations are often interrupted by traffic signals at intersections, which slows down trams and degrades tram reliability.
Many methodologies have been proposed to optimize traffic signal timing to improve vehicle mobility and safety. For example, Webster [2] developed a method to optimize green time allocation at an isolated intersection. Robertson [3] and Little et al. [4] developed delay-based and bandwidth-based methods, respectively, to coordinate traffic signal timings in a corridor. To capture the interaction between traffic control and drivers’ behavior, traffic assignments have been taken into account in traffic control optimization [5]. Sheffi et al. [6] developed a traffic signal timing optimization model considering equilibrium assignment. The combination of traffic control and traffic assignment in the case of emergency evacuation was specifically considered in Marciano et al. [7].
Transit Signal Priority (TSP) strategies are recognized as an effective way to alleviate the negative impacts of traffic signals on transit operations. TSPs are usually classified as either active or passive priority [8]. In active priority schemes, a transit vehicle sends out a priority request as it approaches a signalized intersection and the signal controller responds such that the vehicle could pass the intersection without stopping. Traditional active TSP strategies include “green extension”, “red truncation”, and “phase insertion” [9]. The strategy implemented depends on the predicted arrival times of transit vehicles at an intersection and multiple priority requests from different approaches [10]. The active TSPs could potentially reduce travel times of transit vehicles and improve schedule adherence [11]. But they may incur additional delays to traffic on side streets, especially when traffic demand is close to the intersection capacity [12]. Traditional active TSPs have been implemented in several tram systems, such as Melbourne, Toronto, and Utah [13–15]. Recently, Li et al. [16] and Yang et al. [17] developed methodologies to jointly optimize offsets and green times in the active priority scheme. Shi et al. [1] considered the activation of TSP strategies in tram timetable development.
Passive TSP strategies synchronize traffic signal timings at adjacent intersections offline based on transit vehicle volumes and operational characteristics of transit vehicles. Typical passive TSPs originated from MAXBAND, which was developed for general vehicles [4]. Most related studies focused on buses. They maximized the bandwidths for buses by adjusting the offsets and phase sequences to smooth bus movements. Additionally, bus dwell time [18], dwell time variation, and capacity at bus stops [19] were considered in bus bandwidth optimization. Well-designed passive TSPs could enhance the reliability of transit operations [20] and improve transit mobility [21].
The bandwidth-maximization methods are suitable for bus systems. The band designed for buses could be well utilized since there are multiple bus lines on a corridor and bus frequencies are relatively high. Nevertheless, the bandwidth that is maximized for trams may be wasted since there is usually one tram line on a corridor and trams operate with relatively low frequencies. For example, tram frequencies in many cities in China, such as Shenyang and Suzhou, are above 10 mins.
Jeong et al. [22] proposed a model to maximize the bandwidths for general vehicles given a fixed tram bandwidth based on the MAXBAND model. Their model may result in relatively long tram travel times. San Diego implemented a practical strategy to instruct trams to pass multiple signalized intersections without stopping [23]. A tram dwells at a station until the next green light at the immediately downstream intersection starts. Then the tram departs within five-second departure window. If the tram misses the departure window, it must wait for the next green light. The strategy ensures that the tram will receive green lights at the following intersections until it reaches the next station if the tram leaves the station during the departure window.
Assuming that signal timing plans that have been optimized for general vehicles are given, we propose a methodology to resynchronize traffic signal timings in a subsystem to favor tram movements. Trams are instructed to follow the tram progressions optimized by the proposed methodology, leading to smooth and safe tram movements. Our work falls into the category of passive TSPs. It is different from existing studies mainly in the following.
(1) The objective of existing studies is to maximize the bandwidth for general vehicles or transit vehicles. But our objective is to balance the operational needs between minimizing bidirectional tram travel times and reducing the likelihood of activating the green extensions. Few studies have considered the active TSPs in transit progression design.
(2) We depict both tram and vehicle progressions in one signal timing plan, making it possible to control the influence of signal timing resynchronization on through traffic.
This paper is organized as follows. Section 2 describes the proposed methodology and Section 3 evaluates the effectiveness of the proposed methodology in a real-world case study. Section 4 summarizes the paper and discusses possible directions for future research.
2. Methodology
2.1. Scenario Descriptions
We considered tram and vehicle progressions in a subsystem as shown in Figure 1. Trams run on exclusive median rail lane. The left-turn lane for general vehicles is adjacent to the median rail lane. The two stations beyond the subsystem are referred to as major stations. Stations within the subsystem are referred to as minor stations. For simplicity, the proposed methodology only considered straight tram movements. It could be applied to right- or left-turn tram movements with minor revisions. Intersections in the subsystem adopt the same cycle length. But the green times for trams differ across intersections.
Assuming that signal timing plans that have been optimized for general vehicles are given, the proposed methodology optimized tram progressions by adjusting the offset and phase sequence at each intersection. Tram and vehicle progressions are represented by tram and vehicle bands, respectively. The left side of a tram band depicts ideal tram movements. Trams following planned tram progressions could go through signalized intersections without stopping.
A tram driver was informed of suggested speeds via an on-board user interface to follow the planned tram progression. Tram locations and traffic signal timings were assumed to be available in real time. The suggested speed was dynamically updated based on the deviation of a tram from the left side of the planned tram band. When a tram was ready to depart from a station, we estimated the speed for the tram to reach the immediately downstream intersection at the left side of the planned tram band. Let represent the maximum tram speed. The suggested speeds were produced based on the following rules.
(1) If , the suggested speed was set as .
(2) If , but the tram could reach the immediately downstream intersection before the right side of the planned tram band with the speed , the suggested speed was set as .
(3) Otherwise, the tram was instructed to wait until the appearance of the next tram band.
When a tram was running along the road, the suggested speed was updated with or at a fixed frequency, depending on which one was smaller.
2.2. Model Formulation
The parameters and variables used in this study are summarized in Table 1. All time related variables are in unit of the cycle length, , for convenience.
Figure 2 illustrates the parameters and variables representing vehicle and tram progressions. A node in the vertical axis represents an intersection () or a tram station (). The vehicle and tram bandwidths in direction are represented by and , respectively. At intersection node , the green times for straight-moving vehicles and trams in direction are represented by and , respectively. Their lengths may differ, depending on the phase sequence. The time lags from the right side of the green phase to the right edges of and are denoted by and , respectively. The time to clear vehicle queues at intersection node in direction is represented by .
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Vehicle and tram travel times in direction from intersection to its immediately downstream intersection is represented by and , respectively. Let denote tram dwell time at a station between intersection and its immediately downstream intersection in direction .
We used initialized phase time to represent signal coordination in a subsystem. For convenience, the initialized phase times are defined based on directions and transportation modes. The initialized phase times and denote the starting time of the first green phase for straight-moving vehicles and trams, respectively, at intersection in direction .
Tram travel time, , between two major stations consists of two components: the time from the upstream major station to the first intersection of the subsystem, , and the time from the first intersection of the subsystem to the downstream major station . The first component depends on the time when the tram is ready to depart from the upstream major station. Thus, it varies across trams. The second component is constant since we assumed that trams could follow the planned tram progressions. The expectation of can be obtained as follows:where denote the minimum tram travel time, in the unit of a cycle length, from the upstream major station to its immediately downstream intersection. Let represent the earliest arrival time of a tram at the intersection and represent the distribution of over a cycle length. is influenced by scheduled tram headway, the cycle lengths of the given subsystem, and the upstream subsystem. When no a priori information about is available, it is reasonable to assume that follows uniform distribution and (1) is reduced to
We developed a mixed integer model to optimize tram bands by adjusting the offset and phase sequence at each intersection. The objective function consists of two components:The objective balanced the operational needs between minimizing bidirectional tram travel times and reducing the frequency of activating the green extensions. The first component represents the expected bidirectional tram travel times. The second component is the penalty for activating the green extensions, where is a binary variable indicating whether the tram band in direction requires activating the green extensions at intersection node . If so, = 0. The negative impact of activating the green extensions is quantified by the value of , which ranges from zero to one. For example, indicates that one unit of the cycle length would be added to the objective function for activating the green extensions at an intersection. If , the green extensions would be activated whenever needed.
The optimization is subject to a series of constraints regarding signal timings, vehicle, and tram progressions.
Phase Sequence Constraints
The binary variables , ., and in constraints (4)-(7) define four-phase sequence patterns as illustrated in Figures 3(a)–3(d). “Lead” and “Lag” herein refer to the relative sequence of the phase for outbound and inbound left-turn vehicles, respectively. For example, if , , and , the lead-lag sequence is adopted (Figure 3(a)). Outbound straight-moving vehicles can use both the first and second phases. But outbound straight-moving trams can only use the second phase, since they will conflict with left-turn vehicles in the first phase. If , , and , the lead-lead sequence is adopted (Figure 3(c)). The green times for straight-moving vehicles and trams are equal. Constraint (8) defines the relationship between the green times for straight-moving vehicles and trams.
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Initialized Phase Time Constraints
Constraints (10)-(12) define the relationship of the initialized phase times at one intersection, which reconcile vehicle and tram progressions in one signal timing plan. The relationship depends on the phase sequence adopted. Constraint (10) defines the relationship between the initialized phase times for straight-moving vehicles in outbound and inbound directions. Constraint (11) defines the relationship between the initialized phase times for straight-moving vehicles and trams in the outbound direction. Constraint (12) indicates that the initialized phase times for straight-moving trams in both directions are equal. Once one of the initialized phase times is determined, the others could be derived based on the phase sequence.
Vehicle Progression Constraints
Constraints (14)-(15) depict vehicle progression in the subsystem. Constraint (14) ensures that vehicle progression is within the green time window at each intersection. Constraint (15) describes vehicle progression from one intersection to its immediately downstream intersection, where is an integer variable. Constraint (16) is specified to balance the needs of granting tram priority and improving vehicular mobility. Bandwidth was predetermined based on vehicle flows. The parameter is the trade-off factor. If , the signal timings would be adjusted without sacrificing vehicle bandwidth. Smaller may result in shorter tram travel times, but would incur additional delays to through traffic.
Tram Progression Constraints
Constraints (18)–(21) regulate tram progression in the subsystem. Constraint (18) ensures that tram progression is within the green time window at each intersection. Constraint (19) describes tram progression from one intersection to its immediately downstream intersection, where is an integer variable. Constraints (20)-(21) restrict tram travel times along the roads and dwell times at stations, respectively. Constraints (22) and (23) require the tram band to initialize immediately when the green phase starts at the intersections immediately downstream of major stations. As a result, trams waiting at major stations could start moving as early as possible.
Active TSP Constraints
Constraint (25) defines the relationship between variables and . represents the time needed for a tram to pass intersection safely. If , the tram band does not require activating the green extensions and . Otherwise, and the green extensions would be activated to ensure safe tram movements.
The proposed model can be solved using commercial software of CPLEX [24].
3. Case Study
The proposed methodology was evaluated numerically on Tram Line T1 in Songjiang District of Shanghai. Line T1 is 11.6 km in length and serves 18 stations including terminals. It was designed to operate on exclusive median rail lanes. The evaluation considered a subsystem on Rongle road, which is illustrated in Figure 4. There are three stations – within the subsystem. Traffic volumes in the evening peak hours and traffic signal timings that were optimized based on traffic flows are presented in Table 2. Traffic signal parameters are presented in the unit of the cycle length, which was 110 s. Vehicle progressions were optimized using the MAXBAND model. The resulting vehicle bandwidths were 16 and 15 s for the outbound and inbound directions, respectively.
The proposed model was applied given the signal timing plan optimized for general vehicles. Key parameters used in the proposed model are listed as follows:(i)The speed of general vehicles was 30 km/h based on empirical data.(ii)Maximum tram speed was 45 km/h(iii)The minimum tram dwell time at all stations was 30 s and the maximum was 90 s.
We evaluated and compared the operations of trams and general vehicles in deterministic and stochastic environments in three scenarios. Scenarios 1 and 2 were based on the original traffic signal timings. Scenario 3 resynchronized traffic signal timings using the proposed model to favor tram movements. Vehicle bandwidths are equal in three scenarios.
The proposed model was applied to produce tram bands without modifying traffic signal timing plan in Scenarios 1 and 2. The parameter was set to be one in Scenario 1, indicating that the resulting tram bands would not require activating the green extensions. In contrast, the parameter was set to be zero in Scenario 2, indicating that the green extensions would be activated whenever needed. In Scenario 3, the parameter was set to be one. Scenario 3 was the proposed approach and Scenario 2 was considered as the state-of-the-art practice approach.
3.1. Deterministic Evaluation
Given deterministic tram travel times and dwell times, Figure 5 presents the resulting vehicle and tram progressions in three scenarios. It was revealed that the green extensions in Scenario 2 and signal timing resynchronization in Scenario 3 could effectively reduce tram travel times. In Scenario 1, dwell time at station for outbound trams were relatively long, which was reduced by the green extensions in Scenario 2 and by signal timing resynchronization in Scenario 3. The outbound tram travel times in three scenarios were 550, 440, and 424 s, respectively, and the inbound tram travel times were 470, 470, and 375 s, respectively.
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Sensitivity analyses were performed to investigate the influence of the bandwidth control parameter and the penalty coefficients on tram travel times in Scenario 3. Figure 6 presents the expected bidirectional tram travel times (BTTT) between two major stations given various combinations of and . Generally, tram travel time increased with and . The results are understandable. Smaller values of η and ω mean a lager feasible region where the proposed model could search for the optimal results. Nevertheless, the gain obtained by reducing the values of η and ω could be small. For example, given that , tram travel time was only reduced by 1% if we decreased the value of η from 1 to 0.8.
3.2. Stochastic Evaluation
To further illustrate the applicability of the proposed model and explore the influence of the signal timing resynchronization on general vehicles, we employed TransModeler [25] to evaluate the operations of trams and general vehicles in the stochastic environment. After fine-tuning simulation parameters, traffic flows were well represented. For example, the simulated turning movements were close to the field measurements in Table 2. The resulting mean absolute percentage error (MAPE) equaled 6%, which was relatively low.
The simulation produced tram trajectories using the following setups:(1)Tram acceleration and deceleration rates follow uniform distribution between 0.7 and 1.3 m/s2, which was based on the field survey on a tram line of Suzhou, China.(2)Tram dwell time follows a nonnegative truncation of normal distribution with a mean of 30 s and a standard deviation of 18% of the mean [26].(3)To reflect the fact that drivers may not be able to follow the suggested speeds on the on-board user interface, it was assumed that the actual speed follows a nonnegative truncation of normal distribution with a mean of suggested speed and a standard deviation of 7% of the suggested speed.(4)Maximum green extension was 10 s.(5)Scheduled tram headway was 10 mins for each direction.
Tram progressions were produced in all scenarios and drivers were instructed to follow planned tram progressions. Green extensions were not allowed in Scenario 1, but they would be granted in Scenarios 2 and 3 when necessary. Although tram progressions in Scenario 3 did not require activating green extensions, green extensions were allowed to prioritize tram movements since trams may not be able to strictly adhere to the planned progressions in the stochastic environment. For each scenario, fourteen simulation experiments were executed with different random seeds. Each experiment emulated traffic operations in two-hour evening peak period of a day.
Figure 7 illustrates simulated tram trajectories in Scenario 3. Due to the randomness of tram speeds and dwell times, trams may not be able to adhere to the planned progressions. For example, the trajectory represented by black dashed line in Figure 7(a) was behind the planned progression because it was stopped by the red phase at intersection . The trajectory represented by black dashed line in Figure 7(b) waited for the next tram band since it was delayed be excessively long dwell time at station . Among 168 tram runs in fourteen experiments, 3 and 5 tram runs were behind the planned progressions in the outbound and inbound directions, respectively.
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Figure 8 presents the boxplot of tram travel times in three scenarios. The green circles represent the expected travel times in the deterministic environment. The outliers indicated by red crosses were tram runs behind the planned progressions. The results in the stochastic environment were consistent with those in the deterministic environment. Specifically, in the outbound direction, the mean tram travel times in Scenarios 2 and 3 were 19% and 22% lower than those in Scenario 1. In the inbound direction, the mean tram travel times in Scenarios 1 and 2 were close, which were reduced by 17% in Scenario 3. The effects of the green extensions in Scenario 2 and signal resynchronization in Scenario 3 on tram travel time variation were limited. In the outbound direction, the standard deviations of tram travel times were 32, 42, and 35 s in Scenarios 1, 2, and 3, respectively, while they were 41, 36, and 40 s in the three scenarios, respectively.
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We conducted the Wilcoxon signed-rank test [27] to evaluate the effects of the green extensions in Scenario 2 and signal timing resynchronization in Scenario 3 on tram mobility. The Wilcoxon signed-rank test is commonly used to test for a difference of paired observations, where the observations are taken before and after an action. The null hypothesis is that the median of the probability distribution of the difference equals zero. The alternative hypothesis is that the median is less than zero. If the null hypothesis is rejected, we conclude that the action is effective.
Table 3 summarizes the results of the Wilcoxon signed-rank test on tram travel time. The resulting -values are all close to zero, suggesting the effectiveness of the green extensions in Scenario 2 and signal timing resynchronization in Scenario 3 in reducing tram travel times. Tram travel times in Scenario 3 were significantly lower than those in Scenario 2, demonstrating the robustness of the proposed methodology.
Figure 9 presents the boxplots of travel times for through vehicles. Since vehicle bandwidths in three scenarios are equal, vehicle travel times in three scenarios were close, suggesting limited effects of green extensions in Scenario 2 and traffic signal timing resynchronization in Scenario 3 on the mobility of through vehicles.
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Green extensions were allowed in Scenarios 2 and 3, which may incur additional delays to traffic on side streets. Figure 10 quantifies vehicle delays on side streets. Since no green extensions were triggered at Guyang-Rongle and Songdong-Rongle intersections, vehicle delays at Guyang Road and Songdong Road in three scenarios were close, which was shown in Figures 10(a) and 10(d). Figure 10(b) shows that the median vehicle delays in three scenarios were close at Husong road, which was due to the fact that the frequency of green extension activations was relatively low in both Scenarios 2 and 3. Nevertheless, the green extensions increased the variation of vehicle delays and resulted in relatively long delays in some signal cycles. At Fangta-Rongle intersection, the green extensions in Scenario 2 were activated more frequently than in Scenario 3. As a result, the median and maximum delays in Scenario 2 were 18% and 38% higher than those in Scenario 1, respectively. They were 12% and 26% higher than those in Scenario 3, respectively.
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Table 4 summarizes tram travel time, travel time of through vehicles, and vehicle delay on side streets. The comparison of tram travel time revealed that tram mobility was improved by green extensions in Scenario 2, which was further improved by signal timing resynchronization in Scenario 3. But the green extensions in Scenario 2 would increase vehicle delays on side streets. Signal timing resynchronization in Scenario 3 reduced the negative impacts on traffic on side streets by decreasing the frequency of the green extension activations. The effects of the green extensions and signal timing resynchronization on through traffic were limited. Compared with the state-of-the-art practice approach (Scenario 2), the proposed approach (Scenario 3) reduced tram travel time by 10% with lower negative impacts on vehicles on side streets.
4. Conclusion and Future Research
We developed a mixed integer model to resynchronize traffic signal timings to favor tram movements. The objective of the developed model balanced the operational needs between minimizing bidirectional tram running times and reducing the likelihood of activating the green extensions. Trams following the tram bands produced by the proposed model are prevented from being stopped by red phases at signalized intersections. The effectiveness of the developed model was demonstrated in a real-world case study. Compared with the state-of-the-art practice approach, the developed model reduced tram travel time by 10% with lower negative impacts on traffic on side streets. The reduction in tram travel time was obtained without sacrificing the mobility of through traffic.
Empirical studies have revealed that transit drivers would respond positively to real-time information to keep on schedule [28]. That is, drivers will adjust speeds along the roadways and dwell times at stations to keep on schedule. The on-board user interface provides suggested speeds to facilitate tram drivers. The deployment of Automatic Vehicle Location (AVL) systems and the positive responses of transit drivers to real-time information make it feasible to apply the proposed model in practice.
The results presented demonstrated that the proposed methodology is promising. Investigating the value of the proposed methodology in a pilot study is the next step we are pursuing. In addition, our model was developed assuming deterministic tram travel times and dwell times. How to consider the variations of tram operations in the model is reserved for future research.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (71671124) and Shanghai Science and Technology Committee (17DZ1204404).