INCREASING CAPACITY OF INTERSECTIONS WITH TRANSIT PRIORITY

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INTRODUCTION
Provision of highly reliable transit services has been widely realized as an effective way to reduce traffic pressure on congested urban roads [1].To improve the reliability of transits, transit signal priority (TSP) and dedicated bus lane (DBL) were proposed to optimize the allocation of space and time resources on the road.The DBLs provide dedicated lanes to avoid mixing buses with general traffic, and with TSP buses can pass the intersection with a requested priority signal phase.Due to their low-cost feature and effectiveness for mitigating traffic congestion, DBL and TSP have become pervasive throughout the major cities in the world.Especially in China, among the 70 major cities, more than 80% of them have DBLs and 40% have bus rapid transit (BRT) that are often provided with TSP.
Properly designed DBL and TSP can significantly improve transit reliability in terms of reducing delay and travel time fluctuation of buses caused by general traffic and signal control.However, most of the implementations failed to consider the capacity and delay of general traffic.The method used to provide a highly efficient TSP usually changed the signal control timing dramatically, which brought negative impacts on general traffic regarding the dimension of time.The DBL occupied the lanes for buses only, which disturbed the general traffic regarding the dimension of space.To overcome the drawbacks of transit priority, this paper focused on providing bus priority on dedicated bus lanes as well as maximizing the capacity of intersections.An integrated optimization (IO) method for lane assignment and traffic signal control has been developed in this paper based on the concept of using one or more of general traffic lanes as variable lanes for both left-turn traffic and through traffic.The optimization model is formulated as a Mix-Integer-Non-Linear-Program (MINLP), in which the lane assignment, main-signal timing, pre-signal timing and transit priority are integrated.The applicability and performance of the proposed IO model is tested by numerical experiments and VISSIM simulation under different geometric layouts, bus priority demands and flow rates of general traffic.
The rest of this paper is organized as follows.The literature is reviewed on research of DBL, TSP, variable lanes and pre-signal control.The development of IO method is introduced, including the concept of the IO scenario, the MINLP model and the solution

INCREASING CAPACITY OF INTERSECTIONS
WITH TRANSIT PRIORITY Y. Hao, J. Teng, Y. Wang, X. Yang: Increasing Capacity of Intersections with Transit Priority method.The evaluation methods and test results of the proposed method are then presented.Conclusions and recommendations for the future work are given in the final section.

LITERATURE REVIEW
Smith Wilbur conducted the earliest bus pre-emption experiment, which gained significant reduction on buses travel time [1].A large number of studies followed and focused on TSP strategies.Many of these studies focused on TSP timing strategies [2,3], and the comparison and evaluation of the proposed strategies using either simulation or field data [4][5][6][7][8][9].Conditional transit signal priority strategies were provided according to the bus schedule and provided only if the buses were behind the schedule [10][11][12][13], which would reduce the ineffective signal priority.Ma et al. [12] established a linear model to generate TSP strategies in case the priority was effective.Hao et al. [13] proposed a TSP method based on transit schedule, which provided priority between two successive bus stops, and calculated bus priority time from the last bus stop to the first one in order to keep the TSP effective.Besides TSP, DBL is also an effective measure in providing transit priority and improve transit reliability [3,6,14].Combined with intermittent bus lane (IBL), DBL shows its benefit in transit priority as well as in reducing disturbance to general traffic flows [15,16].
Variable lane is an effective measure addressing the problem of traffic flow changing at different time periods and the problem of congestion.Daganzo et al. [17] introduced the concept and implementation method of dynamic lane assignment for freeway congestion mitigation as part of the California PATH Program of the University of California.Lam et al. [18] developed a model integrating both the design of land-use and signal-phase control at isolated intersections and performed the optimization simultaneously to produce a better overall result.Zhao et al. [19] developed a two-step optimization model for dynamic lane assignment at isolated signalized intersections to solve the congestion problem.The arm-group-based model overcame the shortcoming of static lane marking and fully utilized the existing traffic facilities.Zhao et al. [20] also proposed an unconventional approach, which used the exit lanes for left-turn traffic dynamically, to increase the capacity of heavily congested intersections.Li et al. [21] developed a variable lane control strategy that attributed lanes and generated signal control method according to the detector data.The results of the simulation experiment showed that the strategy could increase capacity and reduce delay.The implementations of the variable lane assignment began in France in the 1960s, and two important trails are currently under way in the Paris region and the Grenoble conurbation.The aim of the trails is to gain profit overall rather than locally [22].
Pre-signal as one of the means to provide TSP has been studied and implemented widely.The pre-signals have been used in Europe to provide transit priority for a long time [23].Wu and Hornsell [24] proposed three categories of pre-signal based on a number of pre-signals that have been installed in the U.K., particularly in London.Buses in this study are subjected to the pre-signal as well as general traffic, which would provide partly priority to buses.Long [25] proved that the pre-signal control for buses in a real intersection performed better than passive bus priority, in terms of delay per person and vehicle.Xuan & Daganzo et al. [26] proposed a method that reorganized all the lanes upstream of an intersection using a mid-block pre-signal.The result shows that if drivers behave properly, the increasing of the capacity could be significant.This method has proven to be efficient but with a flaw that it requires the interval length of two successive intersections.Wang [27] developed a lane-based optimization model of signal timings, which considered lane markings and signal timings at the same time to improve the operation of intersections.
In summary it can be concluded that the TSP and DBL strategies have been widely explored; the impacts of TSP and DBL on general traffic have been evaluated, but not considered in the optimization method in previous studies.There have been a significant number of studies focusing on using the variable lane and pre-signal strategies to improve the performance of intersection operation.The primary contribution of this paper is to provide bus signal priority on dedicated bus lanes as well as maximizing the capacity of intersections by integrating lane assignment and traffic signal control.

Basic concept of IO control
Both TSP and DBL strategies cause reduction of the capacity for general traffic.The basic idea of the IO model is to use one or more general lanes as variable lanes, which is designed to respond to stochastic traffic demand and alleviate capacity reduction when transit priority is provided.As shown in Figure 1, pre-signals are set on the upstream of the variable lane to announce the right-of-way.The variable lane function is activated only when an approaching bus is requesting signal priority.Bus detector should be located no later than the exit lane of the upstream intersection, and the earlier the better.When bus priority is requested, the IO model will provide an optimized signal-timing plan, including pre-signal and main signal.
The pre-signal plan includes the use plan of variable lane and the duration time for each movement.The strategy used to calculate the bus priority time was proposed in a previous work [13], which provide effective bus priority under the constraints of bus schedule adherence or headway stability at intersections between two successive bus stops.Figure 2 shows the process of the IO control method when bus priority is requested.The variable lanes may be used more than once in one cycle according to the optimized method.In a conventional intersection with four arms, the variable lanes can be used as both left-turn traffic lane and through traffic lane.Arrows are given as pre-signals for the announcement of the right-of-way.The green arrow for each movement will light up according to the corresponding phases in the main signal, and other movements receive a red arrow.Lane clearance is also considered when exchanging the right-of-way.

Model assumptions
In this paper, buses operate in a dedicated bus lane and only one transit priority request can be served in one cycle.The first-come-first-serve rule is applied for multiple priority requests situation.The speed of buses is considered as a constant.To decrease the impacts of pedestrians and bikes on buses, only the bus lane in the middle of the road is considered in this paper.With the correction factor of bikes and pedestrians considered, the model can be applied to both setting form of bus lanes, bus lanes in the middle of the road or on the side of the road.
The assumptions are suitable for DBL operation, bus rapid transit (BRT) system and tram with dedicated lanes.The model is still suitable under the mixed traffic situation with the bus travel time model amended.The amendment should consider the bus speed fluctuation and queue length caused by general traffic.
The notations used hereafter are summarized in

METHODOLOGY 4.1 Objective function
Three criteria are usually employed for the optimization of signal settings: capacity maximization, cycle length minimization and delay minimization.The purpose of this paper is to maximize the capacity of intersections during provision of TSP.Therefore, the capacity maximization is chosen as the optimization objective.Adopting the commonly used assumption that the turn proportions would remain constant [27], maximizing the reserve capacity is equivalent to maximizing the common flow multiplier, μ.

Reserved capacity constraint
The traffic demand is allocated to all the lanes in every arm.The maximum traffic volume that still allowed the intersection to operate reasonably well is defined as the traffic demand Q multiplied by the flow multiplier μ.The sum of the flow assigned to each lane equals the maximum traffic volume., , , ..., , , , ...,

Lane assignment constraints
1) Movement permitted on one lane Each traffic lane can permit only one movement to avoid conflict with buses on the dedicated bus lane, which can be specified as: Bus travel time to the intersection from the time bus detected for bus movement j in arm i q ijk Assigned flow of movement j on k lane in arm i (veh/h) ' q ijk Assigned flow of movement j on k lane in arm i when bus priority provided (veh/h) 2) Prohibited movement Only the movement permitted on a lane can be assigned to it, which can be specified as: , , , ..., , , , ..., , , ,..., where M is an arbitrary large positive constant.If δ ijk =0, the movement is prohibited, and the assigned flow should be 0. Otherwise, if δ ijk =1, the assigned flow can be any positive value as long as it satisfies other constraints.

4) Maximum permitted movements at the exit lanes
The number of exit lanes should be no less than the maximum number of entrance lanes that are assigned the flow to the exit lanes, which can be specified as the following equation: ( , , ,..., , , ,..., , )

Signal timing constraints
1) Intersection main-signal timing Cycle length: Let the maximum and minimum cycle lengths of the intersection be C max and C min , respectively.Instead of defining the cycle length directly as a variable, its reciprocal ζ, is used as the variable to preserve the linearity in the mathematical formulation.
Start of green: since the signal timing at the intersection is cyclic in nature, the starts of green can be picked arbitrarily along the time axis as long as they satisfy other relevant constraints.However, for linearity reason, all of the starts of the green variables are confined to the range of [0,1]., , , ..., , , , ..., The start of green for the pair of movements from arm i to arm j and from arm j to arm i should be equal when they are in the same direction with the bus operation (ij b ).
Duration of green: the duration of green is subjected to the minimum green time g ij , to satisfy the safety and comfortable requirement and can be set as: , , , ..., , , , ..., The duration of green for the pair of movement from arm i to arm j and from arm j to arm i should be equal when they are in the same direction with the bus operation (ij b ).
Sequence of signal phase: designate the set of incompatible signal groups as s } .For any two conflict signal phases, (lm) and (ij) in s } , the order of the signal displays is governed by a successor function [28], Ω ijlm ; where Ω ijlm =0 represents the start of green of signal group (lm) following that of the signal group (ij), and Ω ijlm =1 stands for the opposite.The following constraints can be set for the successor functions: , ( ), ( ) ij ml Lane signal settings: the signal timing of the movement in a lane should be the same with the intersection signal timings set for the movement, which can be specified as: 2) Pre-signal timing The pre-signals are employed only when bus priority is requested, which can be described as e b =1.
Cycle length: For coordination purpose, the cycle length of the pre-signal should be the same as the main signal.
Clearance time of variable lane: The clearance time is needed to empty the variable lane for different movements.For convenience, the ratio of the clearance time in a cycle is used as the constraint.
Start of green: The start of green of the movement j on variable lane in arm i can be calculated as follows, as long as they satisfy other relevant constraints.

( )( ) ( )(
) Duration of green: To keep the operation order of intersections, the green duration of movement j on the variable lane in arm i should not be larger than the main-signal.

Flow ratio constraints 1) Flow ratio for normal lanes
The flow ratio of a lane is the ratio of the flow rate to the saturation flow rate, which can be expressed as: , , , ..., , ,..., y s 2) Flow ratio for variable lanes Since frequent changing of the right-of-way on variable lanes may cause capacity reduction, the flow saturation of the variable lane should be reduced by coefficient c ., , , ..., , ,..., y s 3) Maximum acceptable degree of saturation The maximum acceptable degree of saturation that ensures the intersection to perform well is defined as Y max .The degree of saturation of each traffic lane should not be greater than the maximum acceptable degree of saturation.

Transit signal priority constraints 1) Location of variable lanes
When the approaching bus is requesting priority and the function of variable lane is activated, the location of the variable lane is decided as follows: , , , ,..., , , , ..., The first lane with the same direction to bus lane is set to be the variable lane.

2) Bus travel time
A bus detector should be set at the exit bus lane of the upstream intersection.The travel time of the bus from where it is detected to the intersection can be calculated as: , , , ..., , , , ..., 3) Available priority strategies Two types of transit signal priority strategies are employed in this paper, which are green extension and red truncation.The priority strategies can be specified as follows: , , ,

P Green extension
No strategy Red truncation

4) Priority strategies
The priority time of a bus in a certain intersection is generated by the optimization model proposed in the previous work [13].The optimization model provided effective bus priority at intersections between two successive bus stops according to the bus schedule.
To avoid severe changing of signals in intersections, coefficient η is used to keep the available priority time in a relatively small range.For example, as shown in Figure 3, , ,  T T T   d d d 1 2 3 are the bus detected time, η is the available priority time.Suppose the buses are detected at T d 1 , no strategy will be taken, while if they are detected at T d 2 or T d 3 , the strategies will be calculated as Equation 25.Solutions 1-4 in Figure 3 show the probable solutions of providing bus priority.
The priority time of a bus is decided by their detected time, travel time to the intersection and the available priority time, which can be specified as follows: , , , ' , , '

5) Pre-signal signal timing
The pre-signal timing should be recalculated using the following equation after bus priority has been provided:

Solution method
There are two main parts of the integrated optimization (IO) model: the signal timing part, including both main-signal and pre-signal, and the transit priority part.The IO model is a Mix-Integer-Non-Linear-Program (MINLP) with the objective function as shown in Equation 1 and subjects to Equations 2-33.It is hard to obtain a global solution for this kind of problems.However, after examining the constraints, the variables that make the whole IO model non-linear have been found, including the length of the variable lane Li c , and the length from the bus detected location to the intersection Lij b , which is often the length of the link between two successive intersections.For implementation, the link length Lij b is often predetermined, and the length of the variable lane can be ranged in a set, which can be tried and put into the model by the brute force solution.The simplified method made the IO model turn to a Mix-Integer-Linear-Program (MILP) which can be solved by the standard branch-and-bound technique.
The signal timings without bus priority are calculated by Equations 2-22.When bus priority is required, the signal timings and bus priority strategy should be recalculated by Equations 23-34.The bus priority strategy can be calculated as shown in Figure 4.

Numerical analysis
This section presents the evaluation of the performance of the proposed model by numerical analysis.
Consider a four-arm intersection with dedicated bus lane, as shown in Figure 5. Three cases with different numbers of through and left-turn lanes are studied as shown in Figure 5.
Common data input in all of the three cases is shown in Table 2.A waving area exists in the upstream of the variable lane.To maximize the adverse impact of the waving area and test the performance of the proposed model under a worse situation, the length of the variable lane is set at 50 meters and shorter than the queue length of that arm.
The flow volume inputs are presented in Table 3.To identify the applicability of the IO model, the conventional priority method and the IO method have been compared.Variable lane strategy has been only employed in the IO method, and bus priority has been employed in both IO and conventional priority method.The maximum cycle length is set to be 120s.

1) Evaluation under different geometry layouts
Represented in Figure 6 is the capacity comparison between the IO model and the conventional priority method under different geometry layouts in the three cases.The priority time for buses is described as the proportion to a cycle time which is in the range of [5%, 30%] and the performance of the IO model is tested along the priority proportion axis.The capacity ratio axis is the ratio of the IO model to the conventional model.The conventional model differs from the IO model in the geometry layout part, which means that no variable lanes are considered in the conventional model while the IO model considered all factors described in Section 2.
As illustrated in Figure 6, the IO strategy shows its effectiveness in increasing the capacity of intersections, especially in the intersections with high volume and several traffic lanes.The IO method can increase the capacity of the intersection when bus priority is provided and performs better than the conventional priority method when the bus priority time is longer.The dissatisfactory performance of IO model shown in Figure 6c is analyzed to be caused by the reduction coefficient multiplied with the flow ratio of variable lanes.

2) Evaluation under different flow ratios
Under the same geometry layout in Case 1, changing the ratio of the left-turn to through traffic and keeping the demand flow and cycle length the same are the basic concepts in this evaluation section.
The flow ratio of left-turn to through traffic differs from 1:1 to 1:3 as shown in Figure 7.As the left-turn demand increases, the capacity of the intersection firstly increases and then decreases.This phenomenon can be explained as follows: Capacity can be increased for vehicles travelling in the same direction as the buses on the basis of the bus signal priority.The redundant time saved by TSP, the variable lanes, and the reduction of through traffic can be reallocated to the left-turn and other phases, so as to increase the capacity of the intersection.
Along with the increase of the left-turn traffic (impaired by TSP) and decrease of the through traffic (benefit from TSP), the utilization of variable lane increases in the proposed IO method, showing higher capacity multiplier.

Simulation evaluation
The geometry layout, traffic volume and signal plan in Case 2 in the numerical analysis are chosen to be the simulation environment.The cycle length is set to be 120 seconds and the variable lane is set to be 50 metres long.Red truncation and 20 seconds are implemented as the priority strategy and priority time, respectively.The frequency of buses is in the range of 2 minutes to 20 minutes.Using the control variable method and keeping the priority strategy and time unchanged, the bus frequency changed in the increments of 2 minutes according to the cycle length.The control variable method can compare the performance of the IO model to the OC model under the same bus priority strategy.
Three lines in Figure 8 stand for the proposed integrated optimization (IO) method, conventional priority method and the original control (OC) situation, respectively.As shown in Figure 8, IO method and conventional priority method increase the capacity of intersection along with the bus frequency decrease, and the IO method performed better in maximizing the capacity of intersections.When the bus frequency is lower than 16 minutes/vehicle, the capacity with IO method could be larger than the OC situation, which means the capacity reduction caused by TSP can be regained by IO method under certain conditions.Figure 8 also shows the delay of the intersection under different methods, which decreases when bus frequency decreases.The delay of intersection under IO method is 3.1% greater than the OC situation when bus frequency is 12 minutes/vehicle, and less than 3% when the bus frequency decreases.Figure 9 shows the performance of the IO method on capacity increase.The horizontal axis stands for the vehicles input while the vertical axis stands for the vehicles output.Both IO and conventional priority method performed well when the vehicles input is low, but the IO model increases more than 8% of capacity under high saturation degree situation.

CONCLUSION AND FUTURE WORK
This paper presents an integrated optimization model of lane assignment and traffic signal control for transit progression with DBL.Geometry layout, main-signal timing, pre-signal timing and transit priority are integrated in a unified framework.These problems have been formulated as MINLP and transformed into a series of MILP problem, which can be solved by the standard branch-and-bound technique.Finally, numerical experiments and simulation tests validate the effectiveness of the proposed IO model.Test results indicate that the proposed IO method can increase the capacity of intersection in comparison with the conventional bus priority method.A better performance can be achieved when bus frequency is low and left-turn vehicles occupy a relatively large proportion.The intersection delay also receives reduction by the IO method comparing to the conventional priority method.
In the future work, more extensive numerical experiments, simulation tests can be conducted to assess the performance of the model under various demands and different driving behaviours.Another important work considers the arterial coordination situation in the model to provide integrated optimization in an arterial.The problem of multi-requested priority could be another extension in the future work.

Figure 1 -
Figure 1 -Concept of integrated optimization control

Figure 2 -
Figure 2 -Operating process of IO model

Y
. Hao, J. Teng, Y. Wang, X. Yang: Increasing Capacity of Intersections with Transit Priority

Figure 7 -
Figure 7 -Numerical analysis results: different left-turn and through flow ratio in Case 1

Figure 9 -
Figure 9 -Simulation test results: throughput and bus frequency impact

ACKNOWLEDGEMENT
This research was made possible through the funding by the National Natural Science Foundation of China (Grant No. 61174185 & Grant No. 51238008).城市道路，有效的公交信号优先，公交专用道，动 态车道划分

Table 1 -
Notations and variables c Length of variable lanes in arm i Lij b Bus travel length from the location bus detected to the intersection for bus movement j in arm i N j EL Number of exit lanes in arm j P i Transit priority strategies i Y. Hao, J. Teng, Y. Wang, X. Yang: Increasing Capacity of Intersections with Transit Priority Duration of green for movement j in arm i when bus priority requested x Bus arrival time at intersections for bus movement p in arm q i c ~The ratio of clearance time of variable lanes in arm i ij z Duration of green for movement j in arm i 'ij z ic H Start of green for variable lanes in arm i ik H Start of green on k lane in arm i ic U Duration of green for variable lanes in arm i ik U Duration of green on k lane in arm iΩ ijlmOrder of signal phase for a pair of mutually incompatible traffic movements (i, j) and (l, m), if the start of green of signal phase (l, m) follows that of signal phase (i, j), and for the reverse order

Table 2 -
Common data input

Table 3 -
Traffic demand under conventional situation