# A Group-Based Signal Timing Optimization Model Considering Safety for Signalized Intersections with Mixed Traffic Flows

Authors: Fen Wang; Keshuang Tang; Keping Li; Zhigang Liu; Lin Zhu
Journal: Journal of Advanced Transportation
 (2019)

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## Abstract

The conventional stage-based signal control approach with uniform phase structure has been dominantly applied at signalized intersections in China. However, this approach cannot efficiently handle mixed traffic flows with unbalanced volumes. Moreover, this signal control approach has resulted in many safety issues, such as traffic conflicts (a) between the right-turning motorized vehicles and the straight-through bicycles and (b) at the change of phases due to bicycles’ clearance failure. Hence, the objective of this paper is to propose a group-based signal optimization model that considers both safety and delay for the intersections with mixed traffic flows. In the proposed model, safety was evaluated based on the traffic conflicts during the inter-green period and was incorporated into the signal timing procedure. A probabilistic approach was developed to estimate the probability of occurrence of conflicts, with a novel safety indicator combining postencroachment time and kinetic energy for measuring conflict severity. The average delay per person, according to the Highway Capacity Manual 2010 method, was adopted in this paper. Then, the multiobjective optimization issue was formulated as a nonlinear program and solved by a Nondominated Sorting Genetic Algorithm. A numerical study was performed to demonstrate the applicability and performance of the proposed model. Results indicated that the proposed model can provide an effective tool for researchers and practitioners to simultaneously optimize traffic safety and efficiency in signal planning. It may also overcome the disadvantages of most of the conventional models, which are incapable of quantifying safety in the optimization process.


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## Body

## 1. Introduction

The conventional stage-based signal control approach having uniform phase structure has been dominantly applied at signalized intersections in China until now. However, this kind of approach cannot efficiently handle the mixed traffic flows consisting of unbalanced volumes of motorized and nonmotorized vehicles. The above-mentioned approach, along with unruly road user behaviour, has resulted in many safety issues, such as traffic conflicts between the right-turning motorized vehicles and the straight-through bicycles, as well as traffic conflicts at the change of phases due to the failure of bicycles to clear the intersection. According to traffic police department statistics, approximately 70% of nationwide intersection accidents involved bicycles and 90% of them occurred during the change of phases at signalized intersections in China [1].Figure1 illustrates a typical signal plan implemented at many signalized intersections in China. As illustrated in the figure, the bicycles ordinarily share the same signal indications with the motorized vehicles of the same direction and the right-turning motorized vehicles can make a right-turn on red. Such a signal plan can possibly lead to two dangerous situations. The first one is that the right-turning motorized vehicles and the straight-through bicycles are given the right of way at the same green intervals, resulting in conflicts between these two incompatible movements. The second one is that the signal changes and clearance intervals are usually set according to the speed of the motorized vehicles. However, compared to the motorized vehicles, the bicycles require longer clearance time. Thus, bicyclists often fail to safely clear the intersection before the onset of the next phase.Figure 1
A typical stage-based signal phasing plan and its resulting traffic conflicts at signalized intersections in China.Therefore, in some Chinese cities, bicycle-specific traffic signals were introduced years ago, to improve the safety of bicycles at signalized intersections. With separate bicycle signals, bicycle movements can be released concurrently with the compatible motorized traffic movements by setting different intergreen intervals, to provide additional clearance time for the cyclists. At the same time, it is also possible to eliminate or reduce the traffic conflicts between the right-turning motorized vehicles and the straight-through bicycles, by using a lagging onset of green for the right-turning motorized vehicles.The use of bicycle-specific signals and probe vehicle data for queue length estimation [2] form the basis of the group-based signal control approach for mixed traffic flows. Unlike the stage-based approach, the duration of each signal group is considered independently. This allows the consideration of additional constraints such as intergreen periods and the minimum green intervals [3]. In addition, the group-based signal control approach has the potential to separate all the incompatible movements based on the intergreen period matrix to benefit the safety of cyclists, while maintaining the operational efficiency of the entire intersection. Although safety is a vitally important parameter in signalized intersection design, most of the existing signal timing methods focus only on the optimization of operational efficiency, without accounting for safety at the same time [4].


## 2. Literature Review

Group-based signal control approach is a method that can freely combine the compatible movements at intersections. The first group-based signal timing model was proposed by Improta and Cantarella in 1984. Traffic movements that cannot move together and concurrently fulfil safety requirements are regarded as incompatible groups and adequate intergreen or clearance periods have to be provided to separate them well within a signal cycle [5]. In 1988, Gallivan and Heydecker designed a new algorithm that did not restrict group formations and there might be multiple appearances for an identical stage occurring in single stage sequence [6]. Subsequently, in 1992, Heydecker improved this model by grouping all the possible cycle structures into a much smaller number of equivalent classes [7]. A successor function was introduced in the formulation to represent each of the signal groups, which was suitable for all phase-based formulations. Wong [8–10], Silcock [11] and Allsop [12] further extended the group-based signal timing model to account for area-wide signal control and proposed new solution algorithms. Recently, a lane-based optimization method combining the design of lane markings and signal timings for isolated intersections was also developed [13]. Wong and Heydecker further improved the model in order to handle the traffic demands more efficiently. They considered the numbers of approach lane in traffic arms as new integer variables and optimized them to obtain the ideal lane arrangement in different arms of an intersection [14].However, these group-based or lane-based signal timing models were mainly developed for single-objective optimization issues, such as either to minimize delay (or the number of stops) or to maximize capacity and throughput of the intersection. As discussed previously, signal phasing and timing are often complicated due to the existence of mixed traffic flows at signalized intersections in China. Thus, an enhanced group-based signal control model is required to accommodate the mixed traffic flows and, at the same time, account for both safety and delay.In addition, in the group-based signal control approach, most of the traffic conflicts occur at the phase change, such as the period between the intergreen intervals. However, due to stochastic characteristics of the traffic flow and road user behaviour, some potential conflicts may still exist even if intergreen periods are set properly based on local conditions. The problem becomes more intense in the case of mixed traffic flows since a larger number of incompatible traffic movements exist. Ignoring the potential traffic conflicts at the change of phases and optimizing the operational efficiency only based on capacity or delay may lead to unsafe solutions.Hence, the objective of this study is to develop a group-based signal control optimization model capable of accommodating the mixed traffic flows and to account for both safety and delay in the planning stage. To facilitate the application of the proposed model, a probabilistic approach was developed to estimate the occurring probability and severity of traffic conflicts between motorized vehicles and bicycles during the intergreen intervals. The traffic conflicts between the right-turning motorized vehicles and the pedestrians are reduced by adopting a lagging onset of green for the right-turning motorized traffics. Unlike most of the existing models, this optimization considered the average control delay per person for all the traffic modes (estimated using the Highway Capacity Manual 2010 method [15]), instead of just the average delay period for motorized vehicles. The applicability and advantages of the proposed model were demonstrated by a numerical study at an intersection located in Shanghai.


## 3. Safety Estimation

As mentioned earlier, the main safety estimation methods that are currently available to the practitioner include accident analysis (AA) and traffic conflict technique (TCT). The AA method assesses safety by using accident data as a direct measure of safety. The TCT approach assesses safety through conflict opportunity and severity represented by certain indices such as post encroachment time (PET) and time to collision (TTC) [16, 17]. The TCT is more appropriate for the signal timing stage since accident data are ordinarily not available.In the group-based signal control approach, most of the traffic conflicts occur at the change of phases. Tang and Nakamura proposed a probabilistic approach and a PET based indicator to estimate safety for the intergreen intervals [18]. They found that intersection safety during the intergreen intervals could be improved by reasonably setting intergreen times for a variety of signal groups of incompatible traffic movements. In the study, a conflict at the change of phases was defined as a consecutive pass of the last clearing vehicle entering the intersection at the conflict area in the previous phase and the first entering vehicle of the conflicting movement released in the subsequent phase.Therefore, the traffic conflicts between any two incompatible groupsi and j at the alteration of phases at the intersection Rij, can be estimated based on (a) the crossing probability of the last clearing vehicle of groupKi (Pi_clearing), (b) the existence probability of the first entering vehicle from movement groupKj (Pj_entering), and (c) the traffic conflict severity (Sij). This can be expressed as(1)Rij=Sij×Pi_clearing×Pj_enteringThe calculation ofSij, Pi_clearing, and Pj_entering is demonstrated in the following sections.


### 3.1. Traffic Conflict Severity (Sij)

In the case of mixed traffic, it is difficult to distinguish the severity of motor-motor conflicts from motor/bicycle conflicts when only time-based indicators are used. For example, 28 parameters grouped into four body regions are applied for frontal offset crashworthiness evaluation in the United States [19]. Therefore, in this paper, a new safety index combining PET and the kinetic energy rejection was proposed to represent the traffic conflict severity. According to the law of physics, the kinetic energy of an object is the energy it possesses due to its motion [20]. In classical mechanics, the kinetic energy of a nonrotating object of mass m traveling at a speed v equals 1/2mv2. Instead of PET alone, the safety index can be used to evaluate the entire intersection more comprehensively, as presented in (2)Sij=ΔKEijePETij(3)PETij=Iij′+tej-xi+ScivciwhereΔKEij is the kinetic energy rejection during the crash, J;tej represents entrance time of the first vehicle after onset of green, s;xi is the distance to stop line for the last vehicle of group Ki at the onset of yellow, m;Sci stands for the clearance distance of signal group Ki, m;Iij′ refers to the intergreen period between group Ki and group Kj, s;vci represents the clearing speed, m/s.The underlying rationale of the proposed traffic conflict measure is that it has a negative, exponential relationship with PET. Therefore, it becomes equal to the total kinetic energy rejection when the value of PET is zero (in case a real collision occurs).


### 3.2. Crossing Probability of the Last Clearing Vehicle from GroupKi (Pi_clearing)

A driver’s stop/pass decision at the onset of yellow is highly dependent upon the vehicle’s distance to the stop-line. Therefore, for the reliable estimation of the crossing probability of the last clearing vehicle, it is necessary to define the distance of the last clearing vehicle to the stop-line. If the distance of the vehicle is away from the stop-line, the driver will more likely choose to stop and a low probability of traffic conflicts will occur at the alteration of phases.By assuming a random arrival process and supposing the distance range to be[0,x0], the probability that a vehicle is located between [0,x0] at the onset of yellow can be estimated as follows: (4)Pi_x0=numix0ie-numi(5)numi=x0iαivi3.6wherePi_x0 represents the existence probability of the clearing vehicle within range of x0i from the stop line at the onset of yellow indication;numi represents the number of vehicles within the range of x0i;αi refers to the traffic density, veh/m;x0i stands for the distance to the stop line at the onset of yellow indication, m;vi refers to the approaching speed of the vehicle, m/s.Provided that a driver is located atxi in the range of [0,x0] at the onset of yellow of signal group Ki, the crossing probability of the driver can be estimated based on a binary logistic regression model. (6)Pi_cross=11+ezi(7)zi=a0i+a1ix0iwherePi_cross is the probability of the last driver to choose to cross at the onset of yellow signal;zi represents the linear function of x0i;a0i is the constant of the linear function;a1i stands for the coefficient of x0i.Then, the crossing probability of the last clearing vehicle (Pi_clearing) can be expressed as(8)Pi_clearing=Pi_cross×Pi_x0=λivi3.6e-x0iλivi/3.6×11+ea0i+a1ixi


### 3.3. The Existence Probability of the First Entering Vehicle from Group Kj (Pj_entering)

By following the above assumption of Poisson arrival, the probability that there is at least one vehicle of the conflicting groupKj to be released in the next phase at the onset of green can be estimated as follows: (9)Pj_entering=1-Pj_entering0=1-e-μjC1-φjwherePjentering(0) is the probability that there is no vehicle waiting behind the stop line;μj represents the flow rate of movement uj, veh/s;C refers to the cycle length, s;φj represents the green signal duration of movement uj.


## 3.1. Traffic Conflict Severity (Sij)

In the case of mixed traffic, it is difficult to distinguish the severity of motor-motor conflicts from motor/bicycle conflicts when only time-based indicators are used. For example, 28 parameters grouped into four body regions are applied for frontal offset crashworthiness evaluation in the United States [19]. Therefore, in this paper, a new safety index combining PET and the kinetic energy rejection was proposed to represent the traffic conflict severity. According to the law of physics, the kinetic energy of an object is the energy it possesses due to its motion [20]. In classical mechanics, the kinetic energy of a nonrotating object of mass m traveling at a speed v equals 1/2mv2. Instead of PET alone, the safety index can be used to evaluate the entire intersection more comprehensively, as presented in (2)Sij=ΔKEijePETij(3)PETij=Iij′+tej-xi+ScivciwhereΔKEij is the kinetic energy rejection during the crash, J;tej represents entrance time of the first vehicle after onset of green, s;xi is the distance to stop line for the last vehicle of group Ki at the onset of yellow, m;Sci stands for the clearance distance of signal group Ki, m;Iij′ refers to the intergreen period between group Ki and group Kj, s;vci represents the clearing speed, m/s.The underlying rationale of the proposed traffic conflict measure is that it has a negative, exponential relationship with PET. Therefore, it becomes equal to the total kinetic energy rejection when the value of PET is zero (in case a real collision occurs).


## 3.2. Crossing Probability of the Last Clearing Vehicle from GroupKi (Pi_clearing)

A driver’s stop/pass decision at the onset of yellow is highly dependent upon the vehicle’s distance to the stop-line. Therefore, for the reliable estimation of the crossing probability of the last clearing vehicle, it is necessary to define the distance of the last clearing vehicle to the stop-line. If the distance of the vehicle is away from the stop-line, the driver will more likely choose to stop and a low probability of traffic conflicts will occur at the alteration of phases.By assuming a random arrival process and supposing the distance range to be[0,x0], the probability that a vehicle is located between [0,x0] at the onset of yellow can be estimated as follows: (4)Pi_x0=numix0ie-numi(5)numi=x0iαivi3.6wherePi_x0 represents the existence probability of the clearing vehicle within range of x0i from the stop line at the onset of yellow indication;numi represents the number of vehicles within the range of x0i;αi refers to the traffic density, veh/m;x0i stands for the distance to the stop line at the onset of yellow indication, m;vi refers to the approaching speed of the vehicle, m/s.Provided that a driver is located atxi in the range of [0,x0] at the onset of yellow of signal group Ki, the crossing probability of the driver can be estimated based on a binary logistic regression model. (6)Pi_cross=11+ezi(7)zi=a0i+a1ix0iwherePi_cross is the probability of the last driver to choose to cross at the onset of yellow signal;zi represents the linear function of x0i;a0i is the constant of the linear function;a1i stands for the coefficient of x0i.Then, the crossing probability of the last clearing vehicle (Pi_clearing) can be expressed as(8)Pi_clearing=Pi_cross×Pi_x0=λivi3.6e-x0iλivi/3.6×11+ea0i+a1ixi


## 3.3. The Existence Probability of the First Entering Vehicle from Group Kj (Pj_entering)

By following the above assumption of Poisson arrival, the probability that there is at least one vehicle of the conflicting groupKj to be released in the next phase at the onset of green can be estimated as follows: (9)Pj_entering=1-Pj_entering0=1-e-μjC1-φjwherePjentering(0) is the probability that there is no vehicle waiting behind the stop line;μj represents the flow rate of movement uj, veh/s;C refers to the cycle length, s;φj represents the green signal duration of movement uj.


## 4. The Proposed Optimization Model

### 4.1. Decision Variables

Figure2 illustrates a typical four-arm signalized intersection with bicycle lanes and bicycle-specific signals in China. As presented in the figure, motorized vehicle groups and bicycle signal groups are numbered in a clockwise direction.Figure 2
Signal groups of a typical four-leg signalized intersection.The right of way of each movementui is specified by two variables expressed as a fraction of the cycle length C in the group-based control approach: the start of green, θi, and the duration of green signal, φi. For each pair of incompatible movements ui and uj, the intergreen time Iij is defined as the shortest time interval following which the movement uj receives right of way after the green signal of movement ui is terminated. Note that the intergreen time Iij is lane dependent, based on the geometry of the intersection.Then, the terms C,θi and φi are used as decision variables during the optimization process.


### 4.2. The Objective Functions

The mathematical description of the multiobjective model is as follows:(10)minΦs=fconflictsfdelaysfconflicts=fφ,Cfdelays=fθ,φ,Cs.t.s:θ,φ,C∈SwhereS is the variable collections used in this method;fconflict represents the objective function for safety maximization;fdelay is the objective function for delay minimization.By assuming that there are two persons in each vehicle, the objective of the proposed model is to minimize traffic conflicts per hour and to minimize average delay per person.(11)fconflict=3600C×∑i=1,j=1NM+NBRij2(12)fdelay=2∑i=1NMqMidMi+∑j=NM+1NM+NBqBjdBj2∑i=1NMqMi+∑j=M+1NM+NBqBjwherefconflict is the objective function for safety maximization;fdelay represents the objective function for delay minimization;NM stands for the number of signal groups for motorized vehicle;NB equals the number of signal groups for bicycle;Rij indicates the traffic conflicts severity between any two incompatible signal groupsKi andKj at the change of phases;dMi is the delay of the motorized vehicles, s;dBj is the delay of the bicycles, s;qMi represents the flow rate of signal group i for motorized vehicles, vehicles/s;qNi stands for the flow rate of signal group j for bicycles, vehicles/s.The average control delay of motorized vehicles in each signal group can be estimated according to Webster’s delay formula (21). (13)dMi=0.9sMi1-λMi2×C2sMi-qMi+qMi2sMiλMisMiλMi-qMiwheresMi represents the saturated flow rate, vehicles/s;λMi stands for the proportion of the effective green with respect to cycle length.The average control delay of bicycles in each signal group can be estimated based on the method recommended in the Highway Capacity Manual 2010 (22).(14)dBi=0.5·C·1-gBi/C21-gBi/C·min⁡qBi/cBi,1wheregBi is the effective green time for the bicycle lane, s;cBi represents the capacity of the bicycle lane, bicycles/s;


### 4.3. The Constraints

#### 4.3.1. Start of Green Signal

The start of green Signal can be quite arbitrary since it satisfies other constraints in the problem. Asθi is the fraction of cycle lengthC, it must be within the range of (0, 1).(15)1≥θi>0∀i=1,2,⋯,NM+NB


#### 4.3.2. Duration of Green Signal

The duration of green signal should be subject to a minimum value, to avoid sudden stop-and-go motion and to provide sufficient time for road users to prepare for the change of signal states. In the case of mixed traffic flows, the minimum green duration must also ensure sufficient time for pedestrians to cross the street.(16)1≥φi≥φimin∀i=1,2,⋯,NM+NBwhereφimin represents the minimum green duration.According to the Traffic Signal Timing Manual [21], the total time required for the pedestrian movements is the sum of the start-up time and the period required for a person to traverse the crosswalk.


#### 4.3.3. Capacity Requirement

In the planning stage, the effective green duration should be sufficient to accommodate the traffic flow of each signal group.(17)φi+eCsi≥qi∀i=1,2,⋯,NM+NBwheree is time difference between actual and effective green signal.


#### 4.3.4. Intergreen Time

For any pair of incompatible groups, the intergreen time constraints are necessary only when both of these groups are adjacent in the cycle.As illustrated in Figure3, the intergreen times Iij′ can be expressed as follows.Figure 3
Definition of intergreen times between incompatible groups.
(a)
θ i ≤ θ j (b)
θ i ≥ θ jIfθi≤θj,(18)Iij′=θj-θi+φi×CIfθi≥θj,(19)Iij′=1+θj-θi+φi×CIn this study, the minimum intergreen timesIij are determined according to the method recommended by RilSA (FGSV-Verlag, 2003) as described below. It is considered as a more flexible and efficient technique for the group-based signal control approach [18]. (20)Iij=tu+tc-teThe crossing timetu is usually assumed to be 3 seconds, 4 seconds or 5 seconds for through-ahead vehicles depending on the speed limit and these rates are often used as yellow times. The difference of the second and third terms periods, (tc-te), is then used as the all-red time. A more detailed description of the formula can be found in RiLSA.The constraints should be set as(21)Iij′≥IijIfθi≤θj and Iij>0(22)θj-θi+φi+IijC≥0,∀i=1,2,…M+N,j=1,2,…M+NIfθi≥θj and Iij>0(23)θi+φi+IijC-1+θj≤0,∀i=1,2,…M+N,j=1,2,…M+N


#### 4.3.5. Lagging Onset of Green Signal for Right-Turning Motorized Vehicles

As mentioned above, conflicts between the right-turning motorized vehicles and the through-ahead bicycles exist at the intersections with mixed traffic flows. Hence, it is very important to temporarily separate these two conflicting traffic movements. In the proposed model, the signal groups of right-turning motor-vehicles (RM) are postponed by a period following the onset of the adjacent straight-through bicycles (SB), to release the queued bicycles. Then, the constraints in signal timing can be set asIfθRM≤θSB(24)θRM+φRM+IMBC≤θSBIfθRM≥θSB and IBM>0(25)1-φSB×qSB≤θSB+1-θRM×sRM,


#### 4.3.6. Cycle Length

To ensure reasonable solutions, a boundary must be set for the cycle length as follows:(26)Cmin≤C≤CmaxwhereCminCmax represents the minimum (maximum) cycle length (s).


### 4.4. Solution Algorithm

Recently, researchers have introduced many multiobjective models to explore the characteristics of traffic and transportation [22–24]. Considering the drawbacks of the existing approaches, new numerical search algorithms are becoming popular, such as Genetic Algorithms (GAs). The enhanced GAs such as the Nondominated Sorting Genetic Algorithm (NSGA-II) are similar to the simple GAs in the selection, crossover and mutation operators, aiming to find the Pareto frontier of compromise solutions for all objectives. NSGA-II obtained a close approximation to the Pareto set by using an analytical formula to determine delay and stops. Sun et al. applied NSGA-II to optimize delay and stops for an isolated intersection under two-phase control [25]. Abbas et al. applied NSGA-II to a small three-signal network [26]. Stevanovic et al. proposed a traffic signal timing optimization model based on surrogate measures of safety [27]. They adopted a multiple-objective genetic algorithm to identify the optimal compromise between two competing objectives: surrogate safety and traffic efficiency. Ma et al. [28, 29] used the multiobjective optimization model to improve the safety and convenience of pedestrian signals, including the exclusive pedestrian phase, two-stage midblock crosswalk, and the conventional two-way crossing at a single intersection.Thus, the NSGA-II algorithm was adopted to solve the proposed multiobjective model, which follows a standard flowchart as presented in Figure4 [30]. The NSGA-II is similar to GAs. However, the algorithm ranks the whole population based on all objectives before the selection. All individuals with rank One are removed from consideration and all other individuals are ranked again and are assigned a rank of Two. A crowding distance is calculated for all individuals after each of them are assigned a rank. The selection operator is then applied while assigning higher fitness to individuals with higher ranks and crowding distances [26]. The algorithm ensures elitism by combining the parent population with the children population before the crossover and mutation operators are applied.Figure 4
Flow chart of the solution algorithm based on NSGA-II.


## 4.1. Decision Variables

Figure2 illustrates a typical four-arm signalized intersection with bicycle lanes and bicycle-specific signals in China. As presented in the figure, motorized vehicle groups and bicycle signal groups are numbered in a clockwise direction.Figure 2
Signal groups of a typical four-leg signalized intersection.The right of way of each movementui is specified by two variables expressed as a fraction of the cycle length C in the group-based control approach: the start of green, θi, and the duration of green signal, φi. For each pair of incompatible movements ui and uj, the intergreen time Iij is defined as the shortest time interval following which the movement uj receives right of way after the green signal of movement ui is terminated. Note that the intergreen time Iij is lane dependent, based on the geometry of the intersection.Then, the terms C,θi and φi are used as decision variables during the optimization process.


## 4.2. The Objective Functions

The mathematical description of the multiobjective model is as follows:(10)minΦs=fconflictsfdelaysfconflicts=fφ,Cfdelays=fθ,φ,Cs.t.s:θ,φ,C∈SwhereS is the variable collections used in this method;fconflict represents the objective function for safety maximization;fdelay is the objective function for delay minimization.By assuming that there are two persons in each vehicle, the objective of the proposed model is to minimize traffic conflicts per hour and to minimize average delay per person.(11)fconflict=3600C×∑i=1,j=1NM+NBRij2(12)fdelay=2∑i=1NMqMidMi+∑j=NM+1NM+NBqBjdBj2∑i=1NMqMi+∑j=M+1NM+NBqBjwherefconflict is the objective function for safety maximization;fdelay represents the objective function for delay minimization;NM stands for the number of signal groups for motorized vehicle;NB equals the number of signal groups for bicycle;Rij indicates the traffic conflicts severity between any two incompatible signal groupsKi andKj at the change of phases;dMi is the delay of the motorized vehicles, s;dBj is the delay of the bicycles, s;qMi represents the flow rate of signal group i for motorized vehicles, vehicles/s;qNi stands for the flow rate of signal group j for bicycles, vehicles/s.The average control delay of motorized vehicles in each signal group can be estimated according to Webster’s delay formula (21). (13)dMi=0.9sMi1-λMi2×C2sMi-qMi+qMi2sMiλMisMiλMi-qMiwheresMi represents the saturated flow rate, vehicles/s;λMi stands for the proportion of the effective green with respect to cycle length.The average control delay of bicycles in each signal group can be estimated based on the method recommended in the Highway Capacity Manual 2010 (22).(14)dBi=0.5·C·1-gBi/C21-gBi/C·min⁡qBi/cBi,1wheregBi is the effective green time for the bicycle lane, s;cBi represents the capacity of the bicycle lane, bicycles/s;


## 4.3. The Constraints

### 4.3.1. Start of Green Signal

The start of green Signal can be quite arbitrary since it satisfies other constraints in the problem. Asθi is the fraction of cycle lengthC, it must be within the range of (0, 1).(15)1≥θi>0∀i=1,2,⋯,NM+NB


### 4.3.2. Duration of Green Signal

The duration of green signal should be subject to a minimum value, to avoid sudden stop-and-go motion and to provide sufficient time for road users to prepare for the change of signal states. In the case of mixed traffic flows, the minimum green duration must also ensure sufficient time for pedestrians to cross the street.(16)1≥φi≥φimin∀i=1,2,⋯,NM+NBwhereφimin represents the minimum green duration.According to the Traffic Signal Timing Manual [21], the total time required for the pedestrian movements is the sum of the start-up time and the period required for a person to traverse the crosswalk.


### 4.3.3. Capacity Requirement

In the planning stage, the effective green duration should be sufficient to accommodate the traffic flow of each signal group.(17)φi+eCsi≥qi∀i=1,2,⋯,NM+NBwheree is time difference between actual and effective green signal.


### 4.3.4. Intergreen Time

For any pair of incompatible groups, the intergreen time constraints are necessary only when both of these groups are adjacent in the cycle.As illustrated in Figure3, the intergreen times Iij′ can be expressed as follows.Figure 3
Definition of intergreen times between incompatible groups.
(a)
θ i ≤ θ j (b)
θ i ≥ θ jIfθi≤θj,(18)Iij′=θj-θi+φi×CIfθi≥θj,(19)Iij′=1+θj-θi+φi×CIn this study, the minimum intergreen timesIij are determined according to the method recommended by RilSA (FGSV-Verlag, 2003) as described below. It is considered as a more flexible and efficient technique for the group-based signal control approach [18]. (20)Iij=tu+tc-teThe crossing timetu is usually assumed to be 3 seconds, 4 seconds or 5 seconds for through-ahead vehicles depending on the speed limit and these rates are often used as yellow times. The difference of the second and third terms periods, (tc-te), is then used as the all-red time. A more detailed description of the formula can be found in RiLSA.The constraints should be set as(21)Iij′≥IijIfθi≤θj and Iij>0(22)θj-θi+φi+IijC≥0,∀i=1,2,…M+N,j=1,2,…M+NIfθi≥θj and Iij>0(23)θi+φi+IijC-1+θj≤0,∀i=1,2,…M+N,j=1,2,…M+N


### 4.3.5. Lagging Onset of Green Signal for Right-Turning Motorized Vehicles

As mentioned above, conflicts between the right-turning motorized vehicles and the through-ahead bicycles exist at the intersections with mixed traffic flows. Hence, it is very important to temporarily separate these two conflicting traffic movements. In the proposed model, the signal groups of right-turning motor-vehicles (RM) are postponed by a period following the onset of the adjacent straight-through bicycles (SB), to release the queued bicycles. Then, the constraints in signal timing can be set asIfθRM≤θSB(24)θRM+φRM+IMBC≤θSBIfθRM≥θSB and IBM>0(25)1-φSB×qSB≤θSB+1-θRM×sRM,


### 4.3.6. Cycle Length

To ensure reasonable solutions, a boundary must be set for the cycle length as follows:(26)Cmin≤C≤CmaxwhereCminCmax represents the minimum (maximum) cycle length (s).


## 4.3.1. Start of Green Signal

The start of green Signal can be quite arbitrary since it satisfies other constraints in the problem. Asθi is the fraction of cycle lengthC, it must be within the range of (0, 1).(15)1≥θi>0∀i=1,2,⋯,NM+NB


## 4.3.2. Duration of Green Signal

The duration of green signal should be subject to a minimum value, to avoid sudden stop-and-go motion and to provide sufficient time for road users to prepare for the change of signal states. In the case of mixed traffic flows, the minimum green duration must also ensure sufficient time for pedestrians to cross the street.(16)1≥φi≥φimin∀i=1,2,⋯,NM+NBwhereφimin represents the minimum green duration.According to the Traffic Signal Timing Manual [21], the total time required for the pedestrian movements is the sum of the start-up time and the period required for a person to traverse the crosswalk.


## 4.3.3. Capacity Requirement

In the planning stage, the effective green duration should be sufficient to accommodate the traffic flow of each signal group.(17)φi+eCsi≥qi∀i=1,2,⋯,NM+NBwheree is time difference between actual and effective green signal.


## 4.3.4. Intergreen Time

For any pair of incompatible groups, the intergreen time constraints are necessary only when both of these groups are adjacent in the cycle.As illustrated in Figure3, the intergreen times Iij′ can be expressed as follows.Figure 3
Definition of intergreen times between incompatible groups.
(a)
θ i ≤ θ j (b)
θ i ≥ θ jIfθi≤θj,(18)Iij′=θj-θi+φi×CIfθi≥θj,(19)Iij′=1+θj-θi+φi×CIn this study, the minimum intergreen timesIij are determined according to the method recommended by RilSA (FGSV-Verlag, 2003) as described below. It is considered as a more flexible and efficient technique for the group-based signal control approach [18]. (20)Iij=tu+tc-teThe crossing timetu is usually assumed to be 3 seconds, 4 seconds or 5 seconds for through-ahead vehicles depending on the speed limit and these rates are often used as yellow times. The difference of the second and third terms periods, (tc-te), is then used as the all-red time. A more detailed description of the formula can be found in RiLSA.The constraints should be set as(21)Iij′≥IijIfθi≤θj and Iij>0(22)θj-θi+φi+IijC≥0,∀i=1,2,…M+N,j=1,2,…M+NIfθi≥θj and Iij>0(23)θi+φi+IijC-1+θj≤0,∀i=1,2,…M+N,j=1,2,…M+N


## 4.3.5. Lagging Onset of Green Signal for Right-Turning Motorized Vehicles

As mentioned above, conflicts between the right-turning motorized vehicles and the through-ahead bicycles exist at the intersections with mixed traffic flows. Hence, it is very important to temporarily separate these two conflicting traffic movements. In the proposed model, the signal groups of right-turning motor-vehicles (RM) are postponed by a period following the onset of the adjacent straight-through bicycles (SB), to release the queued bicycles. Then, the constraints in signal timing can be set asIfθRM≤θSB(24)θRM+φRM+IMBC≤θSBIfθRM≥θSB and IBM>0(25)1-φSB×qSB≤θSB+1-θRM×sRM,


## 4.3.6. Cycle Length

To ensure reasonable solutions, a boundary must be set for the cycle length as follows:(26)Cmin≤C≤CmaxwhereCminCmax represents the minimum (maximum) cycle length (s).


## 4.4. Solution Algorithm

Recently, researchers have introduced many multiobjective models to explore the characteristics of traffic and transportation [22–24]. Considering the drawbacks of the existing approaches, new numerical search algorithms are becoming popular, such as Genetic Algorithms (GAs). The enhanced GAs such as the Nondominated Sorting Genetic Algorithm (NSGA-II) are similar to the simple GAs in the selection, crossover and mutation operators, aiming to find the Pareto frontier of compromise solutions for all objectives. NSGA-II obtained a close approximation to the Pareto set by using an analytical formula to determine delay and stops. Sun et al. applied NSGA-II to optimize delay and stops for an isolated intersection under two-phase control [25]. Abbas et al. applied NSGA-II to a small three-signal network [26]. Stevanovic et al. proposed a traffic signal timing optimization model based on surrogate measures of safety [27]. They adopted a multiple-objective genetic algorithm to identify the optimal compromise between two competing objectives: surrogate safety and traffic efficiency. Ma et al. [28, 29] used the multiobjective optimization model to improve the safety and convenience of pedestrian signals, including the exclusive pedestrian phase, two-stage midblock crosswalk, and the conventional two-way crossing at a single intersection.Thus, the NSGA-II algorithm was adopted to solve the proposed multiobjective model, which follows a standard flowchart as presented in Figure4 [30]. The NSGA-II is similar to GAs. However, the algorithm ranks the whole population based on all objectives before the selection. All individuals with rank One are removed from consideration and all other individuals are ranked again and are assigned a rank of Two. A crowding distance is calculated for all individuals after each of them are assigned a rank. The selection operator is then applied while assigning higher fitness to individuals with higher ranks and crowding distances [26]. The algorithm ensures elitism by combining the parent population with the children population before the crossover and mutation operators are applied.Figure 4
Flow chart of the solution algorithm based on NSGA-II.


## 5. Case Study

### 5.1. Study Site and Data Preparation

A typical four-arm intersection in Shanghai (at Cao’an and Jiasong Road) was selected as a case study. A speed limit of 80 km/h was introduced on Cao’an Road and a 60 km/h limit on Jiasong Road. A conventional four-phase signal plan was adopted for this intersection, as illustrated in Figure1, including a cycle length of 145 s, an intergreen interval of 5 s for phases 1 and 3 and an intergreen interval of 6 s for phases 2 and 4. Signal group settings were the same as those shown in Figure 2. Field surveys based on videotaping method were conducted on January 17th - 18th and July 4th - 5th, 2013. Traffic volumes, geometric parameters and relevant driver behavioural parameters such as speed, stop/pass decision and reaction time were collected and used for delay and safety estimations, as summarized in Table 1. The intergreen times were determined based on the RiLSA method introduced earlier and the calculated intergreen time matrix is presented in Table 2.Table 1
Observed traffic volumes driver behavioural parameters as well as estimated stop/pass decision models at the study site.
(a)
Motorized vehicles
Signal Group K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 Traffic demand in pcu/h 44 328 184 235 486 100 200 311 155 212 573 25  saturation flow in pcu/h 1800 2000 1600 1800×2 2000×3 1600 1800 2000 1600 1800 2000×3 1600  Clearing Speed∗ in km/h 35 47 35 35 47 35 35 47 35 35 47 35  Stop/Pass Decision Model Left-turning Straight-through Right-turning P c i = 1 / ( 1 + e x p ( a 0 - a 1 x ) ) a 0 = 0.8 , a 1 = - 0.028 a 0 = 2.005 , a 1 = - 0.43 a 0 = 0.8 , a 1 = - 0.028  (b)
Bicycles
Signal Group K13 K14 K15 K16 K17 K18 K19 K20 Traffic demand in pcu/h 264 230 315 120 400 450 505 136  saturation flow∗∗ in pcu/h 2000 3000 2000 3000 2000 3000 2000 3000  Clearing Speed∗ in km/h 15 15 15 15 15 15 15 15  Stop/Pass Decision Model Left-turning Straight-through P c i = 1 / ( 1 + e x p ( a 0 - a 1 x ) ) a 0 = 0.45 , a 1 = - 0.008 a 0 = 1.03 , a 1 = - 0.25 ∗For the sake of safety, take the 15 percentile clearing speed to make sure that most of the vehicles could clear the intersection. ∗ ∗According to HCM2010, because service quality deteriorates at flow levels well below capacity, the concept of capacity has little utility in the design and analysis of bicycle paths and other facilities. Thus it is hard to obtain the saturation flow of the bicycles during a unique or unambiguous theoretical method. In this case, results from one previous study on the saturation flow rate of bicycles in China were adopted.Table 2
Intergreen Matrix.
Entering Vehicles K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 K13 K14 K15 K16 K17 K18 K19 K20  Clearing Vehicles K1    5 5   5  5 5    5   5 5 5 K2    5 5  5   5 5  5  5 5 5  5 5 K3             5 5   5   5 K4 5 5     5 5   5  5 5   5   5 K5 5 5     5 5  5   5 5 5  5 5 5 K6              5 5 5   5 K7  5  5 5     5 5   5 5 5   5 K8 5   5 5     5 5  5  5 5 5  5 5 K9             5   5 5 5 K10 5 5   5  5 5     5   5 5 5 K11 5 5  5   5 5     5 5 5  5 5 5 K12               5   5 5 5 K13  7 7 7 7   7 7 7 7    6   6 6 6 K14   7 7 7 7 7    7    6 6 6   6 K15 7 7   7 7 7 7   7 7 6 6   6   6 K16  7     7 7 7 7    6   6 6 6 K17  7 7 7 7   7 7 7 7   6 6 6   6 K18 7    7     7 7 7 6   6   6 6 K19 7 7   7 7 7 7   7 7 6   6 6 6 K20 7 7 7 7    7     6 6 6   6According to the weights of cars of similar size to Toyota Camry, the average mass of the motorized vehicle was set as 1600 kg (approximately 3500 lb). The average weight for an adult in China is considered to be 70 kg and the average mass of a bicycle equals 15 kg. The kinetic energy rejection for any two incompatible movements was then calculated based on intersection geometry and the results are presented in Tables3 and 4.Table 3
List oftej-Sci/vci.
Entering Groups K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 K13 K14 K15 K16 K17 K18 K19 K20  Clearing Groups K1    2.0 2.5   2.0  2.0 2.0    2.5   2.5 2.5 2.5 K2    2.0 2.0  2.0   2.0 2.0  2.5  2.5 2.5 2.5  2.5 2.5 K3             2.5 2.5   2.5   2.5 K4 2.0 2.0     2.0 2.0   2.0  2.5 2.5   2.5   2.5 K5 2.0 2.0     2.0 2.0  2.0   2.5 2.5 2.5  2.5 2.5 2.5 K6              2.5 2.5 2.5   2.5 K7  2.0  2.0 2.0     2.0 2.0   2.5 2.5 2.5   2.5 K8 2.0   2.0 2.0     2.0 2.0  2.5  2.5 2.5 2.5  2.5 2.5 K9             2.5   2.5 2.5 2.5 K10 2.0 2.0   2.0  2.0 2.0     2.5   2.5 2.5 2.5 K11 2.0 2.0  2.0   2.0 2.0     2.5 2.5 2.5  2.5 2.5 2.5 K12               2.5   2.5 2.5 2.5 K13  1.5 1.5 1.5 1.5   1.5 1.5 1.5 1.5    2.0   2.0 2.0 2.0 K14   1.5 1.5 1.5 1.5 1.5    1.5    2.0 2.0 2.0   2.0 K15 1.5 1.5   1.5 1.5 1.5 1.5   1.5 1.5 2.0 2.0   2.0   2.0 K16  1.5    1.5 1.5 1.5      2.0   2.0 2.0 2.0 K17  1.5 1.5 1.5 1.5   1.5 1.5 1.5 1.5   2.0 2.0 2.0   2.0 K18 1.5    1.5    1.5 1.5 1.5 1.5 2.0   2.0   2.0 2.0 K19 1.5 1.5   1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.0   2.0 2.0 2.0 K20 1.5 1.5 1.5 1.5    1.5    1.5 2.0 2.0 2.0   2.0Table 4
List of kinetic energy rejection for each collision.
K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 K13 K14 K15 K16 K17 K18 K19 K20 K1    340000 66987   933013  340000 933013    31034   52003 31034 52003 K2    1145692 890000  1145692   314308 890000  36830  103932 70381 103932  36830 70381 K3             18927 13912   10064   22815 K4 340000 933013     340000 66987   933013  31034 52003   31034   52003 K5 314308 890000     1145692 890000  1145692   36830 70381 36830  103932 70381 103932 K6              22815 18927 13912   10064 K7  933013  340000 933013     340000 66987   52003 31034 52003   31034 K8 1145692   197180 890000     1145692 890000  103932  36830 70381 36830  103932 70381 K9             10064   22815 18927 13912 K10 340000 66987   933013  340000 933013     31034   52003 31034 52003 K11 1145692 890000  1145692   314308 890000     103932 70381 103932  36830 70381 36830 K12               10064   22815 18927 13912 K13  8091 34303 18160 8091   60515 2433 18160 27473    7841   14466 7841 28933 K14   6758 33887 34303 2433 33887    37060    14466 7841 14466  1216 7841 K15 18160 60515   8091 31001 18160 8091   60515 5319 7841 14466   7841   14466 K16  34303    31001 33887 34303 2433 18160    7841   14466 7841 14466 K17  60515 8091 18160 60515   8091 22860 60515 8091   14466 7841 14466   14891 K18 33887    34303    31001 31001 14891 2433 14466   14891   14466 7841 K19 18160 8091   60515 31001 18160    8091 31001 7841   14466 7841 14466 K20 33887 34303 18160 33887    34303    31001 14466 7841 14466   7841


### 5.2. Results Analysis

The NSGA-II program was run in MATLAB with a population size of 200 and a maximum number of generations of 1,500. The convergence of safety and delay against the number of generations is presented in Figure5(a), where the traffic conflicts and delays of every 10 generations were aggregated and demonstrated together. It was concluded that both traffic conflicts and delays tend to be stable after 1,300 generations. When the optimal solutions reach convergence, the minimum delays are approximately 60 s. Figure 5(b) presents the solutions from the multiobjective optimization, which belongs to the Pareto frontier, clearly demonstrating the trade-off relationship between the two objectives.Figure 5
Converging trend and Pareto front of the solution algorithm.
(a)
Converging trend (b)
Pareto frontThrough MATLAB processing, it was found that the signal timing plan is optimized to reduce conflict intensity. The optimized cycle length is longer than the original signal timings, since it is not possible to reduce vehicular conflicts without sacrificing part of the traffic efficiency. Additionally, Stevanovic demonstrated a similar result in his research [27].Figure5(a) illustrates all evaluated solutions in the safety-delay domain during the multiobjective optimizations. Figure 5(b) presents the solutions from multiobjective optimization which belong to the Pareto Frontier, efficiently demonstrating the trade-off between the two objectives. By adopting any of the signal timing plans representing these solutions, a traffic signal engineer can improve safety (by reducing the intensity of traffic conflicts) without compromising the efficiency (delay) of the current signal-timing plan.Figure6 presents the flexibility of sequencing and combination of the signal groups. By comparing the delay and the safety intensity (Table 1), it is found that the safety index is improved by almost 35 times, while the delay is doubled using the new method. According to (8), the safety index Sij is sensitive to the duration of PET, due to the use of the exponential function. This could explain why the safety performance was strongly improved while the efficiency was decreased.Figure 6
Optimal singal setting from one of the solutions.


## 5.1. Study Site and Data Preparation

A typical four-arm intersection in Shanghai (at Cao’an and Jiasong Road) was selected as a case study. A speed limit of 80 km/h was introduced on Cao’an Road and a 60 km/h limit on Jiasong Road. A conventional four-phase signal plan was adopted for this intersection, as illustrated in Figure1, including a cycle length of 145 s, an intergreen interval of 5 s for phases 1 and 3 and an intergreen interval of 6 s for phases 2 and 4. Signal group settings were the same as those shown in Figure 2. Field surveys based on videotaping method were conducted on January 17th - 18th and July 4th - 5th, 2013. Traffic volumes, geometric parameters and relevant driver behavioural parameters such as speed, stop/pass decision and reaction time were collected and used for delay and safety estimations, as summarized in Table 1. The intergreen times were determined based on the RiLSA method introduced earlier and the calculated intergreen time matrix is presented in Table 2.Table 1
Observed traffic volumes driver behavioural parameters as well as estimated stop/pass decision models at the study site.
(a)
Motorized vehicles
Signal Group K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 Traffic demand in pcu/h 44 328 184 235 486 100 200 311 155 212 573 25  saturation flow in pcu/h 1800 2000 1600 1800×2 2000×3 1600 1800 2000 1600 1800 2000×3 1600  Clearing Speed∗ in km/h 35 47 35 35 47 35 35 47 35 35 47 35  Stop/Pass Decision Model Left-turning Straight-through Right-turning P c i = 1 / ( 1 + e x p ( a 0 - a 1 x ) ) a 0 = 0.8 , a 1 = - 0.028 a 0 = 2.005 , a 1 = - 0.43 a 0 = 0.8 , a 1 = - 0.028  (b)
Bicycles
Signal Group K13 K14 K15 K16 K17 K18 K19 K20 Traffic demand in pcu/h 264 230 315 120 400 450 505 136  saturation flow∗∗ in pcu/h 2000 3000 2000 3000 2000 3000 2000 3000  Clearing Speed∗ in km/h 15 15 15 15 15 15 15 15  Stop/Pass Decision Model Left-turning Straight-through P c i = 1 / ( 1 + e x p ( a 0 - a 1 x ) ) a 0 = 0.45 , a 1 = - 0.008 a 0 = 1.03 , a 1 = - 0.25 ∗For the sake of safety, take the 15 percentile clearing speed to make sure that most of the vehicles could clear the intersection. ∗ ∗According to HCM2010, because service quality deteriorates at flow levels well below capacity, the concept of capacity has little utility in the design and analysis of bicycle paths and other facilities. Thus it is hard to obtain the saturation flow of the bicycles during a unique or unambiguous theoretical method. In this case, results from one previous study on the saturation flow rate of bicycles in China were adopted.Table 2
Intergreen Matrix.
Entering Vehicles K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 K13 K14 K15 K16 K17 K18 K19 K20  Clearing Vehicles K1    5 5   5  5 5    5   5 5 5 K2    5 5  5   5 5  5  5 5 5  5 5 K3             5 5   5   5 K4 5 5     5 5   5  5 5   5   5 K5 5 5     5 5  5   5 5 5  5 5 5 K6              5 5 5   5 K7  5  5 5     5 5   5 5 5   5 K8 5   5 5     5 5  5  5 5 5  5 5 K9             5   5 5 5 K10 5 5   5  5 5     5   5 5 5 K11 5 5  5   5 5     5 5 5  5 5 5 K12               5   5 5 5 K13  7 7 7 7   7 7 7 7    6   6 6 6 K14   7 7 7 7 7    7    6 6 6   6 K15 7 7   7 7 7 7   7 7 6 6   6   6 K16  7     7 7 7 7    6   6 6 6 K17  7 7 7 7   7 7 7 7   6 6 6   6 K18 7    7     7 7 7 6   6   6 6 K19 7 7   7 7 7 7   7 7 6   6 6 6 K20 7 7 7 7    7     6 6 6   6According to the weights of cars of similar size to Toyota Camry, the average mass of the motorized vehicle was set as 1600 kg (approximately 3500 lb). The average weight for an adult in China is considered to be 70 kg and the average mass of a bicycle equals 15 kg. The kinetic energy rejection for any two incompatible movements was then calculated based on intersection geometry and the results are presented in Tables3 and 4.Table 3
List oftej-Sci/vci.
Entering Groups K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 K13 K14 K15 K16 K17 K18 K19 K20  Clearing Groups K1    2.0 2.5   2.0  2.0 2.0    2.5   2.5 2.5 2.5 K2    2.0 2.0  2.0   2.0 2.0  2.5  2.5 2.5 2.5  2.5 2.5 K3             2.5 2.5   2.5   2.5 K4 2.0 2.0     2.0 2.0   2.0  2.5 2.5   2.5   2.5 K5 2.0 2.0     2.0 2.0  2.0   2.5 2.5 2.5  2.5 2.5 2.5 K6              2.5 2.5 2.5   2.5 K7  2.0  2.0 2.0     2.0 2.0   2.5 2.5 2.5   2.5 K8 2.0   2.0 2.0     2.0 2.0  2.5  2.5 2.5 2.5  2.5 2.5 K9             2.5   2.5 2.5 2.5 K10 2.0 2.0   2.0  2.0 2.0     2.5   2.5 2.5 2.5 K11 2.0 2.0  2.0   2.0 2.0     2.5 2.5 2.5  2.5 2.5 2.5 K12               2.5   2.5 2.5 2.5 K13  1.5 1.5 1.5 1.5   1.5 1.5 1.5 1.5    2.0   2.0 2.0 2.0 K14   1.5 1.5 1.5 1.5 1.5    1.5    2.0 2.0 2.0   2.0 K15 1.5 1.5   1.5 1.5 1.5 1.5   1.5 1.5 2.0 2.0   2.0   2.0 K16  1.5    1.5 1.5 1.5      2.0   2.0 2.0 2.0 K17  1.5 1.5 1.5 1.5   1.5 1.5 1.5 1.5   2.0 2.0 2.0   2.0 K18 1.5    1.5    1.5 1.5 1.5 1.5 2.0   2.0   2.0 2.0 K19 1.5 1.5   1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.0   2.0 2.0 2.0 K20 1.5 1.5 1.5 1.5    1.5    1.5 2.0 2.0 2.0   2.0Table 4
List of kinetic energy rejection for each collision.
K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12 K13 K14 K15 K16 K17 K18 K19 K20 K1    340000 66987   933013  340000 933013    31034   52003 31034 52003 K2    1145692 890000  1145692   314308 890000  36830  103932 70381 103932  36830 70381 K3             18927 13912   10064   22815 K4 340000 933013     340000 66987   933013  31034 52003   31034   52003 K5 314308 890000     1145692 890000  1145692   36830 70381 36830  103932 70381 103932 K6              22815 18927 13912   10064 K7  933013  340000 933013     340000 66987   52003 31034 52003   31034 K8 1145692   197180 890000     1145692 890000  103932  36830 70381 36830  103932 70381 K9             10064   22815 18927 13912 K10 340000 66987   933013  340000 933013     31034   52003 31034 52003 K11 1145692 890000  1145692   314308 890000     103932 70381 103932  36830 70381 36830 K12               10064   22815 18927 13912 K13  8091 34303 18160 8091   60515 2433 18160 27473    7841   14466 7841 28933 K14   6758 33887 34303 2433 33887    37060    14466 7841 14466  1216 7841 K15 18160 60515   8091 31001 18160 8091   60515 5319 7841 14466   7841   14466 K16  34303    31001 33887 34303 2433 18160    7841   14466 7841 14466 K17  60515 8091 18160 60515   8091 22860 60515 8091   14466 7841 14466   14891 K18 33887    34303    31001 31001 14891 2433 14466   14891   14466 7841 K19 18160 8091   60515 31001 18160    8091 31001 7841   14466 7841 14466 K20 33887 34303 18160 33887    34303    31001 14466 7841 14466   7841


## 5.2. Results Analysis

The NSGA-II program was run in MATLAB with a population size of 200 and a maximum number of generations of 1,500. The convergence of safety and delay against the number of generations is presented in Figure5(a), where the traffic conflicts and delays of every 10 generations were aggregated and demonstrated together. It was concluded that both traffic conflicts and delays tend to be stable after 1,300 generations. When the optimal solutions reach convergence, the minimum delays are approximately 60 s. Figure 5(b) presents the solutions from the multiobjective optimization, which belongs to the Pareto frontier, clearly demonstrating the trade-off relationship between the two objectives.Figure 5
Converging trend and Pareto front of the solution algorithm.
(a)
Converging trend (b)
Pareto frontThrough MATLAB processing, it was found that the signal timing plan is optimized to reduce conflict intensity. The optimized cycle length is longer than the original signal timings, since it is not possible to reduce vehicular conflicts without sacrificing part of the traffic efficiency. Additionally, Stevanovic demonstrated a similar result in his research [27].Figure5(a) illustrates all evaluated solutions in the safety-delay domain during the multiobjective optimizations. Figure 5(b) presents the solutions from multiobjective optimization which belong to the Pareto Frontier, efficiently demonstrating the trade-off between the two objectives. By adopting any of the signal timing plans representing these solutions, a traffic signal engineer can improve safety (by reducing the intensity of traffic conflicts) without compromising the efficiency (delay) of the current signal-timing plan.Figure6 presents the flexibility of sequencing and combination of the signal groups. By comparing the delay and the safety intensity (Table 1), it is found that the safety index is improved by almost 35 times, while the delay is doubled using the new method. According to (8), the safety index Sij is sensitive to the duration of PET, due to the use of the exponential function. This could explain why the safety performance was strongly improved while the efficiency was decreased.Figure 6
Optimal singal setting from one of the solutions.


## 6. Conclusions and Future Work

This paper presented a group-based signal control optimization model for mixed traffic flows, which could account for both safety and delay. The proposed model has three main advantages compared to previous models:(i)
It is able to incorporate safety into the signal timing procedure of intersections, which overcomes the drawbacks of the current signal optimization models;(ii)
It equally considers the motorists and bicyclists in the delay and safety estimation;(iii)
It can produce optimized solutions that can significantly reduce traffic conflicts between the motorized vehicles and bicycles.A numerical study was performed to demonstrate the applicability and effectiveness of the proposed model, based on the data collected at an intersection located in Shanghai. The results indicated that the proposed model could aid researchers and practitioners in the signal timing stage to reach a desired balance between safety and operational efficiency for the intersections with mixed traffic flows. The proposed model can also be easily extended to the coordinated signalized intersections, by adjusting the arrival pattern of vehicles in the estimation of safety during the change of phases.It is necessary to perform a few more studies in the future to further enhance the proposed model.(i)
Safety reliability, indicated by the probability of occurrence of extremely dangerous situations, will be incorporated into the model as an optimization objective. This would allow signal timing schemes to be generated according to the acceptable risk levels of signal operators.(ii)
An online optimization model will be developed to dynamically handle the mixed traffic flows, based on the detected vehicle and bicycle volumes.(iii)
The enlargement of cycle length may result in the increased waiting time of bicyclists and pedestrians and cause more traffic violations.These effects will be considered to improve the model in the future.



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*Source: 2747569-2019-01-15.xml*