# Control Strategy of Signal Transition after Emergency Vehicle Signal Preemption

Authors: Haibo Mu; Linzhong Liu; Yubo Song; Na Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Emergency vehicle (EV) plays an important role in evacuations or rescues when emergencies occur. To insure that an EV can transfer people in danger to emergency shelters or medical assistance organizations as soon as possible, EV signal preemption (EVSP) strategy is usually adopted. After EV has passed through the intersection, traffic signal has to transfer back to normal signal timing scheme. This paper focuses on the control strategy of EV signal transitioning from EVSP back to normal operation. Considering both efficiency and fairness, the maximum vehicles passing through in per unit time during the transition period and the minimum difference between the maximum and the minimum queue length after transition are selected as objectives, and a multi-objective optimization model is presented. A nondominated sorting genetic algorithm II (NSGA-II) is designed to solve the optimization model and unique encoding and decoding methods are presented. The established model and designed algorithm are verified and the control effect is analyzed. Simulation results indicate that by adopting the control strategy obtained by the presented model, the number of vehicles passing through in per unit time during the transition period is increased and the difference of vehicle length in different directions is reduced significantly, from which we can conclude that the control method proposed in this paper has good performance.


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## Body

## 1. Introduction

In modern society, emergency events, such as natural disasters, accident disasters, public health incidents, and social security incidents occur frequently and bring huge losses to people’s lives and property security. Since emergency vehicle (EV) plays an important role in evacuation and rescue, it is hoped that it can reach the rescue or medical sites as soon as possible. However, the traffic congestion prevailing in modern society makes it difficult for EVs to reach their destination quickly and safely. Moreover, the operation of EV may further aggravate traffic congestion and even lead to new accidents such as vehicle collisions at intersections. In order to effectively reduce the response time of EV, improve the efficiency of emergency traffic and emergency rescue, and ensure that the EV pass through intersections safely and rapidly, EV signal preemption (EVSP) strategies have been put forward. Since EVSP was first suggested in 1929, the concept of it is not new [1]. Thanks to the development and application of Intelligent Transportation System (ITS), EVSP problems can now be dealt with in real time [2–10].After the EV has passed through the intersection, traffic signal has to transfer from EVSP back to normal operation in a safe and expedient manner. However, most of the existing researches on signal transition schemes aim at nonemergency situations, and mainly focus on the transition from one timing plan to another so as to adapt to the change of traffic flow. When the signal control scheme changes, it is necessary to determine the transition scheme between the old and the new control schemes so as to realize the reasonable transition between different schemes in different time periods and ensure the smooth operation of traffic flow. The suppliers of traffic signal controller and the developers of traffic simulation software CORSIM have developed a series of practical methods to achieve the transitions of signal planning control. Although they use different names in different controller vendors, these methods can usually be divided into dwell, add, subtract, shortway, immediate, two-cycle and three-cycle. However, the classical algorithms mentioned above cannot meet the demand of real-time changing traffic conditions, and result to some deficiencies in signal control effect. In order to evaluate the performance of the above transition methods, some scholars carried out simulation studies and obtained the advantages, disadvantages and applicable conditions of various methods [11–15]. In addition to the classical methods mentioned above, some scholars used prewritten transition algorithms to control signal transition process [16–18]. Other scholars studied this problem by establishing optimization models. To obtain the optimal timing scheme, they constructed performance functions to seek the optimum performance indexes during signal transition period, most of which were the shortest transition time or the least impact on social vehicles [19–24].However, the transition strategy of how to transfer traffic signal back to normal signal after EV signal preemption is not well researched. Obenberger et al. [25] evaluated the effect of four commonly used transition strategies to illustrate which strategy can exit preemption control and switch back to normal signal timing scheme most effectively. To determine how to choose the best coordinated recovery strategy at the end of signal preemption so as to minimize the interference to normal signal operation, Yun et al. [26] employed hardware-in-the-loop simulation to study signal preemption problem and obtained the evaluation results of various EV preemption strategies for EVs arriving in a single way in coordinated traffic signal system. However, both of them mainly focused on analyzing the effect of various commonly used transition strategies, and effective signal preemption strategies were not put forward. Qin [27] proposed two signal preemption control strategies to shorten the response time of EVs. The first one was adopted to enable signal transitioning from normal operation to EVSP, and the second one, implemented by an optimal control algorithm, was used for the signal transitioning from the EVSP back to normal operation. Yao et al. [28] proposed a multi-objective signal recovery transition optimization model of emergency traffic based on power function and solved it by genetic algorithm. However, the goal of the second transition stage was to make the queue length of each line after EV passing through the intersection to not exceed the queue length under normal operation, and the other indexes, such as the number of passing vehicles, or the equilibrium or fairness of vehicles in all directions, were not considered.As can be seen from the literatures mentioned above that nowadays few studies have focused on the problem of how to make EVs transfer back to normal signal timing scheme, and the effects of various commonly used transition strategies on exit signal preemption are emphatically analyzed. Model optimization method adopted to solve this problem is even fewer, and the objective function is either the shortest queue length or the shortest time to complete the signal transition in most cases. Therefore, this paper focuses on the optimization model method to study the control strategy of EV signal transitioning from EVSP back to normal operation. We believe that as long as the number of vehicles passing through the system in all directions during the transition period is large enough and the efficiency of the system is high enough, a slightly longer transition time has little impact on the system in fact. What is more, the equilibrium of different directions is also considered in this paper so as to obtain sufficiently small difference between the numbers of vehicles queued in all directions at the end of the transition period. The maximum numbers of vehicles passing through the system per unit time during the transition period and the minimum difference between the maximum and the minimum length after the transition are selected as the objectives, and a multi-objective signal transition optimization model is presented. An optimization algorithm based on Nondominated sorting genetic algorithm II (NSGA-II) is designed to solve the optimization model to obtain the optimal signal transition strategy.The goal of this research is to develop and evaluate a signal transition strategy based on model optimization method. We limit the scope of this paper to signal transition strategy from the EVSP back to normal operation, and the strategy from normal operation to EVSP will not be discussed further in this paper. In support of this goal, the research will seek to achieve the following objectives: (1) develop a new transition method based on a multi-objective optimization model, (2) design a solving algorithm for the presented optimization model, and (3) carry out simulation calculation and verify the efficiency of the proposed signal transition strategy.The remainder of the paper is organized as follows. Firstly, some related time parameters are calculated in Section2. Secondly, a multi-objective programming model for traffic signal transitioning from the EVSP back to normal operation is presented in Section 3. The NSGA-II algorithm for solving the proposed transition model is investigated in Section 4. Finally, simulation analysis is carried out in Section 5 and conclusions are presented in Section 6.


## 2. Calculation of Related Time Parameters

Suppose that all the intersections in the traffic network are typical four-leg intersections and controlled by two-phase signal with fixed normal cycle length, as shown in Figure1.Figure 1
Phase diagram of traffic signal.For the intersection discussed in this paper, we employws0, d1 and d2 to denote the starting time of the first phase, the duration time of the first phase and the duration time of the second phase, respectively. Suppose that EV passes through or clears intersection at time t0. If ws0, d1 and d2 are determined, then the current phase j and the elapsed green time of phase j can be calculated correspondingly. Provided that C denotes the cycle length of the intersection, and then we have:(1)D0=0,Dj=∑k=1jdk,j=1,2,(2)a=t0−ws0C,b=t0−ws0modC,where mod is a function whose result is the remainder oft0−ws0 divided by C. If it satisfies that Du,i−1≤b<Du,i, then the current phase is j and the elapsed green time of phase j is t0−ws0−a×C−∑k=0j−1Dk, and the remaining cycle time can be denoted by RC=C−b.Lettsafe be the safety time interval that must be kept between the last vehicle in the queue on the EV approach and the EV so as to avoid collision between EV and social vehicles. The time duration of transition, denoted by Rd, can be written as:(3)Rd=RC+nC−tsafe,n=0,1,2,…,wheren stands for the number of additional normal cycles so as to realize the objectives given in the next section. To avoid too long transition time, the maximum value of n is set to 3, that is to say, at most three additional signal cycles can be added.In order to obtain EVSP signal control strategy,Rd is divided into N time steps on average, and the length of each time step, denoted by T, can be expressed as follows:(4)T=RdN=RC+nC−tsafeN.Each time stepT is the combination of green time and yellow time or there is only green time in T. If there is no signal switchover in T, then T contains at most only one yellow time. Otherwise, it is stipulated that only one signal switchover can occur in T, and T contains two or one yellow time.Assume that there exists no all red time for traffic signal at each intersection. Letgjk, Yjk be the green time and yellow time of phase j at time step k, respectively; and αjk be a 0-1 variable, where αjk=1 indicates that a signal switchover occurs in phase j at time step k and αjk=0 otherwise. Then, we have:(5)NT=RC+nCi−tsafe=∑k=1Ng1k+g2k+Y1k×α1k+Y2k×α2k.In the following optimization model, we will determine the value ofgjk and αjk to obtain the Pareto optimal solution.


## 3. The Multi-Objective Transition Optimization Model

Letqfk be the queue length in direction f at the beginning of time step k at each intersection, f=1,2,3,4; afk be the number of vehicles arriving in time step k, f=1,2,3,4; f=1,2,3,4 represents the direction from west to east, from east to west, from north to south and from south to north, respectively. If qf−tk, qf−rk, and qf−lk denotes the number of vehicles going straight, turning right and turning left in qfk, respectively, then we have:(6)qfk=qf−tk+qf−rk+qf−lk,f=1,2,3,4.LetAf−t, Af−r and Af−l be the proportion of straight, right-turn, and left-turn vehicles in direction f, respectively; Sgfi and Syfi, i=t,r,l be the saturation discharge rate during green time and yellow time for straight, right-turn, and left-turn vehicles in direction f, respectively; gjmin  and gjmax be the minimum and maximum green time of phase j, respectively.The proposed transition model includes two important decision variables that control the transition process. The first decision variable is a 0-1 variableαjk, which can be determined dynamically according to traffic condition. The second decision variable is gjk, denoting the green time of phase j during time step k.The first objective is to maximize the number of vehicles passing through the intersection in unit time during the transition period, which is the sum of the number of vehicles going straight, turning right and turning left in four directions in the green interval and yellow interval of all time steps. The second objective, which is measured at the end of the transition period, is to minimize the difference between the largest and the smallest queue length in all directions at the end of the transition period. The number of queued vehicles in each direction is composed of straight, right-turn and left-turn vehicles. If the number of straight, right-turn and left-turn vehicles in directionf at the beginning of time step k is known, then the queue length of subsequent time step k+1,k+2,…,N can be calculated. Taking time step k+1 as an example, we have(7)qf−ik+1=qf−ik+afk×Af−i−minqf−ik+afk×Af−i,Sgfi×gik+Syfi×Yjk×αjkk=0,1,2,…,N−1,i=t,r,l.At the end of time stepN, the queue length in direction f is as follows:(8)qfN=qf−tN+qf−rN+qf−lN.Ifqmax, qmin be the maximum and minimum value of qfN, then the second objective function can be obtained by calculating the difference between qmax and qmin.As a multi-objective optimization model, the proposed signal transition optimization model is as follows:(9)Maximize∑f=14∑k=1N∑j=12Sgft+Sgfr+Sgfl×gjk+Syft+Syfr+Syfl×Yjk×αjkRd,(10)MinimizemaxqfN+1,f=1,2,3,4−minqfN+1f=1,2,3,4.Subject to:(11)qf−ik+1=qf−ik+afk×Af−i−minqf−ik+afk×Af−i,Sgfi×gjk+Syfi×Yjk×αjkk=0,1,2,…,N−1,i=t,r,l,(12)qfk+1=∑i=tlqf−ik+1,k=0,1,2,…,N−1,(13)∑k=1NTk=RC+nCi−tsafe,(14)Tk=g1k+g2k+Y1k×α1k+Y2k×α2k,k=0,1,2,…,N−1,(15)gjmin≤gjk≤gjmax,j=1,2,(16)g1min+g2min+Y1k+Y2k≤Tk≤g1max+g2max+Y1k+Y2k,k=0,1,2,…,N−1.In the above optimization model, Equations (9) and (10) are the objective functions. Equation (9) represents maximizing the number of vehicles passing through the intersection in unit time during the transition period, which is the quotient of the sum of vehicles going straight, turning right and turning left in four directions in the green interval and yellow interval in N time steps divided by the time duration of transition. Equation (10) represents minimizing the difference between the largest and the smallest queue length in all directions at the end of the transition period. Equations (11) and (12), namely the first and second constraint equations, denote the calculation method of straight, right-turn and left-turn queue length in each direction, and the calculation method of total queue length in each direction. Equation (13) ensures that the sum of all time steps equal to the time duration of transition. Equation (14) defines the calculation method of the green interval of time step k. Equations (15) and (16) denotes the range of the green time of phase j at time step k and that of the length of each time step k.


## 4. Design of Solving Algorithm Based on NSGA-II

Many evolutionary optimization methods have been used to solve multi-objective optimization problems, such as multi-objective genetic algorithm (MOGA), multi-objective differential evolution algorithm (MODEA) and nondominated sorting genetic algorithm II (NSGA-II), etc. [29]. NSGA-II adopts an elite strategy to accelerate the convergence speed. The proposed crowding distance sorting can better guarantee the diversity of the population, and the adoption of fast nondominated sorting method reduces the time complexity [30]. In order to improve the solving efficiency, NSGA-II algorithm is adopted in this paper to solve the established multi-objective optimization model.LetTmin be the minimum step size during the signal transition. Let n take values from 1 to 3, respectively. If we have known the value of RC, we can calculate the value of time duration Rd according to Equation (3). Let N=intRd/Tmin and T=Rd/N. If T is not an integer, the value of (Rd mod N) is evenly distributed to the foremost (Rd mod N) time steps in unit of 1 s. For example, if Rd=107s and Tmin=24s, then N=4, T=Rd/N=26 and Rd mod N=3s. Then the length of the foremost 3 time steps will increase 1 s, respectively, and then we have T1=T2=T3=27s and T4=26s. Different values of n correspond to different values of N, which in turn results in different T and different Ti. The best green time of each phase and whether there exists a phase transition in this time step will be found out according to different Ti values.


### 4.1. Chromosome Design

Suppose that the value of transition durationRd is divided into N time steps. For each time step, it is necessary to determine whether phase transition exists, how many phase transitions have taken place (1 or 2 times) and the duration of the two phases. To express all of the information mentioned above, four gene bits need to be designed at each time step, as shown in Figure 2.Figure 2
Four gene positions per time period.Wherel1 and l2 represent the duration time of the first phase and second phase, respectively; the value of δ1 (δ2) is 1 or 0, where δ1=1 (δ2=1) indicates that there exists a signal transition from phase 1 (phase 2) to phase 2 (phase 1), and δ1=0 (δ2=0) otherwise. When expressed in this way, the values of δ1 and δ2 determine the subsequent phase. If δ1=1, then the subsequent phase can only be phase 2. Otherwise, the subsequent phase is still phase 1. Similarly, if δ2=1, the subsequent phase can only be phase 1, and phase 2 otherwise. Basing on this representation method, once crossover and mutation are performed, a large number of infeasible solutions will appear and the computational efficiency will be reduced greatly. For example, if the original values of the four gene bits are [15, 15, 1, 1], it indicates that the traffic signal will transfer to phase 2 after phase 1 has executed 15 s, and phase 2 will also execute 15 seconds, and then phase transition will occur again, namely traffic signal will transfer from phase 2 to phase 1 again. If mutation occurs and the values of four gene bits change to [15, 15, 0, 1], it means that phase transition does not occur after phase 1 has executed 15 s, and phase 2 will not execute correspondingly. But we can infer from the third gene that phase 2 still executes 15 s, which is inconsistent with the actual situation and thereby corresponds to an infeasible solution.In order to avoid the occurrence of the above infeasible solutions, when design chromosome, we make each time step correspond to two gene bits. If the length of each time step isT and the transition duration Rd is divided into N time steps, then the length of each chromosome is 2N. The two gene bits of each time step are expressed as . The first gene is described by two quantities: the phase represented by this gene (phase 1 or phase 2) and the duration of the represented phase during this time step. The second gene is a flag indicating whether a phase transition occurs when the signal is transferred from the current time step to the next time step, and its value is 1 if a phase transition occurs and 0 otherwise. Since the value 1 or 0 is generated randomly, it can represent different situations.


### 4.2. Encoding

The concrete encoding method of this paper is shown in Figure3. The transition duration Rd is divided into N time steps, and each time step i, i=1,2,…,N corresponds to two gene bits. Thus there are 2N gene bits in each chromosome. For the convenience of description, the positions of each gene bit of time step i in the whole chromosome are called the odd position and the even position of time step i.The odd position of time step i is described by two quantities, namely pi and li−pi where pi represents the corresponding phase and li−pi represents the duration time of the phase corresponding to pi. It represents the first phase (phase 1) if pi=0 and the second phase (phase 2) if pi=1. If li−pi>0, the value of li−pi may be the green time or the sum of the green time and the yellow time. Also, li−pi may be equal to 0. The even position of time step i is the flag bit, indicating whether there exists a phase transition or not, and it is described by δi. If δi=1, it indicates that phase transition occurs when traffic signal transfers from the current time step to the next time step, and δi=0 otherwise.Figure 3
Real number encoding method.There are some restrictions on the duration of the phase represented by the first gene bit in each time period so as to ensure the feasibility of solution. Whether phase transition occurs during this time period can be implicitly denoted by the duration of the phase represented by the first gene bit. It is assumed in this paper that the EV passes through the intersection from west to east in the first phase, i.e.,p1=0. Generate the value of δ1 randomly. If δi=1, it indicates that phase transition occurs from time step 1 to time step 2. If the first phase is performed all the time in the first time step, then l1−p1=T1. If phase switchover occurs in the first time period, namely the second phase is executed immediately after the first phase is executed, then the value of l1−p1 needs to ensure that the second phase can execute the minimum green time. If δ1=0, it means that phase transition from time step 1 to time step 2 does not occur, and the value of l1−p1 is unlimited as long as it does not exceed T1.For anyi>1, whether pi represents the first phase or the second phase is related to all the quantities corresponding to the two gene bits in i−1 time step, and it will be described in the following decoding process. Let gmin be the shortest green time of the first phase and the second phase, and Y be the yellow time of each phase. The value of li−pi is limited by δi−1. If δi−1=1, it means that the latest phase executed in time step i−1 will be executed continuously in time step i, and the value of li−pi can be 0, Ti or between gmin+Y,Ti−gmin−Y. If δi−1=0, then it is similar to the first time step, and the value of li−pi is required to not exceed Ti.


### 4.3. Decoding

#### 4.3.1. Determination of the Phase Represented by Odd Position of Each Time Period

Since it is deterministic thatp1=0 for the first time step, then the phase represented by p2 can be determined directly. The chromosome shown in Figure 4 is taken as an example to illustrate the decoding idea of the phase represented by the odd position when i=2. Figure 4 only shows the concrete value of each gene bit in the first two time steps by assuming Ti=30s, i=1,2,…,N. Since p1=0 and l1−0=14s, it indicates that the first phase is executed for 14 s and then traffic light transfers to the second phase. We can indicate that l1−1=30−14=16s, that is, the second phase is executed for 16 s. Since δ1=1, it indicates that phase transition occurs when transferring from time step 1 to time step 2, and the first phase will execute firstly in time step 2, namely p2=0 and l2−0=15s, indicating that after the first phase has been executed for 15 s the second phase will be executed continuously for l2−1=30−15=15s. Since δ2=0, it indicates that phase transition does not occur when entering time step 3, that is, the second phase is executed continuously at the beginning of time step 3, and so on.Figure 4
Example of normal chromosome.For the chromosome shown in Figure4, if l2−p2=0 then it will be changed to Figure 5, and the result of decoding will be different. According to the values of the first two gene bits, the first phase will be executed firstly in time step 2, and the green time of the first phase should not be zero. But now l2−p2=0 means that the green time of this phase is 0 s. To avoid this obvious conflict, when decoding under this condition, it is obliged that p2=1. That is, the odd position of time step 2 represents the second phase and l2−1=0. Therefore, the first phase is executed for 30 s namely l2−0=30. If l2−p2=30s, it means that the green time of phase p2 is equal to T2, then p2=0, l2−0=30 s and l2−1=0. Although both l2−p2=0 and l2−p2=30s indicate that the first phase is executed for 30 s in time step 2, the phase represented by p2 is different. Similarly, when δ1=0 or l1=T1, a similar method can be used for analysis.Figure 5
Example of special chromosome.Ifi>2, the phase corresponding to the odd position of the previous time step might be the first phase or the second phase. Therefore, when decoding it, the main work is to determine whether the phase corresponding to the odd position of the current time step is the same as that corresponding to the odd position of the previous time step. At this time, it can be analyzed in a similar way as i=2.


#### 4.3.2. Determination of the Green Time and Yellow Time of Each Phase

We can conclude from the knowledge mentioned above that the value ofli−pi and li−1−pi denote the green time of phase pi and phase 1−pi in time step i or the sum of green time and yellow time. Since the green time and yellow time must be known clearly when calculating the two objective functions, it is necessary to determine whether the yellow time is included in li−pi and li−1−pi, which depends on the phase switchover in that time step. After the yellow time Yi−pi of phase i and the yellow time Yi−1−pi of phase i−1 have been determined, the green time of the two phases are also determined simultaneously. The green time of phase pi and phase 1−pi is li−pi−Yi−pi and li−1−pi−Yi−1−pi, respectively. The green time of each phase is the value of decision variable gjk.Letα be the phase represented by the odd position of time step i. It is deterministic that α=0 for the first time step, and the value of α may be 0 or 1 for others. The calculation method of the yellow time of each phase in each time step is as follows.(1)
0
<
l
i
−
p
i
<
T
iIfδi=1, then phase transitions occur both in time step i and in the period from time step i to time step i+1, and both phases include yellow time. That is, Yi−pi=Y and Yi−1−pi=Y; If δi=0, then phase transition occurs in time step i but not in the period from time step i to time step i+1, and yellow time is included only in phase pi but not in phase 1−pi, that is, Yi−pi=Y and Yi−1−pi=0.(2)
l
i
−
p
i
=
T
iIfδi=1, then phase transition occurs in the period from time step i to time step i+1 but not in time step i, and yellow time is included only in phase pi but not in phase 1−pi. That is, Yi−pi=Y and Yi−1−pi=0; If δi=0, then phase transition does not occur both in time step i and in the period from time step i to time step i+1, and neither phase includes yellow time n, that is, Yi−pi=0 and Yi−1−pi=0.(3)
l
i
−
p
i
=
0Ifδi=1 then phase transition occurs in the period from time step i to time step i+1 but not in time step i. Since li−1−pi=T, then yellow time is included only in phase 1−pi but not in phase pi, that is, Yi−pi=0 and Yi−1−pi=Y; If δi=0, then there is phase transition neither in time step i nor in the period from time step i to time step i+1, and neither phase includes yellow time, that is, Yi−pi=0 and Yi−1−pi=0.


### 4.4. Selection, Crossover and Mutation Operator

#### 4.4.1. Selection Operator

Since there are more than one optimization objectives in a multi-objective optimization model, the selection method of single-objective optimization problem cannot be used for the selection operation of multi-objective optimization problem. Instead, the selection operation is performed by two steps: chromosome sorting and the same-level nondominated solution aggregation density sorting. Chromosome sorting is a ranking method based on the Pareto optimal concept. In this paper, an improved fast sorting method is adopted in chromosome sorting [31, 32].The method of chromosome sorting makes it possible to preferentially select chromosomes with lower level to participate in evolution. It is also needed to determine a selection strategy for chromosomes with the same level so as to realize the selection operation of such chromosomes. In order to maintain the distribution and diversity of the evolutionary population, aggregation density strategy is adopted to select chromosomes with the same level. In the process of selecting chromosomes with the same level, chromosomes with small aggregation density are usually selected to participate in the next generation. Crowding distance is introduced to determine the aggregation density of chromosomes, and the chromosomes with larger crowding distance have a small aggregation density. Assume that chromosomei has r sub-targets, and Pidistance, the crowding distance of chromosome i is represented by the sum of the distances of the former chromosome i−1 and the latter chromosome i+1 on each sub-target. The calculation equation is:(17)Pidistance=∑k=1rPi+1.fk−Pi−1.fk,wherePi.fk is the function value of the kth sub-target of chromosome i.


#### 4.4.2. Crossover Operator

The partially matched crossover method is used in this paper for crossover [33]. First, two parent chromosomes are selected randomly according to the crossover probability and several gene bits are selected randomly from 2N gene bits of a pair of parent chromosomes. Then, the positions of the two groups of genes are exchanged. The concrete operation process is shown in Figure 6.Figure 6
Partly matched crossover.


#### 4.4.3. Mutation Operator

Randomly select a chromosome based on the mutation probability and generate two random integer numberse, f from 1 to N, and exchange the values of the even position of the two gene bits. The specific operation process is shown in Figure 7.Figure 7
Mutation.


### 4.5. Specific Flow of Algorithm

Now, the specific flow of NSGA-II algorithm adopted to find the control schemes is given as follows.Step 1.
Given the values of intersection parameters and traffic flow parameters such asws0, d1, d2, C, t0, tsafe, Tmin; gjmin and gjmax, j=1,2; Af−i, Sgfi, and Syfi, i=t,r,l, etc. Calculate the values of a and baccording to formula (2), Rd according to formula (3), and N, Ti, i=1,2,⋯,N according to the method shown in Section 4.Step 2.
Initialize parameters of genetic algorithm such as population size, maximum number of iterations, crossover probability and mutation probability, and define chromosome structure.Step 3.
Create an initial population according to the method shown in Section4.2 on the basis of the value of gmin, N, and Ti, i=1,2,⋯,N. Obtain the value of Yi−pi and Yi−1−pi according to the value of δi, li−pi and li−1−pi of the initial population, and then determine the value of decision variable gjk and αjk. Calculate values of the two objective functions of each chromosome.Step 4.
Sort the initial population with nondominated sorting method and calculate the crowding distancePidistance of each chromosome.Step 5.
Determine whether the current number of iterations is greater than the maximum number of iterations. If it is, output the current Pareto-optimal solutions and end the algorithm; otherwise, go to Step6.Step 6.
Select chromosomes by binary tournament selecting method to take-on crossover operation according to the crossover probability, generate offspring chromosomes and calculate values of the two objective function of each offspring chromosome.Step 7.
Select chromosomes by binary tournament selecting method to mutate their structures according to the mutation probability, and calculate values of the two objective functions of each mutated chromosome.Step 8.
Combine offspring and parent populations together, calculate crowding distancePidistance of each chromosome and sort all the chromosomes according to each chromosome’s values of the two objective functions and crowding distance.Step 9.
Cut the combined population to a given number according to the sorting results, sort the population after cutting with nondominated sorting method and calculate crowding distancePidistance of each chromosome.Step 10.
Record the current Pareto-optimal solutions, add 1 to the current number of iterations and go to Step5.


## 4.1. Chromosome Design

Suppose that the value of transition durationRd is divided into N time steps. For each time step, it is necessary to determine whether phase transition exists, how many phase transitions have taken place (1 or 2 times) and the duration of the two phases. To express all of the information mentioned above, four gene bits need to be designed at each time step, as shown in Figure 2.Figure 2
Four gene positions per time period.Wherel1 and l2 represent the duration time of the first phase and second phase, respectively; the value of δ1 (δ2) is 1 or 0, where δ1=1 (δ2=1) indicates that there exists a signal transition from phase 1 (phase 2) to phase 2 (phase 1), and δ1=0 (δ2=0) otherwise. When expressed in this way, the values of δ1 and δ2 determine the subsequent phase. If δ1=1, then the subsequent phase can only be phase 2. Otherwise, the subsequent phase is still phase 1. Similarly, if δ2=1, the subsequent phase can only be phase 1, and phase 2 otherwise. Basing on this representation method, once crossover and mutation are performed, a large number of infeasible solutions will appear and the computational efficiency will be reduced greatly. For example, if the original values of the four gene bits are [15, 15, 1, 1], it indicates that the traffic signal will transfer to phase 2 after phase 1 has executed 15 s, and phase 2 will also execute 15 seconds, and then phase transition will occur again, namely traffic signal will transfer from phase 2 to phase 1 again. If mutation occurs and the values of four gene bits change to [15, 15, 0, 1], it means that phase transition does not occur after phase 1 has executed 15 s, and phase 2 will not execute correspondingly. But we can infer from the third gene that phase 2 still executes 15 s, which is inconsistent with the actual situation and thereby corresponds to an infeasible solution.In order to avoid the occurrence of the above infeasible solutions, when design chromosome, we make each time step correspond to two gene bits. If the length of each time step isT and the transition duration Rd is divided into N time steps, then the length of each chromosome is 2N. The two gene bits of each time step are expressed as . The first gene is described by two quantities: the phase represented by this gene (phase 1 or phase 2) and the duration of the represented phase during this time step. The second gene is a flag indicating whether a phase transition occurs when the signal is transferred from the current time step to the next time step, and its value is 1 if a phase transition occurs and 0 otherwise. Since the value 1 or 0 is generated randomly, it can represent different situations.


## 4.2. Encoding

The concrete encoding method of this paper is shown in Figure3. The transition duration Rd is divided into N time steps, and each time step i, i=1,2,…,N corresponds to two gene bits. Thus there are 2N gene bits in each chromosome. For the convenience of description, the positions of each gene bit of time step i in the whole chromosome are called the odd position and the even position of time step i.The odd position of time step i is described by two quantities, namely pi and li−pi where pi represents the corresponding phase and li−pi represents the duration time of the phase corresponding to pi. It represents the first phase (phase 1) if pi=0 and the second phase (phase 2) if pi=1. If li−pi>0, the value of li−pi may be the green time or the sum of the green time and the yellow time. Also, li−pi may be equal to 0. The even position of time step i is the flag bit, indicating whether there exists a phase transition or not, and it is described by δi. If δi=1, it indicates that phase transition occurs when traffic signal transfers from the current time step to the next time step, and δi=0 otherwise.Figure 3
Real number encoding method.There are some restrictions on the duration of the phase represented by the first gene bit in each time period so as to ensure the feasibility of solution. Whether phase transition occurs during this time period can be implicitly denoted by the duration of the phase represented by the first gene bit. It is assumed in this paper that the EV passes through the intersection from west to east in the first phase, i.e.,p1=0. Generate the value of δ1 randomly. If δi=1, it indicates that phase transition occurs from time step 1 to time step 2. If the first phase is performed all the time in the first time step, then l1−p1=T1. If phase switchover occurs in the first time period, namely the second phase is executed immediately after the first phase is executed, then the value of l1−p1 needs to ensure that the second phase can execute the minimum green time. If δ1=0, it means that phase transition from time step 1 to time step 2 does not occur, and the value of l1−p1 is unlimited as long as it does not exceed T1.For anyi>1, whether pi represents the first phase or the second phase is related to all the quantities corresponding to the two gene bits in i−1 time step, and it will be described in the following decoding process. Let gmin be the shortest green time of the first phase and the second phase, and Y be the yellow time of each phase. The value of li−pi is limited by δi−1. If δi−1=1, it means that the latest phase executed in time step i−1 will be executed continuously in time step i, and the value of li−pi can be 0, Ti or between gmin+Y,Ti−gmin−Y. If δi−1=0, then it is similar to the first time step, and the value of li−pi is required to not exceed Ti.


## 4.3. Decoding

### 4.3.1. Determination of the Phase Represented by Odd Position of Each Time Period

Since it is deterministic thatp1=0 for the first time step, then the phase represented by p2 can be determined directly. The chromosome shown in Figure 4 is taken as an example to illustrate the decoding idea of the phase represented by the odd position when i=2. Figure 4 only shows the concrete value of each gene bit in the first two time steps by assuming Ti=30s, i=1,2,…,N. Since p1=0 and l1−0=14s, it indicates that the first phase is executed for 14 s and then traffic light transfers to the second phase. We can indicate that l1−1=30−14=16s, that is, the second phase is executed for 16 s. Since δ1=1, it indicates that phase transition occurs when transferring from time step 1 to time step 2, and the first phase will execute firstly in time step 2, namely p2=0 and l2−0=15s, indicating that after the first phase has been executed for 15 s the second phase will be executed continuously for l2−1=30−15=15s. Since δ2=0, it indicates that phase transition does not occur when entering time step 3, that is, the second phase is executed continuously at the beginning of time step 3, and so on.Figure 4
Example of normal chromosome.For the chromosome shown in Figure4, if l2−p2=0 then it will be changed to Figure 5, and the result of decoding will be different. According to the values of the first two gene bits, the first phase will be executed firstly in time step 2, and the green time of the first phase should not be zero. But now l2−p2=0 means that the green time of this phase is 0 s. To avoid this obvious conflict, when decoding under this condition, it is obliged that p2=1. That is, the odd position of time step 2 represents the second phase and l2−1=0. Therefore, the first phase is executed for 30 s namely l2−0=30. If l2−p2=30s, it means that the green time of phase p2 is equal to T2, then p2=0, l2−0=30 s and l2−1=0. Although both l2−p2=0 and l2−p2=30s indicate that the first phase is executed for 30 s in time step 2, the phase represented by p2 is different. Similarly, when δ1=0 or l1=T1, a similar method can be used for analysis.Figure 5
Example of special chromosome.Ifi>2, the phase corresponding to the odd position of the previous time step might be the first phase or the second phase. Therefore, when decoding it, the main work is to determine whether the phase corresponding to the odd position of the current time step is the same as that corresponding to the odd position of the previous time step. At this time, it can be analyzed in a similar way as i=2.


### 4.3.2. Determination of the Green Time and Yellow Time of Each Phase

We can conclude from the knowledge mentioned above that the value ofli−pi and li−1−pi denote the green time of phase pi and phase 1−pi in time step i or the sum of green time and yellow time. Since the green time and yellow time must be known clearly when calculating the two objective functions, it is necessary to determine whether the yellow time is included in li−pi and li−1−pi, which depends on the phase switchover in that time step. After the yellow time Yi−pi of phase i and the yellow time Yi−1−pi of phase i−1 have been determined, the green time of the two phases are also determined simultaneously. The green time of phase pi and phase 1−pi is li−pi−Yi−pi and li−1−pi−Yi−1−pi, respectively. The green time of each phase is the value of decision variable gjk.Letα be the phase represented by the odd position of time step i. It is deterministic that α=0 for the first time step, and the value of α may be 0 or 1 for others. The calculation method of the yellow time of each phase in each time step is as follows.(1)
0
<
l
i
−
p
i
<
T
iIfδi=1, then phase transitions occur both in time step i and in the period from time step i to time step i+1, and both phases include yellow time. That is, Yi−pi=Y and Yi−1−pi=Y; If δi=0, then phase transition occurs in time step i but not in the period from time step i to time step i+1, and yellow time is included only in phase pi but not in phase 1−pi, that is, Yi−pi=Y and Yi−1−pi=0.(2)
l
i
−
p
i
=
T
iIfδi=1, then phase transition occurs in the period from time step i to time step i+1 but not in time step i, and yellow time is included only in phase pi but not in phase 1−pi. That is, Yi−pi=Y and Yi−1−pi=0; If δi=0, then phase transition does not occur both in time step i and in the period from time step i to time step i+1, and neither phase includes yellow time n, that is, Yi−pi=0 and Yi−1−pi=0.(3)
l
i
−
p
i
=
0Ifδi=1 then phase transition occurs in the period from time step i to time step i+1 but not in time step i. Since li−1−pi=T, then yellow time is included only in phase 1−pi but not in phase pi, that is, Yi−pi=0 and Yi−1−pi=Y; If δi=0, then there is phase transition neither in time step i nor in the period from time step i to time step i+1, and neither phase includes yellow time, that is, Yi−pi=0 and Yi−1−pi=0.


## 4.3.1. Determination of the Phase Represented by Odd Position of Each Time Period

Since it is deterministic thatp1=0 for the first time step, then the phase represented by p2 can be determined directly. The chromosome shown in Figure 4 is taken as an example to illustrate the decoding idea of the phase represented by the odd position when i=2. Figure 4 only shows the concrete value of each gene bit in the first two time steps by assuming Ti=30s, i=1,2,…,N. Since p1=0 and l1−0=14s, it indicates that the first phase is executed for 14 s and then traffic light transfers to the second phase. We can indicate that l1−1=30−14=16s, that is, the second phase is executed for 16 s. Since δ1=1, it indicates that phase transition occurs when transferring from time step 1 to time step 2, and the first phase will execute firstly in time step 2, namely p2=0 and l2−0=15s, indicating that after the first phase has been executed for 15 s the second phase will be executed continuously for l2−1=30−15=15s. Since δ2=0, it indicates that phase transition does not occur when entering time step 3, that is, the second phase is executed continuously at the beginning of time step 3, and so on.Figure 4
Example of normal chromosome.For the chromosome shown in Figure4, if l2−p2=0 then it will be changed to Figure 5, and the result of decoding will be different. According to the values of the first two gene bits, the first phase will be executed firstly in time step 2, and the green time of the first phase should not be zero. But now l2−p2=0 means that the green time of this phase is 0 s. To avoid this obvious conflict, when decoding under this condition, it is obliged that p2=1. That is, the odd position of time step 2 represents the second phase and l2−1=0. Therefore, the first phase is executed for 30 s namely l2−0=30. If l2−p2=30s, it means that the green time of phase p2 is equal to T2, then p2=0, l2−0=30 s and l2−1=0. Although both l2−p2=0 and l2−p2=30s indicate that the first phase is executed for 30 s in time step 2, the phase represented by p2 is different. Similarly, when δ1=0 or l1=T1, a similar method can be used for analysis.Figure 5
Example of special chromosome.Ifi>2, the phase corresponding to the odd position of the previous time step might be the first phase or the second phase. Therefore, when decoding it, the main work is to determine whether the phase corresponding to the odd position of the current time step is the same as that corresponding to the odd position of the previous time step. At this time, it can be analyzed in a similar way as i=2.


## 4.3.2. Determination of the Green Time and Yellow Time of Each Phase

We can conclude from the knowledge mentioned above that the value ofli−pi and li−1−pi denote the green time of phase pi and phase 1−pi in time step i or the sum of green time and yellow time. Since the green time and yellow time must be known clearly when calculating the two objective functions, it is necessary to determine whether the yellow time is included in li−pi and li−1−pi, which depends on the phase switchover in that time step. After the yellow time Yi−pi of phase i and the yellow time Yi−1−pi of phase i−1 have been determined, the green time of the two phases are also determined simultaneously. The green time of phase pi and phase 1−pi is li−pi−Yi−pi and li−1−pi−Yi−1−pi, respectively. The green time of each phase is the value of decision variable gjk.Letα be the phase represented by the odd position of time step i. It is deterministic that α=0 for the first time step, and the value of α may be 0 or 1 for others. The calculation method of the yellow time of each phase in each time step is as follows.(1)
0
<
l
i
−
p
i
<
T
iIfδi=1, then phase transitions occur both in time step i and in the period from time step i to time step i+1, and both phases include yellow time. That is, Yi−pi=Y and Yi−1−pi=Y; If δi=0, then phase transition occurs in time step i but not in the period from time step i to time step i+1, and yellow time is included only in phase pi but not in phase 1−pi, that is, Yi−pi=Y and Yi−1−pi=0.(2)
l
i
−
p
i
=
T
iIfδi=1, then phase transition occurs in the period from time step i to time step i+1 but not in time step i, and yellow time is included only in phase pi but not in phase 1−pi. That is, Yi−pi=Y and Yi−1−pi=0; If δi=0, then phase transition does not occur both in time step i and in the period from time step i to time step i+1, and neither phase includes yellow time n, that is, Yi−pi=0 and Yi−1−pi=0.(3)
l
i
−
p
i
=
0Ifδi=1 then phase transition occurs in the period from time step i to time step i+1 but not in time step i. Since li−1−pi=T, then yellow time is included only in phase 1−pi but not in phase pi, that is, Yi−pi=0 and Yi−1−pi=Y; If δi=0, then there is phase transition neither in time step i nor in the period from time step i to time step i+1, and neither phase includes yellow time, that is, Yi−pi=0 and Yi−1−pi=0.


## 4.4. Selection, Crossover and Mutation Operator

### 4.4.1. Selection Operator

Since there are more than one optimization objectives in a multi-objective optimization model, the selection method of single-objective optimization problem cannot be used for the selection operation of multi-objective optimization problem. Instead, the selection operation is performed by two steps: chromosome sorting and the same-level nondominated solution aggregation density sorting. Chromosome sorting is a ranking method based on the Pareto optimal concept. In this paper, an improved fast sorting method is adopted in chromosome sorting [31, 32].The method of chromosome sorting makes it possible to preferentially select chromosomes with lower level to participate in evolution. It is also needed to determine a selection strategy for chromosomes with the same level so as to realize the selection operation of such chromosomes. In order to maintain the distribution and diversity of the evolutionary population, aggregation density strategy is adopted to select chromosomes with the same level. In the process of selecting chromosomes with the same level, chromosomes with small aggregation density are usually selected to participate in the next generation. Crowding distance is introduced to determine the aggregation density of chromosomes, and the chromosomes with larger crowding distance have a small aggregation density. Assume that chromosomei has r sub-targets, and Pidistance, the crowding distance of chromosome i is represented by the sum of the distances of the former chromosome i−1 and the latter chromosome i+1 on each sub-target. The calculation equation is:(17)Pidistance=∑k=1rPi+1.fk−Pi−1.fk,wherePi.fk is the function value of the kth sub-target of chromosome i.


### 4.4.2. Crossover Operator

The partially matched crossover method is used in this paper for crossover [33]. First, two parent chromosomes are selected randomly according to the crossover probability and several gene bits are selected randomly from 2N gene bits of a pair of parent chromosomes. Then, the positions of the two groups of genes are exchanged. The concrete operation process is shown in Figure 6.Figure 6
Partly matched crossover.


### 4.4.3. Mutation Operator

Randomly select a chromosome based on the mutation probability and generate two random integer numberse, f from 1 to N, and exchange the values of the even position of the two gene bits. The specific operation process is shown in Figure 7.Figure 7
Mutation.


## 4.4.1. Selection Operator

Since there are more than one optimization objectives in a multi-objective optimization model, the selection method of single-objective optimization problem cannot be used for the selection operation of multi-objective optimization problem. Instead, the selection operation is performed by two steps: chromosome sorting and the same-level nondominated solution aggregation density sorting. Chromosome sorting is a ranking method based on the Pareto optimal concept. In this paper, an improved fast sorting method is adopted in chromosome sorting [31, 32].The method of chromosome sorting makes it possible to preferentially select chromosomes with lower level to participate in evolution. It is also needed to determine a selection strategy for chromosomes with the same level so as to realize the selection operation of such chromosomes. In order to maintain the distribution and diversity of the evolutionary population, aggregation density strategy is adopted to select chromosomes with the same level. In the process of selecting chromosomes with the same level, chromosomes with small aggregation density are usually selected to participate in the next generation. Crowding distance is introduced to determine the aggregation density of chromosomes, and the chromosomes with larger crowding distance have a small aggregation density. Assume that chromosomei has r sub-targets, and Pidistance, the crowding distance of chromosome i is represented by the sum of the distances of the former chromosome i−1 and the latter chromosome i+1 on each sub-target. The calculation equation is:(17)Pidistance=∑k=1rPi+1.fk−Pi−1.fk,wherePi.fk is the function value of the kth sub-target of chromosome i.


## 4.4.2. Crossover Operator

The partially matched crossover method is used in this paper for crossover [33]. First, two parent chromosomes are selected randomly according to the crossover probability and several gene bits are selected randomly from 2N gene bits of a pair of parent chromosomes. Then, the positions of the two groups of genes are exchanged. The concrete operation process is shown in Figure 6.Figure 6
Partly matched crossover.


## 4.4.3. Mutation Operator

Randomly select a chromosome based on the mutation probability and generate two random integer numberse, f from 1 to N, and exchange the values of the even position of the two gene bits. The specific operation process is shown in Figure 7.Figure 7
Mutation.


## 4.5. Specific Flow of Algorithm

Now, the specific flow of NSGA-II algorithm adopted to find the control schemes is given as follows.Step 1.
Given the values of intersection parameters and traffic flow parameters such asws0, d1, d2, C, t0, tsafe, Tmin; gjmin and gjmax, j=1,2; Af−i, Sgfi, and Syfi, i=t,r,l, etc. Calculate the values of a and baccording to formula (2), Rd according to formula (3), and N, Ti, i=1,2,⋯,N according to the method shown in Section 4.Step 2.
Initialize parameters of genetic algorithm such as population size, maximum number of iterations, crossover probability and mutation probability, and define chromosome structure.Step 3.
Create an initial population according to the method shown in Section4.2 on the basis of the value of gmin, N, and Ti, i=1,2,⋯,N. Obtain the value of Yi−pi and Yi−1−pi according to the value of δi, li−pi and li−1−pi of the initial population, and then determine the value of decision variable gjk and αjk. Calculate values of the two objective functions of each chromosome.Step 4.
Sort the initial population with nondominated sorting method and calculate the crowding distancePidistance of each chromosome.Step 5.
Determine whether the current number of iterations is greater than the maximum number of iterations. If it is, output the current Pareto-optimal solutions and end the algorithm; otherwise, go to Step6.Step 6.
Select chromosomes by binary tournament selecting method to take-on crossover operation according to the crossover probability, generate offspring chromosomes and calculate values of the two objective function of each offspring chromosome.Step 7.
Select chromosomes by binary tournament selecting method to mutate their structures according to the mutation probability, and calculate values of the two objective functions of each mutated chromosome.Step 8.
Combine offspring and parent populations together, calculate crowding distancePidistance of each chromosome and sort all the chromosomes according to each chromosome’s values of the two objective functions and crowding distance.Step 9.
Cut the combined population to a given number according to the sorting results, sort the population after cutting with nondominated sorting method and calculate crowding distancePidistance of each chromosome.Step 10.
Record the current Pareto-optimal solutions, add 1 to the current number of iterations and go to Step5.


## 5. Simulation Analyses

Assume that the EV passes through the intersection from west to east. For the base case, namely normal operation, the following signal timing was used. Cycle timeC=84s; the green time of phase 1 and phase 2 is 50 s and 30 s, respectively; the yellow time is 2 s for both phases; the starting time of the first phase ws0=0s; d1=52s, d2=32s, tsafe=2s; g1min=g2min=10s, g1max=g2max=60s, Tmin=24s; Sgft=0.5pcu/s, Sgfr=0.95Sgft, Sgfl=0.85Sgft, Syft=0.25pcu·s−1, Syfr=0.95Syft, Syfl=0.85Syft, f=1,2,3,4; Turning movement ratios are set to 20%, 70%, and 10% for right, through, and left turns, respectively in west-east and east-west direction, and 30%, 60%, and 10% in north-south and south-north direction.The NSGA-II algorithm was run for 200 generations in all cases. The control parameters used in the NSGAII -based optimization process are as follows(i) Population size: 100;(ii) Crossover probability: 0.8;(iii) Mutation probability: 0.02;(iv) Number of generations: 200.In this paper, we defined four scenarios. The timet0 at which EV passed through the intersection was set to 8 o’clock, 9 o’clock, 10 o’clock, and 18 o’clock, respectively, and runs for each scenario were simulated. In different scenarios, EV passed through the given intersection in the first or the second phase of the background cycle, and the elapsed green time of the first or the second phase of the background cycle for each scenario was different.Whent0 is 9 o’clock or 10 o’clock, we assume the initial queue lengths of straight, right-turn and left-turn for west-east and east-west direction are 10, 8, and 6, respectively, and 8, 7, and 6, respectively for north-south and south-north direction. The arrival rate is a random variable and it is 1440–1800 pcu/h for west-east and east-west direction and 1260–1440 pcu/h for north-south and south-north direction. The Pareto solution set of different number of additional normal cycle (n) for 9 o’clock is shown in Figure 8.Figure 8
Pareto solution set corresponding to differentn for 9 o’ clock.We can see from Figure8 that corresponding to different n the number of time step N is different when t0 is 9 o’clock and this conclusion is applicable to other time as well. The number of the Pareto solutions is 5, 3, and 3 respectively for one, two and three additional normal cycles, and the green time of phase 1 and phase 2 for different Pareto solution is also different.The Pareto frontier with differentn for 9 o’clock and 10 o’clock are shown in Tables 1 and 2. The final Pareto frontier for 9 o’clock and 10 o’clock are shown in Figures 9 and 10.Table 1
Pareto frontier with differentn when t0 is 9 o’clock.
Corresponding cycle
f
1 (pcu/s)
f
2 (pcu)
n
=
1
2.71
5.1
2.64
4.5
2.60
3.8
2.57
2.7
2.25
1.8
n
=
2
2.69
6.2
2.67
4.5
2.59
3.3
n
=
3
2.59
4.7
2.52
2.8
2.45
1.1Table 2
Pareto frontier with differentn when t0 is 10 o’clock.
Corresponding cycle
f
1 (pcu/s)
f
2 (pcu)
n
=
1
2.73
6.2
2.63
3.6
2.31
2.1
2.30
1.7
2.27
0.2
n
=
2
2.62
6.1
2.57
3.0
2.39
1.9
n
=
3
2.52
3.4
2.49
1.9
2.45
0.0Figure 9
Pareto frontier whent0 is 9 o’ clock.Figure 10
Pareto frontier whent0 is 10 o’ clock.As can be seen from Tables1 and 2 that, for each solution with different n in the Pareto solution set, with the reduction of one objective, the other objective is reduced simultaneously, which implies that the decrease in the difference between the maximum queue length and the minimum queue length in all directions (f2) is at the cost of the reduction of the number of vehicles passing through the intersection per unit time during the transition period (f1). From Figures 9 and 10 we can conclude that, the Pareto frontier is composed of nondominant solutions selected from all Pareto solutions corresponding to different n. In the six Pareto solutions shown in Figure 9, three solutions come from n=1, two solutions come from n=2 and only one solution comes from n=3. What is more, the solutions for n=1 and n=2 appear alternately, while the two objectives of the solution from n=3 are smaller than those of n=1 and n=2. Unlike the results in Figure 9, the solutions in Figure 10 show that a larger n corresponds to a smaller difference between the maximum queue length and the minimum queue length in all directions, while a smaller n has a bigger number of vehicles passing through the intersection per unit time. The difference between Figures 9 and 10 tells us that if EV passes through the intersection in different phases of the background cycle or it passes through the intersection in the same phase but the elapsed green time of this phase is different, the Pareto solutions with different n will be diverse, correspondingly.To verify the control effect of the transition method presented in this paper under the condition of much larger traffic volume, we assume that 8 o’clock is the morning peak period and 18 o’clock is the evening peak period. Suppose that for both periods the arrival rate is 1980–2160 pcu/h for west-east and east-west direction and 1800–1980 pcu/h for north-south and south-north direction, respectively. The initial queue lengths in all directions are doubled compared with the initial queue length at 9 o’clock and 10 o’clock. The Pareto optimal set of the morning peak period (8 o’clock) and of the evening peak period (18 o’clock) is shown in Tables3 and 4, respectively.Table 3
Pareto frontier of the morning peak period.
Number
f
1 (pcu/s)
f
2 (pcu)
Correspondingn
1
2.80
8.1
1
2
2.72
7.0
1
3
2.70
6.2
1
4
2.64
5.2
1
5
2.83
10.4
2
6
2.57
4.3
2
7
2.56
4.1
3
8
2.53
3.0
3Table 4
Pareto frontier of the evening peak period.
Number
f
1 (pcu/s)
f
2 (pcu)
Correspondingn
1
2.79
8.0
1
2
2.66
7.7
1
3
2.61
6.3
1
4
2.59
5.8
1
5
2.57
5.4
1
6
2.51
3.5
2
7
2.36
2.7
3From Tables3 and 4, we can infer that although there are many queued vehicles at each direction and the arrival rate exceeds the saturation flow rate, the signal transition method can still guarantee both fair and efficiency, which is embodied in the smaller difference of the queue length in all directions and the larger number of vehicles passing through the intersection in unit time. Also we can conclude that if we want to have larger passing through vehicles in unit time, we can choose to add one additional normal cycle. Adding two or three additional normal cycle is suitable for obtaining smaller difference of queue length in all directions. The results shown in Tables 3 and 4 also indicate that the signal preemption control method is effective when the arrival rate is much larger in the morning peak period.To check the effectiveness of the control strategy of EV signal transitioning from EVSP back to normal operation presented in this paper, comparison work between this control strategy and traditional classical transition method named smooth transition is carried out. Simulations of smooth transition control method under each of the given four scenarios are executed 20 runs respectively, and the simulation results are shown in Table5. The results of Pareto frontier with the worst f1 (but the best f2), and the worst f2 (but the best f1) at each time point obtained by this method are also listed in Table 5.Table 5
Simulation results of smooth transitioning control method.
Time
This method with the worstf1 (the best f2)
This method with the worstf2 (the best f1)
Smooth transition
f
1 (pcu/s)
f
2 (pcu)
f
1 (pcu/s)
f
2 (pcu)
f
1 (pcu/s)
f
2 (pcu)
8 o’clock
2.53
3.0
2.80
8.1
2.76
27.3
9 o’clock
2.25
1.8
2.69
6.2
2.03
14.9
10 o’clock
2.27
0.2
2.62
6.1
1.87
16.4
18 o’clock
2.36
2.7
2.79
8.0
2.75
25.5It can be concluded from Table5 that there is little difference in the number of vehicles passing through the intersection in unit time obtained by the two methods, and the results with the best f1 (the worst f2) are superior to the average f1 obtained by smooth transition method in all scenarios. But there is obvious disparity in the difference of queue length obtained by the two methods. The difference of queue length obtained by smooth transition method is obviously larger than that obtained by the method presented in this paper, and the difference is especially large during the morning and evening peak period because of the large traffic volume. Compared with the average value of f2 obtained by smooth transition method, the shorten rates of the worst f2 at each time point obtained by this method is 70.33%, 58.39%, 62.80% and 68.63%, respectively. Therefore, the proposed signal transition method can give consideration to both efficiency and fairness of each direction.When more than one EV arrives, the signal transition method proposed in this paper is also applicable. If the arrival interval of adjacent EV is relatively long, traffic signal can be transferred to normal operation as long as each EV passing through the intersection by adopting this method. If the arrival interval of adjacent EV is very short, they can be seen as one EV and signal transition will not be carried out until they have both passed through the intersection.


## 6. Conclusions

Control strategy of EV signal transitioning from EVSP back to normal signal timing scheme was discussed in this paper. A multi-objective transition optimization model was presented considering both efficiency and fairness, and a solving method based on nondominated sorting genetic algorithm II was designed to solve the optimization model. In order to avoid a large number of infeasible solutions in the algorithm and considering the nature of the problem, we designed a unique chromosome structure and put forward special encoding and decoding methods.Simulation was carried out in this paper to verify the control effect of the proposed strategy, and the value of the two objectives was calculated. The Pareto solution set of different number of additional normal cycle and the Pareto frontier for different scenarios was obtained. The data were compared with those obtained by smooth transition method. The results show that the control strategy obtained by the multi-objective programming model in this paper is superior to that obtained by smooth transition method in all cases.We assume that the arrival rate is a random variable in this paper. But perhaps the distribution function of arrival rate under emergency events is not close enough to the frequency. Since the arrival rate has some influence on the stability of control strategy, future research will focus on signal transition strategy under this condition and study it in uncertainty environments.



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*Source: 1382415-2020-01-27.xml*