J. Cent. South Univ. (2017) 24: 432-441

DOI: 10.1007/s11171-017-3445-0

Optimization of traffic signal parameters based on distribution of link travel time

LI Mao-sheng(黎茂盛), XUE Hong-li(薛红丽), SHI Feng(史峰)

School of Transportation and Traffic Engineering, Central South University, Changsha 410075, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: In order to make full use of digital data, such as data extracted from electronic police video systems, and optimize intersection signal parameters, the theoretical distribution of the vehicle’s road travel time must first be determined. The intersection signal cycle and the green splits were optimized simultaneously, and the system total travel time was selected as the optimization goal. The distribution of the vehicle’s link travel time is the combined results of the flow composition, road marking, the form of control, and the driver’s driving habits. The method proposed has 15% lower system total stop delay and fewer total stops than the method of TRRL (Transport and Road Research Laboratory) in England and the method of ARRB (Australian Road Research Board) in Australia. This method can save 0.5% total travel time and will be easier to understand and test, which establishes a causal relationship between optimal results and specific forms of road segment management, such as speed limits.

Key words: travel time distribution; signalized intersection; signal cycle time; green split

1 Introduction

Intersections are the bottlenecks of the transportation network on urban roads, and time spend waiting at intersections tends to account for 20%- 50% of total travel time within cities [1]. Many methods had been implemented to improve the traffic capacity of urban road intersections, such as broadening intersection approach, canalization, adding left turning lane and optimization of intersection signal parameters. The intersection signal parameters include the duration of signal cycle and green splits.

Duration of the signal cycle is the length of the total time that elapses for the signal light to cycle through all its different colors. The optimization of the signal cycle covers vehicle mean delay, queuing length, number of stop, resource consumption, pollutant emission, comfort, and etc. The main optimization methods of cycle duration are the TRRL (Transport and Road Research Laboratory) method in British, ARRB (Australian Road Research Board) method in Australia, and the HCM (Highway Capacity Manual) method in USA. WEBSTER [2] proposed a classical model of signal timing designed to minimize the mean delay per vehicle. The method has been widely adopted for its simple form, few parameters, strong accuracy, and early introduction. The WEBSTER arithmetic is mainly applied to low saturation. If the duration is too long, the signal cycle is visibly too long. As a result, the vehicle delay extended considerably. SUN et al [3] proposed an optimization model of the signal timing, based on supersaturated signal intersection. AKCELIK [4] established optimization evaluation indexes by introducing a stopping compensation factor considering the number of stop and overall traffic delay, which corresponds to the ARRB method in Australia. The U.S. Highway Capacity Manual (HCM) proposed a formula for the calculation of the length of the signal cycle [5], and many scholars believe it produces the shortest signal cycle. CHANG and LIN [6] proposed one disperse dynamic optimization model to solve the traffic flow supersaturated condition and used two-stage control system to produce optimal cycle and green splits. PAIK [7] built a random signal optimization model to optimize the green splits, cycle length, and phase difference at the same time. Many other scholars have studied methods of optimization control, and some have used artificial intelligence for traffic control. Among these, common methods include fuzzy control [8, 9], genetic algorithms [10, 11], ant colony algorithms [12, 13], and neural networks [14].

A green split is the ratio of effective green time to cycle in a given phase. In general, there are two ways in which this the ratio could be optimized, saturation distribution and unequal saturation distribution. WEBSTER [2] indicated that if the vehicle delay is at a minimum, the total time spend at intersections is also at a minimum. Then, the split ratio is in proportion to traffic flow ratio of the each phase, here called as qual-saturation distribution. AKCELIK [4] proposed a method of calculating unequal saturation split, and used it on conditions of different requirements for each saturation phases. The SCATS split introduced the concept of similar saturation, realizing the special traffic requirement using regulation surplus, to facilitate roughly equal saturation. The TRANSYS system and SCOOT system optimized the green ratio based on equal saturation, taking into account the queuing length, degree of crowding, delay, and number of stop.

Although these methods have certain advantages and are widely used, their goal is not rendering the traffic of vehicles traversing the section of road near the intersection more efficiently. The efficiency of traffic is directly related to the average vehicle delay, queue length, amount of time spend standing, consumption, pollutant emissions, and driver comfort. The distribution of vehicle’s link travel time is caused by composite factors such as the road vehicle composition, the basic channel of road, the control mode of the intersection, and the driving habits of the drivers. There are many advantages in using the distribution of travel time to optimize the parameters of intersection traffic signal. The digital data recorded by the electronic police system were used to determine the travel time distribution of vehicles, which takes a photo of every vehicle when it passes through the intersection and records its characteristics (such as, license plate number, time at which it went through the intersections). The goal of this work is increasing the efficiency of traffic flow by optimizing two parameters of the signal cycle length and green signal ratio simultaneously.

2 Theoretical distribution of link travel time

Suppose that the stop line before the intersection is the starting position of the next road, and the travel time of the vehicle at the intersection is included in the next road traveling time. The road segments in the traffic network can be divided into 3 types by whether there are intersections adjacent to before or after each one. Here, (0, 1), (1, 0) and (1, 1) are three basic types of link road (Fig. 1). It assumed that all the intersections are signalized intersections, 1 represents that the road segment is adjacent to the signal intersection while 0 opposites.

Road type (1, 0) is the last road segment of a path. On this road segment, the differences in drivers’ driving behavior cause fluctuations in travel time. The distribution of link travel time in this type of segment issubjected to the probability density function described in Section 2.1. The main difference between type (0, 1) and type (1, 1) is that the inflow rate of type (1, 1) is affected by the signalized intersection. After determining the probability density function of the inflow rate, the analytical method of type (1, 1) can be considered consistent with the method of calculation of type (0, 1). Here, only the theoretical method of travel time distribution of type (0, 1) is summarized; this is to maintain the integrity of the work [15].

Fig. 1 Classification of road type

2.1 Speed limits

It is here assumed that the distribution of link travel time is subject to lognormal distribution in the absence of other factors, and its probability density function f ^*(t) is as follows:

        (1)

where t is larger than zero which coincides with the physical meaning of link travel time; expectation and standard variance of t are described by parameters μ and σ, respectively.

Under the influence of speed limits [16], travel time distribution on the link can be described using the limited depend variable model with shifted characters. The possibility of time travel in the interval can be migrated to the interval , then the actual link travel time distribution f(t) after being migrated is as follows:

 (2)

where δ is a sufficiently small positive number; t₀=L/v_{max_Pan}; L is the length of urban road k; v_{max_Pan} is the punishment speed for the road speed limits.

2.2 Traffic lights and traffic volume

A cycle signal interval [0, T) was divided into g small segments, and the length of each small segment was S=T/g. The study period was broken down with {i, j} representing the jth small segment in the ith cycle (i=1, 2, …, I; j=1, 2, …, g). A probability density function fi^j(t) was used to represent the possibility when a vehicle entering road L during {i, j} period without the impacts of intersection traffic lights and road traffic volume. The function f ^ij(t) can be expressed as

(3)

1) Cases with no queue. Intersection light phase determines whether the vehicle passes through the intersection. Specifically, vehicles must stop at red light and wait until the next green light. Using a probability density function represents a vehicle entering road L during period {i, j}, and reaching to the stop line of the intersection C during the period {k, n}.

        (4)

In this case, the influence of the intersection lights, the probability density function can be obtained from function f ^ij(t) by translating the possibility of the vehicle reaching the stop line of the intersection C during the {k, n} period to the next green light segment, then is expressed as follows:

  (5)

where Q_kn is the number of vehicles in the queue during the {k, n} period; b is the number of small segments contained in the green light, b=floor(T_g/S); T_g is the length of the green light. It is necessary to translate all the possibility for every {k, n} period (k=I, I-1, …, 1; n=g, g-1, …, 1) in reverse, and still using probabilitydensity function represents the result after translation. At start time, the function equals zero, namely

2) Queue. After the queue dissipates, vehicles can pass through the intersection during the green light. At this time, the probability density function can also be obtained from function f ^ij(t) by translating the possibility of a vehicle reaching to the intersection’s stop line during the (k, n) period to the first green light segment after queue dissipates. The function is expressed as

       (6)

where (If orif means that t takes any value in (k, n) period. It is necessary to translate all possibilities for every {k, n} period(k=I, I-1, …, 1; n=g, g-1, …, 1) in reverse and to use probability density function to represent the results after translation. Q_kn is the number of vehicles in the queue in front of intersection C during the {k, n} period; h is the average headway time between vehicles (for example, 2 s); T_g is the length of the green light; b is the number of small segments contained in the green light, b=floor(T_g/S); V is the maximum number of vehicles that can pass through the intersection during every green light, u is the traffic outflow rate per cycle of the green phase, u=V/T_g.

Assume that during the period {i, j}, the weighting ratio between the number of vehicles entering the road segment to the number passing through during the entire study period is ξ_ij. There is ξ_ij=N_ij/N_z, where N_ij is the traffic volume when entering road L during the period {i, j}, and N_z is the total volume during the study period. The probability density function can be found by shifting function left-hand units, which represents the possibility when a vehicle entering road L during {i, j} period which is affected by intersection traffic lights and road traffic volume. For this reason, use that a probability density function represents the probability density function of mean travel time when a vehicle enters road L with the impacts of intersection traffic lights and road traffic volume, and then can be expressed as follows:

 (7)

3 Measuring travel time

Faced with the fluctuating travel environment, different road users use different criteria to choose route. For risk neutral users, the mean travel time (MTT) is sole route choice criterion. According to Ref. [17], risk-averse users focus on travel time reliability, so they use travel time budget (TTB) as their route choice criterion. CHEN and ZHOU [18] considered two aspects of travel time reliability and unreliability, and mean-excess travel time (METT) served as the route choice criterion. The mean-excess travel time (METT) is here defined as the conditional expectation of the exceed TTB on the corresponding path. In the present work, the mean-excess travel time (METT) and travel time budget (TTB) were selected as the route choice criteria for optimal intersection signal parameters.

It was here assumed that L_m is the stochastic variable of travel time on the link L_m. ξ_m is the travel time budget on the link L_m under α reliability demand.Based on the definition of mean-excess travel time, the expectation η_m(α) of the mean-excess travel time (METT) on the link L_m is

(8)

where the travel time budget (TTB) is as follows:

              (9)

The probability density function f(t_m) of link travel time is known in this work, so the expectation η_m(α) is as follows:

          (10)

4 Method of optimizing traffic signal parameters

The theoretical distribution of the vehicle’s road travel time, the mean-excess travel time (METT) and the travel time budget (TTB) are used to optimize traffic signal parameters. The length of traffic cycle and green ratio of the intersection whose crossing obstructs of traffic flow are excluded by traffic phase arrangement and only merging and diverging conflicts take place at the intersection. The cross network (Fig. 2) is used to study the parameter optimization problem of the intersection traffic signal. The cross network includes five nodes, four links, four paths and four ODs (r₁s₁, r₁s₂, r₂s₁, r₂s₂); the corresponding traffic demands are represented by respectively. It meets and contains a signal intersection C₁ (Fig. 2). The link properties are listed in Table 1. The vehicles’ mean velocity through the example transportation network is 48 km/h and the punishment velocity is 55 km/h. The length of the signal cycle of the signalized intersection C₁ is T with two phases, and vehicles right turnning are not subject to the traffic signal.The corresponding signal phase diagram is shown in Fig. 3.The green ratio of traffic phase one is λ₁, the green ratio of phase two is λ₂, and they meet λ₁+λ₂=1. The end-points S₁ and S₂ can be considered a virtual intersection. Both signal cycles are 1 min long and the green ratios are 1.

Fig. 2 Cross network diagram

Table 1 Link properties

Fig. 3 Signal phase diagram of intersection C₁

First, in the case of known traffic demand , the signal cycle of T, and the green ratio of phase one is λ₁. These are used with the method introduced in Section 2 to calculate the vehicles’ link travel time distribution on four links, and then the mean-excess travel time and travel time budget criteria values are calculated. Finally, the traffic volume on the link is multiplied by the mean-excess travel time or travel time budget criterion value with link traffic volume to produce the system’s total travel time under the corresponding measurement. By changing traffic demand, signal cycle and green ratio parameters, the system’s total travel time can be determined for the different parameters. The optimal signal cycle of T and the green ratio of λ₁ parameter values can be found for a given traffic demand.

The optimal method can be expressed as

           (11)

where is link traffic flow and is the measure of travel time. Hill-climbing search, genetic algorithm, simulated annealing, tabu search and harmony search can be used to solve the optimal problem 11 [19]. Here, hill-climbing search is employed to solve the optimal model due to relatively easy to be coded and its fast convergence.

Hill-climbing search algorithm is degribed as below.

Step 0: The start point of variables T and λ₁ are given, such as T=0 and λ₁=0.1, and then λ₂=1-λ₁. Value of variables S_L and S_S are given and they satisfy conditions of mod(180, S_L)=0, mod(180, S_S)=0 and S_L>S_S. Let the search step of variable λ₁ be 0.1, and the large search step of variable T be S_L. Set Temp=0 and .

Step 1: Find the minimum value at the large search step S_L of variable T.

Step 1.1) Under the given value set of variables T, λ₁ and λ₂, calculate the value If and then set Temp_T=T and

Step 1.2) If λ₁<1 and T<180, let λ₁:=λ₁+0.1; goto Step 1.1. If λ₁≥1 and T<180, let T:=T+S_L and set variable λ₁ as 0.1; goto Step 1.1.

Step 1.3) If λ₁=1 and T=180, goto Step 2.

Step 2: Find the minimum value in the neighborhood at the small search step S_s of variable T. Here has mod(10, S_s)=0, and values of variable T and λ₂ are set as Temp_T-10 and 0.1 respectively. Let λ₂=1-λ₁.

Step 2.1) Under the given value set of variables T, λ₁ and λ₂, calculate the value If and then set Temp_T=T and

Step 2.2) If λ₁<1 and TT+10, let λ₁:= λ₁+0.1; go to Step 2.1. If λ₁≥1 and TT+10, let T:+T+S_s and set variable λ₁ as 0.1; goto Step 2.1.

Step 2.3) If λ₁=1 and T=Temp_T+10, goto Step 3.

Step 3: The optimal signal cycle of T and the green ratio of λ₁, λ₂ parameter values are output and the minimum value Temp of objective function is also shown. End.

4.1 Signal cycle length

It was here assumed that the traffic volume ratios of link 1 and link 2 to total traffic demand q are 0.4 and 0.6, respectively. The traffic volume ratios of link 3 and link 4 are both 0.5, and the green ratio of the signalized intersection C₁ is 0.5. When the cycle length of the intersection signal C₁ changes, the total travel time of the mean-excess travel time (METT), and the mean travel time (MTT) are shown in Table 2. Table 2 also gives the results of Vissim simulation under the corresponding parameters, and the corresponding system users’ total travel times of traditional users equilibrium model with the different signal cycle are also listed in Table 2. Under the influence of the intersection signal cycle, the trend of total travel time of traffic network is shown in Fig. 4.

Table 2 Total system travel time under different signal cycle lengths

Fig. 4 Trends in changes in total system travel time with different signal cycles:

As shown in Fig. 4, when the traffic volume is large (q=2500 pcu/h), the optimal signal cycle is 1.5 min; when the traffic volume is medium (q=2000 pcu/h), the optimal signal cycle is 1 min; when the traffic volume is small (q=1500 pcu/h), the optimal signal cycle is 0.5 min. In conclusion, when the traffic volume is large, longer signal cycle can improve the traffic capacity. However, to a certain extent, the delay time is longer. In this way, the optimal signal cycle can reduce the total travel time in the system to a minimum. The system’s optimal signal cycle length was found to be 0.5 min by using Vissim simulation when the traffic system was under medium traffic demand (q=2000 pcu/h). However, the optimal signal cycle length was 1 min as calculated using mean-excess travel time or travel time budget criteria value. This phenomenon drew the authors’ attention. It is caused by the following model adopted in Vissim simulation software. When the link traffic demand is low, the vehicles’ space headway is large, and the influence of the front car on the latter car is small. Vehicles maintain a maximum speed and maximum possible acceleration in the simulation, and the difference in the manner of driving among vehicles is the minimal. Large degrees of vehicle synchronization are difficult to achieve in the practical road traffic flow and the simulation is merely an idealized demonstration of actual road traffic flow.

4.2 Impact of green ratio

The signal period T₁ of intersection C₁ is set to 1.5 min, and the traffic volume ratios of link 1 and link 2 are 0.4 and 0.6, respectively, while the traffic volume ratios of section 3 and section 5 are both 0.5, and the green ratio of the intersection C₁ increases from 0.2 to 0.8 by step 0.1. The variations in total travel time are showed in Table 3, and its variation with green ratio is shown in Fig. 5.

From Table 3, it can be concluded that the total travel time for the entire system reaches its minimum value, when the optimal value of the green ratio is 0.5. So the best green ratio of the intersection is 0.5 under the conditions of the current traffic demand. As showed in Fig. 5, total travel time for the whole system increases with improper green ratio, which can cause traffic jams and other traffic problems.

4.3 Signal timing

Without exception, the famous methods such as TRRL, ARRB and HCM all optimize the signal period before calculating the green ratio. However, the optimal signal period and the green ratio are not independent of each other. The system’s total travel time here served as the goal of optimization and the signal period length and the green ratio are optimized simultaneously. The impact on the system’s total travel time of signal timing under different level of traffic demand and its trends are shown in Figs. 6-8 (T is the signal period, λ₁ is green ratio of phase one). The optimal signal period and green ratio can be obtained from optimal problem 11 by the hill- climbing search, which is listed in Table 4.

4.4 Comparative analysis

Table 4 shows a comparison of the optimal signal period length and green ratio as computed using the TRRL method, ARRB method, HCM method and the TTD-METT method proposed in this work. These

optimal signal period length and green ratio were used in the simulation, and then the total stop delay, total travel time, total queue length and total stops were calculated as shown in Table 5. However, the optimal signal period length and green ratio calculated using the HCM method were also simulated to produce results, these are only displayed and are not included in the comparison due to it produces the shortest signal cycle in the case of large degrees of vehicle synchronization, but it is difficult to achieve in the practical road traffic flow.

Table 3 Impacts of green ratio on total system travel time

Fig. 5 Trends in changes in total system travel time with different splits:

Most of criterions results, such as the total stop delay, total travel time, total queue length and total stops simulated by the Vissim software using the optimal signal period length and green ratio calculated using the TTD-METT method proposed, are the best among others (bold numbers in the METT column of Table 5). These also proved that the proposed method can decrease the total stop delay by 8.85%-14.73% relative to the TRRL

method and 19.97%-31.94% relative to the ARRB method. It also reduces the system’s total travel time by 0.3%-0.61% and 0.28%-0.90%, and decreases the total number of stops by 11.39%-19.03% and 5.43%-17.59%, respectively. However, it increases the total queue length detectably. Due to the distribution of vehicle’s link travel time captures the impact of composite factors such as the road vehicle composition, the basic channel of road, the control mode of the intersection, and the driving habits of the drivers.

Fig. 6 Total travel time and trends in changes in signal timing (q=2500 pcu/h):

Fig. 7 Total travel time and trends in changes in signal timing (q=2000 pcu/h):

Fig. 8 Total travel time and trends in changes in signal timing (q=1500 pcu/h):

Table 4 Results of comparison of 5 kinds of signal timing methods

Table 5 Simulation results under several kinds of signal timing schemes

5 Conclusions

Based on the distribution method of link travel time, which has been verified by the digital data of the electronic police system, this paper proposed a method for optimizing intersection signal parameters, which indicated the connection between the results of optimization with traffic signals, traffic volume, and road management, such as speed limits on urban road. In contrast to widely used methods, such as the TRRL method, ARRB method, and HCM method, the method proposed in this paper optimized the signal period and the green ratio simultaneously by setting the total travel time as an optimization objective, which uses the link travel time distribution. The proposed method was compared to TRRL and ARRB, and results showed that it can decrease system delay, number of stops, and total travel time according to the simulation, and those decrements indicate the advantages of the present method.

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(Edited by DENG Lü-xiang)

Cite this article as: LI Mao-sheng, XUE Hong-li, SHI Feng. Optimization of traffic signal parameters based on distribution of link travel time [J]. Journal of Central South University, 2017, 24(2): 432-441. DOI: 10.1007/s11171- 017-3445-0.

Foundation item: Project(14BTJ017) supported by National Social Science Foundation Project of China; Project supported by the 2014 Mathematics and Interdisciplinary Science Project of Central South University, China

Received date: 2016-03-11; Accepted date: 2016-06-16

Corresponding author: LI Mao-sheng, Associate Professor, PhD; Tel: 86-731-82655640; E-mail: maosheng.li@csu.edu.cn