Dynamic Illumination Method for Rural Highway Intersections with Traffic Flow Changes

Abstract

The conflicts between vehicles and vulnerable road users are frequent at rural highway intersections. Improving lighting conditions at intersections can significantly reduce the likelihood of nighttime accidents. This study aims to develop a dynamic illumination method for rural highway intersections with traffic flow changes. First, the safety design principles of a lighting scheme were formulated for minimum illuminance, vehicle speed, and traffic risk at intersections. In addition, a traffic risk quantification model for intersections was developed with illuminance and traffic flow parameters as inputs, incorporating gap acceptance theory and the braking safety distance model. Second, combining safety design principles and a traffic risk quantification model, a calculation method for optimal illuminance at intersections was designed with minimum illuminance as the optimization objective, and considering the risk at intersections not exceeding the safety threshold as the constraint. Finally, a case study was conducted in a simulated environment. Evaluation results demonstrate that limiting the maximum nighttime vehicle speed at intersections to 50 km/h can maximize the energy-saving rate of the lighting system while ensuring traffic safety and transport efficiency. The advantage of the method is that it can calculate the safe lighting scheme for highway intersections without any historic accident data and save lighting energy consumption. It increases the feasibility of promoting lighting facilities at rural highway intersections.

Keywords

operations vehicle-highway automation safety safety performance and analysis highway safety manual

Compromised vision significantly contributes to the high number of nighttime accidents at intersections ( 1 ). At rural highway intersections, conflicts between vehicles on highways and vulnerable road users (VRUs), such as pedestrians and cyclists on both sides of rural roads, are particularly severe ( 2 ). However, because of the significant energy consumption after installing street lights, most of the world’s highways are not typically illuminated ( 3 ). In the absence of lighting facilities and relying solely on the limited illumination range of vehicle headlights, it is difficult for drivers to observe VRUs emerging from bypasses on both sides of the intersection in time, thereby increasing the risk of nighttime fatal accidents ( 4 ). By comparing accident data at intersections before and after the installation of street lights, researchers have demonstrated that improved illumination can dramatically reduce the likelihood of nighttime accidents ( 5 ). Nevertheless, the brightness of the street lights in the traditional road lighting system cannot be adjusted in real-time in response to the changing state of the road traffic flow, resulting in a significant amount of wasted electricity, which is an important reason why the current road lighting system is not widely used at rural highway intersections ( 6 ). Therefore, it is recommended to implement a lighting system that can detect the traffic flow status and adjust the lighting scheme to improving lighting conditions at rural highway intersections.

The smart street light is a composite application product that integrates numerous information devices on conventional street lamp poles ( 7 ). Roadside sensors are installed on both sides of the lamppost to detect traffic data such as vehicle type, position, and speed. The illumination is automatically adjusted based on the correlation between traffic data and lighting ( 8 ). Smart street lights provide efficient power savings potential compared with traditional road-lighting systems ( 9 ). However, the existing smart street lights generally configure the lighting scheme according to the road speed limit, which ensures the safety of vehicles traveling at the speed limit, but wastes energy when vehicles travel at a lower speed. In addition, the situation of traffic flow varies significantly at different time periods and regions. Therefore, it is necessary to design a dynamic matching scheme between illuminance and traffic flow status, through which matching can not only reduce the likelihood of nighttime accidents at intersections but also reduce the power consumption of the lighting system, thereby enhancing the feasibility of promoting lighting systems at rural highway intersections.

Literature Review

For many years, it has been a topic of concern to improve the service quality and intelligence of smart street lights by optimizing the road lighting scheme. Currently, scholars develop road lighting schemes primarily from two categories.

The first category is the road lighting schemes considering energy conservation. Zhao et al. proposed a fuzzy control system that conserved energy for highway tunnel lighting ( 10 ). The fuzzy control model was created using tunnel exterior environment luminance, traffic volume, and vehicle speed as inputs, and tunnel interior luminance as outputs. According to the results of tests conducted on tunnels in Guangxi Province, China, the system can save approximately 41% of daily lighting energy ( 10 ). Wang et al. designed an intelligent control system for tunnel lighting that enables lighting to move with vehicles ( 11 ). When a roadside camera detects a vehicle, the street lights within 400 m in front of the vehicle can be automatically adjusted to the appropriate luminance, and the remaining street lights maintain 10% of maximum luminance ( 11 ). Based on data mining of vehicle speed distribution, Qin et al. designed a dynamic regulation method for tunnel lighting ( 12 ). Using the traffic flow data of five highway tunnels in Jilin Province, China, they conducted an example analysis and determined that the average energy-saving rate (ESR) of this method is 55% ( 12 ). Hu et al. proposed an adaptive dimming algorithm for smart street lights based on K-means and the directed acyclic graph support vector machine ( 13 ). This algorithm can cluster traffic volume data and develop a six-category dimming model to achieve intelligent dimming of street lights. Experimental results demonstrated that the algorithm’s ESR is 68.43% ( 13 ).

The second category is the road lighting schemes considering traffic safety. Researchers have analyzed the effect of road lighting on the visual performance of drivers, which provides a theoretical foundation for the practical design of road lighting schemes. On the one hand, improving road lighting can enhance drivers’ nighttime perception of vehicle speed and surrounding targets, reducing the likelihood of collisions ( 14 ). On the other hand, visual comfort is also a crucial criterion for assessing the quality of road lighting ( 15 ). Johansson et al. used the perceived outdoor lighting quality (POLQ) as an evaluation index of roadway lighting quality ( 16 ). Using factor analysis, they determined that POLQ comprises two main components: the perceived intensity quality and the perceived comfort quality. The former is related to the light intensity and direction of the light source; the latter is related to the color temperature and glare of the light source ( 16 ). Lin et al. analyzed the influence of uncomfortable glare on eye movement and pupil area utilizing eye movement velocity as a metric ( 17 ). Davidovic et al. conducted questionnaires which revealed that LED streetlights with a color temperature of 3,000 K provide a superior visual experience ( 18 ). Wang et al. proposed a method for designing a gradient illumination scheme along lighting transition zones at highway intersections ( 19 ). The core of this method was an optimization model that utilized the safety threshold of pupil area change rate as the constraint, and determined the optimal illuminance increase rates along the transition zones ( 19 ).

From the above, we can find that there are a lot of studies on intelligent road lighting schemes and there are two features in current studies. First, dynamic road lighting control algorithms considering energy conservation only focus on the traffic flow status within a tunnel or roadway segment. This can accommodate the illumination needs of vehicles traversing a tunnel or roadway segment. Nevertheless, for the intersection with more conflict points, it is essential to consider the influence of traffic flow status in intersecting lanes on traffic safety and to design the lighting scheme according to the degree of influence. Second, raising the level of road lighting to the greatest extent within the glare threshold conditions can improve drivers’ night perception and visual comfort, thereby decreasing the likelihood of traffic accidents. However, in designing lighting schemes at highway intersections, it is crucial to ensure traffic safety while reducing the illuminance to achieve a balance between energy conservation and traffic safety. In the meantime, it can prevent the harmful influence of the light and dark adaptation process on drivers when vehicles enter and exit intersection illumination areas.

Objectives and Contributions

The aim of this study is to develop a dynamic illumination method for intersections formed by highways and rural roads that considers changes of traffic flow. First, the safety design principles of the lighting scheme are formulated for minimum illuminance, vehicle speed, and traffic risk at intersections. In addition, a traffic risk quantification model for intersections was developed with illuminance and traffic flow parameters as inputs. Second, combining safety design principles and a traffic risk quantification model, a calculation method for optimal illuminance at intersections is designed. Finally, a cross-shaped highway intersection is used as the experimental scenario, and conflicting events are designed using conflicts among pedestrians, cyclists, motorcyclists, and vehicles. The simulated experimental platform is used for data collection and case analysis.

Relative to previous studies on intelligent road lighting schemes, the contributions of this study include the following two aspects.

A traffic risk quantification model for highway intersections is developed with traffic flow parameters of vehicles and VRUs and illuminance at intersections as inputs. The risk quantification model can calculate the probability of conflict between vehicles and VRUs by analyzing their relative position relationship and assesses the traffic safety risk at intersections. Therefore, the safety lighting scheme for highway intersections can be calculated without historical accident data.

A calculation method for optimal illuminance at intersections is designed. The method with minimum power as the optimization objective and considering risk not exceeding the safety threshold as the constraint can further reduce the illuminance while ensuring the traffic safety at intersections. In addition to saving energy, reducing the illuminance at intersections can reduce the light and dark adaptation time for drivers when vehicles enter and leave the lighting area, and reduce the safety hazards. Thus, the feasibility of promoting lighting facilities at rural highway intersections is increased.

The remainder of the paper is organized as follows. The next section develops a dynamic illumination optimization model for rural highway intersections considering traffic safety and energy-saving. The section after that details the driving simulation platform construction and experimental data collection. The penultimate section provides a case study of the proposed method. The final section provides conclusive comments on this study.

Illuminance Optimization Model for Intersections

An illuminance optimization model for intersections that considers safety and energy saving is developed in this section. The model optimizes the lowest energy consumption of the street lights while ensuring that the traffic risk at intersections does not exceed an acceptable risk value. By selecting the reasonable illuminance, the intersection accident rate is maintained below the safety threshold, and the energy consumption of street lights is reduced while maintaining the traffic safety at intersections.

Safety Design Principles of Lighting Scheme at Intersections

Because of significant energy consumption, insufficient highway lighting has become a worldwide issue, leading to highway intersections becoming accident-prone areas at night. Reducing the energy consumption while ensuring traffic safety is the key factor to increase the installation rate of lighting facilities at rural highway intersections.

In the road lighting system, the power P of street lights can be calculated by Equation 1 ( 20 ):

P = \frac{E_{s} \cdot w \cdot s}{η \cdot U_{s} \cdot ρ \cdot λ}

(1)

where

$E_{s}$ = the illuminance of street lights at intersections (lx),

w = the total width of the road (m),

s = the installation spacing of street lights (m),

$U_{s}$ = utilization coefficient of street lights, with a recommended value of 0.55 (Chinese Standard for lighting design of urban road CJJ45-2015, 2016) ( 20 ),

ρ = the maintenance coefficient of street lights, with a recommended value of 0.7 (CJJ45-2015) ( 20 ), and

λ = the arrangement coefficient of street lights, with a recommended value of 2 when street lamps are installed symmetrically (CJJ45-2015) ( 20 ).

Since w, s, η, $U_{s}$ , ρ, and λ in Equation 1 are difficult to modify, dynamic road lighting systems are typically regulated using road traffic flow parameters, which adjust the lighting power of street lights to improve the energy efficiency of the lighting system. The traffic flow parameters for controlling $E_{s}$ are as following: time headway of vehicles $t_{hv}$ , vehicle speed v, and time headway of VRUs $t_{hb}$ . The time headway is the time interval between the front ends of two vehicles/VRUs passing a section in a queue of vehicles/VRUs traveling in the same lane ( 21 ).

The primary objective of installing street lights at highway intersections is to reduce the nighttime accident rate; therefore, it is necessary to assure nighttime traffic safety when dynamically tuning the illuminance of street lights. The following three safety design principles must be satisfied when applying traffic flow parameters to control $E_{s}$ , and the optimal value of $E_{s}$ for satisfying all three constraints simultaneously is $E_{m}$ .

First, when vehicles enter the lighting area at the intersection, if the illumination is too low, it will increase the drivers’ mental stress and affect the drivers’ visual search ability to identify VRUs at the intersection, which increases safety hazards ( 11 ). Therefore $E_{s}$ should not be less than the lower limit of illuminance $E_{\min}$ . According to the Chinese Standard for Lighting Design of Urban Road CJJ45-2015 for illuminance requirements at intersections, the illuminance of the intersection pavement analyzed in this study must be at least 20 lx; thus, the recommended value of $E_{\min}$ is 20 lx ( 20 ).

Second, the speed of the vehicle passing through the intersection is changeable. Generally, the vehicle speed is higher when the vehicle is far from the intersection and lower when the vehicle is close to the intersection ( 22 ). Therefore, it is necessary to ensure that the illuminance of street lights does not change significantly when the vehicle speed changes, to avoid potential safety hazards caused by the driver adjusting to the changes in illuminance. In addition, it is necessary to accommodate the illumination demands of all vehicles traveling at different speeds near the intersection. Consequently, the value of vehicle speed v that input into the lighting control system should not be less than the maximum value V of the speed data of all vehicles on all positions inside the lighting area. The value of V can be determined using road speed detection equipment.

Last, after the installation of street lamps at the highway intersection, the rate of accidents between highway vehicles and rural road VRUs should be kept below acceptable ranges. Traffic risk is defined as the probability of accidents occurring in a traffic scenario if the actual situation deviates from the intended aim because of random factors ( 23 ). In this paper, the value of traffic risk R at the intersection is utilized as an evaluation index for the lighting scheme. The R-value between vehicles and VRUs must not exceed the acceptable risk value $R_{acp}$ with a 99% confidence interval.

Combining with the above constraints, the $E_{m}$ is related to $E_{\min}$ , V, R, and $R_{acp}$ . The $f_{E} (\cdot)$ is used to define the relationship function between $E_{m}$ and the above variables. The $E_{m}$ can be expressed according to Equation 2, as follows:

E_{m} = f_{E} (E_{min}, V, R \leq R_{acp})

(2)

For the third constraint, if a vehicle and a VRU are involved in a collision at an intersection, both parties should satisfy two conditions:

The vehicle and the VRU experience a traffic conflict, whereby the traffic conflict is an event in which multiple road users approach one other in time and space, and at least one of them must change their travel trajectory to avoid a collision ( 24 ).

Following a traffic conflict between a vehicle and a VRU, the driver fails to observe the VRU in time to apply the brakes and change the vehicle’s trajectory. Consequently, the traffic risk R can be represented as Equation 3:

R = P_{C} \cdot P_{A | C}

(3)

where

$P_{C}$ = the probability of conflict between vehicles and VRUs, and

$P_{A | C}$ = the probability that drivers fail to brake in time after the vehicle-VRU conflict occurs.

The $P_{C}$ in Equation 3 relies on $t_{hv}$ , $t_{hb}$ , and other variables. The $P_{A | C}$ is determined by the illuminance E of the intersection, the vehicle speed v, and the brake deceleration $a_{b}$ applied by vehicle drivers after observing VRUs. $f_{C} (\cdot)$ and $f_{A | C} (\cdot)$ are used to describe the relationship function between $P_{C}$ , $P_{A | C}$ , and variables. Therefore, $P_{C}$ and $P_{A | C}$ can be expressed by Equations 4 and 5:

P_{C} = f_{C} (t_{hb}, t_{hv}, \dots)

(4)

P_{A | C} = f_{A | C} (E, v, a_{b}, \dots)

(5)

In Equation 5, the recommended value of $a_{b}$ is 4.5 m/s² ( 21 ); E combines street lamp illuminance $E_{s}$ and automobile headlamp illuminance $E_{v}$ . According to the superposition principle of illuminance, the relationship among E, $E_{s}$ and $E_{v}$ as shown in Equation 6 ( 25 ):

{\begin{matrix} E = E_{s} + E_{v} \\ E_{v} = \frac{n \cdot Φ_{v} \cdot U_{v}}{w \cdot \bar{v} \cdot \bar{t_{hv}}} \end{matrix}

(6)

where

n = the number of highway lanes,

$\bar{v}$ = average vehicle speed (m/s),

$\bar{t_{hv}}$ = average time headway of vehicles (s),

$Φ_{v}$ = luminous flux of vehicle headlamp, with a recommended value of 3,000 lumen ( 26 ), and

$U_{v}$ = the utilization coefficient of vehicle headlamp, with a recommended value of 0.35 ( 26 ).

In combination with Equations 2 to 6, the optimal illuminance $E_{m}$ of street lights with traffic flow parameters as variables can be expressed as follows:

{\begin{matrix} E_{m} = f_{E} (E_{min}, V, P_{C} \cdot P_{A | C} \leq R_{acp}) \\ P_{C} = f_{C} (t_{hb}, t_{hv}, \dots) \\ P_{A | C} = f_{A | C} (E, v, a_{b}, \dots) \\ E = E_{m} + \frac{n \cdot Φ_{v} \cdot U_{v}}{w \cdot \bar{v} \cdot \bar{t_{hv}}} \end{matrix}

(7)

According to Equation 7, $P_{C}$ and $P_{A | C}$ are crucial parameters for calculating $E_{m}$ . In the following section, the calculation models $f_{C}$ and $f_{A | C}$ of the two parameters will be developed, and the optimal illuminance $E_{m}$ will be calculated using these two models.

Calculation Method for Optimal Illuminance

Probability of Conflict between Vehicles and VRUs (P_C) Modeling

There are two circumstances of vehicles crossing VRUs and VRUs crossing vehicles at highway intersections. The priority of VRUs is generally higher than that of vehicles at unsignalized intersections. When a conflict between vehicles and VRUs occurs at the intersection, vehicles should slow down and give the way to VRUs to ensure their safe passage. Consequently, this paper focuses on the circumstance of vehicles crossing VRUs.

The gap acceptance theory is typically used to determine the probability of a traffic collision involving two categories of road users with different priorities, such as vehicles and VRUs. Gap acceptance theory is a sophisticated method for analyzing competing interference between road users with different priorities, and it is widely applied at unsignalized intersections ( 27 ). Time headway and critical gap are essential parameters in gap acceptance theory, and the critical gap is the average minimum headway acceptable to road users when crossing the traffic flow ( 28 ). The core principle of gap acceptance theory is to analyze the relationship between headway time and critical gap for traffic flows with multiple conflict points, and then to determine whether the traffic flow with lower priority can successfully cross the traffic flow with higher priority, that is, whether vehicles can cross the VRUs.

The relative positions of vehicles and VRUs at a highway intersection is shown in Figure 1. The VRUs of the east entrance and the vehicles of the south entrance are used as an example. The relationship between the headway time and the critical gap is investigated when the vehicles and the VRUs conflict at the intersection. The yellow area of Figure 1 is shown as Figure 2.

In the process of the vehicle enters the intersection, the vehicle will cross VRUs if the traffic flow of VRUs can provide a sufficient crossing gap for the vehicle; otherwise, the vehicle will stop and give way until VRUs provide a sufficient crossing gap, at which time a traffic conflict occurs between them. Therefore, the probability of conflict between vehicles and VRUs is calculated as follows: Calculate the probability $P_{t}$ of vehicles crossing VRUs successfully using the gap acceptance theory, and then indirectly calculate the probability $P_{C}$ that the vehicle cannot cross VRUs, that is, the probability of conflict.

P_{C} = 1 - P_{t}

(8)

When a vehicle can pass through VRUs smoothly, the following two conditions must be met simultaneously: 1) The time headway $t_{hb}$ of VRUs is greater than the critical gap $t_{cv}$ required for the vehicle to pass through VRUs; 2) When the driver begins to judge whether they can pass through VRUs smoothly, the difference between the time required for VRUs and the vehicle to travel from the current position to the conflict area should be greater than $t_{cv}$ .

Condition 1 can be expressed as Equation 9:

t_{hb} > t_{cv}

(9)

For condition 2, assuming that the driver starts to judge whether they can pass through VRUs when driving △t away from $t_{hv}$ , then the time $t_{rv}$ and $t_{rb}$ required for the vehicle and VRUs to travel to the conflict area from that moment can be expressed as follows:

t_{rv} = t_{hv} - Δ t + t_{v}

(10)

t_{rb} = t_{hb} - Δ t + t_{b}

(11)

After using Equations 10 and 11 to calculate the time difference between $t_{rv}$ and $t_{rb}$ , Condition 2 can be expressed as Equation 12:

t_{rb} - t_{rv} = t_{hb} + t_{b} - t_{hv} - t_{v} > t_{cv}

(12)

It can be known that Equation 13 should be satisfied when vehicles can pass VRUs smoothly by combining Equations 9 and 12:

{\begin{matrix} t_{hb} > t_{cv} \\ t_{hb} - t_{hv} > t_{v} - t_{b} + t_{cv} \end{matrix}

(13)

According to Equation 13, the joint distribution probability $P_{t}$ of vehicles passing through VRUs can be expressed as Equation 14:

\begin{matrix} P_{t} = P (t_{hb} > t_{cv} \cap t_{hb} - t_{hv} > t_{v} - t_{b} + t_{cv}) \\ = \int_{t_{cv}}^{+ \infty} [\int_{0}^{t_{hb} + t_{b} - t_{cv} - t_{v}} f_{1} (t_{hv}) d t_{hv}] f_{2} (t_{hb}) d t_{hb} \end{matrix}

(14)

where

$f_{1} (t_{hv})$ = the probability distribution function of vehicles’ time headway, and

$f_{2} (t_{hb})$ = the probability distribution function of VRUs’ time headway.

$f_{1} (t_{hv})$ and $f_{2} (t_{hb})$ can be determined by counting the time headway of vehicles and VRUs at intersections.

According to Equations 8 and 14, the probability of conflict between vehicle and VRU can be expressed as Equation 15:

P_{C} = 1 - \int_{t_{cv}}^{+ \infty} [\int_{0}^{t_{hb} + t_{b} - t_{cv} - t_{v}} f_{1} (t_{hv}) d t_{hv}] f_{2} (t_{hb}) d t_{hb}

(15)

Probability of Driver Failing to Brake in Time under Conflict (P_A|C) Modeling

The relative position between a vehicle and VRUs that occur conflicts at a highway intersection is shown in Figure 3, where point A and point B are the conflict points between the vehicle and the VRUs entering the intersection from the left and right sides, respectively, in the vehicle driver’s visual field.

The conflict point B is taken as an illustration; the process of a collision between a vehicle and a VRU can be separated into the following three stages:

The vehicle travels into the intersection at speed v.

When the vehicle reaches a distance X from the conflict point, the driver observes the VRU emerging from the intersection and takes braking measures. Hereinafter, X is referred to as “driver visual recognition distance.”

After the driver takes braking measures, the safety braking distance required for the vehicle from braking to stop is S, and the distance between the center of mass and the conflict point when the vehicle stops is $D_{s}$ .

If $D_{s}$ is greater than the safety distance d from the conflict point after the vehicle stopped, it indicates that the vehicle has not collided with the VRU; otherwise, it means that the vehicle failed to brake in time. Therefore, the criteria for judging that the vehicle driver fails to brake in time is:

D_{s} < d

(16)

where the recommended value of d is 5 m ( 29 ).

According to the above analysis, when the conflict occurs between a vehicle and VRUs, whether the vehicle can brake in time mainly depends on the driver visual recognition distance X to the VRUs and the vehicle’s safety braking distance S.

1) Driver Visual Recognition Distance (X)

During nighttime driving, drivers perceive external traffic information primarily through vision, and illumination and vehicle speed affect the drivers’ visual characteristics (such as sight distance, vision field, and attention span), thereby affecting the drivers’ visual recognition distance to VRUs ( 30 , 31 ). Finally, it will affect the drivers’ ability to brake in time. In addition, the shape and speed of pedestrians and cyclists varied significantly, resulting in differing visual recognition distances for different types of VRUs operating at the same speed and under the same illumination ( 32 ).

According to the above analysis, the illuminance E of the lighting area at the intersection, the speed of vehicles v, the type of conflict events M, and the direction k of VRUs in the driver’s visual field will all have an impact on X, so the class conditional probability density function of X will be expressed as p(X|E, v, M, k). When the control variables E, v, M, and k are independent of each other and the influence of additional variables (such as the driver’s mental state and driving experience, etc.) on X is relatively weak, X generally obeys the normal distribution ( 33 ). Assuming that the mean value of X under E, v, M, k is μ(E, v, M, k), and the standard deviation of X is σ(E, v, M, k), then p(X|E, v, M, k) can be expressed as:

\begin{matrix} p (X | E, v, M, k) \\ = \frac{1}{\sqrt{2 π} σ (E, v, M, k)} \exp (- \frac{{[X - μ (E, v, M, k)]}^{2}}{2 σ {(E, v, M, k)}^{2}}) \end{matrix}

(17)

2) Vehicle’s Safety Braking Distance (S)

After the driver observes VRUs, the safety braking distance S determines whether the vehicle can brake in time. According to the theory of vehicle dynamics, during the braking process, the vehicle needs to go through three stages: the braking coordination stage, the growth stage of deceleration, and the continuous braking stage.

a) In the braking coordination stage, the brake needs to pass the free travel time $t_{1}$ before it can produce braking force, so the vehicle can still be regarded as a uniform motion at speed v. Then, the braking distance $S_{1}$ of the vehicle in this stage can be expressed as:

S_{1} = v \cdot t_{1}

(18)

where the recommended value of $t_{1}$ is 0.03 s ( 21 ).

b) After the generation of braking force, the vehicle enters the growth stage of deceleration. During this stage, the deceleration will increase from 0 to $a_{b}$ , and, in this process, $f_{a} (t)$ is used to express the relationship of deceleration with time t. The braking distance $S_{2}$ and vehicle speed $v_{k}$ after this stage can be expressed as follows:

S_{2} = \int_{0}^{t_{2}} (v - \int_{0}^{t} f_{a} (t) d t) d t = v \cdot t_{2} - \int_{0}^{t_{2}} (\int_{0}^{t} f_{a} (t) d t) d t

(19)

v_{k} = v - \int_{0}^{t_{2}} f_{a} (t) d t

(20)

where the recommended value of $t_{2}$ is 0.3 s ( 21 ).

c) In the continuous braking stage, the vehicle takes $v_{k}$ as the initial velocity and $a_{b}$ as the deceleration to perform uniform deceleration until the vehicle speed is 0. Then, the braking distance $S_{3}$ of the vehicle in this stage can be expressed as:

S_{3} = \int_{0}^{\frac{v_{k}}{a_{b}}} (v_{k} - a_{b} \cdot t) d t = \frac{{(v - 0.5 a_{b} \cdot t_{2})}^{2}}{2 a_{b}}

(21)

Combining Equations 18, 19, and 21, S can be expressed as:

\begin{matrix} S = S_{1} + S_{2} + S_{3} = v (t_{1} + t_{2}) \\ - \int_{0}^{t_{2}} (\int_{0}^{t} f_{a} (t) d t) d t + \frac{{(v - 0.5 a_{b} \cdot t_{2})}^{2}}{2 a_{b}} \end{matrix}

(22)

3) Probability of Driver Failing to Brake in Time (P_A_|C)

It can be seen from Figure 3 that the relationship between $D_{s}$ , X, and S is:

D_{s} = X - S

(23)

According to Equations 16 and 23, the driver fails to brake in time when X and S meet the following relationship:

X < S + d

(24)

Combining Equations 17, 22, and 24, it can be determined that, given E, v, M, and k, the probability that the driver fails to adopt timely braking measures when the vehicle and the VRU are involved in a traffic dispute is $\int_{- \infty}^{S + d} p (X < S + d | E, v, M, k) d X$ .

In addition, since the driver cannot determine the type of VRU and direction of appearance at the intersection in advance, the probability of the driver failing to brake in time under the maximum conflict under different VRU types M and direction k is $P_{A | C}$ , as shown in Equation 25:

{\begin{matrix} P_{A | C} = max_{M, k} [\int_{- \infty}^{S + d} p (X < S + d | E, v, M, k) d X] \\ S = v (t_{1} + t_{2}) - \frac{a_{b}}{6} {t_{2}}^{2} + \frac{{(v - 0.5 a_{b} \cdot t_{2})}^{2}}{2 a_{b}} \end{matrix}

(25)

Solution Algorithm for Optimal Illuminance (E_m)

After developing the calculation models of parameters $P_{C}$ and $P_{A | C}$ , the solution algorithm for the optimal illuminance of intersections is designed by using these two parameters. The solution algorithm is described below.

First, the probability $P_{C}^{i, j}$ of the traffic conflict between vehicles from the entrance i and VRUs from the entrance j at the intersection, and the probability $P_{A | C}^{i, j}$ of the driver failing to take braking measures in the conflict are calculated. According to Equations 15 and 25, the calculation models of $P_{C}^{i, j}$ and $P_{A | C}^{i, j}$ can be expressed by Equations 26 and 27:

P_{C}^{i, j} = f_{C} (f_{1} (t_{hv}^{i}), f_{2} (t_{hb}^{j}), t_{v}^{i, j}, t_{b}^{i, j}, t_{cv}^{i, j})

(26)

P_{A | C}^{i, j} = f_{A | C} (E^{j}, V^{i})

(27)

where

$E^{j}$ = the illuminance at entrance j of the intersection, and

$V^{i}$ = the maximum speed of the imported vehicle at entrance i of the intersection.

Then, combining Equation 26 and Equation 27 establishes the constraints of the lighting optimization model based on the traffic risk quantification model of Equation 7, which can be expressed as:

f_{C} (f_{1} (t_{hv}^{i}), f_{2} (t_{hb}^{j}), t_{v}^{i, j}, t_{b}^{i, j}, t_{cv}^{i, j}) \cdot f_{A | C} (E^{j}, V^{i}) \leq R_{acp}

(28)

Finally, with the minimum lighting power as the optimization objective, the optimal illuminance $E_{m}$ of street lights is calculated. The specific process is as follows.

First, the minimum illuminance $E_{rm}^{i, j}$ of entrance j that satisfies the constraint condition that the traffic risk between vehicles at entrance i and VRUs at entrance j does not exceed $R_{acp}$ is obtained by solving Equation 28. Since the street lamps at entrance j need to meet the lighting requirements of entrance i and its opposite entrance r, the maximum value of $E_{rm}^{i, j}$ and $E_{rm}^{r, j}$ is used as the minimum illuminance $E_{rm}^{j}$ of entrance j. In addition, to ensure the uniformity of the illuminance of each entrance lane in the intersection area, the maximum value of all $E_{rm}^{j}$ is used as the optimal illuminance $E_{rm}$ of the lighting area at intersections, with $E_{rm}$ consisting of street lamps’ illuminance and vehicle headlights’ illuminance. Then, the minimum value $E_{op}$ of street light is calculated based on $E_{rm}$ using Equation 6. Finally, the minimum value of comparison $E_{op}$ and $E_{op}$ is used as $E_{m}$ . The above solution procedure of $E_{m}$ can be expressed as:

{\begin{matrix} E_{rm}^{j} = \max (E_{rm}^{i, j}, E_{rm}^{r, j}) \\ E_{rm} = max_{j} (E_{rm}^{j}) \\ E_{op} = E_{rm} - \frac{n \cdot Φ_{v} \cdot U_{v} \cdot K_{v}}{1000 w} \\ E_{m} = {\begin{matrix} E_{op} & E_{op} > E_{\min} \\ E_{\min} & E_{op} \leq E_{\min} \end{matrix} \end{matrix}

(29)

Experimental Design and Platform Construction

To determine the parameters of the driver visual recognition distance distribution model in the section “Probability of Driver Failing to Brake in Time under Conflict (PA|C) Modeling.” under different illuminances, vehicle speeds, and conflict event types, and then to calculate the probability of drivers failing to brake in time when a traffic conflict between a vehicle and a VRU, we will collect experimental data using an indoor driving simulation experiment platform.

Experimental Design

The experiment aims to collect drivers’ visual recognition distances for pedestrians, cyclists, and motorcyclists as the vehicle approaches the intersection under different illuminances and speeds.

A two-way, four-lane and two-way, two-lane major-minor intersection was chosen as the study object. At the intersection in question, the speed limit for the major road was 70 km/h, and that for the minor road was 50 km/h. The maximum illuminance of the street lamps at the intersection was 36 lx. Within the maximum illuminance, four lighting schemes were chosen for the experimental scenario. The values of illuminance E for the four lighting schemes were 0, 12, 24, and 36 lx. Under each lighting scheme, the vehicle speeds were set to 30, 40, 50, 60, and 70 km/h. For each illuminance and speed, the types of conflict events were determined by random conflicts between pedestrians, cyclists, motorcyclists, and vehicles from the left and right side of the intersection, and the speeds of pedestrians, cyclists, and motorcyclists were set to 4, 12, and 50 km/h, respectively. Therefore, each participant performed 120 experiments (four illuminances × five vehicle speeds × six conflict events).

In each experiment, when the driver maneuvers the vehicle to the conflict trigger position before the conflict point, the emergency will move from its initial position to the conflict point at a constant rate. If the driver discovers an emergency, they must immediately apply the brakes until the vehicle stops. Assuming that the distance between the conflict trigger position and the conflict point is S_T, the distance between the initial position of emergencies and the conflict point is S_h, and the speed of emergencies is v_h (km/h). The respective speeds of pedestrians, cyclists, and motorcyclists are 4, 12, and 50 km/h. The values of S_T at various vehicle speeds and the values of S_h at various emergency speeds are shown in Tables 1 and 2, respectively.

Table 1.

The Distance between the Conflict Trigger Position and the Conflict Point (S_T) at Different Vehicle Speeds (v)

v (km/h)	$S_{T}$ (m)
70	389
60	333
50	278
40	222
30	167

Table 2.

The Distance between the Initial Position of Emergencies and the Conflict Point (S_h) and the Speed of Emergencies (v_h) of Pedestrian, Cyclist, and Motorcyclist

Emergency	$v_{h}$ (km/h)	$S_{h}$ (m)
Motorcyclist	50	278
Cyclist	12	67
Pedestrian	4	22

A total of 100 drivers participated in this study (67 men and 33 women) and every driver needed take each task. Therefore, a total of 12,000 experiments were completed (120 experiments/driver × 100 drivers). Drivers were aged between 22 and 35 years (mean = 25.88, standard deviation = 2.63). Their visual acuity scores were all greater than 1.0, and their minimum driving experience was 2 years. The procedure for the experiment is as follows.

Step 1: The participant drives the vehicle at the speed in the experiment scheme, and a VRU would be triggered when reaching the conflict trigger position before the intersection. If the participant encounters an emergency, they must immediately apply the brakes until the vehicle stops.

Step 2: The data acquisition module of the driving simulator calculates and records the visual recognition distance of the participant based on the brake pedal opening degree and the position of the vehicle.

Step 3: After the participant completes one experiment trip, they must rest for 5 min before beginning the next test. Subsequently, they repeat Steps 1 and 2 to complete the remaining experiments.

Experimental Platform Construction

There are a few issues in the real-road experiment, such as traffic safety risks, unadjusted positions, and heights of lamps; therefore, we develop an indoor simulation platform to conduct this experiment. The indoor simulation experiment platform is shown in Figure 4. The platform incorporates a numerically controlled lighting device and driving simulator to provide safe and controlled research scenarios.

The driving simulator consists of two components: simulator software and simulator hardware. The simulator’s software is UC-win/Road. The simulator’s hardware consists of a high-performance computer, three LCD screens displaying driving scene video information, a Logitech G29 steering wheel with an accelerator, and brake pedal kit. The numerically controlled lighting system comprised a thyristor module, two LED lamps, an ATmega328 controller, and control software. LabVIEW was used to design the software interface, which controls the rate of change of illuminance near the driver’s eyes (INDE). The numerically controlled lighting device aims to simulate the effect of street lamp illumination on the driver’s visual performance similar to INDE in the real-road experiment. According to our experimental data, the values of INDE were 0.5, 4, 8, and 12 lx when the values of E were 0, 12, 24, and 36 lx, respectively. We have provided a detailed introduction to the parameter calibration process of indoor simulation experiment platform in a previous published article. Therefore, we did not extensively discuss this part of the content in this article. Readers can refer to the published article ( 34 ).

Discussion

Investigation into Traffic Flow Parameters of Vehicles and VRUs

The lighting period of the selected intersection is 19:00–6:00, and the vehicle speed is in the range of 30–70 km/h and obeys uniform distribution. In the original lighting scheme, when vehicles approach the intersection, the lighting system sets the illuminance of 36 lx, and the illuminance will not be adjusted according to the traffic flow status.

According to traffic volumes of vehicles and VRUs at the intersection, the whole lighting period is divided into three time periods: 19:00–21:00, 21:00–23:00, and 23:00–6:00. The commonly used headway distribution models include: negative exponential distribution, shifted negative exponential distribution, log-normal distribution, Erlang distribution, and so forth. Among them, the log-normal distribution is widely used as the headway distribution model for highway continuous traffic flow or interrupted traffic flow, and it has a better fitting result for the headway time distribution of traffic flow in our research scenario ( 35 ). Therefore, the log-normal distribution is used to describe the headway of vehicles and VRUs, and the primary form of the probability density function of headway time distance $t_{h}$ is shown in Equation 30. The headway distribution parameters of vehicles and VRUs in different time periods are shown in Table 3.

f (t_{h}) = \frac{1}{t_{h} σ_{h} \sqrt{2 π}} \exp (- \frac{\ln t_{h} - μ_{h}}{2 {σ_{h}}^{2}})

(30)

where

$μ_{h}$ = the mean of log $t_{h}$ ,

$σ_{h}$ = the standard deviation of log $t_{h}$ , and

$t_{h}$ = headway time distance.

Table 3.

The Time Headway Distribution Parameters of Vehicles and Vulnerable Road Users (VRUs) under Different Traffic Flow Status

Time period	Road user	Traffic volumes (vph)	μ_h (s)	σ_h (s)
19:00–21:00	Vehicle	915	0.93	0.94
19:00–21:00	VRU	710	1.51	0.48
21:00–23:00	Vehicle	480	1.62	0.92
21:00–23:00	VRU	450	1.94	0.55
23:00–6:00	Vehicle	50	3.92	0.95
23:00–6:00	VRU	45	4.21	0.52

Note: $μ_{h}$ = the mean of log $t_{h}$ ; $σ_{h}$ = the standard deviation of log $t_{h}$ , $t_{h}$ = headway time distance; vph = vehicles per hour.

The average time $t_{b}$ for VRUs to reach the conflict zone from the stop line is 1.53 s, and the average time $t_{v}$ for vehicles to reach the conflict zone from the stop line is 2.12 s. We use linear growth function to express the relationship of deceleration with time t in the growth stage of deceleration, tht is, $f_{a} (t) = a_{b} t / t_{2}$ .

Calculation and Interpolation of P_A|C

First, the K-S test is used to test the normality of the driver’s visual recognition distance X of the VRU collected in the Section “Experimental design and platform construction” under different illuminances, vehicle speeds, and conflict event types. The test results indicate that all data obey the normal distribution (p < 0.05). Then, the mean value and standard deviation of X for each experimental scenario are calculated. Finally, we use Equation 25 to calculate the driver’s failure to brake in time for the VRUs that appear from the left and right sides of the intersection. Nevertheless, experimental data is restricted, so the interpolation method must be used to build the continuous function $f_{A | C}$ in Equation 27 based on the existing discrete data, so that this continuous curve passes through all the given discrete data points. Since the input of $f_{A | C}$ includes two variables, E and v, we adopt the most common bi-cubic interpolation algorithm in two-dimensional space to interpolate the probability of a vehicle not braking in time under a conflict. The algorithm has the following three characteristics ( 36 ):

1) The algorithm adopts a segmented interpolation method, and the data in each segmented interval can be expressed by the binary cubic equation shown in Equation 31:

P_{A | C} = f_{A | C} (E, v) = \sum_{p = 0}^{3} \sum_{q = 0}^{3} a_{pq} v^{p} E^{q}

(31)

2) Each discrete data must pass through the interpolation result.

3) The fitting curve is smooth, which means that the first and second derivative of $f_{A | C}$ in both E and v directions exist and are continuous.

The data analysis software MATLAB is used to interpolate the original data.

Influence of the Critical Gap on E_m

As an essential parameter in the intersection risk quantification model, the critical gap $t_{cv}$ is easily affected by factors including the road lighting and traffic flow status of intersections, which in turn affects the probability of conflict between vehicles and VRUs ( 37 ). Meanwhile, probability of conflict between vehicles and VRUs is one of the parameters for calculating the optimal illumination $E_{m}$ of street lights at intersections. Therefore, a feedback effect is established between the input $t_{cv}$ and output $E_{m}$ in the lighting system, susceptible to external environmental influence, resulting in issues of illumination oscillation and time delay. To solve the above problems, it is necessary to analyze the influence of $t_{cv}$ on $E_{m}$ and select the appropriate fixed value of $t_{cv}$ as input of the lighting system. Previous studies have shown that, when a vehicle passes through straight-going VRUs, the range of vehicle drivers’ $t_{cv}$ is 2.5–7 s ( 38 ). We will analyze the effect of $t_{cv}$ on $E_{m}$ in this range, and the $t_{cv}$ is taken as an independent variable. The values of $E_{m}$ under the three time periods when the vehicle speed is 30–70 km/h are calculated. The relationship between $E_{m}$ and $t_{cv}$ under different time periods and vehicle speeds can be expressed by the linear model shown in Equation (32), and the fitting degrees R² are all greater than 0.9:

E_{m} = K t_{cv} + B

(32)

The fitting results of the linear model show that, when the vehicle speed is lower than 50 km/h, the change rate K of $E_{m}$ with $t_{cv}$ is 0 lx·s⁻¹, indicating that the calculation result of $E_{m}$ is not affected by $t_{cv}$ . When the range of vehicle speed is 50–70 km/h, the range of K values is 0–0.106 lx/s, which demonstrates that the influence of $t_{cv}$ on $E_{m}$ is negligible. Therefore, when the lighting environment of the intersection changes, the influence of lighting factors on $t_{cv}$ can be ignored, and $t_{cv}$ in the model is taken as a fixed value of 7 s to ensure that the calculated $E_{m}$ can guarantee driving safety in the entire range of $t_{cv}$ .

Method Evaluation

According to the time headway distribution parameters of vehicles and VRUs in Table 3, we calculate the optimal illuminance $E_{m}$ at different vehicle speeds (30–70 km/h) in three time periods, as shown in Table 4.

Table 4.

The Optimal Illuminance (E_m) at Different Vehicle Speeds (V) in Three Time Periods

V (km/h)	$E_{m}$ (lx)
V (km/h)	19:00–21:00	21:00–23:00	23:00–6:00
30	20.00	20.00	20.00
35	20.00	20.00	20.00
40	20.00	20.00	20.00
45	20.00	20.00	20.00
50	20.00	20.00	20.00
55	20.41	21.43	20.83
60	22.92	23.93	24.09
65	25.94	26.87	27.10
70	31.24	32.00	31.75

It can be seen from Table 4 that, in the three time periods, the average values of $E_{m}$ at different vehicle speeds are 25.13, 26.06, and 25.94 lx, respectively, which shows that, when the traffic volume is low, as the traffic volume increases, the illuminance gradually increases. This result is partly because, when traffic volume increases, so does the risk of accidents, making it necessary to improve illumination to ensure driving safety. However, as the traffic volume continues to increase, the density of the vehicles increases. In this case, the vehicles’ headlights have a more significant impact on the ambient lighting. When the traffic volume of vehicles is large, the probability of accidents between vehicles and VRUs decreases; therefore, the illuminance of street lights shows a downward trend.

According to the original lighting scheme and the optimized lighting scheme in Table 4, Equation 1 is used to calculate the power $P_{o}$ of the original lighting scheme and the power $P_{\min}$ of the optimized lighting scheme. Then, Equation 33 is used to calculate the ESR of the lighting system. The values of ESR corresponding to different vehicle speeds in the three time periods are shown in Table 5.

ESR = (1 - \frac{P_{min}}{P_{o}}) \times 100 %

(33)

To compare the energy-saving effect of the optimized lighting scheme with the original lighting scheme more intuitively, the average energy-saving rate (AESR) and the daily average energy-saving rate (DAESR) of the lighting system are calculated by using the data in Table 5.

Table 5.

Values of Energy-Saving Rate (ESR) Corresponding to Different Vehicle Speeds (V) in Three Time Periods

V (km/h)	ESR (%)
V (km/h)	19:00–21:00	21:00–23:00	23:00–6:00
30	44.44	44.44	44.44
35	44.44	44.44	44.44
40	44.44	44.44	44.44
45	44.44	44.44	44.44
50	44.44	44.44	44.44
55	40.31	40.48	42.15
60	36.34	33.52	33.09
65	27.94	25.35	24.74
70	13.21	11.11	11.80

According to the values of ESR in Table 5, the AESR and DAESR were calculated over the speed range 30–70 km/h during the three time periods. It can be obtained by the calculation that the AESRs of the optimized lighting scheme in the three periods are 38.11%, 36.96%, and 37.11% respectively, and the DAESR is 37.27%. The results indicate that the method proposed in this paper can effectively reduce the power consumption of the intersection lighting system to ensure traffic safety. In addition, according to Table 5, the energy-saving effect of the lighting scheme grows steadily as the vehicles’ speeds decrease. However, when the vehicle speed is less than 50 km/h, the change in speed has no impact on the power consumption on illumination. Therefore, limiting the maximum nighttime vehicle speed at intersections to 50 km/h can maximize the ESR of the lighting system while ensuring traffic efficiency. To ensure that vehicles’ travel speed can be regulated below 50 km/h, it is necessary to inform drivers of the speed limit at intersections by placing speed limit signs ahead of the approaching intersection. In this way, even if the travel speed is often higher than usual when traffic volume is low at night, they should slow down as they approach the intersection. Usually, the speed limit signs are placed approximately 30–100 m ahead of the intersection. Because of the need to set the speed limit at the intersection to 50 km/h, the speed limit sign should be positioned about 50 m ahead of the intersection.

Conclusions

In this paper, a dynamic illumination method for rural highway intersections with traffic flow changes was developed to improve nighttime traffic safety. First, the safety design principles of the lighting scheme were formulated for minimum illuminance, vehicle speed, and traffic risk at intersections. In addition, a risk quantification model for highway intersections was developed incorporating gap acceptance theory and a braking safety distance model. Second, according to safety design principles and the traffic risk quantification model, a calculation method for optimal illuminance at intersections was designed. Finally, a case study was conducted in a simulated environment. The following conclusions can be drawn from this study.

For the intersection scenario studied in this paper, the range of vehicle drivers’ critical gap $t_{cv}$ is 2.5–7 s when the vehicle conflicts with VRUs. The range of optimal illuminance $E_{m}$ change rate with $t_{cv}$ is 0–0.106 lx/s within the range of $t_{cv}$ , indicating that $t_{cv}$ has a very negligible impact on $E_{m}$ . Therefore, when the lighting of intersections changes, the impact of critical gap on illuminance can be ignored, and $t_{cv}$ in the illuminance optimization model needs to be taken as a constant value of 7 s to ensure that the calculated $E_{m}$ can maintain traffic safety.

According to traffic volumes of vehicles and VRUs at the intersection, the whole lighting period is divided into three periods: 19:00–21:00, 21:00–23:00, and 23:00–6:00. The AESRs of optimal illuminance are 38.11%, 36.96%, and 37.11% for the three periods, respectively. The DAESR is 37.27%, suggesting that the illuminance scheme can provide traffic safety while reducing the energy consumption of lighting system at intersections.

With the decrease of vehicle speed, the ESR of optimal lighting scheme increases. However, when the speed is below 50 km/h, the change of speed has little effect on the energy consumption of the lighting system. Therefore, limiting the maximum nighttime vehicle speed at intersections to 50 km/h can maximize the ESR of the lighting system while maintaining traffic efficiency.

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: H. Li, L. Wang; data collection: H. Li; analysis and interpretation of results: Y. Bie; draft manuscript preparation: H. Li. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported in part by the National Natural Science Foundation of China under Grant 71971097, in part by the National Key R&D Program of China 52131203, in part by the Youth Program of National Natural Science Foundation of China 52002143, and in part by the National Natural Science Foundation of China under Grant 52172385.

Data Accessibility Statement

Some or all data, models, and code generated or used during the study are available from the corresponding author by request.

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Dynamic Illumination Method for Rural Highway Intersections with Traffic Flow Changes

Abstract

Keywords

Literature Review

Objectives and Contributions

Illuminance Optimization Model for Intersections

Safety Design Principles of Lighting Scheme at Intersections

Calculation Method for Optimal Illuminance

Probability of Conflict between Vehicles and VRUs (PC) Modeling

Probability of Driver Failing to Brake in Time under Conflict (PA|C) Modeling

Solution Algorithm for Optimal Illuminance (Em)

Experimental Design and Platform Construction

Experimental Design

Experimental Platform Construction

Discussion

Investigation into Traffic Flow Parameters of Vehicles and VRUs

Calculation and Interpolation of PA|C

Influence of the Critical Gap on Em

Method Evaluation

Conclusions

Footnotes

Author Contributions

Declaration of Conflicting Interests

Funding

Data Accessibility Statement

References

Probability of Conflict between Vehicles and VRUs (P_C) Modeling

Probability of Driver Failing to Brake in Time under Conflict (P_A|C) Modeling

Solution Algorithm for Optimal Illuminance (E_m)

Calculation and Interpolation of P_A|C

Influence of the Critical Gap on E_m