Dynamic combination control of phase movement at signal intersection under the automated vehicle environment

Abstract

With the continuous perfection of the technology of automated vehicles (AV), data exchange can be conveniently carried out between different vehicles and infrastructures, which makes it easier to collect different types of traffic parameters. Therefore, under AV environment, the vehicle status can be determined to obtain the periodic arrival rate of movements and a more efficient control strategy can be designed. The combination styles of phase movement (PM), an important factor of the signal control, will also become more complicated for intersection signal control. The current methods about the PM combination styles only considered two kinds of movement combination styles, and cannot get the extensive phase combination (PC) schemes in AV environment. This paper documents a new PM combination method by fractionalized movement compatibility relations, and uses discrete mathematics to calculate overall PC schemes. Then, a PM dynamic combination control method is proposed to optimize cyclically signal control. The analysis results of numerical tests showed that the average vehicle of the proposed method is reduced by 6.9 % and 14.5 % for 20 signal cycles, respectively, and the total throughput can be increased by 4.3% and 7.8%, respectively, compared with the dynamic timing control mode and the fixed control mode. Results show that the proposed method could significantly improve intersection control effectiveness.

Keywords

Intersection control movements compatible phase combination schemes phase movement dynamic combination vehicle delay

1 Introduction

With the rapidly growing of urban traffic demand, traffic congestion has become a crucial problem especially in urban locations, which causing many serious problems, such as waste of time, excessive fuel consumption, and air pollution [1]. To alleviate these problems, methods such as widening existing roads and constructing new roads have been employed. However, these methods cannot satisfy the increasement of traffic demand. Signalized intersections have been proved as the bottlenecks in urban traffic networks. Many researchers pointed out that signal timing optimization at signalized intersections can effectively reduce urban traffic congestion and improve traffic efficiency [2, 3].

The development of traffic signal control has experienced three stages, that are, fixed-time control, actuated control and adaptive traffic control [4, 5]. Fixed-time control utilizes historical traffic data or assume that the traffic flow of intersection movements is fixed arrival patterns to create timing plans for different time of a day to serve a certain time period. However, this strategy cannot without capability of responding to short-term traffic demand and pattern changes. Actuated control employs point detectors to modify a fixed timing plan, such as, occasionally skips a phase if a vehicle is present or shortens a phase split when vehicles are not being served. The researches about the actuated control mainly concentrate on the optimal max-minimum green time and unit extension time, its phase sequence is fixed [6]. Adaptive control attempts to adjust the signal timing to accommodate the changes of traffic volume. Since it has more flexibility on signal operations than actuated control, adaptive control offers more potential improvements [7]. Currently, researches on adaptive control mainly focus on the real-time allocation of green time [8, 9]. In summary, the most research efforts on traffic control focused on the signal timing decision and attempted to serve the fluctuation of traffic flow by adjusting the timing scheme.

However, for control scheme, it includes phase sequence and signal timing scheme. Signal timing scheme optimization is not the only method for enhancing the control efficiency of an intersection, optimal phase sequence is another means, but few studiers concentrate it. Zheng [10] used four parameters (phase sequence, minimum green, unit extension and maximum green) to improve the performance of signal control. And the phase sequence was determined by optimizing the phase splits. Feng [11] proposed an algorithm to optimize the phase sequence and phase duration by minimization of total vehicle delay and minimization of queue length. They considered the phase sequence to optimize control scheme, but the phase sequence was based on the ring barrier controller structure (RBCS), as shown in Fig. 1.

Fig. 1

Ring barrier controller structure.

For intersections, there are compatible phases and conflict zones between different movements. The compatible phase means that there is no conflict point between two movements. According to the right rules, the left turn lane is located at the left of the through lane. The types of compatible phases include: diffluence compatible, opposite compatible, and confluence compatible. the diffluence compatible indicates that the movements are non-conflict and locate in the same approach. The opposite compatible indicates that the movements are non-conflict and locate in the opposite approach. The confluence compatible indicates that the movements are non-conflict, locate in different approach, and flow into the same exit at the end. The RBCS perfectly combines the diffluence and opposite compatible relation of the movements, achieves flexible movement lapping between different phases. Under unautomated vehicles environment, the vehicles of confluence relation movement flowing into the same exit of intersection may lead to potential traffic problems, so the RBCS do not consider the confluence compatible.

However, with the development of automated technology, automated vehicles could use mechatronics technology to drive autonomously and safely according to the predetermined lane trajectory [12, 13], So the confluence relation between movements can be fractionalized as compatible relation under some situation. It shows that the PC schemes obtained by RBCS would not fit the control mode of the AV environment.

In current research about the AV environment, the researches mainly focus on vehicle data collection, information transmission and traffic flow characteristics at intersection to optimize signal timing [14, 15]. Therefore, this paper uses discrete mathematics to calculate overall PC schemes by fractionizing movement compatibility relations, and determines the vehicle status to obtain the periodic arrival rate of the approach movements based on the AVs data. Finally, the PM dynamic combination method is established to further optimize the signal control cyclically.

The remainder of this paper is organized as follows: Section 2 introduces how to determine the vehicle status of movements in an AV environment and how to determine the PC scheme. The dynamic combination method of PM is presented in Section 3. Section 4 studies the PM dynamic combination method. Evaluation of the proposed model performance is shown in Section 5 through the numerical tests. Section 6 concludes the current study and further work.

2 Determining the vehicle status of movements under AV environment

To obtain the periodic arrival rate of movements in an intersection approach, the location and speed of each vehicle near the intersection needs to be determined from available automated vehicles. Generally speaking, the road segment of one movement near an intersection approach can be divided into three regions: queuing region, slow-down region, and free-flow region (see Fig. 2). Due to the vehicles on the arterial road are limited by the signal and other vehicles, it is difficult to remains in a free flow state for them. Therefore, there are only two regions left need to be considered for one movement. In this paper, the following two different algorithms are applied to determine the location and speed of each unequipped vehicle [11].

Fig. 2

Three regions of movements.

For the queuing region, the speed of each vehicle is assumed to be zero; thus, only the location parameter needs to be determined. Under AV environment, the locations and stop time of every stopped vehicle in queue can be directly acquired, and according to the location of the vehicle in the queue, the number of vehicles in the queue region N_que can be calculated. So, the status of vehicles in queue region can be determined.

For the slow-down region, the Wiedemann’s car following model divides the state of a moving vehicle into four stages: free flow, following, closing and emergency. Under an automated vehicles (AV) environment, the relative position, speed and acceleration of vehicles can be obtained. And the state of vehicles can be determined according to these parameters, the evaluation process is shown in Fig. 3.

Fig. 3

Vehicle state determination process.

Δv is relative velocity, Δu is headway, the meaning and values of each parameter are listed in Table 1.

Table 1

Vehicle state determination parameters definition

Notation	Description	Value used	Unit
OPDV	increasing speed difference	-2.25SDV	m/s
ABX	desired minimum headway at low Δv	AX+BX	m
SDV	decreasing speed difference	((Δu-AX)/CX)²	m/s
SDX	maximum flowing distance	AX+EXBX	m
AX	Minimum headway	6.56	m
BX	Calibration factor	2.5 $\sqrt{v}$	-
CX	Calibration factor	40	-
EX	Calibration factor	1.5	-
v	Min speed of vehicle and leader	-	m/s

Vehicles in the slow-down region are not in a free-flow state, and the emergency state also rarely occurs. Following and closing states are more common states for vehicles in the slow-down region. According to Fig. 3, the state of all vehicles can be determined, thus the number of all vehicles of movement σ_ij in the slow-down region $N_{ij}^{slo}$ and the slow-down region length Γ_slo can be calculated. Therefore, the density of movement σ_ij in the slow-down region $k_{ij}^{slo} {= N}_{ij}^{slo} / γ_{ij}^{slo}$ . Based on the basic relationship among three tffic flow pameters, the arrival flow rates of movement σ_ij in the slow-down region $Q_{ij}^{slo} {= k}_{ij}^{slo} {\bar{v}}_{ij}^{slo}$ , ${\bar{v}}_{ij}^{slo}$ is the average vocity oment σ_ij in the slow-down region.

Therefore, according to the available automated vehicle data, the vehicle status of movement, include the location and speed of each vehicle in different regions can be determined and related parameters of all vehicles can be collected in real time and accurately. In summary, the number of vehicle movements σ_ij in the queue region $N_{ij}^{que}$ and the arrival flow rates of movement σ_ij in the slow-down region $Q_{ij}^{slo}$ can be determined precisely, and the dynamic combination control of PM for a signal intersection will have the data resources.

3 Phase combination scheme

To facilitate model presentation, notations used hereafter are summarized in Table 2.

Table 2
Parameter symbols and definitions

Sets, parameters and variables

i Intersection approach, i = 1, 2, 3, 4

σ _ij Intersection Movement

M Movements’ set of the intersection

e _ij The presence of the movement σ_ij, e_ij = 0 or 1

Ψ The set of the combination of diffluence movements

Ψ _α The movement set of diffluence relation

φ _α Describe the compatibility relation of movements in Ψ_α, φ_α or 1

Φ The set of the combination of opposite movements

Φ _β The movement set of opposite relation

ϕ _β Describe the compatibility relation of movements in Φ_β, ϕ_β = 0 or 1

Θ The set of the combination of confluence movements

Θ _γ The movement set of confluence relation

θ _γ Describe the compatibility relation of movements in Θ_γ, θ_γ = 0 or 1

Λ The set of the combination of compatible movements

υ The identifier of the phase combination (PC) scheme

m _υ The phase number of the corresponding PC scheme

P _υ The υ th PC scheme

τ Non zero natural number, τ ∈ N⁺

δα, ϖ The element identifiers of the array Char p_v [m_v] , δ, α, ϖ ∈ N

w The number of the feasible phase combination (FPC) scheme

m _w The phase number of the corresponding FPC scheme

F _w The wth FPC scheme

h Cycle

${\bar{D}}_{ij}^{w} (h)$ The movement σ_ij average delay of the F_w in the hth cycle

$ξ_{ij}^{w} (h)$ . Thee movement σ_ij volume-to-capacity ratio of the F_w in the hth cycle

${Ge}_{ij}^{w} (h)$ e movement σ_ij effective green time of the F_w in the hth cycle

C^w (h) The cycle time length of the F_w in the hth cycle

T _h The calculation time of the hth cycle,

S _ij The saturation flow rate of the movement σ_ij

A The yellow time

$G_{ij}^{w} (h)$ e movement σ_ij green time of the F_w in the hth cycle

$R_{1, ij}^{w} (h)$ Front-red time

$R_{2, ij}^{w} (h)$ Back-red time

$Ω_{ij}^{w} (h)$ The movement σ_ij traffic capacity of the F_w in the hth cycle

N _ij The lane number of movement σ_ij

$S_{ij}^{\ln}$ One lane saturation flow rate of the movement σ_ij

N _i The number of movements in arm i

G _min The minimal green time

G _max The maximal green time

l The phase

T^w (h, l) The phase l length for the F_w in the hth cycle

$η_{ij}^{w}$ The number of elements that contain movement σ_ij in F_w,

k,k The sorting of the phases that contain movement σ_ij,

$x_{ij}^{w} (k)$ The identifier of the element that contains movement σ_ij in Char f_w [m_w]

$l_{ij}^{w} (k)$ The identifier of the phase that contains movement σ_ij in F_w.

${Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$ The effective green time of phase $l_{ij}^{w} (k)$ or the F_w in the hth cycle

$T^{w} {(h, l}_{ij}^{w} (k))$ The phase $l_{ij}^{w} (k)$ length for the F_w in the hth cycle

T _s The limit time, which is the acceptable waiting time for a driver

Sets, parameters and variables
i	Intersection approach, i = 1, 2, 3, 4
σ _ij	Intersection Movement
M	Movements’ set of the intersection
e _ij	The presence of the movement σ_ij, e_ij = 0 or 1
Ψ	The set of the combination of diffluence movements
Ψ _α	The movement set of diffluence relation
φ _α	Describe the compatibility relation of movements in Ψ_α, φ_α or 1
Φ	The set of the combination of opposite movements
Φ _β	The movement set of opposite relation
ϕ _β	Describe the compatibility relation of movements in Φ_β, ϕ_β = 0 or 1
Θ	The set of the combination of confluence movements
Θ _γ	The movement set of confluence relation
θ _γ	Describe the compatibility relation of movements in Θ_γ, θ_γ = 0 or 1
Λ	The set of the combination of compatible movements
υ	The identifier of the phase combination (PC) scheme
m _υ	The phase number of the corresponding PC scheme
P _υ	The υ th PC scheme
τ	Non zero natural number, τ ∈ N⁺
δα, ϖ	The element identifiers of the array Char p_v [m_v] , δ, α, ϖ ∈ N
w	The number of the feasible phase combination (FPC) scheme
m _w	The phase number of the corresponding FPC scheme
F _w	The wth FPC scheme
h	Cycle
${\bar{D}}_{ij}^{w} (h)$	The movement σ_ij average delay of the F_w in the hth cycle
$ξ_{ij}^{w} (h)$ .	Thee movement σ_ij volume-to-capacity ratio of the F_w in the hth cycle
${Ge}_{ij}^{w} (h)$	e movement σ_ij effective green time of the F_w in the hth cycle
C^w (h)	The cycle time length of the F_w in the hth cycle
T _h	The calculation time of the hth cycle,
S _ij	The saturation flow rate of the movement σ_ij
A	The yellow time
$G_{ij}^{w} (h)$	e movement σ_ij green time of the F_w in the hth cycle
$R_{1, ij}^{w} (h)$	Front-red time
$R_{2, ij}^{w} (h)$	Back-red time
$Ω_{ij}^{w} (h)$	The movement σ_ij traffic capacity of the F_w in the hth cycle
N _ij	The lane number of movement σ_ij
$S_{ij}^{\ln}$	One lane saturation flow rate of the movement σ_ij
N _i	The number of movements in arm i
G _min	The minimal green time
G _max	The maximal green time
l	The phase
T^w (h, l)	The phase l length for the F_w in the hth cycle
$η_{ij}^{w}$	The number of elements that contain movement σ_ij in F_w,
k,k	The sorting of the phases that contain movement σ_ij,
$x_{ij}^{w} (k)$	The identifier of the element that contains movement σ_ij in Char f_w [m_w]
$l_{ij}^{w} (k)$	The identifier of the phase that contains movement σ_ij in F_w.
${Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$	The effective green time of phase $l_{ij}^{w} (k)$ or the F_w in the hth cycle
$T^{w} {(h, l}_{ij}^{w} (k))$	The phase $l_{ij}^{w} (k)$ length for the F_w in the hth cycle
T _s	The limit time, which is the acceptable waiting time for a driver

For intersection, the corresponding movement σ_ij (i = 1, 2, 3, 4 ; j = 1, 2, 3) symbol of the approach i is showed in Fig. 4. Generally speaking, the intersection sets exclusive right-turn lanes for right turn movement through channelization according to the right rules, so the right turn movement is not considered. When j = 2, σ_i2 is the movement generated by a shared lane that represented a movement of two directions; other symbols represented movement of a single direction.

Fig. 4

Movement symbols.

The intersection in Fig. 4 may not include all movements, e_ij represents if a movement is included in the movement set of intersection (M). $e_{ij} = {\begin{matrix} {0 σ}_{ij} \notin M \\ {1 σ}_{ij} \in M \end{matrix}$ (1)

The diffluence movements means the two movements have the same intersection approach. Thus, Ψ is defined as a set of the combination of diffluence movements. ${Ψ = {Ψ;}_{α}, α \in N^{+}}$ (2)

N⁺ is non-zero natural numbers, for intersections, α is an approach of the intersection. Ψ_α is a set of all movements in one approach of the intersection. $Ψ_{α} = {σ_{α j} {| σ}_{α j} \in M}$ (3)

To better describe the compatibility movements in Ψ_α, the φ_α parameter is adopted. $φ_{a} = {\begin{matrix} 0 & incompatible \\ 1 & compatiale \end{matrix}$ (4)

When the movements in Ψ_α are compatible, φ_α = 1 otherwise φ_α = 0. Ψ is calculated according to the definition of the movement compatibility relation, and all movements in Ψ_α are compatible when the left turn lane is located on the left side of the through lane. Thus, φ_α ≡ 1, ∀ α.

Tosite movement means that two opposite movements are located in two contrary approaches of an intersection. Φ is defined as the set of the combination of opposite movements. $Φ = {Φ_{β}, β \in N^{+}}$ (5)

The movements in Φ_β are considered as a compatible relation. However, when aent iΦ_β shares a lane with another movement, the movements in Φ_β are considered to be an incompatibelation. $φ_{β} = {\begin{matrix} 0 & σ_{ij | e_{i 2} = 1} \in Φ_{β} \\ 1 & σ_{ij | e_{i 2} = 1} \notin Φ_{β} \end{matrix} \forall j$ (6)

When the movements in Φ_βare compatible, φ_β = 1 otherwise φ_β = 0. σ_{ij|e_i2=1} is a movement in approach i with a shared lane. For example, if approach 1 has the shared lane (i = 1, e₁₂ = 1), σ_1j|e₁₂=1 = σ_1j.

For the confluence movement relation, it means that the two movements are located in different approach of intersection and flow into the same exit of intersection at the end. Under unautomated vehicle environment, the confluence movement is usually considered to be a conflict relationship. Thus, θ_γ ≡ 0. However, under AV environment, the vehicles can collect previously unobtainable and high-fidelity traffic data, and the drivers can receive the predetermined lane trajectory to perform driving functions safely [16]. So, the confluence relation between movements can be fractionalized as compatible relation under some situation.

Thus, Θ is defined as a combination of confluence movements. $Θ = {Θ_{γ}, γ \in N^{+}}$ (7)

Θ_γ is a combination of two confluence movements, and γ is the number of the se exits of two confluence movements. To describe the compatibility of movements in Θ_γ, the θ_γ parameter is introduced. $θ_{γ} = {\begin{matrix} 0 & incompatible \\ 1 & compatible \end{matrix}$ (8)

When the movements in Θ_γ are compatible, θ_γ = 1 otherwise θ_γ = 0.

If σ_{ij|e_i2=1} ∈ Θ_γ, then θ_γ = 0, ∀ γ. However, if σ_{ij|e_i2=1} ∉ Θ_γ, the two movements ie not always compatible, and other conditions need to be considered. If the two movements in Θ_γ are compatible, the following two conditions must be simultaneously satisfied. ➀ The two movements in Θ_γ do not share lanes with other movements. ➁ The total lane number occupied by two confluence movements at intersection should match with the lane number of the corresponding exit.

For condition ➀. $σ_{{ij | e}_{i 2} = 1} \notin Θ_{γ}$ (9)

For condition ➁, the vehicles are assumed to pass through the intersection according to driving rules (yield and no changing lanes). Therefore. $N_{Θ_{γ} (1)} {+ N}_{Θ_{γ} (2)} ⩽ N_{γ}$ (10)N_{θ_γ(1)} and N_{θ_γ(2)} are the corresponding lane numbers of two movements in Θ_γ. N_γ is the lane number of the corresponding exit γ of two movements in Θ_γ.

To better understand Equation (10), the instructions are presented in Fig. 5(a) - (c), which assumes that the two movements σ_a and σ_b are the confluence relation, N_{σ
_a} and N_{σ
_b} are the lane numbers of the movements σ_a and σ_b, respectively, and N_a&b is the number of the two movements (σ_a and σ_b) that corresponds to the exit lane. In Fig. 5(a), N_{σ
_a} = 1, N_{σ
_b} = 1, N_a&b = 3, and N_{σ
_a} + N_{σ
_b} < N_a&b. Thus, σ_a and are compable aording to Equation (10). In Fig. 5(b), N_{σ
_a} = 2, N_{σ
_b} = 1, N_a&b = 3, and N_{σ
_a} + N_{σ
_b} = N_a&b. Thus, σ_α and σ_b are also compatible according to Equation (10). In Fig. 5(c), N_{σ
_a} = 3, N_{σ
_b} = 1, N_a&b = 3, and N_{σ
_a} + N_{σ
_b} > N_a&b. Therefore, a conflict point exists between σ_α and σ_b; thus, σ_α and σ_b are incompatible.

Thus, if movements in Θ_γ are compatible, Equation (11) should be satisfied, and θ_γ = 1. ${\begin{matrix} σ_{{ij | e}_{i 2} = 1} \notin Θ_{γ} \\ N_{Θ_{γ} (1)} {+ N}_{Θ_{γ} (2)} ⩽ N_{γ} \end{matrix}$ (11)

Fig. 5

Compatibility of confluent movements.

The compatible movement combination can be obtained by judging the compatibility of the movements as ${Λ = {Ψ;}_{α | φ_{α} = 1}} \cup {{Φ}_{β | ψ_{β} = 1}} \cup {Θ_{γ | θ_{γ} = 1}}$ (12)

Λ is the set of the combination of compatible movements, Ψ_{α|φ_α=1} is the combination of φ_α = 1 that corresponds to Ψ_α, and Φ_{β|ψ_β=1} and Θ_{γ|θ_γ=1} are similar.

The purpose of signal control is to ensure the vehicles crossing the intersection safely by adjusting the signal scheme. The movements in one phase must have a non-conflict relation without considering a permission phase. According to the definition of the movement compatibility relation above, each combination ompatible movements can be employed as one phase. Each element of the Λ can be as one phas t number of usable signal phases at intersections are the number of Λ elements. The intersection PC scheme is composed of some usable signal phases that are arranged in a certain order according to phase sequence requirement. Thus, an array is introduced to describe the PC scheme.

The number of elements in the Λ represents the usable signal phase number of intersections, and a series of PC schemes can be obtained by randomly sampling each element. Each PC scheme is expressed by the character array, namely, Charp_υ [m_υ], as follows: $P_{υ} {= {Charp}_{υ} {[m}_{υ}], υ \in N^{+} {andm}_{υ} \in N^{+}}$ (13)

m_υ is the phase number of the corresponding PC scheme, that is, the elements number of the array Charp_υ [m_υ].

Furthermore, $\max υ = \sum_{τ = 1}^{| Λ |} C_{| Λ |}^{τ} A_{τ}^{τ}, τ \in N^{+}$ (14) $\max m_{υ} = | Λ |$ (15)

When all signal PC schemes have been determined, some PC schemes still cannot satisfy the phase setting requirements, therefore, obtaining the feasible phase combination (FPC) scheme is pretty necessary. The specific setting requirements are afollows: ➀ The PC scheme must contain all movements in the intersection. ➁ If one movement occupies multiple phases (i.e., the lapping movement), these phases must be adjacent to each other. The setting of the condition ➁ ensures the continuous release of movements during a cycle. The FPC scheme is the PC scheme that satisfies these two conditions.

Condition ➀ $\cup_{δ = 0}^{m_{υ} - 1} p_{υ} [δ] = M δ \in N .$ (16)

Condition ➁

The array Charp_υ [m_υ] with two situations satisfies Condition ➁.

1) Common movements among other elements does not exist, with the exception of the adjacent elements. $[ϱ] \cap p_{υ} [ϖ] = \emptyset \forall | ϱ - ϖ | ⩾ 2$ (17)

2) In addition, the adjacent elements permit the existence of common movements. Thus, if the two elements separated by one element have common movements, it needs to be satisfied that there is the same movement between the two elements and the intermediate elements is necessary, and there is not the same movement should exist between other elements. ${\begin{matrix} p_{υ} [ϱ] \cap p_{υ} [ϖ] \neq \emptyset \\ p_{υ} [ϱ] \cap p_{υ} [\frac{ϱ + ϖ}{2}] \cap p_{υ} [ϖ] \neq \emptyset \end{matrix} \forall | ϱ - ϖ | = 2$ (18a) Additionally, $p_{υ} [ϱ] \cap p_{υ} [ϖ] = \emptyset \forall | ϱ - ϖ | ⩾ 3$ (18b)

In Equations (17)-(18b), ϱ ≠ ϖ, ϱ and ϖ are the element identifiers of Charp_υ [m_υ]; thus, ϱ ∈ N, ϖ ∈ N and ϱ ∈ [0, m_υ - 1], ϖ ∈ [0, m_υ - 1].

Thus, the array Charp_υ [m_υ] of the signal PC scheme must simultaneously satisfy Equations (16) and (17) or Equations (16), (18a) and (18b) to be an FPC scheme.

APC schemes of the intersection can be determined according to this method, which can be expressed as $F_{w} {= {Charf}_{w} {[m}_{w}], w \in N^{+} {and m}_{w} \in N^{+}}$ (19)

m_w is the phase number of the corresponding FPC scheme or the elements number of the array Char f_w [m_w].

The number of FPC schemes obtained by this method is more comprehensive, includes all phase combinations that the RBCS can obtain and perfectly considers the movement lapping between two signal phases.

4 PM Dynamic combination method

After PC schemes are determined, a PM dynamic combination method should be presented to optimize cyclically the control scheme by combining the vehicle status of movements. Firstly, the beginning of each cycle is selected as the calculation time. It should be assumed that the time of calculation at h th cycle is T_h, according to the second section, the number of movements σ_ij of queued vehicles and the follow-up arrival rate of the movements of σ_ij at T_h can be acquired, they are denoted as $N_{ij}^{que} (h)$ and $Q_{ij}^{slo} (h)$ . Then, the optimal model in h th cycle needs to be established by combining with the PC schemes to achieve a PM dynamic combination.

Wu [17] pointed out that the average delay and capacity can be used to evaluate a signal scheme. Therefore, the average delay and traffic capacity are also used in this paper to test the new optimization model of signal control scheme. PM dynamic combination is considered but without the movement priority. The calculation model in the Highway Capacity Manual [18] takes the lane group as a research unit, which divides each approach of the intersection into several lane groups. According to the concept of lane group, a relationship exists between the movement and the lane group. That is, when the approach of the intersection does not have the shared lane, each movement is a lane group. Otherwise, all movements in an approach are one lane group.

According to the vehicle of approach located region, the vehicle status of movements σ_ij be determined, including $N_{ij}^{que} (h)$ and $Q_{ij}^{slo} (h)$ . Assumed traffic flow arrivals and dissipation process are a steady-state patterns, and traffic flows are unsaturated condition. So the cumulative arrival condition of vehicle for movements can be illustrated as shown in Fig. 6.

Fig. 6

Cumulative arrival condition of vehicle for movement

In Fig. 6, when the phase of the movement is in the middle of the cycle, the red time of the movement is divided into two parts—front-red time and back-red time which are denoted as $R_{1, ij}^{w} (h)$ and $R_{2, ij}^{w} (h)$ , respectively. When the phase of the movement is the first phase, $R_{1, ij}^{w} (h) = 0$ or the last phase, $R_{2, ij}^{w} (h) = 0$ .

Therefore, the delay can be determined by summing the area of the trapezoids that compose the polygon BCDE in Fig. 6. The area of a given trapezoid or triangle can be determined by the queue at the start of the cycle $N_{ij}^{que} (h)$ , the follow-up arrival rate $Q_{ij}^{slo} (h)$ , and some time parameters. Thus, the total delay can be computed by association with a given trapezoid or triangle as following $\begin{matrix} D_{ij}^{w} (h) = 0 {. 5 N}_{ij}^{que} (h) \cdot R_{2, ij}^{w} (h - 1) + 0 {. 5 [N}_{ij}^{que} (h) + \\ {(N}_{ij}^{que} {(h) + Q}_{ij}^{slo} (h) \cdot {(R}_{1, ij}^{w} (h) + L))] \cdot {(R}_{1, ij}^{w} (h) + L) + \\ 0 {. 5 (N}_{ij}^{que} {(h) + Q}_{ij}^{slo} (h) \cdot {(R}_{1, ij}^{w} {(h) + L))}^{2} / {(S}_{ij} {- Q}_{ij}^{slo} (h)) \\ = 0 {. 5 [N}_{ij}^{que} (h) \cdot {(R}_{2, ij}^{w} {(h - 1) + 2 (R}_{1, ij}^{w} (h) + L)) \\ + Q_{ij}^{slo} (h) \cdot {(R}_{1, ij}^{w} {(h) + L)}^{2}] + 0 {. 5 (N}_{ij}^{que} (h) + Q_{ij}^{slo} (h) \\ \cdot (R_{1, ij}^{w} (h) + L))^{2} / (S_{ij} - Q_{ij}^{slo} (h)) \end{matrix}$ (20)

So the average delay of movement σ_ij for the F_w in the hth cycle can be calculated ${\bar{D}}_{ij}^{w} (h) = \frac{D_{1, ij}^{w} (h)}{N_{ij}^{que} {(h) + Q}_{ij}^{slo} (h) \cdot {(R}_{1, ij}^{w} (h) + L + {Ge}_{ij}^{w} (h))}$ (21)

The calculation model of the capacity is expressed as follows: $Ω_{ij}^{w} (h) = \frac{N_{ij} \cdot S_{ij}^{\ln} \cdot {Ge}_{ij}^{w} (h)}{C^{w} (h)}$ (22)

The average delay and traffic capacity model functions under the premise of a single movement as a lane group. An approach generally contains several movements; thus, the average delay and capacyf eh approach in the F_w during the hth cycle can also be calculated as

${\begin{matrix} {\bar{D}}_{i}^{w} (h) = \frac{\sum_{j = 1}^{N_{i}} {[N}_{ij}^{que} {(h) + Q}_{ij}^{slo} (h) \cdot {(R}_{1, ij}^{w} (h) + L + {Ge}_{ij}^{w} (h))] \cdot {\bar{D}}_{ij}^{w} (h)}{\sum_{j = 1}^{N_{i}} {[N}_{ij}^{que} {(h) + Q}_{ij}^{slo} (h) \cdot {(R}_{1, ij}^{w} (h) + L + {Ge}_{ij}^{w} (h))]} \\ Ω_{i}^{w} (h) = \sum_{j = 1}^{N_{i}} Ω_{ij}^{w} (h) \end{matrix}$ (23)

However, when the approach of the intersection has a shared lane, all movements in an approach will be considered as one lane group. The average delay and traffic capacity of a lane group can be also calculated accorng to this model.

After defined the average delay and capacity of every approach, the average delay and capacity of the entire intersection in the F_w during the hth cycle can be calculated as ${\begin{matrix} {\bar{D}}^{w} (h) = \frac{\sum_{i = 1}^{I} Q_{i} (h) \cdot {\bar{D}}_{i}^{w} (h)}{\sum_{i = 1}^{I} Q_{i} (h)} \\ Ω^{w} (h) = \sum_{i} Ω_{i}^{w} (h) \end{matrix}$ (24)

Where I is the number of approaches in the intersection, and Q_i (h) is the vehicle passing volume of the approach i during the hth cycle.

The average vehicle delay is expected to have a minimum value, whereas the traffic capacity is expected to have a maximum value. To achieve the PM dynamic combination, the multi-objective optimization problem needs to be transformed into a single objective optimization problem to optimize signal control scheme. The single objective function can be described as

$\begin{matrix} Min O^{w} (h) = {\bar{D}}^{w} (h) + \frac{3600}{Ω^{w} (h)} \\ = \frac{\sum_{i = 1}^{I} Q_{i} (h) \cdot {\bar{D}}_{i}^{w} (h)}{\sum_{i = 1}^{I} Q_{i} (h)} + \frac{3600}{\sum_{i} Ω_{i}^{w} (h)} \end{matrix}$ (25)

The variables in the objective function Equation (25) are the effective green time of the movement ${Ge}_{ij}^{w} (h)$ , which needs to satisfy the corresponding constraints of actual traffic situation. The following constraints are common restrictions for the variables of the objective function. $G_{ij}^{w} (h) ⩾ G_{\min}, G_{ij}^{w} (h) ⩽ G_{\max}$ (26)

$\begin{matrix} C^{w} (h) & = R_{1, ij}^{w} {(h) + L + Ge}_{ij}^{w} (h) {+ R}_{2, ij}^{w} (h) \\ = \sum_{l = 1}^{m_{w}} T^{w} (h, l) \end{matrix}$ (27) ${Ge}_{ij}^{w} (h) {= G}_{ij}^{w} (h) + A - - L$ (28) ${h, w, m}_{w}, l \in N^{+}$ (29)

Equation (26) requires that the green time for each movement should satisfy the minimal green time but not exceed the maximal green time. The sum of the red times and the green time for a single movement or the sum of the time duration for all phases at the intersection should be equal to the cycle length, as illustrated by Equation (27). The relationship between the effective green time and the display green time is expressed in Equation (28).

According to the above FPC schemes, the following symbols are introduced to conveniently itrate the relations among the signal control parameters: $η_{ij}^{w}$ is the number of elements that contain ment σ_ij in F_w, k is the sorting of the phases that contain movement σ_ij, $x_{ij}^{w} (k)$ is the identifier of the element that contains movement σ_ij in Char f_w [m_w], and $l_{ij}^{w} (k)$ is the identifier of the phase that contains movement σ_ij in F_w. The following equations illustrate the correspondence between different symbols. $k = {\begin{matrix} 1 & if η_{ij}^{w} = 1 \\ [1, 2] & if η_{ij}^{w} = 2 \\ [1, 3] & if η_{ij}^{w} = 3 \end{matrix}$ (30) $l_{ij}^{w} {(k) = x}_{ij}^{w} (k) + 1$ (31) $x_{ij}^{w} {(1) < x}_{ij}^{w} {(2) < x}_{ij}^{w} (3)$ (32)

To facilitate the model formulation, the relations between the time parameters of the movements and phases should satisfy the following equations: ${Ge}_{ij}^{w} (h) = \sum_{k = 1}^{η_{ij}^{w}} {Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$ (33)

$\begin{matrix} T^{w} {(h, l}_{ij}^{w} (k)) & = T_{ij}^{w} {(h, l}_{ij}^{w} (k)) \\ = {\begin{matrix} {Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k)) + L & if k = 1 \\ {Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k)) & if k = 2 or 3 \end{matrix} \end{matrix}$ (34) $T_{ij}^{w} {(h, l}_{ij}^{w} {(k)) = T}_{μ λ}^{w} {(h, l}_{μ λ}^{w} {(k^{'})) ifl}_{ij}^{w} {(k) = l}_{μ λ}^{w} (k^{'})$ (35) $R_{1, ij}^{w} (h) = {\begin{matrix} 0 & {if l}_{ij}^{w} (1) = 1 \\ \sum_{l = 1}^{l_{ij}^{w} (1) - 1} T^{w} (h, l) & {if l}_{ij}^{w} (1) \neq 1 \end{matrix}$ (36) $R_{2, ij}^{w} (h) = {\begin{matrix} 0 & {if l}_{ij}^{w} (η_{ij}^{w}) {= m}_{w} \\ \sum_{{l = l}_{ij}^{w} (η_{ij}^{w}) + 1}^{m_{w}} T^{w} (h, l) & {if l}_{ij}^{w} (η_{ij}^{w}) \neq m_{w} \end{matrix}$ (37) $k, k^{'}, η_{ij}^{w} {, l}_{ij}^{w} (k) \in N^{*}$ (38)

Where ${Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$ is the effective green time of phase, $l_{ij}^{w} (k)$ is the kth phase that contains the movement σ_ij in the F_w during the hth cycle; $T^{w} {(h, l}_{ij}^{w} (k))$ is the phase $l_{ij}^{w} (k)$ length in the F_w during the hth cycle; μ and i, Λ and j have the same definition.

When movement σ_ij and σ_μλ are the lapping movements ( $η_{ij}^{w} ⩾ 2$ and $η_{μ λ}^{w} ⩾ 2$ the same phase, the following constraint is needed to ensure the presence of the phase that contains the two movements.

$\begin{matrix} T_{ij}^{w} {(h, l}_{ij}^{w} {(k)) = T}_{μ λ}^{w} {(h, l}_{μ λ}^{w} (k^{'})) \neq 0 if \\ l_{ij}^{w} {(k) = l}_{μ λ}^{w} (k^{'}) and η_{ij}^{w} ⩾ {2, η}_{μ λ}^{w} ⩾ 2 \end{matrix}$ (39)

To avoid the driver that waits for a long red-light time, the time that the driver has waited and the follow-up waiting time should be satisfied as. $R_{2, ij}^{w} {(h - 1) + R}_{1, ij}^{w} (h) ⩽ T_{s}$ (40)

Where, T_s is the acceptable waiting time for a driver.

In summary, the proposed calculation model is formulated with the objective function a the constraints, which contains main variables and correlation variables.

1) Main variables:

Effective green time of phase: ${Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$

2) Correlation variables:

The movement σ_ij green time of the F_w in the hth cycle: $G_{ij}^{w} (h)$

The movement σ_ij effective green time of the F_w in the hth cycle: ${Ge}_{ij}^{w} (h)$

The cycle time length of the F_w in the hth cycle: C^w (h)

Front-red time: $R_{1, ij}^{w} (h)$

Back-red time: $R_{2, ij}^{w} (h)$

The phase $l_{ij}^{w} (k)$ length for the F_w in the hth cycle: $T^{w} {(h, l}_{ij}^{w} (k))$

All correlation variables can be transformed into a corresponding expression containing only the main variables ${Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$ . This transformation can be realized by Python programming, the partial program codes are shown in the Fig. 7.

Fig. 7

Partial program codes of Python.

So, the proposed model will be the expression containing only the main variables and constants. After determining the proposed model expression by Python programming, it can be found that the objective function is a quadratic function of independent variables ${Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$ , which is a convex function. All constraints are linear combinations of independent variables ${Gep}_{ij}^{w} {(h, l}_{ij}^{w} (k))$ , so the region formed by all constraints iconvex set. Therefore, the proposed model is the mixed-integer-non-liner-program, and also a convex programming, which can be solved by the branch-and-bound algorithm [19].

For the branch-and-bound algorithm, it is a common algorithm about the mixed-integer-non-liner-program, belonging to enumeration algorithm, which can search solution space systematically, analyze all feasible solutions and adopt necessary restrictions to eliminate the non-optimal regions in feasible region, so this paper uses the branch and bound algorithm to solve the proposed model. In the process of solving the model, we firstly determine the specific expression of the model by Python programming, and then use the toolbox of standard branch-and-bound technique to solve the optimization objective and all the decision variables. Different PC schemes correspond to different values of the objective and decision variables. Finally, the scheme that corresponds to the minimum objective value will be selected as the optimal control scheme at the calculation time of the hth cycle (T_h). Thus, the PM dynamic combination control for a signalized intersection can be achieved by rolling the calculation process cycle by cycle from the beginning time.

5 Numerical tests

The intersection chosen by this study and the movement symbols are shown in Fig. 8.

Fig. 8

Movement layout of example intersection.

According to the third section, M = {σ₁₁, σ₁₂, σ₁₃, σ₂₁, σ₂₃, σ₃₁, σ₃₃, σ₄₁, σ₄₃} and e₁₁ = e₁₂ = e₁₃ = e₂₁ = e₂₃ = = e₃₃ = e₄₁ = e₄₃ = 1, e₂₂ = e₃₂ = e₄₂ = 0. And movement compatibility then can be determined, the values of movement correlation parameters are listed in Table 3.

Table 3

Values of the movement correlation parameters

Parameter	Value
Ψ _α	Ψ₁ = {σ₁₁, σ₁₂, σ₁₃} , Ψ;₂ = {σ₂₁, σ₂₃} , Ψ;₃ = {σ₃₁, σ₃₃} , Ψ;₄ = {σ₄₁, σ₄₃}
Ψ	{{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₃₁, σ₃₃} , {σ₄₁, σ₄₃}}
φ _α	φ₁ = 0, φ₂ = 0, φ₃ = 1, φ₄ = 1
Ψ _{α\|φ_α=1}	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₃₁, σ₃₃} , {σ₄₁, σ₄₃}
Φ _β	Φ₁ = {σ₁₁, σ₃₁} , Φ₂ = {σ₁₃, σ₃₃} , Φ₃ = {σ₂₁, σ₄₁} , Φ₄ = {σ₂₃, σ₄₃}
Φ	{{σ₁₁, σ₃₁} , {σ₁₃, σ₃₃} , {σ₂₁, σ₄₁} , {σ₂₃, σ₄₃}}
ϕ _β	ϕ₁ = 0, ϕ₂ = 0, ϕ₃ = 1, ϕ₄ = 1.
Φ _{β\|ϕ_β=1}	{σ₂₁, σ₄₁} , {σ₂₃, σ₄₃}
Θ _γ	Θ₁ = {σ₁₁, σ₂₃} , Θ₂ = {σ₁₃, σ₄₁} , Θ₃ = {σ₂₁, σ₃₃} , Θ₄ = σ₃₁
Θ
Λ

After all FPC schemes are determined, each FPC scheme is consist of character arrays, which are denoted as Char f_w [m_w]. Each array represents one FPC scheme. In this paper, 400 FPC schemes are obtained for the intersection in Fig. 8 by programming calculations. The four-phase scheme has 48 schemes, the five-phase scheme has 264 schemes, and the six-phase scheme has 88 schemes; the partial FPC schemes are listed in Table 4.

Table 4

Partial FPC schemes

		FPC schemes	Phases
1	4	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₃₁, σ₃₃} , {σ₄₁, σ₄₃}	4
2	4	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₄₁, σ₄₃} , {σ₃₁, σ₃₃}
3	4	{σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₂₃} , {σ₄₁, σ₄₃}
4	4	{σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₄₁, σ₄₃} , {σ₂₁, σ₂₃}
5	4	{σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₂₃, σ₄₃}.
6	4	{σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₃, σ₄₃} , {σ₂₁, σ₄₁}
⋮	⋮	⋮
49	5	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₃₁, σ₃₃} , {σ₃₁, σ₄₃} , {σ₄₁, σ₄₃}.	5
50	5	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₄₁, σ₄₃} , {σ₃₁, σ₄₃} , {σ₃₁, σ₃₃}
51	5	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₃₁, σ₃₃} , {σ₃₁, σ₄₃}
52	5	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₄₁, σ₄₃} , {σ₃₁, σ₃₃}
53	5	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₂₁, σ₃₃} , {σ₃₁, σ₄₃}
54	5	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₃₁, σ₄₃} , {σ₃₁, σ₃₃}
⋮	⋮	⋮
313	6	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₄₁, σ₄₃} , {σ₃₁, σ₄₃} , {σ₃₁, σ₃₃}	6
314	6	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₂₁, σ₃₃} , {σ₃₁, σ₃₃} , {σ₃₁, σ₄₃}.
315	6	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₃, σ₄₃} , {σ₄₁, σ₄₃} , {σ₃₁, σ₄₃} , {σ₃₁, σ₃₃}
316	6	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₃₃} , {σ₃₁, σ₃₃} , {σ₃₁, σ₄₃} , {σ₄₁, σ₄₃}
317	6	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₄₁, σ₄₃} , {σ₃₁, σ₄₃}
318	6	{σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₄₁, σ₄₃}
⋮	⋮	⋮
389	6	{σ₃₁, σ₄₃} , {σ₂₃, σ₄₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₄₁} , {σ₂₁, σ₃₃} , {σ₁₁, σ₁₂, σ₁₃}
399	6	{σ₃₁, σ₄₃} , {σ₂₃, σ₄₃} , {σ₂₁, σ₂₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₁₁, σ₁₂, σ₁₃}.
400	6	{σ₃₁, σ₄₃} , {σ₂₃, σ₄₃} , {σ₄₁, σ₄₃} , {σ₂₁, σ₄₁} , {σ₂₁, σ₃₃} , {σ₁₁, σ₁₂, σ₁₃}

The collection of the actual data of vehicles is difficult since some vehicles do not have the communication device. This paper uses a numerical method to test the proposed method and assumes that the specific arrival of vehicles can be obtained. In current research about traffic flow, the arrival of traffic flow at the approach of an intersection is often regarded as a certain distribution rule, such as the Poisson distribution, the binomial distribution, or the negative binomial distribution. However, for a signalized intersection, the arrival of vehicles has a certain randomness between signal cycles [20]. Therefore, the numerical example stochastically assumes the traffic volumes listed in Table 5, including basic volume and random variation.

Table 5

Vehicle arrival situation of movements

Movements		Traffic flow (pcu/h)		Movements		Traffic flow (pcu/h)
	Volume	Variation				Volume	Variation
i = 1	Left	550	±100	i = 3	Left	400	±150
	Through	550	±100		Through	750	±150
i = 2	Left	400	±100	i = 4	Left	750	±100
	Through	700	±150		Through	450	±100

The extensively employed control modes include the fixed control (FC) mode and the dynamic timing control (DTC) mode. The PC scheme and signal timing are fixed for the FC mode. For the DTC mode, the PC scheme is fixed, but the signal timing can be adjusted slightly during different the signal cycles. To verify the feasibility of the proposed method, a numerical example is used to compare the PM dynamic combination control (PMDC) mode with the FC and DTC. In the process of calculation, 20 signal periods are selected as the research period, and every cycle length is 100 s. The arrival of vehicles in the same cycle is the same for different control modes to ensure that the result of the contrast is persuasive. In addition, other parameters involved in the calculation process must be consistent. Under the AV environment, the specific parameters are as follows: the minimum green time G_min = 7s, the maximum green time G_max = 60s, the saturation flow rate S_ij = 2000veh/h/lane, the movement lost time L = 1.8 s, the yellow time A = 4 s, and the acceptable waiting time for a driver, T_s= 120 s. [4, 2 22].

According to the movement arrival of each cycle, the objective value of all FPC schemes in each cycle can be calculated by the PMDC mode. All FPC schemes of a partial cycle that correspond to the objective value are shown in Fig. 9. The FPC scheme with a minimum objective value for each cycle is selected as the optimal control scheme of the current cycle by comparing the size of the objective value. Thus, the optimal control scheme of 20 cycles can be obtained. The optimal PC schemes of each cycle are listed in Table 6, and the movement timing scheme of 20 cycles for PMDC are listed in Table 7.

Fig. 9

Target value of the FPC schemes for a partial cycle.

Table 6

Optimal PC schemes of every cycle

h	w	PC schemes	Phases
1	365	{σ₃₁, σ₃₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₄₁, σ₄₃} , {σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃}	6
2	235	{σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ₄₁}	5
3	365	{σ₃₁, σ₃₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₄₁, σ₄₃} , {σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃}	6
4	235	{σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ₄₁}	5
5	27	{σ₃₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₁₁, σ₁₂, σ₁₃} , {σ₂₃, σ₄₃}	4
6	12	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₃, σ₄₃} , {σ₂₁, σ₄₁} , {σ₃₁, σ₃₃}	4
7	40	{σ₂₁, σ₄₁} , {σ₃₁, σ₃₃} , {σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃}	4
8	43	{σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₄₁}	4
9	27	{σ₃₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₁₁, σ₁₂, σ₁₃} , {σ₂₃, σ₄₃}	4
10	99	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₃, σ₄₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₃₃} , {σ₂₁, σ}	5
11	27	{σ₃₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₁₁, σ₁₂, σ₁₃} , {σ₂₃, σ₄₃}	4
12	104	{σ₁₁, σ₁₂, σ₁₃} , {σ₂₃, σ₄₃} , {σ₂₁, σ₄₁} , {σ₂₁, σ₃₃} , {σ₃₁, σ₃₃}	5
13	40	{σ₂₁, σ₄₁} , {σ₃₁, σ₃₃} , {σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃}	4
14	236	{σ₂₃, σ₄₃} , {σ₁₁, σ₁₂, σ₁₃} , {σ₂₁, σ₄₁} , {σ₂₁, σ₃₃} , {σ₃₁, σ₃₃}	5
15	225	{σ₂₁, σ₄₁} , {σ₄₁, σ₄₃} , {σ₂₃, σ₄₃} , {σ₃₁, σ₃₃} , {σ₁₁, σ₁₂, σ₁₃}	5
16	166	{σ₃₁, σ₃₃} , {σ₁₁, σ₁₂, σ₁₃} , {σ₂₃, σ₄₃} , {σ₄₁, σ₄₃} , {σ₂₁, σ₄₁}	5
17	47	{σ₂₃, σ₄₃} , {σ₂₁, σ₄₁} , {σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃}	4
18	65	{σ₁₁, σ₁₂, σ₁₃} , {σ₃₁, σ₃₃} , {σ₂₁, σ₄₁} , {σ₄₁, σ₄₃} , {σ₂₃, σ₄₃}	5
19	375	{σ₂₁, σ₄₁} , {σ₂₁, σ₂₃} , {σ₂₃, σ₄₃} , {σ₃₁, σ₄₃} , {σ₃₁, σ₃₃} , {σ₁₁, σ₁₂, σ₁₃}	6
20	161	{σ₃₁, σ₃₃} , {σ₁₁, σ₁₂, σ₁₃} , {σ₄₁, σ₄₃} , {σ₂₁, σ₄₁} , {σ₂₁, σ₂₃}	5

Table 7

Movement timing scheme of 20 cycles for PMDC

h	w	Optimization phase timings											O^w (s/pcu)
		Legs	1			2		3		4
		Movements	σ ₁₁	σ ₁₂	σ ₁₃	σ ₂₁	σ ₂₃	σ ₃₁	σ ₃₃	σ ₄₁	σ ₄₃
1	365	Start of green(s)	65	65	65	12	41	0	0	13	33	83.334
		Duration of green(s)	31	31	31	17	20	8	9	24	28
		End of green(s)	96	96	96	29	61	8	9	37	61
2	235	Start of green(s)	16	16	16	57	0	27	27	58	0	86.577
		Duration of green(s)	7	7	7	39	12	26	27	38	12
		End of green(s)	23	23	23	96	12	53	54	96	12
3	365	Start of green(s)	57	57	57	14	26	0	0	15	25	93.790
		Duration of green(s)	39	39	39	7	27	10	11	7	28
		End of green(s)	96	96	96	21	53	10	11	22	53
4	235	Start of green(s)	14	14	14	64	0	35	35	68	0	101.883
		Duration of green(s)	17	17	17	32	10	25	29	28	10
		End of green(s)	31	31	31	96	10	60	64	96	10
5	27	Start of green(s)	27	27	27	16	58	0	0	16	58	80.099
		Duration of green(s)	27	27	27	7	38	12	12	7	38
		End of green(s)	54	54	54	23	96	12	12	23	96
6	12	Start of green(s)	0	0	0	36	15	66	66	36	15	72.387
		Duration of green(s)	11	11	11	26	17	30	30	26	17
		End of green(s)	11	11	11	62	32	96	96	62	32
7	40	Start of green(s)	67	67	67	0	38	17	17	0	38	79.807
		Duration of green(s)	29	29	2	13	25	17	17	13	25
		End of green(s)	96	96	96	13	63	34	34	13	63
8	43	Start of green(s)	14	14	14	66	0	35	35	66	0	104.197
		Duration of green(s)	17	17	17	30	10	27	27	30	10
		End of green(s)	31	31	31	96	10	62	62	96	10
9	27	Start of green(s)	26	26	26	15	57	0	0	15	57	110.127
		Duration of green(s)	27	27	27	7	39	11	11	7	39
		End of green(s)	53	53	53	22	96	11	11	22	96
10	99	Start of green(s)	0	0	0	59	14	30	30	62	14	114.648
		Duration of green(s)	10	10	10	37	12	25	28	34	12
		End of green(s)	10	10	10	96	26	58	58	96	26
11	27	Start of green(s)	27	27	27	16	59	0	0	16	59	107.630
		Duration of green(s)	28	28	28	7	37	12	12	7	37
		End of green(s)	55	55	55	23	96	12	12	23	96
12	104	Start of green(s)	0	0	0	35	14	66	63	35	14	102.587
		Duration of green(s)	10	10	10	27	17	30	33	24	17
		End of green(s)	10	10	10	62	31	96	96	59	31
13	40	Start of green(s)	59	59	59	0	27	16	16	0	27	91.830
		Duration of green(s)	37	37	37	12	28	7	7	12	28
		End of green(s)	96	96	96	12	55	23	23	12	55
14	236	Start of green(s)	14	14	14	25	0	63	60	25	0	97.785
		Duration of green(s)	7	7	7	34	10	33	36	31	10
		End of green(s)	21	21	21	59	10	96	96	56	10
15	225	Start of green(s)	58	58	58	0	16	43	43	0	12	98.368
		Duration of green(s)	38	38	38	8	23	11	11	12	27
		End of green(s)	96	96	96	8	39	54	54	12	39
16	166	Start of green(s)	15	15	15	67	36	0	0	63	36	91.000
		Duration of green(s)	17	17	17	29	23	11	11	33	27
		End of green(s)	32	32	32	96	59	11	11	96	63
17	47	Start of green(s)	26	26	26	15	0	60	60	15	0	97.510
		Duration of green(s)	30	30	30	7	11	36	36	7	11
		End of green(s)	56	56	56	22	11	96	96	22	11
18	65	Start of green(s)	0	0	0	30	61	15	15	30	58	102.346
		Duration of green(s)	11	11	11	24	35	11	11	27	38
		End of green(s)	11	11	11	54	96	26	26	57	96
19	375	Start of green(s)	65	65	65	0	15	35	36	0	16	102.559
		Duration of green(s)	31	31	31	12	16	26	25	11	16
		End of green(s)	96	96	96	12	31	61	61	11	32
20	161	Start of green(s)	16	16	16	57	59	0	0	27	27	106.510
		Duration of green(s)	7	7	7	39	37	12	12	28	26
		End of green(s)	23	23	23	96	96	12	12	55	53

The PC structure of FC and DTC mode is the RBCS at present, the timing schemes are solved by the corresponding timing model. For the FC mode, the control scheme primarily employs the Webster timing model, which is based on the traffic volume of the critical movement. Thus, the optimal control scheme for the FC mode can be acquired according to the basic traffic volume; the parameters of the specific optimal control scheme are listed in Table 8. For the DTC mode, the signal timing scheme of each cycle can be solved using the optimal timing model. The vehicle arrival of all movements in each cycle are different due to its randomness; thus, the optimal timing schemes of different cycles are different. The movement timing scheme of the DTC mode for 20 cycles is shown in Table 9.

Table 8

Movement timing scheme of the FC mode

w	Optimization phase timings (s)										Cycle length(s)
	Legs	1			2		3		4
	Movements	σ ₁₁	σ ₁₂	σ ₁₃	σ ₂₁	σ ₂₃	σ ₃₁	σ ₃₃	σ ₄₁	σ ₄₃
60	Start of green(s)	0	0	0	50	50	25	25	74	75	100
	Duration of green(s)	21	21	21	21	20	21	21	22	21
	End of green(s)	21	21	21	71	70	46	46	96	96

Table 9

Movement timing scheme of the DTC mode for 20 cycles

h	Optimization phase timings
	Legs	1			2		3		4
	Movements	σ ₁₁	σ ₁₂	σ ₁₃	σ ₂₂	σ ₂₃	σ ₃₁	σ ₃₃	σ ₄₁	σ ₄₃
1	Start of green(s)	0	0	0	43	43	26	26	57	58
	Duration of green(s)	22	22	22	11	10	13	13	39	38
	End of green(s)	22	22	22	54	53	39	39	96	96
2	Start of green(s)	0	0	0	46	46	23	23	71	74
	Duration of green(s)	19	19	19	24	21	19	19	25	22
	End of green(s)	19	19	19	70	67	42	42	96	96
3	Start of green(s)	0	0	0	47	47	24	24	71	75
	Duration of green(s)	20	20	20	24	20	19	19	25	21
	End of green(s)	20	20	20	71	67	43	43	96	96
4	Start of green(s)	0	0	0	48	48	24	24	73	74
	Duration of green(s)	20	20	20	22	21	20	20	23	22
	End of green(s)	20	20	20	70	69	44	44	96	96
5	Start of green(s)	0	0	0	51	51	24	24	75	76
	Duration of green(s)	20	20	20	21	20	23	23	21	20
	End of green(s)	20	20	20	72	71	47	47	96	96
6	Start of green(s)	0	0	0	49	49	24	24	73	76
	Duration of green(s)	20	20	20	23	20	21	21	23	20
	End of green(s)	20	20	20	72	69	45	45	96	96
7	Start of green(s)	0	0	0	47	47	22	22	71	75
	Duration of green(s)	18	18	18	24	20	21	21	25	21
	End of green(s)	18	18	18	71	67	43	43	96	96
8	Start of green(s)	0	0	0	48	48	23	23	74	75
	Duration of green(s)	19	19	19	23	22	21	21	22	21
	End of green(s)	19	19	19	71	70	44	44	96	96
9	Start of green(s)	0	0	0	48	48	24	24	72	75
	Duration of green(s)	20	20	20	23	20	20	20	24	21
	End of green(s)	20	20	20	71	68	44	44	96	96
10	Start of green(s)	0	0	0	46	46	22	22	71	72
	Duration of green(s)	18	18	18	22	21	20	20	25	24
	End of green(s)	18	18	18	68	67	42	42	96	96
11	Start of green(s)	0	0	0	48	48	22	22	73	76
	Duration of green(s)	18	18	18	24	21	22	22	23	20
	End of green(s)	18	18	18	72	69	44	44	96	96
12	Start of green(s)	0	0	0	50	50	23	23	75	76
	Duration of green(s)	19	19	19	22	21	23	23	21	20
	End of green(s)	19	19	19	72	71	46	46	96	96
13	Start of green(s)	0	0	0	49	49	23	23	73	75
	Duration of green(s)	19	19	19	22	20	22	22	23	21
	End of green(s)	19	19	19	71	69	45	45	96	96
14	Start of green(s)	0	0	0	47	47	23	23	70	71
	Duration of green(s)	19	19	19	20	19	20	20	26	25
	End of green(s)	19	19	19	67	66	43	43	96	96
15	Start of green(s)	0	0	0	47	47	25	25	71	72
	Duration of green(s)	21	21	21	21	20	18	18	25	24
	End of green(s)	21	21	21	68	67	43	43	96	96
16	Start of green(s)	0	0	0	48	48	25	25	72	73
	Duration of green(s)	21	21	21	21	20	19	19	24	23
	End of green(s)	21	21	21	69	68	44	44	96	96
17	Start of green(s)	0	0	0	48	48	25	25	72	73
	Duration of green(s)	21	21	21	21	20	19	19	24	23
	End of green(s)	21	21	21	69	68	44	44	96	96
18	Start of green(s)	0	0	0	47	47	24	24	72	73
	Duration of green(s)	20	20	20	22	21	19	19	24	23
	End of green(s)	20	20	20	69	68	43	43	96	96
19	Start of green(s)	0	0	0	48	48	24	24	73	74
	Duration of green(s)	20	20	20	22	21	20	20	23	22
	End of green(s)	20	20	20	70	69	44	44	96	96
20	Start of green(s)	0	0	0	47	47	22	22	72	73
	Duration of green(s)	18	18	18	22	21	21	21	24	23
	End of green(s)	18	18	18	69	68	43	43	96	96

The average delay of vehicles and throughput are selected to compare the superiority of different control modes. The indexes value for different control modes can be calculated according to the different control schemes. The indexes comparison for different control modes are shown in Fig. 10.

Fig. 10

Indexes comparison for different control modes.

According to the result, the PMDC mode can effectively reduce the average vehicle delay and increase throughput at the signalized intersection when compared with the DTC and FC modes. For the 20 signal cycles, the average vehicle delay for the PMDC mode is 96.249 s/pcu. However, the average vehicle delay for the DTC mode and FC mode are 103.419 s/pcu and 110.153 s/pcu, respectively. So, the average delay of the PMDC mode can be reduced by 6.9% and 14.5%, respectively then the DTC and FC modes. In addition, for the index of throughput, the total throughput of three control modes (PMDC, DTC and FC) for the 20 signal cycles are 862 pcu, 825pcu and 795pcu, respectively. So, the total throughput of the PMDC mode can be increased by 4.3% and 7.8%, respectively then the DTC and FC modes. Therefore, the results show that the proposed method can improve the control mode of intersection than the current control modes.

6 Conclusions and future work

A new dynamic combination control method of PM for the signal intersection in the AV environment with considering the comprehensive styles of PC schemes is proposed in this paper. The proposed method uses discrete mathematics to calculate overall PC schemes and determines the vehicle status to obtain the arrival characteristics of the movement based on the data source of the AV environment. Then the signalized intersection is optimized cyclically according to the combination the PC scheme and vehicle arrival of the movement. This paper introduced numerical tests to verify the feasibility and effectiveness of the proposed method. The results show that the proposed method can effectively improve the control mode for intersection.

The conclusions of this paper are listed as below:

The PC scheme of the proposed method is more comprehensive than the conventional PC scheme. A number of the PC schemes presented in this paper include a series of other combination schemes in addition to the RBCS (Table 4).

For one cycle, the different PC schemes have a significant influence on the signal control at the intersection. For example, the optimal objective function values significantly vary for different PC schemes in cycle 3 of Fig. 9(a). The min-maximum objective values are 93.7901 s/pcu, and 259.2157 s/pcu, respectively. The corresponding number w for the FPC schemes are 365 and 220. The maximum objective value is two times of the minimum, which indicates that PC style has a significant influence on the effect of the control scheme. So the selection of a reasonable phase combination scheme is important.

The proposed method can reduce effectively the average vehicle delay and increase throughput for the signalized intersection. For the 20 signal cycles, the average vehicle delay of the proposed method can be reduced by 6.9 % and 14.5 %, respectively, and the total throughput of the proposed method can be increased by 4.3% and 7.8%, respectively, compared with the DTC mode and FC mode (Fig. 10).

In conclusion, the proposed method can improve the signal control effect by considering PM dynamic combination, and provides a new idea for signal control dynamic optimization under the AV environment. However, the method considers only the compatibility relationship between the vehicle movements. The priority level of all movements is the same when determining the PC scheme of each cycle. Future researches will increase pedestrian movements and non-motorized vehicle movements, and consider setting a movement priority level to prioritize movements by special vehicles (emergency vehicle, special service vehicle, and bus).

Footnotes

Acknowledgments

This study was supported by the National Natural Science Foundation of China under Grant Nos. 71771183 and Nos. U1664262.

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