# Coordinated Control Strategy of Short-Turning Bus and Full-Length Bus Based on Stable Headway

Authors: Huasheng Liu; Yu Li; Yanqiu Zhang; Jin Li; Yi Wang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Under the condition of dynamically collecting vehicle operation information, bus dynamic dispatching is an effective means to stabilize the vehicle operation according to the traffic plan and improve the stability of the system by giving real-time response and continuous dynamic control to the vehicle running state. Speed control does not increase the extra parking time at the station, does not waste parking resources, does not significantly prolong the waiting time of passengers at the station, and is more easily accepted by passengers. In this paper, by using the speed control method, the problem of how to realize the coordinated control between the short-turning bus and the full-length bus is studied in order to stabilize that the headway between the full-length buses is equal to the departure interval, and the headway between the short-turning bus and the full-length bus is equal to the expected headway. The problem of ensuring the coordinated and stable operation of the short-turning bus and the full-length bus is studied. The result of an example shows that the coordinated control strategy can make the whole system run more stable.


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## Body

## 1. Introduction

When the bus runs on the line according to the traffic plan, it is easily affected by the external environment, such as road conditions, weather, and other factors, resulting in delays at road sections and intersections, resulting in delays when the buses are running to the station ahead. As a result, the arrival interval of buses at the front station is extended, and the passenger flow is concentrated. Therefore, the boarding and alighting time of passengers and the parking time of buses at the station will be extended. It aggravates the delay of the bus arriving at the next stop. Because of the persistence and dissemination of the delay, with the continuation and spread of the delay, the headway between the bus and the front of the bus will gradually increase, resulting in a large interval, and the headway between the front of the bus and the rear will gradually decrease, resulting in the phenomenon of bus crosstalk. When the buses at the station do not arrive correctly, the phenomenon of bus crosstalk and large interval leads to the uneven distribution of passenger flow among buses, some buses are overcrowded, passengers' comfort decreases, and the service level of the system is reduced. It increases the passenger travel time cost and the bus enterprise operation cost, reduces the punctuality of the bus, and reduces the attractiveness of the bus to the residents. In view of the above problems, scholars have proposed a variety of bus scheduling strategies, which are divided into static scheduling and dynamic scheduling [1].Furth [2] put forward a combined dispatching strategy of the combination of short-turning bus and short-cutting bus and studied the method of setting the operation interval of this strategy. Ceder [3] studied the method of inserting deadheading buses during the operation of short-turning buses and full-length buses, so as to minimize the size of the fleet. Vijayaraghavan [4] studied the combined scheduling strategy to optimize the fleet size by inserting short-cutting buses and short-turning buses into the full-length bus operation line. Dell Site and Filippi [5] took the lowest cost (the sum of the passenger cost and the operator cost) as the goal and determined the operation range and fleet size of the short-turning line, as well as the departure frequency of the short-turning bus and the full-length bus. Ulusoy et al. [6] analyzed the difference of temporal and spatial distribution of passenger demand, established the total cost objective function, and optimized the coordinated departure frequency of the full-length bus and short-turning bus and short-cutting bus. Cortés et al. [7] adopted the scheduling strategy of combining short-turning buses with deadheading buses and determined the optimal departure frequency. Canca et al. [8] studied the combined scheduling of short-turning buses and full-length buses when the waiting time was prolonged due to interference and inadequate infrastructure. Teng and Jin [9] developed a bus operation control system to dynamically adjust bus speed, bus dwell time, and traffic signal timings along the running path. Zhang et al. [10] studied a dynamic short-term bus control that can significantly reduce total cost and improve transit service reliability. Argote-Cabanero et al. [11] propose a dynamic control method to overcome bunching and improve the regularity of fixed-route transit systems. The method uses a combination of dynamic holding and en-route driver guidance to achieve its objectives.In the research of bus control strategy by stabilizing the headway, Xuan et al. [12] consider the headway of the bus, a kind of dynamic control strategy is proposed, which uses the deviation between the bus arrival and the virtual timetable at the control point. A virtual schedule is introduced regardless of whether the system runs on a published schedule or not. The results show that by using this method, the bus can not only strictly abide by the published timetable but also maintain the normal departure interval without too much slack. Ma et al. [13] proposed a nonlinear optimal control model with passenger demand and disturbance uncertainties for the real-time bus regulation in urban rapid transit. The aim of this model is to assure the stability of bus headway in the line, reduce passengers’ waiting time, and improve bus service level. Liu [14] proposed a real-time speed control method based on the stability of the bus headway, which aims to minimize the deviation between the actual headway and the expected headway of each departure bus on a single line.In summary, static dispatching refers to the comprehensive analysis and compilation of the departure schedule of each bus line according to the temporal and spatial distribution law of bus passenger flow, the travel demand of bus passenger flow, the transport capacity of bus enterprises, the operating cost, and the social benefits of bus enterprises, making the bus scheduling plan and the scheduling of the sales staff [15]. The commonly used static scheduling methods include short-cutting bus scheduling, bus release scheduling, cross-line intermodal transport scheduling, and short-turning bus scheduling [16, 17]. At present, the research [18, 19] on short-turning buses generally focuses on the fleet size, departure frequency, and interval control between short-turning buses and full-length buses. The combined dispatching of short-turning buses and full-length buses or short-turning buses and short-cutting buses is also limited to the study of the departure frequency and form of each fleet, but there is little research on the coordinated control of short-turning buses in the whole operation process. Because of the random characteristics of the bus operation process, it often can not achieve the service reliability goal of the expected traffic plan. Therefore, the static scheduling mode which only controls the departure interval between the full-length bus and the short-turning bus can not completely guarantee the coordinated and stable operation of the system.Compared with static scheduling, dynamic scheduling can give real-time response to some essentially unpredictable factors, which is of great significance to ensure the maximization of static scheduling. Dynamic scheduling is based on static scheduling technology, combined with ITS technology, according to the traffic flow and the actual situation to take corresponding scheduling measures to deal with various situations encountered in bus operation [20]. According to the control purpose, dynamic scheduling is divided into two types: headway-based control and timetable-based control; according to the range of control space, dynamic scheduling is divided into two types: interstation control and station control. Speed control is an interstation control strategy, which does not increase the extra parking time at the station, does not waste parking resources, does not significantly prolong the waiting time of passengers at the station, and is more easily accepted by passengers. At the same time, with the gradual application of intelligent networks and self-driving vehicles, this method is also easier to achieve. This paper mainly uses the speed control strategy to realize the coordinated operation of the short-turning bus and the full-length bus.The short-turning bus scheduling strategy was originally proposed to solve the problem that the bus passenger flow is excessively concentrated in a continuous section in some lines during the peak period, and the passenger flow distribution within and outside the section is obviously unbalanced. By setting up the short-turning bus and the full-length bus to meet the passenger travel needs inside and outside the larger passenger flow, respectively, the problem of waste of transport capacity can be solved effectively. The short-turning bus and the full-length bus complement each other, and the instability of the road system often leads to the uncoordinated operation of the two, resulting in a larger and smaller interval, and the system can not achieve the expected reliability goal. In this paper, firstly, the control strategy of the full-length bus is studied to ensure its stability of it, and on this basis, the running state of the short-turning bus is controlled, so as to realize the overall coordination and stable operation of the short-turning bus and the full-length bus.


## 2. Full-Length Bus Control Strategy Based on Stabilizing Headway

### 2.1. Control Train of Thought

The schematic diagram of the full-length bus operation is shown in Figure1, in which the stations S0–S1 and S2–S3 are the short-turning bus operation interval. When the time t0 is defined, the number of running buses is n on the line, and the bus number closest to the departure station is set to 1. The remaining buses are numbered according to the order of distance 1 from near to far; at this time, the position of each bus is indicated by xi (the position of the departure station of the line is defined as 0 km, and the position of each bus is relative to the departure station); the running speed of each bus is vi, and the departure interval of the operating speed is Ci; departure interval is Hi (the bus i and the bus i-1); the headway is hi (the bus i and the bus i-1).Figure 1
schematic diagram of the full-length bus operation.


#### 2.1.1. Control Train of Thought

The speed control method is adopted to control the full-length bus for the purpose of stabilizing that the headway between the adjacent full-length buses is equal to the departure interval. First of all, based on the position of bus1 in the expected running state, the guiding speed of bus 1 is given, and the stable bus 1 runs according to the driving plan; bus 2 is based on bus 1, and the headway between stable bus 1 and bus 2 is equal to the departure interval, and bus 3 is based on bus 2. The headway between stable bus 3 and bus 2 is equal to the departure interval, and so on, to ensure that the full-length bus runs as far as possible according to the predetermined driving plan.


#### 2.1.2. Control Flow

Take a regulation as an example, as shown in Figure2. First of all, according to the position x1 of bus 1, it is judged whether bus 1 runs according to the driving plan, and if it runs normally, it will not be regulated; otherwise, the guiding speed of bus 1 is calculated; then, starting from the order of near and far from bus 1, judge the degree to which the headway hi deviates from the departure interval Hi between bus i and bus i-1 one by one, and given fluctuation parameter is β. If it is within the fluctuation range, the guiding speed of the bus i is taken as the guiding speed of the bus i-1. Otherwise, the running speed of bus i is calculated. Because the control starts from bus 1, the stability of bus 1 is also the basis of the stability of the whole line.Figure 2
One-time control flow chart of the full-length bus.The train of thought to judge whether bus1 runs according to the driving schedule: calculate the expected position X1 from the bus start to time t0 when bus 1 is running on time, calculate the operation deviation based on the actual position, and set the deviation range parameter α, if, in the range, bus 1 is in the expected running state and does not need to be regulated. Otherwise, the guidance speed of bus 1 is calculated according to the formula of the guiding speed of bus 1.


### 2.2. Guide Speed Model of the Full-Length Bus

Operating speed, also known as transport speed, is the ratio of the total distance of the bus to the running time, including the stopping time at the station and the running time between stations [21]. The running speed refers to the ratio of the total distance of the bus to the running time excluding the parking time at the station. Assuming that the bus lanes are arranged throughout the line, the relationship between operating speed and running speed can be found in (1)XiT−xi=CiT·T60,XiT−xi=ViTT60−b3600·λ·XiT−xiH+hi+12·60,⇒CiT·T60=ViTT60−b3600·λ·XiT−xiH+hi+12·60,⇒CiT=ViT1+b/3600·λ·ViT·hi+1+H/2·60,ViT=CiT1−b/3600·λ·CiT·hi+1+H/2·60.In equation (1), t0 is the initial time of single speed control; T is the single speed control time length, min; XiT is the the expected location of the bus i at t0+T time, km; xi is the actual position of the bus i at t0 time, km; CiT is, during the period from t0 to t0+T, the guided operating speed of the bus i, km/h; ViT is, during the period from t0 to t0+T, the guided running speed of the bus i, km/h; b is the unit passenger boarding time, s; λ is the passenger arrival rate per unit distance per unit time, pax/km⋅h; H is the line departure interval, min; hi is the headway between bus i and bus i-1 at t0 time, min.The first bus (bus1) guides the operating speed as (2)X1T=Ct·t0−D1+T60,⇒C1T=60·X1T−x1T=60·Ct·t0−D1+T/60−x1T.In (2), D1 is the bus 1 departure time; Ct is the bus expected operating speed, km/h. Other buses guide operating speed as in(3)CiT=CiT·T/60+xi−Ci−1T·T/60+xi−1H/60,⇒CiT=60xi−xi−1−Ci−1T·TH−T.By using the model of the relationship between operating speed and running speed, the theoretical guided running speedviT of the bus can be obtained. However, due to the influence of vehicles, roads, and other factors, the actual guide running speed viT should meet the following constraints:(4)ViT=Vmin,viT<Vmin,viT,Vmin≤viT≤Vmax,Vmax,viT>Vmax.In (4), viT is, during the period from t0 to t0+T, the theoretical calculation of the guiding running speed of the bus i, km/h; Vmax is the maximum allowable running speed on the road where the line is located, km/h; Vmin is the minimum allowable running speed on the road where the line is located, km/h.


### 2.3. Evaluation Index of the Control Effect of the Full-Length Bus

The deviation degree of the first bus position isΔx1, the average headway deviation is selected as the evaluation indexes of the control strategy, and the average headway deviation is defined as the average of the absolute value of the difference between the headway and the departure interval of each bus running on the line.The degree of deviation from the position of the first bus is(5)Δx1=x1−X1X1·100%,=x1−t0−D1·Ctt0−D1·Ct·100%.The deviation degree of the average headway of the full-length bus is(6)Δh1T¯=Δh1TnT,=∑i=1nhiT−HnT,whereΔh1T¯ is the average headway deviation at t0+T time, min/veh; Δh1T is the deviation of the headway at t0+T time, min; hiT is the headway between bus i and bus i−1 at t0+T time, min; nT is the number of buses running on the line at t0+T time.


## 2.1. Control Train of Thought

The schematic diagram of the full-length bus operation is shown in Figure1, in which the stations S0–S1 and S2–S3 are the short-turning bus operation interval. When the time t0 is defined, the number of running buses is n on the line, and the bus number closest to the departure station is set to 1. The remaining buses are numbered according to the order of distance 1 from near to far; at this time, the position of each bus is indicated by xi (the position of the departure station of the line is defined as 0 km, and the position of each bus is relative to the departure station); the running speed of each bus is vi, and the departure interval of the operating speed is Ci; departure interval is Hi (the bus i and the bus i-1); the headway is hi (the bus i and the bus i-1).Figure 1
schematic diagram of the full-length bus operation.


### 2.1.1. Control Train of Thought

The speed control method is adopted to control the full-length bus for the purpose of stabilizing that the headway between the adjacent full-length buses is equal to the departure interval. First of all, based on the position of bus1 in the expected running state, the guiding speed of bus 1 is given, and the stable bus 1 runs according to the driving plan; bus 2 is based on bus 1, and the headway between stable bus 1 and bus 2 is equal to the departure interval, and bus 3 is based on bus 2. The headway between stable bus 3 and bus 2 is equal to the departure interval, and so on, to ensure that the full-length bus runs as far as possible according to the predetermined driving plan.


### 2.1.2. Control Flow

Take a regulation as an example, as shown in Figure2. First of all, according to the position x1 of bus 1, it is judged whether bus 1 runs according to the driving plan, and if it runs normally, it will not be regulated; otherwise, the guiding speed of bus 1 is calculated; then, starting from the order of near and far from bus 1, judge the degree to which the headway hi deviates from the departure interval Hi between bus i and bus i-1 one by one, and given fluctuation parameter is β. If it is within the fluctuation range, the guiding speed of the bus i is taken as the guiding speed of the bus i-1. Otherwise, the running speed of bus i is calculated. Because the control starts from bus 1, the stability of bus 1 is also the basis of the stability of the whole line.Figure 2
One-time control flow chart of the full-length bus.The train of thought to judge whether bus1 runs according to the driving schedule: calculate the expected position X1 from the bus start to time t0 when bus 1 is running on time, calculate the operation deviation based on the actual position, and set the deviation range parameter α, if, in the range, bus 1 is in the expected running state and does not need to be regulated. Otherwise, the guidance speed of bus 1 is calculated according to the formula of the guiding speed of bus 1.


## 2.1.1. Control Train of Thought

The speed control method is adopted to control the full-length bus for the purpose of stabilizing that the headway between the adjacent full-length buses is equal to the departure interval. First of all, based on the position of bus1 in the expected running state, the guiding speed of bus 1 is given, and the stable bus 1 runs according to the driving plan; bus 2 is based on bus 1, and the headway between stable bus 1 and bus 2 is equal to the departure interval, and bus 3 is based on bus 2. The headway between stable bus 3 and bus 2 is equal to the departure interval, and so on, to ensure that the full-length bus runs as far as possible according to the predetermined driving plan.


## 2.1.2. Control Flow

Take a regulation as an example, as shown in Figure2. First of all, according to the position x1 of bus 1, it is judged whether bus 1 runs according to the driving plan, and if it runs normally, it will not be regulated; otherwise, the guiding speed of bus 1 is calculated; then, starting from the order of near and far from bus 1, judge the degree to which the headway hi deviates from the departure interval Hi between bus i and bus i-1 one by one, and given fluctuation parameter is β. If it is within the fluctuation range, the guiding speed of the bus i is taken as the guiding speed of the bus i-1. Otherwise, the running speed of bus i is calculated. Because the control starts from bus 1, the stability of bus 1 is also the basis of the stability of the whole line.Figure 2
One-time control flow chart of the full-length bus.The train of thought to judge whether bus1 runs according to the driving schedule: calculate the expected position X1 from the bus start to time t0 when bus 1 is running on time, calculate the operation deviation based on the actual position, and set the deviation range parameter α, if, in the range, bus 1 is in the expected running state and does not need to be regulated. Otherwise, the guidance speed of bus 1 is calculated according to the formula of the guiding speed of bus 1.


## 2.2. Guide Speed Model of the Full-Length Bus

Operating speed, also known as transport speed, is the ratio of the total distance of the bus to the running time, including the stopping time at the station and the running time between stations [21]. The running speed refers to the ratio of the total distance of the bus to the running time excluding the parking time at the station. Assuming that the bus lanes are arranged throughout the line, the relationship between operating speed and running speed can be found in (1)XiT−xi=CiT·T60,XiT−xi=ViTT60−b3600·λ·XiT−xiH+hi+12·60,⇒CiT·T60=ViTT60−b3600·λ·XiT−xiH+hi+12·60,⇒CiT=ViT1+b/3600·λ·ViT·hi+1+H/2·60,ViT=CiT1−b/3600·λ·CiT·hi+1+H/2·60.In equation (1), t0 is the initial time of single speed control; T is the single speed control time length, min; XiT is the the expected location of the bus i at t0+T time, km; xi is the actual position of the bus i at t0 time, km; CiT is, during the period from t0 to t0+T, the guided operating speed of the bus i, km/h; ViT is, during the period from t0 to t0+T, the guided running speed of the bus i, km/h; b is the unit passenger boarding time, s; λ is the passenger arrival rate per unit distance per unit time, pax/km⋅h; H is the line departure interval, min; hi is the headway between bus i and bus i-1 at t0 time, min.The first bus (bus1) guides the operating speed as (2)X1T=Ct·t0−D1+T60,⇒C1T=60·X1T−x1T=60·Ct·t0−D1+T/60−x1T.In (2), D1 is the bus 1 departure time; Ct is the bus expected operating speed, km/h. Other buses guide operating speed as in(3)CiT=CiT·T/60+xi−Ci−1T·T/60+xi−1H/60,⇒CiT=60xi−xi−1−Ci−1T·TH−T.By using the model of the relationship between operating speed and running speed, the theoretical guided running speedviT of the bus can be obtained. However, due to the influence of vehicles, roads, and other factors, the actual guide running speed viT should meet the following constraints:(4)ViT=Vmin,viT<Vmin,viT,Vmin≤viT≤Vmax,Vmax,viT>Vmax.In (4), viT is, during the period from t0 to t0+T, the theoretical calculation of the guiding running speed of the bus i, km/h; Vmax is the maximum allowable running speed on the road where the line is located, km/h; Vmin is the minimum allowable running speed on the road where the line is located, km/h.


## 2.3. Evaluation Index of the Control Effect of the Full-Length Bus

The deviation degree of the first bus position isΔx1, the average headway deviation is selected as the evaluation indexes of the control strategy, and the average headway deviation is defined as the average of the absolute value of the difference between the headway and the departure interval of each bus running on the line.The degree of deviation from the position of the first bus is(5)Δx1=x1−X1X1·100%,=x1−t0−D1·Ctt0−D1·Ct·100%.The deviation degree of the average headway of the full-length bus is(6)Δh1T¯=Δh1TnT,=∑i=1nhiT−HnT,whereΔh1T¯ is the average headway deviation at t0+T time, min/veh; Δh1T is the deviation of the headway at t0+T time, min; hiT is the headway between bus i and bus i−1 at t0+T time, min; nT is the number of buses running on the line at t0+T time.


## 3. Coordinated Control Strategy of the Short-Turning Bus and Full-Length Bus

### 3.1. Dispatching Form of the Short-Turning Bus

Through the analysis of the relationship between the departure frequencyfA and fB of the full-length bus and the short-turning bus, it can be determined that there are the following three modes of departure [22]:(i)
WhenfA/fB=1, that is, a full-length bus starts alternately with a short-turning bus, set HAB1/fA=1/fB, the headway between the short-turning bus and its adjacent full-length bus is HAB/2(ii)
WhenfA/fB=g and g>1, that is, the alternating departure of the number of full-length buses is g and a short-turning bus, set 1/fA=HA, 1/fB=HB, Hb/Ha=g. Taking the situation g=3 as an example, the expected headway QH between the short-turning bus and the nearest full-length bus is analyzed; let us mark that the departure time of the bus is 0 in Figure 2; send out a short-turning bus after a full-length bus h0 time; a full-length bus will be departed after h1 time. Then, the mark bus and the analysis bus, the position information at 3h0+h1 time, and the headway relationship between short-turning buses and full-length buses are shown in Figure 3.The headway relationship between the short-turning bus and the full-length bus is(7)QH+h′=HAh′+h″=3HAh″=3Cth0+h1−60S0+Cth1Ct,⇒QH=h0+3−12HA−60S0Ct=h0+4HA−60S0Ct.By analogy, the expected headway between the short-turning bus and the nearest full-length bus is(8)QH=h0+g−12HA−60S0Ct.In (8), S0 is the location of departure station of short-turning buses, km; h’ is the headway between the short-turning bus and the rear full-length bus, min; h’‘is the headway between the short-turning bus and the third full-length bus in front of it, min;other parameters are the same as before.(iii)
WhenfB/fA=g and g>1, it is in the form of alternating departure of the number of short-turning buses that are g and a full-length bus; in the actual operation process, considering the endurance of passengers, the scheduling scheme of departing two or more short-turning buses in succession is not generally adopted.Figure 3
Schematic diagram of the departure frequency relationship between short-turning buses and full-length buses (g = 3).


### 3.2. Control Train of Thought

The running diagram of the short-turning bus is shown in Figure4, a short-turning bus and the number of full-length buses g are departed alternately, the headway between the short-turning bus and the nearest full-length bus should be equal to the expected headway QH, and the expected headway of the nearest rear full-length bus should be in accordance with h′=HA−QH. Therefore, in this paper, a short-turning bus in the operating range and the number of full-length buses are g in front of it which are regarded as a group, and the whole line is divided into several groups. The short-turning bus control mode is based on the group as the unit, and the speed control is carried out on the basis of the full-length bus nearest to the short-turning bus in the group. Ensure that the headway between the full-length buses is HA, and the headway between the short-turning bus and the nearest full-length bus running in front is equal to QH. Based on the guidance speed of the full-length bus, the guidance speed of the short-turning bus is determined.Figure 4
Schematic diagram of short-turning bus operation.


#### 3.2.1. Control Process

For the short-turning bus system, on the basis of the guide speed of the full-length bus, stabilize the headway between the short-turning busi and the nearest full-length bus j; the expected headway is QH. According to the nearest full-length bus k in the group of short-turning bus i+1, the guiding speed of bus i+1 is given; it is stable where the headway between the bus i+1 and the full-length bus k is the expected headway which is QH, and so on. It should be noted that the full-length bus control needs to be carried out in advance before the short-turning bus system is controlled. Set the regulation period of the full-length bus as t1,t2t1,t2 and the regulation period of the short-turning bus as t3,t2t3,t2; then, the relationship between the three is t1<t3<t2. Each regulation of the short-turning bus system in t3,t2 is carried out at the same time as the full-length bus control.Set a certaint time; the number of short-turning buses is n1 running on the line; that is to say, the number of groups is n1 composed of full-length buses and short-turning buses; the coding of the group adopts the coding of the short-turning bus in the group; that is, it is coded according to the order from near to far from the originating station. qhi indicates the headway between the short-turning bus and the nearest full-length bus in the group i; QCi indicates the guided operating speed of bus i in the section fleet. Take group i as an example; the control flow of the short-turning buses group is shown in Figure 5.Figure 5
Control flow of the short-turning buses group.


### 3.3. Guidance Speed Model of the Short-Turning Bus

Suppose that the number of full-length buses isn and the number of short-turning buses is m running on the line at t0 time, and the guidance speed of the full-length bus has been given; therefore, the number of groups is m formed on the line, and the short-turning bus i is taken as an example for speed control.The guidance operating speed of short-turning busi is(9)QCiT=CjT,qhi∈QH·1−θ,QH·1+θxj−qxi+CjT·TQH+T/60,qhi∉QH·1−θ,QH·1+θ..The expected guidance running speed of short-turning busi is(10)qvi=QCiT1−b/3600·λ·QCiT·qhi+1+QH/2·60.The final guidance running speed of short-turning busi is(11)QViT=Vmin,qviT<Vmin,qvi1T,Vmin≤qviT≤Vmax,Vmax,qviT>Vmax.In equations (9)–(11), qxi is the short-turning bus position at t0 time (the relative position that still takes the departure station of the full-length bus as the origin), km; xj is located in front of the short-turning bus i; the full-length bus j, which is the nearest to it, is located at t0 time, km; qhi is the headway between the short-turning bus i and the full-length bus j at t0 time, min; QH is the expected headway between the short-turning bus and its nearest full-length bus (running front), min; CjT is the guidance operating speed of the full-length bus j at t0+T time, km/h; QCiT is the guidance operating speed of the short-turning bus at t0+T time, km/h; qviT is the expected guidance running speed of the short-turning bus at t0+T time, km/h; QViT is the guidance running speed of the short-turning bus at t0+T time, km/h.


### 3.4. Evaluation Index of Coordinated Control Effect

The whole line fleet is divided into the full-length bus fleet and the short-turning bus fleet, and the average headway deviation of the whole line is defined as the sum of the average headway deviation of the full-length bus system and the average headway deviation of the short-turning bus system. The average headway deviation of the short-turning bus system is defined as the average value of the absolute difference between the headway of all the running short-turning buses on the line and the nearest full-length bus (running front) and the expected headway. The average headway deviation model of the whole line is(12)ΔhT¯=Δh1T¯+Δh2T¯=Δh1TnT+Δh2TmT=∑j=1nhjT−HnT+∑i=1mqhiT−QHmT.In (12), Δh¯ is the headway deviation of the whole system at t0+T time, min/veh; Δh is the deviation value of the headway of the whole system at t0+T time, min; Δh1T¯ is the deviation value of the average headway of the full-length bus at t0+T time, min/veh; Δh1T is the deviation value of the headway of the full-length bus at t0+T time, min; Δh2T¯ is the deviation value of the average headway of the short-turning bus at t0+T time, min/veh; Δh2T is the deviation value of the headway of the short-turning bus at t0+T time, min; qhiT is the headway between the short-turning bus i and i-1, min; hiT is the headway between the short-turning bus j and j-1, min; nT is the number of full-length buses running on the line at t0+T time; mT is the number of short-turning buses running on the line at t0+T time.


## 3.1. Dispatching Form of the Short-Turning Bus

Through the analysis of the relationship between the departure frequencyfA and fB of the full-length bus and the short-turning bus, it can be determined that there are the following three modes of departure [22]:(i)
WhenfA/fB=1, that is, a full-length bus starts alternately with a short-turning bus, set HAB1/fA=1/fB, the headway between the short-turning bus and its adjacent full-length bus is HAB/2(ii)
WhenfA/fB=g and g>1, that is, the alternating departure of the number of full-length buses is g and a short-turning bus, set 1/fA=HA, 1/fB=HB, Hb/Ha=g. Taking the situation g=3 as an example, the expected headway QH between the short-turning bus and the nearest full-length bus is analyzed; let us mark that the departure time of the bus is 0 in Figure 2; send out a short-turning bus after a full-length bus h0 time; a full-length bus will be departed after h1 time. Then, the mark bus and the analysis bus, the position information at 3h0+h1 time, and the headway relationship between short-turning buses and full-length buses are shown in Figure 3.The headway relationship between the short-turning bus and the full-length bus is(7)QH+h′=HAh′+h″=3HAh″=3Cth0+h1−60S0+Cth1Ct,⇒QH=h0+3−12HA−60S0Ct=h0+4HA−60S0Ct.By analogy, the expected headway between the short-turning bus and the nearest full-length bus is(8)QH=h0+g−12HA−60S0Ct.In (8), S0 is the location of departure station of short-turning buses, km; h’ is the headway between the short-turning bus and the rear full-length bus, min; h’‘is the headway between the short-turning bus and the third full-length bus in front of it, min;other parameters are the same as before.(iii)
WhenfB/fA=g and g>1, it is in the form of alternating departure of the number of short-turning buses that are g and a full-length bus; in the actual operation process, considering the endurance of passengers, the scheduling scheme of departing two or more short-turning buses in succession is not generally adopted.Figure 3
Schematic diagram of the departure frequency relationship between short-turning buses and full-length buses (g = 3).


## 3.2. Control Train of Thought

The running diagram of the short-turning bus is shown in Figure4, a short-turning bus and the number of full-length buses g are departed alternately, the headway between the short-turning bus and the nearest full-length bus should be equal to the expected headway QH, and the expected headway of the nearest rear full-length bus should be in accordance with h′=HA−QH. Therefore, in this paper, a short-turning bus in the operating range and the number of full-length buses are g in front of it which are regarded as a group, and the whole line is divided into several groups. The short-turning bus control mode is based on the group as the unit, and the speed control is carried out on the basis of the full-length bus nearest to the short-turning bus in the group. Ensure that the headway between the full-length buses is HA, and the headway between the short-turning bus and the nearest full-length bus running in front is equal to QH. Based on the guidance speed of the full-length bus, the guidance speed of the short-turning bus is determined.Figure 4
Schematic diagram of short-turning bus operation.


### 3.2.1. Control Process

For the short-turning bus system, on the basis of the guide speed of the full-length bus, stabilize the headway between the short-turning busi and the nearest full-length bus j; the expected headway is QH. According to the nearest full-length bus k in the group of short-turning bus i+1, the guiding speed of bus i+1 is given; it is stable where the headway between the bus i+1 and the full-length bus k is the expected headway which is QH, and so on. It should be noted that the full-length bus control needs to be carried out in advance before the short-turning bus system is controlled. Set the regulation period of the full-length bus as t1,t2t1,t2 and the regulation period of the short-turning bus as t3,t2t3,t2; then, the relationship between the three is t1<t3<t2. Each regulation of the short-turning bus system in t3,t2 is carried out at the same time as the full-length bus control.Set a certaint time; the number of short-turning buses is n1 running on the line; that is to say, the number of groups is n1 composed of full-length buses and short-turning buses; the coding of the group adopts the coding of the short-turning bus in the group; that is, it is coded according to the order from near to far from the originating station. qhi indicates the headway between the short-turning bus and the nearest full-length bus in the group i; QCi indicates the guided operating speed of bus i in the section fleet. Take group i as an example; the control flow of the short-turning buses group is shown in Figure 5.Figure 5
Control flow of the short-turning buses group.


## 3.2.1. Control Process

For the short-turning bus system, on the basis of the guide speed of the full-length bus, stabilize the headway between the short-turning busi and the nearest full-length bus j; the expected headway is QH. According to the nearest full-length bus k in the group of short-turning bus i+1, the guiding speed of bus i+1 is given; it is stable where the headway between the bus i+1 and the full-length bus k is the expected headway which is QH, and so on. It should be noted that the full-length bus control needs to be carried out in advance before the short-turning bus system is controlled. Set the regulation period of the full-length bus as t1,t2t1,t2 and the regulation period of the short-turning bus as t3,t2t3,t2; then, the relationship between the three is t1<t3<t2. Each regulation of the short-turning bus system in t3,t2 is carried out at the same time as the full-length bus control.Set a certaint time; the number of short-turning buses is n1 running on the line; that is to say, the number of groups is n1 composed of full-length buses and short-turning buses; the coding of the group adopts the coding of the short-turning bus in the group; that is, it is coded according to the order from near to far from the originating station. qhi indicates the headway between the short-turning bus and the nearest full-length bus in the group i; QCi indicates the guided operating speed of bus i in the section fleet. Take group i as an example; the control flow of the short-turning buses group is shown in Figure 5.Figure 5
Control flow of the short-turning buses group.


## 3.3. Guidance Speed Model of the Short-Turning Bus

Suppose that the number of full-length buses isn and the number of short-turning buses is m running on the line at t0 time, and the guidance speed of the full-length bus has been given; therefore, the number of groups is m formed on the line, and the short-turning bus i is taken as an example for speed control.The guidance operating speed of short-turning busi is(9)QCiT=CjT,qhi∈QH·1−θ,QH·1+θxj−qxi+CjT·TQH+T/60,qhi∉QH·1−θ,QH·1+θ..The expected guidance running speed of short-turning busi is(10)qvi=QCiT1−b/3600·λ·QCiT·qhi+1+QH/2·60.The final guidance running speed of short-turning busi is(11)QViT=Vmin,qviT<Vmin,qvi1T,Vmin≤qviT≤Vmax,Vmax,qviT>Vmax.In equations (9)–(11), qxi is the short-turning bus position at t0 time (the relative position that still takes the departure station of the full-length bus as the origin), km; xj is located in front of the short-turning bus i; the full-length bus j, which is the nearest to it, is located at t0 time, km; qhi is the headway between the short-turning bus i and the full-length bus j at t0 time, min; QH is the expected headway between the short-turning bus and its nearest full-length bus (running front), min; CjT is the guidance operating speed of the full-length bus j at t0+T time, km/h; QCiT is the guidance operating speed of the short-turning bus at t0+T time, km/h; qviT is the expected guidance running speed of the short-turning bus at t0+T time, km/h; QViT is the guidance running speed of the short-turning bus at t0+T time, km/h.


## 3.4. Evaluation Index of Coordinated Control Effect

The whole line fleet is divided into the full-length bus fleet and the short-turning bus fleet, and the average headway deviation of the whole line is defined as the sum of the average headway deviation of the full-length bus system and the average headway deviation of the short-turning bus system. The average headway deviation of the short-turning bus system is defined as the average value of the absolute difference between the headway of all the running short-turning buses on the line and the nearest full-length bus (running front) and the expected headway. The average headway deviation model of the whole line is(12)ΔhT¯=Δh1T¯+Δh2T¯=Δh1TnT+Δh2TmT=∑j=1nhjT−HnT+∑i=1mqhiT−QHmT.In (12), Δh¯ is the headway deviation of the whole system at t0+T time, min/veh; Δh is the deviation value of the headway of the whole system at t0+T time, min; Δh1T¯ is the deviation value of the average headway of the full-length bus at t0+T time, min/veh; Δh1T is the deviation value of the headway of the full-length bus at t0+T time, min; Δh2T¯ is the deviation value of the average headway of the short-turning bus at t0+T time, min/veh; Δh2T is the deviation value of the headway of the short-turning bus at t0+T time, min; qhiT is the headway between the short-turning bus i and i-1, min; hiT is the headway between the short-turning bus j and j-1, min; nT is the number of full-length buses running on the line at t0+T time; mT is the number of short-turning buses running on the line at t0+T time.


## 4. Case Analysis

Taking a bus express line in Beijing as an example, this paper uses MATLAB programming to verify the effectiveness of the coordinated control strategy in this paper: the whole 81 km of the line and the whole 73 km of the short-turning bus line. The bus line station information is shown in Table1.Table 1
Bus line stop information.
Serial numberFull-length busShort-turning busDongzhimen to Miyun bus stationMiyun bus station to DongzhimenWangjing subway station to Miyun bus stationMiyun bus station to Wangjing subway station1Dongzhimen hub stationDongzhimen hub station————2ZuojiazhuangZuojiazhuang————3Sanyuanqiao west stationSanyuanqiao west station————4Xiba riverXiba river————5——Sun palace bridgeWangjing west subway stationWangjing west subway station6Taiyang homeTaiyang homeTaiyang homeTaiyang home7Miyun Children's palaceMiyun Children's palaceMiyun Children's palaceMiyun Children's palace8Miyun west bridgeMiyun west bridgeMiyun west bridgeMiyun west bridge9Miyun orchard districtMiyun orchard districtMiyun orchard districtMiyun orchard district10Miyun drum towerMiyun drum towerMiyun drum towerMiyun drum tower11Miyun Binhe bridgeMiyun Binhe bridgeMiyun Binhe bridgeMiyun Binhe bridge12Lakeside communityLakeside communityLakeside communityLakeside community13Miyun shaheMiyun shaheMiyun shaheMiyun shahe14Binyang BeiliBinyang Beili————15Binyang districtBinyang districtBinyang districtBinyang district16Binyang XiliBinyang XiliBinyang XiliBinyang Xili17Miyun old city wallMiyun old city wall————18Guangyang districtGuangyang districtGuangyang districtGuangyang district19The great wall surrounds the islandThe great wall surrounds the islandThe great wall surrounds the islandThe great wall surrounds the island20Miyun new countrysideMiyun new countrysideMiyun new countrysideMiyun new countryside21Miyun bus stationMiyun bus stationMiyun bus stationMiyun bus station


### 4.1. Parameter Determination

In this paper, the track data of buses on a certain day on the line are collected, and the vehicle information that has been departed at the initial time of the simulation of the full-length bus and the short-turning bus (according to the order from near to far from the departure station) is shown in Table2, and the basic parameters of coordinated control strategy simulation are shown in Table 3. In the practical application of the control strategy, if the guidance speed is given too frequently, it is easy to give the next indication if the execution has not reached the expected state. Therefore, in order to make each simulation time sufficient and as short as possible to achieve the desired goal, this paper selects the minimum time distance of all running full-length buses as the simulation time. At the same time, in order to make the simulation effect closer to the actual situation, the guidance speed plus random number φφ∈−1,1 is adopted to reflect the execution effect of the bus to the regulation and control instruction.Table 2
The bus information has been departed at the initial time of the simulation of the full-length bus and the short-turning bus.
Full-length busShort-turning busHA (min)5HB (min)6Initial position (km)[0.85, 1.62, 2.9, 3.9, 5.28, 5.95, 7.69,8.33, 10.07, 10.95, 14.1, 24.12, 34.04]Initial position (km)[0.9,2.7,5.5,7.3]Initial headway (min)[5,3.2, 3.1,3, 4.4, 1.9, 4.9, 2,5.4, 2.5,8,16.9, 13.6]S0 (km)0Operating speed (km/h)[50.7, 14.5, 24.5, 19.7, 18.8, 21, 21.5,19.2, 19.5, 21.2, 23.5, 35.5, 43.7]S1 (km)73Table 3
Basic parameters of coordinated control strategy simulation.
Parameter nameb (s)αβθCtCmaxCmint0t1QH (min)Parameter value50.050.050.0326 km/h60 km/h26 km/h5 : 275 : 392.5


### 4.2. Result Analysis

#### 4.2.1. Analysis of the Output Result of the Full-Length Bus System

In this paper, the first bus control effect and the whole line bus control effect are analyzed. The simulation results of the deviation degree of the first bus position and the average headway deviation of the full-length bus under the control strategy are shown in Table4.Table 4
Simulation output results under the control strategy of the full-length bus system.
TimeAverage headway deviationmin/vehThe position of the first bus deviates%TimeAverage headway deviationmin/vehThe position of the first bus deviates%5 : 29 : 306.010.846 : 23 : 001.043.455 : 30 : 004.573.826 : 25 : 000.632.405 : 30 : 304.520.196 : 30 : 000.662.405 : 31 : 004.410.106 : 35 : 000.682.565 : 32 : 004.350.066 : 40 : 000.682.115 : 32 : 304.190.056 : 45 : 000.650.315 : 33 : 004.100.696 : 50 : 000.660.435 : 34 : 004.000.586 : 55 : 000.642.425 : 34 : 303.850.507 : 00 : 000.642.485 : 35 : 003.503.087 : 05 : 001.251.595 : 36 : 003.370.407 : 07 : 000.600.675 : 36 : 303.250.237 : 09 : 000.580.145 : 37 : 003.150.167 : 11 : 000.611.205 : 37 : 303.040.427 : 14 : 000.582.065 : 39 : 302.930.357 : 19 : 000.651.155 : 40 : 002.800.307 : 24 : 000.661.605 : 41 : 002.660.577 : 29 : 000.612.475 : 42 : 002.411.577 : 34 : 000.590.075 : 43 : 002.230.377 : 39 : 000.580.925 : 44 : 002.010.257 : 42 : 000.600.075 : 45 : 001.770.177 : 47 : 000.610.115 : 46 : 001.292.507 : 52 : 000.551.615 : 46 : 201.312.437 : 56 : 000.511.905 : 46 : 301.272.538 : 01 : 000.493.335 : 47 : 001.202.368 : 06 : 000.493.175 : 48 : 001.091.658 : 10 : 000.493.725 : 49 : 000.912.168 : 15 : 000.531.225 : 51 : 300.842.658 : 21 : 000.480.255 : 56 : 000.790.418 : 25 : 000.933.466 : 01 : 000.781.828 : 28 : 000.470.016 : 06 : 000.753.718 : 32 : 000.490.846 : 11 : 000.731.978 : 37 : 000.431.126 : 15 : 000.772.628 : 42 : 000.423.846 : 20 : 001.570.548 : 48 : 000.443.45


#### 4.2.2. Analysis of the Control Effect of the First Bus

The deviation degree of the first bus position of the full-length bus with and without control strategy is shown in Figure6. The average deviation of the first bus position under the control strategy is 1.48%, which is 0.71% lower than that without the control strategy. It shows that the control strategy can better stabilize the operation of the first bus and can better achieve the control purpose when controlling other buses based on the first bus.Figure 6
Deviation trend chart of the first bus position of the full-length bus under controlled and uncontrolled conditions.


#### 4.2.3. Control Effect of the Full-Length Bus

The variation trend of the average headway deviation of the full-length bus system with and without control strategy is shown in Figure7.Figure 7
Variation trend of average headway deviation under controlled and uncontrolled conditions.The following analytical conclusions can be drawn:(i)
After adopting the control strategy, the average headway deviation of the full-length bus shows a gradually decreasing trend and finally converges within the 1min/veh(ii)
The average headway deviation of the full-length bus with control strategy begins to stabilize at 5 : 46(iii)
The average deviation of the average headway under the control strategy is 61.47% lower than that under the uncontrolled condition(iv)
Under the control strategy, the average headway deviation curve is basically below the uncontrolled average headway deviation curveIn summary, the full-length bus speed control strategy is effective, which can better stabilize the running state of the system and can effectively prevent the occurrence of too small or too large bus intervals.


#### 4.2.4. Analysis of Simulation Results of the Short-Turning Bus

The average headway deviation of the short-turning bus system with control strategy is shown in Table5.Table 5
Simulation output results under the fleet control strategy of short-turning buses.
TimeAverage headway deviationmin/veh5 : 39 : 301.66355 : 40 : 001.65465 : 41 : 001.65455 : 42 : 001.64075 : 43 : 001.64355 : 44 : 001.29955 : 45 : 001.40975 : 46 : 001.31685 : 46 : 201.30385 : 46 : 301.31745 : 47 : 001.32725 : 48 : 001.11425 : 49 : 001.21185 : 51 : 301.08535 : 56 : 000.99496 : 01 : 000.91766 : 06 : 000.86356 : 11 : 000.58586 : 15 : 000.50546 : 20 : 000.29496 : 23 : 000.20926 : 25 : 000.13026 : 30 : 000.10636 : 35 : 000.09086 : 40 : 000.10136 : 45 : 000.09256 : 50 : 000.10856 : 55 : 000.11217 : 00 : 000.09817 : 05 : 000.21647 : 07 : 000.14217 : 09 : 000.17257 : 11 : 000.29147 : 14 : 000.28407 : 19 : 000.14597 : 24 : 000.11387 : 29 : 000.09577 : 34 : 000.11097 : 39 : 000.12927 : 42 : 000.12637 : 47 : 000.11447 : 52 : 000.09537 : 56 : 000.08738 : 01 : 000.09998 : 06 : 000.10788 : 10 : 000.10668 : 15 : 000.12418 : 21 : 000.10348 : 25 : 000.23488 : 28 : 000.34138 : 32 : 000.16718 : 37 : 000.11978 : 42 : 000.11188 : 48 : 000.1161The average headway deviation of the short-turning bus system with and without control strategy is shown in Figure8.Figure 8
Variation trend of average headway deviation of short-turning buses under controlled and uncontrolled conditions.The following analysis results can be obtained:(i)
By adopting the control strategy, the average headway deviation decreases gradually and tends to be stable.(ii)
The average headway deviation of the short-turning bus with control strategy begins to stabilize at 6 : 30, while that of the short-turning bus without control strategy begins to expand after this time.(iii)
By adopting the control strategy, the average deviation of the average headway of the short-turning bus system is 69.37% lower than that of the uncontrolled one.(iv)
Without adopting the control strategy, the average headway deviation of the short-turning bus gradually expands. After the control, the average headway deviation of the short-turning bus decreases gradually and finally converges within the 0.2min/veh.In summary, the effect of adopting the control strategy for the short-turning bus based on the full-length bus is remarkable, which can better stabilize the running state of the short-turning bus fleet, and can effectively prevent the short-turning bus and the full-length bus from being too large or too small.


#### 4.2.5. Analysis of Simulation Results of the Whole System

The average headway deviation of the whole system composed of short-turning buses and full-length buses with control strategy is shown in Table6.Table 6
Simulation output results under the whole system control strategy.
TimeAverage headway deviationmin/veh5 : 29 : 306.01445 : 30 : 004.56855 : 30 : 304.52165 : 31 : 004.40835 : 32 : 004.35335 : 32 : 304.19025 : 33 : 004.09805 : 34 : 004.00445 : 34 : 303.85475 : 35 : 003.50205 : 36 : 003.36505 : 36 : 303.24645 : 37 : 003.14525 : 37 : 303.04485 : 39 : 304.59275 : 40 : 004.45505 : 41 : 004.31335 : 42 : 004.05135 : 43 : 003.87415 : 44 : 003.31265 : 45 : 003.17665 : 46 : 002.60335 : 46 : 202.60995 : 46 : 302.58455 : 47 : 002.53175 : 48 : 002.20765 : 49 : 002.12285 : 51 : 301.92115 : 56 : 001.78636 : 01 : 001.69276 : 06 : 001.61486 : 11 : 001.32076 : 15 : 001.27766 : 20 : 001.86126 : 23 : 001.24826 : 25 : 000.76076 : 30 : 000.76666 : 35 : 000.76896 : 40 : 000.77766 : 45 : 000.73806 : 50 : 000.76426 : 55 : 000.75127 : 00 : 000.73947 : 05 : 001.46367 : 07 : 000.74117 : 09 : 000.75417 : 11 : 000.90377 : 14 : 000.86767 : 19 : 000.79507 : 24 : 000.76947 : 29 : 000.70507 : 34 : 000.69987 : 39 : 000.71227 : 42 : 000.73087 : 47 : 000.72737 : 52 : 000.65017 : 56 : 000.59628 : 01 : 000.59268 : 06 : 000.60058 : 10 : 000.60028 : 15 : 000.65608 : 21 : 000.58358 : 25 : 001.16958 : 28 : 000.81278 : 32 : 000.65338 : 37 : 000.54828 : 42 : 000.53328 : 48 : 000.5528————Under the condition of adopting control strategy and not adopting control strategy, the average headway deviation of the whole system is shown in Figure9.Figure 9
Variation trend of average headway deviation of the whole system under controlled and uncontrolled conditions.The results of the analysis are as follows:(i)
By adopting the control strategy, the average headway deviation of the whole system shows a gradually decreasing trend and finally converges within the 1min/veh(ii)
By adopting the control strategy, the average headway deviation of the whole system tends to be stable at 6 : 23(iii)
By adopting the control strategy, the average deviation of the average headway of the whole system is 69.34% lower than that of the uncontrolled one(iv)
Under the control strategy, the average headway deviation curve is basically below the uncontrolled average headway deviation curveIn summary, the coordinated control strategy of the full-length bus and the short-turning bus proposed in this paper is effective, which can better stabilize the running state of the whole system, and can effectively prevent the excessive deviation of the headway within and between the system.


## 4.1. Parameter Determination

In this paper, the track data of buses on a certain day on the line are collected, and the vehicle information that has been departed at the initial time of the simulation of the full-length bus and the short-turning bus (according to the order from near to far from the departure station) is shown in Table2, and the basic parameters of coordinated control strategy simulation are shown in Table 3. In the practical application of the control strategy, if the guidance speed is given too frequently, it is easy to give the next indication if the execution has not reached the expected state. Therefore, in order to make each simulation time sufficient and as short as possible to achieve the desired goal, this paper selects the minimum time distance of all running full-length buses as the simulation time. At the same time, in order to make the simulation effect closer to the actual situation, the guidance speed plus random number φφ∈−1,1 is adopted to reflect the execution effect of the bus to the regulation and control instruction.Table 2
The bus information has been departed at the initial time of the simulation of the full-length bus and the short-turning bus.
Full-length busShort-turning busHA (min)5HB (min)6Initial position (km)[0.85, 1.62, 2.9, 3.9, 5.28, 5.95, 7.69,8.33, 10.07, 10.95, 14.1, 24.12, 34.04]Initial position (km)[0.9,2.7,5.5,7.3]Initial headway (min)[5,3.2, 3.1,3, 4.4, 1.9, 4.9, 2,5.4, 2.5,8,16.9, 13.6]S0 (km)0Operating speed (km/h)[50.7, 14.5, 24.5, 19.7, 18.8, 21, 21.5,19.2, 19.5, 21.2, 23.5, 35.5, 43.7]S1 (km)73Table 3
Basic parameters of coordinated control strategy simulation.
Parameter nameb (s)αβθCtCmaxCmint0t1QH (min)Parameter value50.050.050.0326 km/h60 km/h26 km/h5 : 275 : 392.5


## 4.2. Result Analysis

### 4.2.1. Analysis of the Output Result of the Full-Length Bus System

In this paper, the first bus control effect and the whole line bus control effect are analyzed. The simulation results of the deviation degree of the first bus position and the average headway deviation of the full-length bus under the control strategy are shown in Table4.Table 4
Simulation output results under the control strategy of the full-length bus system.
TimeAverage headway deviationmin/vehThe position of the first bus deviates%TimeAverage headway deviationmin/vehThe position of the first bus deviates%5 : 29 : 306.010.846 : 23 : 001.043.455 : 30 : 004.573.826 : 25 : 000.632.405 : 30 : 304.520.196 : 30 : 000.662.405 : 31 : 004.410.106 : 35 : 000.682.565 : 32 : 004.350.066 : 40 : 000.682.115 : 32 : 304.190.056 : 45 : 000.650.315 : 33 : 004.100.696 : 50 : 000.660.435 : 34 : 004.000.586 : 55 : 000.642.425 : 34 : 303.850.507 : 00 : 000.642.485 : 35 : 003.503.087 : 05 : 001.251.595 : 36 : 003.370.407 : 07 : 000.600.675 : 36 : 303.250.237 : 09 : 000.580.145 : 37 : 003.150.167 : 11 : 000.611.205 : 37 : 303.040.427 : 14 : 000.582.065 : 39 : 302.930.357 : 19 : 000.651.155 : 40 : 002.800.307 : 24 : 000.661.605 : 41 : 002.660.577 : 29 : 000.612.475 : 42 : 002.411.577 : 34 : 000.590.075 : 43 : 002.230.377 : 39 : 000.580.925 : 44 : 002.010.257 : 42 : 000.600.075 : 45 : 001.770.177 : 47 : 000.610.115 : 46 : 001.292.507 : 52 : 000.551.615 : 46 : 201.312.437 : 56 : 000.511.905 : 46 : 301.272.538 : 01 : 000.493.335 : 47 : 001.202.368 : 06 : 000.493.175 : 48 : 001.091.658 : 10 : 000.493.725 : 49 : 000.912.168 : 15 : 000.531.225 : 51 : 300.842.658 : 21 : 000.480.255 : 56 : 000.790.418 : 25 : 000.933.466 : 01 : 000.781.828 : 28 : 000.470.016 : 06 : 000.753.718 : 32 : 000.490.846 : 11 : 000.731.978 : 37 : 000.431.126 : 15 : 000.772.628 : 42 : 000.423.846 : 20 : 001.570.548 : 48 : 000.443.45


### 4.2.2. Analysis of the Control Effect of the First Bus

The deviation degree of the first bus position of the full-length bus with and without control strategy is shown in Figure6. The average deviation of the first bus position under the control strategy is 1.48%, which is 0.71% lower than that without the control strategy. It shows that the control strategy can better stabilize the operation of the first bus and can better achieve the control purpose when controlling other buses based on the first bus.Figure 6
Deviation trend chart of the first bus position of the full-length bus under controlled and uncontrolled conditions.


### 4.2.3. Control Effect of the Full-Length Bus

The variation trend of the average headway deviation of the full-length bus system with and without control strategy is shown in Figure7.Figure 7
Variation trend of average headway deviation under controlled and uncontrolled conditions.The following analytical conclusions can be drawn:(i)
After adopting the control strategy, the average headway deviation of the full-length bus shows a gradually decreasing trend and finally converges within the 1min/veh(ii)
The average headway deviation of the full-length bus with control strategy begins to stabilize at 5 : 46(iii)
The average deviation of the average headway under the control strategy is 61.47% lower than that under the uncontrolled condition(iv)
Under the control strategy, the average headway deviation curve is basically below the uncontrolled average headway deviation curveIn summary, the full-length bus speed control strategy is effective, which can better stabilize the running state of the system and can effectively prevent the occurrence of too small or too large bus intervals.


### 4.2.4. Analysis of Simulation Results of the Short-Turning Bus

The average headway deviation of the short-turning bus system with control strategy is shown in Table5.Table 5
Simulation output results under the fleet control strategy of short-turning buses.
TimeAverage headway deviationmin/veh5 : 39 : 301.66355 : 40 : 001.65465 : 41 : 001.65455 : 42 : 001.64075 : 43 : 001.64355 : 44 : 001.29955 : 45 : 001.40975 : 46 : 001.31685 : 46 : 201.30385 : 46 : 301.31745 : 47 : 001.32725 : 48 : 001.11425 : 49 : 001.21185 : 51 : 301.08535 : 56 : 000.99496 : 01 : 000.91766 : 06 : 000.86356 : 11 : 000.58586 : 15 : 000.50546 : 20 : 000.29496 : 23 : 000.20926 : 25 : 000.13026 : 30 : 000.10636 : 35 : 000.09086 : 40 : 000.10136 : 45 : 000.09256 : 50 : 000.10856 : 55 : 000.11217 : 00 : 000.09817 : 05 : 000.21647 : 07 : 000.14217 : 09 : 000.17257 : 11 : 000.29147 : 14 : 000.28407 : 19 : 000.14597 : 24 : 000.11387 : 29 : 000.09577 : 34 : 000.11097 : 39 : 000.12927 : 42 : 000.12637 : 47 : 000.11447 : 52 : 000.09537 : 56 : 000.08738 : 01 : 000.09998 : 06 : 000.10788 : 10 : 000.10668 : 15 : 000.12418 : 21 : 000.10348 : 25 : 000.23488 : 28 : 000.34138 : 32 : 000.16718 : 37 : 000.11978 : 42 : 000.11188 : 48 : 000.1161The average headway deviation of the short-turning bus system with and without control strategy is shown in Figure8.Figure 8
Variation trend of average headway deviation of short-turning buses under controlled and uncontrolled conditions.The following analysis results can be obtained:(i)
By adopting the control strategy, the average headway deviation decreases gradually and tends to be stable.(ii)
The average headway deviation of the short-turning bus with control strategy begins to stabilize at 6 : 30, while that of the short-turning bus without control strategy begins to expand after this time.(iii)
By adopting the control strategy, the average deviation of the average headway of the short-turning bus system is 69.37% lower than that of the uncontrolled one.(iv)
Without adopting the control strategy, the average headway deviation of the short-turning bus gradually expands. After the control, the average headway deviation of the short-turning bus decreases gradually and finally converges within the 0.2min/veh.In summary, the effect of adopting the control strategy for the short-turning bus based on the full-length bus is remarkable, which can better stabilize the running state of the short-turning bus fleet, and can effectively prevent the short-turning bus and the full-length bus from being too large or too small.


### 4.2.5. Analysis of Simulation Results of the Whole System

The average headway deviation of the whole system composed of short-turning buses and full-length buses with control strategy is shown in Table6.Table 6
Simulation output results under the whole system control strategy.
TimeAverage headway deviationmin/veh5 : 29 : 306.01445 : 30 : 004.56855 : 30 : 304.52165 : 31 : 004.40835 : 32 : 004.35335 : 32 : 304.19025 : 33 : 004.09805 : 34 : 004.00445 : 34 : 303.85475 : 35 : 003.50205 : 36 : 003.36505 : 36 : 303.24645 : 37 : 003.14525 : 37 : 303.04485 : 39 : 304.59275 : 40 : 004.45505 : 41 : 004.31335 : 42 : 004.05135 : 43 : 003.87415 : 44 : 003.31265 : 45 : 003.17665 : 46 : 002.60335 : 46 : 202.60995 : 46 : 302.58455 : 47 : 002.53175 : 48 : 002.20765 : 49 : 002.12285 : 51 : 301.92115 : 56 : 001.78636 : 01 : 001.69276 : 06 : 001.61486 : 11 : 001.32076 : 15 : 001.27766 : 20 : 001.86126 : 23 : 001.24826 : 25 : 000.76076 : 30 : 000.76666 : 35 : 000.76896 : 40 : 000.77766 : 45 : 000.73806 : 50 : 000.76426 : 55 : 000.75127 : 00 : 000.73947 : 05 : 001.46367 : 07 : 000.74117 : 09 : 000.75417 : 11 : 000.90377 : 14 : 000.86767 : 19 : 000.79507 : 24 : 000.76947 : 29 : 000.70507 : 34 : 000.69987 : 39 : 000.71227 : 42 : 000.73087 : 47 : 000.72737 : 52 : 000.65017 : 56 : 000.59628 : 01 : 000.59268 : 06 : 000.60058 : 10 : 000.60028 : 15 : 000.65608 : 21 : 000.58358 : 25 : 001.16958 : 28 : 000.81278 : 32 : 000.65338 : 37 : 000.54828 : 42 : 000.53328 : 48 : 000.5528————Under the condition of adopting control strategy and not adopting control strategy, the average headway deviation of the whole system is shown in Figure9.Figure 9
Variation trend of average headway deviation of the whole system under controlled and uncontrolled conditions.The results of the analysis are as follows:(i)
By adopting the control strategy, the average headway deviation of the whole system shows a gradually decreasing trend and finally converges within the 1min/veh(ii)
By adopting the control strategy, the average headway deviation of the whole system tends to be stable at 6 : 23(iii)
By adopting the control strategy, the average deviation of the average headway of the whole system is 69.34% lower than that of the uncontrolled one(iv)
Under the control strategy, the average headway deviation curve is basically below the uncontrolled average headway deviation curveIn summary, the coordinated control strategy of the full-length bus and the short-turning bus proposed in this paper is effective, which can better stabilize the running state of the whole system, and can effectively prevent the excessive deviation of the headway within and between the system.


## 4.2.1. Analysis of the Output Result of the Full-Length Bus System

In this paper, the first bus control effect and the whole line bus control effect are analyzed. The simulation results of the deviation degree of the first bus position and the average headway deviation of the full-length bus under the control strategy are shown in Table4.Table 4
Simulation output results under the control strategy of the full-length bus system.
TimeAverage headway deviationmin/vehThe position of the first bus deviates%TimeAverage headway deviationmin/vehThe position of the first bus deviates%5 : 29 : 306.010.846 : 23 : 001.043.455 : 30 : 004.573.826 : 25 : 000.632.405 : 30 : 304.520.196 : 30 : 000.662.405 : 31 : 004.410.106 : 35 : 000.682.565 : 32 : 004.350.066 : 40 : 000.682.115 : 32 : 304.190.056 : 45 : 000.650.315 : 33 : 004.100.696 : 50 : 000.660.435 : 34 : 004.000.586 : 55 : 000.642.425 : 34 : 303.850.507 : 00 : 000.642.485 : 35 : 003.503.087 : 05 : 001.251.595 : 36 : 003.370.407 : 07 : 000.600.675 : 36 : 303.250.237 : 09 : 000.580.145 : 37 : 003.150.167 : 11 : 000.611.205 : 37 : 303.040.427 : 14 : 000.582.065 : 39 : 302.930.357 : 19 : 000.651.155 : 40 : 002.800.307 : 24 : 000.661.605 : 41 : 002.660.577 : 29 : 000.612.475 : 42 : 002.411.577 : 34 : 000.590.075 : 43 : 002.230.377 : 39 : 000.580.925 : 44 : 002.010.257 : 42 : 000.600.075 : 45 : 001.770.177 : 47 : 000.610.115 : 46 : 001.292.507 : 52 : 000.551.615 : 46 : 201.312.437 : 56 : 000.511.905 : 46 : 301.272.538 : 01 : 000.493.335 : 47 : 001.202.368 : 06 : 000.493.175 : 48 : 001.091.658 : 10 : 000.493.725 : 49 : 000.912.168 : 15 : 000.531.225 : 51 : 300.842.658 : 21 : 000.480.255 : 56 : 000.790.418 : 25 : 000.933.466 : 01 : 000.781.828 : 28 : 000.470.016 : 06 : 000.753.718 : 32 : 000.490.846 : 11 : 000.731.978 : 37 : 000.431.126 : 15 : 000.772.628 : 42 : 000.423.846 : 20 : 001.570.548 : 48 : 000.443.45


## 4.2.2. Analysis of the Control Effect of the First Bus

The deviation degree of the first bus position of the full-length bus with and without control strategy is shown in Figure6. The average deviation of the first bus position under the control strategy is 1.48%, which is 0.71% lower than that without the control strategy. It shows that the control strategy can better stabilize the operation of the first bus and can better achieve the control purpose when controlling other buses based on the first bus.Figure 6
Deviation trend chart of the first bus position of the full-length bus under controlled and uncontrolled conditions.


## 4.2.3. Control Effect of the Full-Length Bus

The variation trend of the average headway deviation of the full-length bus system with and without control strategy is shown in Figure7.Figure 7
Variation trend of average headway deviation under controlled and uncontrolled conditions.The following analytical conclusions can be drawn:(i)
After adopting the control strategy, the average headway deviation of the full-length bus shows a gradually decreasing trend and finally converges within the 1min/veh(ii)
The average headway deviation of the full-length bus with control strategy begins to stabilize at 5 : 46(iii)
The average deviation of the average headway under the control strategy is 61.47% lower than that under the uncontrolled condition(iv)
Under the control strategy, the average headway deviation curve is basically below the uncontrolled average headway deviation curveIn summary, the full-length bus speed control strategy is effective, which can better stabilize the running state of the system and can effectively prevent the occurrence of too small or too large bus intervals.


## 4.2.4. Analysis of Simulation Results of the Short-Turning Bus

The average headway deviation of the short-turning bus system with control strategy is shown in Table5.Table 5
Simulation output results under the fleet control strategy of short-turning buses.
TimeAverage headway deviationmin/veh5 : 39 : 301.66355 : 40 : 001.65465 : 41 : 001.65455 : 42 : 001.64075 : 43 : 001.64355 : 44 : 001.29955 : 45 : 001.40975 : 46 : 001.31685 : 46 : 201.30385 : 46 : 301.31745 : 47 : 001.32725 : 48 : 001.11425 : 49 : 001.21185 : 51 : 301.08535 : 56 : 000.99496 : 01 : 000.91766 : 06 : 000.86356 : 11 : 000.58586 : 15 : 000.50546 : 20 : 000.29496 : 23 : 000.20926 : 25 : 000.13026 : 30 : 000.10636 : 35 : 000.09086 : 40 : 000.10136 : 45 : 000.09256 : 50 : 000.10856 : 55 : 000.11217 : 00 : 000.09817 : 05 : 000.21647 : 07 : 000.14217 : 09 : 000.17257 : 11 : 000.29147 : 14 : 000.28407 : 19 : 000.14597 : 24 : 000.11387 : 29 : 000.09577 : 34 : 000.11097 : 39 : 000.12927 : 42 : 000.12637 : 47 : 000.11447 : 52 : 000.09537 : 56 : 000.08738 : 01 : 000.09998 : 06 : 000.10788 : 10 : 000.10668 : 15 : 000.12418 : 21 : 000.10348 : 25 : 000.23488 : 28 : 000.34138 : 32 : 000.16718 : 37 : 000.11978 : 42 : 000.11188 : 48 : 000.1161The average headway deviation of the short-turning bus system with and without control strategy is shown in Figure8.Figure 8
Variation trend of average headway deviation of short-turning buses under controlled and uncontrolled conditions.The following analysis results can be obtained:(i)
By adopting the control strategy, the average headway deviation decreases gradually and tends to be stable.(ii)
The average headway deviation of the short-turning bus with control strategy begins to stabilize at 6 : 30, while that of the short-turning bus without control strategy begins to expand after this time.(iii)
By adopting the control strategy, the average deviation of the average headway of the short-turning bus system is 69.37% lower than that of the uncontrolled one.(iv)
Without adopting the control strategy, the average headway deviation of the short-turning bus gradually expands. After the control, the average headway deviation of the short-turning bus decreases gradually and finally converges within the 0.2min/veh.In summary, the effect of adopting the control strategy for the short-turning bus based on the full-length bus is remarkable, which can better stabilize the running state of the short-turning bus fleet, and can effectively prevent the short-turning bus and the full-length bus from being too large or too small.


## 4.2.5. Analysis of Simulation Results of the Whole System

The average headway deviation of the whole system composed of short-turning buses and full-length buses with control strategy is shown in Table6.Table 6
Simulation output results under the whole system control strategy.
TimeAverage headway deviationmin/veh5 : 29 : 306.01445 : 30 : 004.56855 : 30 : 304.52165 : 31 : 004.40835 : 32 : 004.35335 : 32 : 304.19025 : 33 : 004.09805 : 34 : 004.00445 : 34 : 303.85475 : 35 : 003.50205 : 36 : 003.36505 : 36 : 303.24645 : 37 : 003.14525 : 37 : 303.04485 : 39 : 304.59275 : 40 : 004.45505 : 41 : 004.31335 : 42 : 004.05135 : 43 : 003.87415 : 44 : 003.31265 : 45 : 003.17665 : 46 : 002.60335 : 46 : 202.60995 : 46 : 302.58455 : 47 : 002.53175 : 48 : 002.20765 : 49 : 002.12285 : 51 : 301.92115 : 56 : 001.78636 : 01 : 001.69276 : 06 : 001.61486 : 11 : 001.32076 : 15 : 001.27766 : 20 : 001.86126 : 23 : 001.24826 : 25 : 000.76076 : 30 : 000.76666 : 35 : 000.76896 : 40 : 000.77766 : 45 : 000.73806 : 50 : 000.76426 : 55 : 000.75127 : 00 : 000.73947 : 05 : 001.46367 : 07 : 000.74117 : 09 : 000.75417 : 11 : 000.90377 : 14 : 000.86767 : 19 : 000.79507 : 24 : 000.76947 : 29 : 000.70507 : 34 : 000.69987 : 39 : 000.71227 : 42 : 000.73087 : 47 : 000.72737 : 52 : 000.65017 : 56 : 000.59628 : 01 : 000.59268 : 06 : 000.60058 : 10 : 000.60028 : 15 : 000.65608 : 21 : 000.58358 : 25 : 001.16958 : 28 : 000.81278 : 32 : 000.65338 : 37 : 000.54828 : 42 : 000.53328 : 48 : 000.5528————Under the condition of adopting control strategy and not adopting control strategy, the average headway deviation of the whole system is shown in Figure9.Figure 9
Variation trend of average headway deviation of the whole system under controlled and uncontrolled conditions.The results of the analysis are as follows:(i)
By adopting the control strategy, the average headway deviation of the whole system shows a gradually decreasing trend and finally converges within the 1min/veh(ii)
By adopting the control strategy, the average headway deviation of the whole system tends to be stable at 6 : 23(iii)
By adopting the control strategy, the average deviation of the average headway of the whole system is 69.34% lower than that of the uncontrolled one(iv)
Under the control strategy, the average headway deviation curve is basically below the uncontrolled average headway deviation curveIn summary, the coordinated control strategy of the full-length bus and the short-turning bus proposed in this paper is effective, which can better stabilize the running state of the whole system, and can effectively prevent the excessive deviation of the headway within and between the system.


## 5. Conclusions

In this paper, speed control is adopted, and the coordinated control strategy of the full-length bus and the short-turning bus is put forward. First of all, for the purpose of stabilizing the headway of full-length bus equal to the departure interval, the full-length bus control strategy based on stabilizing headway is put forward, the full-length bus guidance speed model is established, and the evaluation index of the full-length bus control strategy is given. Secondly, based on the analysis of the relationship between the full-length bus and the short-turning bus, the coordinated control strategy is put forward for the purpose to stabilize the headway between the short-turning bus and its nearest full-length bus in the form of a group. The guidance speed model of the short-turning bus is established, and the evaluation index of the full-length bus control strategy is given. Taking a bus express line in Beijing as an example, the implementation effect of the full-length bus control strategy based on stabilizing headway and the coordinated control strategy of the short-turning bus and full-length bus is verified, the implementation effect is verified by MATLAB programming, the results show that the application effect of the control strategy is remarkable, and it can stabilize the running state of the whole system, which verifies the effectiveness of the control strategy in this paper. However, the composition of bus stop time and its influencing factors are complex. In this paper, only the passenger boarding time and getting off time are considered in the study of passenger travel time cost, and the time cost caused by other factors is not taken into account. It is assumed that when the bus arrives, all passengers can get on the bus, which has a certain gap with the reality. In addition, the research background of this paper is based on the bus lines running on the roads with bus lanes. Therefore, further research will be carried out in the following aspects in the future. First, in the study of passenger travel time cost, more factors can be taken into account according to the research conditions to make the study closer to the actual situation; the second is to expand the scope of the study to bus lines on non-bus lanes.



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*Source: 3468743-2022-04-07.xml*