# Bus Predictive-Control Method considering the Impact of Traffic Lights

Authors: Maosheng Li; Yao Li
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

Amid the COVID-19 pandemic, many travelers have switched from public transit to other modes. How to maintain the stability and service quality of the bus system under regular pandemic prevention and control, so as to maintain the attractiveness of the bus, is an important research direction. Predicting operation states and adopting appropriate control measures for running buses are effective means of improving the bus system’s schedule reliability and service quality. Focusing on the impacts of intersection traffic lights on the link’s travel time durations, we establish a probabilistic prediction model for bus headways, classifying the bus headways into three states: bunching, stable, and big gap states. Based on the prediction of bus headways, the most suitable control strategy is selected by the proposed method from the plan set, such as holding control, speed-adjusting control, and stop-skipping control to minimize the bus headway deviation. Simulation experiments were employed to verify the effectiveness of the proposed method. Compared with the no-control situation, the expected headway variation, average passenger waiting time, and bus bunching frequency for 100 simulations by the proposed method are reduced by 77.73%, 41.66%, and 87.11%, respectively. Compared with some control methods without prediction, the proposed method is more robust, maintains good control performance, and reduces bus bunching despite significant variations in environmental parameters. In addition, the model still performs well when considering the execution errors of bus drivers.


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## Body

## 1. Introduction

The worldwide pandemic of COVID-19 has had a great impact on the world economy and production as well as people’s lifestyle [1]. At the beginning of the pandemic, countries imposed strict restrictions on traffic, resulting in a sharp drop in bus ridership [2–4]. On the other hand, failing to provide safe travel conditions resulted into solo driving [5, 6]. In the middle of the development of the pandemic, after some measures [7, 8], such as reducing bus capacity, wearing masks, and frequent disinfection, were implemented, the pandemic was brought under control effectively, people regained confidence in the safety of buses, and bus ridership began to recover. At present, many people have been vaccinated and the pandemic is fully under control and has entered the stage of normalized prevention and control. In China, people use the health code and the travel code for nationwide information sharing. Green of the health code means that the person is healthy, and he/she is allowed to enter public places and take public transportation. This method can guarantee the daily travel of healthy people, as long as they have a green health code. By requiring passengers to display the health code and measure body temperature when boarding the bus, bus travel has basically resumed as before. But these actions will lead to longer boarding time durations, which can lead to longer bus dwell time durations at stops and an increased risk of bus bunching. Therefore, ensuring that the stable operation of buses under regular pandemic prevention and control is important, this article aims to study the reliability of the bus system in the later stage of the pandemic.More passengers gather at a bus stop while bus arrivals lagged due to traffic congestion or intersection traffic lights, which induces extra passenger boarding time and exacerbates the delay of this bus. In turn, the following bus with fewer passengers served may depart from the bus stop earlier before the scheduled time [9]. Two or more buses may arrive at the same bus stop at almost the same time, resulting in bus bunching (or bus cluster) [10–12], which reduces the operation reliability and service quality of the bus system [13]. Therefore, it is necessary to predict the likelihood of bus bunching so that effective control measures can be implemented to distribute buses more evenly along the bus route and alleviate bus bunching.Establishing a predictive-control framework is a common idea for solving the bus bunching problem [14]. Researchers have developed various prediction models to identify the bus bunching phenomenon [15, 16]. Some fit bus link travel time durations with historical data by regression analysis or machine learning [17, 18]. The others combine real-time data with historical data to achieve real-time predictions [19, 20]. Existing studies often use a random term to represent the effect of intersection traffic lights and other disturbances such as traffic congestion, driver operational variability, and unforeseen circumstances on bus link travel time durations [21]. All of the existing research studies do not consider bus delays at intersections adequately. Traffic signal control is a common means of traffic flow control in the urban transportation system. It breaks off the continuous flow for meeting traffic demand in different directions, eliminating intersection conflicts, and has a massive impact on bus link travel time durations, so it should not be overlooked. To improve prediction accuracy, we consider the influence of intersection traffic lights and establish a headway probability prediction model of buses to calculate the bus headway probabilities and identify bus operation states.Over the past few decades, researchers have proposed and validated a variety of practical measures to address bus bunching [9], such as stop-skipping strategies [22], limited-boarding strategies [23], static or dynamic holding strategies [24–28], speed-adjusting strategies [29], transit signal priority strategies [30], bus substitution strategies [31, 32], and combined strategies [33, 34]. Holding strategies can effectively alleviate bus bunching, but they often lead to confusion and dissatisfaction among passengers on board and increase bus operating time durations and costs [35]. Although easy to implement, stop-skipping strategies are prone to a decline in passenger satisfaction. They are usually implemented when there are significant delays on the bus, no passengers needing to get off the bus, and few passengers waiting at the stop. In contrast, speed-adjusting strategies can avoid these problems and achieve similar control effects as holding and stop-skipping strategies. Studies have shown the effectiveness and practicality of speed-adjusting strategies under different traffic disturbances [29]. However, the speed-adjusting strategies are limited by the road environment and traffic conditions. When there are many disturbances in the link’s flow environment, the speed-adjusting control may not be able to restore the stability of bus headways. Therefore, to enhance the control effect and improve the robustness of the control strategy, this paper proposes a combined control method in which speed-adjusting control is adopted prior to holding control and stop-skipping control.This paper aims to determine the optimal control parameters and implement real-time control to resolve the bus bunching by calculating the bus headway probabilities and then selecting a suitable control strategy based on the results calculated. The paper is structured as follows: Section2 introduces the probability prediction method of bus headways. In Section 3, a combined control method is presented based on the prediction of the bus headway state. Section 4 conducts simulation experiments and results analysis. Some conclusions and suggestions for future research are presented in Section 5.


## 2. Probability Prediction Method of the Bus Headway

On the same bus route with the total bus stopJ, buses will follow the departure schedule to run away from the starting station. Buses can be considered to operate independently of each other, i.e., will not depart immediately after arriving at the bus stop J. Still, they will follow the departure schedule of stop J [29]. So we assume that the bus route is a one-way traffic corridor with J bus stops (Figure 1). Bus stops divide the bus route into J−1 links, and the endpoints of the link zj,j+1 are the stop j and the stop j+1, respectively. Depending on whether links contain intersection r,r=1,2,…,R, the set K of links of a bus route can be divided into two sets K′ and K″, satisfying K=K′∪K″ and K′∩K″=∅. zj,j+1∈K′ means that the link zj,j+1 contains intersections and zj,j+1∈K″ means that the link zj,j+1 does not contain intersections.Figure 1
Bus route properties.The link’s predictive-control model [29] is extended in the paper. We consider that the bus arriving at a bus stop is the beginning of the predictive-control cycle. When the bus arrives at a bus stop, the state of its headway at the next stop is predicted and a combined control will be implemented based on the prediction results. When the controlled bus arrives at the next stop, a new round of predictive control is started and so on.The innovative prediction method that considers the influence of intersection traffic lights is shown in Figure2. The prediction process is divided into three steps: bus departure time prediction, bus arrival time prediction, and bus headway probability prediction. The bus departure time prediction takes into account the impact of the passenger arrival rate on the bus dwell time at the stop, and the bus arrival time prediction fully considers the impact of the traffic lights on the link travel time. At the same time, the headway probability prediction divides the bus headways into three states, bunching, stable, and big gap states, and then calculates the probabilities of the three states, respectively. The steps of the prediction process are as follows.Figure 2
Principle of the headway probability prediction method.Firstly, the arrival time of the busi at the stop j is obtained from GPS data, and then, the number of boarding passengers Bji at the stop j and its probability PBji are predicted based on the average passenger arrival rate and the average number of alighting passengers at the stop j and the departure time dji and its probability Pdji of the bus i at the stop j are calculated based on these predicted results. Secondly, all possible travel time instances of the bus i on the link zj,j+1 affected by the intersection traffic lights and all possible arrival time instances aj+1ik,k=1,2,…,n of bus i at stop j+1 and their probability Paj+1ik,k=1,2,…,n can be calculated based on the discretized average link speeds vk∈v1,v2,…,vn. Thirdly, the probability distribution of different headways, hji, between bus i and the preceding bus at stop j+1 can be predicted based on the expected departure time instances and arrival time instances, and the probability of bus headway hji in the bunching state Psj+1i, in the stable state Pmj+1i, and in the big gap state Plj+1i are calculated, respectively. The final output of the prediction model is the probability of the three states of bus headway hji of the bus i at the bus stop j.


### 2.1. Predicting Bus Departure Time Instances

The bus dwell time at bus stops without control can generally be divided into two parts. The first part is the fixed timec, i.e., the door opening and closing time of buses after arriving at bus stops. The second part is the passenger’s boarding and alighting time, related to the number of passengers to be served at the stop [36]. Assume that the average passenger boarding time is presented by b1 sec per person, the symbol b2 sec per person is used as the average passenger alighting time, and the bus passenger capacity is described as N people per vehicle. Considering the impact of COVID-19, passengers need to show a health QR (Quick Response) code to the bus driver and measure their body temperature when boarding the bus. These operations will prolong the average boarding time of passengers. b1 is usually 2∼3 seconds per person, but if the impact of the pandemic is considered, it will be 5∼10 seconds per person. In addition, in order to maintain the social distance between passengers, the bus capacity is limited to 50% and the value of N is 60∼80 in normal time, which is 30∼40 during the pandemic. If the arrival time of the bus i at the stop j is illustrated by aji, then the dwell time sji and the departure time dji without control can be obtained as follows:(1)sji=c+maxb1×Bji,b2×Aji.(2)dji=aji+sji,where Aji represents the number of alighting passengers and Bji is the number of boarding passengers.If the number of onboard passengers of busi is Qji when it arrives at stop j and the onboard passengers get off at the stop j according to a specific rate, the number of alighting passengers at the stop j for the bus i can be calculated as follows:(3)Aji=Qji×αj,where αj represents the passenger alighting rate.Assuming that the passenger arrival pattern conforms to the Poisson distribution, the average passenger arrival rate isμ person per minute. Based on the passenger actual arrival rate of μji and the bus headway hji, the number of boarding passengers and its probability can be calculated as follows:(4)Bji=minμjihji60+Mji−1,N−Qji+Aji.(5)PBji=Pμji=k=μkk!×e−μ,where Mji−1 represents the rejected passengers by the bus i−1 at the stop j and Mji−1=max0,μji−1hji−1−N+Qji−1−Aji−1.The number of onboard passengers of the busi when it leaves the stop j is calculated based on the number of boarding and alighting passengers, which is equal to the number of onboard passengers of the bus i when it arrives at the stop j+1.(6)Qj+1i=Qji−Aji+Bji.The probability of the bus departure timedji is equal to the probability that the number of boarding passengers is Bji, which can be obtained as follows:(7)Pdji=PBji,where the departure time probability Pd1i of each bus at the first stop is 1 for all.Considering buses departing at the first stop with a fixed headwayH, the departure time of each bus at the first stop satisfies the following relationship:(8)d1i+1=d1i+i×H,where d11=0.


### 2.2. Predicting Bus Arrival Time Instances

Without interference, the bus speed conforms to the normal distribution; then, the probability density function of the bus speed under idealistic conditions is(9)pv=12πσe−v−V¯2/2σ2,where σ represents the standard deviation of bus speeds and V¯ represents the mean of bus speeds. Considering that other vehicles will affect buses and the speed limits of the general urban roads, we assume that the actual speed of the bus is within the interval Vmin,Vmax and then the speed probability of the interval −∞,Vmin and Vmax,+∞ can be superimposed on the interval Vmin,Vmax, so the actual probability of the bus speeds is obtained as follows:(10)p˜v=0,v∈−∞,Vmin∪Vmax,+∞,pv+∫v−δv+δpmdm∫VminVmaxpmdm∫−∞Vminpmdm+∫Vmax+∞pmdm,v∈Vmin,Vmax,where the interval Vmin,Vmax is divided into n segments and each segment length is 2δ=Vmax−Vmin/n, v+δ,v−δ is the range of neighborhoods of v, and δ is a sufficiently small number. The median vk,k=1,2,…,n of each segment is taken as the actual speed of the segment, and the integral of the probability density function p˜v of each segment is the probability of the speed vk. The set of the possible bus speeds is v1,v2,…,vn, and the probability of the bus speeds satisfies the following relationship:(11)Pvk=∫vk−Vmax−Vmin/2nvk+Vmax−Vmin/2np˜mdm,k=1,2,…,n.The length of the linkzj,j+1 is Lj,j+1, and the average speed of the bus i on the link zj,j+1 is vj,j+1i. If the influence of intersection traffic lights and vehicle queue length is not considered, the probability density of bus link travel time satisfies the following relationship:(12)ft=p˜Lj,j+1t,t∈Lj,j+1Vmax,Lj,j+1Vmin,0,t∈others,The departure time of the busi at the stop j is dji, and the location of the stop j is xj. If the link zj,j+1 contains n′ intersections, these intersections are indexed as rk,k=1,2,…,n′, their locations are xr1,xr2,…,xrn′, the number of vehicles queuing at intersection r is Zri, the average length of the vehicles is u meters, and the average headway distance is y meters, and the distance between the bus and the intersection rk when it stops is xrk−u+yZrki. The time of bus i arriving at the intersection rk is equal to the departure time dji of the bus i at the stop j plus the travel time xrk−u+yZrki−xj/vj,j+1i and the delay γ′∑m=1k−1ej,j+1irm of the link zj,j+1, which satisfies the following relationship:(13)dji+γ′∑m=1k−1ej,j+1irm+xrk−u+yZrki−xjvj,j+1i=Crk×τ+ω.(14)γ′=1,n′>1andk=2,3,…,n′,0,n′=1andk=1,where Crk represents the signal circle length of the intersection rk, τ is the natural number, ω is numeric in the interval 0,Crk, and ej,j+1irk represents the delay of the bus i at the intersection rk on the link zj,j+1. The intersection traffic lights considered in this paper have fixed circle length, and the yellow-light time durations are included in the red-light phase, with the green-light phase as the starting phase and βrk as the green time ratio. Then, ω≤Crkβrk represents that the bus i arrives at the intersection rk at the time when the traffic light is green; otherwise, it is red.However, the impact of intersection traffic lights on the bus link travel time cannot be ignored. There are two situations. The first is the bus arrives at the stop line when the traffic light is green. The bus can directly pass the intersection without stopping. The intersection delay is 0, and the probability of bus link travel time is the same as formula (12). The second situation is that the bus arrives at the intersection in the red-light period, needs to stop, and wait. The intersection delay depends on the vehicle queue length and the moment the bus arrives at the intersection. If the number of queuing vehicles is 0, the intersection delay is the remaining red-light time in the current cycle of the signal. If the number of queuing vehicles exceeds 0, the intersection delay is the remaining red-light time when the bus stops plus the red-light time caused by waiting for the vehicle ahead to cross the intersection. Therefore, the delay at the intersection rk is calculated as follows:(15)ej,j+1irk=0,ω≤Crkβrk,Cr−ω+Crk1−βrku+yZrkiCrkβrkvj,j+1i,Crkβrk<ω<Crk.The link delay is the sum of all intersection delays of the link, which can be obtained as follows:(16)ej,j+1i=∑m=1n′ej,j+1irm.Thus, the travel time of the busi on the link zj,j+1 can be obtained to satisfy the following relationship:(17)tj,j+1i=Lj,j+1vj,j+1i+ej,j+1i,where vj,j+1i belongs to the set v1,v2,…,vn, so tj,j+1i also has n possible values.According to intersection delays, the bus link travel timetj,j+1i for a bus travelling at the speed vj,j+1i will increase by ej,j+1i. Related studies have shown that link travel time durations offer migration patterns under the influence of intersection traffic lights [37]. Concerning this idea, the distribution of the bus link travel durations under the influence of intersection traffic lights can be obtained as follows:(18)f˜tj,j+1i=ftj,j+1i,tj,j+1i≤Lj,j+1Crβr+τ−dji−γ′ej,j+1ixrk−xj−u+yZrki+ej,j+1i,ftj,j+1i+ftj,j+1i−ej,j+1i,tj,j+1i∈others,where tj,j+1i≤Lj,j+1Crβr+τ−dji−γ′ej,j+1i/xrk−xj−u+yZrki+ej,j+1i is derived from ω≤Crkβrk of formula (15).Based on the departure timedji and the possible bus link travel time tj,j+1i, the possible arrival time aj+1i of the bus i at the stop j+1 and its probability Paj+1i can be predicted and calculated as follows:(19)aj+1i=dji+tj,j+1i.(20)Paj+1i=f˜aj+1i−dji=f˜tj,j+1i.


### 2.3. Predicting Probabilities of Different Bus Headways

The bus headway at each stop can be expressed as the difference between the bus departure time of itself and its preceding bus at the same stop [22], which can be calculated as follows:(21)hji=dji−dji−1.According to formula (7) and formula (20), we can calculate the probability of bus headways as follows:(22)Phj+1i=Pdji×Paj+1i=PBji×f˜tj,j+1i.The bus departure interval isH seconds. In this paper, the headway deviation coefficients φ1 and φ2 are set in the range of 0<φ1<1<φ2. φ1 is the minimum stable headway coefficient, which is generally taken as 0.6∼0.8 [13], and φ2 is the maximum stable headway coefficient, which can be set according to the bus route and bus departure schedule. We define that the headways in the interval −∞,φ1H indicate the bus will bunch, in the interval φ2H,+∞ indicate the bus will cause a big gap, and in the interval φ1H,φ2H indicate the bus is in the stable state. The cumulative probabilities of bus headways for these three intervals can be calculated as follows:(23)Psj+1i=Phji<φ1H.(24)Pmj+1i=Pφ1H≤hji≤φ2H.(25)Plj+1i=Phji>φ2H,where Psj+1i represents the bunching probability, Pmj+1i represents the stability probability, and Plj+1i represents the big gap probability.


### 2.4. Probabilistic Analysis of Bus Headways in the Bunching State

To investigate the impacts of the maximum speed limitVmax and the average passenger arrival rate μ on the bus headway probability in the bunching state, a bus route with 18 bus stops is established for simulation. The bunching probability and the variance of the bunching probability at each stop are calculated on different parameters. In Figure 3, it is found that the bunching probability of each stop on the bus route varies and the maximum speed limit has a negligible impact on the variance of bunching probability. In contrast, the average passenger arrival rate has an enormous impact, as shown in Figure 4. With the increase in the passenger arrival rate, the variance of bunching probability tends to be stable. In addition, the cumulative bunching probability of stops along the route decreases as the maximum speed limit increases. At the same time, no clear pattern is shown for the different average passenger arrival rates.Figure 3
Probability and variance of bus headways in the bunching state for differentVmax.Figure 4
Probability and variance of bus headways in the bunching state for differentμ.


## 2.1. Predicting Bus Departure Time Instances

The bus dwell time at bus stops without control can generally be divided into two parts. The first part is the fixed timec, i.e., the door opening and closing time of buses after arriving at bus stops. The second part is the passenger’s boarding and alighting time, related to the number of passengers to be served at the stop [36]. Assume that the average passenger boarding time is presented by b1 sec per person, the symbol b2 sec per person is used as the average passenger alighting time, and the bus passenger capacity is described as N people per vehicle. Considering the impact of COVID-19, passengers need to show a health QR (Quick Response) code to the bus driver and measure their body temperature when boarding the bus. These operations will prolong the average boarding time of passengers. b1 is usually 2∼3 seconds per person, but if the impact of the pandemic is considered, it will be 5∼10 seconds per person. In addition, in order to maintain the social distance between passengers, the bus capacity is limited to 50% and the value of N is 60∼80 in normal time, which is 30∼40 during the pandemic. If the arrival time of the bus i at the stop j is illustrated by aji, then the dwell time sji and the departure time dji without control can be obtained as follows:(1)sji=c+maxb1×Bji,b2×Aji.(2)dji=aji+sji,where Aji represents the number of alighting passengers and Bji is the number of boarding passengers.If the number of onboard passengers of busi is Qji when it arrives at stop j and the onboard passengers get off at the stop j according to a specific rate, the number of alighting passengers at the stop j for the bus i can be calculated as follows:(3)Aji=Qji×αj,where αj represents the passenger alighting rate.Assuming that the passenger arrival pattern conforms to the Poisson distribution, the average passenger arrival rate isμ person per minute. Based on the passenger actual arrival rate of μji and the bus headway hji, the number of boarding passengers and its probability can be calculated as follows:(4)Bji=minμjihji60+Mji−1,N−Qji+Aji.(5)PBji=Pμji=k=μkk!×e−μ,where Mji−1 represents the rejected passengers by the bus i−1 at the stop j and Mji−1=max0,μji−1hji−1−N+Qji−1−Aji−1.The number of onboard passengers of the busi when it leaves the stop j is calculated based on the number of boarding and alighting passengers, which is equal to the number of onboard passengers of the bus i when it arrives at the stop j+1.(6)Qj+1i=Qji−Aji+Bji.The probability of the bus departure timedji is equal to the probability that the number of boarding passengers is Bji, which can be obtained as follows:(7)Pdji=PBji,where the departure time probability Pd1i of each bus at the first stop is 1 for all.Considering buses departing at the first stop with a fixed headwayH, the departure time of each bus at the first stop satisfies the following relationship:(8)d1i+1=d1i+i×H,where d11=0.


## 2.2. Predicting Bus Arrival Time Instances

Without interference, the bus speed conforms to the normal distribution; then, the probability density function of the bus speed under idealistic conditions is(9)pv=12πσe−v−V¯2/2σ2,where σ represents the standard deviation of bus speeds and V¯ represents the mean of bus speeds. Considering that other vehicles will affect buses and the speed limits of the general urban roads, we assume that the actual speed of the bus is within the interval Vmin,Vmax and then the speed probability of the interval −∞,Vmin and Vmax,+∞ can be superimposed on the interval Vmin,Vmax, so the actual probability of the bus speeds is obtained as follows:(10)p˜v=0,v∈−∞,Vmin∪Vmax,+∞,pv+∫v−δv+δpmdm∫VminVmaxpmdm∫−∞Vminpmdm+∫Vmax+∞pmdm,v∈Vmin,Vmax,where the interval Vmin,Vmax is divided into n segments and each segment length is 2δ=Vmax−Vmin/n, v+δ,v−δ is the range of neighborhoods of v, and δ is a sufficiently small number. The median vk,k=1,2,…,n of each segment is taken as the actual speed of the segment, and the integral of the probability density function p˜v of each segment is the probability of the speed vk. The set of the possible bus speeds is v1,v2,…,vn, and the probability of the bus speeds satisfies the following relationship:(11)Pvk=∫vk−Vmax−Vmin/2nvk+Vmax−Vmin/2np˜mdm,k=1,2,…,n.The length of the linkzj,j+1 is Lj,j+1, and the average speed of the bus i on the link zj,j+1 is vj,j+1i. If the influence of intersection traffic lights and vehicle queue length is not considered, the probability density of bus link travel time satisfies the following relationship:(12)ft=p˜Lj,j+1t,t∈Lj,j+1Vmax,Lj,j+1Vmin,0,t∈others,The departure time of the busi at the stop j is dji, and the location of the stop j is xj. If the link zj,j+1 contains n′ intersections, these intersections are indexed as rk,k=1,2,…,n′, their locations are xr1,xr2,…,xrn′, the number of vehicles queuing at intersection r is Zri, the average length of the vehicles is u meters, and the average headway distance is y meters, and the distance between the bus and the intersection rk when it stops is xrk−u+yZrki. The time of bus i arriving at the intersection rk is equal to the departure time dji of the bus i at the stop j plus the travel time xrk−u+yZrki−xj/vj,j+1i and the delay γ′∑m=1k−1ej,j+1irm of the link zj,j+1, which satisfies the following relationship:(13)dji+γ′∑m=1k−1ej,j+1irm+xrk−u+yZrki−xjvj,j+1i=Crk×τ+ω.(14)γ′=1,n′>1andk=2,3,…,n′,0,n′=1andk=1,where Crk represents the signal circle length of the intersection rk, τ is the natural number, ω is numeric in the interval 0,Crk, and ej,j+1irk represents the delay of the bus i at the intersection rk on the link zj,j+1. The intersection traffic lights considered in this paper have fixed circle length, and the yellow-light time durations are included in the red-light phase, with the green-light phase as the starting phase and βrk as the green time ratio. Then, ω≤Crkβrk represents that the bus i arrives at the intersection rk at the time when the traffic light is green; otherwise, it is red.However, the impact of intersection traffic lights on the bus link travel time cannot be ignored. There are two situations. The first is the bus arrives at the stop line when the traffic light is green. The bus can directly pass the intersection without stopping. The intersection delay is 0, and the probability of bus link travel time is the same as formula (12). The second situation is that the bus arrives at the intersection in the red-light period, needs to stop, and wait. The intersection delay depends on the vehicle queue length and the moment the bus arrives at the intersection. If the number of queuing vehicles is 0, the intersection delay is the remaining red-light time in the current cycle of the signal. If the number of queuing vehicles exceeds 0, the intersection delay is the remaining red-light time when the bus stops plus the red-light time caused by waiting for the vehicle ahead to cross the intersection. Therefore, the delay at the intersection rk is calculated as follows:(15)ej,j+1irk=0,ω≤Crkβrk,Cr−ω+Crk1−βrku+yZrkiCrkβrkvj,j+1i,Crkβrk<ω<Crk.The link delay is the sum of all intersection delays of the link, which can be obtained as follows:(16)ej,j+1i=∑m=1n′ej,j+1irm.Thus, the travel time of the busi on the link zj,j+1 can be obtained to satisfy the following relationship:(17)tj,j+1i=Lj,j+1vj,j+1i+ej,j+1i,where vj,j+1i belongs to the set v1,v2,…,vn, so tj,j+1i also has n possible values.According to intersection delays, the bus link travel timetj,j+1i for a bus travelling at the speed vj,j+1i will increase by ej,j+1i. Related studies have shown that link travel time durations offer migration patterns under the influence of intersection traffic lights [37]. Concerning this idea, the distribution of the bus link travel durations under the influence of intersection traffic lights can be obtained as follows:(18)f˜tj,j+1i=ftj,j+1i,tj,j+1i≤Lj,j+1Crβr+τ−dji−γ′ej,j+1ixrk−xj−u+yZrki+ej,j+1i,ftj,j+1i+ftj,j+1i−ej,j+1i,tj,j+1i∈others,where tj,j+1i≤Lj,j+1Crβr+τ−dji−γ′ej,j+1i/xrk−xj−u+yZrki+ej,j+1i is derived from ω≤Crkβrk of formula (15).Based on the departure timedji and the possible bus link travel time tj,j+1i, the possible arrival time aj+1i of the bus i at the stop j+1 and its probability Paj+1i can be predicted and calculated as follows:(19)aj+1i=dji+tj,j+1i.(20)Paj+1i=f˜aj+1i−dji=f˜tj,j+1i.


## 2.3. Predicting Probabilities of Different Bus Headways

The bus headway at each stop can be expressed as the difference between the bus departure time of itself and its preceding bus at the same stop [22], which can be calculated as follows:(21)hji=dji−dji−1.According to formula (7) and formula (20), we can calculate the probability of bus headways as follows:(22)Phj+1i=Pdji×Paj+1i=PBji×f˜tj,j+1i.The bus departure interval isH seconds. In this paper, the headway deviation coefficients φ1 and φ2 are set in the range of 0<φ1<1<φ2. φ1 is the minimum stable headway coefficient, which is generally taken as 0.6∼0.8 [13], and φ2 is the maximum stable headway coefficient, which can be set according to the bus route and bus departure schedule. We define that the headways in the interval −∞,φ1H indicate the bus will bunch, in the interval φ2H,+∞ indicate the bus will cause a big gap, and in the interval φ1H,φ2H indicate the bus is in the stable state. The cumulative probabilities of bus headways for these three intervals can be calculated as follows:(23)Psj+1i=Phji<φ1H.(24)Pmj+1i=Pφ1H≤hji≤φ2H.(25)Plj+1i=Phji>φ2H,where Psj+1i represents the bunching probability, Pmj+1i represents the stability probability, and Plj+1i represents the big gap probability.


## 2.4. Probabilistic Analysis of Bus Headways in the Bunching State

To investigate the impacts of the maximum speed limitVmax and the average passenger arrival rate μ on the bus headway probability in the bunching state, a bus route with 18 bus stops is established for simulation. The bunching probability and the variance of the bunching probability at each stop are calculated on different parameters. In Figure 3, it is found that the bunching probability of each stop on the bus route varies and the maximum speed limit has a negligible impact on the variance of bunching probability. In contrast, the average passenger arrival rate has an enormous impact, as shown in Figure 4. With the increase in the passenger arrival rate, the variance of bunching probability tends to be stable. In addition, the cumulative bunching probability of stops along the route decreases as the maximum speed limit increases. At the same time, no clear pattern is shown for the different average passenger arrival rates.Figure 3
Probability and variance of bus headways in the bunching state for differentVmax.Figure 4
Probability and variance of bus headways in the bunching state for differentμ.


## 3. Combined Control Method

This section will propose a real-time bus control method based on the headway probability prediction, which is a two-layer decision process, as shown in Figure5.Figure 5
Principle of the combined control method.The upper layer is the control strategy selection layer. Only the speed-adjusting control is used when the bus headway at the next stop is predicted to be in a stable state. In the bunching state, the holding control and the speed-adjusting control are combined to use. In the big gap state, the stop-skipping control and the speed-adjusting control are combined.The lower layer is the control parameter optimization layer. The optimal holding time or adjusted speed and whether to skip the next stop should be determined according to the objective function at this layer.


### 3.1. Combined Control Objective Function

To make buses more evenly distribute on the bus route and avoid bus bunching, we set the deviation reduction of bus headway from the bus departure interval as the control objective and the objective function is as follows:(26)Cji=hj+1i−H.


#### 3.1.1. Stable State

If the conditionPmj+1i>maxPsj+1i,Plj+1i is satisfied, the speed-adjusting control is implemented and the driver is informed of the recommended speed in the following link. The adjustable range of speeds is assumed as interval VA,VB. The bus’ optional travel speed v˜j,j+1i on the following link satisfies inequation VA≤v˜j,j+1i≤VB and v˜j,j+1i∈v1,v2,…,vn. The value of the objective function for the corresponding case of the possible speed v˜j,j+1i is calculated, and the speed v˜j,j+1i with the smallest objective function value Cji is selected as the proposed driving speed vˇj,j+1i for the following link.


#### 3.1.2. Bunching State

If the conditionPsj+1i>maxPmj+1i,Plj+1i is satisfied, a bus bunching warning is issued to the bus driver and the driver is informed of the holding time for the current stop and the recommended speed for the following link. Assume that the holding time is in the range of interval 0,W. All possible holding time instances w˜ji at bus stops can be obtained in steps. The objective function values Cji are calculated for all possible holding time instances w˜ji and optional travel speed v˜j,j+1i. The control parameters with the smallest Cji are selected as the optimal holding time wˇji and the proposed driving speed vˇj,j+1i for the following link.


#### 3.1.3. Big Gap State

If the conditionPlj+1i>maxPsj+1i,Pmj+1i is satisfied, a big gap warning is issued to the driver and the driver is informed of the recommended driving speed for the following link and whether the next stop needs to skip. Firstly, the proposed speed vˇj,j+1i and the link travel time tˇj,j+1i of the following link are determined by minimizing the objective function, and the bus headway hˇj+1i after implementing the speed-adjusting control is calculated. If the inequation hˇj+1i≥φ2H is satisfied, that is, the bus headway at the next stop, which is being controlled, is still in a big gap state, then the bus skips the next stop and the bus dwell time of the next stop is 0. Otherwise, the bus will dwell at the next stop, so passengers can board and alight the bus at the next stop.


### 3.2. System State Update

Figure6 illustrates the predictive-control principle. The prediction module outputs the cumulative probabilities of the three states of bus headway, and the control module outputs the control parameters based on the prediction results and uses them to update the state of the bus. The updated states are then fed into the prediction module, and a new prediction and control cycle round begins.Figure 6
Description of the predictive-control framework.The control module will output the holding timewˇji, the driving speed vˇj,j+1i, and whether to skip the next station. According to these instructions, the departure time dˇji at the stop j, the arrival time at the stop j+1, and the headway at the stop j+1 of the bus i are updated as follows:(27)dˇji=aji+sji+wˇji.(28)aˇj+1i=dˇji+tˇj,j+1i.(29)hˇj+1i=aˇj+1i−aj+1i−1.


### 3.3. Evaluation Indicators

#### 3.3.1. Coefficient of Headway Variation

The coefficient of headway variation (CHV) is selected as the main evaluation index of the control performances, which can compare the variety of bus headways in different mean cases.(30)CHV=ShDh,where Sh represents the standard deviation of bus headways and Dh represents the mean of bus headways.


#### 3.3.2. Average Passenger Waiting Time

Passenger waiting time can reflect the impact of the control strategy on passengers’ service qualification and is divided into two parts: the waiting time of newly arrived passengers and the waiting time of passengers rejected by the previous bus [38]. The total passenger waiting time PWT can be calculated as follows:(31)PWT=12∑j=1J∑i=1Iμjihji2+∑j=1J∑i=1IMji−1hji.The average passenger waiting timePWT¯ is the total passenger waiting time divided by the sum of the boarding passengers and the passengers rejected by the previous bus.(32)PWT¯=PWT∑j=1J∑i=1IBji+Mji=∑j=1J∑i=1Iμjihji2+2Mji−1hji2∑j=1J∑i=1IBji+Mji.


#### 3.3.3. Total Number of Passengers Served

The number of passengers served by buses during the operation period is an essential indicator of the bus company’s operating revenue, so we count the total number of passengers served,TNP, as the third evaluation indicator.(33)TNP=∑j=1J∑i=1IBji.


#### 3.3.4. Average Running Time

The average running timeART is used to evaluate the impact of control strategies on the operating costs of bus companies.(34)ART=∑i=1IdJi−d1iJ.


#### 3.3.5. Bus Bunching Frequency

Reducing bus bunching frequency,BBF, is the primary goal of the predictive-control method, so this article counts the bus bunching frequency to evaluate the effectiveness of control strategies as follows:(35)BBF=∑j=1J∑i=1Ibbfji,where hˇji<μ1H represents that the bus headway remains in the bunching state after control; then, bbfji=1. Otherwise, bbfji=0.


## 3.1. Combined Control Objective Function

To make buses more evenly distribute on the bus route and avoid bus bunching, we set the deviation reduction of bus headway from the bus departure interval as the control objective and the objective function is as follows:(26)Cji=hj+1i−H.


### 3.1.1. Stable State

If the conditionPmj+1i>maxPsj+1i,Plj+1i is satisfied, the speed-adjusting control is implemented and the driver is informed of the recommended speed in the following link. The adjustable range of speeds is assumed as interval VA,VB. The bus’ optional travel speed v˜j,j+1i on the following link satisfies inequation VA≤v˜j,j+1i≤VB and v˜j,j+1i∈v1,v2,…,vn. The value of the objective function for the corresponding case of the possible speed v˜j,j+1i is calculated, and the speed v˜j,j+1i with the smallest objective function value Cji is selected as the proposed driving speed vˇj,j+1i for the following link.


### 3.1.2. Bunching State

If the conditionPsj+1i>maxPmj+1i,Plj+1i is satisfied, a bus bunching warning is issued to the bus driver and the driver is informed of the holding time for the current stop and the recommended speed for the following link. Assume that the holding time is in the range of interval 0,W. All possible holding time instances w˜ji at bus stops can be obtained in steps. The objective function values Cji are calculated for all possible holding time instances w˜ji and optional travel speed v˜j,j+1i. The control parameters with the smallest Cji are selected as the optimal holding time wˇji and the proposed driving speed vˇj,j+1i for the following link.


### 3.1.3. Big Gap State

If the conditionPlj+1i>maxPsj+1i,Pmj+1i is satisfied, a big gap warning is issued to the driver and the driver is informed of the recommended driving speed for the following link and whether the next stop needs to skip. Firstly, the proposed speed vˇj,j+1i and the link travel time tˇj,j+1i of the following link are determined by minimizing the objective function, and the bus headway hˇj+1i after implementing the speed-adjusting control is calculated. If the inequation hˇj+1i≥φ2H is satisfied, that is, the bus headway at the next stop, which is being controlled, is still in a big gap state, then the bus skips the next stop and the bus dwell time of the next stop is 0. Otherwise, the bus will dwell at the next stop, so passengers can board and alight the bus at the next stop.


## 3.1.1. Stable State

If the conditionPmj+1i>maxPsj+1i,Plj+1i is satisfied, the speed-adjusting control is implemented and the driver is informed of the recommended speed in the following link. The adjustable range of speeds is assumed as interval VA,VB. The bus’ optional travel speed v˜j,j+1i on the following link satisfies inequation VA≤v˜j,j+1i≤VB and v˜j,j+1i∈v1,v2,…,vn. The value of the objective function for the corresponding case of the possible speed v˜j,j+1i is calculated, and the speed v˜j,j+1i with the smallest objective function value Cji is selected as the proposed driving speed vˇj,j+1i for the following link.


## 3.1.2. Bunching State

If the conditionPsj+1i>maxPmj+1i,Plj+1i is satisfied, a bus bunching warning is issued to the bus driver and the driver is informed of the holding time for the current stop and the recommended speed for the following link. Assume that the holding time is in the range of interval 0,W. All possible holding time instances w˜ji at bus stops can be obtained in steps. The objective function values Cji are calculated for all possible holding time instances w˜ji and optional travel speed v˜j,j+1i. The control parameters with the smallest Cji are selected as the optimal holding time wˇji and the proposed driving speed vˇj,j+1i for the following link.


## 3.1.3. Big Gap State

If the conditionPlj+1i>maxPsj+1i,Pmj+1i is satisfied, a big gap warning is issued to the driver and the driver is informed of the recommended driving speed for the following link and whether the next stop needs to skip. Firstly, the proposed speed vˇj,j+1i and the link travel time tˇj,j+1i of the following link are determined by minimizing the objective function, and the bus headway hˇj+1i after implementing the speed-adjusting control is calculated. If the inequation hˇj+1i≥φ2H is satisfied, that is, the bus headway at the next stop, which is being controlled, is still in a big gap state, then the bus skips the next stop and the bus dwell time of the next stop is 0. Otherwise, the bus will dwell at the next stop, so passengers can board and alight the bus at the next stop.


## 3.2. System State Update

Figure6 illustrates the predictive-control principle. The prediction module outputs the cumulative probabilities of the three states of bus headway, and the control module outputs the control parameters based on the prediction results and uses them to update the state of the bus. The updated states are then fed into the prediction module, and a new prediction and control cycle round begins.Figure 6
Description of the predictive-control framework.The control module will output the holding timewˇji, the driving speed vˇj,j+1i, and whether to skip the next station. According to these instructions, the departure time dˇji at the stop j, the arrival time at the stop j+1, and the headway at the stop j+1 of the bus i are updated as follows:(27)dˇji=aji+sji+wˇji.(28)aˇj+1i=dˇji+tˇj,j+1i.(29)hˇj+1i=aˇj+1i−aj+1i−1.


## 3.3. Evaluation Indicators

### 3.3.1. Coefficient of Headway Variation

The coefficient of headway variation (CHV) is selected as the main evaluation index of the control performances, which can compare the variety of bus headways in different mean cases.(30)CHV=ShDh,where Sh represents the standard deviation of bus headways and Dh represents the mean of bus headways.


### 3.3.2. Average Passenger Waiting Time

Passenger waiting time can reflect the impact of the control strategy on passengers’ service qualification and is divided into two parts: the waiting time of newly arrived passengers and the waiting time of passengers rejected by the previous bus [38]. The total passenger waiting time PWT can be calculated as follows:(31)PWT=12∑j=1J∑i=1Iμjihji2+∑j=1J∑i=1IMji−1hji.The average passenger waiting timePWT¯ is the total passenger waiting time divided by the sum of the boarding passengers and the passengers rejected by the previous bus.(32)PWT¯=PWT∑j=1J∑i=1IBji+Mji=∑j=1J∑i=1Iμjihji2+2Mji−1hji2∑j=1J∑i=1IBji+Mji.


### 3.3.3. Total Number of Passengers Served

The number of passengers served by buses during the operation period is an essential indicator of the bus company’s operating revenue, so we count the total number of passengers served,TNP, as the third evaluation indicator.(33)TNP=∑j=1J∑i=1IBji.


### 3.3.4. Average Running Time

The average running timeART is used to evaluate the impact of control strategies on the operating costs of bus companies.(34)ART=∑i=1IdJi−d1iJ.


### 3.3.5. Bus Bunching Frequency

Reducing bus bunching frequency,BBF, is the primary goal of the predictive-control method, so this article counts the bus bunching frequency to evaluate the effectiveness of control strategies as follows:(35)BBF=∑j=1J∑i=1Ibbfji,where hˇji<μ1H represents that the bus headway remains in the bunching state after control; then, bbfji=1. Otherwise, bbfji=0.


## 3.3.1. Coefficient of Headway Variation

The coefficient of headway variation (CHV) is selected as the main evaluation index of the control performances, which can compare the variety of bus headways in different mean cases.(30)CHV=ShDh,where Sh represents the standard deviation of bus headways and Dh represents the mean of bus headways.


## 3.3.2. Average Passenger Waiting Time

Passenger waiting time can reflect the impact of the control strategy on passengers’ service qualification and is divided into two parts: the waiting time of newly arrived passengers and the waiting time of passengers rejected by the previous bus [38]. The total passenger waiting time PWT can be calculated as follows:(31)PWT=12∑j=1J∑i=1Iμjihji2+∑j=1J∑i=1IMji−1hji.The average passenger waiting timePWT¯ is the total passenger waiting time divided by the sum of the boarding passengers and the passengers rejected by the previous bus.(32)PWT¯=PWT∑j=1J∑i=1IBji+Mji=∑j=1J∑i=1Iμjihji2+2Mji−1hji2∑j=1J∑i=1IBji+Mji.


## 3.3.3. Total Number of Passengers Served

The number of passengers served by buses during the operation period is an essential indicator of the bus company’s operating revenue, so we count the total number of passengers served,TNP, as the third evaluation indicator.(33)TNP=∑j=1J∑i=1IBji.


## 3.3.4. Average Running Time

The average running timeART is used to evaluate the impact of control strategies on the operating costs of bus companies.(34)ART=∑i=1IdJi−d1iJ.


## 3.3.5. Bus Bunching Frequency

Reducing bus bunching frequency,BBF, is the primary goal of the predictive-control method, so this article counts the bus bunching frequency to evaluate the effectiveness of control strategies as follows:(35)BBF=∑j=1J∑i=1Ibbfji,where hˇji<μ1H represents that the bus headway remains in the bunching state after control; then, bbfji=1. Otherwise, bbfji=0.


## 4. Simulation Experiments

The previous sections describe the principle of the predictive-control method, and this section will implement simulation experiments to verify the effectiveness of the predictive-control method proposed in this paper. Meanwhile, the proposed method is compared with the holding control and the speed-adjusting control to analyze the control performance.


### 4.1. Simulation Environment and Parameters

A bus route with 18 bus stops and 8 intersections is constructed. The information about the links and intersections is shown in Tables1 and 2.Table 1
Link information.
LinksOrigin stopsDestination stopsOrigin locationDestination locationLength (m)z1,2120600600z2,3236001100500z3,43411001700600z4,54517002500800z5,65625003100600z6,76731003700600z7,87837004300600z8,98943004800500z9,1091048005500700z10,11101155006100600z11,12111261006800700z12,13121368007400600z13,14131474008000600z14,15141580008600600z15,16151686009100500z16,17161791009700600z17,181718970010300600Table 2
Intersection information.
IntersectionsLocationsRelated linksDistance from link origin (m)Distance from link destination (m)1900z2,330020022000z4,530050033400z6,730030044000z7,830030055100z9,1030040067700z12,1330030078900z15,16300200810000z17,18300300The signal cycle length is 120 seconds, the ratio of green time to cycle length is 0.6, and the starting phase is the green-light phase. In the simulation experiment, the average speed isV¯=6m/s, the variance of speed is σ2=30, the adjustable speed range is 4.74,7.14m/s, the maximum holding time is W=60s, and the departure interval between buses is H=300s. The average boarding time is b1=5 second per person, and the average alighting time b2=1 second per person. The actual passenger arrival rate is randomly generated, with a desirable range of 0,6 people per minute, and the average passenger arrival rate for all stations is μ=3 people per minute. The passenger alighting rate at the origination stop is set to 0, at the destination stop is set to 1, and at stops between them is set to a random number of [0, 1]. The headway deviation coefficients φ1 and φ2 are taken as 0.8 and 2, respectively, and the passenger capacity of buses is N=40 people.


### 4.2. Numerical Experiments

#### 4.2.1. Prediction Based on Links

Based on the arrival and departure information of the previous bus, the arrival time of the current bus, and the number of passengers on board, the headway of the current bus at the next bus stop can be predicted (Figure7).Figure 7
Individual prediction process.Assume that the arrival time of the busm at the stop k is akm=473s, the number of passengers in the bus m, Qkm, is 20, and the passenger alighting rate at the stop αk is 0.2. According to formula (3), the alighting passenger number of the bus m at the stop k is Akm=Qkm×αk=20×0.2=4. The arrival time of the bus m−1 at the stop k is akm−1=153s. According to formula (21), the headway of the bus m at the stop k is hkm=akm−akm−1=473−153=320s. If the actual arrival rate of passengers is μkm=2 people per minute, the number of passengers rejected by the bus m−1 at the stop k is Mkm−1=0. According to formula (4) and formula (5), the number of boarding passengers of the bus m at the stop k and its probability can be calculated as Bkm=10 people and PBkm=0.0842. According to formula (6), the number of passengers on board when the bus m arrives at the stop k+1 is Qk+1m=Qkm−Akm+Bkm=26 people. According to formula (1) and formula (2), we can calculate the dwell time and the departure time of the bus m at the stop k as skm=25s and dkm=akm+skm=498s, respectively. According to formula (7), the probability of the bus departure time can be calculated as Pdkm=0.0842.Suppose the length of the linkzk,k+1 is 600 m, containing an intersection that is 300 m from the link origin. According to formulas (9)∼(20), the link travel time tk,k+1m, the possible arrival time instances of the bus m at the stop k+1, and the probabilities Pak+1m can be calculated for all possible speeds. If the arrival time of the bus m−1 at the stop k+1 is ak+1m−1=313 s, according to formulas (21) and (22), the bus headway hk+1m and its probability Phk+1m can be obtained, as shown in Figure 8.Figure 8
Bus headways and their probabilities for different speeds.According to formulas (23)∼(25), the probabilities that the bus headway is in bunching, stable, and big gap states are calculated as Psk+1m=0.0046, Pmk+1m=0.0725, and Plk+1m=0.0071, respectively.


#### 4.2.2. Combined Control

Comparing the value of the probabilitiesPsk+1m, Pmk+1m, and Plk+1m, it is found that the probability Pmk+1m that the bus m is in the stable state at the stop k+1 is the largest, so the speed-adjusting control is selected.Calculating the bus headwayh˜k+1m and the objection C˜ji (Figure 9) for all possible adjusted speeds v˜j,j+1i∈v1,v2,…,vn and VA≤v˜j,j+1i≤VB, it can be found that when the adjusted speed takes the value of 5.58m/s, the bus headway is 292.5269s, while the control objective function value is the smallest, 7.4731. Therefore, the proposed speed of the following link for the bus m is chosen as vˇk,k+1m=5.58m/s, and the holding time at the stop k is wˇkm=0. According to formulas (27)∼(29), we update the departure time at the stop k, the arrival time at the stop k+1, and the bus headway for bus m at the stop k+1 after control: dˇkm=akm+skm+wˇkm=498, aˇj+1i=dˇji+tˇj,j+1i=605.5269s, and hˇj+1i=aˇj+1i−ak+1m−1=292.5269s.Figure 9
Headways and control objective function values for different adjusted speeds.


### 4.3. Other Control Strategies

#### 4.3.1. Holding Control

If the bus headwayhji of the target bus is less than the departure interval H, then the target bus is controlled at the stop with the holding time minΔhji,W. Otherwise, no control is implemented.


#### 4.3.2. Speed-Adjusting Control

The speed can be adjusted in the range ofVA,VB. The deviation of bus headway and the departure interval after the implementation of each adjusted speed is calculated, and the speed with the slightest variation of bus headway, Δhji, is selected as the optimal adjusted speed.


### 4.4. Simulation Results and Discussion

Operational data from 20 buses were selected for analysis. Bus trajectories with no control (NC), holding control (HC), speed-adjusting control (SC), and the proposed combined control (CC) for the same parameter settings are presented in Figure10.Figure 10
Bus trajectories. (a) NC. (b) HC. (c) SC. (d) CC.
(a)(b)(c)(d)Figure10(a) shows that buses without control are subject to severe bunching. Still, the amount of bunching can be significantly reduced and the stability of the headways can be improved by implementing control, whether the holding control, the speed-adjusting control, or the combined control, as shown in Figures 10(b)–10(d). However, bus bunching and a big gap occur in both HC and SC, both of which reduce the amount of bunching to some extent compared to the no-control case, but did not maintain a stable state of bus headways for the whole operation period. Figure 10(b) shows that when bus headways are greater than the bus departure interval, HC cannot maintain the stability of the bus system. Figure 10(c) indicates that SC cannot avoid bus bunching completely when the speed of the preceding bus varies widely. Bus bunching occurs in both HC and SC. But, CC under the same conditions can predict the bus headway state and its probability and then control the bus to avoid bunching, as shown in Figure 10(d). Therefore, the CC based on headway probability prediction is not only effective in reducing bus bunching and restoring headway stability, but it is also more effective and robust than the HC and the SC. The predictive-control method maintains good performance under both normal conditions and high disturbance conditions.The distribution of bus headways of the four cases is illustrated in Figure11. The results show that the headway distribution in the no-control case is most dispersed, with a large proportion of headways in the bunching and the big gap states, which indicates that the headways vary dramatically and are extremely unstable. In contrast, the headway distribution in the other three cases is more concentrated, with more headways in the stable state, which indicates that the headway stability can be effectively improved by controlling the bus. HC and SC effectively control the bus headway at the front stops on the bus route, and the bus headway can be maintained in a stable state. However, as the control errors accumulate, the headways at the rear stops in the bunching and the big gap states increase significantly, indicating that the performance of HC and SC will gradually weaken. In addition, HC cannot take the bus to catch up with the previous bus when there is a big gap between them, and the bus headway cannot restore stability in such cases. SC is more effective than the holding control, but it cannot avoid bus bunching entirely due to the adjustable speed range limitations. In contrast, CC can effectively respond to various situations and keep the bus headways in the stable state, avoiding the bus bunching phenomenon in most cases.Figure 11
Bus headway. (a) NC. (b) HC. (c) SC. (d) CC.
(a)(b)(c)(d)Figure12 shows the mean, median, and variance of the bus headways for the four cases, and it can be found that the mean and median of the bus headway for the four cases are relatively close. However, the variances of bus headways have a significant difference. The bus headway variance in NC is the largest, which improved to a more considerable extent after control. The bus headway variance and the 1.5 IQR in CC are small, which indicates that the distribution of bus headways in CC is the most concentrated and stable.Figure 12
Analysis of bus headway deviations.In order to quantify and analyze the effects of different control strategies on bus headways, passengers, operating costs, and bus bunching under various scenarios, the expected values of the performance indicators for 100 simulations are calculated as shown in Table3. The statistical results show that CC can effectively reduce the coefficient of headway variation CHV, the average passenger waiting time PWT¯, the total number of passengers served TPN, the average running time ART, and the bus bunching frequency BBF with 77.73%, 41.66%, 3.25%, 8.23%, and 87.11% improvement, respectively, and has better performance than the HC and the SC. SC has the smallest average running time ART, indicating that SC has the best operating cost savings performance.Table 3
Performance of different methods.
Control methodsNo controlHolding controlSpeed controlCombined controlECHV0.98240.59400.62230.2187EPWT¯/s300.9550231.9353230.1811175.5721ETNP/person1998202420552063EART/h1.09181.07990.98711.0020EBBF/time162.6104.1114.0420.96In addition, the total holding time and the variation of the adjusted speed were calculated separately, as shown in Table4. The results show that CC saves 52.05% of the total holding time compared with HC, effectively reducing the operating time. The variation of the adjusted speed for both SC and CC are relatively small, and the difference is negligible. The variation of the adjusted speed for CC is slightly lower than that of SC, which indicates that it is less complicated for bus drivers to operate under CC and more beneficial for bus drivers to implement control commands.Table 4
Total holding time and variance of the adjusted speed.
Control methodsTotal holding time (s)The variation of the adjusted speedHolding control9034.20—Speed control—4.0546Combined control4332.194.0121The previous simulation experiments have verified the validity and robustness of the predictive-control model during the pandemic on a theoretical level. However, in the actual operation process, the bus driver may not be able to fully execute the control instructions due to the external environment or personal operations, resulting in execution errors. In order to explore the influence of the execution error on the predictive-control model, an execution error coefficientep is set in the next experiment, that is, the optimal adjusted speed and the optimal holding time are multiplied by 1+ep as the actual values and the bus system state is updated according to the actual adjusted speed and holding time. For the command of stop-skipping, it is assumed that the bus driver can execute it completely, and the bus trajectories under the no-control (NC) situation and the predictive-control situation with different execution errors (−80%, 0, and 80%) are shown in Figure 13.Figure 13
Bus trajectories. (a) NC. (b) ep = −80%. (c) ep = 0. (d) ep = 80%.
(a)(b)(c)(d)It can be found that although the execution error reaches −80% and 80%, the predictive-control model still improves the instability of bus operation and reduces the bus bunching greatly. When the execution errorep takes a negative value, that is, the actual running speed and holding time are smaller than the optimal speed and holding time, the performance of the model is worse than when ep is greater than 0, indicating that the predictive-control model is more sensitive to the negative value of the execution error.Figure14 shows the improvement of the evaluation indicators under different execution errors compared to the no-control situation (NC). It can be found that the improvement of BBF and CHV is relatively stable when the execution error changes, maintaining more than 90% and 60%, respectively. However, when ep is negative, PWT, TPN, and ART show a relatively large reduction, indicating that the negative execution error will have a greater impact on passengers.Figure 14
Index improvement under different execution errors (ep).


## 4.1. Simulation Environment and Parameters

A bus route with 18 bus stops and 8 intersections is constructed. The information about the links and intersections is shown in Tables1 and 2.Table 1
Link information.
LinksOrigin stopsDestination stopsOrigin locationDestination locationLength (m)z1,2120600600z2,3236001100500z3,43411001700600z4,54517002500800z5,65625003100600z6,76731003700600z7,87837004300600z8,98943004800500z9,1091048005500700z10,11101155006100600z11,12111261006800700z12,13121368007400600z13,14131474008000600z14,15141580008600600z15,16151686009100500z16,17161791009700600z17,181718970010300600Table 2
Intersection information.
IntersectionsLocationsRelated linksDistance from link origin (m)Distance from link destination (m)1900z2,330020022000z4,530050033400z6,730030044000z7,830030055100z9,1030040067700z12,1330030078900z15,16300200810000z17,18300300The signal cycle length is 120 seconds, the ratio of green time to cycle length is 0.6, and the starting phase is the green-light phase. In the simulation experiment, the average speed isV¯=6m/s, the variance of speed is σ2=30, the adjustable speed range is 4.74,7.14m/s, the maximum holding time is W=60s, and the departure interval between buses is H=300s. The average boarding time is b1=5 second per person, and the average alighting time b2=1 second per person. The actual passenger arrival rate is randomly generated, with a desirable range of 0,6 people per minute, and the average passenger arrival rate for all stations is μ=3 people per minute. The passenger alighting rate at the origination stop is set to 0, at the destination stop is set to 1, and at stops between them is set to a random number of [0, 1]. The headway deviation coefficients φ1 and φ2 are taken as 0.8 and 2, respectively, and the passenger capacity of buses is N=40 people.


## 4.2. Numerical Experiments

### 4.2.1. Prediction Based on Links

Based on the arrival and departure information of the previous bus, the arrival time of the current bus, and the number of passengers on board, the headway of the current bus at the next bus stop can be predicted (Figure7).Figure 7
Individual prediction process.Assume that the arrival time of the busm at the stop k is akm=473s, the number of passengers in the bus m, Qkm, is 20, and the passenger alighting rate at the stop αk is 0.2. According to formula (3), the alighting passenger number of the bus m at the stop k is Akm=Qkm×αk=20×0.2=4. The arrival time of the bus m−1 at the stop k is akm−1=153s. According to formula (21), the headway of the bus m at the stop k is hkm=akm−akm−1=473−153=320s. If the actual arrival rate of passengers is μkm=2 people per minute, the number of passengers rejected by the bus m−1 at the stop k is Mkm−1=0. According to formula (4) and formula (5), the number of boarding passengers of the bus m at the stop k and its probability can be calculated as Bkm=10 people and PBkm=0.0842. According to formula (6), the number of passengers on board when the bus m arrives at the stop k+1 is Qk+1m=Qkm−Akm+Bkm=26 people. According to formula (1) and formula (2), we can calculate the dwell time and the departure time of the bus m at the stop k as skm=25s and dkm=akm+skm=498s, respectively. According to formula (7), the probability of the bus departure time can be calculated as Pdkm=0.0842.Suppose the length of the linkzk,k+1 is 600 m, containing an intersection that is 300 m from the link origin. According to formulas (9)∼(20), the link travel time tk,k+1m, the possible arrival time instances of the bus m at the stop k+1, and the probabilities Pak+1m can be calculated for all possible speeds. If the arrival time of the bus m−1 at the stop k+1 is ak+1m−1=313 s, according to formulas (21) and (22), the bus headway hk+1m and its probability Phk+1m can be obtained, as shown in Figure 8.Figure 8
Bus headways and their probabilities for different speeds.According to formulas (23)∼(25), the probabilities that the bus headway is in bunching, stable, and big gap states are calculated as Psk+1m=0.0046, Pmk+1m=0.0725, and Plk+1m=0.0071, respectively.


### 4.2.2. Combined Control

Comparing the value of the probabilitiesPsk+1m, Pmk+1m, and Plk+1m, it is found that the probability Pmk+1m that the bus m is in the stable state at the stop k+1 is the largest, so the speed-adjusting control is selected.Calculating the bus headwayh˜k+1m and the objection C˜ji (Figure 9) for all possible adjusted speeds v˜j,j+1i∈v1,v2,…,vn and VA≤v˜j,j+1i≤VB, it can be found that when the adjusted speed takes the value of 5.58m/s, the bus headway is 292.5269s, while the control objective function value is the smallest, 7.4731. Therefore, the proposed speed of the following link for the bus m is chosen as vˇk,k+1m=5.58m/s, and the holding time at the stop k is wˇkm=0. According to formulas (27)∼(29), we update the departure time at the stop k, the arrival time at the stop k+1, and the bus headway for bus m at the stop k+1 after control: dˇkm=akm+skm+wˇkm=498, aˇj+1i=dˇji+tˇj,j+1i=605.5269s, and hˇj+1i=aˇj+1i−ak+1m−1=292.5269s.Figure 9
Headways and control objective function values for different adjusted speeds.


## 4.2.1. Prediction Based on Links

Based on the arrival and departure information of the previous bus, the arrival time of the current bus, and the number of passengers on board, the headway of the current bus at the next bus stop can be predicted (Figure7).Figure 7
Individual prediction process.Assume that the arrival time of the busm at the stop k is akm=473s, the number of passengers in the bus m, Qkm, is 20, and the passenger alighting rate at the stop αk is 0.2. According to formula (3), the alighting passenger number of the bus m at the stop k is Akm=Qkm×αk=20×0.2=4. The arrival time of the bus m−1 at the stop k is akm−1=153s. According to formula (21), the headway of the bus m at the stop k is hkm=akm−akm−1=473−153=320s. If the actual arrival rate of passengers is μkm=2 people per minute, the number of passengers rejected by the bus m−1 at the stop k is Mkm−1=0. According to formula (4) and formula (5), the number of boarding passengers of the bus m at the stop k and its probability can be calculated as Bkm=10 people and PBkm=0.0842. According to formula (6), the number of passengers on board when the bus m arrives at the stop k+1 is Qk+1m=Qkm−Akm+Bkm=26 people. According to formula (1) and formula (2), we can calculate the dwell time and the departure time of the bus m at the stop k as skm=25s and dkm=akm+skm=498s, respectively. According to formula (7), the probability of the bus departure time can be calculated as Pdkm=0.0842.Suppose the length of the linkzk,k+1 is 600 m, containing an intersection that is 300 m from the link origin. According to formulas (9)∼(20), the link travel time tk,k+1m, the possible arrival time instances of the bus m at the stop k+1, and the probabilities Pak+1m can be calculated for all possible speeds. If the arrival time of the bus m−1 at the stop k+1 is ak+1m−1=313 s, according to formulas (21) and (22), the bus headway hk+1m and its probability Phk+1m can be obtained, as shown in Figure 8.Figure 8
Bus headways and their probabilities for different speeds.According to formulas (23)∼(25), the probabilities that the bus headway is in bunching, stable, and big gap states are calculated as Psk+1m=0.0046, Pmk+1m=0.0725, and Plk+1m=0.0071, respectively.


## 4.2.2. Combined Control

Comparing the value of the probabilitiesPsk+1m, Pmk+1m, and Plk+1m, it is found that the probability Pmk+1m that the bus m is in the stable state at the stop k+1 is the largest, so the speed-adjusting control is selected.Calculating the bus headwayh˜k+1m and the objection C˜ji (Figure 9) for all possible adjusted speeds v˜j,j+1i∈v1,v2,…,vn and VA≤v˜j,j+1i≤VB, it can be found that when the adjusted speed takes the value of 5.58m/s, the bus headway is 292.5269s, while the control objective function value is the smallest, 7.4731. Therefore, the proposed speed of the following link for the bus m is chosen as vˇk,k+1m=5.58m/s, and the holding time at the stop k is wˇkm=0. According to formulas (27)∼(29), we update the departure time at the stop k, the arrival time at the stop k+1, and the bus headway for bus m at the stop k+1 after control: dˇkm=akm+skm+wˇkm=498, aˇj+1i=dˇji+tˇj,j+1i=605.5269s, and hˇj+1i=aˇj+1i−ak+1m−1=292.5269s.Figure 9
Headways and control objective function values for different adjusted speeds.


## 4.3. Other Control Strategies

### 4.3.1. Holding Control

If the bus headwayhji of the target bus is less than the departure interval H, then the target bus is controlled at the stop with the holding time minΔhji,W. Otherwise, no control is implemented.


### 4.3.2. Speed-Adjusting Control

The speed can be adjusted in the range ofVA,VB. The deviation of bus headway and the departure interval after the implementation of each adjusted speed is calculated, and the speed with the slightest variation of bus headway, Δhji, is selected as the optimal adjusted speed.


## 4.3.1. Holding Control

If the bus headwayhji of the target bus is less than the departure interval H, then the target bus is controlled at the stop with the holding time minΔhji,W. Otherwise, no control is implemented.


## 4.3.2. Speed-Adjusting Control

The speed can be adjusted in the range ofVA,VB. The deviation of bus headway and the departure interval after the implementation of each adjusted speed is calculated, and the speed with the slightest variation of bus headway, Δhji, is selected as the optimal adjusted speed.


## 4.4. Simulation Results and Discussion

Operational data from 20 buses were selected for analysis. Bus trajectories with no control (NC), holding control (HC), speed-adjusting control (SC), and the proposed combined control (CC) for the same parameter settings are presented in Figure10.Figure 10
Bus trajectories. (a) NC. (b) HC. (c) SC. (d) CC.
(a)(b)(c)(d)Figure10(a) shows that buses without control are subject to severe bunching. Still, the amount of bunching can be significantly reduced and the stability of the headways can be improved by implementing control, whether the holding control, the speed-adjusting control, or the combined control, as shown in Figures 10(b)–10(d). However, bus bunching and a big gap occur in both HC and SC, both of which reduce the amount of bunching to some extent compared to the no-control case, but did not maintain a stable state of bus headways for the whole operation period. Figure 10(b) shows that when bus headways are greater than the bus departure interval, HC cannot maintain the stability of the bus system. Figure 10(c) indicates that SC cannot avoid bus bunching completely when the speed of the preceding bus varies widely. Bus bunching occurs in both HC and SC. But, CC under the same conditions can predict the bus headway state and its probability and then control the bus to avoid bunching, as shown in Figure 10(d). Therefore, the CC based on headway probability prediction is not only effective in reducing bus bunching and restoring headway stability, but it is also more effective and robust than the HC and the SC. The predictive-control method maintains good performance under both normal conditions and high disturbance conditions.The distribution of bus headways of the four cases is illustrated in Figure11. The results show that the headway distribution in the no-control case is most dispersed, with a large proportion of headways in the bunching and the big gap states, which indicates that the headways vary dramatically and are extremely unstable. In contrast, the headway distribution in the other three cases is more concentrated, with more headways in the stable state, which indicates that the headway stability can be effectively improved by controlling the bus. HC and SC effectively control the bus headway at the front stops on the bus route, and the bus headway can be maintained in a stable state. However, as the control errors accumulate, the headways at the rear stops in the bunching and the big gap states increase significantly, indicating that the performance of HC and SC will gradually weaken. In addition, HC cannot take the bus to catch up with the previous bus when there is a big gap between them, and the bus headway cannot restore stability in such cases. SC is more effective than the holding control, but it cannot avoid bus bunching entirely due to the adjustable speed range limitations. In contrast, CC can effectively respond to various situations and keep the bus headways in the stable state, avoiding the bus bunching phenomenon in most cases.Figure 11
Bus headway. (a) NC. (b) HC. (c) SC. (d) CC.
(a)(b)(c)(d)Figure12 shows the mean, median, and variance of the bus headways for the four cases, and it can be found that the mean and median of the bus headway for the four cases are relatively close. However, the variances of bus headways have a significant difference. The bus headway variance in NC is the largest, which improved to a more considerable extent after control. The bus headway variance and the 1.5 IQR in CC are small, which indicates that the distribution of bus headways in CC is the most concentrated and stable.Figure 12
Analysis of bus headway deviations.In order to quantify and analyze the effects of different control strategies on bus headways, passengers, operating costs, and bus bunching under various scenarios, the expected values of the performance indicators for 100 simulations are calculated as shown in Table3. The statistical results show that CC can effectively reduce the coefficient of headway variation CHV, the average passenger waiting time PWT¯, the total number of passengers served TPN, the average running time ART, and the bus bunching frequency BBF with 77.73%, 41.66%, 3.25%, 8.23%, and 87.11% improvement, respectively, and has better performance than the HC and the SC. SC has the smallest average running time ART, indicating that SC has the best operating cost savings performance.Table 3
Performance of different methods.
Control methodsNo controlHolding controlSpeed controlCombined controlECHV0.98240.59400.62230.2187EPWT¯/s300.9550231.9353230.1811175.5721ETNP/person1998202420552063EART/h1.09181.07990.98711.0020EBBF/time162.6104.1114.0420.96In addition, the total holding time and the variation of the adjusted speed were calculated separately, as shown in Table4. The results show that CC saves 52.05% of the total holding time compared with HC, effectively reducing the operating time. The variation of the adjusted speed for both SC and CC are relatively small, and the difference is negligible. The variation of the adjusted speed for CC is slightly lower than that of SC, which indicates that it is less complicated for bus drivers to operate under CC and more beneficial for bus drivers to implement control commands.Table 4
Total holding time and variance of the adjusted speed.
Control methodsTotal holding time (s)The variation of the adjusted speedHolding control9034.20—Speed control—4.0546Combined control4332.194.0121The previous simulation experiments have verified the validity and robustness of the predictive-control model during the pandemic on a theoretical level. However, in the actual operation process, the bus driver may not be able to fully execute the control instructions due to the external environment or personal operations, resulting in execution errors. In order to explore the influence of the execution error on the predictive-control model, an execution error coefficientep is set in the next experiment, that is, the optimal adjusted speed and the optimal holding time are multiplied by 1+ep as the actual values and the bus system state is updated according to the actual adjusted speed and holding time. For the command of stop-skipping, it is assumed that the bus driver can execute it completely, and the bus trajectories under the no-control (NC) situation and the predictive-control situation with different execution errors (−80%, 0, and 80%) are shown in Figure 13.Figure 13
Bus trajectories. (a) NC. (b) ep = −80%. (c) ep = 0. (d) ep = 80%.
(a)(b)(c)(d)It can be found that although the execution error reaches −80% and 80%, the predictive-control model still improves the instability of bus operation and reduces the bus bunching greatly. When the execution errorep takes a negative value, that is, the actual running speed and holding time are smaller than the optimal speed and holding time, the performance of the model is worse than when ep is greater than 0, indicating that the predictive-control model is more sensitive to the negative value of the execution error.Figure14 shows the improvement of the evaluation indicators under different execution errors compared to the no-control situation (NC). It can be found that the improvement of BBF and CHV is relatively stable when the execution error changes, maintaining more than 90% and 60%, respectively. However, when ep is negative, PWT, TPN, and ART show a relatively large reduction, indicating that the negative execution error will have a greater impact on passengers.Figure 14
Index improvement under different execution errors (ep).


## 5. Conclusion

This paper proposes a combined control method based on headway probability prediction to solve the bus bunching problem. The main findings of this paper are as follows:(1)
Compared to the no-control case, the proposed method effectively avoids the bus bunching while considering the impact of the COVID-19 pandemic, improving the coefficient of headway variationCHV, the average passenger waiting time PWT¯, the total number of passengers served TPN, the average running time ART, and the bus bunching frequency BBF by 77.73%, 41.66%, 3.25%, 8.23%, and 87.11%, respectively.(2)
The proposed method is robust with a wider range of adaptations and better performance than holding control and speed-adjusting control. It can effectively reduce the total holding time compared to holding control, saving 52.05% of bus operating time. Compared to speed-adjusting control, the variance of the adjusted speed is lower, which facilitates the bus driver to implement control commands.(3)
The proposed method shows good robustness to the bus driver’s execution errors, and the improvement of bus bunching frequencyBBF can be maintained above 90%.However, this study treats the departure interval as a constant value. The proposed model implements real-time control based on the predicted results and does not avoid bus bunching at the planning level. Therefore, future research needs to consider the factor of departure interval and pay attention to the phenomenon of less passenger demand, increased passenger boarding time, and reduced bus capacity during the pandemic, improving the bus system by adjusting the departure interval and implementing real-time prediction control to stabilize the bus system and meet the passenger demands.



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*Source: 5280109-2022-04-29.xml*