# Stop Plan of Express and Local Train for Regional Rail Transit Line

Authors: Qin Luo; Yufei Hou; Wei Li; Xiongfei Zhang
Journal: Journal of Advanced Transportation
 (2018)

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## Abstract

The urban rail transit line operating in the express and local train mode can solve the problem of disequilibrium passenger flow and space and meet the rapid arrival demand of long-distance passengers. In this paper, the Logit model is used to analyze the behavior of passengers choosing trains by considering the sensitivity of travel time and travel distance. Then, based on the composition of passenger travel time, an integer programming model for train stop scheme, aimed at minimizing the total passenger travel time, is proposed. Finally, combined with a certain regional rail line in Shenzhen, the plan is solved by genetic algorithm and evaluated through the time benefit, carrying capacity, and energy consumption efficiency. The simulation result shows that although the capacity is reduced by 6 trains, the optimized travel time per person is 10.34 min, and the energy consumption is saved by about 16%, which proves that the proposed model is efficient and feasible.


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## Body

## 1. Introduction

With the continuous expansion of urban scale and the development of urban rail transit network, the round-trip passenger flow between city centers and suburbs grows gradually. As the central city is saturated basically, more and more companies and residential buildings tend to transfer to the suburbs, which further promotes the swelling of the traffic flow. Besides, due to the extension of urban rail transit line, it forms long-distance regional rail transit lines [1]. If the train stops frequently, it will inevitably cause the increase of travel time, thereby reducing the operational efficiency and being not conductive to the efficient development of rail transit. And long-distance commuters want to arrive their destinations as soon as possible and passengers at stations along the route wish to reach the destinations without transfer. Therefore, cities need to use express and local train operating mode to meet the demands for the commuters [2]. On the one hand, from the passenger's point of view, this model can shorten the travel time of commuting passenger flow and improve the passenger service level, but at the same time, it will increase the travel time of some local passengers to a certain extent and affect their travel experience. On the other hand, for operating enterprises, the mode will save the train energy consumption, thereby enhancing operational efficiency, but the system carrying capacity will be reduced. For this reason, how to ensure the stop scheme of express trains reasonably and save the passenger travel time to an extreme is a significant project at present.In view of the demand for commuters, scholars mainly study the mode of the express and local train and the plan of the full-length and short-run route. Based on the express and local train operation mode, the literature mainly considers the travel demand of passengers. For instance, L. Wang [3] considered the shortest total travel time of the passengers to build up an optimization model. R. Tang [4] constructed a linear programming model with the goal of maximizing passenger satisfaction, which is reducing average passenger waiting time. Hassannayebi E et al. [5] studied the optimal time table of express/local train and built up a model in order to reduce the passenger waiting time by using variance and penalty function. Chun J H et al. [6] come up with the express/local train mode, which can solve the problem of passenger crowding and train delay with building the model of minimizing the number of serviceable cars. Y Lai et al. [7] took transfer passengers into account and then established an innovative network structure to study the train stop plan. Y. Gao et al. [8] proposed a biobjective programming model to achieve the simultaneous minimization of the energy consumption and the travel time. However, the above literature sources do not consider the travel process comprehensively. In the combination mode, there is a situation that express trains cross through local trains. And this part of time should not be ignored. Therefore, Canca D et al. [9, 10] also mainly considered meeting the travel demand of the passenger flow from the passenger angle by adjusting the train schedule including the overtaking factor. However, they ignored the commuting passengers' sensitivity to travel time, whose travel purpose is to arrive the destination quickly. Therefore, D. Di et al. [11, 12] established a passenger path-choice model to determine the passenger principle in path choice by considering perception difference of time without considering the effect of travel distance.In the condition of the full-length and short-run route, the recent studies mainly focused on the selection of short-run route, service level, traffic capacity, and rolling stock [13]. For example, J. Q. Liao [14], respectively, devised the operation scheme in condition of the single route and the full-length and short-run route aiming at the maximum travel saving time of all passengers. Q. Hu et al. [15] analyzed the section of short-run route from the use of the rolling stock and capacity. However, the operation scheme plan of mixed-routing cannot save the travel time for long-distance passengers and solve the problem of commuting effectively.To sum up, based on the analysis of the composition of passenger travel time and passenger selection behavior, this paper makes a reasonable stop scheme of express trains aiming to reduce the total travel time to maximize the operational efficiency of express/local train for regional rail transit line.The remainder of the paper is organized as follows. Section2 analyzes the passenger flow characteristics and the effect of travel time and carrying capacity in the condition of express and local train operating mode. In Section 3, a stopping plan optimization model of trains based on minimizing the total passenger travel time is established, and then the model is solved by genetic algorithm. In Section 4, an example of a regional line in Shenzhen is provided to optimize the train stop plan by using the model and algorithm above and evaluate it by the travel time, the capacity, and the energy consumption compared with the general plan. The final section draws the conclusions and proposes some issues for future.


## 2. Analysis of Passenger Flow Characteristics and Effects

### 2.1. Passenger Flow Characteristics

The long-distance regional rail transit line is connected in series with urban areas, suburbs, and satellite towns, and the intercity passenger flow is not only large, but also imbalanced in time direction. During the morning rush, the traffic flow moves from satellite towns or suburbs to the downtown, and at the evening peak, the situation is opposite. Meanwhile, with different levels of development along the rail transit, the passenger volume of boarding, and alighting at stations are also inconsistent, and overall, they show the disproportion of the section flow. Consequently, two types of the fluctuating passenger flow have demands for regional express rails: commuters prefer to take the through train from downtown area to the suburban, while short-distance passengers want to reach any two stations directly.


### 2.2. Travel Time Cost

Although operating express/local train prolongs the total trip time of passengers who take local train because of overtaking, it saves the time of those passengers who take express train.


### 2.3. Carrying Capacity

Express and local train scheme will reduce the capacity of lines [16]. In the traditional pattern, the line capacity nr is only related to the minimum headway I, and its calculating method is as formula (1). However, the carrying capacity of express/local train takes the deduction coefficient method for calculation and analysis. The deduction coefficient of the express train refers to the number of the local cars deducted from the parallel operation chart because of drawing an express train, and the formula is as (2) [17].(1)nr=3600I(2)εe=1+tdI-kwhere εe is the deduction coefficient of the express train, td is the difference of travel time of express/local train, and k means the times which are crossed of slow trains.


### 2.4. Train Energy Consumption

In daily operation, in addition to the maintenance of infrastructure, the energy consumption during the train operation also accounts for a large proportion [18]. And compared to the normal train mode, where trains stop at all stations, the express train only stops at some stations, so the operating scheme of the express and local train reduces the energy consumption of traction and braking at start and stop, and it can achieve the energy saving and emission reduction. In terms of the operating enterprise, it has great benefit [19]. And the equations of energy consumption calculation and parameters (Table 1) are as follows: (3)E=Et+Eb(4)Et=∑m=1f1+f2∑i=1n∫0ltFtdl(5)Eb=∑m=1f1+f2∑i=1n∫0lbFbdl-Er(6)Er=∑m=1f1+f2∑i=1n∑k=m+1f1+f2xm,k·∫0ltFbdl·ξ·tctbwhere (3) means the total energy consumption, (4) is the train traction energy consumption, (5) is the energy consumption of train braking, and (6) shows the energy saved in regenerative braking.Table 1
Energy consumption model parameters.
Parameters (unit) Explanation E (KJ) The total energy consumption E t (KJ) The traction energy consumption E b (KJ) The braking energy consumption E r (KJ) The regenerating energy l t (m) The distance of traction F t (KN) The net force of traction process xm,k Boolean variable. When themth and kth train are in the same power supply section, the value is 1; otherwise, the value is 0. l b (m) The distance of braking F b (KN) The net force of braking process ξ The energy transformation ratio t c (s) The overlapping time of the acceleration time of themth and the braking time of the kth. t b (s) The braking time of thekth train.


## 2.1. Passenger Flow Characteristics

The long-distance regional rail transit line is connected in series with urban areas, suburbs, and satellite towns, and the intercity passenger flow is not only large, but also imbalanced in time direction. During the morning rush, the traffic flow moves from satellite towns or suburbs to the downtown, and at the evening peak, the situation is opposite. Meanwhile, with different levels of development along the rail transit, the passenger volume of boarding, and alighting at stations are also inconsistent, and overall, they show the disproportion of the section flow. Consequently, two types of the fluctuating passenger flow have demands for regional express rails: commuters prefer to take the through train from downtown area to the suburban, while short-distance passengers want to reach any two stations directly.


## 2.2. Travel Time Cost

Although operating express/local train prolongs the total trip time of passengers who take local train because of overtaking, it saves the time of those passengers who take express train.


## 2.3. Carrying Capacity

Express and local train scheme will reduce the capacity of lines [16]. In the traditional pattern, the line capacity nr is only related to the minimum headway I, and its calculating method is as formula (1). However, the carrying capacity of express/local train takes the deduction coefficient method for calculation and analysis. The deduction coefficient of the express train refers to the number of the local cars deducted from the parallel operation chart because of drawing an express train, and the formula is as (2) [17].(1)nr=3600I(2)εe=1+tdI-kwhere εe is the deduction coefficient of the express train, td is the difference of travel time of express/local train, and k means the times which are crossed of slow trains.


## 2.4. Train Energy Consumption

In daily operation, in addition to the maintenance of infrastructure, the energy consumption during the train operation also accounts for a large proportion [18]. And compared to the normal train mode, where trains stop at all stations, the express train only stops at some stations, so the operating scheme of the express and local train reduces the energy consumption of traction and braking at start and stop, and it can achieve the energy saving and emission reduction. In terms of the operating enterprise, it has great benefit [19]. And the equations of energy consumption calculation and parameters (Table 1) are as follows: (3)E=Et+Eb(4)Et=∑m=1f1+f2∑i=1n∫0ltFtdl(5)Eb=∑m=1f1+f2∑i=1n∫0lbFbdl-Er(6)Er=∑m=1f1+f2∑i=1n∑k=m+1f1+f2xm,k·∫0ltFbdl·ξ·tctbwhere (3) means the total energy consumption, (4) is the train traction energy consumption, (5) is the energy consumption of train braking, and (6) shows the energy saved in regenerative braking.Table 1
Energy consumption model parameters.
Parameters (unit) Explanation E (KJ) The total energy consumption E t (KJ) The traction energy consumption E b (KJ) The braking energy consumption E r (KJ) The regenerating energy l t (m) The distance of traction F t (KN) The net force of traction process xm,k Boolean variable. When themth and kth train are in the same power supply section, the value is 1; otherwise, the value is 0. l b (m) The distance of braking F b (KN) The net force of braking process ξ The energy transformation ratio t c (s) The overlapping time of the acceleration time of themth and the braking time of the kth. t b (s) The braking time of thekth train.


## 3. Optimization Model and Algorithm

### 3.1. Problem Description

First, operating express/local train combination plan is mainly for commuting problem, so this paper chooses the morning peak as the optimization period to study. Second, two kinds of trains own the same technical parameters. Third, although the line is divided into the upstream and downstream direction, the model only discusses the one-way situation where the passenger flow is larger. And then, the regional rail transit line which this paper studies can meet the demand of passenger flow without the situation of passenger retention. The last assumption is that passengers’ arriving can all be considered to follow the normal distribution.


### 3.2. Modeling

Passenger travel timet includes waiting time tw and running time tr like formula (7), in which the total passenger waiting time consists of two parts: the first part is the normal passenger waiting time when the foregoing local train is not overtook, and the second part is the increased passenger waiting time at stations caused by overtaking. And tr includes the interval running time and station dwell time.(7)t=tw+tr.


#### 3.2.1. Analysis of Passenger Selecting Behavior for Train

The research object is commuting passenger flow with early morning rush hour. Considering that passengers are generally unwilling to transfer, this paper assumes that passengers at most experience one transfer in the same line and same direction. Besides, when OD are both express train stations, passengers will not choose the transfer way from express to local train [20]. Therefore, according to the different nature of origin and destination, all kinds of the travel selection behavior are shown in Table 2.Table 2
Passenger selection behavior.
OD Local train station Express train station Local train station Local train Local train Local train→Express train  Express train station Local train Local train Local train→Express train Express train→Local train Express trainThe behavior of passengers selecting trains can be analyzed by Logit model like formula (8) [21]. And (9) shows the passenger flow who selects some kind of travel mode [22]. (8)pijk=exp⁡-θ·cijk∑k∈Sexp⁡-θ·cijkwhere pijk is the passenger ratio choosing the mode k to travel from station i to j, θ is the passenger expectation variable based on travel impedance which is inversely proportional to the passenger expectation, cijk is the impedance choosing the mode k to travel from station i to j, and S is collection of travel modes. (9)qk=qij·Pijkwhere qk is the passengers choosing mode k to travel and qij is the passengers flowing from i to j.Passenger travel impedance is mainly affected by both the train running timety and travel distance l, and the combined impedance is determined by the weight of both.(1) Train Running Time. When passengers travel from local to express station or just the reverse, they need to think about transferring. The principles are as follows.( 1 )   t t r a n s > t l, where ttrans is the total time of changing trains and tl only refers to the time of local train. In addition, the transfer time is as (10)ttrans=tw1+tr1+tw2+tr2where tw1 and tr1, respectively, represent the waiting time and running time including dwell time of the first train for passenger; tw2 and tr2, respectively, represent the same meaning for the second train.( 2 ) When passengers consider taking local train to express, they not only need to take the total time into account, but also need to think of whether the middle has express station to supply to transfer.(2) Travel Distance. The operation ratio of express and local train can be estimated by the number of passengers whose travel distance is greater than five sections and less than five sections, like the following formula [23]:(11)p=∑i-j>5qij∑i-j≤5qij.From the above two parts, the combined impedance is as (12) and the flow chart of passengers train-choice behavior is as shown in Figure 1.(12)cijk=μ1·ty+μ2·lwhere μ1 and μ2 are the weight of train running time and travel distance, and they depend on the specific situation.Figure 1
The flow chart of passenger train-choice behavior.


#### 3.2.2. Objective Function

Total travel timet is calculated as formula (13) and the model parameters are listed in Table 3. (13)t=∑m=1f1qij-ml+xq·∑m=1f2qij-me·T2·c1f1+c2f2+∑k=1i-1c1hkxk+tij+∑w=i+1j-1ts+c1xwhw.Table 3
Model parameters.
Parameters (unit) Explanation f 1 (time/h) The operating frequency of the local train f 2 (time/h) The operating frequency of the express train x q Boolean variable. When thei station and j station are both local stations, the value is zero. q i j - m l (person) The number of passengers waiting for themth train which is the local train q i j - m e (person) The number of passengers waiting to themth train which is the local train T (min) The length of the optimized period c1/c2 Boolean variable. When passengers take the local train, the value ofc1 is one, c2 is zero, and when passengers take the express train, the value of c1 is zero, c2 is one. h k (min) The increased delay time because the slow train is crossed by the express train at preceding stations x k Boolean variable. When the slow train is crossed by the express train at thekth station, the value is one; otherwise, the value is zero. t i j (min) The interval running time of trains t s (min) The dwelling time at stations x w Boolean variable. When the local train is crossed by the express train at thewth station, the value is one; otherwise, the value is zero. h w (min) The increased dwell time in the trains because the local train is crossed by the express train at this station q i j - m (min) The number of passengers waiting for themth train ηmax Maximum section load factor a Passenger capacity of the trainIn conclusion, when the number of stations isn, the paper establishes the following objective function to achieve the minimization of the total passenger travel time as follows:(14)min⁡Z=min⁡∑i=1n-1∑j=i+1nts.t.qij-m≤ηmax·afd≥1d=1,2,3.


### 3.3. Solution Algorithm

The evolutionary algorithm, which is represented by genetic algorithm, uses crossover and mutation operator to realize the information exchange and local search between individuals in the group and provides each individual with the opportunity of optimization [24]. Through the competitive selection mechanism of survival of the fittest, the evolutionary algorithm guides the population to the better. Exactly, the unique binary coding rule of genetic algorithm is suitable for the value of 0-1 of the train stop strategy. Besides, instead of starting from a point and optimizing according to a fixed route, it searches the whole feasible solution space at the same time, so it can avoid falling into a local optimum and find the global optimal results. The steps are as follows.( 1 ) Population initialization. It generates N binary coded individuals randomly, and the length is the number of stations, where in code, 1 means that express train will stop at the corresponding station and 0 means no stopping. What is more, because the express train must stop at the original and terminal stations, the encoding is 1.( 2 ) Ensuring the fitness function. It constructs the fitness function as formula (15). The greater the fitness value, the better the individual.(15)ux=M-fxwhere M means the total time of general mode and fx is the time of combination mode with every plan.( 3 ) Taking the operations of selecting, crossover, and mutation. The selection operation adopts roulette method, the crossover operator is uniform crossover, and the mutation operator is a single point mutation.( 4 ) After an iteration, the worst individual is replaced by the best individual in the current generation, that is also called “survival of the fittest”.( 5 ) With the requirement of iterations, steps 4 and 5 are repeated.


## 3.1. Problem Description

First, operating express/local train combination plan is mainly for commuting problem, so this paper chooses the morning peak as the optimization period to study. Second, two kinds of trains own the same technical parameters. Third, although the line is divided into the upstream and downstream direction, the model only discusses the one-way situation where the passenger flow is larger. And then, the regional rail transit line which this paper studies can meet the demand of passenger flow without the situation of passenger retention. The last assumption is that passengers’ arriving can all be considered to follow the normal distribution.


## 3.2. Modeling

Passenger travel timet includes waiting time tw and running time tr like formula (7), in which the total passenger waiting time consists of two parts: the first part is the normal passenger waiting time when the foregoing local train is not overtook, and the second part is the increased passenger waiting time at stations caused by overtaking. And tr includes the interval running time and station dwell time.(7)t=tw+tr.


### 3.2.1. Analysis of Passenger Selecting Behavior for Train

The research object is commuting passenger flow with early morning rush hour. Considering that passengers are generally unwilling to transfer, this paper assumes that passengers at most experience one transfer in the same line and same direction. Besides, when OD are both express train stations, passengers will not choose the transfer way from express to local train [20]. Therefore, according to the different nature of origin and destination, all kinds of the travel selection behavior are shown in Table 2.Table 2
Passenger selection behavior.
OD Local train station Express train station Local train station Local train Local train Local train→Express train  Express train station Local train Local train Local train→Express train Express train→Local train Express trainThe behavior of passengers selecting trains can be analyzed by Logit model like formula (8) [21]. And (9) shows the passenger flow who selects some kind of travel mode [22]. (8)pijk=exp⁡-θ·cijk∑k∈Sexp⁡-θ·cijkwhere pijk is the passenger ratio choosing the mode k to travel from station i to j, θ is the passenger expectation variable based on travel impedance which is inversely proportional to the passenger expectation, cijk is the impedance choosing the mode k to travel from station i to j, and S is collection of travel modes. (9)qk=qij·Pijkwhere qk is the passengers choosing mode k to travel and qij is the passengers flowing from i to j.Passenger travel impedance is mainly affected by both the train running timety and travel distance l, and the combined impedance is determined by the weight of both.(1) Train Running Time. When passengers travel from local to express station or just the reverse, they need to think about transferring. The principles are as follows.( 1 )   t t r a n s > t l, where ttrans is the total time of changing trains and tl only refers to the time of local train. In addition, the transfer time is as (10)ttrans=tw1+tr1+tw2+tr2where tw1 and tr1, respectively, represent the waiting time and running time including dwell time of the first train for passenger; tw2 and tr2, respectively, represent the same meaning for the second train.( 2 ) When passengers consider taking local train to express, they not only need to take the total time into account, but also need to think of whether the middle has express station to supply to transfer.(2) Travel Distance. The operation ratio of express and local train can be estimated by the number of passengers whose travel distance is greater than five sections and less than five sections, like the following formula [23]:(11)p=∑i-j>5qij∑i-j≤5qij.From the above two parts, the combined impedance is as (12) and the flow chart of passengers train-choice behavior is as shown in Figure 1.(12)cijk=μ1·ty+μ2·lwhere μ1 and μ2 are the weight of train running time and travel distance, and they depend on the specific situation.Figure 1
The flow chart of passenger train-choice behavior.


### 3.2.2. Objective Function

Total travel timet is calculated as formula (13) and the model parameters are listed in Table 3. (13)t=∑m=1f1qij-ml+xq·∑m=1f2qij-me·T2·c1f1+c2f2+∑k=1i-1c1hkxk+tij+∑w=i+1j-1ts+c1xwhw.Table 3
Model parameters.
Parameters (unit) Explanation f 1 (time/h) The operating frequency of the local train f 2 (time/h) The operating frequency of the express train x q Boolean variable. When thei station and j station are both local stations, the value is zero. q i j - m l (person) The number of passengers waiting for themth train which is the local train q i j - m e (person) The number of passengers waiting to themth train which is the local train T (min) The length of the optimized period c1/c2 Boolean variable. When passengers take the local train, the value ofc1 is one, c2 is zero, and when passengers take the express train, the value of c1 is zero, c2 is one. h k (min) The increased delay time because the slow train is crossed by the express train at preceding stations x k Boolean variable. When the slow train is crossed by the express train at thekth station, the value is one; otherwise, the value is zero. t i j (min) The interval running time of trains t s (min) The dwelling time at stations x w Boolean variable. When the local train is crossed by the express train at thewth station, the value is one; otherwise, the value is zero. h w (min) The increased dwell time in the trains because the local train is crossed by the express train at this station q i j - m (min) The number of passengers waiting for themth train ηmax Maximum section load factor a Passenger capacity of the trainIn conclusion, when the number of stations isn, the paper establishes the following objective function to achieve the minimization of the total passenger travel time as follows:(14)min⁡Z=min⁡∑i=1n-1∑j=i+1nts.t.qij-m≤ηmax·afd≥1d=1,2,3.


## 3.2.1. Analysis of Passenger Selecting Behavior for Train

The research object is commuting passenger flow with early morning rush hour. Considering that passengers are generally unwilling to transfer, this paper assumes that passengers at most experience one transfer in the same line and same direction. Besides, when OD are both express train stations, passengers will not choose the transfer way from express to local train [20]. Therefore, according to the different nature of origin and destination, all kinds of the travel selection behavior are shown in Table 2.Table 2
Passenger selection behavior.
OD Local train station Express train station Local train station Local train Local train Local train→Express train  Express train station Local train Local train Local train→Express train Express train→Local train Express trainThe behavior of passengers selecting trains can be analyzed by Logit model like formula (8) [21]. And (9) shows the passenger flow who selects some kind of travel mode [22]. (8)pijk=exp⁡-θ·cijk∑k∈Sexp⁡-θ·cijkwhere pijk is the passenger ratio choosing the mode k to travel from station i to j, θ is the passenger expectation variable based on travel impedance which is inversely proportional to the passenger expectation, cijk is the impedance choosing the mode k to travel from station i to j, and S is collection of travel modes. (9)qk=qij·Pijkwhere qk is the passengers choosing mode k to travel and qij is the passengers flowing from i to j.Passenger travel impedance is mainly affected by both the train running timety and travel distance l, and the combined impedance is determined by the weight of both.(1) Train Running Time. When passengers travel from local to express station or just the reverse, they need to think about transferring. The principles are as follows.( 1 )   t t r a n s > t l, where ttrans is the total time of changing trains and tl only refers to the time of local train. In addition, the transfer time is as (10)ttrans=tw1+tr1+tw2+tr2where tw1 and tr1, respectively, represent the waiting time and running time including dwell time of the first train for passenger; tw2 and tr2, respectively, represent the same meaning for the second train.( 2 ) When passengers consider taking local train to express, they not only need to take the total time into account, but also need to think of whether the middle has express station to supply to transfer.(2) Travel Distance. The operation ratio of express and local train can be estimated by the number of passengers whose travel distance is greater than five sections and less than five sections, like the following formula [23]:(11)p=∑i-j>5qij∑i-j≤5qij.From the above two parts, the combined impedance is as (12) and the flow chart of passengers train-choice behavior is as shown in Figure 1.(12)cijk=μ1·ty+μ2·lwhere μ1 and μ2 are the weight of train running time and travel distance, and they depend on the specific situation.Figure 1
The flow chart of passenger train-choice behavior.


## 3.2.2. Objective Function

Total travel timet is calculated as formula (13) and the model parameters are listed in Table 3. (13)t=∑m=1f1qij-ml+xq·∑m=1f2qij-me·T2·c1f1+c2f2+∑k=1i-1c1hkxk+tij+∑w=i+1j-1ts+c1xwhw.Table 3
Model parameters.
Parameters (unit) Explanation f 1 (time/h) The operating frequency of the local train f 2 (time/h) The operating frequency of the express train x q Boolean variable. When thei station and j station are both local stations, the value is zero. q i j - m l (person) The number of passengers waiting for themth train which is the local train q i j - m e (person) The number of passengers waiting to themth train which is the local train T (min) The length of the optimized period c1/c2 Boolean variable. When passengers take the local train, the value ofc1 is one, c2 is zero, and when passengers take the express train, the value of c1 is zero, c2 is one. h k (min) The increased delay time because the slow train is crossed by the express train at preceding stations x k Boolean variable. When the slow train is crossed by the express train at thekth station, the value is one; otherwise, the value is zero. t i j (min) The interval running time of trains t s (min) The dwelling time at stations x w Boolean variable. When the local train is crossed by the express train at thewth station, the value is one; otherwise, the value is zero. h w (min) The increased dwell time in the trains because the local train is crossed by the express train at this station q i j - m (min) The number of passengers waiting for themth train ηmax Maximum section load factor a Passenger capacity of the trainIn conclusion, when the number of stations isn, the paper establishes the following objective function to achieve the minimization of the total passenger travel time as follows:(14)min⁡Z=min⁡∑i=1n-1∑j=i+1nts.t.qij-m≤ηmax·afd≥1d=1,2,3.


## 3.3. Solution Algorithm

The evolutionary algorithm, which is represented by genetic algorithm, uses crossover and mutation operator to realize the information exchange and local search between individuals in the group and provides each individual with the opportunity of optimization [24]. Through the competitive selection mechanism of survival of the fittest, the evolutionary algorithm guides the population to the better. Exactly, the unique binary coding rule of genetic algorithm is suitable for the value of 0-1 of the train stop strategy. Besides, instead of starting from a point and optimizing according to a fixed route, it searches the whole feasible solution space at the same time, so it can avoid falling into a local optimum and find the global optimal results. The steps are as follows.( 1 ) Population initialization. It generates N binary coded individuals randomly, and the length is the number of stations, where in code, 1 means that express train will stop at the corresponding station and 0 means no stopping. What is more, because the express train must stop at the original and terminal stations, the encoding is 1.( 2 ) Ensuring the fitness function. It constructs the fitness function as formula (15). The greater the fitness value, the better the individual.(15)ux=M-fxwhere M means the total time of general mode and fx is the time of combination mode with every plan.( 3 ) Taking the operations of selecting, crossover, and mutation. The selection operation adopts roulette method, the crossover operator is uniform crossover, and the mutation operator is a single point mutation.( 4 ) After an iteration, the worst individual is replaced by the best individual in the current generation, that is also called “survival of the fittest”.( 5 ) With the requirement of iterations, steps 4 and 5 are repeated.


## 4. Case Analysis

### 4.1. Line Profile

Taking a certain line in Shenzhen as an example (Figure2), this paper analyzes and verifies the optimization of the express and local train stop problem. The data required in the example analysis is shown in Table 4, the OD passenger flow during the morning peak (8:00~9:00) is shown in Table 5, and the length of interval and pure runtime of trains are followed in Table 6. Besides, according to the OD data, the passenger flow of boarding and alighting and the distribution of section flow of up direction (GXB→HZN) and down direction (HZN→GXB) are, respectively, shown in Figures 3, 4, and 5.Table 4
Values of case parameters.
Parameters (unit) Value Parameters (unit) Value Length (km) 53.07 Train composition A8 The number of stations 18 Dwell time (s) 30 Minimum headway (s) 120 Maximum running time (km/h) 120 Passenger capacity (person) 2480 Maximum section load factor 100% Passenger expectation variableθ 0.9 Weightμ1 0.6 Weightμ2 0.4Table 5
The value of OD passenger flow during the morning rush hour.
Station GXB HMG QSH BJ SYL SL AB DY NY BL SH PSW ZYK KZ ST NHL DH HZN GXB 0 1133 1708 1464 597 420 179 2661 261 161 525 391 224 111 118 66 63 131 HMG 2480 0 624 1179 751 462 211 2509 282 185 580 419 238 120 122 74 147 132 QSH 5399 1085 0 1473 1941 543 325 1883 361 322 569 435 276 130 158 129 461 173 BJ 5914 3702 2044 0 621 961 514 1740 440 358 858 629 342 142 172 85 172 183 SYL 1763 1686 2959 945 0 954 488 1190 453 422 658 463 284 115 148 97 66 156 SL 935 807 777 843 829 0 307 1334 338 370 390 263 206 93 126 128 71 164 AB 444 391 395 417 337 246 0 1024 215 242 271 195 139 63 85 87 52 111 DY 3774 2956 1565 1030 528 939 1225 0 1054 1204 1717 918 739 204 456 230 146 474 NY 526 468 446 380 322 267 242 1193 0 180 318 284 162 68 98 93 57 126 BL 293 252 250 217 182 155 140 782 96 0 231 254 122 56 74 71 48 97 SH 1844 1725 1405 1229 766 561 524 3484 628 757 0 337 338 302 622 365 195 475 PSW 1116 1074 999 721 628 396 427 2680 669 853 391 0 302 317 489 433 247 643 ZYK 932 833 679 575 362 288 248 1549 316 389 270 280 0 206 353 240 129 324 KZ 428 375 419 259 186 284 247 868 279 349 568 913 430 0 324 317 175 471 ST 444 382 399 267 183 227 204 892 234 295 767 714 469 211 0 387 245 674 NHL 431 367 421 226 150 272 208 790 243 270 443 707 321 222 443 0 183 497 DH 562 576 971 426 196 352 259 980 307 328 511 852 385 333 834 572 0 610 HZN 226 186 192 156 83 117 90 443 108 116 188 354 167 174 603 274 2138 0Table 6
The length of interval and pure runtime of trains.
Interval Length (km) Runtime (min) Interval Length (km) Runtime (min) GXB-HMG 2.905 2.563 BL-SH 4.213 2.922 HMG-QSH 4.279 3.363 SH-PSW 1.566 1.521 GSH-BJ 2.494 2.113 PSW-ZYK 2.007 1.714 BJ-SYL 3.535 2.84 ZYK-KZ 4.926 3.339 SYL-SL 6.364 4.173 KZ-ST 3.229 2.367 SL-AB 3.428 2.503 ST-NHL 1.623 1.634 AB-DY 2.413 1.915 NHL-DH 1.745 1.795 DY-NY 6.037 4.35 DH-HZN 1.238 1.241 NY-BL 2.072 1.756Figure 2
The regional rail transit line in Shenzhen.Figure 3
The passenger flow of boarding and alighting of up direction.Figure 4
The passenger flow of boarding and alighting of down direction.Figure 5
The distribution of section passenger flow.


### 4.2. Solution

This paper uses the genetic algorithm to solve the model. Different parameters will affect the speed and results of the algorithm. The paper uses two of the population sizes, crossover probability and mutation probability, as the independent variables and the other two as the dependent variables to analyze the influence of the values on algorithm results. When the values of the population size were from 30 to 150, the results showed that the final fitness value did not change with the population size, while the rate of convergence will be affected as shown in Figure6. And according to the current literature that the range of the crossover rate is 0.4 to 0.99 and 0.001 to 0.1, respectively [25], the paper supposes that the crossover rate steps and the mutation rate steps are 0.5 and 0.005. The results show that although the number of iterations changes, iterations have little difference from each other and the final fitness value is invariable as Figures 7 and 8 show. Therefore, the final algorithm parameter values are shown in Table 7.Table 7
Values of algorithm parameters
Parameters Value Parameters Value Population 50 Crossover probability 0.8 Iterations 150 Mutant probability 0.05Figure 6
The relationship between the population size and number of iterations.Figure 7
The relationship between the crossover rate and number of iterations.Figure 8
The relationship between the mutation rate and number of iterations.According to the case data, the maximum cross-section passenger flow appears at the BJ station in the downward direction, with 49333 people, so that the number of departure times during the optimization period is 20 pairs/h. As the downward direction of the cross-section of the passenger flow is much larger than the upward direction, it is designed to operate the express and local train mode in the down direction. And in this situation, the optimal result is different with the different proportion of the express/local train. Based on the OD in Table5 and formula (11), the ratio is approximately 1:2. However, in order not to leave out all the plans, the paper takes 3:1, 2:1, 1:1, 1:2, 1:3 as alternatives.The case is calculated and analyzed according to the algorithm parameter in Table7 and fitness function set as (15). Figure 9 shows the changing of travel time value. When the proportion is 1:2, that is to say, operating 6 express trains and 14 local trains, the passenger travel time is optimal. Corresponding express station stop scheme is shown in Table 8, in which 1 stands for stopping at station and 0 for no stopping, and express train overtakes the local at station ST and PSW without stopping.Table 8
The best express train stop scheme.
Station HZN DH NHL ST KZ ZYK PSW SH BL NY DY AB SL SYL BJ QSH HMG GXB Plan 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1Figure 9
Variation of total travel time.


### 4.3. Results Analysis

#### 4.3.1. Time Benefit Analysis

Compared with the general mode which only operates the local trains, the combination saves the passenger travel time with taking the express trains, whereas part of the passengers’ time is increased because of slow trains. In the condition of the same train frequency, comparing the total travel time before and after the plan, like Table9, we can see that in the specific passenger flow conditions, the time benefit is quite obvious.Table 9
Total travel time comparison.
Compared content (unit) General mode Combination mode Time benefit Per capita benefit Total travel time (min) 3 638 656 2 469 317 1 169 339 10.34 min


#### 4.3.2. Capacity Analysis

It has an effect on carrying capacity due to operating the express/local train. According to the parameters in the foregoing case, when the minimum headway is 120s and the ratio of express and local train is 6:14 (roughly 1:2), where the capacity of general mode is 30 pairs/h, while in condition of the combination mode, the capacity depresses to 24 pairs/h.From the stop scheme, we draw a train graph like Figure10. With not considering the train turnaround temporarily, here we only draw the sketch of two sets of express/local train combination which run from the station HZN to the BL, and the start order is the local train, local train, and the express train, respectively. In a triple of the combination, the first local train is crossed at station PSW, and the second is overtaken at ST without others.Figure 10
The sketch of train operation.


#### 4.3.3. Energy Consumption Comparison

The difference between the energy consumption of express trains and slow trains is mainly caused by the express train running directly at some stations without stopping and experiencing a kind of uniform speed at a longer distance in the interval. The train traction curve, brake curve, and speed curve diagrammatic drawing are as shown in Figures11, 12, and 13, respectively. Traction energy consumption and braking energy consumption are calculated by (4) and (5), and (6) represents the regenerative braking energy. Above all, the total consumption is as formula (3). In this case, there are three substations along the rail line. Under the above optimal stopping plan and the series calculation of train energy consumption, the results are compared in Table 10. It shows that the express/slow train mode has great advantages in energy saving and ensures the efficiency of the operating enterprises.Table 10
Train energy consumption comparison.
Compared content (unit) General mode Combination mode benefit Benefit rate Energy consumption (KJ) 1.41 × 1 0 8 1.19×108 2.26×107 16%Figure 11
The sketch of traction curve.Figure 12
The sketch of brake curve.Figure 13
The sketch of speed curve.


## 4.1. Line Profile

Taking a certain line in Shenzhen as an example (Figure2), this paper analyzes and verifies the optimization of the express and local train stop problem. The data required in the example analysis is shown in Table 4, the OD passenger flow during the morning peak (8:00~9:00) is shown in Table 5, and the length of interval and pure runtime of trains are followed in Table 6. Besides, according to the OD data, the passenger flow of boarding and alighting and the distribution of section flow of up direction (GXB→HZN) and down direction (HZN→GXB) are, respectively, shown in Figures 3, 4, and 5.Table 4
Values of case parameters.
Parameters (unit) Value Parameters (unit) Value Length (km) 53.07 Train composition A8 The number of stations 18 Dwell time (s) 30 Minimum headway (s) 120 Maximum running time (km/h) 120 Passenger capacity (person) 2480 Maximum section load factor 100% Passenger expectation variableθ 0.9 Weightμ1 0.6 Weightμ2 0.4Table 5
The value of OD passenger flow during the morning rush hour.
Station GXB HMG QSH BJ SYL SL AB DY NY BL SH PSW ZYK KZ ST NHL DH HZN GXB 0 1133 1708 1464 597 420 179 2661 261 161 525 391 224 111 118 66 63 131 HMG 2480 0 624 1179 751 462 211 2509 282 185 580 419 238 120 122 74 147 132 QSH 5399 1085 0 1473 1941 543 325 1883 361 322 569 435 276 130 158 129 461 173 BJ 5914 3702 2044 0 621 961 514 1740 440 358 858 629 342 142 172 85 172 183 SYL 1763 1686 2959 945 0 954 488 1190 453 422 658 463 284 115 148 97 66 156 SL 935 807 777 843 829 0 307 1334 338 370 390 263 206 93 126 128 71 164 AB 444 391 395 417 337 246 0 1024 215 242 271 195 139 63 85 87 52 111 DY 3774 2956 1565 1030 528 939 1225 0 1054 1204 1717 918 739 204 456 230 146 474 NY 526 468 446 380 322 267 242 1193 0 180 318 284 162 68 98 93 57 126 BL 293 252 250 217 182 155 140 782 96 0 231 254 122 56 74 71 48 97 SH 1844 1725 1405 1229 766 561 524 3484 628 757 0 337 338 302 622 365 195 475 PSW 1116 1074 999 721 628 396 427 2680 669 853 391 0 302 317 489 433 247 643 ZYK 932 833 679 575 362 288 248 1549 316 389 270 280 0 206 353 240 129 324 KZ 428 375 419 259 186 284 247 868 279 349 568 913 430 0 324 317 175 471 ST 444 382 399 267 183 227 204 892 234 295 767 714 469 211 0 387 245 674 NHL 431 367 421 226 150 272 208 790 243 270 443 707 321 222 443 0 183 497 DH 562 576 971 426 196 352 259 980 307 328 511 852 385 333 834 572 0 610 HZN 226 186 192 156 83 117 90 443 108 116 188 354 167 174 603 274 2138 0Table 6
The length of interval and pure runtime of trains.
Interval Length (km) Runtime (min) Interval Length (km) Runtime (min) GXB-HMG 2.905 2.563 BL-SH 4.213 2.922 HMG-QSH 4.279 3.363 SH-PSW 1.566 1.521 GSH-BJ 2.494 2.113 PSW-ZYK 2.007 1.714 BJ-SYL 3.535 2.84 ZYK-KZ 4.926 3.339 SYL-SL 6.364 4.173 KZ-ST 3.229 2.367 SL-AB 3.428 2.503 ST-NHL 1.623 1.634 AB-DY 2.413 1.915 NHL-DH 1.745 1.795 DY-NY 6.037 4.35 DH-HZN 1.238 1.241 NY-BL 2.072 1.756Figure 2
The regional rail transit line in Shenzhen.Figure 3
The passenger flow of boarding and alighting of up direction.Figure 4
The passenger flow of boarding and alighting of down direction.Figure 5
The distribution of section passenger flow.


## 4.2. Solution

This paper uses the genetic algorithm to solve the model. Different parameters will affect the speed and results of the algorithm. The paper uses two of the population sizes, crossover probability and mutation probability, as the independent variables and the other two as the dependent variables to analyze the influence of the values on algorithm results. When the values of the population size were from 30 to 150, the results showed that the final fitness value did not change with the population size, while the rate of convergence will be affected as shown in Figure6. And according to the current literature that the range of the crossover rate is 0.4 to 0.99 and 0.001 to 0.1, respectively [25], the paper supposes that the crossover rate steps and the mutation rate steps are 0.5 and 0.005. The results show that although the number of iterations changes, iterations have little difference from each other and the final fitness value is invariable as Figures 7 and 8 show. Therefore, the final algorithm parameter values are shown in Table 7.Table 7
Values of algorithm parameters
Parameters Value Parameters Value Population 50 Crossover probability 0.8 Iterations 150 Mutant probability 0.05Figure 6
The relationship between the population size and number of iterations.Figure 7
The relationship between the crossover rate and number of iterations.Figure 8
The relationship between the mutation rate and number of iterations.According to the case data, the maximum cross-section passenger flow appears at the BJ station in the downward direction, with 49333 people, so that the number of departure times during the optimization period is 20 pairs/h. As the downward direction of the cross-section of the passenger flow is much larger than the upward direction, it is designed to operate the express and local train mode in the down direction. And in this situation, the optimal result is different with the different proportion of the express/local train. Based on the OD in Table5 and formula (11), the ratio is approximately 1:2. However, in order not to leave out all the plans, the paper takes 3:1, 2:1, 1:1, 1:2, 1:3 as alternatives.The case is calculated and analyzed according to the algorithm parameter in Table7 and fitness function set as (15). Figure 9 shows the changing of travel time value. When the proportion is 1:2, that is to say, operating 6 express trains and 14 local trains, the passenger travel time is optimal. Corresponding express station stop scheme is shown in Table 8, in which 1 stands for stopping at station and 0 for no stopping, and express train overtakes the local at station ST and PSW without stopping.Table 8
The best express train stop scheme.
Station HZN DH NHL ST KZ ZYK PSW SH BL NY DY AB SL SYL BJ QSH HMG GXB Plan 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1Figure 9
Variation of total travel time.


## 4.3. Results Analysis

### 4.3.1. Time Benefit Analysis

Compared with the general mode which only operates the local trains, the combination saves the passenger travel time with taking the express trains, whereas part of the passengers’ time is increased because of slow trains. In the condition of the same train frequency, comparing the total travel time before and after the plan, like Table9, we can see that in the specific passenger flow conditions, the time benefit is quite obvious.Table 9
Total travel time comparison.
Compared content (unit) General mode Combination mode Time benefit Per capita benefit Total travel time (min) 3 638 656 2 469 317 1 169 339 10.34 min


### 4.3.2. Capacity Analysis

It has an effect on carrying capacity due to operating the express/local train. According to the parameters in the foregoing case, when the minimum headway is 120s and the ratio of express and local train is 6:14 (roughly 1:2), where the capacity of general mode is 30 pairs/h, while in condition of the combination mode, the capacity depresses to 24 pairs/h.From the stop scheme, we draw a train graph like Figure10. With not considering the train turnaround temporarily, here we only draw the sketch of two sets of express/local train combination which run from the station HZN to the BL, and the start order is the local train, local train, and the express train, respectively. In a triple of the combination, the first local train is crossed at station PSW, and the second is overtaken at ST without others.Figure 10
The sketch of train operation.


### 4.3.3. Energy Consumption Comparison

The difference between the energy consumption of express trains and slow trains is mainly caused by the express train running directly at some stations without stopping and experiencing a kind of uniform speed at a longer distance in the interval. The train traction curve, brake curve, and speed curve diagrammatic drawing are as shown in Figures11, 12, and 13, respectively. Traction energy consumption and braking energy consumption are calculated by (4) and (5), and (6) represents the regenerative braking energy. Above all, the total consumption is as formula (3). In this case, there are three substations along the rail line. Under the above optimal stopping plan and the series calculation of train energy consumption, the results are compared in Table 10. It shows that the express/slow train mode has great advantages in energy saving and ensures the efficiency of the operating enterprises.Table 10
Train energy consumption comparison.
Compared content (unit) General mode Combination mode benefit Benefit rate Energy consumption (KJ) 1.41 × 1 0 8 1.19×108 2.26×107 16%Figure 11
The sketch of traction curve.Figure 12
The sketch of brake curve.Figure 13
The sketch of speed curve.


## 4.3.1. Time Benefit Analysis

Compared with the general mode which only operates the local trains, the combination saves the passenger travel time with taking the express trains, whereas part of the passengers’ time is increased because of slow trains. In the condition of the same train frequency, comparing the total travel time before and after the plan, like Table9, we can see that in the specific passenger flow conditions, the time benefit is quite obvious.Table 9
Total travel time comparison.
Compared content (unit) General mode Combination mode Time benefit Per capita benefit Total travel time (min) 3 638 656 2 469 317 1 169 339 10.34 min


## 4.3.2. Capacity Analysis

It has an effect on carrying capacity due to operating the express/local train. According to the parameters in the foregoing case, when the minimum headway is 120s and the ratio of express and local train is 6:14 (roughly 1:2), where the capacity of general mode is 30 pairs/h, while in condition of the combination mode, the capacity depresses to 24 pairs/h.From the stop scheme, we draw a train graph like Figure10. With not considering the train turnaround temporarily, here we only draw the sketch of two sets of express/local train combination which run from the station HZN to the BL, and the start order is the local train, local train, and the express train, respectively. In a triple of the combination, the first local train is crossed at station PSW, and the second is overtaken at ST without others.Figure 10
The sketch of train operation.


## 4.3.3. Energy Consumption Comparison

The difference between the energy consumption of express trains and slow trains is mainly caused by the express train running directly at some stations without stopping and experiencing a kind of uniform speed at a longer distance in the interval. The train traction curve, brake curve, and speed curve diagrammatic drawing are as shown in Figures11, 12, and 13, respectively. Traction energy consumption and braking energy consumption are calculated by (4) and (5), and (6) represents the regenerative braking energy. Above all, the total consumption is as formula (3). In this case, there are three substations along the rail line. Under the above optimal stopping plan and the series calculation of train energy consumption, the results are compared in Table 10. It shows that the express/slow train mode has great advantages in energy saving and ensures the efficiency of the operating enterprises.Table 10
Train energy consumption comparison.
Compared content (unit) General mode Combination mode benefit Benefit rate Energy consumption (KJ) 1.41 × 1 0 8 1.19×108 2.26×107 16%Figure 11
The sketch of traction curve.Figure 12
The sketch of brake curve.Figure 13
The sketch of speed curve.


## 5. Conclusion

This paper studied the problem of the train stopping schedule in the condition of express and local train operation mode. With the analysis of the passengers’ requirement to minimize the travel time, travel distance, and mainly the stations, the paper established a passenger train-choice model and used the entropy method to calculate the weight coefficient scientifically. On this basis, a 0-1 integer programming mathematical model aimed at minimizing the total travel time was set up, and the genetic algorithm was designed to solve the model. Finally, the algorithm result was insensitivity with the parameters. Besides, the travel time and energy consumption were both saved effectively through the case analysis, resulting in a considerable social benefits. However, the combination of express and local train operation reduced the line capacity, and the actual operation can change the ratio of express/local train to get relatively large capacity.



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*Source: 3179321-2018-06-25.xml*