# Optimization for Metro Operation Scheme of Suburban Lines: A New Method for Dealing with the Imbalanced Passenger Flow

Authors: Xiaobing Ding; Gan Shi; Jiangang Jin; Lixing Yang; Zhigang Liu
Journal: Journal of Advanced Transportation
 (2023)

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## Abstract

The development of some central cities tends to be saturated, so some mega cities try to adopt the express and local mode of the metro. Suburban lines with imbalanced passenger flow between stations will lead to extremely crowded passenger flow in some stations, while scarce passenger flow in other stations. To balance the train’s carrying capacity with the imbalanced passenger flow, this study explores the collaborative optimization of train operation under the new mode of express and local. First, a time-stamped data cropping strategy based on fine-grained time zones as the grid index is developed, which is used to deeply explore the OD spatiotemporal representation of passenger flow correlation mapping mechanism based on multisource data; then, the calculation model of the proportion of trains matching with the OD temporal and spatial characteristics of suburban passenger flow are constructed, and an efficient solution algorithm is developed to solve the problem of interest. Finally, a set of numerical experiments with operation data from Shanghai Metro Line 16 are conducted to verify the performance and effectiveness of the proposed model and algorithm. The experimental results show that the proposed approach can effectively realize the collaborative optimization of passenger OD prediction, train proportion, stop scheme, and travel time, so as to provide decision-making support and method guidance for the optimization of metro organizations in megacities.


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## Body

## 1. Introduction

As China’s urbanization continues to advance, the passenger flow between central cities and satellite cities is increasing. In addition, the development of central cities has been or is close to saturation, and a large number of enterprises and institutions have moved to the suburbs, so the suburban passenger flow between urban areas and suburbs has also increased. In this situation, it will lead to uneven passenger flow in sections, which exhibits obvious tidal characteristics. The imbalance of passenger flow between stations makes it difficult for the train capacity to meet the demand at stations with high passenger flow, while there is more capacity left at stations with small passenger flow, resulting in the phenomenon that the mismatch rate between capacity and passenger flow continues to be high. The passenger flow from the suburbs to the urban area is very large in the morning peak period, so is large from the urban area to the suburbs in the evening peak period [1, 2]. Although the express and local operation mode can solve the problem mentioned above to a certain extent, it will simultaneously bring some urgent new technical problems.To solve the problem of train operation schemes under the unbalanced passenger flow of suburban lines, many researchers try to explore different train scheduling strategies in metro operation systems [3, 4]. For example, during the morning peak period, due to the large passenger flow in some stations, the operation enterprises make temporary adjustments to the train operation scheme, such as train skip-stop, large and small routing, flow restriction outside the station, and other measures [5], but the effect is not obvious, and it increases the complexity of the metro operation organization.From a new perspective, this study aims to propose a new mode to realize the high-precision matching between passenger flows at the stations. We mainly study running the express and local trains based on the traditional operation mode, in which the local trains still stop at all stations, but the express trains stop at crossing stations according to the passenger flow characteristics of the stations. The new operation mode will bring the following new optimization problems that need to be studied in depth: the determination method of the operation proportion of express and local trains; the determination of the stop scheme of express trains based on passenger flow OD; the optimization of the total travel time of passengers [6–8]; and the discount optimization of the train carrying capacity, etc. By solving the new problems above, we can finally realize the optimal operation of express and local modes. In the literature, this is a new mode for the suburban passenger transport strategy, and no relevant study has been found so far. The following discussion will elaborate on this new mode in detail.


### 1.1. Literature Review

The suburban lines of some developed countries have adopted the express and local mode, which can be better adapted to the demand of long-distance passenger flow, and it also has better social and economic benefits, which can better meet the needs of long-distance passengers.First, experts both at home and abroad have studied the conditions for running express and local trains on subway lines and the basic conditions for construction. The subway lines in the central city usually run in parallel in pairs, which is mainly the backbone line to meet the internal traffic demand of the city. They are mainly underground lines, with high passenger flow density and short departure intervals [9]. Suburban lines are obviously different from urban lines. In terms of the applicability of suburban ones, some experts made a preliminary study on the applicability of express and local trains. For example, Pan et al. [10] mainly analyzed the operation organization mode of express lines, the specific implementation mode, characteristics, and implementation effects of the operation mode, which is an early exploration of the mode of running express trains between the suburbs and the city center. Later, Xu and Liu [11] proposed that the operation express and local mode can fully consider the unbalanced characteristics of the space-time distribution of passenger flow and the technical characteristics of the nonparallel operation diagram, which can maximize the operation benefits. After affirming the advantages of the express and local modes, Liu and Zhang [12] conducted a basic study on the construction of the Tokyo rail transit network and gave the passenger flow characteristics suitable for the express and local modes as well as the hardware construction requirements of the line. The above researches on express and local mode mainly focus on the applicable conditions. Some scholars also discussed the construction requirements for suburban rail lines, laying the foundation for the early research on suburban lines.However, in order to truly implement the suburban express and local mode in megacities, it is also necessary to study the optimization and mutual coordination of new indicators under the new mode, in which there are the following two important aspects: the OD prediction of passenger flow and the stop scheme of the express train.(1)
The OD mining of passenger flowIn express and local modes, the stop scheme and the determination of the proportion of express and local trains are closely related to the passenger flow of stations. It is important to explore how to predict passenger flow at a specific time by analyzing the time distribution characteristics of passenger flow OD. Among many models for passenger outflow distribution, the gravity model and the growth coefficient method is used widely. Stouffer first proposed the intervention opportunity model. Then, Casey proposed the gravity mode, which obtained NT in terms of the maximum entropy principle and the maximum likelihood principle. Later, Cordone and Redaelli [13] proposed a nonfamily travel model, and Richard [14] established a special model to predict OD. Bowman [15] established a model to predict the occurrence of trips by taking the traveler’s activity arrangement as the entry point. These scholars have made more in-depth research on the model of passenger flow forecasting and distribution, laying a theoretical foundation. In recent years, there are also some research studies worth learning in this respect in China, for example, Xie and Wu [16] proposed a polynomial distributed lag model for short-term traffic flow prediction, which comprehensively considers the influence factors of the traffic state time series other than its own lag and the model calculation is accurate. Subsequently, Ding and Xu [17] proposed an algorithm for predicting passenger flow detention in subway stations based on angle cost, and studied the calculation of detained passenger flow in subway stations from a new perspective, laying a foundation for the formulation of the proportion of express and local trains and the stop scheme. These research results mainly focus on the exploration of passenger flow forecasting methods and the improvement of the accuracy of forecast results. But the passenger flow OD under the express and local modes should focus on exploring the prediction method of station passenger flow, especially the stranded passenger flow, which will determine the stop scheme of express trains.(2)
The stop scheme of express and slow modeThe scholar who first began to study the express and local train stop scheme was Salzburg. Since then, most researchers have studied the optimization of express train stop schemes and train operation schemes. For example, Ghoseiri et al. [18] began to concentrate on the study of interstation stop schemes and proposed that the interstation stop mode can reduce the travel time of express passengers, which directs further research. Then, Li and Lo [19] proposed the following two important indicators for the express train stop scheme: the time cost and congestion cost incurred during the travel of passengers in the section. Besides, Tang and Xu [20] analyzed the construction mode and operating characteristics of suburban railways, based on OD passenger flow data, and established a model of train stop schemes for major stations. Then, in order to obtain a more optimized stop plan, Zhao et al. [21, 22] took the minimum train stop mode as the goal and studied the optimization of the express train stop combination.To sum up, the research results show that there are few relevant literature on the optimization of subway stop plan at present, and some sporadic studies are mainly concentrated in the railway direction. With the rapid increase of suburban passenger flow in mega cities and the rapid development of metro suburban lines, it is urgent to optimize the operation of express and local trains; In the above literature, many scholars have studied the optimization modeling of the stop plan, but there are still many difficulties in solving the model. Basically, the nonlinear integer programming model is converted into a linear programming model to reduce the difficulty of solving the model. However, the scale of the solution is too large, and it is still impossible to reduce the scale of the problem, so it is more difficult to obtain accurate solutions; This paper starts with the optimization of new indicators generated by express fast and local trains, involving passenger flow OD, stop the plan of fast trains, running proportion of fast and slow trains, and line capacity loss caused by fast and slow train mode, and establishes optimization model and designs solution algorithm.


### 1.2. The Focus of This Study

Generally speaking, in order to realize the balanced transportation of passenger flow on suburban lines, it is necessary to consider the matching of passenger flow and the remaining carrying capacity of coming trains, the proportion of express and local trains, the on-the-way time of trains, and the train carrying capacity. To solve the above optimization problems, the core work is to establish the optimization model and solve them on the basis of considering the metro operation logic and constraints.The remainder of this paper is organized as follows: Section2 is the problem description; Section 3 is about the collaborative optimization modeling based on the new mode; Section 4 is the case study based on the model constructed in Section 3 to realize the collaborative optimization of new operation indicators under the express and local mode, which mainly includes the operation proportion of express and local trains, the stop scheme of express trains, and the train carrying capacity, which realizes the collaborative optimization of relevant indicators of express and local train operation scheme on suburban lines. Some conclusions and further research directions are presented in the last section.


## 1.1. Literature Review

The suburban lines of some developed countries have adopted the express and local mode, which can be better adapted to the demand of long-distance passenger flow, and it also has better social and economic benefits, which can better meet the needs of long-distance passengers.First, experts both at home and abroad have studied the conditions for running express and local trains on subway lines and the basic conditions for construction. The subway lines in the central city usually run in parallel in pairs, which is mainly the backbone line to meet the internal traffic demand of the city. They are mainly underground lines, with high passenger flow density and short departure intervals [9]. Suburban lines are obviously different from urban lines. In terms of the applicability of suburban ones, some experts made a preliminary study on the applicability of express and local trains. For example, Pan et al. [10] mainly analyzed the operation organization mode of express lines, the specific implementation mode, characteristics, and implementation effects of the operation mode, which is an early exploration of the mode of running express trains between the suburbs and the city center. Later, Xu and Liu [11] proposed that the operation express and local mode can fully consider the unbalanced characteristics of the space-time distribution of passenger flow and the technical characteristics of the nonparallel operation diagram, which can maximize the operation benefits. After affirming the advantages of the express and local modes, Liu and Zhang [12] conducted a basic study on the construction of the Tokyo rail transit network and gave the passenger flow characteristics suitable for the express and local modes as well as the hardware construction requirements of the line. The above researches on express and local mode mainly focus on the applicable conditions. Some scholars also discussed the construction requirements for suburban rail lines, laying the foundation for the early research on suburban lines.However, in order to truly implement the suburban express and local mode in megacities, it is also necessary to study the optimization and mutual coordination of new indicators under the new mode, in which there are the following two important aspects: the OD prediction of passenger flow and the stop scheme of the express train.(1)
The OD mining of passenger flowIn express and local modes, the stop scheme and the determination of the proportion of express and local trains are closely related to the passenger flow of stations. It is important to explore how to predict passenger flow at a specific time by analyzing the time distribution characteristics of passenger flow OD. Among many models for passenger outflow distribution, the gravity model and the growth coefficient method is used widely. Stouffer first proposed the intervention opportunity model. Then, Casey proposed the gravity mode, which obtained NT in terms of the maximum entropy principle and the maximum likelihood principle. Later, Cordone and Redaelli [13] proposed a nonfamily travel model, and Richard [14] established a special model to predict OD. Bowman [15] established a model to predict the occurrence of trips by taking the traveler’s activity arrangement as the entry point. These scholars have made more in-depth research on the model of passenger flow forecasting and distribution, laying a theoretical foundation. In recent years, there are also some research studies worth learning in this respect in China, for example, Xie and Wu [16] proposed a polynomial distributed lag model for short-term traffic flow prediction, which comprehensively considers the influence factors of the traffic state time series other than its own lag and the model calculation is accurate. Subsequently, Ding and Xu [17] proposed an algorithm for predicting passenger flow detention in subway stations based on angle cost, and studied the calculation of detained passenger flow in subway stations from a new perspective, laying a foundation for the formulation of the proportion of express and local trains and the stop scheme. These research results mainly focus on the exploration of passenger flow forecasting methods and the improvement of the accuracy of forecast results. But the passenger flow OD under the express and local modes should focus on exploring the prediction method of station passenger flow, especially the stranded passenger flow, which will determine the stop scheme of express trains.(2)
The stop scheme of express and slow modeThe scholar who first began to study the express and local train stop scheme was Salzburg. Since then, most researchers have studied the optimization of express train stop schemes and train operation schemes. For example, Ghoseiri et al. [18] began to concentrate on the study of interstation stop schemes and proposed that the interstation stop mode can reduce the travel time of express passengers, which directs further research. Then, Li and Lo [19] proposed the following two important indicators for the express train stop scheme: the time cost and congestion cost incurred during the travel of passengers in the section. Besides, Tang and Xu [20] analyzed the construction mode and operating characteristics of suburban railways, based on OD passenger flow data, and established a model of train stop schemes for major stations. Then, in order to obtain a more optimized stop plan, Zhao et al. [21, 22] took the minimum train stop mode as the goal and studied the optimization of the express train stop combination.To sum up, the research results show that there are few relevant literature on the optimization of subway stop plan at present, and some sporadic studies are mainly concentrated in the railway direction. With the rapid increase of suburban passenger flow in mega cities and the rapid development of metro suburban lines, it is urgent to optimize the operation of express and local trains; In the above literature, many scholars have studied the optimization modeling of the stop plan, but there are still many difficulties in solving the model. Basically, the nonlinear integer programming model is converted into a linear programming model to reduce the difficulty of solving the model. However, the scale of the solution is too large, and it is still impossible to reduce the scale of the problem, so it is more difficult to obtain accurate solutions; This paper starts with the optimization of new indicators generated by express fast and local trains, involving passenger flow OD, stop the plan of fast trains, running proportion of fast and slow trains, and line capacity loss caused by fast and slow train mode, and establishes optimization model and designs solution algorithm.


## 1.2. The Focus of This Study

Generally speaking, in order to realize the balanced transportation of passenger flow on suburban lines, it is necessary to consider the matching of passenger flow and the remaining carrying capacity of coming trains, the proportion of express and local trains, the on-the-way time of trains, and the train carrying capacity. To solve the above optimization problems, the core work is to establish the optimization model and solve them on the basis of considering the metro operation logic and constraints.The remainder of this paper is organized as follows: Section2 is the problem description; Section 3 is about the collaborative optimization modeling based on the new mode; Section 4 is the case study based on the model constructed in Section 3 to realize the collaborative optimization of new operation indicators under the express and local mode, which mainly includes the operation proportion of express and local trains, the stop scheme of express trains, and the train carrying capacity, which realizes the collaborative optimization of relevant indicators of express and local train operation scheme on suburban lines. Some conclusions and further research directions are presented in the last section.


## 2. Problem Formulation

### 2.1. Problem Description

In the studied problem, the special channel for rapid transportation between suburban satellite towns and central urban areas is called express and local modes, which can connect the urban backbone network to special channels with them. The frame structure of suburban lines is shown in Figure1.Figure 1
Schematic diagram of the urban backbone network and suburban line structure.Adopting the new modes of express and local trains can solve the problem of passenger flow transportation between suburban lines to a certain extent, but it will bring about some new problems simultaneously, such as (i) how to ensure the high-precision matching of transport capacity and passenger flow during the rush hours; (ii) how to determine the proportion of express trains and local trains; (iii) how to determine the stop scheme of the express trains; (iv) the impact of the adoption of express and local train mode on the reduction of carrying capacity; and (v) the result of the same line transfer of some passengers, which will increase the complexity of station passenger transport organization and train dispatching in the state of delay.In the problems above, for instance, there are 5 stations marked as A–E, and the average passenger flow can be known from the operation history, while in the traditional mode, due to the unbalanced passenger flow, some stations are in the insufficient state, some stations are in the moderate state, still, others are in the surplus state. When in the insufficient state, the station is in danger as shown in Figure2(a) in which, n1 represents the number of passengers detained at the platform after train2 leaves Station A with passengers, n2 represents the number of passengers detained at the platform after train2 leaves Station B with passengers, n3 represents the number of passengers detained at the platform after train2 leaves Station C with passengers, n4 represents the number of passengers detained at the platform after train2 leaves Station D with passengers; m1 represents the number of passengers detained at the platform after train3 leaves Station A with passengers, m2 represents the number of passengers detained at the platform after train3 leaves Station B with passengers, m3 represents the number of passengers detained at the platform after train3 leaves Station C with passengers, m4 represents the number of passengers detained at the platform after train3 leaves Station D with passengers. Therefore, it is necessary to study the method of determining the stop scheme of an express train to realize the rapid matching of carrying capacity and passenger flow. The objective function of the stop scheme optimization model is to minimize the stranded passenger flow in the station. Besides, it is necessary to study the proportion of express and local trains m : n. In Figure 2(b), the application of the proposed optimization model can alleviate the waste of carrying capacity caused by the imbalance of passenger flow or the long waiting time of passenger flow in some stations, so the state of stations becomes safe.Figure 2
The passenger flow status under different metro operation modes: (a) passenger flow in traditional operation mode; (b) passenger flow in express and local mode.
(a)(b)As discussed in the above descriptions, the operation mode of express and local trains on suburban lines is relatively complex, and there will be many new situations, such as the passenger flow between express and local trains will transfer at the same station, and the other situation is to take the express train first and then transfer to the local train to reach the destination station, which is shown in Figure3.Figure 3
Express and local train operation mode of the suburban line of rail transit.Given the problems arising from the new operation mode of express and local trains, such as the stop scheme of express trains, the running proportion of express and local trains, and the loss of line carrying capacity, it is necessary to establish a mathematical model with the research goal of minimizing the travel time of passengers and giving consideration to the minimum cost of enterprises. The research boundary can be defined as the suburban lines of the subway, the obvious space-time imbalance, and the tidal characteristics of the passenger flow. The decision variables of the model involve the full load rate of trains, the number of allocated passenger trains, and the minimum departure interval. The collaborative optimization between new indicators is studied in depth. The reason why this optimization model is a collaborative optimization is that the mining of passenger flow OD will determine the stop scheme of express trains, and the stop scheme will also affect the travel time of passengers. The OD of passenger flow also determines the running proportion of express and local trains, and the running proportion of trains determines the loss capacity of the line quantitatively. Therefore, this is a collaborative optimization model, which is closely linked.


### 2.2. Symbols Formulation

To formulate the problem, some main related notations and parameters in the formulating process are listed in Table1 for the completeness of the research.Table 1
Notations and parameters used in the paper.
CinTravel time and cost of passengers on express and local lines, yuanaTime-consuming weighttwWaiting time in the platform, mintzTravel time in the sections, mindDwell time, minYCongestion cost, yuanfkThe departures of class k trains within one hour, trainsjlDistance between stationsj and station l, kmz′The sections which the trainK passesvkThe speed of traink, km/hSkThe departure stationekThe arrival stationqkThe passengers who choose classk of trains in the sectionsbkTrain formation of classkzkThe train capacity of classksThe distance of sectionsvThe running speed of traincprsThe travel cost of sectionr-s for route P, yuanc¯The average travel cost of all routes, yuanxprsThe probability of passengers choosing routepfprsThe passenger flow of routep, personqrsOD passenger flow inR–S section, personxaSection passenger flow, personδa,prsWhen section a belongs to thep-th path of R-S, it is 1, otherwise, it is 0tijThe running time in sectionsi-j, minhi1The start additional time of local trains in stationi, minhi2The dwell time of local trains in stationi, minxiWhether the express train stop at stationi, if it stops i = 1, if not i = 0nThe number of stationsQijkThe passengers who select trainK from station i to station jzw1Total travel time of passengers in the upward direction, minzw2Total travel time of passengers in the downward direction, minTThe length of optimize period, minf1The departure frequency of express trains, train/hf2The departure frequency of local trains, train/hlijThe travel length from stationi to station j, kmlw1The total running distance of uplink trains in the optimized period, kmlw2The total running distance of the downlink train in the optimized period, kmlwThe total cost of train operation, yuancrunRunning cost per kilometer, yuan/kmnw1The total dwell stops of uplink trains in the optimized periodnw2The total dwell stops of downlink trains in the optimized periodcwThe total cost of train stopping, yuancdThe stop cost of every station, yuan/timetrInterval running time(s)ttThe transfer time(s)∆tWaiting time of trains overtaketm,nlIn traditional operation mode, the whole travel time is from stationm to station n when the local train runstm,neThe travel time of an express train from stationm to station nTm,nlThe travel time of a local train from stationm to station nTm,nleThe travel time for passengers fromm to n by taking a local train and then transferring to an express trainTm,nelThe travel time for passengers fromm to n by taking an express train and then transferring to a local trainsjA station for an express trainsThe first station where an express stops when a local train transferring to an express trains′The last station where an express train stops when transferring from a local trainTcUnit passenger route time, sTtripTotal travel time of passengers, sTtransferIf there is a transfer, transfer time for passengers to transfer to other lines sToutThe exit time of passenger, sTctTotal travel time, sTceTotal travel time by express trains, sTclTotal travel time by local trains, s


### 2.3. Assumptions

According to the needs of the research problem, some assumptions are made in this paper, which are listed as follows:Assumption 1. In the new mode, due to the different speeds of the express train and local train, the express train may need to overtake the local train in a running interval, which requires the line to have certain conditions. In the modeling process, it is assumed to be in an ideal state. Assuming that all stations on the line have overtaking conditions, the local train ahead can wait for a little at the nearest station, and the express train will overtake quickly.Assumption 2. Although some metro operators are trying to dynamically adjust the train dwell time according to the passenger flow to shorten the whole operation time, it is still in the stage of theoretical research and has not been implemented. In the new mode, in order to compare the improvement degree between the optimized scheme and the original scheme, the dwell time is regarded as the same value.Assumption 3. During the normal operation mode, the section running speed is usually constant without considering the additional time of train start and stop. Only when the train is in a state of delay, the driver will accelerate to make up for the time loss to match with the train scheme as much as possible. This situation is not within the scope of this paper. In order to compare the running time of the train in each section and the calculation of the total running time, the running speed of the train in the section is regarded as the same value.Assumption 4. According to the operation practice of urban rail transit, the allocated passenger trains run circularly in an operation cycle, therefore, it can be assumed that the number of allocated passenger trains is sufficient in one operation day, that is, it is not necessary to consider the complete shortage of trains in the optimization modeling.


## 2.1. Problem Description

In the studied problem, the special channel for rapid transportation between suburban satellite towns and central urban areas is called express and local modes, which can connect the urban backbone network to special channels with them. The frame structure of suburban lines is shown in Figure1.Figure 1
Schematic diagram of the urban backbone network and suburban line structure.Adopting the new modes of express and local trains can solve the problem of passenger flow transportation between suburban lines to a certain extent, but it will bring about some new problems simultaneously, such as (i) how to ensure the high-precision matching of transport capacity and passenger flow during the rush hours; (ii) how to determine the proportion of express trains and local trains; (iii) how to determine the stop scheme of the express trains; (iv) the impact of the adoption of express and local train mode on the reduction of carrying capacity; and (v) the result of the same line transfer of some passengers, which will increase the complexity of station passenger transport organization and train dispatching in the state of delay.In the problems above, for instance, there are 5 stations marked as A–E, and the average passenger flow can be known from the operation history, while in the traditional mode, due to the unbalanced passenger flow, some stations are in the insufficient state, some stations are in the moderate state, still, others are in the surplus state. When in the insufficient state, the station is in danger as shown in Figure2(a) in which, n1 represents the number of passengers detained at the platform after train2 leaves Station A with passengers, n2 represents the number of passengers detained at the platform after train2 leaves Station B with passengers, n3 represents the number of passengers detained at the platform after train2 leaves Station C with passengers, n4 represents the number of passengers detained at the platform after train2 leaves Station D with passengers; m1 represents the number of passengers detained at the platform after train3 leaves Station A with passengers, m2 represents the number of passengers detained at the platform after train3 leaves Station B with passengers, m3 represents the number of passengers detained at the platform after train3 leaves Station C with passengers, m4 represents the number of passengers detained at the platform after train3 leaves Station D with passengers. Therefore, it is necessary to study the method of determining the stop scheme of an express train to realize the rapid matching of carrying capacity and passenger flow. The objective function of the stop scheme optimization model is to minimize the stranded passenger flow in the station. Besides, it is necessary to study the proportion of express and local trains m : n. In Figure 2(b), the application of the proposed optimization model can alleviate the waste of carrying capacity caused by the imbalance of passenger flow or the long waiting time of passenger flow in some stations, so the state of stations becomes safe.Figure 2
The passenger flow status under different metro operation modes: (a) passenger flow in traditional operation mode; (b) passenger flow in express and local mode.
(a)(b)As discussed in the above descriptions, the operation mode of express and local trains on suburban lines is relatively complex, and there will be many new situations, such as the passenger flow between express and local trains will transfer at the same station, and the other situation is to take the express train first and then transfer to the local train to reach the destination station, which is shown in Figure3.Figure 3
Express and local train operation mode of the suburban line of rail transit.Given the problems arising from the new operation mode of express and local trains, such as the stop scheme of express trains, the running proportion of express and local trains, and the loss of line carrying capacity, it is necessary to establish a mathematical model with the research goal of minimizing the travel time of passengers and giving consideration to the minimum cost of enterprises. The research boundary can be defined as the suburban lines of the subway, the obvious space-time imbalance, and the tidal characteristics of the passenger flow. The decision variables of the model involve the full load rate of trains, the number of allocated passenger trains, and the minimum departure interval. The collaborative optimization between new indicators is studied in depth. The reason why this optimization model is a collaborative optimization is that the mining of passenger flow OD will determine the stop scheme of express trains, and the stop scheme will also affect the travel time of passengers. The OD of passenger flow also determines the running proportion of express and local trains, and the running proportion of trains determines the loss capacity of the line quantitatively. Therefore, this is a collaborative optimization model, which is closely linked.


## 2.2. Symbols Formulation

To formulate the problem, some main related notations and parameters in the formulating process are listed in Table1 for the completeness of the research.Table 1
Notations and parameters used in the paper.
CinTravel time and cost of passengers on express and local lines, yuanaTime-consuming weighttwWaiting time in the platform, mintzTravel time in the sections, mindDwell time, minYCongestion cost, yuanfkThe departures of class k trains within one hour, trainsjlDistance between stationsj and station l, kmz′The sections which the trainK passesvkThe speed of traink, km/hSkThe departure stationekThe arrival stationqkThe passengers who choose classk of trains in the sectionsbkTrain formation of classkzkThe train capacity of classksThe distance of sectionsvThe running speed of traincprsThe travel cost of sectionr-s for route P, yuanc¯The average travel cost of all routes, yuanxprsThe probability of passengers choosing routepfprsThe passenger flow of routep, personqrsOD passenger flow inR–S section, personxaSection passenger flow, personδa,prsWhen section a belongs to thep-th path of R-S, it is 1, otherwise, it is 0tijThe running time in sectionsi-j, minhi1The start additional time of local trains in stationi, minhi2The dwell time of local trains in stationi, minxiWhether the express train stop at stationi, if it stops i = 1, if not i = 0nThe number of stationsQijkThe passengers who select trainK from station i to station jzw1Total travel time of passengers in the upward direction, minzw2Total travel time of passengers in the downward direction, minTThe length of optimize period, minf1The departure frequency of express trains, train/hf2The departure frequency of local trains, train/hlijThe travel length from stationi to station j, kmlw1The total running distance of uplink trains in the optimized period, kmlw2The total running distance of the downlink train in the optimized period, kmlwThe total cost of train operation, yuancrunRunning cost per kilometer, yuan/kmnw1The total dwell stops of uplink trains in the optimized periodnw2The total dwell stops of downlink trains in the optimized periodcwThe total cost of train stopping, yuancdThe stop cost of every station, yuan/timetrInterval running time(s)ttThe transfer time(s)∆tWaiting time of trains overtaketm,nlIn traditional operation mode, the whole travel time is from stationm to station n when the local train runstm,neThe travel time of an express train from stationm to station nTm,nlThe travel time of a local train from stationm to station nTm,nleThe travel time for passengers fromm to n by taking a local train and then transferring to an express trainTm,nelThe travel time for passengers fromm to n by taking an express train and then transferring to a local trainsjA station for an express trainsThe first station where an express stops when a local train transferring to an express trains′The last station where an express train stops when transferring from a local trainTcUnit passenger route time, sTtripTotal travel time of passengers, sTtransferIf there is a transfer, transfer time for passengers to transfer to other lines sToutThe exit time of passenger, sTctTotal travel time, sTceTotal travel time by express trains, sTclTotal travel time by local trains, s


## 2.3. Assumptions

According to the needs of the research problem, some assumptions are made in this paper, which are listed as follows:Assumption 1. In the new mode, due to the different speeds of the express train and local train, the express train may need to overtake the local train in a running interval, which requires the line to have certain conditions. In the modeling process, it is assumed to be in an ideal state. Assuming that all stations on the line have overtaking conditions, the local train ahead can wait for a little at the nearest station, and the express train will overtake quickly.Assumption 2. Although some metro operators are trying to dynamically adjust the train dwell time according to the passenger flow to shorten the whole operation time, it is still in the stage of theoretical research and has not been implemented. In the new mode, in order to compare the improvement degree between the optimized scheme and the original scheme, the dwell time is regarded as the same value.Assumption 3. During the normal operation mode, the section running speed is usually constant without considering the additional time of train start and stop. Only when the train is in a state of delay, the driver will accelerate to make up for the time loss to match with the train scheme as much as possible. This situation is not within the scope of this paper. In order to compare the running time of the train in each section and the calculation of the total running time, the running speed of the train in the section is regarded as the same value.Assumption 4. According to the operation practice of urban rail transit, the allocated passenger trains run circularly in an operation cycle, therefore, it can be assumed that the number of allocated passenger trains is sufficient in one operation day, that is, it is not necessary to consider the complete shortage of trains in the optimization modeling.


## 3. Collaborative Optimization Modeling Based on New Mode

### 3.1. The Calculation of Passengers’ Total Travel Time

The total travel time of passengers is affected by the stop schemes, dwell time, section running speed, and different proportions of express-local trains.Assuming that in a group of express and local trains, the operation proportion is 1 :u, the trains’ departure is regular with a balanced manner, and the operation scheme in the same interval is also regularly laid out, this section will discuss the calculation of passengers’ travel time when the number of local trains exceeds that of express trains, that is, when the operation proportion is 1 : u.In express and local train mode, it is assumed that passengers get on at stationm and get off at station n, and the number of crossing stations is c. According to the actual operation of express and local trains, the total travel time T of passengers can be expressed as the following formula:(1)T=tw+ts+tr+tt.(1)
When the local proportion is 1 :u and u ≤ c, then in the m-n interval, the departure interval is h2, and the calculation model of local train travel time is constructed, as shown in the following formula:(2)Tmnl=∑x=mn−1tr+∑x=m+1n−1ts−∑i=m+1c−11−αkiβits+∑i=2c−1h2−hi′+2Iβi+2βiI+h22.(2)
When the express proportion is 1 :u and u ≤ c, then in the m-n interval, the departure interval is h2′, the calculation model of express train travel time is constructed, as shown in the following formula:(3)Tmne=∑x=mn−1tr′+∑x=m+1n−1αxts+h2′2.(3)
Assuming that there are express stations betweenn-M travel intervals, when passengers choose to take the local train first and then transfer to the express train, and get off or transfer at the first express station, which the express trains pass through, or taking express train first, then transfer to local one, so the expression can be shown in the following formula:(4)Tmnle=∑x=ms′−1tYx′+∑x=m+1s′−1αxtTx′+h2′2+∑x=ms′−1tQYx+∑x=2s′tTx′−∑i=1c−11−αkiβsi′tTki+∑i=2c−1h2−hi′+2Iβsi′−∑x=1s′−1tYx′+∑x=2s′−1αxtTx+βs′i′−1h2+h1+h2∑i=1cβs′i′+∑x=s′n−1tYx+∑x=s′+1n−1tTx−∑i=1c−11−αkiβs′i′tTki+2βs′i′+∑i=2c−1h2−hi′+2Iβsi′.In whichβi, γj, βsi, βsi′, βs′i, and βs′i′ take the following piecewise function values, respectively:(5)βi=1,ki∈m,n,0,other,γj=1,sj∈m,n,0,other,βsi=1,ki∈m,n,0,other,βsi′=1,ki∈m,n,0,other,βs′i=1,ki∈s′,n,0,other,βs′i′=1,ki∈1,s,0,other.The above analysis shows that the express and local modes can improve the service level of long-distance trips, but as the express and local train pair is balanced regular grid,h2 is determined by the interval of maximum distance between the two overtaking stations and long running time, so in this case, it will cause local trains in other overtaking stations to wait for a longer time. Therefore, Shanghai Metro Line 16 has suspended express and local train modes. With the increasing number of express trains overtaking, if the distance between overtaking stations is not balanced, it will cause passengers of local trains to wait too long in some overtaking stations. Given the overtaking station, an interval is not balanced and cannot be changed in the short term, the number of times that express trains overtake local trains should be reduced as quickly as possible.The metro operation of the express and local trains on suburban lines is typically dense, and the traffic density is also large, so the arrival rule of passengers can be regarded as independent of the train schedule. Under the condition of a short interval, the average waiting time of the whole passenger flow tends to be half of the departure interval. In the express and local mode (ignoring the temporary transfer of some local passengers to the express train and then to the local train at last), the average waiting time of passengers on the express train is expressed astw=1/2Ti/fi, and that local trains can be expressed as tw=θi+1−θi/mi×1/2Ti/fi, θi is the sharing rate of passengers who originally plan to take the local train being attracted by the express train, and its value θi=1/1+mi, mi is the passenger flow of the interval. Passenger transit time Tc and Tct are shown in the following formulas:(6)Tc=tw+Ttrip+Ttransfer,Transfer0,Notransfer+Tout,(7)Tct=Tcl+Tce.The aboveTce and Tcl can be calculated from the following formulas:(8)Tce=Ti2fi×∑k=1N∑j=1NDi,k,j−ηi×∑k=hshe−1∑j=k+1heDi,k,j+∑k=hs+1he∑j=hsk−1Di,k,j,(9)Tcl=θi+1−θimi×12Tifi×ηi×∑k=hshe−1∑j=k+1heDi,k,j+∑k=hs+1he∑j=hsk−1Di,k,j.


### 3.2. Optimization of Model for Metro Operation Based on Express and Local Mode

#### 3.2.1. The Route Selection of Passengers

In the express and local mode, the decision variable of the model is 0-1 variable, and the passenger topology will change under different express stop schemes. So, a passenger path allocation model based on generalized path cost is constructed to obtain the passenger flow of each path and section in the offline network under different stop schemes.(1) The Construction of the Cost Function Model. Passengers always need to transfer from express and local lines to other lines or from other lines to express and local lines. The generalized travel cost consists of “express and local lines” and “nonexpress and local lines” and the functions of passengers for two sections are constructed, respectively.The broad travel cost of passengers on express and local modes can be divided into the sum of waiting time and passengers’ on-the-way time. The sum of passengers’ travel time in each section is taken as the sum of passengers’ time spent on the operation line, which can be converted into travel cost to characterize the impact of the travel time cost on passengers’ route line choice. The path cost function is shown in the following formula:(10)Cin=atw+Tz+d+Y.The time parameters can be calculated as follows:(1)
Waiting time.In the broad travel cost of passengers, the section running time and the dwell time of the train will not change with the departure frequency trains, while the waiting time will change with departure interval. If the departure interval varies greatly, the waiting time will become the main factor that affects the choice of express or local trains.Assuming that the arrival of passengers follows the Poisson distribution, without considering the detention of passengers, the waiting time of passengers at the platform can be approximately regarded as 1/2 of the departure interval [12]. The waiting time cost can be expressed as follows:(11)W=602×fk.(2)
The train’s running time.The train’s running time can be divided into running time into sections and dwell time, as shown in the following formula:(12)Tz=60×∑z∈z′szvk+d,k=1,2.(3)
The time cost of train congestion.On the express and local lines, passengers will choose whether to take the train according to the load rate of the train, which will have a certain impact on the selection of express or local trains. The degree of congestion can be expressed by the proportion of passenger demand for express or local trains’ capacity. The greater the proportion, the higher the degree of congestion, and the higher the congestion cost will be.e,e+1 can be regarded as the passenger flow of sections, and the congestion coefficient of class k train lines can be shown in formula (13), and the train congestion cost is obtained by multiplying the operation time of the train section and the congestion degree coefficient, which is shown in the following formula:(13)ηk=∑i=skr∑j=r+1ekfi,j,kbkxkzk,(14)Y=0,ηk≤1,α⋅sv⋅ηkβ,ηk>1.In which, the value ofα is 0.15 and the value of β is 4.According to the passenger flow characteristics of suburban lines, combined with the train congestion cost function, formula (14) can be transformed into the following formula:(15)Y=Qk=∑i=re∑j=e+1sfi,j,k,0,Qk≤bkfkzk,∑e=ij−1θ⋅SV⋅Qkbkfkzkβ,Qk>bkfkzk.(2) Passenger Flow Route Assignment Model and Algorithm. The OD choice of r−s section can be determined by the path cost, and the generalized path cost can determine the random utility of the path choice. Due to the judgment error caused by passenger route selection, the random errors are independent of each other and obey the same Gumbel distribution [13], then the path selection probability can be calculated by the Logit model.It is assumed that there arem alternative travel paths for passengers, then we can get the probability of selecting path P in OD passenger flow, the passenger flow of path P, and the passenger flow of each section, which is expressed in the following formula:(16)xprs=exp−cprs/c¯∑p∈mexp−cprs/c¯,fprs=qrsxprs,xa=∑r∑s∑pfprsδa,prs.(3) The Construction of Path Selection Algorithm. The section passenger flow allocation model includes a balanced allocation model and an unbalanced allocation model. The final allocation result of the balanced allocation model can make the travel costs of all utilized routes between station OD reach the same state, so as to balance the passenger flow allocation of all the metro networks. Compared with the unbalanced allocation model, the algorithm logic of the balanced allocation model is more rigorous.Beckmann traffic balance model allocation method is used to obtain the unique passenger flow allocation result of each section of express and local lines, as shown in formula (17), in which Caxa is the generalized time cost of the section.(17)minZx=∑a∫0xaCaϖdϖ,s.t.∑lflrs=qrs,flrs≥0,xa=∑r∑s∑lflrsδa,lrs.In it, when the value ofδa,krs is 1, it indicates the L-th path of section a belongs to r and s; when the value of δa,krs is 0, it indicates that section a does not belong to the L-th path of r and s; Caϖ: the generalized cost of the section.The solving algorithm of this model is shown as follows:Step 1. Bring the initial value into formula (10) to calculate the initial section cost: c0, and allocate 0-1 to obtain xa1, let n = 1, w and Tz can be calculated from formular (11) and (12).Step 2. Renewal of the section cost: ca=caxan;Step 3. Search the descent direction of the objective function shown in formula (17), renew the line resistance, then a set of additional flow yan is obtained by 0-1 distribution method;Step 4. According to the formula ∑ayan−xancaxan+λyan−xan=0, calculating the step length: λ;Step 5. Updating the formula xan+1=xan+λyan−xan;Step 6. Convergence judgment, if the formula (18) is satisfied, the calculation is ended; otherwise, return to step 2.(18)∑axan+1−xan2∑axan<ε.Step 7. Stopping the iteration, and getting the final section passenger flow xa.


#### 3.2.2. Collaborative Optimization Modeling Based on New Indicators

The passenger travel time under express and local modes can be finely divided into uplink and downlink directions, and then the total passenger travel time can be obtained as the objective function of the optimization model.The up-down direction travel time of passengers can be divided into the travel time of express trains, the travel time of local trains, and the additional time of train start and stop, and superimposed them, which is shown in the following formula:(19)zw1=∑i=1n−1∑j=i+1nQij1⋅xi⋅xj⋅tij+hi1+hi2+d+∑i=1n−1∑j=i+1nQij2⋅xi⋅xj⋅j−i×tij+hi1+hi2+d,zw2=∑i=j+1n∑j=1n−1Qij1⋅xi⋅xj⋅tij+hi1+hi2+d+∑i=j+1n∑j=1n−1Qij2⋅xi⋅xj⋅j−i×tij+hi1+hi2+d,minz1=zw1+zw2.The operation cost of the enterprise is determined by the running mileage and the number of stops of trains. The running kilometers of the up and down direction of trains, and the total cost of the trains’ operation are shown in the following formula:(20)lw1=∑i=1n−1∑j=i+1nxi⋅xj⋅lij⋅T⋅f1+l1n⋅T⋅f2,lw2=∑i=j+1n∑j=1n−1xi⋅xj⋅lij⋅T⋅f1+l1n⋅T⋅f2,lw=lw1+lw2×crun.Therefore, the optimization objective of the number of up and down stops, the total stop cost, and the lowest enterprise operation cost are shown in the following formula:(21)nw1=∑i=1n−1∑j=i+1nxi⋅T⋅f1+T⋅f2⋅n−2,nw2=∑i=j+1n∑j=1n−1xi⋅T⋅f1+T⋅f2⋅n−2,cw=nw1+nw2×cd,minz2=lw+cw.The stop scheme mainly includes reasonable times of stops and dwell time, so that passengers can choose the appropriate train number according to OD, and passengers can transfer between different types of trains. However, unreasonable stops will reduce the speed of trains, increase the running time and energy consumption, and extend the travel time of passengers. Travel time is an important factor affecting passengers’ choice of travel mode, so the stop scheme can be optimized from passengers’ travel time.(1) The Objective Function. In express and local modes, the travel time of passengers is complicated. Passengers’ travel time on the express train is shortened, while time on the local train travel is lengthened by avoidance. Therefore, the total variable of the travel time of express and local trains is selected as the research objective, and the minimum consumption of the average total travel time of passengers taking express and local trains is taken as the objective, that is, the maximum objective is taken.(1)
Average waiting time of passengers.The average waiting time is used to describe the total average waiting time of passengers, as shown in the following formula:(22)twait=uij×n×Tz2nij.In which,μij: number of passengers from station i to station j;Tz: operation cycle time;nij: number of trains from i to j;n: total number of cycles in a running scheme.(2)
The dwell time.After the overall stop objective scheme is determined, it is necessary to study the refined optimization. It is mainly considered from the following four aspects: (i) the number of stations that the express train stops; (ii) the travel time that passengers take express and local trains; (iii) the objective function of minimizing operation costs; (iv) running logarithmic (the minimum needs) objective function, etc., especially when facing more complex constraints. This express stop scheme optimization problem abstraction for multiple phases is an uncertain integer programming optimization model, and the complex constraint condition can be assumed as a Lagrange relaxation process, so it needs to design the subgradient algorithm model based on the label correction algorithm of the optimal lower bound to obtain the approximate optimal solution. The basic research framework and structure are shown in Figure4:Figure 4
The optimized stop scheme.The relevant time of train stop includes the dwell time and the additional time of start and stop. The total time consumption of the train stop is shown in the following formula:(23)tstation=μpqh+τt+τq∑rRxijrfrnij.Then the objective function of minimizing the total travel time of passengers is shown in the following formula:(24)minTtotal=∑twait+∑tstation.The constraints are shown in the following formulas:(25)∑r=1Rμijr=μij≥Dij,(26)∑p=si−1∑q=izμpq≤Qfr,(27)∑i=s+1zxijr≤z−sxszr,(28)fr≤Mxszr,(29)μijr≤Mxijr,(30)k≤∑i=1nxijr≤k′,(31)∑r=1Rfr≤E,(32)∑r=1R∑i=2n−1∑p=si−1∑q=izμpqxijrQfr≥λ.Formula (25) refers to the total transport capacity of all types of trains greater than or equal to passenger flow demand. Dij is the passenger flow between I and J; formula (26) ensures that the total number of passengers cannot exceed the train capacity under a certain full capacity proportion, where Q is the number of train passengers; formula (27) and (28) are the structural constraints of the train scheme. M is an arbitrarily large number. Formula (29) indicates that the train does not stop at the station, so there are no passengers getting on or getting off. Formula (30) represents the lower limit k and upper limit k′ of train stop times; formula (31) represents the line and station capacity constraints, which means that the average load rate of trains in each section meets a certain threshold. Formula (32) indicates that the proportion between the total passenger service transport of the train and the carrying capacity at a certain full capacity rate is greater than the Poisson distribution strength of passengers entering the station, indicating that the system is in a steady state.


#### 3.2.3. The Constraints of the Optimization Model

(i)
The constraint of section passenger flow load.In the actual passenger-carrying process, the train section has a certain limit on the number of passengers. The maximum number of passengers in the train section is represented by the section's full load rate, which should satisfy the limit of the section and full load rate, shown in (1) and (2) of formula (33).(ii)
The constraints of station passenger flow demand.While meeting the OD constraints of section passenger flow, the train stop scheme needs to meet the constraints of station passenger flow at the same time, so that the passenger flow in the station can be transported in time.Settingl, i, and j as station label l<i<j, then the stations need to meet the passenger flow demand constraints, shown in (3) of formula (33);(iii)
The train departure times constraint in the optimization period.In order to ensure the safety of the train, a certain departure interval is required between the front and rear trains, thus, the total number of departures per unit time is limited, and the total number of departures in the optimization period shall be met, shown in (4) of formula(33).Combine the above three constraints into a model constraint set, as shown in the following formula:(33)1Qk=∑i=re∑j=e+1sQijk,2Qkfk⋅bk⋅zk≤βmax,3∑l<i∑j>iQljk⋅xl⋅xj+∑j>iQijkT⋅fk≤bk⋅zk,4∑kfk≤60Imin.In which,βmax: the upper limit of full load proportion of express and local trains, generally, the value is 1.2 [14];bk: the train formation of train k;zk: the capacity of train k;Imin: the minimum departure interval of the train per min.


### 3.3. Solution Algorithm and Numerical Experiment

The model is double objective programming, and each station has two variables “express train stops” and “local train stops,” it is assumed that there aren stations for the express trains to decide whether to stop or not, so there are 2n kinds of stop schemes and the decision variables include both express and local train departure interval f1 and f2, which increases the complexity of the problem.In the solving algorithm, the setsni and si, ni are defined as the number of solution individuals that dominate the individual in the population; si is the set of solutions individuals dominated by individual i. For each individual i in the current set Fn, when the function gets the extreme value, the crowding degree of individual i tends to infinity; when the function obtains a nonextreme value, the crowding degree is id=∑j=1mfji+1−fji−1.In which,id: the crowding degree of individual i;fji+1: The j-th optimization target value of individual i+1;j: The j-th optimization target value of individual i−1;m: the number of optimization objectives.The process of the improved nondominated genetic algorithm calculation codes is shown in Appendix A, and the elite strategy is as follows:Step 1: Chromosome coding, the decision variables are the running frequencyf1 and f2, and the stop sequence of express trains, the chromosome coding method is 000100︸f1000010︸f2010000︸n0, in which, f1 and f2 is binary coding, n0 indicates the number of stations that express trains need to stop. In the stop sequence, 1 indicates that the station at which the express train stops, while 0 indicates not stop. Randomly initialize the parent population, the population number is 200, and the number of iterations is set to 200;Step 2: The offspringQt and the parent Pt were merged into Rt, and the population size was set to 2N;Step 3: GeneratingFn,n=1 according to the nondominated sorting method;Step 4: If the population number inFn is less than N, go to the next gene stop the iteration, and get the final section passenger flow xa, make n = n + 1, return to Step3, if Fn is more than N, go to step5;Step 5: Calculating the individual crowding degreeid in Fn, the individuals with high crowding degrees are preferentially selected for the next generation until the number of individuals in the next generation reaches N;Step 6: Selection, crossover, and mutation, among which, roulette is adopted for selection, a single-point crossover is adopted, the crossover probability is 0.8 and the mutation probability is 0.04;Step 7: If the number of iterations does not reach 200, return to step 2; when the number of iterations reaches 200, go to step 8;Step 8: We output the optimal value;Step 9: Decoding the optimal value.


### 3.4. Evaluation and Analysis of Optimization Results

The analysis and evaluation of the optimization results of the operation scheme can be obtained from the travel time of passengers and the number of trains needed.(1)
The travel time of passengersThe primary purpose of optimizing the stop scheme of express trains is to match the demand of passengers as much as possible, so as to minimize the transfer of passengers between the express and local trains, to save passengers’ travel time, and promote travel convenience. Meanwhile, consider the overall travel time change of all passengers along the line, so as to achieve the optimization goal of shortening the overall travel time of passengers across the line.(2)
The number of trains neededTrains needed are the number of trains that can be normally put into operation to complete the transportation task, the number of trains per hour is calculated as shown ind.(34)Ny=∑kbkfkθk60.In which,Ny: the number of trains required, train;θk: the turnaround time of the train, min.In whichθk=260⋅∑zSz/Vk+∑i∈i′xi⋅d+tz,k=1,260⋅∑zSz/Vk+n⋅d+tz,k=2.(3)
The calculation of carrying capacityThe express and local mode adopts a nonparallel operation schedule, and the carrying capacity can be calculated by graphic method or analytical method. Although the graphic method has high precision, the calculation process is more complicated, and the method of combining analysis and graphics is usually adopted. As shown in Figure5, assuming that only station 5 has the condition of crossing line (the four stations from station 1 to station 4 do not have the condition of crossing line), when an express train overtakes local stations without stopping, the shortest time required to occupy the interval for laying down an express and local train can be calculated by the following formula:(35)Ien+Iln≥2hmini+ntstation+∆t−tstation.(1)
When the proportion of express and local trains isk : m = 1 : 3.When the running proportion of express and local trains is 1 : 3, each combination of express and local trains can run 4 trains, then the number of passengers that can be transported in each combination cycle.Pcycle = 1296 ∗ 3 + 648 = 4536, based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/4536 = 1.6, cycle time: Tcycle = 3600/1.6 = 2250, the average departure interval between four trains in each combination of express and local trains: Td = 2250/4 = 562.5 s. Td+tre−trl=562.5+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(2)
When the proportion of express and local trains isk : m = 1 : 2.When the proportion of express and local trains is 1 : 2, each combination of express and local trains can run 3 trains, then the number of passengers that can be transported in each operating cycle:Pcycle = 1296 ∗ 2 + 648 = 3240, based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/3240 = 2.25, cycle time: Tcycle = 3600/2.25 = 1650 s, the average departure interval between three trains in each combination of express and local trains is Td = 2250/3 = 533 s. Td+tre−trl=533+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(3)
When the proportion of express and local trains isk : m = 1 : 1.When the proportion of express and local trains is 1 : 1, each combination of express and local trains can run 2 trains, then the number of passengers that can be transported in each operating cycle:Pcycle = 1296 ∗ 1 + 648 = 1944, the total number of express and local train groups required to operate during peak hours, and the total number of express and local train groups is n = = 7290/1944 = 3.75, cycle time: Tcycle = 3600/3.75 = 960 s, the average departure interval between two trains in each combination of express and local trains is Td = 960/2 = 480 s. Td+tre−trl=480+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(4)
When the proportion of express and local trains isk : m = 2 : 1.When the proportion of express and local trains is 2 : 1, each combination of express and local trains can run 3 trains, then the number of passengers that can be transported in each operating cycle:Pcycle = 648 ∗ 2 + 1296 = 2592, based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/2592 = 2.81, cycle time: Tcycle = 3600/2.81 = 1280 s, the average departure interval between three trains in each combination of express and local trains is Td = 1280/3 = 426 s. Td+tre−trl=426+2123−3082<Icatch=90s. Therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(5)
When the proportion of express and local trains running isk : m = 3 : 1.Figure 5
The operation scheme of express and local mode.When the running proportion of express and local trains is 3 : 1, each combination of express and local trains can run 4 trains, then the number of passengers that can be transported in each combination cycle.Pcycle = 648 ∗ 3 + 1296 = 3240. Based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/3240 = 2.25, cycle time: Tcycle = 3600/2.25 = 1600 s, the average departure interval between four trains in each combination of express and local trains is Td = 1600/4 = 400 s. Td+tre−trl=400+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.


## 3.1. The Calculation of Passengers’ Total Travel Time

The total travel time of passengers is affected by the stop schemes, dwell time, section running speed, and different proportions of express-local trains.Assuming that in a group of express and local trains, the operation proportion is 1 :u, the trains’ departure is regular with a balanced manner, and the operation scheme in the same interval is also regularly laid out, this section will discuss the calculation of passengers’ travel time when the number of local trains exceeds that of express trains, that is, when the operation proportion is 1 : u.In express and local train mode, it is assumed that passengers get on at stationm and get off at station n, and the number of crossing stations is c. According to the actual operation of express and local trains, the total travel time T of passengers can be expressed as the following formula:(1)T=tw+ts+tr+tt.(1)
When the local proportion is 1 :u and u ≤ c, then in the m-n interval, the departure interval is h2, and the calculation model of local train travel time is constructed, as shown in the following formula:(2)Tmnl=∑x=mn−1tr+∑x=m+1n−1ts−∑i=m+1c−11−αkiβits+∑i=2c−1h2−hi′+2Iβi+2βiI+h22.(2)
When the express proportion is 1 :u and u ≤ c, then in the m-n interval, the departure interval is h2′, the calculation model of express train travel time is constructed, as shown in the following formula:(3)Tmne=∑x=mn−1tr′+∑x=m+1n−1αxts+h2′2.(3)
Assuming that there are express stations betweenn-M travel intervals, when passengers choose to take the local train first and then transfer to the express train, and get off or transfer at the first express station, which the express trains pass through, or taking express train first, then transfer to local one, so the expression can be shown in the following formula:(4)Tmnle=∑x=ms′−1tYx′+∑x=m+1s′−1αxtTx′+h2′2+∑x=ms′−1tQYx+∑x=2s′tTx′−∑i=1c−11−αkiβsi′tTki+∑i=2c−1h2−hi′+2Iβsi′−∑x=1s′−1tYx′+∑x=2s′−1αxtTx+βs′i′−1h2+h1+h2∑i=1cβs′i′+∑x=s′n−1tYx+∑x=s′+1n−1tTx−∑i=1c−11−αkiβs′i′tTki+2βs′i′+∑i=2c−1h2−hi′+2Iβsi′.In whichβi, γj, βsi, βsi′, βs′i, and βs′i′ take the following piecewise function values, respectively:(5)βi=1,ki∈m,n,0,other,γj=1,sj∈m,n,0,other,βsi=1,ki∈m,n,0,other,βsi′=1,ki∈m,n,0,other,βs′i=1,ki∈s′,n,0,other,βs′i′=1,ki∈1,s,0,other.The above analysis shows that the express and local modes can improve the service level of long-distance trips, but as the express and local train pair is balanced regular grid,h2 is determined by the interval of maximum distance between the two overtaking stations and long running time, so in this case, it will cause local trains in other overtaking stations to wait for a longer time. Therefore, Shanghai Metro Line 16 has suspended express and local train modes. With the increasing number of express trains overtaking, if the distance between overtaking stations is not balanced, it will cause passengers of local trains to wait too long in some overtaking stations. Given the overtaking station, an interval is not balanced and cannot be changed in the short term, the number of times that express trains overtake local trains should be reduced as quickly as possible.The metro operation of the express and local trains on suburban lines is typically dense, and the traffic density is also large, so the arrival rule of passengers can be regarded as independent of the train schedule. Under the condition of a short interval, the average waiting time of the whole passenger flow tends to be half of the departure interval. In the express and local mode (ignoring the temporary transfer of some local passengers to the express train and then to the local train at last), the average waiting time of passengers on the express train is expressed astw=1/2Ti/fi, and that local trains can be expressed as tw=θi+1−θi/mi×1/2Ti/fi, θi is the sharing rate of passengers who originally plan to take the local train being attracted by the express train, and its value θi=1/1+mi, mi is the passenger flow of the interval. Passenger transit time Tc and Tct are shown in the following formulas:(6)Tc=tw+Ttrip+Ttransfer,Transfer0,Notransfer+Tout,(7)Tct=Tcl+Tce.The aboveTce and Tcl can be calculated from the following formulas:(8)Tce=Ti2fi×∑k=1N∑j=1NDi,k,j−ηi×∑k=hshe−1∑j=k+1heDi,k,j+∑k=hs+1he∑j=hsk−1Di,k,j,(9)Tcl=θi+1−θimi×12Tifi×ηi×∑k=hshe−1∑j=k+1heDi,k,j+∑k=hs+1he∑j=hsk−1Di,k,j.


## 3.2. Optimization of Model for Metro Operation Based on Express and Local Mode

### 3.2.1. The Route Selection of Passengers

In the express and local mode, the decision variable of the model is 0-1 variable, and the passenger topology will change under different express stop schemes. So, a passenger path allocation model based on generalized path cost is constructed to obtain the passenger flow of each path and section in the offline network under different stop schemes.(1) The Construction of the Cost Function Model. Passengers always need to transfer from express and local lines to other lines or from other lines to express and local lines. The generalized travel cost consists of “express and local lines” and “nonexpress and local lines” and the functions of passengers for two sections are constructed, respectively.The broad travel cost of passengers on express and local modes can be divided into the sum of waiting time and passengers’ on-the-way time. The sum of passengers’ travel time in each section is taken as the sum of passengers’ time spent on the operation line, which can be converted into travel cost to characterize the impact of the travel time cost on passengers’ route line choice. The path cost function is shown in the following formula:(10)Cin=atw+Tz+d+Y.The time parameters can be calculated as follows:(1)
Waiting time.In the broad travel cost of passengers, the section running time and the dwell time of the train will not change with the departure frequency trains, while the waiting time will change with departure interval. If the departure interval varies greatly, the waiting time will become the main factor that affects the choice of express or local trains.Assuming that the arrival of passengers follows the Poisson distribution, without considering the detention of passengers, the waiting time of passengers at the platform can be approximately regarded as 1/2 of the departure interval [12]. The waiting time cost can be expressed as follows:(11)W=602×fk.(2)
The train’s running time.The train’s running time can be divided into running time into sections and dwell time, as shown in the following formula:(12)Tz=60×∑z∈z′szvk+d,k=1,2.(3)
The time cost of train congestion.On the express and local lines, passengers will choose whether to take the train according to the load rate of the train, which will have a certain impact on the selection of express or local trains. The degree of congestion can be expressed by the proportion of passenger demand for express or local trains’ capacity. The greater the proportion, the higher the degree of congestion, and the higher the congestion cost will be.e,e+1 can be regarded as the passenger flow of sections, and the congestion coefficient of class k train lines can be shown in formula (13), and the train congestion cost is obtained by multiplying the operation time of the train section and the congestion degree coefficient, which is shown in the following formula:(13)ηk=∑i=skr∑j=r+1ekfi,j,kbkxkzk,(14)Y=0,ηk≤1,α⋅sv⋅ηkβ,ηk>1.In which, the value ofα is 0.15 and the value of β is 4.According to the passenger flow characteristics of suburban lines, combined with the train congestion cost function, formula (14) can be transformed into the following formula:(15)Y=Qk=∑i=re∑j=e+1sfi,j,k,0,Qk≤bkfkzk,∑e=ij−1θ⋅SV⋅Qkbkfkzkβ,Qk>bkfkzk.(2) Passenger Flow Route Assignment Model and Algorithm. The OD choice of r−s section can be determined by the path cost, and the generalized path cost can determine the random utility of the path choice. Due to the judgment error caused by passenger route selection, the random errors are independent of each other and obey the same Gumbel distribution [13], then the path selection probability can be calculated by the Logit model.It is assumed that there arem alternative travel paths for passengers, then we can get the probability of selecting path P in OD passenger flow, the passenger flow of path P, and the passenger flow of each section, which is expressed in the following formula:(16)xprs=exp−cprs/c¯∑p∈mexp−cprs/c¯,fprs=qrsxprs,xa=∑r∑s∑pfprsδa,prs.(3) The Construction of Path Selection Algorithm. The section passenger flow allocation model includes a balanced allocation model and an unbalanced allocation model. The final allocation result of the balanced allocation model can make the travel costs of all utilized routes between station OD reach the same state, so as to balance the passenger flow allocation of all the metro networks. Compared with the unbalanced allocation model, the algorithm logic of the balanced allocation model is more rigorous.Beckmann traffic balance model allocation method is used to obtain the unique passenger flow allocation result of each section of express and local lines, as shown in formula (17), in which Caxa is the generalized time cost of the section.(17)minZx=∑a∫0xaCaϖdϖ,s.t.∑lflrs=qrs,flrs≥0,xa=∑r∑s∑lflrsδa,lrs.In it, when the value ofδa,krs is 1, it indicates the L-th path of section a belongs to r and s; when the value of δa,krs is 0, it indicates that section a does not belong to the L-th path of r and s; Caϖ: the generalized cost of the section.The solving algorithm of this model is shown as follows:Step 1. Bring the initial value into formula (10) to calculate the initial section cost: c0, and allocate 0-1 to obtain xa1, let n = 1, w and Tz can be calculated from formular (11) and (12).Step 2. Renewal of the section cost: ca=caxan;Step 3. Search the descent direction of the objective function shown in formula (17), renew the line resistance, then a set of additional flow yan is obtained by 0-1 distribution method;Step 4. According to the formula ∑ayan−xancaxan+λyan−xan=0, calculating the step length: λ;Step 5. Updating the formula xan+1=xan+λyan−xan;Step 6. Convergence judgment, if the formula (18) is satisfied, the calculation is ended; otherwise, return to step 2.(18)∑axan+1−xan2∑axan<ε.Step 7. Stopping the iteration, and getting the final section passenger flow xa.


### 3.2.2. Collaborative Optimization Modeling Based on New Indicators

The passenger travel time under express and local modes can be finely divided into uplink and downlink directions, and then the total passenger travel time can be obtained as the objective function of the optimization model.The up-down direction travel time of passengers can be divided into the travel time of express trains, the travel time of local trains, and the additional time of train start and stop, and superimposed them, which is shown in the following formula:(19)zw1=∑i=1n−1∑j=i+1nQij1⋅xi⋅xj⋅tij+hi1+hi2+d+∑i=1n−1∑j=i+1nQij2⋅xi⋅xj⋅j−i×tij+hi1+hi2+d,zw2=∑i=j+1n∑j=1n−1Qij1⋅xi⋅xj⋅tij+hi1+hi2+d+∑i=j+1n∑j=1n−1Qij2⋅xi⋅xj⋅j−i×tij+hi1+hi2+d,minz1=zw1+zw2.The operation cost of the enterprise is determined by the running mileage and the number of stops of trains. The running kilometers of the up and down direction of trains, and the total cost of the trains’ operation are shown in the following formula:(20)lw1=∑i=1n−1∑j=i+1nxi⋅xj⋅lij⋅T⋅f1+l1n⋅T⋅f2,lw2=∑i=j+1n∑j=1n−1xi⋅xj⋅lij⋅T⋅f1+l1n⋅T⋅f2,lw=lw1+lw2×crun.Therefore, the optimization objective of the number of up and down stops, the total stop cost, and the lowest enterprise operation cost are shown in the following formula:(21)nw1=∑i=1n−1∑j=i+1nxi⋅T⋅f1+T⋅f2⋅n−2,nw2=∑i=j+1n∑j=1n−1xi⋅T⋅f1+T⋅f2⋅n−2,cw=nw1+nw2×cd,minz2=lw+cw.The stop scheme mainly includes reasonable times of stops and dwell time, so that passengers can choose the appropriate train number according to OD, and passengers can transfer between different types of trains. However, unreasonable stops will reduce the speed of trains, increase the running time and energy consumption, and extend the travel time of passengers. Travel time is an important factor affecting passengers’ choice of travel mode, so the stop scheme can be optimized from passengers’ travel time.(1) The Objective Function. In express and local modes, the travel time of passengers is complicated. Passengers’ travel time on the express train is shortened, while time on the local train travel is lengthened by avoidance. Therefore, the total variable of the travel time of express and local trains is selected as the research objective, and the minimum consumption of the average total travel time of passengers taking express and local trains is taken as the objective, that is, the maximum objective is taken.(1)
Average waiting time of passengers.The average waiting time is used to describe the total average waiting time of passengers, as shown in the following formula:(22)twait=uij×n×Tz2nij.In which,μij: number of passengers from station i to station j;Tz: operation cycle time;nij: number of trains from i to j;n: total number of cycles in a running scheme.(2)
The dwell time.After the overall stop objective scheme is determined, it is necessary to study the refined optimization. It is mainly considered from the following four aspects: (i) the number of stations that the express train stops; (ii) the travel time that passengers take express and local trains; (iii) the objective function of minimizing operation costs; (iv) running logarithmic (the minimum needs) objective function, etc., especially when facing more complex constraints. This express stop scheme optimization problem abstraction for multiple phases is an uncertain integer programming optimization model, and the complex constraint condition can be assumed as a Lagrange relaxation process, so it needs to design the subgradient algorithm model based on the label correction algorithm of the optimal lower bound to obtain the approximate optimal solution. The basic research framework and structure are shown in Figure4:Figure 4
The optimized stop scheme.The relevant time of train stop includes the dwell time and the additional time of start and stop. The total time consumption of the train stop is shown in the following formula:(23)tstation=μpqh+τt+τq∑rRxijrfrnij.Then the objective function of minimizing the total travel time of passengers is shown in the following formula:(24)minTtotal=∑twait+∑tstation.The constraints are shown in the following formulas:(25)∑r=1Rμijr=μij≥Dij,(26)∑p=si−1∑q=izμpq≤Qfr,(27)∑i=s+1zxijr≤z−sxszr,(28)fr≤Mxszr,(29)μijr≤Mxijr,(30)k≤∑i=1nxijr≤k′,(31)∑r=1Rfr≤E,(32)∑r=1R∑i=2n−1∑p=si−1∑q=izμpqxijrQfr≥λ.Formula (25) refers to the total transport capacity of all types of trains greater than or equal to passenger flow demand. Dij is the passenger flow between I and J; formula (26) ensures that the total number of passengers cannot exceed the train capacity under a certain full capacity proportion, where Q is the number of train passengers; formula (27) and (28) are the structural constraints of the train scheme. M is an arbitrarily large number. Formula (29) indicates that the train does not stop at the station, so there are no passengers getting on or getting off. Formula (30) represents the lower limit k and upper limit k′ of train stop times; formula (31) represents the line and station capacity constraints, which means that the average load rate of trains in each section meets a certain threshold. Formula (32) indicates that the proportion between the total passenger service transport of the train and the carrying capacity at a certain full capacity rate is greater than the Poisson distribution strength of passengers entering the station, indicating that the system is in a steady state.


### 3.2.3. The Constraints of the Optimization Model

(i)
The constraint of section passenger flow load.In the actual passenger-carrying process, the train section has a certain limit on the number of passengers. The maximum number of passengers in the train section is represented by the section's full load rate, which should satisfy the limit of the section and full load rate, shown in (1) and (2) of formula (33).(ii)
The constraints of station passenger flow demand.While meeting the OD constraints of section passenger flow, the train stop scheme needs to meet the constraints of station passenger flow at the same time, so that the passenger flow in the station can be transported in time.Settingl, i, and j as station label l<i<j, then the stations need to meet the passenger flow demand constraints, shown in (3) of formula (33);(iii)
The train departure times constraint in the optimization period.In order to ensure the safety of the train, a certain departure interval is required between the front and rear trains, thus, the total number of departures per unit time is limited, and the total number of departures in the optimization period shall be met, shown in (4) of formula(33).Combine the above three constraints into a model constraint set, as shown in the following formula:(33)1Qk=∑i=re∑j=e+1sQijk,2Qkfk⋅bk⋅zk≤βmax,3∑l<i∑j>iQljk⋅xl⋅xj+∑j>iQijkT⋅fk≤bk⋅zk,4∑kfk≤60Imin.In which,βmax: the upper limit of full load proportion of express and local trains, generally, the value is 1.2 [14];bk: the train formation of train k;zk: the capacity of train k;Imin: the minimum departure interval of the train per min.


## 3.2.1. The Route Selection of Passengers

In the express and local mode, the decision variable of the model is 0-1 variable, and the passenger topology will change under different express stop schemes. So, a passenger path allocation model based on generalized path cost is constructed to obtain the passenger flow of each path and section in the offline network under different stop schemes.(1) The Construction of the Cost Function Model. Passengers always need to transfer from express and local lines to other lines or from other lines to express and local lines. The generalized travel cost consists of “express and local lines” and “nonexpress and local lines” and the functions of passengers for two sections are constructed, respectively.The broad travel cost of passengers on express and local modes can be divided into the sum of waiting time and passengers’ on-the-way time. The sum of passengers’ travel time in each section is taken as the sum of passengers’ time spent on the operation line, which can be converted into travel cost to characterize the impact of the travel time cost on passengers’ route line choice. The path cost function is shown in the following formula:(10)Cin=atw+Tz+d+Y.The time parameters can be calculated as follows:(1)
Waiting time.In the broad travel cost of passengers, the section running time and the dwell time of the train will not change with the departure frequency trains, while the waiting time will change with departure interval. If the departure interval varies greatly, the waiting time will become the main factor that affects the choice of express or local trains.Assuming that the arrival of passengers follows the Poisson distribution, without considering the detention of passengers, the waiting time of passengers at the platform can be approximately regarded as 1/2 of the departure interval [12]. The waiting time cost can be expressed as follows:(11)W=602×fk.(2)
The train’s running time.The train’s running time can be divided into running time into sections and dwell time, as shown in the following formula:(12)Tz=60×∑z∈z′szvk+d,k=1,2.(3)
The time cost of train congestion.On the express and local lines, passengers will choose whether to take the train according to the load rate of the train, which will have a certain impact on the selection of express or local trains. The degree of congestion can be expressed by the proportion of passenger demand for express or local trains’ capacity. The greater the proportion, the higher the degree of congestion, and the higher the congestion cost will be.e,e+1 can be regarded as the passenger flow of sections, and the congestion coefficient of class k train lines can be shown in formula (13), and the train congestion cost is obtained by multiplying the operation time of the train section and the congestion degree coefficient, which is shown in the following formula:(13)ηk=∑i=skr∑j=r+1ekfi,j,kbkxkzk,(14)Y=0,ηk≤1,α⋅sv⋅ηkβ,ηk>1.In which, the value ofα is 0.15 and the value of β is 4.According to the passenger flow characteristics of suburban lines, combined with the train congestion cost function, formula (14) can be transformed into the following formula:(15)Y=Qk=∑i=re∑j=e+1sfi,j,k,0,Qk≤bkfkzk,∑e=ij−1θ⋅SV⋅Qkbkfkzkβ,Qk>bkfkzk.(2) Passenger Flow Route Assignment Model and Algorithm. The OD choice of r−s section can be determined by the path cost, and the generalized path cost can determine the random utility of the path choice. Due to the judgment error caused by passenger route selection, the random errors are independent of each other and obey the same Gumbel distribution [13], then the path selection probability can be calculated by the Logit model.It is assumed that there arem alternative travel paths for passengers, then we can get the probability of selecting path P in OD passenger flow, the passenger flow of path P, and the passenger flow of each section, which is expressed in the following formula:(16)xprs=exp−cprs/c¯∑p∈mexp−cprs/c¯,fprs=qrsxprs,xa=∑r∑s∑pfprsδa,prs.(3) The Construction of Path Selection Algorithm. The section passenger flow allocation model includes a balanced allocation model and an unbalanced allocation model. The final allocation result of the balanced allocation model can make the travel costs of all utilized routes between station OD reach the same state, so as to balance the passenger flow allocation of all the metro networks. Compared with the unbalanced allocation model, the algorithm logic of the balanced allocation model is more rigorous.Beckmann traffic balance model allocation method is used to obtain the unique passenger flow allocation result of each section of express and local lines, as shown in formula (17), in which Caxa is the generalized time cost of the section.(17)minZx=∑a∫0xaCaϖdϖ,s.t.∑lflrs=qrs,flrs≥0,xa=∑r∑s∑lflrsδa,lrs.In it, when the value ofδa,krs is 1, it indicates the L-th path of section a belongs to r and s; when the value of δa,krs is 0, it indicates that section a does not belong to the L-th path of r and s; Caϖ: the generalized cost of the section.The solving algorithm of this model is shown as follows:Step 1. Bring the initial value into formula (10) to calculate the initial section cost: c0, and allocate 0-1 to obtain xa1, let n = 1, w and Tz can be calculated from formular (11) and (12).Step 2. Renewal of the section cost: ca=caxan;Step 3. Search the descent direction of the objective function shown in formula (17), renew the line resistance, then a set of additional flow yan is obtained by 0-1 distribution method;Step 4. According to the formula ∑ayan−xancaxan+λyan−xan=0, calculating the step length: λ;Step 5. Updating the formula xan+1=xan+λyan−xan;Step 6. Convergence judgment, if the formula (18) is satisfied, the calculation is ended; otherwise, return to step 2.(18)∑axan+1−xan2∑axan<ε.Step 7. Stopping the iteration, and getting the final section passenger flow xa.


## 3.2.2. Collaborative Optimization Modeling Based on New Indicators

The passenger travel time under express and local modes can be finely divided into uplink and downlink directions, and then the total passenger travel time can be obtained as the objective function of the optimization model.The up-down direction travel time of passengers can be divided into the travel time of express trains, the travel time of local trains, and the additional time of train start and stop, and superimposed them, which is shown in the following formula:(19)zw1=∑i=1n−1∑j=i+1nQij1⋅xi⋅xj⋅tij+hi1+hi2+d+∑i=1n−1∑j=i+1nQij2⋅xi⋅xj⋅j−i×tij+hi1+hi2+d,zw2=∑i=j+1n∑j=1n−1Qij1⋅xi⋅xj⋅tij+hi1+hi2+d+∑i=j+1n∑j=1n−1Qij2⋅xi⋅xj⋅j−i×tij+hi1+hi2+d,minz1=zw1+zw2.The operation cost of the enterprise is determined by the running mileage and the number of stops of trains. The running kilometers of the up and down direction of trains, and the total cost of the trains’ operation are shown in the following formula:(20)lw1=∑i=1n−1∑j=i+1nxi⋅xj⋅lij⋅T⋅f1+l1n⋅T⋅f2,lw2=∑i=j+1n∑j=1n−1xi⋅xj⋅lij⋅T⋅f1+l1n⋅T⋅f2,lw=lw1+lw2×crun.Therefore, the optimization objective of the number of up and down stops, the total stop cost, and the lowest enterprise operation cost are shown in the following formula:(21)nw1=∑i=1n−1∑j=i+1nxi⋅T⋅f1+T⋅f2⋅n−2,nw2=∑i=j+1n∑j=1n−1xi⋅T⋅f1+T⋅f2⋅n−2,cw=nw1+nw2×cd,minz2=lw+cw.The stop scheme mainly includes reasonable times of stops and dwell time, so that passengers can choose the appropriate train number according to OD, and passengers can transfer between different types of trains. However, unreasonable stops will reduce the speed of trains, increase the running time and energy consumption, and extend the travel time of passengers. Travel time is an important factor affecting passengers’ choice of travel mode, so the stop scheme can be optimized from passengers’ travel time.(1) The Objective Function. In express and local modes, the travel time of passengers is complicated. Passengers’ travel time on the express train is shortened, while time on the local train travel is lengthened by avoidance. Therefore, the total variable of the travel time of express and local trains is selected as the research objective, and the minimum consumption of the average total travel time of passengers taking express and local trains is taken as the objective, that is, the maximum objective is taken.(1)
Average waiting time of passengers.The average waiting time is used to describe the total average waiting time of passengers, as shown in the following formula:(22)twait=uij×n×Tz2nij.In which,μij: number of passengers from station i to station j;Tz: operation cycle time;nij: number of trains from i to j;n: total number of cycles in a running scheme.(2)
The dwell time.After the overall stop objective scheme is determined, it is necessary to study the refined optimization. It is mainly considered from the following four aspects: (i) the number of stations that the express train stops; (ii) the travel time that passengers take express and local trains; (iii) the objective function of minimizing operation costs; (iv) running logarithmic (the minimum needs) objective function, etc., especially when facing more complex constraints. This express stop scheme optimization problem abstraction for multiple phases is an uncertain integer programming optimization model, and the complex constraint condition can be assumed as a Lagrange relaxation process, so it needs to design the subgradient algorithm model based on the label correction algorithm of the optimal lower bound to obtain the approximate optimal solution. The basic research framework and structure are shown in Figure4:Figure 4
The optimized stop scheme.The relevant time of train stop includes the dwell time and the additional time of start and stop. The total time consumption of the train stop is shown in the following formula:(23)tstation=μpqh+τt+τq∑rRxijrfrnij.Then the objective function of minimizing the total travel time of passengers is shown in the following formula:(24)minTtotal=∑twait+∑tstation.The constraints are shown in the following formulas:(25)∑r=1Rμijr=μij≥Dij,(26)∑p=si−1∑q=izμpq≤Qfr,(27)∑i=s+1zxijr≤z−sxszr,(28)fr≤Mxszr,(29)μijr≤Mxijr,(30)k≤∑i=1nxijr≤k′,(31)∑r=1Rfr≤E,(32)∑r=1R∑i=2n−1∑p=si−1∑q=izμpqxijrQfr≥λ.Formula (25) refers to the total transport capacity of all types of trains greater than or equal to passenger flow demand. Dij is the passenger flow between I and J; formula (26) ensures that the total number of passengers cannot exceed the train capacity under a certain full capacity proportion, where Q is the number of train passengers; formula (27) and (28) are the structural constraints of the train scheme. M is an arbitrarily large number. Formula (29) indicates that the train does not stop at the station, so there are no passengers getting on or getting off. Formula (30) represents the lower limit k and upper limit k′ of train stop times; formula (31) represents the line and station capacity constraints, which means that the average load rate of trains in each section meets a certain threshold. Formula (32) indicates that the proportion between the total passenger service transport of the train and the carrying capacity at a certain full capacity rate is greater than the Poisson distribution strength of passengers entering the station, indicating that the system is in a steady state.


## 3.2.3. The Constraints of the Optimization Model

(i)
The constraint of section passenger flow load.In the actual passenger-carrying process, the train section has a certain limit on the number of passengers. The maximum number of passengers in the train section is represented by the section's full load rate, which should satisfy the limit of the section and full load rate, shown in (1) and (2) of formula (33).(ii)
The constraints of station passenger flow demand.While meeting the OD constraints of section passenger flow, the train stop scheme needs to meet the constraints of station passenger flow at the same time, so that the passenger flow in the station can be transported in time.Settingl, i, and j as station label l<i<j, then the stations need to meet the passenger flow demand constraints, shown in (3) of formula (33);(iii)
The train departure times constraint in the optimization period.In order to ensure the safety of the train, a certain departure interval is required between the front and rear trains, thus, the total number of departures per unit time is limited, and the total number of departures in the optimization period shall be met, shown in (4) of formula(33).Combine the above three constraints into a model constraint set, as shown in the following formula:(33)1Qk=∑i=re∑j=e+1sQijk,2Qkfk⋅bk⋅zk≤βmax,3∑l<i∑j>iQljk⋅xl⋅xj+∑j>iQijkT⋅fk≤bk⋅zk,4∑kfk≤60Imin.In which,βmax: the upper limit of full load proportion of express and local trains, generally, the value is 1.2 [14];bk: the train formation of train k;zk: the capacity of train k;Imin: the minimum departure interval of the train per min.


## 3.3. Solution Algorithm and Numerical Experiment

The model is double objective programming, and each station has two variables “express train stops” and “local train stops,” it is assumed that there aren stations for the express trains to decide whether to stop or not, so there are 2n kinds of stop schemes and the decision variables include both express and local train departure interval f1 and f2, which increases the complexity of the problem.In the solving algorithm, the setsni and si, ni are defined as the number of solution individuals that dominate the individual in the population; si is the set of solutions individuals dominated by individual i. For each individual i in the current set Fn, when the function gets the extreme value, the crowding degree of individual i tends to infinity; when the function obtains a nonextreme value, the crowding degree is id=∑j=1mfji+1−fji−1.In which,id: the crowding degree of individual i;fji+1: The j-th optimization target value of individual i+1;j: The j-th optimization target value of individual i−1;m: the number of optimization objectives.The process of the improved nondominated genetic algorithm calculation codes is shown in Appendix A, and the elite strategy is as follows:Step 1: Chromosome coding, the decision variables are the running frequencyf1 and f2, and the stop sequence of express trains, the chromosome coding method is 000100︸f1000010︸f2010000︸n0, in which, f1 and f2 is binary coding, n0 indicates the number of stations that express trains need to stop. In the stop sequence, 1 indicates that the station at which the express train stops, while 0 indicates not stop. Randomly initialize the parent population, the population number is 200, and the number of iterations is set to 200;Step 2: The offspringQt and the parent Pt were merged into Rt, and the population size was set to 2N;Step 3: GeneratingFn,n=1 according to the nondominated sorting method;Step 4: If the population number inFn is less than N, go to the next gene stop the iteration, and get the final section passenger flow xa, make n = n + 1, return to Step3, if Fn is more than N, go to step5;Step 5: Calculating the individual crowding degreeid in Fn, the individuals with high crowding degrees are preferentially selected for the next generation until the number of individuals in the next generation reaches N;Step 6: Selection, crossover, and mutation, among which, roulette is adopted for selection, a single-point crossover is adopted, the crossover probability is 0.8 and the mutation probability is 0.04;Step 7: If the number of iterations does not reach 200, return to step 2; when the number of iterations reaches 200, go to step 8;Step 8: We output the optimal value;Step 9: Decoding the optimal value.


## 3.4. Evaluation and Analysis of Optimization Results

The analysis and evaluation of the optimization results of the operation scheme can be obtained from the travel time of passengers and the number of trains needed.(1)
The travel time of passengersThe primary purpose of optimizing the stop scheme of express trains is to match the demand of passengers as much as possible, so as to minimize the transfer of passengers between the express and local trains, to save passengers’ travel time, and promote travel convenience. Meanwhile, consider the overall travel time change of all passengers along the line, so as to achieve the optimization goal of shortening the overall travel time of passengers across the line.(2)
The number of trains neededTrains needed are the number of trains that can be normally put into operation to complete the transportation task, the number of trains per hour is calculated as shown ind.(34)Ny=∑kbkfkθk60.In which,Ny: the number of trains required, train;θk: the turnaround time of the train, min.In whichθk=260⋅∑zSz/Vk+∑i∈i′xi⋅d+tz,k=1,260⋅∑zSz/Vk+n⋅d+tz,k=2.(3)
The calculation of carrying capacityThe express and local mode adopts a nonparallel operation schedule, and the carrying capacity can be calculated by graphic method or analytical method. Although the graphic method has high precision, the calculation process is more complicated, and the method of combining analysis and graphics is usually adopted. As shown in Figure5, assuming that only station 5 has the condition of crossing line (the four stations from station 1 to station 4 do not have the condition of crossing line), when an express train overtakes local stations without stopping, the shortest time required to occupy the interval for laying down an express and local train can be calculated by the following formula:(35)Ien+Iln≥2hmini+ntstation+∆t−tstation.(1)
When the proportion of express and local trains isk : m = 1 : 3.When the running proportion of express and local trains is 1 : 3, each combination of express and local trains can run 4 trains, then the number of passengers that can be transported in each combination cycle.Pcycle = 1296 ∗ 3 + 648 = 4536, based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/4536 = 1.6, cycle time: Tcycle = 3600/1.6 = 2250, the average departure interval between four trains in each combination of express and local trains: Td = 2250/4 = 562.5 s. Td+tre−trl=562.5+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(2)
When the proportion of express and local trains isk : m = 1 : 2.When the proportion of express and local trains is 1 : 2, each combination of express and local trains can run 3 trains, then the number of passengers that can be transported in each operating cycle:Pcycle = 1296 ∗ 2 + 648 = 3240, based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/3240 = 2.25, cycle time: Tcycle = 3600/2.25 = 1650 s, the average departure interval between three trains in each combination of express and local trains is Td = 2250/3 = 533 s. Td+tre−trl=533+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(3)
When the proportion of express and local trains isk : m = 1 : 1.When the proportion of express and local trains is 1 : 1, each combination of express and local trains can run 2 trains, then the number of passengers that can be transported in each operating cycle:Pcycle = 1296 ∗ 1 + 648 = 1944, the total number of express and local train groups required to operate during peak hours, and the total number of express and local train groups is n = = 7290/1944 = 3.75, cycle time: Tcycle = 3600/3.75 = 960 s, the average departure interval between two trains in each combination of express and local trains is Td = 960/2 = 480 s. Td+tre−trl=480+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(4)
When the proportion of express and local trains isk : m = 2 : 1.When the proportion of express and local trains is 2 : 1, each combination of express and local trains can run 3 trains, then the number of passengers that can be transported in each operating cycle:Pcycle = 648 ∗ 2 + 1296 = 2592, based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/2592 = 2.81, cycle time: Tcycle = 3600/2.81 = 1280 s, the average departure interval between three trains in each combination of express and local trains is Td = 1280/3 = 426 s. Td+tre−trl=426+2123−3082<Icatch=90s. Therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.(5)
When the proportion of express and local trains running isk : m = 3 : 1.Figure 5
The operation scheme of express and local mode.When the running proportion of express and local trains is 3 : 1, each combination of express and local trains can run 4 trains, then the number of passengers that can be transported in each combination cycle.Pcycle = 648 ∗ 3 + 1296 = 3240. Based on the maximum section passenger flow of Line 16 at peak hours Pmax, the combined number of express and local train groups required to operate is n = Pmax/Pcycle = 7290/3240 = 2.25, cycle time: Tcycle = 3600/2.25 = 1600 s, the average departure interval between four trains in each combination of express and local trains is Td = 1600/4 = 400 s. Td+tre−trl=400+2123−3082<Icatch=90s, therefore, when the combination of express and local trains is this proportion, overtaking will occur when the train runs on the line.


## 4. Case Study

To test and verify the performance of the proposed model, this section mainly verifies the accuracy and feasibility of the optimization scheme constructed in the third and fourth parts through examples. Figure6 is the flow chart of the whole experimental verification.Figure 6
The flow chart of the case study for the experimental verification.


### 4.1. The Basic Information of Shanghai Metro Line 16

Shanghai Metro Line 16 starts from Longyang Road Station in Pudong New Area in the West and ends at Dishuihu station in the East. The whole line is located in the Pudong New Area of Shanghai. It connects the central area of Pudong in the north and Nanhui new town in the south, connecting Zhoupu Town, Hangtou Town, Xinchang Town, and Huinan Town. The whole line is 59.334 km long, of which the underground line is 6.734 km long and the overhead line is 52.6 km long. There are 13 stations, including 3 underground stations and 10 elevated stations. The train adopts 6-train marshaling of A-type trains.(1)
Shanghai Metro Line 16 was selected as the experimental verification object, passenger flow video and WIFI probe data were collected, 5-minute AFC passenger flow data were obtained, and the large probability OD representation of passenger flow was calculated by the passenger flow OD mining module preprogrammed based on AFC data.(2)
Passenger flow of fine granularity time prediction, and express stop scheme optimization, based on traffic during high precision matching train overtaking operation scale quantitative calculation, calculation scheme based on stop and time train overtaking operation scale line through actual ability, the coordination of three indexes determination of optimal dispatching plan.(3)
According to the changes in station scale, passenger flow level estimation of the metro network was made, and taking passenger travel time as input characteristics, the multilayer self-limiting feedback model for coordinated optimization of stations is studied, which mainly involves controlling the total passenger flow of stations, temporarily adjusting train operation and traffic organization schemes, such as train skip-stop, adjustment of passenger transport organization mode, and multilines transfer.


### 4.2. The Optimization Modeling and Solution of Train Scheme Based on Express-Local Mode

#### 4.2.1. Optimization of the Stop Scheme for Express Trains

Through questionnaire field survey and online survey, the travel situation of passengers in stations along the rail transit is obtained, and the OD travel situation, travel time period, weekly travel times, total travel time, and other parameters are obtained, as shown in Table2.Table 2
Passenger flow time distribution of Shanghai rail transit line 16.
Leaving stationArrive stationTravel periodDepart timeOff timeTimes a weekTime in a weekLingang AvenueLuoshan RoadBefore 6:008:3018:306818.97 minHangtoudongLuoshan Road7:00–7:308:0017:004540.7 minLingang AvenueLuoshan Road6:30–7:008:0016:301223.74 minEast ZhoupuLuoshan Road11:00–14:008:3017:00193.2 minHeshahangchengLuoshan Road7:00–7:309:0017:001163.04 minHuinanLuoshan Road7:00–7:307:0017:002259.08 minEast ZhoupuLuoshan Road8:30–9:009:0018:301138.31 minXinchangLuoshan Road7:00–7:308:3017:301129.89 minEast ZhoupuLuoshan Road8:00–8:309:0018:002198.07 minHuinanLuoshan Road7:00–7:307:3016:302187.69 min………………………………A multiobjective model based on passenger flow, travel time, and enterprise operation cost is established as follows: three A-type trains are grouped, with 216 seats and an upper limit of 250 (15.7% overcapacity), and 7–9 people/m2 is used for overloading. The minimum departure interval is 4 minutes.An optimization model of a double-objective train stopping scheme is established, and it is assumed that Shanghai Metro Line 16 hasn stations and m sections, N=S,E, S is the collection of stations, s=s1,s2…sn; E is the set of running sections, E=e1,e2…e3; Sr is the stop assembly of train R, and a multiobjective optimization model based on the total travel time of passengers and enterprise operation cost is established. The main objectives are as follows: the maximum change in the total travel time of passengers ∆T is based on the total change in the total travel time of passengers as the goal; the minimum objective function of train operation costs, which explains the method of optimizing the travel time of passengers from different dimensions.(1)
The total transit timeZ1 of passengers taking express and local trains is the smallest, as shown in the following formula:(36)minZ1=∑i=1n∑j=1nρijetij+∑k=i+1j−1hk1+hk2xk+d12+ρijltij+∑k=i+1j−1hk1+hk2+d22.(2)
The optimal goalZ1 of train operation cost is the minimum, as shown in the following formula:(37)minZ2=c1θe+θlTinterval+c2×lij×m1+m2+c3×m1×n−2+m2×∑i=2n−1xi.The optimization model of formula (36) and formula (37) is constructed, and the optimization objective function of the stop scheme is established. The 8 constraints in 4.2 should be met.The ant colony algorithm based on pheromone dynamic change is used to search for the optimal solution many times. Set C is set as the set of all stations when initializing information.CG (min): the optimal value of each ant colony search;CA (min): indicates the optimal target value.Ant colony algorithm solution steps are as follows:(1)
Initialization of ant colony and path. M ants were placed in the computing center of the station along Line 16. Initial timet = 0, the sum of initial information and pheromone on all distribution paths is zero.(2)
Set the tabu index of the ant colony as 1; and put the number of the first stop visited by all ants from the initial departure center into its corresponding taboos list;(3)
For each search, the number of cycles increases by 1;(4)
According to the state transition probability, the individual ant of each ant colony selects the number of the next station to arrive;(5)
Modify the index number of the tabu table, move the ants selected to the new number, and insert the number into the tabu table;(6)
If all the numbers in set C have not been traversed, skip to step (4) to continue, otherwise stop the operation;(7)
If an ant has found the destination site, calculate the total path cost (travel time) of the ant, update the minimum cost path currently obtained, record the corresponding value of the minimum cost path at this time, and update the pheromone concentration on the path;(8)
Calculate the amount of information on each path;(9)
If the maximum number of cycles is not reached, clear all taboos and jump (1); otherwise, jump (7);(10)
Update the stop scheme of the found minimum cost (travel time) and write down the minimum cost value. If the difference between CG (min) and CA (min) is less than the preset value, the minimum cost line is obtained and the whole program is terminated; otherwise, switch to (3) and continue the execution.MATLAB was used to solve the ant colony algorithm mentioned above. The main calculation codes are shown in Appendix B, and the schematic diagram of the iterative process is shown in Figure7. Through the MATLAB algorithm, the optimized model is calculated, and the stop scheme of the operation scheme is finally obtained as shown in Figure 8. The detailed stop scheme and dwell time are shown in Table 3.Figure 7
Variation of optimal fitness with the number of express and local trains.Figure 8
Schematic diagram of stop optimization scheme of Shanghai rail transit line 16.Table 3
Optimization scheme of Shanghai Metro Line 16 stop scheme.
Stop schemeStationsLong Yang roadHuaxia Middle roadLuoshan roadEast ZhoupuHesha Hang ChengEast HangtouXin ChangWild Animal ParkHuinanEast HuinanShu YuanLingang AvenueDishui LakeExpress stop scheme●〇●〇〇〇●〇●〇〇●●Dwell time (s)4503000040035003530Local stop scheme●●●●●●●●●●●●●Dwell time (s)40353530303035303530303035As shown in Figure7, when the number of iterations is between 5 and 8, the degree of fit is almost unchanged, close to 0, indicating that this interval cannot be the optimal value. With the gradual increase in the number of iterations, the degree of fit continues to increase, reaching the maximum value at 13–16, and approaching stability. When the number of iterations continues to increase, the degree of fit shows a downward trend. From the whole iterative optimization process, the optimal value can be the optimal value at 13–16.In Table3, the express train stops at Long Yang Road for 45 seconds, Luoshan Road for 30 seconds, Xinchang for 40 seconds, Huinan for 35 seconds, Lingang Avenue for 35 seconds, and Dishui Lake for 30 seconds. Besides, the local train stops and dwell times are listed in Table 3. The optimized train stop plan can save the total travel time of passengers, and the concept of “passengers first” can also be reflected in the actual operation, which is of good practical significance.Figure8 shows the specific form of the optimized express and local train stop scheme, the transfer between express and local trains, as well as the transfer of transfer stations on other lines in the Metro network, which is marked with blue color. Through the express and local train operation mode, passengers can not only take the express train to the destination station quickly but also transfer between the express and local trains, as well as enter other lines in the Metro network through transfer, which can save travel time.


#### 4.2.2. Carrying Capacity of Express Train under Overtaking Condition

It is known that Wild Animal Park Station, Huinan East Station, and Hangtou East Station in Line 16 have overtaking conditions, and overtaking conditions are set in every 1 or 2 line intervals. Therefore, according to the line conditions and passenger flow, the station locations of express and local trains overtaking in the scheme can be determined according to the traffic density among Luoshan Road, Hangtou East Station, Wildlife Park Station, Huinan East Station, or Lingang Avenue.According to the stop scheme, the express train stops at 7 stations. By calculation, the time saved by the express overtaking is about 231 s. In the actual calculation, the minimum running interval H is the current minimum starting interval of Line 16, 8 min = 480 s.(1)
Whenk : m = 1 : 3.The theory of carrying capacity isNmax=n×4=6.4The actual calculation results are as follows: 1.6 columns for the express train and 3.8 columns for the local train, which are usually large integers, namely, 2 columns for the express train and 4 columns for the local train.Since two local trains run in this period, the average value of localT should be taken as follows:(38)Tdl=Tcycle3=22503=750,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−562.5750=0.65=1.Line carrying capacity is calculated.N=3600−se×ttrain/h=3600−1.6×231/480=6.73. The value is a small integer, and the carrying capacity is 6 columns/hour.(2)
Whenk : m = 1 : 2.In order to meet the demand of passenger flow, the line carrying capacity should reach:Nmax=n×3=6.75≈7Express train: 2.25 ≈ 3, local train: 4.5 ≈ 5.Since two local trains run in this period, the average value of local trains can be calculated as follows:(39)Tdl=Tcycle2=16002=800,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−533800=0.645=1.Line carrying capacity is calculated asN=3600−se×ttrain/h=3600−2.25×231/480=6.4(3)
Whenk : m = 1 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×2=3.75×2=7.5≈8Express train: 4 columns, local train: 4 columns.(40)Tdl=Tcycle=960,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−450960=0.593=1.Line carrying capacity can be calculated as(41)N=3600−se×tTrainh=3600−3.75×231480=5.7.(4)
Whenk : m = 2 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×3=2.812×3=8.44≈9Express train: 6 columns, local train: 3 columns.(42)Tdl=Tcycle=1280,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−4261280=0.486=1.Line carrying capacity can be calculated asN=3600−se×ttrain/h=3600−5.62×231/480=4.8. The number of trains carrying through is 5.(5)
Whenk : m = 3 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×4=2.25×4=9Express train: 6.75 columns, local train: 2.25 columns.(43)Tdl=Tcycle=1600,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−4001600=0.406=1.Line carrying capacity can be calculated as follows:(44)N=3600−se×tTrainh=3600−6.75×231480=4.25.Analysis of the calculation results based on the assumption of the passenger flow in peak hours, the carrying capacity, and the number of express and local trains; the parameters shown in Table4 are calculated under the 5 given proportions.Table 4
Line indicators under the proportion of express and local modes.
k : m3 : 12 : 11 : 11 : 21 : 3Pcycle (person times)32402592194432404536N (train times)2.252.8123.752.251.6Tcycle (second)1600128096016002250Tinterval (s)400426480533562.5Nmax98.447.56.756.4h (s)400426480533562.5k (train)6.755.623.752.251.6m (train)2.252.813.754.53.8N (train)4.254.85.76.416.73N/Nmax47.22%56.7%76%94.96%105.16%In Table4, by comparing the number and locations of overtaking stations of express and local trains and the calculation results carrying capacity, it can be found that when the actual carrying capacity of the express train is less than that of local trains, the analysis is as follows:(1)
When only the proportion of express and local trains is determined.(1)
When the number of local trains is more than that of express trains, the line carrying capacity increases as the proportion of the number of local trains in the combination cycle of express and local trains increases, indicating that the driving of express trains will have an adverse effect on the line carrying capacity;(2)
In the case that the number of express trains is more than that of local trains, the less cycle time used in each combination and the more trains included, the greater the line carrying capacity in that cycle.(3)
When the proportion of express and local trains is 1 : 1, the combination number of express and local trains is the most, and the combination cycle time is the least; the more trains used in the cycle, the less the combination of express and local trains. Because the types of express and local trains used in this scheme are different, the line carrying capacity under the 1 : 1 proportion is not the lowest.In order to represent the changing trend of the data in Table4, the data of 3 : 1 is taken as the reference data in Table 5.(2)
The carrying capacity under the different proportions of express and local trains.(1)
As the proportion of express trains decreases, the capacity of the line increases gradually, and it shows that the express train will cause the loss of the line carrying capacity under the express and local modes.(2)
When the number of local trains gradually increases, the line carrying ability after crossing will gradually increase, which is closer to the theoretical carrying ability under the condition of no crossing, which also shows that the line carry ability will decrease from the other hand.(3)
When the running proportion of express and local trains isk : m = 1 : 3, the line carrying capacity under the condition of overcrossing exceeds that under the condition of not overcrossing. This is because the number of local trains on this line is larger than that of express trains, so the transport capacity is also larger than that of express trains. Therefore, when only the running proportion is considered under this proportion, the combined number of express and local trains will decrease. These results show that the carrying capacity Nmax and N/Nmax become larger.After comprehensively considering the theoretical carrying capacity under the operation proportion of express and local trains, the following suggestions are proposed:(1)
The operation proportion of express and local trains on Shanghai rail transit line 16 should be 1 : 3;(2)
The crossing location of express and local trains should be selected from Luoshan Road, Hangtou East Station, Wild Animal Park Station, Huinan East Station, and Lingang Avenue.(3)
By adopting this operation scheme, the distance between express and local trains is large, so there is a large space to adjust the distance between express and local trains in the combination cycle of express and local trains, which is conducive to the regulation of traffic scheduling under operation conditions. In addition, because there is only one station for the express trains to cross over the local trains in the scheme, the choice of crossing point location is large, which is also conducive to adjust the operation of express and local trains at any time.Table 5
Normalization of line index data under the proportion of express and local mode.
k : m3 : 12 : 11 : 11 : 21 : 3Pcycle (person)10.800.601.001.40N (train)11.251.671.000.71Tcycle (s)10.800.601.001.41Nmax (train)10.940.830.750.71h (s)11.071.201.331.41k (train)10.830.560.330.24m (train)11.251.672.001.69N (train)11.131.341.511.58N/Nmax11.201.612.012.23


### 4.3. Analysis of Optimization Results of Train Scheme

According to the initial operation scheme of Line 16, the express train stopped at Huinan station and Xinchang station, and passengers repeatedly complained that the stop setting of the express stations was unreasonable. At the overtaking station, the waiting time of the local train is about more than 5 minutes, which is generally unacceptable to passengers. The operators have been looking for solutions. Through a series of optimization methods studied in this paper, multidimensional decision support can be provided for the organization of express and local trains.In Figure9, after optimization, the key indicators are as follows:(1)
The express stops at Longyang Road, Luoshan Road, Xinchang, Huinan Station, Lingang Avenue, and Dishui Lake.(2)
The interval between the express train is 7 minutes, the interval between the local train is 7 minutes, and the interval between express and local trains is 5 minutes, during the peak hours (evening and morning).(3)
The proportion of express and local trains in the morning peak should be 1 : 1; while during the flat period, it should be: 1 : 3.(4)
To obtain the maximum line carrying capacity, the operation proportion is 1 : 4, but the gap is too large, which is not conducive to traffic organization and management. It is suggested to take into account passenger flow and carrying capacity, and the operation proportion of express and local trains is 1 : 3.(5)
The appropriate number of trains is 6–8. After the optimization of the operation schedule, the train operation map is drawn (the red station is the main stop station, the green station is the crossing condition station, and the black station is the common station).Figure 9
Optimization proposal of Shanghai rail transit line 16 train operation (part). Note: In Table5, the red stations are the express stations, the green stations are the stations with crossing conditions, and the rest are ordinary stations.


## 4.1. The Basic Information of Shanghai Metro Line 16

Shanghai Metro Line 16 starts from Longyang Road Station in Pudong New Area in the West and ends at Dishuihu station in the East. The whole line is located in the Pudong New Area of Shanghai. It connects the central area of Pudong in the north and Nanhui new town in the south, connecting Zhoupu Town, Hangtou Town, Xinchang Town, and Huinan Town. The whole line is 59.334 km long, of which the underground line is 6.734 km long and the overhead line is 52.6 km long. There are 13 stations, including 3 underground stations and 10 elevated stations. The train adopts 6-train marshaling of A-type trains.(1)
Shanghai Metro Line 16 was selected as the experimental verification object, passenger flow video and WIFI probe data were collected, 5-minute AFC passenger flow data were obtained, and the large probability OD representation of passenger flow was calculated by the passenger flow OD mining module preprogrammed based on AFC data.(2)
Passenger flow of fine granularity time prediction, and express stop scheme optimization, based on traffic during high precision matching train overtaking operation scale quantitative calculation, calculation scheme based on stop and time train overtaking operation scale line through actual ability, the coordination of three indexes determination of optimal dispatching plan.(3)
According to the changes in station scale, passenger flow level estimation of the metro network was made, and taking passenger travel time as input characteristics, the multilayer self-limiting feedback model for coordinated optimization of stations is studied, which mainly involves controlling the total passenger flow of stations, temporarily adjusting train operation and traffic organization schemes, such as train skip-stop, adjustment of passenger transport organization mode, and multilines transfer.


## 4.2. The Optimization Modeling and Solution of Train Scheme Based on Express-Local Mode

### 4.2.1. Optimization of the Stop Scheme for Express Trains

Through questionnaire field survey and online survey, the travel situation of passengers in stations along the rail transit is obtained, and the OD travel situation, travel time period, weekly travel times, total travel time, and other parameters are obtained, as shown in Table2.Table 2
Passenger flow time distribution of Shanghai rail transit line 16.
Leaving stationArrive stationTravel periodDepart timeOff timeTimes a weekTime in a weekLingang AvenueLuoshan RoadBefore 6:008:3018:306818.97 minHangtoudongLuoshan Road7:00–7:308:0017:004540.7 minLingang AvenueLuoshan Road6:30–7:008:0016:301223.74 minEast ZhoupuLuoshan Road11:00–14:008:3017:00193.2 minHeshahangchengLuoshan Road7:00–7:309:0017:001163.04 minHuinanLuoshan Road7:00–7:307:0017:002259.08 minEast ZhoupuLuoshan Road8:30–9:009:0018:301138.31 minXinchangLuoshan Road7:00–7:308:3017:301129.89 minEast ZhoupuLuoshan Road8:00–8:309:0018:002198.07 minHuinanLuoshan Road7:00–7:307:3016:302187.69 min………………………………A multiobjective model based on passenger flow, travel time, and enterprise operation cost is established as follows: three A-type trains are grouped, with 216 seats and an upper limit of 250 (15.7% overcapacity), and 7–9 people/m2 is used for overloading. The minimum departure interval is 4 minutes.An optimization model of a double-objective train stopping scheme is established, and it is assumed that Shanghai Metro Line 16 hasn stations and m sections, N=S,E, S is the collection of stations, s=s1,s2…sn; E is the set of running sections, E=e1,e2…e3; Sr is the stop assembly of train R, and a multiobjective optimization model based on the total travel time of passengers and enterprise operation cost is established. The main objectives are as follows: the maximum change in the total travel time of passengers ∆T is based on the total change in the total travel time of passengers as the goal; the minimum objective function of train operation costs, which explains the method of optimizing the travel time of passengers from different dimensions.(1)
The total transit timeZ1 of passengers taking express and local trains is the smallest, as shown in the following formula:(36)minZ1=∑i=1n∑j=1nρijetij+∑k=i+1j−1hk1+hk2xk+d12+ρijltij+∑k=i+1j−1hk1+hk2+d22.(2)
The optimal goalZ1 of train operation cost is the minimum, as shown in the following formula:(37)minZ2=c1θe+θlTinterval+c2×lij×m1+m2+c3×m1×n−2+m2×∑i=2n−1xi.The optimization model of formula (36) and formula (37) is constructed, and the optimization objective function of the stop scheme is established. The 8 constraints in 4.2 should be met.The ant colony algorithm based on pheromone dynamic change is used to search for the optimal solution many times. Set C is set as the set of all stations when initializing information.CG (min): the optimal value of each ant colony search;CA (min): indicates the optimal target value.Ant colony algorithm solution steps are as follows:(1)
Initialization of ant colony and path. M ants were placed in the computing center of the station along Line 16. Initial timet = 0, the sum of initial information and pheromone on all distribution paths is zero.(2)
Set the tabu index of the ant colony as 1; and put the number of the first stop visited by all ants from the initial departure center into its corresponding taboos list;(3)
For each search, the number of cycles increases by 1;(4)
According to the state transition probability, the individual ant of each ant colony selects the number of the next station to arrive;(5)
Modify the index number of the tabu table, move the ants selected to the new number, and insert the number into the tabu table;(6)
If all the numbers in set C have not been traversed, skip to step (4) to continue, otherwise stop the operation;(7)
If an ant has found the destination site, calculate the total path cost (travel time) of the ant, update the minimum cost path currently obtained, record the corresponding value of the minimum cost path at this time, and update the pheromone concentration on the path;(8)
Calculate the amount of information on each path;(9)
If the maximum number of cycles is not reached, clear all taboos and jump (1); otherwise, jump (7);(10)
Update the stop scheme of the found minimum cost (travel time) and write down the minimum cost value. If the difference between CG (min) and CA (min) is less than the preset value, the minimum cost line is obtained and the whole program is terminated; otherwise, switch to (3) and continue the execution.MATLAB was used to solve the ant colony algorithm mentioned above. The main calculation codes are shown in Appendix B, and the schematic diagram of the iterative process is shown in Figure7. Through the MATLAB algorithm, the optimized model is calculated, and the stop scheme of the operation scheme is finally obtained as shown in Figure 8. The detailed stop scheme and dwell time are shown in Table 3.Figure 7
Variation of optimal fitness with the number of express and local trains.Figure 8
Schematic diagram of stop optimization scheme of Shanghai rail transit line 16.Table 3
Optimization scheme of Shanghai Metro Line 16 stop scheme.
Stop schemeStationsLong Yang roadHuaxia Middle roadLuoshan roadEast ZhoupuHesha Hang ChengEast HangtouXin ChangWild Animal ParkHuinanEast HuinanShu YuanLingang AvenueDishui LakeExpress stop scheme●〇●〇〇〇●〇●〇〇●●Dwell time (s)4503000040035003530Local stop scheme●●●●●●●●●●●●●Dwell time (s)40353530303035303530303035As shown in Figure7, when the number of iterations is between 5 and 8, the degree of fit is almost unchanged, close to 0, indicating that this interval cannot be the optimal value. With the gradual increase in the number of iterations, the degree of fit continues to increase, reaching the maximum value at 13–16, and approaching stability. When the number of iterations continues to increase, the degree of fit shows a downward trend. From the whole iterative optimization process, the optimal value can be the optimal value at 13–16.In Table3, the express train stops at Long Yang Road for 45 seconds, Luoshan Road for 30 seconds, Xinchang for 40 seconds, Huinan for 35 seconds, Lingang Avenue for 35 seconds, and Dishui Lake for 30 seconds. Besides, the local train stops and dwell times are listed in Table 3. The optimized train stop plan can save the total travel time of passengers, and the concept of “passengers first” can also be reflected in the actual operation, which is of good practical significance.Figure8 shows the specific form of the optimized express and local train stop scheme, the transfer between express and local trains, as well as the transfer of transfer stations on other lines in the Metro network, which is marked with blue color. Through the express and local train operation mode, passengers can not only take the express train to the destination station quickly but also transfer between the express and local trains, as well as enter other lines in the Metro network through transfer, which can save travel time.


### 4.2.2. Carrying Capacity of Express Train under Overtaking Condition

It is known that Wild Animal Park Station, Huinan East Station, and Hangtou East Station in Line 16 have overtaking conditions, and overtaking conditions are set in every 1 or 2 line intervals. Therefore, according to the line conditions and passenger flow, the station locations of express and local trains overtaking in the scheme can be determined according to the traffic density among Luoshan Road, Hangtou East Station, Wildlife Park Station, Huinan East Station, or Lingang Avenue.According to the stop scheme, the express train stops at 7 stations. By calculation, the time saved by the express overtaking is about 231 s. In the actual calculation, the minimum running interval H is the current minimum starting interval of Line 16, 8 min = 480 s.(1)
Whenk : m = 1 : 3.The theory of carrying capacity isNmax=n×4=6.4The actual calculation results are as follows: 1.6 columns for the express train and 3.8 columns for the local train, which are usually large integers, namely, 2 columns for the express train and 4 columns for the local train.Since two local trains run in this period, the average value of localT should be taken as follows:(38)Tdl=Tcycle3=22503=750,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−562.5750=0.65=1.Line carrying capacity is calculated.N=3600−se×ttrain/h=3600−1.6×231/480=6.73. The value is a small integer, and the carrying capacity is 6 columns/hour.(2)
Whenk : m = 1 : 2.In order to meet the demand of passenger flow, the line carrying capacity should reach:Nmax=n×3=6.75≈7Express train: 2.25 ≈ 3, local train: 4.5 ≈ 5.Since two local trains run in this period, the average value of local trains can be calculated as follows:(39)Tdl=Tcycle2=16002=800,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−533800=0.645=1.Line carrying capacity is calculated asN=3600−se×ttrain/h=3600−2.25×231/480=6.4(3)
Whenk : m = 1 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×2=3.75×2=7.5≈8Express train: 4 columns, local train: 4 columns.(40)Tdl=Tcycle=960,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−450960=0.593=1.Line carrying capacity can be calculated as(41)N=3600−se×tTrainh=3600−3.75×231480=5.7.(4)
Whenk : m = 2 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×3=2.812×3=8.44≈9Express train: 6 columns, local train: 3 columns.(42)Tdl=Tcycle=1280,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−4261280=0.486=1.Line carrying capacity can be calculated asN=3600−se×ttrain/h=3600−5.62×231/480=4.8. The number of trains carrying through is 5.(5)
Whenk : m = 3 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×4=2.25×4=9Express train: 6.75 columns, local train: 2.25 columns.(43)Tdl=Tcycle=1600,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−4001600=0.406=1.Line carrying capacity can be calculated as follows:(44)N=3600−se×tTrainh=3600−6.75×231480=4.25.Analysis of the calculation results based on the assumption of the passenger flow in peak hours, the carrying capacity, and the number of express and local trains; the parameters shown in Table4 are calculated under the 5 given proportions.Table 4
Line indicators under the proportion of express and local modes.
k : m3 : 12 : 11 : 11 : 21 : 3Pcycle (person times)32402592194432404536N (train times)2.252.8123.752.251.6Tcycle (second)1600128096016002250Tinterval (s)400426480533562.5Nmax98.447.56.756.4h (s)400426480533562.5k (train)6.755.623.752.251.6m (train)2.252.813.754.53.8N (train)4.254.85.76.416.73N/Nmax47.22%56.7%76%94.96%105.16%In Table4, by comparing the number and locations of overtaking stations of express and local trains and the calculation results carrying capacity, it can be found that when the actual carrying capacity of the express train is less than that of local trains, the analysis is as follows:(1)
When only the proportion of express and local trains is determined.(1)
When the number of local trains is more than that of express trains, the line carrying capacity increases as the proportion of the number of local trains in the combination cycle of express and local trains increases, indicating that the driving of express trains will have an adverse effect on the line carrying capacity;(2)
In the case that the number of express trains is more than that of local trains, the less cycle time used in each combination and the more trains included, the greater the line carrying capacity in that cycle.(3)
When the proportion of express and local trains is 1 : 1, the combination number of express and local trains is the most, and the combination cycle time is the least; the more trains used in the cycle, the less the combination of express and local trains. Because the types of express and local trains used in this scheme are different, the line carrying capacity under the 1 : 1 proportion is not the lowest.In order to represent the changing trend of the data in Table4, the data of 3 : 1 is taken as the reference data in Table 5.(2)
The carrying capacity under the different proportions of express and local trains.(1)
As the proportion of express trains decreases, the capacity of the line increases gradually, and it shows that the express train will cause the loss of the line carrying capacity under the express and local modes.(2)
When the number of local trains gradually increases, the line carrying ability after crossing will gradually increase, which is closer to the theoretical carrying ability under the condition of no crossing, which also shows that the line carry ability will decrease from the other hand.(3)
When the running proportion of express and local trains isk : m = 1 : 3, the line carrying capacity under the condition of overcrossing exceeds that under the condition of not overcrossing. This is because the number of local trains on this line is larger than that of express trains, so the transport capacity is also larger than that of express trains. Therefore, when only the running proportion is considered under this proportion, the combined number of express and local trains will decrease. These results show that the carrying capacity Nmax and N/Nmax become larger.After comprehensively considering the theoretical carrying capacity under the operation proportion of express and local trains, the following suggestions are proposed:(1)
The operation proportion of express and local trains on Shanghai rail transit line 16 should be 1 : 3;(2)
The crossing location of express and local trains should be selected from Luoshan Road, Hangtou East Station, Wild Animal Park Station, Huinan East Station, and Lingang Avenue.(3)
By adopting this operation scheme, the distance between express and local trains is large, so there is a large space to adjust the distance between express and local trains in the combination cycle of express and local trains, which is conducive to the regulation of traffic scheduling under operation conditions. In addition, because there is only one station for the express trains to cross over the local trains in the scheme, the choice of crossing point location is large, which is also conducive to adjust the operation of express and local trains at any time.Table 5
Normalization of line index data under the proportion of express and local mode.
k : m3 : 12 : 11 : 11 : 21 : 3Pcycle (person)10.800.601.001.40N (train)11.251.671.000.71Tcycle (s)10.800.601.001.41Nmax (train)10.940.830.750.71h (s)11.071.201.331.41k (train)10.830.560.330.24m (train)11.251.672.001.69N (train)11.131.341.511.58N/Nmax11.201.612.012.23


## 4.2.1. Optimization of the Stop Scheme for Express Trains

Through questionnaire field survey and online survey, the travel situation of passengers in stations along the rail transit is obtained, and the OD travel situation, travel time period, weekly travel times, total travel time, and other parameters are obtained, as shown in Table2.Table 2
Passenger flow time distribution of Shanghai rail transit line 16.
Leaving stationArrive stationTravel periodDepart timeOff timeTimes a weekTime in a weekLingang AvenueLuoshan RoadBefore 6:008:3018:306818.97 minHangtoudongLuoshan Road7:00–7:308:0017:004540.7 minLingang AvenueLuoshan Road6:30–7:008:0016:301223.74 minEast ZhoupuLuoshan Road11:00–14:008:3017:00193.2 minHeshahangchengLuoshan Road7:00–7:309:0017:001163.04 minHuinanLuoshan Road7:00–7:307:0017:002259.08 minEast ZhoupuLuoshan Road8:30–9:009:0018:301138.31 minXinchangLuoshan Road7:00–7:308:3017:301129.89 minEast ZhoupuLuoshan Road8:00–8:309:0018:002198.07 minHuinanLuoshan Road7:00–7:307:3016:302187.69 min………………………………A multiobjective model based on passenger flow, travel time, and enterprise operation cost is established as follows: three A-type trains are grouped, with 216 seats and an upper limit of 250 (15.7% overcapacity), and 7–9 people/m2 is used for overloading. The minimum departure interval is 4 minutes.An optimization model of a double-objective train stopping scheme is established, and it is assumed that Shanghai Metro Line 16 hasn stations and m sections, N=S,E, S is the collection of stations, s=s1,s2…sn; E is the set of running sections, E=e1,e2…e3; Sr is the stop assembly of train R, and a multiobjective optimization model based on the total travel time of passengers and enterprise operation cost is established. The main objectives are as follows: the maximum change in the total travel time of passengers ∆T is based on the total change in the total travel time of passengers as the goal; the minimum objective function of train operation costs, which explains the method of optimizing the travel time of passengers from different dimensions.(1)
The total transit timeZ1 of passengers taking express and local trains is the smallest, as shown in the following formula:(36)minZ1=∑i=1n∑j=1nρijetij+∑k=i+1j−1hk1+hk2xk+d12+ρijltij+∑k=i+1j−1hk1+hk2+d22.(2)
The optimal goalZ1 of train operation cost is the minimum, as shown in the following formula:(37)minZ2=c1θe+θlTinterval+c2×lij×m1+m2+c3×m1×n−2+m2×∑i=2n−1xi.The optimization model of formula (36) and formula (37) is constructed, and the optimization objective function of the stop scheme is established. The 8 constraints in 4.2 should be met.The ant colony algorithm based on pheromone dynamic change is used to search for the optimal solution many times. Set C is set as the set of all stations when initializing information.CG (min): the optimal value of each ant colony search;CA (min): indicates the optimal target value.Ant colony algorithm solution steps are as follows:(1)
Initialization of ant colony and path. M ants were placed in the computing center of the station along Line 16. Initial timet = 0, the sum of initial information and pheromone on all distribution paths is zero.(2)
Set the tabu index of the ant colony as 1; and put the number of the first stop visited by all ants from the initial departure center into its corresponding taboos list;(3)
For each search, the number of cycles increases by 1;(4)
According to the state transition probability, the individual ant of each ant colony selects the number of the next station to arrive;(5)
Modify the index number of the tabu table, move the ants selected to the new number, and insert the number into the tabu table;(6)
If all the numbers in set C have not been traversed, skip to step (4) to continue, otherwise stop the operation;(7)
If an ant has found the destination site, calculate the total path cost (travel time) of the ant, update the minimum cost path currently obtained, record the corresponding value of the minimum cost path at this time, and update the pheromone concentration on the path;(8)
Calculate the amount of information on each path;(9)
If the maximum number of cycles is not reached, clear all taboos and jump (1); otherwise, jump (7);(10)
Update the stop scheme of the found minimum cost (travel time) and write down the minimum cost value. If the difference between CG (min) and CA (min) is less than the preset value, the minimum cost line is obtained and the whole program is terminated; otherwise, switch to (3) and continue the execution.MATLAB was used to solve the ant colony algorithm mentioned above. The main calculation codes are shown in Appendix B, and the schematic diagram of the iterative process is shown in Figure7. Through the MATLAB algorithm, the optimized model is calculated, and the stop scheme of the operation scheme is finally obtained as shown in Figure 8. The detailed stop scheme and dwell time are shown in Table 3.Figure 7
Variation of optimal fitness with the number of express and local trains.Figure 8
Schematic diagram of stop optimization scheme of Shanghai rail transit line 16.Table 3
Optimization scheme of Shanghai Metro Line 16 stop scheme.
Stop schemeStationsLong Yang roadHuaxia Middle roadLuoshan roadEast ZhoupuHesha Hang ChengEast HangtouXin ChangWild Animal ParkHuinanEast HuinanShu YuanLingang AvenueDishui LakeExpress stop scheme●〇●〇〇〇●〇●〇〇●●Dwell time (s)4503000040035003530Local stop scheme●●●●●●●●●●●●●Dwell time (s)40353530303035303530303035As shown in Figure7, when the number of iterations is between 5 and 8, the degree of fit is almost unchanged, close to 0, indicating that this interval cannot be the optimal value. With the gradual increase in the number of iterations, the degree of fit continues to increase, reaching the maximum value at 13–16, and approaching stability. When the number of iterations continues to increase, the degree of fit shows a downward trend. From the whole iterative optimization process, the optimal value can be the optimal value at 13–16.In Table3, the express train stops at Long Yang Road for 45 seconds, Luoshan Road for 30 seconds, Xinchang for 40 seconds, Huinan for 35 seconds, Lingang Avenue for 35 seconds, and Dishui Lake for 30 seconds. Besides, the local train stops and dwell times are listed in Table 3. The optimized train stop plan can save the total travel time of passengers, and the concept of “passengers first” can also be reflected in the actual operation, which is of good practical significance.Figure8 shows the specific form of the optimized express and local train stop scheme, the transfer between express and local trains, as well as the transfer of transfer stations on other lines in the Metro network, which is marked with blue color. Through the express and local train operation mode, passengers can not only take the express train to the destination station quickly but also transfer between the express and local trains, as well as enter other lines in the Metro network through transfer, which can save travel time.


## 4.2.2. Carrying Capacity of Express Train under Overtaking Condition

It is known that Wild Animal Park Station, Huinan East Station, and Hangtou East Station in Line 16 have overtaking conditions, and overtaking conditions are set in every 1 or 2 line intervals. Therefore, according to the line conditions and passenger flow, the station locations of express and local trains overtaking in the scheme can be determined according to the traffic density among Luoshan Road, Hangtou East Station, Wildlife Park Station, Huinan East Station, or Lingang Avenue.According to the stop scheme, the express train stops at 7 stations. By calculation, the time saved by the express overtaking is about 231 s. In the actual calculation, the minimum running interval H is the current minimum starting interval of Line 16, 8 min = 480 s.(1)
Whenk : m = 1 : 3.The theory of carrying capacity isNmax=n×4=6.4The actual calculation results are as follows: 1.6 columns for the express train and 3.8 columns for the local train, which are usually large integers, namely, 2 columns for the express train and 4 columns for the local train.Since two local trains run in this period, the average value of localT should be taken as follows:(38)Tdl=Tcycle3=22503=750,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−562.5750=0.65=1.Line carrying capacity is calculated.N=3600−se×ttrain/h=3600−1.6×231/480=6.73. The value is a small integer, and the carrying capacity is 6 columns/hour.(2)
Whenk : m = 1 : 2.In order to meet the demand of passenger flow, the line carrying capacity should reach:Nmax=n×3=6.75≈7Express train: 2.25 ≈ 3, local train: 4.5 ≈ 5.Since two local trains run in this period, the average value of local trains can be calculated as follows:(39)Tdl=Tcycle2=16002=800,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−533800=0.645=1.Line carrying capacity is calculated asN=3600−se×ttrain/h=3600−2.25×231/480=6.4(3)
Whenk : m = 1 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×2=3.75×2=7.5≈8Express train: 4 columns, local train: 4 columns.(40)Tdl=Tcycle=960,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−450960=0.593=1.Line carrying capacity can be calculated as(41)N=3600−se×tTrainh=3600−3.75×231480=5.7.(4)
Whenk : m = 2 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×3=2.812×3=8.44≈9Express train: 6 columns, local train: 3 columns.(42)Tdl=Tcycle=1280,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−4261280=0.486=1.Line carrying capacity can be calculated asN=3600−se×ttrain/h=3600−5.62×231/480=4.8. The number of trains carrying through is 5.(5)
Whenk : m = 3 : 1.In order to meet the demand of passenger flow, the line carrying capacity should reachNmax=n×4=2.25×4=9Express train: 6.75 columns, local train: 2.25 columns.(43)Tdl=Tcycle=1600,soverrun=tl+Icatch−te−TdTdl=3082+90−2123−4001600=0.406=1.Line carrying capacity can be calculated as follows:(44)N=3600−se×tTrainh=3600−6.75×231480=4.25.Analysis of the calculation results based on the assumption of the passenger flow in peak hours, the carrying capacity, and the number of express and local trains; the parameters shown in Table4 are calculated under the 5 given proportions.Table 4
Line indicators under the proportion of express and local modes.
k : m3 : 12 : 11 : 11 : 21 : 3Pcycle (person times)32402592194432404536N (train times)2.252.8123.752.251.6Tcycle (second)1600128096016002250Tinterval (s)400426480533562.5Nmax98.447.56.756.4h (s)400426480533562.5k (train)6.755.623.752.251.6m (train)2.252.813.754.53.8N (train)4.254.85.76.416.73N/Nmax47.22%56.7%76%94.96%105.16%In Table4, by comparing the number and locations of overtaking stations of express and local trains and the calculation results carrying capacity, it can be found that when the actual carrying capacity of the express train is less than that of local trains, the analysis is as follows:(1)
When only the proportion of express and local trains is determined.(1)
When the number of local trains is more than that of express trains, the line carrying capacity increases as the proportion of the number of local trains in the combination cycle of express and local trains increases, indicating that the driving of express trains will have an adverse effect on the line carrying capacity;(2)
In the case that the number of express trains is more than that of local trains, the less cycle time used in each combination and the more trains included, the greater the line carrying capacity in that cycle.(3)
When the proportion of express and local trains is 1 : 1, the combination number of express and local trains is the most, and the combination cycle time is the least; the more trains used in the cycle, the less the combination of express and local trains. Because the types of express and local trains used in this scheme are different, the line carrying capacity under the 1 : 1 proportion is not the lowest.In order to represent the changing trend of the data in Table4, the data of 3 : 1 is taken as the reference data in Table 5.(2)
The carrying capacity under the different proportions of express and local trains.(1)
As the proportion of express trains decreases, the capacity of the line increases gradually, and it shows that the express train will cause the loss of the line carrying capacity under the express and local modes.(2)
When the number of local trains gradually increases, the line carrying ability after crossing will gradually increase, which is closer to the theoretical carrying ability under the condition of no crossing, which also shows that the line carry ability will decrease from the other hand.(3)
When the running proportion of express and local trains isk : m = 1 : 3, the line carrying capacity under the condition of overcrossing exceeds that under the condition of not overcrossing. This is because the number of local trains on this line is larger than that of express trains, so the transport capacity is also larger than that of express trains. Therefore, when only the running proportion is considered under this proportion, the combined number of express and local trains will decrease. These results show that the carrying capacity Nmax and N/Nmax become larger.After comprehensively considering the theoretical carrying capacity under the operation proportion of express and local trains, the following suggestions are proposed:(1)
The operation proportion of express and local trains on Shanghai rail transit line 16 should be 1 : 3;(2)
The crossing location of express and local trains should be selected from Luoshan Road, Hangtou East Station, Wild Animal Park Station, Huinan East Station, and Lingang Avenue.(3)
By adopting this operation scheme, the distance between express and local trains is large, so there is a large space to adjust the distance between express and local trains in the combination cycle of express and local trains, which is conducive to the regulation of traffic scheduling under operation conditions. In addition, because there is only one station for the express trains to cross over the local trains in the scheme, the choice of crossing point location is large, which is also conducive to adjust the operation of express and local trains at any time.Table 5
Normalization of line index data under the proportion of express and local mode.
k : m3 : 12 : 11 : 11 : 21 : 3Pcycle (person)10.800.601.001.40N (train)11.251.671.000.71Tcycle (s)10.800.601.001.41Nmax (train)10.940.830.750.71h (s)11.071.201.331.41k (train)10.830.560.330.24m (train)11.251.672.001.69N (train)11.131.341.511.58N/Nmax11.201.612.012.23


## 4.3. Analysis of Optimization Results of Train Scheme

According to the initial operation scheme of Line 16, the express train stopped at Huinan station and Xinchang station, and passengers repeatedly complained that the stop setting of the express stations was unreasonable. At the overtaking station, the waiting time of the local train is about more than 5 minutes, which is generally unacceptable to passengers. The operators have been looking for solutions. Through a series of optimization methods studied in this paper, multidimensional decision support can be provided for the organization of express and local trains.In Figure9, after optimization, the key indicators are as follows:(1)
The express stops at Longyang Road, Luoshan Road, Xinchang, Huinan Station, Lingang Avenue, and Dishui Lake.(2)
The interval between the express train is 7 minutes, the interval between the local train is 7 minutes, and the interval between express and local trains is 5 minutes, during the peak hours (evening and morning).(3)
The proportion of express and local trains in the morning peak should be 1 : 1; while during the flat period, it should be: 1 : 3.(4)
To obtain the maximum line carrying capacity, the operation proportion is 1 : 4, but the gap is too large, which is not conducive to traffic organization and management. It is suggested to take into account passenger flow and carrying capacity, and the operation proportion of express and local trains is 1 : 3.(5)
The appropriate number of trains is 6–8. After the optimization of the operation schedule, the train operation map is drawn (the red station is the main stop station, the green station is the crossing condition station, and the black station is the common station).Figure 9
Optimization proposal of Shanghai rail transit line 16 train operation (part). Note: In Table5, the red stations are the express stations, the green stations are the stations with crossing conditions, and the rest are ordinary stations.


## 5. Conclusions and Further Studies

With the development of suburban metro lines and the increase of the long-distance travel passenger demand, the research of express and local modes is essential to improve the passengers' service level.This paper presents the collaborative optimization of the unique indicators of metro operation under the new mode, mainly including passenger flow OD prediction, stop scheme of express trains, the proportion of express and local trains, passenger travel time, and optimization of the train carrying capacity. Through the data mining of AFC passenger flow of suburban lines, the distribution of passenger flow OD is predicted, and then the passenger flow travel characteristics of suburban lines are finely described, which provides good support for the decision-making of the stop scheme of express trains. The express train stop scheme is the core decision-making under the express and local modes. A linear programming model based on minimizing the total travel time of passengers and maximizing the operation cost of enterprises is established. Combined with the constraints of the subway operation business, a highly accurate solution algorithm is designed to obtain the express train stop scheme matching the passenger flow demand, so as to achieve the matching of transport capacity and passenger flow. The express and local train modes will cause loss to the line capacity, the calculation model of the train capacity based on the optimization of the train stop scheme and the optimization of the total travel time of passengers is established, and the carrying capacity under different operation proportion of express and local trains is discussed. Finally, the collaborative optimization modeling of train operation organization under express and local train mode is studied, and a set of feasible theories and methods for express and local train operation scheme optimization are formed.As the express and local train operation mode is a complex problem, the theory and optimization method proposed in this paper still have some deficiencies in practical application, such as there is a strong interaction between indicators, and it is necessary to further study the deep-seated interaction among indicators, which will be the main direction of future research.



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*Source: 7686208-2023-03-16.xml*