Research on an Optimization Method for Metro Train Formation Based on Virtual Coupling Technology

Abstract

This study addresses the issues of unbalanced capacity allocation and rigid train formations in urban metro systems under tidal passenger flow conditions. By integrating temporal–spatial passenger demand with real-time dynamic train formation, we propose a virtual formation optimization method driven by carriage load factors. This method enhances the flexibility of train formation strategies by coordinating virtual coupling and decoupling operations between trains traveling in opposite directions. A mixed-integer linear programming (MILP) model is developed, with train unit allocation and turnover scheduling as the main decision variables. The model aims to minimize total passenger waiting time and system operating costs, while incorporating constraints related to unit allocation, turnover, and passenger assignment. The model can be efficiently solved using commercial solvers such as CPLEX. To evaluate the proposed method, a case study is conducted on a metro line in a major city. Numerical experiments demonstrate that, compared with a fixed 6-car formation scheme, the proposed method reduces total passenger waiting time by approximately 4.2% and operating costs by 11.6%. When compared to a fixed 8-car formation scheme, it achieves a 48.8% reduction in operating costs with only a 4.3% increase in passenger waiting time. These results highlight the potential of the proposed virtual formation strategy to enhance operational efficiency and resource utilization in urban metro systems, offering both practical value and implementation feasibility.

1. Introduction

With the rapid development of urban metro systems, how to effectively cope with the increasing passenger flow and complex operation demand has become an important problem to be solved in the field of transportation. Due to their fixed and single nature, traditional train formations cannot flexibly adapt to peak and tidal passenger flows, resulting in a mismatch between the train capacity and the actual passenger flow demand. In recent years, virtual grouping technology, as an innovative train scheduling mode, has received widespread attention. Virtual grouping technology allows train units to be dynamically ungrouped and coupled in real time to achieve more flexible and efficient capacity allocation. This technology effectively alleviates the pressure of passenger flow during peak hours, while optimizing train operating costs and improving the overall economy and resource utilization of the system. Virtual grouping technology was firstly proposed by Bock et al. [1], who proposed the idea of virtual grouping trains, where the connection between trains is not a physical connection, but a short tracking distance is realized through wireless communication. In 2020, CAF successfully implemented a virtual coupling of two trams operating at 20 km/h [2], which took an important step from the theory to the engineering practice. Virtual grouping technology is still in the development stage.

Virtual grouping technology is still in the development stage, and the current research mainly focuses on train operation control and vehicle technology. In terms of train operation control, several advanced control strategies have been used for virtual formation control, including feedback control, sliding mode control, Model Predictive Control (MPC), train following control, optimization and optimal control, and Petri nets, etc. The control architectures can be divided into two categories: centralized and distributed. Yan et al. [3] and Li et al. [4] proposed a distributed cooperative optimization method based on the MPC framework by using an ant colony algorithm and alternating direction multiplier method, respectively. Felez et al. [5] introduced the concept of self-driving vehicle formation control into the virtual formation of trains, and verified that it is better than the mobile occlusion under the MPC framework in terms of operational efficiency and safety. Wu et al. [6] further considered the problems of line speed limitation and collision avoidance on the basis of the MPC framework. Zhang Youbing et al. [7] constructed a virtual formation train control system based on vehicle–vehicle communication, breaking through the restrictions of traditional occlusion, realizing autonomous adjustment of speed and interval of trains, effectively compressing the spacing of trains and enhancing the transportation capacity. Zhang YG et al. [8] establish train tracking simulation model based on cellular automata to explore the impact of different grouping strategies and blockage types on train density and scheduling efficiency. Luo Xiaolin et al. [9], on the other hand, proposed a tracking control method based on distributed robust MPC (RMPC), which realizes train unit spacing compression and system robustness enhancement under interference environment.

In addition to these technical contributions, recent review papers have also provided a comprehensive overview of virtual coupling research. For example, Felez and Vaquero-Serrano [10] summarized the state of the art and identified key challenges for implementing virtual coupling in practice, while Xun et al. [11] surveyed control methods and operational frameworks for railway virtual coupling. These reviews highlight the rapid development of virtual coupling technology but also indicate that most existing work emphasizes control strategies and technical feasibility. This gap motivates the present study to focus on demand-driven optimization of train formation.

In the field of operational organization and management, Yin et al. [12] developed a train rescheduling optimization model that comprehensively considers passenger delays, energy consumption, and regenerative braking characteristics. Passenger flow was simulated using a non-homogeneous Poisson process, and a multi-stage stochastic optimization was achieved via an Approximate Dynamic Programming (ADP) algorithm. Shi et al. [13] proposed a train rescheduling method under metro disturbances, incorporating passenger dynamic behavior and energy consumption, significantly enhancing scheduling flexibility. Zhou et al. [14] focused on optimizing train timetables, unit circulation, and passenger flow control strategies under tidal flow conditions, and introduced a mixed-capacity train formation model to improve system adaptability.

To address the spatiotemporal imbalance of transport capacity, Tai Guoxuan et al. [15] proposed a flexible grouping-based timetable optimization approach, aiming to minimize average waiting time and operating distance, and designed a three-stage heuristic algorithm. Empirical results demonstrated its effectiveness in improving load factors and operational efficiency. Yu Dandan et al. [16] developed a train scheduling scheme based on flexible formation, starting from the perspectives of generalized passenger travel cost and operator cost, and achieved systematic design of formation selection and time-slot aggregation via a four-stage optimization process. You et al. [17], Yang et al. [18], and Han et al. [19] confirmed through empirical studies the effectiveness of flexible capacity adjustment strategies in improving service quality and reducing operational costs. Fu Jiale et al. [20] integrated online flexible formation with short-long route strategies and built a bi-objective optimization model with travel cost and operating mileage. The NSGA-II algorithm was applied to jointly optimize departure frequency, train unit configurations, and terminal settings, striking a balance between service performance and cost efficiency. Addressing the mismatch between tidal passenger flow and rigid formations, Zhou Housheng [21] proposed an MILP model for the flexible use of virtual train units. By coordinating virtual coupling and decoupling between opposite-direction trains, the model achieved dual optimization of passenger waiting time and operational cost. Zhou et al. [22] further proposed a flexible formation strategy enabling coupling/decoupling at both ends of metro lines, jointly optimizing the timetable and rolling stock utilization using a MILP model efficiently solved by a Variable Neighborhood Search algorithm. Song Zhanpeng et al. [23] considered timetable design, rolling stock allocation, and flexible formation in an integrated manner. Through coupling and decoupling operations at turnback stations, dynamic grouping adjustments were implemented to minimize passenger waiting time and operational costs.

In addition to operational optimization, it is also important to consider advancements at the structural level of railway vehicles. Structural optimization of car bodies and load-bearing components not only reduces weight, but also lowers energy consumption, facilitates maintenance, and extends service life, thereby complementing operational strategies aimed at improving efficiency. Recent studies have investigated advanced design approaches and the use of innovative lightweight materials, demonstrating how structural improvements can contribute to both performance and sustainability [24,25,26]. Other works have explored composite materials and multidisciplinary design methods for lightweight carbody development [27,28]. Integrating these two perspectives—operational optimization and structural optimization—broadens the outlook of current research, providing a more comprehensive vision of how rail systems can achieve efficiency, reliability, and sustainability.

Despite the notable progress in train scheduling and formation optimization, most existing studies focus on single-line and static scenarios, with limited exploration of multi-line coordination and response to complex passenger flow patterns. Achieving efficient resource allocation and service quality assurance in dynamic environments remains a considerable challenge. To address this gap, this study proposes a dynamic virtual formation optimization method based on a full load rate threshold, aiming to enhance the real-time adaptability of train formations and provide more precise dynamic scheduling solutions for urban metro systems. Building on this objective, the contributions of this paper are threefold: it introduces a novel load factor-driven optimization framework that explicitly connects passenger demand dynamics with formation strategies; it develops a mixed-integer linear programming (MILP) model that simultaneously coordinates unit allocation, turnover scheduling, and virtual coupling/decoupling operations; and it validates the proposed approach through a real-world case study, demonstrating significant improvements in both passenger waiting time and operating costs compared with conventional fixed-formation schemes. Together, these contributions extend the literature on virtual coupling by shifting the focus from technical feasibility to demand-oriented optimization, thereby providing new insights for sustainable metro operations.

2. Methods

2.1. Description of the Problem

This study focuses on a bidirectional metro line with depots at both terminal stations, where trains can rapidly perform virtual decoupling and coupling procedures at designated turnback stations. During operation, train units are capable of rapidly performing virtual decoupling and coupling procedures. As illustrated in Figure 1, when a downbound train reaches a designated turnback station, it may undergo virtual coupling and decoupling operations. The decoupled units can be virtually coupled to an upbound train, while the remaining units continue to operate toward the downstream terminal station. The number of units involved in the turnback process is determined dynamically based on the real-time full-load rate of carriages.

This operational strategy breaks away from the rigidity of traditional fixed train formations, allowing train units to be regrouped dynamically across opposite directions. It enhances the flexibility of capacity allocation and effectively addresses the spatial-temporal imbalances in passenger demand that are common in urban metro systems. During the optimization process, passenger flow intensities at each station must be comprehensively considered to facilitate the rational allocation of train units. Leveraging the structural characteristics of the rail line, virtual coupling and decoupling operations are implemented at eligible stations, which are selected based on three main criteria: (i) engineering feasibility, meaning sufficient track length and signaling capacity to support safe operations; (ii) operational feasibility, ensuring that coupling and decoupling can be scheduled without disrupting service; and (iii) demand relevance, where the station is located in a section with significant passenger flow imbalances, making such operations beneficial. Consequently, this approach yields train unit deployment strategies that align with the distribution of passenger demand while also offering improved economic efficiency.

To provide a clear overview of the methodological design, Figure 2 presents the overall research framework of this study. As shown, the framework consists of five stages: (1) the input of passenger flow data, timetable, and formation constraints; (2) passenger flow distribution analysis; (3) load factor calculation; (4) MILP model formulation and optimization with solvers such as CPLEX or Gurobi; and (5) the output of optimized train unit allocation and virtual coupling/decoupling strategies. This diagram bridges the problem description and the subsequent mathematical modeling, helping readers better understand the logic and reproducibility of the proposed approach.

2.2. Mathematical Modeling

2.2.1. Model Assumptions

Assumption 1:

The base train schedule (with arrival/departure times at each station) is developed in advance and used as an input parameter to the model.

Assumption 2:

Each train unit is equipped with complete power, brake, communication and control units, supporting independent automatic driving and vehicle–vehicle communication, and meeting the requirements of cooperative operation and safe spacing under virtual grouping.

Assumption 3:

Virtual de-coupling and virtual coupling operations are only permitted at the turnaround station. Train units traveling in the same direction may perform either de-coupling or coupling, but not both. This prevents frequent switching of operation modes in the same direction, which could result in scheduling disorder.

Assumption 4:

Before turning back, any train unit designated for detachment must be cleared of passengers. The cleared passengers will be guided by the passenger information and guidance system to other appropriate train units for boarding.

2.2.2. Parameter Definition

To ensure the standardization and clarity of the subsequent model formulation,

Table 1. Representation and definition of parameter symbols.

Notation	Definitions
$S_d$	The set of stations in direction $d$, $S_d=\{1,2,\dots ,S\}$, $i,i^{\prime },i^{″}$ is the station index, $i,i^{\prime },i^{″}\in S_d$,
$K_d$	Set of all train units in direction $d$, $K_d=\{1,2,\dots ,K\}$, $K$ is the total number of trains in direction $d$, and $k,k^{\prime },k^{″}$ is the train index, $k,k^{\prime },k^{″}\in K$
$T$	The set of discrete time points $T\in \{1,2,\cdot \cdot \cdot ,T\}$, $t$ is the index of the discrete time point, $t\in T$
$D$	Set of train travel directions, $D\in \{-1,1\}$, where −1 and 1 represent the upbound and downbound directions, respectively, $d$ denotes the index of the downbound direction, while $-d$ denotes the index of the upbound direction.
$T_{\min },T_{\max }$	Minimum and Maximum Turnback Time
$S_d^{before}(i)$	The set of all stations before station $i$ in direction $d$, including station $i$
$S_d^{after}(i)$	The set of all stations after station $i$ in direction $d$, excluding station $i$
$F_{k,i,d}$	Departure time of train $k$ from station $i$ in direction $d$
$A_{k,i,d}$	Arrival time of train $k$ at station $i$ in direction $d$
$C_{i,d}$	Whether station $i$ in direction $d$ is available for decoupling and coupling operations; takes the value 1 if available, and 0 otherwise.
$L_{\min },L_{\max }$	The set of minimum/maximum numbers of train units per train
$V_{i,t,d}$	Number of passengers arriving at station $i$ iii in direction $d$ during time interval $(t-1,t]$
$M$	A sufficiently large positive constant (used for linearizing logical constraints) $M={10}^6$
$E$	Energy cost per unit distance for an individual train unit
${\sigma }_{k,k^{\prime },i}$	Whether trains $k$ and $k^{\prime }$ satisfy the operation time requirements for virtual coupling or virtual decoupling at the station; takes the value 1 if satisfied, and 0 otherwise.
${\theta }_{i^{\prime },i^{″},t,d}$	Proportion of passengers arriving at station $i^{\prime }$ in direction $d$ during time interval $(t-1,t]$ whose destination is station $i^{″}$
${\lambda }_{k,i,d}$	Train load factor
$W_{i,d}$	Distance between stations $i$ and $i+1$ in direction $d$ (km)
$Q$	Rated passenger capacity of a train unit (persons per unit)
$O_{i,d}$	Downstream distance from station $i$ in direction $d$

and

Table 2. Representation and Definitions of Model Variables.

Notation	Definitions
$x_{k,i,d}$	Number of train units of train $k$ in direction $d$ upon arrival at station $i$. (integer variable)
$u_{k,d}$	The number of units to be decoupled at the decoupling station for train $k$ in direction $d$ (integer variable)
$v_{k,d}$	The number of units to be coupled for train $k$ at the coupling station in direction $d$. (integer variable)
$N_{k,i,d}$	The number of passengers remaining on board when train $k$ in direction $d$ arrives at station $i$; the variable takes a value of 1 if detachment occurs, and 0 otherwise.(integer variable)
${\eta }_{k,k^{\prime }}^{d,-d}$	Indicates whether the train unit decoupled from train $k$ in direction $d$ is coupled to train $k^{\prime }$ in the opposite direction $-d$. (0–1 variable)
${\alpha }_{k,d}$	Indicates whether the train in direction $d$ is performing virtual coupling or virtual decoupling; 1 for coupling, 0 for decoupling. (0–1 variable)
${\beta }_{k,d}^1$	Indicates whether train $k$ in direction $d$ decouples 2 train units under condition ${\lambda }_{k,i,d}<0.6$; the variable takes a value of 1 if detachment occurs, and 0 otherwise. (0–1 variable)
${\beta }_{k,d}^2$	Indicates whether train $k$ in direction $d$ decouples 1 train unit under condition $0.6\le {\lambda }_{k,i,d}<0.9$; the variable takes a value of 1 if detachment occurs, and 0 otherwise. (0–1 variable)
${\beta }_{k,d}^3$	Indicates whether train $k$ maintains its original formation in direction $d$ without performing ungrouping under condition ${\lambda }_{k,i,d}\ge 0.9$; the variable takes a value of 1 if detachment occurs, and 0 otherwise. (0–1 variable)
$p_{k,i,d,t}$	Indicates the number of passengers who board train $k$ among those arriving at station $i$ in direction $d$ during time interval $(t-1,t]$. (continuous variable)

provide a unified definition of all parameters and variables used throughout the paper.

2.2.3. Objective Function

Passenger waiting time is one of the key indicators for evaluating the service quality of urban metro systems. This study considers two main optimization objectives: passenger service level and system operating cost. Specifically, the service level is assessed using passenger waiting time, while the system operating cost is evaluated based on the energy consumption cost incurred by train units during operation.

For passengers arriving at a station within a given time interval $(t-1,t]$ in direction $d$, their individual waiting time is defined as the difference between the actual boarding time $F_{k,i,d}$ and the arrival time $t$. The total passenger waiting time can be calculated using the following formula:

$$Z_1={\displaystyle \sum _{d\in D}{\displaystyle \sum _{t\in T}{\displaystyle \sum _{k\in K_d}{\displaystyle \sum _{i\in S_f}(F_{k,i,d}-t)·p_{k,i,d,t}}}}}$$

(1)

The enterprise operation cost consists of two components: the first is the energy consumption cost incurred by train units during operation along the mainline section; the second is the energy-saving benefit resulting from the reduced travel distance after the train units successfully perform turnback operations at terminal stations. These two components correspond to the first and second terms of Equation (2) in the model, respectively. In terms of parameter definitions, $E$ represents the unit energy consumption cost per kilometer for a single train unit (calculated as the product of electricity price and average traction energy consumption per vehicle-kilometer), with units of CNY/(unit·km); $W_{i,d}$ denotes the distance between stations $i$ and $i+1$ in direction $d$, measured in kilometers; and $O_{i,d}$ refers to the distance from station $i$ to the terminal station in direction $d$, also in kilometers. The coefficient “2” in the second term is designed to capture the energy-saving effect of eliminating round trips, since the train unit no longer needs to travel to the line’s end and return after completing the turnback operation.

$$Z_2={\displaystyle \sum _{d\in D}{\displaystyle \sum _{i\in S_d}{\displaystyle \sum _{k\in K_d}{W}_{i,d}\cdot E\cdot x_{k,i,d}}}}-{\displaystyle \sum _{k^{\prime }\in K_{-d}}{\displaystyle \sum _{k\in K_d}{\displaystyle \sum _{d\in D}2\cdot E\cdot }{O}_{i,d}\cdot u_{k,d}\cdot {\eta }_{k,k^{\prime }}^{d,-d}}}$$

(2)

2.2.4. Restrictive Conditions

In this paper, three types of constraints are considered: train unit allocation constraints, load factor-based train unit turnover constraint, and passenger allocation constraints.

(1)

Virtual Formation Train Unit Allocation Constraints

Constraint (3) imposes limits on the size of train formations, ensuring that the number of train units per train remains within predefined minimum and maximum bounds throughout operation. The lower bound $L_{\min }$ guarantees that, even after a de-coupling operation, trains operating in their original direction can still meet the service level requirements of the line. Conversely, the upper bound $L_{\max }$ restricts the formation length to within the physical capacity of station platforms, thereby ensuring that trains can safely and reliably stop at any platform at any time.

$$L_{\min }\le x_{k,i,d}\le L_{\max },\forall k\in K_d,i\in S_d,d\in D$$

(3)

In order to simplify operational organization and reduce scheduling complexity, Assumption 3 stipulates that trains in the current operating direction can only select either virtual coupling or virtual uncoupling. When train $k$ on direction $d$ chooses virtual coupling, the number of train units follows a non-decreasing trend, namely, when ${\alpha }_{k,d}=1$, $x_{k,i,d}\le x_{k,i+1,d}$, which is described by Constraint (4), where $M$ is a sufficiently large positive number. Similarly, when train $k$ on direction $d$ chooses virtual uncoupling, the number of train units follows a non-increasing trend, namely, when ${\alpha }_{k,d}=0$, $x_{k,i,d}\ge x_{k,i+1,d}$, as captured by Constraint (5).

$$x_{k,i,d}\le x_{k,i+1,d}+(1-{\alpha }_{k,d})\cdot M,\forall k\in K_d,i,i+1\in S_d,d\in D$$

(4)

$$x_{k,i,d}\ge x_{k,i+1,d}-{\alpha }_{k,d}\cdot M,\forall k\in K_d,i,i+1\in S_d,d\in D$$

(5)

Constraint (6) specifies the conservation of train units. Whether train $k$ in direction $d$ performs a virtual coupling or decoupling operation at station $i$ is determined by the change in the number of train units $(x_{k,i,d}-x_{k,i-1,d})$. If $x_{k,i,d}-x_{k,i-1,d}>0$, it indicates that train k has performed a virtual coupling, and the coupled train units originate from train $k^{\prime }$ in the opposite direction $-d$ which has previously performed a virtual decoupling. Conversely, if $x_{k,i,d}-x_{k,i-1,d}<0$, it indicates that train $k$ has performed a virtual decoupling, with the detached train units assigned to train $k^{\prime }$ in direction −d for subsequent virtual coupling. If $x_{k,i,d}-x_{k,i-1,d}=0$, train $k$ neither couples nor decouples virtually, and the number of train units remains constant. This logic is captured in constraint (6), where ${\sigma }_{k^{\prime },k,i}$ denotes whether train $k$ and train $k^{\prime }$ satisfy the required operation time window for virtual coupling or decoupling at station $i$, taking a value of 1 if the condition is met and 0 otherwise.

$$x_{k,i,d}-x_{k,i-1,d}={\displaystyle \sum _{k^{\prime }\in K_{-f}}{\sigma }_{k^{\prime },k,i}(x_{k^{\prime },S+1-(i-1),-d}-x_{k^{\prime },S+1-i,-d})},\forall k\in K_d,i,i-1\in S_d,d\in D$$

(6)

In the practical operation of urban metro systems, not all stations are capable of supporting virtual coupling and decoupling operations due to engineering or structural constraints. In this paper, the parameter $C_{i,d}$ is defined to indicate whether station $i$ in direction $d$ has the capability for decoupling or coupling, with a value of 1 representing capability and 0 otherwise. Accordingly, constraint (7) specifies that if $C_{i,d}=0$, the station does not support virtual coupling or decoupling, and the number of train units must remain unchanged, i.e., $x_{k,i,d}=x_{k,i-1,d}$. To achieve linearization of constraint (7), the big-M method is introduced, resulting in its reformulation as constraints (8) and (9).

$$x_{k,i,d}=x_{k,i-1,d},if C_{i,d}=0,\forall k\in K_d,d\in D,i\in S_d$$

(7)

$$x_{k,i,d}\le x_{k,i-1,d}+C_{i,d}\cdot M,\forall k\in K_d,d\in D,i\in S_d$$

(8)

$$x_{k,i,d}\ge x_{k,i-1,d}-C_{i,d}\cdot M,\forall k\in K_d,d\in D,i\in S_d$$

(9)

(2)

Load Factor-Based Train Unit Turnover Constraint

To intuitively demonstrate the virtual decoupling and coupling mechanism based on train load factors, Figure 3 illustrates the operational rules at the turnaround station. The number of train units to be decoupled is determined by segmented load factor thresholds: when the load factor is low, more units are decoupled; when it is medium, fewer units are decoupled; and when it is high, no decoupling is performed. This concise rule design ensures that train formations are dynamically adjusted in accordance with passenger density.

Given that the optimization model proposed in this study adopts train formation units as the fundamental capacity units, the Load Factor is introduced as the key metric to evaluate capacity utilization. This indicator is defined as the ratio of the number of passengers who need to continue riding on a train to the total passenger capacity of the corresponding formation. The specific calculation formula is as follows:

$${\lambda }_{k,i,d}={\displaystyle \frac{N_{k,i,d}}{x_{k,i,d}\cdot Q}}$$

(10)

where the variable $N_{k,i,d}$ denotes the number of passengers who still need to continue traveling to the next or subsequent stations when train $k$ arrives at station $i$, and $Q$ represents the capacity of a single train unit. The load factor can thus be interpreted as the ratio of the actual number of onboard passengers to the theoretical carrying capacity of the train, aligning with traditional passenger density indicators in terms of crowd perception and operational response logic. To avoid inconsistencies in the model arising from nonlinear fractional expressions, the associated constraints are linearized in this study.

Where the variable $N_{k,i,d}$ denotes the actual number of passengers onboard Train $k$ as it departs from Station $i$ in Direction $d$. This value is calculated based on the distribution of passenger flow assigned to Train $k$, in proportion to the origin–destination (OD) passenger matrix, and is defined as follows:

$$N_{k,i,d}={\displaystyle \sum _{i^{″}\in S_d^{after}(i)}{\displaystyle \sum _{i^{\prime }\in S_d^{before}(i)}{\displaystyle \sum _{t\le F_{k,i,d},t\in T}{p}_{k,i,d,t}\cdot {\theta }_{i^{\prime },i^{″},t,d}}}},\forall k\in K_d,i\in S_d,d\in D$$

(11)

where $p_{k,i,d,t}$ denotes the number of passengers boarding at station $i$ during time $t$ and assigned to train $k$ in direction $d$, and ${\theta }_{i^{\prime },i^{″},t}$ represents the proportion of these passengers whose destination station during time $t$ is $i^{″}$. The station sets $S_d^{before}(i)$ and $S_d^{after}(i)$ denote the collections of stations located before and after station $i$ in direction $d$, respectively. This formula only accounts for passengers who boarded the train prior to its arrival at station $i$ and who have not yet disembarked, thereby serving as the basis for calculating the subsequent train load factor and triggering the decoupling mechanism.

In order to maintain the integer linear property of the model structure, this paper adopts an integer linear constraint based on segmented load factor intervals. Referring to the subway design specification (GB50157-2013) and relevant domestic subway design standards, the load factor is segmented into a comfortable zone ($\lambda <0.6$), a medium-load zone ($0.6\le \lambda <0.9$), and a crowded zone ($\lambda \ge 0.9$). Specifically, when the train arrives at the ungrouping station, its on-board passenger density determines which ungrouping mode to trigger. The specific definitions and values of the corresponding 0–1 variables ${\beta }_{k,d}^1$, ${\beta }_{k,d}^2$, and ${\beta }_{k,d}^3$ are detailed in

Table 3. Virtual formation uncoupling trigger conditions and adjustment strategies.

Load Factor Interval	Crowding Level	Trigger Condition	Number of Uncoupled Units	Uncoupling Status
${\lambda }_{k,i,d}<0.6$	Comfortable/Spacious	${\beta }_{k,d}^1=1$	2	Uncoupling Two Consist Units
$0.6\le {\lambda }_{k,i,d}<0.9$	Normal/Acceptable	${\beta }_{k,d}^2=1$	1	Uncoupling One Train Unit
${\lambda }_{k,i,d}\ge 0.9$	Crowded/Overloaded	${\beta }_{k,d}^3=1$	0	Maintaining the Original Formation Without Uncoupling

.

To represent the linkage control relationship between the train’s load factor status and its decoupling behavior, this paper introduces a set of piecewise linear inequality constraints to enable the dynamic triggering of train formation modes. When the train’s load factor at the designated turnback station is within the low interval (${\lambda }_{k,i,d}<0.6$), the mode of detaching two train units is activated, corresponding to constraint (12). When the load factor falls within the medium interval ($0.6\le {\lambda }_{k,i,d}<0.9$), the system activates the detachment of one train unit, as described by constraint (13). If the load factor is in the high interval (${\lambda }_{k,i,d}\ge 0.9$), the original consist is maintained without any decoupling, which triggers constraint (14). Constraint (15) is further designed to ensure that only one decoupling mode is triggered at a given time, thereby avoiding operational disruptions.

$$N_{k,i,d}<0.6\cdot x_{k,i,d}\cdot Q+M\cdot (1-{\beta }_{k,d}^1),\forall k\in K_d,i\in S_d,d\in D$$

(12)

$$0.6\cdot x_{k,i,d}\cdot Q-M\cdot (1-{\beta }_{k,d}^2)\le N_{k,i,d}<0.9\cdot x_{k,i,d}\cdot Q+M\cdot (1-{\beta }_{k,d}^2),\forall k\in K_d,i\in S_d,d\in D$$

(13)

$$N_{k,i,d}\ge 0.9\cdot x_{k,i,d}\cdot Q-M\cdot (1-{\beta }_{k,d}^3),\forall k\in K_d,i\in S_d,d\in D$$

(14)

$${\beta }_{k,d}^1+{\beta }_{k,d}^2+{\beta }_{k,d}^3=1$$

(15)

To calculate the number of train units to be decoupled under different formation modes, this paper employs the big-M method to formulate Equations (16)–(18), which, respectively, constrain the feasible number of decoupled train units under low, medium, and high load factor conditions, thereby ensuring consistency with the triggered decoupling modes.

$$2-M\cdot (1-{\beta }_{k,d}^1)\le u_{k,d}\le 2+M\cdot (1-{\beta }_{k,d}^1),\forall k\in K_d,i\in S_d,d\in D$$

(16)

$$1-M\cdot (1-{\beta }_{k,d}^2)\le u_{k,d}\le 1+M\cdot (1-{\beta }_{k,d}^2),\forall k\in K_d,i\in S_d,d\in D$$

(17)

$$0\le u_{k,d}\le M\cdot (1-{\beta }_{k,d}^3),\forall k\in K_d,d\in D$$

(18)

In order to ensure that the train can realize matching connection with the train in the opposite direction after the train is unraveled in the middle of the way, this paper introduces the decision variable ${\eta }_{k,k^{\prime }}^{d,-d}$ to indicate whether the unraveling unit completes the coupling at the same station, and ensures that the number of unraveling unit and coupling unit is balanced by the constraints (19) and (20) to prevent the competition of coupling for multiple trains at the same time and to meet the feasibility requirements. At the same time, constraints (21) and (22) control the minimum and maximum folding time of the train unit connection operation in the folding station, so as to take into account the connection continuity and scheduling feasibility of the coupling operation.

$$u_{k,d}=v_{k,d},if {\eta }_{k,k^{\prime }}^{d,-d}=1$$

(19)

$${\displaystyle \sum _{k^{\prime }\in K_{-d}}{\eta }_{k,k^{\prime }}^{d,-d}\le 1,\forall k\in K_d,d\in D}$$

(20)

$$A_{k^{\prime },S+1-i,-d}-F_{k,i,d}\le T_{\max }+M\cdot (1-{\eta }_{k,k^{\prime }}^{d,-d}),\forall k\in K_d$$

(21)

$$A_{k^{\prime },S+1-i,-d}-F_{k,i,d}\ge T_{\min }-M\cdot (1-{\eta }_{k,k^{\prime }}^{d,-d}),\forall k\in K_d$$

(22)

(3)

Passenger Allocation Constraints

Constraint (26) ensures that the waiting demand of all passengers at a station can be reasonably allocated to trains. To this end, this paper introduces the passenger assignment variable $p_{k,i,d,t}$, which represents the number of passengers boarding at station $i$ during time period $(t-1]$ and assigned to train K in direction $d$.

$${\displaystyle \sum _{k\in K_{d,}{F}_{k,i,d}\ge t}{p}_{k,i,d,t}=V_{i,t,d}},\forall i\in S_d,t\in T,d\in D$$

(23)

Constraint (27) stipulates that the number of passengers carried by a train must not exceed its maximum capacity in order to prevent the risk of overloading. For train $k$, its actual passenger load upon departing station $i$ is equal to the sum of all passengers who boarded the train at stations prior to station $i$ and whose destinations are located beyond station $i$, calculated as follows: ${\displaystyle \sum _{i^{″}\in S_d^{after}(i)}{\displaystyle \sum _{i^{\prime }\in S_d^{before}(i)}{\displaystyle \sum _{t\le D_{k,i,d},t\in T}{p}_{k,i,d,t}\cdot {\theta }_{i^{\prime },i^{″},t,d}}}}$. Here, ${\theta }_{i^{\prime },i^{″},t}$ indicates the proportion of passengers arriving at station $i^{\prime }$ during time period $(t-1,t]$ whose destinations are station $i^{″}$. The capacity of train k is determined by multiplying the number of train units ($x_{k,i,d}$) by the rated capacity per unit ($Q$).

$${\displaystyle \sum _{i^{″}\in S_d^{after}(i)}{\displaystyle \sum _{i^{\prime }\in S_d^{before}(i)}{\displaystyle \sum _{t\le F_{k,i,d},t\in T}{p}_{k,i,d,t}\cdot {\theta }_{i^{\prime },i^{″},t,d}}}}\le x_{k,i,d}\cdot Q,\forall k\in K,i\in S,d\in D$$

(24)

To facilitate model solving, this paper introduces weighting coefficients ${\mu }_1$ and ${\mu }_2$, which combine the dual objectives of total passenger waiting time and system operating cost into a single-objective optimization problem through a linear weighting method. The resulting objective function is expressed as follows.

$$minZ={\mu }_1{Z}_1+{\mu }_2{Z}_2$$

(25)

2.2.5. Model Analysis

In summary, a mixed-integer linear programming (MILP) model is constructed in this paper, which takes into account the constraints of grouping adjustment, coupling matching, turnaround time, and capacity resources in order to coordinate and optimize the operating cost and passenger service level. By introducing weighting coefficients, the model integrates passenger travel time and operating cost into a single-objective optimization problem, which is easy to be solved efficiently by calling optimizers such as CPLEX or GUROBI via MATLAB.

$$\left\{\begin{array}{c}MinZ={\mu }_1{Z}_1+{\mu }_2{Z}_2\\ st.\mathrm{equation} (1)–(6),(7)–(27)\end{array}\right.$$

(26)

3. Results

In this paper, a metro line in China is taken as the research object, and numerical experiments are conducted to verify the effectiveness of the proposed method. All instances are solved using CPLEX through MATLAB 2021A.

3.1. Parameter Settings

To verify the feasibility and effectiveness of the proposed method within a real-world urban metro system, this study selects a representative metro line as the case study and designs three groups of numerical experiments for simulation and analysis. The selected line comprises 10 stations in both upbound and downbound directions. A weekday train timetable is adopted as the scheduling input, and the simulation time horizon is set from 5:00 to 15:00, discretized at 1 min intervals. The afternoon peak period was not included because the studied metro line exhibits more concentrated and representative tidal flows in the morning, making this period sufficient to validate the method. Nevertheless, the proposed optimization framework is general and can be equally applied to evening peak operations or other demand scenarios.

Figure 4 illustrates the three-dimensional distribution of passenger arrivals at stations, based on processed passenger flow data across both time and station indices. As shown in the figure, passenger density peaks significantly around Stations 5–6 during the morning rush hour (7:00–9:00), followed by a sharp decline post-peak—demonstrating clear temporal concentration and directional characteristics. In contrast, the downbound direction exhibits a slightly lower peak, with several stations experiencing moderate passenger volumes during the peak period, resulting in a more evenly distributed and sustained demand profile.

The main model parameters are configured as follows: the train formation size is constrained between 4 and 8 units, with the minimum and maximum allowable formation sizes denoted as $L_{\min }=4$ and $L_{\max }=8$, respectively. Station 6 in the upbound direction (corresponding to Station 5 in the downbound direction) is designated as capable of executing virtual decoupling and coupling operations (denoted as $C_{6,d}=1$). For turnaround operations, the minimum and maximum permissible turnaround times are set to 3 and 10 min, respectively $T_{\min }=3$, $T_{\max }=10$). The rated capacity of a single train unit is set to 310 passengers, and the energy cost incurred per unit per kilometer is assumed to be 1.61 CNY/(unit·km) [29]. The distances between adjacent mainline stations are as follows: [2.03, 1.94, 1.96, 2.07, 2.14, 1.92, 2.19, 2.07, 2.12] kilometers. In the multi-objective optimization model, the weighting coefficients for operating cost and energy savings are set as $\mu =0.6$ and $\mu =0.4$, respectively.

3.2. Analysis of Results

To analyze the allocation of train units as well as the execution of virtual detachment and coupling operations, Figure 5 and Figure 6 illustrate the train unit distribution results for the upbound and downbound directions, respectively. The numbers within the rectangular boxes indicate the number of train units when the train departs from the corresponding station. Gray-shaded boxes denote stations where changes in train unit numbers occur.

As shown in Figure 4 and Figure 5, during off-peak hours, both upbound and downbound trains are generally allocated with fewer units (four train units). However, during peak hours, the number of units allocated to upbound trains increases significantly. Although the passenger demand in the downbound direction remains relatively low, some trains are still assigned more than five train units. This is due to the need for upbound trains to perform virtual detachment at Station 4, with the detached units virtually coupled to upbound trains to enhance overall capacity in the upbound direction. Take, for example, upbound Train 10 and downbound Train 11, highlighted with a blue dashed rectangle in Figure 5. Train 10 detached two units at Station 6, which were then virtually coupled to Train 11, increasing its capacity from 4 to 6 units (see dashed boxes in Figure 5 and Figure 6).

Furthermore, all trains involved in virtual detachment or coupling operations are marked with bold rectangular boxes in Figure 5 and Figure 6. In Figure 5, a total of 7 trains performed virtual coupling, while in Figure 6, 12 train units were involved in coupling operations. This indicates that the units coupled at Station 6 in the upbound direction (Figure 5) originated from detachments at Station 4 in the downbound direction (Figure 6), and vice versa. These results confirm that the train unit conservation constraint is satisfied.

To further validate the effectiveness of the load factor-based train formation adjustment strategy, Figure 7 and Figure 8 present the results of formation adjustments at the turnback station, driven by real-time load factor measurements. Figure 7 corresponds to the upbound direction, while Figure 8 illustrates the downbound direction. In both figures, the blue and orange bars represent the load factors at the decomposing station, and the dashed and solid lines indicate the number of train units before and after adjustment, respectively.

As shown in Figure 7, the overall load factor of upbound trains is relatively low. In particular, trains 10, 11, 16, and 17 exhibit load factors below 0.8, falling within the low-to-medium density operational range. To improve the utilization efficiency of transportation capacity, the system initiates virtual detachment operations for these trains, reducing the number of train units for subsequent trips. For instance, trains 10 and 17, with relatively low load factors, had their formations reduced from 8 to 6 units; similarly, trains 11 and 13 were adjusted from 7 to 6 units. The released train units were then reallocated to downbound trains via virtual coupling to alleviate capacity constraints in the opposite direction.

Figure 8 presents the adjustments for the downbound trains. Compared to their upbound counterparts, downbound trains exhibit generally higher load factors. Some trains (e.g., trains 11 and 13), after receiving additional train units from the upbound direction, increased their formations from 4 or 5 units to 6 or 8 units, thereby better accommodating high passenger demand. In contrast, certain trains (e.g., train 14) maintained their original configurations, indicating that the system is capable of differentiated and targeted capacity deployment based on actual load conditions.

In summary, Figure 7 and Figure 8 demonstrate the flexibility and practicality of the load factor-driven virtual formation strategy. By enabling dynamic adjustments at the train-unit level, this strategy facilitates real-time reallocation and optimized configuration of train units between directions. It effectively mitigates resource waste caused by unbalanced capacity distribution under tidal passenger flow conditions, highlighting the model’s adaptive scheduling capability and operational efficiency.

3.3. Comparison of Operating Results Under Different Train Consist Configurations

In order to more clearly demonstrate the effectiveness of the proposed load rate-driven virtual consist optimization method for urban metro train, this paper calculates the total passenger waiting time and system operating cost under fixed consist configurations (4-car, 6-car, and 8-car consists), and compares them with the results of the proposed virtual consist method. The comparison results are shown in

Table 4. Comparison of results under different consist configurations.

Serial Number	Consist Scheme	Total Waiting Time (s)	System Operating Cost (CNY)	Waiting Time Deviation	Operating Cost Deviation
1	Fixed 4-Car Consist	38,704	13,775	+25.2%	−25.6%
2	Fixed 6-Car Consist	32,225	20,663	+4.2%	+11.6%
3	Fixed 8-Car Consist	29,606	27,551	−4.3%	+48.8%
4	Virtual Consist	30,919	18,511	0	0

.

4. Conclusions and Discussion

4.1. Conclusions

Aiming to address the rigidity of capacity allocation and the inefficient utilization of resources in urban metro systems under tidal passenger flow conditions, this paper proposed a virtual consist optimization method driven by segmented load rates, and developed a corresponding mixed-integer linear programming model to achieve intelligent matching between train formation strategies and passenger flow dynamics.

The case study results verified the effectiveness of the proposed method. Compared with the fixed 6-car consist scheme, average passenger waiting time and system operating costs were reduced by 4.2% and 11.6%, Compared with the fixed 8-car consist scheme, the proposed method reduces operating costs by 48.4% while increasing waiting time by only 4.3%. These outcomes demonstrate that the proposed method can effectively improve train unit utilization, optimize energy use, and reduce the waste of redundant transportation resources, thereby enhancing the sustainable operation of urban metro systems.

Beyond empirical results, the study contributes to the literature by integrating load factor thresholds into a rigorous optimization framework, offering a transparent and flexible decision-making approach that links passenger demand patterns to train formation strategies. With its good applicability, the method shows promising potential for extension to multi-depot, multi-line, and real-time scheduling scenarios, providing a valuable decision-support tool for operators in increasingly complex environments.

4.2. Discussion

This paper proposed a load factor-driven virtual formation optimization method for urban metro systems. The MILP model coordinates unit allocation, turnover scheduling, and coupling/decoupling operations to minimize both waiting time and operating costs.

Numerical experiments verified that the method significantly improves efficiency while maintaining service quality, highlighting its feasibility for practical application. In addition, the study contributes a new perspective on dynamic capacity allocation under tidal passenger flow conditions.

Despite these contributions, some limitations should be acknowledged. First, the case study was conducted on a single metro line, which may not fully capture the interactions and transfer effects in multi-line networks. Second, the model assumes stable and predictable passenger demand, whereas real-world operations often face disturbances such as delays, equipment failures, or sudden passenger surges. Third, the optimization focused mainly on operational cost and service level; other aspects, including energy efficiency, passenger comfort, and system resilience, were not explicitly considered. In addition, the current model calculates load factors at the level of the entire train. In real operations, passengers usually board the nearest cars according to station access points, leading to uneven crowding along the train. For example, even if the average load factor of a train is 0.88, some cars may remain relatively empty, and uncoupling them could increase the effective crowding of the remaining cars above 0.90. This simplification highlights a limitation of our framework, and future research should consider car-level passenger distribution that accounts for the spatial layout of station entrances and exits.

In practice, there are also other commonly used strategies to cope with tidal passenger flows. One approach is physical coupling and decoupling of train units at stations, which can achieve flexible capacity allocation but often requires longer dwell times and specialized infrastructure, limiting its feasibility in high-frequency metro operations. Another widely adopted method is short-turning or reversing trains at intermediate turnback stations, which redistributes capacity but may reduce service coverage for passengers traveling to terminal stations and cause unbalanced rolling stock utilization. Compared with these alternatives, the virtual coupling strategy explored in this paper provides greater operational flexibility and avoids the extended dwell times and infrastructure modifications required for physical coupling, while still maintaining through-service to terminal stations. Nevertheless, we acknowledge that a more systematic comparison with these approaches, such as running numerical experiments with full-length trains and turnback scenarios, would offer deeper insights into the relative benefits of virtual coupling. This constitutes a promising avenue for future research.

Another limitation lies in the structural assumptions of the train units. In this study, we assumed that the train can be fully composed of detachable units. In reality, however, it may be difficult to ensure continuous gangways between all units, which could obstruct passenger movement along the train. In such cases, passengers located in the rear units may face longer walking distances at turnback stations due to uneven distribution and overcrowding of adjacent cars, potentially extending dwell times and reducing the operational benefits of optimization. Addressing this issue would require incorporating dwell time extensions and structural design constraints into the model, which will be considered in future research.

Future research will refine the framework by extending it to multi-line systems, integrating real-time passenger flow forecasting and disruption management, and exploring advanced solution approaches such as metaheuristics or reinforcement learning to improve computational efficiency. Incorporating broader objectives, such as sustainability and passenger comfort, will further enhance the robustness and practical value of the proposed method.