Train Operational Plan Optimization for Urban Rail Transit Lines Considering Circulation Balance

Abstract

The passenger demand of urban rail transit (URT) lines often present asymmetric tidal time-varying characteristics. To match the demand fluctuation, the train operational plan (TOP) generally has asymmetric bi-directional frequency/headway setting and imbalanced circulation, leading to high operation cost. This paper incorporates circulation balance into TOP optimization to balance the bi-directional arrival, departure, circulation, and resource utilization, and reduce the overall operation cost. Based on time-varying section demand and predetermined service level, bi-directional stepped maximum headway functions are collaboratively constructed, and then the circulation process is described by the trip flow circulation network that is formulated as a cost-oriented integer linear programming model. Using the optimized frequency setting, the final TOP is obtained by a two-stage approach to successively solve the schedule and rolling stock circulations at terminals. The case study based on an URT line in Shenzhen indicates that the proposed approach can not only ensure the required service level for travel demand, but also improve the efficiency of circulation and utilization, and effectively reduce the overall operation cost. The proposed approach provides an effective technique to keep balanced, stable and sustainable operation for URT lines.

1. Introduction

Due to the rapid urbanization process of the society, the urban populations and automobile ownership experience a sharp rise. A number of cities are facing the challenge to tackle the traffic problems, such as public transit service, traffic congestion, air pollution, and so on [1,2,3,4]. The application and development of urban rail transit (URT) systems provide a good way to mitigate the above problems efficiently, economically, and safely. In recent years, more and more studies focus on the planning and operation of URT systems.

In general, the main output of URT planning is the train operational plan (TOP) including the train schedule and rolling stock circulation, which is then applied to the daily operation and service [2]. In a TOP, the train schedule is aimed to provide the required service level for passengers, while the rolling stock circulation determines the feasibility of the whole plan [1]. It means that the design of the train schedule should match the demand fluctuation, whereas the rolling stock circulation results in the final operation cost. Since a number of studies have made efforts to the optimization of TOP, next we provide a review of the state of the art.

The train schedule represents the arrival and departure times at stations along the route of a trip, as well as the headways between subsequent services. Passengers take trains according to the schedule, which influences the behaviors in passengers’ journeys, such as arrival, waiting, boarding, transit, and alighting, and the corresponding travel cost and service level. Because travel demand have large variations during a day, the train scheduling should match the demand fluctuation characters well to provide high-quality service for passengers, and improve the utilization of transit capacity [1,2].

Early researchers studied the scheduling optimization with regular headways [5,6,7,8,9], but it cannot fit the dynamics of time-dependent demand well, which means it is needed to design flexible demand-sensitive schedules [4,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. Wu et al. [4] focused on mitigating the unfairness of waiting time of time-dependent individual passengers by adjusting schedules with skip-stopping pattern. An approximate general model was constructed with first-in-first-out principle and capacity constraints, which was solved by exact methods and a variable neighborhood search algorithm for cases with different scales. Considering the time-dependent origin-destination (O-D) demand in an oversaturated URT line, Niu and Zhou [10] established a nonlinear programming model to optimize the schedule with limited train fleet availability, with the objective of minimizing the waiting time of passengers. Barrena et al. [11] studied three formulations of train timetabling under a dynamic demand environment for rapid train services to minimize average passenger waiting time, where the objective function was converted to a linear representation by introducing flow variables and the model was solved by a branch-and-cut algorithm. Barrena et al. [12] further generalized the non-periodic timetabling problem by two formulations that focused on passenger welfare, which was solved by a fast adaptive large neighborhood search. Niu et al. [15] proposed a nonlinear model with linear constraints to adjust timetables with time-dependent demand and given skip-stopping patterns. Hassannayebi et al. [16] dealt with the timetabling problem for a double-track URT system by a nonlinear formulation with the objective of minimizing the average waiting time, where a Lagrangian relaxation-based decomposition approach was adopted to solve it. Zhang et al. [18] focused on the comprehensive timetabling optimization for URT systems with essential realistic conditions, such as train capacity and operation parameters, by two non-convex formulations that were distinguished by the capacity constraint and solved by the adaptive large neighborhood search algorithm. Shang et al. [19] addressed the time-dependent passenger demand-driven timetable synchronization and optimization problem for URT networks by a mixed-integer programming model to minimize passenger transfer and total waiting times. They adopted a binary variable determination method for reducing constraints and solved the model by the genetic algorithm. To fit the traffic in different days, Jamili and Pourseyed-Aghaee [22] proposed a robust approach to optimize the skip-stopping patterns with uncertain boarding and alighting demand by a nonlinear model, then it was converted into a linear one and solved by a decomposition-based algorithm and a simulated annealing algorithm. Wang et al. [27] incorporated skip-stopping pattern into the scheduling optimization problem by a bilevel framework and solved the model in a rolling horizon way. Yang et al. [28] integrated the timetabling and stop planning by a multi-objective mixed integer linear programming model, considering the minimization of the total dwelling time and total delay between the real and expected departure times from origin station.

The generated schedules need to be executed by practical resources [31], such as rolling stocks, crews, and energy, so the output of a demand-sensitive schedule does not mean the feasibility of the final TOP. Thus, rolling stock circulation must be considered to ensure sufficient available resources. Conventional approaches include sequential optimization of rolling stock circulation based on the designed schedule [2,32,33,34], and collaborative optimization of rolling stock circulation with scheduling [1,3,35,36,37,38,39,40]. Zhao et al. [1] studied the integrated planning of train scheduling and rolling stock circulation with skip-stopping pattern, incorporating the fairness in train scheduling and workload balance in rolling stock circulation. A mixed-integer nonlinear programming model was established with time-dependent O-D demand, first-come-first-serve (FCFS) principle, and rigid capacity constraints, which was solved by an iterative searching approach. Shi et al. [2] addressed the TOP optimization by two subproblems, i.e., timetabling by headway setting and rolling stock circulation based on the given timetable, where the timetable can ensure the required service level by adopting the maximum headway function (MHF), while the rolling stock circulation was solved by finding the optimal assignments among trips. Wang et al. [3] combined train scheduling and rolling stock circulation planning under time-varying section demand by a mixed integer nonlinear programming model considering load factor, headway deviations and depot operations, and designed an iterative nonlinear programming approach to solve it. Wang et al. [34] proposed a two-stage method to generate the train schedule and circulation plan for an URT line. A bi-level model and a mixed integer linear model were established to formulate the two problems respectively, which were solved in a sequential way. Wang et al. [36] optimized the train schedule and train circulation plan simultaneously based on a given service pattern from the demand analysis and line planning; two mixed integer nonlinear programming models were established and transformed into linear models for solving. Mo et al. [38] proposed an integrated model of train schedule and rolling stock circulation plan to maximize the utilization of regenerative energy, with explicit consideration of train turn-around constraints, train circulation constraints, and dynamic passenger demands. The original nonlinear model was reformulated as an equivalent linear model and solved by linear programming solvers.

In the mentioned literatures, the scheduling optimization is mainly aimed to fit the fluctuating passenger demand by headway setting or skip-stopping strategies, while the rolling stock circulation focuses on reducing the operation cost as much as possible. The operation cost in URT planning mainly includes the number of used rolling stocks and the number of running trips. Rolling stock circulation is essentially the assignment of trip sequences to available rolling stocks, which is influenced by the distribution of arrival and departure times of trips at terminal stations, as well as connection principles [1]. In daily operation, the passenger demand of URT lines often present asymmetric tidal time-varying characteristics, resulting in the asymmetric bi-directional headway setting and frequency setting, as well as the imbalanced arrival and departure of trips at the two terminals. Due to the imbalanced distribution between the two directions, the departing trips cannot be effectively connected by arriving trips at terminal stations, then more rolling stocks would drive out of the depot to execute the departing trips, leading to higher operation cost. Moreover, the asymmetric bi-directional frequency setting would also cause the imbalanced circulation and utilization of rolling stocks between the two terminal depots. Because the terminal depots have finite parking capacity, it would lead to the storage pressure and resource waste of the starting depot in the direction with sparse demand, which is actually another hidden operation cost. Thus, although demand-sensitive scheduling is beneficial to reducing the number of running trips, the overall operation cost may be still high if the circulation balance is ignored in TOP optimization.

The core of the above problem is the imbalanced distribution of bi-directional departure and arrival caused by independent frequency/headway setting in the two directions. To tackle this issue, we need to make the frequency/headway setting more balanced to add effective connections by involving bi-directional circulation balance. In this way, more departing trips would be connected by ending trips and the resource utilization at the two terminals would be more balanced (or symmetric), which is beneficial to reducing the overall operation cost and improving the bi-directional resource utilization efficiency. It is an effective technique for stability and sustainability of URT operation.

The main contribution of this paper is to incorporate circulation balance into TOP optimization, with explicit consideration of time-varying section demand and predetermined service level, where the bi-directional arrival, departure, circulation, and resource utilization are balanced by specific circulation balance strategies including collaborative division, trip flow circulation network and balance constraint, and then the overall operation cost is reduced. Additional contributions are stated as follows.

• Based on the time-varying section demand and predetermined service level, we collaboratively construct the bi-directional stepped MHFs to connect the two directions by sequences and obtain the space-time zones of trips departing in each divided interval.
• Using the division and stepped MHFs, we construct the trip flow circulation network to describe the relaxed circulation process of bi-directional trips, which is formulated by an integer linear programming (ILP) model to minimize the overall operation cost and obtain the optimized frequencies in each divided interval.
• The TOP is solved by a two-stage approach to sequentially obtain the schedule and rolling stock circulations at terminals based on the optimized frequencies and the corresponding stepped headways.
• The proposed approach is evaluated by an URT line in Shenzhen, China. Combined with the comparison with the practical TOP and the existing approach, the performance of the solved solution is measured in terms of service level, operation cost, as well as the efficiency of circulation and utilization. The impacts of involved circulation balance strategies on TOP optimization are further investigated.

The paper is organized as follows. Section 2 presents the overall TOP problem, briefly introduces the construction of MHF and analyzes the impact of circulation balance. The methodology is presented in Section 3, Section 4 and Section 5: Section 3 proposes the collaborative construction method of bi-directional stepped MHFs, the trip flow circulation is formulated by an ILP model in Section 4 and the solution method is developed in Section 5. Section 6 presents the results and discussion of the case study. Section 7 concludes the paper.

2. Problem Statement

In this paper, we consider a double-track URT corridor (Figure 1) with $N$ stations denoted by $s_1,s_2,\cdots ,s_N$ in down direction (termed by $\mathrm{D}$), and the reverse order is the up direction (termed by $\mathrm{U}$). The two terminal stations $s_A, s_B$ are exactly $s_1,s_N$, of which each is adjacent to a depot $p_A,p_B$ respectively. The parking capacities of the two terminal stations $s_A, s_B$ are denoted by $C^A,C^B$; when the parking capacity of one terminal station is reached, the arriving trips need to drive into the corresponding depot. A departing trip can be executed by a new rolling stock driving from the depot or connected by an ending trip from the reverse direction, while an ending trip can continue to connect a departing trip in the reverse direction or return the depot to finish its daily tasks. The connections should satisfy the turning-around time at terminal stations ${\tau }^A,{\tau }^B$. The operation period (period of starting trips) is denoted by $\left[T_1,T_2\right]$, the total travel time is $\tau \left(s_A,s_B\right)$, and the maximum and minimum headway between any two consecutive trips in the same direction are $h_{\max },h_{\min }$. To facilitate the presentation of our studied problem in this paper, the following assumptions are made.

Assumption 1 (A1).

All trips stop at every intermediate station, and the travel time of each trip is fixed.

Assumption 2 (A2).

Trips run along the line in order, and the overtaking at stations or sections is not allowed.

Assumption 3 (A3).

All trips are provided by the two depots, and return the depots after finishing one day’s operation.

Assumption 4 (A4).

The connections at the two terminals follow the FCFS principle. The turning-around times at terminal stations satisfy ${\tau }^A\le h_{min},{\tau }^B\le h_{min}$because trips can depart only if the connecting ending trips have turned around.

Assumption 5 (A5).

The parking capacity of a terminal depot is finite, though it may have some redundancy during peak hours. Thus, it cannot store too many rolling stocks from the other depot; otherwise, it would increase the storage pressure and hidden organization cost.

Assumption 6 (A6).

All trips have the same fixed capacity and together can meet the whole travel demand. The composition of rolling stocks does not change in the daily operation, so the decomposition and composition of rolling stocks are not included in the rolling stock circulation planning [1,2,3].

Assumption 7 (A7).

The fixed cost of a rolling stock is far larger than the cost of executing trips and connections.

2.1. Maximum Headway Function

In the TOP optimization of an URT system, the schedule is to satisfy the required level of service. To generate a demand-sensitive schedule, Shi et al. [2] proposed the MHF to ensure the occupancy rates in different periods, and try to satisfy the travel and comfort of passengers. Next, we take the down direction as an example to briefly introduce the generation of MHF.

We denote $Q_k^{\mathrm{D}}\left(t\right)$ as the time-varying section demand of the $k^{\mathrm{th}}$ section between $s_k$ and $s_{k+1}$ in the down direction, $k=1,2,\cdots ,N-1,t\in \left[T_1,T_2\right]$. The capacity of a trip is denoted by $C$ and the ideal occupancy rate is denoted by $\alpha$. As for station $s_k$, the departure time of trip $i$ is denoted by $x_i^k$ (for simplicity, the departure time from the starting station is denoted by $x_i=x_i^1$), and the detained demand left behind trip $i$ is denoted by ${\sigma }_i^k$.

• During off-peak hours

During off-peak hours, the ideal occupancy rate can be guaranteed. The headway between trip $i$ and next trip is increased from $h_{\min }$ until the waiting demand at one station is equal to $\alpha C$ or the headway is equal to $h_{\max }$. For any station $s_k, k=1,2,\cdots ,N-1$, if there exists a time $t_i^k\ge x_i^k+h_{\min }$ satisfying

$${{\displaystyle \int }}_{x_i^k}^{t_i^k}{Q}_k^{\mathrm{D}}\left(t\right)+{\sigma }_i^k=\alpha C$$

(1)

then $t_i^k-x_i^k$ is the headway to ensure the service level for section demand at station $s_k$. The headway between trip $i$ and next trip is

$$F^{\mathrm{D}}\left(x_i\right)=\min \left\{h_{\max },\min \left\{t_i^k-x_i^k|k=1,2,\cdots ,N-1\right\}\right\}$$

(2)

and the detained demand at station $s_k, k=1,2,\cdots ,N-1$ is ${\sigma }_{i+1}^k=0$.

2.

During peak hours

During peak hours, although the headway is set as $F^{\mathrm{D}}\left(x_i\right)=h_{\min }$, the occupancy rate still exceeds $\alpha$, so the detained demand at station $s_k, k=1,2,\cdots ,N-1$ is

$${\sigma }_{i+1}^k=\max \left\{0,{{\displaystyle \int }}_{x_i^k}^{x_i^k+h_{\min }}{Q}_k^{\mathrm{D}}\left(t\right)+{\sigma }_i^k-C\right\}$$

(3)

Let denote the departure time of an up-direction trip $j$ from the starting station by $y_j$, then the deduction of $F^{\mathrm{U}}\left(y_j\right)$ is similar. Based on the deduction process, the bi-directional MHFs are constructed through the methods of curve fitting and interpolation [2]. Using MHFs, the generated schedule can not only guarantee the required service level without considering travel demand, but also benefit to reducing operation cost.

2.2. Circulation Imbalance in TOP Optimization

The rolling stock circulation determines the feasibility of the whole TOP and influences the operation cost. As stated in Section 1, rolling stock circulation and the overall operation cost are influenced by circulation balance. In a TOP, circulation balance involves two aspects: distribution balance between arriving and departing trips at terminals, and resource utilization balance between depots. Balanced distribution of arriving and departing trips can add effective connections at terminals and reduce the used rolling stocks, whereas the resource utilization balance is beneficial to mitigating the storage pressure of depots, reducing organization complexity, extending the lifespan of rolling stocks, and improving the operation sustainability. Next, we present a scenario to illustrate the impact of circulation imbalance to TOP optimization.

Assume that the operation period is from 7:00 to 8:00, and the total travel time of a trip is 30 min. In the original TOP (shown in Figure 2a), there are 7 down-direction trips with a headway of 10 min and 21 up-direction trips with a headway of 3 min. In the figure, the rolling stock circulation is illustrated by broken lines and triangles at the two terminals: the broken lines represent the connections at terminal stations, the hollow triangles represent the connections at the corresponding depot, and the solid triangles represent the rolling stocks driving out of the depot for the first time or into the depot for the last time. Due to the difference of bi-directional headway setting, the distribution of arriving and departing trips is imbalanced. There are only 3 effective connections at each terminal, then the used rolling stock numbers from $p_A,p_B$ are respectively 4 and 18. The total number of used rolling stocks is 22, and the difference of used rolling stocks between the two depots is 14. In this way, the final storage number of rolling stocks of $p_A$ would add 14, while that of $p_B$ reduces 14, which is not beneficial to resource utilization and storage.

In contrast, if we add the frequency in down direction, as shown in Figure 2b where there are 10 down-direction trips with a headway of 3 min and 5 down-direction trips with a headway of 6 min, then the effective connections at the two terminals would increase, i.e., 5 connections at $s_A$ or $p_A$, and 10 connections at $s_B$. The used rolling stock numbers from $p_A,p_B$ are respectively 10 and 11, the total number of used rolling stocks is reduced to 21, and the difference of used rolling stocks between the two depots is reduced to 1. The final storage number of rolling stocks of $p_A$ would add 6, while that of $p_B$ reduces 6, then the storage pressure is mitigated and the resource utilization of the two depots is balanced. The comparisons of the mentioned indexes between the two TOPs are shown in

Table 1. Index comparisons between the two TOPs.

TOP	TTN	NRS	DRS	CDS
TOP 1	28	22	14	$\pm 14$
TOP 2	36	21	1	$\pm 6$

TTN: total trip number; NRS: number of used rolling stocks; DRS: difference of used rolling stocks between the two depots; CDS: change of depot storage.

.

It can be seen from Table 1 that although the TTN increases in TOP 2, the NRS, DRS and CDS are all reduced because the frequency/headway adjustment in down direction makes the distributions of arriving and departing trips and the resource utilization at the two terminals more balanced, and the effective connections are added. According to (A5) and (A7), the cost of a rolling stock is much higher than that of running trips, so the overall operation cost of TOP 2 is lower than that of TOP 1.

It indicates that circulation imbalance results from the imbalanced frequency/headway setting, and incorporating circulation balance into TOP optimization is beneficial to reducing the overall operation cost. Thus, we should not only design demand-sensitive schedules to reduce running trips, but also consider the circulation balance by properly collaborating the bi-directional frequency/headway setting to add effective connections, mitigate the resource utilization imbalance and reduce the overall operation cost.

In this paper, we first construct the bi-directional MHFs based on time-varying section demand and service level to collaboratively divide the space-time zone and obtain the bi-directional stepped MHFs; based on the collaborative division, we formulate the circulation process by a network model and obtain the optimized frequency/headway setting; according to the optimized results, we can generate the schedule and calculate the rolling stock circulations at terminals by connection principle. In this way, we can obtain the TOP. The thought of the proposed approach is illustrated in Figure 3.

3. Collaborative Construction of Bi-Directional Stepped Maximum Headway Function

In practical operation, operators generally divide the operation period into several intervals according to demand fluctuation, and the headways during a certain interval should be stable for organization convenience. As mentioned in Section 2, MHFs can ensure the required service level for travel demand, so we can construct stepped MHFs based on MHFs, which satisfy ${\overline{F}}^{\mathrm{D}}\left(x\right)\le F^{\mathrm{D}}\left(x\right), {\overline{F}}^{\mathrm{U}}\left(y\right)\le F^{\mathrm{U}}\left(y\right),x,y\in \left[T_1,T_2\right]$. In this way, the headway setting is no longer than that determined by MHFs, then the service level would still be guaranteed.

The whole operation period and the URT line form the space-time zone of TOP. Based on the stepped MHFs, the space-time zone of TOP would be divided into several zones by the divided intervals of the stepped MHFs. Each divided space-time zone is the running range of the trips departing from the specific interval and direction, and the bi-directional space-time zones are overlapped. To add effective connections at terminals, we collaboratively divide the bi-directional departing periods and space-time zones to obtain a more balanced space-time zone division. It is actually the combination of different connection sequences between the two directions. Assume that there are $m$ connection sequences from the two directions and each sequence has $p_u,u=1,2,\cdots ,m$ intervals, i.e., $\left\{{\mathsf{\Omega }}_1^u,{\mathsf{\Omega }}_2^u,\cdots ,{\mathsf{\Omega }}_{p_u}^u\right\}$. The time difference between each two neighbor intervals is $\tau \left(s_A,s_B\right)$. In either direction, the period after $T_1+\tau \left(s_A,s_B\right)$ would be divided by previous connection sequences, so the first interval of each connection sequence is within $\left[T_1,T_1+\tau \left(s_A,s_B\right)\right]$.

Figure 4 shows a connection sequence from down direction. In this connection sequence, the odd-numbered intervals are the divided departure intervals from down direction, while even-numbered intervals are those from up direction.

Let denote the length of ${\mathsf{\Omega }}_v^u=\left[t_{v,l}^u,t_{v,r}^u\right],v=1,2,\cdots ,p_u,u=1,2,\cdots ,m$ by $L_v^u$. If ${\mathsf{\Omega }}_v^u$ is an odd-numbered interval, the stepped MHF is

$${\overline{F}}^{\mathrm{D}}\left(x\right)=\min \left\{F^{\mathrm{D}}\left(x\right)|x\in {\mathsf{\Omega }}_v^u\right\}$$

(4)

and the error between the stepped MHF and MHF is

$${\epsilon }_v^u=\frac{1}{L_v^u}{{\displaystyle \int }}_{t_{v,l}^u}^{t_{v,r}^u}\left(F^{\mathrm{D}}\left(x\right)-{\overline{F}}^{\mathrm{D}}\left(x\right)\right)\mathrm{d}x$$

(5)

otherwise, the stepped MHF is

$${\overline{F}}^{\mathrm{U}}\left(y\right)=\min \left\{F^{\mathrm{U}}\left(y\right)|y\in {\mathsf{\Omega }}_v^u\right\}$$

(6)

and the error between the stepped MHF and MHF is

$${\epsilon }_v^u=\frac{1}{L_v^u}{{\displaystyle \int }}_{t_{v,l}^u}^{t_{v,r}^u}\left(F^{\mathrm{U}}\left(y\right)-{\overline{F}}^{\mathrm{U}}\left(y\right)\right)\mathrm{d}y$$

(7)

The error of the connection sequence is

$${\epsilon }^u=\max \left\{{\epsilon }_v^u|v=1,2,\cdots ,p_u\right\}, u=1,2,\cdots ,m$$

(8)

To trade off accuracy and complexity, we set the maximum connection sequence error ${\epsilon }_{\max }$ to determine proper division granularity for each connection sequence. The division process of connection sequences from down direction is shown in Algorithm 1. Specifically, when $u=0$, $t_{1,l}^u=$ $t_{1,r}^u=T_1$.

Algorithm 1: Division of connection sequences from down direction

Input Bi-directional MHFs $F^{\mathrm{D}}\left(x\right),F^{\mathrm{U}}\left(y\right)$, the operation period $\left[T_1,T_2\right]$ and the error threshold ${\epsilon }_{\max }$

Output ${\epsilon }^u,\left\{{\mathsf{\Omega }}_1^u,{\mathsf{\Omega }}_2^u,\cdots ,{\mathsf{\Omega }}_{p_u}^u\right\},u=1,2,\cdots ,m$ and the corresponding stepped MHFs

Begin

$u=0$;

while $t_{1,r}^u<T_1+\tau \left(s_A,s_B\right)$

$u=u+1,v=1$;

$t_{v,l}^u=t_{v,r}^{u-1},t_{v,r}^u=T_1+\tau \left(s_A,s_B\right)$;

while $t_{v,l}^u+\tau \left(s_A,s_B\right)\le T_2$ do

$v=v+1$;

$t_{v,l}^u=t_{1,l}^u+\left(v-1\right)\tau \left(s_A,s_B\right),t_{v,r}^u=t_{1,r}^u+\left(v-1\right)\tau \left(s_A,s_B\right)$;

end while

$p_u=v$;

Obtain ${\epsilon }^u$ by Equations (4)–(8);

while ${\epsilon }^u>{\epsilon }_{\max }$ do

$v=1$;

$t_{v,r}^u=\left(t_{v,l}^u+t_{v,r}^u\right)/2$;

while $t_{v,l}^u+\tau \left(s_A,s_B\right)\le T_2$ do

$v=v+1$;

$t_{v,l}^u=t_{1,l}^u+\left(v-1\right)\tau \left(s_A,s_B\right),t_{v,r}^u=t_{1,r}^u+\left(v-1\right)\tau \left(s_A,s_B\right)$;

end while

$p_u=v$;

Obtain ${\epsilon }^u$ by Equations (4)–(8);

end while

end while

End

The division of connection sequences from up direction is similar to the above process. Using the division process, we can obtain the divided intervals of both directions, as well as the corresponding space-time connection zones and stepped MHFs.

4. Network Flow Model Formulation for Circulation Process

In this section, we construct the trip flow circulation network to describe the circulation process of bi-directional trips; based on the network, we formulate the TOP optimization by a capacitated minimum cost flow model.

4.1. Construction of Trip Flow Circulation Network

Based on the connection sequences, we can obtain the space-time zones of trip flow circulation, where the headway of each zone is within a given range. We can regard the divided bi-directional intervals as space-time nodes, the transmissions of trips between the terminals as space-time arcs. Combined with the waiting, connection and in-/out-depot operations, we can form the trip flow circulation network, as shown in Figure 5.

The trip flow circulation network in Figure 5 is a directed graph, denoted by $G=\left(V,A\right)$ where $V$ includes space-time nodes and source nodes, and $A$ includes out-depot arcs, transmission arcs, waiting arcs and in-depot arcs. The corresponding definitions are as follows.

• Space-time nodes represent the divided bi-directional intervals with specific time lengths. Down-direction space-time nodes are denoted by $d_i,i=1,2,\cdots ,M_1$ and up-direction space-time nodes are denoted by $u_j,j=1,2,\cdots ,M_2$, where $M_1,M_2$ are respectively the numbers of divided intervals from the two directions.
• Source nodes represent the two terminal depots $p_A,p_B$. Because trips all depart from depots and would return depots after one day’s operation, the source nodes are also sink nodes in this network.
• Out-depot arcs represent the process that new rolling stocks drive out of depots to execute trips, denoted by $\left(p_A,d_i\right),i=1,2,\cdots ,M_1$ and $\left(p_B,u_j\right),j=1,2,\cdots ,M_2$ without capacity constraint or cost.
• Transmission arcs represent bi-directional trips’ running process between the two terminals, i.e., the connection sequences. The bi-directional transmission arcs and sets are denoted by $\left(d_i,u_j\right)\in A_1$ and $\left(u_j,d_i\right)\in A_2$, which are determined by the specific incidence relationship in connection sequences. Since each divided interval has a specific stepped MHF, the flow lower bound of a transmission arc $L\left(d_i,u_j\right)$ or $L\left(u_j,d_i\right)$ is the frequency with the corresponding stepped MHF, while the flow upper bound of a transmission arc $U\left(d_i,u_j\right)$ or $U\left(u_j,d_i\right)$ is the frequency with the minimum headway $h_{\min }$. The cost factor of a transmission arc is $c_1$, indicating the cost of executing a running trip.
• Waiting arcs, denoted by $\left(d_i,d_{i+1}\right),i=1,2,\cdots ,M_1-1$ or $\left(u_j,u_{j+1}\right),j=1,2,\cdots ,M_2-1$, connect each two neighbor space-time nodes in the same direction and involve the waiting and connection process of trips at terminal stations and depots. In fact, waiting arcs only mean a relaxed circulation process because they cannot indicate specific connections between bi-directional trips or the corresponding connection locations. Thus, waiting arcs also have no capacity constraint or cost.
• In-depot arcs represent the process that trips return the depots after one day’s operation, denoted by $\left(d_{M_1},p_A\right)$ and $\left(u_{M_2},p_B\right)$. The number of used rolling stocks from either depot cannot exceed the storage capacity of a depot $U_{depot}$. The cost factor of an in-depot arc is $c_2$, indicating the operation cost of one used rolling stock.

The trip flow circulation network describes a relaxed circulation process of bi-directional trips, including out-/in-depot operation, running process, and relaxed waiting and connection process. Using this network, we can collaboratively optimize the frequencies and headways of bi-directional divided intervals to add effective connections and balance the resource utilization of the two depots.

4.2. Network Flow Model Formulation

The trip flow circulation network is a cyclic flow network, so the inflow of each node should be equal to its outflow. Since the transmission arcs and in-depot arcs all have capacity constraints and cost factors, the illustrated relaxed TOP optimization can be regarded as a capacitated minimum cost flow problem. The optimization target is to minimize the overall operation cost, including the operation cost of running trips and used rolling stocks, as well as the hidden cost of resource utilization imbalance between the two terminals. The problem can be formulated by an ILP model as follows.

$$\min Z=c_1\left({{\displaystyle \sum }}_{\left(d_i,u_j\right)\in A_1}f\left(d_i,u_j\right)+{{\displaystyle \sum }}_{\left(u_j,d_i\right)\in A_2}f\left(u_j,d_i\right)\right)+c_2\left(f\left(d_{M_1},p_A\right)+f\left(u_{M_2},p_B\right)\right)+c_3\left|f\left(d_{M_1},p_A\right)-f\left(u_{M_2},p_B\right)\right|$$

(9)

$${{\displaystyle \sum }}_{\left(p,q\right)\in A}f\left(p,q\right)-{{\displaystyle \sum }}_{\left(q,p\right)\in A}f\left(q,p\right)=0, p\in V$$

(10)

$$f\left(d_i,u_j\right)\ge L\left(d_i,u_j\right), \left(d_i,u_j\right)\in A_1$$

(11)

$$f\left(u_j,d_i\right)\ge L\left(u_j,d_i\right), \left(u_j,d_i\right)\in A_2$$

(12)

$$f\left(d_i,u_j\right)\le U\left(d_i,u_j\right), \left(d_i,u_j\right)\in A_1$$

(13)

$$f\left(u_j,d_i\right)\le U\left(u_j,d_i\right), \left(u_j,d_i\right)\in A_2$$

(14)

$$f\left(d_{M_1},p_A\right)\le U_{depot}$$

(15)

$$f\left(u_{M_2},p_B\right)\le U_{depot}$$

(16)

$$-\theta U_{depot}\le f\left(d_{M_1},p_A\right)-f\left(u_{M_2},p_B\right)\le \theta U_{depot}$$

(17)

$$f\left(p,q\right)\in \mathbb{N}, \left(p,q\right)\in A$$

(18)

In this model, Equation (9) is the objective function to represent the overall operation cost: the first item is the operation cost of running trips, the second item is the operation cost of used rolling stocks, and the third item is the hidden operation cost caused by resource utilization imbalance between the two terminals that includes the cost of storage pressure, organization complexity, etc. Equation (10) is the flow conservation constraint for each node in the network; constraints (11)–(14) are the capacity constraints for transmission arcs; constraints (15) and (16) are the capacity constraints for in-depot arcs; constraint (17) is the balance constraint to control the imbalance degree of resource utilization between the two terminals by the redundancy ratio $\theta$; constraint (18) is the non-negative integer constraint for arc flows.

5. Solution Method

Since TOP consists of the schedule and rolling stock circulations at the two terminals, we adopt the conventional thought of “two-stage” to optimize the two sub-processes successively [2,32,33,34]. The two-stage approach first generates the schedule based on the obtained frequency setting, and then solves the rolling stock circulations at the two terminals based on the schedule.

5.1. Schedule Generation

By solving the ILP model, we can obtain the frequency setting of each divided interval from the two directions, then we can solve the departure times of bi-directional trips from the terminals with a uniform headway distribution in each divided interval. In this way, we can obtain the down-direction departure times denoted by $X=\left\{x_i|i\in I\right\}$ and up-direction departure times denoted by $Y=\left\{y_j|j\in J\right\}$. Using the travel time, the final schedule can be further deducted.

5.2. Calculation of Rolling Stock Circulation

Using the given schedule, we can deduct the rolling stock circulations at the two terminals based on the approach in [1,2]. Next, we take the terminal station $s_A$ as an example to show the solving of rolling stock circulation. The solving of rolling stock circulation at $s_B$ is similar.

At the terminal station $s_A$, we need to deduct the connections between the ending up-direction trips and the starting down-direction trips, which is an assignment of $J$ to $I$, i.e., $M\subset J\times I$. If the connection $\left(j,i\right)\in M$, then it satisfies two conditions [2]:

(1)

if $\left(j^{\prime },i^{\prime }\right),\left(j^{″},i^{″}\right)\in M,j^{\prime },j^{″}\in J,i^{\prime },i^{″}\in I$, then $j^{\prime }\ne j^{″}$,$i^{\prime }\ne i^{″}$;

(2)

$y_j+\tau \left(s_A,s_B\right)+{\tau }^A\le x_i$.

Furthermore, since $s_A$ has finite parking capacity $C^A$, when the parking capacity is reached, the arriving ending trip has to enter the depot to execute the subsequent starting trip.

To reduce the number of used rolling stocks, we should obtain as many connections as possible. It means the assignment should be a maximum cardinality assignment based on the schedule. The connections are solved by ordering, reversing and exchange operations as shown in Algorithm 2. For details, please refer to [1,2].

Algorithm 2: Solving procedure of rolling stock circulation at $s_A$

Input Down-direction departure time $x_i,i\in I$, up-direction arrival time $y_j+\tau \left(s_A,s_B\right),j\in J$

Output Rolling stock circulation at $s_A$

Step 1  Ordering operation. As for up-direction trip $j=1,2,\cdots ,\left|J\right|$, find the earliest feasible down-direction trip $i$ that satisfy the above two conditions according to FCFS principle, and obtain the ordering assignment.

Step 2  Reverse ordering operation. As for down-direction trip $i=\left|I\right|,\left|I\right|-1,\cdots ,1$, find the latest feasible up-direction trip $j$ that satisfy the above two conditions according to FCFS principle, and obtain the reverse ordering assignment.

Step 3  Connection at the terminal station. As for the starting trips in the ordering assignment, select the earliest feasible ending trip in the reverse ordering assignment to generate connections at the terminal station following the FCFS principle, combined with the parking capacity constraint.

Step 4  Connection at the depot. As for the trips that cannot make connections at the terminal station, execute the ordering assignment to generate the connections at the depot. The remaining trips would be executed by the rolling stocks driving from the depot or return the depot.

6. Case Study

In this section, we apply the proposed approach to the case based on an URT line in Shenzhen, China, to evaluate the proposed approach and measure the performance of the solved results.

6.1. Input Data Setting

The URT line has 30 stations, of which the total travel time $\tau \left(s_A,s_B\right)$ is 70 min. The passenger arriving period is 5:00~24:00. Figure 6 illustrates the space-time distributions of bi-directional section demand. It can be seen from Figure 6 that bi-directional travel demand present obvious tidal phenomenon in a day. The up direction is the commuting direction of most commuters in the morning, a huge number of passengers gather during 7:00~11:00 and the morning peak occurs from 8:00 to 10:00; after the flat peak, passengers begin their returning travel in down direction and the evening peak occurs from 17:00 to 20:00.

To satisfy the travel demand, the operation period (starting trips) is from 5:00 to 24:00, then the whole period of one day’s operation is 5:00~2:00 (next day); the minimum and maximum headway are $h_{\min }=2.5 \min ,h_{\max }=15 \min$. The nominal capacity of a train is $C=1200$ persons per train, the ideal occupancy rate is $\alpha =0.75$. As for a terminal station, the turning-around time is ${\tau }^A={\tau }^B=2 \min$, and the parking capacity is $C^A=C^B=2$ trains. The storage capacity of a depot $U_{depot}=40$ trains, the redundancy ratio $\theta =0.25$, and the maximum connection sequence error ${\epsilon }_{\max }=3.5 \min$. According to the proportional relation in practical operation, the three cost factors are set as: $c_1=1,c_2=1000,c_3=1$.

6.2. Performance Discussion

According to the collaborative construction approach in Section 3, the whole space-time zone is divided by 3 connection sequences: 2 connection sequences from down direction with 17.5 min and 52.5 min respectively and 1 connection sequence from up direction with 70 min. The operation periods from the two directions are both divided into 27 intervals. Based on the trip flow circulation network, we establish the ILP model according to (9)–(18) to collaboratively optimize the bi-directional frequency of each divided interval, which is solved by LINGO 11.0. Based on the obtained frequency setting, we successively solve the schedule and the rolling stock circulations at the two terminals.

6.2.1. Headway Distribution of the Solved TOP

Given the optimized schedule, we calculate the actual headways among bi-directional trips, and compare them with the generated stepped MHFs. The comparison is illustrated in Figure 7. Each point in Figure 7 is the headway between the current trip (departing at the current time) and next trip.

At the beginning and end of the operation period (5:00~6:00 and after 23:30), the sparse demand lead to the long headway in these periods, and the bi-directional stepped MHFs overlap with each other. Because the morning peak occurs in the up direction, the up-direction stepped MHF is much smaller than down-direction stepped MHF in the early period (before 15:00); due to the tidal phenomenon of daily commuters, the down direction presents the evening peak with smaller stepped MHF after 15:00. As for actual headways, we can find that during the early period in a day (before 15:30), actual up-direction headways are generally approximate to the up-direction stepped MHF, while the actual down-direction headways are smaller than the corresponding down-direction stepped MHF and distribute around the up-direction stepped MHF, especially before 9:00; after that, the actual headways present a reversal: actual down-direction headways gradually trend to the down-direction stepped MHF, while actual up-direction headways generally deviate from the up-direction stepped MHF and distribute around the down-direction stepped MHF. This is because our approach collaboratively optimizes the bi-directional frequency/headway setting to add effective connections and balance the bi-directional resource utilization, then the down-direction headways would be properly narrowed to connect the dense departing up-direction trips at $s_B$ to satisfy the huge amount of up-direction travel demand during morning peak hours, while the up-direction headways should be adjusted to fit the high service level for down-direction commuters during evening peak hours. During other periods, actual headways also need to distribute around the smaller stepped MHF to ensure the required service level and add effective connections. It is indicated that our approach optimizes the TOP based on the required service level, and then the incorporation of collaborative division, trip flow circulation and hidden balance cost can further contribute to reaching the balanced distribution of arriving and departing trips and stable connections at terminals, which can benefit reducing the overall operation cost.

6.2.2. Rolling Stock Circulation of the Solved TOP

In the solved TOP, illustrated in Figure 8, there are both 166 trips running from the two directions. At the terminal $s_A/p_A$, the number of arriving and departing trips is 332, 15 rolling stocks are used for circulation, and 76 depot operations are required (including the in-/out-depot operations of 15 rolling stocks and 23 connections executed at the depot). At the terminal $s_B/p_B$, the number of arriving and departing trips is 332, 25 rolling stocks are used for circulation, and 80 depot operations are required (including the in-/out-depot operations of 25 rolling stocks and 15 connections executed at the depot). The total number of used rolling stocks is 40; due to the limit of the redundancy ratio, the difference of used rolling stocks between the two depots is 10.

Rolling stocks run between the two terminals by connecting bi-directional trips. Due to the connection principle as well as the distribution of bi-directional arriving and departing trips, there would exist some exchanges of rolling stocks between the two depots. We list the circulation of each rolling stock driving from each depot, as shown in

Table 2. Circulation of rolling stocks from $p_A$.

Rolling Stock	Connection Sequence
1	1-7′-22->71′-70-93′88-113′-120-136′-147->165′ (returning $p_A$)
2	2-8′-23-47′-53-68′-67-88′-83-108′-110-128′ (returning $p_A$)
3 *	3-9′-24-74′-72-95′-90-115′-123-138′-148-152′-164 (returning $p_B$)
4	4-10′-25-48′->115-132′ (returning $p_A$)
5	5-12′-27-49′-54-69′-68-90′-85-110′-114-131′-144-164′ (returning $p_A$)
6 *	6-13′-28-50′-55-70′-69-92′-87-112′-118-134′-146-151′-162 (returning $p_B$)
7 *	7-14′-29-80′-77-102′-98 (returning $p_B$)
8 *	8-16′-31-83′-79-104′-102 (returning $p_B$)
9	9-17′-32-52′-56-72′-131-143′-152-155′ (returning $p_A$)
10 *	10-18′-33->86′-81-106′-106 (returning $p_B$)
11 *	11-19′-34-53′-57-73′-71-94′-89-114′-121 (returning $p_B$)
12 *	12-21′-36->89′-84-109′-112 (returning $p_B$)
13 *	13-22′->97-121′-134 (returning $p_B$)
14	14-25′->99-122′-135-145′-154-158′ (returning $p_A$)
15	21-46′->113-130′ (returning $p_A$)

and

Table 3. Circulation of rolling stocks from $p_B$.

Rolling Stock	Connection Sequence
1	1′-15-29′-41->96′-91-116′-125 (returning $p_B$)
2	2′-16-33′-44->100′-95 (returning $p_B$)
3 *	3′-17-37′->107-126′-140-147′-157-160′ (returning $p_A$)
4 *	4′-18-41′-49-64′-64-84′->141-163′ (returning $p_A$)
5	5′-19-44′-51-66′-128 (returning $p_B$)
6	6′-20-45′-52-67′-66-87′-82-107′-108 (returning $p_B$)
7 *	11′-26-77′-74-98′-93-118′-129-142′-151-166′ (returning $p_A$)
8	15′-30-51′-117-133′-145-150′-161 (returning $p_B$)
9 *	20′-35-54′->119-135′-155-157′ (returning $p_A$)
10	23′-37-55′-58-75′-73-97′-92-117′-127-141′-150-154′-166 (returning $p_B$)
11*	24′-38-56′-59-76′->133-144′-153-156′ (returning $p_A$)
12	26′-39->91′-86-111′-116 (returning $p_B$)
13 *	27′-40-57′->122-137′->158-162′ (returning $p_A$)
14 *	28′->101-123′-137-146′-156-159′ (returning $p_A$)
15	30′-42-58′-60-78′-75-99′-94-119′-130 (returning $p_B$)
16	31′->103-124′-138 (returning $p_B$)
17	32′-43-59′-61-79′-76-101′-96-120′-132 (returning $p_B$)
18 *	34′->105-125′-139->161′ (returning $p_A$)
19	35′-45-60′->124-139′->163 (returning $p_B$)
20	36′-46-61′-62-81′->136 (returning $p_B$)
21	38′-47-62′-63-82′-78-103′-100 (returning $p_B$)
22	39′-48-63′->126-140′-149-153′-165 (returning $p_B$)
23	40′->109-127′-142-148′-159 (returning $p_B$)
24	42′-50-65′-65-85′-80-105′-104 (returning $p_B$)
25	43′-111-129′-143-149′-160 (returning $p_B$)

. In Table 2 and Table 3, a down-direction trip is noted as $i,i\in I$, while an up-direction trip is noted as $j^{\prime },j\in J$; the symbol “-” shows the connections at terminal stations, and the symbol “->” shows the connections at depots. The symbol “*” means the rolling stock returns the opposite terminal after one day’s operation. It is shown that there are both 8 rolling stocks returning the opposite depot in both directions: rolling stock 3, 6–8, and 10–13 from $p_A$, and rolling stock 3, 4, 7, 9, 11, 13, 14, and 18 from $p_B$. In this way, the number of stored rolling stocks in each depot can recover after one day’s operation, resulting in a balanced state in resource utilization.

6.2.3. Comparison between the Practical TOP and the Solved TOP

In the practical TOP, operators set frequencies for either direction according to the demand fluctuation. In either direction, the headway during the beginning and end of the operation period is 10 min, the headways during peak hours and off-peak hours are 3 min and 8 min respectively, and for smooth and stable operation, the headway during the transition between different periods is set as 6 min. The comparisons of the operation indexes between the practical TOP and the solved TOP are listed in

Table 4. Comparisons of operation indexes between the practical TOP and the solved TOP.

TOP	TTN	NRS (Total/$p_A$/$p_B$)	NDO (Total/$p_A$/$p_B$)	DRS
Practical TOP	328	47/8/39	146/64/82	31
Solved TOP	332	40/15/25	156/76/80	10

TTN: total trip number; NRS: number of used rolling stocks; NDO: number of depot operations; DRS: difference of used rolling stocks between the two depots.

.

It can be seen from Table 4 that, the TTN in the practical TOP is 328, a little less than that of the solved TOP, but the NRS and DRS are larger than those in the solved TOP. This is because the practical plan only considers to satisfy the travel demand in either direction but ignores the connection balance at terminals, leading to high operation cost, whereas our approach properly adds the frequency in the direction with sparse demand to collaborate the bi-directional frequency/headway setting, then more departing trips can be effectively connected and the resource utilization can reach a balanced state, resulting in the reduced NRS and DRS. The NDO in the solved TOP is larger than that in the practical TOP because the added connections would make it easier for the parking capacity of terminal stations to reach the saturated state and more connections have to be executed at depots. Therefore, although the running trips in the solved TOP slightly increase, the used rolling stocks apparently decrease and the resource utilization becomes more balanced, which means the overall operation cost is effectively reduced. It indicates our approach can ensure the circulation balance in TOP optimization, balance the distributions of bi-directional arriving and departing trips and resource utilization, improve connection efficiency and effectively reduce the overall operation cost.

6.2.4. Comparison between the Proposed Approach and the Existing Approach

In this part, we make comparisons between the proposed approach and the existing approach based on [2]. In [2], MHF is adopted to reduce running trips and match the demand fluctuation, while the rolling stock circulation is just the output with the given schedule. We obtain the results of the TOP solved by the existing approach. The comparisons of the operation indexes between the practical TOP and the solved TOP are listed in

Table 5. Comparisons of operation indexes between different approaches.

TOP	TTN	NRS (Total/$p_A$/$p_B$)	NDO (Total/$p_A$/$p_B$)	DRS
Existing approach	292	76/5/71	250/108/142	66
Proposed approach	332	40/15/25	156/76/80	10

TTN: total trip number; NRS: number of used rolling stocks; NDO: number of depot operations; DRS: difference of used rolling stocks between the two depots.

.

It can be seen from Table 5 that the existing approach can effectively reduce running trips according to demand fluctuation, but the circulation balance is completely ignored, resulting in the huge imbalance of the distribution between departure and arrival at terminals. It fails to realize effective connections, leading to the large NRS, NDO and DRS. Particularly, the number of used rolling stocks from $p_B$ even exceeds the storage capacity of the depot. It means this is an infeasible TOP, no matter how much the TTN is improved. Thus, it is shown that although our approach increases TTN, the bi-directional trips can run in a balanced way and the overall operation cost is effectively reduced. It indicates the importance of circulation balance in TOP optimization.

6.3. Impact of Circulation Balance Strategies on TOP Optimization

In this paper, we incorporate circulation balance into TOP optimization, with explicit consideration of collaborative division, trip flow circulation and balance constraint, which all impact the operation indexes. To investigate the impact of the involved circulation balance strategies on TOP optimization, we mainly focus on the balance constraint by sensitivity analysis on the redundancy ratio $\theta$, while the impact of collaborative division and trip flow circulation is studied by removing the balance constraint.

Table 6. Comparisons of operation indexes in the solved TOP with different $\theta$.

$\theta$	TTN	NRS (Total/$p_A$/$p_B$)	NDO (Total/$p_A$/$p_B$)	DRS
0	342	40/20/20	174/86/88	0
0.25	332	40/15/25	156/76/80	10
0.50	326	40/12/28	160/78/82	16
0.75	318	40/8/32	160/70/90	24
1	318	40/8/32	156/66/90	24
-	318	40/8/32	156/68/88	24

TTN: total trip number; NRS: number of used rolling stocks; NDO: number of depot operations; DRS: difference of used rolling stocks between the two depots.

lists the solved operation indexes with different redundancy ratio $\theta$, where the last row shows the solved results without considering balance constraint. When $\theta =0$, although the TTN is the most, the bi-directional trips, used rolling stocks and depot operations are almost symmetrical, reaching the best circulation balance. With the increase of $\theta$, TTN and NDO gradually decreases, while DRS increases. It means that the resource utilization balance is ensured by adding the frequency in the direction with sparse travel demand. When $\theta \ge 0.75$, the results reach a stable state because the balance constraint is relaxed and the change of bi-directional frequency distributions is small. The NRS remains unchanged with the increase of $\theta$, which indicates that the balance constraint mainly controls the resource utilization balance, but the impact on NRS is slight.

Without the balance constraint (the last row in Table 6), the results are approximate to those when $\theta =1$. It means the collaborative division and trip flow circulation mainly influence TTN and NRS (most of the operation cost). This is because the overall distribution of arriving and departing trips is determined by the collaborative division and trip flow circulation to derive the connections and operation cost, although the balance constraint can also slightly adjust the frequency setting to ensure the resource utilization balance.

Thus, the collaborative division and trip flow circulation can rebalance the distribution of bi-directional arriving and departing trips to add effective connections and contribute to the most cost reductions, while the balance constraint mainly aims to ensure the resource utilization balance between the two depots to mitigate the storage pressure and organization complexity. The combination of collaborative division, trip flow circulation and balance constraint can balance the arrival and departure of bi-directional trips, keep stable connections, and effectively reduce the overall operation cost.

7. Conclusions

In this paper, we incorporate circulation balance into TOP optimization to reduce the overall operation cost. Based on time-varying section demand and predetermined service level, we collaboratively divide the whole operation space-time zone by sequences and construct the bi-directional stepped MHFs. The trip flow circulation network is constructed based on the collaborative division to describe the relaxed circulation process of bi-directional trips, and then the TOP optimization is transformed into a capacitated minimum cost flow problem that is formulated by an ILP model, involving the balance constraint. Using the two-stage approach, we obtain the schedule with stepped headways and the rolling stock circulations at the two terminals are solved by ordering, reverse ordering, and exchange operations.

The performance of the proposed approach is evaluated by an URT line in Shenzhen. In the solved TOP, the final actual headways generally distribute around the stepped MHF with dense travel demand to ensure the required service level and realize effective connections. The incorporation of circulation balance can also ensure balanced exchange of rolling stocks between the two terminals, which is beneficial to resource utilization. The comparison between the practical TOP and the solved TOP shows that the proposed approach balances the distributions of bi-directional arriving and departing trips and resource utilization by properly adding the frequency in the direction with sparse travel demand to improve connection efficiency and effectively reduce the overall operation cost. The comparison between the proposed approach and the existing approach shows that although the TOP solved by the existing approach has fewer running trips, it wastes more resource and becomes infeasible due to the ignorance of circulation and utilization, while our approach can better balance the circulation process and reduce the overall operation cost. It further demonstrates the importance of involving circulation balance in TOP optimization.

We further investigate the impact of the circulation balance strategies on TOP optimization. It indicates that the collaborative division and trip flow circulation can rebalance the distribution of bi-directional arriving and departing trips to add effective connections and contribute to the most cost reductions, while the balance constraint mainly aims to ensure the resource utilization balance between the two depots to mitigate the storage pressure and organization complexity. The incorporation of circulation balance can make the operation and organization trend to a balanced and symmetric state, which can effectively improve the efficiency of circulation and utilization and reduce the overall operation cost. It is an effective way for stable and sustainable URT operation.

It is suggested that URT operators should properly increase the frequency in the direction with sparse demand to ensure collaborative bi-directional frequency and headway setting and add effective connections at terminals. Furthermore, operators should also trade off demand-sensitive scheduling and circulation balance in daily operation according to the practical targets.

Our future research can focus on diverse operation orientation, strategies, and restrictions in TOP optimization for URT systems. First, we can extend the proposed approach to time-varying O-D demand or individual passengers to obtain more accurate TOPs and satisfy diverse orientations. Second, different URT structures and short-loop operations can be involved because they would impact the circulation process of trip flows and the resource limitation of depots.