Traffic Modeling and Validation for Intersecting Metro Lines by Considering the Effect of Transfer Stations

This paper proposes a nonlinear discrete event state-space model for intersecting metro lines considering the effect of transfer stations. Disturbances such as technical problems in the rolling stock and signaling systems can cause deviations in the predefined train departure times. Any delay in the metro traffic system will increase over time and propagate to other trains, leading to instability which will reduce the efficiency of the system. Transfer stations in metro networks are designed to transfer passengers between trains on different intersecting metro lines. Therefore, traffic modelling of the metro transportation system requires consideration of the effect of such transfer stations. After introducing a discrete event nonlinear model for intersecting metro lines with one or two transfer stations, the accuracy and effectiveness of the introduced model to describe the dynamic behavior of metro traffic system has been evaluated and verified using the results of simulations based on real data from two intersecting lines on the Tehran metro network.

Delay rate at platform k + 1 on line . λ c,k+1 Delay rate at platform k + 1 of transfer station on line . ϕ k+1 Time required for a passenger to board on the train in nominal dwell time at platform k + 1 on line .

,i k+1
Nominal effect of passengers function of train i from platform k + 1 on line . ψ ,i k+1 Actual effect of passengers function of train i from platform k + 1 on line .

I. INTRODUCTION
The metro is the backbone of public transportation in large cities because it is fast, efficient, and safe. Along with the other modes of transportation, the metro provides a public transportation network to move passengers from any point of origin to any destination in the city. Metro systems consist of intersecting lines and transfer stations that link the key points of the metropolis [1].
In the metro networks, all trains depart according to a predefined train schedule, called the nominal schedule. Unwanted disturbances in the metro system will cause deviation of the train departure times from the nominal schedule [2]. Because the metro traffic system is high-frequency and inherently unstable, the delays will increase in time and propagate to other trains, which decreases the efficiency of a metro traffic system [2]. Consequently, delay recovery and traffic regulation in the metro transportation system are important aspects to improve the quality of transformation services and increase the passenger satisfaction.

A. LITERATURE REVIEW
Campion et al. [2] and Van Breusegem et al. [3] introduced a valuable model for a metro traffic system. The dynamic model was introduced based on deviations of the departure time from the nominal schedule for open lines and loop lines in a metro traffic system. They proposed an optimal state feedback approach to delay recovery which guaranteed the stability of high-frequency metro traffic systems. Murata [4] and Goodman [5] considered a metro traffic regulation problem and introduced a mathematical evaluation function that considers the effect of passenger expectations. An on-line optimization procedure was used to find the optimum arrival and departure times by minimizing the proposed penalty function. Fernandz et al. [6] proposed a predictive traffic regulation model for metro loop lines. They used a convex quadratic programming model to minimize the corresponding cost function in the presence of operational constraints. The main advantages of their proposed approach were its ability to manage constraints and the solvability of the real-time optimization problem.
Berbey et al. [7] presented a Lyapunov-based stability analysis method for a metro traffic system modeled using the definition of the departure time deviation from the nominal schedule. They defined a new stability index to evaluate the effect of saturation on metro lines and predict the need for rescheduling. Also, rescheduling is another method to delay recovery. Gao et al. [8] proposed a real-time method to reschedule an over-crowded metro line using a skip-stop pattern during the recovery period to minimize the deviation from the timetable after disruption and reduce the passengers' total waiting time for increasing the passenger satisfaction.
Some researchers have considered the dynamic effect of passengers in the metro traffic model. Lin and Sheu [9], [10] introduced two adaptive optimal control (AOC) and dual heuristic programming (DHP) for delay recovery in a metro traffic system. The results show advantages for the AOC over the DHP when dealing with modelling error. Li and Shutter et al. [11] developed a state-space model that considered the safety constraints for the train traffic system. They designed a robust model predictive controller for the traffic regulation to guarantee disturbance attenuation.
Moaveni and Karimi [12] presented a model for a metro traffic system by considering the effect of the number of passengers on the platform and in the trains when calculating the dwell time. In this model, deviations in the departure time of trains and in the number of passengers on the trains from an initial value are defined as state-space variables. They applied model-based predictive control (MPC) to minimize the cost function that included passenger demands in the presence of constraints. Moaveni and Najafi [13], [14] proposed a new nonlinear state-space model that used a knock-on delay concept to modify the transmission delay between sequential trains in an open-loop railway traffic system. They designed a robust model predictive controller (RMPC) to delay recovery and to increase passenger satisfaction.
All of the mentioned studies considered the metro lines to be independent of each other. However, metro lines are clearly connected through transfer stations and, if a delay occurs in one of these lines, passengers can transfer the delay from one line to another line through the transfer stations [15].
Goverde [16] studied complex railway networks when passengers change trains at transfer stations by considering the intermodal connection of a bus service to railway service. He presented optimal buffer times in timetables for scheduled connections by minimizing the total expected transfer waiting times of passengers at a transfer station. Schutter et al. [17], [18] considered the effect of the transfer station to model a railway network. They used the switching max-plus method to describe a discrete event model. For delay recovery, they designed an optimal controller for the system by defining a cost function that kept the trains running on schedule and breaking connections. Although transfer stations are considered in this model, it is not suitable for describing the traffic dynamics in metro networks due to VOLUME 10, 2022 the difference in the distance between the platforms and the running time of trains.
In some researches, the effect of a transfer station was considered as a constraint on the control system design. Li et al. [19] proposed a distributed optimal control by considering transfer coordination constraints to synchronize trains' departure at the transfer stations. Since passengers are the main cause of delay transmission between lines through the transfer station, providing appropriate methods to control passenger flow can prevent the spread of delays and alleviate the traffic congestion in the metro network. Yuan et al. [20] also use the passenger flow control method to reduce or avoid traffic congestion inside stations. They formulated a model of coordinated passenger flow control as a mixed-integer linear programming model by discretizing the time horizon and minimizing the total waiting time of passengers, including outside stations and on platforms. Wang et al. [21] introduced a mixed-integer programming model based on the equivalent time interval to minimize the total number of stranded passengers on a whole metro line by considering the effect of transfer stations. The main shortcoming of the above mentioned studies is that in these studies, the effect of the transfer stations has not been considered in the open-loop dynamic.

B. MAIN CONTRIBUTION
In this paper, the effect of transfer stations in metro traffic modelling to increase the accuracy of a model for high-frequency metro systems is considered. Providing a traffic model for each line of the metro network based on the time deviation method by considering the type of stations on the line can be a solution for decentralized control of the system so that the effect of transfer stations on the lines is also considered. The current study introduces a nonlinear discrete event model for a metro traffic system with intersecting metro lines by considering the effect of transfer stations. The model considers the concept of knock-on delay and buffer times. Based on the proposed approach, the time deviations of trains on lines 2 and 4 in Tehran metro are modeled. The proposed model has been verified using actual data on train departure times on lines 2 and 4.
The rest of this paper is categorized as follows: Section 2 presents the proposed nonlinear discrete event model and includes a state-space model and the effect of the number of passengers at a platform on the model. Section 3 compares the simulation results with actual data. In the last section, the conclusion is presented.

II. PROPOSED METRO TRAFFIC MODEL
In this section, a discrete event nonlinear model for two intersecting metro lines by considering the effect of transfer stations on the traffic dynamic is presented. Modelling has been done according to the assumption that trains have sufficient capacity for transportation. The two intersecting lines have at least one transfer station at which passengers can change lines to travel to their desired destinations. The intersecting lines could include one or more transfer stations, as shown in Fig. 1. Note that the maximum number of transfer stations in Tehran metro network is two stations on intersecting lines. The mathematical model is driven by defining the departure on line as: The running and dwell times are defined as (2) and (3), respectively.
By defining u ,i k as: and substituting (2), (3) and (4) into (1), the dynamic departure time of train i on line at platform k +1 can be expressed as: As well, the nominal departure times of train i on line at platforms k + 1 satisfy (6).
Therefore, by defining the time deviation as t ,i k = t ,i k −T ,i k and using (5) and (6), the time deviation of train i on line at platform k + 1 from the nominal departure time can be determined as: It is known that, in a metro traffic system, any delay in a train departure time will propagate along the line when the delay time is longer than the buffer time of that line (t L B , L ∈ { , }). The decision function for transferring delay, f ,i k , is introduced to allow the delay to be correctly transmitted along a line. Also, ψ ,i k+1 and ,i k+1 are the actual and nominal functions to show the effect of passengers on determining the dwell time of the train i at the platform k + 1 on line , respectively. Moreover, these two functions are determined based on the type of platform: located at the transfer station or not. Therefore, (7) should be considered in the two following cases: In this case, ψ ,i k+1 and ,i k+1 , are defined as: where, and λ k+1 is the delay rate. The delay rate for a platform which is not in a transfer station is defined as [14]: where, P Act and d k+1 are the actual number of passengers at the platform and the dwell time, respectively. Using (8) and (9), (7) can be rewritten as: where, In this case, for determining the f ,i k , the two following conditions are presented: In this condition, the delay will not be transferred to the next train on the line. Therefore, β k+1 t ,i−1 k+1 + α k+1 f ,i k in (11) which generates the delay of the next train must be zero as: So, the decision function for transferring delay can be obtained as (14), using (12) and (13).
By employing (14), (11) can be rewritten as: In this condition, the delay equal to t ,i−1 k+1 − t ,i k − t B is transmitted to the next train on the line. Therefore: (16) and the decision function for transferring delay is presented as: Using (17), (11) can be rewritten as: Using (14) and (17), when a platform k +1 is not a platform of a transfer station, the decision function for transferring delay, f ,i k , can be defined as: where, In platforms of transfer stations, there are two sets of passengers: one set is moving toward the platform from the station entrance on line , P Ave_p,k+1 , and a second set is transfer passengers, P Ave_c,k+1 , which are coming from the another platform of the transfer station. Therefore, ψ ,i k+1 and ,i k+1 , as the actual and nominal function of the effect of passengers are defined as: where, λ k+1 and λ c,k+1 are defined as (10) and (21), respectively.
Equation (20) shows that any delay in one line results in changing the departure times on the another line at the transfer station, because the number of passengers at the platforms are changed. In other words, any delays in intersecting lines can be transferred to the other lines at the transfer stations. The dynamic equation of departure time deviation is as (22) using (7), (9) and (20). where, and f ,i k , are defined in four following conditions: k < t B In this condition, the train delays will not be transmitted to the next trains on both lines. Therefore, (22) which generates the delay of the next train must be zero as: and the decision function for transferring delay, f ,i k , is obtained as: Equation (22) can be rewritten as (26), using (23) and (25).
k > t B In this condition, just the delay on line will be transferred, thus: and, the decision function for transferring delay, f ,i k , is determined as: Using (28) and (23), (22) can be rewritten as (29).
k < t B In this condition, the delay just will be transferred between the trains on line , so: and the decision function for transferring delay, f ,i k , is determined as: Therefore, the dynamic equation of departure time deviation is as (32), using (31) and (23).
k > t B In this condition, delays between trains on both lines must be considered. Therefore: and the decision function for transferring delay, f ,i k , is: By considering (34) and (23), (22) can be rewritten as: Consequently, when a platform k + 1 is a platform of the transfer station using (25), (28), (31), and (34), the decision function for transferring delay can be determined for both lines as follows: where, In order to prevent hard nonlinearity in the traffic dynamics in step function g(δ) and g(δ ), this function is approximated asĝ(δ) andĝ(δ ) in (37).
It is clear that if a large value is selected for G, then the behavior ofĝ(δ) andĝ(δ ) is closer to the behavior of g(δ) and g(δ ). The platform k + 1 is not a platform of a transfer station: where, The platform k + 1 is a platform of a transfer station: A nonlinear state-space model for M trains and N platforms on line and M trains and N platforms on line can be developed by considering time deviation (11) and (22) for line , and (38) and (40) for line as follows: where, (k, x(k), u(k), f(k, x(k)), w(k)) represents the nonlinear dynamics of the system and, x(k), u(k), f(k, x(k)) and w(k) are state, control, decision function for transferring delay, and disturbance vector, respectively for M L trains at N L platforms where L ∈ { , } and M L < N L .

III. SIMULATION RESULTS AND DISCUSSION
This section provides the simulation results of the proposed nonlinear metro traffic model by considering the properties of two intersecting lines of Tehran metro network. Fig.2 shows that lines 2 and 4 of the Tehran metro network contained two transfer stations. The actual departure time of the trains of this network was used to validate the model. Data received for this network from the control center is for the movement of the trains from Farhangsara platform, P 2 1 , to Sadeghiyeh platform, P 2 22 , on line 2, and from Kolahdoz platform, P 4 1 , to Eram-Sabz platform, P 4 19 , on line 4. Table 1 presents the parameters of lines 2 and 4 in Tehran metro network. Fig.3 shows the minimum and maximum number of passengers in all platforms on lines 2 and 4 when the nominal headway is 4 min from 16 : 00 to 18 : 00 at working days. Also, Fig.4 shows the minimum and maximum delay rates calculated for all platforms on lines 2 and 4 by using the data of Fig.3.
The platforms 11 and 19 on line 2, and 7 and 13 on line 4 are platforms of two transfer stations on the intersecting lines 2 and 4 of Tehran metro network. Evidently, the number of passengers on the platforms of transfer stations are higher in comparison with the other platforms because, in addition to passengers arriving from the entry gates, some passengers also are coming from the platform of another line to this platform (Fig.3). Consequently, the delay rates of the platforms of the transfer stations are higher than the other platforms as it is shown in Fig.4.
Simulations of this section have been performed using the data of Fig.3 and Fig.4. The simulation results are presented for two scenarios. In the first scenario, the effect of transfer stations on the traffic modelling of two intersecting lines of the metro network is studied. Moreover, second scenario is expressed with the aim of validating the introduced model with comparing the actual values of departure times and its simulation results.

A. SCENARIO 1-METRO TRAFFIC MODELLING IN THE PRESENCE OF TRANSFER STATIONS
In this scenario, the simulation results in two following conditions are shown and compared: by considering the effect    of transfer stations in the model and without considering the effect of transfer stations. The delays have occurred for train 10 at platforms 10 and 11 on line 2, w 2,10 10 = 240sec, and w 2,10 11 = 240sec. According to the maximum and minimum delay rates in Fig. 4, their average value is used in the simulations. Also, λ 2 k+1 = 0.041 and λ 4 k+1 = 0.038 are considered for the simulations without considering the effect of the transfer station. The time deviations for the departure times on metro lines 2 and 4 are shown in Figs. 5 and 6. Evidently, when the effect of the transfer station is not considered in the model, the delay on line 2 has no effect on the time deviations of the train departure times on line 4. As well as, on line 2, the transferred delay coefficient between platforms when the effect of transfer stations has not been considered is smaller in comparison with the condition that the effect of transfer stations has been considered in the model.

B. SCENARIO 2-VALIDATING THE INTRODUCED MODEL USING THE ACTUAL DATA
To validate the introduced model for metro traffic system by considering the effect of transfer stations, a set of data has   delay rates, λ L k+1 shown in Fig. 4. It can be seen that the simulation results include the actual data, which confirms the accuracy of the model. The length of a delay on line 2   Table 2 indicate that this ocurred delay in line 2 has been transmitted from platform 11 at the transfer station to train 10 on line 4, Table 3. After the first delay for train 10 at platform 10, this train arrives to the platform 11 with more passengers compared to the normal condition. Hence, the accumulation of transferred passengers at platform 7 of the transfer station on line 4 caused a delay for train 10 on this line.

IV. CONCLUSION
In this paper, a nonlinear discrete event model for the metro traffic systems of two intersecting lines with regard to the transfer stations has been introduced. The introduced model considers the buffer time and effect of transfer stations in the dynamic equations. By studying the effect of transfer stations on the metro traffic system, it was shown that passengers play a major role in the delay transmission between the intersecting metro lines.
The proposed model was validated by employing the actual data from the Tehran metro network. The introduced model has been simulated by considering the effect of uncertainty of delay rates (λ L k+1 ) and a delay scenario in Tehran metro lines 2 and 4. Comparing the simulation results and actual departure times confirmed the accuracy of the proposed model.
Further research can be done by considering more than two intersecting metro lines in modelling and designing the control system. From 2009 to 2015, he was an Assistant Professor and from 2015 to 2018, he was an Associate Professor with the Department of Control and Signaling, School of Railway Engineering, Iran University of Science and Technology, Tehran. Since 2018, he has been an Associate Professor with the Systems and Control Engineering Group, K. N. Toosi University of Technology. He is also a member of the Center of Excellence for Modelling and Control of Complex Systems. He has authored or coauthored more than two books and more than 80 articles. His current research interests include large-scale control systems, control configuration selection, robust control systems, estimation theory, and automatic traffic control systems. VOLUME 10, 2022