Automatic train regulation of complex metro networks with transfer coordination constraints: A distributed optimal control framework

Transportation Research Part B 117 (2018) 228–253

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Automatic train regulation of complex metro networks with transfer coordination constraints: A distributed optimal control framework Shukai Li a,∗, Xuesong Zhou b, Lixing Yang a, Ziyou Gao a,∗ a b

State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ 85281, USA

a r t i c l e

i n f o

Article history: Received 12 March 2018 Revised 3 September 2018 Accepted 4 September 2018

Keywords: Metro networks Automatic train regulation Dual decomposition Distributed optimal control

a b s t r a c t In designing the automatic train regulation strategy of metro networks subject to frequent disturbances, it is essential to coordinate the trains at the transfer stations among different lines to facilitate passengers transferring. In this paper, we systematically investigate the distributed optimal control method framework for automatic train regulation of largescale complex urban metro networks with the transfer coordination constraints. A dynamic train traffic model of metro networks is elaborately developed in the form of the statespace equation. In case frequent disturbances happen, a dynamic optimization problem is developed to minimize the timetable and headway deviations for each line of the metro network under the interaction constraints of different lines on the transfer coordination. By regarding each line as a subsystem of the whole network, the optimization problem is formulated to coordinate a number of subsystems coupled by the state constraints. To satisfy the real-time control requirement, according to the dual decomposition technique, a new distributed optimal control method based on the distributed message passing mechanism is designed, which effectively decomposes the original large-scale problem spatially and temporally into multiple small-scale optimization subproblems that can be computed completely in parallel on a single computing platform to speed up the solution procedure, and thereby reduces the computational burden of centralized implementations for the large-scale urban metro networks. Numerical examples are given to illustrate the effectiveness of the proposed method. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Urban metro traffic plays an important role in relieving the traffic pressure in many modern large cities, which is regarded as an environmentally friendly transportation mode with high capacity, good punctuality and low energyconsumption (Goodman and Murata, 2001; Assis and Milani, 2004; Mannino and Mascis, 2009; Ye and Liu, 2016). With the development of urban railway networks and the increase of the passenger demand, passengers will need to make several interchanges between different lines to arrive at their destinations. As a survey in Beijing metro network shows, more than forty percent of passengers need to experience the transfer activities during their trips, which calls for the transfer co∗

Corresponding authors. E-mail addresses: [email protected] (S. Li), [email protected] (Z. Gao).

https://doi.org/10.1016/j.trb.2018.09.001 0191-2615/© 2018 Elsevier Ltd. All rights reserved.

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ordination among different metro lines to improve the transfer efficiency of passengers. Transfer synchronization or transfer coordination of different metro lines affects the service quality of passengers to a large extent. The transfer coordination problems for the bus transit have been well studied in Dessouky et al. (2003), Wu et al. (2015), Wu et al. (2016)s, and Fonseca et al. (2018). Correspondingly, it is important to take into account the transfer coordination of different lines in designing the train scheduling/rescheduling strategy of metro networks so as to shorten the total travel time and waiting time of passengers transferring among different lines (Liebchen, 2008). Specifically, for the high-frequency metro networks, any deviation with respect to the nominal schedule of a given train will be amplified with the time due to the accumulation of passengers (Van Breusegem et al., 1991). When an inevitable disturbance happens on one metro line, such as infrastructure failures or signal errors, the train on this line will be delayed and train delay increases from one station to the next one with the accumulation of passengers on this line. Meanwhile, the train delays on one line will be propagated and further affect its transfer coordinations with other lines of metro networks at the transfer stations. In order to reduce the train delays caused by the disturbances on each line, the train regulation (train rescheduling) is essential to recover train delays and to suppress such instability of the metro line operation. Moreover, to enhance the performance of transfer coordination between different metro lines, it is necessary to ensure the transfer coordination for the train regulation design of metro networks. In addition, the provision of real-time information leads to a new realm of control strategies and opportunities to influence the train traffic dynamic. The real-time railway disruption management has become an active area of railway management currently (Cacchiani et al., 2014). The large-scale and real-time nature of the problem calls for the development of advanced control methodology to reflect the issue of computational efficiency. Based on the above considerations, the research scope of this paper is to design the real-time distributed optimal control for automatic train regulation of complex metro networks by taking into account the transfer coordination constraints, so as to reduce the train delays and the passenger waiting time of each line and meanwhile shorten the total waiting time of passengers transferring among different lines of the network. 1.1. Literature review In recent years, with the development of urban railway networks, there are many research focusing on the train scheduling and rescheduling problems of metro networks with the consideration of schedule synchronization or transfer coordination. By taking into account the passenger transfer behavior and transfer waiting times, the train timetabling of metro network were intensively studied. Liebchen (2008) developed a timetable to optimize the arrival and departure times at the transfer stations of the metro network in Berlin so that the passenger transfer times between the lines were minimized. Wong et al. (2008) developed a mixed integer programming optimization model to minimize all passengers’ transfer waiting times in certain railway system for the timetable synchronization problem. The problem in Shafahi and Khani (2010) was also formulated as a mixed integer programming model that gave the departure time of vehicles in lines so that passengers could transfer between lines at transfer stations with the minimum waiting time. Sels et al. (2016) derived a PESP model to minimize the total passengers’ travel time in cyclic timetabling, and applied macroscopic simulations to generate a robust railway timetable. A last-train network transfer model was established by Kang et al. (2015) to maximize passenger transfer connection headways, which reflects the last-train connections and transfer waiting time, where a genetic algorithm was designed to test a numerical example to verify its effectiveness. By considering the origin-destination passenger demands, the train scheduling problem for an urban rail transit network was studied in Wang et al. (2015b) with the passenger transfer behavior. Guo et al. (2017) studied the multiperiod-based timetable optimization problem for metro transit networks to enhance the transfer synchronization performance between different rail lines. In highly interconnected timetables or dense railway traffic, a single delayed train can cause a domino effect of secondary delays throughout the entire network. Therefore, if a disturbance or a disruption occurs, the railway system must be rescheduled (Cacchiani et al., 2014). Based on macroscopic models, the delay management problems were systematically investigated by Schöbel (2001), Schöbel (2007), Dollevoet et al. (2012) to determine which trains should wait for delayed feeder trains and which trains should depart on time. The microscopic railway traffic rescheduling problems were addressed by Tomii et al. (2005), Sato et al. (2013) with the aim to minimize the passenger dissatisfaction or inconvenience. Dollevoet et al. (2014) studied the delay management approach with the microscopic models in an iterative optimization framework to optimize the passengers delays. Corman et al. (2017) proposed a new comprehensive and detailed mathematical model for the microscopic delay management problem by incorporating both the traffic regulations and the passenger rerouting options to minimize the passenger travel time. In addition, Kroon et al. (2008) proposed a stochastic optimization model to allocate the time supplements and the buffer time in a given timetable so that the timetable is robust against the stochastic disturbances. Liebchen et al. (2010) proposed a delay-resistant periodic timetable in which the objective was to optimize the transfer time between any two adjacent trains to guarantee successful transferring and meanwhile minimize total travelling cost. Kecman et al. (2013) aimed at developing a global network-scale optimization tool to optimize the actual state over all network and controls traffic from a global perspective with adjustments to the timetable. Corman et al. (2012) developed a decision support system for the traffic management of large and busy railway networks in case of entrance delays and blocked tracks. Caimi et al. (2012) proposed a model predictive control framework for railway traffic management in bottleneck areas, which manages traffic by retiming and rerouting of trains as well as partial speed profile coordination. In particular, they proposed a closed-loop discrete-time control system. Corman et al. (2014) gave a novel approach to solve the problem of coordinating the tasks of multiple dispatchers in the disturbances. The problem

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was formulated as a bi-level program with the objective of minimizing delay propagation. Kang et al. (2015) proposed a rescheduling model for last trains of metro network with the train delays caused by incidents that occurred in train operations, so as to optimize the running time and the dwell time, and the average transfer redundant time, and meanwhile to minimize the difference between the original timetable and the rescheduled one. The real-time railway disruption management is currently an active area of railway management (Cacchiani et al., 2014). The traditional buffer time allocations are static, which cannot be used dynamically from a system-wide point of view and reduces the system utilization. With the development of the modern signalling systems, automatic train regulation (ATR) technique has been proposed to dynamically adjust the running and dwell time to recover the train delays subject to frequent minor disruptions in real-time. Van Breusegem et al. (1991) developed a traffic model of metro lines based on the discrete-event and applied a linear quadratic regulator approach to design a state feedback algorithm to minimize a given system performance. Fernandez et al. (2006) constructed a traffic regulation model for metro loop lines to optimize a given cost along a time horizon and proposed a regulation strategy by adjusting train run times to minimize the timetable and headway deviations. Based on a dual heuristic dynamic program, the automatic train regulation method was proposed by Lin and Sheu (2010, 2011) to handle the non-linear and stochastic characteristics of metro lines. By using a model predictive control method, the global on-line rescheduling problem for the entire railway networks was investigated by Kersbergen et al. (2016). Based on an iterative convex programming, Wang et al. (2015a) studied the real-time train control for the urban rail transit systems to determine the train scheduling in a higher level and the speed profile in a lower control layer. Moreover, by considering the uncertain and time-varying passenger arrival flow, the robust train regulation of metro line based on state-feedback control method were studied by Li et al. (2016a,b). Most existing literature on the automatic train regulation to recover train delays are devoted to the single metro line (Van Breusegem et al., 1991; Fernandez et al., 2006; Lin and Sheu, 2011; Li et al., 2017b), which did not involve the transfer coordination among different lines of the metro network. For the urban railway networks with increased passenger demands, it is necessary to investigate the automatic train regulation of complex metro networks by considering the transfer coordination among different lines. The train regulation problems for metro lines are usually formulated as an optimization problem and solved by using the nonlinear programming methods. In particular, for large-scale metro networks, the existing results about the train rescheduling problem are usually based on a centralized optimization way (Corman et al., 2012; Kang et al., 2015), in which the formulated nonlinear optimization problem has a high computation complexity that leads to the problem intractable in real-time. In order to reduce the computation complexity and hence to develop solutions to be applied on-line in large networks, the distributed model predictive control methods were developed by breaking the original large-scale optimization problem into multiple small-scale optimization subproblems, where the computational complexity is significantly reduced, because one controller could determine the control actions for its subsystem by solving a low-dimensional optimization problem (Langbort et al., 2004; Wakasa et al., 2008; Boyd et al., 2011). The distributed control algorithms have been designed for the transportation systems (Zhao et al., 2003; Li et al., 2017a; Zhou et al., 2017) and other systems by different methods, such as the dual decomposition technique (Li et al., 2015), the economic approaches based on computational markets (Vasirani and Ossowski, 2011), the Benders’ decomposition method (Morosan et al., 2011) and the alternating direction method of multipliers (Timotheou et al., 2015), which have been verified that the distributed control solution strategy is scalable to large transportation topologies and is suitable for online execution. However, the above distributed control methods usually need the information exchange process from all the subsystems to the master, which increases the communication time. Therefore, by regarding each line as a subsystem of the whole network, it is essential to design the completely distributed optimal control framework for automatic train regulation of metro networks in such a way as to achieve fast and efficient computational performance. 1.2. Proposed approach and contributions Considerable research over the past decade have been done on the train rescheduling problems for railway networks, which are usually solved by various nonlinear optimization method based on the centralized optimization way, providing an effective way to find the solutions. However, this makes that the formulated nonlinear optimization problem has a high computation complexity and becomes difficult to implement in real-time. In addition, with the development of the modern signalling systems for metro system, automatic train regulation technique has been proposed to dynamically adjust the running and dwell time of each train to recover the original timetable from disturbances in real-time. However, most studies on the automatic train regulation methods are confined to the single metro lines, which are limited by the lack of matching powerful methodological and algorithmic constructs, especially for real-time control in large-scale traffic systems. To meet the above challenges, this study will make meaningful inroads into the complicated issues associated with the distributed optimal control framework for automatic train regulation of complex metro networks, especially to obtain efficient solution methodology applied to the realistic large-scale network in the context of the real-time information. Firstly, for the frequent disturbances, a dynamic optimization problem is developed to minimize the timetable and headway deviations for metro network under the interaction constraints for different lines on the transfer coordination. Then, the distributed optimal control algorithm is designed by decomposing the original problem spatially and temporally and solving the produced subproblems iteratively by individual controller of each line, where the spatial decomposition is achieved by dividing the whole metro network into different lines, and the temporal decomposition is achieved by separating the considered time horizon into small receding horizon. In this way, the optimization problem of metro network is formulated to coordinate

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Table 1 The comparison of different characteristics of related models and methods. Characteristics

Traffic model

Research problem

Optimization model

Solution methodology

Van Breusegem et al. (1991) Fernandez et al. (2006) Lin and Sheu (2010) Li et al. (2017b) Liebchen (2008) Corman et al. (2012)

Single line Single line Single line Single line Multiple lines Railway networks

Linear quadratic regulator Quadratic programming Dynamic programming Model predictive control Mixed-integer-programming Bilevel programming

Centralized Centralized Centralized Centralized Centralized Centralized

Kang et al. (2015) This paper

Metro networks Metro networks

Automatic train regulation Automatic train regulation Automatic train regulation Automatic train regulation Train scheduling Real-time train rescheduling Train rescheduling Automatic train regulation

Mixed-integer-programming Distributed model predictive control

Centralized optimization Distributed optimization

optimization optimization optimization optimization optimization optimization

a number of metro lines coupled by state constraints on the transfer coordination, where the computational complexity is significantly reduced. Specifically, the contributions of this paper are presented as follows. 1. With the development of urban railway networks and the increase of the passenger demand, the distributed optimal control framework for automatic train regulation of metro networks is systematically investigated in this study. The existing studies only confined to the automatic train regulation of single metro line (Van Breusegem et al., 1991; Fernandez et al., 2006; Lin and Sheu, 2011; Li et al., 2017b). In contrast, the analysis and design of automatic train regulation of complex metro networks have become more complicated, which not only need to minimize the timetable and headway deviations of each line, but also to shorten the total travel time and waiting time of passengers transferring among different lines of the networks. It is essential to design the automatic train regulation of metro networks with the transfer coordination, rather than of the single metro line. 2. By regarding each line as a subsystem of the whole network, the optimization problem is formulated to coordinate a number of subsystems coupled by the state constraints. Compared to the train rescheduling for railway networks with the centralized optimization (Corman et al., 2012; Kang et al., 2015), a new distributed optimal control method based on the distributed message passing mechanism is designed, which effectively breaks the original large-scale optimization problem into multiple small-scale optimization subproblems spatially and temporally that can be computed in a parallel fashion on a single computing platform to speed up the solution procedure, thereby reduces the computational burden of centralized implementations and can be implemented in real-time. The proposed distributed optimal control algorithm and framework can also be used in other large-scale transportation problems. The main features of our paper are summarized in Table 1 according to the traffic model, research problem, optimization model and solution methodology as compared to the related studies. The rest of this paper is organized as follows. In Section 2, a dynamic train traffic model for metro networks is developed. In Section 3, the distributed optimal control for automatic train regulation of metro networks is designed. In Section 4, numerical examples are provided to demonstrate the effectiveness of the proposed methods. We conclude this paper in Section 5. 2. Problem formulation Let us consider a metro network that consists of Z operating lines denoted by the set K = {k|k = 1, 2, . . . , Z }, where for each line k, there are Mk stations denoted by the set J = { j| j = 1, 2, . . . , Mk } and Nk trains are operating on line k denoted by the set I = {i|i = 1, 2, . . . , Nk } to transport the passengers from the origin to the destination. Specifically, in a metro network, a station is referred to as a transfer station if more than one line crosses at the station. In reality, different metro lines are usually constructed separately in the physical space, thus, a set of transfer stations are needed for the convenience of passengers transferring, and passengers may make several transfers between different lines to reach their destinations. An illustration of the metro network with the transfer stations is schematically given in Fig. 1, where the networks includes three lines and the single direction of the trains is considered for each line. There are two transfer stations i and j, which allow passengers to transfer from station 2 to station 1 at station i, and from station 3 to station 2 at station j, respectively. The proper transfer time between two connected lines is necessary to shorten waiting time of passengers transferring among different lines. In the practical operations of metro transport, the disturbances and disruptions, such as equipment failures or passenger demand variations, are inevitable. The original train timetable cannot keep the optimized objective, and the train regulation strategy is required to be implemented to reduce the train delays. The existing train regulation strategies are confined to a single metro lines based on the train delays information of this line, which do not consider the propagation effect of the train delays to other lines. In particular, for the metro network with a number of transfer stations, when the disturbance or a small disruption happens on one line, the trains on this line will be delayed, which will further affect the transfer coordination of this line with other lines for the passengers transferring, and thus increase the waiting time of passengers. We take an example to clearly illustrate this phenomenon. A proper transfer schedule of lines 1 and 2 at the transfer station i is plotted in Fig. 2 (a), where the dotted line represents the scheduled timetable, while the solid line denotes the actual timetable. Fig. 2 plots the train timetable for line 1 and 2 at one transfer station of these two lines, where the

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Fig. 1. The illustration of the metro network with transfer stations.

axis Time-Line 2 denotes the train timetable for line 2, while the axis Time-Line 1 denotes the train timetable for line 1. From Fig. 2 (a), we can observe that, at the first stage, there is a sufficient transfer time for the passengers transferring from line 2 to line 1 with the scheduled timetable, which achieves the transfer coordination of two lines of metro network. However, when a disturbance happens for the trains on line 2 at the second stage, there is no sufficient transfer time for the passenger, which results in that the passengers disembarking from the train of line 2 can not embark on the current arrival train of line 1. The passengers aiming for line 1 have to wait for the next train of line 1, and the waiting time of the passengers for transferring is thus increased. Specifically, for the high-frequency metro lines, the train delay will be propagated from one station to the next one, which further negatively affects the transfer efficiency of the passengers during the metro network at the third stage. By comparison, as shown in Fig. 2 (b), the train regulation strategy is designed to simultaneously adjust the dwell time (e.g., holding) and running time (e.g., speed up) of trains on both line 1 and line 2 so as to recover train delays and keep the sufficient transfer time between two lines from the first stage to the third stage. Therefore, for the frequent disturbances or small disruptions of the metro network, it is important to design train regulation strategy to not only reduce the train delays on one line, but also to coordinate the trains of different lines at the transfer stations to improve the transfer efficiency from the passenger perspective, especially for a number of transfer stations. In this study, we will investigate the automatic train regulation design problem of metro networks by considering the transfer coordination time. To address this problem, the dynamic evolution equation for the train traffic of metro networks and the transfer coordination constraint among different lines are first constructed, and then other safety constraints and control constraints are given. Moreover, by regarding the minimization of the timetable and headway deviations of each line as the objective, the optimal control model for the automatic train regulation of metro network is presented. A distributed optimal control algorithm is designed to break the original large-scale optimization problem into multiple small-scale optimization subproblems to reduce the computational burden. In order to facilitate the analysis, we make the following assumptions. (1) For the metro network, each metro line is operated independently, i.e., the trains that operate on one line are not allowed to run on the other lines of the network;

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Fig. 2. The illustration of the transfer coordination of metro network.

(2) A physical metro line with two directions (up-track and down-track) is regarded as two separate lines, where the trains of different lines are operated separately. The multiple directional transfers among different lines are considered for this study. (3) For each high-frequency metro line, trains follow each other strictly according to the given sequence, and each station can only accommodate one train at any time, and overtaking and crossing operations are prohibited at any positions. Similarly to the existing literature (Wang et al., 2015b; Guo et al., 2017), in this study, we also assume that each metro line is operated independently on the network as Assumption (1). Assumption (2) has been well considered by (Wang et al., 2015b; Guo et al., 2017), which facilities us to cope with the multi-directional transfers. Under Assumption (2), multiple transfer directions of different lines at the transfer station are considered in this study, e.g., there are eight transfer directions at Crisscross transfer station and sixteen transfer directions at Triple-line transfer station (Guo et al., 2017). Moreover, with Assumption (2), there are sufficient train services used in the terminal and the delays do not affect the operations on the opposite direction via turnround times. Assumption (3) is reasonable considering the fact that the lines on urban rail transit systems usually have double tracks, and train overtaking and crossing are normally not allowed during the operations. In addition, the dwell time of each train in this study is assumed to be affected by the number of passengers entering and exiting train, where the delay rate for the dwell time is estimated by the time-dependent origin-destination demand (Fernandez et al., 2006; Lin and Sheu, 2010; Eberlein et al., 2001), unlike the existing studies that directly used a timedependent origin-destination demand to determine the train scheduling (Niu and Zhou, 2013; Yin et al., 2016). Under this assumption, the arriving train takes all the waiting passengers at the station during the time interval between the departure instants of two successive trains, i.e., all the passengers can board the first arriving train. This assumption naturally holds

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S. Li et al. / Transportation Research Part B 117 (2018) 228–253 Table 2 Indices and parameters used in the paper. k = 1, 2, . . . , Z: i = 1, 2, . . . , Mk : j = 1, 2, . . . , Nk : G = (V, E ): V: E: Nk : System parameters Hk : lj, m, n : Ri, j, k : α i, j, k : tmin , k : Dj, k : u1 min , k : u1 max , k : u2 min , k : u2 max , k : pk , qk , yk : State Variables Ti, j, k : ti, j, k : ei, j,k = ti, j,k − Ti, j,k : ri, j, k : si, j, k : w1 i, j, k : w2 i, j, k : Decision variables u1 i, j, k : u2 i, j, k :

indices of the lines of metro network; indices of the trains on each line k; indices of the stations on each line k; the directed graph; the set of nodes; the set of all directed edges between nodes; the set of neighbors of metro line k; the the the the the the the the the the the

scheduled headway on line k; coordinated transfer time from line m to n at station j of line n; nominal running time of train i between station j and j + 1 on line k; delay rate for the dwell time of train i at station j on line k minimum allowable safety headway on line k; minimal staying time at station j for line k; minimum allowable value for the train dwell time adjustment on line k; maximum allowable value for the train dwell time adjustment on line k; minimum allowable value for the train running time adjustment on line k; maximum allowable value for the train running time adjustment on line k; given positive weights for line k;

the the the the the the the

scheduled departure time of train i from station j on line k; actual departure time of train i from station j on line k; error variable; actual running time of train i between station j and j + 1 on line k; actual dwell time of train i at station j on line k; disturbance to the running time of train i between stations j and j + 1 on line k; disturbance to the dwell time of train i between stations j and j + 1 on line k;

the running time adjustment for train i between stations j and j + 1 on line k; the dwell time adjustment for train i on station j of line k.

for the stations during the unrush hours and may become restrictive to some busy stations in the rush hours. Specifically, in the busy city like Beijing, to improve service quality and alleviate passengers crowding, the passenger flow control strategy has been adopted outside the stations during peak hours to limit the entering passengers, which may make this assumption less restrictive. For the path choices of passengers (Wong et al., 2008; Wang et al., 2015b), we also assume that passengers choose their paths in the system by only two criteria: the number of interchanges and the number of stops on the trip. The passengers choose a path with few interchanges and stops. This assumption enables us to compute the number of transfer passengers at each station and thereby easily to determine the coordination transfer time. The train regulation in this study is to cope with the timetable perturbations, without the need for train cancellations or rerouting. The considered train regulation is to adjust train running and dwell time, which belongs to the microscopic delay managements. Throughout this paper, we summarize the symbols and parameters, which are listed in Table 2. 2.1. The train dynamic of metro networks We first construct the train traffic dynamic on the line of the metro network, which constitutes the state transition equation for the dynamic optimization problem of the train regulation method design. The departure time of train i at station j on line k is regarded as the state variable, which is denoted by ti, j, k . Based on the discrete-event approach proposed by Van Breusegem et al. (1991), the train traffic model of metro network is given as follows:

ti, j+1,k = ti, j,k + ri, j,k + si, j+1,k ,

(1)

where ri, j, k is the running time of train i from station j to j + 1 on line k and si, j+1,k is the dwell time of train i at station j + 1 on line k. This equation describes the dynamic evolutions of train dynamic on each line independently. Moreover, to describe the transferring of passengers among different lines of the whole metro network, we further apply a directed graph G = (V, E ) to describe the transfer relationship for different lines of metro network, where V = {1, 2, . . . , Z } is the set of nodes (the metro lines) and the edge E ⊂ V × V is the set of all directed edges between nodes in the graph. The set E is defined in the following way: If there are transfer stations between metro line i and j, which allow the passengers to transfer from line i to j, then we have (i, j ) ∈ E. If (i, j ) ∈ E, we say that lines i and j are neighbors, and we denote by Nk = { j : (k, j ) ∈ E ∪ ( j, k ) ∈ E } the set of neighbors of metro line k. For example, regarding the metro network in Fig. 1, we have V = {1, 2, 3}, E = {(2, 1 ), (3, 1 )} and N1 = {2, 3}. For the practical metro network, to ensure the transfer coordination at the transfer stations, we often set a proper transfer time between the departure time of two trains at the transfer station of different lines to shorten the waiting time of passengers transferring during the metro network. Then, an edge between two connected nodes (metro lines) denotes a coupling term in the constraints of the transfer coordination associated with the nodes (metro lines).

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Fig. 3. The illustration of transfer coordination constraint.

To provide the proper transfer time for passengers from one line to another one at the transfer station, we consider that the metro network has the following transfer coordination constraints represented by linear equalities

ti, j,n − ti , j ,m = l j,m,n = l¯j,m,n + δ,

(m, n ) ∈ E,

(2)

where lj, m, n is the coordinated transfer time (including transfer waking time and transfer waiting time) for passengers transferring from line m to n at one transfer station (station j of line n and station j of line m). lj, m, n specifies a relative difference between the departure time of two trains at the transfer station. In practice, the frequent minor disruptions lead to the deviation of the actual transfer time between two different lines from the desired transfer time. If the difference between the departure time of two trains is smaller than the desired one, there will be some passengers who can not embark on the current arrival train at the transfer line. Otherwise, the waiting time of the passenger to embark on the current arrival train at the transfer line will be increased. Thus, this coupling constraint is necessary to ensure the transfer coordination among different lines in designing the train regulation under the frequent minor disruptions. The illustration diagram for the transfer coordination constraint is plotted in Fig. 3. In particular, it should be pointed that, for the real-time train regulation problem in this study, the coordinated transfer time l j,m,n = l¯j,m,n + δ can be updated in real-time at each decision stage according to the train delay and passenger demand feedback information, where l¯j,m,n is a constant scheduled transfer time and δ is a comparatively smaller variable, which improves the robustness and flexibility of the train regulation algorithm. For example, if there is a sudden increase in passenger demand at the transfer station, we can set a comparatively bigger value lj, m, n to satisfy the transferring of large passenger flow. The proposed optimal control algorithm in the next section allows one to handle this dynamic constraints on the plant variables. In practice, some connections among different lines exist only as realised one, nobody would like to keep them. In this case, we can cancel these connections and do not consider the corresponding coordination constraint for these connections in the following formulated optimization model. Specifically, the running time ri, j, k is formulated as

ri, j,k = Ri, j,k + u1 i, j,k + w1 i, j,k ,

(3)

where Ri, j, k is the scheduled running time, u1 i, j, k is the control variable to adjust the running time of train i between stations j and j + 1 on line k, and w1 i, j, k is the disturbance variable in the running time. The dwell time si, j+1,k is modeld as

si, j+1,k = αi, j+1,k (ti, j+1,k − ti−1, j+1,k ) + D j+1,k + u2 i, j+1,k + w2 i, j+1,k ,

(4)

where αi, j+1,k is the delay rate denoting the effect of the time interval between the departure instants of two successive trains at station j + 1 for line k, which is related to the entering and exiting passengers flow at the station for the arriving train. The higher passenger arrival rate implies a bigger value of αi, j+1,k . A larger passenger arrival flow will lead to a longer dwell time of the train. αi, j+1,k is varying with the different stations and time. According to the existing historical result (Van Breusegem et al., 1991), the usual values for αi, j+1,k are in the range of 0.01 to 0.05, u2 i, j+1,k is the control variable to adjust the dwell time of train i at station j + 1 on line k, and w2 i, j+1,k is the disturbance variable for the dwell time. Here it should be noted that, for the train regulation design problem, the passenger OD demand for the entering and exiting passengers flow is converted to a delay rate αi, j+1,k , which affects the dwell time of the train. For the train dwell time of each line, as an approximation we mainly consider the effect of the passengers entering and leaving the stations of this line according the passenger OD demand, not involving the effect of transfer passengers from other lines particular for the transfer stations. This approximation provides a simple and easy way to describe the effect of the passenger flows to dwell time of trains. By substituting Eqs. (3) and (4) into (1), the train traffic dynamic of the whole metro network under the frequent minor disruptions is described by

ti, j+1,k = ti, j,k + αi, j+1,k (ti, j+1,k − ti−1, j+1,k ) + Ri, j,k + D j+1,k + u1 i, j,k + u2 i, j+1,k + wi, j,k ,

(5)

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where wi, j,k = w1 i, j,k + w2 i, j+1,k denotes the frequent minor disruption, u1 i, j, k and u2 i, j+1,k are the decision variables of the train regulation strategy to be designed. The dynamic Eq. (5) explicitly describes the dynamic evolution of the departure time of train of the network under the minor disruptions, which facilitates us to design the train regulation strategy to recover the train dynamics from the minor disruptions. For the above train dynamic model, we only consider the effect of the entering and exiting passenger flows to the dwell time of the train. The dynamic changing for number of passengers entering and exiting the train is not accurately considered, and thereby the vehicle capacity is also not taken into account for the train dynamic model. 2.2. The system constraints In addition, for each metro line, we consider the following safety and control constraints according to the operational requirement. (1) Safety constraint for trains on each line: To ensure the safety distance between two adjacent trains, for each metro line k, we have the following constraints.

ti, j,k − ti−1, j,k ≥ tmin,k ,

(6)

where tmin , k is the minimum allowable safety headway for line k, and the value of tmin , k can be different for different lines. (2) Control constraints of the train regulation strategy: For the practical limits for the control input, we consider the following control constraints

u1 min,k ≤ u1 i, j,k ≤ u1 max,k ,

(7)

u2 min,k ≤ u2 i, j,k ≤ u2 max,k ,

(8)

where the first constraint is related to the train running time adjustment and the second constraint is about the train dwell time adjustment. For each line k, u1 min , k and u2 min , k are the minimum allowable values, and u1 max , k and u2 max , k are the maximum allowable values. 2.3. The dynamic optimization problem To reduce the train delays and improve the headway regularity of metro network, we further consider a given scheduled train dynamics of metro network. For a specific duration of operating hours, a scheduled train traffic model can be constructed as follows.

Ti, j+1,k = Ti, j,k + Ri, j,k + αi, j+1,k (Ti, j+1,k − Ti−1, j+1,k ) + D j+1,k ,

(9)

where the scheduled timetable is characterized by a scheduled headway Hk between two adjacent trains for each line, i.e., Hk = Ti, j+1,k − Ti−1, j+1,k , which is determined by the commercial requirement and the passenger flow of the operating hours. For example, the scheduled headway Hk has a small value during the peak hours to satisfy the larger passenger flow. In this study, we mainly consider the uniform headways for each line and across different lines, for which the transfer coordination constraints among different lines can be easily determined. The following proposed optimal control model and method is capable to the uniform headways. Especially, during the peak hours, the scheduled headway Hk for different metro lines have a small difference, and most metro lines have the same scheduled headway Hk during the peak hours for Beijing metro networks. The proposed control method can be directly and efficiently applied to the automatic train regulation of Beijing metro networks during the peak hours. In addition, for the more general case of multiple headways across different lines, we need to further determine the complex connections of trains among different lines in order to formulate new transfer coordination constraints, unlike the uniform case with the evident transfer coordination constraint. Under this case, the formulated optimal control problem may become more complicated for the complex connection constraints, which needs to be investigated in our future work. Based on the given scheduled timetable, we define the error variable as ei, j,k = ti, j,k − Ti, j,k , which represents the deviation of the actual timetable from the scheduled one. By subtracting (9) from (5), it can easily obtain the error dynamic for train traffic model of metro networks as follows

ei, j+1,k = ei, j,k + αi, j+1,k (ei, j+1,k − ei−1, j+1,k ) + u1 i, j,k + u2 i, j,k + wi, j,k ,

(10)

which explicitly describes the dynamic evolution of the train delays under the minor disruptions wi, j, k . Additionally, other methods have been proposed to predict the evolution of train delays. Oneto et al. (2018) applied the most recent big data technologies to predict train delays. Based on Bayesian networks, Corman and Kecman (2018) proposed a stochastic model to effectively predict the propagation of train delays. In the following, we will adopt the error dynamic model (10) to further study the train regulation problem. Moreover, in the form of the error state ei, j, k , the transfer coordination constraint (2) can be rewritten as

ei, j,n − ei , j ,m = l j,m,n − (Ti, j,n − Ti , j ,m ),

(m, n ) ∈ E,

(11)

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237

where Ti, j, n and Ti , j ,m are given values. The safety constraint (6) can be rewritten with the error state form as

ei, j,k − ei−1, j,k ≥ tmin,k − Hk ,

(12)

where Hk is a given headway for each line k. Then, for the train regulation problem of metro networks, we consider the following objective function

J=

 i

j



pk e2i, j,k + qk (ei, j,k − ei−1, j,k )2 + yk (u1 2i, j,k + u2 2i, j,k ) ,

(13)

k

where pk , qk , and yk are given positive weights for each metro line k. The first term in (13) is the sum of the deviations of the actual departure time from the scheduled time, which is used to reduce train delays. The second term is about the headway deviation of the trains, which is adopted to improve the headway regularity so as to reduce the average waiting time for the passengers on each line. The third term denotes the amplitude of the control action, which is used to reduce the control costs. The weighted matrices pk and qk depend on the practical control purpose and reflect the trade-off between the regulation objectives. For the train traffic model of metro networks (10) with the objective function (13) and constraints (7)–(8) and (11)–(12), the train regulation problem of metro network is formulated to solve the following optimal control problem:

min

u1 i, j,k ,u2 i, j,k

 i

j

pk e2i, j,k + qk (ei, j,k − ei−1, j,k )2 + yk (u1 2i, j,k + u2 2i, j,k )



(14)

k

s.t. ei, j+1,k = ei, j,k + αi, j+1,k (ei, j+1,k − ei−1, j+1,k ) + u1 i, j,k + u2 i, j,k + wi, j,k , ei, j,n − ei , j ,m = l j,m,n − (Ti, j,n − Ti , j ,m ),

(m, n ) ∈ E,

ei, j,k − ei−1, j,k ≥ tmin,k − Hk , u1 min,k ≤ u1 i, j,k ≤ u1 max,k , u2 min,k ≤ u2 i, j,k ≤ u2 max,k . In (14), the objective takes a quadratic term, the first constraint is the state transition equation, the second is the coupling constraint for the transfer coordination among different lines, the third is the safety constraints and the last two are the control constraints. The minor disruption wi, j, k and the parameters αi, j+1,k and lj, m, n are time-dependent, which are updated with the time. For the real-time updated system parameters, an existing dynamic programming method become hard to deal with the this optimal control problem. To deal with this difficulty, we will apply a model predictive control (MPC) algorithm with the character of on-line optimization to solve the above optimal control problem (14). Additionally, for the metro network with a number of transfer stations in practice, the number of the state and control variables become more large and the coupling constraints among different lines become more complex. As a result, the computation complexity of the dynamic optimization problem increases rapidly under the centralized optimization. To reduce the computation complexity, we will further design the distributed optimal control algorithm based on the dual decomposition technique to split the dual of the original optimization problem into a sequence of smaller problems, so as to effectively reduce the computational burden. 3. Real-time distributed optimal control algorithm In this section, we first construct the real-time model for the error dynamic of train traffic model (10) and design the model predictive control schemes for the optimization problem with the updated system parameters and disturbances. Then, to further reduce the computational burden, we will develop a distributed optimal control approach. In such an approach, the overall optimization problem could be decomposed into several subproblems, and the computational complexity is significantly reduced since one controller could determine the control actions for its subsystem by solving a low-dimensional optimization problem. Moreover, because the decisions for each subproblem are taken independently, this approach could prevent the breaking down of integrated system from the failure of one subsystem, although the resulting solution will be suboptimal. 3.1. The formulated real-time optimal control model The train traffic model (10) describes the state evolution for each train at the stations of each line, which however can not directly express the real-time state evolution of the trains at all the stations of all the lines, and thereby is not able to act as the state transfer equation for the real-time control. Thus, to achieve the real-time control, we need to transform the original train traffic model (10) into a real-time train traffic model. Then, based on the framework of the real-time model proposed by Van Breusegem et al. (1991), the equivalently real-time model for the train traffic model (10) can be formulated as the vector and matrix form. For each line k, associated with the error state variable ei, j, k , we consider the error state vector as e j,k = [e j−1,1,k , e j−2,2,k , . . . , e j−Nk ,Nk ,k ]T with the dimension Nk being the number of the station in line k. The subscript j of ej, k can be regarded as the stage of the state evolution of each line k, and the error state vector ej, k

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represents the error state of the trains at all the stations of line k. Because of the traffic security requirements for the metro line (e.g. at most one train at a time in a section between two successive stations), for each line k, the deviations e j−1,1,k (train j − 1 at station 1), e j−2,2,k (preceding train at the next station 2), . . . , e j−Nk ,Nk ,k are known in a short time or can be easily predicted by a short-term predictor (at most a two step predictor). For example, for the extreme case of no train between two stations at a time, to obtain all the information for the state vector ej, k , we need to predict the departure time of one train that did not yet arrive at this station, which can be worked by two step predictor. The process for two step predictor is worked as: the first step is to predict the running time of this train between the preceding station and this station and the second step is to predict the dwell time of the train at this station, so as to obtain the departure time of the train at this station. Therefore, all the components of ej, k are known or can be predicted in a short time, which allows for a complete on-line feedback control (Van Breusegem et al., 1991). Additionally, the other extreme case of more than one train in a section at a time is not allowed for the formulated real-time train traffic model in this study. For convenience, the state vector is further denoted as ej, k ek (j), i.e., ek ( j ) = [e j−1,1,k , e j−2,2,k , . . . , e j−Nk ,Nk ,k ]T , where k = 1, 2, . . . , Z. Then, associated with the error state vector ek (j), the real-time train traffic network model of (10) can be equivalently formulated with the vector and matrix form as follows.

ek ( j + 1 ) = Ak ( j )ek ( j ) + Bk ( j )u1 k ( j ) + Bk ( j )u2 k ( j ) + Bk ( j )wk ( j ), k = 1, 2, . . . , Z,

(15)

where j denotes the stage of the metro network, the control vectors u1 k ( j ) = [u1 j,0,k , u1 j−1,1,k , . . . , u1 j−Nk +1,Nk −1,k

]T

and

u2 k ( j ) = [u2 j,0,k , u2 j−1,1,k , . . . , u2 j−Nk +1,Nk −1,k ]T , the disturbance vector wk ( j ) = [w j,0,k , w j−1,1,k , . . . , w j−Nk +1,Nk −1,k ]T , Ak (j) and Bk (j) are ⎡ the system parameters, which are given as ⎤ −α j−1,1,k 1−α j−1,1,k ⎢ ⎢ 1−α 1 ⎢ j−2,2,k

0

0

0

−α j−2,2,k 1−α j−2,2,k

···

0

0

···

···

···

···

0

···

0

1 1−α j−N ,N ,k k k

1 1−α j−1,1,k

0

0

···

1 1−α j−2,2,k

0

···

··· ···

···

Ak ( j ) = ⎢

⎣

⎡ ⎢ ⎢ ⎣

Bk ( j ) = ⎢

0

··· ···

0

1 1−α j−N ,N ,k k k

A.

⎤

−α j−N ,N ,k k k 1−αN ,k k

⎥ ⎥ ⎥ ⎦

⎥ ⎥ ⎥ ⎥ ⎦

, Nk ×Nk

. The detailed derivation of (15) from (10) is given in Appendix Nk ×Nk

Moreover, based on model (15), we can obtain the equivalently real-time optimal control problem of (14) as follows.

min

u 1 k ( j ),u 2 k ( j )

jf Z   

eTk ( j )Pk ek ( j ) + (ek ( j + 1 ) − ek ( j ))T Qk (ek ( j + 1 ) − ek ( j )) + u1 k ( j ) Yk u1 k ( j ) T

k=1 j= j0

+u2 k ( j ) Yk u2 k ( j ) T



(16)

s.t. ek ( j + 1 ) = Ak ( j )ek ( j ) + Bk ( j )u1 k ( j ) + Bk ( j )u2 k ( j ) + Bk ( j )wk ( j ), Em,n en ( j ) − E¯m,n em ( j ) = Lm,n − Tm,n , (m, n ) ∈ E, ek ( j − 1 ) − ek ( j ) ≤ (Hk − tmin,k )INk ×1 , u1 min,k INk ×1 ≤ u1 k ( j ) ≤ u1 max,k INk ×1 , u2 min,k INk ×1 ≤ u2 k ( j ) ≤ u2 max,k INk ×1 , k = 1, 2 . . . , Z, where INk ×1 is a matrix with Nk × 1 dimension, and all the elements equal to 1. In the above model, j0 and jf are the initial and end stage, respectively. The states of the whole network are changing from stage j0 to jf with the time. The first constraint for the state transfer equation is in fact a time-varying discrete dynamic system, where the system parameters Ak (j) and Bk (j) and the disturbance wk (j) are chancing with stage j for every lines. For the second constraint, Em, n and E¯m,n are matrixes, and Lm, n and Tm, n are vectors, all of them can be directly obtained according to the second constraint of (14). Due to the system parameters and the disturbances that are updated in real-time for the optimal control problem (16), the traditional existing programming method will be difficult to cope with the above optimal control problem. Alternatively, we apply a model predictive control algorithm with the property of on-line optimization to deal with the formulated optimal control problem in the next part. 3.2. The model predictive control schemes We formulate the original optimization problem into a model predictive control scheme, which implements repeatedly the optimal control in a rolling horizon way (Sirmatel and Geroliminis, 2018). At each sampling step j, an optimal control input is computed to minimize a given objective over a pre-specified prediction horizon M. Then, only the first element of the computed sequence is effectively used and the overall procedure is repeated at the next sampling step according to

S. Li et al. / Transportation Research Part B 117 (2018) 228–253

239

the so-called receding horizon (or moving horizon) principle. Under the model predictive control schemes, the optimization problems are repeatedly solved online in a rolling horizon way with the real-time updated parameters and disturbances. Specifically, according to (16), at each sampling step j, for each line k, we have the following performance in the M step finite prediction horizon.

Jk ( j ) =

M−1 



ek ( j + i + 1 ) Pk ek ( j + i + 1 ) + (ek ( j + i + 1 ) − ek ( j + i ))T Qk (ek ( j + i + 1 ) − ek ( j + i )) T

i=0



+u1 k ( j + i ) Yk u1 k ( j + i ) + u2 k ( j + i ) Yk u2 k ( j + i ) . T

T

(17)

Moreover, associated with (17) is the following optimization problem to compute the control input for the whole network. Z 

min

uk ( j+i ),i=0,1,2,...,M−1

Jk ( j )

(18)

k=1

s.t. ek ( j + i + 1 ) = Ak ( j + i )ek ( j + i ) + Bk ( j + i )u1 k ( j + i ) + Bk ( j + i )u2 k ( j + i ) + Bk ( j + i )wk ( j + i ), Em,n en ( j + i ) − E¯m,n em ( j + i ) = Lm,n − Tm,n , (m, n ) ∈ E, ek ( j + i − 1 ) − ek ( j + i ) ≤ (Hk − tmin,k )INk ×1 , u1 min,k INk ×1 ≤ u1 k ( j + i ) ≤ u1 max,k INk ×1 , u2 min,k INk ×1 ≤ u2 k ( j + i ) ≤ u2 max,k INk ×1 , k = 1, 2 . . . , Z. Under the predictive control framework (17)–(18), the transfer coordinated constraint for the formulated optimal control problem (14) is a dynamic constraint, which can be dynamically changed according to the practical requirement in each decision stage. When some connections exist only as realized one and nobody would like to keep them in practice, they can be broken, i.e., the corresponding existing transfer coordinated constraints are removed in some decision stages. Traditionally, the above optimization problem can be solved by the centralized model predictive control method where there are complex transfer coordination constraints among different lines, which, however, makes that the formulated optimization problem has a high computation complexity and leads to the problem being intractable in real-time, especially for the complex metro network with a number of transfer stations. Therefore, it is necessary to apply a distributed model predictive control approach to decompose the original large-scale optimization problem for online control. For the above optimization problem (18), it is obvious that the objective function can be decomposed into the individual performance index for each line k. Although the objective function is decomposed into the individual performance index for each subsystem, the optimization problem is not solved individually for each subsystem because of the coupling equality constraint on the state variables. Therefore, the optimization problem cannot be directly solved in a distributed manner for each line because of the coupling through the coordination constraints among different lines. To address this problem, we will present a distributed algorithm for solving the optimization problem in the next section. 3.3. Dual decomposition technique In this section, we will design a distributed algorithm for solving the optimization problem by using a dual decomposition technique. The motivation for using dual decomposition is to coordinate a number of subsystems (metro lines) coupled by state and input constraints (transfer coordination constraint). Each subsystem is equipped with a local model predictive controller while a centralized entity (master) manages the subsystems via prices associated with the coupling constraints (Samar et al., 2007; Boyd et al., 2011). The illustration of the dual decomposition technique is shown in Fig. 4, where the dashed lines illustrate the necessary communication, which shows that the master needs information from each line (subsystem) in order to update the prices and communicate these prices to the subsystems to satisfy the coupling constraints for the transfer coordination among different lines. It is important to note that the master needs no information of local subsystem constraints, objectives or dynamics, which lead to a decentralized algorithm, computationally inexpensive and highly desirable from a practical viewpoint. In order to handle these coupling constraints for the transfer coordination of the optimization problem (18), the dual decomposition method is introduced to move the coupling constraints into the objective function in the form of adding the Lagrangian multipliers to guarantee the satisfaction of interaction terms. The Lagrangian function of the overall optimization problem (18) can be written as

L=

Z  k=1

Jk ( j ) +

M−1 



λm,n ( j + i )(Em,n en ( j + i ) − E¯m,n em ( j + i ) − Lm,n + Tm,n ).

(19)

i=0 (m,n )∈E

where λm,n ( j + i ) is the Lagrangian multiplier. Then, according to the theory of duality, the optimization problem of whole metro network is equivalent to its dual problem

max

min

λm,n ( j ) uk ( j+i ),i=0,1,2,...,M−1

L

(20)

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S. Li et al. / Transportation Research Part B 117 (2018) 228–253

Fig. 4. The illustration of the dual decomposition technique.

s.t. ek ( j + i + 1 ) = Ak ( j + i )ek ( j + i ) + Bk ( j + i )u1 k ( j + i ) + Bk ( j + i )u2 k ( j + i ) + Bk ( j + i )wk ( j + i ), ek ( j + i − 1 ) − ek ( j + i ) ≤ (Hk − tmin,k )INk ×1 , u1 min,k INk ×1 ≤ u1 k ( j + i ) ≤ u1 max,k INk ×1 , u2 min,k INk ×1 ≤ u2 k ( j + i ) ≤ u2 max,k INk ×1 , k = 1, 2 . . . , Z. Once the Lagrangian multipliers λm,n ( j + i ) are fixed, the overall dual control problem (20) can be divided into Z optimization subproblems, where the kth subproblem is presented as follows.

min

uk ( j+i ),i=0,1,2,...,M−1

Lk = Jk ( j ) +

M−1  i=0

+



(



λm,k ( j + i )(−E¯m,k ek ( j + i ))

m

λk,n ( j + i )(Ek,n ek ( j + i ) − Lk,n + Tk,n ))

(21)

n

s.t. ek ( j + i + 1 ) = Ak ( j + i )ek ( j + i ) + Bk ( j + i )u1 k ( j + i ) + Bk ( j + i )u2 k ( j + i ) +Bk ( j + i )wk ( j + i ), ek ( j + i − 1 ) − ek ( j + i ) ≤ (Hk − tmin,k )INk ×1 , u1 min,k INk ×1 ≤ u1 k ( j + i ) ≤ u1 max,k INk ×1 , u2 min,k INk ×1 ≤ u2 k ( j + i ) ≤ u2 max,k INk ×1 , k = 1, 2 . . . , Z. It is obvious that the above optimization subproblem (21) of each line k is a quadratic programming (QP) problem, which can be solved very efficiently. Moreover, we will give the equivalent standard quadratic programming formulation of (21). For each line k, define Ek = [eTk ( j + 1 ), eTk ( j + 2 ), . . . , eTk ( j + M )]T , U1k = [uT1k ( j ), uT1k ( j + 1 ), . . . , uT1k ( j + M − 1 )]T and U2k = [uT2k ( j ), uT2k ( j + 1 ), . . . , uT2k ( j + M − 1 )]T . Then, at the sampling stage j with the measured state ek (j), according to the state transfer equation of (21), we can get the state prediction state Ek during the M step finite horizon as follows:

Ek = Fk ek ( j ) + k (U1k + U2k ), k = 1, 2 . . . , Z,

⎡

(22)

⎤

Ak ( j ) Ak ( j + 1 )Ak ( j ) ⎥ ⎦, ··· Ak ( j + Z − 1 )Ak ( j + Z − 2 ) . . . Ak ( j ) ⎡ Bk ( j ) 0 Bk ( j + 1 ) ⎢A k ( j + 1 ) B k ( j ) k = ⎣ ··· ··· Ak ( j + 1 ) · · · Ak ( j + Z − 1 )Bk ( j ) Ak ( j + 2 ) · · · Ak ( j + Z − 1 )Bk ( j + 1 ) Based on (22), the equivalent standard quadratic programming formulation problem (21) of each line k is given as

⎢ where Fk = ⎣



min

U1k ,U2k

U1k U2k

T

I

I



T

(Tk P¯k + Tk GT1 Q¯ G1 k )

I

I

U 1k

U2k



U1k + U2k

¯ Q T

⎤

0 ··· 0 ··· ⎥ ⎦. ··· ··· · · · Bk ( j + Z − 1 ) at the sampling step j for the optimization

0

0 Q¯

 T

U1k U2k



S. Li et al. / Transportation Research Part B 117 (2018) 228–253



U1k +2 U2k

⎡ s.t.

T

k

⎢ I2Nk M ⎣−I

2Nk M

I

I

 T

241

1 2

(Tk P¯Fk ek ( j ) + Tk GT1 Q¯ G1 Fk ek ( j ) − Tk GT1 Q¯ G2 + Tk Tk )

(23)

⎤   U (Hk − tmin,k )INk M×1 − G2 + G1 Fk ek ( j ) ⎥ 1k Uk,max , ⎦ U2k ≤ −Uk,min

where the weighted matrixes P¯ , Q¯ and R¯ can be directly obtained from the cost function of (21), k = [−G1 k − G1 k ], Uk,max = [u1 max,k , . . . , u1 max,k , u2 max,k , . . . , u2 max,k ]T2N M×1 , Uk,min = [u1 min,k , . . . , u1 min,k , u2 min,k , . . . , u2 min,k ]T2N M×1 , k is rek

k

lated to the Lagrangian multiplier, which can also be directly obtained from the cost function of (21). The matrices G1 , G2 and the detailed derivation of (23) from (21) are given in Appendix B. Then, with the standard quadratic programming formulation (23), at each decision stage j, the Z subproblems can be efficiently solved in parallel by a quadratic programming method, e.g., the quadprog function from the MATLAB optimization tool box. Moreover, the Lagrange multipliers can be updated by using a subgradient method. We can summarize the distributed optimal control algorithm to solve the control problem (16) by using the dual decomposition technique and the corresponding detailed descriptions are presented as Algorithm 3.1. Algorithm 3.1. •

•

Step 1. At the samplingstep j, get the measured state ek ( j ) for the error dynamic model (10) with the updated system parameters and disturbances of each metro line k. Step 2. In the predicted horizon M, the iteration optimization process to make the interactionconstraints for the transfer coordination satisfied is illustrated as follows: (1) Set the iteration step s = 1 and the initial given Lagrange multipliers λm,n ( j + i )(s ), i = 0, 1, . . . , M − 1, for metro networks. (2) For the Lagrange multipliers λm,n ( j + i )(s ) at iteration step s, solve the Z subproblems (21) in parallel by the quadratic programming method and obtain the control strategy and the error state for metro network in the predictive horizon. (3) Update the Lagrange multipliers by the subgradient method, which is given as

λm,n ( j + i )(s + 1 ) = λm,n ( j + i )(s ) + c · d ( j + i )(s )

(24)

where c is a given constant representing the updated step length, and d ( j + i )(s ) is the search direction, which is given by

d ( j + i )(s ) = Em,n en ( j + i ) − E¯m,n em ( j + i ) − Lm,n + Tm,n , (m, n ) ∈ E,

(25)

where en ( j + i ) and em ( j + i ) are obtained from the step s. (4) Move to the next iteration s + 1 and repeat (1)–(3)until the interaction balance constraints are satisfied, or thetermination condition is reached, for example, d ( j + i )(s ) 2 < ε for some ε > 0. •

•

Step 3. Get the optimal train regulation control u1k ( j ) and u2k ( j ) satisfying the interactionconstraints and implement it to the dynamic error model (10). Step 4. According to the measured value ek ( j + 1 ) at the next step j + 1 for each line k of metro network, repeat Steps 1–3 until the step horizon j f .

It is clear that Algorithm 3.1 alternates sequentially between the minimization of Z subproblems (21) and the update of the Lagrange multipliers (24), which allows the minimization of Z subproblems (21) to be done in parallel. However, at each iteration, the update of the Lagrange multipliers (24) (master) need the information of all the subproblems (subsystems), which thereby increases the communication time from the subsystems to the master. Moreover, based on the coupling structure of the metro networks, we can find that the minimization of the kth subproblem (21) only needs the partial Lagrange multipliers λk,n ( j + i ) and λm,k ( j + i ), where m, n ∈ Nk (the set of neighbors of metro line k). So the update of the Lagrange multipliers λm,n ( j + i )(s ) can be further decoupled at iteration step s. Then we further modified Algorithm 3.1 based on a distributed message passing mechanism, which does not need the information exchange process from all the subsystems to the master, where each subsystem k depends only on information from its neighbors Nk at each iteration, achieving fast overall computation times. The improved distributed optimal control algorithm based on message passing is given as follows. Algorithm 3.2. •

•

Step 1. At the samplingstep j, get the measured state ek ( j ) for the error dynamic model (10) with the updated system parameters and disturbances of each metro line k. Step 2. In the predicted horizon M, set the iteration step s = 1 and the initial given Lagrange multipliers λm,n ( j + i )(s ), i = 0, 1, . . . , M − 1.For each metro line k in parallel: repeat

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S. Li et al. / Transportation Research Part B 117 (2018) 228–253

Fig. 5. The illustration of the proposed distributed optimal control Algorithm 3.2.

(1) At iteration step s, solve the k subproblem (21)by the quadratic programming method based on the partial Lagrange multipliers λk,n ( j + i ) and λm,k ( j + i ), where m, n ∈ Nk . Obtain the control strategy and the error state in the predictive horizon for line k. (2) Communicate error state ek ( j + i ) of line k to its neighboring lines Nk . (3) Update its partial Lagrange multipliers for line k

λk,n ( j + i )(s + 1 ) = λk,n ( j + i )(s ) + c · dk,n ( j + i )(s ),

(26)

λm,k ( j + i )(s + 1 ) = λm,k ( j + i )(s ) + c · dm,k ( j + i )(s ) m, n ∈ Nk ,

(27)

where dk,n ( j + i )(s ) = Ek,n en ( j + i ) − E¯k,n ek ( j + i ) − Lk,n + Tk,n and dm,k ( j + i )(s ) = Em,k ek ( j + i ) − E¯m,k em ( j + i ) − Lm,k + Tm,k , m, n ∈ Nk . •

•

until the interaction balance constraints are satisfied, or thetermination condition is reached. Step 3. Get the optimal train regulation control u1k ( j ) and u2k ( j ) satisfying the interactionconstraints and implement it to the dynamic error model (10). Step 4. According to the measured value ek ( j + 1 ) at the next step j + 1 for each line k of metro network, repeat Steps 1–3 until the step horizon j f .

The illustration of Algorithm 3.2 is plotted as Fig. 5. During this distributed optimal control algorithm, each metro line computes its own objective. Once it has computed its regulation strategy, it communicates this information to other lines,

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Fig. 6. A small metro networks.

receives their regulation strategies and then computes the resulting deviations, which is completely decentralized: each system is updated only based on information obtained from its neighbors, and it does not exchange any information with the other systems. Therefore, Algorithm 3.2 decomposes the original large-scale optimization problem into multiple smallscale optimization subproblems that can be computed completely in parallel on a single computing platform to speed up the solution procedure, and thereby reduces the computational burden of centralized implementations for the large-scale urban metro networks. As a stopping criterion, we can use the number of iterations, the convergence tolerance of the dual variable, the norm of the subgradients, and so on. To obtain the best feasible solution for all lines of the metro network, the terminal conditions are set to be the same for all lines. It should be noted that, if a new metro line is added to the metro network, i.e., a new agent is added to the overall system, the algorithm at the group involving the new agent is only modified. This is one of the advantages of the proposed distributed optimal control Algorithm 3.2. In addition, we suppose that all the exchanges are perfect between the agents, there is no delay induced by the communications, and there is no loss of information during the exchanges. 4. Numerical examples 4.1. Network description and simualtion setup To demonstrate the performance of the distributed optimal control algorithm for automatic train regulation of metro networks, we first consider a representative metro network, which is chosen from one part of Beijing metro networks and shown in Fig. 6, where Line 1, 9, 10 denotes the actual number of metro lines. In the proposed train traffic model framework, the actual metro line numbers 1, 9, 10 are converted to the model numbers 1, 2 and 3. Then, this metro network includes three lines, where the black line represents Line 1, the yellow one denotes Line 2 and the blue one is Line 3. We consider the single train operational direction of each line, which are denoted in the figure. The set K = {k, k = 1, 2, 3} and the number of stations for each line are given as M1 = 7, M2 = 8, and M3 = 8. The transfer relationship among these lines are given as E =

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S. Li et al. / Transportation Research Part B 117 (2018) 228–253 Table 3 The disturbances to the running time and dwell time of trains.

Stage 5

Stage 9

Stage 13

Station

1

2

3

4

5

6

7

8

w1 (5) w2 (5) w3 (5) w1 (9) w2 (9) w3 (9) w1 (13) w2 (13) w3 (13)

0 0 0 0 0 0 0 0 0

5 0 10 5 10 15 8 15 20

18 5 10 15 5 10 16 8 10

20 30 15 10 20 25 10 18 10

10 30 15 30 20 35 33 25 30

0 20 8 5 15 10 8 19 8

0 25 15 0 5 5 0 15 6

null 0 0 null 0 0 null 0 0

{(1, 2 ), (1, 3 ), (2, 3 )} according to the transfer stations of the network, for example, (1,2) means that there are passengers transferring from station 4 of line 1 to the station 7 of line 2. For the stochastic passenger arrival rate, in the simulations, the delay rate α i, j, k at station j for train i on metro line k are randomly chosen from the range [0.02,0.06] at different decision stages, which are updated in real-time. Consider the fact that the uniform headways between trains of different lines can result in better coordination in transfer stations (Shafahi and Khani, 2010; Kang et al., 2015) and under this case the coordination transfer times at transfer stations among different lines can be easily determined by the average walking time of passengers. For simplicity, we adopt the uniform headways between trains for each line k, which are set as Hk = 140 s and the minimum allowable safety headway is tmin,k = 120 s, k = 1, 2, 3. To ensure the transfer coordination among different lines from the passenger viewpoint, we consider the transfer coordination constraints among different lines. Under the uniform headways, the transfer coordination constraint among different lines (i.e., the nominal coordination transfer time) can be easily determined, where the walking/connecting time can be set as the nominal coordination transfer time. As shown in the literature (Wong et al., 2008; Guo et al., 2017), the walking/ connecting time can be assumed to an average constant for simplification under the un-delayed operations, which can be collected by survey and calculated through mathematical statistics. Under this assumption, the nominal transfer time for each connections can be chosen as an average constant through mathematical statistics. Therefore, we adopt a constant nominal transfer time in the text case with the uniform headways between trains of different lines for simplicity. We set the scheduled transfer time l1, 2, 7 for trains between line 1 and line 2 as 30 s, and similarly, l1,3,7 = 30 s, and l2,3,4 = 30 s. In the cost function (17), the weight matrices Pk , Qk and Yk are set to be the same, which are chosen as diag{0.1, 0.1, . . . , 0.1}. The control constraints for the train regulation strategy are set as u1 min,k = −15 s, u2 min,k = −15 s, u1 max,k = 15 s and u2 max,k = 15 s, which means that the increase of the adjusting running time and dwell time is not allowed to exceed 15 s, respectively, and the decrease is not to exceed 15 s, respectively. The proposed distributed train regulation algorithm is verified by MATLAB R2016a on a PC (2.6-GHz processor speed and 8-GB memory size) with the platform of Windows 7. For each suboptimization problem, the formulated quadratic programming problem is solved with the quadprog function in the MATLAB optimization tool box in each sampling step of the decision process to find the optimal value. The considered time step horizon is chosen as T = 15 and the prediction step horizon is chosen as M = 3. We consider that at the initial stage j0 = 1, the initial condition of the departure time of all the trains at three lines are given as e1 (1 ) = [0, 5, 20, 30, 25, 0, 0], e2 (1 ) = [0, 0, 10, 5, 40, 10, 10, 0] and e3 (1 ) = [0, 0, 10, 20, 20, 10, 40, 0], which shows that at the initial stage, the trains at stations 2–5 of line 1 are delayed, and the trains at stations 3–7 of line 2 are delayed, and the trains at stations 3–7 of line 3 are delayed. For the initial train delays, it is necessary to design train regulation strategy to first reduce train delays of each lines. At the same time, the train delays on one line also affect the passenger transferring from this line to other lines. For example, under the initial delays, the difference between the departure time of the train from station 4 of line 1 and the train from station 7 of line 2 is delayed by 20 s compared to the previous sufficient transfer time 30 s. Then there is no sufficient transfer time for the passenger to transfer, and the waiting time of the passengers for transferring is increased. Thus, to ensure the sufficient transfer time under the delay disturbances, we should design the train regulation strategy by simultaneously adjusting the dwell time (e.g., holding) and running time (e.g., speed up) of trains on both line 1 and line 2 to recover train delays and keep the sufficient transfer time between two lines (as shown in descriptions of Fig. 2), so as to reduce the broken transfers in the next stages caused by the initial disturbances. In addition, considering the fact that the metro system is subject to frequent minor disruptions caused by random disturbances, there are three times of disturbances to the operation of the trains on three lines of the metro network, which are given in Table 3. 4.2. Compared with other control strategies In this section, under the initial train delays and real-time disturbances, we compare the following three control strategies to validate the effectiveness of the proposed distributed train regulation strategy in this study. (1) Case 1: No control (NC). In this case, for the train delays, no control for the train regulation strategy is applied to the train traffic model of the metro networks.

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Table 4 The total train delays under three different methods. Line 1 Station NC(s) SC(s) DC(s) NC−SC (% ) NC NC−DC (% ) NC

4 159.2 126.2 134.4 20.7 15.5

Line 2 5 247.9 173.0 186.0 30.2 24.9

6 244.5 78.4 83.9 67.9 65.6

7 236.1 31.9 34.5 86.4 85.3

4 124.3 116.8 115.9 6.0 6.7

Line 3 5 221.9 200.6 189.6 9.6 14.5

6 277.4 158.4 152.3 42.9 45.1

7 314.0 125.6 118.9 60.0 62.1

4 153.1 117.2 143.0 23.4 6.6

5 230.9 171.2 180.5 25.8 21.8

6 256.3 110.5 114.1 56.8 55.4

7 312.3 134.9 152.8 56.8 51.0

(2) Case 2: Separate control (SC). In this case, the control for the train regulation strategy of each line is made separately, which does not consider the transfer coordination constraints between the two connected lines. With the objective   2 2 2 2 pk ei, j,k + qk (ei, j,k − ei−1, j,k ) + rk (u1 i, j,k + u2 i, j,k ) and constraints (6)–(8) for each line k, the control strategy for i

j

each line is calculated by the model predictive control method based on the real-time system updated information. (3) Case 3: Distributed control (DC). In this case, the proposed distributed optimal control is implemented for the automatic train regulation of metro networks. By solving the optimization problem (21) according to Algorithm 3.2, we can obtain the train regulation of each line by taking account of the transfer coordination constraints between two lines. Then, under the different control strategies as cases 1–3, we can calculate the corresponding train delays, realised transfer time, and headway deviations under the initial train delays and disturbances, respectively. At first, the total train delays during the considered time horizon from station 4 to station 7 of three lines under different control strategies are calculated as summarized in Table 4. From Table 4, we can observe that for the case (NC), there are larger accumulated train delays for all the stations of the three lines under the initial train delays and disturbances, which negatively affect the operational efficiency of the metro networks. By comparison, under the cases (SC) and (DC), the train delays for all the stations are effectively reduced. For the case (SC), the total train delays are reduced from 6.0% to 86.4% compared to the (NC) case, and for the case (DC), the total train delays are reduced from 6.6% to 85.3%. The control strategies of (SC) and (DC) both reduce the train delays and improve the operational efficiency of the metro network. In particular, the reduced train delays in some stations for case (SC) are better than the results of the case (DC). This is because that the (DC) strategy further consider the transfer coordination constraint to allow the transfer time for the passenger transferring from one line to anther line, which may lead to the increase of train delays in some stations. Moreover, for the metro network with the transfer stations, it is necessary to consider the transfer coordination between two connected lines to improve the transfer efficiency of the passengers. If the actual transfer time is smaller than the nominal one, some passengers can not embark to the current arrival train at the transfer line. Otherwise, it will increase the waiting time of the passenger to embark the current arrival train at the transfer line. So it is desirable to keep the nominal transfer time of two connected lines. Under the initial train delays and disturbance, the actual transfer time between two connected lines will deviate from the nominal time. We choose representative connection lines 1 and 3. The corresponding results of the realised transfer time under three cases are illustrated in Fig. 7, which shows that after disturbances, the realised transfer time is changing for different stages. From Fig. 7, we can observe that for the case (NC), the realised transfer times between lines (1,3) are all larger than the nominal transfer time in each stage due to the disturbance to the train delays that increases the waiting time of the passenger for transferring. For the case (SC), the train regulation strategy can reduce the realised transfer time compared to the case (NC) after disturbances in some stages. However, under the case (SC), the realised transfer time still deviates from the nominal one due to the fact that it does not consider the transfer coordination. By comparison, the case (DC) effectively keeps the actual transfer time with the nominal one after the disturbances, which provides the proper transfer time for the passengers to transfer between two connected lines. The realised transfer time under cases 1 and 2 are all larger than that under case 3 with the proposed method. Therefore, compared to the control strategies (NC) and (SC), the proposed method (DC) not only reduces the train delays, but also reduces the transfer time for passengers transferring among different lines. Additionally, a smaller headway will lead to a shorter waiting time for passengers. However, a smaller headway needs strict management and high technology even though it can increase the capacity. Thus, we need to keep the headway regularity, i.e., the minimization of the headway deviations from the nominal one. Under three different control strategies, the headway deviations at station 5 of each line of the metro networks are plotted in Fig. 8. From Fig. 8, we can find that under the case (NC), there are bigger fluctuations of the headway deviations from zero for each line under the initial train delays and disturbances, which thereby increase the waiting time of the passengers. By comparison, under the cases (SC) and (DC), the headway deviations(HD) of the train at station 5 for each line are both effectively reduced and converged to zero after the disturbances. According to results in Fig. 8, the average of absolute for the headway deviations of lines 1–3 under different cases 1–3 are calculated as summarised in Table 5. It is clearly shown from Table 5 that the average of absolute for headway deviations of case (SC) and (DC) are smaller than that of case (NC) for each line. Thus, for each line, the methods of (SC) and (DC) both reduce the headway deviations of the trains, which further reduce the waiting time of the passengers for each line.

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Fig. 7. The realised transfer time between two connected lines 1 and 3, where (a) is for the case 1, (b) is for the case 2 and (c) is for the case 3.

Fig. 8. The headway deviations (HD) at station 5, where (a) is for case (1), (b) is for case (2) and (c) is for case (3).

The proposed distributed train regulation algorithm breaks the original large-scale optimization problem into multiple small-scale optimization subproblems. To illustrate the convergence speed for the proposed algorithm clearly, the evolution of the norm of the subgradients for each line in the computation process is plotted in Fig. 9, where the black, blue and red lines denote the subgradients evolutions for metro lines 1–3, respectively. It shows that the best feasible solution can be obtained after 35 iterations. Consider that the maximum computation time for each iteration is only 0.002 s. The best feasible solution at each decision stage can be obtained by 0.07 s, which reduces the computational burden of centralized implementations for the metro networks and satisfies the real-time requirement in practice. 4.3. Robustness of the proposed distributed train regulation algorithm In this section, the robustness of the proposed distributed train regulation algorithm, i.e. its sensitivity to the system parameters will be further investigated. This is relevant because in practice, the real time data could be inaccurate or in-

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Table 5 The average of absolute for the headway deviations of lines 1–3.

Line 1 Line 2 Line 3

Case 1

Case 2

Case 3

9.8410 9.5058 7.4490

6.6225 7.8914 6.8348

6.5749 7.9328 6.7933

Fig. 9. The convergence of the computation process. Table 6 The total train delays for different cases of line 1. Case

αi, j+1,k

Station 1

Station 2

Station 3

Station 4

Station 5

Station 6

Station 7

1 2 3 4 5 6 7 8 9 10

0.02 0.024 0.028 0.032 0.036 0.04 0.044 0.048 0.055 0.06

1.1238 1.1317 1.1396 1.1475 1.1554 1.1634 1.1714 1.1795 1.1936 1.2038

33.5405 33.5384 33.5365 33.5347 33.5331 33.5318 33.5306 33.5297 33.5286 33.5281

105.9687 105.8940 105.8199 105.7462 105.6731 105.6006 105.5289 105.4577 105.3345 105.2474

134.4161 134.2799 134.1440 134.0086 133.8736 133.7395 133.6067 133.4743 133.2438 133.0798

186.0989 185.9690 185.8396 185.7106 185.5822 185.4543 185.3273 185.2009 184.9810 184.8249

83.9048 83.8269 83.7497 83.6730 83.5969 83.5215 83.4470 83.3732 83.2455 83.1556

34.5133 34.4677 34.4224 34.3772 34.3321 34.2874 34.2429 34.1986 34.1216 34.0669

complete. Nevertheless, the proposed distributed algorithm should have a relatively stable performance despite these uncertainties. Therefore, the robustness of the proposed train regulation algorithm is important. Consider that the real-time updated delay rate αi, j+1,k could be inaccurate or incomplete. We will conduct the simulations for the changing of the realized value αi, j+1,k to show the robustness of the proposed method to the uncertain parameters. Without loss of generality, the realized value of αi, j+1,k is supposed to be the same at each station j of the line k. A set of 10 different cases of the realized value αi, j+1,k for all the stations of the lines are generated from the range (0.02,0.06), which is shown in Table 6. The initial train delays and the traffic disturbances wj are chosen as the same to that in Section 4.2. To show the results clearly, we choose line 1 as a representative line of the network. Then, under the proposed distributed train regulation strategy, the total train delays from station 1 to station 7 for line 1 of the metro network under cases 1–10 are calculated in Table 6. From Table 6, we can observe that the total train delay for each station of line 1 are all approximately equivalent for 10 cases with different system parameters, and the maximum difference of train delays in each station for all the cases is only 1.274, i.e., for changing of the system parameter from 0.02 to 0.06, the difference of train delays for different system parameters are controlled in the range of 1.274. Moreover, the corresponding results of the total train delays for lines 1–3 at different stations are plotted as Fig. 10, which shows that the train delays are all approximately equivalent for all the lines. It indicates the robustness of the proposed algorithm to the train delays with the uncertain system parameters.

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Fig. 10. The total train delays of 10 cases for line 1 (a), line 2 (b) and line 3(c).

Fig. 11. The headway deviations (HD) of station 5 of line 1.

In addition, by calculation, under the changing system parameters of cases 1–10, we find that the headway deviations for all the stations of three lines are all kept the approximate equivalent values. In particular, the evolution of the headway deviations for station 5 of line 1 is plotted in Fig. 11, which shows that with the changing system parameters, the difference among the headway deviations is very small. Moreover, with the changing system parameters, the transfer coordination constraint among different lines are all satisfied under cases 1–10. Thus, the proposed train regulation algorithm is also robustness to the headway regularity of each line of the metro network. In sum, the proposed distributed train regulation algorithm is robust to the uncertain system parameters for the real-time updated delay rate αi, j+1,k , which further reveals that the proposed method has a relatively stable performance despite these uncertainties.

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Fig. 12. The Beijing metro network.

4.4. More experiments with respect to large-scale networks To further test the effectiveness of the proposed distributed train regulation method, this set of experiments will be implemented in a large-scale metro network. For the practical Beijing metro networks, there are 17 two-way lines (including urban lines and suburban lines) and 270 stations and the number of connections among different lines for transferring is more than 60. If double-direction connections are considered, the number will be increased more than 480. During the morning peak hours, most urban lines are operating with the same or similar headway, while the suburban lines are operating with a longer headway. In this test case, we mainly consider the urban lines of metro network (as shown in Fig. 12) that consists of 12 lines, 199 physical stations and 44 transfer stations during the morning peak hours, where the uniform headways for different lines are chosen as 140 s. There are a large number of transfer stations in this metro network, and a great amount of passengers need to make several interchanges between different lines to arrive at their destinations, especially during the peak hours. Therefore, it is important to ensure the transfer coordination among the transfer stations under the case of the frequent minor disturbances. Traditionally, the centralized algorithm for the train regulation will be implemented for the large-scale urban metro networks, which however has a large computational burden. Under the proposed distributed train regulation Algorithm 3.2, the original large-scale optimization problem is decomposed into 12 subproblems for each line, and the computational complexity can be significantly reduced. For simplicity, we mainly consider one transfer direction at each transfer station between two connected stations of the metro network in the optimization process of this experiment. In this experiment of metro network, the system parameters α i, j, k at station j for train i on metro line k are randomly chosen from the range [0.02,0.06] at different decision stages. We also consider the single direction of each line and the single transfer direction at the transfer station, where the considered direction of each line is marked in Fig. 12. The scheduled transfer time for all the lines are chosen as 30 s. The initial train delays at the initial stage and the real-time updated disturbances are chosen from 10 s to 70 s. The weights for the cost function and the control constraints are chosen to be the same as before. To clearly illustrate the experiment results, we also choose some representative transfer stations. First, we consider the case that the train dynamic is without train regulation strategy. Under this case, the evolutions for transfers time from line 2 to line 1, line 4 to line 1, line 8 to line 1 and line 5 to line 2 are illustrated in Fig. 13 (a)–(d), respectively, which indicate that, under the initial train delays and disturbances, the transfer time among different lines at the transfer station all deviates from the nominal transfer time, which negatively affect the transfer time of the passengers. By comparison, under the proposed distributed train regulation, the corresponding transfer time among different lines are plotted in Fig. 13 (e)–(h), which shows that under the initial train delays and disturbances, the proposed train regulation strategy makes that the actual transfer times among different lines satisfy the nominal transfer time. The realised transfer times are effectively reduced under the proposed control method, which thereby facilities passenger transfers.

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Fig. 13. The transfer time between two connected lines. Table 7 The comparison results between the case of NC and the case of DC.

Line 1 Line 2 Line 3 Line 4 Line 5 Line 6

Station

6

7

8

9

10

11

12

NC DC NC DC NC DC NC DC NC DC NC DC

291.6 216.8 219.0 90.3 343.6 179.1 296.2 82.8 253.0 181.6 286.8 129.4

287.0 130.9 263.9 99.51 471.8 289.2 315.4 85.1 283.7 138.3 303.2 82.4

330.2 113.1 318.9 206.6 456.2 241.5 395.8 162.1 283.6 121.6 284.1 44.5

394.2 174.2 342.9 145.2 470.5 95.1 390.1 50.8 269.6 64.9 276.0 16.2

354.9 134.2 458.8 288.4 454.8 112.4 378.5 60.4 326.3 170.3 288.3 34.1

361.6 71.2 482.5 170.8 461.2 100.2 323.2 24.7 370.0 171.1 311.5 67.3

358.8 87.7 428.3 61.9 551.1 120.6 410.5 173.5 425.7 194.8 377.8 167.8

Moreover, under the initial train regulation and disturbances, we will give the comparison results of the train delays under the case without train regulation (NC) and the case with the proposed distributed train regulation (DC). We choose the total train delays along the considered time horizon from station 6 to station 12 of lines 1–6, where the number of the stations of each line can be distinguished according to the line direction as shown in Fig. 12. The corresponding computation results for the total train delays are given in Table 7. From Table 7, we can find that the train delays without train regulation are all larger than the case with the proposed train regulation, which negatively affect the operational efficiency of the metro networks. The proposed train regulation effectively reduces the train delays of each line of the metro network, which greatly improves the operational efficiency of the large-scale metro networks. In particular, for the large metro network, at each decision stage, the maximum computation time for the subproblem is only 0.052 s and the best feasible solution for the entire network can be obtained by 0.208 s, which satisfies the real-time requirement in practical applications. In addition, in this test, we consider the single transfer direction for metro lines for simplicity. By considering double-direction connections, the transfer relationship among different direction of differen lines become more complicated and the coordination transfer time should be further determined by the off-line optimization for train timetable synchronization problem. When the coordination transfer time at transfer stations among different lines are available, the proposed control method can also be applied to automatic train regulation problem with double-direction connections. 5. Conclusion With the development of urban railway networks and the increase of the passenger demand, transfer coordination among different metro lines is essential to improve the transfer efficiency of passengers. This paper presents a practically-useful distributed optimal control framework for automatic train regulation of the large-scale urban metro networks coupled by the transfer coordination constraints, which provides insights of critical importance to the design of automatic train regulation strategy and generate results of fundamental significance to real-time railway disruption management theories for metro networks. Specifically, to satisfy the large-scale and real-time nature of the problem, the computational efficiency of the methodology is a principal focus of this research. Accordingly, this paper designs a new distributed optimal control method based on the distributed message passing mechanism to break the original large-scale optimization problem into multiple small-scale optimization subproblems that can be computed completely in parallel on a single computing platform and

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effectively reduces the computational burden. The proposed distributed optimal control method is completely decentralized: each optimization subsystem is updated only based on information obtained from its neighbors, and it does not exchange any information with the other systems, which provides a foundation for addressing more realistic large-scale metro network in the context of the real-time information. Numerical examples are given to demonstrate the performance of the distributed optimal control algorithm for automatic train regulation of metro networks, which show that, under the proposed train regulation method, the transfer coordination among different lines is not only satisfied, but also the train delays and headway deviations are effectively reduced, which greatly improves the operational efficiency of the large-scale metro networks. The proposed distributed optimal control algorithm has a low computational burden, which achieves fast and efficient computational performance and satisfies the real-time requirement in practical applications, especially for the large-scale urban metro networks. To accurately consider the number of transferring passengers to the dwell time of the trains at the transfer stations, we should further construct the dynamic transfer model for the passenger number among different lines, which lead to a more complex coupled train and passenger dynamic dynamic model, especially for the large-scale metro networks, which needs to be investigated in our future research. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 71771017, 71621001), the Beijing Natural Science Foundation (No. L171006), and the Research Foundation of State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (No. RCS2018ZZ001). Appendix A. Derivation of (15) from (10) First, Eq. (10) can be further rewritten as

(1 − αi, j+1,k )ei, j+1,k = ei, j,k − αi, j+1,k ei−1, j+1,k + u1 i, j,k + u2 i, j,k + wi, j,k .

(28)

Dividing both sides of Eq. (28) by (1 − αi, j+1,k ), Eq. (28) is further obtained as

ei, j+1,k = For

αi, j+1,k 1 1 e − e + (u1 i, j,k + u2 i, j,k + wi, j,k ). 1 − αi, j+1,k i, j,k 1 − αi, j+1,k i−1, j+1,k 1 − αi, j+1,k

each

line

k

with

Nk

stations,

we

let

ek ( j ) = [e j−1,1,k , e j−2,2,k , . . . , e j−Nk ,Nk ,k ]T ,

u2 k ( j ) = [u2 j,0,k , u2 j−1,1,k , . . . , u2 j−Nk +1,Nk −1,k ]T ,

[u1 j,0,k , u1 j−1,1,k , . . . , u1 j−Nk +1,Nk −1,k ]T ,

(29)

and

u1 k ( j ) = wk ( j ) =

[w j,0,k , w j−1,1,k , . . . , w j−Nk +1,Nk −1,k ]T . According to equation (29), we can obtain that

⎡ −α j−1,1,k

⎢ ⎢ ek ( j + 1 )=⎢ ⎢ ⎣

1−α j−1,1,k 1 1−α j−2,2,k

0

⎡ ⎢ ⎢ +⎢ ⎢ ⎣

1 1−α j−1,1,k

0 0

0 −α j−2,2,k 1−α j−2,2,k

··· ···

0

0 ··· 0

0 ···

⎥ ⎥ ⎥ek ( j ) ⎥ −α j−N ,N ,k k k ⎦ ···

1

1−α j−N

k ,Nk ,k

0

0

···

1 1−α j−2,2,k

0 ··· ···

··· ···

··· ···

⎤

···

0

1 1−α j−N

k ,Nk ,k

(30)

1−αN

k ,k

⎤ ⎥ ⎥ ⎥(u1 k ( j ) + u2 k ( j ) + wk ( j )). ⎥ ⎦

Appendix B. Derivation of (23) from (21) First, the objective function of (21) can be rewritten as

EkT P¯ Ek + (G1 Ek − G2 )T Q¯ (G1 Ek − G2 ) + U1Tk RkU1k + U2Tk RkU2k + k Ek −

⎡

I ⎢−I where G1 = ⎣ ··· 0

0 I ··· ···

0 0 ··· −I

⎤

··· · · ·⎥ ⎦ ··· I N

k M×Nk

⎡

ek ( j ) ⎢ 0 , and G2 = ⎣ ··· 0 M

m −1   i=1

⎤ ⎥ ⎦

λk,n ( j + i )(Lk,n − Tk,n ),

(31)

n

. Nk M×1

Then, by substituting Ek = Fk ek ( j ) + k (U1k + U2k ) into (31), we can obtain that (31) is equivalent to

(Fk ek ( j ) + k (U1k + U2k ))T P¯ (Fk ek ( j ) + k (U1k + U2k ))

(32)

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+(G1 (Fk ek ( j ) + k (U1k + U2k )) − G2 )T Q¯ (G1 (Fk ek ( j ) + k (U1k + U2k )) − G2 ) +U1Tk RkU1k + U2Tk RkU2k + k (Fk ek ( j ) + k (U1k + U2k )) −

m −1   i=1

λk,n ( j + i )(Lk,n − Tk,n )

n

= (U1k + U2k )T (Tk P¯ k + Tk GT1 Q¯ G1 k )(U1k + U2k ) + U1Tk RkU1k + U2Tk RkU2k +2(U1k + U2k )T (Tk P¯ Fk ek ( j ) + Tk GT1 Q¯ G1 Fk ek ( j ) − Tk GT1 Q¯ G2 +

1 T T   ) + k , 2 k k

where k = eTk ( j )FkT P¯ Fk ek ( j ) + eTk ( j )FkT GT1 Q¯ G1 Fk ek ( j ) + GT2 Q¯ G2 − 2eTk ( j )FkT G1 Q¯ G2 + k Fk ek ( j ) −

m −1  i=1 n

λk,n ( j + i )(Lk,n − Tk,n ) is

a constant. Thus, the minimization of the objective function in (21) is converted equivalently to minimize the objective function in (23). In addition, the second constraint of (23) is equivalent to G2 − G1 Ek ≤ (Hk − tmin,k )INk M×1 which, along with Ek = Fk ek ( j ) + k (U1k + U2k ), can be rewritten as U1k + U2k ≤ (Hk − tmin,k )INk M×1 − G2 + G1 Fk ek ( j ).





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