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Asymptotic average train frequency as a function of the number of running trains and of the average passenger arrival rates to the platforms (symmetric passenger arrival).
Source publication
We present in this article traffic flow and control models for the train dynamics in metro lines. The first model, written in the max-plus algebra, takes into account minimum running, dwell and safety time constraints, without any control of the train dwell times at platforms, and without consideration of the passenger travel demand. We show that t...
Contexts in source publication
Context 1
... rate is varied in order to derive its effect on the train dynamics and on the physics of traffic. The results are given in the left side of Table II and in Figure 5. The left side of Table II shows the increasing of the train time-headways (degradation of the train frequencies) due to increases in the passenger arrival rates. ...
Context 2
... left side of Table II shows the increasing of the train time-headways (degradation of the train frequencies) due to increases in the passenger arrival rates. Figure 5 gives a three dimensional illustration of this effect. As shown in Theorem 5.3, the control law proposed here guarantees train dynamic stability, for every level of passenger demand. ...
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Citations
... The latest works on metro traffic modeling and control from Farhi et al. [7]- [10] and Schanzenbächer et al. [17]- [22] have relaxed the constraint cited above. A first version exists for a metro line without junction, with minimum dwell and run times [8]. ...
... A second one studies the effect of the passenger demand on the train dynamics [9]. A summary of these can be found in [7], [10]. Schanzenbächer et al. have extended the work in [8] to a version for metro lines with a junction, at first with static dwell and run times [17] -the paper this article is based on. ...
... Finally, the authors of this paper have proposed in [20] a dwell time control minimizing the variance of the train time-headway in a linear metro line. The models in [7], [8], [10], [17], [19], [21] are based on a Max-plus algebra modeling approach. For more details on Max-plus linear modeling and on the main results used for the approach considered here, the reader is referred to [1], [3]. ...
We propose in this article macroscopic control laws for the number of trains running on a metro line with a junction, in feedback of the train run and dwell times as well as the required train frequency (responding to a given level of the passenger travel demand). Often, on dense traffic metro lines, human experience-based traffic control strategies do not efficiently stabilize the train dynamics or respond to the passenger travel demand. Our work is based on a mathematical model of the train dynamics permitting closed-form solutions of the traffic phases on the metro line. We first give a complete description of the derived traffic phases of the train dynamics. Second, we focus on the traffic phases where the train frequency is optimal with respect to given parameters, and identify set points which we use for the macroscopic control of the number of running trains. The objective of the control is to derive from the set points of the phase diagram, the optimal number of running trains with regard to possible variations and/or perturbations of the metro line parameters (train run times, train dwell times and the required train frequency). Finally, based on infrastructure parameters, passenger travel demand, and run and dwell times, we simulate a case study on the metro line 13 of Paris (France) demonstrating how the feedback traffic control practically works.
... We distinguish three parts of the metro line: a central part and two branches crossing at the junction. The metro line is discretized in space into a number of segments as in [7,8,9,10,13,14,15], all following the same discrete event modeling approach. A first model for the train dynamics on a metro line with a junction has been proposed in [13], with train dwell and run times respecting given lower bounds, i.e. they are constant and independent of the passenger travel demand. ...
This paper is an extended abstract on a traffic model for the train dynamics on a metro line with a junction. The dynamic model includes control laws on the dwell and on the run time, taking into account the passenger travel demand. Our model extends two existing traffic models: 1. a model that describes the train dynamics on a metro line with a junction but without taking into account the travel demand, i.e. dwell and run times are fixed, and 2. another model that controls the train dynamics on a linear metro line (without junction) taking into account the passenger travel demand. We present the phase diagram of the train dynamics in the metro line with a junction. Train time-headway and frequency are functions of 1. the number of trains on the line, 2. the difference between the number of trains on the two branches, and 3. the passenger demand level. The derived diagram can be used for the optimization of the traffic control on the metro line and at the junction.
... Starting with a recently developed traffic model for metro lines, with static dwell and run times, in [4], the authors of [10] have extended this approach to a metro line with a junction. Applied to a RATP metro line in Paris, this traffic model has precisely reproduced the microscopic characteristics of the line, e.g. the bottleneck segment of the line, the average frequency depending on dwell, run, safe separation times and the number of segments and trains. ...
... We consider a linear metro line without junction, as in Fig. 1. The line is discretized in space into a number of segments where the length of every segment is bigger than the train length, as in [4]. The train dynamics is modeled here by applying run and dwell time control laws under train speed and safe separation constraints. ...
... The average on k and j of the quantities above are denoted r, w, t, g, h and s. We have the following relationships [4]. ...
We present here a real-time control model for the train dynamics in a linear metro line system. The model describes the train dynamics taking into account average passenger arrival rates on platforms, including control laws for train dwell and run times, based on the feedback of the train dynamics. The model extends a recently developed Max-plus linear traffic model with demand-dependent dwell times and a run time control. The extension permits the elimination of eventual irregularities on the train time-headway. The resulting train dynamics are interpreted as a dynamic programming system of a stochastic optimal control problem of a Markov chain. The train dynamics still admit a stable stationary regime with a unique average growth rate interpreted as the asymptotic average train time-headway. Moreover, beyond the transient regime of the train dynamics, our extension guarantees uniformity in time of the train time-headways at every platform.
... Starting with a recently developed traffic model for metro lines, with static dwell and run times, in [4], the authors of [10] have extended this approach to a metro line with a junction. Applied to a RATP metro line in Paris, this traffic model has precisely reproduced the microscopic characteristics of the line, e.g. the bottleneck segment of the line, the average frequency depending on dwell, run, safe separation times and the number of segments and trains. ...
... We consider a linear metro line without junction, as in Fig. 1. The line is discretized in space into a number of segments where the length of every segment is bigger than the train length, as in [4]. The train dynamics is modeled here by applying run and dwell time control laws under train speed and safe separation constraints. ...
... The average on k and j of the quantities above are denoted r, w, t, g, h and s. We have the following relationships [4]. ...
We present here a real-time control model for the train dynamics in a linear metro line system. The model describes the train dynamics taking into account average passenger arrival rates on platforms, including control laws for train dwell and run times, based on the feedback of the train dynamics. The model extends a recently developed Max-plus linear traffic model with demand-dependent dwell times and a run time control. The extension permits the elimination of eventual irregularities on the train time-headway. The resulting train dynamics are interpreted as a dynamic programming system of a stochastic optimal control problem of a Markov chain. The train dynamics still admit a stable stationary regime with a unique average growth rate interpreted as the asymptotic average train time-headway. Moreover, beyond the transient regime of the train dynamics, our extension guarantees uniformity in time of the train time-headways at every platform.
... Although we are interested in traffic control and optimization, we here adopt the approach of Farhi et al. [12]- [14], which permits the understanding of the physics of traffic in a metro line, and in particular the effect of the passenger demand on the traffic phases of the train dynamics. Fundamental traffic diagrams similar to the ones derived in the road traffic (see [6]- [11]) are derived in [12]- [14]. ...
... Although we are interested in traffic control and optimization, we here adopt the approach of Farhi et al. [12]- [14], which permits the understanding of the physics of traffic in a metro line, and in particular the effect of the passenger demand on the traffic phases of the train dynamics. Fundamental traffic diagrams similar to the ones derived in the road traffic (see [6]- [11]) are derived in [12]- [14]. As mentioned earlier, we propose a control law for the train dwell times as functions of the passenger arrival rates onto the platforms, and derive the effect of the passenger demand on the physics of traffic. ...
... where h is built from f, such that h satisfies additive 1-homogeneity, monotonicity, and connectivity properties needed by Theorem 1. The whole proof is available in [12]. ...
We propose a traffic flow and control model for the train dynamics in a linear metro line without junction. The model takes into account time constraints such as minimum interstation running times, minimum train dwell times at platforms, and minimum safe separation times between successive trains. Moreover, it includes a control law that sets the train dwell times at platforms based on the feedback of the train time-headways and of the passenger arrival rates at platforms. We show that the dynamic system converges to a stable stationary regime with a unique average growth rate, and derive, by numerical simulation, the traffic phases of the train dynamics. We compare the obtained traffic phases with the ones derived with an existing max-plus algebra traffic model, and derive the effect of the passenger travel demand on the train dynamics. Finally, we draw some conclusions and discuss perspectives of the proposed approach.
... We base this work on the traffic models presented in [4]- [6], where the physics of traffic in a metro line system without junction are entirely described. It is a discrete event modeling approach, where we use train departure times as the main modeling variables. ...
... It is a discrete event modeling approach, where we use train departure times as the main modeling variables. Two cases have been studied in [4]- [6]. ...
... We extend the approach developed in [4]- [6] by modeling a metro line with a junction. Let us consider a metro line with one junction as shown in Figure 1 below. ...
This paper presents a mathematical model for the train dynamics in a mass-transit metro line system with one symmetrically operated junction. We distinguish three parts: a central part and two branches. The tracks are spatially discretized into segments (or blocks) and the train dynamics are described by a discrete event system where the variables are the $k^{th}$ departure times from each segment. The train dynamics are based on two main constraints: a travel time constraint modeling theoretic run and dwell times, and a safe separation constraint modeling the signaling system in case where the traffic gets very dense. The Max-plus algebra model allows to analytically derive the asymptotic average train frequency as a function of many parameters, including train travel times, minimum safety intervals, the total number of trains on the line and the number of trains on each branch. This derivation permits to understand the physics of traffic. In a further step, the results will be used for traffic control.
... In Section 2, we give a short review on the dynamic programming systems. In Section 3, we extend the max-plus algebra model of [19], [20] in order to take into account the effect of passengers on the train dynamics. We briefly review the natural instability of the train dynamics, when it is not controlled. ...
... We recall here, the assumptions and notations we considered in [19], [20]. We consider a metro line of N platforms as shown in Figure 1. ...
... We only give here a sketch of it. For the whole proof, see [19]. The proof consists in showing that the dynamic system (13), or, more precisely, its explicit form, is additive 1-homogeneous, monotone, and connected. ...
... The model is able to explain the physics of traffic in the case where the train dwell times on platforms take into account only the train dynamics (safety times, time-headways, etc.) independent of the passenger volumes and destinations. The objective here is to wholly understand the physics of traffic in this case, in order to extend the approach to the case where dwell times are set in feedback on both train dynamics and passenger arrivals [17], [18]. ...
... As mentioned above, the effects of the passenger demand on the train dynamics, in particular on the train dwell times on platforms, are not modeled here. However, we show in [17], [18] that our approach is extensible to traffic control models that set the train dwell times on platforms as functions of the passenger demand. ...
... The model does not take into account the effect of passengers on the train dwell times on the platforms. However, the approach we present here is extensible to traffic control models where the train dwell times are set in feedback on the traffic state including the train positions and the passenger arrivals; see [17], [18]. Let us consider a metro line of N platforms as shown in Figure 1. ...
On a metro line with a junction, the merge of the line is a key point for the operation. This paper proposes a discrete event traffic model for a metro line with a junction. It is based on the line’s existing signaling system and determines trains’ departure times at all line nodes. Furthermore, the model estimates trains’ average headway and frequency. We apply our model to Paris metro line 13, which is currently not operated with a FIFO (first-in–first-out) rule on the merge. Thus, we seek to evaluate the effect of a FIFO rule on the nominal frequency. Compared to the current line operation, we also test the FIFO rule as a control strategy for daily disturbances. At the steady state, we show that the train frequency is maximized regardless of the distribution of the trains on the three parts of the line (the central part and two branches). Then, we study the train frequency in two disturbed situations that often occur on a metro line. We show that the FIFO rule reduces the impact of these disturbances in time and intensity. This research is motivated by the Paris metro operator, as the considered line is expected to be fully automated in the coming years. The operator wants to better understand the potential benefits of new operating rules and how operating protocols can improve the throughput and reliability of the system. This work is the first step in seeing frequency evolution on a line with a junction.
