A Real-Time Timetable Rescheduling Method for Metro System Energy Optimization under Dwell-Time Disturbances

. Automatic Train Systems (ATSs) have attracted much attention in recent years. A reliable ATS can reschedule timetables adaptively and rapidly whenever a possible disturbance breaks the original timetable. Most research focuses the timetable rescheduling problem on minimizing the overall delay for trains or passengers. Few have been focusing on how to minimize the energy consumption when disturbances happen. In this paper, a real-time timetable rescheduling method (RTTRM) for energy optimization of metro systems has been proposed. The proposed method takes little time to recalculate a new schedule and gives proper solutions for all trains in the network immediately after a random disturbance happens, which avoids possible chain reactions that would attenuate the reuse of regenerative energy. The real-time feature and self-adaptability of the method are attributed to the combinational use of Genetic Algorithm (GA) and Deep Neural Network (DNN). The decision system for proposing solutions, which contains multiple DNN cells with same structures, is trained by GA results. RTTRM is upon the foundation of three models for metro networks: a control model, a timetable model and an energy model. Several numerical examples tested on Shanghai Metro Line 1 (SML1) validate the energy saving effects and real-time features of the proposed method.


Introduction
With rapid development of intelligent transportation, Automatic Train Systems (ATSs) have attracted much attention in recent years.A reliable high-level ATS ensures the entire railway network to function safely, cost-e ectively and e ciently under sudden disturbances [1].Disturbances in metro networks such as temporary platform blockages make o ine schedules suboptimal for use.Hence, various methods of Train Timetable Rescheduling (TTR) [2][3][4][5][6] have been proposed to handle unforeseen events which may disturb timetable.Prevailing methods for timetable rescheduling can be classi ed into two categories: passenger-oriented and train-oriented.e passenger-oriented research focuses on minimizing the total delay time of passengers [4], while the train-oriented research focuses on minimizing the overall delays of all trains [7,8].However, short delays in one train may not necessarily arouse block con icts of the route ahead, still, it will inevitably a ect the energy consumption of the overall metro network.Since a metro network consumes a large amount of electric energy every day, there is tremendous potential of energy conservation in metro transportations.So far, there is little research that mentions the topic on how to save energy when a metro network encounters disturbances.
Di erent from the traditional TTR problem, the energy-efcient TTR problem consists of two steps.e rst step is to build an o ine timetable for optimal use of the total energy of the metro network, which is called Energy-E cient Train Timetabling (EETT) [9].In this step, all trains in the network run with con rmed dwell time and no emergencies happen [8].e second step is to build a real-time timetable to cope with sudden disturbances.In this step, proper travel time in di erent sections and di erent trains will be determined and modi ed in real time to minimize the system energy consumption a er disturbances happen.
Energy-e cient TTR problem is a challenging task because of its real-time demand and randomness of disturbances.Any disturbance in a metro network can bring a series of chain reactions that make the o ine schedule not optimal any more.Rescheduling timetable is a good solution to avoid the chain reactions and save more energy.EETT methods that are nonreal-time methods cannot be used in the energy-ecient TTR problem.Moreover, disturbances occur randomly at di erent stations on di erent trains, and the length of delay is also stochastic.It requires the method having a self-adaptability that makes the appropriate choice presciently according to the disturbances.
Energy-e cient TTR problem has been rarely mentioned in recent years.Gong et al. [10] proposed an integrated Energy-e cient Operation Methodology (EOM) to compensate dwell time disturbances in real-time.By reducing the travel time of the delaying train in the section following a disturbance, the schedule can be recovered to the original one which is optimized by GA. e cost of EOM is that more energy will be consumed before the recovery of the schedule.
In this paper, a real-time timetable rescheduling method (RTTRM) for energy optimization is proposed.Di erent from EOM, this method reschedules all trains in the network a er a disturbance happens, rather than reschedules the delaying train only.e proposed method can nish making decisions immediately a er a disturbance happens, attributed to the combinational usage of GA and DNN.Firstly, GA provides a series of corresponding energy-optimal timetables for random disturbances.en, a decision system based on DNN learns from the connection between the random disturbances and the optimal timetable which are deduced by GA.Finally, the well trained decision system provides proper solutions in different cases of disturbances in real-time.
e remainder of the paper is organized as follows.Section 2 reviews the previous contributions related to TTR and EETT problem.Section 3 presents the description of a control model, a timetable model and an energy model in metro system.Section 4 introduces GA and a decision system to solve the energy-e cient TTR with dwell-time disturbances.In section 5, several experiments based on a real-world network of Shanghai Metro Line 1 (SML1) is presented to validate the proposed method.e nal section concludes the paper.

Literature Review
Pioneering theoretical works related to TTR problem were rst carried out by Carey's group [11][12][13].ey eliminated the distribution and propagation of the train delay by using the sequential solution procedures.Törnquist and Persson [14] presented a Mixed Integer Programming (MIP) model that took into account the reordering and rerouting of trains.D' Ariano et al. [15] computed a con ict-free train timetable compatible with the actual status of the railway network and proposed a branchand-bound algorithm for minimizing the global secondary delay.Khan and Zhou [16] developed a stochastic optimization formulation with the purpose of minimizing the total trip time in a published timetable and reducing the expected schedule delay.Cacchiani and Toth [17] classi ed approaches of timetable rescheduling into six categories: stochastic optimization [18], light robustness [19], recovery robustness [20], delay management [21], bicriteria and Lagrangian-based approaches [22] and meta-heuristics [23].Dündar and Sahin [24] developed a Genetic Algorithm (GA) to reschedule the trains on a single track railway line in Turkey and determined the meets and passes of the trains in the opposite direction.ey designed benchmarking experiments among an Arti cial Neural Network (ANN), a MIP model and two kinds of GA. ey also validated that GA outperforms the other methods.Šemrov et al. [25] used a rescheduling method based on reinforcement learning, more speci cally Q-learning, on a real-world railway network in Slovenia.ey illustrated that Q-learning leaded to rescheduling solutions that were at least equivalent and o en superior to the simple First-In-First-Out (FIFO) method and the random walk method.Xu et al. [7] proposed a Mixed-Integer Linear Program (MILP) model for the quasi-moving block signaling system to reduce the nal delay and to solve a real-world instance in China.ey optimized tra c in transition from a disordered condition to a normal condition and analyzed delays in di erent transition phases.Ortega et al. [4] proposed a biobjective optimization method for timetable rescheduling during the end-ofservice period of a subway, in order to minimize the total transfer waiting time for all transfer passengers and the deviation from the scheduled timetable.
Albrecht and Oettich [26] rst discussed EETT problem.ey used a Dynamic Programming (DP) to calculate the optimal timetable.e quality criteria of the multiobjective optimization function were the overall waiting time and the energy consumption.Albrecht [27] considered to change the additional running time to the synchronize acceleration and the regenerative braking in order to minimize total energy consumption and power peaks.It was the rst to study EETT with the regenerative braking.Yang et al. [28] rst described EETT problem in a mathematical programming mode and solved this problem by maximizing time overlaps of nearby accelerating and braking trains.Sun et al. [29] developed a bi-objective timetable optimization model to minimize the total passenger waiting time and energy consumption.

Formulation of Model for Metro Network
In this section, three models are formulated: a control model, a timetable model, and an energy model, all of which are used to describe the operation process for a metro network.For the single-train control model, state variables of trains, such as the position and the speed, are formulated.For the timetable model, the departure time and the arrival time for each single train are calculated.For the energy model, energy exchange among trains running in the same network is formulated.

Parameter and Variable.
e notation system which includes parameters and variables, is presented as follows: 1 : conversion e ciency of the train traction system (from electricity to mechanical energy).
, : interstation running time of train running from station to station + 1. total travel time.: constant traction force in the constant force accelerating phase.: constant braking force in the constant force braking phase.v : speed in switching point (from the constant force accelerating phase to the constant power accelerating phase).v : speed in switching point (from the constant power braking phase to the constant force braking phase).v , : cruising speed of train running from station to station + 1. v lim1 : the maximum speed limit between stations.v lim2 : the pull-in speed limit and the pull-out speed limit.
: current position of train .: current time of train .v : current driving speed of train .
: traction or braking force per unit mass of train .: resistance per unit mass.g: gravity per unit mass.

Assumptions
(i) e network in the model has narrow spacing (for example Shanghai Metro Line 1).Trains drive in each section in a sequence of accelerating-cruising-braking, without the repeated accelerating and braking.(ii) e adjacent trains run with enough spacing, so we do not consider the blocking con ict during operation.(iii) e disturbances are small enough which will not lead to network disruption.(iv) Only one disturbance happens during a complete test procedure from the rst train's departure to the last train's arrival.(v) e disturbances will not happen at the starting station.
In Equation (2), = (v) meets Davis's Equation [30]   where 1 , 2 and 3 are constant parameters.g = g( ) repre- sents the gravity per unit mass, which is a piecewise linear function.= (v, ) ∈ − (v), + (v) represents the force per unit mass or acceleration.We assume that is positive in the acceleration phase and negative in the braking phase.Figure 1 shows the bound curves of , where − (v) is the lower bound and + (v) is the upper bound.
In Figure 1, formulated in advance which obey the motor characteristic of a train.According to the theories proposed by the SCG group [31], the maximum acceleration-speedholdmaximum braking strategy is chosen as the optimal driving strategy in a period.e traction force meets = + (v) in the accelerating phase, and the braking force meets = − (v) in the braking phase.In the speedhold phase, the traction force changes with gradient to keep the driving speed constant, which meets the constraint − (v) < < + (v).According to the functional curves in Figure 1, the expression of variable in six di erent phases is concluded as follows. (

Constant force braking
Constant power braking  where also represents the total simulation period.

Energy Model for Multiple
Trains.Before considering a case of multitrain driving in one network, one should rst consider energy regeneration from braking trains.e braking process converts mechanical energy into electric energy through the overhead contact line.Taking full advantage of this feedback energy can save a large amount of energy.In most common cases, the feedback energy from braking trains is fully utilized by accelerating trains at the same time period.ere is another case that energy supply exceeds the demand, and extra energy is consumed by the heating resistors to avoid the voltage increase on the DC side.With the theory shown in Figure 3, it is possible to build an optimal timetable to maximize the utilization of the feedback energy.
Overall energy consumption of a network is derived as follows.Calculate the required traction energy and the regenerated braking energy for train .e traction power at the time can be calculated by where the rst condition denotes train driving in the accelerating phase or in the speedhold phase, and the second condition denotes train driving in the braking phase or in the dwelling phase.Hence, by summing over the traction power for train and integrating it for time, total required traction energy for all the trains in the network is acquired e braking energy from train at time is e numerical examples based on the data of Shanghai Metro Line 1, a network with narrow spacing [10].Trains drive in each section in a sequence of accelerating-cruising-braking, without the repeated accelerating and braking.e cruising speed v , has a one-to-one mapping from the periodic time , .e relation is de ned as , = v , .As shown in Figure 2, the area enclosed by the red curve and the axis represents the section spacing.If the cruising speed increases, the periodic time will decrease to keep the section spacing constant.Hence, the arrival time can be deduced from the cruising speed.
e derivation process of , = v , is shown in Appendix A.

Timetable Model for a Single Train.
e timetable of a metro-line operation can be modeled as [32] where , = ℎ at = 1.
If a disturbance occurs in a dwell time +1, , then the departure instants will be e maximum travel time for whole trains in the metro-line is stochastic search method for optimization problems inspired by the process of natural selection.Holland [33] rst used it to generate high-quality solutions to optimization problems.With extensive generality and practical applicability, it has obtained considerable success in providing satisfactory solutions to many management problems for railway tra c.In this paper, GA is used to solve the energy optimization model.

4.1.1.
Genotype.e decision variable in this energy optimization problem is v , , a continuous variable changing from v min to v max .It is di cult for a genotype covering the range of the continuous decision variable.Hence, we discrete the v min , v max section and de ne a genes set g = g ᐈ ᐈ ᐈ ᐈ g = 0, 1, 2, . . ., to correspond the possible values of the decision variable, where is a positive integer.e possible values of the decision variable is 4.1.2.Chromosomes.A ( − 1) × -dimensional matrix is used to de ne a chromosome, in which ( , ) is selected from the genes set g.

4.1.3.
Initialization.An integer number pop_size is de ned as the population size of the initial population.pop_size decreases during each successive generation with genetic operators like crossover and mutation.( = 0) is de ned as the tness function of every individual.An evolutionary computation framework based on python named DEAP is used, which contains GA and other advanced evolutionary strategies.e process of crossover and mutation in GA can be referred to the o cial documents on Github [34].

Energy Optimization under Disturbance
Situation.e optimization model mentioned in the second step is formulated as follows: where is not zero but a stochastic variable.GA is also used to solve the energy optimization problem based on (16).In this situation, the chromosome is not a ( − 1) × dimen- sional matrix.e dimension of the chromosome is determined by the value of 0 , 0 .A delay occurs during the time interval , , , + +1, + , so the timetable rescheduling task begins a er the instant , + +1, + .e optimal inde- pendent variables are taken as the reference.Before the delay happens, the original v , deduced from (13) are retained.v , a er the delay are regarded as the new independent variables for (16).
e total regenerated energy utilized at time is the minimum between the braking energy feedback and the traction energy, which is e minimum operator is used to distinguish two di erent conditions.If the traction energy is larger than the regenerative braking energy, the regenerative braking energy will be fully utilized; otherwise if the traction energy is smaller than the regenerative energy, the extra regenerative braking energy will be consumed by the heating resistors.e total regenerative energy for all the trains in the network is The net energy consumption for all the trains in the network is e objective of energy optimization is to minimize ( ).

Energy Optimization with Dwell Time Disturbance
e objective of the energy-e cient TTR problem is to minimize energy consumption by taking full advantage of the regenerative braking energy when random disturbances happen.In this paper, a real-time timetable rescheduling method (RTTRM) is proposed to solve the energy-e cient TTR problem, which combines GA with a decision system, and gives a proper decision immediately a er a random disturbance happens.RTTRM solves the energy-e cient TTR problem in three steps.Firstly, it solves the energy-optimal timetable under a no-disturbance situation.Secondly, by introducing a stochastic variable into the dwell time, ensembles of energy-optimal timetable under di erent disturbances are traversed.A decision system based on DNN is trained by the state variables which are sampled under di erent disturbances.In this step, the schedule before the disturbance instant should not be modi ed.Lastly, the trained decision system is used to judge which case the disturbance belong to and give an energy-optimal solution accordingly.

Energy Optimization under No-Disturbance Situation.
e energy optimization model mentioned in the rst step can be formulated as follows: is is a one-objective optimization problem with complex constraints that can be appropriately solved by GA.GA is a  ( 13) selector, a decision network and a voter.Cruising speed decision system comes into e ect a er a disturbance occurs.At the departure instant of each train, the decision system decides the cruising speed of that train in the next section.e state variables of other running trains are put into the input layers.Selector is used to select corresponding DNN cell according to the train number and the station number at present of the departing train, which e ectively simpli es the structure of DNN cell.Finally, voter decides the cruising speed of the departing train.Figure 4(b) shows the detailed structure for one of the DNN cell, which is trained by GA results at each disturbance case.DNN cell is structured according to the existing sample volume.eoretically, the disturbance cases are in nite, so the samples obtained by GA are in nite.Hence, DNN but not SNN is needed to ensure the sample capacity of the decision system.

Experimental Validation
In order to validate the real-time timetable rescheduling method, a series of numerical experiments are set.e environment for experiments is set in Shanghai Metro Line 1

Deep Neural Network.
Deep Neural Network (DNN) [35] has strong generalization ability, and has been widely applied to elds including computer vision, speech recognition, natural language processing, audio recognition, and board game programs, where they have produced results comparable to and in some cases superior to human experts.Compared with Shallow Neural Network (SNN) [36], DNN has an excellent ability of feature learning, and the acquired characteristics can describe the nature of the samples, which is bene cial to visualization or classi cation.e sample capacity and ability of DNN have been greatly improved than SNN.
In the previous section, a series of sampling data by GA are obtained, which contains the state variables of all trains in the network, sampling at each departure instant in di erent disturbances, and also contains a series of cruising speed as the target of decisions.ese data can be used to train a decision system based on DNN which gives wise choice of cruising speed when disturbances happen.Figure 4 displays a structure of the decision system, which decides cruising speed at each departure instant.
Figure 4(a) shows the structure of the cruising speed decision system, which contains four parts: an input layers, a iterations, the population distribution gradually centralizes.Finally a er 15 generations, the population converges at a tness value of 286.48 kWh, which is the optimal energy consumption without disturbances happen.
Example 5.2. is example is based on a two-train network in the 6 sections of SML1.A disturbance is random selected from the discrete point set of (1 , 2 , . . ., 6s), and the disturbance occurs when Train No. 2 (T2) running at the 2nd station (S2) or at the 3rd station (S3).In each travel, only a disturbance happens.e optimization model is based on Equation ( 16).In Figure 6, indexes such as energy saving and time consumption are compared between GA and RTTRM.
e initial population size of GA is set as 200.A er 5, 8 and 11 generations, the population size still keeps 200 and the tness values gradually improve.Figure 6 (le ) shows the population distribution in these generations.GA_5gen, GA_8gen and GA_11gen represent the 5th, the 8th and the 11th generations o spring, respectively.Figure 6 (right) shows the calculation time in di erent methods.
As shown in Figure 6(a), disturbance occurs in T2S2.e 5th generation o spring of GA which distributes below zero, proposes a worse strategy than no action.By comparison, the 11th generation has a better performance, which distributed upside to the 5th generation and the 8th generation.By comparison between the results of RTTRM and the best individual of the 11th generation by GA, RTTRM averagely saves 0.840% in di erent delays, which is lower than GA that saves 1.115% on average.As shown in Figure 6(b), RTTRM in the T2S3 case has the best performance that saves 0.858% energy on average, which is equal to the best individual of the 11th generation by GA.Comparing from the calculation time both in the Figures 6(a Example 5.3.In this numerical example, RTTRM is validated in the three-train network in the 13 sections of SML1.In each travel test, a disturbance randomly selected from (1 , 2 , . . ., 6s) occurs when the train No.1 stops at the 2nd (SML1), which is one of the oldest metro lines in China.ere are totally 28 stations with a daily ridership of over 1,000,000 passengers [37].According to the statistical data in peak hours of workday, metro network implement the tight schedule, where the average travel time during a section is only 2 min, and the uniform interval time between trains is 164 s.At rst, a simple case that two trains run over 6 sections of SML1 which from Xinzhuang to Shanghai Indoor Stadium is analyzed.en a more complex case that three trains run over 13 sections of SML1, which is from Xinzhuang to Xinzha Road, is analysed.
e rst numerical example validates GA in the energy optimization problem.e experiment is set in the 6 sections of SML1.In this case, the headway is set as 120 s and the dwell time is set as 20 s.Assuming two trains driving without disturbance, the energy optimization model is based on Equation (13).A er 15 generations by GA, the optimal timetable is obtained in Table 1.
Figure 5 shows the population distribution of the 1st, 8th and 15th generations given by GA. e population distribution of the 1st generation is discrete and stochastic.A er 8  comparing Figures 6 and 7, it shows that to increase the number of trains in the network, more energy will be saved.According to the time consumption statistics, the calculation time to iterate a suboptimal strategy for the RTTRM is only 0.25 s on average.By comparison, GA takes 4500 s to reach similar e ects.
Example 5.4. is numerical example is based on a threetrain network.e changing process of energy consumption during the 13 sections of SML1 among three di erent station.Both the GA and the RTTRM are used to generate the timetable a er the delay happens, and then the nal energy consumption of the metro system is calculated in each strategy and in each case of delay.e results are shown in Figure 7.
As can be seen from Figure 7, RTTRM can save 4.354% energy on average in the three-train network, while GA in the 11th generation saves 3.212% energy on average, both of which save more energy percentage than the two-train network.By proven to achieve good results in energy saving, but it takes too much time to give a decision.Compared with the two methods above, RTTRM saves 3.261% of the total energy in this case, and can ful ll the real-time requirement in the energy-e cient TTR problem.
Table 2 gives the computation time and the total energy consumption of these methods.
e computation time is measured in the CPU Intel i7-4720HQ.

Conclusion
In this paper, a real-time timetable rescheduling method for minimizing the overall energy consumption of metro systems under disturbances is proposed.Di erent from the prevailing methods for TTR problem focusing on minimizing the overall delay, the method focuses on solving energy-e cient TTR problems that maximizing the usage of regenerative energy.
e di culties of energy-e cient TTR problems are the demand of real-time response and the randomness of disturbances.Combining GA and DNN, the proposed method can reschedule the timetable in a very short response time and have a self-adaptability that is able to make decisions for different cases of disturbances.e method is based on three methods is compared.e disturbance is set as 3.7 s when Train No.1 stops at the 2nd station.In the rst method, the train does not take any action when delay occurs.In the second method, GA is used to give an o ine strategy to cope with delay.In the third method, RTTRM is used to generate an optimal timetable in real time.Figure 8 gives the energy consumption of the whole journey in di erent methods.
As can be seen from Figure 8, the energy consumption with no-action method maintains as the highest from the moment when the delay occurs to the end.e GA method is models: a control model, a timetable model, and an energy model.e energy model formulates the optimization equations in both cases with and without disturbances.e randomness of dwelling-time disturbances is described in the energy model.e most important part of the method is a GA optimizer and a decision system.GA is used to search an optimal timetable for each case of the delay in order to minimize the energy consumption as well as extracting the state variables in each optimal strategy into a training set.e decision system based on DNN is proposed to give a wise advice for the cruising speed according to the state variables samples from GA.
Several experimental studies are conducted on Shanghai Metro Line 1 to validate the method.Both two-train network and three-train network are simulated in the experiments.e results indicate that the decision system is e ective to select suboptimal cruising speed for the energy saving concern.RTTRM saves 0.840% energy on average in two-train network in the 6 sections of SML1, and saves 4.354% energy on average in three-train network in the 13 sections of SML1.e method only takes 0.010 s in the two-train network and 0.25 s in threetrain network on average, which is much faster than GA.A whole journey comparison between RTTRM and GA is done under a 3.7 s disturbance situation.RTTRM saves 3.261% energy, which is almost equal to GA, but the computation time of the RTTRM is only 0.203 s.

F 1 :
e bound curves of accelerating phase and braking phase.

F 2 :
e relation between the cruising speed and the periodic time.

F 4 :
Cruising speed decision system.(a) Structure of decision system and (b) Structure of the DNN cell.
) and 6(b), RTTRM has absolute advantage in order of magnitude.

F 6 :
Energy saving and time consumption in two-train network.(a) Random delays occur when Train No.2 running at the 2nd station and (b) Random delays occur when Train No. 2 running at the 3rd station.

F 7 :
Energy saving and time consumption in three-train network.

F 8 :
Energy consumption during the train driving process in di erent strategies in the case of a random delay happens.

T 2 :
Comparison of computation time and energy consumption among di erent strategies.

T 1 :
Optimal timetable of the test segment without disturbance.