Method for Energy-efficient Timetable Rescheduling for Small-scale Delays

Abstract

We proposed a method for energy-efficient timetable rescheduling for small-scale delays using mathematical optimization. In the developed algorithm, the timetable rescheduling is converted for reducing the powering energy and increasing the regenerative energy. We considered passenger convenience under the condition that the total time from the starting station to the terminal station of each train did not change. This paper describes the details of the method for energy-efficient timetable rescheduling and reports the results of a case study using actual railway line data.

1. Introduction

These days, the energy issues are attracting a great deal of attention because of concerns about the stability of fossil fuel supplies and increase in global energy consumption. In addition, global environmental issues have led companies in various industrial sectors to work towards the Sustainable Development Goals (SDGs) and achieve carbon neutrality. Over recent decades, Japanese railways, for example, have introduced ground equipment such as energy storage systems and regenerative inverters [1], energy-efficient rolling stock [2] and energy-efficient transport operations [3] to save energy.

There are two major approaches to saving energy in transport operations: the first is through train-driving strategy, to minimize energy consumption during train operation [4]. The second focuses on timetabling, to optimize the running time to minimize powering energy [5, 6]. However, these studies assume that trains can run according to the planned timetable, and do not assume train delays.

Therefore, we proposed a method for energy-efficient timetable rescheduling, in the event of small-scale delays of a few minutes and the need to stop trains at stations [7]. Energy-efficient timetable rescheduling is achieved with a short calculation time of about one minute, based on current timetable rescheduling which adjusts operating intervals caused by train delays. Currently, timetables are usually rescheduled without considering energy conservation (“basic timetable rescheduling”). Consequently, an energy-efficient rescheduled timetable created in the light of each specific delay situation contributes to more energy-efficient train operation. The proposed energy-efficient timetable rescheduling is a method for adjusting the arrival and departure times using mathematical optimization, so that powering energy can be reduced by running slowly and regenerative power interchange can be promoted by coordinating the timing of powering and braking.

In this paper, we report the details of the method for energy-efficient timetable rescheduling and present the results of a case study on actual railway line data. In the case study, we simulated the energy-efficient timetable rescheduling using the Train Operation Power Simulator [8] developed by RTRI and calculated the energy-saving effect.

2. Method for creating energy-efficient timetable rescheduling

2.1 Basic concept of energy-efficient timetable rescheduling

In general, extending the running time can reduce the maximum speed of a train by shortening the powering time, thereby reducing the energy consumption for train operation. In addition, if the regenerative power can be interchanged to a nearby powering train, the energy supplied from the substation can be reduced. Focusing on these two points, we will reduce the energy consumption for train operation by running at low speeds and by coordinating the timing of powering and braking.

When creating energy-efficient rescheduled timetables, the calculation time is aimed at less than one minute in order to update timetable rescheduling according to the delay situation.

Figure 1 illustrates the energy-efficient timetable rescheduling process. The timetable targets the event of a small-scale delay (from dotted line (planning timetable) to solid red line (when small delays occur)), as shown in Fig. 1(a). Then, as shown in Fig. 1(b), trains are sometimes required to have extended stopping time at stations (from dotted line to blue and green solid line) in order to prevent spikes in passenger numbers and also to avoid having trains slow down or come to a stop between stations. This is basic timetable rescheduling.

Fig. 1　An example of creating the energy-efficient timetable rescheduling

In this paper we focus on the extended stopping time at station (“suspension time”), in the black dotted circle in Fig. 1(b). As shown in the dotted red and purple circles in Fig. 1(c), the suspension time is allocated to the running time and the stopping time for energy efficient, by reducing the powering energy with running slowly to the next station and by effective use of regenerative power with coordinating the timing of powering and braking. This is the proposed energy-efficient timetable rescheduling.

The terms used in this paper describes: “powering energy” is the energy supplied by the overhead contact line minus the auxiliary energy; “auxiliary energy” is the energy used for purposes other than train running, such as lighting and air conditioning; “regenerative energy” is the energy returned to the overhead contact line by the braking train; and “energy consumption” is the value of powering energy minus the regenerative energy, adding the auxiliary energy. The Train Operation Power Simulator [8] consists of a feeding circuit calculation component, rolling stock calculation component and speed profile calculation component, and enables highly accurate calculation of various energies.

2.2 Passenger convenience

In general, there is a trade-off between energy consumption to operate trains and passenger convenience, so it is necessary to find a balance between the two.

As shown in Fig. 1, the number of trains, the stations where each train stops, and the number of cars in trainsets are not changed from the planned timetable, so the impact on passengers is kept to a minimum.

Passenger convenience was considered on the condition that the total time from departure to arrival of each train does not change from the basic timetable rescheduling.

In large stations with a large number of passengers or stations with connections to other lines, it may be more desirable to maintain the stopping time at the station rather than aim to improve energy efficiency. For such stations, the arrival and departure times or stopping time can be set to be the same as the basic timetable rescheduling.

2.3 Algorithm for powering energy reduction

Powering energy can be reduced by extending the running time and operating at lower speeds. The extent of this reduction varies depending on the trains and the sections between stations, as it is influenced by factors such as distance, gradients, curves, and other conditions. Therefore, a mathematical optimization algorithm was developed to determine the running time of each train and the sections between stations in order to minimize the total powering energy.

First, we plot the amount of powering energy as a function of the extended running time for each train and section between stations as shown in Fig. 2 (“WT plot”). As shown in Fig. 2, the decrease in powering energy tends to saturate as a function of the extended running time. Therefore, rather than extending the running time of a specific train or section between stations by a large margin, it would be more energy efficient to gradually extend the running time of multiple trains or sections between stations.

Fig. 2　An example of WT plot

Next, based on the basic timetable rescheduling, under the condition that the total time from the departure to the arrival of each train does not change, running time of sections between stations and stopping time at stations of each train are determined with the optimization calculation that minimizes the sum of powering energy. The running and stopping time should be values in multiples of the unit time (e.g., 5 seconds) for creating the train timetable. The formulation of this powering energy reduction algorithm is described in Chapter 3.

2.4 Algorithm for effective use of regenerative power

We developed the algorithm that promotes regenerative power interchange by coordinating the timing of powering and braking. First, we defined the “Regenerating-powering overlap index” as the overlap time between braking and powering times of two trains, as shown in Fig. 3. Figure 3(a) shows the timetable with color-coded driving operation. Figure 3(b) shows the Regenerating-power overlap indexes between braking of train 3 and powering of train 2, train 1, zooming in on the blue hatched area of Fig. 3(a).

Fig. 3　Definition of the regenerating-powering overlap index

Considering the Regenerating-powering overlap index there are five possible cases of overlapping relationship between braking time (represented as o_start~o_end) and powering time (represented as c_start~c_end), as shown in Fig. 4. An example of Regenerating-powering overlap indexes between train 3 and train 2, and between train 3 and train 1 in Fig. 3(b) are equivalent to Case (3) and Case (2) in Fig. 4, respectively.

Fig. 4　Divided cases of braking train and powering trains

When the braking time and powering time do not overlap, the Regenerating-powering overlap index (represented as z) is equal to 0. Otherwise, the Regenerating-powering overlap index is z_end − z_start, where z_end = min{o_end, c_end} and z_start = max{o_start, c_start}. So, the Regenerating-powering overlap index z is expressed by (1).

  

$$z=\max \left\{0,z_{\mathit{end}}-z_{\mathit{start}}\right\}$$(1)

When considering the Regenerating-powering overlap index, there is a concern that the calculation time will increase if all combinations for one braking train with other all trains are considered. In addition, even if the timing of powering and braking are adjusted for combinations of one braking train with multiple powering trains, the number of powering trains which can actually interchange regenerative power are limited by distance between trains and voltage conditions of the overhead contact lines. Therefore, powering and braking trains are paired in blocks delimited by distance and time as shown in Fig. 5. In a DC feeding system, the regenerative power interchange generally occurs only between a braking train and its nearby powering train. We divided the area by substations (“substation block,” or distance separation, as shown in Fig. 5), and pair trains between stations in the same substation block. In addition, we limit pairings of trains between stations to those braking and powering within a certain period of time (time separation as shown in Fig. 5).

Fig. 5　Candidate pair for timing alignment between powering and braking trains

The running time and stopping time of each train at and section between stations are determined with the optimization calculation to maximize the sum of the Regenerating-powering overlap indexes. The formulation of this effective use of regenerative power algorithm is described in Chapter 3.

Note that the speed profile changes as the running time is extended, and the powering and braking times also change. It is necessary to prepare powering and braking time data for each extended running time.

2.5 Compatibility between reducing powering energy and effective use of regenerative power

Figure 6 shows an example of the effect of adjustments to make effective use of regenerative power during powering. As shown in Fig. 6, the arrival and departure times of train 3 is adjustable and the arrival and departure times of trains 1 and 2 are fixed. In order to increase the sum of the Regenerating-powering overlap indexes in Fig. 6(a), the running time of train 3 from station A to station B is shortened as shown in Fig. 6(b). As a result, the sum of the Regenerating-powering overlap indexes will increase, and so will powering energy. Therefore, it is necessary to make the powering energy reduction algorithm and effective use of regenerative power algorithm as compatible as possible.

Fig. 6　Difficulty of balancing between reducing powering energy and effective use of regenerative power

We introduce a weight parameter, W [kWh/s], to adjust the balance between reducing powering energy and effective use of regenerative energy. The weight parameter, W, represents the expected value of the regenerative energy per second of the Regenerating-powering overlap index, and the value obtained by the sum of the Regenerating-powering overlap indexes multiplied by the weight parameter is considered to have a positive correlation with the regenerative energy. So, based on the algorithms proposed in Sections 2.3 and 2.4, the objective function was set to maximize the sum of the total reduction in powering energy and the total Regenerating-powering overlap indexes multiplied by W, to balance reducing powering energy and effective use of regenerative power. If W increases, increasing the total Regenerating-powering overlap indexes will be preferred over total reduction in powering energy. It is possible to maximize energy efficiency by adjusting W to balance reduction in powering energy and effective use of regenerative power.

3. Formulation as a mathematical optimization problem

The algorithms described in sections 2.3 and 2.4 are combined and formulated as a mathematical optimization problem to allow balancing between powering energy reduction and effective use of regenerative power. This chapter describes the formulation with details of the objective function and an overview of the constraints. Table 1, 2 and 3 show the set, constant, variable notation for the formulation. Based on these formulations, calculations were performed using the mathematical optimization solver Gurobi Optimizer (ver. 9.5.1).

Table 1　Set notation

Set	Explanation
Aadjust	A set of trains of adjusting the running time and stopping time (Represent the element as aadjust)
Sadjust (aadjust)	A set of adjusting sections between stations of adjusting train aadjust (Represent the element as sadjust)
D(aadjust, sadjust)	A set of candidates for the running extension time of sections between stations sadjust for train aadjust (Reperesent the element as ∆d) The extension time is a multiple of the unit time for creating the timetable
M(aadjust, sadjust)	A set of candidates for the stopping extension time of sections between stations sadjust for train aadjust (Reperesent the element as ∆m) The extension time is a multiple of the unit time for creating the timetable
Bpower (aadjust, sadjust)	A set of powering trains which considers the Regenerating-powering overlap index when train aadjust is braking between stations sadjust (Represent the element as bpower)
Spower (aadjust, sadjust, bpower)	A set of sections between stations of train bpower∈Bpower (aadjust, sadjust) which considers the Regenerating-powering overlap index when train aadjust is braking between stations sadjust (Represent the element as spower)

Table 2　Constant notation

Set	Explanation
∆e(aadjust, sadjust, ∆d)	The reduction of the powering energy [kWh] when the running time of section between stations sadjust of train aadjust is extended by ∆d

Table 3　Variable notation

Variable	Explanation
x(aadjust, sadjust, ∆d)	A variable that is 1 if the running extension time of train aadjust between stations sadjust is ∆d, and 0 otherwise
y(aadjust, sadjust, ∆m)	A variable that is 1 if the stopping extension time of train aadjust between stations sadjust is ∆m, and 0 otherwise
z(aadjust, sadjust, bpower, spower)	A non-negative variable representing the Regenerating-powering overlap index for braking train aadjust between stations sadjust and powering train bpower between stations spower
W	The weight parameter [kWh/s] for adjusting the balance between reducing powering energy and effective use of regenerative power

Objective function is maximizing the sum of the total powering energy reduction and the total Regenerating-powering overlap indexes multiplied by W, as represented by (2).

The following constraints were considered.

(a) Redistribute the suspension time of the basic rescheduled timetable to the running time and stopping time of the energy-efficient rescheduled timetable.

(b) The departure time of the energy-efficient rescheduled timetable should not be earlier than the planning timetable.

4. A case study using actual railway line data

We report the results of a case study applying the method to actual railway line data to create an energy-efficient rescheduled timetable, as described in Chapters 2 and 3.

  

$$\mathrm{Max}{\sum }_{a^{\mathit{adjust}}}{\sum }_{s^{\mathit{adjust}}}{\sum }_{\mathrm{∆}d}\left(∆e\left(a^{\mathit{adjust}},s^{\mathit{adjust}},∆d\right)\times x(a^{\mathit{adjust}},s^{\mathit{adjust}},∆d)\right)+W\times {\sum }_{a^{\mathit{adjust}}}{\sum }_{s^{\mathit{adjust}}}{\sum }_{b^{\mathit{power}}}{\sum }_{s^{\mathit{power}}}z\left(a^{\mathit{adjust}},s^{\mathit{adjust}},b^{\mathit{power}},s^{\mathit{power}}\right)$$(2)

4.1 Calculation conditions

The target railway line has 38 stations and is approximately 69 km long. The target train timetable covers one hour during an evening rush hour with 62 trains and 612 trips between stations. Auxiliary power is set according to the ambient temperature of equivalent to 15℃. The planning timetable of the case study is shown in Fig. 7(a). Based on the actual data of past delays, the initial delay of the first train was set to 290 seconds and is indicated by the red line in Fig. 7(a). Figure 7 shows only the area that includes the stations to be adjusted due to the delay.

Fig. 7　Created energy-efficient timetable rescheduling

Figure 7(b) shows the basic timetable rescheduling created by calculating the delay propagation for each train, specifying the required headway time, minimum stopping time at each station and the running operation at the planned running time, and assuming that trains have extended stopping time at stations. The trains whose stopping time at stations are extended are indicated by the bold lines in Fig. 7(b). The four trains that preceded the first delayed train were assumed to have had extended stopping time at stations to prevent spikes in passenger numbers, and the three trains that followed the first delayed train were assumed to have had extended stopping time at stations to avoid trains slowing down or stopping between stations. By adjusting the 7 trains and 50 trips between stations that had extended stopping time, an energy-efficient rescheduled timetable was created. The suspension time for each train ranged from 30 to 195 seconds, and the total suspension time for all trains was 745 seconds.

The data on the relationship between the extended running time and the reduction of powering energy for each train and section between stations (WT plot) used in the powering energy reduction algorithm is prepared using the Train Operation Power Simulator [8]. The data on the powering and braking time of the extended running time, used in the effective use of regenerative power algorithm is also the same.

We describe the calculation conditions for creating an energy-efficient rescheduled timetable. Candidate extended running times (∆d in chapter 3) and candidate extended stopping times (∆m in chapter 3) are set to 5, 10, 15, ..., 40 seconds for all trains. The weight parameter W for adjusting the balance between powering energy reduction and effective use of regenerative power was varied to 0, 0.05, 0.1, 0.15, 0.2, 0.25 kWh/s.

4.2 The energy-efficient timetable rescheduling created by the proposed method

The energy-efficient timetable rescheduling was created using the calculation conditions described in section 4.1. To confirm the running extension time by the powering energy reduction algorithm, the energy-efficient timetable rescheduling created with W set to 0 is shown in Fig. 7(c). It can be seen that the suspension time in the basic timetable rescheduling as shown in Fig. 7(b) is allocated to running extension time in the energy-efficient timetable rescheduling as shown in Fig. 7(c) as indicated by the red dotted circles, which can be seen more clearly in the enlarged images in the lower right-hand part of the figures.

4.3 Estimation of the energy-saving effect by simulation

The basic timetable rescheduling and energy-efficient timetable rescheduling were simulated with the Train Operation Power Simulator [8], and various kinds of energies were estimated for one hour (from 18:30 to 19:30 as shown in Fig. 7).

First, the speed profiles of train G with W set to 0 is shown in Fig. 8, comparing the basic timetable rescheduling (dotted line) and the energy-efficient timetable rescheduling (solid line). Figure 8 shows that the total time does not change, and that the powering energy is efficiently reduced by reducing the powering time and the maximum speed.

Fig. 8　Comparison of the train performance curve of train G(W = 0)

Next, the powering and regenerative energy of adjusted trains and between stations (7 trains and 50 between stations) for each W is shown in Fig. 9. Increasing the value of W means that increasing regenerative energy is preferred over decreasing the powering energy. It is confirmed that as the value of W increases, regenerative energy increases and powering energy also increases, as shown Fig. 9. In other words, by adjusting the weight parameter W, the balance between reducing powering energy and effective use of regenerative power can be adjusted.

Fig. 9　Comparison of powering and regenerative energies

In addition, the energy-saving effect for each W calculated based on the energy consumption of the entire one-hour timetable is shown in Fig. 10. Those energy-saving effects were calculated by comparing this with the energy consumption of the basic timetable rescheduling. Under the calculation conditions of this case study, there is almost no difference between the energy-saving effects due to the value of W, the maximum and minimum energy-saving effects were 2.2% and 1.9%, respectively.

Fig. 10　Energy-saving effect

Table 4 shows the calculation time for creating the energy-efficient timetable rescheduling. The calculation time was within the target of 1 minute when W was set in the range of 0 to 0.1 kWh/s.

Table 4　Calculation time taken to create the energy-efficient timetable rescheduling

W[kWh/s]	0	0.05	0.1	0.15	0.2	0.25
Calculation time [s]	0.1	13.0	59.9	101.2	459.8	479.7

5. Conclusion

In this study, we proposed a method for timetable rescheduling that can achieve energy-saving effects in the event of small-scale delays lasting a few minutes, creating a need for trains to have extended stopping time at stations. By adjusting the arrival and departure times of each train at stations, we achieved a balance between reducing powering energy and effective use of regenerative power, while taking passenger convenience into consideration.

As a result of a case study, it was confirmed that an energy-efficient timetable rescheduling could be created within one minute and that an energy-saving effect of up to 2.2% could be achieved.

In the future, we plan to apply the developed method to timetable planning and to create an energy-efficient timetable that balances the reduction of powering energy and the effective use of regenerative power.

Acknowledgment

This study is financially supported in part by the Japanese Ministry of Land, Infrastructure, Transport and Tourism.

References

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Authors

	Aiko KUNISAKI Assistant Senior Researcher, Transport Operation Systems Laboratory, Signalling and Operation Systems Technology Division Research Areas: Energy-efficient of Transport Operation, Mathematical Optimization
	Yoko TAKEUCHI Senior Chief Researcher, Head of Transport Operation Systems Laboratory, Signalling and Operation Systems Technology Division Research Areas: Train Operation Simulation, Energy Simulation and Control, Mathematical Optimization