Maximization of Energy Efficiency by Synchronizing the Speed of Trains on a Moving Block System

Abstract

The publication deals with the problem of the effect of interference with the movement of the “preceding” train on the movement of the “following” train in the case of efforts to reduce the distance between trains on a moving block interval. The paper presents results of the simulations for train of the ED250 type for a set of 135 traffic situations: for three contexts of reducing the speed of the “preceding” train, for five variants of the method of controlling the “following” train and for nine initial distances between these trains. The results confirmed the possibility of gaining time and energy benefits by implementing an appropriate method of controlling the “following” train, as well as providing insights into the area of shortening the gaps between trains, including the pursuit of synchronizing the speed of trains and possibly coupling them into a so-called virtual couples.

1. Introduction

Background of the Issue

Modern railroad traffic control systems have not exhausted all possible improvements and optimizations from the point of view of increasing the capacity of the railroad line, improving the fluidity of train traffic, and minimizing losses of time and energy efficiency.

Based on the technical documentation, i.e., the system requirement specifications for ERTMS/ETCS [1], solutions are applied to fundamentally guarantee the safety of train traffic, including the establishment of the minimum permissible distances between trains.

ERTMS/ETCS Level 3 introduces the concept of operating trains with moving block distances, in which all trackside equipment is removed and trains are separated by an absolute braking distance. The movement of trains conducted according to the moving block distance [2] consists of determining the free path for the “following” train to the place of the last reporting of the identified end of the “preceding” train. On the basis of the dynamic speed profile, taking into account the train braking curve, the selection of the maximum permissible, at a given moment, speed of the “following” train is determined, taking into account the distances to the end of the free path [3].

However, such a speed profile of the “following” train is not necessarily the most advantageous from the perspective of travel time and energy consumption. In the event of an unforeseen disturbance to the “preceding” train, such as a brief reduction in speed, the preservation of the maximum speed by the “following” train may adversely affect the conservation of the train’s kinetic energy and thus the maintenance of the scheduled running time.

2. Related Work

Consideration of aspects of the benefits of appropriate train control of the “following” train is the subject of numerous studies and publications. It should be noted that these benefits are multidimensional.

The authors of the report [4,5] present the concept of virtual coupling as a solution to allow trains to congest the line and increase its capacity. This solution is also considered by the authors of [6], where requirements on tolerated latency associated with the channels used for train-to-trackside and train-to-train communications are indicated. Key parameters identified include the need to equalize the speed between trains and to establish the distance between them (the time needed for coupling). It was also pointed out that an important aspect of applying a technical solution is to carry out studies on the possible benefits of increasing train throughput and punctuality, taking into account the dynamics of moving trains and scenarios of disturbed operations.

The study in [7] also takes into account Virtual Coupling and analyzes the harmony between the rail transit periodic operation characteristics or system constraints and Virtual Coupling technology. It proposes a cooperative speed and distance adjustment strategy that satisfies the virtual coupling condition.

New technologies, such as Virtual Coupling, carry many risks that theoretical considerations may not show. Therefore, the authors of [8] point to the need to prepare appropriate test benches in laboratory conditions, which can identify problems in real systems.

An interesting study providing an overview of train speed profiling solutions for optimizing energy consumption is [9]. The authors indicate that the energy efficiency of running a single train can only be improved in terms of coasting, as previously studied in [10]. On the other hand, the authors of [11] point to traffic fluidity as one criterion that is relevant for the evaluation of the quality of rail transport services and directly translates into the energy efficiency of train running. Another possibility for increasing the energy efficiency of train running is reported in [12]. A strategy where the energy obtained by recuperation is immediately used by other trains is considered. The authors of [13] investigate the energy-efficient train timetabling and rolling stock circulation planning problem for metro lines based on flexible selections of train operation levels, which involve running levels (various running times and speed profiles) and dwell levels (various dwell times at stations). In [14], the focus is on a methodology for energy-efficient train operation involving Q-Learning-based driving profile shaping. The basic idea of the Energy Distribution Based Method (EDBM), which transforms the eco-driving problem into a finite Markovian decision process, is presented. The topics of the article [15] are outside the railway industry but deal with the study of the eco-driving problem for hybrid electric vehicles (PHEVs) to minimize the total cost of energy consumption, which seems to coincide with railway issues. In [16], the authors deal with a speed trajectory optimizer for a partial eco-driving optimization problem. Pontryagin’s maximum principle is used to derive the necessary conditions for the optimal solution. Further issues on eco-driving can be found in [17,18,19,20,21,22,23]. As is clear, these topics are popular, and new ideas for improving energy efficiency in this way are constantly emerging.

Currently, further automation and digitization of railway solutions is being pursued [24,25,26], allowing more information to be transmitted [27] and processed into key information for more optimal train control.

In the late 1990s, Transrail started a research project to develop a C-DAS system, the first large-scale tests of which, called Computer-Aided Train Operation (Cato), were carried out with mining operator LKAB on heavy haulage lines in northern Sweden in 2010; Transrail launched the technology in 2015.

Cato-Train calculates the optimum speed profile required for a train to reach its destination, within a specified time, based on the train’s traction characteristics, timetable and the driver’s adherence to previously given recommendations.

The optimal speed profile, displayed as a pointer on the traction vehicle driver’s dashboard (DMI), can be determined based on five advisory strategies:

• application of the minimum braking,
• application of the “experienced driver” profile,
• keeping a constant minimum speed to a target with a specified driving time,
• keeping a constant speed and driving without the application of propulsion to a target with a specific driving time,
• optimal speed profile, taking into account track profile, speed limits and time to destination.

Publication [28] collates information on the driver assistance systems used, including variants of the system architecture. Variants of locating the data processing unit on the train or in the Traffic Control Centre (TCC) are considered.

The basis for determining the optimal speed profile is the following data: geography of the network and location of the time points (timetable), gradients of the track gradients, permissible train running speed, train characteristics including traction and resistance characteristics of a given train set, semaphore indications, location and speed of a given train.

In the case of a system located in a TCC, there is a possibility of detecting traffic conflicts and calculating new target train running times to avoid these conflicts.

This paper is a continuation of the authors’ work [29] on the issue of energy efficiency and train smoothness on a railway line with fixed block distance. In this publication, the conclusions of a simulation of train movement in terms of controlling the “following” train on the basis of augmented speed information of the “preceding” train are presented. As noted, the results were strongly influenced by the adopted traffic contexts of the “preceding” train and the moment (distance between trains) when the control of the “following” train was initiated.

The application of augmented speed and location information of the preceding train to the speed profile selection algorithm of the following train in terms of energy consumption is an innovative aspect of the work.

The present publication, on the other hand, deals with the issue of train movement but on a line with moving block distance (e.g., as a result of the ERTMS/ETCS L3 system), where it is crucial to achieve higher throughput than is possible on a railway line equipped with fixed block distance.

The use of moving spacing is intended to reduce the distance between trains, but this can cause some risks and problems. If the movement of the “preceding” train is disturbed, e.g., by a brief permanent reduction in its speed, the smoothness of movement of the “following” train may be impaired due to approaching the “preceding” train too quickly. On the other hand, the approach of the trains at a distance of the braking distance implies the necessity to implement braking which, contrary to expectations, may result in the extension of the distance between the trains and a significant increase in the energy consumption of the “following” train due to excessive loss of kinetic energy.

The object of the authors’ work is therefore to analyze the possibility of reducing the impact of the primary disturbance from the “preceding” train to the “following” train. The proposed idea is therefore based on an early selection of the speed of the “following” train—identification of the key distance between trains at the point where the benefits of controlling the “following” train can be obtained.

The application of appropriate control of the movement of the “following” train, including the benefits of minimal mechanical energy consumption and the smoothness of its movement, is also considered in the work from the point of view of creating a convoy of trains, and the so-called application of virtual coupling between trains.

3. Methodology

3.1. Abbreviations Used in the Publication

In the remainder of the paper, the following abbreviations are used to simplify the description:

• P_P—”preceding” train,
• P_N—“following” train,
• P_W—“model” train (P_N moving in undisturbed traffic),
• V_P—the speed of the “preceding” train,
• V_N—the speed of the “following” train,
• D₀—initial distance (at the beginning of the simulation) between trains,
• T—simulation duration.

3.2. Movement Situations

Traffic situations refer to the movement of two consecutive ETR 610 (ED250 series) trains, moving on the same straight railroad track (with zero gradient) for a given initial distance between each other D₀, with an initial maximum speed: V_max = 200 km/h.

It is assumed that the movement of P_P is disturbed, as a result of which its speed is reduced from the maximum value to the limitation speed V_ogr (braking a_h = −0.5 m/s²); then, after a 10 s constant limitation speed (shutdown of the braking system of about 3–6 s, control actions of about 5 s [30]), the maximum allowable acceleration is activated to regain the maximum speed, which occurs at t = T.

The railroad line is equipped with a traffic control system that allows trains to operate at the distance of the road, on the basis of maintaining a moving block distance.

Separation of trains at the distance of the road is to provide between trains such a length of clear path so that it is possible to brake the head of the train following the “following” train before the dangerous place, i.e., the end of the “preceding” train.

In order to ensure safety, the system controlling the movement of trains at the distance of the road must have information on the position of the train, which is then processed by the dependency system and influences information on further driving conditions, transmitted directly to the driver’s console in the form of signal images (applies, among others, to vehicles equipped with ETCS).

Conducting traffic in the so-called moving block distance is based on maintaining an appropriate distance between trains (moving road spacing), where the boundaries of this block spacing change depending on the current traffic situation on the controlled section of road, and the distance between trains is a variable value, which is the result of a set of current information on the position of individual trains and their speed.

In practice, moving block distance is the equivalent of running a train on what is known as electric visibility, where the train moves in the braking distance based on continuous, up-to-the-minute information provided in the form of a movement authority. In the case of ERTMS/ETCS L3 [31], fixed reference points are realized by balises located in the track, which allows verification of the train’s position as a function of time, while the measurement of the distance traveled over time is determined by odometer calculations on board the traction vehicle.

In the simulation, the change in the distance between the trains is determined by the initial value of D₀ and, in subsequent moments of time, the difference in the distance traveled by the two trains. To denote the changing distance, the following notation is used: D(t), which varies during the simulation in the range of t: 0 < t < T.

The following evaluation criteria were adopted in the simulation for traffic situations:

• maintaining a safe braking distance for P_N,
• minimizing driving time for P_N,
• minimizing mechanical energy consumption for P_N.

3.3. Simulation

The software and simulations were realized in SCILAB [32].

The simulation is implemented for 3 traffic contexts—different constraint speeds P_P, for 9 different initial distances between trains—D₀ = 1000, 1500, 2000, 2500, 3000, 3500, 4000, 4500, 5000 (m) at which distance, the control P_N is implemented for 5 different variants of the method.

It is worth noting that Variant 5 was added to the simulation on the basis of the analysis of the results obtained for Variants 3 and 4.

An exemplary and illustrative description of the model for the purpose of presentation is presented in Figure 1 (below).

Figure 1. An overview diagram of the traffic model (source: own development).

The simulation continues from the moment t = 0, when P_N is point A, and P_P starts braking at point O, until the moment t = T, when P_P obtains again the maximum speed—depending on the context of movement—at the points B1, B2, B3.

The “following” train at time t = T has speed V(T) and is at distance S(T) from point A, and at distance D(T) from end point P_P.

The parameters of P_N, S(T)—the distance traveled, V(T)—the speed of the train, D(T)—the distance to the preceding train, and the mechanical energy used, Z(T), are the key values for evaluating the control variants used.

The possibility of using appropriate speed control P_N allows the train speed to be corrected and adjusted in advance, taking into account the changed traffic conditions on the section of the railroad line ahead of this train.

3.3.1. Contexts

The behavior of P_P is determined by the traffic situation, as well as the interaction of the traction vehicle’s systems with those of the trackside equipment.

For the purpose of the study, it is necessary to extract a few characteristic situations from the entire area of possible traffic scenarios, which allow the evaluation of solutions in the broadest possible perspective (speed changes P_P).

In this work, a disturbance in the movement of P_P, of a short-term nature, causes a reduction in the speed of P_P; then, with a 10 s time delay for the switching of the braking system to the acceleration system, the recovery of maximum speed is considered. The time T after which the recovery of the maximum speed of P_P will occur is the moment of termination of the simulation and depends on the limiting speed P_P adopted in the given traffic context: V_P = V_ogr.

For the three traffic contexts, three speed limits P_P were adopted: context 1 V_ogr = 160 km/h, context 2 V_ogr = 100 km/h, context 3 V_ogr = 40 km/h.

• Context 1 represents a speed reduction of 20% of the maximum speed and can refer to situations of temporary signal loss in the ETCS system.
• Context 2 assumes a value of 50% of the maximum speed and may result from the occurrence of a speed restriction based on speed grading in the automatic linear interlocking system.
• Context 3 represents a speed reduction of 80% of the maximum speed and can represent the situation of transition in ETCS between SR and FS modes.

Contexts are also based on the assumption that trains following each other on a section of the railroad line, traveling at maximum speed, are subject to the study: V_max = 200 km/h. On the other hand, the adopted way of conducting train traffic is not disturbed by predetermined traffic operations, e.g., stopping a passenger train at the platform, passing a train through a group of tracks (turnouts) or changing the order of trains on a section of the railroad line. This is because such traffic situations should be taken into account in the train running plan with an appropriate time reserve, and the adoption of the situation of stopping P_P does not allow optimization of the control of P_N without knowledge about the required time of stopping P_P.

The contexts are based on real operating situations of applied speed restrictions, i.e., speed reduction to 160 km/h when ETCS is switched off and the train is running without system supervision; speed reduction to 100 km/h when a train passes a junction; speed reduction of 40 km/h when ETCS operation is changed from “Full Supervision” to “Staff Responsible”.

As mentioned above, contexts have been assumed as contingencies, with a short-term effect and a temporary reduction in speed. This is because the occurrence of a disruption to P_P traffic should generate minimal secondary disruptions to the next train, especially when there are moving block distances between trains and an effort to minimize the distance between them.

3.3.2. Variants

P_N control was implemented for 5 variants, where variant 1 is a variant of the current train operation.

The options are based on the possible ways of transmitting and using train and infrastructure information managed in the ERTMS control system.

• Variant 1

In variant 1, the speed of P_N is limited by the allowable distance to the end of P_P, resulting from the maintenance of a safe stopping distance. In the absence of a speed limit, P_N moves or accelerates to the maximum speed: V_P = V_max.

P_N control is based on the indications of the control system on the driver’s desktop (in the ETCS-DMI system) and in addition, the driver’s observation of the signaling on the route; the minimum required visibility of semaphore indications from a distance of 500 m has been established.

If the need to reduce speed arises, the driver implements “service” braking with a_h = 0.5 m/s² for the situation of exceeding the safe speed (resulting from the distance between trains) and has “emergency” braking applied with a_H = 1.5 m/s².

The model adopted the following gradations of braking (Table 1):

Table 1. Graduation of braking force.

Distance to Stopping Point (m)	Train Speed (km/h)	Deceleration Delay (m/s2)
1800	V > 160	−0.5
1500	V > 140	−1.0
1200	V > 100	−1.0
900	V > 60	−1.0
100	V > 0	−1.5

The maximum distance between trains when P_N variant 1 is forced to implement braking (speed reduction) will be referred to as the limit distance and denoted as D_gr.

• Variant 2

The speed and location (end of train) information P_P is sent to P_N. Since the information undergoes a delay from the current moment as a result of transmission and processing, the model assumes the use of P_P information with a delay of T_sp = 10 s.

P_N control is based on the comparison of speeds between trains: when V_N > V_P, the driver shuts down the drive, and the train moves by momentum, thus reducing traction energy consumption.

If V_N ≤ V_P, driving at maximum speed continues.

Maintaining a safe distance between trains is implemented as in variant 1, i.e., excessive approach of P_N to the end of P_P results in the implementation of braking according to the gradation of braking force as in Table 1.

• Variant 3

Information about the speed and location (end of train) of P_P (T_sp as in variant 2) is sent to P_N.

P_N control involves synchronizing the speed between trains, i.e., based on the speed information P_P, the required maximum allowable delay or acceleration P_N needed to equalize the speed of the two trains is recalculated.

Maintaining a safe distance between trains is implemented as in variant 1, i.e., excessive approach of P_N to the end of P_P results in the implementation of braking according to the gradation of braking force as in Table 1.

• Variant 4

Information about the speed and location (end of train) of P_P (T_sp as in variant 2) is sent to P_N.

P_N control involves comparing the speed and distance D(t) between the trains. Based on this information, the required maximum permissible delay is recalculated as ah, in order to obtain the speed of the two trains aligned over a certain distance.

For V_N > V_P, the braking force is calculated from the amount of energy required to change the kinetic energy P_N according to the following relation:

$$m{a}_{h}D=m\frac{V_P^2}{2}-m\frac{V_N^2}{2}$$

(1)

$$a_h=\frac{V_P^2-V_N^2}{2D}$$

(2)

In the case of V_N ≤ V_P, driving at maximum speed continues.

Maintaining a safe distance between trains is implemented as in variant 1, i.e., excessive approach of P_N to the end of P_P results in the implementation of braking according to the gradation of braking force as in Table 1.

• Variant 5

Information about the speed and location (end of train) of P_P (Tsp as in variant 2) is sent to P_N.

The control of P_N is identical to variant 4, the difference is a different way of determining the braking force, taking into account the resistance to movement.

For V_N > V_P, the determined path is based on the following relationship:

$$S_h= m_f\underset{V_N}{\overset{V_P}{{\displaystyle \int }}}\frac{V}{\left(k_1{V}^2+k_{2}V+k_3+a_h\right)} dV$$

(3)

where mf—rotating mass of the train, V_N—speed P_N, V_P—speed P_P, k₁—coefficient in the equation of resistance to motion, depending on the square of the speed, k₂—coefficient in the equation of resistance to motion, depending on the speed, k₃—coefficient in the equation of resistance to motion, independent of speed.

Selection of the best value of ah is made for the criterion: (D − Sh) → 0.

The recalculation of values is repeated at a cycle of T_c = 10 s.

In the case of V_N ≤ V_P, driving at maximum speed continues.

Maintaining a safe distance between trains is implemented as in variant 1, i.e., excessive approach of P_N to the end of P_P results in the implementation of braking according to the gradation of braking force as in Table 1.

The comparison of results from each variant refers to the passage of the “reference” train (undisturbed movement of the “following” train), normalizing the parameters of speed loss and travel time extension in each variant (the relationship shown in Section 3.4). The calculated mechanical energy consumption is due to the need to recover the lost speed, distance and time between P_N and P_W.

3.4. Input Data

The model is based on the characteristics established for the ETR610 type ED250.

The train model was described, with traction characteristics based on the parameters given in the publication [33].

For the acceleration of a train on a straight and level track, with a normal load (weight of 427 tons) and 100% of the available traction power, the following accelerations are used:

• ar = 0.49 m/s²—average acceleration from 0 km/h to 40 km/h,
• ar = 0.49 m/s²—average acceleration from 0 km/h to 120 km/h,
• ar = 0.36 m/s²—average acceleration from 0 km/h to 160 km/h,
• ar = 0.07 m/s²—residual acceleration at 250 km/h.

The traffic resistance was determined based on the parameters specified for the trainsets in the publications [34,35] and is described by the characteristics: F = 8V² + 130V + 4000 (N) for speeds specified in m/s.

The 187.4-m-long ETR610 train has a weight of 427 tons, converted to a weight that includes rotating components, is 452 tons.

The model assumed the characteristics of the ETR610 train (type ED250) for acceleration from 0 km/h to 200 km/h, braking from 200 km/h to 0 km/h in service mode (−0.5 m/s²) or in emergency mode (−1.5 m/s²), and braking due to the effect of traffic resistance only (section on zero gradient).

The consumption of mechanical energy is calculated as a fractional sum of the energy required to overcome the resistance to movement; if the speed is increased, in addition, the difference in kinetic energy within the limits of the speed change is considered.

For the energy required to overcome the resistance to movement, the drag force was determined for the average speed over a one-second time interval and the distance traveled in that time:

$$Z_{opr}={\displaystyle \sum }_{i=1}^{T_n}{F}_{\text{ś}r}\Delta S_i$$

(4)

where i is the number of consecutive 1 s time slices such that i: {i = 1, …, T_n}, and T_n is the duration of the analyzed change.

For accelerated motion, the kinetic energy consumption in time T from speed V₁ to speed V₂ is:

$$Z_{\Delta \mathrm{V}}=\underset{V1}{\overset{V2}{{\displaystyle \int }}}{m}_{f}VdV$$

(5)

While the total mechanical energy consumption for accelerated motion is:

$$Z_m=Z_{opr}+Z_{\Delta V}$$

(6)

Below, Figure 2, Figure 3, Figure 4 and Figure 5 show the simulated running of the ETR610 train for the maximum allowable acceleration of the train and different ways of braking from maximum speed to stop.

Figure 2. Acceleration characteristics of the ETR610 (ED250) train.

Figure 3. Emergency braking characteristics of the ETR610 (ED250) train.

Figure 4. Braking characteristics of the ETR610 (ED250) service train.

Figure 5. Braking characteristics due to resistance to motion of the ETR610 (ED250) train.

The presented traction characteristics of acceleration and braking were compared and confirmed with the actual values obtained with an ETR610 train of the ED250 series on railway line No. 9 Warsaw–Gdansk and railway line No. 4 Warsaw–Krakow.

3.5. Verification of Results

Based on the second-by-second changes in speed (acceleration/deceleration) obtained in the simulation, the data and final results were verified in Excel. In one-second successive steps, changes were calculated in distance, speed and mechanical energy consumption. The final results obtained were related to the values obtained in the simulation.

Verification was done for several cases of the obtained runs of P_N.

3.6. Results and Their Normalization

The simulation results in different magnitudes of P_N traffic parameters, i.e., train speed, distance traveled, and mechanical energy consumed. In order to compare the results obtained for different variants of running P_N, normalization was applied, which consists of converting the lost speed, distance and extended running time of P_N against P_W.

We consider the “reference” train to be the hypothetical movement of the “following” train, without the occurrence of interference and restrictions on the speed of its movement, which will travel from point A to point B during simulation T (Figure 6), using mechanical energy only to overcome the resistance to movement at a constant speed of travel V = V_max.

Figure 6. Normalization of simulation results.

The considered movement P_N (Figure 6) from point A to point O travels a route in simulation time T. In contrast, the compared quantities of movement parameters P_N are considered at point O.

The standardization process consists of the following:

• Aligning the speed of the “following” train with the “reference” train.
• The required time T_O-s1 and mechanical energy Z(∆V) necessary for P_N to regain maximum speed V = V_max (section O–s1 in Figure 6) are determined.
• Alignment of the covered distance of the “following” train with the “model” train.
• The necessary time T_s1–B and the mechanical energy Z(∆S) needed to travel the missing distance to where P_W was at time t = T (section s1–B in Figure 6) are determined.
• Elimination of the resulting delay time of the “following” train to the “model” train.
• The mechanical energy Z(∆T) required for the so-called “catch-up” P_W (section B–C in Figure 6) is determined, which is possible by hypothetically raising the speed above a fixed maximum speed: $V_{\mathscr{H}}$ > V_max.

Driving time and mechanical energy consumption P_W on section A–B (simulation duration):

$$T{w}_{A–B}=\frac{S_{AB}}{V_{max}}=T$$

(7)

$$Z{w}_{A–B}=F_{opr,v}\ast S_{AB}$$

(8)

Driving time and mechanical energy consumption PW at B–C:

$$T{w}_{B–C}=\frac{S_{BC}}{V_{max}}$$

(9)

$$Z{w}_{B–C}=F_{opr,v}\ast S_{BC}$$

(10)

Normalization of driving time and mechanical energy consumption P_N from simulation (section A–O):

$$T_{A–O}=T_{symul}$$

(11)

$$Z_{symul}={\displaystyle \sum }_i^{A–O}{F}_{opr,v}\ast \Delta S_i$$

(12)

Normalization of driving time and mechanical energy consumption of the “following” train on the O–s1 section:

$$T_{O–s1}=\underset{Vogr}{\overset{Vmax}{{\displaystyle \int }}}\frac{m_f}{\left(F_{tr,v}+F_{opr,v}\right)}dV$$

(13)

$$Z\left(\Delta V\right)={\displaystyle \sum }_i^{O–s1}{F}_{opr,v}\ast \Delta S_i+\underset{Vogr}{\overset{Vmax}{{\displaystyle \int }}}{m}_{f}VdV$$

(14)

where F_opr,v—train speed-dependent drag force; F_tr,v—train speed-dependent traction force.

Normalization of driving time and mechanical energy consumption of the “following” train on the s1–B section:

$$T_{s1–B}=\frac{S_{s1–B}}{V_{max}}$$

(15)

$$Z\left(\Delta S\right)=F_{opr,v}\ast S_{s1–B}$$

(16)

Normalization of driving time and mechanical energy consumption of the “following” train on section B–C:

$$T_{B–C}=\underset{Vmax}{\overset{V_{\mathscr{H}}}{{\displaystyle \int }}}\frac{m_f}{\left(F_{tr,v}+F_{opr,v}\right)}dV+\underset{V_{\mathscr{H}}}{\overset{Vmax}{{\displaystyle \int }}}\frac{m_f}{\left(F_{opr,v}+F_h\right)}dV$$

(17)

where F_h—braking force.

$$Z\left(\Delta T\right)={\displaystyle \sum }_i^{\left(B–C\right)/2}{F}_{opr,v}\ast \Delta S_i+\underset{Vmax}{\overset{V_{\mathscr{H}}}{{\displaystyle \int }}}{m}_{f}VdV$$

(18)

Loss of time:

$$uT=T_{symul}+T_{O–s1}+T_{s1–B}-T_{symul}=T_{O–s1}+T_{s1–B}=T{w}_{B–C}-T_{B–C}$$

(19)

Loss of mechanical energy:

$$u{Z}_m=Z_{symul}+Z\left(\Delta V\right)+Z\left(\Delta S\right)+Z\left(\Delta T\right)-Z{w}_{A–B}-Z{w}_{B–C}$$

(20)

Algorithm for Converting Time Loss into Mechanical Energy

Assumption: identical acceleration and deceleration P_N on B–C: ar = ah = a, P_N and P_W travel S_N = S_W = S on B–C with time difference ∆T = T_w − T_n:

$$S=V_{max}{T}_w=V_{max}\frac{T_n}{2}+\frac{a{\left(\frac{T_n}{2}\right)}^2}{2}+V_{\mathscr{H}}\frac{T_n}{2}-\frac{a{\left(\frac{T_n}{2}\right)}^2}{2}=\left(V_{max}+V_{\mathscr{H}}\right)\frac{T_n}{2}$$

(21)

because

$$T_n=T_w-\Delta T=\frac{S}{V_{max}}-\Delta T$$

(22)

After transformations, we obtain:

$$V_{\mathscr{H}}=\frac{\left(S+\Delta T{V}_{max}\right)}{\left(\frac{s}{V_{max}}-\Delta T\right)}$$

(23)

because the change in kinetic energy is equivalent to the action of a force along a given path:

$$F\frac{S}{2}=\frac{m{V}_{\mathscr{H}}{}^2}{2}-\frac{m{V}_{max}{}^2}{2}$$

(24)

can be written as

$$S=\frac{V_{\mathscr{H}}{}^2-V_{max}{}^2}{a}$$

(25)

After substituting (19) into (17) and transformations, we obtain the dependence of $V_{\mathscr{H}}$ on a, ∆T and V_max:

$$V_{\mathscr{H}}{}^3-V_{\mathscr{H}}{}^2{V}_{max}-V_{max}{}^2{V}_{\mathscr{H}}+V_{max}{}^3-a\Delta T\left(V_{max}{}^2+V_{max}{V}_{\mathscr{H}}\right)=0$$

(26)

The consumption of mechanical energy is determined by the change in kinetic energy of the train and the balancing energy of the drag force. The drag force is averaged for the speed of the arithmetic mean of $V_{\mathscr{H}}$ and V_max and is assumed to act on half of the section B–C, where traction force is applied to change the speed from V_max to $V_{\mathscr{H}}$.

$$Z\left(\Delta T\right)=\frac{\left(F_{opr, \mathscr{H}}+F_{opr,max}\right)}{2}\frac{S}{2}+\frac{m\left(V_{\mathscr{H}}{}^2-V_{max}{}^2\right)}{2}$$

(27)

The difference in energy expended by P_N and energy expended by P_W on the B–C section is a measure of the mechanical energy lost to compensate for the travel time between trains:

$$Z_{\Delta T}=Z\left(\Delta T\right)-Z{w}_{B–C}=\frac{\left(F_{opr, \mathscr{H}}+F_{opr,max}\right)}{2}\frac{S}{2}+\frac{m\left(V_{\mathscr{H}}{}^2-V_{max}{}^2\right)}{2}-F_{opr,max}S$$

(28)

4. Simulation Results

4.1. Description

In the simulation, the results of the change in speed and location of P_N were obtained for contexts and variants of train running for 135 traffic situations (3 contexts × 5 variants × 9 distances).

In the form of tables and figures, the resultant (final) values for all traffic situations at time t = T (s) concerning the speed obtained by P_N: speed, distance traveled and mechanical energy consumed, and calculations from the normalization of the results, allow for a final comparison of the extension of travel time P_N and the loss of mechanical energy with respect to the reference train P_W.

From the simulations, key movement situations where the case of boundary distance (described in Section 3.3) arises were extracted and are shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 for each movement context of the relationships: V(t), S(t). In addition, for each context and each initial distance, the relationship D(t) is shown in Figure 13, Figure 14 and Figure 15, where the smallest distance between trains conducive to the use of virtual coupling is sought. For the selected distance, the V(t) relationship is also shown in Figure 16 and Figure 17, which allows us to assess whether the selected train control option also allows for smooth train movement.

The values presented in the following tables (Table 2, Table 3 and Table 4) should be interpreted as follows:

Table 2. Simulation results for context 1.

	Symulation Results					Normalization Results
	Variant	T [s]	S [m]	V [km/h]	Zm [kWh]	Loss S [m]	Loss T [s]	Zm + Comp. S [kWh]	Comp. T [kWh]	Σ Zm [kWh]
D = 5000 m	1	97	5385	200	60.918	5	1	8.179	22.211	30.390
	2	97	5040	175	0	493	9	−6.108	74.569	68.461
	3	97	4785	194	88.708	615	12	53.752	88.357	142.109
	4	97	5012	181	17.441	470	9	3.377	74.569	77.946
	5	97	5114	194	36.622	286	6	−1.616	59.031	57.415
D = 4500 m	1	97	5385	200	60.918	5	1	8.179	22.211	30.390
	2	97	5040	175	0	493	9	−6.108	74.569	68.461
	3	97	4785	194	88.708	615	12	53.752	88.357	142.109
	4	97	5004	181	18.077	480	9	4.112	74.569	78.681
	5	97	5114	194	36.622	286	6	−1.616	59.031	57.415
D = 4000 m	1	97	5385	200	60.918	5	1	8.179	22.211	30.390
	2	97	5040	175	0	493	9	−6.108	74.569	68.461
	3	97	4785	194	88.708	615	12	53.752	88.357	142.109
	4	97	4996	181	19.261	487	9	5.366	74.569	79.935
	5	97	5114	194	36.622	286	6	−1.616	59.031	57.415
D = 3500 m	1	97	5385	200	60.918	5	1	8.179	22.211	30.390
	2	97	5040	175	0	493	9	−6.108	74.569	68.461
	3	97	4785	194	88.708	615	12	53.752	88.357	142.109
	4	97	4983	181	20.777	500	9	7.012	74.569	81.581
	5	97	5114	194	36.622	286	6	−1.616	59.031	57.415
D = 3000 m	1	97	5385	200	60.918	5	1	8.179	22.211	30.390
	2	97	5040	175	0	493	9	−6.108	74.569	68.461
	3	97	4785	194	88.708	615	12	53.752	88.357	142.109
	4	97	4964	180	22.897	521	10	9.335	79.319	88.654
	5	97	5114	194	36.622	286	6	−1.616	59.031	57.415
D = 2500 m	1	97	5385	200	60.918	5	1	8.179	22.211	30.390
	2	97	5040	175	0	493	9	−6.108	74.569	68.461
	3	97	4785	194	88.708	615	12	53.752	88.357	142.109
	4	97	4943	181	26.399	539	10	12.048	79.319	91.367
	5	97	5114	194	36.622	286	6	−1.616	59.031	57.415
D = 2000 m	1	97	4807	180	69.046	682	13	58.061	92.709	150.770
	2	97	4616	151	0	1258	23	32.887	131.760	164.647
	3	97	4785	194	88.708	615	12	53.752	88.357	142.109
	4	97	4746	176	46.129	777	14	40.923	96.926	137.849
	5	97	4808	184	66.578	650	12	49.463	88.357	137.820
D = 1500 m	1	97	4379	169	45.985	1233	23	56.866	131.760	188.626
	2	97	4169	144	0	1817	33	45.877	166.355	212.232
	3	97	4380	169	69.829	1233	23	80.709	131.760	212.469
	4	97	4375	170	56.514	1224	23	65.386	131.760	197.146
	5	97	4383	169	67.659	1230	23	78.514	131.760	210.274
D = 1000 m	1	97	3740	166	100.798	1907	35	121.265	172.968	294.233
	2	97	2985	93	0	4068	74	107.212	291.312	398.524
	3	97	3757	167	102.129	1881	34	122.352	169.671	292.023
	4	97	3757	166	101.948	1883	34	122.182	169.671	291.853
	5	97	3757	167	102.129	1881	34	122.353	169.671	292.024

Table 3. Simulation results for context 2.

	Symulation Results					Normalization Results
	Variant	T [s]	S [m]	V [km/h]	Zm [kWh]	Loss S [m]	Loss T [s]	Zm + Comp. S [kWh]	Comp. T [kWh]	Σ Zm [kWh]
D = 5000 m	1	188	10,436	200	116.958	65	2	12.851	32.142	44.993
	2	188	9210	156	0	1634	30	−20.365	156.287	135.922
	3	188	7883	194	173.578	2573	47	107.716	211.055	318.771
	4	188	8672	164	45.806	2057	38	20.199	182.720	202.919
	5	188	9474	194	72.573	982	18	−9.158	113.047	103.889
D = 4500 m	1	188	10,436	200	116.958	65	2	12.851	32.142	44.993
	2	188	9210	156	0	1634	30	−20.365	156.287	135.922
	3	188	7883	194	173.578	2573	47	107.716	211.055	318.771
	4	188	8600	164	50.067	2125	39	25.148	185.913	211.061
	5	188	9462	194	73.583	994	18	−8.025	113.047	105.022
D = 4000 m	1	188	9905	180	126.904	636	12	64.061	88.357	152.418
	2	188	9210	156	0	1634	30	−20.365	156.287	135.922
	3	188	7883	194	173.578	2573	47	107.716	211.055	318.771
	4	188	8523	165	55.883	2190	40	30.662	189.103	219.765
	5	188	9412	194	77.529	1044	19	−3.582	116.880	113.298
D = 3500 m	1	188	9479	168	98.21	1192	22	58.247	128.112	186.359
	2	188	9210	156	0	1634	30	−20.365	156.287	135.922
	3	188	7883	194	173.578	2573	47	107.716	211.055	318.771
	4	188	8435	166	62.788	2262	41	36.369	192.294	228.663
	5	188	9397	194	78.538	1059	20	−2.426	120.699	118.273
D = 3000 m	1	188	8963	170	117.557	1687	31	80.621	159.684	240.305
	2	188	8742	142	0	2346	43	2.496	198.623	201.119
	3	188	7883	194	173.578	2573	47	107.716	211.055	318.771
	4	188	8334	168	71.883	2340	43	43.361	198.623	241.984
	5	188	8966	171	68.433	1669	31	29.393	159.684	189.077
D = 2500 m	1	188	8511	166	102.981	2193	40	76.835	189.103	265.938
	2	188	7968	123	0	3479	63	30.583	259.255	289.838
	3	188	7883	194	173.578	2573	47	107.716	211.055	318.771
	4	188	8225	171	83.677	2415	44	53.05	201.730	254.780
	5	188	8471	170	95.093	2180	40	63.073	189.103	252.176
D = 2000 m	1	188	7968	169	139.478	2691	49	113.517	217.225	330.742
	2	188	6907	99	0	5077	92	64.085	342.654	406.739
	3	188	7883	194	173.578	2573	47	107.716	211.055	318.771
	4	188	7848	171	97.627	2789	51	69.758	223.337	293.095
	5	188	7992	169	124.052	2676	49	98.896	217.225	316.121
D = 1500 m	1	188	7469	169	134.317	3190	58	113.334	244.435	357.769
	2	188	6089	89	0	6108	110	79.184	393.168	472.352
	3	188	7477	170	157.659	3174	58	135.558	244.435	379.993
	4	188	7320	170	116.645	3327	60	95.102	250.386	345.488
	5	188	7474	169	150.923	3186	58	129.899	244.435	374.334
D = 1000 m	1	188	6974	169	158.176	3693	67	143.172	271.007	414.179
	2	188	5214	84	0	7098	128	91.762	443.186	534.948
	3	188	6982	168	166.058	3695	67	152.029	271.007	423.036
	4	188	6632	167	133.047	4053	73	123.547	288.433	411.980
	5	188	6991	167	163.966	3695	67	150.889	271.007	421.896

Table 4. Simulation results for context 3.

	Symulation Results					Normalization Results
	Variant	T [s]	S [m]	V [km/h]	Zm [kWh]	Loss S [m]	Loss T [s]	Zm + Comp. S [kWh]	Comp. T [kWh]	Σ Zm [kWh]
D = 5000 m	1	260	12,188	170	192.301	2462	45	123.194	204.848	328.042
	2	260	11,641	126	0	3755	68	−8.231	273.912	265.681
	3	260	9105	194	214.805	5350	97	135.775	356.772	492.547
	4	260	10,520	155	77.564	4338	79	46.176	305.692	351.868
	5	260	12,046	194	129.141	2409	44	20.772	201.730	222.502
D = 4500 m	1	260	11,683	170	184.895	2971	54	120.855	232.421	353.276
	2	260	11,267	121	0	4226	77	0.787	299.979	300.766
	3	260	9105	194	214.805	5350	97	135.775	356.772	492.547
	4	260	10,369	157	85.969	4458	81	52.939	311.422	364.361
	5	260	11,680	170	105.495	2971	54	41.458	232.421	273.879
D = 4000 m	1	260	11,146	175	184.533	3452	63	118.565	259.255	377.820
	2	260	9772	88	0	6465	117	44.391	412.654	457.045
	3	260	9105	194	214.805	5350	97	135.775	356.772	492.547
	4	260	10,207	160	95.929	4582	83	61.314	317.135	378.449
	5	260	11,204	169	136.908	3464	63	79.709	259.255	338.964
D = 3500 m	1	260	10,667	172	202.972	3959	72	145.918	285.526	431.444
	2	260	9360	86	0	6918	125	49.456	434.893	484.349
	3	260	9105	194	214.805	5350	97	135.775	356.772	492.547
	4	260	10,030	163	107.616	4714	85	70.523	322.834	393.357
	5	260	10,663	172	144.041	3964	72	87.029	285.526	372.555
D = 3000 m	1	260	10,180	171	194.072	4461	81	143.949	311.422	455.371
	2	260	7548	52	0	9566	173	87.068	567.786	654.854
	3	260	9105	194	214.805	5350	97	135.775	356.772	492.547
	4	260	9755	165	120.554	4957	90	82.058	337.015	419.073
	5	260	10,166	173	171.893	4445	81	117.763	311.422	429.185
D = 2500 m	1	260	9684	170	184.164	4966	90	140.35	337.015	477.365
	2	260	7056	51	0	10,079	182	91.845	592.795	684.640
	3	260	9105	194	214.805	5350	97	135.775	356.772	492.547
	4	260	8995	160	128.056	5788	105	104.516	379.184	483.700
	5	260	9678	171	169.915	4964	90	124.805	337.015	461.820
D = 2000 m	1	260	9180	170	173.152	5470	99	134.049	362.387	496.436
	2	260	6573	50	0	10,572	191	97.569	617.780	715.349
	3	260	9105	194	214.805	5350	97	135.775	356.772	492.547
	4	260	8502	160	129.806	6282	114	111.186	404.316	515.502
	5	260	9187	171	169.484	5456	99	129.282	362.387	491.669
D = 1500 m	1	260	8684	170	168.723	5967	108	134.576	387.587	522.163
	2	260	6072	50	0	11,076	200	102.542	640.094	742.636
	3	260	8694	170	198.806	5957	108	164.563	387.587	552.150
	4	260	8003	160	141.15	6781	123	127.51	429.346	556.856
	5	260	8691	171	184.573	5952	108	149.318	387.587	536.905
D = 1000 m	1	260	8185	170	193.479	6465	117	164.305	412.654	576.959
	2	260	5517	51	0	11,616	210	107.203	640.094	747.297
	3	260	8200	169	208.691	6469	117	181.468	412.654	594.122
	4	260	7497	160	151.462	7288	132	142.876	454.292	597.168
	5	260	8174	172	197.752	6453	117	165.573	412.654	578.227

Results obtained in the simulation:

T—simulation time,

S—the distance traveled by P_N at time T,

V—speed P_N at time t = T,

Z_m—consumed mechanical energy by P_N at time T.

The results obtained in the normalization process (according to the formulas given in Section 3.1):

Loss S—the difference of the distance traveled P_N to P_W when P_N obtains speed V_max,

Loss T—the required time P_N to equalize the speed to V_max and get to the place Sw(T), where P_W is at t = T

Z_m + Comp. S—the difference of the consumed mechanical energy P_N to P_W (Z_m + compensation for the lost path Loss S), calculated from the beginning of the simulation until P_N is in place S_w(T)—when P_W was at the end of the simulation t = T. In the case where there was a slight reduction in the speed of P_N in the simulation (less resistance to movement), a negative value is possible, which indicates that P_N moved according to a less energy-intensive characteristic at the expense of increased travel time (Loss T),

Comp. T—the amount of mechanical energy required for P_N to offset the lost time to P_W,

∑Z_m—the difference of consumed mechanical energy P_N to P_W (Z_m + Comp. S), taking into account the compensation of lost time P_N (Comp. T), which will allow P_N to be in the same place, time and speed as P_W.

In the cell, the colors indicate the gradation of values: green least (most favorable)—red most (least favorable).

4.2. Result

From the figures shown in the above tables for total mechanical energy consumption due to changes in speed and loss of distance and increase in travel time P_N, one can see how shortening the distance between trains increases energy consumption.

The following boundary distances were determined in the simulation, Context 1—2000 m, Context 2—4000 m, Context 3—5000 m, and represent an important magnitude from the point of view of the possibility of applying P_N speed control and optimizing train energy consumption.

For the distance between trains exceeding the limit distance, in terms of energy consumption P_N, variant 1 is the most favorable, which is due to the possibility of running the train at maximum speed, while the length of the distance allowed buffering the disruption of the movement speed P_P.

For distances between trains smaller than the boundary distance, in terms of energy consumption, variant 5 is then most advantageous, where the train control takes into account the speed P_P, the distance between trains, and the resistance of traffic on the distance between trains.

The estimated difference in mechanical energy consumption of variant 1 to the most favorable variant is:

• Context 1, D_gr = 2000 m: w1 − w5 = 12.950, w2 − w5 = 26.827, w3 − w5 = 4.289, w4 − w5 = 0.028 (kWh);
• Context 2, D_gr = 4000 m: w1 − w5 = 39.120, w2 − w5 = 22.625, w3 − w5 = 205.473, w4 − w5 = 106.467 (kWh);
• Context 3, D_gr = 5000 m: w1 − w5 = 105.540, w2 − w5 = 43.178, w3 − w5 = 270.045, w4 − w5 = 129.366 (kWh).

Further shortening the distance between trains to a distance of 1000 m, the difference between the variants of the control disappears, which does not allow us to identify the single most favorable variant: often, variant 4 appears, there are variants 2 and 3, and once there is a variant 1. It is important that for a distance of less than 2000 m (in any context) the idea of control loses its meaning—the curves of running P_N for the different variants are close to each other, except for variant 2, the use of which is unfavorable due to a significant increase in the time of running P_N.

The following figures show the P_N run curves for boundary distances. The dashed curve in black refers to the run shape of the preceding train, which has been shifted to the left in the figures of the V(t) relationship for illustrative purposes, by the boundary distance.

For context 1:

Figure 7. Velocity change V(t) in simulation by context 1 for distance D₀ = D_gr = 2000 m.

Figure 7. Velocity change V(t) in simulation by context 1 for distance D₀ = D_gr = 2000 m.

Figure 8. Change in distance traveled S(t) in simulation by context 1 for distance D₀ = D_gr = 2000 m.

Figure 8. Change in distance traveled S(t) in simulation by context 1 for distance D₀ = D_gr = 2000 m.

For context 2:

Figure 9. Velocity change V(t) in simulation by context 2 for distance D₀ = D_gr = 4000 m.

Figure 9. Velocity change V(t) in simulation by context 2 for distance D₀ = D_gr = 4000 m.

Figure 10. Change in distance traveled S(t) in simulation by context 2 for distance D₀ = D_gr = 4000 m.

Figure 10. Change in distance traveled S(t) in simulation by context 2 for distance D₀ = D_gr = 4000 m.

For context 3:

Figure 11. Velocity change V(t) in simulation by context 3 for distance D₀ = D_gr = 5000 m.

Figure 11. Velocity change V(t) in simulation by context 3 for distance D₀ = D_gr = 5000 m.

Figure 12. Change in distance traveled S(t) in simulation by context 3 for distance D₀ = D_gr = 5000 m.

Figure 12. Change in distance traveled S(t) in simulation by context 3 for distance D₀ = D_gr = 5000 m.

In the presented Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 you can see how the movement of P_N is shaped for Variant 5. With respect to other variants, the control in Variant 5 allows, during the whole simulation, obtaining a high train speed (but not too high, causing excessive approach to P_P), which allows us to obtain a running distance and small time losses with respect to the “reference” train.

4.3. Results in Terms of Creating a Virtual Coupling

In order to identify the best solution for creating a virtual coupling, the following assumptions were made:

(1)

A constant distance D(t) is maintained between P_N and P_P, which means the speed is equalized between the trains;

(2)

The solution is chosen for the shortest distance D₀;

(3)

The V(t) curve for the P_N waveform has a smooth character (small changes in speed), which indicates the smoothness of the P_N movement.

Figure 13, Figure 14 and Figure 15 show for each movement context the curves of distance change D(t) train P_N, depending on the assumed initial distance: D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000, 1500, 1000 m.

Changes in the distance between P_N and P_P over the course of the simulation for context 1:

Figure 13. Summary of distance curves D(t) for D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000, 1500, 1000 (m) in simulation for context 1.

Figure 13. Summary of distance curves D(t) for D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000, 1500, 1000 (m) in simulation for context 1.

Changes in the distance between P_N and P_P over the course of the simulation for context 2:

Figure 14. Summary of distance curves D(t) for D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000, 1500, 1000 (m) in simulation for context 2.

Figure 14. Summary of distance curves D(t) for D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000, 1500, 1000 (m) in simulation for context 2.

Changes in the distance between P_N and P_P over the course of the simulation for context 3:

Figure 15. Summary of distance curves D(t) for D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000, 1500, 1000 (m) in simulation for context 3.

Figure 15. Summary of distance curves D(t) for D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000, 1500, 1000 (m) in simulation for context 3.

For context 1, the run P_N, which is characterized by maintaining a constant distance in the simulation, occurs only for variant 3 at distances D₀ = 3500, 3000, 2500, 2000 m (Figure 13: row 2 and row 3, column 1).

Given the assumption of choosing the shortest distance D₀ and keeping a constant distance between trains, the most favorable solution for achieving virtual coupling is to control according to variant 3 at a distance D₀ = 2000 m, which is also the limiting distance for this context.

For context 2, the course of P_N that is characterized by maintaining a constant distance over the length of the entire simulation, can also be observed only for variant 3 at distances: D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000 m (Figure 14: row 1, row 2 and row 3, column 1).

Given the assumption of choosing the shortest distance D₀ and the smoothness of movement P_N based on the V(t) curve (Figure 16), the most favorable solution for achieving virtual coupling is to control according to variant 3 at a distance D₀ = 2000 m.

Figure 16. Velocity change V(t) in simulation by context 2 for distance D₀ = 2000 m.

Figure 16. Velocity change V(t) in simulation by context 2 for distance D₀ = 2000 m.

For context 3, the course of P_N that is characterized by maintaining a constant distance over the length of the entire simulation can also be observed only for variant 3 at distances: D₀ = 5000, 4500, 4000, 3500, 3000, 2500, 2000 m (Figure 14: row 1, row 2 and row 3, column 1).

Given the assumption of choosing the shortest distance D₀ and the smoothness of movement P_N based on the V(t) curve (Figure 17), the most favorable solution for achieving virtual coupling is to control according to variant 3 at a distance D₀ = 2000 m.

Figure 17. Velocity change V(t) in simulation by context 3 for distance D₀ = 2000 m.

Figure 17. Velocity change V(t) in simulation by context 3 for distance D₀ = 2000 m.

From the above comparisons, it can be seen that for each context, the best solution for creating a virtual coupling is variant 3 with a distance between trains of 2000 m.

However, the use of this option carries the choice of a more energy-intensive solution:

• For context 1, the difference in energy consumption is: w3 − w2 = 4289 kWh,
• For context 2: w3 − w4 = 25,676 kWh,
• For context 3: w3 − w5 = 0.878 kWh,

Which, however, is not a significant figure, considering that it represents less than 10% of the energy lost.

4.4. Sensitivity Analysis

A sensitivity analysis was performed to assess the impact of different contexts on the presented parameters for the evaluation of the options (Table 5).

Table 5. Sensitivity analysis of the impact on performance of the different contexts.

Context 1	VW (km/h)	SW (m)	T (s)	Zm(PW) (kWh)
PW	200	5389	97	54.83
	VN/VW	SN/SW	Loss T/T	∑Zm/Zm(PW)
PN (D0 = 4 km)	97%	95%	6%	105%
PN (D0 = 2 km)	92%	89%	12%	251%
Context 2	VW (km/h)	SW (m)	T (s)	Zm(PW) (kWh)
PW	200	10,444	188	106.28
	VN/VW	SN/SW	Loss T/T	∑Zm/Zm(PW)
PN (D0 = 4 km)	97%	90%	10%	107%
PN (D0 = 2 km)	85%	77%	26%	297%
Context 3	VW (km/h)	SW (m)	T (s)	Zm(PW) (kWh)
PW	200	14,444	260	146.98
	VN/VW	SN/SW	Loss T/T	∑Zm/Zm(PW)
PN (D0 = 4 km)	85%	78%	24%	231%
PN (D0 = 2 km)	86%	64%	38%	335%

The reference values are the running parameters of a reference train (undisturbed) moving at maximum speed during the realized simulation, consuming the specified mechanical energy over the distance travelled.

The values were related to the running of a P_N train running according to variant 5, for an initial distance of 4 km and a distance of 2 km (Table 2, Table 3 and Table 4).

The analysis carried out shows the following relationships:

• The magnitude of the speed reduction of the P_P train with respect to the P_N parameters is non-linear. The difference in speed and distance loss between contexts 1 and 2 is smaller than between contexts 2 and 3.
• The factor most susceptible to context variation is the path loss (speeds stabilize at 85%), which has a key impact on the final value of time loss and ultimately on energy consumption.
• Reducing the initial distance affects energy consumption more than the context change itself. This can be seen by comparing the percentage of energy consumption for context 3 at D₀ = 4 km compared to contexts 1 and 2, at D₀ = 2 km. A factor contributing to this condition is the lack of smoothness of the train (dynamic speed changes).

5. Conclusions and Discussion

The results obtained in the simulation confirm the possibility of increasing the energy efficiency of the train by using information about the “preceding” train.

The amount of energy consumption of the “following” train is not only influenced by the traffic context of the “preceding” train (the amount of speed reduction), but the distance between the trains is also key.

In the simulation, a boundary distance was established for each traffic context, for which it is reasonable to control the “following” train. This justification is based on obtaining the shortest increase in train travel time, the longest running distance and the highest speed at the end of the interference from the “preceding” train.

Using the normalization of the results (Section 3.6), the obtained values from the simulation for each variant of the “following” train movement can be converted into a single comparative parameter—mechanical energy consumption. For the indicated boundary distances, the most favorable results, related to minimizing the loss of mechanical energy, are obtained for variant 5. In this variant, the mechanical energy consumption with respect to variant 1 is as follows:

• Context 1: ∆Z_m (w1 − w5) = 12.950 kWh;
• Context 2: ∆Z_m (w1 − w5) = 39.120 kWh;
• Context 3: ∆Z_m (w1 − w5) = 105.540 kWh.

The control of P_N in variant 5 consists of comparing the speed and distance between trains, taking into account traffic resistance, and based on this, calculating the required maximum permissible delay a_h so as to achieve an alignment of the speeds of P_N and P_P over the indicated distance. However, the application of this option may be problematic due to the need to determine the traffic resistance to which P_N will be subjected over the distance approaching P_P.

On the basis of the sensitivity analysis carried out, it was also observed that the excessive reduction in the distance between trains is more important for the energy efficiency of the train than the magnitude of the speed reduction in individual contexts. For such cases, the loss of distance has less impact, while the dynamic change in speed of the “following” train is important.

From the point of view of the implementation of virtual coupling, the most advantageous is variant 3 applied at a distance of 2000 m (regardless of the context). Control according to this variant allows you to guide P_N using information about the speed of P_P. As a result of adjusting the acceleration and braking of the train, it is possible to maintain a constant distance between trains and the smoothness of the movement of the train, which is important for the coupling time of trains.

Further studies should take into account the impact on the control of external conditions, i.e., lowering the power in the overhead line, increasing the resistance of traffic to the incline of the track or changes in the operating conditions of equipment and control systems, such as changes in the time of transmission and processing of information. Such an analysis will make it possible to determine the susceptibility of individual control variants to external factors and to select the most favorable variant for synchronizing train speeds.