## Abstract

Virtual coupling of railway trains is an emerging technology that has the potential to significantly increase railway operational efficiency by reducing the train following distance from absolute braking distances to relative braking distances. Current research in this topic is mainly focused on passenger trains and uses distance-based headways. This paper studied virtual coupling for heavy haul freight trains and demonstrated that the distance headway scheme was challenging and sometimes impractical for heavy haul trains to achieve virtual coupling. A time-based headway scheme was then proposed to set the follower train to be a certain time behind the schedule of the leader train rather than a distance headway. The time-based headway required the follower train to reproduce the leader train's operational status at the same track location. This also allowed the follower train to copy any optimized train driving strategies from the leader train. Demonstrative simulations were carried out without the consideration of communication errors and train localization errors. The results show that a conventional distance headway simulation had maximum distance and speed errors of 716 m (36%, reference 2 km) and 24 km/h (66%, reference 36 km/h), respectively. A time-based headway simulation reduced the maximum distance and speed errors to 0.07 m (0%, reference 2 km) and 0.1 km/h (9%, reference 1.18 km/h), respectively.

## 1 Introduction

Virtual coupling [1,2] represents a cutting-edge advancement in railway technology, redefining the traditional concept of train connectivity. Unlike conventional physical coupling [3], where train wagons (cars) are linked by mechanical couplings, virtual coupling relies on a sophisticated system of communication and control technologies [4]. In this system, trainsets operate as a closely coordinated platoon, maintaining a predesigned, electronically controlled distance without the need for physical attachment [5,6]. This is achieved through real-time data exchange and advanced algorithms that manage speed, braking, and distance, ensuring synchronized operation. Virtual coupling not only enhances the operational flexibility and capacity of rail networks but also promises significant improvements in terms of safety, energy efficiency, and line capacity. By enabling trains to run more closely yet independently, this technology paves the way for a more efficient, responsive, and adaptable rail transport system, meeting the increasing demands of modern freight and passenger transportation. One of the most important features of virtual coupling is that the train following distance between adjacent trains is reduced from the traditional absolute braking distance to a relative braking distance [7,8].

Railway virtual coupling is regarded as a derivation of the concept of road vehicle platooning [9–12]. However, this concept presents unique challenges when applied to the heavier and more complex domain of rail transport, especially for heavy haul freight trains [3,4,13]. Firstly, the sheer weight and length of heavy haul freight trains significantly impact the feasibility of maintaining a fixed distance. These trains, often composed of hundreds of loaded wagons, can stretch for many kilometers, and weigh more than several thousand tons. The momentum and inertia associated with such mass make it difficult to achieve the precise, responsive control required for fixed-distance operations. Rapid acceleration or deceleration, which is relatively straightforward in light road vehicles, is far more complex and riskier with heavy freight trains. Additionally, the variability in rail topography presents another layer of complexity. Unlike the similar road grades found underneath adjacent vehicles in car platooning scenarios, due to the longer train length and headway, rail tracks can vary greatly in elevation and curvature over the related distances. This variability can affect train handling and braking performance, making it challenging to maintain a constant distance between virtually coupled trainsets. For instance, a leader train descending a steep grade would require more braking effort and distance than a follower train on an ascending grade, complicating the synchronization needed for fixed-distance virtual coupling. Such a case also applies to train traction capabilities as shown in Fig. 1.

These factors contribute to why maintaining a fixed distance in virtually coupled trainsets, particularly in heavy haul freight operations, presents significant technical and operational challenges. The system must account for the unique dynamics of each train, the varying track conditions, and the inherent complexities of handling heavy, long trains. This necessitates the development of advanced control algorithms and robust communication systems capable of adapting to the diverse and dynamic nature of railway operations.

Recently, several comprehensive surveys regarding railway virtual coupling controllers have been presented [14–16]. Typical control algorithms that have been used included Consensus-Based Control [17–19], Optimal Predicative Control [20–22], Sliding Mode Control [23–25], Artificial Intelligence Based Control [26,27], and Constraints-Following Control [28,29]. Wu et al. [15] also reviewed different designs of headways used for virtual coupling. The review [15] showed that the headway designs used for virtual coupling were mostly speed-dependent distance headways. In other words, these headway designs included a constant minimum headway for operational safety and a dynamic component that increased with the operational speeds. However, none of the headway designs were time based. Time based headway indicates that the follower train is a certain time behind the leader train, for example, 2 min, instead of a fixed distance headway. This paper will show that the time headway is more suitable for freight train virtual coupling.

Previous reviews [14–16] and a report released by the Federal Railroad Administration [30] have also indicated that the research about virtual coupling control has so far been focused on passenger and light trains. Virtual coupling control for heavy haul trains is rare in public literature [31,32]. This paper introduces a time headway control scheme for virtual coupling of heavy haul freight trains. Section 2 introduces the basics of longitudinal train motion modeling. Section 3 introduces the virtual coupling controller used in this paper. Sections 4 and 5 present simulations and comparisons about simulations by using a distance headway scheme and a time headway scheme. Section 6 concludes this paper.

## 2 Nonlinear Longitudinal Motion of Heavy Haul Freight Trains

For railway network and signaling simulations, train longitudinal motion models without the consideration of coupler forces are often used [33]. The fundamental principle in modeling longitudinal train motion lies in understanding the external forces acting on a train. These include tractive force generated by the locomotive, brake force generated from the locomotive and wagons, resistance forces such as curving resistance, aerodynamic drag and rolling resistance, and gravitational forces on inclines or declines. The interaction of these forces dictates the train's acceleration and velocity.

A critical aspect of this modeling is the nonlinear nature of these forces. Tractive force, for example, varies nonlinearly with speed and engine characteristics. Similarly, resistance forces depend on factors like train speed and vehicle types in a nonlinear manner. This nonlinearity is typically captured using equations that describe the relationship between these forces, train mass, and velocities. For tractive forces and dynamic brake (DB) forces, notch-less locomotive driving is often assumed for automated train driving. Figure 2 shows typical force limits for the traction and DB forces. The maximum traction force that a locomotive can exert is fundamentally constrained by two key factors: the electrical current limit and the power limit. The traction motors of a locomotive are powered by electrical current. There is a maximum current that these motors can handle without overheating or suffering damage. This current limit determines the maximum force at lower speeds. The power limit is defined by the maximum amount of electrical power that the locomotive's generator or power source can provide. Power is the product of traction force and speed. The power limit determines the maximum force at higher speeds.

where $Fb$ is the air brake force; $Pb$ is the brake cylinder pressure; $Aps$ is the brake cylinder cross-sectional area; $Fps$ is the cylinder piston spring force; $\mu b$ is the coefficient of friction on the brake shoes; $\delta rigging$ is the rigging factor of the mechanical leverages; $\delta efficiency$ is the mechanical transmission efficiency; $\delta adhesionlimit$ is the reduction factor due to wheel-rail adhesion; $nb$ is the number of brake cylinder; $rb$ is the radius at the brake shoe attaching point on the wheel; $rw$ is the nominal wheel running radius; $Ff$ is the cylinder piston friction force; and $Vkm$ is the vehicle speed in the unit of km/h.

where $Frg$ is the combination of resistance force and gradient force; $mw$ is the mass of the mass of the wagon in tonnes; $ma$ is the axle load of the wagon in tonne; $R$ is the curing resistance; and $\u03d1$ is the track gradient. The last two terms of Eq. (3) were for curving resistance and track gradient force, respectively. The rest of the equation was for rolling resistance.

Having reached this step, the refined location $x2(i)$ and velocity $v2(i)$ are used as the final results and saved as the initial position $x0(i)$ and velocity $v0(i)$ for the next time-step.

## 3 Virtual Coupling Controller Design

This section presents the design of the desired virtual coupling controller. The longitudinal train motion model used for the controller design will be presented first. Then the details of the control law will be discussed.

### 3.1 Longitudinal Train Motion Model.

*T*/

*B*) force, and $Ftrg(i)$ denotes the resistance and gradient force on the $ith$ train, respectively, $\tau $ represents the actuation delay, and $uttb(i)$ denotes the computed control input by the VC controller. Defining a normalized control input $ut(i)=uttb(i)/M$ and an unknown disturbance-like input $wt(i)$=$\u2009(F\u02d9trg(i)+Ftrg(i))/M$, we then arrive at $\tau a\u02d9t(i)+at(i)=ut(i)+wt(i)$, which can be rewritten in a compact state-space form as follows:

where $ut(N+1)$ stands for the commanded leader control input that is unknown to the follower trains.

### 3.2 An Adaptive Virtual Coupling Controller.

where $yt(i)\u2208Rny,(ny\u22643)$ represents the local noisy sensor measurement from on the $ith$ train with $nt(i)$ being the unknown measurement noise and $C\u2208Rny\xd73$ being the prescribed measurement matrix. The above sensor measurement model can be tailored to accommodate different measurement scenarios. For example, when only the noisy train position can be measured, one can set $C=[1,0,0]$; when the longitudinal train position is not available, but the train velocity and acceleration can be directly measured, the matrix $C$ then can be chosen as $C=[0,0,0;0,1,0;0,0,1]$.

for any $i=1,2,\u2026,N$, where the estimated uncertainty $w\u0302t(i)$ by the designed state observer is adopted to compensate the effects of the unknown input $wt(i)$, and $et(i)$ denotes the train tracking error with $x\u0302t(i\u22121)$ being the estimated train motion state from the preceding train $i\u22121$ via train-to-train (T2T) communication and $dt$ being the desired gap reference for two consecutive trains and $x\u0302t(i\u22121)|i=1=xt(N+1)$, $\alpha t(i)$ represents the adaptive gain whose value is adjusted in an online manner, and $G3$ is the controller gain matrix to be designed.

It is clear from Eqs. (12) and (13) that the design of the proposed observer-based adaptive VC controller lies in the determination of the gain matrices $G1,G2,\u2009G3$. Once they are determined, the observer and controller can be implemented in real-time by each train $i$ in the platoon based on the information collected from its local sensor (namely, the sensor measurement $yt(i)$) and the information received from its predecessor train (namely, the transmitted train data $x\u0302t(i\u22121)$ via wireless T2T communication). The design and implementation details are outlined in the following algorithm.

/% Offline parameter design %/ |

Step 1. Design the observer gain matrices$G1$and$G2$. Given the system and measurement matrices $A,B,C,$ and two positive scalars $c1,c2,$ solve the following linear matrix inequality $[Q1,\u2009Q2;\u2009Q2\u22a4,\u2009\u2212c0I]<0$ to find real matrices $P1>0,\u2009P2,$ and a scalar $c0>0,$ where $Q1=[Q11,Q12;\u2009Q12\u22a4,\u22122c2B\u22a4B],\u2009Q2=[\u2212P2,c2B;0,\u2212c1\u22121],\u2009Q11=\u2009P1A+A\u22a4P1+P2C+C\u22a4P2\u22a4+c1c22(BB\u22a4A+A\u22a4BB\u22a4),\u2009Q12=\u2009P1B\u2212c2A\u22a4B+c1c22BBTB.$ If the above inequality is feasible, then one has that $G1=P1\u22121P2$ and $G2=c1c2B\u22a4P1\u22121P2$; otherwise, reset $c1,c2$ and repeat Step 1. |

Step 2. Design the controller gain matrix$G3$. Given the system matrices $A,B$, and a real matrix $X>0$, solve the following algebraic Riccati equation $G3A+A\u22a4G3\u22122G3BB\u22a4G3=\u2212X$ to find a feasible solution for a real matrix $G3>0$. If not feasible, reset $X$ and repeat Step 2. |

/% Online implementation %/ |

each train $i=1,2,\u2026,\u2009N$ do for Collect the available sensor measurement $yt(i)$ at current sampling time $t$. Step 3. Receive the data $x\u0302t(i\u22121)$ at time $t$ from the preceding train via T2T communication. Step 4. Update the observer (12) and controller (13) to obtain the desired control input $ut(i)$. Step 5. Apply the T/B force $uta(i)=M\xb7ut(i)$ to update the dynamics (8) for the $ith$ train. Step 6. Repeat Step 7.- Step 3– at the next sampling time $t+=t+\Delta t.$Step 6 |

end |

/% Offline parameter design %/ |

Step 1. Design the observer gain matrices$G1$and$G2$. Given the system and measurement matrices $A,B,C,$ and two positive scalars $c1,c2,$ solve the following linear matrix inequality $[Q1,\u2009Q2;\u2009Q2\u22a4,\u2009\u2212c0I]<0$ to find real matrices $P1>0,\u2009P2,$ and a scalar $c0>0,$ where $Q1=[Q11,Q12;\u2009Q12\u22a4,\u22122c2B\u22a4B],\u2009Q2=[\u2212P2,c2B;0,\u2212c1\u22121],\u2009Q11=\u2009P1A+A\u22a4P1+P2C+C\u22a4P2\u22a4+c1c22(BB\u22a4A+A\u22a4BB\u22a4),\u2009Q12=\u2009P1B\u2212c2A\u22a4B+c1c22BBTB.$ If the above inequality is feasible, then one has that $G1=P1\u22121P2$ and $G2=c1c2B\u22a4P1\u22121P2$; otherwise, reset $c1,c2$ and repeat Step 1. |

Step 2. Design the controller gain matrix$G3$. Given the system matrices $A,B$, and a real matrix $X>0$, solve the following algebraic Riccati equation $G3A+A\u22a4G3\u22122G3BB\u22a4G3=\u2212X$ to find a feasible solution for a real matrix $G3>0$. If not feasible, reset $X$ and repeat Step 2. |

/% Online implementation %/ |

each train $i=1,2,\u2026,\u2009N$ do for Collect the available sensor measurement $yt(i)$ at current sampling time $t$. Step 3. Receive the data $x\u0302t(i\u22121)$ at time $t$ from the preceding train via T2T communication. Step 4. Update the observer (12) and controller (13) to obtain the desired control input $ut(i)$. Step 5. Apply the T/B force $uta(i)=M\xb7ut(i)$ to update the dynamics (8) for the $ith$ train. Step 6. Repeat Step 7.- Step 3– at the next sampling time $t+=t+\Delta t.$Step 6 |

end |

One salient feature of the proposed observer-based adaptive VC controller is that it enables one to perform a formal stability analysis of the derived VC control system model. More specifically, let $x\u0303t(i)=x\u0302t(i)\u2212xt(i)$ and $w\u0303t(i)=w\u0302t(i)\u2212wt(i)$ be the resulting individual estimation errors and $s\u0303t(i)=xt(i)\u2212xt(i\u22121)\u2212Dt$ be the individual spacing error for the $ith$ train. The following theorem states the sufficient condition that ensures the uniform ultimate boundedness of the estimation errors and spacing errors for each train. The proof follows the similar procedure in Ref. [36] and is thus omitted here for brevity.

Theorem 1. For the leader and follower trains modeled by Eqs. (10) and (9), if there exist the gain matrices $G1,G2,\u2009\u2009and\u2009G3$ under Step 1 and Step 2 in Algorithm 1, then the designed observer Eq. (12) and adaptive VC controller Eq. (13) guarantee that the resulting individual estimation errors $x\u02dct(i),\u2009w\u02dct(i),$ and the individual spacing error $s\u02dct(i)$ are uniformly ultimately bounded, namely, there exists small positive scalars $\sigma 1,\sigma 2,\sigma 3$ such that $limt\u2192\u221e||x\u02dct(i)||\u2264\sigma 1,\u2009limt\u2192\u221e||w\u02dct(i)||\u2264\sigma 2,\u2009limt\u2192\u221e||s\u02dct(i)||\u2264\sigma 3$ for any $i=1,2,\u2026,N$.

This controller represents one way to achieve virtual coupling. As reviewed in Refs. [14–16], many other controllers are also able to achieve the same basic function. Comparatively, the controller used in this paper has the following advantages: (1) fast computing speed thanks to the simple structure of the final controller and (2) adaptive feature allows the controller to be used for different trains and different track conditions without changing the controller gains.

## 4 Platooning Performance Using Distance Headway

Simulations were first carried out to model two heavy haul trains using a fixed distance headway, i.e., the objective of the follower train is to keep a fixed distance between it and the leader train. Figure 3 shows the elevation and curvature information of the simulated track section. The track section is taken from a real-world heavy haul railway. The track elevation is represented by the thick orange line on the graph; it fluctuates over the track location range. The elevation values range from below 0 to above 600 m. There is a long section of downhill grade from about 270 to 320 km. After that, the track shows a landscape that has variable grades. The curvature is represented by the thinner blue line on the graph. The values of curvature range from approximately −0.0025 to 0.0025 per meter, i.e., minimum curve radius of 400 m. This can be regarded as a tight curve radius from a typical freight railway point of view. Both simulated trains had the configuration of 1 locomotive + 109 wagons. Key train information is presented in Table 1.

Train information | Values |
---|---|

Train configurations | 1 locomotive + 109 wagons |

Locomotive mass | 162 tons |

Wagon mass | 108 tons |

Total train mass | 11,934 tons |

Locomotive length | 23 m |

Wagon length | 11 m |

Total train length | 1222 m |

Train information | Values |
---|---|

Train configurations | 1 locomotive + 109 wagons |

Locomotive mass | 162 tons |

Wagon mass | 108 tons |

Total train mass | 11,934 tons |

Locomotive length | 23 m |

Wagon length | 11 m |

Total train length | 1222 m |

The locomotive had traction and DB characteristics as shown in Fig. 2. The conversion factor from air brake pressures to brake shoe normal forces was set to be a constant at 441.6 N/kPa. The brake shoe coefficient of friction was expressed by Eq. (2) and the resistance forces were expressed by Eq. (3). A Proportional–Integral–Derivative controller was used to generate traction and brake forces for the leader train. Target speeds for the leader trains were set to be 80 km/h at the start of the simulation and 0 km/h from the track location of 381 km to stop the train. The leader train started at the location of 300 km with zero speed.

For the distance headway simulation, the follower train used the controller presented in Sec. 3 and started at the location of 298 km. In other words, the fixed headway was set to be 2 km. This value was selected to make the physical following distance of the follower train to be 788 m considering the leader train length is 1222 m. As discussed previously, one of the features of virtually coupled trainsets is that their train physical following distance should be shorter than the absolute braking distance. For such type of freight trains, the absolute braking distance at emergency from 80 km/h is about 800 m on flat track and the 788 m following distance satisfies this requirement. In addition, it is also safe for the most extreme case when the leader train is braking on a 1.2% grade uphill (braking distance 579 m) and the follower train is braking on a 1.2% grade downhill (braking distance 1293 m). The relative braking distance is 714 m.

Figure 4 shows the simulated headway, speed difference, and leader train speed. In this figure, the simulated headway indicates the distance between the leader and the follower train, and the speed difference indicates that between the leader and the follower train. It is noted that, for the fixed distance headway scheme, the objective of the follower train is to keep the same speed as the leader train with a fixed distance of 2 km. Therefore, the speed difference in Fig. 4 and the deviation of headway from 2 km can be regarded as the control errors. The results show that the maximum headway error and speed error reached 716 m and 24 km/h, respectively, during the simulation. These represent a 36% error and a 66% error respectively for references of 2 km and 36 km/h when the maximum errors were recorded. The exact reason for the errors will be discussed later in this paper.

Figure 5 shows the simulated traction and brake forces for the leader and follower train. In this figure, negative forces indicate brake forces while positive forces indicate traction force. Results show that the leader train had force variations from −1292 to 196 kN. From the start of the simulation to about 328 km, the leader train had mainly brake forces. However, it is noted that there was a brief traction period near the 308-km location. And the traction force had reached the corresponding maximum traction force as shown in Fig. 2. From 328 km to the end of the simulation, the leader train mainly had traction forces. Also considering the information for Fig. 2, it can be seen that the leader locomotive was operating at the corresponding maximum traction force most of the time. The operational states with maximum traction forces that were observed at the leader locomotive also influenced the operational states of the follower locomotive. As explained previously, if the leader train is already operating at its maximum traction force state, then the follower train needs to have lower or at least the same track gradient force to be able to match the speed of the leader locomotive. Otherwise, for example, if the follower train is experiencing a steeper uphill grade, the required distance headway cannot then be able to be kept. Specifically in Fig. 5, near the location of 308 km, the leader train had a case of operating at its maximum traction force. This had disturbed the operation of the follower train. To be able to keep the required headway, the follower train had to conduct some extreme traction and brake forces as shown in Fig. 5 in the section 308–322 km. These extreme traction and brake forces would cause significant impacts and in-train forces within the heavy haul train.

Figure 6 is a demonstration of possible in-train forces for sudden changes of traction and brake forces at large magnitudes. To obtain these results, a longitudinal train dynamics simulation [3] was carried out with the considerations of individual wagon models, coupler force models, and distributed brake forces. It is noted that such longitudinal train dynamics models are different from the longitudinal train motion models presented in Sec. 2. The difference is that the longitudinal train motion models do not consider in-train forces. And the purpose of the longitudinal train dynamics simulation here is to demonstrate the possible in-train forces. A train model the same as described at the start of this section was modeled with frictional draft gear and 10-mm uniform coupler slacks [3]. The maximum traction and brake forces were set to be the same as shown in Fig. 5. In the longitudinal train dynamics simulation, the traction and brake forces were alternating every 20 s. In Fig. 6, positive forces indicate compressive coupler forces whilst negative forces indicate tensile coupler forces. The results show three representative coupler positions. It can be seen that the maximum in-train forces were over 1200 kN for compressive and tensile forces. From a typical heavy haul train operational practice point of view, these force magnitudes are acceptable for coupler systems. However, they can be regarded as high forces for a heavy haul train given its length and mass.

Figure 4 shows that the headway error significantly increased after the track location of 330 km. The reason for this error had been briefly discussed before. It can be seen from Fig. 5 that, after the track location of 330 km, the leader train was mainly operating at its maximum traction state. This could have made the follower train unable to keep up with the leader's speed. If the follower experienced any larger resistance forces and/or track gradient forces, the headway would increase. Whenever the leader train moved away from its maximum traction force state, the leader train needed to operate at maximum traction force state to catch up with the leader train. As a result, the headway error also fluctuated during the process as shown in Fig. 4.

## 5 Platooning Performance Using Time Headway

The time headway simulation used the same track section and train models that have been used in Sec. 4. The difference here is that the train platoon started with a distance headway (2 km) and then switched to a time headway when the leader train reached 309 km. The time headway was set to be 94 s, which means the follower train was running 94 s behind the schedule of the leader train. In this case, the 94 s was determined by the train speed at the switching point, i.e., 21.27 m/s and the distance headway (2 km). By matching the product of time headway and train speed with the distance headway, two schemes can have zero impacts on the controller. In the time headway scheme, the follower train did not need to maintain the same speed of the leader train in the time domain. The objective was for the follower train to achieve the same train speed as the leader train at the same track location. In other words, the follower train was trying to copy the leader train's historical data with a designed time delay. It is noted that the time headway scheme cannot be used for train start and stop. This is because, with zero train speed, regardless how long is the time headway, the production of the time headway and zero train speed is zero. If used for train start and stop, the distance headway between two trains would become zero, which is not practical. Therefore, during the simulation, the platoon switched back to a distance headway scheme after 381 km to carry out train stops. The distance headway for the stop process was set to be 1633 m.

Figure 7 shows the leader train speed, location difference between the leader and the follower trains, as well as the speed difference between these two trains. The results show that the platoon achieved good performance during the start and stop stages. The distance headways were well maintained around the designed values: 2 km and 1.63 km for start and stop, respectively. During the time headway operations, as the leader train speed varied, the actual distance headway was roughly the production of the leader train speed and the time headway (94 s). Therefore, the actual distance headway, as expected, varied during the process. Due to the same reason, the speed difference between the two simulated train also varied during the process. It is noted that during the time headway operations, the corresponding distance headway can be shorter than 2 km. This is because the 2 km headway was set for 80 km/h. When train speeds are lower, the distance headway can be shorter and still keep the save relative braking distance. Figure 8 shows the actual space and speed errors outputted from the controller, i.e., $et(i)$ from Eq. (13). It can be seen that the controller in this case achieved excellent performance, the maximum space and speed errors being 0.07 m and 0.1 km/h, respectively. It is noted that both maximum errors were recorded before the switch point. When these maximum errors were recorded the reference space and speed were 2 km and 1.18 km/h, respectively. These indicate 0% and 9% errors for the space and speed, respectively. The space error and speed error during time-heady operations indicate the differences between the follower train's current status (location and speed) and the leader train's historical status (location and speed) 94 s ago.

It is noted that these results were obtained without the consideration of communication delays, communication errors, train location errors, etc. [37–39]. The design of the simulation was for the demonstration of the advantages of the time headway scheme. The simulated performance does not represent realistic controller performance. Per the authors' previous simulation experience like [39], train location errors have more direct and lower influences on the controller performance. For example, if the maximum train localization error is 10 m, this number will be mostly directly reflected at space errors, i.e., the absolute space error will be increased by 10 m. However, influences caused by the communication delays will be amplified by train speeds. For example, if the communication delay is about 2 s and the trains are running at 36 km/h (10 m/s), then the communication delay will increase the space error by 10 m/s X 2 s = 20 m. However, if the trains are running at 72 km/h (20 m/s), then the space error will be increased by 40 m.

Figure 9 shows the simulated traction and brake force of the platoon along the track location. It can be seen that, during the start and stop stages (distance headway), two trains had different traction and brake forces due to the fact that they had different target speeds at the same track location. However, during the time headway stage, the two trains had almost the same traction and brake forces at the same location of the track. This phenomenon agreed with the objectives of the controller design. More importantly, the time headway scheme avoided the outcome of high traction and brake forces as shown in Fig. 5.

## 6 Conclusion

Virtually coupled trainsets, or virtual coupling, use wireless communications to connect multiple trainsets as a network. These trainsets operate as a platoon in a coordinated manner. It is an emerging railway technology that has the potential to significantly increase railway operational efficiency.

The concept of railway virtual coupling is believed to be derived from road vehicle platooning. Current research in this topic is mainly focused on passenger trains and uses distance-based headways. However, this concept presents unique challenges when applied to heavy haul freight trains. For example, due to the longer train length and headway, rail tracks can vary greatly in elevation and curvature over the related distances. This variability makes it challenging to maintain a constant distance between virtually coupled trainsets. In this scenario, a time-based headway scheme is more suitable. The time time-based headway required the follower train to reproduce the leader train operational status at the same track location. This also allowed the follower train to copy any optimized train driving strategies from the leader train.

Demonstrative simulations were carried out without the consideration of communication errors and train localization errors. Two heavy haul trains were simulated; each had the configuration of 1 locomotive + 109 wagons (11,934 tons; 1222 m). A real-world heavy haul railway track section was simulated. A trip of 80 km long was simulated. The results showed that a conventional distance headway simulation had maximum distance and speed errors of 716 m and 24 km/h, respectively. These represent a 36% error and a 66% error from the references of 2 km and 36 km/h respectively when these maximum errors were recorded. A time-based headway simulation reduced the maximum distance and speed errors to 0.07 m and 0.1 km/h, respectively. These represent nearly 0% error for the space (reference 2 km) and a 9% error for the speed (reference 1.18 km/h). Future work will be directed to add the considerations of communication delays and train localization errors. Meaningful works can also be done in terms of network capacity differences when different headway schemes are used.

## Acknowledgment

The editing contribution of Mr. Tim McSweeney (Adjunct Research Fellow, Centre for Railway Engineering) is gratefully acknowledged.

## Funding Data

Australian Research Council Discovery Early Career Award funded by the Australian Government (Project No. DE210100273).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## References

**72**, pp.