On a Theory for Potential Capacity Gains due to Connected andAutonomous Trains
E. J. Moreya,∗ , R. E. Wilsona and K. Galvinba Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom

E.Morey@bristol.ac.uk, RE.Wilson@bristol.ac.uk
b Research, Technology & Innovation, Thales UK, Reading, United Kingdom

kevin.galvin@uk.thalesgroup.com∗ Corresponding author
Extended abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

March 18, 2022
Keywords: line capacity; fundamental diagram; block-based signalling; autonomous trains

1INTRODUCTION

This paper continues the work carried out in “Fundamental diagrams and emergent dynamics
of mainline rail operations” (Morey et al., 2021) and explores modelling improvements in capacity that could be available from the digitalisation of rail operations by increasing levels of
the European Train Control System (ETCS) (Stanley et al., 2011). Technological advances incommunications and receiving data (e.g., via 5G radio-based systems) may enable real-time monitoring and analysis (Theeg & Vlasenko, 2020), which could help increase line capacity, via useof moving-block signalling. In particular, trains might share position and speed directly with
their immediate neighbours, which might achieve better line capacity by implementing what wecall train-following models (TFMs) —– analogous to the car-following models (CFMs) that areused to describe highway traffic.

2SUMMARY OF LEGACY FIXED-BLOCK THEORYIn Morey et al. (2021), we investigated the line capacity of mainline rail operations that use legacy
fixed-blocked signalling rules, summarised by Pachl (2020) and the multi-author volume Theeg& Vlasenko (2020). In summary, the safety-critical principles are that (i) track is partitioned
into blocks; (ii) each block should only be occupied by at most one train at any time; (iii) eachtrain must be able to come to a stop safely within the blocks ahead (known as the ‘aspects’)that its driver knows to be clear, see Figure 1(a); and (iv) each train must be able to come to astop safely ahead of any block ahead that its driver knows to be occupied by another train, seeFigure 1(b).
Constant acceleration rules were then applied to Figure 1(a) to derive a maximum safe speed2vmax:= 2b(αLB − LM ),(1)where b is the braking rate, α is an integer-valued aspect parameter, LB is block length (whichfor simplicity is assumed the same for all blocks), and LM  LB is a safety margin.
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E. J. Morey, R. E. Wilson and K. Galvin2

Direction of Travel
i=1

i=2

i=3

(2)

(3)

(a)
(1)

LB

LBLB
Max. Stopping DistanceLM(b)Stopping Distance LMsLB

Figure 1 – Safety limit cases: represents α = 3 (four aspect) fixed block signalling, where (a) thefollower train must be able to stop safely within the blocks it knows are free, also (b) net spacing,s, must be sufficient for the follower to come to a stop before the leader’s block.
In contrast, constant acceleration formulae applied to Figure 1(b) yield a bound for speedv 2 ≤ 2b(s − LB − LM ),(2)

which involves the net-spacing s. Relation (2) is analogous to the equilibrium (uniform-flow)
speed-spacing relationship in a CFM (Wilson, 2001). It thus forms the basis for deriving a
set of fundamental diagrams (FDs) that describe smooth running scenarios via relations between
speed, density, and flow (corresponding to timetabled frequency). Trade-offs between the various
parameters were examined in some detail by Morey et al. (2021). The starting point for thisanalysis is to write ρ := 1/(s + LT ) (where LT is the train length) and to employ a jam density
ρJ := 1/(LT +LB +LM ) to find the speed-density relationship"v = V (ρ) := min v†

1 − ρ̃ρ̃1/2#, vmax ,(3)where v† := (2b/ρJ )1/2 and ρ̃ := ρ/ρJ , with 0 ≤ ρ̃ ≤ 1, defines a re-scaled density.
The resulting FDs are summarised in Figure 2 (black lines), with parameter choices LT =400m < LB = 1600m, LM = 100m, α = 2, b = 0.65 ms−2 and a = 0.4 ms−2 , justified in Moreyet al. (2021).

3CONNECTED AND/OR AUTONOMOUS TRAINS

Our new work is to consider potential capacity improvements achievable by the introduction ofconnected and/or autonomous trains (CATs). We assume that these CATs share each others’
positions and speeds directly and do not have to obey fixed-block rules. Moreover, they are
considered to be omniscient, rendering the limit described in Figure 1(a) and (1) redundant.
In contrast, the effect of continuous position knowledge is to set the block length LB to zero.
Thus (3) is modified by setting vmax = ∞ (although other practical physical limits might applyinstead) and by using an increased jam density ρJ := 1/(LT +LM ). This gives rise to a new set ofFDs indicated by the blue lines in Figure 2, quantifying the capacity gains. In summary, at any
given spacing (alternatively, density), the CATs will be faster than legacy trains. In consequence,flow is increased — the parameters we have chosen suggest that a doubling of capacity is possible.

4CONNECTION TO GIPPS’S CAR-FOLLOWING MODEL

The modelling so far has focused on safe operations under constant braking assumptions. This isalso the premise of the classical Gipps CFM (Gipps, 1981), which (translating into the notation
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E. J. Morey, R. E. Wilson and K. Galvin3(b) Speed-Density(a) Speed-Spacing
80800010000

Spacing, s (m)vmax

Speed, v (m/sec)

Speed, v (m/sec)

vmax

00

Density, ρ (trains/m) 0.002(c) Flow-Density(d) Speed-Flow
80

vmax

00

vmax

Speed, v (m/sec)

Flow, q (trains/sec)

0.03

Density, ρ (trains/m) 0.002

00Flow, q (trains/sec)

0.03

Figure 2 – Fundamental diagrams (FDs) for mainline rail traffic. Black lines show legacy operations and blue lines show the improved performance of connected and/or autonomous trains.
(a) Speed v versus net spacing s. (b) Speed v versus density ρ. (c) Flow q versus density ρ. (d)Speed v versus flow q.
of this paper) has an equilibrium speed-spacing function (Wilson, 2001) that takes the formv
 u(u11 )2s b −ub̂
−1 + t 1 +.

2

v=

τ +θ
1
1b −b̂

!

(τ + θ)(4)

Here b̂ is the maximum assumed braking rate of the lead vehicle, and τ and θ are parametersthat model a safety margin in terms of reaction-time delays. For trains, we effectively assume
the worst case scenario where the leader may undergo a ‘brick-wall’ stop. Hence we set b̂ = ∞.
Exploiting definitions of v† and ρJ , we may thus obtain

v = −v‡ +v‡2 + v†2

1 − ρ̃
ρ̃ 12,(5)

where v‡ := b(τ + θ) describes the additional velocity reduction that might have been achieved
if there were no reaction-time delay. By setting v‡ = 0, we recover the earlier result (3).
We might use (5) a starting point to model the reduction in capacity due to communication
and actuation delays. However, supposing that τ + θ is of the order of several seconds, then v‡ isof the order of just 1 or 2 ms−1 , far less than typical mainline operating speeds v† . So the effectof these delays on capacity is likely to be rather small.
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E. J. Morey, R. E. Wilson and K. Galvin

4

There are a couple of further avenues for investigation. Firstly, (i) Gipps’s model (4) with b̂
finite would allow us to relax the ‘brick-wall’ stop assumption of the earlier theory — although
this seems at odds with current railway safety thinking. Secondly (ii) it is known (Morey et al.,2021, Wilson, 2001) that these Gipps-based models can display dynamical instabilities and other
undesirable behaviour. Finally (iii) one should consider mixtures of legacy trains — one mightsay Driver Operated and/or Guided trains (DOGs) — and CATs. However, the benefits of CATs
are only realised if the CATs follow other CATs — which might be very difficult to organise given
practical timetabling constraints. We have a proverb in English: it is very difficult to herd CATs!
Of course, line capacity is only one part of the broader question of operational rail capacity,which also involves node capacity, fleet utilisation, crew provision etc. These points will be
expanded upon in our presentation at the TRISTAN meeting.

References
Gipps, P. G. 1981. A behavioural car-following model for computer simulation. Transportation ResearchPart B: Methodological, 15(2), 105–111.
Morey, E. J., Wilson, R. E., & Galvin, K. 2021. Fundamental diagrams and emergent dynamics ofmainline rail operations.
Pachl, J. 2020. Railway Signalling Principles. Technische Universität Braunschweig.
Stanley, P., Hagelin, G., Heijnen, F., Löfstedt, K., Poré, J., Suwe, K. H., & Zoetardt, P. 2011. ETCS forEngineers. Hamburg, Germany: DVV Media Group.
Theeg, G., & Vlasenko, S. 2020. Railway Signalling & Interlocking. 3 edn. Leverkusen: PMC MediaInternational Publishing.
Wilson, R. E. 2001. An analysis of Gipps’s car-following model of highway traffic. IMA Journal of Applied
Mathematics, 66(5), 509–537.

5Acknowledgements

This work is funded and delivered in partnership between the Thales Group and the Universityof Bristol, and with the support of the UK Engineering and Physical Sciences Research Council
Grant Award EP/R004757/1 entitled ‘Thales-Bristol Partnership in Hybrid Autonomous Systems
Engineering (T-B PHASE)’.

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