How does capacity change with speed?

[Ironside]

One thing that confuses me is trying to understand how the number of trains that can use a line changes as the speed of the line increases.

When trains are running slower presumably they can get closer to the train in front but take longer to clear a given section. So if trains are run at 400 kph rather than 200 kph does the number of paths per hour change?

 

[AM9]

One thing that confuses me is trying to understand how the number of trains that can use a line changes as the speed of the line increases.

When trains are running slower presumably they can get closer to the train in front but take longer to clear a given section. So if trains are run at 400 kph rather than 200 kph does the number of paths per hour change?

Looking at it from a rolling stock point of view. Each train on a 400km travelling at a maximum of 200km/h might take 2.5 hrs non-stop end to end with a 20 min turnaround. If the speed was increased to 400km/h it might take 1.5 hrs end to end with the same turnaround.
If there was a requirement to carry 3000 passengers per hour, and each train could seat 1000 passengers then a 20 minute service frequency would be required. To provide that service on a 200kph line, it would require 17 trains yet only 11 trains on a 400kph line. Apart from the capital cost of the additionsl trains, there would be extra storage sidfings, more staff and probably more maintenance despite their lower speeds.

 

[Searle]

I would expect the frequency to decrease. Take a mainline with 3 state signalling (red, single yellow, green), the linespeed can't go over 75mph~, as the trains won't be able to stop in one signal length of they go any faster. You can run trains two signals apart for them both to see green all the way.

Imagine if it was the upgraded to a 125mph to railway now, so it would have 4 state signalling (red, single yellow, double yellow, green). The trains will be only be able to run three signals apart now, so you have an evident reduction in capacity.

It's all to do with braking distances in short.

(Please note that I've made most of the terms up and the speeds as well because I don't work on the railway!)

 

[Voglitz]

One thing that confuses me is trying to understand how the number of trains that can use a line changes as the speed of the line increases.

When trains are running slower presumably they can get closer to the train in front but take longer to clear a given section. So if trains are run at 400 kph rather than 200 kph does the number of paths per hour change?

The line capacity is lower at 400 km/h. There is an FOI request, overdue at the time of writing.

 

[AM9]

I would expect the frequency to decrease. Take a mainline with 3 state signalling (red, single yellow, green), the linespeed can't go over 75mph~, as the trains won't be able to stop in one signal length of they go any faster. You can run trains two signals apart for them both to see green all the way.

Imagine if it was the upgraded to a 125mph to railway now, so it would have 4 state signalling (red, single yellow, double yellow, green). The trains will be only be able to run three signals apart now, so you have an evident reduction in capacity.

It's all to do with braking distances in short.

(Please note that I've made most of the terms up and the speeds as well because I don't work on the railway!)

If we are talking 400km/h then visual signals would be useless (they aren't allowed above 225km/h). Any new infrastructure designed for speed will have moving block and in-cab signalling.

 

[Voglitz]

If we are talking 400km/h then visual signals would be useless (they aren't allowed above 225km/h). Any new infrastructure designed for speed will have moving block and in-cab signalling.

Is HS2 going to be equipped with moving block signalling?

 

[HSTEd]

The number of paths will reduce as the train speed increases.
However this is a surprisingly gradual process - the stopping distance increases as the train speed increases, but the time taken to traverse the stopping distance decreases - so whilst the spacial separation of the trains climbs the temporal separation only does so more slowly.

The limiting factor is not the emergency stopping distances of stacked trains - it is in fact that timetabled trains will end up timetabled in any order and could potentially end up out of order. As a result there must be sufficient spacing between trains for a train to clear a diverging junction, for the interlocking to confirm this and order the points moved to the opposite position, then allow the points to move and for the system to gain interlock before it transmits the movement authority to the incoming train.
All this must be achieved before the train exceeds its previous movement authority and preferably without requiring the train to slow to an unnecessary degree.

If we are merely considering maximum stopping distances, from 340km/h Japanese trainsets [as a result of the Fastech360Z programme] have been designed to stop in roughly 4000m in an emergency scenario, without using the fancy air brakes they have designed.
That translates to a headway of 42 seconds.
The bulk of the headway for the signalling system is the interlocking equipment I mentioned before - which in time terms is independent of train speed.
--- old post above --- --- new post below ---

Is HS2 going to be equipped with moving block signalling?

No, because moving block signalling on a main line railway does not exist.
It will be computer controlled block signalling with lots of track circuits.

 

[Voglitz]

No, because moving block signalling on a main line railway does not exist.
It will be computer controlled block signalling with lots of track circuits.

18 trains per hour on a 360 km/h railway, also does not exist.

 

[HSTEd]

18 trains per hour on a 360 km/h railway, also does not exist.

Perhaps not, but things exist that are a hell of a lot closer to it than todays signalling is to moving block.
13-14tph on HSLs is common, and 360km/h operation has been trialled extensively abroad.

 

[AM9]

18 trains per hour on a 360 km/h railway, also does not exist.

Yet. I assume the OP has a view for the future, not one that labours the shortcomings of current conventional signalling.

 

[edwin_m]

The HS2 assumption is indeed for fixed block signalling, but I imagine they could go to moving block if they are convinced the technology is available and reliable by the time they need to decide. Euston will probably still constrain the capacity of the route as a whole, so this could be an opportunity to get some through trains onto HS1 or some other non-London destination.

At very low speeds the capacity reduces due to the length of time each train takes to pass over the line, but at higher speed it reduces because the braking distance increases. The optimum speed for maximum capacity depends on a range of factors but I think it's typically around 60mph.

There used to be a simple analysis on the HS2 website of the effect of a lower-speed junction - not sure if it has survived the transition to .gov.uk. For the benefit of McCoy there were and probably still are also some documents demonstrating that 18 high speed trains per hour is feasible with a fixed block system.

 

[AM9]

The number of paths will reduce as the train speed increases.
However this is a surprisingly gradual process - the stopping distance increases as the train speed increases, but the time taken to traverse the stopping distance decreases - so whilst the spacial separation of the trains climbs the temporal separation only does so more slowly.

The limiting factor is not the emergency stopping distances of stacked trains - it is in fact that timetabled trains will end up timetabled in any order and could potentially end up out of order. As a result there must be sufficient spacing between trains for a train to clear a diverging junction, for the interlocking to confirm this and order the points moved to the opposite position, then allow the points to move and for the system to gain interlock before it transmits the movement authority to the incoming train.
All this must be achieved before the train exceeds its previous movement authority and preferably without requiring the train to slow to an unnecessary degree.

If we are merely considering maximum stopping distances, from 340km/h Japanese trainsets [as a result of the Fastech360Z programme] have been designed to stop in roughly 4000m in an emergency scenario, without using the fancy air brakes they have designed.
That translates to a headway of 42 seconds.
The bulk of the headway for the signalling system is the interlocking equipment I mentioned before - which in time terms is independent of train speed.
--- old post above --- --- new post below ---


No, because moving block signalling on a main line railway does not exist.
It will be computer controlled block signalling with lots of track circuits.

So by having many small fixed blocks, can they be grouped into larger ones at high speed, effectively giving some of the benefits of moving block?

 

[HSTEd]

So by having many small fixed blocks, can they be grouped into larger ones at high speed, effectively giving some of the benefits of moving block?

Yes, indeed HS2 proposes to do this at junctions for this very reason.
Block lengths measured in a few hundred metres at 250km/h or 360km/h will pass rapidly.

Also this document makes very interesting reading - and I am glad I decided to save it and other stuff at the time. Gov.uk is hopeless.

 

[Voglitz]

Yet. I assume the OP has a view for the future, not one that labours the shortcomings of current conventional signalling.

The OP asked, in the 'Future High Speed Rail' forum, how does capacity change with speed?

I provided an accurate answer.

 

[MarkyT]

An important thing to consider is that with conventional lineside signalling the signal positioning is fundamentally tied to braking distance of the highest speed trains envisaged to use any particular line. There is usually around one standard braking distance (at max speed and adjusted for gradient) between 3-aspect signals, and around half a standard braking distance between 4-aspect signals. This gives maximum headway capacity at the highest speed but constrains capacity for a series of trains following each other at any speed lower than that maximum.

By contrast with a modern in-cab system such as ETCS, the spacing of block markers is no longer directly linked to braking capability and there are usually more individual shorter blocks sections within the braking distance. Hence without the complexities of non-existent moving block technology, following trains can adjust their spacing more readily, keeping as many free blocks ahead of them as they need at their current speed.

The greater number of shorter blocks within a braking distance can be thought of as being more 'aspects'. The problem is that with more aspects the closer the actual train spacing approaches the theoretical braking distance:

In 3-aspect signalling ideal unconstrained spatial separation of trains H (headway distance) is as follows:

H3 = 2d + S + O + L, where d = braking distance, S = sighting distance, O = overlap, L = train length

In 4-aspect:

H4 =1.5d + S + O + L

In a notional 5 aspect (like Japanese conventional signalling):

H5 = 1.333d + S + O + L

etc.

With a very high-resolution cab based fixed block or moving block system the spacing could approach:

H = d + O + L

That's great for following traffic capacity but incorporates no margin for 'closing up' behind a preceding train slowing down for a junction divergence for instance. What I mean by this is that headway normally assumes drivers seeing greens at a nominal sighting point (the S in the expressions above often assumed to be around 800m on approach and well before the AWS). In practice that means in 4-aspect territory , for instance, if the double yellow ahead of you doesn't step up to green, then you've got around 800m decision distance before you must start braking. That gives time for the preceding train to turn off, the route to reset for the main and a sequence of greens to magically reappear in front of your eyes. The closer a cab signalling system approaches the magical 1d spacing, the smaller this margin becomes, and the more quickly a slowdown of one train affects the ones following it.

 

[The Planner]

I would expect the frequency to decrease. Take a mainline with 3 state signalling (red, single yellow, green), the linespeed can't go over 75mph~, as the trains won't be able to stop in one signal length of they go any faster. You can run trains two signals apart for them both to see green all the way.

Imagine if it was the upgraded to a 125mph to railway now, so it would have 4 state signalling (red, single yellow, double yellow, green). The trains will be only be able to run three signals apart now, so you have an evident reduction in capacity.

It's all to do with braking distances in short.

(Please note that I've made most of the terms up and the speeds as well because I don't work on the railway!)

Good job you put the disclaimer at the end, as we run 125mph on 3 aspect!

 

[HSTEd]

Another advantage of having short block stations is that in stations, like OOC, you can potentially have a train entering a platform before the previous train has completely cleared it.
You can do this because as the train pulls out each section in turn can be added to the movement authority of the train behind, allowing it to decelerate more slowly than if it had to stop and then draw forward into the platforms.

 

[The Planner]

To be fair we try and do that now under conventional signalling to get reoccupation down, we are not very good at it though!

 

[krus_aragon]

With a very high-resolution cab based fixed block or moving block system the spacing could approach:

H = d + O + L

(where d = braking distance, S = sighting distance, O = overlap, L = train length)

I found myself thinking along similar lines. As signal block lengths reduce, approaching a limit of zero (e.g. ideal moving block) you end up looking at a situation similar to that of the Highway Code's stopping distances for cars:

So let's drop the signals and work on an idealised railway with full moving block signalling.

At higher speeds the spacing required is largely dictated by the braking distance. This increases in proportion to the square of the speed (check how it changes between 20mph and 40mph for the car above). So while a train travelling twice as fast could make twice as many return trips in a day, you'd need more than twice as much space between trains: up to four times as much.) So unless you change the rules (e.g. the vastly improved brakes on the HST) going twice as fast can actually halve your capacity.

Lengthening your trains, however, will have a negligible effect on braking distance (assuming a distributed traction EMU, where extra carriages don't reduce acceleration), and gives extra capacity. Doubling the length of your high-speed EMU will almost double your passenger capacity.


Given the above, it's not surprising that HS2 is being designed for trains up to 400m long: lots of capacity! But why operating at such high speeds, when that reduces capacity? There are a number of external influences. Long-distance railways are in competition with airlines, and people tend to prefer a quick journey allowing more time at their destination.

Sir Richard Moon (chairman of the LNWR) felt that 40mph was more than fast enough for a long-distance express passenger train, and it was far more economical in coal to go that slow. Only when the East Coast route to Scotland started competing in the Race to the North did they speed up to avoid losing custom!

 

[Rich McLean]

I would expect the frequency to decrease. Take a mainline with 3 state signalling (red, single yellow, green), the linespeed can't go over 75mph~, as the trains won't be able to stop in one signal length of they go any faster. You can run trains two signals apart for them both to see green all the way.

Imagine if it was the upgraded to a 125mph to railway now, so it would have 4 state signalling (red, single yellow, double yellow, green). The trains will be only be able to run three signals apart now, so you have an evident reduction in capacity.

It's all to do with braking distances in short.

(Please note that I've made most of the terms up and the speeds as well because I don't work on the railway!)

There are stretches of 125mph with 3 aspect signalling, with sections long enough to bring a HST to a stop

 

[NotATrainspott]

I found myself thinking along similar lines. As signal block lengths reduce, approaching a limit of zero (e.g. ideal moving block) you end up looking at a situation similar to that of the Highway Code's stopping distances for cars:

So let's drop the signals and work on an idealised railway with full moving block signalling.

At higher speeds the spacing required is largely dictated by the braking distance. This increases in proportion to the square of the speed (check how it changes between 20mph and 40mph for the car above). So while a train travelling twice as fast could make twice as many return trips in a day, you'd need more than twice as much space between trains: up to four times as much.) So unless you change the rules (e.g. the vastly improved brakes on the HST) going twice as fast can actually halve your capacity.

Lengthening your trains, however, will have a negligible effect on braking distance (assuming a distributed traction EMU, where extra carriages don't reduce acceleration), and gives extra capacity. Doubling the length of your high-speed EMU will almost double your passenger capacity.


Given the above, it's not surprising that HS2 is being designed for trains up to 400m long: lots of capacity! But why operating at such high speeds, when that reduces capacity? There are a number of external influences. Long-distance railways are in competition with airlines, and people tend to prefer a quick journey allowing more time at their destination.

Sir Richard Moon (chairman of the LNWR) felt that 40mph was more than fast enough for a long-distance express passenger train, and it was far more economical in coal to go that slow. Only when the East Coast route to Scotland started competing in the Race to the North did they speed up to avoid losing custom!

If HS2 at 400km/h had half the capacity of a 200km/h railway, then that 200km/h railway would be running at 40tph, higher than the Victoria Line or the DLR. I think the rate at which capacity drops decreases as you get faster. The fast lines into various South London rail termini could only be 160km/h at most (third rail speed limit) and they run at 22tph in the peaks, which is around what HS2 would be limited to at Birmingham Junction once you've got full captive running everywhere, even at 360-400km/h.

 

[Metrailway]

If HS2 at 400km/h had half the capacity of a 200km/h railway, then that 200km/h railway would be running at 40tph, higher than the Victoria Line or the DLR. I think the rate at which capacity drops decreases as you get faster. The fast lines into various South London rail termini could only be 160km/h at most (third rail speed limit) and they run at 22tph in the peaks, which is around what HS2 would be limited to at Birmingham Junction once you've got full captive running everywhere, even at 360-400km/h.

Frequencies of 40+ tph have been run on the Underground in the past with conventional signalling but lower speeds. 40tph was the 'standard' peak frequency for LT its early years.

IIRC the Moscow Metro still runs these frequencies with conventional signalling.

 

[HSTEd]

At higher speeds the spacing required is largely dictated by the braking distance. This increases in proportion to the square of the speed (check how it changes between 20mph and 40mph for the car above). So while a train travelling twice as fast could make twice as many return trips in a day, you'd need more than twice as much space between trains: up to four times as much.) So unless you change the rules (e.g. the vastly improved brakes on the HST) going twice as fast can actually halve your capacity.

The braking distance is however not the only component of the technical headway - for example you have to account for point interlocking times and the like at diverging junctions.
These quantities are purely times and thus remain the same no matter what the train speed is.
There are also reaction times for the trains computer system to receive an updated movement authority and for the computer to allow time for the driver or ATO system to respond.
Additionally you have to account for the fact that we don't have perfect train location systems and still have to rely on track circuit blocks, so the length of blocks becomes an important factor - at higher speeds you clear blocks more rapidly so they can be released for the train behind to use.

Lengthening your trains, however, will have a negligible effect on braking distance (assuming a distributed traction EMU, where extra carriages don't reduce acceleration), and gives extra capacity. Doubling the length of your high-speed EMU will almost double your passenger capacity.

Actually longer trains reduces stopping distance - Japanese studies indicate that in wet conditions with degraded traction, vehicles further down the train have better braking performance as they "see" a dry rail as the water has been driven off by about the 20th Axle.

 

[SpacePhoenix]

Does having distributed traction make much difference with regard to regenerative breaking? A train (7+2) with two power cars might have say 4 traction motors and a train with distributed traction (9 car), if it's got a traction motor per bogie that's 18 TMs per train.

 

[HSTEd]

Does having distributed traction make much difference with regard to regenerative breaking? A train (7+2) with two power cars might have say 4 traction motors and a train with distributed traction (9 car), if it's got a traction motor per bogie that's 18 TMs per train.

Regenerative braking performance is drastically better with distributed traction - as all the brake force applied can be regenerative.
With power cars, only they can apply brakeforce with regenerative brakes, so they can brake far less harshly without causing the wheels to slip.

 

[edwin_m]

Regenerative braking doesn't affect headway on plain line.

In theory trains are separated by minimum braking distance so that in the unlikely event of something happening to the train in front, the following one can brake to a stop. Because this rarely happens in practice, it is normal to calculate the headway based on maximum service brake.

In cases where the train actually slows down, it is better to base the headway calculations on the optimum rather than the maximum braking rate. This is likely to be the use of regenerative brake on its own, which will therefore affect calculations of headway around stations and junctions. The attainable rate of regenerative braking depends principally on the proportion of the train's weight that is sitting on motored axles.

 

[krus_aragon]

The braking distance is however not the only component of the technical headway - for example you have to account for point interlocking times and the like at diverging junctions.
These quantities are purely times and thus remain the same no matter what the train speed is.
There are also reaction times for the trains computer system to receive an updated movement authority and for the computer to allow time for the driver or ATO system to respond.
Additionally you have to account for the fact that we don't have perfect train location systems and still have to rely on track circuit blocks, so the length of blocks becomes an important factor - at higher speeds you clear blocks more rapidly so they can be released for the train behind to use.

All valid points, hence why I was speaking an idealised (mathematical) manner. But my main point was this:

While the above are fixed O(1) or have a linear O(x) relationship to the speed, the braking distance has a square O(x[sup]2[/sup]) relationship. For higher values of x (speed), the changes in braking distance will be greater than the other two combined, so one can model the spacing as a square O(x[sup]2[/sup]) relationship. (Please forgive the use of computational complexity notation: I'm having a mental block and can't think of the normal mathematical notation.)

To calculate actual headway, yes, you'll need to calculate all the values. but I was just trying to show the general pattern.

Actually longer trains reduces stopping distance - Japanese studies indicate that in wet conditions with degraded traction, vehicles further down the train have better braking performance as they "see" a dry rail as the water has been driven off by about the 20th Axle.

I thought that might be the case, but didn't want to make any claims without knowing.
--- old post above --- --- new post below ---

If HS2 at 400km/h had half the capacity of a 200km/h railway, then that 200km/h railway would be running at 40tph, higher than the Victoria Line or the DLR. I think the rate at which capacity drops decreases as you get faster. The fast lines into various South London rail termini could only be 160km/h at most (third rail speed limit) and they run at 22tph in the peaks, which is around what HS2 would be limited to at Birmingham Junction once you've got full captive running everywhere, even at 360-400km/h.

This was based on an idealised moving block system, with no consideration of diverging points, station stops, etc. While I'm not familiar with the routes into South London, I'd assume that signal spacing, stopping patterns and acceleration are the main issues in calculating headways.

 

[edwin_m]

While the above are fixed O(1) or have a linear O(x) relationship to the speed, the braking distance has a square O(x[sup]2[/sup]) relationship. For higher values of x (speed), the changes in braking distance will be greater than the other two combined, so one can model the spacing as a square O(x[sup]2[/sup]) relationship.

Just to complete the story, the important figure is the time interval between trains, which determines the number of trains a route can carry. If the distance separation varies approximately with the square of speed, the time variation varies approximately linearly with speed because a train going faster covers the same distance in less time.

 

[EM2]

Just seen this from a colleague on Twitter:

'Does anyone know where to find info on how increasing line speed on a single track branch increases frequency?'

I know he's doing a presentation about Braintree, which has a maximum line speed of 50mph, and a service pattern of 1tph each way.
How much would speed have to increase to double the frequency? And how would you do it?

 

[The Planner]

Not au fait with Braintree, but i'll guess its one train in section? What are they assuming as the turnaround at each end? 9 minutes as now? What is the plan with the LCs, as they appear to put a spanner in it? I take it no new infrastructure is to be built and this is cheap as chips? The distances between the stations means that you aren't going to get a lot, if anything, from linespeed to be honest. They need to look at the Witham end and the 10mph to start with IMO.