Speed that maximizes throughput of a track

On some forum or other, I read that the maximum throughput (in, say, trains per hour or passengers per hour) was reached at something like 40 miles per hour. That is, as you raise the speed of the trains, at some point, you reduce the maximum number of passengers that can be carried per hour from point A to point B on a track, and that this point is reached way below any speed that most of us would call "high speed rail." In other words, HSR is a less efficient (in terms of throughput) use of tracks than lower-speed trains.

I was surprised at this, so I put together a rather simple-minded mathematical model.

First:

Capacity = (Passengers per unit time) = N v / s

where

N = passengers per train
v = speed
s = train spacing.

Next, I assumed that trains have to have at least one braking distance between them. That is, the spacing has to be such that if the train in front of you were stopped (or were to stop instantly), you could come to a complete stop without hitting it. That gives:

s = L + (v^2)/(2 B)

where

L = the amount of track space a train takes up when stopped (that's bigger than the actual train lenght, BTW)
B = the maximum braking deceleration that it is safe to assume you have.

and get:

capacity = N v / ( L + (v^2)/(2 B) )

A little calculus shows that this is maximized when

L = (v^2)/(2 B)

That is, the maximum transport is reached when your braking distance is equal to the space that a train occupies when stopped (or is very slow.)

My first reaction was that this must be a very slow speed. I realize that I have no idea what numbers to use for L or B, but I have a really hard time believing that a 40 mph train can stop in, say, 8 car lengths.

Does my analysis make sense? Is my model at all reasonable?

And does anyone have a better idea what stopping lengths are for trains, or how much space railroads have to give to a typical passenger train before allowing for the increased stopping distance due to speed?

On some forum or other, I read that the maximum throughput (in, say, trains per hour or passengers per hour) was reached at something like 40 miles per hour. That is, as you raise the speed of the trains, at some point, you reduce the maximum number of passengers that can be carried per hour from point A to point B on a track, and that this point is reached way below any speed that most of us would call "high speed rail." In other words, HSR is a less efficient (in terms of throughput) use of tracks than lower-speed trains.

I am sure you are mistaken. The key to throughput is signalling systems.

Consider Eurotunnel where the working speed is about 100 mph. They use moving block signalling which takes into account the length of trains (very long), the speed and other conditions.

Or the British West Coast mainline where the number of trains is per hour is very high, with minutes between them. The working speed there is 125 mph with the possibility of higher speeds.

Japanese subways operate with as little as 2-3 minutes of headway.

Japanese subways operate with as little as 2-3 minutes of headway.

Certainly the freight railroads argue that a fast passenger train takes up more track space than a slow freight train . but i think theyre only looking at one side of the equation.

Yes, indeed. To a high speed rail operator, a slow freight takes up a lot of track space, as it takes time to clear a block due to its very slow acceleration and length (at least in the N.A. situation).

But addressing amm's comments/questions, you have to take into account the station spacing, track conditions and so forth (as D.B. mentions), then the signal spacing ("blocks") and speed limits are determined. Perhaps 40mph top speed allows for highest passenger throughput- but to say this makes HSR less efficient is incorrect- as you are comparing apples to oranges. Subways serve high density areas with large passenger loads over relatively short distances. HSR carries large passenger loads over intercity (i.e. long) distances.

As for stopping length for trains, railways in Japan allow 130kmh top speed on lines with grade crossings, as this is the top speed from which a train can stop within 600 meters. Of course at lower speeds trains can stop at a shorter distance. Lines without grade crossings and lineside signaled for high speed running (typically an additional block added as a "cushion"), can run at 160kmh, cab signalling allows even higher speeds (like shinkansen)- and like the Eurotunnel, signaling and train control systems allow more fine tuning to increase train frequency by reducing braking distance.

george matthews wrote:I am sure you are mistaken. The key to throughput is signalling systems.

Can you expand on this?

My model didn't consider signaling systems as such, but it did assume that any train must have a full stopping distance between it and the next train. I believe this is the principle on which conventional block signals are based.

The only way I could see to reduce this spacing while maintaining spacing would be to factor in the front train's speed and minimum possible stopping distance versus the back train's speed and the stopping distance one can safely assume it is capable of. (This is what drivers on highways and roads try to do -- if they're trying to drive safely.) This would require more sophisticated signalling systems than conventional block signals.

george matthews wrote:.. the British West Coast mainline where the number of trains is per hour is very high, with minutes between them. The working speed there is 125 mph with the possibility of higher speeds.

Not sure what you mean by "minutes" exactly. 5 minutes at 125 mph works out to 10 miles spacing, which sounds to my inexpert ear like plenty of space to come to a complete stop from 125 mph. I'm hoping someone who actually knows these numbers (stopping distances, train spacing distances, etc.) will speak up, though.

kaitoku wrote:... Perhaps 40mph top speed allows for highest passenger throughput- but to say this makes HSR less efficient is incorrect- as you are comparing apples to oranges.

Are you objecting to my using the word "efficient" to mean "highest passenger throughput"? If so, it is understandable, but is a different discussion. (E.g., what is the most appropriate way to define "efficiency" for HSR?) If you prefer, I will stop using the word "efficiency" in this context.

If, however, there is a way in which HSR obtains its maximum passenger throughput at a substantially higher speed than conventional rail (say, at 180 kph instead of 65), I'd love to hear how this is obtained. Since I'm a techie, I'd love to hear the technical details.

And if there is a way that you can (theoretically) get whatever throughput you want by simply making the speed high enough, I'd be even more interested.

kaitoku wrote:As for stopping length for trains, railways in Japan allow 130kmh top speed on lines with grade crossings, as this is the top speed from which a train can stop within 600 meters.

(130 km/hr = ~ 80 mi/hr. -- interesting coincidence.)

I'd always heard that USA (freight?) trains take over a mile to stop, and they generally run a lot slower than this. Is the 600 meters (~ 0.4 mi.) figure only for passenger trains? Do passenger trains have substantially better braking deceleration than freights?

With the exception of maybe some of the long distance trains, Amtrak and commuter trains generally don't exceed 10 cars, with one or sometime two locomotives. A freight train may have dozens or over a hundred cars, which means a much longer stopping distance.

Is the 600 meters (~ 0.4 mi.) figure only for passenger trains? Do passenger trains have substantially better braking deceleration than freights?

No, the 600m is for all trains. JR Freight's M250 "super rail cargo" container train runs at 130km/h on some sections of track, but otherwise the fastest freights (all container trains) are restricted to 100 or 110km/h.

http://en.wikipedia.org/wiki/M250_series

And yes, passenger trains are better than freights at braking, due to plain physics (shorter and lighter) and modern passenger trains are fitted with electronic controlled braking (instantaneous response) as well as disk brakes, rather than the locomotive supplied sequential air brakes and friction (brake shoe) brakes.

what is the most appropriate way to define "efficiency" for HSR?

Perhaps by comparison with other (non HSR) passenger trains on the same route, or competing forms of transport (car, bus, airplane).

amm in ny wrote:On some forum or other, I read that the maximum throughput (in, say, trains per hour or passengers per hour) was reached at something like 40 miles per hour. ... I was surprised at this, so I put together a rather simple-minded mathematical model. ... That is, the maximum transport is reached when your braking distance is equal to the space that a train occupies when stopped (or is very slow.)

40 miles/hour is the rule of thumb for the speed at which maximum highway throughput is maximized. Maybe it's the same for railways, but I'd be wary of accepting this number without a source.

It is true that HSR is a more inefficient use of tracks, in the sense of lower throughput. If there's only one track and it's impossible to build another one, then of course you want to get the absolute maximum use out of it.

But throughput measures only the rate at which vehicles pass by a given point: x cars passed by this point in an hour. It doesn't capture the fact that the people in the vehicles are traveling slower in order to achieve this maximum throughput -- and therefore getting to their destination slower. Their time is also worth something and should not be ignored.

Perhaps the value of that lost time exceeds the amortized cost to build a second track, and run both tracks at half of their maximum throughput -- but higher speed. In other words, throughput is the wrong number to maximize in a transportation system.

taoyue wrote:40 miles/hour is the rule of thumb for the speed at which maximum highway throughput is maximized. Maybe it's the same for railways, but I'd be wary of accepting this number without a source.

It was a posting at Railroad.net, I think it was the NJ Transit forum. It was by an engineer (NJ Transit?), or at least that's what (s)he said. IIRC, the figure was 45 mph.

taoyue wrote:But throughput measures only the rate at which vehicles pass by a given point: x cars passed by this point in an hour. It doesn't capture the fact that the people in the vehicles are traveling slower in order to achieve this maximum throughput -- and therefore getting to their destination slower. Their time is also worth something and should not be ignored.

I agree. I wasn't arguing that lower throughput per track meant that HSR was a bad idea.

But, until I had done the calculation, it had not occurred to me that increased speed might, at some point, decrease throughput. Or that that point might be so quickly reached. It just seemed counter-intuitive. It just seemed so obvious that higher speeds would result in more trains per hour. I keep thinking that there must be some factor that I've left out.

On the French LGV (High Speed Line) Jonction which acts as a belt line around the east side of Paris, peak traffic in each direction of a double track railway is 26 trains per hour, or one train every 2' 20". If every train was a double TGV Duplex, then each train would have 1100 seats giving 28600 seats per hour in one direction or 57200 seats per hour past any given point along the line. Now not every train consists of Double Duplex pairs, only about a third at the present. TGVs have four braking systems, Tread Brakes, Multi-Disc Brakes, Regenerative Brakes, and Eddy-current Brakes. The tread brakes are only used at low speeds, and the Eddy-current brakes are only used in an Emergency.

For an explanation of a signalling system capable of protecting High-speed trains see this Wikipedia article covering the German LZB system;

http://en.wikipedia.org/wiki/LZB

What you model doesn't consider is signal block length which typically is 2-miles for North American freight railroads, but can easily be as short as half a kilometer in places on European High Speed lines, though one kilometer is standard. Stopping distance for a French TGV or German ICE from 300 kph (186mph) is about 9 kilometers (5.59 miles), without using the Emergency brakes.

JayBee wrote:On the French LGV (High Speed Line) Jonction which acts as a belt line around the east side of Paris, peak traffic in each direction of a double track railway is 26 trains per hour, or one train every 2' 20"....Stopping distance for a French TGV or German ICE from 300 kph (186mph) is about 9 kilometers (5.59 miles),...

You don't say what you are trying to show by giving these figures, but I am grateful to have numbers that are used in real life. Anyway, I thought I'd see how the train spacing compares with the time to come to a complete stop (9 km)

300 kph = 1/12 km / sec.
2' 20"= 140 sec / train = ~12 km train spacing.

This looks like one stopping distance, with a safety factor thrown in. It also sounds like they are getting close to the theoretical maximum capacity of one track.

Of course, if the trains ran only 150 kph, the stopping distance would only be 2.25 km. Adding 1.5 km or so for safety, you could run one train every 90-100 seconds. This is what I meant by "increased speed leads to decreased throughput."

JayBee wrote:What you model doesn't consider is signal block length which typically is 2-miles for North American freight railroads, but can easily be as short as half a kilometer in places on European High Speed lines, though one kilometer is standard.

I'm not sure why everyone keeps throwing in "block lengths" and "improved signaling." AFAIK, the point of blocks and signals is to insure that trains never get closer than some minimum safe distance (which I've assumed to be the stopping distance) to each other. No matter what kind of system you use, you can't/mustn't get them closer than that. A poor system may force trains to be further apart, but if it lets them get closer, the railway is going to get shut down entirely (I hope!)

My calculation implicitly assumes an optimal distance-maintaining mechanism, and it doesn't matter what it is (1cm blocks, radio control, mental telepathy, whatever.)

What would make a difference would be if you could have a minimum safe spacing that didn't scale as the square of the speed. For automobiles on a highway, we are taught a minimum safe spacing rule that is linear in speed (in the USA: 1 car-length per 10 miles/hour.) Doing the same calculation as before, but with this rule, you get that as you increase the speed, you increase the throughput, but that increase drops off to nothing as you approach some limiting number of cars per second.

Disclaimer: I am aware that there are lots of other considerations besides theoretical maximum throughput. This is just one consideration that I could apply some back-of-the-envelope math to.