Setting Speed Zones
At the Boston meetup two days ago, I was asked about what tools I use to generate timetables, for example for my New York commuter rail posts. The answer is that I use speed zones and then run this code on them – but then the question is how to figure out speed zones. I hope that this sequence of steps will help advocates who are interested in rail modernization.
Generating curve radii
The most difficult element to fix on mainline rail is the right-of-way geometry. Most other things that can restrict a train’s speed can be fixed with more modern maintenance, but right-of-way geometry doesn’t change without physical construction, often in constrained areas – if they weren’t constrained, the curves would have been built wider in the first place.
The best case scenario is that there exist track maps with exact curve radii. I have these for large chunks of the Northeast Corridor, but not all. For example, here is Metro-North (with thanks to The Korot). Curves on such maps are denoted as circles or bumps deviating from a line, with the direction of the circle indicating the direction of the curve’s curvature. On this and other American maps, the radius is listed in degrees, and the cant (see the section below) in inches.
To convert the radius from degrees to more usual units, set one degree to be 1,746 meters, and note that degrees measure curvature and not radius, so a two degree curve has half the radius of a one degree curve. More precisely, the formula is that degrees measure the change in azimuth over 100 feet; 100 feet are 30.48 meters, and converting 30.48 from degrees to radians gives 1,746.37536… meters.
For example, on the Metro-North chart, let’s look at Harrison, New York. It’s on PDF-p. 24 of the chart; Harrison is sandwiched between two curves with opposite orientations, with the platforms on tangent (uncurved) track. The curve just west of Harrison has radius 1° 58′ 30″, which is 884.24069… meters; the curve just east has radius 2° 2′ 15″, which is 857.11674… meters.
Converting curve radii to speeds
The formula for the speed of a train, in SI units, is
I wrote about lateral acceleration, cant, and cant deficiency two and a half years ago. In short, lateral acceleration, in m/s^2, is the centrifugal force coming from the action of the train rounding the curve at speed. For the purposes of the formula, it is measured in the horizontal plane. To reduce the centrifugal force felt by the passengers (for comfort and safety) as well as that felt by the train body (for safety and maintenance costs), the tracks will typically be banked so that the inner rail is lower than the outer rail, which is called cant or superelevation, and is written in units of distance, such as mm or inches.
The speed of a train on canted track is typically higher than the perfect balancing speed, where the force of gravity counteracts that of centrifugal force; thus, on a fast train there is a residual force pointing to the outside of the train, which can be written down as lateral acceleration in the plane of the tracks (in m/s^2), but is more typically written down in the same units as superelevation, representing the additional superelevation required for the speed to perfectly balance, which is called cant deficiency or underbalance.
The conversion rate between cant (or cant deficiency) is the track gauge measured between the middle of the two rails, divided by the gravitational constant (9.8 m/s^2). Track gauge is typically given as inner rail to inner rail; standard gauge is 1,435 mm inner rail to inner rail. The relevant quantity to superelevation calculations is a few cm more; on standard gauge, it’s taken to be about 1,470-1,500 mm, so the conversion rate is 1 m/s^2 = 150 mm of cant or cant deficiency. The two quantities, cant and cant deficiency, are additive.
The American track charts that I have specify the actual cant. However, the values tend to be too conservative. Again with the example of Harrison, the slightly wider western curve has 5″ cant and the slightly tighter eastern curve has 4.125″ cant. Regulations for maximum cant depend on the country and maintenance standards. The absolute maximum cant I am aware of on any standard-gauge railway is 200 mm on the Tokaido and Tohoku Shinkansen. The reasons not to raise cant further include maintenance difficulties and the risk of a train running at lower speed or even stopping on the track. On lines that are not captive to just high-speed trains, the highest cant I am aware of is 180 mm, in Germany, and this is rare; 160 mm is more common. The American limit is 7″, but frequent inspections are required at that point to ensure that the tracks don’t get bent out of shape to produce higher cant; 8″ is a do-not-exceed level, and in practice track irregularities may lead to exceeding it if there isn’t regular track maintenance.
In practice, raising the cant is usually easy – it can be done with a track geometry machine automatically. However, in one case, it is not: that of S-curves, which have exactly the shape implied by the letter S. Our example of Harrison has two reverse curves in close proximity, but is not an S-curve, as there are hundreds of meters of tangent track between the two curves. Other places do have S-curves, and there, the maximum cant must be lowered somewhat; regulations vary on this, but in Europe, the maximum change in cant is 30-55 mm per second depending on the country (there’s a secondary regulation on mm per meter, but on the Northeast Corridor, the binding rule is mm/s, not mm/m). For example, if we take 45 mm/s, and 180 mm of cant, then it takes four seconds to reverse a curve; note that it is four and not eight, because half of the increase in cant, called a superelevation spiral, is within the curve. Harrison’s current curves impose a hard limit of about 150 km/h, at which point the hundreds of meters of tangent track make it trivially easy to have full superelevation. However, other places, most infamously among Northeastern railfans Elizabeth, there is an actual S-curve, forcing lower cant and lower speeds.
Finally, the maximum cant deficiency depends on the track, the train, and the regulations. Traditionally, American regulations limited most passenger trains to 3″ of cant deficiency, which is ridiculously conservative; in the 2000s, a waiver allowing 5″ in some cases was derided as the “magic high-speed rail waiver” because it applied not just to higher-speed tracks but also to lower-speed lines that had through-service to higher-speed tracks. Since then, FRA regulations have changed, and now the practical limit in the US, with extensive testing, is 6″, or 150 mm cant deficiency, at most speeds. European limits tend to be around 130-150 mm; high-speed trains are at the lower end of this range unless they are tilting trains, which nearly all trains are not. Cant deficiency, like cant, requires its own superelevation spiral on S-curves, but the limits are in practice looser than for cant, and in some cases trains can change superelevation abruptly, with no spiral, for example on switches.
The upshot is that in the typical case, the most aggressive assumption should be 180 mm cant, 150 mm cant deficiency, for a lateral acceleration in the horizontal plane of 2.2 m/s^2. Most lines will not have this pair of aggressive assumptions: 180 mm is only viable when it’s guaranteed that trains will not stop on a canted curve, which is a reasonable assumption on a reliable high-speed line and even on a German high-speed line. Moreover, if there is any freight on the line, superelevation must fall drastically: slower trains would be at cant excess, and freight trains have high center of mass (diesel locomotives and double-stacked containers both have higher center of mass than electric passenger trains) and therefore have tight cant excess limits. Aggressive assumptions are viable on the Northeast Corridor and on controllable commuter lines with no or almost no freight, such as the LIRR, but not everywhere else.
Finding curve radii
In some cases, curve radii are spelled out in a chart. In others, they are not, and must be figured out. Our program’s schedule writer, Devin Wilkins, tells me she has just found a track chart for SEPTA, but otherwise, I have no such charts south of New York. There, the dirty, imperfect method of estimating curve radii must be used.
For that, I use Google Earth. Nowadays, Google Earth Pro is free, and comes with a circle tool. In theory, I can fidget with the radius of a circle until I find that it approximates the arc of a curve well. This requires paying special attention to how the drawn curve compares with not just the broad outline of the curve but also the exact arc of each track or even each rail: the drawn curve should be at the same relative position to the rails, such as following one rail of one track, or right in the center of one track, or right between the track centers, and so on.
In cases of uncertainty, it’s also possible to use Google Earth line tools, which state the azimuth of each line. If I can find the exact start and end points of each curve, and the azimuths of the tangents on both side, then I can draw the chord with a line tool, verifying that its azimuth is the exact arithmetic mean of the azimuths of the two tangents; if it is not the mean, then either I made an error (more likely) or the curve is not a perfect circle (possible but less likely). The radius of the curve is approximately the length of the chord times 180/pi divided by change in azimuth; more precisely, the radius is
This method is error-prone, especially for short, sharp curves. Computing the start and end points of the curve will always have errors, and if the change in azimuth is small, then these will lead to large errors. The circle tool method suffers from the same drawback: it’s easier to use it to estimate the radius of a curve with 60 degree change in azimuth than that of one with 10 degree change.
Over time I’ve gotten this method down to the point that my errors from what I later find with track charts such as that of Metro-North are fairly small, and not very biased in the larger-radius direction. But it takes time and practice and ideally you should avoid it for short, sharp curves.
Update 10-28: Ari Ofsevit has a third method, using chord lines.
Other speed limits
Speed limits on intercity trains mostly come from curves. But there are other things to keep in mind, not all of which are fixable:
- Tunnels increase air resistance to the point that unless the tunnels are constructed with large enough radius to have a lot of free air (modern tunnels are, legacy ones aren’t), or unless the trains are pressurized, the speed limit has to be lower just to avoid popping passengers’ ears at entry and exit.
- Switches generally have low speed limits – they have tight curves and no superelevation – making all complex junctions and major stations slow.
- Terminal stations have another set of speed limits coming from the bumper tracks. American limits are very conservative – 10 mph where a ramp down from 40 or 50 km/h is more normal in Europe – but even 50 km/h is not 200 km/h.
- Some pieces of infrastructure are so shoddy that they limit the dynamical axle load of the train, which is derived from both static axle load, which is a function of train mass, and speed. As usual, American limits on this are conservative, assuming high static axle load and fixing a low speed on some very old bridges, instead of permitting lighter trains to run faster. But it is sometimes a real problem.
Pedestrian Observations 
One other consideration is speed limits due to level crossings. While these should be avoided as much as possible, on some legacy lines, they would be expensive to mitigate, especially near stations.
The regulatory requirements vary, but systems either require a lower speed limit (no more than about 140 km/h for automated crossings with significant remaining risk of collision) or positive verification that the crossing is clear (up to ~200 km/h). In the latter case, the speed limit is due to the need to hold the crossing closed for the time it takes to both close and check the crossing and for a train to stop if the crossing doesn’t clear. At higher speed rail crossings, this takes several minutes, being incompatible with frequent trains and/or reliable road access.
Yeah, good point – I wasn’t thinking about it becase my use case is either grade-separated lines (the Northeast Corridor) or commuter lines where the grade crossings are dead letter (in US law, four quadrant gates are good up to 110 mph, which is faster than most commuter trains can even get to).
Modern inverters can do all sorts of magic. Commuter trains are optimized for acceleration. And there will be another station in a few miles/kilometers, they don’t have time to get to high speeds. In most cases, there are exceptions in places where there is enough demand for local and express service.
Doesn’t the NEC still have Grade Crossings though?
I guess part of the problem is that track alignment drawings are not readily available except to the folk who use them, the railroads and whoever they employ to maintain and/or update the track alignment. I recall back in the late 1980’s when I was involved in preparing the Parliamentary pans for what was then known as Channel Tunnel Rail Link. not hS1 but a predecessor, the largest volume of drawings were the alignment drawings, that included every Point of Curve, curve radius, transition curve detail etc, together with the cant and deficiency. Every other rail project I’ve worked on since then has detailed alignment drawings that contain all that information. But I guess its not easy to access.
Bottom line is detailed track plans are always developed for rail work. Your Google Earth solution is a good fix, but every curve has two components, start of transition, end of transition/start of radius and the same on the exit. So dependent on which point you use your radius could be a little out, perhaps not enough to impact your speed calculations as in theory the transition should be designed to bring the track into the constant radius curve.
The NEC grade crossings are in an area that any reasonable high-speed rail plan would bypass one way or another; even the medium-speed alternative chosen by the NEC Future process bypasses that area on I-95 because of the movable bridges.
The track charts aren’t always up to date, but because right-of-way geometry is so difficult to change, curve radii are reliable even in a 2015 map. The superelevations may have changed, yes, but that’s because they’re not hard to change. (For example: the chart I have for the Providence Line asserts that there’s a string of 1° curves with 4″ cant between Mansfield and Canton, which means that at the Acela’s 7″ cant deficiency, the maximum speed is 125 mph – and even 120 on the 1° 4″ curve south of the Canton Viaduct; but the actual maximum is 130 mph per the charts I have, because the superelevation is somewhat more than just 4″.)
The transition spiral issue is more complicated, yeah, but you can more or less model the spiral as a line segment followed by a circular arc, with half the spiral on the line and half on the curve; that’s how I model S-curves.
True, hence why we don’t trust the track plans and always rescan (LIDAR, Dibit) everything before we start new design work so we know where everything is!
The few grade crossings are all in Connecticut where the line squiggles along the coast. The squiggles limit the speed.
But really this sort of thing is kindergarten mathematics.
Alon, I’ve wasted plenty of time reading German track geometry regulations (RiL800.0110 and prEN 13803 FTW!) so if you have any questions about the lengths and offsets of horizontal transition spirals (simple cubic approximations to parabolas work just fine for your purposes, though fancier transitions can be employed) or of vertical superelevation transitions I can shoot you some outlines and scanned diagrams. But really for open track mainline just going with circular arcs is going to be fine.
My recommendation is always to totally ignore what the fucking neaderthals of the FRA (captive “regulators” of UPRR/CSX/BNSF/etc) pulled out of their rectums for coal trains pulled by steam locomotives in 1930 and just do whatever adults elsewhere in the world do. (I can’t read Asian languages, so I go with European norms, and as German is easist for me to find and read, and doesn’t seem very much different from what I’ve pieced together from other European sources, I just go with that.)
Re S-curves, or other back-to-back horizontal curves: they’re really not that huge a deal! The Germans even have a word for this, “Gleisschere”, and you just continue your superelevation change continuously from positive to negative (left rail/right rail higher) at the point where the adjacent horizontal transition spirals touch where R=+∞/-∞) If one can’t do that (and one often can in practive, by going with gentler=longer transitions), the Germans like a section of straight track of length L = 0.4 V (L in metres, V in kmh, eg track speed 200kmh → Zwischengerade 500m)
I have that book too at home, I mine it for standards whenever I feel like I may be suffering from allism…
I’m puzzled about that S-curves bit, too – the regulations on the maximum cant gradient (both mm/s and mm/m) apply to any change in cant, not just S-curves.
With some exception that on mainline railways are normally only really relevant in conjunction with turnouts, the length of the cant ramp is always identical to the length of the transition curve, so in the first instance it is fixed.
That means that the situation is quite simple (and it doesn’t depend on whether we’re talking about S-curves or simple curves adjacent to straight track): Either you’re already at the limit of the maximum permissible cant gradient, or else you’re not. If you aren’t, you can indeed “simply” send through a tamper to raise the cant until you hit any of the relevant maximum limits (either maximum absolute cant, or the maximum cant gradient given the available transition curves).
If you’re already at the maximum cant gradient, the only way of raising the cant further is by lengthening the transition curve itself, which means changing the track geometry in a more significant way. The most important consequence is that lengthening the transition curve leads to laterally offsetting the circular bit of the curve. If we’re talking about only a few mm or cm at most, this might still be easily doable, but if you need to significantly lengthen the transition curve, on tighter curves the offset can easily change by a metre or more.
(There is one trick you can do: If you keep the lateral offset of the circular curve constant, replacing the regular clothoid transition curve [curvature varies strictly linearly along the length of the transition curve] with a Bloss curve allows lengthening the transition curve by ~√(5/3) while at the same time also permitting somewhat more generous mm/s cant gradient limits [because the Bloss curve provides a smoother transition than the clothoid, where the rate of change of cant (and cant deficiency) suddenly varies at the start and end of the transition curve]. This means you gain some reserves for further raising the cant without having to laterally shift the whole curve.)
As for the 180 mm maximum cant in Germany – that is the strict legal do-not-exceed value, so the normal planning limits are 160 mm for ballasted track and 170 mm for ballastless track. In some very rare cases those planning limits might be allowed to be slightly exceeded by perhaps 5 mm or so, but going up to the legal 180 mm limit would be very very unusual, if indeed it ever happened at all.
All correct. Thanks!
Anyway I now see this has come up here before, just a couple years ago. Time is meaningless for US railfans’ butwhataboutism!
https://pedestrianobservations.com/2016/07/30/northeast-corridor-95-cheaper-frankford-junction/#comment-102661 (even including Bloss spiral in the tiny calculations.)
https://pwayblog.com/2016/07/23/the-orphan-rule-of-cant-design-over-a-reverse-transition/: everybody outside the USA understands that back-to-back transitioned S curves are nothing at all special.
major stations slow.
Everything is going to be stopping at major stations. The train has to be going slow so it can stop.
Yes, but, separately from that, such stations impose speed limits beyond just acceleration limits, due to constrained urban environments, interlockings, etc.
everything will be decelerating to stop or accelerating from a stop.
Google Maps says it’s 800 meters from 7th Ave. and 31st St. to 10th Ave and 31st. How fast could a westbound train be going in 800 meters? Though the western end of the train is 200, 300, 400 meters farther west.
About 120 km/h using either Velaro Novo or FLIRT technical data.
A very simplistic view: Round it up to kilometer it would take 30 seconds to traverse it. Slow it down to 60 it would take a minute. Billions and billions of dollars to extend the interlockings to save 30 seconds isn’t worth it. Spend the billions someplace else.
… convert the 25Hz to 60Hz they would save money by avoiding conversion losses and the new catenary would support higher speeds which saves minutes, on a New York to Washington D.C. trip, not seconds.
… two short tunnels in Elizabeth New Jersey. Trains not-stopping in Elizabeth don’t need to be between the platforms.
Nah, the tunnel between Baltimore and West Baltimore is probably the highest priority. It’s a pity they didn’t ask you, in 1840, how to arrange things.
If the speed is 60 km/h, then yeah, it’s 30 seconds and it’s not worth much.
The problem is that the Penn Station speed limit now is 15 mph and the Grand Central speed limit is 10 mph. On current timetables, Metro-North does 125th-42nd in the same time as the subway or even slower, despite using higher-performance trains making two fewer stops. Grand Central in particular can save not 30 seconds but 3.5-4 minutes by doing switch improvements within right-of-way to waive the mile-long 10 mph restriction.
Plus it affects pretty much all trains and a large percentage of all passengers.
It’s too bad they didn’t put Penn Station’s tunnel portals at 11th Ave or 5th Ave. Or that they didn’t start spreading out Grand Central’s tracks at 72nd. But they didn’t and you have to work with what you have, not what works in Sim City
Alon, I’ll never understand why you engage with the logorrheic trollbot.
Anyway, station approach speeds are, for obvious kindergarten mathematics reasons, often the most effective investment of capital cost per saved mintute of trip time.
Trains should be entering platforms at the fastest possible speed that allows reliable and safe stopping, not crawling for km in advance of them. While 80kmh (or greater speed) diverging turnouts (760m radius) are almost major mainline junction things in the USA, one increasingly finds them being placed right up close to platforms as part of line rebuilds and upgrades in those parts of the world that care about speed, throughput and reliability.
PS https://en.wikipedia.org/wiki/Utrecht_Centraal_railway_station#Redesign_of_the_track_layout and links from it are worth a quick read.
I don’t see any billion dollar skyscrapers in this view.
How does high super elevation interact with infrequent maintenance trains which are diesel powered? Does high super elevation mean all track work must be performed by electric locomotives?
Not at all.
No.
Traction power makes no difference and maintenance vehicles are infrequent visitors on any track, but any slow train will ride the inside rail in a highly canted curve. This will lead to increased rail wear due to increased wheel slip and possibly flange contact. This is why infrastructure managers will usually try to balance the cant according to the traffic mix. Thus, you don’t cant tracks to the maximum allowed value if you also have lots of slow, heavy freight trains trundling along.