Interlude: Physics of train acceleration

This past week I’ve been working on a slide deck that people can use to present the Regional Rail concept to non-transportation-nerd audiences, and one of the things that I wanted to include, just as an aside, was a couple of slides explaining the “stop penalty” and why it’s so bad for traditional diesel-hauled commuter rail like we have in Boston. This led me down a horrible rabbit-hole for the better part of two days, trying to figure out a closed-form expression that would allow plotting comparisons showing speed over time, or distance over time, or speed over distance. I am quite certain that there actually is such a solution, but I couldn’t come up with one that looked believable when I plotted it out, so I went back to square one and wrote a simple numerical simulator that would just generate the data points I needed to make my plots rather than trying to come up with the general formula.

In doing this simulation, there are three different regimes we have to consider, which derive from different limits (mechanical, physical, or legal) on the train system. At high speeds, there is a regulatory speed limit: the train is not allowed to go faster than a certain speed, because of track conditions, grade crossings, or the sort of signal system it has. At low speeds, the limits are mechanical: the motors can only apply so much force without the wheels spinning (or the axle shearing, or the gears stripping, or the motor windings burning up). In between these two regimes, however, physical law — Conservation of Energy — directly limits the acceleration that can be achieved. Any train will have the ability to generate (or draw from batteries or overhead wires) a certain maximum amount of power, and even without accounting for all the losses in the system (electrical resistance, inefficiency in the motor controllers and the motors themselves, friction, drag, and so on) this limits the acceleration attainable. In fact, at constant power, the conservation law tells us that

\displaystyle a_{max}(v) = \frac{P}{m\cdot{}v}.

If we assume that you always want your train to accelerate as fast as it can until it reaches the speed limit (or until it has to slow down for the next station), then a \leq a_{MAX} when we are in this power-limited regime, and a \leq F_{max}/m (where F_{max} is the maximum force the motor can exert in the forward direction, called “tractive effort”) when it hasn’t reached maximum power yet. In both equations, m is the mass of the entire train, including locomotives, coaches, fuel, and all the passengers. Note that in the energy-limited regime, acceleration is a function of velocity — in fact, it’s inversely proportional to velocity. Acceleration is the first derivative of velocity, which means this is a differential equation; luckily it’s one that has a solution, although as I said I had trouble figuring this out (because I never took Diff. Eq. in college, most likely) and couldn’t figure out how to do the integration piecewise — because what we really want to figure is speed as a function of time, and then integrate that to get distance traveled as a function of time. But just to make things clearer, here’s a plot showing the three different regimes for three different train configurations that I set up in my numerical model.

The image shows acceleration as the dependent variable and time as the independent variable, with different regimes identified depending on what the cause of the limit on acceleration is (mechanical, power/energy, or legal)

The three train configurations modeled are two trains with 500 passengers, all seated — one with my favorite articulated EMU (Electric Multiple Unit), the Stadler FLIRT, and one with a diesel locomotive of the type the MBTA uses and three bi-level coaches — and a third diesel train with the same locomotive but a nine-coach consist and a crush load of 1600 passengers (the highest load and the longest train currently operated on the MBTA). I assume that all coaches are bi-levels because I don’t have a source for the mass of a single-level coach: in actual operation, at least one single-level coach is required in every train for accessibility.

One thing is abundantly clear: that 1600-passenger “monster” load is extremely slow to accelerate. In the formula above, you’ll note that the mass of the whole consist is in the denominator of the acceleration equation, which is a consequence of Newton’s Second Law (F=ma). Naturally, the heavier the train, the slower it can accelerate, unless you have a way to add power somehow. With Multiple Units, you do have a way to add power: a single FLIRT trainset has two 1,000-kW electric motors, for a total power of 2 megawatts, and if you couple two FLIRTS together, all four motors work together at a system power of 4 megawatts. (You can couple up to three, but in most cases where you might want that many seats, you should be running more frequent trains instead.) The poor commuter-rail train has only one diesel engine with a power output of just over 2.2 megawatts no matter how many cars you couple to the end. (Yes, that does mean that a diesel locomotive on its own could accelerate faster than the electric train, because it weighs less and has more power — but that doesn’t help move any passengers!) One thing is clear: the heavy train takes inordinately long to reach the speed limit of 79 mi/h that I’ve set on this simulation. (On the Providence Line, the speed limit is 125 mi/h, but Regional Rail trains making more frequent station stops would not reach that speed, and it wouldn’t make sense to buy trains that were designed to go that fast — 99 mi/h is easily as fast as is useful on commuter trains.) This is even more clear if you look at the acceleration as a function of distance:

A line graph compares acceleration of three different train configurations; this is the same as the last graph except that the independent variable has been changed to distance traveled.

Look at that heavy train: it takes about 4.5 kilometers (more than 2¾ miles) to reach the speed limit that’s typical on the current MBTA commuter lines! Remember, that’s just acceleration; you still have to decelerate for the next station stop, and if the stations are less than 9 km (5½ miles) apart, that train is never going to reach the maximum speed allowed on the line. That’s why the current service pattern favors expresses: if you can go twelve miles without having to stop or slow down significantly, you don’t have to pay this “acceleration penalty”. (That’s only part of the full “stop penalty” — the other part is the actual time the train has to spend stopped at a station while passengers board and alight, the “dwell time”. Regional Rail aims to solve both of these issues at once with better design, EMUs, and level boarding.)

Another way to visualize this is to look at how long it take to travel a given distance. The graph below shows the same three train configurations, but the x-axis is distance and the y-axis is time, the exact opposite of how you’d normally look at this sort of physics problem, but very useful when you’re thinking about train schedules:
A line graph comparing three different train configurations accelerating from a stop, with distance as the independent variable and time as the dependent variable

The two-trainset FLIRT consist reaches the speed limit first, for obvious reasons (it has nearly double the power), at about 4500 feet of travel, and the three-coach diesel train hits the speed limit at about 6500 feet; after this point, they are traveling at the same speed and those two lines are parallel, separated by about 13 seconds that is the residual advantage of the electric train’s quicker start. The nine-coach diesel barely reaches the speed limit by the right-hand side of the plot, which is at a distance of three miles. If you assume that a train can decelerate exactly as fast as it can accelerate (which is not unreasonable for dynamic or regenerative braking, I’m not so sure about friction brakes), then the long heavy diesel shouldn’t be stopping more than once every six miles or more — which is far from what that train currently does, and that’s why even the “express” trains are quite slow (unless you’re getting on or off right before or after, respectively, the express segment). On the inner parts of many of the commuter lines, stations are spaced much more closely together — as little as a mile or two — and on lines that don’t have stops like that now, those are the exact places where Regional Rail envisions adding “infill” stops to provide better service to residential and commercial areas in the urban core. (Ideally, every Providence Line train would stop at both Forest Hills and Ruggles, for example — but this would only be acceptable to Mansfield commuters if the stop penalty were effectively mitigated so their commutes were not lengthened.)

The code for the simulator is in my GitHub repo, in the file physics.rb. The slide deck will be posted here once I finish writing it.

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