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Copyright 2006 by David S. Lawyer. Feel free to make copies but commercial use of it is prohibited. For example, you can't (except to an insignificant degree) combine it with advertising on the Internet. Please let me know of any errors or suggestions for improvement.

Trains, trucks, and autos could save a great deal of energy if they would coast more. There are various types of coasting and it's different for railroads and autos, but much of the mathematics is the same.

One form of auto coasting, commonly used for obtaining much better fuel economy, has become known as "pulse and glide". One starts the cycle at say 55 mph and accelerates to 70 mph. Then one coasts down to 55 mph again, with the engine either shut off or at idle. Then this pulse (acceleration) and glide (coasting) is repeated. It saves energy because the auto engine is much more efficient at the high torque required during acceleration. It's been publicized for use with hybrid autos.

Railroads have been practicing coasting for likely over a century, but few railroads in the US promoted it (in the early 21st century). It's not pulse and glide, but it's often coasting to slow down the train before a stop or speed restriction. A train has such low rolling resistance that power can often be cut off several miles before the next stopping point and the train coasts till it gets almost to the next stop. Then it applies the brakes to finally come to a halt.

The author has developed mathematical optimization methods for train coasting as have others. It's actually part of the problem of optimal train trajectories. A train trajectory is the plot of the speed versus the distance (on the horizontal x-axis). Due to limitations of train power and speed limits, not all trajectories are feasible.

In the U.S., where freight train traffic is much greater than passenger train traffic, there's obviously more potential for energy savings by more coasting of freight trains. In Western Europe, where much of the freight traffic has been lost to trucks, there's more potential savings for the coasting of passenger trains. For Russia, there's significant potential for costing for both freight and passenger traffic and information about it was published during the Soviet period.

Instead of dissipating the kinetic energy of a train by friction brakes, one can utilize this kinetic energy by simply coasting and let the train slow down by coasting instead of braking. This policy often results in a train shutting off power a few miles before reaching it's next stop. Instead of using up fuel under power, the train coasts and uses no fuel at all except possibly the fuel used for idling the engines for diesel (non-all-electric) trains.

Coasting is part of the larger problem of optimal train speeds. There are various formulations of the problem and various means of mathematical solutions. While it's likely been known for over a century that railroad coasting saves energy, research on the mathematical aspects goes back at least to the 1960's.

A seemingly simplified problem is just to find the optimal steady speed on a section of track. This solution doesn't cover the acceleration, coasting or braking phases. But once one finds the optimal speed for each segment of track, then one tries to reach and cruise at this speed and coast down to the optimal speed if the train is going too fast..

Another related problem is to find how many trains one should run in a day. The more trains, the less passengers or freight per train and the higher the wind force per ton (since there are more head-ends of the train to push thru the still air). But with more trains, better service is provided due to less waiting for the next train. Thus the problem of train size is coupled with the problem of optimal speed. See Optimal Velocity

Optimal train speeds and trajectories are problems in the trade-off between time and energy. The faster one goes, the more time one saves but at a cost of increased energy consumption. One can claim that there are other costs besides just time and energy but it's reasonable to subsume some of these other costs under the categories of time and energy.

For example, the depreciation cost of a train can be dividing into energy costs and time costs. A train depreciates over time as it become technological obsolete. It also depreciates due to mileage but one could instead use energy consumption as a substitute for mileage. If the train uses more energy for propulsion, it likely either traveled more miles or went faster (or both) and thus endured more wear. Higher speeds mean harsher bumps on uneven track, resulting in more wear. More energy use should mean more engine wear. The assumption that wear on the train is directly proportional to the energy it uses is far from exact, but the assumption often made that wear is proportional to mileage is not exact either. One may also assume that the mechanical maintenance costs are proportional to the energy used.

The problem of optimal train trajectories may be stated as an optimal control problem. For a level track the state variables are x the train location from kilometer post 0, and v, the velocity of the train. The control variable u is the traction force (per unit of train mass) exerted by the train motors. The differential equations of motion for a unit mass of train are:

dv/dt = u - (a + bv +cv^2) dx/dt = v

The cost to minimize is:

Integral from 0 to s: (d/v + eu)dx

where d is the cost per unit time per unit mass and e is the price of energy. There is an upper limit to velocity based of safe speed limits. There is also an upper bound to u also which depends on v.

The soulution, based on optimal control theory, is to accelerate at maximum u until reaching optimal crusing speed. For stopping or slowing, it's optimal to coast down to a certain speed and then do maximal safe braking.

Optimal Train Velocity and Size, by David S. Lawyer in Bulletin of the International Railway Congress Association, January 1969. This is only for the optimal constant velocity and neglects the acceleration and coasting phases.