Philip G. Howlett

The Train Control Problem

In 1977–78 Milroy [1] considered the problem of driving a train from one station to the next along a level track within a given allowable time in such a way that energy consumption is minimised. He used the energy flows in the traction and braking systems of the train to derive state variable equations with time as the independent variable and position and speed as the dependent state variables. He used an heuristic application of the Pontryagin Principle to conclude that the optimal driving strategy consisted of a maximum acceleration-coast-brake control sequence. Subsequent studies confirmed the optimality of this control sequence for short journeys, and showed that a speed-hold phase should be included on longer journeys.

Modelling the Train Control Problem

The train control problem was originally formulated with applied acceleration as the control, and with journey cost measured by the mechanical energy required to drive the train. The mechanical energy model is physically sound, but does not properly describe the real control mechanism and does not represent the real financial cost of a journey. The fuel consumption model was designed to model the control mechanism of a typical diesel-electric locomotive, and uses the total fuel consumption to measure the cost of a journey.

A More General Model

So far we have assumed that power is constant for each fuel supply rate, and hence that the applied acceleration is inversely proportional to the speed of the train. In this chapter we use a more general model of applied acceleration, and obtain results similar to those of the previous two chapters. The more general model can be used to describe tractive effort and dynamic braking curves such as those shown in Figure 2–1 and Figure 2–3.

Necessary Conditions for an Optimal Strategy

With an appropriate formulation of the train control problem, we have already shown that an optimal driving strategy exists. In this chapter, necessary conditions of the Fritz-John type will be obtained for the optimal strategy, and these conditions will be used to find a Hamiltonian function and to demonstrate the validity of the Pontryagin Principle for this problem. This chapter was originally published in 1988 as a report to the School of Mathematics and Computer Studies at the South Australian Institute of Technology [34]. The methods used are an extension of the methods developed by Craven [18].

Practical Driving Strategies

There are two aspects of train control that can be addressed before we consider the optimisation problems. First, we can show that the discrete control mechanism of the fuel consumption model is not a practical limitation, since any speed profile can be followed as accurately as we please using alternate coast-power pairs. Second, by considering the energy balance equations we can show that speed-holding, where possible, is the most efficient driving mode.

Non-Constant Gradient

In this chapter we use the fuel-consumption model to formulate the train control problem on non-level track. Once again, an optimal strategy no longer exists. That is, there is no feasible strategy that minimises fuel consumption. We will show that for a given sequence of control settings, there exist optimal switching points that define a strategy of optimal type. This strategy minimises fuel consumption for the given control sequence. We find key equations that define necessary conditions for a strategy of optimal type, and present an algorithm for solution of these equations. On non-steep track an approximate speed-holding strategy is best. On steep track, speed-holding may be disrupted by segments of maximum power around steep inclines, or by segments of coasting around steep declines.

Determination of Optimal Driving Strategies

In this chapter the Pontryagin Principle will be used to find the nature of the optimal strategy for the mechanical energy model with the cost functional

Minimisation of Fuel Consumption

J(u,\,v)\, = \,\int\limits_O^T {p\,[u(t)]\,q\,[v(t)]\,dt}