  We investigate analytic solutions to the problem of finding driving
strategies which minimize total tractive energy consumption subject to
constraints that ensure safe separation for a fleet of trains travelling on the
same track in the same direction and minimize line occupancy time span for the
fleet. It is known that a fleet of trains can be safely separated if
intermediate signal times are prescribed by inequality constraints for each
train at each signal location. The optimal driving strategy for each train is
then uniquely defined by an optimal driving speed on each actively timed
segment. The problem of finding optimal intermediate times has been solved for
a fleet of two trains but for more than two trains the problem rapidly becomes
intractable as the number of trains and signals increases. The main difficulty
is in distinguishing between active and inactive constraints. This is
exacerbated by the fact that an inactive constraint with one set of prescribed
times may become active if the set of prescribed times changes. The curse of
dimensionality means it is not feasible to check every different combination of
designated active and inactive constraints and to optimize the corresponding
prescribed times. Nevertheless we can formulate a substantial optimization
problem with intermediate signal times prescribed by active equality
constraints for all trains in the fleet. We find necessary and sufficient
conditions on the optimal times and demonstrate viable solution algorithms to
find the associated optimal speed profiles for each train. Compared to the
schedule required for safe separation when each train uses the classic
single-train optimal strategy our method provides a substantial reduction in
the line occupancy time span with only a minimal increase in journey costs.
