  This article presents a novel scheduling approach to minimise the energy
consumption of a machine during its idle periods. In the scheduling domain, it
is common to model the behaviour of the machine by defining a small set of
machine modes, e.g. "on", "off" and "stand-by". Then the transitions between
the modes are represented by a static transition graph. In this paper, we argue
that this type of model might be too restrictive for some types of machines
(e.g. the furnaces). For such machines, we propose to employ the complete
time-domain dynamics and integrate it into an idle energy function. This way,
the scheduling algorithm can exploit the full knowledge about the machine
dynamics with minimised energy consumption encapsulated in this function. In
this paper, we study a scheduling problem, where the tasks characterised by
release times and deadlines are scheduled in the given order such that the idle
energy consumption of the machine is minimised. We show that this problem can
be solved in polynomial time whenever the idle energy function is concave. To
highlight the practical applicability, we analyse a heat-intensive system
employing a steel-hardening furnace. We derive an energy optimal control law,
and the corresponding idle energy function, for the bilinear system model
approximating the dynamics of the furnace (and possibly other heat-intensive
systems). Further, we prove that the idle energy function is, indeed, concave
in this case. Therefore, the proposed scheduling algorithm can be used.
Numerical experiments show that by using our approach, combining both the
optimal control and optimal scheduling, higher energy savings can be achieved,
compared to the state-of-the-art scheduling approaches.
