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.

中文翻译：

---

最小化闲置能耗的多项式时间调度方法：在工业炉中的应用

本文提出了一种新颖的调度方法，以最大限度地减少机器在空闲期间的能耗。在调度域中，通常通过定义一小组机器模式来对机器的行为进行建模，例如“开”、“关”和“待机”。然后模式之间的转换由静态转换图表示。在本文中，我们认为这种类型的模型对于某些类型的机器（例如熔炉）可能过于严格。对于此类机器，我们建议采用完整的时域动力学并将其集成到空闲能量函数中。通过这种方式，调度算法可以利用封装在该函数中的最小能耗的机器动力学的全部知识。在本文中，我们研究了一个调度问题，其中以发布时间和截止日期为特征的任务按给定的顺序进行调度，从而最大限度地减少机器的空闲能量消耗。我们表明，只要空闲能量函数是凹的，就可以在多项式时间内解决这个问题。为了突出实际适用性，我们分析了一个采用钢淬火炉的热密集系统。我们推导出能量最优控制律和相应的空闲能量函数，用于近似熔炉动态的双线性系统模型（以及可能的其他热密集系统）。此外，我们证明了空闲能量函数在这种情况下确实是凹的。因此，可以使用所提出的调度算法。数值实验表明，通过使用我们的方法，结合最优控制和最优调度，