智能电网中可延展需求的电源板封装
Power Strip Packing of Malleable Demands in Smart Grid
摘要：
我们考虑了一个智能电网框架中的一组$\mathcal{N}$客户的供电问题。每个客户需要一定量的电能，该电能必须在时间间隔$[0,1]$期间供应。我们假设每个需求都必须不间断地供应，$\ell$和$r$之间可能有持续时间，这是给定的系统参数（$\ell\le r$）。在每个时刻，电网的功率是该时刻供应的需求的所有消耗率的总和。我们的目标是找到一个分配，最大限度地减少{ \it功率峰值} -最大功率超过$[0,1]$ -同时满足所有的要求。要做到这一点，首先我们找到最佳功率峰值的下限。我们发现，这个问题取决于是否对$\ell, r$属于一个“好”的区域$\mathcal{G}$。如果是，则最优分配几乎完美地“填充”矩形$time \times power = [0,1] \times [0, A]$，其中$A$是所有能量需求的总和，从而实现最优功率峰值$A$。相反，如果$\ell, r$不属于$\mathcal{G}$，我们确定的下限$\bar{A} >A$的最佳值的功率峰值，并介绍了一个简单的线性时间算法，几乎完美地安排在一个矩形$[0, A /\bar{A}] \times [0, \bar{A}]$的所有需求，并表明它是渐近最优的。
英文摘要：
We consider a problem of supplying electricity to a set of $\mathcal{N}$ customers in a smart-grid framework. Each customer requires a certain amount of electrical energy which has to be supplied during the time interval $[0,1]$. We assume that each demand has to be supplied without interruption, with possible duration between $\ell$ and $r$, which are given system parameters ($\ell\le r$). At each moment of time, the power of the grid is the sum of all the consumption rates for the demands being supplied at that moment. Our goal is to find an assignment that minimizes the {\it power peak} - maximal power over $[0,1]$ - while satisfying all the demands. To do this first we find the lower bound of optimal power peak. We show that the problem depends on whether or not the pair $\ell, r$ belongs to a "good" region $\mathcal{G}$. If it does - then an optimal assignment almost perfectly "fills" the rectangle $time \times power = [0,1] \times [0, A]$ with $A$ being the sum of all the energy demands - thus achieving an optimal power peak $A$. Conversely, if $\ell, r$ do not belong to $\mathcal{G}$, we identify the lower bound $\bar{A} >A$ on the optimal value of power peak and introduce a simple linear time algorithm that almost perfectly arranges all the demands in a rectangle $[0, A /\bar{A}] \times [0, \bar{A}]$ and show that it is asymptotically optimal.
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