Power Strip Packing of Malleable Demands in Smart Grid

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

Asymptotically optimal scheduling of random malleable demands in smart grid

We study the problem of scheduling random energy demands within a fixed normalized time horizon. Each demand has to be serviced without interruption at a constant intensity, while its duration is bounded by a pair of malleability constraints. Such constraints are assumed to be characterized by an i.i.d random vector that follows a general distribution. At each time instance, the total power consumption is computed as the sum of the intensities of all demands being serviced at that moment. Our objective is to minimize both the maximum and the total convex cost of the power consumption of the grid. The problem is considered in the asymptotic regime. In this regime, the number of demands is assumed to be large, and their (random) energy requirements are inversely proportional to the number of demands. Such setting allows us to introduce a linear-time scheduling policy and shows its asymptotic optimality with respect to both cost crit

Asymptotic convex optimization for packing random malleable demands in smart grid

We consider a problem of scheduling electric power demands in a smart-grid framework. Our model consists of n energy requirements {Ai, ℓi, ri}n i=1, needed to be scheduled in time interval [0,1]. Here Ai is the amount of energy, while ℓi and n are respectively, the left and right constraints on the length of the time period, during which Ai has to be supplied without interruption. The triples are assumed to be i.i.d. random vectors, with A distributed according to some general distribution G, and pair (ℓ, r) distributed uniformly in the region {0 ≤ ℓi ≤ r ≤ 1}. Our goal is to find a scheduling policy minimizing the power peak - maximal power over [0,1] - and/or the operational convex cost of the system while satisfying all the demands. The problem becomes very complicated as the number n of demands increases. To address this issue, we consider an asymptotic approach, in which the average amount of energy in each demand is inversely proportional to n, thus keeping the total scheduled amount stable. In this paper we first introduce lower bounds for both types of costs and then introduce a scheduling algorithm, asymptotically optimal in the sense that its cost converges to a corresponding lower bound almost surely, as n increases to infinity. Moreover, the algorithm is on-line (each demand is scheduled at the time its parameters become known) and has fully linear running time