Minimizing total cost over time

In some cases, minimizing peak power demand may not be good enough. Although the peak is minimized, there might be many timeslots with high power demand and hence the total cost is still high. We focus on another objective in the smart grid scheduling problems which is to minimize the total electricity cost. There were some results in minimizing total cost [27, 45, 65]. We first elaborate the relation between the two objectives, minimizing total cost and minimizing peak demand.

Relating to minimizing peak demand. For the same input instance, the ob- jective of minimizing peak or minimizing total cost might leads to different optimal schedule. That is, minimizing the peak demand does not necessarily minimize the to- tal cost, and vice versa. Example 3.1 shows an instance and schedules with respect to minimizing peak and minimizing total cost. The example shows that a schedule with minimized peak may have a higher cost. In other way round, a schedule with minimized cost may have higher peak demand.

Example 3.1. Consider positive integer x and J = {J1, J2, J3} where w(J1) = w(J2) =

x, h(J1) = h(J2) = 1, w(J3) = 1, h(J3) = 2, I(J1) = I(J2) = [0, 2x) and I(J3) =

[x − 1, x) (see Figure 3.1a). Figure 3.1b shows a schedule Sc with minimum total cost,

which is 3α+ 2x − 1. Figure 3.1c shows a schedule Sp with minimum peak, which has

cost (x + 1) · 2α. It is easy to see that the peak in Sc is 3, which is higher than the peak

in Sp, while the cost in Sc is lower than Sp when x > 3

α₋₂α₋₁

x-1 x 0 2x J2 J3 J1 1 1 1 x x 2

(a) Illustration of the three jobs in J x-1 x 0 2x J2 J3 J1

(b) Schedule Sc with minimum

total cost x-1 x 0 2x J2 J3 J1

(c) Schedule Sp with minimum

peak Figure 3.1: An illustration of Example 3.1.

Intuitively, when α is big enough, the cost at the peak hour will dominate the total cost. In fact, it was shown in [55] that there is a polynomial time reduction of the decision version of the min-peak problem to that of the min-cost problem for a large enough α:

Lemma 3.1 ([55]). A grid scheduling problem with objective to minimize the maximum power request can be reduced to a grid scheduling problem with objective to minimize the total cost by setting α > (τ − 1)(2P

J∈J h(J) + 1), and the solution of min-cost problem

under this setting is a solution of the corresponding min-max problem.

Previously there were some results in minimizing total cost in smart grid model [27, 45, 65].

Koutsopoulos and Tassiulas [45] studied a similar problem to the GRID problem. Comparing to our problem, their cost function can be an arbitrary convex function while our cost function is an α-power function of load. Moreover, they studied stochastic model and aimed to minimize the expectation of long-term cost while we aim to minimize the total cost and guarantee the worst case performance.

The authors [45] claimed that for instance where jobs can be preempted, the GRID problem is equivalent to a load balancing problem (and hence NP-hard) and proposed an iterative load balancing algorithm to find an optimal schedule of preemptive jobs. For non-preemptive case of the GRID problem, the authors claimed that it is NP-hard.

For the online setting, the authors devised a stochastic model and focused on mini- mizing the long-term average cost. The authors proposed two strategies and proved one of them to be asymptotically optimal by deriving a lower bound for the performance of all policies.

Instead of minimizing the long-term average cost, Feng et al. [27] investigated the worst case competitive ratio of the GRID problem under online list setting. In the model time is divided into integral timeslots, each job has unit power request and unit time duration and is released with arbitrary feasible timeslots, which can be non-contiguous. Moreover, the authors consider the case where the cost function is quadratic (that is, α = 2). They investigated the greedy strategy and claimed that it is 2-competitive.

However, Liu et al. [55] proved that greedy strategy is no better than 3-competitive when α = 2 by showing an adversary and hence the precise competitiveness of the greedy algorithm is still unknown.

Narayanaswamy et al. [65] considered a more practical model that other than the power generated by the generator (with quadratic cost function), there is also a renewable resource. The renewable power resource (for example, wind power) varies over time and can only be predicted accurately before short period of time. The generator has to decide how much power to generate such that the generated power together with the renewable power satisfy the time-varying power demand. The objective is to generate sufficient amount of power with minimum regret, which is the difference between the online and the optimal costs. The authors applied recent work in online optimization and proved the algorithm is asymptotically good by deriving bounds in terms of the generator parameters.

Another flourishingly studied problem about optimizing power demand with convex cost function is Dynamic Voltage/Speed Scaling problem. We will discuss in details in Section 3.2