Energy Management of Internet Data Centers in Smart Grid by Tao Jiang Liang Yu & Yang Cao

Author:Tao Jiang, Liang Yu & Yang Cao
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

3.3.1 Relaxed Problem

Defining the time-averaged expectation of charging and discharging power of the battery at IDC under any feasible control policy of P3.1 as follows,

(3.25)

(3.26)

Since battery energy levels update according to (3.15), summing over all , taking expectation of both sides and dividing both sides with and letting , we have . Then, a relaxed problem of P3.1 is given as follows, called P3.2.

(3.27a)

(3.27b)

(3.27c)

Let and be the optimal solution of P3.1 and P3.2, respectively. As mentioned before, any feasible solution to P3.1 is also a feasible solution to P3.2. Therefore, we have . Obviously, P3.2 is easier to solve than P3.1 due to the removal of “time coupling” property. From the framework of Lyapunov optimization [34], we have the following Theorem for the solution to P3.2:

Theorem 1

If are i.i.d. over slots, then there exists a stationary and randomized policy that takes control decisions , , and every time slot as a pure (possibly randomized) function of the observed while satisfying the constraints of P3.2 and providing the following guarantees.

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