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Unformatted text preview: nergy stored in the battery in period t. For simplicity, we neglect
energy loss (although this is easily handled as well), so we have qt+1 = qt + ct , t = 1, . . . , T − 1,
where ct is the charging of the battery in period t; ct < 0 means the battery is discharged. We will
require that q1 = qT + cT , i.e., we ﬁnish with the same battery charge that we start with. With
the battery operating, the net consumption in period t is ut + ct ; we require this to be nonnegative
(i.e., we do not pump power back into the grid). The total cost is then pT (u + c).
The battery is characterized by three parameters: The capacity Q, where qt ≤ Q; the maximum
charge rate C , where ct ≤ C ; and the maximum discharge rate D, where ct ≥ −D. (The parameters
Q, C , and D are nonnegative.)
(a) Explain how to ﬁnd the charging proﬁle c ∈ RT (and associated stored energy proﬁle q ∈ RT )
that minimizes the total cost, subject to the constraints.
(b) Solve the problem instance with data p and u given...
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 Fall '13
 F.Borrelli
 The Aeneid

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