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Unformatted text preview: nt of performance of the heat pump and η ∈ (0, 1] is the eﬃciency constant. The
eﬃciency is typically around 0.6 for a modern unit; the theoretical limit is η = 1. (When Tt = Ttout ,
we take γt = ∞ and Et = 0.)
Electrical energy prices vary with the hour, and are given by Pt > 0 for t = 1, . . . , 24. The total
energy cost is C = t Pt Et . We will assume that the prices are known.
Discomfort is measured using a piecewiselinear function of temperature,
Dt = (Tt − T ideal )+ ,
where T ideal is an ideal temperature, below which there is no discomfort. The total daily discomfort
4
is D = 2=1 Dt . You can assume that T ideal < Ttout .
t
To get a point on the optimal costcomfort tradeoﬀ curve, we will minimize C + λD, where λ > 0.
The variables to be chosen are T1 , . . . , T24 ; all other quantities described above are given.
Show that this problem has an analytical solution of the form Tt = ψ (Pt , Ttout ), where ψ : R2 → R.
The function ψ can depend on the consta...
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 Fall '13
 F.Borrelli
 The Aeneid

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