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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 piecewise-linear 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 cost-comfort trade-off 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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