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Mathematics 2022, 10, 3167 15 of 27 • Objective function: The function J to be minimized in the optimization problem includes a term related to economic cost of cooling power production, as well as an additional term related to the charge ratio, that allows promotion of the TES tank charging/discharging according the operating mode scheduling, to be proposed once the cooling demand profile and the energy price variation have been analysed. Both terms included in the objective function J are detailed below: J = Jcost + Jratio, (22a) ̇ref ̇ref ̇ref – Latent zone: The PCM cylinders are not desired to reach subcooled solid or superheated liquid state at any instant, thus the charge ratio constraints indicated in Equation (20) are also imposed: γTES(t+k)≥γmin >0 TES γTES(t+k)≤γmax <1 TES ∀k ∈ [1, PH] . (20) As mentioned in Sections 2.3 and 3.1, the overall charge ratio can be computed from the specific enthalpies of the cylindrical layers included in the TES tank vector xTES defined in Equation (1). Specifically, Equation (21) allows computation of γTES from some of the state variables included in xTES: Umax − UTES γTES= TES , PH Jcost = ∑ we,sec(t−1+k) Qe,sec(t−1+k) + wTES(t−1+k) QTES(t−1+k) + wTES,sec(t−1+k) QTES,sec(t−1+k), (22b) k=1 5. Case Study Umax = n TES Umin = n TES Umax , pcm pcm Umax − Umin TES TES UTES = npcm Upcm Umax = pcm Umin = pcm Upcm = γTES = Umin pcm pcm , , hlat+ , (21) V ρ pcm pcm pcm V ρ hlat− , pcm pcm pcm nlay mlay ∑ hpcm,k , k=1 Umax −U pcm pcm . Umax − Umin pcm pcm PH Jratio = ∑ cγTES (t + k) γTES(t + k) . (22c) k=1 This section is devoted to the analysis of a case study, focusing on the scheduling problem. A challenging load profile is proposed, where the need for operating mode scheduling is highlighted. The presented simulation results compare the performance of the proposed scheduling strategy to a that of the same refrigeration cycle without TES, regarding economic cost and constraint meeting. A sensitivity analysis is also performed, where the performance of the proposed strategy is assessed when significant parametric uncertainty in the prediction model is considered.PDF Image | Refrigeration Systems with Thermal Energy Storage
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