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Mathematics 2022, 10, 3167 20 of 27 as well as the distribution of the temperature field of the PCM cylinders in the cooling, charging, and discharging processes. Figure 12 shows that the refrigeration cycle without energy storage is not able to satisfy the cooling demand peak from t = 5 h to t = 8 h, thus incurring in some cooling power deficit, which is remarked in Figure 12. However, the TES-backed-up refrigeration cycle operated according to the proposed scheduling strategy is shown to provide the required cooling power to the secondary fluid by combining the power produced at the evaporator and the TES tank contribution during the peak period. To do this, the TES tank is charged during the off-peak periods, as shown in Figure 13b. The charge ratio is shown to be kept within the safety limits, as depicted in Figure 14, whereas a great deal of the energy capacity of the TES tank is used along the day. Notice in Figure 15a that, during the charging processes, the temperature of the intermediate fluid is below Tlat , being over Tlat in the pcm pcm discharging process; in the stand-by processes, the thermal inertia of the intermediate fluid causesthat,whileapproachingthethermalequilibriumwiththePCMcylindersatTlat , the latter continues charging/discharging while the residual cold energy is transferred. The distribution of temperatures within the PCM cylinders represented in Figure 15b show how the layers are quitting the latent zone from the outermost layer to the innermost, both in charging and discharging processes, as their latent energy depletes. The performance of the low-level cooling power controller is not analysed in this work for the sake of brevity, since it is analysed in depth in [25], thus no plots of the actual control actions are included in this work. In any case, some important issues concerning this control layer will be remarked upon. The settling time of the cooling power closed loop turns out to be small enough to assume, given the scheduling time, that the required cooling powers are almost immediately provided, as long as the computed references are feasible, which is ensured by the power constraints met by the scheduler. The hypothesis about the separation between the time scales considered in the modelling stage is confirmed in the aforementioned work. Concerning the charging and discharging cooling powers, constant cold energy supply/release is achieved by increasing the mass flow of the refrigerant/secondary fluid as the thermal resistance caused by the cylindrical shell in sensible zone grows. Further details about the performance of the cooling power controller can be found in [25]. Regarding economic cost, the overall energy costs along the day of both the proposed scheduling strategy for the TES-backed-up cycle and the original system are gathered in Table 5, as well as the achieved reduction. Notice that, in the case of the refrigeration cycle without storage, it is assumed that the energy deficit shown in Figure 12 must be bought at the market price detailed in Figure 9. pcm 900 800 700 600 500 400 300 200 100 Cooling demand satisfaction Cooling demand Evaporator, NMPC Evaporator, without TES TES, NMPC Deficit, without TES 0 0 2 4 6 8 10 12 Time [h] Figure 12. Comparison of the cooling demand satisfaction between the proposed NMPC-based scheduling strategy and the refrigeration cycle without energy storage. Power [W]PDF Image | Refrigeration Systems with Thermal Energy Storage
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