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Mathematics 2022, 10, 3167 8 of 27 Regarding the inputs, the description of the inlet flows includes extensive (m ̇ TES and m ̇ TES,sec) and intensive variables (the {PTES,in – hTES,in} pair for the refrigerant and TTES,sec,in for the secondary fluid). The latter describe the thermodynamic state of the corresponding fluid, whereas the surroundings temperature Tsurr represents a measurable disturbance. Concerning the outputs, the thermodynamic state of the outlet flows is determined by TTES,sec,out and the {PTES,out − hTES,out} pair. The variables Q ̇ TES and Q ̇ TES,sec refer to the cooling powers transferred between the refrigerant/secondary fluid and the intermediate fluid. The TES tank state vector xTES is made up of the temperature of the intermediate fluid Tint and an enthalpy description of the cold energy stored inside every PCM cylinder. Notice that the cold-energy storage ratio γTES (also called as charge ratio) is not actually a state variable, since it can be computed from the cold-energy distribution of the PCM cylinders. 3. Modelling 3.1. Simplified Dynamic Modelling In this section a simplified dynamic model describing the dominant plant dynamics as straightforwardly as possible is intended to be developed, so that it can be used as the prediction model in the cooling power scheduling strategy. It is to be noted that the simplified model is used only for the sake of scheduling, while the plant is simulated using the detailed and more realistic model described in [25]. In that work, it was shown that two different time scales arise in the combined system, one related to the faster intrinsic dynamics of the refrigeration cycle, and a slower one related to the states of the TES tank. Indeed, since it is the intermediate fluid that transfers heat to the refrigerant pipes, the secondary fluid pipes, and the PCM cylinders, its low heat capacity is the main cause of those slower dynamics. A detailed model of the combined system, focused on the fastest dynamics caused by the refrigerant circulation, was proposed in [25]. That model also describes the slower dynamics related to heat transfer within the TES tank, but the sampling time needed to be small enough to properly describe the refrigerant dynamics. Since the scheduling strategy is intended to be performed using a much greater sampling time that fits the dynamics of the cooling demand profile and the TES tank, such a detailed model becomes unsuitable and computationally unaffordable for the scheduling strategy. Therefore, in this work we are interested in developing a simplified version of the previous detailed model, focusing on the thermal behaviour of the intermediate fluid and the energy distribution of the PCM cylinders, that are indeed the slow TES states of the detailed model. Concerning the slower time scale, since the refrigeration cycle intrinsic dynamics become negligible, the whole TES-backed-up vapour-compression cycle can be statically modelled. Then, the intermediate fluid and its heat transfer to the PCM cylinders repre- sent the dominant dynamics. As a consequence, in the simplified model, the TES tank charging Q ̇ TES and discharging Q ̇ TES,sec cooling powers are considered as virtual manipu- lated inputs, whereas the refrigeration cycle is implicit and supposed to somehow provide these cooling powers, within some feasible ranges. The simplified model is schemati- cally described in Figure 4, where the surroundings temperature Tsurr is considered as a measurable disturbance. Figure 4. Simplified model of the TES-backed-up refrigeration system. Then, the simplified model just describes the non-linear dynamics of the intermediate fluid and its heat transfer to the PCM cylinders. The highly time-efficient dynamic mod- Virtual manipulated variables Disturbance Q ̇ T E S ̇ QT ES,sec Tsurr Simplified TES tank dynamic model xTESPDF Image | Refrigeration Systems with Thermal Energy Storage
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