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Energies 2020, 13, 340 6 of 21 The energy stored at the successive time step k + 1 in the storage device is then expressed as: Ek+1 = (1 − uklk) Ek + ηchPch,k∆tk − Pdch,k ∆tk (1) ηdch The energy Ek also corresponds to the state of charge (SoC) of the storage system, which may be expressed in per units (or in per cent) by dividing it by the energy capacity Eˆ of the storage system: SoCk = Ek; SoC%k =100 Ek (2) Eˆ Eˆ By introducing the power capacity Pˆ of the storage system, the constraints on the charge and discharge power values are given by: Pch,k ≤ ukPˆ (3) Pdch,k ≤(1−uk)Pˆ (4) The impact of the ramp constraint Rk (in per units) on the charge and discharge during the time step ∆tk can be expressed in terms of the SoCk as in [39]: − Rk∆tk ≤ SoCk+1 − SoCk ≤ Rk∆tk (5) The idle operation of the TES is modelled by assuming uk = 1 and Pch,k = 0, so that the reduction in the energy stored is represented by the effect of the per-unit internal losses lk. Furthermore, the minimum limit EˇK and the maximum limit EˆK are imposed on the final energy stored in the TES at the end of the study period (k = K): EˇK≤EK ≤EˆK (6) The general form (6) to represent the final energy stored in the TES is extended with respect to the constraint of having the same energy at the beginning and the end of the study period, used in various references, e.g., [40–42]. This extended form is more suitable to be incorporated into calculation tools that use uncertain or predicted values, for example, within an approach based on model predictive control (MPC). Detailed TES models have been formulated in [43,44] by considering heat transfer constraints. These models can be used for sensible heat storage and latent heat storage (see Section 3.1) with different parameters. The presence of non-linear heat transfer constraints is solved with iterative calculations. The results show that latent heat storage can provide more flexibility than sensible heat storage, in particular with low initial thermal energy level. For a TES composed of a water tank, the model shown in [45] considers the volume of water in the tank as the relevant variable for the energy balance, assuming that the temperatures of the hot and cold water are constant during the time step of interest. A further detailed model of a TES composed of a high-temperature section and a low-temperature section is described in [38]. 3. Types of Thermal Energy Storage Technologies 3.1. Categorisation of the TES Technologies 1. 2. 3. TES technologies can be partitioned into three categories: Sensible heat storage Latent heat storage Thermochemical storagePDF Image | Thermal Energy Storage for Grid Applications
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