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Several conductor designs featuring High Temperature Superconducting (HTS) stacked tapes for fusion coils are being proposed. These conductors are planned to operate in time-varying magnetic field and current; thus, the estimation of AC losses is fundamental for the conductor design and the accurate analysis of its performance in operation. The case study of an HTS conductor proposed for the hybrid (HTS-LTS) Central Solenoid coil for the EU DEMO tokamak is considered in this work. Here, a numerical model based on the finite element method (FEM) and the H-formulation is used, in order to estimate the hysteresis losses. The FEM model is first benchmarked against available analytical formulae as well as available literature data. Then it is applied to the real case operational scenario. It is shown that for the conductor design analyzed, the coupling losses are orders of magnitude lower than the hysteresis ones. The impact of the hysteresis+coupling losses on the temperature margin of the coil is assessed with a thermal-hydraulic model. It is shown that the heat generated in the HTS layers is partially transferred to the LTS layers, leading these layers to quench. An alternative conductor concept is also analyzed, showing that, however, in the top and bottom modules of the CS coil, due to the bending of the magnetic field, a too large heat deposition is present.