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For HTS electro-technical devices overload currents due to faults in grids are the operational reality. Overloads are working modes for fault current limiters as well. In this work we reproduce experimentally overload conditions in different HTS 1-G and 2-G tapes by use rectangular current pulses with different amplitudes and durations. Voltages and temperatures along tested samples have been measured. These experiments permitted us to follow the change of heat developments during overloads due to a difference of cooling regimes in the liquid nitrogen. Measuring of characteristic times of a heat development in different modes provides the data about possible surviving time of HTS devices and bearable current overloads. We also studied an influence of different shunting or reinforcing methods of HTS tapes on heat development. The obtained data can be used to analyze overload conditions at HTS power cables and HTS resistive FCL.