Apparatuses and techniques are described for reducing peak current consumption in a memory device when performing a word line voltage refresh operation or a read operation. When a word line voltage refresh operation or read operation is performed for the first time after a memory device powers up, the operation is performed with a power-saving technique such as reducing a ramp up rate of a voltage pulse, ramping up the voltage pulse in multiple steps, initiating the ramp up for different groups of word lines in a block at different times, initiating the ramp up for different blocks of word lines at different times, and reducing the number of blocks which are refreshed concurrently. When an additional word line voltage refresh operation or read operation is subsequently performed, the power-saving technique can be omitted.