A wire electrical discharge machining apparatus includes a unit capable of separately opening and closing each of a high impedance path and a low impedance path, a unit that sets an open/close pattern in which a combination of closing one of the feeding paths and opening another one of the feeding paths is designated for switching power feeding between the high impedance path and the low impedance path, a unit that changes pulse energy per feeding pulse in a present feeding path to reduce a difference in discharge pulse energy applied to an inter-electrode gap from a machining power supply between at a time of high-impedance-path feeding and at a time of low-impedance-path feeding, and a unit that controls opening and closing of the path open/close unit in accordance with the changed open/close pattern.