LOS ANGELES—A power outage cut electricity Thursday to a large swath of Southern California, as well as parts of Arizona and Mexico, causing a nuclear-power plant to shut down and exacerbating strains from a heat wave.

More than one million utility customers in San Diego County, parts of Orange County, parts of Arizona and the Mexican state of Baja California all lost power after a line transporting electricity from Arizona power plants to California customers failed, according to the California Independent System Operator.

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