South Australia is the country's fifth most populous state, with 1.7 million people and Adelaide as its capital, and is a major wine producer and traditional manufacturing hub.
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The Bureau of Meteorology said a vigorous cold front was moving across the state with an intense low pressure system due on Thursday.
"We'll have gale force winds and large seas (across the south of the country); also heavy rain and thunderstorms, which will lead to renewed river rises," it said on its website.
SA Power Networks said repairs to its transmission network were underway.
"There were more than 21,000 lightning strikes recorded over a 12-hour period from midday yesterday on the West Coast, and as a result it is likely some damage has occurred to our distribution network," it said.
The state had been brought to a standstill, with ports closed, trains and trams stopped, traffic lights out and long commuter delays, state agencies said.
South Australia was relying on accessing power from Australia's populous east coast via a power interconnector with the neighboring state of Victoria when there was a failure on Wednesday.
No power was flowing from Victoria into South Australia, said a spokesman for the Australian Energy Market Operator, which operates the power systems in southern and eastern Australia.
When the state tried to compensate, it experienced what is known as a "voltage collapse," Simon Emms, executive manager of network services at network operator ElectraNet, told ABC Radio, due to storm damage to power lines. This led to a statewide outage.
A spokeswoman for Electranet said power was being restored to some areas of Adelaide, but could not say when the lights would go on across the state.
"Now, clearly, questions will be raised," Federal Minister for the Environment and Energy Josh Frydenberg told Sky News. "Serious questions will be raised that need to be answered as to how this extreme weather event could take out the whole of the electricity supply across a major state such as South Australia."
The impact was wide-ranging, with traffic coming to a standstill in Adelaide while power supplies were disrupted to BHP Billiton's Olympic Dam copper-uranium mine, a huge mining operation more than 500 km (300 miles) to the northwest.
A BHP spokesman said back-up power generation was being used to run critical infrastructure. (Writing by Jonathan Barrett; Editing by Nick Macfie)