National Grid said it expects all its customers, except those in Norfolk and Plymouth counties, to have power back Sunday.

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As of midnight, the utility had 76,333 customers without electricity following the weekend’s storm. Norfolk, Plymouth and Bristol Counties were among the communities in National Grid’s service territory most battered by the storm.

“We are working hand in hand with state and local officials so that we can restore power as quickly and safely as possible,” said Kathy Lyford, vice president of National Grid’s New England operations, in a statement late this afternoon.

NStar, meanwhile, still had more than 198,500 customers in the dark.

Crews with both utilities have begun to assess damage wrought by the storm, but the companies say it will take time to come up with power restoration estimates and are still warning customers that outages could last several days in some places.

“This blizzard has caused devastating damage to our electric system in the southern part of our service area and we are working to push forward with our restoration effort as soon as it’s possible to do so,” Craig Hallstrom, president of NStar Electric, said in a statement.

Both National Grid and NStar declared the snowstorm a “Level 5” event—the highest emergency designation—meaning the utility companies expect that the worst outages could stretch over three days or more.

During an 11 a.m. conference call today, National Grid’s Massachusetts president Marcy Reed said that nearly two-thirds of the utility’s outages were caused by damage to tranmission lines, which are the backbone of the company’s power system. Approximately 1,400 wires were down in the storm’s path, as well, she said.

“We understand that customers who spent the night without power are quite anxious to get that restored,” Reed said. “It goes without saying we’ll be working around the clock.”

NStar and National Grid said difficult weather conditions are keeping most crews from handling all but emergency and fire calls for downed wires and outages. Both utilities had crews patrolling and assessing damage Saturday in anticipation of repair work.

NStar said its hardest-hit areas were the South Shore, Cape Cod, and the South Coast around Greater New Bedford. When possible the utility used its system’s “remote switching” technology to reroute and restore power to homes that had gone dark.

“The biggest challenge we are facing right now is conditions are still not safe for us to have our crews on the road,” said Michael Durand, a spokesman for Northeast Utilities, the parent company of NStar. “A combination of the weather still being very bad in the areas where we have damage, and the high amounts of snow that we will have to travel through—it’s going to make the repairs difficult and time consuming.”

In Quincy, which was hit by outages Friday night, some had begun to shovel out, but others reported being trapped by snow and worried about how to get to a shelter the city had opened. One resident said Hancock Street, a major travel artery, had an 8-foot bank of snow.

By midnight, however, fewer than 5,000 customers in Quincy were still without power.

A call to the city’s emergency operations center was not returned.

Quincy resident Rakiesha Chase-Jackson and her husband spent the first part of Saturday morning in their Volkswagen GTI, charging phones so they could communicate via Twitter, listening to NPR, and—most importantly—warming up.

The storm knocked out power to couple’s apartment in the Wollaston Beach area sometime around 10 p.m. Friday, cutting the electricity that operates the thermostat.

“Without the thermostats,” Chase-Jackson said, “we hunkered down in the [heated] car. Sadly there’s no portable kitchen, so we had to come back inside to make food.”

Meanwhile, Unitil and Western Massachusetts Electric Co., which is also part of Northeast Utilities, saw very few outages Friday night into Saturday morning.

“We expect winds to remain strong throughout the day, so we still have a full complement of crews ready to respond should we get outages,” Unitil spokeswoman Carol Valianti said via email. “The one thing we are urging customers to do is keep their gas meters and vents clear of snow.”