The second open enrollment period for buying health insurance under the Affordable Care Act opened Nov. 15. (Don Ryan/AP)

HealthCare.gov and online insurance marketplaces in more than a dozen states opened for a second year on Saturday without the widespread computer troubles that frustrated consumers and the Obama administration when the exchanges debuted 13 months ago.

At public libraries, health centers and other community hubs around the country, people shopping for health plans — and a small army of enrollment workers and volunteers — found that the Web site for the federal marketplace generally worked and that a federal call center for advice had little or no waiting time.

Not all customers had smooth experiences. Some had forgotten their passwords for online accounts they created a year ago. Others discovered that they still couldn’t find affordable plans. And one state-run exchange — Washington state’s — took down its enrollment system within hours of its launch after discovering it was spitting out incorrect subsidy amounts.

But as opening day unfolded for the second sign-up period under the Affordable Care Act, it became clear that the technical flaws that dominated the start of HealthCare.gov last year had faded, giving way to new questions: Can the administration and its allies convince 7.1 million people already insured through the marketplaces to shop again for a 2015 health plan that might suit them better? And can more than 20 million uninsured Americans, who were eligible for the coverage last time but did not sign up, somehow be coaxed to buy health plans now?

At enrollment centers, people who are uninsured appeared to outnumber existing customers renewing their coverage. Workers said that relatively few consumers were finishing applications in one sitting, with most going home to consider health plans’ prices and benefits.

In a broad summary of the most popular ”Silver” level plans across each state in the federal exchange, 25 states had premium increases and nine states had declines in the new exchange that opened Saturday, according to an analysis by The Washington Post.

The second round of the insurance marketplaces began less than 24 hours after federal health officials made public a huge set of data on the premiums of the health plans being sold through the exchanges for the coming year. Overall, the monthly premiums are increasing 7 percent on average for existing plans that are being sold in the federal marketplace a second year, according to an analysis of the data by The Washington Post.

The existing plans are, on balance, slightly less expensive than new ones being sold through the marketplace for the first time, the analysis shows. But premiums vary significantly around the country; for example, the average premium for a “silver” plan, the most popular level of coverage, has gone up in 25 states that use the federal marketplace and down in nine states.

On Saturday, Obama administration officials urged consumers to act quickly, pointing out that this year’s open enrollment period will last for only three months, half as long as last year.

“We are up and running,” Health and Human Services Secretary Sylvia Mathews Burwell said as she stopped in at a large community health center in Northern Virginia’s Prince William County, a jurisdiction with a relatively large proportion of residents who remain uninsured.

Speaking to several dozen families and enrollment workers gathered in a brightly lit clinic waiting room festooned with balloons, Burwell announced that about 23,000 people around the country had completed applications for health plans during the first eight hours after HealthCare.gov opened for enrollments shortly after 1 a.m. Saturday. During the entire first day of enrollment last year, only six people managed to enroll because of serious computer defects.

Burwell also said that 1.2 million people had gone “window shopping” — which allowed consumers to look at plans but not buy them — on HealthCare.gov since last Sunday night. When HealthCare.gov opened last year, consumers could not explore health plans without going through the computer system’s registration process — a feature that federal health officials changed for this year.

Burwell did not mention — and White House spokeswomen declined to say — how many people visited HealthCare.gov on the first day, a figure that was 2.8 million by late afternoon of the Web site’s first day a year ago.

President Obama encouraged Americans to take advantage of open enrollment for the Affordable Care Act in his weekly address on Saturday. (Reuters)

With tables of cookies and face-painting for children, workers at the Greater Prince William Community Health Center in Manassas welcomed people such as Fernando Cabana, 42, who came with his wife and their 6-year-old daughter. Cabana, a construction worker who is uninsured, is the kind of person the administration wants to attract to health plans available under the 2010 health-care law that created the insurance marketplaces.

Cabana used to have family coverage, whose premiums cost $800 per month, through United Healthcare, he said. To save nearly $400 a month, he dropped coverage for himself and kept it for his wife and child.

“I’ve been waiting . . . for this day,” Cabana said Saturday. He did not expect to make a decision immediately but wanted to get information about his choices through the federal health exchange. In the end, however, an enrollment worker had trouble getting the computer system to recognize his Social Security number.

Virginia is one of about three dozen states relying on the federal exchange, while Maryland and the District operate their own marketplaces. DC Health Link opened Saturday with insurance brokers and three dozen enrollment assisters gathered to help people sign up at the Martin Luther King Jr. Memorial Library. As of 6 p.m. Saturday, 66 people had chosen health plans. The Maryland Health Connection, which was rebuilt to try to correct major computer problems, is opening in stages between Saturday and Wednesday.

The scene in Manassas resembled those in many other cities and towns Saturday. Some enrollment sites were busier than others. At Columbus Metropolitan Library in Ohio, just half the people who had made appointments to meet with enrollment helpers had shown up by late morning, according to Zachary Reat of the Ohio Association of Foodbanks, an organization that received a sizable federal grant to help coordinate enrollment activities around the state. To fill empty appointments, volunteers called uninsured people and urged them to come in, Reat said.

In Austin, lines were long even before the doors opened at an enrollment drop-in center run by the local nonprofit group Foundation Communities in an old storefront at a run-down shopping mall. Texas has a large number of uninsured Latino residents. The state was a focus of the administration’s campaign last year to coax people to sign up for coverage — and it will be again.

“It’s good. The word has gotten out,” said Elizabeth Colvin, director of Insure Central Texas, a program of Foundation Communities that helped about 15,000 people during the open enrollment last year. For this year, the group has not only Spanish speakers among its enrollment workers, but also speakers of Chinese, Arabic, Vietnamese and French.

Still, language was not the only potential barrier. David and Maria Medrano of Austin came to the mall enrollment center hoping to buy a plan but were disappointed for the second year in a row. As before, their income was too little for them to qualify for subsidies to buy coverage through the insurance exchange; under the health-care law, anyone earning less than 138 percent of the federal poverty level is supposed to be covered under Medicaid instead. But they are unsure whether they are poor enough to get Medicaid in Texas, one of about two dozen states that have chosen not to expand that program under the law.

Two years ago, David Medrano, 61, shattered both of his heels and ankles on the job as an independent construction contractor, and his wife had to quit her job to help him.

“We hawked everything to survive,” he said.

The couple have been relying on community health services and sliding pay scales. He has just been diagnosed with cancer and began chemotherapy and radiation a few days ago. On Saturday, they made an appointment for the coming week to get help in finding out whether they are eligible for Medicaid.

As in Austin, multiple languages could be heard Saturday on the edge of Golden Gate Park in San Francisco as several hundred people waited for help at an enrollment event sponsored by a state legislator and Covered California, the state’s insurance exchange. Most seemed to be enrolling for the first time and said they know little about what is available but felt desperate for help paying medical bills.

The crowd skewed toward older Asian residents, though there were also some young people, gay couples and a few young Latino families. After meeting with counselors, applicants were directed to certified insurance agents or navigators at plastic tables lining the large room. There was a large crowd at the start of the event, but after two hours, the plastic chairs for waiting were mostly empty.

In Massachusetts, officials were determined to make up for a debacle last year. The federal health-care law was largely modeled on a law enacted in Massachusetts six years ago. At first, the Massachusetts insurance marketplace, known as the Health Connector, worked well. But last year, its computer program collapsed during open enrollment.

The state scrapped the computer program, switched vendors and rolled out new software for Saturday’s enrollment window. Massachusetts Connector officials sent out a quarter-million mailings, are making 70,000 phone calls during the enrollment period and have barnstormed the state, offering apologies for the past and pleading with people to trust the system this time. A local supermarket chain, Market Basket, is dropping fliers in customers’ grocery bags.

The emphasis is on keeping people enrolled, rather than finding new uninsured to sign up, in part because Massachusetts has proportionately the fewest uninsured residents in the country, about 3 percent.

On Saturday, a home-health worker, an immigrant from Brazil, sat on the edge of her chair, gripping a plastic envelope filled with papers, at a community center in Cambridge, Mass. An adviser, Leonilde Gomes, eased her into the first barrage of questions on the insurance application.

“The first thing we are going to do is create an account,” Gomes explained carefully.

“Name?”

“Monica Souza.”

“Name of best friend?”

“Favorite color?”

Souza, 44, paused. “It’s to make sure you are identified,” Gomes said softly. “It’s for security.”

Souza was in the offices of the Massachusetts Alliance of Portuguese Speakers in a busy section of Cambridge that reflects the tide of immigration. The warren of offices is surrounded by the Pino Bros. Tattoo shop, Ribeiro DeSousa Insurance, Fernandes Fish Market and Grillo’s “World’s Best Pickles.”

Souza is uninsured. “I tried to get it before. I don’t know why I couldn’t. There were all sorts of questions,” she explained. She had made an appointment at the community center on the first day of insurance enrollment.

“Everybody needs health insurance,” she explained. “You don’t think so, until you do.”

Spencer Hsu in Manassas, Doug Struck in Boston, Karen Brooks Harper in Austin and Kari Lydersen in San Francisco contributed to this report.