WASHINGTON (AP) — The second sign-up season under President Barack Obama's health overhaul opened, with hopes that this time consumers will have a positive experience. Obama says about a half-million people logged into the federal government's website on Saturday, the first day of open enrollment.

But the fear is that entrenched political opposition and renewed legal challenges may yet collapse the program that's bringing health care to millions of previously uninsured Americans. The administration can't afford another technology meltdown.

With 7 million paying customers in new insurance markets, the Affordable Care Act has shown it is helping to reduce the number of uninsured. Insurers, not known for altruism, have stuck with the fledgling program despite ongoing technical headaches with the HealthCare.gov website. More companies are participating for 2015, a sign they see a business opportunity.

Obama said Sunday that more than 500,000 people had successfully logged in to the website on the first day.

"HealthCare.gov works really well now," he said at a news conference in Brisbane, Australia, where he attended a global economic summit.

Yet, "Obamacare" is still struggling to win hearts and minds. The latest Associated Press-GfK poll finds that, if forced to choose between repealing the law and implementing it as written, 56 percent of Americans would repeal it completely. Only 41 percent would carry it out.

However, most don't see the law going away. Sixty-one percent said they expect it to be implemented in its current form, or something near that.

Health and Human Services Secretary Sylvia M. Burwell, a management expert assigned to save what's been a problem child of social programs, says she's confident the sign-up season will be successful, even if it's only half as long as last year's: three months, through Feb. 15.

Will the law remain on the books after Republicans gain full control of Congress in January?

"The idea of repeal ... is not something this administration will let happen," said Burwell.

What about the legal challenge the Supreme Court has just agreed to hear, calling into question the law's tax credits that make premiums affordable for millions?

"Nothing has changed," Burwell said, adding that the tax credits "will be continuing." At least for the time being. The Supreme Court isn't likely to hear the case until the spring, after 2015 open enrollment is over.

Website outages are not out of the question this year, but a full-scale meltdown seems less likely. HealthCare.gov has been revamped to handle last season's peak loads and beyond. The federal website will serve as the online portal for coverage in 37 states, while the remaining states run their own insurance exchanges. Consumers can also apply in person or through call centers.

The pool of potential customers is an estimated 23 million to 27 million people who don't have access to affordable coverage on the job.

For most newcomers, the online application has been simplified, cut to 16 computer screens from 76. Navigation is easier. Window shopping is available without first having to create an account.

Premiums for 2015 are a wild card. Nationally, the average increase is expected to be modest. But prices can vary dramatically from state to state, even within regions of a state. Many returning customers could end up facing premium increases if they don't shop around.

The administration is seeking to lower expectations. Burwell said her target is a total enrollment of 9.1 million people in 2015, a 28 percent increase. The nonpartisan Congressional Budget Office had estimated the number would nearly double, to 13 million people enrolled next year.

The administration is facing several new tests.

This sign-up period will be the first time that renewal has been tried for current customers, and also overlaps for the first time with tax-filing season.

For those already signed up, coverage will renew automatically if they do nothing, but that may not produce the best result. The returning customers could miss out on lower-premium options and get stuck with outdated and possibly incorrect subsidies. In most cases, they have until Dec. 15 to update their income information or switch insurance plans, in order to have the changes take effect on Jan. 1.

The tax issues will emerge during next year's filing season.

Current customers who got tax credits this year will have to file new tax forms to prove they got the right amount. Too much subsidy and their tax refunds will be reduced. Too little, and the government will pay them.

People who remained uninsured risk a penalty that will be deducted from their tax refunds. Millions may qualify for waivers, but getting exemptions could involve a paperwork ordeal.

Community-based counselors helping uninsured people say interest remains strong, but they worry about this year's abbreviated sign-up season.

Nathalie Milias, who works with Haitian groups in Miami, said she has been feeling the demand since the summer as uninsured people approached her.

"They see me in the store, they call," said Milias. "The locations I go to are already calling and saying, 'How many appointments do you want for Monday?'"

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Associated Press writer Kelli Kennedy in Miami contributed to this report.

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(Photo: AP Photo/Ann Heisenfelt)