The National Transportation Safety Board has opened an investigation into the fire.
Passengers have had limited cellphone service because of the power failure, but many were able to make calls to friends and family when the Triumph rendezvoused with another Carnival ship that dropped off food and supplies. The other ship had a working cellular antenna.
Tilley said late Wednesday that she had received an update from Carnival saying the Triumph had again rendezvoused with another cruise line, taking on more supplies and food, and that a third tugboat had also arrived to bring the ship to port.
Meanwhile, officials in Mobile are preparing a cruise terminal that has not been used for a year to help passengers go through customs after their ordeal.
The Triumph is expected to arrive Thursday afternoon.
The company has disputed the accounts of passengers who describe the ship as filthy, saying employees are doing everything to ensure people are comfortable.
Passengers are supposed to receive a full refund and discounts on future cruises, and Carnival announced Wednesday they would each get an additional $500 in compensation.
"We know it has been a longer journey back than we anticipated at the beginning of the week under very challenging circumstances," Carnival President and CEO Gary Cahill said. "We are very sorry for what our guests have had to endure."
Carnival spokesman Vance Gulliksen acknowledged the Triumph's recent mechanical woes, explaining that there was an electrical problem with the ship's alternator on the previous voyage. Repairs were completed Feb. 2.
Testing of the repaired part was successful and "there is no evidence at this time of any relationship between this previous issue and the fire that occurred on Feb. 10."
But according to the email sent to passengers Jan. 28, the issue affected the ship's cruising speeds, delaying its arrival in Galveston. The email also informed Smedley and other passengers that the propulsion problem would prevent them from docking at two ports.
Jay Herring, a former senior officer with Carnival Cruise Lines who worked on the Triumph from 2002 to 2004, said the ship was not problematic when he was on it.
The Triumph takes five generators — with one on backup — to power the ship, and 80 percent of that energy is needed to simply push the massive vessel through the water, Herring said.
Each of those generators is the size of a bus, so it's unrealistic to think that the ship could have enough backup power on board to run services when the engines die, Herring added.
"It's one of their bigger ships. It's certainly on the top end of Carnival's fleet," he said of the Triumph. "There are so many moving parts and things that can go wrong."
Associated Press writer Bob Johnson contributed to this report from Montgomery, Ala.
Plushnick-Masti can be followed on Twitter at https://twitter.com/RamitMastiAP .