02 Santa Fe hesitating while driving

Hopefully, someone can give me some insight.. I have a 2002 Santa Fe that runs great - I've never had any issues with it. A couple of weeks ago, I was driving home and the car started jerking as I was driving.
The car never died, and the next morning when I started it, I again had no problem.
I took it into the dealer anyway, and they said they found an "insufficient power" code, but they didn't know what that meant. They tightened the battery cable, since I had had the battery replaced a couple of months earlier and said that should do it.
Yesterday, I was driving to another state for the holiday, and after about 2 and a half hours, it started doing it again. I stopped for about 15 minutes and when I started going again it was fine for another hour and a half before starting again. Another 20 minute stop and it was fine for the remaining hour and a half.
No consistency that I can see that's causing it. The first time, I was driving about 35 in town and had only been in the car about 20 minutes. The second two times, I was going 65-70. Again today, the car is having no problem, so I don't think the dealer is going to be much help. I'm supposed to go out of town again at the end of December, so I'd really like to get the problem fixed.
Thanks for any help!
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I'm at least a little confused. I thought the "insufficinet power" code was only used on the 3.5 liter engine, which wasn't available until late in the 2003 model year.
Any chance you have the code number? I can probably find out what it means if you have the code number.
If this code refers to the system voltage being too low (code P0562), then your alternator is probably failing.
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