The power cable must be capable of handling the maximum amount of power that the PSU will ever pull from the wall outlet. I once knew of a chart out on the internet that I can't find anymore for some reason that shows which cables should be used. I learned from this chart that you can't just use any old cable, especially for PSUs that have a large wattage capacity.
Since I can't find this chart anymore though (probably because I don't know what to look for), I can't say whether a 1000W PSU needs a special-sized cable or not. However, since you're in the U.S. and your input voltage is likely 115-120V, 15A should provide at least 1725 to 1800W. So, a 15A PSU power cable should be quite good.
Edit: I found it thanks to Firefox's history searching feature (but also thanks to me stumbling upon words that helped me find it), and it's not like I thought, but it's still informative!
I think I was trying to learn more about the C19 connector (the 3rd one from the bottom).
Edited by TwoCables - 5/20/15 at 6:58pm