If they are out at sea, probably not a lot. Tsunamis are only a problem when they come to shore. Offshore the wave height may only be inches.

Yeah, but I was responding to Barrett808's post where he said that he guessed that the contraptions "will float just offshore." On the open ocean tsunamis wouldn't be much of a problem; regular ole wind-churned waves would be quite sufficient to wreck a "solar island."

I'd be much more concerned about hurricanes. For example, I don't think anything like this could be done off the Atlantic shoreline. The Pacific would probably be okay, though.