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I purchased the Belkin 4-port USB3 hub that is sold at the Apple store. Since making this change, I have not experienced the problem again. I have literally put my iMac to sleep *many* times just to prove that the problem is really solved. Now, when it goes to sleep, the hub and drives shut down, as expected (all of the lights on the hub shut off). When the iMac returns from sleep, the hub reactivates (all the lights turn back on) a moment before the iMac comes back to life, and the hard drives are visible and ready to use. I have had no further disconnect messages.

Have you run into the situation where the Belkin hub seems to get into a wake/sleep cycle that gives the Mac insomnia?

I have 10.8.4 on a rMBP that seems to cycle (sometimes, sometimes well behaved). It doesn't wake up the whole way, but is maintenance wakeup that also wakes up my third party display (TB-DP connection). System logs says hub port 2 is waking up, which is a recent OWC USB3 raid.. but I can turn off that Raid and sometimes the rMBP gets into the same cycle regardless. Blue Tooth is turned off. I may try swapping cables.
 
I finally figured this out .....

I'm no engineer. New iMac early this year & I purchased a new surge protector, hooked everything up including my G-Drive. It never worked right from that point. Whenever my computer went to sleep, the G-Drive ejected & I would get the pop-up warning: improperly ejected. I spent hours on the phone, on forums, looking for a solution and resorted to backing up once a week. Finally I took it to the Genius Bar yesterday where it worked perfectly. Even so, to be certain, I got a new USB. Happily I connected up at home it started all over again until I finally looked at the surge protector which, turns out, was an HP Monster with Green Power Energy Savings (the outlets switch off automatically & I hadn't paid any attention.) I ditched Monster, plugged it up to an older surge, and it's working perfectly.
 
Plugged Directly into your Mac, or through USB hub?

If you are using a USB hub then that is most likely your problem.

It was connected directly. My previous external HDD did not get detected by the mac at all but it worked perfectly well with my brother's windows laptop. So I gave it to him. This is the next one I bought and now I've stopped using it also.

First I had the drive encrypted and I didn't save the password to keychain, so I had to manually mount it each time coming back from sleep. Now the password is saved, but still I see that the HDD is getting disconnected.
 
See my post #43. If you're directly connected it's likely your drive is incompatible with the USB3 bus. Contact the manufacturer, and if you can try another brand drive to make sure.
 
I've been having this problem since I got my 2013 MBA, I never had it with my 2011 MBA. My external drive is a LaCie 2Big, attached via thunderbolt to my TBD.

I tried unchecking the "put network drives to sleep when possible" box. I'll see if that works.
 
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