Is it normal to lose Plex share library after Apple TV loses power?

When I unplug my Apple TV (e.g. move power outlets), I have to re-link my Plex server to Infuse. I realize that power loss is not frequent so this is not a terrible inconvenience. But is this normal behavior?

My guess would be that when you power down the ATV and then power back up it gets reassigned a new IP address by your router. Then plex needs to see the new IP address.

You can try assigning a static IP to your ATV in your router settings and see if that helps.