you don't need to power down the router to change ip address only need router to drop internet connection and you will normally get a new ip address when router reconnects to internet
Yes it's easy really, but out of your control.
When you connect to the BT network the mux has to find you an available IP address from several available blocks of addresses. When the router/modem handshake takes place a series of test tones are used to determine linespeed and connection method. Because this action is totally dynamic it can connect you to any one of a series of RAS servers and indeed metro centres to get you connected ... the paths of which may vary somewhat in quality and speed. Even though balancing does take place constantly across the network, some nodes will always be busier than others, it's a massive network. And that is why there's always some inconsistency with connection sync rates. A healthy line should see better consistency, but it depends on your location and that of your nearest point of connection and traffic.