The screenshot below is the method provided by the Asus RT-N66W model. Each router vendor provides a different UI to achieve this, but once you login to the Administrative Console of the router you should be able to locate the reconnect button. Most routers allow you to disconnect from upstream ISP, and reconnect via a point-and-click from the Administrative Console.
Here are a few ways to change IP address of your router. So, if you want the world to see you with a different IP address, you'll have to change IP address of the router. When computers connected to the router visits the Internet, the world views your computer with a public IP address of your router. In home network with dynamically assigned IP address, a router is assigned a public IP address and all the computers connected to the router are assigned private IP addresses. A router is a layer 3 networking device that connects multiple computers to the Internet.