Scan IP address of ESP8266
The de-facto standard way for devices to broadcast their IP address to clients is through the Multicast DNS system (mDNS). In this system the server sends a UDP message to a specific multicast group address and clients listen for it so they can work out what is where. 

mDNS is also known as Zeroconf or Bonjour depending on who you talk to.

The ESP8266 Arduino core (if you are programming the ESP8266 directly) has mDNS examples with it. If you are using the AT command interface instead, then you will need to read the manual to find how to send UDP packets to multicast addresses.