I am sorry that you had to wait that long, but I was on sick-leave and was not able to respond.
As for your question in general if you have moved within the same city you dont need to change anything, but inform the insurance company etc..
Even if you have moved nearby you can still keep your old registration number. But if you want to change it:
1. find our which registration office if responsible for you
2. the car re-registraion costs in average of 26 Euros
3. the new registraion plate costs are to be counted in addition( average 30 euros)
I hope I could assist you with the answer, if there further questions please use are community and we are happy to assist you.