You can transfer PPF from one Bank to account, and they can't say that you have to close this PPF account.
I don't think premature closing is allowed in PPF.
If your PPF account have completed 15 years, then the new account which you open will have lock in period of 15 years.
First check in Tamil Nadu SBI branch if PPF account option is available in that branch, if yes, then write a letter to your SBI Branch in AP requesting for transfer
How to transfer PPF account from one branch to the other of ( bank or post office)?
http://mutual-funds-personalfin.blogspot.in/2011/06/how-to-transfer-ppf-account-from-one.html
Format:
To,
The Bank Manager/Post Master (General)
………………………………………
………………………………………
Subject: Request for transfer of PPF A/c No…………………………………….
Dear Sir,
I am holding the above mentioned PPF Account in your Bank/Post office. I hereby approach you with the following request(s):
1. I have shifted my residence. My new residential address is as under:
Address
……………………………………………………………….
………………………………………………………………….
………………………………………………………………….
I request you to kindly make necessary change in the records.
2. Kindly transfer the balance in my above mentioned account with your bank to (name of the bank) at (name of the branch). The address of branch is as under:
Address
……………………………………………………………….
………………………………………………………………….
………………………………………………………………….
The original Bank/Post office PPF Pass Book for the entries made from …………(date)
to ……………………..(date) is enclosed herewith.
Thanking you,
Yours faithfully,
(………………………….)
Enclosure: Original Pass Book