LG 60PK550 suddenly shutdown,when i turn it ON the relay work and then shutdown without any signal or any thing on screen

The LCD has quit working and the screen is no longer working for your TV.It would be more easier to purchase a new TV than to try and fix the TV that you have that is broken.

any sound out from your TV??

That's not the function of the ASD relay. The name is misleading. The relay is to turn things on for running the engine. It is controlled by the pcm. It will only signal that relay when the ignition key is turned to off.

check first the psu unplugging the Y-sus from it if the voltages stay on then you have a Y-sus failure if the psu keeps closing then its most likely the psu itself or your main board

Contact the authorized service center for LG. Your TV need a through checkup. Gopakumar Gopalan Google

BAD is a very vague description but certainly a plasma screen can produce a poorer picture with age, such things as "sparkly" patches on the screen or a lack of sharpness from the picture. Of course there is also the potential for the screen to have an image burn.

LG 60PK550 only outputs the audio via the optical out port on the back of the TV the HDMI ports don't have arc capabilities

No,the power supply board have problems.Those power supply transformers on the power supply board,causing this,high voltage flow going through those transformers.Tries websites like Shopjimmy.com,Ebay.com to buy a refurbish power supply board for the replacement.

Without seeing the unit itself and opening it up/running tests I couldn't tell you exactly what is happening, however I can offer some suggestions as to what it could be.
1) your power supply is bad. That clicking sound is the power relay coming on and going off. 
2) your main board is bad. It handles a lot of your signals for powering on your set. If it's bad, the tv won't turn on correctly as it doesn't recieve all the right signals.
3) your Y or Z boards are bad. These could cause the power supply to go into shutdown if the voltages on these things are too far out of whack. 

Easiest way to tell if you're going to go into your tv is to try to unplug your Y and Z boards and turn it on. If it turns on and stays on chances are one of those two boards are bad. 

There might be an internal short circuit /loose connector , it happens that when it short circuits the electricty gets dicharged causing the phone to turn off without reason ,Try exhanging it from the warrenty or if the store allows you to exhange it