When looking at the P&ID , I was confused the use of drilled threaded plug or solid threaded plug at the end of drain or vent line. Please advise what occations are suitable for drilled plug valve.

We use drilled plugs in only one case.  If I have a leak-off line that would normally be passing a flow of steam, but I do not want liquid water to be carried along, I drill a weep hole in a plug at the low point of the line so the water can drip out onto the base but the majority of the steam is carried on to an elevated location to vent further away from personnel. We do this on leak-off lines from steam turbine carbon ring housings. We also do this on the drain ports of some mechanical seals if a steam quench might cause a buildup of liquid water on the atmosphere side of the seal.  In this case, I don't want to leave the port completely open in the case of a catastrophic seal failure.  The open port would allow greater leakage. The drilled plug forces most of the leakage to pass through the close clearance bushing (sometimes called a disaster bushing).  

I have looked at your P&ID and I (with more than 45 years in piping) still do not have a reasonable answer for you.  After consulting with a colleague (with more than 34 years in piping) and we both recommendthe following.

You should find the process engineer that made the P&ID or was responsible for creating the P&ID and ask them what is the purpose of, and function of, the through hole drilled in the threaded plug.  

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