You Do Need power plug adapter while traveling from Bhutan to Luxembourg.

Here is Why You Need An Adapter

The outlet and plug type used in Luxembourg are C and F and in Bhutan are C, D and G. This is how they look like.

Bhutan

In Bhutan, outlets of type C, D and G are in use.

Type C

Type D

Type G

Luxembourg

In Luxembourg, outlets of type C and F are in use.

Type C

Type F