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

Here is Why You Need An Adapter

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

Luxembourg

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

Type C

Type F

Bhutan

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

Type C

Type D

Type G