Is power set of a power set of a set equal to the power set of the same set?

You are right. You can start by $A=\{a\}$.

$P(A) = \{\emptyset, \{a\}\}$

$P(P(A)) = \{\emptyset, \{\emptyset\}, \{\{a\}\}, \{\emptyset, \{a\}\}\}$

To reduce confusion of $\emptyset$, you can do $B = \{a,b\}$. However $P(P(B))$ will be length to list and I am too lazy to do this. (Actually making clear on all those sets involves empty set will make you clearer on understanding power set of a power set.)