What are the IUPAC names for singly- and doubly-protonated piperazine? If ammonia, $\ce{NH3}$, is protonated, it becomes ammonium: $\ce{NH4+}$.

If piperidine is protonated, it becomes piperidinium:



If piperazine is protonated once, what does it become? By analogy, I would assume piperazinium:



If the resulting monocation is protonated again, what does it then become?  The best I've come up with are piperazine dication and piperazin-dium, but that latter seems really awkward:
The relevant rule in the current (corrected) version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) is:

P-73.1.1.2 General rule for systematically naming cationic centers in parent hydrides
A cation derived formally by adding one or more hydrons to any position of a neutral parent hydride (listed in Chapter P-2), or whose degree of hydrogenation has been modified (see P-31) is named by replacing the final letter ‘e’ of the parent hydride name, if any, by the suffix ‘ium’, preceded by multiplying prefixes ‘di’, ‘tri’, etc. to denote the multiplicity of identical cationic centers. (…)

Therefore, the preferred IUPAC names are piperazin-1-ium and piperazine-1,4-diium, respectively.