  Motivated by super Poisson-Lie (PL) symmetry of the Wess-Zumino-Witten (WZW)
model based on the $(C^3+A)$ Lie supergroup of our previous work [A. Eghbali
{\it et al.} JHEP 07 (2013) 134], we first obtain and classify all Drinfeld
superdoubles (DSDs) generated by the Lie superbialgebra structures on the
$({\cal C}^3+ {\cal A})$ Lie superalgebra as a theorem. Then, introducing a
general formulation we find the conditions under which a two-dimensional
$\sigma$-model may be equivalent to a WZW model. With the help of this
formulation and starting the super PL symmetric $(C^3+A)$ WZW model, we get a
hierarchy of WZW models related to super PL T-duality, in such a way that it is
different from the super PL T-plurality, because the DSDs are, in this process,
non-isomorphic. The most interesting indication of this work is that the
$(C^3+A)$ WZW model does remain invariant under the super PL T-duality
transformation, that is, the model is super PL self-dual.
