  We revisit the question of whether and how the steady states arising after
non-equilibrium time evolution in integrable models (and in particular in the
XXZ spin chain) can be described by the so-called Generalized Gibbs Ensemble
(GGE). It is known that the micro-canonical ensemble built on a complete set of
charges correctly describes the long-time limit of local observables, and
recently a canonical ensemble was built by Ilievski et. al. using particle
occupation number operators. Here we provide an alternative construction by
considering truncated GGE's (tGGE's) that only include a finite number of well
localized conserved operators. It is shown that the tGGE's can approximate the
steady states with arbitrary precision, i.e. all physical observables are
exactly reproduced in the infinite truncation limit. In addition, we show that
a complete canonical ensemble can in fact be built in terms of a new (discrete)
set of charges built as linear combinations of the standard ones.
  Our general arguments are applied to concrete quench situations in the XXZ
chain, where the initial states are simple two-site or four-site product
states. Depending on the quench we find that numerical results for the local
correlators can be obtained with remarkable precision using truncated GGE's
with only 10-100 charges.
