  The simulations of the solidification of ternary Al-Cu-Ni alloys by means of
a general multi-phase-field model for an arbitrary number of phases reveal that
the real microstructure can be generated by coupling the real thermodynamic
parameters of the phases and the evolution equations. The stability
requirements on individual interfaces for model functions guarantee an absence
of "ghost" phases in a $n$-dimensional phase-field space. The special
constructed thermal noise terms disturb the stability and can produce the
heterogeneous nucleation of product phases in accordance to the energetic and
concentration conditions. Of particular interest is that in triple points the
nucleation of the forth phase occurs without additional noise. Another
observation is the growth of the eutectic-like or peritectic-like structure in
various alloys.
