A five-dimensional (5D) cluster model of the S1 superstructure phase of decagonal Al-Ni-Co quasicrystals is proposed based on the results of the structure refinement of the basic-Ni phase with the same chemical composition. This model leads to a structure composed of fivefold symmetric clusters at vertices of the Penrose pentagon tiling (PPT) with an edge length of about 20 angstrom. The clusters with two different orientations are located at the vertices of the PPT. The structure has the 5D color space group P10(5)/mc(')m(') which explains the satellite reflections characteristic of the S1 phase and forces to locate clusters with different orientations at most adjacent vertices of the PPT. The model is obtained from that for the basic-Ni phase by a small number of atom flips which can be regarded as interlayer phason flips. This suggests that the phase transition from the basic-Ni to S1 phases occurs by the interlayer phason flips without long-range atom diffusion.