  Localization and delocalization of quantum diffusion in time-continuous
one-dimensional Anderson model perturbed by the quasi-periodic harmonic
oscillations of $M$ colors is investigated systematically, which has been
partly reported by the preliminary letter [PRE {\bf 103}, L040202(2021)]. We
investigate in detail the localization-delocalization characteristics of the
model with respect to three parameters: the disorder strength $W$, the
perturbation strength $\epsilon$ and the number of the colors $M$ which plays
the similar role of spatial dimension. In particular, attentions are focused on
the presence of localization-delocalization transition (LDT) and its critical
properties. For $M\geq 3$ the LDT exists and a normal diffusion is recovered
above a critical strength $\epsilon$, and the characteristics of diffusion
dynamics mimic the diffusion process predicted for the stochastically perturbed
Anderson model even though $M$ is not large. These results are compared with
the results of time-discrete quantum maps, i.e., Anderson map and the standard
map. Further, the features of delocalized dynamics is discussed in comparison
with a limit model which has no static disordered part.
