‘Current advances in the density functional perturbation theory allow for first-principle modeling of electron-phonon interaction in crystals. This interaction determine such physical processes as temperature dependence of bandstructure, electronic transport, polaron formation and the latter is studied in the present work. We examine recent developments that pave the way to complete ab initio many-body calculations of polarons in real materials. Based on the work of Sio and coauthors [PRB 99, 235139 (2019)] we redefine their approach and derive variational polaron equations that allow as to employ more efficient minimization algorithm. These equations are applied to the classic 2D and 3D Frohlich model and reproduce its results. We also investigate generalized Frohlich model by considering anisotropic effective masses and obtain more accurate results that the ones were recently obtained with gaussian ansatz approach [arXiv:2109.12594]. As the next step we plan to include full band degeneracy, phonon dispersion and electron-phonon interaction and also consider higher-order interactions that was neglected by the authors of the original model‘ (c)