In this paper, a fast and realistic approach is proposed for the DC modeling of superconducting coils made of first-generation (1G) high temperature superconducting (HTS) tapes, taking into consideration the non-uniformity of the electric current density distribution in the HTS tapes cross-section, by using a power minimization criterion with the dependence of the critical current density on the magnetic flux density (MFD). Integral equations are used to evaluate the MFD, enabling to discretize only the active parts of the system. The model is then combined with an inverse procedure for the characterization of non-inductive HTS coils using measurements.