Dynamical simulations of polaron transport in conjugated polymers in the presence of an external time-dependent electric field have been performed within a combined extended Hubbard model (EHM) and Su-Schrieffer-Heeger (SSH) model. Nearly all relevant electron-phonon and electron-electron interactions are fully taken into account by solving the time-dependent Schrodinger equation for the pi electrons and the Newton's equation of motion for the backbone monomer displacements by virtue of the combination of the adaptive time-dependent density matrix renormalization group (TDDMRG) and classical molecular dynamics (MD). We find that after a smooth turn-on of the external electric field, the polaron is accelerated at first and then moves with a nearly constant velocity as one entity consisting of both the charge and the lattice deformation. An ohmic region (3 <= E-0 <= 9 mV/angstrom) where the stationary velocity increases linearly with the electric field strength is observed for the case of U=2.0 eV and V=1.0 eV. The maximal velocity is well above the speed of sound. Below 3 mV/angstrom, the polaron velocity increases nonlinearly, and in high electric fields with strengths of E-0 >= 10.0 mV/angstrom, the polaron will become unstable and dissociate. The relationship between electron-electron interaction strengths and polaron transport is also studied in detail. We find that the on-site Coulomb interactions U will suppress the polaron transport, and small nearest-neighbor interactions V values are also not beneficial to the polaronic motion while large V values favor the polaron transport.