Novel engineered step-chirped gratings (SCG) for broadband frequency doublers based on quasi-phase matched second harmonic generation with and without singly pump-resonance in lithium niobate waveguides have been theoretically modeled and simulated taking pump depletion into account, for the first time to our knowledge. It is shown that apodized SCGs flatten the conversion bandwidth which can be improved extensively with apodization ratio. Also, it is demonstrated numerically for an ASCG that by means of singly pump-resonant waveguide, the conversion efficiency of quasi-continuous waves shown with the envelope of responses increases substantially especially for low loss. The optimized values of back facet reflectivity and input power to achieve the maximum efficiency, for several waveguide loss parameters have also been calculated. Furthermore, the ASCG structure may provide a more convenient method for fabrication and poling as it increases the chirp period and duty cycle.