To deal with plant model mismatches in control practice, this paper proposes two variations of an offset-free framework which integrates nonlinear model predictive control (NMPC) and moving horizon estimation (MHE). We prove that the proposed method achieves offset-free regulatory behavior, even in the presence of plant model mismatches. If the plant uncertainty structure is known, the MHE can be tuned to estimate uncertainty parameters, to remove the plant model mismatch online. In addition, we incorporate the advanced step NMPC (as-NMPC) and the advanced step MHE (as-MHE) strategies into the proposed method to reduce online computational delay. Finally, the proposed method is applied on a large scale air separation unit, and the steady state offset-free behavior is observed.