Phase-field model has been proven to be a powerful tool in describing the complex pore-structure evolution and the intricate multi-physics in the powder bed fusion additive manufacturing process. However, as one of the diffuse-interface approaches, the models employ a finite interface width in representing the transient microstructure. They have to be projected asymptotically onto their corresponding sharp-interface equations to guarantee their quantitative validity. Even though this issue has been solved for liquid-solid interfaces via the development of the quantitative solidification phase-field model, there is no related work addressing the interfaces in powder bed fusion. In this work, we developed a variational quantitative phase-field model to overcome the issues of quantitative validity in powder bed fusion simulations. The model eliminates artificial interface effects caused by the diffuse-interface description of the interfaces, regardless of the finite interface width. Moreover, the model is derived in a variational manner and consistent with non-equilibrium thermodynamics. Cross-coupling terms between the conserved kinetics (i.e., mass and thermal transfer) and the non-conserved one (grain growth), which are typically neglected in conventional models, are considered in the evolution equations. These cross-couplings derived in terms of phase-field parameters via asymptotic analysis are instrumental in ensuring the elimination of interface effects. Furthermore, to enforce the quantitative validity of the model, it is apparent that anisotropic interpolation of the kinetic mobilities is essential. Numerically, we demonstrate the importance of the cross-couplings and the anisotropic interpolations, obtaining good convergence of results with respect to interface width. Also, we make comparisons of the microstructural results obtained using the model to the ones obtained via existing models.