We consider the application of a four-dimensional variational data assimilation method to a numerical model, which employs local mesh refinement to improve its solution. We focus on structured meshes where a high-resolution grid is embedded in a coarser resolution one, which covers the entire domain. The formulation of the nested variational data assimilation algorithm was derived in a preliminary work (Int. J. Numer. Meth. Fluids 2008; under review). We are interested here in complementary theoretical aspects. We present first a model for the multi-grid background error covariance matrix. Then, we propose a variant of our algorithms based on the addition of control variables in the inter-grid transfers in order to allow for a reduction of the errors linked to the interactions between the grids. These formulations are illustrated and discussed in the test case experiment of a 2D shallow water model.