In this paper, a 3-dimensional mathematical model for one cell of an anode-supported SOFC (solid oxide fuel cells) is presented. The model is derived from the partial differential equations representing the conservation laws of ionic and electronic charges, mass, energy, and momentum. The model is implemented to fully characterize the steady state operation of the cell with countercurrent flow pattern of fuel and air. The model is also used for the comparison of countercurrent with concurrent flow patterns in terms of thermal stress (temperature distribution) and quality of operation (current density). Results reveal that the steady-state cell performance curve and output of simulations qualitatively match experimental data of the literature. Results also demonstrate that countercurrent flow pattern leads to an even distribution of temperature, more uniform current density along the cell and thus is more enduring and superior to the concurrent flow pattern. Afterward, the thorough 3-dimensional model is used for state estimation instead of a real cell. To estimate states, the model is simplified and changed to a 1-dimensional model along flow streams. This simplified model includes uncertainty (because of simplifying assumptions of the model), noise, and disturbance (because of measurements). The behaviors of extended and ensemble Kalman filter as an observer are evaluated in terms of estimating the states and filtering the noises. Results demonstrate that, like extended Kalman filter, ensemble Kalman filter properly estimates the states with 20 sets. (C) 2015 Elsevier Ltd. All rights reserved.