The stability of a solid oxide fuel cell (SOFC) stack is strongly dependent on the magnitude and profile of the internal chemical potential of the solid electrolyte. If the internal partial pressure is too high, the electrolyte can be delaminated from the electrodes. The formation of high internal pressure is attributed to a negative cell voltage, and this phenomenon can occur in a bad cell (with higher resistance) in a stack. This fact implies that the internal chemical potential plays an important role in determining the lifetime of a stack. In the present work, we fabricate planar type anode-supported cells ($25cm^2$) with a bi-layer electrolyte (with locally increased electronic conduction at the anode side) to prevent high internal pressure, and we test the fabricated cells under a negative voltage condition. The results indicate that the addition of electronic conduction in the electrolyte can effectively depress internal pressure and improve the cell stability.