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In this article, we address the phenomenon of temporal, self-sustained oscillations for a solid oxide fuel cell which utilises a humidified methane fuel stream. Our objective is to uncover the fundamental mechanisms giving rise to current oscillations that have been observed experimentally. To this end, we develop a model based on the fundamental chemical kinetics and transfer processes which take place within the fuel cell. This leads to a three-dimensional dynamical system, which, under typical operating conditions, is rationally reducible to a planar dynamical system. This planar dynamical system was studied in (Sands et al., Proc. R. Soc. Lond. A 470 (2014)) for the case where the parameter ¯c0 ⪡ 1, corresponding with a weakly humidified fuel stream. In the present article, the structural dynamics of the planar dynamical system for the case where the parameter ¯c0 = O(1), which corresponds with a humidified fuel stream, are studied in detail. Self-sustained oscillations are shown to arise through Hopf bifurcations in this planar dynamical system, and the key parameter ranges for the occurrence of such oscillations are identified. Fuel stream humidification is also shown to significantly alter the fuel cell dynamics, leading to hysteresis. [ABSTRACT FROM AUTHOR]