In this paper, we address the problem of periodic task scheduling in a sensor node powered with energy harvester. The scheduler can be occasionally forced to skip jobs because of energy shortage or processing overload. Every task executes jobs in conformance with the so-called Skip-Over model where blue jobs may be aborted at any time in contrast to red ones that should complete before deadline. The work presented here aims to consider stability and robustness issues for the Skip-Over model in a uniprocessor energy harvesting system. We present two scheduling strategies, called Green-BWP-LF and Green- BWP-MS specifically adapted to that context. A simulation study shows that these policies outperform the conventional Green- BWP algorithm based on the classical Earliest Deadline rule.