Energy-aware scheduling policies must be integrated into a sophisticated design process in order to design Energy Harvesting (EH) sensor nodes with real-time capabilities. These systems should achieve energy neutrality by efficiently managing energy resources, ensuring continuous operation even in situations with intermittent energy availability while respecting timeliness constraints. An overview of the ineffectiveness of classical real-time scheduling techniques is given. This motivates more advanced research studies to propose and validate new energy harvesting aware schedulers. The paper presents EED-H, a scheduler which efficiently specifies how processing time and energy should be allocated to the software tasks in order to ensure energy neutrality whenever feasible. The paper finally draws attention to a few technical issues with the scheduler's actual implementation, mainly in relation to measuring energy production, energy consumption and energy prediction.