Wolfgang Dreyer, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany Joint work with Clemens Guhlke We study reversible storage systems serving to store electrical or chemical energy for later use. Particularly, we consider storage systems that consist of an ensemble of many interconnected storage particles as they appear in lithium-ion batteries. During charging and discharging of the battery, one observes a phase transition with two coexisting phases and hysteretic behavior. There are two regimes for fast and slow charging with different storage mechanisms. In this lecture we describe in detail the slow charging regime, where the time to approach equilibrium in a single storage particle is much smaller than the time for full charging of the ensemble. Here the observed phase transition is a many-particle effect and happens in the ensemble instead within a single particle. Its evolution is modelled by a nonlocal and nonlinear conservation law of Fokker-Planck type. There are two parameter that control if the ensemble transits the 2-phase region along a Maxwell line, along a hysteresis path, or if the ensemble shows the same behaviour as its constituents.