The dynamics of power consumption constitutes an essential building block for planning and operating sustainable energy systems. Whereas variations in the dynamics of renewable energy generation are reasonably well studied, a deeper understanding of the variations in consumption dynamics is still missing. Here, we analyse highly resolved residential electricity consumption data of Austrian, German and UK households and propose a generally applicable data-driven load model. Specifically, we disentangle the average demand profiles from the demand fluctuations based purely on time series data. We introduce a stochastic model to quantitatively capture the highly intermittent demand fluctuations. Thereby, we offer a better understanding of demand dynamics, in particular its fluctuations, and provide general tools for disentangling mean demand and fluctuations for any given system, going beyond the standard load profile (SLP). Our insights on the demand dynamics may support planning and operating future-compliant (micro) grids in maintaining supply-demand balance.