The analysis of load curves collected from smart meters is a key step for many energy management tasks ranging from consumption forecasting to customers characterization and load monitoring. In this contribution, we propose a model based on a functional formulation of nonnegative tensor factorization and derive updates for the corresponding optimization problem. We show on the concrete example of multi-sites load curves disaggregation how this formulation is helpful for 1) exhibiting smooth intraday consumption patterns and 2) taking into account external variables such as the outside temperature. The benefits are demonstrated on simulated and real data by exhibiting a meaningful clustering of the observed sites based on the obtained decomposition.