The energy transition towards renewable resources and distributed generation challenges our energy systems. Given the fact that many renewable resources are intermittent, a more flexible demand side may help to balance the energy generation and consumption. One way towards flexibilization is the introduction of energy management systems that optimize the utilization of energy in an automated way using measures of demand side management. So far, this flexibilization of the demand side focuses mainly on the provision and utilization of electricity. More promising is the introduction of smart grids that optimize the energy flows across all energy carriers. Multi-modal energy management is not limited to the provision of energy but includes also the utilization on the demand side. This paper addresses an integrated multi-commodity optimization of all relevant energy carriers. Hybrid appliances are investigated that may utilize not only electricity but also an additional energy carrier, such as hot water or natural gas, and thus provide additional potential for flexibility regarding the utilization of energy. This paper analyzes the effe ... mehrcts of hybrid appliances in residential buildings that are equipped with electrical heating elements in combination with photovoltaic systems. Firstly, an overview of the different notions used in the context of multi-energy systems is given and appropriate terms are proposed. Secondly, the results of detailed bottom-up simulations of households are presented having automated building energy management systems that optimize the local utilization, conversion, storage, and provision of energy. The analysis includes self-consumption and self-sufficiency rates as well as possible cost reductions.