This article implements an integrated electrical and thermal energy management of a smart home in real time, taking into account the residents' comfort conditions. The smart home can benefit from a multicarrier energy system. The energy resources include electrical network, photovoltaic generating unit and storage unit, as well as a hybrid water heating system supplied from solar irradiation and natural gas network. Additionally, electrical and thermal loads are categorized into three types: time flexible; power flexible; and fixed loads. Taking advantage of smart grid technology, an energy management system receives required economic, technical, and climatic information in real time, and decides the energy schedules time by time. The integrated energy management problem is formulated to optimize economic and discomfort metrics. A novel metric of discomfort degree-day is proposed to measure the residents' dissatisfaction with the indoor temperature, which considers both magnitude and duration of deviation from a desired temperature in a day. Additionally, the uncertainties in solar irradiation and outdoor temperature are presented by uncertainty sets, and handled through a robust optimization model. The out-of-sample results through a realistic case study indicate that a more robust schedule of energy resources and loads (e.g., ranging from 20% to 100% robustness) is obtained with negligible change (e.g., $<$ 1%) in the operation cost, but at the expense of 48.82% and 58.97% increment of dissatisfaction under the uncertainty sets with the confidence levels of 0.6 and 0.8, respectively.