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This paper investigates the real-time scheduling of residential appliances in the home energy management system (HEMS), including electric water heater (EWH), air conditioner (AC), clothes dryer (CD), electric vehicle (EV), photovoltaic cell (PV) and battery. A demand response (DR) mechanism is proposed to stimulate the household to participate in the DR services. The conditional value-at-risk (CVaR) is applied to make a trade-off between the expected costs and the risk of exposure to uncertainties of the electricity price and other factors. Fuzzy logic controller is utilized to determine the output power of the battery, and two sets of rules are designed to adjust the charging and discharging operations of the battery according to the DR request from the grid. Stochastic aspects of the electricity price, outdoor temperature, hot water usage, PV generation, and loads are considered. Simulation results show that the proposed DR mechanism can decrease the cost of the household considerably, and the proposed real-time scheduling algorithm can optimize the operation of appliances close to the ideal deterministic results, even when the gap between the forecast electricity price and the real price is large.