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This paper evaluates the real-time price-based demand response (DR) management for residential appliances via stochastic optimization and robust optimization approaches. The proposed real-time price-based DR management application can be imbedded into smart meters and automatically executed on-line for determining the optimal operation of residential appliances within 5-minute time slots while considering uncertainties in real-time electricity prices. Operation tasks of residential appliances are categorized into deferrable/non-deferrable and interruptible/non-interruptible ones based on appliances' DR preferences as well as their distinct spatial and temporal operation characteristics. The stochastic optimization adopts the scenario-based approach via Monte Carlo (MC) simulation for minimizing the expected electricity payment for the entire day, while controlling the financial risks associated with real-time electricity price uncertainties via the expected downside risks formulation. Price uncertainty intervals are considered in the robust optimization for minimizing the worst-case electricity payment while flexibly adjusting the solution robustness. Both approaches are formulated as mixed-integer linear programming (MILP) problems and solved by state-of-the-art MILP solvers. The numerical results show attributes of the two approaches for solving the real-time optimal DR management problem for residential appliances.