This paper presents a dynamic programming approach to optimize energy cost of multiple interacting household appliances such as air conditioning systems and refrigerators with temperature flexibility, under time varying electricity price signals. We adopt a first order differential equation model with a binary (ON-OFF) switching control function for each load. An energy cost minimization problem is then formulated with a pair of constraints on the temperature lower and upper bounds, as well as an equality condition on the initial and final temperature states. We use dynamic programming to compute cost-optimal control inputs and temperature trajectories for a given electricity price profile and ambient temperature condition. To account for temperature deviation from its desired setpoint, a quadratic temperature deviation penalty is added to the cost function. Moreover, to minimize the control input chattering for equipment protection, the cost function is expanded to also minimize the number of on-off switching events. Results for the different weighting combinations of the optimization objectives provide useful insights on the optimal operation of individual and multiple interacting HVAC loads. In particular, we observe that the loads are desynchronized under the cost-optimal operation, in the presence of local (renewable) power generation. The presented optimization algorithm and observed results can lead to the development of novel model predictive and rule-based feedback control policies for optimal energy management in households.

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