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With the increase in the number of distributed energy resources and the amount of intelligence in electricity infrastructures, the possibilities for minimizing costs of household energy consumption increase. Household systems are hybrid systems, in the sense that they exhibit both continuous and discrete dynamics. In this paper the mixed-logical dynamic framework is used to construct a dynamic model of a household system equipped with distributed energy resources. A model predictive controller (MPC) is then proposed that uses the mixed-logical dynamic model to control the energy flows inside the household. In simulation studies we assess the performance of the proposed controller, and we illustrate how additional profits can be obtained by increasing the decision freedom of the controller.