This paper formalizes a demand response task as an optimization problem featuring a known time-varying engineering cost and an unknown (dis)comfort function. Based on this model, this paper develops a feedback-based projected gradient method to solve the demand response problem in an online fashion, where: i) feedback from the user is leveraged to learn the (dis)comfort function concurrently with the execution of the algorithm; and, ii) measurements of electrical quantities are used to estimate the gradient of the known engineering cost. To learn the unknown function, a shape-constrained Gaussian Process is leveraged; this approach allows one to obtain an estimated function that is strongly convex and smooth. The performance of the online algorithm is analyzed by using metrics such as the tracking error and the dynamic regret. A numerical example is illustrated to corroborate the technical findings.
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