Personalized Demand Response via Shape-Constrained Online Learning

Abstract

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. √x x. A vector of zeros is represented by 0 and a vector of ones by 1, with the corresponding dimensions. O refers to the big O notation; that is, given two positive sequences {a k } ∞ k=0 and {b k } ∞ k=0 , we say that

Distributed Personalized Gradient Tracking with Convex Parametric Models

“…In the proposed personalized gradient tracking strategy, the dynamic gradient tracking update is interlaced with a learning mechanism to let each node learn the user's cost function U i (x), by employing noisy user's feedback in the form of a scalar quantity given by y i,t = U (x i,t ) + i,t , where x i,t is the local, tentative solution at time t and i,t is a noise term. It is worth pointing out that in this paper, we consider convex parametric models, instead of more generic non-parametric models, such as Gaussian Processes [12,[16][17][18][19][20][21], or convex regression [22,23]. The reasons for this choice stem from the fact that (i) user's functions are or can be often approximated as convex (see, e.g., [24,25] and references therein), which makes the overall optimization problem much easier to be solved; (ii) convex parametric models have better asymptotical rate bounds 2 than convex non-parametric models [22], which is fundamental when attempting at learning with scarce data; and (iii) a solid online theory already exists in the form of recursive least squares (RLS) [26][27][28][29][30][31].…”

“…As mentioned, letting user's utility functions enter in optimization problems to drive the systems to a "comfortable" solution is fundamental in data-driven control and optimization problems involving both humans and machines (see [33,34]). Further motivating examples include, e.g., optimizing the operations of a smart grid from engineering perspectives, while taking into account the user's preferences [21].…”

“…This represents a first step towards generic parametric models 3 . Non-parametric approaches in the literature to learn unknown functions are e.g., (shape-constrained) Gaussian processes [16,21,35] and convex regression [23,36]. As said, we prefer here parametric models for their faster asymptotical rates, cheap online computational load, and ease of introducing convexity constraints 4 .…”