Socially Optimal Energy Usage via Adaptive Pricing

We study the coordination of the electricity usage of a group of users by an operator. This question has been studied extensively by the community, for example, in the context of demand response, customer aggregation, and virtual power plants (see [pinson2014benefits, zhang2017robust] and the references within). The common setup is where each user is endowed with a cost (or utility) function, and the operator seeks to minimize a global cost function that is made up of the individual costs of the users and social welfare considerations.

A central challenge in these problems is that the cost functions of the users may not be known to the operator. Furthermore, users themselves may not be able to provide an analytical description, for example, if its consumption profile arises from applying a learning algorithm. In most existing works, however, the operator needs to somehow learn the cost functions of the users [Li19]. This restricts the users to having simple (typically quadratic) cost functions and is often too restrictive to be implemented in practice.

In this paper, we overcome this challenge by deploying a two-time-scale incentive mechanism that alternatively updates between the operator and the users. More specifically, based on a price, users solve their individual cost optimization problem; based on the actions of the users, the price is updated. This setup can accommodate a large class of user cost functions (e.g., they need not be convex or differentiable) and the operator does not need to learn what they are. The key is to use an ”externality” term that captures the difference between the optimal solutions of individual users and the socially optimal solution [maheshwari2022inducing]. We present the model and state the high-level results and some preliminary simulation results in this abstract.

We consider a system with $N$ users. Let $x_{i}\in\mathbb{R}^{T}$ denote the electricity consumption of user $i$. We assume $x_{i}$ takes value in a compact set $\mathcal{X}_{i}$. The cost for player $i$ is $f_{i}(x_{i})$. The operator is interested in minimizing a global cost function:

$$\Phi(x)=\sum_{i}f_{i}(x_{i})+g\left(\sum_{i}x_{i}\right).$$

(1)

The first term in $\Phi$ represents the sum of the users’ costs, while the second term represents a social or system cost. For example, if $g\left(\sum_{i}x_{i}\right)=||\sum_{i}x_{i}||^{2}$, the system is then penalizing the total power used by all the users.

To guide the users to the minimizing solution of $\Phi(x)$, the system operator broadcast a price $p$, and user $i$ solves

$$\min_{x_{i}\in\mathcal{X}_{i}}\quad c_{i}(x_{i})=f_{i}(x)+p^{T}x_{i}.$$

(2)

We define the externality $e$ as the marginal social cost arising from the term $g$:

$$e(x)=\nabla_{z}g(z)|_{z=\sum_{i}x_{i}}.$$

(3)

We use an iterative algorithm to find the price that would incentivize minimizing the social cost $\Phi$. An important assumption we make is that given any $p$, there is a unique solution to (2), denoted by $x_{i}(p)$. But otherwise are not concerned with how user $i$ solves (2). Then we update the price $p$ as

$$p_{k+1}=(1-\beta_{k})p_{k}+\beta_{k}e(x(p_{k})).$$

(4)

In the full paper, we will show that under mild assumptions, the dynamic in (4) will converge to a unique $p^{*}$, such that $x(p^{*})$ is the minimizer of $\Phi$. This price update scheme is attractive because the operator doesn’t need to know or to learn the $f_{i}$’s. As long as the decisions $x_{i}$’s are observed, a unique $p^{*}$ can be found to induce globally optimal behavior.

We consider two examples here to show the performance of the update in (4). The first is where $x_{i}\in\mathbb{R}$, $f_{i}(x_{i})=\frac{1}{2}(x_{i}-\bar{x}_{i})^{2}$ and $g(\sum x_{i})=\frac{\lambda}{2}(\sum x_{i})^{2}$ for some $\lambda$.

The second example shows that $f_{i}$’s need not be differentiable or convex. This is exemplified through a group of water heaters where $x_{i}\in\mathbb{R}^{T}$ represents the 1-day load profile, $f_{i}$ represents user discomfort, and $g$ is the same as previous.