Optimizing Individualized Incentives from Grid Measurements and Limited Knowledge of Agent Behavior

TL;DR

This paper proposes feedback-based optimization algorithms to solve the problem of controllability of distributed energy resources, and shows that each converges to an asymptotically stable incentive with (near)-optimality guarantees given mild assumptions on the problem.

Abstract

As electrical generation becomes more distributed and volatile, and loads become more uncertain, controllability of distributed energy resources (DERs), regardless of their ownership status, will be necessary for grid reliability. Grid operators lack direct control over end-users' grid interactions, such as energy usage, but incentives can influence behavior -- for example, an end-user that receives a grid-driven incentive may adjust their consumption or expose relevant control variables in response. A key challenge in studying such incentives is the lack of data about human behavior, which usually motivates strong assumptions, such as distributional assumptions on compliance or rational utility-maximization. In this paper, we propose a general incentive mechanism in the form of a constrained optimization problem -- our approach is distinguished from prior work by modeling human behavior (e.g., reactions to an incentive) as an arbitrary unknown function. We propose feedback-based optimization algorithms to solve this problem that each leverage different amounts of information and/or measurements. We show that each converges to an asymptotically stable incentive with (near)-optimality guarantees given mild assumptions on the problem. Finally, we evaluate our proposed techniques in voltage regulation simulations on standard test beds. We test a variety of settings, including those that break assumptions required for theoretical convergence (e.g., convexity, smoothness) to capture realistic settings. In this evaluation, our proposed algorithms are able to find near-optimal incentives even when the reaction to an incentive is modeled by a theoretically difficult (yet realistic) function.

Figures (10)

• Figure 1: An annotated example of a step $g$ function at a single bus $j$. Each step corresponds to a "controllable device".
• Figure 2: Min. nodal voltage magnitude (top) and incentive cost (bottom) vs. iterations for quadratic-convex stationary $g$ function experiment.
• Figure 3: Min. nodal voltage magnitude (top) and incentive cost (bottom) vs. iterations for stationary $g$ step function experiment ($D$ indicates # of controllable devices at each bus).
• Figure 4: Min. nodal voltage magnitude (top) and incentive cost (bottom) vs. iterations for quadratic-convex time-varying $g$ function experiment.
• Figure 5: Min. nodal voltage magnitude (top) and incentive cost (bottom) vs. iterations for time-varying $g$ step function experiment ($D = 6$ controllable devices on average).

Theorems & Definitions (13)

• Definition 2.3: Linear approximation $\overline{g}_{{\mathbf{u}}^\star}( {\mathbf{i}})$
• Theorem 3.2
• Theorem 3.4
• Theorem 3.7
• Theorem 3.10
• Lemma 4.1
• Theorem 4.2
• Lemma 4.4
• Theorem 4.5
• proof : Proof of \ref{['thm:DAIOconvergence']}