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Predictability and Fairness in Load Aggregation with Deadband

F. V. Difonzo, M. Roubalik, J. Marecek

TL;DR

The paper addresses predictability and fairness in load aggregation with deadband by modeling the DER ensemble as a stochastic set-valued (Filippov) system that captures AC losses and controller discontinuities. It extends prior ergodicity results to non-smooth dynamics via Filippov convexification, establishing conditions under which the closed-loop admits a unique Filippov invariant measure, ensuring long-run consistency independent of initial conditions. Through simulations with Matpower, the authors show that deadband increases mixing time and that lag-based controllers yield ergodic regulation, while traditional PI controllers may fail to guarantee ergodicity. The work provides a rigorous framework for assessing fairness in non-smooth, nonlinear smart-grid controllers and suggests directions for expanding the theory to more general Filippov systems and alternative convexifications.

Abstract

Virtual power plants and load aggregation are becoming increasingly common. There, one regulates the aggregate power output of an ensemble of distributed energy resources (DERs). Marecek et al. [Automatica, Volume 147, January 2023, 110743, arXiv:2110.03001] recently suggested that long-term averages of prices or incentives offered should exist and be independent of the initial states of the operators of the DER, the aggregator, and the power grid. This can be seen as predictability, which underlies fairness. Unfortunately, the existence of such averages cannot be guaranteed with many traditional regulators, including the proportional-integral (PI) regulator with or without deadband. Here, we consider the effects of losses in the alternating current model and the deadband in the controller. This yields a non-linear dynamical system (due to the non-linear losses) exhibiting discontinuities (due to the deadband). We show that Filippov invariant measures enable reasoning about predictability and fairness while considering non-linearity of the alternating-current model and deadband.

Predictability and Fairness in Load Aggregation with Deadband

TL;DR

The paper addresses predictability and fairness in load aggregation with deadband by modeling the DER ensemble as a stochastic set-valued (Filippov) system that captures AC losses and controller discontinuities. It extends prior ergodicity results to non-smooth dynamics via Filippov convexification, establishing conditions under which the closed-loop admits a unique Filippov invariant measure, ensuring long-run consistency independent of initial conditions. Through simulations with Matpower, the authors show that deadband increases mixing time and that lag-based controllers yield ergodic regulation, while traditional PI controllers may fail to guarantee ergodicity. The work provides a rigorous framework for assessing fairness in non-smooth, nonlinear smart-grid controllers and suggests directions for expanding the theory to more general Filippov systems and alternative convexifications.

Abstract

Virtual power plants and load aggregation are becoming increasingly common. There, one regulates the aggregate power output of an ensemble of distributed energy resources (DERs). Marecek et al. [Automatica, Volume 147, January 2023, 110743, arXiv:2110.03001] recently suggested that long-term averages of prices or incentives offered should exist and be independent of the initial states of the operators of the DER, the aggregator, and the power grid. This can be seen as predictability, which underlies fairness. Unfortunately, the existence of such averages cannot be guaranteed with many traditional regulators, including the proportional-integral (PI) regulator with or without deadband. Here, we consider the effects of losses in the alternating current model and the deadband in the controller. This yields a non-linear dynamical system (due to the non-linear losses) exhibiting discontinuities (due to the deadband). We show that Filippov invariant measures enable reasoning about predictability and fairness while considering non-linearity of the alternating-current model and deadband.
Paper Structure (8 sections, 1 theorem, 30 equations, 5 figures, 1 algorithm)

This paper contains 8 sections, 1 theorem, 30 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions and Related Work
  3. Related Models
  4. Related Results
  5. Reasoning about Predictability and Fairness with Deadband
  6. The Main Result
  7. Simulating the Impact of Deadband
  8. Conclusions and Further Work

Key Result

Theorem 1

Consider the feedback system depicted in Figure system. Assume that each agent $i \in \{1,\cdots,N\}$ has a state governed by the non-linear iterated function system eq:nonlinear-disc where, for some event $C^2$-functions $h_W,h_H:\mathbb{R}^n\to\mathbb{R}$, with $0$ as a regular value, both eq:W an Moreover, let us assume that for every invariant measures $\nu^\pm$ of the feedback loop relative t

Figures (5)

  • Figure 1: An illustration of a closed-loop model, taken from marecek2021predictability. In this paper, we extend the reasoning to discontinuous dynamical systems obtained by considering deadband in the controller $\mathcal{C}$.
  • Figure 2: An illustration of the Filippov convexification in three easy steps. On the top of the column, (a) displays two trajectories of discontinuous dynamics of active power output over time, given by deadband. Below, (b) constructs neighbourhoods around the continuous subsets of the same trajectories. At the bottom, (c) suggests the convexification.
  • Figure 3: Mixing time as function of deadband expressed in percentage of reference value $r$.
  • Figure 4: Results of simulations on the first test case: The state of the controllers (right) and the state of agents (left) utilising probability function $g_{i1}$ of \ref{['eq:prob-func']} as functions of time, for the two controllers and two initial states of each of the two controllers.
  • Figure 5: Results of simulations on Standard IEEE 30-bus test case: The state of the controllers (right) and the state of agents (left) utilising probability function $g_{i1}$ of \ref{['eq:prob-func']} as functions of time, for the two controllers and two initial states of each of the two controllers.

Theorems & Definitions (10)

  • Definition 1: Discrete-time inclusion, smirnov
  • Definition 2: Comparison functions
  • Definition 3: Incremental ISS, angeli2002lyapunov
  • Definition 4: Filippov convexification, filippov
  • Definition 5: Filippov solution, filippov
  • Definition 6: Filippov invariant measure
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2