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Distributed equilibrium seeking in aggregative games: linear convergence under singular perturbations lens

Guido Carnevale, Filippo Fabiani, Filiberto Fele, Kostas Margellos, Giuseppe Notarstefano

TL;DR

The paper addresses distributed Nash equilibrium seeking in aggregative games where agents lack direct access to the global aggregative variable. It introduces TRADES, a fully distributed gradient-based algorithm with a consensus-based tracking mechanism for the aggregative variable, analyzed through a singular perturbation lens that separates fast consensus from slow optimization. Under strong monotonicity and local constraints, the method achieves linear convergence to the NE with constant stepsizes, and it supports a generalized aggregative variable beyond the standard arithmetic average. The approach is validated on a smart-grid voltage-support case, demonstrating convergence and improved voltage profiles due to coordinated reactive power injections, highlighting practical scalability for networked control applications.

Abstract

We present a fully-distributed algorithm for Nash equilibrium seeking in aggregative games over networks. The proposed scheme endows each agent with a gradient-based scheme equipped with a tracking mechanism to locally reconstruct the aggregative variable, which is not available to the agents. We show that our method falls into the framework of singularly perturbed systems, as it involves the interconnection between a fast subsystem - the global information reconstruction dynamics - with a slow one concerning the optimization of the local strategies. This perspective plays a key role in analyzing the scheme with a constant stepsize, and in proving its linear convergence to the Nash equilibrium in strongly monotone games with local constraints. By exploiting the flexibility of our aggregative variable definition (not necessarily the arithmetic average of the agents' strategy), we show the efficacy of our algorithm on a realistic voltage support case study for the smart grid.

Distributed equilibrium seeking in aggregative games: linear convergence under singular perturbations lens

TL;DR

The paper addresses distributed Nash equilibrium seeking in aggregative games where agents lack direct access to the global aggregative variable. It introduces TRADES, a fully distributed gradient-based algorithm with a consensus-based tracking mechanism for the aggregative variable, analyzed through a singular perturbation lens that separates fast consensus from slow optimization. Under strong monotonicity and local constraints, the method achieves linear convergence to the NE with constant stepsizes, and it supports a generalized aggregative variable beyond the standard arithmetic average. The approach is validated on a smart-grid voltage-support case, demonstrating convergence and improved voltage profiles due to coordinated reactive power injections, highlighting practical scalability for networked control applications.

Abstract

We present a fully-distributed algorithm for Nash equilibrium seeking in aggregative games over networks. The proposed scheme endows each agent with a gradient-based scheme equipped with a tracking mechanism to locally reconstruct the aggregative variable, which is not available to the agents. We show that our method falls into the framework of singularly perturbed systems, as it involves the interconnection between a fast subsystem - the global information reconstruction dynamics - with a slow one concerning the optimization of the local strategies. This perspective plays a key role in analyzing the scheme with a constant stepsize, and in proving its linear convergence to the Nash equilibrium in strongly monotone games with local constraints. By exploiting the flexibility of our aggregative variable definition (not necessarily the arithmetic average of the agents' strategy), we show the efficacy of our algorithm on a realistic voltage support case study for the smart grid.

Paper Structure

This paper contains 10 sections, 3 theorems, 32 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup and case study
  3. Problem Setup
  4. Case study: voltage support in smart grids
  5. Nash equilibrium seeking algorithm
  6. Algorithm description and main result
  7. Proof of Theorem \ref{['th:convergence']}
  8. Voltage support example revisited
  9. Conclusion
  10. Acknowledgments

Key Result

Theorem III.4

Consider eq:global_update. There exist constants $\bar{\gamma}, \bar{\delta}, a_1, a_2 > 0$ such that, for any $\delta \in (0,\bar{\delta})$ and $\gamma \in (0,\bar{\gamma})$ and $(x^0,z^0) \in X \times \mathbb{R}^{Nd}$ such that $\mathbf{1}_{N,d}^\top z^0 = 0$, it holds that

Figures (3)

  • Figure 1: Convergence of the TRADES algorithm to the NE $x^\star\in\prod_{i=1}^{321} X_i$.
  • Figure 2: Error in the distributed estimation $\phi_i(x_i^t)+z_i^t$ of the bus voltage vector $\sigma^t\in\mathbb{R}^{N_bT}$, across iterates $t$. The shaded area refers to all $N$ agents, a median case is shown by the solid red line.
  • Figure 3: Variation of bus voltages due to the additional EV loads in the network (green circles), corresponding to the time interval where most EV charge occurs (12--1am). The resulting voltage distribution improves over the base load case (triangle markers), following reactive power injection from the smart EV chargers. The aggregate reactive injections at each bus are represented by asterisks; these are consistent with the considered network topology, where buses 57 and 83 are significantly loaded, thus requiring additional effort to compensate for voltage drop.

Theorems & Definitions (4)

  • Definition II.1: Nash equilibrium
  • Theorem III.4
  • Lemma III.5: carnevale2022tracking
  • Lemma III.6: carnevale2022tracking