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Stochastic Generalized Dynamic Games with Coupled Chance Constraints

Seyed Shahram Yadollahi, Hamed Kebriaei, Sadegh Soudjani

TL;DR

This work addresses stochastic generalized dynamic games with coupling safety constraints under uncertain dynamics by proposing a concentration-of-measure–based convex under-approximation that replaces non-convex coupling chance constraints with expected constraints. It establishes existence of SGNE for the under-approximated game and shows that its equilibria yield an $\varepsilon$-SGNE for the original game, linking the two formulations. A semi-decentralized, sampling-based algorithm is developed to compute SGNE without requiring distributional knowledge, and convergence is proven under monotonicity and Lipschitz conditions. The approach is validated through a microgrid demand-side-management scenario with a shared battery, demonstrating convergence, constraint satisfaction with high confidence, and practical effectiveness in coordinating distributed agents under uncertainty.

Abstract

Designing multi-agent systems with safety constraints and uncertain dynamics is a challenging problem. This paper studies a stochastic dynamic non-cooperative game with coupling safety chance constraints. The uncertainty is assumed to satisfy a concentration of measure property. Firstly, due to the non-convexity of chance constraints, a convex under-approximation of chance constraints is given using constraints on the expectation. Then, the conditions for the existence of the stochastic generalized Nash equilibrium (SGNE) of the under-approximated game are investigated, and the relation between the $\varepsilon-$SGNE of the original game and the under-approximated one is derived. A sampling-based algorithm is proposed for the SGNE seeking of the under-approximated game that does not require knowing the distribution of the uncertainty nor the analytical computation of expectations. Finally, under some assumptions on the game's pseudo-gradient mapping, the almost sure convergence of the algorithm to SGNE is proven. A numerical study is carried out on demand-side management in microgrids with shared battery to demonstrate the applicability of the proposed scheme.

Stochastic Generalized Dynamic Games with Coupled Chance Constraints

TL;DR

This work addresses stochastic generalized dynamic games with coupling safety constraints under uncertain dynamics by proposing a concentration-of-measure–based convex under-approximation that replaces non-convex coupling chance constraints with expected constraints. It establishes existence of SGNE for the under-approximated game and shows that its equilibria yield an -SGNE for the original game, linking the two formulations. A semi-decentralized, sampling-based algorithm is developed to compute SGNE without requiring distributional knowledge, and convergence is proven under monotonicity and Lipschitz conditions. The approach is validated through a microgrid demand-side-management scenario with a shared battery, demonstrating convergence, constraint satisfaction with high confidence, and practical effectiveness in coordinating distributed agents under uncertainty.

Abstract

Designing multi-agent systems with safety constraints and uncertain dynamics is a challenging problem. This paper studies a stochastic dynamic non-cooperative game with coupling safety chance constraints. The uncertainty is assumed to satisfy a concentration of measure property. Firstly, due to the non-convexity of chance constraints, a convex under-approximation of chance constraints is given using constraints on the expectation. Then, the conditions for the existence of the stochastic generalized Nash equilibrium (SGNE) of the under-approximated game are investigated, and the relation between the SGNE of the original game and the under-approximated one is derived. A sampling-based algorithm is proposed for the SGNE seeking of the under-approximated game that does not require knowing the distribution of the uncertainty nor the analytical computation of expectations. Finally, under some assumptions on the game's pseudo-gradient mapping, the almost sure convergence of the algorithm to SGNE is proven. A numerical study is carried out on demand-side management in microgrids with shared battery to demonstrate the applicability of the proposed scheme.
Paper Structure (11 sections, 12 theorems, 66 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 12 theorems, 66 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Notations and Preliminaries
  4. Modeling Framework and Problem Statement
  5. Convex Under-Approximation of the SGDG
  6. Characterization of SGNE in $\mathcal{G}_{2}$
  7. SGNE of $\mathcal{G}_{2}$ as an $\varepsilon-$SGNE of $\mathcal{G}_{1}$
  8. SGNE Seeking Algorithm for $\mathcal{G}_{2}$
  9. Convergence analysis of Algorithm \ref{['alg: suggested algorithm']}
  10. Simulation Results
  11. Conclusion

Key Result

Proposition 1

If Assumptions ass: cost function in system model--ass: second assumption about cost function hold, the solution set of $\mathrm{SVI}(\mathcal{U}_{\mathcal{G}_{2}},\mathbb{F}_{\mathcal{G}_{2}})$ is not empty.

Figures (4)

  • Figure 1: A visual representation of the convergence of $u^{i}_{9},u^{i}_{17},u^{i}_{21}$ during the execution of Algorithm \ref{['alg: suggested algorithm']}.
  • Figure 2: A visual representation of the convergence of $\mathrm{res}(z)$ and $\mathrm{res}(\tilde{z})$ during the execution of Algorithm \ref{['alg: suggested algorithm']}.
  • Figure 3: Aggregative demand and profile of power exchange of all the households with the grid.
  • Figure 4: Profile of the mean of renewable energy and profile of power exchange of all the households with the battery.

Theorems & Definitions (31)

  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 21 more