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Adaptive Pricing for Optimal Coordination in Networked Energy Systems with Nonsmooth Cost Functions

Jiayi Li, Jiale Wei, Matthew Motoki, Yan Jiang, Baosen Zhang

TL;DR

This paper addresses coordinating distributed energy users under privacy and network constraints by embedding DCOPF into the system cost $J(\vec{x})$, which may be nonsmooth. It introduces a two-timescale framework with a generalized-gradient-based price update $\dot{\vec{p}}\in \partial J(\vec{x}^*(\vec{p})) - \vec{p}$, ensuring convergence to the social-welfare optimum while preserving user privacy. The main contributions are (i) a vectorized, nonsmooth-aware pricing rule, (ii) proofs of existence, uniqueness, and strong asymptotic stability of the equilibrium, and (iii) numerical validation on networked grids demonstrating effective alignment of individual and social objectives under DCOPF constraints. The results enable scalable, privacy-preserving decentralized coordination in realistic grid settings, with applicability to practical distributed energy resources and responsive loads.

Abstract

Incentive-based coordination mechanisms for distributed energy consumption have shown promise in aligning individual user objectives with social welfare, especially under privacy constraints. Our prior work proposed a two-timescale adaptive pricing framework, where users respond to prices by minimizing their local cost, and the system operator iteratively updates the prices based on aggregate user responses. A key assumption was that the system cost need to smoothly depend on the aggregate of the user demands. In this paper, we relax this assumption by considering the more realistic model of where the cost are determined by solving a DCOPF problem with constraints. We present a generalization of the pricing update rule that leverages the generalized gradients of the system cost function, which may be nonsmooth due to the structure of DCOPF. We prove that the resulting dynamic system converges to a unique equilibrium, which solves the social welfare optimization problem. Our theoretical results provide guarantees on convergence and stability using tools from nonsmooth analysis and Lyapunov theory. Numerical simulations on networked energy systems illustrate the effectiveness and robustness of the proposed scheme.

Adaptive Pricing for Optimal Coordination in Networked Energy Systems with Nonsmooth Cost Functions

TL;DR

This paper addresses coordinating distributed energy users under privacy and network constraints by embedding DCOPF into the system cost , which may be nonsmooth. It introduces a two-timescale framework with a generalized-gradient-based price update , ensuring convergence to the social-welfare optimum while preserving user privacy. The main contributions are (i) a vectorized, nonsmooth-aware pricing rule, (ii) proofs of existence, uniqueness, and strong asymptotic stability of the equilibrium, and (iii) numerical validation on networked grids demonstrating effective alignment of individual and social objectives under DCOPF constraints. The results enable scalable, privacy-preserving decentralized coordination in realistic grid settings, with applicability to practical distributed energy resources and responsive loads.

Abstract

Incentive-based coordination mechanisms for distributed energy consumption have shown promise in aligning individual user objectives with social welfare, especially under privacy constraints. Our prior work proposed a two-timescale adaptive pricing framework, where users respond to prices by minimizing their local cost, and the system operator iteratively updates the prices based on aggregate user responses. A key assumption was that the system cost need to smoothly depend on the aggregate of the user demands. In this paper, we relax this assumption by considering the more realistic model of where the cost are determined by solving a DCOPF problem with constraints. We present a generalization of the pricing update rule that leverages the generalized gradients of the system cost function, which may be nonsmooth due to the structure of DCOPF. We prove that the resulting dynamic system converges to a unique equilibrium, which solves the social welfare optimization problem. Our theoretical results provide guarantees on convergence and stability using tools from nonsmooth analysis and Lyapunov theory. Numerical simulations on networked energy systems illustrate the effectiveness and robustness of the proposed scheme.

Paper Structure

This paper contains 8 sections, 7 theorems, 30 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation
  3. Planner's Optimization Problem
  4. User's Optimization Problem
  5. Adaptive Price Update under Nonsmoothness
  6. Nonsmooth Stability Analysis
  7. Numerical Illustrations
  8. Conclusions and Outlook

Key Result

Lemma 1

Under Assumption ass:cost, the demand update $\boldsymbol{x}^*_i(\boldsymbol{p}_i)$ is a bijection given by a continuously differentiable and strictly decreasing function which naturally has the properties that $\nabla\boldsymbol{x}^*_i(\boldsymbol{p}_i)<0$ and, $\forall \tilde{\boldsymbol{p}}_i, \hat{\boldsymbol{p}}_i \in \mathbb R$, if $\tilde{\boldsymbol{p}}_i\neq\hat{\boldsymbol{p}}_i$, then

Figures (2)

  • Figure 1: Two-timescale design of incentive pricing mechanism.
  • Figure 2: Convergence of price signals and user demands of our proposed mechanism in IEEE 14-bus system with $14$ users, where the dashed lines represent the optimal demand profile to the global social welfare problem.

Theorems & Definitions (16)

  • Remark 1: Linear generation cost assumption
  • Definition 1: Generalized gradient Clarke1998Nonsmooth
  • Remark 2: Relation to gradient
  • Lemma 1: Bijective demand update
  • proof
  • Theorem 1: Unique equilibrium with incentive aligned
  • proof
  • Theorem 2: Asymptotic stability
  • Lemma 2: Property preservation in composition
  • proof
  • ...and 6 more