[Paper Review] A Stackelberg Game Approach for Two-Level Distributed Energy Management in Smart Grids

Juntao Chen, Quanyan Zhu|arXiv (Cornell University)|Aug 29, 2016

Smart Grid Energy Management28 references3 citations

TL;DR

This paper proposes a Stackelberg game-theoretic framework for two-level distributed energy management in smart grids, where generators act as leaders and microgrids as followers to jointly optimize power generation and injection under physical constraints. A PMU-enabled distributed algorithm ensures convergence to a unique Stackelberg equilibrium using only local voltage angle measurements, preserving privacy and ensuring system stability.

ABSTRACT

The pursuit of sustainability motivates microgrids that depend on distributed resources to produce more renewable energies. An efficient operation and planning relies on a holistic framework that takes into account the interdependent decision-making of the generators of the existing power grids and the distributed resources of the microgrid in the integrated system. To this end, we use a Stackelberg game-theoretic framework to study the interactions between generators (leaders) and microgrids (followers). Entities on both sides make strategic decisions on the amount of power generation to maximize their payoffs. Our framework not only takes into account the economic factors but also incorporates the stability and efficiency of the smart grid, such as the power flow constraints and voltage angle regulations. We develop three update schemes for both generators and microgrids, respectively, and among which a fully distributed algorithm enabled by phasor measurement units is presented. The distributed algorithm merely requires the information of voltage angles at local buses for updates, and its convergence to the unique equilibrium is shown. We further develop the implementation architectures of the update schemes in the smart grid. Finally, case studies are used to corroborate the effectiveness of the proposed algorithms.

Motivation & Objective

• To model strategic interactions between generators (leaders) and microgrids (followers) in smart grids using a Stackelberg game framework.
• To incorporate physical power system constraints—such as power flow and voltage angle regulations—alongside economic objectives in the decision-making process.
• To develop distributed algorithms that enable generators and microgrids to compute equilibrium strategies without sharing private information.
• To ensure convergence to a unique Stackelberg equilibrium using only local measurements from phasor measurement units (PMUs).
• To design an implementation architecture leveraging PMU and wireless communication infrastructure for real-world deployment.

Proposed method

• Formulate a two-level Stackelberg game where generators set power generation levels first, and microgrids respond optimally based on the generators' decisions.
• Integrate physical constraints into the payoff functions, including power flow equations and voltage angle regulations via the bus admittance matrix.
• Propose three update schemes for microgrids and three algorithms for generators, including a fully distributed algorithm relying solely on local voltage angle measurements from PMUs.
• Use the Schur complement and matrix invertibility conditions to prove convergence of the distributed algorithm to a unique Stackelberg equilibrium.
• Derive sufficient conditions for convergence by analyzing the Hessian matrix and system Jacobian, ensuring strict convexity and full rank of key submatrices.
• Implement the framework using a PMU-based communication architecture that avoids exchange of sensitive data like generation levels or regulation parameters.

Experimental results

Research questions

• RQ1 How can strategic interactions between generators and microgrids in a smart grid be modeled to reflect both economic incentives and physical power system constraints?
• RQ2 What distributed algorithm enables generators and microgrids to reach a unique Stackelberg equilibrium without centralized coordination or sharing of private information?
• RQ3 Under what conditions does the proposed PMU-based distributed algorithm converge to the equilibrium, and how can convergence be mathematically guaranteed?
• RQ4 How do physical constraints such as power flow and voltage angle regulation affect the equilibrium outcomes in the Stackelberg game framework?
• RQ5 What system architecture supports the practical deployment of the proposed distributed energy management algorithms in real smart grids?

Key findings

• The proposed PMU-enabled distributed algorithm converges to a unique Stackelberg equilibrium using only local voltage angle measurements, ensuring privacy and scalability.
• The convergence of the algorithm is guaranteed under sufficient conditions involving the invertibility of key submatrices, including the Schur complement of the bus admittance matrix.
• The Hessian matrix of the optimization problem is shown to be positive definite, ensuring strict convexity and uniqueness of the equilibrium solution.
• Matrix invertibility is proven through Sylvester’s criterion and properties of the bus admittance matrix, particularly the non-positivity of off-diagonal elements and positive diagonal entries.
• The system architecture supports decentralized implementation by leveraging existing PMU and wireless communication infrastructure, minimizing communication overhead.
• Case studies confirm the effectiveness of the proposed algorithms in achieving stable, efficient, and economically optimal energy management across the integrated grid.