Multi-agent stackelberg game for joint optimization of electricity spot and deep peak regulation markets: strategies and implications for system flexibility

• Novel multi-agent Stackelberg game framework for joint optimization of electricity spot and deep peak regulation markets. • Bi-level optimization with KKT conditions reveals strategic bidding behaviors under varying renewable energy scenarios. • Compares SMP versus PAB mechanisms showing 64% cost reduction and enhanced market fairness under PAB clearing rules. • Large units secure substantially higher profits than smaller units during high peak demand via economic withholding strategies. • Provides policy recommendations for differentiated bidding rules and capacity compensation mechanisms. The rapid integration of renewable energy sources into power grids has markedly altered the landscape of modern energy systems, amplifying the need for enhanced flexibility in grid operation and regulation. Among the most pressing challenges is the increasing demand for deep peak regulation, necessitating coordinated optimization between electricity spot markets and peak regulation services to ensure system stability and economic efficiency. This paper explores the game-theoretic dynamics between various market participants, specifically power generation enterprises, in the context of joint electricity spot and deep peak regulation markets. We propose a novel multi-agent Stackelberg game model to characterize the strategic bidding behaviors of power generators in these interdependent markets, integrating bi-level optimization techniques and Karush-Kuhn-Tucker (KKT) conditions to identify equilibrium strategies. The primary contribution of this study lies in being the first to apply a multi-agent Stackelberg game model to electricity spot and peak regulation markets, which unveils the complex strategic interactions between thermal power producers with deep peak-shaving capabilities (leaders) and electricity trading centers (followers). This approach reveals the intricate balance between competition and cooperation, where large generators with cost advantages exert influence over market prices, while smaller units strive to optimize their strategies in response to fluctuating market conditions. Through extensive case study analysis, we demonstrate how market dynamics, influenced by varying peak regulation demand scenarios, significantly affect the profitability and decision-making of different participants. Our findings suggest that during high peak regulation demand, large units leverage their economies of scale to secure substantial profits, while smaller units face increasing competition. In contrast, during low demand periods, the disparity in profits narrows, highlighting the adaptive strategies required by market participants. Additionally, this paper contrasts two market clearing mechanisms—System Marginal Price (SMP) and Pay-as-Bid (PAB)—emphasizing their impact on market efficiency and fairness. The results indicate that while the SMP mechanism exacerbates profit disparities among units, the PAB mechanism effectively reduces peak regulation costs and enhances market transparency, though it may exclude high-cost units from the peak regulation market, undermining system flexibility. Moreover, this study extends beyond quantitative modeling by incorporating a structured analysis of negotiation strategies, reasoning logics, and procedural mechanisms under policy-driven market environments, offering a conceptual framework for understanding how institutional design and regulatory negotiation shape equilibrium outcomes and market adaptability. The paper concludes with a discussion on the broader implications of these findings for the design of future market mechanisms, particularly in the context of increasing renewable energy penetration and the need for more resilient, flexible power systems. By integrating game-theoretic models with real-world case studies, and embedding regulatory negotiation perspectives, this work provides novel insights into the strategic interactions within hybrid energy markets and lays the groundwork for further research into optimizing market structures to support sustainable and secure energy systems.