Mean Field Games for Renewable Energy Development

Abstract

We propose a mean field game (MFG) framework to model the evolution of renewable energy production in competitive electricity markets. Producers interact through the spot price while optimising their profits under production, installation, and capacity adjustment costs, as well as the generation uncertainty. We first formulate the market as an $N$-player stochastic differential game and analyse its mean field game limit as $N\to\infty$. We characterise the representative producer's optimal control via forward-backward stochastic differential equations (FBSDEs) derived from the stochastic maximum principle and determine the corresponding equilibrium spot price. We establish existence and uniqueness of solutions to the FBSDEs and prove that the MFG admits a unique equilibrium. We then extend the model to a Stackelberg mean field game to incorporate the role of a social planner. The planner's optimisation problem leads to an extended Hamilton-Jacobi-Bellman (HJB) system, for which we prove existence and uniqueness of viscosity solutions. Finally, we implement a deep learning-based numerical scheme to approximate the equilibrium and investigate the impact of policy interventions on capacity dynamics. Our results highlight how optimal subsidy design depends on prevailing market conditions and can mitigate both capacity shortages and overproduction.

Figures (20)

• Figure 1: $\mu_0^X = 1000 MWh$, $D = 1500 MWh$, $\sigma^0 = 100$, $T=1$, $r = 1$.
• Figure 2: $\mu_0^X = 1000 MWh$, $D = 1500 MWh$, $\sigma^0 = 100$, $T=2$, $r = 1$.
• Figure 3: $\mu_0^X = 1000 MWh$, $D = 1500 MWh$, $\sigma^0 = 1$, $T=2$, $r = 1$.
• Figure 4: $\mu_0^X = 1000 MWh$, $D=1500MWh$, $\sigma^0 = 100$, $T=2$, $r = 2$.
• Figure 5: $\mu_0^X = 2000 MWh$, $D = 1500 MWh$, $\sigma^0 = 100$, $T=1$, $r=1$.

Theorems & Definitions (33)

• Remark 1
• Definition 1
• Definition 2
• Corollary 2
• proof
• Proposition 3
• proof
• Theorem 4
• proof
• Corollary 5