Luciano Campi, Zhuoshu Wu
6 min
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.
Alex: Each farm optimizes assuming the market average, and the math shows they converge on the same stable path. It explains why markets might overproduce on sunny days—prices crash from the collective average.
Sam: To see what that looks like in practice, they ran computer tests with real solar costs. Starting with low capacity below demand, farms add panels quickly at first—high prices make profits beat costs—but slow and stop once balanced, fixing shortages in one or two years. With excess starting capacity, growth halts as prices fall, leaving overproduction.
Alex: And bigger shared noise spreads paths but keeps the trend?
Sam: Yes. Higher volatility widens spreads without changing directions. Shortages fill fast, but gluts persist—suggesting subsidies to nudge the mean path closer to demand.
Alex: So simulations confirm subsidies could prevent persistent mismatches. To design those precisely, the paper adds a social planner overseeing the system?
Sam: Yes—like a coach setting team incentives, knowing how players will respond. The planner picks subsidies first, anticipating producers' reactions. It's a leader-follower setup, modeled as a Stackelberg extension to the mean field game. The planner minimizes costs from capacity-demand mismatches plus subsidy expenses.
Alex: Planner goes first, based on expected group average. How does the math find the best policy?
Sam: It couples to the producers' forward-backward equations. The shadow price becomes a function of time and average capacity, satisfying a partial differential equation. The planner's minimal cost follows a Hamilton-Jacobi-Bellman equation. The paper proves a unique viscosity solution exists, ensuring a clear optimal subsidy path.
Alex: And that shows subsidies reducing overproduction?
Sam: Simulations suggest targeted subsidies nudge capacity to demand, easing curtailments like in California, North Germany's wind strains, or South Australia's solar issues.
Alex: But how do they solve that on a computer for the planner's policy?
Sam: They rewrite functions as running sums to the end, targeting zero at the finish. Neural networks—like video game AIs learning from scenarios—guess the functions and noise sensitivities. Time is discretized into steps using the Euler-Maruyama method, like mile markers on a road trip, adding drifts and noise jumps. Networks output values, and training minimizes end errors over simulated paths.
Alex: So this lets them test subsidy impacts numerically. In low-capacity starts, subsidies cover installs to ramp supply quickly. For excess, initial taxes slow growth until balance?
Sam: Yes. This grounds the theory in policy design.
Alex: What are the model's limits?
Sam: It assumes risk-neutral producers focused on expected profits, Markov controls depending only on current time and average, and bounded subsidies. It skips spatial constraints like transmission bottlenecks or learning from past shocks.
Alex: That keeps the math clean but might miss real caution or geography. Still, a meaningful step for planning.
Sam: The work offers a rigorous way to compute optimal subsidies or taxes as renewables grow. It proves unique equilibria and numerical reliability for stable grids.
Alex: That's a clear path from theory to practice for decarbonizing markets. Thanks, Sam—this has been insightful.
Sam: My pleasure, Alex. Thanks for listening to ResearchPod.