Stochastic Modeling and Dynamic Optimization for the Long-Term Sustainability of Energy Savings in Demand-Response Systems

The global energy transition critically depends on the effectiveness and long-term persistence of energy-saving measures. However, traditional evaluation models tend to be deterministic, failing to capture the stochastic nature of phenomena such as consumer fatigue and the “rebound effect.” This study proposes a rigorous mathematical framework based on stochastic optimal control. We introduce the Dynamic Sustainability Coefficient (DSC), ψ(t), which is modeled using an Ornstein-Uhlenbeck-type Stochastic Differential Equation (SDE). The net savings maximization problem is solved using the Hamilton-Jacobi-Bellman (HJB) equation. The numerical results, based on calibrated synthetic data, show that the adaptive optimal control strategy outperforms static strategies, increasing the project's Net Present Value (NPV) by 20.1%. A clear methodological workflow is provided to facilitate the model's reproducibility and application in energy policy.