InFlow: entropic stochastic dual dynamic programming with HJB cross-certification for seasonal energy storage
Seasonal reservoir operation requires optimization methods that remain reliable under strongly periodic and uncertain inflow dynamics. We present InFlow, a verification-oriented framework combining periodic stochastic dual dynamic programming (SDDP), entropic robustness, control-space Bellman annealing, and an independently discretized Hamilton-Jacobi-Bellman (HJB) benchmark. The entropic transition operator is interpreted as a KL-penalized worst-case expectation and linked to fixed-radius KL ambiguity sets through Lagrange duality. We establish a uniform soft-Bellman approximation bound and prove the validity of Gibbs-tilted SDDP cuts. The benchmark relies on a normalized seasonal Cox-Ingersoll-Ross inflow process whose positivity condition fails during the driest weeks, requiring explicit treatment of the degenerate boundary. Independent verification shows strong agreement between HJB storage gradients and HiGHS storage-balance duals, with water-value correlations of 0.995 and 0.996 for the standard and exact-cut-only implementations. Refining the HJB grid from 61^2 to 81^2 states changes the annual mean water value by only 0.6% and the peak value by less than 0.1%. Out-of-sample simulations based on 3,000 independent trajectories show that a robustness level of gamma=5 increases nominal mean cost by 4.6% while reducing CVaR90 by 11.2%. Under a dry-season stress scenario with inflows reduced by 40%, the mean-cost difference becomes statistically insignificant whereas CVaR90 improves by 11.5%. A robustness sweep confirms monotonicity of the risk-adjusted value. Finally, a two-reservoir cascade reproduces grid-oracle marginal water values with correlation 0.999 and recovers the predicted upstream-to-downstream value ratio of two. InFlow is intended as a transparent methodological benchmark rather than a calibrated operational reservoir model.