This paper develops a continuous-time stochastic control framework for the joint
operation of gas-fired generation, wind power, and energy storage under correlated
electricity and gas price uncertainty.
The operator’s decision problem is formulated as a finite-horizon
Hamilton--Jacobi--Bellman (HJB) equation, capturing the trade-off between immediate
revenues and the continuation value of storage under physical and operational
constraints.

To address the resulting high-dimensional control problem, we employ a mesh-free
neural approximation based on physics-informed and Deep Galerkin methods.
An analytical linear--quadratic (LQ) formulation is derived as a benchmark,
providing structural insight and a reference point under simplified assumptions.

Numerical experiments demonstrate stable convergence of the neural HJB solver and
recovery of economically interpretable policy structures.
When calibrated to historical electricity and gas price data and evaluated under
realistic transaction costs, the learned policy exhibits sparse, threshold-driven
storage operation with extended no-trade regions.
In these regimes, optimal behavior leaves the storage inactive despite ongoing
generation, reflecting the option-like and highly state-dependent value of
operational flexibility.

Overall, the results show that neural HJB solvers provide an economically
consistent and transparent framework for analyzing hybrid energy systems.
By linking stochastic price dynamics, operational constraints, and realized
storage decisions, the approach clarifies when flexibility is actively exercised
and when it remains economically dormant in low-carbon power systems.