A Hybrid Heuristic–Benders Method for Wind–Hydrogen Investment Planning with Non-Analytical Cost Functions

This paper studies capacity planning for a wind–hydrogen integrated energy system under scenario-based uncertainty in wind generation, hydrogen demand, and electricity prices. The model is formulated as a two-stage stochastic program in which first-stage investment decisions are selected before uncertainty is realized and second-stage hourly operation is optimized for each representative scenario. The main methodological difficulty is that part of the first-stage hydrogen-storage investment cost may be available only through a non-analytical evaluator, such as supplier quotation logic, simulation software, or a data-driven estimator, while the operational recourse model remains linear. To address this setting, a hybrid heuristic–Benders framework, denoted as GSOA-Benders, is developed by coupling the General-Soldiers Optimization Algorithm for derivative-free first-stage search with Benders cuts generated from linear programming subproblems. The framework is not presented as a replacement for commercial solvers on explicit convex or mixed-integer models; rather, it is intended for cases where exact algebraic reformulation of the first-stage cost is unreliable or unavailable. In the black-box case study with 500 scenarios, the method converges in 35.86 s and obtains an investment plan expressed as x=[1,0.53,23.23,0], corresponding to wind-farm construction, a 0.53 MW electrolyzer, a 23.23 MWh hydrogen tank, and no fuel-cell investment. Additional discussion is provided on stability-gap interpretation, benchmark limitations, component lifetime assumptions, hydrogen losses, and environmental extensions.