The day-ahead scheduling of multi-stack Power-to-X (PtX) plants must simultaneously cope with stack degradation under variable loading and with compound uncertainty in renewable generation and electricity prices. Existing scheduling frameworks address these two challenges in isolation, since degradation-aware models remain deterministic and stochastic models treat the electrolyser as a constant-efficiency device. This work develops a degradation-aware two-stage stochastic mixed-integer linear programming (MILP) framework that closes this gap. First-stage binaries fix the commitment and startup decisions of every stack, while second-stage scenario-indexed variables capture the dispatch, the hydrogen output, the shortfall, and the load-dependent and start–stop cycling degradation cost monetised at the stack level through a piecewise linear epigraph. Joint wind price uncertainty is represented by a Gaussian copula fitted on empirical CDF marginals and reduced to twenty representative scenarios via k-medoids clustering. The framework is fully implemented in MATLAB R2024a with the Optimization Toolbox, using the built-in intlinprog and linprog solvers. On a 100 MW reference plant with ten heterogeneous PEM stacks, out-of-sample evaluation against four formal benchmarks demonstrates the lowest LCOH at EUR 24/kg, the highest demand reliability at 85.0%, the highest hydrogen delivery at 7.68 t/day, and up to 50% total cost reduction over deterministic baselines, with end-to-end runtime under two minutes on standard workstation hardware.