Large-scale power-to‑hydrogen (P2H) stations with multi-stack configurations, are emerging as valuable flexible resources for the power grid. The energy management strategy (EMS) determines multi-stack operation statuses. Nonetheless, existing EMS focus on production without adequately addressing the implications for grid-side power factor (PF) and potential security concerns. This paper addresses this limitation by presenting a model that characterizes the PF of a multi-stack P2H system across varying operation statuses defined by current and temperature. Through this model, we highlight a clear trade-off between the PF constraint and production targets in multi-stack scheduling. Subsequently, we introduce an improved EMS framework for multi-stack P2H that seeks a balance between PF, production, and security. This EMS is organized as a two-layer execution structure to guarantee control accuracy and tractability, which includes a model-based robust multi-stack scheduling programming and a rule-based real-time increment correction algorithm in series. Case studies compare multi-stack scheduling strategies under the proposed EMS with the traditional production-oriented strategy. The effectiveness of the extended PF and security dimensions is verified to comprehensively improve the responsiveness to power instructions. Furthermore, we outline five representative cluster-level scheduling strategies aligned with different load scenarios, offering insights for practical industrial implementations.