Operational boundary of flow network

Flow networks can represent complex systems and exhibit generality. The operational boundary (OB) can describe the secure operational range of complex systems. It helps to enhance network safety to determine the OB of flow networks. This paper proposes the OB for flow networks to completely represent the maximum range the network's feasible flow can reach. Firstly, the concept and mathematical model of the OB are proposed. As a subset of the feasible flows, the set of all the blocking flows is defined as the OB. The OB forms the boundary that separates the feasible flows from the infeasible flows. The feasible flows in critical states are located on the OB, and the maximum flows are located on the outermost layer of the OB. Secondly, an analytical algorithm and a sampling algorithm are proposed to calculate the OB. The former one can completely represent the OB, while the latter one better reflects the state of the boundary points. Moreover, the OB-based safety management framework, which includes security assessment and security control, has been developed to improve the safety and reliability of flow networks. Finally, the effectiveness of the proposed method is demonstrated in multiple-source multiple-sink networks.