Modeling Minimum Cost Network Flows With Port‐Hamiltonian Systems

Abstract

AbstractWe give a short overview of advantages and drawbacks of the classical formulation of minimum cost network flow problems and solution techniques, to motivate a reformulation of classical static minimum cost network flow problems as optimal control problems constrained by port‐Hamiltonian systems (pHS). The first‐order optimality system for the port‐Hamiltonian system‐constrained optimal control problem is formally derived. Then we propose a gradient‐based algorithm to find optimal controls. The port‐Hamiltonian system formulation naturally conserves flow and supports a wide array of further modeling options as, for example, node reservoirs, flow dependent costs, leaking pipes (dissipation) and coupled sub‐networks (ports). They thus provide a versatile alternative to state‐of‐the art approaches towards dynamic network flow problems, which are often based on computationally costly time‐expanded networks. We argue that this opens the door for a plethora of modeling options and solution approaches for network flow problems.