  We formulate an economic optimal control problem for transport of natural gas
over a large-scale transmission pipeline network under transient flow
conditions. The objective is to maximize economic welfare for users of the
pipeline system, who provide time-dependent price and quantity bids to purchase
or supply gas at metered locations on a system with time-varying injections,
withdrawals, and control actions of compressors and regulators. Our formulation
ensures that pipeline hydraulic limitations, compressor station constraints,
operational factors, and pre-existing contracts for gas transport are
satisfied. A pipeline is modeled as a metric graph with gas dynamics partial
differential equations on edges and coupling conditions at the nodes. These
dynamic constraints are reduced using lumped elements to a sparse nonlinear
differential algebraic equation system. A highly efficient temporal
discretization scheme for time-periodic formulations is introduced, which we
extend to develop a rolling-horizon model-predictive control scheme. We apply
the computational methodology to a pipeline system test network case study. In
addition to the physical flow and compressor control solution, the optimization
yields dual functions that we interpret as the time-dependent economic values
of gas at each location in the network.
