  We formulate a nonlinear optimal control problem for intra-day operation of a
natural gas pipeline network that includes storage reservoirs. The dynamics of
compressible gas flow through pipes, compressors, reservoirs, and wells are
considered. In particular, a reservoir is modeled as a rigid, hollow container
that stores gas under isothermal conditions and uniform density, and a well is
modeled as a vertical pipe. For each pipe, flow dynamics are described by a
coupled partial differential equation (PDE) system in density and mass flux
variables, with momentum dissipation modeled using the Darcy-Wiesbach friction
approximation. Compressors are modeled as scaling up the pressure of gas
between inlet and outlet. The governing equations for all network components
are spatially discretized and assembled into a nonlinear differential-algebraic
equation (DAE) system, which synthesizes above-ground pipeline and subsurface
reservoir dynamics into a single reduced-order model. We seek to maximize an
objective function that quantifies economic profit and network efficiency
subject to the flow equations and inequalities that represent operating
limitations. The problem is solved using a primal-dual interior point solver
and the solutions are validated in computational experiments and simulations on
several pipeline test networks to demonstrate the effectiveness of the proposed
methodology.
