  Natural gas consumption by users of pipeline networks is subject to
increasing uncertainty that originates from the intermittent nature of electric
power loads serviced by gas-fired generators. To enable computationally
efficient optimization of gas network flows subject to uncertainty, we develop
a finite volume representation of stochastic solutions of hyperbolic partial
differential equation (PDE) systems on graph-connected domains with nodal
coupling and boundary conditions. The representation is used to express the
physical constraints in stochastic optimization problems for gas flow
allocation subject to uncertain parameters. The method is based on the
stochastic finite volume approach that was recently developed for uncertainty
quantification in transient flows represented by hyperbolic PDEs on graphs. In
this study, we develop optimization formulations for steady-state gas flow over
actuated transport networks subject to probabilistic constraints. In addition
to the distributions for the physical solutions, we examine the dual variables
that are produced by way of the optimization, and interpret them as price
distributions that quantify the financial volatility that arises through demand
uncertainty modeled in an optimization-driven gas market mechanism. We
demonstrate the computation and distributional analysis using a single-pipe
example and a small test network.
