Stochastic Active Discretizations for Accelerating Temporal Uncertainty Management of Gas Pipeline Loads

TL;DR

The adaptivity of the method in refining discretization based on error metrics enables computationally tractable evaluation of intertemporal uncertainty in order to support decisions about timing and quantity of pipeline operations to maximize delivery under transient and uncertain conditions.

Abstract

We propose a predictor-corrector adaptive method for the simulation of hyperbolic partial differential equations (PDEs) on networks under general uncertainty in parameters, initial conditions, or boundary conditions. The approach is based on the stochastic finite volume (SFV) framework that circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. The initial boundary value problem (IBVP) on a set of network-connected one-dimensional domains that represent a pipeline is represented using active discretization of the physical and stochastic spaces, and we evaluate the propagation of uncertainty through network nodes by solving a junction Riemann problem. The adaptivity of our method in refining discretization based on error metrics enables computationally tractable evaluation of intertemporal uncertainty in order to support decisions about timing and quantity of pipeline operations to maximize delivery under transient and uncertain conditions. We illustrate our computational method using simulations for a representative network.

Figures (4)

• Figure 1: Refinement paradigms on a given edge $i\in \mathcal{E}$ of the gas network under uncertainty. (a) Isotropic refinements in the physical and stochastic spaces. (b) Anisotropic refinements in the physical and stochastic spaces. Anisotropic refinements can capture non-smooth flows with targeted, directional allocations of computational resources for more efficient UQ-informed predictive control.
• Figure 2: Minimum growth in the number of degrees of freedom (NDoFs) with respect to the dimension of the stochastic space upon refinement of a stochastic control volume. Anisotropic refinements have flexibility in allocating new resources, whereas isotropic refinements are constrained to exponential growth.
• Figure 3: The joint push-forward probability density function of the density $\rho$ and mass flux $q$ in the middle of terminal edge of the test network at hour 12.
• Figure 4: Push-forward probability density function for the density $\rho$ in the middle of the terminal edge at selected simulation times $t$ in hours. Prior to hour 4, the flows are deterministic. As the maximum possible timing of the uncertain withdrawal (12 hours) is passed, the network stabilizes and the uncertainty narrows from complicated, multi-modal distributions.

Theorems & Definitions (1)

• Definition 1: Reconstruction, Def. (3.7, harmon2022)