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Stochastic Finite Volume Method for Uncertainty Management in Gas Pipeline Network Flows

Saif R. Kazi, Sidhant Misra, Svetlana Tokareva, Kaarthik Sundar, Anatoly Zlotnik

Abstract

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.

Stochastic Finite Volume Method for Uncertainty Management in Gas Pipeline Network Flows

Abstract

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.
Paper Structure (11 sections, 29 equations, 8 figures)

This paper contains 11 sections, 29 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Steady-State Optimal Gas Flow Problem
  3. Gas Flow Modeling
  4. Pipeline Network Modeling
  5. Deterministic Optimal Gas Flow
  6. Chance-Constrained Optimal Gas Flow
  7. Computational Studies
  8. Single Pipe with Compressor
  9. 8-node Pipeline Test Network
  10. Economic Interpretation of Lagrange Multipliers
  11. Conclusion

Figures (8)

  • Figure 1: Single pipe test system and component parameters.
  • Figure 2: Single pipe example probability distribution functions $q_3\sim \mathcal{U}[200,300]$ and $q_3\sim N(200,50/3)$ with truncated tails.
  • Figure 3: Pressure Probability Distribution at Withdrawal Node $J3$. Top: Truncated Normal Uncertainty $q_3\sim N(200,50/3)$; Bottom: Uniform Uncertainty $q_3\sim \mathcal{U}[200,300]$
  • Figure 4: 8-node Gas Pipeline Network System
  • Figure 5: Pressure probability distributions for 8-node case study. Top: Node 3; and Bottom: Node 5.
  • ...and 3 more figures