Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

TL;DR

This paper addresses solving large, nonlinear, potential-driven steady-state network flow equations whose convergence is not guaranteed for general topologies. It introduces a hierarchical partitioning method based on generalized block-cut trees, enabling solving smaller nonlinear sub-systems in a sequence that preserves equivalence with the original network equations NF_$\pi$. The approach leverages replication of articulation points and a tree-structured Hierarchical Solution Algorithm to achieve scalable solutions for large networks, with theoretical guarantees of equivalence and finite termination. Empirical results on GasLib and Texas networks show substantial partitioning benefits (largest sub-network ≲ one-third of the full network) and successful recovery of the full-network solution. The work paves the way for robust large-scale network flow computation and suggests future exploration for transient dynamics and optimization applications.

Abstract

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology and in general there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. This article proposes and describes such a procedure, an hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

Figures (4)

• Figure 1: The slack vertex is coloured green while the replicated vertices are shown in red. The chosen vertex (coloured orange) of the original network in (a) is replicated as shown to create a new, equivalent network in (b).
• Figure 2: We can visually discern four sub-networks (blocks shown in blue colour) connected to each other (through the articulation point) in the form of a tree (shown inset, which is the block-cut tree). Using the cumulative injections, can determine flow in each edge of the tree. If we then solve the system \ref{['eq:NF-pi']} for slack sub-network $N_1$, can subsequently solve $N_2$, $N_3$ and $N_4$ to determine solution of system \ref{['eq:NF-pi']} for the full network.
• Figure 3: An illustration of Step \ref{['step:2']} in the proof of Theorem \ref{['thm:equivalence-generalized-block-cut-system']}.
• Figure 4: The Texas-2451 network is shown with the blocks that have more than two nodes colored.

Theorems & Definitions (22)

• Proposition 1
• proof
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5
• Lemma 1
• proof