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Scalable Sensor Placement for Cyclic Networks with Observability Guarantees: Application to Water Distribution Networks

J. J. H. van Gemert, V. Breschi, D. R. Yntema, K. J. Keesman, M. Lazar

TL;DR

The paper addresses scalable sensor placement for state estimation in large-scale cyclic networks with parametric uncertainties, focusing on water distribution networks. It develops a structural observability framework and a spanning-tree based graph algorithm that converts cyclic graphs to trees, guaranteeing observable configurations with $n_y$ sensors such that the system is observable for all $A \in \mathcal{P}(\mathcal{A})$ and $C \in \mathcal{P}(\mathcal{C})$. The approach is implemented on EPANET benchmarks (Hanoi, AnyTown, Net3, D-town, L-town) and achieves sensor placements in under 0.1 seconds for the largest network. This yields scalable, guaranteed observability without requiring exact parameter knowledge, though it does not optimize sensor costs and may yield non-unique solutions.

Abstract

Optimal sensor placement is essential for state estimation and effective network monitoring. As known in the literature, this problem becomes particularly challenging in large-scale undirected or bidirected cyclic networks with parametric uncertainties, such as water distribution networks (WDNs), where pipe resistance and demand patterns are often unknown. Motivated by the challenges of cycles, parametric uncertainties, and scalability, this paper proposes a sensor placement algorithm that guarantees structural observability for cyclic and acyclic networks with parametric uncertainties. By leveraging a graph-based strategy, the proposed method efficiently addresses the computational complexities of large-scale networks. To demonstrate the algorithm's effectiveness, we apply it to several EPANET benchmark WDNs. Most notably, the developed algorithm solves the sensor placement problem with guaranteed structured observability for the L-town WDN with 1694 nodes and 124 cycles in under 0.1 seconds.

Scalable Sensor Placement for Cyclic Networks with Observability Guarantees: Application to Water Distribution Networks

TL;DR

The paper addresses scalable sensor placement for state estimation in large-scale cyclic networks with parametric uncertainties, focusing on water distribution networks. It develops a structural observability framework and a spanning-tree based graph algorithm that converts cyclic graphs to trees, guaranteeing observable configurations with sensors such that the system is observable for all and . The approach is implemented on EPANET benchmarks (Hanoi, AnyTown, Net3, D-town, L-town) and achieves sensor placements in under 0.1 seconds for the largest network. This yields scalable, guaranteed observability without requiring exact parameter knowledge, though it does not optimize sensor costs and may yield non-unique solutions.

Abstract

Optimal sensor placement is essential for state estimation and effective network monitoring. As known in the literature, this problem becomes particularly challenging in large-scale undirected or bidirected cyclic networks with parametric uncertainties, such as water distribution networks (WDNs), where pipe resistance and demand patterns are often unknown. Motivated by the challenges of cycles, parametric uncertainties, and scalability, this paper proposes a sensor placement algorithm that guarantees structural observability for cyclic and acyclic networks with parametric uncertainties. By leveraging a graph-based strategy, the proposed method efficiently addresses the computational complexities of large-scale networks. To demonstrate the algorithm's effectiveness, we apply it to several EPANET benchmark WDNs. Most notably, the developed algorithm solves the sensor placement problem with guaranteed structured observability for the L-town WDN with 1694 nodes and 124 cycles in under 0.1 seconds.

Paper Structure

This paper contains 12 sections, 2 theorems, 8 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Basic notation
  3. Preliminaries and problem statement
  4. Basic graph notions
  5. Instrumental graph definitions
  6. Colorability rules for graphs
  7. Problem statement
  8. Main results
  9. Sensor placement in tree graphs using colorability
  10. Sensor placement in cyclic graphs using spanning trees and colorability
  11. Implementation on EPANET benchmarks
  12. Conclusions

Key Result

Lemma III.2

Let Assumption Ass: C and Assumption Ass: ATree hold, and let $C\in\mathbb{R}^{n_y\times n_x}$ satisfy with $n_y = n_e - 1$ and $n_e,\mathcal{V}_e$ introduced in Definition Def: Intersection_Extreme_Node. Then, the linear system eq: LTI is observable.

Figures (8)

  • Figure 1: Subfigure (a) illustrates an example of a bidirected cyclic network with the structure of $A_{\mathrm{adj}}$ in \ref{['eq:IncAdjWdn']}. Subfigure (b) presents two possible spanning trees derived from (a), and subfigure (c) displays a graph with arbitrary edges.
  • Figure 2: Subfigures (a)-(f) illustrate the sequential application of the color change rule on the graph $\mathcal{G}(\mathcal{A}^\top,\mathcal{C}^\top)$. The round nodes represent the nodes associated with $\mathcal{A}^\top\!$, while the red hexagon nodes indicate the sensor nodes corresponding to $\mathcal{C}^\top\!$.
  • Figure 3: Building blocks for developing a scalable sensor placement algorithm.
  • Figure 4: Subfigures (a)-(f) illustrate the sequential application of the color change rule for cyclic graphs. The round nodes represent the nodes associated with $\mathcal{A}^\top\!$, while the red hexagon nodes indicate the sensor nodes corresponding to $\mathcal{C}^\top\!$.
  • Figure 5: The Hanoi network from EPANET.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition II.1: kalman1960general
  • Definition II.2
  • Definition II.3
  • Definition II.4
  • Definition II.5
  • Definition II.6
  • Definition II.7
  • Definition II.8: jia2020unifying
  • Lemma III.2
  • proof
  • ...and 5 more