Control Node Placement and Structural Controllability of Water Quality Dynamics in Drinking Networks

TL;DR

Addresses booster station placement in water distribution networks to maintain chlorine residuals under time-varying hydraulics. Introduces a controllability-driven CBSP formulation that combines the water-quality controllability Gramian with a submodular set-function, solved via a forward greedy algorithm and augmented by a structural-controllability weighting across hydraulic scenarios. Validated on networks of varying size and topology, showing near-optimal placements and tradeoffs between trace and log-determinant metrics, with guidance for backup/mobile injections. Provides scalable, operator-oriented insights and outlines future work on multi-species dynamics and budgeted deployments to extend applicability.

Abstract

Chlorine, the most widely used disinfectant, needs to be adequately distributed in water distribution networks (WDNs) to maintain consistent residual levels and ensure safe water. This is performed through control node injections at the treatment plant via booster stations distributed across the WDNs. While previous studies have applied various optimization-based approaches for booster station placement, many have failed to consider the coverage of the station injections and the dynamic nature of WDNs. In particular, variations in hydraulics and demand significantly impact the reachability and efficacy of chlorine injections which then impact optimal placement of booster stations. This study introduces a novel formulation that combines control- and graph-theoretic approaches to solve the booster station placement problem. Unlike traditional methods, our approach emphasizes maximizing the system's ability to control disinfectant levels with minimal control energy, taking into account the time-varying hydraulic profiles that lead to different optimal station placements. We propose a simple weighting technique to determine the placements by assessing the structural controllability of each configuration, based on the network's topology, independent of specific parameters like decay rates or pipe roughness. This method ensures effective chlorine coverage across the network. Our approach is validated on different networks, demonstrating its operational effectiveness, scalability, and practicality.

Figures (13)

• Figure 1: Flowchart of the chlorine booster station placement problem solution.
• Figure 2: Test networks: (a) Three-node network, (b) Net1, (c) FOS, (d) Net3, and (e) C-Town network.
• Figure 3: Tank TK1 volume of the Three-node network plotted with a darker color for windows when the tank is filling and lighter color for windows when it is emptying, over a total duration of 24 hours.
• Figure 4: Booster stations placement results for the Three-node network by solving the (a) $\mathop{\mathrm{trace}}\nolimits$- and (b) $\log \det$-based CBSP problems for a simulation period of 24 hours with $\Delta t_\mathrm{WQ}=10$ sec and $\Delta t_\mathrm{H} = 1$ hour.
• Figure 5: Four case scenarios based on different (a) base demands, (b) demand patterns for nodes, and (c) the corresponding TK2 volume in Net1.

Theorems & Definitions (1)

• Definition 1