Comprehensive Framework for Controlling Nonlinear Multi-Species Water Quality Dynamics

TL;DR

The paper tackles real-time control of disinfectant and contaminants in water distribution networks governed by nonlinear, multi-species 1-D advection-reaction PDEs. It advances a comprehensive MOR-based framework that combines linear and nonlinear MOR paths (POD, BPOD, NLPOD with DEIM/Gappy) with Model Predictive Control and McCormick relaxations to enable tractable, network-wide regulation. Core contributions include detailed linearization, multiple MOR strategies, explicit handling of nonlinear terms, and a generalized control framework validated through extensive case studies across networks of varying size and hydraulics. The work demonstrates that reduced-order models can achieve accurate, bounded-control performance with real-time computation, significantly improving scalability for water quality management in real networks.

Abstract

Tracing disinfectant (e.g., chlorine) and contaminants evolution in water networks requires the solution of 1- D advection-reaction (AR) partial differential equations (PDEs). With the absence of analytical solutions in many scenarios, numerical solutions require high-resolution time- and spacediscretizations resulting in large model dimensions. This adds complexity to the water quality control problem. In addition, considering multi-species water quality dynamics rather than the single-species dynamics produces a more accurate description of the reaction dynamics under abnormal hazardous conditions (e.g., contamination events). Yet, these dynamics introduces nonlinear reaction formulation to the model. To that end, solving nonlinear 1-D AR PDEs in real time is critical in achieving monitoring and control goals for various scaled networks with a high computational burden. In this work, we propose a novel comprehensive framework to overcome the large-dimensionality issue by introducing different approaches for applying model order reduction (MOR) algorithms to the nonlinear system followed by applying real-time water quality regulation algorithm that is based on an advanced model to maintain desirable disinfectant levels in water networks under multi-species dynamics. The performance of this framework is validated using rigorous numerical case studies under a wide range of scenarios demonstrating the challenges associated with regulating water quality under such conditions.

Figures (16)

• Figure 1: Conceptual framework of the paper.
• Figure 2: Implicit and Explicit Upwind discretization schemes for Pipe $i$ connecting Junctions 1 and 2. Each scheme calculates concentration $c^\mathrm{P}_i(s,t+\Delta t)$ at segment $s$ (colored in maroon) depending on concentrations at the segments/nodes included in its frame. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
• Figure 3: (a) Linear and (b) nonlinear MOR methods configuration.
• Figure 4: An illustrative example of applying the Greedy sampling algorithm to construct the measurement matrix $\boldsymbol{K}$ for the case of $n_r=5$.
• Figure 5: (a) Discrete MPC prediction horizon scheme, and (b) graphical representation of McCormick envelope relaxation.