Model Order Reduction of Nonlinear Multi-Species Water Quality Dynamics

Tracing disinfectant (e.g., chlorine) and contaminants evolution in water networks requires the solution of 1-D advection-reaction (AR) partial differential equations (PDEs). While analytical solutions are non-existent in most cases, numerical solutions require high-resolution time- and space-discretizations. This results in large model dimensions even for small networks thereby leading to a physical-driven model that is intractable when considering constrained control and water quality regulation algorithms. In addition, considering multi-species rather than single-species dynamics produces a more accurate description of the reaction dynamics. The multi-species model can be used to model disinfectant interactions with other species (e.g., microbial, chemical) in the bulk flow, attached to the pipe walls, or contamination events. Yet, it adds complexity by introducing nonlinear reaction dynamics 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 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 feedback control to maintain desirable disinfectant levels in water networks. These approaches focus on reducing model dimension by orders of magnitude while preserving important properties of the original system (e.g., controllability). That is, this framework is constructed and tested considering different variables and parameters including hydraulics, network size, events nature, etc. Moreover, it is considered a generalized scalable framework in the sense that simplifications are included to consider single-species water dynamics and differentiations are suggested to consider different chlorine linear/nonlinear decay and reaction models. The performance of this framework is validated using rigorous numerical case studies under a wide range of scenarios demonstrating the challenges associated with controlling chlorine levels under multi-species water quality dynamics.