Model Order Reduction of The Time-Dependent Advection-Diffusion-Reaction Equation with Time-Varying Coefficients: Application to Real-Time Water Quality Monitoring

Advection-Diffusion-Reaction (ADR) Partial Differential Equations (PDEs) appear in a wide spectrum of applications such as chemical reactors, concentration flows, and biological systems. A large number of these applications require the solution of ADR equations involving time-varying coefficients, where analytical solutions are usually intractable. Numerical solutions on the other hand require fine discretization and are computationally very demanding. Consequently, the models are normally not suitable for real-time monitoring and control purposes. In this contribution, a reduced order modeling method for a general ADR system with time-varying coefficients is proposed. Optimality of the reduced order model regarding the reduction induced error is achieved by using an
$\mathcal{H}_{2}$
-norm reduction method. The efficacy of the method is demonstrated using two test cases. Namely, a case for an ADR with arbitrary dynamics varying coefficients and a second case including the modeling of an exemplary water quality distribution path with randomly generated demand. The reduced order models are evaluated against high fidelity simulations using MATLAB's finite element method PDE tool-box. It is shown that the reduction can achieve a significant computational speedup allowing for the usage of the model for real-time applications with sampling times in milliseconds range. Moreover, the constructed ROM is shown to achieve high prediction accuracy with the normalized mean square error below
$2.3 \%$
for a real-world water quality simulation test case.