Parallel hybrid ordinary differential equation for modeling biological phosphorus removal modified for enhanced...

Parallel hybrid ordinary differential equations (ODEs) include a novel technique for operational management support of wastewater treatment plants committed to carbon neutrality, human footprint reduction, and ecosystem service conservation in water environments. These equations seamlessly integrate mechanistic and data-driven components operating together. However, simultaneously improving their predictive performance and physical interpretability is a major challenge. In a previous work, this problem was resolved using a parallel hybrid ODE that integrates Activated Sludge Model No. 2d (ASM2d) as the mechanistic component with an artificial neural network (ANN) as the data-driven component. Supporting data are incorporated for modeling biological phosphorus removal. However, the supporting data-enhanced hybrid ODE (SH-ODE) does not involve mass conservation in the ANN component and automatic calibration in the ASM2d. This study modifies the previously developed SH-ODE to enhance its physical consistency. First, a stoichiometry defined in the ASM2d was embedded into the ANN component as hard physical constraints to preserve mass conservation. Operational data were acquired from three anaerobic/aerobic sequencing batch reactors, and phosphate release activity was evaluated from anaerobic batch experiments to serve as supporting data. The predictive performance for phosphate achieved a coefficient of determination (R²) value of 0.90. Second, a serialized ANN was incorporated into the SH-ODE with mass conservation for the automatic calibration of five selected parameters. Owing to the pivotal contributions of the supporting data, the R² value further improved to 0.93. Moreover, the supporting data contributed to phosphate estimation by the ASM2d through biomass compensation achieved by the parallelized ANN. A novel index was developed to quantify physical interpretability based on the contribution of the ASM2d for estimating phosphate dynamics. Results showed that the modified SH-ODE achieved a high predictive performance and physical interpretability.