Exploring stochastic differential equation for analyzing uncertainty in wastewater treatment plant-activated sludge modeling

The management of wastewater treatment plant (WWTP) and the assessment of uncertainty in its design are crucial from an environmental engineering perspective. One of the key mechanisms in WWTP operation is activated sludge, which is related to the biological oxygen demand (BOD) parameter. In the modeling of BOD, the conventional approach utilizing ordinary differential equations (ODEs) fails to incorporate the stochastic nature of this parameter, leading to a considerable level of uncertainty in the design of WWTP. To address this issue, this study proposes a stochastic model that utilizes stochastic differential equations (SDEs) instead of ODE to simulate BOD activities of microorganisms and wastewater flow rate (Q). The SDEs and integral Ito are solved using the Euler-Maruyama method for a period of 15 sequential days and the timespan of 2019-2020 for a WWTP in Tabriz City. SDE improves the design of WWTP facilities by identifying uncertainties and enhancing reliability. This, in turn, increases the reliability of the technical structures within the WWTP and improves the performance of its biological system. By considering the randomness of the problem, the proposed method significantly improves the results, with an enhancement of 11.47 and 10.11% for the BOD and Q models, respectively.