Abstract
A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.
Highlights
The use of suspended microorganisms to remove undesired components, including organic carbon, nitrogen, and phosphorus species, is referred to as activated sludge processes and is widely used around the world in municipal wastewater treatment plants (WWTPs) [1] [2] [3]
A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness
The results presented were obtained by running the executive version of the C++ code on a desktop computer with a quad-core Intel® 2.67 GHz Xenon® CPU and 8 Gb RAM
Summary
The use of suspended microorganisms to remove undesired components, including organic carbon, nitrogen, and phosphorus species, is referred to as activated sludge processes and is widely used around the world in municipal wastewater treatment plants (WWTPs) [1] [2] [3]. Non-linear reaction rates based on Monod Kinetic cause a system of highly nonlinear ODEs. In some applications, the computation time of the ASM is critical due to a long simulation period, the need for online simulation [10], and the use of parameter estimation algorithms that require numerous simulations. In addition to the non-linearity of the system of ODEs comprising ASMs, they typically cover a wide range of biochemical reaction rate scales, ranging from seconds (for example, oxygen transfer rate) to days (for example, microbial growth rates) and result in a mixed (stiff/nonstiff) system of ODEs [12] that generally requires small time-steps when conventional ODE solver algorithms are used. Due to the inherent fluctuating behavior of the external forcing vectors, including variation in the time series of the influent rate and the characteristics in wastewater treatment streams, the optimal time-step size can vary greatly during the course of a simulation
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