Second-order cone optimization of the gradostat *

The gradostat consists of multiple chemostats interconnected by mass flows and diffusion. It has been used to model biochemical systems such as wastewater treatment networks and microbial activity in soil. In this paper we maximize the production of biogas in a gradostat at steady state. The physical decision variables are the water, substrate, and biomass entering each tank and the flows through the interconnecting pipes. Our main technical focus is the nonconvex constraint describing microbial growth. We formulate a relaxation and prove that it is exact when the gradostat is outflow connected, its system matrix is irreducible, and the growth rate satisfies a simple condition. The relaxation has second-order cone representations for the Monod and Contois growth rates. We extend the steady state models to the case of multiple time periods by replacing the derivatives with numerical approximations instead of setting them to zero. The resulting optimizations are second-order cone programs, which can be solved at large scales using standard industrial software. (c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The study focuses on maximizing biogas production in a gradostat, with a main technical emphasis on the nonconvex constraint describing microbial growth. By formulating a relaxation and using numerical approximations, steady state models are extended to multiple time periods, resulting in second-order cone programs solvable with standard industrial software.