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Second-order cone optimization of the gradostat

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.