Periodic optimal control for biomass productivity maximization in a photobioreactor using natural light

TL;DR

This paper develops an optimal control strategy for maximizing long-term biomass productivity in photobioreactors under natural light cycles, using a simple model and Pontryagin's maximum principle, with practical bang-bang control solutions.

Contribution

It introduces a novel optimal control framework considering day/night cycles and biomass constraints, providing explicit control laws and bifurcation analysis for photobioreactor productivity.

Findings

Optimal control laws depend on strain parameters and light conditions.

Bifurcation diagram illustrates control regime transitions.

A simple sub-optimal strategy nearly matches optimal productivity.

Abstract

We address the question of optimization of the microalgal biomass long term productivity in the framework of production in photobioreactors under the influence of day/night cycles. For that, we propose a simple bioreactor model accounting for light attenuation in the reactor due to biomass density and obtain the control law that optimizes productivity over a single day through the application of Pontryagin's maximum principle, with the dilution rate being the main control. An important constraint on the obtained solution is that the biomass in the reactor should be at the same level at the beginning and at the end of the day so that the same control can be applied everyday and optimizes some form of long term productivity. Several scenarios are possible depending on the microalgae's strain parameters and the maximal admissible value of the dilution rate: bang-bang or bang-singular-bang control or, if the growth rate of the algae is very strong in the presence of light, constant maximal dilution. A bifurcation diagram is presented to illustrate for which values of the parameters these different behaviors occur. Finally, a simple sub-optimal bang-bang strategy is proposed that numerically achieves productivity levels that almost match those of the optimal strategy.