Separating two species of microalgae in photobioreactors in minimal time

Abstract

This paper proposes a strategy to separate two strains of microalgae in minimal time. The control is the dilution rate of the continuous photobioreactor. The microalgae dynamics is described by the Droop's model, taking into account the internal quota storage of the cells. Using Pontryagin's principle, we develop a dilution-based control strategy that leads to the most ecient species separation in minimal time. A numerical optimal synthesis based on direct methods is performed throughout the paper, in order to determine the structure of the optimal feedback-control law, which is bang-singular. Our numerical study reveals that singular arcs play a key role in the optimization problem since they allow the optimal solution to be close to an associated static optimal control problem. A resulting turnpike-like behavior, which characterizes the optimal solution, is highlighted throughout this work.

Optimal control separating two microalgae species competing in a chemostat

“…are not equivalent to the occurrence of a singular arc starting at some time t c < t f (contrary to what was suggested earlier in [12] and [6]). However, from extensive numerical simulations using direct optimization approaches (see Sect.…”

“…V, but also [12], [11]), we note thatΦ andΦ vanish in some significantly large time interval [t c ,t f ] depending on the initial condition. To summarize, combining the theoretical analysis of (OCP) together with the numerical simulations of Sect.…”

“…where s in is the constant input concentration of the substrate. The chemostat dilution rate D(•) is a bounded nonnegative control function such that D(t) ∈ [0, D max ], where D max > 0 is the maximum capacity of the feeding pump, above the maximum effective growth rate of the two species as illustrated in Figure 1 (see also [12]). In addition, for i = 1, 2, ρ i is a real-valued function quantifying the rate of substrate absorption, i.e.…”

“…It appears that microalgae are better represented by Droop's model (leading to a five-dimensional controlled system) which accounts for the internal accumulation of the limiting nutrient ( [9], [10]). The minimal-time problem with Droop's model has been tackled by [11], [12] in a simplified case in which the model dimension has been partially reduced. In this paper, we focus on the problem of selecting the species of interest in minimal-time with two Droop-species.…”

Modelado y control de la producción de microalgas en fotobiorreactores industriales

“…El objetivo fundamental, que es la maximización del rendimiento en la producción de microalgas, sólo se puede alcanzar en base a un detallado análisis de las principales variables del cultivo y su optimización a lo largo de las escalas de tiempo existentes. Por tanto, el diseño de estrategias de control avanzadas para conseguir llevar el sistema de producción de microalgas cerca de sus valores teóricos se ha convertido en una línea de investigación de candente actualidad (Fernández et al, 2016b;de Andrade et al, 2016a;Djema et al, 2020).…”