# Model Based Optimal Control of the Photosynthetic Growth of Microalgae in a Batch Photobioreactor

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## Abstract

**:**

## 1. Introduction

_{2}, salts of nitrogen, salts of phosphorous, etc. Water photolysis is one of the most energy-demanding reactions in nature. The water is split, the hydrogen is used in the photosynthesis transport chain and the O

_{2}is released as residue. All of these reactions are named light-dependent reactions. CO

_{2}is required to form primary metabolites (e.g., carbohydrates, lipids, etc.), reactions that do not require light but require the protons and electrons produced by the photodissociation of water. Due to the simple metabolic requirements needed for the growth of microalgae, the photosynthetic metabolism is very attractive.

_{2}flow rate [28]. This paper focuses on the batch mode operation of artificially illuminated PBRs, the control variable being the incident light intensity on the PBR surface.

## 2. Mathematical Modeling of the Photosynthetic Growth Process

#### 2.1. The Radiative Model

^{−2}·s

^{−1}(to simplify μmol·m

^{−2}·s

^{−1}) and hereafter named incident light intensity. The incident light intensity that touches the culture of microalgae at $z=0$ attenuates inside the PBR due to absorption phenomena and it is named irradiance and is denoted by $I$. $z$ is the depth of the culture, $z\in \left[0,L\right]$ and $L$ is the depth of the PBR. The propagation of the light inside the PBR depends on a series of factors such as medium properties, reactor geometry, the shape of cells, the concentration of pigments [21], etc. The attenuation of light inside of a microalgae culture (i.e., the irradiance) is generally described by the Lambert-Beer law, which relates the irradiance to the depth of the culture, $z$, and the properties of the liquid environment:

^{−1}so that the exponential factor is dimensionless.

^{2}·kg

^{−1}(same as ${E}_{a}$ and ${E}_{s}$ as reported in Table 1).

#### 2.2. The Kinetic Growth Model

- -
- -
- local photosynthetic responses, $\mu \left(I\left(z\right)\right)$, can be calculated for any depth $z$ of the culture. These local photosynthetic responses are simply averaged into an average photosynthetic response, $\langle \mu \rangle $ (or average specific growth rate) [31]. The $\langle \rangle $ denote an averaged value.

^{−1}), the light penetrates less than a quarter of the culture, leaving the rest of it in dark. In the dark zone of the culture the specific growth rate is negative, ${\mu}_{p}<{\mu}_{d}$. The average specific growth rate is very low (i.e., 0.002 h

^{−1}) for this concentration of biomass, and an incident light intensity lower than 500 μmol·m

^{−2}·s

^{−1}leads to the decrease of the biomass concentration. The average of the specific growth rates is justifiable because, in a CSTR, a cell is migrating continuously between the light and the dark zones.

#### 2.3. The Mass Balance Model

^{−1}can be obtained:

^{−1}, at the beginning of the batch ($t=0$), and $V$ is the working volume of the PBR.

^{−6}is needed to express $Q$ in moles, where 3600 converts seconds to hours and 10

^{−6}converts micromoles to moles, i.e., $Q=A\xb7q\xb73600\times {10}^{-6}$).

^{−1}, ${x}_{2}$ is the newly produced biomass in grams, ${x}_{3}$ is the amount of light consumed in moles of photons and ${x}_{4}$ is the biomass yield on light energy in grams per moles of photons. The fourth equation is in fact algebraic but is expressed as a differential equation. The integration of the mass balance model must be done with an integrator able to solve DAEs (differential-algebraic equations) such as ode15s or ode23t in Matlab [39].

Parameter | Value | Unit |
---|---|---|

${E}_{a}$ | 200 | m^{2}·kg^{−1} |

${E}_{s}$ | 870 | m^{2}·kg^{−1} |

$b$ | 0.0008 | - |

${\mu}_{o}$ | 0.17 | h^{−1} |

${K}_{S}$ | 135 | μmol·m^{−2}·s^{−1} |

${K}_{I}$ | 2500 | μmol·m^{−2}·s^{−1} |

${\mu}_{d}$ | 0.01 | h^{−1} |

$V$ | 5.74 × 10^{−3} | m^{3} |

$A$ | 10.2 × 10^{−2} | m^{2} |

$L$ | 0.054 | m |

## 3. The Optimal Control Problem

^{−2}·s

^{−1}. To calculate the amount of light consumed, $Q$, the model requires the incident light intensity, ${I}_{0}$, the lighted surface of the PBR, and the current process time. By dividing the newly produced biomass by the amount of light consumed, the biomass yield on light energy can be easily calculated ($Y={X}_{T}/Q$).

^{−2}·s

^{−1}, it can be found that the biomass yield on light energy, $Y$, is a unimodal unidimensional function whose maximum gives the optimal incident light intensity, ${I}_{0\left(opt\right)}$ (Figure 4). The optimization of $Y$ can be done with unidimensional optimization methods [41] such as the golden section search, the parabolic interpolation, etc. The constraints to which the input variable ${I}_{0}$ is subjected have been selected from a technological point of view, 1 μmol·m

^{−2}·s

^{−1}to avoid division by zero (Equation (7)), and 2000 μmol·m

^{−2}·s

^{−1}is the maximal sun radiation on a summer day.

^{−1}, ${X}_{T}\left(0\right)=4.4690$ g (not displayed in Figure 4), $Q\left(0\right)=9.18$ g·mol photon

^{−1}, which means that $Y\left(0\right)={X}_{T}\left(0\right)/Q\left(0\right)=0.4868$.

- -
- a lower bound, ${I}_{0\left(min\right)}$, under which the biomass would decrease. Below ${I}_{0\left(min\right)}$ the specific growth rate is negative (Figure 4), and
- -
- an upper bound, ${I}_{0\left(max\right)}$, which is set in Figure 4 at 80% of the maximum growth rate.

- -
- the control horizon: ${t}_{0}\le t\le {t}_{f}$, where ${t}_{0}$ and ${t}_{f}$ are the initial and the final time of the control horizon (the batch period),
- -
- the initial conditions: $x\left({t}_{0}\right)={x}_{0}$,
- -
- the set of lower and upper bounds: ${I}_{0\left(min\right)}\le {I}_{0}\left(t\right)\le {I}_{0\left(max\right)}$, with ${t}_{0}\le t\le {t}_{f}$. The lower bound, ${I}_{0\left(min\right)}$, is critical because the biomass decreases under this value, while the upper bound is not, and can remain constant (e.g., 2000 μmol·m
^{−2}·s^{−1}). Even though ${I}_{0\left(max\right)}$ can inhibit the microalgae growth, it is attenuated inside the culture. The lower and the upper bounds are required by most of the unidimensional optimization methods [41], e.g., fminbnd function in Matlab.

^{−2}·s

^{−1}. As has been the case previously (Figure 4), the results are the final values of the states, after the integration of the model for one hour with each ${I}_{0}\in \left[1,2000\right]$ μmol·m

^{−2}·s

^{−1}. Optimization on three horizons is presented, 1, 84, and 168 h, taking a seven-day cultivation process as a reference. The initial condition for biomass is $X\left(0\right)=0.3$ g·L

^{−1}, while the other states have been set to zero ($Q\left(0\right)$ should be set to a very small positive value to avoid division by zero). It can be observed that the optimal incident light intensity is similar, regardless of the length of the optimization horizon. Comparing Figure 4 with 5, it can be observed that higher concentrations of biomass require higher concentrations of light to reach an optimal yield, $X\left(0\right)=0.3$ g·L

^{−1}requires ≈ 130 μmol·m

^{−2}·s

^{−1}while $X\left(0\right)=1.1386$ g·L

^{−1}requires ≈ 331 μmol·m

^{−2}·s

^{−1}. This points out the need for an increase in the incident light intensity as the biomass grows. Working with constant light throughout the photosynthetic growth process could require more light energy to obtain the same amount of biomass compared to an increasing light profile. This problem has been addressed in [32] by controlling the light-to-microalgae ratio, but the present paper takes it a step further by investigating the optimization of a lumostatic batch (i.e., a batch with variable incident light intensity).

## 4. The Closed-Loop Control System

#### 4.1. The Control Structure

^{−2}·s

^{−1}. The current lower and upper bounds, ${I}_{0\left(min\right)}^{k}\le {I}_{0\left(opt\right)}^{k}\le {I}_{0\left(max\right)}^{k}$, are calculated with the process model before the optimization procedure.

#### 4.2. Simulation Results

^{−1}, while the other three states have been set to zero (same as for Figure 5).

^{−1}. Further increasing the sampling period results in significantly lower concentrations of biomass, as can be seen in Table 2. A constant incident light intensity of 133.17 μmol·m

^{−2}·s

^{−1}throughout the entire batch (see also Figure 5) results in a biomass concentration 35.48% lower compared with the 1-h sampling period.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The normalized irradiance attenuation with the depth of the culture, at an incident light intensity of 500 μmol·m

^{−2}·s

^{−1}.

**Figure 2.**(

**A**). Front view of an airlift PBR with a LED growth light panel and their main components [37]. (

**B**). The lateral view of the PRB showing the attenuation of the light inside the microalgae culture.

**Figure 3.**The decrease of the specific growth rate with the depth of the culture, at an incident light intensity of 500 μmol·m

^{−2}·s

^{−1}.

**Figure 4.**The response of the system for ${I}_{0}\in \left[1,2000\right]$ μmol·m

^{−2}·s

^{−1}on a one-hour horizon.

**Figure 5.**The responses of the system for ${I}_{0}\in \left[1,2000\right]$ μmol·m

^{−2}·s

^{−1}, with three different optimization horizons.

**Figure 7.**Simulation results on a seven-day batch with different optimization horizons (1-h horizon—blue, 20-h horizon—red and 50-h horizon—yellow).

**Figure 8.**Simulation results on a 7-day batch with different sampling periods (1-h sampling period—blue and 12-h sampling period—red).

**Figure 9.**Simulation results on a seven-day batch with different weighing factors (see Equation (15)). With blue $w=1$, with red $w=0.7$ and with yellow $w=0.4$.

Length of the Sampling Period | Biomass Concentration [g·L^{−1}] | Decrease [%] |
---|---|---|

1-h | 1.460 | |

12-h | 1.418 | −2.88 |

24-h | 1.374 | −5.89 |

168-h | 0.942 | −35.48 |

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**MDPI and ACS Style**

Ifrim, G.A.; Titica, M.; Horincar, G.; Antache, A.; Baicu, L.; Barbu, M.; Guzmán, J.L.
Model Based Optimal Control of the Photosynthetic Growth of Microalgae in a Batch Photobioreactor. *Energies* **2022**, *15*, 6535.
https://doi.org/10.3390/en15186535

**AMA Style**

Ifrim GA, Titica M, Horincar G, Antache A, Baicu L, Barbu M, Guzmán JL.
Model Based Optimal Control of the Photosynthetic Growth of Microalgae in a Batch Photobioreactor. *Energies*. 2022; 15(18):6535.
https://doi.org/10.3390/en15186535

**Chicago/Turabian Style**

Ifrim, George Adrian, Mariana Titica, Georgiana Horincar, Alina Antache, Laurențiu Baicu, Marian Barbu, and José Luis Guzmán.
2022. "Model Based Optimal Control of the Photosynthetic Growth of Microalgae in a Batch Photobioreactor" *Energies* 15, no. 18: 6535.
https://doi.org/10.3390/en15186535