# Robust Control Based on Modeling Error Compensation of Microalgae Anaerobic Digestion

^{1}

^{2}

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^{*}

## Abstract

**:**

_{2}sequestration tasks. In the case of anaerobic digestion of microalgae, biogas can be produced from mainly proteins and carbohydrates. Anaerobic digestion is a complex process that involves several stages and is susceptible to operational instability due to various factors. Robust controllers with simple structure and design are necessary for practical implementation purposes and to achieve a proper process operation despite process variabilities, uncertainties, and complex interactions. This paper presents the application of a control design based on the modeling error compensation technique for the anaerobic digestion of microalgae. The control design departs from a low-order input–output model by enhancement with uncertainty estimation. The results show that achieving desired organic pollution levels and methanogenic biomass concentrations as well as minimizing the effect of external perturbations on a benchmark case study of the anaerobic digestion of microalgae is possible with the proposed control design.

## 1. Introduction

_{2}) mitigation [6].

_{2}in AD can be used to grow photoheterotrophic microalgae populations in conjunction with either artificial or solar light as a source of energy [23].

_{2}sequestration applications [33,34,35,36]. Numerical simulations show that the proposed controller may achieve robust regulation of the organic pollution level and the methanogenic biomass concentration. Thus, our paper’s main contribution is the introduction of a robust and practical controller for the continuous AD of microalgae that achieves good closed-loop performance despite model uncertainties and external perturbations.

## 2. Materials and Methods

#### 2.1. AD of Microalgae

#### 2.2. MAD Model

_{1}(sugar and lipids excluding nitrogen), S

_{2}(proteins), and S

_{I}(inert substrate). (ii) S

_{1}and S

_{2}are degraded to VFAs (S

_{3}) via hydrolysis–acidogenesis–acetogenesis by the bacterial populations X

_{1}and X

_{2}, respectively. (iii) S

_{3}is converted to methane by methanogenic population X

_{3}. (iv) The specific growth rates for the hydrolysis–acidogenesis–acetogenesis reactions are modeled as Contois functions. (v) The methanogenesis-specific growth rate is modeled with a Haldane function with a multiplicative ammonia inhibition term. (vi) pH values for the AD of C. vulgaris are in the range of 6.0 < pH < 7.5. (vii) The total inorganic carbon concentration (C) is the sum of the dissolved carbon dioxide concentration CO

_{2}and the bicarbonate concentration HCO

_{3}. (viii) The total inorganic nitrogen (N) is the sum of free ammonium and ammonium ions. Ammonium ions are consumed in the hydrolysis–acidogenesis–acetogenesis of S

_{1}and the methanogenesis of VFAs. (ix) The anaerobic digester is a 1-L (V

_{liq}) continuous perfectly stirred reactor with 0.1-L headspace (V

_{gas}). (x) The microalgae fed is composed by fractions of sugars-lipids β

_{1}, proteins β

_{2}, and inerts β

_{I}. (xi) The operation temperature (T

_{op}) is maintained constant. (xii) All of the produced methane is transferred to the headspace.

_{i}are the stoichiometric parameters (i = 1,..., 12). S

_{in}, N

_{in}, C

_{in}, and z

_{in}are the input concentrations of organic matter, inorganic nitrogen, inorganic carbon, and alkalinity, respectively. P

_{CO2}and P

_{CH4}are the partial pressures of CO

_{2}and CH

_{4}, and ρ

_{CO2}and ρ

_{CH4}are their liquid–gas transfer rates.

_{v}as the pipe resistance coefficient and P as the atmospheric pressure.

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}, x

_{8}, x

_{9}, x

_{10}, x

_{11}, x

_{12}]

^{T}= [S

_{1}, S

_{2}, S

_{I}, S

_{3}, X

_{1}, X

_{2}, X

_{3}, N, C, P

_{CO}

_{2}, P

_{CH}

_{4}, z]

^{T}. The control input is the dilution rate, i.e., u = D. The nominal operation is simulated with the following parameter values [28]: S

_{in}= 29.5 gCOD/L, β

_{1}= 0.3, β

_{2}= 0.4, β

_{I}= 0.3, μ

_{1,max}= 0.3 d

^{−1}, ks

_{1}= 2.11 gCOD/L, μ

_{2,max}= 0.053 d

^{−1}, ks

_{2}= 0.056 gCOD/L, μ

_{3,max}= 0.14 d

^{−1}, ks

_{3}= 0.02 gCOD/L, k

_{I3}= 16.4 gCOD/L, k

_{N}= 1.1 × 10

^{−9}, k

_{INH3}= 1.1 × 10

^{−9}, α

_{1}= 12.5, α

_{2}= 0.0062, α

_{3}= 11.5, α

_{4}= 0.03, α

_{5}= 9.1, α

_{6}= 8.1, α

_{7}= 0.054, α

_{8}= 0.03, α

_{9}= 20, α

_{10}= 0.062, α

_{11}= 0.3, α

_{12}= 0.2, k

_{L}a = 5 d

^{−1}, k

_{v}= 5e4 L/d bar, R = 8.314 × 10

^{−2}bar/M K, T

_{op}= 308.15 K, pH = 7, P

_{atm}= 1.01325 bar, V

_{liq}= 1 L, V

_{gas}= 0.1 L, N

_{in}= 0.011 M, C

_{in}= 0.019, z

_{in}= 0.017.

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}, x

_{8}, x

_{9}, x

_{10}, x

_{11}, x

_{12}]* = [ 0.289, 1.097, 8.85, 0.065, 0.6848, 1.176, 0.8668, 0.0648, 0.0748, 0.405, 0.608, 0.017] is obtained with the above parameter values, and a dilution value base of u = 0.05 d

^{−1}. It can be noted that the nominal biomass productivity (i.e., u∙x

_{6}) is around 0.0433 gCOD L

^{−1}d

^{−1}. Figure 1 shows the effect of a step change on the dilution rate at t = 500 d. It is noted that a smooth response is obtained for both methanogenic biomass and the organic pollution level, defined as the sum of the microalgae organic components S

_{1}and S

_{2}and the S

_{3}produced in the microalgae AD. On the other hand, an initial inverse response is observed for both the VFA concentration and the biogas flow.

#### 2.3. Robust Control Design Based on MEC

_{p}and τ

_{0}are the steady-state process gain and the process time constant. The corresponding first-order input–output model in the time domain is enhanced with lumped model uncertainties η(t), including structural uncertainties with bounded variation due to the model reduction ξ(y(t)), and constant or persistent external perturbations π(t), i.e.,

_{e}is an observer parameter, denoted as the estimation time constant, that modulates the convergence rate of the estimation of the real uncertain term. After algebraic manipulations, the reduced observer can be written as

_{ref}(t) is the desired set-point, e(t) = y(t) − y

_{ref}(t), is the regulation or tracking error, and τ

_{c}is a controller parameter, denoted as the closed-time constant, that modulates the closed-loop convergence rate to the desired set-point. Based on the process time constant, τ

_{0}, tuning of parameters τ

_{c}and τ

_{e}follows the rule [31,32,47]: 0 < τ

_{e}< τ

_{c}< τ

_{0}.

**Assumption 1.**The dilution rate is the control input, i.e., u = D.

**Assumption 2.**The control input is subjected to a saturation nonlinearity, i.e., u

_{min}≤ u ≤ u

_{max}.

**Assumption 3.**The controlled variable is available for control design purposes.

- A.
- For optimization and control purposes of AD processes, usually in practice, only a relatively limited number of control actions are possible. These are mostly restricted to the input flow rate or the input of a particular substrate in the feed [19,20]. Therefore, this paper selects the dilution rate (directly related to the input flow rate) as the control input variable.
- B.
- The minimum and maximum control inputs are selected following previous studies on the operational behavior of the MAD model [30]. The maximum input flow rate must be chosen to prevent the washout condition.

## 3. Results

_{ref}change at t = 1000 d and an external perturbation of +20% on the input substrate feed S

_{in}at t = 1500 d.

#### 3.1. Control of the Organic Pollution Level

#### 3.1.1. Control Problem

_{1}, S

_{2}, and the produced VFAs S

_{3}.

#### 3.1.2. Numerical Results

_{p}= 515 and τ

_{0}= 75.

^{−1}, which leads to a steady-state value of 2.16 gCOD/L and a methanogenic biomass concentration of 0.86 gCOD/L. (ii) Once the controller is activated at t = 500 d, to achieve the desired reference of the organic pollution level of 2 gCOD/L, a slight decrease in the dilution rate to 0.0495 d

^{−1}is performed to increase the degradation of the organic pollution level via an increase in the HRT. (iii) At t = 1000 d, the dilution rate is further decreased to around 0.0457 d

^{−1}to achieve the desired reference of 1.5 gCOD/L. (iv) Finally, at t = 1500 d, when the disturbance in the substrate input occurs, dilution is also decreased because high HRT is required to degrade the increase of the input substrate. It is also noted that the methanogenic biomass concentration significantly increases due to the additional substrate input.

#### 3.2. Control of the Methanogenic Biomass Concentration

#### 3.2.1. Control Problem

#### 3.2.2. Numerical Results

_{p}= −22.72 and τ

_{0}= 75, which are computed based on the input–output response shown in Figure 1. Figure 4 shows the closed-loop performance of the MEC controller for three sets of controller parameters, as well as the comparison against a conventional PI controller tuned with IMC rules.

^{−1}, to allow the increase of the methanogenic biomass concentration from the nominal value of 0.866 gCOD/L to 0.9 gCOD/L. (ii) The second set-point associated with a lower methanogenic biomass concentration is achieved with an increase in the dilution rate to 0.0518 d

^{−1}. (iii) Finally, when the disturbance in the substrate input occurs, dilution is slightly increased because less HRT is required to maintain the same desired microalgae biomass concentration.

## 4. Discussion

- Dilution and S
_{in}values: The AD operation is markedly influenced by the dilution rate and the substrate feed [19,20]. The observed values in the base and controlled process numerical simulation correspond to the region of lower organic pollution level and higher methanogenic biomass concentration according to the in-depth study presented by Khedim et al. [30] for the selected parameter values. As the substrate input feed increases and the dilution rate decreases, an increase in the methanogenic biomass concentration can achieve, as is shown in Figure 3 when the external perturbation is applied. Khedim et al. [30] suggest that the optimum yield of the MAD model in terms of biogas production was obtained for the following ranges [0.001–0.05] d^{−1}, [0.03–30] gCOD/L of D, and S_{in}, respectively. Low dilution rates correspond to high HRT, allowing the active biomass population to remain in the reactor and not limiting the hydrolysis step. - MEC closed-loop performance: The numerical results of the proposed controller on the benchmark MAD model demonstrate the capabilities and versatility of the MEC control approach when controlling the complex operation of AD. It is also noted that only two papers have addressed control designs for the AD of microalgae for the organic pollution level control problem [22,43]. In both cases, considering a possible error in that references in the time units (from hours to days) and sight differences between the values of some variables, the magnitude of the computed dilution rate is similar to the numerical assessment of two proposed nonlinear controllers based on feedback linearization and robust adaptive controllers. However, since both contributions include state and uncertain kinetic estimators, a fair comparison is not possible.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Base numerical simulations of two main MAD model variables (VFA and X

_{3}concentrations in gCOD/L), biogas flow (L/d), and organic pollution level (gCOD/L), including the effect of a step change in the dilution rate at t = 500 d.

**Figure 2.**Effect of the dilution on the organic pollution level and methanogenic biomass concentrations.

**Figure 4.**Closed-loop performance of the MEC controller for the methanogenic biomass concentration control problem.

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**MDPI and ACS Style**

Rodríguez-Jara, M.; Velasco-Pérez, A.; Vian, J.; Vigueras-Carmona, S.E.; Puebla, H.
Robust Control Based on Modeling Error Compensation of Microalgae Anaerobic Digestion. *Fermentation* **2023**, *9*, 34.
https://doi.org/10.3390/fermentation9010034

**AMA Style**

Rodríguez-Jara M, Velasco-Pérez A, Vian J, Vigueras-Carmona SE, Puebla H.
Robust Control Based on Modeling Error Compensation of Microalgae Anaerobic Digestion. *Fermentation*. 2023; 9(1):34.
https://doi.org/10.3390/fermentation9010034

**Chicago/Turabian Style**

Rodríguez-Jara, Mariana, Alejandra Velasco-Pérez, Jose Vian, Sergio E. Vigueras-Carmona, and Héctor Puebla.
2023. "Robust Control Based on Modeling Error Compensation of Microalgae Anaerobic Digestion" *Fermentation* 9, no. 1: 34.
https://doi.org/10.3390/fermentation9010034