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Optimization-based control strategy for a large-scale polyhydroxyalkanoates production in a fed-batch bioreactor using a coupled PDE–ODE system

  • Atthasit Tawai EMAIL logo , Malinee Sriariyanun and Chanin Panjapornpon
Published/Copyright: March 7, 2023
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Green Processing and Synthesis
From the journal Green Processing and Synthesis Volume 12 Issue 1

Abstract

Control strategy development for fed-batch bioreactor (FBBR) plays an important role in the improvement of polyhydroxyalkanoate (PHA) production. To develop a feeding strategy for PHA production in a large-scale FBBR, an optimization-based control scheme that considers nutrient dispersion is proposed in this work. A coupled partial differential equations and ordinary differential equation model is proposed to describe the axial-dispersed nutrient and well-dispersed microbial dynamics with process constraints. An analytical model predictive control (AMPC) method that applies integrated variables of nutrients is employed to develop the real-time control system. The control objective is to regulate the PHA concentration at the updated set points by adjusting the nutrient feed rates; a process disturbance is introduced to evaluate the control robustness. Simulation experiments of a fed-batch operation are conducted to investigate the performance of the developed controller; the controlled output is designed to track the updated set points corresponding to the biomass concentration. Results of closed-loop and regulatory systems showed that the proposed control strategy could provide more productivity (33–38%) compared to the applied PI controller. The performance test demonstrates that the developed control system could apply the biomass concentration for updating set points, provide the optimal control actions that promote PHB accumulation and handle the disturbance effectively.

Keywords: polyhydroxyalkanoates production; fed-batch bioreactor; coupled PDE–ODE model; analytical model predictive control; nonlinear optimization-based control

Nomenclature

A, B

constant matrices

D

differential operator

F N

feed rate of nitrogen source (L·h−1)

F S

feed rate of substrate (L·h−1)

h

nonlinear functions

H

medium height (cm)

k 1, k 2, k 3, k 4, k 5

kinetic constants

K sr

saturation constant

N

nitrogen concentration (g·L−1)

N f

inlet concentration of nitrogen source (g·L−1)

n k

dimensionless exponent

Pe

Peclet number

S

substrate concentration (g·L−1)

S f

inlet concentration of substrate (g·L−1)

S m

carbon/nitrogen ratio for μ = 0

t

time (h)

u

vector of manipulated input

V

working volume (L)

W N, W C

weighting factors

X

total biomass concentration (g·L−1)

y

output variables

y sp

output set points

z

spatial coordinates

Greek letters

ε

tuning parameter

η

error dynamic parameter

λ

tuning parameter

μ m

maximum growth rate (h−1)

v o ( k )

compensated output set point

ξ p

state variables depending on spatial coordinate and time

ξ o

state variables depending time

ξ ˜ p

lumped variable of PDE subsystem

τ

mean residence time (h)

Subscript

lb

lower bound

sp

set point

ub

upper bound

Superscript

–

measured variable

∼

lumped variable

^

estimated state

1 Introduction

Polyhydroxyalkanoates (PHAs) are intracellular-storage biopolymers synthesized by a wide range of microorganisms, which present biodegradable thermoplastic properties. The biopolymers in the family of PHAs, such as polyhydroxybutyrate (PHB) and polyhydroxyvalerate (PHV), present many beneficial properties for such applications as medical uses and food packaging [1,2,3,4]. Although biodegradable polymers have been considered as a potential solution to handle the plastic waste problem, the large-scale production of biopolymers is limited because of the high production cost – which is about 4–10 times higher than that of petrochemical-based polymers [5]. PHAs are energy-storage polymers of microbial cells; PHA accumulation can be promoted under unfavorable growth conditions [6,7]. Since cell growth and polymer accumulation involve influencing factors, the process design of optimal operation for the effective production of PHAs is one of the major challenges.

The process operation of fed-batch fermentation is determined by the mechanism of PHA synthesis. There are two main stages for PHA production: microbial growth with adequate nutrients and PHA accumulation with nutrient limitation [8,9]. Because cell growth and polymer production could be inhibited by a high substrate concentration, fed-batch fermentation has been applied to improve PHA productivity [10]. Literature works have proposed prescribed feeding policies to enhance PHA production in a fed-batch bioreactor (FBBR) [11,12,13]. The feeding rates of limiting nutrients were optimized and implemented to improve the PHB productivity in fed-batch cultivation [11]. This work indicated that model-based optimization could be particularly suitable for the provision of complicated limiting nutrient conditions for PHB production. Khanna and Srivastava developed a mathematical model to optimize an FBBR; two different nutrient-feeding strategies are selected offline and are conducted for a fed-batch operation [12]. The experimental results showed that the fed-batch operation could increase the PHB productivity by 2.6 times compared to the batch. The developed policies were based on experimental implementations; the obtained results demonstrated a considerable increase in PHA productivity with regard to other previous works. To contemplate the uncertain kinetics and unknown perturbations that are inevitable in online operations, process models have been developed and applied in model-based fed-batch cultivation [14,15,16]. However, the mathematical models developed by assuming fully dispersed broth in FBBR may be absent in the effect of dispersion in large-scale production. A dispersion model of a large fed-batch bioreactor was proposed for PHB production by Ralstonia eutropha [17]. In this study, experiments were conducted to validate the developed kinetic model, and the results showed the potential for improving the PHB production in large-scale bioreactors by considering the dispersion effect.

During the last decades, some control strategies have been proposed to improve the control performance and closed-loop robustness of FBBR for PHA production [17,18,19,20,21]. A dynamic optimization technique was applied to develop optimal feed rate profiles of carbon and nitrogen sources that can improve PHB productivity in a fed-batch bioreactor [18]. The adaptive control schemes that applied the constraint discrete-time optimal control algorithm were developed for PHB production [19]. Recently, there have been control strategies proposed based on input/output (I/O) linearization and real-time optimization to develop advanced feeding strategies for fed-batch fermentation [21,22]. Srisuddee et al. proposed a control method based on the I/O linearization technique for a fed-batch bioreactor described by an ODE model [20]. Ochoa et al. applied a soft-sensor prediction block based on previous work to develop the real-time optimization and control for a fed-batch production of PHAs in a large-scale bioreactor [21,22].

However, in large-scale production, the feed substrate and nitrogen source with high concentrations in a fed-batch bioreactor are placed on the upper surface of the broth while the microbial cells are well dispersed in the agitated broth. During the process operation, the changing of substrate and nitrogen concentration in the broth affects the carbon-to-nitrogen (C/N) ratio that could promote the cell and polymer productions. Though some studies have shown the effects of nutrient dispersion on the PHA production in large-scale bioreactors, most of the control strategies have been developed based on fully dispersed behavior. Therefore, the performance of traditional controllers that assume full dispersion of nutrients may be limited by the complex interaction between the dispersed nutrients and microbial cells [23,24,25,26]. Thus, this work aims to develop an alternative optimization-based control strategy that considers the axial dispersion of the substrate and nitrogen source coupled with the well dispersion of biomass. The control objective is to control the PHA concentration at the updated set points by adjusting the substrate and nitrogen feed rates; a process disturbance is introduced to evaluate the control robustness. To investigate closed-loop responses, the dynamics of substrate and nitrogen-source concentrations are considered as distributed variables and can be described by PDEs, while that of the well-dispersed biomass and PHA concentrations are modeled by ODEs. The control strategy is based on the analytical model predictive control (AMPC) method that applies a coupled PDEs–ODEs model for a fed-batch bioreactor. The AMPC method has presented the advantage of optimal manipulation to prevent an aggressive control action for the continuous reactor system described by PDEs–ODEs model in some previous studies [27]. For the fed-batch reactor, the optimal control action could promote polymer production and avoid the effect of substrate inhibition that reduces microbial growth simultaneously. A large-scale fed-batch bioreactor for PHB production is considered a case study in this work. The developed AMPC-based controller is combined with a nonlinear state estimator and integrator to estimate the unmeasured variables and handle process disturbances. The main contributions of this work, beyond the novelty of the problem formulation, lie in integrating the dispersion effects of concentrated substrates of the controller, compensating unknown perturbations that are inevitable in the real-time operation, and providing control actions that optimize the nutrient feed rates to enhance the biomass and PHA production.

2 Materials and methods

2.1 Problem formulation

Consider the system of a mathematical model for a fed-batch bioreactor used for large-scale polyhydroxyalkanoates production in Eq. 1. Dynamics of the state variables are described by a coupled parabolic partial differential equations and ordinary differential equation (PDE–ODE) system as follows:

(1) ∂ ξ p ( z , t ) ∂ t = A ∂ 2 ξ p ( z , t ) ∂ z 2 + B ∂ ξ p ( z , t ) ∂ z + F p ( ξ p ( z , t ) , ξ o ( t ) ) ∂ ξ o ( t ) ∂ t = F o ( ξ o ( t ) , ξ ˜ p ( t ) , u ( t ) ) y = h ( ξ p ( z , t ) , ξ o ( t ) ) ∣ z = H

where

ξ ˜ p ( t ) = ∫ 0 H ξ p ( z , t ) d z

with the following initial and boundary conditions:

ξ p ( z , 0 ) = ξ 0 p , ξ o ( 0 ) = ξ 0 o

z = 0 : ∂ ξ p ( z , t ) ∂ z = 0

z = H : ∂ ξ p ( z , t ) ∂ z = 0

where ξ p ( z , t ) denotes the vector of states depending on the spatial coordinate (axial liquid level) and time, ξ o ( t ) denotes the vector of time-dependent state variables, ξ ˜ p ( t ) denotes the integrated state of ξ p , u ( t ) denotes the vector of manipulated variables, y denotes the vectors of the output variables, F p , F o and h are nonlinear functions, z ∈ [ 0 , H ] is the spatial coordinate, t ∈ [ 0 , t f ] is the time, and A and B are matrices.

2.2 Control system design

Considering the polymer production in the fed-batch bioreactor, the agitated cells with intracellular PHAs consume the dispersed substrate and nitrogen source in the broth. To illustrate the effects of nutrient dispersion in the fed-batch bioreactor, an open-loop simulation of PHA production is conducted. A set of constant nutrient feed rates is applied; the substrate and nitrogen concentrations are considered distributed variables. The open-loop profile of substrate concentration along the axial distance (dispersed broths) for 5, 10, 15, and 20 h is shown in Figure 1. More details of the mathematical model and operating conditions are given in the next sections.

Figure 1 Spatial dispersion of substrate concentration.
Figure 1

Spatial dispersion of substrate concentration.

The concentrated substrate is fed to the bioreactor and is consumed by the microbial cells. The figure illustrates that a large-scale fed-batch operation intensifies the lack of homogeneity of the broths; the agitated cells at different positions in the bioreactor are affected by nutrient dispersion [17]. For the coupled PDE–ODE system, the dynamic behaviors of the states ξ p and ξ o are bidirectional interconnection. Consider a compact form of the system in Eq. 1:

(2) ξ ̇ o = F o ( ξ o , ξ ˜ p , u ) y o = ξ o

where u is the vector of the manipulated inputs. An analytical model predictive control strategy based on I/O linearization, which has been successfully applied to continuous reactor systems, is modified to formulate the optimization problem for the fed-batch bioreactor in this work [27,28,29,30]. The nonlinear optimization is established to provide the optimal control actions by minimizing the squared error between the existing state and the updated target in each time step. The constraint optimization problem can be formulated as the following equation; then, it is solved for the manipulated input at each time step ( t ∈ [ t 0 , t f ] ):

(3) min u ( ε D + 1 ) r y o − v o ε 2

subject to

ξ ̇ o = F o ( ξ o , ξ ˜ p , u ) u lb ≤ u ≤ u ub v lb o ≤ v o ≤ v ub o

where D is the differential operator (i.e., D = d d t ), ε is the tuning parameter that is constant and adjustable. y o is the vector of outputs, u lb and u ub denote the lower bound and upper bound of the manipulated input u , respectively, v o is the reference output setpoint obtained from the real-time calculation at the current time. The relative order (r) of the controlled outputs y o with respect to manipulated input u of the dynamic behaviors of the states ξ o is investigated for the smallest integer that provides ∂ [ d r y o / d t r ] / ∂ u ≠ 0 [27]. To design the optimization-based control scheme for the system in Eq. 2, the differential operator is applied to the controlled output y o , of which a relative order number is equal to 1. Then, the optimization problem in Eq. 3 can be rewritten as the following equation:

(4) min u ε F o ( ξ o , ξ ˜ p , u ) + y o − v o ε 2

subject to:

ξ ̇ o = F o ( ξ o , ξ ˜ p , u ) u lb ≤ u ≤ u ub v lb o ≤ v o ≤ v ub o

The solutions from the optimization problem are obtained by applying the distributed process model, and the cell growth and polymer accumulation can be employed to provide the optimal control actions for the fed-batch bioreactor.

2.3 Mathematical model of a fed-batch bioreactor for PHB production

Among the family of PHAs, PHB is an important biodegradable polymer that presents beneficial properties and can be synthesized by various microorganisms. With the mechanical properties (e.g., tensile strength, flexural modulus) that are similar to polypropylene, the PHB market is predicted to reach $98 million by 2024 [31]. In this work, a fed-batch bioreactor of PHB production is considered a case study (see Figure 2 for a schematic diagram of the fed-batch bioreactor).

Figure 2 Schematic diagram of a typically PHB fed-batch bioreactor.
Figure 2

Schematic diagram of a typically PHB fed-batch bioreactor.

In this example, the fermentation is operated in the fed-batch bioreactor for 30 h. The prepared inoculum (Cupriavidus necator) is supplied into a 500 L bioreactor. The culture medium is composed of 75 vol% of sugarcane vinasses and 25 vol% of molasses [22,32]. The process is initially performed in batch operation for 2 h; microbial cells are well dispersed in the agitated medium. Then, the process is performed in a fed-batch operation, where the substrate (molasses) and nitrogen source are fed to the bioreactor to manipulate the PHB production. In recent years, some research articles have proposed the online monitoring of substrate, biomass, and PHB by using sensors, gas chromatography, infrared spectroscopy, or estimated by a reliable measurement [19,21,33,34]. Thus, the substrate, biomass and PHB concentrations, and working volume are assumed to be measurable in real-time during the operation. However, the nitrogen concentration cannot be measured online [22]; therefore, a nonlinear state estimation is applied in this work. A nonlinear first-principles model for the fed-batch bioreactor is developed, with additional simplified assumptions considered in the one-dimensional PHB bioreactor model:

  1. The biomass and PHB concentrations in the bioreactor are assumed to a follow fully dispersed behavior.

  2. The dispersion of substrate and nitrogen sources is considered to be one-dimensional.

  3. The microorganisms consume substrate and nitrogen sources to produce biomass and PHB. By-product production is neglected.

  4. The temperature, pH, and dissolved oxygen of the bioreactor are effectively regulated in a range [22]; the impact of these factors on microbial growth can be ignored.

  5. The substrate concentration, total biomass concentration, PHB concentration, and working volume are measurable in real time during the operation.

The governing equations of the fed-batch bioreactor for PHB production are composed of dynamic behaviors of the substrate concentration, nitrogen concentration, biomass concentration, PHB concentration, and working volume.

2.3.1 Mass balance of the distributed variables

The process model for fed-batch fermentation was reformulated in previous work to include dispersion terms of nutrients with Pe = 20 [25,26]. The previous studies showed a significant effect of axial dispersion on the kinetics of PHB formation; more details and equations are available in the literature. The conservation of substrate and nitrogen sources is distributed along the liquid level of the fed-batch bioreactor. The dynamics of substrate and nitrogen concentrations can be described as follows:

(5) τ ∂ S ( z , t ) ∂ t = 1 Pe ∂ 2 S ( z , t ) ∂ z 2 − ∂ S ( z , t ) ∂ z − τ S f F S V ( t ) − k 1 d X ( t ) d t − k 2 X ( t ) − S ( z , t ) V d V ( t ) d t τ ∂ N ( z , t ) ∂ t = 1 Pe ∂ 2 N ( z , t ) ∂ z 2 − ∂ N ( z , t ) ∂ z − τ N f F N V ( t ) − k 3 d X ( t ) d t − N ( z , t ) V d V ( t ) d t

with the boundary and initial conditions:

z = 0 : ∂ S ( z , t ) ∂ z = 0 ∂ N ( z , t ) ∂ z = 0

z = H : ∂ S ( z , t ) ∂ z = 0 ∂ N ( z , t ) ∂ z = 0

t = 0 : S ( z , 0 ) = S 0 N ( z , 0 ) = N 0

where S(z, t) denotes substrate concentration, N(z, t) denotes nitrogen concentration, X(t) denotes total biomass concentration, V(t) denotes the working volume, τ denotes the mean residence time, z ∈ [ 0 , H ] denotes spatial distance (dispersed broths), Pe denotes the Peclet number, k 1, k 2, and k 3 denote kinetic constants, S f and N f denote the inlet concentration of substrate and nitrogen source, and F S and F N denote the feed rate of substrate and nitrogen source, respectively.

2.3.2 Mass balance of the lumped variables

In general, the process is initially operated in batch operation, and the inoculated cells are well dispersed in the agitated medium. After the process is performed in fed-batch operation, concentrated substrate and nitrogen source are fed to the bioreactor, which results in a complex interaction between the dispersed nutrients and microorganisms. The dynamics of the total biomass concentration, accumulated PHB concentration, and working volume can be written as follows:

(6) d X ( t ) d t = μ m N ˜ ( t ) S ˜ ( t ) N ˜ ( t ) S ˜ ( t ) + K sr 1 − N ˜ ( t ) S ˜ ( t ) S m n k X ( t ) − X ( t ) V ( t ) d V ( t ) d t d P ( t ) d t = k 4 d X ( t ) d t + k 5 X ( t ) − P ( t ) V ( t ) d V ( t ) d t d V ( t ) d t = F S ( t ) + F N ( t )

where

S ˜ ( t ) = ∫ 0 H S ( z , t ) d z and N ˜ ( t ) = ∫ 0 H N ( z , t ) d z

with the initial conditions:

t = 0 : X ( 0 ) = X 0 P ( 0 ) = P 0 V ( 0 ) = V 0

where μ m denotes the maximum growth rate, S ˜ ( t ) and N ˜ ( t ) are integrated states of substrate and nitrogen-source concentration, K sr denotes saturation constant, S m denotes carbon/nitrogen ratio for μ = 0, n k denotes dimensionless exponent, and k 4 and k 5 are kinetic constants. The process parameter values are presented in Table 1 [22,26].

Table 1

Parameters of the fed-batch bioreactor for PHB production

Symbol Parameter Value
μ m Axial dispersion coefficient 0.437 h−1
S m Carbon/nitrogen ratio for μ = 0 0.3 g N/g S
K sr Saturation constant 0.0697
n k Dimensionless exponent 1.490
Pe Peclet number 20
k 1 Kinetic constant 1.831
k 2 Kinetic constant 0.067
k 3 Kinetic constant 0.6511
k 4 Kinetic constant 0.300
k 5 Kinetic constant 0.008

2.4 Formulation of the control system

Since the growth and PHB accumulation of microbial cells depending on the dispersed carbon and nitrogen source, the control objective of this example process is to regulate the PHB concentration (P) at the updated set points by manipulating the feed rates of the nitrogen source ( F N ) and the substrate ( F S ) . The application of process control requires the online state estimator for unmeasurable states. In this work, the integrated state of nitrogen concentration ( N ˜ ) can be estimated by applying a dynamic simulation of a finite-element model. More details of the state estimation are given in the next subsection. Because the cell growth can be inhibited by a high carbon-to-nitrogen (C/N) ratio, an optimum C/N ratio is defined for the fed-batch operation [16,35]. The process models of Eqs. 5 and 6 are applied to the optimization problem of Eq. 4, and the constrained optimization problem for the manipulated input ( u = F N ) is solved at each time instant:

(7) min u J [ P ¯ , X ¯ , N ˜ , S ¯ , V ¯ , F S , u , y sp ]

J = w N ε 2 k 4 μ X ¯ − X ¯ V ¯ ( F S + F N ) + k 5 X ¯ − P ¯ V ¯ ( F S + F N ) + P ¯ − y sp 2

where

μ = μ m N ˜ S ¯ N ˜ S ¯ + K sr 1 − N ˜ S ¯ S m n k

F S = w C N f u S f and y sp = P d X ¯

w C = w C , 1 , S ¯ / N ˜ ≤ R CN w C , 2 , S ¯ / N ˜ > R CN

subject to:

X ̇ = φ 1 ( N ˜ , S ˜ , X , P , V ) P ̇ = φ 2 ( X , P , V ) u lb ≤ u ≤ u ub y sp,lb ≤ y sp ≤ y sp,ub

where S ¯ , P ¯ , and X ¯ are the measured states of substrate concentration, PHB concentration, and total biomass concentration, respectively. N ˜ is the integrated state of nitrogen concentration, V ¯ is the current volume, F N is the manipulated input (u), y sp is the set point, w N and w C are weighting factors, R CN and P d are the defined carbon/nitrogen (C/N) ratio and PHB productivity, respectively, and ε is tuning parameter. Note that the defined C/N ratio and PHB productivity are based on the experimental results in previous works [17,22,35]. The set point calculator applies the defined PHB productivity to create the set point trajectory that relies on the current biomass concentration.

2.5 The control approach

For the control application, real-time state estimation for the unmeasurable variables is required for providing the state feedback variables. In this work, the unmeasurable state and the state derivatives are estimated by applying a finite-based, nonlinear state estimator, and the process model of Eqs. 5 and 6. A schematic diagram of the proposed control structure is shown in Figure 3.

Figure 3 Schematic diagram of the developed control system for the fed-batch bioreactor.
Figure 3

Schematic diagram of the developed control system for the fed-batch bioreactor.

The experimental simulation of the example process is conducted by computing software such as MATLAB and COMSOL Multiphysics. A compact equation of the nonlinear state estimator can be written as follows:

(8) ξ ˆ ̇ p = F ˆ p ( ξ ˆ p , ξ ˆ o ) ξ ˆ ̇ o = F ˆ o ( ξ ˆ p , ξ ˆ o , u ) y ˆ = h ˆ ( ξ ˆ p , ξ ˆ o )

where the vectors of estimated states are ξ ˆ p = [ S ˆ N ˆ ] T , ξ ˆ o = [ X ˆ P ˆ ] T , and y ˆ = [ P ˆ ] . The nonlinear state estimator in Eq. 8 requires the manipulated input u from the AMPC controller to calculate the estimated state N ˆ at each time instant. Additionally, first-order integral action is combined to compensate for unmeasured disturbances as presented in the following equations [36]:

(9) η ̇ = v − η λ ν = y sp − η

where η is the integral of error of the controlled output y. λ is a positive constant, and ν is the compensated set point. To estimate the unmeasured state, state derivatives and ensure the closed-loop stability of the developed control system, the integrator and finite-element-based state estimator are combined with the AMPC controller. The optimization problem for the controller can be written in a function of variables as follows:

(10) min u J [ P ¯ , X ¯ , N ˆ , S ¯ , V ¯ , F S , u , y s p ]

subject to:

X ̇ = φ 1 ( N ˜ , S ˜ , X , P , V ) P ̇ = φ 2 ( X , P , V ) u lb ≤ u ≤ u ub y sp,lb ≤ y sp ≤ y sp,ub

Implementation of the developed control system requires feedback measurements from the available sensors and state estimator. To avoid a high substrate concentration caused by substrate accumulation, the input constraints of the feed streams need to be specified.

3 Results and discussion

The process behaviors of the fed-batch bioreactor are simulated by applying the coupled PDE–ODE system in Eqs. 5 and 6. In the simulation, the initial conditions are applied: S ( z , 0 ) = 32 g·L−1, N ( z , 0 ) = 2.1 g·L−1, X ( 0 ) = 8.75 g·L−1, P ( 0 ) = 1 g·L−1, and V ( 0 ) = 280 L. For the feedstock, molasses is used as the feed substrate with a sugar concentration ( S f ) of 160 g·L−1 while the feed nitrogen concentration ( N f ) is 28 g·L−1. The temperature, pH, and dissolved oxygen of the fed-batch bioreactor are controlled at 30°C, 7, and 40%, respectively [22]. The PHB concentration is controlled by manipulating the nitrogen and substrate feed rates of the closed-loop system. A traditional PI controller is applied to compare the control performance. The PI controller employs the updated set point, and the internal model control (IMC) tuning method is applied to tune the PI controller. In the closed-loop simulations, the process parameters are applied: ε = 20, u lb = 0.01 L·h−1, u ub = 10 L·h−1, P d = 0.4, w N = 0.7, w C , 1 = 7, w C , 2 = 0.1, R CN = 20, λ = 0.001, y sp,ub = 3 g·L−1, and y sp,ub = 12 g·L−1 [17,22,37]. The PI controller is simulated with a set of tuning parameters, K p = −0.3 and K I = −0.003. The closed-loop response of the PHB concentration, response of the substrate and nitrogen concentration, response of biomass concentration, the profile of the substrate, and nitrogen feed rate are shown in Figures 4–7, respectively.

Figure 4 PHB concentration response of the closed-loop system.
Figure 4

PHB concentration response of the closed-loop system.

Figure 5 Substrate and nitrogen concentration profiles corresponding to the closed-loop system of Figure 4.
Figure 5

Substrate and nitrogen concentration profiles corresponding to the closed-loop system of Figure 4.

Figure 6 Biomass concentration profile corresponding to the closed-loop system of Figure 4.
Figure 6

Biomass concentration profile corresponding to the closed-loop system of Figure 4.

Figure 7 Profile of substrate and nitrogen-source feed rates corresponding to the closed-loop system of Figure 4.
Figure 7

Profile of substrate and nitrogen-source feed rates corresponding to the closed-loop system of Figure 4.

At the beginning of the fed-batch operation, the proposed controller increases the feed rates of substrate and nitrogen sources to support the growth of microorganisms. An increase in microbial growth, itself resulting from the adequate substrate and nitrogen source, increases the cells for polymer accumulation, which increases the PHB concentration [8]. Then, the control actions are optimally adjusted to minimize the error between the response of the PHB concentration and the updated set points. As shown in Figure 4, the PHB accumulation increases rapidly after the bioreactor is regulated with a low-nitrogen supply condition. The lack of a nitrogen source decreases the intracellular enzyme for cell growth, and the substrate is consumed for the enhancement of PHB production [9]. The different results corresponding to cultivations carried out with the various types of cultures and substrates are reported in previous research works. For the experimental studies reporting PHA production by C. necator from a vinasses–molasses mixture, cultivation in lab- and pilot-scale bioreactors have been performed in some works. Cultivations were carried out in a 5-L bioreactor using vinasse with nitrogen and mineral salts [38]. The PHB content (26% of total biomass) was reported as the best value when compared to the application of pure vinasse and vinasse with nitrogen. García et al. reported a value of productivity for PHA production (0.27 g·L−1·h−1) from fermentation in a 5-L bioreactor using vinasses–molasses mixture [21]. An extended study was conducted in a 500-L bioreactor using similar substrates [22]. The feeding policy for fed-batch operation is developed based on a lumped model; a soft sensor is applied for substrate prediction. Among their developed control strategies, the best value reported for productivity (0.364 g·L−1·h−1) was obtained in the fermentation that applied the productivity and profitability-oriented optimization. To compare the control performance of the proposed system with similar substrates, cultures, and conditions in this work, the modified PI controller is applied with the updated set points [39,40]. The PHB concentration, content of PHB (% of total biomass), substrate concentration, and nitrogen concentration are shown in Figures 8–11, respectively.

Figure 8 PHB concentration of the closed-loop system.
Figure 8

PHB concentration of the closed-loop system.

Figure 9 PHB content of total biomass.
Figure 9

PHB content of total biomass.

Figure 10 Substrate concentration of the closed-loop system.
Figure 10

Substrate concentration of the closed-loop system.

Figure 11 Nitrogen concentration of the closed-loop system.
Figure 11

Nitrogen concentration of the closed-loop system.

As shown in the figures, the proposed control strategy achieved higher performance than the applied PI controller. The PHB concentrations at the end of the operation obtained from the proposed controller and modified PI controller are 11.27 and 8.45 g·L−1, while the PHB contents (% of total biomass) are 40.12% and 34.70%, respectively. As explained above, the changing of substrate and nitrogen concentration in the broth during the operation could selectively promote cell and polymer productions. The proposed controller applied the integrated variables of substrate and nitrogen concentrations to update the set points and provide the optimal control actions to promote PHB accumulation. Additionally, the low final substrate and nitrogen-source concentrations shown in Figures 10 and 11 demonstrate that the proposed feeding strategy has the ability to utilize the nutrients effectively. The developed control strategy is evaluated for a regulatory problem to investigate the control robustness. The regulatory test is conducted by using the same initial conditions as the open-loop simulation. An unmeasured step disturbance is introduced by decreasing the feed nitrogen concentration by 10% after 12 h [37]. The responses of the performance test are shown in Figures 12 and 13, where it can be seen that the developed control scheme successfully rejects the effect of the disturbance and compensates for the reduced feed nitrogen concentration. The PHB concentration and productivity of the performance test are shown in Figures 14 and 15, respectively.

Figure 12 Substrate and nitrogen concentration profiles under performance test.
Figure 12

Substrate and nitrogen concentration profiles under performance test.

Figure 13 Profile of substrate and nitrogen-source feed rates under performance test.
Figure 13

Profile of substrate and nitrogen-source feed rates under performance test.

Figure 14 PHB concentration under regulatory test.
Figure 14

PHB concentration under regulatory test.

Figure 15 PHB Productivity of the closed-loop and regulatory system.
Figure 15

PHB Productivity of the closed-loop and regulatory system.

The fed-batch operation with a change in nutrient feed concentration could lead to inconsistency in the polymer production rate (unfortunately, there is no report of the investigation for the PHA production by vinasses/molasses) [37]. Though the disturbance affects PHA production, the proposed strategy updates the trajectory path for set point tracking while the first-order integrator compensates for the error, quickly rejects the disturbance, and improves the process responses. It can be seen from the results that are shown in Figure 15 that the proposed control system could provide more productivity (33–38%) compared to the applied PI controller. From the results, it is clear that the proposed control strategy that considers nutrient dispersion has the ability to enhance PHB production and improve the performance of the fed-batch bioreactor.

4 Conclusions

This article has proposed an optimization-based control strategy to enhance PHA production in the fed-batch bioreactor described by a coupled PDEs–ODEs model. The AMPC-based controller is combined with a nonlinear state estimator and integrator to estimate the unmeasured variables and handle process disturbances; the constrained optimization problem of the controller applies the set point trajectory corresponding to the biomass concentration. In simulation experiments, a change in feed concentration was introduced to investigate control robustness. The results showed that the developed control scheme successfully provides the optimal control actions that enhance the polymer production and follow the updated set point. Control performance of the control strategy under a regulatory test demonstrated the capability to handle the process disturbance and provide adequate control actions efficiently. Additionally, further applications of the control strategy to industrial-scale PHA processes with a greater dispersion effect are expected to obtain better control performance and improve process efficiency.

  1. Funding information: This work was supported by the Sirindhorn International Thai-German Graduate School of Engineering, King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-63-KNOW-30). Support from these sources is gratefully acknowledged.

  2. Author contributions: Atthasit Tawai: writing – original draft; Malinee Sriariyanun: writing – review and editing; Chanin Panjapornpon: formal analysis.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-07-25
Revised: 2023-01-24
Accepted: 2023-02-12
Published Online: 2023-03-07

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/gps-2022-8084
Keywords for this article
polyhydroxyalkanoates production; fed-batch bioreactor; coupled PDE–ODE model; analytical model predictive control; nonlinear optimization-based control
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  27. Recovery of critical metals from carbonatite-type mineral wastes: Geochemical modeling investigation of (bio)hydrometallurgical leaching of REEs
  28. Photocatalytic properties of ZnFe-mixed oxides synthesized via a simple route for water remediation
  29. Attenuation of di(2-ethylhexyl)phthalate-induced hepatic and renal toxicity by naringin nanoparticles in a rat model
  30. Novel in situ synthesis of quaternary core–shell metallic sulfide nanocomposites for degradation of organic dyes and hydrogen production
  31. Microfluidic steam-based synthesis of luminescent carbon quantum dots as sensing probes for nitrite detection
  32. Transformation of eggshell waste to egg white protein solution, calcium chloride dihydrate, and eggshell membrane powder
  33. Preparation of Zr-MOFs for the adsorption of doxycycline hydrochloride from wastewater
  34. Green nanoarchitectonics of the silver nanocrystal potential for treating malaria and their cytotoxic effects on the kidney Vero cell line
  35. Carbon emissions analysis of producing modified asphalt with natural asphalt
  36. An efficient and green synthesis of 2-phenylquinazolin-4(3H)-ones via t-BuONa-mediated oxidative condensation of 2-aminobenzamides and benzyl alcohols under solvent- and transition metal-free conditions
  37. Chitosan nanoparticles loaded with mesosulfuron methyl and mesosulfuron methyl + florasulam + MCPA isooctyl to manage weeds of wheat (Triticum aestivum L.)
  38. Synergism between lignite and high-sulfur petroleum coke in CO2 gasification
  39. Facile aqueous synthesis of ZnCuInS/ZnS–ZnS QDs with enhanced photoluminescence lifetime for selective detection of Cu(ii) ions
  40. Rapid synthesis of copper nanoparticles using Nepeta cataria leaves: An eco-friendly management of disease-causing vectors and bacterial pathogens
  41. Study on the photoelectrocatalytic activity of reduced TiO2 nanotube films for removal of methyl orange
  42. Development of a fuzzy logic model for the prediction of spark-ignition engine performance and emission for gasoline–ethanol blends
  43. Micro-impact-induced mechano-chemical synthesis of organic precursors from FeC/FeN and carbonates/nitrates in water and its extension to nucleobases
  44. Green synthesis of strontium-doped tin dioxide (SrSnO2) nanoparticles using the Mahonia bealei leaf extract and evaluation of their anticancer and antimicrobial activities
  45. A study on the larvicidal and adulticidal potential of Cladostepus spongiosus macroalgae and green-fabricated silver nanoparticles against mosquito vectors
  46. Catalysts based on nickel salt heteropolytungstates for selective oxidation of diphenyl sulfide
  47. Powerful antibacterial nanocomposites from Corallina officinalis-mediated nanometals and chitosan nanoparticles against fish-borne pathogens
  48. Removal behavior of Zn and alkalis from blast furnace dust in pre-reduction sinter process
  49. Environmentally friendly synthesis and computational studies of novel class of acridinedione integrated spirothiopyrrolizidines/indolizidines
  50. The mechanisms of inhibition and lubrication of clean fracturing flowback fluids in water-based drilling fluids
  51. Adsorption/desorption performance of cellulose membrane for Pb(ii)
  52. A one-pot, multicomponent tandem synthesis of fused polycyclic pyrrolo[3,2-c]quinolinone/pyrrolizino[2,3-c]quinolinone hybrid heterocycles via environmentally benign solid state melt reaction
  53. Green synthesis of silver nanoparticles using durian rind extract and optical characteristics of surface plasmon resonance-based optical sensor for the detection of hydrogen peroxide
  54. Electrochemical analysis of copper-EDTA-ammonia-gold thiosulfate dissolution system
  55. Characterization of bio-oil production by microwave pyrolysis from cashew nut shells and Cassia fistula pods
  56. Green synthesis methods and characterization of bacterial cellulose/silver nanoparticle composites
  57. Photocatalytic research performance of zinc oxide/graphite phase carbon nitride catalyst and its application in environment
  58. Effect of phytogenic iron nanoparticles on the bio-fortification of wheat varieties
  59. In vitro anti-cancer and antimicrobial effects of manganese oxide nanoparticles synthesized using the Glycyrrhiza uralensis leaf extract on breast cancer cell lines
  60. Preparation of Pd/Ce(F)-MCM-48 catalysts and their catalytic performance of n-heptane isomerization
  61. Green “one-pot” fluorescent bis-indolizine synthesis with whole-cell plant biocatalysis
  62. Silica-titania mesoporous silicas of MCM-41 type as effective catalysts and photocatalysts for selective oxidation of diphenyl sulfide by H2O2
  63. Biosynthesis of zinc oxide nanoparticles from molted feathers of Pavo cristatus and their antibiofilm and anticancer activities
  64. Clean preparation of rutile from Ti-containing mixed molten slag by CO2 oxidation
  65. Synthesis and characterization of Pluronic F-127-coated titanium dioxide nanoparticles synthesized from extracts of Atractylodes macrocephala leaf for antioxidant, antimicrobial, and anticancer properties
  66. Effect of pretreatment with alkali on the anaerobic digestion characteristics of kitchen waste and analysis of microbial diversity
  67. Ameliorated antimicrobial, antioxidant, and anticancer properties by Plectranthus vettiveroides root extract-mediated green synthesis of chitosan nanoparticles
  68. Microwave-accelerated pretreatment technique in green extraction of oil and bioactive compounds from camelina seeds: Effectiveness and characterization
  69. Studies on the extraction performance of phorate by aptamer-functionalized magnetic nanoparticles in plasma samples
  70. Investigation of structural properties and antibacterial activity of AgO nanoparticle extract from Solanum nigrum/Mentha leaf extracts by green synthesis method
  71. Green fabrication of chitosan from marine crustaceans and mushroom waste: Toward sustainable resource utilization
  72. Synthesis, characterization, and evaluation of nanoparticles of clodinofop propargyl and fenoxaprop-P-ethyl on weed control, growth, and yield of wheat (Triticum aestivum L.)
  73. The enhanced adsorption properties of phosphorus from aqueous solutions using lanthanum modified synthetic zeolites
  74. Separation of graphene oxides of different sizes by multi-layer dialysis and anti-friction and lubrication performance
  75. Visible-light-assisted base-catalyzed, one-pot synthesis of highly functionalized cinnolines
  76. The experimental study on the air oxidation of 5-hydroxymethylfurfural to 2,5-furandicarboxylic acid with Co–Mn–Br system
  77. Highly efficient removal of tetracycline and methyl violet 2B from aqueous solution using the bimetallic FeZn-ZIFs catalyst
  78. A thermo-tolerant cellulase enzyme produced by Bacillus amyloliquefaciens M7, an insight into synthesis, optimization, characterization, and bio-polishing activity
  79. Exploration of ketone derivatives of succinimide for their antidiabetic potential: In vitro and in vivo approaches
  80. Ultrasound-assisted green synthesis and in silico study of 6-(4-(butylamino)-6-(diethylamino)-1,3,5-triazin-2-yl)oxypyridazine derivatives
  81. A study of the anticancer potential of Pluronic F-127 encapsulated Fe2O3 nanoparticles derived from Berberis vulgaris extract
  82. Biogenic synthesis of silver nanoparticles using Consolida orientalis flowers: Identification, catalytic degradation, and biological effect
  83. Initial assessment of the presence of plastic waste in some coastal mangrove forests in Vietnam
  84. Adsorption synergy electrocatalytic degradation of phenol by active oxygen-containing species generated in Co-coal based cathode and graphite anode
  85. Antibacterial, antifungal, antioxidant, and cytotoxicity activities of the aqueous extract of Syzygium aromaticum-mediated synthesized novel silver nanoparticles
  86. Synthesis of a silica matrix with ZnO nanoparticles for the fabrication of a recyclable photodegradation system to eliminate methylene blue dye
  87. Natural polymer fillers instead of dye and pigments: Pumice and scoria in PDMS fluid and elastomer composites
  88. Study on the preparation of glycerylphosphorylcholine by transesterification under supported sodium methoxide
  89. Wireless network handheld terminal-based green ecological sustainable design evaluation system: Improved data communication and reduced packet loss rate
  90. The optimization of hydrogel strength from cassava starch using oxidized sucrose as a crosslinking agent
  91. Green synthesis of silver nanoparticles using Saccharum officinarum leaf extract for antiviral paint
  92. Study on the reliability of nano-silver-coated tin solder joints for flip chips
  93. Environmentally sustainable analytical quality by design aided RP-HPLC method for the estimation of brilliant blue in commercial food samples employing a green-ultrasound-assisted extraction technique
  94. Anticancer and antimicrobial potential of zinc/sodium alginate/polyethylene glycol/d-pinitol nanocomposites against osteosarcoma MG-63 cells
  95. Nanoporous carbon@CoFe2O4 nanocomposite as a green absorbent for the adsorptive removal of Hg(ii) from aqueous solutions
  96. Characterization of silver sulfide nanoparticles from actinobacterial strain (M10A62) and its toxicity against lepidopteran and dipterans insect species
  97. Phyto-fabrication and characterization of silver nanoparticles using Withania somnifera: Investigating antioxidant potential
  98. Effect of e-waste nanofillers on the mechanical, thermal, and wear properties of epoxy-blend sisal woven fiber-reinforced composites
  99. Magnesium nanohydroxide (2D brucite) as a host matrix for thymol and carvacrol: Synthesis, characterization, and inhibition of foodborne pathogens
  100. Synergistic inhibitive effect of a hybrid zinc oxide-benzalkonium chloride composite on the corrosion of carbon steel in a sulfuric acidic solution
  101. Review Articles
  102. Role and the importance of green approach in biosynthesis of nanopropolis and effectiveness of propolis in the treatment of COVID-19 pandemic
  103. Gum tragacanth-mediated synthesis of metal nanoparticles, characterization, and their applications as a bactericide, catalyst, antioxidant, and peroxidase mimic
  104. Green-processed nano-biocomposite (ZnO–TiO2): Potential candidates for biomedical applications
  105. Reaction mechanisms in microwave-assisted lignin depolymerisation in hydrogen-donating solvents
  106. Recent progress on non-noble metal catalysts for the deoxydehydration of biomass-derived oxygenates
  107. Rapid Communication
  108. Phosphorus removal by iron–carbon microelectrolysis: A new way to achieve phosphorus recovery
  109. Special Issue: Biomolecules-derived synthesis of nanomaterials for environmental and biological applications (Guest Editors: Arpita Roy and Fernanda Maria Policarpo Tonelli)
  110. Biomolecules-derived synthesis of nanomaterials for environmental and biological applications
  111. Nano-encapsulated tanshinone IIA in PLGA-PEG-COOH inhibits apoptosis and inflammation in cerebral ischemia/reperfusion injury
  112. Green fabrication of silver nanoparticles using Melia azedarach ripened fruit extract, their characterization, and biological properties
  113. Green-synthesized nanoparticles and their therapeutic applications: A review
  114. Antioxidant, antibacterial, and cytotoxicity potential of synthesized silver nanoparticles from the Cassia alata leaf aqueous extract
  115. Green synthesis of silver nanoparticles using Callisia fragrans leaf extract and its anticancer activity against MCF-7, HepG2, KB, LU-1, and MKN-7 cell lines
  116. Algae-based green AgNPs, AuNPs, and FeNPs as potential nanoremediators
  117. Green synthesis of Kickxia elatine-induced silver nanoparticles and their role as anti-acetylcholinesterase in the treatment of Alzheimer’s disease
  118. Phytocrystallization of silver nanoparticles using Cassia alata flower extract for effective control of fungal skin pathogens
  119. Antibacterial wound dressing with hydrogel from chitosan and polyvinyl alcohol from the red cabbage extract loaded with silver nanoparticles
  120. Leveraging of mycogenic copper oxide nanostructures for disease management of Alternaria blight of Brassica juncea
  121. Nanoscale molecular reactions in microbiological medicines in modern medical applications
  122. Synthesis and characterization of ZnO/β-cyclodextrin/nicotinic acid nanocomposite and its biological and environmental application
  123. Green synthesis of silver nanoparticles via Taxus wallichiana Zucc. plant-derived Taxol: Novel utilization as anticancer, antioxidation, anti-inflammation, and antiurolithic potential
  124. Recyclability and catalytic characteristics of copper oxide nanoparticles derived from bougainvillea plant flower extract for biomedical application
  125. Phytofabrication, characterization, and evaluation of novel bioinspired selenium–iron (Se–Fe) nanocomposites using Allium sativum extract for bio-potential applications
  126. Erratum
  127. Erratum to “Synthesis, characterization, and evaluation of nanoparticles of clodinofop propargyl and fenoxaprop-P-ethyl on weed control, growth, and yield of wheat (Triticum aestivum L.)”
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