Stochastic optimization of enzymatic hydrolysis of lignocellulosic biomass

Computers and Chemical Engineering 135 (2020) 106776

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Stochastic optimization of enzymatic hydrolysis of lignocellulosic biomass F. Fenila, Yogendra Shastri∗ Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

a r t i c l e

i n f o

Article history: Received 4 October 2019 Revised 20 January 2020 Accepted 7 February 2020 Available online 8 February 2020 Keywords: Hydrolysis Uncertainties Optimization Optimal control,

a b s t r a c t Uncertainties in feedstock composition and kinetic parameters impact the performance of enzymatic hydrolysis of lignocellulosic biomass. Operating strategies need to be accordingly modified to achieve higher yield of glucose. Therefore, stochastic optimization and optimal control problems were formulated to maximize glucose concentration and minimize variance. The optimal temperature for maximization of glucose concentration reduced in the presence of uncertainties. For uncertainties in kinetic parameters, the optimal temperature reduced from 323.34 K to 317.07 K for a batch time of 12 h, and the glucose concentration increased by 7.7% when the stochastic optimal temperature was used instead of deterministic temperature. Similarly, for the objective of minimization of variance in glucose concentration at the end of the batch, stochastic optimization reduced the variance by 89% as compared to deterministic optimization. Stochastic optimal control resulted in up to 60% improvement in mean glucose concentration in comparison to the deterministic optimal control. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Lignocellulosic ethanol is a potential substitute for liquid transportation fuels and does not pose competency as a source of food (Nigam and Singh, 2011). The sources of lignocellulosic biomass include agricultural residues such as corn stover, corn cob, sugarcane bagasse, sugarcane tops, and dedicated energy crops such as switchgrass (Berndes et al., 2010). The conversion of these raw materials to ethanol is a multiple-step process, including preprocessing, pretreatment, hydrolysis, fermentation, and purification. Hydrolysis has been identified as one of the crucial steps in the production of lignocellulosic ethanol (Morales-Rodriguez et al., 2011). The amount of glucose available for fermentation is determined by hydrolysis. Therefore, this work focusses on enzymatic hydrolysis of pretreated lignocellulosic biomass where the cellulosic and hemicellulosic polymers are broken down to fermentable sugars like glucose and xylose. The enzymatic hydrolysis is preferred over other chemical and physicochemical methods due to mild operating conditions and less formation of inhibitors (Wahlstrom and Suurnakki, 2015). The optimal operation of enzymatic hydrolysis is necessary for improving the glucose yield. The operating conditions for enzymatic hydrolysis include hydrolysis temperature, time, enzyme

∗

Corresponding author. E-mail address: [email protected] (Y. Shastri).

https://doi.org/10.1016/j.compchemeng.2020.106776 0098-1354/© 2020 Elsevier Ltd. All rights reserved.

loading, and solid loading (Newman et al., 2013; Ruangmee and Sangwichien, 2013). The optimal operating conditions depend on the source of enzyme Steffien et al. (2014), thermal stability, and the activity of the enzyme. Since experimental evaluation is cost and time-intensive, the optimal operating conditions can be derived using mathematical models validated against experimental data. For example, Fenila and Shastri (2016) derived the optimal temperature to maximize the glucose concentration and minimize batch time using a reaction kinetics model. Similarly, Hodge et al. (2009) carried out model-based optimization studies to derive the optimal feeding strategies to maintain constant level of insoluble solids throughout the reaction time in a fed-batch reactor. Tai and Keshwani (2014) derived optimal control strategy to improve glucose concentration and accumulated cellulose conversion for fed-batch reactor using a modified epidemic model. In these studies, the optimal operating conditions were derived, assuming deterministic values of feedstock composition and kinetic parameters. However, the feedstock composition and kinetic parameters are uncertain in reality. For example, the cellulose, hemicellulose, and lignin content in rice straw can vary from 32 to 41%, 21.5 to 28% and 9.9 to 19.64% respectively (Van Dyk and Pletschke, 2012; Bhandari et al., 2014). The uncertainty in feed composition occurs due to feedstock variety, environment, harvest time, weather, and agronomic factors (Kenney et al., 2013). The proportion of the feedstock components such as cellulose, hemicellulose, and lignin vary during crop growth. For instance,

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F. Fenila and Y. Shastri / Computers and Chemical Engineering 135 (2020) 106776

Nomenclature C CB G ri Ei kir E1B Rs KiIG2 KiIG KiIX K3M Eimax Kiad EiF T C0 tf

Concentration of cellulose (g/kg) Concentration of cellobiose (g/kg) Concentration of glucose (g/kg) Reaction rate (i= 1,2,3) (g/kgh) Concentration of enzyme (i= 1,2,3) (g/kg) Kinetic parameter (i= 1,2,3) (kg/gh) Bound Enzyme (g/kg) Substrate Reactivity Inhibition constant for cellobiose (i= 1,2,3) (g/kg) Inhibition constant for glucose (i= 1,2,3) (g/kg) Inhibition constant for xylose (i= 1,2,3) (g/kg) Substrate (cellobiose) saturation constants (g/kg) maximum mass of enzyme that can adsorb onto a unit mass of substrate (gprotein/gcellulose) Dissociation constant for enzyme adsorption (i= 1,2,3) (gprotein/gcellulose) Free Enzyme (i= 1,2 ) (g/kg) Temperature (K) Initial concentation of cellulose(g/kg) Batch time (h)

literature (Verma et al., 2017). However, the optimal operating decisions were not determined in literature considering the uncertainty in feedstock composition and kinetic parameters. Moreover, uncertainty in parameters related to thermal deactivation kinetics of enzymes were also ignored in the previous studies (Fenila and Shastri, 2016). Since optimal temperature is dependent on the rate of enzyme deactivation, it is important to consider those in optimization studies. In this work, we have addressed the impact of uncertainties on the optimal operating decisions. The optimization problems formulated were solved for both deterministic and stochastic parameters. The objective functions were calculated for kinetic parameter and feed composition uncertainty using the deterministic and stochastic optimal operating decisions. This article is organised as follows: Section 2 explains the model adapted for this study and the range for feed composition and model parameters. Section 3 explains the stochastic optimization approach used in this work. Section 4 explains the problem formulation and approach used for solving stochastic optimization and optimal control problems. The results of problem formulation explained in Section 4 are explained in Section 6. The article is concluded in Section 7. 2. Modelling of enzymatic hydrolysis of lignocellulose

Pordesimo et al. (2005) observed a drop in glucan content and rise in lignin and xylan content of corn stover during crop development. Apart from the above factors related to the choice of biomass type, the blending of biomass types to meet the demand also contributes to feed composition uncertainty (Ou et al., 2018). The feedstock composition uncertainty can also arise due to factors related to the experimental determination of feedstock composition such as analysis of samples in different laboratories, differences in the method used and the instrument used for measurement (Templeton et al., 2010). Parameters estimated with pure lignocellulosic materials will also contribute uncertainty in kinetic parameters. Significant errors in the estimation of parameters from experimental data can arise due to the complexity of models itself, and data from experiments being not informative enough (Sin et al., 2010). Since these uncertainties affect the conversion as well as optimal operating decisions, the optimization problems should be solved considering uncertainty in these parameters. The effect of uncertainties on enzymatic hydrolysis of lignocellulose was previously addressed in literature. Verma et al. (2017) quantified the impact of uncertainties in kinetic parameters and feed composition on acid pretreatment and enzymatic hydrolysis of lignocellulosic biomass. The variation in output glucose concentration was calculated by stochastic simulation by propagating the uncertainty in input parameters through a kinetic model (Diwekar, 2003). The sensitive uncertain parameters were identified by a global sensitivity analysis (Verma et al., 2017). Prunescu and Sin (2013) had performed uncertainty and identifiability analysis of enzymatic hydrolysis of lignocellulosic biomass for feed composition and kinetic parameter uncertainty. They had identified that the uncertainty in kinetic parameters impact the conversion of lignocellulose more than uncertainty in feedstock composition. Morales-Rodriguez et al. (2012) performed optimization under uncertainty for various process configurations for lignocellulosic ethanol production and developed an optimization framework for a lignocellulosic biorefinery in the presence of uncertainties. Ou et al. (2018) studied the impact of biomass blending on the uncertainty of hydrolysed sugar yield using corn stover, switchgrass, and grass clippings. The blending ratio was optimized to meet the target sugar concentrations, sugar yield, and sugar yield per unit feedstock. The impact of uncertainty in feedstock composition and kinetic parameters were quantified in

A semi-mechanistic model proposed in literature (Kadam et al., 2004) and modified for thermal deactivation kinetics in our previous work is adapted for this work. Kadam et al. (2004) represented the change in concentration of cellulose (C), cellobiose (CB), glucose (G) by Eqs. (1) to (3).

dC = −r1 − r2 dt

(1)

dCB = 1.065r1 − r3 dt

(2)

dG = 1.111r2 + 1.053r3 dt

(3)

The conversion rate of cellulose to cellobiose is represented by (4). Similarly, the conversion rate of cellulose to glucose, and cellobiose to glucose is represented in Eqs. (5) and (6), respectively.

r1 = r2 = r3 =

[1 +

k1r E1B Rs C CB + KG1IG + KX1IX K1IG2

]

k2r ( E1B + E2B ) Rs C [1 +

CB K2IG2

+

G K2IG

+

X K2IX

]

k3r (E2F )CB K3M [1 +

G K3IG

+

X K3IX

] + CB

(4)

(5)

(6)

In Eq. (4), Rs refers to substrate reactivity factor and is defined as a ratio of substrate at time ’t’ to initial substrate concentration Eq. (7).

Rs = α

C C0

(7)

The bound fraction of enzymes (E1 and E2 ) are calculated as expressed in Eq. (8)

EiB =

Eimax Kiad EiF C 1 + Kiad EiF

(8)

The kinetic model proposed by Kadam et al. (2004) represented the relationship between temperature and reaction rate using the Arrhenius relationship. However, the thermal deactivation of enzymes was ignored. In order to use the model to find the optimal

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Table 1 Parameters values used for modelling of enzymatic hydrolysis of lignocellulose. Parameter

Unit

Value

Parameter

Unit

Value

k1r0 k3r0 K2ad E2max K2IG K1IG2 K1IX K3IX

(h−1 ) (h−1 )

144000 4.636e+04 0.1 0.01 0.04 0.015 0.1 201.0 1 0.1316

k2r0 K1ad E1max K1IG K3IG K2IG2 K2IX K3M Ea kp2

(h−1 )

46360 0.38 0.06 0.0995 3 132 0.2 24.9 5495 0.4480

α

(g/g substrate) (g/g substrate) (g/kg) (g/kg) (g/kg) (g/kg) -

kp1

temperature, the model should also be able to capture the effect of thermal deactivation. Therefore, the thermal deactivation of enzymes was also captured in the modified model. The validation of the modified model incorporating thermal deactivation is explained by Fenila and Shastri (2016). The thermal deactivation of enzymes E1 and E2 are given by Eqs. (9) and 10. The rate of deactivation kd1 and kd2 for enzymes E1 and E2 are expressed as a function of temperature (T) in Eqs. (11) and 12, respectively.

dE1 = −kd1 E1 dt

(9)

dE2 = −kd2 E2 dt

(10)

kd1 = 10−20 (exp(k p1 T ))

(11)

kd2 = 10−65 (exp(k p2 T ))

(12)

The values of parameters appearing in the model are tabulated in Table 1. The model has been validated previously by Fenila and Shastri (2016), and the validation results are not reported here. 3. Impact of uncertainties on enzymatic hydrolysis of lignocellulose The uncertainty in the composition of feedstock impacts the composition of pretreated lignocellulosic material (Verma et al., 2017), which further impacts the output glucose concentration from enzymatic hydrolysis. In this section, the necessity of performing stochastic optimization is discussed by highlighting variability in output glucose concentration due to uncertainty in feedstock composition and kinetic parameters. The impact of uncertainty in feedstock composition and the kinetic parameter is demonstrated by stochastic simulation. The steps followed for stochastic simulation are explained as follows: •

•

•

•

Among the parameters in the model adapted for this work, the uncertain kinetic parameters to be studied are selected based on global sensitivity analysis carried out in Verma et al. (2017). The input range of uncertain parameters such as feedstock composition and kinetic parameters selected are defined. N samples from the input range are drawn using Latin Hypercube Sampling (LHS). Latin Hypercube sampling is a stratified sampling technique. In this method, the input distribution is divided into non overlapping intervals of equal length and values are selected randomly from each interval (Huntington and Lyrintzist, 1998). LHS requires less number of samples than Monte Carlo sampling. However, LHS is limited by the fact that the stratification scheme used in this method is onedimensional (Diwekar and Kalagnanam, 1997). The samples drawn from the input range are propagated through the hydrolysis model.

•

(g/g substrate) (g/g substrate) (g/kg) (g/kg) (g/kg) (g/kg) (g/kg) cal/mol

The variation in the output concentration and the distribution are studied.

Batch times between 12 h and 168 h and temperature of 318 (K) are considered for stochastic simulation. The feedstock is assumed to be composed of cellulose, and background sugars such as glucose and xylose formed during pretreatment. The range of cellulose concentration is assumed to be 45–52.5 (g/kg). The range of background glucose and xylose are assumed to be 12.97-16.72 g/kg and 10.95-11.75 g/kg, respectively. The sensitive kinetic parameters identified by Verma et al. (2017) such as k1r0 , k3r0 , K1IG , K3M , K3IG , K3IG , K3IG , Ea , α and parameters related to deactivation such as kp1 and kp2 are considered uncertain. The kinetic parameters are assumed to vary by ± 5%. The feedstock composition and kinetic parameters are assumed to be uniformly distributed. 500 samples are drawn from the input range defined using Latin Hypercube Sampling (LHS). The output distribution of glucose concentration followed a similar distribution for all batch times in the presence of feedstock composition uncertainty. It is expected that the output distribution would be similar to the input distribution since the feedstock composition appears linearly in the model equations. Therefore, the uncertainty gets linearly transferred to the model output. For feedstock composition uncertainty, although the output glucose distribution did not change much with batch time, it is important to note that the mean glucose concentration increased from 27.64 to 41.56 g/kg when batch time increased from 12 h to 168 h. This was a result of higher batch time and independent of uncertainty consideration. The distribution of glucose concentration for feedstock composition uncertainty for various batch times is given in the supplementary information. For kinetic parameter uncertainty, the distribution of output glucose concentration followed normal distribution for 12 h even though the input distribution was uniform. This was due to the fact that kinetic parameters appeared in nonlinear form in the model equations. Another important point to note was that the skewness of the distribution of output glucose concentration shifted from right to left as the reaction time increased from 12 h to 24 h Fig. 1. This signifies the improvement in conversion. For a batch time of 12 h, the standard deviation in glucose concentration for feedstock composition and kinetic parameter differed only by 0.45. However, for a batch time of 168 h, the standard deviation differed by 6.75. This shows that the type of uncertainty affects both mean as well as the standard deviation in output glucose concentration. Moreover, the kinetic parameter uncertainty has more impact on the output glucose concentration than the feedstock composition uncertainty. The mean glucose concentration varied between 12.77 to 38.55 g/kg for batch times between 12 h to 168 h. The standard deviation varied between 3.48 to 10.36 Fig. 1. In summary, the results showed that uncertainties in feedstock composition and kinetic parameters might significantly affect the product yield. Moreover, the impact varied with batch time. Since the optimal operating strate-

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Fig. 1. Distribution of glucose concentration for enzymatic hydrolysis with kinetic parameter uncertainty for various batch times (a) Batch time: 12 h (b) Batch time: 24 h (c) Batch time: 48 h (d)Batch time: 72 h (e) Batch time: 168 h.

gies will be a function of all these factors, these results emphasized the necessity of systematically considering uncertainty in the optimization framework. This implies the formulation of stochastic optimization and optimal control problems. These problem formulations are explained in the next section. 4. Optimization and optimal control problem formulation The steps followed in formulating and solving optimization problems in the presence of various uncertainties are explained as follows: 1. The optimal operating temperature for enzymatic hydrolysis problem is determined by solving a deterministic optimization problem. The optimal temperature, thus derived, is referred to as “deterministic optimum”. 2. A stochastic optimization problem by considering N LHS samples of the uncertain parameters is solved to determine the corresponding optimal operating temperature. The optimal temperature thus derived is referred to as “stochastic optimum”. 3. The deterministic optimum is applied to the stochastic model simulated with N samples derived using LHS from the input range, and the objective functions are calculated. This is referred to as the “conventional approach”. 4. The stochastic optimum is applied to the stochastic model simulated with N samples derived using LHS from the input range, and the objective functions are calculated. This is referred to as the “proposed approach”. 5. The benefit of using the proposed approach over the conventional approach is calculated based on the percentage difference in the objective function obtained from the conventional and proposed approach. The steps followed for comparing the conventional and proposed approach is represented in Fig. 2. For the application of stochastic optimal control, a similar approach is followed, the only difference being that the optimal solutions are temperature profiles instead of a single temperature value for the complete batch. The various optimization problems formulated and solved in this work such as maximization of glucose concentration (deterministic case)

and maximization of mean glucose concentration (stochastic case), and minimization of variance are listed in Table 2. 4.1. Maximization of glucose concentration (optimization) The objective of the hydrolysis process is to convert cellulose to glucose. The higher yield of glucose would result in higher glucose available for the production of ethanol in fermentation. Therefore, an optimization problem is formulated to maximize the glucose concentration at the end of the batch with reactor temperature as the decision variable. The maximization of the glucose concentration problem is solved for batch times in the range of 12 h to 168 h. For this particular objective function, three different cases are considered: •

•

•

Case 1: In this case, the kinetic parameters, as well as feedstock composition, are considered deterministic. Case 2: In this case, the kinetic parameters are considered to be uncertain. However, feedstock composition is considered to be deterministic. Case 3: In this case, the feedstock composition is considered to be uncertain, whereas, the kinetic parameters are considered to be deterministic.

4.1.1. Objective function: Case 1 The objective function for maximization of glucose concentration for the deterministic case is defined in Eq. (13) and the complete formulation is given by Eqs. (13) to (14).

J = max[G(t f )] T

(13)

subject to:

x˙ = f (x(t ), t , T );

(14)

x(t0 ) = x0 Here, x˙ in Eq. (14) refers to the model equations given in Eqs. (1) to (10). G(tf ) refers to the glucose concentration at time tf .

F. Fenila and Y. Shastri / Computers and Chemical Engineering 135 (2020) 106776

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Fig. 2. Approach used in this work for various deterministic and stochastic optimization problems. Table 2 List of various optimization problems formulated and solved for improving enzymatic hydrolysis of lignocellulose. S.No.

Problem

1.

Maximization of glucose concentration Case 1 Case 2 Case 3 Minimization of variance Case 1 Case 2

2.

Feedstock composition

Kinetic parameters

Deterministic Deterministic Stochastic

Deterministic Stochastic Deterministic

Deterministic Stochastic

Stochastic Deterministic

4.1.2. Objective function: Case 2 and 3 The optimal operating temperature for carrying out enzymatic hydrolysis is a trade-off between faster reaction kinetics and slower enzyme deactivation. Therefore, any change in parameters related to reaction kinetics and enzyme deactivation will impact the optimal operating temperature. Even though the uncertainty associated with other kinetic parameters will not change the optimal operating temperature, it will impact the amount of glucose produced. The optimal operating temperature for each of the possible combinations of these parameters can be estimated similar to the approach used for finding the deterministic optimal temperature. However, it is not possible to estimate the value of all the parameters accurately from batch to batch. Therefore, a stochastic optimization problem is formulated to determine the temperature to maximize the expected value of glucose concentration for the range of uncertain parameters defined. The expected value of glucose concentration at the end of batch time tf is chosen as the objective function for stochastic optimization in the presence of kinetic parameter and feed composition uncertainty Eq. (15) and the complete formulation is given by Eqs. (15) to (16).

J = max[E[G(t f )]] T

(15)

Batch time (h)

Decision variable

12 to 72 and 168

Temperature

12 to 72 and 168

Temperature

from hydrolysis is fed to fermentation. The fermentation is affected by initial glucose concentration resulting in reduced ethanol yield and higher purification cost. From an operational standpoint, it would be useful to have a consistent feed to the fermentation processing step. Therefore, minimization of variance problem with the objective of minimization of glucose concentration variance at the end of the batch is formulated for feedstock composition and kinetic parameter uncertainty. The average glucose concentration obtained by solving stochastic maximization of glucose concentration in Section 4.1 is taken as the target glucose concentration (G¯ ) for minimization of variance problem. The minimization of the variance problem was solved for two cases. •

•

Case 1: The kinetic parameters are considered uncertain. However, the feedstock composition is assumed to be deterministic. Case 2: The feedstock composition is considered uncertain. However, the kinetic parameters are assumed to be deterministic.

The objective function formulated for minimization of variance with temperature as the decision variable is given in Eq. (17) and the complete formulation is given by Eqs. (17) to (18).

J = min(E[(G(t f ) − G¯ )2 ] ) T

subject to:

(17)

subject to:

x˙ = f (x(t ), t , T , θ )

(16)

x(t0 ) = x0 x˙ in Eq. (16) refers to the model equations given in Eqs. (1) to (10), and θ refers to set of uncertain parameters. 4.2. Minimization of variance (optimization)

x˙ = f (x(t ), t , T , θ )

(18)

x(t0 ) = x0 x˙ in Eq. (18) refers to the model equations given in Eqs. (1) to (3) and (9) to (10). θ refers to set of uncertain parameters. 4.3. Maximization of glucose concentration (optimal control)

From Verma et al. (2017), it is evident that the uncertainty in the feed composition, as well as kinetic parameters, results in significant variation in the output glucose concentration. The output

The optimal operating policies obtained from optimization problems do not change with time and have a constant value

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F. Fenila and Y. Shastri / Computers and Chemical Engineering 135 (2020) 106776

for the entire batch time. Even though static temperature profiles are easier to implement, dynamic temperature profiles are useful for reactions with catalyst sensitive to temperature. For example, Ho and Humphrey (1970) determined optimal temperature profiles for maximization of 6-aminopenicillanic acid yield. Dynamic values of control variables can be obtained by solving an optimal control problem. A number of optimal control studies are available in the literature for the batch process. For example, Benavides and Diwekar (2013) formulated and solved an optimal control problem for the production of biodiesel in a batch reactor. Similarly, Verma and Shastri (2019) derived optimal control profiles for maximization of yield, minimization of deviation from target yield, and minimization of batch time for acid pretreatment of sugarcane bagasse in a batch reactor. The objective function of optimal control problem for maximization of glucose concentration is given in Eq. (19).

this work. The additional model equations included for kinetic parameters is given in supplementary information. The stochastic optimal control problem is formulated using Pontryagin’s maximum principle with temperature as the control variable. The methodology explained in Benavides and Diwekar (2012) is used for the formulation of stochastic optimal control problem. The objective function for the stochastic optimal control problem is given in Eq. (25).

J = max[G(t f )]

H=

(19)

The Hamiltonian for deterministic optimal control problem with maximization of glucose concentration as the objective function is given in Eq. (20).

H = p1 F1 + p2 F2 + p3 F3 + p4 F4 + p5 F5

(20)

In Eq. (20), p1 , p2 , p3 , p4 , and p5 are Lagarange multipliers. F1 , F2 , F3 , F4 , and F5 are RHS of model equations given in Eqs. (1) to (3) and Eqs. (9) to (10). The costate equations and necessary conditions were derived as given in Eqs. (21) and (22).The temperature obtained from this problem is referred as “deterministic optimal control” in this paper.

dp ∂H =− dt ∂x

(21)

∂H =0 ∂T

(22)

The static uncertainties gets translated to dynamic ones when propagated through a dynamic process (Benavides and Diwekar, 2012). The dynamic propagation of uncertainties can be represented by a stochastic model. The stochastic model can be further used to formulate the stochastic optimal control problem. For example, Acikgoz and Diwekar (2010) derived optimal control profiles to regulate blood glucose level in insulin dependent diabetic patients using a stochastic model. Similarly, Yenkie and Diwekar (2013) derived optimal control profiles for seeded batch crystallizer using a stochastic model. The stochastic model for time-dependent variation kinetic parameters is represented using the mean-reverting Ito process (Verma et al., 2017). In the mean-reverting Ito process, even though the uncertain variables vary in the defined range, they are reverted to the mean value at a rate of ζ . The generic representation of mean-reverting Ito process for an uncertain parameter Ki is given in Eq. (23).

Ki = ζi (μi − Ki ) + gi t



t

(23)

In Eq. (23), ζ i refers to the rate of reversion, μi is the mean value of uncertain parameter.  t refers to the random number from normal distribution with zero mean and standard deviation as one. gi is the variance parameter Eq. (24).



gi =

var (Ki ) t

(24)

The stochastic model reported in Verma et al. (2017) for enzymatic hydrolysis has been developed without considering the thermal deactivation kinetics of hydrolytic enzymes. The stochastic model reported in Verma et al. (2017) along with parameters related to thermal deactivation (kp1 and kp2 ) of enzymes is used in

J = max(E[G(t f )] )

(25)

The adjoint equations and necessary conditions are derived by using Ito’s lemma (Diwekar, 2003; Benavides and Diwekar, 2012). Similar to the stochastic maximization of glucose problem formulated in Section 4.1, the expected glucose concentration for batch time (tf ) was taken as the objective function. The Hamiltonian, adjoint equations are derived as given in Eqs. (26)–(28). 16 

zi Fi +

i=1



16  g2i wi 2 i=1

 dz j 1 ∂ Fi = − Z − dt ∂xj i 2 16

i=1

(26)





∂ g2i dx j

  wi

16  ∂ 2 g2i 1 dw j ∂ Fi = −2wi − zi − dt dx j 2 ∂ x2j i=1

(27)



∂ 2 ( gi ) 2 dx2j

  wi

(28)

In Eqs. (26)–(28), zi and wi are adjoint variables, Fi refers to RHS of model equations (Eqs. (1) to (3) and (9) to (10)). The complete description of Hamiltonian and adjoint equations for stochastic optimal control problem are given in the supplementary information. The above-formulated problem was solved by the method of steepest ascent. Due to the non-convexity of the stochastic optimal control problem, it was solved for different initial guesses of temperature. The stochastic optimal control problem was terminated when the difference in the objective function between two consecutive iterations became negative (Ji+1 − Ji < 0). The temperature obtained from this problem is referred to as “stochastic optimal control” in this paper. 5. Solution methodology The optimization and optimal control problems formulated in Section 4 are coded in MATLAB 2018®. The model equations explained in Section 2 are simulated using ode45. The optimization problems formulated for maximization of glucose concentration and minimization of variance in Sections 4.1 and 4.2, respectively are solved using active-set method in fmincon solver. Temperature is used as the decision variable for both the objective functions chosen. No bounds on temperature are considered for optimization problems. For stochastic optimization problems, 100 samples were drawn from the input range using Latin Hypercube Sampling (LHS). The stochastic optimization problems are solved by an approach similar to deterministic optimization problems. The deterministic optimal control problems formulated using Pontryagin’s maximum principle was solved using the method of steepest ascent. For stochastic optimal control problem, the dynamic propagation of uncertainties is represented by a stochastic model. In the stochastic model, each of the uncertain kinetic parameter considered is represented as state equations. The Hamiltonian is formulated for the objective function chosen using Pontryagin’s maximum principle. The adjoint equations and necessary condition were derived using Ito’s lemma. The optimal decision variables were derived using the method of steepest ascent. The deterministic optimization, stochastic optimization, and stochastic optimal control problems were solved for multiple guesses due to the non-convexity of the problems.

F. Fenila and Y. Shastri / Computers and Chemical Engineering 135 (2020) 106776

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Fig. 3. (a): Optimal temperature for maximization of glucose concentration (b): Percentage improvement in glucose concentration when proposed approach was used instead of conventional approach for kinetic parameter uncertainty.

6. Results and discussion The results of optimization and optimal control problems formulated in Section 4 are discussed in this section. 49 g/kg of cellulose, 14.85 g/kg of glucose, and 11.35 g/kg of xylose were taken as the initial composition. The sensitive kinetic parameters (k1r0 , k3r0 , K1IG , K3M , K3IG , K3IG , K3IG , Ea , α ) identified by global sensitivity analysis Verma et al. (2017) and deactivation parameters (kp1 , kp2 ) were considered for stochastic optimization and optimal control studies. All the selected parameters were assumed to be 5% uncertain. 100 samples were taken from the range defined using Latin Hypercube Sampling (LHS). 6.1. Maximization of glucose concentration (optimization) The results for maximization of glucose concentration problem for deterministic and stochastic values of feedstock composition and kinetic parameters are explained in this section. 6.1.1. Case 1 The optimal temperature decreased from 323.34 (K) to 318 (K) when batch time increased from 12 h to 168 h Fig. 3(a). This is attributed to the sensitivity of enzymes to temperature. Prolonged exposure of enzymes to a higher temperature will lead to loss of enzymes due to thermal deactivation, resulting in reduced glucose formation. The amount of enzyme (49 mg/g cellulose of E1 and 4.16 mg/g cellulose of E2 ) and substrate (49 g/kg of cellulose) considered is also the same for all the batch times. Therefore, if the same temperature is chosen for all batch times, the available enzyme will be more for smaller batch times due to less deactivation. 6.1.2. Case 2 The maximization of glucose concentration under kinetic parameter uncertainty was solved for 5% and 10% uncertainty. The results of 5% and 10% uncertainty were represented as case 2(a) and 2(b), respectively. For case 2(a) the glucose concentration improved up to 7.7% when the proposed approach was used instead of conventional approach Fig. 3(b). The mean glucose concentration increased from 53.2 g/kg to 57.3 g/kg for a batch time of 168 h when the proposed approach was used instead of the conventional approach. The decrease in the enzyme concentration when the conventional approach was used instead of proposed approach as represented in Fig. 4 confirms that the improvement in glucose concentration in the proposed approach was due to less deactivation

of enzymes. For example, the concentration of E1 was 0.509 g/kg for 168 h when conventional approach was used whereas, it improved to 0.995 g/kg for proposed approach due to reduction in temperature by 6 (K) Fig. 4(a). Similar improvement in enzyme concentration was observed in E2 also for batch times all the batch times considered Fig. 4(b). Similar to the case 1, the optimal temperature reduced with batch time for case 2(a) also Fig. 3(a). The glucose concentration improved up to 6.8% when the proposed approach was used instead of conventional approach for case 2(b) Fig. 3(b). The mean glucose concentration obtained for case 2(b) for a batch time of 168 h was 49.27 g/kg for conventional approach and 52.65 g/kg for the proposed approach. It can be noted that the mean glucose concentration has reduced for case 2(b) when compared with case 2(a). This is attributed to the fact that when the range of uncertain parameters is increased from 5% to 10%, the impact of deactivation parameters would also be high. This trend can be explained based on the distribution of output glucose concentration for both cases. The distribution of glucose concentration for both conventional as well as proposed approach for case 2(b) as shown in Fig. 5 were skewed right. The distribution of glucose concentration in Fig. 5 were obtained using same temperature for conventional approach. This indicate that the low conversion in case 2(b) is due to higher deactivation. It can be seen from Fig. 6(a) and (b) the range and frequency of glucose concentration overlap for both conventional as well as the proposed approach for a batch time of 12 h. Therefore, the improvement in glucose concentration is less (0.7%) for case 2(b). On the other hand, the range and frequency of glucose concentration for conventional and proposed approach differ more in case 2(a) Fig. 5. Therefore, there was a 2% improvement in glucose concentration for case 2(a) for a batch time of 12 h. The benefit of using proposed approach instead of conventional approach increased with batch times for both case 2(a) and 2(b) Fig. 3(b) since the difference in the optimal temperature was also more. The mean glucose concentration for case 2(a) and 2(b) using the conventional and proposed approach are tabulated in Table 3. 6.1.3. Case 3 Even though the uncertainty in feed composition causes variability in the output glucose concentration, the optimal temperature to maximize glucose did not change because the optimal temperature is a function of the reaction kinetics and deactivation kinetics Fig. 3(a). Therefore, the mean glucose concentration changes

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Fig. 4. Enzyme concentration at the end of batch various batch times using deterministic and stochastic optimum for maximization of glucose concentration (a): E1 (b): E2 .

Fig. 5. Distribution of glucose concentration for maximization of glucose concentration with kinetic parameter uncertainty for a batch time of 12 h for 5% uncertainty (a): Conventional approach (b): Proposed approach.

in the presence of feed composition uncertainty but the optimal temperature remains the same. For example, for a batch time of 12 h the mean glucose concentration obtained using conventional approach is 30.36 g/kg. The glucose concentration obtained in for case 1 for 12 h is 30.34 g/kg. 6.2. Minimization of variance (optimization) For minimization of variance problem, the objective function values are compared with the deterministic and stochastic opti-

mum for maximization of glucose concentration problem. From, Table 3 it can be seen that the mean glucose concentration reduced in presence of uncertainties. Therefore, the mean value obtained from stochastic maximization of glucose problem was considered as the target value for minimization of variance problem. The variance decreased by 89% when optimal temperature to minimize variance was used instead of deterministic optimal temperature to maximize glucose concentration. However, the variance decreased by 16% when the optimal temperature to minimize variance was used instead of stochastic optimal temperature to maxi-

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Fig. 6. Distribution of glucose concentration for maximization of glucose concentration with kinetic parameter uncertainty for a batch time of 12 h for 10% uncertainty (a): Conventional approach (b): Proposed approach.

Fig. 7. (a): Objecive function values calculated for minimization of variance using optimal temperature from case 1 and case 2 (a) of maximization of glucose problem (TMG−case1 and TMG−case2(a ) ) and minimization of variance problem (TMV ) (b): Optimal temperature for minimization of variance in presence of kinetic parameter uncertainty (case 1) and feedstock composition uncertainty (case 2). Table 3 Glucose concentration for maximization of glucose problem for deterministic values of kinetic parameters (case 1), stochastic values of kinetic parameters (case 2), and stochastic model with kinetic parameter uncertainty. Time (h)

12 24 48 72 168

Glucose concentration for maximization of glucose problems (g/kg) Case 1

Case 2 (a)

Case 2 (b)

Deterministic optimum

Deterministic optimum

Stochastic Optimum

Deterministic optimum

Stochastic Optimum

Deterministic optimum

Deterministic optimal control

Stochastic Optimum

30.34 38.15 46.43 50.89 58.23

27.44 33.91 41.08 45.7 53.2

27.7 34.85 43.27 48.24 57.3

26.29 31.99 38.53 42.32 49.27

26.38 32.06 38.69 43.04 52.65

24.84 28.2 31.28 32.86 34.26

24.96 28.37 31.66 33.23 36.52

25.94 31.19 37.14 40.8 49.03

mize glucose concentration Fig. 7(a). It can be noted from Fig. 7(b) that the optimal temperature to minimize variance for all batch times were less than 312 K below which thermal deactivation is less pronounced. The reduction in variance can be attributed to both less deactivation as well as slower kinetics, which eliminates extreme values of maximum and minimum. From Fig. 8, it can be seen that the objective function value had reduced significantly when the optimal temperature from the minimization of variance problem was used instead of maximization of glucose concentration problem. The range of glucose concentration reduced from 6 to 31 g/kg to 11 to 30 g/kg for a batch time of 24 h Fig. 9. Similar changes in glucose concentration were observed for other batch times also. Similar to the maximization of glucose prob-

Stochastic model Stochastic Optimal Control 26.95 32.73 40.37 45.32 54.88

lem for the deterministic and stochastic case, the optimal temperature for minimization of variance reduced with increase in batch time. The optimal temperature to minimize variance further reduced in comparison to the deterministic and stochastic optimum to maximize glucose concentration problems for all the batch times. For example, for a batch time of 12 h, the deterministic and stochastic optimal temperature for maximization of glucose concentration were 323.34 (K) and 317.07 (K), respectively. From Fig. 7(a) it is evident that the objective function for minimization of variance problem reduced significantly when optimal temperature to minimize variance was used instead of optimal temperature to maximize glucose (deterministic and stochastic optimum).

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Fig. 8. Distribution of objective function glucose concentration for minimization of variance problem for a batch time of 24 h. (a) Using deterministic optimum for maximization of glucose concentration (b) Using stochastic optimum for maximization of glucose concentration with kinetic parameter uncertainty (c) Using optimal temperature for minimization of variance with kinetic parameter uncertainty.

Fig. 9. Distribution of glucose concentration for minimization of variance problem for a batch time of 24 h. (a) Using deterministic optimum for maximization of glucose concentration (b) Using stochastic optimum for maximization of glucose concentration with kinetic parameter uncertainty (c) optimal temperature for minimization of variance with kinetic parameter uncertainty.

Even though the optimal temperature to maximize glucose concentration did not change in the presence of feed composition uncertainty, the minimization of variance around a targeted value problem was solved for feed composition uncertainty since it affects the amount of glucose produced. The optimal temperature did not change in the presence of uncertainty in the composition of feedstock up to 20% variation. For 30% variation in feed composition, there was a reduction in optimal temperature Fig. 7(b).

The higher concentration of background glucose resulting in more product inhibition and hence necessitates performing hydrolysis reaction at reduced temperature to conserve more enzymes. Therefore, there is a slight reduction in the optimal temperature in the presence of feed composition uncertainty. Possibility of 30% variation in feedstock composition for the same feedstock is less. However, the blending of crops with a higher difference in composition due to unavailability of the desired crop would result in

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Fig. 10. Concentration of enzymes for selected batch time (12 h) using stochastic optimum and stochastic optimal control (a) Concentration of E1 (b) Concentration of E2 .

a significant change in both composition and optimal operating conditions.

6.3. Maximization of glucose concentration (optimal control) Batch times 12 h to 168 h were considered for deterministic and stochastic optimal control. The stochastic optimal control problem formulated in Section 4.3 were solved for an initial guess between 318 to 326 (K). The state equations were solved by forward integration using solver ode45. The adjoint equations were solved by backward integration using solver ode 45. The stochastic optimal control problem was solved for batch times 12 h to 168 h. The average glucose concentration from the temperature profile obtained using the stochastic optimal control problem was compared with deterministic and stochastic optimization problem and deterministic optimal control. For the stochastic model, the mean glucose concentration was calculated by using four optimal temperatures, namely deterministic optimum, stochastic optimum, deterministic optimal control, and stochastic optimal control. Among the four different temperatures compared, the stochastic optimal control profiles yielded the highest concentration of glucose for all the batch times. The application of stochastic optimal control resulted in up to 60% improvement in glucose concentration when compared with deterministic optimum. For a batch time of 168 h, 34.26 g/kg was formed using deterministic optimum, whereas 54.88 g/kg was formed using stochastic optimal control (Table 3). The deterministic optimal control temperatures yielded more glucose concentration than deterministic optimum (Table 3). However, the stochastic optimal control resulted in up to 50.27% improvement in glucose concentration than deterministic optimal control. For example, for a batch time of 168 h, deterministic, and stochastic optimal control yielded 36.52 and 54.88 g/kg of glucose, respectively. Up to 11.93% improvement in glucose concentration was obtained when stochastic optimal control was used instead of stochastic optimum. The significant reduction in glucose concentration for deterministic optimum and deterministic optimal control was due to the fact that the presence of uncertainties was not considered in the derivation of optimal operating temperature.

For hydrolysis carried out for a batch time of 12 h, at the deterministic optimum (323.34 K), the glucose concentration of 24.84 g/kg was achieved whereas at the stochastic optimum (317.07 K) 25.94 g/kg was produced (Table 3). In comparison, the mean value of glucose concentration obtained using stochastic optimal control was 26.25 g/kg (Table 3). The glucose concentration using stochastic optimal control is 5.67% higher than deterministic optimization. Similarly, the glucose concentration using stochastic optimal control is 0.11% higher than stochastic optimum. Similar to 12 h, for other batch times also the stochastic optimal control profiles yielded higher glucose concentration than the optimal temperature from stochastic optimization problems. The range, as well as the distribution of glucose concentration, changed for stochastic optimum and stochastic optimal control (distribution of glucose concentration using stochastic optimum and stochastic optimal control is given in supplementary information). The glucose concentration improved up to 60% and 11.93% in comparison to deterministic and stochastic optimum 168 h. From Fig. 10 it can be observed that the concentration of enzymes (E1 Fig. 10(a) and E2 Fig. 10(b) are higher when stochastic optimal control was used instead of stochastic optimum. This confirms that the improvement in glucose concentration is due to less degradation of enzymes for stochastic optimal control. The temperature profiles for deterministic and stochastic optimal control problem started with a lower temperature, followed by a gradual increase. The variation in the stochastic optimal control profiles was due to the uncertainties propagated in the stochastic model. For example, for a batch time of 12 h, the initial temperature was 306 (K) which reached 316.5 (K) at the end of 12 h. The lower temperature in the initial hours of hydrolysis ensures lower deactivation of enzymes. The increase in temperature as the reaction proceeds is to increase the reaction rate as the enzymes are not recycled after the completion of the reaction. The variation in temperature profile at every time step is due to the variation in the kinetic parameters. A similar trend was observed for the optimal temperature profiles obtained for other batch times also Fig. 11. The optimal control profiles reached temperatures as low as 270 (K) when solved without any bounds on the decision variable. However, there are no reported values of hydrolysis performed at temperatures such as 270 (K). Moreover, the enzyme activity will also be not stable at such low temperatures.

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Fig. 11. Optimal temperature profile for different batch times from optimal control problems for maximization of glucose problem. (a) Deterministic optimal control (b) Stochastic optimal control.

Since the literature on enzymatic hydrolysis available for 298 (K), a lower bound of 298 (K) was given for control variable. The time steps where the temperature reached below 298 (K), the stochastic optimal control profiles showed a constant value of 298 (K). For example, the temperature profile for the batch time of 168 h remained at 298 (K) for first 44.6 h.

7. Conclusion The impact of uncertainties in feed composition as well as kinetic parameters on mean glucose concentration and the variation in the glucose concentration were studied in this work. The optimal operating temperature reduced with increase in batch time owing to the thermal deactivation of enzymes. The optimal temperature for maximization of glucose concentration reduced by 8.1 (K) and 20.8 (K) for 5 and 10% uncertainty in kinetic parameters. The approach proposed in this work improved glucose concentration by 7.7 and 6.8% for 5 and 10% uncertainty. The decrease in variance was higher for smaller batch times in comparison to stochastic optimum for maximization of glucose. Significant changes in the temperature profile and glucose concentration were observed when uncertainties were considered dynamic instead of static. The stochastic optimal control profiles obtained from this work improved the glucose concentration by 60% and 11.93% when used instead of deterministic and stochastic optimal temperature for maximization of glucose concentration. Based on the results obtained in this work, it can be concluded that the uncertainty in kinetic parameters has a significant impact on the optimal operating variables as well as the performance of the enzymatic hydrolysis process. Moreover, the incorporation of uncertainties in deriving optimal decision variables could reduce the impact of these variables. The results presented in this work is applicable to the range of uncertainty considered here. Therefore, if the uncertainty range is observed to be larger in practice, the optimization problems will need to be re-solved considering the new range. Moreover, the applicability of this method is also limited by inherent model assumptions such as assumption of perfectly well mixed reactor, and no mechanical degradation of enzymes. In this work, the impact of uncertainties on economics were not considered. Therefore, this work could be extended to study the impact of uncertainties on the economics of enzymatic hydrolysis.

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