Modeling and Parameter Updating for Nosiheptide Fed-Batch

Article pubs.acs.org/IECR

Modeling and Parameter Updating for Nosiheptide Fed-Batch Fermentation Process Dapeng Niu,*,† Long Zhang,‡ and Fuli Wang† †

College of Information Science and Engineering, Northeastern University, Shenyang 110819, China 606 Institute, Chinese Aeronautical Establishment, Shenyang 110015, China

‡

ABSTRACT: Nosiheptide is a sulfur-containing peptide antibiotic obtained through fermentation. It can be used as feed additives because of its relative safety and good effect. However, nosiheptide fermentation does not have a high yield. Keeping the fermentation environment or operating conditions optimum through optimization is an effective way to improve nosiheptide’s yield, while accurate and reliable process models are the basis to achieve process optimization. Based on the reaction mechanism of the nosiheptide fed-batch fermentation process, we establish its mechanism models. Fermentation processes have slow time-varying characteristics and the conditions usually change due to disturbances during the production process, so the accuracy of established models tends to decline. Thus, models do not match the actual process gradually, leading to model mismatch, which have bad effects on the optimization and control of the process. Therefore, it is necessary to update the process models in time. In this paper, we update process models through identification of model parameters. Because there are many parameters in nosiheptide fed-batch fermentation process models, the calculation cost is quite large to correct all the parameters simultaneously. Considering that different parameters have different effects on process model output, we propose a model updating method based on parameter sensitivity analysis. During the model updating process, we identify and correct the parameters having a main impact on the models, while ignoring the parameters having secondary effects. Thus, we realize updating of the mechanism models for nosiheptide fed-batch fermentation process. Simulation results show that the proposed model updating method improves the models’ accuracy.

1. INTRODUCTION Nosiheptide is a sulfur-containing peptide antibiotic produced by Streptomyces actuosus.1 As a new nonabsorbing feed additive, nosiheptide is generally obtained through fermentation, but its yield is not high. Optimization of nosiheptide fermentation process is an effective way to solve such a problem, while process models are the basis of process optimization. Therefore, how to build accurate and reliable process models is one of the key issues of fermentation process optimization.2−4 Existing models of fermentation processes include mechanism models, black box models,5−7 etc. Black-box models are usually the mapping relation between biochemical variables and environment variables obtained through neural networks or other techniques. When there are plenty of modeling data, we can achieve accurate black box models. Unfortunately, it is quite difficult to obtain enough data for establishment of black box models during the nosiheptide fermentation process. Mechanism models can intuitively reflect the basic knowledge of the fermentation process, so they are widely used in fermentation processes. In this paper, we take the characteristics of the nosiheptide fed-batch fermentation process into full consideration and establish the mechanism models according to fermentation kinetics and mass balance. The models will the lay © XXXX American Chemical Society

foundation for the control and optimization of the process. However, nosiheptide fermentation process has slow timevarying characteristics and uncertainty.8 When fermentation conditions or production processes change, the built models tend to deviate from the actual process. This will affect the consequent control and optimization study on the process. Hence, we must update the established model in a timely manner. Model updating is known as the adjustment process of model parameters or its structure.9 It aims to make the output of the model consistent with the actual measurements and to improve the model’s practicality in applications. In this paper, we use a fixed model structure and adjust model parameters to achieve model updating for the nosiheptide fed-batch fermentation process. And we update the process models based on the recent past batches, considering that model mismatch is usually gradually caused by the slow time-varying disturbances and process characteristics during the fermentation production process. Received: April 1, 2016 Revised: July 11, 2016 Accepted: July 11, 2016

A

DOI: 10.1021/acs.iecr.6b01245 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

electrodes, dissolved oxygen (DO) electrodes, and other testing equipment. We can measure several variables on line, such as fermentation broth temperature (T), tank pressure (p), air flow rate (Q), stirring speed (n), fermentation broth pH value, DO concentration, and the exhaust gas concentration of oxygen and carbon dioxide. Streptomyces actuosus, provided by Shenyang Pharmaceutical University, is used as the experimental bacteria. The fermentation culture is composed of starch (55 g/L), soybean flour (40 g/L), yeast extract (2 g/L), sodium chloride (4 g/L), ammonium sulfate (1 g/L), potassium nitrate (0.5 g/L), and calcium carbonate (4 g/L). The pH value of the culture is 7.0. The initially loaded culture volume is 50 L. The nosiheptide fed-batch fermentation experiment is carried on using the provided bacteria and culture. After pasteurization is accomplished using the steam, the 100 L fermentor is loaded with 50 L of culture and 6 L of bacteria solution. With ventilation of 3.0 m3/h, stirring speed of 400 rotations/min, tank pressure of 35 Pa, and broth temperature of 29 °C, the fermentation process continues for 96 h. During the process, new culture may be fed into the fermentation tank according to the progress. Physical variables can be obtained directly through the data acquisition system, while the chemical variables such as biomass, substrate, and nosiheptide concentrations are measured offline through analysis of samples every 6 h.

There are many parameters in nosiheptide fed-batch fermentation process models. If we update all parameters at the same time, the calculation amount is quite large and overfitting is liable to happen. Through careful analysis, we find that different parameters have different effects on model output values of the state variables. For example, some parameters influence several state variables, while other parameters only influence one certain state variable; some parameters influence one state variable strongly, while other parameters almost have no effects on that variable. Considering this situation, we propose a model updating strategy based on parameter sensitivity analysis. During the process of model updating, we analyze and identify the main parameters in the model and ignore the other parameters, to reduce the calculation amount and to improve the efficiency of model identification and updating. Sensitivity analysis, often known as SA, is a way of evaluating the relative importance of model parameters.10,11 It can effectively analyze the influence of model parameters on the state variables, so as to provide a basis for model parameter correction and process optimization. SA has been widely used in atmospheric science, economics, ecology, and such other fields.12We carry on SA to the constructed models of the nosiheptide fed-batch fermentation process and then update the models so as to lay a reliable model foundation for subsequent simulation and optimization control research. The remaining parts of this paper are organized as follows: Section 2 describes the nosiheptide fermentation experiment. Section 3 builds the kinetic models for nosiheptide fed-batch fermentation process based on analysis of the main factors affecting the process, and a differential evolutionary algorithm is used for identification of unknown parameters. Section 4 updates the model parameters based on parameter sensitivity analysis; Section 5 concludes the work.

3. MODELING OF NOSIHEPTIDE FED-BATCH FERMENTATION PROCESS During the nosiheptide fermentation process, the bacteria grow in the broth with suitable pH and temperature. The growth and metabolism pathway is extremely complex. For such a complex process, we would simplify it to build its models. Basic simplifications for the fermentation process include the following: stirring system inside the reactor ensures ideal mix, avoiding the differences between different regions of temperature, pH, and concentration of the substance; temperature, pH, and other environmental conditions can be controlled to remain stable, so that the corresponding kinetic parameters are also kept steady; cells have inherent chemical compositions, not changing with the fermentation time and conditions; and various descriptive variables for fermentation dynamics have no significant reaction lag to fermentation condition changes. Under these simplified conditions, we will establish the models for biomass growth, substrate consumption, dissolved oxygen consumption, and product formation according to the reaction mechanism and material balance. 3.1. Biomass Growth Model. The model of biomass growth can be expressed as13

2. NOSIHEPTIDE FED-BATCH FERMENTATION EXPERIMENT We use a 100 L stirred fermentation tank with control system for nosiheptide fermentation. Other related systems include the air supply system, the steam treatment system, the water heating system, and their corresponding control systems. The fermentation equipment is shown in Figure 1. Our laboratory also has exhaust-gas analyzers, mass spectrometers, pH

dX = μg X − μd X dt

(1)

where X is biomass concentration, g/L; μg is specific growth rate, h−1; μd is specific death rate, h−1. During the batch fermentation process, the cell growth is mainly affected by temperature, pH, dissolved oxygen, substrate concentration, and the concentration of cells itself. These factors have important influence on biomass specific growth rate and mortality ratio.14−17 Combining them with the series theory, we can get

Figure 1. Main body of fermentation equipment system. B


Article

Industrial & Engineering Chemistry Research μg =

Ag exp[−Eg /R(273 + T )] −pH

−pH

+ 10 1 + K1/10 ⎛ ⎞ X ⎜1 − ⎟X X ⎝ max ⎠

/K 2

·

dC O 1 dX = KLa ·(C* − CO) − mO·X − · YX /O dt dt

CO S · · KSX + S K OX + CO

⎛ CO ⎞ μd = Ad exp[−Ed /R(273 + T )]·⎜1 − ⎟ Kd + CO ⎠ ⎝

− (2)

(3)

KLa = 0.1322·

dV dt

P00.36·n0.18·d 0.56·Q 0.3992 V 0.4·D

(7)

where P0 is the input power under nonaerobic conditions, W; n is stirring rate, rotations/min; d is agitator diameter, m; Q is ventilation volume, m3/h; and D is fermentation tank diameter, m. Substitute eq 7 into eq 6, and then the dissolved oxygen model can be obtained as dC O P 0.36·n0.18·d 0.56·Q 0.3992 = 0.1322· 0 ·(C* − CO) dt V 0.4·D 1 dX 1 dP F − mO ·X − · − · − CO YX /O dt YP / O dt V

Ag exp[−Eg /R(273 + T )] CO dX S = · · · −pH −pH dt + 10 /K 2 KSX + S K OX + CO 1 + K1/10 ⎛ X ⎞ ⎜1 − ⎟X − Ad exp[−Ed /R(273 + T )]· X max ⎠ ⎝

where F =

(6)

where KLa is volumetric oxygen transfer coefficient, h−1; C* is saturation oxygen concentration, g/L; CO dissolved oxygen concentration of the broth, g/L; mO is maintenance coefficient on oxygen, (g/g)/h; YX/O is yield constant (biomass/dissolved oxygen), g/g; and YP/O is yield constant (product/dissolved oxygen), g/g. The volumetric oxygen transfer coefficient KLa is modeled as18

where μg biomass specific growth rate, h−1; Ag is Arrhenius growth constant; Eg is growth activation energy, kJ/mol; R is universal gas constant, J/(mol·K); T is broth temperature, °C; K1 and K2 are constant; pH is pH value of the broth; S is substrate concentration, g/L; KS is substrate Contois saturation constant, g/L; X is biomass concentration, g/L; CO is dissolved oxygen concentration, g/L; KO is Contois saturation constant of dissolved oxygen, g/L; Xmax is the maximum biomass concentration, g/L; μd is cell mortality ratio, h−1; Ad is Arrhenius death constant; Ed is death activation energy, kJ/mol; and Kd is Monod constant, g/L. Substituting eqs 2 and 3 into eq 1 and taking changes in the broth volume into account caused due to feeding, we can get the following growth model

⎛ CO ⎞ F ⎜1 − ⎟− X Kd + CO ⎠ V ⎝

1 dP F · − CO YP / O dt V

(8)

3.4. Product Formation Model. The composition process of microbial metabolites is very complex. And biosynthetic pathways and metabolic regulation mechanisms change with different strains and products. According to the relationship between product formation rate and biomass growth rate, product formation processes can be classified into three types, i.e., growth-coupled type, partly growth-coupled type, and nongrowth-coupled type. Nosiheptide is a secondary metabolite, and its formation process belongs to non-growth-coupled type. The Luedeking−Piret model is a classical model commonly used to describe the formation of microbial metabolites. It takes the productivity as a function of cell growth rate and cell concentration. The model can be expressed as19

(4)

is the feed flow rate of substrate, m3/h and V is

culture volume, m3. 3.2. Substrate Consumption Model. During the nosiheptide fermentation process, the consumed substrate is mainly used in three ways: to maintain the basic life activities of bacteria, to provide nutrients needed for cell growth and reproduction, and to produce metabolites. Taking feeding into account, according to the mass balance, the substrate consumption model can be expressed as

dP dX =α + βX dt dt

(9)

where α is the product formation coefficient associated with cell growth rate, g/g and β is the product formation coefficient associated with biomass concentration, g/(g·h). Its kinetic equation can be simplified to

dS 1 dX 1 dP F = −mS ·X − · − · − (S − SF ) dt YX / S dt YP / S dt V (5)

dP = βX dt

where mS is maintenance coefficient on substrate, (g/g)/h; YX/S is yield constant (biomass/glucose), g/g; YP/S is yield constant (production/glucose), g/g; P is production concentration, g/L; SF is feed substrate concentration, g/L. 3.3. Dissolved Oxygen Model. The fermentation tank pressure is kept almost invariant through regulating the exhaust gas in nosiheptide fed-batch fermentation process. So we assume that the pressure is constant during the entire fermentation process. Under this assumption, according to the law of mass balance, the dissolved oxygen model is obtained as

(10)

Nosiheptide production belongs to the noncoupled type and it is regardless of the cell growth rate, so α = 0. At the same time, nosiheptide is a small molecule compound and it is sensitive to hydrolysis. The fermentation broth volume change caused by feeding can also affect the product concentration. Therefore, the nosiheptide production model is dP F = βX − KhP − P dt V

(11) −1

where Kh is hydrolysis rate constant, h . C


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where a and b are randomly selected distinct integers from 1 to Np. (3) Cross operations: Carry on binomial distribution crossG+1 G+1 over between the generated ẑG+1 = [ẑG+1 i 1i , ẑ2i , ..., ẑni ] in step (2) and the individual in the current population, namely,

3.5. Model Parameter Estimation. Equations 4, 5, 8, and 11 constitute the kinetic models of the nosiheptide fed-batch fermentation process. Parameters to be determined include Ag, Eg, R, K1, K2, KS, KO, Xmax, Ad, Ed, Kd, mS, YX/S, YP/S, P0, d, D, C*, mO, YX/O, YP/O, β, and Kh. The parameters Ag, Eg, R, Ad, and Ed can be determined with chemical experimental techniques. P0, d, and D are decided by the fermentation tank type. Through calculation with the measured data, C* can also be obtained. The other parameters are to be determined through estimation method. Fermentation kinetic parameter estimation is a highly nonlinear optimization problem, and there are a number of local extreme points. The traditional method for parameter identification of nonlinear systems is often unable to obtain global optimal solution. With the rise of a simulated annealing and genetic algorithm, modern optimization algorithms are widely used in fermentation processes for solving optimization problems. Differential evolution (DE) in 1995 proposed by Storn and Price, with the genetic algorithm as the basic framework, uses differential operations for real-coded genetic individuals to achieve crossover and mutation.20 It is fast and robust and has strong search capabilities on the real field. The algorithms has been applied to many classic optimization problems, including system identification and parameter optimization, and it has achieved great success.21,22 In this paper, differential evolution algorithm is used to estimate the parameters of fermentation kinetic models. When kinetic parameters are known, we can calculate the average relative error between mechanism model outputs and experimental values23 with J=

1 Ne +

z ̅jiG + 1

⎡ X (r ) − X (r ) S(r ) − Se(r ) e + ⎢ Xe(r ) Se(r ) r=1 ⎣ Ne

(12)

where Ne is the number of all sampling points for multiple batches of fermentation; Xe(r), Se(r), COe(r), and Pe(r) are the experimental values of biomass concentration, substrate concentration, product concentration, and dissolved oxygen concentration for the rth sample, respectively; and X(r), S(r), CO(r), and P(r) are the model output values of biomass concentration, substrate concentration, product concentration, and dissolved oxygen concentration for the rth sample, respectively. The basic principle to estimate the kinetic parameters is to make the relative error of the eq 12 minimum. That is the objective function of the optimization problem. Assuming that the number of pending parameters z1, z2, ..., zn is n, we can use DE for parameter estimation, writing the ith (1 ≤ i ≤ Np, Np is the size of the population) individual of the population as an ndimensional vector zi = [z1i, z2i, ..., zni]. The steps of DE are as follows: (1) Initialization: Generate the initial population z0i (i = 1, ..., Np), determine the mutation rate FM, the crossover probability factor CR, and the maximum evolution generation Gm, and let G = 0. (2) Mutation operation: Generate variation individual according to the following equation zî G + 1 = ziG + FM(zaG − zbG)

j = 1, ..., n (14)

where CR ∈ [0,1] is the crossover probability factor and PC is a random number between [0,1]. (4) Competition operation: Compare the generated zG+1 i̅ with zGi , replace zGi with zz̅ G+1 if the latter is better i than the former, or keep zGi into the next generation, otherwise. (5) Repeat steps 2−5 until the optimal solution is obtained or G > Gm. With eq 12 as the objective function, we use the DE algorithm to estimate the unknown parameters of nosiheptide fed-batch fermentation models. We first acquire the data set for parameter estimation from nosiheptide fermentation experiments. The online measurable variables are automatically obtained with sensors, and the sampling period is 6 min. Since the biomass concentration, substrate concentration, and product concentration are collected through laboratory analysis of samples off-line, the sampling period, 6 h in this work, is relatively long. To supplement the “lost” experimental data, we synchronize the biochemical data with the online measurable variables. We use the interpolation fitting method to deal with the biochemical data. The fermentation process is a slow process, and there is little probability of sudden changes in the two samples, so we adopt the cubic smoothing spline fitting method to process the data so as to obtain all the modeling data. We in total get 30 batches of nosiheptide fermentation experimental data, with 27 batches for model parameter identification and 3 batches left for verification. The result of model parameter estimation probably would be affected by process noise or faults, so we removed noise by normalization and supposed that the process is carried on normally without serious faults. DE algorithm parameter settings are population size NP = 140, crossover probability factor CR = 0.9, mutation rate FM = 0.6, and the maximum evolution generation 1000. Kinetic parameters obtained are summarized together with the known parameters in Table 1. Using the validation data to test the mechanism model, we can get the comparison results of experimental measurements and model predictions for biomass concentration, substrate concentration, dissolved oxygen concentration, and product concentration, as shown in Figure 2. In order to quantitatively describe the performance of the models, we use the average relative error (ARE), maximum relative error (MRE), and mean square deviation (MSD) to illustrate the predicting effects of the models as shown in Table 2. Table 2 shows the errors between the model output values and the experimentally measured values. Among them, the average relative error and the maximum relative error for product concentration, 11.59% and 13.37%, respectively, are largest, followed by biomass concentration and substrate concentration. Overall, the established process models for nosiheptide fed-batch fermentation are able to follow the

∑⎢

CO(r ) − COe(r ) P(r ) − Pe(r ) ⎤ ⎥ + ⎥⎦ Se(r ) Pe(r )

⎧ zG , P > C c R ⎪ ji =⎨ , G 1 + ⎪ zjî , otherwise ⎩

(13) D


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Industrial & Engineering Chemistry Research Table 1. Values of Model Parameters

Table 2. Errors between Experimental Data and Predictions

parameters

values

Arrennius constant for growth: Ag activation energy for growth: Eg universal gas constant: R constant: K1 constant: K2 substrate Contois saturation constant: KS Contois saturation constant of dissolved oxygen: KO maximum biomass concentration: Xmax Arrennius constant for growth: Ad activation energy for cell death: Ed Monod constant: Kd maintenance coefficient on substrate: ms yield constant (biomass/glucose): YX/S yield constant (nosiheptide/glucose): YP/S stirring power under nonaerobic conditions: P0 agitator diameter: d fermentation tank diameter: D saturation oxygen concentration: C* maintenance coefficient on oxygen: mO yield constant (biomass/dissolved oxygen): YX/O yield constant (product/dissolved oxygen): YP/O constant: β Hydrolysis rate constant: Kh

0.12 h−1 60 kJ/mol 8.31 J/(mol·K) 10−10 1.3 × 10−4 0.18 g/L 0.035 g/L 0.87 g/L 0.0019 h−1 340 kJ/mol 0.037 0.062 (g/g)/h 0.25 g/g 0.68 g/g 1500 W 0.01 m 0.5 m 0.037 g/L 0.47 (g/g)/h 0.028 g/g 0.091 g/g 0.051 g/(g·h) 0.0004 h−1

biomass concentration (g/L) substrate concentration (g/L) dissolved oxygen concentration (%) product concentration (g/L)

ARE

MRE

MSD

1.11% 0.46% 0.25% 11.59%

2.59% 0.87% 0.64% 13.37%

0.008 0.204 0.002 0.021

4. MODEL UPDATING FOR NOSIHEPTIDE FED-BATCH FERMENTATION PROCESS BASED ON PARAMETER SENSITIVITY ANALYSIS With the continuity of fermentation experiments, large quantities of experimental data will be preserved. How to make effective use of these experimental data is important for model correction of the nosiheptide fermentation process. We use the sliding window technology to obtain the sample data required for model calibration. During model calibration, the current state of the fermentation process is mainly depicted by the past N batches of data in the sliding window. When a new batch of sample data is collected, the earliest batch of sample data in the window is removed from the N batches, and the new data is added immediately. Thus, with the progress of nosiheptide fermentation experiments, the sample data in the sliding window is updated continuously, which will not only avoid the growing amount of data but also effectively renew the collected sample data. We carry on model calibration through adjusting the model parameters. However, there are many model parameters in nosiheptide fed-batch fermentation process models. If we

experimental measurements, laying foundation for subsequent model calibration and process optimization.

Figure 2. Prediction of established models (model output is plotted as dash line, while * represents experimental data). E


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Figure 3. Verification of updated models (model output is plotted as dashed line, while * represents experimental data).

calibrate all the parameters in the meantime, a large amount of calculation will be needed. Through careful analysis of these parameters, we can see that different parameters take different effects on the fermentation process. Some parameters have big influences on several state variables and have very small influences on other variables. So, we propose a strategy of model updating based on parameter sensitivity analysis, correcting the main parameters affecting the model and ignoring the secondary parameters. Sensitivity analysis is a method for evaluating the impact of parameter changes on model output. We can use it to analyze the impact of parameters on model state variables of the nosiheptide fermentation process. We divide the variables and parameters in nosiheptide fermentation models into four categories, i.e., state variables, fixed parameters, nonfixed parameters, and operating variables. Then, we write the established models in a unified expression of differential equations as follows. Y ̇ = f (Y , Pa , Pb , u)

S pyi = bj

∂yi ∂pbj

(16)

then the average value for sensitivity during the period of Δt can be written as S ̅pyi = bj

1 Δt

t 0 +Δt

∫t 0

S pyi dt bj

(17)

Equation 17 is calculated numerically, and the period Δt is 6 min. Normalize the average sensitivity coefficient, and we can get the relative sensitivity value of each parameter to each state variable. In the calibration process, we select the specific calibration parameters, considering not only the relative sensitivity values but also the errors between the model output and the experimental measurements. Thus, the process becomes a process of information fusion for centralized decision-making. We use the voting method for decisionmaking information fusion to calculate the integrated value of the parameter sensitivity. The adopted voting principle in this work is “one person one vote”. Each state variable is assigned a weight of 0.25, and its vote attitude to each state variable is determined by the relative error of the model output and experimental measurements:

(15)

where Y is state variable vector, Y = {X, S, CO, P}; Pa is fixed parameter vector, Pa = {R, Xmax, P0, d, D}; Pb is nonfixed parameter, Pb = {Ag, Eg, K1, K2, KS, KO, Ad, Ed, Kd, mS, YX/S, YP/S, C*, mO, YX/O, YP/O, β, Kh}; u is operating variable vector, u = {F, T, pH, Q, n, SF}, and f = {f1, f 2, f 3, f4}. Define the sensitivity of a certain parameter pbj to a particular state variable yi as24

Ey = i

1 N

N

∑ l=1

|yil* − yil̂ | y* il

(18)

where Eyi is the error of state variable yi, y*i is experiment value, ŷi is model estimation value, and N is the number of samples. F


Article

Industrial & Engineering Chemistry Research On the basis of eqs 16 and 17, we give the synthetic sensitivity coefficient of pbj as following

Table 3. Errors between Experimental Data and Predictions

4

Sp = bj

∑ (0.25 × S ̅px Ex )

biomass concentration (g/L) substrate concentration (g/L) dissolved oxygen concentration (%) product concentration (g/L)

i

bj

i

i=1

(19)

Similarly, we can get the synthetic sensitivity coefficients for each parameter of the nosiheptide fermentation process. According to the actual situation, we decide the sensitivity threshold of parameter correction as S*. The parameters, whose corresponding synthetic sensitivity coefficients are bigger than S*, need to be modified, while other parameters remain unchanged. When a new batch of sample data slides into the sliding window, the calibration procedure is started. First, the accuracy of the mechanism models corresponding to the new batch is calculated. If the model accuracy is high enough, the system will wait for new sample data until the model accuracy is smaller than a set value. Then, the error of each state variable is analyzed. And parameters in the nosiheptide fed-batch fermentation process, needing to be corrected, should be determined based on the synthetic sensitivity coefficients. At last, the process model is corrected and updated with historical data in the sliding window, aiming to make the model reach the highest precision. For the nosiheptide fermentation process, the precision threshold is set as θ*. When a new batch of sample data, whose precision θ is less than θ*, slides into the sliding window models, the established models will be corrected. In practical applications, precision of the nosiheptide fermentation process model is expressed as follows 1 θ= 4 × Ns

⎛ |yij* − yiĵ | ⎞ ⎜ ⎟ × 100% ∑ ∑ ⎜1 − * ⎟ y i=1 j=1 ⎝ ⎠ ij 4

ARE

MRE

MSD

1.1% 0.49% 1.31% 4.55%

2.61% 1.54% 2.84% 6.11%

0.0082 0.2635 0.0112 0.0080

fed-batch fermentation process models after correction can follow the experimental measurements quite well.

5. CONCLUSION The nosiheptide fermentation process is a complex biochemical process. The parameter identification problem for its models is a highly nonlinear optimization problem with many local extreme points. On the basis of the mechanism models, we use a differential evolution algorithm for parameter identification to determine unknown parameters of nosiheptide fed-batch fermentation kinetic models. Because the fermentation process has characteristics of variability and uncertainty, the established models often deviate from the true values of the actual process when directly applied to the actual process, resulting in model mismatch. This may severely affect subsequent control and optimization research. Hence, it is necessary to update the established models in timely manner. This paper presents a parameter synthetic sensitivity analysis method based on errors of state variables. In the model calibration process, we identify and correct the main parameters affecting the model, ignoring secondary parameters, to ensure model updating speed and its precision. Simulation results show that the established fermentation model and model updating method proposed are both effective.

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N

AUTHOR INFORMATION

Corresponding Author

(20)

*(D.N.) E-mail: [email protected]. Tel.: 86-013889160713.

where yij* is experimental values, ŷij is model output, and Ns is the number of samples. On the basis of eq 19 for calculation of synthetic sensitivity coefficients, we can get the coefficients of the nosiheptide fermentation process. With sensitivity threshold S* set as 0.0002, we decide 7 parameters to be corrected as follows, Ag, Eg, mS, YX/S, C*,β, and Kh. The size of the sliding window W is determined through experiments. Since a big size may reduce the efficiency of model updating, we set W as 1, 4, 7, 10, 13, 16 for model updating, respectively, and calculate the average ARE for another 12 batches of experiments besides the originally obtained 30 batches. All the experiments are carried out under normal production conditions, and process noise is removed through normalization. When W is set as 10, the average ARE get the smallest value. So, we choose W = 10 as the size of the sliding window. Figure 3 shows model output curves of biomass concentration, substrate concentration, product concentration, and dissolved oxygen concentration. ARE, MRE, and MSD between model output and experimental values are shown in Table 3. From the table, we can see that the average relative error and maximum relative error reach a maximum of 4.55% and 6.11%, respectively, followed by dissolved oxygen concentration. Substrate concentration has the largest variance, reaching 0.2635, followed by dissolved oxygen concentration, biomass concentration, and product concentration. Overall, nosiheptide

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Nature Science Foundation of China (No. 61304121) and the Fundamental Research Funds for the Central Universities (No. N150404017).

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REFERENCES

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Modeling and Parameter Updating for Nosiheptide Fed-Batch

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