Simultaneous hybrid modeling of a nosiheptide fermentation process using particle swarm optimization☆

Qiangda Yang *,Hongbo Gao ,Weijun Zhang Huimin Li

1 School of Metallurgy,Northeastern University,Shenyang 110819,China

2 Department of Electromechanical Engineering,Liaoning Provincial College of Communications,Shenyang 110122,China

1.Introduction

Nosiheptide,produced by Streptomyces actuosus during fermentation,is a sulfur-containing peptide antibiotic.It has been widely used as a feed additive for animal growth because of its non-toxic and residue-free properties[1].However,most industrial nosiheptide production currently has problems of low yields and high costs.The optimization of the nosiheptide fermentation process could help mitigate the above problems,and availability of a process model is often the basis and premise.

Models of fermentation processes are classically developed on the basis of balance equations together with rate equations for microbial growth,substrate consumption and product formation[2].However,due to the complexity of many fermentation processes(including the nosiheptide fermentation process),the underlying physicochemical phenomena are seldom fully understood,and the development of mechanistic models is costly,time-consuming and tedious.As a result,empirical approaches have been widely employed to develop fermentation process models[3].In empirical modeling,the model is developed exclusively from the historical data without invoking the process phenomenology[4,5].Thus,the expensive,time-consuming and tedious search for a mechanistic model can be avoided.

Recently,hybrid modeling approaches have been investigated as an attractive alternative to develop fermentation process models[6].The hybrid model of a fermentation process commonly consists of a set of nonlinear differential equations that incorporates the available priori knowledge about the process under consideration,and some empirical models that each estimates one of the unknown variables in the differential equations[7–13].Moreover,hybrid models have been shown,in many applications to fermentation processes,to have better properties than pure empirical models[9,10,12];they have better generalization ability,are easier to analyze and interpret,and require significantly fewer training examples.

In this paper,we consider the use of a hybrid model to represent the dynamic behavior of a lab-scale nosiheptide batch fermentation process.Artificial neural networks are often used for the empirical part of a hybrid model due to their powerful nonlinear mapping ability[14].In this study,we utilize multilayer feed for ward neural networks(MFNNs)to develop the empirical model part,considering that such networks have been successfully applied in many fermentation processes[10–13].One major challenge for hybrid modeling of fermentation processes is the lack of empirical model target outputs,because there are at present no proper sensors or methods to directly measure the unknown variables such as the specific growth rate,specific consumption rate,and specific production rate.To address this problem,most of the above mentioned works first estimate the unknown variables from the off-line sparse measurements of state variables,and then train the empirical models using the estimation values as their target outputs.However,the reliability and accuracy of unknown variable estimation values are often not guaranteed,and this would considerably influence the generalization ability of the derived hybrid model.Therefore,in this paper we propose a simultaneous hybrid modeling approach that transforms the training of empirical models into a dynamic system parameter identification problem.This allows the in direct training of empirical models without kno wing their target out puts,and thus overcomes the shortcoming of existing hybrid modeling approaches that must use unreliable and inaccurate estimation data for the training of the empirical model part.

The resulting parameter identification(hereafter referred as RPI)is a complex optimization problem.Traditional optimization algorithms cannot be used efficiently to solve it.In recent years,with the development of intelligence optimization algorithms,particle swarm optimization(PSO)has been widely applied to various optimization problems due to its easy implementation and fast convergence[15–17].However,the standard PSO(SPSO)can be easily trapped in a local optimum,especially when the optimization problem is complex and has many dimensions,as is the above RPI problem[18].Therefore,an improved PSO,called AEPSO,is proposed in this paperby introducing escaping and adaptive inertia weight adjustment strategies into SPSO.The former strategy is to avoid being trapped in a local optimum,and the latter is to improve the convergence speed.Further,AEPSO is employed to solve the RPI problem,and thereby accomplish the training of the empirical model part.In addition,the selection of the empirical model structure,namely the topology of the corresponding neural networks is also addressed.

2.Materials and Methods

2.1.Microbial strain and culture condition

In this study,the simultaneous hybrid modeling approach and AEPSO algorithm are tested using a lab-scale nosiheptide fermentation carried out in a 100 L stirring bioreactor.The bioreactor is equipped with some sensors,such as the temperature,pH,pressure,dissolved oxygen and foaming sensors,and is operated at batch mode with an about 96-h production period.The strain used to produce nosiheptide is Streptomyces actuosus 17–30.To achieve enough data,a set of experiments has been conducted.In each, first,70 L of fermentation medium is prepared,sterilized at 121°C for 30 min and then cooled down to room temperature.Then,2.5 L of inoculum is added to the fermentation mediumand cultured.The temperature is maintained by automatic adjustment of the flow rates of hot and cold water in the jacket of the bioreactor,and the pH is controlled by the automatic addition of acid/base solutions.The air flow rate and stirring rate are automatically adjusted to maintain the desired dissolved oxygen concentration.The pressure is kept at 35 Pa(gauge)through the automatic adjustment of escaping air flow.Soybean oil is utilized as an anti-foaming agent.The experiments are repeated at various values of temperature,pH and dissolved oxygen concentration within the range of 27–32 °C,6.3–8.1 and 2.9–5.6 mg·L?1respectively.It should be noted that special details regarding medium composition and analytical methods have been omitted for reasons of confidentiality.

2.2.Modeling approach

2.2.1.Existing hybrid modeling approaches

According to fermentation kinetics and the law of mass conservation,the dynamic behavior of a fermentation process can be generally described by the following set of differential equations[19,20].

where x=[x1,x2,··,xm]Tdenotes the state variable vector, μ =[μ1,μ2,··,μp]Tdenotes the unknown variable vector,and u=[u1,u2,··,un]Tdenotes the control variable vector.F=[F1,F2,··,Fm]Tand γ =[γ1,γ2,··,γp]Tare two groups of functions,among which often only the former can be derived from first principles.The central idea of hybrid modeling is as follows.First,the mechanistic model structure,namely the expressions of F=[F1,F2,··,Fm]Tare derived on the basis of first principles.Next,the unknown functions γ=[γ1,γ2,··,γp]Tare approximated using empirical modeling approaches.Finally,the two parts are combined to form an overall model.

As noted by Psichogios and Ungar[10],deriving the mechanistic model structure of a fermentation process is a straightforward task.Therefore,for hybrid modeling,the key is to develop the empirical model part.However,as mentioned above,there is a lack of target outputs of empirical models.To solve this problem,most existing hybrid modeling approaches take a step-by-step modeling strategy to develop the empirical models of unknown variables.This process includes(i)estimating the derivatives of state variables by their off-line measurements and proper interpolation functions,(ii)estimating the unknown variables based on the estimation values of state variable derivatives and Eq.(1),and(iii)developing the empirical models with the unknown variable estimation values as their target outputs.Although this procedure is simple,the sampling interval of state variables is relatively long(up to several hours).This inevitably has detrimental influence on the unknown variable estimation values,leads to their reliability and accuracy being difficult to guarantee,and eventually impacts the generalization ability of the whole hybrid model.This paper presents an improved hybrid modeling approach,which can indirectly train the empirical models without knowing their target outputs,namely the values of unknown variables.This approach can be seen to estimate and model unknown variables simultaneously,and therefore we define it the simultaneous hybrid modeling approach.The basic description of this approach is given below.

2.2.2.Proposed simultaneous hybrid modeling approach

As is stated above,this paper uses MFNNs to develop the empirical models of unknown variables that can be described by the following equation:

where γEdenotes the MFNNs based empirical models of unknown variables used to approximate the unknown functional relationshipsγ,and θ is the parameter vector of empirical models,namely the weights and thresholds of MFNNs.

The training of empirical models is essentially the identification of the vector θ.The simultaneous hybrid modeling approach fulfills this task in three steps.Firstly,Eq.(3)is substituted into Eq.(1),and then a dynamic system model is obtained as follows:

Secondly,θ is regarded as a parameter vector to be identified,and thus the empirical model training problem is transformed into a dynamic system parameter identification problem.Finally,θ is identified by minimizing the objective function defined in Eq.(5)using only the measurements of state and control variables.This accomplishes the training of empirical models,and meanwhile avoids the need of unknown variable values.The objective function is

where b=1,2,…,B denotes the b th batch of training data,B is the total number of batches,h=1,2,…,Hbdenotes the h th set of of fline measurements of state variables,Hbis the total number of state variable samples in the b th batch of training data,m=1,2,…,M denotes the m th state variable,M is the total number of state variables,and x and^x denote the measured and predicted values of state variables,respectively.

There are two main concerns with regard to the development of a hybrid model for a fermentation process using the proposed approach.The firstis how to solve the RPI problem and then obtain the parameters of each empirical model.The second is how to selecta suitable structure for each empirical model.

In the following subsections, first,an AEPSO algorithm with escaping and adaptive inertia weight adjustment strategies is proposed.Then,AEPSO is employed to identify the parameters of each empirical model.Finally,the structure of each empirical model is selected using the uniform design method together with the leave-one-out cross validation technique.

2.3.Optimization algorithm

2.3.1.Standard PSO

PSO is proposed by Kennedy and Eberhart[21],inspired by the social behavior of bird flocking.In PSO,candidate solutionsare called particles,each of which is characterized by its position zi=(zi1,zi2,…,ziD)and velocity vi=(vi1,vi2,…,viD).Here i=1,2,…,Sw(the swarm size)and D is the search space dimension.Let pbesti=(pbesti1,pbesti2,…,pbestiD)and gbest=(gbest1,gbest2,…,gbestD)denote the best positions found by the i th particle and the whole swarm,respectively.In SPSO,from iteration g to g+1,the d th dimensions of the velocity and position of the i th particle are updated as follows:

where ω is the inertia weight controlling the influence of the old velocity on the new one,c1and c2are two positive acceleration coefficients,and r1and r2are two uniformly distributed random numbers in the range[0,1].SPSO is randomly initialized by a set of candidate solutions,and then tries to find the optimal solution by performing repeated applications of the above updating equations until a termination criterion is reached,such as reaching the maximum iteration time.

2.3.2.Proposed AEPSO

Here,we introduce the proposed adaptive escaping PSO(AEPSO)algorithm in detail;it is based on two search techniques including an escaping strategy and an adaptive inertia weight adjustment strategy.

2.3.2.1.Escaping strategy.In SPSO,the swarm of particles would quickly converge to gbest[22].However,if gbest is located in a local optimum,then it would be difficult to jump out of the local optimum once all particles converge to this position,as there is no repellent available.Therefore,an escaping strategy is proposed to avoid being trapped in a local optimum,which enables each particle to escape from gbest with a certain probability by randomly choosing the sign of c2.When using the escaping strategy,the velocity updating equation becomes

where the±sign is the direction operator of movement,and defines from iteration g to g+1 whether the i th particle should escape from or converge to gbest.The sign is chosen to be(?)if rand＜Pe,otherwise it is chosen to be(+),where rand is a random number with uniform distribution in the range[0,1]and Peis a pre-specified escaping probability.

2.3.2.2.Adaptive inertia weight adjustment strategy.Considering that the escaping strategy may lead to a relatively low convergence speed,an adaptive inertia weight adjustment strategy is also proposed.The starting point is that if a particle driven by the current velocity can obtain a better position than its previous position,we should increase the influence of this particle's inertia part—namely,increase its inertia weight,and vice versa.As a result,the inertia weight of each particle is adjusted adaptively at each step according to Eq.(9)

whereis the inertia weight of the i th particle at iteration g,ωminand ωmaxare the lower and upper limits of the inertia weight,α is a positive constant,and(for minimization problems),whereanddenote the fitness function values of the i th particle at iterations g?1 and g respectively.In addition,is calculated using Eq.(9)only if g≥1,whereas if g=0,is randomly generated in the range[ωmin,ωmax].

2.3.2.3.Steps of AEPSO.The proposed AEPSO is constructed by incorporating the above two strategies into SPSO;it can be summarized in the following steps,as shown in Fig.1.

2.4.Identification of empirical model parameters

The parameter vector θ of empirical models is obtained by using AEPSO to solve the RPI problem.During the search process,the quality of each particle is evaluated by taking Eq.(5)as the fitness function.For the i th particle,the calculating steps of the value of its fitness function at iteration g,denoted by,are shown in Fig.2.

The flowchart to identify the empirical model parameters using AEPSO is shown in Fig.3.

2.5.Selection of empirical model structure

The structure of empirical models refers to the topology of the corresponding neural net works including the number of neurons in the input layer,the number of neurons in the out put layer,and the number of hidden layers and neurons in these layers.MFNN with one hidden layer can simplify the selection of empirical model structure,and approximate any nonlinear function with arbitrary precision[23],so in this paper MFNN with one hidden layer is used.In general,the number of input neurons and output neurons is decided by the problem itself,so the only tunable parameter is the number of hidden neurons.

There is currently no theoretical guidance for the selection of the hidden neuron number,so it is mostly selected by trial and error.Assume that the number of unknown variables is p,the number of hidden neurons of each empirical model has hid1,hid2,…,hidpoptions,respectively.Then it would need to attempt hid1× hid2×…× hidptimes to select a set of optimal structure.In this way,obviously,when there are a relatively large number of unknown variables and the optional range of the hidden neuron number of each empirical model is relatively wide,the selection of empirical model structure will face a huge workload.

Fig.1.The proposed AEPSO algorithm.

Uniform design can effectively reduce the number of experiments while providing sufficient information[24].Therefore,this method is applied to locate limited but sufficient experiments for selecting the empirical model structure in this paper(the detailed description is provided in Section 3.2).

3.Results and Discussion

3.1.Experimental data

In all,thirteen normal batches of experimental data,which correspond to 427 state variable samples and 14875 control variable samples,have been collected.Among these data,twelve batches with 394 state variable samples and 13726 control variable samples are selected randomly to develop the hybrid model,and the rest is used to test the performance of this model.In addition,it should be noted that the data of control variables are obtained online through a distributed control system with a sampling interval of 5 min;whereas the data of state variables are obtained offline through manual sampling and laboratory testing with a sampling interval of around 3 h.

3.2.Development of hybrid model

3.2.1.Determination of mechanistic model structure

According to our technologists'experience about the nosiheptide batch fermentation process and the research results of related literature[19,20,25,26],the mechanistic model structure can be described by the following three differential equations:

Fig.2.The calculation of the i th particle's fitness function value at iteration g.

Fig.3.The flowchart to identify the empirical model parameters using AEPSO.

where x1,x2and x3are state variables(g·L?1)and denote respectively the biomass,substrate and nosiheptide concentrations.μ1,μ2and μ3are unknown variables(h?1)and denote respectively the specific growth rate,specific consumption rate and specific production rate.u1,u2and u3are control variables and denote respectively the temperature(°C),pH and dissolved oxygen concentration(g·L?1).β is the hydrolysis rate constant of nosiheptide with the value of 0.0004 h?1obtained through experiments.γ1,γ2and γ3are the corresponding functional relationships with their concrete expressions being unknown.

3.2.2.Development of unknown-variable empirical models

After having determined the mechanistic model structure,three empirical models for μ1,μ2and μ3are developed using the proposed simultaneous hybrid modeling approach.The structure of each empirical model is schematically shown in Fig.4,where bias=?1 is designed for the introduction of thresholds.

3.2.2.1.Parameter settings for AEPSO.Swis determined by the empirical formula:and c2are selected to be 2,and ωminand ωmaxare set to be ωmin=0.4 and ωmax=0.9 respectively,which are widely used in conventional literature.Peand α are determined by experiments,and they are set to 0.2 and 0.05,respectively.The method for determining Peand α is omitted for reasons of space.

3.2.2.2.Structure selection of unknown-variable empirical models.Three steps are carried out to select an optimal number of hidden neurons for each empirical model by utilizing the uniform design method together with the leave-one-out cross-validation technique as described below.

Firstly,we assume the number of hidden neurons of each empirical model to be in the range of6–25 through experience,and then carry out a uniform experimental design using the hidden neuron number(HNN)of each empirical model as factor at twenty levels.Table 1 shows the uniform design table(U20(203))including 20 groups of hidden neuron numbers.

After establishing the above uniform design table,experiments are carried out for each group of hidden neuron numbers in this table,and the leave-one-out cross-validation technique is used to find the optimal group.Fig.5 shows how the twelve batches of modeling data are divided into training and validation subsets.In each trial,one integrated mean relative error is calculated,denoted by IMRE(b)(b=1,2,…,12).It is the average of the mean relative errors of x1,x2and x3for batch b that is leftout for validation.After these errors are obtained for all the twelve batches,IMRE is calculated to represent the average of IMRE(b)(b=1,2,…,12).To obtain valid statistical information,20 experiments are run repeatedly for each group of hidden neuron numbers.In this paper,we usewhich represents the average of IMRE for all of the runs as the final experimental result,and list it in the last column of Table 1.The flowchart of this procedure is illustrated in Fig.6.

Finally,the combination with the minimumis selected and adopted as the acquired optimal combination of hidden neuron numbers of the three empirical models.Table 1 indicates that the 9th group of combination shown in bold has the best modeling effects.Hence the empirical model structure of μ1,μ2and μ3has been selected to be 5×14×1,6×10×1 and 4×11×1,respectively.

3.2.2.3.Parameter identification of unknown-variable empirical models.After the structure of the three empirical models has been selected,we use all twelve batches of modeling data to identify the parameters of these empirical models using the method described in Section 2.4.

3.3.Testing results

The prediction capability of the hybrid model is evaluated by the testing batch.Fig.7 illustrates the predicted results of the biomass,substrate and nosiheptide concentrations with this model.It can be seen from this figure that the model developed by the simultaneous hybrid modeling approach can predict the three state variables with high accuracy.

3.4.Comparison of modeling effects

Fig.4.Schematic representation of MFNN based empirical models with respect to μ1,μ2 and μ3.

To con firm the advantage of the simultaneous hybrid modeling approach,we also use the existing hybrid modeling approach to develop the model of the nosiheptide batch fermentation process with the same twelve batches of modeling data.The basic steps are as follows.(i)Estimate the unknown variables using the off-line measured data of state variables,suitable interpolation functions(we test two different commonly used interpolation functions,the cubic spline[13]and the quintic polynomial[11])and Eqs.(10)–(12).(ii)Select MFNNs to develop the empirical models of μ1, μ2and μ3and use AEPSO to determine the parameters of these models.The structure of each empirical model and the algorithm parameters of AEPSO are the same as used in Section 3.2.

Table 1 The uniform design table of U20(203)and experimental results

The dashed lines and dot–dashed lines in Fig.7 graphically illustrate the predicted results for the testing batch with the models developed by the existing hybrid modeling approach using the cubic spline interpolation function and the quintic polynomial interpolation function,respectively.Table 2 summarizes the mean relative errors and root mean square errors for the prediction of the biomass,substrate and nosiheptide concentrations in the testing batch.Boldface text in Table 2 indicates the best results among the models.

From Fig.7,we find that the models developed by the above two modeling approaches are all able to predict the variation trends of the three state variables over time in the nosiheptide batch fermentation process.From the data shown in Table 2,it can be seen that the model developed by the simultaneous hybrid modeling approach has significantly better generalization ability than those developed by the existing hybrid modeling approach.

3.5.Comparison of optimization effects

To evaluate the performance of AEPSO as well as the effectiveness of its two associated strategies,we compare it to the SPSO,the adaptive PSO(APSO)with only the adaptive inertia weight adjustment strategy,and the escaping PSO(EPSO)with only the escaping strategy.These are used to solve the RPI problem in the simultaneous hybrid modeling of the nosiheptide batch fermentation process.The modeling data and structure of each empirical model are the same as used in Section 3.2.In the four PSO algorithms,the relevant parameters are the same as AEPSO in Section 3.2,except that the inertia weight of each particle in SPSO or EPSO is maintained at 0.729.Each algorithm is repeated 20 times,and the termination criterion is that the iteration time reaches 5000.The convergence curves in terms of the mean best fitness function values are plotted in Fig.8.It can be seen from this figure that(i)APSO has a faster convergence speed compared to SPSO,and its optimization precision is also somewhat improved,(ii)EPSO has higher optimization precision compared to SPSO,but its convergence speed is relatively slow,and(iii)AEPSO has an obvious advantage in optimization precision and roughly the same convergence speed when compared with SPSO.The above experimental results show that(i)the adaptive inertia weight adjustment strategy is able to enhance the search efficiency and convergence speed,(ii)the escaping strategy is able to improve the global search ability and avoid local optima,and(iii)AEPSO is able to obtain strong global search ability in parallel with a fast convergence speed by the introduction of the two strategies.

Fig.5.Demonstration of the leave-one-out cross-validation technique and division of the twelve batches to training and validation subsets.

To further indicate the advantages of AEPSO,we also compare it to two other recently proposed PSO variants,the chaotic PSO(CPSO)[15]and quantum-behaved PSO(QPSO)[28],as well as another intelligence optimization algorithm typically used in parameter identification problems,the genetic algorithm(GA).They are tested on the RPI problem in the simultaneous hybrid modeling of the nosiheptide batch fermentation process,with the twelve batches of modeling data and three groups of different empirical model structures(the 6th,15th and 19th groups)selected randomly from Table 1.In CPSO and QPSO,the relevant parameters are set according to the source references[15,28].In GA,the roulette wheel selection,one-point crossover,uniform mutation and elitist strategy are utilized with the crossover and mutation probabilities set to be 0.95 and 0.05,respectively.The swarm size of each algorithm is set according to the empirical formula,and the termination criterion is to reach the maximum iteration time,5000.All three parameter identification problems are solved 20 times.The mean best fitness function values(MBFFV)and median iteration times(MIT)required to reach a pre-specified optimization precision are presented in Table 3.The optimization precision specified for the 6th,15th and 19th groups are 0.2,0.25 and 0.15 respectively.Boldface text in Table 3 indicates the best results among the algorithms.

From the results in Table 3,we observe that AEPSO surpasses the other three algorithms from aspects of global search ability and convergence speed for all of the above three parameter identification problems.This further validates the excellent performance of the new AEPSO algorithm for solving complex and high-dimensional optimization problems like the RPI problem in this paper.

4.Conclusions

Fig.7.The predicted results of(a)biomass concentration,(b)substrate concentration and(c)nosiheptide concentration.

The optimization of the nosiheptide fermentation process needs a sufficiently accurate model.This paper presents a simultaneous hybrid modeling approach and employs it to model a lab-scale nosiheptide batch fermentation process.The proposed approach has an advantage over existing hybrid modeling approaches in that it does not require estimation data(i.e.,the estimated empirical model target outputs)to train the empirical model part of a hybrid model.Instead,the proposed approach transforms the training of empirical models into a dynamic system parameter identification problem,and thus allows trainingindirectly the empirical models without knowing their target outputs.An AEPSO algorithm with escaping and adaptive inertia weight adjustment strategies is proposed to solve the resulting parameter identification problem,and thereby obtain the parameters of empirical models.The optimal structure of empirical models is selected using the uniform design method together with the leave-one-out crossvalidation technique.The experimental results show that(i)the proposed modeling approach outperforms existing ones in terms of the generalization ability of the developed models,and(ii)the new AEPSO algorithm is able to avoid local optima effectively,while maintaining high convergence speed.

Table 2 The prediction errors of state variables

Fig.8.The convergence curves of mean best fitness function values.

Table 3 The MBFFV and MIT of AEPSO,CPSO,QPSO and GAon the RPI problems with three groups of different empirical model structures

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