## 4 Conclusions and Future Works  
In this paper, a new mathematical model of two-stage anaerobic digestion of corn steep liquor has been proposed. The model has been presented as a set of five ordinary differential equations and two algebraic equations, representing seven dependent state variables (substrate concentration (\(S_{1}\)) in BR\({}_{1}\); microbial biomass concentration in BR\({}_{1}\) (\(X_{1}\)) and BR\({}_{2}\) (\(X_{2}\)); acetate formation in BR\({}_{1}\) (\(Ac_{1}\)) and BR\({}_{2}\) (\(Ac_{2}\)); flow rate of the hydrogen (\(Q_{H_{2}}\)) in the gas phase of BR\({}_{1}\) and methane flow rate \(Q_{CH_{4}}\) in BR\({}_{2}\)). The mathematical model consists of nine unknown model parameters divided into two groups, \(p_{1}=[\mu_{1 max}\)\(K_{S_{1}}\)\(Y_{1}\)\(Y_{2}\)\(Y_{H_{2}}]\) and \(p_{2}=[\mu_{2 max}\)\(K_{S_{2}}\)\(Y_{3}\)\(Y_{CH_{4}}]\). Different metaheuristic algorithms, some of the popular algorithms of the swarm intelligence domain, have been applied to estimate the unknown parameters based on experimental data. The feasibility of the Genetic algorithm, Firefly algorithm, Cuckoo search algorithm, and Coyote Optimization Algorithm applied to a parameter identification problem has been highlighted. The chosen metaheuristic algorithms have been adapted and implemented here for the first time for parameter estimation of a newly proposed nonlinear mathematical model of TSAD of corn steep liquor. To confirm the superiority of some of the algorithms, a comparison between the techniques has been made--using the observed numerical results, graphical results, and statistical analysis. Figure 4: [PERSON]’s test results. The simulation results of the parameters' identification of the proposed mathematical models show that GA, FA, CS, and COA techniques can predict the experimental results. The four metaheuristics have achieved similar results in modelling the process dynamics in BR\({}_{1}\). In the case of modelling the process dynamics in BR\({}_{2}\), the lowest value of the objective function has been achieved by GA (\(J\) = 0.0761). A better description of the process dynamics trend, however, has been obtained by FA (\(J\) = 0.1075). The observed objective function values of CS and COA are \(J\) = 0.0913 and \(J\) = 0.0940, respectively. It has been demonstrated that the considered metaheuristics are efficient and are also powerful algorithms for parameter identification of complex nonlinear models. As a result, a mathematical model of TSAD of corn steep liquor with a high degree of accuracy is proposed. The model can be used for a simulation of the process behaviour to gather enough information for planning subsequent laboratory experiments. The mathematical model can also be used to optimize the process based on a designed control system. A good model in control loop synthesis is required to obtain optimal control. In addition, the numerical results have been compared based on box plots--a standard technique for analysis of the distribution of the obtained model estimates. The statistical results show that the parameter estimates \(Y_{3}\) and \(Y_{CH_{4}}\) are more easily obtained compared to the parameter estimates \(\mu_{2 max}\) and \(K_{S_{2}}\). Such results are interesting, and it is worth conducting further research on the sensitivity of the model parameters to optimize the identification process. Although some good results have been achieved with the application of the considered metaheuristics, there are still some limitations that need to be improved in future. First, the presence of several sets of raw experimental data is important for model validation. This is crucial, especially in the case of modelling non-linear, time-varying processes with interdependent variables, such as the one considered in this paper. Additional laboratory experiments are planned to be carried out. Second, it is well known that to achieve better performance, the input parameters of metaheuristic algorithms should be adjusted according to the problem domain.