## 1 Introduction  
For many years, hydrogen has been considered an alternative to fossil fuels. Scientists have been aiming to develop an environmentally friendly method to produce hydrogen from biomass by optimizing Anaerobic Digestion (AD) systems [1]. In the AD process, microorganisms decompose biomass in the absence of oxygen. Through AD, food and animal wastes are recycled to produce hydrogen gas which can be subsequently converted into biogas (methane) [2, 3]. The utilization of wastewater for hydrogen production through AD is promising as it results in the generation of value-added products and reduces the organic load of the wastewater. For example, Ref. [4] investigates the evaluation of hydrogen production with microbial consortia. The significant variables are optimized using a central composite design, resulting in two mathematical models. The resulting environmental benefits can be found in the removal of 50% of the COD, which could be further improved with the recovery of the other metabolites produced, mainly acetic and butyric acids. The Two-Stage Anaerobic Digestion (TSAD), in which the hydrogen and methane production takes place in two separate bioreactors, has several advantages over the conventional single-stage process [5; 6]. The TSAD permits the selection and the enrichment of different bacteria in each anaerobic digester and increases the stability of the whole process by controlling the acidification phase in the first digester and hence preventing the overloading and/or the inhibition of the methanogenic population in the second digester [5; 6]. The production of corn starch and starch-derived products results in large amounts of aqueous by-products [7; 8; 9]. Corn Steep Liquor (CSL) is a concentrated liquid by-product derived from water and used in the initial stage of the corn wet milling process. Globally, large amounts of CSL are produced daily and discharged into waterways. Despite its high nutritional value and relatively low cost, currently, CSL is not widely used internationally as a feedstock in the fermentation industry. This is primarily because most corn producers sell off their products to dry milling operations, where the corn is processed for various corn-derived products such as corn flour. CSL has been used recently as a substrate for AD [2]. An experimental study of two-stage anaerobic biodegradation of corn extract has been performed with mesophilic temperatures in both bioreactors. An automatic mode has been implemented using the developed computer system for monitoring and control. The so obtained experimental data are used here to develop a mathematical model. For many industrially relevant processes, detailed models are not available due to an insufficient understanding of the underlying phenomena. Mathematical models, which naturally could be incomplete and inaccurate to a certain degree, can still be very useful and effective in describing the effects essential for control, optimization, or understanding of the process. To the best of our knowledge, no mathematical models of TSAD of corn steep liquor have been published in the literature yet. In this work, a mathematical model of TSAD of corn steep liquor is proposed. Metaheuristic algorithms are considered for model parameter estimation. The main advantage of metaheuristic techniques, which have been proven to be a good alternative to conventional optimization methods, is that they provide a satisfactory solution for a reasonable computational time [10]. Metaheuristics do not depend on the initial conditions of the model. This advantage allows efficient scanning of a large search space to reach a global extremum. Two categories of metaheuristics are known: single-based and population-based solution methods [11]. In addition, some authors [12; 13] proposed eight different groups--biology-based, physic-based, swarm-based, social-based, music-based, chemistry-based, sport-based, and math-based metaheuristic methods.