Modeling the Efficiency of Biogas Plants by Using an Interval Data Analysis Method

Abstract

This article considers the task of developing mathematical models and their computer implementation that would establish the dependence of pH (acidity of the environment) on the volume and structure of raw materials for daily loading, as well as on the operating parameters of temperature and humidity based on the interval analysis of experimental data obtained during BGP research of a given type. In the process of research, based on the developed interval models, it was established that this indicator depends on the volume and structure of raw materials, as well as on the temperature and humidity of the substrate in the bioreactor. To build this mathematical model, it is proposed to use the method of interval data analysis and the method of identification of model parameters based on multidimensional optimization. The results of experimental studies for a specific type of biogas plant are given, and interval models with guaranteed prognostic properties that characterize the pH of the environment depending on the specific type of bio-raw material of solid and liquid fractions, temperature, and humidity are obtained. Based on the use of different types of raw materials, the developed models, based on experimental data, describe different configurations of the structure and volumes of raw materials for daily loading. The obtained mathematical models are an algebraic nonlinear equation that can be applied to control the level of pH of the environment in the bioreactor by determining the optimal volumes of raw materials of each type during the loading period depending on the temperature and humidity of the substrate in the bioreactor.

1. Introduction

Biogas production has become an alternative to traditional energy sources such as coal. It contributes to the reduction in greenhouse gas emissions and improves air quality. Additionally, it helps reduce dependence on non-renewable energy sources, particularly as global consumer energy needs grow [1]. The production of biogas can also be associated with the use of biological waste, such as organic solid household waste or products of agricultural activity [2,3]. Using biological waste for biogas production can help solve the problem of waste disposal and reduce its negative environmental impact. It also allows agriculture to produce energy, thereby reducing dependence on fuel imports [4].

The main element in biogas production technology is the biogas plant. A biogas plant is a complex of buildings and technological equipment integrated into a single automatic system for controlled methane fermentation [5]. The efficiency of a biogas plant depends on many factors, particularly the type of raw material or its mixture, temperature, humidity, etc. One critical factor affecting the efficiency of methane fermentation is the pH level of the fermentation environment. Deviations from the norm decrease the efficiency of biogas production, and at critical values, the process can stop altogether [6].

One way to increase the efficiency of biogas production is through the development and implementation of mathematical models and software for modeling and managing processes in a biogas plant. These models adapt the plant’s parameters and the technological process to the determined structure of raw materials. This approach requires developing mathematical models for a specific type of biogas plant (BGP), correlating the acidity of the environment with the volume and structure of raw materials, temperature, and humidity. An inductive approach can be used to build mathematical models that reflect these relationships in the form of algebraic equations. As is known [7,8], the inductive approach is based on analyzing data obtained through measurements of process characteristics during BGP operation to identify a mathematical model for researching and improving BGP efficiency.

However, experimental data collection and the mathematical model itself can introduce errors. Since the mathematical model is a simplification, it generates inaccuracies. To describe these errors, interval data analysis can be used. Unlike stochastic data analysis methods, interval data analysis does not require significant data sampling and provides guaranteed prognostic properties for the mathematical model [9,10,11]. The accuracy of the model will be lower in this case.

Thus, the goal of the article is to develop a mathematical model and its computer implementation that correlates the acidity of the environment with the volume and structure of raw materials and parameters such as temperature and humidity, based on interval data analysis obtained from studying a specific type of BGP. The use of interval data analysis methods to identify a mathematical model will require nonlinear optimization methods [12].

2. A Review of Methods of Modeling Processes in Biogas Plants

Modeling processes in biogas plants is a significant and complex task, requiring the integration of knowledge from various disciplines, including biology, chemistry, engineering, and other scientific fields. Biogas plants utilize biological processes to convert organic materials, such as wastewater, plant residues, and food waste, into biogas—a mixture of methane and carbon dioxide [13,14,15,16,17].

Modeling the kinetics of biochemical reactions in biogas plants is crucial for understanding and predicting methanogenesis processes and the anaerobic decomposition of organic compounds. This approach is based on kinetic models, such as the Monod or Michaelis–Menten models [15,18], which consider the dependence of the reaction rate on the concentration of substrates and inhibitors, as well as the impact of temperature and pH on microorganism activity.

Mass transfer and heat transfer models in biogas plants are based on equations of substance movement and heat transfer equations. They account for the hydrodynamic conditions in the bioreactor, such as temperature, concentration, and substance distribution, as well as thermal flows arising from biochemical reactions and heat exchange with the surrounding environment [19,20,21].

The development of control and regulation systems is based on mathematical models of dynamic processes in biogas plants. These models consider the system dynamics, the dependence of output parameters on input signals, and control algorithms designed to ensure optimal conditions for biochemical processes [22,23].

Gas flow models in biogas plants take into account gas dynamic processes, such as the production and removal of biogas, its distribution, and storage. They also consider the impact of various factors, such as pressure, temperature, and humidity, on the properties of the gas mixture.

For assessing the economic efficiency of biogas plants, models were developed that considered construction and operational costs, revenues from the sale of biogas and other products, as well as other factors such as risks and investment levels. These aspects of modeling are important for the design and optimization of biogas plants, ensuring their efficiency, stability, and economic viability.

There is still insufficient knowledge about the anaerobic digestion process and its response to changes in operational parameters such as feedstock, temperature, and acidity. Adequate monitoring alone cannot help implement a predictive subsystem in biogas plants. Therefore, it is necessary to analyze in detail the approaches to modeling anaerobic digestion processes.

The first model for simulating biogas plants was created by Andrews in 1969, and the most complex model is the Anaerobic Digestion Model No. 1 (ADM1), which simulates a variety of substrates among all models [15,23]. ADM1, created by Bastone et al. (2002), has been the most widely used model in recent years [15]. Modeling methods for biogas plant processes are detailed in

Table 1. Methods of mathematical modeling of processes in biogas plants, “first generation”.

Model	Model Description
Andrews, (1969)	This model demonstrates that modeling the rate-limiting step provides information on the entire process. Bacterial inhibition can be explained by the accumulation of acid [24].
Andrews and Graef, (1971)	Dynamic modeling of the process of enzymatic hydrolysis of complex organic compounds has been carried out [25].
Hill and Barth, (1977)	This model was created to ensure stability in the process of anaerobic digestion of animal husbandry wastes. Taking into account mass balances between volatile compounds, volatile acids, soluble organics, two groups of bacteria, cations, nitrogen, and carbon dioxide, the pH values were calculated [26].
Heyes and Hall, (1981)	A dynamic model was developed to represent hydrogen inhibition of acetogenesis and pH inhibition of methanogenesis using glucose as a substrate [27].
Hill, (1983)	The model was developed to simulate the steady-state methane productivity (qualitative and quantitative) in the process of anaerobic digestion of animal husbandry wastes [28].
Mosey, (1983)	Four bacterial groups were identified in the model of biogas production through anaerobic digestion of glucose. Acetogenesis is determined as the limiting step in the model [29].

,

Table 2. Methods of mathematical modeling of processes in biogas plants, “second generation”.

Model	Model Description
Costello, (1991)	The reactor process, physico-chemical system, and biological composition were utilized in the system to create a mathematical model. Additionally, the model includes the accumulation of lactic acid, product inhibition, and pH acidity [30].
Angelidaki, (1993)	The model was developed to simulate the anaerobic degradation of complex organic materials, including the enzymatic hydrolytic stage, four bacterial stages, and 12 chemical compounds [31].
Vavilin, (1996)	The model was developed to simulate the hydrolytic (limiting) stage of anaerobic digestion. The model includes surface colonization of particles by hydrolytic bacteria and surface degradation [32].
Husain, (1998)	Monod functions were used to determine the mortality rate of acidogens and methanogens [33].

,

Table 3. Methods of mathematical modeling of processes in biogas plants, “third generation”.

Model	Model Description
Bernard (2001)	A mass balance model was developed to determine parameters at the stages of acidogenesis and methanogenesis in the process. Electrochemical equilibrium is used to incorporate alkalinity into the model [34].
Siegrist (2002)	The rate of hydrolysis, acetoclastic methanogenesis, and propionate degradation were the specific focus of the mathematical model created, which simulated the dynamic behavior of mesophilic and thermophilic fermentation [35].
Batstone (2002)	ADM1 includes both biochemical and physico-chemical processes. In this comprehensive model, 26 dynamic state variable concentrations, 8 implicit algebraic variables, and 32 state variable concentrations are utilized [36].
Zaher (2009)	The model was created to understand microbial activity based on the availability of macro-elements ($C,H,N,O,P,\mathrm{and}S$) and the thermodynamics of acidogenesis and methanogenesis [37].
Rajendran (2014)	46 reactions (for inhibition, kinetic rate, pH, ammonia, volume, loading rate, and retention time) are carried out in the model to predict biogas production from any substrate and under any operating conditions using Aspen Plus V 7.3.2 [38].
Arzate (2015)	This model combines life cycle assessment characteristics and a mathematical model of process productivity, which can help reduce the environmental impact of fermentation processes [39].

and

Table 4. Methods of mathematical modeling of processes in biogas plants, other models.

Model	Model Description
Barampouti, (2005)	The model was created to forecast biogas production by examining 17 parameters from two years of data from a wastewater treatment plant [40].
Nopharatana, (2007)	The model was created to simulate biological reactions in a reactor with solid municipal waste considering them in two fractions: soluble and insoluble. The model incorporates Contois, Monod, and Gompertz equations [41].
Yusuf and Ify, (2011)	The model was created to predict the maximum and final biogas yield as well as the final methane output during the co-fermentation of cow manure and water hyacinth based on a first-order kinetic model [42].
Syaichurrozi and Sumardino, (2013)	A kinetic model for determining biogas production was developed using the modified Gompertz equation. The influence of the COD/N ratio on the kinetic model was investigated [43].
Brule, (2014)	The model was created to optimize analyzes of raw materials. It provides quality control of analyses, interpretation of reaction kinetics and assessment of methane yield [44].
Dyvak, Gural (2018)	The model for estimating the daily output of methane during anaerobic microbiological fermentation [45].

and further explained in [15].

Table 1 provides the characteristics of the models that form the “first generation”. These models are characterized by attempts to model the basic microbiological processes.

Table 2 presents models that constitute the “second generation” and are aimed at addressing issues encountered in the studied classes of physico-chemical systems.

Table 3 presents models constituting the “third generation” aimed at comprehensively addressing issues in anaerobic microbiological fermentation.

In Table 4, models constituting a specific class of models are presented. Within this class of models, the interval approach, under which this research was conducted, is to be highlighted.

The analyzed models are based on the properties of biochemical transformations and have a deterministic nature, which limits their application in developing a prototype biogas plant for automatic process tuning in biogas production and purification in the biogas plants (Figure 1).

These models cannot account for possible deviations at any stage of biogas production and purification in the technological parameters of these processes and do not consider the type and structure of bioresources. Therefore, to address these issues, it is advisable to use mathematical modeling methods based on interval analysis.

3. Materials and Methods

3.1. Principle of Functioning of BGP

The basic configuration of a biogas plant based on nodes and structures is shown in Figure 2 [5].

Let us briefly consider the principle of operation of a biogas plant. Waste enters the receiving tank (1). In it, their preliminary accumulation, heating (2), and thorough mixing (3) take place. Raw materials are fed into the fermenter (5) 4–6 times a day using a special pump for liquid and gelatinous substrates. The fermenter (5) is a gas-tight, hermetic tank. To maintain a stable temperature inside, the fermenter is equipped with a heating system for the bottom and walls (2). The substrate is constantly stirred with the help of mechanical stirrers (3). Discharge of the fermented substrate occurs automatically with the same frequency as loading. The operation of the entire biogas plant is managed by the automation system (11). Biogas accumulates in the gasholder (6), which is used as a gas-tight covering for the fermenter. The removal of biogas takes place through the pipeline (8). From the gas holder (6) there is a continuous supply of biogas to a cogeneration plant or a biogas purification system. After installation, the processed substrate is fed to the separator (9). The mechanical separation system operates 4–6 times a day and separates the fermentation residues after the fermenter into solid and liquid biofertilizers. All equipment is controlled by the automation system (11).

The fermentation process is a key stage in the production of biogas and includes the decomposition of organic materials using microorganisms. To achieve effective fermentation, several important parameters are controlled [19,20,21,46,47]:

• The temperature, since different types of microorganisms reproduce optimally at different temperatures. Usually, the temperature regime for fermentation can be from 35 °C to 55 °C;
• The pH level, which indicates the concentration of hydrogen ions (H⁺) in a grout, which determines its acidity or alkalinity. Most of the microorganisms involved in fermentation reproduce optimally in a certain range of acidity in the environment. Usually, it can be from 6.5 to 8.5. Regulation of the pH level helps to provide optimal conditions for the growth of microorganisms;
• The fermentation time, which is determined by the decomposition of organic substances and the formation of biogas. It can vary from several days to several weeks, depending on the conditions and type of raw materials;
• The concentration and structure of raw materials: different types of raw materials can be used for biogas production, such as organic waste, biomass, agricultural residues, and others. At the same time, the quantity and quality of the raw material base are of great importance for the efficiency of the production process;
• The raw material humidity, which affects gas formation and mass balance of the process. Usually, the humidity of raw materials should be in the range from 70% to 80%;
• The C/N (carbon/nitrogen) ratio, which is determined by the ratio of the mass of carbon to the mass of nitrogen in the raw material. This is an important parameter, as too little or too much ratio can affect the efficiency of fermentation;
• The intensity of mixing ensures an even distribution of microorganisms and raw materials, which helps to avoid stagnation zones and improves fermentation efficiency.

All these parameters interact and determine the efficiency of the fermentation process in biogas production. Consequently, their control allows for achieving optimal conditions for the microorganisms’ activity and obtaining the maximum volume of produced biogas [47].

Controlling the pH level is a crucial element in optimizing the fermentation process. Automated control systems can regulate and monitor the required concentration of acid or alkali added to the fermenter to maintain the desired pH level [5,6]. By adding acids or alkalis to the reactor, the pH level can be precisely maintained or adjusted. Additionally, it is possible to manage the pH level based on the supply of raw materials. The optimal ratio of solid to liquid fractions in the raw materials directly affects the efficiency of the fermentation process and methane production. Furthermore, maintaining the optimal ratio of solid to liquid fractions helps avoid problems with uneven distribution of raw materials and ensures effective mixing for uniform fermentation.

3.2. The Task Statement and Its Solution

Let the mathematical model that relates the pH level of the environment in the BGP fermenter with the volume and structure of raw materials, as well as the temperature and humidity of the environment, be described as an algebraic expression [48,49]:

$$Y\left(\overrightarrow{\beta },X\right)=f_1\left(\overrightarrow{\beta },X\right)+,\dots ,+f_m\left(\overrightarrow{\beta },X\right)$$

(1)

where $Y\left(\overrightarrow{\beta },X\right)$ is the modeled value of pH of the environment; $\overrightarrow{\beta }$ is the vector of model parameters; and $f_1\left(\overrightarrow{\beta },X\right)+,\dots ,+f_m\left(\overrightarrow{\beta },X\right)$ is the set of basic functions from the matrix of input variables $X$ (which determine the structure of the biomass that is loaded into the bioreactor and process parameters—temperature and humidity of the environment), as well as from the vector of model parameters $\overrightarrow{\beta }$. It should be noted that the parameters of the model in the general case also depend on the input variables, i.e., $\overrightarrow{\beta }=\overrightarrow{\beta }\left(X\right)$.

The results of the experiment, which are necessary for the identification of the parameters of nonlinear (in the general case) model (2), are presented in the following form:

$${\overrightarrow{X}}_i\to \left[Y_i^{-};Y_i^{+}\right],i=1,\dots ,N,$$

(2)

where $\left[Y_i^{-};Y_i^{+}\right]$ are the lower and upper limits of the experimentally obtained values of the pH level of the environment for the specified $i$-th conditions of the measurement, which are determined by the vector ${\overrightarrow{X}}_i$, for each of $i=1,\dots ,N$ measurements, where $N$—number of measurements. In this case, the task of identification of the model in the form of expression (2) is to calculate the estimates $\overrightarrow{\widehat{\beta }}$ of the parameters $\overrightarrow{\beta }$, which may also depend on the values of the matrix $X$. In the case when these estimates are calculated, the mathematical model takes the following form:

$$\widehat{Y}\left(\overrightarrow{\widehat{\beta }},X\right)=f_1\left(\overrightarrow{\widehat{\beta }},X\right)+,\dots ,+f_m\left(\overrightarrow{\widehat{\beta }},X\right)$$

(3)

where $\widehat{Y}\left(\overrightarrow{\widehat{\beta }},X\right)$ are the calculated values of estimates of the modeled pH level of the environment in the BGP fermenter.

Based on the condition that the modeled pH values of the environment should belong to the interval values obtained experimentally [49]

$${\widehat{Y}}_i\left(\overrightarrow{\widehat{\beta }},X\right)\to \left[Y_i^{-};Y_i^{+}\right],$$

(4)

we obtain a mathematical task for determining the estimates $\overrightarrow{\widehat{\beta }}$ of the parameter vector $\overrightarrow{\beta }$:

$$Y_i^{-}\le f_1\left(\overrightarrow{\widehat{\beta }},{\overrightarrow{X}}_i\right)+,\dots ,+f_m\left(\overrightarrow{\widehat{\beta }},{\overrightarrow{X}}_i\right)\le Y_i^{+},i=1,\dots ,N.$$

(5)

The obtained system (5) is an interval system of nonlinear algebraic equations (ISNAE) relatively to unknown estimates of the vector of model parameters $\overrightarrow{\widehat{\beta }}$. The set of ISNAE solutions Ω determines the vector of model parameter estimates $\overrightarrow{\widehat{\beta }}$. Since the methods of solving ISNAE (5) have a combinatorial computational complexity, in practice, point solutions of the system are used to determine $\overrightarrow{\widehat{\beta }}$.

In this case, to find parameter estimates, the optimization problem is solved in the following form [48]:

$$\delta \left(\overrightarrow{\widehat{\beta }}\right)\stackrel{\hspace{1em}\overrightarrow{\widehat{\beta }},{\alpha }_i\hspace{1em}}{\to }min$$

(6)

$${\alpha }_i\in \left[\mathrm{0,1}\right],i=1,\dots ,N,$$

(7)

where ${\alpha }_i$ are the coefficients of a linear combination that determine a point within the experimental data $\left[Y_i^{-};Y_i^{+}\right].$

In expression (6), the objective function $\delta \left(\overrightarrow{\widehat{\beta }}\right)$ is formed taking into account the limitations of the system of nonlinear interval Equation (5). The objective function is a criterion for minimizing the squared error [49]:

$$\begin{array}{c}\delta \left(\overrightarrow{\widehat{\beta }}\right)={{\displaystyle \sum }}_{i=1}^N{\left({\widehat{Y}}_i\left(\overrightarrow{\widehat{\beta }},{\overrightarrow{X}}_i\right)–P\left(\left[Y_i^{-};Y_i^{+}\right],{\alpha }_i\right)\right)}^2=\\ ={{\displaystyle \sum }}_{i=1}^N{\left(f_1\left(\overrightarrow{\widehat{\beta }},{\overrightarrow{X}}_i\right)+,\dots ,+f_m\left(\overrightarrow{\widehat{\beta }},{\overrightarrow{X}}_i\right)–\left({\alpha }_i\cdot Y_i^{-}+\left(1-{\alpha }_i\right)\cdot Y_i^{+}\right)\right)}^2.\end{array}$$

(8)

Accordingly, gradient methods are used to solve the optimization problem (6) [48,49].

The results of biogas production measurements were used to develop mathematical models of the pH of the environment in the biogas plant. The data were provided by Teofipol Energy Company LLC (Teofipol, Ukraine), which produces biogas based on BGP basic equipment. Measurement data for the months of August, October, November, and December are shown in

Table 5. Results of experimental studies on the characteristics and parameters of the fermentation process at BGP during August.

Control Point Number, i	1	2	3	4	5	6	7	8	9	10
Pulp loading mass, 1000 kg, $x_1$	72	56	80	58	54	144	130	78	58	87
Cattle manure loading mass, 1000 kg, $x_2$	10	10	10	10	40	20	10	5	5	10
Pulp with a straw loading mass, 1000 kg $x_3$	4	4	8	16	0	16	0	0	0	0
Bard loading volume, m3, $x_4$	150	40	80	140	90	0	160	90	120	150
Urea loading volume, m3, $x_5$	59	14.4	59	0	72	14.4	59	14.4	14.4	43
Humidity, %, $x_6$	97.9	97.8	97.2	96.525	96.9	96.291	97.1	96.837	96.75	96.73
Temperature in the bioreactor, °C, $x_7$	47.9	48.2	48	47.6	48.3	47.3	47	47.2	46.6	44.4
The lower limit of the measured Y, $Y_i^{-}$	7.8012	8.1279	7.8804	8.0091	7.9299	8.0784	7.900	8.059	7.92	8.0388
The upper limit of the measured Y, $Y_i^{+}$	7.9588	8.2921	8.0396	8.1709	8.0901	8.2416	8.0598	8.2214	8.08	8.2012

,

Table 6. Results of experimental studies on the characteristics and parameters of the fermentation process at BGP during October.

Control Point Number, i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Bard loading volume, m3, $x_1$	120	180	120	220	200	10	180	120	200	340	110	120	130	250	100	260
Pulp loading mass, 1000 kg, $x_2$	117	244	156	209	137	195	223	137	127	109	84	102	96	78	170	106
Cattle manure loading mass, 1000 kg, $x_3$	30	40	20	40	20	25	40	30	20	40	20	30	30	20	50	40
Urea loading volume, m3, $x_4$	25	0	58.8	28.6	25	34.8	50	50	38.4	36	0	0	7.2	0	0	0
Humidity, %, $x_5$	97.1	98	96.6	97	96.4	96.9	96.2	96.8	95.9	96.1	96.2	96.6	96.5	96.7	96.8	97
Temperature in the bioreactor, °C, $x_6$	40.6	40.9	41	40.8	41	41.6	41	40.5	40.5	40.3	40.5	402	40.2	42.5	43.4	43.5
The lower limit of the measured Y, $Y_i^{-}$	8.039	7.901	8.059	8.039	8.069	7.979	8.029	7.949	7.959	7.959	7.920	7.880	7.999	8.058	8.029	7.969
The upper limit of the measured Y, $Y_i^{+}$	8.201	8.059	8.221	8.201	8.232	8.141	8.191	8.110	8.120	8.120	8.080	8.039	8.161	8.221	8.191	8.131

,

Table 7. Results of experimental studies of the characteristics and parameters of the fermentation process at BGP during November (fermenter 1).

Control Point Number, i	1	2	3	4	5	6	7	8	9	10	11	12	13
Bard loading volume, m3, $x_1$	20	265	560	90	130	120	180	110	50	120	170	200	200
Pulp loading mass, 1000 kg, $x_2$	78	80	130	174	222	170	245	210	96	63	208	54	137
Cattle manure loading mass, 1000 kg, $x_3$	40	20	40	40	40	40	60	50	50	30	40	20	30
Dry bard loading volume, 1000 kg, $x_4$	8	0	24	18	24	12	30	0	0	0	0	0	0
Urea loading volume, m3, $x_5$	0	0	0	0	50	25	0	10.8	25	27.6	18	18	10.2
Humidity, %, $x_6$	96.794	96.2	96.3	97.019	96.5	96.8	96.5	96.906	96.1	96.305	96.771	96	96.4
Temperature in the bioreactor, °C, $x_7$	43.5	44	44.3	44.4	44.4	44.6	44.4	44.5	42.3	42.6	43.6	45	45.1
The lower limit of the measured Y, $Y_i^{-}$	8.1279	7.9893	8.0982	8.0982	8.0289	8.0784	8.1972	7.9893	8.0685	8.0091	8.1477	7.8507	8.0388
The upper limit of the measured Y, $Y_i^{+}$	8.2921	8.1507	8.2618	8.2618	8.1911	8.2416	8.3628	8.1507	8.2315	8.1709	8.3123	8.0093	8.2012

,

Table 8. Results of experimental studies of the characteristics and parameters of the fermentation process at BGP during November (fermenter 2).

Control Point Number, i	1	2	3	4	5	6	7	8	9	10	11	12	13
Bard loading volume, m3, $x_1$	20	265	560	90	130	120	180	110	50	120	170	200	200
Pulp loading mass, 1000 kg, $x_2$	78	80	130	174	222	170	245	210	96	63	208	54	137
Cattle manure loading mass, 1000 kg, $x_3$	40	20	40	40	40	40	60	50	50	30	40	20	30
Dry bard loading volume, 1000 kg, $x_4$	8	0	24	18	24	12	30	0	0	0	0	0	0
Urea loading volume, m3, $x_5$	0	0	0	0	50	25	0	10.8	25	27.6	18	18	10.2
Humidity, %, $x_6$	95.88	96.3	96.55	96	95.868	96.156	96.1	96.3	95.9	96.4	96.478	96.4	95.987
Temperature in the bioreactor, °C, $x_7$	41.2	41.4	41.9	41.2	41.4	41.6	41.4	41.2	39.5	39.7	41.2	43.8	44.7
The lower limit of the measured Y, $Y_i^{-}$	7.9497	7.9299	8.1477	8.0289	7.9497	7.9992	8.0883	7.9497	8.0289	8.0784	8.1279	7.9398	8.1279
The upper limit of the measured Y, $Y_i^{+}$	8.1103	8.0901	8.3123	8.1911	8.1103	8.1608	8.2517	8.1103	8.1911	8.2416	8.2921	8.1002	8.2921

,

Table 9. Results of experimental studies of the characteristics and parameters of the fermentation process at BGP during December (fermenter 1).

Control Point Number, i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Bard loading volume, m3, $x_1$	130	70	80	130	200	150	30	110	320	120	130	210	220	40	220
Pulp loading mass, 1000 kg, $x_2$	129	159	147	120	101	97.5	102	112.5	69	100	0	37.5	51	33	137.5
Urea loading volume, m3, $x_3$	0	0	0	39.4	0	16.5	14.4	14.4	0	14.4	0	14.4	14.4	0	14.4
Treacle loading volume, m3, $x_4$	0	0	14.4	10	0	0	10	15	5	0	0	0	0	11	10
Humidity, %, $x_5$	96	96.6	96.1	96.635	96.5	97	96.4	97	96.7	96.9	96.8	97.2	96.3	96.696	96.8
Temperature in the bioreactor, °C, $x_6$	34	34.7	32.1	35.4	34.8	35.8	35.9	36	36.3	36.1	36.1	36	36.9	38	38.2
The lower limit of the measured Y, $Y_i^{-}$	7.8447	7.8842	8.0127	7.9732	7.8546	7.8941	8.1312	7.9633	7.9336	7.9633	7.9139	7.9435	7.8447	8.0028	8.0522
The upper limit of the measured Y, $Y_i^{+}$	8.0353	8.0758	8.2073	8.1668	8.0454	8.0859	8.3288	8.1567	8.1264	8.1567	8.1061	8.1365	8.0353	8.1972	8.2478

and

Table 10. Results of experimental studies of the characteristics and parameters of the fermentation process at BGP during December (fermenter 2).

Control Point Number, i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Bard loading volume, m3, $x_1$	130	70	80	130	200	150	30	110	320	120	130	210	220	40	220
Pulp loading mass, 1000 kg, $x_2$	129	159	147	120	101	97.5	102	112.5	69	100	0	37.5	51	33	137.5
Urea loading volume, m3, $x_3$	0	0	0	39.4	0	16.5	14.4	14.4	0	14.4	0	14.4	14.4	0	14.4
Treacle loading volume, m3, $x_4$	0	0	14.4	10	0	0	10	15	5	0	0	0	0	11	10
Humidity, %, $x_5$	96.2	96.3	96.1	96.782	96.8	96.737	97	96.8	97	96.9	96.3	96.2	96.7	96.2	96.8
Temperature in the bioreactor, °C, $x_6$	44.6	44.3	43.8	43.3	42.9	42.8	42.5	42.3	42.1	42.2	41.6	40.6	39.6	38.7	38.2
The lower limit of the measured Y, $Y_i^{-}$	8.062	8.062	8.1318	8.102	7.933	8.102	8.24	8.013	8.102	8.052	7.973	8.082	7.963	8.072	7.934
The upper limit of the measured Y, $Y_i^{+}$	8.258	8.258	8.329	8.298	8.126	8.298	8.44	8.207	8.298	8.248	8.167	8.278	8.157	8.268	8.126

.

The pH of the environment was measured with a device with an error of 1%. The given data characterize the use of different amounts and types of solid and liquid raw materials, which will allow us to simulate and analyze the effect of combining various raw materials on the pH environment in BGP fermenters.

To determine the analytical expression of the dependence of the pH of the environment on the volume and structure of the raw material, as well as on the temperature and humidity of the environment, indicator functions were used as basic functions. This was justified by the following considerations:

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In the study of statistical data, the non-linear nature of the dependencies was revealed, and the indicator functions allowed modeling a wide range of non-linear dependencies between variables. Power values can define various curve shapes, including polynomial, exponential, and others, which can be adapted to a specific context and produce models that more adequately reflect the properties of the data;

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Compared to some other nonlinear functions, exponential functions can be relatively simple to interpret;

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The use of power functions can allow the building of complex nonlinear dependencies with a smaller number of parameters compared to other nonlinear models.

Based on physical considerations, the structure of the nonlinear algebraic equation describes the pH dependence on volume and raw material structure. It also includes the temperature and humidity of the environment. The equation is given in a specific form:

$$\widehat{Y}\left(\overrightarrow{\widehat{\beta }},X\right)={\beta }_0+{\left({\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^s{\beta }_i·x_i}{1+{{\displaystyle \sum }}_{j=1}^l{\beta }_j·x_j}}\right)}^{{\beta }_{k+5}}+{\beta }_{k+1}·{x_{k+1}}^{{\beta }_{k+2}}+{\beta }_{k+3}·{x_{k+2}}^{{\beta }_{k+4}},$$

(9)

where $x_i$ is a volume in m³ of daily loading in the corresponding period of the $i$-th type of raw material of the solid fraction (dry bard, sugar beet pulp, straw, etc.); $x_j$ is a mass of 1000 kg of daily loading in the corresponding period of the $j$-th type of raw material of the liquid fraction (urea, treacle, etc.); $x_{k+1}$ is humidity in %; $x_{k+2}$ is temperature, C, in the fermentation environment; $k=s+l$ is the number of types of raw materials, where $s$ is the number of types of raw materials of the solid fraction and l the number of types of raw materials of the liquid fraction.

The application of the expression ${{\displaystyle \sum }}_{i=1}^s{\beta }_i·x_i/\left(1+{{\displaystyle \sum }}_{j=1}^l{\beta }_j·x_j\right)$ in the equation of mathematical model (9) is justified by the need to take into account the ratio of the proportion of raw materials of the solid fraction to the raw materials of the liquid fraction for the substrate in the BGU bioreactor.

4. Results and Discussion

Based on the materials and methods described above, nonlinear models were obtained for all cases of combining types of raw materials. The MATLAB R2023b Update 6 environment was used to solve the optimization problem (6) based on the given data [50,51,52,53]. For data for the month of August, the following vector of estimates $\overrightarrow{\widehat{\beta }}$ of the model parameters was obtained

$${\overrightarrow{\widehat{\beta }}}^T=\left(\begin{array}{c}9.997,17.304,321.69,-264.728,-0.116,251.675,\\ 3176.194,-1.493,215.389,-1.32,3.608\end{array}\right).$$

Now, let us substitute the obtained identification results into expression (9):

$$\widehat{Y}=9.997+{\left({\displaystyle \frac{17.30·x_1+321.69·x_2-264.72·x_3}{1-0.116·x_4+251.675·x_5}}\right)}^{3.61}-3176.194·{x_6}^{-1.49}+215.389·{x_7}^{-1.32}$$

(10)

The mathematical model, expressed as Equation (10), explains how the pH of the environment relates to the volume and structure of raw materials, as well as temperature and humidity. As raw materials for the solid fraction, pulp, cattle manure, and a mixture of pulp with straw were used, while the raw materials for the liquid fraction were bard and urea. Figure 3 shows the results of a comparison of experimental data ($\left[Y_i^{-};Y_i^{+}\right]$) and modeled pH values ($\widehat{Y}\left(\overrightarrow{\widehat{\beta }},X\right)$) of the substrate environment in the bioreactor based on the constructed models. In particular, Figure 3a demonstrates that the modeled results belong to the intervals of the measured pH values, which indicates the adequacy of the model.

Accordingly, based on data for the month of October, estimates ($\overrightarrow{\widehat{\beta }}$) of the vector of model parameters were calculated in the following form:

$${\overrightarrow{\widehat{\beta }}}^T=\left(2.982,-2.456,8.669,102827876.087,9.326,57.5192,-0.536,3.494,-0.864,1.993\right).$$

A nonlinear model was obtained based on the types of raw materials of the solid (pulp, cattle manure) and liquid (bard, urea) fractions used in this case, in the form of such a dependence:

$$\widehat{Y}=2.982+{\left({\displaystyle \frac{-2.456·x_2+8.669·x_3}{1-102827876.087·x_1+9.326·x_4}}\right)}^{1.993}+57.5192·{x_5}^{-0.536}+3.494·{x_6}^{-0.864}.$$

(11)

The adequacy of the model is shown in Figure 3b by the appropriateness of the modeled pH value to the interval value of the measurement results.

In November, two fermenters were used with the same raw material loading parameters and different temperature and humidity parameters. The structure of the raw materials of the solid fraction consists of pulp, cattle manure and dry bard, of the liquid fraction—bard and urea. Accordingly, based on the data for the first fermenter, we obtained the following estimates ($\overrightarrow{\widehat{\beta }}$) of the vector of the parameters: ${\overrightarrow{\widehat{\beta }}}^T=(4.986,36.114,-151.306,191.582,136.827,561.936,-180.749,-1.517,124.524,-0.968,1.887)$. Nonlinear dependence was found in the following form:

$$\widehat{Y}=4.986+{\left({\displaystyle \frac{36.114·x_2-151.306·x_3+191.58·x_4}{1+136.827·x_1+561.936·x_5}}\right)}^{1.88}-180.749·{x_6}^{-1.52}+124.52·{x_7}^{-0.97}.$$

(12)

The obtained vector of parameter estimates ${\overrightarrow{\widehat{\beta }}}^T=(8.064,0.0024,-0.0028,\mathrm{0.017,346.0068},10157.327,-463.569,-79.703,-891.169,$$-29.228,0.238)$ for the second fermenter in the month of November determined a nonlinear model of dependence between the pH of the environment and the volume and structure of raw materials, as well as with temperature and humidity of this type:

$$\widehat{Y}=8.064+{\left({\displaystyle \frac{0.0024·x_2-0.0028·x_3+0.017·x_4}{1-346.0068·x_1+10157.327·x_5}}\right)}^{0.24}-463.569·{x_6}^{-79.70}-891.169·{x_7}^{-29.23}$$

(13)

Figure 3c,d show graphs of the modeled value and experimental data for the specified period. The modeled values $\widehat{Y}$ belong to the intervals of the measured values $\left[Y_i^{-};Y_i^{+}\right]$ at all measurement points, which indicates the adequacy of the models with the selected structure.

The results of modeling based on the data for the month of December, in which two fermenters also operated with the same loading of raw materials (pulp—solid fraction, bard, urea and treacle—liquid fraction) are given below.

Vector of estimates of the model parameter for the first fermenter ${\overrightarrow{\widehat{\beta }}}^T=(8.852,35.613,3369.81,-4807.85,31.38,-9.061,-1.19,-13.97,-0.78,0.624)$ and analytical dependence:

$$\widehat{Y}=8.852+{\left({\displaystyle \frac{5.613·x_2}{1-3369.81·x_1-4807.859·x_3+31.38·x_4}}\right)}^{0.624}-9.061·{x_5}^{-1.19}-13.97·{x_6}^{-0.78}.$$

(14)

Vector of estimates of the model parameter for the second fermenter ${\overrightarrow{\widehat{\beta }}}^T=(7.769,9.613e-07,30.477,-111.19,68.74,8.891,-0.595,-176.716,-1.729,0.153)$ and model:

$$\widehat{Y}=7.769+{\left({\displaystyle \frac{9.613e-07·x_2}{1+30.477·x_1-111.19·x_3+68.74·x_4}}\right)}^{0.153}+8.891·{x_5}^{0.595}-176.716·{x_6}^{-1.72}.$$

(15)

The models are adequate and reflect the dependence of the pH level on the parameters of the technological parameter with the specified accuracy (Figure 3e,f).

To study the dependence of the pH level on various factors, numerical experiments were conducted based on the constructed models. Study results show the impact of the solid and liquid fraction ratio of raw materials at fixed temperature and humidity values based on model (10). This is illustrated in Figure 4a.

Figure 4a shows that in August, at the maximum daily loading of raw materials of the solid fraction, the share of raw materials of the liquid fraction stabilizes the value of the pH level in the range of 7.9–8. At the same time, with the maximum daily load of raw materials of the liquid fraction, a change in the share of raw materials of the solid fraction causes a decrease in the value of the pH level from 8.4 to 8 units. Analysis of the structure of raw materials for August showed that pulp is the dominant type of solid raw materials.

The pulp includes a large number of organic acids, and accordingly, its addition leads to the production of a larger amount of organic acids in the fermentation process. At the same time, the produced organic acids reduce the pH level in the system. The liquid fraction raw material consists of bard and urea, which contain organic substances. These substances decompose, forming organic acids and ammonia. In particular, urea contains ammonia compounds, for example, ammonia (NH₃), which reacts with water to form ammonium and hydroxide ions, which increases the pH level.

The same trend is observed for the structure of raw materials (pulp, liquid bard, urea and in small amounts of treacle) in December (Figure 4b). At the same time, a slight change in the pH level is observed, which is explained by the properties of the pulp to stabilize the pH level. These properties help maintain the pH level at a certain point despite variations in the reaction environment.

The combined effect of raw materials of different fractions and the humidity of the environment was also researched based on the data and the model for the month of August, which is shown in Figure 5.

The analysis of research results (Figure 5) showed that a change in humidity in the range of 30–60 percent (optimum is usually 35–55%) with a minimum daily load of raw materials of the solid fraction provides a change in the pH level by one unit (Figure 5a). At the same time, the maximum daily load of raw materials of the dry fraction (Figure 5b) allows for a lower pH level by one unit at the same humidity parameters.

The results of the study of the combined effect of the raw material structure and temperature on the pH level based on the data and the model for August are shown in Figure 6. An increase in temperature in the anaerobic fermentation process of solid fractions (the main component of pulp and cattle manure) leads to an increase in the pH level within the unit (Figure 6a). At the same time, regulation of the daily share of the solid fraction of raw materials from the minimum to the maximum within the sample for August allows lowering the pH level at an average temperature of 47 ºC from 7.9 to 6.9, i.e., by one unit (Figure 6a,b).

For the raw materials of the liquid fraction, there is a tendency to decrease the pH level, which, in our opinion, is due to the dominance of liquid bard in the raw materials, because an increase in temperature leads to the rapid decomposition of organic substances and the production of organic acids. At the same time, the daily share of raw materials of the liquid fraction within the sample for August has an insignificant effect on the pH level.

The difference between the minimum and maximum daily loading of raw materials of the liquid fraction at the average daily loading of solid fractions is 0.2 units (Figure 6c,d). To improve the balance in the structure of raw materials, it is recommended to increase the proportion of urea.

5. Conclusions

One of the main characteristics of a biogas plant in the production of biogas is the pH indicator of the fermented substrate. In the course of research, it was established that this indicator depends on the volume and structure of raw materials, as well as on the temperature in the bioreactor and the humidity of the substrate. To study and increase the efficiency of the functioning of the biogas plant during the production of biogas, it is proposed to use a mathematical model that relates the specified characteristic and the factors influencing it. Based on the inductive approach, it is proposed to use the method of interval data analysis and the method of identification of parameters of the model based on multidimensional optimization to build this mathematical model.

On these grounds, interval models with guaranteed prognostic properties characterizing the pH environment were obtained for specific types of solid and liquid fractions of biosubstrates, substrate temperature, and humidity dependencies. The prognostic properties of the obtained models are characterized by accuracy within the limits of the experimental data error, which does not exceed 1%. The mathematical models obtained are algebraic nonlinear equations (from the standpoint of parameter identification tasks) that can be applied to determine the optimal volumes during the loading period of each type of raw material substrate, depending on the humidity and temperature of the multicomponent substrate in the bioreactor.

It is worth noting that based on the obtained models, an analysis of the pH environment dependency on factors characterizing the raw material composition and technological parameters has been conducted. The experiments conducted showed permissible variations in pH values depending on the ratio of dry and liquid fractions of raw material for a multicomponent substrate. In particular, it has been established that under given optimal temperature conditions (35–55%) and humidity (not less than 96%), the model ensures pH environment control by regulating the ratio of dry and liquid fraction raw material components within the unit range, enabling pH environment control within optimal limits (6.5–8.5) and ensuring fermentation process stability.

The results of the research presented in the article also showed the presence of an element in the model reflecting the ratio of dry and liquid fractions of raw material components. In models (10)–(15), such a ratio describes the raw material structure comprising different components for the experiments conducted. This means that the proposed model is universal for any type of bioreactors using multicomponent substrate and only requires parameter identification based on a data sample describing the technological process and raw material components.