Zahir Barahmand*
| Gamunu Samarakoon
© 2023 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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Growing concern about global climate change has led to considerable interest in investigating renewable energy sources such as the biological conversion of biomass to methane in an anaerobic environment. Through a series of complicated biochemical interactions, it uses various bacterial species to degrade biodegradable material in the feedstock. Due to the complex and interacting biochemical processes, anaerobic digestion has nonlinear dynamics. Anaerobic digestion is highly at risk of instabilities and uncertainties because of its dynamic and nonlinear behavior, uncertain feedstock quality, and sensitivity to the process’s environmental conditions. Therefore, effectively operating a biogas production unit necessitates a thorough understanding of the system’s uncertainties. The present study aims to identify and assess the sources and methods of coping with the uncertainties in anaerobic digestion processes through a narrative review. Moreover, the knowledge gap is also investigated to reveal the challenges and opportunities to have a robust model. The results indicate that the unpredictability of model parameters and input variables were the most significant source of uncertainty, and the Monte Carlo technique, confident interval, and interval observers, as well as sensitivity analysis were the most frequently used tools to cope with these uncertainties.
ADM1, anaerobic digestion, biogas reactor, narrative review, uncertainty, WWTP
Global population growth, industrialization, and a rising quality of living are all driving up energy demand [1]. Still, fossil fuels are the most common energy source, accounting for 79.9% of total demand [2]. Hence, alternative energy strategies for creating and using clean, renewable, dependable and sustainable energy sources are being sought. In contrast with the aerobic treatment of organic waste, anaerobic digestion (AD) can produce usable biogas utilized for energy production [3]. Biogas production by the AD process has been demonstrated to be a sustainable, renewable, and carbon-neutral energy source that can help reduce the world’s reliance on fossil fuels and carbon footprint [4]. Being a multi-step biological process, AD degrades organic materials and converts them into biogas [5]. The main constituents of biogas are methane (50-70%) and CO2 (30-50%). It is a multi-step complex process that includes the following phases: hydrolysis, acidogenesis, acetogenesis, and methanogenesis [6]. Different groups of microorganisms mediate the different steps of the AD process, forming a complex microbial community [5].
Furthermore, changes in the feed composition are inherent to this process, and they can readily lead to the accumulation of intermediate products such as volatile fatty acids (VFAs), hydrogen, ammonia, and other chemicals [7]. This accumulation may inhibit microorganisms, resulting in process failure or reactor instability [8]. Additionally, various disturbances might be added to the process by raising the reactor’s load. As a result, operating the AD reactors at modest feed rates is preferable to keep the operation stable [7] while maintaining optimum biogas production. Therefore, determining the ideal substrate feeding rate in the AD process is quite challenging [9].
As a result, modeling the biogas process is quite challenging. On the other hand, depending on the preferences, timing, and framing of the alternative scenarios, among other factors, designers might cope with the model uncertainty in a variety of ways [10]. A distinction must be made between sources of uncertainty, such as data, choices and relations, and types of uncertainty, such as data variability and inconsistency across alternatives [11]. There is uncertainty when there is a lack of information, either there is no data available or the data that are available is incorrect or vague. In contrast, variability may also be regarded as a necessary aspect of heterogeneity, which is the quality of data [12] or multiple values of the same quantity at different times, places, or instances [13]. Barahmand and Eikeland [14] provided a comprehensive overview of different types and sources of uncertainty. Uncertainty can be quantified in a variety of ways. Different techniques can be used depending on the type of model (mathematical, control, etc.). Barahmand et al. [14, 15] described the most used uncertainty modeling techniques. The key difference between these approaches is how they explain ambiguity in input parameters.
Among the most common methods for estimating uncertainty is the Monte Carlo technique [16]. Statistical modeling and random sampling are used in Monte Carlo simulations to estimate mathematical functions and simulate the behavior of complex systems [17]. The purpose of this technique is to develop probabilistic models for real-world processes to estimate specific average properties, such as mathematical expectations, variances, and correlations [16]. A Monte Carlo simulation involves generating random numbers, simulating the random values with a more complicated distribution and performing calculations [18]. In addition to being used directly for uncertainty quantification, the Monte Carlo approach can also be used indirectly in other methods, including global sensitivity analysis (GSA).
Even though sensitivity and uncertainty analysis are often thought of as one activity, they are actually two separate concepts, each with its own purpose, that is often explored in con- junction with one another. A sensitivity analysis (SA) attempts to determine the relationship between variations in input values and variations in outputs [19]. The SA technique is used in numerical models to estimate the effect of uncertainty on one or more input variables on output variables [20]. Additionally, SA can serve as a guide in conducting experimental analyses, reducing models, and estimating parameters [16]. SAs can be either local or global. A local SA examines how small perturbations affect the model’s output. Conversely, a global SA is used to investigate the effects of large variations in model parameters on model outputs [21]. The critical aspect of a complex model (such as ADM1) with several input parameters and output parameters is identifying the most influential input parameter or variable on the model’s outputs [22]. There is a significant fluctuation in the results when specific assumptions are changed when sensitivity is high; these assumptions must be extremely well established. Changing specific assumptions when sensitivity is high results in significant changes in the output; these assumptions should be extremely well established [15, 23].
Up to this point, all the uncertainties and models mentioned above belonged to mathematical models. Physical systems cannot be fully represented by a single mathematical model, and there may be uncertainties in the system as a result of disturbance signals or changes in system parameters [24]. A good control system in such a scenario should be robust and pro- vide consistent results. In order to cope with the uncertainty of the system, different observers and controllers have been developed. A system observer is a digital algorithm that combines sensor outputs with knowledge of the system to provide results that are superior to those provided by traditional structures, which are entirely dependent on sensors [25]. For example, an interval observer’s task is to estimate the system states at each instant in time within certain intervals (upper and lower estimates) to deal with the uncertainty and disturbances that are large but bounded [26].
Modeling the biogas production process mathematically can be very complex, and the digestion process is usually modeled as a black box due to its complexity [27]. By analyzing big data and extracting internal information, machine learning (ML) models can solve non- linear classification and regression tasks [28, 29]. It should be noted that this method is entirely dependent on readily available online data or historical recordings of the process itself [30]. ML involves a cycle of three steps [31, 32]:
As much as ML algorithms are capable of managing complex multivariate data, predicting nonlinear connections and managing missing data, choosing the most appropriate algorithm for a given task is critical for achieving the best results [33, 34].
Through a narrative review, the present study aims to identify the primary sources of uncertainty in the AD process and assess the tools and methods used to cope with them to reveal the challenges and opportunities to have a robust model. The present literature review provides a comprehensive, critical, and objective assessment of the current state of knowledge regarding the topic [35]. A narrative review takes a less formal approach than a systematic review [36]. Due to the limited number of sources in the literature, this method was chosen. The study’s proposed research questions among the recent studies are:
(1) What are the sources and types of uncertainty in these processes?
(2) What are the tools and frameworks for dealing with uncertainties?
What is the most critical knowledge gap in the research?
2.1 AD: Global status
In large piles of organic waste, combustible gas is generated. This fact has been known for centuries. AD technology developed during the previous century was predominantly used to stabilize putrescible solids in domestic wastewater [37]. As a result, heated, fully mixed reactors were developed, which are still widely used for the digestion of sewage sludge and animal manure. As fossil fuel prices declined and their accessibility increased, biogas pro- duction gained popularity during the first half of the twentieth century. However, it declined after the 1950s as a result of the low cost of fossil fuels. In the 1970s, AD regained popularity as a renewable energy source [37]. However, implementation slowed down due to high costs, ineffective management, and high failure rates of the digesters as a result of limited knowledge, design issues, and a lack of appropriate management [38]. At present, most biogas is produced in Europe and the United States, but Asia has installed the largest number of digesters, which are mostly small scale [39]. As opposed to other technologies, AD is possible at a wide range of scales, ranging from small-scale digesters making just enough biogas to power a single household to large centralized biogas plants with digesters capable of producing several thousand cubic meters of biogas per day [38, 40, 41].
2.2 ADM1 theoretical background
2.2.1 Description and basic concept
Developed by the International Water Association (IWA), the Anaerobic Digestion Model No.1 (ADM1) [42] is the most common platform for the modeling and simulation of AD processes [43]. The ADM1 is a structured mathematical model, which covers complex processes involved in converting organic substrates into (biogas) methane, CO2, and inert byproducts [44]. Although the ADM1 does not cover all of the processes that take place during the AD process (such as precipitation of solids and sulfate reduction), the goal of ADM1 is to be a tool to provide as accurate predictions as possible that can be used in all phases of AD plants such as process development, operation, and optimization [45].
ADM1 includes several steps: Disintegrating complex solids into soluble and particulate inerts, carbohydrates, proteins, lipids, and inert substances, a nonbiological step, is the first step in ADM1 [46]. Next, the products of disintegration are hydrolyzed in the enzymatic step. The products are sugars, amino acids, and long-chain fatty acids. Then, in the acidogenesis process, sugars and amino acids are fermented to produce VFA, H2, and CO2. Finally, in both hydrogenotrophic (cleavage of acetate to methane) and acetoclastic (reduction of CO2 by molecular hydrogen to produce methane) methanogenesis processes, methane is produced [47]. Figure 1 shows the reaction path in ADM1 [42].
Figure 1. The reaction paths described in Batstone et al. [42]
2.2.2 Growth kinetics
Many kinetic expressions that define substrate conversion rates in terms of substrate concen- trations and rate constants explain substrate conversion processes [42]. First-order kinetics explain the disintegration of influent substrates as well as the hydrolysis of carbohydrates, proteins, and lipids [46]. All inter-cell biochemical processes of substrate uptake in acido- genic and acetogenic steps are based on Monod-type growth kinetic expressions with pH inhibition and noncompetitive inhibition by total volatile fatty acids (TVFAs), free ammonia, and hydrogen [47]. Dead biomass is retained in the system as composite particulate material, and again first-order kinetics describe endogenous decay processes. The metabolism of biomass to preserve its life is known as endogenous decay [48].
2.2.3 Basic equations
The original ADM1 consists of 108 different equations, which can be classified into differential equations, algebraic equations, process rates, and process inhibition equations [49]. The outputs of ADM1 are the solutions to the 35 differential algebraic equations (DAEs) to model the concentration rate of various species in the liquid and gas phases. Table 1 provides a simple classification of these equations [50].
Table 1. Classification of the ADM1 equations (Source: Jeppsson and Rosen, 2006)
|
Model equations |
No. of equations |
Model equations |
No. of equations |
|
Process rates |
28 |
Process inhibition equations |
15 |
|
Biochemical process rates |
19 |
Differential equations |
35 |
|
Acid-base rates |
6 |
Water-phase equations |
32 |
|
Gas transfer rates |
3 |
Soluble matter |
12 |
|
Algebraic equations |
30 |
Particulate matter |
12 |
|
Soluble matter |
14 |
Cations and anions |
2 |
|
Inhibition |
5 |
Ion states |
6 |
|
Ion states |
4 |
Gas-phase equations |
3 |
|
Gas-phase equations |
7 |
|
|
Jeppsson and Rosen [49] implemented ADM1’s ordinary differential equation model. Barahmand [50] grouped these 35 DAEs into five classes. As seen in the following, all the DAEs are nonlinear with order and degree of unity. For soluble matters, the general form of the differential Eqns. (1)-(12) can be written as:
$\begin{aligned} & \frac{d S_z}{d t}=\frac{q_{\text {in }}}{V_{\text {lia }}}\left(S_{z, \text { in }}-S_z\right)+ \\ & \mathcal{F}\left(\rho_a, f_{\text {product, substrate }}, N_c, Y_d, C_e\right)\end{aligned}$ (1)
where, Sz is the soluble substrate concentrations in kg.COD.m-3), qin is the influent volumetric flow rate in m3d-1, Vliq is the volume of the liquid in m3, Sz,in is the concentration of the influent soluble substrate in kg.COD.m-3, Ƒ represents different functions and variables defined in [49], and ρa represents 19 biochemical process rates, 6 acid-based rates, and 3 gas transfer rates in d-1. fproduct,substrate is the yield (catabolism only) of the product on a substrate in kg.COD. (kg.COD)-1, Nc is the nitrogen content of component c in, k.mole.N. (kg.COD)-1, Yd is the yield of biomass on a substrate in kg.COD. (kg.COD)-1, and Ce is the carbon content of component e in k.mole.C. (kg.COD)-1.
The general form of differential Eqns. (13)-(24) associated with particulate matters is:
$\frac{d X_z}{d t}=\frac{q_{\text {in }}}{V_{\text {liq }}}\left(X_{z, \text { in }}-X_z\right)+\mathcal{F}\left(\rho_a, f_{\text {product,substrate }}, Y_d\right)$ (2)
where, Xz is the concentration of particulate component (biomass) z in kg.COD.m-3, and Xz,in is the concentration of influent biomass z in kg.COD.m-3. The general form of differential Eqns. (25)-(26), cations and anions, does not include Ƒ term.
$\frac{d S_{c a t^{+} / a n^{-}}}{d t}=\frac{q_{i n}}{V_{l i q}}\left(S_{c a t^{+} / a n^{-}, i n}-S_{c a t^{+} / c n^{-}}\right)$ (3)
where, Scat+ is the concentration of cation, San- is the concentration of anion, and Scat+,in and San-,in are input cation and anion all in k.mole.m-3 Moreover, there are six differential Eqns. (26)-(32) for ion states. Their general form is:
$\frac{d S_w}{d t}=-\rho_{A . i}$ (4)
where, S is the concentration, ρ is the acid-base rates, and w and i are defined in Table 2.
Table 2. Definitions of w and i in (4)
|
W |
i |
|
va- |
4 |
|
bu- |
5 |
|
pro- |
6 |
|
pro- |
7 |
|
ac- |
10 |
|
nh3 |
11 |
The last three differential Eqns. (33)-(35) are related to the gas phase with the general form of:
$\frac{d S_{\text {gas.u }}}{d t}=-\frac{S_{\text {gas.u }} q_{\text {gas }}}{V_{\text {gas }}}+\rho_{T, l} \frac{V_{\text {liq }}}{V_{\text {gas }}}$ (5)
where Sgas.u is the concentration of gases (H2. CH4. CO2) in kg.COD.m-3, ρT,l is the gas trans- fer rates in d-1, and qgas is the gas flow in Nm3 d-1, and Vgas is the gas volume in m3. Except for these differential equations, there are numerous intermediate equations such as biochemical processes, acid-based and gas transfer rates, inhibition-related equations, and algebraic equations.
2.2.4 Inhibition functions
Different functions were proposed, emphasizing modeling the effects of pH, hydrogen, inad- equate nitrogen inhibition, and free ammonia inhibitions [42]. Continuing this way, Jeppsson and Rosen [49] implemented and updated inhibition functions.
2.3 Sources of uncertainty
AD is extremely vulnerable to instabilities and uncertainty due to its dynamic and nonlinear behavior. ADM1 model consists of numerous parameters and input state variables directly affecting the process performance. Table 3 provides a classification of these based on [49].
Table 3. Classification and number of parameters and variables in ADM1
|
Parameters |
|
|
Stoichiometry |
41 |
|
Biochemical |
36 |
|
Physiochemical |
23 |
|
Physical |
2 |
|
State variables |
|
|
Input state variables |
28 |
Among over 100 parameters, although most of them have crisp and deterministic values, several are uncertain, such as the influent substrate composition (uncertain feedstock qual- ity). On the other hand, input state variables also suffer considerable ambiguity in real-life applications. Therefore, effectively operating a biogas production unit necessitates a thor- ough understanding of the system’s uncertainties.
An article review entails locating, reading, understanding, and presenting the published research and theory in an organized manner [51]. Literature reviews can be classified into different types depending on their purpose and methodology. Grant and Booth [52] reviewed the 14 most com- mon literature review methods. Narrative or traditional review, scoping review (i.e. [14, 16, 22, 53]), systematic review, critical review, rapid review, and umbrella review are among com- monly used methods. To address specific research questions, a narrative literature review was conducted. The goal is to find a few pieces of research that describe a particular problem. There is no preconceived research question or search technique in narrative reviews compared with scoring or systematic reviews, and it is less formal than them [54]. A narrative literature review was conducted in three phases and nine steps to address the review question [55]. The three phases are planning, conducting, and reporting. The planning phase covers topic selection, forming the research questions and defining the objectives, the development and validation of a review protocol, and searching the literature and selecting the literature. The conducting phase includes analyzing and synthesizing; the reporting phase consists of concluding and reporting [56]. Following is a step-by-step description of how this review was conducted.
Step 1: topic selection-the topic selected for this study is the application of uncertainty modeling in biogas production through AD.
Step 2: forming the research questions and defining the objectives-the objective of this study was to explore and describe the sources of uncertainty in the AD process and methods to cope with them. The purpose of this study was to address the following review question:
Q1) What are the sources and types of uncertainty in these processes?
Q2) What are the tools and frameworks for dealing with uncertainties?
Q3) What is the most critical knowledge gap in the research?
Table 4. Search string in different databases
|
Database |
String |
|
Scopus and WoS |
((anaerobic AND digestion) OR ADM1 OR WWTP) AND (uncertain* OR robust*) |
|
Google Scholar and ScienceDirect |
((anaerobic AND digestion) OR ADM1 OR WWTP) AND (uncertain OR uncertainty OR robust OR robustness) |
Step 3: the development and validation of a review protocol -similar to research design in empirical research, this step consists of a predetermined plan for how all other steps of the research process would be conducted.
Step 4: search the literature - to search for relevant literature, four online databases were selected based on a purposive sampling method [57], namely Google Scholar, ScienceDirect, Scopus, and Web of Science (WoS). The strings used in the databases are listed in Table 4.
On the one hand, this study focused primarily on AD. ADM1, on the other hand, is a very popular and well-known method, and according to the authors’ experience, many studies included only ADM1 in the title. As a result, ADM1 has been added to the string with the “OR.” Similarly, the search string has been amended to include WWTP, which is stand for wastewater treatment plant. The asterisk (*) can be used with word stems to get different versions of a phrase with less typing. For example, uncert* can find uncertain, uncertainty, uncertainties, etc. For Scopus, WoS, and ScienceDirect, the strings were applied to the titles only with no other limitation on the year or type of the publications.
Step 5: selecting the literature-the purpose of this step is to determine which articles will be included in the analysis or excluded [55]. For Scopus, WoS, and ScienceDirect, the strings were applied to the titles only. Moreover, this search was limited to studies in English. No other limitations were applied to the year or type of publication.
Step 6: content analysis and synthesis reading selected articles and analyzing them is part of this step. Furthermore, the step involves categorizing and grouping similar data following concepts and themes. Synthesizing allows the grouped data to be arranged into a specific form.
Step 7: conclusion and reporting - a demonstration of how this study extends existing research findings is included in this step.
A total of 56 documents were found to be relevant after completing steps 3-5 in Section 3. By applying the search strings in Table 4, Scopus, WoS, and ScienceDirect were able to locate 25, 32, and 12 documents, respectively. In order to eliminate duplicates, Microsoft Excel® was used and 36 documents remained. A further 20 relevant documents were added from Google Scholar, bringing the total number of studies to 56. In the final step and after a full-text screening of all studies, five studies were found to be nonrelevant. As a result, this literature review continued with 51 selected documents (Table 5).
Table 5. Selection of the related studies
|
|
No. of documents |
|
Result of search in titles: Scopus |
25 |
|
Result of search in titles: WoS |
32 |
|
Result of search in titles: ScienceDirect |
12 |
|
After deleting duplicates: |
36 |
|
Adding 20 document from Google Scholar |
56 |
|
Removing five documents after full-text screening |
(5) |
|
Number of final selected files |
51 |
4.1 Year-wise analysis
The year-wise analysis gives an overall picture of the research progress. The present study examined 51 studies that were published between 2000 and 2022 (with no limit on year). An overview of the number of publications and the overall trend is shown in Figure 2. In 2012, the highest number of publications was recorded with eight publications, whereas in some years, such as 2006 and 2007, no studies were recorded. Meanwhile, the overall trend shows a steady increase in interest and the number of studies. Despite the large number of uncertain- ties associated with AD modeling, a limited number of studies have been conducted on this topic. Among the reasons for this would be the challenges associated with modeling uncer- tainty in complex systems.
Figure 2. Number of selected studies and overall trend between 2000-2021
4.2 Co-occurrence analysis
Analysis of co-occurrences within a collection of units consists of counting the number of objects that occur together [58]. Based on extracted keywords from the citation context, VOSviewer® was employed to generate keyword co-occurrence networks using the full count- ing method. In this analysis, the minimum number of occurrences of the keyword was set to five, and from 473 keywords in 51 selected studies, 14 keywords met the threshold (see Figure 3).
Figure 3. Co-occurrence analysis of the selected studies using VOSviewer®
5.1 Content-based study
To address the research questions, this section investigates the selected 51 studies to reveal the sources of uncertainty, methods to cope with the model uncertainties, and their applications. Moreover, by identifying the knowledge gap, the present study provides some suggestions for further studies. In the following, the selected studies were reviewed and cat- egorized based on frequently used methods and tools to deal with the system uncertainty. Table 6 summarizes different tools and their frequencies among selected studies. As seen among 16 different tools, the Monte Carlo simulation dominates and has the highest contri- bution. Few of the articles used several methods in combination.
Table 6. Tools and methods to deal with the system uncertainties in selected documents
|
Tools/Methods |
Freq. |
References |
|
Monte Carlo Simulation (MCS) |
14 |
[3, 59-71] |
|
Confidence Interval / Interval Observer (CI/IO) |
11 |
[72-82] |
|
Global Sensitivity Analysis (GSA) |
3 |
[62, 65, 83] |
|
Morris Screening (MS) |
3 |
[62, 65, 83] |
|
Artificial Neural Network (ANN) |
3 |
[84-86] |
|
Luenberger Observer (LO) |
3 |
[87-89] |
|
Local Sensitivity Analysis (LSA) |
3 |
[63, 65, 90] |
|
Standardized Regression Coefficients (SRC) |
2 |
[62, 65] |
|
Random Forest (RF) |
2 |
[85, 86] |
|
Bifurcation Analysis (BA) |
1 |
[91] |
|
Detrended Fluctuation Analysis (DFA) |
1 |
[92] |
|
Min-Max Method (MMM) |
1 |
[93] |
|
Scenario Analysis (ScA) |
1 |
[94] |
|
Fuzzy Decision-Making (FDM) |
1 |
[7] |
|
Expected Utility Theory (EUT) |
1 |
[66] |
|
Multi-Criteria Analysis (MCA) |
1 |
[67] |
|
Uncertainty Quantification (UQ) |
1 |
[83] |
|
Dempster-Shafer Theory of evidence (DST) |
1 |
[95] |
|
Support Vector Regression (SVR) |
1 |
[86] |
|
Simple Observer (SO) |
1 |
[96] |
|
Deep Deterministic Policy Gradient (DDPG) |
1 |
[97] |
|
Neural Network Sliding Mode Control (NNSMC) |
1 |
[98] |
|
Variance-Based Sensitivity Analysis (VSA) |
1 |
[99] |
|
Nonlinear State Observers (NSO) |
1 |
[100] |
|
Gramian based Fixed-time Convergent Observer (GFCO) |
1 |
[101] |
|
Asymptotic Observer (AO) |
1 |
[101] |
|
Super-Twisting Observer (STO) |
1 |
[101] |
|
External validation of Near-infrared (NIR) calibrations |
1 |
[102] |
|
Robust H Theory (RHT) |
1 |
[103] |
|
Output Feedback Observer (OFO) |
1 |
[104] |
|
Extreme Learning Machine (ELM) |
1 |
[85] |
|
Support Vector Machine (SVM) |
1 |
[85] |
|
Genetic Programming (GP) |
1 |
[85] |
|
Multiplicative White Noise (MWN) |
1 |
[105] |
5.2 Monte Carlo simulation
In a Monte Carlo simulation, random sampling and statistical modeling are used to estimate mathematical functions and simulate the behavior of complex systems [106]. This technique develops probabilistic models for real-world processes to estimate specific average proper- ties, such as mathematical expectations, variance, and covariance. The main steps to performing a Monte Carlo simulation are random number generation; simulation of the ran- dom values with more complicated distribution and calculations [107]. As seen in the following, the Monte Carlo approach was used directly for uncertainty quantification or indi- rectly in other methods such as GSA.
Neba et al. [60] presented a systematic technique to establish the process performance objectives by combining practical identifiability, uncertainty quantification, and attainable region principles, which are particularly useful when exact kinetic coefficients are unknown. A Monte Carlo simulation was used to assess the uncertainty bands on the model states and attainable regions to construct high degrees of uncertainty for the case of the modified Hill model. Asadi and McPhedran [59] employed different uncertainty modeling tools to maximize biogas production. This study used a Genetic Algorithm (GA) combined with data-driven modeling and uncertainty analysis to determine the optimal operating parameter values for a municipality waste treatment plant’s AD process. This study utilized an Artificial Neural Network (ANN), Adaptive Network-based Fuzzy Inference System (ANFIS) using Monte Carlo Simulation for uncertainty analysis and nonlinear regression models. Sharara et al. [61] employed the Monte Carlo simulation methodology to establish manure AD systems in the face of uncertainty. In this probabilistic approach, parameters were sampled repeatedly from their respective distributions.
Ramin et al. [62, 108] applied a GSA using the Benchmark Simulation Model No. 2 (BSM2) platform for a secondary settling tank in a wastewater treatment plant. Two methods were employed to apply this SA: (1) standardized regression coefficients method using linear regression of Monte Carlo simulations and (2) Morris screening. The Monte Carlo simulation was employed to examine the impact of parameter uncertainty on the stated plant perfor- mance criteria. Kil et al. [3], based on ADM1, explored and calculated the links between the influent components’ concentrations and the process’s steady-state using process output information to estimate waste characteristics. The system uncertainty was evaluated using the Monte Carlo simulation. Xu [63] developed a method for calibrating parameters to optimize ADM1 for an industrial-scale plant. With a series of Monte Carlo simulations, local SA and a multivariate regression technique, Xu proposed a partial least square method to validate the calibrated parameter set. Gehring et al. [64] examined the substrate characterization uncer- tainty influence on the system results. The Monte Carlo simulation with a 20% uncertainty range for the recorded feedstock data was used in AMD1 to mathematically model the ther- mophilic mono-fermentation of maize silage.
Solon et al. [65] utilized a GSA to analyze the impact of uncertainty in substrate composi- tion, kinetics, stoichiometry, and mass transfer parameters in ADM1. Two methods (Morris screening method and standardized regression coefficients) were employed to perform the GSA. The Monte Carlo simulations were used in the standardized regression coefficients approach, with Latin Hypercube Sampling [109], which randomly selected a set of parame- ters for each simulation. Južnič-Zonta et al. [66] provided a multi-criteria assessment technique for establishing biochemical wastewater treatment plant operating strategies based on an ADM1 and under an uncertain environment. The Monte Carlo simulation and the Expected Utility Theory (EUT) [110] were used to cope with the alternatives among risky operating methods with multi-dimensional outcomes. Since World War II, EUT has been regarded as the preeminent paradigm in decision-making. EU theory remains the leading method for modeling risky decision-making [111]. Benedetti et al. [67] conducted a scenario analysis based on an open-loop version of BSM2 using Monte Carlo simulations and mul- ti-criteria analysis. Even though MCSs were not an iterative optimization approach, the parameter space was investigated in great depth with potentially fewer simulations than a multi-objective genetic algorithm [112].
This review revealed that the Monte Carlo simulation was the most frequently used tech- nique (almost 28 percent of the methods) to address the uncertainty in the AD process.
5.3 Confidence interval
Confidence intervals are used to determine a range of possible values for a target parameter, and a confidence factor indicates that the real value is included within that interval [113].
Using a confidence interval observer, Alcaraz-Gonzalez et al. [77] demonstrated how to manage an AD pilot plant for treating wine vinasses using a novel Single-Input-Single-Out- put (SIMO) control approach. In these studies, three primary sources of uncertainty have been introduced to develop a robust control observer. These sources include poorly known kinetics, a lack of sensors to measure all the online variables and unknown influent composition. Girault et al. [72] developed a procedure for degradation kinetics to divide a substrate’s total Chemical Oxygen Demand (COD) into each ADM1’s input state variables. A confidence range of 95% and an F distribution on the primary source of uncertainty were used. The primary source of uncertainty in COD fractionation was reported as temporal variability in substrate properties. Montiel-Escobar et al. [78] proposed a robust state estimation technique for AD processes described by ADM1 to approximate important variables under uncertain scenarios. The interval observers were used to cope with the uncertainty of the process inputs, unknown specific growth rates, and reaction kinetics were considered.
Batstone et al. [73] used an interval observer to evaluate the degradability extent, the lumped apparent first-order coefficient values and their uncertainties based on nonlinear parameter approximation. The observed parameter confidence regions were quite nonlinear, especially for continuous systems, implying that estimating genuine parameter uncertainty requires iterative or sampling approaches. Lübken et al. [74] modified the ADM1 and estimated kinetic parameters to simulate the performance of a mono-substrate two-stage agricultural biogas plant. The model’s uncertainty was approximated using confidence intervals of parameter estimates. The same procedure was done by Batstone et al. [75] for parameter estimation in modeling AD.
This technique was the second most frequently used method to cover the process uncer- tainty. Almost 18 percent of the methods belonged to this technique.
5.4 Other tools
Ghanavati et al. [7] maximized the ADM1-based process and combined a fuzzy supervisory control methodology with an Adaptive Model Predictive Controller (AMPC). The AMPC was based on the Auto-Regressive Moving Average (ARMA) model, which updates its parameters at each sample period to make the controller more resilient to uncertainties and external stresses. Restrepo et al. [91] performed a comparative performance analysis of types of controllers applied to an Upflow Anaerobic Sludge Blanket (UASB). By employing bifurcation analysis (BA) to discover and compare the stability regions and the impact of identification uncertainty over these regions in the process parameter space, these controllers were applied to a bioreactor. López-Pérez et al. [93] used a Min-Max method (+7% of the nominal values) to design a software sensor based on the state observer for a continuous bioreactor in a heavy metal removal application. They analyzed the system’s observability properties at a local level, taking parametric uncertainties into account. Lardon et al. [95] offered a methodological framework based on Dempster-Shafer’s theory of evidence to use a pilot size fixed-bed AD reactor to explore how different faults may be controlled efficiently in an experimental setting. The evidence theory’s key benefit is that it may also be used to discover inconsistencies in the expert rules related to each module. Das et al. developed a novel index to evaluate the performance of state estimation tech- niques. Hurst exponent [114], a dimensionless estimator for a time series’ self-similarity, was calculated using the Detrended Fluctuation Analysis (DFA) approach to estimate the internal variables from a nonlinear AD model and available measurements. There were different methods such as SA, Morris screening standardizes regression coefficient, scenario analysis, fuzzy logic, and multicriteria analysis, among others. SA (global and local) accounted for almost 15 percent of the methods.
More information about sources, methods, and applications of the uncertainty in AD processes can be found in Tables 7 and 8.
Table 7. Sources of uncertainties and methods to cope with them in the selected studies
|
Tools |
Sources of Uncertainty |
Ref |
|
MCS |
Input variables |
[59] |
|
MCS |
Kinetic coefficients and parameter estimation |
[60] |
|
MCS |
Model parameters |
[61] |
|
BA |
Controller proportional constant and gain-temporal parameter |
[91] |
|
DFA |
State estimation |
[92] |
|
MMM |
Parametric uncertainties |
[93] |
|
DFT |
Poor online data |
[95] |
|
CI |
Lack of adequate sensors, poorly known kinetics, and uncertain influent |
[76] |
|
CI |
Poorly known kinetics and substrate composition |
[77] |
|
GSA: MCS + SRC + MS |
Model parameters |
[62] |
|
MCS |
Component composition |
[3] |
|
CI |
Model parameters |
[47] |
|
MCS + LSA |
Model parameters |
[63] |
|
MCS |
Substrate composition |
[64] |
|
CI |
Process inputs, reactions, and specific growth rates |
[78] |
|
ScA |
Model parameters |
[94] |
|
FL |
External disturbances in feed composition and model uncertainty |
[7] |
|
GSA: MCS + SRC + MS + LSA |
Model parameters |
[65] |
|
CI |
Model parameters |
[73] |
|
MCS + EUT |
Model parameters |
[66] |
|
MCS + MCA |
Model parameters |
[67] |
|
ANN |
Model parameters |
[84] |
|
CI |
Model parameters |
[74] |
|
GSA: QU + MS |
Model parameters |
[83] |
|
MCS |
Model parameters |
[68] |
|
CI |
Model parameters |
[75] |
|
LO |
Process kinetics and variations in the influent composition |
[87] |
|
MWN |
Model’s biological uncertainty |
[105] |
|
LO |
Input constraints, parameter uncertainty, and load disturbances |
[88] |
|
IO |
Uncertainties of the inputs, load disturbances, and nonlinearity of the system |
[79] |
|
IO |
Unknown inputs, disturbances in the state, and COD of the input, as well as parametric uncertainties |
[76] |
|
RF + ANN + ELM, SVM, and GP |
Sensor noise and delay |
[85] |
|
OFO |
Several load disturbances, severe uncertainties, and typical operational failures. |
[104] |
|
RHT |
Feedstock conditions, time lags, and the lack of reliable measurements |
[103] |
|
IO |
Parameter uncertainty and load disturbances |
[80] |
|
IO |
Disturbances, the dynamics of the main state variables, lack of reliable and cheap sensors of the key process variables |
[81] |
|
NIR |
Model parameters |
[102] |
|
LO |
Unmeasured variables (such as VFA) and unknown microbial growth kinetics |
[89] |
|
GFCO + AO + STO |
Uncertainties in inlet chemical oxygen demand (COD) and the acidogenic bacteria population, methanogenic bacteria population, and time-varying parameters to estimate the inlet volatile fatty acid (VFA) concentration |
[101] |
|
NSO |
Differences in the rate of biochemical reactions, process uncertainties, and the consequences of interconnection between the two bioreactors |
[100] |
|
NNSMC |
Model parameters |
[98] |
|
DDPG |
External perturbations and parametric model uncertainty |
[97] |
|
SO |
Noise and variations of the COD influent concentration |
[96] |
|
MCS |
Model parameters |
[69] |
|
MCS |
Input data, model structure, and parameter values |
[70] |
|
SVR + ANN + RF |
Regression model uncertainties |
[86] |
|
MCS |
Stoichiometric, bio-kinetic, and influent parameters, hydraulic behavior of the plant and mass transfer parameters, and the combination of them |
[71] |
|
VSA |
Subsampling, i.e., taking subsets of the database of WWTPs and evaluating the potential changes in the overall index |
[99] |
|
CI |
Influent data |
[82] |
|
LCA |
External disturbances in the influent and kinetic/stoichiometric model parameters |
[90] |
|
Literature review |
|
[115] |
Table 8. Applications and year of publication of the selected studies
|
Ref |
Application |
Year |
|
[59] |
Maximizing biogas production |
2018 |
|
[60] |
Self-optimizing operation of anaerobic digesters |
2016 |
|
[61] |
A techno-economic optimization |
2004 |
|
[91] |
UASB controller |
2018 |
|
[92] |
Optimal anaerobic digestion processes monitoring |
2017 |
|
[93] |
Cadmium removal in an anaerobic process |
2016 |
|
[95] |
Online diagnosis in biological processes |
2004 |
|
[76] |
Anaerobic digester with highly nonlinear dynamics |
2001 |
|
[77] |
Interval-based regulation for anaerobic digestion processes |
2005 |
|
[62] |
Importance of secondary settling tank models |
2012 |
|
[3] |
Continuous anaerobic digestion based on ADM1 |
2017 |
|
[47] |
Degradation kinetics |
2012 |
|
[63] |
Optimizing ADM1 |
2019 |
|
[64] |
Mathematical simulation of maize silage |
2013 |
|
[78] |
state estimation scheme for ADM1 |
2012 |
|
[94] |
A biodegradability and modeling |
2013 |
|
[7] |
Control of an anaerobic bioreactor on the ADM1-based virtual plant. |
2021 |
|
[65] |
Modeling of anaerobic digesters (ADM1) |
2015 |
|
[73] |
Estimation of Parameters in Anaerobic Digesters |
2008 |
|
[66] |
Multi-criteria analyses of bio-processes (ADM1) |
2012 |
|
[67] |
Wastewater treatment plant design and control |
2010 |
|
[84] |
Two Stages Anaerobic Digestion Process (ADM1) |
2018 |
|
[74] |
Simulation of a biogas reactor (ADM1) |
2015 |
|
[83] |
ADM1-based AD Model |
2021 |
|
[68] |
Wastewater treatment plant (WWTP) control strategies |
2012 |
|
[75] |
Anaerobic sequencing batch reactor |
2004 |
|
[87] |
The control scheme for the stability of continuous anaerobic digestion processes |
2010 |
|
[105] |
Controlling anaerobic digestion processes at an industrial scale |
2003 |
|
[88] |
A pilot plant up-flow fixed-bed reactor that is treating industrial wine distillery wastewater |
2008 |
|
[79] |
Chemical and biochemical processes |
2005 |
|
[76] |
Industrial wine distillery wastewater |
2005 |
|
[85] |
Monitoring of volatile fatty acids in anaerobic digestion processes |
2020 |
|
[104] |
Control of an anaerobic digestion pilot plant |
2016 |
|
[103] |
Anaerobic digestion stable operation |
2019 |
|
[80] |
Operational stability in anaerobic digestion processes |
2013 |
|
[81] |
Highly uncertain anaerobic digestion processes |
2012 |
|
[102] |
Estimation of process parameters of anaerobic digestion |
2012 |
|
[89] |
Highly uncertain continuous anaerobic digestion processes |
2021 |
|
[101] |
Monitoring anaerobic digestion |
2022 |
|
[100] |
A two-stage anaerobic digestion process |
2022 |
|
[98] |
A biomass anaerobic digestion |
2019 |
|
[97] |
Digestion systems of tequila vinasses |
2022 |
|
[96] |
Stabilization in anaerobic digestion process |
2001 |
|
[69] |
Eindhoven wastewater treatment plant upgrade |
2013 |
|
[70] |
WWTP model |
2020 |
|
[86] |
Energy benchmarking to compare WWTPs, identify targets, and improve their performance |
2016 |
|
[71] |
Model-based design of a wastewater treatment plant |
2009 |
|
[99] |
44 different wastewater treatment plants |
2020 |
|
[82] |
Activated sludge models to full-scale wastewater treatment plants |
2012 |
|
[90] |
Activated sludge model 2D calibration with full-scale WWTP data |
2014 |
|
[115] |
Literature review |
2021 |
There are several sources of uncertainty, such as uncertainty in input variables, kinetic coefficients, parameter estimation, model parameters, reaction, and specific growth rates, substrate composition, controller proportional constant, gain and temporal parameters, among others. As found in this review, Monte Carlo simulation was the most frequently used technique (nearly 28 percent of the methods) to address the uncertainty associated with the AD process. The Confident Interval technique and SA were the second and third most fre- quently used methods for addressing the uncertainty of processes, and they accounted for almost 18 and 15 percent of the methods. According to the results, the most significant source of uncertainty was the unpredictability of model parameters and input variables. As a result of the knowledge gap and mathematical sense, fuzzy set theory is an important technique that eliminates costly and time-consuming simulations such as Monte Carlo or GSA.
Finding the sources of uncertainty, defining appropriate fuzzy values and conducting SA are the most important steps in modeling uncertainty using fuzzy set theory to have a robust model.
The current study identifies and analyzes uncertainty caused by AD processes and tools and strategies to cope with the uncertainties through a narrative review. Four databases have been selected and were used for the search (Scopus, Web of Science, ScienceDirect, and Google Scholar). After the full-text screening, this number dropped to 51 articles as the final selected studies. Among different methods and tools to cope with the uncertainty, 34 different methods have been addressed in selected studies. The highest contribution belongs to Monte Carlo simulation, followed by a statistical approach using confidence intervals/ interval observer and GSA. Uncertainty in model parameters such as reaction kinetics, substrate composition, and parameter estimates are the primary sources of uncertainty in biogas modeling processes. Based on this review, the following approaches have been sug- gested. These suggestions are associated with quantifying the uncertainty in AD process methodologies. It is recommended that all published AD studies contain a comprehensive uncertainty study. More attention should be paid to uncertainty in all model parameters. Finally, as a knowledge gap, it is suggested to use powerful mathematical tools such as the fuzzy set theory to quantify the model uncertainties. In comparison with Monte Carlo simulations, this method is much less costly.
[1] Selormey, G.K., Barnes, B., Kemausuor, F., Darkwah, L. (2021). A review of anaerobic digestion of slaughterhouse waste: Effect of selected operational and environmental parameters on anaerobic biodegradability. Reviews in Environmental Science and Bio/Technology, 20(4): 1073-1086. https://doi.org/10.1007/s11157-021-09596-8.
[2] REN21. (2020). Renewables 2020 Global Status Report. https://www.ren21.net/gsr-2020, accessed on Mar. 7, 2022.
[3] Kil, H., Xi, Y., Li, D. (2017). A new waste characterization method for the anaerobic digestion based on ADM1. Chemical Engineering Communications, 204(12): 1428-1444. https://doi.org/10.1080/00986445.2017.1367670
[4] Taherdanak, M., Zilouei, H. (2016). Biogas production from agricultural residues: A review. Current Biochemical Engineering (Discontinued), 3(3): 188-199.
[5] Jung, H., Kim, D., Choi, H., Lee, C. (2022). A review of technologies for in-situ sulfide control in anaerobic digestion. Renewable and Sustainable Energy Reviews, 157: 112068. https://doi.org/10.1016/j.rser.2021.112068
[6] Lohani, S.P., Havukainen, J. (2018). Anaerobic digestion: Factors affecting anaerobic digestion process. Waste Bioremediation, 343-359. https://doi.org/10.1007/978-981-10-7413-4_18
[7] Ghanavati, M.A., Vafa, E., Shahrokhi, M. (2021). Control of an anaerobic bioreactor using a fuzzy supervisory controller. Journal of Process Control, 103: 87-99. https://doi.org/10.1016/j.jprocont.2021.05.010
[8] Meegoda, J.N., Li, B., Patel, K., Wang, L. B. (2018). A review of the processes, parameters, and optimization of anaerobic digestion. International Journal of Environmental Research and Public Health, 15(10): 2224. https://doi.org/10.3390/ijerph15102224
[9] Gaida, D., Wolf, C., Bongards, M. (2017). Feed control of anaerobic digestion processes for renewable energy production: A review. Renewable and Sustainable Energy Reviews, 68: 869-875. https://doi.org/10.1016/j.rser.2016.06.096
[10] Heijungs, R., Lenzen, M. (2014). Error propagation methods for LCA - A comparison. The International Journal of Life Cycle Assessment, 19: 1445-1461. https://doi.org/10.1007/s11367-014-0751-0
[11] Cherubini, E., Franco, D., Zanghelini, G.M., Soares, S. R. (2018). Uncertainty in LCA case study due to allocation approaches and life cycle impact assessment methods. International Journal of Life Cycle Assessment, 23(10): 2055-2070. https://doi.org/10.1007/s11367-017-1432-6
[12] Heijungs, R., Huijbregts, M.A.J. (2004). A review of approaches to treat uncertainty in LCA. In Proceedings of the 2nd International Congress on Environmental Modelling and Software, Osnabrück, Germany, pp. 1-6.
[13] Begg, S., Bratvold, R., Welsh, M. (2014). Uncertainty vs. variability: What’s the difference and why is it important? SPE Hydrocarbon Economics and Evaluation Symposium, Houston, Texas. https://doi.org/10.2118/169850-MS
[14] Barahmand, Z., Eikeland, M.S. (2022). Techno-economic and life cycle cost analysis through the lens of uncertainty: A scoping review. Sustainability, 14(19): 12191. https://doi.org/10.3390/su141912191
[15] Barahmand, Z., Jayarathna, C., Ratnayake, C. (2022). Sensitivity and uncertainty analysis in a circulating fluidized bed reactor modeling. In Scandinavian Simulation Society (Ed.), Proceedings of the 23rd Scandinavian Simulation Society Conference, pp. 350-359. https://doi.org/10.3384/ecp21185350
[16] Barahmand, Z., Eikeland, M.S. (2022). Life cycle assessment under uncertainty: A scoping review. World, 3(3): 692-717. https://doi.org/10.3390/world3030039
[17] Zang, T.A. (2002). Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles. National Aeronautics and Space Administration, Langley Research Center.
[18] Lemaire, M. (2014). Mechanics and Uncertainty. John Wiley & Sons.
[19] Hoops, S., Hontecillas, R., Abedi, V., Leber, A., Philipson, C., Carbo, A., Bassaganya-Riera, J. (2016). Ordinary differential equations (ODEs) based modeling. Computational Immunology, 63-78. https://doi.org/10.1016/B978-0-12-803697-6.00005-9
[20] Pichery, C. (2014). Sensitivity analysis. In P. Wexler (Ed.), Encyclopedia of Toxicology, 3rd ed., pp. 236-237. https://doi.org/10.1016/B978-0-12-386454-3.00431-0
[21] Zi, Z. (2011). Sensitivity analysis approaches applied to systems biology models. IET Systems Biology, 5(6): 336-346. https://doi.org/10.1049/iet-syb.2011.0015
[22] Barahmand, Z., Samarakoon, G. (2022). Sensitivity analysis and anaerobic digestion modeling: A scoping review. Fermentation, 8(11): 624. https://doi.org/10.3390/fermentation8110624
[23] Barahmand, Z. (2021). Design of an Industrial Chlorination Reactor Using CPFD Simulations (Master's thesis). University of South-Eastern Norway.
[24] Das, A., Sharma, V. (2017). Modelling of uncertainty in control systems and design of a robust controller using H∞ method. 2017 International Conference on Intelligent Computing and Control Systems (ICICCS), 1008-1013. https://doi.org/10.1109/ICCONS.2017.8250617
[25] Ellis, G. (2002). Control systems and the role of observers. Observers in Control Systems, 1-4. https://doi.org/10.1016/B978-012237472-2/50002-X
[26] Khan, A., Xie, W., Zhang, B., Liu, L.W. (2021). A survey of interval observers design methods and implementation for uncertain systems. Journal of the Franklin Institute, 358(6): 3077-3126. https://doi.org/10.1016/j.jfranklin.2021.01.041
[27] Olatunji, K.O., Madyira, D.M., Ahmed, N.A., Adeleke, O., Ogunkunle, O. (2022). Modeling the biogas and methane yield from anaerobic digestion of Arachis hypogea shells with combined pretreatment techniques using machine learning approaches. Waste and Biomass Valorization, 14: 1123-1141. https://doi.org/10.1007/s12649-022-01935-2
[28] Huang, R., Ma, C., Ma, J., Huangfu, X., He, Q. (2021). Machine learning in natural and engineered water systems. Water Research, 205: 117666. https://doi.org/10.1016/j.watres.2021.117666
[29] Xu, R.Z., Cao, J.S., Feng, G., Luo, J.Y., Feng, Q., Ni, B. J., Fang, F. (2022). Fast identification of fluorescent components in three-dimensional excitation-emission matrix fluorescence spectra via deep learning. Chemical Engi- neering Journal, 430: 132893. https://doi.org/10.1016/j.cej.2021.132893
[30] Kazemi, P., Bengoa, C., Steyer, J.P., Giralt, J. (2021). Data-driven techniques for fault detection in anaerobic digestion process. Process Safety and Environmental Protection, 146: 905-915. https://doi.org/10.1016/j.psep.2020.12.016
[31] Fanourgakis, G.S., Gkagkas, K., Tylianakis, E., Froudakis, G. (2020). A generic machine learning algorithm for the prediction of gas adsorption in nanoporous materials. Journal of Physical Chemistry C, 124(13): 7117-7126. https://doi.org/10.1021/acs.jpcc.9b10766
[32] Marsland, S. (2011). Machine Learning: An Algorithmic Perspective. Chapman and Hall/ CRC: New York, 406. https://doi.org/10.1201/9781420067194
[33] Sahni, N., Simon, G., Arora, R. (2018). Development and validation of machine learning models for prediction of 1-year mortality utilizing electronic medical record data available at the end of hospitalization in multicondition patients: A proof-of-concept study. Journal of General Internal Medicine, 33(6): 921-928. https://doi.org/10.1007/s11606-018-4316-y
[34] Alzubi, J., Nayyar, A., Kumar, A. (2018). Machine learning from theory to algorithms: An overview. Journal of Physics: Conference Series, 1142(1): 012012. https://doi.org/10.1088/1742-6596/1142/1/012012
[35] Bowman, T. (2022). Library guides: Literature review: Traditional or narrative literature reviews. https://libguides.csu.edu.au/review/Traditional.
[36] Jahan, N., Naveed, S., Zeshan, M., Tahir, M.A. (2016). How to conduct a systematic review: A narrative literature review. Cureus, 8(11): e864. https://doi.org/10.7759/cureus.864
[37] Abbasi, T., Tauseef, S.M., Abbasi, S.A. (2012). A brief history of anaerobic digestion and 'biogas'. In T. Abbasi, S.M. Tauseef, & S.A. Abbasi (Eds.), Biogas Energy, 11-23. https://doi.org/10.1007/978-1-4614-1040-9_2
[38] Vasco-Correa, J., Khanal, S., Manandhar, A., Shah, A. (2018). Anaerobic digestion for bio-energy production: Global status, environmental and techno-economic implications, and government policies. Bioresource Technology, 247: 1015-1026. https://doi.org/10.1016/j.biortech.2017.09.004
[39] REN21. (2016). Renewables 2016 Global Status Report. REN21 Secretariat, Paris.
[40] Angelidaki, I., Ellegaard, L. (2003). Codigestion of manure and organic wastes in centralized biogas plants. Applied Biochemistry & Biotechnology, 109(1): 95-105. https://doi.org/10.1385/ABAB:109:1-3:95
[41] Sure Surendra, K.C., Takara, D., Hashimoto, A.G., Khanal, S.K. (2014). Biogas as a sustainable energy source for developing countries: Opportunities and challenges. Renewable and Sustainable Energy Reviews, 31, 846-859. https://doi.org/10.1016/j.rser.2013.12.015
[42] Batstone, D.J., Keller, J., Angelidaki, I., Kalyuzhnyi, S. V., Pavlostathis, S.G., Rozzi, A., Sanders, W.T.M., Siegrist, H., Vavilin, V. A. (2002). The IWA anaerobic digestion model no 1 (ADM1). Water Science and technology, 45(10): 65-73. https://doi.org/10.2166/wst.2002.0292
[43] Samarakoon, G., Dinamarca, C., Nelabhotla, A., Winkler, D., Bakke, R. (2020). Modelling bio-electrochemical CO2 reduction to methane. https://www.sintef.no/globalassets/project/tccs-10/dokumenter/c4/nelabhotla-presentation-tccs.pptx.pdf.
[44] Parker, W.J. (2005). Application of the ADM1 model to advanced anaerobic digestion. Bioresource Technology, 96(16): 1832-1842. https://doi.org/10.1016/j.biortech.2005.01.022
[45] Ramírez, I. (2012). ADM1 applications for a hybrid up-flow anaerobic sludge-filter bed reactor performance and for a batch thermophilic anaerobic digestion of thermally pretreated waste activated sludge. Revista Facultad de Ingeniería Universidad de Antioquia, 65: 167-179.
[46] Tassew, F.A., Bergland, W.H., Dinamarca, C., Bakke, R. (2019). Influences of temperature and substrate particle content on granular sludge bed anaerobic digestion. Applied Sciences, 10(1): 136. https://doi.org/10.3390/app10010136
[47] Fezzani B., Cheikh, R.B. (2008). Implementation of IWA anaerobic digestion model No. 1 (ADM1) for simulating the thermophilic anaerobic co-digestion of olive mill wastewater with olive mill solid waste in a semi-continuous tubular digester. Chemical Engineering Journal, 141(1): 75-88. https://doi.org/10.1016/j.cej.2007.10.024
[48] Droste, R.L. (1998). Endogenous decay and bioenergetics theory for aerobic wastewater treatment. Water Research, 32(2): 410-418. https://doi.org/10.1016/S0043-1354(97)00281-9
[49] Jeppsson, U., Rosen, C. (2006). Aspects on ADM1 Implementation within the BSM2 Framework. https://www.iea.lth.se/WWTmodels_download/TR_ADM1.pdf.
[50] Barahmand, Z. (2022). Comprehensive local sensitivity analysis of the ADM1-based anaerobic digestion process. Sustainable Development And Planning 2022, 235-246. https://doi.org/10.2495/SDP220201
[51] Brink, H., van der Walt, C., van Rensburg, G. (2012). Fundamentals of Research Methodology for Healthcare Professionals (3rd ed.). Juta Academic.
[52] Grant, M., Booth, A. (2009). A typology of reviews: An analysis of 14 review types and associated methodologies. Health Information and Libraries Journal, 26: 91-108. https://doi.org/10.1111/j.1471-1842.2009.00848.x
[53] Barahmand, Z., Eikeland, M.S. (2022). A scoping review on environmental, economic, and social impacts of the gasification processes. Environments, 9(7): 92. https://doi.org/10.3390/environments9070092
[54] Demiris, G., Oliver, D.P., Washington, K.T. (2018). Behavioral Intervention Research in Hospice and Palliative Care: Building an Evidence Base. Academic Press.
[55] Juntunen, M., Lehenkari, M. (2021). A narrative literature review process for an academic business research thesis. Studies in Higher Education, 46(2): 330-342. https://doi.org/10.1080/03075079.2019.1630813
[56] Sehularo, L. A., Molato, B.J., Mokgaola, I.O., Gause, G. (2021). Coping strategies used by nurses during the COVID-19 pandemic: A narrative literature review. Health SA Gesondheid, 26, Article no. 0. https://doi.org/10.4102/hsag.v26i0.1652
[57] Palinkas, L.A., Horwitz, S.M., Green, C.A., Wisdom, J.P., Duan, N., Hoagwood, K. (2015). Purposeful sampling for qualitative data collection and analysis in mixed method implementation research. Administration and Policy in Mental Health and Mental Health Services Research, 42(5): 533-544. https://doi.org/10.1007/s10488-013-0528-y
[58] Zhou, X., Zhou, M., Huang, D., Cui, L. (2022). A probabilistic model for co-occurrence analysis in bibliometrics. Journal of Biomedical Informatics, 128: 104047. https://doi.org/10.1016/j.jbi.2022.104047
[59] Asadi, M., McPhedran, K. (2021). Biogas maximization using data-driven modelling with uncertainty analysis and genetic algorithm for municipal wastewater anaerobic digestion. Journal of Environmental Management, 293: 112875. https://doi.org/10.1016/j.jenvman.2021.112875
[60] Neba, F.A., Tornyeviadzi, H. M., Østerhus, S. W., & Seidu, R. (2020). Self-optimizing attainable regions of the anaerobic treatment process: Modeling performance targets under kinetic uncertainty. Water Research, 171: 115377. https://doi.org/10.1016/j.watres.2019.115377
[61] Sharara, M.A., Owusu-Twum, M.Y., Runge, T.M., Larson, R. (2020). Planning methodology for anaerobic digestion systems on animal production facilities under uncertainty. Waste Management, 104: 262-269. https://doi.org/10.1016/j.wasman.2020.01.028
[62] Ramin, E., Flores-Alsina, X., Sin, G., Gernaey, K.V., Jeppsson, U., Mikkelsen, P.S., Plósz, B.G. (2012). Relative importance of secondary settling tank models in WWTP simulations-A global sensitivity analysis using BSM2. In Proceedings of the sixth biannual meeting of the International Environmental Modelling and Software Society, Leipzig, Germany, July 1-5, 2012, pp. 588-595.
[63] Xu, Z. (2019). ADM1 parameter calibration method based on partial least squares regression framework for industrial-scale anaerobic digestion modelling (Thesis). Stellenbosch University. https://scholar.sun.ac.za:443/handle/10019.1/107258, accessed onMar. 8, 2022.
[64] Gehring, T., Lübken, M., Koch, K., Wichern, M. (2013). ADM1 simulation of the thermophilic mono-fermentation of maize silage - Use of an uncertainty analysis for substrate characterization. https://www.semanticscholar.org/paper/ADM1-simulation-of-the-thermophilic-of-maize-silage-Gehring-L%C3%BCbken/bf190dff6f15557f897748c31e75435d0c06b06d, accessed on Mar. 8, 2022.
[65] Solon, K., Flores-Alsina, X., Gernaey, K., Jeppsson, U. (2015). Effects of influent fractionation, kinetics, stoichiometry and mass transfer on CH4, H2 and CO2 production for (plant-wide) modeling of anaerobic digesters. Water Science & Technology, 71: 870-877. https://doi.org/10.2166/wst.2015.029
[66] Južnič-Zonta, Ž., Kocijan, J., Flotats, X., Vrečko, D. (2012). Multi-criteria analyses of wastewater treatment bio-processes under an uncertainty and a multiplicity of steady states. Water Research, 46(18): 6121-6131. https://doi.org/10.1016/j.watres.2012.08.035
[67] Benedetti, L., De Baets, B., Nopens, I., Vanrolleghem, P. A. (2010). Multi-criteria analysis of wastewater treatment plant design and control scenarios under uncertainty. Environmental Modelling & Software, 25(5): 616-621. https://doi.org/10.1016/j.envsoft.2009.06.003
[68] Benedetti, L., Batstone, D.J., De Baets, B., Nopens, I., Vanrolleghem, P.A. (2012). Uncertainty analysis of WWTP control strategies made feasible. Water Quality Research Journal, 47(1): 14-29. https://doi.org/10.2166/wqrjc.2012.038
[69] Benedetti, L., Belia, E., Cierkens, K., Flameling, T., De Baets, B., Nopens, I., Weijers, S. (2013). The incorporation of variability and uncertainty evaluations in WWTP design by means of stochastic dynamic modeling: the case of the Eindhoven WWTP upgrade. Water science and technology, 67(8): 1841-1850. https://doi.org/10.2166/wst.2013.064
[70] Lindblom, E., Jeppsson, U., Sin, G. (2020). Identification of behavioural model input data sets for WWTP uncertainty analysis. Water Science and Technology, 81(8): 1558-1568. https://doi.org/10.2166/wst.2019.427
[71] Sin, G., Gernaey, K., Neumann, M., van Loosdrecht, M., & Gujer, W. (2009). Uncertainty analysis in WWTP model applications: A critical discussion using an example from design. Water Research, 43(11): 2894-2906. https://doi.org/10.1016/j.watres.2009.03.048
[72] Girault, R., Bridoux, G., Nauleau, F., Poullain, C., Buffet, J., Steyer, J. P., Sadowski, A.G., Béline, F. (2012). A waste characterisation procedure for ADM1 implementation based on degradation kinetics. Water research, 46(13): 4099-4110. https://doi.org/10.1016/j.watres.2012.04.028
[73] Batstone, D.J., Tait, S., Starrenburg, D. (2009). Estimation of hydrolysis parameters in full-scale anaerobic digesters. Biotechnology and Bioengineering, 102(5): 1513-1520. https://doi.org/10.1002/bit.22163
[74] Lübken, M., Koch, K., Gehring, T., Horn, H., Wichern, M. (2015). Parameter estimation and long-term process simulation of a biogas reactor operated under trace elements limitation. Applied Energy, 142: 352-360. https://doi.org/10.1016/j.apenergy.2015.01.014
[75] Batstone, D.J., Torrijos, M., Ruiz, C., Schmidt, J.E. (2004). Use of an anaerobic sequencing batch reactor for parameter estimation in modelling of anaerobic digestion. Water Science and Technology, 50(10): 295-303. https://doi.org/10.2166/wst.2004.0663
[76] Alcaraz-González, V., Harmand, J., Rapaport, A., Steyer, J.P., González-Álvarez, V., Pelayo-Ortiz, C. (2005). Robust interval-based regulation for anaerobic digestion processes. Water Science and Technology, 52(1-2): 449-456. https://doi.org/10.2166/wst.2005.0552
[77] Alcaraz-Gonzalez, V., Maloum, A., Harmand, J., Rapaport, A., Steyer, J.P., González-Alvarez, V., Pelayo-Ortiz, C. (2000). Robust interval-based SISO and SIMO regulation for a class of highly uncertain bioreactors: application to the anaerobic digestion. In Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No. 00CH37187), 5: 4532-4537. https://doi.org/10.1109/CDC.2001.914625
[78] Montiel-Escobar, J.L., Alcaraz-González, V., Méndez-Acosta, H.O., González-Álvarez, V. (2012). ADM1-based robust interval observer for anaerobic digestion processes. CLEAN-Soil, Air, Water, 40(9): 933-940. https://doi.org/10.1002/clen.201100718
[79] Alcaraz-González, V., Harmand, J., Rapaport, A., Steyer, J. P., González-Alvarez, V., Pelayo-Ortiz, C. (2005). Application of a robust interval observer to an anaerobic digestion process. Developments in Chemical Engineering and Mineral Processing, 13(3-4): 267-278. https://doi.org/10.1002/apj.5500130308
[80] Alcaraz-González, V., Fregoso-Sanchez, F.A., Mendez-Acosta, H.O., Gonzalez-Alvarez, V. (2013). Robust regulation of alkalinity in highly uncertain continuous anaerobic digestion processes. CLEAN - Soil, Air, Water, 41(12): 1157-1164. https://doi.org/10.1002/clen.201100736
[81] Alcaraz-González, V., López-Bañuelos, R.H., Steyer, J.P., Méndez-Acosta, H.O., González-Álvarez, V., Pelayo-Ortiz, C. (2012). Interval-based diagnosis of biological systems-a powerful tool for highly uncertain anaerobic digestion processes. CLEAN - Soil, Air, Water, 40(9): 941-949. https://doi.org/10.1002/clen.201100721
[82] Cierkens, K., Plano, S., Benedetti, L., Weijers, S., de Jonge, J., Nopens, I. (2012). Impact of influent data frequency and model structure on the quality of WWTP model calibration and uncertainty. Water Science and Technology, 65(2): 233-242. https://doi.org/10.2166/wst.2012.081
[83] Trucchia, A., Frunzo, L. (2021). Surrogate based global sensitivity analysis of ADM1-based anaerobic digestion model. Journal of Environmental Management, 282: 111456. https://doi.org/10.1016/j.jenvman.2020.111456
[84] Camberos, S.U.A., Gurubel, K.J., Sanchez, E.N., Aguirre, S.A., Perez, R.G. (2018). Neuronal modeling of a two stages anaerobic digestion process for biofuels production. IFAC-PapersOnLine, 51(13): 408-413. https://doi.org/10.1016/j.ifacol.2018.07.313
[85] Kazemi, P., Steyer, J.P., Bengoa, C., Font, J., Giralt, J. (2020). Robust data-driven soft sensors for online monitoring of volatile fatty acids in anaerobic digestion processes. Processes, 8(1): 67. https://doi.org/10.3390/pr8010067
[86] Torregrossa, D., Schutz, G., Cornelissen, A., Hernandez-Sancho, F., Hansen, J. (2016). Energy saving in WWTP: Daily benchmarking under uncertainty and data availability limitations. Environmental Research, 148: 330-337. https://doi.org/10.1016/j.envres.2016.04.010
[87] Méndez-Acosta, H.O., Palacios-Ruiz, B., Alcaraz-González, V., González-Álvarez, V., García-Sandoval, J.P. (2010). A robust control scheme to improve the stability of anaerobic digestion processes. Journal of Process Control, 20(4): 375-383. https://doi.org/10.1016/j.jprocont.2010.01.006
[88] Méndez-Acosta, H.O., Palacios-Ruiz, B., Alcaraz-González, V., Steyer, J.P., González-Álvarez, V., Latrille, E. (2008). Robust control of volatile fatty acids in anaerobic digestion processes. Industrial and Engineering Chemistry Research, 47(20): 7715-7720. https://doi.org/10.1021/ie800256e
[89] Alcaraz-González, V., Fregoso-Sánchez, F.A., Alvarez-González, V., Steyer, J.P. (2021). Multivariable robust regulation of alkalinities in continuous anaerobic digestion processes: Experimental validation. Processes, 9(7): 1153. https://doi.org/10.3390/pr9071153
[90] Machado, V., Lafuente, J., Baeza, J. (2014). Activated sludge model 2d calibration with full-scale WWTP data: Comparing model parameter identifiability with influent and operational uncertainty. Bioprocess and Biosystems Engineering, 37(7): 1271-1287. https://doi.org/10.1007/s00449-013-1099-8
[91] Rodriguez, G.A., Obregon, L.G., Restrepo, J.B. (2018). Linear and nonlinear control of an upflow anaerobic sludge blanket reactor: Effects of IMC-PID tuning and uncertainty. Chemical Engineering Transactions, 70: 1855. https://doi.org/10.3303/CET1870310
[92] Das, L., Kumar, G., Rani, M.D., Srinivasan, B. (2017). A novel approach to evaluate state estimation approaches for anaerobic digester units under modeling uncertainties: Application to an industrial dairy unit. Journal of Environmental Chemical Engineering, 5(4): 4004-4013. https://doi.org/10.1016/j.jece.2017.07.039
[93] López-Pérez, P., Neria-González, M., Aguilar-López, R. (2016). Concentrations monitoring via software sensor for bioreactors under model parametric uncertainty: Application to cadmium removal in an anaerobic process. AEJ-Alexandria Engineering Journal, 55(2): 1893-1902. https://doi.org/10.1016/j.aej.2016.03.013
[94] Astals, S., Esteban-Gutiérrez, M., Fernández-Arévalo, T., Aymerich, E., García-Heras, J.L., Mata-Alvarez, J. (2013). Anaerobic digestion of seven different sewage sludges: A biodegradability and modelling study. Water Research, 47(16): 6033-6043. https://doi.org/10.1016/j.watres.2013.07.019
[95] Lardon, L., Punal, A., Steyer, J.P. (2004). On-line diagnosis and uncertainty management using evidence theory-Experimental illustration to anaerobic digestion processes. Journal of Process Control, 14: 747-763. https://doi.org/10.1016/j.jprocont.2003.12.007
[96] Mailleret, L., Bernard, O. (2001). A simple robust controller to stabilise an anaerobic digestion process. IFAC Proceedings Volumes, 34(5): 207-212. https://doi.org/10.1016/S1474-6670(17)34221-0
[97] Mendiola-Rodriguez, T.A., Ricardez-Sandoval, L.A. (2022). Robust control for anaerobic digestion systems of Tequila vinasses under uncertainty: A deep deterministic policy gradient algorithm. Digital Chemical Engineering, 3: 100023. https://doi.org/10.1016/j.dche.2022.100023
[98] Yan, Y.C., Li, H.K. (2019). Neural network sliding mode control for biomass anaerobic digestion system with uncertain parameters (Environmental problem). Ekoloji, 28(107): 4167-4171.
[99] Mauricio-Iglesias, M., Longo, S., Hospido, A. (2020). Designing a robust index for WWTP energy efficiency: The ENERWATER water treatment energy index. Science of the Total Environment, 713: 136642. https://doi.org/10.1016/j.scitotenv.2020.136642
[100] Tawai, A., Sriariyanun, M. (2022). Nonlinear optimization-based robust control approach for a two-stage anaerobic digestion process. Journal of Chemistry, 2022: 8966350. https://doi.org/10.1155/2022/8966350
[101] Rueda-Escobedo, J.G., Sbarciog, M., Moreno, J.A., Van Impe, J., Wouwer, A.V. (2022). Robust state and input estimation with enhanced convergence rate for monitoring anaerobic digestion. Journal of Process Control, 117: 169-180. https://doi.org/10.1016/j.jprocont.2022.07.014
[102] Krapf, L.C., Gronauer, A., Schmidhalter, U., Heuwinkel, H. (2012). Evaluation of agricultural feedstock-robust near infrared calibrations for the estimation of process parameters in anaerobic digestion. Journal of Near Infrared Spectroscopy, 20(4): 465-476. https://doi.org/10.1255/jnirs.1013
[103] Flores-Estrella, R.A., Alcaraz-González, V., García-Sandoval, J.P., González-Álvarez, V. (2019). Robust output disturbance rejection control for anaerobic digestion processes. Journal of Process Control, 75: 15-23. https://doi.org/10.1016/j.jprocont.2018.12.012
[104] García-Sandoval, J.P., Méndez-Acosta, H.O., González-Álvarez, V., Schaum, A., Alvarez, J. (2016). VFA robust control of an anaerobic digestion pilot plant: Experimental implementation. IFAC-PapersOnLine, 49(70): 973-977. https://doi.org/10.1016/j.ifacol.2016.07.328
[105] Mailleret, L., Bernard, O., Steyer, J.P. (2003). Robust regulation of anaerobic digestion processes. Water Science and Technology, 48(6): 87-94. https://doi.org/10.2166/wst.2003.0364
[106] Harrison, R.L. (2010). Introduction to monte carlo simulation. AIP Conference Proceedings, 1204: 17-21. https://doi.org/10.1063/1.3295638
[107] Ross, J., Marshak, A. (1991). Monte carlo methods. In Photon-Vegetation Interactions: Applications in Optical Remote Sensing and Plant Ecology, 441-467. https://doi.org/10.1007/978-3-642-75389-3_14
[108] Ramin, E., Flores-Alsina, X., Sin, G., Gernaey, K.V., Jeppsson, U., Mikkelsen, P.S., Plósz, B.G. (2014). Influence of selecting secondary settling tank sub-models on the calibration of WWTP models-A global sensitivity analysis using BSM2. Chemical Engineering Journal, 241: 28-34. https://doi.org/10.1016/J.CEJ.2013.12.015
[109] Iman, R.L., Conover, W.J. (1982). A distribution-free approach to inducing rank correlation among input variables. Communications in Statistics-Simulation and Computation, 11(3): 311-334. https://doi.org/10.1080/03610918208812265
[110] Peasgood, T. (2014). Expected utility theory. In Encyclopedia of Quality of Life and Well-Being Research, 2092-2096. https://doi.org/10.1007/978-94-007-0753-5_962
[111] List, J.A., Haigh, M.S. (2005). A simple test of expected utility theory using professional traders. Proceedings of the National Academy of Sciences, 102(3): 945-948. https://doi.org/10.1073/pnas.0408022101
[112] Zolpakar, N.A., Lodhi, S.S., Pathak, S., Sharma, M.A. (2020). Application of multi-objective genetic algorithm (MOGA) optimization in machining processes. In Optimization of Manufacturing Processes, 185-199. https://doi.org/10.1007/978-3-030-19638-7_8
[113] Nandy, A., Chakraborty, S., Chakraborty, J., Venture, G. (2021). Statistics and computational intelligence in clinical gait analysis. Modern Methods for Affordable Clinical Gait Analysis, pp. 17-24. https://doi.org/10.1016/B978-0-323-85245-6.00009-6
[114] Hurst, H.E. (1951). Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers, 116(1): 770-799. https://doi.org/10.1061/TACEAT.0006518
[115] Sevillano, C.A., Pesantes, A.A., Peña Carpio, E., Martínez, E.J., Gómez, X. (2021). Anaerobic digestion for producing renewable energy-the evolution of this technology in a new uncertain scenario. Entropy, 23(2): 1-23. https://doi.org/10.3390/e23020145
