## 2 Materials and Methods
### Model Categorization  
Mathematical process models can generally be classified by regarding their level of theoretical depth. While experimental models describe a process by experimental input-output analyses, all physical dependencies are described by differential equations in theoretical models. This requires very detailed process understanding and knowledge of all influencing parameters. The different model approaches can be categorized in black box, gray box and white box modeling, with increasing model complexity from white to black box approaches (Figure 4).  
White box models consist of linear or nonlinear differential equations and know their physics, structures, and parameters well without any restrictions made by the physical system they describe [34]. They are the most detailed models and close to a full description of a real system, providing results near to the actual behavior of the real system. However, as they do not contain approximations, a complex design and implementation of all influencing parameters is necessary [35].  
Black box models on the other hand create a system response without knowing any physical dependencies of the input variable and without considering the complexity of the system. Experimental data or results of a physical system train these models, such as  
Figure 4: Detail level of mathematical models categorized in white box, gray box and black box models, modified according to [34].  
artificial neural networks (ANN) or statistical models. They describe the functional relationship between model inputs and outputs and are efficient at representing the behavior of a process as a trend [36]. By using only a set of rules and equations, these models are fast in terms of computing and processing, but show a lack of flexibility as they cannot adept to slightly different physics [35].  
As a mixture of white and black box models, gray box models, such as fuzzy models or differential equations, know physical dependencies of input and output data, but not any generally valid parameters that describe the physical system. They use physical representations and approximations with model parameters, which have to be gained for each physical system they describe, i.e., by experiments. Accordingly, their application is restricted to one well-defined physical system and its estimated parameters [34].  
Within this paper, literature on biogas modeling is classified according to its view on the biogas plant and the modeling approach. Regarded views on the biogas plant are according to Figure 1: energy system, regional potential, greenhouse gas (GHG) emissions, and biogas production.  
Depending on the complexity of the AD model within each publication, it is further assigned to the categories of white box, gray box, or black box models. Biogas modeling approaches which include a detailed mathematical description of all physical and biological process steps of the anaerobic digestion are thus classified as white box models (e.g., the ADM1 model), simple systems of equations or linear relationships are classified as black box models (e.g., the calculation of methane potential based on the organic solids of a feedstock), and simplified dynamic relationships of biogas production with respect to the feedstock are classified as gray box models (e.g., simplifications of the ADM1 model).  
Moreover, the classification into top-down and bottom-up models presented in [37] is transferable to modeling approaches of biogas production. Providing an aggregate view of the energy sector and economy when simulating economic trends and energy demand supply, these models are classified as top-down model accordingly. In comparison, modeling detailed information about processes, technical systems and time-dependent conditions is the result of bottom-up modeling approaches of biogas production.