## 3 Results
### Dynamic Biogas Process Modeling
#### 3.4.2 Gray Box Biogas Process Models  
The ADM1 is seldom used to optimize and control full-scale plants due to its high complexity. It needs an extensive dataset regarding the feedstock, which leads to a complex feedstock analysis as a result of missing data available in the literature [93]. Several studies modify the ADM1 or generate simplified AD models. These so-called gray box models reduce the complexity by omitting non-limiting reaction steps and bridge a gap between simplicity and accuracy. This type of model creates the possibility of using practical data of the digester input substrate without knowing all parameters necessary for complex AD models [94], but it requires a parameterization of the mathematical functions obtained and modeled for a specific physical problem. Based on the idea of flexibly generating energy and, thus, flexible biogas production, Reference [95] modeled an AD process as two differential equations, four parameters of two reaction steps (a fast one for the digestion of carbohydrates and a slow one for the digestion of lipids and proteins). The biogas volume flow rate was calculated as the sum of the flow rates resulting from the two reaction mechanisms. This overall volume flow rate was modeled as transfer function in the Laplace domain with the feedstock as input variable. This model was used to observe the AD behavior under fluctuating input feedstock, leading to a variation of biogas production rates. Using experimental data of a 1.5 m\({}^{3}\) digester (1 m\({}^{3}\) digestion volume, 0.5 m\({}^{3}\) gas storage volume), the simulation results were also compared with the ADM1 and showed a sufficient compliance of both AD process models. Using a systematical simplification of the ADM1 presented in [96], Reference [97] opened the possibility of relying on four reduced (\"R\") AD models, ADM1-R1 to ADM1-R4. The authors implemented these simplifications in a dynamic biogas plant model programmed with dynamic gas storage and CHP models. The simulation was subjected to the German electricity and its European power exchange (EPEX) spot market. Flexible feeding of a biogas plant was investigated to obtain a maximum total revenue on the EPEX spot market under an additional variation of gas storage capacity. The feeding control inside the model was realized by a model predictive controller [98]. This work is still under investigation; a verification of this model with a research biogas plant of the Deutsches Biomasserntum was started at the time of writing this review. Reference [99] used a time-dependent biogas production flow rate, modeled as exponential function. The biogas volume flow rate depended on feedstock properties, e.g., the organic fraction, its biogas production potential and maximum biogas production rate. The biogas production was modeled and simulated under flexible feedstock input using MATLAB(r) Simulink. A CHP unit as a balance equation with electrical load dependent efficiencies was included, as was a dynamic gas storage system. The model was verified by real data (daily biogas and electrical energy production and overall thermal energy production of December 2018) of a biogas plant located in Bornholm, Denmark, and could predict its electrical and thermal output well considering the real production data of the biogas plant. The deviation between the real data and the simulation showed averages of 7.7, 0.78 and 1.99% for biogas, electrical and thermal energy production, respectively. The possibility of utilizing biomass to produce electrical energy, on the one hand, and fuel, on the other hand, is shown in [100], with a model of a combined electrical energy and fuel supply system with compressed natural gas (CNG) as vehicle fuel. A combined MATLAB(r) and Microsoft Excel(r) simulation was modeled to emulate the impact of the EPEX spot market regarding the electrical energy production in addition to the impact of filling cycles of CNG for different types of vehicle fleets regarding the CNG production.