## 3 Results
### Dynamic Biogas Process Modeling
#### 3.4.3 Black Box Biogas Process Models  
Some fields of research deal with the implementation of artificial neural networks (ANNs) as black box models to predict the behavior of AD without the need for AD mechanisms or the physical laws influencing the model due to the high complexity of AD (white box ADM1 and simplified gray box AD models). Reference [109] used ANNs to predict and optimize the biogas production from one specific digester located in Jordan fed with wastes from slaughterhouses, restaurants and the dairy industry as well as fruit and vegetables. Data measured (total solid, volatile solid, pH value, temperature) were collected over a period of 177 days to train the ANN using the back-propagation method implemented in the ANN toolbox embedded in MATLAB(r). The ANN was able to predict the CH\({}_{4}\) ratio of biogas of this plant with an accuracy of R\({}^{2}\) = 0.87.  
Another ANN, published in [110], was developed with experimental data of 25 digesesters in lab-scale with a volume of 15 L. A total of 10 L of their volume was fed with hydrothermal pretreated waste, consisting of cow dung, rice bran, banana stem and sawdust in random ratios between 5 and 30%. Measurements of the biogas production rates, pH values and input ratios trained the ANN to predict the biogas production rate of a digester regarding different input substances. In order to realize an ANN that can predict the behavior of the AD of different substrates, it was trained in [111] with randomized datasets as results of the original ADM1, which was modeled in MATLAB(r) based on [112]. The model produced 12,000 datasets with variations of its protein, carbohydrate and lipid ratios and dilution rates as input parameters. This ANN was built and trained within the Python packages \"scikit-lern,\" \"NumPy,\" \"PyTorch\" and \"BFGS.\" It could predict equilibrium states of a methane production rate with an accuracy of 99% and dynamic states with an accuracy of 96.7% regarding different input substrates.  
Reference [113] used different sizes of the electrical power units on biogas plant sites combined with different biogas storage sizes for a black box model of a biogas plant with renewable energy dispatch simulation. The variation or flexibilization of the feedstock was not investigated; a constant biogas production with constant characteristics was accepted inside the model. The gas storage system model was composed of an overall mass balance equation. A cost minimization by selling electrical power on the German energy spot market with a flexible premium (Renewable Energy Act 2012) was calculated with time-dependent earnings. Increasing the CHP capacity installed from 600 kWel by a factor 3.3 to 2 MWel with simultaneously increased biogas storage capacity (factor 1.4) led to the highest profit on the spot market, but additional benefits cannot be expected without a flexible premium.  
Investigating not only additional gas storage systems but also additional heat storage capacity of an existing flexible biogas power plant (1360 kWel flexible) without feedstock effects is done in [114; 115] inside the simulation program IPSEpro(r) as flowchart-based modeling software. The model includes tanks (hygienization, mixing, fermentation and digestate storage), a gas storage and upgrade system, and a CHP. The biogas production was integrated as a timeline with real one-year data as output information of the fermentation tank. The CHP model consisted of energy balance equations. Gas and heat storage systems were modeled by mass and energy balance equations. The implementation of a gas upgrade system was done as a black box model. Additional heat storage (water) led to additional income. The existing gas storage capacity (4800 m\({}^{3}\)) could provide control energy reserves and biomethane production simultaneously.