# Strategic Resource Planning for Sustainable Biogas Integration in Hybrid Renewable Energy Systems
## 2 Materials and Methods
### Biogas Production and Management
The biogas production at any given time \(t\) can be described using Equation (3):  
\[Q(t)=Q_{max}\times exp(-k\times(t-1)) \tag{3}\]  
Here, \(Q(t)\) is the biogas production at time \(t\), \(Q_{max}\) is the initial production rate (calculated previously), and \(k\) is a decay constant that governs how quickly biogas production declines. This exponential decay is essential because it reflects the actual behavior of anaerobic digesters: biogas production peaks shortly after the biomass is introduced and slowly tapers off as digestion continues. Therefore, understanding the time dynamics of biogas production is crucial for ensuring a consistent energy supply, especially when the system is integrated with other renewable energy sources, like solar and wind, which are less controllable. Due to the use of animal waste from local farms, including rabbits, pigs, and chickens, the decay constant for biogas production is estimated to be around 0.135 [33]. This value reflects the relatively fast degradation of organic material in animal waste, leading to a quicker process than other biomass types. Figure 2 shows the biogas production process for 1 kg of biomass over 30 days. The top graph illustrates the biogas production rate (m\({}^{3}\)/day), which starts high and gradually decreases following an exponential decay as the available biomass is consumed. By the  
end of the period, the production rate approaches zero. The bottom graph presents the cumulative biogas production (m\({}^{3}\)), which steadily increases over time before flattening out as production slows. This indicates that most of the biogas is generated in the early days of the process, with diminishing returns as the digestion process progresses. As seen in Figure 2, the retention time of the biomass is estimated to be 30 days. Retention time refers to the period during which the biomass remains in the digester for biogas production. The 30-day period ensures that most biodegradable material is fully broken down, maximizing biogas yield. This duration allows optimal digestion while preventing excess accumulation of undegraded material, ensuring efficient system operation. As has been shown in Figure 2, the representation of cumulative biogas production over a given period is calculated using Equation (4):  
\[Q_{Cumulative}=\sum_{t_{i}}^{t_{i}+T}Q(t) \tag{4}\]  
In this case, \(Q_{Cumulative}\) represents the cumulative biogas produced between the starting time \(t_{i}\) and the end time \(t_{i}+T\). \(T\) corresponds to retention time. Converting daily biogas production equations into hourly values provides a more detailed and precise understanding of the system's dynamics. Using hourly rates makes monitoring and managing fluctuations in biogas production throughout the day easier, especially when integrating biogas generation with other renewable energy sources like solar and wind. This allows for better matching energy supply with real-time demand, optimizing generator operation, and ensuring more efficient use of biogas, particularly during peak energy consumption or reduced availability of other energy sources. The hourly biogas production is analyzed with greater granularity, providing insights into the fluctuations in generation rates across different hours. This detailed perspective is crucial  
Figure 2: The biogas production rate for 1 kg over 30 days. for optimizing energy supply and ensuring that demand is met consistently throughout the day, as described in Equation (5):  
\[Q(t)=\frac{m_{Biomass}\times f_{VS}\times\eta_{Conversation}\times k_{Methanogenic }\times exp\left(\frac{-k}{730}\times(t-1)\right)}{24} \tag{5}\]  
The cumulative biogas production must account for each input over its retention time if there are different biomass inputs at various times. In this case, the equation would sum the biogas production from each input separately over the relevant hourly period.