Storages
Storages have one incoming and one outgoing Flow with a charging and discharging efficiency.
A storage has a state of charge \(c(\text{t}_i)\) which is limited by its size \(\text C\) and relative bounds \(\eqref{eq:Storage_Bounds}\).
\[ \label{eq:Storage_Bounds}
\text C \cdot \text c^{\text{L}}_{\text{rel}}(\text t_{i})
\leq c(\text{t}_i) \leq
\text C \cdot \text c^{\text{U}}_{\text{rel}}(\text t_{i})
\]
Where:
- \(\text C\) is the size of the storage
- \(c(\text{t}_i)\) is the state of charge at time \(\text{t}_i\)
- \(\text c^{\text{L}}_{\text{rel}}(\text t_{i})\) is the relative lower bound (typically 0)
- \(\text c^{\text{U}}_{\text{rel}}(\text t_{i})\) is the relative upper bound (typically 1)
With \(\text c^{\text{L}}_{\text{rel}}(\text t_{i}) = 0\) and \(\text c^{\text{U}}_{\text{rel}}(\text t_{i}) = 1\), Equation \(\eqref{eq:Storage_Bounds}\) simplifies to
\[ 0 \leq c(\text t_{i}) \leq \text C \]
The state of charge \(c(\text{t}_i)\) decreases by a fraction of the prior state of charge. The belonging parameter $ \dot{ \text c}_\text{rel, loss}(\text{t}_i)$ expresses the "loss fraction per hour". The storage balance from \(\text{t}_i\) to \(\text t_{i+1}\) is
\[
\begin{align*}
c(\text{t}_{i+1}) &= c(\text{t}_{i}) \cdot (1-\dot{\text{c}}_\text{rel,loss}(\text{t}_i) \cdot \Delta \text{t}_{i}) \\
&\quad + p_{f_\text{in}}(\text{t}_i) \cdot \Delta \text{t}_i \cdot \eta_\text{in}(\text{t}_i) \\
&\quad - \frac{p_{f_\text{out}}(\text{t}_i) \cdot \Delta \text{t}_i}{\eta_\text{out}(\text{t}_i)}
\tag{3}
\end{align*}
\]
Where:
- \(c(\text{t}_{i+1})\) is the state of charge at time \(\text{t}_{i+1}\)
- \(c(\text{t}_{i})\) is the state of charge at time \(\text{t}_{i}\)
- \(\dot{\text{c}}_\text{rel,loss}(\text{t}_i)\) is the relative loss rate (self-discharge) per hour
- \(\Delta \text{t}_{i}\) is the time step duration in hours
- \(p_{f_\text{in}}(\text{t}_i)\) is the input flow rate at time \(\text{t}_i\)
- \(\eta_\text{in}(\text{t}_i)\) is the charging efficiency at time \(\text{t}_i\)
- \(p_{f_\text{out}}(\text{t}_i)\) is the output flow rate at time \(\text{t}_i\)
- \(\eta_\text{out}(\text{t}_i)\) is the discharging efficiency at time \(\text{t}_i\)