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 $\text{(1)}$.

$$\begin{array}{}\text{(1)}& \text{C}\cdot {\text{c}}_{\text{rel}}^{\text{L}}({\text{t}}_i)\le c({\text{t}}_i)\le \text{C}\cdot {\text{c}}_{\text{rel}}^{\text{U}}({\text{t}}_i)\end{array}$$

Where:

• $\text{C}$ is the size of the storage
• $c({\text{t}}_i)$ is the state of charge at time ${\text{t}}_i$
• ${\text{c}}_{\text{rel}}^{\text{L}}({\text{t}}_i)$ is the relative lower bound (typically 0)
• ${\text{c}}_{\text{rel}}^{\text{U}}({\text{t}}_i)$ is the relative upper bound (typically 1)

With ${\text{c}}_{\text{rel}}^{\text{L}}({\text{t}}_i)=0$ and ${\text{c}}_{\text{rel}}^{\text{U}}({\text{t}}_i)=1$, Equation $\text{(1)}$ simplifies to

$$\begin{array}{}\text{(2)}& 0\le c({\text{t}}_i)\le \text{C}\end{array}$$

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

$$\begin{array}{rl}c({\text{t}}_{i+1})& =c({\text{t}}_i)\cdot (1-{\dot{\text{c}}}_{\text{rel,loss}}({\text{t}}_i)\cdot \mathrm{\Delta }{\text{t}}_i)\\ & +p_{f_{\text{in}}}({\text{t}}_i)\cdot \mathrm{\Delta }{\text{t}}_i\cdot {\eta }_{\text{in}}({\text{t}}_i)\\ \text{(3)}& & -\frac{p_{f_{\text{out}}}({\text{t}}_i)\cdot \mathrm{\Delta }{\text{t}}_i}{{\eta }_{\text{out}}({\text{t}}_i)}\end{array}$$

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
• $\mathrm{\Delta }{\text{t}}_i$ 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 ${\text{t}}_i$