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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

\[ \begin{align*} c(\text{t}_{i+1}) &= c(\text{t}_{i}) \cdot (1-\dot{\text{c}}_\text{rel,loss}(\text{t}_i))^{\Delta \text{t}_{i}} \\ &\quad + p_{f_\text{in}}(\text{t}_i) \cdot \Delta \text{t}_i \cdot \eta_\text{in}(\text{t}_i) \\ &\quad - p_{f_\text{out}}(\text{t}_i) \cdot \Delta \text{t}_i \cdot \eta_\text{out}(\text{t}_i) \tag{3} \end{align*} \]

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 \(\text{t}_i\)

Mathematical Patterns Used

Storage formulation uses the following modeling patterns:

  • Basic Bounds - For charge state bounds (equation \(\eqref{eq:Storage_Bounds}\))
  • Scaled Bounds - For flow rate bounds relative to storage size

When combined with investment parameters, storage can use: - Bounds with State - Investment decisions (see InvestParameters)

Implementation

Python Class: Storage

Key Parameters: - capacity_in_flow_hours: Storage capacity \(\text{C}\) - relative_loss_per_hour: Self-discharge rate \(\dot{\text{c}}_\text{rel,loss}\) - initial_charge_state: Initial charge \(c(\text{t}_0)\) - minimal_final_charge_state, maximal_final_charge_state: Final charge bounds \(c(\text{t}_\text{end})\) (optional) - eta_charge, eta_discharge: Charging/discharging efficiencies \(\eta_\text{in}, \eta_\text{out}\)

See the Storage API documentation for complete parameter list and usage examples.

See Also