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Storage

A Storage accumulates energy over time — charge now, discharge later.

Basic: Charge Dynamics

\[ c(t+1) = c(t) \cdot (1 - \dot{c}_{loss})^{\Delta t} + p_{in}(t) \cdot \Delta t \cdot \eta_{in} - p_{out}(t) \cdot \Delta t / \eta_{out} \]
battery = fx.Storage(
    label='battery',
    charging=fx.Flow(label='charge', bus=elec_bus, size=50),
    discharging=fx.Flow(label='discharge', bus=elec_bus, size=50),
    capacity_in_flow_hours=200,  # 200 kWh
    eta_charge=0.95,
    eta_discharge=0.95,
)
# Round-trip efficiency: 95% × 95% = 90.25%

Charge State Bounds

\[ C \cdot c_{rel}^{min} \leq c(t) \leq C \cdot c_{rel}^{max} \]
fx.Storage(...,
    relative_minimum_charge_state=0.2,  # Min 20% SOC
    relative_maximum_charge_state=0.8,  # Max 80% SOC
)

Initial & Final Conditions

fx.Storage(..., initial_charge_state=100)  # Start at 100 kWh

Must end where it started (prevents "cheating"):

fx.Storage(..., initial_charge_state='equals_final')

fx.Storage(...,
    minimal_final_charge_state=50,
    maximal_final_charge_state=150,
)

Adding Features

tank = fx.Storage(...,
    relative_loss_per_hour=0.02,  # 2%/hour loss
)

Optimize storage size:

battery = fx.Storage(...,
    capacity_in_flow_hours=fx.InvestParameters(
        minimum_size=0,
        maximum_size=1000,
        specific_effects={'costs': 200},  # €/kWh
    ),
)

Different charge/discharge rates:

fx.Storage(
    charging=fx.Flow(..., size=100),     # 100 MW pump
    discharging=fx.Flow(..., size=120),  # 120 MW turbine
    ...
)

Reference

Symbol Type Description
\(c(t)\) \(\mathbb{R}_{\geq 0}\) Charge state at timestep \(t\)
\(C\) \(\mathbb{R}_{\geq 0}\) Capacity (capacity_in_flow_hours)
\(p_{in}(t)\) \(\mathbb{R}_{\geq 0}\) Charging power (from charging flow)
\(p_{out}(t)\) \(\mathbb{R}_{\geq 0}\) Discharging power (from discharging flow)
\(\eta_{in}\) \(\mathbb{R}_{\geq 0}\) Charge efficiency (eta_charge)
\(\eta_{out}\) \(\mathbb{R}_{\geq 0}\) Discharge efficiency (eta_discharge)
\(\dot{c}_{loss}\) \(\mathbb{R}_{\geq 0}\) Self-discharge rate (relative_loss_per_hour)
\(c_{rel}^{min}\) \(\mathbb{R}_{\geq 0}\) Min charge state (relative_minimum_charge_state)
\(c_{rel}^{max}\) \(\mathbb{R}_{\geq 0}\) Max charge state (relative_maximum_charge_state)
\(\Delta t\) \(\mathbb{R}_{> 0}\) Timestep duration (hours)

Classes: Storage, StorageModel