Flow¶

A Flow is the primary optimization variable — the solver decides how much flows at each timestep.

Basic: Bounded Flow Rate¶

Every flow has a size \(P\) (capacity) and a flow rate \(p(t)\) (what the solver optimizes):

\[ P \cdot p_{rel}^{min} \leq p(t) \leq P \cdot p_{rel}^{max} \]

# 100 kW boiler, minimum 30% when running
heat = fx.Flow(label='heat', bus=heat_bus, size=100, relative_minimum=0.3)
# → 30 ≤ p(t) ≤ 100

Cannot be zero

With relative_minimum > 0, the flow cannot be zero. Use status_parameters to allow shutdown.

Adding Features¶

StatusVariable SizeFlow EffectsFixed Profile

Allow the flow to be zero with status_parameters:

\(s(t) \cdot P \cdot p_{rel}^{min} \leq p(t) \leq s(t) \cdot P \cdot p_{rel}^{max}\)

Where \(s(t) \in \{0, 1\}\): inactive or active.

generator = fx.Flow(
    label='power', bus=elec_bus, size=50,
    relative_minimum=0.4,
    status_parameters=fx.StatusParameters(
        effects_per_startup={'costs': 500},
        min_uptime=2,
    ),
)

See StatusParameters.

Optimize the capacity with InvestParameters:

\(P^{min} \leq P \leq P^{max}\)

battery = fx.Flow(
    label='power', bus=elec_bus,
    size=fx.InvestParameters(
        minimum_size=0,
        maximum_size=1000,
        specific_effects={'costs': 100_000},
    ),
)

See InvestParameters.

Add effects per energy (flow hours) moved:

gas = fx.Flow(
    label='gas', bus=gas_bus, size=150,
    effects_per_flow_hour={'costs': 50},  # €50/MWh
)

Flow hours: \(h(t) = p(t) \cdot \Delta t\)

Lock the flow to a time series (demands, renewables):

\(p(t) = P \cdot \pi(t)\)

demand = fx.Flow(
    label='demand', bus=heat_bus, size=100,
    fixed_relative_profile=[0.5, 0.8, 1.0, 0.6]  # π(t)
)

Optional Constraints¶

Load FactorFlow Hours

Constrain average utilization:

\(\lambda_{min} \leq \frac{\sum_t p(t)}{P \cdot n_t} \leq \lambda_{max}\)

fx.Flow(..., load_factor_min=0.5, load_factor_max=0.9)

Constrain total energy:

\(h_{min} \leq \sum_t p(t) \cdot \Delta t \leq h_{max}\)

fx.Flow(..., flow_hours_min=1000, flow_hours_max=5000)

Reference¶

Symbol	Type	Description
\(p(t)\)	\(\mathbb{R}_{\geq 0}\)	Flow rate at timestep \(t\)
\(P\)	\(\mathbb{R}_{\geq 0}\)	Size (capacity) — fixed or optimized
\(s(t)\)	\(\{0, 1\}\)	Binary status (with `status_parameters`)
\(p_{rel}^{min}\)	\(\mathbb{R}_{\geq 0}\)	Minimum relative flow (`relative_minimum`)
\(p_{rel}^{max}\)	\(\mathbb{R}_{\geq 0}\)	Maximum relative flow (`relative_maximum`)
\(\pi(t)\)	\(\mathbb{R}_{\geq 0}\)	Fixed profile (`fixed_relative_profile`)
\(\Delta t\)	\(\mathbb{R}_{> 0}\)	Timestep duration (hours)

Classes: Flow, FlowModel