Flows¶

A flow represents energy transfer on a bus. This page covers the full mathematical model: sizing, bounds, fixed profiles, and effect contributions.

Flow Rate Variable¶

Each flow \(f\) has a non-negative rate variable \(P_{f,t}\) at each timestep:

\[ P_{f,t} \geq 0 \quad \forall \, f \in \mathcal{F}, \; t \in \mathcal{T} \]

Sizing¶

The nominal capacity \(\bar{P}_f\) (Flow.size) sets the scale for all relative parameters. When no capacity is specified (\(\bar{P}_f = \infty\)), the flow is unbounded above.

Bounds¶

Sized Flows¶

When a flow has a nominal capacity \(\bar{P}_f\), the flow rate is bounded by relative minimum and maximum profiles:

\[ \bar{P}_f \cdot \underline{p}_{f,t} \leq P_{f,t} \leq \bar{P}_f \cdot \bar{p}_{f,t} \quad \forall \, f, t \]

By default, \(\underline{p}_{f,t} = 0\) and \(\bar{p}_{f,t} = 1\), so the bounds simplify to \(0 \leq P_{f,t} \leq \bar{P}_f\).

Unsized Flows¶

When no capacity is specified (\(\bar{P}_f = \infty\)), the flow is unbounded above:

\[ 0 \leq P_{f,t} \quad \forall \, f, t \]

Fixed Profile¶

When Flow.fixed_relative_profile (\(\pi_{f,t}\)) is set, the flow rate is fixed to a profile scaled by the capacity:

\[ P_{f,t} = \bar{P}_f \cdot \pi_{f,t} \quad \forall \, f, t \]

This is implemented by setting both lower and upper bounds equal to the profile value.

Sizing (Investment)¶

When Flow.size is a Sizing object, the fixed capacity is replaced by a decision variable. See Sizing for the full formulation.

Status (On/Off)¶

When Flow.status is set, binary on/off behavior is added. The flow becomes semi-continuous: \(\{0\} \cup [\underline{P}, \bar{P}]\). See Status for the full formulation.

Effect Contributions¶

Each flow can contribute to tracked effects (cost, emissions, ...). The per-timestep contribution of flow \(f\) to effect \(k\) is:

\[ c_{f,k,t} \cdot P_{f,t} \cdot \Delta t_t \]

where \(c_{f,k,t}\) is the effect coefficient per flow-hour (Flow.effects_per_flow_hour). These contributions feed into the effect tracking equations.

Parameters¶

Symbol	Description	Reference
\(P_{f,t}\)	Flow rate variable	`flow_rate[flow, time]`
\(\bar{P}_f\)	Nominal capacity	`Flow.size`
\(\underline{p}_{f,t}\)	Relative lower bound	`Flow.relative_minimum`
\(\bar{p}_{f,t}\)	Relative upper bound	`Flow.relative_maximum`
\(\pi_{f,t}\)	Fixed relative profile	`Flow.fixed_relative_profile`
\(c_{f,k,t}\)	Effect coefficient per flow-hour	`Flow.effects_per_flow_hour`
\(\Delta t_t\)	Timestep duration	dt

See Notation for the full symbol table.

Examples¶

Sized Flow with Minimum Load¶

A boiler with capacity \(\bar{P} = 10\) MW, minimum load \(\underline{p} = 0.3\), maximum load \(\bar{p} = 1.0\):

\[ 10 \times 0.3 \leq P_t \leq 10 \times 1.0 \quad \Rightarrow \quad 3 \leq P_t \leq 10 \; \text{MW} \]

Fixed Demand Profile¶

A demand of 100 MW capacity with profile \(\pi = [0.4, 0.7, 0.5, 0.6]\):

\[ P_t = 100 \cdot \pi_t \quad \Rightarrow \quad P = [40, 70, 50, 60] \; \text{MW} \]

Effect Contribution¶

A gas flow with cost coefficient \(c = 0.04\) €/MWh, rate \(P = 5\) MW, duration \(\Delta t = 1\) h:

\[ 0.04 \times 5 \times 1 = 0.2 \; \text{€} \]