Carrier Balance

Formulation

Every carrier \(b\) must be balanced at every timestep — total outflow equals total inflow:

\[ \sum_{f \in \mathcal{F}_b^{\text{out}}} P_{f,t} - \sum_{f \in \mathcal{F}_b^{\text{in}}} P_{f,t} = 0 \quad \forall \, b \in \mathcal{B}, \; t \in \mathcal{T} \]

where:

• \(\mathcal{F}_b^{\text{out}}\) — flows that produce into carrier \(b\) (imports of ports and outputs of converters)
• \(\mathcal{F}_b^{\text{in}}\) — flows that consume from carrier \(b\) (exports of ports and inputs of converters)

The sign convention uses coefficients: \(+1\) for flows producing into the carrier and \(-1\) for flows consuming from the carrier. The constraint is then:

\[ \sum_{f \in \mathcal{F}_b} \text{coeff}_{b,f} \cdot P_{f,t} = 0 \quad \forall \, b, t \]

Parameters

Symbol	Description	Reference
\(\mathcal{F}_b^{\text{out}}\)	Flows producing into carrier \(b\)	carrier_coeff[f.id] = +1 (port imports, converter outputs, storage discharging)
\(\mathcal{F}_b^{\text{in}}\)	Flows consuming from carrier \(b\)	carrier_coeff[f.id] = -1 (port exports, converter inputs, storage charging)
\(P_{f,t}\)	Flow rate variable	flow_rate[flow, time]
\((b\!:\!n)\)	Compound carrier-node coordinate	e.g., heat:A, heat:B
\(\mathcal{F}_{b:n}\)	Flows assigned to node \(n\) of carrier \(b\)	Subset of \(\mathcal{F}_b\) with matching node
\(\text{coeff}_{b:n,f}\)	Coefficient of flow \(f\) in node \(n\)'s balance	+1 or -1, same convention as single-node

See Notation for the full symbol table.

Example

A thermal carrier with a boiler output (3 MW) and a demand input (3 MW):

\[ \underbrace{P_{\text{boiler\_th},t}}_{+1 \times 3} + \underbrace{(-1) \cdot P_{\text{demand},t}}_{-1 \times 3} = 0 \quad \checkmark \]

Multi-Node Carriers

A carrier can be split into independent nodes, each with its own balance equation. This models spatially separated subsystems on the same physical medium — e.g., separate heat networks in different buildings.

Formulation

Each node \(n\) of carrier \(b\) gets its own balance constraint:

\[ \sum_{f \in \mathcal{F}_{b:n}} \text{coeff}_{b:n,f} \cdot P_{f,t} = 0 \quad \forall \, (b, n), \; t \in \mathcal{T} \]

The compound coordinate \(b\!:\!n\) (e.g., heat:A, heat:B) identifies each node in the carrier dimension. Flows on different nodes never interact — their balance equations are fully independent.

Example

Two independent heat nodes A and B, each with its own supply and demand:

\[ \begin{aligned} \text{Node A:}\quad & P_{\text{src\_a(heat:A)},t} - P_{\text{sink\_a(heat:A)},t} = 0 \\ \text{Node B:}\quad & P_{\text{src\_b(heat:B)},t} - P_{\text{sink\_b(heat:B)},t} = 0 \end{aligned} \]

Node A's supply serves only node A's demand. Node B is balanced independently.

Usage

from fluxopt import Flow, Port

# Flows declare their node
supply_a = Flow('heat', node='A', size=100)
demand_a = Flow('heat', node='A', size=100, fixed_relative_profile=[0.5])

supply_b = Flow('heat', node='B', size=100)
demand_b = Flow('heat', node='B', size=100, fixed_relative_profile=[0.8])

The flow id auto-includes the node: Flow('heat', node='A') gets id='heat:A', which qualifies to src_a(heat:A) after component qualification.