Flows

A flow represents energy transfer on a bus. Each flow \(f\) has a non-negative rate variable \(P_{f,t}\) bounded by an optional nominal capacity \(\bar{\mathrm{P}}_f\).

Bounds

When Flow.size is set, the rate is bounded by relative minimum and maximum profiles:

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

By default \(\underline{\mathrm{p}}_{f,t} = 0\) and \(\bar{\mathrm{p}}_{f,t} = 1\), so the bounds simplify to \(0 \leq P_{f,t} \leq \bar{\mathrm{P}}_f\). With no capacity (\(\bar{\mathrm{P}}_f = \infty\)) the flow is just \(P_{f,t} \geq 0\).

Fixed Profile

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

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

Implemented by setting both bounds equal to the profile value.

Aggregate Bounds

Aggregate bounds constrain the flow-hours \(H_{f,p}\) — the flow rate summed over the horizon — within each period independently:

\[ H_{f,p} = \sum_{t \in \mathcal{T}} P_{f,t,p} \cdot \Delta t_t \]

Flow-hour bounds (flow_hours_min / flow_hours_max) — absolute [MWh]:

\[ \underline{\mathrm{H}}_f \leq H_{f,p} \leq \bar{\mathrm{H}}_f \quad \forall \, p \]

Load factor bounds (load_factor_min / load_factor_max) — utilization relative to capacity, with \(T = \sum_t \Delta t_t\) the period duration:

\[ \underline{\lambda}_f \cdot \bar{\mathrm{P}}_f \cdot T \leq H_{f,p} \leq \bar{\lambda}_f \cdot \bar{\mathrm{P}}_f \cdot T \quad \forall \, p \]

When Flow.size is a Sizing or Investment object, \(\bar{\mathrm{P}}_f\) is the size variable \(S_{f,p}\) and the products stay linear (constant \(\lambda \cdot T\) times a variable). Load factor bounds therefore require a size; flow-hour bounds do not.

Cross-period budgets are not expressed here — route them through an effect with total_min / total_max.

Ramp Limits

Ramp limits (ramp_up_per_hour / ramp_down_per_hour) cap the rate change between consecutive timesteps, as a fraction of capacity per hour:

\[ P_{f,t} - P_{f,t-1} \leq \mathrm{r}^{+}_{f,t} \cdot \bar{\mathrm{P}}_f \cdot \Delta t_t \qquad P_{f,t-1} - P_{f,t} \leq \mathrm{r}^{-}_{f,t} \cdot \bar{\mathrm{P}}_f \cdot \Delta t_t \]

Constraints apply from the second timestep onward; periods are independent (no ramp coupling across period boundaries). Like load factors, ramps are relative to size — when Flow.size is a Sizing or Investment object, \(\bar{\mathrm{P}}_f\) is the size variable and the constraint stays linear.

Interaction with Status

For flows with status, the ramp does not bind across on/off transitions: the startup (shutdown) binary relaxes the ramp-up (ramp-down) constraint in the transition step, so a unit may always jump to its semi-continuous minimum and drop back to zero:

\[ P_{f,t} - P_{f,t-1} \leq \mathrm{r}^{+}_{f,t} \cdot \bar{\mathrm{P}}_f \cdot \Delta t_t + \bar{\mathrm{P}}_f \cdot \text{startup}_{f,t} \]

Transitions are pinned to actual state changes (status|exclusive), so the relaxation cannot be exploited between running steps. Component-status flows relax via their component's transition binaries. Custom (tighter) startup/shutdown ramp rates are future work.

Effect Contributions

Each flow contributes to tracked effects (cost, emissions, …). Per-timestep:

\[ \mathrm{c}_{f,k,t} \cdot P_{f,t} \cdot \Delta t_t \]

\(\mathrm{c}_{f,k,t}\) is the per-flow-hour coefficient (Flow.effects_per_flow_hour). Units cancel: e.g. €/MWh × MW × h = €. Contributions feed into the effect tracking equations.

See also

• Sizing — when Flow.size is a Sizing object instead of a fixed value.
• Status — when Flow.status is set, the flow becomes semi-continuous \(\{0\} \cup [\underline{\mathrm{P}}, \bar{\mathrm{P}}]\).
• Carrier Balance — Multi-Node — when Flow.node targets a specific node.

Parameters

Symbol	Description	API
\(P_{f,t}\)	Flow rate variable	flow--rate[flow, time]
\(\bar{\mathrm{P}}_f\)	Nominal capacity	Flow.size
\(\underline{\mathrm{p}}_{f,t}\)	Relative lower bound	Flow.relative_rate_min
\(\bar{\mathrm{p}}_{f,t}\)	Relative upper bound	Flow.relative_rate_max
\(\pi_{f,t}\)	Fixed relative profile	Flow.fixed_relative_profile
\(\mathrm{c}_{f,k,t}\)	Effect coefficient per flow-hour	Flow.effects_per_flow_hour
\(\underline{\mathrm{H}}_f\)	Minimum flow-hours per period	Flow.flow_hours_min
\(\bar{\mathrm{H}}_f\)	Maximum flow-hours per period	Flow.flow_hours_max
\(\underline{\lambda}_f\)	Minimum load factor per period	Flow.load_factor_min
\(\bar{\lambda}_f\)	Maximum load factor per period	Flow.load_factor_max
\(\mathrm{r}^{+}_{f,t}\)	Max ramp up, fraction of size per hour	Flow.ramp_up_per_hour
\(\mathrm{r}^{-}_{f,t}\)	Max ramp down, fraction of size per hour	Flow.ramp_down_per_hour
\(\Delta t_t\)	Timestep duration (h)	dt

See Notation for the full symbol table and Indexing Convention for how indices broadcast.