Status (On/Off Constraints)¶

Overview¶

Status adds binary on/off behavior to flows. The core variables are a binary status indicator and transition indicators (startup/shutdown). Duration tracking variables enforce minimum and maximum consecutive up- and downtime.

Variables¶

Symbol	Code	Domain	Description
\(\sigma_{f,t}\)	`flow_on[flow, time]`	\(\{0, 1\}\)	On/off indicator
\(\tau^+_{f,t}\)	`flow_startup[flow, time]`	\(\{0, 1\}\)	Startup event indicator
\(\tau^-_{f,t}\)	`flow_shutdown[flow, time]`	\(\{0, 1\}\)	Shutdown event indicator
\(D^{\text{up}}_{f,t}\)	`uptime[flow, time]`	\(\geq 0\)	Consecutive uptime [h]
\(D^{\text{down}}_{f,t}\)	`downtime[flow, time]`	\(\geq 0\)	Consecutive downtime [h]

Semi-Continuous Flow Rates¶

With fixed size, the on/off indicator gates the flow rate bounds:

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

When \(\sigma_{f,t} = 0\): \(P_{f,t} = 0\). When \(\sigma_{f,t} = 1\): \(P_{f,t} \in [\bar{P}_f \underline{p}, \bar{P}_f \bar{p}]\).

This gives the semi-continuous behavior \(\{0\} \cup [\underline{P}, \bar{P}]\).

Switch Transitions¶

Startup and shutdown indicators are linked to status changes:

\[ \tau^+_{f,t} - \tau^-_{f,t} = \sigma_{f,t} - \sigma_{f,t-1} \quad \forall \, t > 0 \]

At the first timestep with known previous state \(\sigma_{f,0}^{\text{prev}}\):

\[ \tau^+_{f,0} - \tau^-_{f,0} = \sigma_{f,0} - \sigma_{f,0}^{\text{prev}} \]

The previous state is derived from Flow.prior: on if the last prior value > 0, off otherwise. Without prior, the initial transition is unconstrained.

Duration Tracking¶

Duration tracking uses a Big-M formulation to count consecutive hours in a state.

Uptime¶

The uptime variable \(D^{\text{up}}_{f,t}\) tracks consecutive on-hours:

Reset when off:

\[ D^{\text{up}}_{f,t} \leq \sigma_{f,t} \cdot M \quad \forall \, t \]

Forward accumulation:

\[ D^{\text{up}}_{f,t+1} \leq D^{\text{up}}_{f,t} + \Delta t_t \quad \forall \, t \]

Backward tightening (force accumulation when on):

\[ D^{\text{up}}_{f,t+1} \geq D^{\text{up}}_{f,t} + \Delta t_t + (\sigma_{f,t+1} - 1) \cdot M \quad \forall \, t \]

where \(M\) is the total horizon length (Big-M constant).

Downtime¶

Downtime tracking uses the same formulation applied to the inverted state \((1 - \sigma_{f,t})\).

Minimum Duration¶

Minimum uptime is enforced at shutdown transitions — the accumulated duration must meet the minimum before turning off:

\[ D^{\text{up}}_{f,t} \geq D^{\text{up,min}} \cdot (\sigma_{f,t} - \sigma_{f,t+1}) \quad \forall \, t < |\mathcal{T}| \]

The term \((\sigma_{f,t} - \sigma_{f,t+1})\) equals 1 only at shutdown (on → off), enforcing \(D^{\text{up}}_{f,t} \geq D^{\text{up,min}}\).

Minimum downtime follows the same pattern on \((1 - \sigma)\).

Maximum Duration¶

Maximum duration is enforced as an upper bound on the duration variable itself.

Previous Duration Carryover¶

When Flow.prior provides historical state, the previous duration is computed by counting consecutive matching timesteps at the end of the prior. At \(t=0\):

\[ D^{\text{up}}_{f,0} = \sigma_{f,0} \cdot (D^{\text{up,prev}} + \Delta t_0) \]

If the previous uptime hasn't yet met the minimum, the unit is forced to stay on:

\[ \sigma_{f,0} \geq 1 \quad \text{if } 0 < D^{\text{up,prev}} < D^{\text{up,min}} \]

Effect Contributions¶

Running Costs¶

Running costs are charged per hour of operation, multiplied by the on-indicator:

\[ \Phi_{k,t}^{\text{running}} = \sum_{f} r_{f,k,t} \cdot \sigma_{f,t} \cdot \Delta t_t \]

where \(r_{f,k,t}\) is Status.effects_per_running_hour[k].

Startup Costs¶

Startup costs are charged per startup event (no duration scaling):

\[ \Phi_{k,t}^{\text{startup}} = \sum_{f} u_{f,k,t} \cdot \tau^+_{f,t} \]

where \(u_{f,k,t}\) is Status.effects_per_startup[k].

Both feed into the per-timestep effect equation.

Interaction with Sizing¶

When a flow has both Status and Sizing, the on/off indicator and the size variable are decoupled via Big-M constraints:

\[ P_{f,t} \leq \sigma_{f,t} \cdot M^+ \qquad \text{(on-indicator gates flow)} \]

\[ P_{f,t} \leq S_f \cdot \bar{p}_{f,t} \qquad \text{(rate limited by size)} \]

\[ P_{f,t} \geq (\sigma_{f,t} - 1) \cdot M^- + S_f \cdot \underline{p}_{f,t} \qquad \text{(minimum when on)} \]

where \(M^+ = S^+ \cdot \bar{p}_{f,t}\) and \(M^- = S^+ \cdot \underline{p}_{f,t}\), using the maximum possible size \(S^+\) as Big-M.

An additional constraint prevents the unit from being "on" with zero size:

\[ \sigma_{f,t} \leq S_f \quad \forall \, t \]

Parameters¶

Symbol	Description	Reference
\(\sigma_{f,t}\)	On/off binary	`flow_on[flow, time]`
\(\tau^+_{f,t}\)	Startup indicator	`flow_startup[flow, time]`
\(\tau^-_{f,t}\)	Shutdown indicator	`flow_shutdown[flow, time]`
\(D^{\text{up}}_{f,t}\)	Consecutive uptime	`uptime[flow, time]`
\(D^{\text{down}}_{f,t}\)	Consecutive downtime	`downtime[flow, time]`
\(D^{\text{up,min}}\)	Minimum uptime	`Status.min_uptime`
\(D^{\text{up,max}}\)	Maximum uptime	`Status.max_uptime`
\(D^{\text{down,min}}\)	Minimum downtime	`Status.min_downtime`
\(D^{\text{down,max}}\)	Maximum downtime	`Status.max_downtime`
\(r_{f,k,t}\)	Running cost coefficient	`Status.effects_per_running_hour`
\(u_{f,k,t}\)	Startup cost coefficient	`Status.effects_per_startup`
\(M\)	Big-M (horizon length)	computed

See Notation for the full symbol table.