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
\(\mathrm{D}^{\text{up}}_{f,t}\)	`uptime[flow, time]`	\(\geq 0\)	Consecutive uptime [h]
\(\mathrm{D}^{\text{down}}_{f,t}\)	`downtime[flow, time]`	\(\geq 0\)	Consecutive downtime [h]

Semi-Continuous Flow Rates¶

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

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

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

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

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

Reset when off:

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

Forward accumulation:

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

Backward tightening (force accumulation when on):

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

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:

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

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

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

Maximum Duration¶

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

Duration values are in hours. With sub-hourly timesteps (e.g., dt=0.5), a min_uptime=2 means the unit must stay on for 4 consecutive timesteps.

Previous Duration Carryover¶

Flow.prior_rates provides the flow rates from timesteps before the optimization horizon. This lets the solver know the initial on/off state and how long the unit has been running or idle:

The prior rates determine:

Initial on/off state: last value > 0 means on, = 0 means off
Previous duration: consecutive hours in the current state at the end of the prior, used for duration constraint carryover

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

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

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

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

Effect Contributions¶

Running Costs¶

A per-hour cost while the unit is on, independent of the flow rate:

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

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

Startup Costs¶

A one-time cost charged each time the unit switches from off to on:

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

where \(\mathrm{u}_{f,k,t}\) 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:

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

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

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

where \(M^+ = \mathrm{S}^+ \cdot \bar{\mathrm{p}}_{f,t}\) and \(M^- = \mathrm{S}^+ \cdot \underline{\mathrm{p}}_{f,t}\), using the maximum possible size \(\mathrm{S}^+\) 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]`
\(\mathrm{D}^{\text{up}}_{f,t}\)	Consecutive uptime	`uptime[flow, time]`
\(\mathrm{D}^{\text{down}}_{f,t}\)	Consecutive downtime	`downtime[flow, time]`
\(\mathrm{D}^{\text{up,min}}\)	Minimum uptime	`Status.min_uptime`
\(\mathrm{D}^{\text{up,max}}\)	Maximum uptime	`Status.max_uptime`
\(\mathrm{D}^{\text{down,min}}\)	Minimum downtime	`Status.min_downtime`
\(\mathrm{D}^{\text{down,max}}\)	Maximum downtime	`Status.max_downtime`
\(\mathrm{r}_{f,k,t}\)	Running cost coefficient	`Status.effects_per_running_hour`
\(\mathrm{u}_{f,k,t}\)	Startup cost coefficient	`Status.effects_per_startup`
\(M\)	Big-M (horizon length)	computed

See Notation for the full symbol table.