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Duration Tracking

Duration tracking allows monitoring how long a binary state has been consecutively active. This is essential for modeling minimum run times, ramp-up periods, and similar time-dependent constraints.

Consecutive Duration Tracking

For a binary state variable \(s(t) \in \{0, 1\}\), the consecutive duration \(d(t)\) tracks how long the state has been continuously active.

Duration Upper Bound

The duration cannot exceed zero when the state is inactive:

\[\label{eq:duration_upper} d(t) \leq s(t) \cdot M \quad \forall t \]

With: - \(d(t)\) being the duration variable (continuous, non-negative) - \(s(t) \in \{0, 1\}\) being the binary state variable - \(M\) being a sufficiently large constant (big-M)

Behavior: - When \(s(t) = 0\): forces \(d(t) \leq 0\), thus \(d(t) = 0\) - When \(s(t) = 1\): allows \(d(t)\) to be positive

Duration Accumulation

While the state is active, the duration increases by the time step size:

\[\label{eq:duration_accumulation_upper} d(t+1) \leq d(t) + \Delta d(t) \quad \forall t \]
\[\label{eq:duration_accumulation_lower} d(t+1) \geq d(t) + \Delta d(t) + (s(t+1) - 1) \cdot M \quad \forall t \]

With: - \(\Delta d(t)\) being the duration increment for time step \(t\) (typically \(\Delta t_i\) from the time series) - \(M\) being a sufficiently large constant

Behavior: - When \(s(t+1) = 1\): both inequalities enforce \(d(t+1) = d(t) + \Delta d(t)\) - When \(s(t+1) = 0\): only the upper bound applies, and \(d(t+1) = 0\) (from equation \(\eqref{eq:duration_upper}\))

Initial Duration

The duration at the first time step depends on both the state and any previous duration:

\[\label{eq:duration_initial} d(0) = (\Delta d(0) + d_\text{prev}) \cdot s(0) \]

With: - \(d_\text{prev}\) being the duration from before the optimization period - \(\Delta d(0)\) being the duration increment for the first time step

Behavior: - When \(s(0) = 1\): duration continues from previous period - When \(s(0) = 0\): duration resets to zero

Complete Formulation

Combining all constraints:

\[ \begin{align} d(t) &\leq s(t) \cdot M && \forall t \label{eq:duration_complete_1} \\ d(t+1) &\leq d(t) + \Delta d(t) && \forall t \label{eq:duration_complete_2} \\ d(t+1) &\geq d(t) + \Delta d(t) + (s(t+1) - 1) \cdot M && \forall t \label{eq:duration_complete_3} \\ d(0) &= (\Delta d(0) + d_\text{prev}) \cdot s(0) && \label{eq:duration_complete_4} \end{align} \]

Minimum Duration Constraints

To enforce a minimum consecutive duration (e.g., minimum run time), an additional constraint links the duration to state changes:

\[\label{eq:minimum_duration} d(t) \geq (s(t-1) - s(t)) \cdot d_\text{min}(t-1) \quad \forall t > 0 \]

With: - \(d_\text{min}(t)\) being the required minimum duration at time \(t\)

Behavior: - When shutting down (\(s(t-1) = 1, s(t) = 0\)): enforces \(d(t-1) \geq d_\text{min}(t-1)\) - This ensures the state was active for at least \(d_\text{min}\) before turning off - When state is constant or turning on: constraint is non-binding

Implementation

Function: ModelingPrimitives.consecutive_duration_tracking()

See the API documentation for complete parameter list and usage details.

Use Cases

Minimum Run Time

Ensuring equipment runs for a minimum duration once started:

# State: 1 when running, 0 when off
# Require at least 2 hours of operation
duration = modeling.consecutive_duration_tracking(
    state_variable=on_state,
    duration_per_step=time_step_hours,
    minimum_duration=2.0
)

Ramp-Up Tracking

Tracking time since startup for gradual ramp-up constraints:

# Track startup duration
startup_duration = modeling.consecutive_duration_tracking(
    state_variable=on_state,
    duration_per_step=time_step_hours
)
# Constrain output based on startup duration
# (additional constraints would link output to startup_duration)

Cooldown Requirements

Tracking time in a state before allowing transitions:

# Track maintenance duration
maintenance_duration = modeling.consecutive_duration_tracking(
    state_variable=maintenance_state,
    duration_per_step=time_step_hours,
    minimum_duration=scheduled_maintenance_hours
)

Used In

This pattern is used in: - OnOffParameters - Minimum on/off times - Operating mode constraints with minimum durations - Startup/shutdown sequence modeling