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OnOffParameters

OnOffParameters model equipment that operates in discrete on/off states rather than continuous operation. This captures realistic operational constraints including startup costs, minimum run times, cycling limitations, and maintenance scheduling.

Binary State Variable

Equipment operation is modeled using a binary state variable:

\[\label{eq:onoff_state} s(t) \in \{0, 1\} \quad \forall t \]

With: - \(s(t) = 1\): equipment is operating (on state) - \(s(t) = 0\): equipment is shutdown (off state)

This state variable controls the equipment's operational constraints and modifies flow bounds using the bounds with state pattern from Bounds and States.

State Transitions and Switching

State transitions are tracked using switch variables (see State Transitions):

\[\label{eq:onoff_transitions} s^\text{on}(t) - s^\text{off}(t) = s(t) - s(t-1) \quad \forall t > 0 \]
\[\label{eq:onoff_switch_exclusivity} s^\text{on}(t) + s^\text{off}(t) \leq 1 \quad \forall t \]

With: - \(s^\text{on}(t) \in \{0, 1\}\): equals 1 when switching from off to on (startup) - \(s^\text{off}(t) \in \{0, 1\}\): equals 1 when switching from on to off (shutdown)

Behavior: - Off → On: \(s^\text{on}(t) = 1, s^\text{off}(t) = 0\) - On → Off: \(s^\text{on}(t) = 0, s^\text{off}(t) = 1\) - No change: \(s^\text{on}(t) = 0, s^\text{off}(t) = 0\)

Effects and Costs

Switching Effects

Effects incurred when equipment starts up:

\[\label{eq:onoff_switch_effects} E_{e,\text{switch}} = \sum_{t} s^\text{on}(t) \cdot \text{effect}_{e,\text{switch}} \]

With: - \(\text{effect}_{e,\text{switch}}\) being the effect value per startup event

Examples: - Startup fuel consumption - Wear and tear costs - Labor costs for startup procedures - Inrush power demands

Running Effects

Effects incurred while equipment is operating:

\[\label{eq:onoff_running_effects} E_{e,\text{run}} = \sum_{t} s(t) \cdot \Delta t \cdot \text{effect}_{e,\text{run}} \]

With: - \(\text{effect}_{e,\text{run}}\) being the effect rate per operating hour - \(\Delta t\) being the time step duration

Examples: - Fixed operating and maintenance costs - Auxiliary power consumption - Consumable materials - Emissions while running

Operating Hour Constraints

Total Operating Hours

Bounds on total operating time across the planning horizon:

\[\label{eq:onoff_total_hours} h_\text{min} \leq \sum_{t} s(t) \cdot \Delta t \leq h_\text{max} \]

With: - \(h_\text{min}\) being the minimum total operating hours - \(h_\text{max}\) being the maximum total operating hours

Use cases: - Minimum runtime requirements (contracts, maintenance) - Maximum runtime limits (fuel availability, permits, equipment life)

Consecutive Operating Hours

Minimum Consecutive On-Time:

Enforces minimum runtime once started using duration tracking (see Duration Tracking):

\[\label{eq:onoff_min_on_duration} d^\text{on}(t) \geq (s(t-1) - s(t)) \cdot h^\text{on}_\text{min} \quad \forall t > 0 \]

With: - \(d^\text{on}(t)\) being the consecutive on-time duration at time \(t\) - \(h^\text{on}_\text{min}\) being the minimum required on-time

Behavior: - When shutting down at time \(t\): enforces equipment was on for at least \(h^\text{on}_\text{min}\) prior to the switch - Prevents short cycling and frequent startups

Maximum Consecutive On-Time:

Limits continuous operation before requiring shutdown:

\[\label{eq:onoff_max_on_duration} d^\text{on}(t) \leq h^\text{on}_\text{max} \quad \forall t \]

Use cases: - Mandatory maintenance intervals - Process batch time limits - Thermal cycling requirements

Consecutive Shutdown Hours

Minimum Consecutive Off-Time:

Enforces minimum shutdown duration before restarting:

\[\label{eq:onoff_min_off_duration} d^\text{off}(t) \geq (s(t) - s(t-1)) \cdot h^\text{off}_\text{min} \quad \forall t > 0 \]

With: - \(d^\text{off}(t)\) being the consecutive off-time duration at time \(t\) - \(h^\text{off}_\text{min}\) being the minimum required off-time

Use cases: - Cooling periods - Maintenance requirements - Process stabilization

Maximum Consecutive Off-Time:

Limits shutdown duration before mandatory restart:

\[\label{eq:onoff_max_off_duration} d^\text{off}(t) \leq h^\text{off}_\text{max} \quad \forall t \]

Use cases: - Equipment preservation requirements - Process stability needs - Contractual minimum activity levels

Cycling Limits

Maximum number of startups across the planning horizon:

\[\label{eq:onoff_max_switches} \sum_{t} s^\text{on}(t) \leq n_\text{max} \]

With: - \(n_\text{max}\) being the maximum allowed number of startups

Use cases: - Preventing excessive equipment wear - Grid stability requirements - Operational complexity limits - Maintenance budget constraints

Integration with Flow Bounds

OnOffParameters modify flow rate bounds by coupling them to the on/off state.

Without OnOffParameters (continuous operation): $$ P \cdot \text{rel}\text{lower} \leq p(t) \leq P \cdot \text{rel}\text{upper} $$

With OnOffParameters (binary operation): $$ s(t) \cdot P \cdot \max(\varepsilon, \text{rel}\text{lower}) \leq p(t) \leq s(t) \cdot P \cdot \text{rel}\text{upper} $$

Using the bounds with state pattern from Bounds and States.

Behavior: - When \(s(t) = 0\): flow is forced to zero - When \(s(t) = 1\): flow follows normal bounds

Complete Formulation Summary

For equipment with OnOffParameters, the complete constraint system includes:

  1. State variable: \(s(t) \in \{0, 1\}\)
  2. Switch tracking: \(s^\text{on}(t) - s^\text{off}(t) = s(t) - s(t-1)\)
  3. Switch exclusivity: \(s^\text{on}(t) + s^\text{off}(t) \leq 1\)
  4. Duration tracking:
    • On-duration: \(d^\text{on}(t)\) following duration tracking pattern
    • Off-duration: \(d^\text{off}(t)\) following duration tracking pattern
  5. Minimum on-time: \(d^\text{on}(t) \geq (s(t-1) - s(t)) \cdot h^\text{on}_\text{min}\)
  6. Maximum on-time: \(d^\text{on}(t) \leq h^\text{on}_\text{max}\)
  7. Minimum off-time: \(d^\text{off}(t) \geq (s(t) - s(t-1)) \cdot h^\text{off}_\text{min}\)
  8. Maximum off-time: \(d^\text{off}(t) \leq h^\text{off}_\text{max}\)
  9. Total hours: \(h_\text{min} \leq \sum_t s(t) \cdot \Delta t \leq h_\text{max}\)
  10. Cycling limit: \(\sum_t s^\text{on}(t) \leq n_\text{max}\)
  11. Flow bounds: \(s(t) \cdot P \cdot \text{rel}_\text{lower} \leq p(t) \leq s(t) \cdot P \cdot \text{rel}_\text{upper}\)

Implementation

Python Class: OnOffParameters

Key Parameters: - effects_per_switch_on: Costs per startup event - effects_per_running_hour: Costs per hour of operation - on_hours_total_min, on_hours_total_max: Total runtime bounds - consecutive_on_hours_min, consecutive_on_hours_max: Consecutive runtime bounds - consecutive_off_hours_min, consecutive_off_hours_max: Consecutive shutdown bounds - switch_on_total_max: Maximum number of startups - force_switch_on: Create switch variables even without limits (for tracking)

See the OnOffParameters API documentation for complete parameter list and usage examples.

Mathematical Patterns Used: - State Transitions - Switch tracking - Duration Tracking - Consecutive time constraints - Bounds with State - Flow control

Used in: - Flow - On/off operation for flows - All components supporting discrete operational states

Examples

Power Plant with Startup Costs

power_plant = OnOffParameters(
    effects_per_switch_on={'startup_cost': 25000},  # €25k per startup
    effects_per_running_hour={'fixed_om': 125},  # €125/hour while running
    consecutive_on_hours_min=8,  # Minimum 8-hour run
    consecutive_off_hours_min=4,  # 4-hour cooling period
    on_hours_total_max=6000,  # Annual limit
)

Batch Process with Cycling Limits

batch_reactor = OnOffParameters(
    effects_per_switch_on={'setup_cost': 1500},
    consecutive_on_hours_min=12,  # 12-hour minimum batch
    consecutive_on_hours_max=24,  # 24-hour maximum batch
    consecutive_off_hours_min=6,  # Cleaning time
    switch_on_total_max=200,  # Max 200 batches
)

HVAC with Cycle Prevention

hvac = OnOffParameters(
    effects_per_switch_on={'compressor_wear': 0.5},
    consecutive_on_hours_min=1,  # Prevent short cycling
    consecutive_off_hours_min=0.5,  # 30-min minimum off
    switch_on_total_max=2000,  # Limit compressor starts
)

Backup Generator with Testing Requirements

backup_gen = OnOffParameters(
    effects_per_switch_on={'fuel_priming': 50},  # L diesel
    consecutive_on_hours_min=0.5,  # 30-min test duration
    consecutive_off_hours_max=720,  # Test every 30 days
    on_hours_total_min=26,  # Weekly testing requirement
)

Notes

Time Series Boundary: The final time period constraints for consecutive_on_hours_min/max and consecutive_off_hours_min/max are not enforced at the end of the planning horizon. This allows optimization to end with ongoing campaigns that may be shorter/longer than specified, as they extend beyond the modeled period.