Energy Cost Function

Maps electricity pricing to specific time windows, allowing the scheduler to shift loads to cheaper periods. The function penalizes consumption during peak hours and incentivizes it during off-peak or renewable surplus windows.

• Input: Timestamp, grid price signal
• Mechanism: Multiplies instantaneous power draw by a time-varying cost coefficient
• Example: Charging a fleet at 2:00 AM when rates are $0.05/kWh vs. 2:00 PM at $0.25/kWh

Assigns a virtual cost to actions that accelerate capacity fade and internal resistance growth. This transforms battery health from a constraint into an economic term.

• Key Factors: Depth of Discharge (DoD), C-Rate, temperature at charge initiation
• Mechanism: Adds a nonlinear cost derived from a battery degradation model (e.g., Arrhenius-based aging equations)
• Result: The optimizer avoids deep discharges and fast charging unless the operational value justifies the lifetime cost

A negative cost term that rewards energy recovery during deceleration events. This allows the function to accurately model kinetic energy recapture.

• Input: Planned deceleration profiles from the route, vehicle mass, motor efficiency map
• Mechanism: Subtracts the estimated recovered watt-hours from the total trip cost
• Impact: Routes with frequent stops may be penalized less than a pure kinematic model would suggest, favoring efficient driving patterns

Adjusts the cost of consuming energy based on the current usable reserve in the battery. This prevents the scheduler from treating all watt-hours as equal.

• Mechanism: A dynamic coefficient that increases as SoE drops below a comfort threshold
• Purpose: Creates a soft constraint that discourages draining a battery near its minimum charge threshold unless absolutely necessary
• Example: The 100th Wh drawn from a nearly empty battery costs 10x more than the 100th Wh drawn from a full one, preserving an energy buffer for contingencies

Links energy cost to a battery thermal model to prevent overheating. Charging or discharging at high C-Rates generates heat, which degrades the cell.

• Mechanism: The cost function includes a term that penalizes actions predicted to push cell temperature beyond safe limits (e.g., >45°C)
• Integration: Requires real-time telemetry from the Battery Management System (BMS) API
• Outcome: The scheduler may slow down a charge rate or delay a high-power task to allow for passive cooling, trading time for longevity

Balances the energy cost term against other scheduling objectives like tardiness and throughput using scalarization weights.

• Mechanism: The total cost function is a weighted sum: J = w_energy * J_energy + w_time * J_time
• Tuning: Operations managers adjust w_energy to shift the fleet's behavior from purely time-optimal to energy-optimal
• Example: A high w_energy might cause a robot to wait 5 minutes for an off-peak window rather than charging immediately, accepting a minor delay for a 40% cost reduction