Mixed-Integer Linear Programming (MILP)

MILP introduces integer variables that represent yes/no or on/off decisions—such as whether a charging station is active at a specific time slot. Unlike pure linear programming, these discrete variables allow the model to capture real-world constraints like binary charging states, vehicle-to-charger assignment, and minimum power thresholds below which chargers cannot operate efficiently.

The objective function defines what the optimization seeks to minimize or maximize, expressed as a linear combination of decision variables. Common EV charging objectives include:

• Minimizing total electricity cost under time-of-use tariffs
• Minimizing peak load to avoid demand charges
• Maximizing renewable energy utilization by aligning charging with solar generation
• Minimizing battery degradation by penalizing high C-rates

Constraints enforce the physical and operational limits of the charging system. MILP handles multiple constraint types simultaneously:

• Power balance constraints ensuring total load does not exceed transformer capacity
• State of Charge (SoC) constraints guaranteeing vehicles reach target charge levels by departure time
• Sequential constraints linking SoC across consecutive time intervals
• Mutual exclusion constraints preventing conflicting charger assignments

The dominant solution method for MILP problems, branch-and-bound systematically explores the solution space by:

• Solving linear programming relaxations where integer variables are temporarily treated as continuous
• Branching on fractional variables to create subproblems with tighter bounds
• Bounding by pruning subproblems that cannot improve upon the current best solution
• Modern solvers like Gurobi and CPLEX use advanced presolve, cutting planes, and heuristics to accelerate convergence

EV charging optimization typically uses time-indexed MILP formulations where the planning horizon is discretized into uniform intervals (e.g., 15-minute slots). Each interval has its own set of decision variables, creating a temporal coupling where the charging decision at time t directly affects the state at time t+1. This structure enables precise modeling of time-of-use pricing windows and dynamic grid signals.

MILP is NP-hard, meaning solution time can grow exponentially with problem size. For large-scale EV fleet optimization, practitioners employ:

• Rolling horizon approaches that solve shorter time windows sequentially
• Aggregation techniques grouping similar vehicles to reduce variable count
• Heuristic warm starts providing good initial solutions to accelerate solver convergence
• Decomposition methods like Lagrangian relaxation for geographically distributed charging networks

An advanced process control algorithm that solves a finite-horizon optimization problem at each time step to determine optimal charging schedules. Unlike static MILP formulations, MPC continuously re-optimizes as new data arrives, making it ideal for real-time EV fleet management where energy prices and grid conditions fluctuate.

• Uses a receding horizon approach: optimize over a future window, implement only the first step, then repeat
• Incorporates forecasted inputs like solar generation and electricity prices
• Handles multi-variable constraints including transformer capacity and battery degradation
• Often uses MILP as the internal solver for each time-step optimization

A real-time power allocation algorithm that distributes available electrical capacity across multiple charging points to prevent circuit breaker trips and minimize infrastructure upgrade costs. While MILP solves the scheduling problem, dynamic load balancing executes the millisecond-level power adjustments.

• Prevents phase imbalance in three-phase installations
• Operates within the constraints calculated by higher-level MILP optimization
• Essential for sites with limited grid connection capacity
• Reduces peak demand charges by up to 40% in commercial fleet depots

An optimization technique that limits the peak power draw from the grid during a billing interval to reduce substantial demand charges levied on commercial EV fleet operators. MILP is the primary mathematical tool for solving this problem because it requires discrete decisions about which vehicles charge and when.

• Demand charges can constitute 30-70% of a commercial electric bill
• MILP models incorporate non-linear battery charging curves as piecewise linear approximations
• Integrates with time-of-use energy pricing for combined cost minimization
• Typical optimization window spans a full billing demand interval (15 or 30 minutes)

An empirical or physics-based mathematical representation of capacity fade and internal resistance growth in lithium-ion cells as a function of cycling and calendar aging. MILP formulations incorporate these models as cost functions to balance charging speed against battery longevity.

• Cycle aging: degradation proportional to energy throughput and depth of discharge
• Calendar aging: time-dependent degradation accelerated by high state of charge and temperature
• Linearized degradation costs enable integration into MILP objective functions
• Critical for V2G applications where bidirectional cycling accelerates wear

A load management strategy that reduces grid power consumption during periods of highest electricity demand by utilizing stored energy from batteries or curtailing flexible loads. MILP determines the optimal discharge schedule by solving a mixed-integer problem where battery discharge decisions are discrete and power levels are continuous.

• Reduces or eliminates costly demand charges
• Can incorporate vehicle-to-grid (V2G) discharge as a peak shaving resource
• Often combined with behind-the-meter solar generation forecasting
• Typical commercial implementations target a fixed peak threshold (e.g., 500 kW)

The application of time-series machine learning models to predict the aggregate power demand of EV fleets hours or days in advance. These forecasts serve as input parameters to MILP scheduling models, and forecast accuracy directly impacts optimization quality.

• Uses historical charging data, weather, and fleet telematics as features
• Common approaches: LSTM networks, gradient boosting, and transformer-based architectures
• Probabilistic forecasts provide confidence intervals for robust optimization
• Forecast errors propagate through MILP, potentially causing constraint violations