Mixed-Integer Linear Programming (MILP)

The defining characteristic of MILP is the use of binary integer variables (0 or 1) to represent discrete switch states in the distribution network.

• Open switch: Represented as 0, indicating no current flow through that branch
• Closed switch: Represented as 1, allowing power to flow along that path
• Tie switches: Modeled as binary variables that can change state during optimization
• Sectionalizing switches: Similarly represented to enable feeder reconfiguration

This binary formulation directly maps the physical reality of switchgear—a switch is either open or closed, with no intermediate state. The solver explores combinations of these binary variables to find the optimal topology.

MILP formulations enforce linearized power flow physics to ensure all solutions respect electrical engineering principles while maintaining computational tractability.

• DistFlow equations: Linear approximations of branch flow equations specifically derived for radial distribution networks
• Voltage drop constraints: Linear relationships between power flow and voltage magnitude differences along feeders
• Current limits: Thermal capacity constraints expressed as linear inequalities on branch flows
• Kirchhoff's laws: Node balance equations ensuring power injected equals power consumed plus losses

The linearization is essential—true AC power flow is non-convex and nonlinear, making global optimization computationally prohibitive for real-time grid operations.

A critical constraint in MILP-based reconfiguration is the radiality requirement—the solution must produce a network topology without any closed loops.

• Spanning tree constraints: Mathematical conditions ensuring the resulting graph connects all nodes without cycles
• Single-parent rules: Each load bus must receive power from exactly one upstream source
• Connectivity guarantees: Constraints preventing unintentional islanding of customers
• Substation separation: Rules ensuring feeders remain electrically distinct under normal operation

Radiality is enforced for protection coordination—radial networks allow simple overcurrent protection schemes with predictable fault current paths, eliminating the complexity of directional relaying required in meshed systems.

Unlike heuristic methods such as the Branch Exchange Method, MILP solvers provide a mathematical guarantee of finding the globally optimal solution within the defined search space.

• Branch-and-bound: The core algorithm systematically partitions the solution space and prunes suboptimal regions
• Optimality gap: Solvers report the percentage difference between the best found solution and the theoretical lower bound
• Proven convergence: Commercial solvers like Gurobi and CPLEX can close the gap to 0.0%, certifying optimality
• Benchmarking value: MILP solutions serve as ground truth for evaluating faster heuristic algorithms

This guarantee is crucial for regulatory compliance and investment planning, where utilities must demonstrate they are operating the grid at maximum efficiency.

MILP frameworks naturally extend to multi-objective optimization, allowing grid operators to balance competing goals through weighted objective functions or constraint-based approaches.

• Loss minimization: Reducing I²R losses by shortening current paths and balancing feeder loading
• Switching cost minimization: Penalizing unnecessary switch operations to preserve equipment lifespan
• Voltage profile improvement: Minimizing deviations from nominal voltage across all buses
• Load balancing: Equalizing utilization percentages across parallel transformers and feeders

The solver generates a Pareto optimal front—a set of solutions where improving one objective necessarily degrades another, enabling operators to make informed trade-off decisions based on current operational priorities.

MILP problems are NP-hard in general, meaning solution time can grow exponentially with the number of binary variables. Practical grid applications require careful formulation strategies.

• Tight big-M values: Using the smallest possible constants in disjunctive constraints to improve solver performance
• Symmetry breaking: Adding constraints to eliminate mathematically equivalent but permuted solutions
• Warm starting: Providing the solver with a known feasible topology to accelerate convergence
• Time limit settings: Configuring solvers to return the best solution found within operational time windows (typically 30-300 seconds)
• Decomposition techniques: Breaking large networks into smaller subproblems solved sequentially

Modern commercial solvers leverage cutting planes and presolve reductions to dramatically reduce solve times, making MILP viable for networks with hundreds of switches.