Model Predictive Control (MPC)

MPC continuously solves a finite-horizon optimization problem at each control interval, but only the first control action in the sequence is executed. The horizon then shifts forward one step and the problem is resolved with updated measurements.

• Prediction Horizon: Typically spans 15 minutes to 4 hours for distribution reconfiguration, capturing multiple load cycles
• Control Horizon: The first 1-3 switching steps that are actually implemented before re-optimization
• Recursive Feedback: New SCADA and PMU measurements refresh the initial state at every time step, correcting for model mismatch

This rolling mechanism provides inherent robustness against forecast errors, as the controller never commits to a long sequence of actions without course correction.

At the core of MPC lies an explicit mathematical model of the distribution grid that predicts future states as a function of control inputs. This model captures:

• DistFlow Equations: Recursive power flow relationships that compute voltage magnitudes and branch flows along radial feeders
• Load Dynamics: Time-varying ZIP load models and thermostatically controlled load behavior, including Cold Load Pickup (CLPU) effects after outages
• DER Integration: Forecasted solar irradiance and wind speed translated into active power injections at distributed generation nodes
• Switch State Representation: Binary integer variables encoding the open/closed status of sectionalizing and tie switches

The fidelity of this internal model directly determines the quality of the optimized switching schedule.

MPC explicitly enforces operational constraints within the optimization formulation, ensuring that every candidate switching sequence respects the physical and regulatory limits of the grid:

• Radiality Constraint: The network must remain a spanning tree with no closed loops, enforced through graph-theoretic constraints or spanning tree enumeration
• Voltage Limits: All bus voltages must stay within ANSI C84.1 bounds (typically ±5% of nominal)
• Thermal Limits: Line currents and transformer loading must not exceed rated capacities
• N-1 Criterion: The optimized topology must survive the loss of any single feeder or transformer without cascading violations
• Switching Frequency: Limits on the number of daily operations to preserve switch mechanical life

Constraint handling is what distinguishes MPC from heuristic reconfiguration methods that may produce infeasible topologies.

The MPC optimizer minimizes a weighted sum of competing objectives, generating a Pareto optimal trade-off between operational goals:

• Loss Minimization: Reducing I²R losses across feeders, typically the primary objective during normal operation
• Load Balancing: Equalizing feeder utilization to release headroom capacity and reduce thermal stress on aging transformers
• Switching Cost: Penalizing each open/close operation to avoid excessive wear on switchgear and transient disturbances
• Conservation Voltage Reduction (CVR): Minimizing service voltage within allowable bounds to reduce energy consumption
• Restoration Speed: During fault scenarios, maximizing the number of customers re-energized per switching step

Weighting coefficients are tuned by utility operators to reflect operational priorities and regulatory incentives.

The MPC optimization is typically cast as a Mixed-Integer Linear Programming (MILP) problem to leverage powerful commercial solvers like Gurobi or CPLEX:

• Binary Variables: Switch statuses (0=open, 1=closed) are integer-constrained
• Continuous Variables: Bus voltages, branch power flows, and generator dispatches are real-valued
• Linearized Power Flow: The non-linear DistFlow equations are linearized using approximations like the LinDistFlow model, which assumes negligible line losses and small voltage angle differences
• Big-M Method: Logical constraints linking switch status to power flow paths are encoded using large constant disjunctive formulations

MILP guarantees global optimality within the linearized model, unlike heuristic methods that may converge to local minima. Solve times for distribution-scale problems are typically under 30 seconds.

MPC provides a structured framework for transitioning from normal optimization to emergency Service Restoration (SR) mode when faults are detected:

• Fault Isolation: The internal model is updated to reflect faulted line segments as permanently open, removing them from the feasible topology space
• Cold Load Pickup Forecasting: The prediction model estimates the elevated demand that will occur upon re-energization, preventing sequential overloads during restoration
• Intentional Islanding: When distributed generation is present, MPC can compute islanding boundaries that balance local generation with load to sustain isolated microgrids
• Receding Horizon Recovery: After fault clearance, the controller smoothly transitions back to loss-minimizing topology over multiple control steps

This dual-mode capability allows a single control framework to handle both economic optimization and emergency response without switching between disparate algorithms.

The defining mechanism of MPC where the optimization is solved repeatedly over a finite prediction horizon that shifts forward at each time step. Only the first control action is implemented before the horizon recedes and the problem is re-solved with updated state measurements. This provides inherent feedback to counteract model inaccuracies and disturbances.

• Prediction horizon: typically 15-60 minutes for distribution grids
• Control horizon: shorter segment where actual switching commands are committed
• Re-optimization interval: often 5-15 minutes for quasi-steady-state grid control

The mathematical representation of grid dynamics as a set of first-order differential or difference equations: x(k+1) = Ax(k) + Bu(k). The state vector x captures bus voltages, line currents, and energy storage levels. The control input u represents switch states and tap positions. This compact form enables efficient numerical optimization.

• Observable states: measured via SCADA and PMU sensors
• Disturbance inputs: forecasted renewable generation and load changes
• Output equation: maps states to measurable quantities for feedback correction

The scalar objective J that the MPC solver minimizes over the prediction horizon. For grid reconfiguration, this typically penalizes:

• Line losses: I²R heating in conductors, weighted by real-time electricity price
• Switching effort: number of breaker operations to preserve equipment life
• Voltage deviation: squared error from nominal 1.0 pu at each bus
• Constraint violation: soft penalties on thermal limits and radiality

Multi-objective weighting factors are tuned via Pareto analysis to balance competing goals.

MPC explicitly enforces operational limits as hard constraints in the optimization problem, unlike unconstrained controllers. Critical grid constraints include:

• Radiality constraint: network must remain a spanning tree with no closed loops
• Thermal limits: line ampacity and transformer MVA ratings
• Voltage bounds: ANSI C84.1 range of ±5% from nominal
• Switch interlocking: physical restrictions on certain breaker combinations

Constraint satisfaction is guaranteed for the entire prediction horizon, providing operational security guarantees.

MPC's predictive capability depends on accurate forecasts of external disturbances over the horizon. For grid applications, this requires:

• Load forecasts: short-term demand predictions from historical AMI data and weather inputs
• PV generation forecasts: solar irradiance predictions using sky imagers and numerical weather models
• Wind forecasts: turbine output estimates from mesoscale meteorological data

Forecast uncertainty is handled via robust MPC formulations that guarantee constraint satisfaction within a bounded uncertainty set.

Grid reconfiguration MPC requires solving a Mixed-Integer Linear Program (MILP) or Mixed-Integer Quadratic Program (MIQP) at each time step. Switch statuses are binary variables (0=open, 1=closed), while voltages and flows are continuous. Commercial solvers like Gurobi, CPLEX, or HiGHS use branch-and-bound algorithms to find globally optimal solutions.

• Typical problem size: 100-1000 binary variables for a medium feeder
• Solution time target: < 30 seconds for real-time implementation
• Warm-starting: previous solution used to accelerate convergence