Model Predictive Control (MPC)

MPC solves an open-loop optimal control problem over a finite prediction horizon at each sampling instant, but only the first control action is implemented. The horizon then recedes forward one step, and the optimization is repeated. This provides inherent feedback to compensate for model inaccuracies and unmeasured disturbances.

• Prediction Horizon (Np): The number of future time steps over which the system output is predicted.
• Control Horizon (Nc): The number of future time steps over which control moves are computed (typically Nc ≤ Np).
• Re-optimization: The optimization problem is solved anew at each time step, incorporating the latest measurements.

Unlike unconstrained controllers, MPC systematically incorporates hard constraints on both manipulated variables (MV) and controlled variables (CV) directly into the optimization problem. This is critical for grid applications where violating voltage limits or thermal ratings is unacceptable.

• Input Constraints: Generator ramp rate limits, battery state-of-charge boundaries, and inverter capacity limits.
• Output Constraints: Bus voltage magnitude limits (e.g., ANSI C84.1 Range A: 0.95–1.05 pu) and line current thermal limits.
• Constraint Prioritization: Soft constraints with slack variables can be used to ensure feasibility when hard constraints conflict.

The core of MPC is an explicit mathematical model that predicts the future evolution of the system. For smart grids, this model captures the electromechanical dynamics of generators, the state-of-charge dynamics of batteries, and the power flow relationships across the network.

• State-Space Representation: The system is typically modeled as x(k+1) = Ax(k) + Bu(k), where x represents states like voltage angles and u represents control inputs like generator setpoints.
• Disturbance Modeling: Forecasted renewable generation and load profiles are treated as measured disturbances fed into the prediction model.
• Linearization: Nonlinear AC power flow equations are often linearized around an operating point for computational tractability in real-time control.

The control objectives are encoded as a mathematical cost function that the optimizer minimizes. This allows MPC to balance competing goals like minimizing generation cost, tracking a reference voltage profile, and minimizing control effort.

• Quadratic Cost: A common formulation is J = Σ (y - y_ref)ᵀ Q (y - y_ref) + Σ Δuᵀ R Δu, where Q penalizes output deviations and R penalizes aggressive control moves.
• Economic MPC: The cost function directly represents operational costs (e.g., fuel cost, battery degradation cost) rather than tracking a setpoint.
• Multi-Objective Tuning: The weighting matrices Q and R are tuned to balance the trade-off between tight regulation and smooth control action.

For large-scale grids, a single centralized MPC becomes computationally intractable. Distributed MPC decomposes the overall problem into smaller subproblems solved by local controllers that coordinate through information exchange.

• Decomposition Strategies: The grid can be partitioned by geographical area, voltage level, or ownership boundary.
• Coordination Protocols: Local controllers exchange boundary variable predictions (e.g., tie-line power flows) and iterate to achieve consistency, often using algorithms like ADMM.
• Resilience: Distributed architectures eliminate the single point of failure inherent in centralized control and reduce communication bandwidth requirements.

MPC performance critically depends on the quality of predictions for uncontrollable inputs. In smart grids, MPC is tightly coupled with machine learning forecasts of renewable generation and load demand over the prediction horizon.

• Solar Irradiance Forecasts: Short-term (5–60 minute) sky-imager or satellite-based predictions feed the disturbance model.
• Load Forecasts: Time-series models predict the aggregate and nodal load evolution.
• Scenario-Based MPC: Instead of a single deterministic forecast, multiple probabilistic scenarios can be incorporated using stochastic programming to generate control actions robust to forecast uncertainty.