Unit Commitment

Unit Commitment is fundamentally formulated as a Mixed-Integer Programming (MIP) problem. The on/off status of each generator is represented by a binary variable (0 or 1), while the power output is a continuous variable. This mathematical structure makes the problem computationally complex (NP-hard) but allows solvers to find globally optimal schedules that respect all physical constraints.

Unlike simpler dispatch problems, Unit Commitment is defined by intertemporal constraints that link decisions across time periods. Key examples include:

• Minimum Up/Down Times: Once started, a thermal unit must run for a minimum duration before shutting down, and vice-versa.
• Ramp Rate Limits: The maximum speed at which a unit can increase or decrease output between hours.
• Start-Up Costs: A non-linear cost function that depends on how long the boiler has been cooling, making the commitment decision path-dependent.

Modern Unit Commitment is almost always Security-Constrained (SCUC). This means the optimization must simultaneously solve for a generation schedule that remains feasible not just under normal conditions, but also during N-1 contingencies—the unexpected loss of any single transmission line or generator. This ensures the resulting schedule is both economically efficient and meets NERC reliability standards for operating reserves.

To handle renewable energy variability, Unit Commitment has evolved beyond a single load forecast:

• Deterministic UC: Uses a single, best-guess load and wind/solar forecast, augmented with conservative reserve margins.
• Stochastic UC: Explicitly models a scenario tree of possible future net-load outcomes. The solver finds a single commitment schedule that minimizes the expected cost across all probabilistic scenarios, often yielding significant savings by reducing unnecessary reserve procurement.

For large-scale systems with thousands of units, solving a single monolithic MIP is intractable. Lagrangian Relaxation (LR) decomposes the problem by relaxing the coupling constraints (like the power balance equation) and attaching a Lagrange multiplier (price signal) to them. This breaks the problem into independent, single-unit dynamic programming sub-problems that can be solved in parallel, iterating until the global constraints are satisfied.

In restructured electricity markets (like PJM, ERCOT, or CAISO), Unit Commitment is the computational engine that clears the Day-Ahead Market. It simultaneously processes generator offers, demand bids, and virtual transactions to produce hourly Locational Marginal Prices (LMPs) and a financially binding schedule for the next operating day, ensuring that enough capacity is secured to meet the forecasted peak load.

The fixed fuel and wear-and-tear expense incurred when bringing a thermal generating unit from an offline state to synchronization and minimum load. This cost is a critical input to the UC objective function.

• Cold start: Unit offline >48 hours, highest cost due to thermal stress
• Warm start: Unit offline 8-48 hours, moderate cost
• Hot start: Unit offline <8 hours, lowest cost
• Includes auxiliary power, boiler firing, and turbine rolling costs
• Creates the intertemporal coupling that makes UC a dynamic optimization problem

Temporal constraints that prevent a generating unit from being turned off immediately after starting (minimum up time) or restarted immediately after shutting down (minimum down time). These constraints enforce thermal and mechanical integrity.

• Min up time: Typically 4-24 hours for coal, 1-4 hours for gas turbines
• Min down time: Prevents hot restarts that cause rotor stress
• Modeled as integer constraints in the Mixed-Integer Linear Programming (MILP) formulation
• Directly impacts the look-ahead horizon required for feasible schedules

An enhanced UC formulation that simultaneously enforces N-1 transmission security constraints alongside generator scheduling. SCUC ensures the resulting commitment schedule remains feasible under contingency conditions.

• Incorporates line flow limits and contingency analysis
• Uses shift factors (PTDF matrices) to model transmission impacts
• Computationally intensive due to the explosion of constraint sets
• Standard practice in ISO/RTO day-ahead markets (PJM, MISO, ERCOT)
• Often solved using Benders decomposition to separate generation and network subproblems

The primary input signal to the Unit Commitment problem, representing the predicted system demand over the scheduling horizon. Forecast accuracy directly determines UC solution quality and reliability.

• Day-ahead forecasts: Hourly resolution, drives base UC decisions
• Short-term forecasts: 5-15 minute resolution, used for intra-day adjustments
• Errors propagate: under-forecasting causes scarcity pricing, over-forecasting causes unnecessary starts
• Modern approaches use gradient boosting machines and LSTM networks trained on weather, calendar, and economic features

The dominant mathematical optimization framework used to solve the Unit Commitment problem. MILP handles the binary on/off decisions (integer variables) alongside continuous power output variables within a linear objective and constraint structure.

• Objective: Minimize total production cost + start-up cost + no-load cost
• Binary variables: Unit commitment status (0=offline, 1=online)
• Continuous variables: MW output per unit per time period
• Solved using branch-and-bound algorithms with commercial solvers (CPLEX, Gurobi)
• Lagrangian relaxation historically used as an alternative decomposition method