Stochastic Unit Commitment

SUC replaces a single deterministic load forecast with a scenario tree or a set of weighted Monte Carlo trajectories. Each scenario represents a distinct, plausible realization of wind, solar, and load uncertainty. The optimization engine must find a single, non-anticipative day-ahead commitment schedule that minimizes the expected cost across all scenarios, ensuring physical feasibility for every sampled outcome.

• Non-anticipativity: Commitment decisions are identical across all scenarios for the same time period.
• Recourse actions: Dispatch of committed units adapts to the specific scenario realization.

Unlike deterministic methods that use heuristic reserve margins (e.g., N-1 or percentage of load), SUC endogenously co-optimizes energy and reserves. The model calculates the precise amount of upward and downward flexibility required to cover the distribution of forecast errors. This links the procurement of spinning reserves directly to the statistical properties of renewable generation uncertainty.

• Upward reserve: Covers scenarios where net load exceeds the central forecast.
• Downward reserve: Covers scenarios of surplus renewable generation and minimum load violations.

Standard SUC minimizes expected cost, which is risk-neutral. Advanced formulations incorporate risk measures like Conditional Value at Risk (CVaR) to penalize high-cost tail scenarios. This prevents the optimizer from selecting a cheap commitment that fails catastrophically during low-probability, high-impact events such as simultaneous wind droughts and load spikes.

• Weighted CVaR: Balances expected cost against the average cost of the worst (1-α)% of scenarios.
• Robust optimization: An alternative approach that optimizes against the worst-case realization within an uncertainty set.

SUC must respect complex intertemporal constraints under uncertainty. Ramping limits, minimum up/down times, and startup/shutdown trajectories must be feasible for every transition between time steps across all scenarios. This creates a large-scale, mixed-integer programming problem where the commitment of a slow-start thermal unit in hour 1 restricts the feasible dispatch space in hour 6.

• Startup costs: Trajectory-dependent costs that vary with the unit's offline duration.
• Energy storage: State-of-charge dynamics couple decisions across the entire optimization horizon.

The sheer scale of the SUC mixed-integer program—often millions of variables and constraints—necessitates decomposition. Benders decomposition separates the unit commitment master problem from the scenario-specific economic dispatch subproblems. Progressive hedging decomposes by scenario, iteratively penalizing deviations from a consensus commitment until non-anticipativity is enforced.

• Lagrangian relaxation: Relaxes coupling constraints to solve independent subproblems.
• Column generation: Dynamically generates candidate commitment schedules.

The quality of an SUC solution depends critically on the input scenarios. Raw Monte Carlo sampling from ARIMA or Gaussian Process forecast models is followed by scenario reduction algorithms. These techniques, such as fast forward selection or k-means clustering, select a tractable subset of representative scenarios that preserve the statistical moments and tail behavior of the original distribution.

• Moment matching: Ensures the reduced set preserves mean, variance, and skewness.
• Kantorovich distance: A metric for quantifying the information loss from reduction.

The foundational mathematical framework for SUC. First-stage decisions (here-and-now) commit generating units before uncertainty is realized. Second-stage decisions (wait-and-see) represent recourse actions like dispatch adjustments, reserve deployment, or load curtailment once the actual net load is known. The objective minimizes commitment costs plus the expected value of future operating costs.

SUC requires a tractable representation of uncertainty. Techniques include:

• ARIMA models for temporal correlation in load and wind forecast errors
• Copula theory to capture spatial dependence between wind farms
• Latin Hypercube Sampling for efficient coverage of the input space
• K-means clustering or backward reduction to distill thousands of scenarios into a representative subset while preserving statistical moments

An alternative to scenario-based SUC where constraints are expressed probabilistically. Instead of modeling explicit scenarios, a chance constraint ensures that load is met with a specified probability (e.g., 95%). This directly links to reliability metrics like Loss of Load Probability (LOLP). The formulation is often solved by transforming the probabilistic constraint into a deterministic equivalent using the inverse cumulative distribution function of the forecast error.

Beyond minimizing expected cost, SUC incorporates risk to avoid catastrophic tail events:

• Conditional Value at Risk (CVaR) quantifies the expected cost in the worst (1-α)% of scenarios, penalizing solutions with extreme shortfalls
• Variance penalties add a weighted term for cost dispersion across scenarios
• Robust optimization minimizes cost against the worst-case realization within an uncertainty set, providing a guaranteed floor on performance

Large-scale SUC problems are computationally intractable as a single monolithic optimization. Benders decomposition separates the problem into a master problem (unit commitment) and subproblems (economic dispatch per scenario). Progressive hedging decomposes by scenario, solving each independently and iteratively penalizing deviations in first-stage decisions until consensus is reached. Lagrangian relaxation dualizes coupling constraints to enable parallel solution.

SUC explicitly co-optimizes energy and ancillary services to ensure ramping capability. Key products include:

• Regulation reserve for automatic frequency control (seconds)
• Spinning reserve for rapid contingency response (10 minutes)
• Flexibility ramping product to ensure sufficient inter-interval ramping capacity against steep net load changes, particularly during morning and evening ramps driven by solar generation