Robust Unit Commitment Methods

• Robust Unit Commitment is a framework for optimizing generator start-up and shutdown decisions under worst-case uncertainties.
• It employs two-stage decision models with first-stage commitments and adaptive second-stage dispatch actions to ensure feasibility.
• Advanced computational strategies like column-and-constraint generation and decomposition methods enable scalable solutions for large power systems.

The robust unit commitment (RUC) problem refers to the optimization of generator start-up and shutdown scheduling over a planning horizon, ensuring system reliability and cost efficiency under worst-case realizations of uncertain parameters such as renewable generation, load, and network contingencies. Unlike stochastic or deterministic unit commitment, RUC models guarantee feasibility and bounded system cost across a prescribed uncertainty set, often constructed to reflect data-driven statistical properties, physical correlations, or operational risk criteria.

1. Formal Structure of the Robust Unit Commitment Problem

Let $x$ denote first-stage (here-and-now) decisions, typically generator on/off status, start-up/shut-down binaries, and day-ahead scheduled output or reserve levels. The second-stage (“recourse”) decisions $y(\xi)$ adapt dispatch, curtailment, and possibly load shedding after the realization of uncertainty $\xi$ (which may represent renewable errors, demand, contingencies, or other exogenous drivers). The RUC adopts a two-stage or multistage robust optimization formulation: $$\min_{x \in \mathcal X} \left\{c_1^\top x + \max_{\xi \in \mathcal U} Q(x, \xi)\right\},$$ where $Q(x, \xi)$ is the minimum recourse cost for scenario $\xi$, and $\mathcal U$ is a convex, polyhedral, or otherwise representable uncertainty set. Variants include distributionally robust UC, where the expectation is taken over the worst-case probability law in an ambiguity set (e.g., Wasserstein or KL balls), yielding formulations

$$\min_{x \in \mathcal X} \left\{c_1^\top x + \max_{P \in \mathscr P} \mathbb{E}_{\xi \sim P}[Q(x, \xi)] \right\}.$$

Robust models demand solution feasibility for all $\xi \in \mathcal U$.

2. Uncertainty Set Construction and Properties

Polyhedral/Budgeted Sets

A fundamental design in RUC is the Bertsimas-Sim box/budget set, where uncertainty variables $\xi$ are constrained within deterministic bounds, and a “budget” caps the sum of deviations (spatial or temporal aggregation). For wind, e.g.: $$\mathcal U = \left\{\xi:\ \underline \xi_{m,t} \leq \xi_{m,t} \leq \overline \xi_{m,t},\ \sum_t (v_{m,t}^+ + v_{m,t}^-) \leq \Gamma^T,\ \sum_m (v_{m,t}^+ + v_{m,t}^-) \leq \Gamma^S\right\},$$ where $v_{m,t}^\pm$ are binary indicators of maximum deviation in time and space (Wang et al., 2015).

Correlated and Dynamic Sets

To avoid conservativeness from box sets, several works introduce dynamic uncertainty sets that capture both spatial and temporal correlation structures: $$p^{r}_{it} = f_{it} + g_{it} u_{it},\quad u_t = \sum_{l=1}^L A^l u_{t-l} + B v_t,\quad \Vert v_t \Vert \leq \Gamma,$$ where $A^l$ and $B$ encode correlation via empirical statistical estimation (Lorca et al., 2016). Budget parameters and principal component analysis (PCA) often guide fitting to historical data.

High-Confidence and Statistical Guarantee Sets

Data-driven approaches create uncertainty sets with statistical guarantees. For example, constructing ellipsoidal error sets from training/validation data so that, with high probability, actual errors fall within the set at a prescribed service level: $$\mathcal E = \{e:\ (e - \mu)^\top \Sigma^{-1} (e - \mu) \leq \alpha\}$$ and

$$\mathcal U_1 = \{u:\ \underline U \leq u \leq \overline U,\ u - \hat u(w) \in \mathcal E\}$$

where ensemble forecasts are optimally weighted for the UC objective and the coverage of $\mathcal E$ is dimension-free (Xie et al., 5 Nov 2024).

Distributionally Robust Ambiguity Sets

RUC can be formulated with ambiguity sets defined by Wasserstein metric balls or relative entropy (KL-divergence) balls around empirical distributions (Cho et al., 2022, Yurdakul et al., 2020). Moment-based ambiguity sets require all laws $P$ matching empirical moments up to second order (Zheng et al., 2019).

3. Decision Rules and Affine/Implicit Policies

Second-stage (recourse) policies in RUC can be:

• Causal affine decision rules: Recourse actions (e.g., generator dispatch, reserve, curtailment) are enforced to depend affinely on contemporaneously revealed uncertainty, typically parameterized as

$$x_{it}^g(\xi_t) = a_{it}^{g1} \sum_{j} \xi_{jt} + a_{it}^{g0}$$

so that causal nonanticipativity is preserved. These policies substantially reduce problem size and can regularize solutions, yielding improved out-of-sample robustness (Cho et al., 2022, Lorca et al., 2016).

• Piecewise-linear and implicit rules: Piecewise-linear rules (PLDR) can be deployed for better coverage of asymmetric tail behavior, at a moderate computational penalty (Zugno et al., 2015). Implicit nonanticipative policies, enforced by auxiliary per-period upper and lower bounds with ramping constraints, offer scalability for large networks (Li et al., 2018).
• Multistage frameworks and approximate dynamic programming: The cost-to-go structure of the multistage robust UC is exploited in primal–dual iterative schemes, using McCormick relaxations and simplex-vertex initialization to maintain feasible and near-optimal cost-to-go approximations as uncertainty unfolds (Lan et al., 2023).

4. Computational Strategies

Column-and-Constraint Generation (C&CG)

C&CG is fundamental for solving robust two-stage UC: iteratively solve a master problem for the current scenario set, then identify (via a subproblem) the worst-case realization over the full uncertainty set. Tightened master scenarios and feasibility cuts ensure convergence. For models with bilinear recourse constraints (e.g., left-hand-side uncertainty), special relaxations and binary-extreme policies are necessary (Wang et al., 2022).

Benders-Type Decomposition and SDP Reformulations

In distributionally robust UC, Benders decomposition is applied: the master solves for commitment and dual variables, while subproblems (often LP or SDP) calculate scenario-wise recourse costs or cut planes. For moment-constrained ambiguity sets, semidefinite programming (SDP) formulations based on S-lemma duality allow exact enforcement and convergence (Zheng et al., 2019, Yurdakul et al., 2020).

Data-Driven Constraint Screening

Constraint screening, particularly for network constraints (e.g. line flow limits), incorporates historical data via PCA and regression-based cost bounds, using umbrella constraint discovery to eliminate redundant constraints in the robust feasible region (Awadalla et al., 2023).

Hybrid and Hierarchical Decomposition

Scenario clustering and partitioning interpolate between stochastic and robust UC, allowing practitioners to tune the conservativeness of the solution, leading to tractable solutions on large networks via parallelization (Blanco et al., 2016).

5. Model Enhancements and Applications

Integration of Risk Constraints

Risk-constrained RUC formulations treat the operational loss for out-of-band realizations as an explicit term in the objective, with adjustable uncertainty sets sized to balance cost and risk. Piecewise linear risk proxies are imposed as constraints, and the risk limit is a tunable parameter controlling risk aversion (Wang et al., 2015).

Strategic Wind Generation Curtailment

First-stage strategic curtailment of wind, by selecting less than the full forecast for commitment, can endogenously reduce the uncertainty band, decrease flexibility reserve requirements, and render the system feasible at high penetration levels where full-absorb policies become infeasible (Wang et al., 2015).

Network and Transmission Constraints

RUC models for transmission-constrained systems (TCUC) handle robust feasibility under uncertain nodal injections, employing budgeted uncertainty sets and specialized preprocessing such as the column-merging method (CMM) to aggregate uncertain nodes and reduce the scenario tree’s exponential complexity (Li et al., 2018).

Correlated and Complex Uncertainty

Recent works address robust UC with both right- and left-hand-side uncertainty, notably generator efficiency as a function of uncertain temperature, and enforce budget and correlation constraints across time for both temperature and demand. Relaxations based on binary-extreme worst-cases allow tractable C&CG while preserving robustness (Wang et al., 2022).

Distributionally Robust Microgrid and Quantum Methods

KL-divergence-based DRUC with clustered scenario support yields tractable convex MINLPs via duality, and Pareto-superior out-of-sample performance compared to stochastic UC for moderate divergence tolerances (Yurdakul et al., 2020). Quantum reinforcement learning has been applied to UC, where MDP-based formulations blend two-stage UC with scalable policy optimization, showing competitive solution quality and computational efficiency (Wei et al., 28 Oct 2024).

6. Comparative Performance and Scalability

Recent numerical studies demonstrate:

• Affine WDRO models yield 5–10% cost reductions relative to classic RUC with full reliability, and achieve flat computational scaling in data sample size (N) (Cho et al., 2022).
• Multistage robust methods with dynamic, correlated uncertainty sets and implicit nonanticipativity enable large-scale systems (e.g., Polish 2736-bus, 2383-bus, with 60–100 stochastic resources) to be solved efficiently—1–2 hours per run—while outperforming deterministic or static-robust baselines in cost and risk (Lorca et al., 2016, Li et al., 2018, Lan et al., 2023).
• Data-driven uncertainty sets with statistical guarantees, combined with predict-and-optimize frameworks, outperform standard ensemble or MSE-based forecast-driven robust UC by 1–2% in cost and simultaneously satisfy reliability targets (Xie et al., 5 Nov 2024).
• Moment-based DRUC (MI-SDP) yields lower real-time cost and less wind curtailment than deterministic or classic RUC while incurring marginally higher day-ahead commitment cost (Zheng et al., 2019).
• Strategic wind curtailment in RUC makes high-penetration operation feasible when traditional RUC fails due to excessive uncertainty (Wang et al., 2015).

7. Extensions and Current Developments

RUC research continues to extend toward:

• Integrated forecast-optimization loops with statistical guarantees and data-driven uncertainty set refinement (Xie et al., 5 Nov 2024).
• Higher-fidelity modeling of joint uncertainties (load, renewables, contingencies), network topology outages, and complex system physics (ac-OPF, security constraints) (Zhao et al., 2019, Wang et al., 2022).
• Hybrid methods combining stochastic and robust paradigms via scenario clustering and convex combinations, tuned to specific reliability/cost trade-offs (Blanco et al., 2016).
• Scalability improvements via pre-processing (constraint screening, node-aggregation), advanced decomposition, and new high-performance computational paradigms (quantum RL, advanced surrogate models) (Awadalla et al., 2023, Wei et al., 28 Oct 2024).

Robust unit commitment is central to reliable, cost-competitive integration of stochastic resources in modern power systems. The field is marked by active methodological development, growing data-driven integration, and continuing advances in computational tractability and solution quality.