Frequency-Constrained Unit Commitment

• Frequency-Constrained Unit Commitment is a scheduling framework that embeds post-contingency frequency-security criteria such as RoCoF and nadir constraints directly into the optimization model.
• It employs diverse modeling techniques including MILP-based linearization, Bernstein polynomial approximations, and neural-network surrogates to capture explicit time-domain dynamics.
• FCUC enables proactive strategies such as corrective UFLS and adaptive reserve allocation, crucial for maintaining grid stability in low-inertia and island power systems.

Searching arXiv for recent and foundational work on frequency-constrained unit commitment, including the specified papers. Frequency-constrained unit commitment (FCUC) denotes the class of unit commitment formulations in which post-contingency frequency-security requirements are embedded directly into scheduling decisions, rather than checked only after commitment and dispatch have been determined. In the cited literature, these requirements range from aggregate constraints on Rate-of-Change-of-Frequency (RoCoF), frequency nadir, and quasi-steady-state deviation to explicit time-domain differential models, locational RoCoF limits, underfrequency load shedding (UFLS), and continuous-time trajectories. FCUC has become particularly salient in low-inertia systems and island power systems, where increasing renewable penetration reduces rotating inertia and frequency control capacity, thereby changing both the feasible UC region and the cost-security trade-off (Sarvarizadeh et al., 7 Jul 2025).

1. Core formulation and security metrics

The canonical FCUC problem augments the standard UC objective with constraints that enforce acceptable frequency behavior following a credible disturbance, typically the largest generator outage or a worst-case $N-1$ contingency. In aggregate formulations, the largest power infeed may be treated as a fixed contingency size or as a decision variable. In the stochastic UC formulation for Great Britain’s 2030 power system, the largest infeed is explicitly made a decision variable through

$P_L \geq P_g \quad \forall\, g \in \mathcal{G},$

and frequency security is enforced through RoCoF, quasi-steady-state, and nadir constraints derived from the swing equation (Badesa et al., 2018).

These constraints are typically expressed in compact algebraic form. Representative examples are

$$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$

$$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$

and a nadir constraint that couples inertia, fast response, primary response, and damping through a nonconvex relation subsequently inner-approximated for MILP compatibility (Badesa et al., 2018). Other formulations use the largest outage $\Delta P$, aggregate inertia $M$, damping $D$, and droop $R_g$ to constrain RoCoF, steady-state frequency error, and nadir directly in the UC model (Paturet et al., 2019).

A basic conceptual distinction in FCUC is between preventive and corrective formulations. Preventive FCUC requires the pre-contingency schedule itself to satisfy the security criterion. Corrective FCUC allows explicitly modeled post-contingency actions, most notably UFLS, to enter the optimization. The corrective frequency-constrained UC (C-FCUC) for island power systems is formulated so that it can be converted into a preventive FCUC (P-FCUC) or the standard UC; in that sense, the C-FCUC is a generalization of both (Rouco et al., 2023).

2. Analytical FCUC architectures

A large part of the FCUC literature begins from reduced analytical frequency-response models. These models derive tractable expressions for the security metrics and then linearize or approximate the resulting constraints so that they can be embedded in MILP-based UC or SCUC.

One influential line uses a uniform system frequency response model and treats wind power and contingency uncertainty through a scenario tree. In this setting, RoCoF and steady-state error constraints are linear, whereas the nadir constraint is highly nonlinear. To preserve the mixed-integer linear structure, one paper proposes a computationally efficient bound extraction method that replaces direct piecewise linearization of the nadir surface by linear lower bounds on aggregate variables such as $M$, $R_g$, and $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$0. For medium-sized networks, this method is reported as computationally more efficient than a piece-wise linearization method adapted from the literature, while the inclusion of inertia-related constraints increases total costs by forcing more synchronous machines online (Paturet et al., 2019).

A second analytical line extends the reserve model itself. In the formulation that optimizes primary frequency control (PFC) droop gains and reserve capacities, the risk of frequency insecurity is allowed to vary with the operating condition. The UC model co-optimizes droop gains, PFC reserves, and time-varying secondary frequency control reserve requirements. Copula theory is used to establish the joint distribution model among frequency control performance, secondary frequency control reserve capacities, and power fluctuations, and distributionally robust optimization is then used to determine SFC reserve requirements under ambiguity in the probability model (Liu et al., 2021). This shifts FCUC from static reserve heuristics toward adaptive reserve quantification.

A third analytical strand addresses spatial heterogeneity. The location based RoCoF constrained security constrained unit commitment (LRC-SCUC) explicitly embeds nodal RoCoF constraints triggered by $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$1 contingencies. It monitors multiple time windows, captures the Fiedler-mode contribution to spatial RoCoF, and employs an effective piecewise linearization technique to linearize the nonlinear function representing the location based RoCoF constraints in SCUC. In the reported simulations, the inclusion of inertia-related constraints substantially improves system stability at the cost of higher operation cost, while virtual inertia reduces total cost and improves market efficiency (Tuo et al., 2021). This literature departs from center-of-inertia surrogates by recognizing that frequency security can be locational rather than purely aggregate.

3. Explicit time-domain dynamics and Bernstein polynomial formulations

A distinct development in FCUC replaces scalar frequency metrics by explicit time-domain dynamics. The 2025 formulation based on a general-order differential equation model introduces a second-order frequency-response representation in which system frequency deviation and reserve trajectories are modeled explicitly: $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$2

$P_L \geq P_g \quad \forall\, g \in \mathcal{G},$3

Traditional FCUC models are described there as typically relying on simplified first-order assumptions or scalar metrics such as frequency nadir, whereas the new formulation explicitly models time-domain frequency response using second-order dynamics (Rajabdorri et al., 2 Sep 2025).

The common tractability device in this branch is Bernstein polynomial approximation. A continuous function $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$4 over $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$5 is approximated as

$P_L \geq P_g \quad \forall\, g \in \mathcal{G},$6

with $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$7. Because derivatives and integrals of Bernstein polynomials can be written algebraically in terms of the coefficients, the underlying ODEs or DAEs can be transformed into linearizable algebraic systems, then embedded into UC as MILP constraints (Rajabdorri et al., 2 Sep 2025).

A closely related formulation studies frequency-secured UC under significant wind uncertainty by incorporating a DAE with dead band and coordinating thermal units and wind farms for primary frequency response. It adopts distributionally robust chance constraints based on a Wasserstein ambiguity set and applies Bernstein polynomials to derive tight inner approximation of the DAE, obtaining mixed-integer linear constraints solvable by off-the-shelf solvers. The paper reports that the nadir error can be reduced below $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$8 relative to high-fidelity simulation when the number and placement of Bernstein segments are increased appropriately, and that optimizing the wind inverter droop factors lowers cost by up to $P_L \geq P_g \quad \forall\, g \in \mathcal{G},$9 relative to fixed droop settings (Zhou et al., 2022).

Bernstein polynomials have also been used to move FCUC from hourly stepwise scheduling to a fully continuous-time framework. In the continuous-time FCUC for La Palma, all continuous variables are represented by order-3 Bernstein polynomials, startup and shutdown trajectories are modeled explicitly, and zero- and first-order continuity are imposed through coefficient relationships across intervals. Because the analytical frequency nadir constraint is highly nonlinear in the Bernstein-coefficient space, a data-driven frequency nadir constraint is introduced instead. The reported results show that when $$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$0 Hz is enforced, no violations occur, the model is solved timely, and intra-hour nadir variations of up to $$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$1 Hz that are invisible to discrete-hourly models are mitigated (Rajabdorri et al., 2023).

The 2025 general-order Bernstein formulation adds a further criterion: the duration of frequency excursions below a threshold. It defines an area-based quantity

$$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$2

and constrains this area so as to control the maximum time frequency spends below the critical threshold. The area-duration relation is calibrated data-drivennly using simulated frequency trajectories from more than $$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$3 outage scenarios (Rajabdorri et al., 2 Sep 2025). This is a notable shift from nadir-only security toward protection-aware time-under-threshold criteria.

4. Corrective FCUC and UFLS-aware formulations

Corrective FCUC makes UFLS an endogenous recourse variable rather than an exogenous failure mode. In the analytical C-FCUC for island power systems, the critical outage size before the nadir threshold is violated is computed analytically,

$$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$4

and the required UFLS is then

$$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$5

The reserve constraint is relaxed accordingly, and the objective includes a load-shedding cost term. By setting a high UFLS cost, the model behaves as preventive FCUC; omitting the UFLS constraints recovers standard UC (Rouco et al., 2023).

A later corrective formulation estimates optimal UFLS through a Tobit model driven by the initial RoCoF. In that model,

$$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$6

with fitted parameters such as $$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$7 and $$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$8, after k-means clustering is used to reduce dataset size. The reserve requirement becomes

$$\left|\text{RoCoF}\right| = \frac{P_L \cdot f_0}{2H} \leq \text{RoCoF}_{\text{max}},$$9

and the objective co-optimizes generation costs and expected UFLS cost. In the reported Spanish island study, the corrective FCUC is shown to reduce system operation costs without jeopardizing the quality of the frequency response in terms of UFLS occurrence (Sarvarizadeh et al., 7 Jul 2025).

The broadest comparison of island FCUC formulations evaluates analytical preventive, data-driven preventive, analytical corrective, and data-driven corrective approaches against a base UC. Two performance indices are defined: $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$0 measuring extra cost and extra CPU time per MW of UFLS reduction. In that case study, data-driven corrective FCUC attains operation cost $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$1 k€, total UFLS $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$2 MW, operation cost index $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$3 €/MW, and computational cost index $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$4 s/MW; it is identified as having the most advantages among the compared formulations (Sarvarizadeh et al., 7 Jul 2025). The result does not imply that corrective FCUC is universally superior, but it establishes UFLS-aware scheduling as a competitive design point in island systems where UFLS is inevitable for sufficiently large disturbances.

5. Data-driven and neural-embedded FCUC

Another major direction replaces explicit reduced-order analytical constraints by learned surrogates trained on dynamic simulation or operational data. In one generic framework, deep neural networks are trained to predict frequency nadir and dynamic stability from commitment, dispatch, demand, and worst-case contingency features, and then reformulated exactly as mixed-integer linear constraints using ReLU big-$$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$5 formulations. The data generation stage uses region-of-interest active sampling to concentrate labels near the UFLS threshold. On the IEEE 39-bus system, the active sampling strategy reduces mean absolute error to $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$6 Hz with $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$7 near the threshold, and full-order PSS/E simulations verify that the resulting FCUC commits frequency-secure schedules (Zhang et al., 2021).

A related literature emphasizes conservative constraint learning. The conservative sparse neural network embedded FCUC with converter-based DERs augments the loss function with a positive prediction error penalty so that the learned surrogate does not overestimate frequency security. It also prunes the network topology to accelerate MILP solution and constrains optimization inputs to remain inside convex hulls of sampled stable operating points to avoid extrapolation. The case study reports MAPE $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$8, $$\left|\Delta f^{\text{ss}}\right| = \frac{P_L - R_{\mathcal{S}} - R_{\mathcal{G}}}{D \cdot P_D} \leq \Delta f^{\text{ss}}_{\text{max}},$$9 negative rate, and up to $\Delta P$0 speedup relative to dense neural embeddings, while maintaining stable system operation against frequency violation under contingency (Sang et al., 2023).

Sparse learning and selective linearization are pushed further in the active linearized sparse neural network-based FCUC. There, a multi-output predictor learns maximal frequency deviation and the highest locational RoCoF simultaneously from high-fidelity simulation data, sparse computation is used to avoid dense matrix multiplications, active sampling maintains the bindingness of the learned constraints, and only a subset of ReLU activations are linearized with binaries. On the IEEE 24-bus system, computational time falls from $\Delta P$1 s for DNN-FCUC to $\Delta P$2 s for ALSNN-FCUC, a $\Delta P$3 reduction, while time-domain simulation confirms RoCoF $\Delta P$4 Hz/s and maximal frequency deviation $\Delta P$5 Hz under the enforced worst-case contingency (Tuo et al., 2023).

Not all data-driven FCUC is neural. The continuous-time Bernstein framework uses a linear regression surrogate for nadir (Rajabdorri et al., 2023); corrective island FCUC uses Tobit regression (Sarvarizadeh et al., 7 Jul 2025); and the comparative island study embeds regression-tree logic directly in MILP for data-driven corrective FCUC (Sarvarizadeh et al., 7 Jul 2025). The common feature is not a particular learning architecture, but the translation of simulation-derived frequency behavior into optimization-compatible constraints.

6. Spatial, technology-specific, and system-level extensions

The recent literature increasingly rejects the assumption that center-of-inertia frequency is an adequate security proxy. In the nodal frequency stability-constrained UC and ED framework for renewable-dominated systems, nodal post-contingency trajectories are checked during the branch-and-bound process, and violating incumbent solutions are cut off through additional MILP constraints. Two mitigation strategies are proposed: requiring more thermal generation, or constraining the maximum output of the generator with the largest incumbent output. Both approaches are reported to eliminate dispatch scenarios that resulted in instantaneous frequencies below $\Delta P$6 Hz, while the second approach minimizes the difference in production cost values from the non-stability-constrained case. The paper further states that all violations of nodal frequency were missed if only COI trajectory was monitored (Trujillo et al., 14 Jun 2026). This directly challenges the misconception that aggregate frequency metrics are always sufficient.

Technology-specific FCUC has also expanded. A recent FCUC model for the Chilean power system incorporates hydro-reservoir dynamics, including the water-hammer effect, through a data-driven nadir constraint. In the dynamic analysis for 2024, the reported nadir improvement per MW satisfies $\Delta P$7 MW gas CCGT $\Delta P$8 MW coal $\Delta P$9 MW hydro-reservoir, while in the 2035 carbon-free scenario hydro-reservoirs become the main provider of frequency regulation. Relative to the industry standard, the proposed FCUC yields operational cost reductions of about $M$0 in 2024 dry autumn, $M$1 in 2024 wet spring, $M$2 in 2035 dry autumn, and $M$3 in 2035 wet spring. The same study reports that $M$4 MW GFM $M$5 MW hydro-reservoir $M$6 MW SC in nadir support, and its financial analysis finds that GFM inverters significantly outperform synchronous condensers in terms of investment return (Aravena et al., 7 Oct 2025).

Island systems remain the dominant empirical testbed across much of the literature. The 2025 general-order Bernstein formulation is validated on a real-world Spanish island system and reports enhanced frequency security with a moderate increase in operational cost (Rajabdorri et al., 2 Sep 2025). The broader island comparative studies likewise center on Spanish systems (Rouco et al., 2023, Sarvarizadeh et al., 7 Jul 2025, Sarvarizadeh et al., 7 Jul 2025). This concentration suggests, without proving universally, that FCUC has been driven first by the operating conditions in which low inertia, limited interconnection, and protection sensitivity make ex post frequency validation especially inadequate.

Across these formulations, the central methodological divide is between scalar security proxies and explicit dynamic security modeling. Traditional FCUC often enforces only nadir or RoCoF, frequently through first-order or aggregate approximations. Newer formulations instead model higher-order dynamics, continuous-time trajectories, protection-aware excursion duration, locational instability, technology-specific response, or data-driven surrogates learned from high-fidelity simulation. The literature therefore defines FCUC not as a single model class, but as a family of optimization architectures that differ mainly in how faithfully and tractably they encode post-contingency frequency behavior.