RoCoF-Based UFLS Control Scheme

• The paper demonstrates a novel UFLS control scheme based on RoCoF, which accelerates and refines load shedding decisions for grid stability.
• It details diverse methodologies including PMU-based algorithms, cooperative game theory, and data-driven modeling to optimally allocate load shedding.
• Practical implementations reveal significant performance improvements over conventional UFLS methods, with reduced load shedding and quicker frequency restoration.

A RoCoF-based Under-Frequency Load Shedding (UFLS) control scheme is a frequency stability countermeasure in power systems leveraging the Rate of Change of Frequency (RoCoF) as a rapid indicator of system disturbance severity. Such schemes supplant classical frequency-threshold logic by providing faster, more precise calculation of required load decoupling and improved adaptation to network conditions—especially critical in low-inertia environments with high renewable penetration. RoCoF-based UFLS approaches span from distributed relay strategies driven directly by PMU measurements to centralized allocation rooted in system-wide swing equation analysis and optimized via cooperative game theory or data-driven modeling.

1. Core RoCoF-UFLS Principles and Mathematical Formulation

RoCoF-based UFLS schemes rely on the physical coupling between system inertia, active power disturbance, and the initial post-contingency RoCoF as given by the multi-machine swing equation. Considering $M$ synchronous machines, the equivalent center of inertia (COI) frequency $f_c$ and its derivative are defined as: $$f_c = \frac{\sum_{i=1}^{M} H_i f_i}{\sum_{i=1}^{M} H_i}$$

$$\Delta P_{shed} = 2 H_{eq} \frac{df_c}{dt} \frac{1}{f_n}$$

where $H_i$ is inertia constant, $H_{eq}$ is total system inertia, $f_n$ is nominal frequency, and $\Delta P_{shed}$ is the required load shedding to arrest frequency decline (Gautam et al., 2021).

PMU-based methods estimate $f$, $df/dt$ using windowed synchrophasor signal models, most commonly via (i) static models such as (enhanced/interpolated) DFT, or (ii) dynamic models incorporating phasor derivatives (cs-TFM). The dynamic approach offers better tracking of ramps, as per: $$x(t) = \Re\Big\{ (X + \dot X t) e^{j(\omega_0 t + \phi)} \Big\}$$ Instantaneous frequency and RoCoF are then

$$\hat{f} = \frac{1}{2\pi} \Im\left\{ \frac{\dot{X}}{X} \right\}, \quad \widehat{\mathrm{ROCOF}} = \frac{d\hat{f}}{dt}$$

with practical schemes employing finite-difference smoothing for noise robustness (Frigo et al., 2019).

Advanced RoCoF estimation can exploit geometric/differential methods. For a three-phase voltage vector $v(t)$, quasi-steady-state (QSS) frequency and RoCoF are formulated: $$\omega_{qss} = \frac{1}{T} \oint_T v \cdot dv/d\tau\, d\tau, \quad \bar{\mathrm{RoCoF}}(t) = \frac{1}{\Delta \tilde{t}_w} \int_{t-\Delta t_w}^t T(\tau) \frac{d \omega_{qss}}{d\tau} d\tau$$ where $T(\tau)$ is a boolean mask for physically meaningful “closed-loop” signal intervals (Gutierrez-Florensa et al., 5 Nov 2025).

2. Detection, Thresholds, and Distributed Implementation

Each load bus is equipped with a PMU and local relay logic. At every reporting interval (e.g., 20 ms, 50 fps), relays receive instantaneous $f$ and $df/dt$, applying filtering (e.g., 500 ms moving average) to RoCoF signals to suppress noise and poorly conditioned oscillatory artifacts (Derviškadić et al., 2018).

The load-shedding trigger engages when filtered RoCoF drops below a bank of negative thresholds, typically in the range of $-0.2$ to $-1.3$ Hz/s, corresponding to sequentially larger portions of local load (see table below for typical settings):

LS Stage	RoCoF Threshold (Hz/s)	Load Shedding Block (%)
1	--	5
2	–0.2	10
3	–0.4 / –0.3	10
...	...	...

After shedding at a given stage, a prescribed time delay (e.g., 500 ms), ensures no immediate re-triggering. Restoration is managed by a parallel frequency threshold sequence (e.g., 49.75/49.60 Hz etc.) with a reconnection delay (e.g., 5 s) (Derviškadić et al., 2018).

A key operational feature is that decision-making is strictly decentralized—each relay observes only its local measurements, so no message exchange or central coordinator is required.

3. Advanced UFLS Allocation via Game Theory and Data-driven Modeling

Beyond distributed fixed-threshold logic, quantitative allocation of $\Delta P_{shed}$ can be cast as an optimization or cooperative game. In the two-stage strategy (Gautam et al., 2021), the first stage computes the total required shed based on COI RoCoF. The second stage uses the Shapley value from cooperative game theory to allocate shedding among candidate load buses. The characteristic function $v(S)$, evaluated via time-domain simulation, measures marginal frequency benefit (steady state and transient) of tripping coalition $S$. The equivalent Shapley value for each bus determines the fraction of $\Delta P_{shed}$ assigned: $D_k = \frac{\Phi_k^{eq}}{\sum_j \Phi_j^{eq}}$

$p_k = D_k \cdot P_d$

where $p_k$ is the amount shed at bus $k$. This approach ensures that buses whose shedding yields the largest frequency stabilization effect receive higher assignment, directly reflecting network topology, inertia distribution, and disturbance characteristics.

Alternatively, data-driven models fit empirical mappings from observed initial RoCoF to required load shedding using censored-linear (Tobit) regression: $$p_{\ell, t}^{UFLS} = \begin{cases} 0, & \Delta \dot f_{initial} \le a^{UFLS} \ b^{UFLS} (\Delta \dot f_{initial} - a^{UFLS}), & \Delta \dot f_{initial} > a^{UFLS} \end{cases}$$ Typical fit parameters: $a^{UFLS}=1.66$ Hz/s (threshold), $b^{UFLS}=3.246$ MW/(Hz/s) (Sarvarizadeh et al., 7 Jul 2025). This model can be integrated into corrective frequency-constrained unit commitment (MILP) formulations, permitting co-optimization of spinning reserves and anticipated UFLS actions.

4. RoCoF Estimation Algorithms and Practical Issues

RoCoF estimation fidelity is central to the dependability of the scheme, particularly given PMU model limitations during non-nominal or high-distortion conditions. The comparative performance of static (e-IpDFT, i-IpDFT) versus dynamic (cs-TFM) PMU models indicates that:

• M-class (≥5 cycle, 100 ms) windows are required for <0.5 Hz/s RoCoF resolution (suitable for UFLS).
• Finite-difference smoothing is preferred over direct derivatives due to lower noise sensitivity.
• Dynamic phasor models yield significantly lower relative frequency error (RFE) under interharmonic and oscillatory conditions: e.g., $|RFE| \leq 0.38$ Hz/s, $96\%$ correlation for cs-TFM (M-class) (Frigo et al., 2019).

Transient epochs (step changes, islanding) challenge the validity of the synchrophasor model. Transient detection via nRMSE is used to identify such intervals, blocking RoCoF-based commands and reverting to last valid frequency or conservative logic. Geometric estimation methods employing a QSS closed-loop check further improve robustness by automatically excluding "physically meaningless" intervals, thus preventing false trips and permitting shorter detection windows (down to 0.25 s after post-fault signal recovery) (Gutierrez-Florensa et al., 5 Nov 2025).

Parameter selection is system-specific, but general recommendations are:

Parameter	Typical Value/Setting
PMU window (RoCoF)	M-class: ≥100 ms (5 cycles)
RoCoF threshold for action	–0.5…–2 Hz/s
Filtering/delay window	0.25–0.5 s (QSS: as low as 0.1 s)
Circulation threshold (QSS)	ε = 0.01–0.1 pu

5. Performance, Scaling, and Benchmarking

RoCoF-based UFLS schemes have demonstrated markedly superior performance to conventional frequency-threshold approaches. Key findings:

• In distributed PMU-based relaying, for a 1 GW loss (IEEE-39 bus, scenario 1):
  • Frequency-only: 48.77 Hz nadir, 15% maximum shed, duration 40.9 s, 4 MWh curtailed.
  • RoCoF-based Case A: 48.72 Hz nadir, 5% shed, 10.5 s duration, 0.9 MWh curtailed energy (Derviškadić et al., 2018).
• Cooperative game allocation on WECC-9 yields precise bus-level shedding assignments (e.g. for $P_d \approx 85$ MW, buses 5/6/8: [34, 24, 27] MW) and refloats system frequency to 60 Hz in <2 seconds with computational time $<1$ ms (Gautam et al., 2021).
• In real-time HIL, dynamic RoCoF-based UFLS achieves lower "Expected Energy Not Served" (down to 11.6 MWh) than static models and avoids blackouts where frequency-only logic fails (Frigo et al., 2019).
• In MILP-embedded Tobit modeling (islanded grid, 40 MW peak), operation cost reductions of 1–2% were achieved with UFLS causality and adequacy preserved (Sarvarizadeh et al., 7 Jul 2025).

RoCoF-based UFLS scales to large networks without central coordination, and computation is not a dominant bottleneck.

6. Limitations, Guidelines, and Future Directions

Limitations include:

• Reliance on valid synchrophasor or QSS conditions; heavy distortion, deep unbalance, or slow recovery may impair RoCoF validity.
• Necessity for accurate, high-sampling PMUs or voltage digitization (≥1–2 kHz) and tight clock synchronization.
• Sensitivity to parameter tuning; RoCoF thresholds, delays, and model fits require offline studies for robust operation across grid states.

Recommended practices:

• Employ dynamic phasor (cs-TFM) with finite-difference for window-based PMU RoCoF; M-class windows are baseline.
• Incorporate transient detection logic to block RoCoF action during strong model violation.
• In centralized schemes, characterize candidate load buses via network simulation for proper marginal frequency benefit allocation.
• When using QSS geometrical methods, set the circulation threshold ε at least one order above observed measurement noise.

Future work may extend to virtual inertia emulation, coordination with frequency support from IBRs, and integration of AI/ML for improved parameter adaptation.

7. Comparative Assessment to Conventional UFLS

RoCoF-based UFLS offers several demonstrated advantages relative to fixed-schedule under-frequency relays:

• Rapid, physics-based estimation of required shedding size adapts to disturbance magnitude/inertia in real time.
• Quantitative allocation methods—via Shapley value or data-driven regression—permit network- and scenario-adaptive trip scheduling, reducing the risk of over/under-curtailment.
• Decentralized logic with minimal communication requirements; robust to single-point failures.
• Faster restoration, minimized load disconnection, and avoidance of full system collapse in simulated low-inertia, high-disorder contingencies (Gautam et al., 2021, Frigo et al., 2019, Gutierrez-Florensa et al., 5 Nov 2025).

A plausible implication is that future UFLS designs in converter-dominated grids will universally adopt RoCoF-based and model-driven trip logic, further integrating disturbance estimation and optimal resource allocation to assure frequency security.