⚡ electrical engineering

Probabilistic Frequency Hazard Analysis: Adapting the Seismic Hazard Framework to Power System Frequency Exceedance Risk

This paper introduces Probabilistic Frequency Hazard Analysis (PFHA), a novel framework adapting the mathematical architecture of seismic hazard assessment to quantify power system frequency exceedance risks with formal uncertainty quantification, source-level disaggregation, and continuous hazard curves, as validated by its application to the Great Britain power system.

Original authors: Sewedo Todowede

Published 2026-04-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sewedo Todowede

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A New Way to Predict Power Grid "Earthquakes"

Imagine the power grid as a giant, high-speed race car. For decades, this car had a heavy, spinning flywheel (called synchronous inertia) that kept it stable. If the engine hiccupped, the flywheel's momentum smoothed it out, and the car kept driving straight.

But today, we are replacing the heavy flywheel with lightweight, high-tech electric motors (renewables like wind and solar). The car is faster and cleaner, but it's also much more sensitive. A tiny bump in the road (a generator tripping off) can now send the car swerving wildly.

The author, Sewedo Todowede, is worried: How do we know how likely it is for this car to crash (frequency drop) in the future?

Existing methods are like a mechanic looking at the car and saying, "It looks okay, but I'm not 100% sure." This paper introduces a new, super-rigorous method called Probabilistic Frequency Hazard Analysis (PFHA).

The Core Idea: Borrowing from Earthquake Science

The brilliant twist in this paper is borrowing a tool from Earthquake Engineering.

The Old Way (Seismology): For earthquakes, scientists use PSHA (Probabilistic Seismic Hazard Analysis). They don't just guess; they calculate the odds of an earthquake of a certain size hitting a specific spot, based on thousands of past quakes, fault lines, and soil types.
The New Way (Power Grids): The author realized that a power grid losing power is mathematically identical to an earthquake hitting a building.
- The Earthquake = A generator or wind farm suddenly shutting down.
- The Ground Shaking = The system frequency dropping (e.g., from 50Hz down to 49Hz).
- The Building Damage = Lights going out or equipment breaking.

The paper says: "Let's stop guessing about power grid crashes and start calculating them exactly like we calculate earthquake risks."

How the New System Works (The "Recipe")

The author built a massive mathematical engine that mixes four main ingredients:

1. The "Guest List" (Source Catalogue)

Imagine a party where 51 specific guests (generators and interconnectors) are invited. The system knows exactly who they are, how big they are, and how often they tend to leave early (trip).

Analogy: Instead of saying "someone might leave," the system says, "There is a 10% chance the giant wind farm leaves, and a 2% chance the nuclear plant leaves."

2. The "Weather Report" (System State)

The risk depends on the conditions. If the grid is "low on inertia" (like the car having a light flywheel), a small trip causes a big crash. If it's "high inertia," the same trip is a minor bump.

Analogy: The system looks at 70,000 past "days" (half-hour periods) to see what the weather was like when things went wrong. It realizes that low inertia days are the most dangerous, even if they are rare.

3. The "Crash Simulator" (Frequency Response)

The system runs a simulation: "If Guest X leaves while the weather is Y, how much does the frequency drop?"

The author uses two different simulators: a quick, rough math formula (like a back-of-the-napkin calculation) and a detailed physics engine (like a high-end video game). They weigh them together to get the most accurate prediction.

4. The "Safety Net" (Controls)

The grid has safety nets:

Dynamic Containment (DC): Like a parachute that opens instantly to slow the car down.
LFDD (Low-Frequency Demand Disconnection): Like a circuit breaker that cuts off power to some houses to save the whole grid (similar to a fire door closing to stop smoke).
The Paper's Superpower: It can calculate exactly how much risk is removed by these safety nets. It can tell the grid operator: "If we buy 100 more MW of parachutes, we cut the crash risk by 77%."

The Results: What Did We Learn?

The author tested this new method on the Great Britain power system and compared it to the official government report.

It Matches Reality: The new method agreed with the official government report within a factor of 1.5. This is a huge success because the two methods were built completely differently.
It Finds the "Hidden" Risks: The old methods often looked at "average" conditions. This new method zooms in on the rare, scary days (low inertia, high wind) where the real danger lies. It found that 60% of the risk comes from less than 25% of the time.
It Breaks Down the Blame: It can tell you exactly who is causing the risk. For example, at the deepest danger levels, it's not just one wind farm; it's often two big things failing at the exact same time (like the 2019 event where a wind farm and a gas turbine failed together).

Why Does This Matter?

Think of this paper as upgrading the power grid's risk management from a crystal ball (guessing based on feelings) to a weather forecast (based on data, physics, and probability).

For the Grid Operator: It tells them exactly how much "safety gear" (parachutes/DC) they need to buy to keep the lights on.
For the Public: It means the grid is being managed with more precision, reducing the chance of blackouts as we switch to green energy.
For the World: It's the first time this specific "Earthquake Math" has been applied to power grids, setting a new standard for how we measure safety in a changing world.

In short: The author took the most rigorous math used for earthquakes, adapted it for electricity, and proved that we can now predict power grid crashes with much higher accuracy, helping us navigate the transition to renewable energy safely.

1. Problem Statement

The transition to renewable energy in power systems (specifically Great Britain) is leading to a decline in synchronous inertia. This reduction increases the system's sensitivity to generation and interconnector disturbances, resulting in higher Rates of Change of Frequency (RoCoF) and deeper frequency nadirs.

Current Limitations: Existing risk assessment methods, such as the National Energy System Operator's (NESO) Frequency Risk and Control Report (FRCR), rely on deterministic settlement-period enumeration. While valuable, these methods lack:
- Formal quantification of uncertainty (both aleatory and epistemic).
- Continuous hazard curves (risk as a function of threshold).
- Source-level disaggregation (identifying specific contributors to risk).
- Systematic quantification of risk reduction from specific control measures.
The Gap: There is a need for a probabilistic framework that can rigorously quantify the annual exceedance rate of frequency thresholds, account for complex system dynamics, and provide actionable insights for risk management.

2. Methodology: Probabilistic Frequency Hazard Analysis (PFHA)

The paper introduces PFHA, a framework that adapts the mathematical architecture of Probabilistic Seismic Hazard Analysis (PSHA)—the standard in earthquake engineering—to power system frequency risk.

A. The Hazard Integral

The core of PFHA is a hazard integral derived from the total probability theorem, calculating the annual rate ( $\lambda$ ) of frequency deviations exceeding a threshold ( $\delta$ ):
$\lambda(\Delta f > \delta) = \sum_{i=1}^{N_{src}} \nu_i \int_{\Delta P} \int_{s} P(\Delta f > \delta | \Delta P, s) \cdot f_i(\Delta P) \cdot g(s) \, d\Delta P \, ds$
Where:

$\nu_i$ : Annual trip rate of source $i$ (estimated via Bayesian methods).
$f_i(\Delta P)$ : Empirical probability mass function (PMF) of loss size for source $i$ .
$g(s)$ : Joint probability density of the system state vector $s = (H, D, R)$ , representing Inertia, Demand, and Frequency Response holdings.
$P(\Delta f > \delta | \Delta P, s)$ : Conditional exceedance probability provided by a Frequency Response Prediction Equation (FRPE), modeled using a log-normal distribution.

B. Key Components

Source Catalogue (51 Sources): Includes individual generators (CCGTs, Nuclear, Wind, etc.) and interconnectors.
- Loss Distributions: Constructed empirically from 24 months of settlement-period generation data (B1610), capturing real operational profiles (e.g., bimodal interconnector flows).
- Occurrence Rates: Estimated using a Bayesian Gamma-Poisson conjugate model to avoid zero-rate issues for sources with no observed trips in the historical window.
- Multi-Layer Events: Includes Layer A (single trips), Layer B (simultaneous pairs of large sources), and Layer C (compound cascades triggered by high RoCoF disconnecting Distributed Energy Resources).
Frequency Response Prediction Equation (FRPE): Analogous to the Ground Motion Prediction Equation (GMPE) in seismology.
- Dual Architecture: A logic tree combines an Analytical SFR model (40% weight, requires bias correction) and a Physics-based lookup model (60% weight, no bias correction).
- Aleatory Variability ( $\sigma$ ): Calibrated from residuals of 283 historical event replays.
System State Integration: Uses 71,476 empirical half-hourly settlement periods (2022–2026) to create a non-parametric joint distribution of Inertia, Demand, and Response. This captures the correlation between low inertia and high renewable output.
Control Modeling: Explicitly models Dynamic Containment (DC) and Low-Frequency Demand Disconnection (LFDD). LFDD is modeled with a "strict inequality" logic: a stage at 48.8 Hz cannot prevent a breach at 48.8 Hz, only deeper thresholds.
Uncertainty Quantification: A 324-path logic tree (6 branches) captures epistemic uncertainty regarding model choices (FRPE type, bias correction, occurrence models, control effectiveness).

3. Key Contributions

First PSHA-style Formulation for Frequency: This is the first published work to explicitly formulate a PSHA-style hazard integral for bulk power-system frequency exceedance.
Source-Level Disaggregation: The framework decomposes risk by source, loss magnitude, system state, and "epsilon" (standard deviations above the median), identifying exactly which scenarios drive risk at different severity levels.
Marginal Risk Reduction Quantification: Unlike existing methods, PFHA can isolate the specific risk reduction contribution of individual controls (e.g., DC vs. LFDD) by running the hazard integral under different control configurations.
Bayesian Rate Estimation: Solves the "zero-rate" problem for rare events by using technology-class priors, ensuring all credible sources contribute to the hazard even without historical trip data.
Dual-Model Architecture: Combines analytical speed with physics-based accuracy, weighted to manage epistemic uncertainty.

4. Results and Validation

The framework was applied to the Great Britain power system using four years of operational data.

Cross-Validation with FRCR:
- At the 49.2 Hz threshold, PFHA results agree with the independent FRCR within a factor of 1.54 (0.213/yr vs. 0.138/yr).
- At 48.8 Hz, PFHA estimates are lower (0.020/yr vs. 0.039/yr), attributed to PFHA's continuous exceedance probability vs. FRCR's binary breach comparison.
Risk Reduction Analysis:
- Combined controls (DC + LFDD) reduce the exceedance rate by 77% at 49.2 Hz and 92% at 48.8 Hz.
- LFDD provides the majority of risk reduction at deep thresholds, while DC provides uniform reduction across all thresholds.
Disaggregation Insights:
- State Dependence: Low-inertia periods ( $H < 140$ GVA⋅s), representing <25% of operational time, contribute >60% of the hazard at 49.2 Hz.
- Dominant Scenarios: At 49.2 Hz, the dominant risk comes from ~1200 MW losses (e.g., IFA interconnector or Nuclear) occurring during low-inertia periods with moderate demand.
Validation against History:
- The model correctly identifies the 9 August 2019 event as the only historical breach of 48.8 Hz.
- Temporal split testing (training on 2014–2024, testing on 2025) showed stable results with <20% variance, confirming no overfitting.

5. Significance and Conclusion

Methodological Bridge: PFHA successfully bridges geophysical hazard analysis and power system reliability, offering a rigorous, auditable, and extensible framework.
Operational Utility: The ability to generate continuous hazard curves and decompose risk by source and control measure provides system operators with actionable data for:
- Setting minimum inertia constraints.
- Optimizing frequency response procurement (e.g., determining the marginal value of additional Dynamic Containment).
- Validating the effectiveness of protection schemes like LFDD.
Future Applicability: The framework is not limited to GB; it can be adapted to other synchronous areas (e.g., EirGrid, Continental Europe, Australian NEM) by updating the source catalogue and state distributions.

In conclusion, this paper establishes a new standard for probabilistic frequency risk assessment, moving beyond deterministic snapshots to a comprehensive, uncertainty-aware view of system stability in the energy transition era.