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Comparative Analysis on Inertia Estimation Algorithms (IEAs) in Providing Proper Frequency Response
Karl M. H. Lai, Yunhe Hou, Kwunhang Wong
TL;DR
Declining and uncertain inertia from IBRs and HVDC-connected systems makes accurate real-time estimation important, but noisy measurements and parameter changes complicate the task. The paper comparatively evaluates measurement-, model-, and data-based estimators and examines their use in wind-based inertia emulation. SR-KF and RLS provide the most accurate and fastest estimates for frequency-control activation, while smoother alternatives better suit monitoring and dispatch.
Problem
Real-time inertia estimation is needed for modern frequency control, but noisy frequency measurements, event-driven parameter jumps, and locational transient responses make it challenging.
Method
The paper compares measurement-based methods, Kalman-filter model-based methods, and LMS/RLS data-based parameter-estimation techniques, including their use in wind-generation inertia extraction.
Results
SR-KF and RLS provide the most accurate and speedy inertia estimates for frequency-control and activation purposes, while UKF, MMAE, polynomial fitting, or small-step LMS better suit smoother monitoring and dispatch.
Takeaways & Limitations
Estimator choice should follow the application: prioritize speed and accuracy for control activation, or smoother estimates for monitoring and dispatch.
Takeaways & Limitations
Performance depends on persistent excitation, and the studies lack outlier or faulty data while often assuming linear frequency-control relationships.
Abstract
from arXiv · showhide
The inertia constant H[s] is a fundamental indicator of power system resilience, linking power imbalance between generation and load to frequency deviation. It is essential in frequency reserve dispatch under stability-constrained optimal power flow (OPF), demand response (DR) in ancillary service, system decoupling and frequency control in modern power system. While the inertia constant is traditionally defined as the intrinsic kinetic energy of synchronous generators on bar normalized to the power base, this neglects the releasable power under nonlinear dynamics and control inside HVDC and Inverter-based Resources (IBRs). Accurate real-time inertia estimation is therefore essential to perform proper frequency control and to indicate the risks of failure in frequency restoration. It, however, is challenging with noisy frequency measurement under event-driven parameter jumps and locational transient responses. This paper presents a systematic comparative analysis on inertia estimation algorithms (IEAs) for frequency response applications. Classical methods such as filtering and fitting under measurement-based methods are benchmarked against data-based parameter estimation techniques such as recursive least squares (RLS) and model-based methods such as Kalman filtering (KF). The main contributions are: (i) a holistic review of model- and data-based inertia estimation methods, (ii) exploration on the effect of IEA to wind-based inertia emulation strategies. The findings underscore the need for robust, adaptive, and data-driven estimation frameworks to ensure secure operation of future low-inertia grids.
1. Introduction – Frequency Disturbance and Inertia in Modern Power Systems
Modern grids face greater frequency risk as inverter-based resources and HVDC links reduce dependable inertia, making real-time estimation important for monitoring, dispatch, and frequency control. The paper compares measurement-, model-, and data-based inertia estimation methods against challenges from noisy, transient, and locational measurements.
- Inverter-based resources may not provide inertia, while virtual inertia, droop, and battery-based responses remain constrained by intermittent input or limited state of charge.
- Reduced inertia associated with IBRs and HVDC imports can increase ROCOF, lower frequency nadirs, and raise cascade-tripping and UFLS-failure concerns.Accurate ROCOF measurement with a PMU requires at least 2.5 cycles.
- Real-time inertia monitoring supports stability-constrained dispatch, demand-response participation, protection coordination, and decentralized frequency-response control.Precise estimates are also needed for inertia extraction from IBR plants under varying energy availability.
- Inertia estimation remains difficult because monitored quantities vary among inertia constant, available energy, and ROCOF, while measurement-based workflows require frequency preprocessing and event identification.Additional challenges include real-time power-imbalance calculation, ROCOF denoising, and locational measurement effects.
- The paper evaluates measurement-based fitting and filtering, stochastic model-based Kalman filtering, and data-based LMS and RLS approaches using the swing equation.These methodologies span generator-parameter summation and event-driven analyses under high renewable penetration.
2. Inertia Estimation – Measurement-based Approach
Measurement-based inertia estimation applies filtering, fitting, and event-driven calculations to infer inertia from frequency and power deviations, but its accuracy depends strongly on disturbance size, noise handling, and load effects. Comparative simulations show difficulty tracking time-varying inertia, especially near zero ROCOF and sign-changing load steps.
- Earlier methods: The classical Inoue approach approximates ROCOF with a polynomial linear term while treating nonlinear local terms as relatively small.It was developed to suppress noise in average-frequency measurement and differentiation for small, noisy signals.
- Event-driven estimation: Only disturbances large enough to exceed the CSUM threshold produced a true H estimate in the sliding-window approach.The method improves numerical stability, but its event threshold and window-related settings depend on system experience, noise level, and disturbance size.
- Load-effect compensation: 5.48% mean error with 46.60% variance was reported for the R-, V-, and RV-methods under generator-loss disturbances in the Nordic system.These methods add load-effect compensation for frequency and voltage deviations and use polynomial fitting with Butterworth low-pass filtering.
- Simulation and preprocessing: The comparison evaluates exponential, polynomial-fit, Savitzky-Golay, Butterworth, median, and averaged filtering strategies under random load steps and added Gaussian frequency noise.The simulation uses load-step variance 0.008 pu every 2 seconds, changing inertia values, and frequency-measurement noise with standard deviation 0.001.
- Measurement-based performance: Measurement-based estimation does not track varying H well because differentiator noise is amplified by the 1/ROCOF calculation, especially near zero ROCOF.Using absolute power deviation and ROCOF, retaining the previous value, or returning to a default value can avoid some unstable conditions.
3. Inertia Estimation – Model-based Approach
Model-based inertia estimation avoids direct ROCOF computation, improving stability under small disturbances while using Kalman-filter variants to address noisy measurements and parameter jumps.
- Model-based estimation: Kalman-filter methods avoid ROCOF in inertia computation, producing more stable estimates under small ROCOF values caused by negligible load steps.The filters model process and measurement noise to accommodate parameter jumps and frequency-measurement errors.
- SRKF and Information Filter: Square-root Kalman filtering propagates covariance through Cholesky factorization, improving numerical stability and reducing round-off error in ill-conditioned estimation.For inertia estimation, SRKF also handles small frequency and power-deviation signals without numerical instability.
- SRKF and Information Filter: SRKF achieved lower MSE than the Information Filter, 0.015 versus 0.018, while IF had lower bias and a mean estimate of 2.366 closer to the true value.SRKF initialization produced poorer early estimates, whereas IF was more sensitive to measurement noise; both methods had the same clipped maximum error.
- Adaptive UKF: Adaptive UKF enlarges error and measurement covariances to capture parameter jumps, while Joseph-form updates prevent covariance shrinkage from causing non-positive-definite matrices.UKF performance depends on covariance tuning and the spread of sigma points.
- MMAE: MMAE averages estimates from fixed-parameter UKF models using model probabilities, improving initialization robustness and interpretability at higher computational cost.Its fixed-state observer logic can limit performance when actual dynamics vary, and adaptive enhancement is suggested.
- Comparative performance: AD-UKF and MMAE performed worse than standard KF methods in the linear or mildly linear test, with UKF MSE about 0.08 versus 0.02 for polynomial fitting.AD-UKF was more precise near the true initialization, whereas MMAE was more robust to poor initialization; SRKF and IF were more reliable overall.
4. Inertia Estimation – Parameter Estimation Approach (LMS vs RLS)
The section compares LMS and RLS parameter-estimation approaches for inertia, finding that RLS tracks abrupt parameter changes more accurately while both methods require excitation and introduce estimation delay.
- LMS behavior: LMS performs better when parameters vary slowly, whereas its step size controls movement toward the solution and smaller values slow convergence.Underfitting can avoid collinear measurements when damping is insignificant and the measurement matrix is rank-deficient.
- RLS behavior: RLS tracks abruptly changing parameters faster than LMS, but its forgetting factor trades tracking speed against peaky fluctuations and numerical stability.Persistent excitation is also required for RLS to track the parameter vector properly.
- LMS vs RLS: RLS achieves lower MSE and variation than LMS, with a much smaller maximum error during parameter jumps.LMS MSE is 0.042–0.070 with variation ±0.030–0.035, versus RLS MSE 0.009–0.012 with variation ±0.003–0.004; RLS maximum error is approximately 0.39.
- Operational constraints: Both LMS and RLS require sufficient frequency-changing samples, so their estimates are delayed by learning and forgetting dynamics.The delay can affect the timing of frequency-control activation.
- Comparative performance: RLS outperforms measurement-based, ADUKF, and MMAE alternatives overall, while SR-KF and IF are robust but slightly less accurate.RLS requires tuning of forgetting factors and initialization to avoid instability or peaky measurements.
- Operational constraints: Without persistent excitation from power and frequency deviation, all evaluated algorithms—including RLS and SR-KF—perform poorly.The section recommends coordinating estimation timing with control activation and avoiding controllers that remove small ripples until frequency change is sufficiently large.
5. Effect of Inertia Estimation in Inertia Extraction with Wind Generation - MPC
This section applies inertia estimation to wind-generation inertia extraction using MPC. Gain-scheduled QP-MPC updates its model with RLS estimates, and the updated controller achieves lower cost under parameter mismatch while preserving frequency support.
- Operational constraint: Inertia estimation and MPC activation face a structural trade-off because effective MPC can remove the deviations needed to excite the estimators.The proposed operating condition keeps the power setpoint constant under MPP or VSG operation while deviations originate from the external grid.
- GS-QP-MPC design: GS-QP-MPC uses RLS-estimated inertia to select an updated MPC evolution model instead of relying on a fixed inertia value.The estimator operates while MPC is inactive, and MPC activates when frequency, ROCOF, or power-deviation thresholds are exceeded.
- Simulation setup: The simulation creates persistent excitation with 0.005 pu load fluctuations and introduces an inertia jump at 50 s when a high-inertia, low-load generator trips.The aggregated wind farm has 746 MW output, and the simulation uses an infinite sink that does not change inertia during wind-generator control activation.
- Control results: Both QP-MPC and GS-QP-MPC produce similar power outputs and secure the frequency change during the wind-farm event.The controllers differ more clearly in torque-setpoint behavior than in aggregate power output.
- Parameter mismatch: Using an incorrect inertia of 2.8 s causes QP-MPC to oscillate slightly and initially release less output than GS-QP-MPC using the updated estimate near 2.5 s.Under the true inertia range [2.5, 2.3] s, the mismatched controller later requires substantially more release after recognizing larger ROCOF.
- Control results: GS-QP-MPC returns consistently lower cost than fixed-parameter QP-MPC when the true inertia changes from 2.5 to 2.3 s.Both costs decrease monotonically over time, while the fixed model remains suboptimal under parameter mismatch.
6. Conclusion
The paper compares measurement-, model-, and data-based inertia estimators and finds SR-KF and RLS most accurate and fast for frequency-control applications. It also introduces GS-QP-MPC and identifies persistent-excitation, tuning, data-quality, and linearity limits.
- Estimator conclusions: SR-KF and RLS provide the most accurate and fastest inertia estimates without clipping, smoothing, or other numerical-stability techniques.For monitoring and dispatch, slower but smoother UKF, MMAE, polynomial-fitting, or small-step LMS alternatives may be preferable.
- Application contribution: The paper introduces GS-QP-MPC to update model parameters, improve objective-function performance, and avoid oscillations in wind-turbine torque input.Its control application requires a deadband so the estimator can operate.
- Limitations and future work: Future work should develop tune-free estimators, evaluate outlier and faulty data, combine estimates across locations, and address nonlinear large-signal frequency control.The paper also calls for improved secondary power-frequency control setpoints.