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Online Estimation of Power System Inertia Using Dynamic Regressor Extension and Mixing

Johannes Schiffer, Petros Aristidou, Romeo Ortega

arXiv:1812.00201v2eess.SYcs.SY

TL;DR

As power-electronic-interfaced generation reduces inertia and can accelerate frequency dynamics, real-time system inertia monitoring is important for power-system operation. The paper develops a DREM-based online estimator and tests it on an ENTSO-E system; across 25 scenarios, 21 estimates are within 15% of the COI inertia constant, and the estimated-inertia system response differs by only a few mHz.

  • Problem

    Replacing synchronous generation with power-electronic-interfaced devices decreases available inertia and can lead to faster frequency dynamics that stress protection and control schemes.

    Source: A. Motivation and Existing Literature · p. 1

  • Method

    The estimator combines a nonlinear aggregated power-system model with DREM to estimate inertia and aggregated mechanical power setpoint from PFC-generator measurements.

    Source: B. Contributions · p. 2, A. Center of Inertia Frequency Dynamics · p. 2

  • Results

    In 25 test scenarios, 21 estimates had error below 15% versus the COI inertia constant, and all estimated-inertia system responses matched simulations within a few mHz.

    Source: §VI · p. 8

  • Takeaways & Limitations

    The estimator is also applicable during rescheduling events, which occur frequently in deregulated power systems.

    Source: §VI · p. 8

    Accurate estimation requires sufficiently large system excitation; the evaluated scenarios comprise 25 generator outages and a rescheduling event.

    Source: B. Contributions · p. 2

Abstract

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The increasing penetration of power-electronic-interfaced devices is expected to have a significant effect on the overall system inertia and a crucial impact on the system dynamics. In the future, the reduction of inertia will have drastic consequences on protection and real-time control and will play a crucial role in the system operation. Therefore, in a highly deregulated and uncertain environment, it is necessary for Transmission System Operators to be able to monitor the system inertia in real time. We address this problem by developing and validating an online inertia estimation algorithm. The estimator is derived using the recently proposed dynamic regressor extension and mixing procedure. The performance of the estimator is demonstrated via several test cases using the 1013-machine ENTSO-E dynamic model.

I. INTRODUCTION

Lower inertia and faster frequency dynamics increase pressure on protection and control, while existing estimation approaches often rely on offline data or restrictive event conditions. The paper proposes a DREM-based online estimator using an aggregated system model and evaluates it on a large ENTSO-E system.

A. Motivation and Existing Literature

  • Replacing synchronous generators with power-electronic-interfaced devices decreases available inertia and can produce faster frequency dynamics that stress protection and control schemes.The paper notes that these dynamics may also endanger legacy components.Source: A. Motivation and Existing Literature · p. 1
  • Many existing estimators use offline data; online alternatives may require generator frequency derivatives, precise disturbance timing, probing signals, or broad generator monitoring.These constraints motivate an approach suited to real-time use with limited measurements.Source: A. Motivation and Existing Literature · p. 1

B. Contributions

  • The proposed DREM-based algorithm estimates system inertia and aggregated mechanical power setpoint using a first-order nonlinear aggregated model.It is demonstrated on a 1013-machine ENTSO-E system; the proposed method uses frequency and electrical-power measurements from PFC generators, a subset of system machines.Source: B. Contributions · p. 2

A. Center of Inertia Frequency Dynamics

  • The paper represents primary-controlled system frequency dynamics with a reduced-order model centered on the center-of-inertia speed and aggregated system quantities.The model defines the total inertia from generator inertia constants and power ratings and describes the principal frequency dynamics with an aggregated swing equation.Source: A. Center of Inertia Frequency Dynamics · p. 2

B. Problem Formulation

The formulation approximates center-of-inertia frequency dynamics using measurements available from primary-frequency-controlled generators. It targets total system inertia and their aggregated mechanical power setpoint, including when frequency changes reflect rescheduling or outages.

Problem formulation

  • The formulation approximates center-of-inertia frequency by average PFC frequency and expresses the power balance using PFC-unit measurements.The paper assumes average PFC frequency, total PFC mechanical power, and PFC electrical power are measurable.Source: B. Problem Formulation · pp. 2–3
  • The estimation problem is to identify total inertia Htot and aggregated PFC mechanical power Pm,PFC from average frequency, total PFC mechanical power, and PFC electrical power.Htot includes both PFC and non-PFC units, so the target is system-wide despite partial system measurements.Source: B. Problem Formulation · p. 3
  • Estimating aggregated PFC mechanical power is useful when frequency varies because that power changes, including during rescheduling or a PFC-generator outage.The algorithm can also be run offline with time-stamped data, but then estimation is delayed rather than real time.Source: B. Problem Formulation · p. 3

A. System Parametrization

The system model is recast into a standard dynamic-parameter-estimation form, then filtered and rewritten as a regression for estimator development.

A. System Parametrization

  • The system model is brought into standard form for dynamic parameter estimation, and that parametrization is used to construct the regression and develop the estimator.Source: A. System Parametrization · p. 3

B. Regression Construction

  • Applying the operator αp/(p + α), with α > 0, to the model yields a filtered relation with an exponentially decaying term from filter initial conditions.Source: B. Regression Construction · p. 3
  • The variables ξ1 and ξ2 are introduced to rewrite the filtered relation compactly as a regression.Source: B. Regression Construction · p. 3

C. DREM Parameter Estimator

The estimator applies DREM to a one-dimensional regression with two unknown parameters, producing decoupled scalar regressions for gradient-based estimation. The filtering operator and convergence condition shape how the estimator uses the available excitation.

C. DREM Parameter Estimator

  • DREM extends the one-dimensional regression with two unknown parameters using a linear, L∞-stable operator to construct additional regression information.Source: C. DREM Parameter Estimator · p. 3
  • The operator may be an exponentially stable LTI filter or a delay; the paper selects the delay operator based on its reported success in power-engineering applications.The delay magnitude can shape the determinant signal ∆ and is selected with transient performance in mind.Source: C. DREM Parameter Estimator · pp. 3–4
  • Piling the original and filtered regressions yields an extended system, whose premultiplication by the adjugate matrix produces scalar regression equations.Source: C. DREM Parameter Estimator · p. 4
  • The decoupled regressions feed a gradient algorithm that estimates the parameter vector using positive tuning gains.The paper derives error dynamics for this estimator and neglects exponentially decaying filter-initialization terms in subsequent analysis.Source: C. DREM Parameter Estimator · p. 4
  • DREM requires the determinant of the extended regressor matrix not to be square integrable, unlike conventional persistency of excitation, which is a property of the regressor.Source: C. DREM Parameter Estimator · p. 4

IV. NUMERICAL VERIFICATION ON 1013-MACHINE ENTSO-E SYSTEM: NOMINAL TEST CASE

The evaluation uses the 1013-machine ENTSO-E system and first checks whether an aggregated model captures its main frequency dynamics and PFC injection. The DREM estimator is then used to identify overall inertia.

IV. Numerical Verification on 1013-Machine ENTSO-E System: Nominal Test Case

  • The ENTSO-E test system has 21,382 buses, 1,013 synchronous generators, and 133,997 differential-algebraic states.PFC units are connected at 871 buses; the remaining generators have constant active-power setpoints.Source: §IV · p. 4
  • The evaluation first tests whether model (6) captures the system’s main frequency dynamics and whether an aggregated turbine-governor model approximates total PFC injection.Source: §IV · p. 4
  • After validating the models, the study uses DREM estimator (18) to identify overall system inertia from RAMSES time-domain simulations.Source: §IV · p. 4

A. Aggregated

The study validates an aggregated model against the full ENTSO-E model, then applies the DREM estimator to estimate inertia in a nominal test scenario. The reported final estimation error is below 7%, with a lower error when measured PFC injection is supplied directly at the cost of more measurements.

A. Aggregated

  • Model (21) is considered sufficiently accurate for PFC effects when steam power plants predominate; substantial hydro or gas participation calls for modifying the model.The extension for those dynamics is left for future research.Source: A. Aggregated · p. 5

B. Validation of Aggregated Model

  • The aggregated reduced-order model approximates the full-order ENTSO-E model well, including total PFC injection, though frequency nadir differs slightly.The discrepancy is attributed to non-PFC units not being explicitly represented in model (6), (21).Source: B. Validation of Aggregated Model · p. 5

C. Online-Inertia Estimation: Nominal Test Scenario

  • The estimated parameters converge to constant values after initial transients, and the final estimation error is below 7%.The reported behavior was similar across initial estimates scaled by α_i ∈ [0, 30].Source: C. Online-Inertia Estimation: Nominal… · p. 5
  • The final estimation error is below 7% with aggregated-model inputs, versus a stationary error of 1% when measured PFC injection is supplied directly.The model-based approach reduces the number of required measurements, while the use of direct PFC-injection measurements requires more.Source: C. Online-Inertia Estimation: Nominal… · pp. 5–6
  • The delay d shapes the truncated L2-norm of the regressor, which rises after disturbance and settles as frequency and PFC-injection transients decay.The authors report best results with d ∈ [1, 8] s; estimator gains trade off convergence speed against noise sensitivity.Source: C. Online-Inertia Estimation: Nominal… · p. 6

V. FURTHER TEST CASES: GENERATOR OUTAGE AND RESCHEDULING

Further test scenarios use the same ENTSO-E system, simulation software, and estimator gains to assess performance and accuracy across operating scenarios.

V. Further Test Cases: Generator Outage and Rescheduling

  • The further tests retain the ENTSO-E system, simulation software, and estimator gains to assess whether one tuning works across diverse operating scenarios.Source: §V · p. 6

A. Online-Inertia Estimation: Further Generator Outages

Across additional generator outages, estimation error was 15% or lower in most scenarios. In a representative high-error case, the estimated inertia produced a frequency response closer to the full-system simulation than the nominal inertia did, at least with model (6), (21).

A. Online-Inertia Estimation: Further Generator Outages

  • In the ES917736 outage, stationary estimation error was 7%, falling to 2% when total PFC injections came directly from RAMSES rather than model (21).Source: A. Online-Inertia Estimation: Further… · p. 6
  • The 23 additional randomly selected outages spanned eight countries and 500–1200 MW of lost generation; each disconnected generator’s inertia was removed from the final total.Consequently, final total system inertia differed across outage scenarios.Source: A. Online-Inertia Estimation: Further… · p. 6
  • 21 of 25 outage scenarios had estimation error at or below 15%, with some below 1%; four exceeded 15% for disturbances above 800 MW in Germany, France, or Italy.The results suggest neither outage location nor size alone is decisive for estimator performance.Source: A. Online-Inertia Estimation: Further… · p. 6, Figure 5 · p. 7
  • With 27.5% relative estimation error in η1 for the DE912342 outage, estimated-inertia frequency closely resembled RAMSES, while nominal-inertia frequency showed larger discrepancies.This may indicate that, at least with model (6), (21), estimated inertia better represents the true response and effective inertia than nominal inertia.Source: Figure 6 · p. 7, A. Online-Inertia Estimation: Further… · p. 7

B. Online-Inertia Estimation: Rescheduling Events

The authors examine rescheduling as a frequent, scheduled source of frequency variation for inertia estimation and report error below 10% in this scenario. They also note that rescheduling changes generator injections beyond the modeled dynamics.

B. Online-Inertia Estimation: Rescheduling Events

  • Rescheduling events are frequent, recurring, and scheduled, while usual operating frequency variation lacks sufficient excitation and outages are infrequent and unscheduled.The authors identify rescheduling as another useful source of frequency variation for inertia estimation.Source: B. Online-Inertia Estimation:… · p. 7
  • The example uses a frequency trajectory reproduced in RAMSES with scheduled generator and load setpoint changes, then feeds frequency and total PFC power to estimator (18).Fig. 8 depicts a typical system-frequency evolution during rescheduling.Source: B. Online-Inertia Estimation:… · p. 8, Figure 8 · p. 8
  • The rescheduling scenario’s maximum average relative estimation error is below 10%, and the authors conclude it provides useful data for the DREM-based estimator.The parameter estimates converge to a band around the nominal value of 1.Source: B. Online-Inertia Estimation:… · p. 8
  • Generator power injections in this scenario also reflect rescheduling sequences, so using model (21) offers no significant advantage and is omitted.The rescheduling-related PFC power variation is denoted Pres,PFC.Source: B. Online-Inertia Estimation:… · p. 8

VI. CONCLUSIONS

The paper presents a real-time inertia estimator based on an aggregated power system model and DREM. Across 25 test scenarios, it reports estimation and frequency-response results, and identifies measurement resolution and noise as topics for further investigation.

VI. CONCLUSIONS

  • The real-time algorithm estimates a large-scale system’s inertia constant and aggregated mechanical power setpoint of PFC generators.It uses a first-order nonlinear aggregated power system model with the DREM procedure.Source: §VI · p. 8
  • Across 25 test scenarios, 21 have estimation error below 15% against COI inertia, while estimated-inertia system responses match simulations with errors of only a few mHz.The estimator is also applicable to rescheduling events, which the authors say occur numerous times daily in deregulated power systems.Source: §VI · p. 8
  • Future work will explore applications in power system protection and real-time control and investigate how measurement resolution and noise affect estimation performance.Source: §VI · p. 8
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