[Paper Review] Inertia Estimation Through Covariance Matrix

Federico Bizzarri, Davide del Giudice+4 more|arXiv (Cornell University)|Jun 28, 2022

Power System Optimization and Stability4 citations

TL;DR

This paper proposes an online inertia estimation technique for power systems using ambient measurements and the covariance matrix of system variables. By formulating a least-squares optimization problem that fits a stochastic differential equation model to measured voltage magnitudes, the method accurately estimates both conventional and virtual inertia, as well as equivalent damping and droop coefficients, with errors below 5.42% in real-world systems.

ABSTRACT

This work presents a technique to estimate on-line the inertia of a power system based on ambient measurements. The proposed technique utilizes the covariance matrix of these measurements and solves an optimization problem that fits such measurements to the synchronous machine classical model. We show that the proposed technique is adequate to accurately estimate the actual inertia of synchronous machines and also the virtual inertia provided by the controllers of converter-interfaced generators that emulate the behavior of synchronous machines. We also show that the proposed approach is able to estimate the equivalent damping of the classical synchronous machine model. This feature is exploited to estimate the droop of grid-following converters, which has a similar effect of the swing equation equivalent damping. The technique is comprehensively tested on a modified version of the IEEE 39-bus system as well as on a dynamic 1479-bus model of the all-island Irish transmission system.

Motivation & Objective

• To develop an online, continuous inertia estimation method that does not require artificial disturbances or probing signals.
• To estimate both conventional inertia of synchronous machines and virtual inertia from converter-interfaced generators that emulate synchronous behavior.
• To simultaneously estimate equivalent damping and droop coefficients of grid-following converters using the same measurement framework.
• To validate the method on large-scale, real-world power systems with high penetration of renewables and dynamic components.
• To enable performance-based rewards for frequency support by quantifying the effective inertia contribution of non-synchronous resources.

Proposed method

• The method models the power system using stochastic differential equations derived from the classical swing equation, with load and wind fluctuations modeled as Ornstein-Uhlenbeck processes.
• It uses the covariance matrix of ambient measurements—specifically voltage magnitude at generator buses—to estimate system parameters via a least-squares optimization.
• The state-space model is derived from the power system's dynamic model, and the covariance matrix is computed from time-series measurements collected at 15-minute intervals.
• The optimization problem fits the model to measurements by minimizing the difference between theoretical and empirical covariance matrices.
• The method estimates inertia constants, equivalent damping, and droop coefficients by solving a linear system derived from the covariance structure of the state variables.
• Filtering and preprocessing of measurements are applied to enhance signal-to-noise ratio and improve estimation accuracy.

Experimental results

Research questions

• RQ1 Can the inertia of synchronous machines be accurately estimated using only ambient voltage magnitude measurements and no direct rotor speed or current data?
• RQ2 Can the proposed method estimate virtual inertia from converter-interfaced generators that emulate synchronous behavior?
• RQ3 To what extent can the method simultaneously estimate equivalent damping and droop coefficients of grid-following converters?
• RQ4 How accurate is the method in large, real-world power systems with hundreds of buses and dynamic components?
• RQ5 What is the impact of measurement filtering and sampling strategy on estimation accuracy in complex systems?

Key findings

• The proposed method achieved a relative percentage error in inertia estimation of 0.22% for generator G7 and up to 5.42% for G10 in the 1479-bus Irish transmission system.
• The method successfully estimated both conventional inertia and virtual inertia from converter-interfaced generators, including those with synthetic inertia control.
• The technique accurately estimated the equivalent damping and droop coefficients of grid-following converters, enabling performance-based assessment of their frequency support.
• The method demonstrated robustness in a real-world system with 1479 buses, 245 loads, 22 synchronous plants, and 169 wind farms, with normalized estimates closely centered around the true values.
• The results showed that the method is suitable for continuous, online estimation without requiring system disturbances or special signal injection.
• The method’s accuracy is maintained even when only voltage magnitude measurements are used, without requiring rotor speed or current measurements.