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Cubature Kalman Filter as a Robust State Estimator Against Model Uncertainty and Cyber Attacks in Power Systems

Tohid Kargar Tasooji, Sakineh Khodadadi

TL;DR

This paper investigates robust nonlinear state estimation in power systems subject to model uncertainty and cyber attacks, comparing the cubature Kalman filter (CKF) against the extended Kalman filter (EKF) and its square-root variant. It demonstrates that CKF provides improved estimation accuracy over EKF in certain regimes and reliably supports attack detection across random, DoS, replay, and false data injection attacks, using a fourth-order synchronous machine as a case study. Two detectors, a χ²-based residue detector and a Euclidean detector for FDI, are analyzed for their effectiveness under these adversarial conditions. The findings suggest CKF as a robust estimator for cyber-physical power systems, with future work targeting event-triggered CKF implementations to enhance efficiency and resilience.

Abstract

It is known that the conventional estimators such as extended Kalman filter (EKF) and unscented Kalman filter (UKF) may provide favorable performance; However, they may not guarantee the robustness against model uncertainty and cyber attacks. In this paper, we compare the performance of cubature Kalman filter (CKF) to the conventional nonlinear estimator, the EKF, under the affect of model uncertainty and cyber-attack. We show that the CKF has better estimation accuracy than the EKF under some conditions. In order to verify our claim, we have tested the performance various nonlinear estimators on the single machine infinite-bus (SMIB) system under different scenarios. We show that (1) the CKF provides better estimation results than the EKF; (2) the CKF is able to detect different types of cyber attacks reliably which is superior to the EKF.

Cubature Kalman Filter as a Robust State Estimator Against Model Uncertainty and Cyber Attacks in Power Systems

TL;DR

This paper investigates robust nonlinear state estimation in power systems subject to model uncertainty and cyber attacks, comparing the cubature Kalman filter (CKF) against the extended Kalman filter (EKF) and its square-root variant. It demonstrates that CKF provides improved estimation accuracy over EKF in certain regimes and reliably supports attack detection across random, DoS, replay, and false data injection attacks, using a fourth-order synchronous machine as a case study. Two detectors, a χ²-based residue detector and a Euclidean detector for FDI, are analyzed for their effectiveness under these adversarial conditions. The findings suggest CKF as a robust estimator for cyber-physical power systems, with future work targeting event-triggered CKF implementations to enhance efficiency and resilience.

Abstract

It is known that the conventional estimators such as extended Kalman filter (EKF) and unscented Kalman filter (UKF) may provide favorable performance; However, they may not guarantee the robustness against model uncertainty and cyber attacks. In this paper, we compare the performance of cubature Kalman filter (CKF) to the conventional nonlinear estimator, the EKF, under the affect of model uncertainty and cyber-attack. We show that the CKF has better estimation accuracy than the EKF under some conditions. In order to verify our claim, we have tested the performance various nonlinear estimators on the single machine infinite-bus (SMIB) system under different scenarios. We show that (1) the CKF provides better estimation results than the EKF; (2) the CKF is able to detect different types of cyber attacks reliably which is superior to the EKF.

Paper Structure

This paper contains 15 sections, 6 theorems, 45 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Nonlinear State Estimators
  3. EXTENDED KALMAN FILTER
  4. CUBATURE KALMAN FILTER
  5. SQUARE-ROOT CUBATURE KALMAN FILTER
  6. Model Uncertainty
  7. Attack Model
  8. ATTACK/FAILURE DETECTOR
  9. Case Study
  10. Estimator's performance under normal conditions
  11. Estimator's Performance with Noisy Measurements
  12. Estimator's performance under model uncertainty
  13. Estimator's Performance under Cyber Attacks
  14. Cyber-attack detection
  15. Conclusion

Key Result

Lemma 3.1

Consider the stochastic nonlinear system with parameter uncertainties given by eq:equation8 and let Assumption Ass:1 and Assumption Ass:2 be satisfied. Then the upper bound for the estimation error in the CKF, $e_{k|k}^c$, satisfies the following bound: provided that the initial estimation error satisfies $\left\|e_{0|0}\right\| \le \epsilon$, and $\xi^{c}$ is sufficiently small. Note that $\xi^{

Figures (13)

  • Figure 1: Diagram of nonlinear state estimator for a synchronous machine.
  • Figure 2: Block diagram of estimation and detection mechanism for the synchronous machine.
  • Figure 3: State estimation results under normal conditions.
  • Figure 4: State estimation results with noisy measurements.
  • Figure 5: State estimation results under model uncertainty.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 4 more