A Theoretical Framework of Robust H-Infinity Unscented Kalman Filter and Its Application to Power Sy

A Theoretical Framework of Robust H-Infinity Unscented Kalman Filter and Its Application to Power System Dynamic State Estimation

Abstract: This paper presents a new theoretical framework that, by integrating robust statistics and robust control theory, allows us to develop a robust dynamic state estimator of a cyber physical system. This state estimator combines the generalized maximum-likelihood-type (GM)-estimator, the unscented Kalman filter (UKF), and the H-infinity filter into a robust H-infinity UKF filter in the Krein space, which is able to handle large system uncertainties as well as suppress outliers while achieving a good statistical efficiency under Gaussian and non-Gaussian process and observation noises. Specifically, we first use the statistical linearization approach to build a linear-like regression model in the Krein space. Then, we show that the H-infinity UKF is just the Krein space Kalman filter that exhibits a bounded estimation error in presence of system uncertainties while minimizing the least squares criterion; consequently, it suffers from a lack of robustness to outliers and non-Gaussian noise. Because the GM-estimator is able to handle outliers, but it may yield large estimation errors in presence of system uncertainties, we propose to combine it with the H-infinity UKF in a robust H-infinity UKF. We carry out a theoretical analysis to demonstrate the connections that our filter has with the

Hinfinity UKF and the GM-UKF. The good performance of the new filter is demonstrated via extensive simulation performed on the IEEE 39-bus power system.

Existing system: It turns out that most of the existing DSEs assume an exact dynamical system model, accurate measurements, and known Gaussian system process and measurement noises. However, due to unknowable system inputs, including noise, parameter variations and actuator failures and inaccuracies of the model parameter values of the synchronous generators, the loads, the lines, and the transformers, to name a few, the system model is subject to uncertainties. Furthermore, the process and the observation noises of the system nonlinear dynamic models are nonGaussian as verified by the Pacific Northwest National Lab (PNNL). Finally, outliers can occur that may corrupt the state estimates. Two types of outliers are commonly seen in power systems, namely innovation and observation outliers.

Proposed system: To bound the influence of the model uncertainties and large system process and measurement noises, the first-order Taylor series approximations-based H-infinity extended Kalman filter and the unscented transformation-based H-infinity unscented Kalman filter (UKF) have been proposed. However, they are vulnerable to any types of outliers and exhibit large biases and variances in presence of nonGaussian noise. To suppress gross errors in the PMU measurements, several robust DSEs are developed. For instance, in , Rouhani and Abur propose a distributed two-stage robust UKF-based DSE using the least-absolute-value (LAV) estimator that can handle observation outliers. However, the authors do not address the vulnerability of the DSE to innovation outliers. To handle both types of outliers, a generalized maximum-likelihood type iterated EKF and UKF are developed .

Advantages: This was precisely the case in the 1996 blackout, where the model being used predicted system stability while in reality the system was undergoing numerous cascading failures, which resulted in a rapid system collapse that occurred within minutes.

Innovation outliers may also be induced by the approximations in the state prediction model or by process impulsive noise. To bound the influence of the model uncertainties and large system process and measurement noises, the firstorder Taylor series approximations-based H-infinity extended Kalman filter and the unscented transformation-based H-infinity unscented Kalman filter (UKF) have been proposed.

Disadvantages: A theoretical analysis is performed to demonstrate that, in the absence of nonGaussian noise and outliers, the proposed robust H-infinity UKF reduces to the Hinfinity UKF, on the other hand, if the tuning parameter of the H-infinity criterion tends to infinity, the proposed robust H-infinity UKF resembles the GM-UKF. This provides the justification why both the benefits of the H-infinity filter and the GM-estimator are maintained in the proposed robust H-infinity UKF. Modules: Contributions and Paper Organization: To address the aforementioned challenges, this paper presents a novel theoretical framework that integrates both robust statistics and robust control theory to develop a robust dynamic state estimator. Specifically, it yields the following contribution. The GM-estimator, the UKF, and the H-infinity filter are for the first time integrated into a unified framework to yield the robust H-infinity filter for general nonlinear systems in the Krein space. The statistical linearization is used to build a linear like batch-mode regression form in the Krein space. Specifically, we show that the H-infinity UKF is just the Krein space Kalman filter that minimizes the least squares criterion and has a bounded estimation error in presence of system uncertainties. However, it is vulnerable to outliers and non-Gaussian noise. It is worth pointing out that the H-infinity unscented Kalman filter was firstly proposed in following the framework of linear Hinfinity filter. This paper provides an alternative way of deriving the H-infinity unscented Kalman filter in the Krein space.

Dynamic state estimator:

THE enhancement of the reliability, security, and resiliency of electric power systems depends on the availability of a fast, accurate, and robust dynamic state estimator (DSE) that processes both model information and online measurements obtained from phasor measurement units (PMUs). A DSE provides a better widearea situation awareness of the system dynamics, leading to improved system oscillation monitoring and controls, enhanced dynamic security assessment and system protection schemes, among others. Therefore, it is of a critical importance that any power system DSE is robust to gross errors on the measurements and the model parameter values while providing good state estimates in the presence.

Least – absolute – value: To bound the influence of the model uncertainties and large system process and measurement noises, the first-order Taylor series approximations-based H-infinity extended Kalman filter and the unscented transformation-based H-infinity unscented Kalman filter (UKF) have been proposed. However, they are vulnerable to any types of outliers and exhibit large biases and variances in presence of nonGaussian noise. To suppress gross errors in the PMU measurements, several robust DSEs are developed. For instance, in Rouhani and Abur propose a distributed twostage robust UKF-based DSE using the least-absolute-value (LAV) estimator that can handle observation outliers. However, the authors do not address the vulnerability of the DSE to innovation outliers. To handle both types of outliers, a generalized maximum-likelihood type iterated EKF and UKF are developed.

Derivation of the Linear-Like Regression Form of the Nonlinear UKF: The main idea of the UKF is to use a deterministic sampling technique known as the unscented transform to choose a set of sample points, termed sigma points, which have the same means and covariance matrices as the a priori state statistics under the Gaussian assumption . These sigma points are then propagated through the non-linear functions f and h, yielding an estimation of the a posteriori state statistics by using the Kalman filters structure, i.e., the mean and covariance estimates. Consequently, no calculation of Jacobian matrices is required, which for complex functions can be a difficult task by itself, requiring complicated derivatives if done analytically or being computationally costly if done numerically.