Yili Xia

Adaptive Frequency Estimation in Smart Grid Applications: Exploiting Noncircularity and Widely Linear Adaptive Estimators

Accurate estimation of system frequency in real time is a prerequisite for the future smart grid, where the generation, loading, and topology will all be dynamically updated. In this article, we introduce a unified framework for the estimation of instantaneous frequency in both balanced and unbalanced conditions in a three-phase system, thus consolidating the existing approaches and providing next-generation solutions capable of joint adaptive frequency estimation and system fault identification. This is achieved by employing recent developments in the statistics of complex variables (augmented statistics) and the associated widely linear models, allowing us to benefit from a rigorous account of varying degrees of noncircularity corresponding to different sources of frequency variations. The advantages of such an approach are illustrated for both balanced and unbalanced conditions, including voltage sags, harmonics and supply-demand mismatch, all major obstacles for accurate frequency estimation in the smart grid.

Widely Linear Adaptive Frequency Estimation of Unbalanced Three-Phase Power Systems

A novel technique for online estimation of the fundamental frequency of unbalanced three-phase power systems is proposed. Based on Clarkes transformation and widely linear complex domain modeling, the proposed method makes use of the full second-order information within three-phase signals, thus promising enhanced and robust frequency estimation. The structure, mathematical formulation, and theoretical stability and statistical performance analysis of the proposed technique illustrate that, in contrast to conventional linear adaptive estimators, the proposed method is well matched to unbalanced system conditions and also provides unbiased frequency estimation. The proposed method is also less sensitive to the variations of the three-phase voltage amplitudes over time and in the presence of higher order harmonics. Simulations on both synthetic and real-world unbalanced power systems support the analysis.

An Augmented Echo State Network for Nonlinear Adaptive Filtering of Complex Noncircular Signals

A novel complex echo state network (ESN), utilizing full second-order statistical information in the complex domain, is introduced. This is achieved through the use of the so-called augmented complex statistics, thus making complex ESNs suitable for processing the generality of complex-valued signals, both second-order circular (proper) and noncircular (improper). Next, in order to deal with nonstationary processes with large nonlinear dynamics, a nonlinear readout layer is introduced and is further equipped with an adaptive amplitude of the nonlinearity. This combination of augmented complex statistics and enhanced adaptivity within ESNs also facilitates the processing of bivariate signals with strong component correlations. Simulations in the prediction setting on both circular and noncircular synthetic benchmark processes and real-world noncircular and nonstationary wind signals support the analysis.

An Adaptive Diffusion Augmented CLMS Algorithm for Distributed Filtering of Noncircular Complex Signals

An adaptive diffusion augmented complex least mean square (D-ACLMS) algorithm for collaborative processing of the generality of complex signals over distributed networks is proposed. The algorithm enables the estimation of both second order circular (proper) and noncircular (improper) signals within a unified framework of augmented complex statistics. The analysis shows that the performance advantage of the widely linear D-ACLMS over the strictly linear D-CLMS increases with the degree of noncircularity while maintaining similar performance for proper data. Simulations on both synthetic benchmark and real world noncircular data support the approach.

An augmented affine projection algorithm for the filtering of noncircular complex signals

An augmented affine projection adaptive filtering algorithm (AAPA), utilising the full second order statistical information in the complex domain is proposed. This is achieved based on the widely linear model and the joint optimisation of the direct and conjugate data channel parameters. The analysis illustrates that the use of augmented complex statistics and widely linear modelling makes the AAPA suitable for the processing of both second order complex circular (proper) and noncircular (improper) signals. The derivation is supported by the analysis of convergence in the energy conservation setting. Simulations on both benchmark and real-world noncircular wind signals support the analysis.

Quaternion-Valued Echo State Networks

Quaternion-valued echo state networks (QESNs) are introduced to cater for 3-D and 4-D processes, such as those observed in the context of renewable energy (3-D wind modeling) and human centered computing (3-D inertial body sensors). The introduction of QESNs is made possible by the recent emergence of quaternion nonlinear activation functions with local analytic properties, required by nonlinear gradient descent training algorithms. To make QENSs second-order optimal for the generality of quaternion signals (both circular and noncircular), we employ augmented quaternion statistics to introduce widely linear QESNs. To that end, the standard widely linear model is modified so as to suit the properties of dynamical reservoir, typically realized by recurrent neural networks. This allows for a full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances, and a rigorous account of second-order noncircularity (improperness), and the corresponding power mismatch and coupling between the data components. Simulations in the prediction setting on both benchmark circular and noncircular signals and on noncircular real-world 3-D body motion data support the analysis.

Application of multivariate empirical mode decomposition for seizure detection in EEG signals

We present a method for the analysis of electroencephalogram (EEG) signals which has the potential to distinguish between ictal and seizure-free intracranial EEG recordings. This is achieved by analyzing common frequency components in multichannel EEG recordings, using the multivariate empirical mode decomposition (MEMD) algorithm. The mean frequency of the signal is calculated by applying the Hilbert-Huang transform on the resulting intrinsic mode functions (IMFs). It has been shown that the mean frequency estimates for the ictal and seizure-free EEG recordings are statistically different, and hence, can serve as a test statistic to distinguish between the two classes of signals. Simulation results on real world EEG signals support the analysis and demonstrate the potential of the proposed scheme.

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks.

Steady state analysis of the CLMS and augmented CLMS algorithms for noncircular complex signals

The recently introduced augmented complex least mean square (ACLMS) algorithm is shown to be suitable for the processing of both second order circular (proper) and noncircular (improper) signals, by virtue of the underlying widely linear model. In theory, both the linear CLMS and widely linear ACLMS achieve the same mean square error for propers signals, whereas the ACLMS exhibits lower mean square error for improper signals. However, improperness can arise due to the system noise, input, or channel model and to shed more light on the convergence and steady state properties of ACLMS and CLMS in these cases we here employ the energy conservation principle. Simulations in adaptive prediction and system identification settings for signals with different probability distributions and degrees of noncircularity support the the analysis.

Augmented MVDR Spectrum-Based Frequency Estimation for Unbalanced Power Systems

A robust technique for online estimation of the fundamental frequency of both balanced and unbalanced three-phase power systems is proposed. This is achieved by extending the recently introduced iterative frequency estimation method based on minimum variance distortionless response (MVDR) spectrum , in order to enhance its robustness in unbalanced system conditions. The approach is made optimal for the second-order noncircular nature of the unbalanced complex-valued system voltage, by combining the iterative MVDR (I-MVDR) frequency estimation and the complete available (augmented) second-order statistics. Such an approach makes it possible to eliminate the otherwise unavoidable estimation bias in unbalanced system conditions. It is also shown that the proposed method approaches the theoretical Cramer-Rao lower bound (CRLB), which we rigorously derive for the vector parameter in power systems. Simulations over a range of unbalanced conditions, including voltage sags, the presence of higher-order harmonics, and for real-world unbalanced power systems, support the analysis.