Abstract-A novel technique for online estimation of the fundamental frequency of unbalanced three-phase power systems is proposed. Based on Clarke's 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.Index Terms-Augmented complex least mean square (CLMS) (ACLMS), complex noncircularity, frequency estimation, unbalanced three-phase voltage, widely linear modeling.

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions often require the calculation of the gradient and Hessian, however, real functions of quaternion variables are essentially non-analytic. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized HR (GHR) calculus, thus making possible efficient derivation of optimization algorithms directly in the quaternion field, rather than transforming the problem to 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 proposed quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are elaborated, and the results illuminate the usefulness of the GHR calculus in greatly simplifying the derivation of the quaternion least mean squares, and in quaternion least square and Newton algorithm. The proposed gradient and Hessian are also shown to enable the same generic forms as the corresponding real-and complex-valued algorithms, further illustrating the advantages in algorithm design and evaluation.

Signal processing is a crucial technology for the efficient use of limited and intermittent power resources in the smart grid of the future, and a number of challenges remain to be met. One major issue, as we move towards distributed energy production and use (microgrid) is real time estimation of power quality parameters (frequency, voltages, power factor). The accurate knowledge of frequency is a key parameter of a power system, and its optimal estimation becomes critical in the future smart grid, where the generation, loading and topology are all dynamically updated. In this work, we first consolidate the existing approaches to real-time frequency estimation in a three phase system, and then provide a unified framework for the estimation of the instantaneous frequency in both balanced and unbalanced conditions of a three phase power system. This is achieved by using recent developments in the statistics of complex variables (augmented statistics), by employing the associated widely linear models, and by rigorously accounting for the different degrees of noncircularity associated with various natures of frequency variations in real-world conditions. The usefulness of the proposed framework for frequency tracking in smart grids is illustrated in the context of two major issues in power quality control, namely the tracking of false frequency perturbations in the presence of unbalanced voltage sags (here both synthetic and real-world) and in adaptive frequency tracking in microgrids and islands where there is mismatch between production and consumption. THE NEED FOR FREQUENCY ESTIMATION IN SMART GRIDGovernments, utilities and consumers are all interested in making the ways we produce and use energy more efficient and sustainable. For the electrical power grid this involves fundamental paradigm shifts as we build a smart grid, adopt more renewable energy sources, and promote more energy efficient practices. A smart grid delivers electricity from suppliers to users using digital technology and has a number of properties, including incorporating all forms of energy generation and storage, using sensor information, enabling active participation by end users, being secure and reliable, and using optimization and control to make decisions [1]. This will require the interplay between sensor networks, generation systems, and the power grid, with key technologies from signal processing.It is estimated that the financial loss due to outages in the US economy approaches USD $45.7 billion annually, with power quality issues costing USD $6.7 billion annually [2,3]. Among them, voltage sags, that is, an increase in load current over up to few hundred cycles, are the most frequent problem [4] that severely affects medical centres, semiconductor plants, and broadcasting stations, among others [5]. Voltage sags are typically followed by frequency variations and occur due to switching between the main grid and microgrids, short circuits, motor starting, transformer inrush, fast reclosing of circuit breakers, unexpectedly large or ...

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.