Two modified DFT‐based algorithms for fundamental phasor estimation

This paper investigates two primary algorithms for estimating phasor information used in Phasor Measurement Units (PMU). Power systems behavior and responses are more complicated nowadays and subsequently their supervision, control and protection are more critical and challenging. Various supervisory systems should be employed for protective and control purposes. All these supervisory systems rely on PMU technology with data stamps synchronized by Global Positioning System (GPS). A PMU's accuracy and performance in phasor estimation during different power system working conditions to track the actual signal has an impact on the system state estimation and subsequently in the ability in fault and disturbances diagnosis. Many algorithms are proposed recently to be employed for phasor estimation in PMU devices. It is showed that Discrete Fourier based algorithms are the best methods in digital world to implement in PMU devices. Such methods are Discrete Fourier Transform (DFT) and Smart DFT that are compared and investigated in this article. Simplicity and low computational burden of these methods resulted in attracting many literatures to employ them in phasor estimation algorithms. However, their accuracy in off-nominal frequency and performance during different working condition are controversial which are studied in this article. Extensive simulations are performed in MATLAB Simulink to study and compare DFT and SDFT algorithms under different system disturbances.

The design of both area-efficient and reliable VLSI arrays for computation of the complex N-point discrete Fourier transform (DFT) is considered. C.D. Thompson's VLSI model of computation (1979) is used to quantify the area required by wiring and the achievable period. Four architectures using the fast two-dimensional (2-D) algorithm for the DFT that achieve the maximum throughput per chip area are presented. The first two use N-element 2-D meshes requiring N+2N/sup 3/2/ multiplications and 1+2 square root N periods per DFT. The first uses in place data I/O and can be made systolic with Goertzel's algorithm. The second is systolic with row parallel I/O. The third and fourth architectures both use pipelined DFT blocks of length square root N connected by a pipelined matrix transposer. The third uses systolic 2-D meshes for the short DFTs with period square root N. The fourth uses pipelined butterfly networks implementing standard FFT algorithms for the short DFTs, but only N(1+log/sub 2/ N) multiplications per DFT, thus preserving maximum throughput per unit area. The technique of algorithm-based fault-tolerance applies directly to all four architectures with only fractional redundant overhead. >

To assess the insulation condition of capacitive apparatus, on-line monitoring of dielectric loss angle is an effective measure. Harmonic analysis algorithm is used to calculate dielectric loss angle traditionally, this method can effectively suppress the DC (Direct Current) offset and harmonic interference. However, when the power system frequency fluctuates, the synchronous sampling condition is difficult to meet, so, traditional harmonic analysis method can produce leakage of spectrum and picket-fence effect, which greatly affects the accuracy of dielectric loss angle measurement. In order to reduce this adverse effect, this paper proposes a method called hamming windowed interpolated DFT (Discrete Fourier Transform) algorithm based on modified ideal sampling frequency. Time series analysis based on modified ideal sampling frequency can be used to modify the time series in non-synchronous sampling condition to meet synchronous sampling condition. Hamming windowed interpolated DFT algorithm can inhibit leakage of spectrum and picket-fence effect. Firstly this paper analyzes the principle of traditional DFT algorithm and the reason that causes leakage of spectrum and picket-fence effect in non-synchronous sampling condition. Secondly, the principle of hamming windowed interpolated DFT algorithm based on modified ideal sampling frequency is elaborated. Finally this paper presents the simulation result which shows that the improved approach has higher accuracy.

This paper presents a shared architecture design for computations of Discrete Fourier Transform (DFT), Inverse DFT (IDFT), and Inverse Modified Cosine Transform (IMDCT) algorithms. The proposed method has following advantages, such as 1) Low hardware costs compared with conventional recursive DFT (RDFT) and recursive IDFT (RIDFT) algorithms. For hardware implementation, 2 real multipliers and 8 real adders are required in the proposed architecture. Compared with previous approaches, it can greatly reduce 80% of number of multipliers and 53% of number of adders. 2) The proposed RDFT-based IMDCT architecture has four times data throughput more than the other existing recursive algorithm 3) Computational cycles of the proposed RDFT-based IMDCT algorithm can be greatly reduced by 50% more than our previous recursive IMDCT algorithms. Hence, the proposed RDFT / RIDFT / RDFT-based IMDCT design is more suitable for portable devices in Digital Radio Mondiale (DRM) system.

Data window selection for removal of decaying DC component in current phasor estimation for distance protection

In today’s era, it is important to analyze and utilize various signals in industrial or laboratory applications. Measured signals provide critical information about the controlled system, which can be contained precisely within a narrow frequency range. Many methods and algorithms exist to process such signals in both the time and frequency domains. In particular, signal processing in the frequency domain is primary in industrial practice because dominant components within a specific narrow frequency band are sought. The discrete Fourier transformation (DFT) algorithm is the tool used in practice to find these frequency components. The DFT algorithm provides the full frequency spectrum with a higher number of calculation steps, and its spectrum frequency resolution is low. Therefore, research has focused on finding a method to achieve high-frequency spectrum resolution. An important factor in selecting the technique was that such an algorithm should be implementable on a microprocessor-based system under harsh industrial conditions. Research results showed that the DFT ZOOM method meets these requirements. The frequency zoom has many advantages but requires some modification. It is implemented in high-performance analyzers, but a thorough and detailed description of the respective algorithm is lacking in technical articles and literature. This article mathematically and theoretically describes the modified frequency zoom algorithm in detail. The steps of the frequency zoom, from creating an analytical signal through frequency shifting and decimation to the frequency analysis of the signal, are realized. The algorithm allows for the analysis of a signal with high-frequency resolution in a limited frequency band. A significant modification of DFT ZOOM is that of using the Hilbert transform to create an analytic signal. This resolves the aliasing issue caused by the overlap between fundamental and sideband spectra. Results from processing deterministic and stochastic signals using the modified DFT ZOOM are presented. The presented experimental results contribute to a more detailed frequency analysis of the signal. As part of this scientific research, the issues of frequency zoom were thoroughly addressed, solving the partial problems of this algorithm, both in theory and in the context of signal theory.

This paper proposes an adaptive phasor estimation algorithm based on a least square method that can suppress the adverse effect of an exponentially decreasing DC offset component in a phasor estimation process. The proposed algorithm is composed of three stages: a basic least squares model, a time constant calculation, and an adaptive least squares model. First, we use the basic least squares model to estimate the parameter of the DC offset component in the fault current signal. This model is designed to incorporate fundamental frequency, and harmonic and constant components. Second, we use the estimated parameter to calculate the time constant of the DC offset component. Third, we redesign a least squares model that incorporates fundamental frequency, harmonic components, and exponential function of the DC offset component. Since this model incorporates the exponential function of the DC offset component contained in the fault current signal, it estimates the phasor of the correct fundamental frequency component without influence of the DC offset component. We evaluated the performance of the proposed algorithm using computer generated signals and EMTP simulation signals. The evaluation results show that the proposed algorithm can effectively suppress the adverse influence of the exponentially decaying DC offset component.

Discrete Fractional Fourier Transform (DFRFT) has received lots of attention in last two decades because of its superior benefits and wide applications in various fields. In this study we present a comparative analysis of the most famous algorithms for the computation of DFRFT. Analysis is done on the parameters like time complexity, accuracy, consistency, basic mathematical properties and generalization of Discrete Fourier Transform (DFT) and approximation of continuous Fractional Fourier Transform (FRFT). Main objective of the research is to portray the major advantages and disadvantages of the previously proposed algorithms so that appropriate algorithm may be selected as per requirements. On the basis of study it has been observed that there exist several definitions and algorithm for computing the DFRFT. These algorithms are based on different techniques, such as eigenvectors, chirp convolution, spectral decomposition, non-Fresnel integral, or orthogonal projection. Each of these algorithms has its own advantages and disadvantages. Despite of these developments, we still feel that there has been dire need of a standard definition and computing method for accurate and efficient computation of DFRFT.

This paper presents an application of the discrete Fourier transform (DFT) to calculate neural activities efficiently in simulating large biologically motivated neural nets. The experimental results demonstrate the DFT technique is more superior in performing calculation of the neural activity which reduces the time complexity to a theoretical order of O(nlog/sub 2/, n), n being the number of neural units at each iteration. Our study also found that although the computational speed is improved drastically, there are tradeoffs involving: (1) the error generated from the transform, (2) initial setting up time, and (3) the memory storage requirement when using the DFT algorithm. More specifically, we outline criteria and conditions under which the DFT method will yield optimal results in large software neural simulations. >

In addition to high‐order harmonic and noise components, the fault current may contain decaying DC components which highly affect the estimated fundamental frequency phasor. This study presents a new hybrid method which aims to calculate and remove the unwanted effect of decaying DC components on fundamental phasor estimation. The main novelty of the proposed method is that the proposed method is independent of the dynamic behaviour of the current transformer (CT). In other words, unlike previously published papers, this method analytically calculates the CT time constant. To such aim, utilising frequency modulation, fault current signal is shifted. Utilising simple but comprehensive formulations based on integration and Prony‐complex theory, the magnitude and time constant of decaying DC components are calculated in one cycle. After that, utilising a modified discrete Fourier transform the fundamental phasor component is calculated. Several test signals and also practical fault signals are applied to the proposed method. The simulation results illustrate the proposed method simply but comprehensively remove the unwanted effects of the decaying DC components. Also, it has quick speed as well as a high‐precision response in the estimation of fundamental phasor under different simulation scenarios.

This paper proposes a novel fast and accurate method to mitigate the decaying DC (DDC) component from the fault current signal. At the first stage, it starts to identify the DDC parameters by integrating six sub-cycle samples of current signal considering the symmetry feature of odd harmonics. By receiving the next samples, the estimation of DDC parameters is updated and improved. This procedure continues until the integral of a full cycle current signal is calculated. To improve the accuracy of phasor estimation procedure, in each stage, by subtracting the DDC component from original current signal, the discrete Fourier transform method is applied to estimate the fundamental phasor. Performance of the proposed method for different DDC parameters and fault conditions is verified through various test studies, and its robustness against harmonics, noise, off-nominal frequency, and multiple DDCs conditions is also investigated. Besides, to demonstrate the reliability of proposed method in terms of dependability and security, it is implemented on the phasor estimation block of a distance relay. Three types of data including mathematical, simulated, and real power system signals are used to evaluate the performance of proposed method. Obtained results confirm the desirable performance of proposed method in various operating conditions.

A novel methodology to estimate the state-of-health and remaining-useful-life of a Li-ion battery using discrete Fourier transformation

The signal processing technique is one of the principal tools for diagnosing power quality (PQ) issues in electrical power systems. The Discrete Fourier Transform (DFT) is a frequency analysis technique used to process power system signals and identify PQ problems. However, the DFT algorithm may lead to spectral leakage and picket-fence effect problems for asynchronously sampled signals that contain harmonic, inter-harmonic, and flicker components. To resolve this shortcoming, a hybrid method for frequency estimation based on a second-level DFT approach and a frequency-domain interpolation algorithm to obtain the accurate fundamental frequency of a power system is proposed in this paper. This method uses a second-level DFT to compute the cosine and sine parts for the fundamental frequency components of the acquired signals. Then, a frequency-domain interpolation approach is adopted to determine the amplitude ratio for the cosine and sine parts of the system’s fundamental frequency. A set of mixed signals with harmonic, inter-harmonic, and flicker components with the fundamental frequency deviation is used. The evaluation results demonstrate the superiority of the new method over other approaches for assessing asynchronously sampled signals contaminated with noise, harmonic, inter-harmonic, and flicker components.

Deteriorated measurements due to the saturation of current transformers (CTs) are a major challenge in digital power system protection schemes. Failing to resolve this issue effectively can have serious consequences for the accuracy of measured components in phasor-based digital relaying algorithms. Potentially, this may compromise the secure and reliable operation of the protection system and, hence, the entire power system. Accurate fault detection or classification can be achieved by applying an algorithm that can deal with the CT saturation effects. In this paper, a method is presented that can accurately estimate the fundamental current phasor during CT saturation. Reconstruction of the deteriorated measured waveform is accomplished by applying a supervised machine learning algorithm, namely the support vector machine (SVM). The least squares (LS) method is integrated with the SVM-based algorithm to reduce the complexities of waveform reconstruction regressions. A modified discrete Fourier transform (DFT), robust to decaying DC components, is then applied to the reconstructed waveform to extract the required phasor components. The proposed approach is validated by evaluating its classification and phasor estimation performance using standard metrics over numerous simulated cases and field measurements. The results demonstrate the high accuracy of the proposed method to classify different levels of CT saturation and ensure precise estimation of the fundamental phasor component. • Saturated CT signals are reconstructed using supervised time-series prediction. • A combined least squares–SVM model performs fast waveform reconstruction. • An online recursive LSSVM approach overcomes batch-learning limitations. • Modified DFT removes decaying DC and improves phasor estimation accuracy. • Accurate phasor estimation is achieved within one cycle for all CT saturation levels.

The concept of Wide-Area Measurement Systems (WAMS) envisages the transmission of current and voltage phasor samples from remote Phasor Measurement Units (PMUs) to control centres, relays or controllers. Compression of the high frequency samples of sinusoidal waveforms into phasors is usually done using Discrete-Fourier Transform(DFT)-like algorithms. The phasor data is normally useful only when the system is in sinusoidal steady state, or nearly so. It is worthwhile to explore means to get reasonably accurate transient information as well, even if it means that some additional compressed information needs to be transmitted from the PMUs. Transient reconstruction has practical utility in the analysis of disturbances (at a remote control centre) and for flagging the initiation and clearing of network disturbances, which may be required for some WAMS analytics. In this paper, it is shown that one-cycle-delayed reconstruction of power system transient waveforms can be done with samples of the dc, fundamental and harmonic phasors reported at a rate of one sample per cycle obtained from the DFT algorithm. For power system transients, a few harmonics are usually adequate to reconstruct the transient waveforms with reasonable accuracy. The paper explains the reconstruction method, presents illustrative examples and discusses possible applications and practical issues.