Recursive Discrete Fourier Transform Research Articles

Sample-by-sample Power Quality Disturbance classification based on Sliding Window Recursive Discrete Fourier Transform

Power Quality Disturbances (PQDs) are caused by large-scale grid connections of nonlinear loads and Distributed Generations (DG). They affect the electrical and electronics equipment performance and may cause serious accidents and economic losses. The classification of PQDs becomes a key issue for end-users in order to enhance the Power Quality (PQ) and comprises an important step in monitoring the Electric Power System (EPS). Accurate PQDs classification is crucial for improving the quality of power supply, monitoring the condition of power equipment, and mitigating power grid issues. In the literature, several methods for detecting and classifying PQDs have been proposed. However, these methods consume a significant amount of processing time due to the detection and sample storage steps required to classify each disturbance class. In this work, we propose a method for classifying PQDs on a sample-by-sample basis. The innovation of this method lies in its ability to classify PQDs for each individual sample, thereby eliminating the need for window-based (batch) processing. To do so, it is applied the Sliding Window Recursive Discrete Fourier Transform (SWRDFT) to real-time estimation of PQ parameters. For the classification stage, decision tree, Artificial Neural Networks (ANN), Ensemble and the Naive Bayes classifiers were employed. The proposed method using Ensemble achieved the most satisfactory results for six different types of PQDs (oscillatory transients, harmonics, notches, sags, spikes and swells), in which oscillatory transients and spikes were identified in the first monitored samples (sample 2 and 11 respectively), sags were identified in the 14th sample counting from the beginning of the disturbance, notches were captured from the 35th sample, and harmonics and swells were identified after the 113th sample.

A Generalized Harmonic Extraction Algorithm Based on Multi-Window Average Filter Under Synchronous Rotating Frame

Various harmonic extraction algorithms have been proposed to improve the steady-state and dynamic performance for active power filter. However, it is difficult to select a suitable method for a certain load current scenario. This article analyses the general characteristics of the harmonic extraction based on the “frequency shift” characteristic by Park transform, and it can be transformed into a low-pass filter under the synchronous rotating frame (SRF) for design. Starting from the typical average filter, a general FIR filter structure called multi-window average (MWA) is proposed. According to the actual load current components, aiming at the shortest delay, the design process and guideline are given. Combined with SRF, the MWA structure can generalize recursive discrete Fourier transform (RDFT), cascade delayed signal cancellation (CDSC) and their derived algorithms, which proves the equivalence between time-based harmonic extraction methods and frequency-based ones. Moreover, using the MWA structure can effectively reduce the complexity of the CDSC algorithm. The experiment proves that the typical RDFT and CDSC harmonic extraction can be equivalently realized by using the SRF-based structure. Under the presence of 6 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> + 1 and 12 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> − 2 harmonics, the proposed harmonic extraction time can reach 1/4 fundamental cycle. While greatly reducing the dynamic response time, the MWA structure reveals inherent unity among various harmonic extraction algorithms for the first time.

Portable RDFT-Based EIS System Design With a Low-Complexity Impedance Calculation

This paper presents a novel low-complexity impedance calculation based on a compact recursive discrete Fourier transform (RDFT) for electrical impedance spectroscopy (EIS) system design. Compared to a traditional fast Fourier transform, an arbitrary-length RDFT is not limited by the power-of-two transform length. By choosing a suitable integer ratio of the desired frequency/frequency resolution, the spectrum leakage can be significantly reduced by 4-Hz resolution per bin, which enables a lower magnitude relative error (MRE) and a phase absolute error (PAE) in impedance calculations. Here, the practical measurement results show that the maximum MRE and PAE are below 0.1% and 0.5°, respectively, making this approach superior to previously reported ones. Compared with the state-of-the-art Goertzel filter-based EIS system, the proposed method provides more frequency bins (+43.75%) within the same frequency band, and the operations in terms of square root, division, and arctangent are drastically reduced by 50%, 33%, and 50%, respectively. In addition, the proposed system only costs $474 USD and can serve as a suitable solution for various EIS applications in the future.

SWRDFT Controller for Single Phase Grid Tied PV system with Low Voltage Ride through Capability

Voltage control and power quality (PQ) management are crucial aspects for the control of grid connected voltage source converters (VSC). In this paper, a novel control technique for solar photovoltaic (PV) tied single-phase grid is described based on a Sliding Window Recursive Discrete Fourier Transform (SWRDFT). This article considers a single-phase, double-staged PV interconnection. The major goal of control is to operate the VSC as per the standard grid codes and ensure quality power is delivered to the grid even under abnormal grid conditions. The system is designed to work at unity power factor during typical operation. Reactive power injection is mandated as per a country's grid code and helps to regulate the voltage profile and low voltage ride through (LVRT) during grid voltage fluctuations. Furthermore, the paper focuses on two important aspects viz. load compensation and LVRT operation carried out in the event of grid voltage sags. During night time operation, in the absence of PV, the VSC and dc link capacitor operate as a shunt compensator. The proposed controller completes both the tasks as demonstrated by simulation and hardware results. Additionally, a comparison study using state of the art fundamental extraction techniques based on Second / Third order generalized Integrator (SOGI/TOGI) is presented.

An impedance identification-based islanding detection method for droop control inverters based on RDFT and positive–negative notch filters

Droop control inverters are widely used because of their plug-and-play characteristics, no need for communication, and adaptability to weak power grids. However, because of the difficulty of islanding detection, after the grid is restored, drooping control inverters may cause security problems. In order to address this issue, this paper proposes an islanding detection method for droop control inverters based on impedance identification. By injecting current disturbance into common coupling points (PCC), the impedance is identified by the current response, and then the islanding can be determined by the impedance change. On the one hand, a signal extraction method based on positive–negative notch filters is proposed to reduce the content and impact on the power quality of the disturbance injection. On the other hand, this paper uses a signal processing method based on recursive discrete Fourier transform (RDFT), which can effectively reduce the computational burden. Moreover, the proposed method can eliminate the non-detection zone of the island detection. The experimental results verify the effectiveness of the proposed method.

Fast Real-Time RDFT- and GDFT-Based Direct Fault Diagnosis of Induction Motor Drive

This paper presents the theoretical analysis and experimental verification of a direct fault harmonic identification approach in a converter-fed electric drive for automated diagnosis purposes. On the basis of the analytical model of the proposed real-time direct fault diagnosis, the fault-related harmonic component is calculated using recursive DFT (RDFT) and Goertzel DFT (GDFT), applied instead of the full spectrum calculations required in the most popular FFT algorithm. The simulation model of an inverter sensorlessly controlled induction motor drive is linked with the induction machine rotor fault model for testing the sensitivity of the GDFT- and RDFT-based fault diagnosis to state variable estimation errors. According to the presented simulation results, the accuracy of the direct identification of a fault-related harmonic is sensitive to the quality of fault harmonic frequency estimation. The sensitivity analysis with respect to RDFT and GDFT algorithms is included. Based on the experimental setup with a sensorlessly controlled induction motor drive with the investigated rotor fault, fault diagnosis algorithms were implemented in the microprocessor by integration with the control system in one microcontroller and experimentally verified. The RDFT and GDFT approach has shown accurate and fast direct automated fault identification at a significantly decreased number of arithmetical operations in the microcontroller, which is convenient for the frequency-domain fault diagnosis in electric drives and supports fault-tolerant control system implementation.

Improvement of Recursive DFT for APF With Higher Switching Frequency to Suppress Wideband Harmonics

Shunt active power filter (APF), with higher switching frequency, has a potential market for suppressing wideband harmonics in the more electric aircraft (MEA) or high-speed train (HST) power grid. In this case, the selective extraction of harmonic current reference is essential. Recursive discrete Fourier transform (RDFT) is widely adopted for its excellent selectivity and low computational burden. However, RDFT is tough to implement in the higher frequency APF with the shorter switching cycle, and suffers from the demerit of slow dynamics. To alleviate those, three improved RDFTs are proposed in this paper. Firstly, it is revealed that RDFT can be regarded as a series of DFT and inverse-DFT (IFT), and its settling time depends on the comb filter of DFT. Then, by rearranging the sampling frequency of DFT according to the extracted harmonic, the computational burden of twice-sampling RDFT (TS-RDFT) is reduced while satisfying the same extraction accuracy. Furtherly, to improve the dynamics of TS-RDFT while retaining the merits of simplicity and selectivity, the comb filter is reconfigured based on the load current spectrum. For balanced loads, the settling time of TS-RDFT can be shortened from 1 to 1/6 fundamental cycle in the synchronous reference frame. For unbalanced loads, it is shortened to 1/2 cycle in the $\alpha \beta $ frame. Finally, the simulations with Matlab for harmonic extraction in 400Hz grid and the experiments with the APF prototype for harmonic suppression in 50Hz grid are carried out to verify the effectiveness of the proposed methods.

Phase-Locked Loop Independent Second-Order Generalized Integrator for Single-Phase Grid Synchronization

Second-order generalized integrator (SOGI) requires fast and accurate grid voltage frequency information for quadrature signal generation (QSG) in single-phase systems. Conventionally, synchronous reference frame phase-locked loop (PLL) (SRFPLL) is used in a cascade manner and provides frequency feedback to SOGI. PLL is a non-linear closed-loop control system that requires adequate gain parameter design trade-off to provide accurate grid synchronization. Additionally, it creates interdependent loops while estimating the grid frequency and phase-angle. This results in unwanted frequency disturbances during grid transients, such as voltage sag and phase-angle jumps (PAJs) which are fed back to the SOGI as well. Thus, this article proposes two PLL independent frequency-feeding approaches for SOGI: Type-A and -B. The frequency is estimated using the normalized single-phase grid voltage signal in both types. In Type-A, the normalization is done using cascaded recursive discrete-Fourier transform (RDFT) and inverse RDFT (IRDFT). On the other hand, in Type-B, SOGI outputs are used for normalization. The grid voltage amplitude is estimated using the absolute value of the in-phase and in-quadrature components of the proposed frequency-adaptive SOGI. Furthermore, the third-order polynomial approximation of the arctangent function is implemented for phase-angle estimation in both types. The performance of both the types during transients and steady-state grid voltage disturbances (dc offset and harmonics) are compared with SOGIPLL using both simulations and experiments.

Applying Directly Modified RDFT Method in Active Power Filter for the Power Quality Improvement of the Weak Power Grid

The recursive discrete Fourier transformation (RDFT) method can be used for grid voltage phase-locking and harmonic current detection in a shunt active power filter (SAPF). However, in weak power grids such as microgrids, significant errors might occur in the amplitude and phase detection due to grid frequency deviation. In this study, to resolve this problem, a directly modified RDFT (DMRDFT) method is proposed for SAPF weak grid application. Through theoretical analysis, the errors of phase and amplitude detection were found to consist of fixed error and fluctuating error. The fixed error is only determined by frequency deviation, whereas the fluctuating error is also related to the recursive pointer and the initial phase. The DMRDFT algorithm can obtain the real grid frequency through the calculation of the phase angle difference for two consecutive periods. Then it can employ the grid frequency deviation and the recursive pointer value to directly correct the detection results gathered by the conventional RDFT algorithm. As a result, DMRDFT can yield accurate amplitude and phase information of the grid voltage or current with a simple calculation. Simulation results verify the high precision of the proposed DMRDFT method in both steady and dynamic situations. Experimental results show that the DMRDFT method can significantly increase the SAPF compensation performance when grid frequency shifts.

Research on three-phase four-wire active power filter based on LCLCL coupling structure

In this paper, the main circuit topology of the three-phase four-wire active power filter (APF) and its harmonic current detection and control algorithm are studied. An active power filter system combining LCLCL filter and a three-phase H-bridge structure is proposed. This paper has applied harmonic detection algorithm based on an improved synchronous phase-locked loop (SRF-PLL) and recursive discrete Fourier transform (RDFT) while using triangular carrier modulation control based on PI adjustment to realize the harmonic compensation of the power filter to the grid current. The system has a notch effect on the harmonic at the switching frequency. It can detect the fundamental component and each harmonic component of the load current separately with a small amount of calculation, good real-time performance. Finally, a 5KVA single-phase APF prototype was built to test the superiority of the main circuit topology and the algorithm of current detection and control.

DSP-Based SiC-MOSFET SAPF With 100-kHz Sampling and Switching Frequency for Wideband Harmonic Suppression

In this article, 100-kHz sampling and switching frequency shunt active power filter (SAPF), which is based on a dual-core DSP TMS320F28377D, is designed for the potential market of high-frequency harmonic suppression in more electric aircraft (MEA) or high-speed train (HST). By utilizing the multithreading and multiprocessing method, different computation segments, such as analog-to-digital conversion, recursive discrete Fourier transform (RDFT), and inner loop current and outer loop voltage control, can be rearranged as several parallel operation groups. Compared with the traditional single pipeline structure of the mostly 10–20 kHz switching frequency commercial SAPF, this parallel execution speeds up the signal processing so that the highly increased sampling frequency makes the current control loop reaches wider bandwidth even with simple proportional integral (PI) regulator. Meanwhile, for wideband harmonic suppression, a multiresolution RDFT strategy is proposed to extend the range of the selected harmonic without sacrificing its extraction accuracy. After raising the sampling frequency to 100 kHz, the proposed SAPF with much easier current controller could suppress wider frequency harmonics than other 10–20 kHz SAPF. A 10- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text {kV}\cdot \text {A}$ </tex-math></inline-formula> prototype with CREE SiC power module CCS050M12CM2 is set up to verify its effectiveness. The result complies with the IEEE standard 519-2014.

A Design of Input-Decimation Technique for Recursive DFT/IDFT Algorithm

In this paper, an input-decimation technique for the recursive discrete Fourier transform (RDFT)/inverse DFT (RIDFT) algorithm is proposed for the high-speed broadband communication systems. It is worth noting that the input-decimation approach is presented to decrease the number of input sequences for the recursive filter so that the computation cycle of RDFT/RIDFT can be shortened to meet the computing time requirement ( $3.6~{\mu }s$ ) for the high-speed broadband communication systems. Therefore, the input-decimation RDFT/RIDFT algorithm is able to carry out at least 55.5% reduction of the total computation cycles compared with the considered algorithms. Furthermore, holding the advantages of input-decimation technique, the computational complexities of the real-multiplication and -addition are reduced to 41.3% and 22.2%, respectively. The area and the power consumption can be minimized by employing the cost-efficient constant multiplier with the refined signed-digit expression of twiddle factors. Finally, the physical implementation results show that the core area is $0.37\times 0.37$ mm2 with $0.18~\mu \text{m}$ CMOS process. The power consumption is 5.16 mW with the supply voltage of 1.8 V and the operating clock of 40 MHz. The proposed design can achieve 258 million of computational efficiency per unit area (CEUA) and really outperform the previous works.

An Adaptive Recursive Wavelet Based Algorithm for Real-Time Measurement of Power System Variables During Off-Nominal Frequency Conditions

An adaptive wavelet-based method of phasor and frequency estimations and its application for online monitoring, control, and protective equipment are presented in this paper. The method is recursive with a low computational burden, suited for real-time implementations, and does not need any preprocessing stages. It computes the frequency of the analyzing signal accurately during nominal and off-nominal frequency conditions. To reduce computational errors of the phase angle and amplitude estimations that occur during off-nominal frequency conditions, the proposed method adapts itself according to the frequency drift. The proposed recursive adaptive method along with three other approaches based on recursive wavelet transform, discrete Fourier transform (DFT), and recursive DFT is implemented and compared using MATLAB, considering dynamic and static conditions. The methods are also employed for real-time computation of the phasor and frequency of the recorded data of the South West Interconnected System (SWIS) in Western Australia, and their efficiencies are assessed and compared. In addition, a real 33.3 MVA, 132/23.3 kV transformer located in SWIS is modeled in the electromagnetic transients program (EMTP) software, and the recorded signals have been employed for performance evaluation of the above-mentioned methods. The method is also implemented on a computer-based system in the laboratory and its performance is appraised accordingly. Extensive MATLAB simulations, laboratory implementation, field data, and EMTP results show that the proposed adaptive method has better static and dynamic performances in comparison to the other approaches.

VLSI Architecture for Novel Hopping Discrete Fourier Transform Computation

The hopping discrete Fourier transform (HDFT) is a new method applied for time-frequency spectral analysis of time-varying signals. In the implementation of HDFT algorithms, the updating vector transform (UVT) plays a key role, and therefore a novel recursive DFT-based UVT formula is introduced in the proposed design for a HDFT algorithm and its architecture. The perceived advantages can be summarized as: 1) the proposed algorithm reduces the number of multiplications, additions, and coefficients by 42.3%, 33.3%, and 50%, respectively, compared with Park’s method under the settings of an M-sample complex input sequence (M = 256), and an N-point recursive DFT computation scheme (N = 64) for time hop L (L = 4); 2) by adopting the hardware-sharing scheme and the register-shifting concept, the proposed design only takes nine multipliers and 12 adders for realization. The proposed hardware accelerator can be implemented using a field-programmable gate array, which can operate at 48.33 MHz clock rate. The resource utilization of combinational logic lookup tables (LUTs) and digital signal processing (DSP) blocks reduced by 11.7% and 42.5% compared with Juang et al. ’s work. For very-large-scale integration realizations, the proposed design would be more powerful than other existing algorithms in future applications focusing on DSP, filtering, and communications.

R–DFT-based Parameter Estimation for WiGig

In this paper we present parameter estimation methods for IEEE 802.11ad transmission to estimate the frequency offset value and channel impulse response. Furthermore a less known low complexity signal processing architecture – the Recursive Discrete Fourier Transform (R-DFT) – is applied which may improve the estimation results. The paper also discusses the R-DFT and its advantages compared to the conventional Fast Fourier Transform.

Design of a recursive single-bin DFT algorithm for sparse spectrum analysis

Discrete Fourier transform (DFT) is the basic means of spectrum analysis in the field of digital signal processing, and the fast Fourier transform (FFT) has become the most popular algorithm which decreases the computational complexity from quadratical to linearithmic. However, engineers are often challenged to detect a single or just a few of the frequency components. For this kind of sparse spectrum analysis, the FFT no longer has advantage because it always computes all the frequency components. This paper proposes a recursive single-bin DFT (RSB-DFT) algorithm to compute one specific frequency spectrum, whose theoretical derivation is elaborated and implementation steps are given as a flow diagram. A 16-point RSB-DFT calculation example is also given to exhibit computation process of the algorithm. An application example for bioimpedance spectroscopy (BIS) measurement demonstrates that the proposed RSB-DFT algorithm can compute specific single spectral lines accurately. The computation efficiency of the proposed RSB-DFT algorithm is demonstrated as the highest compared with the DFT, FFT, Goertzel algorithm, which means that the RSB-DFT algorithm has the potential to become an alternative and efficient tool for sparse spectrum analysis.

Harmonic impedance measurement based on short time current injections

This work presents an alternative method of measuring the equivalent impedance of an electric power network by injecting a short time current waveform based on Modulated Gaussian Signal. The frequency domain measurement of the equivalent impedance of an electric network is an important issue to correctly design harmonic filters and capacitor banks, detect resonance and antiresonance frequencies and evaluate the harmonic penetration and emission limits. The method proposed in this paper uses the Sliding Window Recursive Discrete Fourier Transform (SWRDFT) to extract the harmonic components of the measured voltage due to the injection of the short time current into the network. The proposed method was tested with different electric system topologies and the estimated impedance was compared with the impedance estimated by other methods found in the literature. Real time simulation using Hardware-in-the-Loop (HIL) methodology, as well as experimental results are presented in this paper.

System identification of the quadrotor with inner loop stabilisation system

In order to build the dynamic model of quadrotors with inner loop stabilisation system, a real-time system identification technique is developed in frequency domain with real flight data. Firstly, a second order transfer function is adopted as the low order equal system (LOES) for each channel. Then, a recursive discrete Fourier transformation (DFT) is detailed, which helps to calculate the recursive frequency spectrum of the inputs and outputs. Based on the latest frequency spectrum, the unknown parameters in LOES are calculated by a least squares method and then the observable canonical form of state space equations is given. Finally, the performance of the identified model is analysed in time-domain by calculating Theil's inequality coefficient and in frequency-domain by plotting Bode diagrams. The identified model is utilised for the design of tracking controllers. The experimental result validates the proposed method.

System identification of the quadrotor with inner loop stabilisation system

In order to build the dynamic model of quadrotors with inner loop stabilisation system, a real-time system identification technique is developed in frequency domain with real flight data. Firstly, a second order transfer function is adopted as the low order equal system (LOES) for each channel. Then, a recursive discrete Fourier transformation (DFT) is detailed, which helps to calculate the recursive frequency spectrum of the inputs and outputs. Based on the latest frequency spectrum, the unknown parameters in LOES are calculated by a least squares method and then the observable canonical form of state space equations is given. Finally, the performance of the identified model is analysed in time-domain by calculating Theil's inequality coefficient and in frequency-domain by plotting Bode diagrams. The identified model is utilised for the design of tracking controllers. The experimental result validates the proposed method.

Recursive and non-recursive algorithms for power system real time phasor estimations

This work presents some of the computational algorithms used for phasor estimations in Electrical Power Systems. The IEEE C37.118.1 standard establishes the phasor estimation structure and performance, but does not define the algorithm itself to be used. Considering this, various methods can be adopted provided that the standard precision is met. Some estimation algorithms presented in the literature were compared in this paper, and their behavior was evaluated for some test cases. The methods: Discrete Fourier Transform, Recursive Discrete Fourier Transform, Least Squares, Recursive Least Squares, Discrete Wavelet Transform and Recursive Wavelet Transform were tested using synthetic signals, evaluating the Total Vector Error, response and delay times, as well as overshoot. The algorithms were also embedded in hardware and evaluated by simulated signals, using the outputs of a Real Time Digital Simulator.