Harmonic Phasors Research Articles

Extended C37.118.1 PMU Algorithms for Joint Tracking of Fundamental and Harmonic Phasors in Stressed Power Systems and Microgrids

This paper extends synchrophasor algorithms approximating C37.118.1 filtering requirements to provide phasor measurement units (PMUs) with the capability of accurately tracking single-phase harmonic phasors subject to varying nominal frequency and out-of band interharmonic interference. The fastest solution is built on a Kalman filter (KF) bank responding with notches at harmonic frequencies, while the most accurate solution relies on a five-cycle finite-impulse-response filter with more than 80-dB harmonic rejection. Highly distorted standardized test signals following WECC and Hydro-Québec experiences for stressed transmission systems and IEC recommendations for medium- and low-voltage system distortions are used to demonstrate the good performance of the two algorithms in tracking nonstationary fundamental and harmonics quantities. The two schemes are compared advantageously in terms of computation speed and performance with a four-cycle short-time fast Fourier transform algorithm. Finally, the effectiveness of harmonic phasors-enabled PMUs is demonstrated on a generation-rich microgrid subjected to a severe fault, followed by an offnominal frequency operation in islanded mode and subsequent grid re synchronization.

Shifting Window Average Method for Phasor Measurement at Offnominal Frequencies

A shifting window average method (SWAM) is proposed for phasor measurement at offnominal frequencies. The proposed SWAM is not only applicable for fundamental frequency phasor measurement, but also for harmonic phasor measurement as well. SWAM is based on a mathematical analysis of the errors induced in the commonly used DFT method at offnominal frequency inputs. First, a comprehensive derivation of magnitude and phase-angle measurement errors of discrete Fourier transform (DFT) is proposed. It is indicated that the errors are caused by modulations between different exponential components in the signal, and can be modeled by quasisinusoids with frequencies equal to the frequency differences between modulated components. SWAM is then introduced to eliminate the errors by examining the analytic form of the errors. Simulation tests have been performed on several benchmark signals. Simulation cases have validated that SWAM achieves higher accuracy compared with the DFT-based method and the three-sample average filter method. Experimental tests were also performed to validate the accuracy of the proposed method. Due to its high accuracy and reasonably low processing effort, SWAM is a valuable candidate for online phasor measurement in power systems.

New Algorithms for the Estimation of Distorted Time-varying Power System Signal Parameters in Noise

This article proposes an improved two-step augmented Gauss-Newton type algorithm for the estimation of distorted power system signals. The proposed algorithm is capable of accurately estimating the amplitude, phase, and frequency of the harmonics and interharmonics present in the power system signal in the presence of high additive white Gaussian noise. The estimation of all the parameters is divided into two steps. In the first step, the frequency of the signal is computed using an improved recursive Newton-type algorithm; in the second step, the amplitude and phase of the power system voltage or current are computed using a recursive Gauss-Newton-type algorithm. Further, to reduce the computational complexity of the standard Gauss-Newton algorithm, certain approximations are carried out, making the algorithm suitable for on-line applications. Also, a variable forgetting factor is used to improve the speed of convergence and accuracy of the algorithm. To demonstrate the robustness and accuracy in the estimation of the distorted fundamental and harmonic phasors using the proposed algorithm, the results obtained from computer simulations are presented.

Harmonic Phasor Analysis Based on Improved FFT Algorithm

An approach for power system harmonic phasor analysis under asynchronous sampling is proposed in this paper. It is based on smoothing sampled data by windowing the signal with the four-term fifth derivative Nuttall (FFDN) window, and then calculating harmonic phasors in the frequency domain with an improved fast Fourier transform (IFFT) algorithm. The applicable rectification formulas of the IFFT are obtained by using the polynomial curve fitting, dramatically reducing the computation load. The FFDN window can effectively inhibit the spectral leakage and the picket fence effect can be modified by the IFFT algorithm under asynchronous sampling, and the overall algorithm can easily be implemented in embedded systems. The effectiveness of the proposed method was analyzed by means of simulations and practical experiments for multifrequency signals with the fluctuation of the fundamental frequency and with the presence of white noise and interharmonics.

Dynamic Harmonic Analysis Through Taylor–Fourier Transform

A new dynamic harmonic estimator is presented as an extension of the fast Fourier transform (FFT), which assumes a fluctuating complex envelope at each harmonic. This estimator is able to estimate harmonics that are time varying inside the observation window. The extension receives the name “Taylor-Fourier transform (TFT)” since it is based on the McLaurin series expansion of each complex envelope. Better estimates of the dynamic harmonics are obtained due to the fact that the Fourier subspace is contained in the subspace generated by the Taylor-Fourier basis. The coefficients of the TFT have a physical meaning: they represent instantaneous samples of the first derivatives of the complex envelope, with all of them calculated at once through a linear transform. The Taylor-Fourier estimator can be seen as a bank of maximally flat finite-impulse-response filters, with the frequency response of ideal differentiators about each harmonic frequency. In addition to cleaner harmonic phasor estimates under dynamic conditions, among the new estimates are the instantaneous frequency and first derivatives of each harmonic. Two examples are presented to evaluate the performance of the proposed estimator.

A PMU for the Measurement of Synchronized Harmonic Phasors in Three-Phase Distribution Networks

Owing to the extension of electric distribution networks, monitoring and control issues rely on large-scale distributed measurement systems, which are able to carry out simultaneous measurements of electrical quantities in several points of the monitored systems. In this paper, the implementation of digital procedures that are suitable for the evaluation of the synchronized harmonic phasors in a flexible phasor measurement unit (PMU) based on PXI modular hardware is presented. The results of the experimental tests are shown to characterize the measurement system to evaluate the behavior of the designed instrument under real operating conditions on three-phase electric distribution networks.

GPS-Based System for the Measurement of Synchronized Harmonic Phasors

A measurement system, based on high-performance Global Positioning System (GPS) receivers and general-purpose acquisition (DAQ) boards, for the evaluation of the synchronized harmonic phasors in the nodes of an electric distribution network, is presented. To meet the requirements of different fields of application, two measurement procedures have been implemented: One is based on a fixed observation window, whereas, in the other one, the observation interval is a function of the actual power system frequency. Experimental tests are presented to characterize the measurement system: The results show that the proposed procedures allow the accuracy that is usually required in power quality monitoring of distribution systems to be reached.

Adaptive resonator-based method for power system harmonic analysis

A simple algorithm for the harmonic estimation, in a wide range of frequency changes, with benefits in a reduced complexity and computational efforts is prescribed. This implementation is based on a recently introduced common structure for recursive discrete transforms and contemplated as an implementation of finite-impulse-response (FIR) and infinite-impulse-response (IIR) filter transfer functions to reduce computational efforts. This structure consists of digital resonators in a common negative feedback loop. The structure of the estimation algorithm consists of two decoupled modules: the first one for an adaptive filter of input signal with harmonic amplitude and phase calculation, the second one for an external frequency estimation. A very suitable algorithm for frequency and harmonic phasor estimations is obtained. To demonstrate the performance of the developed algorithm, computer-simulated data records are processed. Simulation results show that this algorithm is applicable to detect the harmonic amplitudes of steady-state, varying and decaying sinusoidal signals. It has been found that the proposed method really meets the needs of online applications. This technique provides accurate amplitude estimates in about one period.

A New PMU-Based Fault Detection/Location Technique for Transmission Lines With Consideration of Arcing Fault Discrimination—Part I: Theory and Algorithms

A new fault detection/location technique with consideration of arcing fault discrimination based on phasor measurement units for extremely high voltage/ultra-high voltage transmission lines is presented in this two-paper set. Part I of this two-paper set is mainly aimed at theory and algorithm derivation. The proposed fault detection technique for both arcing and permanent faults is achieved by a combination of a fault detection index |M| and a fault location index |D|, which are obtained by processing synchronized fundamental phasors. One is to detect the occurrence of a fault and the other is to distinguish between in-zone and out-of-zone faults. Furthermore, for discriminating between arcing and permanent faults, the proposed technique estimates the amplitude of arc voltage by least error squares method through the measured synchronized harmonic phasors caused by the nonlinear arc behavior. Then, the discrimination will be achieved by comparing the estimated amplitude of arc voltage to a given threshold value. In addition, in order to eliminate the error caused by exponentially decaying dc offset on the computations of fundamental and harmonic phasors, an extended discrete Fourier transform algorithm is also presented.

A New PMU-Based Fault Detection/Location Technique for Transmission Lines With Consideration of Arcing Fault Discrimination—Part II: Performance Evaluation

The theory and algorithms of the proposed technique have been presented in Part I of this two-paper set. In Part II of this two-paper set, the proposed technique is evaluated by considerable simulation cases simulated by the Matlab/Power system Blockset simulator. For the proposed fault detector, the trip time achieved can be up to 3.25 ms and the average value of trip times is about 8 ms for both permanent and arcing faults on transmission lines. For the proposed fault locator, the accuracy can be up to 99.99% and the error does not exceed 0.45%. Moreover, the proposed arcing fault discriminator can discriminate between arcing and permanent faults within four cycles after fault inception. It has proven to be an effective tool to block reclosing on the permanent faults in the computer simulations. The simulation results also demonstrate that the presented extended discrete Fourier transform algorithm eliminates effectively the error caused by exponentially decaying dc offset on fundamental and harmonic phasor computations. Finally, a test case using the real-life measured data proves the feasibility of the proposed technique.

Vectorial summation of probabilistic current harmonics in power systems: From a bivariate distribution model towards a univariate probability function

AbstractThis paper extends the investigation into the bivariate normal distribution (BND) model which has been widely used to study the asymptotic behaviour of the sum of a sufficiently large number of randomly‐varying harmonic phasors (of the same frequency). Although the BND model is effective and applicable to most problems involving harmonic summation, its main drawback resides in the computation time required to extract the probability density function of the harmonic magnitude from the two‐dimensional BND model. This paper proposes a novel approach to the problem by assimilating the generalized Gamma distribution (GGD) model to the marginal distribution (the magnitude) of the BND using the method of moments. The proposed method can accurately estimate the parameters of the GGD model without time‐consuming calculation. A power system containing ten harmonic sources is taken as an example where the comparison of the Monte‐Carlo simulation, the BND model and the GGD model is given and discussed. The comparison shows that the GGD model approximates the BND model very well.

Online Tracking of Changing Harmonics in Stressed Power Systems: Application to Hydro-Quebec Network

This paper applies, to a practical test case, two schemes for determining harmonics in power systems that are not limited to stationary waveforms, but can equally estimate harmonic phasors in waveforms with time-varying amplitudes and changing fundamental frequency. The first scheme relies on the short-time Fourier analysis performed using the fast Fourier transform (FFT) with time-domain windowing and frequency-domain interpolation. Another scheme combines fundamental-frequency tracking with a Kalman filter-based harmonic analyzer, yielding a uniformly sampled, self-synchronizing harmonics tracker. The performance of these advanced measurement schemes for changing harmonics is illustrated convincingly on nonstationary waveforms generated in the EMTP using a realistic series-compensated transmission network.

A Newton solution for the harmonic phasor analysis of AC/DC converters

The steady state equations that describe the power converter and DC system in the harmonic domain are solved by means of Newton's method, in a manner suitable for embedding in an iterative harmonic analysis of the complete power system. The solution includes the interaction of the power converter with the DC system, and the effect of variation in the firing and end of commutation angles caused by AC voltage and DC current harmonics. The convergence of Newton's method is investigated, and methods for accelerating the solution are implemented. Finally, the solution obtained is validated by means of time domain simulation.

Determining the joint moments of a motor drive's harmonic current phasors

It has been previously demonstrated that if a sufficiently large number of harmonic current sources are operating, a complete probabilistic characterization of the resulting power system-wide harmonic voltages can be found with knowledge of only the first and second order joint moments of each harmonic current's rectangular components. However, little consideration has been given to the problem of determining these joint moments for specific types of harmonic sources. This paper presents a method for analytically determining the joint moments of the resolved harmonic current components produced by two motor drives which utilize a phase-controlled rectifier. It is shown that when given the load torque as a function of the randomly varying motor speed, the means, variances and covariance of each harmonic current phasor's rectangular components can be expressed in terms of the probability density function of the motor speed.

Newton solution for the steady-state interaction of AC/DC systems

The steady-state interaction of a six-pulse converter with a Thevenin equivalent of the three-phase AC and DC systems is solved. The converter is represented by harmonic phasor mismatch equations, and the interaction between the converter and AC and DC systems is embodied in the solution of harmonic mismatch equations for the converter terminal conditions. Since there are a large number of simultaneous nonlinear equations to be solved, a sparse Newton-type solution is developed that exploits the harmonic three-port nature of the converter, and the relatively weak interaction between switching angles and terminal harmonics. The resulting solution is fast, and closely matches a time domain simulation of the test system.

On the assessment of harmonic pollution [of power systems

This paper analyses the interactions between the incremental changes of current harmonic phasors injected by a power system NL (nonlinear load) and the resulting variations of harmonic voltages and powers. It proves that it is not always possible, following simple measurements of individual harmonic powers, to decide if a certain harmonic current is harmful or useful. It is suggested that the NL "distortion" be evaluated with the help of a quantity called non60 Hz (nonfundamental) apparent power.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Newton-type algorithms for the harmonic phasor analysis of nonlinear power circuits in periodical steady state with special reference to magnetic nonlinearities

Computer algorithms of the Newton-Raphson type are derived for the harmonic analysis of systems containing nonlinear dynamic components in periodic steady state. The problem is formulated in the complex conjugate multiharmonic space that inherently represents the harmonic coupling between the different harmonic frequencies. The theory is applied to single-phase systems, including magnetic nonlinearities. >

Steady-state analysis of nonlinear dynamic systems with periodic excitation based on linearization in harmonic space

This paper considers a very general formulation of the differential equations of a dynamic system in periodic steady state. These are linearized around an operating point, in algebraic form in terms of incremental harmonic phasor components. The solution is iterative, either Newton-type with quadratic convergence in the neighbourhood of the solution, or it has linear convergence if the Jacobian is not updated at each iteration.

Six Phase Synchronous Machine with AC and DC Stator Connections, Part I: Equivalent Circuit Representation and Steady-State Analysis

This paper describes a detailed circuit representation of a six phase synchronous machine wherein mutual leakage couplings between the two sets of three phase stator windings are included. The main modes of power transfer of a six phase machine with sinusoidal inputs are examined. A harmonic phasor method of steady-state analysis to obtain the current, voltage and torque waveforms of a six phase machine with mixed AC-DC stator connections is described. It is shown that the output waveforms for various operating conditions from a digital computer program using the harmonic phasor method agree remarkably well with those obtained from an established detailed analog simulation.

Six-Phase Synchronous Machine with ac and dc Stator Connections, Part I: Equivalent Circuit Representation and Steady-State Analysis

This paper describes a detailed circuit representation of a six phase synchronous machine wherein mutual leakage couplings between the two sets of three phase stator windings are included. The main modes of power transfer of a six phase machine with sinusoidal inputs are examined. A harmonic phasor method of steady-state analysis to obtain the current, voltage and torque waveforms of a six phase machine with mixed AC-DC stator connections is described. It is shown that the output waveforms for various operating conditions from a digital computer program using the harmonic phasor method agree remarkably well with those obtained from an established detailed analog simulation.