Unbalanced Power Systems Research Articles

A total least squares enhanced smart DFT technique for frequency estimation of unbalanced three-phase power systems

In this paper, a robust technique is proposed for the estimation of fundamental frequency in unbalanced three-phase power systems. This is achieved by firstly justifying the modelling mismatch inherently overlooked in the conventional complex-valued least squares enhanced smart discrete Fourier transform method, and then applying the total least squares framework to minimise the error vectors corresponding to both the independent and dependent consecutive voltage measurements and, eventually, to achieve higher immunity to both noise and harmonic pollution. Simulations on both synthetic and real-world noisy unbalanced power systems demonstrate the performance advantage of the proposed algorithm over other smart discrete Fourier transform based ones.

A Simple Frequency Estimator for Power Systems

Fast and accurate frequency estimation of an unbalanced three-phase power system in the presence of measurement offset is a challenging task. A simple to implement and tune method is proposed in this work through a delay-based linear-regression framework. Using two challenging scenarios, we verify the performance of our procedure in the presence of severe grid abnormalities, including voltage unbalance, harmonics, frequency jump, and dc offset. Comparative hardware-in-the-loop experimental results demonstrate the suitability of the proposed approach.

Robust Frequency and Phase Estimation for Three-Phase Power Systems Using a Bank of Kalman Filters

In this paper we propose a powerful frequency, phase angle, and amplitude estimation solution for an unbalanced three-phase power system based on multiple model adaptive estimation. The proposed model utilizes the existence of a conditionally linear and Gaussian substructure in the power system states by marginalizing out the frequency component. This substructure can be effectively tracked by a bank of Kalman filters where each filter employs a different angular frequency value. Compared to other Bayesian filtering schemes for estimation in three-phase power systems, the proposed model reformulation is simpler, more robust, and more accurate as validated with numerical simulations on synthetic data.

Improved “Augmented MVDR Spectrum-Based Frequency Estimation Algorithm” for Unbalanced Power Systems

This article proposes mymargin an improved augmented iterative minimum variance distortionless response (AI-MVDR) frequency estimation algorithm which resolves a significant performance degradation of the AI-MVDR algorithm at off-nominal frequency of unbalanced power signals. This problem of the AI-MVDR algorithm is caused by the omitted terms in computing the augmented covariance matrix which is suggested to compensate the unbalance effect. In this article, a new frequency estimator is analytically derived considering the effect of the omitted terms by replacing a fixed nominal frequency parameter with a variable frequency parameter in obtaining the augmented covariance matrix. The performance of the derived equation is also verified through computer simulations.

An Analytical Model for Small Signal Stability Analysis in Unbalanced Electrical Power Systems

This paper presents a general model to carry out a small signal stability analysis in electrical power systems (EPSs) that operate in unbalanced conditions. The classic traditional approach is based on a phase representation where it is supposed that the triphasic electric grid does not suffer any variations during its operation. With the presence of unbalances in transmission lines and loads, it is necessary to develop a general model that allows answering the needs and challenges with which modern electric systems must deal. The present work firstly intends to address the three-phase representation of an EPS, including the controllers. The proposed model is applied to a classical test system, a three machine-nine bus system, considering all the dynamic and algebraic variations associated with angular stability analysis. The proposed approach to small signal stability analysis shows results that differ from the classical analysis. The results are backed up with time domain simulations, and therefore, these results could be used in the calculation of the controllers that operate in unbalanced multimachine systems.

An Enhanced IEEE 33 Bus Benchmark Test System for Distribution System Studies

The transformation of passive distribution systems to more active ones thanks to the increased penetration of distributed energy resources, such as dispersed generators, flexible demand, distributed storage, and electric vehicles, creates the necessity of an enhanced test system for distribution systems planning and operation studies. The value of the proposed test system, is that it provides an appropriate and comprehensive benchmark for future researches concerning distribution systems. The proposed test system is developed by modifying and updating the well-known 33 bus distribution system from Baran & Wu. It comprises both forms of balanced and unbalanced three-phase power systems, including new details on the integration of distributed and renewable generation units, reactive power compensation assets, reconfiguration infrastructures and appropriate datasets of load and renewable generation profiles for different case studies.

A Novel Low-Complexity Frequency Estimation Algorithm for Industrial Internet-of-Things Applications

In this article, we present a novel autocorrelation-based frequency estimation algorithm for single-tone sinusoidal signals. In comparison to other state-of-the-art frequency estimation methods, the proposed one provides a better tradeoff between accuracy, complexity, and estimation range. In particular, the algorithm is able to achieve the Cramer–Rao lower bound for moderate and high signal-to-noise ratio, and its implementation is feasible even in resource-constrained microcontrollers, as those commonly used in the Industrial Internet-of-Things (IIoT) applications and low-cost instrumentation. Finally, we investigate the performance of the algorithm in the case of a practical IIoT application, i.e., frequency estimation of unbalanced three-phase power systems, showing that it outperforms several other autocorrelation-based estimators.

Phase angle error‐based maximum correntropy adaptation for frequency estimation of three‐phase power system

A robust technique based on maximum correntropy criterion (MCC) is proposed for frequency estimation in an unbalanced power system using error in phase angle. This is achieved by replacing the magnitude error used in the standard cost function of the MCC with the phase error and by extending the concept of the augmented (widely linear) complex statistics for the processing of unbalanced voltages and other abnormal system conditions. The idea emanates from the facts that: (i) useful information is primarily conveyed over the phase in power system frequency estimation, and (ii) the performance degradation of the well-known minimum mean squared error criterion based adaptive filters in impulsive noise environments can be overcome by the MCC due to the non-quadratic and higher order moments imbedded in its cost function. The proposed phase error-based MCC method also utilises all the second-order information within the complex valued system voltage through the Clarke's transformation. Simulation studies conducted using both synthetic and experimental data verify that the proposed algorithm provides better estimations compared to the considered algorithms.

Estimation of Grid Frequency in Disturbed Converter-Based Power Systems by PLL State Variable Feedback

Fast real-time estimation of the grid frequency is essential for stable operation of renewable converter-based sources in future power systems. Therefore this paper presents a new phase-locked loop for the estimation of time-dependent frequencies in unbalanced power systems with harmonics. The proposed frequency estimation method consists of two signal processing steps: In the first step, a least mean square estimator reconstructs the fundamental sinusoidal signal from the measured three-phase grid voltage and splits it into positive, negative and zero sequence components. In the second step, the resulting first harmonic three-phase positive sequence is converted into the synchronous reference frame in the form of a phase-locked loop using a state feedback controller scheme to reconstruct the current grid frequency. Here the controller output is equivalent to the signal to be reconstructed. The feedback controller design is based on linear matrix inequalities where the requirements are explicitly considered. The capability of proposed state feedback phase-locked loop is demonstrated by full scaled electro magnetic transient simulations.

Adaptive Three-phase Estimation of Sequence Components and Frequency Using H∞ Filter Based on Sparse Model

The estimation of sequence or symmetrical components and frequency in three-phase unbalanced power system is of great importance for protection and relay. This paper proposes a new H ∞ filter based on sparse model to track the sequence components and the frequency of three-phase unbalanced power systems. The inclusion of sparsity improves the error convergence behavior of estimation model and hence short-duration non-stationary PQ events can easily be tracked in the time domain. The proposed model is developed using l 1 norm penalty in the cost function of H ∞ filter, which is quite suitable for estimation across all the three phases of an unbalanced system. This model uses real state space modeling across three phases to estimate amplitude and phase parameters of sequence components. However, frequency estimation uses complex state space modeling and Clarke transformation generates a complex measurement signal from the unbalanced three-phase voltages. The state vector used for frequency estimation consists of two state variables. The proposed sparse model is tested using distorted three-phase signals from IEEE-1159-PQE database and the data generated from experimental laboratory setup. The analysis of absolute and mean square error is presented to validate the performance of the proposed model.

A Generalized Index for Static Voltage Stability of Unbalanced Polyphase Power Systems Including Thévenin Equivalents and Polynomial Models

This paper proposes a Voltage Stability Index (VSI) suitable for unbalanced polyphase power systems. To this end, the grid is represented by a polyphase multiport network model (i.e., compound hybrid parameters), and the aggregate behavior of the devices in each node by Th\'evenin Equivalents (TEs) and Polynomial Models (PMs), respectively. The proposed VSI is a generalization of the known L-index, which is achieved through the use of compound electrical parameters, and the incorporation of TEs and PMs into its formal definition. Notably, the proposed VSI can handle unbalanced polyphase power systems, explicitly accounts for voltage-dependent behavior (represented by PMs), and is computationally inexpensive. These features are valuable for the operation of both transmission and distribution systems. Specifically, the ability to handle the unbalanced polyphase case is of particular value for distribution systems. In this context, it is proven that the compound hybrid parameters required for the calculation of the VSI do exist under practical conditions (i.e., for lossy grids). The proposed VSI is validated against state-of-the-art methods for voltage stability assessment using a benchmark system which is based on the IEEE 34-node feeder.

Evaluation of harmonic contribution to unbalance in power systems under non-stationary conditions using wavelet packet transform

Unbalance and harmonic distortion are two significant power quality (PQ) problems in modern power systems. Current research is focused on the contribution to unbalance of only the fundamental frequency component, separating the unbalance from the harmonics and ignoring the unbalance distortion existed in all harmonic components. This paper proposes a new methodology to accurately evaluate the contribution to unbalance of every harmonic component of voltages and currents using wavelet packet transform (WPT). WPT provides the advantage of handling non-stationary disturbances in compare with other signal-processing techniques, such as fast Fourier transform (FFT). The proposed WPT methodology offers an effective generalization to unbalanced distorted three-phase systems by maintaining a sufficient distinction between the effects of unbalance and distortion. The IEEE Standard 1459-2010 definitions are adapted, and the reformulation of the concept of symmetrical harmonic components using WPT is introduced to evaluate the balanced, first unbalanced and second unbalanced components of all harmonic orders. Synthetic waveforms including noise, a simulated three-phase system, and an experimental test are examined, considering stationary and non-stationary disturbances. The results are compared with their true values and those by FFT demonstrating the accuracy of the proposed WPT methodology and its superiority over FFT under non-stationary conditions.

Dynamic Frequency and Amplitude Estimation for Three-Phase Unbalanced Power Systems Using the Unscented Kalman Filter

We present a dynamic frequency and amplitude estimation algorithm for unbalanced three-phase power systems with harmonics. To develop the algorithm, we first construct a nonlinear dynamic model of the complex representation of the three-phase voltage with harmonic content and then apply an unscented Kalman filter to estimate magnitudes and frequencies of the fundamental component and its harmonics. Performance evaluation of the algorithm via Monte Carlo simulation shows the accurate tracking and fast convergence time when the frequency and amplitude of the grid voltage waveform change continuously or abruptly. Specifically, we benchmark the proposed algorithm with two types of input amplitudes and two types of frequency changes.

DFIG wind turbine under unbalanced power system conditions using adaptive fuzzy virtual inertia controller

The Doubly-Fed Induction Generator (DFIG) based Wind Turbines Generator (WTG) with traditional Maximum Power Point Tracking (MPPT) control provides no inertia response under system frequency events. Recently, the DFIG wind turbines have been equipped with the Virtual Inertia Controller (VIC) to enhance the frequency stability of the power system. However, the conventional VICs with fixed gain have negative effects on the inter-area oscillations of regional networks. To cope with this drawback, this paper proposes a novel adaptive VIC to improve both the inter-area oscillations and frequency stability. In the proposed scheme, the gain of the VIC is dynamically adjusted using fuzzy logic. The effectiveness and control performance of the adaptive fuzzy VIC is evaluated under different frequency events such as loss of generation and three-phase fault with load shedding. The simulation studies are performed on a generic two-area network integrated with a DFIG wind farm, and the comparative results are presented for these three cases: DFIG without VIC, DFIG with fixed gain VIC, and DFIG with adaptive fuzzy VIC. The results confirm the ability of the proposed adaptive fuzzy VIC in improving both the interarea oscillations and frequency stability of the system.

A Data Analytics Perspective of Power Grid Analysis-Part 2: Teaching Old Power Systems New Tricks [Lecture Notes

In Part 1 of this "Lecture Notes" article [1], we introduced a modern perspective on the standard tools for power system analysis-the Clarke and related transforms-through the lens of data analytics. We also indicated their limitations when dealing with unbalanced power system conditions.

Robust frequency estimation of unbalanced power system using a phase angle error based least mean kurtosis algorithm

In this paper, an augmented (widely linear) complex Least Mean Kurtosis (WL-LMK) algorithm that uses the negated kurtosis of the phase angle error instead of the conventional magnitude error is proposed for robust frequency estimation of three-phase power systems. The motivation of the proposed method is based on two underlying facts. First, it is primarily the phase rather than the amplitude that conveys useful information in frequency estimation. Second, the negated kurtosis based cost function is computationally efficient and most celebrated in applications where the noise contamination degrades the performance of the classical adaptive filtering algorithms. To further achieve robust frequency estimation for both balanced and unbalanced conditions, the proposed method makes use of the full second-order information within the complex-valued system voltage through the Clarke’s αβ transformation and the concept of augmented complex statistics and widely linear modelling. The estimation performance of the proposed method is evaluated for several critical cases that often arise in power system. Simulation results of both synthetic cases and experimental studies show that the proposed method achieves more accurate frequency estimates than its developed widely linear counterparts in the literature.

A new distributed measurement index for the identification of harmonic distortion and/or unbalance sources based on the IEEE Std. 1459 framework

The number and power of time-varying nonlinear loads connected to the electricity network has increased progressively in recent years, resulting in voltage and current disturbances. In the current, deregulated electricity market, with the proliferating presence of Distributed Generation (DG), it is becoming increasingly important to identify the source of such power quality disturbances. This paper presents a new index based on distributed measurements that allows us to identify the location of sources of harmonic distortion and/or unbalance in distribution networks. The novelty of the proposal lies in the use of terms taken from a different decomposition of apparent power for harmonic distortion and unbalance in power systems under IEEE Std. 1459. From these new terms, a series of indicators have been defined that facilitate the evaluation of the harmonic distortion and voltage unbalance of each user with respect to the electric power supply. The proposed index is validated by means of an experimental platform in which different working conditions have been developed, including load with capacitor banks for power factor compensation.

A modified backward/forward sweep-based method for reconfiguration of unbalanced distribution networks

<span lang="EN-US">A three-phase unbalanced power flow method can provide a more realistic scenario of how distribution networks operate. The backward/forward sweep-based power flow method </span><span lang="EN-AU">(BF-PF)</span><span lang="EN-US"> has been used for many years as an important computational tool to solve the power flow for unbalanced and radial power systems. However, some of the </span><span lang="EN-AU">few </span><span lang="EN-US">available research tools produce many errors when </span><span lang="EN-AU">they </span><span lang="EN-US">are used for </span><span lang="EN-AU">network </span><span lang="EN-US">reconfiguration </span><span lang="EN-AU">because the </span><span lang="EN-US">topology change</span><span lang="EN-AU">s</span><span lang="EN-AU">after multiple switch actions</span><span lang="EN-US"> and the nodes are disorganized continually. </span><span lang="EN-AU">T</span><span lang="EN-US">his paper presents </span><span lang="EN-AU">a modified</span><span lang="EN-AU">BF-PF for </span><span lang="EN-US">three-phase unbalanced radial </span><span lang="EN-AU">distribution networks</span><span lang="EN-US"> that is capable </span><span lang="EN-AU">of arranging</span><span lang="EN-US"> the system topology when reconfiguration changes the branch connections. A binary search is used to determine the connections between nodes, allowing the algorithm to avoid those problems when reconfiguration is carried out, regardless of node numbers. Tests are made to verify the usefulness of the proposed algorithm in both the IEEE 13-node test feeder and the 123-node test feeder, converging in every run where constraints are accomplished. This approach can be used easily for a large-scale feeder network reconfiguration.</span><span lang="EN-AU"> The full version of this modified </span><span lang="EN-US">backward/forward sweep</span><span lang="EN-AU"> algorithm is available for research at MathWorks</span><span lang="EN-US">.</span>

Magnitude‐cum‐phase angle error‐based WL adaptation for frequency estimation of three‐phase power system

Motivated by the need for robust frequency estimation for both balanced and unbalanced power system conditions and the fact that most of the existing techniques perform poorly in the presence of external interferences, a more noise-resilient widely linear least mean kurtosis (WL-LMK) algorithm is proposed that uses both the magnitude and phase angle errors for online estimation of three-phase power system frequency. The proposed method makes use of the full second-order information within the complex-valued system voltage through Clarke's α β transformation and the concept of augmented complex statistics and WL modelling. The effectiveness of the proposed scheme is verified through simulation studies of synthetic signals and experimental data.

Application of Multi-Resonator Notch Frequency Control for Tracking the Frequency in Low Inertia Microgrids Under Distorted Grid Conditions

Frequency variations are much faster and more significant in microgrids with low-inertia inverters and other control devices for distributed generation. Accordingly, it is imperative to utilize a concise frequency estimation method for analyzing the transient behavior of microgrids. This paper proposes a robust algorithm which is based on multi-resonator and adaptive notch filter and is referred to as the multi-resonant notch frequency control for improving the frequency estimation of perturbed power system. In general, power system distortions such as harmonics, inter-harmonics, and faults may affect the accuracy of frequency estimation. In contrast to conventional methods for frequency estimator, the proposed method is less sensitive to distortions and unbalanced power system conditions. The proposed method uses the information in all three-phase signals which is proved to be more robust against signal variations than single-phase methods. Simulation results demonstrate the accuracy and the speed of the proposed method for frequency estimation in both grid-connected and islanded mode of low inertia microgrids under grid faults.