Norton Equivalent Model Research Articles

Multi-Inverter Resonance Modal Analysis Based on Decomposed Conductance Model

The Norton equivalent model based on the transfer function and the frequency domain analysis method for inverter resonance analysis lacks a comprehensive analysis of the resonant characteristics, and more information about the resonant key components and the degree of participation cannot be obtained. In this paper, a decomposed conductance model is proposed to characterize the resonance characteristics of the multi-inverter grid-connected system and the effect of the equivalent control link of the inverter on the resonance in more detail by combining the modal analysis method and the sensitivity analysis method. Firstly, based on αβ coordinates, the conductance division is carried out for the dual-loop inverter control link with the voltage external loop and current internal loop using capacitor-current feedback damping, and the inverter model based on the decomposition conductance is derived. The mathematical model of the multi-inverter grid-connected system is then established. Secondly, the resonance characteristics of the system are analyzed by combining the modal and frequency domain analysis methods when the number of inverters, inverter parameters, and grid-side impedance are changed. Thirdly, the degree of involvement of the system components, especially the equivalent control link of the inverter in resonance conditions, is determined in combination with the proposed model and the sensitivity analysis method, which is the basis for proposing an effective suppression strategy. Finally, a simulation model is built to verify the proposed method and the analysis results.

Estimation of harmonic impedance and harmonic contribution with harmonic complex power in the absence of harmonic phase angle

Estimation of harmonic impedances and harmonic contributions is beneficial to harmonic control, and harmonic phasors data is necessary for the existing estimation methods. However, the phase angles of harmonic phasors are not recorded in some of current power quality measuring instruments (such as Fluke 1760 etc.), which makes the existing methods unavailable. To solve this problem, the Norton equivalent circuit model under harmonic frequency is improved and a new model composed of harmonic complex power sources and harmonic impedances is proposed. Based on the model, the harmonic impedance and harmonic contribution can be estimated by using independent component analysis algorithm, and the harmonic phasor data is no longer required. The superiority of the proposed model is verified through simulation and field cases.

Stability Evaluation of Grid-Connected Microgrid Clusters in Asymmetrical Grids

Stability of grid-connected microgrids could be to a high extent guaranteed by designing a robust control system for inverters. However, this is not necessarily the case when microgrids are connected to an asymmetrical grid, where the grid impedance could have a different value in each phase. In this condition, the asymmetry of the grid impedance causes couplings between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">αβ</i> - or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> -frames in the average model, and therefore, the control system could not be simplified as a single-input single-output system anymore. In this case, the system turns into a multi-input multi-output system, and therefore, stability evaluation of such a system consisting of several microgrids that form microgrid clusters with multiparallel inverters could be a challenging task, which needs heavy mathematical derivations. In this article, a straightforward method is used for modeling the grid-connected microgrid clusters, in which the asymmetry of the grid impedance has been considered. Based on this approach, all the inverters in a microgrid are lumped in a frequency-domain-based model using the inverters’ Norton equivalent models. Then, based on the derived model, the generalized Nyquist criterion is used to evaluate the stability of the whole system. The proposed method considerably reduces the mathematical burden, while accurately predicts the system stability. The experimental results in a small-scale test bench in the laboratory validate the accuracy of the proposed method.

Parameter Determination of Coupled and Decoupled Admittance Matrix Methods of the Norton Equivalent Model for an Air Extractor

Context: Studies carried out in low voltage networks have explored the modeling of linear single-phase loads (such as motors) and non-linear ones (such as those based on power electronics). However, induction motors exhibit non-linear characteristics between voltage and current due to the saturation of their magnetic parts. Therefore, it is necessary to study induction motors in the frequency domain with a model that allows reviewing the characteristic nonlinearity of their voltage-current interaction. Methodology: This article presents the frequency domain modeling of a single-phase induction motor used as a silent air extractor (127 V, 60 Hz, 66 W), which presents a capacitive behavior (fp = 0,93 in leading) and harmonic distortion due to a third-order component (7,0%) when fed with a pure sinusoidal voltage of 127 V. Results: This work establishes the parameters of two approaches to the Norton equivalent model (coupled and decoupled admittance matrix) which are used to estimate the distorted current signal and the values ​​of consumed active and non-active power. The results show comparisons errors of P, Q, THDi, and NRMSE indices of less than 7, 4, 14, and 3%, respectively. Conclusions: The parameters of the Norton equivalent model estimated for the single-phase induction motor allow calculating the current signal with a high degree of precision. This signal exhibits nonlinear characteristics and a capacitive behavior due to the permanent presence of a capacitor aiding the start and operation of the engine.

Analysis of the Performance of the Norton Equivalent Model of a Photovoltaic System Under Different Operating Scenarios

The massive integration of photovoltaic systems to distribution networks has affected their operation in terms of harmonic distortion in voltage and current signals. This is due to the non-linear characteristic of photovoltaic inverters. For this reason, it is necessary to characterize these systems in different operating scenarios in order to analyze in detail their impacts on electrical grids. However, only a few studies use frequency domain models in order to study their real behavior. Therefore, this work seeks to estimate the parameters of the Norton equivalent model of the coupled and decoupled admittance matrix of a photovoltaic system that injects active power into the low voltage network of a university building. It is proposed to analyze the performance of the model with three operating scenarios based on solar irradiance, active power generated and the THDi index. Measurements at the common coupling point between the photovoltaic system and the electrical network have been taken into account in order to obtain the models and validate them. The results obtained show the analysis of the performance of the models for the three proposed operating states, in terms of powers, power factor, estimation of the waveforms of the current signal and the THDi index.

Differential Mode Noise Estimation and Filter Design for Interleaved Boost Power Factor Correction Converters

Interleaved power factor correction (PFC) is widely used circuit topology due to good efficiency and power density for single-switch boost PFC. As the differential mode (DM) electromagnetic interference (EMI) noise magnitude depends upon the input current ripple, this research details a comprehensive study of DM EMI filter design for interleaved boost PFC with the aim of minimizing the component size. It is also demonstrated that the different numbers of interleaved stages and switching frequency influence the filter attenuation requirement and, thus, the EMI filter size. First, an analytical model is derived on the basis of the Norton equivalent circuit model for the differential mode noises of interleaved boost PFC within the frequency range of 9–500 kHz. The derived model can help identify the proper phase shifting among the interleaved boost converters in order to minimize the considered differential mode noises at the filter design frequency. So, a novel phase-shift method is developed to get a minimized attenuation required by a filter in Band B. Further, a volume optimization of the required DM filter was introduced based on the calculated filter attenuation and volumetric component parameters. Based on the obtained results, unconventional and conventional phase shifts have demonstrated a good performance in decreasing the EMI filter volume in Band B and Band A, respectively. A 2-kW interleaved PFC case study is presented to verify the theoretical analyses and the impact of phase-shifting on EMI filter size.

Impact of digital time delay on the stable grid‐hosting capacity of large‐scale centralised PV plant

In view of the trend towards extensive application of digital controllers in the PV inverter of large-scale centralized photovoltaic (LSCPV) plant and the increasing number of grid-connected LSCPV plants, this paper investigates in detail the influence of the digital time delay of the inverter digital controller on the stable grid-hosting capacity of LSCPV plant. The studies are based on the Norton equivalent model of the grid-connected LSCPV system considering the digital time delay when modelling the digital control system of the PV inverter. Taking into account the actual situation in LSCPV plant, the stable grid-hosting capacity of LSCPV is discussed for the cases in which the PV inverters in the LSCPV plant have the same and different digital time delay values by using the root-locus method. Simulation results of the digitally controlled grid-connected LSCPV system model validates the theoretical analysis.

High-Q perfect absorption induced by the coupling of LSP and SPP modes

At optical frequencies, we design an anisotropic metasurface to realize high-Q perfect absorption in numerical calculation. By establishing the Norton equivalent network model, we first reveal that the root cause of this phenomenon is the coupling excitation of the localized surface plasmon (LSP) mode and the surface plasmon polariton (SPP) mode. This absorber can realize bifunctional switching and is expected to realize the energy conversion of photons–SPP–photons. Meanwhile, we analyze the control of the structural design on high-Q perfect absorption. This research will provide theoretical guidance for the further development of applications related to high-Q perfect absorption such as absorption filters with specific wavelengths and narrow band photon emitters.

Assessing Time-Varying Harmonic Interactions in a Wind Park

The harmonic interaction mechanism in a wind park is examined in this paper. The paper investigates the feasibility of a solution to the yet challenging harmonic contribution estimation from multiple sources in a wind park with limited available information, via an extension of a simple modeling approach as well as from detailed analysis of field measurements. The paper has two distinct objectives in assessing harmonic interactions (a) one to extend the classical Norton equivalent model to a multi-measurement wind park system and to suggest potential areas for further model developments from field measurement analysis, and (b) second to draw inferences from field measurements and to develop a new independent concept of analysis from long-term field measurements in the wind park. From practical experience, a new concept of analysis with a ‘harmonic interaction break-even point’ is introduced. With the help of it, one could identify whether the primary emission (emission from considered source) or secondary emission (emission from a distant source) dominates in the analysis period. In this way, the highest responsibility between different interacting time-varying harmonic sources is evaluated. It was concluded that from long-term measurements one can define a magnitude of power production where a certain harmonic order is canceled or reaches its lowest magnitude. If one finds this cancellation point, one can define a level of secondary/primary emission or at least a feasible range. This knowledge is a step forward towards harmonic contribution analysis.

Determination of parameters and performance analysis of load models for fluorescent recessed lightings before power supply signal variations

The high utilization of nonlinear devices in buildings increases harmonic distortion in the electrical voltage and current signals, which has led to the requirement of characterizing and predicting the impact of such devices on low-voltage networks using load models. One of the charges that still performs an important role in the energy consumption of commercial, industrial, and educational buildings is the recessed fluorescent light with electronic ballast, which is commonly used in Colombia. However, information regarding the modeling of their behavior in the case of distorted power supply voltages is scarce. Therefore, this work presents the parameter configuration and performance analysis of two load models in the time domain, i.e., ZIP and exponential, and one model in the frequency domain, i.e., Norton equivalent model to the coupled admittance matrix method, for two commonly used recessed fluorescent lights.

Novelty method for the utility harmonic impedance estimation based on Gaussian mixed model

The accurate estimation of harmonic impedance on the utility side of a point of common coupling (PCC) is much important to harmonic control, harmonic emission level, and harmonic responsibility analysis. In this study, a novel estimation method for the utility harmonic impedance is proposed, which employs the Gaussian mixed model (GMM) to fit the measured data at a PCC, because the authors find that the statistical distribution of the measured data at a PCC is close to the Gaussian distribution. Also, in theory, GMM can approach the probability distribution of any data. In this method, the Norton equivalent circuit model at a PCC is first expressed as a GMM, and the utility harmonic impedance in the GMM is estimated by expectation maximisation iterative method. Compared to the existing estimation methods, an outstanding advantage of this method is that it is less affected by background harmonics; therefore, higher accuracy can be acquired in this method. The effectiveness and accuracy of this method are verified by simulation experiments and field data.

A New Method for Determination of Harmonic Responsibility Based on Quality Engineering Theory

At present, the essence of traditional methods for determination of harmonic responsibility among different parts in power system is mainly based on the contribution of relevant physical quantity on the point of common coupling (PCC) of both utility side and customer side. However, theoretical basis of various methods is of much difference, leading to a large difference in the final calculation result of harmonic responsibility. Therefore, there is no reliable approach applied in actual cases. This project aims to fill the current gap in literature, proposing a new method to determine harmonic responsibility from the perspective of economic loss. Firstly, according to power theory of IEEE Std.1459-2010 and quality engineering theory, the index set of harmonic pollution evaluation on PCC is established, and the expressions of calculation of economic loss are obtained. Secondly, collecting relevant data on PCC, the parameters of Norton equivalent models of utility side and customer side are obtained by the reference impedance method, then contributions of corresponding physical quantity on PCC can be analyzed. Based on relevant expressions of economic loss, calculation results of economic loss are all obtained. At the same time, values of harmonic responsibility of utility side and customer side are calculated. Finally, new adjustment scheme of rewarding-penalizing electricity charge (RPEC) is discussed and compared with other traditional schemes. After comparison, Results of comparative analysis show that the new adjustment scheme can not only increase the penalty for harmonics but also reflect commodity characteristic of electricity energy. In this manner, calculation result of electricity charge is more reasonable. Hence, electricity environment on PCC can be improved.

Comprehensive Harmonic Responsibility Calculation Based on Different Weighting Methods

Dividing the responsibility of harmonics between customer and utility sides is necessary to improve the objectivity, rationality, and scientificalness of power measurement and power quality evaluation. On the basis of expert experience and customer needs and considering the background harmonic fluctuation, this study proposes two reasonable conditions to evaluate harmonic responsibility. At the same time, the reference index of harmonic responsibility calculation and the problem of different frequency harmonics are considered in the comprehensive calculation. On the basis of the IEEE Std. 1459–2010 power theory, the index set of harmonic responsibility is established, and several common subjective and objective weighting methods are used to weigh each index. On the basis of optimization theory, an optimization model is established by constructing a Lagrange function in finding the condition extremum to unify the subjective and objective information. Finally, the calculation method is verified by a Norton equivalent model, and the harmonic responsibility of each index is calculated by the harmonic current vector method; thus, calculating the actual impedance of customer and utility sides is unnecessary, making the calculation simple and effective. Results of the comparative analysis show that the comprehensive evaluation method of harmonic responsibility with the combinatorial weighting method can not only meet the different needs of different groups for harmonic indicators but also reflect the background harmonic fluctuations sensitively. In this manner, the subjective and objective information are unified, and the shortcomings of a single weighting method are overcome. Hence, the evaluation results are accurate, practical, and reasonable.

Symmetric-component decoupled control of grid-connected inverters for voltage unbalance correction and harmonic compensation

In this paper, a strategy is investigated for control of distributed generators to additionally support the local grid on top of conventional electricity generation. The strategy decouples unbalance and harmonic compensation in the phase sequences and the frequency domain for a grid-connected inverter with local unbalanced and distorting loads. Unlike conventional control schemes for grid-connected converters, the proposed strategy is designed to be sequence-asymmetric for the purpose of unbalanced and harmonic local voltage correction. A frequency-domain Norton equivalent model is derived to illustrate the working principle of the strategy. Accordingly, it is shown that following a frequency-domain decoupled method the fundamental positive-sequence, the harmonic symmetrical sequence, the fundamental zero-sequence and the fundamental negative-sequence components can be regulated independently. Consistent to the model analysis, experiments validate the reduction of the local voltage total harmonic distortion and unbalanced factor when the converter is programmed for both harmonic sinking and voltage unbalance correction.

An Improved Antenna Scattering Model: An equivalent model based on the reciprocity theorem

The Thevenin and Norton equivalent models of a receiving antenna cannot be used to calculate the antenna's scattered power because they are too simplistic. Based on the reciprocity theorem, a more elaborate equivalent model, consisting of a passive distributed transformer, is proposed for wire antennas in both the transmit and receive modes. The model parameters are derived from the antenna's transfer admittance parameters, the latter obtained under short circuit loading conditions in transmit mode only. A method is then applied to determine the new admittance parameters when the antenna is perturbed by an arbitrary load, which allows the receive-mode current distribution on the antenna to be calculated. Once the receive-mode current is known, scattered power can be calculated with a formula that avoids integration over the far-field sphere. The method has been applied to o a l/2-dipole, l-dipole, and three-element Yagi antenna for resistive loads spanning up to nine decades. Calculated scattered-power values are shown to agree almost perfectly with values obtained using a commercial simulator, thus demonstrating that it is possible to calculate scattered power based on the characterization of the antenna in transmit mode only. Results indicate that the aperture efficiency of a dipole antenna can exceed 50% when it is connected to a mismatched load. Finally, a newly defined transducer efficiency is presented, which can be used to choose an optimum load to obtain the best compromise between the aperture and scattering areas.

Norton Equivalent Circuit for Pulsed Photoconductive Antennas—Part II: Experimental Validation

This second part of two papers’ sequence presents the experimental validation of the Norton equivalent circuit model for pulsed photoconductive antennas (PCAs) provided in the first paper of the sequence. To this goal, different prototypes of photoconductive antenna sources have been manufactured and assembled. The average powers radiated and their pertinent energy spectral densities have been measured. In order to obtain a validation of the original equivalent circuit proposed, an auxiliary electromagnetic analysis of the complete setup, including the quasi-optical (QO) link for the signals from the antenna feeds to the detectors had to be developed. By using the combined theoretical model (circuit and quasi-optics), an excellent agreement is achieved between the measured power and the power estimated. This agreement fully validates the circuit model, which can now be used to design new PCAs, including optical and electrical features of the semiconductor materials, as well as the details of the antenna gaps and the purely QO components.

Impact of phase-locked loop on stability of active damped LCL-filter-based grid-connected inverters with capacitor voltage feedback

Active damped LCL-filter-based inverters have been widely used for grid-connected distributed generation (DG) systems. In weak grids, however, the phase-locked loop (PLL) dynamics may detrimentally affect the stability of grid-connected inverters due to interaction between the PLL and the controller. In order to solve the problem, the impact of PLL dynamics on small-signal stability is investigated for the active damped LCL-filtered grid-connected inverters with capacitor voltage feedback. The system closed-loop transfer function is established based on the Norton equivalent model by taking the PLL dynamics into account. Using an established model, the system stability boundary is identified from the viewpoint of PLL bandwidth and current regulator gain. The accuracy of the ranges of stability for the PLL bandwidth and current regulator gain is verified by both simulation and experimental results.

Measurement-Based Compartmental Modeling of Harmonic Sources in Traction Power-Supply System

Harmonic pollution of the traction power-supply system (TPSS) has become a vital concern for its adverse impacts on the tractive drive system of the train and power-quality issues of TPSS and connected power system. Harmonic modeling is the precondition to evaluate the harmonic pollution for TPSS. This paper systematically discusses the main harmonic sources in TPSS as well as their characteristics based on accurate field measurements. Moreover, the background harmonic voltage of the connected utility system and the harmonic of the train are compartmentalized and modeled with consideration of the stochastic and statistical features. This paper studies the background harmonic voltage with time slicing and probability statistics technique. Meanwhile, the harmonic of the train is described with the Norton equivalent model, which considers the interaction between harmonic voltage and harmonic current. Moreover, the influence of the working conditions and power output on the harmonic distribution is mentioned in the low-order harmonic modeling of the train. The results suggests the proposed methods can effectively restore the distribution and characteristics of the harmonics in TPSS.

Linear equivalent models at the maximum power point based on variable weather parameters for photovoltaic cell

In order to sweep completely the obstacle to the whole linearization of photovoltaic (PV) system with non-linear PV cell, in this paper, the voltage-current characteristic of PV cell at the maximum power point (MPP) is linearized and two linear equivalent models including Thevenin equivalent model and Norton equivalent model are proposed. On the basis of this work, the whole linearization of PV system is workable and reasonable, and then the conventional linear theories or laws can be used to study PV system conveniently. Meanwhile, in this work, the direct relationships between three linear model parameters and variable weather parameters (VWP) were found, which ensures the strong adaptation of these proposed models to the varying weather conditions. Finally, some simulation experiments verify that these proposed models are feasible and available in practical application, illustrate that the characteristics of three linear model parameters are influenced by varying weather conditions and unaffected by varying load, and show that PV system using the proposed Thevenin equivalent model has the same maximum power point tacking (MPPT) stead-state performance and similar MPPT transient-state performance with the conventional four-parameter model under fast varying weather conditions.

Power-Quality Impact Assessment for High-Speed Railway Associated With High-Speed Trains Using Train Timetable—Part I: Methodology and Modeling

The proliferation of the voltage-source converter-based high-speed trains has resulted in significant distortions in voltage and current waveforms in both traction power supply system and the utility system. The dynamic behaviors of the high-speed trains (HSTs) make the assessment of such power-quality (PQ) problems quite difficult. There is an urgent need for techniques that can quantify the collective PQ impacts of modern trains during a 24-h period. Dynamic behavior modeling of the modern trains during the operating duty period between two station stops is studied here for PQ assessment. The 24-h profiles of the train timetable and rail infrastructure are entered to compute the information, including train positions, speeds, power consumptions, etc. Moreover, six sets of the measurement-based Norton-equivalent model under different operations are implemented to represent the dynamic harmonic behaviors of the train. In addition, the systemic modeling of the utility system, traction lines, and Scott-connection transformer is also described. After comparing the results of calculations and measurements, the proposed model is ideally effective for analyzing the consequences of HST's dynamic behavior and system topology that are involved in fundamental power flow and harmonic power flow in order to evaluate the comprehensive PQ impacts in a companion paper.