Norton Equivalent Circuit Research Articles

An Efficient Half-Bridge MMC Model for EMTP-Type Simulation Based on Hybrid Numerical Integration

Electromagnetic transient (EMT) simulation is critical and fundamental in the design and operation of the modular multilevel converter (MMC). This paper proposes a highly efficient EMT simulation model for MMC based on hybrid numerical integration. The topology and operational principle of MMC are elaborated first. Based on this, an efficient simulation model for MMC is proposed. Each arm of the MMC is reduced to a two-node Norton equivalent circuit in the main network simulation. The computation of the capacitor dynamic equations is decoupled from that of the arm inductor dynamic equation. They are computed in a leapfrog manner. This makes the equivalent conductance of the MMC arm constant and greatly improves the simulation efficiency. Moreover, the dynamic equations of the capacitors are also decoupled from each other. Various operational conditions of the MMC are also considered, and accuracy and adaptability are thus ensured. Extensive studies on power systems with different scales verify the accuracy and improved efficiency of the proposed EMT simulation model for MMC.

Unified Real-Time Simulation Method for DC/DC Conversion Systems Consisting of Cascaded Dual-Port Submodules

The DC/DC conversion systems are constructed from multiple cascaded dual-port submodules in series and/or parallel at both the input port and output port. The complexity of the series-parallel structures brings great challenges to real-time simulation. This paper proposes a unified real-time simulation method of series-parallel DC/DC conversion systems, which unifies the input-series-output-series (ISOS), input-series-output- parallel (ISOP), input-parallel-output-parallel (IPOP) and input-parallel-output-series (IPOS) structures. First, based on the nodal analysis method, the dual-port equivalent model of each submodule in the series-parallel DC/DC conversion system is obtained. Second, by cascading each submodule, the unified form of the system dual-port equivalent model of the series-parallel DC/DC conversion system is constructed. For the series connection, the port is equivalent to Thevenin's equivalent circuit. For the parallel connection, the port is equivalent to Norton's equivalent circuit. The real-time simulation model of a specific series-parallel DC/DC conversion system can be obtained by combining the input/output port equivalent circuits. Besides, the high-frequency waveforms of the internal circuit can also be obtained through parallel calculation steps. The real-time simulation of the proposed unified model is carried out with a 250ns time-step on Xilinx K-7 Series FPGA, which verifies the accuracy and simulation efficiency of the unified model.

Numerical Stability and Dispersion Analysis of the 2-D FDTD Method Including Lumped Elements

The numerical stability and dispersion analysis of the extended two-dimensional finite-difference time-domain (2D-FDTD) method are systematically studied. Particularly, three different passive linear lumped elements including the resistor, inductor, and capacitor are analyzed, respectively. Moreover, three different formulations of the explicit, semi-implicit, and implicit schemes are discussed, respectively. Furthermore, by combining the von Neumann technique and Jury criterion, the numerical stability of the extended 2D-FDTD method is analyzed, which has not been reported thus far. Theoretical results show that: 1) For the resistor, the stability condition is same as the FDTD method unloaded case. 2) For the inductor, in the explicit and implicit schemes, the stability is connected with the value of the inductance; for the semi-implicit scheme, the stability is independent of the value of the inductance. 3) For the capacitor, the stability relationship is related to both the mesh size and the value of the capacitance. On the other hand, based on the Norton equivalent circuit, the analysis of the numerical dispersion of the extended 2D-FDTD is presented; and some interesting theoretical results are deduced. Finally, the microstrip circuits including the three lumped elements are simulated to demonstrate the validity of the theoretical results.

Time-Domain Modelling of Pulsed Photoconducting Sources—Part I: The Norton Equivalent Circuit

In the circuit theory, the Norton and Thevenin equivalent generators are tools that simplify the solutions of networks involving passive or active components. They have been extensively used in the frequency domain to describe time-harmonic sources. A time-stepped evolution is instead typically used to include transient sources. As a particular case of the latter, the Norton equivalent circuit is extended here to investigate pulsed photoconducting sources, where a dc bias voltage and a pulsed optical laser are combined to generate terahertz (THz) bursts. The proposed derivation relies on the application of the electromagnetic (EM) equivalence theorem. The main conclusion of this derivation is the understanding that, from the three different spectral regions (dc, THz, and optics), only the THz radiation is to be explicitly included in the equivalent circuit. The theory is validated by a campaign of measurements reported in a connected paper.

Settling Angle-Based Stability Criterion for Power-Electronics-Dominated Power Systems

Impedance-based analysis methods are widely applied to the small-signal stability analysis of grid-connected converters. As the initial step to carry out the impedance-based stability analysis, it is critical to select whether to represent the converter as a Norton or Thevenin equivalent circuit. However, there is lack of definition on the representation of converters, and inappropriately selected impedance model choice leads to improper stability analysis. Moreover, existing impedance-based stability analysis methods, e.g., return-ratio matrix (RRM)-based method, only apply to minimum phase systems where the RRM does not contain right-half-plane poles. In complex and power-electronics-dominated power systems, interactions between multiple converters lead to nonminimum phase power systems, where current methods can no longer be directly applied. To address these problems, a stability criterion is proposed by introducing the concept of settling angle. As the main contribution of this article, the settling angle of total admittance is proven as the intrinsic hallmark of power systems and directly reflects the system stability. The proposed settling-angle-based stability criterion is validated by control hardware-in-the-loop experimental results for a three-terminal system that integrates grid-forming and grid-following converters through a common bus.

Time-Domain Modelling of Pulsed Photoconducting Sources—Part II: Characterization of an LT GaAs Bow-Tie Antenna

Drude’s description of the response of low-temperature gallium arsenide to optical pulse excitation is used to evaluate the components of a time-domain Norton equivalent circuit of a photoconductive antenna (PCA) source. The saturation of the terahertz (THz) radiated power occurring at large optical excitation levels was previously associated by the scientific community to radiation and charge screening of the bias. With the present circuit, we are able to model accurately the measured saturation as only due to the EM feedback from the antenna to the bias. The predicted THz radiated power is shown to match very accurately the measurements when the circuit is combined with an accurate description of the experimental conditions and the modeling of the THz quasi-optical (QO) channel.

A Design and Validation of 400 W PV Emulator Using Simple Equivalent Circuit for PV Power System Test

In this paper, we propose a photovoltaic emulator (PVE) composed of a series of connected power diodes with a single constant current source, rather than a power converter-based PVE. Accordingly, this enables readily verifying the operation because a the simple hardware structure and lack of a complex control algorithm. The proposed PVE can be intuitively implemented using the proposed selection criteria for the power diode and equivalent resistances. Since there is no feedback control based on complex analog/digital controls and sensors, the control response can be very fast. In addition, the proposed PVE can be easily integrated with a PV power system such as a power optimizer, to allow testing in a simple and flexible manner. Spice simulation was performed based on the electrical characteristics provided by the solar panel manufacturer, and this was utilized to validate the applied emulator circuit model, the Norton Equivalent Circuit. During the design process, the simulation helped to manufacture the PVE within error ranges satisfying the desired I–V and P–V curves, as well as the maximum power point (MPP). Partial shading could be easily implemented through use of the multiple series connection of individual PVEs, demonstrating local MPP and global MPP. A 400 W class PVE was built through appropriate power diode selection, with a thermal design to increase the output power. Moreover, its performance and feasibility were verified through intensive experiments. The measured efficiency, transient response time, and maximum transient error of the partial shading tests were 91%, 22 μs, and 5.8%, respectively.

Implementation of an Alternative Frequency-Dependent Three-Phase Transmission Line Model Based on the Folded Line Equivalent Model in MatLab-Simulink

This paper proposes an alternative multiconductor transmission line model that combines the folded line equivalent with the modal transformation. The folded line equivalent decomposes the nodal admittance matrix of a transmission line into its open-circuit and short-circuit contributions. These contributions are fitted to rational functions, which are associated with Norton equivalent circuits based on their state space models. The proposed model uses an orthogonal matrix to transform voltages and currents from the phase domain to the folded line equivalent domain and vice versa. Because the transformation matrix is orthogonal, we represent it using ideal transformers in simulation software. First, we use a circuit representation of Clarke’s matrix to decompose a transmission line into its modes. Then, each mode is decomposed into its open-circuit and short-circuit contributions using a circuit implementation of the proposed matrix. The proposed approach can accurately represent short lines in simulations with time steps equal to or greater than the propagation time of the transmission line. We compare the results obtained with the proposed approach to those obtained with power systems computer-aided design/electromagnetic transients including the DC universal line model.

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.

Supraharmonic Emissions from DC Grid Connected Wireless Power Transfer Converters

Power converters for wireless power transfer (WPT) and, in general, for electrical vehicle charging are evolving in terms of nominal power and performance, bringing along non negligible emissions in the supraharmonic range (2 kHz to 150 kHz). The large installed power and the high concentration with a relatively short separation distance can be addressed by feeding the converters through a DC grid for better dynamic response and lower impedance. The prediction of conducted emissions in real supply conditions requires carrying out measurements with low impedance values, lower than those available in line impedance stabilization networks (LISNs) for AC grids. This work proposes an approach to extrapolate converter emissions in an ideal 0 Ω condition, that together with the input impedance curve (determined by a least mean square approach) form a Norton equivalent circuit of the converter. The interaction of the converters with the DC grid and superposition of emissions can be then thoroughly evaluated by means of a general ladder grid scheme to which the Norton equivalents are connected. Such a grid model is suitable for Monte Carlo simulation aimed at assessing the degree of compensation between sources of emissions and the overall network distortion. Results using a Simulink model are provided considering emissions aggregation and compensation under random phase conditions for the following cases: close-by and separated sources (5 m and 100 m cable separation, respectively); increased number of sources studying scenarios with 3 and 10 sources; and using different resolution bandwidth values (200 Hz and 500 Hz) against a random change of the frequency of the emission components.

Efficient Implementation of Multi-Port Frequency Dependent Network Equivalents for Electromagnetic Transient Studies using Norton Equivalent Circuits

Power system electromagnetic transient studies require that a small part of the electrical network be modelled in detail. The rest of the system is represented by a network equivalent taking into account the frequency dependence of the system components (FDNE). With the purpose of getting a FDNE, this paper presents a procedure for including high-order rational models in EMTP-type simulation programs based on Norton equivalent circuits. This approach has the advantage of using few conductance branches compared to those obtained by fitting any particular circuit structure.In relation to the work carried out, this paper presents the following issues: a) a procedure for calculating the admittance matrix of the rest of the system as a function of frequency b) this matrix is synthesized in the form of multi-terminal π-equivalent circuits, whose branches are fitted by rational functions in the pole-residue form c) each partial fraction is converted into a differential equation in time domain. The trapezoidal rule of integration is applied to that differential equation, resulting in a conductance in parallel with a history term current source. Considering all the partial fractions corresponding to a branch, a Norton equivalent circuit is obtained for that branch.This paper also shows how the procedure is implemented in three different ways in an EMTP-type program, along with the validation of such approach through simulations in an electric utility 500 kV transmission system.

Optimization Configuration of Grid-Connected Inverters to Suppress Harmonic Amplification in a Microgrid

This paper provides insight into the optimal configuration scheme of the grid-connected inverters based on harmonic amplification suppression. The connection of the inverters changes the natural resonance frequencies of the grid. Hence, a reasonable configuration of grid-connected inverters can optimize the impedance distribution and shift the natural resonance frequencies to frequency bands farther away from the harmonic sources. We proposed a scheme of site selection and determination of the number of inverters to suppress harmonic amplification. The resonance frequencies and modal frequency sensitivities (MFSs) were obtained by the resonance modal analysis (RMA). Moreover, the concepts of security region and insecurity region of resonance frequency were illustrated. The grid-connected sites can be obtained by calculating the participation factors (PFs) of the resonance frequencies in the insecurity region. Furthermore, the optimal number was determined by building the Norton equivalent circuit of the inverter and evaluating the output impedance at each frequency. Finally, simulations in Matlab/Simulink based on a modified IEEE-9 bus microgrid were utilized to verify the effectiveness of the proposed scheme.

Research on modeling and grid connection stability of large-scale cluster energy storage power station based on digital mirroring

With the large-scale integration of renewable energy into the grid, its randomness and intermittent characteristics will adversely affect the voltage, frequency, etc. of the new power system, and even cause partial system collapse. However, the above problems can be solved by configuring large-scale clustered energy storage in the new power system. This paper proposes the structure and technical points of the digital mirroring system of large-scale clustered energy storage power station, and conducts mathematical modeling for the lithium-ion battery system and grid-connected system of Power Conversion System (PCS for short) of large-scale clustered lithium-ion battery energy storage power station. Based on this, This paper proposes a control strategy that introduces virtual resistance, establishes a Norton equivalent circuit with multiple PCSs operating in parallel, and then analyzes the stability of multiple PCSs in parallel operation, and gives the value range and influence parameters of the virtual resistance when the system is operating stably. Finally, experiments and simulation analysis verify the rationality and applicability of the conclusions and methods of this paper.

Python supervised co-simulation for a day-long harmonic evaluation of EV charging

To accurately simulate electric vehicle DC fast chargers' (DCFCs') harmonic emission, a small time step, i.e., typically smaller than 10 μs, is required owing to switching dynamics. However, in practice, harmonics should be continuously assessed with a long duration, e.g., a day. A trade-off between accuracy and time efficiency thus exists. To address this issue, a multi-time scale modeling framework of fast-charging stations (FCSs) is proposed. In the presented framework, the DCFCs' input impedance and harmonic current emission in the ideal grid condition, that is, zero grid impedance and no background harmonic voltage, are obtained based on a converter switching model with a small timescale simulation. Since a DCFC's input impedance and harmonic current source are functions of the DCFC's load, the input impedance and harmonic emission at different loads are obtained. Thereafter, they are used in the fast-charging charging station modeling, where the DCFCs are simplified as Norton equivalent circuits. In the station level simulation, a large time step, i.e., one minute, is used because the DCFCs' operating power can be assumed as a constant over a minute. With this co-simulation, the FCSs' long-term power quality performance can be assessed time-efficiently, without losing much accuracy.

Mathematical modeling of PMSG-based wind power plants (WPPs) for harmonic resonance studies and analytical assessment of wind turbines converters controls on WPP harmonic resonance response

In this paper harmonic resonance analysis of the grid comprising wind power plants (WPPs) with permanent magnet synchronous generators (PMSGs) is presented. Usually, Norton equivalent circuit is used as the wind turbine (WT) model for harmonic resonance studies, in which, WTs are simply modeled as ideal current and voltage sources and impacts of converters controls on the WTs output impedances are neglected. This paper deals with the harmonic resonance analysis of study WPP system taking the impacts of converters controls, and current and voltage measurement filters into account. In this way, harmonic impedance models of the WTs are presented for simple and detailed cases, and then WT harmonic impedance is extracted for the mentioned cases. Next, results of frequency scan and harmonic mode resonance analyses for different cases are given and compared, and then effects of the current and voltage measurements filters and WT converter control on harmonic resonance analysis of the study WPP system are presented. It is shown that increasing the bandwidth of the grid-side converter current control loop can not shift the resonant frequencies, but enhances the system damping, and consequently reduces the driving point impedances in the resonant frequencies.

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.

Nonlinear modeling and performance testing of high-power electromagnetic energy harvesting system for self-powering transmission line vibration deicing robot

This paper proposed the nonlinear modeling and a full-scale testing method for an electromagnetic energy harvester (EMEH) used to power-up a vibration de-icing robot used in power transmission line. The aim of the proposed work is to have an detailed analytical model of the EMEH on MATLAB environment which is capable of considering the EMEH ferromagnetic core material nonlinear characteristic in the design stage and can provide a fast and accurate analytical model with different design parameters for manufacturing. The nonlinear model is derived by considering Norton equivalent circuit of the EMEH with a nonlinear Norton equivalent inductance. With the nonlinear model design data, a prototype is manufactured and different performance tests such as the validation for the Norton equivalent circuit parameters, AC-DC conversion, and maximum power transfer from the EMEH using the discontinuously conducting DC-DC converters are conducted.

DQ impedance‐decoupled network model‐based stability analysis of offshore wind power plant under weak grid conditions

This study presents a dq impedance-decoupled network modelling method of offshore wind power plant (OWPP) for stability analysis. DQ impedance frequency responses of grid-connected inverters (GCIs) and long transmission cables (LTCs) are first measured by frequency scanning method, which are fitted as transfer function matrices using matrix fitting algorithm. Then, the GCIs are modelled as Norton equivalent circuits. In addition, per-unit-length electrical parameters of the LTCs are extracted from the measured dq impedance frequency responses of a specific LTC, based on which dq impedance-decoupled two port network models of the LTCs are established. DQ impedance-decoupled network model of the whole OWPP is then established based on connection relationships of these GCIs and LTCs. Finally, generalised Nyquist criterion (GNC) is performed in all the dq impedance-decoupled subsystems, and the subsystems where the GNC is not satisfied are identified as instability sources. Compared with conventional impedance-based stability analysis methods of OWPP, the proposed dq impedance-decoupled modelling method is able to facilitate the application of GNC and further instability source identification based on partitioning the whole OWPP into several decoupled subsystems. The effectiveness of the network modelling method is validated based on time-domain simulation results in Matlab/Simulink and real-time verification results in OPAL-RT.

Theoretical modeling of triboelectric nanogenerators (TENGs)

Triboelectric nanogenerators (TENGs), using Maxwell's displacement current as the driving force, can effectively convert mechanical energy into electricity. In this work, an extensive review of theoretical models of TENGs is presented. Based on Maxwell's equations, a formal physical model is established referred to as the quasi-electrostatic model of a TENG. Since a TENG is electrically neutral at any time owing to the low operation frequency, it is conveniently regarded as a lumped circuit element. Then, using the lumped parameter equivalent circuit theory, the conventional capacitive model and Norton's equivalent circuit model are derived. Optimal conditions for power, voltage, and total energy conversion efficiency can be calculated. The presented TENG models provide an effective theoretical foundation for understanding and predicting the performance of TENGs for practical applications.

Multi-Rate Real-Time Simulation Method Based on the Norton Equivalent

For the problem of poor accuracy of the existing multi-rate simulation methods, this paper proposes a multi-rate real-time simulation method based on the Norton equivalent, compared with multi-rate simulation method based on the ideal source equivalent. After the Norton equivalence of the fast subsystem and the slow subsystem are established, they are solved simultaneously at the junction nodes. In order to reduce the amount of the simulation calculation, the Norton equivalent circuit is obtained by incremental calculation. The data interaction between the fast subsystem and the slow subsystem is realized by extrapolation method. For ensuring the real-time performance of the simulation, the method of the slow subsystem calculates ahead of the fast subsystem is given for the slow subsystem with a large amount of calculation. Finally, the AC/DC hybrid power system was simulated on the real-time simulation platform (FPGA-based Real-Time Digital Solver, FRTDS), and the simulation results were compared with the single-rate simulation, which verified the correctness and accuracy of the proposed method.