Norton Equivalent Research Articles

High-Speed EMT Modeling of MMCs With Arbitrary Multiport Submodule Structures Using Generalized Norton Equivalents

In order to improve features, such as fault current blocking and capacitor voltage balancing, modular multilevel converter (MMC) topologies incorporating multiport submodules (SMs) are being considered as candidates for HVdc transmission applications. This paper presents high-speed and accurate electromagnetic transient (EMT) models for MMCs composed of such multiport SMs. The approach uses the Schur's complement technique to recursively eliminate internal nodes of the converter structure to create a multiport Norton equivalent that connects to the external network. Thus, the final admittance matrix seen by the EMT solver has a dimension orders of magnitude smaller than that of the unreduced structure. As with previously developed approaches for MMCs with single-port SMs, all internal information, such as individual SM capacitor voltages, is preserved and can be output by the program if needed. This increases the bookkeeping effort, but the overall reduction in matrix size more than compensates for any resulting time penalty. Approximately two to three orders of magnitude speedup over a straightforward implementation in an EMT program is achieved.

Shifted Frequency Modeling of Hybrid Modular Multilevel Converters for Simulation of MTDC Grid

When investigating voltage and current stresses in critical main circuit components during faults of the converter, a detailed equivalent model capable of representing the balancing control strategies of the capacitor voltages on the submodule level, along with blocking and delocking, is always necessary. Among previously proposed equivalent models of the modular multilevel converter (MMC), only submodule averaged models (SAMs) can capture interested inner dynamics inside each arm. However, the simulation accuracy of SAMs is not always satisfactory, especially when the time step is larger than 10 $\mu$ s. In order to further improve the simulation accuracy with guaranteed simulation efficiency, the shifted frequency modeling of the half- and full-bridge hybrid MMC is proposed in this paper. Therein, each submodule is represented by Thevenin equivalents derived by submodule dynamic phasors. The arm of the MMC is represented by Norton equivalents to guarantee the efficiency, considering both normal and dc-blocking conditions. The effectiveness of the proposed model in terms of accuracy and efficiency is validated by simulating an MMC-based HVdc transmission.

Norton Equivalent Circuit for Pulsed Photoconductive Antennas–Part I: Theoretical Model

A novel equivalent circuit for pulsed photoconductive sources is introduced for describing the coupling between the photoconductive gap and the antenna. The proposed circuit effectively describes the mechanism of feeding the antenna by the semiconductor when this latter is illuminated by a laser operating in a pulsed mode. Starting from the classical continuity equation, which models the free carriers’ density with respect to the laser power pump and the semiconductor features, a Norton equivalent circuit in the frequency domain is derived. According to the Norton theorem, the equivalent source representation is decoupled from the antenna. In particular, for photoconductive antennas (PCAs), the Norton circuit takes into account of the electrical and optical properties of the semiconductor material, the features of the laser excitation, as well as the geometrical dimensions of the gap. The presence of the electrodes around the gap is part of the antenna and, therefore, it is taken into account in the antenna impedance. The proposed circuit allows the analysis of the coupling between the photoconductive source and the antenna, providing a tool to analyze and design PCAs.

A Multirate EMT Co-Simulation of Large AC and MMC-Based MTDC Systems

This paper proposes a novel multirate co-simulation method to improve simulation efficiency of large-scale AC plus multiterminal DC (MTDC) grids. In this method, the whole system is partitioned into different AC and MTDC subsystems. They each are simulated with different time steps according to their requirements of accuracy. Unlike the existing methods where simplified models are adopted, the proposed approach fully reserves both the detailed behavior of converters and the nonlinear dynamics of large-scale AC systems. To realize the coordination between different subsystems, updating and discretization of the interface models are significant to guarantee the overall numerical accuracy and stability of the co-simulation method. Accordingly, the interface models of the partitioned AC and DC networks are represented by time-varying Thevenin and Norton equivalents. To eliminate the aliasing or time-delay errors, their parameters are derived using moving-window prediction, stepwise correction, and averaging techniques. A numerical oscillation suppression algorithm is developed for the discretization of the interface model. Thus, numerical stability is well guaranteed. The variables inside each subsystem as well as the interfaces, which are derived by augmented network equations (ANEs), are calculated simultaneously. The rate ratio of large AC and MTDC systems is determined based on our proposed criterion of electromagnetic transient simulation accuracy. The effectiveness of the cosimulation method is validated on a practical system integrating large AC and modular-multilevel-converter-based MTDC grids.

Frequency‐dependent modeling of transmission lines using bergeron cells

This paper proposes using Bergeron's equivalent circuit with traveling time equal to the simulation time step as an element for frequency‐dependent modeling of transmission lines for electromagnetic transient (EMT) simulations of power systems. According to the simulation time step used, a transmission line is divided into aforementioned Bergeron's equivalents, each of which is called a ‘Bergeron cell’ in this paper. In this way, the traveling‐wave nature of a line is represented by the cascaded Bergeron cells. Then, the frequency‐dependent loss nature of the line is represented by a matrix partial fraction expansion, and this is inserted at each connection point of the Bergeron cells in the form of a multiphase Norton equivalent. Since the frequency‐dependent loss is modeled in the dimension of impedance, the change of the line length is easily taken into account by a simple multiplication. This methodology thus allows variable‐length modeling and completely avoids modal decomposition in both model identification and EMT simulation stages. The proposed methodology is applied to the frequency‐dependent modeling of overhead and submarine‐cable transmission lines, and its accuracy is assessed.

Norton Equivalent Modeling of Current Source MMC and Its Use for Dynamic Studies of Back-to-Back Converter System

The current-source modular multilevel converter (CSMMC) is a good alternative for high-power applications with medium-voltage and high-current requirements. In such applications, to obtain the multilevel current output and for current scaling, a large number of submodules are required in CSMMC. The detailed representation of such systems in electromagnetic transient-type simulation programs brings computational complexity with a large computing burden. This paper presents a computationally efficient equivalent model of CSMMC based on the Norton equivalents. Each arm of the converter is represented by a two-node Norton equivalent in this model, which eliminates the internal intermediate nodes and, hence, significantly reduces computing time. The proposed model is implemented in PSCAD/EMTDC by using the module component coded in FORTRAN and results are validated with the full-size traditional model in PSCAD/EMTDC. Moreover, hybrid models are presented and validated for the submodule level studies such as fail-safe functionality with redundant submodules. This paper also focuses on the control method and dynamic studies of the CSMMC-based back-to-back system. The results of these studies are compared with the proposed equivalent model, and accurate matching of the results confirms the usefulness of the proposed model in such studies.

Model based prediction of sound pressure at the ear drum for an open earpiece equipped with two receivers and three microphones

In future hearing systems, one or more microphones and one or more receivers located within the earmold or ear canal are feasible. In order to predict the sound pressure at the ear drum in such a scenario, a one-dimensional electro-acoustic model of a prototype open earpiece with two integrated receivers and three integrated microphones was developed. The transducers were experimentally characterized by their (frequency-dependent) sensitivity (microphones) and Norton equivalents (receivers). The remaining acoustical system was modeled by 12 frequency-independent parameters which were fitted using a training set-up with well-defined loads at both sides of the ear piece. Put on an individual subject, the model could then be used to determine the acoustic impedance at the medial end of the earpiece, based on measured transfer functions between the integrated components. Subsequently, a model of the ear canal and its termination was estimated from the measured ear canal impedance, which could eventually be used to predict the drum pressure in the individual subject. Comparison to probe tube measurements of the drum pressure in 12 human subjects showed good agreement (less than ±3 dB up to 3 kHz, less than ±5 dB up to 6…8 kHz).

Hysteresis and Oscillation in High-Efficiency Power Amplifiers

Hysteresis in power amplifiers (PAs) is investigated in detail with the aid of an efficient analysis method, compatible with commercial harmonic balance. Suppressing the input source and using, instead, an outer-tier auxiliary generator, together with the Norton equivalent of the input network, analysis difficulties associated with turning points are avoided. The turning-point locus in the plane defined by any two relevant analysis parameters is obtained in a straightforward manner using a geometrical condition. The hysteresis phenomenon is demonstrated to be due to a nonlinear resonance of the device input capacitance under near optimum matching conditions. When increasing the drain bias voltage, some points of the locus degenerate into a large-signal oscillation that cannot be detected with a stability analysis of the dc solution. In driven conditions, the oscillation will be extinguished either through synchronization or inverse Hopf bifurcations in the upper section of the multivalued curves. For an efficient stability analysis, the outer-tier method will be applied in combination with pole-zero identification and Hopf-bifurcation detection. Departing from the detected oscillation, a slight variation of the input network will be carried out so as to obtain a high-efficiency oscillator able to start up from the noise level. All the tests have been carried out in a Class-E GaN PA with measured 86.8% power-added efficiency and 12.4-W output power at 0.9 GHz.

Interfacing <formula formulatype="inline"><tex Notation="TeX">${\rm k}$</tex></formula>-Factor Based White-Box Transformer Models With Electromagnetic Transients Programs

White-box transformer models are used by transformer manufacturers during the dielectric design of windings. The models are often based on constant parameters (RLCG matrices) with the high-frequency losses accounted for by a scaling of the dc resistance ( k-Factor). We show an efficient procedure for interfacing such models with Electromagnetic Transients Program (EMTP)-type circuit simulators via state equations and a Norton equivalent. The approach makes no approximations except for the discretization in the time domain. Diagonalization is utilized for achieving high computational efficiency. Proprietary information about internal voltages is optionally hidden from the user. Internal surge arresters are handled by the EMTP circuit solver by declaring their connection points as external nodes. The model interface includes automated initialization from 50/60-Hz initial conditions. The proposed interfacing capability permits manufacturers to apply their models in complete network simulations, and to share their models with users.

Enhancement of Voltage Stability by optimal location of UPFC using MPSO and Power Flow Analysis using ECI Algorithm

Voltage instability and voltage collapse have been considered as a major threat to present power system networks due to their stressed operation. It is very important to do the power system analysis with respect to voltage stability. Flexible AC Transmission System (FACTS) is an alternating current transmission system incorporating power electronic-based and other static controllers to enhance controllability and increase power transfer capability. A FACTS device in a power system improves the voltage stability, reduces the power loss and also improves the load ability of the system. FACTS have made the power systems operation more flexible and secure. Amongst the several FACTS controllers, the Unified Power Flow Controller (UPFC) is most effective to improve the enhancement of voltage stability and reduces the power loss. This study investigates the application of Particle Swarm Optimization (PSO) to find sizing of Unified Power Flow Controller (UPFC) device to minimize the voltage stability index, total power loss, load voltage deviation, and cost of FACTS devices to improve voltage stability in the power system. A new model is proposed in this thesis to improve existing power-based model by using the Norton Equivalent Theorem. The proposed model can be integrated with the Equivalent Current Injection (ECI) power flow model easily. By ECI algorithm, it is much quickly and precisely to implement power flow calculations. Finally It is observed from the results that the voltage stability margin is improved, the voltage profile of the power system is increased and real power losses also reduced by optimally sizing UPFC device in the power system. Index Terms: Voltage stability, Unified Power Flow Controller (UPFC), Equivalent -Current Injection (ECI), Modified particle swarm optimization (MPSO)

Online Valuing the Loadability of Transmission Line

Based on dynamic thermal rating technologies and taking into account of voltage drop and steady-statestability limitation, the model and algorithm of online valuing loadability of transmission line under operation are established. Considering the generator units voltage regulation characteristics, the variable parameter dual-port Norton equivalent model based on expanded power ∞ow is derived, and the online tracking of the equivalent parameters of the system is implemented. Based on the historical samples, time series method is used to analyze the variation law of the equivalent parameters to make the maximum loadability of transmission line under operation closer to the actual situation. Studies have shown that the dual-port Norton equivalence can very well re∞ect the efiect of parallel ∞ow on the transmission line loadability in interconnected systems, and the changes in the equivalent parameters re∞ect the nonlinearity and time-varying properties of power systems, and by the equivalent parameters variation, the time-varying nature of the electromagnetic power is re∞ected. The feasibility and efiectiveness of the equivalent model and the algorithm are conflrmed by the analysis of the loadability of Weihai 220 kV transmission line under operation.

Variable Parameter Equivalent Model of Transmission Line Based on Dual-port Norton Equivalence

Firstly, this paper proposes a variable parameter equivalent model based on dual-port Norton equivalence for the loadability of transmission line under operation condition, which can very well reflect the non-linearity and timevarying properties of power systems. And the changes in the equivalent parameters reflect the effect of parallel flow on the transmission line loadability in interconnected system. This equivalent model is inferred firstly in theory. And then, a practical method is given for forecasting equivalent parameters and obtaining the loadability of transmission line under operation. At last, the feasibility and effectiveness of the equivalent model and the method are testified by continuous power flow in IEEE-39 system.

A New Method for Finding the Thevenin and Norton Equivalent Circuits

The paper presents a new pedagogical method for finding the Thevenin and Norton equivalent circuits of a linear electric circuit (LEC) at the n-different pairs of terminals simultaneously, regardless of the circuit topology and complexity. The proposed method is appropriate for undergraduate electrical and electronic engineering students leading to straightforward solutions, mostly arrived at by inspection, so that it can be regarded as a simple and innovative calculation tool for Thevenin equivalents. Furthermore, the method is easily adapted to computer implementation. Examples illustrating the method’s scientific and pedagogical reliability, as well as real test results and statistically-sound data assessing its functionality are provided.

Implications for Distribution Networks of High Penetration of Compact Fluorescent Lamps

The desire to reduce electrical loading by using energy efficient lighting has resulted in a high level of interest in replacing conventional incandescent lamps with compact fluorescent lamps (CFL). This has resulted in the New Zealand Government, through the Electricity Commission of New Zealand, running campaigns to install five CFLs in every home by, subsidizing the cost of suitable CFLs. CFLs are however, a nonlinear load hence inject harmonics into the electrical network. The CFL use electronic ballasts and the design of these have an enormous impact on the electrical performance of the CFL. In the past, the harmonics injected into the network by CFLs has been ignored as each is very small as the typically CFL is only 20 Watts. However, if widespread adoption of CFLs occurs, the combined effect of all these small sources can be just as detrimental as one large source, and is even harder to mitigate due to their distributed nature. This paper presents the results of a study to quantify the effect widespread adoption of CFLs will have on a typical distribution network. The two aspects investigated were harmonic distortion and system losses. To enable 28 800 homes to be represented a methodology for modelling a large number of distributed loads using Norton equivalents was developed and applied. In order to assess losses, a fundamental frequency power-flow was also incorporated.

Malware-Propagative Mobile Ad Hoc Networks: Asymptotic Behavior Analysis

In this paper, the spreading of malicious software over ad hoc networks, where legitimate nodes are prone to propagate the infections they receive from either an attacker or their already infected neighbors, is analyzed. Considering the Susceptible-Infected-Susceptible (SIS) node infection paradigm we propose a probabilistic model, on the basis of the theory of closed queuing networks, that aims at describing the aggregated behavior of the system when attacked by malicious nodes. Because of its nature, the model is also able to deal more effectively with the stochastic behavior of attackers and the inherent probabilistic nature of the wireless environment. The proposed model is able to describe accurately the asymptotic behavior of malware-propagative large scale ad hoc networking environments. Using the Norton equivalent of the closed queuing network, we obtain analytical results for its steady state behavior, which in turn is used for identifying the critical parameters affecting the operation of the network. Finally, through modeling and simulation, some additional numerical results are obtained with respect to the behavior of the system when multiple attackers are present, and regarding the time-dependent evolution and impact of an attack.

A general cosimulation approach for coupled field-circuit problems

An indirect procedure to couple transient finite-element simulation with circuit simulation is proposed. The procedure is based on extracting lumped parameters from the field simulation and Norton equivalents from the circuit simulation. This approach provides more stability and accuracy because both winding currents and terminal voltages across coupling branches are free to change. It is also more flexible since the finite-element equations and the circuit equations are solved separately and allows complicated system level simulation

The Thevenin—Norton Equivalent Circuit

The Thevenin-Norton (TN) equivalent of a one-port network is a combination of the Thevenin and Norton equivalents and contains both a voltage source and a current source. The concept of the TN equivalent is explained and its usefulness illustrated with examples.

Powerful Pedagogical Approaches for Finding Thevenin and Norton Equivalent Circuits for Linear Electric Circuits

This paper presents two powerful pedagogical approaches for finding Thevenin and Norton equivalent circuits for d.c. linear electric circuits. These approaches are based solely on mesh analysis (for planar circuits) and on nodal analysis (for planar or nonplanar circuits) and simultaneously provide all equivalent circuit parameters from one circuit.

Analytical Method for Predicting the Air-Gap Flux of Interior-Type Permanent-Magnet Machines

This paper presents an analytical method to calculate the air-gap flux of interior-type permanent-magnet (IPM) machines taking into account the assembly gap and saturation in stator and rotor. Special considerations are given to the calculation of leakage flux in the nonlinear short circuit or the magnetic bridges. An equivalent circuit was developed, and a computer-aided graphic analysis program calculated the magnet operating point. The Norton equivalent of flux concentration structures was studied. Agreements have been obtained between the results of the analytical model, finite-element analysis, and test results on prototype motors.

An efficient approach for contingency ranking based on voltage stability

This paper presents an efficient approach for line outage contingency ranking based on voltage stability concerns. This is achieved through non-iterative fast calculation of reactive support and voltage security indices using Norton's equivalent. Earlier approach required full AC load flow for every contingency simulation. Results for IEEE, 30 bus, 57 bus and an Indian 91 Bus Power Systems have been obtained to demonstrate the effectiveness of the algorithm.