Diego Bellan

This work provides a theoretical/methodological contribution to the transient analysis of asymmetrical faults in three-phase systems. Transient analysis of three-phase systems is usually performed by resorting either to the instantaneous Symmetrical Component Transformation (SCT) or to numerical methods. In this paper, an analytical methodology based on the time-domain Clarke transformation is presented for the transient analysis of the most common asymmetrical faults. For each kind of asymmetrical fault, a specific circuit coupling between the Clarke αβ0 circuits is derived. Two main advantages are obtained over the SCT approach. First, the Clarke circuits involve real-valued voltages/currents, instead of complex variables as with the SCT. Second, the Clarke circuits αβ0 are not all coupled to each other. Therefore, the dynamic order of the Clarke equivalent circuits is lower than that of the SCT circuits. This property can be of interest in both the derivation of analytical and numerical solutions. A simple radial system is used to exemplify the proposed methodology.

This work deals with the time-domain analysis of asymmetrical faults in three-phase systems. Conventional three-phase analysis provides steady-state solutions for asymmetrical faults. Transient analysis, however, is usually performed by resorting either to oversimplified approximate circuits, or to numerical methods. In this paper, a rigorous analytical methodology based on the time-domain Clarke transformation is presented for the most common asymmetrical faults in three-phase systems. In particular, it is shown that asymmetrical faults result in circuit coupling in the Clarke equivalent circuits. Circuit representation of coupling is also derived in the paper. Coupled equivalent circuits allow rigorous analytical solution of transients in case of asymmetrical faults. The analytical results derived in the paper are validated through proper numerical simulation of faulted radial systems.

In this paper, the theoretical properties of transients in three-phase networks, including the fourth wire and unbalanced source, are systematically investigated by resorting to several analytical tools. First, a Cartesian space vector is introduced to provide the geometrical three-dimensional representation of a three-phase voltage/current transient. It is shown that a voltage/current transient can be always represented as a three-dimensional trajectory leaving one plane (corresponding to the previous steady state) towards another plane (corresponding to the new steady state at the end of the transient). The quantity driving the trajectory from one plane to another is the Park/Clarke zero component. Second, the Clarke transformation is used to study the original three-phase circuit as a superposition of two independent transients: the transient of the Clarke space vectors and the transient of the Clarke zero components. Third, the Park transformation is used to evaluate the dq0 components of transient voltages/currents. Since the Park transformation is related to the Clarke transformation through a simple frequency shift, the effects on the frequency content of the steady-state and transient waveforms are put into evidence. Three case studies are presented and solved to prove the theoretical results.

This work deals with the statistical analysis of additive noise impact on the space-vector ellipse parameters used to detect and classify three-phase voltage sags. In fact, since voltage waveforms are always corrupted by additive noise and harmonics, the space vector is pre-processed through the Discrete Fourier Transform to extract the power frequency components. Thus, harmonics can be readily discarded, but additive noise can still have significant impact on the elliptical trajectory of the space vector on the complex plane. Therefore, by modeling the ellipse parameters (i.e., the shape index and the inclination angle) as random variables, the related statistical characterization is derived in the paper. In particular, the main results and the novelty of the paper are given by the analytical derivation in closed-form of the probability density function, cumulative distribution function, mean value, and variance of the ellipse parameters as functions of the additive noise variance. Since the ellipse shape index and inclination angle are commonly used to detect and classify voltage sags, the results derived in the paper are useful for both uncertainty propagation analysis, and assessment of detection capability in case of voltage dips close to the minimum value defined in the IEEE Standard 1159. Analytical results are validated through numerical simulation of noisy voltage sags.

This work deals with the use of the Clarke transformation for the theoretical derivation of circuit models for the analysis of asymmetrical transients in three-phase circuits. Asymmetrical transients occur when only one or two phases of a three-phase power system are involved in a switch operation. Such a condition is critical from a theoretical viewpoint since the Clarke transformation is based on the assumption of circuit symmetry between the three phases. If the symmetry assumption is not met, the equivalent circuits in the transformed variables α, β, and 0 are not uncoupled. The literature concerning numerical approaches for asymmetrical transient analysis is very rich, but a comprehensive and rigorous analytical investigation of circuit models within the framework of the Clarke transformation is still lacking. Contrary to numerical approaches, analytical solutions provide deeper insight into the phenomenon and allow for theoretical interpretation and better understanding of the transient behavior. The proposed circuit models show that the β variables are always uncoupled with α and 0 variables, whereas coupling between α and 0 variables can be properly represented by an ideal transformer. Moreover, in the case of single-line switching, the β variables have no transient, i.e., they keep the steady-state behavior. Transient properties can be readily and effectively observed by representing the trajectory of the space vector on the complex plane. All the analytical results have been checked numerically through the Simulink (Matlab R2020a, The MathWorks, Inc., Natick, MA, USA) implementation of a specific three-phase circuit introduced to illustrate the main theoretical issues.

This work deals with circuit modeling of noise mode conversion due to system asymmetry in a three-phase motor drive system. In fact, it is well-known that in case of system asymmetry (e.g., slightly asymmetrical LC filter parameters), differential-mode noise can convert into common-mode noise, resulting in increased level of conducted electromagnetic interference. This phenomenon has been observed with measurements and reported in previous works, but a clear and rigorous analytical description is still a challenging point. The main novelty proposed in the paper is a rigorous analytical description of differential-to-common-mode noise conversion based on the Clarke transformation and the eigenvalue analysis. In particular, the magnitude and the frequency location of the differential-mode resonances injected into the common-mode circuit are derived in closed form. Moreover, since system asymmetry is usually uncontrolled (e.g., component tolerance and parasitic elements), a statistical analysis is also presented by treating the parameters of the LC filter as random variables. Thus, a second contribution proposed in the paper is the analytical derivation in closed form of the probability density function, the mean value, and the standard deviation of the random frequency location of the resonance peaks injected into the common-mode circuit. The importance of the analytical results derived in the paper is two-fold. First, a deep theoretical understanding of the phenomenon in terms of circuit theory concepts is achieved. Second, the impact of differential-to-common-mode noise conversion is described in quantitative terms. Thus, the obtained analytical results can be used to predict or explain the noise conversion impact on the frequency-domain measurements of common-mode currents. Theoretical derivations are validated through a time-domain Simulink implementation of a three-phase motor drive system, and a frequency-domain analysis through the discrete Fourier transform.

This work investigates the statistical radiatedsusceptibility of an electrically-short transmission line (TL)consisting of a single wire over a ground plane. The angularparameters of the impinging plane wave and the height ofthe wire over the ground plane are modelled as randomvariables with Gaussian/Uniform distributions. Thestatistical properties of the current in the TL terminations(i.e., mean value, standard deviation, and cumulativedistribution function) are derived through a properlydefined numerical methodology consisting in a modifiedversion of the conventional Stochastic Reduced-OrderModel (SROM) approach. The proposed methodologyconsists in a straightforward approximation of the inputcontinuous random variables with small-size discreterandom variables. In contrast with conventional SROM, nonumerical pre-processing is needed. The modified SROMproposed in this paper demonstrates high efficiency whencompared with classical Monte Carlo approach. Theproposed technique can be applied to the statistical analysisof much more complex systems whose input/outputrelationship requires a huge computational burden and forwhich the conventional Monte Carlo approach is notsuitable

This paper provides the circuit model of a three-phase inverter as a four-port network. Such model is used to derive equivalent circuits for differential and common mode noise emissions when parasitic capacitances are taken into account. Moreover, a rigorous methodology based on the symmetrical component transformation is proposed to investigate the interaction between the three-phase side and the single-phase side. This allows a proper understanding of interaction between differential and common mode noise. Furthermore, the relationship between the DC side differential mode noise and the three-phase load is made clear. A more complete and rigorous circuit representation is obtained with respect to the conventional behavioral modeling available in the relevant literature.

This paper deals with the analytical evaluation of the transient current in single-line-to-ground fault in a three-phase power system. The proposed methodology, in contrast with conventional numerical techniques, is based on the Clarke transformation working in the time domain. Due to the system asymmetry introduced by the single line fault, the modal circuits alpha, beta, and zero, related to the Clarke transformation, result in coupled circuits. A simple circuit representation of such coupling is derived, allowing straightforward evaluation of the fault current in the Laplace domain. Time domain behavior obtained through inverse Laplace transform is validated by means of Simulink time-domain simulation of single line faults.

This work investigates from a circuit viewpoint the phenomenon of voltage unbalance emission due to load asymmetry in three-phase power systems. Voltage unbalance emission occurs whenever a power system shows a lack of structural symmetry between the three phases. In such a case, in fact, the transformation methods, such as the symmetrical component transformation, provide coupled modal circuits. As a result, in a practical three-phase three-wire power system, the positive-sequence circuit injects current into the negative-sequence circuit and therefore the load voltage shows a negative-sequence component due to modal circuits interaction. Since in many cases of practical interest the asymmetry level in a three-phase load can be regarded as a small perturbation with respect to the ideal symmetrical condition, in this paper a weak-coupling approach between sequence circuits is proposed. More specifically, load asymmetry is investigated, and analytical conditions for circuit weak-coupling are derived. In this case an approximate equivalent circuit for the estimation of voltage unbalance emission is derived. Analytical results are validated by means of numerical simulation of a radial power system with asymmetrical three-phase load.