Clarke Transformation Research Articles

Application of Three-Dimensional Unbalanced Coordinate Transformation to Stand-Alone Four-Leg Voltage-Source Inverter

Three-phase four-leg voltage-source inverter has been extensively investigated in recent years for its compactness, small size, and high efficiency, and it has been proved to be the best solution for providing transformer-less neutral connection to three-phase asymmetric loads. The main purpose of this article is to apply 3-D unbalanced coordinate transformation to the four-leg inverter, which supplies stand-alone asymmetric loads, to maintain the balance of the output capacitor voltages, and obtain simpler control structure and better control performance. First, a double closed-loop PI control structure can be designed based on the Clarke and Park transformations. Second, the phases and amplitudes of the capacitor voltages are locked based on the 3-D unbalanced coordinate transformation. Furthermore, the Clarke transformation oriented on the inductor currents is replaced by a 3-D unbalanced coordinate transformation, and the corresponding unbalance indices are continuously adjusted according to the unbalance indices of the capacitor voltages, so as to recover the capacitor voltages to the balanced condition. The simulation and experimental results of the whole control system for the four-leg inverter are provided in the end, so as to verify the feasibility and effectiveness of the 3-D unbalanced coordinate transformation for application to the power electronic systems.

MODAL TRANSFORM FEATURES OF CABLE LINE ELECTRICAL VALUES IN A TRAVELING WAVE FAULT LOCATOR

It is known that during a short circuit on an overhead line, the traveling waves in the phases contain components of two aerial and one ground modes. Due to the different propagation velocity along the line of these components at the locator installation place they arrive at different times. This leads to the fact that the traveling wave front in the phases becomes less pronounced, and, consequently, the accuracy of determining their arrival time and accuracy of the fault location decrease. To divide the phase values ​​of an overhead line into independent components of modes, invariant transformations – Clark transformation, Karrenbauer transformation, or Wedepohl transformation – are used. However, these transformations in their classical form cannot be applied to the electrical values of a cable line, since its design differs from that of an overhead power line. The purpose of the article is to illustrate the use of modal transformation on a cable line. As a result of the study, the following conclusions were obtained: currents in the cable line cores do not affect each other, therefore, in the wave fault locator that measures current waves, it is necessary to use directly the phase currents of the cores without modal transformation; if the wave fault locator measures the currents or voltages of the cable screens, the modal transformation should be performed, and it must be taken into account that the three modes formed by the screens are similar to the three modes of the overhead line; if a power transformer is adjacent to the cable line, the current traveling waves in the cores will be completely reflected from the measurement place, in this case in the wave fault locator it is necessary to use voltage traveling waves in the mode between the core and the screen of each cable of the three-phase group.

Rotor Winding Short-Circuit-Fault Protection Method for VSPSGM Combining the Stator and Rotor Currents

Rotor winding short circuit faults are common faults for variable-speed pumped-storage generator-motors (VSPSGM). At present, the exciting rotor fault protection of VSPSGM is simple and has low sensitivity. It can only act when the instantaneous value of the rotor phase current reaches three times the rated current. Therefore, it is difficult to cover some rotor winding short-circuit faults with weak fault characteristics. It is urgent to study a novel rotor winding short-circuit-fault protection method for VSPSGM. In this paper, a protection method that combines the stator and rotor currents with different frequencies is proposed. The characteristics of the stator and rotor currents before and after the fault is analyzed by using Clark transformation. On this basis, a specific protection criterion is constructed based on the discrete integral operation, which is easy to implement and not affected by the change of rotor speed. Then, the calculation method of the protection setting is proposed, considering the effect of unbalanced voltage and sensor measurement error. Simulation results show that the proposed method can reliably realize the protection of rotor winding faults. It has faster protection action speed than other methods in the same field. The protection coverage rate is over 90%.

Remedy Strategy for Five-Phase FTPMMs Under Single-Phase Short-Circuit Fault by Injecting Harmonic Currents From Third Space

This article proposes a new fault-tolerant control for five-phase fault-tolerant permanent-magnet motors under single-phase short-circuit fault. In the proposed method, the effect on torque performance caused by a short-circuit fault is attributed into two parts: the loss of the short-circuited phase, which is equivalent to a phase open-circuit fault, and the resultant short-circuit current. The remedial action for loss of a phase is realized in the fundamental space, while the torque ripple caused by the short-circuit current is compensated by the harmonic currents in the third space. To decouple the control of the fundamental and third space, the reduced-order Clarke and Park transformation matrixes for the two spaces are obtained separately under single-phase short-circuit fault. Afterward, the fundamental currents with minimum joule losses or equal joule losses for loss of a phase, and the harmonic currents in the third space for torque ripple reduction are derived. Hence, the proposed method can realize torque ripple-free operation under single-phase short-circuit fault. Finally, the effectiveness of the proposed method is verified by experimental results under various operational conditions.

Generalized Clarke Transformation and Enhanced Dual-Loop Control Scheme for Three-Phase PWM Converters Under the Unbalanced Utility Grid

Three-phase grids imbalance causes serious alternating input current distortion in three-phase pulsewidth modulation (PWM) rectifiers where the conventional control scheme is employed. In this article, a generalized Clarke transformation is proposed to convert three-phase unbalanced sinusoidal quantities to two orthogonal sinusoidal quantities with equal amplitudes. Then, these two orthogonal sinusoidal quantities could be converted to two direct quantities by the Park transformation. Furthermore, based on the generalized Clarke transformation, an enhanced dual-loop control (EDLC) scheme is proposed to control the three-phase PWM rectifier in the synchronous rotating frame. Finally, a 5-kW Vienna-type rectifier prototype was built to verify the proposed generalized Clarke transformation and the proposed EDLC control scheme. Experimental results show that, by using the proposed EDLC scheme, the low current total harmonic distortion is achieved even both the amplitude and phase of the utility grid voltages are unbalanced.

Clarke Transform Based Fast Assessment of Switching Overvoltages in an MV Distribution Network

The paper investigates the use of the Clarke transform in time-domain simulations, namely the evaluation of transient overvoltages in medium voltage (MV) distribution networks. A simplified Clarke equivalent circuit can be set up using readily available symmetrical components data used in steady-state studies, such as line series impedances and the aggregate MV network capacitance. Attention was focused on an existing 20 kV radial distribution system, a large mixed cable/overhead network operated with ungrounded neutral. The worst-case scenario of a single-line-to-ground (1LG) fault occurring at the end of a long overhead line was simulated by means of equivalent lumped-parameter Clarke circuits which were implemented and solved by means of ATP-EMTP. An extensive transient simulation parametric study was carried out using Clarke circuits with different degrees of simplification, and results were compared with those yielded by a complete 3-phase ATP model, evidencing a very good agreement and a drastic reduction of computation times.

Three-Phase PLL Based on Adaptive Clarke Transform Under Unbalanced Condition

This letter proposes a three-phase phase-locked loop (PLL) algorithm relying on an adaptive Clarke transform (CT) under amplitude and/or phase angle unbalanced condition. Analytical expressions for coefficients of matrix associated with the adaptive CT are derived and used to generate orthogonal signals. An auxiliary algorithm is also proposed to track the actual amplitudes and phase angle deviations. Unlike the conventional CT relying on constant coefficients, the adaptive CT can generate accurate orthogonal signals from the three-phase voltages including phase angle and/or amplitude imbalances. As a result, the proposed PLL is not affected by the unbalanced phase angles with/without amplitude imbalances. The advantages of the proposed adaptive CT-based PLL method is illustrated by both simulation and experimental results.

Benefits of Fuzzy Logic on MPPT and PI Controllers in the Chain of Photovoltaic Control Systems

This paper presents a comparative study between two maximum power point tracking (MPPT) algorithms, the incremental conductance algorithm (InC) and the fuzzy logic controller (FLC). The two algorithms were applied to a low photovoltaic power conversion system, and they both use different PI controllers and grid synchronization techniques. Moreover, both InC and FLC methods have Clarke and Park Transformation. To some extent, the incremental conductance and fuzzy logic controller approaches are similar, but their control loops are different. Therefore, the InC has classic Proportional Integrative (PI) controllers with simple phase-locked loops (PLL). At the same time, the FLC works with fuzzy logic PI controllers linked with the Second Order Generalized Integrator (SOGI). The proposed techniques examine the solar energy conversion performance of the photovoltaic (PV) system under possible irradiance changes and constant temperature conditions. Finally, a performance comparison has been made between InC and FLC, which demonstrates the effectiveness of the fuzzy controller over the incremental conductance algorithm. FLC turns to convert photovoltaic power easily, decreasing fluctuations, and it offers a quick response to the variation of solar irradiance (shading effect). The simulation results show a superior performance of the controller with fuzzy logic, which helps the inverter convert over 99% of the power generated by the photovoltaic panels. In comparison, the incremental conductance algorithm converts around 80%.

Classification of multi-stage voltage sags and calculation of phase angle jump based on Clarke components ellipse

Multi-stage voltage sag is one of the problems in the power system that requires classification and separation of stages to recognize this phenomenon. An algorithm for classification of multi-stage voltage sags based on the ellipse drawn by Clarke's transformation components is presented in this paper. The proposed algorithm has the ability to detect if the voltage sag is single-stage or multi-stage, and separates the various stages of multi-stage voltage sag using the point-on-wave of initiation and recovery parameters that is the novelty of this paper. Also a new method for calculation the phase angle jump by considering the zero Clarke component in drawing an ellipse is presented. The proposed method has a better accuracy than the previous method, which did not consider the zero component of the Clarke's transform. For validation, the method is tested with real data provided by the Department of Energy and the Electric Power Research Institute and compared with the previous method. A 12-bus distribution network has also been used to evaluate the performance of the multi-stage voltage sags classification algorithm accurately. This 33 kV network has distributed generation sources and various types of multi-stage voltage sags have been created in this network that show the efficiency of the algorithm. The results show that the proposed algorithm is able to classify any type of multi-stage voltage sag.

Improving the Differential Protection of Power Transformers Based on Clarke Transform and Fuzzy Systems

Improving the Differential Protection of Power Transformers Based on Clarke Transform and Fuzzy Systems

VSC-based LVDC distribution network with DERs: Equivalent circuits for leakage and ground fault currents evaluation

In this paper a new, general methodology for leakage and ground fault currents evaluation in grid-connected, low-voltage DC microgrids is introduced. Modified sequence equivalent circuits are considered, generalizing traditional AC power systems analysis. For three-phase systems, a modified Clarke transformation and, for two-wire systems, a modified modal transformation are introduced. These transformations, applied to a hybrid AC/DC system, allow building a modal equivalent circuit of the whole system, constituted by two coupled sub-circuits. The proposed equivalent circuits are intended as general time-domain models and are suitable for both steady-state and transient analysis. This approach highlights the whole system topology and allows evaluating the influence of active and passive components on leakage and ground fault currents. The proposed methodology is experimentally verified and, successively, applied to a hybrid AC/DC microgrid including DERs and storage devices.

An Evaluation Method for Bus and Grid Structure Based on Voltage Sags/Swells Using Voltage Ellipse Parameters

Voltage sags and swells are significant issues of power quality. In order to alleviate voltage sags and swells on sensitive loads, a method is proposed to evaluate the bus and grid structure based on voltage sags/swells using voltage ellipse parameters. The voltage matrix is established for voltage sags/swells caused by four types of short-circuit faults. The voltage matrix can be converted into a minor axis matrix and a major axis matrix using the Clarke transformation, and accordingly a voltage sag matrix and a voltage swell matrix can be obtained with the interaction between voltage sags/swells and voltage ellipse parameters. The key indicator and the vulnerability indicator of bus as well as the indicator of grid structure are defined according to the voltage sag matrix and swell matrix. The buses are sorted using the indicators proposed, and the result of bus sorting provides a basis for selecting the buses that need to be managed to alleviate voltage sags/swells. The experiment results in IEEE 30-bus system and IEEE 57-bus system show that these indicators facilitate the selection of the access point for newly-added sensitive loads, and provide a theoretical guidance to the mitigation of voltage sags and swells, and help to accomplish the planning, operation and transformation of power grid.

Fast Estimation of Phase Angle for Three-Phase Voltage Systems Under Distorted Conditions

This article proposes a fast and robust technique for instantaneous phase angle estimation of three-phase voltage systems under unbalanced and distorted conditions. It mainly relies on the Clarke transformation, the Teager energy operator, and the delayed signal cancellation (DSC) approach. Unlike an adaptive DSC-based synchronous-reference frame phase-locked loop (PLL), the proposed technique avoids interdependent loops, thus increasing the overall stability and easing the tuning process. Moreover, it removes the phase loop of the PLL and, hence, eliminates the requirement of an in-loop filter (low- pass filter) and/or a loop filter (proportional–integral controller). As a result, the tuning of these filters is not required, and a quicker response is produced. The proposed technique also avoids inverse trigonometric function operation. In addition, it shows harmonics immunity at steady state and can generate a fast dynamic response with a settling time less than one nominal fundamental cycle. Both simulation and experimental results are presented to reveal the benefits of the technique.

Space Vector Transients of Three-phase Transformers

This paper is devoted to the transient analysis of three-phase transformers by means of the space vector tool. Space vector definition is based on the Clarke transformation, operating in the time domain. Thus, the space vector is better suited to transient analysis when compared with approximate approaches based on the phasor symmetrical-component transformation. Space-vector equivalent circuits are derived for three-phase transformers with Wye and Delta connections. A case study, consisting in the transient due to a capacitor bank insertion, shows that the proposed space-vector approach can clearly evidence the asymmetrical transient behavior of the phase variables in terms of different peak levels and oscillations amplitude.

Clarke Transformation Solution of Asymmetrical Transients in Three-Phase Circuits

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.

Circuit Modeling and Statistical Analysis of Differential-to-Common-Mode Noise Conversion Due to Filter Unbalancing in Three-Phase Motor Drive Systems

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.

Teager Energy Operator for Fast Estimation of Three-Phase Grid Frequency

This article proposes a fundamental frequency estimation technique for three-phase voltage systems under unbalanced and harmonic conditions. It is based on the Clarke transformation, recursive discrete Fourier transform, and Teager energy operator. The proposed technique is computationally efficient and relatively simple to implement, thus suitable for low-cost applications. It does not require real-time evaluation of the computationally demanding inverse trigonometric functions. The proposed method does not suffer from the accumulation error and is also not affected when the voltage signal is sampled at zero or close to zero. It is immune to various voltage harmonics and can produce quick transient response with a time of less than 75% of one fundamental period. When compared with the competing techniques based on phase-locked loops and three consecutive samples, the proposed method can produce faster response under dynamic conditions. The advantages of the proposed technique are confirmed by simulated and experimental results in the laboratory.

Proposing a new method to improve the longitudinal differential relay performance using the Clarke transformation: Theory, simulation, and experiment

Longitudinal differential protection is among the most important protection mechanisms in the short transmission lines. During energization, some factors might disrupt the performance, and on-line fault detection using the longitudinal differential relay might result in an undesired interruption. Therefore, in this paper, a new method is proposed to detect the fault current from the switching currents, and other transient and loading conditions accurately. For this purpose, a new mathematical and performance method is proposed based on the Clarke transformation to extract the features, classification, and instantaneous patterns of the calculations. Therefore, a unique operating characteristic in Iα–Iβ framework is suggested to detect the internal fault from the external fault and other transient conditions by point -to -point checking and tracking the trajectory Iα with respect to Iβ. A typical power transmission network is simulated in the MATLAB software to validate the proposed method. Furthermore, the actual fault on the real transmission line GT918 from Bushehr network in Iran's power grid is analyzed. Simulation and practical test results indicated high speed, stability, reliability, and accuracy of the proposed method in detecting the internal fault in the short transmission lines. The internal fault is accurately detected fewer than 2 ms in all the tests.

Investigation of voltage regulation in grid–connected PV system

<p>In the present scenario the power demand on the load side is increasing day by day, so to balance the power demand and power supply various renewable energy comes to picture as the additional source of electricity generation. The power generated by various renewable resources such as solar, wind, tidal energy and geothermal sources is environmentally clean and have a less emission impact. Out of which PV system draws more attention because it generates energy with a much lower level of carbon dioxide emissions. In the proposed work the objective is to investigate the synchronisation of the grid-connected PV system in terms of voltage and frequency. It includes the P-V characteristics under the circumstances of MPPT technique such as perturb & observe (P&O) method can able to track the local maximum point. The proposed inverter is a voltage source H-Bridge inverter which is controlled using a Clarke and Park transformation to drive a controlled current into the grid to maintain the THD value within the standards. As the grid frequency is fluctuating between SRF-PLL is generally used to fix the output frequency and phase of the grid. It also includes with the design of a three-phase H-bridge inverter as an interface between PV system and grid system. The proposed work is designed and simulated in MATLAB SIMULINK 2017b environment.</p>

Design and simulation of three-level SVPWM based on FPGA

After expounding the overall design of the three-level SVPWM principle and algorithm, the SVPWM top level module using FPGA is created. And A/D conversion module, the Clarke transformation module, the adjacent vector function time module and the triangle carrier module are also established. Quartus II is used to simulation. And compared with the simulation results of MATLAB, the simulation waveforms are consistent, which verifies the correctness of the simulation results. Taking the diode clamped three-level inverter as the research object, the simulation waveforms of the different modulation M are compared, and the correctness of the three-level SVPWM control strategy is further verified.