Methods for stability analysis of unbalanced three-phase systems

Under ideal sinusoidal operating conditions, the definition of power factor for single-phase and balanced three-phase systems is unique and meaningful. However in non-sinusoidal situations and/or unbalanced three-phase system operation, different power factors have been proposed to deal with these situations. In this paper a new fuzzy based Representative Quality Power Factor (RQPF) is introduced to represent three recommended power factors, Fundamental Positive Sequence Power Factor (FPSPF), Transmission Efficiency Power Factor (TEPF) and Oscillation Power Factor (OSCPF). In addition, the problem of defining power factor for three-phase system is formulated and the RQPF module is explained. In order to test the validity of the proposed fuzzy based module, the RQPF is applied to different cases, balanced, unbalanced, linear, nonlinear, sinusoidal and non-sinusoidal. The obtained results reveal that the new RQPF is meaningful and accurately represents the existing power factors in all cases and in all situations. Taking into consideration the advantages of fuzzy systems, this factor is useful for power quality evaluation, cost-effective analysis of PQ mitigation techniques and as well billing purposes in these situations.

Abstract- Based on Fast Fourier Transform (FFT), power quality indices (PQIs) definitions contained in the standards cannot be successfully applied for non-stationary signals analysis in unbalanced three-phase systems with non-sinusoidal conditions. Unlike FFT, Discrete Wavelet Transform (DWT) gives more accurate definitions preserving time-frequency information and reduces the computational effort. This paper proposes a comparative analysis of definitions employing these two instruments for unbalanced three-phase systems case in sinusoidal and nonsinusoidal situations using waveforms captured with data acquisition system which contains Hall voltage and current transducers. The experimental testing of harmonic distortion was made for a RL unbalanced three-phase load, a three-phase rectifier with diodes (TPRD) and a rectifier with near sinusoidal currents (RNSIC). I. I NTRODUCTION For the electric power quality monitoring, the IEEE working group has analysed the customers expectations concerning electric power delivery, the susceptibility limits of the electric and electronic equipments connected at the electric network and thus obtained the standards regarding the evaluation methods of electric power quality, the parameters that define this concept and the functioning classes of the measurement tools [1-3]. In the last years electric power quality became un important issue in the power systems domain because of the increased usage of nonlinear loads which are producing great harmonic distortion in power supply [4-12, 33-36]. Voltage and current quality can be determined based on the deviations between the real voltage/current and the ideal voltage/current waveforms [13]. These deviations are counted power quality disturbances and can be classified as stationary (with time unchangeable characteristics) or non-stationary (with time changeable characteristics) and each of them as sinusoidal or non-sinusoidal. The Fast Fourier Transform (FFT) approach in Power Quality Indices (PQIs) definition is replaced in this study with a time-frequency domain approach using Discrete Wavelet Transform (DWT). This method eliminates the FFT drawbacks as high computational effort, time related information loss, spectral leakage and provides representations for any non-stationary waveforms [14-17]. DWT is a useful tool in the waveforms analysis due to its accuracy and sensitivity at signal irregularities [5], [32]. The waveforms of voltage and current are obtained using a data acquisition system with linear/nonlinear loads composed of a Digital Signal Processor (DSP) board with high-speed 12 bit Analog to Digital Converters (ADC) coupled to a three-phase system using the Hall transducers. This paper presents a comparative analysis of DWT based measurements of power quality indices and the IEEE Standard 1459-2000 based values using unbalanced linear three-phase systems and nonlinear power loads. The power quality disturbances are calculated with a RL unbalanced three-phase load, a three-phase rectifier with diodes (TPRD) and a rectifier with near sinusoidal currents (RNSIC). II. T

This paper investigates the effectiveness of implementing the un-decimated wavelet transform (UWT) in the computation of reactive power in three-phase system subject to time-varying disturbance. The advantages of this decimation-free wavelet transform are highlighted and the formulation of the reactive power in the un-decimated wavelet domain is presented based on two reactive power metering approaches namely; 1) “non-quadrature” and 2) “quadrature” approaches. Various case studies including different loading conditions and/or time-variant/time-invariant disturbances are presented to exemplify the effectiveness of using the un-decimated wavelet in the reactive power measurement in both balanced and unbalanced three-phase systems. The results of the analysis are compared to the reactive power values based on the decimated wavelet transform (DWT) and conclusions are presented.

Generalized Nyquist Criterion (GNC) based on d- q frame two-by-two impedance matrices is widely used in the three-phase balanced systems for small-signal stability at the AC interface of power electronics devices, which is not applicable for the three-phase unbalanced systems. An algorithm is developed to model the terminal impedance of the source and the load in the three-phase ungrounded unbalanced system at the AC interface and to predict the small-signal stability after the connection based on the measured d-q frame four-by-four impedance matrices. This algorithm can be used in all kinds of three-phase unbalanced systems including distribution systems in utility, microgirds and three-phase electrical systems on planes. The four-by-four impedance matrix measurement process is designed according to the single-phase d-q frame impedance measurement method. Time domain simulation and scaled-down experiments have been done to prove the accuracy and feasibility of the proposed method.

The problem of off-nominal frequency estimation in unbalanced three-phase power systems is addressed from the frequency domain perspective. It is first established that the original Smart discrete Fourier transform (SDFT) technique designed for real-valued single-phase voltage can be extended to deal with complex-valued αβ transformed voltage. By observing that the underlying time series relationship among the consecutive DFT fundamental components employed by SDFT technique does not hold when noise or unexpected higher order harmonics are present, resulting in suboptimal estimation performances, the least squares framework is then built upon the underlying relationship among the consecutive DFT fundamental components to minimise the mean square model error. The benefits of the proposed LS-SDFT over the time-domain widely linear least squares (WL-LS) frequency estimator are verified by simulations for unbalanced power system conditions in the presence of noise and higher order harmonic pollution, as well as for real-world measurements.

This paper presents harmonic and reactive power compensation based on a generalized theory of instantaneous reactive power for three-phase power systems. This new theory gives a generalized definition of instantaneous reactive power, which is valid for sinusoidal or nonsinusoidal and balanced or unbalanced three-phase power systems with or without zero-sequence currents and/or voltages. The properties and physical meanings of the newly defined instantaneous reactive power are discussed in detail. A harmonic and reactive power compensator based on the new theory for a three-phase harmonic-distorted power system with zero-sequence components in the load current and/or source voltage is then used as an example to show harmonic and reactive power measurement and compensation using the new theory. Simulation and experimental results are presented.

Phasor estimation is an important task in many monitoring and control applications in power systems. Fast Fourier transform (FFT) and Park transform are the most common methods to calculate voltage and current phasors in three-phase systems. Park transform has a small delay but could not work in unbalanced conditions. FFT works well in unbalanced systems but introduces a delay of at least one fundamental cycle. In this paper a new method is presented to calculate magnitude and angle of each phase using Park transform. The delay of phasor estimation is half a cycle and it works well in unbalanced conditions. Several simulations are performed in PSCAD/EMTDC environment to prove the feasibility of this new method.

Reactive and harmonic current compensation for unbalanced three-phase systems using the synchronous detection method

This paper proposes two complex vector models to characterize three-phase unbalanced systems with four wires and containing PWM converters. In the first case, the three-phase system is decomposed in orthogonal odq components. A complex vector model is then built from the dq components and o component. In the second case a complex vector model is introduced by decomposing the system in three single-phase systems. In both cases, a second orthogonal simulated model is introduced to handle a single-phase system, such as the homopolar system or each of the three single-phase systems that compose a three-phase four wires system. From the vector models, a vector controller that employs two different reference frames is proposed and emulated on the stationary reference frame. A digital version for such a controller is also presented. As an example, a current control in a PWM-VSI system is presented.

The transformation of passive distribution systems to more active ones thanks to the increased penetration of distributed energy resources, such as dispersed generators, flexible demand, distributed storage, and electric vehicles, creates the necessity of an enhanced test system for distribution systems planning and operation studies. The value of the proposed test system, is that it provides an appropriate and comprehensive benchmark for future researches concerning distribution systems. The proposed test system is developed by modifying and updating the well-known 33 bus distribution system from Baran & Wu. It comprises both forms of balanced and unbalanced three-phase power systems, including new details on the integration of distributed and renewable generation units, reactive power compensation assets, reconfiguration infrastructures and appropriate datasets of load and renewable generation profiles for different case studies.

100:1 single-core prototype current transformer is described to measure line currents simultaneously in balanced or unbalanced three-phase systems. The transformer core comprises a toroidal and a wye-shaped inner core accommodating primary line conductors and secondary detecting windings. Mutual effects of line current fluxes on current measurements are eliminated by three air-gap slits introduced along the toroidal section of the core. Calibration and performance of the transformer are discussed for burdens 0.1 Ω and 1 Ω in the current range 0–50 A.

Frequency is one of the most significant parameters in the smart grid systems. Thus, accurate frequency estimation becomes an essential task for monitoring, controlling and protecting a real-time smart grid system. In this chapter, we present an overview of the frequency estimation methods in the smart grid system with a focus on real-time adaptive estimation algorithms. Primarily, in Sects. 5.1 and 5.2, the importance of the frequency estimation in the smart grid systems and the challenges encountered in its real-time applications are introduced in detail. In Sect. 5.3, a three-phase power system is then formulated as a two-phase system in the complex domain by using the well-known Clarke’s transformation so as to be able to estimate the frequency of the smart grid system in the real time. For this purpose, the adaptive real-time frequency estimation algorithms are comparatively presented as strictly and widely linear algorithms in Sect. 5.4. The strictly linear algorithms yield optimal solutions only under balanced three-phase systems, whereas the widely linear algorithms give a better solution under both balanced and unbalanced conditions due to the fact that they take into account all statistical information of the system. Considering smart grid applications in real time, the mentioned properties of these algorithms under both balanced and unbalanced conditions are proven in Sect. 5.5.

This paper introduces the application of an enhanced phase-locked loop (EPLL) system to the estimation of synchrophasors in a phasor measurement unit (PMU). The major concern is accurate estimation within off-nominal frequency operation of the system. The well-known technique based on discrete Fourier transform (DFT) can provide accurate estimation of the phasors in a three-phase balanced system. However, the negative-sequence component causes errors to the DFT estimates. The DFT cannot accomplish this task in a single-phase system. The EPLL is shown to be a solution for those shortcomings of the DFT technique, both in single-phase and in unbalanced three-phase systems, at the expense of some more complicated structure. Extensive steady-state and dynamic tests are performed and the results are presented and discussed.

In recent years, the use of renewable energy (RE) in distributed power generation has been increasing. However, a key drawback of RE is unstable power output that causes systemic effects. Moreover, with the increasing penetration of RE in three-phase distribution systems, the problem of unbalanced three-phase systems becomes increasingly challenging. This paper proposes a novel control strategy for a grid-connected three-phase four-wire smart inverter. By using the proposed power control strategy and the positive- and negative-sequence decoupling controls, the smart inverter can provide not only frequency but also voltage regulation to maintain a balanced and stable system. To verify the feasibility of the proposed method, a 2 kW grid-connected three-phase four-wire prototype inverter was implemented, which was connected to a three-phase AC source for analysis. The experimental results indicate that the proposed control strategy is a feasible scheme. System frequency regulation and phase voltage compensation can be achieved to support the system under an unbalanced or unstable condition.

The traditional method used to detect and localize power quality disturbances by Fast Fourier Transform (FFT) can provide only amplitude - frequency information, but looses time information. In evaluating power supply quality, the FFT-based indices present different limitations. This paper proposes a set of four indices based on Discrete Wavelet Transform (DWT), such as total harmonic distortion, equivalent distortion index, K- factor, and crest factor that can be also applied for non- stationary signals analysis, in unbalanced three-phase systems with non-sinusoidal conditions. In the comparative analysis of Power Quality Indices (PQIs) definitions, these two instruments were used for the three-phase power systems with non-linear loads: three rectifier topologies with near sinusoidal input currents (RNSIC). Keywords—Discrete Wavelet Transform (DWT), data acquisition system, Power Quality Indices (PQIs), rectifier with near sinusoidal currents (RNSIC), harmonics.