2300 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 4, OCTOBER 2009 Wavelet Packet Transform-Based Power Quality Indices for Balanced and Unbalanced Three-Phase Systems Under Stationary or Nonstationary Operating Conditions Walid G. Morsi, Student Member, IEEE, and M. E. El-Hawary, Fellow, IEEE Abstract—Three-phase power-quality indices (PQIs) can be used to quantify and hence evaluate the quality of the Electric Power System (EPS) waveforms. The recommended PQIs are defined based on the fast Fourier transform (FFT) which can only provide accurate results in case of stationary waveforms, however in case of nonstationary waveforms even under sinusoidal oper- ating conditions, the FFT produces large errors due to spectral leakage phenomenon. Moreover, FFT is incapable of providing any time-related information which is a required property when dealing with time-evolving waveforms since it can provide only an amplitude-frequency spectrum. Since wavelet packet trans- form (WPT), which is a generalization of the wavelet transform, can represent EPS waveforms in a time-frequency domain, it is used in this study to define and formulate three-phase PQIs. In order to handle the unbalanced three-phase case, the concept of equivalent voltage and current is used to calculate those indices. The results of four numerical examples considering stationary and nonstationary, balanced and unbalanced three-phase systems in sinusoidal and nonsinusoidal situations indicate that the new WPT-based PQIs are closer to the true values. In addition, phase and overall crest factors are redefined in the time-frequency domain using WPT while a new crest factor is introduced in this paper. The redefined crest factors and the new crest factor help identifying and quantifying the waveform impact based on the time-frequency information obtained from the WPT. New crest factor can only be determined via WPT, which proves the powerful of this method and its suitability to define three-phase PQIs in nonstationary operating conditions. Index Terms—Nonstationary waveforms, power-quality (PQ) indices (PQIS), three-phase systems, wavelet packet transform (WPT). I. INTRODUCTION P OWER QUALITY (PQ) is defined as a combination of voltage quality and current quality [1]. Voltage quality can be defined based on how much deviations exist between the ac- Manuscript received November 26, 2007; revised April 22, 2009. Current version published September 23, 2009. This work was supported by the Izaak Walton Killam Memorial Pre-doctoral Scholarship. Paper no. TPWRD-00721- 2006. W. Morsi is with the Department of Electrical and Computer Engineering at the University of New Brunswick, Fredericton, NB E3B 5A3, Canada (e-mail: wmorsi@unb.ca). M. E. El-Hawary is with the Department of Electrical and Computer Engineering at Dalhousie University, Halifax, NS B3J 2X4, Canada (e-mail: elhawary@dal.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRD.2009.2027496 Fig. 1. PQ disturbances classifications based on waveform characteristics. tual voltage (or current) and the ideal voltage (or current) wave- forms. The ideal voltage (or current) should have sinusoidal waveshape with fixed magnitude and fixed frequency as their nominal values. Voltage and/or current deviations are considered as power quality disturbances. They may be classified as stationary or nonstationary, sinusoidal or nonsinusoidal as shown in Fig. 1. Stationary PQ disturbances are defined as those waveforms where their statistical characteristics do not change with time while nonstationary disturbances are those waveforms where their statistical characteristics change with time. Stationary PQ disturbances may result from the presence of nonlinear loads connected to the electric networks, such as per- sonal computers (PCs), power-electronic devices, flexible ac transmission systems (FACTS), and uninterruptible power sup- plies (UPS). Those nonlinear loads draw nonsinusoidal currents from the network that is stationary in nature even if it is supplied from the sinusoidal voltage source. On the other hand, nonstationary PQ disturbances, such as voltage dips, swells, or transients are due to abnormal oper- ating conditions in the electric power system. For example, dips can result from short-circuit faults, motors starting, and trans- formers energization, voltage swell can be due to changes in large loads or power-line switching. Transients may be due to lightening strokes or switching processes. Since during electric power system deregulation, the con- sumers especially those owing sensitive equipments will be more concerned of choosing their own suppliers based on the quality of the electric service they can provide, therefore there is a need to evaluate the electric power quality through quantifying the deviations that can exist in the electric voltage and currents waveforms. In order to evaluate the electric PQ, PQ indices that were rec- ommended in [2] and [3] can be used for that purpose. However, 0885-8977/$26.00 © 2009 IEEE