IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 3, JULY 2006 1743 A Simple Energy Operator Computational Method for Voltage Flicker Assessment Mohamed Amin Eldery, Student Member, IEEE, Ehab F. El-Saadany, Senior Member, IEEE, and Magdy M. A. Salama, Fellow, IEEE Abstract—In this paper, a novel algorithm, based on calculating the energy operator of a sinusoidal waveform to track the voltage flicker, is presented. The mathematical derivation of the proposed algorithm different circuit designs required to realize it are de- scribed. The proposed algorithm is fast and robust and requires only a few samples to calculate the energy. The simulation and the experimental results prove that the new algorithm can successfully track the envelope. There are no spikes in the envelope even when the voltage signal is suddenly changed at its peak value. Index Terms—Digital signal processor (DSP), envelope tracking, power quality monitoring, sag, swell, teager energy operator, voltage flicker. I. INTRODUCTION M ODERN end-use equipment is very sensitive to voltage fluctuation and flicker. In many cases, mitigating the voltage flicker helps to prevent equipment malfunction. Flicker mitigation techniques depend on injecting a certain amount of reactive power, defined by the difference between, the reference value and the measured voltage. As a result, the measurement accuracy is crucial. Voltage flicker measurement depends prin- cipally on the accuracy of the envelope tracking algorithm. The envelope tracker should be accurate, robust, and fast with the least mathematical burden. The voltage envelope is either calculated or estimated. Dif- ferent signal processing algorithms, such as, the Fast Fourier Transform (FFT) [1], [2] and the Wavelet [3], [4] are adopted to calculate the voltage envelope. However, these algorithms intro- duce a lag in the envelope tracking that is equal to the length of the used window. It is this delay that imposes some limitations on the on-line application of these algorithms. The envelope of the voltage signal can be estimated by var- ious estimation methods such as the Kalman Filter (KF) [5], the Least Absolute Value (LAV) [6], the Simulated Annealing (SA) [7], and the ADAptive LInear NEuron (ADALINE) [8]. Although the KF is fairly accurate, it has a high mathematical burden which limits its use for on-line tracking. The LAV and SA algorithms require that the flicker waveform is known in ad- vance which is not a realistic assumption. Although the ADA- LINE is efficient and has a fast convergence, compared to other estimation algorithms, the ADALINE is still considered mathe- matically cumbersome. Manuscript received December 27, 2004; revised December 24, 2005. Paper no. TPWRD-00621-2004. The authors are with the Department of Electrical and Computer Engi- neering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: maaelder@uwaterloo.ca). Digital Object Identifier 10.1109/TPWRD.2006.874106 In [9], it is proven that the energy contained in a sinusoidal signal is proportional to the square of the signal amplitude. Thus, it is proposed in [10] to use the energy operator to track the envelope by tracking the amplitude of the signal. The Teager Energy Operator (TEO) is successfully applied to achieve this task since the TEO calculates the energy, and in turn, the amplitude, by only three samples [11]. Since, the algorithm does not depend on any kind of estimation or optimization, the algorithm is considered suitable for on-line tracking. Moreover, the delay between the actual envelope and the tracked envelope is within two samples only. However, the algorithm’s practical application indicates that large spikes are embedded in the tracked envelope which is a fatal drawback in using these measurements for flicker mitigation techniques. In addition, these spikes reach very high values when the level of the amplitude is suddenly changed [10], [12]. In this paper, tracking the fluctuation in the voltage waveform is achieved by a novel algorithm to calculate the energy oper- ator. The proposed algorithm attains the same TEO advantages: simple, fast and independent of estimation techniques. Yet, the new algorithm overcomes TEO drawbacks, insensitive to noise and immune to severe spikes in voltage magnitude. This paper is organized as follows: Section II describes the mathematical modeling of flicker. Section III presents the TEO along with the mathematical derivation of the proposed algo- rithm. Different circuit designs of the proposed algorithm are discussed in Section IV. In Section V, the proposed algorithm is used to track the envelope of various signals modulated by different waveforms. Section VI shows the experimental verifi- cation of the proposed algorithm. Finally, Section VII concludes the paper. II. VOLTAGE FLICKER GENERATION Voltage flicker is characterized by slow changes in the root mean square (rms) of the voltage that cause the light flickering. Voltage flicker can be represented by either a modulated signal or generated experimentally by utilizing nonlinear loads. A. Flicker Simulation The slow changes in the are represented by the following amplitude modulation: (1) where is the amplitude of the fundamental, is the funda- mental frequency and is the modulating signal that can have different shapes such as square wave, sinusoidal wave, triangle 0885-8977/$20.00 © 2006 IEEE