2858 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 8, AUGUST 2010 A Space-Vector Discrete Fourier Transform for Unbalanced and Distorted Three-Phase Signals Francisco A. S. Neves, Member, IEEE, Helber E. P. de Souza, Fabrício Bradaschia, Marcelo C. Cavalcanti, Member, IEEE, Mario Rizo, and Francisco J. Rodríguez, Member, IEEE Abstract—In this paper, a space-vector discrete-time Fourier transform is proposed for fast and precise detection of the fundamental-frequency and harmonic positive- and negative- sequence vector components of three-phase input signals. The discrete Fourier transform is applied to the three-phase signals represented by Clarke’s αβ vector. It is shown that the complex numbers output from the Fourier transform are the instantaneous values of the positive- and negative-sequence harmonic compo- nent vectors of the input three-phase signals. The method allows the computation of any desired positive- or negative-sequence fundamental-frequency or harmonic vector component of the input signal. A recursive algorithm for low-effort online im- plementation is also presented. The detection performance for variable-frequency and interharmonic input signals is discussed. The proposed and other usual method performances are compared through simulations and experiments. Index Terms—Active filters, amplitude and phase estimation, discrete Fourier transforms (DFTs), industrial power system harmonics. I. I NTRODUCTION T HE fast and precise detection of fundamental-frequency and also harmonic components of three-phase voltages and currents is nowadays a very important task for implement- ing grid-connected voltage-source converters (VSCs). Among the many applications of these detection algorithms are se- ries and shunt active filters, uninterruptible power supplies, adjustable speed drives, and renewable-energy-generation sys- tems connected to the grid through VSCs [1]–[7]. Since grid-connected converters are generally controlled us- ing a reference frame oriented by the fundamental-frequency positive-sequence (FFPS) voltage vector, many algorithms for obtaining this voltage vector have been proposed recently. The first methods for determining the magnitude and phase angle of the grid voltage vector are based on a synchronous reference Manuscript received December 23, 2008; revised July 23, 2009 and September 18, 2009; accepted October 26, 2009. Date of publication November 24, 2009; date of current version July 14, 2010. This work was supported in part by “Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—CAPES” (Brazil) and in part by “Ministerio de Educación y Ciencia” (Spain). F. A. S. Neves, F. Bradaschia, and M. C. Cavalcanti are with the Department of Electrical Engineering, Federal University of Pernambuco, Recife 50730- 540, Brazil (e-mail: fneves@ufpe.br). H. E. P. de Souza is with the Department of Electrical Engineering, Federal University of Pernambuco, Recife 50730-540, Brazil, and also with the Depart- ment of Industry, Federal Institute for Education, Science and Technology of Pernambuco, Pesqueira 55200-200, Brazil. M. Rizo and F. J. Rodríguez are with the Department of Electronics, University of Alcalá, 28805 Alcalá de Henares, Spain (e-mail: franciscoj. rodriguez@uah.es). Digital Object Identifier 10.1109/TIE.2009.2036646 frame phase-locked loop (SRF-PLL) [8]–[10]. The SRF-PLL is adequate for tracking the instantaneous grid voltage vector, but not for detecting the FFPS component, since it is very sensitive to unbalances and distortions present in the three- phase signals. Some methods were proposed for obtaining the desired positive- and negative-sequence vectors in unbalanced systems [11]–[15]. However, the performance of these schemes may be severely affected by harmonics present in the three- phase signals. In order to overcome this problem, some au- thors proposed schemes in which the harmonic components are filtered out from the input signals. In [16], second-order generalized integrators (SOGIs) [17] are used for computing two filtered pairs of orthogonal signals from the α and β com- ponents of the three-phase input signals. The orthogonal filtered signals are used for determining the positive- and negative- sequence vector components. However, these detected signals may be affected by low-order harmonics and subharmonics. Furthermore, the SOGI needs the grid frequency estimation for its correct operation. This frequency can be obtained from an output SRF-PLL, but due to the complexity of the resulting nonlinear closed-loop scheme, it is difficult to design the pa- rameters in order to ensure stability or some desired transient performance. An alternative to this scheme is proposed in [18], based in modulating functions for computing the grid frequency to be used in the SOGI blocks. In [19] and [20], discrete Kalman filters (DKFs) are used for detecting the FFPS signals in three-phase systems. The DKF gains may be adjusted for rejecting the effects of low-order harmonics or negative-sequence components, but the response is slow as compared with the previously mentioned techniques. Fourier transform has been used for many years for separat- ing fundamental and harmonic components in single-phase sig- nals. However, for three-phase signals, there is not an algorithm based on Fourier transform to directly determine the positive- and negative-sequence fundamental-frequency and harmonic components of three-phase signals. Some methods based on fast Fourier transform (FFT) for three-phase systems were recently proposed [21]–[23], but none of them allows the direct separation of the sequence components of each harmonic three- phase unbalanced signal. For online harmonics detection, these methods generally use a recursive discrete Fourier transform (RDFT) algorithm. An active filter that uses an FFT scheme for detecting the FFPS component in balanced three-phase three- wire systems is proposed in [21]. In [22], the phase detection error caused by off-nominal frequency is discussed. Further- more, a scheme for detecting the phase angle of the FFPS voltage vector in three-phase systems eliminating the influence of the negative-sequence fundamental-frequency components is 0278-0046/$26.00 © 2010 IEEE