Determination of Active and Reactive Power in Multi-Phase Systems through Analytical Signals Associated Current and Voltage Signals

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1 56 ACA ELECROEHNICA Deterination of Active and Reactive Power in ulti-phase Systes through Analytical Signals Associated Current and Voltage Signals Gheorghe ODORAN, Oana UNEAN and Anca BUZURA Suary - his paper presents a ethod for calculating the active and reactive power in ulti-phase systes through analytical signals associated voltage and current signals. Analytical signals are coplex signals, vary over tie. he iaginary part of the analytic signal is the Hilbert conjugate real part. he two coponents are orthogonal over a period signal. his property leads to useful consequences in the calculation of active and reactive power. Applications are ade in single phase AC circuits as well as ulti-phase circuits. he ethod is suitable for digital signal processing, for situations where easureent signals are acquired in the for of discrete sequences. Keywords: reactive power, analytic signals, Hilbert transfor, ulti-phase systes.. INRODUCION easureent of active and reactive power in single-phase and ulti-phase circuits is a research apparently closed that nothing new can be brought. his assertion can be contradicted, however, if we take into account the ultitude of research papers appearing in this field. uch of this research relates to techniques for signal acquisition and processing. In the easureent signal processing in the past decades have ade perforing algoriths, which are highlighted iportant inforation on power quality electrical energy quality refers to the present of distortions in ulti-phase syste. Distortions refer to the present current and voltage haronics due to nonlinear consuers. Another aspect of network distortions is the deviation fro syetry of ulti-phase syste. Currently one of the ost used techniques for analysis of ulti-phase syste, is, pq technique, introduced by Akagi and his collaborators [2]. A successful technique is the technique of Hilbert spaces, which extends the analysis to ultiphase systes voltages and currents [],[4],[7],[9]. his paper proposes a ethod to calculate the active and reactive power based on the concept of analytic signal. Signal voltages and currents of a ulti-phase syste are nonsinusoidal periodic signals. hese signals support the developent of coplex Fourier series. runcated coplex Fourier series, the only nonero positive frequency is in fact a analytical signal. Also nuerical sequence of discrete Fourier transfor, anuscript received ay 8, 203. considered only in the positive-frequency space is discrete Fourier transfor of the analytical signal. With these observations, analysis technique based on the analytical signal, the paper proposed is easy to use. he paper is organied as follows: in the second part of the paper presents definitions and properties of analytic signal, useful properties in the analysis below. In Part three relationships active and reactive power calculation, using analytical signal for a ultiphase syste voltages and currents are presented. 2. ANALYICAL SIGNALS. DEFINIIONS. PROPERIES. Let x(t) be a real signal in the frequency spectre of which, the continuous current coponent is absent. We shall further note: xˆ H ( x) () xt ˆ( ) is also called the Hilbert conjugate of the x(t) signal. A real function x(t) and its Hilbert transfor xt ˆ( ) are related to each other in such a way that they together create a so called analytic signal. he analytical signal associated to the x(t) signal is the following coplex signal: x jxˆ (2) he Hilbert transfor can be easily deterined using the reverse Fourier transfor: xˆ H ( x) F ( ( j) X ( j)) (3) where ( ) X j is the Fourier transfor of the x(t) signal: 203 ediaira Science Publisher. All rights reserved.

2 Volue 54, Nuber 2, and : X ( j) F( x ) (4) H( j) j sgn( ) (5) he xt ˆ( ) signal has its spectral coponents shifted by ( / 2) in the doain of positive frequencies and by ( / 2) in the doain of negative frequencies. his odification in the phase spectru, the shifting by ( / 2), is fulfilled by the transfer function ( j), also called the transfer function of the Hilbert transforer. ( ( j), is also called the transfer function of the filter with phase shifting by ( / 2). ). It is useful to observe that: H ( x) x (6) he analytical signal odule is: ( x xˆ ). (7) he arguent of the analytical signal t () is: xt ˆ( ) arg( ) tan. (8) xt () he following properties are useful: If x(t) is a periodical signal with a period, the signals xt () and xt ˆ( ) are orthogonal on the interval (0,): ( ) ˆ x t x 0 (9) he above relation is obvious, taking into account that the haronics of the pairs of signals xt () and xt ˆ( ) are orthogonal. his orthogonality is not always realied in applications because of truncations in nuerical calculations. he energy of the signals xt () and xt ˆ( ) is the sae: 2 2 ( ) ˆ x t x (0) he deonstration is as follows. We shall build the signals: g x xˆ and gˆ xˆ x aking into account the orthogonality relation g gˆ 0 further on we obtain by replaceent: 2 2 xˆ x 0 which leads to relation (0). hus, a function and its Hilbert transfor has the sae energy and therefore the energy can be used to easure the calculation accuracy of the approxiated Hilbert transfor. Let us consider he periodical signals u(t) and i(t) the voltage and current at the bornes of a circuit. We shall have the following useful relations: he average reactive power Q: () Q uˆ i u iˆ he deonstration is as follows: we shall build the signals: h u i and hˆ uˆ iˆ We have: h hˆ 0 which can be developed as: u uˆ uˆ i u iˆ i iˆ 0 aking into account the orthogonality relation, the first and last integrals are ero and in the end we obtain relation (). he average active power can be calculated through one of the relations below: P u i uˆ iˆ Deonstration: we shall build the signal: (2) f u iˆ and fˆ uˆ i. We have: f fˆ 0 which can be developed as: u uˆ u i uˆ iˆ i iˆ 0 and in the end we obtain the relation (2). o the signals u(t) and i(t) we associate the analytical signals: u u j uˆ i i j iˆ An iportant relation for the calculus of coplex power is: S u t i t P jq 2 Deonstration: after replaceent, we obtain: * ( ) ( ) (3) * u ( ) ( ) ( ( ) ˆ( ))( ( ) ˆ t i t u t j u t i t j i ) u i uˆ iˆ j uˆ i u iˆ P P j( Q Q) In relation (3), P, and Q are the average active and reactive power.

3 58 ACA ELECROEHNICA 3. HE CALCULUS OF HE POWER IN A ULI-PHASE SYSE USING ANALYICAL SIGNALS. A. Analysis of the ulti-phase syste in a nonsinusoidal regie, using the concept of analytical signals. Let us consider the ulti-phase syste in nonsinusoidal regie, as described by a set of signals ( x) ( x, x2,..., x ) of arbitrary for. o these signals we shall associate a set of analytical signals: ( ) (,,..., ) (4) 2 We shall extend the Fortescue transforation to the signals ( ). We shall propose a foral analogy between the sequences of syetrical phasor coponents (for the non-syetrical sinusoidal regie) and the sequences of syetrical coponents of analytical signals (for the non-syetrical nonsinusoidal regie). We shall deterine a syste fored of sets of syetrical coponents of analytical signals: ( ),( ),...,( ) which we shall call succession s0 s s signals 0,,...,-. We shall further use the rotation 2 j operator e e j. he direct Fortescue transforation is: l ( ), 0, (5) l he coponents of the ( ) set are [4]:,,,,2,3, 2l,..., l ( ) l (6) he signal, is the reference signal of the ( ) set and shall be further on sybolied as. Each of the coplex signals of the set have the sae odule and are shifted fro each other by the angle l l(2 / ). he ( ) set fors a star whose rays are equal, whose odule varies in tie and which are shifted by an angle lβ. It is a star which spins by a variable angular speed ( d, where tan I ). Re For the sets of l=0,,2,...,- succession, we have such stars. We have the relation: ( ) ( ) ( )... ( ) (7) s0 s s In the above relations, we don't refer to the phasor rotation, but to the shifting of analytical signals. here is a significant difference between phasors and analytical signals: phasors are coplex nubers, while analytical signals are coplex values which vary over a period of tie. he inverse Fortescue transfor is: ( ) l ( ),, (8) l0 Coing back to relation (5), we shall obtain sets of succession ( l 0, ) real signals: ( x ) Re ( ), ( l 0, ) (9) We have the relation: ( x) ( x ) ( x )... ( x ) (20) s0 s s he coponents of the set ( x ) are: x Re{ }, x Re{ }, x x,2,3, 2l Re{ },..., ( ) l Re{ } At the sae tie, we have sets of real signals ( x ˆ ), ( l 0, ) ( xˆ ) I ( ), ( l 0, ). (2) he set of signals ( x ˆ ) is the Hilbert conjugate of the signal set ( x ). Coing back to the original signals, we obtain that: x Re, (, ) (22) And respectively: xˆ I, (, ) (23) B. he calculus of the power in a ulti-phase syste using analytical signals in a succession of syetrical coponents Let us consider a ulti-phase syste of voltages ( u ) ( u, u2,..., u ) and currents ( i ) ( i, i2,..., i ). We shall associate to these values two sets analytical signals ( u ) ( u, u2,..., u ) and, respectively, ( i ) ( i, i2,..., i ). As a result of the Fortescue analysis applied to these signals, we can obtain two systes of syetrical coponents: ( u ), and respectively ( i ), l 0,( ) he active power P l and the reactive power Q which are due to the succession l signals is: or: * Pl Re u i 2 (24) * Ql I u i 2 (25) l

4 Volue 54, Nuber 2, Pl u i (26) Ql uˆ i (27) Where, u Reu, uˆ Iu, i Rei. Also, the instantaneous powers can be calculated as follows. he instantaneous powers corresponding to succession l: or: p u i (28) q u i (29) ˆ he total active power is: ( ) * 2 l0 (30) P Re u i ( ) * 2 l0 (3) Q I u i ( ) P u i (32) l0 ( ) Q uˆ i (33) l0 P and Q represent total active and reactive powers, as defined by Budeanu, due to all voltage and current haronics (pairs of the sae order) of the two sets of voltage and current of the ulti-phase syste. If the power transitted by a certain haronic h (voltage, current) is of interest, then a haroncal analysis of the signals ( ) u and ( i ). should be perfored. he calculus relations are siilar. he above relations are easy to ipleent by processing in discrete tie. In that case, the instruents are the discrete Fourier transfor and the discrete Hilbert transfor. 4. CONCLUSIONS he paper presents a ethod for calculating the active and reactive power in ulti-phase systes, through analytical signals associated voltage and current signals. his new type of analysis consists in using the analytical signals associated to real signals of the ulti-phase syste. By applying the direct Fortescue transforation to these analytical signals, we can deterine sequences of syetrical coponents also expressed as analytical signals. hese output analytical signals are fored of pairs of real signals which are interconnected through the Hilbert transfor. he real coponent of the output analytical signal corresponding to the sae Fortescue succession, also sus up all the haronical coponents of the sae succession. he iaginar coponent of the output analytical signal corresponding to the sae Fortescue succession, also sus up all the haronical coponents of the sae succession, but shifted by ( / 2). hus, we can deterine the active and reactive power coponents corresponding to a certain succession as well as the total active and reactive powers, as defined by Budeanu. At the sae tie, we can deterine the power coponents for certain haronics of interest. he ethod has the advantage of accoplishing the processing of instantaneous signals of the ultiphase syste, no atter their for, obtaining output sequences of instantaneous signals of syetrical coponents signals which su up all the haronics of the sae succession. Finally, it is iportant to note that the ethod can be easily ipleented by processing in discrete tie. REFERENCES. J.L. Willes, A New Interpretation of the Akagi-Nabae Power Coponents for Nonsinusoidal hree-phase Situations, IEEE rans. Instru. eas, vol. 4, Aug. 992, pp Akagi,H; Watanabe, E.H.; Aredes,.,. Instantaneous Power heory and Aplications to Power Conditioning, New Jersey IEEE Press / Wiley-Interscience, 2007., ISBN Sanjit K. itra Digital Signal Procesing. A Coputer-Based Approach. cgraw-hill International edition ISBN G. Grandi, G. Serra and A. ani, General Analysis of ulti- Phase Systes Based an Space Vector Approach. EPE-PEC 2006, Potoro, Slovenia, pp C.I. Budeanu, he Different Options and Conceptions Regarding Active Power in Nonsinusoidal Systes, Instytut Roain de l Energie, pub. No. 4, Bucharest, IEEE rial-use Standard Definitions for the easureent of Electric Power Quantities Under Sinusoidal, Non-sinusoidal, Balanced Conditions, IEEE Std , Jan Hanoch Lev-Ari, Aleksandar. Stankovic, Hilbert Space echniques for odelling fnd Copensation of Reactive Power in Energy Processing Sytes IEEE transactions and systes Fundaental theory and applications, vol.50.no.4,april David G. Long, Ph.D. Coents on Hilbert ransfor Based Signal Analysis ERS echnical Report ERS odoran, Gh. he ± α Phase Shift Filter Cobination between the Fourier ransfor and Hilbert ransfor. Acta electrotehnica,volue 48, no.3, 2007.pp.240., ISSN

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