A New Algorithm for Reactive Electric Power Measurement

Transcription

1 A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 7-25, 27 A ew Algorith for Reactive Electric Power Measureent Adalet Abiyev Girne Aerican University, Departernt of Electrical Electronics Engineering, Mersin, Turkey Abstract In this paper a Walsh function (WF) based new algorith for reactive power easureent is presented. The proposed algorith allows the convenient calculating process to obtain a reactive power fro the entire instant power signal without tie delay between current and voltage signal. To test the validity of the suggested approach the siulation tool has been developed by use of Matlab 6.5 software environent. Keywords: Discrete Walsh function, active power, reactive power, phase shift Introduction The reactive power directly influences the power factor and, as a result, overloads the connecting cables between the electrical energy sources and energy user devices and plays a vital role in the stable operation of power systes (Fairney 994). The extension of the wavelet transfor to the easureent of power coponents (reactive power and active power) through the use of a broad-band quadrature phase-shift networks is deonstrated in (Yoon & Devaney 2). The proposed wavelet-based power etering syste requires the phase shift of the input voltage signal. According to Purkayastha & Savoie (99) the aplitude-pulse odulation together with phase shift operation is used to easure reactive power in the frequency range of fro 5 to 7 Hz. An electronic shifter based on stochastic signal processing for siple and cost-effective digital ipleentation of a reactive power and energy eter has been proposed by Djokic et al. (2). Toral et al. (2), suggest a coputer algorith for calculating reactive (quadrature) power. In scientific papers the averaging of the value of the product of the current saples and the voltage saples with shifting to the quarter one of the saples (current or aabiyev@gau.edu.tr 7

2 Algorith for Electric Power Measureent voltage) relatively to another is used. The Fourier transforation including fast Fourier transforation based digital or analogue filtering algoriths, used to easure reactive power, involves quite coplex coputations. In single-phase circuits an instant electric power (EP) can be evaluated by using different ethods depending on load type. In a siple case, when a source voltage, u(t), and a current flowing through load, i(t), are the pure sinusoidal signals, the instant EP is defined by Abijev (26). p () t P [ P cos 2ω t + Q sin 2ω t] = () For applying a discrete WF for easuring EP coponents the instant power, p () t, is written in the discrete for as 2π 2π P( n) = P P cos 2 t n + Q sin 2 t n (2) T T Where t is the tie interval between neighbourhood saples, t =T/, is the nuber of saples within observation period of T, n=, 2,...-. Considering these equalities, final expression for power is written as P 4π 4π ( n) = P P n + Qsin n cos (3) Thus we have suitable expressions (Eq. 2 and Eq. 3) allowing for siultaneous easuring of active and reactive coponents of the EP by use of WF. Analogue Measureent Approach This approach is based on analogue signal processing theory and iplies the integral of the product of instant power, p ( t), and corresponding order of analogue WF: S( ) = p( t) Wal(, t) dt (4) 8

3 A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 7-25, 27 where is an order of WF, n is the required integer nuber of averaging periods T. The higher the n the lower the rando error influence to the easureent results. When =, considering Eq. 2 we get fro Eq. 4: S( ) = {[ P ( P cos 2ω t + Q sin 2ω t)] Wal(, dt (5) Since zero-order WF, Wal(, t) has only + value in the noralized period of T[7], Eq.(5) results in average(active) power, P: S( ) = Pdt = P (6) When =3, considering Eq. 2 we get fro Eq.4: S( 3) = {[ P ( P cos 2ω t + Qsin 2ω t)] Wal(3, dt (7) Third-order WF, Wal( 3, t) is an odd function with noralized cycling period of T/2 and is orthogonal with the P cos 2ω t ter as it should be (figure ). So P cos 2ω twal(3, t) dt = The average power, P has constant value within a period of T, therefore: PWal(3, dt =. So, S( 3) = Q sin 2ω twal(3, dt (8) As can be seen fro figure, the product of Q sin 2ω t Wal(3, dt results in rectification of the reactive coponent of the entire instant power signal, p () t. As a result integral of Eq.(8) gives the average value of the reactive power, Q. 9

4 Algorith for Electric Power Measureent Figure. Graphical interpretation of the rectifying effect Digital Measureent Approach For digital easuring of the active and reactive power coponents we use discrete expression of the WF (Abiyev & Aliyev23, Abijev 26). Wal ( i, ) = ( ) k ω + k ω k β k k = β (9) where i is order of WF in the WF syste, i=,,2,,-, β k is arguent of WF and defines the bit(digit) coefficients of β k represented in binary code, β = ( β, β 2... β k ), β =, 2 k, ω is the bit(digit) coefficients of ω represented in binary code, ω = ( ω, ω, ω2... ω ), ω =, 2, is a in binary representation of highest-order WF serial nuber in the WF syste. For exaple, if nuber of saples, =64 the diension of WF syste would also be 64. So fro = 2 we get = 6. 2

5 A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 7-25, 27 ow we can write general expression for digitally easuring of the EP by use of equalities of Trautan (975) and Abiyev (26): S( i) = n= P( n)( ) ( ω k+ ω k ) k= For easuring the reactive coponent of the EP we use the third- order WF, Wal ω ( = 3, β k ). For the third-order Walsh function ω = 3 therefore only ω = 6. The reaining bit coefficients of and 5 ω, =,2,3, 4 are equal to the zero: 3 =. In this case the third-order WF is ( ) 2 βk () ( ω 5 ω 4 ) β 2 ( ) β 2 ) β 2 W (3, β ) = (-) = (-) = ( () 2 The arguent, β 2 changes depending on noralized tie of T=.2s as shown in figure 2. Figure 2. Tie representation of final three bit coefficients of the binary representation of β k 2

6 Algorith for Electric Power Measureent Figure 3 depicts the β k and third-order discrete WF, W ( 3, β 2 ) = ( ) β 2. So for the third-order coponent of the EP fro Eq.() and Eq.(3) we obtain the equality β2 () 3 = S P( n) ( ) (2) n= Figure 3. Tie representation of β k and third-order discrete WF Considering Eq.(3) we have 4π 4π β 2 () 3 = S cos sin P P n + Q n ( ) (3) n= The next ters of this su are equal to zero: β P( ) 2 = n = 4 2 and () 3 = π β S P cos n ( ) = = n Considering these, Eq. (3) becoes(see figure 3): 22

7 A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 7-25, 27 / 4 / 2 3 / 4 () = + 4π 4π 4π 4π S 3 Qsin n Qsin n Qsin n Qsin n (4) n= n= / 4 / 2 n= 3 / 4 The analysis of intervals indicated in the figure 3 shows that since the function of sin( 4π n / ) has negative values at the intervals of [/4, /2-] and [3/4, -], then the Eq. (4) results in: S () 3 = n= 4π Qsin n (5) 4π This expression defines the average value of the signal of Qsin n and is proportional to the average reactive value of the EP in the investigated circuit. During experiental studying the input voltage, u (t), and the current, i (t), signals were taken as u( t) = U sin( ω t) and i( t) = I sin( ω t ϕ), where I =2A, U =4V, ω = 2πf, f = 5 is the linear frequency in Hz, ω =34 is frequency in rad/sec, ϕ -phase shift between the voltage, u (t) and the current, i (t) signals. During experiental studies the phase shift,ϕ between the voltage, u (t) and the current, i (t) signals has been varied in the interval of ϕ = 9. The signal proportional to the instant value of the power, p(t) which is applied to the first nputs of the pair of ultipliers is represented in figure 4 and is written as p(t) = 8sin(34t) sin( 34t ϕ) The tie representation of the signals S (t) and S 3 (t) are shown in the figure 4. The essential advantages of the proposed ethod for the easuring of the reactive power have been verified by experiental studies. One of these advantages is that, in contrast to the known existing ethods, the proposed ethod does not require the tie delay of the current signal to the π / 2 with respect to the voltage signal. The tie delaying process requires the corresponding hardwire which ay result in the additional easureent error. 23

8 Algorith for Electric Power Measureent Figure 4. Coponent output signals versus tie representation Conclusion The evaluation and easureent of coponents of EP with application of a Walsh function siplifies the volue of coputing operations on soe order in coparison with sets of algoriths based on decoposition of signals on haronics (trigonoetric coponents). Measuring of the reactive EP Walsh functions results in certain advantages: - during the processing of the signals on the base of Walsh functions the tieshifting of the signals acts on the structure of the signals. This influence becoes 24

9 A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 7-25, 27 useful during the evaluation of the power coponents allowing the obtaining of extra knowledge concerning the phase-shifts on the haronics of the input signals; - during digital signal processing the saple values of the signals ultiplication by Walsh functions is replaced by suing of the saples with the corresponding +or - signs. References Abiyev A, 26. The Walsh Function based electric power easuring ethod. Proceedings of the 3rd International Syposiu on Electrical, Electronics, and Coputer Engineering, oveber 23-25, 26. icosia, orth Cyprus, Abiyev A, Aliyev IM, 23. An electric power easureent based on discrete Walsh transforation. Proceedings of the International Conference Interactive Systes: The Probles of Huan-Coputer Interaction, Septeber 23, Ulyanovsk, Russia, Djokic B, So E, Bosnjakovic P, 2. A high perforance frequency insensitive quadrature phase shifter and its application in reactive power easureents. IEEE Trans. Instru. Meas., 49, Fairney W, 994. Reactive power real or iaginary? Power Engineering Journal, 8 (2), Purkayastha I, Savoie PJ, 99. Effect of haronics on power easureent. IEEE Transactions on Industry Applications, 26 (5), Toral SL, Quero JM, Franquelo LG, 2. Reactive power and energy easureent in the frequency doain using rando pulse arithetic. IEE Proc.-Sci. Technol., 48 (2), Trachtan AM, Trachtan VA,975. Fundaentals of the discrete signals theory on the finite intervals. - M.: Sov. Radio, 28 p. Yoon W-K, Devaney MJ, 2. Reactive power easureent usinq the Wavelet Transfor. IEEE Trans. Instru. Meas., 49,