Evaluating Power Quality through the Sample Entropy Algorithm

Transcription

1 Evaluating Powe Quality though the Saple Entopy Algoith S. Valencia Raíez, J. H. Estada, E. A. Cano Plata Abstact This pape pesents a bief suay of pevious wok, whee a new ethod to easue the powe quality though electoagnetic fields adiated by distibution systes, was poposed. It has been shown that is possible to easue the wave fo iegulaity of electical signals adiated by electical syste by eans of the saple entopy algoith. This wok shows the behavio of entopy when it is used to evaluate the phenoena that deteioate the powe quality. Distotions, tansfoes aging and oveloadings ae detected by eans of the entopy concept. Index Tes Entopy, electoagnetic fields, powe quality, distotions, iegulaity I I. INTRODUCTION N ELECTRICAL ENGINEERING, the study of inteuptions and wave distubances in the electical supply ceates the concept of powe quality. This appoach has gained inceasing inteest in powe distibution copanies. Phenoena such as haonics, swells, sags, flicke, etc. ae analyzed though voltage and cuent signals and cuently the powe systes ae peanently onitoed by data acquisition systes. The diffeent electical distubances affect eleents of the distibution syste, specially the electic tansfoe []. The electical syste onitoing is fundaental to guaantee a eliable and quality supply. Cuently thee exist diffeent kind of equipent, which ae able to obtain powe quality indexes by invasive o non-invasive ethods, which eans, soe of the equie to be connected to the syste and soe of the does not [][3]. The voltage and cuent easueents ae done though diect inteventions with invasive equipent on electical installations fo detecting incipient faults in the netwoks eleents (tansfoes, geneatos, filtes, loads). This kind of easueents could Date subitted fo eview 9 th Jun. Date subitted fo nd eview 8 th Oct. This wok was suppoted by the Univesidad Nacional de. S. Valencia Raíez is with the Electical, Electonic and Copute Engineeing Depatent, Univesidad Nacional de, K7 via to Aipot, Manizales, (e-ail: svalencia@ unal.edu.co). J.H. Estada is with the Electical, Electonic and Copute Engineeing Depatent, Univesidad Nacional de, K7 via to Aipot, Manizales, (e-ail: jhestadae@ unal.edu.co). E.A. Cano - Plata is with the Electical, Electonic and Copute Engineeing Depatent, Univesidad Nacional de, K7 via to Aipot, Manizales (e-ail: eacanopl@ unal.edu.co). be highly expensive if it is needed to evaluate equipent that is found fa away fo load centes of electical syste. On the othe hand, the non-invasive ethods can be bette candidates fo evaluating an electical syste in a fast and pactical way, the acoustic eission and the infaed ethod ae two of the. These last ae a good solution fo the detection of possible powe quality issues though patial onitoing of electical and echanics signals poduced by the syste. The signals ae ead by sensos located neaby to the eleents that ae pat of the netwok, aking the safe and pactical to use [4][]. Estada et al [6] have been developing a new ethod that allows to obtain a powe quality index. This index is taken fo the electoagnetic waves adiated by lines and tansfoes, as a non-invasive ethod. Estada et al deonstated that the entopy of agnetic and electical signals, ises when the espective wave suffes defoations owing to distotions poduced by haonics. The pevious woks have focused in taking easueents fo powe and distibution systes. The pesent wok goes futhe in the study of the entopy behavio on seveal signals siulated unde distubances of peiodical type. II. ENTROPY Entopy is a concept that has been applied in diffeent disciplines, basically it easues the tendency of a syste towad the chaos, (destuction, disode, iegulaity, insolvent systes) [7]. A. Saple Entopy SapEn Richan and Mooan [8] took the studies ade by Pincus [9] and poposed the Saple Entopy as an algoith fo evaluating the coplexity of chaotic systes, which is based on a fast calculation of the vaiability fo non-lineal tie seies. The SapEn algoith easues tie seies saples pobability of being located at a cetain distance. The Saple Entopy of a signal is obtained in the following way []:. Fo a tie seies X x,..., xi,... xn of length N the following vectos ae obtained

2 U. () i x, x,..., x i, i N i i ni. is defined as the nube of vectos U ( j) that ae () close to the vecto U() i. The distance that sepaates the vectos is calculated as the axiu absolute value of the diffeence between vecto s coponents. d x(), i x( j) ax Ui ( k) U ( j k) 3. Then C () i is calculated as the egulaity of any vecto U ( j ) is to be close the vecto U () i within a toleance. k,..., () Let X x,..., x i,... x N ( ) seies y whose values ae aveages of the consecutive eleents of X without any ovelaps, whee is the scale facto. Each eleent geneated fo the tie seies is calculated though thee equation: y j j i ( j) then it is constucted a new tie x, i j Whee co () esponds to y and this is the oiginal tie seies. Each seies length is the X length divided by the facto. This opeation is appeciated in Figue. N () C () i n () i N () 4. It is consideedd the natual logaith of each C () i and then it is aveaged fo i. N N i. It is inceased the diension to to 4 ae epeated. 6. So, the SapEn is calculated as: ln C ( i) SapEn(, ) li ln N The SapEn estiated value depends on and. In this wok and is estiated as. SD x, whee SD x is the standad deviation of the oiginal data sequence, X. This values ae suggested by Pincus [9] and ae validated by Richan[8] and Madalena [ ]. B. Multi-scale Entopy MSE Multi-Scale entopy MSE is a tool that can evaluate the tie seies in uppe scales. The MSE poposed by Madalena Costa et al [] has been widely used in studies of physiological signals in which it allows to ecognize between healthy and unhealthy oganiss. One of the ost ipotant featues of the MSE is that it can diffeentiatee diffeent kind of signals, as fo instance, the pink noise that is pesent in a lot of phenoena such as eathquakes, stock aket behavio, distibution of ountains, etc. [] [4]. Madalena poposed to conside the stuctue and oganization of the tie seiess in uppe scales. and t the steps fo (3) (4) Fig. Uppe scales calculation fo a tie seies. Finally, each new tie seiess epesents a new scale functionn of the facto, which willl be pocessed by SapEn. This pocedue is called Multi-scales entopy. It is ipotant nott to foget that on the paaetes and all the calculations depend, that ae chosen as well as the data saple fequency. Theses paaete should not be changedd fo the evaluation of a study case because the esults entopyy will be biased. A ipotant featue of the entopy is that thei values ae not affected by the aplitude of the signal, the SapEn only evaluates the wave fo that the signal analyzed has along its seies []. This eans that a signal of V and anothe signal of 44V, which have the sae level of haonic distotion, will have the sae entopy value. III. E ENTROPY MEASUREMENTS A. SapEn It iss possible to evaluate the iegulaity that an electical wave has, though the SapEn algoith, the geathe the iegulaity, the highe the entopy. In ode to appeciate what was said befoe, fou signals, which ae descibed in Table I, wee constucted using Matlab, with a fequency of 6Hz and a saple ate of 4 Hz, then the signals ae pocessed by the Saple Entopy algoith.

3 Table II shows the esults of the entopy value obtained fo each of the signals. Signal is an ideal sinusoidal wave; its SapEn value is salle than othe signals. Signal has a highe iegulaity, this is due to the 3th haonic added to the wave, which causes an incease in the entopy value. The othe signals even ae oe iegula than the fist one, which causes highe entopy values. TABLE I HARMONICS CONSTRUCTED FOR ENTROPY CALCULATIONS Tie Haonics Funtion seies Signal Signal Signal 3 3 d 3 d + th +7 th sin( wt ). sin( wt). sin(3 wt ). sin( wt). sin(3 wt ).sin( wt).sin(7 wt) Signal 4 Gaussian sin( wt) N White noise wee pesent in the pevious signal. See table III. Figue 3 shows how the signal is gets distoted as haonic coponents ae inceased. TABLE III NUMBERR OF HARMONICS IN EACH SIGNAL Wavee Haonic signal signal nd signal 3 nd +3 d signal 4 nd +3 d +4 t signal nd +3 d +4 t signal 6 nd +3 d +4 + th +6 th signal 7 nd +3 d +4 + th +6 th +7 th th th + th Figue shows the constucted signals: Fig. 3 Gaphic epesentation of the signals with haonic distotion to be teated by entopy Fig. Gaphic epesentation of the signals to be teated by entopy TABLE II SAMPEN RESULTS FROM FIG. Tie seies Signal Signal Signal 3 Signal 4 SapEn [nats] These seven signals wee pocessed by eans of the SapEnn algoith, with the uppe scales fo the tie seies of each signal; the MSE cuves ae shown in Figue 4. The esults indicatee that the oe coplex the signal, eaning, the oe distoted it is a by haonic coponents, the highe the entopy value. B. MSE In ode to obseve the entopy behavio when it coes to the evaluation of powe quality phenoena, this wok just tied with steady state distubances: Haonics, flicke and notching. This is the case because the entopy evaluates the iegulaity that the signal has along the tie seies. Alll the signals wee coputed and evaluated though the MSE algoith (This algoith calculates uppe scales). ) Haonics In ode to evaluate haonics distubances though the entopy, seven signals wee geneated. Figue 3 shows the signals constucted, signal coesponds to an ideal sinusoidal wave, and the est of the wee constucted by adding a new haonic coponent, but conseving the distubances that Fig. 4 Multi-scale entopy cuves, obtained fo the signals descibed in Table III The MSE esults can vay fo a signal to anothe, this is due to the scaling change whee the saples that ake the signal ae aveaged and the signal suffes a new discetization of its tepoal seies. This discetization can poduce diffeent behavios to the oiginal MSE cuve. To obtain a esult that integates all MSE values, the scales of each signal ae aveaged, see Figue.

4 TABLE IV SIGNAL WITH DIFFERENT PERCENTAGE OF VOLTAGE DROP CAUSED BY FLICKER AND MSE RESULTS Signal Aplitude deviation[%] MSE aveage [Nats] Fig. Entopy inceases diectly to the injection of haonic coponents to the fundaental 6 Hz wave. Obseving the MSE aveage cuves in Figue it can be seen a eaction of the entopy when the signal is oe and oe coplex, due to the incease of haonics in the signal. ) Flicke The voltage fluctuations ae anothe peiodic phenoenon that altes the wave fo in electical systes. This distubance has two vaiables that poduce a highe coplexity wave-fo, which ae its aplitude and its fequency vaiation. These vaiations poduce peiodical changes in the voltage agnitude. In the sae way of that the detection of haonics iegulaities is done, the MSE can be used to detect Flicke iegulaities. Two scenaios ae analyzed:. Flicke with vaiations in voltage agnitude and fixed fequency of Hz.. Flicke with vaiations in fequency and fixed voltage dops up to % of the oiginal signal aplitude. Fo each scenaio the signals wee constucted with a fequency of 6 Hz, saple ate of 4.Hz, and a total nube of cycles to be evaluated equal to 3. Scenaio : Fo this siulation, 6 waves wee constucted with diffeent vaiation voltages caused by flicke and a fixed fequency of HZ fo each case; the fist signal coesponds to a sinusoidal wave without flicke. A bief epesentationn of these signals is shown in Figue 6, and the esults ae pesented in Table IV A siila esult to the one obtained fo the case of haonic distotion iss given, as soon as the wave is affected by the vaiation in the voltage aplitude, an incease in the signal entopy value stats to develop, the geate the signal iegulaity, the highe the entopy value. As it is shown in Figue 7, the signal with % of deviation fo the peak value, has a oe chaotic behavio when copaed with othe signals. The oe distant is the signal fo the ideal case, the highe the entopy value. Fig. 7 Aveage MSE fo each signal VS. dop level caused by flicke Scenaio : In thiss scenaio the MSE behavio is obseved when the flicke fequency is vaied, sustaining a fixed dop voltage of % fo each siulation case. The chosen fequency values and esultant entopy ae shown in Table V. TABLE V SIGNALS WITH DIFFERENT FREQUENCY BY FLICKER AND MSE RESULTS Signal 3 4 Fequency [HZ]. MSE aveage [ Nats] The signal with a Hz fequency epesents the wave with the lowest entopy; this can be appeciated in the Figue 8. The next signal in ode of iegulaity is the wave with a flicke of.hz; this is due to the vaiation of the cycles along the tie seies. An inteesting fact is that fo fequencies highe than Hz, thee MSE values ae siila between each othe. Fig. 6 Constucted signals with vaiations due to Flicke

5 So fa, it has been obseved thatt the entopy is a paaete that vaies in a diect elationship with signal iegulaity. Pevious woks [6], [], developed by the authos of this pape have easued the electoagnetic fields fo industial and dwelling cicuits. The industial cicuit entopy was highe than the entopy value of dwelling cicuit. This is caused due to the iegulaities in the wave-fo due to the high quantity of non-lineal poducing lage distubances when copaed to the loads connected to the industial netwok, dwelling netwok. Fig. 8 Aveage MSE fo each signal VS. Fequency flicke 3) Notching Anothe peiodical phenoenon is the voltage notching, which siila to haonic and flicke phenoena, affects the powe quality with peiodical distotions in the wave-fo. In ode to siulate the notching phenoenon it was decidedd to constuct eight signals with diffeent pecentage in the notches depth, which ae defined in Table VI. Evey case has a notch occuence with a fequency of Hz. TABLE VI SIGNALS WITH DIFFERENT PERCENTAGE IN THE DEPTH OF THE NOTCHES AND MSE RESULTS Signal Notch Depth [%] MSE aveage [Nats] Likewise as the above cases, it is confied the fact that whateve the iegulaity pesent in the wave-fo, it will be detected by the entopy value and the highe the levels of distotion, the highe entopy values. As it is shown in Figue 9, the signal with the highest entopy value is the wave with a notch depth of 7%. Applying the saee techniques, Estada et al exploed the tansfoe behavio when aging effects affect it. Figue poposes that electical stess causes the entopy of agnetic adiatedd waves to decease []. Fig. The oldest tansfoe shows the lowest entopy value. Based on this kind of elation between age and entopy it could be feasible to pedict the tansfoe lifespan. Up to now, these woks pesent an oppotunity fo developent of a new ethod to easue the electic syste conditions fo the point of view of haonic distotion and aging of tansfoes though the analysis of the adiated electoagnetic signals. It is possible to estiate powe quality issues that thee electical cuent signal can pesent and to deteinate it by eans of agnetic field easueents, in the sae way it is possible to estiate the voltage signal distotion though thee electic field easueents. The fields ae teated sepaately because the distance that exists between the obseve and the eission point is less than a wave length [6]. Fig. 9 Aveage MSE cuve tendency fo diffeent pecentages of notch depth. IV. CONCLUSIONS The entopy is a stong candidate to be a new indicato of the powe quality fo the viewpoint of signal iegulaity. Distotions like haonics, flicke and notching contibute to deflect the signal fo its ideal sinusoidal shape. When the wave loses its ideal fo, the egulaity of the tie seies is affected, which leads to injuious wave shapes. These iegulaities can be detected setting an ideal value of entopy. If the iegulaities appea in the signal, the entopy will do the sae. The geat advantage of the Saple Entopy ethod is that It

6 does not equied knowing the signal aplitude, it is only needed to know the signal wave fo. This is a otivation to continue developing this ethodology in ode to achieve a ean of obtaining electical signals though noninvasive tools, which would peit to evaluate the iegulaity of the signal though of adiated signals of the cicuit In ode to find initiatives fo the developent of new ways to easue the powe quality and to detect incipient faults, eseaches have been done, which have consisted in field easueents though non-invasive ethods. Pevious woks caied out by the authos of this pape have focused in studies about adiated electoagnetic signals. It has been deteined that entopy is a tool that easues the wave-fo iegulaity pooted by phenoena that deteioate it. [4] C. Gell-Mann, Muay and Tsallis, Nonextensive entopy- Intedisciplinay applications, Nonextensive Entopy- Intedisciplinay Appl., vol., 4. [] E. A. C. P. J.H. Estada, S. Valencia Raíez, C.L. Cotés, Magnetic Flux Entopy as a Tool to Pedict Tansfoe s Failues, IEEE Tans. Magn., vol. VOL. 49, N,. [6] J. A. Toes, Javie Ignacio and Salaza, Modelaiento y siulación de capo agnético a fecuencia exteadaente baja en cicuitos secundaios, Sci. Tech., vol. 3,. REFERENCE [] T. Ise, Y. Hayashi, and K. Tsuji, Definitions of powe quality levels and the siplest appoach fo unbundled powe quality sevices, Haonics and Quality of Powe,. Poceedings. Ninth Intenational Confeence on, vol.. pp vol.,. [] R. M. Hais, X. Hu, M. D. Judd, and P. J. Mooe, Detection and location of acing faults in distibution netwoks using a noncontact appoach, in Powe Modulato and High Voltage Confeence (IPMHVC), IEEE Intenational,, pp [3] L. Weixuan and X. Zixiang, Online onitoing and fault diagnosis syste of Powe Tansfoe, Powe and Enegy Engineeing Confeence (APPEEC), Asia-Pacific. pp. 4,. [4] H. Malik, R. K. Jaial, A. Azee, and A. K. Yadav, Application of Moden Technology fo Fault Diagnosis in Powe Tansfoes Enegy Manageent, in Counication Systes and Netwok Technologies (CSNT), Intenational Confeence on,, pp [] C.-C. Kuo, Atificial ecognition syste fo defective types of tansfoes by acoustic eission, Expet Syst. Appl., vol. 36, no. 7, pp. 34 3, 9. [6] E. A. C. S. Valencia, J.H. Estada, C.L. Cotes, Haonics detection in tansfoes by Entopy of electoagnetic signals adiated.. [7] H. W. Boe and F. Takens, Dynaical Systes and Chaos. Spinge,. [8] J. S. Richan and J. R. Mooan, Physiological tie-seies analysis using appoxiate entopy and saple entopy, A. J. Physiol. Cic. Physiol., vol. 78, no. 6, pp. H39 H49,. [9] S. Pincus, Appoxiate entopy (ApEn) as a coplexity easue, Chaos An Intediscip. J. Nonlinea Sci., vol., no., pp. 7, 99. [] M. Costa, A. Goldbege, and C.-K. Peng, Multiscale Entopy Analysis of Coplex Physiologic Tie Seies, Phys. Rev. Lett., vol. 89, no. 6, p. 68, Jul.. [] Calos Jesus Gacía Meza, La teoía de Caos: Algunas Iplicaciones en el Áea de la Metodología en la Ciencia, Univesidad Autonoa de Nuevo León, 993. [] J. Maasoui, Esfandia and Racine, Entopy and pedictability of stock aket etuns, J. Econo., vol. 7, pp. 9 3,. [3] R. E. Phillips, Steven J and Andeson, Robet P and Schapie, Maxiu entopy odeling of species geogaphic distibutions, Ecol. Modell., vol. 9, pp. 3 9, 6.