Estimation of voltage sags patterns with k-means algorithm and clustering of fault zones in high and medium voltage grids

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1 REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338) Estmato of voltage sags patters wth kmeas algorthm ad clusterg of fault zoes hgh ad medum voltage grds Estmacó de patroes de hudmetos e tesó co el algortmo kmeas y agrupacó de zoas de falla e redes de alta y meda tesó Mguel Romero, Lus Gallego ad Adrés Pavas 3 AbstractThs paper proposes kmeas clusterg algorthm to detfy voltage sags patters ad group fault zoes wth smlar mpact hgh ad medum voltage electrc. The proposed methodology comprses three stages. Frst, etwork modelg ad faults smulato were performed order to get formato about voltage sags caused by faults the trasmsso system. Voltage sags patters were detfed at the secod stage by meas of a kmeas clusterg algorthm, allowg the determato of fault zoes. Usg the power qualty measuremets data base of the major electrcty utlty of Bogotá, voltage sags were classfed accordg to the prevously determed voltage sags patters. At the thrd stage of the methodology a comparso betwee smulated ad measured sags s performed, allowg the detfcato of sags caused by faults. Keywords: Sags classfcato, patters voltage sags, Kmeas algorthm. Resume E este artículo se propoe el uso del algortmo Kmeas para detfcar patroes de hudmetos e tesó y agrupar zoas de falla co mpacto smlar e redes de alta y meda tesó. La metodología propuesta comprede tres etapas. Prmero, se realza u modelo de la red de trasmsó y dstrbucó y se smula u barrdo de todo tpo de fallas, obteedo formacó sobre los hudmetos e tesó. E segudo lugar, se detfca patroes de hudmetos e tesó usado el algortmo kmeas y se determa dferetes zoas de falla para cada uo de los patroes. Falmete, se usa los patroes ecotrados para clasfcar formacó real de hudmetos e tesó regstrados e Bogotá y se establece las zoas de falla para grupos de hudmetos e tesó. Palabras Clave: Clasfcacó de hudmetos e tesó, patroes de hudmetos e tesó, algortmo kmeas.. INTRODUCTION Power qualty (PQ) assessmet s really mportat for utltes ad users for detfyg some crtcal areas ther systems ad apply correctve actos to mprove the PQ codtos (Bolle,003). The detfcato of trastory dsturbaces Electrcal Egeer ad MSc. electrcal egeerg, PhD studet Natoal uversty of Colomba. Research assstat the Research group PAASUN. mfromerol@ual.edu.co Electrcal Egeer, MSc ad PhD electrcal egeerg, Natoal uversty of Colomba. Researcher the Research group PAASUN. Assocate professor, Natoal Uversty of Colomba. lgallegov@ual.edu.co. 3 Electrcal Egeer ad MSc Natoal Uversty of Colomba. Caddate to PhD degree the same sttuto. Researcher the Research group PAASUN. Assocate professor, Natoal Uversty of Colomba. fapavasm@bt.ual.edu.co such as voltage sags ad swells requres cotuous measuremet of PQ ad techques for aalyss of large amout of formato. Accordg to above, some methodologes for detecto ad classfcato of dsturbaces are proposed o (Bswal et al, 009; Mokhls et al, 009; Romero et al, 00). O the other had, power qualty measuremet ad assessmet has take relevace sce the publcato of the CREG Resoluto Colomba (CREG 04, 005). The resoluto demads the realzato of power qualty measuremets o bus bars wth voltage levels greater tha kv have to be performed. I Bogotá cty 90 power qualty measurg devces were stalled o the above metoed bus bars of the dstrbuto system, whch record dsturbaces lke voltage sags, swells, ubalace ad flcker, amog others accordg to stadard IEC (IEC , 009). That formato s set to a cotrol ceter, processed ad subsequetly reported to the regulatory body CREG (PAASUN, 009). Ths sttuto wll establsh the lmts for voltage sags from these reports the ear future. The etwork operators are terested o assessg voltage sags to determe ther cause, wth the am of explorg sutable solutos ad establsh resposbltes betwee customers ad etwork operators (Cajamarca et al, 006). Accordg to above, ths paper proposes a methodology that cossts of four stages:. Network modelg. Network dstrbuto system 5kV ad 0kV of the all Colomba system s modeled usg symmetrcal compoets. Ths model s made order to smulate all possble faults. O ths model, ay possble localzato of faults s smulated as well.. Fault smulato. Dfferet types of faults o several locatos are performed. Iformato of voltage sags the whole electrc system of Bogotá s obtaed for every 3

2 ESTIMATION OF VOLTAGE SAGS PATTERNS WITH KMEANS ALGORITHM AND CLUSTERING OF FAULT ZONES IN HIGH AND MEDIUM VOLTAGE GRIDS Fg.. Dsturbg ad terest zoe the Colomba dstrbuto system. 3. smulated fault. 4. Idetfcato of voltage sags patters. Voltage sags formato s aalyzed by meas of prcpal compoets aalyss, afterwards patters of voltage sags are detfed usg the kmeas algorthm. The results are clusters of voltage sags wth a respectve occurrece zoe. 5. Classfcato of real voltage sags. Real voltage sags recorded Bogotá from 008 to 00 are classfed resortg to the prevously determed clusters. Fally, zoes where real faults occur ad cause voltage sags Bogotá are determed.. NETWORK MODELING I order to smulate faults o the dstrbuto system ad fd the relatoshp betwee voltage sags ad faults, a symmetrcal compoets model s developed. Network dstrbuto system 5kV ad 0kV of the etre Colomba system (756 bus bars) s modeled symmetrcal compoets order to get all possble localzato of faults the Bogotá s electrc etwork (Romero, 00). Ths model cossts of postve, egatve ad zero sequece matrx. The terested zoe s defed lke the zoe the Colomba dstrbuto system where voltage sags occurred by faults are evaluated. I ths case, terested zoe cosst o the bus bars of the dstrbuto system o Bogotá cty closed o gray cotour Fg.. Not all faults the Colomba dstrbuto system cause voltage sags Bogotá, therefore a dsturbg zoe s detfed lke the zoe where the occurrg faults ca cause voltage sags o the bus bars of a specfc zoe. I order to detfy the dsturbg zoe, a ew matrx Voltage sags matrx s calculated from the symmetrcal compoets matrces. Ths matrx has formato about voltage of all bus bars whe fault occurs o every bus, as descrbed (Goswam et al, 008). To calculate the matrx of sags caused by threephase faults, the equato for theoretcal faults (Aderso, 973) s used: V Z k = () Z kk Where: V : Voltage o bus bar whe fault occurs o bus bar k. Z k : Mutual mpedace betwee ad k bus bars. Z kk : Self mpedace k bus bar. From (), the matrx of sags s calculated as follow: V sags = [] Z[ DagZ] () Where: V sags : Matrx of voltage sags o all buses whe faults occur o every bus bar. Z: Postve sequece mpedace matrx. The term [DagZ] of the equato s a matrx calculated from the dagoal of the postve sequece mpedace matrx 3 REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338)

3 ROMERO, GALLEGO, PAVAS Z. For usymmetrcal faults (sgle, dualphase ad dualphase to groud) a smlar procedure s performed, as explaed detal (Romero et al, 00). The sags matrx s modfed by removg the bus bars whch faults occur but do ot cause voltage sags, especally at the set of bus bars of terest. The result of ths procedure determes whch bus bars of the Colomba system are the dsturbg zoe. I the case of ths paper, the zoe of terest s Bogotá (gray zoe), the dsturbg zoe are all bus bars Fg., cludg the bus bars the gray zoe. That shows that may voltage sags observed the Bogotá's system may be caused by faults sde the cty or by faults located the earby parts of Colomba system. 3. FAULT SIMULATION To determe the voltage sags profle every bus sde the zoe of terest, smulatos of faults all bus bars ad sectos les of the dsturbg zoe are performed. For ths, every 0% of le dfferet types of faults (sgle, dual ad three phase fault) are smulated the dsturbg zoe. The smulato procedure s descrbed the followg:. A vector wth formato regardg faults occurrg o les s geerated (L), cotag percetage of le (%T), type of fault (Tf) ad mpedace of le (Z).. Symmetrcal compoets matrces are modfed by removg the faled le betwee odes A ad B. 3. Impedace (Z) of faulted le s splt to two (Z $ ad Z ) accordg to (%T). 4. Z s added to ode A of matrces sequece geeratg a fcttous ode C. 5. Z s added betwee the fcttous ode C ad ode B of the sequece matrces. 6. Depedg o the type of fault (Tf) voltages of all odes are calculated whe fault occur fcttous ode C. 7. The agles of voltages symmetrcal compoets are modfed due trasformer coectos. 8. Values of voltage symmetrcal compoets are trasformed to values of voltage phase compoets. The locato of faults ad voltage values obtaed each bus the area of terest ca be orgazed two arrays: cause ad effect, as show Table. Table.Orgazed formato for detfyg voltage sags patters. Locato ad type of faults Le % of le Type of fault Voltage Sags magtude (pu) Va Vv Vc Va Vb Vc ,6 0,8,00 0,05 0,79, ,83,00,00 0,8,00, ,78 0,07 0,00 0,78 0,00 0,00 I the ext secto, smulated voltage sags are grouped accordg to the patter detfcato. Because of the causal relatoshp betwee smulated faults ad voltage sags, dfferet obtaed clusters are a drect classfcato of faults. That s, faults that occur dfferet parts the system ad geerate smlar profles of voltage sags are grouped the same cluster. 4. DETECTION OF VOLTAGE SAGS PATTERNS BY MEANS OF KMEANS ALGORITHM Wth the voltage sags formato caused by faults, patters of voltage sags are detfed ad grouped. The, the relatoshp betwee the locato of faults ad the occurrece of such patters s determed. Kmeas algorthm s a tool to put observatos to dfferet clusters accordg to the level of smlarty (Quepo, 00; Ramos, 009). Some advatages of kmeas algorthm for clusterg data were detfed prevous works (Mora et al, 009); (Camargo et al, 009). A example of that algorthm s show Fg., where observatos o two dmesos (X,Y) are grouped o three dfferet clusters. Fg.. Groupg of observatos by kmeas algorthm Fg. shows the performace of kmeas method, however axes uts have ot a quattatve or physc meag. Kmeas algorthm cossts of the follows steps:. A tal k value of clusters s defed,. k cetrods (+) are located radomly o the sample space, 3. dstaces betwee observatos ad cetrods are calculated, 4. each observato s assged to the earest cetrod, 5. the posto of cetrods s update to the average of the observatos assged to each cetrod, 6. Several teratos are performed from step order to mmze the dstace betwee observatos ad cetrods. REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338) 33

4 ESTIMATION OF VOLTAGE SAGS PATTERNS WITH KMEANS ALGORITHM AND CLUSTERING OF FAULT ZONES IN HIGH AND MEDIUM VOLTAGE GRIDS I ths case kmeas algorthm s mplemeted to detfy voltage sags patters matrces of Table I. I voltage sags matrx (44 x 6355) every row belog to a observato the tme of 44 voltage values (3 phases of 48 bus bars). For applyg kmeas algorthm to voltage sags matrx some lmtatos about the algorthm ad hgh dmesoalty of the matrx are evaluated: The kmeas algorthm has some mportat lmtatos: Iteratve algorthms of classfcato lke kmeas are defcet for problems o a large scale. Kmeas algorthm eeds a lot of tme to optmze the clusterg process. Because of hgh dmesoalty of the matrx, the possble tal locatos of cetrods crease expoetally, so lkelhood to stop a local maxmum creases correspodgly. The utlzato of the above metoed algorthm has also lmtato regardg to the hgh dmesoalty of the voltage sags matrx: Matrx ca have redudat formato the 44 varables. Matrx ca have rrelevat formato wthout capacty of dscrmato clusters. Matrx ca have segmeted formato, ts meas useful formato ca be dstrbuted o several varables. Accordg to these udesred characterstcs, reducg dmesoalty of voltage sags matrx s ecessary before patters are detfed. For ths purpose, prcpal compoets aalyss s mplemeted the ext secto. A. Prcpal Compoets Aalyss The utlzato of prcpal compoets aalyss (PCA) s proposed to reduce the dmesoalty of the voltage sags matrx. I ths aalyss, the formato of observatos ad p dmesos are represeted wth r<p dmesos. The ew dmesos are leal combatos (o correlated) of the tal oes (Melédez et al, 007). A example of the prcpal compoets aalyss (PCA) s show Fg. 3. I that case, the am s reduce the dmesoalty (p=) of the observatos 4. I Fg. 3 the am s to fd a subspace wth dmeso smaller tha p, such that by projectg each observato, ths retas ther structure wth the least dstorto as possble. 4 A example of prcpal compoets aalyss s llustrated Fg 3, however axes uts have ot a quattatve or physc meag Fg. 3. Prcpal compoets aalyss for dmesos observatos. The subspace wth dmeso smaller s represeted by a le whch has the followg codto: The sum of the dstaces betwee the orgal observatos ad ther projectos oto the le should be as short as possble. To expla the above, the projecto of the observato X o the drecto a Fg. 3 s the scalar: Xˆ aˆ projx = (3) aˆ = z aˆ The vector z a represets the projecto of X oto the le ad r represets the dstat betwee X ad the le. The the purpose s to mmze the square of the sum of the dstaces r : Mmze r = = = x z a I Fg. 3 the projecto of each observato oto le forms a tragle. By the Pythagorea Theorem, ext equatos are deduced: (4) x = z + r (5) x ' x = z + r (6) by the sum of all observatos =... = = x' x = z + r (7) = The frst term of Eq. (9) s costat, thus mmze r s equvalet to maxmze z, whch meas maxmzg the sum of the square of the projectos. At the same tme, t s equvalet to maxmze the varace (Peña, 00). 34 REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338)

5 ROMERO, GALLEGO, PAVAS Accordg to the above, the best le to represet the observatos a sgle dmeso s oe whch maxmze the varaces of the data. Ths crtera s exteded to dmesoal space, where <p. I order to fd the ma compoets of a voltage sags matrx from the covarace matrces, the Matlab fucto prcomp s used. The result of ths fucto s a matrx wth 44 ucorrelated varables orgazed a way that the frst varables have the greatest varace. It meas that frst varables have the most formato from the tal matrx. Table. Varace percetage of prcpal compoets. Number of compoets Varace percetage (%) 64,34 79,6 3 84, ,3 5 89,45 6 9,9 The Table II shows the percetage of varace accordg to the umber of prcpal compoets. The frst fve prcpal compoets represet the 89.4% of the varace of the orgal data, t meas early 90% of the total formato. Fally, these fve compoets are selected, so the tal matrx (6355 x 44) s reduced to ts ma compoets (6355x5) for applyg kmeas algorthm. B. Clusterg of voltage sags ad fault zoes for each cluster The ext step s to determe the optmal umber of clusters for groupg the observatos of voltage sags (Daves ad Bould, 979). For ths purpose, some dexes are calculated for each value k of clusters, the the best value of the dcator shows the optmal umber of clusters to group the formato. For ths case three dfferet dcators are used ad results are show below.. Idex. Square of the sum of the dstaces betwee observatos ad cetrods. (Optmal result k=0). Ths dex shows the varato of the sum of the dstaces betwee the data from each of the clusters ad ther cetrods.. Idex. Slhouette dex. (Optmal result k=0, 55 y 85). I ths dex a value betwee ad s assged to each observato, whch measures the smlarty of the data the same cluster ad compares t wth the smlarty of data from other clusters. The closer the dex to, the better the data are grouped together. 3. Idex 3. Relatoshp tertra cluster dstaces. (Optmal result k=60). Ths dex estmates the proporto betwee the average dstace of the data to the respectve cetrods ad the mmum dstace betwee cetrods. The results of the proposed dcators obta dfferet k value for a optmal clusterg. Ths mples the observatos of voltage sags are ot aturally grouped, t meas there are ot clearly dfferetated clusters. Gve that the results of the dcators were coclusve, t s possble to group the observatos a approprate umber k of clusters defed by the goal of the groupg. Therefore the goal of groupg s defed as follow: Clusterg of voltage sags represets a drect faults classfcato (locato ad type) accordg to ther mpact (sags profle). Thus, the umber of clusters s related to the sze of the zoes where faults occur ad cause smlar profles of voltages sags. Therefore, few umber of clusters meas few very bg zoes ad s ot possble to dscrmate the place of occurrece of faults wth dfferet mpact o bus bars. O the other had, a lot umber of clusters meas very small zoes ad the classfcato s effcet. After tryg varous amouts of clusters, the sze of the resultg zoes was evaluated for dfferet umber of clusters, fally 50 clusters are selected. Wth ths umber of clusters sze of zoes s cosdered approprate. By applyg kmeas to the prcpal compoets obtaed above, a vector C(6355 x ) s obtaed ad t dcates whch of the 50 clusters are classfed each observato of matrx voltage sags ad matrx faults. By groupg the locatos of faults by the vector C, zoes whch faults have smlar mpact are determed. I Fg. 4 the locato of faults that geerate voltage sags wth smlar mpact classfed cluster 48 s detfed (gray zoe). Smlarly, by groupg the types of smulated faults by the vector C types of faults of each cluster are determed. 5. CLASSIFICATION OF REAL VOLTAGE SAGS OCCURRED ON BOGOTÁ. Clusterg of smulated voltage sags s ow used as a classfer for real voltage sags occurrg the 5kV bus bars of Bogotá cty. For ths, the recorded formato of voltage sags each of the bus bars betwee Jauary 008 ad December 009 s processed a cetralzed database. REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338) 35

6 ESTIMATION OF VOLTAGE SAGS PATTERNS WITH KMEANS ALGORITHM AND CLUSTERING OF FAULT ZONES IN HIGH AND MEDIUM VOLTAGE GRIDS Fg. 4. Dsturbg zoe for voltage sags grouped clusters 48. For obtag a matrx of N observatos x 44 varables, the voltage sags formato s processed takg to accout the followg assumptos: Voltage sags that occur wth a tme wdow of mute are caused by the same evet5. Several voltage sags the same wdow the same bus are caused by the same evet. If voltage sags are ot recorded a bus a tme wdow, the voltage sags does ot occur for the same tme wdow.. observato of 44 dmesos are calculated, observatos are assged to the earest cetrod. The fal result of the classfcato s show Table III, whch ca be aalyzed as follows: O the other had, accordg to the smulatos, faults the 5kV ad 0kV trasmsso system ot cause voltage sags oly oe bus, or voltage sags bus bars sulated from each other, so that formato s excluded. As a result, from 7580 measured voltage sags all 5kV bus bars of Bogotá caused by 955 evets, 3509 voltage sags are fltered ad they are attrbutable to 360 faults the trasmsso ad dstrbuto system. It meas a real formato matrx wth (360 x 44) dmesos. The formato of voltage sags attrbutable to faults the trasmsso ad dstrbuto system s classfed wth the cetrods of each cluster foud the prevous secto as follows:. 360 evets attrbuted to faults are classfed to dfferet clusters, cluster 4 s the oe wth more evets (5). For clusters 4, 33, 35 ad 48 do ot have evdece of the type of fault occurs. Faults that produce more voltage sags are grouped to clusters 48, 4 ad wth 986, 848 ad 77 voltage sags respectvely. Faults clusters, 7, 4 ad 37 are ot threephase faults. Faults clusters 9 ad 47 are twophase or twophase to groud. Faults Clusters 7 ad 4 occur few tmes but ther mpact o the bus bars s the hghest wth 47 ad 46 bus bars affected by fault respectvely. Faults cluster 4 occur frequetly, buy they have the least mpact wth oly 4 bus bars affected by fault. Fgs. 5 ad 6 dsplay dsturbg zoes for clusters 4 ad 33 where more umber of faults occurred, ad caused voltage sags the cty of Bogotá. Dstaces betwee the 50 cetrods ad every real 5 Ths s because the data records do ot have a better resoluto to provde a smaller tme wdow sze. 36 REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338)

7 ROMERO, GALLEGO, PAVAS Fg. 5. Dsturbg zoe for voltage sags cluster 4. Fg 6. Dsturbg zoe for voltage sags cluster 33. REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338) 37

8 ESTIMATION OF VOLTAGE SAGS PATTERNS WITH KMEANS ALGORITHM AND CLUSTERING OF FAULT ZONES IN HIGH AND MEDIUM VOLTAGE GRIDS Cluster umber Table 3. Clusterg of voltage sags occurred o 5kV bus bars of Bogotá Number of faults Percetage of faults Type of faults Total umber of sags 5,83%,, ,50%,,3, ,8%,, ,83% ,8%,, ,8% ,8%, ,06%,,3, ,%,,3, ,8%,, ,8%, ,00%,,3, Bus bars affected/fault 6. CONCLUSIONS Ths paper proposed a methodology for estmatg patters of smulated voltage sags by meas of kmeas algorthm ad for determg the locato of the faults that cause voltage sags dstrbuto bus bars of the system. The Colomba system was modeled ad fault smulato was performed all sectos of the system to geerate voltage sags. Subsequetly smulated voltage sags were clustered usg prcpal compoets aalyss ad kmeas algorthm, allowg the detfcato of zoes of faults occurrece for each cluster. Voltage sags clusters were used to classfy real voltage sags occurred the system. The methodology was appled to the 5kV ad 0kV system of Bogotá ad real areas where faults occur ad caused voltage sags Bogotá were detfed. 7. ACKNOWLEDGMENT The authors would lke to thak CODENSA, COLCIENCIAS ad the Uversdad Nacoal de Colomba for provdg facal support for the developmet of ths research project. 8. REFERENCES Aderso P. Aalyss of faulted power systems. IEEE Press Seres o Power Egeerg. Paul M. Aderso seres Edtor 973. Bswal, B., Dash, P.K., Pagrah, B.K, Power Qualty Dsturbace Classfcato Usg Fuzzy CMeas Algorthm ad Adaptve Partcle Swarm Optmzato, Idustral Electrocs, IEEE Trasactos o vol.56, o., pp.0, Ja Bolle, M., Uderstadg Power Qualty Problems Voltage sags ad terruptos. IEEE Press Cajamarca., Torres., Pavas., Urruta., Gallego. ad Delgadllo., Impact Assessmet of Power Qualty Lmts Colomba: a Regulatory Approach. IEEE Lat Amérca Trasm. ad Dstrb. Coferece. Caracas (Veezuela) Camargo, M.; Jmeez, D. ad Gallego, L.;, Usg of Data Mg ad Soft Computg Techques for Modelg Bddg Prces Power Markets, Itellget System Applcatos to Power Systems, 009. ISAP '09. 5th Iteratoal Coferece o, vol., o., pp.6, 8 Nov. 009 CREG, Comsó Reguladora de Eergía y Gas. Resolucó CREG 04 abrl 6 de 005. Modfcacó de las ormas de caldad de la poteca eléctrca aplcables a los servcos de Dstrbucó de Eergía Eléctrca Daves D ad Bould D, A cluster separato measure. IEEE Trasacto o patter aalyss ad mache tellgece vol Goswam, A.K., Gupta, C.P., Sgh, G.K. Area of vulerablty for predcto of voltage sags a aalytcal method Ida dstrbuto systems. Dept. of Electr. Eg., Ida Ist. of Techol., Roorkee. Ida Coferece, 008. INDICON 008. Aual IEEE. IEC, Iteratoal Electrotechcal Commsso, IEC. PIEC Electromagetc compatblty (EMC) Part 430: Testg ad measuremet techques, Power qualty measuremet methods Melédez, J. Berjaga, X. Herraz, S. Sáchez, S. Castro, M. Classfcato of voltage sags based o knn the prcpal compoet space. Isttut d Iformatca Aplcacos ext., Uverstat de Groa (Spa) 007. Mokhls, H., Khald, A.R., L, H.Y., Voltage sags patter recogto techque for fault secto detfcato dstrbuto etworks, PowerTech, 009 IEEE Bucharest, vol., o., pp.6, Jue 8 009July 009. MoraFlorez, J. CormaeAgarta, G. OrdoezPlata, Kmeas algorthm ad mxture dstrbutos for locatg faults power systems, Electrc Power Systems Research, Volume 79, Issue 5, May 009, Pages 747. PAASUN., Iovacó tecológca e detfcacó y medcó de clusters de caldad de poteca para Bogota. Proyecto de Ivestgacó facado por COLCIENCIAS, CODENSA S.A. E.S.P. y la Uversdad Nacoal de Colomba Peña, D., Aálss de datos multvarates. Mc Graw Hll, Uversdad Carlos III de Madrd, 00. Quepo, N. Ptos, S. Fudametos de data mg y sus aplcacoes. Clasfcacó o supervsada, 00. Publcacó e dapostvas ppt lk: Ramos A. Aálss cojuto. Curso de doctorado, tutoral, 009, Materal e Dapostvas. Lk: _ágel\_ra\ewlemos\_domíguez/ Romero, M., Pavas, A., Cajamarca, G., Gallego, L., A ew methodology for the comparatve aalyss of sags amog substatos a dstrbuto etwork Colomba, Harmocs ad Qualty of Power (ICHQP), 4th Iteratoal Coferece o, vol., o., pp.8, 69 Sept. 00 Romero, Mguel F. Dseño de ua metodología para el aálss de sags de tesó e redes de dstrbucó. Uversdad Nacoal de Colomba. Maestría e geería eléctrca. Tess de grado REVISTA INGENIERÍA E INVESTIGACIÓN Vol. 3 Suplemeto No. (SICEL 0) OCTUBRE DE 0 (338)