COMBINED MODELLING OF LONG, SHORT INTERRUPTIONS AND VOLTAGE DIPS: A MARKOVIAN SOLUTION

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1 C I R E D 8 th Internatonal Conference on Electrcty Dstrbuton Turn, 6-9 June 25 COMBINED MODELLING OF LONG, SHORT INTERRUPTIONS AND VOLTAGE DIPS: A MARKOVIAN SOLUTION Massmlano GIORGIO, Roberto LANGELLA, Teresa MANCO, Alfredo TESTA Second Unversty of Naples - Italy massmlano.gorgo@unna2.t, roberto.langella@eee.org, teresa.manco@unna2.t, alfredo.testa@eee.org SUMMARY A model of an electrcal system node s presented. It allows predctng the effects of long and short nterruptons as well as of voltage dps on fnal users startng from statstcal nput data. The model s developed usng an extenson of the homogeneous mult-state Marov process to tae nto account the non exponental nature of the falure-repar process. Case studes demonstrate the usefulness of the model for practcal applcatons. I. INTRODUCTION In the last years, wth the lberalzaton of energy marets, regulaton of electrcty supply qualty has taen on growng nterest. Utltes have to assure contnuty of supply and voltage qualty, n order to meet Authortes rules and customer satsfacton [-2]. The detrmental effects of long nterruptons affect all customers. Moreover, short nterruptons and voltage dps may cause functonal problems to many nds of customers because of the senstvty of electrcal and electronc equpments, such as process controllers, adjustable speed drves, programmable logc controllers, personal computers, etc. The quantfcaton of the damages caused by these dsturbances s very mportant because t allows evaluatng the opportunty of adoptng local actons to reduce totally or partally the users senstvty [3-5]. Unfortunately, these damages are dffcult to estmate because of: ) the random nature of the abovementoned phenomena and ) the fact that dsturbance effects depend on the fnal user [6-7]. The use of Power Qualty montors and statstc ndcators by tself does not solve the problem. In ths paper, a model of an electrcal system node s developed extendng the homogeneous mult-state Marov process [8-9]. Ths extenson allows consderng the non exponental nature of the recovery tme consequent to long and short nterruptons as well as to voltage dps. The resultng model, even f apparently complex, allows tang nto account the user senstvty smply drawng border lnes between the states whose effects are relevant and the others. In the followng sectons, after a bref descrpton of the standard dsturbance characterzaton, the proposed model s presented. Fnally, wth reference to representatve casestudes, the parameter settng s dscussed n order to demonstrate the flexblty and the powerfulness of the methodology. II. DISTURBANCE CHARACTERIZATION An electrcal node behavour s usually characterzed separately n terms of Long Interruptons (LI), Short 2 Interruptons (SI) and Voltage Dps (VD). Ths characterzaton s made accordng to the ndexes reported n the Standards [2]. Typcal data structures are reported n Table, 2 and 3 wth reference to LI, SI and VD, respectvely. The sources of the table data are the Italan Authorty for Energy (22) and the results of a measurement campagn on an Italan bus bar performed over 346 days. The nterrupton ndexes used are the well nown: - CAIDI (Customer Average Interrupton Duraton Index): average duraton of LI per customer per year. - SAIFI (System Average Interrupton Frequency Index): average number of LI per customer per year. - MAIFI (Momentary Average Interrupton Frequency Index): average number of short (momentary) nterruptons per customer per year. Voltage dps are characterzed each by a par of data, duraton and ether retaned voltage or depth; for ths reason, ther average number per year s gven n a duraton-depth table. TABLE - LONG INTERRUPTIONS CAIDI Class SAIFI mn L 2.93 TABLE 2 - SHORT INTERRUPTIONS DURATION Class MAIFI Freq. <.5 s S s S s S s S s S TOT S 6.73 % TABLE 3 - VOLTAGE DIPS DURATION DEPTH VD 3% VD>3% Class Num. Freq. Class Num. Freq. - ms DL DH ms DL DH s DL s DL s DL s DL TOT DL 27 % DH % Interrupton: condton n whch the voltage on the delvery pont of electrcal energy for end-user s less than % of declared voltage. The nterruptons are classfed nto long nterruptons (duraton > 3 mn) and short (and momentary) nterruptons (duraton 3 mn). 2 Voltage dps: varaton of nomnal voltage > % wth duraton ncluded nto the nterval ms-8s. CIRED25 Sesson No 5

2 C I R E D 8 th Internatonal Conference on Electrcty Dstrbuton Turn, 6-9 June 25 It s worthwhle to note that each row of the tables ndvduates the dsturbances n terms of type (Class symbol) and level (Class number) that wll be used as one of the possble states of the system (L(.), S(.), DL(.),.DH(.)). In Table 2 and 3 also the relatve level frequency s gven. In other words, the electrcal node s ether n a normal state (Node OK) or n one of the fourteen states that together wth the normal state consttute a set of ffteen exhaustve and mutually exclusve states. III. MODELLING In Fg. a frst general Marovan model that can be used to nclude, at the same tme, long and short nterruptons and voltage dps s represented. Accordng to ths model, no mportance to the duraton of SI and VD and to the depth of VD s gven. The tme spent n each of the states s exponentally dstrbuted, so the falure-repar process can be vewed as an homogeneous Marov process (see Appendx). λ and µ represent the transton rates between the state Node OK and the state n whch the -th dsturbance s present. Intal values for these transton rates (see the next secton for detal about parameter settng) can be calculated from the data contaned n Tables -3; for example, wth reference to LI, λ 3 =SAIFI [transton per year] and µ 3 =/CAIDI [transton per mn] are obtaned. The other four transton rates can be evaluated n a smlar way, wth some lttle mathematcal complcatons. Ths smplfed model does not provde the opportunty of dstngushng among the levels of the dfferent type of dsturbances. So, the dfferent customer senstvtes can not be properly modelled. NODE OK 2 3 VOLTAGE DIPS SHORT LONG INTERRUPTIONS INTERRUPTIONS III.A. Structure In order to solve the aforementoned problem, a ffteen (4+) states model accordng to the classes defned n the rows of Tables, 2 and 3 has been frstly used. But, n spte of the ncreased number of states, ths model s only able to consder exponental dstrbuted dsturbance duraton (tme to recovery). Unfortunately, the dsturbance duraton, for each state, can not realstcally be consdered an exponental random varable. To solve ths problem, the method of stages has been used (see Appendx). It has allowed authors developng dfferent complex, but easy to handle, models. For the sae of brevty, n the followng part of the paper, reference s made only to the model consstng of a pure parallel of seres stages, represented n Fg. 2. Accordng to ths model, the OK state s connected wth fourteen states/columns (see rows of Tables -3). The parallel columns -6 are used to model VD of depth 3%; the columns 7-8 are used to model VD of depth > 3%; the columns 9-3 are used to model SI and, fnally, the last column s used to model LI. Each column consstng of a seres of stages. For example, dl - s the number of stages of the frst state, DL (frst column) and dl 6 -dl 5 s the number of stages of the 6-th state, DL6. For the sae of clarty, only the frst and the last stage of each state are represented. In Fg. 2, t s possble to observe two dashed border lnes: A s representatve of users senstve only to short and long nterruptons and B of users senstve to short nterruptons wth duraton greater than.5 s and to long nterruptons. Each border lne separates, for the users they represent, the states that do not cause damage to the users (.e. those on the left of the border lne) from the states that cause damage to the users (.e. those on the rght of the border lne). Gven the user, the probablty to be up ( down ) can be calculated as the sum of the state probabltes of the states that are on the left (rght) of the border lne. Fg. - Smplfed model of the electrcal system node. DL DL2 DL3 DL4 DL5 DL6 DH DH2 S S2 S3 S4 S5 L A B > DL DL2 DL3 DL4 DL5 DL6 DH DH2 S S2 S3 S4 S5 L Fg. 2 - Proposed model of the electrcal system node consttuted by a pure parallel of seres stages. CIRED25 Sesson No 5

3 C I R E D 8 th Internatonal Conference on Electrcty Dstrbuton Turn, 6-9 June 25 III.B. Parameter Settng The expermental data reported n Secton II have been used to set all the transton rates, λ (.) and µ (.) as well as the number of stages for each of the fourteen states (columns) of the model n Fg. 2 (here, µ (.) s an ext transton rate equal for each of the stages appertanng to the gven state (.)). Intal values of the transton rates λ (.) and of the MTTR (Mean Tme To Repar) can be obtaned usng drectly the number of events per year reported n Tables -3. For example, wth reference to SI (Tab.2), t s possble to set λ S5 =MAIFI S5 =.83 [transtons per year]. Smlarly, an ntal value of the ext transton rate, µ S5, can be calculated as: (s5- s4)/mttr S5, beng (s5-s4) and MTTR S5 the number of stages and the mean traversng tme assocated to state S5, respectvely. The MTTR S5 can be set to 2 s, that s the centre of the nterval duraton of the correspondng class. To complete the ntal parameter settng, t s necessary to assgn the number of stages for each of the fourteen states. Gven the ntal values of the transton rates λ (.) and µ (.) the number of stages for the dfferent columns have been heurstcally chosen wth the am of obtanng a tme-to-repar (or recoverng tme) dstrbuton wth characterstcs as close as possble to those of the real dstrbutons reported n Tables, 2 and 3. The ndex used to evaluate ths closeness s the Root Mean Square Error (RMSE): RMSE = mf rf rf where rf (,2,,) are the real relatve frequences reported n Tables, 2 and 3 and mf the correspondng frequences calculated on the bass of the model. Here and n the followng of ths secton, the subscrpt s used to mae reference to the -th state on the rght sde of the border lne. Gven the structure of the model n Fg 2, the pdf of the tmeto-repar s a mxture of the pdfs (5 for border B and 6 for A) of the tme spent to traverse each column on the rght sde of the border lne. Each pdf refers to a seres of stages, so t s a Specal Erlangan dstrbuton of parameters µ and N, beng N the number of stages of the -th state. The mxture can be obtaned by: f = MIX 2 () w f. (2) where s the number of states on the rght sde of the border lne and w s a set of weghts so that Σ w =. The value depends on the nd of fnal user under consderaton and can reach the maxmum value of 4. The weght w s the rato between the transton rate from the State OK to the -th column and the summaton of the transton rates extended to the states under consderaton. The authors experence has demonstrated that (gven the ntal settng of the rates λ (.) and µ (.) ) just changng the number of stages, t s not possble to obtan optmal results. Sensble mprovements of the soluton can be produced ntroducng some lttle adjustments of the transton rates λ and µ prevously calculated. Obvously, these adjustments must be done assumng a constant value for the real global MTTR of all the DOWN states and for the MTTF (Mean Tme To Falure), whch the steady state soluton depends on. These MTTR and MTTF can be calculated usng the data n Tables, 2 and 3. Ths calbraton can be accomplshed adoptng the followng relatons: * * * λ = w λg = w λ (3) and * MTTR * µ = µ (4) MTTR where the asters denotes the transton rates obtaned after the adjustment, and MTTR and the MTTR * can be calculated by the fallowng formulas: * * MTTR = w MTTR ; MTTR = w MTTR (5) beng MTTR the mean traversng tme of the -th state on the rght sde of the border lne. Equaton (3) allows changng the weghts wthout changng the global MTTF of the node (t s suffcent to use the ntal λ g and to respect the condton Σ w* =). Equaton (4) allows restorng the MTTR of the model, changed due to the weghts adjustment, to the ntal real global MTTR value. IV. APPLICATIONS Two case studes correspondng to the two nds of users whose senstvtes are represented by the borderlnes of Fg. 2 are developed. IV.A. Case-Study A Reference s made to users senstve only to short and long nterruptons. Frst of all, usng ntal values of the transton rates, a startng set of number of stages for each class has been found. Then, for several adjusted sets of weghts, the relatve errors between the pdf expermentally obtaned and the pdf reproduced by the model have been calculated for each class of nterruptons. The resultng best combnaton of weghts and transton rates, heurstcally found, are reported n Table 4. The RMSE s equal to.65. TABLE 4 Case Study A: Best combnaton of weghts and transton rates heurstcally found and characterzed by an RMSE=.65 for the startng set of stages. S S2 S3 S4 S5 L N () w µ Error % CIRED25 Sesson No 5

4 C I R E D 8 th Internatonal Conference on Electrcty Dstrbuton Turn, 6-9 June 25 Fgure 3 reports a comparson between the real pdf of the tme-to-repar obtaned from Tables,2 and 3 and the pdf reproduced by the model usng the values ndcated n Table RMSE=.65 RMSE=.34 RMSE=.25 RMSE=.6 pdfs - probablty densty functons [s - ] t - tme Fg. 3 - Case Study A: Comparson of the real pdf of the tmeto-repar obtaned and the pdf reproduced by the model (Table 4). Fgure 4 reports the absolute value of the relatve errors, class by class, for three dfferent combnatons of weghts and transton rates all characterzed by the number of stages of Table 4. The RMSE value correspondng to each combnaton s also reported. 5 ERRORS % 5 IS S IIS S2 S3 IIIS S4 IVS S5 VS L IL ES Fg. 5 - Case Study A: Relatve errors, class by class, for three dfferent combnatons of weghts and transton rates all characterzed by the number of stages of Table 6. IV.B. Case Study B As done n the prevous Sub-secton, some smulatons have been run for users senstve to short nterruptons wth duraton greater than.5 s and to long nterruptons. The resultng best combnaton of weghts and transton rates, heurstcally found, are reported n Table 6. The RMSE s equal to.27. TABLE 6 Case Study B: Best combnaton of weghts and transton rates heurstcally found and characterzed by an RMSE=.27. ERRORS % 5 RMSE=.65 RMSE=.53 RMSE=.73 S2 S3 S4 S5 L N 5 5 w µ Error % Fgure 6 reports a comparson between the real pdf of the tme-to-repar obtaned from Tables,2 and 3 and the pdf reproduced by the model usng the values reported n Table 6. IS S IIS S2 S3 IIIS IVS S4 S5 VS LIL ES Fg. 4 - Case Study A: Relatve errors, class by class, for three dfferent combnatons of weghts and transton rates all characterzed by the number of stages of Table 4. After the frst calbraton, the soluton found (TABLE 4) has been compared to other solutons, obtaned varyng only the number of stages (TABLE 5). A better soluton wth RMSE=.6 has been found. The correspondng absolute values of the relatve errors are drawn n Fg.5. TABLE 5 Case Study A: Dfferent sets (,2,3,4) of values of stage number (and correspondng RMSE) used n combnaton wth the set of weghts of Tab. 4. S S2 S3 S4 S5 L RMSE N () N (2) N (3) N (4) pdfs- probablty densty functons [s - ] t tme Fg. 6 - Case Study B: Comparson of the real pdf of the tme-torepar obtaned and the pdf reproduced by the model (Table 6). CIRED25 Sesson No 5

5 C I R E D 8 th Internatonal Conference on Electrcty Dstrbuton Turn, 6-9 June 25 VI. CONCLUSIONS In ths paper, a model of an electrcal system node has been presented. The model has been developed extendng the homogeneous mult-state Marov process to tae nto account the non exponental nature of the recovery tme process consequent to long and short nterruptons as well as to voltage dps. The resultng model, even f apparently complex, has allowed tang nto account the user senstvty smply drawng border lnes between states whose effects are relevant and the others. By means of the ntroduced models t s possble: - assessng the economcal damages due to nterruptons and voltage dps; - evaluatng the convenence of local actons to mprove relablty and Power Qualty. VII. APPENDIX - Remars on the Marov Approach and ts extenson The Marov approach s based on the fallowng hypothess: the predcton of the future states of the system, based on the present state alone, does not dffer from that formulated on the bass of the whole hstory of the system (Marov property). Whether ths transton probablty does not depend on the age of the system (tme) the Marov process s called homogeneous. In a homogeneous Marov process the tme between successve transactons has an exponental dstrbuton. In the applcatons to sngle two state components n whch both tme to falure and tme-to-repar are exponentally dstrbuted, the falure-repar process can be vewed as a two state homogeneous Marov process. When the random varables tme to falure and/or tme-torepar can not be assumed exponentally dstrbuted, extensons of the homogeneous Marov process have to be adopted. A smple, but powerful soluton conssts n dvdng a state nto sub-states, each beng defned as a stage. If two or more exponentally dstrbuted stages are combned, the tme spent n the resultng state s non-exponentally dstrbuted. In partcular: - f the N stages (see Fg.7) are traversed n a sequental order (seres) and have constant and equal transton rate, µ, the tme spent to pass through the state s a Specal Erlangan random varable of parameters µ and N havng pdf: f = ( t) ( N )! N µ µ µ t e, (6) STATE STAGE STAGEN w λ STAGE VIII. BIBLIOGRAPHY [] Fg. 7 - Stages n seres. STATE µ µ N w N λ Fg. 8 - Stages n parallel. STAGE N UP DOWN UP DOWN [2] Jont Worng Group Cgré C4.7 / CIRED, 8- Oct. 23, Recommendng power qualty ndces and objectves n the context of an open electrcty maret, CIGRE/IEEE PES Internatonal Symposum on Qualty and Securty of Electrc Power Delvery Systems. [3] G. Brauner, C. Hennerbchler, ClRED 8-2 June 2, Voltage Dps and Senstvty of Consumers n Low Voltage Networs, Conference Publcaton No. 482 IEE 2. [4] Hngoran, N.G., June 995, Introducng custom power. Spectrum, IEEE, Volume: 32 Issue: 6, Page(s): [5] Krby, B.; Hrst, E., June 996, Unbundlng Electrcty: Ancllary Servces, Power Engneerng Revew, IEEE, Volume: 6 Issue: 6, Page(s): 5. [6] Bollen, M.H.J.; Qader, M.R.; Allan, R.N.,, 27 Jan. 998, Stochastcal and statstcal assessment of voltage dps, Tools and Technques for Dealng wth Uncertanty (Dgest No. 998/2), IEE Colloquum on Page(s): 5/ - 5/4. [7] M.H.J. Bollen, T. Tatjasanant, G. Yalqnaya, Nov/Dec 997, Assessment of the number of voltage sags experenced by a large ndustral customer, IEEE Transactons on Industry Applcatons. [8] S.M. ROSS, 983, Stochastc Processes, Wley. [9] R. Bllngton, R.N. Allan, 992, Relablty evaluaton of engneerng systems, 2 nd edton, Plenum Press. - f the N (see Fg.8) stages are n parallel (.e.: the state can be traversed by traversng one of ts stages) gven the transton rates µ (,2,,N) of the -th stage and the probablty, w λ, of traversng the state passng across the -th stage, the probablty densty functon of the tme spent n the state s: N t f = w µ e µ (7) CIRED25 Sesson No 5