Estimating Peak Load of Distribution Transformers

Transcription

1 Estmatng eak Load of Dstrbuton Transformers K. Halcka * A. Jurcuk * J. Naarko* Z.A. Stycyńsk** W. Zalewsk* Abstract - In dstrbuton system, bus load estmaton s complcated because system load s usually montored at only a few ponts. As a rule recevng nodes are not equpped wth statonary measurng nstruments so measurements of loads are performed sporadcally. In general, the only nformaton commonly avalable regardng loads, other than major dstrbuton substatons and equpment nstallatons, s bllng cycle customer kwh consumpton. Ths paper presents possbltes of applcaton of the statstcal approach and the fuy set theory to electrcal load estmaton. The frst of them s grounded on the applcaton of dversty factor and converson factor. In the second of them unrelable and naccurate nput data have been modeled by means of fuy numbers. A regresson model, expressng the correlaton between a substaton peak load and a set of customer features (explanatory varables), exstng n the substaton populaton, s determned. 1. Introducton Controllng the utlty load curve shape s the most pror task of load management. The prncpal reason of consderng these actvtes s to acheve a better match between the current and future capacty of supplement, transmsson, dstrbuton system and a customer demand. From the utlty perspectve, the magntude and tmng of load changes are both mportant. Utltes want the greatest load reducton to occur around the tme of the system peak. In other words reducng of end-user demand should be agree wth dstrbuton system peak perods. Demand Sde Management (DSM) ams to control load growth (especally peak load) and alter shape of the load curve. These actons allow to reduce captal expendtures (avoded costs), mprove load factor and system effcency. Controllng and management of electrcal energy demand needs precse modellng and forecastng of peak loads. From the power utlty t s seems the most mportant part of applyng demand sde actvtes. The man dffcultes n the modelng of peak loads at recevng buses n dstrbuton systems result from the random nature of loads, dversfcaton of load shapes on dfferent parts of the system, the defcency of measured data and the fragmentary and uncertan character of nformaton on loads and customers. The mathematcal estmaton of the loads at the system buses seems to be most realstcally possble accordng to ncomplete prmary nformaton. It demands earler determnaton of the stable relatons between bus loads and easer avalable data. The statstcal methods could be used to load estmaton n dstrbuton transformers. These methods requre a great number of load data obtaned from measurement experment. Tradtonal systems used to drect the power dstrbuton system operatons are not adapted for collectng and processng a vast amount of data and do not allow a dspatcher to utle the nformaton effectvely. In ths case the most renowned method for expressng the uncertanty n load models s fuy set theory. For the purpose of smplcty of mathematcal operaton the trapeodal and trangular forms of fuy numbers are usually used. The fuy set theory could be used as an effectve tool of load estmaton. Ths paper presents two approaches to load estmaton - statstcal analyss and fuy regresson method. These approaches are llustrated by numercal examples. * Balystok Techncal Unversty ul. Wejska45A, Balystok, oland. E-mal: jnaarko@cksr.ac.balystok.pl " 1EH Unversty of Stuttgart faffenwaldrng 47, D Stuttgart, Germany. E-mal: sty@ehsun.e-technk.un-stuttgart.de 87

2 2. Statstcal approach 2.1 roblem formulaton The most popular coeffcent factors used by engneers to estmate the load on the dstrbuton power system are the dversty factor (D f ) and KWh-to-eak-kW or converson factor (Q)[l]. The dversty factor s defned to be the sum of ndvdual non-concdent customer peak demands dvded by the concdent peak demand of customer as a group. The KWh-to-eak-KW s defned to be sum of all ndvdual customers peaks dvded by the total energy usage of the group over a gven perod. (1) Load research data provde an mportant nput to load estmaton. Wth the wde avalablty of electronc demand recorders, t s now possble for utltes to automatcally gather hourly test data for a large sample of dverse classes of customers. The load Research Department of Arkansas ower and Lght Company mantans demand recorders on several hundred randomly selected customers throughout Arkansas. The devces store hourly KWHR measurements for each test customer. For each test customer a total of 8760 data ponts are taken annually [1]. Although D f and C f factors have been used for many years, they are stll not statstcally confrmed. Ths work presents the statstcal propertes of the two factors. 2.2 Statstcal propertes of dversty factor The dversty factor was calculated several tmes for dfferent samples. Two such procedures, useful for evaluaton of statstcal ndependence and underlyng trends, are the run test and cumulatve perodogram. The run test s defned as a sequence of dentcal observatons that s followed and preceded by dfferent observatons or no observatons at all. The run test counts the number of runs that are completely above or below the medan [2]. The computatonal results gven n [3] ndcate that the samples taken for calculatng monthly dversty factor for dfferent months are ndependent. The dversty factor estmated from load research data was examned for normal dstrbuton. The average dversty for each number of customers n a group represents a random varable. The calculatons appled to 60 dfferent sets of randomly selected 2, 5, 10,15, 20,25 and 30 customers n group. One of the most convenent non-parametrc test for normalty s the ch-square goodness of ft test. Another wdely used goodness of ft test s Kolmogorov-Smrnov test [2]. The results of the tests confrmed that, on sgnfcance level 0.05, there was no reason to reject the hypothess that dstrbuton of D f factor can be well approxmated by normal dstrbuton [3]. 2.3 Statstcal propertes of KWh-to-eak-kW factor To exam f the average KWh-to-eak-KW factor dsplay any underlyng trend, a test whch analyse the resduals between the average of C f values and the medan was carred out. Fgure 1 shows the relatonshp of the C f factor n dfferent number of customers wth the months, whch also confrm that no effect of the number of customers on the C f factor. (2)

3 Fg. 1. The average monthly KWh-to-eak-KW factor versus number of customers for weekday perod The KWh-to-eak-kW factor was calculated several tmes for dfferent samples. It was assumed that the samples were ndependent of each other. Ths assumpton was tested by the run test above. Computatonal results confrmed that the samples taken for calculatng C f factor were ndependent. The shape of plots of frequency hstograms of KWh-to-eak-KW factors calculated for dfferent groups of customers and also n monthly dmenson clearly ndcated that they had normal dstrbuton (Fg. 2). Ths hypothess was tested usng Ch-square and Kolmogorov-Smrnov tests. The results of the tests confrmed that, on the sgnfcant level 0.05, there was no reason to reject the hypothess that dstrbuton of C f factor may be well approxmated by normal dstrbuton. 2.4 Substaton load estmaton In general, the only nformaton commonly avalable regardng loads at locatons, other than dstrbuton substatons and major equpment nstallaton, s bllng cycle customer kwh consumpton. Ths energy data has been used to formulate estmates of loads on dstrbuton crcuts by applyng dversty and KWh-to-eak-KW converson factors. In order to estmate the demand of class of customer for the peak load day (ether weekday or weekend) of any month t s necessary to the number of customers and the total kwh consumpton of the class durng the month. The peak demand may be obtaned by usng the followng equaton C f ( m, n, k) ( m, n, k) = E (3) D ( N, m, n, k) where: - peak demand, E - kwh consumpton for customer group, C f - KWh-to-eak-KW converson factor, D f - dversty factor, m - month, n - type of day, k - class, N - number of customers. Four substatons were consdered wth the developed algorthm [1]. ercentage errors between actual and estmated peak values for all months are presented n table f

4 Table 1. Results of daly peak load estmaton Month mest estm % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Note: mest, - measured peak load of the peak day of month, estm - estmated peak load of the peak day of month, - estmaton error, % - percentage errors. 3. Fuy regresson approach 3.1 Intaton n the fuy regresson The general regresson model s gven by the followng equaton [4]: Y = ZA + e where: Y - vector of output varables, Z - matrx of ndependent varables, A - vector of parameters, e - vector of unobservable errors. Two cases can be dscrmnated dependng on the type of output varable. The frst when the output varable (daly peak load - d ) s a real number and the second when the output value s an nterval d є < d L, d R >- where: The frst case s descrbed n ths secton. It can be represented n the form: d = Z A (5) d = a 0 + a a k k = l, 2,..., n (6) ( Z ) + The lnear fuy regresson model (6) s represented usng symmetrc trangular fuy parameters aj=[a c,a r ] as follows: (4) d dc dr ( ) = [ a ( ) = a ( ) = a 0c 0c 0r, a 0r + a + a 1c 1r ] + [ a 1 1 1c, a 1r a a ] kc kr [ a k k kc, a kr ] k where: dc, a c - center parameters of fuy numbers (membershp functon µ = 1), dr, a r - spreads of fuy numbers (geometrcally the spread s a half of the base of the trangle). The parameters aj of the vector A of the lnear fuy regresson model are determned by a soluton of a lnear programmng (L) problem whch s to mnme the sum of spreads dpr ( ) of elements of vector d [5].

5 From (7) - (9), the L problem (10) - (12) can be wrtten as follows: The parameters a j = [a jc, a jr ] of vector A are determned as the optmal soluton of the L problem (13) - (15). Snce the L problem always has feasble solutons, the fuy parameters are obtaned from the L problem for any data. 3.2 Applcatons of the fuy regresson analyss to the electrcal load estmaton The loads on dstrbuton transformers are the nstantaneous summatons of the ndvdual demands of many customers. Snce the pattern of electrcal demand of each customer cannot be determned precsely, t s usually necessary to calculate system loadngs on an estmaton bass. The probablstc models are wdely used to estmate system loads. In order to develop the relevant types and parameters of probablty dstrbuton, large numbers of recorded consumpton values are requred. To obtan the above data a specal measurement project has to be consdered. Many relatonshps between output quanttes (peak load, load flow, losses of power and energy) and descrbng values comng from measurements can be represented by a regresson model. Results of nvestgatons made on the bass of expermental desgn show that the energy consumpton s the most correlated factor wth the peak load demand [6]. The use of statstcal methods s not always possble due to occurrence of a large defct of measurements. The fuy set theory s a convenent mathematcal tool that allows us to partally elmnate unrelablty from nput nformaton and to lmt the nfluence of defct of measurements. The daly 15-mnutes peak power consumpton for a gven substaton may be found on the bass of kwh consumpton usng fuy regresson models 3.3. Numercal example To verfy the proposed method of peak load estmaton the measurements of daly energy consumpton A d and daly peak load d at selected fve dstrbuton substatons n Balystok ower Dstrbuton Utlty Co. were made n January. Investgatng objects are substatons wth transformers wth 15/0.4 kv rato of transformaton and power rato from 160 to 400 kva. On the ground of measurements the fuy lnear regresson models (6) were determned. For the general fuy model presented n form: d = [a oc, a Or ] + [a 1c, a 1r ]A d where: d - the daly 15-mnutes peak power consumpton, A d - the daly energy consumpton, the L problem correspondng to the gven data was formulated from (13) - (15). By solvng ths one, the followng model was obtaned: d = [18.90, ] + [0.0529, 0.019]-A d (17) On the bass of model (17) daly peak loads at the tested substaton n February were estmated. The results are shown on Fg. 3 together wth the correspondng measurement data. 91

6 4. Conclusons In ths paper possbltes of applcaton of the statstcal approach and the fuy set theory to electrcal load estmaton was presented. The dversty and KWh-to-eak-KW factors have stable statstcal results. The averages are normally dstrbuted. The means of dversty and converson factors for all calculated months have narrow confdence ntervals. Calculatons and statstcal analyss ndcate that the value of dversty and KWh-to-eak- KW factors depend on months, type of day and class of customer, whle the number of customer n a group does not have any effectve nfluence on evaluaton of the Cj factor. The results of nvestgatons and calculatons allow us to conclude that the dversty and converson factors are a useful tool used n practce by electrcal engneers. To obtan more general results t s preferable to collect a substantal amount of data for more wde varety of classes and large number of customers. In case when realaton of extensve measurement experment s mpossble, the use of the fuy approach allows to perform estmaton process. The proposed fuy regresson analyss allows us to estmate daly 15-mnutes peak power demand at dstrbuton transformers durng normal state condtons, on the bass of kwh consumpton. The method does not handle the nstantaneous loads changes caused for example by swtchng operatons or lne outages. The spread of the fuy models depends on maxmum and mnmum value of a gven data. It does not depend on sample se. In standard regresson models the wdth of confdence ntervals depends on sample se, standard devaton and sgnfcance level. In case of small sample se, standard devaton was become excessve. The applcaton of the fuy approach elmnates ths dffculty. It s seen from the consderatons and relatonshps descrbed above that the fuy set approach to electrcal load estmaton puts a new qualty nto the system analyss n uncertan condtons. The authors see usefulness of applyng of presented methods to problems of load forecastng and load estmaton n power dstrbuton systems. The presented approaches may be a useful tool supportng plannng dstrbuton engneers n plannng and desgn to estmate the load of dstrbuton networks. Acknowledgement Ths paper reports work sponsored by State Commttee for Scentfc Research (KBN) under contract S/IZM/1/99. References [ 1] Broadwater R., Sargent A., Yaral A., Shaalan H., Naarko J.: Estmatng Substaton eaks from Load Research Data. IEEE Transactons on ower Delvery, Vol. 12, No. 1, January [ 2] Bendat J.S., ersol A.G.: Random Data Analyss and Measurements rocedures. New York, John Wly & Sons, [ 3] Naarko J., Tawalbeh N.I.: Statstcal ropertes of Dversty and Converson Factors and ther Use to Load Estmaton. In "Dstrbuted Energy storage for ower Systems" (eds. K. Feser, Z.A. Stycynsk) Verlag Man, Aachen, [ 4] Vaderman S.B.: Statstcs for Engneerng roblem Solvng. WS ublshng Company. Boston, [ 5] Tanaka H., Uejma S., Asa K.:,,Lnear Regresson Analyss wth Fuy Model". IEEE Transactons on Systems, Man and Cybernetcs, Vol. 12, No. 6, December 1982, pp [ 6] Naarko J., Zalewsk W.:,,The Fuy Regresson Approach to eak Load Estmaton n ower Dstrbuton Systems". IEEE Transactons on ower Systems, E-036-WRS (To be publshed). 92