Fuzzy Set Approach To Asymmetrical Load Balancing In Distribution Networks

Transcription

1 Fuzzy Set Approach To Asymmetrcal Load Balancng n Dstrbuton Networks Goran Majstrovc Energy nsttute Hrvoje Por Zagreb, Croata goran.majstrovc@ehp.hr Slavko Krajcar Faculty of electrcal engneerng and computng Zagreb, Croata slavko.krajcar@fer.hr Abstract - Feeder load balancng and transformer load balancng are used to decrease an overload rsk because of the load changng that are happenng all the tme n the system. So, t s possble to decrease the fault duraton and frequency as an overload consequence. The dspatchers usually don t consder every transformer and feeder wth the same mportance. Transformer overload usually needs an urgent nterventon. The problem becomes more complcated when we deal wth 3-phase nonsymmetrcal system. t s obvous that load balancng s complcated problem that can be solved by multple objectve decson and compromse. Keywords: dstrbuton network, load, fuzzy set, membershp functon, load balancng. NTRODUCTON Besdes techncal reasons (falure duraton and frequency, t s necessary to solve load balancng problem because of economcal reasons. t s possble to save the great amount of energy (money by optmal usage of every element n the system. The man ssue s to balance the system,.e. to form the system n whch every transformer and every feeder s loaded equally. t means that load ndex of each element s equal to the system load ndex. But, because of the dscretonal nature of the load, t s almost mpossble to reach the exact value. So, the compromse s necessary. The man dea s presented here. 2. PROBLEM FORMULATON The suppostons that wll be used n ths problem solvng are: - every sngle feeder load s known, - loads are 3-phase and NONSYMETRCAL, - loads are presented by constant current value. The task s to balance the system.e. to form the system n whch every transformer and every feeder wll be loaded equally by adequate swtchng operaton. t means that load ndex of each element should be equal to the system load ndex. Analogue, each phase has to be loaded equally, whch means that phase balancng has to be done, also. Total load and phase load balancng has to be done wth a mnmum number of swtchng operatons due to swtchgear lfetme expectancy. So, three condtons are defned and every condton wll be presented usng fuzzy sets. Untl ths moment lots of statstcal methods have been used for load balancng analyss. The results have always been close to the optmal soluton. These methods were developed at the end of 8 s by Chnese and Japanese engneers, especally Aok, Hso, Chen, Kanesh [5,6,9]. Castro has developed smple exploraton technque [8]. He has used databases and current dspatchng data. There s also specal procedure that deals wth load balancng between two transformers (those wth the hghest and the lowest load. Transformer loads can be almost equalzed by correct swtcher operaton. The same procedure s used for feeder load balancng (loop openng. Baran and Wu have developed gradaton load balancng method that s based on the exploraton process [7]. Chen and Cho have ntroduced optmal swtchng operatons based on every hour load samples []. Crtcal swtch closng and reclosng s defned by the best sample exploraton. These methods deal wth separate consderaton of transformers and feeders. Ln and Chn [2] also gve fuzzy approach to preventve and correctve swtchng n order to get total load and phase load balance. Roytelman and Shahdehpour analyzed practcal aspect of dstrbuton automaton [5]. Naga Raj descrbes fuzzy framework of ths ssue [6]. He defnes target load transfer of every element and descrbes t by means of membershp functon. A fuzzy approach to three dfferent condtons at the same tme: total load balancng, phase load balancng, and mnmal swtchgear lfetme abrdgement s presented here. 3. LOAD BALANCNG Basc method steps are: - defnton of the most overloaded element, - defnton of all swtchng operaton that could decrease overload, - fuzzy set theory usage whle choosng the best swtchng to avod overload. System load ndex s value that descrbes load n the system accordng to the followng equaton: OS = T mn( naz( T, SNOF ( T ( naz (T - nomnal transformer load (A (T - actual transformer load (A SNOF (T - sum of nomnal feeder loads connected to transformer T (A

2 Balanced transformer and feeder load are as follows: rav( T = OS naz( T rav( f = OS naz( f (2 Load balancng method starts wth maxmal phase current evaluaton. t s necessary to dvde elements n two groups: donors (elements that decrease load by swtchng and acceptors (elements that ncrease load. Let s suppose that sngle-pole crcut breakers are present n the system. n that case maxmal phase load wll be: dav prm dav el (a (a prm ( el = max( el ( a ( b, el ( c ( el = max( el ( a + ( b, el ( c + ( a, el ( b ( c ( a, el ( b + ( c (3 - maxmal donor load - maxmal acceptor load - actual phase a load of element - actual phase load transferred by swtchng from donor to acceptor Load balance membershp functon descrbes new transformer/feeder load balancng. Every element s descrbed by ts own membershp functon, whch defnes how close new element load s to ts balanced load defned n (2. Analytcally, membershp functon s defned as follows: ( el rav ( el ( el ( ( rav el µ el =.2naz( el rav ( el ( el < rav ( el rav ( el < ( el <.2 naz( el.2naz( el < el Graphcally, membershp functon s defned n Fgure. µ (4 n that way every element s descrbed by ts own membershp functon. Snce, load balancng has to satsfy every sngle element that s ncluded n load transfer ( AND element, AND element 2,., AND element n, logc functon AND s used. n set theory t s represented as mnmum functon. Accordng to that, every possble swtchng operaton (load transfer s descrbed by ts own membershp functon as follows: µ uravn (x = mn( µ el,..,, µ el µ 2 el (5 n x=,,t µ uravn(x - membershp functon that descrbes load transfer by swtchng x; x=,..,t t - number of possble dfferent load transfers by swtchng n - total number of element ncluded n load transfer For fnal soluton t s possble to choose OR swtchng, OR swtchng 2, OR swtchng n. Logc functon OR s used. n set theory t s represented by maxmum. So, fnal soluton s evaluated wth maxmal membershp functon. µ rj = max( µ uravn(, µ uravn(2,.., µ uravn(t (6 Target functon can also be defned as sum of relatve square devaton: = N el rav(el 2 S ( (7 = rav(el Balancng process can be safe and mproved f t s: combned heurstc and fuzzy rules, frst observed remote controlled crcut breaker, transferred load from potental acceptor to neghborng element to get more acceptor s capacty avalable, f at frst acceptor s not able to accept donor s load, checked protecton system after load transfer, accepted part of a soluton, f whole swtchng soluton s not acceptable. n some cases t s not possble to solve the problem because: t results wth acceptor s overload, avalable swtchng doesn t mprove load balancng. Ιrav.2Ι Naz Ι(Α Fgure Load balance membershp functon Decreasng of acceptor load, as mentoned above, can solve the frst problem. Proposed method analyses only one, the most overloaded element. So, the second problem can be solved by transfer load analyss of all overloaded elements.

3 4. PHASE LOAD BALANCNG f t s possble to manpulate every phase load separately (.e. wth sngle-pole crcut breakers, t s possble to balance phase loads, too. n that case every element that partcpate n phase load balancng s descrbed as follows: m( el = max( el( a el( b, el( a el( c, el( b el( c (8 m (el j -maxmal phase load dfference (A; j=,2,..,n; n -total number of elements, el j -element j phase load (A Usng those values every sngle element can be descrbed wth phase balance membershp functon. m m B B µ sm = (9 m B µ sm - phase balance membershp functon, m - maxmal phase load dfference (A, - base current (A B The most convenent swtchng for phase load balancng s defned as follows: µ sm = max( µ sm(s, µ sm(s2,.., µ sm(st ( 5. NFULENCE OF SWTCHNG OPERATONS The lfetme of every swtchng devce (swtchgear, crcut breaker, swtcher (SD s more or less defned wth number of swtchng operaton. For every load balancng swtchng operatons are necessary, consequently load balancng has ts nfluence on SD lfetme expectancy. To decrease that nfluence t s mportant to defne whch SDs are the most convenent for that purpose. n ths paper fuzzy set wll be used to descrbe lfetme of SD. Maxmum number of swtchng operatons n lfetme s known value (from producer for every type of devce. Dspatchers should know approxmate recent workng condton of devces n ther area. Accordng to that, membershp value based on subjectve evaluaton of lfe condton s gven to every sngle SD. The membershp curve of devce s shown n Fgure 3. µ p( S x µ sm TSO(S x ZSO(S x NSO Fg. 3. Membershp functon of swtchng devce B m(a Fg. 2. Phase balance membershp functon Base current ( B defnes declvty or severty of phase balance calculaton. Usually, nomnal thermal current or ground relay current are set as base current. The phase balance membershp functon s shown n Fgure 2. The same theory used n load balancng procedure s used to choose the most convenent swtchng for phase balancng. The best soluton wll be defned by combnaton of mnmum and maxmum operators. Every swtchng operaton s defned wth the followng membershp functon: µ sm ( = mn( µ sm (el, µ sm (el2,., µ sm (eln ( NSO - number of swtchng operaton, ZSO - number of swtchng operaton n lfetme expectancy, TSO - number of swtchng operaton n lfetme (tll ths moment. Membershp functon for number of swtchng durng analyzed load transfer s defned as follows: MNSO NSO NSO 2 µ P = NSO 2 (2 + 2 NSO MNSO 2 MNSO - maxmal allowable number of swtchng operaton n actual load transfer, - number of swtchng operaton n actual load transfer

4 Or graphcally: µ P Fnally, all three condtons (total load balance, phase load balance and number of swtchng operaton can be observed at the same tme usng membershp functons. That s very mportant fuzzy set advantage comparng to the other approaches. Usng fuzzy sets t s possble to represent dfferent condtons wth the same type of functon. Fnal soluton wll be gven as mnmum of all (three membershp values. µ uk = mn( µ rj, µ sm, µ skl (4 2 MNSO NSO µpfg. 4. Membershp functon for number of swtchng operaton Every load transfer s realzed wth at least two swtchng operatons (open/close. Two operatons are the most acceptable concernng SD lfetme, so the membershp functon s equal to. More swtchng operatons, lower membershp functon. f total number of swtchng operatons s hgher than allowable (MNSO, membershp functon s equal to. Procedure descrbed n chapter 3. and 5. s used to get the most convenent swtchng concernng SD lfetme expectancy and allowable number of swtchng operatons. µ skl = mn( µ p, µ p( (3 µ skl - total swtchng membershp value, µ p (S x - membershp value of SD ; x=,2,..,p p - number of SD µ p - membershp value of number of swtchng for concrete load transfer Every aspect can be more or less emphaszed by the defnton of ts µ functon. That s the other possblty of fuzzy set theory that can be very useful n solvng nonsymmetrcal load balancng problem. Ths approach wll be presented on a smple example gven n the followng chapter. 6. METHOD APPLCATON n ths example nonsymmetrcal load balancng s analyzed, takng nto account number of swtchng operaton. The dstrbuton network of kv s shown n Fgure 5. Load data at the begnnng and n the followng steps are presented n Fgure 6. The procedure used here s presented n chapters above. All three condtons are taken nto account. Load ndex s rato between actual and nomnal load. Phase load balancng s also calculated, as well as nfluence of number of swtchng. Table presents total load ndex. Every step s confrmed by sum of squared devaton value. Obvously, ths procedure gves better load allocaton concernng load balance and t avods overload. S S2 S35 S36 S33 S32 S3 T f f 2 S S2 S2 S3 S4 S23 S34 S44 S42 S3 S4 f 3 f 4 T2 S22 S24 S43 S4 Fg. 5. Example network scheme

5 .2 Membershp value Transformer Transformer 2 Feeder Feeder 2 Feeder3 Feeder STEP Fg. 6. Load changes durng load balancng process 7. CONCLUSON Feeder load balancng and transformer load balancng are used to decrease the overload rsk due to load changes. Accordng to that fault frequency and duraton are also decreased as an overload consequence. Ths paper deals wth dstrbuton load balancng method based on fuzzy set theory. Thereby transformer load balancng, feeder load balancng and phase balancng are analyzed smultaneously. Swtchgear lfetme s defned by number of swtchng operatons. Load balancng should be performed by the mnmum number of swtchng operatons. Presented fuzzy set approach s presented on a general example. Fgure 6. shows membershp values tendng to equalze. t proves that the balancng process s successful. REFERENCES [] L. A. Zadeh, Fuzzy Sets (nformaton and Control, 965. [2] D. Ðonlagæ, Osnove projektranja nezrazth (fuzzy regulacjskh sustava, KoREMA, Zagreb, 994. [3] E. H. Mamdan, Applcaton of Fuzzy Algorthms for Control of Smple Dynamc Plant, EE 2, 975, pp [4] H. J. Zmmerman, Fuzzy Set Theory and ts Applcatons (Kluwer Njhoff Applcaton, 985. [5] K. Aok, H. Kuwabara, T. Satoh & M. Kanesh, An Effcent Algorthm for Load Balancng of Transformers and Feeders by Swtch Operaton n Large Scale Dstrbuton Systems, EEE Transmsson on Power Delvery, Vol. PWRD-3, Oct. 988, pp [6] K. Aok, K. Nara, etc., A new algorthm for servce restoraton n dstrbuton systems, EEE Trans. on Power Delvery, Vol. 4, No. 3, July 989, pp [7] M. E. Baran & F.F. Wu, Network Reconfguraton n Dstrbuton Systems for Loss Reducton and Load Balancng, EEE Trans. on Power Delvery, Vol PWRD- 3, Aprl 989, pp [8] C.H.Castro, J.B.Bunch, & T.M.Topka, Generalzed Algorthms for Dstrbuton Feeder Deployment and Sectonalton, EEE Trans. on Power Aparatus and Systems, Vol. PAS-99, Mar/Aprl 98,pp [9] Y.Y.Hsu, Y.Jwo -Hwo, S.S.Lu, Y.W.Chen, H.C.Feng & Y.M.Lee, Transformer and Feeder Load Balancng Usng a Heurstc Search Approach, EEE Trans. on Power Systems, Vol. PWRS-8, Feb. 993, pp []C.S. Chen & M.Y.Cho, Determnaton of Crtcal Swtches n Dstrbuton Sytems, EEE Trans on Power Systems, Vol. PWRD-8, Feb. 993, pp []G.Chang, J.Zrda & J.D.Brdwell, Knowledge Based Dstrbuton System Analyss and Reconfguraton, EEE Trans. on Power Delvery, Vol. PWRS-5, Aug. 99, pp [2]W.M.Ln & H.C.Chn, Preventve and Correctve Swtchng for Feeder Contgences n Dstrbuton Systems, EEE Transacton on Power Delvery, Vol. 2. No. 4. Oct 997. [3]S.Steven, & L.Tanmoto, The Elements of Artfcal ntellgence (Computer Scence Press, 987. [4]C.S.Chang, & M.Y.Chung, An Expert System for On- Lne Securty-Economc Load Allocaton on Dstrbuton Systems, EEE Trans. on Power Delvery, Vol. PWRD- 5, Jan. 99, pp [5]M.Roytelman, & S.Shahdehpour, Practcal Aspect of Dstrbuton Automaton n Normal and Emergency Condtons, EEE Transacton on Power Delvery, Vol.8, No. 4, Oct 993. [6]B.Naga Raj, &K.S.Prakasa Rao, A New Fuzzy Reasonng Approach for Load Balancng n Dstrbuton System, EEE Transacton on Power Systems, Vol., No. 3, Aug 995, pp