OPTIMIZATION OF VOLTAGE PROFILE IN DISTRIBUTION FEEDERS BY US- ING CAPACITORS BANKS

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1 OPTMATON OF OLTAE PROFLE N DTRUTON FEEDER Y U- N CAPACTOR AN MANOEL FRMNO DE MEDERO JR., ARRHENU NÍCU DA C. OLERA Laboratory o Computer Eng. and Automaton, Dept. o Computer Eng., Unv. Fed. do Ro. do Norte P CODE Natal, RN, RAL E-mals: rmno@leca.urn.br, arrhenus@leca.urn.br JOÉ ADRANO DA COTA Companha Energétca do Ro. do Norte - COERN P CODE Natal, RN, RAL E-mal: jadc@bol.com.br ATRACT Ths wor presents a methodology or the optmal allocaton o capactors n eeders o electrc power dstrbuton networs. The optmaton algorthm s based on load low calculaton accordng to the power summaton method, ncludng shunt admttances n the equaton or voltage correctons. The proposed methodology s appled to a real system. The results obtaned by applcaton o the method encourage ts use as a practcal tool or plannng tass o dstrbuton networs. EYWORD Optmaton, capactors bans, voltage prole, gradent method.. POWER UMMATON METHOD - NCLUON OF HUNT ADMTTANCE N DERATON The power summaton algorthm starts by reducng a system o several nodes to a model wth only two nodes. The voltage at the rst node s nown and the powers real and reactve are represented at the second node, beng equal to the sum o all the powers o the ollowng nodes. R +j R +j R 3 +j 3 3. NTRODUCTON n any electrc power dstrbuton system, there are many actors whch can reduce the qualty o power supply. The losses, or nstance, can eercse an mportant nluence, because they cause undesrable declnes n the voltage levels o eeders and secondary crcuts. Loong orward to mnme that nluence, reactve power s njected through capactor bans n the networ, wth the objectve o regulatng the voltage prole. n order to determne the capacty and locaton o the bans, an algorthm was bult, based on the method o steepest descent, whose senstvty parameters are derved rom the equatons o power summaton load low []. Furthermore, the load low equatons were moded, n the sense o consderng eactly mpedances to the ground. Another relevant contrbuton o the present wor conssts o dervng an epresson or calculatng the step o the optmaton. Ths s carred out by means o an "eact lne search over the gradent drecton. ased on the proposed algorthms, a computatonal program was developed, n order to smulate practcal stuatons. P +j P +j P 3 +j 3 Fg.. Eample o complete system y applyng the method o power summaton, t s possble to obtan a smpled system whose setch s presented n Fg.. R +j P Fg.. Reduced system, equvalent to the system o Fg. The relatonshp between the voltage at node and the voltage at node s s: 3 R P R. P ntendng to obtan more precse results by the load low calculatons, consumed powers are separated nto two types: loads o constant power, that request power ndependently o the voltage, and loads o constant mpedance, 3 J

2 whose power consumpton vares wth the square o the voltage. Those loads can be modeled through shunt admttances, accordng to setch o Fg. 3. Fg. 3. Complete modelng o loads Applyng rchho s voltage law to the crcut o Fg. 3 leads to: R j The comple power at the node s gven by equaton 3: P j 3 The current lowng through the mpedance to the ground s gven by: j ubsttuton o n 3 leads to: P j j 5 ubsttuton o 5 n and urther smplcatons results n: Y R R P P R +j voltage to be calculated; s voltage at the prevous node wth respect to the node, already nown; P actve equvalent power sum o powers at node ; reactve equvalent power sum o powers at node ; R resstance among the nodes; reactance among the nodes; conductance n the node lned to the ground; shunt susceptance at node. P +j +j 6 Ater voltage s calculated, t s possble to nd the losses at the stretch between the two nodes. The epressons or these calculatons are: P P R 7 losses * P 8 losses * Repeatng the use o equatons 6, 7 and 8 permts to mplement a load low calculaton to supply normaton about the voltage prole o the system. tartng rom that, the method o steepest descent gradent s used to acheve optmaton o the prole. 3. OPTMATON The uncton to be optmed s presented below: OJ n 9 med Where med s an medum value between the slac voltage and the lowest lmt. Consderng the nature o the equaton that relates the voltages n the eeder, that are dened or lne stretches, t s complcated to ntroduce methods based on the nowledge o Hessan matr. Reerence [] presents the dervaton o such a method, although adoptng the total losses o the networ as objectve uncton and assumng some smplcatons by the determnaton o Hessan. For the objectve uncton proposed here, t seems the most vable, the use o a smpler computaton method, as the steepest descent gradent, whose equatons are derved below. The search equaton o gradent method s: c - control varable; - estmate nde; c - step o the method; - gradent o the uncton n the teraton. The calculaton o gradent vector s carred out by dervng the objectve uncton wth respect to all control varables. Let =c, c,..., cn represent the susceptances o the capactors to be connected at the networ. That way, the gradent vector wll be:

3 Cn C C. The choce o the step c s undamental or a good method convergence; t should not be ether so large to cause method dvergence, nor so small so that the process becomes very slow. n the present wor, the calculaton o the step was accomplshed startng rom a quadratc appromaton o objectve uncton. Wthout usng second order normaton Hessan, ths methodology depends on a test step guessng, n order to begn the process. tartng rom the test step, an optmed step s calculated, beng ths last one, updated at each teraton. The equatons or mplantaton o that method are: ' g c c a a g c ' Flowng 9 wth respect to a capactors ban allocated n a generc poston j, t s obtaned the ollowng epresson: n av OJ 3 Accordng to eq. 3, t can be observed that the voltage n any node o the system wll depend o a capactors ban connected at any node o the networ. The dervaton o gradent s accomplshed startng rom dervatve wth respect to a capactors ban allocated at a specc node. Dependng on where that ban s allocated, the equaton o the dervatve assumes derent characterstcs, as presented below. A. Calculaton o the voltage dervatve at node wth respect to a capactors ban allocated to the same node : n order to obtan the correspondng epresson, t s necessary to solate the varable as a uncton o the other parameters n equaton 6. Ths results n: R R P P Now, eq. can be used to determne the nluence o a capactors ban located at node over the voltage at the same node. Dervng ths equaton leads to: Rg P R d d Rg P R d d u 5 R P u R P P R susceptances correspondng to allocaton o capactors ban at node ; voltage at node ; voltage at node.. Calculaton o the voltage dervatve at node wth respect to a capactors ban at a node j, located beore node accordng to power low drectons.

4 Dervng eq. 6 wth respect to the susceptance o a capactors ban allocated to a node, localed beore node, results the epresson presented as: d d Y R u d d P Y R P Y R 6 C. Calculaton o the voltage dervatve at node wth respect to a capactors ban allocated to a node j, placed ater the node. The epresson 6 just relates voltages n adjacent nodes. To solve the problem o calculatng ths dervatve t would be necessary to nd an epresson that relates the nodes and j. Ths s carred out usng an algorthm based on the power summaton method, accordng to the ollowng dervaton. Consder that: UM um CTE CTE j um j CTE CTE um LOE 7 UM summaton o reactve power at node ; CTE - reactve constant power at node ; CTE - reactve power o constant mpedance at node ; um LOE - summaton o all losses among the nodes and j; um CTE j - summaton o all reactve constant powers among the nodes and j; um CTE j - summaton o all reactve powers o constant mpedance among the nodes and j....ecause 7 depends on voltages, t s necessary to derve the resultng power at node wth respect to a capactors ban connected to node j. Ths dervaton s presented below. OMA j 8 OMA OMA j OMA 9 Fnally, one can use 6 to calculate the voltage dervatve at node wth respect to the equvalent reactve power at node. Ths results n: OMA u R. REULT - P 8 R The developed method presented n ths paper was d n several real systems by usng a computatonal program, developed or that purpose. n the ollowng, the results obtaned or a eeder o 3,8, wth 5 nodes and an ntal loadng o 353 A are summared. ntally, a load low calculaton was carred out n order to determne the voltage prole o the system, wthout consderng allocaton o reactve power capactors n the networ base-case. Later on, the algorthm descrbed above, based on gradent method, was used to optme the voltage prole. Consderng slac node voltage equal to.5 p.u., lowest voltage lmt equal to 5 p.u. and a medum voltage o med = 65 p.u., a senstve mprovement n the voltage prole was observed, assurng that no voltage lmt was volated. The value o med was adjusted by a process o tral-and-error, startng rom the lowest lmt. That way, only three trals were enough to reach the nal value. n order to mplement the lne search procedure by optmaton, a quadratc appromaton o the objectve uncton was dealed, allowng calculaton o an optmal step to each teraton. A comparson between both voltage proles, beore and ater optmaton, s presented n g.. ubsttutng 8 nto 9, t comes to, ater epanson and smplcaton:

5 oltage Level p.u. oltage Level p.u..5 5 ystem wthout bans ystem wth bans Node Number Fg.. Comparson o the proles o voltage o the system Table llustrates the behavor o the optmal step, as well as the value o objectve uncton, wth the teraton, consderng as ntal test step equal to. TALE teraton Functon objectve tep The proles shown n the g. were obtaned usng the capactors bans as contnuous varable theoretcal values. Loong or a practcal mplemetaton, t s necessary to adapt these results to commercal values o bans. uch values are: 5 Ar, 3 Ar and 6 Ar. To accomplsh that procedure, the same methodology as presented n []-[] s adopted. The crteron used to allocate a ban n a specc node s to observe n whch node the gradent presents larger value at the begnnng o the process. Followng, table presents the nal ratngs o allocated bans. TALE Nodes contaned Poston o the ban alue o the ban Ar,, 3, 6 6 5, 5, 6, 7, 8, 9,, 7 6,, 5 3,, 5, 6, 7, 8 6 7, 8, 9, 9, 3, 3, 3, , 3, 35, 36,5,,, 37, 38, 39, 5 6,,, 3,, 5 3,, 5, 6, 7, 8, 9, 5 6 Consderng ths new conguraton o bans, the nal voltage prole s shown n g Node Number 5. CONCLUON Fg. 5. New voltage prole o the system n ths wor a more complete verson was presented or the load low calculaton, that consders shunt mpedances to the ground. Ths new load low model served as support or the applcaton o the steepest descent method gradent, seeng to optme the voltage prole along a eeder. That way, equatons or gradent calculaton were derved. Later on smulatons were carred out to very the eectveness o the method. The obtaned results showed a great mprovement o the voltage prole along the eeder, mantanng the voltage at all the nodes o the networ wthn the assgned band.. As uture perspectves or ths wor, can be mentoned: the need o the dervaton o the Hessana matr n order to allow mplementaton o second order methods. However, results and computatonal tmes obtaned by gradent method are compatble wth practcal tass o networ planng. REFERENCE [] Manoel Frmno de Mederos Jr., Ma Chanca P. Flho, Optmal Power Flow n Dstrbuton Networs by Newton's Optmaton Method, CA '98 - nternatonal ymposum on Crcuts and ystems, EEE Monterey - CA,UA. [] Ma Chanca P. Flho, Otmação do luo de carga pelo método da soma de potêncas, Tese de Mestrado, Unversdade Federal do Ro rande do Norte, Natal, 997. [3] Alson de oua arbosa, Fluo de potênca em sstemas de dstrbução: Aplcações Prátcas, Tese de Mestrado, Unversdade Federal da Paraíba, Campna rande,995. [] Washngton, E. de Macedo, Trndade, Walmeram Jr., Fluo de Potênca em stemas de Dstrbução: olução Alternatva. [5] M. Frmno de Mederos Jr.; Ma C. Pmentel Flho, Mamação do Lucro com a Dstrbução de Energa elétrca através da Alocação de Capactores em almentadores de 3,8. CA, Floranópols C, etembro de.