A New Approach to Probabilistic Load Flow

Transcription

1 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 73, DECEMBER 79, 837 A New Appoach o Pobablsc Load Flow T K Basu, R B Msa ad Puob Paoway Absac: Ths pape descbes a ew appoach o modellg of asmsso le uceaes usg complex adom vaables. Fo example, f hee s a sysem havg geeag us, I asmsso les ad m umbe of load dsbuos, all possble cogeces ca epeseed by as may as ( X l X m saes. The mehod poposed hee cosdes all he ( X l X m cogeces a elega mae usg he fs fou momes based o complex adom vaables. Geeao, asmsso le ouage eves ad load vaao modellg s descbed usg he complex adom vaable appoach. The CLF algohm would have o be u ( X l X m mes volvg pohbve amou of calculaos wh added dffculy of aalyzg ad syheszg he soluo. The poposed PLF ca ackle all he dffee cogeces a sgle load flow. Soluo of he sochasc load flow equaos s caed ou he mome doma of a compose powe sysem. The pu vaables such as he oal powe geeaed by all he geeag us ad he sysem demad ae cosdeed complex adom vaables. Soluos o sochasc volage ad powe equaos have bee foud cosdeg all possble cogeces of geeao ad asmsso le ouages, load demad vaao usg he momes of he odal admace max self. The algohm yelds he values of he mpoa chaacescs of he oupu vaables (bus volages, le cues ad geeaed powe ems of he pu vaables (geeag capaces of he dffee geeag us a each bus, load capaces of dffee load buses ad asmsso le avalables. Key wods: pobably, capacy, loss of load pobably, pobably desy fuco, pobably mass fuco, complex adom vaables, momes, cumulas, slack bus, PV bus, P Q bus, load flow soluo, bulk powe asmsso ad sysem plag. A. INTRODUCTION The load flow aalyss s udoubedly he mos useful sudy made fo desgg ad opeag ew powe sysems as also fo plag fuue exesos o mee ceased load demads ad defe opeag pacces. The algohm assesses, he seady sae behavo ad espose of he sysem beg suded fo whch, adequae daa defg he opeag T K Basu s wh Eleccal Egeeg Depame a Ida Isue of Techology, Khaagpu, 7 3 INDIA (elephoe: 386, emal: kb@ee.kgp.ee.. R B Msa s wh Relably Egeeg Depame a Ida Isue of Techology, Khaagpu, 7 3 INDIA (elephoe: 38399, emal: av@ee.kgp.ee.. Puob Paoay was wh Eleccal Egeeg Depame a Ida Isue of Techology, Khaagpu, 7 3 INDIA codos have o be povded. The mos commo echques used he pas fo deemg he geeag capacy o sasfy he sysem demad ad o have suffce capacy o pefom coecve ad peveve maeace geeao facles wee deemsc aue. Deemsc mehods do o cosde he sysem composo (umbe, sze ad foced ouage aes o he load demad chaacescs. The use of fxed deemsc ceo plag capacy eseve, esuls a vaable sk level ove he plag hozo wheeas, pobablsc echques ake o accou he adom aue of boh demad ad supply of powe. Ths esues ha he acual facos ha fluece he sysem elably ae ake o accou. The coveoal load flow (CLF aalyss s deemsc aue ad has seveal dawbacks, some of whch may be saed as, a Powe demads ae modeled by sgle values, usually epeseg meas of cea lms ad gves fomao abou a sgle opeag po. b Uavalably of geeag ad asmsso equpme s egleced. c Calculaos volve soluo of a gve powe sysem ewok ude codos accodg as s opeag ude balaced ad ubalaced codos. Theefoe, umeous load flows ude boh omal ad abomal (ouages of geeag us, asmsso les ad uceaes of load opeag codos wll be eeded. As opposed o he deemsc mehods, he pobablsc echques ake o accou he foced ouages, maeace equemes, vaably of hydo flows ec. Ohe facos such as uceay load foecass ad sascal vaaos hydo u eegy avalables ca also be cosdeed. Also, pobablsc load flow (PLF deemes how he vaably age fo he pus affecs he age of vaaos of he oupu quaes. I pacce, fom he pobably dsbuo of he pus, he PLF povdes fomao o he coespodg dsbuos fo he oupu. The pobably of adequae asmsso capacy each cofguao ca be foud afe pefomg a load flow sudy o each, usg he appopae load model. Thus he CLF algohm would have o be u ( X l X m mes volvg pohbve

2 838 NATIONAL POWER SYSTEMS CONFERENCE, NPSC amou of calculaos wh added dffculy of aalyzg ad syheszg he soluo. These pos ca be vey sasfacoly ackled by applyg he pobablsc load flow algohm. Aohe oewohy po s ha he vey aue of pu vaables (geeag ad asmsso equpmes ad load demads s adom ad o deemsc ad hece s mpeave ha pobablsc echques be appled Load Flow Aalyss o evaluae he effecs of pu uceaes o he seadysae behavo of a powe sysem. A pobablsc load flow algohm esseally asfoms, he pu adom vaables defed ems of pobably desy fucos (pdf o ems of oupu adom vaables smlaly defed. B. PROPOSED METHOD I ay powe sysem, loads ae complex quaes as alog wh acve compoes (MW hee ae eacve compoes (MVAR as well, he aue of whch depeds o he ype of load coeced ad he asmsso sysem feedg. Thus, eve f MW demad emas fxed he MVAR demad of he sysem may chage due o chages powe sysem (of he loads coeced ad he asmsso sysem supplyg powe o he sysem heeby asg o loweg he MV A demad. Fomulao of he poblem usg complex adom vaables mgaes he complexy of cosdeg eal ad magay compoes sepaaely ode o oba he ue pcue. C. The coceo of CRY A CRY ca be expessed he fom Z=X jy whee boh X ad Y ae eal adom vaables. If Z has a pobably mass fuco f (Z, he ca also be defed he same pobably space wh a jo pobably mass fuco, pmf f (x, y. If Z, akes up hee dffee values say, Z l = x l jy l, Z = x jy ad Z 3 = X 3 jy 3 wh pobables of occuece p, p ad p 3 especvely. The p = f (X=x, Y= y fo Z =x jy.i may be oed ha he eal adom vaables X ad Y occu as oe eve.e., X always occus alog wh Y ad obvously p l p p 3 =. Ths ca be show as Fg. Fg Dscee pobably dsbuo of CRY Z Addo ad subaco of hese pdf's ca be caed ou usg he basc pobably coceps o add successve us sequeally o poduce he fal pobably dsbuo fuco. Geeao modellg: Load chages ad swchg o/off of he geeaos (due o falue o maeace ae esposble fo vaaos he geeaed powes. The geeao plas of a sysem ae modelled usg geeal dscee dsbuo ad he bomal dsbuo. Fg epeses a ypcal dscee dsbuo of a geeag pla. [] p p 4 p 3 p p 5 p p 6 p c c c c 3 c 4 c 5 c 6 c 7 c p 7 powe Fg Typcal Dscee Dsbuo of a geeao pla A dsbuo wh a sgle mpulse, wh pobably, he model fo he deemsc poblem, s a pacula case of hs dsbuo. The mome s ae calculaed usg he expesso as below, m = = C p p ( whee, m = mome of ode, C = geeag capacy of a u ( MV A; C = P j Q ad p = pobably of havg C (MV A The bomal dsbuo has bee poposed o model geeao plas wh seveal decal us. The pobably dsbuo fuco of he CRV ca be foud by cosdeg he followg oucome of a obsevao: u k s wokg (pobably p o s o wokg (pobably p. If hee ae geeag us a a geeag po he pobably ha R = of hem ae wokg sae ( sevce s gve by, p ( R = = ( C p ( p ( whee, p = pobably of havg a u sevce (equal fo evey u ad p = q = pobably of havg a u o wokg. The b mome sasfes he ecusve elao,

3 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 73, DECEMBER 79, 839 = m = [ ( ] p a m. (3 j= j j whee, he coeffces a j ae he coeffces of he expaso (l((3 Appedx of [] Load Modellg: If x ad y ae he acve ad eacve compoes of load powes (a ay po such ha, Z = S D =x jy he, he momes of load ae calculaed by, D m ( S = S p. (4 = D whee, m = mome of ode, S D = load a a pacula bus ( hs case s MV A; S D = P D j Q D ad p = pobably of havg S D (MVA Tasmsso Le Modellg: Tasmsso les ae epeseed by he sees admaces (akg o accou chagg admaces also. The admace of he le s assumed o be a complex adom vaable Y L, havg he Beoull dsbuo, Y L = wh pobably p = wh pobably (I p If hee ae m e les he he complex adom vaable Y L wll have bomal dsbuo wh paamees p. Thus, p m m ( YL = = ( C p ( p. (5 = O,,...m Hee oo momes of each asmsso le s calculaed by, L m ( Y = Y p (6 = L whee, m = mome of ode, Y L = sees admace of le (akg o accou chagg capacaces ad asfome appg Ad p = pobably of havg Y L Alhough he dsbuos peseed hee ae he mos commoly used oes, he mehod of cumulas s compleely geeal ad ay ohe dsbuo could be used o model ay pa of he sysem, such as he bomal dsbuo, bea dsbuo, he, he ucaed omal, ec. I a ypcal model of a powe sysem, hee mgh be dffee combaos of dsbuo fucos.e., geeao ca be bomal ad loads may be epeseed by omal ad dscee dsbuos. The kowledge of momes, whe hey all exs, s fo all paccal puposes equvale o kowledge of he dsbuo fuco. The mplcao of usg momes ad cumulas powe sysem aalyss s ha he fs fou momes gve a dea of he mea, sadad devao, skewess ad kuoss, whch coas he complee hsoy of ay load o geeao dsbuo. Fuhe, we ca develop he pobably desy fuco wh he kowledge of momes. The h mome m s gve by, m = E [ Z ] = Z p ; =,,3 (7 = whee Z s a CRV ad P s pobably mass. The cumulas ae lea combaos of he sascal momes ad abou a abay po. The elaoshp bewee momes ad cumulas [9] s gve as, m l = k l ; m = k k ; m 3 = k 3 3k k k 3 ; (8 m 4 = k 4 4k 3 k l 6k k 3k k 4 ; Covesely, k = m l ; k = m m l ; k 3 = m 3 3m m l m 3 ; (9 k 4 = m 4 4m 3 m l m m 3m 6m 4 ; I pacula, abou he mea, k l = ; k = m ; k 3 = m 3 ; ( k 4 = m 4 3m ; Two mpoa popees of momes ad cumulas ae used fo he addo ad poduc of depede CRV's. If Z l ad Z wee wo depede CRV's, he he mome of he poduc of wo CRV's s equal o he poduc of he mome of each. m ( ZZ m ( Z. m ( Z = ( Also, sum of depede adom vaables ae chaacezed by cumulas, whch ae he sum he dvdual CRY cumulas.

4 84 NATIONAL POWER SYSTEMS CONFERENCE, NPSC k ( Z Z I geeal, Z = k ( Z k ( ( m ( Z = m ( Z =,,3...(3 ad, k ( Z = k ( Z ;=,,3 (4 Slack Bus Repeseao: The pobably dsbuo fuco of slack bus volage s as show Fg 3. The momes of slack bus ae gve by, m ( V = ( V ; =,,3,.... (5 s s Is cumulas ae gve by, k V = m ( V = V = V ( s s s k ( =, fo. (6 V s f(v s. V s Fg 3 PDF of slack bus volage The slack bus s heefoe, assumed capable (by meas of geeag u coeced o o maa a cosa volage fo all possble cogeces of he geeag us, asmsso les, load vaao ad uceay. Ths eeds a %elable geeao a he slack bus wh a ag such ha ca povde eacve powe fo all cogeces. Load (PO bus epeseao: I s equed o specfy oly P D ad Q D a such a bus as a a load bus, volage ca be allowed o vay wh he pemssble lms of 5 %. As load model cosdeed hee s of he dscee dsbuo ype hece, he pdf of oal load S D = P D jq D coeced o a bus, s obaed by fs calculag he momes usg equao (4 ad he he especve cumulas usg he se of equaos (9 ad addg hem all usg equao (. Smlaly, he pdf of oal avalable capacy a a bus S G = P G jq G may be obaed by covoluo of he dvdual pdf's of he us coeced o he bus queso, usg he cumula mehod. V s The cumulas of jeced powe S = P jq a a bus s obaed fom he kowledge of cumulas of S G ad S D by covolvg he pdf of geeao wh he pdf of load demad (egaded as egave geeao. Sce hese CRV's ae assumed o be depede, he cumulas add as saed equao (. Theefoe, k ( G D S = k ( S k ( S (7 The momes ad cumulas of he bus volages fo all possble saes of asmsso les ae obaed fom he soluo of he basc sochasc odal max equaos as dealed he ex seco. Volage Coolled (P V bus epeseao: A he k h bus, he momes of Q k ae obaed fom he kowledge of he momes of volage ad admaces of he odal admace max akg o accou boh pesece of shu admaces ad apped asfomes. Soluo of Sochasc Nodal Admace Max equaos: I ems of complex adom vaables, he ewok powe flow equaos, whch ae bascally deemsc aue, wll be efeed o as he sochasc powe flow equaos. The fs sep he soluo of he pobablsc load flow poblem s o fd he e jeced powe a each bus of he sysem excep he efeece o slack bus. A ay ode hee ca be dffee kds of dsbuos ehe dscee o couous fo vey lage sysems. I s foud ha f a sysem s vey lage he dscee dsbuo of sysem capacy ouages ca be appoxmaed by a couous dsbuo. Such a dsbuo appoaches he omal dsbuo as he sysem sze ceases. I he poposed mehod, depedece bewee he pu adom vaable loads ad geeaos s assumed, alhough s possble o cosde some degee of lea depedece bewee dvdual adom vaables o goups of hem. Fo a slack bus, volage ad phase agle emas cosa. Fo a PQ bus, he momes of bus volages ae calculaed eavely such ha he (ul h eao gves he mome of volage a bus as, ( u m ( V = m P ( u ( V I jq k ( u ( u Y k V Y k V = = k m ( Y (8 The eave pocess s coued ll he chage magude of bus volage,?v (u bewee wo cosecuve eaos s less ha a cea oleace fo all bus volages,

5 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 73, DECEMBER 79, 84 V = m ( u ( ( V V ( u u (9 I ode o calculae he paamees fo he soluo of hs equao, s ecessay, fs, o deeme he elemes of he bus admace max Y BUS fom he asmsso le ad le chagg admaces wh goud as efeece. Also, efleco of he effec of ap chagg ude load asfomes o he powe flow he ecoeced ewok s ake cae of he odal admace max. Fo PV bus, he values of Q ae o be updaed evey eao. The momes of S k ae obaed as, m (S = m (P jq = m (V.m ( Y k V k k = The momes of PV bus volages ae gve by, m ( sp ( V = ( V m ( δ =,,3 ( The evsed values of m (d u ae foud fom he equao gve below: = m ( δ = m ( V u ( u P jq m ( ( V u m ( Y k ( u YkV Yk = = k V ( u ( Fo a PV bus, he uppe lm, Q (max ad lowe lm, Q (m of Q o hold he geeao VAR wh lms ae also usually gve.e., Q (max Q Q (m.theefoe, fs he calculaed value of m l (Q (u fom equao (3.7 s checked f s wh he gve lms.e., Q (max m l (Q (u Q (m. If he value of m l (Q (u s wh he saed bouds he he ew value of m (d u s fs calculaed usg equao (3.9 subsug he values of m V sp ad calculaed value of m l (Q he equao. Fom hs he ew value of V s calculaed usg pevously specfed value ( V sp ad calculaed value of m (d u. Now eag hs ew value he ex bus s ake up. If m l (Q (u < Q (m, he subsue m l (Q (u = Q (m o fd he ew value of S,eag he h bus as a P Q bus ad coug compuaos smla o a PQ bus. Smlaly, f m l (Q (u > Q (max he same mae akg he eacve powe mome m l (Q (u = Q (max he ew value of S s aga compued ad eag he h bus as a PQ bus coug compuaos smla o a PQ bus ae caedou. The le flows ae calculaed wh he fal bus volages ad he gve le admaces ad le chagg usg equao gve below: y k m ( P k jq k = m ( V ( V V k y k V V (3 Fally, he slack bus powe ca be deemed by summg he flows of he les emag a he slack bus. Illusave example: The mehod fo solvg he load flow poblem s llusaed he followg sample powe sysem gve Fg 4 (Hee acceleao facos of.4 ad.4 ad oleaces of pe u fo he eal ad magay compoes of volage ae used. The asmsso le mpedaces pe u o a,kva base ae gve Table. The scheduled geeao ad loads ad he assumed pe u bus volages ae gve Table. Bus Bus 3 3 L 3 Bus 3 Lflg.4 A hee bus sample sysem fo load flow soluo Table Le Admaces fo he sample svsem L Bus Admaces shuadm p q =lp code pq Y pq. j j j Table Scheduled geeao ad loads ad assumed bus volages fo sample sysem. G Bus Assumed Geeao Load Type pobably code Bus MW MVAR MW MVAR p

6 84 NATIONAL POWER SYSTEMS CONFERENCE, NPSC Volage.99 j. slack. 5 j. 7 3 pv. 3. j. 8 pq. P ow e P 7. jq j. m m m3 m4 49. j. 37 j j384 6 The dsbuo wh pobably s he deemsc case of he poblem. Fs he pobablsc load flow usg he sochasc powe equaos s caed ou. The momes of Y BUS max elemes ae gve Table 3 ad he momes of bus volages, jeced powes a bus ad le flows ae gve Table 4, Table 5 ad Table 6 especvely. P jq P3 jq 3.4 j..8 j..6 j.8.6 j.3. j..664 j j j.496 Table 3 Momes of Y BUS max elemes fo he sample sysem Elem es Y.3 j5.85 Y.5 j.383 Y 33.6 j 9.84 Y.6 j3. Y 3.7 j. Y 3.9 j7. m m m 3 m j8. 9.5j j j j j j j j j j j jl j jl j j j 6. Table 4 Momes of Bus Volages fo he sample sysem Volages m m m 3 m 4 V.99 j V.95 j.8 V j j j.6.56 j.9.97 j..875 j..46 j j.7 j j.948 Table 5 Momes of jeced powe a bus fo he sample svsem Table 6 Momes of Le Flows fo he sample sysem Le m m m m Flow s.63 j j j j j j j j j j j j.6.76 j j j j5.548 j j j j j j j j I he above ewok cofguao he dffee possbles of adequae asmsso capably ca be show by he followg codoal cofguaos keepg md he fac ha hee wll be 3 = 8 possble cofguaos. I he coveoal load flow (CLF aalyss, he pobably of adequae asmsso capably each of hese 8 cofguaos ca be foud afe pefomg a load flow sudy o each cofguao usg appopae load models. I s assumed ha load ad avalable geeao emas fxed ad hee ae o asmsso cualme cosas such ha couy s he sole ceo. Table 7 shows he asmsso sae pobables fo o asmsso cosas. Table 7 Tasmsso sae pobables

7 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 73, DECEMBER 79, 843 Saes Les ou Pobably = p (B j p p p 3 =.54 q p p 3 =.6 3 p l q p 3 = p p q 3 =.56 5, q l q p 3 =.54 6,3 q l p q 3 =.4 7,3 p l q q 3 =.4 8,,3 q l q q 3 =.6 Noe:,,3 efe o les,3 ad 3 I s see ha eve assumg ha hee s o load vaao ad ouage of geeag us.e., salled capacy ad demad beg fxed he coveoal load flow (CLF has o be caed ou 8 mes. Table 8 shows he bus volages fo each of he egh dffee cofguaos he load flow caed ou sepaaely fo each case. Smlaly, Y BUS max elemes ad le flows ca be obaed. Table 8 Bus Volages fo all egh le cogeces V l V V 3 es ou.99 j.4987 j j j.49 j j j.4987 j j j,.99 j,3.99 j,3.99 j,,3.99 j j j j.4554 j.555 j j j 8 j.6985 j 4 j 8 The fs mome o expeced value of bus volage a each bus fo all he dffee le saes s obaed as, Ex( V V p (4 = = Thus, he value fo he above fo he bus volages a bus, ad 3 ae, Ex(V = (.4987 j.6489 X.54 (.49 j.4554 X.6 (.4987j j.6489 X.6 ( j.5936 X.56 (.49 j.4554 X.54 (.573 j.6985 X.4 =.9 j.4 Value obaed by pobablsc load flow, m l (V =.9 j.8. Ex(V 3 = (.4797 j.5793 X.54 (.4736 j.456 X.6 (.483 j.795 X.6 ( j.5 X.56 (.353 j.555 X.54 ( j.5 X.4 =.97 j.699 Value obaed by pobablsc load flow, m l (V 3 =.986 j.6 Fom he calculao above s obseved ha he expeced value of each bus volage each of he deemsc cases ca be obaed fom he fs mome of he bus volages he pobablsc load flows aalyss. Smlaly, he same s ue fo he jeced powes a buses ad le flows. Theefoe, he egh load flows fo each abomal codo of les s uecessay f kowledge of he complee hsoy of he oupu dsbuo fucos ca be obaed fom oe load flow. II. CONCLUSION The mehod fo soluo of he sochasc load flow poblem wh he complex adom vaable modellg usg he mehod of momes as show heefoe educes pohbve amou of calculaos combed wh added dffculy of aalyzg ad syheszg he soluo he case of vey lage sysems. The algohm s applcable whe abay fucos ae used o model loads of he sysem. The pu vaables ae cosdeed o be complex adom, heeby a moe ealsc epeseao of he sysem s obaed. The modelg mehod ca be used wh he segmeao of load mehod, gve [3] o calculae he loss of load pobably ad he expeced geeao by he maches (he wo mos mpoa paamees equed fo geeao plag fom he kowledge of he fs ad he zeo h momes of he load mpulses befoe ad afe loadg of he maches. Thus, he applcao of he pese modelg pocedue wh he load segmeao mehod poposed by Ghosh ad Msha would gve a moe accuae epeseao of he powe sysem ad hece moe accuae esuls ca be expeced.