ENHANCING THE COMPUTATIONAL PERFORMANCE OF NEWTON- RAPHSON POWER FLOW CODES IN POWER SYSTEM PROBLEMS

Transcription

1 Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) ENHANCIN THE COMUTATIONAL ERFORMANCE OF NEWTON- RAHSON OWER FLOW CODES IN OWER SYSTEM ROLEMS ARASAD RAJU Assoc rofessor EEE Dept Sree Chataa Isttute of Techologcal Sceces LMD ColoKaragarA 50557Ida prasadraj55@galco ROF J AMARNATH EEE Dept J N T U H College of Egeerg Hderabad A Ida jaaarath@ahooco ROFD SUARAYUDU EEE Dept ulla Redd Egeerg College Kurool-0 A Ida Abstract : ower flow studes of power sstes gve the portat coputatos procedures to fd the preset operatoal characterstcs ad loadg predctos whch are useful for future plag of the sste accordg to the growg dead Newto Raphso ethod s a ver popular oe to copute the soluto of power flow proble due to ts fast covergece few teratos Jacobs of the Newto Raphso ethod to be solved a sple faable ad relable aer to evaluate the accurate values Ths paper descrbes a powerful techque ow as Autoatc Dfferetato (AD) appled to copute frst order partal dervatves or Jacob eleets that occur Newto Raphso ethod whe solvg power flow proble Evaluato of the dervatves obtaed for the epressos specfed the for of coputer progra a wa such that code s dfferetated rather tha epresso ule the tradtoal fte dfferetato Ma progra of Newto Raphso power flow ethod s eepted fro the tedous tas of coputg error full Jacoba partal dervatves ad aes the calls to a well defed AD tool to do ths tas accuratel wheever requred Ths loose couplg betwee a progra ad AD code ehaces the coputatoal perforace ad create a fleble soluto evroet to copute the load flow solutos effcetl I ths paper Autoatc Dfferetato techque s eplaed cotet to power flow ad the deostrated o a -bus test sste ad sulato results have bee copared ad aalzed Kewords: Newto Raphso power flow Jacobas Autoatc Dfferetato (AD) AD Too ADMAT Itroducto Forato of ower flow proble aes the use of teratve process to provde the fal soluto about voltage agtudes agles actve power flows ad reactve power flows wth the etwors Newto Raphso ethod s oe of the best sutable soluto approach of power flow proble whch posses the qualt of solvg the sste of olear equatos wth the strog covergece propertes[-] artal dervatves ow as Jacoba eleets whch are the fuctos of the adttace atr ad bus voltages foud the Newto Raphso ethod ust be solved accuratel ad effcetl usg a sple approach Tradtoal Fte Dfferetato ethod used to solve frst order partal dervatves gves approated values of the Jacoba eleets wth trastoal errors If these accurate values used for further coputatos load flows gves the ufar values of voltages agles ad powers of gve sste To overcoe ths drawbac there s a great wa to copute dervatos of a gve fucto ver accuratel up to ache precso Ths sple ad effcet approach s ow as Autoatc Dfferetato Operators of power sste eed accurate studes to operate etwors as close as possble to ther lts wth the securt trats Optal power flow proble s a powerful approach; geerall use the actve reactve power flows voltages ad agles represetato b asg precse evaluato of frst order dervatves of etwor equatos[] Wth the help of Autoatc Dfferetato tools le ADMAT ADJaC the cople error proe tas of coputg partal dervatves becoes sple ad accurate ISSN : ol No05 Ma 0 09

2 Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) Forato of ower flow Model To appl the popular Newto Raphso ethod to solve the ower flow equatos der a power sste etwor wth a fte uber of buses Let sa that power trasfer s occurrg betwee th bus ad th bus as show Fg The power satch at the th bus gve as: Fg Represetato of sall porto of power sste etwor Δ K LK cal 0 () ΔQ Q Q Q 0 () K LK cal Where Δ ΔQ descrbes the actve ad reactve power satch Q are the actve reactve power geerato ad L Q L refer s the actve reactve power loads specfed at th bus respectvel Fall cal Q cal are the uspecfed actve reactve powers trastted betwee th ad th bus ow as power flow equatos gve wth respect to odal voltages ( ) phase agles ( ) ad le pedaces as show: Q cal cal [ ( ) ( s [ s( ) ( () () Where b as b s uber of buses ad bus s slac bus Applg Newto Raphso ethod to the soluto of power flow proble power satch equato s or power balace equatos of eq () & () are wrtte as show below Δ Δ Δ (5) ΔQ Q Q F(X (-) ) J(X (-) ) X () It s clear fro eq (5) that the ters represetg the J(X (-) ) are the Jacoba eleets whch are the artal dervatves of real ad reactve powers wth respect to the ow state varables X ( ad ) roper selecto of tal values for X s portat factor to reduce teratve process whch wll be cotued utl X gve eq (6) attas the value that should be wth the prescrbed tolerace Ad at ths popt X() ( s s) gve eq (7) gves the feasble soluto of load flow proble ( ) ( ) ) ( ) Q ΔX J ( X ) F( X ) (6) () ( ) () X X AX (7) Coputg repetedl to upgrade the values of X () usg fq (7) utl ΔX () tolerace value gves fal values for feasble soluto Ad also t requres accurate coputato of Jacoba eleets (frst order dervatves) for each terato[5-6] ISSN : ol No05 Ma 0 09

3 Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) Autoatc Dfferetato: As the covetoal dfferetatg approaches have the a draw bacs le occurrece of approated values wth trucato errors hgh coputatoal epeses calculatg dervatves leads towards ufar solutosi further stages the terpla betwee dervatves coputato large scale o ler power sste applcato ad uercal optzato develops a effcet varet fal soluto Autoatc Dfferetato (AD) o the other had facltates effectve wa of accurate evaluato of dervatves of fuctos represeted the for of progra usg hgh level prograg laguages[7] Soe of the advatages of AD over other dfferetato techques are () It coputes the dervatves of the code whch represets the fucto rather tha fucto drectl b cobg the dervatves of eleetar operatos whose dervatves are alread ow It reduces the coplet ad eor allocato whch leads towards reducto t of coputato ad wor load of coputato () AD coputes the eact values of dervatves ver accuratel up to ache precso ad relved fro the usace of havg approated values of dervatves wth trucato errors ()The a progra of power flow s releved fro the burde of coputg cople dervatves ad assged t to powerful AD tool the loose couplg betwee these two codes creases the fleblt ad effcec of the coputg evroet (v) arallel coputg of dervatves s possble usg ultthreadg cocepts of Hgh level prograg laguages stead of seral coputg whch further reduces the processg te as well as t of the coputato creasg the perforace of the dervatve coputato owerful AD Tools were developed dfferet hgh level prograg laguages le Fortra C C MATLA Java to evaluate Frst Order Secod Order dervatve effcetl Soe of the popular AD tools are ADIFOR ADOL ADMAT ad ADJaC etc Techque of Autoatc Dfferetato: I Autoatc Dfferetato ever fucto s epressed as coputer progra b decoposg t as a log sequece of sple eleetar operatos (such as ultplcato Addto ad subtracto) ad trsc fuctos (le s ep etc) the for of stateets whose dervatves are alread ow Cha rule of dfferetato allows these stepwse dervatves to be cobed to gve the whole dervatve of a gve fucto whch s coded as a progra usg hgh level laguages such as C C MATLA ad Java[8-9] The cha rule whch allows decoposto of dfferetals s gves as: f ( g( t) ) t f ( s) s g( t) t t t0 s g ( t0 ) t t0 (8) Let der a fucto f() where f:r R the the subroute show Fg eplas that the fucto { } f f() s a coposto of eleetar fuctos such that f s fucto wth alread coputed quattes j j J where j s are ow as the teredate varables Fg Represetato of ters of varables j FORWARD for ed for ed f e f ( j J p : p ( Fg Ipleetato of Forward ode AD j f ) : p j ) j J f j Two well ow strateges of Autoatc Dfferetato are () Forward ode () Revege ode Here ths paper the scope s restrcted to deostrate the forward ode Autoatc Dfferetato ol I the forward ode approach gradet object s assocated wth each scalar varable the code such that alwas accoodate the dervatves of wth respect to the teredate varables j as show the below gve code sppet of Fg ISSN : ol No05 Ma 0 095

4 Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) Applcato of AD Techque to ower Flow Fucto Cosder a sall part of ower Sste Networ cotas two buses ad as show Fg The the actve ower trasfer cal s gves as equato Ths ca be rewrtte as cal Idepedet varables of equato (9) are: [ ( ) ( s [ ] (0) (9) Coded decoposed fucto stateets Table Fdg gradet vector of eq (9) wr t the varables of eq (0) Augeted Frst order dervatve (or) gradet stateets [ 000 ] [ 000] [ 000] [ 000] 5 [ 000] [ ] [ 00] 8 ( ) [ ] 9 s( 8 ) 9 88 [ 00( ) ( 0 ( 9 ) 0 s 88 [ 00 s( ) s( 9 9 [ 00 ( ) ( [ 00 s( ) ( s s( ) s( ) ( ) 7 7 ( ) Cos 7 ( ) ( ) ( ) 7 7 s( ) s 5 ( ) 5 [ s ( ( ) s( )) ( s( ) ( )) ( s( ) ( ) [ ( ( ) s( )) ( ( ) s( )) ( s( ) ( )) ( s( ) ( ) ISSN : ol No05 Ma 0 096

5 Substtutg eq (0) gves: [ ] s cal () The gradet vector s gve as: cal () Usg basc eleetar arthetc rules ad eleetar fucto eq () ca be decoposed to sple ad sall eleetar fuctos as show frst colu of Tables ts dervatve or gradet also coputed as show the secod colu of table such that accurate values of the fucto ad dervatves were provded sultaeousl statl ad ssteatcall as gve the Table Fro table t s observed that fal dervatve vector or gradet vector of output s gve as ( ) ( ) s s [ ( ) ( s s () Substtutg eq (0) eq () we get ( ) ( ) s s [ cal ( ) ( s s (5) I the case of dfferetato of ultple fuctos of a sste of f() where f: R R the (6) I ths cotet for a ode power sste wth varables le odal voltages (v s) ad agles ( s) ad wth the real power flow s assug ode as slac bus the real power dervatves are v v (7) Eq (7) represets the Jacobea of Nodal powers whch ca be solved b the Autoatc Dfferetato techque as eplaed above ver spl accuratel statl sultaeousl ad ssteatcall Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) ISSN : ol No05 Ma 0 097

6 Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) Slarl for reactve power gradets also we ca develop equatos as usg the cocepts dscussed above Ad the fal dervatve vector s slar to the eq(7) s gve below b eq(8) Q Q Q Q v Q v (8) Q Q Q Q Q Eq (8) represets the Jacobea of Nodal powers Q Q Q whch ca be solved b the Autoatc Dfferetato techque as eplaed above ver spl accuratel statl sultaeousl ad ssteatcall 6 Deostrato of AD Techque o a Test Sste Autoatc Dfferetato techque s appled to a bus test sste to deostrate ts effectveess coputg frst order dervatves e Jacoba eleets occurs ower flow soluto to solve odal voltages agles real ad reactve power at each bus usg Newto Raphaso ethod The test sste s as show the Fg ad put data for the gve bus sste s gve table To fd the feasble soluto of ower flow proble Newto Raphaso approach gve b equato (9) for bus sste to be solved assug approprate tal values for depedet varables le voltages (v s) ad load agles ( s) assug bus as the slac bus It s clear fro eq(9) that the Jacoba atr cotag the dervatve eleets should be solved for ever terato utl the covergece to acheve the fal soluto of ower flow proble Fg Four us sste wth le adttace data Table Iput Data for the gve four bus sste us Q Tpe of us <0 Slac Q Q Q Autoatc Dfferetato Forward ode approach s appled to solve Jacoba eleets usg the popular AD tool ADMAT wrtte MATLA prograg laguage Ma progra of Newto Raphaso ower flow soluto also developed MATLA prograg evroet such that t s releved fro the tedous coputatoal wor load of calculatg dervatve eleets ad cocetrate ol o the coputato of power satch values Y bus forato cotrollg geerator lts ad atr operatos etc Coputato of Jacoba eleets s atteded b AD tool ADMAT upo gettg call fro a progra wheever at a pot t s requred Ad well developed fuctos of ADMAT wll coplete ths assged tas spl effcetl ad returs the accurate error less values of dervatves to the a progra for further usage ISSN : ol No05 Ma 0 098

7 Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) Δ Δ Δ Δ Q Δ Q Δ Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Δ Δ Δ Δ Δ Δ Jacoba eleets (9) 7 Results ad Dscussos The bus sste show Fg 5 s sulated to solve the power flow proble the MATLA usg both tradtoal Fte Dfferetato Techque ad ovel Autoatc Dfferetato Techque oth the results are preseted ad copared as show below Table Approated values of Jacoba eleets of eq(9) coputed usg Fte Dfferetato (FD) ethod Table Accurate values of Jacoba eleets of eq(9) coputed usg Autoatc Dfferetato (AD) ethod After seve teratos of covergece t s wtess fro the above tables that Jacoba atr obtaed b cople ad tedous Fte Dfferetato ethod gves the approated values of the dervatves wth trucato errors as show table Whereas the Jacoba atr obtaed usg Autoatc Dfferetato ethod gves the eact ad accurate values of the dervatves as show table ISSN : ol No05 Ma 0 099

8 Arasad Raju et al / Iteratoal Joural of Egeerg Scece ad Techolog (IJEST) Usg the jacoba values obtaed table ad table b both Fte Dfferetato ad Autoatc Dfferetato for further calculatos of real power reactve power voltage ad load agle at each bus of the gve test sste usg the eq(9) wll cotrbutes followg fal results of Newto Raphso power flow soluto Table 5 Fal ower flow soluto of gve bus sste due to the Jacoba eleets obtaed b the Fte Dfferetato us s Q s Q δ Table 6 Fal ower flow soluto of gve bus sste due to the Jacoba eleets obtaed b the Autoatc Dfferetato us s Q s Q δ It s observed fro the table 5 that because of the accurate approated Jacoba values obtaed b Fte Dfferetato show b table the fal power flow soluto presets ufar values of real power reactve power voltage agle at each bus as show table 5 cotrast to ths t s observed fro the table 6 that owg to the accurate Jacoba values obtaed b AD show b table the fal power flow soluto provdes accurate ad otceabl proved values of real power reactve power voltage agle at each bus as show table 6 It s also otced fro the table 6 that the power s atches are derabl ver sall ad alost equal to zero due to the proved values of powers resulted fro accurate values of Jacobas obtaed b Autoatc Dfferetato Coclusos A odel has bee forulated for a four bus power sste etwor the the techque of Autoatc Dfferetato tested pleeted usg that sste ad the effectveess of ths techque s deostrated Refereces [] Kudur (99) ower Sste Stablt ad Cotrol Mcraw-Hll [] Arrllaga J ad Arold C(990) Coputer Aalss of ower Sstes Joh Wle & Sos [] Stott(97): Revew of Load-flow Calculato Methods IEEE roceedgs vol 6 pp [] Alguacl Nad JCoejo (000): Multperod optal power flow usg beders decoposto IEEE Trasactos o power sstes pp 96-0 [5] Da Costa M N Marts ad JLR erera (999): Developets the Newto Raphso ower flow forulato based o curret jectos IEEE Trasactos o power sstes pp 0-6 [6] Da CostaM N Marts ad JLRerera (00): A augeted Newto Raphso ower flow forulato based o curret jectos It J Electrcal ower ad eerg Sstes ol pp 05- [7] A rewa(000) Evaluatg Dervatves: rcples ad Techques of Algorthc Dfferetato Froters Appled Matheatcs SIAM hladelpha A [8] T F Colea ad Aera(998) ADMAT: A Autoatc Dfferetato Toolbo for MATLA Techcal report Coputer Scece Departet Corell Uverst [9] Rchard D Nedger (00): Itroducto to Autoatc Dfferetato ad MATLA Object-Oreted rograg SIAM REIEW ol 5 No ISSN : ol No05 Ma 0 00