Voltage Sensitivity Analysis in MV Distribution Networks

Transcription

1 Proceedgs of he 6h WSEAS/IASME I. Cof. o Elecrc Power Sysems, Hgh olages, Elecrc Maches, Teerfe, Spa, December 6-8, olage Sesvy Aalyss M Dsrbuo Neworks S. CONTI, A.M. GRECO, S. RAITI Dparmeo d Igegera Elerca, Eleroca e de Ssem Uversy of Caaa ale Adrea Dora, Caaa ITALY Absrac: - The paper preses a sudy o he assessme of ode volage sesvy dsrbuo eworks wh respec o varaos of ode acve ad reacve powers. The work provdes a aalycal ool o quafy ode volages varao due o ecos of ode powers a ay M dsrbuo ework eco po. Smple aalycal expressos have bee developed o lk ode volages o ode acve ad reacve powers hrough ework elecrcal parameers, order o defe ad calculae approprae sesvy coeffces. A useful graphcal represeao s also gve for he derved expressos, provdg mmedae access o qualave ad quaave formao o ode volages sesvy. The proposed examples are referred o ypcal coducor secos used wh udergroud cable les ad usulaed overhead les M dsrbuo eworks. Key-Words: - Dsrbuo Sysems, Node olages, Sesvy Aalyss, olage Regulao. Iroduco The am of he work s o provde a smple aalycal ool o quafy ode volage varaos due o ecos of acve ad reacve power a oe or more odes of M dsrbuo eworks. I oher words, he paper preses a sudy o he assessme of ode volages sesvy wh respec o varaos of ode powers. olage sesvy aalyss s he base for he soluo of varous power sysem opmsao problems, relaed, for example, o volage regulao, loss reduco, ework expaso plag, opmal placeme of reacve sources ad geeraors, ec. [], [2], [3]. I parcular, he framework of he proposed sudy s he research of ew soluos for ovave maageme of acve dsrbuo eworks order o mpleme real me corol of power fluxes presece of dsrbued geerao (DG). The eed for corollg acve ad reacve power ecos o guaraee correc dsrbuo operao presece of DG calls for aalycal ools ha are suable o be used by approprae auomac corol algorhms. Ths s, for example, he case of o-le corol sysems used o avod ha volage ad curre cosras [4] are volaed durg ormal operao of dsrbuo eworks. Of course, he aalycal expressos ha wll be preseed are o eded o subsue exsg powerful load flow programs, whch are able, amog oher fucos, o perform ework sesvy assessme. The cosdered expressos are useful o be egraed o ad hoc opmsao ools o corol ode acve ad reacve powers, e.g. auomac volage regulao procedures or DG sallao plag dsrbuo eworks. I such a coex, s requred o kow ework sesvy order o assess he effecveess of possble corbuo of dsrbued geeraors o volage regulao procedures by corollg her power oupu. I Seco 2, he mehod used o oba mahemacal expressos ha lk ode volages o ode acve ad reacve powers s descrbed. Furher, learsed equaos are descrbed o derve useful closed aalycal expressos o calculae ode volages. The dfferece he resuls provded by he wo formulaos (lear ad o lear) s small due o he fac ha volage drops are small as well dsrbuo eworks. Cosequely, learsed expressos ca be used appropraely assessg sesvy coeffces for ode volages hs coex. These coeffces are he elemes of sesvy marces, called [S P ] ad [S ], whch coa measures of volage varaos due o, respecvely, acve ad reacve ode powers. I Seco 3, a graphcal represeao of he derved volage sesvy coeffces s gve akg o accou praccal examples referred o M eworks. Such a approach s useful o ge mmedae percepo of volage sesvy varao as ode dsace from he org vares. Furher, ca easly be hghlghed how seco ad ype of coducor (overhead le or udergroud cable le) fluece ework sesvy.

2 Proceedgs of he 6h WSEAS/IASME I. Cof. o Elecrc Power Sysems, Hgh olages, Elecrc Maches, Teerfe, Spa, December 6-8, Aalycal Model I hs seco he aalycal model used o perform he sesvy aalyss wll be descrbed. Le us cosder a hree-phase symmercal, radal dsrbuo ework wh odes ad braches, where we defe as odes he pos of load coeco, he pos of le characerscs chage ad he ucos, ad as braches he coducor segmes bewee wo odes. The odes ca be umbered accordg o he followg rule [5]: he org of he ework (ypcally a H/M prmary subsao) akes he umber 0, whle he oher odes are umbered sequeally mposg ha a recevg ode akes a umber hgher ha he sedg ode earer o. The erms recevg ad sedg are used uder he assumpo ha a radoal radal ework,.e. whou dsrbued geeraors, he power flow s dreced from a lower o a hgher umber. The braches are defed by he same umber as her recevg ode, as show Fg I 4 I Fg.. Oe-le dagram of a hree-phase symmercal, radal dsrbuo ework. Ths umberg mehod allows a smple sorage of he ework srucure a sgle square marx (called cdece marx, [A]) whose dmeso s (x). I parcular, he rows correspods o he braches ad he colums o he odes. The elemes of [A] descrbe he ework opology ad are equal o f he ode correspodg o colum s fed hrough he brach correspodg o row, 0 oherwse. The calculao of he brach flows s easly obaed applyg he mesh mehod for ework aalyss. I ca be easly show ha: J = [ A] I () [ I ] s he vecor of he load curres, dmeso (x); [ J ] s he vecor of he brach curres, dmeso (x). The ework complex mpedace s equal o: Z = A Z A (2) [ ] [ ] [ ] [ ] b 6 6 I 6 3 I 3 where [ Z b ] s he dagoal marx, dmeso (x), whose elemes are he complex mpedaces of he correspodg braches. The ma dagoal elemes of [ Z ], ( Z ), are equal o he sum of he brach mpedaces formg he pah from he org o he ode. The off-dagoal elemes, ( Z ), are equal o he sum of he brach mpedaces formg he pah from he org o he commo ode of he pahs formed by he org ad he odes ad, respecvely. Le us cosder ode h ad s volage phasor, h. We wll assume ha 0 s kow. Le h be he volage drop across brach h ad U h he oal volage drop from ode 0 o ode h : U h = 0 h (3) [ U ] = [ A] [ ] (4) [ U ] s he colum vecor ( x ) whose elemes are he volage drops dcaed by U h ; [ ] s he colum vecor ( x ) whose elemes are he volage drops dcaed by h. ecor [ ] ca be expressed as follows: [ ] = [ Z ] [ J ] (5) As kow, cross parameers ca usually be egleced aalyss of dsrbuo eworks. Subsug expresso (5) (4), we oba he b followg expresso for vecor [ U ] [ U ] = [ A] [ Z ] [ J ] : b (6) he, f we cosder expresso (), we also have: [ U ] = [ A] [ Z b ] [ A] [] I (7) Cosderg he defo of [ Z ] gve by (3), expresso (7) s equvale o: [ U ] = [ Z ] [ I ] (8) Node volages vecor [ ] s he gve by he followg: [ ] = [ 0 ] [ U ] = [ 0 ] [ Z ] [ I ] (9) For he -h ode, he complex power S s defed as: S = I = P + (0) s he volage phasor a ode ; I s he curre phasor a ode ; I s he complex cougae of I ; P s he e real power he -h ode; s he e reacve power he -h ode. If we express (0) by meas of he correspodg

3 Proceedgs of he 6h WSEAS/IASME I. Cof. o Elecrc Power Sysems, Hgh olages, Elecrc Maches, Teerfe, Spa, December 6-8, marxes we oba: [ S] [ ] = I = [ P] + [ ] () where [ P ] ad [ ] are he colum vecors, dmeso ( x ), whose elemes are, respecvely, he ode acve ad reacve powers. The complex cougae of [ S ] s gve by: S = [] I = [ P] [ ] (2) From hs equao, s possble o oba a expresso for ode curres vecor [ I ]: P I = (3) [] [ ] [ ] [ ] Cosequely, subsug expresso (3) (9), [ ] ca be wre as: [ ] [ ] [ ] [ ] [ ] [ ] P = 0 Z Fally, cosderg ha [ Z ] [ R] + [ X ] ad [ X ] are ( ) (4) =, where [ R ] x marxes, respecvely, real ad magary par of ework mpedace marx, we : oba he followg expresso for [ ] ( ) [ P] [ ] [ ] 0 (5) [ ] = [ ] [ R] + [ X ] 2. Lear Expressos ad Sesvy Coeffces Smplfed lear expressos ca be derved from (5) uder he followg hypoheses (commoly acceped dsrbuo eworks aalyss): - he phase dfferece bewee ode volages s eglgble ad, as a cosequece, f phasor 0 s chose o he real axs, oly he real par of volage [ ] = real[ ] s cosdered; - cosa curre models are cosdered for loads (ode powers are referred o sysem omal volage sead of acual ode volage). Cosequely, expresso (5), ca be wre as: [ R] [ P] + [ X ] [ ] = 0 (6) [ ] [ ] om The rms value of volage a ode,, ca expressed as follows: N N = + 0 R P X (7) om = = Such expressos have bee compared o he oes derved from o lear equaos (5). The dfferece he resuls provded by he wo formulaos (lear ad o lear) s small due o he fac ha volage drops are small as well dsrbuo eworks. Cosequely, learsed expressos ca be used appropraely assessg sesvy coeffces for ode volages hs coex. As obvous, he volage a he -h ode o oly depeds o he -h ode powers, bu also o he powers eced or absorbed a he oher ework odes: = ( P, P2,, P,, 2,, ) (8) The oal dffereal of fuco s gve by: d = dp + d (9) = = where we fd he sesvy coeffces,, whch, from (7), be expressed as: = = wh, =,2,,. om om R X ad (20) The above dervaves ca be regarded as volage sesvy coeffces wh respec o ode powers varao. Ther physcal meag s he followg: - for = provdes volage varao a he -h ode due o uy varao of he -h power; - for provdes volage varao a he -h ode due o uy varao of he -h power. Cosderg he equaos gve by expresso (9) we have: dp d dp = (2) d d d We ca defe a sesvy marx, [ S ], ( x 2 ), whose elemes are he sesvy coeffces defed by (20): [ S] = (22) Marx [ S ] ca be wre as:

4 Proceedgs of he 6h WSEAS/IASME I. Cof. o Elecrc Power Sysems, Hgh olages, Elecrc Maches, Teerfe, Spa, December 6-8, [ S ] = [ S P S ] (23) whch he wo sub-marxes [ P ] ( x ), are hghlghed: S ad [ ] [ ] S = = [ R] p [ ] S = = [ X ] om om S, (24) Assessme of he elemes of sesvy marx [ ] dcaed by ad ( SP ) = = R om ( S ) = = X om S, (25) (26) allows o quafy volage varaos a each ework ode due o acve ad reacve power varaos a ay oher ode. Expressos (25) ad (26) are of geeral valdy ad ca be appled o all radal dsrbuo eworks or radally operaed dsrbuo eworks. Sesvy coeffces ( ) p S ad ( ) S ca be expressed as fucos of le logudal parameers, (le ressace ad reacace per klomere) by roducg he followg quaes: R r = (27) L ad X x = (28) L R ad X are, respecvely, real ad magary par of mpedace Z = R + X ; L s: - for = he sum of he brach leghs formg he pah from he org (ode 0) o ode ; - for he sum of he brach leghs formg he pah from he org o he commo ode of he pahs formed by he org ad odes ad. Parameers r ad x represes he weghed average, wh respec o braches legh, of he logudal parameers per klomere of he braches belogg o he commo pah from he org o odes ad. Parameer r s a fuco of: - braches legh; - braches coducor seco; - braches coducor ressvy. Parameer x s a fuco of: - braches legh; - braches coducor seco; - frequecy; - geomerc characerscs of elecrcal les. Subsug expressos (27) ad (28) (25) ad (26) we oba: ( S P ) = om L r (29) ( S ) = om L x (30) Such expressos show ha he sesvy coeffces deped o ework exeso (pah legh) ad, respecvely, o r ad x. Such expressos show ha he mos sesve eworks are he oes wh exeded les ad/or characersed by hgh value logudal parameers S ad (r ad x ). Noe ha he rao bewee ( P ) ( ) S s gve by r / x. Assumg he same seco ad ype of coducor for all he ework braches, expressos (27) ad (28) ca be wre as: X = L x ad R = L r where r ad x are le ressace ad reacace per klomere. Sce expressos (29) ad (30) ca be wre as: ( S P ) = L r (3) om ( S ) = L x (32) om he ( S P ) r = (33) S ( ) x 3 Graphc represeao of (S P ) ad (S ) Assumg he same seco ad ype of coducor for all he ework braches, s possble o provde a eresg graphcal represeao of ( S ad ( S ). Plog sesvy coeffces vs. ) P L, we oba sragh les whose slop s, respecvely, r om ad x. om

5 Proceedgs of he 6h WSEAS/IASME I. Cof. o Elecrc Power Sysems, Hgh olages, Elecrc Maches, Teerfe, Spa, December 6-8, Such a graphcal represeao s useful o ge mmedae percepo of sesvy varao wh ode dsace from he org. Furher, ca be easly hghlghed how seco ad ype of coducor (overhead le or udergroud cable le) fluece he ework sesvy. I he followg, he graphcal represeao of volage sesvy coeffces (Sp) ad (S ) for a 20 k ework s gve, cosderg boh cases of udergroud cable les ad usulaed overhead les wh coducor seco of 35mm 2 (Fg. 2) ad 95 mm 2 (Fg. 3). The followg parameers have bee used: seco=35mm 2 : cable le r = Ω/km cable le x = 0.2 Ω/km overhead le r = 0.59 Ω/km overhead le x = Ω/km seco= 95mm 2 :cable le r = Ω/km cable le x = 0.8 Ω/km overhead le r = 0.93 Ω/km overhead le x = Ω/km From boh fgures ca be oed ha he greaer he ode dsace from he org, he hgher he volage sesvy. As for maxmum dsaces cosdered for perpheral odes, 5 km has bee ake as a realsc value for acual dsrbuo eworks. I Fg. 2 s appare ha sesvy wh respec o acve power eco s greaer ha he oe wh respec o reacve power eco. Ths s especally rue for cable les, whch r/x = 3.38 for he cosdered case. I he odes ha are 5 km far from he org he sesvy coeffces reach he hghes values: cable le: S P = k/mw; S = k/mar; overhead le: S P = k/mw S = k/mar Furher, ca be oed ha a usulaed overhead le s more sesve ha a udergroud cable le wh respec o reacve power ecos, vce versa wh respec o acve power ecos. For he overhead les cosdered he example we have r/x =.34. I cocluso, cosderg a coducor seco of 35mm 2, passg from cable o overhead les, S P reduces by abou 20%, whle S creases by 94%. Fg. 3 shows ha cable les are more sesve o ecos of acve power ha o ecos of reacve power (r/x=.38). O he oher had, overhead les are more sesve o ecos of reacve power ha o ecos of acve power (r/x=0.54). Furher, cable les are more sesve ha overhead les wh respec o acve power ecos, whle overhead les are more sesve ha cable les wh respec o reacve power ecos. Noe ha a greaer coducor seco deermes lower values of volage sesves. (S P ) [k/mw], (S ) [k/mar] 0,000-0,025-0,050-0,075-0,00-0,25-0,50-0,75-0,200-0,225-0,250-0,275-0,300-0,325-0,350-0,375 (SP) - Cable Le - 35 mmq - 20 k -0,400 (S) - Cable Le - 35 mmq - 20 k -0,425 (SP) - Usulaed Overhead Le - 35 mmq - 20 k -0,450-0,475 (S) - Usulaed Overhead Le - 35 mmq - 20 k -0,500-0,525-0, L [km] Fg. 2. Graphcal represeao of (S P ) ad (S ) for a 20 k ework wh udergroud cable les or usulaed overhead les, coducor seco 35mm 2.

6 Proceedgs of he 6h WSEAS/IASME I. Cof. o Elecrc Power Sysems, Hgh olages, Elecrc Maches, Teerfe, Spa, December 6-8, ,000-0,025-0,050-0,075 (S P ) [k/mw], (S ) [k/mar] -0,00-0,25-0,50-0,75-0,200-0,225-0,250-0,275 (SP) - Cable Le - 95 mmq - 20 k (S) - Cable Le - 95 mmq - 20 k (SP) - Usulaed Overhead Le - 95 mmq - 20 k (S) - Usulaed Overhead Le - 95 mmq - 20 k -0, L [km] Fg. 3. Graphcal represeao of (S P ) ad (S ) for a 20 k ework wh udergroud cable les or usulaed overhead les, coducor seco 95mm 2. 4 Coclusos The paper preseed a smple aalycal ool o quafy ode volage varaos due o ecos of acve ad reacve powers a oe or more odes of M dsrbuo eworks. Sesvy coeffces have bee derved for ode volage wh respec o varaos of bus power. A graphcal represeao of he coeffces has bee dscussed hrough praccal examples. Such a represeao clearly hghlghs how ode dsace from he org, seco ad ype of coducor (overhead le or udergroud cable le) fluece he ework sesvy. Refereces: []. Kumar, I. Gupa, H.O. Gupa, C.P. Agarwal, olage ad Curre Sesves of Radal Dsrbuo Nework: a New Approach, IEE Proc. o Geerao Trasmsso ad Dsrbuo, ol. 52, No.6, November [2] H.N. Ng, M.M.A. Salama, A.Y. Chkha, Capacor Placeme Dsrbuo Sysems Usg Fuzzy Techque, I Proc. of IEEE CCECE'96, pp [3] G. Ramakrsha, N.D. Rao, Implemeao of a Fuzzy Logc Scheme for / Corol Dsrbuo Sysems, IEEE PES 999 Wer Meeg, ol.2, pp [4] S. Co, S. Ra, G. Ta, U. aglasd, Dsrbued Geerao L Dsrbuo Neworks: olage ad Thermal Cosras, Proc. of he 2003 IEEE PowerTech Cof., Jue 2003, Bologa, Ialy. [5] M. Papadopoulos, N.D. Hazargyrou, M.E. Papadaks, Graphcs Aded Ieracve Aalyss of Radal Dsrbuo Neworks, IEEE Trasacos o Power Delvery, ol. PWRD-2, No. 4, Ocober 987.