Static Voltage Collapse Studies In Power Systems. D. Vijaya Kumar, B.Manmadha Kumar, I.Ramesh, A.Jagannadham

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1 Internatonal ournal of Scentfc & Engneerng Research, olume 5, Issue 4, Aprl Statc oltage Collapse Stues In ower Systems D. jaya Kumar, B.Manmaha Kumar, I.Ramesh, A.agannaham Abstract- Ths paper concentrates on the statc voltage stablty stues on power systems. The behavor of the system when subjecte to graual an steay ncrease n system loang s stue. The tool le contnuaton loa flow usng the moal analyss technque s use n a systematc way so as to prect the system behavor at fferent operatng contons. The metho evse s a worthy tool to use n power system plannng stues from the vew pont of voltage stablty. The man objectve s to evce sutable computer programs so as to conuct plannng stues on a power system from the vew pont of voltage stablty. To o the loa flow an entfy the wea buses n the system usng Egen value analyss. Inex Terms Moel analyss technque oltage stablty stues, Newton- Raphson Metho, an Branch artcpaton, Egen value analyss. I INTRODUCTION Once a reactve power source has reache ts maxmum ORE an more voltages relate problems are beng lmt, t can no longer regulate the voltage. Therefore, experence by moern power systems n Ina as well sustane loa growth results n accelerate voltage ecay Mas n other foregn countres. It s well nown that the an hence greater reactve power requrements. Ths may voltage problems are closely relate to the management of force other voltage regulatng evces to ther lmts, wth reactve power resources an flows n the system. It has been subsequent further acceleratons n the rate of eclne of establshe that qute a few of the gr falures experence by oltage. If loa s ncrease further, the pont of voltage fferent electrc utltes have been cause by the voltage collapse woul soon be reache. Ths phenomenon has come collapse phenomenon [10]. The stuy of a system for voltage to be nown as voltage collapse. nstablty/collapse contons wll have to be one both from A voltage collapse process may consst of the followng the plannng as well as from the operaton vewponts [13]. steps: [10] The present stuy manly concentrates on the plannng a) The crtcal sturbance by ncreasng transmsson lne aspects. loang woul evelop an untenable ncrease of seres Lac of aequate reactve power resources n a power system s a major contrbutng factor to the process of voltage collapse. As loas n a power system ncrease, voltage across the networ tens to ecrease an reactve power, losses ncrease. The ncrease reactve power eman woul be supple by voltage regulatng evces such as generators or reactve power losses I X. b) Ths woul cause a sharp voltage reucton at a number of transmsson loa substatons. c) By reucng loas an provong over exctaton n surrounng unts, the voltage reucton woul create an ntate stablty. statc var compensators, f possble [1]. However, ue to ) But the voltage reucton woul also ntate a powerful physcal lmtatons, such evces cannot supply unlmte establzng force, automatc on-loa transformer tap amounts of reactve power. Often sustane loa growth changng, whch woul progressvely ncrease the loas, an wll result n some source of reactve power, or perhaps a number of such sources, encounterng lmts,.e. reachng a physcal lmtatons n the amount of reactve power that they can supply. may even provoe loa overshoot. e) Ths loa ncrease cause by each tap changng step woul progressvely reuce the transmsson voltage levels. f) Each voltage reuctons woul ncrease seres reactve power losses nversely to the voltage cube. g) Ths changng stuaton coul only mantan stablty untl the frst unts over exctatons protecton functons. Djaya Kumar s rofessor n Department of E.E.E, Atya Insttute of h) Once ths occurs, the voltage falls so sharply that Technology an Management, Teal, Anhra raesh, Ina. system collapse s nevtable. B.Manmaha Kumar s Senor Assstant rofessor n Department of ) The sustane excess, reactve eman woul srupt E.E.E, Atya Insttute of Technology an Management, Teal, A., Ina. Emal.ID: bomann@yahoo.co.n nter unt co-ornatons so leang to angle nstablty as the fnal cause of breaown. I.Ramesh an A.agannaham are Assstant rofessors n Department of E.E.E, Atya Insttute of Technology an Management, Teal, A., Ina.

2 Internatonal ournal of Scentfc & Engneerng Research, olume 5, Issue 4, Aprl MATHEMATICAL MODELLING FOR OLTAGE STABILITY ANALYSIS The objectve of ths paper s to evelop the necessary bacgroun for the analyss of statc voltage collapse problem, establshng the concepts of -/Q curves (nose curves). oltage senstvty to reactve power, number of solutons at fferent loa levels, the pont of collapse (oc) an the conton whch occur at voltage collapse n a smple, sngle lne loss-less power system, an then to generalze to a multbus system. The ssues of senstvty of voltage to loa power on the upper an lower sectons of the nose curve an the change of number of solutons as the loa power s change s explore n ths paper. Conser a sngle lne, loss-less system wth constant loa an fxe senng en voltage, E. Ths assumpton of fxe senng en voltage amounts to lmt-less reactve power supply from the generator to eep the generator termnal voltage fxe at E at fferent loa contons. Ths secton wll establsh the nose curves, voltage Senstvty to reactve power, number of solutons at fferent loa levels an the pont of collapse n the two-bus system. ) Solutons for < max ) 1 Soluton at max ) Soluton for > max Fg.1. Lossless Two System The actve an reactve power balance equatons are: c E Sn θ (1) X E Q Q + Cos θ () X X From euqaton (1) E Sn θ X E max X at θ 0 45 Fg.. Dagram (3) (4) The Fg. llustrates the followng eas. a) Senstvty: 1 < 0 Ths shows that an ncrease n at 1 wll cause a ecrease n voltage an a ecrease n wll cause an ncrease n voltage. Ths s an expecte behavor n normal power system operaton an s usually terme as stable soluton, upper branch soluton or hgh voltage soluton.[1, ]. > 0 The senstvty of to at soluton pont s opposte to that at 1. Ths s calle an unstable soluton, lower branch soluton or low voltage soluton. b) Number of Solutons: It coul be seen that the number of solutons varous as s ncrease. It can be vsualze from fgure that there wll be: c) ont of Collapse (o C): The pont ( max, C) on the agram, s the ont of Collapse Ths ncates that voltage s nfntely senstve to at c. Smlar results can be obtane for Q also. ) oltage Stablty Crtera: For a sngle lne networ, the crteron for voltage stablty s state as follows > 0 Q > 0 (5) Where - an Q -Q. The pont (Qmax, C) s the oc on Q agram. The conton for voltage collapse s thus; Q (or) Q 0 (or) 0 Thus an alternate statement for voltage collapse can be gven as: A ower system n steay state s at a pont of voltage collapse f some Δ j s unboune as Q tens to zero.

3 Internatonal ournal of Scentfc & Engneerng Research, olume 5, Issue 4, Aprl OWER FLOW EQUATIONS Usng bus frame of reference, the bus current at th bus s expresse as [3]: I n 1 Y ; For 1,,... n (6) The complex power at th bus * * * S I Y ; For 1,,... n (7) efne e jѳ ; θ θ - θ ; Y G + jb; S S + jq S Then [ G [ Cos(θ θ ) + B sn(θ θ )] (8) 3 OLTAGE COLLASE CONDITION IN MULTI- BUS SYSTEMS The power flow equaton n the case of mult-bus system can be wrtten as (θ,, λ) ((θ, ) S λ * 0 (14) Q(θ,, λ) Q((θ, ) Q S λ Q* 0 (15) Where λ s the loang parameter an * p p an Q * λ λ Lnearzaton of equatons (14) an (15) yels, D p θ + q v Where s gven by λ λ 0 (16) p * λ λ λ (17) q Q * Qλ λ Therefore t gves the recton of loang from equaton (16) [ ] λ (18) θ 1 v λ So λ can be ncrease untl becomes sngular. The pont of voltage collapse therefore correspons to the sngularty of,.e.,et 0 OLTAGE SENSITIITY (x) The lnearse power flow equaton s re-wrtten n a mofe form as: Q (x) G Sn(θ θ ) B sn(θ θ )] Δp θ Δθ For 1,,3, n. (19) Δq Δ qθ Q Defne msmatch vector for real an reactve power as: (x) (x) - S Usng equaton (16) an assumng that there s no angular (9) nstablty,.e., et θ 0 Q(x) Q(x) - Q S (10) If λ s set equal to E, the generator termnal voltage Equatons (9) an (10) can be expresse n compact form as: - QS -1 (Qe - Qθ θ -1 e) E (0) p(x) f (x) q 0 (11) Where Qe an q(x) e E E If λ s set equal to b, the susceptance of SC at the loa bus Soluton of f(x) 0 by Newton Raphson Metho (Q -b ) an Let X Xe + Δx, then by Taylor seres expanson. QS -1 (Qb - Qθ θ -1 b) b - QS -1 Qb b (1) F(x) f(xe +Δx) f(xe)+(xe)δx + hgher orer terms (1) q Choose Δ x such that f(x) 0 an gnorng the hgher orer As pb 0 an G b b b terms of the expanson, Thus, n sngle lne case, n aton to the conton gven by Δx - -1 (xe) f(xe) (13) equaton (5) for voltage stablty, the followng are also to be Where (xe) the acoban evaluate at Xe, Expresse as an satsfe. teratve scheme, the next estmate of state vector s X+1 X + ΔXI, the teratve metho of soluton contnues untl msmatch > 0, > 0 () functons satsfy a pre-etermne tolerance between E b successve teratons. In a mult bus system the suffcent conton for voltage stablty s that all elements of Qs -1, -( Qe - Qθ θ -1 e) an (Qb - Qθ θ -1 b) are postve when there s no angular stablty problem. Angle Senstvty to Change n arameter The relatonshp between voltage senstvty to parameter λ, A smlar relatonshp can be establshe between θ an λ. Usng equaton (15): O θ λ + Q0 Q v qλ λ 0 (3) From equaton (3)

4 Internatonal ournal of Scentfc & Engneerng Research, olume 5, Issue 4, Aprl Q -1 (Qθ θ + Qλ λ) (4) θ - (S -1 (λ Q -1 Qλ) λ (5) If et Q 0, then et 0 mples et S 0 an equaton (5) shows that angular senstvty to parameter λ s nfnte when et 0. Thus t s seen that n general angle an voltage stablty ssues are nterrelate. 4 MODEL ANALYSIS FOR OLTAGE STABILITY STUDIES A ower system s voltage stable, at a gven operatng conton, f for every bus n the system, bus voltage magntue ncreases as reactve power njecton at the same bus s ncrease. A system s voltage unstable f, for at least one bus n the system, bus voltage magntue ecreases as the reactve power njectons at the same bus s ncrease. In other wors, a system s voltage stable f -Q senstvty s postve for every bus an unstable f -Q senstvty s negatve for at least one bus. The lnearse steay state system power voltage equatons are gven by equaton (19) Δ θ Δθ (6) Q θ Qv Δ Where Δ ncremental change n bus real power ncremental change n bus reactve power njecton. Δθ ncremental change n bus voltage angle. Δθ1max maxj 1j, Δ ncremental change n bus voltage magntue. Δθgmax Max g, System voltage stablty s affecte by both & Q. However, at each operatng pont s ept constant an voltage stablty s evaluate by conserng the ncremental relatonshp between Q an. Although ncremental changes n are neglecte n the formulaton, the effects of change n system loa or power transfer levels are taen nto account by stuyng the ncremental relatonshp between Q an at fferent operatng contons. let Δ 0 then, (qv - qθ -1 pθ pv) Δ (7) R Δ An Δ -1 R (8) R s calle the reuce acoban matrx of the system; R s the matrx whch rectly relates the bus voltage magntue an bus reactve power njectons. Elmnatng the real power an angle part from the system steay state equatons allows one to concentrate on the stuy of the reactve eman an supply problem of the system as well as to mnmze computatonal efforts. ARTICIATION FACTORS artcpaton Factor:- The relatve partcpaton of bus n moe s gven by bus partcpaton factor. ζ η Where ncates the contrbuton of the th egen value to the _ Q senstvty at bus. The bgger the value of, the more λ contrbutes n etermnng Q senstvty at bus. For all the small egen values, bus partcpaton factors etermne the areas close to voltage nstablty. partcpaton factors etermne the areas assocate wth each moe. The sze of the bus partcpaton n a gven moe ncates the effectveness of remeal actons apple at that bus n stablzng the moe. The sum of all the bus partcpaton for each moe s equal to unty. Branch artcpatons:- When the change n reactve power njectons n Δ Qm, the resultng voltage varatons s Δm an the moel angle varaton s, Δθm - -1 pθ pv Δm Wth Δ an Δθ nown, the lnearse reactve loss varaton across transmsson branch 1j, 1j, an the lnearse reactve power output varaton at generator g, g, can be calculate, Let, The partcpaton factor of branch j to moe I s efne as, 1j 1j 1max (9) Branch artcpatons thus ncate, for each moe, branches consume the most reactve power for a gven for a ncremental change s reactve loa. Branches wth hgh 1j are those whch cause moe I to be wea. The branch partcpaton thus proves valuable nformaton regarng.. Remeal actons n terms of transmsson branch enhancement an restrbutng the power flow to allevate the loang on that branch,. Crtera for contngency selecton, The partcpaton factor of generator g to moe s efne as, ncremental change n system reactve loang.e g (30) gmax Generator partcpatons prove mportant nformaton s regarng proper strbuton of reactve reserves among all

5 Internatonal ournal of Scentfc & Engneerng Research, olume 5, Issue 4, Aprl machnes n orer to mantan an aequate voltage stablty margn. 5 TEST SYSTEM AND RESULTS A Computer program has been evelope to conuct the moal analyss of the system. The program s teste wth the small system shown n Fg. 3 the test s conucte n such a way that, the operatng ponts are vare by mplementng a step change n reactve power at bus 3 (n steps of 0.05 p.u), startng from 0 p.u an ncrementng untl, the loa flow fals to converge. REFERENCES [1].Kunur, ower Systems stablty an Control, Mc.Graw Hll Inc. Fg.3. Four Two Generator Test System New Yor (Boo). [] C.W. Taylor, ower system voltage stablty, Mc.Graw Hll Internatonal Etons; New Yor, 1994 (Boo) The results obtane at fferent operatng contons are [3] A.R. Bergen, ower System Analyss, rentce Hall Inc. presente n the Table.1. Englewoo Clffs, New ersey 0763, 1986 (Boo) [4] I.A. Hsens, D..Hll, Energy Functons, transent stablty an OERATING OINT voltage behavor n power systems wth non-lnear loas, IEEE TABLE I Moel Analyss Results at Dfferent Operatng onts Q pu Q pu oltage Senstvty oltage Senstvty BUS BUS BUS BUS Egen alue Egen alue Egen alue Egen alue BUS ARTICIIATION BUS ARTICIIATION art. art. art. art BRANCH ARTICIIATION BRANCH ARTICIIATION. art. art. art. art CONCLUSION The funamental am behn ths paper s to evelop some tools whch help n prectng the behavor of a complex system wth respect to system we voltage relate performance as the loang on encounter ther lmts. The thess wor one establshng of effect of reactve power on the voltage stablty of a power system. The wor one s establshe on a sample 4 bus test system. The Egen value analyss metho normally nown as the moal analyss s use for ths stuy. It s observe that as an when the reactve power s ncrease on a Q bus, the mnmum value of the Egen values s ecreasng whch ncates that the stablty of the system s ecreasng an when the system s unstable the Egen value becomes zero. The stuy was supporte by many other factors such as the bus an branch partcpaton factor an voltage senstvty whch gves a measure of the system stablty. Trans on ower Systems, ol.4,.4, Oct 1989, [5].A. Lof, et.al., Fast calculatons of a voltage stablty nex, IEEE Trans. On ower Systems, ol. 7,.1 February, 199. [6] B.Scott, Revew of loa flow calculatons methos, IEEE roceengs ol. 6.7, pp , uly [7] B.Gao, G.K.Marrson,.Kunur, oltage Stablty Evaluaton usng moal Analyss, IEEE Trans an power systems, ol.7,. 4, pp , vember, 199. [8] oltage stablty of power systems: Concepts analytcal tools an nustry experence, Report prepare by IEEE worng group on voltage stablty, 1990(Boo). [9] K.Walve, Moellng of power systems components at severe sturbances, CIGRE report 38-18, ars, August, 7-September,4, [10] W.R.Lachs, oltage Collapse n EH ower Systems, IEEE paper A 78057, 1978 IEEE/ES Wster meetng, New Yor, an/feb [11].Ajarappu an C. Chrsty, The Contnuaton ower Flow: A tool for steay state voltage stablty analyss, IEEE ICA conference proceengs, pp , May, [1] N. Flatabo, R.Ogeneal, an T. Carlsen, oltage Stablty contons n a power transmsson systems calculate by senstvty methos, IEEE Trans. On power systems, ol. 5,.4, pp , vember, [13] Y.Tamura, H.Mor an S. Iwamoto, Relatonshp between voltage nstablty an multple an flow solutons n electrc power

6 Internatonal ournal of Scentfc & Engneerng Research, olume 5, Issue 4, Aprl systems;, IEEE Trans. On power systems. ol. AS10,.5, pp, , May,