POWER FLOW IN COMPLEX FORM BASED LOADABILITY ASSESSMENT

Transcription

1 54 Power flow n complex form base loaablty assessment POWER FLOW I COMPLEX FORM BASED LOADABILITY ASSESSMET PhD Iu. Axente PhD prof. I. Stratan nversty of Western Ontaro Lonon Techncal nversty of Molova ITRODCTIO There was a tme when electrc utltes coul affor bulng oversze transmsson systems. However ths state of affars has apprecably change as a result of power system restructurng an eregulaton. owaays workng uner economcal an envronmental constrants utltes rarely resort to transmsson system expanson. Due to ths an some other factors n the last three ecaes transmsson systems have been operate much closer to ther voltage stablty lmt (or loaablty lmt). Several factors are responsble for ths []: envronmental pressures on transmsson expanson ncrease electrcty consumpton n heavy loa areas (where t s not feasble or economcal to nstall new generatng plants) new system loang patterns ue to the openng up of the electrcty market etc. A number of voltage nstablty ncents have been experence aroun the worl []. As a consequence voltage stablty has become a major concern n power system plannng an operaton. From the power system securty pont of vew knowlege of the crtcal power an voltage s very mportant as the operatng voltage an power at the system noes shoul be kept as far away as possble from ther crtcal values. Onlne montorng of the stablty status s essental when operatng the system near ts loaablty lmt. As expecte sgnfcant research efforts have been evote to construct a more comprehensve unerstanng of voltage stablty ssues an evelop methos for recognzng an resolvng them. Varous loaablty assessment methos have been propose n the lterature []-[6] most of them beng sutable only for offlne system analyss. sually these methos are base on power flow analyss an propertes of the assocate Jacoban matrx. However n a real power system the sze of the problem s so large that the computaton s too tmeconsumng; therefore most of these methos are unsutable for onlne loaablty assessment. Perhaps the smplest loaablty assessment metho s to repeately perform power flow computatons graually ncreasng the loa untl the loaablty lmt s slghtly volate an convergence to any real soluton s no longer possble. However ths metho has an nherent convergence problem n close proxmty of loaablty lmt pont because the power flow Jacoban matrx becomes sngular at ths pont. The contnuaton power flow metho []-[6] overcomes ths problem by reformulatng the power flow equatons so that they reman well-contone at all possble loang contons. Although the contnuaton metho s robust an flexble t s very slow an tme-consumng. A metho base on contnuaton power flow usng moel trees s propose n [7] whch has an ncrease computaton spee. In [8]-[] an [4] the problem of maxmum loaablty lmt etermnaton s formulate as an optmzaton problem. The latter paper proposes a rect nternal pont optmzaton metho whch s very effectve n ealng wth a great number of power flow unsolvable cases when the Jacoban matrx s ll-contone. eural networks base approaches for onlne precton of the closest loaablty margn are propose n [3]-[4]. Another approach for real-tme etermnaton of loaablty lmt successfully nstalle an proven n several Energy Management Systems s presente n [5]. An approach for maxmum loaablty assessment n a probablstc framework s propose n [6]. In ths paper a new approach for loaablty assessment s propose whch s base on power flow n complex form. It s well-known [0] (page 98) that the complex power flow equatons o not meet Cauchy-Remann contons [] (page 35); therefore the ewton-raphson metho cannot be rectly apple n ths case. Conventonally ths problem s overcome by splttng each complex power flow equaton nto two real equatons. The approach propose n ths paper conssts n performng some manpulatons on conventonal complex power flow equatons an as a result obtanng a new system of complex power flow equatons whch meet Cauchy- Remann contons. Thus now the calculaton of ervatves wth respect to complex varables s possble consequently the ewton-raphson metho can be rectly apple to complex power flow equatons. Moreover these new complex power flow equatons have solutons beyon the loaablty lmt

2 Power flow n complex form base loaablty assessment 55 pont whch can be successfully employe n loaablty assessment. An approach for solvng the power flow equatons n complex form has been prevously reporte n [7]-[9] but t nvolves solvng twce as many complex power flow equatons as usual an has some sgnfcant rawbacks scusse n secton V. Also ths paper proposes a new approach for loaablty assessment name parabola approxmaton approach whch s base on some propertes of the solutons obtane beyon the loaablty lmt pont. It conssts n performng three computatons of power flow n complex form beyon the loaablty lmt pont an then base on obtane solutons the magnary part of curve s approxmate by a parabola the constant term of whch s the noe power lmt. Snce only three power flow computatons have to be performe the propose approach s not much tme-consumng an can be successfully apple n onlne loaablty assessment.. POWER FLOW I COMPLEX FORM Frst the concept of Power Flow n Complex Form s explane usng a two-noe system. Then ts applcaton to a mult-noe system s presente. A. Two-noe System For a two-noe system t s possble to analytcally obtan the power flow solutons an etermne the loaablty lmt. Therefore the two-noe system analyss presente below wll be useful for ganng nsght nto the propose concepts. The power flow computaton for a two-noe system shown n Fgure nvolves solvng the followng equaton: ( Y + Y ) S () Alternatvely the followng equaton whch s the conjugate of () can be solve: ( Y + Y ) S () The well known ewton-raphson technque cannot be rectly apple to solve () because t oes not satsfy the Cauchy Remann contons [] (page 35) hence preventng the applcaton of ervatves n complex form [0] (page 98). Therefore the conventonal way of solvng () s to splt t nto two real equatons an solve them smultaneously applyng the ewton-raphson metho. It s well known that () has no solutons beyon the loaablty lmt pont therefore the conventonal power flow verges. Also the ewton- Raphson technque s prone to vergence n the proxmty of loaablty lmt pont (because of sngularty of the Jacoban matrx) although solutons exst. Thus sometmes t s not clear whether the vergence s cause by system overloang or t s ue to mperfectons of the soluton technque use n power flow computaton. Z P + jq Y Y Fgure. Two-noe power system. P + jq In orer to be able to apply the ewton-raphson metho n complex form the followng approach s propose. From () the expresson for calculatng s obtane: Y S + (3) Y Y ow after substtutng (3) n ()() an performng some manpulatons the followng quaratc equaton s obtane: A + B + C 0 (4) Where: A Y Y B Y S Y S Y Y C Y S. Equaton (4) has the followng propertes whch have been entfe through analytcal an numercal nvestgatons: ) It satsfes the Cauchy Remann contons so the ewton-raphson metho n complex form can be apple for solvng t. ) Its scrmnant s a real number whch s postve up to the loaablty lmt pont an negatve beyon t. 3) It has two solutons both up to an beyon the loaablty lmt pont. 4) p to the loaablty lmt one of the solutons correspons to the stable steay-state (upper part of -curve) an the other one to the unstable steay-

3 56 Power flow n complex form base loaablty assessment state (lower part of -curve). 5) In the pont of loaablty lmt the scrmnant s equal to zero an there s only one soluton. 6) Solutons obtane up to the loaablty lmt pont satsfy () an (). 7) A soluton obtane up to the loaablty lmt beng substtute n (3) wll result n ts conjugate. 8) Solutons obtane beyon the loaablty lmt pont mathematcally satsfy the apple moel (equaton (4)) but they cannot physcally exst n a real power system. 9) Any soluton obtane beyon the loaablty lmt pont beng substtute n (3) wll not result n ts conjugate. Thus beyon the loaablty lmt pont calculate wth (3) s not equal to the conjugate of an ths fact can serve as crteron that the power enforce n noe s above the loaablty lmt. The voltage magntue n noe V can be calculate wth the followng formula: V (5) Snce up to the loaablty lmt pont s equal to the conjugate of V s a real number. As mentone above beyon the loaablty lmt pont (calculate wth (3)) s not equal to the conjugate of therefore n ths case V s a complex number. Thus beyon the loaablty lmt pont V has the magnary part an the two solutons obtane for V are complex conjugates. These two solutons obtane beyon the loaablty lmt pont mathematcally satsfy the apple moel but they cannot physcally exst n a real power system. However the fact that V calculate wth (5) s a complex number ncates that the power enforce n noe s above the loaablty lmt. ow let us exemplfy conserng a two-noe system wth the followng parameters: 6 Z j34.7 Ω Y j S 6 V. For a two-noe system the power lmt can be calculate analytcally by equatng the scrmnant of (4) to zero an solvng the resultng equaton for P prove that cosθ (power factor) s known. In the two-noe system uner conseraton assumng that cosθ 0.85 (laggng) the analytcally calculate power lmt n noe s: P lm MW. Stressng the system by graually ncreasng the consumpton n noe from 0 to 00 MW (n steps of 0.MW)whle keepng cosθ const 0.85 (laggng) a seres of solutons for V has been obtane. sng these solutons the curve shown n Fgure has been rawn. From Fgure we can see that up to the loaablty lmt pont the voltage magntue n noe V s a real number (the magnary part s zero). Beyon the loaablty lmt pont V s a complex number (the magnary part s fferent from zero). Also the two solutons obtane for V are complex conjugates. V kv V cr real part magnary part P MW Fgure. curve for the two-noe system uner conseraton. B. Mult-noe System The ea explane n prevous subsecton for a two-noe system s equally applcable to a mult-noe power system. For convenence an ease of explanaton the system noes are numbere as follows: number of -noes; number of -noes; + + total number of noes; -noes; + + -noes; Slack-noe. For each -noe ( ) the followng equaton s wrtten: Y S (6) k k k For each -noe ( + + ) the followng equaton s wrtten: -P lm

4 Power flow n complex form base loaablty assessment 57 k k k Y m + m m Y P (7) Also for each -noe ( ) the followng equaton s wrtten: S Y k k (8) k Fnally for each -noe ( + + ) the followng equaton s wrtten: V (9) Equatons (8) an (9) combne together an presente n matrx form look as follows: Y [ 0 ] Y S I slack (0) [ 0] [ E] [ 0 ] V 0 3 Where: Zero matrxes [ 0 ] [ 0 ] [ 0 ] 0 3 an entty matrx [ E ] have the followng szes: an respectvely. Y Y Y Y Y Y Y Y Y Y S S 0 S 0 0 S Y + Y + Y + Y + Y + Y + Y Y + Y + Y + V V 0 + V 0 0 V Y Y I slack Y Y Y The nverse of matrx s calculate [ 0] [ E] as follows: Y Y Z Z Z () [ 0] [ E] [ 0] [ E] Where: Z Y an Z Y Y. After the left multplcaton of both ses of (0) by Ẑ the followng equaton s obtane: Z Z [ 0 ] [ E] () S [ 0 ] I slack [ 0 ] V 0 3 ow the followng notatons are ntrouce: Z [ 0 ] Z S A A A [ 0] [ E] [ 0] V A A (3) Where: A Z S A Z V A [ 0] an A V. Z Z I slack B B (4) [ 0] [ E] Where B Z I slack. Conserng (3) an (4) () s rewrtten as follows: A A B A 0 A 3 (5) A B From (5) the followng formula s erve for calculatng the conjugate noe voltages:

5 58 Power flow n complex form base loaablty assessment A k + B k + k (6) Substtutng (6) n (6) an (7) the followng equatons are obtane: For -noes ( ): + A k Δ S B Y kk S k k k (7) I S 0 For -noes ( + + ): + A k Δ P B Y kk+ k k k + + A mk + Y m Bm + Y P m (8) k k I + I P 0 Where: I Y kk an I Y mm. k m The Jacoban matrx of the obtane system of nonlnear complex equatons (7) an (8) s: ΔŜ H J (9) ΔP L The elements of the Jacoban matrx are calculate wth (0) () an (). ΔS A j H j Y j I. (0) Where: j j an j + ΔP A L Y I + Y k Ak + I k Where ΔP + j j k k j j j k () L Y Y A () Where: + + j + an j. Voltage correctons an new voltages at the k th teraton are calculate respectvely wth (3) an (4). ΔS H S Δ J Δ (3) [ ΔP] L [ ΔP] ( k) ( k ) ( k) + Δ (4) Moelng of FACTS evces s covere n [9] an [3].. LOADABILITY ASSESSMET SIG PARABOLA APPROXIMATIO a. Parabola Approxmaton Technque Examnng the magnary part of the curve shown n Fgure t can be observe that beyon the loaablty lmt pont ths curve looks lke a parabola. Therefore t s propose here to approxmate ths curve wth the followng secon-orer polynomal: P a mag ( V) + a mag ( V) + a0 (5) In orer to etermne the polynomal coeffcents a a an a 0 t s suffcent to have nformaton on three ponts from the magnary part of curve. Ths nformaton can be obtane by performng three power flow computatons n complex form for three fferent P beyon the loaablty lmt pont. Then the polynomal coeffcents are etermne solvng the followng set of lnear equatons: () () mag( V ) mag( V ) () a P ( ) ( ) ( ) mag( V ) mag( V ) a P (6) ( 3 a ) ( 3) ( 3) 0 P mag( V ) mag( V ) Where the upper nex enclose by parentheses s the pont number: an 3. Actually only a 0 s of nterest here so only ths polynomal coeffcent has to be calculate. It can be easly observe that the constant term a 0 of the approxmaton polynomal (5) correspons to the lmt power P lm so generally a0 P lm. Thus performng only three power-flow computatons beyon the loaablty lmt pont we can estmate the loaablty lmt. b. umercal results for a two-noe system The same two-noe system as escrbe n secton III has been use here for exemplfyng the above propose parabola approxmaton technque. umercal results for fve loaablty assessment case stues are presente n Table.

6 Power flow n complex form base loaablty assessment 59 Table. Loaablty assessment usng parabola approxmaton : two-noe system case stuy GS GS Case no Power flow results Pont P no. Imag(V ) MW kv Estmate P lm MW Error % e e e e e-0 Base on results presente n Table the followng conclusons can be nferre for a two-noe system: ) The loaablty lmt etermne usng parabola approxmaton technque can be consere as one precsely calculate. ) The accuracy of the loaablty lmt etermne usng parabola approxmaton approach practcally oes not epen on postons of those three ponts use for calculatng the parabola coeffcents. 3) Therefore not only the magnary part of curve looks lke a parabola but t s actually a parabola. c. umercal results for a mult-noe system The sngle-lne agram of a three-machne nnebus meshe system uner conseraton s shown n Fgure 3. The power flow results are presente respectvely n Table an Table. All the values are n per unt on 00 MW base. The buses are numerate n the followng orer: frst -noes then -noes an fnally the slack-bus. The smulatons have been carre out usng Matlab software. A program has been wrtten n Matlab to mplement the power flow n complex form technque presente n secton III. 6 Fgure 3. Three-machne nne-bus meshe system Transmsson lne ata an bus ata together wth Table. Transmsson lne ata for the system n fg.3 From Bus umber To Bus umber GS Seres Resstanc e R s (pu) 5 9 Seres Reactanc e X s (pu) 4 Shunt Susceptanc e B/ (pu) Table 3. Bus ata an power flow results for the system n Fgure 3 Bus Voltage P G Q G P L Q L Typ Magnt Angle o. (pu) (pu) (pu) (pu) e. (pu) (eg.) Slack oe has been chosen for loaablty assessment. The system has been stresse by graually ncreasng the loa n noe (from 0 to 8 pu n steps of 0.00 pu) whle keepng the power factor constant. The lower part of curve has been bult up by movng n the opposte recton (ecreasng P L from 8 to 0 pu). A seres of power flow computatons usng the above mentone program have been performe n orer to bul up the curve for noe. The frst power flow has been compute usng flat start. The subsequent

7 60 Power flow n complex form base loaablty assessment power flow computatons have been performe startng wth the preceng power flow soluton. The resultng curve s shown n Fg. 4. The followng conclusons can be rawn from Fg. 4: ) The magnary part of V s zero for P L P lm an s fferent from zero beyon the loaablty lmt pont ( P L > ). P lm ) Lke n two-noe system case beyon the loaablty lmt pont there are two complex conjugate solutons. 3) Therefore the magnary part of curve s symmetrc about the P-axs an looks lke a parabola. 4) Thus the parabola approxmaton technque propose above can also be apple to mult-noe systems. V pu cr P L pu P Llm 4.67 Real part Imagnary part Fg. 4. curve for bus of mult-noe system The parabola approxmaton technque has been apple to the mult-noe system uner conseraton (shown n Fgure 3) for estmatng the loaablty lmt n noe. umercal results for seven loaablty assessment case stues usng parabola approxmaton technque are presente n Table 4. Base on results presente n Error! ot a val bookmark self-reference. the followng conclusons can be nferre for a mult-noe system: ) The accuracy of the loaablty lmt etermne usng parabola approxmaton technque s qute satsfactory. The error s below 5%. ) The accuracy slghtly epens on postons of those three ponts use for calculatng the parabola coeffcents. A better precson s acheve when at least one pont s stuate close to the loaablty lmt pont. Table 4. Loaablty assessment usng parabola approxmaton: nne-noe meshe system case stuy Case no Power flow results Pont Imag(V no. P L pu ) pu Estmate P Llm pu Estmaton error % ) Thus performng only three power flow computatons n complex form beyon the loaablty lmt pont we can estmate the loaablty lmt wth an error below 5%. 4) In orer to obtan a better accuracy the same proceure has to be repeate. Ths tme the postons of those three ponts use for calculatng the parabola coeffcents are chosen close to the loaablty lmt obtane n prevous step. Performng two steps of parabola approxmaton the loaablty lmt can be etermne very accurately (error below 0.5%). 3. SPARSITY COVERGECE AD ILL-CODITIOIG ISSES a. Sparsty It was mentone earler that an approach for solvng the power flow equatons n complex form has been prevously reporte n [7]-[9]. Ths metho has the avantage that the sparsty of the Jacoban matrx s preserve whch s a key avantage n terms of computng tme an memory storage n case of large power systems. Ths avantage s obvous when performng power flow computatons up to the loaablty lmt pont for large power systems. However the metho presente n [7]-[9] has also savantages. The problems wth ths metho appear when performng power flow computatons beyon the loaablty lmt pont. In

8 Power flow n complex form base loaablty assessment 6 ths case the computaton tme ncreases conserably ue to the ncrease number of teratons. The number of teratons epens on ntal voltage values mpose at the begnnng of ewton-raphson algorthm. Beyon the loaablty lmt pont t s very ffcult to prect the ntal voltage values that wll ensure a small number of teratons especally for large power systems. Therefore t s practcal to use the flat start whch results n an ncrease number of teratons wth ths metho. Smulatons show that the number of teratons ncreases up to 70 an more. Thus although the computaton tme of a sngle teraton s reuce ue to sparsty of the Jacoban matrx the overall computaton tme ncreases conserably because of an ncrease number of teratons. The metho propose n ths paper oes not have the above mentone savantage. When performng power flow computatons beyon the loaablty lmt pont usng the metho propose n ths paper an flat start the convergence s ensure n 7-9 teratons. Although the computaton tme of a sngle teraton s ncrease comparatve wth the metho presente n [7]-[9] the overall computaton tme s reuce because of a small number of teratons. As regars to memory storage requrements nowaays the computer memory s not an ssue. b. Convergence Another sgnfcant savantage of the metho presente n [7]-[9] s that t s prone to convergence towars unwante solutons when performng power flow computatons beyon the loaablty lmt pont. Ths metho s very senstve to ntal voltage values mpose at the begnnng of ewton-raphson algorthm. In orer to rect ths metho to converge towars the approprate solutons the ntal voltage values have to be chosen very carefully that n case of large power systems s not a trval ssue an t s also tme consumng. Otherwse napproprate solutons can be obtane whch are useless (cannot be use for loaablty lmt etermnaton). The power flow n complex form algorthm propose n ths paper relably converges towars the approprate solutons from flat start an there s no nee n guessng the ntal voltage values. c. Ill-contonng Although the metho propose n ths paper has the savantage that the sparsty (nherent n the orgnal set of power flow equatons) s estroye the Jacoban matrx s better contone comparatve wth the metho presente n [7]-[9]. Ths consttutes an avantage when performng power flow computatons n proxmty of the loaablty lmt pont where the Jacoban matrx s llcontone but also beyon the loaablty lmt pont. The problem of ll-contonng usually appears n proxmty of the loaablty lmt pont. The loaablty lmt assessment approach propose n ths paper s base on complex power flow computatons beyon the loaablty lmt pont where the Jacoban matrx s better contone than n ts proxmty. umerous smulatons have been carre out usng the metho propose n ths paper an no convergence problems have been encountere.. Comparatve analyss A comparatve stuy has been performe usng the IEEE 300 Bus Power Flow Test Case [] The smulatons have been carre out on a Sony VG- W0D laptop usng Matlab. In orer the comparson to make sense the metho propose n ths paper an that presente n [7]-[9] have been teste uner absolutely entcal contons. All the power flow computatons have been performe usng flat start. The solutons beyon the loaablty lmt pont have been obtane by mposng a loa of 9000 MW n noe all the other power flow ata remanng unchange. The smulaton results are presente n Tabel 5. Table 5. Comparatve analyss: IEEE 300 bus power flow test case Metho Presente n [7] [9] Power flow conton Computaton tme secons umber of teratons Conton number Convergence p to the loaablty lmt pont Beyon the loaablty lmt pont Propose n the present paper p to the Beyon the loaablty loaablty lmt pont lmt pont Converge to approprate solutons Converge to unwante solutons Converge to approprate solutons Converge to approprate solutons From Table 5 t can be seen that the metho propose n ths paper has a better performance comparatve wth the metho presente n [7]-[9] when the power flow computatons are carre out beyon the loaablty lmt pont. To be mentone

9 6 Power flow n complex form base loaablty assessment that the loaablty lmt assessment approach propose n ths paper s base on complex power flow computatons beyon the loaablty lmt pont. sng for ths purpose the metho presente n [7]- [9] can lea to erroneous results an ncrease computaton tme ue to rawbacks explane earler. 4. OMECLATRE complex voltage at bus an ts conjugate * ; V voltage magntue at bus ; S Ŝ complex power at bus an ts conjugate; P actve power at bus ; Y j Y element from row an column j of the j bus amttance matrx an ts conjugate. * s the conjugate of only up to the loaablty lmt pont. Beyon t ths oes not hol true an shoul be treate as an nepenent varable. COCLSIOS Ths paper proposes an approach for solvng the power flow equatons n complex form.e. wthout resortng to splttng each complex equaton nto two real equatons. The conventonal complex power flow equatons are mofe to meet the Cauchy-Remann contons an then are solve n complex numbers by applyng the ewton-raphson technque for complex varables (complex noe voltages). The propose power flow n complex form converges beyon the loaablty lmt pont where the conventonal power flow verges. Although the solutons obtane beyon the loaablty lmt pont are fcttous ones they can be employe n loaablty assessment. Beyon the loaablty lmt pont the noe voltage magntues calculate wth (5) are complex numbers ncatng that the power enforce n one or more noes s above the loaablty lmt. Also beyon the loaablty lmt pont the solutons obtane for noe voltage magntues appear n complex conjugate pars. Snce beyon the loaablty lmt pont the noe voltage magntue s a complex number the curve conssts of two parts: real an magnary (see Fgure an Fg. 4). p to the loaablty lmt pont the magnary part of curve s zero. Beyon the loaablty lmt pont the magnary part of curve s fferent from zero an t looks lke a parabola. Therefore t s propose to approxmate the magnary part of curve wth a parabola. The constant term of the approxmaton polynomal correspons to the lmt power. sng the parabola approxmaton approach propose n ths paper the loaablty lmt can be etermne after performng only three computatons of power flow n complex form. Therefore we fn ths technque sutable for onlne loaablty assessment. The accuracy of the loaablty lmt etermne usng parabola approxmaton technque slghtly epens on postons of those three ponts use for calculatng the parabola coeffcents but t s qute satsfactory (typcally the error s below 5%). The precson s hgher when at least one pont s stuate close to the loaablty lmt pont. References. T. Van Cutsem C. Vournas. Voltage Stablty of Electrc Power Systems. orwell: Kluwer Acaemc Publshers C. W. Taylor. Power System Voltage Stablty (Internatonal Eton). ew York: McGraw-Hll P. Kunur. Power System Stablty an Control. ew York: McGraw-Hll C. A. Canzares F. L. Alvarao. Pont of collapse an contnuaton methos for large AC/DC systems IEEE Trans. on Power Systems vol. 8 no. pp V. Ajjarapu C. Chrsty. The contnuaton power flow: a tool for steay stage voltage stablty analyss IEEE Trans. on Power Systems vol. 7 no. pp K. Iba H. Suzuk M. Egawa T. Watanabe. Calculaton of crtcal loang conton wth nose curve usng homotopy contnuaton metho IEEE Trans. on Power Systems vol. 6 no. pp P. Patar J. Sharma Loaablty margn estmaton of power system usng moel trees n Proc. of 006 IEEE Power Ina Conf. ew Delh Ina G. D. Irsarr X. Wang J. Tong S. Mokhtar Maxmum loaablty of power systems usng nteror pont nonlnear optmzaton metho" IEEE Trans.

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