A Modified Approach for Continuation Power Flow

Transcription

1 212, TextRoad Publcaton ISSN Journal of Basc and Appled Scentfc Research A Modfed Approach for Contnuaton Power Flow M. Beragh*, A.Rab*, S. Mobaeen*, H. Ghorban* *Department of Engneerng, Abhar Branch, Islamc Azad Unversty, Abhar, Iran 1. Nomenclature x ABSTRACT One of the conventonal statc voltage stablty analyss methods s PV curve, whch s obtaned by contnuaton power flow (CPF) n regular manner. Ths method s robust; however has some weakness n large electrc power system consderng generators reactve power lmts. Ths paper takes the advantage of a predctor/corrector scheme to obtan generators reactve lmt httng ponts as well as determnng the type of bfurcaton (saddle node or lmt nduced), and then uses these as the gven data n contnuaton power flow algorthm. Ths wll elmnate the weakness of contnuaton power flow n handlng generators reactve lmts. The proposed method s tested on IEEE 118 bus system trough numercal examnaton. Results showed good performance and robustness of the proposed method. Keywords: voltage stablty, saddle node bfurcaton, lmts nduced bfurcaton, contnuaton power flow, generators reactve power lmt. Loadng Parameter State varables K L Varaton Coeffcent of load at bus K Gj Varaton Coeffcent of generaton at bus j P L Base actve load at bus Q L Base reactve load at bus P Gj Base actve generaton at bus j P L ( ) Actve load of bus as a functon of Q L ( ) Reactve load of bus as a functon of P Gj ( ) Actve generaton of bus j as a functon of F(x, ) Power flow equatons set Maxmum reactve power of the generator G max Q G s V G AVR set pont of the generator G V G Termnal voltage of the generator G Operatng pont z z The k th CEP n the stable operatng regon k I k The nterval whch s correspondng to z k F (x, ) Equalty constrant set n k G (x, ) Inequalty constrant set n I k G ( k) g (x, ) The th element of (x, ) x (x, ) Dervatve of F ( k) (x, ) wth respect to x F F ( k) (x, ) Dervatve of F ( k) (x, ) wth respect to n Prmary correcton teraton pc n Secondary correcton teraton sc 2. INTRODUCTION Voltage stablty s a very complex subject that has been challengng power system engneers n the past two decades [1]. Voltage collapse typcally occurs n power systems whch are heavly loaded. It assocates wth reactve *Correspondng Author: S. Mobaeen, Department of Engneerng, Abhar Branch, Islamc Azad Unversty, Abhar, Iran, E-mal: mobaeen@gmal.com, Tel:

2 Beragh et al., 212 power demands not beng met because of lmtatons on the producton or transmsson of reactve power and s usually ntated by 1) a contnuous load ncrease and/or, 2) a major change n network topology resultng from a crtcal contngency [2]. A perfect remedal acton to prevent voltage collapse requres an effectve voltage stablty assessment method as well as a good offlne system plannng process. A wde varety of modelng prncples of computaton and control methods have been developed for power system voltage stablty assessment and control. The research s manly based on analytcal methods such as statc analyss and dynamc smulaton. Statc studes such as P-V curves have been used for many years. The conventonal method to obtan P-V curves s contnuaton power flow (CPF). Generators reactve power lmts are a key factor n voltage nstablty. Therefore, t s necessary to consder the reactve capablty of generators n voltage stablty analyss of a power system. In a large power system wth many generators, consderng generators reactve power lmts may leads to several dffcultes n usng the conventonal CPF method, e.g., f some of generators ht ther reactve power capablty lmts n the predcton step, then the correcton step may not act properly n determnng exactly whch generator and by whch loadng parameter has ht the lmt. By usng teratve step length selecton or very small step length n the predcton step, ths weakness s elmnated; however, computaton tme wll become very longer. Relatng to generators reactve power lmt ponts, an nterestng method frst had been proposed n [3-4] and then has been extended n [5], where the basc dea to obtan generators reactve power lmt httng pont (whch s named as constrant exchange pont or CEP n those papers) s a predctor/corrector scheme. The reference [5-6] demonstrated the computaton of CEPs, stablty judgment of the obtaned operatng condton, and the type of nstablty. However, some key ssues such as correcton of predcton error whch leads to an unrealstc CEP, and handlng the dvergence of correcton step whch may occurred because of predcton error near SNB pont were not yet solved. The term predcton error s defned later. In addton to above mentoned problems, n order to dentfy the SNB, the pont of collapse (PoC) [7-9] or optmzaton method must be used together wth proposed method n that paper. Ths paper frst extends the prevous method n [5] and resolves above problems. Proposed predctor scheme s almost the same as proposed one n [5], except defnng a prorty to ht lmt lst (PHL lst) for generators n predcton of each CEP. However, the corrector s dvded nto prmary and secondary correcton steps n ths paper. The proposed correcton algorthm s robust and fast n handlng the predctor error as well as the occurrence of dvergence. Obtaned CEPs by the proposed predctor/corrector scheme are then used as the gven data n contnuaton power flow. The combnaton of these two methods (predctor/corrector scheme and contnuaton power flow) s named as modfed contnuaton power flow (MCPF) n ths paper. Ths wll elmnate the weakness of CPF n a large power system wth multple generators and hence multple reactve power constrants. In addton, nstabltes of SNB or LIB type are exactly recognzed by MCPF method and there wll not be any need for other methods such as PoC or optmzaton based methods. The organzaton of the paper s as follow: Secton 3 ntroduces the contnuaton power flow, proposed predctor/corrector scheme, and MCPF approach prncples. The effectveness and performance of the MCPF s demonstrated by numercal examnaton trough IEEE 118 bus system n Secton 4. Results show good performance and robustness of the algorthm. Fnally, conclusons are drawn n Secton Proposed Method 3.1 Contnuaton Power Flow (CPF) The algorthm of the contnuaton method smply consders a set of nonlnear equatons ncludng one or more parameters. In order to use contnuaton method, the power flow equatons must become reformulated to nclude a load parameter by usng the generaton and load varaton scenaros whch s represented n (1)-(3), whle equaton (4) stands for power flow equatons. The power flow equaton n ths technque s solved for contnuous changes n load parameter usng a predctor/corrector scheme that remans well condtonedat and around crtcal pont, and s named bfurcaton pont n the lterature. PL ( ) (1.K L ). PL (1) Q L ( ) (1.K L ). Q L (2) P ( ) (1.K ). P (3) Gj Gj Gj F(x, ) (4) Fg.1 llustrates the teratve process of a typcal CPF method. The upper porton of the curve s the stable regon, whle the lower porton s the unstable regon. The algorthm starts from a known soluton A whch corresponds to the power flow soluton at the current operatng pont. Then, t takes advantage of a predctor to estmate a soluton 1547

3 B correspondng to an ncreased value of the load parameter, and fnally t uses a corrector to fnd the exact soluton C by means of the Newton-Raphson technque. The advantage of CPF method s that addtonal nformaton regardng the behavor of some system varables can be obtaned durng the soluton. Detals of the contnuaton power flow approach are dscussed n [1]. Although CPF algorthm s very robust, t s computatonally expensve for a large system wth multple constrants such as generators reactve power lmts. c Fg.1: Predctor-corrector scheme n the CPF method 3.2 Predctor-corrector scheme for obtanng generators reactve power lmt ponts The man objectve n ths approach s to obtan generators reactve power lmt ponts usng a predctor/corrector scheme. Generally the constrant of a generator vares dependng on the value of loadng parameter. For each generator of a power system, a par of equalty and nequalty constrants exsts and ther roles are exchanged to each other when the generator reaches reactve power capablty lmt. Therefore, the pont at whch a generator hts ts reactve power capablty lmt s named constrant exchange pont (CEP). The par of equalty and nequalty constrants for a generator before and after the CEP s represented n (5) and (6) respectvely. s v VG VG max (5) q QG QG s v VG VG max (6) q QG QG The power flow soluton s obtaned under a set of equalty and nequalty constrants for a specfed load parameter as follow: N1 N F(x, ) F : R R (7) G(x, ) G : R N1 R M (8) For a gven load parameter, the soluton of (7) and (8), f exsts, wll be unque. In a mult machne system there are several CEPs each of whch corresponds to the reactve power lmt of a specfc generator. Consderng z (x, ) as a stable operatng pont whch s shown n Fg.2, the CEPs are expressed as z 1, z 2,, n order along the stable operatng regon of the power system. On the other hand, the ntervals whch are dvded by CEPs are defned as I 1, I 2,..., such that the constrant exchange pont z k belongs to the nterval I k. At the nterval I k the equalty and nequalty constrants wll be expressed as follow: N1 N F (x, ) F : R R (9) G (x, ) G : R N1 R M (1) Startng from an operatng pont n the nterval I k, the goal s to fnd the mnmum loadng parameter for a specfc load and generaton varaton scenaros so that at least one of the nequalty constrants wll be exchanged to an equalty constrant. At ths pont, whch s denoted as zk 1, the equalty constrant would be as follow: F k ) h ( (x, ) (11) mn{g (x, )} (12) 1548

4 Beragh et al., 212 (k Where g ) (x, ) represents the th element of nequalty set G ( k) (x, ) at the nterval I k and equalty constrant. z z 1 z2 I h stands for new I 1 I 2 I( m1) z m I m 1 2 Fg.2: CEPs and correspondng ntervals m c If we show the dfferental change of nequalty and equalty constrants at the pont zk 1 wth h and h respectvely, then the stablty of the new CEP, zk 1 wll be determned accordng to table 1 [5]. Each CEP may be stable (n the upper porton of P-V Curve), unstable (n the lower porton of P-V curve), or crtcal (boundary between the stable and unstable portons). The crtcal CEP, f exsts, corresponds to the lmt nduced bfurcaton n power system. In table 1, h s obtaned usng equatons (13) and (14), assumng that s postve. Also, equatons (15) and (16) are applcable to calculate h for. x F x (x, ) F (x, ) h zk1 (13) x h zk1 (14) (k1) x F (x, ) zk1 (15) x h zk1 (16) h x (k1) F x (x, ) h h x Table 1: dentfcaton the type of a CEP CEP Type ( k ) h ( k ) h Stable Crtcal (LIB) Unstable Predcton of the next CEP and prortzaton of generators Startng from an operatng pont n the nterval I k, the nearest CEP s frst estmated by a predctor. Assume that an operatng pont has been obtaned n the nterval I k as z, whch may or may not be a CEP. The frst step s to calculate the tangent vector at ths pont by the followng equaton: x F 1 x F. R (17) 1 z In equaton (1) the notaton z * mples the evaluaton of * at the pont z.the tangent vector s denoted as z ( x, ), whch has become normalzed. Then, the locus of the soluton of equaton (9) s approxmated usng tangent vector as follow: z z s. z s R (18) g Let the th element of nequalty constrant be shown as g (x, ). Then, by usng equaton (18) the element (x, ) wll be approxmated as follow: 1549

5 ( k ) ( k ) ( k ) ( k ) g g x g ( x, ) g ( x, ). s. x (19) z The value of s whch makes the above equaton null s represented by s and s calculated as: g (x, ) s (2) g g x. x z Above s s defned for all the elements of G ( k) (x, ). The predcted loadng parameter whch corresponds to reactve power lmt pont of the th generator s shown by and could be obtaned as follow: s. (21) ( k ) g g 1 g 3 g Fg. 3: reactve power reserve as a functon of loadng parameter It s expected that generators wth lower values of (or s ) ht the reactve power lmt before those wth hgher (or s ). Thus, for each generator, an ndex called prorty to ht lmt (PHL) s defned and generators are sorted accordng to ths ndex. A generator wth lower, has a hgher PHL and vce versa. The sorted lst s named PHL lst and the frst generator n PHL lst, s estmated to be the correspondng CEP for the next nterval. Fg. 4: proposed corrector scheme Correcton of the predcted CEP Snce the relatonshp between voltage and reactve power s extremely nonlnear, the lnear predcton may lead to error n dentfcaton of whch generators are assocated wth the CEP, e.g., varaton of reactve power reserves of 155

6 Beragh et al., 212 three generators as a functon of loadng parameter s shown n Fg.3. At ntal pont whch s depcted by, t s predcted that the frst generator corresponds to CEP, however the thrd one reaches ts reactve capablty lmt before others. We refer to ths as predcton error n ths paper. To overcome predcton error, secondary corrector step may be needed. The secondary corrector step s defned and explaned n Fg.4. The proposed method for secondary corrector step contans two correcton loops: 1) The power flow equatons are converged: n ths case, f there s one or more generators n the obtaned CEP by prmary corrector step whch are volatng ther reactve power lmts, then the PHL lst s updated and only these generators are sorted n PHL lst and the equatons (22) and (23) are solved agan. 2) The power flow equatons are not converged: dvergence of power flow equatons n obtanng the predcted CEP s caused by sngularty n Jacoban matrx. In ths case, updatng of PHL lst s done by deleton of the frst element and the equatons (22) and (23) are solved agan. If the updated PHL lst becomes empty, then there s no more CEP n orderalong stable operatng regon Computatonal procedure of CEPs: The procedure of the proposed method n order to obtan all CEPs n the stable operatng regon s summarzed as follow: Step 1) (Intal computaton) Determne ntal values for state varable by solvng power flow equatons at loadng parameter. Step 2) (Computaton of the tangent vector) Compute tangent vector usng equaton (16). If then the pont s SNB and stop, otherwse go to step 3. Step 3) (Assessment of nequalty constrants) In the nequalty constrant set, f h whch s obtaned by equaton (12), fulflls n the condton h, Step 4) Step 5) Step 6) Step 7) Step 8) then the pont s CEP and go to step 4, otherwse go to step 5. (Determnng the type of the CEP) Determne the type of the obtaned CEP accordng to table 1. If the CEP s stable then go to step 5, otherwse go to step 7. (Estmatng the next CEP) Calculate s and z m usng equaton (2) and (24) respectvely, then obtan PHL lst for the next CEP and Go to step 6. (Correcton of the estmated CEP) Compute the exact CEP usng corrector scheme n fgure 5. If CEP exsts go to step 2, otherwse go to step 8. (Identfcaton of crtcal CEP) If the CEP s crtcal then the type of nstablty s LIB and stop. Otherwse go to step 8. (SNB detecton) If the CEP doesn t exst or s n the unstable operatng regon then the type of nstablty s SNB and stop. If the algorthm reaches step 8, t ndcates the exstence of SNB. In ths step the obtaned CEP s unstable or the computaton of correcton step s dverged. The dvergence of the soluton s newly consdered and we demonstrated that n some stuatons where there s not any unstable CEP near the crtcal pont, t leads to nablty of computaton of unstable CEPs by predctor/corrector scheme. However, the am here s to dentfy all of the CEPs n the stable operatng regon and also the type of nstablty whch s properly done by proposed predctor/corrector scheme. 3.3 Modfed Contnuaton Power Flow (MCPF) As mentoned before consderng generators reactve power lmts n a large electrc power system may lead to several dffcultes n usng the conventonal CPF method. The proposed predctor/corrector based method calculates CEPs more accurate and faster than the CPF approach. Then, the obtaned CEPs are used as known solutons n the CPF method n order to trace P-V curves and voltage stablty analyss. Ths elmnates the weakness of contnuaton power flow n consderng the reactve power lmt of generators. Ths method s called modfed contnuaton power flow (MCPF) method n ths paper. Startng from a known soluton whch corresponds to the current operatng condton n the nterval I the procedure s summarzed as follow: Step 1) (Identfcaton of CEPs and type of nstablty) Frst obtan CEPs usng predctor/corrector scheme. The type of nstablty s also dentfed n ths step. Step 2) (Prmary determnaton of the ntervals) 1551

7 Step 3) Step 4) Step 5) Step 6) Step 7) Step 8) Defne current and the next nterval by ncreasng loadng parameter. Prmarly the ntervals I and I 1 s selected as current and next ntervals whch s shown by I C and I N respectvely. (Predcton usng the CPF approach) Approxmate the next soluton n the current nterval usng the predctor of the contnuaton power flow approach. (Interval dentfcaton) Indentfy the nterval to whch the approxmated soluton belongs. If the predcted soluton doesn t belong to the current nterval go to step 6, otherwse go to step 5. (Correcton of the approxmated soluton) Correct the approxmated soluton to obtan exact soluton usng Newton-Raphson method, go to step 8. (Selectng the CEP as the soluton) Select the CEP whch corresponds to the next nterval as the exact soluton. Set the next nterval as the current nterval, go to step 7. (Updatng the equatons) Exchange the role of equalty and nequalty constrants of the correspondng generator to each other, go to step 8. (Evaluaton of stoppng rule) If the sgn of loadng parameter n CPF method becomes negatve, then stop otherwse go to step 3. The computaton process conssts of CEPs dentfcaton and the contnuaton power flow. The computaton tme for CEPs dentfcaton s almost equvalent to 1 n ).m. T, where T stands for sngle teraton tme of the ordnary power ( pc flow computaton,m equals to overall number of CEPs along the stable operatng regon, and npc s average of overall prmary correcton teraton step. The computaton tme for the contnuaton power flow depends on the step length; however no step length reducton (whch would be n an teratve manner) s needed for obtanng generators reactve power lmt ponts, also these ponts are computed more accurately and faster. Ths mproves the performance of contnuaton of power flow n a large power system analyss. 4. Case Study 4.1 Study system A modfed IEEE 118 bus system s used to test the proposed method. In ths system, 54 buses nclude generators (PV bus) and 64 buses are load buses (PQ bus). The total actve and reactve power n operatng pont s 4,242 (MW) and 1,436 (MVar) respectvely and the 69 th bus s consdered as the reference bus. In the load varaton scenaro, the actve and reactve power of each PQ bus s ncreased proportonal to the load of operatng pont. However, the loads of PV are assumed to be fxed. Also, generaton varaton scenaro s defned so that all generators except the reference one provde the load varaton proportonal to ther operatng pont generatons and the reference generator compensates the losses. In other words, the load varaton coeffcent K of actve and reactve power for PQ and PV buses equals to 1 and respectvely. Where, the generaton varaton coeffcent K G generators except the reference, and calculated from the equaton (22). K Gn PQ K jref L P.P Gj D n PV (22) L s the same for all 4.2 The computaton of CEPs The obtaned CEPs of IEEE 118 bus system n normal operatng condton, are lsted n table 2. In the developed program, the maxmum teraton of both prmary and secondary correcton steps equals to 5. As t can be seen, the computaton of CEPs by predctor/corrector scheme s fast and accurate. The average teraton of overall prmary correcton step s 1.72 and the teraton of secondary correcton for all CEPs, except the 32 th CEP, equals to 1. In computaton of the 32 th CEP, the secondary correcton step ncludes 2 teratons, because of the error of the predctor n estmatng the correspondng generator for ths CEP. The frst fve elements of the PHL lst for estmatng the 32 th CEP s shown n table 3. The generator of the bus 113 s frst estmated to be CEP. After prmary correcton the generators of buses 66 and 1 volate ther reactve power lmts. Thus n secondary correcton step, the generator of bus 66 s estmated as CEP and then the prmary correcton confrms the recttude of ths predcton. In table 4, the PHL lst for CEP 33 s shown. The generator of bus 1 s estmated as CEP whch s accepted after the prmary correcton step. Ths pont s the last CEP n the stable operatng 1552

8 Beragh et al., 212 regon whch s dentfed as LIB pont or crtcal CEP n table 2. In other words, the type of nstablty n normal operatng condton for the consdered load and generaton varaton scenaro would be the lmt nduced bfurcaton and occurs n total loadng of 7,85 MW. The computed CEPs for contngency condton, where two lnes are faulted, are lsted n table 5. It s assumed that the lne between buses 44 and 45 and also the lnes from bus 15 to 17 are faulted. The type of nstablty n ths condton wll be a saddle node bfurcaton (whch s dentfed by proposed corrector scheme) and the last CEP n the stable operatng regon corresponds to the generator of the bus 14. The average of overall teraton of prmary correcton step n the consdered contngency condton wll be equal to The reason for ths ncrease n comparson to the table 3 s the exstence of 7 and 25 teraton n computaton of the CEPs 24 and 25 respectvely. Table 2: Obtaned CEPs n the normal operatng condton No. P total (MW ) Generator Bus CEP Type n sc n pc 1 4, Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , (LIB) 1 2 Average

9 Table 3: PHL lst for the 32 th CEP n normal operatng condton Rank Generator Bus Table 4: PHL lst for the 33 th CEP n normal operatng condton Rank Generator Bus Table 5: Obtaned CEPs n the consdered contngency condton No. P total (MW ) Generator Bus CEP Type n sc n pc 1 4, Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable , Stable Stable SNB 5 25 Average The generator of bus 55 has the hghest prorty n the PHL lst of CEP 24 whch s shown n table 6. By selectng ths generator, the prmary correcton step leads to dvergence of soluton. Then n the secondary correcton step, the 1554

10 Beragh et al., 212 generator of bus 14 wll be selected and prmary correcton confrms that. After dentfcaton of the 24 th CEP, the PHL lst for CEP 25 wll be accordng to table 7. Generators of buses 55, 4, 32, 11, and 113 have the frst to ffth prorty respectvely. Choosng each of these generators as CEP and performng prmary correcton leads to dvergence of the soluton and t s concluded that there s no more CEP n the stable operatng regon. In other words the type of nstablty would be saddle node bfurcaton. As explaned above the man reason for the ncrease n the average teraton of prmary and secondary correcton steps s the dvergence of computaton n CEPs 24 and 25. Our experence showed that the maxmum obtaned teraton of secondary correcton would be 2. Also the teraton of the prmary correcton step for a good predcted CEP does not exceed 3 teratons. Neglectng the teraton of computng the 25 th CEP (whch does not exst), the average teraton of overall prmary correcton step would be equal to 2.4. Also the computaton tme of CEPs for consdered contngency condton s almost equals to 73 T ( (1 2.4) 24 T) where T s the computaton tme of an ordnary power flow equatons. Table 6: PHL lst for the 24 th CEP n the consdered contngency condton Rank Generator Bus Rank Table 7: PHL lst for the 25 th CEP n the consdered contngency condton Generator Bus Fg. 5: PV Curves n normal operatng condton 4.3 Tracng P-V curves usng MCPF Usng MCPF algorthm, traced P-V curves for normal and consdered contngency condtons are shown n fgures 5 and 6 respectvely. In fgure 5, the voltage versus loadng parameter for buses 3 and 1 s shown. Both of these buses are PV, at whch the crtcal CEP corresponds to the generator of bus 1. As t can be seen from fgure 5, n normal operatng condtons the type of nstablty s LIB whch occurs n total loadng of 7,85 and ths s n accordance wth lsted CEPs of table 2. The voltages of buses 43 and 44, whch are both PQ buses, have been shown n fgure 6. In the consdered contngency condton, as mentoned before n table 5, the type of nstablty s dentfed as SNB and usng MCPF, the total loadng of SNB pont n ths condton s 6,695 MW. 1555

11 Fg. 6: PV Curves n the consdered contngency condton 4.4 Comparson between CPF and MCPF Conventonal contnuaton power flow method provdes all CEPs and maxmum loadng pont on the P-V curves, however, n order to obtan the exact CEPs, teratve step length selecton or very small step length s needed. Ths s very tme consumng n a large electrc power system wth many generators. In proposed method all CEPs and the type of nstablty are frst dentfed and then used as the gven data n contnuaton power flow. As the result shows, the MCPF algorthm computes all CEPs as well as the type of nstablty, exactly and as fastest as possble, whch s because no step length reducton or teratve step length selecton s needed n tracng P-V curves. However, n computaton of other ponts whch are not CEP, CPF and MCPF act exactly the same. 5. Conclusons In ths paper a new method of tracng P-V curves consderng generators reactve power lmts s presented. The proposed method, whch s named as MCPF, frst computes the generators reactve power httng pont usng predctor/corrector scheme and then uses these pont as gven data n contnuaton power flow. Ths wll elmnate the weakness of conventonal CPF n a large electrc power system wth multple generators, whch very small step length or teratve selecton of step length s needed n ths method. The proposed predctor/corrector scheme s an extenson of proposed method n [5] by defnng a PHL lst for each CEP, and optmzng the corrector step n order to handle the predctor error whch may lead to an unrealstc CEP or dvergence of equatons. The results show a good performance of the proposed method. The aspraton for our future work s to compute other constrants of power system n a predctor/corrector manner. For example, transmsson lnes thermal capablty lmts are another key factor n voltage stablty analyss and can be ncluded n MCPF algorthm. 6. REFERENCES [1] Begovc, M., Voltage collapse mtgaton IEEE Power System Relayng Commttee.Workng Group K12, IEEE publcaton No.93, THO596-7PWR. [2] Abed, A. M., WSCC voltage stablty crtera, load sheddng strategy, and reactve power reserve montor methodology. Proc. IEEE Power engneerng socety summer meetng, Edmonton, AB, Canada, 1: [3] Balamourougan, V., T.S.Sdhu, M.S.Sachdev, 24. Technque for onlne predcton of voltage collapse. IEE Proc.-Gener. Transm. Dstrb., 151(4). [4] Hskens, A., B.B.Chakrabart, Drect calculaton of reactve power lmt ponts.elect. Power Energy Syst., 18 (2): [5] Yorno, N., H.Q., L, H. Sasak, 25. A Predctor/Corrector Scheme for obtanng Q Lmt Ponts for Power Flow Studes. IEEE Transacton on Power systems. [6] Dobson, I., L.Lu, 1993.New methods for computng a closest saddle node bfurcaton and worst case load powermargn for voltage collapse. IEEE Trans. Power Syst., 8 (3): [7] Rosehart, W. D., C.A.,Canzares, V.H.Quntana, Optmal power flow ncorporatng voltage collapse constrants. Proc PES Summer Meetng, pp: [8] Canzares, C. A., F.L., Alvarado, 1993.Pont of collapse and contnuaton method for large AC/DC systems. IEEE Trans. Power Syst., 8(1):1 8. [9] Canzares, C. A., F.L. Alvarado, C.L.,DeMarco, I.,Dobson, W.F., Long, 1992.Pont of collapse methods appled to AC/DC power systems. IEEE Trans. Power Syst., 7(2): [1] Ajjarapu, V., Identfcaton of steady-state voltage stablty n power systems. Int. J. Energy Syst., 11(1):