ADVANCED PROBABILISTIC POWER FLOW METHODOLOGY

Transcription

1 ADVANCED ROBABILITIC OWER FLOW METHODOLOY eorge tefopoulos, A.. Melopoulos an eorge J. Cokknes chool of Electrcal an Computer Engneerng eorga Insttute of Technology Atlanta, eorga 333-5, UA Abstract A comprehensve moel for stochastc loa flow analyss n electrc power systems s propose for the purpose of estmatng the statstcs of bus voltage magntues, crcut currents an transmsson lne power flows. The propose metho s rven by a realstc nonconformng stochastc electrc loa moel, efne n terms of few nepenent stochastc processes; t s base on the quaratze power flow formulaton an t can account for the major operatng practces of electrc power grs, such as economc spatch, congeston management or other network constrants. The statstcs of crcut currents, bus voltage magntues, an lne power flows are compute from lnearze moels of these quanttes wth respect to the nepenent stochastc loa varables at multple ponts. The metho s valate va Monte Carlo smulatons n whch the problem s fully solve for each ranom sample, ncorporatng nonlneartes resultng from the AC power flow equatons an operatng constrants. The propose metho s emonstrate wth the IEEE Relablty Test ystem (RT) an results are presente n the paper. Keywors: Monte-Carlo smulaton, Mult-pont lnearzaton, robablstc power flow, enstvty analyss, tochastc loa flow INTRODUCTION Loa flow analyss s the most wely use tool for steay-state stues n power systems. Its applcaton s base on the assumpton that the system loang s precsely known. That s, the electrc loa an generaton are etermnstcally known quanttes. However, n many cases the loa nees to be assume stochastc n nature an therefore the power system operaton has to be stue base on estmates of ths eman an takng nto conseraton the probablstc nature of the loa. Ths s performe usng the stochastc loa flow analyss, also referre to as probablstc power flow analyss. robablstc power flow s a term that refers to power flow analyss methos that rectly treat the uncertanty of electrc loa an generaton. The frst notons of probablstc power flow analyss appeare n the early 97s [] [3]. In [] an [3] a smplfe probablstc loa flow s propose, base on the assumptons that the actve bus loas are nepenent ranom varables an the transmsson system s represente by the DC network moel (thus the reactve power flow s neglecte). The generaton spatch proceure s moele wth an arbtrary functon whch allocates the varaton of the total electrc loa to specfc generaton buses. nce the varables of the actve electrc loa at each bus are assume nepenent, the probablty ensty functon of the crcut flows can be compute wth a seres of convolutons. Later, ths basc metho was extene an t was also apple to the AC network moel [4,5]. A conventonal etermnstc power flow analyss s performe ntally, assumng net bus loas equal to ther mean values. Ths soluton etermnes the operatng pont about whch the loa flow equatons are subsequently lnearze. The assumpton of nepenence of the noal electrc loas s, however, qute unrealstc. In [6] a lnear epenence between noal powers n probablstc loa flow s suggeste. Ths assumpton s also mple by Allan et al. n [7]. In [8], a lnear epenence moel of electrc loas s propose, agan, along wth a metho whch combnes Monte Carlo smulaton an convolutons, usng a lnearze power flow moel. In [9], Dopazo et al. use a metho whch moels the correlaton between the loas at any two buses. Ther propose metho assumes that crcut flows an bus voltage magntues are aussan strbute an, thus, only ther varance shoul be compute. However, Monte Carlo smulatons ncate that t s unrealstc to assume, a pror, aussan strbutons of crcut flows an bus voltages [7,8,]. For ths reason, auer an Heyt [] have propose the use of hgher moments (thr an fourth) for accurate representaton of the probablty strbuton functons. The topc of probablstc power flow was further nvestgate n the m an late 8 s. ome typcal extensons an mprovements can be foun n [-5]. An extene lterature revew on power system probablstc analyss (untl the late 8 s) can be foun n [6]. An effcent metho for treatng the correlaton among bus loas an the generaton spatch proceure s propose n [7]. The moel assumes aussan strbuton of bus loas an a lnearze economc spatch moel. The crcut flows an bus voltages are expresse as a lnear combnaton of the bus loas only, an are assume to be normally strbute. The lnearze equatons are utlze to etermne the moments of probablty ensty functon of crcut flows an bus voltages. The ncluson of ths moel n a relablty analyss metho resulte n more accurate representaton of the electrc loa at reuce computatonal requrements [7]. Whle ths approach moels the economc respatch of generatng unts ue to electrc loa varatons, t s base on the lnearze power flow equatons an the lnearze economc spatch moel. In [8] the metho was further extene aressng the ssues of the economc spatch of the generatng unts, the effects of nonlneartes of the power system moel an the uncertanty assocate wth 5th CC, Lege, -6 August 5 esson 7, aper, age

2 the avalablty of the generatng unts. In [9] the basc eas from [7] an [8] were extene to the problem of transmsson loss evaluaton. Furthermore the a pror assumpton that the crcut loangs an bus voltages are normally strbute was wave. Ths work extens some of the eas presente n [7], [8] an [9]. More specfcally ths paper apples a stochastc loa flow methoology to the quaratze power flow moel [-]. In aton t ntrouces a nonconformng stochastc electrc loa representaton. Detals on the quaratc power flow formulaton can be foun n [-]. Ths paper focuses on the non-conformng loa moel whch represents the electrc loa accurately an realstcally wth a few nepenent stochastc processes, [ν ]. The parameters of these processes are etermne from hstorcal bus-loa ata. The propose methoology computes statstcs of quanttes uner stuy from ther lnearze moels wth respect to the nepenent loa varables [ν ], aroun the expecte value of the nonconformng loa. Emphass s gven n the ablty of takng nto conseraton major operatng practces such as economc spatch an possbly congeston management or observaton of network constrants. The ntene applcaton of the propose methoology s operatons plannng. However, t s our belef that t can be easly extene to long term plannng or on-lne operaton. ROBLEM TATEMENT Conser an electrc power system consstng of the power gr, the loas, an the generatng unts. ven the probablstc loa moel, t s esre to calculate the statstcs of the voltage magntue at each bus an of the current magntue an power flow at each system branch conserng the operatng constrants an optmzaton practces of the system. More specfcally t s esre to compute the statstcs of the current magntue I km (or the apparent power flowt km ) of any crcut connectng buses k an m, or the voltage magntue at any bus k, V k. The operaton of the system s constrant by crcut capactes, voltage magntue regulaton an the nee to optmze the operaton of the system (mnmze operatng cost, economc spatch). We vew crcut flows an voltage magntues as functonals of the parameters affectng the operaton an whch account for major operatng practces such as economc spatch an congeston management. 3 ROOED MODEL DECRITION 3. Non-conformng electrc loa stochastc moel The electrc loa representaton s a key ssue n a probablstc loa flow methoology. Total nepenence of bus loas has been extensvely assume n lterature [,3,5,7,9,]. Ths assumpton, n combnaton wth lnearze power flow equatons results n the soluton of the probablstc loa flow beng a sum of nepenent ranom varables weghte by senstvty coeffcents. Therefore, t can be obtane by a seres of convolutons. However, ths assumpton of nepenence s qute unrealstc [4,6,8,8,9], snce there are varous reasons for correlaton to exst between bus loas [8]. A typcal probablstc loa moel s a conformng electrc loa moel,.e. a specfc bus loa s a fxe percentage of the total system loa. tatstcally, ths means that the bus loas are correlate one hunre percent. Ths practce fals to represent the fact that the actual loas are not fully correlate. For a more realstc representaton of the electrc loa, t s necessary to represent the bus electrc loa as a non-conformng loa. We propose a generalze nonconformng electrc loa moel whch s efne n terms of n nepenent an zero mean ranom processes (typcally no more than three to fve). At a certan nstance of tme (or certan tme nterval) the stochastc processes become ranom varables wth specfc probablty strbuton functons an of zero mean. The characterzaton of the ranom varables an thus the etermnaton of the non-conformng loa moel can be obtane from bus hstorcal ata usng loa forecastng methos, whch are, however, outse the scope of ths paper. Here we assume that the moel of the nepenent stochastc processes s gven. If the number of bus loas n the system s L the entre system actve an reactve loang can be represente as two L - mensonal vectors: r r r = v, () r r Q = A, () where r r :L -mensonal actve loa vector, = [ k ], r r :L -mensonal base case loa vector, = [ k ], v r :n -mensonal vector of stochastc loa varables, :nxl matrx of partcpaton coeffcents, = [ p k ], =,... n an k =,..., L, Q r : L -mensonal reactve loa vector, A :L x L agonal matrx of proportonalty constants A = ag( a k ) k =,,L. It s assume that the loa at each bus k mantans a constant power factor; therefore the reactve power consumpton at bus k s proportonal to the actve power consumpton, wth a constant of proportonalty a k. Ths assumpton may be rather lmtatve, however, t s a commonly mae assumpton s many cases (nclung actual utlty practces an stues) an proves a goo startng pont. It wll be wave n future work by slghtly complcatng the loa moel. pecfcally, the parameters ak wll be replace wth stochastc processes, whch wll enable realstc representaton of power factor varatons. The presente non-conformng loa moel assumes varable correlaton between the varous bus loas. Note that n a conventonal conformng loa moel the bus loas are fully correlate. If v = for =,... n then the 5th CC, Lege, -6 August 5 esson 7, aper, age

3 non-conformng loa moel becomes a smple conformng loa. The loa varatons at every bus have the exact same statstcs, mpose by the sngle ranom varable v. If n then the above moel becomes a nonconformng loa moel. The ablty of ths moel to represent the varable correlaton between bus loas s llustrate wth the very smple system n Fgure. BU BU3 BU BU4 Fgure. ystem for llustraton of the correlaton among bus loas when usng non-conformng loa representaton We assume that the bus loas are gven by: = 4v v, = 5 v 5, (3) v 3 = 8v, where v ~ N (,) an v ~ N (,). Computaton of the correlaton of the bus-loas gves: Cor[, 3 ] =.577 Cor[, ] =.943 (4) Cor[, ] = whch ncates that there s varable correlaton among the loas. Note that f a conformng loa moel of one ranom varable was use then every bus loa woul be % correlate wth each other. 3. eneraton moel The probablstc nature of the loa mposes varatons to the total generaton requre by the system. Ths s manly ue to the varaton of the power eman, an to a lesser egree on changes of transmsson losses as the power eman changes. The total change n generaton that s to be spatche among the unts s set equal to g L n = γ p v, (5) k= = k where the nvolve quanttes are g : total generaton change, γ : coeffcent accountng for the change n system losses (compute at the base case), v : th ranom varable from the set of n nepenent zero mean loa ranom varables, p : partcpaton coeffcent for the th ranom varable k v (prove by the loa moel), L : total number of constant power loas n the system. Here coeffcent γ expresses the change n losses as a percentage of the change n the loa eman. It s a coeffcent compute at the base-case operatng pont. In orer to stngush between the cases of loa ncrease an loa ecrease (.e., g > or g < ) two atonal ranom varables, w an w, are ntrouce, representng the total loa ncrease an total loa ecrease, respectvely. The stncton between loa ncrease an ecrease s necessary snce we wll assume a lnearze unt spatch an the partcpaton factors for ncrease an ecrease n loa are n general fferent for each unt, especally f a unt s operatng close to one of ts lmts. mlarly, two coeffcents for the losses, γ an γ are efne. Therefore, equaton (5) can be rewrtten as L n w w = p k v, (6) γ γ k = = wth the atonal constrants: w, w, (7) w w =. (8) Equaton (8) mposes that only one of w or w s nonzero an both cannot be nonzero at the same tme. Equaton (6) proves a lnear relaton between the nepenent ranom varables, v s, an the epenent varables w an w. The re-spatch s performe lnearly among the unts base on the values of the partcpaton factor of each generator, at the base case operatng pont. Each unt k s assgne one partcpaton factor for loa ncrease an one for loa ecrease, enote as p an p respectvely. In general these two have fferent values, espe- cally close to the upper or lower lmt respectvely. If a unt s operatng at ts upper or lower lmt then the corresponng partcpaton factor s zero whle the other one s non-zero. Thus the actve power generaton of each unt k s = p w p w, (9) where the symbols are efne as : new actve power proucton of unt k, : base case actve power proucton of unt k, p : partcpaton factor of unt k for total eman ncrease, p : partcpaton factor of unt k for total eman ecrease, w : epenent ranom varable of total eman ncrease, w : epenent ranom varable of total eman ecrease. 5th CC, Lege, -6 August 5 esson 7, aper, age 3

4 The values of the partcpaton factors are calculate by the economc spatch algorthm. However, n ths paper they are assgne base on how close to ts upper or lower lmt respectvely each unt s operatng. Therefore the partcpaton factors are proportonal to the fference between the actual proucton at base case an the unt lmts. For the generate reactve power of unts operatng n Q moe, we assume that the power factor s constant, therefore the change n ther reactve proucton s proportonal to the change n the actve power proucton, so that the power factor s kept constant. 3.3 Transmsson network moel The network quanttes (crcut currents, power flows or bus voltages) are also treate as ranom varables that epen on the stochastc loa varables, v s. Lnear epenence s assume for small evatons aroun an operatng pont, an therefore crcut currents, flows an bus voltages are lnearze wth respect to the loa varables, v. pecfcally, a network quantty F (.e. lne current, power flow or bus voltage) s expresse as a lnear combnaton of the loa ranom varables as: n ( c v ) F = F, () = where F : Base case value of F, c : Lnearzaton coeffcent of quantty F wth respect to v. nce v s are nepenent ranom varables the statstcs of the quantty F are erve from the statstcs of the v varables by smply performng a seres of convolutons on the probablty ensty functons of v s. The propose metho s base on the concept of utlzng a lnearze moel aroun a specfc value of electrc loa as efne by a sample of the electrc loa moel varables. The lnearze moel s constrane wth the operatonal practces such as economc spatch, congeston management an others. The performance of the system n terms of strbutons of bus voltage magntues, crcut currents an flows s erve from the lnearze moel. To further mprove on the accuracy of the propose metho the non-conformng loa moel s sectonalze nto a small number of segments as t s epcte n Fgure. A specfc combnaton of a segment from each varable efnes an electrc loa event. uch an event C s llustrate n Fgure. We then conser the contonal probablty of the electrc loa gven that the electrc loa belongs to the electrc loa event C. ubsequently, we conser the contonal expecte value of the electrc loa gven that the loa belongs to event C. The operatng contons of the system at the contonal expecte value are etermne by smulaton of the electrc power network. Then the lnearze moel of the system s compute aroun ths operatng pont. Fnally, the contonal (gven event C ) performance of the system n terms of strbutons of bus voltage magntues, crcut flows an total operatng cost are erve from the lnearze moel an the known contonal electrc loa moel. The proceure s apple to all possble electrc loa events an the results are weghte wth the probabltes of the electrc loa events an summarze nto an overall probablstc moel. It s mportant to note that at each expecte value of a loa event congeston management or any other type of remeal actons can be apple, f necessary, as well as the effect of possble contngences can be accounte for. The bass of ths ea s epcte n Fgure. Three stochastc loa varables are assume, namely v, v an v followng some probablty strbuton n the nterval 3 [a,b]. Ths nterval s ve nto three sectons, an the expecte value of each loa ranom varable s calculate n each secton. The trplet of such expecte values of each loa varables efnes a specfc loa profle an therefore a base case operatng pont. Furthermore, a trplet of such sectons, one for each varable, efnes a possble event,.e. event ε of Fgure s efne as: = v, b v, v () C v v v 3 { ( ) ( ) ( )} 3 3, event C a b a a Fgure. chematc representaton of non-conformng loa sectonalzaton For each electrc loa event, the base case contons of the electrc power network are compute as well as the lnearze moel aroun the base case contons. Assumng n ranom processes an m sectons for each process the total number of events to be consere s m. n Although the number of events can be very large, leang to bulky computatons, n practce we antcpate that no more than three to fve ranom processes wll be neee an three to fve sectons for each varable can prove aequate results. Ths results n several hunres or maybe a few thousan events, an therefore loa flow computatons. Ths s stll typcally one or two orers of magntue better than Monte Carlo smulaton, where several tenths or even hunres of thousans of trals mght be necessary n a typcal stuaton to get creble results. It shoul be clarfe, however, that snce lnearzaton are nvolve there s always some approxmaton nvolve n the results, whch s greater as the egree of 3 b b 3 5th CC, Lege, -6 August 5 esson 7, aper, age 4

5 E Q E Q E Q E Q E Q nonlnearty ncreases. Mult-lnearzaton mtgates ths effect, but cannot elmnate t. 4 MONTE CARLO IMULATION The propose multple-pont-lnearzaton base methoology s valate wth an nepenent metho base on Monte Carlo smulaton. pecfcally, the same problem s also solve va Monte Carlo smulaton n whch each ranom sample s fully solve, thus ncorporatng nonlneartes resultng from the AC power flow equatons an major operatng practces such as economc spatch an congeston management or other necessary remeal actons. Both the lnearzaton soluton an the Monte Carlo approach are base on the ngle hase Quaratc ower Flow moel. The basc ea of Monte Carlo smulaton s to smulate a specfe system wth a reasonable number of ranom raws of all possble system states accorng to ther probablty strbutons. In ths moel, the crcut currents, lne power flows an bus voltages are compute for a system state etermne from a ranom raw of the state of electrc loa, that s, by a ranom raw of the values of the stochastc loa varables. Economc spatch an congeston management, or other possble remeal actons take place at any state that t s necessary, n orer to brng the system to an acceptable an secure operatonal conton. In the Monte Carlo smulaton a large number of ranom raws are generate. The process generates the statstcs of crcut currents an the bus voltages. pecfcally the cumulatve probablty strbuton functon of these quanttes s generate. From ths strbuton, other quanttes, such as the probablty ensty functon, the expecte value, the varance, the stanar evaton, etc. can be compute. 5 RELIMINARY REULT The propose metho has been apple to the IEEE 4- Bus Relablty Test ystem (RT-4), llustrate n Fgure 3. The system s operatng at peak loang, at base case. The etale system ata for each system component can be foun n [3]. The nonconformng loa moel conssts of two zero mean stochastc loa varables namely v an v, whch are assume to be normally strbute wth zero mean an varance of.. Therefore, each bus loa s expresse as a lnear functon of these two ranom varables,.e., k = k pk v pk v, () where k : actve power eman (loa) at bus k, : base case loa value at bus k k, p : partcpaton coeffcent for the st ranom varable, k p : partcpaton coeffcent for the n ranom varable. k Three cases are consere for the propose multlnearzaton approach. The stochastc loa varables are segmente nto one, three an fve sectons. Comparatve results from all the cases are presente. For valaton purposes an for the small IEEE RT-4 Monte Carlo trals are use. nce Monte Carlo smulaton s use only for valaton purposes, at ths stage, no attempt was mae to formally estmate the requre number of trals, for a partcular confence level. BU7 BU8 BU BU BU6 BU9 BU BU5 BU4 BU3 BU4 BU BU4 BU BU9 BU BU5 BU3 BU BU Fgure 3. The IEEE 4 Bus Relablty Test ystem BU3 BU6 BU8 BU7 Tables an prove a comparson between the mean value, stanar evaton an thr an fourth moments for several network quanttes, as calculate usng the propose approach an usng the Monte Carlo () smulaton results. A close agreement between the results s observe for the voltage magntues of the Q system buses, even when usng sngle pont lnearzaton. A very close agreement, even when usng sngle lnearzaton, s also observe for the majorty of the crcut currents, as shown n Table. Fgure 4 shows the probablty ensty an strbuton functon for the current magntue at crcut -3 as obtane usng the propose multpont-lnearzaton metho an Monte Carlo smulaton. In only few of the crcuts was there a sgnfcant msmatch between the propose metho wth sngle lnearzaton an the smulaton results. Ths msmatch s mnmze or even elmnate when mult-lnearzatons are use. Results from these crcuts are shown n Table 3 an probablty strbuton plots of such a case are presente n Fgure 5. 5th CC, Lege, -6 August 5 esson 7, aper, age 5

6 ropose Metho Bus Ln. 3 Ln. 5 Ln. Mean (kv) t. Dev. (kv) kewness Kurtoss Mean (kv) t. Dev. (kv) kewness Kurtoss Mean (kv) t. Dev. (kv).9... kewness Kurtoss Mean (kv) t. Dev. (kv) kewness Kurtoss Table : Comparson of propose lnearzaton metho an Monte-Carlo smulaton results for Q bus voltage magntues ropose Metho Crcut Ln. 3 Ln. 5 Ln. Mean (A) t. Dev. (A) kewness Kurtoss Mean (A) t. Dev. (A) kewness Kurtoss Mean (A) t. Dev. (A) kewness..9.. Kurtoss Mean (A) t. Dev. (A) kewness....3 Kurtoss Table : Comparson of propose lnearzaton metho an Monte-Carlo smulaton results for crcut currents CDF DF Ln Ln 3 Ln Crcut Current (A).8 Ln.6 Ln 3 Ln Crcut Current (A) Fgure 4. Comparson of propose metho an Monte Carlo smulaton results. robablty Densty an Cumulatve Dstrbuton Functon of Crcut -3 current ropose Metho Ln. 3 Ln. 5 Ln. Mean (A) t. Dev. (A) kewness Kurtoss Mean (A) t. Dev. (A) kewness Kurtoss Mean (A) t. Dev. (A) kewness Kurtoss Mean (A) t. Dev. (A) kewness Kurtoss Mean (A) t. Dev. (A) kewness Kurtoss Table 3: Comparson of propose lnearzaton metho an Monte-Carlo smulaton results for crcut currents DF CDF 3 ngle Lnearzaton an 5 Lnearzatons Monte Carlo Ln Ln 3 Ln Crcut Current (A) Ln Ln 3 Ln 5 ngle Lnearzaton. Monte Carlo an Mult-lnearzatons Crcut Current (A) Fgure 5. Comparson of propose metho an Monte Carlo smulaton results. robablty Densty an Cumulatve Dstrbuton Functon of Crcut 7-8 current 6 CONCLUION Ths paper presente a new comprehensve methoology for probablstc power flow analyss. The methoology s base on a stochastc non-conformng electrc loa moel an t s apple on the quaratze power flow moel of the system. It s capable of ncorporatng the operatng practces an constrants of the power system. The methoology s useful n computng the expecte performance of the system efne n terms of probablstc strbutons of bus voltage magntues, crcut flows an operatng cost. The propose metho has been valate wth an nepenent metho base on Monte Carlo smulaton. The current evelopment an mplementaton of the methoology only takes nto conseraton loa uncertanty. However, n the presente mult-lnearzaton 5th CC, Lege, -6 August 5 esson 7, aper, age 6

7 framework other uncertantes can be easly ncorporate lke generaton uncertanty, effects of possble network contngences or uncertantes n system parameters. The methoology s very useful n assessng the relablty of the electrc power system. The recent several massve falures of the electrc power gr wth the most notceable August 3 blackout n U have generate renewe nterest n the relablty of the power gr, because of the complexty of the system moel an the complexty of the operatng practces an constrants, t s necessary to evelop methoologes that ncorporate these ssues. The propose methoology s very promsng towars ths goal. REFERENCE [] R. R. Booth, ower ystem mulaton Methos Base on robablty Analyss, IEEE Trans. ower Apparatus an ystems, vol. A-9, pp. 6 69, Jan./Feb. 97 [] B. Borkowska, robablstc loa flow, IEEE Trans. ower Apparatus an ystems, vol. A-93, No. 3, pp , May/June 974 [3] R. N. Allan, B. Borkowska, C. H. rgg, robablstc analyss of power flows, roc. of the IEE, vol., No., pp , Dec. 974 [4] R. N. Allan, C. H. rgg, D. A. Newey, R. F. mmons, robablstc power flow technques extene an apple to operatonal ecson makng, roc. of the IEE, vol. 3, No., pp , Dec. 976 [5] R. N. Allan, M. R.. Al-hakarch, robablstc technques n AC loa flow analyss, roc. of the IEE, vol. 4, No., pp. 54 6, Feb [6] R. N. Allan, M. R.. Al-hakarch, Lnear epenence between noal powers n probablstc AC loa flow, roc. of the IEE, vol. 4, No. 6, pp , June 977. [7] R. N. Allan, A. M. Lete a lva, an R. C. Burchett, Evaluaton methos an accuracy n probablstc loa flow solutons, IEEE Trans. ower Apparatus an ystems, vol. A-, No. 5, pp , May 98 [8] A. M. Lete a lva, V. L. Arent, R. N. Allan, robablstc loa flow conserng epenence between nput noal powers, IEEE Trans. ower Apparatus an ystems, vol. A-3, No. 6, pp , June 984 [9] J. F. Dopaso, O. A. Kltn, A. M. asson, tochastc loa flows, IEEE Trans. ower Apparatus an ystems, vol. A-94, No., pp , Mar./Apr. 975 []R. N. Allan an A. M. Lete a lva, robablstc loa flow usng multlnearsatons, roc. of the IEE, vol. 8, t. C, No. 5, pp. 8-87, ep. 98 []. W. auer,. T. Heyt, A generalze stochastc power flow algorthm, presente at the IEEE/E ummer Meetng, July 978, aper A []A. M. Lete a lva, R. N. Allan,. M. oares, an V. L. Arent, robablstc loa flow conserng network outages, roc. of the IEE, vol. 3, t. C, pp , May 985 [3]A. M. Lete a slva, V. L. Arent, an M. H. Barbosa, robablstc technques n loa flow analyss A practcal applcaton, roc. of st Int. ymp. yst. (IMAE), Toronto, Canaa, pp , July 986 [4]A. M. Lete a lva,. M.. Rbero, V. L. Arent, R. N. Allan, an M. B. Do Coutto Flho, robablstc loa flow technques apple to power system expanson plannng, IEEE Trans. ower ystem, vol. 5, No. 4, pp , Nov. 99 [5]A. M. Lete a lva an V. L. Arent, robablstc loa flow by multlnear smulaton algorthm, roc. of the IEE, vol. 37, t. C, No 4, pp. 76-8, July 99 [6]M. Th. chllng, A. M. Lete a lva, R. Bllnton, an M. A. El-Kay, Bblography on power system probablstc analyss (96-988), IEEE Trans. ower ystems, vol. 5, No., pp., Feb. 99 [7]A.. Melopoulos, A.. Bakrtzs, R. Kovacs, ower system relablty evaluaton usng stochastc loa flows, IEEE Trans. ower Apparatus an ystems, vol. A-3, No. 5, pp. 84 9, May 984 [8]A.. aks Melopoulos, eorge J. Cokknes, Xng Yong Chao, A new probablstc power flow analyss metho, IEEE Trans. ower ystems, vol. 5, No., pp. 8 9, Feb. 99 [9]A.. Melopoulos, X. Chao, eorge J. Cokknes, R. Monsalvatge, Transmsson loss evaluaton base on probablstc power flow, IEEE Trans. ower ystems, vol. 6, No., pp , Feb. 99. []A... Melopoulos,. Kang,. J. Cokknes an R. Dougal, Anmaton an vsualzaton of spot prces va quaratze power flow analyss, n roc. 36th Annu. Hawa Int. Conf. on ystem cences, 6-9 Jan., 3, pp []A... Melopoulos,. J. Cokknes an R. Lasseter, A multphase power flow moel for µr analyss, n roc. 36th Annu. Hawa Int. Conf. on ystem cences, Jan. 6-9, 3, pp []A... Melopoulos, W. ao,. L,. J. Cokknes an R. Dougal, Quaratze nucton motor moel for power flow analyss, n roc. n IATED Int. Conf., EuroE, Crete, reece, June 5-8.,, pp [3]IEEE Commttee Report, IEEE Relablty Test ystem, IEEE Trans. ower Apparatus an ystems, vol. A-98, No. 6, pp , Nov./Dec th CC, Lege, -6 August 5 esson 7, aper, age 7