A Three-Phase State Estimation in Unbalanced Distribution Networks with Switch Modelling

Transcription

1 A Three-Phase State Estmaton n Unbaanced Dstrbuton Networks wth Swtch Modeng Ankur Majumdar Student Member, IEEE Dept of Eectrca and Eectronc Engneerng Impera Coege London London, UK ankurmajumdar@mperaacuk Bkash C Pa Feow, IEEE Dept of Eectrca and Eectronc Engneerng Impera Coege London London, UK bpa@mperaacuk Abstract State estmaton has become an mportant task n modern energy/ dstrbuton management systems However, the state estmaton s not very popuar n modern unbaanced threease dstrbuton systems Ths paper proposes a method for three-ase state estmaton wth detaed three-ase modeng of system components ncudng swtches and star and deta coected oads Ths method s then tested on a standard IEEE 3-bus system and the resuts are compared wth oad fow resuts Keywords state estmaton, dstrbuton management systems DMS), three-ase modeng, weghted east squares WLS), swtch modeng I INTRODUCTION Wth the nfux of asor measurement unts PMUs), ntegent meterng etc n transmsson systems and smart meters wth nformaton and communcaton technoogy ICT) nfrastructure n dstrbuton systems, power systems now-adays need to be montored and controed effcenty To enabe ths, the states of the system need to be observed propery Ths woud hep to nfuence the operatona decsons and thus, to avod contngency and cascaded trppng It s done through an energy/ dstrbuton management system EMS/DMS) functon- the state estmaton SE) 2, 3 Ths functon estmates the bus votages and anges based on the avaabe measurements, network data and topoogy nformaton Fgure I shows a typca DMS archtecture In transmsson systems, the state estmaton concept s we estabshed but n dstrbuton systems due to the absence of suffcent measurements and unbaanced and asymmetrc nature of the system, t was not mandatory to have a state estmaton functon But wth growng number of controabe devces and the ncorporaton of smart meters n the system, state estmaton s becomng mportant n dstrbuton network operaton As a startng pont, the souton methodoogy many focuses on weghted east squares WLS) estmaton technque 4 But the majorty of dstrbuton systems operate under varyng degrees of unbaance Moreover, unke the transmsson system, the dstrbuton system s rada n nature and has a hgher R/X rato Therefore, the fast decouped method causes numerca nstabty when apped to dstrbuton systems 5 Hence, ths has paved the way for the need of unbaanced three-ase state estmaton rather than sngease state estmaton Over the years, there have been research on three-ase oad fow n dstrbuton systems 6 However, most of the operatona e, contro and contngency, decsons based on state estmaton have been apped to dstrbuton system consderng to be baanced 7 Moreover, the oads consdered are constant power and Y-coected oads and the dfferent status cosed/open) of the swtches are not consdered 8, 9 Measurements vector Fgure State Estmator and Bad Data Detecton State estmates Dstrbuton Optma Power Fow Basc EMS/DMS functons Notfy DNO Dstrbuton Network Operator Contro setponts Power Grd A typca dstrbuton management system archtecture To acheve the accurate estmates of the state varabes, ths paper presents a weghted east squares based estmator wth the detaed modeng of the system components and dfferent types of oads and aso consderng the dfferent operatona status of the swtches The paper s organzed as foows Secton II descrbes the formuaton of the probem Secton III dscusses the modeng of dfferent components of the dstrbuton system and Secton IV demonstrates the resuts and dscussons of the technque apped to 3-bus system Secton V concudes the paper II PROBLEM FORMULATION The state estmaton s a process whch estmates reatme states of the system votage magntudes and anges)

2 The probem can be ooked at as a constraned non-near optmzaton probem wth the foowng objectve functon Subject to: J = z meas h func x) T R z meas h func x) ) c eq x) = 0 c neq x) 0 2a) 2b) x State varabes such as votage magntudes and anges m Number of measurements per ase R Measurement error covarance matrx, z meas = z a z b z c z a z b zc zm a zm b zm c z Measured vaue of th measurement h func x) vector of measurement as a functon of state x c eq x) vector of zero njecton measurements c neq x) vector of nequaty constrants In three ase system, x = δ δ δn Vn where, = V a V b V c, δ = δ a δ b δ c are the three-ase votage magntude and votage ange at bus respectvey The measurements are usuay consdered to be wth random errors due to bases, drfts or wrong coectons of the measurement devces, e meters It s assumed that the measurement errors are dentcay and ndependenty dstrbuted Hence, the covarance matrx of the errors are gven by R=Cove)=Eee T )=dagσ,σ 2 2,,σ2 m), where σ = σ a σ b σc and σ 2 s the varance of the th measurement error In three-ase system, the rea power njecton and reactve power njecton equatons at bus for ase can be wrtten as: = +B, = n = j= j sn n = j= B, j cos ) G, j cos δ δj δ δ j ) ) G, j sn δ δj δ δ j Where G+jB s the system admttance matrx, n s number of buses and s the number of ases that can be, 2 or 3 ase The branch rea power fow j and reactve power fow j equatons from bus to bus j for ase can be ) 3) 4) wrtten as foows: j = = = j = = = ) ) V G, j cos δ δ +B, j sn δ δ ) ) G, j cos δ δj +B, j sn δ δj 5) ) ) V G, j sn δ δ B, j cos δ δ ) ) G, j sn δ δj B, j cos δ δj V Votage magntude of ase at bus δ Ange of ase n bus A Equaty constrants c eq x) The equaty constrants are the set of equatons correspondng to vrtua measurements 6) 0 = G P D P 7) 0 = G Q D Q 8) Where G andq G are the rea and reactve power njected at bus respectvey, the oad demand at the same bus s represented by D and Q D Indces n s number of buses and s the number of ases whch can be, 2 or 3 ase B Inequaty constrants c neq x) These are the set of constrants of state varabes that represent the system operatona and securty mts, such as settng upper and ower mts for contro varabes The constrants are as foows: Bus votage - Votage magntudes at each bus n the network: mn, V max, Bus ange - The bus ange at each bus n the network: δ mn, δ δ max, Eq ) and 2) can be soved by Newton s method, whch transates nto sovng the foowng equaton at each teraton Gx k ) x k+ = H T R z hx k ) 9) where, n s the tota number of buses n the system x k+ = x k+ x k and Hx k ) = h x s the Jacoban matrx and Gx k ) = H T x k )R Hx k ) s the Gan matrx n the k th teraton

3 III THREE-PHASE DISTRIBUTION SYSTEM MODELLING The dstrbuton system conssts of unsymmetrca network components and unbaanced oad Hence, the snge ne representaton for an unbaanced dstrbuton system does not work Therefore, the exact three-ase modeng of the network components s necessary The foowng subsectons descrbe the three ase modeng of varous components of the network such as ne, transformers and swtches A Lne Modeng The dstrbuton system conssts of untransposed overhead nes and underground cabes whch can be three-ase or snge and/or two-ase ateras Ths combned wth the unbaanced oads snge, two or three-ase oads) contrbute to the unbaanced nature of the system Due to the untransposed nature of the nes, t s essenta to compute the mpedance of the nes accuratey A modfed Carson s equaton s apped to compute the sef and mutua mpedance of the nes 0 ) Z = r j02 Ln Ω/me GMR 0) Z j = 0095+j02 Ln ) Ω/me ) D j Z Sef-mpedance of conductor n Ω/me Z j Mutua mpedance between conductors and j n Ω/me r Resstance of conductor n Ω/me GMR Geometrc mean radus of conductor n feet D j Dstance between conductors and j n feet The modfed Carson s equaton aso takes nto the ground return path neutra conductor) for the unbaanced currents The modfed Carson s equatons 0) and ) for a three ase overhead or underground crcut whch conssts of neut neutra conductors makes the resutng mpedance matrx 3 + neut) 3 + neut) However, for most appcatons, t s necessary to have the 3 3 ase mpedance matrx Therefore, 3+neut) 3+neut) mpedance matrx s broken down to 3 3 matrx by Kron s reducton as gven n 2) In ths approach, a the nes w be modeed by 3 3 ase mpedance matrx and for two ase and snge ase nes the mssng ases are modeed by settng the mpedance eement to zero Z j = Z j Z n Z nj Z 2) Therefore, for each ne between two nodes, there w be a 3 3 matrx nstead of a snge eement for a snge ase baanced system Hence, the resutant Y -bus matrx of the system w be of n 3) n 3) The structure of the Y -bus matrx s shown n 3) Y aa Y ab Y ac Yn aa Yn ab Y ba Y bb Y Yn ba Yn bb Y ca Y cb Y cc Yn ca Yn cb Y = Yn aa Yn ab Yn ac Y aa Y ab Y ba Yn ba Yn ca Y bb n Y cb n B Transformer Modeng Y n Y cc n Y ca Y bb Y cb Yn ac Yn Yn cc Y ac Y Y cc 3) The dstrbuton system generay conssts of feeder and dstrbuton transformers whch provde the fna votage transformaton to the oads The three ase transformers are modeed by admttance matrx whch depends on the coecton type A transformer can be Y-Y, Y-, - etc In the anayss of the dstrbuton feeder, t s requred to mode the varous three ase transformer coectons correcty The comprehensve cacuatons of three ase transformers and ther varous coectons can be found n reference 0, Whe formng the Y -bus, a transformer can be consdered as one eement between two nodes of the system Therefore, the transformer contrbutes to 6 6 bock n the Y -matrx C Swtch Modeng Swtches are consdered as branches wth zero mpedance It s assumed that the status of the swtches, e cosed or open, are known beforehand The operatona constrants for the swtches are consdered as equaty constrants as gven by c eq = 0 n equaton 2) of the orgna probem formuaton When the swtch between bus and bus j s assumed cosed for branch -j, the votages and anges for bus and bus j and ase for a the three ases are equa j = 0 δ δ j = 0 4) When the swtch s assumed open between bus and bus j, the actve and reactve power fow to the swtch w be zero D Load Modeng j = 0 j = 0 5) The oads n dstrbuton systems are generay unbaanced The oads are three-ase, two-ase or snge-ase They can be coected n grounded Y or ungrounded confguraton from the pont of vew of eectrcty usage, oads can be broady cassfed as constant power, constant mpedance or constant current oads They are commony represented as power consumed per ase and consdered to be L-N for Y- oads and L-L for -oads The typca ZIP modes for wye and oads are shown n 6) and 8)

4 2 L 2 L L = P n L = Q n = 2 n = 2 n c P +c P 2 c Q +cq 2 c P +c P 2 c Q +cq 2 2 = ab,,ca I a I b I c Fgure 2 Vab V V ) +c P 3 ) +c Q 3 ) V 2 ) V 2 6) 7) ) V 2 V )+c ) 3Vn 3Vn ) V 2 )+c Q V ) 3Vn 3Vn I ab I V I ca Vca Deta-coected three-ase oad Fgure 2 shows a typca -coected three ase oad The votage magntudes are ne-to-neutra for the state estmaton formuaton gven n Secton II Therefore, n case of deta oads, the equvaent wye powers are cacuated at each teraton n order to cacuate the actve and reactve power at each node Ths s ustrated n the foowng steps Cacuate ne-to-neutra votage for deta oads V ab V V a δa Vb δb = V b V ca δb Vc δc 20) V c δc Va δa Read the actve and reactve power of deta oads Cacuate the ne currents of deta oads ) Pab +jq ab I ab = 2) V ab δ ab Cacuate the current at each ase I a 0 I b = 0 I c 0 I ab I I ca 22) Cacuate the equvaent ne-to-neutra actve and reactve powers V a I a = P a +jq a V b I b = P b +jq b 23) V c I c = P c +jq c For the ases where the oads are non-exstent, the actve and reactve power vaues are set to zero for those partcuar ases E Measurements The dstrbuton system normay covers a arge geograca area Hence, t s not possbe to pace meters at every node and nes Hence, the redundancy of dstrbuton systems are usuay far ess than that of transmsson systems However, t s requred to make the system observabe n order to sove the state estmaton Therefore, the oad data taken from hstorca oad data profes are taken as pseudo measurements and zeronjecton buses are consdered as vrtua measurements A Smuaton Resuts IV CASE STUDY AND DISCUSSIONS A standard IEEE-3 bus dstrbuton system has been used n ths paper The feeders are sma yet they show some nterestng characterstcs The system represents a typca dstrbuton system wth votage magntude measurements ony at the substaton, wth more branch current measurements than power fow measurements, and a oads are consdered as pseudo measurements Short and reatvey hgh oaded for a 46 kv feeder One substaton votage reguator consstng of three snge ase unts coected n wye Both overhead and underground nes are present wth a varety of asng It has shunt capactors It has one transformer: grounded wye-grounded wye Unbaanced spot and dstrbuted oads are present The oads are of constant power, constant current and constant mpedance type and are star and/or deta coected Fgure 3 shows a typca IEEE 3-bus system A WLS state estmator s coded n Matab and tested on the standard IEEE- 3 bus system and run on a system wth Inte Xeon and 2 GB RAM The data for the system are gven n 2 and 3 Fgure IEEE-3 bus unbaanced dstrbuton system The overhead nes and underground cabes are modeed by modfed Carson s equatons The oads are modeed n ZIP mode and the three-ase transformer s confgured as a grounded Y-Y coecton Measurements have been generated usng norma dstrbuton curve wth oad fow vaues as true or mean vaues and standard devaton Each measurement s taken from the dstrbuton curve randomy and ths experment 7

5 s performed a number of tmes n a Monte Caro approach One such case has been shown here, n the resuts It s assumed that the measurement errors are ndependent and dentcay dstrbuted The zero njectons are consdered as equaty constrants The votage magntudes are set to operate wthn 5% of the nomna vaues and the votage anges wthn 30 to +30 The swtch between buses 9 and 0 s assumed to be cose Therefore, the equaton 4) are aso equaty constrants n the state estmaton formuaton The oads on nodes 2, 9 and 0 are deta-confgured oads The constant mpedance oads are on nodes 2 and 0, whe the nodes 0 and 7 have constant current oads Votage pu) TABLE I STATE ESTIMATES Ange estmatesn degrees) Votage estmatesn pu) Bus No a b c a b c True Va True Vb True Vc Estmated Va Estmated Vb Estmated Vc Bus Number Fgure 4 True and estmated votages for IEEE 3 bus system Tabe I present the estmated vaues of the state varabes The error n rea measurement s assumed to be 3% and the Votage of ase a pu) True V a Estmated V a wth 20% error Estmated V a wth 40% error Estmated V a wth 50% error Bus number Fgure 5 True and estmated votages for ase a wth 20% 40% and 50% error error n pseudo measurement s assumed to be 20% Fgure 4 shows the true and estmated votage magntudes of the three ases for the IEEE-3 bus mode The mssng ases have been taken to be of votage magntude vaue equa to However, the state estmaton process s performed wth 40% and 50% error n pseudo measurements as we Fgure 5 shows that for cases when the error n pseudo measurement s arge the state estmates are ess accurate The fgure shows that when the error s 20% the estmates for ase a are cosest to the true vaue compared to other cases In Tabe I, the mssng ases have been represented by dashes There s a cosed swtch between bus9and0 Fgure 4 and Tabe I show that the votage magntude vaues and votage anges reman the same across the cosed swtch The obtaned resuts have been found to satsfactory wthn the aowabe toerance V CONCLUSIONS Ths paper presents a WLS based three-ase state estmaton wth detaed modeng of the dfferent components of three ase system consderng both the star and detaconfgured oads The method acheved a reabe souton to the state estmaton probem Smuaton resuts on IEEE 3-bus dstrbuton system showed the effectveness of the approach and compared wth the oad fow resuts The reabe state estmaton resuts provde the bass for contro and montorng of modern dstrbuton systems REFERENCES C Gomez-Ques, A Gomez-Exposto, and A de a Va Jaen, State estmaton for smart dstrbuton substatons, IEEE Transactons on Smart Grd, vo 3, pp , June S S S R Depuru, L Wang, and V Devabhaktun, Smart meters for power grd: Chaenges, ssues, advantages and status, Renewabe and sustanabe energy revews, vo 5, no 6, pp , 20 3 P Koponen, L D Saco, N Orchard, T Vorsek, J Parsons, C Rochas, A Z Morch, V Lopes, and M Togeby, Defnton of smart meterng and appcatons and dentfcaton of benefts, Deverabe D3 of the European Smart Meterng Aance ESMA avaabe at www esmahome eu, members area), A Abur and A Exposto, Power System State Estmaton, Theory and Impementaton CRC Press, A Montce, State Estmaton n Eectrc Power Systems, A Generazed Approach Luwer s power Eectroncs and power Systems Seres, J Martnez and J Mahseredjan, Load fow cacuatons n dstrbuton systems wth dstrbuted resources a revew, n Power and Energy Socety Genera Meetng, 20 IEEE, pp 8, Juy 20 7 R Sngh, B Pa, and R Jabr, Statstca representaton of dstrbuton system oads usng gaussan mxture mode, IEEE Transactons on Power Systems, vo 25, pp 29 37, Feb C Lu, J Teng, and W-H Lu, Dstrbuton system state estmaton, IEEE Transactons on Power Systems, vo 0, pp , Feb D Thukaram, J Jerome, and C Surapong, A robust three-ase state estmaton agorthm for dstrbuton networks, Eectrc Power Systems Research, vo 55, no 3, pp 9 200, W H Kerstng, Dstrbuton system modeng and anayss CRC press, 202 P Xao, D Yu, and W Yan, A unfed three-ase transformer mode for dstrbuton oad fow cacuatons, IEEE Transactons on Power Systems, vo 2, pp 53 59, Feb Dstrbuton test feeders testfeeders/ndexhtm 3 W H Kerstng, Rada dstrbuton test feeders, n Power Engneerng Socety Wnter Meetng, 200 IEEE, vo 2, pp , IEEE, 200