Chapter 3 ROBUST TOPOLOGY ERROR IDENTIFICATION. (1970), is based on a super-bus network modeling that relates the power and voltage

Transcription

1 Chapte 3 ROBUST TOOLOGY ERROR IDENTIFICATION 3. Motvaton As entoned n Chapte 2, classc state estaton, deved by Schweppe and Wldes 970, s based on a supe-bus netwo odelng that elates the powe and voltage easueents though a nonlnea equaton nvolvng the nodal voltage agntudes and phase angles, teed the state vaables. It s assued that the topology of the netwo s copletely nown thout eo whch banches ae actve/nactve, and that the paaetes of the lnes ae also nown the esstance, eactance, and capactance. An ncoect assued status of a lne leads to a topology eo, whch then leads to ultple confong bad data and ncoect state estates. Reseach n topology eo dentfcaton s well docuented ove the last two decades and nueous ethods have been poposed. Most ethods fall nto the categoy of post-pocessng ethods. That s, classc state estaton s caed out, and then analyss s done on the esduals. If outles n the esduals coespond to a bus o a banch, then the topology aound that aea s n queston. Fo eaples of vaous post-pocessng ethods, see

2 Cleents and Davs 988, Wu, and Lu 989, o Costa and Leão 993. oweve, due to the confong natue of these easueents, especally f n lage nube, these ethods ay fal n detectng whee the topology eos occued. Thus thee ests a need fo a pe-pocessng ethod that s able to dentfy topology eos and dscnate between bad and good data. 3.2 Netwo Modelng The poposed odel uses Kchhoff s Cuent Law KCL as a poweful foundaton. KCL states that the algebac su of all the cuent aound a node bus s equal to zeo. Equvalently, the powe gong n the netwo aound a node s equal to the powe leavng that node. Fgue 3. depcts an eaple of KCL. ee, thee lnes et fo bus j. The ccles epesent easueents on the lnes flows and at the bus njecton, and the specfc values n pe unt pu. As one can see, the powe enteng the bus 3.4 pu equals the powe leavng the bus.6 pu pu pu. KCL ll enable us to dentfy outles, easueents that do not confo to KCL. Fo eaple, f the value of 2.6 n Fgue 3. wee changed to 7.6, that easueent would be labeled as an outle snce KCL s volated. Also, KCL ll enable us to estate powe on lnes that do not have a dect coespondng easueent. Fo eaple, f the value of 2.6 was eoved, we can estate the powe on that lne algebacally. To use KCL, the atheatcal odel ust encopass all lnes of the netwo, espectve of the assued status. Ths would nclude shot ccuts that ay splt a bus nto ultple sub buses called substatons. Modelng these types of powe lnes would help dentfy whch way a bus s splt. In Fgue 3.2, a bus s odeled at the substaton level. If the ccut beae dectly connectng bus 2 and bus 0 was closed, then we can epesent ths as just one supe-bus, egadless of the beaes statuses between the two buses. If ths wee the case, 24

3 Fgue 3.. Eaple of Kchhoff s Cuent Law Fgue 3.2. Eaple of a Substaton 25

4 ths would ean that bus 0-2 s connected to all fou adjacent buses, 03, 09,, and 2. oweve, f we assue that the beae s closed when n fact t s open, as depcted n Fgue 3.2, then a topology eo has occued at the substaton level. In actualty, bus 0 s connected to buses 03 and 09 and bus 2 s connected to buses and 2. Theefoe all shot ccut lnes need to be odeled to fnd out whch lnes ae enegzed, thus whch way the bus s splt. 3.3 Modelng the Real owe The ntent s to deve a odel that elates the eal powe easueents, z, to the unnown state vaables, p. Each eleent n p coesponds to a lne n the netwo, and evey lne n the netwo has a coespondng vaable n p. We stat the devaton by consdeng the Ohc losses that ae dsspated n a banch. The powe that tavels though a banch leavng node and enteng node l, say, ae equvalent f the lne has no esstance. If thee s esstance assocated th lne -l, then soe powe s lost dung the couse of tavel va fcton. Ths powe loss s called the Ohc loss. As an eaple, f the powe enteng a banch s 3.0 pu, and the powe leavng the banch s 2.8 pu, then the Ohc loss fo that banch s 0.2 pu. Fo the eal powe, powe that s leavng the banch s a negatve value, thus the Ohc loss fo the lne can be found by just addng the powe on each end. Foally, ths quantty can be wtten as L l 2 2 G V + V 2V V cos θ, 3. l l l l 26

5 whee V s the voltage agntude, G s the nown sees conductance as defned n Append A, and θ l θ θl s the dffeence n voltage angles between bus and bus l. If we let l be the powe flow on the sde of the banch dected towad bus l, and assue that we have a decoupled odel, then we can appoate θ l as the poduct of the nown eactance of the lne, X l, and the state vaable, l. That s, θ l X l. A decoupled odel ples that the eal l powe,, and eactve powe,, ae analyzed ndependently of each othe. Substtutng nto 3. yelds 2 2 G V + V 2V V cos X. 3.2 Ll l l l l l Fnally, n pe unt, the voltage agntudes at each bus vay slghtly above and below one. So, f we set V V and substtute nto 3.2, we get as an appoaton to the tue Ohc losses, l Ll G cos X, l l l and n 3.3, the only unnown quantty s. We ll ncopoate ths equaton nto ou l odel. We stat by defnng the unnown state vaable, as the powe assocated th a l patcula banch. Thus, we can epess the eal powe easueents as a functon of the unnown eal powe. Matheatcally, we have l + ε l l, and l + L + ε l l l + G l l cos X l + ε l l 2,

6 whee l s the powe easueent on the lne nea node equal to the tue powe on the lne, and l s the powe nea node l equal to the negatve tue powe on the lne, plus Ohc losses. Fo the njectons at bus and bus l, we obtan usng Kchhoff s Cuent Law l and l N, l l N l whee N and Nl ae the set of all banches connected fo node and node l, espectvely. Ths s atheatcally statng that the njecton powe at a node s the su of the powe on the lnes leavng that patcula node. As a ule fo state vaable assgnent, f we obtan one obsevaton fo banch -l, then that obsevaton, l, s assocated th the state vaable, In essence, we ae pcng a decton of the state, o a decton of the powe flow. Fgue 3.3 shows ths fo just one easueent on the lne. So, egadless of whethe the easueent was taen on the sendng end whee the powe s cong fo o the ecevng end whee the powe s gong to, the easueent s assgned to a sendng end, even f the easueent was taen on the tue ecevng end. In suay, fo just one easueent on the lne, l Fo zeo o two easueents on a lne, the decton of the state s chosen abtaly. Now, assue we have n b banches and eal powe easueents, n < b. l. l. We can elate the easueents to the nonlnea equatons defned n 3.4, whch can be epessed atheatcally as z h + e, whee z s an vecto of easueents, h s an vecto of equatons elatng the state vaables to the easueents, s the n b vecto of unnowns, and e 28

7 s an vecto of ando eos th ean zeo and a nown covaance at R dag σ,..., σ 2 2. We can fo the Jacoban at of patal devatves by defnng h, and specfcally, we have l l l, + 2X lgl sn X l, and l l l l N l l. -l l l Fgue 3.3. State Vaable Assgnent fo One Obsevaton 3.4 Modelng the Reactve owe The odel fo the eactve powe s vey sla to the odel deved fo the eal powe n Secton 3.3. Agan, evey banch n the netwo has a coespondng unnown state vaable, and hee t s the eactve powe flong fo bus to bus l. As th the eal powe, we can epess the eactve powe on both sdes of the banch as +, l, l l, 3.5 l l, l l L l l N N l 29

8 whee N, Nl ae defned as befoe, and The loss of eactve powe s epessed as L s the loss of eactve powe fo the banch -l. l L l B V + V B V + V 2V V cos θ, s l l l l l l whee B epesents the nown susceptance of the lne as defned n Append A, V s the voltage agntude, and θ θ θ s the dffeence n voltage angles between bus and bus l. The l l subscpt s-l on the fst susceptance te stands fo the shunt of lne -l. The shunt does not have any esstance, thus fo the eal powe loss descbed n Secton 3.3, a te G does not s l est. Eactly as th the eal powe loss, substtutng θ l th X l and settng V l Vl pu yelds Ll B 2B cos X s l l l l Notce that ths loss s dependent on the tue eal powe flow, l. We estate the eal powe flow th ˆ and substtute t bac nto 3.6 to obtan ˆ 2B 2B cos X ˆ. l Ll s l l l l Theefoe, the eactve powe on the l- sde of the lne s now epessed as l + ˆ. l L l Though sulatons, we wee pleased to see that ˆ L s qute accuate at estatng the tue l eactve powe loss Ll, snce ths s an appoaton to an appoaton. Because t s ndependent of the state vaable obtaned. l, t can be calculated as soon as the estate of eal powe s 30

9 The devaton of the eactve odel agan follows sut th the eal powe odel. We assue we have n b banches and eactve powe easueents, n < b. We can elate the easueents to the nonlnea equatons defned n 3.5, whch can be epessed atheatcally as z h + e, whee z s an vecto of easueents, h s an vecto of equatons elatng the state vaables to the easueents, s the n b vecto of unnowns, and e s an vecto of ando eos th ean zeo and a nown covaance at R dag σ,..., σ 2 2. We can fo the Jacoban at of patal devatves by defnng h, and specfcally, we have l l l,, and l l l N l l. It should be noted that any leveage ponts do not nfluence the eal and eactve odels deved fo detectng topology eos. Leveage ponts ae coon n classcal state estaton and ae due to the agntude of the eactance of a lne. Whethe t s vey lage o vey sall, t can have a daatc effect on the odel at. In ou odel, the entes of the odel at ae bascally s, 0 s, and s n the desgn at the negatve entes can be slghtly salle o lage than. 3

10 3.5 The ube Estato and IRLS Algoth ete ube ntoduced the concept of M-estaton 964 fo estatng a locaton paaete th non-noal eos. e developed an estato that pefos le a least squaes estato fo sall esduals below a specfc cutoff value, λ and pefos le a least absolute value estato fo lage esduals above λ. Thus, the ube estato s a two-pat functon of the esduals and we dese to nze the su of the functons of the esduals. Specfcally, f we let J be the objectve functon that we would le to nze, then the soluton that aes J a nu s the estate of, ˆ. In geneal, J s gven by whee pecsely fo the ube estato, J ρ, 2, λ 2 ρ. 2 c λ, > λ 2 ee, / σ s the th escaled esdual, z h s the th esdual, and λ s an abtay cutoff value whch can ange fo.0 to 3.0. At the Gaussan dstbuton, the lowe the cutoff value, the oe obust the estato s n the pesence of outlyng data; the hghe the cutoff value, the oe effcent the estato s th egad to the vaance of the estates. Also, as the cutoff value deceases, the oe the estato becoes an to the least absolute value estato; and as the cutoff value nceases, the oe the estato becoes an to the least squaes estato. The estate ˆ s the soluton to J / 0, and n patcula, 32

11 J ρ ρ ψ z h ψ ψ σ z h ψ " ψ 0. σ We can ultply each sde by to get d of the negatve sgn. Fnally, we have, " ψ 0, 3.7 whee T " s the th ow of the weghted Jacoban at, R, and ψ s ube s ψ- functon, the devatve of ρ th espect to. Ths s epessed as, λ ψ. λ, > λ One ethod to solve 3.7 s to use the teated eweghted least squaes algoth IRLS. To set the foundaton fo ths algoth, we fst ultply and dvde 3.7 by. Ths yelds σ " ψ " q 0, whee q ψ / s a weght functon bounded between 0 and. We can ewte ths n at notaton as T R 0, 3.8 whee z h and s the dagonal at of weghts, q. We can appoate h by a fst ode Taylo sees epanson about the pont by 33

12 34 h h +, whee s the estate of at the th teaton. Substtutng nto 3.8 gves, at the st + teaton, 0 + T h z R, o 0 + T T R h z R, usng fo shothand notaton, A A, fo a at A dependent on, T T R h z R +, T T R R. 3.9 In 3.9, +, and h z. Convegence s eached when all of the, b,...,n, ae below soe ctcal theshold o toleance, ε. The IRLS algoth poved nuecally stable and showed good convegence ates n all sulatons pefoed. Once the state vecto s estated, the etee outles ae dentfed, deleted fo the easueent set and one eecuton of the IRLS algoth s pefoed fo the pevous step. We delete the outlyng easueents to cancel out any nfluence on the estates of the state. Due to possble low edundancy, only those easueents that have a weghted esdual th absolute apltude of, say, at least 6.0, ae copletely deleted fo the easueent set. 3.6 Testng the Estates Once the powe flows ae estated, a statstcal test needs to be appled n ode to vefy a lne s status. We ae not nteested n testng the ndvdual eleents n the estated state

13 vecto, pe se, but athe the apltudes of powe on the lne. To be oe specfc, we dese to test the powe at the sendng end. Fo eaple, the powe could leave node, the sendng end, and ente node l, the ecevng end, but we only have an obsevaton at the ecevng end. Thus the unnown state vaable s assocated th that easueent on the ecevng end, as descbed n Secton 3.3. If the lne -l has a esstance and/o eactance assocated th t, then powe has been lost ove the couse of tavel though the lne due to fcton. As a consequence, ou estates of the eal and eactve powe fo that lne ay be undeestatng the tue values. oweve, afte obtanng the estates of eal powe flow, ˆ, we can estate any loss of powe on the lne, both eal and eactve. Once we obtan ths estated loss, we add t bac to the state estate of powe fo banch -l, f necessay, theeby obtanng an estate of powe fo the sendng end. The state vaable fo banch -l ay aleady be assocated th the tue sendng end, as descbed n Secton 3.3. If ths s the case, then we do not need to add bac possble losses. In suay, we dese to estate an obsevaton on the sendng end, whee the powe just entes the lne, whethe a easueent was taen thee o not. By addng bac any possble losses that have accuulated, we ncease the statstcal powe of the statstcal test, whch n tun deceases the nube of cases whee the lne s wongly tagged as dsconnected. In essence, we assue nothng about the topology of the netwo and let the easueents estate the topology. To contnue, let ẑ be the vecto of estated apltudes of powe flow, the sendng end estates, and let the standadzed estates of apltude fo banch j, say, be z ˆ j / sˆ, j,,, whee z j z j ŝ s the standad eo fo the flow estate, ẑ j. To deve the standad eo, we fst wte ẑ as a functon of the state vecto ˆ. That s, ẑ h ˆ, and h ˆ 35

14 s the coespondng nonlnea vecto functon fo each of the values of ẑ. If we pefo a fst ode Taylo sees epanson of h about ˆ, we get h h ˆ + ˆ ˆ, o h ˆ h ˆ ˆ, whee ˆ h / evaluated at ˆ. Replacng ths epesson fo ẑ h ˆ, we get z ˆ h ˆ ˆ. Thus, the covaance at of ẑ can be found by Cov zˆ Cov h ˆ ˆ, Cov ˆ ˆ, Cov ˆ ˆ T Cov T S. 3.0 Fo M-estates, ube 973 deved the asyptotc covaance at of ˆ and t s gven by Cov ˆ sˆ R 2 T, and ŝ s a scale estate based on the weghted esduals and s calculated by 2 ψ 2 s ˆ, 2 ψ whee 2 ψ s the squae of the ψ-functon, and ψ s the fst devatve th espect to the esdual,. So, s ˆ2 z j s the j th dagonal eleent of S n 3.0. The standadzed flow estates found by z ˆ / ˆ ae then copaed to a ctcal value whch can vay between 2.5 and 3.0. If j s z j the estate s salle than the ctcal value, the lne s labeled as off, and f the estate s 36

15 lage than the ctcal value, the lne s labeled as on. Monte Calo sulatons showed that the scale estate anged anywhee fo..4 fo dffeent systes, but fo pactcal puposes we set 2 s ˆ Zeo-Injectons One poble that occus n pactce deals th the concept of edundancy. As entoned n Chapte 2, edundancy s the ato of the nube of obsevatons elatve to the nube of state vaables paaetes to be estated. In powe systes, the edundancy can be qute low fo the classcal state estaton odel when not accountng fo any zeo-njectons. Low edundancy can cause pobles n obsevng the netwo whethe all state vaables can be estated, as well as pobles th adequate estaton of eo fo statstcal dagnostcs. Fo the obust topology eo dentfcaton ethod just descbed, the nube of unnown state vaables s about.5 tes geate than the nube of vaables n classcal state estaton. Ths, n tun, deceases the edundancy fo the topology eo ethod by about 33%. Fo soe utlty copanes, eta easueents necessay to suffcently estate the topology ay be vey had to acque. One technque to ncease the nube of easueents s to use zeo-njectons. Fo ou ethodology, zeo njectons ae an ecellent addton to the easueent set. A zeo njecton s the addton of one powe easueent, assgned the value of zeo, to all banches n the netwo that we now ae not enegzed. Ths appoach ads th estaton and asssts th the convegence of the algoth va addtonal obsevatons. Snce these ae pefect easueents.e. no eo, Kchhoff s Cuent Law s now stengthened aound that aea of the netwo fo the dentfcaton of outles as well as fo accuate estaton of powe on banches not 37

16 assocated th any easueents. Ths s due to the addtonal weght gven n the at R fo those zeo easueents, although ths s not the only way to handle these types of easueents. 3.8 Sulaton Results We appled the topology eo dentfcaton to a 20-bus syste that was deved fo the standad IEEE 8-bus syste. Specfcally, we alteed bus and bus 24 by ang the two buses that wee connected by a sngle bus couple. So, bus 9 s coupled th bus and bus 20 s coupled th bus 24. These buses only splt n one way.e. the couple s on o off, thus the buses ae connected o not. We odel the shot ccut banches, the bus couples, as actual lnes n ou odel, and odel all othe buses as supe-nodes. Restatng, a supe-node s a bus that s actually a congloeate of salle buses. In Fgue 3.2, f we consdeed bus 0 and bus 2 to be connected, then t would consttute a supe-node. The easueent confguaton conssted of 240 pas of and easueents along th 9 zeo njectons fo a total of 259 pas of obsevatons. The dffeence between a egula easueent and a zeo njecton s that the zeo njecton s a pefect easueent.e. no eo and consequently s gven a uch hghe weght n R, the dagonal at of nown easueent vaances. The netwo has 8 banches, plyng thee ae 8 pas of powe flows that need to be estated, the state vaables fo the and odels. Seveal cases of topology eos and goss easueents wee ntoduced to the syste, and we ll go nto specfc detal th one patcula case. We ntoduced 9 topology eos and 8 bad easueents n the syste 4 bad -easueents and 4 bad -easueents. Fo the topology eos, we assued that lnes 27-5, 28-9, and ae connected when n fact they 38

17 ae dsconnected. In addton we assue that lnes 5-9, 49-54, and ae dsconnected when n fact they ae connected. We also assued that the bus couple between bus and bus 9 s closed whle n fact t s open and the bus couple between s open whle n fact t s closed. Fnally, we assued that the couple between buses 0 and 2, as depcted n Fgue 3.2, s closed t was odeled as a supe-node. The njecton easueent at bus 0 s only assocated th the powe on lnes connected to bus 03 and bus 09. Now, snce bus 0 s epesented as a supe-node, thus the njecton s the su of powe assocated th lnes to buses 03, 09,, and 2, we epect that the ube estato ejects the eal and eactve powe njecton easueents at that bus as bad data. Ths s due to Kchhoff s Cuent Law. It does that effectvely by povdng these njecton easueents th escaled esduals of -7.6 and 5.56, espectvely. In the second step of the pocedue, the substaton assocated th bus 0 s epesented n detal as shown n Fgue open ccut beae closed ccut beae zeo njecton powe easueent Fgue 3.4. Detaled Substaton 39

18 We odel the ccut beaes n the substaton based on the poneeng schee poposed by Ivng and Stelng 982 coupled th the wo by Montcell 993 whee the ccut beaes ae odeled as lnes connected to zeo njecton buses. Ths now leads to a syste th 25 buses and 90 banches. The ube estato s then eecuted. The easueents th absolute weghted esduals lage than 3 ae dsplayed n Table 3.. We obseve n Table 3.2 that all of the 8 bad data have been popely ejected. The etee bad data th / > 6.0 ae then deleted and one teaton of the IRLS algoth was caed out. The σ esults of the statstcal test appled to the flow estates ae suazed n Table 3.. We see that the ethod dentfes coectly whch banches ae connected and whch ae not. Fo nstance, t dentfes that banches 0-2, 0-22, 0-23, 2-24, and 2-25 ae off, evealng that bus 0 splts n two sepaate buses. Table 3.. owe Rescaled Resduals fo the 25-Bus Syste. Measueent Eact value Est. value Weght. Resd. FL FL FL FL FL IN FL FL FL IN IN Note: Bold epesents the outlyng data ntoduced nto the syste 40

19 Table 3.2. Standadzed Flows and Banch Statuses Banch l / s l z l / s l z Status ON ON ON OFF OFF OFF ON OFF OFF OFF ON OFF ON ON OFF ON OFF 4