State Estimation, Observability and BdD Bad Data

Transcription

1 State Estimation, Obseability and BdD Bad Data EE 5 Analysis of Powe Systems Cen-Cing Liu, Boeing Distinguised Pofesso Wasington State Uniesity

2 Concept of State Estimation Wood, Wollenbeg M mete 65 MW M3 6 MW 4 MW M3 MW 5 MW 35 MW UE VALUE Use te measuements M 3 = 5 MW =.5 p.u. and M 3 = 4 MW =.4 p.u..5 p.u p.u. 3 3, we can find 3., 3. wo measuements ae sufficient. is eample assumes metes ae pefectly accuate.

3 If all 3 metes ae some eos. Fo eample, M = 6 MW M 3 = 6 MW M 3 = 37 MW If only M 3 & M 3 ae used, use DC load flow to sow M 58.5 MW 9.55 MW 64.5 MW 65 M3 6 MW 5 37 MW 4 MACED UE Values in paenteses MACED M3 3 3 MW 35

4 If M & M3 used instead, flows sown in boes M mete MW 95.5 M MACED M3 37 MW 37 4 MACED We need a pocedue tat uses info aailable fom all 3 metes to poduce te best estimate t

5 Maimum Likeliood + X tue - olts meas tue Want to estimate te alue of te oltage souce tue Note { mean alue of Z meas }=Z tue if Note { mean alue of Z meas }=Z tue if as eo mean

6 Nomal Distibution wit Zeo Mean meas tue Pobability density function PDF is nomal andom eo PDF ep - standad deiation aiance PDF Small bette measuement quality

7 So te PDF of Z meas PDF ep meas If is know, ten PDF ep tue meas Want to find an estimate tat maimies te pobability density tat te obseed measuement Z meas would occu

8 eefoe, we can meas PDF log ma meas log - ma meas o ma min meas o min

9 Find est by meas meas est o

10 State Estimation eal-time measuements: MW, MVA line flows MW, MVA bus injections Bus oltage magnitudes Line cuents Data Eos: Failues in measuing o telemety equipment Eos in te measuing instuments N Noise in te communication system Delays in te data tansmission Basic Steps: ypotesie a system model Estimate te state aiables Detect te pesence of bad data Identify te bad data

11 Benefits of State Estimation: Can educe # of tansduces and emote teminal units. Bad data detection and identification.

12 State Estimation: Poblem Fomulation Measuements Stuctue S Paamete Values p SAIC SAE ESIMAO Compute Algoitms Estimated system state & model Notation: t: continuous, t n : discete n=,, Unknown independent aiables: y ~ t Node injection Comple, nonstationay, enionmentally dependent ecto-alued stocastic pocess t ~ ue stuctue Binay type stocastic pocess t ~ ue paamete alues Line, tansfome, mete data

13 Unknown State Vaiables: Node Voltages e powe flow equation now eads y ~ t f t, t, t Known quantities: Measuements ~ ~ tn ~ line flows, bus injections, oltage magnitudes Pseudo measuements nontelemeteed infomation on line flows, injections, oltages. Paamete alues p t n note tat p t n t n t n ~ t n : Paamete alue eo Stuctue S t n eo S t n t n C t n

14 Model fo measuement: ~ ~, ~, ~ t t t t t t ~ n n n n n tn b Wee : Eo due to mete constaints, diffeence tn between [ ] and te actual powe system, mete nonlineaities, miscalibation noises, A/D conesion eos, communication noises, etc. b tn : bad data due to tansient swings, g, majo mete-communication failues. n

15 Static State Estimato: ypotesie model: detemine. ypotesie model: detemine,, t t n n p s and te measuement eo coaiance mati t t E. State estimate alue of wic minimies te t t n n E : ^ t n esidual t t t n n n 3. Detection: est to decide if bad data o stuctual eo eist 4. Identification:est to decide wic measuements o stuctual data ae incoect

16 Fou Basic Opeations of Static State Estimation SCADA Paamete p Measuement Stuctue s ypotesie model assume c, b obtain,, p Y END ^ Estimation: i Find minimie i i Detection: Ceck assumptions c, b, by testing if is small enoug No Identification: Logic to detemine location of bad data & stuctual eo Modify input

17 Estimation: Weigted least Squae WLSAlgoitm M t M d l Measuement Model sym ~ ~ E b Suppose no bad data b=, ten ~ ~ ~ ~ o minimie te esidual We minimie ~ ~ ] [ ] [ J

18 Wy: Assume tat te n components of ae jointly Gaussian, ten te joint pobability density function is gien by y ~ function is gien by ep / / y y y y f m m y p det / / ~ y y y y y f m m y n wee is te coaiance mati and is te mean of m y ~ y ~

19 Conside ~ ep det / / ~ f n o maimie, we need to minimie ~ f ] [ ] [ J ma likeliood estimate ] [ ] [ J PDF PDF ^

20 J wee Necessay condition fo optimality J Wee is te Jacobian o sole, te following iteatie algoitm is used 3 ] [ ] [ G i i i i i Note tat if i= coneges to ten ^, i i ^ Note tat if, i=,,coneges to ten i ] [

21 Question: ow to coose G i Gain mati ^ ^ Suppose, ten 3&4 yields aylo seies 4 i i i ^ i ^ i i i C it 3 i i Compae wit 3 i i i G Infomation mati

22 An Eample of Weigted Least Squae Algoitm M MW 65 MW M3 M3 35 MW X =. X 3 =.4 X 3 =.5 3 Mete Caacteistics: full scale alue MW accuacy 3MW

23 Intepetation: Nomal distibution density N, -3-3MW Mean Pob{-3 3} = MW e mete will gie a eading witin inteal fo 99% of te time. = MW =. p.u.

24 DC Load Flow 3 3 B B B B B B B B Mati Coaiance Mati Coaiance 4-3 3

25 WLS Estimate fom eq ^ ^ ^ esidual J, [ ] [.4 ^ ^ ]

26 eading: Powe Geneation, Opeation & Contol Jon Wiley & Sons, A. Wood, B. Wollenbeg, 996, Capte.

27 Netwok Obseability Wu and Monticelli DC load flow p Bθ Line flows θ θ f ij B ij i j Let Y=diag{B ij }, ten te ecto of line flow measuements can be witten as θ e tue line flows ae gien by f YA θ Wee Y=diag{B ij } j, A include te datum node uneduced

28 A netwok is said to be obseable if any flow in te netwok can be obseed by some sot of indication i in te set of measuements. In ote wods, wenee tee is any noneo In ote wods, wenee tee is any noneo flow in te netwok, at least one of te measuements sould be noneo.

29 Definition: A netwok is said to be obseable if fo all suc tat te line flows f f YA Any state fo wic Is called an unobseable state., A Fo an unobseable state Let A, if ten banc i, j is an ij unobseable banc.

30 Q:ow to ceck te obseability of te netwok? We will sow tat e netwok is obseable if and only if as a full ank, wee is obtained fom by deleting any column. A Wy: A:uneduced wee any eal numbe and = [,, ] Ift te netwok kis obseable, bl ten A

31 So we obtain i t l i i.e., te column sum is eo Define = o sow as a full ank Now suppose Let ten o ence is of full ank. o

32 On te ote and, if is of full ank, and, ten Z k k k Now k Now te column sum of is always equal to eo wy y ence o obseabil ity teefoe, A k

33 k k = k k k = - k Squae Squae k k

34 Eample Li Line 4 Li Line Line i Line Node Incidence Mati 3 A

35 Delete one column of ank = is not of full ank

36 as a ank of

37 A Now suppose eo measuements 4 4 en, and Line flows Unobseable bances Unobseable state: = 4, = 3

38 Solability of state estimation s Netwok Ob bilit C id l t l it Obseability. Conside least squae algoitm J J ^ O Now if ^ n ef

39 Now suppose te netwok is obseable, ten is of full ank, teefoe eists, and is soled uniquely. Solability Pseudo measuements ae often used fo estimation of unobseable states.

40 Identification of Bad Data Weigted Least Squae J e WLS estimate satisfies m i i i e WLS estimate satisfies d dj J bi t i d now Jacobian te is wee d b d I t mete is bad b d: sie of bad data

41 Define te state estimation eo Suppose is small, and &, en,, Also, fo,coaiance mati wee wy? E X

42 Now fo te esidual ecto I Wee te esidual sensitiity mati W I W

43 Fo, te esidual coaiance E W E wy? Now define te diagonal mati diag D Weigted esidual g Nomalied esidual w Nomalied esidual D N

44 en fo, coaiances Uw w E W And U N E N D D N Note tat diagonal elements of U N ae all equal to to.

45 Nomalied esidual sensitiity mati W N D W Suppose W N, jk is te j, k element. en Now so so W j, k, W ofw, kk W N, N jk j, W k column ow D W k t ow jt column element of W k t ow j t column, jk /,,, jk j kk jk j

46 wit and SOW W - W= - W wy? J W Z If is nomal, ten as Ci-squaed distibution wit J K= m n degees of feedom m is # of measuements, n is # of states. As K become lage K>>3, te standadied andom aiables J K K become eo mean and nomal N,.

47 Bad data detection ypoteses : no bad data o stuctual eos : is not tue Let P e : pob. of ejecting wen is actually tue False alam P d : pob. of accepting wen is tue detection ti

48 . J - testt Accept if eject accept, otewise wen is nomal, K is lage and is tue, ten is N,. So = coesponds to P e =.5

49 . N test Accept if eject otewise N, k, k,..., m 3. w test Accept is eject otewise w, k, k,,..., m

50 Fo single o multiple non-inteacting Bad data Odeed esidual seac: N weigted nomalied esiduals w N put into a descending ode of magnitude. e measuement i coesponding to te lagest esidual ma w ma n is emoed fist, etc. Gouped esidual seac: te p lagest esiduals in w o N emoed simultaneously and ten put back one afte anote until bad data is detected.

51 efeences [] Bad Data Analysis fo Powe System State Estimation, E. andscin et al. IEEE ans. Powe Appaatus and Systems, Ma/Apil 975, pp [] A, Monticelli, Electic Powe System State t Estimation, Poceedings of te IEEE, Feb., pp. 6-8.