State Estimation. Ali Abur Northeastern University, USA. Nov. 01, 2017 Fall 2017 CURENT Course Lecture Notes

Transcription

1 State Estmaton Al Abu Notheasten Unvesty, USA Nov. 0, 07 Fall 07 CURENT Couse Lectue Notes

2 Opeatng States of a Powe System Al Abu NORMAL STATE SECURE o INSECURE RESTORATIVE STATE EMERGENCY STATE PARTIAL OR TOTAL BLACKOUT OPERATIONAL LIMITS ARE VIOLATED

3 Enegy Management System Applcatons SCADA / EMS Confguaton Al Abu Measuements State Estmaton Topology Pocesso Secuty Montong Load Foecastng Extenal Equvalents Emegency Contol Restoatve Contol Contngency Analyss On-lne Powe Flow Secue Y STOP N Secuty Constaned OPF Peventve Acton

4 State Estmaton and Related Functons Weghted Least Squaes (WLS Estmato Al Abu Analog Measuements P, Q, P f, Q f, V, I, θ k, δ k Topology Pocesso State Estmato (WLS V, θ Bad Data Pocesso Load Foecasts Geneaton Schedules Netwok Obsevablty Analyss Pseudo Measuements [ njectons: P, Q ] Ccut Beake Status Assumed o Montoed

5 Powe System State Estmaton Poblem Statement Al Abu [] : Measuements P-Q njectons P-Q flows V magntude, I magntude [x] : States V, θ, Taps (paametes EXAMPLE: [] [ P; P3; P3; P; P; P3; V; Q; Q3; Q3; Q; Q; Q3 ] m 3 (no. of measuements [x] [ V; V; V3; θ; θ3 ] n 5 (no. of states

6 Netwok Model Bus/banch and bus/beake Models Al Abu Bus/Beake Bus/Banch Topology Pocesso

7 Measuements Bus/banch and bus/beake Models Al Abu Bus/banch Bus/Beake V

8 Measuement Model [ m ] [h([x]] + [e] Al Abu +e +e State Estmato 3 +e 3 : tue measuement e : measuement eo e e s + e systematc andom

9 Measuement Model Al Abu Assumptons e ~ N ( 0, σ olds tue f: e s 0, e ~ N ( 0, σ If e s 0, then E(e 0,.e. SE wll be based!

10 Maxmum Lkelhood Estmato (MLE Lkelhood Functon Al Abu Consde the andom vaables X, X,, X n p.d.f of f(x θ, whee θ s unknown. wth a The jont p.d.f of a set of andom obsevatons x { x, x,, x n } wll be expessed as: f n ( x θ f (x θ f(x θ f(x n θ Ths jont p.d.f s efeed to as the Lkelhood Functon. The value of θ, whch wll maxme the functon fn( x θ wll be called the Maxmum Lkelhood Estmato (MLE of θ.

11 Maxmum Lkelhood Estmato (MLE Maxmum Lkelhood Estmato Al Abu } ( exp{ ( σ µ πσ f ( ( ( ( m m m m m f f f f m m m m m f f L log log ( ( log ( log σ π σ µ Nomal (Gaussan Densty Functon, f( Lkelhood Functon, f m ( Log-Lkelhood Functon, L

12 Maxmum Lkelhood Estmato (MLE Weghted Least Squaes (WLS Estmato Al Abu Gven the set of obsevatons,,, n MLE wll be the soluton to the followng: Mnme Subject to m W h Maxme OR Mnme ( x + f m m ( µ σ ( Defnng a new vaable, measuement esdual:,.., m W σ µ E( h ( x The soluton of the above optmaton poblem s called the weghted least squaes (WLS estmato fo x.

13 Maxmum Lkelhood Estmato (MLE Weghted Least Squaes (WLS Estmato Al Abu Q Q Q P P P V x x x x x h x x x h θ Decoupled Lnea Models ( ( ( ( ( Usng Taylo expanson :

14 Maxmum Lkelhood Estmato (MLE Weghted Least Squaes (WLS Estmato Al Abu ] [ ] [ ] [ ] Subject to [ Mnme x W m + } { W dag W W σ Lnea ncemental poblem: Iteatvely Solve fo x: ε + + x Untl x] [ ] [x ] [x ] [ ] [ ] [ ] [ x] [ k k W G T ] [ ] [ ] [ ] [ W G T

15 Powe Flow and State Estmaton Compason Al Abu Powe Flow State Estmaton Temnaton cteon Bus P,Q msmatch ΔX State ncement Fomulaton Detemnstc Stochastc Soluton Depends on choce of slack bus Independent of efeence bus selecton Bus Types Impotant Ielevant Loads Modeled Not used Geneato Lmts Modeled Not used Tansmsson System Modeled Modeled

16 Measuement Model Al Abu Gven a set of measuements, [] and the coect netwok topology/paametes: [] [h ([x] ] + [e] Measuements: Contan eos Tue System States: Unknown! Measuement Eos: Unknown!

17 Measuement Model Al Abu Followng the state estmaton, the estmated state wll be denoted by [ xˆ ]: [] [h ([ xˆ ] ] + [] Measuements: Contan eos Estmated System States Measuement Resduals: Computed

18 Smple Example 3 4 Z h θ + e Z.0 / 0.0 / θ SLOPEθ* h h h 3 h 4 h : ESTIMATED MEASUREMENT : MEASURED VALUE : MEASUREMENT RESIDUAL Z h θ* Al Abu

19 Weghted Least Squaes (WLS Estmaton Al Abu Mnme + ω + ω33 ω44 ω + What ae weghts, w? ω.0 σ ow ae they chosen? σ Assumed eo vaance of measuement.

20 Netwok Obsevablty Defntons Al Abu Fully obsevable netwok: A powe system s sad to be fully obsevable f voltage phasos at all system buses can be unquely estmated usng the avalable measuements.

21 Netwok Obsevablty Necessay and Suffcent Condtons Al Abu n mn m n n m p Z θ θ θ 3 3 m n NECESSARY BUT NOT SUFFICIENT EXAMPLE: m, n, UNOBSERVABLE SYSTEM Rank( n SUFFICIENT [ 3 ] θ θ State Vecto ] [ ] [ ] [ ] [ ˆ P T Z W G θ Sngula Matx! Cannot be nveted.

22 Measuement Classfcaton Types of Measuements Al Abu. CRITICAL MEASUREMENTS If they ae lost o tempoaly unavalable, the system wll no longe be obsevable, thus state estmaton can not be executed If they have goss eos, they can not be detected Measuement esduals wll always be equal to eo,.e. ctcal measuements wll be pefectly satsfed by the estmated state. REDUNDANT MEASUREMENTS CAN BE REMOVED WITOUT AFFECTING NETWORK OBSERVABILITY

23 Netwok Obsevablty Defntons Al Abu Unobsevable banch: If the system s found not to be obsevable, t wll mply that thee ae unobsevable banches whose powe flows can not be detemned. Obsevable sland: Unobsevable banches connect obsevable slands of an unobsevable system. State of each obsevable sland can be estmated usng any one of the buses n that sland as the efeence bus.

24 Netwok Obsevablty Defntons Al Abu Obsevable Islands RED LINES: Unobsevable Banches

25 Megng Obsevable Islands Pseudo-measuements Al Abu If the system s found unobsevable, use pseudo-measuements n ode to mege obsevable slands. Pseudo-measuements: Foecasted bus loads Scheduled geneaton Select pseudo-measuements such that they ae ctcal. Eos n ctcal measuements do not popagate to the esduals of the othe (edundant measuements.

26 Obsevable Islands Unobsevable Banches Al Abu ISLAND ISLAND ISLAND 3

27 Robust (eslent Estmaton Reslency: A Smat Gd Requement Al Abu If an estmato emans nsenstve to a fnte numbe of eos n the measuements, then t s consdeed to be obust. Example: Gven { 0.9, 0.95,.05,.07,.09 }, estmate usng the followng estmatos:.. Xˆ Xˆ mean{ } 5 medan{ }, Soluton: Replace 5.09 by an nfntely lage numbe 5. 5 The new estmate wll then be: ˆ 5 Ths estmato s NOT obust. Replace both 5 and 4 by nfnty. a b The new estmate wll then be: Ths s a moe obust estmato than the one above. 5,...,5 X a X ˆ b.05 (fnte

28 Robust Estmaton M-Estmatos Al Abu M-Estmatos (ube 964 Consde the poblem: Mnme Subject to m ρ( h( x + Whee ρ( s a chosen functon of the measuement esdual In the specal case of the WLS state estmaton: ρ ( σ

29 Robust Estmaton M-Estmatos Al Abu Some Examples of M-Estmatos othewse ( a a σ σ σ ρ othewse ( a a a σ σ σ σ ρ Quadatc-Constant Quadatc-Lnea ρ( Least Absolute Value (LAV

30 Robust Estmaton LAV Estmato Example Al Abu Measuement Model: A x + A x + e,..., 5 Measuements: Z A A LAV estmate fo x and measuement esduals: x T T [3.005; 4.00] [ 0.0; 0.05; 0.0; 0.0; 0.0] CANGE measuement 5 fom 5.0 to 5.0 ( Smulated Bad Datum : T x [3.0; 4.0] LAV estmate fo x and measuement esduals: T [ 0.0; 0.0; 0.045; 0.0; 9.98 ]

31 Robust Estmaton LAV Estmato Example Al Abu Measuement Model: A x + A x + e,..., 5 Measuements: Z A A LAV estmate fo x and measuement esduals: x T T [3.005; 4.00] [ 0.0; 0.05; 0.0; 0.0; 0.0] CANGE measuement 5 fom 5.0 to 5.0 ( Smulated Bad Datum : T x [3.0; 4.0] LAV estmate fo x and measuement esduals: T [ 0.0; 0.0; 0.045; 0.0; 9.98 ]

32 Bad Data Detecton Ch-squaes χ Test Al Abu Consde X, X, X N, a set of N ndependent andom vaables whee: X ~ N(0, Then, a new andom vaable Y wll have a feedom,.e.: χ N X Y ~ χ N dstbuton wth N degees of

33 Bad Data Detecton Al Abu Now, consde the functon f ( x and assumng: e m N R e N m ~ ~ N e R (0, m ( N e f(x wll have a χ dstbuton wth at most (m-n degees of feedom. In a powe system, snce at least n measuements wll have to satsfy the powe balance equatons, at most (m-n of the measuement eos wll be lnealy ndependent.

34 Bad Data Detecton Detecton Algothm χ --Test Al Abu Solve the WLS estmaton poblem and compute the objectve functon: m ( h ( x J ( x Look up the value coespondng to p (e.g. 95 % pobablty and (m-n degees of feedom, fom the Ch-squaes dstbuton table. Let ths value be χ ( mn, p ee: p P{ J( x χ σ ( mn, p } Test f J ( x χ( mn, p If yes, then bad data ae detected. Else, the measuements ae not suspected to contan bad data.

35 Bad Data Identfcaton Popetes of Measuement Resduals Al Abu Lnea measuement model: K s called the hat matx. Now, the measuement esduals can be expessed as follows: whee S s called the esdual senstvty matx. (, ˆ ˆ R R K K x T T R R x T T ( ˆ Se e K I e x K I K I + ] [Note that K ( ( ( ( ˆ

36 Bad Data Identfcaton Dstbuton of Measuement Resduals Al Abu The esdual covaance matx Ω can be wtten as: E[ S T ] Ω S E[ e e ] S R S T S ence, the nomaled value of the esdual fo measuement wll be gven by: R N Ω The ow/column of S coespondng to a ctcal measuement wll be eo. If thee s a sngle eo n the measuement set (povded that t s not a ctcal measuement the lagest nomaled esdual wll coespond to that eo. R T S T

37 Bad Data Identfcaton Lagest Nomaled Resdual Test Al Abu. Compute the nomaled esduals. Fnd k such that kn s the lagest among all N,,,m.. 3. If kn > c3.0, then the k-th measuement wll be suspected as bad data. Else, stop, no bad data wll be suspected. 4. Elmnate the k-th measuement fom the measuement set and go to step.

38 Use of Synchophaso Measuements Al Abu Gven enough phaso measuements, state estmaton poblem wll become LINEAR, thus can be solved dectly wthout teatons teatve Non Z R R X e X Z Measuements Phaso Iteatve Z R R X e X h Z Measuements Conventonal T T + + ( ˆ ( ˆ (

39 Placng PMUs: Al Abu : Powe Injecton : PMU : Powe Flow : Voltage Magntude

40 Explotng eo njectons Al Abu : Powe Injecton : PMU : Powe Flow : Voltage Magntude

41 Use of Synchophaso Measuements Al Abu Gven at least one phaso measuement, thee wll be no need to use a efeence bus n the poblem fomulaton Gven unlmted numbe of avalable channels pe PMU, t s suffcent to place PMUs at oughly /3 d of the system buses to make the ente system obsevable just by PMUs. Systems No. of eo njectons Numbe of PMUs Ignong eo Injectons Usng eo njectons 4-bus bus bus 0 3 9

42 Pefomance Metcs Al Abu State Estmaton Soluton Accuacy: Vaance of State nvese of the gan matx, [G] - E[ (x x* (x x* ] Convegence: Condton Numbe Rato of the lagest to smallest egenvalue Lage condton numbe mples an ll-condtoned poblem.

43 Pefomance Metcs Al Abu Measuement Desgn Ctcal Measuements: Numbe of ctcal measuements and the types Local Redundancy Numbe of measuements ncdent to a gven bus (N- Robustness Capablty of the measuement confguaton to ende a fully obsevable system dung sngle measuement and banch losses

44 Summay Al Abu State Estmaton and ts elated functons ae evewed. Impotance of measuement desgn s llustated. Commonly used methods of dentfyng and elmnatng bad data ae descbed. Impact of ncopoatng phaso measuements on state estmaton s befly evewed. Metcs fo state estmaton soluton and measuement desgn ae suggested.

45 Powe Educaton Toolbox (P.E.T Powe Flow and State Estmaton Functons Al Abu Fee softwae to: Buld one-lne dagams of powe netwoks Run powe flow studes Run state estmaton

46 Refeences Al Abu F.C. Schweppe and J. Wldes, ``Powe System Statc-State Estmaton, Pat I: Exact Model'', IEEE Tansactons on Powe Appaatus and Systems, Vol.PAS-89, Januay 970, pp.0-5. F.C. Schweppe and D.B. Rom, ``Powe System Statc-State Estmaton, Pat II: Appoxmate Model'', IEEE Tansactons on Powe Appaatus and Systems, Vol.PAS-89, Januay 970, pp F.C. Schweppe, ``Powe System Statc-State Estmaton, Pat III: Implementaton'', IEEE Tansactons on Powe Appaatus and Systems, Vol.PAS-89, Januay 970, pp A. Montcell and A. Gaca, "Fast Decoupled State Estmatos", IEEE Tansactons on Powe Systems, Vol.5, No., pp , May 990. A. Montcell and F.F. Wu, ``Netwok Obsevablty: Theoy'', IEEE Tansactons on PAS, Vol.PAS-04, No.5, May 985, pp

47 Refeences Al Abu A. Montcell and F.F. Wu, ``Netwok Obsevablty: Identfcaton of Obsevable Islands and Measuement Placement'', IEEE Tansactons on PAS, Vol.PAS-04, No.5, May 985, pp G.R. Kumphol, K.A. Clements and P.W. Davs, ``Powe System Obsevablty: A Pactcal Algothm Usng Netwok Topology'', IEEE Tans. on Powe Appaatus and Systems, Vol. PAS-99, No.4, July/Aug. 980, pp A. Gaca, A. Montcell and P. Abeu, ``Fast Decoupled State Estmaton and Bad Data Pocessng'', IEEE Tans. on Powe Appaatus and Systems, Vol. PAS-98, pp , Septembe 979. Xu Be, Yeojun Yoon and A. Abu, Optmal Placement and Utlaton of Phaso Measuements fo State Estmaton, 5th Powe Systems Computaton Confeence Lège (Belgum, August -6, 005.