DRAF and INCOMPLEE able of Contents fom A. P. Saks Melopoulos Powe System Modelng, Analyss and Contol Chapte 7 3 Opeatng State Estmaton 3 7. Intoducton 3 7. SCADA System 4 7.3 System Netwok Confguato 7 7.4 State Estmaton 8 7.4. Least Squaes Soluton 3 7.4. Least Absolute Devaton Soluton (L l Appoach) 8 7.4.3 Chebyshev o Mn-Ma Soluton (L Appoach) 7.4.4 Summay of the hee State Estmaton Appoaches 5 7.4.5 Qualty of the State Estmate 6 7.4.6 Summay and Dscusson 34 7.5 Detecton and Identfcaton of Bad Data 35 7.5. Detecton of Bad Data 35 7.5. Identfcaton of Bad Data 36 7.5.3 Summay and Dscusson 47 7.6 Sequental State Estmatos 48 7.7 State Estmato Obsevablty 55 7.7. Algebac Obsevablty 55 7.7. opologcal Obsevablty 56 7.7.3 Ctcal Measuements 59 7.8 Lmtatons and Bases of State Estmaton 6 7.8. Bas Fom Unbalanced Opeaton 63 7.8. Bas Fom System Asymmety 64 7.8.3 Bas Fom Systematc Measuement Eos 67 7.8.4 Bas fom Measuement me Skews 69 7.9 Synchonzed Measuements 7 7. Fomulaton of the hee-phase State Estmaton 7 7.. hee-phase System State 7 7.. hee-phase Measuements 73 7..3 Least Squaes Estmaton 73
7..4 hee-phase Powe System Model 74 7..5 Obsevablty Analyss hee Phase State Estmaton 75 7..6 Qualty of hee-phase State Estmato 77 7..7 Dscusson of the hee-phase State Estmato 77 7. Hybd hee-phase State Estmato 77 7. Summay and Dscusson 78 7.3 Poblems 79 Page Copyght A. P. Saks Melopoulos 99-6
Chapte 7 7. Intoducton Opeatng State Estmaton Effectve contol and opeaton of electc powe systems eques accuate and elable knowledge of the system model and the opeatng state of the system n eal tme. Fo ths pupose, moden powe systems ae equpped wth an etensve data acquston system. Local analog and status quanttes, such as voltage magntude, eal and eactve powe flows, loads, status of beakes (open/close) etc., ae measued and tansmtted to a cental locaton. he measuements ae smple, equng smple nstumentaton. ypcal analog measuements ae: (a) voltage magntudes, (b) eal and eactve powe flows and (c) cuent magntude measuements. Recent technology based on GPS (Global Postonng System) has made t possble to measue voltage phase angles as well. ypcal status measuements ae: (a) beake status, (b) dsconnect swtch status, etc. All measuements ae taken evey one to seveal seconds. hey ae tansmtted to a cental locaton (the Enegy Management System (EMS) o the Enegy Contol Cente) whee they ae pocessed to yeld the opeatng state of the system. he pocess conssts of two analyss poblems: (a) detemnaton of netwok topology, and (b) detemnaton of opeatng state. he netwok topology s constucted fom the status of beakes and dsconnect swtches. he opeatng state of the system s constucted fom analog measuements by means of two dstnct computatonal pocedues: (a) on-lne powe flow and (b) state estmaton. On-lne powe flows utlze a subset of avalable measuements whch ae enough to defne the powe flow poblem. Soluton of the powe flow poblem yelds the opeatng state of the system. Because measuements ae usually coupted wth eo (esultng fom P o C naccuaces, nstument eo, tansmsson eo, etc.), the eo s dectly tansmtted to the computed opeatng state. It s also possble that one of the measuements may nclude a lage eo (goss eo - due to mete malfunctonng o communcaton eos) esultng n a non-solvable powe flow poblem o n a soluton that may be qute dffeent than the actual opeatng condton of the system. In a eal tme envonment, t s mpotant to have the ablty to dentfy wong measuements o eos n measuements (bad data). Fo ths pupose, t s necessay to take advantage of edundant measuements. he edundant measuements ae utlzed to compute the best estmate of the opeatng state of the system wth statstcal methods. he computatonal pocedue s called state estmaton. As we shall dscuss, state estmaton povdes the mechansm to: (a) detemne whethe the system state can be computed fom estng data (obsevablty); (b) flte out usual measuement eos and, theefoe, compute the system state wth mnmum eo; (c) dentfy and eject bad data; and (d) detemne the degee of confdence on the estmated state of the system. he conceptual vew of the Copyght A. P. Saks Melopoulos 99-6 Page 3
pocess s llustated n Fgue 7.. he two appoaches,.e. on-lne powe flow and state estmaton, ae dentfed. In addton othe applcatons ae possble usng the collected data. hese applcatons ae paamete estmaton and emote calbaton of the metes. he consttuent pats of ths pocess wll be dscussed net. Bad Data Identfcaton & Rejecton Obsevablty Analyss State Estmaton Status Data Analog Data opology Pocesso Goss Bad Data Detecton by Lmt Checks Consstency Checks On-Lne Powe Flow Soluton Paamete Estmaton Remote Calbaton Best Estmate of System State & Model System State Fgue 7. Conceptual Vew of Real me Powe System Modelng and State Estmaton he objectve of ths chapte s to ntoduce the models nvolved n the state estmaton poblem and to dscuss ts vaous etensons and applcatons. Because nomally thee ae edundant measuements, t s possble to use the measuements fo the pupose of detemnng o mpovng the system model paametes o to emotely calbate the metes. 7. SCADA System he necessay hadwae to enable eal tme modelng of a powe system s collectvely efeed to as SCADA system. SCADA stands fo Supevsoy Contol And Data Acquston system. he supevsoy contol subsystem conssts of hadwae and softwae whch (a) collect status data (.e. beake status open/close) and analog data (.e. measuements of voltage magntude, powe etc.) and tansmt these data to a cental locaton fo pocessng and dsplay (b) allow emote tppng of beakes, changes of tansfome tap, etc. In most cases, supevsoy contol s a manual functon,.e., the dspatche at the contol cente wll ntate a command to open/close a beake, etc. he data-acquston subsystem conssts of emote temnal equpment fo ntefacng wth powe system nstumentaton and contol devces; ntefaces wth communcaton channels; and equpment fo ntefacng wth the system contol cente. Snce the SCADA system tansmts data fom the feld to a cental locaton and vce vesa, communcaton meda, potocols and communcaton speeds ae vey mpotant. In Page 4 Copyght A. P. Saks Melopoulos 99-6
the past t was customay to have sepaate communcaton channels fo the Supevsoy Contol and Data Acquston. oday, howeve t s a unfed system shang a common two way communcaton channels that may consst of seveal physcal layes. Communcatons ae ntegated n the emote temnal unt (RU) whch manages data collecton, contol functons and communcaton wth a maste staton. he maste staton has multple communcaton channels to emote temnal unts. Many tmes a dedcated channel s assgned to each emote temnal unt. In othe cases, thee ae less channels than emote temnal unts equng moe than one emote to shae a channel. Analog data s scanned peodcally, typcally evey one second to a few seconds. Each scan s tggeed by the system contol cente at the pescbed nteval by usng a equest to all emote statons to send n data. he amount of data collected and tansmtted s vey lage fo typcal powe systems. hese data s tanspoted va communcaton channels. In ode to mnmze communcaton taffc some fom of data compesson s utlzed. Fo eample, fo status data one can send only changes of status data. hs appoach mnmzes amount of data tansfe and the amount of pocessng needed at the maste staton. Analog data can be also compessed wth a numbe of methods. Independently of system confguaton, SCADA system manufactue, communcaton softwae and compute confguaton, the end esult of the SCADA system functon wll be the collecton of a set of system data evey samplng peod. he data conssts of: Beake status Dsconnect swtch status ansfome tap settng MW flow measuements MVAR flow measuements Voltage magntude (kv) measuements Cuent magntude (ka) measuements phase angle dffeence measuements etc. A smplfed vew of a SCADA system s llustated n Fgues 7.a and 7.b. Copyght A. P. Saks Melopoulos 99-6 Page 5
G G MW Flow Measuement MVAR Flow Measuement kv Measuement Dsconnect Swtch Status Beake Status RU Communcaton Lnk wth Contol Cente Fgue 7.a Smplfed Vew of a SCADA System - Suvey Ponts Contact Inputs Analog Inputs Contact Outputs Analog Outputs RU Data Commands Maste Staton Fgue 7.b Smplfed Vew of a SCADA System - Confguaton At the cental locaton (enegy management system o contol cente), the data ae managed wth the data acquston softwae. hese softwae pefom the followng tasks: (a) ntate the collecton of data and place them n compute memoy, (b) goss eocheckng, (c) conveson to engneeng unts, (d) lmt-checkng, and (e) geneaton of a data base whch s ntefaced wth applcaton pogams. Page 6 Copyght A. P. Saks Melopoulos 99-6
he data ae utlzed to fom the system model (Netwok Confguato) and to estmate the system opeatng state (state estmaton). he net sectons descbe these applcatons. 7.3 System Netwok Confguato Data collected wth the SCADA system ae utlzed n two ways. Status data (ccut beake status, nteupt swtch status, tansfome tap settng, etc.) ae utlzed to fom the system netwok confguaton and model. he softwae whch take the status data and computes the system netwok confguaton and model s known as system netwok confguato. A typcal task pefomed by the system netwok confguato s llustated n Fgues 7.3a and 7.3b. he nfomaton eceved wth the SCADA system detemnes the status of the beakes. he system netwok confguato uses pestoed nfomaton and the beake/swtch status to detemne a bus oented model,.e. whch ccuts ae connected to whch bus and what s the model of each ccut. hs task s llustated n Fgue 7.3. ypcally, ths pocedue s eecuted only when a change n status data occus. he system netwok confguaton and model s net combned wth the analog data fo the pupose of detemnng the opeatng condtons of the system. ypcally, thee ae edundant measuements whch ae used to obtan the best (n some sense to be dscussed late) estmate of the opeatng state of the system. he computatonal pocedue whch pefoms ths task s known as the state estmato and wll be dscussed n the net paagaph. AutoBank 5kV/3kV G AutoBank 5kV/3kV G Fgue 7.3a Beake Oented Model Pestoed Netwok Data Copyght A. P. Saks Melopoulos 99-6 Page 7
SG SG Fgue 7.3b Bus Oented Netwok Model of the System of Fgue 7.3a 7.4 State Estmaton State estmaton s a computatonal pocedue whch uses a edundant set of measuements and a bus oented netwok model to compute a statstcal estmate of the system opeatng state. In a powe system, the opeatng condtons ae unquely defned by the set of vaables consstng of all bus voltage magntudes and angles ecept the phase angle at an abtaly selected bus whch s set equal to zeo. We efe to ths set of vaables as the state of the system. As an eample, the opeatng condton of the system of Fgue 7.4 s defned wth thee state vaables: V, V, and δ. Knowledge of these thee vaables s suffcent to detemne othe quanttes of nteest, fo eample, P, Q. Fo ths eample, assume that a set of edundant measuements s taken as t s shown n Fgue 7.4. (fve measuements: V, V, P, P, and Q ) A subset of these measuements s enough to povde the state vaables. Fo eample assume that the followng thee measuements ae selected: V, P, and V, whee P s the eal powe flow on ccut -. Fom these thee measuements, the state vaables V, V, and δ can be computed. -j5. Voltage Magntude Measuement MW Flow Measuement MVA Flow Measuement Fgue 7.4 A Smplfed Eample System fo Powe System State Estmaton In geneal, a measuable quantty such as P, Q, P, Q, V, etc., can be epessed as a functon of the system state. Let z denote a measued quantty. hen: Page 8 Copyght A. P. Saks Melopoulos 99-6
z h () (7.) whee s the system state and h s a functon specfc to the measued quantty z. Assume that m measuements ae taken. hen: whee z h() (7.) z h s the system state - an n vecto s a vecto of measued quanttes - an m vecto s a vecto functon - an m vecto functon. ypcally moe measuements ae taken than the numbe of state vaables to be detemned,.e. m>n. In ths case, the set of Equatons (7.) epesents an ovedetemned set of nonlnea equatons n eal vaables. In geneal an ovedetemned set of equatons, such as (7.) does not have a soluton fo. Only f the system model epessed wth the functons h() s eact and the measuements z ae etemely accuate, then equatons (7.) have a unque soluton fo. hs of couse s unlkely n a eal system. Yet t s possble to obtan a soluton fo va a pocedue known as the state estmaton. hs pocedue wll be fst ntoduced by an eample and then t wll be dscussed n moe detal. Eample E7.: Consde the smplfed powe system of Fgue E7.. he followng measuements ae taken (all n p.u.): z V., z V.99 z 3 P -.56, z 4 Q. Fomulate the state estmaton poblem as an ovedetemned set of nonlnea equatons. ~ j V Ve ~ jδ V V e -j5. Soluton: he state vecto s: Voltage Magntude Measuement MW Flow Measuement MVA Flow Measuement Fgue E7. Copyght A. P. Saks Melopoulos 99-6 Page 9
V V δ he vecto functon h() s gven as: h ( ) V h ( ) V h 3( ) 5.V V snδ h4 ( ) 5.V 5.V V cosδ he state estmaton poblem s fomulated as the followng ovedetemned set of equatons:. V.99 V.56 5.V V sn δ. 5.V 5.V V cosδ In summay, the state estmaton poblem s fomulated as an ove detemned set of nonlnea equatons. Specfcally, the poblem s posed as follows: Gven whee: b h() (7.3) b h() m > n s a known m vecto s an m vecto of known functon s the unknown n vecto Compute the vecto. hs poblem s known as the nonlnea estmaton poblem. In case that the functons h ae lnea, then the poblem collapses to the lnea estmaton poblem defned as follows: Gven whee: b H (7.4) b s a known m vecto H s a known m n mat s the unknown n vecto m > n. Page Copyght A. P. Saks Melopoulos 99-6
Compute the vecto. In geneal, the ovedetemned set of equatons (7.3) o (7.4) does not have a soluton,.e. a vecto does not est whch satsfes all equaton (7.3) o (7.4). In ths case, t s epedent to defne the esdual vecto: o h( ) b (nonlnea case) (7.5) H b (lnea case) (7.6) and compute a soluton whch wll mnmze the esdual vecto n some sense. Fo ths poblem, thee ae thee appoaches: (a) the least squaes soluton, (b) the least absolute devaton soluton(l appoach), and (c) the Chebyshev o mn-ma soluton(l appoach). hese methods wll be befly dscussed net. An Altenatve Intoducton of the State Estmaton Poblem: An altenatve way to ntoduce the state estmaton poblem s by consdeng the hadwae used fo obtanng the measuements. Any meteng devce compses an nstumentaton channel, a tansduce and an A/D convete (n ecent systems the tansduces ae omtted snce they ae not necessay). he ntumentaton channel compses nstument tansfomes (Potental ansfomes (P), Cuent ansfomes (C), Optcal ansfomes, etc.), nteconnectng cables and possbly attenuatos. he tansduce may be an analog devce that convets the nput sgnal nto a DC output sgnal popotonal to the quantty measued. he A/D convete samples the sgnal and convetes t nto a dttal fom. he sgnal can be the output of the tansduce o t can be the output of the nstumentaton channel. In the latte case, an ntellgent devce s equed to etact the appopate nfomaton,.e. ms value of the voltage wavefom, eal powe flow, etc. A pctoal vew of ths aangement s shown n Fgues 7.5a and 7.5b. P Contol Cable ansduce Output - V DC Fgue 7.5a Physcal System fo Analog Measuements - Use of ansduces Copyght A. P. Saks Melopoulos 99-6 Page
P Contol Cable A/D Conveson Dgtal Inteface Fgue 7.5b Physcal System fo Analog Measuements - ansduce-less echnology Measuements obtaned wth the system of Fgue 7.5 wll contan a measuement eo. he statstcs of ths eo can be quantfed fom the paametes of the nstumentaton channel, the tansduce and the A/D convete. hs pocess s comple and fo typcal powe system nstumentaton channels [???] the statstcs may be based. As a convenent smplfcaton, we assume that the statstcs of the measuement eo ae unbased, they obey a gaussan dstbuton wth a known standad devaton, the epected mean value s zeo and the eo of a measuement s uncoelated to the eo of any othe measuement. Specfcally, f the eo of measuement s epesented wth η ι, then { } E η (7.7) { } σ { η }, j E η (7.8) E η (7.9) j Consde a specfc measuement b. hs measuement s elated to the state of the system va a known functon h (). We have named ths functon the model of the measument. We postulate that the dffeence between the model and the measuement s the o of the measuement,.e.: η h ( ) b (7.) It should be ecognzed that the measuement eo s the same as the measuement esdual we have ntoduced aleady. he dffeence s that now we can assocate statstcs ths esdual. In subsequent paagaphs we shall use equatons (7.5) and (7.) ntechangeably and wll assume that the esduals o the eos h have the statstcs stated wth equatons (7.7), (7.8) and (7.9). Page Copyght A. P. Saks Melopoulos 99-6
7.4. Least Squaes Soluton he least squaes soluton of the ovedetemned system (7.5) o (7.) s the vecto whch mnmzes the sum of the squaes of the components of the esdual vecto o the vecto of the measuement eos η. Mathematcally, ths s epessed as follows: m Mnmze J η η (7.) A vaaton of ths method s the weghted least squaes method whch mnmzes the sum of the weghted squaes of the components of the esdual vecto o the vecto of the measuement eos η. Mathematcally, ths s epessed as follows: whee: m Mnmze J w W η Wη (7.) w : the weght fo the esdual W : a dagonal mat, the dagonal elements beng the weghts w. A most usual case of weghted least squaes s defned as follows. We postulate that we want to compute the state of the system that mnmzes the sum of the squaes of the nomalzed measuement eos, s. he nomalzed measument eo s defned wth: s η (7.3) σ In ths case, the poblem s stated as follows: whee: Mnmze J m m h ( ) z s σ η Wη (7.4) W dag,,..., σ σ σ m Note that ths s a weghted least squaes fomulaton wth the weghts defned as the nvese of the squaed standad devatons. It should be also noted that the fomulaton n tems of equaton (7.) s equvalent to assumng that all standad devatons of the eo of all measuements ae equal. In subsequent paagaphs we wll consde the weghted least squaes appoach. Copyght A. P. Saks Melopoulos 99-6 Page 3
Usng the nonlnea and lnea model equatons, the nonlnea and lnea state estmaton poblem s epessed as follows: Mnmze J ( h( ) b) W( h( ) b), fo the nonlnea case (7.5) Mnmze J ( H b) W( H b), fo the lnea case (7.6) he unknown vecto s obtaned fom the soluton of the necessay condtons, whch n mat notaton ae epessed as follows: dj d (7.7) he above poblem s fst solved fo the lnea case and then fo the nonlnea case. Lnea Case: Dect dffeentaton of equaton (7.6) wth espect to, we obtan: dj d d [( H b) W ( H b)] H W ( H b) d (7.8) Upon soluton of last equaton fo the state vecto : ( H WH ) H Wb (7.9) Equaton (7.9) povdes the soluton to the lnea estmaton poblem (7.6). Nonlnea Case: o obtan the soluton to the nonlnea estmaton poblem (7.5), assume that an ntal guess of the vecto s known. he nonlnea model equatons (7.5) ae lneazed aound the pont yeldng: h( ) η h( ) ( ) h. o. t. b Whee h.o.t. denotes hghe ode tems. Assumng that the vecto s vey close to the soluton, then the hghe ode tems (h.o.t.) ae neglgbly small and ae omtted fom above equaton, yeldng: h( ) η ( ) h( ) b (7.) Let h( ) ' H, and b h( ) H b Page 4 Copyght A. P. Saks Melopoulos 99-6
Obseve that the vecto (7.) becomes: ' b h( ) H b s known (o computable). Now equaton η H b ' Now the poblem s dentcal to the lnea estmaton poblem. hus, the soluton s: ' ( H WH ) H Wb Upon substtuton of the b vecto: ( H WH ) H W ( H h( ) b) ( H WH ) H W ( h( ) b) he last equaton s genealzed nto the followng teatve equaton: ( H WH) H W ( h( ν ν ν ) b) (7.) ν whee H s the mat h ( )/ computed at. hs s the Jacoban of the vecto functon h(). In summay, the least squaes soluton of the lnea estmaton poblem s gven by Equaton (7.9) and the least squaes soluton of the nonlnea estmaton poblem can be obtaned wth the teatve algothm (7.). An eample wll llustate the method. Eample E7.: Consde the ovedetemned set of equatons deved n Eample E7.. Compute the soluton [ V V δ ], usng as an ntal guess [..99. ] and least squaes estmaton. Assume that all weghts ae equal to.. Soluton: he poblem wll be solved wth the teatve algothm (7.). Snce the weghts ae all. the weght mat W s the dentty mat. he objectve functon s: J ( V.) ( V.99) ( 5V V snδ.56) ( 5V 5V V cos. ) δ Mn J 4 w ( h ( ) z ) Copyght A. P. Saks Melopoulos 99-6 Page 5
he Jacoban mat H s: H 5V snδ 5V cosδ 3V 5V snδ 5V cosδ 5V V cosδ 5V V snδ he computatons follow: st Iteaton h ) ( b...56.585.. H ( ). 4.85... 4.7.. 4.85. H H.55 8.95. 8.95 7.9....55 H.3537.33 3.66 ( H H ) H.538.53.55 nd Iteaton..538.9946.99.53.9953..55.55 Page 6 Copyght A. P. Saks Melopoulos 99-6
h( ) b.54.53.3.83 H ( )...5655 4.847...5644 5... 4.767.557 H H 3.89.589.647.589 9.4 46.49.647 46.49.489 H ( H.3.369.838 H ).496.4988.5.4988.563.68.5.68.7 ( H H ) H.8.73.96.9974.996.5 hs completes the soluton. It s epedent to compute the esduals (o measuement eos) usng the above computed state: V..6 V.99.6 3 5.V V snδ.56.69 4 5.V 5.V V cosδ..63 Copyght A. P. Saks Melopoulos 99-6 Page 7
Note that n ths case the esduals o measuement eos ae vey small. 7.4. Least Absolute Devaton Soluton (L l Appoach) he least absolute devaton soluton of the ovedetemned system (7.5) and (7.6) s the vecto whch mnmzes the sum of the absolute devatons of the components of the esdual vecto. Mathematcally, ths s epessed as follows: Mnmze J m A vaaton of ths method s the weghted least absolute devaton method whch mnmzes the sum of the weghted sum of the absolute devatons of the components of the esdual vecto. Mathematcally, ths s epessed as follows: Mnmze J w m he weghts can be selected as dscussed n the pevous method. Agan we wll consde the soluton of the weghted least absolute devaton appoach. hs equement tanslates nto the followng optmzaton poblem: Mnmze J w Subject to: m fo the nonlnea poblem, o: h ( ) b,,,..., m Mnmze J w Subject to: m m hj j b,,,..., m j fo the lnea estmaton poblem. Both of the above poblems ae easly tanslated nto a lnea pogammng poblem. he pocedue wll be demonstated fo the nonlnea case. Fo ths pupose, the nonlnea equatons ae lneazed aound a gven pont to yeld: h ' ( ) b H( ) h( ) b H b Page 8 Copyght A. P. Saks Melopoulos 99-6
whee: b ' b h ( ) Now the poblem becomes Mnmze J w Subject to: m m hj j j ' b,,,..., m he above poblem s tansfomed nto an optmzaton poblem of the lnea pogammng vaety, by eplacng the vaables and wth a pa of nonnegatve vaables:,,,, he lnea pogammng poblem s: Mnmze Subject to: J m w ( ) ' H( ) b,,, An eample wll llustate the method. Eample E7.3: Consde the poblem of Eample E7.. Agan assume the weghts to be equal to.. Solve ths poblem usng the least absolute devaton method. Soluton: Assumng as a statng soluton the pont [..99.] lneazng aound ths pont, the L poblem s stated as follows: Mnmze 4 ( ) V V. V V. 3 3 4.85 δ 4.85 δ J Subject to.56 4 4 4.85 V 4.85 V 4.7 V 4.7 V.585, and All vaables ae non-negatve. Note that snce the voltage magntudes ae non-negatve Copyght A. P. Saks Melopoulos 99-6 Page 9
by defnton, thee was no need to ntoduce and vaables fo the voltage magntudes. Upon soluton of the above lnea pogam we obtan the followngs: V.67 δ.55.67 and all othe s equal to zeo hus, the new state vaables and esduals ae: V.98933 V. 99 δ.55 3 3 3 4 4 4...67. Note that only one esdual s nonzeo. In geneal, the LAV soluton wll esult n m-n mamum numbe of non-zeo esduals. In ths case m4 and n3, thus 4-3 esduals wll be nonzeo. On the othe hand, usng the nonlnea model equatons and the above computed state, the esduals ae computed to be: V..67 V.99. 3 5.V V snδ.56.954 5.V 5.V V cosδ. nd Iteaton 4.838 Lneazaton of the model aound the new soluton (fom the fst teaton) yelds the followng lnea pogam: Mnmze J 4 ( ) V V V V Subject to. 67. Page Copyght A. P. Saks Melopoulos 99-6
3 3.557 V.557 V.556 V.556 V 4.6δ 4.6δ.954 4 4 4.768 V 4.768 V 4.943 V 4.943 V.545δ.545δ.838 All the vaables n above lnea pogam ae non-negatve. Upon soluton of above poblem, we obtan the followng state and esduals: V.559 V l.9949 V. V.99 δ -.74 δ -.58 3 3 3 4 4 4.5... Usng the above computed state and the nonlnea model equatons, the actual esduals ae computed to be: V..5 V.99. 3 5.V V snδ.56. 5.V 5.V V cosδ. 4.5 hs completes the soluton. Note agan the esduals (o measuement eos) ae low n ths case. 7.4.3 Chebyshev o Mn-Ma Soluton (L Appoach) he mn-ma soluton of the ovedetemned system (7.5) o (7.6) s the vecto that mnmzes the absolutely lagest (mamum) component of the esdual vecto. Mathematcally, ths s epessed as follows: Mnmze * (,,..., ) ma whee s the -th component of the esdual vecto. hs equement tanslates nto the followng optmzaton poblem: n Mnmze Subject to * J * w h ( ) b,,,..., m Copyght A. P. Saks Melopoulos 99-6 Page
fo the nonlnea estmaton poblem, o: Mnmze * Subject to m b h w n j j j,...,,, * fo the lnea estmaton poblem. One way to solve ths poblem s by fst tansfomng t nto a lnea pogam and subsequently usng the smple algothm to solve ths poblem. he pocedue wll be demonstated fo the nonlnea pogam. Specfcally, the stated poblem s equvalent to: Mnmze * Subject to m b h w,...,,, ) ) ( ( * m b h w,...,,, ) ) ( ( * Upon lneazaton of the esduals: ' ) ( ) ( ) ( b H b h H b h whee: ) ( ' h b b Upon substtuton: Mnmze * Subject to ' * ) ( b R H W ' * ) ( b R H W whee, * * * * R M an m vecto. Net, eplace the vaable wth nonnegatve vaables:,, Page Copyght A. P. Saks Melopoulos 99-6
and ntoduce the slack (y) and suplus vaables (z). hen, Mnmze * Subject to W * ' ( H H R ) y b W, -, *, y, z * ' ( H H R ) z b Above poblem s a standad lnea poblem. Upon soluton, the state estmaton poblem s solved. An eample wll demonstate the method. Eample E7.4: Consde the poblem stated n Eample E7.. Agan assume the V V, usng the weghts to be equal to.. Compute the soluton [ δ ] Chebyshev method and an ntal guess of [..99.]. Soluton: he fomulaton of the poblem s: Mnmze * Subject to V. * V. 99 5.V * V snδ. 56 5.V 5.V V cosδ. * * Upon emoval of absolute values, lneazaton of the esultng equatons aound the opeatng pont [ V V δ ] [..99.] and ntoducton of slack and suplus vaables, the above poblem s tanslated nto: V.. V.99. 3 5.V V snδ.56.56 5.V 5.V V cosδ. 4 Mnmze * Subject to: V V * V y * V z...585 Copyght A. P. Saks Melopoulos 99-6 Page 3
* V V y. * V V z. *. V. V. V. V 4.85 δ 4.85 δ y3 *. V. V. V. V 4.85 δ 4.85 δ z3 * 4.85 V 4.85 V 4.7 V 4.7 V. δ. δ y 4 * 4.85 V 4.85 V 4.7 V 4.7 V. δ. δ z 4 *, y, z.56.56.585.585 Upon soluton of above lnea pogam, the followng system state s computed: [ V V δ ] [.9948.995. 47] and the followng esduals: l - * y.5 - * y.5 3 - * y 3.5 4 - * y 4.484 Usng the above system state and the nonlnea model, the followng esduals ae computed usng the full nonlnea model: V..5 V.99.5 3 5.V V snδ.56.8 5.V 5.V V cosδ. Second Iteaton: 4.773 Lneazaton of the model aound the new soluton (fom the fst teaton) yelds the followng lnea pogam: Mnmze * Subject to: V V V V * V y * V z * V y * V z.5.5.5.5 Page 4 Copyght A. P. Saks Melopoulos 99-6
*.56 V.56 V.5595 V.5595 V 4.769 δ 4.769 δ y3 *.56 V.56 V.5595 V.5595 V 4.769 δ 4.769 δ z.8 3 * 4.8463 V 4.8463 V 5.57 V 5.57 V.55 δ.55 δ y 4.773 * 4.8463 V 4.8463 V 5.57 V 5.57 V.55 δ.55 δ z 4.773.8 * >, V, V,..., δ, δ, y, z Upon soluton of above poblem, the followng system state and esduals ae obtaned: V.7 V l.9975 V -.7 V.995 δ -.7 δ -.54 l -* y.5 -* y.5 3 -* y 3.5 4 -* y 4.5 Usng the above system state and the nonlnea model, the followng esduals ae obtaned: V..5 V.99.5 3 5.V V snδ.56.3 5.V 5.V V cosδ. 4 Above completes the solutons..7 7.4.4 Summay of the hee State Estmaton Appoaches Hee we summaze the esults of the thee appoaches fo state estmaton, namely least squaes (LS), least sum of absolute values (LAV) and mnmum mamum esdual (mnma). able 7. summaes the soluton and the esduals. able 7. Summay of State Estmaton Solutons Vaable LS LAV mn-ma V V δ.9974.996 -.5 -.6.9949.99 -.58.5.9975.995 -.54.5 Copyght A. P. Saks Melopoulos 99-6 Page 5
3 4.6.69.63...5.5 -.3 -.7 7.4.5 Qualty of the State Estmate Coect contol and opeaton of the system eques accuate knowledge of the opeatng state of the system. State estmaton povdes the opeatng state of the system n eal tme and, n addton, t povdes nfomaton about the qualty of the state estmate n a quanttatve way. Such analyss povdes confdence ntevals fo the computed estmate of the state. Wth espect to the qualty of state estmaton, thee ae two elated poblems. he fst one elates to the valdty of the data (measuements). If the measuements ae polluted wth easonable measuement eo (wthn the specfcatons of the measung nstuments), and assumng thee s enough edundancy, the state estmate wll be easonably accuate. Howeve, f one o moe data have lage eos (due to a numbe of easons), the state estmate wll not be accuate. hus, t s necessay that the state estmato be smat enough to detect and eject bad data. he second poblem elates to the eo tansmtted to the state estmate fom the measuement eo. hs eo s measued wth the standad devaton of the state estmate. It should be epected that n the pesence of statstcally easonable measuement eos, the standad devaton of the state estmate should decease as the edundancy nceases. Consde a measuement of a physcal quantty of an electc powe system. We have dscussed the fact that ths measuement s obtaned va an nstumentaton channel that can be comple. he measuement pocess wll ehbt some eo. Fo smplcty, we ntoduced a numbe of assumptons egadng the measuement eo. Consde the nomalzed eo fo measuement, s : s h ( ) b σ We have assumed that the nomalzed eos ae Gaussan dstbuted wth standad devaton. and zeo coss coelaton. Gven the mete accuacy defned above, two poblems can be defned as follows: (a) what s the pobablty that all data ae located wthn epected bounds (Goodness of Ft) and (b) what s the accuacy of the computed soluton? hese poblems wll be addessed net. Page 6 Copyght A. P. Saks Melopoulos 99-6
7.4.5. Goodness of Ft he goodness of ft s defned as the pobablty that the dstbuton of the measuemenmt eos ae wthn the epected bounds. hs pobablty s computed as follows. Assume that the state estmate $ has been computed wth the least squae appoach. Consde the nomalzed esduals computed at the soluton $. We have postulated that the nomalzed esduals s ae Gaussan andom vaables wth zeo mean and standad devaton. Now consde the followng vaable: χ m s Snce the vaables {s,,,, m}, ae Gaussan andom vaables, the vaable χ s also a andom vaable and t s ch-squae dstbuted [???]. Also snce the vaables {s,,,, m} ae dependent upon only n vaables (the state vaables ) though a set of model functons (functons h()), the ch-squae dstbuted vaable χ has m-n degees of feedom. he ch-squae dstbuton s well known. Fo eample, able 7. tabulates the pobablty dstbuton functon of a geneal ch-squae dstbuted andom vaable, P( α, ν ), wth ν degees of feedom. P( α, ν ) P [ χ α], whee ν s the degees of feedom. able 7. Ch-Squae Pobablty Dstbuton Functon* p P( ζ, ν ) P[ χ ζ ], v degees of feedom] Copyght A. P. Saks Melopoulos 99-6 Page 7
P ν 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3.5..5.5..5.5.75.9.95.975.99.995.393.57.98..77.7.4.676.989.34.73.6.6 3.7 3.57 4.7 4.6 5.4 5.7 6.6 6.84 7.43 8.3 8.64 9.6 9.89.5..8.5 3. 3.8..5.97.554.87.4.65.9.56 3.5 3.57 4. 4.66 5.3 5.8 6.4 7. 7.63 8.6 8.9 9.54..9.5..9 3.6 4.3 5..56.6.484.83.4.69.8.7 3.5 3.8 4.4 5. 5.63 6.6 6.9 7.56 8.3 8.9 9.59.3..7.4 3. 3.8 4.6 5.3 6. 6.8.393.3.35.7.5.64.7.73 3.33 3.94 4.57 5.3 5.89 6.57 7.6 7.96 8.67 9.39..9.6.3 3. 3.8 4.6 5.4 6. 6.9 7.7 8.5.58..584.6.6..83 3.49 4.7 4.87 5.58 6.3 7.4 7.79 8.55 9.3..9.7.4 3. 4. 4.8 5.7 6.5 7.3 8. 8.9 9.8.6..575..9.67 3.45 4.5 5.7 5.9 6.74 7.58 8.44 9.3...9.8 3.7 4.6 5.5 6.3 7. 8. 9. 9.9.8.7.7 3.6 4.5.455.39.37 3.36 4.35 5.35 6.35 7.34 8.34 9.34.3.3.3 3.3 4.3 5.3 6.3 7.3 8.3 9.3.3.3.3 3.3 4.3 5.3 6.3 7.3 8.3 9.3.3.77 4. 5.39 6.63 7.84 9.4..4.5 3.7 4.8 6. 7. 8. 9.4.5.6.7 3.8 4.9 6. 7. 8. 9.3 3.4 3.5 3.6 33.7 34.8.7 4.6 6.5 7.78 9.4.6. 3.4 4.7 6. 7.3 8.5 9.8..3 3.5 4.8 6. 7. 8.4 9.6 3.8 3. 33. 34.4 35.6 36.7 37.9 39. 4.3 3.84 5.99 7.8 9.49..6 4. 5.5 6.9 8.3 9.7..4 3.7 5. 6.3 7.6 8.9 3. 3.4 3.7 33.9 35. 36.4 37.7 38.9 4. 4.3 4.6 43.8 5. 7.38 9.35..8 4.4 6. 7.5 9..5.9 3.3 4.7 6. 7.5 8.8 3. 3.5 3.9 34. 35.5 36.8 38. 39.4 4.6 4.9 43. 44.5 45.7 47. 6.63 9..3 3.3 5. 6.8 8.5..7 3. 4.7 6. 7.7 9. 3.6 3. 33.4 34.8 36. 37.6 38.9 4.3 4.6 43. 44.3 45.6 47. 48.3 49.6 5.9 7.88.6.8 4.9 6.7 8.5.3. 3.6 5. 6.8 8.3 9.8 3.3 3.8 34.3 35.7 37. 38.6 4. 4.4 4.8 44. 45.6 46.9 48.3 49.6 5. 5.3 53.7 *able adopted fom CRC book Fom basc pobablty theoy we know that the epected value of χ s: E [ χ ] ν m n, whee ν s the degees of feedom. Above statstcal popetes can be used to compute the pobablty that the data b s statstcally coect when the state s computed n the least squae sense. We wll call ths popablty the confdence level of the state estmate. he confdence level s computed as follows. Consde the least squaes soluton $. hs soluton mnmzes the sum of the squaes of s,.e. any othe state vecto wll esult n a lage value of χ,.e., whee: m ς s ( ) χ ζ m s ( ˆ ) Page 8 Copyght A. P. Saks Melopoulos 99-6
he pobablty of above event, χ ζ, s gven by the ch-squae dstbuton: [ χ ζ ]. P[ χ ζ ]. P( ζ, ν ) P hs popablty epesses how well the nomalzed esduals s ae dstbuted wthn the epected bounds. A hgh popablty value ndcates that these esduals ae well wthn the statstcal bounds,.e. the nomalzed esduals ae gaussan dstbuted wthn the ange (-. to.). hs means that the actual esduals ae compaable to the epected eos of the measuements. A low popablty value ndcates that the esduals ae hghe than what s statstcally epected. hs s the eason fo callng ths popablty the confdence level fo the estmaton esults. A smple pocedue to compute the confdence level s gven below. Step. Compute the state estmate, $, n the least squaes sense. m m h ( ˆ) b Step. Compute the value ζ s ( ˆ). σ Step 3. Read the pobablty P[ χ ζ ] P( ζ, ν ) fom the tabulated ch-squae popablty dstbuton functon. Note that ntepolaton can be used o moe detaled data of the dstbuton functon. Step 3. Compute the pobablty P[ χ ζ ]. P( ζ, ν ). Note that the confdence level can be computed only fo the least squaes soluton. Howeve, as an appomaton, the above pocedue can be appled to the othe two solutons fo $ (L and L ). P( ζ, ν ) P [ χ ζ ] 7.4.5. Accuacy of Soluton he accuacy of the soluton s epessed wth the covaance mat of the state estmate, ˆ. Specfcally, let be the tue but unknown soluton, and ˆ be the soluton to the poblem (7.5). hs soluton may be the least squaes, L o L soluton. he defnton of the covaance mat s: C E[( ˆ )( ˆ ) ] Note that a lneazed epesson fo ˆ s as follows (whch s obtaned by applyng the state estmaton algothm at pont ) ˆ ( H WH) H W fo least squae soluton Copyght A. P. Saks Melopoulos 99-6 Page 9
whee h( ) b and H s the jacoban mat. he jacoban s supposed to be computed at the tue state. Howeve, snce the tue state s not known, we appomate the jacoban mat at the known best estmate of the state, ˆ. As we dscussed, the esduals epesent the measuement eos,.e. η. he statstcs of the measument eo have been ntoduced and epeated hee [ η] E ηη W. E and [ ] Upon substtuton of above nto the defnton of the covaance mat, the followng s obtaned C E[( H WH) H Wηη W H( H WH) ] Snce the only andom vaables n above equaton s the measuement eos η, then above equaton s ewtten as follows: C { [ ηη ]} W H( H WH ( H WH) H W E ) Now the above equaton s smplfed to yeld: C ( H WH) Once the covaance mat of the soluton has been computed, the standad devaton of a component of the soluton vecto s gven wth whee, σ C (, ) C (, ) s the th dagonal enty of the covaance mat. he covaance mat s also known as the nfomaton mat snce t povdes useful nfomaton on the epected eo of the computed state vaables. We symbolze the nfomaton mat wth I: I ( H WH) Othe measues of accuacy can be also deved such as E( $ b ), Cov( $ b ). hese vaables ae deved net. Consde the estmate of the measuements defned wth: b ˆ h( ˆ) he statstcs of the estmate $ b ae computed as follows: [ bˆ ] h( ) E Page 3 Copyght A. P. Saks Melopoulos 99-6
C bˆ Cov( bˆ) E[( bˆ b )( bˆ b ) ] whee b s the tue value of the measuements. Note that: bˆ b h( ˆ) h( ) H( )( ˆ ) H( ˆ)( ˆ ) Upon substtuton: Cov( bˆ) E Now let s compute the covaance Note that: [ H( ˆ )( ˆ ) H ] H( H WH) H [( bˆ b)( bˆ b ] Cov ( bˆ b) E ) bˆ b ( bˆ b) ( b b ) [ h( ˆ) h( ) ] H( ˆ)( ˆ ) H( H WH) H W Upon substtuton and some staghtfowad manpulatons: Cov( bˆ b) W H ( H WH ) H A summay of the state estmaton statstcs s gven n able 7.3. able 7.3 Summay of Statstcal Popetes of Statc Estmatos [] ˆ Cov( ˆ) C ( H WH) E [ bˆ ] h( ) [ bˆ b] E E E [ J ] m n Cov( bˆ) H ( H WH ) Cov( bˆ b) W H ( H H WH ) [ χ ζ ]. P( ζ, m n) P In summay, the qualty of the state estmate s quantfed as follows: A measue of data valdty s epessed wth the confdence level obtaned fom the ch-squae dstbuton. he accuacy of the estmated state vaables s gven wth the dagonal entes of the nfomaton mat whch epess the squae of the standad devaton. he evaluaton of the qualty of a state estmate s llustated wth an eample. Eample E7.5. Consde the poblem solved n Eample E7.. Compute the confdence H Copyght A. P. Saks Melopoulos 99-6 Page 3
level and the standad devaton of the soluton components (V, V, δ ), as well as the standad devaton of the measuement estmates gven that the eo of the measuements s.,.3,.3, and.4 fo the measuements of V, V, P l and Q, espectvely. Use the least squae appoach. Soluton: Fst, the least squae technque s used to solve the poblem statng wth an ntal guess: st Iteaton [ V V δ ] [..99. ].. h( ) b.56.585.. H ( ). 4.85... 4.7.. 4.85. H W[ h( 47.8 ) b] 456. 574. 437. H WH 36434.. 36434. 3667.... 455. ( H WH) H W[ h(.33 ) b].7376.55..33.9967 ˆ.99.7376.9974..55.55 Page 3 Copyght A. P. Saks Melopoulos 99-6
nd Iteaton.868.359.74.33..56.99..96764.563587.9974.9967 ) ( b h.5636 5.539 4.879 4.89.5677.5688...... ) ( ' H 39.95 79.386 77.865 ] ) ( [ ' b h H W 4586. 454.9 38. 454.9 45479. 3759. 38. 3759. 436. H WH.5.38.7 ] ) ( [ ) ( ' b h W H WH H.54.99364.9984.5.38.7.55.9974.9967 ˆ.475.4.36.6..56.99..475.564.9936.9984 ) ( b h he nfomaton mat s: Copyght A. P. Saks Melopoulos 99-6 Page 33
I ( H WH) 4.7583.7649.5879.7649.8435.596.5879.596.6577 he standad devatons of the system states ae: -4 σ V.7583.66-4 σ V.8435-4 σ δ.6577.686.47 Note that the standad devaton of the state vaables s lowe than those of the measuements. he confdence level s calculated as follows: ζ 4.6.36.4.475 σ..3.3.4 Snce ν m n, we obtan fom able 7.: P [ χ.963] P(.963, ν ). 5 hus, the confdence level s: P [ χ.963]. P(.963, ν ). 885 hs completes the soluton..963 7.4.6 Summay and Dscusson In summay, a state estmato s a computatonal pocedue by whch a statstcal estmate of the state of the system s obtaned. Inputs to the state estmato ae a set of measuements, whch ae chaactezed wth the statstcal popetes, and a mathematcal model descbng the system. he state estmato, n addton to the state estmate, $, computes the covaance of $, the estmates of the measuements, b $, the covaance mat of b $, the esduals b - b $, and the value of the objectve functon. he nputs and outputs of the state estmato ae llustated n Fgue 7.6. Of mpotance s Page 34 Copyght A. P. Saks Melopoulos 99-6
the value of the objectve functon, whch can povde nfomaton about the qualty of the computed state estmate. Fo ths pupose the ch-squae test s utlzed. Measuements & elemety Nose Noseless Measuements System Model State Estmato Measuement Estmate bˆ Covaance of b, ˆ p Resdual b bˆ State Estmate ˆ Covaance of, ˆ I Objectve Functon Value Pobablty of Goodness of Model Ft (Confdence Level) Fgue 7.6 Inputs and Outputs of the State Estmato 7.5 Detecton and Identfcaton of Bad Data he pesence of bad data deteoates the pefomance of the state estmate. It s mpeatve, theefoe, that bad data be detected, dentfed, and ejected. hs objectve s acheved wth nfomaton povded fom edundant measuements, a chaactestc of the state estmato. hee ae two nteelated poblems: the fst poblem s the one of detectng the estence of bad measuements; the second poblem s the one of dentfyng whch data s bad. hese two poblems ae addessed net. 7.5. Detecton of Bad Data Detecton of the estence of bad data can be acheved wth the ch-squae test,.e. by computng the confdence level. If the system of Equatons (7.5) s fee of bad data, the confdence level wll be hgh. In the pesence of one o moe bad data, the confdence level wll decease. hs means that wheneve the confdence level s low, bad data est Copyght A. P. Saks Melopoulos 99-6 Page 35
n the measuement set. Note that the ch-squae test does not ndcate whch datum o data s bad. he dentfcaton of the bad data s acheved wth othe methods to be descbed below. 7.5. Identfcaton of Bad Data Identfcaton of bad data nomally conssts of two steps. In the fst step, bad data may be dentfed by nspecton o smple consstency ules. hs step dentfes the obvously bad data and t s vey much system dependent. As an eample, n powe system state estmaton, measuements of voltages, powe flow, etc., ae known to have specfc anges. If a measuement s out of ths ange, t wll be classfed as a bad measuement o at least as a measuement suspected of beng bad (suspect measuement). In the second step, bad data ae dentfed wth statstcal analyss of the esduals and/o ts effects on confdence level. hs analyss depends on the selected method fo the soluton of Equaton (7.5). In the case of least squae soluton, the possble bad data ae dentfed wth the lage esduals. Howeve, t s known that t s possble that: (a) a measuement wth a lage esdual may not be always a bad measuement and (b) a bad measuement may have a vey small esdual (outles). A athe secue but computatonally demandng way to dentfy a bad datum s by means of hypothess testng. Specfcally, assume that a measuement (o a goup of measuements) has been dentfed as suspect (ths chaactezaton may be due to a lage nomalzed esdual o because of falue to pass a consstency check, etc.). Fo ths pupose, the suspect datum s emoved,.e. the coespondng equaton b h () s emoved fom Equatons (7.5) and the least squae soluton s computed agan. Subsequently, the confdence level s computed. A dastc mpovement n the confdence level ndcates that the data unde consdeaton s bad. hs pocedue tends to be computatonally demandng. On the othe hand, the L l and L solutons tend to be moe vesatle n dentfyng the bad data. Fo eample, the L soluton dectly povdes the measuement wth the lagest esdual. Detecton and dentfcaton of bad data wll be llustated wth eamples. Fst we consde a geneal eample to llustate the effects of outles. hen we dscuss a typcal applcaton of hypothess testng n the famewok of least squaes appoach. he fnal eample llustates bad data detecton and dentfcaton n the contet of the mn-ma appoach. Eample E7.6. Consde the set of ovedetemned equatons below esultng fom fve measuements. It s desed to estmate the state vecto of these equatons, detemne the pesence o absence of bad measuements and dentfy the bad measuements f any. Assume that all measuements have a statstcal eo of standad devaton.. 3. 4 3 4. 7 Page 36 Copyght A. P. Saks Melopoulos 99-6
Soluton: he equatons elatng the measuements to the state of the system ae lnea. heefoe a lnea state estmato povdes the soluton. he model equatons n mat fom ae: 3. b H, whee : H, b 3 4 7 4. he least squaes soluton of above poblem s: ˆ ˆ ˆ he esduals ae: b H.89 ( H WH ) H Wb.6494.64.773.843.36 he sum of the squaes of the esduals s: J 4..5 he pobablty of goodness of ft s obtaned fom able 7. wth m-n degees of feedom: P. hs pobablty ndcates the pesence of bad data n ths measuement set. heefoe bad data have been detected. he net step s to dentfy the bad data. he bad data dentfcaton wll be done wth hypothess testng. Specfcally, the followng thee hypothess wll be eamned: Hypothess : Measuement 3 (lagest esdual) s bad. Hypothess : Measuement (net lagest esdual) s bad. Hypothess 3: Measuement 4 (net lagest esdual) s bad. Hypothess 4: Measuement (net lagest esdual) s bad. Copyght A. P. Saks Melopoulos 99-6 Page 37
he computatons and conclusons follow. Hypothess : Measuement 3 s emoved and the esultng estmaton poblem s solved. he soluton s: ˆ ˆ ˆ.89 ( H WH ) H Wb.36 he esduals, sum of weghted squaed esduals and pobablty of goodness of ft wth m-n degees of feedom (able 7.) ae:.369 4 b H.4355, J 3. 66, P...75 Note that the pobablty s stll low ndcatng that measuement 3 may not be a bad measuement, o thee ae addtonal bad measuements n the set. Hypothess : Measuement s emoved and the esultng estmaton poblem s solved. he soluton s: ˆ ˆ ˆ.5 ( H WH ) H Wb 3.33 he esduals, sum of weghted squaed esduals and pobablty of goodness of ft wth m-n degees of feedom (able 7.) ae:.333 4 b H.46, J 9. 67, P...83 Note that the pobablty s stll low ndcatng that measuement may not be a bad measuement, o thee ae addtonal bad measuements n the set. Hypothess 3: Measuement 4 s emoved and the esultng estmaton poblem s solved. he soluton s: ˆ ˆ ˆ. ( H WH ) H Wb.333 Page 38 Copyght A. P. Saks Melopoulos 99-6
he esduals, sum of weghted squaed esduals and pobablty of goodness of ft wth m-n degees of feedom (able 7.) ae:.333 4 b H.667, J. 667, P. 56..333 Note that the above pobablty ndcates that measuement 4 s a bad measuement and should be emoved fom the set of measuements pemanently. Fo ths eample, thee s no need to poceed futhe. Note that the bad measuement was dentfed n the thd hypothess test. In the ognal state estmaton soluton the esdual of the 4 th measuement was not the lagest (absolutely). In othe wods the soluton of the ntal state estmaton poblem faled to yeld a hgh enough esdual fo ths measuement. Only the hypothess testng was able to dentfy the bad measuement. In ths case the pobablty by whch the bad datum was dentfed s elatvely low (.56) due to the fact that the numbe of edundant measuements s vey low. One can povde a geometcal ntepetaton of ths data, obsevng that the model equatons ae of the fom: b a Gaphng the data on a coodnate system whee the hozontal as epesents a and the vetcal as epesents b, Fgue E7.6 s obtaned. In ths fgue one can clealy see that thee of the data le on an almost staght lne, whle the foth datum les away fom ths lne (outle). In ths case, the gaphcal epesentaton of the data can mmedately yeld the bad datum. 6 5 4 b 3 3 4 5 6 7 a 8 Copyght A. P. Saks Melopoulos 99-6 Page 39
Fgue E7.6 Gaphcal Repesentaton of the Data of Eample E7.6 Eample E7.7. Consde the system of Fgue E7.7. Fve measuements ae taken as follows: V.99, P l.4, Q l.5, P l -.46, Q.. -j5. Voltage Magntude Measuement MW Flow Measuement MVA Flow Measuement Fgue E7.7 A Smple wo Bus Powe System All measuement nstuments ae known to have an accuacy of %. he voltage at bus s. p.u. wth absolute cetanty. Pefom bad data dentfcaton. Soluton: Fst, the state estmaton poblem s solved. Detals ae omtted. he computed state estmate s:.994 ˆ.99497 he covaance mat of the state estmates s: Cov( ˆ) 6.9.9.9.9 he measuement estmates ae: he esduals ae:.99497.48 ˆb.494.48.4 Page 4 Copyght A. P. Saks Melopoulos 99-6
b.497.8 ˆb.85.788.4 he value of the objectve functon at the estmate s: J ( ˆ) 3.688 whch yelds a pobablty of statstcally coect estmate: p. he low (zeo) pobablty means that bad data est n the measuement set (detecton step). Inspecton of the esduals eveals that the second o the fouth measuement may be bad. o detemne whch one s bad (o maybe both ae bad), hypothess testng s employed. hee hypotheses wll be eamned: () the fouth measuement s bad, () the second measuement s bad, and (3) both (second and fouth) measuements ae bad. Hypothess. In ths case, the fouth measuement s emoved. he state estmate becomes:.949 ˆ.99496 and the objectve functon at the estmate s: J ( ˆ).8358 the pobablty that the estmate s statstcally coect s computed to be (m - n ): p.4 Above pobablty s too low. Wth hgh pobablty, ths measuement s not a bad datum. Hypothess. In ths case, the second measuement s emoved. he state estmate becomes:.475 ˆ.99497 Copyght A. P. Saks Melopoulos 99-6 Page 4
and the objectve functon at the estmate s: J ( ˆ).777 he pobablty that the estmate s statstcally coect s computed to be (m - n ): p.95 Obvously, the second measuement s bad. he degadaton of the pobablty of statstcally coect estmate s due to some loss n data edundancy. Note that the edundancy ( m n ) dopped fom 5% to %. Hypothess 3 need not be eamned. Bad data dentfcaton n the mn-ma method can be pefomed wth senstvty nfomaton povded by the LP algothm. he pocedue s outlned wth an eample. Eample E7.8. Consde the poblem of Eample E7.7. Pefom bad data dentfcaton usng the mn-ma method. Soluton: Fst, the state estmate must be computed n the mn-ma sense. Fo ths pupose, we stat fom the known state estmate n the least squae sense.994 ˆ.99497 Fomulaton of the state estmaton poblem n the mn-ma sense n tems of the vaables: yelds: V δ V δ.99497.994 Mnmze * Subject to: * V V y.5 * V V z.5 *.4885 V.4885 V 4.859 δ 4.859 δ y *.4885 V.4885 V 4.859 δ 4.859 δ z * 4.96 V 4.96 V.48 δ.48 δ y3.8.8.9 Page 4 Copyght A. P. Saks Melopoulos 99-6
* 4.96 V 4.96 V.48 δ.48 δ z3.9 *.4885 V.4885 V 4.859 δ 4.859 δ y.789 4 *.4885 V.4885 V 4.859 δ 4.859 δ z4 *.93 V 4.93 V.48 δ.48 δ y *.93 V 4.93 V.48 δ.48 δ z 4 5 4 5.4.4.789 *, y, z Upon soluton of the lnea poblem, the system state s: V -.46 and V.994 δ -.4 δ -.998 and the esduals ae: l -* y -.364 -* y -.8 3 -* y 3 -.689 4 -* y 4 -.798 5 -* y 5.8 Usng the above computed state and the nonlnea model, the esduals ae computed to be: l -.4 -.8 3 -.679 4 -.798 5.787 he second teaton s: Mnmze * Subject to: * V V y.4 * V V z.4 *.4945 V.4945 V 4.78 δ 4.78 δ y *.4945 V.4945 V 4.78 δ 4.78 δ z * 4.954 V 4.954 V.48 δ.48 δ y3 * 4.954 V 4.954 V.48 δ.48 δ z3.8.8.679.679 Copyght A. P. Saks Melopoulos 99-6 Page 43
*.4945 V.4945 V 4.78 δ 4.78 δ y4 *.4945 V.4945 V 4.78 δ 4.78 δ z 4 *.7866 V 4.7866 V.48 δ.48 δ y *.7866 V 4.7866 V.48 δ.48 δ z 4 5 4 5.798.798.787.787 *, y, z Upon soluton of the lnea pogam, the system state s: V -.9 and V.993 δ -. δ -.998 and the esduals ae: l -* y.3 -* y.8 3 -* y 3.69 4 -* y 4.8 5 -* y 5.8 Usng the above system state and the nonlnea model, the esduals ae computed to be: l -.3.8 3 -.693 4 -.8 5.8 he senstvtes of the lagest esdual wth espect to the measuements ae obtaned fom the lnea pogammng poblem (see Append B): d db * d db * 3 d db * 5 * d.,. 5, db * d.,. 5, db. 4 o detemne the qualty of the state estmate, the ch-squae test s pefomed. Recall that the mete accuacy s %. hen: Page 44 Copyght A. P. Saks Melopoulos 99-6
J.3..8..693..8..8. 6.75 Snce m - n 3, the confdence level wll be: P. hs confdence level ndcates the pesence of bad data. Eamnng the above data, t s obseved that the second measuement wll have the lagest effect on the lagest esdual, equal to: d db *.4 hus, the second measuement s suspected to be wong. he second measuement s emoved fom the measuement set and the L state estmaton poblem s solved statng fom the pevous soluton: V.993 δ -.998 Net, the L poblem s solved as follows: st Iteaton Mnmze * Subject to: * V V y.3 * V V z.3 * 4.954 V 4.954 V.48 δ.48 δ y3 * 4.954 V 4.954 V.48 δ.48 δ z3 *.4945 V.4945 V 4.787 δ 4.787 δ y4 *.4945 V.4945 V 4.787 δ 4.787 δ z4 *.7839V 4.7839 V.48 δ.48 δ y.8 *.7839 V 4.7839 V.48 δ.48 δ z 4 5 4 5.693.693.8.8.8 *, y, z Upon soluton of the lnea pogam, the system state s computed to be: Copyght A. P. Saks Melopoulos 99-6 Page 45
V -.476 and V.9957 δ -.459 δ -.439 and the esduals ae computed to be: l -* y -.568 3 -* y 3 -.568 4 -* y 4 -.568 5 -* y 5.86 Usng the computed system state and the nonlnea model, the esduals ae computed to be: nd Iteaton l -.57 3 -.5 4 -.47 5 -.3 Mnmze * Subject to: * V V y.57 * V V z.57 * 4.983 V 4.983 V.5553 δ.5553 δ y3 * 4.983 V 4.983 V.5553 δ.5553 δ z3 *.563 V.563 V 4.8448 δ 4.8448 δ y4 *.563 V.563 V 4.8448 δ 4.8448 δ z4 *.9338V 4.9338 V.5553 δ.5553 δ y.3 *.9338 V 4.9338 V.5553 δ.5553 δ z 4 5 4 5.5.5.3.47.47 *, y, z Upon soluton of the lnea pogam, the system state s computed to be: V -.6 and V.99576 δ -.6 δ -.4364 and the esduals as computed wth the lnea pogam, the esduals as computed wth the nonlnea model, the senstvtes and the effect of measuements on the lagest Page 46 Copyght A. P. Saks Melopoulos 99-6
esdual ae: Resdual LP Model Nonlnea Senstvtes * Model d*/db.576 -.576.93749.474 --- --- --- --- 3.576 -.5.6779.3 4.576 -.5.6473.3 5.56.3.. Note that * s postve fo each measuement. hs means emoval of any measuement wll deteoate the state estmate. o detemne the qualty of the state estmate, a ch-squae test s agan pefomed usng eactly the same pefomance nde. Specfcally, n. agan. J.576..5..5..3..775 Now m - n. Readng on the table, the confdence level s: P.95 he jump n the confdence level fom zeo to.95 ndcates that the second measuement s a bad measuement. Note that the mn-ma appoach mmedately dentfed the bad measuement. Compae ths pefomance wth that of the least squae appoach, whee the bad measuement was dentfed at the second hypothess testng. 7.5.3 Summay and Dscusson In ths secton we dscussed methods fo detectng and dentfyng bad measuements n a set of data. hs pocess can be accomplshed only when thee ae edundant measuements. hus, edundant measuements ae needed to enhance: (a) the ablty to detect and eject bad measuements and (b) to obtan the state vaables wth the smallest possble eo. ypcally, the numbe of measuements s two-to-thee tmes geate than the numbe of states vaables allowng fo a consdeable amount of measuement edundancy ( to 3%). In ths pocess we tactly assumed that the system state s obsevable fom the measuement set. Copyght A. P. Saks Melopoulos 99-6 Page 47
Measuement Set z Pefom State Estmaton Compute Qualty of Estmate Is Estmate Acceptable? NO Detemne Mamum Nomalzed Resdual s j YES SOP Remove Measuement j Pefom State Estmaton Compute Qualty of Estmate Is Pefomance Impovement Substantal? NO Restoe Measuement j YES Dscad Measuement j Fgue 7.. Flow Chat fo Bad Data Detecton, Identfcaton and Rejecton 7.6 Sequental State Estmatos Sequental state estmatos ae mathematcal pocedues by whch the effect of a sngle measuement on the state estmate s evaluated. Usng ths pocedue, the state estmate can be computed by pocessng each measuement ndvdually untl all measuements have been accounted. Sequental state estmatos ae useful n two applcaton aeas: (a) n eal tme applcatons, whee a state estmate has been computed and a set of few new measuements s eceved. One way to ncopoate the new measuements nto the state estmaton pocess s to add the new measuements to the old ones and ecompute the state estmate usng the ente set of measuements. hs appoach s n geneal neffcent fom the computatonal pont of vew. Anothe way s to evaluate the effect of the new Page 48 Copyght A. P. Saks Melopoulos 99-6
measuements on the state estmate wth the use of the sequental state estmato. (b) n bad data dentfcaton and especally n hypothess testng, t s equed to evaluate the effect of emoval of a measuement on the state estmate. hs poblem s deally suted to sequental state estmatos snce wth mnmal computatons one can evalauet the effects of emovng one measuement and theefoe can detemne f the measuement s bad o not. he sequental state estmato s pesented n ths secton. Sequental state estmaton algothms can be developed fo geneal nonlnea systems. Fo smplcty we wll lmt the pesentaton to lnea systems. One can epand the lnea sequental state estmaton algothm to nonlnea systems by lneazaton of the poblem. Consde the lnea system H b Cov( ) W whee b : vecto of measuements; dmenson m : state vecto; dmenson n : eo of measuements; dmenson m. he estmate of n the least squae sense,.e., the one that mnmzes the objectve J W s gven by ˆ ( H WH ) H Wb Now assume that a new scala measuement s added to the set of measuements ' h b E( ' ). E ' ' ( ) ρ ' he new estmate $ ' n the least squae sense s gven by ˆ ' ˆ aφ whee: φ A h Copyght A. P. Saks Melopoulos 99-6 Page 49
φ ρ h a ' WH H A Poof: he addton of the new measuement wll augment the model as follows: ' b' b h H ' ρ W Cov he new state estmate s: ( ' ' ˆ b b W h H h H W h H ρ ρ 7.) Note that ' ρ ρ ρ hh A hh WH H h H W h H A whee A s the old nfomaton mat ' A s the new nfomaton mat. Applcaton of the mat nveson lemma yelds: [ ] ' A hh A h A h A hh A A ρ ρ Also h b Wb H b b W h H ' ' ρ ρ Substtuton of above epessons nto (7.) yelds: [ ] h b Wb H A hh A h A h A ' ˆ' ρ ρ Page 5 Copyght A. P. Saks Melopoulos 99-6
Recall that ˆ A H Wb and defne φ A - h Upon substtuton and staghtfowad manpulatons: hen ˆ ' ' ˆ φ ρ h φ ' he quantty s a scala whch shall be called a. In ths way, equaton (7.4) s ρ h φ obtaned. he above esults suggest the followng sequental lnea state estmato gven a new measuement of b h : Step : Compute φ A - h ' Step : Compute a ρ h φ Step 3 : Compute new estmate $ $ aφ Step 4 : Update the nvese of the nfomaton mat Step 5 : If thee ae moe measuements go to step. A A ρ hh Note that above algothm s not self-statng. Step eques the cuent nfomaton mat I ( A - ). When the algothm stats, the mat I s undefned. Also, step eques the cuent estmate $. hese dawbacks can be ovecome by applyng the followng ntalzaton pocedue: Select n ndependent measuements. hese measuements, defne a detemnstc poblem: n equatons n n-unknowns. Soluton of ths poblem wll povde an ntal state estmate $. At ths soluton, compute the nvese of the nfomaton mat. Use the computed state and nfomaton mat to stat above algothm. he sequental lnea estmato algothm s dectly applcable to the nonlnea state estmaton of powe systems. he basc equatons ae deved as follows. Let b be a set of measuements yeldng the state estmate $. he model of the system s descbed wth the equatons h( ) b E ( ). Cov( ) W Assume a new measuement b a s added wth the followng model a h ( ) b a E ( ). a a Copyght A. P. Saks Melopoulos 99-6 Page 5
Cov( ) ρ a he new state estmate, assumng that one teaton s suffcent, wll be: whee b h( ˆ) W [ H h] ' ˆ ( A' ) ρ ˆ b a h ( ˆ) a A : the new nvese of the nfomaton mat A A ρ - hh A : the old nvese of the nfomaton mat A (H WH) H : the old Jacoban mat h : the contbuton to the Jacoban mat fom the new measuement. Applcaton of the mat nveson lemma as n the case of the lnea state estmato, and notng that A H ( h( ˆ) ), W b yelds: whee ˆ ' ˆ aφ φ A h h' ( ˆ) b' a ρ h φ he above esult s dectly utlzed n a sequental nonlnea state estmato. he pocedue wll be llustated wth an eample. Eample E7.9: Consde the smple electc powe system of Fgue E7.9 wth the llustated measung system. he voltage at bus s known to be. p.u. wthout eo. Othe measuements taken ae b b b V.97, Va( ) (.) P.94, Va( ) (.) 3 P.93 3, Va( 3 ) (.) 4 Q.48 4, Va( 4 ) (.) b : measuement eo of th measuement. Page 5 Copyght A. P. Saks Melopoulos 99-6
~ j V Ve -j. Voltage Magntude Measuement MW Flow Measuement MVA Flow Measuement Fgue E7.9 he state estmate based on above set of measuements s ˆ δ ˆ ˆ V.9988.978 and the nfomaton mat at the estmate s I.8735.839 ( ˆ).839.968 a) Compute the pobablty that the estmate s statstcally coect. b) A new measuement s taken: he eactve powe flow Q s measued to be.7 p.u. he vaance of ths new measuement eo 5 s va( 5 ) (.). Compute the new state estmate $ '. (One teaton s enough.) c) Compute the pobablty that the new estmate $ ' s statstcally coect. Soluton: a) At the computed state estmate, the measuement estmate s: P.VV sn δ.935 P.VV sn δ.935 Q.V.V V cosδ.48 he value of the objectve functon, computed at the estmate $, s 6 J 4 h ( ) σ b.597 Copyght A. P. Saks Melopoulos 99-6 Page 53
Usng able 7., the pobablty that the data ae statstcally coect s: P.94 b) he model of the new measuement Q s: h ( Q V V V δ 5 ).. cos At the pevously computed state estmate h5 ( ) Q.69 hus b ( 5 h5 ).89 he lneazed model s h [ V V snδ - V cosδ ] [-.935-7.93] he vecto φ s computed Also φ ( ˆ) 4.53487 3.59 I h 6 h φ.86868 ρ. a -.89/.86868 -.45 Applcaton of equaton (7.7a) yelds:.9993 ˆ'.974 c) he estmates of the measuements ae computed at the new state estmate. V.974 P.9355 P -.9355 Page 54 Copyght A. P. Saks Melopoulos 99-6
Q -.48446 Q.69599 he value of the objectve functon s computed: J( $ ' ).6645 Usng able 7., the pobablty that the data ae statstcally coect s : P.985 he nceased confdence s due to the addton of one moe good measuement. 7.7 State Estmato Obsevablty One mpotant queston n state estmaton elates to the obsevablty of the system state. Smply stated, gven a set of measuements can the system state be estmated (obseved) fom these measuements? We wll consde ths queston and elated ssues. 7.7. Algebac Obsevablty Fo the state of the system to be obsevable t s necessay that the numbe of measuements be geate o equal to the numbe of states. hs s a necessay condton but not suffcent. o obtan the suffcency condton, consde a system on whch m- measuements wee taken (vecto z) whle the system state s an n-vecto(vecto ). he lneazed model of the measuement s n geneal z H η whee: z s an m-vecto of measuements s an n-vecto of states H s an m n mat η s an m-vecto of measuement nose A suffcent condton to obtan a unque soluton fo n a specfed sense (.e. least squaes sense, LAV sense o mn-ma sense) s that the ank of H be n. hs means that the suffcent condtons fo obsevablty ae: m > n ank(h) n Above condtons ae suffcent condtons fo algebac obsevablty. ypcally, the tem obsevablty always means algebac obsevablty. Copyght A. P. Saks Melopoulos 99-6 Page 55
In geneal, the mat H s not a squae mat. A computatonally smple way to detemne the ank of H s based on the obsevaton that ank( H ) ank( H H) he mat H H s a squae mat. Now the algebac obsevablty test becomes equvalent to detemnng whethe the mat H H has full ank. hs can be acheved wth an LU factozaton pocedue as t has been dscussed n Append A. Specfcally, the mat H H s factoed nto the poduct of two tangula matces, L and U. H H LU Whee: L s a lowe tangula mat, U s an uppe tangula mat Note that whee: { H H} det{} L det{ U} Π l Π u det, the symbol Π means poduct, l epesents entes of the lowe dagonal (L) mat and u epesent entes of the uppe dagonal (U) mat. hus the mat H H s full ank f and only f all the dagonal elements of the L and U matces ae non zeo. Fo futhe nfomaton see Append A. 7.7. opologcal Obsevablty Fo the powe system state estmaton poblem wok by Clements [???] and othes has poven that the topology of the mat H povdes a good measue of the ank of the mat H and theefoe algebac obsevablty. he pocess by whch we deduct the ank of mat H fom ts topology s known as topologcal obsevablty. Computatonally topologcal obsevablty s much moe effcent than algebac obsevablty and t s pefeed. Howeve, t s mpotant to keep n mnd that topologcal obsevablty does not mply algebac obsevablty fo a geneal system. opologcal obsevablty analyss s pefomed as follows. Consde fo eample a measuement z. he model of the system may elate the measuement z to a numbe of system states. Denote these states as the subset of the state vecto entes. hus the measuement z spans the subset X z X z. Now a gven set of measuements z, span a subset Page 56 Copyght A. P. Saks Melopoulos 99-6
X of system states equal to the unon of all subsets coespondng to each z measuement z,.e. X z U X z X z Kumpholz, Clements and Davs [???] have demonstated wth smplfed models, that topologcal obsevablty s equvalent to algebac obsevablty fo pactcal powe system estmaton poblems. hus, one can use ethe appoach to detemne system state obsevablty. A consequence of ths fact s that one can use the topology of the Jacoban mat to detemne system state obsevablty. As a matte of fact we shall dscuss a method whch uses the topology of the nfomaton mat to ntepet t as a connectvty netwok of the system states. hen usng a topology check, the obsevablty can be detemned. he coespondng topologcal obsevablty test can be defned as follows. Fst obseve that the mat H H can be ntepeted as a netwok admttance mat snce t s symmetc. he netwok has a banch between nodes and j f and only f the enty j of the mat H H s non zeo. hus the poston of the off-dagonal non zeo entes of the mat H H detemne the netwok connectvty of the system states. he system s obsevable f and only f ths netwok s fully connected(all states ae connected). Hghly effcent algothms to check connectvty of a netwok est, makng ths appoach pactcal. Algebac and topologcal obsevablty wll be demonstated wth an eample. Eample E7.. Consde the fou bus system of Fgue E7.. A numbe of measuements ae taken as ndcated. 3 4 Voltage Magntude Measuement MW Flow Measuement MVA Flow Measuement Fgue E7. A Smplfed Fou Bus System Copyght A. P. Saks Melopoulos 99-6 Page 57
a) Wte the Jacoban mat H. Only the topology s of nteest. hus set all non zeo entes equal to.. b) Wte the equaton H (z-b) (H H) and daw a ccut that coesponds to above equaton. Soluton: a) he total numbe of measuements s 7. hey ae: [ ] V V V V P Q P Q P z 4 3 4 3 3 he system state s defned as: [ ] 4 4 3 3 V V V V δ δ δ, the voltage phase angle at bus s selected to be the efeence. he Jacoban mat wth each enty eplaced wth. s obtaned as follows: H b) he mat H H s computed to be: 3 6 5 5 5 3 H H he topology of an equvalent ccut s llustated n Fgue E7.a. Page 58 Copyght A. P. Saks Melopoulos 99-6
V δ V δ 3 V 3 δ 4 V 4 Fgue E7.a Equvalent Netwok fo opologcal Obsevablty hs completes the soluton. 7.7.3 Ctcal Measuements State estmaton s a eal tme functon. Many tmes due to malfuncton of equpment o communcaton channels, cetan measuements may be lost esultng to a possbly unobsevable system. In ths case, t s mpotant to dentfy the unobsevable states and to develop pocedues fo the estmaton of the obsevable states. A elated ssue s to dentfy the measuements whch f lost wll cause a state of the system to become unobsevable. hese measuements wll be called ctcal measuements. he teatment of ths subject wll also lead to methods of selectng measuements fo mamzng the elablty of the state estmaton. In the dscusson of these topcs t s epedent to study the popetes of the mat H H. Specfcally, obseve that the mat H H s symmetc and postve semdefnte,.e. Gven any nonzeo > > H H > Consde also the least squae soluton of equaton () H (z-b) (H H) hs equaton can be ntepeted as the nodal equatons of a ccut as follows (H H) : admttance mat : nodal voltages H(z-b) : nodal cuent njectons. Copyght A. P. Saks Melopoulos 99-6 Page 59
Snce H H s symmetc, such a netwok does est. he constucton of ths netwok has been demonstated wth an eample (Eample E7.). When system obsevablty s of nteest the actual values of the entes of the mat H H ae not mpotant but athe the topology of the mat. In ths case t suffces to eplace the non-zeo entes of the mat H wth abtaly selected values such as.. he dentfcaton of the ctcal measuements can be pefomed as follows: Consde a measuement z. Assume that the mat H H has been computed by assgnng the value of. to the non zeo entes of the Jacoban mat H H. Obseve that the measuement z can be epessed as z h whee h s the th ow of the Jacoban mat H. A dect appoach to system state obsevablty wll be to emove the measuement z, ecompute the mat H H and pefom the obsevablty test. Obseve that when the measuement z s emoved, the Jacoban mat H s modfed to H H' H L h L Upon computaton: H H H H h h he obsevablty test wll consst of checkng the topology of the mat H H. hee may be thee cases: Case : he matces H H and H H have the same numbe of nonzeo entes. In ths case, the connectvty of the netwok of states does not change. hus, the measuement z s not a ctcal measuement. Case : he mat H H has at least two less nonzeo entes than mat H H. In ths case, the connectvty of the netwok of states must be eamned to detemne obsevablty. Case 3: he mat H H has odd numbe less nonzeo entes than mat H H. In ths case, the system s not obsevable and the connectvty of the netwok of states must be eamned to detemne the unobsevable states. he pocedue wll be demonstated wth an eample. Page 6 Copyght A. P. Saks Melopoulos 99-6
Eample E7.: Consde the system of eample E7.. Detemne the type of the followng thee measuements: (a) the bus voltage magntude measuement at bus 4. (b) he eal powe flow measuement at ccut - 4. (c) he eactve powe flow measuement at ccut 3 -. Soluton: a) Fo measuement V 4, h [ ]: hh Obseve that the matces H H and H H have the same numbe of nonzeo entes. heefoe, the measuement V 4 s not a ctcal measuement. b) Fo measuement P 4, h [ ]: hh Note that one of the dagonal entes of H H becomes zeo, so P 4 s a ctcal measuement. c) Fo measuement Q 3, h [ ]: hh Copyght A. P. Saks Melopoulos 99-6 Page 6
Obseve that the matces H H and H H have the same numbe of nonzeo entes. heefoe, the measuement Q 3 s not a ctcal measuement. hs completes the soluton. 7.8 Lmtatons and Bases of State Estmaton In pecedng paagaphs we pesented the tadtonal state estmaton pocess as t s appled today n moden enegy management systems. hs applcaton s a specal case of the geneal state estmaton methodology ntoduced by Gauss and Legende (aound 8). he basc dea was to fne-tune state vaables by mnmzng the sum of the esdual squaes. hs s the well-known least squaes (LS) method, whch has become the conestone of classcal statstcs. he easons fo ts populaty ae easy to undestand: At the tme of ts nventon thee wee no computes, and the fact that the LS estmato could be computed eplctly fom the data (by means of some mat algeba) made t the only feasble appoach. Even now, most statstcal packages stll use the same technque because of tadton and computatonal speed. Also, fo one-dmensonal poblems, the LS cteon yelds the athmetc mean of the obsevatons, whch at that tme seemed to be the most easonable estmato. Aftewads, Gauss ntoduced the nomal (o Gaussan) dstbuton as the eo dstbuton fo whch LS s optmal. Snce then, the combnaton of Gaussan assumptons and LS has become a standad mechansm fo the geneaton of statstcal technques. In a eal tme envonment, state estmaton was appled to powe systems by Schweppe and Wldes n the late 96 s [???]. he ntal mplementaton was based on a sngle fequency, balanced and symmetc powe system unde steady state condtons. Ove the past thty plus yeas, the basc stuctue of powe system state estmaton has emaned pactcally the same: Sngle phase model P, Q, V measuement set Non-smultaneousness of measuements Sngle fequency model he above basc stuctue of the powe system state estmaton mples the followng assumptons (whch n tun esult n a based state estmato): all cuent and voltage wavefoms ae pue snusods wth constant fequency and magntude the system opeates unde balanced thee phase condtons the powe system s a symmetc thee phase system whch s fully descbed by ts postve sequence netwok hese assumptons ntoduce devatons between the physcal system and the mathematcal model (bas). Mathematcally, t s known that the least squaes state estmaton pocedue s an unbased estmato f and only f the model s accuate (eact) and the measuemnent eo s statstcally dstbuted. Both of these Page 6 Copyght A. P. Saks Melopoulos 99-6
condtons do not est n a pactcal system and have esulted n pactcal dffcultes manfested by poo numecal elablty of the teatve state estmaton algothm. Substantal effots to fne tune the mathematcal models n actual feld mplementatons ae equed. o allevate the souces of eo, new measuement systems, powe system model and estmaton methods ae needed. Fo eample the fst assumpton can be met by utlzng synchonzed measuements [???]. Synchonzaton s acheved va a GPS (Global Postonng System) whch povdes the synchonzng sgnal wth accuacy of µsec. Assumpton can be met by utlzng thee phase measuements. Fnally assumpton 3 can be met by employng full thee phase models. In ths secton we fst dscuss the bas esultng fom model naccuaces and then we dscuss the effect of measument eos. In patcula model naccuaces esult fom: (a) unbalanced opeatng condtons and (b) asymmetes of powe system models. hen, a state estmato s ntoduced that s based on the followng nfastuctue: Synchonzed measuements of voltage and cuent wavefoms hee phase measuements Use of full thee phase models he state estmaton based on ths system s not subject to the usual bases of the tadtonal state estmaton. hs state estmaton s fomulated n ts geneal fom that allows estmaton of all thee phase voltages esultng n the thee-phase state estmaton. 7.8. Bas Fom Unbalanced Opeaton An actual powe tansmsson system opeates nea balanced condtons. he mbalance may be small o lage dependng on the desgn of the system. As an eample, Fgue 7.7 llustates the thee phase voltages and cuents on an actual system. Note fo eample a % dffeence n the cuents of Phases A and B of tansmsson lne to GILBOA. he voltage n ths case has only a.% dffeence between two phases. Because of mbalance, the measuements may have an eo. We epesent ths as follows: z zt z zt whee s the tue value of the measued quantty (assumng a balanced system), the measuement eo due to mbalance, and z s the measuement. z s Applcaton of the LS state estmaton pocedue, assumng no othe eo souces, yelds: Copyght A. P. Saks Melopoulos 99-6 Page 63
t ( H WH) H W z (7.) t whee s the tue state of the system o the unbased state estmate, and the second tem s the bas esultng fom the mbalance measuement eo. Note that the bas fom unbalanced opeaton depends on the level of mbalance as well as the system paametes (mat H). Fgue 7.7. Actual hee Phase Voltages and Cuents n FRASER Substaton 7.8. Bas Fom System Asymmety An actual powe tansmsson system s neve symmetc. Whle some powe system elements ae desgned to be nea symmetc, tansmsson lnes ae neve symmetc. he mpedance of any phase s dffeent than the mpedance of any othe phase. In many cases, ths mbalance can be coected wth tansposton. Because of cost many lnes ae not tansposed. he asymmety may be small o lage dependng on the desgn of the system. One powe system component that contbutes to the asymmety s the thee phase untansposed lne. As an eample, Fgue 7.8 llustates an actual thee phase lne. Fo the pupose of quantfyng the asymmety of ths lne, two asymmety metcs ae defned: Page 64 Copyght A. P. Saks Melopoulos 99-6
S z ma z z mn S y ma y y mn whee z s the postve sequence sees mpedance of the lne, z ma and z mn ae the ma and mn sees mpedances of the ndvdual phases, y s the postve sequence shunt admttance of the lne, y ma and y mn ae the ma and mn shunt admttances of the ndvdual phases. he above ndces povde n a quanttatve manne the level of asymmety among phases of a tansmsson lne. As a numecal eample, these metcs have been computed fo the lne of Fgue 7.8 and ae pesented n Fgue 7.9. Note that the asymmety s n the ode of 5 to 6%. Copyght A. P. Saks Melopoulos 99-6 Page 65
'-" '-6" 4' 4' 7'-7" 9'-6" 9'-6" 7'-" 7'-7" 9'-6" 58'-" Fgue 7.8. ypcal ansmsson Lne Desgn wth Asymmety Page 66 Copyght A. P. Saks Melopoulos 99-6
.6 Asymmety Facto.4. Sees Admttance Shunt Admttance. 8 66 4 6 Fequency (Hz) Fgue 7.9. Asymmety Indces of the ansmsson Lne of Fgue 7.8 Because of the pesence of non-symmetc components, the state estmate usng sngle phase measument set s based. An estmate of the bas can be computed as follows. Fst obseve that because of powe system component asymmety, the elatonshp of a measuement to the system model wll have an eo. Specfcally: z h( ) h( ) whee h() s the functon elatng the measuement to the state vecto assumng symmetc powe system components, h() s the dffeence between the symmetc model and the asymmetc model. Now the jacoban mat of the measuements becomes: H H s H whee s the jacoban mat assumng symmetc powe system elements. H s Applcaton of the LS state estmaton pocedue, assumng no othe eo souces, yelds: ( t ( H t WH) H W z)( H WH) ( I ( H WH)( H WH) ) ( H WH) (7.) whee s the state of the system assumng a symmetc model, and the othe tems epesent the bas esultng fom the system asymmety. O BE COMPLEED 7.8.3 Bas Fom Systematc Measuement Eos State estmatos ae based on the assumpton that measuement eos ae statstcally dstbuted wth zeo mean. he tadtonal mplementaton of state estmaton uses sensos of V, P and Q. When the sensos ae popely calbated, the measuement eo s vey close to meetng the equements of state estmaton. Howeve, wth so many measuements n a pactcal powe system, thee ae many oppotuntes to have some measuements out of calbaton contbutng always data wth systematc eo. Recent Copyght A. P. Saks Melopoulos 99-6 Page 67
tends esulted n the use of sensoless technology fo powe system measuements. Sensoless technology efes to the use of A/D convete technology to sample the voltage and cuent wavefoms. Once the sampled wavefoms ae avalable, the equed measuements can be eteved wth numecal computatons. hese systems need calbaton and agan the same comments apply fo ths technology as befoe. Independently of the technology used fo measuements, t s mpotant to eamne whethe thee s bas n the measuements. hs can be best acheved by eamnng the ente measuement channel of a typcal powe system nstumentaton [???]. he majo souces of eo (see Fgue 7.5a and 7.5b) ae (a) the nstument tansfomes, (b) the cables connectng the nstument tansfomes to the sensos o A/D convetes and (c) the sensos o A/D convetes. Fgue 7. llustates the tansfe functons of a typcal nstument tansfome. It can be obseved that the chaactestcs of nstument tansfomes nea the powe fequency ae flat. One can conclude that fo powe fequency measuements, thee s no appecable measuement bas fom nstument tansfomes. Howeve, cables and A/D convetes can ntoduced appecable eo at 6 Hz. hs eo wll be a systematc eo. Fgue 7. llustates the tansfe functon of a specfc A/D convete. Note the magntude and phase bas even at powe fequency. It s mpotant to note that the measuement bas s dependent upon the desgn of the A/D convete. he measuement bas esultng fom contol cables s vaable dependng on the total length of the cables. Page 68 Copyght A. P. Saks Melopoulos 99-6
Fgue 7.. Magntude and Phase of Fequency Response of a kv/5/65 Potental ansfome Fgue 7.. Magntude and Phase of Fequency Response of the PMU- 6 Unt he measuement bas can be coected wth softwae. Such methods have been developed [???], but the use n state estmaton s vey lmted. It s mpotant to note that the above souces of eo cannot be coected wth bette (moe accuate) nstumentaton. o avod these souces of eo, thee phase measuements and a thee phase system model s equed. 7.8.4 Bas fom Measuement me Skews he tadtonal SCADA system s based on sequental pollng the Remote emnal Unts and theefoe the measuements eceved wll not be at eactly the same tme. If we assume that the system opeates unde steady state condtons, the measuements ae all consstent and the tme dffeence n obtang one measuement vesus anothe wll not be facto. If, howeve, system condtons change fom the tme the fst measuement was obtaned to the tme of the last measuement, then the data wll be nconsstent. hs nconsstency wll eflect on the pefomance of the state estmato, t may not convege, Copyght A. P. Saks Melopoulos 99-6 Page 69