On the use of PMUs in power system state estimation

On he Use o PMUs in Powe Sys em S a e Es ima ion An onio G´ omez-Exp´ osi o Ali Abu Pa icia Rousseaux Uni e si y o Se ille No heas e n Uni e si y Uni e si y o Li` ege Se ille, Spain Bos on, USA Li` ege, Belgium [email p o ec ed] ab[email p o ec ed] [email p o ec ed] An onio de la Villa Ja´ en Ca alina G´ omez-Quiles Uni e si y o Se ille Uni e si y o Se ille Se ille, Spain Se ille, Spain adela[email p o ec ed] [email p o ec ed] Abs ac - Synch onized phaso measu emen uni s (PMUs) a e becoming a eali y in mo e and mo e powe sys- ems, mainly a he ansmission le el. This pape p esen s, in a u o ial manne , he bene i s ha exis ing and u u e S a e Es ima o s (SE) can achie e by inco po a ing hese de- ices in he moni o ing p ocess. A e a e iew o he ele- an PMU echnological aspec s and he associa ed deploy- men issues (obse abili y, op imal loca ion, e c.), he al e - na i e SE o mula ions in he p esence o PMUs a e e is- i ed. Then, se e al applica ion en i onmen s a e sepa a ely add essed, ega ding he enhancemen s po en ially b ough abou by he use o PMUs. Keywo ds - PMUs, s a e es ima ion, obse abili y, measu emen placemen , dynamic es ima ion, mul i- a ea es ima ion 1 In oduc ion STATE es ima ion (SE) has been o decades one o he essen ial applica ions in Ene gy Managemen Sys ems (EMS), allowing secu e ope a ion o ansmission g ids. Measu emen s ecei ed and p ocessed by he s a e es ima- o s ypically include powe lows, ne powe injec ions and ol age and cu en magni udes [1]. A basic assump- ion behind he SE is ha he measu emen se cons i u es a single snapsho o he sys em being moni o ed, which is no ul illed in p ac ice because i akes a while o emo ely cap u e and cen ally ga he all he in o ma ion o be p o- cessed by he SE. In ac , i would be e y di icul , i no impossible o assu e ha all measu emen s a e synch o- nized (i.e., e e o he same ins an ). Howe e , as long as he ime elapsed be ween he i s and he las componen in he measu emen se is small enough, compa ed o he ime cons an o he sys em load, his assump ion will be accep able in p ac ice. Nowadays, he e is a clea end o b oaden he geo- g aphical scope o many SEs, in acco dance o he needs o egional elec ici y ma ke s, in which long-dis ance en- e gy ansac ions ha e o be accu a ely and pe manen ly moni o ed. In his con ex , he ask o collec ing wide a ea measu emen s and synch onizing he solu ions p o ided by each con ol a ea will be a challenging one [2]. Recen ly, he in oduc ion o mo e sophis ica ed p o- ec ion and measu emen componen s, such as he In elli- gen Elec onic De ice (IED), has p o ided he capabili y o ce ain phase angle di e ences be ween adjacen ( ol - age and cu en ) phaso s o be added a he local a ea o subs a ion le el [3, 4]. While his may bene i o a ce - ain ex en he accu acy o he SE a he TSO le el, i can ha dly be help ul in he mul i-TSO case. Occasionally, a seemingly simple echnical inno a ion can become a majo playe in changing he en i e indus- y. The e a e se e al his o ical examples o such inno- a ions one can iden i y, including he ligh bulb, ansis- o , lase , e c. Global posi ioning sa elli e (GPS) sys em can be included in his ein, pe haps no unlike he in e - ne (o he o iginal DARPA-ne ), which was also c ea ed ini ially o a small g oup o use s, bu hen apidly be- came a uni e sal ool employed by almos anyone wo ld- wide. The GPS sys em p o ides wo impo an bene i s which we e p e iously no eadily o easily acqui ed. One is he abili y o de e mine geog aphical coo dina es and he o he is o ha e global access o a e y accu a e clock allowing o ime s amp measu ed quan i ies i espec i e o he physical coo dina es a which hese measu emen s a e aken. De elopmen and ins alla ion o he GPS sys em was soon ollowed by nume ous enginee ing applica ions, which mainly consis ed o a ecei e and a p ocesso . The ecei e ’s unc ion is o cap u e he signals ansmi ed om a edundan se o sa elli es and hen p ocess hem in a ious di e en ways based on he objec i e o he im- plemen ed applica ion. One example o such an applica- ion is he phaso measu emen uni (PMU). These de ices a e ins alled a subs a ions in elec ic powe sys ems, and hei objec i e is o accu a ely de e mine he equency o he al e na ing cu en and ol age, and also p oduce he phaso ep esen a ion o hese signals de ined wi h espec o he global clock. Deploymen o PMUs s a ed a a slow pace abou a decade ago and accele a ed a e a numbe o successi e blackou s expe ienced in powe sys ems all a ound he globe. Phaso measu emen s a e synch onized wi h espec o he ime e e ence p o ided by he GPS sa elli es. Thanks o his accu a e global ime e e ence, synch ophaso s wi h iden ical ime-s amp ecei ed om a ious subs a- ions allow o c ea e a cohe en pic u e o he sys em s a e a a gi en ins an , elimina ing in his way he need o a i- icially se a phase angle, a bi a ily aken as he e e ence angle in con en ional s a e es ima o s. De ices ha can measu e synch onized phaso s we e de eloped and po en ial bene i s o PMU measu emen s we e ecognized o e wen y yea s ago by Phadke e al. [5, 6]. The i s implemen a ion o GPS-synch onized phase angle measu emen s in an indus ial powe sys em SE was p esen ed in [7]. U iliza ion o PMU measu emen s will impac s a e es ima ion in di e en ways. On he one hand, since he numbe o PMUs ins alled in exis ing powe sys ems is no ye su icien o ca y ou SE exclusi ely based on PMU measu emen s, SE o mula ion and solu ion emains non- linea and i e a i e espec i ely. Mo e imagina i e solu- ions, sequen ially handling con en ional and PMU mea- su emen s in a wo-s ep p ocedu e ha e been also p o- posed [8]. On he o he hand, SE ela ed issues such as ne - wo k obse abili y and measu emen placemen [9, 14], solu ion accu acy and eliabili y (con e gence a e), p o- cessing o bad da a and o he (pa ame e and opologi- cal) ypes o e o s [15, 16] will ha e o be econside ed. Mo eo e , in iew o he highe sampling a es a which PMUs can wo k, he possibili y o es ima ing he dynamic e olu ion o ce ain c i ical a iables is being explo ed [17]. This pape co e s in a succinc manne all o he abo e issues. Then, wi h he help o small u o ial examples, he po en ial bene i s p o ided by he inco po a ion o PMUs in se e al SE en i onmen s a e shown. 2 Phaso Measu emen Uni s APHASOR Measu emen Uni is a digi al de ice p o- iding synch onized ol age and cu en phaso mea- su emen s, e e ed o as synch ophaso s [18]. PMU ea u es we e i s implemen ed in s and alone uni s whose mos ele an unc ion is i s capabili y o p o- ide synch ophaso s. Nowadays, many IEDs (RTUs, p o- ec i e elays, ...) ha e been upg aded o p oduce syn- ch ophaso measu emen s in addi ion o hei own unc- ion. 2.1 Gene al PMU a chi ec u e Mos gene ally, PMUs p o ide mul i-channel inpu so ha in addi ion o he ol age a he ins alla ion bus, cu - en s in mo e han one line and possibly in all inciden lines can be p ocessed by a single uni . The h ee-phase ol ages and cu en s a e con e ed o app op ia e analog inpu s by ins umen ans o me s and an i-aliasing il e - ing. Each analog signal is digi ized by he A/D con e e wi h sampling a e usually a ying om 12 o 128 samples pe cycle o he nominal powe equency. The sampling clock is phase-locked wi h he GPS clock pulse which p o- ides he Uni e sal Time Coo dina ed (UTC) ime e e - ence used o ime- ag he ou pu s. Phaso s o phase ol ages and cu en s a e compu ed om sampled da a by he PMU mic op ocesso using a signal p ocessing echnique as desc ibed in sec ion 2.2 be- low. The calcula ed phaso s a e combined o o m he posi i e sequence phaso measu emen s (as well as neg- a i e and ze o sequences in case o unbalanced condi- ions). O he es ima es o in e es a e equency and a e o change o equency. Compu ed phaso measu emen s a e ansmi ed h ough a digi al communica ion ne wo k o highe le el applica ions a a a e o 10 up o 60 ames pe sec- ond. Many PMUs o e s o age capaci y enabling local exploi a ion o synch ophaso s. Howe e , in many eal- ime applica ions, and in pa icula s a e es ima ion, he phaso measu emen s a e no used locally bu a he a emo e loca ions. Phaso da a om a numbe o PMUs is hen collec ed by a special-pu pose compu e , called Phaso Da a Concen a o (PDC), which co ela es phaso da a by ime-s amp o c ea e a sys em-wide measu emen se . PDCs can p o ide a numbe o specialized ou pu s such as a di ec in e ace o a SCADA, EMS sys em o an uppe le el PDC. 2.2 Sync ophaso s and measu emen echniques The basic de ini ion o he phaso ep esen a ion o a sinusoidal wa e o m a nominal equency 0, bo h in i s pola and ec angula o m, is illus a ed in Fig. 1. The phaso angle is gi en by he angula di e ence be ween he peak o he sinusoid and he e e ence ime = 0. When conside ing synch ophaso , his e e ence ime co - esponds o he ime- ag. I he wa e o m is no a pu e sine signal, he compu ed phaso ep esen s i s undamen- al equency componen . Figu e 1: Sinusoidal wa e o m and i s phaso ep esen a ion The e ha e been se e al digi al algo i hms p oposed o es ima ing he phaso da a. The mos commonly used echnique elies on he Disc e e Fou ie T ans o m (DFT) [19]. The signal phaso is compu ed in a con in- uous p ocess om successi e samples in a mo ing da a window o one o se e al undamen al cycles. Adding he con ibu ion o he new sample while emo ing ha o he oldes one p o ides he mo e compu a ionally e icien e- cu si e DFT algo i hm. The sampling clocks a e usually kep cons an a a mul iple o 0. When equency a ies by a small amoun a ound i s nominal alue, he leakage e o in oduced in phaso es ima es can be compensa ed wi h high accu acy by a pos -p ocessing il e ing. I can also be shown ha he compu ed phaso o a es in he com- plex plane wi h an angula eloci y equal o he di e ence be ween 0and he ac ual equency so ha equency and a e o change o equency es ima es a e gi en by he i s and second de i a i es o he phaso angle. An al e na i e consis s in eplacing he DFT by a wa ele ans o m [20]. A numbe o algo i hms based on nonlinea es ima ion echniques (nonlinea weigh ed leas squa es es ima ion, Kalman il e ing, neu al ne wo ks, e c.) ha e also been p oposed in he li e a u e [21, 22, 23]. Acco ding o hese app oaches, he phase signal is mod- eled as a nonlinea unc ion o he phaso da a (ampli ude, phase angle, equency, a e o change o equency) con- side ed as pa ame e s o be es ima ed om he wa e o m samples. 2.3 PMU pe o mances and s anda ds Pe o mances o a PMU de ice, in e ms o accu acy o p ocessing ime, a e dic a ed by i s componen s, mainly, he ins umen a ion channel, he A/D con e e and he pa- ame e s o he phaso es ima ion algo i hm. Rega ding s a e es ima ion, he accu acy o PMU da a is a e y impo an issue. I is ecognized ha synch opha- so measu emen s a e usually mo e p ecise han con en- ional SCADA ones. Concep ually, PMU da a a e ime agged wi h p ecision be e han 1 mic osecond and mag- ni ude accu acy ha is be e han 0.1%. Howe e , his po en ial pe o mance is no achie ed due mainly o e o s om ins umen a ion channels and sys em imbalances. P esen ly, e alua ion o PMU da a accu acy is s ill a chal- lenging p oblem discussed in he scien i ic li e a u e [24]. PMU a e manu ac u ed by a a ie y o companies de ining speci ica ions o each pa icula PMU de ice. The speci ica ions conce n: he window leng h, sampling a e and ype o phaso es ima ion algo i hm, he phaso es ima e epo ing a e, he communica ion p o ocol and he measu emen accu acy. In o de o achie e in e ope abili y among PMUs, i is essen ial ha hei beha io complies wi h a common s anda d. The mos ecen IEEE C37.118-2005 s an- da d [25] de ines he synch ophaso con en ion and he ime- agging p ocess, p o ides he de ini ion o an accu- acy measu e as well as equi emen s o measu emen pe o mances unde s eady-s a e condi ions. I also de- ines da a communica ion o ma s. Requi emen s o e- sponse o powe sys em ansien s a e no conside ed. 3 Backg ound on S a e Es ima ion Gi en he ollowing measu emen equa ion [26]: z=h(x) + e(1) whe e: xis he s a e ec o (size n= 2N−1), zis he measu emen ec o (size m > n), his he ec o o unc ions, usually nonlinea , ela ing e - o ee measu emen s o he s a e a iables, eis he ec o o measu emen e o s, cus oma ily as- sumed o ha e a No mal dis ibu ion wi h ze o mean and known co a iance ma ix R. When e o s a e independen Ris a diagonal ma ix wi h alues σ2, whe e σis he s anda d de ia ion o he measu e- men e o s. he maximum likelihood es ima e ˆxis ob ained by mini- mizing he Weigh ed Leas Squa es (WLS) unc ion: J= m ∑ i=1 [zi−hi(ˆx)]2/σ2 i The minimum o he scala Jis eached by i e a i ely sol ing he so-called No mal equa ions: Gk∆xk=HT kW[z−h(xk)] (2) whe e: Hk=∂h/∂x is he Jacobian e alua ed a x=xk, Gk=HT kWHkis he gain ma ix, W=R−1is he weigh ing ma ix, ∆xk=xk+1 −xk,kbeing he i e a ion coun e . I e a ions inish when ∆xkis wi hin an app op ia e ole ance. I can be shown ha he co a iance o he es i- ma e is: co (ˆx) = ˆ G−1(3) whe e ˆ Gis he gain ma ix compu ed in he las i e a ion. Upon con e gence, he bad da a p ocessing unc ion is ac i a ed o de ec , iden i y and elimina e bad analog measu emen s. Bad da a de ec ion is accomplished based on he la ges no malized esidual es [27]. I he de ec- ion es ails, hen he measu emen co esponding o he la ges no malized esidual will be decla ed bad and i s alue will be emo ed o co ec ed [1]. In con en ional bus-b anch SE models he s a e ec- o is composed o ol age magni udes and phase angles, whe eas he measu emen ec o ypically comp ises mea- su emen s o powe injec ions, b anch powe lows and ol age magni udes. A lowe ol age le els, hough, line cu en magni udes and bus cu en injec ion magni udes can play a key ole o ob ain a su icien ly edundan sys- em. The inclusion o PMU measu emen s is he subjec o his pape . In he so-called gene alized SE model he s a e ec o is augmen ed wi h powe lows h ough ci cui b eake s (CB) a ce ain subs a ions whe e a opology e o is sus- pec ed, and he measu emen ec o may likewise include exis ing cu en o powe low measu emen s h ough any CB. T ans o me aps and suspec ed ne wo k pa ame e s can also be handled, i su icien edundancy exis s, bo h by con en ional and gene alized SE. 4 Ne wo k obse abili y and measu emen placemen WHEN using con en ional measu emen s ne wo k obse abili y es s can be ca ied ou based on he p ope ies o he measu emen Jacobian Hwhich is he g adien o he nonlinea measu emen equa ion e alua ed a an a bi a y ope a ing poin such as he la s a . I he Jacobian has ull column ank, hen he sys em will be de- cla ed ully obse able. In p inciple, he same app oach will wo k when he Jacobian is modi ied in he p esence o synch onized phaso measu emen s, which a e ei he o ol age o cu en ype. On he o he hand, conside ing he long e m ou look whe e su icien numbe o PMUs a e a ailable o ca y ou s a e es ima ion exclusi ely us- ing PMUs and dis ega ding all he con en ional measu e- men s, ne wo k obse abili y analysis can be o mula ed as a simple g aph co e ing p oblem. Phaso measu emen uni s may ha e di e en numbe o channels, i.e. hey may suppo di e en numbe o inpu s o ol age and cu en signals. I should also be no ed ha each inpu will be connec ed o one phase o a h ee phase ol age o cu en . While all h ee phases o ol age and cu en signals a e moni o ed, mos PMUs will ou pu only he posi i e sequence alues o he co e- sponding h ee phase quan i y. Gi en he ac ha powe sys ems a e spa sely connec ed, i.e. each bus has only a limi ed numbe o neighbo s i espec i e o he sys em size, channel limi s on PMUs may be assumed o be su i- cien o moni o as many signals as needed a a gi en bus. This assump ion will be elaxed la e and i s impac will be in es iga ed. Assuming ha a PMU is placed a bus k, he ollowing quan i ies can hen be assumed o be a ailable: •Vol age phaso a bus k, •Cu en phaso s along all lines/b anches inciden o bus k. Ne wo k model and pa ame e s being known, he abo e in o ma ion will allow compu a ion o phaso ol ages a all he neighbo ing buses as well. Hence, placing a PMU a a gi en bus implies obse abili y o all b anches in- ciden o ha bus. This simple obse a ion will lead o he ollowing in ege p og amming o mula ion o he ne - wo k obse abili y p oblem using only phaso measu e- men s: Minimize CTxi(4) Subjec o AX ≥1(5) whe e Cis he cos ec o o ins alla ion o PMUs, Xis a bina y ec o indica ing he p esence (1) o absence (0) o PMUs a buses, Ais a bina y ma ix mapping nonze o en ies o he bus admi ance ma ix o ones, 1is a ec o o ones. Buses co esponding o he nonze o alues in he solu- ion o (5) will yield he loca ions o place PMUs o ull ne wo k obse abili y [10]. While he con en ional mea- su emen s a e excluded in his o mula ion, equali y con- s ain s can be inco po a ed in o de o educe he numbe o equi ed PMUs. The mos common equali y cons ain s in powe sys ems a e hose p o ided by he ne ze o injec- ions a passi e buses wi h no gene a ion o load. These can be eadily inco po a ed in o he p oblem o (5) as done in [11]. 4.1 Me hods o placing PMUs o di e en objec i es While he ul ima e goal is o popula e powe sys ems wi h enough PMUs o acili a e ull obse abili y based only on PMU measu emen s, his will s ill be a ew yea s away. In he mean ime, as in es men decisions a e o be made whe e and how many PMUs o place in a gi en sys em, di e en objec i es may be conside ed. Fu he - mo e, since PMUs may be conside ed o speci ic ap- plica ions such as special p o ec ion schemes, seconda y ol age con ol, ol age o angle s abili y moni o ing, e c. he e may be al eady a cons ained se o buses whe e PMUs may ha e o be placed. In such cases, seconda y conside a ions in o de o make he bes o hese in es - men s will be impo an . This sec ion will b ie ly e iew wo such cases. 4.2 PMU placemen o de ec opology e o s Topology e o s a e caused by incomple e o w ong in- o ma ion abou one o mo e ci cui b eake s a he subs a- ions. These e o s can be e y di icul o de ec and iden- i y due o he speci ic measu emen con igu a ion a ound he a ec ed subs a ion. A ce ain ype o opology e o ha is e e ed o as b anch opology e o is de ined as he e o in he s a us o a gi en b anch, i.e. whe he o no he b anch is in o ou o se ice [12]. These ypes o e o s a e ela i ely easie o de ec and iden i y compa ed o he mo e complex ones in ol ing se e al b eake s lead- ing o bus spli s o me ge s. De ec abili y o a b anch opology e o is closely linked o he measu emen edundancy and con igu a ion. Hence, i is possible o make a p e iously unde ec able b anch opology e o de ec able by s a egic me e place- men . I a limi ed numbe o PMUs a e being conside ed o be placed in a gi en sys em, one conside a ion may be o imp o e b anch opology de ec abili y. The op i- mal case would be o ha e all b anches opology e o de- ec able, bu e en making a subse o b anches opology e o de ec able would be a welcome imp o emen . This can be accomplished in h ee s eps: •Iden i y all b anches ha a e opology e o unde- ec able, •Fo each iden i ied b anch, de e mine all candida e PMU loca ions so ha i a PMU is placed, his will make he b anch opology e o de ec able, •Se up an op imal selec ion p oblem so ha a mini- mum numbe o he candida es iden i ied in he p e- ious s ep can be selec ed o make opology e o s associa ed wi h all iden i ied b anches in s ep 1, de- ec able. De ails o he p oblem o mula ion along wi h illus a i e examples can be ound in [13]. I should be no ed ha , as shown in [13], phaso measu emen s ha e some unique ad an ages o e he con en ional measu emen s when i comes o opology e o de ec ion and iden i ica ion. 4.3 PMU placemen o measu emen e o de ec ion E e y measu emen sys em may ha e ulne abili y pocke s whe e e o s in one o mo e measu emen s can no be de ec ed. Such measu emen s a e e e ed o as c i - ical and hei measu emen esiduals will be null i espec- i e o hei measu ed alues [15]. A obus measu emen design will add ess hese ulne abili ies and s a egically place me e s in o de o ans o m hese c i ical measu e- men s in o edundan ones whose e o s will be de ec able. Techniques o iden i ying all c i ical measu emen s in a gi en powe sys em exis and can be used o de e mine all such measu emen s. Once hey a e iden i ied, a nume ical ac o iza ion based app oach (whose de ails a e gi en in [15]) can be used o de e mine all possible loca ions whe e PMUs can be placed so ha a gi en c i ical measu emen will be ans o med. These candida e PMU loca ions a e hen conside ed simul aneously and a minimum numbe ha will ans o m all c i ical measu emen s, can be de e - mined again using an in ege p og amming o mula ion. Simula ion esul s shown in [15] clea ly imply depen- dence o he op imal numbe o equi ed PMUs on he ne - wo k opology and exis ing measu emen con igu a ion. Howe e , he e a e cases whe e wi h only a hand ul o PMUs, a e y la ge numbe o c i ical measu emen s can be ans o med, hus d as ically imp o ing he obus ness o s a e es ima ion agains bad da a. 4.4 PMU placemen o pa ame e e o de ec ion and iden i ica ion E e y powe sys em da a base equi es cons an main- enance due o changes in ne wo k pa ame e s ei he due o en i onmen al condi ions such as empe a u e, humid- i y, wind, e c. o due o human e o in en e ing da a co - esponding o equipmen pa ame e s such as ans o me aps, shun capaci o banks, e c. A ypical powe sys em model will ha e a huge numbe o pa ame e s associa ed wi h i s line, ans o me , shun capaci o / eac o models. Hence, use o s a e es ima o as a ool o de ec and iden- i y pa ame e e o s has been a opic o nume ous in es i- ga ions. These in es iga ions mainly ocused on pa ame- e es ima ion based on he assump ion ha a suspec se o pa ame e s ha e al eady been iden i ied. Howe e , selec - ing a suspec se which is gua an eed o con ain e oneous pa ame e s simply based on measu emen esiduals is no always possible. A me hod ha o e comes his limi a ion is ecen ly de eloped and hen applied o he case o s a e- gic placemen o PMUs o pa ame e e o de ec ion and iden i ica ion [16]. I is no ed ha , i s a egically placed, PMUs will en- able e o iden i ica ion o ce ain pa ame e s which a e no possible o iden i y using con en ional measu emen s, no ma e how high o a measu emen edundancy is in- oduced. This esul is alida ed wi h some simple exam- ples in [16]. Along wi h he b anch opology e o de ec- ion p oblem, he pa ame e e o iden i ica ion p oblem cons i u es one o he examples whe e PMUs will ha e a unique edge o e con en ional measu emen s. 5 S a e es ima ion o mula ion in he p esence o PMUs UP o he ad en o he PMU echnology he measu e- men ec o zcon ained only powe ( low and injec- ion), ol age magni ude and, in ce ain pa icula cases, cu en magni ude measu emen s, all o hem aken in a non-synch onized manne . A he local le el [4], phase angle di e ences among adjacen ol age and/o cu en wa e o ms can also be p o ided by he new gene a ion o in elligen elec onic de ices (IEDs). This sec ion discusses se e al modeling issues a ising in he o mula ion o he SE p oblem when PMUs a e o be inco po a ed. 5.1 Measu emen models As s a ed p e iously, PMUs can p o ide bo h ol age and cu en phaso s, collec i ely deno ed as zVand zI e- spec i ely. Depending on whe he he s a e ec o is ep esen ed in pola o ec angula coo dina es, di e en exp essions will esul o he measu emen model, as ollows: •S a e ec o in pola coo dina es: x=[V θ] In his case, he e o - ee models co esponding o each ype o measu emen s ake he o m: z=h(x)(6) zV=Kx (7) zI=hI(x)(8) whe e i is assumed ha zVis ep esen ed in pola co- o dina es so ha Kis a i ial ma ix wi h a single 1 in each ow. I phase angle di e ences, θi−θj, a e also conside ed, hen he espec i e ows in Kwill con ain a 1 and a −1. No e ha he measu emen model as- socia ed wi h zIis nonlinea , i espec i e o zIbeing exp essed in pola o ec angula coo dina es. The ec - angula e sion is p e e able, howe e , owing o he nu- me ical p oblems (unde ined Jacobian e ms) ha may a ise o e y small cu en s. The exp essions o h(.)can be ound elsewhe e [1] while hose co esponding o hI(.)a e gi en in [28]. •S a e ec o in ec angula coo dina es: x =[VRe VIm ] The e o - ee models co esponding o each ype o measu emen s ake he o m: z=h (x )(9) zV =Kx (10) zI =Y x (11) whe e he measu emen ec o s zV and zI a e as- sumed in ec angula coo dina es, zV =[VRe VIm ];zI =[IRe IIm ] because his way he measu emen models (10) and (11) become linea , which is one o he key ad an ages asso- cia ed wi h PMUs. In his case, h (.) educes o a se o quad a ic unc ions, p o ided V2 a he han Vmea- su emen s a e included in z. No e ha , unlike in he pola s a e ec o case, null in- jec ion cons ain s can be linea ly o mula ed i null cu - en s a he han null powe s a e en o ced. The exp essions o h (.)can be ound elsewhe e [1] while hose co esponding o Y a e gi en in [29]. 5.2 Simul aneous SE o mula ion The in o ma ion p o ided by PMUs can and should be handled in heo y a he same ime he con en ional mea- su emen s a e p ocessed by he SE. This equi es modi i- ca ions o he exis ing so wa e in o de o accommoda e he new Jacobian e ms and componen s o he esidual ec o [30]. Depending on he p opo ion o con en ional e sus PMU measu emen s, and he numbe o PMU channels, he pola o ec angula model will be p e e able, as ol- lows: •I zIis emp y, ha is, PMUs p o ide only ol age pha- so s, hen he con en ional pola model will be p e e - able. This equi es mino adap a ions o exis ing SEs o accommoda e phase angle measu emen s (see he sub- sec ion 5.4 below o a discussion on he e e ence angle issue). •In u u e en i onmen s, howe e , whe e PMU mea- su emen s will e en ually eplace con en ional RTU measu emen s, he ec angula model will be p e e - able. E en ually, in he absence o any powe o iso- la ed ol age magni ude measu emen , he esul ing SE model will become ully linea in ec angula coo di- na es, which is a nice ea u e o exploi . In ce ain cases, con en ional aw measu emen s could be p e iously manipula ed so ha hey a e con e ed o PMU-like pseudomeasu emen s. Fo ins ance, a pai o powe ( low o injec ion) measu emen s associa ed wi h a bus whose ol age phaso is p o ided by a local PMU, can be ans o med in o an equi alen cu en phaso , in- c easing in his way he linea i y o he esul ing model. This equi es o cou se ha he co a iance o he pseu- domeasu emen s be compu ed om ha o he aw mea- su emen s. No e also ha , in his app oach, he PMU measu e- men s a e aken in o accoun om he e y beginning du - ing he obse abili y and bad da a analyses. 5.3 Sequen ial SE o mula ion In his scheme, con en ional measu emen s a e i s p ocessed in he usual manne , and hen a new SE is de- signed aimed a imp o ing he ini ial es ima es by inco - po a ing he in o ma ion p o ided by PMUs. A nonlinea ans o ma ion is equi ed in be ween o swi ch be ween pola and ec angula coo dina es [31]. The h ee s ages in ol ed a e as ollows: 1) Dis ega ding PMU measu emen s, ob ain a p elimi- na y es ima e ˜xby sol ing he con en ional nonlinea SE p oblem, gi en by (1)-(2). This equi es ha he en i e ne wo k be obse able in he p esence o jus RTU measu emen s. As a byp oduc , he in e se o he gain ma ix a ising in he las i e a ion p o ides he es- ima e’s co a iance, acco ding o (3): co (˜x) = ˜ G−1 2) The es ima e ˜xis ans o med o ec angula coo di- na es: ˜x = (˜x)(12) whe e he nonlinea unc ions (.) ep esen he well- known ela ionships, ˜ VRe =˜ Vcos ˜ θ ˜ VIm =˜ Vsin ˜ θ In addi ion o ˜x i s co a iance is equi ed. This is ob- ained om: co (˜x ) = ˜ F·co (˜x)·˜ FT(13) whe e ˜ Fis he Jacobian o (.)compu ed o ˜x. No e ha his Jacobian is a 2×2-block diagonal squa e ma- ix. 3) The phaso measu emen s p o ided by PMUs, in ec - angula coo dina es, along wi h he es ima e ˜x , lead o he ollowing linea measu emen model: ˜x =x +εx(14) zV =Kx +εV(15) zI =Y x +εI(16) whe e he co a iance o εVand εIis a known diagonal ma ix and ha o εxis gi en by (13). Acco dingly, he inal es ima e ˆxis he solu ion o he No mal equa ions a ising a his linea s age:   I Kx Y   T  Wx WV WI    I Kx Y  ˆx=   I Kx Y   T  Wx WV WI    ˜x zV zI  (17) whe e he weigh ing ma ices Wx,WVand WIa e he in e se o he espec i e co a iance ma ices. No e ha WVand WIa e cus oma ily conside ed diagonal ma ices, which igno es he ac ha he measu emen noises o he se o “ aw” measu emen s ga he ed by a single PMU a e co ela ed. On he o he hand, he ma ix Wxis gi en by: Wx= co −1(˜x ) = ˜ F−T·˜ G·˜ F−1(18) whe e he in e se o ˜ Fis i ially ob ained by compu - ing he 2×2 in e se o i s cons i u i e diagonal blocks. This ma ix is no longe diagonal bu i s spa si y should be exploi ed. The main ad an age o his al e na i e is ha a con- en ional SE is eso ed o a he beginning, allowing ex- is ing so wa e o be adop ed. Mo eo e , he hi d s age consis s o a linea SE, which implies ha he solu ion is ob ained in a single i e a ion, p e en ing he isk o di e - gence in he p esence o bad da a. Al hough he solu ion eached h ough his h ee-s age app oach is no he op i- mal one, i is su icien ly accu a e o p ac ical pu poses. I s main d awback is ha PMU measu emen s canno be used du ing he i s s ep o po en ially enla ge he obse - able ne wo k and o enhance he bad da a de ec ion and iden i ica ion p ocess. 5.4 Re e ence bus issues S a e es ima ion p oblem is commonly o mula ed by choosing a e e ence bus ( ypically bu no necessa ily he same as he slack bus used o he powe low analysis) and se ing i s ol age phase angle equal o ze o. This also im- plies ha he e e ence phase angle will be excluded om he s a e ec o and he co esponding column o Hwill be emo ed when building he measu emen Jacobian. Al e - na i ely, he e e ence phase angle can be e ained in he s a e ec o bu hen a phase angle pseudo-measu emen o a bi a y alue (ze o o con enience) mus be added o each obse able island. In he absence o any phase angle measu emen , his p ac ice p esen s no p oblems and p o ides a sui able amewo k o de ine he sys em s a e whe e he ac ual alue o he e e ence bus ol age phase angle is i ele- an . Howe e , as he phaso measu emen s s a popula ing he sys ems, he choice o a e e ence bus will no longe be an a bi a y decision. The e a e wo possibili ies: 1) Choose a bus whe e no PMU exis s: This will c ea e inconsis encies be ween he a bi a ily assigned e e - ence angle a he chosen bus and ac ual phase angle measu emen s p o ided by PMUs a o he buses. 2) Choose one o he buses wi h PMUs as he e e ence bus: This will wo k as long as he PMU a he chosen bus unc ions pe ec ly. I he measu emen s p o ided by his PMU con ain e o s, hen hese e o s will no be de ec able and will bias he es ima ed s a e. This issue has been ecognized ea ly on and al e na- i e app oaches we e conside ed. Among hem is a docu- men [32] which is p oduced by he Eas e n In e connec- ion Phaso P ojec (EIPP) g oup. In his documen a i - ual bus angle e e ence, which is compu ed as he a e age o se e al phase angle measu emen s by PMUs loca ed in he icini y o a chosen bus is in oduced. This app oach s ill emains ulne able o e o s in indi idual PMU mea- su emen s despi e he use o a e aging. In he p esence o nume ous PMUs i will be logical o use he absolu e phase angle in o ma ion p o ided by hose de ices. Hence, he measu emen Jacobian will ha e o include columns co esponding o all bus ol age mag- ni udes as well as phase angles, he dimension o he sys- em s a e ec o being wice he numbe o buses [33]. In his case, he sys em will be decla ed obse able i no ze o pi o s a e encoun e ed while ac o izing G. When he e is mo e han one obse able island in he sys em ex- cluding he phaso measu emen s, hen he e has o be a leas one phase angle measu emen in e e y obse able is- land o make he o e all sys em obse able. When he e is only one phase angle measu emen in he sys em, hen his case can be educed o he con en- ional o mula ion wi h an assigned e e ence bus. Since he alue o i s phase angle is i ele an , e o s in his mea- su emen will no a ec he es ima ion esul s (c i ical in- o ma ion). A mo e ealis ic case is when he e a e wo o mo e phaso measu emen s in he sys em. In his case, de- ec ion o phaso measu emen e o s equi es highe e- dundancy as discussed below. Dis ega ding he phaso measu emen s, con en ional ne wo k obse abili y anal- ysis [1] will yield he numbe o obse able islands in a gi en sys em. Ha ing a leas one phaso measu emen s in e e y obse able island will ensu e obse abili y o he en i e ne wo k. In o de o be able o de ec and iden- i y e o s in he phaso measu emen s, hei edundancy should be u he inc eased in hei espec i e obse able islands. De ini ion o c i ical k- uples can be ound in [1]. Following his de ini ion, i can be shown ha wo pha- so measu emen s will ensu e de ec abili y and h ee will be necessa y o iden i ica ion o bad da a associa ed wi h any phaso measu emen in a gi en obse able island. 6 Applica ion en i onmen s IN his sec ion, he imp o emen in SE pe o mance due o he inco po a ion o PMU measu emen s is illus- a ed. Fi s , he e ec o including PMU measu emen s in he accu acy o he TSO-le el es ima o is assessed. Then, he enhanced synch oniza ion capabili y p o ided by PMUs in he mul i-TSO SE case is add essed. Owing o space limi a ions, only small u o ial examples a e con- side ed, bu he main conclusions emain alid o ealis ic ne wo ks. O he issues, such as he imp o ed moni o ing o sma dis ibu ion sys ems by combining SE and PMUs, o he use o he high olume o his o ical da a collec ed by PMUs o de elop eliable load o ecas ing, s ill in hei in ancy, a e no add essed o he sake o b e i y. Fo he simula ions below he s a e a iables a e ex- p essed in pola o m. The complex ol age phaso s p o- ided by PMUs a e assumed in pola o m while he cu - en phaso s a e ep esen ed in ec angula o m in o de o a oid ill-condi ioning p oblems [4]. Unde hese as- sump ions, he PMU measu emen s can be exp essed as ollows: Vm i=Vi+εVi θm i=θi+εθi Im Re,ij =Vi(Asin θi+Bcos θi) + Vj(Csin θj+Dcos θj) + εIRe,ij Im Im,ij =Vi(Esin θi+Fcos θi) + Vj(Gsin θj+Hcos θj) + εIIm,ij whe e he cons an s A o Hdepend on he pa ame e s o he πmodel associa ed wi h b anch i-j[28]. In all simu- la ed scena ios he PMU and con en ional measu emen s ha e been handled simul aneously du ing he es ima ion p ocess. 6.1 U iliza ion o PMUs a he TSO le el The accu acy imp o emen a ising by inco po a ing PMU measu emen s in a con en ional SE a he TSO le el has been assessed wi h he help o he 5-bus ne wo k ske ched in Fig. 2. Di e en scena ios wi h inc eased e- dundancies ha e been conside ed: 1) Con en ional measu emen s only: he se o measu e- men s used o hese simula ions is shown in Fig. 2. The s anda d de ia ion o his ype o measu emen s has been se o 0.01. 2) Inclusion o 1 PMU: in addi ion o he con en ional se o measu emen s a single PMU (‘PMU1’ in Fig. 3) has been loca ed a node 3. This de ice has se e al chan- nels measu ing he complex ol age a bus 3 as well as he cu en phaso s h ough lines 3-4 and 3-5. 3) Inclusion o 3 PMUs: wo mo e PMUs a e added a nodes 4 and 5 (‘PMU2’ and ‘PMU3’ in Fig. 3). These PMUs measu e only he ol age phaso a he co e- sponding node. 1 2 3 4 5 Powe measu emen Vol age magni ude measu emen Figu e 2: 5-bus illus a i e ne wo k wi h con en ional measu emen s 1 2 3 4 5 PMU 2 PMU 1 PMU 3 Complex ec angula cu en measu emen Complex pola ol age measu emen Figu e 3: 5-bus illus a i e ne wo k wi h PMU measu emen s Di e en alues o he s anda d de ia ion associa ed wi h PMU measu emen s ha e been es ed, anging om 0.0005 o 0.01 (same quali y as con en ional measu e- men s). A pa ame e K, ela ing he s anda d de ia ion o con en ional (σc) and PMU (σpmu) measu emen s, has been de ined: K=σc/σpmu Hence, K= 10, o ins ance, means ha he PMU mea- su emen s a e 10 imes mo e accu a e in a e age han con- en ional measu emen s. The measu emen s o he di e en scena ios ha e been c ea ed by adding gaussian noise o he ‘exac ’ mea- su emen s co esponding o a gi en ne wo k s a e. The andomly gene a ed noise has been scaled acco ding o he s anda d de ia ion o he co esponding measu emen , as ollows: zm i=zex i+kiσi(19) whe e zm iis he i- h measu emen , zex i he exac calcu- la ed alue, kia andomly gene a ed gaussian numbe N(0,1) and σi he s anda d de ia ion assumed. Fo each alue o Kone hund ed Mon e Ca lo simu- la ions ha e been pe o med. In o de o e alua e he im- p o emen b ough abou by PMUs, he accu acy o ol - age magni ude and powe low es ima es a e sepa a ely an- alyzed ( hese a e he mos in e es ing magni udes o EMS ope a o s). Fo his pu pose, he ollowing indices ha e been de ined: SV= n ∑ i=1 |˜ Vi−Vex i|/n SP Q = nP Q ∑ i=1 |˜ PQij −PQex ij |/nP Q whe e nand nP Q a e he numbe o nodes and powe low (ac i e and eac i e) measu emen s, espec i ely and PQij ep esen s any powe low measu emen . Fig. 4 shows he esul ing SVindex o he di e en scena ios. I can be obse ed how, as mo e PMUs a e in- co po a ed, lowe alues a e ob ained, which means mo e accu a e es ima es. Mo eo e , he es ima es imp o e as he pa ame e Kinc eases, which happens when he s an- da d de ia ion associa ed wi h he PMU measu emen s de- c eases. Fig. 5 shows simila esul s o he index SP Q. 0 5 10 15 20 0 1 2 3 4 5 6 7x 10−3 K 0PMU 1PMU 3PMU Figu e 4: Simula ion esul s o index SV 0 5 10 15 20 2 3 4 5 6 7 8 x 10−3 K 0PMU 1PMU 3PMU Figu e 5: Simula ion esul s o index SP Q The ollowing ema ks a e in o de : •The inc emen al bene i o adding ol age phaso s (‘PMU2’ and ‘PMU3’) is less no iceable han ha o cu en phaso s (‘PMU1’), pa icula ly when powe lows a e conside ed (Fig. 5). •Fo alues o Kla ge han 10 he accu acy im- p o emen b ough abou by PMUs somewha sa - u a es. In gene al, howe e , his will also depend on he edundancy o he con en ional measu emen se . •F om he poin o iew o es ima e accu acy, i is p obably be e o in es in u he imp o ing he quali y o PMU measu emen s a he han inc eas- ing he numbe o PMUs. Howe e , obus ness agains ailu e o loss o PMU channels is highe when a la ge numbe o PMUs a e ins alled. 6.2 U iliza ion o PMUs a he egional mul i-TSO le el In egional powe sys ems, whe e eal- ime measu e- men s a e ga he ed wi hin neighbo ing a eas by he a - ious con ol cen e s dis ibu ed o e he g id, he Mul i- A ea S a e Es ima ion (MASE) has go enewed in e es . In his en i onmen , he need o p ope ly moni o ing en- e gy ansac ions ac oss TSO bo de s ia la ge in e con- nec ions, while a he same ime p ocessing he eal- ime da a a he mos app op ia e place, make he MASE a good al e na i e. In gene al, MASE elies on some kind o decomposi ion-coo dina ion scheme, aking ad an age o he usually weake geog aphical o measu emen coupling among a eas. The MASE consis s o a sequence o hie a chical SE p ocesses comp ising wo main s ages [2]: 1) each TSO independen ly sol es he s a e es ima ion o i s own a ea, including he ie-lines, bo de nodes and bo de measu e- men s o adjacen a eas; 2) he esul s o hese decoupled es ima o s a e hen used by an ad hoc p ocedu e which coo dina es he es ima e o he en i e sys em. Unde his scheme, when he di e en a eas a e no synch onized in ime wi h PMU measu emen s, each o he a eas se s a local phase angle e e ence o he TSO-le el es ima ion p ocess ( i s s ep). This equi es he in oduc ion o new s a e a iables u, one o each a ea, ela ing he di e en phase angle e e ences o he sys em. These a iables, whose alues a e es ima ed a he second s ep, coo di- na e he esul s o he a eas by e e ing he es ima es o a global phase angle e e ence. When synch onized PMU measu emen s a e a ailable a all a eas, he local phase angle e e ences a e no longe needed. The PMU measu emen s will implici ly coo di- na e he independen es ima es o he Uni e sal Time Co- o dina ed e e ence (UTC). In case some a eas do no ha e PMU measu emen s a ailable, only hose a eas will need o se a local phase angle e e ence, and o each one a u a iable will ha e o be de ined and es ima ed in o de o coo dina e he local es ima es o he UTC. Since he u a iables coo dina e he es ima es o di - e en a eas, hei ole is c ucial in he compu a ion o powe lows h ough ie-lines. The quali y o he ues- ima es will a ec he accu acy o he es ima ed lows h ough he ie-lines and, as a consequence, he es ima es o he ene gy ansac ions among TSOs. I PMUs a e a ailable, he lack o u a iables along wi h he enhanced accu acy usually p o ided by he PMU measu emen s, im- ply be e es ima es o powe lows a ie-lines. Some simula ions ha e been ca ied ou in o de o e alua e he imp o emen o he mul i-a ea s a e es ima- ion in he p esence o PMUs. The ne wo k used in hese es s is made up o h ee IEEE 14-bus es ne wo ks, con- nec ed o each o he as shown in Fig. 6, whe e only he ie-lines and bo de buses a e ep esen ed. Table 1 shows he ie-line pa ame e s adop ed o hese expe imen s. 123 11 2 3 2 3 B A C Figu e 6: Mul i-a ea sys em composed o 3 IEEE 14-bus ne wo ks