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