Kothari D.P., Nagrath I.J. Modern Power Systems Analysis
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Pr'lurs
tu
a
large
system
over
long
transmission
lines.
Voltage
':::i:y,"::,*:,','31^
^,?::^s,112itiu
and
,oto."ungl"
stability
is
basicauy
Effective
counter
Measures
to
prevent
or
contain
voltage
Instability
(i)
Generator
terminal
voltage
shourd
be
raised.
(ii)
Generator
transformer
tap
value
may
be
increased.
(iii)
Q-injection
should
be
carried
out
at
an
appropriate
location.
(iv)
Load-end
oLTc (on-load
tap
changer)
should
be
suitably
used.
(v)
For
under
voltage
conditions,
strategic
load
shedding
should
be
resorted
to'
System
reinforcement
may
be
carried
out
by
instpiling
new
transmission
lines
between
generation
and
load
centres.
series
and
shunt
compensation
may
be
carried
out
and
svcs
(static
var
compensation)
may
be
installed.
Generation
rescheduling
and
s_tarting-up
of
gas
,rrui-"*
,.y
i" .,*.i.,t
out.
Practical
aspects
of
Q-flow
problems
leading
to
voltage
collapse
in
EHv
lines:
(i)
For
lons line.s urifh 'r-^^nrrnu^r L--^^-
"-!'
s'vv'Lr.,'e'
uus's,
recelvlng_end
or
road
voltages
increase
for
light
load
conditions
and
decrease
for
heavy
load
conditions.
(ii)
For
radial
transmission
lines,
if
any
loss
of
a line
takes
place,
reactance
goes
up,
I2x
loss
increases
resulting
in
increase
in
voltage
drop.
This
should
be
suitably
compensated
by
local
e
injection.
of
course
this
involves
cost.
If
there
is
a
shortage
of
rocal
q
,ourrr,
then
import
of
e
wffi;Ewe
desirable.
Unity
power
factor
Nosepoint
(knee)
t
\
Loaw
of V6 and Pr"r;
same
for
const Z load
for
other types of load,
V
degradation
is faster.
PplP6zy
0.2 0.4 0.6
0.8 1.0
1.2
1.4 1.6
Fig. 13.8 PV
curves with
different load
power
factors
Only
the operating
points
above
the critical
points
represent
satisfactory
operating conditions. At
the
'knee'
of the V-P
curve,
the voltage
drops rapidly
with an increase in
load demand.
Power-flow
solution
fails
to
convefge
beyond
this
limit indicating
instability.
Operation at
or near
thel
stability
limit
is
impractical
and a satisfactory
operating
condition
is ensured
by
permitting
sufficient
"power
margin".
SPfti*f
uooern
po@is
I
Voltage
stability
problems
normaliy
occur
in
heavily
stressed
systems.
voltage
stability
and
rotor
angle
(or
synchronous)
stability
are
more
or
less
interlinked'
Rotor
angle
stability,
.a3-.wex
as
voltage
stability
is
affected
by
reactive
power
control.
voltage
stability
is
concern"o
*itt,
load
areas
and
load
Senerator
stability.In
a
large
inter-connected
system,
voltage
coilapse
of
a
load
area
is possible
without
lois
of
synchroni*
or
any
generators.
The
slower
forms
of
voltage
instability
are
often
analysed
as
steady-state
problems'
'snapshots'
in
time
following
an
outage
or
during
load
buildup
are
simulated'
In
addition
to post-disturbancl
load
floivs,
two
other
load
flow
based
methods
are
widely
used:
P-V
curves
and
e-v
curves.
pV
curves
are
used
for
especially
for
radial
systems.
/
curves
(Fig.
13.g),
eV
curves
(Fig.
'ig.
13.8)
and
methods
to quantify
nose
puted.
Power
flow
analysis
cletermines
as
vanous
svstem
parameters
and
""lT:ir'#n#'f
|HI'I,:IJHI#
voltage
exist
for
each
value
of
load.
The
uppe.
on"
indicates
stable
;;i;;;
whereas
lower
one
is
the
unacceptable
value
(multiple
load
flow).
At
limiting
l!ug"
of
voltage
stability
i.e.
at
nose
point
single
ioad
flow
solution
exists.
Nearer
the
nose
point,
lesser
is
the
staLility
maigin,
1400
1200
1000
P>
P,
>
P,, iN
MW
System
characteristics
Capacitor
characteristics
0.9
0.95 1.0 1.05
Vin
pu
Fig. 13.9 System and
shunt
capacitor steady-state
Q-V
characteristics,
capacitor
MVAr shown at rated
voltage
j+Allowable
range
ffuo.
Boo
Voltage
Collapse
Voltage
collapse
is
the
process
by
which
the
seguence
of
events
accompanying
voltage
instability
le_ads
to
unacceptable
voltag6
profile
in
a
significant
part
of
the
power
system.
It
may
be
manifested
in
ieverat
different"ways.
Voltage
collapse
may
be
characterised
as
follows:
(iii)
The
voltage
collapse
generally
manifests
itself
as
a
slow
decay
of
voltage.
It
is
the result
of
an
accumulative
process
involving
the
actions
and
interactions
of
many
devices,
controlJ,
and protective
iyrt"*r.
The
time
frame
of collapse
in
such
cases
would
be
of
the
order
of
several
minutes.
Voltage
collapse
is
strongly
influenced
by
system
conditions
and
characteristics.
(iv)
Reactive
compensation
can
be
made
most
effective
by
the
judicious
choice
of
a
mixture
of
shunt
capacitors,
static
var
system
and possibly
synetuonous
condensers.
Methods
of Improvlng
Voltage
Stabllity
voltage
stability
can
be
improved
by
adopting
the
following
rneans:
(i)
Enhancing
the
localised
reactive
power
support (SVC)
is
more
efl,ective
and
C-banks
are
lnore
economical.
ra-tS
devices
or
synchronous
condenser
may
also
be
used,
(ii)
Compensating
the
line
length
reduces
net
reactance
and power
flow
increases.
(iii)
Additional
transmission
linc
muy
be
crectccl.
It also
improves
reliability.
(iv)
Enhancing
excitation
of generator,
system
voltage
improves
and
e
is
supplied
to
the
system.
(v)
HVDC
tie
may
be
used
between
regional
grids.
(vi)
By
resorting
to
strategic
load
shedding,
voltage
goes
up
as
the
reactive
.
burden
is
reduced.
Future
Trends
and
Challenges
(i\
Ontirnql oifi-- n'F EA/-'n- ,{^*,:^^^
\-./
urrrrr6
vr I
nv
r i,
\rlvv.ltvgD.
(ii)
Better
and probabilistic
load
modelling.
(iii)
Develop
techniques
and
models
for
study
of
non-linear
dynamics
of
largr
siz.e
systems.
For
example,
new
methods
to
obtain
network
equivalentr
suitable
for
voltage
stability
analysis.
M
;
(v)
Post-disturbance
MwA4vAR
margins
should
be ffanslated
to
pre-
disturbance
operating limits
that
operators can
monitor.
(vi)
Training
in voltage stability
basis
(a
training simulator)
for
control centre
and
power plant
operators should
be i
SUIvIMARY
Power system
security
(including
voltage
stability)
is likey
to challenge
planneqs,
analysts; researchers and
operators
for the foreseeable
future. As load
grows,
and as new
transmission
lines and
new
generations
would
be
increasingly
difficult to build or
add,
more and
more
utilities
will face the
security
challenge.
Deregulation and
socio-economic
ffends
compounded
by
technological
developments have
increased the
likelihood
of voltage
instability.
Luckily many
creative
persons
are
working tirelessly
to find
new methods
and innovative
solutions to meet this
challenge.
REFERE
N CES
Books
L l.J. Narguth and
D.P. Kothari, Pttnter
System Engincering,'fata
Mc0raw.Hill,
New
Dclhi, 1994.
2,
C.W,
Taylor, Power System Voltag,e
Stabiliry,
McGraw-Hill,
New
york,
1994.
3. P. Kundur,
Power System
Stability
and Contol, Sections
2.12, ll.2
and Chapter
14, McGraw-Hill,
New York, 1994.
4, T,J.E,
Miller,
Editor, Reactive
Power
Control in Electric
Syslens,
John Wiley
and
Sons, New York, 1982.
5. A. chakrabarti, D.P.
Kothari and
A.K. Mukhopadhyay,
Perforurutnce,
operation
and
Control of EHV Power
Transmission
Systems,
Wheeler
Publishing, New
Delhi. 1995.
6. T.V. Cutsem and
C.
Vournas,
Voltage
Stability of Electric
Power
Syslerns, Kluwer
Academic Publishers,
London, 1998.
7. A.J. Wood and W.F.
Wollenberg, Power
Generation,
Operation,
and
Control,
Znd
Edn,
John
Wiley,
New York,
1996.
E.
John
J. Gratnger and W.D.
Stevenson, Power
SystemAnalysis,
McGraw-Hill, New
York, 1994.
9. G.L.
Kusic,
Computer-Aided Power
Systems Analysis,
Prentice-Hall,
New
Jersey,
1986.
10. G.W. Stagg and A.H. El-Abiad,
Computer Methods
in
Power
System
Analysis,
McGraw-Hill, New York,
1968.
Papers
11. v. Ajjarapu
and B.
Lee,
"Bibliography
on voltage
stability",
IEEE
Trans.
on
Power
Systems,
Vol. 13,
No.
1, February
1998,
pp
lI5-125,
12.
L.D. Arya,
"Security
Constrained
Power
System
Optimization",
PhD
thesis,
IIT
Delhi,
1990.
13.
T.E. Dyliacco, "The
Adaptive
Reliability
Control
System",
IEEE
Trans.
on
pAS,
Vol. PAS-86,
May
1967,
pp
517-531
(This
is
a key
paper
on
system
security
and
energy
control
system)
14.
A.A. Fouad,
"Dynamic
Security
Assessment
Practices
in North
America",
IEEE
Trans. on
Power
Systems, Vol.
3, No.
3,
1988,
pp
1310-1321.
15. B.
Stott,
O. Alsac and
A.J.
Monticelli, "security
Analysis
and
Optimization",
proc
IEEE,
VoL
75, No.
12, Dec.
1987,
pp
1623-1644.
16. Special
issue
of Proc. IEEE,
February
2000.
17. P.R. Bijwe,
D.P. Kothari
and
L.D.
Arya,
"Alleviation
of Line
Overloads
and
voltage violations
by
corrective
Rescheduling",
IEE
proc.
c, vol.
140,
No. 4,
July 1993, pp
249-255.
18. P.R.
Bijwe,
D.P. Kothari
and
L.D. Arya,
"Overload
Ranking
of Line
Outages
with
postourage generation
rescheduling",
Int.
J.
of Electric
Machines
and Power
Systems, Yol.
22, No.
5, 1994,
pp
557-568.
19. L.D. Arya,
D.P.
Kothari
et al,
"Post
Contingency
Line
Switching
for
Overload
Alleviation
or
Rotation",
Int
J. of EMPg
Vol
23.
No.
3, 1995,
pp
345-352.
20. P.R. Bijwe,
S.M. Kelapure,
D.P.
Kothari
and
K.K.
Saxena,
"Oscillatory
Stability
Lirnit Enhancement
by
Adaptive
Control
Rescheduliig,
Int.
J.
of Electric
Power
and Energy
Systems, Vol.
21,
No.
7, 1999, pp
507-514.
21. L.D. Arya,
S.C.
Chaube
and D.P.
Kothari, "Line
switching
for
Alleviating
Overloads under
Line
Outage
Condition
Taking
Bus
Voltage
Limits
into
Account",
Int.
J. of EPES,
Yol. 22,
No. 3, 2000,
pp
213-ZZl.
22.
P.R. Bijwe, D.P.
Kothari
and
S. Kelapure,
"An
Effective
Approach
to Voltage
Security and Enhancement",
'Int.
J. of EPES,
Yol.
22, No
7, 2000, pp
4g3-4g6.
23. L. Fink and
K.
carlsen,
"operating
under
sttress
and
Strain",
IEEE
spectrum,
March 1978,
pp.
48-50.
24.
S.M.
Kelapure,
"Voltage
Security
Analysis
and
Enhancement",
Ph.D.
thesis,
IIT
Delhi,2000.
25.
G.C.
Ejebe,
et.
al,
"Fast
Contingency
Screening
and
Evaluation
for
Voltage
security Analysis",
IEEE
Trans.
on Power
systems,
vol.
3, No.
4, Nov.
l9gg, pp
1582-1590.
26.
T.
Van
Cutsen,
Voltage
Instability:
"Phenomena,
Counter measures,
and
Analysis
Methods", Proc.
IEEE,
Vol.
88, No.
2, Feb.
2000, pp
208-227.
t4
T4.I INTRODUCTION
State
estimation
plays
a very important role in the monitoring and
control of
modern
power
systems.
As
in
case
of load flow analysis,
the aim
of state
estimation
is to obtain
the best
possible
values
of the bus voltage magnitudes
and
angles by
processing
the available
network data. Two
modifications
are,
however, introduced
now
in order to achieve a higher degree
of
accuracy
of the
solution at the cost
of some
additional computations. First, it is recognised
that
the numerical values
of the
data to be
processed
for the
state estimation
are
generally
noisy
due
to
the errors
present.
Second, it is noted that there
are a
larger number
of
variables
in
the
system
(e.9.
P,
Q
line
flows) which
can be
measured
but arc not
utilised
in the load flow analysis.
Thus, the process
involves
imperfect measurements
that are redundant and
the
process
of
estimating
the system
states is based on a statistical criterion that estimates
the
true value of
the
state
variables to
minimize or
maximize the selected
criterion.
A
well known and commonly
used criterion
is
that of minimizing the
sum
of
the
squares of the differences
between the estimated and
"u1le"
(i.e.
measured)
values of
a function.
Most state
estimation
programs
in
practical
use are
formulated as
overdetermined
systems
of non-linear equations and solved as weighted
least-
squares
(WLS)
problems.
State
estimators
may
be both static and dynamic. Both have
been developed
for
power
systems. This
chapter will introduce the basic
principles
of
a static-
state estimator.
ln a
power
system, the state variables
are the voltage magnrtudes
and
phase
angles
at the buses.
The inputs to an estimator
are imperfect
(noisy)
power
system
measurements.
The estimator
is designed to
give
the
"best
estimate"
of
the
system voltage
and
phase
angles
keeping in mind that there
are errors
in
the
measured
quantities
and that there may
be redundant measurements.
The
output
data
are then used
at the energy control centres
for
carrying out
several
::if.-jl:t
or,01-li,r:,
systeT^studics
such
as
cconornic
clisplrch (Chaprcr
7),
strLurl-y
analysts
(Lnapter
lJ).
r4-2
LEAST
souARES
ESTIMATION:
THE
BAsrc
soluTroN
I7l
-
IeI
A lrr
ns
wilf
be
seen
iater
in
Section
r4.3,
the
probrem
of power
system
state
estirnation
is
a
special
case
of
the
more
general
problem
of
estimation
of
a
random
vector
x
from
the
numerical
values
of
anotirer
related
random
vector
y
with
relatively
little
statistical
information
being
available
for
both
x and
y.
In
such
cases,
the
method
of
least-squared-error
estimation
may
be
utilised
with
good
results
and
has
accordingly
been
widely
employed.
Assume
that
x
is
a
vector
of
n
random
variabres
x1t
x2,
...,
x'
that y
is
another
vector
of
m
(>
n)
random
variables
!1,
J2,
...,
J^
and
both
are
related
as
where
H
is
a known
matrix
of
dimension
mx
n
ancl
r
is
a
zero
mean
rancklm
variable
of the
same
din-rension
as
y.
The
vector
x
represents
the
variables
to
be
estimated,
while
the
vector
y
represents
the
variables
whose
numerical
values
are
available.
Equation
(
l4.l)
suggests
that
the
measurement
vector
y
is
linearly
related
to
the
unknown
vector
x
and
in
addition
is
corrupted
by
the
vector
r
(error
vector).
The
problem
is
basically
to
obtain
thc
best
possible
value
of
the
vector
x
from
the given
values
of
the
vector
y.
Since
the
variable
r
is
assumed
to
be
zero
mean,
one
may
take
the
expectation
of Eq.
(14.1)
ancl
get
the
relation
!=Hx+r
f
=
Hx
j'
=
Hf,
(r4.1)
(r4.2)
(r4.3)
whcre
I
,
,
=
cxpcctccl
value
ol'
x
iurd
y,
respec:tively.
This
shows
that
the
load
flow
methods
of
chapter
6
could
be
used
to
estimate
the
mean
values
of
the
bus
voltages.
Howeu"r,
tn.
woulcl
like
to
estimate
the
actual
values
of
bus
voltages
rather
than
their
averages.
One
possible
way
of
obtaining
the
best
possible
estimate
of
the
vector
x
from
y
lies
in
the
use
of
the
method
of
least
square
estimation
(LSE).
To
develop
this
rneth<td,
assllme
that
i
rcpresettts
the
desirecl
estirnate
of
,r
so that
y
given
by
the
equation
represents
the
estimate
of
the
vector
y.
The
effor
!
of
the
estimation
of
y
is
then
given
bv
(r4.4)
The
estimate
i
is
definecl
to irc
thc
I-sE
if
it
is
cornputccl
by
rninirnizing
the
estimation
index
J
given
by
An
lntroduction
to State
Estimation
of Power
Systems
J
=7'V
(14.5)
From
Eqs.
(14.31)
and
(14.4),
one
gets
the following
expression
for
the index:
J
=yt!-
y'H*.- i<'H'y+ *.'HtH*.
For
minimizing J
=
f$),
we must
satisfy the
tollowing
condition.
gradoJ
=
0
It is
easy to
check
(see,
e.g.
[1])
that Eq.
(1a.7)
leads
to the following
result.
HtH*,-H'y-O
(14.8)
This equation
is
called
the
'notmal
equation' and
may be solved
explicitly
for
the LSE
of
the
vector i
as
*.
=
(H,Il)-t
Ht
y
(14.e)
E;il;il]
r.i
ln
orcler to
illustrate
the
method
of LSE,
let us consider
the simple
problem
of
estimating
two
random
variables
x,
and
.rz by using
the data for
a
thret:
dimensional
vector
y.
Weighted
LSE
mI ..
lnq
grsllll|:.iltr
Eilv€il
!rl
D9.
\!+,2)
ls ulrerr
lr-rsrrsu r\,
(ri)
rrru \rr\.rrrr(uJ r1,61ir
squares
estimate
and is obtained
by minimising
the index function
that puts
equal
weightage
to the
effors
of estimation
of all components of
the vector y.
It
is often
desirable
to
put. dift'erent
weightages on the ditt'erent
components
of
y
since
some
of
the measurements
may be
more reliable
and accurate
than
the
others
and these
should be
given
more
importance.
T'o achieve this
we define
the
estimation
index as
(
r4.6)
\r+.t)
l-'
0.1
Assume
H=10
ll
Lr
rl
The
matrix
Ht
F1 is then
given
bY
u'u
=12
I
I
una
its inverse
is
Ll
2)
(H'm-'
=f
''':'
-:':1
l-
u3
213
1
It is
easy
to form
the
vector
Hty and
combining
this
with the itrverse
of
(H'H),
the following
estimate
of x
is obtained.
f rrt lat -. /1 t1\/ r
-l
^
|
\Ltr)/r-\rrt)\/2-Yz)
|
*
=
L-(t
/3)yr*(2/3)yr1(r/3)v-,
l
/ | /
(\
\
r+.J,,
rvtouern
Power
where
W is
a
real
symmetric
weighting
matrix
of dimension
m x
ru.
This
is
often
chosen
as
a diagonal
matrix
for
simplicity.
It
is relatively
straightforward
to extend
the
method
of
LSE
to the
weighted
form
of
"I
and to
derive
the
foilowing
form
of the
normal
equation.
H|WH?_
H'Wy
_0
G4.Ila)
This
leads
to
the
desired
weighted
least
squares
estimate
(WLSE)
I
=
l'wf
*.
=
(H,W
H)-t
H,W y
This
pertains
to
minimization
as
the
hessian
definite.
Some
Properties:
Rewriting
Eq.
(14.ilb)
as
i=ky
k
=
(H'W
H)-'
Ht
W .
(14.11b)
2H|WH
is
a non-negative
(I4.I2a).
(r4.rzb)
/141n\
\
r.i. r
vi,
(14.r3)
(14.r4)
(14.15a)
(14.1sb)
where
Here
the
matrix
k depends
on
the
value
of
H and
the
choice
of
w.
using
Eqs.
(14.1)
and
(r4.lzb)
it
is easy
to get
the
relation
as follows.
r
=f::;,7,
,,,wH)
x
+
kr
Or
X
=r
+
kr
and
E{i}
=E{xl
ln
IJq.
(14.14)
it
is
assumed
that
the
error
r
is
statistically
independent
of
columns
of
H and
the
vector
r
has
a
zero
mean.
An
estimate
that
satisfied
Eq.
(l
4)4)
is
called
an
unbiased
estimate.
This
implies
that
the
estimation
error
ls
zero
on
an
averase.
x
=kr
The
covariance
of
the
error
of
estimation
is
therefore
given
by
P,
=
KRK
where
R is
the
covarianee
o1
the
error
vector
r.
Note
that
the
covariance
p"
is
a
lneasure
of the
accuracy
of
the
estirnaiion
and
a
smaiier
trace
of this
matrix
indicates
a better
estimate.
Eq.
(14.15b)
suggests
that
the
best possible
choice
of the
weighting
matrix
is
to
set
w
-
R-1.
The
optimum
value
of
the
error
covariance'matrix
is
then given
by
P,
-
(HtR-rH)-l
(14
15c)
Assume
that
in
the
Example
I4.1,
we
want
to
obtain
the
WLSE
of
the
variable
x
by
choosing
the
following
weighting
matnx
t---*--
**
1 Et<ample
14;2
An
lntroduction
to
State
Estimation
of
Power
Systems
l"
,3iiSl:
-T--
[o.r
I
tl
w=l
I
I
L
o.tj
The
matrir
HLWH
ts
H,wH
-l'''
o'tl
10.1
1.lJ
and
the
matrix
HtW
is
obtained
as
H'|w=fo't
o
ot-l
L0
1
0.1j
The weighted
least
squares
estimate
of the
vector
x is
then
obtained
as
(from
Eq.
(la.11b))
^
[
(lr/21)
y,
-
(r}/Zt)
y, (t0/2t)
yr
I
*
=
L-
ellr)
y,
(2ot2r)
y,
(r/2\
y,
)
If
this
result
is
compared
with
the
result
in
Example
I4.1,
the
effect
of
introducing
the
weighting
on
the
estimate
is
apparent.
Note
that
the
choice
of
W
in
this
case
suggests
the
data
for
y2
is
considered
more
valuable
and
this
results
in the
components
of
x being
more
heavily
dependent
on
)2.
The
matrix
ft
is in
this
case
found
to
be
(Eq.
(1a.12b))
rc=lrr/21
-ro/2r
ro/zrf
L-
rlzt
20/21
r/21
J
If
the
covariance
of
the
measurement
error
is
assumed
to
be
R
=
L
the
covariance
of
the
estimation
error
is
obtained
as
(Ref.
Eq.
(14.15c))
P,=
(1tr47)
|
*:_
.::1
L-67
r34
)
The
choice
of
W above
yields
unacceptably
large
estimation
error
variances.
Let us
now
choose
the
weighting
matrix
W
=
I.
The
matrix
Kis
then
obtained
AS
*=l''3
-1/3
t/3f
L-
U3
2/3
1/3J
The error
covariance
matrix
is
then given
by
D /r/n\ [
6
-3]
r-x=
\Lt>)
L_
E
6
J
The
error
variances
are
now
seen
to
be
much
smaller
as
is
to
be
expected.
Non-linear
Measurements
The
case
of
special
interest
to
the power
system
state
estimation problem
corresponds
to the
non-linear
measurement
model.
I
536
|
Modern Power System Analysis
I
I
!=h(x)+
r
(14.16)
where
h(x) represents an
rz
dimensional
vector
of
nonlinear functions
of the
variable
x.
It is assumed that the
components
of the
vector h(x) are continuous
in their
arguments
and therefore
may
be differentiated
r.vith respect to
the
components
of x. The
problem is to extend the method of least squares in order
to estimate
the
vector x from the data for the
vector
y
with these
two variables
being related through Eq.
(14.16).
To mimic our treatment of
the linear measurement
case, assume that
i
represents
the
desired estimate so
that the estimate ol the measLlrement
y
could
be obtained using the relation
I
=
h(i)
This
yields
the
error of
estimation of the vectol
y
j =y_
h(3)
In
ordcr
to obtain the
WLSE of ,,r,
we
nrust cho<lse
the
index
of
estimation
J
as
follows.
J=U-
h(fi)l'WlJ
-
h(i)l
(14.18)
The necessary condition
for the index ,I to have
a minimum at .x, is
given
by
Eq.
(14.1e).
ty-h(x)lH(i)=o
(r4.re)
where H
(fr)
is the Jacobian of h
(x)
evaluated
at i. In
general
this
non-linear
equation can not be solved
for the desired estimate,
i. n way out of
this
difficulty is to
make
use
of the linearisa-tion technique.
Let us assume
that
an
a
priori
estimate
xu
of
the vector x
is
available
(say
from the
load flow
solution).
Using Taylor series
approximation, we
get
(1.4.17a)
(r4.r7b)
(14.20)
r11.22)
!=h(xd+
Hu&-x1)+ r
where 1/o stands for the Jacobian
evaluated at x
=
x9 and the noise term
r is
now assurned to
include the effects
of the higher order terms
in the'Taylor
series. Equation
(14.20)
can be rewritten
as:
A)'=y-h
(xo)=
HsAx+r
(14.2I)
where
Ay is the
p^-rturbed
measurement
and Ax is the
perturbed value
of
the
vector
r. An
\\LSE
of
.r is then ea^silv obtained
a-s discussed earlier and
this
leads to the desired expression
for the lineaized solurion of
the non-linear
estjmat.ion
problem.
.i.
=
.rrr
-
[H,,'
I+'HQ]-
t
H()'
]l'
{-r
-
h
t.t,,,)}
Ir is nor likelv
that
the estimate
.i
obtained
from
Eq.
(11.22
r
is
going to be
t-rf Irtucir use silt.'c. irt
tener.ri.
tltr' .r
i,t'rttt't
c'stirlute
.l',.r Il)J\- Ilrrt h'
close
tir thf
(Dt-irnal ydue
rf
rhe
\ect()r
x. Hrt,*'ever,
F,q.
(14.22)
prot'ides us
wirh
a
very
An lrylfgduction
to_State
Estimation
of Power
Systems
I
S3T
I
useful result
in the sense that
it shows
a mechanism
for improving
on the
initial
estimafe
by making
use
of the available
nleasurements.
Having
obtained
new
estimate
i
,
the
process
of linearisation
is repeated
as
many
times
as
desired
and
this
leads
to the
{qllq*tng 4erylle {qryq
qf
the
qqb]ro4
of
thq 4on-linear
estimation problem.
i
(t
+ 1)= i
(t)
+
K(t)
{y
-
h
tt
(/)l}
where
the
matrix K(D
is
deflned as
K(t)
-
LH/
wHil-'
al
w
The
index /
represents the
iteration
number
and
H
lrepresents
the value
of the
Jacobian
evaluated
at
x
=
i
(l).
Usualiy
the iterative process
is
terminated
whenever
the norm
of the
difference
of two
successive
values
of the
estimate
i
1t
+
l)
-
.i
(/)
reaches
a
pre-selecred
rhreshold
level.
A flowchart
for implementing
the
iterative
algorithm
is
shown
in
Fig. 14.7.
A major source
of
computation
in the
algorithm
lies
in the
need
to
update
the
Jacobian
at every stage
of iteration.
As discussed
earlier
in
Chapter
6
(see
Eqs.
(6.86)
and
(6.87))
it is
often
possible
to reduce
the computations
by holding
rhe
value
of
H a constant,
possibly
after
/ exceeds
2
or 3.
T,ris
is
in
general,
permissible
in
view of
the fact that
the
change in
estimate
tends
to be
rather
small after
a couple
of iterations.
(r4.23)
(r4.24)
i
,00"," )
eq.
(+..227
/
I
c"l::'1:
-,=n
)
\,
*
!
-_I
\,,ll
-''!'
./
ls
:-
\-
.-2a.,
No
r
Yes
\
Stcp
rig.
r+.1
',:83--E-,
I
todern
power
System
Analvsis
I
Consider
the
simple
case
o1'a scalar
variable
x and
assunle
that
the
relationship
_x
ls
glven
Dy
Y=
x3
+
r
The
Jacobian
H, is
easily
obtained
in this
case
and
the
iterative
algorithm
takes
the explicit
form
i
(t
+
1)
=
i
(t)
+
t3
i
(Dl-z
{y
-
ti
(Dl3}
where
we
have used
W
=
1.
Let
the
correct value
of
x/ be eclual
to 2
and
assume
that due to
the
effect
of
r
the
measured
value
of
y
is found
to
be
8.5. Also,
assume
that
the
initial
estimate
x
(0)
is taken
to be equal
to 1.
The table
below
gives
the
results
of the
first
few iterations.
,
(t)
1.0
3.5
2.56
2.t6
It is
apparent
that
the
algorithm
would
yield
the
correct
solution
after
several
iterations.
14.3
STATIC
STATE
ESTIMATION
OF POWER
SYSTEMS
lrol-lr2l
As noted
earlier,
for
a system
with
N buses,
the state
vector
x may
be
defined
as thc 2N
-
| vccltlr
tll'lltc
N
--
| vollagc
anglcs
62,
..., 6" and
thc N
voltage
magnitudes
/1,
v2, ..., v".
The load
flow
data,
depending
on
type
of bus,
are
generally
comrpted
by
noise
and the problem
is that
of
processing
an
adequate
set
of available
data
in
order to
estimate
the state
vector.
The
readily
available
data
may
not
provide
enough
redundancy
(the
large
geographical
area
over
which
the system
is
spread
often
prohibits
the telemetering
of all
the available
tnedsurements
to
the central
computing
station).
The
redundancy
factor,
defined
as the
ratio
m./n
should
have
a value
in
the range
1.5 to 2.8
in
order
that
the
computed
value
of the
state
may have
the
desired
accuracy.
It
may
be necessary
to irleh-rde
the data
for
the
power
flows
in
both the
directions
of
some of the
tie
lines in
order to
increase
the
redundancy
factor.
In fact,
some
'psuedo
measurements'
which
represent
the
computed
values
of such quantities
as
the
active
and
reactive injections
at some
remote
buses
may
also be includerl
in
the
vector
y (k).
An
tntroduction
to State
Estimation
of
Power
Systems
|
539
_r-
It
is thus
apparent
that
the
problem of
estimation
of the
power system
state
'is
a non-linear
problenn
and
may
be
solved
using
either
the
batch
pr:ocessing
or
sequential
processing
formula
[see
Section
3.3
of
Reference
1]. Also,
if the
svstem
is assumed
to
have
reached
a
steady-state
condition.
the
voltage angles
problem
is
then
a
static
problem
and the
methods
of
Sec.
14.2
may be
used. if
so
desired.
To
develop
explicit
solutions,
it is
necessary
to start
by
noting the
exact
forms
of
the
model
equations
for
the
components
of the
vector
y
(k).
Let
P, arfi
Qi
denote
the
active
and
reactive
power injections
of
ith bus.
These
are
related
to the
components
of
the
state
vector
through
the following
equations.
I Yj
| |
Vjl lYij
I
cos
(-
6t
+ 6,
* 4ti)
(r4.2s)
(r4.26)
lviP
lY,,l
cos
9,,
(r4.27)
I viP
I Y,, I sin
Iij
(14.28)
l, N
Qn
...
Q7,J
-
1, 7s,
binl
lv1l,
...., I
yN
I
l/
(r4.2e)
The
Jacobian
H
will
theh
have
the
form
Ht
Hz
H3
H4
Hs
H6
H7
Hs
/ru-r
o
o
IN
where
/"
is
the
iclentity
matrix
of
dimension
N,
H, is
the
N
x
(N
-
l; submatrix
of
the
partial
derivatives
of
the
active
power injections
wrt
6's,
H2 is
the
N
P,=
D
j
--r
0
I
2
a
_)
N
e,=
-D
|
%l
I
vjllYijl
sin
(-
6,+
5t+
0;)
j:1
where
I
IU I
repre5ents
the
uragnitude
and
l/U
-
represents
the angle
of the
admittance
of the
line
connecting
the
lth
and
7th
buses.
The
active and
reactive
components
of the
power
flow
from
the ith
to the
7th
bus, on
the
other
hand are
given by
the
tbllowing
relations.
P,j=
|
vil
I Vjl
I
Yij I
cos
(d,
-
6i
* 0,)
-
Q,j=l
%l
I
Vjl
lYijl
sin
(d'
-
6i
*
0t)
-
Let
us
assume
that
fhe vector
y
has
the
general form
J
=
lPt
...
PN
Qt
...
Qru
Prz ...
Pu
-
)2
H_
(14.30)
54q
f
Modern Po
T
N
x
N sub-matrix
of the
partial-derivatives
of the active power
injections
wrt
| 7l' and
so on. Jacobian
H will also
be a sparse
matrix since
I is a
sparse
matrix.
Two special
cases
of interest are
those corresponding
to the
use of only
the
active
and reactive
iniections
and
the use of onl
flows in the
vector
y.
In the first
case,
there are
a total
of 2N components
of
y
compared
to the 2N
-
1 components
of the
state x. There is
thus almost
no
redundancy
of measurements.
However,
this case
is very
close to the
case of
load flow analysis
and
therefore
provides
a
good
measure
of the relative
strengths
of
the
methods
of load flow
and
state estimation.
In the
second case,
it is
possible
to ensure
a
good
enough
redundancy
if there are
enough tie
lines
in
the system.
One can obtain
two measurements
using two separate
meters
at
the two ends
of a single
tie-line.
Since
these two data
should have
equal
magnitudes but
opposite
signs, this arrangement
also
provides
with a
ready
check
of meter malfunctioning.
There
are other
advantages
of this
arrangement
as will be discussed
later.
The Injections
Only Algorithm
In this case, the model
equation
has the
form
!=hlxl+
r
with the
components of
the non-linear
function
given
by
Qa32a)
N
-
D
tv*_illvjllyN_i,;lsin
(6,-
6i+
7ii)
j:1
i=N+1,N+2...,2N
(r4.32b)
The elements of the sub-matrices
Hp H2,
H, and Ho are
then determined
easily as follows.
Ht
(i,
j)
=
lVil lVjl lYij I sin
(r{.
-
6t + 0t)
i
=
1,2,
..., N,
j=I,2,"',N-1
Hz
Q,
i)
=
lvil lYij I cos
(d,
-
6i *
0;)
i
=
7,2, ..., N,
j
=
1,2,
"',
N'
rf /r !\ | rt I I tr | | r, | / C a - n \
.
n A
Ia3
\tt
J)=
-
|
yil
I
vjl
I IUI COS
\Oi-.Oj+
Aij)
I
=
I, Z) ..., lY,
j=I,2,"',N-1
Hq(i,
j)=lVillYijl
sin
({
-
6i+ 0,) i=I,2,...,
N,
j
=
I,2,
"',
N'
(r4.33)
An
tntroduction
to State
Estimation
of
Power
Systems
I
541'
I
Equation
(14.33) may be
used
to
determine
the Jacobian
at
any
specified
value
of
the
system
state
vector.
The
injections
only
state estimation
algorithm
is then
obtained
directly
from
the
results
of sec
14.2.
Since
the
problem is
non-linear,
it is convenient
to employ
the
iterative
algorithm
given in Eq.
(14.22).
ine.
the
submatrices
H.,
and
H^ become
null
with the
result
that
the
linearised
model
equation
rnay be
approximated
as:
^ fH,
o-l
rr
we
parrition,J";,1:'
,,o*^::::
'
f
Ay,1
lArrl l
rol
Av
=
l^'r'r.j'
'=
Lor"l,
'=
L;
j
(r4.34)
(14.35)
(r4.36)
(14.31)
then,
Eq.
(14.34)
may
be
rewritten
in the
decoupled
form as
the following
two
separate
equations
for
the
two
partitioned
components
of
the state
vector
Alp=
H1
Ax5+
ro
Alq=
H4
Axr,
+
rn
Based
on
these
two
equations,
we
obtain
the following
nearly
decoupled
state
estimation
algorithms.
iuQ
+
r)=
xd0
+
tHi
0
wpHt
U)l-'H,
(D
{to-
hrliU)l}
where
the
subscripts
p
and
q are used
to
indicate
the
partitions of
the
weighting
matrix
W
and
the
non-linear
function
h
(.)
which
correspond
to the
vectors
yp
*rd
lerespectively.
As
mentioned
earlier,
if the
covariances
R*
11d
Rn of the
effors
r, and
r(t
are assumed
known,
one
should
select
Wo
=
R
o'
and
Wn
=
Rq"
i
'Note
that
Eqs.
(14.37)
and
(14.38)
are
not truly
decoupled
because
the
partitions
of
the
non-linear
function
depend
on
the
estimate
of
the
entire
state
vector.
It
may
be
possible
to
assume
that
vi
U)
=
I for
all
i and
7
while'
Eq.
(14.37)
is
being
used
in
order
to estimate
the
angle
part of the
state
vector.
similarly
one
may
assume
6i
U)
=
0o
for
all
i and
7
while using
Eq.
(14.38)
in order
to
estimate
the_yllltage
part
of
the
state
vector. Such
approximations
allow
the
two
equations
to be
completely
decoupled
but
may not
yield
very
good
solutions.
A better
way
to decouple
the
two
equations
would be
to use
the load
flow
sglutions
for
x,
and
x6
as
therr
suppose<iiy
consnnt
vaiues in
Eq.
(14.37)
and
Eq.
(14.38)
respectively.
There
are several
forms of
fast
decoupled
estimation
algorithms
based
on
such
considerations
(see
e.g.
[13], tl4l).A
flow
chart
for
one
scheme
of
fast
decoupled
state
estimation
in shown
in Fig.
I4.2.
h,[x)=
D
j:1
.l
=
0,
1,2,
...
\
(14.37)
lVillVjllYijl
cos
(6,-
6i+ 0i), i=I,...,
N
i,
(/ + 1)
=
i,
Q)
+
tHl
u)
w,
Hq
(j)l-'
nl
u)
wo
ur-
hq
ti
(j)ll
j=0
| 2
(14.38)
542
|
Modern
Po*er
System
Analysis
An Introduction
to State
Estimation
of Power Systems
|
543
1_-
*11
Fig. 14.3
I
j
Example
14.4
In order
to
illustrate
an
application
of the
injections
only
algorithm,
consider
the
"imple
2-bus
system
shown
in
Fig.
14.3.
Assuming
lossless
line,
0,j=90".
Also
let
Yrr= jzz=2and
yn=
yzt=
.
The power
relations
in
this
case
would
be
Pt=
-
|
Yr
| | V2l
lYrrl sin
E
Pz=
lVtl
lVzl
I Y,rl
sin
6,
Qt=
iril
i
i Vri'-
|
ynll
yl
|
| Vrl
cos
$
Qz=
lYzzl
lVrl'
-
lyt2ll
yr
| | vrl
cos
6
If we
choose
the
initial
values
lVf
l=lV;
l= 1,
6o2=0",
the
corresponding
power
values
are
P
f
=
Pf
-
0,
Qf
=
Qi
=
1.
The
value
of
the
Jacobian
matrix
evaluated
at the
above
nominal
values
of
the
variables
turns
out
to
be
Application
of
the
LSE
then
yields the
following expressions
for
the
estimates
of the
perturbations
in
the three
state
variables
around their
chosen
initial
values:
-
AP,
-
AP,
=
0.78
AQz
-
0.26
AQr
AV'
=
g39
AQ'
+
0'14
AQr
These equations
should
be
used
in order
to translate
the measured
values of
thc
pcrturhations
in
the active
ancl
reactive
power
injections
into
the
istinrates
of
the
perturbations of
the state
variables.
It
is interesting
to note
that
for lhe
sirnple example,
partitions
Hranrl
Htare
null
matrices
so
that the
decoupied
state
estimators
are the same
as those
given
above.
The
Line Only
Algorithm
This algorithm
has
been
developed
in order to
avoid the need
for solving
a
non-
linear estimation
problem.
which as
seen earlier,
requires
some approximation
or
other.
ln the
line
tlow
only
algorithm,
the data tor the
active and reactive
tie
line
flows
are
processed
in order
to
generate the
vector
of
the voltage difference
across
fte
tie-lines.
Let
z denote
this
vector. A model equation
for this
vector
may
be
expressed
as
z=
Bx
+ r-
(r4.39)
where
B
is the node-elernent
incidence
rnatrix
and r- is the
vector
of
the
errors
in the
voltage
el-ata.
Since this is a
linear
equation. one may
use the
WLSE
technique
to
generate
the estimate
as
*,
=
lB,
wBl-
|
Bt wz
(r4.40)
where
the
weighting
rnatrix
may be
set equai
to the
inverse of the covariance
of
rrif this
is known.
The
main
problem with Eq.
(14.40)
is that the
vector
z
is
not
directly
measurable
but
needs to be
generated
frorn
the tie line flow
data.
I
,o,rlli1i*
1t)-*(j)ll=,'
j
Ai,
AT,
ls
on,^t
,2
an2
>
t\u
-\
\\r---'
1"",
I
,\
(3'9)
Fig.
14.2
r++
|
Moctern
power
System
Analysis
I
V,tdenotes
the
voltage
across
the
line
connecting
the
ith
ancl
theTth
buses,
the
followine
relation
holds.
Vii=
Zii
[@ii--
j
Qi;tV7-
Vi Y,i]
Here
Z,.,stands
for
the
impedance
of the
line.
This
shows
that
the
vector
z
is related
to the
vectors
x
and
ashion
and
one
ffidy
use the
notation
z
=
g
(x, y)
04.42)
In
view
of this
non-linear
relation,
Ecl.
(14.40)
may
be expressecl
in
the
form
i
=
LB,
Wnyr
B,W
I
(i,
y)
(14.43)
This,
being
a non-linear
relationship,
can
not
be solved
except
through
a
numerical
approach (iterative
solution).
The
iterative
form
of
Eq.
(14.43)
is
i
(j
+ r)
=
lBt
wBll
H
w
s
[i
(i),
y],
-/
=
0, 1,2,
...
(r4.44)
Note
that
the
original
problem
of estimation
of
x from
the
data
for
z
is
a linear
problem
so
that
the
solution
given
by
Eq.
(14.40)
is
the
oprinral
solrrtion.
However,
the
data
for
z
need
to
be
generated
using
the
non-linear
transforma_
tion
in
Eq'
(14.42),
which
in
turn
has
necessitatecl
the
use
of
iterative
Eq.
(14.44)'
Compared
to
the
injections
only
iterative
algorithm,
the pr-ese1t
algorithm
has the
advantage
of
a
constant
gain
matrix
[8,
W B]-t
gr
I4z. This
result
in
a
considerable
computational
simplification.
The
concept
of
decoupled
estimation
is
easily
extencled
to
the
case
of the
lirre
flows
[15].
T4.4
TRACKING
STATE
ESTIMATION
OF
POWER
SYSTEMS
t16l
Tracking
the
state
estimation
of
a
given
power
system
is important
for real
tirne
monitoring
of
the
system.
As is
well
known,
the
voltages
of all
real
system
vary
randomly
with
time
and
should
therefore
be
considered
to be
stochastic
processes'
It
is
thus
necessary
to
make
use
of
the sequential
estimation
techniques
of
Ref.
[1]
in
order
to
obtain
the
state
estimate
at
any given
time
point.
The power
relations
in
Eqs.
(14.25)
and,
(74.26)
are
still
valid
bur
must
be
rewritten
after
indicating
that
the
voltage
rnagnitudes
and
angles
are
new
functions
of the
discrete
time
index
ft.
I4.5
SOME
COMPUTATIONAL
CONSIDERATIONS
Brrfh the
qtafin
qnd
fhe trqolzlno ocfimofin- ^1-^*i+L*^ ai^,r^61^-r i- rL^ ---^--r: --
rruvr\rr16
vr)rrrrrqlrulr
Ctr5\Jl
tLllll.lJ
PIEi)t'f
lttrlt
lll
tll€ pICL:CUlllg
sections
are
computationally
intensive,
particularly
for
large power
networks
which
may
have
nlore
than
200
important
buses.
It is,
therefore,
very
important
to pay
attention
fo such
computational
issues
as
illconditioning,
computer
storage
and
tinne
requirements.
However,
we
need
to
first
consider
the
question
of existence
of
a solution
of
the
state
estimation
problem.
An
lntroduction
to
state
Estimation
of
power
systems
|
545
T-
Network
Obsenrability
[17]
Consider
the
static
WLSE
formula
lEq.(14.llb)l
which
serves
as
the
srarting
point
fbr
all
the
algorithrns.
lnverse
of information
matrix
Mn,,
=
Ht wH
should
exist
otherwise
there
is no state
estimate.
This
will
happen
if
rank
of
f/
ls
equal
to n (no.
ot state
vartables).
Since
one
can
always
choose
a non-
singular
l4l,
so
if
lt1 has
a
rank
n,
the
power
network
is
said
to
be
observable.
Problem
of lll-conditioning
Even
if
the
given
power
system
is
an
observable
system
in
terms
of the
measurements
selected
for
the
state
estimation
purposes,
there
is
no guarantee
that
the
required
inversion
of the
information
matrix
will
exist.
During
multiplications
of
the
matrices,
there
is
some
small
but
definite
error
introduced
due
to
the finite
word
length
and
quantisation.
Whether
or
not
these
errors
create
ill-conditioning
of the
information
matrix
may
be determined
from a
knowledge
of the
condition
number
of the
matrix.
This
number
is
defined
as the
ratio
of
the
largest
and
the
smallest
eigen
values
of the
inforrnation
rnatrix.
The
maftix
M
becomes
more
and
more
ill-conditioned
as
its
condition
number
increases
in
magnitude.
Some
detailed
results
on
power
system
state
estimation
using
Cholesky
factorization
techniques
may
be
fbund
in
tl8l.
Factorization
helps
to reduce
ill-conditioning
but
may
not
reduce
the
computational
burden.
A
techrrique
to
reduce
computational
burden
is
describecl
in
Ref.
[]91.
14.6
EXTERNAL
SYSTEM
EOUTVAIENCTNG
[20]
One
of the
widely
practiced
methods
used
for
computational
simplification
is to
divide
the
given
system
into
three
subsystems
as
shown
in
Fig.
14.4.
One of
these
is
referred
to
as
the
'internal'
subsystem
and
consists
of those
buses in
which
we
are really
interested.
The
second
subsystem
consists
of
those
buses
which
are
not
of
direct
interest
to
us
and
is referred
to
as
the
'external,
subsystem'
Finally,
the
buses
which
provide
links
between
these
internal
and
external
subsystems
constitute
the
third
subsystem
referred
to
as
the
'boundary'
subsystem.
For
any given
power
network,
the
identification
of
the
three
subsystems
may
be done
either
in a
natural
or in
an
artiflcial
way.
,/
-t,
/\,
(14.41)
\"
,/
Internal
system
Boundary
system
Flg.
14.4