Voltage
Instability:
Phenometrâ,
Countermeasures,
and
Analysis
Methods
THIERRY
VAN
CUTSEM,
MEMBER.IEEE
Irwited
Paper
A
p-otver
slstem
may
be
subject
to
(rotor)
angle,
frequency
or voltage
,instability.
Voltage
instability
takes
on1n,
i-.
oy o
dramatic
drop
of
transmission
system
voltages,
which
may
liad
to
system
disruption.
During
the
past
wo
dàcades
it has
iecome
a major
threat
for
the
operation
of
many
systems
and,
in
the
prevailing
open access
erwironment,
it
is a
factor
leading
to
limit
4owey
yansfers.
The
objective
of this paper
is to
desuibe
voltage
instability
phenomena,
to
enumerate
preventive
and
curatùe
countermeasures,
and
to
present
in a
unifed
and
coherent
way
various
computer
atwlysis
methods
used
or
proposed.
Keywords-Bifurcations,
load
dynamics,
nonlinear
systems,
power
systems,
security
analysis,
stability.
I. hrrnooucnoN
The
transfer
of
power
through
a transmission
network
is
accompanied
by voltage
drops
between
the generation
and
consumption
points.
In
normal
operating
conditions,
these
drops
are
in
the
order
of
a few percents
of
the nominal
voltage.
One
of
the
tasks
of power
system
planners
and
operators
is to
check
that
under
heavy
stress
conditions
and/or
following
credible
events,
all
bus
voltages
remain
within
acceptable
bounds.
In
some
circumstances,
however,
in
the
seconds
or min-
utes following
a disturbance,
voltages
may
experience
large,
progressive
falls,
which
are
so
pronounced
that
the system
integrity
is
endangered
and power
cannot
be
delivered
cor-
rectly
to
customers.
This
catastrophe
is refened
to as voltage
instability
and
its
calamitous
result
as voltage
collapse.
This
instability
stems
from
the
attempt
of load
dynamics
to resrore
power
consumption
beyond
the amount
that
can
be
provided
by the
combined
transmission
and generation
system.
ln an
increasing
number
of
systems,
voltage
instability
is
recognized
as
a
major
threat
for
system
operation,
at least
as important
as
thermal
overload
and angle
instability
prob-
Manuscript
rcceived
June 3,1999;
revised
August
3,1ggg.
The
author
is with
University
of Liège,
Institut
Montefiore,
84000
Liège,
Belgium
(e-mail:
vct@montefiore.ulg.ac.be).
Publisher
ltem Identifier
S
0018-92
19(00)00837-9.
Iems,
known
for
a longer
time.
Several factors have
con-
tributed
to
this
situation.
It is well
known that the
building
of new
transmission
and
generation
facilities is more and
more
difficult,
often delayed
and
somerimes impossible. The
building
of larger,
remote power
plants
has decreased the
number
of voltage
controlled points
and increased
the elec-
trical
distance
between
generation
and
load
(although
this
might
be partially
compensated
by the
emergence
of
inde-
pendent
power
productions
closer
to
loads).
The
heavy use
of shunt
compensation
to support
the voltage profile
allows
larger power
transfers
but brings
the instability point
closer to
normal
values.
Also,
voltage
instability
is often
triggered by
the
tripping
of
transmission
or
generation
equipments,
whose
probability
of
occurrence
is relatively
large
(compared
for
instance
to
the three-phase
short-circuit
considered in angle
stability
studies).
Last
but not
least, the transmission
open
access
environment
has
created
an economical incentive
to
operate power
systems
closer
to their limits.
More than
ever,
it
becomes
essential
to determine
these operating limits,
in
particular
with
respect
to
voltage
instability.
The
objective
of
this
paper
is
togive adescriptionof
the
phe-
nomena
which
contribute
to
voltage
instability,
to enumerate
countermeasures,
and to present
in
a
(hopefully)
unified
and
coherentwaythecomputeranalysismethods
usedorproposed.
The
incidents
experienced
throughout
the
world
and the
threat
of
other
blackouts
have prompted
significant
research
efforts among
the
power
engineering
community.
The
refer-
ences given
in
this
paper
make
up
only a sample
of
the
vast
literature
devoted
to
the
subject. As
"entry
points"
to this lit-
erature,
let
us
point
out:
l)
early
publications
dealing with
the
subject
tll-tl6l;
2)
a series
of four
seminars
t17l-t201,
which provided
a
forum
for
the presentation
of research
advances
in
the
voltage
stability
area;
3)
the
reports
of several
CIGRE
Task
Forces
[14],
t2lJ-î24)
and IEEE
r#orking
Groups
tZ5)-1271
of-
fering
a
compilation
of techniques
for analyzing
and
counteracting
voltage
instability;
0018-9219/00$10.00
@ 2000
IEEE
PROCEEDINGS
OF
TI{E
IEEE, VOL.
88, NO. 2,
FEBRUARY
2MO
x
î
=vle
I
",o
E
=
pt-o
Fig. 1.
Tvo-bus
system.
4)
more recently, one chapter
of a textbook
[28J
and two
monographs
1291,
t30l
devoted to the subject.
More
details on
the
material presented
in
this
paper
can be
found
in
[30].
More exhaustive
bibliographies are available
in the above reports
and books,
as
well
as in
[31].
II.
VOLTAGE
INSTABILTTY
PHENoMENA
A.
Maximum Load
Power
One of the
primary
causes of
power
system
instability
is
the
transmission of
(large
amounts
of)
power
over long elec-
trical distances. In voltage
stability,
attention
is
paid
to power
transfers
between
generation
and
load
centers.
Let us first
recall some
fundamentals
of the
power
transfer
between
a
generator
and
a
load.
We use the simple
model
of Fig. 1,
in
which
we
consider for simplicity
a
purely
re-
active
transmission
impedance
jX
and we
assume
that the
synchronous
generator
behaves as a constant voltage
source
of magnitude -E
(more
realistic
models will
be discussed in
the sequel).
Under balanced
three-phase,
steady-state
sinusoidal con-
ditions, system
operation is
described
by the
power
flow
or
load
flow
equations
[4],[28],[32)
"""i""..-.,
-i"
"'..'
i
\::
!
"
"....:
ool
Fig.
2. Load voltage venus
active
and rcactive
powers
[30].
on the
"equator"
of the
surface
(where
the two
solutions in
(3)
coalesce,
i.e., the
inner square root vanishes).
The
projec-
tion of
this
limit
curve
onto the
(P,
Q)
plane
is the
parabola
shown
in Fig. 2.
tn the
(P,
Q)
loadpower
space,
this
parabola
bounds
the region where
operation
is feasible.
B. Nose
Curves
In many reasonings
(and
industry practice)
it is common
to consider
the curves which
relate voltage
to
(active
or reac-
tive)
power.
Such
curves,
referred
to
as PV
(or
QV)
curves
or nose
curves
are shown
in Fig.
3,
for
our simple
system.
The curves
depend
on how
Q
varies
with
P; in Fig.
3, a con-
stant
power
factor,
i.e.,
Q
:
tan
/P,
has
been
assumed for
each curve. This
also
corresponds
to
the solid
lines in Fig.
2.
Similarly, one may
consider PV
curves
under constant
Q,
or
QV
curves
under
either
constant
power
factor
or constant
P.
Simply stated, voltage
instability
results
from the attempt
to
operate
beyond
maximum
load power.
This
may
result
from
a severe load
increase
or, more realistically,
from
a large
disturbance
that
increases
X and/or
decreases
.E
to the
ex-
tent that
the
predisturbance
load
demand
can no
longer
be
satisfied. The
latter
scenario is
illustrated
in
the next
section.
which
offers
a deeper
look into
a typical
voltage
instability
mechanism,
relating
the latter
to maximum
load
power
as
well
as system
theoretic
concepts.
C. l,ong-TermVoltage
Instability
lllustrated
on
a Simple
System
The
following
example,
taken
from
[30],
uses
the
simple
system
shown
in Fig.
4(a). Bus
3
represents
a
distribution
feeder. The power
consumed
at
this bus
may correspond
to
a
large number
of
individual
loads fed
through
medium voltage
(MV)
distribution
lines, shunt
capacitors,
etc. We represent
this aggregate
load
by the exponential
model
(widely
used
in
large-scale
stability
studies)
where
Vo is
areference
voltage
and
Po
(resp.
8")
is the
active
(resp.
reactive) power
consumed
under
this voltage.
{.!
4'2
-o.t
,#
p
:
-E]
"ne
(l)
a
=
-ç
+
E],o"o
(z)
where
P
(respectively
Q)
is the active
(respectively
reactive)
power
consumed
by the load,V
the load
bus
voltage
magni-
tude, and
d the
phase
angle difference
between the
load and
the
generator
buses
(see
Fig. l).
Solving
(l),
(2)
with
respect
to
V
yields
V_
Fig. 2 shows how the
terminal voltage
V changes
with
the
load
powers
P,
Q
(dimensionless
variables
are used
in the
figure).
In
"normal"
conditions,
the operating
point
lies on
the
upper
part
of the surface
(corresponding
to the
solution
with
the
plus
sign in
(3),
with
I/ close
to
,E
. Permanent
op-
eration
on the lower
surface,
characterized by
a lower
voltage
and higher
current,
is unacceptable.
The figure
also
confirms
the existence
of a maximum
load
power,
well-known
from
circuit theory
[33].
More
precisely,
the
figure shows
a set of
maximum loadpowerpoints,
located
VAN
CUTSEM:
VOLTACE
INSTABILITY
(3)
P
=
P.
(#.)"
e
=
e"
(#")'
(4)
il,#
lnz
tl +
-
QX
i
rl +
-
x2P2
-
xE2Q.
14
l1
Fig.
3. The
"nose"
curves
[30].
Bus
3 receives its
power
from
the
hansmission
system
through a
transformer
equipped
with
an automatic
load
tap
changer
(LTC).
The objective
of
this device
is to
adjust
the
turn ratio
of the transformer
(in
discrete
steps)
so as
to keep
the
distribution volrage
within
some
deadband
lV"-e
V,+el,
in spite
of voltage
fluctuations
on
the transmission
system
r
[2r),128),
[29], [34],
[3s1.
Most of
the load
power
is provided
by
a
remote
system
(bus
l)
through
a
rather
long
transmission.
The remaining
is supplied locally
by the
generator
at
bus 2. This generator
is
equipped with
an automatic
voltage
regulator
(AVR),
in
order to
keep
the
voltage
at bus
2
(almost)
constant,
and an
overexcitation
limiter
(OE1z;,
whose
role
is to
prevent
the
rotor
(or
freld)
current from
exceeding
a specified
thermal
limit, in case
the
AVR imposes
a
sustained
overexcitation
I2rl, t281,
[29),136),
[37].
We
show and
discuss
hereafter
two unstable
responses
ob-
tained by
simulating
the tripping
(at
t
:
1 s) of
one circuit
between
buses
I
and 4.
Case
l:
The
system evolution,
shown
in Fig.
4(b[d),
starts
with
electromechanical
oscillations
corresponding
to
swings
of the'generator
rotor.
These
transients
die out
soon,
indicating
that
the short-term
dynamics
of the
synchronous
generator3
are
stable.
Thus
a short-term
equilibrium
is
reached
after about l0
seconds, with
Va
settling
down close
to
0.96
per
unit.
The
system response
over the next
minutes
is a typical
example
of long-term
dynamics,
driven
by
the LTC and
OEL.
The operation
of the LTC
starts after
an initial delay
of 20 s
and
continues
at a rate of
one tap
change
each l0 s.
These
changes further
reduce
the transmission-level
voltage
Va.
The
operation
of the
OEL
can be seen
from Fig. 4(c),
showing
the
evolution of
the
generator
field
current. After
the disturbance,
this
current
jumps
to
3
pu,
which
exceeds
the limit, indicated
by the
doned
line. The
OEL has an
in-
lWe
assume
here for
simplicity that
the refer€nce
voltage l,'" and
the LTC
voltage setpoint
are equal.
2The
abbreviation
OXL
is also used.
3Also
referred to
as
"transient"
dynamics.
The term
"transient"
rcfen
to
either the time
period
of a few seconds
after
a disturbance
(like
in
"transient
time constânt") or to large-disturbance
analysis
fiike
in
"transient
(angle)
stability"l. As in
[30],
we
use
"short-term"
to
refer to the time period
unam-
biguously.
zto
verse-time
characteristic,
tolerating
smaller
overloads
to last
longer. Due
to this delay,
the OEL becomes active
at tt
=
70s.
Before this
time,
the
AVR
controls
the
voltage
at bus 2 and
makes
the field cunent
further
increase, in response to the
first
tap changes.
This
conesponds
to a larger and larger re-
active
power
drawn
from
the
generator.
After f
=
70 s, each
attempt
to
increase the field current
is corrected
by the OEL.
With
its field current kept
(almost)
constant,
the machine
be-
haves
as a constant emf behind saturated
synchronous
reac-
tance. The voltage
at bus 2
is no longer
controlled,
and Va
drops dramatically.
This drop
goes
on until
I/a reaches the
unacceptably
low
value
of 0.75
pu.
At this
point,
the LTC
has
reached
its
limit
and
the
voltage
decline stops, since no other dynamic
mech-
anism
is involved.
The evolution
of
the distribution voltage
y3
and
trans-
former
ratio r
are
shown in Fig. 4(d). Before
the
generator
limitation,
each tap movement
produces
the intended
effect
of rising
73, bringing it back
towards its
deadband;
on the
other hand,
after the limitation,
the tap changes have
negli-
gible,
then
reverse effects
t38l-t401.
A
deeper look at the instability mechanism is
provided
by
the PV
curves of
Fig. 4(e), showing voltage l/a as a func-
tion
of the
power
P transmitted to the load. All curves are
drawn
considering the short-term dynamics at
equilibrium.
The
solid
lines are the network P/ curves, similar to those
shown in
Fig. 3. On the other hand, through
(4),
the
load
power
depends on V3, which in tum depends on Va and
r.
The relationship
between P ard
Va,
for a
given
r, is the
short-term load characteristics, shown with
dotted
lines
in
Fig.4(e).
The system
operates initially at
point
A. The disturbance
causes the network
characteristic to
shrink from the right-
most to the
middle curve. After the
electromechanical oscil-
lations,
r has not
changed
yet
and the system operates at the
short-term
equilibrium
point
A'. Subsequent LTC operation
brings
the short-term
equilibria to
point B. At
this
point,
the
generator
field current
gets
limited. Note the further decrease
in maximum load power which
results from the limitation of
the
generator
reactive
power.
The
system
jumps
to
point
B'.
From there
on, the LTC keeps on decreasing the tap
until
it
finally
hits its limit at
point
D.
The vertical
dashed line in Fig. 4(e) is the long-term
char-
acteristic
of the load. Due
to
the
LTC action, a long-term
equilibrium is
such that V3
-
V,
(ignoring
the deadband
te), which
means P
:
Po.ln
other
words,
the
LTC
makes
the
load behave
as constant
power
in the long term.
The
nature of the instability
is revealed by observing
that
this long-term
load
characteristic does not intersect
the final
network
PV curve.
The maximum
power
that the final
con-
Itguration
can deliver is lower
than
what
the LIC
tends to re-
store. The
system becomes
unstable by loosing its
long-terrn
equilibrium.
Case
2: The
system initial
conditions
are modified by
in-
creasing the local generator
active
production.
The response
of the transmission voltage
Va
to the same
disturbance
is
plotted
in Fig.
5(a). In this
case the
gener-
ator field
current
gets
limited
at about t
:
100 s. As the
PROCEEDINGS
OFT}IE
IEEE. VOL.
88. NO.
2. FEBRUARY 2O(n
circuittrippedatt=1s
note
i
un
lza voltage
@
ll.
field current of
generator
G
@
!.2
o.t
.:i:; : :: :::::-:
: : :::::)
iSlfËira
V3 voltage
@
t.6
o.Ë
o.76
rotor
angle
of
generator
G
o
t(r)
Fig. 4. Example
system and simulation
of Case I
(all
quantities
in
per
unit)
[30]
Va
post-disturbance
(after
OEL)
r
=o.ss.rl/r/'
@
,//
///
otl
Fig. 5. Simulation of
Case 2
(all
quantities
in
per
unit or radian)
[50].
LTC keeps on reducing
the ratio, the
generator
eventually
looses
synchronism,
as is evident from
the
rotor
angle
plot
in Fig.
5(b).
The
subsequent
pole-slips
are responsible for
the
voltage
oscillations
in Fig. 5(a). The likely
outcome is
the tripping
of the
generator
by an out-of-step relay and a
blackout of the load area
caused by the tripping of the
over-
loaded lines.
\\e PV curves
arc shown in Fig. 5(c). As in
the fust case,
the
short-term equilibrium
point
follows
the
path
AA'(cir-
cuit tripping), Ats
ÉfC
operation), BB'(generator limita-
tion). Again,
there is no long-term
equilibrium with the
gen-
erator
limited.
However,
a major difference with respect
to
the
previous
case occurs when r reaches
the
value
0.82: the
short-term load
characteristics does no
longer intersect the
network PV cuwe, indicating
that the
system also looses
short-term
equilibrium, which corresponds
to the
above
men-
tioned loss of synchronism.
D. Load Power
Restoration
The
previous
example
has
shown
a typical situation where
the driving force of instability is
the unsuccessful
attempt
of the LTC to restore the load voltage
to its setpoint value
VAN
CUTSEM:
VOLTAGE
INSTABILIry
and thereby the load
power
to its
predisturbance level.
The
intemal
variable of
this
process
if the transformer
ratio r.
We mention hereafter two
other well-identified
load
power
restoration mechanisms.
I
)
Induction Motors
[]51, t281, t291,
[41]
-[43]:
Induction
motors are
present
in many industrial
and commer-
cial loads. When subject
to a step drop
in voltage, the motor
active
power
P first decreases as the square
of the
voltage
V
(constant
impedance behavior),
then recovers
close
to
its
predisturbance
value in
the
time frame of
a second. The in-
temal variable
of
this
process
is the
rotor slip. In fact, a motor
with constant mechanical torque and negligible
stator losses
restores
to constant
active
power.
Taking into account
these
losses
and
more realistic torque behaviors,
there is
a small
steady-state
dependence
of P
with
respect to
V. The steady-
state dependence of
the reactive
power
Q
is a
little more
complex.
Q
fust decreases
somewhat
quadratically with V,
reaches
a minimum,
and then increases up to
the
point
wherc
the motor
stalls
due to low voltage. In large three-phase in-
dustrial motors, the stalling voltage
can
be
as
low as 0.7
pu
while
in smaller appliances
(or
heavily loaded
motors)
it is
higher.
short-term load characteristic'r
=
1.00
Vt
=0.95
A'.'';:
.disturbance
(after
OEL)
...1B'
=0.80
I
t
t'
I
I
l)
@
ii
,,,,
iost-disturbance
(before
OEL)
I
lone-term load
"-
chaïacteristic
P
Va voltage
t.....:
@
t( s)
Load
restoration
by induction
motors
may
play
a
signifi-
cant
role
in systems
having a summer
peak
load,
with a large
amount
of
air conditioning
[29], t441,145).
2)
Thermostatic
Loads
tl3l,
t281,
t291, t461,
[47]:
Another
category
of
self-restoring
load
is the
electrical
heating
controlled by thermostat.
A thermostat
switches the
heating
resistor
according
to
an
on-off cycle,
such that the energy
produced over
a cycle
keeps
the temperature
within a deadband
under
the
given
weather
condition. Following
a decrease
in
voltage, the
re-
sistor
power decreases
as the square
of the
voltage. Over
the
next minutes,
howeveq
the on--off
cycle changes
progres-
sively
since the
resistor
has to stay
on longer
in order to
pro-
duce the
same energy.
Considering
the behavior of a
large
number
of such
devices
over a small
time interval, this in-
crease
in the on
time appears
as a recovery
of
the
power
to
its
predisturbance
value. However,
for a large enough voltage
drop,
the aggregate
load
power
does
not recover to its
pre-
disturbance
value, owing to the
fact that the heaters stay on
p€rmanently,
thus
giving a mere impedance load character-
istic
in
the steady
state.
Also, it has been observed that the
control
cycle of the
(older)
bimetallic thermostat
is itself in-
fluenced
by the
voltage,
making
load
power
restoration faster
than
what could be
expected
from
thermal
inertia
[47].
Clearly,
load self-restoration
by thermostats
is significant
when analyzing
the
winter
peak
load
of systems
with
a
large
amount
of electrical
heating.
It is of concem
mainly when
the
faster
acting
LTC's
do not
restore
voltage, e.g., because
they
reach their
limit.
E. Generic
Models of
Aggregate
Loads
The load seen
by a
bulk
power
delivery
transformer is an
aggregate
of
many individual
loads,
fed through distribution
lines
and
MV/LV
transformers,
compensated
by switched ca-
pacitors, etc.
The
problem of modeling such an
aggregate
load is
not easy
to address.
Indeed,
while typical data can
be obtained
for every
individual equipment
[48],
the real
problem is to determine
the composition of the
load. The
latter
varies from
one bus to
another but also
with the
season,
the
time
of
the day, etc.
Although
a detailed
discussion
of load modeling is out of
scope
of
this
paper, let us mention
an approach
proposed
in
the
recent
years
[47],
t491,
t50l-t521.
The
response
of aggregate
loads to step
decreases in
voltage4 has
been
measured by several
companies. The time
evolution
of
power, over several minutes,
recorded on
the
low
voltage side
of
bulk
power
delivery transformers, is
sketched
in Fig. 6.
The
partial
recovery
originates from
ther-
mostatically
controlled
loads, downstream
(nonmodeled)
LTC's,
voltage
regulators,
and voltage controlled capacitors,
or
even
the consumers
reaction
to the reduced
supplied
power.
The
exponential{ype
recovery shown in Fig.
6
may be
captured
by
generic
models,
for instance
Fig.
6. Load
power
response to a voltage
drop.
Table
I
Multitime-Scale
Powcr Sysæm
Modcl
ù"
=
hc(x,y,zc,za)
generators,
AVRs and PSSs
turbines and
governors
(equivalent)
induction motors
HVDC links
i<
=
r(x,y,z",za) (8)
I
network
Q
=
g(x,y,z",za\ (7)
where
zp is an internal
state variable,
obeying
with
zp'"
<
zp
<
zF"*
(6)
In some
sense,
the
load
obeys an exponential model
which
changes from
the transient
ot to
the steady-state
os expo-
nent. These
exponents can
be
determined from
the initial and
final
power
drops
(see
Fig. 6) through
LP,f P"
LVIU"
while
?p can be obtained from
a least-square fit of the time
response
(dotted
line
in Fig. 6).
Similar
relationships
hold for
the reactive
power.
F. Time-Scale Decomposition
Perspective
Table I
enumerates the components,
phenomena,
con-
trollers, and
protecting
devices which
play
a role in voltage
stability, classified
according to the
time scale of the
corre-
sponding
dynamics.
In stability
studies, an
instantaneous
response is assumed
for
the network, according
to the
quasi-sinusoidal (or
fun-
damental-frequency) assumption
[32].
The
network is thus
described by the algebraic
equations
(7)
derived from the
Kirchhoffs
current law at
each bus, and involving
the
vector
y
of bus voltages magnitudes
and
phase
angles.
aObtained,
for instance, by
changing rhe
transformer tap or by tripping
one of
two
parallel
transformers.
i
rpàp=(i)"
-,"
(l)*
LnlP,
nL--nN
""
-
LVIV"
P
:
zePo
("n)"'
switched shunt
compensation
secondary voltage and frequency
control
(s)
PROCEEDINCS OF TI{E
TEEE,
VOL.
t8. NO.
2. FEBRUARY 2ûN