ELEC0047 - Power system dynamics, control and stability
Long-term voltage stability: fundamentals
Thierry Van Cutsem
t.vancutsem@ulg.ac.be
www.montefiore.ulg.ac.be/~vct
December 2015
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Long-term voltage stability: fundamentals
Introduction
Classification of power system instabilities
2 / 34
Long-term voltage stability: fundamentals
Introduction
Table of contents
Voltage instability results from the inability of the combined transmission
and generation system to provide the power requested by loads
I
Transmission aspects
Generation aspects
Load aspects
3 / 34
Long-term voltage stability: fundamentals
Transmission aspects
One-generator one-load system in steady-state operation
Complex power absorbed by the load:
S = P + jQ = V̄ I¯? = V̄
j
Ē ? − V̄ ?
= (EV cos θ + jEV sin θ − V 2 )
−jX
X
Power flow equations:
P=−
EV
sin θ
X
Q=−
V2
EV
+
cos θ
X
X
After eliminating θ:
V2
2
+ (2QX − E 2 )V 2 + X 2 (P 2 + Q 2 ) = 0
(1)
4 / 34
Long-term voltage stability: fundamentals
Transmission aspects
Feasible region in load power space
To have (at least) one solution:
2
PX
QX
−
− 2 + 0.25 ≥ 0
E2
E
(2)
any P can be reached provided Q is adjusted (but V may be unacceptable !)
dissymmetry between P and Q due to reactive transmission impedance
locus symmetric w.r.t. Q axis; this does no longer hold when transmission
resistance is included
5 / 34
Long-term voltage stability: fundamentals
Transmission aspects
Maximum load power under constant power factor
Under the given load power factor cos φ :
Substituting in (2) gives:
P2 +
Q = P tan φ
E2
E4
tan φ P −
=0
X
4X 2
from which one obtains:
Pmax =
cos φ E 2
1 + sin φ 2X
Qmax =
sin φ E 2
1 + sin φ 2X
E
VmaxP = √ √
2 1 + sin φ
Particular cases:
cos φ = 1 :
cos φ = 0 :
Pmax =
E2
2X
Pmax = 0
Qmax = 0
Qmax =
E2
4X
E
VmaxP = √
2
VmaxP =
E
2
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Long-term voltage stability: fundamentals
Transmission aspects
Load voltage as a function of load power
Solving (1) for V 2 and taking the square root:
s
r
E2
E4
− QX ±
− X 2 P 2 − XQE 2
V =
2
4
7 / 34
Long-term voltage stability: fundamentals
Transmission aspects
PV curves
for a given power :
1 solution with “high” voltage and “low” current (normal operating point)
1 solution with “low” voltage and “high” current
compensating the load increases the maximum power but the “critical”
voltage approaches normal values !
similar curves: QV or SV under constant tan φ, QV under constant P, etc.
8 / 34
Long-term voltage stability: fundamentals
Transmission aspects
Generator reactive power requirement
Reactive power provided by generator:
Qg = Q + XI 2 = Q + X
P 2 + Qg2
E2
Reordering and
with respect to Qg :
ssolving
2
QE 2
E2
E2
±
−
− P2
Qg =
2X
2X
X
normal operation is on the lower part of this curve
Qg increases with P more than linearly (due to losses)
9 / 34
Long-term voltage stability: fundamentals
Transmission aspects
Effect of line capacitance and/or shunt compensation
Thevenin equivalent as seen by the load:
E
Eth =
1 − (Bc + Bl )X
Xth =
X
1 − (Bc + Bl )X
Maximum deliverable power under power factor cos φ:
Pmax =
2
cos φ
Eth
1
cos φ E 2
=
1 + sin φ 2Xth
1 − (Bc + Bl )X 1 + sin φ 2X
corresponding load voltage:
1
E
Eth
√ √
VmaxP = √ √
=
1 − (Bc + Bl )X 2 1 + sin φ
2 1 + sin φ
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Long-term voltage stability: fundamentals
Transmission aspects
Effect of variable shunt compensation
Bl = 0, Bc varied
shunt compensation adjusted to
keep voltage in an interval
smooth change in compensation
by SVC at load bus
V almost constant as long
as SVC is within limits
11 / 34
Long-term voltage stability: fundamentals
Transmission aspects
Sensitivities
Sensitivities can help determine
if operating point is on the upper
part of the PV curve
Example : sensitivity SQg Q of reactive power generation to reactive load power
How many additional Mvar’s
must be produced if the load
consumes 1 Mvar more ?
positive at normal
operating points
tend to infinity as the
maximum load power is
approached
12 / 34
Long-term voltage stability: fundamentals
Transmission aspects
First glance at instability mechanisms
Assume that the load behaves as constant power in steady-state
P = zPo
Q = zQo
When z increases, the load characteristic
changes until it eventually does not intersect the
network characteristic. Equilibrium is lost at the
loadability limit.
The large disturbance causes the network characteristic to shrink so drastically that the
post-disturbance network characteristic does no
longer intersect the (unchanged) load characteristic
13 / 34
Long-term voltage stability: fundamentals
Generation aspects
Table of contents
Transmission aspects
I
Generation aspects
Load aspects
14 / 34
Long-term voltage stability: fundamentals
Generation aspects
Field and armature current limiters
field current limit imposed by OverExcitation Limiter (OEL)
armature (or stator) current limit: seldom enforced by a limiter; most often,
action by operator.
15 / 34
Long-term voltage stability: fundamentals
Generation aspects
Generator QV curves
Region of admissible operating points in the (Q, V ) plane, under constant P
Example of QV curves
50-Hz turbo-generator
rated apparent power: 1200 MVA
rated turbine power: 1020 MW
Xd = 2.051 pu
Xq = 1.966 pu
saturation taken into account
If = 2671 A at no load and nominal
voltage
If = 8300 A after OEL activation
16 / 34
Long-term voltage stability: fundamentals
Generation aspects
Under Automatic Voltage Regulator (AVR) control:
small voltage drop when the reactive power output increases (due to
proportional control often used)
significant for machines with low AVR steady-state open-loop gain (e.g. 30-50
pu/pu)
assuming constant reactive power for a machine under rotor limit is an
approximation (there is some change with voltage)
the reactive power must be updated with the active power output
at low voltages the armature limit is very constraining
usually the limited generator will be tripped by a protection when its voltage
reaches ' 0.85 pu.
17 / 34
Long-term voltage stability: fundamentals
Generation aspects
Effect of generator reactive limits on maximum load power
L closer to G than to G∞
simple PV/PQ model used for G
increase of P of load covered by G∞
maximum load power reached at point M’, not M !
maximum load power significantly reduced by generator reactive limits
most often voltage stability would not be a problem if generators were
unlimited sources of reactive power (i.e. constant voltage sources)
maintain reactive power reserves on generators located near load centers
maximum load power generally reached at higher voltages (more dangerous)
electrical decoupling does no longer hold !
18 / 34
Long-term voltage stability: fundamentals
Generation aspects
Illustration with a power flow calculation
XAB = 0.2, XBL = 0.005, XGB = 0.02 pu, rGB = 1.04, BL = 200 Mvar
PL = 900, PG = 400 MW, QL = 200, QGmax = 250 Mvar, VG = 1.04, VA = 1.05 pu
Operating point beyond maximum load power
Trace of Newton iterations:
iter
max mismatches
1
2
3
4
5
6
Gener G : Q =
457.8 >
6
7
8
9
10
: MW
900.0
199.3
23.7
2.4
0.8
0.3
Qmax =
0.3
50.0
258.6
140.2
304.6
Mvar
445.3
339.1
28.2
1.4
0.4
0.2
250.0. Switched to PQ type
207.8
22.3
19.2
11.3
37.6
divergence of Newton iterations after the generator reactive limit is enforced.
19 / 34
Long-term voltage stability: fundamentals
Generation aspects
Effect of generator reactive limits - another situation
Same system
respective position of the PV curves is
different
maximum load power:
not reached at point M, where reactive capability of G is exceeded (would
cause the OEL to act)
not reached at point M’, where generator voltage VG is higher than setpoint
VG0 (would cause the AVR to regain control)
reached at the breaking point A, where VG = Vg0 and QG = QGmax
20 / 34
Long-term voltage stability: fundamentals
Generation aspects
Illustration with a power flow calculation
same data except QGmax = 350 Mvar
iter
max mismatches
1
2
3
4
5
6
Gener G : Q =
457.8 >
6
7
8
9
Volt of gener G = 1.1418
9
10
11
12
13
Gener G : Q =
458.4 >
13
14
15
16
Volt of gener G = 1.1418
16
17
18
: MW
Mvar
900.0
445.3
199.3
339.1
23.7
28.2
2.4
1.4
0.8
0.4
0.3
0.2
dQg/dQl
Qmax =
350.0. Switched to PQ type
0.3
107.8
13.1
5.0
5.0
0.4
0.2
0.1
dQg/dQl
> setpoint = 1.0500. Back under volt ctl
32.2
496.0
20.6
43.3
1.7
1.5
0.5
0.4
0.2
0.1
dQg/dQl
Qmax =
350.0. Switched to PQ type
0.2
108.4
13.1
5.0
5.1
0.5
0.2
0.1
dQg/dQl
> setpoint = 1.0500. Back under volt ctl
32.2
496.0
20.6
43.3
1.7
1.5 ...
=
2.45 at bus L
= -2.22 at bus L
=
2.46 at bus L
= -2.22 at bus L
21 / 34
Long-term voltage stability: fundamentals
Generation aspects
Sensitivity behaviour in the presence of generator limits
Discontinuities at the points where a generator limit is enforced
case of slide 17
case of slide 19
sensitivity goes to infinity when
approaching max load power
sensitivity does not go to infinity when passing through max load
power
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Long-term voltage stability: fundamentals
Load aspects
Table of contents
Transmission aspects
Generation aspects
I
Load aspects
23 / 34
Long-term voltage stability: fundamentals
Load aspects
Load power restoration
if loads behaved as constant admittances, no voltage instability would occur
(low but steady voltages would be experienced in severe cases)
voltage instability is largely caused by the trend of loads to restore their
pre-disturbance power after a disturbance
this may take place in several time scales :
component
time scale
internal variable
induction motor
' 1 second
motor speed
load tap changer
' few minutes
thermostatically
controlled load
' few minutes
- tens of min.
transformer
ratio
amount of
connected load
equilibrium
condition
mechan. torque
= electrom. torque
controlled voltage
within deadband
temperature
within deadband
others: distribution voltage regulators, consumer reaction to voltage drop
24 / 34
Long-term voltage stability: fundamentals
Load aspects
Load power restoration in induction motor
Steady-state characteristics of motor
large industrial motor
Xs = 0.067, Xm = 3.800, Xr = 0.17, Rs = 0.013, Rr = 0.009 pu
25 / 34
Long-term voltage stability: fundamentals
Load aspects
Load tap changers
also referred to as
on-load tap changers or
under-load tap changers
Automatic adjustment of r to maintain V2 in a deadband
r ' [0.85 − 0.90 1.10 − 1.15]
∆r ' 0.5 - 1.5 %
[V2o − V2o + ]
∆r < 2
“Slow” device → long-term dynamics
Delay between two tap changes:
minimum delay Tm of mechanical origin ' 5 seconds
intentional additional delay: from a few seconds up to 1 − 2 minutes
to let network transients die out before reacting (avoid unnecessary wear)
fixed or variable
delay before first tap change (' 30 − 60 seconds) usually larger than delay
between subsequent tap changes (' 10 seconds)
26 / 34
Long-term voltage stability: fundamentals
Load aspects
Load power restoration through LTC
Assume the load is represented by an exponential model:
P2 (V2 ) = P o (
V2 α
)
V2o
Q2 (V2 ) = Q o (
V2 β
)
V2o
The power balance equations at bus 2 are:
Po(
Qo(
V2 α
)
V2o
V2 β
) − BV22
V2o
V1 V2
sin θ
rX
V2
V1 V2
= − 2 +
cos θ
X
rX
= −
(3)
(4)
27 / 34
Long-term voltage stability: fundamentals
Load aspects
For given values of V1 and r , Eqs. (3,4) can be solved numerically with
respect to θ and V2 (using Newton method for instance)
from which the power leaving the transmission network is obtained as:
P1 = −
V1 V2
sin θ (= P2 )
rX
Q1 =
V12
V1 V2
−
cos θ
r2 X
rX
repeating this operation for various values of V1 and r yields the curves
shown on the next slide.
Numerical example
transformer: 30 MVA, X = 0.14 pu, LTC voltage setpoint = 1 pu
load: α = 1.5, β = 2.4, P2 = 20 MW under V2 = 1 pu,
cos φu = 0.90 (lagging) under V2 = 1 pu
with the compensation capacitor: cos φc = 0.96 (lagging) under V2 = 1 pu
On the SB = 100 MVA base:
V2o
= 1 pu
o
P = 0.20 pu
X = 0.14(100/30) = 0.467 pu
o
Q = P o tan φu = 0.20 × 0.4843 = 0.097 pu
B.12 = Q o − P o tan φc ⇒ B = 0.097 − 0.20 × 0.2917 = 0.039 pu
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Long-term voltage stability: fundamentals
Load aspects
29 / 34
Long-term voltage stability: fundamentals
Load aspects
Response to a 5 % drop of voltage V1 :
in the short term, the operating point changes from A to B
next, three tap changes take place, under the effect of the LTC
and the operating point changes from B to C.
Neglecting the deadband 2, and assuming that LTC does not hit a limit:
the V2 voltage is restored to the setpoint value V2o
hence, the P2 and Q2 powers are restored to their pre-disturbance values
the same holds true for the P1 and Q1 powers, since:
P1
=
P2 (V2 )
Q1
=
Q2 (V2 ) − BV22 + XI22 = Q2 (V2 ) − BV22 + X
P22 (V2 ) + Q22 (V2 )
V22
hence, the load seen by the transmission system behaves in the long-term
(i.e. after the tap changer has acted) as a constant power.
30 / 34
Long-term voltage stability: fundamentals
Load aspects
Thermostatic load power recovery
Heating resistors are switched on/off by thermostats so that the mean power
consumed over a cycle = power required to keep the temperature = Preq
ton
GV 2 = Preq
ton + toff
(5)
If V drops, P = GV 2 drops ⇒ ton increases until (5) is satisfied
For a large number n of identical thermostatically-controlled resistors:
!
n
n
X
ton
1X
2
fi (t) GV 2 ' n
GV 2
P(t) =
fi (t)GV = n
n
ton + toff
i=1
i=1
where fi (t) = 1 if the i-th resistor is on at time t
fi (t) = 0 if it is off.
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Long-term voltage stability: fundamentals
P(t) ' n
Load aspects
ton
GV 2
ton + toff
following a voltage drop, nGV 2 decreases
ton
increases until P(t) recovers to nPreq
but after some time,
ton + toff
thus, the load behaves as constant admittance in the short term and as
constant power in steady state
thermostatically controlled loads are also referred to as “constant-energy”
loads.
However, if the voltage drop is too pronounced, all resistors stay connected
(toff = 0) but Preq cannot be obtained. Then, the load behaves as constant
admittance.
32 / 34
Long-term voltage stability: fundamentals
Load aspects
Generic model of load power restoration
Power consumed by the load at any time t:
βt
αt
V
V
Q(t) = zQ Qo
P(t) = zP Po
Vo
Vo
(6)
zP , zQ : dimensionless state variables associated with load dynamics
αt , βt : short-term (or transient) load exponents
In steady-state, the load obeys:
αs
V
Ps = Po
Vo
Qs = Qo
V
Vo
βs
(7)
αs , βs : steady-state (or long-term) load exponents; usually αs < αt , βs < βt .
The load dynamics are given by:
αs
αt
V
V
T żP =
− zP
Vo
Vo
T żQ =
with zPmin ≤ zP ≤ zPmax and zQmin ≤ zQ ≤ zQmax .
V
Vo
βs
− zQ
V
Vo
βt
(8)
T ' several minutes
33 / 34
Long-term voltage stability: fundamentals
Load aspects
Response of active power P to a step decrease of voltage V
αt '
∆Pt /Po
∆V /Vo
αs '
∆Ps /Po
∆V /Vo
Similar expressions hold true for reactive power
34 / 34