Angle Stability
component failure, etc. However, it is important that system operators do
everything possible to ensure as many regulators are in automatic mode as
possible. This action helps ensure adequate reactive power support and may help
avoid system instability.
7.6
Transient Stability/Instability
This section describes transient stability and instability and presents several
illustrated examples.
7.6.1 Process of Transient Stability
The transient environment is characterized by a system that undergoes a sudden,
severe disturbance. In contrast to the steady state environment where changes
occur gradually, the transient environment involves rapid changes.
Figure 7-13 illustrates the power system used to illustrate transient stability and
instability. This system has remote generation feeding through two high voltage
transmission lines into a large power system.
A remote
generator tied to
a large power
system is used to
keep our
description
simple. The large
power system is
not impacted
significantly by
what happens to
the generator or
the two-line
transmission
system.
Figure 7-13. Power System for Transient Stability & Instability
The top portion of Figure 7-14 contains the initial or pre-disturbance power
system. The remote generator is initially producing 1,000 MW. This MW is
transmitted to a large power system via two transmission lines. The large system
has 101,000 MW of load, 100,000 MW of which is fed from local generation.
A disturbance is created in this power system to study the transient stability of the
system. The circuit breakers at both ends of one of the transmission lines are
opened. From the generator perspective, once a line is opened the generator
suddenly has to transmit its mechanical power input across a much higher
impedance system. The generator must now work harder to transmit its MW
across the transmission system to the load area.
The bottom portion of Figure 7-14 contains power-angle curves for this system.
There are two curves; one for pre-disturbance conditions and one for after the line
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Angle Stability
opens (post-disturbance). Note the pre-disturbance power-angle curve has a
higher PMAX. Once the line is opened the transmission path has sharply higher
impedance that results in a lower PMAX. Note in Figure 7-14 that the postdisturbance curve has a significantly lower peak than the pre-disturbance curve.
Transient
stability is a
rapid event.
From the initial
disturbance to
the peak of the
angle swing is,
at most, a few
seconds.
Figure 7-14. Power-Angle Curves for 1,000 MW Generator Loading
Initially the power system is operating at point “A” in Figure 7-14. When the line
opens the operating point instantly shifts from point “A” to a position on the postdisturbance power-angle curve at point “B”. When at point “B”, the generator
must accelerate since its mechanical power input is now greater than the electrical
power the generator can push out. As the generator accelerates it turns faster than
synchronous speed. The angle increases due to this relative acceleration and
slides from point “B” towards “C”.
When the operating point reaches “C”, the generator’s MW output is again equal
to the mechanical power input. However, the angle cannot stop increasing. The
whole time the generator operated below the mechanical power input line it was
storing energy in its rotor. The generator is now spinning faster than synchronous
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Angle Stability
If the generator
had not slowed
down to
synchronous
speed by the time
it reached point
“F” it would have
been transiently
unstable. This
possibility is
addressed in the
next section.
speed. The angle keeps increasing as long as the generator’s speed is greater than
synchronous speed. The generator must rid itself of the excess stored energy
before it can return to synchronous speed and stop the angle increase.
As the operating point rises above point “C”, the generator starts to slow down.
The generator slows because it is now sending out more electrical power than it is
taking in mechanical power. The generator rises above point “C” until it slows
down to synchronous speed. This occurs at point “D” in Figure 7-14. If the
generator had not slowed down to synchronous speed by the time it reached point
“F” it would have gone unstable.
Equal Area Criterion for Transient Stability
Immediately following the line opening the operating point shifted from point
“A” on the pre-disturbance curve to point “B” on the post-disturbance curve. This
movement represented an immediate reduction in MW transfer due to the opening
of the line. The angle then increased from “B” through “C” and on to “D”. The
angle stopped increasing at “D”. As the operating point moved, there were two
areas created bounded by the points “A-B-C” and “C-D-E”. These two areas are
shaded and labeled “1” and “2” in Figure 7-14.
The two areas represent the accelerating and decelerating periods of the generator.
Area “1” represents the accelerating period of the generator. The size of area “1”
is equivalent to the energy stored in the rotor. Area “2” represents the
decelerating period of the generator. The size of area “2” is equivalent to the
energy removed from the rotor. The operating point rose above point “C” until
area “2” was the same size as area “1”. This is called the equal area criterion for
transient stability. If area “2” is not equal to area “1” by the time point “F” is
reached, the power system enter a period of instability.
Area “1” represents the energy that is absorbed by the rotor as a result of the
reduced MW transfer across the system. This absorbed energy is what causes the
rotor to speed-up and the angle to increase. Area “2” represents the energy
released by the rotor as it decelerates due to the electrical power output exceeding
the mechanical power input. The equal area criterion basically states that the
amount of energy required to slow the rotor to synchronous speed is equal to the
amount of energy that was added to accelerate it from synchronous speed. If the
decelerate area cannot match the accelerate area, instability occurs.
Maximum Angle
Our angle stability descriptions to this point have stated that the angle difference
between any two adjoining points in the power system can never stabilize at a
value greater than 90°. Note that we have emphasized, “stabilize” at a value. The
angle can exceed 90° for short periods of time as long as its final value returns to
less than 90°. This can happen if the angle is oscillating about a point that is less
than 90°.
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Angle Stability
Note that the angle in Figure 7-14 could have swung all the way to point “F” as
long as it did not pass through point “F”. Once the operating point has passed
point “F”, the system is unstable. Once past point “F”, the electrical power output
is again less than the mechanical power input. The angle must increase, as energy
is again stored in the rotor. As the angle is already greater than 90° when past
point “F” the only option is a further angle increase and eventual angle instability.
Figure 7-15 illustrates the same information as Figure 7-14 but in a different
format. Points “A”, “B”, “C” and “D” are labeled on Figure 7-15. These labeled
points correspond to the labeled areas on Figure 7-14. Note that once disturbed
the MW output of the generator oscillates about 1,000 MW before finally settling
down at 1,000 MW.
In this example of transient stability the MW output starts and ends at 1,000 MW.
During the oscillation the output may swing between 1,500 to 500 MW. Note the
time frames in Figure 7-15. Once the angle started to reduce from point “D”, it
indicates that the system was transiently stable. It took approximately 1 second to
determine whether this system was transiently stable or unstable.
It takes only a few seconds to determine if the angle recovers from the first
swings. However, the oscillations that follow the first few swings may last for
many more seconds. When the oscillations finally settle down or dampen, the
operating point is at “C”. The angle at point “C” is greater than the initial angle at
point “A”. This is expected since the system has lost a line and the path
impedance is now greater. For transient stability we are only concerned with the
first several swings. Oscillatory stability concerns itself with subsequent swings.
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Angle Stability
Several points are
labeled “C” in
this figure. This is
intentional as the
operating point
swings through
point “C” several
times before
finally settling at
point “C”.
Figure 7-15. Strip Chart Equivalent of Figure 7-11
7.6.2 Process of Transient Instability
The top of Figure 7-16 is the same power system as was used in Figure 7-14 with
the exception of the loading on the system. Figure 7-14 had a loading of 1,000
MW on the remote generator. When one of the lines was opened the system
remained transiently stable. Figure 7-16 has a loading of 1,500 MW on the
remote generator. One of the two lines is again opened. The shock to the system
is greater since larger amounts of MW flow and a larger initial angle are involved.
The power-angle curve at the bottom of Figure 7-16 illustrates that the operating
point shifts from its initial location at point “A” to “B” once the line is opened.
The remote generator now has more mechanical power input than electrical power
output and must accelerate. As the generator accelerates the angle increases. The
angle keeps increasing until the generator can again return to synchronous speed.
If the generator does not return to synchronous speed by the time the operating
point reaches “F”, the system is transiently unstable.
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Angle Stability
Since the
decelerating
area (area #2) is
smaller than the
accelerating
area (area #1)
this generator
goes out-of-step
with the larger
power system.
Figure 7-16. Power Angle Curve for Transient Instability
A close inspection of Figure 7-16 reveals that this generator goes out-of-step as
predicted by the equal area criterion. The important point to note is that area “2”
is smaller than area “1”. This means there is not enough decelerating energy to
return the generator’s rotor to synchronous speed before reaching point “F”.
Since the generator is still above synchronous speed after point “F” is reached, the
angle continues to increase. The electrical output again drops below the
mechanical power input line. This causes the generator to further accelerate and
go out-of-step.
The system illustrated in Figure 7-16 is transiently unstable. The generator never
recovered from the initial angle swing. Once the generator passed through point
“F” it accelerated at a faster and faster rate. Once the angle passes 180°, the
generator starts motoring (absorbing MW) and accelerates even faster. Generator
protection systems are sometimes designed to detect out-of-step conditions and
trip the generator before damage occurs.
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Angle Stability
7.6.3 Transient Stability Following a Fault
The top of Figure 7-17 illustrates the same system and generation levels as Figure
7-14 but now, instead of just opening a line’s circuit breakers, a fault occurs.
With a fault occurrence there are three power-angle curves instead of two. One
curve is for the pre-fault conditions, one curve is for post-fault conditions, and
one curve is for the period of time during which the fault is applied.
The fault is
cleared by the
transmission
line’s protective
relays.
While the fault is applied to the system, voltages are depressed due to the large
rush of reactive current to the fault. The depressed voltage is represented by the
low peak (low PMAX) of the power-angle curve while the fault is applied.
Immediately after the fault is applied, the operating point shifts from the pre-fault
curve at point “A” to the fault curve at point “B”. The angle then increases along
the fault curve until the fault is cleared at point “J”.
The size of the accelerating area (area “1”) is directly related to the fault clearing
time. When the fault is cleared the operating point shifts to the post-fault curve.
The post-fault curve is smaller then the pre-fault curve because one of the
transmission lines is out-of-service. The power-angle curves of Figure 7-17
indicate that the system is transiently stable as the generator reached synchronous
speed at point “D”. The operating point then oscillates back and forth before
finally settling at point “C”.
Figure 7-17 follows on the next page.
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Angle Stability
The faulted
system curve is
the smallest of
the three curves
as system
voltage is
depressed
during the fault.
Figure 7-17. Transient Stability and a Fault
3F Fault and an Extended Power Angle Curve
The top of Figure 7-18 again illustrates our simple power system. A 3Φ fault is
now applied at the remote generator’s high side bus. Three phase faults are
generally the most severe type of fault. While this 3Φ fault is applied, no MW
can leave the generator. All of the generator’s mechanical power input is stored
in and accelerates the rotor.
The bottom of Figure 7-18 contains an extended power angle curve. Instead of
illustrating only 180° this figure illustrates several alternations of the sine curve.
Note the size of the accelerating area (area “1”) in Figure 7-18. The decelerating
area (area “2”) is no match for the accelerating area and the generator pulls outof-step from the larger power system.
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Angle Stability
Figure 7-18. Extended Power-Angle Curve
After the operating point passes through point “G” the generator becomes a
motor. While motoring the generator absorbs MW from the system, which further
accelerates the generator’s rotor. The combination of the mechanical power input
to the generator and the generator acting as a motor creates a second large
accelerating area “3”. The decelerating area “4” is again no match for the
accelerating area “3” and the generator continues its out-of-step operation.
Throughout our description of transient stability we assumed that the mechanical
power input stays constant. This is a reasonable assumption as it is very difficult
to rapidly change the mechanical power input. In Figure 7-18 note that after a
few seconds mechanical power input is slowly reduced. However, the reduction
is too little too late and the generator is unstable. In the next section several
methods for achieving rapid, and effective reductions in mechanical power input
are described.
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7.6.4 Further Observations with Power-Angle Curves
Benefits of High Speed Reclosing
Several additional concepts can be illustrated on a power-angle curve. One
concept is the benefit of high-speed reclosing following faults. Assume that the
fault in Figure 7-17 was a line-to-ground fault. Further assume that the faulted
transmission line was equipped with high speed reclosing. If this reclosing had
been successful, instead of shifting from the faulted conditions power-angle curve
to the post-fault curve, the system would have shifted back to the larger pre-fault
curve with two lines in service.
With two lines available to draw MW out of the generator the chances of
remaining stable are greatly increased. If high speed reclosing had been
successfully used, area “1” in Figure 7-17 would be slightly smaller. Area #2
would not need to be as large and, if required, area #2 would have more room to
grow. The maximum angle growth (point “D”) would therefore not be as large.
A risk of high speed reclosing is the chance of closing back into the fault and
ending up worse off than just letting the line trip. The effect of closing back into
the fault as seen in Figure 7-17 would be a much larger area “1” with the distance
from points “B” to “J” being greater. As we now know, the larger area “1” is, the
higher the risk of instability.
Need for High Speed Protective Relaying
Power-angle curves also illustrate the need for high-speed protective relays in the
transmission system. The faster the fault is cleared, the smaller the accelerating
area. The smaller the accelerating area, the less the angle grows and the better the
chances to remain angle stable.
Use of Fast Valving
Earlier sections have stated that it is difficult to make rapid changes to the
mechanical power input. While it may be difficult to accomplish, this type of
action could be well worth the effort. Note the position of the mechanical power
input line in Figure 7-18. If following a disturbance, a rapid movement
downward could be made to the position of this line, the accelerating area would
be reduced and the decelerating area increased. A rapid adjustment to mechanical
power input could save a system from transient instability.
Steam turbine/generators may use a process called “fast valving” to achieve a
rapid reduction in mechanical power input. Figure 7-19 illustrates the fast valving
process. The steam flows to the intermediate and low-pressure turbine stages
through the intercept valve. A fast valving system is designed to rapidly shut the
intercept valve when the generator is at risk of transient instability. When the
intercept valve is quickly shut possibly two-thirds of the generator’s mechanical
power input is suddenly removed. This action greatly reduces the accelerating
energy in the system and possibly avoids instability.
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While many
steam units have
the ability to use
fast valving, few
actually
implement the
scheme due to
the possible
harmful impact
to the boiler.
Angle Stability
Following a
severe
disturbance it
may take ½
second to shut the
intercept valve.
The valve then
remains closed
for a few seconds.
Ideally no steam
is vented and the
generator is
rapidly brought
back to initial
loading.
Figure 7-19. Fast Valving in Steam Units
Hydro units cannot utilize fast valving but a hydro-based system can employ a
process with a similar purpose. Braking resistors are sometimes installed in
hydro-based systems. Braking resistors are large resistive loads. The braking
resistor control logic monitors system parameters to determine if the local power
system is accelerating. If all the conditions are met, the braking resistor is placed
in-service. The braking resistor remains in-service for only a short time, perhaps
20 cycles. While it is in-service the braking resistor slows the system down. This
deceleration may be sufficient to avoid instability.
British Columbia
Hydro also has a
braking resistor
installed adjacent
to their northern
most hydro
generation.
Figure 7-20 is a picture of the Bonneville Power Administration (BPA) braking
resistor. This large brake (rated 230 kV & 1,400 MW) is located in the Pacific
Northwest. The brake is inserted if the Pacific Northwest power system
accelerates with respect to the rest of the Western Interconnection. While inserted
the brake slows down the Pacific Northwest system and helps reduce the angle
growth.
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The three towers
in the picture form
the braking
resistor. The
towers contain
approximately 10
miles of ½ inch
stainless-steel
wire stretching
from the towers to
ground. These
wires form the
brake’s
resistance. 230 kV
transmission lines
connect the towers
to the power
system via circuit
breakers. When
the brake is
needed to stop
acceleration, the
circuit breakers
are closed and the
brake is placed inservice for ½
second
Figure 7-20. The BPA Braking Resistor
Generator Dropping
An additional option for rapidly reducing the power system’s accelerating energy
is to quickly trip generation. Generator dropping refers to the intentional tripping
of generating units. Generators may be tripped to avoid an accelerating condition
that could lead to instability.
NERC utilities drop both steam and hydro generation. However the dropping of
hydro generation has definite advantages, as it is often simple and rapid process to
re-synchronize a hydro unit. In contrast, there are many events that could occur
which could delay the re-synchronizing of a steam unit.
7.7
Oscillatory Stability/Instability
This section describes oscillatory stability and instability and presents several
illustrated examples. The oscillatory environment is characterized by a system
that is constantly changing. MW, Mvar, voltage magnitudes, angles, and
frequency may be oscillating.
The study of oscillatory stability is similar to steady state stability in that no
severe triggering event is required. A system may enter into a period of
oscillatory instability as the result of a minor disturbance such as a line switching
operation. Oscillatory instability may be a slowly developing event. A system
may begin a period of oscillations that last for several seconds, minutes, or even
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The causes, effects,
and control of the
oscillations that
accompany
oscillatory stability
and instability are
further addressed
Chapter 8.