Angle Stability (AS) Outline
• Definitions
• Small-disturbance:
– Hopf Bifurcations
– Control and Mitigation
– Practical applications
• Transient Stability (large-disturbance):
– Time domain
– Direct Methods:
• Equal Area Criterion
• Energy Functions
– Practical applications
1
AS Definitions
• IEEE-CIGRE classification (IEEE/CIGRE Joint Task Force on Stability Terms and
Definitions, “Definition and Classification of Power System Stability”, IEEE Trans.
Power Systems and CIGRE Technical Brochure 231, 2003):
Power System
Stability
Rotor Angle Frequency Voltage
Stability Stability Stability
Small -Disturbance Large - Small -
Transient
Angle Stability Disturbance Disturbance
Stability
Voltage Stability Voltage Stability
Short Term Long Term Short Term Long Term
Short Term Long Term
2
AS Definitions
• “Rotor angle stability refers to the ability of
synchronous machines of an interconnected
power system to remain in synchronism after
being subjected to a disturbance. It depends on
the ability to maintain/restore equilibrium
between electromagnetic torque and mechanical
torque of each synchronous machine in the
system.”
• In this case, the problem becomes apparent
through angular/frequency swings in some
generators which may lead to their loss of
synchronism with other generators.
3
Small-disturbance
• “Small-disturbance (or small-signal) rotor angle stability
is concerned with the ability of the power system to
maintain synchronism under small disturbances. The
disturbances are considered to be sufficiently small that
linearization of system equations is permissible for
purposes of analysis.”
• This problem is usually associated with the appearance
of undamped oscillations in the system due to a lack of
sufficient damping torque.
• Theoretically, this phenomenon may be associated with
a s.e.p. becoming unstable through a Hopf bifurcation
point, typically due to contingencies in the system (e.g.
August 1996 West Coast Blackout).
4
Hopf Bifurcations
• For the generator-system example, with
AVR but no QG limits:
5
Hopf Bifurcations
– As previously discussed, the ODE model for this
system for numerical time domain simulations is:
where: X = XL + XTh + X'G; V1r = V1 cos δ1;
V1i = V1 sin δ1 ; and Pm = Pd.
6
Hopf Bifurcations
– The PV curves for M = 0.1, D = 0.1, Kv = 1,
X 'G = 0.25, XTh = 0.25, V = V1o =1 are:
1
XL = 0.5
Operating
0.9 Point
HB
HB
0.8
XL = 0.6
0.7
3
V
0.6
0.5
0.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Pd
7
Hopf Bifurcations
– The eigenvalues for the system with respect to
changes in Pd for XL = 0.5
4
2
Pd
Hopf
1 Bifurcation
Imag.
-1
-2 Pd
-3
-4
-5 -4 -3 -2 -1 0 1 2 3 4
Real
8
Hopf Bifurcations
– There is a Hopf bifurcation at Pd ≈ 1.25:
0.2
0.1
Hopf Bifurcation
0
Real{ev}
-0.1
-0.2
-0.3
-0.4
1.05 1.1 1.15 1.2 1.25 1.3 1.35
Pd
9
Hopf Bifurcations
– A Hopf bifurcation with eigenvalues
µ = ±jβ yields a periodic oscillation of period:
– Hence, for the example:
10
Hopf Bifurcations
– A contingency XL = 0.5 → 0.6 at operating
point Pd = 1.13 yields:
10
∆ f [Hz]
9
δ [rad]
8 E' [pu]
V1 [pu]
7
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t [s]
11
Hopf Bifurcations
– Observe the system oscillations, which are a
“trademark” of Hopf bifurcations and small-
disturbance angle instabilities.
– These oscillations have a period of about 5s,
which is typical in practice, where these kinds
of oscillations are in the 0.1-1 Hz range.
12
Control and Mitigation
• For the IEEE 145-bus, 50-machine test system:
13
Control and Mitigation
– For an impedance load model, the PV curves yield:
14
Control and Mitigation
– Hence, a line 90-92 outage yields:
15
Control and Mitigation
– This has been typically solved by adding Power
System Stabilizers (PSS) to the voltage controllers in
“certain” generators, so that the equilibrium point is
made stable, i.e. the Hopf is removed (FACTS may
also be used to address this problem):
+
_ AVR
+
Vt
Vref
16
Control and Mitigation
– A participation factor analysis in this case yields:
17
Control and Mitigation
– The line 90-92 outage with PSS at generators 93 and
104 yields:
18
Control and Mitigation
• More details regarding this example can
be found in:
N. Mithulananthan, C. A. Cañizares, J. Reeve,
and G. J. Rogers, “Comparison of PSS, SVC
and STATCOM Controllers for Damping
Power System Oscillations,” IEEE
Transactions on Power Systems, Vol. 18, No.
2, May 2003, pp. 786-792.
19
Small Disturbance (SD)
Applications
• In practice, some contingencies trigger plant or
inter-area frequency oscillations in a “heavily”
loaded system, which may be directly
associated with Hopf bifurcations.
• This is a “classical” problem in power systems
and there are many examples of this
phenomenon in practice, such as the August 10,
1996 blackout of the WSCC (now WECC)
system.
20
SD Applications
• Observe that the maximum loadability of the system is reduced by
the presence of the Hopfs; this leads to the definition of a “dynamic”
Operating
ATC value: point
Worst contingency
(N-1 criterion)
1
0.9
HB
HB
0.8
ETC ATC TRM
0.7
TTC
3
V
0.6
0.5
0.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Pd
21
August 10, 1996 WSCC
Blackout
• This information was extracted from a presentation by
Dr. Prabha Kundur, retired President and CEO of
Powertech Labs Inc.
• System conditions:
– High ambient temperatures in Northwest, and high power
transfers from Canada and Northwest to California.
– Prior to main outage, three 500 kV line sections from lower
Columbia River to loads in Oregon were out of service due to
tree faults.
– California-Oregon Interties loaded to 4330 MW north to south.
– Pacific DC Intertie loaded at 2680 MW north to south.
– 2300 MW flow from British Columbia.
22
August 10, 1996 WSCC
Blackout
• Growing 0.23 Hz oscillations caused by tripping of a
series of lines:
3000
2900
2800
2700
2600
2500
2400
23000 3 6 9 12 16 19 22 25 28 31 34 37 40 43 47 50 53 56 59 62 65 68 71 74
Time in Seconds
23
August 10, 1996 WSCC
Blackout
• As a result of the
undamped oscillations,
the system split into
four large islands.
• 7.5+ million customers
experienced outages
from a few minutes to
nine hours.
• Total load loss:
30,500 MW.
24
August 10, 1996 WSCC
Blackout
• PSS solution:
San Onofre
(Addition) Palo Verde
(Tune existing)
25
August 10, 1996 WSCC
Blackout
With existing controls
Eigenvalue = 0.0597 + j 1.771
Frequency = 0.2818 Hz
Damping = -0.0337
With PSS modifications
Eigenvalue = -0.0717 + j 1.673
Frequency = 0.2664
Damping = 0.0429
26
Transient Stability
• “Large-disturbance rotor angle stability or
transient stability, as it is commonly referred to,
is concerned with the ability of the power system
to maintain synchronism when subjected to a
severe disturbance, such as a short circuit on a
transmission line. The resulting system response
involves large excursions of generator rotor
angles and is influenced by the nonlinear power-
angle relationship.”
• The system nonlinearities determine the system
response; hence, linearization does not work in
this case.
27
Transient Stability
• The basic idea and analysis procedures are:
– Pre-contingency (initial conditions): The system is operating in
“normal” conditions associated with a s.e.p.
– Contingency (fault trajectory): A large disturbance, such as a
short circuit or line trip, which is usually associated with the
disappearance of a s.e.p., force the system to move away from
its initial operating point.
– Post-contingency (fault clearance): The contingency usually
forces system protections to try to “clear” the fault; the issue is
then to determine whether the resulting system is stable, i.e.
whether the system remains relatively intact and the associated
time trajectories converge to a “reasonable” operating point.
• Based on nonlinear system theory, this analysis can be
basically viewed as determining whether the fault
trajectory at the “clearance” point is outside or inside of
the stability region of the post-contingency s.e.p.
28
Time Domain Analysis
• Given the complexity of power system models,
the most reliable analysis tool for these types of
studies is full time domain simulations.
• For example, for the generator-load example:
29
Time Domain Analysis
– The ODE for the simplest generator d-axis
transient model and neglecting AVR and
generator limits is:
where
30
Time Domain Analysis
– The objective is to determine how much time an
operator would have to connect the capacitor bank BC
after a severe transmission system contingency,
simulated here as a sudden increase in the value of
the reactance X, so that the system recovers.
– In this case, and as previously discussed in the
voltage stability section, the contingency is severe, as
the s.e.p. disappears.
– Full time domain simulations are carried out to study
this problem for the parameter values
M = 0.1, DG = 0.01, DL = 0.1, τ = 0.01, E’=1,
Pd = 0.7, k = 0.25, BC = 0.5.
31
Time Domain Analysis
– A contingency X = 0.5 → 0.6 at tf = 1s, with BC connection at tc =
1.4s yields a stable system:
x(t)=x(0)=xs1
A(xs1) x(tc)
x(0)=xs1 xs2
x(t)
tc A(xs2)
x(t)
x(tf)=xs1 tf
32
Time Domain Analysis
– If BC is connected at tc = 1.5s, the system is unstable:
x(tc)
x(t) x(t)
xs2
x(t)=x(0)=xs1 x(0)=xs1 A(xs2)
A(xs1) x(tf)=xs1
tc
tf
33
Direct Methods
• Time domain analysis is expensive, so direct
stability analysis technique have been proposed
based on Lyapunov’s stability theory.
• The idea is to define an “energy” or Lyapunov
function ϑ(x,xs) with certain characteristics to
obtain a direct “measure” of the stability region
A(xs) associated with the post-contingency s.e.p.
xs.
• A system’s energy is usually a good Lyapunov
function, as it yields a stability “measure”.
34
Direct Methods
• The rolling ball example can be used to explain the basic
ideas behind these techniques:
– There are 3 equilibrium points: one stable (“valley” bottom), 2
unstable (“hill” tops).
35
Direct Methods
– The energy of the ball is a good Lyapunov or Transient
Energy Function (TEF):
– The potential energy at the s.e.p. is zero, and presents
local maxima at the u.e.p.s (WP1 and WP2 ).
– The “closest” u.e.p. is u.e.p.1, since WP1 < WP2.
36
Direct Methods
– The stability of this system can then be
evaluated using this energy:
• If W < WP1 , the ball remains in the “valley”, i.e. the
system is stable, and will converge to the s.e.p. as
t → ∞.
• If W > WP1 , the ball might or might not converge to
the s.e.p., depending on friction (inconclusive test).
• When the ball’s potential energy WP(t) reaches a
maximum with respect to time t, the system leaves
the “valley”, i.e. unstable condition.
37
Direct Methods
– The “valley” would correspond to the stability
region when friction is “large”.
– In this case, the stability boundary ∂A(xs)
corresponds to the “ridge” where the u.e.p.s
are located and WP has a local max. value.
– The smaller the friction in the system, the
larger the difference between the ridge and
∂A(xs).
– For zero friction, ∂A(xs) is defined by WP1.
38
Direct Methods
• The direct stability test is only a sufficient
but not necessary test:
ϑ(x,xs) < c ⇒ x ∈ A(xs)
ϑ(x,xs) > c ⇒ Inconclusive!
where the value of c is usually a
associated with a local maximum of a
“potential energy” function.
39
Direct Methods
• For the simple generator-infinite bus
example, neglecting limits and AVR:
40
Direct Methods
– The kinetic energy in this system is defined as:
– And the potential energy is:
where δs is the s.e.p. for this system.
41
Direct Methods
– With WP presenting a very similar profile as the rolling
ball example:
42
Direct Methods
– Hence, the system Lyapunov function or TEF is:
– Thus, using similar criteria as in the case of the rolling ball:
• If TEF < WP1 ⇒ System is stable.
• If TEF > WP1 ⇒ Inconclusive for D > 0 (“friction”)
Unstable for D = 0 (unrealistic)
43
Direct Methods
– This is identical to comparing “areas” in the PG vs. δ
graph (Equal Area Criterion or EAC):
44
Direct Methods
Thus, comparing the “acceleration” area:
versus the “deceleration” area:
• If Aa < Ad ⇒ System is stable at tc.
• If Aa > Ad ⇒ Inconclusive for D > 0
Unstable for D = 0 (unrealistic)
45
Direct Methods
• Example: A 60 Hz generator with a 15% transient
reactance is connected to an infinite bus of 1 p.u.
voltage through two identical parallel transmission lines
of 20% reactance and negligible resistance. The
generator is delivering 300 MW at a 0.9 leading power
factor when a 3-phase solid fault occurs in the middle of
one of the lines; the fault is then cleared by opening the
breakers of the faulted line.
1. Assuming a 100 MVA base, determine the critical clearing time
for this generator if the damping is neglected and its inertia is
assumed to be H=5s.
2. Assuming a D = 0.1 s determine the actual critical clearing time.
46
Direct Methods
– Pre-contingency or initial conditions:
where:
47
Direct Methods
48
Direct Methods
– Fault conditions:
where, using a Y-∆ circuit transformation due to the
fault being in the middle of one of the parallel lines:
49
Direct Methods
50
Direct Methods
– Post-contingency conditions:
51
Direct Methods
52
Direct Methods
– During the fault:
53
Direct Methods
– Integrating these equations numerically for
δ(0) = δSpre = 28.82o:
δ(tcc) ≈ 81o
δSpre
tcc ≈ 0.18 s
54
Direct Methods
– For D = 0.1 and a clearing time of tc = 0.27s,
the system is stable:
55
Direct Methods
– For clearing time of tc = 0.28s, the system is
unstable; hence, tcc ≈ 0.275s:
56
Direct Methods
• Generator-motor, i.e. system-system, cases may
also be studied using the EAC method based on
an equivalent inertia M = M1 M2/(M1+M2), and
damping D= M D1/M1 = M D2/M2.
• For the generator-load example neglecting the
internal generator impedance and assuming an
“instantaneous” AVR:
57
Direct Methods
– The “energy” functions, with or without generator limits,
can be shown to be:
– The stability of this system can then be studied using the
same “energy” evaluation previously explained for TEF =
ϑ(x,xo) = WK +WP.
58
Direct Methods
– Thus, for V1 = 1, XL = 0.5, Pd = 0.1, and Qd = 0.25 Pd, the
potential energy well WP(δ,V2) that basically defines the stability
region with respect to the s.e.p. is:
u.e.p.
(saddle)
*
WPmax
s.e.p. 59
(node)
Direct Methods
– Simulating the critical contingency XL = 0.5 → 0.6 for Pd = 0.7 and
neglecting limits, the “energy” profiles are:
tcc
– The “exit” point on ∂A(xs) is approximately at the max. potential energy
point; thus, the critical clearing time is: tcc ≈ 1.42s. A similar value can be
obtained through trial-and-error.
60
Direct Methods
• The advantages of using Lyapunov functions are:
– Allows limited stability analysis.
– Can be used as an stability index.
• The problems are:
– Lyapunov functions are model dependent; in practice, only
approximate “energy” functions can be found.
– Inconclusive if test fails.
– The post-perturbation system state must be known ahead
of time, as the energy function is defined with respect to
the corresponding s.e.p.
• Can only be used as an “approximate” stability
analysis tool.
61
Transient Stability (TS)
Applications
• In practice, transient stability studies are carried
out using time-domain trial-and-error techniques.
• These types of studies can now be done on-line
even for large systems.
• The idea is to determine whether a set of
“realistic” contingencies make the system
unstable or not (contingency ranking), and thus
determine maximum transfer limits or ATC in
certain transmission corridors for given
operating conditions.
62
TS Applications
• Thus, the maximum loadability of the system may be affected by the “size”
of the stability region, leading to the definition of a “true” ATC value:
Operating Worst contingency
point (N-1 criterion)
ETC ATC TRM
TTC
Point at which contingency
leads to instability, as pre-
contingency operating point
is not in the stability region
of the contingency
equilibrium point.
63
TS Applications
• Critical clearing times are not really an issue with
current fast acting protections.
• Simplified direct methods such as the “Extended
Equal Area Criterion” (Y. Xue et al., “Extended
Equal Area Criterion Revisited,” IEEE Trans.
Power Systems, Vol. 7, No. 3 , Aug. 1992, pp.
1012-1022) have been proposed and tested for
on-line contingency pre-ranking, and are being
implemented for practical applications through
an E.U. project.
64
TS Applications
• An example of an application of transient
analysis techniques can be found in L. S. Vargas
and C. A. Cañizares, “Time Dependence of
Controls to Avoid Voltage Collapse,” IEEE
Transactions on Power Systems, Vol. 15, No. 4,
November 2000, pp. 1367-1375.
• This paper discusses the May 1997 voltage
collapse event of the main power system in
Chile:
65
TS Applications
• Main system
characteristics:
SING
Extension: 756 626 km² (800 km)
Inhabitants: 14.5 Mill.
National consumption: 33531
GWh
SIC
National peak load: 5800 MW (2200 km)
Installed capacity: 8000 MW
Frequency: 50 Hz
Trans. level: 66/110/154
/220/500 kV AISEN
Four interconnected systems:
SING, SIC, AISEN,
MAGALLANES MAGALLANES
66
TS Applications
Electricity Demand in Chile
40,000
Santiago
35,000 Extremely dry season
30,000
Others
25,000
GWh
Coal
20,000
Gas
15,000
Hydro
10,000
5,000
0
94
95
96
97
98
99
00
01
02
03
04
05
19
19
19
19
19
19
20
20
20
20
20
20
67
TS Applications
Coal Generators
20% of Load
Main Load Center
50% of Load
Gas & Combined
Cycle Generators 30% of Load
Hydro Generators
68
TS Applications
• Initial state of SIC system:
– 2500 MW load
– Power flow south-north
near 1000 MW (900 MW
through 500 kV lines and
100 MW through 154 kV
lines).
• Events:
1. Line 154 kV trips. 3
2. Major generator in the
south hits reactive limits
and losses voltage control.
3. Operator tries to recover
falling voltages by 1
connecting a capacitor bank
near Santiago.
69
TS Applications
• The line trip and generator limits yield a voltage collapse associated
with a limit-induced bifurcation problem:
Time trajectories
70
TS Applications
PV curves
71
TS Applications
• The connection of the capacitor bank after the generator limits are reached
did not save the system, as the “faulted” system trajectories had “left” the
stability region of the post-contingency operating point:
72
TS Applications
• If the capacitor bank is connect before the generator limits are reached, the
system would have been saved, as the “faulted” system trajectories were
still within the stability of the post-contingency operating point:
73
