postfault stability analysis in power system

Cite this article as: Latiki, L., Medjdoub, A., Taib, N. "Critical Clearing Time and Angle for Power Systems Postfault Stability Assessment", Periodica
Polytechnica Electrical Engineering and Computer Science, 66(3), pp. 277–285, 2022. https://doi.org/10.3311/PPee.19858
https://doi.org/10.3311/PPee.19858
Creative Commons Attribution b|277
Periodica Polytechnica Electrical Engineering and Computer Science, 66(3), pp. 277–285, 2022
Critical Clearing Time and Angle for Power Systems Postfault
Stability Assessment
Lounis Latiki1*
, Abdallah Medjdoub1
, Nabil Taib2
1
Laboratoire de Génie Electrique de Bejaia (LGEB), Faculté de Technologie, Université de Bejaia, 06000 Bejaia, Algeria
2
Laboratoire de Technologie Industrielle et de l'Information (LTII), Faculté de Technologie, Université de Bejaia, 06000 Bejaia,
Algeria
* Corresponding author, e-mail: lounis.latiki@univ-bejaia.dz
Received: 16 January 2022, Accepted: 29 April 2022, Published online: 08 June 2022
Abstract
Transient stability analysis is a very important tool to deal with many behaviors of electrical power systems during and after being
subjected to various disturbances. this paper propose a method for electrical power systems transient stability assessment using
phase plane trajectories. A methodology for computing the critical stability conditions of generators is proposed. The critical conditions
such as critical clearing time (CCT) and critical clearing angle (CCA) were obtained. The computation of CCA and CCT is curried out
step by step using the characteristics of the faulted and postfault trajectories from given initial conditions until their intersection
point. The angle and time values founded represent, by definition, the critical conditions of the system. The proposed algorithm can
be used for complex models since it is based on solving systems of differential equations by iterative methods in the phase plane.
The advantage provided by this method is it's accurate and small time consuming. To demonstrate the effectiveness of the proposed
method, first, critical conditions calculation procedures are given, then the process used in judging power system stability is provided,
finally, simulation results for various test cases of a single machine infinite bus (SMIB) system highlight the proposed methodology.
Keywords
transient stability, phase plane analysis, modified euler's mehod, critical clearing time, critical clearing angle
1 Introduction
Power system stability is the ability of an electric power
system, at a given initial operating state, to restore a state
of stable operating equilibrium after being subjected to
some disturbances. The main kinds of these disturbances
are a changing in loads, a changing in network configura-
tion and a severe faults in the system. In other hand, we can
define power system instability as a loss of synchronism
when he is subjected to a particular disturbance. The main
objective of the stability analysis in power system is to
keep the whole system intact by ensuring the accuracy of
transfer capability of transmission lines and identifying the
potential disturbances that could lead to instabilities.
Stability analysis of power systems involves the compu-
tation of the nonlinear transient dynamic trajectory of the
postfault system, which depends on the initial operating
conditions, the nature and duration as well as magnitude
of the perturbation.
The rotor angle deviation of the synchronous machine
during transient period is used as an index to assess its
ability to maintain or restore equilibrium between electro-
magnetic and mechanical torques by analyzing the electro-
mechanical oscillations inherent in power systems. Several
classes of methods are developed to obtain transient stabil-
ity limits of power systems. Time-domain simulation (TDS)
methods via numerical integration are largely used in tran-
sient stability study. Numerical methods, by solving the
second order nonlinear differential swing equation, using
time-domain numerical integration are the most accurate
and very efficient given their ability to analyze very com-
plex nonlinear mathematical models. This is, because they
take into account all the phenomena present in the system.
The main drawback of these approaches is that they are
time consuming and require the whole system of equations
for assessing stability of large power systems [1–4].
Alternatively, Lyapunov's direct methods (DM), have been
proposed in many research papers [5–10]. The main applica-
tion of these class of methods is to check whether the post-
fault trajectory will converge to an acceptable steady-state

278|Latiki et al.
Period. Polytech. Elec. Eng. Comp. Sci., 66(3), pp. 277–285, 2022
as time proceeds. By using positive-definite Lyapunov func-
tions, these methods, assess stability region of the post-
fault equilibrium without resorting to numerical integration.
Another advantage is that they can evaluate a large scale
power systems stability via total energy present in the sys-
tem using transient energy functions (TEF) [7–10]. However,
these methods suffer from certain drawbacks which have
posed challenges in several researches, such as the absence
of an analytical procedure to find the appropriate TEF and
the critical energy values, which exist only for a small and
limited category of mathematical models. This insufficiency
affect the accuracy of stability assessment.
New development ways are to combine direct method
and time-domain method into an integrated power system
stability program to take the merits of both methods [11–13].
This combination known as hybrid methods, is used for
stability assessment by computing the actual trajectory
using TDS then evaluating the TEF for decision making.
These type of methods are faster than TDS in computation
time and more accurate than DM in estimating stability
margin, but they are relatively slow compared with DM.
Other methods like machine learning (ML) methods have
been proposed for power system transient stability analysis.
In [14], a power system stability is performed via artificial
neural network algorithm, where the TDS and lots parameter
settings are needed for training data acquisition. However,
there is no way to escape from the excessive training time
and complex parametrizations generated by such method.
These methods are complex and time consuming [14–17].
In recent years, phase plane trajectory analysis (PPTA)
methods have been proposed for power system transient sta-
bility analysis [18–21].
In this paper, a power systems transient stability assess-
ment using phase plane trajectories is presented. This
technique use both faulted and critical postfault trajecto-
ries to compute critical conditions such as CCT and CCA.
It is well known that the phase plane representation is
a graphical method used to study the solutions of second
order nonlinear differential equations. The principle of this
method is to visualize the solutions of the equations in the
phase plane, which correspond to system trajectories for dif-
erentinitialconditions.Theobtainedtrajectoriesarecalledthe
phase portrait of the system and have important proprieties.
1.1 Mathematical basic concepts
Considering the autonomous nonlinear second order sys-
tem expressed by Eq. (1):
 
x f x x

, � . (1)
Described in the state representation by Eq. (2):


x x
x f x x
1 2
2 1 2

,
, (2)
where:
x x
1 = , (3)
x x
2 1
=  , (4)
are the state variables.
The solution of Eq. (2) is plotted in a graphic whose
horizontal and vertical coordinates are x1
, x2
, respectively.
A such graphic is called phase plane. This representation
can be found with Eq. (5) derived from Eq. (2) by dividing
the second over the first equations.
dx
dx
f x x
f x x
2
1
1 1 2
2 1 2

,
,
, (5)
where:
f x x f x x
1 1 2 1 2
, ,
, (6)
f x x x
2 1 2 2
,
. (7)
The system Eq. (2) can have more than one equilibrium point
which are also singular points of Eq. (5), and which verify:
 
x x f x x f x x
1 2 1 1 2 2 1 2
0 0

or , , .
For each equilibrium point, the behavior of the system
Eq. (2) in the neighborhood of equilibrium points is deter-
mined by analyzing the eigenvalues λ1
, λ2
and eigenvectors
V1
, V2
of the linearized system given by Eq. (8). For more
details see [20].
J
k x x
x
k x x
x
k x x
x
k x x

1 1 2
1
1 1 2
2
2 1 2
1
2 1 2
2
, ,
, ,
x

(8)
For the stable equilibrium point, the eigenvalues
obtained are both complex with negative real part for the
damped systems, and imaginary with zero real parts for the
undamped systems. For the unstable point, the eigenvalues
obtained λ1
, λ2
are real with opposite signs [22].
Eigenvectors obtained from linearized system have very
important characteristic which determine the direction and
represents the slopes of trajectories in a local small neigh-
borhood of equilibrium points, where those corresponding
to positive eigenvalues are called unstable whereas those
corresponding to negative eigenvalues are called stable
ones which will be used further as initial conditions.

Latiki et al.
Period. Polytech. Elec. Eng. Comp. Sci., 66(3), pp. 277–285, 2022|279
2 Main procedures of the proposed technique
This section aims to determine stability limits of an elec-
tric power system modelled by Eq. (2). These latters are
obtained from solving Eq. (5) for two possible situations
a and b that the system could be in, during fault and after
fault respectively.
For situation a:
dx
dx
g x x
2
1
1 2

, . (9)
For situation b:
dx
dx
h x x
1
2
1 2

, , (10)
g and h are parameterized functions of the situations a and
b respectively.
In power systems, the intersection of the solutions
expressed by equations Eq. (9) and Eq. (10) in the phase
plane, which are called further in the next sections faulted
and postfault dynamics, gives us quantitative information
that are sufficient for stability judgement. To obtain the
intersection point (x1
, x2
) shown in Fig. 1, which gives us
the critical conditions such as CCT and CCA that we detail
in Section 4, we compute:
• The two trajectories Cf(x1
), Cp(x1
), curve fault and
postfault respectively, from starting points (x1,i
, x2,i
)
and (x1,j
, x2,j
) to the intersection point (x1,c
, x2,c
).
This procedure is illustrated in Fig. 1, where the suc-
cession of steps numerated from one to four describe the
steps needed to compute Cf(x1
), Cp(x1
) from (x1,i
, x2,i
) and
(x1,j
, x2,j
) until (x1,c
, x2,c
) point by point.
3 Power system modeling and analysis
3.1 Classical model
Consider a SMIB power system given in Fig. 2.
The mathematical model of such system is expressed
with the second order differential equation given by
Eq. (11):
M D P P
m
 

max Sin . (11)
Eq. (11) can be expressed by a set of first order differen-
tial equations given by Eq. (12), with state variables ẟ–ω,
using Eq. (3) and Eq. (4) as follow:



M P P D
m max Sin
, (12)
where:
ẟ: power angle in radians
ω: relative velocity in (radians per seconds)
Pm
: mechanical power input in (pu)
Pmax
: maximum electrical power output of a synchro-
nous generator in (pu)
D: damping coefficient
M: inertia constant of the generator.
The behavior of power system during transient period
is described in three stages as follows:
Stage 01:
Initially (t 0), the power system is operating at its stable
steady state equilibrium conditions and it is expressed with
a set of differential equations given by Eq. (13):



M P P D
m pre Sin
, (13)
where:
Ppre
: maximum power output of the generator before the
fault in (pu).
With the stable and unstable equilibrium points
(( Sin , ), ( Sin , ))

o m
pre
o ou m
pre
ou
P
P
P
P

1 1
0 0
respectively.
Stage 02:
Assume that at t = 0, a three phase short circuit occurs at
any point in the given system. In this situation, the system
is governed by the fault dynamics given by Eq. (14):
x2,i+1
Cf(x1)
x1
x2 Cp(x1)
zoom
x1,i+1
x2,i
x1,i x1,j
x1,j-1
x2,i+1 x2,j+1
x1,i+1
1
2
3
4
Integration step
fixed
zoom
x1,j+1
(x1,C , x2,C )
Intergation step
Slopes at x1,i and x1,j
Fig. 1 Main procedures of the proposed method
Infinite
bus
TL-2
TL-1
Transformer
P
Q
Pm
Generator
E e 0°
BUS 1
BUS 3
BUS 2
X1
X2
Fig. 2 Single machine infinite bus system

280|Latiki et al.



M P P D
m f Sin
, (14)
where:
Pf
: maximum power output of the generator during the
fault in (pu).
Notethattheinitialconditions ( Sin , )

o m
pre
o
P
P

1
0
of this faulted period are the stable operating conditions of
the pre-fault period.
Stage 03:
When the fault is cleared at t = tcl
by action of protective
system operations, the system is governed by the postfault
dynamics given by Eq. (15):



M P P D
m a Sin
, (15)
where:
Pa
: maximum power output of the generator after the fault
in (pu).
With stable and unstable equilibrium points
( Sin , ), ( Sin , )

s m
a
s u m
a
u
P
P
P
P

1 1
0 0
respectively.
The initial conditions (δ = δcl
, ω = ωcl
) of this postfault
period are the angle and velocity of the generator at the
instant of clearing of the fault.
3.2 Postfault system stability analysis
In order to check either the postfault system is stable or
not, let's define:
• δcr
and ωcr
as the critical angle and critical veloc-
ity, given by the horizontal and vertical coordinates
respectively, obtained from the intersection of the
critical postfault and fault trajectories represented in
the phase plane portrait as shown in Fig. 3.
• The critical trajectory of the postfault system gov-
erned by the postfault dynamics is the path linking
the two points (δcr
, ωcr
), (δu
, ωu
).
According to the representation above, the postfault
system is:
Stable if: δcr
δcl
, case where δcl
= δcl1
.
Unstable if: δcr
δcl
, case where δcl
= δcl3
.
Critical if: δcr
= δcl
.
4 Algorithm of the method
The proposed technique uses an algorithm based on
Modified Euler's Method which compute critical condi-
tions simultaneously, (CCT) and (CCA), using the phase
plane plot of during fault and postfault conditions.
4.1 Phase plane plot of during fault conditions
The faulted period is characterized by:
• The maximum transfer power of the generator (Pf
)
obtained from the change of the network configura-
tion during the fault conditions.
• Initial conditions (δ°, ω°) which are steady state sta-
ble operating conditions of the pre-fault period.
The fault trajectory is obtained step by step using
Eq. (16) built from Eq. (14) in the same manner as Eq. (5):
d
d
M
P P D
m f

Sin
(16)
For a small increment
f
i 1
defined by Eq. (17),
f
i 1
is obtained by applying Modified Euler's method as follow:

f
i
f
i
f
i
1 1
(17)

f
i f
i
d
d
d
d
f
i
f
i
f
i
f
i

1
1
2 1
1
1

(18)

f
i
f
i
f
i
1 1
(19)
4.2 Phase plane plot of the postfault conditions
When the fault is cleared, a change in system configu-
ration occurs and a new maximum transfer power Pa
is
obtained. Then the postfault trajectory is computed step
by step using Eq. (20) built from Eq. (15) in the same man-
ner as Eq. (5).
d
d
P P D
M
m a

Sin
(20)
ẟ (radian)
ẟ𝑢𝑢
ω
(radian/s)
ẟ𝑜𝑜 ẟ𝑠𝑠 ẟ𝑐𝑐𝑐𝑐1ẟ𝑐𝑐𝑐𝑐ẟ𝑐𝑐𝑐𝑐3
Postfault trajectories
Fig. 3 Phase plane trajectories representation

Latiki et al.
For a small increment
a
i 1
defined by Eq. (21),
a
i 1
is obtained by applying Modified Euler's method as follow:

a
i
a
i
a
i
1 1 (21)

a
i a
i
d
d
d
d
a
i
a
i
a
i
a
i

1
1
2 1
1
1

(22)

a
i
a
i
a
i
1 1 (23)
The critical trajectory of the postfault system is plotted
from
a
i u
a
i u

, to (δcr
, ωcr
) with the set of initial
conditions (δu
, ωu
) and d
d u
u

,
where:
d
d u
u

represent the slope of the stable eigenvector obtained
from the linearization of the postfault system around the
saddle equilibrium point (δu
, ωu
).
4.3 CCT calculation from trajectories
During the fault time period, CCT is obtained step by step
from the plot of the faulted trajectory using Eq. (24):
dt
d

(24)
for a small increment
f
i 1
defined by Eq. (19), an inter-
val of time
tf
i 1
is obtained by Eq. (25):

tf
i f
i
f
i
f
i

1
1
1
2

, (25)
where
f
i
f
i

1
2 is the average velocity in the interval

f
i 1
, then the time is given by Eq. (26):
t t t
f
i
f
i
f
i

1 1
. (26)
The procedures for computing critical conditions are
resumed as follows:
First, negatif
a
i 1
is fixed and then
a
i 1
is obtained
by Eq. (22). The increment a
i1
is obtained from Eq. (23).
Second, the obtained increment
a
i 1
is used as an
increment
f
i 1
in Eq. (18) then
f
i 1
is obtained. The
increment
f
i 1
is obtained from Eq. (19).
Finally, the time is obtained directly by Eq. (25) and Eq. (26).
The process is stopped at iteration n when ( )

a
i n
f
i n

achieve its minimum positive value, then the critical condi-
tions are directly obtained and compared to the real clearing
time tcl
for stability judgment. In Fig. 4, a flow chart of the
proposed algorithm shows how the stability decision is taken.
5 Simulation results and analysis
Fig. 5 shows SMIB power system model taken from [23]
used to test the proposed algorithm. It consists of four
555 MVA, 24 kV, 60 Hz units supplying power to an
infinite bus through two transmission lines.
The network reactances shown in Fig. 5 are expressed in
per unit on 2220 MVA, 24 kV base with resistances neglected.
The generators are modeled as SMIB supplying power
to an infinite bus with following parameters expressed in
per units on 2220 MVA, 24 kV base. The SMIB system
parameters are listed in Table 1.
The initial operating conditions of the system are also
given in per unit on 2220 MVA, 24 kV base by:
Find new configuration
network parameter Pf
during the fault period
Find the new parameters
𝑃𝑃𝑎𝑎,ẟ𝑢𝑢
,𝜔𝜔𝑢𝑢
𝑎𝑎𝑎𝑎𝑎𝑎 (
𝑑𝑑𝑑𝑑
𝑑𝑑ẟ
)ቚẟ𝑢𝑢
𝜔𝜔𝑢𝑢
of
the postfault period
Find the critical conditions
ẟ𝑐𝑐𝑐𝑐,𝜔𝜔𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑐𝑐𝑐𝑐
Stable system
Unstable system
Fault ?
At t=tcl the
fault is
cleared
Receive :
- Voltage (V)
- Current(I)
- Power angle (ẟ𝑜𝑜
)
Is tcl tcr
YES
End
Start
NO
YES
NO
NO
YES
Fig. 4 Flow chart of the proposed algorithm
4X555 MVA LV
HV
Infinite
bus
j0.93
j0.5
j0.15
F1 F2
P
Q
Et
EB
Fig. 5 SMIB test system

282|Latiki et al.
P = 0.9 pu, Q = 0.436 pu (overexcited), terminal bus and
infinite bus voltages are Ẽt
= 1.0, ẼB
= 0.90081 with angles
28.23°, 0° respectively.
The algorithm is tested for two faults locations F1 and
F2, with and without damping coefficient D and the crit-
ical conditions are chosen as the limit of stability condi-
tions, beyond this limit the system will be unstable.
The obtained results are compared to those obtained from
conventional fourth order Runge-Kutta simulation method.
In Tables 2–7, the obtained values of CCTs and CCAs
are listed for different fault locations, damping coefficient
values D and number of iterations n. The CCT and CCA
obtained by the Runge-Kutta method is also given for
comparison purposes, and for each case study, the phase
plane trajectories obtained by both the proposed technique
(PM) and Runge-Kutta method (RK) used to find CCA
and CCT are given in Figs. 6–11.
Case 01
Fault location at F1
• for period during fault: δ° = 0.7291 rad, ω° = 0 rad
per second with maximum power transfer Pf
= 0 pu.
• For the postfault period: δu
= π – δs
= 2.1864 rad,
ωu
= 0, Pa
= 1.1024 pu and
d
d u
u

= –5.8604, where
d
d u
u

is the stable eigenvector of the linearized sys-
tem around the equilibrium (δu
, ωu
).
• The Step of integration is obtained by dividing
the interval [δ°, δu
] to n + 1 points and is given by
Δδa
= – (δu
– δ°)/n.
Case 02
Fault location at F2
For period during fault: δ° = 0.7291 rad, ω° = 0 rad per sec-
ond with maximum power transfer Pf
= 0.7307 pu.
For the postfault period: δu
= π – δs
= 2.1864 rad, ωu
= 0,
Pa
= 1.1024 pu and
d
d u
u

= –5.8604.
Table 4 CCT and CCA of a SMIB test in the case of F1 and D = 0.02
Critical
conditions
Proposed method
RK- 4th
order
Distance (PM vs RK)
n1
=
100
n2
=
1000
n3
=
9000
∆t =
10−7
n1
vs
∆t
n2
vs
∆t
n3
vs
∆t
ẟcr
(radian) 1.0090 1.0107 1.0108 1.0075 0.0015 0.0032 0.0033
tcr
(second) 0.1078 0.1097 0.1099 0.1089 0.0011 0.0008 0.0010
Table 5 CCT and CCA of a SMIB test in the case of F2 and D = 0
Critical
conditions
Proposed method
RK- 4th
order
Distance (PM vs RK)
n1
=
100
n2
=
1000
n3
=
9000
∆t =
10−7
n1
vs
∆t
n2
vs
∆t
n3
vs
∆t
ẟcr
(radian) 1.2052 1.2128 1.2128 1.2126 0.0074 0.0002 0.0002
tcr
(second) 0.2142 0.2196 0.2199 0.2199 0.0057 0.0003 0.0000
Table 2 CCT and CCA of a SMIB test in the case of F1 and D = 0
Critical
conditions
Proposed method
RK- 4th
order
Distance (PM vs RK)
n1
=
100
n2
=
1000
n3
=
9000
∆t =
10–7
n1
vs
∆t
n2
vs
∆t
n3
vs
∆t
ẟcr
(radian) 0.9119 0.9118 0.9118 0.9117 0.0002 0.0001 0.0001
tcr
(second) 0.0851 0.0867 0.0868 0.0868 0.0017 0.0001 0.0000
Table 3 CCT and CCA of a SMIB test in the case of F1and D = 0.01
Critical
conditions
Proposed method
RK- 4th
order
Distance (PM vs RK)
n1
=
100
n2
=
1000
n3
=
9000
∆t =
10–7
n1
vs
∆t
n2
vs
∆t
n3
vs
∆t
ẟcr
(radian) 0.9595 0.9596 0.9597 0.9584 0.0011 0.0012 0.0013
tcr
(second) 0.0966 0.0982 0.0984 0.0979 0.0013 0.0003 0.0005
Table 1 SMIB data
Equipment Parameter Value
Generators
S (MVA) 4X555
Rating voltage (kV) 24
Direct axis transient reactance Xd
' (pu) 0.3
Inertia constant H (MW s/MVA) 3.5
Frequency (Hz) 60
Transformers Reactances (pu) 0.15
Transmission
lines
TL-1 Reactance (pu) 0.5
TL-2 Reactance (pu) 0.93
Critical
conditions
Proposed method
RK- 4th
order
Distance (PM vs RK)
n1
=
100
n2
=
1000
n3
=
9000
∆t =
10−7
n1
vs
∆t
n2
vs
∆t
n3
vs
∆t
ẟcr
(radian) 1.3031 1.3155 1.3168 1.3089 0.0058 0.0066 0.0079
tcr
(second) 0.2442 0.2507 0.2514 0.2484 0.0042 0.0023 0.0030
Critical
conditions
Proposed method
RK- 4th
order
Distance
(PM vs RK)
n1
=
100
n2
=
1000
n3
=
9000
∆t =
10−7
n1
vs
∆t
n2
vs
∆t
n3
vs
∆t
ẟcr
(radian) 1.3944 1.4188 1.4192 1.4025 0.0081 0.0163 0.0167
tcr
(second) 0.2742 0.2838 0.2842 0.2774 0.0032 0.0064 0.0072

Latiki et al.
Simulation results obtained in Tables 2–7 show that the
critical conditions obtained by the proposed algorithm are
not conservative and they are very closest to those obtained
by the conventional RK 4th
order numerical simulation
method with very small integration step.
It is well known that, numerical methods perform high
accuracy when the integration step is relatively small. For this
purpose and to show the effectiveness of the proposed method,
the results are discussed for RK 4th
integration step ∆t of 10−7
and for the proposed method iterations number n of 9000.
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
1
2
3
4
5
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
0.05
0.1
0.15
t
(seconds)
t-PM trajectory
t-RK trajectory
CCA-PM = 0.9118 radian
CCA-RK = 0.9117 radian
CCT-RK = 0.0868 s
CCT-PM = 0.0867 s
Fig. 6 Phase plane trajectories used to find CCA and CCT
in the case of F1 and D = 0
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
1
2
3
4
5
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
0.05
0.1
0.15
t
(seconds)
t-PM trajectory
t-RK trajectory
CCT-PM = 0.0982 s
CCT-RK = 0.0979 s
in the case of F1 and D = 0.01
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
1
2
3
4
5
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
0.05
0.1
0.15
t
(seconds)
t-PM trajectory
t-RK trajectory
CCT-PM = 0.1097 s
CCT-RK = 0.1089 s
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
1
2
3
4
5
6
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
t
(second)
t-PM trajectory
t-RK trajectory
CCT-PM = 0.2196 s
CCT-RK = 0.2199 s
in the case of F2 and D = 0

284|Latiki et al.
In Tables 2 and 5, where the coefficient D is set to zero;
and for the two different fault locations F1 and F2, the CCTs
obtained by both methods are 0.0868 seconds and 0.2199 sec-
onds, respectively. For the same situation, the CCA obtained
by the two methods are slightly different and distant from
each other around 10−4
radians and the data presented in
Tables 3–4 and Tables 6–7 shows that as the damping coeffi-
cient increases, the difference in CCT and CCA values pro-
vided by both methods increases but it remains weak.
Figs. 6–11 show the time t-PM and the relative velocity
ω-PM superposed trajectories obtained by proposed method
compared to t-RK and ω-RK obtained by the conventional
RK 4th
order simulation method, where the plots show crit-
ical conditions, CCA and CCT obtained by both proposed
method CCA-PM and CCT-PM, and Runge-Kutta method
CCA-RK and CCT-RK respectively. It is clearly shown that
these critical conditions are very closest to each other.
6 Conclusion
Transient stability analysis is a very useful tool to avoid
the instabilities in power systems networks after being sub-
jected to disturbances. In this paper, a power system tran-
sient stability assessment using phase plane representation
is proposed. The approach uses phase plane faulted and
postfault trajectories to obtain the critical conditions CCA
and CCT simultaneously. The proposed method doesn't
need additional time after clearing time to confirm if the
system remains stable or not after being subjected to per-
turbation. This is because the two trajectories are also com-
puted simultaneously using characteristics of the stable
and unstable equilibriums of the system studied. In other
words, the clearing time of the real system is compared to
the CCT computed by the method then decision is taken.
The critical conditions obtained are very closest to
those obtained by RK 4th
order conventional simulation
method with and without integrating damping coefficient.
Also, obtained results, show that the damping coefficient
has an important effect on transient stability improvement
which can't be shown by direct methods, moreover, the
proposed methodology is faster than time domain simu-
lation methods in computational time and most accurate
than direct methods which perform conservative results.
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
1
2
3
4
5
6
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
t
(seconds)
t-PM trajectory
t-RK trajectory
CCT-PM = 0.2507 s
CCT-RK = 0.2484 s
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
2
4
6
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
(radian)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
t
(seconds)
t-PM trajectory
t-RK trajectory
CCT-PM = 0.2838 s
CCT-RK = 0.2774 s

Latiki et al.
References
[1] Sauer, P. W., Pai, M. A., Chow, J. H. Power System Dynamics and
Stability, Wiley, West Sussex, UK, 2017. ISBN 978-1-119-35577-9
[2] Dong, Y., Pota, H. R. Fast transient stability assessment using large
step-size numerical integration, IEE Proceedings C (Generation,
Transmission and Distribution), 138(4), pp. 377–383, 1991.
https://doi.org/10.1049/ip-c.1991.0047
[3] Padhi, S., Mishra, B. P. Numerical Method Based Single Machine
Analysis for Transient Stability, International Journal of Emerging
Technology and Advanced Engineering, 4(2), pp. 330–335, 2014.
[4] Su, F., Zhang, B., Yang, S., Wang, H. A Novel Termination
Algorithm Of Time-Domain Simulation For Power System
Transient Stability Analysis Based On Phase-Plane Trajectory
Geometrical Characteristic, In: IEEE 5th International
Conference on Electric Utility Deregulation and Restructuring
and Power Technologies, Changsha, China, 2015, pp. 1422–1426.
https://doi.org/10.1109/DRPT.2015.7432455
[5] Owusu-Mireku, R., Chiang, H.-D. A Direct Method for the
Transient Stability Analysis of Transmission Switching Events,
In: 2018 IEEE Power and Energy Society General Meeting
(PESGM), Portland, OR, USA, 2018, pp. 1–5.
https://doi.org/10.1109/PESGM.2018.8586242
[6] Shuai, Z., Shen, C., Liu, X., Li, Z., Shen, Z. J. TransientAngle Stability
of Virtual Synchronous Generators Using Lyapunov's Direct Method,
IEEE Transactions on Smart Grid, 10(4), pp. 4648–4661, 2019.
https://doi.org/10.1109/TSG.2018.2866122
[7] Amte, A. Y., Kate, P. S. Automatic generation of Lyapunov
function using Genetic programming approach, In: 2015 IEEE
International Conference on Energy Systems and Applications,
Pune, India, 2015, pp. 771–775.
https://doi.org/10.1109/ICESA.2015.7503454
[8] Rimorov, D., Wang, X., Kamwa, I., Joós, G. An Approach to
Constructing Analytical Energy Function for Synchronous
Generator Models with Subtransient Dynamics, IEEE
Transactions on Power Systems, 33(6), pp. 5958–5967, 2018.
https://doi.org/10.1109/TPWRS.2018.2829887
[9] Han, D., El-Guindy, A., Althoff, M. Power Systems Transient
Stability Analysis via Optimal Rational Lyapunov Functions, In:
2016 IEEE Power and Energy Society General Meeting, Boston,
MA, USA, 2016, pp. 1–5.
https://doi.org/10.1109/PESGM.2016.7741322
[10] Wu, D., Ma, F., Jiang, J. N. A study on relationship between power
network structure and total transient energy, Electric Power
Systems Research, 143, pp. 760–770, 2017.
https://doi.org/10.1016/j.epsr.2016.10.019
[11] Veerashekar, K., Schuehlein, P., Luther, M. Quantitative transient
stability assessment in microgrids combining both time-domain
simulations and energy function analysis, International Journal of
Electrical Power and Energy Systems, 115, 105506, 2020.
https://doi.org/10.1016/j.ijepes.2019.105506
[12] Kyesswa, M., Çakmak, H. K., Gröll, L., Kûhnapfel, U.,
Hagenmeyer, V. A Hybrid Analysis Approach for Transient
Stability Assessment in Power Systems, In: 2019 IEEE Milan
PowerTech, Milan, Italy, 2019, pp. 1–6.
https://doi.org/10.1109/PTC.2019.8810745
[13] Bonvini,B.,Massucco,S.,Ivlorini,A.,Siewierski,T.AComparative
Analysis of Power System Transient Stability Assessment by
Direct and Hybrid Methods, In: Proceedings of 8th Mediterranean
Electrotechnical Conference on Industrial Applications in
Power Systems, Computer Science and Telecommunications
(MELECON 96), Bari, Italy, 1996, pp. 1575–1579.
https://doi.org/10.1109/MELCON.1996.551253
[14] Shahzad, U. Probabilistic Transient Stability Assessment of Power
Systems Using Artificial Neural Networks, Journal of Electrical
Engineering, Electronics, Control and Computer Science, 8(1),
pp. 35–42, 2022.
[15] Guo, T., Milanović, J. V. Probabilistic Framework for Assessing
the Accuracy of Data Mining Tool for Online Prediction of
Transient Stability, IEEE Transactions on Power Systems, 29(1),
pp. 377–385, 2014.
https://doi.org/10.1109/TPWRS.2013.2281118
[16] Li, Y., Yang, Z. Application of EOS-ELM With Binary Jaya-Based
Feature Selection to Real-Time Transient Stability Assessment
Using PMU Data, IEEE Access, 5, pp. 23092–23101, 2017.
https://doi.org/10.1109/ACCESS.2017.2765626
[17] Huang, D., Yang, X., Chen, S., Meng, T. Wide-area measurement
system-based model-free approach of Postfault rotor angle trajectory
prediction for on-line transient instability detection, IET Generation,
Transmission and Distribution, 12(10), pp. 2425–2435, 2018.
https://doi.org/10.1049/iet-gtd.2017.1523
[18] Su, F., Zhang, B., Yang, S., Wang, H. Power System First-swing
Transient Stability Detection Based on Trajectory Performance
Of Phase-plane, In: 2016 IEEE PES Asia-Pacific Power and
Energy Engineering Conference (APPEEC), Xi'an, China, 2016,
pp. 2448–2451.
https://doi.org/10.1109/APPEEC.2016.7779926
[19] Priyadi, A., Yorino, N., Eghbal, M., Zoka, Y., Sasaki, Y., Yasuda,
H., Kakui, H. Transient Stability Assessment as Boundary Value
Problem, In: 2008 IEEE Canada Electric Power Conference,
Vancouver, BC, Canada, 2008, pp. 1–6.
https://doi.org/10.1109/EPC.2008.4763304
[20] Shrestha, B., Gokaraju, R., Sachdev, M. Out-of-Step Protection
Using State-Plane Trajectories Analysis, IEEE Transactions on
Power Delivery, 28(2), pp. 1083–1093, 2013.
https://doi.org/10.1109/TPWRD.2013.2245684
[21] Sajadi, A., Preece, R., Milanović, J. Identification of transient stabil-
ity boundaries for power systems with multidimensional uncertainties
using index-specific parametric space, In: International Journal of
Electrical Power and Energy Systems, 123, 106152, 2020.
https://doi.org/10.1016/j.ijepes.2020.106152
[22] Marquez, H. J. Nonlinear Control Systems, Wiley, 2003.
ISBN 978-0-471-42799-5
[23] Kundur, P. Power System Stability and Control, McGraw-Hill,
1994. ISBN 0-07-035958-X

postfault stability analysis in power system

More Related Content

Similar to postfault stability analysis in power system (20)

Recently uploaded (20)

postfault stability analysis in power system