IEEE TRANSACTIONS ON POWER SYSTEMS

Intelligent Time-Adaptive Transient Stability
Assessment System

arXiv:1610.08235v2 [] 21 May 2017

James J.Q. Yu, Member, IEEE, David J. Hill, Fellow, IEEE, Albert Y.S. Lam, Senior Member, IEEE, Jiatao Gu,
Student Member, IEEE, and Victor O.K. Li, Fellow, IEEE

Abstract—Online identification of post-contingency transient
stability is essential in power system control, as it facilitates the
grid operator to decide and coordinate system failure correction
control actions. Utilizing machine learning methods with synchrophasor measurements for transient stability assessment has
received much attention recently with the gradual deployment
of wide-area protection and control systems. In this paper, we
develop a transient stability assessment system based on the
long short-term memory network. By proposing a temporal
self-adaptive scheme, our proposed system aims to balance the
trade-off between assessment accuracy and response time, both
of which may be crucial in real-world scenarios. Compared
with previous work, the most significant enhancement is that
our system learns from the temporal data dependencies of the
input data, which contributes to better assessment accuracy. In
addition, the model structure of our system is relatively less
complex, speeding up the model training process. Case studies
on three power systems demonstrate the efficacy of the proposed
transient stability assessment system.
Index Terms—Transient stability assessment, long short-term
memory, phasor measurement units, voltage phasor, recurrent
neural network.

I. I NTRODUCTION
Transient stability is the ability of a power system to retain
synchronism subject to large disturbances [1]. It is a significant
concern in power system design and operation as it is among
the major causes of power blackouts in the past [2]. To meet
the ever-increasing growth of power consumption, the system
loading introduced in the system causes power grids to operate
near the transmission capacities. This leads to more serious
transient stability issues caused by disturbances. Therefore,
real-time assessment of post-disturbance transient stability is
still receiving much attention.
Transient stability assessment (TSA) mathematically corresponds to solving a set of high-dimensional non-linear
differential algebraic equations (DAE) [3]. Transient energy
functions have been employed to assess the system stability
since 1980s (see [2]–[5] for some references). However, this
approach has its drawbacks when employed on practical large
power grids due to the model simplifications required. An
alternative of this direct method is fast time-domain (TD)
simulation [6], where a given set of credible contingencies are
simulated in an off-line manner to guide the design and tuning
of the control system. Meanwhile, accurate TD simulations
require complete information of the grid and the disturbance,
The authors are with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong (e-mail:
{jqyu, dhill, ayslam, jiataogu, vli}@eee.hku.hk).

and impose a heavy computational burden. These properties
hinder TD simulations from being employed for on-line TSA
status prediction. Such assessment results can be significantly
different from the actual system response due to the inaccurate
system model parameters and disturbance estimations.
Nowadays, the synchrophasor technology is among the key
measurement methods in power systems. Phasor measurement
units (PMU) significantly outperform the conventional supervisory control and data acquisition (SCADA) system due to
their capability in sampling system variables in synchronism
from dispersed locations, enabling system-level model validation [7]. With the rapid deployment of PMUs, a collection
of methodologies have been proposed for TSA based on the
real-time system operating state. Utilizing the pre- and postcontingency system dynamics, techniques such as piecewise
constant-current load equivalent method [8], emergency single machine equivalent [9], and post-disturbance trajectory
analysis [10] were developed for on-line TSA. While accurate
assessments can be achieved by these methods, their high computational complexities prevent them from being employed in
practical post-contingency TSA.
To realize fast real-time TSA status prediction, machine
learning and fuzzy logic techniques have been widely adopted
as alternative approaches for TSA in recent years, e.g. decision
tree methods [11], artificial neural networks (ANN)/Support
Vector Machine(SVM) [7], [12], and fuzzy knowledge based
systems [13]. Early work [14], [15] was based on decision
trees and learning methods. Different from the conventional
analytical methods, these machine learning methods extract
the relationship between the system parameters and the corresponding stability conditions utilizing predefined transient
stability datasets. Once such relationships are established,
new transient stability cases can be evaluated with minimal
computational efforts. Thus the real-time computing burden of
the system is alleviated and the assessment can be conducted
in an online manner.
Related work has demonstrated the superiority of these
predictive models in constructing TSA systems. As analyzed
in [13], two competing trends are developed for modern
TSA solutions, namely, the fuzzy-logic rule-based approaches
and the machine learning-based approaches. The former has
the advantage of transparency, and the internal structures
and parameters have physical meanings while the latter has
generally better performance. This leads to a trade-off between TSA transparency and accuracy. Meanwhile, since it
is possible for the instability to propagate over the network
within seconds, remedial actions are usually automated. In

IEEE TRANSACTIONS ON POWER SYSTEMS

However, there is still one major research gap in machine
learning-based TSA techniques. Existing work tends to employ a fixed response time scheme for implementation. The
proposed TSA systems have to wait for a fixed period of
time before evaluating the stability status of the grid. As
the dimension and type of input data have been defined,
such implementations can drastically decrease the complexity
of system design. However, as transient stability issues can
quickly propagate over the whole grid, it is always preferred to
have a faster TSA response so that more time can be reserved
for remedial control actions to take effect. Moreover, it is
possible to identify the issues at the very early stage after fault
clearance, and the existing approaches with fixed response
time cannot dynamically adapt to such cases. Therefore, it
is more advantageous to conduct TSA in an adaptive online
manner. A recent work in [16] reported a novel methodology
to make transient stability assessment at the earliest time.
However, as multiple decision making machines are trained
separately for measurements with different timestamps, the
training process discards the potential causal relationship of
one measurement and its descendant ones.
In order to utilize this information for TSA prediction, in
this work we propose a long short-term memory (LSTM) [17]
based method. The main contributions of this work are as
follows:
•

•

•

•

This work is the first attempt to employ recurrent neural
network (RNN) and LSTM for a time-adaptive TSA
process. The proposed system manages to extract both
spatial and temporal data dependency from the input
power system state for security assessment, which is
novel in stability analysis methodologies.
The system framework is significantly easier to implement than previous time-adaptive TSA [16], resulting in
reduced training time and maintenance effort.
The proposed system is assessed against three small
to large scale power system test cases. The simulation
results demonstrate accurate assessment with significantly
reduced system response time.
Parameter sensitivity tests are carried out to evaluate the
performance of the system.

The remainder of this paper is organized as follows. Section
II briefly introduces the LSTM network employed in this work.
Section III elaborates our proposed LSTM-based TSA System
(TSAS), and Section IV develops a scheme to make timeadaptive assessments. Section V demonstrates numerical results on the benchmark test systems with parameter sensitivity
study. Finally, we conclude this work in Section VI.

75
50

G 03
G 04

G 01
G 02

G 06
G 04

Rotor angle (deg)

such cases, as analyzed in [13], assessment accuracy is critical
for TSA prediction, and machine learning-based techniques
are employed in this work to construct an intelligent TSA
prediction system. Note that we do not intend to compare the
usefulness of the two main trends. In power applications where
a transparent model is necessary, fuzzy-logic-based approaches
may be preferable [13].

G 01

G 07

G 08

G 05
G 06

G 07
G 08

G 09
G 10

G 09

25
0
-25
G 02

-50

0.00

G 03

G 05

0.40

G 10

0.80

1.20

1.60

2.00

Time after fault clearance (s)

Fig. 1. Synchronous generator rotor angle dynamics after a system fault in
New England 10-machine system [19].

II. T RANSIENT S TABILITY A SSESSMENT AND R ECURRENT
N EURAL N ETWORKS
In this section we first introduce TSA, and then give a
brief overview on RNN and LSTM networks. Such networks
are critical in the proposed TSAS which will be presented in
Section III.
A. Transient Stability Assessment
With the gradual deployment of synchrophasor measurement devices, wide area protection and control (WAPaC)
systems have attracted significant attention [18]. To implement
such a response-based protection system, TSA needs to be
conducted after the clearance of system faults to predict the
system stability, i.e., whether the system can maintain its synchronism subject to the disturbance and operating condition.
Fig. 1 depicts the generator rotor angle dynamics of a
system fault in the New England 10-machine system, where
bus 16 experienced a three-phase short circuit, and the fault
is cleared after 0.25 second. In this figure, the number after
“G” in the legend indicates the index of the corresponding
generator. Generator 2 is used as the reference machine. Each
line in the plot represents the rotor angle dynamics of one
synchronous machine. In such transient situations, it is desired
that the system behavior is observed, and the future stability
status of the system is predicted in the earliest possible time.
If the system will lose its synchronism, emergency control
actions will take place to maintain stability. This prediction
process is called TSA, and a faster system response speed is
always preferred to provide extra time for control actions. Here
TSA accuracy and response speed form a trade-off, and it is
optimal to evaluate the trade-off based on the characteristics
of contingencies. We call the schemes with this capability
time-adaptive TSA techniques, as they can adapt to different
contingencies and generate the accurate assessment results at
the earliest possible time.
To design such time-adaptive methods, we investigate the
components that constitutes the response time. Modern fast
TSA based on machine learning techniques can be generalized
in Fig. 2. The post-disturbance system dynamics are digitized
into time-series system dynamic information, which is input
into the TSA system for developing a future system stability
index. It is clear that the response time is mostly influenced by

IEEE TRANSACTIONS ON POWER SYSTEMS

x1(t)
Time-series system
dynamics data

x2(t)

Stable/Unstable

TSA System

x3(t)

y1

y2

y3

σ

σ

σ

Dense

Dense

h1

Fig. 2. Typical design of machine learning TSA systems.

LSTM

%ç?5

Output Gate

Input Gate

x

Dç

LSTM

%ç

+
Eç

Kç

x?

LSTM
C2

x2

C3
x3

Fig. 4. Structure of TSAS. Given a sequence of xt data, the system can
develop corresponding yt predictions.

tanh

Bç

x1

...

h3

C1
Forget Gate

Dense

h2

x

ç

Sigmoid

Sigmoid

tanh

Sigmoid

Dç?5

Dç
šç

Fig. 3. An illustration of LSTM memory cell [23].

two factors: data aggregation time and TSA system computation time. Therefore, in this work we propose a time-adaptive
scheme to minimize the volume of data required, and construct
a computationally efficient system to handle the calculation.

B. Recurrent Neural Network and Long Short-Term Memory
RNN [20] is a special type of neural network and it considers data correlation in the time domain. A typical RNN takes
a temporal sequence of vectors [x1 , x2 , · · · , xT ] as input, and
outputs a sequence of vectors [h1 , h2 , · · · , hT ]. The output is
generated by the following equation for t = 1, 2, · · · , T :
ht = f (Wxt + Hht−1 + b),

(1)

where f (·) is a non-linear activation function, and W, H,
b are the learning parameters. Different from ANNs, RNNs
have additional recurrent connections in the hidden layers,
which provide essential memory capabilities. Such connections facilitate the network to keep previous information for
later use in the time domain, and thus capture the time-domain
dependencies in the input data.
While the RNN architecture utilizes the data correlation
information, typical gradient-based training algorithms suffer
from deteriorated memory performance as the long-range data
correlation is undermined [21]. They suffer from the “vanishing gradient problem” [22], resulting in poor performance
in capturing long temporal dependencies. As an alternative, a
novel RNN architecture was proposed in [17] to address this
problem. This improved model is referred to as long short-term
memory (LSTM).
LSTM, designed to overcome the long temporal dependency
defect in RNN, is a kind of network that implements memory
blocks, containing one or more memory cells [17]. These cells
are employed to maintain a long term memory (state) over
time when combined with a typical RNN. The architecture of
LSTM is presented in Fig. 3.
As illustrated in Fig. 3, LSTM comprises three gates: input,
forget, and output gates. These gates cooperate with a memory

cell Ct to maintain the LSTM cell state:
it = σ(Wi xt + Ui ht−1 + bi )

(2a)

ct = tanh(WC xt + UC ht−1 + bC )

(2b)

ft = σ(Wf xt + Uf ht−1 + bf )

(2c)

Ct = ft ∗ Ct−1 + it ∗ ct ,

(2d)

and update the output representation:
ot = σ(Wo xt + Uo ht−1 + bo )

(3a)

ht = ot ∗ tanh(Ct ),

(3b)
1
1+e−x

where ∗ is the element-wise product, σ(x) =
is a
sigmoid function, and W, U, b are matrices corresponding
to the learning parameters P = [W, U, b]. Numerous LSTMbased applications have been developed due to its simplicity
and efficacy. Interested readers can refer to [24] for a thorough
introduction of the methodology.
III. LSTM- BASED TSAS
The LSTM-based TSAS proposed in this paper is based
on the hypothesis that the post-contingency power system
measurements, e.g., bus voltage phasors, can immediately
indicate the stability of the system after experiencing the
disturbance, see [15], [25] for some references. In addition,
the temporal data dependency of the measurements can reduce
the data quantity required for accurate assessments. Based
on these hypotheses, we establish a non-linear relationship
between the observed measurements and the system stability
by using LSTM as the primary classifier. The structure of the
learning system is presented in Figs. 4.
In the proposed TSAS, two layers of artificial neurons
are employed to extract the characteristics of the input data,
namely, an LSTM memory block layer and a hidden dense
neuron laye. The LSTM layer is utilized to handle input
power system measurements, and outputs ht , sharing the same
dimensionality of xt . As the extracted data feature by the
LSTM layer is not human-readable, an additional dense hidden
layer is appended for dimensionality reduction. In addition, a
final Sigmoid function is employed to normalize the output
solution.
A. Time-Series Simulation
To enable the proposed TSAS for real-time TSA, it needs to
be trained offline. The training data is generated using timeseries simulation of different transient contingencies on the

IEEE TRANSACTIONS ON POWER SYSTEMS

given power system. In this system, we employ the positive
sequence voltage phasors of all buses as input data, which in
real-time can be measured with synchophasor measurement
techniques. Using 50/60-Hz sampling of the simulated measurements, the measured voltage phasor data of an arbitrary
contingency after fault clearance is in the form of


V1,1 V1,2 · · · V1,T
 V2,1 V2,2 · · · V2,T 


 ..
..
..  ,
..
 .
.
.
. 
VB,1

VB,2

···

VB,T

where B and T are the total number of buses and the length
of the post-contingency observation window, respectively.
Vb,t = Vb,t θb,t is the voltage phasor of bus b at time
instance t. While B is determined by the system topology,
T is a control parameter influencing the LSTM complexity
and assessment accuracy; a large T will result in a high
nonlinearity of the TSAS, while a small T may impair the
input data completeness, which in return weakens the accuracy.
In addition, the synchronous machine rotor angle is also
calculated in the time-series simulation, and the maximum
angle deviation of any two machines at any time is recorded,
denoted by δmax . This value is later utilized to determine the
system stability after contingencies.
B. Offline Training
To better simulate the system response after various disturbances, a large set of defined contingencies are utilized in
the time-series simulation. Employing the voltage phasor and
maximum angle deviation data for each training contingency,
the training dataset can be established where the input of each
training case is presented as x = [x1 , x2 , · · · , xT ], where
xt = [V1,t , V2,t , · · · , VB,t , θ1,t , θ2,t , · · · , θB,t ]> .

(4)

Meanwhile, to construct a complete training case, the output
assessment result y is obtained by observing δmax :
(
1 (Stable)
for η > 0
yt =
, t = 1, 2, · · · , T, (5)
0 (Unstable) for η ≤ 0
where η = (360−δmax )/(360+δmax ), and δmax is the maximum
angle deviation of any two generators in the LSTM system at
any time. η is a power angle-based stability index [9] in which
the generator rotor angle difference δ is used to determine if
any generator in the system is out of synchronism. This index
is widely employed as a transient stability index in the previous
literature, see [16], [25] for some references.
Given a collection of N training cases {x(n) , y(n) }N
n=1 ,
the objective of the offline training is to obtain the system
parameters W, U, and b. In this paper, the Adam optimizer
[26] is employed to find the optimal values of these parameters
with the binary cross entropy error function as the objective:

minimize

−

N
X

[y(n) log ŷ(n) +(1−y(n) ) log(1− ŷ(n) )], (6)

n=1

where ŷ(n) is the actual classification (assessment) result of
x(n) on the trained system.

C. Online Assessment
For online assessment of the transient stability status, the
previously trained TSAS is employed to classify the test cases
using post-contingency PMU measurements. With a set of
bus voltage phasor measurements of length T , it is trivial to
calculate the corresponding hT and thus ŷT ∈ (0, 1) using the
trained system parameters. In order to transform the system
output ŷT ∈ (0, 1) into a stability index, a bipartite threshold
δ = 0.5 is defined such that the test cases with ŷT < δ is
considered unstable, and otherwise stable.
IV. T IME - ADAPTIVE TSAS
A. Time Adaptive Implementation
Similar to most existing TSA models, the method proposed
in Section III utilizes a fixed-length observation window,
i.e., the response time is constant. However, this static time
response can be less efficient when dealing with fast transient
instabilities. Moreover, different system models may require
substantially different lengths of observation windows to obtain reliable assessment conclusions.
In this section, we propose a time-adaptive TSAS based on
the LSTM system in Section III. The objective of the timeadaptive scheme is to generate a reliable assessment result as
fast as possible, which is a trade-off between the assessment
speed and accuracy. Consequently, control actions can take
place at an early time to avoid possible system failures.
Contributed by the recurrent connections of LSTM memory
blocks, our proposed TSAS takes the temporal data dependency into consideration when assessing the system stability.
However, as only ŷT is concerned in generating stability
indices, potential information loss happens when ŷt , t =
1, 2, · · · , T −1 values are discarded. Utilizing this information,
we propose a time-adaptive TSAS for fast stability assessment.
The flow chart is presented in Fig 5, where all “TSAS”
blocks are identical with constant trained system parameters.
After the fault clearance, TSA will be conducted immediately
upon the receipt of PMU measurements at the first time
instance. As x1 is available at the system, ŷ1 can be calculated
and utilized to determine the transient stability index. Consequently, the system stability can be assessed as follows, where
(5) is transformed to determine the reliability:


for ŷt > 1 − δ
Stable
Stability = Unstable for ŷt < δ
, t = 1, 2, · · · , T.


Unknown otherwise
(7)
In this equation, the stability threshold δ ∈ (0, 0.5) is used to
manipulate the trade-off between speed and accuracy. A larger
δ may result in an earlier assessment but the accuracy may be
undermined, while a smaller δ will lead to potentially more
accurate result at the sacrifice of assessment speed.
If (7) returns “Stable” or “Unstable”, the system outputs
this conclusion for further control operations. Otherwise, the
system will wait for the next time instance, and the newly
received measurements xt will be included in the calculation
of the corresponding yt values. This process repeats until a

IEEE TRANSACTIONS ON POWER SYSTEMS

t
Fault Clear

x1

x2

TSAS

h1,C1

y1
Reliable?

x3

TSAS

h2,C2

TSAS

y2
N

Reliable?

xT
h3,C3

...

hT-1,CT-1

y3
N

Reliable?

Y

Y

Y

Stop

Stop

Stop

Deadline

TSAS

yT
N

N

Stop

Fig. 5. Flow chart of time-adaptive TSAS.

reliable result is achieved, or the maximum decision-making
time T max is reached.
The proposed TSAS relates to the system in [16] in the
following sense. Both systems can perform online TSA in
a time-adaptive manner. However, due to the difference in
the underlying methods utilized (LSTM versus ELM), our
proposed TSAS extracts additional temporal data-dependency
features over the approach in [16]. Moreover, as we employ
only one LSTM network in the proposed TSAS, the training
complexity is greatly reduced when compared with the multiELM system in [16]. This also yields a simpler yet still
accurate decision making rule (7), whose performance will
be demonstrated in the following sections.

B. Discussion
In the time-adaptive intelligent system, LSTM plays a key
role in predictions based on available time-series data. As a
variant of RNN and ANN, LSTM has two major advantages
that make it suitable for TSA:
•

•

The most computationally expensive process, i.e., training, can be performed in an offline manner. The online
TSA testing process can be run very effectively given the
trained neural network parameters [27].
Neural networks can model transient stability in power
networks with power dynamics data [28]. This allows us
to relax the requirement on accurate knowledge of power
system models in practice.

Hence the problems (requirement of complete power system
model and heavy real-time computational burden as discussed
in Section I) encountered by the traditional non-artificialintelligence TSA techniques can now be overcome with the
proposed system.
Meanwhile, neural networks are often regarded as blackbox
models because the internal network parameters and variables
generally do not carry explicit physical meanings. However,
the lack of these physical meanings does not influence the
practicality of ANN, since only the output value is required
for subsequent analyses and/or calculations. As the interior
structure can be defined by (2) and (3), only algebraic calculations are needed to generate the output from the input power
system dynamics sampled by PMUs. In the case of TSAS, the
output of the neural network holds the transient stability status
of the system.

V. C ASE S TUDIES
The New England 10-machine system [19] is adopted to
evaluate the assessment performance of the proposed TSAS.
All numerical simulations are conducted on a computer with
an Intel Core i7 CPU working at 3.4 GHz and 8 GB RAM.
The LSTM framework is constructed with Theano [29] and
Keras [30]. The proposed system is implemented in Python.
For TSA using supervised learning algorithms, sufficient
previous knowledge that depicts the post-contingency characteristics of the grid is critical [31]. The training cases should
include adequate information to guide the learning of TSAS
to approximate the system behavior under different operation
conditions.
The training and testing cases are generated by time-domain
simulation of post-contingency power system dynamics. In
the simulation, we consider both the nominal power network
topology and the (N − 1) contingencies, in which any one
of the transmission lines or transformers is out of service.
The consumed power is set to 80%, 100%, and 120% of the
basic system load levels, respectively1 . For each load level,
three-phase short-circuit faults are assumed to occur on either
buses, or transmission lines where the faults are located at
20%, 40%, 60%, and 80% of the whole line [32]. Further, the
fault clearance time is set to arbitrary values between 0.1s and
0.4s. Consequently, 5000 TSA contingency cases are generated
with the above configurations. The number of cases is similar
to that of the same network in previous work [16].
For each contingency case, the time-domain simulation is
conducted using TSAT [33], and the calculated bus voltage
phasors are employed as the measurements sampled by PMUs.
In practice, utilities can utilize the historical operation data
to train TSAS (see [13] for an example). By this means
the extensive simulation time for generating power system
dynamics under these contingencies can be avoided. Note
that unless specified, we assume that PMU can accurately
sample power system variables. The impact of inaccurate PMU
measurements will be investigated in Section V-D.
For cross validating and preventing the over-fitting problem,
in this work we randomly divide the generated contingency
cases into training and testing sets in the ratio of 3:1, which
accords with [16], [32]. Only the training cases are employed
to train the TSAS. The testing cases are used to assess the
TSA accuracy when TSAS encounters unknown contingencies.
1 For different system load levels, both generations and loads are scaled by
the ratio.

IEEE TRANSACTIONS ON POWER SYSTEMS

A. TSA Accuracy and Response Time
Table I presents the TSA test results for the New England
10-machine system. The assessment result is presented for
both training cases and testing cases for completeness. In
this table, the values in the “Unknown” column represent
the total number of test instances whose stability indices
cannot be computed at time t after fault clearance. The
simulation results can lead to a conclusion that the majority
of the contingencies can be correctly assessed at a very early
stage after the clearance of contingencies. For both training
and testing cases, around two thirds of all instances can
be assessed at the first post-contingency cycle. In addition,
the early assessment accuracy is perfect for both training
and testing cases. This outstanding performance is potentially
contributed by the deeper neural network structures compared
with the existing TSA with ANN work [7], [16]. Our proposed
TSAS with one layer of LSTM is also tested under the same
simulation environment and input dataset for comparison, and
the assessments are less than 99% accurate from the first postcontingency cycle.
After the first post-contingency cycle, the assessment accuracy to determine those originally “Unknown” cases remains
perfect for testing cases when given more measurements for
assessment, and there is only one wrong assessment among
all 3750 training cases. Moreover, only around 4% of the
contingency cases cannot be assessed with data of two cycles,
and all instances can be correctly assessed within seven postcontingency cycles for testing cases, and eight for training
cases.
Compared with the other time-adaptive approach given in
[16], it can be observed that our proposed TSAS can maintain
a superior accuracy when dealing with the test instances
that require more information for classification. Intuitively
the performance improvement originates from the employed
LSTM network, which extracts the data features from the
temporal dependencies of the training data.
In addition to the assessment accuracy, the response speed
is another major concern when evaluating the performance
of TSA methods. It can be easily derived from Table I
that the average response time (ART) [16] of TSAS on the
New England 10-machine system is 1.448 cycles with 100%
accuracy for testing cases, and 1.412 cycles with 99.97%
accuracy for training cases.

ART
Accuracy

3.5

100

98

3.0
2.5

96

2.0

Accuracy (%)

4.0

ART (cycles)

Thus over-fitting problem (which describes the models learning from both characteristics and random noises in the input
data, leading to poor predicting performance) can be avoided.
Generally over-fitted models will result in inferior accuracy
for testing cases, which is not observed in our simulations.
In order to train the proposed TSAS, control parameters
shall be defined. In this work, two layers of LSTM memory
blocks are employed in the LSTM layer, where each layer has
128 memory blocks. The numbers of input and output nodes
are 2B and 1, respectively. The network is trained for up to
100 epochs, subject to an early stopping method to prevent
overfitting. In the tests, the maximum decision-making time
T max is set to 20. Parameters δ and T are set to 0.4 and 5,
respectively, and their sensitivities will be studied later.

94

1.5
1.0
0.1

0.2

0.3

0.4

92

value

Fig. 6. δ sensitivity test on New England 10-machine system.

To see the performance improvement of TSAS over the
existing TSA techniques, we compare the ART and accuracy
of the methodologies proposed in the literature with TSAS
in Table II. In this table, the simulation performance2 of the
proposed TSAS on the New England 10-machine system and
two large scale test systems to be introduced in Section V-E
are listed. Meanwhile, as the response time of the other timeadaptive methods can be significantly affected by the power
system size, the simulation results for test systems of different
scales are presented. For other non-time-adaptive methods, the
best assessment accuracies are compared.
From the results, it can be concluded that the proposed
TSAS can outperform existing state-of-the-art TSA techniques. TSAS achieves a similar ART performance compared
with the ELM-based system proposed in [16] (1.448 cycles
vs 1.4 cycles for 10-machine system, 2.047 cycles vs 2.8
cycles for 50-machine system), and provides better assessment
accuracy in both of the test systems. While the SVM-based
system [25] and Prediction-based system [34] give accurate
assessment results for all test cases, TSAS achieves the same
performance with much shorter time, which consequently
allows us to reserve more time for subsequent control actions
against unstable situations.
B. δ and T Sensitivity Tests
In the proposed TSAS, two system parameters significantly
influence the assessment performance, namely the stability
threshold δ and training observation window length T . Two
sets of simulations are performed to evaluate their sensitivities,
and the results are depicted in Fig. 6 and Table III. In Fig. 6,
49 δ values in the range [0.1,0.49] with a step size of 0.01 are
simulated, and the ART and accuracy results are plotted. While
a smaller ART is preferred, the system prediction accuracy is
always an important performance metric in TSA. This figure
indicates that δ values around 0.4 are preferred in order to
achieve a perfect assessment accuracy while maintaining a fast
response time.
The sensitivity of the training observation window length
T is presented in Table III. For example, T = 5 means that
the system is trained using the voltage phasors of the first five
2 For fairness of comparison, the simulation results for testing cases are
demonstrated as did in the previous literature [16].

IEEE TRANSACTIONS ON POWER SYSTEMS

TABLE I
TSA ACCURACY AND AVERAGE R ESPONSE T IME ON N EW E NGLAND 10- MACHINE T EST S YSTEM

0
1
2
3
4
5
6
7
8

Unknown
3750
1103
176
105
72
58
29
3
0

Training Set
Correct
Wrong
0
0
2647
0
3574
0
3645
0
3678
0
3692
0
3721
0
3747
0
3749
1

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
99.97%

TABLE II
C OMPARISON WITH PREVIOUS TSA METHODOLOGIES
Method
Proposed TSAS on 10-machine
Proposed TSAS on 162-bus (17-machine)
Proposed TSAS on 145-bus (50-machine)
ELM-based system on 10-machine [16]
ELM-based system on 50-machine [16]
SVM-based system [25]
Prediction-based system [34]
DTFR-based system [35]

ART
1.448 cycles
1.901 cycles
2.047 cycles
1.4 cycles
2.8 cycles
4 cycles
6 cycles
1–2 seconds

Testing Set
Correct
Wrong
0
0
840
0
1196
0
1208
0
1229
0
1229
0
1238
0
1250
0
1250
0

Unknown
1250
410
54
42
21
21
12
0
0

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%

TABLE V
S ELECTED B USES FOR PMU P LACEMENT WITH SBS A LGORITHM

Accuracy
100.00%
100.00%
99.98%
99.10%
99.70%
100.00%
100.00%
95.00%

PMU Count
1
4
7
8
10
15

PMU Positions (Bus No.)
36
1, 17, 35, 36
1, 8, 17, 21, 35, 36
1, 8, 9, 17, 21, 35, 36
1, 5, 8, 9, 17, 21, 25, 27, 35, 36
1, 2, 5, 8, 9, 10, 17, 21, 23, 25, 27, 28, 35, 36, 38

3.5

TABLE III
T S ENSITIVITY T EST ON N EW E NGLAND 10- MACHINE T EST S YSTEM
1
1.009
95.28%
6
1.743
99.92%

2
1.095
98.08%
7
1.819
100.00%

3
1.209
98.80%
8
2.162
100.00%

4
1.342
98.88%
9
1.962
100.00%

5
1.455
100.00%
10
2.231
100.00%

cycles after fault clearance, and tested with continuous timeseries data. Therefore, this test aims to identify the influence
of input data size on the system performance. From the table
it can be observed that while the ART performance decreases
with the increase of T , the assessment accuracy demonstrates
a reverse trend. While a small T can lead to a fast response
time, a medium T value, e.g. 5, is the optimal choice of
the parameter considering the trade-off between ART and
accuracy.
C. PMU Placement Analysis
In TSAS, the real-time voltage magnitudes and angles of
all buses in the power system are employed to assess the
post-contingency stability. Such information requires a large
number of high-speed synchrophasors, like from PMUs, to
maintain a full observability. However, such a large-scale

TABLE IV
S ELECTED B USES FOR PMU P LACEMENT WITH SFS A LGORITHM
PMU Count
1
3
5
10
15

PMU Positions (Bus No.)
33
3, 14, 33
3, 8, 14, 33, 35
1, 3, 4, 8, 10, 14, 25, 33, 34, 35
1, 3, 4, 8, 10, 14, 16, 22, 25, 27, 31, 33, 34, 35, 36

ART (cycles)

T
ART (cycles)
Accuracy
T
ART (cycles)
Accuracy

ART
Accuracy

3.0

100
80

2.5
2.0

60

1.5

40

1.0
20

0.5
0.0

Accuracy (%)

t

5

10

15
20
25
Number of PMUs installed

30

35

0

Fig. 7. PMU data sensitivity analysis with SFS algorithm.

installation may not be economically feasible due to high installation cost of PMUs. Thus it becomes essential to generate
a collection of the most effective PMU installation positions in
the system. This can be achieved by analyzing the sensitivity
of the measurements from different components in the power
system contributing to TSA accuracy and response time. In
this paper we employ the sequential feature selection algorithm
[36] for this analysis. Both sequential forward selection (SFS)
and sequential backward selection (SBS) are adopted to verify
the simulation results, which are demonstrated as follows.
In the SFS algorithm, a number of TSAS with different
number of inputs are constructed. The algorithm first starts
with networks with the voltage phasor measurements of one
bus as input, and the best performing bus is retained in
the next round of selection where the measurements of one
more bus are considered as input. Here the performance
comparison is conducted with respect to the corresponding
assessment accuracy and ART. This process repeats until the
measurements of all buses are included in the input. The SBS
algorithm employs a similar logic as SFS. The difference is
that SBS starts with the measurements of all buses, and these
buses are removed gradually during the selection process [36].

IEEE TRANSACTIONS ON POWER SYSTEMS

TABLE VI
TSA P ERFORMANCE WITH N OISY PMU DATA
Training Set
Correct
Wrong
0
0
2624
0
3558
0
3639
0
3676
0
3694
0
3722
0
3747
1
3749
1

3.5

ART
Accuracy

ART (cycles)

3.0

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
99.97%
99.97%

100
80

2.5
60

2.0
1.5

40

1.0
20

0.5
0.0

5

10

15
20
25
Number of PMUs installed

30

35

0

Fig. 8. PMU data sensitivity analysis with SBS algorithm.

The simulation results are presented in Fig. 7 and Fig. 8, and
the PMU positions are listed in Tables IV and V.
Figs. 7 and 8 show that the response time decreases with the
increase of installed PMUs, and the assessment accuracy also
increases at the same time. This observation is significant when
the total number of PMUs is small (less than 10). After this
initial performance improvement phase, the accuracy remains
at 100% while the ART performance fluctuates around 1.5
cycles, and demonstrates a slight improvement with both of
the algorithms. This leads to a conclusion that a sufficient
number of PMUs need to be employed to provide enough system measurements for reliable assessments. While additional
PMUs can marginally decrease the response time needed for
assessments, additional equipment cost is incurred. Real-world
implementation shall consider this trade-off when addressing
the PMU placement issue.
D. PMU Measurement Accuracy Sensitivity
In previous analyses, similar to [11], [16] and other related
research, we assume that PMUs in power systems can accurately sample system variables at a high frequency. Meanwhile,
in practice PMUs may suffer from sampling errors. According
to IEEE Standard for Synchrophasor Data Transfer for Power
Systems (C37.118.2-2011) [37], the total vector error for all
PMUs complying with the standard should be less than 1%.
Therefore, a numerical simulation is carried out to study the
influence of noisy PMU samples on the performance of TSAS.
In this test, we follow the approach introduced in [38] to
generate noisy test cases based on the TSAT time-domain
simulated power system variables. Specifically, a random noisy
phasor, which satisfies the requirement proposed by [37], is

Unknown
1250
410
60
47
26
24
15
1
0

Ratio of correct assessments (%)

0
1
2
3
4
5
6
7
8

Unknown
3750
1126
192
111
74
56
28
2
0

Accuracy (%)

t

Testing Set
Correct
Wrong
0
0
840
0
1190
0
1203
0
1224
0
1226
0
1234
1
1248
1
1249
1

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
99.92%
99.92%
99.92%

100
80
60

Training Set of 10-machine system
Testing Set of 10-machine system
Training Set of 162-bus system
Testing Set of 162-bus system
Training Set of 145-bus system
Testing Set of 145-bus system

40
20
0

0

2

4

6

8

10

t

Fig. 9. Comparison on the ratio of correct assessment with respect to postcontingency time.

imposed on all voltage phasors in the dataset. The manipulated
noisy data is employed for both training and testing of TSAS,
and all other configurations are identical to previous tests.
The simulation results are summarized in Table VI. From
the table, it can be observed that both ART and accuracy
are slightly influenced by the noise, yet the performance
decrease is minuscule. While there is a wrong assessment in
the testing cases which was correctly identified with accurate
PMU measurements, the training set accuracy remains the
same with previous simulation. Meanwhile, ART for noisy
data (1.423 and 1.466 cycles for training and testing cases
respectively) is slightly longer than that for noiseless data
(1.412 and 1.448 cycles). To conclude, TSAS can achieve
almost the same performance even considering noisy PMU
measurements, and can provide superior TSA performance
when compared with existing techniques whose results are
listed in Table II.
E. Large System Test and Computation Time
Besides the New England 10-machine Test System, we also
study two large power system test cases to analyze the scalability of our proposed TSAS, namely a 17-generator, 162-bus
system and a 50-generator, 145-bus system [39]. The identical
method to generate training and test cases are employed to
develop 20 000 contingencies for each test system. Tests and
analyses are performed with the same simulation environment
as in previous sections, and the results are presented in Tables
VII and VIII.
From these tables we can see that while the majority of the
test cases can still be correctly identified within one cycle only

IEEE TRANSACTIONS ON POWER SYSTEMS

TABLE VII
TSA P ERFORMANCE ON 17- GENERATOR , 162- BUS T EST S YSTEM
t

Unknown
15000
4206
3547
2023
1198
571
94
12
0

0
1
2
3
4
5
6
7
8

Training Set
Correct
Wrong
0
0
10794
0
11453
0
12977
0
13802
0
14429
0
14906
0
14988
0
15000
0

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%

Unknown
5000
1493
1414
876
456
214
42
9
0

Testing Set
Correct
Wrong
0
0
3507
0
3586
0
4124
0
4544
0
4786
0
4958
0
4991
0
5000
0

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%

TABLE VIII
TSA P ERFORMANCE ON 50- GENERATOR , 145- BUS T EST S YSTEM
t
0
1
2
3
4
5
6
7
8
9
10

Unknown
15000
5546
4098
2321
1627
806
70
16
6
4
0

Training Set
Correct
Wrong
0
0
9454
0
10902
0
12679
0
13373
0
14194
0
14930
0
14984
0
14992
2
14994
2
14997
3

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
99.99%
99.99%
99.98%

TABLE IX
S UMMARY OF T RAINING T IME
Test System
New England
162-bus system
145-bus system

Average
37.43 seconds
155.86 seconds
143.16 seconds

Std. Dev.
1.62 seconds
4.75 seconds
4.69 seconds

after the clearance of contingencies, the system requires longer
ART to perform identifications on both of the larger power
grids: 1.901 cycles for the 162-bus system and 2.047 cycles
for the 145-bus system. Meanwhile, the assessment accuracy
for both cases remains perfect except for one test case in the
145-bus system. This observation supports the conclusion that
learning from the temporal data-dependencies of the inputs
contributes to the improvements in assessment accuracy when
compared with the previous state-of-the-art method [16].
We also depict the response speed of all three tested power
systems in Fig. 9. From this figure it can be observed that
the assessment speed is closely related to the system size.
As the New England 10-machine system comprises the least
number of generators (10) and buses (39), most assessments
can be generated at the very early stage. Meanwhile, the 162bus system has more buses than the 145-bus system, but the
number of generators are smaller (17 vs. 50). So they have
similar speed.
Besides the assessment accuracy and response time, training
time is another concern when employing machine learning
methods for TSA problems. Table IX summarizes the training
time of the three included tested systems, where each system
is trained for 100 times for statistical significance. From this
table, our proposed TSAS is computationally efficient during

Unknown
5000
1942
1476
852
639
284
28
6
6
2
0

Testing Set
Correct
Wrong
0
0
3058
0
3524
0
4148
0
4361
0
4716
0
4972
0
4993
1
4993
1
4997
1
4999
1

Accuracy
N/A
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
99.98%
99.98%
99.98%
99.98%

training. The training time is roughly linear with the number
of training cases, and this number is closely related to the
system size.
The fast training speed also makes TSAS robust under
significant changes to the power grid. The system can also
be updated online with new measured system variables for
performance improvement. When the control center detects
power system topology changes, or schedules periodical TSAS
updates, the LSTM system can be efficiently re-trained with
existing system dynamics data. The detailed implementation
of TSAS re-training is beyond the scope of this paper and will
be investigated in future work.
VI. C ONCLUSION AND F UTURE W ORK
In this paper we propose a new TSAS for on-line postcontingency stability assessments by utilizing LSTM networks
to extract data features. A time-adaptive scheme is developed
to facilitate stability assessments at the earliest possible time
instance, reserving more time for the following control actions.
Differently from existing systems, our model learns from
the temporal data dependency and utilizes such features to
achieve a better assessment accuracy while maintaining the
response time performance. Moreover, our model includes
one unique LSTM-based network instead of an ensemble
of machine learning components. This design significantly
reduces the model complexity and renders the training process
less computationally expensive.
In order to evaluate the performance of our proposed TSAS,
three power system test cases are considered, including the
New England 10-machine test system, a 17-generator 162-bus
system, and a 50-generator 145-bus system. All simulation

IEEE TRANSACTIONS ON POWER SYSTEMS

results reveal that the proposed model can achieve a superior
assessment accuracy within a very short period of time after
the clearance of system faults. Meanwhile, the model is
computationally efficient to train using existing measurements,
making the model highly scalable for handling TSA tasks in
large power systems.
Besides the accuracy and response time tests, we also perform preliminary sensitivity tests on two essential parameters
introduced in the proposed system: the stability threshold and
training observation window length. Simulation results suggest
a parameter configuration with good performance to facilitate
future deployments of the system. In addition, the sensitivity
of PMU measurements is studied, and the sequential feature
selection algorithm demonstrates the least number and best
locations of PMUs in the tested grid.
Future work will focus on the availability of PMU measurements, considering information loss scenarios and different
measurement predictors. It is assumed in this paper that
measurements are available for all PMU installation locations,
while a more realistic scenario shall consider cases with
missing PMU measurements due to communication delay or
loss. Besides, the predictor employed in this paper is the
voltage phasors, while a wider range of such predictors are
available for selection [35]. It is interesting to study their
sensitivities as system inputs to achieve a better response
time. The online re-training process is another potential future
research topic.
R EFERENCES
[1] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares,
N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. Van Cutsem,
and V. Vittal, “Definition and classification of power system stability
IEEE/CIGRE joint task force on stability terms and definitions,” IEEE
Trans. Power Syst., vol. 19, no. 3, pp. 1387–1401, Aug 2004.
[2] H.-D. Chiang, Direct Methods for Stability Analysis of Electric Power
Systems: Theoretical Foundation, BCU Methodologies, and Applications. John Wiley & Sons, Mar. 2011.
[3] A.-A. A. Fouad and V. Vittal, Power System Transient Stability Analysis
Using the Transient Energy Function Method. Prentice Hall, 1992.
[4] I. Hiskens and D. Hill, “Energy functions, transient stability and voltage
behaviour in power systems with nonlinear loads,” IEEE Trans. Power
Syst., vol. 4, no. 4, pp. 1525–1533, Nov. 1989.
[5] Y. Xue, Quantitative Study of General Motion Stability and an Example
on Power System Stability.
Nanjing, China: Jiangsu Science and
Technology Press, 1999, in Chinese.
[6] P. Kundur, Power System Stability and Control. McGraw-Hill Education, Jan. 1994.
[7] F. Hashiesh, H. E. Mostafa, A.-R. Khatib, I. Helal, and M. M. Mansour,
“An intelligent wide area synchrophasor based system for predicting and
mitigating transient instabilities,” IEEE Trans. Smart Grid, vol. 3, no. 2,
pp. 645–652, Jun. 2012.
[8] C. Liu and J. Thorp, “Application of synchronised phasor measurements
to real-time transient stability prediction,” IET Generation, Transmission
and Distribution, vol. 142, no. 4, pp. 355–360, Jul. 1995.
[9] M. Pavella, D. Ernst, and D. Ruiz-Vega, Transient stability of power
systems: a unified approach to assessment and control. Norwell: Kluwer
Academic Publishers, 2000.
[10] D. R. Gurusinghe and A. D. Rajapakse, “Post-Disturbance Transient
Stability Status Prediction Using Synchrophasor Measurements,” IEEE
Trans. Power Syst., vol. 31, no. 5, pp. 3656–3664, Sep. 2016.
[11] T. Guo and J. Milanovic, “Probabilistic framework for assessing the
accuracy of data mining tool for online prediction of transient stability,”
IEEE Trans. Power Syst., vol. 29, no. 1, pp. 377–385, Jan. 2014.
[12] J. Geeganage, U. D. Annakkage, T. Weekes, and B. A. Archer, “Application of Energy-Based Power System Features for Dynamic Security
Assessment,” IEEE Trans. Power Syst., vol. 30, no. 4, pp. 1957–1965,
Jul. 2015.

[13] I. Kamwa, S. Samantaray, and G. Joos, “On the accuracy versus
transparency trade-off of data-mining models for fast-response pmubased catastrophe predictors,” IEEE Trans. Smart Grid, vol. 3, no. 1,
pp. 152–161, Mar. 2012.
[14] L. A. Wehenkel, Automatic Learning Techniques in Power Systems.
Norwell, MA, USA: Kluwer Academic Publishers, 1998.
[15] L. Wehenkel, T. V. Cutsem, and M. Ribbens-Pavella, “An artificial
intelligence framework for online transient stability assessment of power
systems,” IEEE Trans. Power Syst., vol. 4, no. 2, pp. 789–800, May
1989.
[16] R. Zhang, Y. Xu, Z. Y. Dong, and K. P. Wong, “Post-disturbance transient stability assessment of power systems by a self-adaptive intelligent
system,” IET Generation, Transmission and Distribution, vol. 9, no. 3,
pp. 296–305, Feb. 2015.
[17] S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural
Computation, vol. 9, no. 8, pp. 1735–1780, Nov. 1997.
[18] M. Begovic, D. Novosel, D. Karlsson, C. Henville, and G. Michel,
“Wide-Area Protection and Emergency Control,” Proc. IEEE, vol. 93,
no. 5, pp. 876–891, May 2005.
[19] A. Pai, Energy Function Analysis for Power System Stability. Boston:
Kluwer Academic Publishers, 1989.
[20] K. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Netw., vol. 1,
no. 1, pp. 4–27, Mar. 1990.
[21] Y. Bengio, P. Simard, and P. Frasconi, “Learning long-term dependencies
with gradient descent is difficult,” IEEE Trans. Neural Netw., vol. 5,
no. 2, pp. 157–166, Mar. 1994.
[22] S. Hochreiter, “Untersuchungen zu dynamischen neuronalen netzen,”
Ph.D. dissertation, Institut f. Informatik, Technische Univ. Munich, 1991.
[23] Understanding LSTM networks. [Online]. Available: http://colah.github.
io/posts/2015-08-Understanding-LSTMs/
[24] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature, vol. 521,
no. 7553, pp. 436–444, May 2015.
[25] F. Gomez, A. Rajapakse, U. Annakkage, and I. Fernando, “Support
Vector Machine-Based Algorithm for Post-Fault Transient Stability Status Prediction Using Synchronized Measurements,” IEEE Trans. Power
Syst., vol. 26, no. 3, pp. 1474–1483, Aug. 2011.
[26] J. B. Diederik Kingma, “Adam: A method for stochastic optimization,”
in Proc. International Conference for Learning Representations, San
Diego, CA, U.S., May 2015, pp. 1–15.
[27] L. K. Hansen and P. Salamon, “Neural network ensembles,” IEEE Trans.
Pattern Anal. Mach. Intell., vol. 12, no. 10, pp. 993–1001, Oct. 1990.
[28] K. S. Narendra and K. Parthasarathy, “Identification and control of
dynamical systems using neural networks,” IEEE Trans. Neural Netw.,
vol. 1, no. 1, pp. 4–27, Mar. 1990.
[29] Theano documentation. [Online]. Available: http://deeplearning.net/
software/theano/
[30] Keras: Theano-based deep learning library. [Online]. Available:
http://keras.io/
[31] Y. Xu, Z. Y. Dong, K. Meng, R. Zhang, and K. P. Wong, “Real-time
transient stability assessment model using extreme learning machine,”
IET Generation, Transmission and Distribution, vol. 5, no. 3, pp. 314–
322, Mar. 2011.
[32] L. Zhu, C. Lu, and Y. Sun, “Time series shapelet classification based
online short-term voltage stability assessment,” IEEE Trans. Power Syst.,
in press.
[33] Dynamic security assessment software. [Online]. Available: http:
//www.dsatools.com/
[34] A. Rajapakse, F. Gomez, K. Nanayakkara, P. Crossley, and V. Terzija,
“Rotor angle instability prediction using post-disturbance voltage trajectories,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 947–956, May
2010.
[35] I. Kamwa, S. Samantaray, and G. Joos, “Development of RuleBased Classifiers for Rapid Stability Assessment of Wide-Area PostDisturbance Records,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 258–
270, Feb. 2009.
[36] P. Devijver and J. Kitler, Pattern Recognition: A Statistical Approach.
NJ: Prentice-Hall, 1982.
[37] “IEEE Standard for Synchrophasor Data Transfer for Power Systems,”
IEEE Std C37.118.2-2011, Dec. 2011.
[38] M. He, V. Vittal, and J. Zhang, “Online dynamic security assessment
with missing pmu measurements: A data mining approach,” IEEE Trans.
Power Syst., vol. 28, no. 2, pp. 1969–1977, May 2013.
[39] V. Vittal, “Transient stability test systems for direct stability methods,”
IEEE Trans. Power Syst., vol. 7, no. 1, pp. 37–43, Feb. 1992.

