INTRODUCTION
Recent blackouts in the USA, some European and Asian countries have illustrated the importance and need of more frequent and thorough power system stability study. Nowadays, power systems have evolved through continuing growth in interconnection, use of new technologies and controls. Due to increased operations which may cause power system to be in highly stressed conditions, the need for dynamic security assessment of power systems is arising. Transient Stability Assessment (TSA) is part of dynamic security assessment of power systems which involves the evaluation of the ability of a power system to remain in equilibrium under severe but credible contingencies. These evaluations aim to assess the dynamic behavior of a power system in a fast and accurate way. Methods normally employed to assess TSA are by using time domain simulation, direct and artificial intelligence methods. Time domain simulation method is implemented by solving the state space differential equations of power networks and then determines transient stability. Direct methods such as the transient energy method determine transient stability without solving differential state space equations of power systems. These two methods are considered most accurate but are time consuming and need heavy computational effort. Presently, the use of Artificial Neural Network (ANN) in TSA has gained a lot of interest among researchers due to its ability to do parallel data processing, high accuracy and fast response.
In transient stability assessment, the Critical Clearing Time (CCT) is a very important parameter in order to maintain the stability of power systems. The CCT is the maximum time duration that a fault may occur in power systems without failure in the system so as to recover to a steady state operation. Earlier ANN works carried out in TSA used the feed forward Multi Layer Perceptron (MLP) with back propagation learning algorithm to determine the CCT of power systems (Pothisarn and Jiriwibhakorn, 2003; Sanyal, 2004). Bettiol et al. (2003) proposed the use of radial basis function networks to estimate the CCT. Another method to assess power system transient stability using ANN is by means of classifying the system into either stable or unstable states for several contingencies applied to the system (Krishna and Padiyar, 2000; Sanyal, 2004). ANN method based on fuzzy ARTMAP architecture has also been used for TSA of a power system (Silveira et al., 2003). Boudour and Hellal (2005) proposed the use of combined supervised and unsupervised learning for evaluating dynamic security of a power system based on the concept of stability margin. Sawhney and Jeyasurya (2004) used ANN to map the operating condition of a power system based on a transient stability index which provides a measure of stability in power systems. Support Vector Machine (SVM) is another ANN method used for TSA (Moulin et al., 2004; Wang et al., 2005) in which the method has several advantages such as automatic determination of the number of hidden neurons, fast convergence rate and good generalization capability.
In this study, two new ANN methods are proposed and developed for transient stability assessment of power systems using PNN and Least Squares Support Vector Machine (LSSVM). Both ANN methods are considered new applications in transient stability assessment of power systems. The procedures of transient stability assessment using PNN and LSSVM are described and the performance of the PNN and LSSVM is compared with the MLPNN so as to verify the effectiveness of these two methods. Both the MLP and PNN networks were developed using the MATLAB Neural Network Toolbox, whereas the LSSVM was developed using the LSSVM Matlab Toolbox (Suykens et al., 2002).
MATHEMATICAL MODEL OF
MULTIMACHINE POWER SYSTEM
The differential equations to be solved in power system stability analysis
using the time domain simulation method are the nonlinear ordinary equations
with known initial values. Using the classical model of machines, the dynamic
behavior of an ngenerator power system can be described by the following
equations:
However,
By substituting (2) in (1), Eq. 1 becomes
Where:
δ_{i} 
= 
Rotor angle of machine i, 
ω_{i} 
= 
Rotor speed of machine i, 
P_{mi} 
= 
Mechanical power of machine i, 
P_{ei} 
= 
Electrical power of machine i, 
M_{i} 
= 
Moment of inertia of machine i. 
Equation 3 is then solved by using a time domain simulation
program through stepbystep integration so as to produce time response of all
state variables.
BACKGROUND ANN THEORY
Here the exolanation of two ANN methods, namely the PNN and the LSSVM are given. Both the ANN methods are used as classifiers to determine the stability of a power system.

Fig. 2: 
PNN pattern layer 
Probabilistic Neural Network: PNN which is a class of Radial Basis Function
(RBF) network is useful for automatic pattern recognition, nonlinear mapping
and estimation of probabilities of class membership and likelihood ratios (Specht,
1992). It is a direct continuation of the work on Bayes classifiers (Burrascano,
1991) in which it is interpreted as a function that approximates the probability
density of the underlying example distribution. The PNN consists of nodes with
four layers namely input, pattern, summation and output layers as shown in Fig.
1. The input layer consists of merely distribution units that give similar
values to the entire pattern layer.
For this work, RBF is used as the activation function in the pattern layer of the PNN.
The dist box shown in Fig. 2 subtracts the input
weights, IW_{1,1}, from the input vector, p and sums the squares of
the differences to find the Euclidean distance. The differences indicate how
close the input is to the vectors of the training set. These elements are multiplied
element by element, with the bias, b, using the dot product (.*) function and
sent to the radial basis transfer function. The output a is given as:
Where, radbas is the radial basis activation function which can be written
in general form as:
The training algorithm used for training the RBF is the orthogonal least squares
method which provides a systematic approach to the selection of RBF centers
(Chen et al., 1991).
The summation layer shown in Fig. 1 simply sums the inputs
from the pattern layer which correspond to the category from which the training
patterns are selected as either class 1 or class 2. Finally, the output layer
of the PNN is a binary neuron that produces the classification decision. As
for this work, the classification is either class 1 for stable cases or class
2 for unstable cases.
Least Squares Support Vector Machine (LSSVM): LSSVM is a reformulation of the standard SVM (Suykens and Vandewalle, 1999). The reformulation leads to solving a set of linear equations which is easier to solve than SVM quadratic equations. The reformulation does not result in SVM losing any of its advantage. LSSVM map input vectors to a higher dimensional space where a maximal separating hyperplane is constructed. Its mathematical formulations are described in this section.
Given the training data set, {x_{k}, y_{k}}^{N}_{k=1}, where, x_{k} ∈ú^{n} represent kth input pattern and y_{k}∈ is the kth output pattern, the LSSVM aims at constructing a classifier of the form,
Where, α_{k} are positive real constant and b is a real constant.
is the RBF kernel which is considered in this study.
The least squares version
to the SVM classifier is done by formulating the classification problem as:
subject to equality constraints,
Where, φ(x_{k}) is a nonlinear function which maps the input space
into a higher dimensional space.
By using the Mercer’;s Theorem, this function
is related to Ψ(x,x_{k}) as follows,
Equation 7 and 8 lead to KarushKuhnTucker
systems and can be written as the solution to the following set of linear equations,
Where:
Z 
= 
[φ(x_{1})^{T} y_{1};...;
φ(x_{N})^{T}y_{N}], 
Y 
= 
[y_{1};…;y_{N}], 

= 
[1;…;1], 
e 
= 
[e_{1};…;e_{N}], 
α 
= 
[α_{1};…;α_{N}]. 
Mercer’;s Theorem can be applied again to the matrix Ω_{kl} = ZZ^{T} where,
Hence, the solution to the classifier as given in Eq. 6 can
be found by solving the linear set of Eq. 10 and 11
instead of using quadratic programming for solving the equation as is the case
with SVM. The LSSVM network developed in this work uses the LSSVM Matlab Toolbox
(Suykens et al., 2002) in which the training of LSSVM is based on the
iterative solver conjugate gradient algorithm.
Performance evaluation of PNN and LSSVM networks: Performance of the developed PNN and LSSVM networks can be gauged by calculating the error of the actual and desired test data. Firstly, error is defined as:
Where:
n 
= 
The test data number. 
The desired output is the known output data or target data used for comparing with the neural network output. Meanwhile, the actual output (AO) is the output obtained from the trained neural network.
From Eq. 12, the mean error can be calculated using,
Where:
N 
= 
The total number of test data. 
The percentage classification error is given by,
MATERIALS AND METHODS
In the PNN and LSSVN methods used for transient stability assessment, the IEEE 9bus test system is used for verification of the methods. Before the PNN and LSSVM implementation, time domain simulations considering several contingencies were carried out for the purpose of gathering the training data sets. Simulations were done by using the MATLABbased PSAT software (Milano, 2005). Time domain simulation method is chosen to assess the transient stability of a power system because it is the most accurate method compared to the direct method. In PSAT, power flow is used to initialize the states variable before commencing time domain simulation. The differential equations to be solved in transient stability analysis are nonlinear ordinary equations with known initial values. To solve these equations, the techniques available in PSAT are the Euler and trapezoidal rule techniques. In this work, the trapezoidal technique is used considering the fact that it is widely used for solving electromechanical differential algebraic equations (Milano, 2007).
The type of contingency considered is the threephase balanced faults created at various locations in the system at any one time. When a threephase fault occur at any line in the system, a breaker will operate and the respective line will be disconnected at the Fault Clearing Time (FCT) which is set by a user. The FCT is set randomly by considering whether the system is stable or unstable after a fault is cleared. According to (Anderson and Fouad 2003), if the relative rotor angles with respect to the slack generator remain stable after a fault is cleared, it implies that FCT < CCT and the power system is said to be stable but if the relative angles go out of step after a fault is cleared, it means that FCT > CCT and the system is unstable.
Transient stability simulation on the test system: Figure
3 shows the IEEE 9bus system in which the data used for this work is obtained
from Anderson and Fouad (2003).

Fig. 3: 
IEEE 9 bus System 
The system consists of three Type2 synchronous generators with AVR Type1,
six transmission lines, three transformers and three loads.
Figure 4 shows examples of the time domain simulation results
illustrating stable and unstable cases. A three phase fault is said to occur
at time t = 1 second at bus 7. In Fig. 4a the FCT is set at
1.08 sec while in Fig. 4b the FCT is set at 1.25 sec. The
relative rotor angles of the generators oscillate and the system is said to
be stable (Fig. 4) whereas the relative rotor angles of the
generators go out of step after a fault is cleared and the system becomes unstable
(Fig. 4b). It can be deduced from Fig. 4
that the FCT setting is an important factor to determine the stability of power
systems. If FCT is set at a shorter time than the CCT of the line, the system
is stable; otherwise the system will be unstable.
Data preprocessing: The simulation on the system for a fault at each line runs for 5 sec at a time step Δt, set at 0.001 sec. The fault is set to occur at one second from the beginning of the simulation. Data for each contingency is recorded in which one steady state data is taken before a fault occurs and 20 sampled data are taken for one second duration after a fault occurs. There are 25 contingencies simulated on the system and this gives a size of 25χ21 or 525 data collected.
The collected data are further analyzed and trimmed down to 468 due to repetitions
of data. The one steady state data taken before all faults occur are reduced
to one since the values will be the same for all faults. Next, the repetitions
are due to faults that occur on the same line. The FCT of the same line are
set at four different times, two for stable cases and two for unstable cases.
At the start of a fault, same values of data are recorded for all the four faults.

Fig. 4: 
Relative rotor angle curves of generators for (a) stable and
(b) unstable cases 
Table 1: 
Input features selected 

A few milliseconds after a fault, the recorded data differ from each other
due to different FCT settings. Due to repetitions of data recorded, one data
out of the four different FCT settings are kept. These data are denoted as data
for stable cases. The data collected are normalized so that they have zero mean
and unity variance.
There are 468 sets of data collected from simulations in which a quarter of the data which is 117 are randomly selected for testing and the remaining 351 data are selected for training the neural network.
Input features selection: The selection of input features is an important
factor to be considered in the ANN implementation. The input features selected
for this study are relative rotor angles (δ_{i1}), motor speed
(ω_{i}), generated real and reactive powers (P_{gen}, Q_{gen}),
real and reactive power flows on transmission line (P_{line}, Q_{line})
and the transformer powers (P_{trans}, Q_{trans}). Overall there
are 29 input features to the ANN (Table 1). The breakdown
of the input features selected for the neural network.
TEST RESULTS
Here, the results obtained from the PNN and LSSVM for transient stability assessment are presented. Initially, the PNN results using 29 input features are given and discussed. Then, results obtained from LSSVM using the same input features as PNN are presented and discussed. For the purpose of evaluating the effectiveness of the PNN and LSSVM, the results of the multi layer perceptron neural network (MLPNN) are also presented. Finally, comparisons are made between the PNN, LSSVM and MLPNN results for transient stability assessment.
PNN results for transient stability assessment: The PNN developed in
this study is used for classifying power system transient stability states in
which the PNN classifies 1 for stable cases and 2 for unstable cases. The architecture
of the PNN is such that it has 29 input neurons, the hidden layer neurons equal
the number of training data which is 351 and with a single output neuron. The
PNN testing results using the 29 input features. The shaded cells in the table
denote the misclassification of test data (Table 2). From
the Table 2 it can be deduced that the false alarm rate is
0.86% and the false dismissal rate is 0.86%. False dismissal rate is the rate
of unstable cases assigned to the stable cases and the false alarm rate is the
rate of stable cases assigned to the unstable cases. Thus, the total error of
misclassification (false alarm rate + false dismissal rate) and the mean error
are both 1.71%.
LSSVM results for transient stability assessment: The developed LSSVM
is also used for classifying power system transient stability states in which
it classifies 1 for stable cases and 2 for unstable cases similar to that of
PNN. The architecture of LSSVM is such that it has 29 input neurons, 351 hidden
neurons which is the same as the number of training data and a single output
neuron. The trained 351 hidden neurons are used to classify the 117 test data.
Table 3 shows the LSSVM testing results in which the shaded
cells in the table denote the misclassification of test data. From the table,
it can be deduced that the false alarm rate is 1.71% and the false dismissal
rate is 1.71. Thus, the percentage error of misclassification and the mean error
are both 3.42%.
Table 2: 
PNN testing results 

MLPNN results for transient stability assessment: The architecture of
the MLPNN is such that it has 29 input neurons representing the 29 input features,
one hidden layer with 13 neurons using the hyperbolic tangent transfer function
and a single output neuron. The mean squared error is used as a goal for training
the neural network which is set at 0.03. The training algorithm used for this
network is the resilient back propagation algorithm (Riedmiller and Braun, 1993).
The performance goal was met at 41,050 epochs with a training time of 25 min
32 sec.
From the Table 4 the calculated mean error is 6%. As shown
in Table 4, some of the MLPNN outputs are not crisp 0 or 1
but in the range 0 to 1, where 0 indicates the system is stable and 1 when the
system is stable. So for classification purpose, a decision rule is used such
that if the MLPNN output is in the range of 0.9 to 1.1 (±10%), it will
indicate that the system is stable (class 1) whereas if the MLPNN output is
in the range of 0.1 to 0.1 (±10%), it means that the system is unstable
(class 2). For MLPNN output outside this range of values, it is considered as
misclassified. The column indicated by C in the table shows the classification
of the converted MLPNN outputs so that they can be easily compared with the
desired outputs to determine the accuracy of the MLPNN. Classes 1 and 2 are
used in column C instead of 1 and 0 for stable and unstable classification so
that the results conformed to the results obtained from PNN and LSSVM. By using
this decision rule the number of misclassified data is 13 out of 117 test data,
which is 11.1%. The shaded cells in the table are the respective misclassified
data which are denoted as x in the column C.
Comparison of neural network results in transient stability assessment:
It can be concluded that the performance of PNN is better compared with LSSVM
and MLPNN in transient stability assessment of the 9 bus power system (Table
5).
Table 3: 
LSSVM testing results 

Table 4: 
MLPNN results using 29 input features 

Table 5: 
Summary of PNN, LSSVM and MLPNN results 

The mean error for PNN is 0.017 compared to 0.0342 for LSSVM network and the
percentage classification errors are also less for PNN (1.71%) compared to 3.42%
for LSSVM, respectively. For MLPNN, there are no false alarms and false dismissals
but the mean error and misclassification percentage are higher than both PNN
and LSSVM which are 0.06 and 11.1% respectively. In terms of training time,
the PNN has the shortest training time (1.32 sec) compared to the time taken
to train the LSSVM (1.7 sec) and MLPNN (25 min 32 sec). The difference in training
time for PNN and LSSVM is insignificant compared to the time taken to train
the MLPNN. In general, the performance of PNN and LSSVM are better compared
to MLPNN and that PNN gives the best performance among the three methods.
CONCLUSION
The use of PNN and LSSVM has been proposed for transient stability assessment of the 9bus power system by means of classifying the system into either stable or unstable states for several three phase faults applied to the system. Time domain simulations were first carried out to generate training data for both neural networks and to determine transient stability state of a power system by visualizing the generator relative rotor angles. The PNN and LSSVM networks are then compared with the MLPNN so as to evaluate its effectiveness in transient stability assessment. The performances of PNN and LSSVM compared to the MLPNN are better in terms of mean and misclassification errors and training time. Results also show that among the three methods used in this work, the PNN gives the best performance in terms of accuracy in classifying the transient stability states. Thus, the PNN and LSSVM networks are promising methods for transient stability assessment of power systems.