1. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING &
ISSN 0976 – 6553(Online) Volume 4, Issue 5, September – October (2013), © IAEME
TECHNOLOGY (IJEET)
ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online)
Volume 4, Issue 5, September – October (2013), pp. 141-145
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VOLTAGE STABILITY OF POWER SYSTEMS USING REGRESSION
MODELS AND K MEANS CLUSTERING
¹P Veeranjaneyulu
Associate Professor, Malla Reddy Institute of Engineering and Technology, Misammaguda,
Secunderabad
²Dr. T Purna Chandra Rao
Professor(retired), NIT, Warangal, India
ABSTRACT
The main purpose of this paper is to present regression models to improve the voltage
stability of entire power system. The proposed method is focused on appropriate modeling of power
system’s voltage stability and computation of most vulnerable voltage stability margins. In
association with regression models, the proposed work also highlights the hierarchical clustering
method called k-means clustering to obtain voltage stability.
Keywords: Problem domain, voltage stability, regression models, distance functions, cluster centers.
1. INTRODUCTION
Voltage stability has become a major concern in many power systems where the reason is the
voltage instability. Voltage stability is concerned with the capability of the power system to maintain
acceptable voltages at all the buses in the system under the normal conditions and after being
subjected to a disturbance. Once associated primarily with weak systems and long lines, voltage
problems are now also a source of concern in highly developed networks as a result of heavier
loadings. The review paper by Ajjarapu and Lee [8] presents an exhaustive list of work done in the
area of voltage stability till 1998. The phenomena which contributes to the voltage stability have
been described, the various countermeasures to avert it have been enumerated and the various
computer analysis methods used or proposed so far have been presented in a coherent way in [16].
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2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 4, Issue 5, September – October (2013), © IAEME
2. MATERIALS AND METHODS USED
Over the years, voltage stability of distribution systems has received a challenging focus with
a need for both analysis and enhancement of the operating conditions. The Voltage Stability problem
of radial distribution system from its single line equivalent has been investigated and the voltage
stability index (VSI) for identifying the node that is most sensitive to voltage collapse has been
developed in [20], [21] and [25]. The determination of the location, size, number and type of
capacitors to be placed are of great significance, as it reduces power and energy losses, increases the
available capacity of the feeders and improves the feeder voltage profile. Numerous methods for
solving this problem in view of minimizing losses have been suggested in the literature [[30]- [34]].
Algorithms for enhancing voltage stability of transmission systems by optimal capacitor placement
have been discussed [[35]- [36]]. A relationship between voltage stability and loss minimization has
been developed and the concept of maximizing voltage stability through loss minimization has been
outlined [[37]- [38]]. Algorithms for enhancing voltage stability of distribution systems by network
reconfiguration that alters the topological structure of the distribution feeders by rearranging the
status of switches have been suggested [[39]- [41]]. However, there is no work till date to improve
the stability of the system as a whole or to improve the stability of particular buses which are in our
interest. In the literature several indices are been proposed to indicate the voltage stability of power
systems. The L-Index method is proposed in [3] which attempts to provide a measure of the stability
of the load buses in a system by ranking them according to a parameter(L-Index). The eigenvalues
and eigenvectors of the power flow jacobian have been used in [22] to characterize the stability
margin in a system. In this paper we are using L-Index [3] and Jacobin matrix [7] to derive the LIndex sensitivity matrix denoted as ( Lq ), which is used to calculate the optimal location of the
Journal of Information capacitors. In this paper the affect of placing a capacitor at a bus on the
remaining buses for radial and meshed systems is found out. L-Index Sensitivities( Lq )matrix which
gives the information of the change in value of L-Index [3] with change in reactive power injection
at any bus in the system has been proposed. A new method is developed to improve the stability of
the system using L-Index sensitivities approach( Lq ) which is applicable to improve the stability of
radial and meshed systems. We have used linear programming optimization technique to get location
and amount of reactive power to be injected.
The objective of this paper is to develop a method which is applicable for voltage stability
based on regression techniques.
3. REGRESSION METHOD
To solve real world applications, we apply precise models like problem domain knowledge of
the system and proper functioning of the system. Generally a model can be defined as a combination
of structure and parameters.
Model = Structure + Parameter.
Models can be broadly classified as linear and non linear. The main focus of this paper is on
linear regression model. A linear model is linear regarding its parameters.
Regression model is the statistical methodology for gauging values of outputs from a
collection of input values.
The output regression model is expressed as
y= Xw+€, where y is output vector, X is the data matrix w is the parameter vector and € is the error
vector. The output is a linear function of the parameters. The data matrix describes the modeling
function. The error distribution is normal distribution. General regression models use the below
listed assumptions.
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3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 4, Issue 5, September – October (2013), © IAEME
•
•
•
Expected value of errors is zero
Errors do not correlate with each other
Error variance S is a constant.
70
60
50
40
30
20
10
0
-10
-20
-5
0
2
3
4
5
6
7
8
Fig 1: Extrapolation of the polynomial model
The above table describes the bad extrapolation capacity of a polynomial model when there is
insufficient data.
The linear regression model is used to determine if the linear model has the enough flexibility
to approximate function between the pre disturbance operation point and the most critical voltage
stability margin. The regression model parameters are solved by least squares estimation. The
estimate minimizes the sum square error of the residual vector.
The minimum of sum squares error is achieved by solving the normal equation of the least
squares estimation problem. If the columns of the data matrix are independent then there is an
explicit solution for the problem.
The output and parameter estimate are specified as follows.
y=Xw
The outputs vary slightly due to errors. In order to compute variance of parameters, an estimate of
error variance and correlation matrix are needed
.
400
350
300
250
200
150
100
50
0
Transfer2to4
-2
1
0
1
3
4
Fig 2: Distribution of Tie line flows
K Means clustering is based on the concept of input vector classification of distance functions
and also on reduction of sum of squared distances from all points in a cluster domain to the cluster
center.
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4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 4, Issue 5, September – October (2013), © IAEME
4. K MEANS ALGORITHM
1. Choose K initial cluster centers. They are the first k samples.
2. At the kth iteration distribute the input vectors among K cluster domains using x€s(k)
3. Compute new cluster centers such that sum of the squared distances from all points in s(k) to
the new center is minimized.
4. If Z(k+1)=Z(k) then the algorithm is converged.
Clusters
Fig 3: Clustering environment
5. RESULTS
The above graph shows the distribution of tie-line flows at maximum loading point. Only few
cases cross 3pu transfer in the post distribution matrix loading point. Ie the maximum transfer limit
can be as low as 2.8pu. The wide distribution of the power transfers is due to the changes.
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
Fig 4: Relation between principle components and their variance
The number of cluster centers for active line flows, reactive line flow and voltage are 10, 15
and 15 respectively. Here k means cluster algorithm is fast can manage large amounts of data.
6. CONCLUSION
The proposed method described the most critical post disturbance voltage stability margin.
Regression models and K means clustering were used to reduce the number of inputs and to improve
model generalization capability. The results prove the ability of the proposed method to approximate
the voltage stability margin.
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5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 4, Issue 5, September – October (2013), © IAEME
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