40220130405014 (1)

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40220130405014 (1)

  1. 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 © IAEME: www.iaeme.com/ijeet.asp Journal Impact Factor (2013): 5.5028 (Calculated by GISI) www.jifactor.com IJEET ©IAEME 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]. 141
  2. 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. 142
  3. 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. 143
  4. 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. 144
  5. 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 7 REFERENCES 1. Adibi MM & Milanicz DP: Reactive capabillity limitations of synchronous machines- IEEE transaction on power systems Vol 9, No 1, Feb 1994, pp. 29-35. 2. Ajjarapu V & Christ C : The continuation of power flow: a tool for steady state voltage stability analysis. IEEE Transaction on power system, Vol 7, NO 1, Feb 1992, PP. 416-443. 3. Sami Repo: online voltage stabillity assessment of power system – An approach to the Black Box modelling. 4. Power system stability and control Prabha S Kundur ,(Book) – EPRI, Power system Engg Series. 5. Kessel. P &Glavitsch.H(July 1986) Estimating the voltage stability of power system, IEEE Transactions on Power delivery. 6. Kundur. P D, Lee C, Bayne J.P. & Dandeno. P.L(June 1985) Impact of Turbine Generator Controls on Unit. 7. Performance under System Disturbance Conditions, IEEE Transactions on Power Apparatus and Systems. 8. Thukkaram D & Abraham Lomi(May 2000). Selection of static VAR compensator Location and size for SystemVoltageStabilityImprovement, Electrical Power Systems Research. 9. Deepika Khurana and Dr. M.P.S Bhatia, “Dynamic Approach To K-Means Clustering Algorithm”, International Journal of Computer Engineering & Technology (IJCET), Volume 4, Issue 3, 2013, pp. 204 - 219, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. 10. Suresh J. Thanekar, Waman Z. Gandhare and Anil P. Vaidya, “Voltage Stability Assessment of a Transmission System - A Review”, International Journal of Electrical Engineering & Technology (IJEET), Volume 3, Issue 2, 2012, pp. 182 - 191, ISSN Print : 0976-6545, ISSN Online: 0976-6553. 11. Champa Nandi, Sumita Deb and Minakshi Debbarma, “Voltage Stability Improvement using Static Synchronous Compensator in Power System with Variable Load Impedance”, International Journal of Electrical Engineering & Technology (IJEET), Volume 1, Issue 1, 2010, pp. 108 - 117, ISSN Print : 0976-6545, ISSN Online: 0976-6553. 145

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