Radial basis function networks for fast contingency ranking

Transcription

1 Eectrica Power and Energy Systems ) 387±395 Radia basis function networks for fast contingency ranking D. Devaraj a, *, B. Yegnanarayana b, K. Ramar a a Department of Eectrica Engineering, Indian Institute of Technoogy, Madras , India b Department of Computer Science and Engineering, Indian Institute of Technoogy, Madras , India Received 1 December 2000; revised 28 February 2001; accepted 2 Apri 2001 Abstract This paper presents an arti cia neura network-based approach for static-security assessment. The proposed approach uses radia basis function RBF) networks to predict the system severity eve foowing a given ist of contingencies. The RBF networks are trained off-ine to capture the noninear reationship between the pre-contingency ine ows and the post-contingency severity index. A method based on mutua information is proposed for seecting the input features of the networks. Mutua information has the advantage of measuring the genera reationship between the independent variabes and the dependent variabe as against the inear reationship measured by the correation-based methods. The performance of the proposed approach is demonstrated through contingency ranking in IEEE 30-bus test system. q 2002 Esevier Science Ltd. A rights reserved. Keywords: Static-security; Radia basis function network; Contingency ranking; Mutua information 1. Introduction Security assessment is one of the key issues in power system operation and panning. Assessment of static-security of a power system enabes us to detect, through simuation, any potentia ine ow vioations or an out-of-imit votage foowing a given ist of contingencies. For arge power systems, due to the time constraint invoved in reatime operation, those contingency cases which are potentiay harmfu to the system must be identi ed and detaied anaysis are carried out for these cases aone. This process of seecting the contingencies according to their severity is referred to as contingency seection. Over the years a number of agorithmic contingency seection methods have been proposed to speed up the process of contingency anaysis. They can be broady cassi ed into two categories: sensitivity-based ranking methods [1,2] and screening methods [3,4]. Ranking methods utiize an approximate system-wide scaar performance index PI) to quantify the severity of each contingency. The PI is used to rank a the contingencies. Screening methods use approximate or partia network soutions. Some of the soutions suggested incude DC oad ow, one iteration of AC oad ow, oca soution methods, etc. Sensitivity-based * Corresponding author. Te.: E-mai address: deva@neuron.iitm.ernet.in D. Devaraj). ranking methods are ef cient but vunerabe to misrankings, whie screening methods are accurate but inef cient. Recenty, arti cia neura networks ANN) have been proposed as toos for contingency screening and ranking [5±9]. Most of the authors have used feedforward neura networks with sigmoida noninearities, for mode deveopment. Any continuous function can be approximated to within an arbitrary accuracy by carefuy choosing the parameters in the network provided the network structure is suf cienty arge. But the shortcoming of this network is that it takes ong time for training. Aso, feedforward network with sigmoida activation function in the hidden nodes has no inherent abiity to detect the outiers. Even though training is done in off-ine, short training time is preferred as one may have to retrain the networks on a reguar basis as the topoogy or the system condition changes. Outiers can occur in practice, because it is hard to produce a compete training set representing a possibe operating conditions of a power system. In this paper, we propose radia basis function RBF) networks [10] to capture the noninear reationship between the pre-contingency system state and the post-contingency severity eve foowing a contingency. RBF networks take ess time for training and the distance-based activation function used in the hidden nodes gives the abiity to detect the outiers during estimation. Input feature seection pays an important roe in ANNbased approaches. Various statistica methods have been /02/$ - see front matter q 2002 Esevier Science Ltd. A rights reserved. PII: S )

2 388 D. Devaraj et a. / Eectrica Power and Energy Systems ) 387±395 contingency to ine overoad is expressed by the foowing scaar PI: proposed for feature seection in contingency seection modes. Pang et a. [11] used cass separation capabiity secure/insecure) of the variabes as the criterion to seect the input features of their statistica security cassi er. The foowing index was used to measure the intercassdistance: F i ˆ Fig. 1. Schematic representation of the earning stage. m S i s S 2 i 2 m I i 1 s I 2 i where m S i and m I i are the mean of the variabe i for the secure and insecure cass and s S i and s I i are their variances. In Refs. [5,9] aso the authors have used the same intercass-distance measure for feature seection in their ANN-based contingency seection modes. Ghosh and Chowdhury [7] used the correation coef cient between the input variabes and the output severity measure to seect the input features for their feedforward neura network mode. The intercass-distance measure assumes Gaussianity in the input domain. If a variabe x has norma or Gaussian distribution, its distribution is competey characterized by its mean and variance. If this assumption is not true, serious errors may occur in feature seection. Aso, methods based on inear dependence ike correation) cannot measure arbitrary reations between the independent and dependent variabes. In this paper, we propose mutua information [12] between the dependent and independent variabes as the criterion to seect the input features of the RBF networks. 2. Proposed methodoogy for contingency seection The proposed method for contingency seection is based on RBF neura networks. The objective is to estimate the severity eves for each contingency. The study presented in this paper focuses on singe ine outages. The severity of a 1 " # PI MVA ˆ XN S post 2m 2 ˆ1 S max where S post is the post-contingency MVA ow in ine, S max the MVA rating of ine, N the no of ines in the system and m is the integer exponent. The contingencies can then be ranked according to the order of their severity. Foowing a contingency, any ine which is overoaded wi make a contribution of greater than unity to the PI, whereas a ine whose ow is beow its rating wi make a contribution of ess than unity. A sma vaue of m in Eq. 2) woud resut in `masking effect' and a very high vaue of m may cause the na ordering to become worse due to increased noninearity [7]. For the IEEE 30-bus system considered in this paper, we have xed the vaue of m as 5. These PI vaues wi be used as target vaues for the ANN during training. It is possibe to train a singe network to estimate the PI vaues of a the contingencies by taking the system state variabes as the input and the system severity eves as the output of the network. But, as the dimension of the input vector increases, the number of basis functions hidden ayer nodes) required to approximate the given function rises exponentiay [10]. Such arge networks are inef cient and sensitive to over tting and exhibit poor performances. So, the probem of estimating the post-contingency severity eve using ANN is decomposed into severa networks, with each one deaing with one contingency. Whie training a network dedicated to a contingency, the ocaized nature of contingencies coud be expoited in the form of dimension reduction through proper feature seection. The schematic representation of the earning stage of the mode for PI estimation is shown in Fig. 1. For mode deveopment, a arge number of training data is generated through off-ine power system simuation. Pre-contingency state power ows are the input to the modes and the PI vaue foowing a contingency is the output of the mode. A mutua information-based feature seection technique is appied to identify the reevant attributes from the set of system state variabes for each contingency mode. By seecting ony the reevant variabes as input features and excuding irreevant ones, higher performance is expected with smaer computationa effort. The seected input features and the output are normaized between 0and 1 and presented to the RBF networks for training. Once the networks are trained, they are ready for contingency ranking at various oad conditions. The detais of mutua information-based feature seection and the architecture and training of RBF network are presented in the foowing sections. 3. Feature seection One of the important issues in ANN-based approach is the

3 D. Devaraj et a. / Eectrica Power and Energy Systems ) 387± proper seection of input features to the mode. The probem of feature seection is stated as foows: given an initia set of n features, nd the subset with k, n features that is `maximay informative' about the output. As most of the contingencies are ocaized in nature, a the variabes in the input vector may not exert equa in uence on the post-contingency PI vaues. Irreevant and redundant attributes in the input not ony compicate the network structure, but aso degrade the performance of the networks. By seecting ony the reevant variabes as input features and excuding irreevant ones, higher performance is expected with smaer computationa effort. This section presents the detais of input feature seection based on mutua information. known and the best we can do is to use the histogram of the data. The steps invoved in cacuating the mutua information from the histogram of the data are given beow: 1. Arrange a the PI vaues in the descending order and divide them into N y casses equay. 2. Cacuate the initia entropy using Eq. 3). 3. Sort the data points in the rst input variabe in the descending order. Divide the sorted patterns into N x groups equay. 4. Compute the conditiona entropy, given the input vector 1 using Eq. 4) and cacuate the mutua information using Eq. 5). 5. Repeat steps 3 and 4 for the remaining variabes aso De nition of mutua information Consider a stochastic system with input X and output Y. Let the discrete variabe X has N x possibe vaues and Y has N y possibe vaues. Now the initia uncertainty about Y is given by the entropy H Y) which is de ned as [13] H Y ˆ2 XN y p j og p j where p j ˆ P Y ˆ y j is the probabiity of occurrence of the event Y ˆ y j : The amount of uncertainty remaining about the system output Y after knowing the input X is given by the conditiona entropy H YuX) which is de ned as 0 1 H YuX ˆ2 XN x X N y p p ji og p ji A 4 iˆ1 where p i ˆ P X ˆ x i is the probabiity of occurrence of the event X ˆ x i and p ji is the conditiona probabiity for output y j given the input x i. Now the difference H Y) 2 H YuX) represents the uncertainty about the system output that is resoved by knowing the input. This quantity is caed the mutua information between the random variabes X and Y. Denoting it by I Y;X), we may thus write I Y; X ˆH Y 2 H YuX The mutua information is therefore the amount by which the knowedge provided by the feature vector decreases the uncertainty about the output Seecting the features with the mutua information Mutua information between two random variabes measures the amount of common information contained in these variabes. The probem of seecting input features which contain much of the information of output can be soved by computing the mutua information between each variabe and output and seecting those variabes having higher mutua information vaues. To compute mutua information the probabiity distribution function of variabes is needed which in practice is not RBF networks RBF network is a cass of singe hidden ayer feed forward neura network [10,14]. The input nodes pass the input to the hidden nodes directy and the rst ayer connections are not weighted. The transfer functions in the hidden nodes are simiar to the mutivariate Gaussian density function f j x ˆexp 2 ix 2 m ji 2! 6 2s 2 j where m j is the vector determining the center of basis function f j and s j are their widths. Each RBF unit has a signi cant activation over a speci c region determined by m j and s j ; thus each RBF represents a unique oca neighborhood in the input space. The connections in the second ayer are weighted and the output nodes are inear summation units. The vaue of the kth output node y k is given by y k x ˆXh w kj f j x 1 w k0 where w kj is the connection weight between the output and jth hidden node and w k0 is the bias term. The training in RBF networks can be decomposed quite naturay and the earning is done in three sequentia stages as against the singe optimization procedure foowed in backpropagation network training. The rst stage of the earning consists of determining the unit centers m j by the K-means custering agorithm, see Appendix A). Next, we determine the unit widths s j using a heuristic approach that ensures the smoothness and continuity of the tted function. The width of any hidden node is taken as the maximum Eucidean distance between the identi ed centers. Finay, the weights of the second ayer connections are determined by inear regression using a east-squares objective function. RBF networks can be viewed as an aternative too for earning in neura networks. Whie RBF networks exhibit 7

4 390 D. Devaraj et a. / Eectrica Power and Energy Systems ) 387±395 Fig. 2. IEEE 30-bus test system. the same properties as backpropagation networks such as generaization abiity and robustness, they aso have the additiona advantage of fast earning and abiity to detect outiers during estimation. 5. Simuation resuts To demonstrate the appicabiity of the proposed RBF network-based approach for contingency ranking, IEEE 30-bus system shown in Fig. 2 is seected as the test system. The transmission ine parameters, generator ratings and base oad condition are given in Ref. [15]. The system has six generators and 41 transmission ines. Forty singe ine outages except ines 25±26 are chosen for contingency anaysis. The various steps invoved in deveoping and evauating the system for contingency ranking and seection are presented beow Training data generation In machine earning approaches training data is the ony avaiabe information to buid the mode, and so they shoud represent the compete operating conditions of the system. For contingency anaysis mode deveopment, input±output patterns are generated as per the foowing procedure: a) First, a range of situations is generated by randomy perturbing the oad at a buses between 70and 140% of their base case vaue and by adjusting the generator output in proportion to the output in the base case condition. b) For each oad-generation pattern pre-contingency ine ows are obtained by soving the oad ow equations using Newton±Raphson agorithm. c) Aso, for each oad-generation pattern, the singe ineoutages speci ed in the contingency ist are simuated sequentiay and their PI vaues evauated by conducting AC oad ow. Based on the above simuation procedure, a training set consisting of 750input±output pairs was created. Additionay, a test set of 250data pairs was generated in order to evauate the earning and generaization abiities of the networks. In a the 1000 patterns it was noticed that the PI MVA vaues are very ow for 31 contingencies. These contingencies are not considered for mode deveopment and the modes are deveoped for the remaining nine cases aone Feature seection Pre-contingency ine ows in a the ines are chosen as the input to the networks and they are 41 in number. The PI corresponding to a contingency is the output of the network.

5 D. Devaraj et a. / Eectrica Power and Energy Systems ) 387± with wide ranges than to those with narrow ranges. To overcome this probem, input data is normaized before presenting it to the custering agorithm. The input data is normaized between 0and 1 using x n ˆ x 2 x min 8 x max 2 x min where x min and x max are the minimum and maximum of the variabe x. Simiary, output data is aso normaized between 0and Network training and evauation Fig. 3. Mutua information for variabes in mode 1±3. To seect the input features of each mode, input feature space is partitioned into ve and the output is divided into three groups. Mutua information of each variabe with respect to the output is evauated foowing the steps given in Section 3.2. For iustration, the mutua information between the input variabes and the output for contingency mode 1±3 is shown through a bar graph in Fig. 3. From this gure it is evident that ony a few variabes are having signi cant information to estimate the PI, and the remaining variabes have very ess amount of information. The rst few variabes that have high mutua information vaue are seected as features to train the networks. The features seected for satisfactoriy training the networks are given in Tabe Data normaization The rst stage of RBF network earning is the identi cation of the custer centers through K-means custering agorithm, which uses Eucidean distance as a measure of dissimiarity. Distance norms are sensitive to variations in the numerica ranges of the different features. For exampe, the Eucidean distance assigns more weighting to features For the prediction of PI for each ine outage, separate RBF network is trained. The seected variabes after normaization are presented to the network. Twenty iterations of the custering agorithm foowed by inear regression are performed to estimate the parameters of the network. As the vaue of h is not known in advance, a tria-and-error procedure is foowed to seect the optimum number of basis functions. After training, the generaization performance of the networks are evauated with the 250test data. The resuts of training and testing phase for a the nine modes are presented in Tabe 1. The resuts ceary show that the training of the RBF networks has been successfu and the correct estimation of PI has been achieved by the RBF network even for previousy unseen data. Tabe 2 presents the PI vaues estimated for one particuar oad condition with the ranking of the contingencies given in the parenthesis. For comparison, the actua vaues of PI cacuated from AC oad ow study are aso presented. The resut shows the agreement between the actua ranking and the ranking based on the output of the RBF networks Comparison with mutiayer perceptron network To compare the performance of the proposed RBF network-based approach with the commony used neura network architecture, mutiayer perceptron MLP) networks are deveoped to estimate the PI vaues. The networks are trained with the conjugate gradient agorithm [10] to reach the same error eve achieved by the RBF Tabe 1 Resuts of RBF networks S. No. Line outage Seected features S ; ˆ 1; 2; ¼; 41 No. of basis functions Training time s) Testing error mse) ,2,4, : ,2,4, : ,2,4, : ,5,8, : ,2,4,6,7, : ,5,8, : ,14,18,21,27, : ,14,18,22,23,24, : ,36,37,38,39, :

6 392 Tabe 2 PI estimation Line outage PI vaue D. Devaraj et a. / Eectrica Power and Energy Systems ) 387±395 Tabe 3 Resuts of MLP networks Line outage Hidden nodes Training time s) Testing error mse) networks during training. After training, the networks are tested with the test data. The resuts of training and testing for the MLP networks are presented in Tabe 3. Based on the information presented in Tabes 1 and 3, it is observed that RBF networks take ess time for training, but they require more number of hidden nodes as compared to MLP networks. Apart from that RBF network exhibits better generaization performance than the MLP network in most of the cases. 6. Concusions This paper has presented a neura network-based fast contingency seection method for power system staticsecurity assessment. A set of RBF networks has been trained to map the noninear reationship between the pre-contingency operating state and the post-contingency security indices. An effective feature seection method has been proposed to reduce the dimension of the input patterns. Simuation resuts on the IEEE 30-bus test system shows the proposed RBF network-based approach that provides an accurate estimation of post-contingency PI vaues for various operating conditions. When compared with MLP networks trained with backpropagation agorithm, the proposed approach signi canty reduces the deveopment time with improved estimation accuracy. Appendix A A.1. K-means custering agorithm The agorithm partition the data points x n, n ˆ 1; 2; ¼; N; into K disjoint custers C j containing N j data points, in such a way as to minimize the sum-of-squares custering function given by J ˆ XK X n[c j ix n 2 m j i 2 RBFN output Load ow resut ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) where m j is the mean of the data points in custer C j. A : : : : : : : : : The agorithm iterativey determines the custer centers m j as foows: 1. Initiaize the custer centers m j, j ˆ 1; 2; ¼; K by randomy seecting K data points from among a of the data points. 2. Generate a new partition by assigning each pattern to its cosest custer center. 3. Compute new custer centers as the centroids of the custers. 4. Compute the cost function according to Eq. A1). Stop if either it is beow a certain toerance vaue or its improvement over previous iteration is beow a certain threshod, otherwise go to step 2. References [1] Ejebe GC, Woenberg BF. Automatic contingency seection. IEEE Trans Power Apparatus Syst 1979;PAS-98 1):97±109. [2] Irisarri G, Sasson AM, Levner D. Automatic contingency seection for on-ine security anaysis-rea-time tests. IEEE Trans Power Apparatus Syst 1979;PAS-98 5):1552±9. [3] Brandwajn V. Ef cient bounding method for inear contingency anaysis. IEEE Trans Power Syst 1988;3 1):38±43. [4] Brandwajn V, Lauby MG. Compete bounding method for A.C. contingency screening. IEEE Trans Power Syst 1989;4 2):724±9. [5] Weerasooriya S, E-Sharkawi MA, Damborg M, Marks RJ. Towards static-security assessment of a arge-scae power system using neura networks. IEE Proc-C 1992;139 1):64±70. [6] Ray D, Chakravorti S, Mukherjee PK. An appication of arti cia neura network for power system security evauation. Inst Engr India) 1994;75:131±5. [7] Ghosh S, Chowdhury BH. Design of arti cia neura network for fast ine ow contingency ranking. Eectrica Power Energy 1996;18 5):271±7. [8] Testa A, Menniti D, Picardi C, Sorrent N. Steady state security prediction in presence of oad uncertainty. Eur Trans Eectric Power 1998;8 2):97±104. [9] Lo KL, Peng LJ, Macqueen JF, Ekwue AO, Cheng DTY. Fast rea power contingency ranking using a counterpropagation network. IEEE Trans Power Syst 1998;13:1259±64. [10] Bishop CM. Neura networks for pattern recognition. Oxford: Oxford University Press, [11] Pang CK, Prabhakara FS, E-Abiad AH. Security evauation in power systems using pattern recognition. IEEE Trans Power Apparatus Syst 1974;PAS-93 2):969±76.

7 D. Devaraj et a. / Eectrica Power and Energy Systems ) 387± [12] Battiti R. Using mutua information for seecting features in supervised neura net earning. IEEE Trans Neura Networks 1994;5 4):537±50. [13] Haykin S. Neura networks Ð a comprehensive foundation. New Jersey: Prentice Ha, [14] Yegnanarayana B. Arti cia neura networks. India: Prentice Ha, [15] Asac O, Scott B. Optima oad ow with steady state security. IEEE Trans Power Syst 1974;PAS-93:745±51.