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A complete fault location formulation for distribution systems using the k-nearest neighbors for regression and classification G. Morales-España, J. Mora-Flórez, Member IEEE, G. Carrillo-Caicedo
Abstract—This paper presents an alternative to the traditional impedance based fault location methods, using a simple technique of the learning approaches called k-Nearest Neighbors (k-NN), where besides the fault location distance, the multiple estimation problem is also addressed. This approach only uses the single end measurements of voltage and current available at the power substation. As principal advantage, considering the classical approaches, this alternative has not dependency on the power system model and also considers the spacial characteristics of the distribution systems. Furthermore, the multiple estimation problem, typical of all fault location approaches, is addressed. According to the proposed tests, faults location in different nodes and values of fault resistances are successfully determined, having an average error rate lower than 1.5% and 13% in distance estimation and zone identification respectively. Index Terms—Classification, fault location, k-Nearest Neighbors, learning approaches, multiple estimation, power distribution systems, regression.
I. I NTRODUCTION
I
N case of events at the power network, it is important to identify and locate the faults, because this makes possible a fast system restoration and consequently it improves the power quality and specially the service continuity. Faults in power systems are events caused by storms, damaged insulators, trees, animals, among others. Most of the currently cited fault location methods for distribution systems are based on those previously applied in transmission lines, but the complexity of the first networks causes errors and consequently a low performance [1]. Recently, several important techniques have been proposed to locate faults in radial power systems [2]-[11]. These techniques use the estimation of the equivalent impedance, as seen at the power substation, during fault circumstances by using the fundamental component of voltage and current, considering fault and prefault stages. The estimated impedance is basically compared with the computed using the model of the power feeder, and as a result, the faulted line section is estimated. The main disadvantage of these methods are related to multiple estimation of the fault location and the high dependence on the power system model. Additionally,
G. Morales-España is with the Institute for Research in Technology (IIT), Universidad Pontificia Comillas, Madrid, España (e-mail:
[email protected];
[email protected]). J. Mora-Flórez is with the Universidad Tecnológica de Pereira, Pereira, Colombia (e-mail:
[email protected]). G. Carrillo-Caicedo is with the Universidad Industrial de Santander, Bucaramanga, Colombia (e-mail:
[email protected]).
the special characteristics of the distribution systems as tapped loads, single phase laterals, multiple conductor gauges, system unbalance, among others, are frequently ignored. In this paper, the k-Nearest Neighbors (k-NN) is proposed as an integrated strategy used to locate faults in power distribution systems by merging the applications presented in [12] and [13]. This helps to overcome the individual drawbacks of each method, as the multiple estimation of the fault location, in case of the regression approach; and the possible ambiguity in the zone location, due to the zone size, in case of the classification approach. Additionally, the use of learning techniques reduce the model dependency, because these are mainly based on information obtained from fault databases. As contents, in section II the importance of fault location (zone and distance) is briefly described. Next, in section III the basic aspects of the k-NN technique (for classification and regression) are presented. In section IV the proposed fault locator is exposed. In section V a description of the tests is given and the obtained results are also analyzed. Finally, the last section is devoted to present the main conclusions of the proposed approach. II. I MPORTANCE OF DISTANCE AND ZONE FOR FAULT LOCATION
The purpose of determining the fault location is explained by considering the power distribution system proposed as example in [14] and shown in Fig. 1. This is a typical distribution system where two normally closed sectionalizers (S1 and S2) are used to supply the system from the power substation S/S1. There is an additional normally open sectionalizer (S3) used to give an additional source to the loads (S/S2) in case of faults, as an economical method to improve service continuity on distribution lines. In the example proposed and by using a distance estimation method to locate the fault F1, there is a decision problem because it is possible to obtain five different locations (multiple estimation), at the same electrical reactance Xd, as presented in Fig. 1a. To solve this problem, a strategy based on the zone division of the power system and the latter identification of the faulted zone by using a learning method, as graphically shown in Fig. 1b, is proposed. By identifying the zone five as the faulted, the multiple estimation problem is avoided. Additionally, once the faulted zone is identified, it is possible to perform switching actions (Opening S1 and S2, and closing S3) to immediately restore
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one neighbor helps to have immunity to noise, in addition, it smooths the curve estimation [18].
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A. k-NN: classification
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k-NN for classification can be easily illustrated with an example. Fig. 2a shows two type of data, triangles and squares, and the objective is to know the label of a new sample, indicated by an X. Therefore, the procedure to follow is to find the closer sample to the new one (X), and assign its label (triangle), as presented in Fig. 2b.
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Fig. 1: Distance estimation and zone division and identification to reduce the multiple estimation problem. (a) Data distribution and an new (un- (b) Prediction of the class of a known) sample. new sample, considering the nearest neighbors.
the service in unfaulted lines, although the fault remains unattended. By the simple switching actions index SAIDI is improved by the immediately restoration of load located in zones six and seven, reducing thus the non supplied load. Considering the problem and the proposed solution, the concept of zone is introduced as one subdivision of the power system. Such zone is latter identified by using a classification strategy based on a learning algorithm. The inputs of such algorithm are characteristics or features, obtained from the measurements of the fundamental component of current and voltage at the power substation. The problem here shown could be presented as a regression and a classification problem, where the fault distance can be obtained by using the regression strategy and finally the classification strategy is used to decide and choose one of the multiple fault locations. The k-NN is a simple and constrained method used in this paper to demonstrate the congruency of defining the faulted zone location as a classification problem [13] and the distance as a regression problem [12]. Having obtained a simple solution, by using k-NN algorithm, helps to conclude that the problem of fault location is well characterized to find the fault distance and unique location. III. L EARNING - BASED METHOD FOR CLASSIFICATION AND REGRESSION
One of the simplest methods of nonparametric approximation is the one based on the nearest neighbor rule [15], [16]. This method consists in estimating the value of an unknown sample from features of the nearest point based on a measure of similarity or distance. This rule is very attractive because it has well-established statistical properties and its simplicity when it is implemented for real applications [17]. The method based on the nearest neighbor can be extended using not only one but a set of closer points to predict the value of a new sample. This method is commonly known as the k-Nearest Neighbors (k-NN), where considering more than
Fig. 2: Location of a new (unknown) sample in a set of well known data. The case of noise is considered by the inclusion of a square within the data corresponding to the triangles. Therefore, classifying the new point (X) using the closest sample, as shown in Fig. 3a, leads to a possible error.
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Fig. 3: Prediction of the class of a new (unknown) sample using one and five nearest neighbors and considering noisy data. It may be noted that the new sample is classified as a square due to noise. To address this problem, the classification algorithm is modified by using a larger number of neighbors and the new label is then obtained by using the simple majority. This generalization of the method is called k-Nearest Neighbors [17]. Using k = 5, for this example, it is possible to see that the new sample belongs to the triangle class, as shown graphically in Fig. 3b. The noise problem is then solved with this new approach. The larger k is, the more robust the classification against noise is. However, the value of k has a limit because in case of high values, any new sample will always belong to the class with more data in the training set. For example, for the case presented in Fig. 3, if k = 31 then the new sample will always be classified as triangles, because there are 18 triangles and 13 squares.
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B. k-NN: regression The method of k-NN can be easily adapted to regression of continuous-valued functions [17]. The algorithm assumes that all data belong to Rp , and by some measure of distance, in this space, the closest k samples to the new sample x(q) are determined. This is to approximate to a function f : Rp → R, based on the already selected k samples. This function corresponds to the average of the labels of the closest k samples. If we consider the arithmetic mean (all data within the group have an equal degree of importance), the approximate function can be written as (1). k
f (xi ) (1) k As an important aspect, all data should be scaled in order to prevent that characteristics quantified with higher values dominate in the distance calculation. As an example, it is considered the construction of a curve using 50 samples linearly spaced in the interval [−5.5], which values correspond to the sinc(x) function, where the distance measure used is the classical Euclidean [18]. The solid line in Fig. 4 shows the regression curve using just the nearest neighbor (1-NN), it can be noticed the stepwise shape of the regression curve with respect to the reference curve. When applying k-NN method with arithmetic average, as shown in Figure 4, for k = 20, there is a smooth curve, but this improvement in the smoothness, in the regression curve, is uncompensated with a flatter trend that the curve takes as k increases. This is because the calculated value for a new sample correspond to the average value of the k closest samples. fˆ(xq )
←
Where KER is a Kernel function that determines the weighting of each sample based on the distance to the reference point given by the function d. The Kernel function corresponds to the weight given to the samples to obtain a weighted average, therefore it should vary inversely with distance, ensuring that the closest points have greater weight (more importance). Incorporating the weighting by distance solves the problem of making flatter the curve with large values of k, since the sample further away will have little effect on the value to predict. Using a weighting equal to the inverse square of the distance, equation (4) is considered. ωi ≡
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when xq = xi , which results in a zero denominator for ωi , then f (xi ) is assigned for the value of fˆ (xq ). However, although the kernel function used is perfectly consistent with the definition of the weighting, it has characteristics that gives the regression an over-training, since the curve is adjusted in such a way that always pass across the values of all training data, making the regressor vulnerable to noise. To solve this problem, the Kernel functions must consider a finite weighting at zero distance [19]. An example of this type of weighting, and the most common, is the Gaussian kernel defined as presented in (5). 2 ωi ≡ e−d(xq , xi ) (5) 1.2 Datos entrenamiento Curva real Regresion 10−NN ponderación gaussiana Regresion 10−NN
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Fig. 5: Regression of sinc(x) function using 10 nearest neighbors with and without weighing. 50 data with noise are used.
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Fig. 4: Regression of the sinc(x) function using one and 20 nearest neighbors. Due that k-NN method is based on distances, an obvious improvement, for the regression and the classification case, is to consider a weighted average which gives more importance to the data which are closer to the new sample [18]. Therefore, an approximation for the prediction function (1) takes the form of (2). K ωi · f (xi ) ˆ (2) f (xq ) ← i=1 k i=1 ωi ωi ≡ KER (d (xq , xi ))
(3)
Fig. 5 shows the application of the Gaussian kernel for regression using 10-NN, where normally distributed noise has been added with standard deviation of 3% of the signal amplitude, and the special feature that each 5 samples the noise has greater intensity (standard deviation 15%). The application of Gaussian kernel in this case improves the performance of the regression curve, mitigating sudden changes presented with the weighting used according to (4). Even though the regression curve, using the Gaussian kernel, has a high error in the evaluation of the training data, its behavior follows the variation of the real curve preserving its characteristic tendency. The value of k should be chosen in such a way that the regression curve results as similar as possible to the original,
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this can be achieved by using cross-validation [20]. Using cross validation k = 35 is obtained and the resulting curve is presented in Fig. 6. This last curve has a high degree on similarity to the curve obtained with k = 20 presented in Fig. 5.
D. Training
1.2 Datos entrenamiento Curva real Regresion 35−NN ponderación gaussiana
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length. It is recommended to subdivide laterals and long feeders into several zones, according to the presence of protective devices. In presence of laterals it is also recommended to define one zone for each lateral, and thus effectively avoid the multiple estimation problem.
This step is performed by using a training set of disturbances characterized by features as inputs, and the labels associated to the corresponding faulted zone and fault distance as output. This training set has to be complete and balanced. Complete in the sense that the training data set covers the whole search space and balanced to avoid a biased training on specific zones.
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Fig. 6: Regression of sinc(x) function with 35 nearest neighbors and weighing. 50 data with noise are used.
Having trained the learning based fault locators by using the entire training set, as presented in step four, an additional precision test must be carried out by using new data (not used in training phase). The error percentages are obtained using (6) and (7) for distance estimation and zone identification respectively.
IV. P ROPOSED STRATEGY %errorR =
A. Fault database acquisition Measurements used to train the learning method are registered at the power substation but also simulations of faults could be used, then in the last case a model is needed. One important advantage of the fault locator is given by the use of the available model and, as a consequence, it is not constrained to adapt the power system model to any particular need of a specific method (As it is normally the case for fault location methods [21]).
C. Definition of zones The power distribution network is subdivided into significant zones to be identified in the fault location process. The zone definition criterion considers the following aspects: system topology, presence of protective devices, feeder lengths and other maintenance crew criteria as the maximum search
(6)
N umber of wrong classif ied f aults × 100 T otal number of f aults (7) where lengthest is the estimated length to the fault point by the proposed method, lengthreal is the real length and Lengthtotal is the total length of the branch. %errorC =
V. T EST SYSTEM AND RESULT ANALYSIS
B. Signal processing This stage consists in obtaining a set of attributes that represent the fault. Each set of attributes is associated with a label that corresponds to the distance and zone related with the fault position. In this case, only attributes which are easy to obtain are considered, then the proposed approach is not conditioned by the equipment limitations. These attributes are related to the fault location and are based on the variations of the fundamental component of phase current (I), phase voltage (V ). These are defined as the subtraction of rms values during the fault and prefault steady states [14]. Three phase descriptors are used, to consider the mutual coupling between phases. Finally, the input descriptors are scaled to the interval [0,1] in order order to prevent that characteristics quantified with higher values dominate in importance (see Section III).
Lengthest − Lengthreal × 100 Lengthtotal
The proposed fault locators based on k-NN are tested in a power distribution system of SaskPower of 25 kV of Saskatchewan (Canada) presented in [7] (see Fig. 7). This power system has been used in several fault location studies, as presented in [7], [21], [22], [23]. Zona 1
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Fig. 7: 25 kV radial power distribution system form SaskPower in Saskatchewan (Canada),
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Fig. 8: Test errors of fault distance estimation using just voltage (V ). 12
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Fig. 10: Test errors of fault distance estimation using voltage and current. TABLE I: Average errors for zone and distance estimations Signals Va,b,c Ia,b,c Va,b,c , Ia,b,c
% Classification error 15.97 30.83 12.08
% Distance error 1.483 1.437 1.542
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Fig. 9: Test errors of fault distance estimation using just the current (I). A. Description of the tests To test the proposed fault location approach, faults in all of the nodes of the test power system are simulated, using 21 different fault resistances, which varies from 0Ω to 40Ω considering a step of 2Ω [24]. The zones are defined in order to avoid the multiple estimation as presented in Fig. 7. The data used for the training stage correspond to faults in all the nodes with 11 different fault resistances from 0Ω to 40Ω in steps of 4Ω. The remaining data are used for testing the proposed algorithms. In this paper, the results for single phase to ground fault are presented, considering that it is the most common event and also taking into account that this fault presented the highest errors in location. All of the test were performed using a toolbox developed by the authors using Matlab. B. Analysis of the obtained results The fault location results for single phase faults (b-g) are shown in this section, results for the other phases are similar. The errors for distance and zone estimations are calculated using (6) and (7) respectively. Figs. 8, 9 and 10 show the errors of fault distance estimation in the case of using just voltage, current, and both voltage and current, respectively. It can be observed, in general, that errors for the fault distance estimation are lower, in average, than 2% in magnitude. In the case of distance estimation, just the current is the signal that provides better results (see Table I).
Table I shows the average errors for zone and distance estimations, where the signals with the best performance, in the zone identification case, are both, voltage and current. It is important to highlight the good performance of using just voltage, with errors lower than 16% and 1.5% for classification and distance estimation respectively, because there are still some power substations where only this measurement is available. VI. C ONCLUSIONS In this paper, a fault location method which merges a classification and a regression strategy based on the k-NN is proposed, analyzed and tested. In tests, only measurements of current and voltage obtained at the power substation under different conditions of fault resistance and location are used. This proposed fault locator gives high performance, compared with the classical reactance based methods, and additionally, it offers the flexibility of a low dependency on the system model. Furthermore, it also overcomes the multiple estimation problem. In this specific application case, the locator is trained using 53% of the fault database and the remaining data was used in tests. As results, errors lower than 1.5% and 13% in distance estimation and zone identification respectively, are obtained. The proposed methodology is an economic and easy alternative to reduce considerably the restoration time, maintaining low continuity indexes in power facilities and avoiding penalties caused by compensations due to deficient power quality. R EFERENCES [1] IEEE Guide for Determining Fault Location on AC Transmission and Distribution Lines, IEEE Std 37.114, Power System Relaying Committee 2004. [2] A. Warrington and C. Van, Protective relays. Their theory and practice. Chapman and Hall Ltd, London, 1968, vol. 1.
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[3] K. Srinivasan and A. St-Jacques, “A new fault location algorithm for radial transmission lines with loads,” IEEE Transactions on Power Delivery, vol. 4, no. 3, pp. 1676–1682, 1989. [4] A. Girgis, C. Fallon, and D. Lubkeman, “A fault location technique for rural distribution feeders,” IEEE Transactions on Industry and Applications, vol. 26, no. 6, pp. 1170–1175, 1993. [5] J. Zhu, D. Lubkeman, and A. Girgis, “Automated fault location and diagnosis on electric power distribution feefers,” IEEE Transactions on Power Delivery, vol. 12, no. 2, pp. 801–809, 1997. [6] R. Aggarwal, Y. Aslan, and A. Johns, “New concept in fault location for overhead distribution systems usings superimposed components,” IEE Proceedings. Generation, Transmission and Distribution, vol. 144, no. 3, pp. 309–316, 1997. [7] R. Das, “Determining the locations of faults in distribution systems,” Ph.D. dissertation, University of Saskatchewan, Saskatoon, Canada, 1998. [8] D. Novosel, D. Hart, Y. Hu, and J. Myllymaki, System for locating faults and estimating fault resistence in distribution networks with tapped loads, 1998, no. US Patent number 5,839,093. [9] L. Yang, One terminal fault location system that corrects for fault resistance effects. US Patent number 5,773,980, 1998. [10] M. Saha and E. Rosolowski, Method and device of fault location for distribution networks. US Patent number 6,483,435, 2002. [11] M. Choi, S. Lee, D. Lee, and B. Jin, “A new fault location algorithm using direct circuit analysis for distribution systems,” IEEE Transactions on Power Delivery, vol. 19, no. 1, pp. 35–41, 2004. [12] G. Morales-España, J. Mora-Florez, and H. Vargas-Torres, “k-nn based regression strategy used to estimate the fault distance in radial power systems,” Revista Facultad de Ingeniería (ISSN 0120-6230), vol. 45, pp. 100–108, September 2008, (In Spanish). [Online]. Available: http://ingenieria.udea.edu.co/grupos/revista/revistas/nro045/100_108.pdf [13] J. Mora-Florez, G. Morales-España, and R. Barrera-Cárdenas, “Evaluating a k-nearest neighbours-based classifier for locating faulty areas in power systems,” Revista Ingeniería e Investigación (ISSN 0120-5609), vol. 28, no. 3, pp. 81– 86, December 2008, (In Spanish). [Online]. Available: http://www.revistaingenieria.unal.edu.co/Resumenes/28_3/11_794.pdf [14] J. Mora-Florez, G. Morales-Espana, and S. Perez-Londono, “Learningbased strategy for reducing the multiple estimation problem of fault zone location in radial power systems,” IET Generation, Transmission & Distribution, vol. 3, no. 4, pp. 346–356, Apr. 2009. [Online]. Available: http://ieeexplore.ieee.org/xpls/abs_all.jsp? isnumber=4806229&arnumber=4806233&count=8&index=3 [15] T. Cover and P. Hart, Nearest neighbor pattern classification. IEEE Transactions on Information Theory, 1967. [16] D. Aha, D. Kibler, and M. Albert, Instance-based learning algorithms. Machine Learning, 1991. [17] F. Moreno, Clasificadores eficaces basados en algoritmos rapidos de busqueda del vecino mas cercano. Alicante, España: Departamento de lenguajes y sistemas informaticos. Universidad de Alicante, 2004. [18] A. Guerra, “Aprendizaje automatico: Otras tecnicas estadisticas de discriminación,” Maestria en Inteligencia Artificial. Universidad Veracruzana. [19] N. Drakos, Computer Based Learning Unit. England: University of Leeds, 1996. [20] W. L. MArtinez and A. R. Martinez, Computational statistics handbook with MATLAB. Chapman and Hall/CRC, 2002. [21] J. Mora-Florez, J. Melendez, and G. Carrillo-Caicedo, “Comparison of impedance based fault location methods for power distribution systems,” Electric Power Systems Research, vol. 78, no. 4, pp. 657–666, Apr. 2008. [22] G. Morales, H. Vargas, and J. Mora, “Impedance based method to fault location in power distribution, considering tapped loads and heavy unbalanced systems,” in Proc. XII encuentro regional Iberoamericano del CIGRÉ, Foz de Iguazú, Brasil, mayo 2007. [23] G. Morales, G. Carrillo, and J. Mora, “Selección de descriptores de tensión para localización de fallas en redes de distribución de energía,” Revista Ingeniería (ISSN 0121-750X), vol. 11, no. 1, pp. 43–50, 2006. [24] J. Dagenhart, “The 40-Ω ground-fault phenomenon,” Industry Applications, IEEE Transactions on, vol. 36, no. 1, pp. 30–32, Jan/Feb 2000.
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Germán Morales-España received the B.Sc. degree in electrical engineering from the Industrial University of Santander (UIS), Colombia, in 2007; the M.Sc. degree in engineering and policy analysis from the Delft University of Technology (TUDelft), The Netherlands, in 2010; and he is pursuing the Erasmus Mundus Joint Ph.D. degree in Sustainable Energy Technologies and Strategies (SETS) delivered by the Universidad Pontificia Comillas (UPCO), Spain, the Royal Institute of Technology (KTH), Sweden, and TUDelft, The Netherlands. He is currently an Assistant Researcher at the Institute for Research in Technology (IIT) at the UPCO. His areas of interest are power quality, protective relaying, economy and regulation of electric sector, and policy analysis. Mr. Morales-España is a member of GISEL (Col) Research Group on Electric Power Systems.
Juan Mora-Flórez (M’ 2009) received the B.Sc. degree in electrical engineering from the Industrial University of Santander (UIS), Colombia, in 1996, the M.Sc. degree in electrical power from UIS in 2001, the M.Sc. degree in information technologies from the University of Girona (UdG), Spain, in 2003, and the Ph.D. degree in information technologies and electrical engineering from UdG in 2006. Currently, he is associated professor at the Electrical Engineering School at the Technological University of Pereira, Pereira, Colombia. His areas of interest are power quality, transient analysis, protective relaying and soft computing techniques. Mr. Mora-Florez is a member of ICE (Col) Research Group on Power Quality and System Stability.
G. Carrillo-Caicedo Received his BSc in Electrical Engineering from UIS, Colombia in 1978; M.Sc. of Engineering from Rensselaer Polytechnic Institute, USA in 1981; Research Specialist from IIT of the Universidad Pontificia Comillas (UPCO), Spain in 1994; Ph.D. from UPCO, Spain in 1995. Currently, he is a laureate titular professor with the Electrical Engineering School at the Industrial University of Santander (UIS), Bucaramanga, Colombia. His areas of interest are Electric Energy markets, Ancillary Services, Power Quality and Technological Management. Mr. Carrillo- Caicedo is a member of GISEL (Col) Research Group on Electric Power Systems
