Ensemble decision trees for high impedance fault detection in power distribution network

Electrical Power and Energy Systems 43 (2012) 1048–1055

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Ensemble decision trees for high impedance fault detection in power distribution network S.R. Samantaray ⇑ School of Electrical Sciences, Indian Institute of Technology Bhubaneswar, Odisha, India

a r t i c l e

i n f o

Article history: Available online 15 July 2012 Keywords: Decision tree (DT) Extended Kalman Filter (EKF) High impedance fault (HIF) Random forest (RF)

a b s t r a c t The paper presents a new technique for high impedance fault (HIF) detection in power distribution network using ensemble decision trees (random forest). Giving the randomness in the ensemble of decision trees (DT) stacked inside the random forest (RF) model, it provides effective decision on HIF detection. The process starts with estimating the amplitude and phase of harmonic contents (fundamental, 3rd, 5th, 7th, 11th and 13th) in the HIF current signal using Extended Kalman Filter (EKF). In the next stage, random forest is trained with the amplitude and phase information of the HIF current signal to build up a highly efficient classifier for HIF detection. While testing, the proposed RF based classifier provides HIF detection with more than 99% reliability, considering extreme operating conditions of the power distribution network. The results indicate that the proposed method can reliably detect HIF in large power distribution network. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction High impedance faults (HIFs) on power distribution feeders are difficult to detect [1,2] using conventional overcurrent, ground fault relays and some versions of distance relaying schemes. Diversity, uncertainties, selectivity, suitability and operational constraints introduce malfunction, limitations and detection errors in case of high impedance faults (HIFs). This is notable when remote source loading, fault resistance non-linearity, capacitive line currents, mutual coupling and back-feed effects are taken into consideration. HIF faults [3,4] are usually characterized by the ripple rich current harmonic content due to non-linearity and thus are abnormal events that frequently occur in distribution feeders. There are two types of HIFs: the active faults and the passive ones. Active faults are followed by electric arc and present currents below the threshold of the protection relays. Normally, these currents decay with time until the complete extinction of the arc [5]. The majority of the techniques used to detect active HIFs make use of signals generated by the electric arc (harmonic and non-harmonic components) [6–9]. However, the arc may vanish even before the detection system gathers enough information to confirm the fault. Passive faults do not present an electric arc. They are more hazardous to people since there is no indication of the energization condition of the conductor. Due to presence of low or no current in HIF, the conventional over-current protection system

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normally fails to detect the same. Thus it is a challenging issue to detect the HIF and isolate the feeder. In recent years a combined wavelet transform and soft computing technique based HIF detection [10–12] has been proposed. This work includes feature extraction using wavelet transform and then classification using soft computing methods. HIF detection using neuro-fuzzy systems [13], uses an artificial neuron set, composed of ‘neuro-fuzzy’ neurons, and is trained to recognize the standard responses. In another work, earth faults with high impedance earthing in electrical distribution networks are characterized [14]. In the occurrence of disturbances, the traces of phase currents, voltages, neutral currents and voltages were recorded at two feeders at two substations. The study dealt with the clearing of earth faults, relation between short circuits and earth faults, arc extinction, arcing fault characteristics, appearance of transients and magnitudes of fault resistances. The above works finds limitations as wavelet transform is highly prone to noise and provides erroneous results even with noise of SNR 30 [15] dB. The fuzzyneural network is sensitive to system frequency-changes, and requires large training sets and training time. The HIF detection technique based on decision tree [16] is one of the recent one providing high accuracy in HIF detection. However, the method has been tested over only 100 cases for HIF detection. The proposed approach based on random forest [17–23] for HIF detection uses harmonic content information derived from EKF [24–28]. In the first stage, magnitude and phase of the harmonic components are derived after preprocessing through EKF, and in second stage the RF is trained to build the classifier for HIF detection. It is found that DT (single decision tree) provides

S.R. Samantaray / Electrical Power and Energy Systems 43 (2012) 1048–1055

low reliability and accuracy compared to proposed random forest (ensemble decision trees), during training and testing with larger data base (29400). The proposed technique provides HIF detection with reliability more than 99% compared to 90% with single DT. The following sections include RF, system studied, results, discussion and conclusions.

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Although random forest is a relatively young data mining tool, people have started recognizing its strengths: (i) It is simple and easy to use, (ii) very high accuracy, (iii) its relatively robust to outliers and noise, (iv) it gives useful internal estimates of error, strength, correlation, (v) not over fitting if selecting large number of trees, (vi) insensitive to choice of split. 2.2. Prediction from ensemble trees

2. Random forests In an ensemble of trees the predictions of all individual trees need to be combined. For classification, the class that most trees vote for is returned as the prediction of the ensemble:

2.1. Background Random forests [17] are a large combination of de-correlated tree predictors such that each tree depends on the values of a random vector sampled independently. Individual trees are noisy and unstable, but since when grown sufficiently deep, they have relatively low bias. Therefore, they are ideal candidates for ensemble growing as they can capture complex interactions, while fully benefit from aggregation based variance reduction. Using a random selection of features to split each node and re-sampling (with replacement) the training set to grow each tree yields error rates that are de-correlated and more robust with respect to noise. The generalization error for forests converges as to a limit as the number of trees in the forest becomes large. The basic idea of most ensemble tree growing procedures is that for the kth tree (k 6 ntree, the number of trees in the ensemble,) a random vector Uk is generated, independent of the past random vectors Uk,. . .Uk1 but with the same distribution, and a single tree is grown using the training set S and the set of attributes in Uk, resulting in a classifier Tk(x, Uk) where x is an input vector. In random split selection, U consists of a number ntry of independent random integers where ntry < na, the number of attributes in S. A random forest consists of a collection of tree-structured classifiers {Tk(x, Uk), k = 1, . . ., ntree}, where {Uk} are independent identically distributed random vectors and each tree casts a unit vote for the most popular class at input x. An algorithmic view the RF growing process is summarized below [17]:

b k ðxÞ; k ¼ 1; . . . ; ntree g b ntree ðxÞ ¼ majority votef C C RF

b k ðxÞ is the class prediction of the kth random forest tree. For where C predicting probabilities, i.e. relative class frequencies, the results of the single trees are averages: ntree X b ntree 2 fS; IgjxÞ ¼ 1 b k 2 fS; IgjxÞ b ntree ð C b T ðU ;SÞ ð C P P RF RF k k ntree 1

138 kV Transmission line

ð2Þ

bT where P denotes the probability associated to an observation x kðUk SÞ by the random forest tree Tk(x, Uk). A traditional decision tree essentially represents an explicit decision boundary, and an instance E is classified into class c if E falls into the decision area (a leaf in the decision tree) corresponding to c [20]. The class probability p(c|E) is typically estimated by the fraction of instances of class c in the leaf into which E falls. This probability estimate is very crude when the tree is pruned because all the instances falling into the same leaf have the same class probability. More accurate probability estimates require unpruned trees [23], which are the backbone of the random forests. Stated otherwise, RF predictor has the additional advantage of providing a stability or instability level of the event through probability-based ranking. Assuming that the probability estimates from individual trees are random variables, each with variance r2, the variance of the 2 average in (2) is nrtree which confirms that the random forest leads seamlessly to improved probability estimates [19]. In addition to the ordinary prediction described above, random forests have a so-called out-of-bag prediction. Remember that each tree is built on a bootstrap sample S , that serves as a learning set for this particular tree. S contains only two third of the out-of-bag observations [17], i.e. those M–N samples not participating to the training of a given tree can serve as ‘‘built-in’’ test sample for computing the prediction accuracy of that tree. The advantage of outof-bag error (OOB) is that more realistic estimate of the error rate can be obtained. If we feed the random forest inducer with S containing only 70% of the original data and keep the rest for testing, giving that each tree is trained on two third of the data only, it turns out that only 50% of the data are actually seen by a given ran-

1. For k = 1 to ntree: a. Draw a bootstrap sample S of size N from the training data S (which contains M > N samples). b. Grow a random forest tree Tk(x, Uk) to the bootstrapped data, by recursively repeating the steps below for each terminal noted of the tree, until the no other split is possible (unpurned tree of maximal depth): i. Select ntry variables from the na WASI features. ii. Pick the best variable/split-point among the ntry. iii. Split the node into two daughter nodes. 2. Output the ensemble of trees {Tk(x, Uk), k = 1, . . ., ntree}.

Generator 50 MVA

ð1Þ

138/25 kV Transformer

25 kV feeders

~

Relaying point

C

HIF model Linear and Rectifier load Fig. 1. Radial distribution network.

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S.R. Samantaray / Electrical Power and Energy Systems 43 (2012) 1048–1055

Rp (Non-linear Resistance)

VP

Rn (Non-linear Resistance)

Vn

Fig. 2. High-impedance fault model.

dom forest tree at learning stage. If the resulting predictor worked fine on the external test set, we have to admit that it is very robust and generalize model.

3. System studied There are two systems considered for the proposed study. One is a simple radial distribution network as shown in Fig. 1. The feeders are from 138/25 kV substation transformer which is connected from a 138 kV transmission line of 100 km line length. The loads (linear and non-linear) and shunt capacitors are also connected as shown in Fig. 1. The HIF faults are created on the distribution feeder (25 kV, 20 km pi sections) as shown in the figure. The parameters of the transmission and distribution lines are similar to that of mesh distribution system. The HIF model is developed

using anti-parallel diodes with non-linear resistance and DC source connected together for each phase as shown in Fig. 2. Both linear and rectifier loads are connected for load switching purpose. The deatils of the HIF model are as follows: Diode: Resistance Ron = 0.001 ohm, Forward voltage = 0.8 V, Snubber resistance = 500 ohm, Snubber capacitance = 250e9 F, Non-linear resistance: Protection voltage 45e3 V, DC voltage source: 15e3 V. Another system is a standard mesh type distribution network as shown in Fig. 3, supplied from two separate 3-phase sources through transmission line (100 km) and transformers. The transmission lines are 138 kV and the transformers are 50 MVA supplying at 138/25 kV to the distribution network. The distribution feeders (pi sections of 20 km each) work at 25 kV and connected with shunt capacitors, linear loads and 2 MVA 6-pulse rectifier load (non-linear load). The resistance, inductance and capacitance of positive and zero sequence of transmission lines are R1 = 0.01273 ohm/km; X1 = 0.9337 mH/km; C1 = 0.0012 lF/km and R0 = 0.3864 ohm/km; X0 = 4.1264 mH/km; C0 = 0.0075 lF/km, respectively. The resistance, inductance and capacitance of distribution lines (pi-section) are R1 = 0.2568 ohm/km; X1 = 2.0 mH/ km; C1 = 0.0086, respectively. The total percentage impedance of the transformers is 6.75%. The simulation models are developed using PSCAD (EMTDC) and the sampling rate chosen is 1.0 kHz on a 50 Hz base frequency (20 samples per cycle). The aforementioned two models are selected to complement each other. One is very simple radial system and the other one is a mesh power distribution network. Thus combining these two, we can get a generalized solution for HIF detection. This is because the data base is including both systems and, the random forest is trained with random data from either of the system. This also improves the robustness of the technique for HIF detection in possible variations in power distribution network. The different simulation conditions taken into consideration include HIF and non-HIF conditions such as three-phase or singlephase load switching, shunt capacitor switching, no-load transformer switching are as follows

Fig. 3. Meshed distribution network.

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amplitude

1.5 1 0.5 0 Fig. 4. Proposed HIF detection scheme.

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 3-Phase load change from 20–60%, 30–70%, 60–110%, 20–110% in forward and reverse way (4  2).  1-Phase load change 30–70%, 20–50%, 40–80%, 40–110%, 20– 100% in forward and reverse way (5  3).  Transformer enrgization at different timings in the cycle slot (16 instances in one cycle).  The above changes are made with change in infinite source phase angle of 0–120°, with a span of 10° (12).  Shunt capacitors are switched on and off (10).  The above changes are made under varying conditions in the HIF model by varying the DC source voltages by (25)% to (25)% in a step of 5%. The central voltage also varies from 2000 to 10000 V with a step of 1000 V (10). From the above operating conditions 5880 (12  (8 + 15 + 16 + 10)  10) cases are simulated for the power distribution network shown in Fig. 1. Similar conditions in the power distribution network shown in Fig. 3 provide 23520 (4  5880). Thus total simulation cases are 29400 (5880 + 23520) including both HIF and non-HIF conditions. From the total cases simulated, 70% is used for training and rest 30% for testing the random forest. Thus, Instead of applying the proposed technique to the two systems separately, we combined the two sets of data in the hope of finding a single, more generalized predictor. The proposed HIF detection scheme is shown in Fig. 4. 4. Proposed technique and result analysis Initially, the harmonic (fundamental, 3rd, 5th, 7th, 11th and 13th) components and corresponding phase are derived by preprocessing the HIF signal using Extended Kalman Filter (Appendix 1). Thus there are 12 features for one HIF or non-HIF event. During training and testing of RF, the target output for HIF and non-HIF are considered as ‘1’ and ‘1’ respectively. These derived features for one cycle, two cycles and three cycles window length after the inception of the HIF/non-HIF events are considered as inputs to train the RF to build an accurate and robust classifier for HIF

phase

1 0 -1 -2

samples Fig. 6a. Estimated amplitude and phase of the fundamental component of HIF current signal using EKF.

detection. Fig. 5 shows the typical HIF current signal (amplitude versus sample (time)). Figs. 6a–6c shows the amplitude and phase of fundamental (a), 3rd (b) and 13th harmonic (c) components estimated using EKF. Similar estimation results are obtained for 7th and 11th harmonic components. After the features are derived using EKF, the RF is trained [18] with 70% of the data sets and rest 30% for testing, which is more generalized for classification algorithm. Thus the RF is trained with 20580 and tested with 8820 data sets. The test results on confusion matrix generated from RF algorithm are on 8820 sets (30% of total data sets) composed of both HIF and non-HIF conditions. Performance indices such as accuracy and reliability are found out to measure the performance of RF for HIF detection and are defined as follows  Accuracy = (No. of HIF and non-HIF cases predicted)/(Total numbers of actual HIF and non-HIF cases).  Reliability = (Numbers predicted HIF cases)/(Numbers predicted HIF cases + Numbers of misdetection). The confusion matrix provides the complete statistics on the actual versus predicted conditions on the test data sets using RF as depicted in Table 1. It is found that during testing 4930 cases are detected as HIF from actual 4946 (4930 + 16) HIF cases. Similarly 3846 are predicted as non-HIF cases against 3874 (3846 + 28). This shows that 16 cases are actual HIF but detected as non-HIF (known as misdetection) and 28 cases are actual non-HIF but detected as

amplitude

0.6 0.4

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samples Fig. 5. Typical HIF current.

100

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samples Fig. 6b. Estimated amplitude and phase of the 3rd harmonic of HIF current signal using EKF.

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S.R. Samantaray / Electrical Power and Energy Systems 43 (2012) 1048–1055 Table 3 Comparison between RF and DT for misdetection and false alarm.

amplitude

0.4 0.3 0.2 0.1 0

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phase

2

Method

Mis-detection

False alarm

RF DT

16 476

28 203

Data set with SNR 20 dB RF DT

38 602

98 391

1 0 -1

Table 4 Comparison between RF and DT for reliability and accuracy.

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samples Fig. 6c. Estimated amplitude and phase of the 13th harmonic of HIF current signal using EKF.

Method

Accuracy (%)

Reliability (%)

RF DT

99.50 92.30

99.67 90.22

Data set with SNR 20 dB RF DT

98.45 88.74

99.27 87.67

Table 1 Comparison of confusion matrix between RF and DT for HIF detection. Predicted

1 (HIF) 1 (NonHIF) Accuracy (%)

RF (70% Train and 30% Test) Actual

DT (70% Train and 30% Test)

1 (HIF)

1 (NonHIF)

1 (HIF)

1 (NonHIF)

4930 16

28 3846

4395 476

203 3746

99.50

92.30

HIF (known false alarm). The overall accuracy obtained with RF is 99.50%. In case of DT (single decision tree), 406 cases are misdetection and 203 cases are false alarm, resulting accuracy of 92.30%. When tested with data sets with SNR 20 dB, the accuracies resulted from RF and DT are 98.45 and 88.74%, respectively as depicted in Table 2. The comparison between RF and DT for misdetection and false alarm is given in Table 3. It is found that the misdetection is 16 compared to 38 in noisy situation for RF. For DT, 476 cases are misdetected compared to 602 in noisy environment. Similar observation is made for false alarm resulted from RF and DT for data sets with and without noise. Reliability is the key performance measure in this study as it provides how many cases are HIF, but predicted as non-HIF. This directly shows the effectiveness of the developed classifier for HIF detection. It is found that RF provides 99.67% reliability compared to 90.61% with DT for HIF detection. Similar observations

Table 2 Comparison of confusion matrix between RF and DT for HIF detection with SNR 20 dB. Predicted

1 (HIF) 1 (NonHIF) Accuracy (%)

RF (70% Train and 30% Test) Actual

DT (70% Train and 30% Test)

1 (HIF)

1 (NonHIF)

1 (HIF)

1 (NonHIF)

5176 38

98 3508

4281 602

391 3546

98.45

88.74

are made for noisy situation, where RF provides 99.27% reliability compared to 87.67% with DT as depicted in Table 4. It is observed that RF provides reliability more than 99% which is substantially high compared to DT. This shows the effectiveness of the proposed RF for HIF detection in power distribution network. The RF is trained for 100 generations of trees and the convergence characteristic (error versus tree generations) is shown in Fig. 7. It is found that after 50 trees the errors are almost constant and thus the RF is suitably trained. There are 3 characteristics in Fig. 7, showing the red line for ‘1’ (Non-HIF events), green for ‘1’ (HIF events) and black for ‘OOB’ error during training.

5. Discussion It is seen from the previous section that RF provides reliability as well as accuracy around 99% compared to 90% (average) in case of DT. Thus RF can reliably detect the HIF in power distribution network with complex configurations and operating conditions. For testing the robustness of the proposed RF, the same has been tested with noisy data with SNR 20 dB and the reliability remains more than 99%. To test the impact of data length (from HIF/non-HIF inception) on the performance of RF, the same has been tested under three conditions such pre-processing one cycle data, two cycle data and three cycle data after the event inception. Table 5 depicts the accuracy and reliability resulted from RF testing. It is found that the reliability is 93.25% with one cycle data pre-processing and improves to 99.50% with two cycle data. This shows substantial improvement from one cycle to two cycle data. But the reliability almost remains same even if the data window is three cycles after the event inception. Similar observation is made for accuracy, which stays almost constant from two cycle and three cycle data length. Thus two cycle data length after HIF inception can reliably detect the HIF from non-HIF conditions. In another comparison, the performance of EKF and FFT is analyzed for two cycle data after HIF inception, as depicted in Table 6. It is found that the reliability and accuracies are around 99% with EKF, compared to 93% (average) with FFT. The performance measures are further degraded in noisy environment with SNR 20 dB. This shows the effectiveness of the EKF over FFT for harmonic component extraction for HIF detection. Further to show the effectiveness of the RF over DT, the comparison between DT and RF for FFT is depicted in Table 7. It is observed that the reliability for HIF detection is around 94% with RF com-

S.R. Samantaray / Electrical Power and Energy Systems 43 (2012) 1048–1055

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Fig. 7. The convergence characteristics of RF during training.

Table 5 Comparison of accuracy and reliability for different data window length with RF. Method

One cycle

Two cycle

Three cycle

Reliability Accuracy

93.25 82.78

99.50 99.67

99.32 99.45

pared to 85% with DT. Also the overall accuracy has got a substantial jump in case of RF compared to DT. This shows the effectiveness of the proposed RF, which is ensemble decision trees (during training, the algorithm runs for large numbers of DTs), compared to DT (the algorithm runs for only single DT) for classification task such as HIF detection. One question remains how to retrain the RF following routine changes in the network operating conditions? The answer is yes, to some extent. Although the RF is robust over a wide range of system conditions and was trained to capture the ‘‘essential’’ concept of system security, as any inductive knowledge, it comes with a guarantee limited to the network states, that result in dynamics ‘‘similar’’ to those included in the learning database. Given the im-

Table 6 Comparison between RF and DT for reliability and accuracy using RF. Method

Accuracy (%)

Reliability (%)

EKF FFT

99.50 93.56

99.67 94.56

Data set with SNR 20 dB EKF FFT

98.45 90.86

99.27 92.56

Table 7 Comparison between RF and DT for reliability and accuracy for FFT. Method

Accuracy (%)

Reliability (%)

DT RF

86.78 93.56

85.56 94.56

Data set with SNR 20 dB DT RF

84.78 90.86

85.86 93.46

proved robustness and reliability of the new predictor, such a function could be called upon infrequently on a yearly, monthly or daily basis using changed scenarios with some uncertainties. An alternative view could be to execute this retrain functionality in real time, at the speed of SCADA information. RF training is inherently fast (only 2-min CPU time on a laptop) and the database update (when required) could take advantage of the computational facilities being deployed for fast real-time simulation and modeling [29].

6. Conclusions The proposed approach provides an effective technique for HIF detection using combined Extended Kalman Filter and random forest. It is found that the proposed technique can detect HIF with reliability more than 99% compared to 90% with existing DT based technique. It is observed that the RF is highly effective for HIF detection with two cycle window after HIF inception. Compared to existing DT based technique with FFT, the proposed RF with EKF is found to be more accurate in noisy environment for HIF detection, and thus showing the robustness of the proposed technique.

Appendix A The Extended Kalman Filter (EKF) is a non-linear time domain stochastic estimator that provides an efficient estimation of the harmonic components of fault currents during a high impedance fault, characterized by the ripple rich current harmonic content due to non-linearity). The advantage of EKF is that it is a recursive means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects and supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The Fast Fourier Transform (FFT) method, which is non-recursive technique, cannot handle signals with partial disturbances (noise), nor can these methods be applied to nonuniformly sampled signals. Let the discrete signal which contains fundamental and harmonics along with a decaying DC component is represented by the model (such components are generated during a high impedance fault) given below

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S.R. Samantaray / Electrical Power and Energy Systems 43 (2012) 1048–1055

Z k ¼ A1 sinðkwT s þ uÞ þ A2 sinð3kwT s þ uÞ þ A3 sinð5kwT s þ uÞ

The Kalman filter gain Kk is obtained as

þ A4 sinð7kwT s þ uÞ þ A5 sinð11kwT s þ uÞ þ A6 sinð13kwT s akT S

þ u13 Þ þ A0 e

ð3Þ

ð4Þ

xk ð3Þ ¼ A2 cos /; xk ð4Þ ¼ A2 sin /; xk ð5Þ ¼ A3 cos /; xk ð6Þ ¼ A3 sin /; xk ð7Þ ¼ A4 cos /; xk ð8Þ ¼ A4 sin /;

ð5Þ

xk ð9Þ ¼ A5 cos /; xk ð10Þ ¼ A5 sin /; xk ð11Þ ¼ A6 cos /; xk ð12Þ ¼ A6 sin / T S

xk ð13Þ ¼ e

xk ð14Þ ¼ A0 e

;

1 0

6 60 6 60 6 6 60 6 6 60 6 60 6 6 60 6 Fk ¼ 6 60 6 6 60 6 60 6 6 60 6 6 60 6 6 40 0

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Rk ¼ kk Rk1 þ ð1  kk Þe2k

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

kk ¼

cosð3kwT s Þ;

sinð5kwT s Þ

cosð5kwT s Þ;

sinð7kwT s Þ

cosð7kwT s Þ;

sinð11kwT s Þ

cosð11kwT s Þ;

sinð13kwT s Þ

cosð13kwT s Þ; 0; 1

ð6Þ

Z k ¼ H k xk

ð7Þ

ð8Þ ð9Þ

where

2

Hk ¼

 @G dx 

k;k1

xð1ÞkT s cosðwkT s Þ  xð2ÞkT s sinðwkT s Þ

ð16Þ

cek ¼ kq  cek1 þ ð1  kq Þ  ek  ek1

ð17Þ

References

Literalizing the above system, the AEKF algorithm is obtained as follows: _

1 1 þ jRðkÞ=R0 j

If ce(k)iceth, Q = Q1 and ce(k)hceth, Q = Q0, where Q0 is the model error covariance and Q1 is a new value of Q and Q1iaQ0, ai1, and ceth is the threshold value of error covariance.

kT s

¼ x k=k þ K k ðZ k  Hk xk=k1 Þ

ð15Þ

where R0 is the initial error covariance R Further, the model error covariance matrix Q is adapted by using a covariance function ce as

cosðkwT s Þ;

sinð3kwT s Þ

ð14Þ

where kk is forgetting factor given by

The observation matrix is given by

Gk ¼ ½sinðkwT s Þ

_

The error covariance R is recursively updated as

0

0

where Q is the covariance matrix and R is the measurement noise covariance. To improve the performance of the Extended Kalman Filter, the measurement error covariance is updated in the following manner. The expression for R is obtained as the error between observed and estimated values of xk as _

and the state transition matrix is given by

2

ð13Þ

R ¼ ðzk  Hk x k ÞT ðzk  Hk x k Þ

kT S

ð12Þ

_

P kþ1=k ¼ Pk=k þ Q

xk ð1Þ ¼ A1 cos /; xk ð2Þ ¼ A1 sin /;

ð11Þ

_

P k=k ¼ Pk=k1  K k Hk P k=k1

where

_ x k=k

_

_

The discrete signal can be represented in state space as

xkþ1 ¼ F k xk

_

K k ¼ P K=K1 HTk ðHk P k=k1 HT þ RÞ1

3

7 6 6 xð3ÞkT s cosð3wkT s Þ  xð4ÞkT s sinð3wkT s Þ 7 7 6 6 xð5ÞkT cosð5wkT Þ  xð6ÞkT sinð5wkT Þ 7 7 6 s s s s 7 6 6 xð7ÞkT s cosð7wkT s Þ  xð8ÞkT s sinð7wkT s Þ 7 7 6 ¼6 7 6 xð9ÞkT s cosð11wkT s Þ  xð10ÞkT s sinð11wkT s Þ 7 7 6 7 6 6 xð11ÞkT s cosð13wkT s Þ  xð12ÞkT s sinð13wkT s Þ 7 7 6 7 6 0 5 4 1 ð10Þ

[1] Johns AT. Correspondence on microprocessor based algorithm for highresistance earth-fault distance protection. Proc Inst Electr Eng, Part C 1985;132:94–5. [2] Yang Q, Morrison I. Microprocessor-based algorithm for high-resistance earthfault distance protection. Proc Inst Electr Eng, Part C 1983;130:306–10. [3] Aucoin BM, Russell BD. Distribution high impedance fault detection using high frequency current components. IEEE Trans Power Appl Syst 1982;PAS101(6):1596–606. [4] Sharaf AM, E1-Sharkawy RM, Talaat HEA, Badr MAL. Novel alpha-transform distance relaying scheme. In: Proc IEEE–CCECE Conf, Calgary, Canada; 1996. [5] Aucoin M, Russell BD. Detection of distribution high impedance faults using burst noise signals near 60 Hz. IEEE Trans Power Del 1987;2(2):342–8. [6] Sultan AF, Swift GW, Fedirchuk DJ. Detection of high impedance arcing faults using a multilayer perceptron. IEEE Trans Power Del 1992;7(4). [7] Sultan AF, Swift GW, Fedirchuk DJ. Detection arcing downed wires using fault current flicker and half cycle asymmetry. IEEE Trans Power Del 1994;9(1):461–70. [8] Russell BD, Chinchali RP. Signal processing algorithm for detecting arcing faults on power distribution feeders. IEEE Trans Power Del 1989;4(1):132–40. [9] Russell BD, Chinchali RP, Kim CJ. Behaviour of low frequency spectra during arcing fault and switching events. IEEE Trans Power Del 1988;3(4):1485–92. [10] Sedighi A-R, Haghifam M-R, Malik OP. Soft computing applications in high impedance fault detection in distribution systems. Electr Power Syst Res 2006. [11] Baqui Ibrahem, Zamora Inmaculada, Mazón Javier, Buigues Garikoitz. High Impedance fault detection methodology using wavelet transform and artificial neural networks. Electr Power Syst Res 2011;81(7):1325–33. [12] Ibrahim Doaa khalil, Tag Eldin El Sayed, Aboul-Zahab Essam M, Saleh Saber Mohamed. Real time evaluation of DWT-based high impedance fault detection in EHV transmission. Electr Power Syst Res 2010;80(8):907–14. [13] Jota Patricia RS, Jota Fábio G. Fuzzy detection of high impedance faults in radial distribution feeders. Electr Power Syst Res 1999;49:169–74. [14] Hanninen Seppo, Lehtonen Matti. Characteristics of earth faults in electrical distribution networks with high impedance earthing. Electr Power Syst Res 1998;44:155–61. [15] Yang Hong-Tzer, Liao Chiung-Chou. A de-noising scheme for enhancing Wavelet-based power quality monitoring system. IEEE Trans Power Del July 2001;16(3):353–60. [16] Sheng Yong, Rovnyak M. Decision tree based methodology for high impedance fault detection. IEEE Trans Power Del 2004;19(2):533–6. [17] Breiman. Random forests. Machine learning, vol. 45; 2001. p. 5–32. .

S.R. Samantaray / Electrical Power and Energy Systems 43 (2012) 1048–1055 [18] Liaw A, Wiener M. Classification and regression by random forest in R, R news, vol. 2, no 3; December 2002. P. 18–22. . [19] Hastie T, Tibshirani R, Friedman J. The elements of statistical learning. 2nd ed. New York: Springer-Verlag; 2009. p. 745. [20] Provost FJ, Domingos P. Tree induction for probability-based ranking. Mach Learn 2003;52(30). [21] Banfield RE, Hall LO, Bowyer KW, Kegelmeyer WP. A comparison of decision tree ensemble creation techniques. IEEE Trans Pattern Anal Mach Intell Nov. 2007;29(1):173–80. [22] Siroky DS. Navigating random forests and related advances in algorithmic modeling. Stat Rev 2009;3:147–63. . [23] Liang H, Zhang H, Yan Y. Decision trees for probability estimation: an empirical study. In: Proc 18th IEEE international conference on tools with, artificial intelligence (ICTAI’06); 2006. p. 1–9. [24] Bak D, Michalik M, Szafran J. Application of kalman filter technique to stationary and non-stationary state observer design. In: Proc IEEE Power Tech Conf, vol. 3, Bologan, Italy; June 2003.

1055

[25] Julier SJ, Uhlmann JK. A new extension of the Kalman filter to non-linear systems. In: Proc 11th international symposium on aerospace/defence sensing, simulation and controls, tracking and resource management II; 1997. [26] Iltis RA. Joint estimation of PN code delay and multipath using extended Kalman filter. IEEE Trans Commun. 1990;38:1677–85. [27] Iltis RA. An EKF-based joint estimator for interference, multipath, and code delay in a DS pread-spectrum receiver. IEEE Trans Commun 1994;42:1288–99. [28] Pecen Recayi, Ula Sadrul, Timmerman Marc. Modeling and imulation of a Kalman filter based control scheme for an AC/DC power system. Int J Electr Power Energy Syst 2004;26(3):173–89. [29] Moslehi K, Kumar ABR, Dehdashti E, Hirsch P, Wu W. Distributed autonomous real-time system for power system operations – a conceptual overview. In: IEEE power systems conference & exposition, New York; 10–13 October 2004.