Detection, classification, and location of faults in power transmission lines

Electrical Power and Energy Systems 67 (2015) 76–86

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Detection, classification, and location of faults in power transmission lines K.R. Krishnanand a, P.K. Dash a,⇑, M.H. Naeem b a b

Siksha O Anusandhan University, Bhubaneswar 751030, India Multimedia University, Cyberjaya, Malaysia

a r t i c l e

i n f o

Article history: Received 22 February 2014 Accepted 15 November 2014

Keywords: Current differential protection CUSUM Spectral energy Scaling Fast Discrete S-Transform Fault classification and location

a b s t r a c t This paper presents a pattern recognition approach for current differential relaying of power transmission lines. The current differential method uses spectral energy information provided through a new Fast Discrete S-Transform (FDST). Unlike the conventional S-Transform (ST) technique the new one uses different types of frequency scaling, band pass filtering, and interpolation techniques to reduce the computational cost and remove redundant information. Further due to its low computational complexity, the new algorithm is suitable for real-time implementation. The proposed scheme is evaluated for current differential protection of a transmission line fed from both ends for a variety of faults, fault resistance, inception angles, and significant noise in the signal using computer simulation studies. Also the fundamental amplitude and phase angle of the two end currents and one end voltage are computed with the help of the new formulation to provide fault location with significant accuracy. The results obtained from the exhaustive computation show the feasibility of the new approach. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction Differential relaying has been already applied for a wide variety of protective systems for the generators, transformers, bus bars, transmission lines and provides high speed fault clearance [1–3]. For transmission lines both the current and power differential protection schemes [4–6] have been used for the detection and classification of faults in the protected zone. Further the implementation of Global Positioning System (GPS) based synchrophasors makes differential protection of transmission lines a practical idea [7,8], even for very long transmission lines [9] in a smart grid environment. A SONET architecture [8] that follows a redundant and reliable asymmetric communication scheme results in the end-to-end time delay of a few milliseconds. The speed of response of the differential scheme is hardly influenced by this communication delay. The concept of using synchrophasors for digital protection has already been attempted [10]. The use of phasor measurement units (PMU) have been suggested [11] for measuring current at the two ends of a transmission line and use these values for determining the restraining and operating quantities for differential protection. However, these schemes suffer from line charging current and CT saturation and thus time-frequency methods have been suggested ⇑ Corresponding author. Tel.: +91 674 2727336. E-mail address: [email protected] (P.K. Dash). http://dx.doi.org/10.1016/j.ijepes.2014.11.012 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

for differential protection of transmission lines. Amongst the various time-frequency techniques, the discrete wavelet transform (DWT) has been introduced [12,13] for a variety of protection of power system elements including the differential energy based approach for differential protection of transmission lines. Although the wavelet transform provides a variable window for low- and high-frequency components in the voltage and current waveforms during faults, special threshold techniques are needed under noisy conditions [14]. Also the wavelet transform uses high pass and low pass filters to splint the power signal into a detail and an approximation repeatedly until a required level of decomposition is achieved. As it only decomposes the signal approximations, it may fail in cases where certain informations belong to the high frequency regions. Thus for more accuracies wavelet packet transform is used for protection purposes [15]. Additionally, the detailed amplitude, instantaneous phase or frequency of the fundamental components, necessary for protection purpose cannot be obtained easily without a complex set of calculations [16]. In recent years STransform (ST) has been used for the protection of transmission lines [14] and power signal disturbance detection due to its superior properties of localizing the time-frequency components. However, unlike the DFT, the ST can vary the amount of time and frequency over which measurements are averaged, depending on the frequency under consideration. This is an important property that differentiates wavelets and the ST from the DFT. A further

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potential advantage of the ST over wavelets is that it utilizes sinusoidal basis functions so ‘‘phase’’ measurements are more directly related to the conventional concept of phase. The phase information associated with the ST makes it as an ideal candidate for the detection and classification of distorted signals, and thus it is used here to localize and detect any power system amplitude or phase changes in the power system signals [17–20]. The physical reality of power system signals is then better analyzed using mathematical methods like S-Transform. Although ST is a powerful tool for power signal disturbance assessment, it involves high computational overhead which is of the order of O(N2logN) using the entire data window for the signal. Thus there has been some attempts [21] to reduce the computational overhead for the calculation of discrete ST for biomedical signal processing, by using a dyadic frequency scaling which may not be suitable for power system protection due to the range and nature of harmonics generated during faults. This paper, therefore, presents intelligent scaling mechanisms which can make the computation of S- Transform faster and suitable for power system protection operations. The scaling techniques act as smart numerical filters for deciding the time domain localizations of significant frequencies. The digital relay based on such methods can intelligently and dynamically decide the online signal calculations for relay operations. Thus with the significant reduction in computational cost, the fast discrete ST (FDST) has a distinct advantage over DWT in giving an unified approach for both fault detection and fault location simultaneously to a very accurate extent. Further the new FDST based current differential protection scheme can include fault location function by simply measuring the differential current and voltage at one end, say the sending end relaying point [22,23]. The paper is organized as follows: Section ‘Spectral energy computations’ [24] describes the spectral energy computation scheme where the energy computations are triggered using a computationally light technique called CuSum algorithm. Section ‘Fast Discrete S-Transform (FDST)’ explains the FDST algorithm and the frequency scaling methods used. It also describes the application of the CuSum and FDST to detect and classify the various kinds of fault. Section ‘Results and discussions’ shows the results obtained from the simulation studies for a practical system taken from the literature. Section ‘Conclusion’ is the conclusion of the study. Spectral energy computations Using the differential current and its energy for operation and the average current and its energy for restraint, a DWT based spectral energy differential relay has been suggested [22] to overcome some of the sensitivity and stability problems of the conventional differential scheme and give suitable protection for both the internal and external faults. Parallel to DWT based protection schemes for power system elements, S-Transform has been used [14] for providing a pattern recognition approach for the protection of both compensated and uncompensated transmission lines. However, before processing the instantaneous difference or average values of the current samples it is necessary to detect any change in the differential current and the average current using the Cumulative Sum technique (CuSum). The CuSum algorithm is described below. CuSum algorithm The CuSum algorithm does cumulative summation of the differences between the reference cycle and the present cycle to detect any sudden change in the signal amplitude. It is useful in reducing the computational load on the digital relay’s processor. It is excellent in change detection of periodic patterns, but it is not useful to

confirm the occurrence of a fault or to perform further computations on the fault signal. Being computationally light, CuSum is suitable to be used as a trigger for initiating the Fast Discrete S-Transform. A cumulated signal for a fundamental cycle is given by Ns X ðSðkÞ  Sðk  k  NsÞÞ

CuSum ¼

ð1Þ

k¼1

where S(k) is the signal amplitude at the kth sample, Ns is number of samples per fundamental cycle and k is the factor determining the time gap between the present amplitude and the reference amplitude. In on-line mode, CuSum is continually computed at each sample. CuSum for the kth sample is given by

CuSumðkÞ ¼ CuSumðk  1Þ þ SðkÞ  Sðk  k  NsÞ

ð2Þ

k is generally taken as 1 considering the simplicity of implementation. The time point at which CuSum(k) exceeds a preset value l is taken as the CuSum Detection Point (CSDP). Effectively, CSDP = k such that CuSum(k) > l for the first time. The CuSum status is cleared later after the detection of the occurrence of the change indicating the occurrence of a fault on the line. Upon the detection of CSDP the algorithm to obtain the time-frequency localized signal energies is initiated. S-Transform Formulations The ST provides a time frequency representation with frequency-dependent resolution while, at the same time, maintaining the direct relationship, through time-averaging, with the Fourier spectrum [15]. The Generalized S-Transform of a time varying signal x(t) is obtained as

Sðs; f Þ ¼

Z

1

xðtÞ  wðs  t; f Þ  expð2piftÞdt

ð3Þ

1

where the window function w(t, f) is chosen as

! 1 t pffiffiffiffiffiffiffi exp wðt; f Þ ¼ 2 rðf Þ 2p ð2rðf Þ Þ and rðf Þ is represented as rðf Þ ¼

ð4Þ

a jf j

ð5Þ

The window is normalized as

Z

1

wðs  t; f Þds ¼ 1

ð6Þ

1

If the continuous time series x(t) is sampled with a sampling period of T sec to obtain N samples, the expression for discrete ST becomes N1   n X n  ði2pNknÞ S jT; xðkTÞ  w ðj  kÞT; ¼ e NT NT k¼0

ð7Þ

n where f ¼ NT , T = sampling interval, and the window function in the discrete domain is chosen as:

!  n c   n c 2   n  a þ bNT   2 2 w jT;  2r ¼ pffiffiffiffiffiffiffi exp ðjTÞ a þ b  NT NT r 2p

ð8Þ

where j = 0, 1, . . ., N  1 represents a time point index, n = 0, 1, . . .(N/2)  1 is a frequency point index of a given cycle, x is the discrete time domain signal, k is the time domain shift required for the convolutive operation with the window w, f is the frequency point computed from the frequency point index and time period NT of the signal. Further r and b are the scaling factors that control the number of oscillations in the window, and a, c are positive constants. When k is increased, the window broadens in the time domain, and hence, frequency resolution is increased in the

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Magnitude

frequency domain. Setting the parameter c at some value from 0 to 1 contributes towards the capture of damped hidden frequencies. The modified S transform also satisfies the normalization condition [20] for S transform windows and hence is invertible. The complex matrix obtained hence can be used to compute the spectral energy by squaring of the absolute values [14].

Selective scaling The selective scaling is dependent on the system studied. It relies on the occurrence of certain frequencies which are found prominently in that system. In power system studies, the harmonics of the fundamental frequency are the major components present. The selective frequencies to be included in the frequency stack are [F, 2F, 3F. . . K.F], where F represents the fundamental frequency and K could be any positive integer. Also, (K.F) 6 (N/2) so that the Nyquist criteria for sampling is adhered to. Since most of the power harmonics are odd harmonics, the frequency stack can be further reduced to [F, 3F, 5F . . . (2n + 1).F]. Since the actual spectrum given by FFT is not an ideal line spectrum, the components exhibit spread in the frequency domain. It would be more appropriate to engage band pass frequency filters at each of the prominent frequencies to capture the spread. Each prominent frequency would act as the mean frequency of its corresponding band pass filter. Fig. 1 shows the frequency filtering through selective scaling. The selected frequencies contribute only to a small part of the discrete S-Transform matrix, but they comprise the majority of the time-frequency localized information required for further processing.

100

200

300

400

500

Magnitude

100

400

500

800

1000

(b)

50 0 0

100

200

300

Frequency (in Hz)

(c)

500

0 0

200

400

600

Fig. 1. Selective scaling (a) the whole frequency spectrum (b) frequency components after selective scaling. The frequency components selected belong to the harmonics of a particular fundamental frequency (c) discrete S-Transform complex matrix representation. Only the darkened portion needs to be computed.

Automatic scaling Since the computation of the discrete S-Transform involves the computation of FFT of the signal, the significant amplitudes in the frequency domain can be found out and the corresponding frequency indices can be added to the frequency stack. This methodology intelligently filters out the less prominent frequency components and hence reduces the number of atomic computations. In automatic scaling, the capture of the significant frequencies and their spread would be automatic because the adding of a frequency to the frequency stack is decided by an amplitude-based filter. The desired cutoff amplitude is set according to the application. The cutoff amplitude can be set depending on statistical parameters such as mean of the amplitudes in the frequency spectrum. Lower the cutoff amplitude more is the probability of increase of frequencies in the frequency stack. Fig. 2 shows the frequency filtering through automatic scaling. The frequencies obtained through automatic scale impart only a minor portion of the discrete S-Transform complex matrix, but they constitute most of the time-frequency localized values to reconstruct the original signal samples. The conventional S-Transform has to perform N ⁄ (N + 2) ⁄ logR(N)/2 additions and N ⁄ (2 ⁄ N + logR(N) ⁄ (1 + N))/2 multiplications using Radix-R FFT and IFFT. The selective and automatic Magnitude

Step 1: Appropriate choice of the frequency scaling is a deciding factor for the fast computation of the algorithm. The scaling technique should intelligently ignore the insignificant frequencies while performing the time localization operation. The choices in frequency scaling could be:

0

Time (in samples)

(a)

100 50 0 0

100

200

300

400

500

400

500

800

1000

Frequency (in Hz) Magnitude

Let xk be an array of samples representing a signal with length N, (k = 1,2,3. . .N).

0

Frequency (in Hz)

(b)

100 50

0 0

100

200

300

Frequency (in Hz) Frequency (in Hz)

FDST matrix formulation for spectral energy

50

Frequency (in Hz)

Fast Discrete S-Transform (FDST) This technique decreases the computational burden of discrete S-Transform by employing an intelligent decision mechanism by numerically filtering out the unwanted frequency information. The prominence of harmonics in the power system makes the high amplitudes in the frequency spectrum continual rather than continuous. Even if the amplitudes are intermittent and have no regular pattern of occurrence, the proposed mechanism can be adjusted in order to automatically choose the frequencies with comparatively higher impact. As an FFT based algorithm exists for evaluation of discrete ST, a frequency stack containing the significant frequencies using the FFT can be formed without much computational burden. Only the frequencies present in that frequency stack need to be subjected to time localization computation. This results in significant reduction in computational requirements of discrete ST. So the frequency values can be chosen so as to evaluate ST only in the vicinity of significant frequency components present in the frequency stack given by the decision mechanism. The proposed FDST uses particular frequency scaling strategy suitable for power system protection computations. This would reduce the number of mathematical calculations involved making it suitable for hardware implementation of a time frequency transform based relaying algorithm in a real-time environment.

(a)

100

500

(c)

0 0

200

400

600

Time (in samples) Fig. 2. Automatic scaling (a) the whole frequency spectrum (b) frequency components after automatic scaling. The frequency components selected have amplitudes higher than particular preset amplitude (c) discrete S-Transform complex matrix representation. Only the darkened portion of the matrix needs to be computed.

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scaling techniques require N ⁄ (1 + K) ⁄ logR(N) additions and N ⁄ (K ⁄ (8 + logR(N)) + logR(N) ⁄ (2 + K))/4 multiplications, where K is the number of frequencies used in the scaling. Other than these scales, many other scales such as dyadic scaling, and logarithmic scaling can be used depending on the application [19]. As opposed to (N/2) time domain computations in discrete S transform, the N dyadic scale N reduces it to log2 2 and the logarithmic scale reduces it to loge 2 . For a sample size of 640, the improvement in speed is nearly 30 times. For one cycle data, the speed is much higher. The intermediate values in the scaling are approximated using different kinds of interpolation techniques. The choice of the interpolation technique is just a tradeoff between required precision and the computational effort in interpolating. Step 2: Calculate the Discrete Fourier Transform of the N signal samples x ¼ ½ x1 . . . xn . . . xN  . Set the width of the Band Pass Filter and this is chosen in accordance to uncertainty principle. The DFT of the signal samples is given by

Y ¼ ½ y1

. . . yn

. . . yN 

ð9Þ

The DFT of the window X is given as

yn ¼

ffi 2pðpffiffiffi N X 1Þðn1Þði1Þ N xi e

ð10Þ

i¼1

The window Y of N frequencies is rotated and concatenated to obtain the H matrix.

2

HMN

y2 6 y 6 3 6 ¼6 6 ... 6 4 yM yMþ1

y3

...

yN

y4

...

y1

...

...

...

yMþ1

. . . yM2

yMþ2

. . . yM1

y1

3

...

wðf 1 ; tn Þ

...

wðf 1 ; t N Þ

...

...

...

7 7 7 . . . wðf m ; t n Þ . . . wðf m ; tN Þ 7 7 7 ... ... ... ... 5 wðf M ; t 1 Þ . . . wðf M ; tn Þ . . . wðf M ; t N Þ

ð12Þ

where each individual value can be represented as



2p2 ðn1Þ2 F

W ðm;nÞ ¼ e

m2





2p2 ðNnþ1Þ2 F

þe



m2

ð13Þ

where m = 1,2, . . ., M and n = 1,2,. . ., N.

W ðm;nÞ ¼ eT 1 þ eT 2 where, T 1 ¼

2p2 ðn1Þ2 F ðaþbmc Þ2

ð14Þ and T 2 ¼

2p2 ðNnþ1Þ2 F ðaþbmc Þ2

Step 4: The window matrix W thus obtained is multiplied with the H matrix element-wise to acquire the windowed frequency domain information.

Gðm;nÞ ¼ Hðm;nÞ  W ðm;nÞ The G matrix thus formed can be represented as

ð15Þ

gðf 1 ; t n Þ

...

gðf 1 ; t N Þ

...

...

...

ð16Þ

Step 4 taking the IDFT of each time localized row representing a pure sinusoid of definite frequency, the Fast Discrete S-Transform matrix can be obtained as given below:

2 SMN

sðf 1 ; t1 Þ

...

sðf 1 ; t n Þ

...

sðf 1 ; t N Þ

...

...

...

...

3

7 7 7 . . . sðf m ; t n Þ . . . sðf m ; tN Þ 7 7 7 ... ... ... ... 5 sðf M ; t1 Þ . . . sðf M ; t n Þ . . . sðf M ; t N Þ

6 6 ... 6 ¼6 6 sðf m ; t 1 Þ 6 4 ...

ð17Þ

where each value in the matrix is given as

Sðm;nÞ ¼

 X 2pðpffiffiffi ffi 1Þðn1Þði1Þ 2 N N Gðm;iÞ e N i¼1

ð18Þ

The S-matrix contains the instantaneous phasor values for each frequency, which can now be used for obtaining features after suitable scaling and interpolation technique. Reduction of computational complexity by scaling Eqs. (12) and (15) can be replaced by the equations given below:

(

Sðm;nÞ ¼

...

...

...

3

7 7 7 . . . gðf m ; tn Þ . . . gðf m ; tN Þ 7 7 7 ... ... ... ... 5 gðf M ; t 1 Þ . . . gðf M ; t n Þ . . . gðf M ; t N Þ

ð11Þ

Step 3: A Gaussian window in frequency domain for N samples is formed. This is a two dimensional window for acquiring localization2in both time and frequency domains. 3

W MN

GMN

gðf 1 ; t1 Þ 6 6 ... 6 ¼6 6 gðf m ; t 1 Þ 6 4 ...

Gðm;nÞ ¼

y2 7 7 7 ... 7 7 7 yM1 5 yM

where M = (N/2) considering Nyquist sampling theorem. For all practical sampling purposes, N is an even positive integer, hence making the time-frequency transforms perform on M discrete and distinct frequencies.

wðf 1 ; t 1 Þ 6 ... 6 6 ¼6 wðf m ; t1 Þ 6 6 ... 4

2

8 N X > < ð2Þ S N

> :

FÞ if ðm 2 b otherwise

Hðm;nÞ  W ðm;nÞ 0

ðm;iÞ

exp



i¼1

0

)

 pffiffiffiffiffi 2pð 1Þðn1Þði1Þ N

ð19Þ 9 > FÞ = if ðm 2 b > ; otherwise

ð20Þ

F represents the set of frequencies obtained from scaling. where b Since some application tasks including pattern recognition demand the fundamental phasors significantly, the instantaneous phasors of the first harmonic frequency can be computed from the matrices independent of the other components. The first row of the H and W matrices can be used for this and the expressions can be reduced to

H1N ¼ ½ yðf 1 ; t1 Þ . . . yðf 1 ; t n Þ . . . yðf 1 ; tN Þ 

ð21Þ

W 1N ¼ ½ wðf 1 ; t1 Þ . . . wðf 1 ; t n Þ . . . wðf 1 ; t N Þ 

ð22Þ

G1N ¼ ½ gðf 1 ; t 1 Þ . . . gðf 1 ; t n Þ . . . gðf 1 ; t N Þ 

ð23Þ

S1N ¼ ½ sðf 1 ; t 1 Þ . . . sðf 1 ; t n Þ . . . sðf 1 ; tN Þ 

ð24Þ

The formulations can be represented in a partially computed form or in a pre-computed form to make it suitable for fast implementation in hardware. Such formulation significantly reduces the computational load on the processor, making fast power quality analyzers and fast adaptive filters for parameter estimation of nonstationary signals possible, which inherit the advantages of FDST. The Fast Discrete S-Transform is capable of giving amplitude information which is resolved simultaneously in time domain and frequency domain.

AMN ¼ FDSTð½ Sð1Þ Sð2Þ . . . . . . SðNÞ Þ

ð25Þ

where M is the number of discrete frequency values and N is the number of time values at which the amplitudes have been localized,

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(a) 20

Maximum FDST Energy (per unit)

10

(b)

0.2 0 -0.2 0.2

0.25

0.3

0.4

(c)

1

Amplitude (per unit)

0.35

0 -1 0.2

0.25

0.3

0.35

0.295

0.3

0.305

0.31

0.305

0.31

0.305

0.31

(b) 0.4 0.2 0 -0.2 0.295

0.3

(c) 0.2 0.1 0 -0.1

0.4

(d)

2 0 -2

Trip

0

0.295

0.3

Time (seconds)

0.2

0.25

0.3

0.35

0.4

(e)

100

Fig. 4. Comparison of maximum differential energy and maximum average energy in each phase during a–g fault (fault distance is 25%, delta is 30° and fault resistance is 10 X) (a). Plots for phase a. (b) Plots for phase b. (c) Plots for phase c.

50 0 0.2

0.25

0.3

0.35

0.4

Time (in seconds)

A is the complex amplitude matrix returned by the FDST after operating on the bunch of signal amplitudes and is given by

½Ai MN

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ realð½Si MN Þ2 þ imagð½Si MN Þ2

ð26Þ

Frequency (Hz)

(a) 150 100 50 0

0.295

0.3

ð27Þ

In online mode, a phase correction factor is introduced to accommodate the error in phase. The actual phase of a signal with frequency f Hz is given as

/corrected ¼ /ST  2pðf =FsÞ

0.305

0.31

100 50 0

0.295

0.3

(c) Amplitude (per unit)

½/i MN ¼ tan fimagð½Si MN Þ=realð½Si MN Þg

0.31

150

The phase matrix is obtained as 1

0.305

(b) Frequency (Hz)

Fig. 3. Schematic diagram of system under study and effect of CuSum on non-fault signal and fault signal. (a) The schematic diagram of system under study. (b) Phase a current signal without fault. (c) CuSum signal of phase a current signal without fault (d) phase a current signal with a–g fault at t = 0.3 s. (e) CuSum signal of the faulted phase a current signal.

20 10

Trip

0 0.295

0.3

0.305

0.31

Time (seconds)

ð28Þ

Out of the M rows in the Si matrix, most of them would have negligible amplitudes and hence they can be regarded as error rather than considering them as a contributing phasor to the original signal. The corresponding rows in Ai and /i matrices do not require computation. Further the computational overhead can be substantially reduced with the help of scaling techniques mentioned earlier. A simple calculation shows that the dyadic scaling is almost 30 times faster than the conventional S-Transform to produce amplitude or phase information. This FDST algorithm is then implemented on a dSPACE Advanced Control Education kit (DS1104 R&D Controller Board) for real-time validation on synthetic signals. The hardware has a 64-bit floating-point processor with a 250 MHz CPU clock and high resolution Analog-to-Digital Converter. The interfacing to the hardware is done using MATLAB/Simulink platform which helps generate C-code for the Simulink blocks used, which is then cross-compiled to dSPACE compatible machine code. Once the code in the real-time hardware starts running, it is interfaced with Control Desk software, allowing the verification of estimations of time-varying harmonics using FDST. The estimations show that FDST exhibits high accuracies even in the presence of reasonable noise.

Fig. 5. Energy contours of a phase during a–g fault (fault distance is 25%, delta is 30° and fault resistance is 10 X). (a) Phase a differential energy contour. (b) Phase a average energy contour. (c) Comparison between maximum average energy and differential energy during fault.

Differential relay tripping The condition for tripping is calculated as follows: 1. The measured current values at both the ends are used to calculate the average and differential values:

I av g ¼

 Is ðkÞ þ Ir ðkÞ 2

Idiff ¼ Is ðkÞ  Ir ðkÞ

ð29Þ ð30Þ

where Is(k) and Ir(k) are the instantaneous current samples at sending end and receiving end, respectively. 2. CuSum technique is used to find the point of surge of differential signal and to determine the CSDP (Cumulative Summation Detection Point).

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vg AaMN

(a) 40

Maximum FDST Energy (per unit)

20

Trip

0.305

0.3

0.31

Idiff ðCSDP  1 þ Ns=2ÞÞ

(b) 0.2 0 0.3

ð32Þ

4. Energy values from the complex amplitude matrix A is obtained by

0.4

0.295

ð31Þ

Adiff MN ¼ FDSTð½Idiff ðCSDP  Ns=2Þ . . . I diff ðCSDPÞ . . .

0 0.295

 ¼ FDST ½Iav g ðCSDP  Ns=2Þ . . . Iav g ðCSDPÞ . . . Iav g ðCSDP  1 þ Ns=2Þ

0.305

vg 2 vg EaMN ¼ jAaMN j

ð33Þ

diff 2 Ediff MN ¼ jAMN j

ð34Þ

0.31

(c) 40 20

Since the energy changes are of high magnitude, only the maximum energy at a given instant of time is required. Hence the energy vector Emax needs to be computed.

Trip

0 0.295

0.305

0.3

0.31

Time (seconds)

Emax 1N ¼ maxðEMN Þ

Fig. 6. Comparison of maximum differential energy and maximum average energy in each phase during a–c fault (fault distance is 55%, delta is 30° and fault resistance is 100 X). (a) Plots for phase a. (b) Plots for phase b. (c) Plots for phase c.

3. Upon the detection of CSDP, half cycles before and after the CSDP are obtained; the set of Iavg and Idiff are passed to the Fast Discrete S-Transform (FDST) to obtain the time-frequency localized signal magnitudes.

ð35Þ

This energy vector can be calculated for both the signals and can be compared to confirm an energy shoot and this corroborates the occurrence of fault. The difference in energy distributions over the time frequency plane is different at fault condition and normal condition. This happens since the differential current is almost absent during normal conditions. The differential energy shoots up during the fault condition due to the reversal of current direction during an internal fault situation. Condition for corroboration of fault is

Table 1 Change in energy of current signals during fault at 25% distance. Faults

For Idiff

For Iavg

Type

Delta

Rf

a

b

c

a

b

c

a–g

30 30 60 60 30 30 60 60 30 30 60 60 30 30 60 60 30 30 60 60 30 30 60 60 30 30 60 60 30 30 60 60 30 30 60 60 30 30 60 60

10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100 10 100

5.1108 0.2859 3.6555 0.3121 0.0003 0.0007 0.0005 0.0007 0.0006 0.0005 0.0006 0.0003 13.8700 11.7230 7.9968 5.7210 11.9180 9.7955 11.4630 10.8050 0.0002 0.0002 0.0000 0.0003 10.5000 2.3700 8.4900 2.1300 9.7400 2.0100 8.4900 1.8000 0.0002 0.0002 0.0000 0.0003 18.3070 18.3070 13.8000 13.8420

0.0003 0.0000 0.0003 0.0001 1.2749 0.5140 1.5549 0.4662 0.0001 0.0002 0.0004 0.0001 7.9400 9.7200 2.6082 4.2100 0.0001 0.0001 0.0002 0.0002 3.2200 3.7200 4.5337 5.0100 9.9700 1.9900 7.7300 1.7800 0.0000 0.0000 0.0000 0.0000 3.2200 3.7200 4.5300 5.0100 5.1200 5.1200 2.9537 2.9500

0.0001 0.0002 0.0001 0.0001 0.0005 0.0004 0.0002 0.0001 2.4800 0.2526 3.2764 0.3370 0.0005 0.0002 0.0002 0.0000 8.1300 10.0030 11.0950 11.8000 3.4500 2.6640 4.1335 3.4900 0.0000 0.0000 0.0000 0.0000 10.3000 2.3900 9.3100 2.1500 3.4500 2.6600 4.1300 3.4900 7.4500 7.4468 8.9772 8.9800

0.4842 0.0717 0.3853 0.1102 0.0060 0.0004 0.0154 0.0076 0.0009 0.0026 0.0169 0.0013 0.9893 0.8517 0.7106 0.5213 0.8705 0.7316 0.9232 0.8735 0.0066 0.0016 0.0016 0.0034 0.8510 0.3390 1.0900 0.5050 0.3570 0.2390 0.3250 0.3690 0.0066 0.0016 0.0016 0.0034 1.2134 1.2134 1.0775 1.0775

0.0094 0.0012 0.0101 0.0047 0.1349 0.1183 0.3107 0.1923 0.0136 0.0043 0.0234 0.0045 0.2495 0.3393 0.0847 0.0498 0.0044 0.0032 0.0116 0.0055 0.2804 0.3629 0.6351 0.7368 0.3180 0.2330 0.2080 0.3650 0.0000 0.0000 0.0000 0.0000 0.2800 0.3630 0.6350 0.7370 0.1332 0.1332 0.3671 0.3671

0.0077 0.0009 0.0221 0.0045 0.0015 0.0019 0.0027 0.0015 0.2437 0.0670 0.5227 0.1235 0.0047 0.0016 0.0135 0.0031 0.5817 0.6811 1.1008 1.1546 0.2461 0.1588 0.4842 0.2896 0.0000 0.0000 0.0000 0.0000 0.8660 0.3410 1.2000 0.5090 0.2460 0.1590 0.4840 0.2900 0.4999 0.4999 0.9471 0.9471

b–g

c–g

ab–g

ac–g

bc–g

ab

ac

bc

abc–g

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K.R. Krishnanand et al. / Electrical Power and Energy Systems 67 (2015) 76–86

(a) R =5 f

R = 10 f

(a)

R = 50 f

R = 100 f

4 2 0 -2 -4

R = 200 f

20 15 10

0

0.29

0.295

0.3

0.305

0.31

0.315

0.32

0.305

0.31

0.315

0.32

0.305

0.31

0.315

0.32

(b) 0

0.2

0.4

0.6

0.8

1

Fault Distance Ratio

2 0

(b)

4 R =5 f

R = 10 f

R = 50 f

R = 100 f

0.285

0.29

0.295

0.3

(c)

R = 200 f

Phase (degrees)

Average Energy

0.285

5

Amplitude (per unit)

Differential Energy

25

3 2

200 0 -200

1

0.285

0.29

0.295

0.3

Times (seconds) 0

0

0.2

0.4

0.6

0.8

1

Fault Distance Ratio Fig. 7. Energy plot of differential and average current signals with respect to various fault distances and for different resistances during a–g fault. (a) Energy plot of differential current signal with respect to various fault distances and for different resistances during a–g fault. (b) Energy plot of average current signal with respect to various fault distances and for different resistances during fault (Rf represents the fault resistance).

Fig. 9. Instantaneous estimation of peak amplitude and phase for fault at 0.3 s (fault distance is 25%, delta is 30° and fault resistance is 10 X). (a) Source side current signal. (b) Peak instantaneous amplitude of the fundamental frequency. (c) Instantaneous phase of the signal.

(a) 100 0

Amplitude (in A)

-100 0.45

0.5

0.55

0.6

0.65

0.6

0.65

0.6

0.65

(b) 20 0 -20 0.45

0.5

0.55

(c) 50

0 0.45

0.5

0.55

Time (in seconds)

Fig. 8. Contour plots for b–c and b–c–g faults at 0.3 s at a fault distance of 25% with a delta of 30°. (a) Contour plot of differential energy of phase a current signal for b–c fault. (b) Contour plot of differential energy of phase a current signal for b–c–g fault having ground resistance of 5 X.

Ediff > Eav g þ Ebias

ð36Þ

The energy bias value (Ebias) accounts for the magnitude of default differential current caused by line charging current. 5. Since maximum energies in the energy matrices for each time point is calculated using Eq. (35), the tripping point is computed as the point after which the data computation shows the maximum differential energy surging with respect to the maximum average energy. The fault can be classified by simultaneous but independent monitoring of all three phases and by checking the energy differences that occur in the faulty cycle detected by the CuSum

Fig. 10. Impact of CT saturation on phase a for an internal a–b–c–g fault (fault at 0.5 s). (a) Differential current during fault (solid curve is the actual signal and the dotted curve is the ideal curve without saturation). (b) Average current during fault (solid curve is the actual signal and the dotted curve is the ideal curve without saturation). (c) Peak amplitudes of the actual signals estimated using FDST (solid curve represents the differential signal amplitude and the dotted curve is the amplitude of the average signal).

algorithm. The algorithm can differentiate fault and load/capacitor switching from the energy differences it causes in the current signals. The rise in differential current occurs from a reversal of current direction. Capacitor switching or load variations would not cause significant reversal of currents. The value of E_Bias also sets the tolerance level below which any energy change can be discarded.

Results and discussions Extensive simulations were done using the system data [22] for different fault resistances, load angles and fault locations using PSCAD/EMTDC software. The system shown in Fig. 3(a) is taken for spectral energy based current differential protection for different fault conditions. The system parameters are given as:

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K.R. Krishnanand et al. / Electrical Power and Energy Systems 67 (2015) 76–86

0.01 0 0.05

Fig. 3(b)–(e) shows the variation in CuSum output in case of a no-fault and fault. These subplots concern the detection of change in signal while constant monitoring through CuSum technique. The figure’s y-axis scales clearly indicate the sudden rise of CuSum signal in case of fault. FDST implementation results Once the waveform surge is detected, FDST is used to corroborate the occurrence of fault using the spectral energy calculations. The misdetection that CuSum could perform due to low frequency components is discarded by the FDST corroboration, if FDST could not confirm a surge in energy. The CuSum value is reset to zero when such a misdetection occurs. Figs. 4–6 show the signal information of the faulted system’s differential and average energy for a period of a single cycle, when processed using Fast Discrete STransform. The system is faulted at 0.3 s for different fault conditions. In the computation performed, discrete energy values for each half cycle before and after the CuSum Detection Point (CSDP) are calculated to check the change in differential energy and average energy. To compare the energy levels of differential current signal and average current signal, the amplitude and hence the energy content of the two signals are localized in time and

0 0.05

0.1

0.15

0.2

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

(c)

0.1

0.15

0.2

0.25

Impact of ground resistances Fig. 7 shows the changes in the maximum energy with respect to fault distance for different ground resistances. The energies for a

(a) CSDP Detection

0.1

0.15

0.2

0.25

0.01 0.25

0.3

0.35

0.4

0.45

0.5

Time (in seconds)

Fig. 11. (a) CuSum of differential current signals of 3 phases. (b) CuSum of average current signals of 3 phases.

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

(b)

0.2 0.1 0 -0.1 0.05

0.1

0.15

0.2

0.25

(c)

300 200 100 0

0.02

0.2

0.5

0.25

0.03

0.15

0.45

frequency using Fast Discrete S-Transform. The highest energy content in all localized frequencies of a given time point is taken for comparison at that instant. It is visually clear from the contour plot the difference in behaviour of the energy of differential current and average current. Since for comparison using a device, we require single values at each instant of time, the maximum energy is considered and plotted. It can be distinctly seen that the differential energy exceeds the average energy after the time of fault for different conditions of fault distances and resistances. In case of no fault, the differential spectral energy is ideally zero, but at the moment of fault, it surges up and exceeds the localized average energy. Half a cycle before the CuSum detection point and half a cycle after that point are used separately to compute the change in energy content of the differential and average signals. These values are provided in Table 1. The energy changes are high in faulted phases, which can be clearly seen in the table.

Amplitude (in p.u.)

Amplitude (in p.u.)

0.25

(b)

0.1

0.4

Time (in seconds)

0.05

0 0.05

0.35

0.02

300 200 100 0

0.01

0.04

0.3

0.25

(b)

0.04

0 0.05

0.02

0.2

0.2

0.02

0.03

0.15

0.15

0.04

(a)

0.1

0.1

Fig. 12. CuSum signal of differential current when compared with that of average current, for each phase (a) comparison for phase a (b) comparison for phase b (c) comparison for phase c.

CuSum implementation results

0 0.05

(a)

0.02

Amplitude (in p.u.)

Parameters of the 400 kV line Considered in the Study: Positive sequence resistance: 0.01537 ohm/km, Positive sequence inductance: 0.8858 mH/km, Positive sequence capacitance: 13.065 nF/km, Zero sequence resistance: 0.04612 ohm/km, Zero sequence inductance: 2.6547 mH/km, Zero sequence capacitance: 4.355 nF/km. Transmission line length: 308 km. Sending end voltage: 400 kV with angle delta, Source inductive impedance: 16 X. Receiving end voltage: 400 kV with zero angle, Receiving inductive impedance: 16 X. The study is done for different fault locations and for different fault resistances using Matlab 7.7 (Simulink) (Intel core 2 duo processor, 3 GHz, 3 GB RAM). The kind of faults simulations done are L-G, LL, LL-G and LLL-G faults. The sampling frequency used is 3.2 KHz or 64 samples per fundamental cycle of 50 Hz. The FDST based algorithm flawlessly provides required energy information and provides tripping signal within 50% of the fundamental time period after the CSDP. All the measurements are done in per unit. Each phase is allotted with its own energy estimator.

0.05

CSDP Detection

0.1

0.15

0.2

0.25

Time (in seconds)

Fig. 13. CuSum signals of differential and average currents for each phase during an a–c–g fault.

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K.R. Krishnanand et al. / Electrical Power and Energy Systems 67 (2015) 76–86

(a)

Amplitude (in p.u.)

300 200 100 0 0.05

CSDP Detection

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.35

0.4

0.45

0.5

0.35

0.4

0.45

0.5

(b) 0.04 0.02 0 -0.02 0.05

0.1

0.15

0.2

0.25

0.3

(c) 300 200 100 0 0.05

CSDP Detection

0.1

0.15

0.2

0.25

0.3

Time (in seconds)

Fig. 14. CuSum signals of differential and average currents for each phase during an a–c–g fault.

certain fault distance and fault resistance combination is computed by finding the maximum energy value in the faulty cycle of the signal, localized both at time and frequency. Such energy values are obtained for various fault resistances and fault distances. It is noticeable that the scale of differential energy variations is much larger than the Average energy variations. Fig. 8 shows the impact of ground in faults. The high frequency components appear in the contour of non-faulted phase only when ground is present in the fault. The experiments with the simulated system reveal that high frequency components with significant amplitudes appear in non faulted phase(s) in the presence of ground in the faulted phase(s). This information could be used to differentiate LL faults and LL-G faults. The high frequency components in the non-faulted phase are the effect of changes in current in the faulted phases and it happens because of the mutual impedance between the phases. The transmission line can be viewed as a simple series RL path. The time constant (L/R) of the fault loop is lesser than the actual transmission line, considering that the fault impedances are almost always resistive. So, the time taken for rise and fall of a signal becomes lesser. During the high resistance faults, the (L/R) ratio becomes further less and causes faster damping of signal components. Since the fundamental component is produced by the generator, it continues to be present, but the temporary high frequency components get damped faster as the fault resistance gets higher. The non-faulted phase(s), in which the time constant has not changed, could sustain the higher frequencies induced by mutual coupling with the faulted phases (see Fig. 9). This information could be used to collect high frequency components through frequency filtering and corroborate a ground fault. The spread of the contour in the time-frequency plane can give the approximate magnitude of ground resistance. As a more effective alternative, the standard deviation value of the first contour can be used. The contour at higher frequencies in the non-faulted phase during a ground fault in the other phases would be present because of the mutual coupling between phases.

can be obtained from FDST, which are otherwise absent in the measured power signal. Fig. 10 shows a case of CT saturation due to an internal a–b–c–g fault at the midpoint of the protected 80% of the line with 0.1 X fault resistance for a delta of 30°. The figure clearly indicates that even though CT saturation causes distortion of the original signals, the magnitude of the differential signals remains higher than the average signals after the fault. The fast action of the relay makes sure that the trip signal is generated before the impact of the CT saturation becomes prominent. The system is also tested for external fault where CT saturation occurred due to an external a–b–c–g fault outside the protected 80% of the line with 0.1 X fault resistance for a delta of 30°. It is observed that the magnitude of the differential current is much lower than that of the average signal. Hence, the relay does not trip at all and the proposed protection scheme remains unaffected by the CT saturations due to external faults. Effect of power swings on FDST based protection In addition, the system has been tested for various delta values, ranging from 0° to 120°. In all cases, the proposed approach provides excellent performance, indicating the capability of the method to execute well even during conditions of power swing. Effect of capacitor/load switching At 0.3 s, a capacitor bank is switched on at the mid-point of the transmission line and the CuSum of average current and differential current signals are obtained for comparison. It is noticeable from Fig. 11 that the signal variations are only from the white Gaussian noise added to them, which account for noisy measurement. The magnitude of CuSum of differential signal would increase significantly with respect to that of average signal only in case of major reversal in direction of current flow as in the case of an internal fault. Thus Capacitor switching or load switching results in line currents significantly lower in magnitude when compared to faults. Therefore, they have no impact on the fault detection procedure and the CuSum does not trigger the Fast Discrete S Transform. The effect of Capacitor switching is clearly visible from Fig. 12, where the differential current is near zero even in the event of capacitor switching. In the case of a fault, the magnitude of the differential current would be much higher than the average current, which will occur in a few milliseconds. Since the CuSum does not trigger the Fast Discrete S Transform in the case of capacitor switching or load switching, variations would not cause significant reversal of currents. The value of E_Bias also sets the tolerance level below which any energy change can be discarded.

(n CSDP ¼

o) ð37Þ

NULL Otherwise Effect of short circuit capacity of the system The three-phase inductive short-circuit power (PSC in VA) is dependent on the internal inductance L of the sources. Varying short circuit capacity involves changes in the source impedances.

Effect of noise and CT saturation on the differential protection

PSC ¼ Even though the effect of noise is negligible in protection of transmission lines, 30 dB of noise was added to the source side signals and the receiver side signals to verify the immunity of the proposed technique to the random noise in the signal. CT saturation can be detected from the unique high frequency contours which

kjCuSumIdiff ðkÞ > CuSumIav g ðkÞ þ CuSumBias

V 2base 2pfL

ð38Þ

In the simulations, fault detections for a–c–g faults are observed and are displayed in Figs. 13 and 14. System parameters are varied to obtain short-circuit powers that are 0.1 times and 10 times the original short-circuit power. The CuSum results shown in Fig. 14

K.R. Krishnanand et al. / Electrical Power and Energy Systems 67 (2015) 76–86 Table 2 Fault location estimations using FDST. Location of the fault (in%)

Location of the fault (in km)

Location estimated by FDST (in km)

Error in estimations (in per unit)

10 20 30 40 50 60 70 80 90

30.8000 61.6000 92.4000 123.2000 154.0000 184.8000 215.6000 246.4000 277.2000

30.7975 61.5267 92.1140 123.4538 155.0756 186.6231 218.7628 251.4882 284.7648

8.0986E06 2.3787E04 9.2850E04 8.2397E04 3.4923E03 5.9191E03 1.0269E02 1.6520E02 2.4561E02

clearly indicate that such variations in short-circuit capacity do not affect the detection since the CuSum signals of differential currents increase to very high values in a very short time. Here, the short circuit power is same as that which corresponds to the original system parameters taken from the reference literature. In Fig. 14, the shortcircuit power is 10 times that of the original system’s short circuit power.

FDST can compute instantaneous amplitude and phase information of the three-phase voltages and currents measured at the source end and hence can be useful in locating the fault on the transmission line. The source side current signal information is computed and plotted in Fig. 9. Voltage signals are also estimated in this manner and used for the purpose of fault location calculation. Hence FDST can act as an excellent tool to achieve the fault location value. The first fault location subroutine is devised in order to locate faults, taking place in the transmission line section between the sending end bus and fault point. Different fault types use separate formulas for apparent impedance computations and are given below: For L-G faults, apparent impedance can be computed as

V  phase   I þI þIphasec Z0 Z1  Iphase þ  Z1   phasea phaseb 3

ð39Þ

For LL faults, apparent impedance is

Z Apparent ¼

V line Iline

ð40Þ

For LLL faults and LLL-G faults, apparent impedance can be given by

Z Apparent ¼

V phase Iphase

ð41Þ

After obtaining the apparent impedances, the fault location is calculated from

Location ¼

imagðZ Apparent Þ X L0

The approximations given in Table 2 are very close to the actual value for practical purposes and the errors are tolerable for such a long transmission line system. Conclusion The Fast Discrete S-Transform is applied successfully to the studied transmission system for differential relaying. The robustness of the transform is tested for various case studies of the system using variation in fault types, fault locations and fault resistances. The capability of Fast Discrete S-Transform (FDST) to give comprehensive information about the signal features can be used to study the fault conditions as well as other minor disturbances in normal conditions of the transmission line. From the computational results presented in the paper it is quite clear that the pattern recognition approach using the Fast Discrete S-Transform is robust to distinguish between internal and external faults, and the nature of the faults. Further, the FDST has the inherent property of estimating the amplitude and phase angles of the voltages and currents and is thus suitable for fault location computations. Acknowledgment

Fault location computation and the condition of external fault

Z Apparent ¼

85

ð42Þ

LLL-G fault, being most severe, requires more attention in fault studies. The results of fault location estimations for a LLL-G fault with a fault resistance of 10 X in the system taken for study are given in Table 1. In the simulation, an ABCG fault is setup at different locations of the 308 km transmission line studied with a fault resistance of 10 X. An additive white Gaussian noise signal of 30 dB is added to the signals.

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