Detection of PQ Events Using Demodulation Concepts - International

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.13(2012) No.1,pp.64-77

Detection of PQ Events Using Demodulation Concepts : A Case Study Rajiv Kapoor2 , Manish Kumar Saini1 ∗ , Prerit Pramod3 2 1

Electronics and Communication Engineering Department, Delhi Technological University, New Delhi. Electrical Engineering Dept., Deenbandhu Chottu Ram University of Science and Technology, Sonepat. 3 Electrical Engineering Department, Delhi Technological University, New Delhi, India. (Received 9 March 2010, accepted 25 May 2010)

Abstract: Amplitude and frequency demodulation concepts have been utilized for the novel real-time analysis of PQ events. The earlier techniques are grounded upon analyzing the few cycles of the power signal, having the computational complexity of the order O(n2 ). The proposed method considers the PQ event formations due to modulation of the power signal as carrier and the event as a modulating signal. The concept of demodulation has been used to separate/segment various event patterns and MUSIC algorithm has been applied to detect the presence of the various frequencies. This technique is well tested and detects transients, sag, and swell (for single PQ events) in real-time. Feedforward neural classifier has been employed for classification of PQ events from the knowledge base. Keywords: PQ events; demodulation; neural network classifier

1

Introduction

The importance of power quality (PQ) event detection and classification is ever increasing due to the wide use of delicate electronic devices. PQ events can occur due to lightning, capacitor switching, motor starting, nearby circuit faults, or accidents, and can lead to power interruptions. Harmonic current due to nonlinear loads throughout the network also degrade the quality of services to the sensitive high-tech customers, such as India’s IT parks in Bangalore, Hyderabad and many other places. The massive rapid transit system, Metro Railways in Delhi and few other places in India have facilitated the massive use of semiconductor technologies in the auto-traction systems, resulting in the increased level of harmonic distortion. The solution to the PQ related problems requires continuous monitoring and the acquisition of large amount of data from the distribution system. In [1], also emphasize the need of an automated PQ detection and classification system to determine the cause of PQ disturbances. Several signal processing and statistical analysis tools have been presented for the detection and classification of PQ events. The survey in [2] cites most of the work presented in PQ events classification using different signal processing tools like Fourier, wavelet, Gabor transforms etc. Recently, authors have presented PQ event detection and classification using higher-order cumulants using quadratic classifier [3], adaptive wavelet network [4], support vector machine [5], covariance based behavior of several voltage waveform features [6], wavelet transform and self organizing learning array system [7], S-transform [8], discrete wavelet transform and artificial neural network with fuzzy logic [9], a support vector machine for the statistical classification of voltage disturbances [10], Hilbert and Clarke transform [11] and S-transform based probabilistic neural network model [12]. All these techniques can detect PQ disturbances but number of samples required was large and hence the complexity of the algorithm is high enough so as not to allow it to work in real-time. The proposed method is based upon considering that PQ events are formations due to modulation of the power signal as carrier and the event as a modulating signal. The main objective of the paper is to separate between single and multiple PQ events. XCF has been used to accomplish the aim. After this stage, two efficient stages are introduced in the proposed methodology. Further, analysis has been performed in two stages; the first stage is based upon demodulating the signal using AM and FM demodulation and second stage is based upon MUSIC algorithm. The information obtained by amplitude, frequency demodulation and dominating ∗ Corresponding

author.

E-mail address: [email protected] c Copyright⃝World Academic Press, World Academic Union IJNS.2012.02.15/576

R. Kapoor et al: Detection of PQ Events Using Demodulation Concepts : A Case Study

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Figure 1: (a) Voltage sag due to sudden switching-on of a large load (b) Output waveform of 1(a).

Figure 2: (a) Generated voltage sag, (b) Generated voltage swell frequency obtained by MUSIC algorithm have been used to from the knowledgebase. The knowledgebase formed has been utilized for training phase of neural network. Feedforward network has been utilized for further identification of PQ events. Section 2 describes the event generation and simulation of events. Section 3 explains the methodology of proposed scheme. Section 4 delineates the simulated results of the Hybrid demodulation concept and spectral harmonics analysis based upon MUSIC algorithm. Section 5 shows the classification results of proposed algorithm at different stages of the power system disturbances. Section 6 accomplishes the feedforward neural network classifier and its corresponding results. Section 7 gives the conclusions of proposed algorithm.

2

Event generation and simulation

A single (or multiple) PQ event means that any one half-cycle of the modulated signal under consideration has only one (or more than one) type of PQ event. For example, if a half-cycle of a modulated signal contains voltage sag and a transient, the modulated signal is said to contain a multiple PQ event. In the absence of the transient, the same event would be called a single PQ event, since it would contain only sag. Voltage sag refers to a fall in the voltage waveform at the receiver’s end for a brief interval of time. Voltage sags are caused due to the sudden switching-on of a large load. A large load means that the device under consideration draws a large input current, which causes a large voltage drop due to the impedance of the line, thus resulting in a net reduction in the voltage at the receiving end. Figure 1(a) represents a circuit which causes the occurrence of voltage sag due to the sudden switching-on of an arbitrary large load. The switch, S represents the sudden switching operation. If the load is connected to the line suddenly at 0.16s, the output thus obtained is shown in Figure 1(b). A voltage swell is a short duration increase in voltage waveform. A voltage swell is a short duration increase in voltage waveform. Transfer of load from the utility source to the standby generator source during loss of utility power. Most facilities contain emergency generators to maintain power to critical loads in case of an emergency. Sudden rejection of loads to a generator could create significant voltage swell. Sudden application of loads (Three phase induction motor 20 KV A, 460 V, 50 Hz, 1440 RP M ) to a generator could create significant voltage sag. Power semiconductor devices act as a controlled switches (i.e. the instant of switching on and off) can be controlled using appropriate control circuits. Thyristors use gate triggering circuits for switching control. Distortion also occurs due to the use of nonlinear loads. AC voltage controllers are thyristor based devices that convert fixed AC voltage to variable AC without a change in the frequency. They are used in heating devices, lighting control, and speed control of motors among many uses. They are also invariably used as fan regulators. The circuit of a 1-ϕ full-wave AC voltage controller used for speed control of a 1-ϕ induction motor is shown in Figure 3(a). The firing angles for thyristors are controlled in such a way so as to obtain the desired RMS voltage at the input terminals of the motor. The waveform of the voltage at the input of the motor is shown in Figure 3(b). Fourier analysis of the voltage waveform shows that only odd-harmonics

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International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 64-77

Figure 3: (a) Circuit representing harmonic distortion occurring due to use of an AC voltage controller, (b) Output waveform of 3(a)

Figure 4: Recorded harmonics of one phase due to fluorescent lamp circuit

are present in the voltage. Harmonic components of voltage have frequencies which are odd integral multiples of the fundamental frequency. Harmonic distortion is also caused by fluorescent lamp. It has highly nonlinear V-I characteristics, which arises mainly due to the magnetic core inductors that are present in its ballast circuit. Thus, the application of a sinusoidal voltage across it gives rise to a distorted current, which has an appreciable content of 3th , 5th and 7th harmonics. The fluorescent lamp circuit is used to record the harmonics in the current in which the current has been converted to an equivalent voltage waveform having RMS value. The distorted current waveform in one of a phase is shown in Figure 4. Generation of all the events has been done in Power System Laboratory of Delhi Technological University, New Delhi (India). To prove the effectiveness and efficiency of any algorithm used to detect and classify PQ events, it is necessary to implement the algorithm on events that occur in the practical circumstances. The algorithm presented in this paper is used on such practical events, which were recorded in the Power Systems Laboratory. The actual method used to accomplish the task of detection and classification of PQ events is described in detail in coming section.

3

Methodology for the detection and classification of events

In this paper, flowchart of the algorithm is shown in Figure 5. In the methodology presented here, a modulated signal is taken as an input to SIA at a sampling rate of 128 samples/cycle and cross-correlation between corresponding half-cycles of input power signal and a standard Sine wave is calculated as in Figure 6. The value of the correlation coefficient so obtained is compared with a preset threshold value, which categorizes the modulated signal on the basis of whether is contains single or multiple PQ events. Next, the amplitude and frequency demodulation of the input wave is obtained. The amplitude demodulated output is further passed through a slope detector that detects the global maxima/minima of the amplitude demodulated output. The frequency demodulated output is used for the obtaining a number of analytical results, that aid the classifier in more precise decision-making. Along with the two demodulation techniques used, the modulated signal is also analyzed simultaneously using the MUSIC algorithm. The MUSIC algorithm determines the harmonics, i.e., frequency components, higher than the fundamental (50Hz) component, present in the modulated signals. All the information so obtained forms the knowledge base of classifier, which classifies PQ event(s) present in the modulated signal. The algorithm is explained through a block diagram in Figure 5. The check box decides whether the modulated signal at the input contains a single or multiple PQ events. Further analysis of the signal, according to chronological order of processing, proceeds in the manner shown in the system of Figure 5.

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R. Kapoor et al: Detection of PQ Events Using Demodulation Concepts : A Case Study

Figure 5: Real-time System for the Implementation of Algorithm (SIA)

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International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 64-77

Figure 6: Calculation of XCF between corresponding half-cycles of modulated signal and standard wave

3.1

Cross-correlation of modulated signal with standard wave

Cross-correlation, in simplest terms, is a measure of similarity between two waveforms. For two continuous functions f (t) and g(t), it is mathematically represented as: ˆ ∞ [f ∗ g](t) = f (Γ)g(t + Γ)dΓ (1) −∞

where “*” stands for operation of cross-correlation and f (Γ) represents the conjugate of f (Γ). In the proposed scheme, cross-correlation is normalized by the square of RMS value of the standard wave. The voltage waveform at every stage is available to the SIA as shown in Figure 5. SIA takes one half-cycle of this voltage waveform as its input power signal. Although the description given here is for one half-cycle at a time, the algorithm works equally well for a larger time lag, but it is mandatory to use an even number of half-cycles or integral number of cycles, as an input to the SIA. For each half-cycle, the block named “Normalized Cross-Correlation”, calculates the normalized cross-correlation coefficient (XCF) between a half-cycle of the modulated signal and the corresponding half-cycle of the standard wave, i.e., between two consecutive zero-crossings, of the modulated signal and corresponding half-cycle of the standard sine wave, as shown in Figure 6. If at any some point of time, a permanent phase shift occurs in the line, Normalized Cross-Correlation block, automatically detects it using a zero-crossing detector, and uses an appropriately phase-shifted standard wave to calculate cross-correlation coefficient (XCFs). The XCF so obtained has a value lying between 0 and 1. As mentioned earlier, the SIA calculates an XCF for each half-cycle of the modulated signal at its input. For those half-cycles, which are close to standard voltage level in magnitude and 50Hz in frequency, the XCF obtained has a value very close to one and if a PQ event is present in the input signal then the XCF has a lower value than one. Different PQ events have different values of the XCF obtained. Since cross-correlation measures the extent of similarity between a wave corrupted with a PQ event and standard wave, those half-cycles of the modulated signal which have more than one type of PQ event, will have a much lower value of XCF with respect to single type of PQ event. The SIA continuously takes one half-cycle of the modulated signal at its input, calculates the XCF for that half-cycle and on the basis of the value of the calculated XCF, sends it to the appropriate next block for further processing, according to the diagram shown in Figure 5. Consider a situation in which SIA is calculating XCF for a half-cycle present at its input (modulated) power signal is as shown in Figure 7(a). If the present time is t = 0s, then at 0.23s i.e., at the end of the first-half of the 11th cycle of the modulated signal, which shows a sag of 0.7 p.u. along with a transient of magnitude 150V and frequency 250Hz superimposed on it, the value of the XCF just calculated by the SIA, would be low as compared to the half-cycles that had occurred at t ≤ 0.22s, because these cycles did not contain PQ event. The values of XCFs calculated for each half-cycle of the modulated signal of Figure 7(a) and the corresponding half-cycles of the standard sine wave by the SIA is shown in Figure 7(b). For the cycles 1-10 and 12-25, the 48 coefficients have values very close to one. This implies that these cycles do not contain PQ event. For the 11th cycle, first and second half-cycle XCF have values 0.42509 and 0.6993 respectively. The XCF for the first half-cycle of the 11th cycle has a lower value than the second because it contains a transient along with sag, whereas the second contains only sag. This shows that it is possible to set threshold values of XCF so as to distinguish single and multiple PQ events from one another. The values for XCFs corresponding to different types of events with waveform are given in Table 1. Note that the values of the XCFs for multiple events given in the Table 1 are for a particular situation and these values can be slightly different if calculated for the same event in some other situation. To further clarify this, consider a half-cycle which has

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R. Kapoor et al: Detection of PQ Events Using Demodulation Concepts : A Case Study

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Figure 7: Modulated signal containing sag and transient in the 11th cycle, (b) Set of XCFs

Figure 8: XCFs obtained for two signal different instances of same event (sag with transient) (a) 0.42509 (b) 0.4271 sag of 0.7 p.u. along with a transient of peak amplitude 80V superimposed on it. For this type of multiple events, there can be many points on the half-cycle where the transient occurs. Two sample cases for such an event are shown in Figure 8. The XCFs calculated for the two cases are mentioned in Figure 8. Note that though there is a finite difference in the two values, the difference is quite small. As mentioned earlier, single and multiple events can be distinguished by setting appropriate threshold values of XCF. The threshold values for the proposed algorithm are shown in Table 2. These values have been set after obtaining XCFs for a number of PQ events of both single and multiple PQ event. The problem of different XCFs for different instances of the same type of multiple event (as exemplified earlier), have also been considered. So, the threshold values given in the Table 2 are exact and XCF for single and multiple events always lie within the specified ranges. The threshold value for a single PQ event is 0.5 and for a multiple PQ event it is 0.1. A value above 0.95 implies that there is no event in the cycle under observation. This means that once the SIA calculates the XCF for a particular half-cycle (of value less than 0.95), the event is categorized as a single PQ event if the value of its XCF is greater than 0.5, “Single Event Analysis” is performed on that half-cycle Figure 5. If the value of XCF lies between 0.1 and 0.5, then the half-cycle is categorized as a multiple PQ event and the SIA performs “Multiple Event Analysis” on it. Any half-cycle having a value of XCF less than 0.1 is categorized as an interruption by the SIA. This concludes the method for the preliminary categorization process for differentiating single and multiple PQ events from one another. The ultimate aim of the complete algorithm is to identify exactly which event i.e., whether sag, swell, transient, impulse or distortion, is present in single and multiple PQ events. To accomplish the task of identification, the input signal of the SIA must be represented as modulated signal, so that demodulation techniques which have been used for extracting information from the signal for the classification task, can be properly justified.

3.2

Representation of PQ events as hybrid amplitude and/or angle modulated signals

When a power signal contains PQ events, it can be considered as a hybrid signal. It means that the signal is amplitude or angle modulated or both at the same time. Amplitude and phase information is extracted from this hybrid signal using the hybrid demodulation[13] approach as explained below. Let the hybrid signal be written as a real function of time as amplitude modulated signal with a linear AM component: s(t) = [1 + a(t)]cos[2πfo t + ϕ], |a(t)| ≤ 1 where fo is the power frequency (carrier frequency), a(t) is the amplitude modulated function, is the phase constant of the power signal and assumed to be 0. The restriction on the magnitude of a(t) prevents over-modulation. If the power signal is simultaneously frequency modulated, then the hybrid power signal can be represented as: s(t) = [1 + a(t)]cos[2πf t]

(2)

where f = fo + ∆f and ∆f represents the instantaneous change in the carrier frequency. Here, ∆f is proportional to instantaneous value of the frequency modulating signal fm (t)such that: ∆f = kf  fm (t)

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(3)

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International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 64-77

where kf is the frequency sensitivity of the modulator. At this point, it is important to introduce another quantity called modulation index, mf , of the modulating signal. It is given by: ∆fmax fmax

mf =

(4)

where ∆fmax represents the maximum frequency deviation and fmax represents the maximum frequency content of the modulating signalfm (t). Now, equation (3) can be written as: s(t) = [1 + a(t)] cos[2πfo t + θ(t)]

´t

(5)

where θ(t) = kf 0 fm (t)dt. If a(t) and θ(t) are slowly varying in comparison with fo , the signal is said to be narrowband and its spectrum is S(f ) given by fourier transform as: ˆ∞ S(f ) =

s(t)e−i2πf t dt

(6)

−∞

and S(f ) contains most of its energy in the vicinity of fo . It is convenient to consider a complex signal for analysis. So, we can also represent the power signal as: zlin (t) = [1 + a(t)]eiθ(t) ei2πfo t , |a(t)| ≤ 1

(7)

The spectrum of a(t) is band-limited to Fa . However, when the signal is angle modulated, its spectrum extends indefinitely on either side of fo , regardless of the spectral extent of θ(t) because of the transcendental nonlinear exponential operation. Fortunately, if the peak phase excursion of θ(t) is limited (to, say, ± π2 or less) and the spectrum of θ(t) is band limited to Fθ , the spectrum of the purely angle modulated signal (i.e. when a(t) is constant) is largely confined to the band fo ± fθ . When modulated simultaneously by a(t) and θ(t), the spectrum of the modulated power signal is extended somewhat by a convolution process to a value larger than that produced by either amplitude or angle modulation alone. Amplitude/angle modulated signal using exponential AM is given as a complex function by zexp (t) = e[a(t)+iθ(t)] ei2πfo t = eh(t) ei2πfo t

(8)

h(t) = a(t) + iθ(t)

(9)

where and the subscript ’exp’ denotes that the signal corresponds to exponential AM. Thus the signal given by (6) is a function of the two real variables a(t) and θ(t), whereas the signal given by (9) is a function of the single complex variable h(t). When the expansion after the demodulation is obtained using a functional Taylor’s series, the use of the exponential form for the input results in a univariate expansion in terms of a single complex variable rather than a bivariate expansion in terms of two real variables. There is little difference in the complexity or difficulty of the two approaches for the first or second term in the expansion. However, higher order terms in bivariate expansion grow much more rapidly in complexity than those from the univariate expansion. In fact, general recursive relationships can be developed in the univariate case to permit writing terms of any desired order explicitly which does not seem to be possible with the bivariate expansion. Of course, the analytical simplicity of using the representation for exponential AM is of little value if the results so obtained cannot easily be related to those for linear AM, which is the form of modulation of practical interest. The relationship between linear and exponential AM can be seen by expanding the factor exp a(t) in (11) in a Taylor’s series and then noting, from (10), that: zexp = [1 + a(t) + o(a2 )]eiθ(t) ei2πfo = zlin (t) + o(a2 )eiθ(t) ei2πfo 2

(10)

2

where o(a ) denotes terms of order a and higher. Thus, if zexp (t) is used as the input signal, the output will contain the desired result represented by zlin (t) plus extraneous terms, all of which will involve factors o(a2 ). Now, consider an arbitrary filter having an impulse response g(t) and a steady-state transfer function G(f ). This is a Fourier pair given by: ˆ ∞ g(t) = G(f )ei2πf t df (11) −∞

ˆ

∞

G(f ) =

g(t)e−i2πf t dt

−∞

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(12)

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Table 1: Values of cross-correlation coefficients (XCF) for the different PQ events S. No.

Events Type

Description of PQ Event

Waveforms of Event

1.

Single Event

Sag

0.6993

2.

Single Event

Swell

0.7

3.

Single Event

Impulse Transient

0.70499

4.

Single Event

Harmonic Distortion(Fluorescent tube)

0.72839

5.

Multiple Event

Sag & Transient

0.42509

6.

Multiple Event

Swell & Harmonics

0.43610

7.

Multiple Event

Sag & Harmonics

8.

Multiple Event

Sag, Distortion & Transient

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XCF

0.43288, 0.30146

0.40815, 0.30146

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Table 3: Threshold values of XCFs Range of XCF Type of PQ Event Threshold Value ≥ 0.95 None 0.95 ∈(0.95,0.50) Single 0.50 ∈ (0.1,0.5) Multiple 0.10 ≤ 0.10 Interruption g(t) is real and for a physically realizable filter, vanishes for t < 0 (i.e., it is causal). G(f ) is complex, in general, with an even real part and an odd imaginary part. Therefore, G(f ) = G(−f )

(13)

where ’G’ denotes the complex conjugate of G. This is the kind of filter of interest. It is called band pass because it passes a band of frequencies preferentially i.e., the magnitude of its response |G(f )| is large in the pass-band and small elsewhere. It is said to have a center frequency fc which may be the frequency at which the response is the greatest, or a frequency that characterizes the center of the pass-band in some sense. The choice is obvious for a symmetric filter but may be arbitrary for any other. In general, it suffices that a center frequency be specified, however chosen. If the width of the pass band is small in comparison with the center frequency, the filter is said to be narrow-band. Just as the bandwidth of the modulated signal is related to the frequency content of its modulating signal relative to carrier frequency, so also is the bandwidth of the band-pass filter related to the apparent frequency content of its impulse response relative to the center frequency. That is, the impulse response can be written as: g(t) = r(t) cos[2πfc t + φ(t)]

(14)

where r(t)and φ(t) are real functions that vary slowly in comparison with fc . Furthermore, just as the spectrum of the real modulated narrow-band signal contains most of its energy in the vicinity of ±fo , the transfer function of the narrow band pass (having a real impulse response) also contains most of its response in the vicinity of ±fc . Finally, just as the complex narrow-band modulated signal z(t) given by (8) is approximately analytic and has negligible spectral content at negative frequencies, so also would the complex impulse response of the narrow band pass filter given by: γ(t) = r(t)eiφ(t) ei2πfc t

(15)

Let γ(t) be approximately analytic and its transfer function Γ(f ) have negligible response at negative frequencies. Note that the odd conjugate relationship for G(f ) given by (11) is not valid for Γ(f ). Rather : Γ(f ) ∼ = 0, f < 0

(16)

with the bulk of the Γ(f ) in the vicinity of fc . When the complex modulated signal given by (8) for a hybrid AM/PM signal having exponential AM is applied as an input to the filter having complex impulse response γ(t) given by (16), the output becomes: ˆ∞

ˆ∞ γ(τ )z(t − τ )dτ = e

x(t) = 0

where

i2πfo t

γ ′ (τ )eh(t−τ ) dτ

(17)

0

γ ′ (t) = r(t)eiφ(t) ei2π(fc −fo )t

(18)

′

and (10) has been used to introduce h(t). The term γ (t) is the complex response of a filter that is the low pass equivalent of the mistuned narrow band pass filter. It is seen from (17) and (19) that: γ ′ (t) = γ(t)e−i2πfo t

(19)

The transfer function Γ′ (f ) of the low-pass equivalent to the mistuned narrow band pass filter is given by: ′

ˆ∞

Γ (f ) =

′

γ (t)e 0

−i2πf t

ˆ∞ dt =

γ(t)e−i2π(f +fo )t dt = Γ(f + fo )

0

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(20)

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Figure 9: (a) Transient PQ event (b) Output of amplitude demodulation and (c) Frequency demodulation As a result, ˆ∞

γ ′ (t)dt = Γ(fo )

(21)

0

The equivalent low-pass filter is not physically realizable, in general, but is approximately physically realizable for a nearly symmetric narrow band-pass filter that is properly tuned. When the filter output x(t) given by (18) is amplitude or phase demodulated, the results are, respectively: √ x(t)x(t)

(22)

f (t) = argx(t) = Im[log x(t)]

(23)

f (t) = magx(t) = or

where ’Im’ denotes the imaginary part and the term ei2πfo t in (15) has been suppressed. For convenience, the primes in (21) and (22) will be dropped henceforth, and the quantities γ(t) and Γ(t) will be understood to refer to the impulse response and transfer function, respectively, of the normalized low-pass equivalent of the real narrow mistuned band pass filter. Equations (23) and (24) give the demodulated output. The demodulator output f (t) given by (23) and (24) are nonlinear functional of the filter output (18), which is itself nonlinear functional. That is, x(t) and, therefore f (t) both depend upon all the values taken by h(t) upto the present time. The functional power series (or Volterra series) can be written as f (t) =

Σ∞ n=1

1 n!

ˆ∞

ˆ∞ dun gn (u1 , ..., un ).Πnr=1 h(t − ur )

du1... 0

(24)

0

wheref (t) is the output,h(t) the input, and the kernels gn (u1 , ..., un ) describe the system. Similarly, the desired functional Taylor’s series expansion is given by ∞ f [h∞ 0 (t)] = Σn=0

1 dn f [ζh∞ 0 (t)] -ζ=0 n! dζ n

(25)

where the notation with superscript ∞ and subscript 0 is used to emphasize that f is a functional of h that depends on its entire past history. For simplicity, following Volterra’s lead, these will be dropped but with the cautionary note that f (h(t)) does not mean the values of f at the current values of h. Dropping the argument t as well permits f (h) = Σ∞ n=0 where δn =

1 δn n!

dn f (ζh)|ζ=0 dζ n

(26)

(27)

The derivatives of the amplitude demodulator output (23) are best found recursively by starting with the square of (23). Writing ζh for t yields The nth derivative of the left side of (29) evaluated at ζ = 0 becomes: n−r dn 2 dr n r d f (ζh)| = Σ C f (ζh)| f (ζh)|ζ=0 = Σnr=0 Cnr δn−r δr ζ=0 ζ=0 r=0 n dζ n dζ n−r dζ r

where (28)was used to introduce δn and Cnr is the Bi-nomial coefficient:

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(28)

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International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 64-77

Cnr =

n (n − r) r

(29)

To facilitate differentiating: ˆ x (ζh) ≡

γeζh

(30)

where (9) has been used for z (t)and where a convolution integral is understood. The nth derivative of the right side of (26) then becomes: ˆ ˆ ¨ dn dn ζh ζh γe γe |ζ=0 = γ1 γ2 (h1 + h2 )n x (ζh) x (ζh) |ζ=0 = n (31) dζ n dζ where the subscripts identify the variables of integration. Equating (30) and (33) yields: ¨ n r Σr=0 Cn δn−r δr = γ1 γ2 (h1 + h2 )n (32) which is the basic recursive generating formula for δn . Evaluating (34) for successive values of n leads to: δ0 = |Γ0 |; δ1 /δ0 = I1 ; δ2 /δ0 = I2 − (δ1 /δ0 )2 = I2 − I12 δ3 /δ0 = I3 − 3(δ1 /δ0 )(δ2 /δ0 ) = I3 − 3I1 I2 + 3I13 δ4 /δ0 = I4 − 4(δ1 /δ0 )(δ3 /δ0 ) − 3(δ2 /δ0 )2 = I4 − 4I1 I3 + 18I12 I2 − 3I22 − 15I14 δ5 /δ0 = I5 − 5(δ1 /δ0 )(δ4 /δ0 ) − 10(δ2 /δ0 )(δ3 /δ0 ) = I5 − 5I1 I4 + 30I12 I3 − 150I13 I2 + 45I1 I22 + 105I15 − 10I2 I3 and so on, where (22) is used to evaluate δ0 and 1 In = 2|Γ0 |2

(33)

¨ γ1 γ2 (h2 + h2 )n , , n ≥ 1

(34)

The expansion for the amplitude demodulation of a filtered hybrid amplitude/angle signal exponential AM (9) is then given by (27) using (35) for the δn and (36) for the I n . As mentioned earlier in (11), the results obtained by using exponential AM can be reduced to those for linear AM by deleting the terms O(a2 ) in the expansion. This can be done noting that (h1 + h2 )n = [(a1 + iθ1 ) + (a2 + iθ2 )]n n(n − 1) (a1 + a2 )2 [i(θ1 − θ2 )]n−2 + ... (35) 2! It is apparent that terms O(a2 ) appear in all but the first two terms. When these undesired terms are deleted, only two leading terms are remained and a term with n(n−1)a1 a2 [i(θ1 −θ2 )]n−2 form the third term. The fourth and the succeeding terms vanish entirely and the result is (h1 + h2 )n = [i(θ1 − θ2 )]n +n(a1 + a2 )[i(θ1 − θ2 )]n−1 + n(n − 1)a1 a2 [i(θ1 − θ2 )]n−2 = [i(θ1 − θ2 )]n + n(a1 + a2 )[i(θ1 − θ2 )]n−1 +

= [n(a1 + a2 ) + i(θ1 − θ2 )][i(θ1 − θ2 )]n−1 + n(n − 1)a1 a2 [i(θ1 − θ2 )]n−2 Then (34) becomes ˜ γ1 γ2 (h2 + h2 ) = I1 = 2|Γ10 |2 In =

1 2|Γ0 |2

1 2|Γ0 |2

˜

(36)

γ1 γ2 [(a1 + a2 ) + i(θ1 + θ2 )]

¨ γ1 γ2 {[n(a1 + a2 ) + i(θ1 − θ2 )].[i(θ1 − θ2 )]n−1 + n(n − 1)a1 a2 [i(θ1 − θ2 )]n−2 }, f or n ≥ 2 (37)

and the expansion for the amplitude demodulation of a filtered hybrid amplitude/angle modulated signal having linear AM (8) is given by (27) using (35) for δn and (39) for In . Once the SIA has calculated the XCF for a half-cycle of the modulated signal at its input, it is sent to the next set of blocks, named “Amplitude Demodulation”, “Frequency Demodulation” and “Music algorithm”, as shown in Figure 5. The next step in the algorithm comprises of estimating the amplitude and frequency demodulation of the modulated signal. This is the most important part of the algorithm because not only do the demodulated outputs provide valuable information for simplifying the classification task, but also provide a number of analytical results which help in forming a much broader knowledge base for the neural network classifier.

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Table 4: Components in transient PQ event shown Figure 9. Quantity Value mf 3 fmax 250 Hz kf 475

Figure 10: The transfer function Γ (f ) of the narrow band pass filter with complex impulse response γ (t)

4

Amplitude and frequency demodulation

In this section, the amplitude and frequency of transient PQ events have been estimated using hybrid demodulation concept. Then, a number of analytical results have been determined after simulation, to support the concept of considering PQ events as hybrid amplitude and angle modulated signals. To calculate the value of kf , the modulation index is calculated using (5). Now, kf is calculated using (4). The values of these quantities for the PQ signal of Figure 9(a) are tabulated in Table 3.

4.1

Design of the narrow band pass filter

The impulse response of the desired band pass filter is computed from (15). The real function r(t) is assumed to be equal to a (t)and φ (t)and is computed from fm (t). Next, the narrow band pass filter with complex impulse response γ (t) is designed using equation (16). From Figure 10, it is clear that the transfer function Γ (f ) of the designed narrow band pass filter has most of its response in the vicinity of 50Hz. The observation in this experiment is in accordance with the actual phenomenon as the magnitude of the change in amplitude of power quality signal is same for both sag and swell, but in case of swell the amplitude increases from the normal value and drops during swell. But in case of swell and transient, the distinction may not be that easy as the amplitude demodulation results a(t) are almost the same. But, the phase information θ(t) differs in this case and is the main cause of distinction of these two events. For values of n greater than 15, the difference between the absolute values of In becomes prominent and the events can be easily categorized. As, for lower values of n, the term n(a1 + a2 ) dominates but as the value of n increases, the term(θ1 + θ2 )m , where m is (n − 1) and (n − 2), starts dominating i.e. θ(t) dominates over a(t), which makes the two events clearly distinguishable. Since, δn depend upon In and upto n ≤ 5, In does not differ much for the given PQ event.

Figure 11: (a) Amplitude envelope of Figure 2(a), (b) Slope detection of Figure 11(a), (c) Frequency demodulation, (d) Pseudo-spectrum.

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International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 64-77

4.2

Multiple signal classification method (MUSIC) algorithm

MUSIC algorithm is a frequency estimation technique as in [14]. MUSIC algorithm gives the frequency content of a signal or autocorrelation matrix using an eigenspace method. This method assumes that a signal, x(n), consists of p complex exponentials in the presence of Gaussian white noise. Given an M × M autocorrelation matrix, Rx , if the eigevalues are sorted in decreasing order, the eigenvectors corresponding to the p largest eigenvalues span the signal subspace. Note that for M = p + 1, MUSIC is identical to Pisarenko’s method. The general idea is to use averaging to improve the performance of the Pisarenko’s estimator. The frequency estimation function for MUSIC is PM U (ejω ) = ∑M

1

i=p+1

where vi are the noise eigenvectors and e = [1 ejω

5

ej2ω

|eH vi |2

(38)

. . . ej(M −1)ω ]T

Results

Power signal is correlated with saved standard sine wave. According to the coefficient of the correlated sequence as in Table 1 the decision is taken about single and multiple PQ events. The results rest of all the stages are shown in Figure 11. Figure 11 shows the event detection as sag with an amplitude demodulation envelope, comprising negative slope detection with no change in frequency demodulation and having MUSIC frequency of 50 Hz.

6

Classifier

Feedforward neural network based classifier is basically Artificial neural networks, employing structures in form of multilayer perceptrons (MLPs), and with supervised learning methodology, are perhaps known as the oldest and most popular ANN based methodologies for solving several complex, real-world problems. The popular supervised learning based MLP, called backpropagation neural networks (BPNN), is known as one of the oldest and most popular neural network architectures, which use supervised learning to determine a complex, nonlinear, multidimensional mathematical fitting. BPNN has been successfully implemented in many application areas, e.g. pattern classification, function approximation, digital signal processing, intelligent control, smart instrumentation, time-series prediction and so on. The original backpropagation networks used to suffer mainly from the drawbacks of slow convergence, because they used to get trapped at local minima. Over the years, different popular, improved variations of BPNN have been proposed to specifically address several important issues, namely, reduction of convergence time, avoiding local minima and arriving at the global minimum, ease of computational burden, reduced memory requirement etc. Some of these prominent variants include backpropagation with adaptive learning rate only, backpropagation with adaptive learning rate and momentum and resilient backpropagation. The present paper utilized a particularly fast variant of BPNN, called Levenberg–Marquardt algorithm, which can provide very fast convergence in the training phase, provided the system can support the memory requirements. Normally, this method is suitable when the system is required to train a maximum of a few hundred free parameters. These are essentially MLPs which employ optimization techniques to train their free, adaptable weights. In a way, they are similar to quasi-Newton algorithms, because the convergence of the cost function can be designed to approach second order training speed. However, unlike quasi-Newton algorithms, they are not required to compute the Hessian matrix. Here the cost function is formulated as the sum of squared errors over all exemplars (N) in the training data set, in a given epoch (k). Then, if J is the Jacobian matrix containing the first order derivatives of the neural network errors with respect to weights and biases and e is the vector of neural network errors, the Hessian matrix can be approximated as: H = J T J and the gradient is given as g = J T e . Then, the Levenberg–Marquardt algorithm can employ weight updates according to the [ ]−1 T J e. The algorithm starts with a high value of l and decreases it after each step, if formula wk+1 = wk − J T J − µI the cost function reduces during that step or increases it only when a tentative step would increase the cost function. This normally ensures that the cost function will decrease after each training iteration and potential a fast training convergence can be achieved.

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Table 5: Performance of SIA for PQ events classification rate using NN classifier S. No. PQ Events NN Classifier 1. Sag 100 2. Swell 100 3. Transient 99.1

7

Conclusion

An efficient SIA system for PQ event analysis and classification is proposed using amplitude demodulation, frequency demodulation and harmonic analyzer for feature extraction based neural classifier. 500 samples of both type i.e. single and multiple PQ events are tested on SIA. This is investigated that the SIA is having 99 % (Approx.) efficiency for single power quality event for NN classifier. Proposed algorithm can be extended for identification of multiple PQ events.

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