Avenues of Application of an Enhanced Algorithm for ... · PDF fileWavelet transform is a choice of signal ... Indian Institute of Technology-Delhi, [email protected] Fifteenth

Abstract—An algorithm incorporating enhancements to the Hilbert-Huang method for time-frequency analysis of signals is reintroduced in this paper. The improved method employs the FFT for discerning closely spaced frequencies in a spectrum, prior to the application of the empirical mode decomposition (EMD). Improvement to the results of Hilbert spectral analysis is achieved by a demodulation technique. Recent successful applications of the enhanced algorithm to power systems signal monitoring and adaptive transfer function estimation are discussed. Further, new avenues of possible application of the said algorithm to power systems are probed.

I. INTRODUCTION

CCURATE determination of magnitude and frequency information in a time-varying waveform is critical to

the characterization of the signal. However, the challenge in extracting reasonably accurate information with regard to non-linearity and non-stationarity in all the three variables (time, frequency, and magnitude) is considerable. The challenge is compounded by the decoupling effect of popular signal processing tools such as the FFT which tend to handle information from one domain only (viz., time-domain or frequency domain) in an instance. The uncertainty principleplays a significant role in the performance of time-frequency distributions [1]. The use of more advanced signal processing tools such as the wavelet transformation offers some advantages to deciphering the time-frequency nature of time-varying waveforms; however, limitations abound [2]. In the realm of emerging electric power systems, with the advent of more disruptive technologies such as FACTS devices,pervasive power electronics interfaces for distributed generation, microgrids, and the requirement of real-time information in the electricity market, there is an imperative need for an advanced signal processing tool that accurately discerns the constituents of a time-varying signal [3, 4]. In Section II of this paper, a brief overview of some techniques for the analysis of time-varying waveforms is provided. Section III describes the enhanced technique that employs the

This work was sponsored in part by the U.S. Office of Naval Research

under Grant No. N0014-02-1-0623, and in part by the US Department of Energy under Award no. DE-FG02-05CH11292. The algorithm and some applications were developed at the Center for Advanced Power Systems, Florida State University, Tallahassee, FL 32310 in 2006-2007 when the authors were employed there.

Hilbert Huang method in an algorithmic manner. Somesuccessful applications and some avenues of future use are discussed in Section IV.

II. SOME SIGNAL PROCESSING TOOLS FOR ESTIMATING NON-STATIONARY NONLINEAR WAVEFORMS

A popular and easy to use tool for decomposing a time-series signal to its constituent frequencies is the fast Fourier transform (FFT) technique. FFT is put to use when the participating modes in the waveform being analyzed conform to linearity and stationarity [5]. However, the FFT spectrum fails to provide accurate information when the analyzed signal possesses non-stationary modes. During such cases, a sliding window FFT may be applied with restrictions in resolution arising from the choice of the fixed window width.

Wavelet transform is a choice of signal processing tool that may be used for characterizing some time-series signals that exhibit non-periodic oscillations, as well as those that evolve in time [3]. The wavelet transform is limited in performance due to its ability to separate individual frequency components that occur sufficiently afar on the frequency scale [3]. The S-transform is a modified wavelet transform that is based on amoving, scalable, localizing Gaussian window [6].

A multi-resolution technique called the empirical mode decomposition (EMD), for the analysis of non-stationary and non-linear waveforms was developed by Huang [7]. Application of Hilbert transform yields the analytic forms, from which their instantaneous amplitudes and frequencies can be extracted. The Hilbert-Huang (HH) method has been successfully applied in the fields of geophysics [8], biomedical engineering [9], image processing [10], in power systems to identify instantaneous attributes of torsional shaft signals [11], to analyze interarea oscillations [12], and to detect and localize transient features of power system events [13]. In this paper, enhancements to the Hilbert Huang transform to aid analysis of closely spaced frequency components with better resolution due to demodulation is reintroduced. Successful applications of this improved algorithm to study (a) generator coherency, (b) time-varying signals in power quality and (c) adaptive estimation of transfer function of a candidate machine, are discussed. Furthermore, some new avenues of possible application of this improved algorithm to problems in power systems are presented. Foremost, for the completeness of this paper, the enhanced algorithm incorporating the Hilbert-Huang

A

Avenues of Application of an Enhanced Algorithm for Analyzing Time-varying

Waveforms Siddharth Suryanarayanan, Member, IEEE, Colorado School of Mines, [email protected] Nilanjan Senroy, Member, IEEE, Indian Institute of Technology-Delhi, [email protected]

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transform, developed by the authors is represented in Section III [3, 14].

III. AN ENHANCED ALGORITHM INCORPORATING THE

HILBERT-HUANG TRANSFORM [15]

The original HH method has two parts. In the first part, a distorted waveform is decomposed using EMD into multiple intrinsic mode functions (IMFs) that possess well-behaved Hilbert transforms. Each IMF is defined by two principal characteristics – (a) its mean is zero; (b) the number of local extrema must be equal to, or differ by at most 1 from the number of zero crossings within an arbitrary time window. The authors refer the interested reader to references [7, 8] for detailed description of the original HH transform method. Prior to discussing the improved (or modified) HH method, it is imperative to mention the motivation for the development of the enhancements. One of the limitations of the otherwise powerful technique of EMD- the backbone of the HH transform – is the compromise in performance when separating frequencies lying within one octave. In order to overcome this obstacle, masking signals may be used. However, the authors are not aware of a particularly scientific method to selection of masking signals. Hence, enhancements to the original HH method by incorporating a pre-processing step that applies FFT to the time-varying signal for determining empirical masking signals and a post processing step of demodulating the results to obtain better information on closely spaced frequency components have been developed [3, 14-16].

The steps involved in extracting the instantaneous amplitudes and frequencies present in a time-varying distorted waveform are presented in an algorithmic fashion below. 1. Perform EMD (given in steps 1.a to 1.g) on the distorted

waveform. 1.a. Identify local maxima and minima of distorted signal,

s(t), 1.b. Perform cubic spline interpolation between the maxima

and the minima to obtain the envelopes eM(t) and em(t), respectively,

1.c. Compute mean of the envelopes, ( ) ( ) ( )( ) 2tetetm mM += ,

1.d. Extract c(t) = s(t) - m(t), 1.e. c(t) is an IMF if the number of local extrema of c(t), is

equal to or differs from the number of zero crossings by one, and the average of c(t) is zero. If c(t) is not an IMF, then repeat steps 1.a -1.d replacing s(t) by c(t), until the new c(t) obtained satisfies the conditions of being an IMF,

1.f. Compute the residue, r(t) = s(t) – c(t), 1.g. If the residue r(t) is above a threshold value of error

tolerance, then repeat steps 1.a-1.e on r(t) to obtain the next IMF.

2. Construct appropriate masking signals (given in steps 2.a to 2.g) to separate oscillatory modes with frequencies in the same octave. 2.a. Perform FFT on the distorted signal, s(t), to estimate

the stationary equivalent frequency components f1,

f2,…, fn, where f1< f2< …< fn, 2.b. Construct n-1 masking signals, mask2, mask3, …,

maskn, to extract the components f2 , f3 ,… fn , where maskk (t) = Mk × sin(2π (fk + fk -1) t). An effective value of Mk (obtained empirically) is 5.5 times the magnitude of fk in the FFT spectrum.

2.c. Obtain two signals s(t)+maskn and s(t) – maskn. Perform EMD (steps 1.a – 1.f) on both signals to obtain their first IMFs only, IMF+ and IMF–. Finally, IMF = (IMF+ + IMF–)/2,

2.d. Obtain the residue R(t) = s(t) – IMF, 2.e. Perform steps 2.c – 2.d iteratively, replacing s(t) with

the residue (R(t)) obtained, until n-1 IMFs containing frequency components f2 , f3 ,… fn are extracted. The final residue R(t) will contain the remaining component f1.

3. Perform amplitude demodulation of a modulated IMF. 3.a. For every IMF obtained, compute instantaneous

amplitude, A(t), and frequency, f(t), by Hilbert transform,

3.b. A(t) may exhibit artificial local maxima and minima due to modulation. Apply cubic spline fitting through these local maxima and minima to obtain two envelopes Γmax(t) and Γmin(t),

3.c. The true amplitudes of the modulating components are

( )( ) ( )

( )( ) ( )

.2

2

minmax2

minmax1

tttA

tttA

Γ−Γ=

Γ+Γ

=

3.d. Let tM be the time at which a local maxima occurs and tm be the time at which a local minima occurs. Solving the equations below, the frequencies f1, f2, of the modulating components can be obtained,

( ) ( )x

ffftf

x

ffftf mM

−

−

+=

+

−

+=

1,

112

112

1 ,

where x = A1/A2. 3.e. The modulated IMF is resolved into its modulating

components of magnitudes A1 and A2 with frequencies f1 and f2 respectively.

Fig.1 depicts the various steps in the improved HH method.

IV. APPLICATIONS OF THE IMPROVED HH METHOD TO

PROBLEMS IN POWER SYSTEMS

A. Instantaneous generator coherency

This section presents an application of the Hilbert-Huang transform to study generator coherency. Generator coherency is an important tool for dynamic reduction of multi-machine power systems. Two generators are said to be coherent, if the difference between their swing curves remains constant for the period of observation. In response to an excitation of a subset of the natural modes of the system, the disturbed generators are monitored. When coherency is detected for the slow modes of the system, the generators are said to be slow coherent.

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Figure 1. Flowchart to performing the improved HH method for time-frequency analysis of waveforms with time-varying frequency components [3, 14].

One way of accomplishing dynamic aggregation of generators on the basis of their slow coherency is by using a linearised model of the complete power system. Gaussian elimination of the eigenbasis matrix, corresponding to the selected slow modes, reveals the generator groups [17]. Such linearised techniques assume linearity and stationarity of generator dynamic behaviour; an assumption not always justified. The alternative method to identify coherent generators is to compare their swing curves obtained by a full-scale time-domain simulation. The swing curves may also be constructed using generator angle measured as part of a wide area measurement system. In this context, comparing the phases of the individual generator swing curves gives an idea of the coherency between them.

A six machine test system is employed to demonstrate the application of the Hilbert-Huang transform to track the coherency between generators [18]. Figure 2 shows the line diagram of the system. An eigenvalue analysis of the linearized system reveals three slow modes at 0.0 Hz, 0.38 Hz, and 0.49 Hz. Corresponding to these modes, the strongly coherent generators are identified as (1, 6) and (4, 5). Additionally, nonlinear time-domain simulations were carried out to capture the swing curves of all the machines for a line 9-12 tripping.

The absolute rotor angles obtained for four selected machines–1, 4, 5, and 6 are shown in Fig. 3a. Empirical mode decomposition on the generator oscillations obtained due to the tripping of line 9-12, revealed 2-3 IMFs. Out of these IMFs, the dominant IMF was identified for each generator. The dominant IMFs obtained from the swing curves ofmachines 4 and 5 are shown in Fig. 3b. Similarly, Fig. 3c shows the dominant IMFs obtained from the swing curves of machines 1 and 6. Hilbert transform on these dominant IMFs reveals their instantaneous phase, and the corresponding phase differences for the machine pairs (4, 5) and (1,6) are shown in Fig. 3d. The instantaneous phase difference remains close to zero for the machine pair (4, 5) indicating that they are coherent during the observation period. Machine pairs (1,6) show an instantaneous phase difference close to zero for the period 0-1.5s. Following this period, the coherency is lost, until it re-establishes at 3s, when the instantaneous phase

difference settles at 360º. In day-to-day power systems operation, random changes in

the system operating conditions can lead to changes in the degree of coherency, which must be effectively tracked to maintain accurate reduced system models. While an instantaneous phase comparison can detect the temporal evolution of the coherency between machines, it would be premature to define coherency using this tool.

Figure 2. Six machine test system

B. Inrush current analysis [3]

This subsection presents results from the application of the modified Hilbert-Huang transform to the analysis of the inrush currents measured at the terminals of a 4160/440 V, 3.75 MVA distribution transformer. The current was sampled at 20 kHz, and later decimated to a sampling frequency of 10 kHz, to reduce the data size.

Figure 4 shows the measured inrush current [3]. From the inset showing the FFT of the current, it is clear that there is a strong dc component along with 2nd harmonics in addition to higher order harmonics. Visual inspection of the current reveals its non-stationarity, and the use of the enhanced algorithm based on Hilbert-Huang transform is appropriate.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-20

-10

0

10

time (s)

roto

r an

gle

(deg

)

absolute rotor angles

0 1 2 3 4 5-4

-2

0

2

4

time (s)

roto

r an

gle

(deg

)

Most significant IMFs of machine angles

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-400

-300

-200

-100

0

100

time (s)

phas

e di

ffer

ence

(de

g)

Instantaneous phase difference

0 1 2 3 4 5-2

-1

0

1

2

time (s)

roto

r an

gle

(deg

)

Most significant IMFs of machine angles

machine 1machine 4

machine 5machine 6

machine 4

machine 5

machine 1

machine 6

machine pair 4 & 5

machine pair 1 & 6

(b)

(a)

(c)

(d)

Figure 3 Instantaneous phase difference between swing curves of coherent machine pairs 1, 6 and 4, 5, for the line 9-12 tripping: (a) absolute rotor angles for the machines 1, 4, 5 and 6, (b) Most significant IMF extracted using EMD from swing curves of machines 4 and 5, (c) most significant IMFs extracted using EMDfrom the swing curves of machines 1 and 6, (d) instantaneous phase difference for the two machine pairs calculated using Hilbert transform.

Figure 4. Inrush current measured on phase ‘c’ of a 4160/440 V, 3.75 MVA distribution transformer [3]. The inset shows the FFT of the current from the time of switching to 1.0 s.

Masking signals of frequencies 540 Hz, 420 Hz, 300 Hz and 180 Hz, were used to extract four IMFs. The first IMF was found to be very small in magnitude and discarded. Application of the Hilbert transform on the remaining three IMFs revealed their instantaneous frequencies and amplitudes. These are presented in Figures 5 and 6 [3].

C.Empirical transfer function estimation [14, 16]

In this section, the application of the Hilbert-Huang technique for online estimation of the transfer function is demonstrated for a notional high-temperature superconducting (HTS) propulsion motor. HTS motors are being proposed for propulsion needs of advanced warships, due to theirsignificantly higher power density and efficiency. Typical HTS

motors are of synchronous type, with a cryogenically cooled superconducting field winding located on the rotor. During operation, varying sea-states give rise to variations in the load torque, and therefore the speed of the rotor. The resultant current variations in the dc field winding must be taken into consideration while designing the cooling system for such motors.

Figure 5. Instantaneous amplitude of modes extracted from the inrush current using the modified Hilbert-Huang transform [3].

The transfer function between the load torque and the motor speed or the field current is not always available, and may have to be periodically estimated from measurements. Traditionally, such empirical transfer function estimation (ETFE) is carried out by subjecting the motor to artificially varying load torques. Comparing the Fourier spectrum of the

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input (load torque) to the outputs (speed or field current variations), gives rise to the empirical transfer function. However, such an exercise would involve taking the motor offline and subjecting it to signals of a wide range of frequencies. Since this is an expensive proposition, a more desirable alternative is to carry out the transfer function estimation while the ship is at sea. This would entail utilizing the normal variations of the sea-states and the related load torques, speed and field current variations. Since such variations would be inherently random in nature, the resultant non-stationarity and possible non-linearities preclude the usage of Fourier based techniques for ETFE.

Figure 6 Instantaneous frequency of modes extracted from the inrush current using the modified Hilbert-Huang transform [3].

The Hilbert Huang technique is ideally suited to study such non-stationary and nonlinear signals. For the purposes of the study, a 40 MW HTS propulsion motor was simulated with associated controls. Figure 7 shows the oscillations produced in the torque, rotor speed and the field current, due to the application of a sea-state [14, 16]. These oscillations were non-stationary and random in nature.

0 5 10 15 20 25 30 35 40 50-0.1

0

0.1

time (s)

torque oscillations during sea-state conditions

per un

it

0 5 10 15 20 25 30 35 40 50-0.05

0

0.05

time (s)

per un

it

rotor speed variations during sea-state conditions

0 5 10 15 20 25 30 35 40 50-0.02

0

0.02

time (s)

per un

it

field current variations during sea-state conditions

Figure 7 Variations in the load torque, rotor speed and field current of a 40 MW HTS propulsion motor, due to the application of sea-states [14, 16].

Using the modified Hilbert Huang transform, the time varying oscillations in the torque and the field current, were decomposed into the individual modes. The instantaneous frequency and amplitude of these modes were used to find the corresponding gains at different frequencies. It was found that the application of the sea-states resulted in oscillations at some

frequencies only. Thus, it was possible to calculate the transfer function gains at these particular frequencies only. The scatter plot in Figure 8 shows the empirically estimated transfer function between the variations in the load torque and the field current. The instantaneous gain between field current variations and torque oscillations, and between speed variations and torque oscillations, is plotted as a scatter plot, because for every sampling time instant there is a calculated instantaneous amplitude gain. The blue solid line in Figure 8 shows the transfer function estimated empirically using the traditional Fourier based techniques. This is an offline technique, wherein a single frequency mode was artificially supplied to the mechanical torque (input), and the resultant field current variation measured (output). This procedure is repeated by sweeping over a range of frequencies of interest. The scatter plot shows reasonable good agreement with the Fourier based ETFE. There are two reasons for the degree of spread in the scatter plot 1. The magnitude of oscillations due to the sea-states is

considerably less at higher frequencies and hence, transfer function gain computation runs into numerical difficulties,

2. The transfer function gain at frequencies close to the natural frequency of the controllers becomes very high. Due to the high magnitudes of oscillations, the system itself approaches nonlinearity, thereby impeding successful estimation of the gain.

Figure 8 Transfer function gain of the field current variations to the torque variations. The scatter plot represents the results from the improved HH method applied during sea-state analysis, while the lines represent results from steady state FFT based analysis [14, 16].

From Fig. 8, it may be concluded that the hybridized technique may not be inherently applicable for estimating the transfer functions of unknown systems. However, the said technique is an efficient tool for updating an already derived estimate of the transfer function, with the additional advantage of being applied in a real-time environment without taking the system offline for conventional testing.

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D.Avenues for future application of the improved algorithm

The potential of the improved Hilbert-Huang algorithm for detecting and characterizing time-varying waveforms in electric power systems is vast. As seen in the above subsections, the method renders useful in a variety of applications including visualization, transfer function estimation, and dynamic modelling of electrical machines. In this subsection, the authors explore some other possible applications of this powerful algorithm in avenues such as SCADA, vulnerability analysis, and quantification of variability in electric power systems. As it is the intent of the authors to encourage the use of this enhanced Hilbert-Huang transform based algorithm to avenues in power systems, a short list of possible applications for the future is listed without going into details. These include: 1) a near real-time SCADA application for monitoring the connection anddisconnection of massively deployed distributed energy resources (DERs) in a meshed distribution system of the future- this information could be vital in deciding the set-points of adaptive protection systems; 2) analysis of reflected waves from special protection systems (SPSs) that are pinged periodically to reveal modes of hidden failures by comparison of the spectrum of the reflected wave with that of a standard SPS – this could facilitate in choosing candidates among SPSs for enhancements and to deter vulnerabilities due to hidden failures that may potentially cascade into a system catastrophe such the two blackout events of 1996 in the WesternElectricity Coordinating Council (WECC, formerly WSCC) system in the western United States; 3) a real-time harmonic load identification tool; 4) an application to quantify the variability in the output of non-dispatchable generation sources such as wind and solar technologies by online monitoring; and 5) a proper orthogonal decomposition tool for power system dynamic reduction. While these are some areas of application of the new algorithm to power systems, the authors realize the existence of numerous other uses within the power systems field as well as in other disciplines where the characterization of time-varying waveform is deemed vital for extracting information.

REFERENCES

[1] Y-J. Shin, A. C. Parsons, E. J. Powers et al., “Time-frequency analysis of power system disturbance signals for power quality,” in 1999 IEEE Power Engineering Society Summer Meeting, Jul. 1999, pp. 402-407.

[2] A. W. Galli, G. T. Heydt, and P. F. Ribeiro, “Exploring the power of wavelet analysis,” IEEE Computer Applications in Power, vol. 9, no. 4, pp. 37-41, Oct, 1996.

[3] N. Senroy, S. Suryanarayanan, and P. F. Ribeiro, “An improved Hilbert-Huang method for analysis of time-varying waveforms in power quality,” IEEE Transactions on Power Systems, vol. 22, no. 4, pp. 1843-1850, Nov, 2007.

[4] M. Steurer, “Real time simulation for advanced time-varying harmonic analysis,” in 2005 IEEE Power Engineering Society General Meeting, Jun 2005, pp. 2250-2252.

[5] A. Oppenheim, R. Schafer, and J. Buck, Discrete-Time Signal Processing, 2nd ed., Englewood Cliffs, NJ: Prentice Hall, 1999.

[6] R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: The S transform,” IEEE Transactions on Signal Processing, vol. 44, no. 4, pp. 998-1001, Apr, 1996.

[7] N. E. Huang, Z. Shen, S. R. Long et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 454, no. 1971, pp. 903-995, Mar, 1998.

[8] N. E. Huang, Z. Shen, and S. R. Long, “A new view of nonlinear water waves: The Hilbert spectrum,” Annual Review of Fluid Mechanics, vol. 31, pp. 417-457, 1999.

[9] H. L. Liang, Q. H. Lin, and J. D. Z. Chen, “Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease,” IEEE Transactions on Biomedical Engineering, vol. 52, no. 10, pp. 1692-1701, Oct, 2005.

[10] J. C. Nunes, Y. Bouaoune, E. Delechelle et al., “Image analysis by bidimensional empirical mode decomposition,” Image and Vision Computing, vol. 21, no. 12, pp. 1019-1026, Nov, 2003.

[11] M. A. Andrade, A. R. Messina, C. A. Rivera et al., “Identification of instantaneous attributes of torsional shaft signals using the Hilbert transform,” IEEE Transactions on Power Systems, vol. 19, no. 3, pp. 1422-1429, Aug, 2004.

[12] A. R. Messina, and V. Vittal, “Nonlinear, non-stationary analysis of interarea oscillations via Hilbert spectral analysis,” IEEE Transactions on Power Systems, vol. 21, no. 3, pp. 1234-1241, Aug, 2006.

[13] Z. Lu, J. S. Smith, Q. H. Wu et al., “Empirical mode decomposition for power quality monitoring,” in 2005 IEEE Power Engineering Society Transmission and Distribution Exposition: Asia and Pacific, 2005, pp. 1-5.

[14] N. Senroy, S. Suryanarayanan, M. Steurer et al., “Application of a Time-Frequency Algorithm for Adaptive Estimation of Transfer Function of a Notional High-Temperature Superconducting Motor,” in 2007 IEEE Electric Ship Technologies Symposium (ESTS 07), Arlington, VA, May 2007, pp. 238-244.

[15] N. Senroy, and S. Suryanarayanan, System and methods for determining masking signals for applying empirical mode decomposition (EMD) and for demodulation functions obtained from application of EMD, USA utility patent application filed, 12/124,843, May 22, 2008.

[16] N. Senroy, S. Suryanarayanan, and M. Steurer, “Adaptive transfer function estimation of a notional high-temperature superconducting ship propulsion motor,” IEEE Transactions on Industrial Application, to be published, May 2008.

[17] J. H. Chow, Time scale modeling of dynamic networks with applications to power systems, New York, NY: Springer-Verlag, 1982.

[18] N. Senroy, “Generator coherency using the Hilbert-Huang transform,” IEEE Transactions on Power Systems, accepted for publication, IEEE Transactions on Power Systems, Jun 2008.

Siddharth Suryanarayanan (S’00, M’04) holds the Ph.D. in electrical engineering from Arizona State University, Tempe, AZ., He is currently an Assistant Professor of Engineering at Colorado School of Mines, Golden, CO, USA with research interests in modelling and simulation of power systems.

Nilanjan Senroy (M’2006) has a Ph.D. from Arizona State University, Tempe, AZ. He is currently an Assistant Professor at the department of electrical engineering, Indian Institute of Technology, New Delhi, India. His research interests include power system stability and signal processing applications in power system.

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