Adaptive Frequency Estimation Using IterativeDESA with RDFT-Based Filter

Sahil Bansal1, Anindita Ghosh2, Chandra Sekhar Seelamantula2, Gurunath Gurrala2 and Prasanta Kumar Ghosh2

1Department of Electrical Engineering, Indian Institute of TechnologyKanpur, India 208016

2Department of Electrical Engineering, Indian Institute of ScienceBangalore, India 560012

bsahil@iitk.ac.in, aninditag@iisc.ac.in, chandrasekhar@iisc.ac.in, gurunath@iisc.ac.in, prasantg@iisc.ac.in

Abstract—This paper proposes a new approach for estimatingfundamental frequency of grid signals. The approach is based ona discrete-time energy separation algorithm (DESA) combinedwith an adaptive bandpass filter (BPF). The BPF is built usinga discrete Fourier transform (DFT) and inverse DFT bothused recursively. The technique is computationally efficient androbust to the harmonics and noise in the signal. The method’sperformance is validated by comparing the results with someexisting algorithms.

Index Terms—Discrete Fourier transform (DFT), frequencyestimation, discrete-time energy separation algorithm (DESA).

I. INTRODUCTION

Accurate estimation of grid voltage frequency is an im-portant problem in several applications of signal processing,digital communication, power system etc. and many suchfields. Frequency indicates dynamic balance in power systembetween power generation and consumption. An accuratefrequency estimation algorithm that is fast and stable un-der various conditions is challenging to design because ofthe dynamic characteristic of power system signals. Severalpower electronic equipment and devices generate harmonicsand noise. A reliable method is needed that can measurefrequency and phase in the presence of these disturbances.The zero-crossing technique [1], phase-locked loop (PLL) [2],frequency-locked loop (FLL), weighted-least-squares, finite-impulse response (FIR) filter [3], Prony method [4] etc. weresome techniques proposed in the literature for fundamentalfrequency estimation.

The accuracy of zero-crossing method is influenced byquantization error, harmonics, noise, etc. [5]. Optimizationmethods such as least squares, least mean square, Newton,Kalman filters [6]–[9] etc. have a common objective ofminimizing the sum of squared error recursively betweenthe estimations and observations. The main issue with theapplicability of these methods in real-time applications isthe convergence speed. The discrete Fourier transform (DFT)along with fast Fourier Transform (FFT) are used in manydigital devices to derive phasors from voltage and currentsignals [10]. An implementation of interpolated DFT on FPGAbased phasor measurement unit (PMU) prototype is mentionedin [11]. The DFT method can lead to inaccurate results due

to spectral leakage phenomenon [12]. It can be improved byadaptively changing the window length but it would lead to asubstantial increase in computational burden.

Various Teager energy operator (TEO) based techniques toestimate fundamental frequency of grid voltages are mentionedin [13]–[17]. In [14] and [15], the authors combine TEO withrecursive discrete Fourier transform (RDFT) based bandpassfilter to calculate the fundamental frequency, where the fun-damental voltage amplitude is assumed to be unity or knownbeforehand. The RDFT filter created by cascading the DFTand inverse DFT techniques can help in rejecting the harmonicand noise components [14], [15]. In [18] an Energy Operator-Synchronization (EO-S) technique to estimate frequency ofa grid signal is proposed. In both EO-S and RDFT-TEOmethods, sudden disturbances in the waveform can result inerroneous outputs.

Discrete Energy Separation Algorithm (DESA) is a pow-erful technique that has been used to resolve the funda-mental issue of evaluating both the amplitude envelope andinstantaneous frequency of an amplitude/frequency modulatedAM-FM signal [19], [20]. TEO and DESA techniques havebeen used together for analyzing amplitude and frequencyvariations in [21]. However, in case of sinusoidal signals, thistechnique cannot reject the effects of the harmonics and noisecomponents. One approach to handle the harmonics and noisecomponents is to apply a filter on the input signal to pass thefundamental frequency components to the DESA technique,which is being explored in this paper.

This paper proposes the iterative approach of DESA tech-nique that combines the accuracy of the discrete-time energyseparation algorithm with the simplicity and robustness of therecursive DFT filter. It is a computationally effective techniquefor the fundamental frequency estimation of a signal (bothsingle-phase and three-phase) under various test conditions.DESA uses the Teager energy operator (TEO) for energytracking purpose and a backward symmetric difference op-erator for averaging purpose. The DESA-1 variation [19] ofthe discrete-time energy separation algorithm is used in theproposed method. The IEEE Std C37.118.1 for synchrophasormeasurements for power systems [22] is followed to stan-dardize the frequency error (FE) of the algorithm. Throughsimulations, it is confirmed that the proposed method provides

978-1-5386-1379-5/17/$31.00 ©2017 IEEE

precise estimation under high level of noise (signal-to-noiseratio = 15 dB).

The rest of this paper is organized as follows. Section IIgives a description of the RDFT-based filter and the DESA-1technique. The combined approach of the two techniques todesign the proposed frequency estimation method is describedin Section III. Simulation results are presented in Section IVwhere the technique is tested for the PMU compliance testsas well as tests introducing noise. The performance of thetechnique is also compared with an existing RDFT-TEO andEO-S methods in the said tests. Finally, Section V gives theconcluding remarks.

II. RDFT-BASED FILTER AND DISCRETE-TIME ENERGYSEPARATION ALGORITHM

A. RDFT-Based Filter

The filter technique begins with an initial guess value offundamental frequency (f̂), which has to be provided in theinput. f̂ is used to calculate the window size (N) to performthe filtering operation on the signal. To obtain the fundamentalfrequency component of a sinusoid x(n) at the N th samplinginstant in frequency domain, the following equation can beused:

X1 =

N∑k=1

x(k)e−j 2πkN (1)

N is calculated as N = Fs/f̂ , where Fs is the samplingfrequency of the signal, and ‘j’ is the imaginary unit. The win-dow size is adaptive, i.e., the window size of the subsequentestimation step varies according to the current estimate of thefrequency (f̂). The frequency resolution (∆f) is equal to f ,where f is the fundamental frequency. For a non-integer valueof N , a rounding function such as ceiling or floor, is generallyused to acquire an integer value for the window size. However,the rounding error produced by these functions prevents theeffective rejection of harmonics from the signal. Accuracy ofthe value of x(N) can be improved by performing a piece-wiselinear interpolation between the samples x(L) and x(L + 1)where L = bNc.The waveform for a single frequency f in time-domain canbe acquired by taking the inverse transform of (1). For theN th sample, this is achieved by using the following inversediscrete transform:

x1(N) =1

NX1e

j2π. (2)

A digital bandpass filter with center frequency f is created us-ing these two equations in series. The fundamental frequencycomponent can be generated starting from the N th sample ofthe signal using (2). From n = (N + 1), the nth sample canbe expressed as follows:

x1(n) = x1(n− 1)ej 2πN +1

N{x(n)− x(n−N)}. (3)

Thus, x1(n) would be the filtered signal having only thefundamental frequency component. For example, consider aninput voltage waveform of 50 Hz fundamental frequency,

Fig. 1. Response of the filter under harmonics as mentioned in Table I.

having unit amplitude and 13% total harmonic distortion asmentioned in Table I. The difference between the original andthe filtered signal, which is devoid of the harmonic distortions,can be observed in Fig. 1.

TABLE ITOTAL HARMONIC DISTORTION

Harmonics Total Harmonic Distortion2nd 3rd 4th 5th

7% 3.5% 6.5% 8.1% 13%

B. Discrete-Time Energy Separation Algorithm-1

DESA-1 is a popular method that uses the discrete-time energy operator [19] for estimating the amplitudeand instantaneous frequency of a signal. The input signalconsisting of an unknown sinusoidal signal and an additivewhite Gaussian noise is represented by

x(n) = A(n) sin(ω(n) + φ) +N(n), (4)

where A(n) is the amplitude, ω(n) is the frequency, φ is thephase. N(n) is an additive white Gaussian noise with mean 0and variance σ2. Frequency is estimated using the TEO (Ψ)operator [23], defined as:

Ψ[x(n)] = x2(n)− x(n− 1)x(n+ 1). (5)

DESA-1 uses the backward symmetric difference for approx-imating the first derivative of x(n). It is obtained as:

y(n) = x(n)− x(n− 1). (6)

Using (5) and (6), the frequency of the signal is obtained asfollows [19]:

ω(n) = cos−1

(1− Ψ[y(n)] + Ψ[y(n+ 1)]

4Ψ[x(n)]

). (7)

However, an averaging operator is needed to reduce the effectof noise [24]. If Φ is the averaging operator and Φn = E[yn],then E[yn] is calculated as,

Φn = αΦn−1 + (1− α)yn, (8)

where 0 < α < 1 is an averaging parameter. The finalexpression for frequency is

ω(n) = cos−1

(1− E[Ψ[y(n)]] + E[Ψ[y(n+ 1)]]

4E[Ψ[x(n)]]

). (9)

Apart from frequency, the amplitude of the sinusoid can beestimated from DESA-1 as

|a(n)| ≈√√√√ E[Ψ[x(n)]]

1−(

1− E[Ψ[y(n)]]+E[Ψ[y(n+1)]]4E[Ψ[x(n)]]

)2 . (10)

III. PROPOSED FREQUENCY ESTIMATION METHOD

The iterative DESA method is performed in three funda-mental steps:

• First, in order to reduce noise, the input signal x(n)is passed through a Butterworth bandpass filter of first-order. It is a discrete infinite impulse response (IIR) filterwith center frequency (f̂) equal to the input frequency.

• Next, the output of the Butterworth filter is passedthrough the RDFT filter that eliminates the effects offurther harmonics and noise.

• Finally, the discrete-time energy separation algorithm(DESA-1) is applied on the RDFT filtered data to de-termine the signal frequency.

The RDFT filter and the DESA-1 technique are used alter-nately in an iterative manner, i.e., the output of DESA-1 ispassed as an updated input frequency to the RDFT filter, thefiltered data is passed again as input to the DESA-1 algorithm,and these two steps are repeated till the value of the outputfrequency converges (to less than ±0.005 away from output ofprevious iteration in our experiments). In the proposed method,DESA-1 provides accurate estimation when the frequencytaken initially is close (±5 Hz) to the actual frequency ofthe signal. The block diagram of the implementation of theiterative DESA technique is shown in Fig. 2.

x(n)1st orderIIR band-pass filter

RDFT-Basedfilter

DESA f̂

Fs/f̂

Fig. 2. Block diagram implementation of iterative DESA technique forfundamental frequency estimation.

IV. SIMULATION RESULTS

The simulation performance of the proposed iterative DESAtechnique is presented in this section. The iterative DESAtechnique is compared with some existing methods that usesthe TEO operator like the RDFT-TEO technique that has beenmentioned in [15] and the Energy Operator Synchronization(EO-S) mentioned in [18]. Tests are performed by focusingmainly on the Phasor Measurement Unit (PMU) compliancetests mentioned in the IEEE Standard C37.118.1 [25] as well

as the modified IEEE Standard C37.118.1a-2014 [26]. Theperformance of the algorithm is recorded as per the errorspecifications of the M class PMU. A first-order discrete IIRButterworth filter with center frequency 50 Hz and a Q-factorof 1.25 is used. The sampling frequency chosen is Fs = 10kHz. The frequency error (FE) is calculated as

FE = |f − f̂ |, (11)

where f̂ is the frequency obtained in the output from theproposed method and f is the actual frequency of the signal.

The tests complying to the phasor measurement unit areof two types: (i) Steady state testing; and (ii) Dynamic statetesting. In steady-state, the absolute frequency error (FE) limitfor a M class PMU is 0.005 Hz [25], [26].

A. Steady-state compliance tests for PMU [27]

• Amplitude scan test: The amplitude of signal (havingother parameters constant) is varied from 20% to 120%.Fig. 3 shows that amplitude scan test does not affect theperformance of the algorithm.

Fig. 3. FE for different amplitudes varying from 20% to 120%.

• Phase scan test: The phase angle of signal is varied from−180◦ to 180◦ keeping other parameters constant. TheFE for different phase values in Fig. 4 shows that changein phase does not affect frequency estimation.

Fig. 4. FE for different phase values between −180◦ to 180◦.

• Frequency scan test: The fundamental frequency ofsignal is varied from 45 Hz to 55 Hz keeping the otherparameters constant. As can be seen from Fig. 5, bothIterative DESA and EO-S perform with better accuraciesunder varying range of frequency.

Fig. 5. FE for different fundamental frequencies varying from 45 to 55 Hz.

• Harmonic rejection test: A signal containing 10% ofharmonics from 2nd to 50th harmonics is taken. Fig. 6shows that varying harmonics do not affect the frequencyestimation.

Fig. 6. FE for harmonics indexed between 2 and 50.

• Out-of-band test: A signal with fundamental frequencyof 50 Hz is taken and a 10% out-of-bound frequencyvarying between 10 Hz to 100 Hz is included in thesignal. Fig. 7 shows the result where FE is calculatedfor each out-of-bound frequencies.

Fig. 7. FE for out-of-bound frequencies varying between 10 Hz and 100 Hz.

All these test indicated that Iterative DESA gives the frequencyerror within the limits set by PMU.

B. Dynamic compliance tests for PMU [26], [28], [29]

The maximum frequency error (FE) in hertz permissible fora modulated signal testing is 0.3 Hz.

• Amplitude modulation test is performed by modulatingthe amplitude of the input signal with a 10% modulationlevel. Therefore, the amplitude oscillates between 0.9 and1.1 per unit (p.u.). The modulated signal is as follows:

x(n) = [1 + 0.1sin(2πfmn

Fs)].cos(

2πfn

Fs), (12)

where fm is the modulation frequency, n is the number ofsamples and Fs is the sampling frequency. The result in

Fig. 8 shows the frequency error for amplitude modulatedsignal. The plot for the RDFT-TEO falls outside the errorlimit so it is only partially visible. The FE for IterativeDESA technique falls within the PMU limit.

Fig. 8. Frequency error (Hz) for amplitude modulated signal.

• Phase modulation test is performed by modulating thephase of the input signal by 0.1 radians. The modulatedsignal is as follows:

x(n) = A.cos(2πfn

Fs+ [0.1sin(

2πfmn

Fs)]), (13)

where fm is the modulation frequency, A is the ampli-tude, n is the number of samples and Fs is the samplingfrequency. Fig. 9 shows the frequency error (Hz) for aphase modulated signal calculated by Iterative DESA fallswithin the limit.

Fig. 9. Frequency error (Hz) for phase modulated signals for modulationfrequencies varying between 0.1 Hz to 5 Hz.

• Frequency ramp test is performed by changing thefrequency linearly with a slope of ±1 Hz/s. The range ofthe frequency is between 45 Hz and 55 Hz . The rampedsignal is as follows:

x(n) = A.cos(2π[f0 + n

Fs]n

Fs), (14)

where f0 is the initial frequency, A is the amplitude, n isthe number of samples and Fs is the sampling frequency.The result of the frequency ramp test is shown in Fig.10. It is seen that Iterative DESA as well as the othertechnique deviates from the original for a frequency ramp.

• Amplitude step test is performed by measuring positiveand negative amplitude steps that occur in dynamic power

Fig. 10. Change of frequency (Hz) for frequency ramp.

systems. For this test, the amplitude of the signal isstepped by 0.1 per unit (p.u.) and the result is shownin Fig. 11. Best accuracy is provided by Iterative DESA.

Fig. 11. Frequency error (Hz) for amplitude step from 0.9 p.u. to 1.1 p.u.

• Phase step test is done by changing the phase of thesignal by ±10 degrees and the absolute frequency erroris shown in Fig. 12, and for Iterative DESA the error iswithin the limit set by PMU standard.

Fig. 12. Frequency error (Hz) for phase step from 0 to 10 degrees.

Another test was performed introducing additive whiteGaussian noise in the signal. A grid signal containing 13%total harmonic distortion as given in Table I is used for thistest. The initial signal is distorted with SNR 30 dB and thefrequency error is shown on log scale in Fig. 13. It is seen thatIterative DESA method is the most effective against noise.

V. CONCLUSIONS

An adaptive frequency estimation technique is explored inthis paper. The method is simple and accurate and compliesto most of the PMU standard requirements. The techniqueis also effective under the presence of harmonics and noise.

Fig. 13. Frequency error for signal distorted with SNR 30 dB

Compared to the existing RDFT-TEO and EO-S method, theproposed algorithm can estimate a varying range of frequen-cies. The simulation results presented in the paper confirm theefficacy of the algorithm for reliable deployment in real-timepower system applications.

ACKNOWLEDGEMENT

This work is supported by the Bosch Research and Technol-ogy Centre, Bangalore, India and the Robert Bosch Centre forCyber Physical Systems, Indian Institute of Science, Banga-lore, India under the Project E-Sense: Sensing and Analyticsfor Energy Aware Smart Campus.

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