Harmonics Estimation in Emerging Power System

1 This is the pre-print version of the accepted paper in EPSR. Permission should be obtained for using any part/whole of the document from the publisher or the authors. Please cite this work as: Sachin K. Jain and S. N. Singh, "Harmonics estimation in emerging power system: Key issues and challenges," Electr. Power Syst. Res., vol. 81, no. 9, pp. 1754-1766, 2011. doi:10.1016/j.epsr.2011.05.004

Harmonics Estimation in Emerging Power System: Key Issues and Challenges Sachin K. Jaina,*, S. N. Singha a

Department of Electrical Engineering, IIT Kanpur, India a

[email protected], [email protected]

ABSTRACT The worldwide increasing applications of nonlinear loads, mostly consisting of power electronics devices, have made the power quality problems a vital issue than ever before. An efficient control and mitigation of power quality parameters are highly dependent on their accurate and timely detection. Harmonics are not only one of the prime culprits that deteriorate power quality of the supply system but it also affects accurate estimation and effective control of power quality problems. In this paper, several methods of power system harmonics estimation are critically reviewed and classified based on the type of analysis tool and applications. Various estimation techniques are discussed in brief; comparison of available approaches are examined and presented. The key issues and challenges in harmonics estimations are highlighted. A vast collection of papers, books and standards are listed in the reference list, which is useful to the researchers, engineers and policy-makers in the area of power quality. Keywords: DFT, Harmonics Measurements, Parametric Methods, Power Quality, Recursive Methods and Total Harmonic Distortion.

1. Introduction Widespread applications of power electronics based non-linear loads have resulted in distortion of supply waveforms. Distorted waveform can be represented as summation of various higher frequency sinusoidal components known as harmonics, which are integer multiple of fundamental frequency. There can be certain frequency components that are not

*

Corresponding author. Tel: +91 9455680207; 9425155406; fax: +91 512 2590063 E-mail: [email protected], [email protected]

2 integer multiple of fundamental component, termed as inter-harmonics. When inter-harmonic frequency is less than fundamental frequency, it is known as sub-harmonic. These harmonics and inter-harmonics have large impact on operational efficiency and reliability of power system, loads and protective relaying [1-4]. The skin effect, eddy current loss and corona loss are the direct function of frequency and increase considerably in the presence of harmonics. Resonance phenomenon not only increases the losses in the system, it sometimes leads to failure of devices like compensating capacitors. Harmonic voltage drop across the system impedances results in voltage disturbances causing other linear load to draw harmonic current. It can also affect the protective relaying characteristics or zero crossing detection, leading to reduced selectivity and reliability of protection system. Communication interference is another serious ill effect of harmonics. Harmonics problem, therefore, demands serious concern for accurate estimation and reliable mitigation. Many standards, guidelines and recommendations including IEEE standard 519-1992 and IEC 61000 series of standards have come into effect in this regard [4-7]. Also, some indices like Total Harmonic Distortion (THD) and Total Demand Distortion (TDD), used respectively for voltage and current harmonics, Group THD, Subgroup THD, Partial Weighted Harmonic Distortion have been developed to determine the amount of distortions present in the original signal [7]. Harmonics estimation, here means, detection of frequency components present in the signal and measurement/ estimation of amplitudes and phases of those frequencies. Discrete Fourier Transform (DFT), which is implemented using Fast Fourier Transform (FFT), has been the preferred choice of the researchers and practitioners over last many decades. But many limitations and demerits of FFT have been reported in the literature out of which picket-fence effect, leakage effect and sensitivity to frequency deviation were primarily taken up by many and as a result several algorithms and methods are available in the literature [8]. Some other non-parametric methods based on Wavelet Transform (WT) and Hilbert-Huang Transform (HHT) are used for harmonics estimation. To achieve higher resolution and better estimation accuracy, many parametric methods such as Autoregressive Moving Average (ARMA), Pisarenko Harmonic Decomposition (PHD), Multiple Signal Classification (MUSIC), Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) and Prony method have been applied in the recent years. Artificial intelligence based tools e.g. Artificial Neural Network (ANN) and recursive techniques e.g. Kalman Filtering (KF), Adaptive Linear Element (ADALINE), Extended Phase Locked Loop (EPLL) are becoming equally popular in harmonics estimation. This paper presents a comprehensive review of harmonic estimation techniques in the power system. The basic introduction of some useful techniques used for harmonics and inter-harmonics estimation are also discussed in brief. Comparisons of various approaches, computational burden, hardware implementation options are presented and the key issues and challenges in the harmonics estimation are highlighted.

3

2. Harmonics Estimation Techniques Many harmonics estimation techniques exist in the signal processing, which are categorized into two broad classes as non-parametric and parametric methods [9-10]. Further sub-divisions, as shown in Fig.1, are reported in the literature [9]; however, authors prefer to classify the available literature in the three categories, two as mentioned above and third, hybrid methods, incorporating advantages of two or more methods as a single solution. Non-parametric methods estimate the spectrum directly from the signal, usually in terms of some coefficients (e.g. amplitude) of a set of known functions, generally referred to as basis function. The estimated signal can be time independent or dependent depending upon the method used. Parametric methods use an appropriate model to represent the signal and then estimate the parameters of the model from the available data points. Estimated parameters are then applied to the selected model to determine harmonic contents in the signal. Recursive methods are special parametric methods, which are adaptive/recursive in nature; therefore overcome many major drawbacks present in the batch signal processing based methods e.g. estimation inaccuracy in the presence of noise. A brief review of some important techniques from each category is presented here. 2.1 Discrete Fourier Transform (DFT) Discrete Fourier Transform [11-12] is a method to transform a periodic, discrete signal from time domain to frequency domain with finite range of data samples. Fast Fourier transform is the widely used efficient and fast algorithm for its evaluation. DFT of a discrete time periodic signal x(n) is defined as follows: N 1

X DF (k )   x(n).e

j

2 kn N

,

for k=0, 1, 2, … , (N-1)

(1)

n0

where, n is nth sample of the data, N is the total number of samples and k is the frequency index. Signal in frequency domain is discrete in nature and resolution depends on the sample window size. Half of the total frequency components are negative (mirror image of positive frequencies) and simply discarded to obtain true spectrum. In order to retain time information, windowed DFT, also known as Short Time Fourier Transform (STFT), is applied. It can be represented as: N 1

X STF (m, f k )   h(n  m).x(n).e

j

2 kn N

, for k=0,1,2,…,(N-1)

(2)

n 0

where, fk is the kth harmonic frequency and m is an integer representing window position on time scale. 2.2 Wavelet Transform (WT) Wavelet Transform [13-14] use wavelets to decompose any signal for detailed analysis with multiple time-frequency resolution. In this method, an appropriate wavelet is compared with the signal over a defined time period and a coefficient is obtained that is basically a correlation of a signal with the wavelet. In Discrete WT (DWT), dilation and translation

4 parameters are kept as a function of an integer parameter m to obtain discrete levels of decomposition. For a discrete signal x(n) it can be defined as: N 1

X DW (m, k )   x(n). n 0

1 m 1 sdo

 *(

m 1 n  kuo sdo ) m 1 sdo

(3)

m 1 m 1 where, k is the index of coefficient at mth level. Here, m gives dilation ( sdo ) and k gives translation parameter ( kuo sdo ).

DWT is usually implemented using series of pair of a low pass and a high pass digital filters with sdo=2 and uo=1, as shown in Fig. 2. Each pair of filters, followed by a down-sampler that basically performs suitable dilation, constitutes a level, and produce detailed (high pass) and approximation coefficients (low pass). These coefficients when arranged together in sequence, starting from last level coefficients, constitute DWT. Since, frequency band is halved at each level; down-sampling improves the efficiency of the transformation. As a result, it provides high time resolution at high frequencies and high frequency resolution at low frequencies. Wavelet packet transform (WPT) is another technique of wavelet theory; implemented using a series of pair of filters similar to DWT. WPT keeps decomposing both outputs of any level, unlike DWT, to achieve improved frequency resolution even at high frequencies. Fig. 3 shows structure of WPT. The coefficients at each level can be given as:

D2mj (k )   g (n) D mj 1 (2k  n)

(4a)

D2mj 1 (k )   h(n) D mj 1 (2k  n)

(4b)

n

n

where, j=0, 1, …, (2(m-1)-1) represents filter nodes at (m-1)th level providing inputs to the 2m nodes at mth level; g(n) and h(n) are the impulse response of low pass and high pass filters, respectively. D00 is same as the input signal xn. 2.3 .Hilbert-Huang Transform (HHT) Hilbert-Huang Transform [15-16] is a relatively new two-step data analysis technique, which utilizes the concept of instantaneous frequency. HHT decompose the data into Intrinsic Mode Functions (IMF) using sifting process, known as Empirical Mode Decomposition (EMD), in the first step. IMFs are the adaptive basis functions derived from the data itself, unlike other well known techniques like DFT and WT where basis functions are chosen before the analysis arbitrary; for which instantaneous frequency is defined at every point. Decomposition process involves successive sifting to arrive at an IMF until stopping criteria is fulfilled. Fig. 4 shows flow chart of complete decomposition process. The decomposition is stopped when either the last obtained component and/or residual is small or residual becomes a monotonic function. The signal can be represented in the form of IMFs as follows: n

x(t )   ci  rn i 1

(5)

5 In the second step, Hilbert Transform (HT) is applied on obtained IMFs according to expression (6) that provides information about amplitude and phase content using (7) in both time and frequency scale.

H (ci (t ))  d i (t ) 

1





ci (t ') dt ' t t' 

P

(6)

where, H(*) represents Hilbert transform of any real valued function, di is Hilbert transform of ith IMF ci, P is the Cauchy principal value. The IMF ci and its Hilbert transform di forms a complex analytical function that can be expressed as:

z(t )  ci (t )  jdi (t )  a(t )e j (t )

(7)

where, a(t) and (t) represents amplitude and phase angle at that instant, respectively. 2.4 Chirp z-transform (CZT) The chirp z-transform [17] is a technique to find out z-transform of a finite length data samples along a general contour

zk  AW k in the z-plane, while overcoming many shortcomings of the FFT. Here, A and W are arbitrary complex numbers as given by (8a) and (8b) respectively. DFT comes out to be a special condition of CZT with A  1 , W  e  j 2 / N and number of points in the z-plane K, equal to the total number of samples N.

A  Ao e j 2o

(8a)

W  Wo e j 2o

(8b)

where, Ao, Wo, o and o are the real numbers. The CZT of the signal x(n), evaluated using two FFT and one inverse FFT, can be expressed using (9) as: N 1

X CZ ( zk )   x(n) A nW nk , for k=0, 1, 2,…, K-1.

(9)

n 0

2.5 Autoregressive Moving Average (ARMA) Autoregressive Moving Average [18] is a model based technique in which the signal x(n) is considered as being the output of an pole–zero system H(z) whose input is the white noise w(n). The transfer function of the system can be written as: H ( z) 

1 Q x(n) b0  b1 z  ...  bQ z  w(n) 1  a1 z 1  ...  aP z  P

(10)

where, P is the number of poles and Q is the number of zeros. Parameters ai and bi of ARMA model are estimated first using the block of samples and then magnitude spectrum M(ej) can be obtained from (10) as:

6

 be  1  a e  Q

M (e j )  H (e j ) 

j

i 0 i P i 1

j

(11)

i

2.6 Prony’s Method Prony’s method [19-20] is a technique for extracting the sinusoid or exponential signals by solving a set of linear equations for the coefficients of the recurrence equation that the signal satisfies. Let, data samples xn be estimated by a function g(x) such that:

g ( x)  Rerx  Sesx  ......  Vevx

(12)

where, R, S,…, V and r, s,…, v are constants suitably chosen for closest match with signal. The function g(x) will satisfy a linear equation of the form (13),

Agnu  Bgnu 1  ......  Ugn  0

(13)

where, u is total number of exponential terms in the function, also known as order of the system. The roots of the algebraic equation (13) gives the exponents of (12), which are used to get the coefficients of (13). Exponent terms are used to evaluate frequencies and damping factors and coefficient part give corresponding amplitudes and phase angles. 2.7 Multiple Signal Classification (MUSIC) MUSIC [10, 21] is a noise subspace based parametric method that uses sinusoidal model (14a) or complex exponential model (14b) to estimate the frequencies present in the signal. K

x(n)   ak cos(2n f k  k )  w(n)

(14a)

k 1 K

x(n)   Ak e j 2 nfk  w(n)

(14b)

k 1

where αk is the magnitude, Ak is complex magnitude, k is the initial phase angle, fk is the harmonic frequency and w(n) is white noise. Covariance matrix of M X M, obtained from available data samples of x(n), is decomposed into two orthogonal subspaces, the noise subspace spanned by the minimum eigenvector and the signal subspace spanned by the eigenvectors of interest. Noise subspace is used in determining pseudo-spectrum (15), peaks of which correspond to those of the dominant frequency components. The unknown frequencies can be obtained either from the frequency locations corresponding to the K highest peaks in the pseudo-spectrum or from the angular position of K roots of pseudo-spectrum that are closest to the unit circle in the z-domain, where K is the dimension of the signal subspace.

7

PX MU (e j ) 

1 N



i  K 1

H

r vi

2

(15)

where, rH represents complex conjugate transpose of signal eigenvector, vi represent noise space eigenvector and M is the dimension of space of data x(n). Amplitudes and phases can be, then, obtained by simultaneously solving K equations [9]: K

P k 1

k

rkH vi

2

 i  ˆ w2 for i=1, 2, ..., K

(16)

2.8 Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) ESPRIT [22-23] is also a parametric method that use sinusoidal (14a) or complex exponential model (14b) and decompose the covariance matrix into the noise subspace and the signal subspace. It, however, estimates the frequencies based on the signal-subspace. Once the frequency components are known, the amplitude and phase angle of each frequency component can be computed from the eigen relationship of the signal. The basic ESPRIT algorithm can be explained in the following steps: 1.

Form the covariance matrix Rx from the given data samples of the signal xn.

2.

Decompose the covariance matrix Rx in to signal space matrix Rxs and noise space matrix Rxn.

3.

Use selection matrices S1 and S2 to find out two submatrices R1 and R2 from signal space Rxsusing (18).

S1  [ I Ns

0ds ]

(17a)

S2  [0ds

I Ns ]

(17b)

where, INs is an identity matrix of size Ns x Ns, Ns=(M-ds) and ds is the distance between two submatrices, which is usually kept equal to 1. For i=1 and 2,

Ri  Si * Rxs

(18)

4.

Use least squares estimation to find the matrix  that relates R1and R2.

5.

The frequency components are obtained from the diagonal elements of rotational matrix ф, which are eigenvalues of matrix .

2.9 Kalman Filtering (KF) Kalman Filter [24] is a set of mathematical equations that uses noisy and inaccurate measurement data and provide an efficient computational (recursive) means to estimate past, present or future values, in a way that minimizes the mean of the squared error. The mathematical model is in the form of state equation (19a) and measurement equation (19b) as:

yn1  n yn  n

(19a)

8

xn  hn yn n

(19b)

where, yn is the state vector and xn is the measurement at time tn. n and n are model and measurement errors respectively. n is state transition matrix that relates previous state to the current state and hn is output matrix that relates the state to the measurement. Kalman filter algorithm starts with some initial estimate, yo, for the state variable. It defines the prior error and then tries to improve the initial estimative by means of a linear blending between the estimate of the state variable and the prediction error. The recursive equation for the update of state variables is given by:

yn  yn0  K n ( xn  hn yn0 )

(20)

The matrix Kn is known as Kalman gain. 2.10 Artificial Neural Network (ANN) Artificial Neural Network [25-27] is a network structure, consisting of a number of artificial neurons connected through directional link, establishing mapping relationship between input and output using set of training data. Harmonics estimation from measured signal can be regarded as a mapping problem; hence ANN applications are possible with different structures such as multi-layer neural network, Back Propagation Network (BPN), ADALINE, Radial Basis Function based Neural Network (RBFNN) etc. The mathematical model assumes signal to be sum of fundamental frequency and all harmonic components and can be represented by (12a). Fig. 5 shows simple three layer, multiple input, single output ANN structure commonly employed in harmonics estimation. BPN algorithm constructs three-layer perceptrons which consist of input, hidden, and output layers and adjusts the weights between neurons. The units are connected from input to output layers in a feedforward way. RBFNN represents a function of interest by using members of a family of compactly supported basis functions to perform curve fitting. 2.11 Phase Locked Loop (PLL) A Phase-Locked Loop [28-29] is basically an oscillator whose frequency is locked onto some frequency component of an input signal. Primarily, PLL was used in power system for estimation of fundamental frequency and phase angle of rotating machines. It was applied in harmonics estimation also, but as a synchronizing system in synchronous reference frame method. Enhanced PLL is proposed in [30] for measurement of harmonics and inter-harmonics. It consists of number of basic units, each working as a nonlinear adaptive notch filter preset for some harmonic frequency that adaptively follows variations in fundamental frequency of the signal.

3. Harmonics Estimation: An Overview Problem of harmonics is as old as the AC system and its causes and effects were studied and reported in the beginning

9 of the twentieth century [31-34]. However, because of low overall distortions very little attention was given to harmonics estimation and control, during that period. With the rapidly increasing percentage of power electronics based non-linear loads towards the last couple of decades of twentieth century, harmonics became serious concern and plenty of research papers and books are published [8-9, 35-41] presenting large number of techniques for accurate and fast measurement/ estimation of harmonics and inter-harmonics. A comprehensive literature survey covering more than 200 articles from reputed journals and international conference proceedings, books and standards is presented that classify available harmonics estimation techniques into three categories as follows: 3.1 Non-parametric Techniques Four methods namely DFT, CZT, WT and HHT are placed in this category. DFT is the most basic spectral analysis technique for harmonics analysis of stationary discrete signals. Direct computation of DFT requires N2 operations, which were reduced considerably by pioneering work of Cooley & Tukey [11] in 1965 who proposed FFT. Many limitations of FFT were uncovered in the literatures [15, 42-43] followed by literatures with remedial suggestions and new techniques. Windowing [44-48], interpolation [48-59] and synchronization [60-64], techniques were proposed to overcome the shortcomings of FFT. Harris [44] examined application of different window functions to limit spectral leakage and demonstrated their usefulness in resolving closely spaced harmonics with large amplitude differences. Jain et al. [49], Grandke [50] and Andria et al. [51] have reported frequency-domain based interpolation algorithms, to improve accuracy by reducing leakage. Sedláček et al. [53] have presented time-domain based interpolation technique and a comparison with frequency-domain interpolation technique. Use of adaptive window width suggested by Hidalgo et al. [65] and Zhu [66] considerably reduces spectral leakage with wide variations in fundamental frequency and presence of interharmonics. IEC Std. 61000-4-7 [7] recommends synchronization of the time window (10 cycles for 50Hz and 12 cycles for 60Hz systems) with the signal frequency to achieve lowest possible spectral leakage. Ferrero et al. [60] presented digital PLL based synchronization of samples with signal fundamental frequency by generating synchronization pulses. Another synchronization technique based on CZT was used by Aiello et al. [61], in which fundamental frequency of the signal is detected to set the sampling frequency accordingly. The two methods were compared by Aiello et al. [63], and it was reported that PLL technique has the more adverse effect of input signal disturbances on the performance whereas CZT technique has the drawback of limited resolution. In such techniques, the estimation accuracy depends upon the exact synchronization. Gallo et al., therefore, came up with two-stage desynchronized technique [67] that uses interpolated FFT. Harmonics and fundamental component are derived from frequency-domain interpolation in first stage and then interharmonics are obtained in second stage after filtering out estimated harmonics from original signal. Wang [68] proposed segmented CZT based technique with the advantages of its ability to handle a very large amount of input data and to limit

10 its calculation to a portion of the frequency spectrum of interest thus providing greatly increased dynamic range and frequency resolution. Daponte et al. [69] also used the segmented CZT along with multiple deep dip window in order to increase the resolvability of low magnitude non-harmonic tones close in frequency to higher magnitude harmonics and to detect very low magnitude high frequency harmonics. Tarasiuk [70-71] used CZT and DFT as the prime tool to propose power quality estimator analyzer. Techniques based on wavelet transform and Hilbert-Huang transform were reported towards the end of the last decade of twentieth century, primarily, for time-varying and nonlinear signals. DWT is capable of decomposing signal into subbands only [72], whereas, CWT offers very high computational burden. Discrete WPT (DWPT) approach is therefore more suitable in harmonics estimation. Pham et al. [73] and Keaochantranond et al. [74]applied DWPT for getting uniform sub-bands in the first step and then harmonic content were obtained using CWT in the next step. Design and implementation of WTP based power quality instrument is reported by Hamid et al. [75] that is capable of directly providing different power quality indices and can also find short duration disturbances. Most of the recent literature [7679] use different filter banks to implement WTP based harmonics estimation techniques with some enhanced features like adaptive filtering [80], linear optimization [81] and harmonics grouping [82]. Vatansever et al. [83] used Hilbert transform on the input sample before applying it to WPT, whereas, Morsi et al. [84] used fuzzy systems to handle the uncertainties associated with the electric power quality evaluation. Recently, Yu and Yang [85], Chen et al. [86] and Zhang et al. [87] proposed HHT based techniques within a year indicating its potential in harmonics estimation. Cho et al. [88] have come up with Gabor-Wigner transform based technique that is capable of time-frequency analysis. It is an operational combination of the Gabor transform, an extended version of STFT; and the Wigner distribution function, which together overcomes the drawbacks of each other. PlatasGarza et al. [89] proposed McLaurin series expansion based new technique for dynamic harmonic analysis, known as Taylor-Fourier transform. This can be implemented using FIR filter banks. Hybrid methods based on non-parametric techniques are discussed in subsection-3.3. 3.2

Parametric Techniques

Frequency resolution is a common problem in almost all non-parametric methods. Inter-harmonics and sub-harmonics detection is, therefore, a challenge for these methods. Many stochastic models based parametric methods [18-20] have been in existence from their applications in non-engineering fields for time-series data analysis before their use began in harmonics estimation [35, 90-92] in late sixties. Subsequently, many data analysis techniques like SVD [93-97], AR/ARMA [98-100] least squares estimation [94, 101-104] etc. were applied to estimate the parameters of suitably chosen model for harmonics detection. Nguyen [105] suggested that the sampled data set can be partitioned in to two sets,

11 namely, training set and test set thereby, ensuring that parametric method does not give inaccurate result in case of improper model order. Many authors have reported application of Prony’s method for power system harmonics and interharmonics estimation [106-112] however it is more popular in estimation and analysis of low frequency oscillations [113114]. Chang et al. [112] have used downsampling technique to identify closely adjacent frequencies using Prony method with noisy data. Schmidt, in his work on determining parameters of multiple wavefronts arriving at an antenna, proposed MUSIC technique based on signal subspace decomposition [21]. Wang et al. [115] presented MUSIC based harmonic extraction algorithm; however, MUSIC is not much popular in power harmonics estimation due to high computational burden. Roy et al. in 1986 proposed signal estimation technique known as ESPRIT [23] for directional-of-arrival estimation, which was successfully applied for harmonics estimation in different papers [116-118]. ESPRIT exploits the shift invariance property of the signal. Sliding window based ESPRIT technique was introduced in [119] that can be applied to nonstationary data also. Bracale et al. [120] applied MUSIC and Prony methods for direct measurements of harmonics groups and subgroups in line with [7]. Liquan et al. [121] have applied improved Complex Maximization of Nongaussianity (CMN) algorithm for harmonics and interharmonics estimation. In the last couple of decades many artificial intelligence based techniques and recursive/adaptive techniques have become quite popular. These have been applied to conventional methods to overcome their shortcomings like inaccuracies on account of incorrect modeling, noise present in the signal and nonlinearity. In 1973, Sharma et al. [122] reported Kalman filter application in harmonics analysis. There has been a substantial gap of over a decade, then Bitmead et al. [123], Dash et al. [124], Girgis et al. [125-126] and many others proposed KF based improved algorithms for online harmonics analysis [127-130]. Extended KF has been used in [131-134] which overcome limitations of KF e.g. effect of mathematical model on accuracy. Least squares estimation based recursive technique is reported in [135] that work well with short length, noisy, non-linear and non-stationary data samples. Köse et al. [136] used combination of extended KF and linear KF for spectral decomposition of distorted supply to obtain harmonics and interharmonics contents. Mori et al. [26, 137] and Osowski [27] suggested ANN application in harmonics detection, independently, almost at the same time. Mori et al. used three layered, backpropagation, feedforward neural network for voltage harmonic prediction, whereas, Osowski considered signal to be consisting of n frequencies of unknown amplitude and phases and then used neural network for parallel processing of many samples for higher estimation speed. Osowski also presented an adaptive estimation that is not as fast as [137] but reduced the circuit complexity and implementation cost relatively. Mathew and Reddy [138] applied feedback type neural network to Pisarenko’s method. Some of the important references of ANN technique in harmonics field are [139-148]. In 1996 Dash et al. [149] proposed new approach for harmonics estimation

12 using Fourier linear combiner realized using an adaptive linear neuron known as ADALINE. This approach is quite different from the backpropagation approach and allows better control the stability and speed of convergence by appropriate choice of parameters of the error difference equation. Recently, Chang et al. [150] proposed two-stage ADALINE that is robust and capable of detecting inter-harmonics and Sarkar et al. [151] proposed self-synchronized SADALINE for enhancing immunity to frequency deviation and noise. Guangjie et al. [152] and Chang et al. [153] presented RBFNN based technique that has simpler structure and is more suitable for learning functions with local variations and discontinuities. Lu et al. [154] applied particle swarm optimization, and Seifossadat et al. [155] applied genetic algorithm based on adaptive perceptron for estimating power system harmonics. Phase locked loop (PLL) has been applied for frequency detection and synchronization for use in applications like aircraft [156-157], machine control, power system [62, 158] and many others. Karimi-Ghartemani et al. [28-30, 159-160] proposed that non linear adaptive filter based on the concept of enhanced PLL can be used for harmonics estimation. It was also shown that EPLL can be used for extracting other signal attributes like peak value, flickering etc. However, it is not a convenient method to estimate any particular harmonics as it uses chain of EPLLs in series, one each for individual harmonic component, each receiving residual of previous stage as input. McNamara et al. [161] proposed a technique that adaptively estimates the fundamental and individual harmonic components of the power signal and tracks their variations over time. It use phase-dictated sinusoid-tracking and extracts sinusoidal components by arranging these in master-slave pattern. J. R. de Carvalho et al. [162] has used three stage algorithm consisting of band pass filter at first stage followed by a downsampler at second stage, to reduce computational burden and finally EPLL to provide amplitude and phase information of different frequency components. 3.3

Hybrid Techniques

With the idea of utilizing the strengths of individual harmonics estimation method while restraining the shortcomings, many hybrid techniques have been reported in the literature [163-188]. Liu and Chen [163] used wavelet transform to shorten the tracking time of Kalman filter based online method by expressing the amplitudes and phases of various harmonics in terms of coefficients of wavelet and scaling functions. Lobos et al. [164] used wavelet transform to identify the transients and then applied Prony method for getting frequency contents. In a hybrid technique presented by Bettayeb et al. [165], amplitude is estimated from linear least squares estimator and phase estimation, which is a nonlinear problem, is achieved using genetic algorithm (GA). Joorabian et al. [166] also used the similar concept of decomposing linear and nonlinear problems and handling them separately. Fuzzy based bacterial foraging optimization technique was used in [167] for phase estimation of the harmonic and fundamental components, whereas, conventional Least Square (LS) method was preferred for amplitude estimation owing

13 to its proven performance. Soliman et al. [168-169] have proposed fuzzy linear regression based technique that accurately estimates frequency deviation and harmonic content in the distorted voltage. Huang et al. [170] applied fuzzy adaptive controller together with extended complex KF for amplitude and frequency estimation of distorted power supply. Hostetter [171-172] has used state variable representation of band limited periodic signal to use DFT in recursive form. Bitmead [173] showed that recursive technique of [171] is equivalent of FIR frequency-sampling filters. Martens [174] presented a recursive technique for DFT calculation known as RCFA (recursive cyclotomic factorization algorithm). Limin et al. [175] proposed another recursive DFT technique recently that is capable of estimating specific harmonic of interest in real time. Recursive WT based technique is suggested by Ren et al. [176], featuring fast response and accuracy. It, however, not only requires high sampling rate but also need one complete cycle data to process it. Many hybrid methods based on wavelet transform and FFT have been proposed in the recent years [177-179]. Bracale et al. [117] presented ESPRIT and DFT based two-stage method. In the first stage ESPRIT is used to estimate fundamental frequency and inter-harmonics, then in second stage DFT is applied with more accurate windowing. The ability of WT to detect transients without any constraint on synchronized and specific number of samples, and robustness and speed of FFT was utilized by Tarasiuk [177] to introduce a hybrid method that can detect transient and harmonics both. Chen et al. [179] used WT for de-noising the signal before processing it with FFT for getting more accurate harmonic spectrum. Wang et al. [180] proposed a WPT based hybrid method, to increase the band pass filtering ability of the empirical mode decomposition (EMD). Another hybrid method was proposed by Costa et al. [181] that use Kalman filter and Prony method. Prony method was used as frequency estimator, while Kalman filter was used to extract the amplitudes and phases of each harmonic. This technique has the tracking capabilities for time-varying harmonics, and also independent of previous knowledge of the harmonic frequencies as they are estimated using Prony's method. In 2009, many hybrid schemes e.g. [150, 166, 182-183] have been proposed that employ ADALINE. Subudhi et al. [182] used ADALINE as adaptive estimator and recursive least squares and Kalman filter to update the weights of that adaptive neural estimator as two separate methods, however KF-ADALINE method performance was reported better than LS-Adaline method.

Sahoo et al. [183] used robust H filter for amplitude estimation and ADALINE for phase

estimation. H filter is based upon state space modeling of the signal with the assumption that number of sinusoids present in the signal is known. Xiong et al. [184] proposed windowed interpolation and Prony based method that is claimed to be capable of detecting inter-harmonics lying close to fundamental or other harmonics. The windowed interpolation method finds harmonic components and identifies the frequency interval in which two adjacent components are located. Prony algorithm is then used to compute these adjacent components. Support vector machine algorithm based technique has been proposed in

14 [185] for harmonics and inter-harmonics estimation using iterative reweighted least squares method. Adaptive particle swarm optimization (PSO) algorithm was used by Dash et al. [186], to optimally select the parameters of unscented KF and measurement error covariances. Unscented KF provides more accurate estimation of the parameters of a nonstationary signal because it does not use linearization for computing the state and error covariance matrices. Adaptive PSO reduces its vulnerability to low SNR signal and any incorrect KF parameter. Zadeh et al. [187] used KF and least error squares techniques to propose new hybrid technique. The Kalman filter was modified to provide precise estimation results insensitive to noise and other disturbances and the least error squares system was arranged to operate in critical transient cases to compensate the delay and inaccuracy identified by the KF. They also presented practical considerations such as the effect of noise, higher order harmonics, and computational issues of the algorithm, supplemented with test results. Sadinezhad et al. [188] proposed Newton method and LS algorithm based optimization technique to measure power system frequency and harmonics adaptively.

4. Comparison of Different Techniques Non-parametric techniques are based on transformation of the given time-series data sequence. These techniques are not capable of incorporating any available information about the system in estimation process. Fourier transform is most commonly used, most fundamental, traditional spectrum analysis technique in harmonics estimation. Although it suffers from many drawbacks but many improved algorithms have made it practically better tool for numerous applications. One critical limitation of FFT is that it requires number of discrete samples equal to two power of an integer, covering integer number of complete cycles of the analyzed data. Failing this, FFT suffers with aliasing, picket-fence and leakage effects, thus making FFT unsuitable for applications where fundamental frequency deviates considerably. Also, the frequency resolution in FFT is direct function of time window size, therefore requiring more number of cycles for interharmonics detection. FFT lead in spectral estimation because it is fast and simple, synchronization and interpolation reduce spectral leakage; however, its inability to retain time information and lower frequency resolution with small data window length make path for other techniques. Comparison of different DFT techniques can be found in [10] and [189]. Wavelet has multi-resolution capability and it also retains time information of the signal, however, low frequency local disturbances are hard to found. It is more suitable for data with gradual frequency changes or transient detection [15] but its high computational burden and interpretation complexities limit its stand-alone application in harmonics estimation. Discrete WPT outperforms DWT because it decomposes the signal into uniform frequency bands and in the regime of [7] it naturally suits better than others. WPT and WT based hybrid methods are gradually becoming popular, but optimal mother wavelet selection still present substantial challenge. HHT use adaptive IMF as the basis functions for

15 decomposition hence allows spectral analysis of even nonlinear and non-stationary data more effectively. It is a posteriori data processing based relatively new technique that requires additional tools for exact interpretation of the results. The accuracy and effectiveness of the technique is highly dependent on the exact spline fitting while creating upper and lower envelops, which is quite difficult. Also, HHT requires over-sampled data for precise definition of instantaneous frequency. Last but not the least; decomposition is not possible for closely located frequency components. This technique is therefore suggested for highly non-linear applications with at most care on the above limitations. The parametric methods need sufficient priori information about the system and analyzed signal for accurate modeling of the system. This is quite difficult for time-varying and non-linear systems. Inaccuracy due to model mismatch is therefore a quite common problem in these techniques, recursive techniques perform better in this sense. Appropriate model order selection is still a tough task. Parametric methods like Prony, ESPRIT, and ARMA etc have very good resolution and capable of detecting inter-harmonics, however these are prone to noise, and accuracy reduces considerably in the presence of noise. These techniques do not require synchronized sampling and are produce estimates free from the spectral leakage and side lobes. Many of these methods give good results for short data sample. MUSIC needs large storage for the array manifold, ESPRIT requires no storage. ANN based techniques are capable of incorporating nonlinearity in the system. These are self adaptive and robust against noise present in the signal, however, it needs fairly large amount of data for training and erroneous results may come when the network sees an unfamiliar waveform. Also, due to multilayered structure and the greedy nature of the back-propagation algorithm, the training process often settles in the undesirable local minima of error surface or it may converge slowly. KF often suffers with the problem of ‘filter dropping off’ and becomes insensitive against sudden changes of state variables if the estimation parameters are not changing for a long time. KF also requires a priori information of the process for accurate modeling. Hybrid techniques although try to pick the pros of basic methods and exhibits good performance; they encounter many practical implementation problems. Table-1 presents performance comparison of ‘harmonics estimation techniques’. 4.1 Computational Burden and Hardware Implementation Computational burden is the key performance assessment criteria that also decide the suitability of spectral estimation techniques for real-time application. The relative computational burden of some techniques is presented in Table-2. Windowing and interpolation in FFT need some additional operations of the order of N. The wavelet packet decomposition requires more multiplications than the regular wavelet decomposition, however, it can be considerably reduced by using special elliptic half-band filters [77]. Eren et al. [76-77] have compared computational complexity of WPT with different filters. HHT having two nested loops demands higher computation, however, Waskito et al. [190]

16 have proposed parallel implementation of HHT to reduce it. Most of the parametric methods have high computational burden, mainly, because it involves either the eigen decomposition or matrix inversion, each having requirements of the order of m2n and n3 operations, respectively, for n X m and n X n matrices. Prony method, for example, needs SVD of Toeplitz matrix followed by finding the roots of the polynomial and then matrix inversion in least squares estimation of Vandermonde matrix [191]. Roy et al. [23] claimed that ESPRIT offers a computational advantage of the order of 105 over MUSIC. Digital Signal Processors (DSP) and Field Programmable Gate Arrays (FPGA) are two obvious options for hardware implementation of any of the discussed spectral estimation techniques. DSP are specialized microprocessors that work on sequential execution of set of instructions thus enabling easy conditional processing of complex mathematical functions. Whereas FPGA is a configurable hardware, in which, millions of logic gates can be programmed to perform given operations. If DSP can fulfill the requirement of execution speed, which most of the recent DSP does, it will be preferred over FPGA, because it provides better precision using floating point operations, better handling of conditional logics and easy programming. FIR filters, the most common basic building block in signal processing, are usually implemented using DSP. FPGA are preferred for ANN and other techniques like HHT, and to some extent CZT, which demand parallel computations. Depending on the requirement, however, both tools have been used for any of the above discussed techniques [192]. Recent literature also suggests hybrid architecture, utilizing both DSP and FPGA, especially for complex techniques like HHT, MUSIC and ESPRIT etc. A twelve states KF algorithm is applied in [193], using an 8-bit microprocessor, for continues real-time tracking of the harmonics in the voltage or current waveforms to obtain in real time the instantaneous values for a maximum of six harmonics as well as the existing harmonic distortion. Various issues related to characterization of power quality instruments are discussed in [194]. Some of the literatures that discuss about hardware implementation of different spectral estimation techniques include: [71, 156-157, 195], [77, 196-197], [71, 198], [190], [199], [200-201], [109], [193], [202], [203-205] respectively for DFT, WT, CZT, HHT, FIR, AR, Prony, KF, ADALINE and ANN. 4.2 Application Remarks FFT is the most versatile tool in spectrum analysis. Its application can range from online measurement tools to offline analysis purposes. FFT with sufficiently large window size, proper number of samples carefully synchronized with supply frequency and small sampling time provides most accurate results with high resolution that can be used as reference for comparison and calibration. WT based methods are suitable for measurement of time-varying signals and disturbance localization whereas HHT technique provides good analysis of non-linear and time-varying signals. Parametric methods such as Prony, ARMA, and ESPRIT etc. are suitable where frequency resolution is more important than computational

17 speed. These are generally employed in sub-harmonics estimation. SVD, LS and MUSIC are especially suitable where only short period data is available. Data, however, should contain as minimum noise as possible for good results. ADALINE and KF being recursive are suitable for online tracking applications where, there is a considerable fundamental frequency deviation and noise. ANN based methods are suitable for systems where only few dominant predefined frequency components are to be measured; may be for monitoring or control purpose. These can be applied to systems, which are difficult to model e.g. non-linear systems; however, enough data should be available for training. The locality of the basic functions makes the RBFNN more suitable for learning functions with local variations and discontinuities.

5. Key Issues and Challenges in Harmonics Estimation The emerging power system will consist of renewable energy sources, smart grid solutions comprising of FACTS devices and non-linear loads like power electronics based equipments, electric vehicles etc. Harmonic estimation is very important to know the harmonic components and their magnitudes for designing the harmonic mitigation devices and other controllers. The harmonics estimation should be fast enough for real time application, highly accurate for better reliability, simple for easy practical implementation and economical too. Power system is highly dynamic in nature, hence, network topology and parameters keep on changing rapidly. The harmonics estimation therefore should be adaptive in nature. It should also be robust against noise and transients present quite often in the measured data. The key issues and challenges in power system harmonics estimation are as follows:  Power system frequency keeps on changing with load variations, making the measured signal highly time-varying. Although, WT, HHT etc. provide time-frequency information, these are computationally demanding. Windowing and interpolation techniques are suggested in the literature for FFT; however, appropriate window size selection is a challenge. Short window size reduces frequency resolution whereas large window size results in errors if frequency and/or amplitude changes during window sampling.  Measured signals captured from monitoring devices are invariably contaminated by some amount of noise, which affect the accuracy of harmonics estimation; especially in case of parametric methods like AR, ESPRIT and Prony etc. Current and potential transformer saturation effects can even change the characteristics of the signal, which is difficult to recover.  Selection of suitable mother wavelet in case of WT, correct spline fitting in HHT and optimal model order in parametric methods and number of hidden layer neurons in ANN based techniques is a difficult task. The workable answer comes only from practical experience or trial and error.  In the restructured power system, prices of the power will be associated to power quality also; hence, data

18 compression becomes an important issue. Techniques like WT offer this additional advantage, however, primary issues such as estimation accuracy and speed needs to be improved.  For application of estimated harmonic components in control algorithms, the problem of measurement and filtering delay is a serious concern. During load dynamics, phase shift in the fundamental frequency is introduced by inherent time lag that may affect control adversely.  For techniques based on ANN, it is difficult to collect sufficient training signal patterns for practical applications because the highly time-varying behavior of nonlinear loads may be unexpected. There is a need for dynamical adjustment of the size of the neural network to effectively search for the minimum estimation errors of the measured signal [153].  Present trend of wide area measurement systems (WAMSs) requires phasor measurement unit for making electric grid more intelligent and efficient. These PMUs devices are very much affected by harmonics and noise. Hence a fast and accurate harmonic estimation technique is required.

6. Conclusion A comprehensive review of available literature on power system harmonics estimation techniques is presented in this paper. Problems of each technique, available solutions and challenges are brought out from plethora of research papers. Fast Fourier transform appears as the most common technique with windowed interpolation and synchronization facilities, providing accuracy within the limit of [7]. Wavelet packet transform is emerging as the key competitor in [7] regime with the availability of advanced hardware for real-time implementation of FIR filter bank architecture, as it can naturally provide harmonics group and subgroup measurements. It is also capable of identifying transient disturbances and data compression, which is becoming important with increasing amount of measured data for storage and transfer purposes. Model based parametric techniques are also competent for data compression; however, these require intensive research to not only reduce the computational burden but also to improve immunity against noise. KF approach is simple and robust; however, accuracy is affected if priori information about noise is not available. For hardware implementation, DSP is first choice; however, FPGA outperforms it in many cases and hence becoming more popular.

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List of Figure Captions

27 Fig. 1. Categorization of harmonics estimation techniques Fig. 2. DWT representation using analysis filter banks Fig. 3. Wavelet packet transformation structure Fig. 4. Flowchart of IMF extraction using empirical mode decomposition Fig. 5. Architecture of multi-layer, multi-output ANN suitable for harmonics estimation

28 TABLE-1 HARMONICS ESTIMATION TECHNIQUES Technique

Key References

Merits

Demerits

Potential Application

FFT

[8], [11-12], [42-43]

+ Simple + Fast

- Leakage & picket-fence effect - Aliasing - Time information is lost hence cannot detect transients.

 Stationary and linear signal with constant fundamental frequency

WI-FFT

[44-48]

+ Reduced spectral leakage + Easy synchronization of data

- Low frequency resolution and some additional computational burden - Estimation accuracy is affected e.g. DC component may appear

 Time-varying signal with frequency deviation and interharmonics

DWT/WPT

[13-14], [72-83]

+ Time-frequency domain + Multi-resolution ability + Easy implementation with filter bank

- Spectrum is in terms of frequency bands - Mother wavelet selection is arbitrary and affects the accuracy

 Disturbance localization  Data Compression

HHT

[15-16], [85-87]

+ Posteriori data processing + Basis function (IMF) is adaptive

- Additional tools needed for interpretation of transformed parameters

 Non-linear and non-stationary signal  Offline analysis

CZT

[17], [61], [6871]

+ Flexibility on data sampling and number of samples

- Computationally more demanding - Additional data storage requirement

 Time interpolation of data from one sample rate to another  Synchronization

AR/ ARMA

[98-100]

+ High resolution and no side lobes + Fairly good accuracy with short data length

- Unable to resolve close frequencies - Model order selection is crucial and affects accuracy

 Transient, peaky signals  Interharmonics analysis

PRONY

[19-20], [106112]

+ Can detect all attributes of signal: frequency, amplitude, phase and damping factor + Very high resolution and no side lobes

- Computationally inefficient - Highly prone to noise and model mismatch

 Suitable for linearly behaving data  Precise offline analysis

MUSIC

[21], [115]

+ Good accuracy with short data length

- Priori knowledge of frequency search range is needed - High computational time

 Offline application  High precision frequency identification

ESPRIT

[22-23], [116119]

+ High resolution + Precise frequency estimates

- Only frequency domain - High computational burden

 Stationary data  Classification

KF

[24], [122-136]

+ Robust against noise + Recursive + Ability to track time varying parameters

- State variable modeling is critical - Filter drooping off - A priory information of noise and process is needed

 Noisy time-varying signal with sufficient knowledge of process  Online control applications

ANN (BPN)

[25-27], [137148]

+ Self-adaptive + Fairly good accuracy with noisy data samples

- Multilayered complex structure - Greedy nature - May be trapped in the local minima

 Only few definite harmonics are critical

ANN(RBFN N)

[152-153]

+ Simpler structure + Fast convergence

- Inaccurate in the presence of noise and transients

 Only few definite harmonics of time- varying signal are more critical

ADALINE

[149-151]

+ Robust against noise + Better convergence

- Inaccurate if signal contain any harmonics that is not included in ADALINE

 Online tracking of timevarying harmonic

EPLL

[28-30], [158160]

+ Adaptive to varying frequency

- Slower dynamic response

 Varying frequency nonstationary signal

29 TABLE-2 COMPUTATIONAL COMPLEXITY COMPARISON Technique

Computational Burden

DFT

N2

FFT

(N/2) log2N

Radix-2

DWT

2NL

Filter approach, infinite series.

CZT

(N+M-1) log2(N+M-1)

Approximate Value

AR

NM

Burg Algorithm

M

2

Levinson-Durbin Algorithm

2

SVD

MN

ANN

(N/2+H)M + (M+H) 2

ADALINE

Remarks

BPN

HM+(N/2) M

RBFNN

2NM

Single-Stage

3NM+1.5NM

Two-Stage

Here, N is number of data samples, M is the order of estimation, L is the length of the filter and H is number of outputs of the ANN.

30

Harmonics Estimation Techniques Parametric Method

Non-parametric Method

Frequency Domain Analysis  Discrete Fourier transform

Time-Frequency Domain Analysis

Stochastic Models

Sinusoidal Models  ESPRIT  MUSIC  KF

 Wavelet transform  Hilbert-Huang transform  Chirp z-transform

 Autoregressive Model (AR)  ARMA  Prony’s 

Fig. 1. Categorization of harmonics estimation techniques.

2

HPF xn

D1 Detail Coefficient, level-1 HPF

Level-1 2

LPF

2

HPF

Level-2 2

LPF

D2 Detail Coefficient, level-2

Fig. 2. DWT representation using analysis filter banks.

HPF xn LPF Level-1

D01

D3

2

A3

Level-3 LPF

D11

2

HPF

D32

LPF

D22

HPF

D223 D 1

LPF

D02

Level-2

Fig. 3. Wavelet packet transformation structure.

31 Start Read data x(t)

Identify extrema then connect maxima and minima separately using cubic spline line to create envelopes Get mean of upper and lower envelopes

Apply sifting process and compute component as difference of data and mean

Consider the component as a data

No

Is a component an IMF? Yes

Store it and compute residual rn using (5)

Anymore IMF possible?

Yes

Consider the residual as a data

No Stop Fig. 4. Flowchart of IMF extraction using empirical mode decomposition.

x1

1

w11 w21

x2

1

2 2

x3

3

n

v2

f(.)

y

v3 w3m

xn

v1

m

bias

wnm

Fig. 5. Architecture of multi-layer, single-output ANN for harmonics estimation.