Maximum Likelihood Frequency and Phasor

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TII.2017.2727529, IEEE Transactions on Industrial Informatics IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS

Maximum Likelihood Frequency and Phasor Estimations for Electric Power Grid Monitoring Zakarya Oubrahim, Student Member, IEEE, Vincent Choqueuse, Member, IEEE, Yassine Amirat, Member, IEEE and Mohamed Benbouzid, Senior Member, IEEE

Abstract—In this paper, a new frequency and phasor estimators are presented. These estimators are based on the maximum likelihood technique and exploit the multidimensional nature of electrical signals. To minimize the likelihood function, we present an optimization algorithm based on the Newton-Raphson technique. While the performance of Fourier-based estimators significantly degrades when the window length is not equal to a multiple of the fundamental half-period, the proposed estimators perform well regardless of the window length. Simulation results show that the proposed estimators clearly outperform the discrete time Fourier transform in terms of Total Vector Error (TVE) and Frequency Error (FE) whatever the signal to noise ratio, harmonic and interharmonic distortion, and off-nominal frequency deviation are. The benefits of the proposed estimators are also illustrated with real power system data obtained from the DOE/EPRI National Database of Power System Events. Index Terms—Frequency and phasor estimations, maximum likelihood estimator, off-nominal conditions, phasor measurement units (PMUs), IEEE C37.118.2011, power grid monitoring.

I. I NTRODUCTION LOBAL climate change, new intelligent computer software, and hardware technologies are the most dominant factors impacting future electrical power systems. These are driving the entire energy system to efficiency and renewable resources electricity [1]–[5]. However, intermittent and discontinuous nature of renewable resources and hierarchical topology of the existing grids may impact power quality and make the balance between energy production and consumption more difficult to maintain. This could have a substantial influence on grid stability and power quality by affecting voltage and frequency control [6]–[8]. To address these challenges, the concept of smart grid has emerged. A smart grid can be considered as a modern electrical power grid infrastructure with enhanced efficiency and reliability through automated control and information communications technology systems. The future smart grid is expected to be able to optimize its structure for maintaining the balance between energy production and consumption, and to detect abnormal events at an early stage [2]–[5]. To meet theses requirements, the promising solution is to use advanced fault diagnosis and fault-tolerant control algorithms [9]. Regarding fault diagnosis, the main existing approaches can be categorized within the following families: 1) model-, 2) signal-, and 3) knowledge-based techniques (data-driven ones) [9]. In this context, knowledge-based techniques (i.e. artificial neural networks, support vector machines, etc.) performances critically depend on a learning stage that requires a training database. These techniques

G

Z. Oubrahim, V. Choqueuse, and M.E.H. Benbouzid are with the University of Brest, FRE CNRS 3744 IRDL, Brest, France (e-mail: [email protected], [email protected], [email protected]). Z. Oubrahim is also with ISEN Brest, Brest, France. M.E.H. Benbouzid is also with the Shanghai Maritime University, Shanghai, China. Y. Amirat is with ISEN Brest, FRE CNRS 3744 IRDL, Brest, France (e-mail: yassine.amirat@isen´ bretagne.fr). This work was supported by Brest Metropole and ISEN Brest.

particularly suffer from computation complexity, which is not the case of signal-based approaches that has definitely a lower computation complexity. If compared to model-based techniques [10], signal-based approaches are usually simpler to implement since they directly utilize measured signals rather than explicit input-output models. In power grid monitoring applications, electrical signals can be recorded and processed by intelligent devices such as Phasor Measurement Units (PMUs). PMUs are predicted to become a vital part for the estimation of the power system state and can be used to validate system performance and control equipment settings [11]. These devices are installed and deployed at the important power systems substations to extract several signal parameters such as the fundemantal frequency and complex phasors. To ensure reliability and interoperability among different manufacturers, the IEEE has released a PMU industry standard dealing with the estimation performance of PMU devices. More precisely, the IEEE standard C37.118.2011 [12] (and its amendment [13]) defines several criteria for the evaluation of the frequency and phasor estimator performances under steady-state and dynamic compliance conditions. These include the Total Vector Error (TVE), Frequency Error (FE), and ROCOF. This standard also defines two performance classes, which are the P- and M-classes. The former is intended for measurement applications that requires fast response to dynamic events, while the latter focuses on the estimation performances and requires TVE and FE smaller than 1% and 5 mHz, respectively. In this study, we will focus on the M-class which is the most challenging in terms of statistical performance. In the literature, several frequency and phasor estimators have been proposed [14]–[18]. Regarding the frequency estimation, the standard IEC 61000-4-30 proposes a simple low-complexity estimator based on the zero-crossing technique. Nevertheless, this technique is quite sensitive to signal distortions and further processing must be considered to minimize the effect of DC components, harmonics, inter-harmonics, and other power quality disturbances. Furthermore, this technique is statistically suboptimal since it does not exploit the multidimensional nature of three-phase electrical signals. To overcome this limitation, several recent studies have proposed to exploit the multidimensional nature of electrical signals using dimensionality reduction techniques. Most of these techniques are based on the Clarke transform such as the widely linear least mean phase techniques in [19] or the least mean squares differentiator in [20]. Nevertheless, while the Clarke transform is a natural technique under balance conditions, its use remains questionable in unbalanced threephase systems. Regarding phasor estimation, this problem can be decomposed into the estimation of the amplitude and phase parameters. For amplitude estimation, the Root Mean Square (RMS) and peak voltage are simple and well-proven techniques. However, these techniques do not provide any estimate of the phase angle parameter and are quite sensitive to noise or distortion. To overcome these issues, many PMU estimation algorithms are based on the Discrete Time Fourier Transform (DTFT) [21], [22]. DTFT can be used to estimate both the amplitude and phase angle parameters. The main benefits

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of this technique are its simple implementation, low computational complexity, accuracy, and immunity against harmonic components. To compensate the interference between positive and negative frequency components in real signals, an enhanced interpolated discrete Fourier transform (IpDFT) algorithm with implementation and validation on a PMU prototype has been proposed in [16]. DTFT algorithms usually exhibit high statistical performances under ideal conditions. However, its performances rapidly degrade under offnominal frequency, amplitude, or phase variations [23]. Recently, several phasor estimators based on the Maximum Likelihood (ML) have been proposed in [24]–[26]. However, these techniques are limited to particular phasor configurations and their performances highly degrade under more general settings. Last but not least, some authors have also proposed more sophisticated techniques to improve the estimation performances, as in [27] where an empirical mode decomposition-based denoising technique has been used. The main drawbacks of the above frequency and phasor estimation techniques are twofolds. First, their statistical performances are inherently suboptimal since they do not fully exploit the three-phase nature of the electrical signals. Then, their performances critically degrade under off-nominal conditions. To overcome these limitations, this paper proposes an improved fundamental frequency and phasor estimator based on the Maximum Likelihood (ML) technique that fully exploits the multidimensional nature of the electrical signals. The proposed algorithm is illustrated by Fig. 1, where X corresponds to the three-phase electrical signal, fb0 corresponds to the estimated fundamental frequency, and Vba , Vbb , and Vbc are the estimated complex phasors. The main contributions of this paper are threefold. •

•

•

While most of previous works in frequency and phasor estimations only exploit single-phase information, we propose a new ML-based frequency and phasor estimator that fully exploits the multidimensional nature of the electrical signals. As compared to our previous work in [28], we present both exact and (low-complexity) approximate estimators. Contrary to other techniques, simulation results show that the proposed estimators achieve good performances even under off-nominal conditions. To maximize the ML cost-function, we present a new optimization algorithm based on the Newton-Raphson method. As illustrated in the simulation section, this algorithm outperforms the downhill simplex algorithm presented in [28] and has lower computational complexity. We perform an in-depth analysis of the estimator performance. Specifically, the Total Vector Error (TVE) and Frequence Error (FE) are evaluated under noisy, harmonic, interharmonic, and off-nominal frequency environment, and are compared with the requirements of the IEEE Std. C37.118.2011.

This paper is organized as follows: Section 2 presents the threephase signal model and the proposed angular frequency and phasor estimators. Section 3 deals with the proposed optimization algorithm and Section 4 focuses on simulation and experimental results.

II. A NGULAR F REQUENCY AND P HASOR E STIMATORS FOR U NBALANCED P OWER S YSTEM

This section presents the three-phase signal and phasor models under noisy environment. Based on the phasor model, we also describe the proposed Maximum Likelihood angular frequency and phasor estimators.

Measured three-phase voltage

X Frequency estimation using ML method

X

fb0

Complex phasor estimation using ML method

Vba

Vbb

Vbc

Characterization process Fig. 1. Flowchart of the proposed algorithm.

A. Signal Model and Phasor Expression under Noisy Environment This study focuses on the analysis of the fundamental component1 . Under noisy environment, the discrete time signal corresponding to the phase m = {a, b, c} can be expressed as follows (see [29]) xm [k] = am cos (kw0 + φm ) + bm [k],

(1)

2πf0 Fs

where w0 = corresponds to the normalized fundamental angular frequency and f0 /Fs is the ratio between the fundamental and sampling frequencies. The parameters am ≥ 0 and φm correspond, respectively, to the amplitude and initial phase for the phase m, and bm [k] refers to the additive noise. In practice, it is usually more convenient to resort to the complex phasor instead of the amplitude and initial phase. Mathematically, the complex phasor for the phase m can be expressed with respect to am and φm as Vm = am ejφm .

(2)

Under nominal conditions, the phase shift between phases is equal to 120◦ and am = 1 pu. If one of these requirements is not satisfied, the system is considered as unbalanced [30], [31]. In practice, the signal parameters are unknown and must be estimated from N consecutive samples of xm [k] (k = 0, 1, · · · , N − 1). Using matrix notations, the signal model in (1) can be expressed as X = G(w0 )S + B,

(3)

where • X and B are N × 3 matrices containing the recorded and noise three-phase samples, respectively. These matrices are defined by   xa [0] xb [0] xc [0]   .. .. .. X= (4) , . . . xa [N − 1] xb [N − 1] xc [N − 1] 1 In practice, the harmonics can be removed by using a simple low pass-filter.



where I22 is the identity matrix of size 2 × 2 and q(w) is defined as

and 

ba [0] .. . ba [N − 1]

 B=

bb [0] .. . bb [N − 1]



bc [0]  .. . . bc [N − 1]

q(w) ,

(5)

N −1 X

e2jωk =

k=0

•

G(w0 ) is a N × 2 real-valued matrix that only depends on the unknown angular frequency w0 . This matrix is given by   1 0   .. .. G(w0 ) =   , (6) . . cos((N − 1)w0 ) sin((N − 1)w0 )

•

S is a 2 × 3 real-valued matrix containing the amplitudes and initial phases for the three phases. This matrix is defined as aa cos(φa ) ab cos(φb ) ac cos(φc ) . (7) S= −aa sin(φa ) −ab sin(φb ) −ac sin(φc )

The main objective of the next subsection is to estimate w0 and Vm (m = {a, b, c}) from the three phase signal X.

sin(N w) jω(N −1) e . sin(w)

(12)

If the number of samples is equal to an integer multiple of the s fundamental half-period i.e. N = lF = lπ (l ∈ N), then 2f ω s , the sin(N w) = sin(lπ) = 0 and so q(w) = 0. When N 6= lF 2f scalar q(w) is generally non-zero but can be neglected as compared to N2 for large N when ω is not near 0 or 1/2 [32]. Using this result, GT (w)G(w) can be approximated by GT (w)G(w) ≈

N I22 . 2

(13)

s Note that this approximation is exact when N = lF . Using (13), 2f the cost-function in (10) can be simplified as follows

J (w) ≈

h i 2 2 T r XT G(w)GT (w)X = kGT (w)Xk2F . N N

(14)

This function can also be expressed in scalar notations as B. Maximum Likelihood Angular Frequency and Phasor Estimators One of the most popular approach for parameter estimation is the Maximum Likelihood Estimator (MLE). As compared to other estimators, the MLE has the distinct advantage of being asymptotically optimal for large enough data records [32]. Under white Gaussian noise, the MLE of w0 and S in (3) is given by {w b0 , b S} = arg min kX − G(w)Sk2F , w,S

(8)

where k.k2F is the Frobenius norm. This particular estimator is also called the Least Squares (LS) estimator. Note that when the noise is not Gaussian, this estimator is also the most natural one since it minimizes the sum of the squared errors between the signal X and the underlying model G(w)S. Initially, the MLE requires the minimization of a cost-function in a 7-dimensional space. However, as the components in S are linearly separable, the estimation of w0 and S can be divided into two steps [33]. The first is the estimation of the nonlinear parameter w0 through the maximization of a 1-dimensional function. The second is the linear estimation of S obtained by replacing w0 with its estimate w b0 . These two steps are described in the next subsections. 1) Angular frequency estimator: Let us denote (.)T and (.)−1 , the matrix transpose and inverse. By replacing S by its estimate (GT (w)G(w))−1 GT (w)X in (8), the ML angular frequency estimator is given by w b0 = arg max J (w), (9)

2 J (w) ≈ N

N −1

X xa [k] cos(kw)

xa [k] sin(kw)

k=0

=2

3 X

xb [k] cos(kw) xb [k] sin(kw)

2 xc [k] cos(kw)

xc [k] sin(kw)

F

Pm (w),

m=1

(15) where Pm (w) corresponds to the periodogram of xm [k] and is defined as N −1 2 1 X −jωk Pm (w) , (16) xm [k]e . N k=0

By maximizing (15) instead of (10), the resulting estimator is called the DTFT angular frequency estimator. 2) Phasor estimator: Under Gaussian noise, the ML estimator of S also corresponds to the LS estimator of S. This estimator is given by [33], [35] −1 b S = G T (w b0 )G(w b0 ) G T (w b0 )X,

(17)

where w b0 corresponds to the ML (or DTFT) estimator of ω0 . Using the definition S, the ML estimator of the phasor is therefore given by [Vba , Vbb , Vbc ] = 1 −j b S. (18)

w

where the cost function J (w) is defined as J (w) , T r[XT G(w)(GT (w)G(w))−1 GT (w)X],

(10)

and T r[.] corresponds to the trace of an N-by-N square matrix, which is equal to the sum of the elements on the main diagonal. It is worth mentioning that the computation of J (w) requires the inversion of GT (w)G(w). Using the expression of G(w) in (6), GT (w)G(w) can be decomposed as [34] N −1 1 X 1 + cos(2kw) sin(2kw) GT (w)G(w) = sin(2kw) 1 − cos(2kw) 2 k=0 (11) N 1 0. For the DTFT estimator (b S = N2 GT (wn )X), the expression of the TVE is simply obtained by replacing M(wn , δ) with MDT F T (wn , δ) , I22 − N2 GT (wn )G(wn + δ) in (44).

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Zakarya Oubrahim was born in Errachidia, Morocco, in 1987. He received the Engineering diploma degree in electrical engineering from the University of Technology of Belfort´ Montbeliard, Belfort, France, in 2013 and the M.Sc. degree in electrical engineering from the University of Lyon I, Lyon, France, in 2014. He is now a PhD degree candidate working on smart grids at the University of Brest, Brest, France. His research interests include signal processing and statistics for power systems monitoring, smart grid solutions, measurement of power quality in multi-phase systems, control of renewable energy systems. Vincent Choqueuse (S08-M09) was born in Brest, France, in 1981. He received the Dipl.Ing. and the M.Sc. degrees in 2004 and 2005, respectively, from the Troyes University of Technology, Troyes, France, and the PhD degree in 2008 from the University of Brest, Brest, France. Since September 2009, he has been an Associate Professor with the Institut Universitaire de Technologie of Brest, University of Brest, Brest, France, and a member of the Institut de ˆ Recherche Dupuy de Lome - IRDL (FRE CNRS 3744). Dr. Choqueuse research interests include signal processing and statistics for power systems monitoring, Smart-Grid, digital communication and digital audio.

Mohamed El Hachemi Benbouzid was born in Batna, Algeria, in 1968. He received the B.Sc. degree in electrical engineering from the University of Batna, Batna, Algeria, in 1990, the M.Sc. and Ph.D. degrees in electrical and computer engineering from the National Polytechnic Institute of Grenoble, Grenoble, France, in 1991 and 1994, respectively, and the Habilitation a` Diriger des Recherches degree, in electrical engineering, from the University of Picardie ”Jules Verne,” Amiens, France, in 2000. After receiving the Ph.D. degree, he joined the Professional Institute of Amiens, University of Picardie ”Jules Verne,” where he was an Associate Professor of electrical and computer engineering. Since September 2004, he has been with the Institut Universitaire de Technologie of Brest, University of Brest, Brest, France, where he is a Professor of electrical engineering. Prof. Benbouzid is also a Distinguished Professor at the Shanghai Maritime University, Shanghai, China. His main research interests and experience include analysis, design, and control of electric machines, variable-speed drives for traction, propulsion, and renewable energy applications, and fault diagnosis of electric machines. Prof. Benbouzid is an IEEE Senior Member. He is the Editor-inChief of the International Journal on Energy Conversion (IRECON). He is also an Associate Editor of the IEEE Transactions on Energy Conversion, the IEEE Transactions on Industrial Electronics, the IEEE Transactions on Sustainable Energy, and the IEEE Transactions on Vehicular Technology.

Yassine Amirat was born in Annaba, Algeria, in 1970. He received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Annaba, Annaba, in 1994 and 1997, respectively. He was a lecturer at Annaba University from 2000 to 2010. He obtained the Ph.D. degree in wind turbine condition monitoring at the University of Brest, Brest, France in 2011. He is currently an Associate Professor of Electrical Engineering at ISEN Brest (France). He is also an affiliated member at the Institut de ˆ Recherche Dupuy de Lome IRDL (FRE CNRS 3744). His main research interests include electrical machines faults detection and diagnosis, fault tolerant control, and signal processing and statistics for power systems monitoring. He is also interested in renewable energy applications: wind turbines, marine current turbines and hybrid generation systems.


Maximum Likelihood Frequency and Phasor

Maximum Likelihood Frequency and Phasor

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