Adaptive linear combiners a robust neural network technique for on-line harmonic tracking

Abdelaziz Zouidi (1), Farhat Fnaiech (1) Senior Member IEEE

Kamal AL-Haddad (2) Fellow Member IEEE and Salem Rahmani(1)

(1) Equipe de recherche: Signal, Image et Commande Intelligente des Systèmes Industriels (SICISI)

Université de Tunis, Ecole Supérieure des Sciences et Techniques de Tunis, 5 Av. Taha Hussein, Tunis 1008, Tunisie Tel: (216) 71 496066; email: abdelazizzouidi@yahoo.com, Fnaiech@ieee.org & Salem.Rahmani@esstt.rnu.tn

(2) Canada Research Chair in Energy Conversion and Power Electronics CRC-ECPE,

Ecole de Technologie Supérieure, 1100 rue Notre-Dame Ouest, Montréal, Québec H3C 1K3, Canada Tel: (514) 396 8874; Fax: (514) 396 8684; email: kamal@ele.etsmtl.ca

Abstract- Intelligent techniques of harmonic detection or estimation are nowadays of a great interest in power system applications, their ability to deal with high non-linearities attract researchers to investigate the performance of these methods mainly based on artificial intelligence namely using artificial neural networks (ANNs). In the literature many harmonic detection or estimation methods were presented, in this paper we focus on adaptive linear neuron (ADALINE) to estimate the fundamental component and selected harmonic content of a distorted signal compared to the fast Fourier transformation (FFT) algorithm. Key words: selective harmonic detection, artificial intelligence, artificial neural networks

1. INTRODUCTION Active power filtering is the up-to-date technique used to decontaminate the electric power circuits by injecting equal but opposite sign of the harmonic signal [1], the most critical phase in this procedure is basically the detection or estimation of the harmonic compensating reference, any problem or malfunction extracting it, will lead certainly to failure. In the literature many algorithms were proposed for this purpose, the up-to-date methods are the intelligent ones based on artificial intelligence techniques namely artificial neural networks. The ANN on-line algorithm’s advantage is certainly their adaptive aspect, however in this case it’s necessary to have a significant mathematical model or representation of the distorted signal such as the discrete Fourier series expansion used in [4-5] where an adaptive linear neuron ADALINE is trained to detect selectively the harmonic individuals of a distorted current. The input pattern of the ADALINE is the expansion functions values, whereas the target values are the corresponding distorted current values. In fact using this algorithm under severe condition may show its reliability which will be the aim of this work.

In section 2 the ADALINE’S algorithm used in [4-5] is presented, with an explicit basic equations presentation and the architecture of the ANN. In section 3 the simulation results of a theoretical and a power system example are presented to conclude in the section 4.

2. THE ADALINE’S ALGORITHM BASIC EQUATIONS

The ADALINE was used in [4-5] to identify the coefficients of the discrete Fourier expansion of a distorted wave form; this allowed the determination of the amplitude and phase of the fundamental component and the selected harmonic components. The load current as a periodic quantity can be expressed by the Fourier series as:

)sin()sin()(2

11 nn

LnLt

dcL tnItIeAti Φ++Φ++= ∑∞

=

− ωωβ (1)

Where,

1LI and are the peak amplitude and the phase of the fundamental component respectively.

1Φ

LnI and are the peak amplitude and the phase of the nth harmonic component (n=2..N) respectively.

nΦ

t

dceA β− is the decaying dc current component where is the peak amplitude and

dcAβ is the decaying coefficient.

Let us consider the following notations:

)(tiLfu = (2) )sin( 11 Φ+tIL ω

)(tiLh = (3) )sin(2

nn

Ln tnI Φ+∑∞

=

ω

( ) tdcLdc eAti β−= (4)

With k as the sampling step and T the sampling period, the discrete time expression of the fundamental can be written as:

)(kTiLfu = (5) kTBkTA LL ωω cossin 11 +Where

111 cosΦ= LL IA (6)

111 sinΦ= LL IB (7) Hence

21

211 LLL BAI += (8)

⎟⎟⎠

⎞⎜⎜⎝

⎛=Φ

1

11 tan

L

LA

BArc (9)

Similarly for the nth harmonic component the discrete time expression will be:

kTnBkTnAkTi Ln

N

nLnLh ωω cossin)(

2+=∑

=

(10)

Where nLnLn IA Φ= cos (11)

nLnLn IB Φ= sin (12)

Consequently,

22LnLnLn BAI += (13)

⎟⎟⎠

⎞⎜⎜⎝

⎛=Φ

Ln

Lnn ABArctan (14)

Using the first order Taylor expansion the discrete time expression of the dc component can be written as:

( ) kTAAkTi dcdcLdc β−= (15) The general discrete expression of the load current is obtained by summing (2), (5) and (6). ( ) )()()( kTikTikTikTi dcLhLfuL ++= (16)

In a matrix form it can be written as:

[ ]⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

=

kTkTNkTNkTkTkTkT

AABABABAkTi dcdcLnLnLLLLL

1cossin

2cos2sin

cossin

...)( 2211

ωωωωωω

α M (17)

Aiming to extract the fundamental and the harmonic components from the load current, an ADALINE (Fig. 1) can be train choosing its input pattern vector x(k)as,

(18) ( )

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

=

kTkTNkTNkTkT

kTkT

kx

1cossin

2cos2sin

cossin

ωωωωωω

M

and it’s training target as the corresponding load current ( )kTIL . Once the training performance was met the weight

vector “w” will be given by:

[ ]tdcdcLnLnLLLL AABABABAw α...2211= (19)

Consequently the amplitudes and the phases of the fundamental and the harmonic components can be deduced according to:

( )nwnwILn 2)12( 22 +−= (20)

( )( )⎟⎠

⎞⎜⎝⎛

−=Φ 122tan nwnwArcn (21)

where, n=1…N. The amplitude and the decaying coefficient of dc component are deduced as:

( )12 += NwAdc (22)

( ))12(

22++= Nw

Nwβ (23)

Fig. 1. ADALINE as on –line selective harmonic component estimator As illustrated in Fig. 1 the harmonic content Adaline estimator is composed of 2N+2 inputs, the learning rule used is the Widro-Hoff delta rule. The training criterion considered for is the mean square error of ANNs outputs with respect to the target values [6].

3. SIMULAION

3.1 Case of theoretical example

In order to investigate the performance of the proposed algorithm we consider the fallowing distorted signal:

)(tf = + )(tf fu )(tfh )18

sin( πω += t

)9

27sin(04.0)6

5sin(07.0)9

3sin(15.0 πωπωπω ++++++ ttt

)18719sin(02.0)313sin(04.0)18

511sin(05.0 πωπωπω ++++++ ttt

(34) Where, and are the fundamental and the harmonic components respectively, given by (35) and (36).

)(tf fu )(tfh

)18sin()( πω += ttf fu (35)

)927sin(04.0)65sin(07.0)93sin(15.0)( πωπωπω +++++= ttttfh

)18719sin(02.0)313sin(04.0)18

511sin(05.0 πωπωπω ++++++ ttt

(36) The angular frequency ω=2πf rad /s where f= 50Hz. The sampling frequency considered was as (50x80) Hz. 3.1.1 Case of theoretical example under deterministic conditions In this case the values of the simulated distorted signal given by (34) are considered as the Adaline’s learning targets without any assumed measurement noise.

Fig. 2. Simulated waveform and its harmonic content

Fig.3. Estimated fundamental amplitude and phase of the fundamental, 3rd and 7th harmonics using the Adaline’s algorithm.

Fig.4. Estimated fundamental amplitude and phase of the fundamental, 3rd and 7th harmonics using the FFT algorithm.

(a) (b) Fig.5. Convergence performances estimating the fundamental amplitude and phase angle, using Adaline (a) and FFT (b)

(c) (d) Fig.6. Convergence performances estimating the 5th harmonic amplitude and phase angle, using the Adaline (c) and the FFT (d) The inspection of the simulation results in figures 3-6 confirms the reliable performance of the Adaline’s algorithm to estimate the fundamental component and its harmonic content compared to the FFT method. 3.1.2 Case of theoretical example under stochastic conditions In order to make sure of the robustness of the Adaline’s selective harmonic detector, simulations were held by adding a random noise of variance 0.002 to the distorted signal given by (34). In the following the registered results by the Adaline and the FFT method are presented.

( e ) ( f ) Fig.7. Estimated fundamental amplitude and phase and 3 using the Adaline (e) and the FFT algorithm (f)

(e1) (f1) Fig.8. Estimated amplitude and phase of the 3rd harmonic using the Adaline (e1) and the FFT algorithm (f1) Obviously when observing the figure above one may deduce the robustness of the adaptive linear combiner (Adaline) in on-line harmonic tracking in sever conditions such as measurement noises and the ability of the neural harmonic tracking system to reject noise. 4.2 Case of a simulated per-phase load current of a three phase diode bridge circuit Aiming to show the reliability of the ANN algorithm in on-line harmonic tracking, simulation were carried on for tracking the harmonic content of distorted load current, the characteristics of the simulated system are represented in table I. In fig. 9 the load current and its harmonic content are viewed whereas fig. 10 show the performance of the Adaline algorithm tracking the load current and its total harmonic content.

TABLE I SIMULATED SYSTEM CHARACTERISTICS

Three-phase System

)(6.155 rmsVVs = , frequency =50Hz

, Ω= 1.0sR mHLs 01.0=

Load Diode bridge through an RL load with

, Ω=12LR mHLL 10=

Fig.9. The load current and its harmonic content

Fig. 10. Performance of the Adaline algorithm estimating the load current and its total harmonic content

Fig. 11. Performances of the Adaline algorithm estimating the fundamental amplitude and phase respectively compared to those using the FFT

Fig. 12. Performances of the Adaline algorithm estimating the 3rd harmonic amplitude and phase respectively compared to those using the FFT The inspection of the simulation results in figures 10-12 confirms the reliability of the Adaline algorithm to estimate the load current fundamental component and its harmonic content.

4. CONCLUSION An artificial neural network based harmonic detection algorithm was presented in this paper; the originality of this work is to focus on the robustness of an ADALINE in on-line tracking of the fundamental and harmonic content of distorted signal. The simulation results obtained confirm success of this algorithm in terms of harmonic detection. In fact this performance has to be tested for active power system purposes experimentally in further work so to make sure of the advantages presented.

REFERENCES

[1] A. Emadi, A. Nasiri and S. B. Bekiarov:”Uninterruptible Power Supplies and Active Filters”, CRC PRESS 2005. [2] A. Zouidi, F. Fnaiech, K. Al-Haddad and S. Rahmani” Artificial Neural Networks as harmonic detectors”, IEEE-IECON 2006 Paris, France, November 7, 2006. [3] N. Pecharanin, H. Mitsui and M. Sone:” Harmonic detection by using neural network”, Neural networks, 1995 Proceedings, IEEE international conference on, Vol. 2, pp. 923-926, 27 Nov. – 1 Dec. 1995. [4] P. K. Dash, D. P. Swain, A. C. Liew and S. Rahman:” An adaptive linear combiner for on-line tracking of power system harmonics”, IEEE trans. On power systems, Vol. 11, No. 4, pp. 1730-1735, Nov. 1996. [5] P. K. Dash, S.K. Panda, B. Mishra, and D. P. Swain,:” Fast estimation of voltage and current phasors in power networks using an adaptive neural network”, IEEE trans. On power systems, Vol. 12, No. 4, pp. 1494-1499, Nov. 1997. [6] B. Widrow and M. A. Lehr:” 30 years of Adaptive neural networks: Perceptron, Madaline and Back propagation», proceeding of the IEEE, Vol 78, No. 9, September 1990.