Paper Title (use style: paper title)

6 downloads 0 Views 134KB Size Report
In this paper the Forgetting Factor RLS (FFRLS) approach has been considered to estimate not only voltage sag,swell,momentary interruption but also the ...
ROBUST HARMONIC ESTIMATION USING FORGETTING FACTOR RLS H.K.SAHOO

POOJA SHARMA

Dept. of Electronics and Telecommunication IIIT, Bhubaneswar India Email: [email protected]

Dept. of Electronics and Telecommunication BIT, Mesra, India Email: [email protected]

N.P.RATH Dept. of Electronics and Telecommunication VSSUT, Burla, India Email: [email protected] Abstract— The prime reasons for power quality degradation include voltage sag, swell and momentary interruptions and also the presence of harmonics. Thus accurate computation of harmonics is really a challenging problem in power system. Many algorithms have been proposed for harmonic estimation to improve the power quality. In this paper the Forgetting Factor RLS (FFRLS) approach has been considered to estimate not only voltage sag,swell,momentary interruption but also the amplitudes and phases of harmonics in case of time varying power signals in presence of White Gaussian Noise. Also comparison results with LMS and NLMS algorithms are presented to show the effectiveness of the proposed RLS algorithm. Keywords- RLS, LMS, NLMS, Harmonic Estimation, White Gaussian Noise I.

INTRODUCTION

The wide spread applications of electronically controlled loads have increased the harmonic distortion in power system voltage and current waveforms. As power semiconductors are switched on and off at different points on the voltage waveform, damped high frequency transients are generated. If switching occurs at the same points on each cycle, the transient becomes periodic. This transient whose frequency is not a multiple of fundamental frequency is non-stationary. Consequently voltage and current waveforms of a distribution or transmission system are not pure sinusoids, but may consist of a combination of fundamental frequency, harmonics and high frequency transients. Also many of power system loads, especially industrial loads are dynamic in nature, which implies time varying amplitude of the current waveform.

In order to provide the quality of the delivered power, it is imperative to know the harmonics parameters such as amplitude and phase. This is essential for designing filter for eliminating or reducing the effects of harmonics in a power system. Different algorithms are proposed to estimate the harmonics[1,2] in a power system.LMS[3,4,5] and NLMS[6] approaches are also quite popular for estimating frequency of distorted sinusoidal signals under noisy conditions In this paper, Fogetting Factor RLS has been proposed for estimating sag, swell, momentary interruption as well as amplitudes and phases of different harmonics [7] of distorted power signals in presence of white noise. II. SIGNAL MODELS FOR POWER QUALITY DISTURBANCES AND HARMONIC ESTIMATION Two types of signal models are proposed to estimate power quality disturbances like voltage sag and swell, notch, momentary interruption as well as amplitudes and phases of different harmonics like fundamental, third and fifth harmonics. A. Signal Model for Power Quality Disturbances Consider a signal d k at time k is a sinusoid yk in the presence of white Gaussian noise vk .

d k = yk + vk

(1)

yk = a1 sin(kω1Ts + φ1 )

(2)

Where

ω1

B. SIGNAL MODELING FOR HARMONIC ESTIMATION

= fundamental of angular frequency;

A static signal and a dynamic power system signal have been considered for estimation, which contains higher harmonics of the 3rd and 5th order.

φ1 = fundamental of phase angle; a1 = fundamental amplitude of the signal;

The static power system signal is given by

Ts = sampling time

yk = 1.2sin(kωTs + π / 6) + 0.5sin(3kωTs + π / 3)

The noise vk is a white Gaussian noise with a zero mean and a variance σ v .

+0.2sin(5kωTs + π / 4)

(8)

2

The dynamic power system signal is given by

yk = (1.5 + a1 (t )) sin( kωTs + π / 6)

So the signal can be modeled as:

+ (0.5 + a3 (t )) sin(3kωTs + π / 3)

yk = xk wkT

(3)

Where

+ (0.2 + a5 (t )) sin(5kωTs + π / 4) Where

xk = [sin(ω1kTs )

cos(ω1kTs )]

(4)

a1 (t ) = 0.15 sin 2π f1kTs + 0.05 sin 2π f 5 kTs a3 (t ) = 0.05 sin 2π f 3 kTs + 0.02 sin 2π f 5 kTs

wk = [ a1 cos(φ1 )

a1 sin(φ1 )]

(5)

Here the desired signal d k and the sinusoid yk are known. Through an iterative process the weights are determined and used to estimate the amplitude and phase of the sinusoid in presence of the disturbance vk .The parameters of amplitude Λ

(9)

Λ

a1 and phase φ1 can be optimally estimated by applying the observed signal d k by passing it through the adaptive filter as follows: Λ

ak = wk2(1) + wk2(2)

(6)

Λ

φk = a tan( wk (2) / wk (1) )

(7)

(10)

a5 (t ) = 0.025 sin 2π f1kTs + 0.005 sin 2π f 5 kTs The static signal can be reconstructed as follows:

yk = xk wkT Where

⎡sin kωTs ⎤ ⎢ ⎥ ⎢ cos kωTs ⎥ ⎢sin 3kωTs ⎥ xk = ⎢ ⎥ ⎢ cos 3kωTs ⎥ ⎢sin 5kωT ⎥ s ⎢ ⎥ ⎢⎣ cos 5kωTs ⎥⎦

T

(11)

⎡ a1 cos Φ1 ⎤ ⎢ a sin Φ ⎥ 1 ⎥ ⎢ 1 ⎢ a3 cos Φ 3 ⎥ wk = ⎢ ⎥ ⎢ a3 sin Φ 3 ⎥ ⎢ a cos Φ ⎥ 5 ⎢ 5 ⎥ ⎢⎣ a5 sin Φ 5 ⎥⎦

III.

T

(12)

PROPOSED APPROACH

Forgetting Factor RLS is a simple approach with a very high rate of convergence. It whitens the input data using inverse correlation matrix of the data. It includes prior information about the input-output mapping. Step1) Initialize weight and inverse correlation matrix Λ

w(0) = 0 And for dynamic signal the weight vector can be defined as:

⎡ (1.5 + a1 (t )) cos Φ1 ⎤ ⎢ (1.5 + a (t )) sin Φ ⎥ 1 1 ⎥ ⎢ ⎢ (0.5 + a3 (t )) cos Φ 3 ⎥ wk = ⎢ ⎥ ⎢ (0.5 + a3 (t )) sin Φ 3 ⎥ ⎢ (0.2 + a (t )) cos Φ ⎥ 5 5 ⎢ ⎥ ⎢⎣ (0.2 + a5 (t )) sin Φ 5 ⎥⎦

T

δ

= small positive constant for high SNR and large positive constant for small SNR (13) Step2)

Λ

Λ

The parameters of amplitudes a1 a3 , a5 and phases , Λ

Step4)

ε (k ) = d (k ) − w(k )T x(k )

Step5)

w(k ) = w(k − 1) + u (k )ε (k )

Step6)

P (k ) = λ −1 P(n − 1) − λ −1u (k ) x(k ) P(k − 1)

Λ

φ1 , φ3 , φ5

π (k ) = P(k − 1) x(k )

Step3) u ( k ) = π ( k ) / (λ + x ( k )π ( k )) Λ

Λ

P(0) = δ −1 I

can be optimally estimated by applying the

Λ

Λ

observed signal d k to the adaptive filter as follows: Λ

ak (1) = wk2(1) + wk2(2) Λ

ak (3) = w

2 k (3)

+w

2 k (4)

Where P ( k ) =Inverse correlation matrix (14)

u (k ) =Gain vector

Λ

ak (5) = wk2(5) + wk2(6)

λ =forgetting factor Λ

φk (1) = a tan( wk (2) / wk (1) ) Λ

φk (3) = a tan( wk (4) / wk (3) )

(15)

IV.

SIMULATION RESULTS

Λ

φk (5) = a tan( wk (6) / wk (5) ) wk =weight vector and xk =input vector

The performance of the filter is shown through MATLAB simulations. For this purpose two test signals are considered. First test signal is taken to show the tracking capability of the filter when the signal is distorted with sag, swell and momentary interruption. The second signal is considered to show the tracking capability of the filter when the signal is distorted with higher order harmonics. All the simulations are performed in presence of White Gaussian Noise using

MATLAB. The distorted signals are sampled at 1.2 KHz and SNR of 30 dB is considered for simulation. Fundamental frequency of 50 Hz and amplitude 1p.u. is considered for the power signal.

RLS LMS NLMS

1.6 1.4

Estimated Amplitude

Test Signal 1:

1.8

A sin(kωTs + φs ) + vk

Condition A: Swell and Sag: The signal with an amplitude of 1 put. was taken and a swell occurs between 0.2 s to 0.4s and a sag occurs between 0.6s to 0.8s. The estimated amplitude is shown in Fig.1.

1.2 1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time in Seconds

yk = (1.5 + a1 (t ))sin(kωTs + π / 6) Test Signal 2:

+(0.5 + a3 (t ))sin(3kωTs + π / 3) +(0.2 + a5 (t ))sin(5kωTs + π / 4) + vk

The estimated amplitudes of fundamental, 3rd and 5th harmonics of test signal 2 and test signal 3 are given in Fig.3, Fig.4 and Fig.5 and estimated phases are given in Fig. 6,Fig.7 and Fig.8.

Figure 2. Signal with ConditionB(Swell and Momentary Interruption)

1.8

Estimated Amplitude(Fundamental)

Condition B: Momentary interruption and swell: Here a swell occurs in between 0.2s and 0.4s and momentary interruption occurs in between 0.6s to 0.8s. The estimated amplitude is shown in Fig.2.

1.6 1.4 1.2

RLS LMS NLMS

1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time in seconds 3.5

0.9

2.5

2

1.5

1

0.5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time in Seconds

1

Estimated Amplitude (Third Harmonic)

Estimated Amplitude

Figure 3. Estimated Fundamental Harmonic Amplitude

LMS RLS NLMS

3

0.8

0.7

0.6

0.5

0.4 LMS RLS NLMS

0.3

0.2

0.1

Figure 1. Signal with Condition A(Swell and Sag)

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time in seconds

Figure 4. Estimated Third Harmonic Amplitude

0.9

1

1.5

Estimated Phase(Fifth Harmonic)

0.9

Estimated Amplitude(Fifth Harmonic)

0.8

0.7

0.6

0.5

RLS NLMS LMS

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

RLS LMS 1

0.5

0

-0.5

-1

1

Time in seconds

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time in seconds Figure 8. Estimated Fifth Harmonic Phase

Figure 5. Estimated Fifth Harmonic Amplitude

V.

Estimated Phase(Fundamental)

1.3 1.2

In case of FFRLS, the estimation accuracy depends on optimal choice of forgetting factor. In this proposed approach, constant forgetting factor is chosen to estimate swell, sag, momentary interruptions as well as harmonic amplitudes and phases.The forgetting factor can be tuned using certain optimization method to improve the tracking capability of the adaptive filter in case of nonstationary signals.

RLS LMS

1.1 1 0.9 0.8 0.7

REFERENCES 0.6 0.5

[1]

.Joorabian, S.S. Mortazavi, A.A. Khayyami, “Harmonic Estimation in a Power System using a novel Hybrid Least Squares-Adaline Algorithm” Elsevier Science Electric Power System Research 79(2009) 107-116.

[2]

P.K.Dash, D.P.Swain, A.Routray, A.C.Liew, “Harmonic Estimation in a power system using adaptive perceptrons”, IEE Proc-Gener. Transm. and Distb. ,Vol. 143(6),pp. 565-574, Nov.1996

[3]

Tandon, A.; Ahmad, M.O.; Swamy, M.N.S.; “An efficient, lowcomplexity, normalized LMS algorithm for echo cancellation”, IEEE workshop on Circuits and Systems, NEWCAS 2004, pp. 161-164, June 2004.

[4]

M. Tarrab and A. Feuer., “Convergence and Performance Analysis of the Normalized LMS Algorithm with Uncorrelated Gaussian Data” IEEE Trans. on Inform. Theory, Vol.34(4) , pp. 680-691, 1988.

[5]

Yegui Xiao and Yoshiaki Tadokoro, “ LMS-based notch filter for the estimation of sinusoidal signals in noise” Volume 46(2), pp.223-231, October 1995.

[6]

Lee, K.A.; Gan,W.S; “Improving convergence of the NLMS algorithm using constrained subband updates,” Signal Processing Letters IEEE, vol. 11, pp. 736-739, Sept. 2004.

[7]

Maamar Bettayeb and Uvais Qidwai, “ Recursive estimation of power system harmonics” Electric Power System Approach Volume 47(2), ,pp.143-152 , October 1998

0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time in seconds Figure 6. Estimated Fundamental Harmonic Phase

1.5

Estimated Phase(Third Harmonic)

CONCLUSION AND DISCUSSION

LMS RLS 1

0.5

0

-0.5

-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time in seconds Figure 7. Estimated Third Harmonic Phase

0.9

1