(2)

where 0

is the notch frequency. Notch filter extracts

fundamental sinusoid from distorted current waveform

without harmfully phase shifting of the high-order har-

monics. The ideal notch filter has zero bandwidth. How-

ever, zero bandwidth cannot be realized in practice.

The most simple type of adaptive n otch digital filter is

adaptive line enhancer (ALE) proposed by B. Widrow

[6]. The structure of ALE is shown in Figure 2.

The adaptation of the finite impulse response (FIR)

filter is realized by using the least mean square (LMS)

algorithm. Disadvantages of this ALE are a relatively

low convergence speed and potential instability.

An infinite impulse response (IIR) filter provides a

sharper magnitude response than the FIR adaptive line

enhancer. Also it requires much smaller filter length,

than the ALE based on FIR filter.

The transfer function of the second order notch IIR

filter is defined as:

21

2

1

1zaz

Hz zaz

2

(3)

where

is the pole zero contracting factor. In general,

should be close to unity to well approximate Equa-

tion (2).

As shown in [7] the transfer function of a single fre-

quency notch filter can be expressed in the form:

11

2

H

zA

z

(4)

where

A

z represents a transfer function of the all-

pass IIR filter.

The structure of notch filter based on all-pass IIR filter

is presented in Figure 3.

A lattice-form realization of all-pass transfer function

is shown in Figure 4.

Figure 2. Structure of ALE.

Figure 3. Structure of the notch filter.

Copyright © 2013 SciRes. EPE

S. A. TEMERBAEV ET AL.

Copyright © 2013 SciRes. EPE

1117

of its low complexity and high-speed convergence. Up-

date of the coefficients 1 and , using gradient algo-

rithm, is given as follows:

k2

k

11

2

11

iiii ii

i

knknenr nenrn

Dn

is an adaptation step. Parameter

i

Dn

where

is

defined: as

Figure 4. All-pass lattice IIR filter.

22

11

1

iiii

nDn en rn

D

In Figure 4 x(n) and y(n) are input and output signals,

respectively. Transfer function of the lattice IIR filter is

the following:

is a forgetting factor: 01

. where

5. MATHLAB-Based Simulation

b/Simulink to

APF is analyzed by consider-

in

21

12

21

212

1

11

Yzzkk zk

Az Xz kzkk z

2

(5) The system was simulated using MathLa

verify the proposed algorithm. Schematic diagram of the

proposed controlled shunt APF is shown in Figure 6.

The linear load is defined as resistance Rlin = 100 Ohms,

non-linear load includes two rectifiers with RL load on

the dc side. Simulation process is divided into steps:

connection of the first rectifier, connection of APF and

connection of the second rectifier. All simulation process

is presented in Figure 7.

The performance of the

The polynomials of nominator and denominator of

Equation (5) have mirror symmetry. Accordingly, lattice

IIR-filter realizes all-pass transfer function with module

equal 1 in the all frequency range.

Transfer function of notch filter, shown in Figure 3 is

presented as:

21

1

2

212

211

1

211

zkz k

Hz kzkk z

2

1

(6) g of the following cases.

where 1 is the adaptive coefficient, which should con-

verge to 0

kcos

to reject a sinusoid with frequency

0

. Frequency suppression of notch filter can be modi-

fied by and stopband widt h by.

1 2

Adaptive IIR filter in Figure 3 is adapted using adap-

tive algorithms related to the lattice FIR filters. The

structure of the lattice second-o rder FIR filter is shown in

Figure 5. In this article gradient lattice algorithm [8] is

used for adaptation purposes. It has been chosen because

k k

Figure 5. FIR lattice filter.

Figure 6. MATHLAB scheme of shunt APF system.

S. A. TEMERBAEV ET AL.

1118

Figure 7. Simulation results.

.1. Case 1

Figure 6 in t = 0.1 sec the first rectifier is

5.2. Case 2

load is increased in t = 0.5 sec. Proposed

6. Conclusions

el adaptive method for grid current

Table 1. THD before and after 0.3 sec.

5

As shown in

connected to the grid. In the 0.3 sec the shunt APF is

connected to the grid and starts compensating harmonics

component of the non-linear load current. Changing of

THD is demonstrated in the Table 1. Grid, linear load

and non-linear load currents are presented in Figure 7.

The nonlinear

technique of calculation compensation signal operates

properly without severe transients at the instants of step

load chang e. THD is presented in Table 2.

In this paper, a nov

harmonic compensation is proposed. The load harmonic

compensation was performed by using the lattice-form

adaptive notch IIR filter. It was shown that adaptive

THD %

Signal nameBefore After

I g 11.29 1.35

I lin 5.44 0.64

I nonl 34.79 40.32

Table 2. THD before and after 0.5 sec.

THD %

Signal nameBefore After

I g 1.35 1.98

I lin 0.64 1.14

I nonl 40.32 40.28

Copyright © 2013 SciRes. EPE

S. A. TEMERBAEV ET AL. 1119

notch filtetive har-

monic filter for the sake of hic mitigation

prooach does not neey training o

notch filter. Performathe proposed l system

is verified by computer simulation. MATLAB/SIM

LINK power system tis used to sime the pro-

posed system. The ults presented

showing the effectiveness of the proposed method.

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Copyright © 2013 SciRes. EPE