apf458

7/25/2019 apf458

1/9

World Applied Sciences Journal 4 (1): 124-132, 2008

ISSN 1818-4952

IDOSI Publications, 2008

Corresponding Author: Majid Nayeripour, Department of Electrical Engineering, Shiraz University of Technology, Shiraz,Iran

124

Nonlinear Sliding Mode Control Design for Shunt Active Power Filter

with the Minimization of Load Current

Majid Nayeripour and Taher Niknam

Department of Electrical Engineering, Shiraz University of Technology, Shiraz, Iran

Abstract: This paper proposes a new method of sliding mode base controller for shunt active power filter

to compensate the harmonic currents of a load. In this controller there isn't any need to PI DC voltage

controller to regulate the DC capacitor voltage of shunt active power filter. In this method, first the

reference currents of shunt active power filter that should be tracked to reduce or eliminate the harmonic

currents of line and power loss are derived based on Lagrange function. In the Lagrange function, the active

components of load currents are minimized with the constraint of three-phase reactive power to zero. Then,

three sliding surfaces are defined such that three-phase currents and voltage of capacitors in shunt active

power filter are gone to reference values. The coefficients of sliding surfaces are derived such that thesliding surfaces go to zero with fast dynamic and more error reduction than conventional sliding mode

controller.

Key words: Shunt active power filter harmonic inverter sliding mode control

INTRODUCTION

Nonlinear loads create current harmonics in the

case of sinusoidal or non-sinusoidal voltages. These

current harmonics cause to voltage harmonic drops and

power loss in the impedance of lines [1]. Active and

Passive filters can be used to comp ensation, if these

harmonics or their powers are known.Passive filters are designed for the elimination of

one current harmonic however; the active filters are

able to compensate total current harmonics.

Current harmonics of a load can be generated by an

active filter comprised of a PWM inverter and a DC bus

capacitor. The capacitors of this bus are charged

through anti parallel diodes with switches, which define

reference capacitor voltages (Fig. 1).

Various methods are presented in the literature to

control of a shunt active filter.

In [2] two sliding surfaces, corresponding to d-axis

and q-axis current errors are used to trace the reference

currents for compensation of current harmonics by

three-phase active power filter with sliding mode

controller.

In [3] the compensation of load current harmonics

is performed using a three phase shunt active power

filter with a null wire and sliding mode controller. In

that paper dq0-axis currents are chosen as the state

variables and three sliding surfaces are defined

corresponding to the errors of these state variables.

Although the actual dqo-axis currents follow their

reference currents, there are no controls on the voltage

of capacitors, which may be lead to large capacitor

voltage fluctuation and instability.

In this paper, two different methods of

compensation are discussed using the definition of

power in three-phase systems. Next with the

minimization of active component of load current, thereactive component of power is eliminated and power

transfer is improved. Finally with a modified sliding

mode controller, three sliding surfaces are defined such

that the error of three-phase currents and voltage of DC

capacitors will be go to zero.

1CV

fci

fbi

fai

R

R

R

L

L

4u

5u

6u

3u

2u

1u

2CV

1u6u

abcV1 abcV

WireNull

abcV

abcI

L

Fig. 1: Three-phase shunt active powe r filter

7/25/2019 apf458

2/9

World Appl. Sci. J., 4 (1): 124-132, 2008

125

POWER DEFINITIONS

In the sinusoidal voltage case, the instantaneous

power, active and reactive powers of a signal phase

linear load are defined as]4[

:

{ }

a a m m

m m

m m m m

p(t) v (t)i (t) v i sin( t)sin( t )

v icos cos(2 t )

2

v i v iP cos , Q sin

2 2

= =

=

= =

(1)

The above equations for a three-phase linear load

with sinusoidal voltages are represented as:

m ma a b b c c

m m m m

v ip(t) v (t)i (t) v (t)i (t) v (t)i (t) 3 cos

2

v i v iP 3 cos , Q 3 sin

2 2

= + + =

= = (2)

In this case the instantaneous power is time

independent and equal to average active power.

In the non-sinusoidal voltage case, the

instantaneous power for signal phase and three-phase

systems are defined as:

a a b b c c

p(t) v(t)i (t)

p(t) v (t)i (t) v (t) i (t) v (t)i (t )

== + +

(3)

Average value of p(t) is defined as the active powerdrawn from the source. The instantaneous power minus

the average power is the power that oscillates between

source and load (p)%

In this case, power definitions can be considered in

the time or frequency domains.

In the frequency domain the rms value of voltage

and current, apparent power, active and reactive powers

are defined as respectively:

1 2 2V v dt VnT

= =

1 2 2I I dt InT= = (4)

S VI

P p V I cosn n n n

Q q V I sinn n n n

== = = =

In this method the reciprocal effects of harmonics

are not considered and for balancing of the powers, the

harmonic power is defined as:

2 2 2D S P Q= (5)

It is clear that using of this method for active or

passive filters is not appropriate because the effect of

voltage and current harmonics with each other are notconsidered [5].

In the time domain, the apparent power, active andreactive powers are defined as:

S VI

1P v(t)i(t)dt

T

2 2Q S P

=

=

=

(6)

In this method, the part of instantaneous power

which doesn't contribute to active power is considered

as the reactive power. This paper uses the time domainanalysis to derive the reference currents.

POWER COMPENSATION

In this paper, we use two methods for power

compensation

A. Compensation of power in dqo frameB.

Compensation with minimization of active

component of load current

In the first method, dqo component of voltage and

current can be obtained from abc-phase by the Clark [5]

transformation.

q a

d b

co

1 10

2 2i i2 3 3

i 1 i3 2 2

i1 1 1

2 2 2

i

=

(7)

Phase voltages can be also transformed dqo-axis

values by similar equations.The instantaneous power is equal to

( ) ( )

( )

a a b b c c

d d q q o o o

d d q q o o

d q q d

T

a b c b c a

c a b

p(t) v (t)i (t) v (t) i (t ) v ( t ) i ( t )

3(v i v i ) 3v i P P

2

3p(t) ( v i v i ), p (t) 3v i

2

1 3p p(t) p(t)dt , q(t) (v i v i )

T 2

v (t) v (t) i (t) v (t) v (t) i (t)1

3 v ( t ) v (t) i (t)

= + +

= + + = +

= + =

= =

+ = +

o

%

(8)

7/25/2019 apf458

3/9

World Appl. Sci. J., 4 (1): 124-132, 2008

126

We can rewrite the above equation as:

q d q

d q d

o oo

v v 0 ip3

q v v 0 i2

p i0 0 2v

=

(9)

In this method one can compensate the p,q% and p0

individually. For example, to compensate the p,q% and

p0 the reference currents of active power filter in a

three-phase system without null wire should be:

~q dcq

02 2

d qcd d q

v vi 2 / 3 p p

v vi v v q

= + (10)

These reference currents in abc-phase will be:

cqca

cb cd

cc

11 0

2ii

2 1 3 1i i

3 2 2 2i 0

1 3 1

2 2 2

=

(11)

In the second method, each phase current is

decomposed into active and reactive components and

minimizes the Lagrange function as fallows:

k wk k i i i K a,b,c= + = (12)

Minimize:

2 2 2a a b b c cL (i i ) (i i ) (i i ) = + + (13)

With the Constraint of:

a b c a a b b c c(i , i ,i ) v i v i v i 0 = + + = (14)

It will be leaded to:

a a a

b b b2 2 2a b c

c cc

i i vp( t)

i i vv v v

i vi

= + +

(15)

wa a

b b2 2 2a b c

cc

i vp(t )

i vv v v

vi

= + +

(16)

a a b b c cp(t) v (t)i (t) v (t) i (t) v (t)i (t )= + +

DYNAMIC EQUATION OF SHUNT ACTIVE

POWER FILTER AND CONTROLLER DESIGN

Figure 1 shows the topology of a three-phase shunt

active power filter. If we use the null wire, the zero

sequence current can be compensated by this null wire.

In the controller design it is assumed that the phase

voltages are symmetrical and if they are not

symmetrical, positive sequence of phase voltages and

reference currents can be determined with a Phase

Locked Loop (PLL) [6]. Control circuit contains the

sliding based on controller that uses three sliding

surfaces to track the three phase currents from their

references. Block diagram of this controller is

shown in Fig. 2.

To eliminate the harmonics of currents, the activepower filter currents fa fb fc(i ,i ,i ) should be following the

reference currents. Switching control variables

a b c(u ,u ,u ) , are defined such that if the upper switch is

closed u=-1, if the lower switch is closed u=1 and if

both switches are open u=0.

Considering the state variables as:

1 fa 2 fb

3 fc 4 c1 5 c2

X X

X X

i , i

i , v , X v

= == = =

(17)

The state equations are shown in (18) and

summarized as (19) [7]:

1 1

2 2

3 3

4 4

5 51 1 1

2 2 2

4 5

4 5

4 5

R 1 10 0

L 2L 2L

R 1 1X X0 0

L 2L 2LX X

R 1 10 0X X

L 2L 2LX X1 1 1

0 0X X2C 2C 2C

1 1 10 0

2C 2C 2C

1(X X ) 0 0

2L 10 (X X ) 0

2L

10 0 (X X )

2L

1

2

= +

+

+

+

& &

& &

& &

& &

& &

a

ab

b

cc

1 2 11 1 3

1 2 32 2 3

Lu

uL

u1 1X X X

C 2C 2C L

1 1 1X X X

2C 2C 2C

+

(18)

X AX B(X)u C

= + + (19)

7/25/2019 apf458

4/9

World Appl. Sci. J., 4 (1): 124-132, 2008

127

In the design of controller for equation of (19),

three sliding surfaces based on PI controller input and

the error of capacitor voltage are defined in (20). In this

design, without any need to regulate the capacitor

voltage by PI controller, all state variables are gone tozero.

[ ]Ta b c = (20)

s ( X X ) I (X X )dta 1 1 1 1 1 1m

s (1 sgnv )(X X )a4 4 4

ms (1 s gnv )(X X )a5 5 5

s (X X ) I2 2 2 2b

m(X X )dt s (1 s gnv (X X )2 2 6 4 4bms (1 sgn v )(X X )

7 5 5b

s (X X )c 3 3 3m

I (X X )dt s (1 sgn v )(X X )c3 8 4 43 3

s (1 sgn v )(Xc 58

= + + + +

= +

+ + +

= + + +

+ mX )5

Indeed~ ~

0 represents a set of nonlinear

differential equation whose unique solution is X-X*0,

with initial conditions, the problem of tracking the state

variable can be constrained to that of keeping the scalar

quantity (t) at zero which depend on m coefficient.In order to evaluate the design parameters, the

RMS value of state variables error signals areminimized. The control variables u are defined as:

T

a b cX X 0u sgn sgn sgn = (21)

These control variable force the state errors

towards the sliding surfaces and hold the state variables

at the o rigin ( )0, 0 = =& .

The necessary condition to reach the sliding

surfaces to zero in a finite time is that:

t( 0) & (22)

For m = 1 we have:

t( 0) S(X X )

S(AX B(X)U C) SX

(SAX SB(X)u SC SX )

=

= + +

= + +

&&

(23)

From (22) and (23) we have:

a b 3 a c

2

2 4 5 2 b a2

2 b b 3 b c1 2

4 4 5 32

1 1t s (X X ) X sgn( )a a51 4 12L 2C1

1 1X sgn( ) X sgn( )2

2C 2C2

1 1s (X X ) X sgn( )

2L 2C

1 1X sgn( ) X sgn( )

2C 2C

1 1s (X X ) X

2L 2C

= + +

+ +

+ + +

+ +

+ + +

&

c b

2 c b 3 c c2 1

sgn( )

1 1X sgn( ) X sgn( ) 0

2C 2C

+ +

(24)

If we divide the first and second part of (24) by

a a( sgn( )) , we have :

( )

( )

( )

( )

b1 4 5 1 2

1 1 a

c3

1 a

sgn1 1 1s (X X ) X X

2L 2C 2C sgn

sgn1X 0

2C sgn

+ +

(25)

So the condition of stability is :

( )

( )

( )

( )b c

1 1 2 31 1 a 1 a 4 5

sgn sgn1 1 1 2Ls X X X

2C 2C sgn 2C sgn (X X )

+

(26)

The same condition is derived for S2and S3.

For m 1, the error of sliding surfaces are differentfrom the error of sliding surfaces for m = 1 and we can

determine the value of m that errors are minimized. In

this case, the sliding surfaces in (20) approach to zero

fast and stability conditions are satisfied too. It is

required that the effect of load variation in the sliding

surfaces (20) is considered. So the coefficient of m can

be evaluated such that the sliding surfaces are

minimized.

Block diagram of a shunt active power filter is

shown in Fig. 2. In this block diagram, the reference

currents are derived using Lagrange method forminimization of load current.

The state variable references and actual values are

the input to the sliding mode controller.

In the proposed method in this paper, the capacitor

voltage error with exponent m and current errors are

considered as the sliding surfaces.

For some specific values of m, the state variable

errors in (20) approaches to zero very fast and the terms

of integral in (20) can be ignored. By eliminating these

7/25/2019 apf458

5/9

World Appl. Sci. J., 4 (1): 124-132, 2008

128

terms from (20), the errors are reduced and dynamic

response is improved.

SIMULATION RESULTS

Figure 3-7 show the simulation results of different

compensation methods. Figure 3 depicts the phase

voltages, load currents and inverter reference currents

for compensation with the minimization method of load

current active component.

Figure 4 shows the phase voltages, load currents

and inverter reference currents, with compensation of

instantaneous active power harmonics and

compensation of instantaneous reactive power with null

wire. Simulation results of Fig. 4 without null wire are

shown in Fig. 5.

Figure 6 and 7 shows the simulation results for

compensation of instantaneous active and reactive

faI

fbI

2cV

asigma

bsigma

csigma

fbIfaI

1c

V2c

V

aU

bU

cU

1cV

fcI

aI

bI

cI

aV

bV

cV

fcI

aVbVcVfcI fbI faI

Fig. 2: Block diagram of shunt active power filter

control

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

Iccref Ics Vc

Icbref Ibs Vb

Icaref Ias Va

Fig. 3: Phase voltages, reference currents and phase

current after compensation with the

minimization of load current active component

power harmonics with null wire and without null wire

respectively.

Figure 8 shows the simulation results similar to

Fig. 4 with null wire, but the compensation of zero

sequence power is included in the control system.Figure 9 depicts the same results without null wire.

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

icaref ias Va

icbref ibs Vb

Iccref Ics V c


currents with compensation of instantaneous

active power harmonics and total reactive power

with null wire

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

Iccref Ics Vc

Icbref Ibs Vb

Icaref Ias Va




without null wire

7/25/2019 apf458

6/9

World Appl. Sci. J., 4 (1): 124-132, 2008

129

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

icaref ias Va

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2Icbref Ibs Vb



active and reactive power harmonics with

null wire

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

Icaref Ias Va

Icbref Ibs Vb

Iccref Ics Vc



active and reactive power harmonics without

null wire

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

Iccref Ics V c

Icbref Ibs V b

Icaref Ias V a




and zero sequence power with null wire

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

Iccref Ics Vc

Icbref Ibs Vb

Icaref Ias Va




and zero sequence power without null wire

7/25/2019 apf458

7/9

World Appl. Sci. J., 4 (1): 124-132, 2008

130

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

Icaref Ias Va

Icbref Ibs Vb

Iccref Ics Vc



active and reactive power harmonics and zero

sequence power with null wire

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

0 0.01 0.02 0.03 0.04 0.05 0.06-2

0

2

time(Sec.)

Icaref Ias Va

Icbref Ibs Vb

Iccref Ics Vc



active and reactive power harmonics and zero

sequence power without null wire

Figure 10 and 11 shows the simulation results for

compensation of instantaneous active and reactive

power harmonics and zero sequence power with null

wire and without null wire respectively.

Fig. 12 shows the total state variable error withdifferent values of coefficient m. It is seen that with

m=4 the error is minimized.

Fig. 13 shows the simulation results of the

proposed controller with minimization of the load

current active component with m=4. The same results

with m=1 are shown in Fig. 14 (conventional sliding

mode control).

Optimized values of S1, S2 and S3 with different

values of mis shown in Fig. 15.

In above simulations the parameters are:

4 5 6 7 8 9

R 0.01 L 0.0024 H C1 C2 1700 f

Vs 220v 1pu

S 1 S 1 S 1 S 1 S 1 S 1

= = = =

= == = = = = =

PI controller coefficients used in the simulations

for m=1 are as:

1 2 3 1 2 3S 3.6,S 8.6,S 5.8,I 3.4,I 4.1,I 3.8= = = = = =

The above coefficients are calculated based on

minimization of state variable RMS error.

The parameters of S1, S2 and S3 are almost robust

against variations of mas shown in Fig. 15 but in order

to achieve the best result, we have used optimum valuesof S1, S2, S3 for each value of m.

Simulation results indicate that with the increasing of

m, PI controller coefficients are less effective and for

m=4, their effects are minimal:

1 2 3 1 2 3S 4.2 S 8 S 5.8 I I I 0= = = = = =

In above simulations load current harmonics in

perunit are:

m1 m3 m5 m7 m9

m11 m13 m15 m17

i 1, i 2/3, i 3/5, i 5/7, i 1/3

i 4/11, i 3/13, i 3/15, i 2/17

= = = = == = = =

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

m

%Error(rms)

Fig. 12: Total state variables error with different values

of coefficient m

time (Sec.)

7/25/2019 apf458

8/9

World Appl. Sci. J., 4 (1): 124-132, 2008

131

0 .005 .01 .015 .02 .025 .03 .035 .04-2

0

2

0 .005 .01 .015 .02 .025 .03 .035 .04-2

0

2

0 .005 .01 .015 .02 .025 .03 .035 .04-2

0

2

time(Sec.)

Ifa Icaref

I fb Icbref

I fc Iccref

Fig. 13: Reference and actual currents with the

minimization of active component of load

current with m=4

0 .005 .01 .015 .02 .025 .03 .035 .04-2

0

2

0 .005 .01 .015 .02 .025 .03 .035 .04-2

0

2

0 .005 .01 .015 .02 .025 .03 .035 .04-2

0

2

time(Sec.)

IfaIcaref

IfbIcbref

IfcIccref

Fig. 14: Reference and actual currents with the

minimization of active component of load

current with m=1. (conventional sliding mode

control)

0 1 2 3 4 5 63.5

4

4.5

5

5.5

6

6.5

m

s1

Optimization of S1

0 1 2 3 4 5 67.5

8

8.5

9

9.5

10

10.5

m

s2

Optimization of S2

0 1 2 3 4 5 65

5.5

6

6.5

7

7.5

8

8.5

9

9.5

m

s3

Optimization of S3

Fig. 15: Optimized values of S1, S2, S3 with different

values of m

CONCLUSIONS

Two method of compensation in shunt active

power filter were considered in this paper.

Using instantaneous reactive power theory it is

possible to compensate the active or reactive power

harmonics, total reactive power or zero sequence

power.

In the minimization method of the activecomponent of load current, the instantaneous reactive

power is zero, which is an advantage of this method.

In the control system design, since the capacitor

voltage error exponent m, PI controller and the error of

state variables are used as the sliding mode controller

inputs, there are some values of m1 that results in fastdynamic response compared to conventional methods.

With this method, we can select m such that the

state variable errors are minimized. This minimization

improves the dynamic response and reduces the state

variable errors. The optimum value of m in the

simulated system was determined to be 4 and it was

shown that m is almost constant due to load variations.

In this controller, there is not any need to use the PI

DC voltage controllers for internal control loop of

capacitor voltages.

REFERENCES

1. Miret, J., and L. Cruz, 2004. A Simple Sliding

Mode Control of an Active Power Filter, proc.of

IEEE Power Electronics Specialists Conference,

pp: 1052-1056.

7/25/2019 apf458

9/9

World Appl. Sci. J., 4 (1): 124-132, 2008

132

2. Marthinus G.F. Gous and Ilendrik J. Beukes, 2004.

Sliding Mode Control for a Three-Phase Shunt

Active Power Filter Utilizing a Four-Leg Voltage

Source Inverter, proc.of IEEE Power Electronics

Specialists Conference, pp: 4609-4615.

3. Mabrouk, N., F. Fnajech, K. Al-Haddad and S.

Rahmani, 2003. Sliding Mode Control of a 3-

Phase-Shunt Active Power Filter, IEEE

Conference, On Industrial Technology, Vol.1,

10-12, pp: 597-601.

4. Aredes, M., 1996. Active Power Line

Conditioners, Ph.D. Thesis, Institut fur

Allgemeine Elektrotechnik, Technical University

of Berlin.

5. Joseph, S. and J.R. Subjak, 1990.

Harmonics_Causes, Effects, Measurments and

Analysis: An Update IEEE Transaction on

Industry Application, Vol. 26. No 6.

6. Mabrouk, N., and F. Fnajech, 2003. Sliding

Mode Control of A 3-Phase-Shunt Active Power

Filter, Proc. ICIT, pp: 597-600.

7. Benhabib, M.C. and S. Saadate, 2005. New

Control Approach For Four-Wire Active Power

Filter Based On The Use of Synchronous

Reference Frame, Electric Power System

Research, Vol. 73, pp: 353-362.

apf458

Documents