A New Method for Calculating the Reference Current ...

A New Method for Calculating the Reference Current of Shunt Active Power Filters Based on Recursive Discrete Fourier Transform H. R. Imani Jajarmi, A. Mohamed, H. Shareef, Subiyanto Abstract – This paper presents a new method based on the fundamental active current component determined from a recursive discrete Fourier transform block. The reference source current and the reference compensating current for shunt active power filters is calculated based on this method to eliminate harmonics, compensate the reactive power of the three-phase diode bridge rectifier with RL load, and to increase power quality. An interval type-2 fuzzy logic controller is used to maintain the DC-link voltage at the reference value. The proposed method is presented and applied to the control system of the voltage source shunt active power filter to evaluate its performance. The results are compared with p-q theory of other reference generation methods. The simulation results confirm the effectiveness of this method in reducing total harmonic distortion and reactive power.

Keywords: Shunt Active Power Filter, Interval Type-2 Fuzzy Logic Controller, Reference Source Current, Recursive Discrete Fourier Transform, Non-Linear Load

I.

Introduction

Power electronic based devices such as converters which are generally used in power supply equipment and control application, draw non-sinusoidal currents and create non-sinusoidal voltage drops in distribution systems [1]. These non-linear converters contribute to harmonic injection into the power system, poor power factor, reactive power burden, imbalance, and other problems that lead to low system efficiency and that create serious power quality problems. Among the options available for mitigating harmonics for power quality improvement, active power filter is greatly accepted and employed as a more flexible and dynamic conditioning device [2], [3]. To mitigate system harmonics, active power filter generates suitable compensating current/voltage signals that cancel the harmonic components of the load with the use of a control algorithm. The control algorithm commonly measures certain variables, such as load currents, DC bus voltage, and source voltages, in order to calculate compensating currents of the APF. By measuring these signals, the harmonic currents/voltages are detected, and the amount of the compensating currents/voltages in the opposite direction of the harmonic currents/voltages required for feeding back into the power system is computed. The techniques for current/voltage reference estimation can be classified as time-domain control and frequency-domain control techniques.

The most widely used frequency domain techniques include the conventional Fourier and fast Fourier transform (FFT) algorithms [4], modified Fourier series techniques [5], discrete Fourier Transform (DFT) [6], and recursive discrete Fourier transform (RDFT) [7]. The time-domain techniques include the DQ method [4], instantaneous reactive power algorithm (p-q) [8], synchronous detection method [9], and synchronous reference frame (SRF) theory [10]. Based on the instantaneous extraction of compensating current/voltage reference from the distorted and harmonic polluted current/voltage signals, the time-domain methods form the control strategy of APFs [11]. Unlike frequency domain approaches that are suitable for both single- and three-phase systems, timedomain approaches that are mostly used for three-phase systems have features such as fast response to changes in the power system, easy implementation, and lower computation burden [12]. However, these techniques are limited to single-node applications and are not suitable for overall network correction because they only take measurements at one point in the power system. Based on Fourier analysis, compensating current/voltage reference comes from the distorted voltage or current signals and forms the basis of the control strategy in the frequency domain [11]. Several harmonic components from the polluted signal are extracted and combined with the Fourier transformation to generate the reference signals. Frequency-domain compensation can be used to address single-node

H. R. Imani Jajarmi, A. Mohamed, H. Shareef, Subiyanto

problems. In addition, it can also be extended over the entire network to reduce harmonic distortion. The biggest challenge of this technique is the need for longer computation time. Considering the increasing amount of highest harmonics that should be eliminated, the volume of necessary calculations needs to be increased to extend response times. However, this problem can be largely prevented based on the current rapid progress in faster processor production. As a result, compensation for high-order harmonics in online applications can more easily take place. An RDFT block is used in this paper to offer a new approach for generating reference currents. An RDFT block enables the component of the fundamental active current component to be determined and multiplied by sin ω t . After using the mean value block, the result is multiplied by 2 and added to the output of the DC capacitor voltage regulator. Aninterval type-2 fuzzy logic controller (IT2FLC) is then used and multiplied again by sin ω t to generate the desired source side currents. Finally, the reference currents of active power filter can be obtained by subtracting the desired source side currents and the load currents. The proposed method is compared with the p-q. The simulation results are presented for both algorithms to compare their effectiveness.

II.

Shunt Active Power Filter

Fig. 1 shows a shunt active power filter (SAPF) [13] that is controlled to supply a compensating current at the point of common coupling (PCC) and to cancelcurrent harmonics on the supply side. The SAPF is controlled to draw/supply a compensated current from/to the utility to eliminate harmonic and reactive currents of the nonlinear load. Therefore, the resulting total current drawn from the mainAC is sinusoidal. The APF should generate sufficient reactive and harmonic current to compensate for the non-linear loads in the line.

generates reference currentsis shown in Fig. 2. These currents must be provided by the power filter to compensate for the reactive power and harmonic currents demanded by the load.

Lf

Rf

Lf

Rf

Lf

Rf

Cdc

g1

g2

g3

g4

g5

g6

i *fa i *fb

Vdc

i *fc

Vdc‐regulated

ifc

i fb

ifa

Fig. 2. Closed-loop fuzzy logic controlled shunt

II.1.1. Instantaneous Active and Reactive p-q Power Theory In instantaneous power theory, three-phase currents ( iLa , iLb , iLc ) and voltages ( vsa , v sb , v sc ) in the a–b–c coordinates are algebraically transformed to the α–β coordinates using Clarke’s transformation, as shown in the following equations [14]:

⎡vα ⎤ ⎢ ⎥= ⎣vβ ⎦

2 3

⎡v ⎤ −1 2 ⎤ ⎢ sa ⎥ ⎡1 −1 2 ⎢ ⎥ vsb 3 2 − 3 2⎦ ⎢ ⎥ ⎣0 ⎢⎣vsc ⎥⎦

(1)

⎡iα ⎤ ⎢ ⎥= ⎣iβ ⎦

2 3

⎡i ⎤ −1 2 ⎤ ⎢ La ⎥ ⎡1 −1 2 ⎢ ⎥ iLb 3 2 − 3 2⎦ ⎢ ⎥ ⎣0 ⎢⎣iLc ⎥⎦

(2)

The instantaneous power is then calculated as follows:

Fig. 1. Connection of shunt active filter with non-linear load

II.1.

Reference Source Currents

The reference source current should be generatedcorrectlyto obtain better APFperformance. The control strategy for a shunt active power filter that

⎡ p ⎤ ⎡ vα vβ ⎤ ⎡iα ⎤ ⎥⎢ ⎥ ⎢ q ⎥ = ⎢v ⎣ ⎦ ⎣⎢ − β vα ⎦⎥ ⎣iβ ⎦

(3)

q = q + q p = p + p,

(4)

The entire reactive power and AC component of the active power are used as the reference power to obtain a sinusoidal current with unity power factor. The reference currents in a–ß coordinates are calculated as follows:


⎡iα' ⎤ ⎡v 1 ⎢ α ⎢ ⎥= ⎢ ' ⎥ v2 + v2 ⎢ α β vβ ⎢⎣iβ ⎥⎦ ⎣

− vβ ⎤ ⎡ − p + ploss ⎤ ⎥ ⎥⎢ ⎢ ⎥ ⎥ vα ⎦ ⎢ − q ⎥⎦ ⎣

(5)

Vsa ( t ) = Vm sin ω t

Here, ploss is the average value of losses in the inverter, which is obtained from the voltage regulator. The DC-link voltage regulator is designed to provide good compensation and excellent transient response. The actual DC-link capacitor voltage ( Vdc ) is measuredby * ), and the error is processed by reference value ( Vdc IT2FLC. The inputs of the IT2FLC are the capacitor voltage deviation and its derivative, whereas its output is the real power ploss requirement for voltage regulation. The reference current is calculated as shown in Eq. (7): 0 ⎤ ⎡ 1 ⎡i*sa ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡iα' ⎤ ⎢* ⎥ 2 ⎢ (6) −1 2 3 2 ⎥ ⎢⎢ ⎥⎥ ⎢isb ⎥ = ⎢ ⎥ ' 3 ⎢ ⎥ i ⎢ ⎥ ⎣⎢ β ⎦⎥ ⎢* ⎥ ⎢ ⎥ ⎢⎣isc ⎥⎦ ⎣ −1 2 − 3 2 ⎦

⎡i*fa ⎤ ⎡i − i* ⎤ ⎢ ⎥ ⎢ La sa ⎥ ⎢i*fb ⎥ = ⎢iLb − i*sb ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ * ⎥ i − i* ⎣i fc ⎦ ⎢⎣ Lc sc ⎥⎦

source voltages by multiplying the unit vectors of respective voltages. Assuming that the power supply voltage is a sine wave: (9)

Load iL ( t ) current can be decomposed to the following equation by Fourier transformation: iL ( t ) =

∞

∑ I n sin ( nω t + Φ n ) = n =1

= I sin (ω t + Φ1 ) +

∞

∑ I n sin ( nω t + Φ n ) =

1 n=2

= I1 sin ω t cos Φ1 + I1 cos ω t sin Φ1 + +

(10)

∞

∑ I n sin ( nω t + Φ n ) = i1 ( t ) + ih ( t ) =

n=2

= i1 p ( t ) + i1q ( t ) + ih ( t )

where, I n ,Φ n representthe peak value and the phase angle of the n-order harmonic current, respectively; n is a positive integer; I1 ,Φ1 representrespectively the value and the phase angle of the fundamental current; I1 p ( t ) is

(7)

the fundamental active current component; I1q ( t ) is the fundamental reactive current component; and ih ( t ) is the current component of harmonics:

This control algorithm is illustrated in Fig. 3.

i1 p ( t ) = I p sin ω t

(11)

i1q ( t ) = I q cos ω t

(12)

ih ( t ) =

Fig. 3. Reference current extraction with conventional p-q theory using IT2FLC

II.2.2. Proposed Method for Harmonics Calculation Ideal compensation occurs when the main current is sinusoidal and is in a similar phase as the source voltage, regardless of the nature of the load. The desired source currents can be presented in the following form after presentation [15]: i*sa

= I sp sin ω t ,

i*sb

= I sp sin (ω t − 120° ) ,

i*sc = I sp sin (ω t + 120° )

(8)

where I sp = I1 cos Φ 1 + I sl is the amplitude of the desired source current. The phase angle can be obtained from the

∞

∑ I n sin ( nω t + Φ n )

(13)

n=2

where I p = I1 cos Φ1 ,I q = I1 sin Φ1 and I p ,I q are the peak values of the fundamental active current and reactive current, respectively. Thus, the value of I p can be determined from I1p . Based on this equation, I1p is multiplied by sin ω t and by the triangle function characteristic, which is rendered as follows: i1 p ( t ) sin ω t = I p sin 2 ω t =

Ip 2

(1 − cos 2ω t )

(14)

It is made up of DC and AC components. A low-pass filter is usually used to separate the DC and AC components. However, the mean value block isimplemented in the simulation of this module because the average value of a sine wave over a full cycle is 0.


Therefore, the DC component I p 2 can also be

III. Recursive Discrete Fourier Transform

worked out by calculating the average value. The value of I p can be obtained (the first component of the

The output of a single DFT (the kth bin of an NpointDFT) is explained by [16]:

amplitude of the desired source current) afterwards after being multiplied by 2. The second component of the AC source current is obtained from the DC capacitor voltage regulator, as shown in Fig. 4. The desired peak current of the AC source can be calculated as follows:

I sp = I1 p + I sl

(15)

The AC source currents must be sinusoidal and in a similar phase with source voltages. Therefore, the desired currents of the AC source can be calculated by multiplying the peak source current with a unity sinusoidal signal. These unity signals can be obtained from Eq. (16). The desired source side currents can be obtained from Eq. (17): iua =

v sa Vm

(16)

i*sa = I sp iua

(17)

Finally, the reference currents of AF can be obtained using Eq. (18): * i*fa = iLa − isa

X0 (k ) =

* Vdc

IT2FLC L.P.F FC=20Hz

(19)

n =0

N-point sequence. The kth order harmonic component is updated by the following equation when the rectangular window of the N-point data is moved forward by one step:

X1 ( k ) =

N −1

∑ x ( n + 1)W −nk

(20)

n =0

Recursive computation of DFT can be obtained from Equations (19) (20): X i ( k ) = W k ⎡⎣ X i −1 ( k ) + x ( i ) − x ( i − N ) ⎤⎦

(21)

This calculation is called recursive DFT [17]. The specific kth order harmonic component is updated by RDFT at each sampling time, wherein one complex is multiplied and the two real sequences are added. If the input signal x ( n ) is a real sequence, then: Re X i ( k ) = Re X i ( N − k ) Im X i ( k ) = − Im X i ( N − k )

The value of I1p in this article is obtained from an

error

∑ x ( n )W −nk , W = e j 2π / N

where X 0 ( k ) is the kth order harmonic component of the

(18)

RDFT block. The control block diagram of shunt active power filter is shown in Fig. 5.

N −1

(22)

RDFT requires the multiplication of M complex, and the 2M real sequence is added to calculate the M ( M ≤ N / 2 ) harmonic components.

Isl

changing of error

Vdc Fig. 4. Structure of fuzzy logic controller

Fig. 5. Reference current extraction with the proposed method (phase a)

IV.

IT2FLC-based DC Bus Voltage Controller

Abbreviation and acronyms should be defined the first time they appear in the text, even after they have already been defined in the abstract. Do not use abbreviations in the title unless they are unavoidable. A conventional type-1 fuzzy logic controller (T1FLC) is commonly used to maintain the DC-link voltage at the reference value. As a result, the inputs of the conventionaltype-1FLC are the capacitor voltage deviation and its derivative and the FLC output are the real power requirement for voltage regulation. A conventional T1FLC is first modeled in this paper, and the T2FLC is designed based on this model. The following seven fuzzy levels or sets are chosenfor T1FLC to convert the inputs and output variables into linguistic variables: NB (negative big), NM (negative medium), NS (negative small), ZE (zero), PS (positive small), PM (positive medium), and PB (positive big).

S H. R. Im mani Jajarmi, A. A Mohamed, H. Shareef, Subiyanto

Membershhip functions are selected for f the inputss and output variabbles, as shownn in Figs. 6 and 7. The rule for f base elem ments of the taable are concluuded based on thee theory that small s errors in i the steady state require fine control and thus requires fine input/ouutput w large errors in thhe transient state variables, whereas require coaarse control and thus requires cooarse input/output variables. Both inpuuts have sevenn subsets. Tabble I [18] shows a fuzzy rule baase formulatedd for the preseent applicationn. The FLC structure is shhown in Fig. 4. 4

FLC requiress deteermined exacctly. Normallly, the T2F proffound computtation due to heavy compu utational loadd at th he type reducttion process. To T simplify the compputation, thee secondaryy mem mbership funcctions can be sset to either 0 or 1 to derivee the T2FLC intervval [20].

Fig.. 8. Type-2 fuzzyy logic controller

Fig. 6. Inputs normaalized membershipp function

Fig. 9. 9 Membership fuunction of IT2FLC C

Fig.. 7. Normalized membership m functtion output

Changing of error c(e) NB NM NP ZE PS PM PB

Suppose S M ruules exist in thhe rule base, each e rule willl have the followinng form:

T TABLE I FUZZY CONTROL RULE Error (e) NB NM NS ZE

PS

PM

PB

NB NB NB NB NM NS ZE

NM NS ZE PS PM PB PB

NS ZE PS PM PB PB PB

ZE PS PM PB PB PB PB

NB NB NB NM NS ZE PS

NB NB NM NS ZE PS PM

NB NM NS ZE PS PM PB

As shownn in Fig. 8, thhe type-2 fuzzzy logic contrroller (T2FLC) alsso contains a fuzzifier, a rule r base, a fuzzy f inference enggine, and an output o processsor that compprises the type redducer and deefuzzifier. Thhese featuress are similar to that t in the structure s of the conventiional T1FLC. T2F FLC was first introduced by b Zadeh in 1970s as an exteension of T1FLC[19]. T In T2FLC, the membership grade for eaach element iss also a fuzzyy set as0 to 1, unlike u the tyype-1 fuzzy set wherein the membership grade is a crissp number of either 0 or 1. The mem mbership functtions of typee-2 fuzzy setss are three-dimenssional and incclude a footprint of uncertaainty, as shown in the shaded region boundedd by the lowerr and upper membbership functioons in Fig. 9. The footprinnt of uncertainty provides p an addditional degrree of freedom m for handling unccertainties. T22FLC can be used at unceertain circumstancees when the membership grades cannoot be

Rule R k : IF x 1 is A1k and x2 is A 2k and…… and x is A k : p

p

(23))

− ⎤ ⎡ TH HEN y is ⎢ wk ,wk ⎥ ⎥⎦ ⎢⎣ −

M;P is the num wheere k=1,2,…M mber of inpu ut variables inn Aik (i=1,22,…,p , k=1,,2,…,M); andd the antecedent part; p −

wk ,w , k are the singleton low wer and upp per weightingg −

facttors of the THEN-part, respectively.

(

Once O a crisp input X = x1 ,x2 ,....,x p

)

T

is applied too

the interval T2FL LC through tthe singleton fuzzifier andd the inference process, the firinng strength of the k th rule,, which is an intterval type-1 set can be obtained ass − ⎤ ⎡ F k = ⎢ f k , f k ⎥ , in which: ⎣⎢ − ⎦⎥

fk =µ −

− A1k

( x1 )* µ ( x2 ) * ...* µ − A 2k

− A kp

( xp )

(24))


−

−

fk =µ where µ ( −

−

A1k

−

( x1 )* µ A ( x2 )* ...* µ A k 2

k p

( xp )

(25)

−

) and µ ( ) denote the grades of the lower and

upper membership functions of IT2FLC, respectively, and * denotes the minimum or product t-norm. The outputs of the inference engine should be typereduced and defuzzified to create a crisp output. A similar configuration as that of the conventional T1FLC is usedfor the design of the IT2FLC. Each set of input/output variables has similar seven linguistic variables, as shown in Figs. 6 and 7, which contain two inputs (error and rate of error) and a single output. The fuzzy labels are NB, NM, NS, Z, PS, PM, and PB. The rules for IT2FLC are similar to those of T1FLC; however, their antecedents and consequents are represented by the IT2FLC. The diagonal rule table, as summarized in Table I, is constructed for the scenario in which the error and change of error approach zero with a fast rise time and without overshoot. The MamdaniIT2FLC is considered in this study,and the popular center-of-sets is assigned for type-reduction methods. The Karnik-Mendel algorithm is then used to obtain the type-reduced set. The embedded MATLAB function block is incorporated into the Simulink modelafter writing the required m-codes in MATLAB for T2FLC.

V.

Carrier-Based PWM Current Control

Figure 10 shows the block diagram of the carrierbased PWM current control (CPWM) scheme, which is a linear control technique.

Vtr

compared with the triangular carrier signals using the limit comparators, to generate the switching pulses of six IGBTs. The switching frequency of this technique is constant and equal to the frequency of the triangular carrier signal [21]. This frequency can be determined by the maximum order of harmonic component considered for elimination. Fig. 11 shows the basic concept of the CPWM current control. According to this technique, * ) in phase a is greater when the reference voltage ( Vca than that of the triangular carrier voltage (Vtr ) the output , of the limit comparator is 1 (s1=1, s2=0).Conversely, the output of the limit comparator is 0 (s1=0, s2=1) when conditions are reversed. Triangular (V ) tr Carrier

Reference Voltage

1 on 0

Switchig Pulse off

Fig. 11. CPWM current control principle

VI.

Simulation Results

Simulations were carried out on the test system, as shown in Fig. 12. This process was conducted to compare the performance of the two SAPF using the p-q and the proposed method for generating reference compensating currents. The test system was made up of a three-phase voltage source, SAPF, and an uncontrolled rectifier with R and L loads. The SAPF was connected to the test system through an inductor, Lf and resistor, Rf. The system parameters are given in Table II.

Triangular carrier

i *f a

* Vca PI controller

i fb

iL

Lf

1

S3

0

Limit Camparator PI controller

i fc

i

Limit Camparator * Vcb PI controller

i*f c

S1 S2

ifa i *f b

1 0

Vcc*

if

Rf

S4

1

S5

0

S6 Limit Camparator

Cdc Vdc

Fig. 10. Block diagram of CPWM current control

The difference between the actual source currents

( i fa ,i fb ,i fc ) and the reference source current (

i*fa ,i*fb ,i*fc

)

is applied to the proportional-integral (PI) controllers to generate the reference voltages. These voltages are then

*

(Vc )

Fig. 12. Test Power System


TABLE II CIRCUIT PARAMETERS OF SAPF Parameter Name Numerical Value Source Voltage 312 V (peak) , 50 Hz Source Resistance and Inductance 0.1 ohm, 1 mH Filter Inductance, Resistance 1.15 mH, 0.1 ohm Load Resistance and Inductance 10 Ω, 5 mH DC Capacitor 2500 µF DC Capacitor Reference Voltage 650V Sample time Ts 10µs

Table III shows the values of total harmonic distortion, power factor, and reactive power measured at the point of common coupling. TABLE III TOTAL HARMONIC DISTORTION, POWER FACTOR, AND REACTIVE POWER Source Current Reactive Method Used Power Factor (phase a)THD% Power(Var) Before any shunt 26.83 0.9896 623.7 compensation Withthe proposed 2.69 1 4.76 method With p-q theory 2.85 1 41.12

VII.

This paper presents a novel control technique for calculating the reference current for three-phase active power filters. SAPF was simulated by MATLAB, and its performance was analyzed in a simple power system. Simulation results prove that the injected harmonics are significantly reduced and the power factor is improved by the proposed method. Moreover, this technique shows better performancecompared with the p-q theory.

References [1]

[2]

[3]

[4]

[5]

[6] Fig. 13. Network without SAPF [7]

[8]

[9] Fig. 14. SAPF with p-q theory [10]

[11]

[12] Fig. 15. SAPF with the proposed method

Figs. 14 and 15 show the load currents after compensation. A comparison of Figs. 13, 14, and 15 suggests that the source currents are made up of fundamental current only and the network with SAPF have lesser harmonic than the network without SAPF. Table III indicates that the harmonic and reactive currents are greatly restrained after compensation. SAPF results using the control system based on this novel technique are better than those of SAPF using the control system based on the p-q theory.

Conclusion

[13]

[14]

[15]

[16]

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Authors’ information Azah Mohamed received her BSc from University of London in 1978 and MSc and PhD from University Malaya in 1988 and 1995, respectively. She is a professor at the Department of Electrical, Electronic and Systems Engineering, University Kebangsaan Malaysia (UKM). Her main research interests are in power system security, power quality and artificial intelligence. She is a senior member of IEEE. H. Shareef received his BSc with honours from IIT, Bangladesh, MS degree from METU, Turkey, and PhD degree from UTM, Malaysia, in 1999, 2002 and 2007, respectively. His current research interests are power system deregulation, power quality and power system distribution automation. He is currently a senior lecturer at the Department of Electrical, Electronic and Systems Engineering, UKM.

Subiyanto received his B.Eng. with rangking I from Diponegoro University, Indonesia, M.Eng degree from Gadjah Mada University, Indonesia, and PhD degree with excelent from UKM, Malaysia, in 1998, 2003 and 2012, respectively. His current research interests are distributed generation, power electronic, power quality and renewable energy. He is currently a post doctoral researcher at the Department of Electrical, Electronic and Systems Engineering, UKM. Hamid Reza Imani Jajarmi is currently a Phd student at Department of Electrical, Electronic and Systems Engineering, UKM.